diff --git "a/magnetization damping/2.json" "b/magnetization damping/2.json" new file mode 100644--- /dev/null +++ "b/magnetization damping/2.json" @@ -0,0 +1 @@ +[ { "title": "0909.3599v1.Resonantly_Damped_Kink_Magnetohydrodynamic_Waves_in_a_Partially_Ionized_Filament_Thread.pdf", "content": "arXiv:0909.3599v1 [astro-ph.SR] 19 Sep 2009RESONANTLY DAMPEDKINK MAGNETOHYDRODYNAMIC WAVES\nINAPARTIALLYIONIZEDFILAMENTTHREAD\nR. Soler, R. Oliver,and J. L.Ballester\nDepartamentdeF´ ısica,Universitatdeles IllesBalears,E -07122,PalmadeMallorca,Spain\n[roberto.soler;ramon.oliver;joseluis.ballester]@uib .es\nABSTRACT\nTransverse oscillations of solar filament and prominence th reads have been fre-\nquently reported. These oscillations have the common featu res of being of short pe-\nriod (2–10 min) and being damped after a few periods. Kink mag netohydrodynamic\n(MHD) wave modes have been proposed as responsible for the ob served oscillations,\nwhereasresonantabsorptionintheAlfv´ encontinuumandio n-neutralcollisionsarethe\nbest candidates to be the damping mechanisms. Here, we study both analytically and\nnumerically the time damping of kink MHD waves in a cylindric al, partially ionized\nfilamentthreadembeddedinacoronalenvironment. Thethrea dmodeliscomposedof\na straight and thin, homogeneous filament plasma, with a tran sverse inhomogeneous\ntransitional layer where the plasma physical properties va ry continuously from fila-\nment to coronal conditions. The magnetic field is homogeneou s and parallel to the\nthread axis. We find that the kink mode is e fficiently damped by resonant absorption\nfor typical wavelengths of filament oscillations, the dampi ng times being compatible\nwiththeobservations. Partialionizationdoesnota ffecttheprocessofresonantabsorp-\ntion, and the filament plasma ionization degree is only impor tant for the damping for\nwavelengthsmuchshorterthanthoseobserved. Toourknowle dge,thisisthefirsttime\nthatthephenomenonofresonant absorptionis studiedin apa rtiallyionizedplasma.\nSubject headings: Sun: oscillations — Sun: magnetic fields — Sun: corona — Sun:\nprominences\n1. INTRODUCTION\nThefine-structuresofsolarprominencesand filamentsarecl early seen inhigh-resolutionob-\nservations. These fine-structures, here called threads, ap pear as very long (5′′−20′′) and thin\n(0′′.2−0′′.6) dark ribbons in H αimages of filaments on the solar disk (e.g., Lin 2004; Linet al .– 2 –\n2005, 2007, 2008, 2009), as well as in observations of promin ences in the solar limb from the\nSolarOpticalTelescope(SOT)aboardtheHinodesatellite( e.g.,Okamotoetal.2007;Berger et al.\n2008; Chaeet al. 2008; Ning etal. 2009). Although statistic al studies show that the orientation\nof threads can significantly vary within the same filament (Li n 2004), vertical threads are more\ncommonlyseeninquiescentprominences(e.g.,Berger et al. 2008;Chae et al.2008)whereashori-\nzontalthreadsareusuallyobservedinactiveregionpromin ences(e.g.,Okamotoet al.2007). From\nthetheoreticalpointof view,filament threads havebeen mod eledas magneticflux tubes anchored\nin the solar photosphere (e.g., Ballester&Priest 1989; Rem pel et al. 1999). In this interpreta-\ntion, only part of the flux tube would be filled with the cool ( ∼104K) filament material, which\nwouldcorrespondtotheobservedthreads. Ithasbeenalsosu ggestedbydifferentialemissionmea-\nsure studiesthat each thread might besurrounded by its ownp rominence-corona transitionregion\n(PCTR) where the plasma physical properties would abruptly vary from filament to coronal con-\nditions (Ciriglianoet al. 2004). The filament material, rou ghly composed by 90% hydrogen and\n10% helium,isonly partiallyionizedfor typicalfilament te mperatures, althoughtheprecise value\nof the ionization degree is not well-known and could probabl y vary in different filaments or even\nindifferent threadswithinthesamefilament (Patsourakos&Vial 20 02).\nOscillations of prominence and filament threads have been fr equently reported since tele-\nscopes with a high time and spatial resolution became availa ble. Early works by Yiet al. (1991)\nandYi&Engvold(1991), witharelativelylowspatialresolu tion(∼1′′), detectedoscillatoryvari-\nations in Doppler signals and He I intensity from threads in q uiescent filaments. Later, H αand\nDopplerobservationswithamuchbetterspatialresolution (∼0.′′2)foundevidenceofoscillations\nand propagating waves along quiescent filament threads (Lin 2004; Linet al. 2007, 2009), while\nobservationsfrom the Hinodespacecraft showed transverse oscillationsofthreadlikestructures in\nboth active region (Okamotoet al. 2007) and quiescent (Ning et al. 2009) prominences. Common\nfeatures of these observations are that the reported period s are usually in a narrow range between\n2 and 10 minutes, and that they are of small amplitude, with th e velocity amplitudes smaller than\n∼3kms−1. Sometheoreticalworkshaveattemptedtoexplaintheseobs ervedoscillationsinterms\nof linear magnetohydrodynamic(MHD) waves supported by the thread body, modeled as a cylin-\ndrical magnetictube(D´ ıazet al. 2002; Dymova&Ruderman 20 05). Thesestudiesconcluded that\nthe so-called kink MHD mode is the best candidate to explain t ransverse, nonaxisymetric thread\noscillations,andisalsoconsistentwiththereportedshor tperiods. Anapplicationofthisinterpreta-\ntionwasperformedbyTerradas et al.(2008),whomadeuseoft hemodelbyDymova&Ruderman\n(2005) and the observations by Okamotoetal. (2007) to obtai n lower limits of the prominence\nAlfv´ en speed. Similarly, Lin etal. (2009) interpreted the ir observations of swaying threads in H α\nsequencies as propagating kink waves and gave an estimation of the Alfv´ en speed. The reader is\nrefereed to recent reviews by Oliver&Ballester (2002); Bal lester (2006); Banerjee et al. (2007);\nEngvold(2008)formoreextensivecomments.– 3 –\nAnother interesting characteristic of prominence oscilla tions is that they seem to be damped\nafterafewperiods. Althoughthisbehaviorwaspreviouslys uggestedbytheresultsofsomeworks\n(e.g., Landmanet al. 1977; Tsubaki&Takeuchi 1986), it was fi rst extensively investigated by\nMolowny-Horaset al. (1999) and Terradas et al. (2002). Thes e authors studied two-dimensional\nDoppler time-series from a quiescent prominence and found t hat oscillations detected in large ar-\neas of the prominence were typically damped after 2–3 period s. Similar results were obtained in\na more recent work by Mashnichet al. (2009). This damping pat tern is also seen in several high-\nresolution Doppler time-series from individual filament th reads by Lin (2004), as well as in the\nHinode/SOT observations by Ninget al. (2009), who reported a maximu m number of 8 periods\nbefore the oscillations disappeared. In the context of the k ink MHD mode interpretation, several\nmechanisms have been proposed to explain this quick damping (see Oliver 2009). Soleret al.\n(2008) studied the damping by nonadiabatic e ffects (radiative losses and thermal conduction) in a\nfully ionized, homogeneous, cylindrical filament thread em bedded in a homogeneouscorona, and\nobtained a kink mode damping time larger than 105periods for typical prominence conditions,\nmeaningthannonadiabatice ffectscannotexplaintheobserveddampingoftransverseosci llations.\nSubsequently, Arreguiet al. (2008) considered a transvers e inhomogeneous transitional layer be-\ntween the fully ionized filament thread and the corona, and in vestigated the kink mode damping\nby resonant absorption in theAlfv´ en continuum. They obtai ned a dampingtimeofapproximately\n3 periods for typical wavelengths of prominence oscillatio ns. Later on, Soler etal. (2009a, here-\nafter Paper I) complemented the work by Arregui etal. (2008) by also considering the damping\nby resonant absorption in the slow continuum, but concluded that this effect is not relevant and\nobtainedsimilarresults tothoseofArreguiet al. (2008). F inally,Soleret al. (2009b, hereafter Pa-\nper II) included the e ffect of partial ionization in a homogeneous filament thread mo del. These\nauthorsfoundthation-neutralcollisionscane fficientlydampthekinkmodeforshortwavelengths,\nalthoughtheyarenote fficient enoughintherangeoftypicallyobserved wavelengths .\nOn the basis of these previous studies, resonant absorption seems to be the most e fficient\ndamping mechanism for the kink mode and the only one that can p roduce the observed damping\ntimes. On the other hand, the e ffect of partial ionization could be also relevant, at least fo r short\nwavelengths. The aim of the present work is to take both mecha nisms into account and to assess\ntheir combined effect on the kink mode damping. To our knowledge, this is the firs t time that the\nphenomenon of resonant absorption is studied in a partially ionized plasma. The filament thread\nmodel assumed here is similar to that of Paper I and is compose d of a homogeneous and straight\ncylinder with prominence-like conditions embedded in an un bounded corona, with a transverse\ninhomogeneoustransitionallayerbetweenbothmedia. Weco nsidertheprominencematerialtobe\npartiallyionized. AsinPaperII,weusethelinearizedMHDe quationsforapartiallyionized,one-\nfluid plasma derived by Fortezaet al. (2007). Since we focus o ur investigationon the kink mode,\nweadopttheβ=0approximation,where βistheratioofthegaspressuretothemagneticpressure.– 4 –\nThis approximation removes longitudinal, slowlike modes b ut transverse modes remain correctly\ndescribed. InthenextSections,weinvestigatethekinkmod edampingbymeansofbothanalytical\nand numericalcomputations,and assessthee fficiency oftheconsidered dampingmechanisms.\nThis paper is organized as follows: Section 2 contains a desc ription of the model configura-\ntion and the basic equations. The results are presented and d iscussed in Section 3. Finally, our\nconclusionsare givenin Section4.\n2. MODEL ANDMETHOD\n2.1. Equilibrium Properties\nWemodelafilamentthreadas aninfiniteandstraightcylindri calmagneticfluxtubeofradius\nasurrounded by an unbounded coronal environment. Cylindric al coordinates are used, namely\nr,ϕ, andz, for the radial, azimuthal, and longitudinal directions, r espectively. In the following\nexpressions, a subscript 0 indicates equilibrium quantiti es while we use subscripts fandcto\nexplicitlydenotefilamentandcoronalquantities,respect ively. Themagneticfieldishomogeneous\nand orientated along the cylinder axis, B0=B0ˆez, withB0=5 G constant everywhere. We\nadopt the one-fluid approximation and consider a hydrogen pl asma composed of ions (protons),\nelectrons,andneutralatoms. Sinceweareinterestedinkin kMHDwavessupportedbythefilament\nthread body, we also assume the β=0 approximation, which neglects gas pressure e ffects and\nremoves longitudinal, slowlike modes. With such assumptio ns (see details in Fortezaet al. 2007,\nand Paper II), the plasma properties are characterized by tw o quantities: the fluid density, ρ0, and\nthe ionizationfraction, ˜ µ0, which givesus informationabout theplasmadegree of ioniz ation. The\nallowed values of ˜µ0range between ˜µ0=0.5 for a fully ionized plasma and ˜ µ0=1 for a neutral\nplasma.\nThe density profile assumed here only depends on the radial di rection and is the same as in\nPaperI(which was adoptedafterRuderman & Roberts 2002), na mely\nρ0(r)=ρf,ifr≤a−l/2,\nρtr(r),ifa−l/2˜ηH, one gets (kza)C<(kza)H, meaning than Cowling’s di ffusion\nbecomes dominant over ohmic di ffusion for smaller wavenumbers than those needed for Hall’s\nterm to become relevant. Next, we have to know whether Cowlin g’s diffusion or Hall’s diffusion\ndominate for kzabeyond both transitional values. To do so, we define the dimen sionless number\nHCas themagnitudeofHall’stermwithrespect to thatoftheamb ipolarterm,\nHC=/vextendsingle/vextendsingle/vextendsingle∇×/bracketleftbigηH(∇×B1)×B0/bracketrightbig/vextendsingle/vextendsingle/vextendsingle\n|∇×{ηA[(∇×B1)×B0]×B0}|∼˜ηH\n˜ηC. (19)\nConsidering again that ˜ ηC>˜ηH, the conditionHC<1 is always satisfied, therefore Cowling’s\ndiffusion is always more important than Hall’s di ffusion. By means of this simple dimensional\nanalysis,weexpectthepresenceofHall’sterm tohaveamino reffect ontheresults.\n2.3. AnalyticalDispersionRelation\nSomeanalyticalprogresscanbeperformedwhennotransitio nallayerispresent,i.e., l/a=0.\nWe showed in Paper II that it is possible to give an analytical dispersion relation when the terms\nwithηandηHare droppedfromtheinductionequation. Insuch asituation ,theinductionequation\n(Equation(6))can bewrittenin acompactform as follows,\n∂B1\n∂t=Γ2\nA\nv2\nA∇×(v1×B0), (20)\nwhereΓ2\nA≡v2\nA−iωηCisthemodifiedAlfv´ enspeedsquared(Forteza et al.2008). N ext,wefollow\nthe standard procedure (see, e.g., Edwin&Roberts 1983; Goo ssenset al. 2009) and obtain the\ndispersion relation of trapped waves by imposing that the ra dial displacement, ξr=ivr/ω, and\nthe total pressure perturbation, pT=B0Bz/µ, are both continuous at the thread boundary, r=a,– 9 –\nand that perturbations must vanish at infinity. So, the dispe rsion relation governing transverse\noscillationsisDm(ω,kz)=0, with\nDm(ω,kz)=nc\nρc/parenleftig\nω2−k2zΓ2\nAc/parenrightigK′\nm(nca)\nKm(nca)−mf\nρf/parenleftig\nω2−k2zΓ2\nAf/parenrightigJ′\nm/parenleftig\nmfa/parenrightig\nJm/parenleftig\nmfa/parenrightig, (21)\nwhereJmandKmaretheBesselfunctionandthemodifiedBesselfunctionofth efirstkind(Abramowitz&Stegun\n1972), respectively,and thequantities mfandncare givenby\nm2\nf=/parenleftig\nω2−k2\nzΓ2\nAf/parenrightig\nΓ2\nAf,n2\nc=/parenleftig\nk2\nzΓ2\nAc−ω2/parenrightig\nΓ2\nAc. (22)\nNotethatEquation(21)is validfor anyvalueof m.\nNext, our aim is to extend this analytical analysis to the cas el/a/nequal0. When an inhomo-\ngeneous transitional layer is present in the equilibrium, t he kink mode is resonantly coupled to\nAlfv´ en continuum modes. The radial position where the Alfv ´ en resonance takes place, rA, can be\ncomputedbysettingthekinkmodefrequency, ωk, equaltothelocalAlfv´ enfrequency, ωA=vAkz.\nTheexpressionof rAcorrespondingtoourdensityprofilewas obtainedinPaperI( Equation(9))1,\nrA=a+l\nπarcsinρf+ρc\nρf−ρc−2v2\nAfk2\nz\nω2\nkρf/parenleftig\nρf−ρc/parenrightig. (23)\nTheideal MHD equations are singularat r=rA. This singularityis removedif dissipativee ffects,\nsuchasmagneticdi ffusionorviscosity,areconsideredinaregionaroundtheres onancepoint,i.e.,\nthe dissipative layer. A method to obtain an analytical disp ersion relation in the presence of an\ninhomogeneous transitional layer is to combine the jump con ditions at the resonance point with\ntheso-calledthinboundary(TB)approximation,whichwasfi rstusedbyHollweg& Yang(1988).\nThe jump conditions were first derived by Sakurai et al. (1991 a) and Goossenset al. (1995) for\nthe driven problem, and later by Tirry& Goossens (1996) for t he eigenvalue problem. They have\nbeen used in a number of papers in the context of MHD waves in th e solar atmosphere (e.g.,\nSakurai et al. 1991b; Goossenset al. 1992; Keppenset al. 199 4; Stenuit etal. 1998; Andrieset al.\n2000; Van Doorsselaereet al. 2004, Paper I among other works ). An important result for the\npresent investigation was obtained by Goossenset al. (1995 ), who proved that the jump condi-\ntionsderivedbySakurai et al.(1991a)inidealMHDremainva lidindissipativeMHD.Thisallows\nustoapplythejumpconditionsderivedbySakurai et al.(199 1a)toourcase. Theassumptionsbe-\nhind the TB approximation and its applications have been rec ently reviewed by Goossens (2008).\n1Note thatthereisa typographicalerrorin Equation(9)ofPa perIsince theterm ρe(whichcorrespondsto our ρc)\nshouldbe−ρe.– 10 –\nIn short,themainassumptionoftheTBapproximationisthat thereisaregionaroundthedissipa-\ntive layer where both ideal and dissipativeMHD applies. In o ur equilibrium, we can assume that\nthe thicknessof the dissipativelayer, namely δA, roughly coincides with the width of theinhomo-\ngeneous transitional layer. This condition is approximate ly verified for thin layers, i.e., l/a≪1.\nSo,wecansimplyconnectanalitycallytheperturbationsfr omthehomogeneouspartofthetubeto\nthoseoftheexternalmediumbymeansofthejumpconditionsa ndavoidthenumericalintegration\nofthedissipativeMHDequationsacross theinhomogeneoust ransitionallayer.\nThe jump conditions for the radial displacement and the tota l pressure perturbation provided\nby Sakurai et al.(1991a)in thecase ofa straightmagneticfie ld are,\n/bracketleftbig/bracketleftbigξr/bracketrightbig/bracketrightbig=−iπm2/r2\nA\n|ρ0∆|rApT,/bracketleftbig/bracketleftbigpT/bracketrightbig/bracketrightbig=0,atr=rA, (24)\nwhere[[X]]=Xc−Xfstandsforthejumpofthequantity X,and∆=d\ndr/parenleftig\nω2−ω2\nA/parenrightig\n. Byconsidering\nthat in our model the magnetic field is straight and constant s o that the variations of the local\nAlfv´ en frequency are only due to the variation of the equili brium density, we can write |ρ0∆|rA=\nω2\nA|∂rρ0|rA. In addition, from the resonance condition we have ω2\nA=ω2\nk. Applying the jump\nconditions(24), wearriveat thedispersionrelationin the TB approximation,\nDm(ω,kz)=−iπm2/r2\nA\nω2\nk|∂rρ0|rA, (25)\nwithDm(ω,kz)defined in Equation (21). Note that in order to solve Equation (25) we need the\nvalue of the kink frequency, ωk. For this reason, we use a two-step procedure. First, we solv e the\ndispersion relation for the case l/a=0 (Equation (21)) and obtain ωk. Next, we assume that the\nreal partofthefrequencyisapproximatelythesamewhenthe inhomogeneoustransitionallayeris\nincluded, allowingus to determine rAfrom Equation (23) and therefore |∂rρ0|rA. Finally, we solve\nthecompletedispersionrelation (Equation(25))withthes eparameters.\n2.3.1. Expressionsin theThin TubeLimit\nEquation(25)isatranscendentalequationthathastobesol vednumerically. AsinPaperI,itis\npossible to go further analytically by considering the thin tube (TT) approximation, i.e., kza≪1.\nThe combination of both the TB and TT approximations has been done in several works (e.g.,\nGoossenset al.1992;Ruderman & Roberts2002;Goossenset al .2002,2009). IntheTTlimit,we\napproximatethekinkmodefrequency as\nωk≈/radicaligg\n2\n1+ρc/ρfvAfkz. (26)– 11 –\nWe put this expression in Equation (23) and obtain that rA≈a. Next we perform a first order,\nasymptoticexpansionforsmallargumentsoftheBesselfunc tionsofEquation(25). Thedispersion\nrelationthen becomes,\nρf/parenleftig\nω2−k2\nzΓ2\nAf/parenrightig\n+ρc/parenleftig\nω2−k2\nzΓ2\nAc/parenrightig\n=iπ/parenleftbiggm\na/parenrightbiggρfρc\n|∂rρ0|rA/parenleftig\nω2−k2\nzΓ2\nAf/parenrightig/parenleftig\nω2−k2\nzΓ2\nAc/parenrightig\nω2\nk.(27)\nWe now write the frequency as ω=ωk+iωI, and the modified Alfv´ en speed squared is approxi-\nmatedbyΓ2\nA≈v2\nA−iωkηC. WeinserttheseexpressionsinEquation(27)andneglectte rmswithω2\nI\nandωIk2\nz. It isstraight-forwardto obtainan expressionfortherati oωI/ωk,\nωI\nωk=−π\n2/parenleftbiggm\na/parenrightbiggρfρc/parenleftig\nρf+ρc/parenrightig1\n|∂rρ0|rA1\n4/parenleftig\nρf−ρc/parenrightig2\nρfρc+1\nv2\nAf+1\nv2\nAcηCfηCc\n2k2\nz\n−1\n2/parenleftig\nρfηCf+ρcηCc/parenrightig\nkz\n/radicalig/parenleftig\nρf+ρc/parenrightig/parenleftig\nρfv2\nAf+ρcv2\nAc/parenrightig. (28)\nThe first term in Equation (28) owes its existence to the facto r in the dispersion relation related\nto the TB approximation and represents the contribution of r esonant absorption. For long wave-\nlengths,thefactor/parenleftig\n1/v2\nAf+1/v2\nAc/parenrightig\nηCfηCck2\nz/2canbeneglected,sothetermrelatedtotheresonant\ndamping is independent of the value of Cowling’s di ffusivity and, therefore, of the ionization de-\ngree. Then, one can see that in the TT case the term of Equation (28) due to the resonant damp-\ning takes the same form as in a fully ionized plasma, which was previously obtained by, e.g.,\nGoossenset al. (1992, Equation (77)) and Ruderman &Roberts (2002, Equation (56)). On the\notherhand, thesecond termin Equation(28)is related to the dampingbyCowling’sdi ffusionand\nis also present in the case l/a=0. This term is proportional to kz, so we also expect it to be of a\nminorinfluenceintheTT regime.\nNext,wetakeintoaccountthatforthepresentsinusoidalde nsityprofile,|∂rρ0|rA≈π/parenleftig\nρf−ρc/parenrightig\n/2l.\nWe insert this expression in Equation (28) and then use it to g ive a relation for the ratio of the\ndampingtimetotheperiod,\nτD\nP=2\nπm/parenleftiggl\na/parenrightigg/parenleftiggρf−ρc\nρf+ρc/parenrightigg\n+2/parenleftig\nρf˜ηCf+ρc˜ηCc/parenrightig\nkza\n/radicalig\n2ρf/parenleftig\nρf+ρc/parenrightig−1\n, (29)\nwherewehaveusedEquation(8)torewriteCowling’sdi ffusivitiesintheirdimensionlessform. To\nperformasimpleapplication,wecompute τD/PfromEquation(29)inthecase m=1,kza=10−2,\nandl/a=0.2, resulting inτD/P≈3.18 for a fully ionized tread (˜ µf=0.5), andτD/P≈3.16\nfor an almost neutral thread (˜ µf=0.95). We note that the obtained damping times are consistent– 12 –\nwiththeobservations. Furthermore, theratio τD/Pdepends veryslightlyon theionizationdegree,\nsuggesting that resonant absorption dominates over Cowlin g’s diffusion. To check this last state-\nment,wecomputetheratioofthetwotermsontheright-hands ideofEquation(29),whichallows\nus to compare the damping times exclusively due to resonant a bsorption, (τD)RA, and Cowling’s\ndiffusion,(τD)C,\n(τD)RA\n(τD)C=/radicaltp/radicalbt\n2/parenleftig\nρf+ρc/parenrightig\nρf/parenleftiggρf˜ηCf+ρc˜ηCc\nρf−ρc/parenrightiggkza\nm(l/a). (30)\nThis last expression can be further simplified by considerin g that in filament threads ρf≫ρcand\n˜ηCf≫˜ηCc, sothat\n(τD)RA\n(τD)C≈√\n2˜ηCfkza\nm(l/a). (31)\nThus, we see that the e fficiency of Cowling’s di ffusion with respect to that of resonant absorption\nincreases with kzaand ˜µf(through ˜ηCf). Considering the same parameters as before, one obtains\n(τD)RA/(τD)C≈2×10−8for ˜µf=0.5, and(τD)RA/(τD)C≈6×10−3for ˜µf=0.95, meaning that\nresonantabsorptionismuchmoree fficient thanCowling’sdi ffusion. FromEquation(31)itisalso\npossible to give an estimation of the wavenumber for which Co wling’s diffusion becomes more\nimportantthan resonantabsorptionbysetting (τD)RA/(τD)C≈1. So, onegets,\nkza≈m(l/a)√\n2˜ηCf. (32)\nConsidering again the same parameters, Equation (32) gives kza≈5×105for ˜µf=0.5, and\nkza≈1.7 for ˜µf=0.95. One has to bear in mind that Equation (31) is valid only for kza≪1, so\nwe expect resonant absorption to be the dominant damping mec hanism in the TT regime even for\nan almostneutral filamentplasma. We willverify theanalyti calestimationsofthepresentSection\nby meansofnumericalcomputations.\n2.4. Numerical Computations\nTo numerically solve the full eigenvalue problem (Equation s (9)–(14)) we use the PDE2D\ncode based on finite elements (Sewell 2005). We follow the sam e procedure as in Papers I and II.\nTheintegrationofEquations(9)–(14)isperformedfromthe cylinderaxis, r=0,totheedgeofthe\nnumerical domain, r=rmax, where all perturbations vanish since the evanescent condi tion is im-\nposedinthecoronalmedium,i.e.,werestrictourselvestot rappedmodes. Theboundaryconditions\natr=0 are imposed by symmetry arguments. To obtain a good converg ence of the solution and\ntoavoidnumericalerrors,weneedtolocatetheedgeofthenu mericaldomainfarenoughfromthe\nfilament thread mean radius to satisfy the evanescent condit ion. We have considered rmax=100a.– 13 –\nWe use a nonuniform grid with a large density of grid points wi thin the inhomogeneous transi-\ntional layer in order to correctly describe the small spatia l scales that develop due to the Alfv´ en\nresonance.\n3. RESULTS\nWefocusourstudyontheratioofthedampingtimetotheperio d,τD/P,whichisthequantity\nthat informs us about the e fficiency of the kink mode damping. In the following sections, w e\ncomputeτD/Pas afunction of thedimensionlesslongitudinalwavenumber ,kza. According to the\nobserved wavelengths (Oliver&Ballester 2002) and thread w idths (Lin 2004), the relevant range\nofkzaof filament oscillations corresponds to 10−30\nwhereas for the perpendicular to the plane easy axis 4 \u0019Meff<0. According\nto Eqs. (1) and (2), 4 \u0019Meff=\u00002Keff=Ms, whereKeffis the e\u000bective\nanisotropy constant de\fned as K?\u00002\u0019M2\ns[15].\n3. Results and discussion\nFigure 1 (e) presents hysteresis loops of the sample with a 1.25 nm thick\nMgO barrier measured in the out-of-plane (red line) and in-plane con\fgu-\nration (black line). The shape of the loops in both directions is nearly the\nsame for each con\fguration as the saturation \felds (of Hs\u00192 kOe) for both\nlayers have nearly the same magnitude with the opposite signs in 4 \u0019Meff.\nEach hysteresis loop is a sum of the loops typical for the easy and hard axis\nand, as explained below, we can infer from magnetization reversals which\nlayer possesses PMA.\nLet us assume that the bottom CoFeB layer (B) has an in-plane easy axis\nand the top layer (T) has a perpendicular to the plane easy axis so that their\nmagnetization directions are orthogonal at remanence. Three con\fgurations\nof a magnetic \feld Happlied for the magnetization measurements are shown\n5-10 -5 0 5 10-101\n Normalized moment\nH (kOe)HTT\nBH\ne.a.e.a.\nB\nHe.a.e.a. T\nBe.a.\ne.a.a) b) c)\nB\nT\nB+T\nd)\n 10 51\n-1\n-10 -5 0e)\nH (kOe)Figure 1: (a)-(c) Con\fgurations used for the magnetic measurements with a magnetic\n\feld applied perpendicular or parallel to the \flm plane. (d) Example of schematic pictures\nof the magnetization reversals of a CoFeB/MgO/CoFeB structure for con\fguration (a).\n(e) Hysteresis loops of a CoFeB/MgO/CoFeB structure measured in con\fgurations (a)\n- black line and (b) - red line. The inset shows schematically the model reversals for\ncon\fgurations (a)-black and (b)-red\n.\nin Figs. 1 (a) - (c). These con\fgurations enable magnetization reversals to be\nobserved with Horiented parallel- (a) (perpendicular- (b)) to the easy axis of\nB (T) layer, respectively, or perpendicular to both easy axes (c). Further, we\nwill refer to these con\fgurations as (a), (b), and (c) con\fgurations. As it is\nschematically shown in Fig. 1 (d), an apparent magnetization reversal of B+T\nfor the con\fguration (a) is a sum of independent magnetization reversals of\nB and T. For the perfectly asymmetric structure with 4 \u0019MB\neff=\u00004\u0019MT\neff\nwith the same thickness (i.e. with the same magnetic moments MSVT;B) the\n6apparent magnetization reversals taken in con\fgurations (a) and (b) would\noverlay. However, as it is seen in Fig. 1 (e) they do not completely overlay\nso that the curve taken in the con\fguration (b) lies a bit higher than that\ntaken in (a). As it is shown in the inset of (e), a simple model explains that\nthe T layer (i.e. the with nominal thickness tof 1.3 nm) possesses an easy\naxis perpendicular to the plane, while the B layer with t= 0:93 nm has an\nin-plane easy axis.\nIn the model, the magnetization reversals in each layer can be approxi-\nmated with a normalized relation [16] M(H;S) = arctan[H=H s\u0002tan(\u0019S=2)]=\narctan[H=H max\u0002tan(\u0019S=2)], where Hsof 2 kOe is a saturation \feld for\nthe hard direction and Sis de\fned as a ratio of remanence to the satura-\ntion moment. For Hkparallel to the easy axis, S= 1 (B layer in Fig. 1\n(d)) and for H?perpendicular to the easy axis (T layer in Fig. 1 (d)),\nS= 0:66 as well as Hmax= 10 kOe are arbitrary chosen for the sake of\nsimplicity. The apparent magnetization curve for con\fguration (a) is a sum\n[tB\u0002M(H;S = 1) +tT\u0002M(H;S = 0:66)]=(tB+tT). For the con\fguration\n(b),tTandtBare reversed in the sum. In order to satisfy the experimental\ndata shown in (e), a ratio tB=tT= 0:79. It is easily seen that if the B layer\nhad an in-plane easy axis and the T layer had an easy axis perpendicular to\nthe plane, a curve taken in con\fguration (b) would lie lower than that taken\nin con\fguration (a). Hence, the thin B layer is that with the in-plane easy\naxis.\nFigures 2 (a) and (b) show typical VNA-FMR spectra of the CoFeB/MgO(1.25\nnm)/CoFeB system measured (see Figs. 1) in con\fguration (a) and (b) , re-\nspectively. Two FMR peaks associated with the bottom and top CoFeB lay-\n76 8ImS21(a.u.)\nH(kOe)topbottom(a) 20GHz\nin-planeconfiguration\n4 6 8 10(b)ImS21(a.u.)\nH(kOe)top\nbottom20GHz\nout-of-planeconfiguration\n( )\n()Figure 2: Typical VNA-FMR spectrum of the as-deposited CoFeB/MgO(1.25 nm)/CoFeB\nstructure with resonance peaks from bottom (B) and top (T) layers measured in the in-\nplane (a) and out-of-plane (b) con\fgurations. Solid red lines represent the Lorentzian\n\fts to the experimental data. (c) Dependence of the FMR \feld on the polar angle \u0002\nof applied \feld in X band (9.1 GHz). The easy axis of magnetization of the B is in the\nin-plane orientation. For the T layer, the out-of-plane direction becomes the easy axis.\ners are clearly visible. To determine the resonance \feld Hrand the linewidth\n\u0001Hat constant frequency with a high precision, the spectra were \ftted with\nLorentzians (marked by solid lines in Fig. 2 (a) and (b)). Figure 2 (c) shows\ndependencies of the X-band (9.1 GHz) resonance \felds of the B and T layers\non the polar angle between the \flm normal and the direction of an applied\n\feld. It is clearly seen that the T layer has 4 \u0019Meff<0 (i.e., a perpendicular\neasy axis) and the B layer with 4 \u0019Meff>0 has an in-plane easy axis. From\nFigs. 2 (a) and (b), we can clearly see that the intensity (area under the FMR\n8peak) of the T layer is higher than that of the B layer. This additionally\ncon\frms that the bottom layer has the lower magnetic moment than that of\nthe top layer.\nA typicalHrvs.fdependence, observed for the CoFeB/MgO(1.25 nm)/CoFeB\nsystem is shown in Fig. 3 (a) and (b) for the in-plane (a) and out-of-plane\n(b) con\fguration, respectively. The observed data points are \ftted using\nEqs. (1) and (2). The values of 4 \u0019Meff, obtained from the \ftting are found\nto be of +2 :2 kG and\u00002:3 kG for the bottom and top layers, respectively.\nThefversusHrdata for the B layer were \ftted assuming Haof 30 Oe as\ncon\frmed by VSM measurements (not shown) in the con\fguration presented\nin Fig. 1(c). The values of gkof the top and bottom layers are equal to 2.04\nand 2.08, respectively, in contrast, the values of g?for these layers are 2.06\nand 2.22. One can notice the di\u000berences in values of g?resulting from clear\ndi\u000berences in the slopes of the f(Hr) dependencies (see, Fig. 3 (b)) for the\nbottom (\r?= 2:88 MHz/Oe) and top ( \r?= 3:11 MHz/Oe) layer, respec-\ntively.\nTo sum up, VSM and FMR measurements con\frmed the presence of or-\nthogonal easy axes in our CoFeB/MgO/CoFeB systems and showed that the\nthickness ratio tB=tT= 0:79 is slightly higher than the ratio of nominal thick-\nness (tB\nnom=tT\nnom= 0:71). The thinner B layer has an in-plane easy axis while\nthe T layer has a perpendicular easy axis. However, keeping in mind our for-\nmer studies of a dead magnetic layer (DML) in the Ta/CoFeB/MgO (B) and\nMgO/CoFeB/Ta (T) structures [13] deposited in the same Timaris system,\nwe estimated DMLB'0:23 nm and DMLT'0:4. With such asymmetric\nDMLs the e\u000bective thickness tB\neff'0:7 nm andtT\neff'0:9 nm which satis\fes\n90 5 1005101520\n0 5 10010203040\nf (GHz)\nHr(kOe)\n(b)(a)\nin-plane configuration\nout-of-plane configuration\nf (GHz)\nHr(kOe)1.25 nm MgO\nbottom\ntop0 2 46 8 10048121620\ntop 1.25 nm MgO \n 1.25 nm MgO \n 0.96 nm MgO\n 0.96 nm MgO\n 0.85 nm MgO\n 0.85 nm MgOf (GHz)\nH (kOe)bottomFigure 3: FMR dispersion relations of the as-deposited CoFeB/MgO(1.25 nm)/CoFeB\nstructure measured in the in-plane con\fguration (a) and out-of-plane con\fguration (b).\nThe solid lines show the \fts given in accordance with Eqs. (1) and (2). Inset in (a) shows\nthat the \ftting parameter practically do not depend on the MgO thickness.\ntB=tT= 0:78. VNA-FMR measurements, which o\u000ber a greater precision than\nVSM measurements, give 4 \u0019Meff=\u00002:3 kG (K?= 10:4\u0002106erg/cm3) and\n4\u0019Meff= +2:2 kG (K?= 7:7\u0002106erg/cm3) for the T and B layers, respec-\ntively. All \ftting parameters for a CoFeB/MgO(1.25 nm)/CoFeB structure\nare juxtaposed in Table 1. As it is shown in the inset of Fig. 3 (a), the thick-\nness of MgO spacer within a range of 0.9 { 1.25 nm had almost no in\ruence\non the \ftting parameters, therefore, the values of \ftting parameters 4 \u0019Meff,\ng,\u000b, and \u0001H0are typical for all samples with various MgO thickness.\n10Table 1: Parameters determined from VNA-FMR spectra for the as-deposited\nCoFeB(0.93 nm)/MgO (1.25 nm)/CoFeB(1.31 nm) for the in-plane and out-of-plane con-\n\fgurations: the in-plane anisotropy \feld ( Ha), the e\u000bective magnetization (4 \u0019M eff), spec-\ntroscopicg-factors for in-plane and out-of-plane con\fguration, Gilbert damping ( \u000b), the\nfrequency-independent FMR linewidth (\u0001 H0). The values of the \ftting parameters do\nnot depend on the MgO thickness. The values of g?are marked by asterisks.\nIn-plane con\fguration\nHa(Oe) 4\u0019Meff(kG)gk,g? \u000b \u0001H0(Oe)\ntop 0 -2.29\u00060.05 2.04\u00060.02 0.018\u00060.002 102\u000622\nbottom 30 2.22\u00060.15 2.08\u00060.03 0.017\u00060.002 69\u000623\nOut-of-plane con\fguration\ntop { -2.3\u00060.01 2.22\u00060.01?0.018\u00060.001 95\u000613\nbottom { 2.19\u00060.04 2.06\u00060.02?0.017\u00060.003 160\u000630\nAlthough it is counter-intuitive that the thinner B layer possesses an in-\nplane easy axis, the same feature has been reported for other Ta/CoFeB(1\nnm)/MgO systems deposited in the same Timaris equipment [17]. Similar ef-\nfect has been recently observed in a substrate/MgO/CoFeB/Ta/CoFeB/MgO\nstructure, where the thicker CoFeB layer exhibits a strong PMA in con-\ntrast to the relatively weak PMA in the thinner CoFeB layer [18, 19]. It is\npossible that the growth mode of the MgO layer in contact with an amor-\nphous CoFeB layer might be responsible. The perpendicular anisotropy in\nthese systems originates from the CoFe/MgO interface [20]. The structure\nof the unannealed CoFeB layers is amorphous regardless of underlying lay-\ners, whereas the MgO barrier deposited on the amorphous CoFeB has an\namorphous structure of up to four monolayers (that is about 0.9 nm) [21].\n11Hence, there are subtle di\u000berences between the CoFeB/MgO (bottom) and\nMgO/CoFeB (top) interfaces; the interface of the bottom CoFeB layer is\nmainly amorphous whereas the interface of the top layer is crystalline, be-\ncause the barrier thickness of the investigated samples is above the transition\nfrom amorphous to crystalline phase. Therefore, di\u000berent structures for the\nCoFeB/MgO interfaces may result in di\u000berent values of anisotropy constant.\nAnother explanation is that the measured dependence Keff\u0002teffvs.teff\nin \flms with PMA is often strongly nonlinear due to either intermixing at\ninterfaces [22] or magnetoelastic e\u000bects [15], with Keff\u0002teffexhibiting a\nmaximum as a function of decreasing teffand with the PMA eventually\nbeing lost for small teffof, for example, 0.7 nm.\nThe values of gfactor yield the ratio of the orbital \u0016Land spin\u0016Smag-\nnetic moments in accordance with equation [9, 11]\n\u0016L\n\u0016S=g\u00002\n2; (3)\nwhere\u0016S=\u0016B. Hence, the di\u000berence between orbital moments \u0001 \u0016Lalong\nthe easy and hard direction in the in-plane [Fig. 1 (a)] and out-of-plane [Fig. 1\n(b)] con\fgurations is proportional to ( g?\u0000gk) and reads \u0001 \u0016L=\u0016B(g?\u0000\ngk)=2. \u0001\u0016Lis of 0.09\u0016Band\u00000:01\u0016Bfor the T and B layer, respectively.\nIn CoFe/Ni multilayers [7], the PMA has been shown to be proportional\nto the orbital moment anisotropy in accordance to Bruno model [23]. How-\never, in the case of the CoFeB/MgO systems this direct relationship between\nthe orbital moment asymmetry and the perpendicular anisotropy is not ful-\n\flled. As can be seen in Table 1, ( g?\u0000gk)\u00190 for the B layer corresponds to\n4\u0019Meff= 2:2 kG. Hence, while ( g?\u0000gk) is negligible, a decrease in 4 \u0019Meff\ndue to PMA from 4 \u0019MS= 15 kG to 2.2 kG is substantial. In contrast,\n12(g?\u0000gk)\u00190:18 is exceptionally large for the T layer, while 4 \u0019Meffmerely\ndecreases to - 2.3 kG. In accordance with the earlier report [24], this con\frms\nthat any relationship between the orbital moment asymmetry and the per-\npendicular anisotropy in CoFeB/MgO systems is highly nonlinear. Of course,\nother factors controlled by annealing such as disorder at interfaces and over-\nor underoxidized interfaces would also play a signi\fcant role in PMA [20].\nFuture work con\frming such a nonlinear relationship for a broad range of\ntCoFeB might resolve this issue.\nAt present, there is no doubt that PMA in MgO/CoFeB structures is\nan interface e\u000bect and it is correlated with the presence of oxygen atoms\nat the interface despite the weak spin-orbit coupling [20, 25]. The origin\nof PMA is attributed to hybridization of the O-p with Co(Fe)-d orbitals at\nthe interface [20] and/or to a signi\fcant contribution of thickness dependent\nmagnetoelastic coupling [15]. A deviation of the g-factor from the 2.0 value\nis expressed by g'2\u00004\u0015=\u0001 , where\u0015 < 0 is the spin-orbit constant for\nFe(Co) and \u0001 is the energy levels splitting in the ligand \feld [11]. While\nthe deviation of the g-factor is inversely proportional to \u0001, PMA (and hence\n4\u0019Meff) is proportional to the enhanced spin-orbit-induced splitting around\nthe Fermi level [20]. This may result in a complex relationship between PMA\nandg-factor anisotropy.\nThe Gilbert damping parameter \u000bis evaluated from the dependence of\nthe linewidth \u0001 Hon the resonance frequency as shown in Fig. 4 for the\nin-plane (a) and the out-of-plane (b) con\fgurations. The lines are linear \fts\nto\n\u0001H=\u000b4\u0019f\n\rk;?+ \u0001H0; (4)\n13/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s50/s48/s48/s52/s48/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48\n/s40/s98/s41/s40/s97/s41\n/s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110/s32/s98/s111/s116/s116/s111/s109/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s55/s32\n/s32/s116/s111/s112/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s56/s32/s32/s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41/s105/s110/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110\n/s32/s98/s111/s116/s116/s111/s109/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s55/s32\n/s32/s116/s111/s112/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s56\n/s32/s32/s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41Figure 4: Linewidth as a function of frequency measured in the in-plane con\fguration (a)\nand out-of-plane con\fguration (b). The \u000bdamping parameter is obtained using Eq. (4).\nThe thickness of MgO was 1.25 nm.\nwhere \u0001H0is the inhomogeneous broadening related to CoFeB layer quality.\nThe values of \u000band \u0001H0are shown in Table 1. The top and the bottom layers\nshow almost the same \u000bof 0.017 - 0.018. This suggests that the damping has\nno relation to PMA. While \u0001 H0for the top layer is almost the same for both\ncon\fgurations, \u0001 H0for the bottom layer at the (b) con\fguration is nearly\ntwice as large as that for the (a) con\fguration. Such a behavior suggests\nthat the layer B is rather inhomogeneous with a large angular dispersion of\nmagnetization across the layer [26, 27].\nSpin pumping to Ta layers (which are a part of the bu\u000ber and cap-\n14ping layers, as shown in Fig. 1 (e)) may also in\ruence the damping in\nCoFeB/MgO/CoFeB systems since magnetization precession induces a spin\ncurrent to the adjacent nonmagnetic Ta layers that result in an enhanced\ndamping [8]. This is an interface e\u000bect and hence scales inversely propor-\ntional to the CoFeB layer thickness. Because the bottom layer with an in-\nplane easy axis is thinner than the top layer with a perpendicular easy axis,\nthe spin pumping e\u000bect a\u000bects it more. To estimate spin pumping e\u000bect the\nstandard equation [28] without back\row is used\n\u0001\u000b=g\u0016Bg#\"\n4\u0019Msteff; (5)\nwhereteffis the e\u000bective thickness of CoFeB and g#\"is the mixing con-\nductance. The measured damping of both layers is of 0.017 - 0.018, while\ndamping of a bulk CoFeB is around 0.004 [12]. Therefore, an increase of \u0001 \u000b\ndue to spin pumping is of 0.014 which gives the mixing conductance g#\"= 0:8\nand 1\u00021015cm\u00002for the e\u000bective thickness 0.7 nm and 0.9 nm of B and\nT layer, respectively. The value of mixing conductance g#\"for Ta/CoFeB\ninterface found in the literature lies in a broad range from 1 :67\u00021014to\n1:4\u00021015cm\u00002[29, 30, 31, 32]. Taking into account our simpli\fcation (the\nlack of back\row), this estimation gives the maximal values of mixing conduc-\ntance. Hence, we can conclude that spin pumping substantially in\ruences\nthe damping in our structures. It is worth mentioning that the measured \u000b\nof 0.017 - 0.018 for CoFeB/MgO/CoFeB systems agrees with \u000b= 0:015 for\nthe Ta/CoFeB(1)/MgO structure reported in [3].\nFinally, we would like to make a further comment on postdeposition an-\nnealing of our CoFeB/MgO/CoFeB systems. We found that annealing at\n330oC for 1 hr, beside increasing Msto 1500 G, enhances also PMA so that\n15both layers possess easy axes perpendicular to the plane. 4 \u0019Meffattains\n-1 kG and -4 kG for the B and T layers, respectively. We found that an\nincrease in K?of 7:7\u0002106erg/cm3equally contributes to both layers and,\nfor example, K?= 17\u0002106erg/cm3for the T layer. On the other hand, the\nlinewidth \u0001 Hstrongly broadens to \u0018400 Oe and\u0018700 Oe for the B layer\nand the T layer, respectively. These values are in agreement with recently\nreported values for a similar systems [17]. Moreover, as it is shown in Fig. 5,\n\u0001Hdoes not follow the linear dependence described by Eq. (4). Therefore,\nit is impossible to determine \u000bprecisely for the annealed systems. Such a\nbehavior of \u0001 Hand the decreased remanence with respect to the saturation\nmagnetization (see, [17]) both con\frm a strong angular dispersion of the easy\nPMA axis in both layers. It has been observed that with increasing PMA\nthe dispersion of anisotropy also increases [6, 7, 27]. As a result, dispersion\nin PMA leads to a large two magnon scattering contribution to the linewidth\nfor in-plane magnetization and to an enhanced Gilbert damping [6]. While\nthe magnetic parameters practically do not depend on the MgO thickness in\nas-deposited structures, the annealed structures show a substantial spread in\n4\u0019Meffas it is shown in Fig. 6, which may imply some di\u000berent CoFeB/MgO\ninterfaces due to, for example, boron di\u000busion [30, 33].\n4. Conclusion\nWe investigated the CoFeB/MgO/CoFeB as-deposited systems with the\nin-plane and out-of-plane orthogonal easy axes due to the substantial dif-\nference in PMA for the bottom (B) and the top (T) CoFeB layers, respec-\ntively. The T and the B layer had comparable Gilbert damping \u000bsuggesting\n16/s53 /s49/s48 /s49/s53 /s50/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48\n/s32/s32\n/s32/s116/s111/s112/s32/s67/s111/s70/s101/s66\n/s32/s98/s111/s116/s116/s111/s109/s32/s67/s111/s70/s101/s66/s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41/s105/s110/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110Figure 5: Linewidth as a function of frequency measured in the in-plane con\fguration for\nthe annealed structure. The thickness of MgO was 1.25 nm.\nthat there is no correlation between the Gilbert damping and PMA. We\nalso showed that 4 \u0019Meffcorrelates with the asymmetry in the g-factor (and\nhence with \u0001 \u0016L) and this correlation is highly nonlinear. Annealing enhances\nPMA in both layers but it has detrimental e\u000bect on the linewidth, however.\nTherefore, despite the Gilbert parameter shows no correlation with PMA, it\nseems that there is some correlation between the linewidth (see Eq. 4) and\nPMA in the annealed systems through a combined e\u000bect between dispersion\nof local anisotropy easy axes in crystallites with a high PMA.\n17/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s52/s56/s49/s50/s49/s54/s50/s48\n/s32/s32\n/s32/s49/s46/s50/s53/s32/s110/s109/s32/s77/s103/s79/s32\n/s32/s49/s46/s49/s48/s32/s110/s109/s32/s77/s103/s79/s32\n/s32/s49/s46/s48/s48/s32/s110/s109/s32/s77/s103/s79\n/s32/s48/s46/s57/s48/s32/s110/s109/s32/s77/s103/s79\n/s32/s49/s46/s50/s53/s32/s110/s109/s32/s77/s103/s79/s32\n/s32/s49/s46/s49/s48/s32/s110/s109/s32/s77/s103/s79/s32\n/s32/s49/s46/s48/s48/s32/s110/s109/s32/s77/s103/s79\n/s32/s48/s46/s57/s48/s32/s110/s109/s32/s77/s103/s79/s102/s32/s40/s71/s72/s122/s41\n/s72/s32/s40/s107/s79/s101/s41/s105/s110/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110Figure 6: FMR dispersion relations of CoFeB/MgO(0.9 { 1.25 nm)/CoFeB annealed struc-\nture measured in the in-plane con\fguration.\nAcknowledgments\nWe acknowledge support from the the project \\Marie Sk lodowska-Curie\nResearch and Innovation Sta\u000b Exchange (RISE)\" Contract No. 644348 with\nthe European Commission, as part of the Horizon2020 Programme, and\npartially by the project NANOSPIN PSPB-045/2010 under a grant from\nSwitzerland through the Swiss Contribution to the enlarged European Union.\n18References\n[1] B. 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Lee, and B.-G. Park, \\Ferro-\nmagnetic resonance spin pumping in CoFeB with highly resistive non-\nmagnetic electrodes,\" Current Applied Physics , vol. 14, no. 10, pp. 1344\n{ 1348, 2014.\n[32] Z. Zhu, S. Chen, B. Zhao, Q. Y. Jin, J. Chen, and Z. Zhang, \\Control\nof perpendicular magnetic anisotropy and spin pumping damping in\nMgO/CoFeB/Ta/Pt structures,\" Journal of Physics D: Applied Physics ,\nvol. 50, no. 35, p. 355001, 2017.\n[33] S. S. Mukherjee, D. MacMahon, F. Bai, C.-L. Lee, and S. K. Kurinec,\n\\Study of boron di\u000busion in MgO in CoFeB/MgO \flm stacks using par-\nallel electron energy loss spectroscopy,\" Applied Physics Letters , vol. 94,\nno. 8, p. 082110, 2009.\n24" }, { "title": "1502.04695v2.Role_of_nonlinear_anisotropic_damping_in_the_magnetization_dynamics_of_topological_solitons.pdf", "content": "Role of nonlinear anisotropic damping in the magnetization dynamics of topological solitons\nJoo-V on Kim\u0003\nInstitut d’Electronique Fondamentale, Univ. Paris-Sud, 91405 Orsay, France and\nCNRS, UMR 8622, 91405 Orsay, France\n(Dated: May 31, 2021)\nThe consequences of nonlinear anisotropic damping, driven by the presence of Rashba spin-orbit coupling\nin thin ferromagnetic metals, are examined for the dynamics of topological magnetic solitons such as domain\nwalls, vortices, and skyrmions. The damping is found to a \u000bect Bloch and N ´eel walls di \u000berently in the steady\nstate regime below Walker breakdown and leads to a monotonic increase in the wall velocity above this transition\nfor large values of the Rashba coe \u000ecient. For vortices and skyrmions, a generalization of the damping tensor\nwithin the Thiele formalism is presented. It is found that chiral components of the damping a \u000bect vortex- and\nhedgehog-like skyrmions in di \u000berent ways, but the dominant e \u000bect is an overall increase in the viscous-like\ndamping.\nPACS numbers: 75.60.Ch, 75.70.Kw, 75.75.-c, 75.78.Fg\nI. INTRODUCTION\nDissipation in magnetization dynamics is a longstanding\nproblem in magnetism [1–3]. For strong ferromagnets such as\ncobalt, iron, nickel, and their alloys, a widely used theoretical\napproach to describe damping involves a local viscous form\ndue to Gilbert for the Landau-Lifshitz equation of motion,\n@m\n@t=\u0000\r0m\u0002He\u000b+\u000b0m\u0002@m\n@t; (1)\nwhich appears as the second term on the right hand side, pro-\nportional to the damping constant \u000b0. This equation describes\nthe damped magnetization precession about a local e \u000bective\nfieldHe\u000b=\u0000(1=\u00160Ms)\u000eU=\u000em, which is given by a variational\nderivative of the magnetic energy Uwith respect to the mag-\nnetization field described by the unit vector m, with\r0=\u00160\r\nbeing the gyromagnetic constant and Msis the saturation mag-\nnetization. Despite the multitude of physical processes that\nunderlie dissipation in such materials, such as the scattering\nof magnons with electrons, phonons, and other magnons, the\nform in Eq. (1) has proven to be remarkably useful for describ-\ning a wide range of dynamical phenomena from ferromagnetic\nresonance to domain wall motion.\nOne feature of the dissipative dynamics described in Eq. (1)\nis that it is local, i.e., the damping torque only depends on the\nlocal magnetization and its time dependence. With the ad-\nvent of magnetic heterostructures, however, this restriction of\nlocality has been shown to be inadequate for systems such\nas metallic multilayers in which nonlocal processes can be\nimportant [4]. A striking example involves spin pumping,\nwhich describes how spin angular momentum can be dissi-\npated in adjacent magnetic or normal metal layers through the\nabsorption of spin currents generated by a precessing magne-\ntization [5, 6]. Early experimental observations of this phe-\nnomena involved iron films sandwiched by silver layers [7]\nand permalloy films in close proximity with strong spin-orbit\nnormal metals such as palladium and platinum [8, 9], where\n\u0003joo-von.kim@u-psud.frferromagnetic resonance line widths were shown to depend\nstrong on the composition and thickness of the adjacent lay-\ners. Such observations also spurred other studies involving\nferromagnetic multilayers separated by normal metal spacers,\nwhere spin pumping e \u000bects can lead to a dynamic coupling\nbetween the magnetization in di \u000berent layers [10, 11]. In\nthe context of damping, such dynamic coupling was shown to\ngive rise to a configuration dependent damping in spin-valve\nstructures [12, 13].\nA generalization of the spin-pumping picture in the context\nof dissipation was given by Zhang and Zhang, who proposed\nthat spin currents generated within the ferromagnetic material\nitself can lead to an additional contribution to the damping,\nprovided that large magnetization gradients are present [14].\nThis theory is based on an sdmodel in which the local mo-\nments (4 d) are exchange coupled to the delocalized conduc-\ntion electrons (3 s), which are treated as a free electron gas.\nThe spin current “pumped” at one point in the material by\nthe precessing local moments are dissipated at another if the\ncurrent encounters strong spatial variations in the magneti-\nzation such as domain walls or vortices – a mechanism that\ncan be thought of as the reciprocal process of current-induced\nspin torques in magnetic textures [15–18]. For this reason,\nthe mechanism is referred to as “feedback” damping since the\npumped spin currents generated feed back into the magnetiza-\ntion dynamics in the form of a dissipative torque. This addi-\ntional contribution is predicted to be both nonlinear and non-\nlocal, and can have profound consequences for the dynamics\nof topological solitons such as domain walls and vortices as a\nresult of the spatial gradients involved. Indeed, recent experi-\nments on vortex wall motion in permalloy stripes indicate that\nsuch nonlinear contributions can be significant and be of the\nsame order of magnitude as the usual Gilbert damping char-\nacterized by \u000b0in Eq. (1) [19].\nAn extension to this feedback damping idea was proposed\nrecently by Kim and coworkers, who considered spin pump-\ning involving a conduction electron system with a Rashba\nspin-orbit coupling (RSOC) [20]. By building upon the\nZhang-Zhang formalism, it was shown that the feedback\ndamping can be expressed as a generalization of the Landau-arXiv:1502.04695v2 [cond-mat.mtrl-sci] 4 Jun 20152\nLifshitz equation [14, 20],\n@m\n@t=\u0000\r0m\u0002He\u000b+m\u0002D LL(m)\u0001@m\n@t; (2)\nwhere the 3\u00023 matrixDLLrepresents the generalized damping\ntensor, which can be expressed as [20]\nDi j\nLL=\u000b0\u000ei j+\u0011X\nk(Fki+˜\u000bR\u000f3ki)\u0010\nFk j+˜\u000bR\u000f3k j\u0011\n:(3)\nHere,\u000b0is the usual Gilbert damping constant, \u0011=\ng\u0016B~G0=(4e2Ms) is a constant related to the conductivity G0\nof the spin bands [14], Fki=(@m=@xk)iare components of\nthe spatial magnetization gradient, ˜ \u000bR=2\u000bRme=~2is the\nscaled Rashba coe \u000ecient,\u000fi jkis the Levi-Civita symbol, and\nthe indices ( i jk) represent the components ( xyz) in Cartesian\ncoordinates. In addition to the nonlinearity present in the\nZhang-Zhang picture, the inclusion of the \u000bRterm results\nin an anisotropic contribution that is related to the underly-\ning symmetry of the Rashba interaction. Numerical estimates\nbased on realistic parameters suggest that the Rashba con-\ntribution can be much larger than the nonlinear contribution\n\u0011alone [20], which may have wide implications for soliton\ndynamics in ultrathin ferromagnetic films with perpendicular\nmagnetic anisotropy, such as Pt /Co material systems, in which\nlarge spin-orbit e \u000bects are known to be present.\nIn this article, we explore theoretically the consequences\nof the nonlinear anisotropic damping given in Eq. (3) on the\ndynamics of topological magnetic solitons, namely domain\nwalls, vortices, and skyrmions, in which spatial gradients can\ninvolve 180\u000erotation of the magnetization vector over length\nscales of 10 nm. In particular, we examine the role of chiral-\nity in the Rashba-induced contributions to the damping, which\nare found to a \u000bect chiral solitons in di \u000berent ways. This ar-\nticle is organized as follows. In Section II, we discuss the\ne\u000bects of nonlinear anisotropic damping on the dynamics of\nBloch and N ´eel domain walls, where the latter is stabilized\nby the Dzyaloshinskii-Moriya interaction. In Section III, we\nexamine the consequences of this damping for vortices and\nskyrmions, and we derive a generalization to the damping\ndyadic appearing in the Thiele equation of motion. Finally,\nwe present some discussion and concluding remarks in Sec-\ntion IV.\nII. BLOCH AND N ´EEL DOMAIN WALLS\nThe focus of this section are domain walls in ultrathin\nfilms with perpendicular magnetic anisotropy. Consider a\n180\u000edomain wall representing a boundary separating two\noppositely magnetized domains along the xaxis, with zbe-\ning the uniaxial anisotropy axis that is perpendicular to the\nfilm plane. We assume that the magnetization remains uni-\nform along the yaxis. The unit magnetization vector m(x;t)\ncan be parametrized in spherical coordinates ( \u0012;\u001e), such that\nm=(sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012). With this definition, thespherical angles for the domain wall profile can be written as\n\u0012(x;t)=2 tan\u00001exp \n\u0006x\u0000X0(t)\n\u0001!\n;\n\u001e(x;t)=\u001e0(t); (4)\nwhere X0(t) denotes the position of the domain wall, \u0001 =pA=K0represents the wall width parameter that depends on\nthe exchange constant Aand the e \u000bective uniaxial anisotropy\nK0, and the azimuthal angle \u001e0(t) is a dynamic variable but\nspatially uniform. The anisotropy constant, K0=Ku\u0000\n\u00160M2\ns=2, involves the di \u000berence between the magnetocrys-\ntalline ( Ku) and shape anisotropies relevant for an ultrathin\nfilm. In this coordinate system, a static Bloch wall is given by\n\u001e0=\u0006\u0019=2, while a static N ´eel wall is given by \u001e0=0;\u0019. A\npositive sign in the argument of the exponential function for\n\u0012in Eq. (4) describes an up-to-down domain wall profile go-\ning along the +xdirection, while a negative sign represents a\ndown-to-up wall.\nTo determine the role of the nonlinear anisotropic damping\nterm in Eq. (3) on the wall dynamics, it is convenient to com-\npute the dissipation function W(˙X0;˙\u001e0) for the wall variables,\nwhere the notation ˙X0\u0011@tX0, etc., denotes a time derivative.\nThe dissipation function per unit surface area is given by\nW\u0010˙X0;˙\u001e0\u0011\n=Ms\n2\rZ1\n\u00001dx˙miDi j\nLL(m) ˙mj; (5)\nwhere mi=mi\u0002x\u0000X0(t);\u001e0(t)\u0003and the Einstein summation\nconvention is assumed. By using the domain wall ansatz\n(4), the integral in Eq. (5) can be evaluated exactly to give\nW=W0+WNL, where W0represents the usual (linear) Gilbert\ndamping,\nW0=\u000b0Ms\u0001\n\r0BBBB@˙X2\n0\n\u00012+˙\u001e2\n01CCCCA; (6)\nwhile WNLis the additional contribution from the nonlinear\nanisotropic damping,\nWNL=Ms\u0001\n\r2666641\n3\u000b3sin2\u001e0(t)˙X2\n0\n\u00012\n+ 2\n3\u000b1\u0006\u0019\n2\u000b2cos\u001e0(t)+\u000b3cos2\u001e0(t)!\n˙\u001e2\n0#\n;(7)\nwhere\u000b1\u0011\u0011=\u00012,\u000b2\u0011\u0011˜\u000bR=\u0001, and\u000b3\u0011\u0011˜\u000b2\nRare dimen-\nsionless nonlinear damping constants. In contrast to the linear\ncase, the nonlinear anisotropic dissipation function exhibits a\nconfiguration-dependent dissipation rate where the prefactors\nof the ˙X2\n0and˙\u001e2\n0terms depend explicitly on \u001e0(t).\nIn addition to the nonlinearity a chiral damping term, pro-\nportional to \u000b2, appears as a result of the Rashba interaction\nand is linear in the Rashba coe \u000ecient\u000bR. The sign of this\nterm depends on the sign chosen for the polar angle \u0012in the\nwall profile (4). To illustrate the chiral nature of this term, we\nconsider small fluctuations about the static configuration by\nwriting\u001e0(t)=\u001e0+\u000e\u001e(t), where\u000e\u001e(t)\u001c\u0019is a small angle.\nThis approximation is useful for the steady state regime below3\nWalker breakdown. For up-to-down Bloch walls ( \u001e0=\u0006\u0019=2),\nthe nonlinear part of the dissipation function to first order in\n\u000e\u001e(t) becomes\nWNL;Bloch\u0019Ms\u0001\n\r266664\u000b3\n3˙X2\n0\n\u00012+ 2\u000b1\n3+Cx\u0019\u000b2\n2\u000e\u001e(t)!\n˙\u001e2\n0377775:(8)\nThe quantity Ci=\u00061 is a component of the chirality vec-\ntor [21],\nC=1\n\u0019Z1\n\u00001dxm\u0002@xm; (9)\nwhich characterizes the handedness of the domain wall. For\na right-handed Bloch wall, \u001e0=\u0000\u0019=2 and the only nonva-\nnishing component is Cx=1, while for a left-handed wall\n(\u001e0=\u0000\u0019=2) the corresponding value is Cx=\u00001. Thus, the\nterm proportional to \u000b2depends explicitly on the wall chiral-\nity. Similarly for up-to-down N ´eel walls, the same lineariza-\ntion about the static wall profile leads to\nWNL;Neel\u0019Ms\u0001\n\r 2\u000b1\n3+Cy\u0019\u000b2\n2+\u000b3!\n˙\u001e2\n0; (10)\nwhere Cy=1 for a right-handed N ´eel wall (\u001e0=0) and\nCy=\u00001 for its left-handed counterpart ( \u001e0=\u0019). Since the\nfluctuation\u000e\u001e(t) is taken to be small, the chiral damping term\nis more pronounced for N ´eel walls in the steady-state velocity\nregime since it does not depend on the fluctuation amplitude\n\u000e\u001e(t) as in the case of Bloch walls.\nTo better appreciate the magnitude of the chirality-\ndependent damping term, it is instructive to estimate numer-\nically the relative magnitudes of the nonlinear damping con-\nstants\u000b1;\u000b2;\u000b3. Following [Ref. 20], we assume \u0011=0:2 nm2\nand\u000bR=10\u000010eV m. If we suppose \u0001 = 10 nm, which is\nconsistent with anisotropy values measured in ultrathin films\nwith perpendicular anisotropy [22], the damping constants can\nbe evaluated to be \u000b1=0:002,\u000b2=0:052, and\u000b3=1:37.\nSince\u000b0varies between 0.01–0.02 [23] and 0.1–0.3 [24] de-\npending on the material system, the chiral term \u000b2is compa-\nrable to Gilbert damping in magnitude, but remains almost an\norder of magnitude smaller than the nonlinear component \u000b3\nthat provides the dominant contribution to the overall damp-\ning.\nThe full equations of motion for the domain wall dynam-\nics can be obtained using a Lagrangian formalism that ac-\ncounts for the dissipation given by W[25, 26]. For the sake\nof simplicity, we will focus on wall motion driven by mag-\nnetic fields alone, where a spatially-uniform magnetic field\nHzis applied along the +zdirection. In addition, we include\nthe Dzyaloshinskii-Moriya interaction appropriate for the ge-\nometry considered [27, 28] when considering the dynamics\nof N ´eel walls. From the Euler-Lagrange equations with the\nRayleigh dissipation function,\nd\ndt@L\n@˙X0\u0000@L\n@X0+@W\n@˙X0=0; (11)\nwith an analogous expression for \u001e0, the equations of motion\nfor the wall coordinates are found to be\n˙\u001e0+\u0012\n\u000b0+\u000b3\n3sin2\u001e0\u0013˙X0\n\u0001=\r0Hz; (12)\n0 5 10 15 20 \nm0Hz (mT)010 20 30 Wall Velocity (m/s) \n0 0.1 0.2 0.3 0.4 \naR (eV nm)0.9 0.95 1vw / v w,0\naR (eV nm)aR = 0 0.05 eV nm 0.1 eV nm 0.15 eV nm (a)\n(b) (c)\n0 0.1 0.2 0.3 0.4 0.15 0.2 0.25 df w / pFIG. 1. (Color online) Bloch wall dynamics. (a) Steady-state domain\nwall velocity,h˙X0i, as a function of perpendicular applied magnetic\nfield,\u00160Hz, for several values of the Rashba coe \u000ecient,\u000bR. The\nhorizontal dashed line indicates the Walker velocity and the arrows\nindicate the Walker transition. (b) The ratio between the Walker ve-\nlocity, vW, to its linear damping value, vW;0, as a function of \u000bR. (c)\nDeviation in the wall angle from rest at the Walker velocity, \u000e\u001eW, as\na function of \u000bR\n˙X0\n\u0001\u0000 \n\u000b0+2\u000b1\n3+\u0019\u000b2\n2cos\u001e0+\u000b3cos2\u001e0!\n˙\u001e0\n=\u0000\r0 \u0019\n2Dex\n\u00160Ms\u0001+2K?\n\u00160Mscos\u001e0!\nsin\u001e0;(13)\nwhere Dexis the Dzyaloshinskii-Moriya constant [28] and\nK?represents a hard-axis anisotropy that results from vol-\nume dipolar charges. The Dzyaloshinskii-Moriya interaction\n(DMI) is present in ultrathin films in contact with a strong\nspin-orbit coupling material [29, 30] and favors a N ´eel-type\nwall profile [31, 32]. The DMI itself can appear as a con-\nsequence of the Rashba interaction and therefore its inclu-\nsion here is consistent with the nonlinear anisotropic damping\nterms used [20, 33, 34].\nResults from numerical integration of these equations of\nmotion for Bloch and N ´eel walls are presented in Figs. 1 and\n2. We used parameters consistent with ultrathin films with\nperpendicular anisotropy, namely \u000b0=0:1,Ms=1 MA /m,\n\u0001 = 10 nm, and K?=\u00160NxM2\ns=2 with the demagnetiza-\ntion factor Nx=0:02 [28]. To study the dynamics of the\nDzyaloshinskii (N ´eel) wall we assumed a value of Dex=1\nmJ/m2, which is much stronger than the volume dipolar in-\nteraction represented by K?and is of the same order of mag-\nnitude as values determined by Brillouin light spectroscopy\nin Pt/Co/Al2O3films [35]. As in the discussion on numeri-\ncal estimates above, we assumed \u0011=0:2 nm2but considered\nseveral di \u000berent values for the Rashba coe \u000ecient\u000bR. The\nsteady-state domain wall velocity, h˙X0i, was computed as a\nfunction of the perpendicular applied magnetic field, Hz. In4\nthe precessional regime above Walker breakdown in which\n\u001e0(t) becomes a periodic function in time, h˙X0iis computed\nby averaging the wall displacement over few hundred periods\nof precession.\nFor the Bloch case [Fig. 1(a)], the Walker field is observed\nto increase with the Rashba coe \u000ecient, which is consistent\nwith the overall increase in damping experienced by the do-\nmain wall. However, there are two features that di \u000ber qual-\nitatively from the behavior with linear damping. First, the\nWalker velocity is not attained for finite \u000bR, where the peak\nvelocity at the Walker transition is below the value reached\nfor\u000bR=0. This is shown in more detail in Fig. 1(b), where\nthe ratio between the Walker velocity, vW, and its linear damp-\ning value, vW;0, is shown as a function of \u000bR. The Walker limit\nis set by the largest extent to which the wall angle \u001e0can de-\nviate from its equilibrium value, \u001e0;eq. By assuming ˙\u001e=0\nin the linear regime, we can determine this limit by rearrang-\ning Eqs. 12 and 13 to obtain the following relationship for the\nBloch wall,\n2Hz\nNxMs=\u0000\u0012\n\u000b0+\u000b3\n3sin2\u001e0\u0013\nsin 2\u001e0: (14)\nThe angle\u001e0=\u001eWfor which the right hand side of this\nequation is an extremum determines the Walker limit. In\nFig. 1(c), we present this limit in terms of the deviation an-\ngle,\u000e\u001eW\u0011j\u001eW\u0000\u001e0;eqj, which is shown as a function of \u000bR.\nAs the Rashba parameter is increased, the maximum wall tilt\npossible in the linear regime decreases from the linear damp-\ning value of \u0019=4, which results in an overall reduction in the\nWalker velocity. Second, the field dependence of the wall ve-\nlocity below Walker breakdown is nonlinear and exhibits a\nslight convex curvature, which becomes more pronounced as\n\u000bRincreases. This curvature can be understood by examining\nthe wall mobility under fields, which can be deduced from Eq.\n(12) by setting ˙\u001e=0,\n˙X0=\r0\u0001\n\u000b0+(\u000b3=3)sin2\u001e0Hz: (15)\nSince the angle \u001e0for Bloch walls varies from its rest value of\n\u001e0;eq=\u0006\u0019=2 at zero field to \u001eWat the Walker field, the sin2\u001e0\nterm in the denominator decreases from its maximum value of\nsin2\u001e0;eq=1 at rest with increasing applied field and therefore\nan increase in the mobility is seen as Hzincreases, resulting\nin the convex shape of the velocity versus field relation below\nWalker breakdown.\nIt is interesting to note that the nonlinear damping terms\na\u000bect the Dzyaloshinskii (N ´eel) wall motion di \u000berently. In\ncontrast to the Bloch case, the Walker velocity for increasing\n\u000bRslightly exceeds the linear damping value, which can be\nseen by the arrows marking the Walker transition in Fig. 2(a)\nand in detail in Fig. 2(b). In addition, the field dependence of\nthe velocity exhibits a concave curvature below breakdown,\nwhich can also be understood from Eq. (15) by considering\nthat\u001e0instead deviates from the rest value of \u001e0;eq=0 or\u0019\nat zero field. As for the Bloch wall case, the deviation angle\nat breakdown is determined by the value of \u001e0that gives an\n0 50 100 150 200 0100 200 300 Wall Velocity (m/s) \n0 0.1 0.2 0.3 0.4 11.0008 aR = 0 0.05 eV nm 0.1 eV nm 0.15 eV nm vw / v w,0\naR (eV nm) aR (eV nm)m0Hz (mT)(a)\n(b)(c)\n0 0.1 0.2 0.3 0.4 0.5 0.51 0.52 0.53 df w / pFIG. 2. (Color online) Dzyaloshinskii (N ´eel) wall dynamics. (a)\nSteady-state domain wall velocity, h˙X0i, as a function of perpendic-\nular applied magnetic field, \u00160Hz, for several values of the Rashba\ncoe\u000ecient,\u000bR. The horizontal dashed line indicates the Walker ve-\nlocity and the arrows indicate the Walker transition. (b) The ratio\nbetween the Walker velocity, vW, to its linear damping value, vW;0, as\na function of \u000bR. (c) The wall angle at the Walker velocity, \u001eW, as a\nfunction of \u000bR\nextremum for the right hand side of\n2Hz\nNxMs=\u0000\u0012\n\u000b0+\u000b3\n3sin2\u001e0\u0013 \u0019Dex\n2K?\u0001cos\u001e0+sin 2\u001e0!\n;(16)\nand is also seen to decrease with increasing Rashba coe \u000e-\ncient [Fig. 2(c)]. In contrast to the Bloch wall case, how-\never, changes in \u001eWhave a comparatively modest e \u000bect on the\nWalker velocity. The shape of the velocity versus field curve\nis consistent with experimental reports of field-driven domain\nwall motion in the Pt /Co (0.6 nm) /Al2O3system [36], which\npossess a large DMI value [35] and harbors N ´eel-type domain\nwall profiles at equilibrium [37].\nAs the preceding discussion shows, the di \u000berences in the\nfield dependence of the wall velocity for the two profiles are\na result of the DMI, rather than the chiral damping term that\nis proportional to \u000b2. This was verified by setting \u000b2=0\nfor the N ´eel wall case with D,0, which did not modify\nthe overall behavior of the field dependence of the velocity. In\nthe one-dimensional approximation for the wall dynamics, the\nDMI enters the equations of motion like an e \u000bective magnetic\nfield along the xaxis, which stabilizes the wall structure by\nminimizing deviations in the wall angle \u001e0(t).\nIII. VORTICES AND SKYRMIONS\nThe focus of this section is on the dissipative dynamics\nof two-dimensional topological solitons such as vortices and\nskyrmions. The equilibrium magnetization profile for these5\nmicromagnetic objects are described by a nonlinear di \u000ber-\nential equation similar to the sine-Gordon equation, where\nthe dispersive exchange interaction is compensated by dipo-\nlar interactions for vortices [38, 39] and an additional uniax-\nial anisotropy for skyrmions [40]. The topology of vortices\nand skyrmions can be characterized by the skyrmion winding\nnumber Q,\nQ=1\n4\u0019\"\ndxdy m\u0001\u0010\n@xm\u0002@ym\u0011\n: (17)\nWhile the skyrmion number for vortices ( Q=\u00061=2) and\nskyrmions ( Q=\u00061) are di \u000berent, their dynamics are quali-\ntatively similar and can be described using the same formal-\nism. For this reason, vortices and skyrmions will be treated\non equal footing in what follows and distinctions between the\ntwo will only be drawn on the numerical values of the damp-\ning parameters.\nA key approximation used for describing vortex or\nskyrmion dynamics is the rigid core assumption, where it is\nassumed that the spin structure of the soliton remains unper-\nturbed from its equilibrium state during motion. Within this\napproximation, the dynamics is given entirely by the position\nof the core in the film plane, X0(t)=[X0(t);Y0(t)], which al-\nlows the unit magnetization vector to be parametrized as\n\u0012(x;y;t)=\u00120[kx\u0000X0(t)k];\n\u001e(x;y;t)=qtan\u00001\"y\u0000Y0(t)\nx\u0000X0(t)#\n+c\u0019\n2; (18)\nwhere qis a topological charge and cis the chirality. An il-\nlustration of the magnetization field given by the azimuthal\nangle\u001e(x;y) is presented in Fig. 3. q=1 corresponds to a\nvortex or skyrmion, while q=\u00001 represents the antivortex or\nantiskyrmion.\nThe dynamics of a vortex or skyrmion in the rigid core ap-\nproximation is given by the Thiele equation,\nG\u0002˙X0+DT\u0001˙X0=\u0000@U\n@X0; (19)\nwhere\nG=Msd\n\r\"\ndxdy sin(\u0012)(r\u0012\u0002r\u001e) (20)\nis the gyrovector and U(X0) is the e \u000bective potential that is ob-\ntained from the magnetic Hamiltonian by integrating out the\nspatial dependence of the magnetization. The damping dyadic\nin the Thiele equation, DT, can be obtained from the dissipa-\ntion function in the rigid core approximation, W(˙X0), which is\ndefined in the same way as in Eq. (5) but with the ansatz given\nin Eq. (18). For this system, it is more convenient to eval-\nuate the dyadic by performing the integration over all space\nafter taking derivatives with respect to the core velocity. In\nother words, the dyadic can be obtained using the Lagrangian\nformulation by recognizing that\nDT\u0001˙X0=Msd\n2\r\"\ndxdy@\n@˙X0\u0010\n˙miDi j\nLL(m) ˙mj\u0011\n: (21)\nc = 0\nq = +1\nq = –1c = 1 c = 2 c = 3 (a)\n(b) (c)\n1\n– 1 0FIG. 3. (Color online) In-plane magnetization fields for vortices and\nskyrmions. (a) Vector fields given by \u001e(x;y) in (18) for di \u000berent\nvalues of qandc. (b) V ortex and (c) skyrmion for spin structure with\nc=1;q=1, where the arrows indicate the in-plane components\n(mx;y) and the color code gives the perpendicular component of the\nmagnetization ( mz).\nBy using polar coordinates for the spatial coordinates, ( x;y)=\n(rcos';rsin'), assuming translational invariance in the film\nplane, and integrating over ', the damping dyadic is found to\nbe\nDT=Msd\n\r \n(\u000b0D0+\u000b1D1+\u000b3D3)I+\u000b2D2\"\na110\n0a22#!\n;\n(22)\nwhereIis the 2\u00022 identity matrix and the dimensionless\ndamping constants are defined as \u000b1\u0011\u0011=r2\nc,\u000b2\u0011\u0011˜\u000bR=rc, and\n\u000b3\u0011\u0011˜\u000b2\nR, in analogy with the domain wall case where the\ncore radius rcplays the role here as the characteristic length\nscale. The coe \u000ecients Didepend on the core profile and are\ngiven by\nD0=\u0019Z1\n0dr \nr(@r\u00120)2+sin2\u00120\nr!\n; (23)\nD1=2\u0019r2\ncZ1\n0dr1\nr(@r\u00120)2sin2\u00120; (24)\nD2=2\u0019rcZ1\n0dr1\nr(@r\u00120)sin\u00120(r(@r\u00120)cos\u00120+sin\u00120);\n(25)\nD3=\u0019Z1\n0dr \nr(@r\u00120)2cos2\u00120+sin2\u00120\nr!\n; (26)\nwhere the expression for D0is a known result but D1;D2and\nD3are new terms that arise from the nonlinear anisotropic\ndamping due to RSOC.\nThe coe \u000ecients a11anda22are configuration-dependent\nand represent the chiral component of the Rashba-induced\ndamping. For vortex-type spin textures ( c=1;3 and q=1),6\nTABLE I. Coe \u000ecients a11anda22of the chiral damping term in\nEq. (22) for di \u000berent vortex /skyrmion charges qand chirality c.\nq=1 q=\u00001\nc 0 1 2 3 0 1 2 3\na11 1 0\u00001 0\u00001\u00001 1 1\na22 1 0\u00001 0 1\u00001\u00001 1\na11=a22=0, which indicates that the \u000b2term plays no role\nfor such configurations. This is consistent with the result for\nBloch domain walls discussed previously, since the vortex-\ntype texture [Fig. 3(b)], particularly the vortex-type skyrmion\n[Fig. 3(c)], can be thought of as being analogous to a spin\nstructure generated by a 2 \u0019revolution of a Bloch wall about\nan axis perpendicular to the film plane. The rigid core approx-\nimation implies that fluctuations about the ground state are ne-\nglected, which is akin to setting \u000e\u001e(t)=0 in Eq. (8). As such,\nno contribution from \u000b2is expected for vortex-type textures.\nOn the other hand, a finite contribution appears for hedgehog-\ntype vortices and skyrmions ( q=1), where a11=a22=1\nforc=0 and a11=a22=\u00001 for c=2. This can be un-\nderstood with the same argument by noting that hedgehog-\ntype textures can be generated by revolving N ´eel-type domain\nwalls. A summary of these coe \u000ecients is given in Table I.\nFor antivortices ( q=\u00001), it is found that the coe \u000ecients\naiiare nonzero for all winding numbers considered. We can\nunderstand this qualitatively by examining how the magneti-\nzation varies across the core along two orthogonal directions.\nFor example, for c=0, the variation along the xandyaxes\nacross the core are akin to two N ´eel-type walls of di \u000berent\nchiralities, which results in nonvanishing contributions to a11\nanda22but with opposite sign. The sign of these coe \u000ecients\ndepends on how these axes are oriented in the film plane, as\nwitnessed by the di \u000berent chiralities cin Fig. 3. Such damping\ndynamics is therefore strongly anisotropic, which may have\ninteresting consequences on the rotational motion of vortex-\nantivortex dipoles, for example, where the antivortex configu-\nration oscillates between the di \u000berent cvalues in time [41].\nFor vortex structures, we can provide numerical estimates\nof the di \u000berent damping contributions \u000biDiby using the Usov\nansatz for the vortex core magnetization,\ncos\u00120=8>>>><>>>>:r2\nc\u0000r2\nr2c+r2r\u0014rc\n0 r>rc: (27)\nLetLrepresent the lateral system size. The coe \u000ecients Diare\nthen found to be D0=\u0019[2+ln(L=rc)],D1=D2=14\u0019=3,\nandD3=\u0019[4=3+ln(L=rc)]. We note that for D0andD3,\nthe system size Land core radius rcappear as cuto \u000bs for the\ndivergent 1=rterm in the integral. By assuming parameters of\n\u000b0=0:1,\u0011=0:05 nm2, and\u000bR=0:1 eV nm, along with\ntypical scales of rc=10 nm and L=1\u0016m, the damping\nterms can be evaluated numerically to be \u000b0D0\u00192:1,\u000b1D1\u0019\n0:0073,\u000b2D2\u00190:19, and\u000b3D3\u00196:4. As for the domain\nwalls, the Rashba term \u000b3D3is the dominant contribution and\nis of the same order of magnitude as the linear damping term,while the chiral term \u000b2D2is an order of magnitude smaller\nand the nonlinear term \u000b1D1is negligible in comparison.\nFor skyrmion configurations, a similar ansatz can be used\nfor the core magnetization,\ncos\u0012\u00120\n2\u0013\n=8>>>><>>>>:r2\nc\u0000r2\nr2c+r2r\u0014rc\n0 r>rc: (28)\nWe note that this di \u000bers from the “linear” profiles discussed\nelsewhere [40], but the numerical di \u000berences are small and do\nnot influence the qualitative features of the dynamics. The ad-\nvantage of the ansatz in Eq. (28) is that the integrals for Di\nhave simple analytical expressions. Because spatial variations\nin the magnetization for a skyrmion are localized only to the\ncore, in contrast to the circulating in-plane moments of vor-\ntices that extend across the entire system, the damping con-\nstants Dihave no explicit dependence on the system size. By\nusing Eq. (28), we find D0=D3=16\u0019=3,D1=496\u0019=15, and\nD2=52\u0019=5. By using the same values of \u000b0,\u0011, and\u000bRas for\nthe vortices in the preceding paragraph, we find \u000b0D0\u00191:7,\n\u000b1D1\u00190:052,\u000b2D2\u00190:43, and\u000b3D3\u00193:3.\nIV . DISCUSSION AND CONCLUDING REMARKS\nA clear consequence of the nonlinear anisotropic damp-\ning introduced in Eq. (3) is that it provides a mechanism by\nwhich the overall damping constant, as extracted from domain\nwall experiments, for example, can di \u000ber from the value ob-\ntained using linear response methods such as ferromagnetic\nresonance [19]. However, the Rashba term can also a \u000bect the\nferromagnetic linewidth in a nontrivial way. To see this, we\nconsider the e \u000bect of the damping by evaluating the dissipa-\ntion function associated with a spin wave propagating in the\nplane of a perpendicularly magnetized system with an ampli-\ntude b(t) and wave vector kjj. The spin wave can be expressed\nasm=\u0002b(t) cos( kjj\u0001rjj);b(t) sin(kjj\u0001rjj);1\u0003, which results in a\ndissipation function per unit volume of\nWsw=Ms\n2\r˙b(t)2\u0010\n\u000b0+\u000b3+\u0011b(t)2kkjjk2\u0011\n; (29)\nwhere a term proportional to the chiral part \u0011˜\u000bRspatially aver-\nages out to zero. The Rashba contribution \u000b3\u0011\u0011˜\u000b2\nRleads to\nan overall increase in the damping for linear excitations and\nplays the same role as the usual Gilbert term \u000b0in this ap-\nproximation, which allows us to assimilate the two terms as\nan e\u000bective FMR damping constant, \u000bFMR\u0019\u000b0+\u000b3. On\nthe other hand, the nonlinear feedback term proportional to\n\u0011is only important for large spin wave amplitudes and de-\npends quadratically on the wave vector. This is consistent\nwith recent experiments on permalloy films (in the absence of\nRSOC) in which the linear Gilbert damping was recovered in\nferromagnetic resonance while nonlinear contributions were\nonly seen for domain wall motion [19]. This result also sug-\ngests that the large damping constant in ultrathin Pt /Co/Al2O3\nfilms as determined by similar time-resolved magneto-optical7\nmicroscopy experiments, where it is found that \u000bFMR=0:1–\n0:3 [24], may partly be due to the RSOC mechanism described\nhere (although dissipation resulting from spin pumping into\nthe platinum underlayer is also likely to be important [42]).\nIncidentally, the nonlinear term \u0011b(t)2may provide a physi-\ncal basis for the phenomenological nonlinear damping model\nproposed in the context of spin-torque nano-oscillators [43].\nFor vortices and skyrmions, the increase in the overall\ndamping due to the Rashba term \u000b3can have important con-\nsequences for their dynamics. The gyrotropic response to any\nforce, as described by the Thiele equation in Eq. (19), depends\non the overall strength of the damping term. This response\ncan be characterized by a deflection angle, \u0012H, that describes\nthe degree to which the resulting displacement is noncollinear\nwith an applied force. This is analogous to a Hall e \u000bect. By\nneglecting the chiral term \u000b2D2, the deflection or Hall angle\ncan be deduced from Eq. (19) to be\ntan\u0012H=G0\n\u000b0D0+\u000b1D1+\u000b3D3; (30)\nwhere G0=2\u0019for vortices and G0=4\u0019for skyrmions. Con-\nsider the skyrmion profile and the magnetic parameters dis-\ncussed in Section III. With only the linear Gilbert damping\nterm (\u000b0D0) the Hall angle is found to be \u0012H=82:3\u000e, which\nunderlies the largely gyrotropic nature of the dynamics. If\nthe full nonlinear damping is taken into account [Eq. (30)],\nwe find\u0012H=68:3\u000e, which represents a significant reduction\nin the Hall e \u000bect and a greater Newtonian response to an ap-\nplied force. Aside from a quantitative increase in the overalldamping, the presence of the nonlinear terms can therefore af-\nfect the dynamics qualitatively. Such considerations may be\nimportant for interpreting current-driven skyrmion dynamics\nin racetrack geometries, where the interplay between edge re-\npulsion and spin torques is crucial for determining skyrmion\ntrajectories [44, 45].\nFinally, we conclude by commenting on the relevance of\nthe chiral-dependent component of the damping term, \u000b2. It\nhas been shown theoretically that the Rashba spin-orbit cou-\npling leading to Eq. (3) also gives rise to an e \u000bective chiral\ninteraction of the Dzyaloshinskii-Moriya form [34]. This in-\nteraction is equivalent to the interface-driven form considered\nearlier, which favors monochiral N ´eel wall structures in ul-\ntrathin films with perpendicular magnetic anisotropy. 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Young supernova remnants (SNRs) exhibit narrow filaments of non-thermal X-ray emission whose widths can be limited\neither byelectronenergy losses or damping of the magnetic fi eld.\nAims.Wewant toinvestigate whether or not di fferent models of these filamentscan be observationally teste d.\nMethods. Usingobservationalparametersoffourhistoricalremnant s,wecalculatethefilamentprofilesandcomparethespectrao fthe\nfilaments with those of the total non-thermal emission. For t hat purpose, we solve an one-dimensional stationary transp ort equation\nfor the isotropic differential number densityof the electrons.\nResults.Wefindthat the difference betweenthe spectra of filamentand totalnon-thermal emission above 1keV ismore pronounced\ninthe damping model than inthe energy-loss model.\nConclusions. A considerable damping of the magnetic field can result in an o bservable difference between the spectra of filament\nand total non-thermal emission, thus potentially permitti ng an observational discrimination between the energy-los s model and the\ndamping model of theX-rayfilaments.\nKey words. acceleration of particles -supernova remnants -X-rays: IS M-ISM:magnetic fields\n1. Introduction\nBased on simple energetic considerations regarding the ene rgy\ndensityof cosmic raysandthe energyrelease per supernovae x-\nplosion, SNRs have long been thought to be sources of galacti c\ncosmicrays. Thispresumptionis supportedby numerousdete c-\ntionsofnon-thermalemissionin the radioandX-rayband(e. g.,\nKoyamaetal. 1995;Slaneetal. 1999,2001;Bambaetal. 2000)\nobserved from the direction of known SNRs and interpreted to\nbe synchrotron radiation of relativistic electrons with en ergies\nupto100TeV.\nHigh-resolution observations of young SNRs performed\nwith the Chandra satellite show that this emission of non-\nthermalradiationisconcentratedinnarrowregionsonthel imbs\n(Vink&Laming2003;Bambaet al. 2005).Theseregionsofin-\ncreased synchrotron emissivity close to the forward shock a re\ncalled filaments and demonstrate the presence of high-energ y\nelectronsaroundtheiraccelerationsites.\nThe most plausible process for the acceleration of elec-\ntrons is diffusive shock acceleration (DSA), which leads to a\npower-law distribution of particles (e.g., Bell 1978; Blan dford\n& Ostriker 1978; Drury 1983). Although no clear evidence\nfor relativistic-ion acceleration exists at shocks, the DS A-\nmechanism is also considered to be responsible for the accel -\neration of cosmic-ray nuclei, as indicated by observations of\nnon-relativistic ion acceleration at solar-wind shocks dr iven by\ncoronal mass ejections (Rouillard et al. 2011). However, ma ny\ndetails of the DSA are still vague such as the maximum energy\nofparticles, the roleof themagneticfield, andhowthe parti cles\nare injected into the accelerationprocess(also referredt o as the\ninjection problem). Apparently, the investigation of the p roper-\nties of the non-thermal filaments may provide key informatio n\nfor a better understanding of the DSA-mechanism. In particu -lar,knowingthemagnetic-fieldstrengthgivesconstraints onthe\nmaximumparticleenergyachievableintheaccelerationpro cess,\nwhich can help answering the question whether SNRs can ac-\ncelerate particles to energies above the knee in the cosmic- ray\nspectrum.\nAccurateanalysisofseveralSNRsshowsthatthefilamentary\nstructuresareverythincomparedwiththeradiioftheremna nts.\nThis limitation of the filament widths is associated with a ra pid\ndecrease of the synchrotronemissivity that can be explaine dby\nenergy losses of the electrons in a locally enhanced magneti c\nfield.Anumberofauthorshaveusedthatmodeltoconstrainth e\nmagnetic-field strength, the degree of turbulence and the ob liq-\nuity (e.g., Bamba et al. 2003; Parizot et al. 2006; Araya et al .\n2010).Arayaetal.(2010),forinstance,haveinvestigated several\nfilaments of the remnant of Cas A and found that the magnetic\nfieldsofthefilamentsarehighlyturbulentandnearlyperpen dic-\nular to the shock normal. Another important result of this an d\nother studies is an estimate of the downstream magnetic-fiel d\nstrength that is higher than simple shock compression would\nsuggest. Such observations indicate an additional amplific ation\nof the magnetic field in the shock region of the SNRs. A possi-\nble amplification process could be a streaming instability i n the\nupstream region as proposed by Lucek & Bell (2000) and Bell\n(2004),ortheeffectsofpreexistingturbulentdensityfluctuations\nonthepropagatingshockfront(Giacalone& Jokipii2007).\nBesides energy losses, also the magnetic field itself can\nlimit the filament widths. Based on the turbulence relaxatio n\ndownstream of the forward shock and neglecting any amplifi-\ncation process, Pohl et al. (2005) have calculated that the t ur-\nbulent magnetic field downstream can decay exponentiallyon a\ndamping-lengthscale ld=1016−1017cmthatissmallenoughto\nproducethenarrowobservablefilaments.Furthermore,from ob-\n1R.Rettig& M. Pohl:Non-thermal X-rayfilaments inyoung supe rnova remnants\nservationsofthepost-shocksteepeningofthesynchrotron spec-\ntrum in Tycho’s SNR it can be seen that also the damping of\nthe magneticfield fairlywell describesthe correspondingX -ray\ndata (Cassam-Chenaï et al. 2007), and thus, may appear withi n\nthefilaments.\nSince the magnetic field controls the radiative cooling of\nelectrons, high magnetic fields lead to strong cooling, and t oo\nfew high-energy electrons remain capable of producing the\ngamma-ray emission, that is observed from regions near the\nedges of numerous SNRs (e.g., Aharonian et al. 2007; Acero\net al. 2010; Abdo et al. 2011). Any gamma rays observed in\nsuch a case are likely to be hadronic in origin. Weak cool-\ning leads to a large number of high-energy electrons and the\npossibility of gamma-ray emission through inverse Compton\nor bremsstrahlung processes. All these implications of the\nmagnetic-fieldstructureontheparticleacceleration,gam ma-ray\nemission and magnetic-field amplification make it necessary to\nunderstandthenon-thermalfilamentsindetail.\nIn this paper we investigate the properties of the filaments\nfor both cases, filaments limited by electron energy losses o r\nby damping of the magnetic field. For that purpose, using ob-\nservational values of some characteristical SNR parameter , we\ncalculatetheX-rayemissionofthe filaments.Theresulting fila-\nment profiles then allow us to make specific predictionsregar d-\ningthemagnetic-fieldstrength.Weadditionallycalculate theto-\ntalnon-thermalemission,whichshallbereferredtoas\"pla teau\",\nand compare their spectra with those of the filaments. It shou ld\nbe noted that in our models we only consider non-thermal syn-\nchrotron emission and restrict ourselves to the evolution o f rel-\nativistic electrons in the downstream region. Furthermore , we\nassumetheelectronstobealreadyacceleratedattheshockf ront\nandtreatourproblemtobe independentof theaccelerationp ro-\ncess. We also do not consider any electron propagationinto t he\nupstream region and simplify the SNRs to be spherical object s\nof constant downstream-velocity profile. Recent hydrodyna mi-\ncalsimulationssuggestthatthisoversimplificationofa co nstant\nvelocityisanacceptableapproximationonlyforSNRsofana ge\nofless thanseveralhundredyears(Telezhinskyet al.2012) ,im-\nplying that our modelsare restricted to SNRs being in the adi a-\nbaticphaseandjustenteringtheSedovphase,respectively .\n2. Modellingthefilaments\nWe calculate the X-ray intensity as a function of the project ed\nradius,rp. It is an integralover the synchrotronemission coe ffi-\ncient,jν,alongthelineofsight, Iν=/integraltext∞\n−∞jνdy.Usingr2=y2+r2p\nandtakingintoaccountthatonlyemissionoriginatinginsi dethe\nSNRcontributes,theX-rayintensitycanbewrittenas\nIν(rp)=2/integraldisplayrs\nrpjν(r)/radicalbigg\n1−r2p\nr2dr, (1)\nwhererandrsdenote the positions inside the SNR and the ra-\ndiusoftheSNR,respectively.Then,obtainingthecorrespo nding\nspectrumofthefilamentsjustinvolvesanintegrationovert heX-\nrayemissionalongtheprojectedradius,whereasthespectr umof\ntheplateauemissioncanbecalculatedasavolumeintegralo ver\nthesynchrotronemissioncoe fficient.Inbothcases,theisotropic\nsynchrotronemissioncoe fficientisneeded,givenby\njν(r)=1\n4π/integraldisplay∞\nN(r,E)Pν(r,E)dE, (2)whereN(r,E) andPν(r,E) are the isotropicdi fferential electron\nnumberdensity and the spectral emissivity per electron, re spec-\ntively.Thus,we needto derivetheappropriateelectrondis tribu-\ntionwithinthefilaments.\n2.1. The electrondistribution\nInthefollowing,wederivetheelectrondistributionthati sneces-\nsary to calculate the synchrotron emissivity. To do this, a t rans-\nport equation describing the dynamics of a distribution of r el-\nativistic electrons a ffected by advection, di ffusion and energy\nlosses needs to be solved. According to simulations describ ed\nin Telezhinsky et al. (2012), we can approximate the advecti on\nvelocityofyoungSNRstobeconstantdownstreamoftheshock ,\nimplyingthat energylosses dueto adiabaticdecelerationc an be\nneglected.Notethat thenon-thermalemission comefroma th in\nspherical shell near the edges of the SNRs. If we restrict our\ntreatment to a region near the shock that is crossed by the ad-\nvection flow on a timescale short compared with the age of the\nSNR, then we approximatethe electron distribution with a on e-\ndimensionalsteady-statesolution.\nIt is convenient to introduce a comoving spatial coordinate ,\nz=rs−r, wherez=0 marks the position of the shock front\nat all times. Hence, restricting ourselves on the downstrea m re-\ngion, the one-dimensional transport equation for the isotr opic\ndifferentialnumberdensity, N=N(z,E),readsasfollows:\n∂\n∂z/bracketleftBigg\nD(z,E)∂N\n∂z/bracketrightBigg\n−v∂N\n∂z−∂\n∂E/bracketleftbigβ(z,E)N/bracketrightbig+Q(z,E)=0.(3)\nIn this equation vdenotesthe constant advectionvelocity of the\nelectrons relative to the forward shock, D(z,E) is the diffusion\ncoefficient,β(z,E)=dE/dtrepresents the electron energy loss\ndue to the emission of radiation, and Q(z,E) is the source term\ndescribing the injection of accelerated electrons into the propa-\ngationprocess.\nSince the electrons are likely accelerated by the DSA-\nmechanism at the forward shock ( z=0), we assume the elec-\ntrons to be injected with a power-law dependence E−s, wheres\nistheinjectionindex.Hence,thesourcetermreads\nQ(z,E)=q0E−sexp/parenleftBigg\n−E\nEcut/parenrightBigg\nδ(z), (4)\nandincludesacut-o ffatenergy Ecut,becausethemaximumpos-\nsible energycan be limited by either the finite acceleration time\noftheSNR(Drury1991)orenergylosses(Webbetal. 1984).\nEq. (3) can be solved using Green’s method, implying that\nthesolutioncanbewrittenintermsofGreen’sfunction,\nN(z,E)=/integraldisplay∞\n0dz′/integraldisplay∞\ndE′g(z,z′;E,E′)Q(z′,E′), (5)\nwhereGreen’sfunction g=g(z,z′;E,E′) satisfies\n∂\n∂z/bracketleftBigg\nD(z,E)∂g\n∂z/bracketrightBigg\n−v∂g\n∂z−∂\n∂E/bracketleftbigβ(z,E)g/bracketrightbig=−δ(z−z′)δ(E−E′).(6)\nAssuming the diffusion coefficient and the energy losses to\nbeseparablein aspatial andin anenergeticpart,\nD(z,E)=d(z)D(E), (7)\nβ(z,E)=−b(z)B(E), (8)\n2R.Rettig& M. Pohl:Non-thermal X-rayfilaments inyoung supe rnova remnants\nand that the spatial dependent terms are inversely proporti onal\ntoeachother,\nd(z)b(z)=α=const., (9)\naswell asintroducingthesubstitutions\ng(z,z′;E,E′)=G(z,z′;E,E′)\nB(E), (10)\nx(z)=/integraldisplayz\n0dy\nd(y), (11)\nλ(E)=1\nα/integraldisplay∞\nEdu\nB(u), (12)\nan analytical solution to Eq. (6) can be found in the literatu re\n(Lerche&Schlickeiser1980,see theirEq.(A20))andreads\nG(x,x′;λ,λ′)=Θ(λ−λ′)\n2α√π/radicalBigg\n1\n/integraltextλ\nλ′D(t)dt\n×exp−[v(λ−λ′)+x′−x]2\n4/integraltextλ\nλ′D(t)dt, (13)\nwhereΘ(λ−λ′) is the step function and λ′=λ(E′). Note, this\nanalyticalsolutionisvalidonlyif Eq.(9)applies.\nIn general, the diffusion coefficient is an unknown param-\neter. But it is often assumed that the di ffusion proceeds in the\nso-called Bohm-regime. Hence, the di ffusion coefficient can be\nwrittenintheextreme-relativisticlimit E≫mc2asD=ηrLc/3,\nwhererL=E/(eB) andη≥1 are the gyroradius and gy-\nrofactor, respectively. Here, mis the mass of the electron, e\nis its charge, cis the speed of light, and Bis the magnetic-\nfield strength. Therefore, we may write D(E)=D0E, where\nD0=ηc/(3eB). Additionally, we assume the emission of syn-\nchrotronradiationtobethemainenergy-lossprocessthati spro-\nportional to the square of the electron energy, B(E)=b0E2,\nwhereb0=4e4B2/(9m4c7). Using these assumptions, Eq. (12)\nrewritesas\nλ(E)=1\nα/integraldisplay∞\nEdu\nb0u2=1\nαb0E,\nsothat\nD(E)=D0\nαb0λ\nandhence,\n/integraldisplayλ\nλ′D(t)dt=D0\nαb0ln/parenleftbiggλ\nλ′/parenrightbigg\n=D0\nαb0ln/parenleftBiggE′\nE/parenrightBigg\n.\nAccording to Eq. (10) and Eq. (13), Green’s function then re-\nducesto\ng(x,x′;E,E′)=θ(E′−E)\n2√παb0D0E2/radicalBigg\n1\nln/parenleftBigE′\nE/parenrightBig\n×exp−/bracketleftBigv\nαb0/parenleftBig1\nE−1\nE′/parenrightBig\n+x′−x/bracketrightBig2\n4D0\nαb0ln/parenleftBigE′\nE/parenrightBig.(14)\nNow,theelectrondistributioncanbeeasilydeterminedusi ng\nEq.(5)andtheappropriatesourcedistribution(4),sothat\nN(x(z),E)=/integraldisplayx(∞)\nx(0)dx′/integraldisplay∞\ndE′q0θ(E′−E)\n2√παb0D0E−2E′−s/radicalBigg\n1\nln/parenleftBigE′\nE/parenrightBig×exp−E′\nEcut−/bracketleftBigv\nαb0/parenleftBig1\nE−1\nE′/parenrightBig\n+x′−x/bracketrightBig2\n4D0\nαb0ln/parenleftBigE′\nE/parenrightBigδ(x′)\n=q0\n2√παb0D0E−2/integraldisplay∞\nEE′−s\n/radicalBig\nln/parenleftBigE′\nE/parenrightBig\n×exp−E′\nEcut−/bracketleftBigv\nαb0E/parenleftBig\n1−E\nE′/parenrightBig\n−x(z)/bracketrightBig2\n4D0\nαb0ln/parenleftBigE′\nE/parenrightBigdE′.(15)\nSubstituting n=E′/E,wecanrewritethedesiredresultas\nN(x(z),E)=q0\n2√παb0D0E−(s+1)/integraldisplay∞\n1n−s\n√ln(n)\n×exp−nE\nEcut−/bracketleftBigv\nαb0E/parenleftBig\n1−1\nn/parenrightBig\n−x(z)/bracketrightBig2\n4D0\nαb0ln(n)dn.(16)\n2.2. Parameterused in themodels\nThe models we use are based on di fferent assumptions on the\nmagnetic field. In the first model, which shall be referred to a s\n\"energy-loss model\", we assume the magnetic field to be spa-\ntially constant, whereas the second one, which is referred t o as\n\"magnetic-fielddampingmodel\",includesthe dampingof mag -\nnetic turbulence and assumes a spatially-dependent magnet ic-\nfieldstrengthdescribedbya profilefollowingtherelation\nB(z)=Bmin+(Bmax−Bmin)exp/parenleftBigg\n−z\nld/parenrightBigg\n, (17)\nwhereldis the damping length and z≥0. Here, we choose\nthe minimum value of the magnetic field to be similar to that\nof the interstellar medium, Bmin=10µG, whereas the maxi-\nmumvalue, Bmax,correspondstothefieldattheshock.Itshould\nbe noted that Eq. (17) describes the averaged amplitude of th e\nmagnetic field in a given volume. Hence, we do not make any\nassumption on the magnetic-field direction and do not distin -\nguish between parallel and transverse di ffusion, as is done in\ndetailed calculations of di ffusion coefficients (e.g., Marcowith\net al.2006).\nAccording to the Rankine-Hugoniot conditions, the down-\nstream advection velocity can be expressed by the shock velo c-\nity,vs,through v=vs/4,ifweconsiderstrongshockswithahigh\nMachnumberand a monatomicgaswith adiabaticindexof 5 /3,\nleadingto a shock compressionratio of 4. Thus, we neglect an y\nnon-linear effects expected to occur with e fficient particle ac-\nceleration, which would modify the shock (Ellison et al. 200 4).\nFurthermore, the filament width, w, is defined as the length, at\nwhich the intensity described by Eq. (1) is reduced by a facto r\n1/eofitsmaximum.\nTo have a realistic model, the values chosen for the shock\nvelocity, filament width, and radius are based on reference v al-\nues of real SNRs. In our case, we consider the young remnants\nofthehistoricalsupernovaeSN1006,CasA,TychoandKepler .\nIt should be noted that the real filament widths found in the li t-\nerature have been measured in a certain X-ray band and not for\nan individual X-ray energy. However, as is shown in the calcu -\nlationdoneinSect. 3,theshapeofthe filamentprofilesdepen ds\non the X-ray energy. Nevertheless, we relate the observatio nal\nvalue of the width to a X-ray energy of 5 keV, since it is an en-\nergy,atwhichnosignificantcontributionfromthermalemis sion\nisexpected.\n3R.Rettig& M. Pohl:Non-thermal X-rayfilaments inyoung supe rnova remnants\nAdditionally,we treat the cut-o ffenergyto be the maximum\nelectronenergythatcanbeachievedintheaccelerationpro cess.\nUnlike cosmic-ray nuclei whose energy is probablyage-limi ted\nduetothefiniteaccelerationtimeavailable,weassumethem axi-\nmumelectronenergytobeloss-limited,sincetheelectrons expe-\nriencesynchrotronlosses duringtheir acceleration.By eq uating\ntheaccelerationtimescaletothesynchrotronlosstime,it ispos-\nsible to derive an expression for the maximum electron energ y\nin terms of the downstream magnetic-field strength and shock\nvelocity(Parizotet al.2006):\nEcut≡Emax≃(8.3TeV)η−1/2/parenleftBiggB(z=0)\n100µG/parenrightBigg−1/2/parenleftBiggvs\n1000km/s/parenrightBigg\n.(18)\nSince different mechanism can account for magnetic-field am-\nplification (Lucek & Bell 2000; Giacalone & Jokipii 2007), th e\nstructure of the magnetic field is generally unknown within t he\nshock region.We thereforesimplify the problemby making th e\nassumptionthatonlythemagnetic-fieldstrengthattheshoc kde-\nterminesthemaximumelectronenergygivenbyEq.(18).Inad -\ndition,we havetakena shockcompressionratioof4.\nAt last, we assume the injection index to be s=2, which\nresultsfromthe Rankine-Hugoniotconditionsforstrongsh ocks\n(Bell1978),aswellasBohmdi ffusion(η=1),whichimpliesthe\nsmallest possible value of the di ffusion coefficient, as the mean\nfreepathofthe particleisequaltothe gyroradius.\nAll parameter used are summarized in Table 1. Note, that it\nispossible that all fourSNRs couldexhibitsimilar shockve loc-\nities. To take the uncertainties of this quantity into accou nt, we\nperformthecalculationofSN 1006andKeplerfortwodi fferent\nshockvelocities.\n3. Results\n3.1. Energy-lossmodel\nIn this model we treat the magnetic field to be constant, B(z)=\nB=const., implying no spatial dependence of the energy-loss\nterm,b(z)=1. Using d(z)=α/b(z), as well as Eq. (11), the\nspatialcoordinate x(z)thenscalesas\nx(z(r))=z\nα=rs−r\nα.\nUsing the parameters given in Table 1, we calculate the X-\nray intensity as a function of the projected radius accordin g to\nEq. (1). Here, the magnetic-field strength is a free paramete r\nwhose value can be chosen so that the filament widths match\nthose found in the observations. In addition, we also determ ine\nthe cut-offenergy of the electron spectrum, which is connected\ntothemagneticfieldthroughEq.(18).\nReproducing the filament width for each SNR at a pho-\nton energy of 5 keV determines the magnetic-field strengths a s\ngiven in Table 2. For our examples the downstream magnetic-\nfield strength ranges from about 100 µG up to about 500 µG.\nRemnants with narrower filaments exhibit a higher downstrea m\nmagnetic field. These magnetic fields then imply cut-o ffener-\ngies of the electronspectra in the energyrangebetween 19 Te V\nand37 TeV. The orderofmagnitudeof the cut-o ffenergiesis in\nagreement with the results obtained from spectral modeling of\nthe radio-to-X-ray spectra of young SNRs whose cut-o ffener-\ngiesoftheirelectrondistributionmustbeintheTeV-band, since\nthe cut-offfrequencies are generally found in the X-ray band\n(Reynolds&Keohane1999).\nIn Fig. (1) we illustrate the profiles of the filaments for\nfour different photonenergiescalculated with the parametersofTycho.Tobenotedfromthefigureisafrequencydependenceof\nthefilamentwidths,whichcanbeexplainedbytheenergyloss of\nthe electrons in a constant magnetic field and by the advectio n\nprocess. The advection length represents the distance cove red\nbytheelectronswithinthesynchrotronlosstime, τsyn=E/|dE\ndt|,\nandisgivenby\nlad=vτsyn=vs\n49m4c7\n4e4B2E\n≃(2×10−2pc)/parenleftBiggB\n200µG/parenrightBigg−2/parenleftbiggvs\n5000kms−1/parenrightbigg/parenleftbiggE\n20TeV/parenrightbigg−1\n.(19)\nSynchrotron radiation usually provides a continuum around a\ncharacteristicsynchrotronfrequency,\nνc=3νLγ2\n2=3eBE2\n4πm3c5\n≃(1.3×1018Hz)/parenleftBiggB\n200µG/parenrightBigg/parenleftbiggE\n20TeV/parenrightbigg2\n, (20)\nwhereνLandγare the Larmor frequency and the Lorentz fac-\ntoroftheacceleratedelectrons,respectively.Hence,the relation\nE∝ν1/2.Therefore,adependenceofthewidthonthefrequency\nof the radiation of the form lad∝ν−1/2would result. But the\nelectronsarealsoa ffectedbydiffusion.WiththeBohmdi ffusion\ncoefficient,D=rLc/3, one thus obtains for the corresponding\ndiffusionlength\nldiff=/radicalBig\nDτsyn=/radicalbigg\n3m4c8\n4e5B3≃(1.3×10−2pc)/parenleftBiggB\n200µG/parenrightBigg−3/2\n,(21)\nwhich does not dependon the electronenergy.Equating the ad -\nvectionanddiffusionlength,therelation\nEc≃(31TeV)/parenleftbiggvs\n5000kms−1/parenrightbigg/parenleftBiggB\n200µG/parenrightBigg−1/2\n(22)\ncan be derived. Electrons with energies E>Eccan stream\nfarther from the shock than advection alone would allow.\nAccording to Eq. (20) and Eq.(22), the characteristic syn-\nchrotronfrequencyfor the electronswith E>Ecis then higher\nthan\nνc(Ec)≃(3.1×1018Hz)/parenleftbiggvs\n5000kms−1/parenrightbigg2\n, (23)\ncorresponding to a X-ray energy of about 13 keV, if vs=\n5000 km/s. Hence, the filament profiles at higher photon ener-\ngies, which need the most energetic electrons, show approxi -\nmately the same behaviour as can be seen in Fig. (1) for the\n5-keVand10-keVprofile.\nIn addition, the advection and di ffusion length also explain\nthe relation between the filament widths and the correspondi ng\nmagnetic fields given in Table 2. Narrower filaments require a\nfaster decreaseinthe synchrotronemissivity,implyinga s horter\nadvection and diffusion length. And according to Eq. (19) and\nEq.(21),thisisgivenforhighermagnetic-fieldstrengths.\nNow, equipped with the X-ray intensity distribution estab-\nlishing the filament profiles, and the volume emissivity, we c al-\nculate the spectra of the filament and plateau for each of our\nexamples. To obtain the filament spectrum, we integrate the i n-\ntensityalongtheprojectedradiusfrom rp=rsuptorp=rs−w.\nBecause we do not know the electron source strength, q0, we\nare not interested in absolute fluxes. However, we can show\nthe qualitative behaviour described by the appropriate pho ton\n4R.Rettig& M. Pohl:Non-thermal X-rayfilaments inyoung supe rnova remnants\nTable 1. Parameters used to derive the filament profiles and spectra. F urthermore, s=2 and Bohm diffusion are assumed in all\nSNRs.\nSNR Age Distance rs vs w B min\n[yr] [kpc] Reference [arcmin]a[pc]b[kms−1] Reference [arcsec] [pc]bReference [µG]c\nSN1006d1000 2.2 1 15 10 4900 2 20 0.2 3 10\n(2900)\nCasA 330e3.4 4 2.5 2.5 5200 5 1.5 0.03 6, 7 10\nTycho 440 2.5 8 4 3 5000 9 5 0.06 6 10\nKeplerf410 4.8 10 1.5 2 5040 11 3.5 0.08 6 10\n(6.4) (3) (6720) (0.11)\nNotes.(a)Taken from Green (2009).(b)Directly inferred using the distance and the appropriate qu antity given in angular units.(c)Only used in\nthe magnetic-field damping model.(d)The shock velocity inthe northwestern limb(value inbracke ts) is used, as well as inthe northeastern limb,\nwherestrongelectronaccelerationappearstooccur.(e)Thesupernova explosionmayhavebeenobservedin1680(Ashw orth1980).Otherwise,the\ndetectionof radioactive44Ti(Hartmannetal.1997) andthe analysis ofthe dynamics oft hisremnant (Fesenetal.2006) alsosuggest anexplosion\ndate in the late 17th century.(f)Distance to this remnant is uncertain. Calculation is perfo rmed for the lower and upper limit (values in brackets)\ntothe distance.\nReferences. (1) Winkler et al. (2003); (2) Ghavamian et al. (2002) +Katsuda et al. (2009); (3) Bamba et al. (2003); (4) Reed et al. (1995);\n(5) Vink et al. (1998); (6) Bamba et al. (2005); (7) Araya et al . (2010); (8) Tian & Leahy (2011); (9) Hughes (2000) +distance; (10) Reynoso &\nGoss (1999); (11) Vink(2008) +distance\nspectral indices, which should be su fficient for the comparison.\nAssuming the photon spectra to show a power-law characteris -\ntic,Nν=Fν/hν∝ν−Γ, whereFνandΓare the flux and photon\nspectralindex,respectively,wecanthendescribethespec trabe-\ntweenthephotonenergies hνandhν′through\nΓ=ln/parenleftBigνFν′\nν′Fν/parenrightBig\nln/parenleftBigν\nν′/parenrightBig. (24)\nHere,wealsowanttocalculatethedi fferencesbetweenthepho-\nton spectral index of the filament spectrum, Γf, and that of the\nplateau emission,Γp, which are given at three di fferent photon\nenergies, Eν, in Table 2. Additionally, in Fig. (2) we show the\nphotonspectralindicescalculatedwiththeparametersofT ycho.\nAs can be seen from Table 2, as well as from Fig. (2), the\nspectra at higher photonenergiesrarely di ffer significantly. The\nplateau shows a steeper spectrum at lower photon energies. U p\ntoaphotonenergyof1keVthedi fferencebetweentheindicesof\nfilamentandplateauisintherange0 .05−0.34,whereasatener-\ngieshigherthan1keVthedi fferenceisalwayssmallerthan0.1.\nThis property can be explained by the e ffective radiation of the\nenergeticelectronsintheenhancedmagneticfield.Accordi ngly,\nthemostenergeticelectronslosealloftheirenergyinside thefil-\naments,implyingthattheregionsfartherfromtheshockpro vide\nalmost no contributionsto the total emission of hard X-rays , so\nthatbothfilamentandplateaushownearlythesamebehaviour .\nIn Table 2 we also show the indicesof the filamentsat three\ndifferent photon energies. One can see that the parameters of\nCasA,Tycho,KeplerandSN1006leadtothesamespectralbe-\nhaviour,if their shockvelocitiesare similar. However,us ing the\nshock velocitiesmeasured in the northwesternlimb of SN 100 6\nand resulting from the upper limit to the distance to Kepler, it\nturnsoutthat the filament spectrumis softerand harder,res pec-\ntively,thanintheformercase.\n3.2. Magnetic-fielddampingmodel\nAs already mentioned above, in the model of magnetic-field\ndampingwe assume themagnetic-fieldstrengthto followa pro - 1\n 2.75 2.8 2.85 2.9 2.95 3Iν(rp) in arbitrary units\nrp in pcfilament profiles in the energy-loss model\n0.1 keV\n1 keV\n5 keV\n10 keV\nFig.1.Non-thermalX-rayintensityasafunctionoftheprojected\nradius calculated for four di fferent X-ray energies with the pa-\nrametersof Tychogivenin Table 1,as well as B=310µG. The\nforwardshockislocatedat rs=3pc.\nfile describedinEq.(17),whichcanalso bewrittenas\nB(z)=Bmin/bracketleftBigg\n1+Bmax−Bmin\nBminexp/parenleftBigg\n−z\nld/parenrightBigg/bracketrightBigg\n.\nThe spatial dependence of the energy-loss term then obeys th e\nrelation\nb(z)=/bracketleftBigg\n1+Bmax−Bmin\nBminexp/parenleftBigg\n−z\nld/parenrightBigg/bracketrightBigg2\n,\ncorrespondingtoa spatialvariationofthedi ffusioncoefficient\nd(z)=α/bracketleftBigg\n1+Bmax−Bmin\nBminexp/parenleftBigg\n−z\nld/parenrightBigg/bracketrightBigg−2\n,\nbecause the product b(z)d(z) must be constant, as required by\nEq. (9). According to Eq. (11), the spatial coordinate x(z) then\n5R.Rettig& M. Pohl:Non-thermal X-rayfilaments inyoung supe rnova remnants\nTable 2.Constraints on the downstream magnetic-field strength, cut -offenergy of the electron spectrum, photon spectral index of\nthefilament,andthedi fferencebetweenthephotonindicesofthespectraoffilamenta ndplateauat threedi fferentphotonenergies,\nEν,forfouryoungSNRs. Thesevaluesarecalculatedusingthe e nergy-lossmodelandtheparametersgiveninTable1.\nSNR B E cut Γf Γp−Γf\n[µG] [TeV] Eν=0.1keV Eν=1keV Eν=10keV Eν=0.1keV Eν=1keV Eν=10 keV\nSN1006a130 36 1.81 2.33 2.71 0.31 0.07 0.02\n(110) (23) (1.99) (2.49) (2.94) (0.25) (0.05) (0.01)\nCas A 520 19 1.83 2.33 2.70 0.31 0.06 0.01\nTycho 310 24 1.82 2.33 2.71 0.32 0.06 0.01\nKeplerb250 26 1.81 2.32 2.70 0.31 0.07 0.01\n(230) (37) (1.75) (2.23) (2.61) (0.34) (0.08) (0.02)\nNotes.(a)The values inbrackets were calculated usingthe shock veloc ityvs=2900 km/s as measured inthe northwestern limb.(b)The values in\nbrackets were calculatedusing the upper limitof 6.4kpc tot he distance.\n 1.8 2 2.2 2.4 2.6 2.8\n 0.1 1 10spectral index\nEν in keVspectral indices in the energy-loss model\nfilament\nplateau\nFig.2.Photon spectral indices of the spectra of filament and\nplateauusingtheparametersofTycho.\nscalesas\nx(z)=1\nαz+ld\n2(Bmax−Bmin)2\nB2\nmin/bracketleftBigg\n1−exp/parenleftBigg\n−2z\nld/parenrightBigg/bracketrightBigg\n+2ldBmax−Bmin\nBmin/bracketleftBigg\n1−exp/parenleftBigg\n−z\nld/parenrightBigg/bracketrightBigg/bracerightBigg\n.(25)\nAgain, using the parameters given in Table 1, we calculate\nthe X-ray intensity as a function of the projected radius. In this\ncase,thedampinglengthandthemaximumfieldstrength, Bmax,\narefreeparametersthatcanbechosensothatthefilamentwid ths\nmatchthoseobserved.AccordingtothecalculationofPohle tal.\n(2005), the damping length should be in the range ld=1016−\n1017cm(ld=0.003−0.03pc).Toalsoinvestigatetheinfluence\nofthedampinglengthontheresults,weperformthecalculat ion\nusing two different values of ldin each of our examples. Here,\nthe larger value used for ldmay describe weak magnetic-field\ndamping, whereas the smaller one may cause a strong damping\nofthefield.But notethatenergylossesarestill included.\nReproducing the filament width of each SNR at a photon\nenergy of 5 keV requires the maximum field strengths given\nin Table 3. Depending on the damping length, the maximum\nfield strength can be found in the range between 50 µG and\n260µG, implying, according to Eq. (18), cut-o ffenergies be-\ntween 27 TeV and 62 TeV. Furthermore, it turns out that an in-\ncreased dampinglength requiresan increased field strength ,be-cause the electronsradiate e fficiently in a larger volume, result-\ning in wider filaments. To retain the observed filament widths ,\nit is then necessary to have a higher magnetic field that, on th e\notherhand,also leadsto a smaller cut-o ffenergyof the electron\ndistribution.\nIf the damping length is too small, the observed filament\nwidths cannot be realized for any maximum field strength. In\nthese cases the intensity first decreases but then increases again\neven for the 5-keV profile, so that the typical shape of the fila -\nment profiles is not given anymore.Therefore, we use damping\nlengthsinthecaseofstrongdamping,forwhichtheprofilesj ust\nstill exhibit the typical shape. For instance, using the giv en pa-\nrametersofSN1006,thedampinglengthusedinthecalculati on\nshouldnotbesmallerthan0.02pc.\nThe filament profiles calculated with the parameters of\nTycho for four different photon energies are illustrated in Fig.\n(3).Ascanbeseenfromthefigure,thereisalsoadependenceo f\nthe filaments on the frequency of the X-rays. This dependence\nis based on the spatial variation of the magnetic-field stren gth.\nOnly in regionsvery close to the shock front the electrons ra di-\nate in fields of high magnitude, so that even the most energeti c\nof them can emit photons of several keV in energy only in a\nsmall volume. The electrons remain energetic when they prop -\nagate into the downstream region, where they radiate at lowe r\nfrequencies.Hence,weexpectincreasedemissionoflow-en ergy\nX-raysinregionsfartherfromtheshock.Forinstance,usin gthe\ncut-offenergyoftheelectronspectruminTychoobtainedforthe\ncase of strong damping, Ecut=45 TeV, one finds that accord-\ning to Eq. (20), even the most energetic electrons located in a\nmagnetic field of B=10µG have their synchrotroncontinuum\naround the characteristic frequency νc(Ecut)=3.3×1017Hz,\ncorrespondingto about 1.4 keV in X-ray energy.This issue ca n\nbeseen forthe 0.1-keVand1-keVprofilein Fig.(3).TheX-ray\nintensity does not decrease with decreasing projected radi us as\nhappens in the energy-loss model, but remains nearly consta nt\nandevenincreases,respectively.\nUsingtheX-rayintensitydistribution,aswell asthevolum e\nemissivity, we nowcalculate the spectra of filament and plat eau\nfor each example. Again, we integrate the intensity along th e\nprojected radius between rp=rsandrp=rs−win order to\nobtainthefilamentspectrum.Thedi fferencebetweenthephoton\nspectralindicesoffilamentandplateau, Γf−Γp, atthreephoton\nenergiesisgiveninTable3,whereasinFig. (4)we illustrat ethe\nphotonspectralindicescalculatedwiththeparametersofT ycho.\nAs can be seen again, the spectrum of the plateau is steeper\nthan that of the filament. However, the di fference between the\nspectra of filament and plateau depends on the chosen damp-\n6R.Rettig& M. Pohl:Non-thermal X-rayfilaments inyoung supe rnova remnants\n 1\n 2.75 2.8 2.85 2.9 2.95 3Iν(rp) in arbitrary units\nrp in pcfilament profiles in the damping model\n0.1 keV\n1 keV\n5 keV\n10 keV\nFig.3.Non-thermalX-rayintensityasafunctionoftheprojected\nradius calculated for four di fferent X-ray energies with the pa-\nrametersofTychogiveninTable1,aswellas ld=0.008pcand\nBmax=85µG. Theforwardshockislocatedat rs=3 pc.\ning length.At relatively largedamping lengths(weak dampi ng)\nthe difference between the indices over the hole energy range\nis smaller than 0.1, whereas at smaller damping lengths (str ong\ndamping)italsotakesvaluesintherange0.1-0.2.Thesmall dif-\nferencesatlargerdampinglengthsareduetothehighermagn etic\nfields that need to be chosenin orderto retain the observedfil a-\nmentwidths.Hence,considerableenergylosseshavetobeta ken\ninto account. Similarly to the energy-loss model, this resu lts in\nanalmostequalbehaviourofthe spectraoffilamentandplate au\nat high photon energies. In contrast, the magnetic fields use d at\nsmallerdampinglengthsarelowenoughto resultinspectrat hat\nshowsignificantdi fferencesamongeachother.\nFinally,onecanalsoseefromTable3thatthefilamentspec-\ntrumbecomessteeperwithdecreasingdampinglength,inpar tic-\nular at small X-ray energies. This is due to the lower magneti c-\nfield strengths used in that case. Although the weaker magnet ic\nfields imply higher cut-o ffenergies, which harden the spectra,\ntheir influence is not su fficient enough to result in spectra simi-\nlar to those found at larger damping lengths. Besides, as in t he\nenergy-lossmodelandindependentlyofthedampinglength, Cas\nA, Tycho, Kepler and SN 1006 show roughly the same spectral\nbehaviour, if the shock velocities are similar, whereas the fila-\nment spectrum obtained from the shock velocity of the north-\nwestern limb of SN 1006 has a steeper profile. In contrast, the\nupper limit to the distance to Kepler implies a harder filamen t\nspectrum.\n4. Conclusions\nCompared to the magnetic-field damping model, the spectra of\nfilament and plateau obtained in the energy-loss model exhib it\nlargerspectralindices.Thiscanbeexplainedbytheconsid erable\nenergy losses leading to the evolution towards a softer elec tron\ndistributionin theenergy-lossmodel,andhence,resultin gin X-\nray spectra that are softer than those obtained in the dampin g\nmodel.\nIn case of a weak magnetic-fielddampingthe di fferencebe-\ntween the spectral indices of filament and plateau over the fu ll\nX-rayspectrumissmallerthan0.1,whichisprobablytosmal lto\nbe detectable.Only if there is a strongdamping,our calcula tion 1.8 2 2.2 2.4 2.6 2.8\n 0.1 1 10spectral index\nEν in keVspectral indices in the damping model\nfilament\nplateau\nFig.4.Photon spectral indices of the spectra of filament and\nplateauusingtheparametersofTycho.\nsuggests a measurable di fference between the spectra in some\nSNRs, since the di fference between the spectral indices of fila-\nmentandplateaucantakevaluesofalmost0.2atX-rayenergi es\nhigherthan1keV.\nIn the energy-lossmodel the di fference between the indices\noffilamentandplateauabovetheX-rayenergyof1keV iseven\nsmaller than 0.1, so that a possible detection can be exclude d\nhere, too. On the other hand, the di fference between the indices\nbelow1keVislargerthan0.1,andataphotonenergyof0.1keV\nit is even approximately 0.3. This might suggest that there i s\na measurable difference in the spectra of filament and plateau\nat small X-ray energies, if the filaments are limited by energ y\nlosses. However,on accountof the interstellar photoelect ricab-\nsorption of the soft X-rays, these di fferent spectral characteris-\ntics are probably not detectable, too. Furthermore, the pla sma\ndownstreamofthe forwardshockisat hightemperature,impl y-\ning also thermal emission contributing to the soft X-ray ban d,\nand thus, complicating a clear identification of the non-the rmal\nemission.\nHence,ifthereisnomeasurabledi fferencebetweenthespec-\ntra of filament and plateau, it is not possible to make definite\npredictions from the comparison of the spectra whether the fi l-\naments are limited by energy losses of the radiating electro ns\nor by damping of the magnetic field. But if a significant di ffer-\nenceappears,ourcalculationsthensuggestthatthefilamen tsare\nlimitedbythemagneticfielditself.\nIt should be noted that our results presented here have been\nderived using Bohm di ffusion. According to Eq. (21), a larger\ndiffusioncoefficient with gyrofactor η>1 would implya larger\ndiffusion length, resulting in significant widening of the fila-\nments, because now, the regions farther from the shock conta in\na sufficient number of high-energyelectrons contributing to the\nintensity.Wideningmustthenbecompensatedbyahighermag -\nnetic field to retain the observed filament widths. Hence, the\nmagnetic-field strengths derived in our models represent lo wer\nlimits for the chosen parameters. The calculation then show s\nthat a larger diffusion coefficient results in softer spectra due\nto a lower cut-offenergy, which decreases with increasing gy-\nrofactor. However, the final results regarding the di fferences in\nspectralindicesdonotchangefundamentally.\nIn a last step we want to compare the predictions derived\nherewithobservations.Atfirst,wenoticethat,independen tlyof\n7R.Rettig& M. Pohl:Non-thermal X-rayfilaments inyoung supe rnova remnants\nTable 3.Constraintsonthemaximummagnetic-fieldstrength,cut-o ffenergyoftheelectronspectrum,photonspectralindexofth e\nfilament,andthedi fferencebetweenthephotonindicesofthespectraoffilamenta ndplateauat threedi fferentphotonenergies, Eν,\nforfouryoungSNRs. Thesevaluesarecalculatedusingthema gnetic-fielddampingmodelandtheparametersgiveninTable 1.\nWeakDamping\nSNR ldBmaxEcut Γf Γf−Γp\n[pc] [µG] [TeV] Eν=0.1keV Eν=1keV Eν=10keV Eν=0.1keV Eν=1keV Eν=10keV\nSN1006 0.03 65 50 1.78 2.12 2.64 0.04 0.08 0.07\n(0.03) (57) (32) (1.93) (2.35) (2.95) (0.05) (0.09) (0.08)\nCas A 0.015 260 27 1.71 1.98 2.51 0.04 0.02 0.01\nTycho 0.02 150 34 1.71 2.00 2.57 0.06 0.04 0.02\nKepler 0.03 135 36 1.71 2.01 2.59 0.05 0.04 0.02\n(0.03) (115) (52) (1.67) (1.93) (2.43) (0.04) (0.05) (0.02)\nStrongDamping\nSNR ldBmaxEcut Γf Γf−Γp\n[pc] [µG] [TeV] Eν=0.1keV Eν=1keV Eν=10keV Eν=0.1keV Eν=1keV Eν=10keV\nSN1006a0.02 64 51 1.81 2.16 2.66 0.04 0.10 0.13\n(0.02) (56) (32) (1.97) (2.39) (2.96) (0.04) (0.11) (0.13)\nCas A 0.004 115 40 1.81 2.12 2.58 0.07 0.16 0.19\nTycho 0.008 85 45 1.82 2.14 2.61 0.05 0.13 0.17\nKeplerb0.01 80 47 1.81 2.14 2.62 0.04 0.11 0.14\n(0.012) (80) (62) (1.74) (2.03) (2.48) (0.03) (0.09) (0.12)\nNotes.(a)The values inbrackets were calculated usingthe shock veloc ityvs=2900 km/s as measured inthe northwestern limb.(b)The values in\nbrackets were calculatedusing the upper limitof 6.4kpc tot he distance.\nthe model, the parameters from the remnants of Cas A, Tycho,\nKepler and SN 1006 lead to nearly the same spectral behaviour\nincaseofsimilarshockvelocities,ascanbeseenfromthesp ec-\ntral indices in Table 2 and Table 3. However, the analysis of\nthe filament spectra of these remnants reported by Bamba et al .\n(2003, 2005) reveals significant di fferences among the spectral\nindices obtained from the fit of an absorbed power-law model.\nComparedtoourspectrawhosecalculationhasbeendoneusin g\nan injection index resulting from an unmodified shock ( s=2),\nthe observation may be an indication for di fferent electron in-\njection indices in these remnants, implying shocks that are dif-\nferently affected by non-linear e ffects due to differences in effi-\nciencyinthe particleacceleration.\nRegardingthemagnetic-fieldstrengths,wetake,asanexam-\npleforcomparisonwithourresults,thenon-thermalfilamen tsof\nCas A analysedby Arayaet al. (2010).Fromthebest-fit param-\netersusedtofittheobservedfilamentspectra,themagneticfi eld\nhas been derived to be in the range (30 −70)µG. These values\nareconsistentwith thosederivedfromthe magnetic-fieldda mp-\ning model, in which the magnetic field varies, according to Eq .\n(17) and the values from Table 3, between the field strengths\n(10−260)µG for weak damping and (10 −115)µG for strong\ndamping, respectively. For comparison, the constant magne tic\nfieldderivedfromtheenergy-lossmodelisseveraltimeshig her,\nB=520µG. This might suggest that the non-thermalfilaments\nofCasA arelimitedbythedampingofthemagneticfield.\nAnother comparison concerns the magnetic fields in SN\n1006 and Tycho. Using the data from radio up to TeV-\nobservations, Acero et al. (2010) have analysed the multi-\nwavelength spectrum of SN 1006 in the framework of a lep-\ntonic and hadronic origin for the gamma-rayemission, givin g a\nmagnetic-fieldof∼30µG inthe leptonicmodelanda magnetic\nfieldof∼120µGinthehadronicmodel,respectively.Moreover,\ncombining radio and X-ray data with recent TeV-observation s\nperformedwith the VERITAS instrument, the magnetic field of\nTychohasbeenestimatedtobe ∼80µGinaleptonic-dominated\nmodel, whereas a hadronic dominated model yields a magnetic\nfield of∼230µG (Acciari et al. 2011).Comparedto our modelpredictions given in Table 2 and Table 3, we notice that the\nmagnetic fields derived from the energy-lossmodel are in goo d\nagreementwiththoseestimatedfromthehadronicmodelused to\ndescribetheobservedspectraofSN1006andTycho.Incontra st,\nthe predictionsfrom the magnetic-field dampingmodel sugge st\nthe leptonic model for the origin of the gamma-ray emission\nfromthese remnants.It shouldbe notedthat currentgamma-r ay\nobservationsdo not reachthe spatial resolutionof those do nein\nX-rays, so that the magnetic fields estimated using gamma-ra y\nobservations of SN 1006 and Tycho are averages over a region\nmuch larger than the filaments, implying that the observed va l-\nuesdonotnecessarilymatchthosefoundforthe filaments.\nTo discriminate between the energy-loss model and\nmagnetic-field damping model, and hence between a leptonic\nand a hadronic origin of TeV-band gamma-ray emission, one\nmay either search for di fferences between X-ray spectra of fil-\naments and plateau, as calculated in this paper, or perform\ngamma-rayobservationswithhigherspatial resolution.\nAcknowledgement\nWe acknowledge support by the \"Helmholtz Alliance for\nAstroparticle Phyics HAP\" funded by the Initiative and\nNetworkingFundoftheHelmholtzAssociation.\nReferences\nAbdo, A.A.,Ackermann, M.,Ajello, M.,et al. 2011, ApJ,734, 28\nAcciari, V.A.,Aliu, E.,Arlen, T.,etal. 2011, ApJ, 730,L20\nAcero, F.,Aharonian, F.,Akhperjanian, A.G.,etal. 2010, A &A,516, A62\nAharonian, F., Akhperjanian, A. 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Polishchuk,2Vladislav Korenivski,2and Arne Brataas1\n1Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, Trondheim, Norway\n2Nanostructure Physics, Royal Institute of Technology, Stockholm, Sweden\nSpin valves form a key building block in a wide range of spintronic concepts and devices from\nmagnetoresistive read heads to spin-transfer-torque oscillators. We elucidate the dependence of the\nmagnetic damping in the free layer on the angle its equilibrium magnetization makes with that in\nthe \fxed layer. The spin pumping-mediated damping is anisotropic and tensorial, with Gilbert- and\nBloch-like terms. Our investigation reveals a mechanism for tuning the free layer damping in-situ\nfrom negligible to a large value via the orientation of \fxed layer magnetization, especially when the\nmagnets are electrically insulating. Furthermore, we expect the Bloch contribution that emerges\nfrom the longitudinal spin accumulation in the non-magnetic spacer to play an important role in a\nwide range of other phenomena in spin valves.\nIntroduction. { The phenomenon of magnetoresistance\nis at the heart of contemporary data storage technolo-\ngies [1, 2]. The dependence of the resistance of a multi-\nlayered heterostructure comprising two or more magnets\non the angles between their respective magnetizations has\nbeen exploited to read magnetic bits with a high spatial\nresolution [3]. Furthermore, spin valves comprised of two\nmagnetic layers separated by a non-magnetic conductor\nhave been exploited in magnetoresistive random access\nmemories [2, 4, 5]. Typically, in such structures, one\n`free layer' is much thinner than the other `\fxed layer'\nallowing for magnetization dynamics and switching in\nthe former. The latter serves to spin-polarize the charge\ncurrents \rowing across the device and thus exert spin-\ntorques on the former [6{9]. Such structures exhibit a\nwide range of phenomena from magnetic switching [5] to\noscillations [10, 11] driven by applied electrical currents.\nWith the rapid progress in taming pure spin cur-\nrents [12{20], magnetoresistive phenomena have found\na new platform in hybrids involving magnetic insulators\n(MIs). The electrical resistance of a non-magnetic metal\n(N) was found to depend upon the magnetic con\fgura-\ntion of an adjacent insulating magnet [21{24]. This phe-\nnomenon, dubbed spin Hall magnetoresistance (SMR),\nrelies on the pure spin current generated via spin Hall\ne\u000bect (SHE) in N [25, 26]. The SHE spin current accu-\nmulates spin at the MI/N interface, which is absorbed\nby the MI depending on the angle between its magne-\ntization and the accumulated spin polarization. The\nnet spin current absorbed by the MI manifests as ad-\nditional magnetization-dependent contribution to resis-\ntance in N via the inverse SHE. The same principle of\nmagnetization-dependent spin absorption by MI has also\nbeen exploited in demonstrating spin Nernst e\u000bect [27],\ni.e. thermally generated pure spin current, in platinum.\nWhile the ideas presented above have largely been ex-\nploited in sensing magnetic \felds and magnetizations,\ntunability of the system dissipation is a valuable, un-\nderexploited consequence of magnetoresistance. Such\nan electrically controllable resistance of a magnetic wire\nFIG. 1. Schematic depiction of the device under investigation.\nThe blue arrows denote the magnetizations. The \fxed layer\nF2magnetization remains static. The free layer F 1magneti-\nzation precesses about the z-axis with an average cone angle\n\u0002\u001c1. The two layers interact dynamically via spin pumping\nand back\row currents.\nhosting a domain wall [28] has been suggested as a ba-\nsic circuit element [29] in a neuromorphic computing [30]\narchitecture. In addition to the electrical resistance or\ndissipation, the spin valves should allow for controlling\nthe magnetic damping in the constituent magnets [31].\nSuch an in-situ control can be valuable in, for example,\narchitectures where a magnet is desired to have a large\ndamping to attain low switching times and a low dissipa-\ntion for spin dynamics and transport [13, 16]. Further-\nmore, a detailed understanding of magnetic damping in\nspin valves is crucial for their operation as spin-transfer-\ntorque oscillators [10] and memory cells [5].\nInspired by these new discoveries [21, 27] and previous\nrelated ideas [31{34], we suggest new ways of tuning the\nmagnetic damping of the free layer F 1in a spin valve\n(Fig. 1) via controllable absorption by the \fxed layer\nF2of the spin accumulated in the spacer N due to spin\npumping [31, 35]. The principle for this control over spin\nabsorption is akin to the SMR e\u000bect discussed above and\ncapitalizes on altering the F 2magnetization direction.\nWhen spin relaxation in N is negligible, the spin lost by\nF1is equal to the spin absorbed by F 2. This lost spin\nappears as tensorial Gilbert [36] and Bloch [37] damp-arXiv:1811.00020v2 [cond-mat.mes-hall] 10 Apr 20192\ning in F 1magnetization dynamics. In its isotropic form,\nthe Gilbert contribution arises due to spin pumping and\nis well established [31{33, 35, 38{40]. We reveal that\nthe Bloch term results from back\row due to a \fnite dc\nlongitudinal spin accumulation in N. Our results for the\nangular and tensorial dependence of the Gilbert damping\nare also, to best of our knowledge, new.\nWe show that the dissipation in F 1, expressed in terms\nof ferromagnetic resonance (FMR) linewidth, varies with\nthe angle\u0012between the two magnetizations (Fig. 3).\nThe maximum dissipation is achieved in collinear or or-\nthogonal con\fgurations depending on the relative size\nof the spin-mixing g0\nrand longitudinal spin glconduc-\ntances of the NjF2subsystem. For very low gl, which\ncan be achieved employing insulating magnets, the spin\npumping mediated contribution to the linewidth vanishes\nfor collinear con\fgurations and attains a \u0012-independent\nvalue for a small non-collinearity. This can be used to\nstrongly modulate the magnetic dissipation in F 1electri-\ncally via, for example, an F 2comprised by a magneto-\nelectric material [41].\nFMR linewidth. { Disregarding intrinsic damping for\nconvenience, the magnetization dynamics of F 1including\na dissipative spin transfer torque arising from the spin\ncurrent lost IIIs1may be expressed as:\n_^mmm=\u0000j\rj(^mmm\u0002\u00160HHHe\u000b) +j\rj\nMsVIIIs1: (1)\nHere, ^mmmis the unit vector along the F 1magnetization\nMMMtreated within the macrospin approximation, \r(<0)\nis the gyromagnetic ratio, Msis the saturation magneti-\nzation,Vis the volume of F 1, andHHHe\u000bis the e\u000bective\nmagnetic \feld. Under certain assumptions of linearity\nas will be detailed later, Eq, (1) reduces to the Landau-\nLifshitz equation with Gilbert-Bloch damping [36, 37]:\n_^mmm=\u0000j\rj(^mmm\u0002\u00160HHHe\u000b) + ( ^mmm\u0002GGG)\u0000BBB: (2)\nConsidering the equilibrium orientation ^mmmeq=^zzz, Eq. (2)\nis restricted to the small transverse dynamics described\nbymx;y\u001c1, while the z-component is fully determined\nby the constraint ^mmm\u0001^mmm= 1. Parameterized by a diagonal\ndimensionless tensor \u0014 \u000b, the Gilbert damping has been in-\ncorporated via GGG=\u000bxx_mx^xxx+\u000byy_my^yyyin Eq. (2). The\nBloch damping is parametrized via a diagonal frequency\ntensor \u0014\n asBBB= \n xxmx^xxx+ \nyymy^yyy. A more familiar,\nalthough insu\u000ecient for the present considerations, form\nof Bloch damping can be obtained by assuming isotropy\nin the transverse plane: BBB= \n 0(^mmm\u0000^mmmeq). This form,\nrestricted to transverse dynamics, makes its e\u000bect as a\nrelaxation mechanism with characteristic time 1 =\n0ev-\nident. The Bloch damping, in general, captures the so-\ncalled inhomogeneous broadening and other, frequency\nindependent contributions to the magnetic damping.\nConsidering uniaxial easy-axis and easy-plane\nanisotropies, parametrized respectively by Kzand\n0 30 60 9000.10.20.30.40.5FIG. 2. Normalized damping parameters for F 1magneti-\nzation dynamics vs. spin valve con\fguration angle \u0012(Fig.\n1). ~\u000bxx6= ~\u000byysigni\fes the tensorial nature of the Gilbert\ndamping. The Bloch parameters ~\nxx\u0019~\nyyare largest for\nthe collinear con\fguration. The curves are mirror symmetric\nabout\u0012= 90\u000e. ~g0\nr= 1, ~gl= 0:01, \u0002 = 0:1,!0= 10\u00022\u0019\nGHz, and!ax= 1\u00022\u0019GHz.\nKx[42], the magnetic free energy density Fmis ex-\npressed as: Fm=\u0000\u00160MMM\u0001HHHext\u0000KzM2\nz+KxM2\nx;with\nHHHext=H0^zzz+hhhrfas the applied static plus microwave\n\feld. Employing the e\u000bective \feld \u00160HHHe\u000b=\u0000@Fm=@MMM\nin Eq. (2) and switching to Fourier space [ \u0018exp(i!t)],\nwe obtain the resonance frequency !r=p\n!0(!0+!ax).\nHere,!0\u0011j\rj(\u00160H0+ 2KzMs) and!ax\u0011j\rj2KxMs.\nThe FMR linewidth is evaluated as:\nj\rj\u00160\u0001H=(\u000bxx+\u000byy)\n2!+t(\nxx+ \nyy)\n2\n+t!ax\n4(\u000byy\u0000\u000bxx); (3)\nwhere!is the frequency of the applied microwave \feld\nhhhrfand is approximately !rclose to resonance, and t\u0011\n!=p\n!2+!2ax=4\u00191 for a weak easy-plane anisotropy.\nThus, in addition to the anisotropic Gilbert contribu-\ntions, the Bloch damping provides a nearly frequency-\nindependent o\u000bset in the linewidth.\nSpin \row. { We now examine spin transport in the\ndevice with the aim of obtaining the damping parame-\nters that determine the linewidth [Eq. (3)]. The N layer\nis considered thick enough to eliminate static exchange\ninteraction between the two magnetic layers [31, 40]. Fur-\nthermore, we neglect the imaginary part of the spin-\nmixing conductance, which is small in metallic systems\nand does not a\u000bect dissipation in any case. Disregarding\nlongitudinal spin transport and relaxation in the thin free\nlayer, the net spin current IIIs1lost by F 1is the di\u000berence\nbetween the spin pumping and back\row currents [31]:\nIIIs1=gr\n4\u0019\u0010\n~^mmm\u0002_^mmm\u0000^mmm\u0002\u0016\u0016\u0016s\u0002^mmm\u0011\n; (4)\nwheregris the real part of the F 1jN interfacial spin-\nmixing conductance, and \u0016\u0016\u0016sis the spatially homogeneous3\nspin accumulation in the thin N layer. The spin current\nabsorbed by F 2may be expressed as [31]:\nIIIs2=g0\nr\n4\u0019^mmm2\u0002\u0016\u0016\u0016s\u0002^mmm2+gl\n4\u0019(^mmm2\u0001\u0016\u0016\u0016s)^mmm2;\n\u0011X\ni;j=fx;y;zggij\n4\u0019\u0016sj^iii; (5)\nwhereglandg0\nrare respectively the longitudinal spin\nconductance and the real part of the interfacial spin-\nmixing conductance of the N jF2subsystem, ^mmm2denotes\nthe unit vector along F 2magnetization, and gij=gji\nare the components of the resulting total spin conduc-\ntance tensor. glquanti\fes the absorption of the spin\ncurrent along the direction of ^mmm2, the so-called longi-\ntudinal spin current. For metallic magnets, it is domi-\nnated by the di\u000busive spin current carried by the itin-\nerant electrons, which is dissipated over the spin re-\nlaxation length [31]. On the other hand, for insulat-\ning magnets, the longitudinal spin absorption is domi-\nnated by magnons [43, 44] and is typically much smaller\nthan for the metallic case, especially at low tempera-\ntures. Considering ^mmm2= sin\u0012^yyy+ cos\u0012^zzz(Fig. 1),\nEq. (5) yields gxx=g0\nr,gyy=g0\nrcos2\u0012+glsin2\u0012,\ngzz=g0\nrsin2\u0012+glcos2\u0012,gxy=gyx=gxz=gzx= 0,\nandgyz=gzy= (gl\u0000g0\nr) sin\u0012cos\u0012.\nRelegating the consideration of a small but \fnite spin\nrelaxation in the thin N layer to the supplemental ma-\nterial [45], we assume here that the spin current lost by\nF1is absorbed by F 2, i.e.,IIIs1=IIIs2. Imposing this spin\ncurrent conservation condition, the spin accumulation in\nN along with the currents themselves can be determined.\nWe are primarily interested in the transverse (x and y)\ncomponents of the spin current since these fully deter-\nmine the magnetization dynamics ( ^mmm\u0001^mmm= 1):\nIs1x=1\n4\u0019grgxx\ngr+gxx(\u0000~_my+mx\u0016sz);\nIs1y=1\n4\u0019\u0014grgyy\ngr+gyy(~_mx+my\u0016sz) +gyz\u0016sz(1\u0000ly)\u0015\n;\n\u0016sz=~gr(lxmx_my\u0000lymy_mx\u0000p_mx)\ngzz\u0000pgyz+gr\u0000\nlxm2x+lym2y+ 2pmy\u0001;\n(6)\nwherelx;y\u0011gxx;yy=(gr+gxx;yy) andp\u0011gyz=(gr+gyy).\nThe spin lost by F 1appears as damping in the magneti-\nzation dynamics [Eqs. (1) and (2)] [31, 35].\nWe pause to comment on the behavior of \u0016szthus ob-\ntained [Eq. (6)]. Typically, \u0016szis considered to be \frst\nor second order in the cone angle, and thus negligibly\nsmall. However, as discussed below, an essential new\n\fnding is that it becomes independent of the cone an-\ngle and large under certain conditions. For a collinear\ncon\fguration and vanishing gl,gzz=gyz= 0 results\nin ~\u0016sz\u0011\u0016sz=~!!1 [38]. Its \fnite dc value con-\ntributes to the Bloch damping [Eq. (6)] [38]. For a\nnon-collinear con\fguration, \u0016sz\u0019\u0000~grp_mx=(gzz\u0000pgyz)\n0 45 90 135 18000.10.20.30.40.50.6FIG. 3. Normalized ferromagnetic resonance (FMR)\nlinewidth of F 1for di\u000berent values of the longitudinal spin\nconductance ~ gl\u0011gl=grof NjF2bilayer. The various parame-\nters employed are ~ g0\nr\u0011g0\nr=gr= 1, \u0002 = 0:1 rad,!0= 10\u00022\u0019\nGHz, and!ax= 1\u00022\u0019GHz.grandg0\nrare the spin-mixing\nconductances of F 1jN and NjF2interfaces respectively. Only\nthe spin pumping-mediated contribution to the linewidth has\nbeen considered and is normalized to its value for the case of\nspin pumping into a perfect spin sink [31].\nand contributes to Gilbert damping via Is1y[Eq. (6)].\nThus, in general, we may express the spin accumulation\nas\u0016sz=\u0016sz0+\u0016sz1[46], where \u0016sz0is the dc value\nand\u0016sz1/_mxis the linear oscillating component. \u0016sz0\nand\u0016sz1contribute, respectively, to Bloch and Gilbert\ndamping.\nGilbert-Bloch dissipation. { Equations (1) and (6) com-\npletely determine the magnetic damping in F 1. However,\nthese equations are non-linear and cannot be captured\nwithin our linearized framework [Eqs. (2) and (3)]. The\nleading order e\u000bects, however, are linear in all but a nar-\nrow range of parameters. Evaluating these leading or-\nder terms within reasonable approximations detailed in\nthe supplemental material [45], we are able to obtain the\nGilbert and Bloch damping tensors \u0014 \u000band\u0014\n. Obtaining\nthe general result numerically [45], we present the ana-\nlytic expressions for two cases covering a large range of\nthe parameter space below.\nFirst, we consider the collinear con\fgurations in the\nlimit of ~gl\u0011gl=gr!0. As discussed above, we obtain\n~\u0016sz0\u0011\u0016sz0=~!!1 and ~\u0016sz1\u0011\u0016sz1=~!!0 [Eq. (6)].\nThus the components of the damping tensors can be di-\nrectly read from Eq. (6) as ~ \u000bxx;yy\u0011\u000bxx;yy=\u000bss=ly;x=\ng0\nr=(gr+g0\nr) = ~g0\nr=(1+~g0\nr);and~\nxx;yy\u0011\nxx;yy=(\u000bss!) =\n\u0000lx;y\u0016sz0=(~!) =\u0000g0\nr=(gr+g0\nr) =\u0000~g0\nr=(1 + ~g0\nr). Here,\nwe de\fned ~ g0\nr\u0011g0\nr=grand\u000bss\u0011~grj\rj=(4\u0019MsV) is the\nGilbert constant for the case of spin-pumping into an\nideal spin sink [31, 35]. Substituting these values in Eq.\n(3), we \fnd that the linewidth, or equivalently damping,\nvanishes. This is understandable since the system we\nhave considered is not able to relax the z component of\nthe spin at all. There can, thus, be no net contribution to4\nFIG. 4. Normalized FMR linewidth of F 1for very small ~ gl.\nThe squares and circles denote the evaluated points while the\nlines are guides to the eye. The linewidth increases from being\nnegligible to its saturation value as \u0012becomes comparable to\nthe average cone angle \u0002. ~ g0\nr= 1,!0= 10\u00022\u0019GHz, and\n!ax= 1\u00022\u0019GHz.\nmagnetic damping. \u0016sz0accumulated in N opposes the\nGilbert relaxation via a negative Bloch contribution [38].\nThe latter may also be understood as an anti-damping\nspin transfer torque due to the accumulated spin [6].\nNext, we assume the system to be in a non-collinear\ncon\fguration such that ~ \u0016sz0!0 and may be disre-\ngarded, while ~ \u0016sz1simpli\fes to:\n~\u0016sz1=\u0000_mx\n!(~gl\u0000~g0\nr) sin\u0012cos\u0012\n~g0r~gl+ ~glcos2\u0012+ ~g0rsin2\u0012; (7)\nwhere ~gl\u0011gl=grand ~g0\nr\u0011g0\nr=gras above. This in turn\nyields the following Gilbert parameters via Eq. (6), with\nthe Bloch tensor vanishing on account of ~ \u0016sz0!0:\n~\u000bxx=~g0\nr~gl\n~g0r~gl+ ~glcos2\u0012+ ~g0rsin2\u0012;~\u000byy=~g0\nr\n1 + ~g0r;(8)\nwhere ~\u000bxx;yy\u0011\u000bxx;yy=\u000bssas above. Thus, ~ \u000byyis\u0012-\nindependent since ^mmm2lies in the y-z plane and the x-\ncomponent of spin, the absorption of which is captured\nby ~\u000byy, is always orthogonal to ^mmm2. ~\u000bxx, on the other\nhand, strongly varies with \u0012and is generally not equal\nto ~\u000byyhighlighting the tensorial nature of the Gilbert\ndamping.\nFigure 2 depicts the con\fgurational dependence of nor-\nmalized damping parameters. The Bloch parameters are\nappreciable only close to the collinear con\fgurations on\naccount of their proportionality to \u0016sz0. The\u0012range over\nwhich they decrease to zero is proportional to the cone\nangle \u0002 [Eq. (6)]. The Gilbert parameters are described\nsu\u000eciently accurately by Eq. (8). The linewidth [Eq.\n(3)] normalized to its value for the case of spin pump-\ning into a perfect spin sink has been plotted in Fig. 3.\nFor low ~gl, the Bloch contribution partially cancels the\nGilbert dissipation, which results in a smaller linewidthclose to the collinear con\fgurations [38]. As ~ glincreases,\nthe relevance of Bloch contribution and \u0016sz0diminishes,\nand the results approach the limiting condition described\nanalytically by Eq. (8). In this regime, the linewidth\ndependence exhibits a maximum for either collinear or\northogonal con\fguration depending on whether ~ gl=~g0\nris\nsmaller or larger than unity. Physically, this change in\nthe angle with maximum linewidth is understood to re-\n\rect whether transverse or longitudinal spin absorption\nis stronger.\nWe focus now on the case of very low ~ glwhich can\nbe realized in structures with electrically-insulating mag-\nnets. Figure 4 depicts the linewidth dependence close to\nthe collinear con\fgurations. The evaluated points are\nmarked with stars and squares while the lines smoothly\nconnect the calculated points. The gap in data for very\nsmall angles re\rects the limited validity of our linear\ntheory, as discussed in the supplemental material [45].\nAs per the limiting case ~ gl!0 discussed above, the\nlinewidth should vanish in perfectly collinear states. A\nmore precise statement for the validity of this limit is\nre\rected in Fig. 4 and Eq. (6) as ~ gl=\u00022!0. For su\u000e-\nciently low ~ gl, the linewidth changes sharply from a neg-\nligible value to a large value over a \u0012range approximately\nequal to the cone angle \u0002. This shows that systems com-\nprised of magnetic insulators bearing a very low ~ glare\nhighly tunable as regards magnetic/spin damping by rel-\natively small deviation from the collinear con\fguration.\nThe latter may be accomplished electrically by employ-\ning magnetoelectric material [41] for F 2or via current\ndriven spin transfer torques [6, 9, 47].\nDiscussion. { Our identi\fcation of damping contribu-\ntions as Gilbert-like and Bloch-like [Eq. (6)] treats \u0016sz\nas an independent variable that may result from SHE,\nfor example. When it is caused by spin pumping cur-\nrent and\u0016sz/!, this Gilbert-Bloch distinction is less\nclear and becomes a matter of preference. Our results\ndemonstrate the possibility of tuning the magnetic damp-\ning in an active magnet via the magnetization of a passive\nmagnetic layer, especially for insulating magnets. In ad-\ndition to controlling the dynamics of the uniform mode,\nthis magnetic `gate' concept [48] can further be employed\nfor modulating the magnon-mediated spin transport in a\nmagnetic insulator [43, 44]. The anisotropy in the result-\ning Gilbert damping may also o\u000ber a pathway towards\ndissipative squeezing [49] of magnetic modes, comple-\nmentary to the internal anisotropy-mediated `reactive'\nsqueezing [50, 51]. We also found the longitudinal accu-\nmulated spin, which is often disregarded, to signi\fcantly\na\u000bect the dynamics. This contribution is expected to\nplay an important role in a wide range of other phenom-\nena such as spin valve oscillators.\nSummary. { We have investigated the angular modu-\nlation of the magnetic damping in a free layer via control\nof the static magnetization in the \fxed layer of a spin\nvalve device. The damping can be engineered to become5\nlarger for either collinear or orthogonal con\fguration by\nchoosing the longitudinal spin conductance of the \fxed\nlayer smaller or larger than its spin-mixing conductance,\nrespectively. The control over damping is predicted to\nbe sharp for spin valves made from insulating magnets.\nOur results pave the way for exploiting magneto-damping\ne\u000bects in spin valves.\nAcknowledgments. { We acknowledge \fnancial support\nfrom the Research Council of Norway through its Centers\nof Excellence funding scheme, project 262633, \\QuSpin\",\nand from the Swedish Research Council, project 2018-\n03526, and Stiftelse Olle Engkvist Byggm astare.\n\u0003akashdeep.kamra@ntnu.no\n[1] Albert Fert, \\Nobel lecture: Origin, development, and\nfuture of spintronics,\" Rev. Mod. Phys. 80, 1517{1530\n(2008).\n[2] S. Parkin, Xin Jiang, C. Kaiser, A. Panchula, K. Roche,\nand M. 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Lett. 116, 146601 (2016).\n[51] Akashdeep Kamra, Utkarsh Agrawal, and Wolfgang\nBelzig, \\Noninteger-spin magnonic excitations in untex-\ntured magnets,\" Phys. Rev. B 96, 020411 (2017).\n[52] A.I. Akhiezer, V.G. Bar'iakhtar, and S.V. Peletminski,\nSpin waves (North-Holland Publishing Company, Ams-\nterdam, 1968).1\nSupplemental material with the manuscript Anisotropic and controllable\nGilbert-Bloch dissipation in spin valves by\nAkashdeep Kamra, Dmytro M. Polishchuk, Vladislav Korenivski and Arne Brataas\nCOLLINEAR CONFIGURATION WITHOUT LONGITUDINAL SPIN RELAXATION\nIn order to appreciate some of the subtleties, we \frst examine the collinear con\fguration in the limit of vanishing\nlongitudinal spin conductance. \u0012= 0;\u0019andgl= 0 imply the following values for the various parameters:\ngxx=gyy=g0\nr; g zz=gyz=p= 0; lx;y=g0\nr\ngr+g0r\u0011l; (S1)\nwhence we obtain:\n\u0016sz\n~=(mx_my\u0000my_mx)\nm2x+m2y; (S2)\n=!0+!ax\n1 +!ax\n2!0[1\u0000cos(2!t)]; (S3)\nwhere we have assumed magnetization dynamics as given by the Landau-Lifshitz equation without damping, and\nthe phase of mxis treated as the reference and set to zero. In order to obtain analytic expressions, we make the\nassumption !ax=!0\u001c1 such that we have:\n\u0016sz=\u0016sz0+\u0016sz2; with (S4)\n\u0016sz0=~\u0010\n!0+!ax\n2\u0011\n; (S5)\n\u0016sz2=~!ax\n4\u0000\ne\u0000i2!t+ei2!t\u0001\n: (S6)\nIn contrast with our assumptions in the main text, a term oscillating with 2 !appears. Furthermore, it yields\ncontributions to the Bloch damping via products such as my\u0016sz, which now have contributions oscillating at !due\nto the\u0016sz0as well as\u0016sz2. We obtain:\n~\u000bxx= ~\u000byy=l; (S7)\n~\nxx=\u0000l!0+3!ax\n4\n!0+!ax\n2and ~\nyy=\u0000l!0+!ax\n4\n!0+!ax\n2; (S8)\nsubstituting which into Eq. (3) from the main text yields a vanishing linewidth and damping. This is expected from\nthe general spin conservation argument that there can be no damping in the system if it is not able to dissipate the\nz-component of the spin. In fact, in the above considerations, \u0016sz2contributed with the opposite sign to ~\nxxand\n~\nyy, and thus dropped out of the linewidth altogether. This also justi\fes our ignoring this contribution in the main\ntext.\nFigure 1 depicts the dependence of the accumulated z-polarized spin and the normalized linewidth for small but\n\fniteglin the collinear con\fguration. The accumulated longitudinal (z-polarized) spin increases with the cone angle\nand the linewidth accordingly decreases to zero [38].\nNUMERICAL EVALUATION\nDespite the additional complexity in the previous section, we could treat the dynamics within our linearized frame-\nwork. However, in the general case, \u0016szhas contributions at all multiples of !and cannot be evaluated in a simple\nmanner. A general non-linear analysis must be employed which entails treating the magnetization dynamics numer-\nically altogether. Such an approach prevents us from any analytic description of the system, buries the underlying\nphysics, and is thus undesirable.\nFortunately, the e\u000bects of non-linear terms are small for all, but a narrow, range of parameters. Hence, we make\nsome simplifying assumptions here and continue treating our system within the linearized theory. We only show2\n10-310-210-100.10.20.30.40.5\nFIG. 1. Ferromagnetic resonance linewidth and the dc spin accumulation created in the spacer as a function of the average\ncone angle in the collinear con\fguration. Depending on ~ gl, there is a complementary transition of the two quantities between\nsmall and large values as the cone angle increases. ~ g0\nr= 1,!0= 10\u00022\u0019GHz, and!ax= 1\u00022\u0019GHz.\nresults in the parameter range where our linear analysis is adequate. Below, we describe the numerical routine for\nevaluating the various quantities. To be begin with the average cone angle \u0002 is de\fned as:\n\u00022=\nm2\nx+m2\ny\u000b\n; (S9)\nwhereh\u0001idenotes averaging over time. The spin accumulation is expressed as \u0016sz=\u0016sz0+\u0016sz1with:\n\u0016sz0=*\n~gr(lxmx_my\u0000lymy_mx\u0000p_mx)\ngzz\u0000pgyz+gr\u0000\nlxm2x+lym2y+ 2pmy\u0001+\n; (S10)\n\u0016sz1=\u0000*\ngrp\ngzz\u0000pgyz+gr\u0000\nlxm2x+lym2y+ 2pmy\u0001+\n~_mx: (S11)\nThe above expressions combined with the equations for the spin current \row (Eqs. (6) in the main text) directly yield\nthe Gilbert and Bloch damping tensors.\nVARIATION WITH ADDITIONAL PARAMETERS\nHere, we discuss the dependence of the FMR linewidth on the easy-plane anisotropy and the spin-mixing conduc-\ntanceg0\nrof the NjF2interface. The results are plotted in Fig. 2. A high easy-plane anisotropy is seen to diminish\nthe con\fguration dependence of the linewidth and is thus detrimental to the dissipation tunability. The easy-axis\nanisotropy, on the other hand, is absorbed in !0and does not need to be examined separately. We also see an increase\nin the con\fguration dependence of the damping with an increasing g0\nr. This is understood simply as an increased\ndamping when the spin is absorbed more e\u000eciently due to a larger g0\nr. The damping is expected to reach the case of\nspin pumping into a perfect spin sink in the limit of ~ g0\nr!1 and\u0012= 0;\u0019.\nEFFECT OF SPIN RELAXATION IN THE SPACER LAYER\nWe now address the role of the small but \fnite spin relaxation in the non-magnetic spacer layer. To this end, we\nconsider that a part of the spin current injected into N by F 1is lost as the \\spin-leakage current\" IIIsl, as depicted in\nFig. 3, such that IIIs1=IIIs2+IIIsl. In order to evaluate the leakage, we consider the spin di\u000busion equation in N which\nreads [31]:\nD@2\nx\u0016\u0016\u0016s=\u0016\u0016\u0016s\n\u001csf; (S12)3\n0 45 90 135 18000.10.20.30.40.50.6\n(a)\n0 45 90 135 18000.20.40.60.81 (b)\nFIG. 2. Normalized ferromagnetic resonance (FMR) linewidth of F 1. (a) Same as Fig. 3 in the main text with additional plots\nfor a large easy-plane anisotropy. (b) Linewidth dependence for di\u000berent spin-mixing conductances of N jF2interface. The\nparameters employed are the same as Fig. 2 in the main text.\nFIG. 3. Schematic depiction of the spin currents \rowing through the device, including the spin-leakage current IIIslthat is lost\non account of a \fnite spin relaxation in the spacer layer N.\nwhereDand\u001csfare di\u000busion constant and spin-\rip time, respectively. We now integrate the equation over the\nthickness of N:\nZ\nd(D@x\u0016\u0016\u0016s) =Zd\n0\u0016\u0016\u0016s\n\u001csfdx: (S13)\nSince the N-layer thickness dis typically much smaller than the spin di\u000busion length in N (e.g., a few nm versus a\nfew hundred nm for Cu), we treat \u0016\u0016\u0016son the right hand side as a constant. Furthermore, in simplifying the left hand\nside, we invoke the expression for the spin current [31]: IIIs= (\u0000~NSD=2)@x\u0016\u0016\u0016s, withNthe one-spin density of states\nper unit volume and Sthe interfacial area. Thus, we obtain\n2\n~NS(IIIs1\u0000IIIs2) =d\n\u001csf\u0016\u0016\u0016s; (S14)\nwhich simpli\fes to the desired relation IIIs1=IIIs2+IIIslwith\nIIIsl=~NVN\n2\u001csf\u0016\u0016\u0016s\u0011gsl\n4\u0019\u0016\u0016\u0016s; (S15)\nwhereVNis the volume of the spacer layer N.\nIt is easy to see that accounting for spin leakage, as derived in Eq. (S15), results in the following replacements to\nEqs. (6) of the main text:\ngxx!gxx+gsl; g yy!gyy+gsl; g zz!gzz+gsl: (S16)4\nSince all our speci\fc results are based on Eqs. (6) of the main text, this completes our assessment of the role played\nby spin relaxation in N. Physically, this new result means that the condition for no spin relaxation in the system,\nwhich was previously treated as gl!0, is now amended to gl+gsl!0. This, however, does not a\u000bect the generality\nand signi\fcance of the key results presented in the main text." }, { "title": "2012.09315v1.Observation_of_anti_damping_spin_orbit_torques_generated_by_in_plane_and_out_of_plane_spin_polarizations_in_MnPd3.pdf", "content": "Observation of anti-damping spin -orbit torque s generated \nby in-plane and out-of-plane spin polarizations in MnPd 3 \n \nMahendra DC1*, Ding -Fu Shao2, Vincent D. -H. Hou3, P. Quarterman4, Ali Habiboglu5, Brooks \nVenuti6, Masashi Miura1,7, Brian Kirby4, Arturas V ailionis8,9, Chong Bi10, Xiang Li1,10, Fen \nXue1, Yen -Lin Huang3, Yong Deng1, Shy -Jay Lin3, Wilman Tsai1, Serena Eley6, Weigang \nWang5, Julie A. Borchers4, Evgeny Y. Tsymbal2, and Shan X. Wang1,10* \n1Department of Materials Science and Engineering, Stanford U niversity, Stanford, California 94305, USA \n2Department of Physics and Astronomy & Nebraska Center for Materials and Nanoscience, University of Nebraska, \nLincoln, NE 68588 -0299 \n3Taiwan Semiconductor Manufacturing Company, Hsinchu , Taiwan \n4NIST Center for Ne utron Research, National Institute of Standards and Technology, 100 Bureau Dr., Gaithersburg, \nMaryland 20899, USA \n5Department of Physics, University of Arizona, Tucson, Arizona 85721, USA \n6Department of Physics, Colorado School of Mines, Golden, C olorado 80401 \n7Graduate School of Science and Technology, Seikei University, 3 -3-1 Kichijoji -kitamachi, Musashinoshi, Tokyo, \n180-8633, Japan \n8Stanford Nano Shared Facilities, Stanford University, Stanford, CA 94305, USA \n9Department of Physics, Kaunas University of Technology, LT -51368 Kaunas, Lithuania \n10Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA \n \n \nHigh spin-orbit torque s (SOT s) generated by topological materials and heavy metals \ninterfaced with a ferromagnetic layer show promise for next generation magnetic memory \nand logic devices . SOTs generated from the in -plane spin polarization along y-axis \noriginated by the spin Hall and Edelstein effect s can switch magnetization collinear with \nthe spin polarization in the absenc e of external magnetic field s. However, an external \nmagnetic field is required to switch the magnetization along x and z-axes via SOT \ngenerated by y-spin polarization . Here, we present that the above limitation can be \ncircumvented by unconventional SOT in magnetron -sputtered thin film MnPd 3. In addition to the conventional in-plane anti-damping -like torque due to the y-spin \npolarization , out-of-plane and in -plane anti-damping -like torque s originat ing from z-spin \nand x-spin p olarizations, respectively have been observed at room temperature . The spin \ntorque efficiency ( 𝜽𝒚) corresponding to the y -spin polarization from MnPd 3 thin films \ngrown on thermally oxidized silicon substrate and post annealed at 400 ℃ is 0.34 - 0.44 \nwhile the spin conductivity ( 𝝈𝒛𝒙𝒚) is ~ 5.70 – 7.30× 105 ℏ𝟐𝒆⁄ Ω-1m-1. Remarkably, we have \ndemonstrated complete external magnetic field -free switching of perpendicular Co layer \nvia unconventional out -of-plane anti-damping -like torque from z-spin polarization. Based \non the density functional theory calculations, we determine that the obser ved x- and z - spin \npolarizations with the in -plane charge current are due to the low symmetry of the (114) \noriented MnPd 3 thin films. Taken together, the new material reported here provide s a path \nto realize a practical spin channel in ultrafast magnetic m emory and logic devices. \n \n*Corresponding author s: mdc2019@stanford.edu and sxwang@stanford.edu \n Efficient control of magnetization at ultra -high speed has been of prim e int erest to the \nspintronics community1. Spin-orbit torque (SOT) has provided efficient and ultrafast control of \nmagnetization in magnetoresistive random access memory (MRAM) and logic devices2,3. SOT \nhas been observed in magnetic semiconductor4 and heavy metals5–7, topological insulators8–12, \nantiferromagnets13–17, and semimetals18–20 interfaced with a ferromag netic layer s. The charge \ncurrent injected into the non-magnetic layer (spin channel) along the x-direction generates a spin \ncurrent along z-direction with spin polarization (𝜎̂) pointing along y-direction (𝜎̂𝑦) in heavy \nmetals due to the bulk spin Hall effect . In the case of topological insulators and Weyl semimetals \nnon-equilibrium spin -density is accumulated at the interface due to the time reversal symmetry \nprotected spin momentum locking21. SOTs exerted on the ferromagnet with in -plane magnetic \nanisotropy (IMA) are in -plane anti-damping -like (𝝉𝑨𝑫𝑳,𝒚∝𝑚̂×(𝜎̂𝑦×𝑚̂)) and out-of-plane \nfield-like (𝝉𝑭𝑳∝𝜎̂𝑦×𝑚̂), where 𝑚̂ is magnetization unit vector . Spin current with 𝜎̂ along z -\ndirection (𝜎̂𝑧) has been observed in transition -metal dichalcogenides18,20,22 and, ferromagnet s \ninterfaced with light metal s23 with the charg e current flow along x-direction . SOT due to 𝜎̂𝑧 \nexerts torque along the out-of-plane direction (𝝉𝑨𝑫𝑳,𝒛∝𝑚̂×(𝜎̂𝑧×𝑚̂)), which will enable \nexternal field free and low power switching of the out-of-plane magnetization24. Recently, 𝜎̂ \nalong the x-direction ( 𝜎̂𝑥) has been reported in the uncompensated antiferromagnet Mn 3GaN in \naddition to 𝜎̂𝑦 and 𝜎̂𝑧 due to the low magnetic symmetry17. SOT due to 𝜎̂𝑥 exerts torque along in-\nplane direction ( 𝝉𝑨𝑫𝑳,𝒙∝𝑚̂×(𝜎̂𝑥×𝑚̂)), which will deterministically switch the magnetization \nalong the x-direction in the absence of external magnetic field. The figure of merit of charge to \nspin conversion is known as the spin torque efficiency 𝜃𝑘∝𝜎𝑖𝑗𝑘\n𝜎𝑥𝑥, where 𝜎𝑖𝑗𝑘 (i, j, and k refers to \nthe spin current flow, charge current flow, and spin polarization directions, respectively) and 𝜎𝑥𝑥 are spin and charge conductivities, respectively . 𝜃𝑘 needs to be high for efficient control of the \nmagnetization. Furthermore, to avoid current shunting through a conducting ferromagnetic layer \nhigh 𝜎𝑖𝑗𝑘 is also required25. Another important requirement for the integration of a spin channel \ninto semiconductor IC technology is the tolerance of SOT materials to thermal annealing at 400 \n℃. However, there has not been a practical spin channel which can handle post annealing at that \ntemperature, and also possesses high 𝝉𝑨𝑫𝑳,𝒚 along with 𝝉𝑨𝑫𝑳,𝒙 and 𝝉𝑨𝑫𝑳,𝒛, enabling deterministic \nswitching of in -plane magnetization alo ng y, in-plane magnetization along x, and out -of-plane \nmagnetization, respectively, without the need of applying an external magnetic field. \nTo achieve high density magnetic memory and logic devices, perpendicular magnetic \nanisotropy (PMA) is desired26. PMA switching via SOT from heavy metals5–7 and topological \ninsulators11,12,27,28 has been reported at the room temperature in the presence of an external \nmagnetic field. Partial PMA switching has been observed on PtMn/[Co/Ni] x13, IrMn/CoFeB14, \nand PtMn/CoFeB/Gd/CoFeB16 stack structures, in the absence of external magnetic field, but \nwith the help of exchange bias. Stray fields from an in -plane magnetic layer present abo ve or \nbelow the spin channel can facilitate external magnetic field -free PMA switching, but current \nshunting and magnetic interference between different magnetic layers pose severe design \nconstraints in this approach29,30. Combination of SOT and spin transfer torque (STT) can also \nswitch PMA in the absence of extern al magnetic field, but STT could lead to reduced endurance \nof magnetic tunnel barrier and slower magnetization switching31. Fast switching of the \nmagnetization with lower critical current densities (𝐽𝑠𝑤 ) can be achieved when the charge \ncurrent flow and magnetization are collinear, however, this geometry still requires an external \nmagnetic field to achieve magnetization switching7. Here, we p resent SOT from sputtered MnPd 3 \nthin films post annealed at 400 ℃ that can generate a spin current with 𝜎̂ along all three axes due to the charge current flow along x-direction, which represents a major advance over the \nliterature and overcomes significant limitations of the existing SOT materials. \n \n \n \nFig. 1. Characterization of MnPd 3 thin film : a, Schematic diagram of D023 MnPd 3 unit cell. \nb, XRD of Si/SiO 2/MnPd 3 (50 nm) film, post -annealed at 400 ºC for 30 min. c, Cross -section \nTEM image of Si/SiO 2/MnPd 3 (10 nm) /CoFeB (5 nm)/MgO (2 nm)/Ta (2 nm) sample. d, Four \nterminal resistance as a function of temperature of Si/SiO 2/MnPd 3 (10 nm) sample. \nThe MnPd 3 thin films were magnetron sputtered at room temperature on 300 nm thick \nthermally oxidized silicon substrates. The thin films wi th the stack structure Si/SiO 2/MnPd 3 (t \nnm)/CoFeB (5 nm)/MgO (2 nm)/Ta (2 nm) with IMA were prepared for the SOT \ncharacterization with t = 4, 6, 8, 10, 12, 16, 20, and 24 nm, respectively. Unless otherwise stated, \nthese films will be labeled MP4 - MP24, i n which the number denotes the MnPd 3 thickness. All \nsamples were post annealed at 400 ℃ for 30 minutes . Using Rutherford backscattering, the \natomic composition of Mn and Pd in MnPd 3 film is 28% and 72% (data not shown), respectively. \nFig. 1a shows the unit cell of MnPd 3. We performed grazin g incidence θ - 2θ X-ray \ndiffraction (XRD) measurements on a Si/SiO 2/MnPd 3 (50 nm) sample, as shown in Fig. 1b. For \nthe grazing incidence XRD measurement Φ and Ω were fixed at 20° and 3.5°, respectively. From \nthe intensity of the peaks and pole figures (Me thods and Extended Data Fig. 1 ), its notable that \nMnPd 3 film has a strong (114) texture. The lattice parameters are estimated to be a = 3.89 Å, b = \n3.88 Å, and c = 15.42 Å indexed by using ref. (32). The cross -section transmission electron \nmicroscopy (TEM) bright image of the MP10 sample is prese nted in Fig. 1c. The high angle \nannular dark field (HAADF) image (data not shown) and the bright image both show that the \nMnPd 3 layer grown on thermally oxidized silicon is polycrystalline. The CoFeB and MgO layers \nare also polycrystalline. The electric an d magnetotransport measurements were performed on \nSi/SiO 2/MnPd 3 (10 nm)/MgO (2 nm)/Ta (2 nm) heterostructure, it will be labelled as MnPd (10 \nnm) sample ( Methods and Extended Data Fig. 2 ). The resistivity shows metallic behavior \ncoinciding with the possibl e transition from a paramagnetic to an antiferromagnetic state below \n50 K, as shown in Fig. 1d. The ordinary Hall resistance as a function of the external magnetic \nfield is non -linear at small fields. From the high -field linear region, we estimate a carrie r \nconcentration of 4.4 × 1022/cm3. At room temperature the valu es of anisotropic magnetoresistance (AMR) and planar Hall resistance (PHR) are estimated to be 0.012% and 20 \nmΩ in MnPd (10 nm) sample, respectively . The Néel temperature of MnPd (10 nm) sample is \napproximately 37 K inferred from using temperature -dependen t magnetometry (Methods and \nExtended Data Fig. 3). Polarized neutron reflectometry (PNR) measurements show weak \nferromagnetism, persisting upto room temperature, in MnPd 3 films possibly originating from \nuncompensated Mn moments, Mn clusters, or local ferro magnetic Mn -based compound \nformation . At room temperature the ferromagnetic component of the magnetization in MnPd 3 is \ndetermined to be 9 ± 1.9 kA/m usi ng polarized neutron reflection (PNR) whereas at 6 K it is ~ 43 \n± 3.4 kA/m (Methods and Extended Data Fi g. 4). \n \n \nFig. 2. SOT characterization using Second Harmonic Hall (SHH) technique on \nSi/SiO 2/MnPd 3 (x nm)/CoFeB (5 nm)/MgO (2 nm)/Ta (2 nm): a, Schematic diagram showing \nin-plane charge current generated spin current with spin polarizations along three axes. The red \nspheres represent electrons and yellow arrows represent spin magnetic moment, respectively. b, \nand c, 𝑅𝑥𝑦1𝜔 and 𝑅𝑥𝑦2𝜔 , respectively, as a function of in -plane magnetic field rotation at a fixed \namplitude of 200 mT in MP12 sample . d, e, and f, Effective spin torque efficiency due to the in -\nplane anti-damping -like, in-plane and out -of-plane a nti-damping -like and field-like torques, \nrespectively, as a function of MnPd 3 film thickness. The Hall bar device used for this \nmeasurement was 10 µm wide and 130 µm long, respectively. \nWe performed SOT measurement s using SHH technique on MP4-MP24 samples, control \nMP (10 nm) and Si/SiO 2/ CoFeB (5 nm)/MgO (2 nm)/Ta (2 nm) samples, and a reference \nSi/SiO 2/Pt (10 nm)/CoFeB (5 nm)/MgO (2 nm)/Ta (2 nm) ( labelled as Pt10 sample) sample33. \nThe MP4 -MP24 samples and reference sample were patterned into Hall bars with length 130 and \nwidth 10 µm, respectively. The d etails of the SHH are presented in Methods and Extended Data \nFig. 5 . The a.c. current injec ted into the Hall bar induces effective spin -orbit field s, which \noscillate the magnetization around its equilibrium position, and as a result, a SHH voltage is \ninduc ed. In the SHH measurement, the sample is rotated in the x-y plane under constant static \nmagnetic field to keep the magnetization in a single domain state. The spin -current with the spin \nmagnetic moment pointing along the negative y-axis and positive x-and z-axes get accumulated \nbetween MnPd 3 and CoFeB layers as shown in Fig. 2a. Thus, accumulated spin -current s exert \n𝝉𝑨𝑫𝑳,𝒙 , 𝝉𝑨𝑫𝑳,𝒚 and 𝝉𝑨𝑫𝑳,𝒛 , 𝝉𝑭𝑳 on the CoFeB layer along the in -plane and out-of-plane \ndirections, respectively . In addition, the Oersted field (\nOeH ) generated due to the a.c. current flow \nin the MnPd 3 layer exerts an Oersted torque ( 𝝉𝑶𝒆) on the CoFeB layer. Fig. 2b shows 𝑅𝑥𝑦1𝜔 as a \nfunction of the in -plane magnetic field angle (𝜑). 𝑅𝑥𝑦1𝜔 fits perfectly to the 𝑠𝑖𝑛2𝜑, indicat ing that \nthe out-of-plane field projection due to the imperfect sample mounting is absent. 𝑅𝑥𝑦2𝜔 as a \nfunction of magnetic field angle is presented in Fig. 3c. Since the torques have diffe rent dependencies with 𝜑, we can extract 𝑅𝑥𝑦2𝜔 due to different types of torques using SHH . In MP10 \nsample, the extracted spin -orbit fields associated with the 𝝉𝑨𝑫𝑳,𝒙, 𝝉𝑨𝑫𝑳,𝒚, 𝝉𝑨𝑫𝑳,𝒛 , and 𝝉𝑭𝑳 are \n(0.02 ± 0.002), (0.132 ± 0.002), ( 0.0036 ± 0.0003) , and ( 0.019 ± 0.00 ) mT per 106 A/cm2, \nrespectively . These values of the 𝝉𝑨𝑫𝑳,𝒚 and 𝝉𝑭𝑳 are comparable or better than the previous \nreports on heavy metals/ferromagnet33, TIs/ferromagnet11,12, and Weyl \nsemimetal/ferromagnets19. In Fig. 2d 𝜃𝑦𝑒𝑓𝑓as a function of MnPd 3 film thickness is presented, \nwhich shows heavy metal like behavior. The simple drift -diffusion model (Eqn. 1) can be \nutilized to extract bulk spin -torque efficiency figure of merit ( 𝜃𝑦(𝑡≈∞)) and spin -diffusion \nlength (𝜆) , \n 𝜃𝑦𝑒𝑓𝑓=𝜃𝑦(𝑡≈∞)(1−sech(𝑡\n𝜆)) (1) \nwhere 𝑡 is MnPd 3 film thickness. Th is drift-diffusion model considers that the spin current \ngenerated by the bulk of thin films is completely absorbed by the ferromagnetic layer without \nany dissipation at the interface and back -flow of th e spin current. 𝜃𝑦(𝑡≈∞) and 𝜆 obtained by \nordinary drift-diffusion model are 0.34 and 6. 30 nm, respectively. Now by considering spin -back \nflow the drift -diffusion model can be modified into34–36: \n𝜃𝑦𝑒𝑓𝑓=𝜃𝑦(𝑡≈∞)(1−sech (𝑡\n𝜆))[1+tanh ((𝑡\n2𝜆)\n2𝜌𝜆𝐺↑↓]−1 (2) \nwhere 𝜌 is bulk resistivity, 𝐺↑↓ is spin-mixing conductivity, respectively. The red line in Fig. 2d \nis a fit to Eqn. (2) with 𝜃𝑦(𝑡≈∞) and 𝜆 as independent fitting parameters. 𝐺↑↓ values of MP4 \nand MP24 samples are estimated to be 4.54 × 1014 m-2 and 3.7 0 × 1015 m-2, respectively. The \nextracted values of 𝜃𝑦(𝑡≈∞) and 𝜆 are 0.44 and 5.88 nm, respectively for 𝐺↑↓value of 4.54 × 1014 m-2. 𝜌 value used for the fitting was 60 µΩcm obtained by measuring four terminal \nresistance of M nPd3 (20 nm) sample. The extracted values of 𝜃𝑦(𝑡≈∞) and 𝜆 are 0.3 6 and 6.30 \nnm, respectively for 𝐺↑↓ value of 3.70 × 1015 m-2. The values of the 𝜎𝑧𝑥𝑦 corresponding to 𝜃𝑦 (0.34 \n- 0.44) are estimated to be (5.67 – 7.33) × 105 ℏ2𝑒⁄ Ω-1m-1. These values of 𝜎𝑧𝑥𝑦 are among the \nlargest for the reported values of the antiferromagnets15,37, heavy metals6,38,39, topological \ninsulators8,11, and Weyl semimetals19. 𝑅𝑥𝑦2𝜔 in the control samples does not show any field or 𝜑 \ndependence indicating that the SOTs in the MP sa mples originat es from the MnPd 3 layer \n(Methods and Extended Data Fig. 5 and Fig. 6 ). The values for the spin-orbit fields associated \nwith 𝝉𝑨𝑫𝑳,𝒚 and 𝝉𝑭𝑳 of the reference Pt10 sample are estimated to be (0.040 ± 0.0 03) and (0. 012 \n± 0.0 01) mT per 106 A/cm2, respectively (Methods and Extended Data Fig. 7 ). The estimated \nvalue of 𝜃𝑦𝑒𝑓𝑓 and 𝜃𝐹𝐿𝑒𝑓𝑓 are (0.07 ± 0.01) and (0.02 ± 0.002), respectively in agreement with the \nprevious reports6,33,40. After post annealing at 400 ℃, the reference Pt10 sample does not show a \nSHH signal, suggesting that its SOT d id not withstand such annealing. \nAs presented in Fig. 2e, 𝜃𝑧𝑒𝑓𝑓 and 𝜃𝑥𝑒𝑓𝑓 do not depend on the MnPd 3 film thickness as \n𝜃𝑦𝑒𝑓𝑓 does. 𝜃𝐹𝐿𝑒𝑓𝑓 shown in Fig. 2d also does not show any specific MnPd 3 film thickness \ndependenc e. The 𝜎𝑧𝑥𝑧,𝑒𝑓𝑓 value in MP12 sample is as large as ~ 0.14 × 105 ℏ2𝑒⁄ Ω-1m-1.𝜎𝑧𝑥𝑧,𝑒𝑓𝑓 in \nMP samples is comparable or better than recent reports on WTe 2/Py18 and Mn 3GaN/Py17. 𝜎𝑧𝑥𝑥,𝑒𝑓𝑓 \nin MP12 sample is 0.77 × 105 ℏ2𝑒⁄ Ω-1m-1. These values of 𝜃𝑘and 𝜎𝑖𝑗𝑘 are largest among the \nreported values as listed in Table 1. The difference in the thickness dependence of the 𝜃𝑥,𝑦,𝑧𝑒𝑓𝑓 \nindicate that their origins are also different. We also performed ST -FMR measurement on MP10 sample as a confirmation of the \nobserved SOT with SHH technique . The details of ST -FMR are presented in Methods and \nExtended Data Fig. 8. 𝜃𝑦𝑒𝑓𝑓 estimated by using ST -FMR of MP10 sample is (0.21 ± 0.01) \nwhereas it is ( 0.22 ± 0.03) estimated using SHH. We also prepared reference samples Si/SiO 2/Pt \n(6 and 10 nm)/CoFeB (5 nm)/MgO (2 nm)/Ta (2 nm) (labelled as Pt6 and Pt10 samples) and \nSi/SiO 2/W (6 nm )/CoFeB (2 nm)/MgO (2 nm)/Ta (2 nm) (labelled as W sample) for the ST -\nFMR measurements. 𝜃𝑦𝑒𝑓𝑓 of as deposited Pt samples is 0.06 ± 0.01 and 0.08 ± 0.01 , respectively \n(Methods and Extended Data Fig. 9). However, after post annealing at 400 ℃ the reference Pt \nsamples do not show a ST -FMR signal , suggesting that SOT did not withstand the annealing \nprocess. 𝜃𝑦𝑒𝑓𝑓 of as deposited W sample is determined to be -0.43 ± 0.03 at 6 GHz excitation \nfrequency. The estimated 𝜎𝑧𝑥𝑦,𝑒𝑓𝑓 is -1.43 × 105 ℏ2𝑒⁄ Ω-1m-1. This value of 𝜃𝑦𝑒𝑓𝑓 of the as \ndeposited W sample is compar able to the previous report41. 𝜃𝑦𝑒𝑓𝑓 of the W sample post annealed \nat 400 ℃ for 30 minutes is estimated to be -0.011 and the corresponding 𝜎𝑦𝑒𝑓𝑓 is -0.13 × 105 \nℏ2𝑒⁄ Ω-1m-1 (Methods and Extended Data Fig. 9 ). \n \n \nTable 1: Summary of 𝜽𝒌, 𝝈𝒊𝒋𝒌, and ρ of different spin channels post annealed at different \ntemperatures and measured at room temperature. Materials ρ \n(µΩcm) 𝜃𝑥𝑒𝑓𝑓 𝜃𝑦 𝜃𝑧𝑒𝑓𝑓 𝜃𝐹𝐿𝑒𝑓𝑓 𝜎𝑧𝑥𝑦(ℏ2𝑒⁄ \n105 Ω-1m-1) Jsw \n(MA/cm2) Switched \nmagnetic \nanisotropy Hx \n(mT) Post \nannealing \ntemp (℃) \nThis work 60 - 95 0.053 0.34-0.44 0.018 0.06 5.7-7.3 11-24.7 IMA -PMA Zero 400 \nPt40 40 - 0.14 - 0.08 3.5 50 PMA 70 As dep. \nW38 188 - -0.22 - -0.022 -1.2 1142 PMA 20 20042-30038 \nMnGaN17 225 -0.013 0.025 0.018 -0.15 0.11 - - - As dep. \nWTe 218 380 - 0.03 0.013 0.034 0.08 - - - As dep. \nFig. 3. Demonstration of external magnetic field -free out-of-plane magnetization sw itching \nvia z-spin polarization generated anti -damping spin-orbit torque: a , The anomalous Hall \nresistance as a function of out-of-plane external field . b, Switching of perpendicular Co layer via \nSOT under the application of -8 mT and -20 m T along x-direction for MP (10 nm )/Co (1 nm ) and \nMP (12 nm)/Co (1 nm ) sample s, respectively . c, and d, Field-free perpendicular Co layer \nswitching via out-of-plane anti-damping -like torque generated by z-spin polarization . The Hall \nbar with a length of 130 μm and a width of 10 μm was used for the magnetization switching \nexperiment. \nIn order to demonstrate external magnetic field-free PMA switching, we prepared \nSi/SiO 2/MnPd 3 (10 and 12 nm)/Co (1 nm)/MgO (2 nm)/Ta (2 nm) sampl es (will be labelled as \nMP10/Co1 and MP12/Co1 samples). MP10/Co1 and MP12/Co1 samples were annealed at 400 \nºC for 30 minutes in vacuum and subsequently field -cooled under the application of an out -of-\nplane magnetic field of 0.45 T. Fig. 3a shows anomalous Hall resistance ( RAHE) as a function of \nout-of-plane magnetic field. The hysteretic RAHE loop confirms PMA is present in the Co layer. \nAlternatively, magnetometry was also used to confirm that PMA is present in MP10/Co1 sample \n(Methods and Extended Data Fi g. 3). Fig. 3b shows current -induced SOT magnetization \nswitching under the presence of negative external magnetic fields. The write d.c. current pulse \nwidth used for PMA switching is 20 ms, which is followed by a read current of 0.4 mA. The full \nswitching of magnetization occurs in MP10/Co1 and MP12/Co1 samples at ~ 10.1 mA and 9.5 \nmA, respectively. Then, in the absence of external magnetic field, the d.c. current in pulses is \nswept from -36 mA to + 36 mA with a step size of 1.06 mA. For MP10/Co1 as shown in Fig. 3c, \npartial switching of the magnetization occurs at a positive current of about 10.78 mA, and \ncomplete swi tching occurs at ~ 32.4 mA (~3 7.0 MA/cm2). In the subsequent reverse sweep \npartial switching of magnetization occurs at ~ -10.87 mA and co mplete switching occurs at ~ - \n31.63 mA. For MP12/Co1 sample as shown in Fig. 3d, partial switching of magnetization occurs \nat a positive current of ~ 11.35 mA. Continuously sweeping the d.c. pulses switches the \nremaining m agnetization at ~ 30.92 mA (~2 4.7 MA/cm2). Subsequently reverse sweeping the \ncurrent pulses, partial switching of magnetization occurs at ~ - 9.60 mA and switching of \nremaining magnetization occurs at ~ – 33.32 mA. Since the RAHE values obtained by field sweep \nand current sweep are clo se, we can conclude that the full switching of PMA has been observed \nin both PMA samples. 𝐽𝑠𝑤 values observed in our PMA samples without external magnetic field \nis comparable or better than the previously reported values in Pt/Co5,6 ( ~23-100 MA/cm2), \nPd0.25Pt0.75/Co43 (~22 MA/cm2), Pt/antiferromagnet44 with external magnetic field. The SOT \nswitching of magnetization in our PMA samples results from the interplay of 𝝉𝑨𝑫𝑳,𝒙, 𝝉𝑨𝑫𝑳,𝒚, 𝝉𝑨𝑫𝑳,𝒛, and 𝝉𝑭𝑳. In the presence of external magnetic field the RAHE vs I loop shows similar \nbehavior as that of positive 𝜃𝑦𝑒𝑓𝑓 such as in the case of Pt/Co/AlOx5,6. In the absence of an \nexternal magnetic field, if there is only 𝝉𝑨𝑫𝑳,𝒛, PMA switching occurs via anti -damping process, \nwhich is confirmed by numerically solving Landau -Lifshitz -Gilbert (LLG) equation (Methods \nand Extended Data Fig. 1 0b). In the presence of a relatively we aker 𝝉𝑨𝑫𝑳,𝒛 and 𝝉𝑨𝑫𝑳,𝒙 and strong \n𝝉𝑨𝑫𝑳,𝒚 the magnetization is partially switched at a lower current (~10 mA), which results in an \nintermediate state. The intermediate state can occur due to an insufficient external magnetic \nfield, which is unable to completely break mirror symmetry45. Previously, intermediate magnetic \nstates were observed below threshold 𝐽𝑠𝑤 (Ref. 46). The external magnetic field-free switching of \nPMA unambiguously demonstrates the presence of 𝜎̂𝑧 generated 𝝉𝑨𝑫𝑳,𝒛 in the MP samples. \nThese experimental results of PMA switching have been qualitatively reproduced by the LLG \nsimulations (Methods and Extended Data Fig. 1 0c and 1 0d). The intermediate states observed \ncould be utilized for n euromorphic computing13. In addition to field -free PMA switching, we \nalso performed field -free magnetization switching of the in -plane CoFeB layer in MP24 sample, \nas detected by using unidirectional spin Hall mag netoresistance ( USMR ) mechanism47,48 \n(Methods and Extended Data Fig. 11). 𝐽𝑠𝑤 is estimated to be ~11.0 MA/cm2 using the parallel \nresistor model. \nWe also performed SHH measurements on the PMA samples, as detailed in Methods and \nExtended Data Fig. 13. The magnitude of the SHH resistance ( 𝑅𝑥𝑦2𝜔) is not symmetric a t up and \ndown magnetization s when the field is swept along y-axis. If there were only 𝝉𝑭𝑳 and 𝝉𝑶𝒆 present \nin MP samples, the magnitude and field dependence of 𝑅𝑥𝑦2𝜔 would remain the same since the \nspin-orbit field associated to them is independent of the magnetization polarity as in the case of \nTa/CoFeB/MgO23. The spin -orbit field associated with 𝝉𝑨𝑫𝑳,𝒙 (𝑯𝑨𝑫𝑳,𝒙 ~(𝜎̂𝑥×𝑚̂)) switches sign as the magnetization switches sign, which results in the un equal and different field \ndependence of 𝑅𝑥𝑦2𝜔. This clearly shows the presence of torque generated by 𝜎̂𝑥 in MP samples. \n𝜃𝑥𝑒𝑓𝑓, 𝜃𝑦𝑒𝑓𝑓, and 𝜃𝐹𝐿𝑒𝑓𝑓 of the MP10/Co1 sample (MP12/Co1 sample) are 0.023 ± 0.001 (0.040 ± \n0.003 ), 0.24 ± 0.0 2 (0.32 ± 0.03 ), and 0.03 ± 0.00 1 (0.12 ± 0.0 1), respectively. These values are \ncomparable to the SHH -measured SOT efficiencies of IMA MP10 and MP12 samples . \n \nFig. 4: Effect of (114) texture on the spin polarization. a, The calcu lated band structure of \nstoichiometric MnPd 3 at room temperature . b, c, and d, The calculated 𝜎𝑧𝑥𝑥, 𝜎𝑧𝑥𝑦, and 𝜎𝑧𝑥𝑧 as a \nfunction of energy for MnPd 3 (114) film, where the x axis is oriented along the [4̅01] direction. \nWe explain the appearance of the non -vanishing 𝜎̂𝑥 and 𝜎̂𝑧 in terms the contribution from \ngrain s of different orientations in our polycrystalline MnPd 3 films. MnPd 3 has crystal space \ngroup I4/mmm [32]. If the film was monocrystalline and (001) oriented with the charge current \nflowing along the [100] direction (the x-direction), only the conventional spin Hall conductivity \n𝜎𝑧𝑥𝑦 (where the spin current flows along the [001] direction (the z-direction) normal to the charge \ncurrent and has spin polarization along the [010] direction (the y-direction)), would be allowed \n-1.0-0.50.00.51.0\n0200400\n010002000\n-0.2 -0.1 0.0 0.1 0.20100200300E-EF (eV)\nΓ X Σ Γ Z s x\nzx\ns z\nzxs y\nzx SHC (ħ/2e Ω cm)-1\nE-EF (eV)a b\nc\nddue to the (001) plane being invariant to all symmet ry operations of this space group. In a \npolycrystalline film, however, other crystal orientations with lower symmetries contribute to the \nspin Hall conductivity. For example, the (114) plane corresponding to the dominant texture of \nour films (reflected by the strongest XRD peak in Fig. 1 (b)) is only invariant with respect to \nmirror symmetry 𝑀[1̅10] and two -fold rotation 𝐶[1̅10]. This allows the appearance of \nunconventional components of the spin Hall conductivity tensor, such as 𝜎𝑧𝑥 𝑥 and 𝜎𝑧𝑥 𝑧, where the \nspin polarization is parallel to the direction of the spin curr ent or the charge current. Moreover, in \na polycrystalline film, the current direction itself varies with respect to the high symmetry \ndirections of different grains, which also influences the shape of the spin Hall conductivity \ntensor. These aspects have b een discussed in Ref. [15]. \nTo quantitatively evaluate the contribution from this mechanism, we perform first -principles \ndensity functional theory calculations of spin Hall conductivity of bulk MnPd 3 assuming a room -\ntemperature paramagnetic phase . Figure 4a shows the calculated band structure of MnPd 3. There \nare several bands crossing the Fermi level ( EF), indica ting the metallic ground state. The small \ngaps between the bands near EF are favorable for the sizable spin Hall conductivity49, which is \ngiven by50 \n𝜎𝑖𝑗𝑘=𝑒2\nℏ∫𝑑3𝑘⃗ \n(2𝜋)3∑𝑓𝑛𝑘⃗ Ω𝑛,𝑖𝑗𝑘(𝑘⃗ ) 𝑛 , (3) \nΩ𝑛,𝑖𝑗𝑘(𝑘⃗ )=−2𝐼𝑚∑⟨𝑛𝑘⃗ |𝐽𝑖𝑘|𝑛′𝑘⃗ ⟩⟨𝑛′𝑘⃗ |𝑣𝑗|𝑛𝑘⃗ ⟩\n(𝐸𝑛𝑘⃗⃗ −𝐸𝑛′𝑘⃗⃗ )2 𝑛′≠ 𝑛 , (4), \nwhere 𝑓𝑛𝑘⃗ is the Fermi -Dirac distribution function for band n and wave vector 𝑘⃗ , Ω𝑛,𝑖𝑗𝑘(𝑘⃗ ) is the \nspin Berry curvature, 𝐽𝑖𝑘=1\n2{𝑣𝑖,𝑠𝑘} is the spin -current operator, 𝑣𝑖 and 𝑠𝑘 are velocity and spin \noperators, respecti vely, and 𝑖,𝑗,𝑘 = 𝑥,𝑦,𝑧. As expected, for MnPd 3 textured in the (001) plane, only the conventional spin Hall conductivity ( 𝜎𝑧𝑥𝑦) is non -vanishing (Table 2 in Extended Data ). \nHowever, for the dominant (114) stacking texture , the unconventional spin Hall conductivities \n(𝜎𝑧𝑥𝑥 and 𝜎𝑧𝑥𝑧) emerge (Table 2 in Extended Data ). Figures 4b -d show the calculated spin Hall \nconductivities for MnPd 3 (114) film as a function of energy when the charge current flows along \nthe [4̅01] direction (the x-direction) . We find a h igh conventional 𝜎𝑧𝑥𝑦~1744 (ℏ\n2𝑒)(Ω cm)−1 and \nsizable unconventional conductivities 𝜎𝑧𝑥𝑥 ~ 279 (ℏ\n2𝑒)(Ω cm)−1 and 𝜎𝑧𝑥𝑧 ~ 166 (ℏ\n2𝑒)(Ω cm)−1 at \nthe Fermi energy. It is evident that the 𝜎𝑧𝑥𝑥 and 𝜎𝑧𝑥𝑧 values are approximately an order of \nmagnitude smaller than 𝜎𝑧𝑥𝑦, which is consistent with our experimental observation. Similarly, \nother MnPd 3 grains with different orientations can also contribute to the unconventio nal spin \nHall conductivity. \nIn summary, we studied anti -damping spin -orbit torques generated by the 𝜎̂𝑥, 𝜎̂𝑦, and 𝜎̂𝑧 \nin MnPd 3/ferromagnet heterostructure. At least two independent characterizations were \nperformed to verify the presence of torques. DFT simulations confirmed the low crystal \nsymmetry present in the (114) oriented MnPd3 thin films as the origin of the observed \nunconventional SOTs. We demonstrated successful growth of conductive MnPd 3 thin films with \nhigh 𝜎𝑧𝑥𝑥, 𝜎𝑧𝑥𝑦, and 𝜎𝑧𝑥𝑧 after post annealing at 400 ℃ for half an hour. Complete external \nmagnetic field-free switching of both IMA and PMA were realized. The observed SOTs were \nrobust against thermal treatment, and compatible with low damping constant o f CoFeB even \nafter post annealing. All of these are key factors for the integration of a practical spin current \nsource based on MnPd 3 into next generation of SOT -based spintronics devices. \n References \n1. Dien y, B. et al. Opportunities and challenges for spintronics in the microelectronics \nindustry. Nat. Electron. 3, 446 –459 (2020). \n2. Grimaldi, E. et al. Single -shot dynamics of spin –orbit torque and spin transfer torque \nswitching in three -terminal magnetic tun nel junctions. Nat. 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Y., Murakami, S., Chen, T. -W. & Nagaosa, N. Intrinsic spin Hall effect in \nplatinum: first -principles calculations. 100, 096401 (2008). \n50. Sinova, J., Valenzuela, S. O., Wunderlich, J., Back, C. H. & Jungwirth, T. Spin Hall \neffects. Rev. Mod. Phys. 87, 1213 –1260 (2015). \nAcknowledgments \nThis research was supported in part by ASCENT, one of six centers in JUMP, a Semiconductor \nResearch Corporation (SRC) program sponsored by DARPA. The authors thank the NSF Center \nfor Energy Efficient Electronics Science (E3S) and TSMC for financial support. Part of this \nwork was performed at the Stanford Nano Shared Facilities (SNSF)/Stanford Nanofabrication \nFacility (SNF), supported by the National Science Foundation under award E CCS -1542152. The \nresearch at the University of Nebraska -Lincoln is supported by the National Science Foundation \nthrough the Nebraska Materials Science and Engineering Center (MRSEC, Grant No. DMR -\n1420645). P. Q. acknowledges support from the National Rese arch Council Research \nAssociateship Program. S.E. and M. B. V. acknowledge funding from NSF award DMR -1905909, and assistance from Randy Dumas at Quantum Design with VSM measurements. The \nauthors would also like to acknowledge Dr. Carlos H. Diaz , Dr. Peng Li, and Dr. Juliet \nJamtgaard for fruitful discussions. \nAuthor contributions \nM. DC conceived, designed , and coordinated the research with contributions from M.M., S.-J.L., \nW.T., and S.X.W. S.X.W supervised the study. M. DC grew thin films, performed XRD \nmeasurement, fabricated Hall bar, ST -FMR device, carried out ST-FMR , SHH, and switching \nmeasurement s with contributions from Y.D., X.L., C.B. , F.X., and Y -L. H. D-F. S. and E. T. \nperformed DFT calculations. V.D.H., A. H. , and W. W. carried out TEM and EDS stu dies. A.V. \nperformed pole figure measurements. M. B.V. and S.E. performed magnetometry measurements . \nP.Q., B.K., and J.B. performed PNR measurements and modelling . M. DC performed LLG \nsimulations. M. DC performed data analysis and wrote the manuscript with contributions from \nD-F. S, P.Q., A.V., and S.X.W. All authors discussed the results and commented on the \nmanuscript. \nOnline content \nAny methods, additional references, Nature Research reporting summaries, source data, extended \ndata, supplementary informa tion, acknowledgements, peer review information; details of author \ncontributions and competing interests; and statements of data and code availability are available \nonline. \nCompeting I nterests \nThe authors declare no competing interests. \nData A vailability \nThe data that support the findings of this study are available from the corresponding authors on \nreasonable request. \n " }, { "title": "1904.11321v3.High_Spin_Wave_Propagation_Length_Consistent_with_Low_Damping_in_a_Metallic_Ferromagnet.pdf", "content": "High Spin-Wave Propagation Length Consistent with Low Damping in a\nMetallic Ferromagnet\nLuis Flacke,1, 2,a)Lukas Liensberger,1, 2Matthias Althammer,1, 2Hans Huebl,1, 2, 3, 4Stephan Geprgs,1Katrin\nSchultheiss,5Aleksandr Buzdakov,5Tobias Hula,5Helmut Schultheiss,5Eric R. J. Edwards,6Hans T. Nembach,6\nJustin M. Shaw,6Rudolf Gross,1, 2, 3, 4and Mathias Weiler1, 2,b)\n1)Walther-Meiner Institute, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany\n2)Physics Department, Technical University of Munich, 85748 Garching, Germany\n3)Nanosystems Initiative Munich, 80799 Munich, Germany\n4)Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany\n5)Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany\n6)Quantum Electromagnetics Division, National Institute of Standards and Technology, Boulder, CO 80305, USA\nWe report ultra-low intrinsic magnetic damping in Co 25Fe75heterostructures, reaching the low 10\u00004regime\nat room temperature. By using a broadband ferromagnetic resonance technique in out-of-plane geometry,\nwe extracted the dynamic magnetic properties of several Co 25Fe75-based heterostructures with varying ferro-\nmagnetic layer thickness. By measuring radiative damping and spin pumping e\u000bects, we found the intrinsic\ndamping of a 26 nm thick sample to be \u000b0.3:18\u000210\u00004. Furthermore, using Brillouin light scattering\nmicroscopy we measured spin-wave propagation lengths of up to (21 \u00061)\u0016m in a 26 nm thick Co 25Fe75\nheterostructure at room temperature, which is in excellent agreement with the measured damping.\nItinerant ferromagnets (FM) are advantageous for\nspintronic and magnonic devices. They bene\ft from, e.g.,\nlarge magnetoresistive e\u000bects and current-induced spin-\norbit torques1. In many magneto-resistive technologies\n(e.g., anisotropic magnetoresistance, giant magnetoresis-\ntance, tunnel magnetoresistance) electronic conductivity\nis indispensable. Moreover, due to high saturation mag-\nnetization in metallic FMs, spin-wave (SW) group ve-\nlocities are in general signi\fcantly higher than in insu-\nlating ferrimagnets2{5. High saturation magnetizations\nin general ease detection. Nevertheless, itinerant FMs\ntypically have considerable magnetic damping6,7. This\nis unfavorable for many applications. For example, low\ndamping is crucial for oscillators based on spin transfer\ntorques and spin orbit torques as well as for achieving\nlarge spin-wave propagation lengths (SWPL)8{10. The\nneed for thin \flm materials with low magnetic damping\nhas triggered the interest in the insulating ferrimagnet\nyttrium-iron garnet (Y 3Fe5O12, YIG)11{13. Although for\nYIG, very small total (Gilbert) damping parameters in\nthe order of \u000bG\u001910\u00005, and large SWPLs of a few tens\nof micrometers (up to \u001825\u0016m) in thin \flms ( \u001820 nm)\nhave been reported5,13,14, its insulating properties and re-\nquirement for crystalline growth are challenges for large\nscale magnonic applications.\nSchoen et al. recently observed ultra-low intrinsic mag-\nnetic damping in Co 25Fe75(CoFe) metallic thin \flms\n(\u000b0= (5\u00061:8)\u000210\u00004)15, and Krner et al. re-\nported PLs of 5 \u0016m\u00008\u0016m in CoFe using time re-\nsolved scanning magneto-optical Kerr microscopy4. This\nmotivated our study on sputter-deposited CoFe-based\nthin \flm heterostructures. We use broadband ferro-\nmagnetic resonance (BB-FMR) spectroscopy16in out-\na)Electronic mail: luis.\racke@wmi.badw.de\nb)Electronic mail: mathias.weiler@wmi.badw.deof-plane (OOP) geometry and Brillouin light scattering\n(BLS) microscopy17and \fnd intrinsic damping param-\neters in the lower 10\u00004regime as well as SWPLs of\nmore than 20 \u0016m. The damping is therefore compara-\nble to YIG/heavy metal (HM) heterostructures18and the\nSWPL is comparable to that of state-of-the-art YIG thin\n\flms5,13. Thin \flm CoFe is a promising candidate for\nall-metal magnonic devices, as it combines low magnetic\ndamping with good electrical conductivity and large sat-\nuration magnetization, while enabling easy fabrication\nby room-temperature processing/deposition, no required\nannealing, polycrystalline structure, and scalability to\nthe nanometer regime.\nFor BB-FMR, Ta(3 nm)/Al(3 nm)/Co 25Fe75(t)/\nAl(3 nm)/Ta(3 nm) heterostructures with di\u000berent\nthicknesstof the CoFe layer were sputter deposited\non a thermally oxidized Si (100) substrate at an Ar\npressure of 5\u000210\u00006bar at room temperature. No\nsubsequent annealing process was performed. The CoFe\nlayer thickness was varied between 1.4 nm < t < 26 nm\nas determined by X-ray re\rectometry.\nThe OOP BB-FMR measurements were performed\nat room temperature with a vector network analyzer\n(VNA). This geometry was chosen to determine the in-\ntrinsic magnetic damping without further damping con-\ntributions due to magnon-magnon scattering19. The\nsamples were placed directly on a coplanar waveguide\n(CPW), with a 80 \u0016m wide center conductor. For the\nmeasurements, the VNA frequency fwas kept con-\nstant and the microwave transmission parameter S21was\nrecorded as a function of applied magnetic \feld H0for a\nrange of frequencies at a VNA output power of 0 dBm.\nA representative set of data as measured of the real\nand imaginary part of S21at 16 GHz for samples with\nt= 1:8 nm andt= 26 nm is shown in Fig. 1 (a) and (b).\nThe magnetic response of the thin \flm FM magne-\ntized out-of-plane is given by the susceptibility \u001fwhicharXiv:1904.11321v3 [cond-mat.mtrl-sci] 30 Aug 20192\nis obtained by solving the Landau-Lifshitz-Gilbert (LLG)\nequation15,20:\n\u001f(H0) =Ms(H0\u0000Hres+He\u000b)\n(H0\u0000Hres+He\u000b+i\u0001H\n2)2\u0000H2\ne\u000b:(1)\nHere,Msis the saturation magnetization, Hresis the\nresonance \feld, He\u000b= 2\u0019f=(\u00160\r) with\rbeing the gy-\nromagnetic ratio and \u0001 H= 2(2\u0019f\u000b)=(\r\u00160) is the full\nwidth at half maximum (FWHM) linewidth of the reso-\nnance. The data in Fig 1 (a) and (b) is \ftted to21\nS21(H0) =S0\n21+iA\u001f(H0)\nMs=S0\n21(1 + \u0001S21);(2)\nwhereS0\n21is the background transmission through the\nCPW without magnetic resonance peak. It is determined\nfrom the \fts as a complex linear background to the data\nS0\n21(H0) =Sa\n21+H0Sb\n21. The factor Ais a complex-valued\nscaling parameter.\nIn the OOP geometry, the resonance condition for thin\n\flms is given by22\n\u00160Hres=\u00160Me\u000b+\u00160He\u000b; (3)\nwhereMe\u000b=Ms\u0000Hkis the e\u000bective magnetization,\nwith the uniaxial out-of-plane anisotropy \feld Hk. In\nFig. 1 (c), we plot the determined Hresvs. the frequency\nf. From the \ft to Eq. (3) (red solid lines in Fig. 1 (c)),\nwe obtainMe\u000band\rof the speci\fc sample.\nThe FWHM linewidth vs. frequency data shown in\nFig. 1 (d) is \ftted to\n\u00160\u0001H=\u00160Hinh+ 2\u00012\u0019f\u000b G\n\r: (4)\nHere,Hinhis the inhomogeneous linewidth broadening\nand\u000bGis the phenomenological Gilbert damping pa-\nrameter23,24.Hinhindicates the presence of long-range\nmagnetic inhomogeneities, which become more relevant\nfor thinner \flms, but do not contribute to our \u000bG.\nSeveral contributions to the measured total damping\n(\u000bG) were extracted from our data. In addition to the\nintrinsic damping of the magnetic material itself ( \u000b0),\nspin pumping ( \u000bsp) contributes signi\fcantly25{27to the\ntotal damping in our thinner heterostructures due to the\nadjacent HM (Ta) layers. Furthermore, we consider addi-\ntional damping contributions from eddy currents ( \u000beddy)\nand radiative damping ( \u000brad)15,21. Due to these contribu-\ntions, the total damping ( \u000bG=\u000b0+\u000bsp+\u000beddy+\u000brad)\ndepends on the FM thickness. We calculated damping\ndue to eddy currents and measured radiative damping\ncontributions to the total damping. The eddy current\ncontribution is given by15\u000beddy =\r\u00162\n0Mst2=16\u001a. Here,\n\u00160Ms= 2:35 T (see SI) and \u001a= 340 n\n m is the esti-\nmated weighted resistivity value of the CoFe \flm derived\nfrom the resistivities of iron and cobalt thin \flms with\nthicknesses of around 20 nm28,29. With these values, we\n\fnd an almost negligible eddy current contribution to the\ntotal damping. A quantitative determination analogous\n1.721 .742 .8322 .8380\n10203040501.41.61.82.02.22.42.62.83.00\n1020304050051015202526 nm1 .8 nmRealI\nmag16 GHzS21 (arb. u.)µ\n0H0 (T)RealI\nmag16 GHzS21 (arb. u.)µ\n0H0 (T)µ0Hres (T)f\n (GHz)26 nm1\n.8 nm2\n6 nm1.8 nmf\n (GHz)µ0/s61508H (mT)FIG. 1. (a) Measured microwave transmission S21at\n16 GHz vs. applied OOP magnetic \feld H0for blanket\nTa(3 nm)/Al(3 nm)/Co 25Fe75(t)/ Al(3 nm)/Ta(3 nm) samples\nwith CoFe thickness t= 1:8 nm ((a) blue symbols) and\nt= 26 nm ((b) black symbols), respectively. The red lines are\n\fts of Eq. (2) to the data. The extracted resonance \felds Hres\nand linewidths \u0001 Has a function of the applied microwave fre-\nquency are shown in (c) and (d), respectively. Here, the error\nbars (smaller than symbol size) are extracted \ft errors from\n(a) and (b). In (c) the red line is a \ft to Eq. (3) to extract the\nLand-factor gand the e\u000bective magnetization Me\u000b. In (d),\nthe linewidth is plotted vs. frequency. The Gilbert param-\neter\u000bGand the inhomogeneous linewidth broadening Hinh\nare extracted by \ftting the data to Eq. (4) (red lines). The\nlinewidth of the t= 26 nm thick sample is shown in Fig. 2 (c)\non an expanded scale.\nto Ref. 21 of the radiative damping is done by analyzing\nthe magnitude of the measured inductance Lof all sam-\nples. The quanti\fcation of this contribution is important\nfor BB-FMR, because it represents a damping by induc-\ntive power dissipation into the CPW and, hence, is not a\nproperty of the sample itself but depends on the setup.\nIn possible applications like, e.g., magnonic waveguides\nor spin-Hall nano-oscillators, this contribution vanishes\nand the damping lowers by \u000brad. With Eq. (2) above and\nEq. (9) from Ref. 21, one obtains:\nL\n\u001f\u0011~L=\u00002Z0A\nMsS0\n21!: (5)\nHere,Z0= 50 \n is the CPW impedance. It has been\nshown, that ~L=~L0+~L1(!);where ~L02Rand ~L12\nC, due to the e\u000bect of inverse spin-orbit torques21. We\nextractLfrom the FMR measurements, and the dipolar\ninductance ~L0from a \ft of ~Lvs.ffor each sample. The3\nradiative damping contribution is then given as15\n\u000brad=1\n4\r\u00160Ms\nZ0~L0: (6)\nThis analysis allows us to determine \u000bradindependently\nof geometrical parameters of the samples or CPWs and\nis used to quantitatively extract the dipolar inductance\nwithout any calibration of the microwave circuit. For the\nthickest sample we obtain \u000brad= (4:69\u00060:05)\u000210\u00004,\nwhich is comparable to previously obtained values15,30.\nThe damping including the spin pumping contribution\n\u000bspis given by\n\u000b0+\u000bsp=\u000b0+ 2\r~g\"#\ne\u000b\n4\u0019Ms1\nt; (7)\nwhereg\"#\ne\u000bis the e\u000bective spin mixing conductance30. We\nsubstract\u000bradand\u000beddyfrom the measured total damp-\ning\u000bG(see Fig. 2 (a) and (b)) and plot the remaining\ndamping\u000b0+\u000bspas a function of 1/ tin Fig. 2 (b) to-\ngether with the total damping \u000bG. From a linear \ft\n(Eq. (7)) to \u000b0+\u000bsp, we obtain ge\u000band\u000b0. Herefore, we\nuseMsas above and \r=2\u0019= 28:65 GHz=T. The \ftted\nge\u000b= (5:5\u00060:3)\u00021018m\u00002is in agreement with liter-\nature values15. They-intercept indicating the extrapo-\nlated intrinsic damping yields \u000b0= (0:91\u00061:69)\u000210\u00004\nhence, the intrinsic damping is below the sensitivity of\nour approach. For the thickest sample t= 26 nm shown\nin Fig. 2 (a), we obtain \u000b0= (3:18\u00060:48)\u000210\u00004(see SI\nfor details). Within the errors, this value lies close to the\nextrapolated value and is the lowest intrinsic damping\nfor a thin \flm ferromagnetic metal reported so far. We\nattribute the slightly reduced intrinsic \u000b0compared to\nRef. 30 to the use of a di\u000berent seed layer, which has a\nsubstantial impact on the damping of CoFe31.\nThe low damping properties of the CoFe heterostruc-\ntures, in combination with the high saturation magneti-\nzation are expected to result in long PLs of dipolar SWs.\nWe use microfocused BLS17to study the SW propaga-\ntion in patterned CoFe samples, which are schematically\ndepicted in Fig. 3 (a) and (b).\nFor our experiments, we fabricated patterned stripes of\na Pt(3 nm)/Cu(3 nm)/Co 25Fe75(t)/Cu(3 nm)/Ta(3 nm)\nheterostructure using laser (sample A) and electron beam\n(sample B) lithography, sputter deposition and a subse-\nquent lift-o\u000b process. This stack sequence was used as\nlower in-plane damping was observed compared to the\nsamples containing Al. Below, we present data on only\ntwo samples with a thickness of t= 5 nm and a width\nw= 1:5\u0016m for sample A and t= 26 nm and w= 5\u0016m\nfor sample B, respectively. An aluminum antenna was\nplaced on top of the CoFe strip to drive spin dynamics\nvia a microwave drive applied to the antenna. For sample\nA we used a simple aluminum strip optimized for excita-\ntion of the uniform (FMR) mode, whereas for the sample\nB we used a CPW antenna optimized for an e\u000ecient ex-\ncitation of SWs with wave number k\u00142\u0016m\u00001.\nIn order to compare the uniform FMR-mode linewidths\nof extended and patterned \flms, we used sample A in\n0.20.40.6012345675\n101520051015200\n.00.51.01.52.02.5/s61537 G/s61537\n0+/s61537sp/s61537 (10-3)1\n/t (nm-1)t = 26 nm/s61537\nG/s61537\n0(a)( b)µ0/s61508H (mT)f\n (GHz)FIG. 2. (a) An expanded view of the linewidth vs. frequency\nplot of the t= 26 nm sample. The total linewidth is shown\nby the blue diamonds, from which the total Gilbert damping\nparameter\u000bGwas extracted. The green circles represent the\nintrinsic linewidth contribution. In (b), the total damping \u000bG\nis plotted for di\u000berent thicknesses tas blue diamonds. We sub-\nstracted the contributions from radiative damping and eddy\ncurrents and show the resulting \u000b0+\u000bspas black squares. The\nred line is an unweighted \ft to Eq. (7) in order to quantify the\nspin pumping contribution within our samples and to be able\nto extrapolate the intrinsic damping of CoFe within our mul-\ntilayer system. For thicker samples, the available frequency\nrange is rather small, leading to an increased uncertainty, as\ndiscussed in Ref. 32.\nbackward volume geometry and placed the laser spot\nclose to the antenna, where the FMR mode is dominantly\nexcited. We recorded BLS spectra for several magnetic\n\felds for each frequency. The BLS intensity is integrated\nand the signal sum is then plotted vs. the external mag-\nnetic \feld in Fig. 3 (c). The FWHM-linewidth \u0001 His\ndetermined by \ftting a Lorentzian (red line). We then\ncompared the \ftted linewidth to the measured in-plane\nBB-FMR linewidth of a blanket \flm, deposited simul-\ntaneously with the structured BLS sample. In the in-\nplane con\fguration the total damping increases due to\nmagnon-magnon scattering19,33and possible anisotropic\ndamping34{37. As shown in Fig. 3 (d), the linewidths\n\u00160\u0001Hdetermined from BB-FMR (black sybmbols) and\nBLS (blue symbols) are very similar, indicating that the\ndamping properties are not a\u000bected by the patterning,\nas expected in a lift-o\u000b process with micrometer feature\nsizes.\nIn the next set of experiments, we investigate the\nSWPL of sample B (see Fig. 3 (b)). Here, the magnitude\nof the external magnetic \feld was \fxed at \u00160H0= 43 mT,\nwhile the \feld was applied perpendicular to the CoFe\nstrip (Damon-Eshbach geometry). The BLS intensity\nwas recorded as a function of position ( x,y) over the\nCoFe strip. The BLS intensity decay in xdirection (i.e.\nthe BLS intensity averaged over the width of the strip\nin order to suppress mode-beating e\u000bects38{40) is shown\nin Fig. 3 (e) for f= 9:5 GHz. The SWPL \u0015prop is ex-\ntracted by a \ft to I=I0exp(\u00002x=\u0015prop)41and plot-\nted vs.fin Fig. 3 (f). From our experiments, we ex-4\n30405060050100150\n0 10 2002468\n0 10 20 30103104\n0.5 1.0 1.5 2.08910111213\n8 10 120481216209.8GHz\nµ0H0(mT)Isum(counts)\nf(GHz)µ0H(mT)\n9.5GHzIntensity(arb.u.)\nx(µm)0Hd=-18mTf(GHz)\nkx(1/µm)λprop(µm)\nf(GHz)µ0H0 = 43 mT\nµ0H0 = 43 mT µ0H0 = 43 mTCoFe\nantennalaserJac\nxy~\nH II xsample A\nlaser-scansample B\n~H II y(a) (b) (c) (d)\n(g) (e) (f)A\nA\nB B BBB-FMR\nBLS\nFIG. 3. (a), (b) Schematic top view of sample A and B, respectively. (c) Integrated BLS intensity vs. \feld of sample A. By use\nof a Lorentzian \ft (red line), the linewidth is extracted. In (d) we compare BLS linewidth (open symbols) to values obtained\nby in-plane BB-FMR on a blanket \flm (closed squares). (e) Respresentative data set of sample B with f= 9:5 GHz. The BLS\nintensity was measured as a function of the position ( x,y) in the area highlighted with the green rectangle in (b). The measured\nsignal was then integrated in ydirection and the exponential decay in xdirection is \ftted (red curve). (f) Propagation length\n\u0015propfor varying frequency f. Depicted error bars are \ft errors. The red curve is based on an analytical model calculation (see\ntext). (g)fvs.kxdispersion determined by phase-resolved \u0016BLS. The red line is a model from Eq. (8).\ntract a maximum SWPL of (21 \u00061)\u0016m, well exceed-\ning previously obtained results for FeNi alloys42and\nCoFe4and very comparable to values found for YIG\nthin \flms5,13. The red curve is the theoretical prediction,\nbased on the analytical Kalinikos-Slavin model detailed\nbelow and using the magnetic parameters determined by\nin-plane BB-FMR ( \u00160Ms= 2:35 T,\u00160Me\u000b= 2:29 T,\n\u000bG\u0000\u000brad= 3:92\u000210\u00003,g= 2:051) for a co-deposited\nreference sample (see SI).\nStarting with a simpli\fed version of Kalinikos and\nSlavin's SW dispersion for the modes with kx?M43,44\nfres=\u00160\r\n2\u0019r\nH0+Hd+Hk+Ms1\u0000exp(\u0000kt)\nkt\n\u0002s\nH0+Hd+Ms\u0012\n1\u00001\u0000exp(\u0000kt)\nkt\u0013\n;(8)\nwe calculated the group velocity vg= 2\u0019@fres=@k. Here,\nk=q\nk2x+k2yis the in-plane wave vector of the travel-\nling SW and \u00160Hk=\u00160Me\u000b\u0000\u00160Ms=\u000060 mT is the\ne\u000bective interface anisotropy \feld. The calculation of the\ntransversal wave vector component ky= 0:31\u0016m\u00001due\nto geometrical con\fnement was shown to be non-trivial\nand is used as a \ftting parameter, as in Ref. 45. Theresonance linewidth is given by46\u0001!=\u000b\u00160\r(Me\u000b=2 +\nH0+Hd) and the lifetime of the SW is \u001c= 1=\u0001!. Here,\n\u000b=\u000bG\u0000\u000brad. The SWPL is \u0015prop=vg\u001c. The demagne-\ntization \feld in y-direction was set to \u00160Hd=\u000018 mT,\nas required for matching Eq. (8) to the SW dispersion\nobtained by phase-resolved \u0016BLS17(see Fig. 3 (g)). This\nvalue forHdis in good agreement with the demagneti-\nzation (\u00160Hd\u0019\u000012 mT) obtained for an ellipsoid with\nthe axes corresponding to the CoFe-stripe dimensions47.\nWe \fnd excellent agreement between this model and our\nexperimental data in Fig. 3 (f).\nIn summary, our sputter-deposited Co 25Fe75layers ex-\nhibit a record low intrinsic damping for metallic thin \flm\nferromagnets of \u000b0.3:18\u000210\u00004in OOP geometry. The\ndamping properties of extended \flms are maintained for\nmicropatterned \flms, and spin-wave propagation lengths\nare in very good agreement with the properties extracted\nfrom BB-FMR. The low magnetic damping, together\nwith the high saturation magnetization, lead to spin-\nwave decay lengths of more than 20 \u0016m at room tem-\nperature, which are the highest reported so far in itiner-\nant magnetic systems. 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The impli-\ncations for the CMB anisotropies and polarization are investigated f or different parameter\nchoices of a non helical stochastic magnetic field. Assuming a Gaussia n smoothing scale de-\ntermined by the magnetic damping wave number at recombination it is f ound that magnetic\nfields with present day strength less than 0.1 nG and negative magne tic spectral indices have\na sizeable effect on the CMB temperature anisotropies and polarizat ion.\nI. INTRODUCTION\nMagnetic fields on large scales are redshifted with the expan sion of the universe. Due to the\nnon trivial interaction of the ionized component of the bary on fluid with the rest of the cosmic\nplasma they suffer additional damping. This dynamics depends on the physical conditions of the\nuniverse. Before recombination baryons and photons are tig htly coupled. Thomson scattering is\nvery efficient at high temperatures and frequent scattering o f photons off free electrons ensures\nthe strong coupling of the baryon and photon fluids. As the tem perature falls Thomson scattering\nbecomes less effective, the two fluids start decoupling introd ucing photon viscosity in the dynamics\nof the baryon fluid. Photons start diffusing and free streaming thereby dragging the baryons out\nof the gravitational potential wells. This leads to a dampin g of the density perturbation spectrum\non scales below the photon diffusion scale, the Silk scale [1, 2 ].\nCosmic magnetic fields present beforedecoupling are tied to the completely ionized baryon fluid.\nThus prior to decoupling radiative viscosity leads to dampi ng of magnetic fields on small scales.\nHowever, contrary to perturbations in a non magnetized plas ma magnetic fields survive damping\ndown to scales shorter than the Silk scale. Perturbations in a magnetized plasma comprise of three\n∗kkunze@usal.es2\ndifferent modes which are the slow and fast magnetosonic modes and the Alfv´ en mode. Whereas\nthe first two are compressional the latter is not. As shown in [ 3, 4] slow magnetosonic modes and\nAlfv´ en modes enter an overdamped regime thereby surviving damping beyond the Silk scale. On\nthe other hand, fast magnetosonic waves are damped at the Sil k scale.\nAfter decoupling radiative viscosity rapidly becomes supp ressed and MHD turbulence can de-\nvelop. Nonlinear interaction between different scales leads to decay of MHD turbulence and dis-\nsipation of the magnetic field energy. Inverse cascade trans fers energy from small to large scales.\nThis has been observed in numerical simulations for helical magnetic fields and recently also for\nnon helical ones [5–7]. Even after recombination matter is n ot completely neutral, but rather there\nis a small fraction of matter which remains ionized. This lea ds to plasma drift1caused by different\nvelocities for the neutrals and ions in a magnetized plasma. The resulting friction force between\nthe two components is of the order of the Lorentz term resulti ng in damping of the magnetic field.\nThe energy liberated by the damping of the cosmic magnetic fie ld leads, on the one hand, to\nheating of matter, and, on the other hand, to spectral distor tions of the photon spectrum. The\nresulting spectral distortions of the cosmic microwave bac kground (CMB) have been calculated in\nthe pre- [10, 11] as well as the post-recombination [11–13] e ra. In [14–16] the effects on the CMB\ntemperature anisotropies and polarization due to the chang e in the thermal evolution in the post\nrecombination universe have been studied. In this case the e volution of the matter temperature\nreceives two additional source functions due to the dissipa tion of magnetic energy by decaying\nMHD turbulence as well as plasma drift. The former being impo rtant for large values of redshift\nclose to decoupling and the latter being dominant in the more recent universe.\nHere the effect of magnetic field dissipation in the pre-recomb ination universe on the CMB\ntemperature anisotropies and polarization is considered. In section II the corresponding angular\npowerspectraarecalculated consideringtheadiabatic, pr imordialcurvaturemodeinthepresenceof\nthe modified thermal and hence ionization history. It is know n that magnetic fields also contribute\nto the final temperature anisotropies and polarization (e.g . [17–20]). However, for the values\nof the magnetic field strengths considered here, these contr ibutions can be neglected as a first\napproximation. In section III our conclusions are presente d.\n1This is often called ambipolar diffusion which, however, str ictly speaking refers to diffusion of electrons and ions\ndue to electrostatic coupling [8, 9]. As pointed out in [9] it is more appropriate to refer to it as plasma drift as the\nions drift w.r.t. the plasma.3\nII. INCLUDING MAGNETIC FIELD DISSIPATION IN THE PRE-RECOMB INATION\nUNIVERSE\nIn [11] the induced CMB spectral distortions in the pre-reco mbination universe have been cal-\nculated and we follow that description here. The magnetic fie ld is assumed to be a non-helical,\nGaussian random field determined by its two-point function i n Fourier space given by\n/angbracketleftB∗\ni(/vectork)Bj(/vector q)/angbracketright= (2π)3δ(/vectork−/vector q)PB(k)/parenleftbigg\nδij−kikj\nk2/parenrightbigg\n, (2.1)\nwhere the power spectrum, PB(k), is assumed to be a power law, PB(k) =ABknB, with the\namplitude, AB, and the spectral index, nB. The ensemble average energy density of the magnetic\nfield is defined using a Gaussian window function so that\n/angbracketleftρB,0/angbracketright=/integraldisplayd3k\n(2π)3PB,0(k)e−2/parenleftBig\nk\nkc/parenrightBig2\n, (2.2)\nwherekcisacertain Gaussiansmoothingscaleanda”0”referstothep resentepoch. Itisconvenient\nto express the magnetic field power spectrum in terms of ρB,0yielding\nPB,0(k) =4π2\nk3c2(nB+3)/2\nΓ/parenleftbignB+3\n2/parenrightbig/parenleftbiggk\nkc/parenrightbiggnB\n/angbracketleftρB,0/angbracketright. (2.3)\nnB= 3 corresponds to the scale-invariant case for which the con tribution to the energy density per\nlogarithmic wavenumber is independent of wave number. The s pectral index depends on the details\nof the generation mechanism of the magnetic field. Whereas in flationary produced magnetic fields\ngenerally have negative spectral indices [21] those genera ted by causal processes such as during the\nelectroweak phase transition [22–28] have positive values . Moreover it was shown in [29] that in\nthe latter case they have to be an even integer with nB≥2.\nThe volume energy injection rate due to damping of magnetoso nic and Alfv´ en waves is deter-\nmined by ˙Q=a−4d(ρBa4)/dtwhere the comoving magnetic energy density is calculated se tting\nthe smoothing scale to be the damping scale kdleading to [11]\na4/angbracketleftρB/angbracketright(z) =a4/integraldisplayd3k\n(2π)3PB(k)e−2/parenleftBig\nk\nkd(z)/parenrightBig2\n=a4\n0/integraldisplayd3k\n(2π)3PB,0(k)e−2/parenleftBig\nk\nkd(z)/parenrightBig2\n. (2.4)\nThis yields the volume energy injection rate\ndQ\ndz=−nB+3\n2ργ,0(1+z)4/parenleftbiggρB,0\nργ,0/parenrightbigg\nk−(nB+3)\nckd(z)nB+5d\ndzk−2\nd(z), (2.5)4\nwhereρB,0\nργ,0= 9.545×10−8/parenleftbigB0\nnG/parenrightbig2forTCMB= 2.725 K. The damping wave number of the magnetic\nfield is given in terms of the photon diffusion wave number kγyielding [3, 4]\nkd=αkγ (2.6)\nwhereαis given by\nα=\n\n1 fast ms modes\n(vAcosθ)−1≃2.6×103\ncosθ(1nG/B0) slow ms & A modes(2.7)\nfor fast, slow magnetosonic (ms) modes and Alfv´ en (A) modes . In the following the angle between\nthe wave vector and the magnetic field vector, θ, will be ignored, hence cos θ= 1. Including\npolarization the photon diffusion scale is given by [2]\nk−2\nγ(z) =/integraldisplay∞\nzdz\n6H(z)(1+R)˙τ/parenleftbigg16\n15+R2\n1+R/parenrightbigg\n. (2.8)\nThe baryon-to-photon density ratio, R, is given by R=3\n4ρb\nργand ˙τis the differential optical depth.\nThe photon diffusion scale upto recombination can be approxim ated in terms of the hypergeometric\nfunction F(α,β;γ;z) (for more details see the appendix A). The explicit form dep ends on the value\nof the redshift, i.e. if it is bigger or smaller than some reds hiftz∗at which the approximate form of\nthe differential optical redshift changes. For the WMAP 9 best -fit parameters z∗≃1486.57 [11, 30].\nForz > z∗it is given by\nk−2\nγ, z≥z∗(z) = 2.16567×107/parenleftbig\nΩr,0h2/parenrightbig−1\n2/parenleftbigg\n1−Yp\n2/parenrightbigg−1/parenleftbig\nΩbh2/parenrightbig−1\n×16\n15/bracketleftbigg1\n3z3F/parenleftbigg1\n2,3;4;−2+zeq\nz/parenrightbigg\n−0.1875\nz4Ωb,0\nΩγ,0F/parenleftbigg1\n2,4;5;−2+zeq\nz/parenrightbigg/bracketrightbigg\nMpc2(2.9)\nand forzdec< z < z ∗by\nk−2\nγ, z z∗≃1486.57 [11]. Thus, the differential optical depth is given by\neq. (A.2) for zdec< z < z ∗and by ˙τ=neσTa\na0forz≥z∗. Thus in the two regimes the photon\ndiffusion scale is given by\nk−2\nγ, z≥z∗(z) = 2.16567×107/parenleftbig\nΩr,0h2/parenrightbig−1\n2/parenleftbigg\n1−Yp\n2/parenrightbigg−1/parenleftbig\nΩbh2/parenrightbig−1\n×/integraldisplay∞\nzdz(1+z)−7\n2(2+z+zeq)−1\n2(1+R)−1/parenleftbigg16\n15+R2\n1+R/parenrightbigg\nMpc2(A.4)\nand inzdec< z < z ∗,\nk−2\nγ, z z∗≃1486.57\nk−2\nγ, z≥z∗(z) = 2.16567×107/parenleftbig\nΩr,0h2/parenrightbig−1\n2/parenleftbigg\n1−Yp\n2/parenrightbigg−1/parenleftbig\nΩbh2/parenrightbig−1\n×16\n15/bracketleftbigg1\n3z3F/parenleftbigg1\n2,3;4;−2+zeq\nz/parenrightbigg\n−0.1875\nz4Ωb,0\nΩγ,0F/parenleftbigg1\n2,4;5;−2+zeq\nz/parenrightbigg/bracketrightbigg\nMpc2(A.6)10\nwhereF(α,β;γ,z) is the hypergeometric function [45], and for zdec< z < z ∗by\nk−2\nγ, zΩieven if at the same time Ω i> ω, or for short wavelengths λ < ρi,\nρi=vTi/Ωi,v2\nTi=κTi/mi. In the case of an inhomogeneous equilibrium, with a density\ngradient perpendicular to the magnetic field vector, in the unpertu rbed state the ions\nmay behave as un-magnetized in case of a low temperature, when th eir diamagnetic drift\nvelocity becomes negligible as compared to electrons [for singly charg ed ionsv∗i/v∗e=\nTi/Te, wherev∗j=κTjn′\nj0/(qjB0nj0), andn′\nj=dnj/dxdenotes the equilibrium density\ngradient]. The same holds in the presence of numerous collisions as ab ove,νi>Ωi, when\ntheir diamagnetic effects are absent too.\nInall these situations, andneglecting theelectron polarizationdrif t (inertia-less limit),\nthe wave will still have the basic properties of the IA mode. Within the two-fluid theory\nsuch a mode in an inhomogeneous plasma [that may be called ion-acoust ic-drift (IAD)\nmode] may in fact become growing [1]-[3] in the simultaneous presence of collisions and\nthe mentioned equilibrium density gradient perpendicular to /vectorB0.\n2Within the kinetic theory the mode is also growing in the presence of th e same density\ngradient and this even without collisions (due to purely kinetic effects ), and the physics\nof the growth rate is similar to the standard drift wave instability [4]. I t requires that the\nwave frequency is below the electron diamagnetic drift frequency ω∗e=v∗ek⊥.\nOn the other hand, keeping the electron inertia results in the instab ility of the lower-\nhybrid-drift (LHD) type [5]-[8]. In some other limits the effects of the same density\ngradient yield growing ionplasma (Langmuir) oscillations [5], or growing electron-acoustic\noscillations [6].\nIn the present manuscript the friction force effects on the IA wav e are discussed, both\nfor fully and partially ionized un-magnetized plasmas, and for inhomog eneous plasmas\nwith magnetized electrons. The latter implies growing modes within bot h the fluid and\nkinetic descriptions, and in the manuscript these two instabilities are compared.\n2 IA wave in fully and partially ionized collisional\nplasmas\nThe equations used further in this section are the momentum equat ions for the ions, the\nelectrons and the neutral particles, respectively:\nmini/parenleftBigg∂\n∂t+/vector vi·∇/parenrightBigg\n/vector vi=−eni∇φ−κTi∇ni−miniνie(/vector vi−/vector ve)−miniνin(/vector vi−/vector vn),(1)\nmene/parenleftBigg∂\n∂t+/vector ve·∇/parenrightBigg\n/vector ve=ene∇φ−κTe∇ne−meneνei(/vector ve−/vector vi)−meneνen(/vector ve−/vector vn),(2)\nand\nmnnn/parenleftBigg∂\n∂t+/vector vn·∇/parenrightBigg\n/vector vn=−κTn∇nn−mnnnνni(/vector vn−/vector vi)−mnnnνne(/vector vn−/vector ve),(3)\nand the continuity equation\n∂nj\n∂t+∇·(nj/vector vj) = 0, j=e,i,n. (4)\nThis set of equations is closed either by using the quasi-neutrality or the Poisson equation.\nThe differences between the two cases are discussed below.\n2.1 Friction in electron-ion plasma\nThe continuity equation (4) yields\nvi1=ωni1/(kn0), ve1=ωne1/(kn0), (5)\n3so that thevelocity difference inthefrictionterm ve−vi≡0if thequasi-neutrality isused.\nThe IA mode propagates without any damping. Hence, the friction f orce in a fully ionized\nplasma in this limit cancels out exactly even without using the momentum balance. The\nphysical reason for this is the assumed exact balance of the pertu rbed densities: what one\nplasma component loses the other component receives, this is valid a t every position in\nthe wave and no momentum is lost.\nA typical mistake seen in the literature is to take the friction force t erm for electrons\nonly, in the form meneνei/vector ve. This comes with the excuse of the large mass difference,\nso that the displacement of the much heavier ion fluid, caused by the electron friction is\nneglected. In the case of a fully ionized electron-ion plasma this yields a false damping of\nthe IA mode within the quasi-neutrality limit:\nω=±k(c2\ns+v2\nTi)1/2−νei/2. (6)\nOn the other hand, if the Poisson equation is used instead of the qua si-neutrality, one\nobtains [9]\nω=±kvs/parenleftBigg\n1−r2\ndek2ν2\nier2\nde\nv2s/parenrightBigg1/2\n−iνier2\ndek2. (7)\nHere, we have used the momentum conservation νie=meνei/miandv2\ns=c2\ns+v2\nTi,\nrde=vTe/ωpe. The physical reason for damping in the present case is the fact th at the\ndetailed balance ni1=ne1does not hold, because of the electric field which takes part\nfor small enough wave-lengths. It can easily be seen that for any r ealistic parameters the\nsecond term in the real part of the frequency in Eq. (7) is much belo w unity and the\nmode is never evanescent. However, in partially ionized plasmas (see below) this may be\ncompletely different.\n2.2 Friction and collisions in partially ionized plasma\nKeeping the quasi-neutrality limit, we now discuss the IA wave damping in plasmas\ncomprising neutrals as well. In view of the results presented above, the electron-ion\nfriction terms in Eqs. (1) and (2) will cancel each other out and in a f ew steps one derives\nthe following dispersion equation containing the collisions of plasma spe cies with neutrals\nand vice versa:\nω3+iω2/parenleftbigg\nνenme\nmi+νin/parenrightbigg/parenleftbigg\n1+mi\nmnn0\nnn0/parenrightbigg\n−k2c2\nsω\n−ik2c2\nsme\nmnn0\nnn0/parenleftbigg\nνen+mi\nmeνin/parenrightbigg\n= 0. (8)\n4/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52\n/s49/s48/s53\n/s49/s48/s54/s49/s48/s51/s49/s48/s52/s49/s48/s53\n/s48 /s50/s48/s48/s48 /s52/s48/s48/s48/s48/s49/s120/s49/s48/s53/s50/s120/s49/s48/s53\n/s110\n/s110/s47/s49/s48/s49/s54\n/s32/s32/s91/s49/s47/s109/s51\n/s93/s32/s32/s32ω\n/s114/s32\n/s32/s32 | γ |\n/s65/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s32/s32/s91/s72/s122/s93\n/s110\n/s110/s32/s47/s49/s48/s49/s54\n/s32/s32/s32/s91/s49/s47/s109/s51\n/s93/s32/s32 ω\n/s114\n/s32/s32/s32/s32/s124 γ |\n/s65/s66/s67\nFigure 1: Frequency ωrand absolute value of the IA mode damping |γ|in terms of the\nnumber density of neutrals. Details of the mode behavior in the regio n A are better seen\nin the linear scale (small figure inside).\nIn the derivation, the ion and neutral thermal terms are neglecte d. The ion thermal terms\nwould give the modified mode frequency ω2=k2c2\ns(1+Ti/Te). Hence, even if Te=Tithe\nwave frequency is only modified by a factor 21/2. The neutral thermal terms are discussed\nfurther in the text. Note that in deriving Eq. (8), the momentum co nservation condition\nνie=meνei/miis nowhere used: the e-i and i-e friction terms exactly vanish in view o f\nEq. (5).\nEquation (8) is solved numerically for a plasma containing electrons, p rotons, and\nneutral hydrogen atoms using the following set of parameters: Te= 4 eV,n0= 1018m−3,\nk= 10 m−1, with [10] σen= 1.14·10−19m−2. The neutral density isvarying inthe interval\n1016−1023m−3. Theionandhydrogentemperaturesaretaken Ti=Tn=Te/20,satisfying\nthe condition of their small thermal effects. This also gives [11], σin= 2.24·10−18m−2.\nThe results are presented in Fig. 1. The IA mode propagates in two d istinct regions A\nand B.\nOnly a limited left part of the region A would correspond to the ’standa rd’ IA wave\nbehavior in a collisional plasma: the mode is damped and the damping is pr oportional\nto the neutral number density. Hence, in this region it may be more o r less appropriate\nto use the approximate expressions for the friction force, like (in t he case of electrons)\nFe≃men0νenve1. However, this domain is very limited because in the rest of the domain\nthe frequency drops and the mode becomes non-propagating for nn0≥3.8·1019m−3(this\nis the lower limit of the region C in Fig. 1).\n5/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s49/s48/s52/s49/s48/s53/s49/s48/s54\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s49/s120/s49/s48/s52/s50/s120/s49/s48/s52\n/s107/s32/s32/s91/s49/s47/s109/s93/s32 ω\n/s114\n/s32/s32/s32| γ |/s107/s99\n/s115\n/s97/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s32/s91/s72/s122/s93\n/s107/s32/s32/s91/s49/s47/s109/s93/s32/s32ω\n/s114\n/s32/s32| γ |\n/s97/s98\n/s99/s107/s99\n/s115\nFigure 2: Frequency ωrand absolute value of the IA mode damping |γ|in terms of the\nwave number. The line kcsshows part of the graph of the ideal mode. Details of the\ndomainaare better seen in the linear scale (small figure inside).\nIncreasing theneutralnumber density, aftersomecriticalvalue (inthepresent casethis\nis around nn0≃1020m−3) the IA mode reappears again in the region B, with a frequency\nstarting from zero. For even larger neutrals number densities, th e mode damping in fact\nvanishes completely and the wave propagates freely but with a freq uency that is many\norders of magnitude below the ideal case kcs≃196 kHz. This behavior can be explained\nin the following manner. For a relatively small number of collisions the IA mode is weakly\ndamped because initially neutrals do not participate in the wave motion and do not share\nthe same momentum. Increasing the number of neutrals, the damp ing may become so\nstrong that the wave becomes evanescent. However, for much la rger collision frequencies\n(i.e., for a lower ionization ratio), the tiny population of electrons and ions is still capable\nof dragging neutrals along and all three components move togethe r. The plasma and the\nneutrals become so strongly coupled that the two essentially differe nt fluids participate\nin the electrostatic wave together. In this regime, the stronger t he collisions are, the less\nwave damping there is! Yet, this a bit counter-intuitive behavior com es with a price: the\nwave frequency and the wave energy flux becomes reduced by sev eral orders of magnitude.\nSimilar effects may be expected by varying the wave-length. The pre vious role of the\nvarying density of neutrals is now replaced by the the ratio of the me an free path of a\nspeciesλfj=vTj/νj(with respect to their collision with neutrals) and the wavelength.\nThis ratio now determines the coupling between the plasma and the ne utrals. The mode\nbehavior is directly numerically checked by fixing nn0= 1020m−3,n0= 1018m−3, and\n6/s49/s48/s49/s53\n/s49/s48/s49/s54\n/s49/s48/s49/s55\n/s49/s48/s49/s56\n/s49/s48/s49/s57/s49/s48/s49/s57/s49/s48/s50/s48/s49/s48/s50/s49/s49/s48/s50/s50/s49/s48/s50/s51/s110\n/s110/s48/s32/s32/s32/s91/s49/s47/s109/s51\n/s93\n/s110\n/s48/s32/s32/s32/s91/s49/s47/s109/s51\n/s93/s110/s111/s32/s112/s114/s111/s112/s97/s103/s97/s116/s105/s111/s110\n/s114/s101/s103/s105/s111/s110\n/s42\n/s42\nFigure 3: The two lines give the lower and upper values of the neutrals ’ density nn0\nbetween which, for the given plasma density n0, the IA mode does not propagate.\nfor other parameters same as above. For these parameters we h aveλfe=vTe/νen= 0.09\nm, andλfi=vTi/νin= 0.004. The numerical results are presented in Fig. 2 for kvarying\nin the interval 0 .2−80 m−1. The mode vanishes in the interval c, between k≃10 m−1\nandk≃25.6 m−1. The explanation is similar as before. Note that for k= 0.2 m−1(in\nthe region a) we have ωr≃390 Hz, and this is about one order below kcs. Compared to\nthe mode behavior in Fig. 1, this implies that the mode in the present do mainais in the\nregime equivalent to the domain B from Fig. 1; here, in Fig. 2, these lar ge wave-lengths\nimply well coupled plasma-neutrals, where the frequency is reduced and the damping is\nsmall. The region ais also given separately in linear scale together with the dotted line\ndescribing the ideal mode kcs. Clearly, in general the realistic behavior of the wave is\nbeyond recognition and completely different as compared to the idea l case.\nAfter checking for various sets of plasma densities, it appears tha t the evanescence\nregion reduces and vanishes for larger plasma densities n0. This is presented in Fig. 3 for\nthe same parameters as above, by taking k= 10 m−1, but for a varying plasma density\nn0. The two lines represent boundary values of the number densities o f neutrals, for the\ngiven plasma density, at which the IA mode vanishes; for the neutra ls densities between\nthe two lines the IA mode does not propagate. The symbols ∗on the two lines denote\nthe boundaries of the region Cfrom Fig. 1. It is seen that for the given case the IA\nmode propagates without evanescence for the plasma densities ab oven0= 3.8·1018m−3.\nPhysical reason for a larger non-propagating domain for low plasma density is obvious,\nnamely the tiny plasma population is less efficient in inducing a synchrono us motion of\n7/s49/s48/s49/s53\n/s49/s48/s49/s54\n/s49/s48/s49/s55\n/s49/s48/s49/s56\n/s49/s48/s49/s57/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s107/s32/s32/s32/s91/s49/s47/s109/s93\n/s110\n/s48/s32/s32/s32/s91/s49/s47/s109/s51\n/s93/s110/s111/s32/s112/s114/s111/s112/s97/s103/s97/s116/s105/s111/s110/s32/s114/s101/s103/s105/s111/s110 /s32\n/s42/s42\nFigure 4: Values of the wave-number, in terms of the plasma density , for which the IA\nwave becomes evanescent. In the region between the lines the mod e does not propagate.\nneutrals. In the other limit, the opposite happens and the forbidde n region eventually\nvanishes.\nA similar check is done by varying the wave-number and the plasma den sity, and the\nresult is presented in Fig. 4 for a fixed nn0= 1020m−3. The lines represent the values\n(n0,k) at which the IA wave becomes evanescent. There can be no wave in the region\nbetween the lines. On the other hand, there is no evanescence for the plasma density\naboven0= 1.2·1019m−3. Here∗denote the boundaries of the region cfrom Fig. 2.\nAll these results clearly indicate that in practical measurements in la boratory and\nspace plasmas, the IA mode can hardly be detected and recognized as the IA mode unless\ncollisions are correctly taken into account (using full friction terms ), and the mode is\nsought in the corresponding domain which follows from our Eq. (8).\n2.3 Thermal effects of neutrals\nKeeping the pressure terms for ions and neutrals yields the following dispersion equation\nω4+iω3/parenleftbigg\nνin+νenme\nmi/parenrightbigg/parenleftbigg\n1+mi\nmnn0\nnn0/parenrightbigg\n−k2/parenleftBig\nv2\ns+v2\nTn/parenrightBig\nω2\n−iωk2/bracketleftbiggn0\nnn0mi\nmnv2\ns/parenleftbigg\nνin+νenme\nmi/parenrightbigg\n+νinv2\nTn/bracketrightbigg\n+k4v2\nTnv2\ns= 0. (9)\nHere,v2\ns=c2\ns+v2\nTi. Without collisions, this yields two independent modes, viz. the\nion-acoustic mode and the gas thermal (GT) mode, ( ω2−k2v2\nTn)(ω2−k2v2\ns) = 0. The\ncollisions couple the two modes, and in order to compare with the prev ious cases we solve\nEq. (9) for k= 10 m−1,n0= 1018m−3,Te= 4 eV, Ti=Te/20, and in terms of the\n8/s53 /s49/s48 /s49/s53 /s50/s48/s48/s52/s48/s48/s48/s56/s48/s48/s48/s49/s50/s48/s48/s48 ω\n/s103/s32/s32/s91/s72/s122/s93\n/s110\n/s110/s48/s47/s49/s48/s49/s57\n/s32/s32/s32/s91/s32/s49/s47/s109/s51\n/s32/s93/s32/s84\n/s110/s61/s84\n/s105/s32/s47/s54/s48/s44/s32/s107/s32/s118\n/s84/s110/s61/s53/s54/s53/s51/s32/s72/s122\n/s32/s84\n/s110/s61/s84\n/s105/s32/s47/s53/s48/s44/s32/s107/s32/s118\n/s84/s110/s61/s54/s49/s57/s50/s32/s72/s122\n/s32/s32/s84\n/s110/s61/s84\n/s105/s32/s47/s52/s48/s44/s32/s107/s32/s118\n/s84/s110/s61/s54/s57/s50/s51/s32/s72/s122/s32\n/s32/s32/s84\n/s110/s61/s84\n/s105/s32/s47/s51/s48/s44/s32/s107/s32/s118\n/s84/s110/s61/s55/s57/s57/s52/s32/s72/s122\nFigure 5: The real part of the frequency of damped gas thermal m ode in terms of the\nnumber density of neutrals and for several temperatures of the neutrals gas.\ndensity and temperature of neutrals. For a low thermal contribut ion of neutrals (i.e., a\nlow neutral temperature, or/and heavy neutral atoms) the pre vious results remain valid.\nLarger values of vTnintroduce new effects, this is checked by varying the temperature Tn.\nThe ion thermal terms do not make much difference, as explained ear lier. The real part of\nthe frequency ωgof the gas thermal mode is presented in Fig. 5, and this only in a limited\nregion that includes the evanescence area Cfrom Fig. 1. The damping is not presented\nbut the mode is in fact heavily damped.\nThe explanation of the figure is as follows. The starting solution for Tn= 0 is in fact\nthe lineωg= 0, and this case would correspond to the the IA mode from Fig. 1. F or\nsome finite Tnthere appears the GT mode. For a low gas temperature the mode be comes\nevanescent for a higher density of neutrals (the dot and dash-do t lines in Fig. 5). This\nevanescence is accompanied with the previously discussed evanesc ence and re-appearance\nof the IA mode (described earlier and no need to be presented here again). However,\nfor still larger Tn, the IA and GT modes become indistinguishable and propagate as one\nsingle mode. This is presented by the two upper (the full and dashed ) lines in Fig. 5,\nthat go up for large enough nn0. Also given are the corresponding ideal values kvTnthat\nappear to be much above the actual wave frequency ωgin such a collisional plasma, but\nthis remains so only until the neutral density nn0exceeds some critical value. After that\nthe wave in fact behaves as less and less collisional and the wave freq uency is increased.\n93 IA wave instability in inhomogeneous partially ion-\nized plasma\n3.1 Fluid description in collisional plasma\nIn the previous text, collisions were shown to yield damping of the IA m ode. However, if\ntheplasma isinhomogeneous, implying thepresence of source offre e energy inthesystem,\na drift-type instability of the IA wave may develop if there is a magnet ic field/vectorB0=B0/vector ez\npresent, and the electrons (ions) are magnetized (un-magnetize d). The magnetic field\nintroduces a difference in the parallel and perpendicular dynamics of the magnetized\nspecies so that the continuity condition in this case can be written as\n∂nj1\n∂t+nj0∇·/vector vj1+/vector vj1·∇nj0= 0. (10)\nHere,∇ ≡ ∇ ⊥+∇z. For the un-magnetized species the direction of the wave plays\nno role so that ∇ →i/vectork,k2=k2\ny+k2\nz. On the other hand, for the equilibrium gradi-\nent along the x-axis and for perturbations of the form ∼f(x)exp(−iωt+ikyy+ikzz),\nwhere|(df/dx)/f|,|(dnj0/dx)/n0| ≪ky, we apply a local approximation, and for ions the\nlast term in Eq. (10) vanishes because of the assumed geometry. T he ions’ dynamics is\nbasically the same as in the previous sections.\nTheelectronmomentum equation(2)willnowincludetheLorentzfor ceterm−ene/vector ve×\n/vectorB. Repeating the derivation from Ref. [3], the total perpendicular ele ctron velocity can\nbe written as\nve⊥=1\n1+ν2enα2/Ω2e/bracketleftBigg1\nB0/vector ez×∇⊥φ+νenα\nΩe∇⊥φ\nB0−v2\nTeνenα\nΩ2e∇⊥ne\nne−v2\nTe\nΩe/vector ez×∇⊥ne\nne/bracketrightBigg\n.\n(11)\nIn the direction along the magnetic field vector, the perturbed elec tron velocity is\nvez1=ikzv2\nTe\nνenω2+ν2\nne\nω2−iνneω/parenleftBiggeφ1\nκTe−ne1\nn0/parenrightBigg\n. (12)\nHere,α=ω/(ω+iνne), and for magnetized electrons, |ν2\nenα2/Ω2\ne| ≪1 in the denominator\nin Eq. (11). Using these equations in the continuity condition (10) fo r electrons one\nobtains\nne1\nn0=ω∗e+iDp+iDz(ω2+ν2\nne)/(ω2−iνneω)\nω+iDp+iDz(ω2+ν2ne)/(ω2−iνneω)eφ1\nκTe, (13)\nDp=νenαk2\nyρ2\ne, Dz=k2\nzv2\nTe/νen, ρe=vTe/Ωe.\n10The term Dpdescribes the effects of collisions on the electron perpendicular dyn amics\nand is usually omitted in the literature. However, as shown in a recent study [3], it can\nstrongly modify the growth rate of the drift and IA-drift wave inst ability in the limit of\nsmall parallel wave-number kz.\nNeglecting the neutral dynamics is equivalent to setting νne= 0. This yields α= 1,\nand Eq. (13) becomes identical to the corresponding expression in Refs. [2, 12]. For a\nnegligible Dp, Eq. (13) becomes the same as the corresponding equation from R ef. [13].\nFor negligible ion thermal effects, the final dispersion equation read s\nk2c2\ns\nω2=ω∗e+iDp+iDz(ω2+ν2\nne)/(ω2−iνneω)\nω+iDp+iDz(ω2+ν2ne)/(ω2−iνneω). (14)\nEquation (14) can be solved numerically keeping in mind a number of con ditions used\nin their derivations, like smallness of the plasma beta to remain in electr ostatic limit,\nsmallness of the parallel phase velocity as compared to the electron thermal speed because\nof the massless electrons limit, also the ratio Dp/Dzshould be kept not too big or too\nsmall in order to have the assumed effects of electron collisions in per pendicular direction.\nWe plan to compare this collisional instability with the kinetic instability du e to the\npresence of the density gradient. Therefore, the wave frequen cy should be below the\nelectron diamagnetic frequency etc.\nWesolveEq.(14)foranelectron-argonplasmainthepresenceofp arentalargonatoms.\nAs an example we take Te= 4 eV,Ti=Tn=Te/30,n0= 1015m−3,B0= 1.2·10−2T,\nk= 500 m−1,Ln= 0.05 m, and take several values for the density of neutrals. The res ult\nin terms of the angle of the propagation θ= arctan( kz/ky) is presented in Fig. 5. The\nthree lines (full for the real part of the frequency, and dashed f or the growth rates) are\nfornn0= 1019,1018,1017m−3. It is seen that i) the instability is angle dependent and\nthere exists an angle of preference and an instability window in terms ofθwithin which\nthe mode is most easily excited, ii) this angle of preference is shifted t owards smaller\nvalues for lower values of the neutral density, and iii) in the same time the instability\nwindow becomes considerably reduced. This shows an interesting po ssibility of launching\nthe IA-drift wave in a certain direction by simply varying the pressur e of the neutral gas.\nVarying the density scale length Ln= (dn0/dx)−1the wave frequency may become\naboveω∗eand in this case the instability vanishes. As an example, this is demonst rated\nin Fig. 7 for the parameters corresponding to the line II from Fig. 6 a nd for the angle θ\nat the maximum growth rate. The growth rate changes the sign for ω≃ω∗e.\n11/s48/s46/s48/s48/s48 /s48/s46/s48/s48/s53 /s48/s46/s48/s49/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s102/s114/s101/s113/s117/s101/s110/s99/s121/s47 ω\n/s42/s101\nθ /s32/s91/s114/s97/s100/s93/s73/s73/s73/s73/s73/s73\n/s73\n/s73/s73\n/s73/s73/s73γ\n/s107\nFigure 6: Real part of the frequency from Eq. (14) (full lines) and the corresponding\ngrowth rates (dashed lines), both normalized to the electron diama gnetic drift frequency,\nfor three values of neutral number density. The lines I, II, III co rrespond (respectively)\ntonn0= 1019,1018,1017m−3. The line γkis the kinetic growth-rate from Eq. (19) (for\nthe same parameters as line II).\n/s54 /s56 /s49/s48 /s49/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s102/s114/s101/s113/s117/s101/s110/s99/s121/s47 ω\n/s42/s101/s32/s32/s32\n/s76\n/s110/s32/s32/s32/s91/s99/s109/s93/s32ω\n/s114\n/s32γ\nFigure 7: The real and imaginary parts of the frequency for the line II from Fig. 6, in\nterms of the characteristic density inhomogeneity scale length Ln= (dn0/dx)−1, and for\nthe angle θat the maximum on Fig. 6.\n123.2 Comparison with collision-less kinetic gradient-driv en IA\nwave instability\nKeeping the same model of magnetized (un-magnetized) electrons (ions), within the ki-\nnetic theory the perturbed number density for electrons can be w ritten as [14]\nne1\nn0=eφ1\nκTe/braceleftBigg\n1+i/parenleftbiggπ\n2/parenrightbigg1/2ω−ω∗e\nkzvTeexp/bracketleftBig\n−ω2/(2k2\nzv2\nTe)/bracketrightBig/bracerightBigg\n. (15)\nIn the derivation of Eq. (15) the electron Larmor radius correctio ns are neglected in terms\nof the type In(b)exp(−b),b=k2\n⊥ρ2\ne, whereIndenotes the modified Bessel function of the\nfirst kind, order n, and only n= 0 terms are kept for the present case of frequencies much\nbelow the gyro-frequency.\nTheionnumberdensitycanbecalculatedusingthekineticdescription forun-magnetized\nspecies, the derivation is straight-forward and it yields [15]\nni1\nni0=−eφ1\nmiv2\nTi/bracketleftbigg\n1−J+/parenleftbiggωi\nkvTi/parenrightbigg/bracketrightbigg\n. (16)\nHere,J(η) = [η/(2π)1/2]/integraltext\ncdζexp(−ζ2/2)/(η−ζ) is the plasma dispersion function, and\nζ=v/vTi. In the case |η| ≫1, and assuming |Re(η)| ≫Im(η), an expansion is used for\nJ(η). This together with the quasi-neutrality yields the kinetic dispersio n equation for\nthe IA-drift wave:\n∆(ω,k)≡1−k2c2\ns\nω2−3k4v2\nTic2\ns\nω4\n+i(π/2)1/2/braceleftbiggω−ω∗e\nkzvTeexp/bracketleftBig\n−ω2/(2k2\nzv2\nTe)/bracketrightBig\n+Te\nTiω\nkvTiexp/bracketleftBig\n−ω2/(2k2v2\nTi)/bracketrightBig/bracerightbigg\n.(17)\nThe real part of Eq. (17) yields the spectrum\nω2\nk=k2c2\ns\n2/bracketleftBig\n1+(1+12 Ti/Te)1/2/bracketrightBig\n. (18)\nThe kinetic growth rate is given by\nγk≃ −Im∆/(∂Re∆/∂ω) =−(π/2)1/2ω3\nk\n2k2c2s×\n×/braceleftbiggωk−ω∗e\nkzvTeexp/bracketleftBig\n−ω2\nk/(2k2\nzv2\nTe)/bracketrightBig\n+Te\nTiωk\nkvTiexp/bracketleftBig\n−ω2\nk/(2k2v2\nTi)/bracketrightBig/bracerightbigg\n.(19)\nHere, the index kis used to denote kinetic expressions. The electron contribution in\nEq. (19) yields a kinetic instability provided that ωk< ω∗e.\nEquation (19) is solved numerically and compared with the growth rat e obtained from\nthe collisional IA-drift mode (8). For a fixed k= 500 m−1as in Figs. 6 and 7, the\n13normalized frequency ωk/ω∗e= 0.485, and the result for the growth rate is presented\nby the line γkin Fig. 6 for the parameters corresponding to the line II from the flu id\nanalysis (i.e., for nn0= 1018m−3). The larger kinetic growth rate appears also to be\nangle dependent, yet with a much wider instability window as compared to the collisional\ngradient driven instability obtained from the fluid theory.\n4 Summary\nThe analysis of the ion acoustic wave presented here shows the impo rtance of collisions in\ndescribing the wave behavior. Without a proper analytical descript ion, the identification\nof the mode in the laboratory and space observations may be rathe r difficult because one\nmight fruitlessly search for the wave in a very inappropriate domain, as can be concluded\nfromthegraphspresentedhere, andinparticularfromFig.2. Not onlythewavefrequency\nmay become orders of magnitude below an expected ideal value, but also the mode may\ncompletely vanish. A similar analysis of the effects of collisions may be pe rformed for\nother plasma modes as well, like the Alfv´ en wave etc, as predicted lon g ago in classic\nRef. [16]. The impression is that these effects are frequently overlo oked in the literature,\nhence the necessity for the quantitative analysis given in the prese nt work that can be\nused as a good starting point for an eventual experimental check of the wave behavior\nin collisional plasmas. Particularly interesting for experimental inves tigations may be\nthe angle dependent mode behavior given in Sec. 3, where it is shown t hat the strongly\ngrowing mode may be expected within a given narrow instability window in terms of the\nangle of propagation. Comparison with the kinetic theory shows a les s pronounced angle\ndependent peak, yet this kinetic effect can effectively be smeared o ut in the presence\nof numerous collisions, that are known to reduce kinetic effects in an y case, and the\nsharp angle dependence that follow from pure fluid effects should be come experimentally\ndetectable.\nAcknowledgements: The results presented here are obtained in the framework of the\nprojects G.0304.07(FWO-Vlaanderen), C 90347(Prodex), GOA/2 009-009(K.U.Leuven).\nFinancial support by theEuropean Commission throughtheSOLAIR E Network (MTRN-\nCT-2006-035484) is gratefully acknowledged.\n14References\n[1] N. A. Krall, in Advances in Plasma Physics , ed. A Simon and W. B. Thompson\n(Interscience, New York, 1968), vol. 1, p. 195.\n[2] A. B. Mikhailovskii, Theory of Plasma Instabilities (Consultants Bureau, New York,\n1974), vol. 2, p. 192.\n[3] J. Vranjes and S. Poedts, Phys. Plasmas 16, 022101 (2009).\n[4] N. A. Krall and D. Book, Phys. Rev. Lett. 23, 574 (1969); ibid., Phys. Fluids 12,\n347 (1969).\n[5] N. A. Krall and P. C. Liewer, Phys. Rev. A 4, 2094 (1971).\n[6] M. Mohan and M. Y. Yu, J. Plasma Phys. 29, 127 (1983).\n[7] M. Bose and S. Guha, Phys. Scripta 34, 63 (1986).\n[8] J. D. Huba, A. B. Hassam, and D. Winske, Phys. Fluids B 2, 1676 (1990).\n[9] J. Vranjes, M. Kono, S. Poedts, and M. Y. Tanaka, Phys. Plasm as15, 092107(2008).\n[10] B. Bedersen and L. J. Kiefer, Rev. Mod. Phys. 43, 601 (1971).\n[11] P. S. Krstic and D. R. Schultz, J. Phys. B: Mol. Opt. Phys. 32, 3485 (1999).\n[12] J. Vranjes, D. Petrovic, B. P. pandey, and S. Poedts, Phys. Plasmas 15, 072104\n(2008).\n[13] J. Vranjes and S. Poedts, Phys. Plasmas 15, 034504 (2008).\n[14] J. Weiland, Collective Modes in Inhomogeneous Plasmas (Institute of Physics Pub.,\nBristol, 2000) p. 59.\n[15] J. Vranjes and S. Poedts, Eur. Phys. J. D 40, 257 (2006).\n[16] B. S. Tanenbaum and D. Mintzer, Phys. Fluids 5, 1226 (1962).\n15" }, { "title": "2306.13013v4.Gilbert_damping_in_metallic_ferromagnets_from_Schwinger_Keldysh_field_theory__Intrinsically_nonlocal_and_nonuniform__and_made_anisotropic_by_spin_orbit_coupling.pdf", "content": "Gilbert damping in metallic ferromagnets from Schwinger-Keldysh field theory:\nIntrinsically nonlocal and nonuniform, and made anisotropic by spin-orbit coupling\nFelipe Reyes-Osorio and Branislav K. Nikoli´ c∗\nDepartment of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA\n(Dated: March 1, 2024)\nUnderstanding the origin of damping mechanisms in magnetization dynamics of metallic ferro-\nmagnets is a fundamental problem for nonequilibrium many-body physics of systems where quantum\nconduction electrons interact with localized spins assumed to be governed by the classical Landau-\nLifshitz-Gilbert (LLG) equation. It is also of critical importance for applications as damping affects\nenergy consumption and speed of spintronic and magnonic devices. Since the 1970s, a variety of\nlinear-response and scattering theory approaches have been developed to produce widely used for-\nmulas for computation of spatially-independent Gilbert scalar parameter as the magnitude of the\nGilbert damping term in the LLG equation. The largely unexploited for this purpose Schwinger-\nKeldysh field theory (SKFT) offers additional possibilities, such as to rigorously derive an extended\nLLG equation by integrating quantum electrons out. Here we derive such equation whose Gilbert\ndamping for metallic ferromagnets is nonlocal —i.e., dependent on all localized spins at a given\ntime—and nonuniform , even if all localized spins are collinear and spin-orbit coupling (SOC) is\nabsent. This is in sharp contrast to standard lore, where nonlocal damping is considered to emerge\nonly if localized spins are noncollinear—for such situations, direct comparison on the example of\nmagnetic domain wall shows that SKFT-derived nonlocal damping is an order of magnitude larger\nthan the previously considered one. Switching on SOC makes such nonlocal damping anisotropic , in\ncontrast to standard lore where SOC is usually necessary to obtain nonzero Gilbert damping scalar\nparameter. Our analytical formulas, with their nonlocality being more prominent in low spatial\ndimensions, are fully corroborated by numerically exact quantum-classical simulations.\nI. INTRODUCTION\nThe celebrated Landau-Lifshitz equation [1] is the\nfoundation of standard frameworks, such as classical mi-\ncromagnetics [2, 3] and atomistic spin dynamics [4], for\nmodelling the dynamics of local magnetization within\nmagnetic materials driven by external fields or currents\nin spintronics [2] and magnonics [3]. It considers localized\nspins as classical vectors M(r) of fixed length normalized\nto unity whose rotation around the effective magnetic\nfieldBeffis governed by\n∂tM=−M×Beff+M×(D ·∂tM), (1)\nwhere ∂t≡∂/∂t. Although spin is a genuine quan-\ntum degree of freedom, such phenomenological equation\ncan be fully microscopically justified from open quantum\nmany-body system dynamics where M(r) tracks the tra-\njectories of quantum-mechanical expectation value of lo-\ncalized spin operators [5] in ferromagnets, as well as in\nantiferromagnets as long as the spin value is sufficiently\nlarge S >1. The presence of a dissipative environment in\nsuch justification invariably introduces damping mecha-\nnisms, which were conjectured phenomenologically in the\nearliest formulation [1], as well as in later renderings us-\ning the so-called Gilbert form of damping [6, 7] written as\nthe second term on the right-hand side (RHS) of Eq. (1).\nThe Gilbert damping Dwas originally considered as a\nspatially uniform scalar D ≡αG, or possibly tensor [8, 9],\n∗bnikolic@udel.edudependent on the intrinsic properties of a material. Its\ntypical values are αG∼0.01 in standard ferromagnetic\nmetals [10], or as low as αG∼10−4in carefully designed\nmagnetic insulators [11] and metals [12]. Furthermore,\nrecent extensions [13–21] of the Landau-Lifshitz-Gilbert\n(LLG) Eq. (1) for the dynamics of noncollinear magneti-\nzation textures find Dto be a spatially nonuniform and\nnonlocal tensor\nDαβ=αGδαβ+ηX\nβ′(M×∂β′M)α(M×∂β′M)β,(2)\nwhere ∂β′≡∂/∂β′, and α, β, β′∈ {x, y, z}.\nIt is generally believed that αGisnonzero only\nwhen SOC [22, 23] or magnetic disorder (or both) are\npresent [15, 24, 25]. For example, αGhas been ex-\ntracted from a nonrelativistic expansion of the Dirac\nequation [22, 23], and spin-orbit coupling (SOC) is vir-\ntually always invoked in analytical (conducted for sim-\nplistic model Hamiltonians) [26–28] or first-principles\ncalculations [24, 25, 29–33] of αGvia Kubo linear-\nresponse [9, 30, 34–36] or scattering [8] theory-based for-\nmulas.\nThe second term on the RHS of Eq. (2) is the\nparticular form [13] of the so-called nonlocal (i.e.,\nmagnetization-texture-dependent) and spatially nonuni-\nform (i.e., position-dependent) damping [13–21, 37]. The\nsearch for a proper form of nonlocal damping has a long\nhistory [19, 37]. Its importance has been revealed by ex-\nperiments [10] extracting very different Gilbert damping\nfor the same material by using its uniformly precessing\nlocalized spins versus dynamics of its magnetic domain\nwalls, as well as in experiments observing wavevector-\ndependent damping of spin waves [38]. Its particulararXiv:2306.13013v4 [cond-mat.mes-hall] 29 Feb 20242\nB\nL lead R leadxyz MnJsd\ne\nee\n(a)\n(b)\n(c)\nFIG. 1. Schematic view of (a) classical localized spins, mod-\neled by unit vectors Mn(red arrows), within an infinite metal-\nlic ferromagnet defined on a cubic lattice in 1D–3D (1D is\nused in this illustration); or (b) finite-size metallic ferromag-\nnet (central region) attached to semi-infinite NM leads termi-\nnating in macroscopic reservoirs, whose difference in electro-\nchemical potentials inject charge current as commonly done\nin spintronics. The localized spins interact with conduction\nelectron spin ⟨ˆs⟩(green arrow) via sd-exchange of strength\nJsd, while both subsystems can experience external magnetic\nfieldB(blue arrow). (c) Nonlocal damping λD\nnn′[Eq. (10)]\nobtained from SKFT vs. distance |rn−rn′|between two sites\nnandn′of the lattice for different dimensionality Dof space.\nform [13] in Eq. (2) requires only noncollinear and non-\ncoplanar textures of localized spins, so it can be nonzero\neven in the absence of SOC, but its presence can greatly\nenhance its magnitude [18] (without SOC, the nonlocal\ndamping in Eq. (2) is estimated [18] to be relevant only\nfor small size ≲1 nm noncollinear magnetic textures).\nHowever, recent quantum-classical and numerically ex-\nact simulations [39, 40] have revealed that αGcan be\nnonzero even in the absence of SOC simply because ex-\npectation value of conduction electron spin ⟨ˆs⟩(r) isal-\nways somewhat behind M(r). Such retarded response of\nelectronic spins with respect to motion of classical lo-\ncalized spins, also invoked when postulating extended\nLLG equation with phenomenological time-retarded ker-\nnel [41], generates spin torque ∝ ⟨ˆs⟩(r)×M(r) [42] and,\nthereby, effective Gilbert-like damping [39–41] that is\nnonzero in the absence of SOC and operative even if\nM(r) at different positions rarecollinear [40]. Including\nSOC in such simulations simply increases [43] the an-\ngle between ⟨ˆs⟩(r) and M(r) and, therefore, the effective\ndamping.\nTo deepen understanding of the origin of these phe-\nnomena observed in numerical simulations, which are\nanalogous to nonadiabatic effects discussed in diversefields where fast quantum degrees of freedom interact\nwith slow classical ones [44–47], requires deriving an an-\nalytical expression for Gilbert damping due to interac-\ntion between fast conduction electrons and slow local-\nized spins. A rigorous path for such derivation is offered\nby the Schwinger-Keldysh nonequilibrium field theory\n(SKFT) [48] which, however, remains largely unexplored\nfor this problem. We note that a handful of studies have\nemployed SKFT to study small systems of one or two\nlocalized spins [49–54] as they interact with conduction\nelectrons. While some of these studies [49, 53, 54] also\narrive at extended LLG equation with nonlocal damp-\ning, they are only directly applicable to small magnetic\nmolecules rather than macroscopic ferromagnets in the\nfocus of our study. It is also worth mentioning that an\nearly work [55] did apply SKFT to the same model we\nare using—electrons whose spins interact via sdexchange\ninteraction with many Heisenberg-exchange-coupled lo-\ncalized spins representing metallic ferromagnet in self-\nconsistent manner—but they did not obtain damping\nterm in their extended Landau-Lifshitz equation, and in-\nstead focused on fluctuations in the magnitude of Mn. In\ncontrast, the vectors Mnare of fixed length in classical\nmicromagnetics [2, 3] and atomistic spin dynamics [4], as\nwell as in our SKFT-derived extended LLG Eq. (9) and\nall other SKFT-based analyses of one or two localized\nspin problems [49–54].\nIn this study we consider either an infinite [Fig. 1(a)],\nor finite [Fig. 1(b)] but sandwiched between two semi-\ninfinite normal metal (NM) leads terminating in macro-\nscopic electronic reservoirs [8, 52, 53], metallic magnet\nwhose localized spins are coupled by ferromagnetic ex-\nchange in equilibrium. The setups in Fig. 1 are of di-\nrect relevance to experiments [10, 38] on external field\n[Fig. 1(a)] or current-driven dynamics [Fig. 1(b)] of lo-\ncalized spins in spintronics and magnonics. Our princi-\npal result is encapsulated by Fig. 1(c)—Gilbert damping,\ndue to conduction electron spins not being able to instan-\ntaneously follow changes in the orientation of classical\nlocalized spins, is always nonlocal and inhomogeneous,\nwith such features becoming more prominent in low-\ndimensional ferromagnets. This result is independently\nconfirmed [Fig. 2] by numerically exact simulations (in\none dimension) based on time-dependent nonequilibrium\nGreen’s function combined with LLG equation (TD-\nNEGF+LLG) scheme [40, 43, 56, 57].\nWe note that conventional linear-response formulas [9,\n30, 34–36] produce unphysical divergent Gilbert damp-\ning [33] in a perfectly crystalline magnet at zero tempera-\nture. In contrast to previously proposed solutions to this\nproblem—which require [58–60] going beyond the stan-\ndard picture of electrons that do not interact with each\nother, while interacting with classical localized spins—\nour formulas are finite in the clean limit, as well as in\nthe absence of SOC. The scattering theory [8] yields a\nformula for αGwhich is also always finite (in the absence\nof SOC, it is finite due to spin pumping [61]). However,\nthat result can only be viewed as a spatial average of our3\nnonlocal damping which cannot produce proper LLG dy-\nnamics of local magnetization [Fig. 3].\nThe paper is organized as follows. In Sec. II we for-\nmulate the SKFT approach to the dynamics of local-\nized spins interacting with conduction electrons within\na metallic ferromagnet. Sections III A and III B show\nhow this approach leads to nonlocal and isotropic, or\nnonlocal and anisotropic, damping in the presence or ab-\nsence of SOC, respectively. The SKFT-derived analyt-\nical results are corroborated by numerically exact TD-\nNEGF+LLG simulations [40, 43, 56, 57] in Sec. III C.\nThen, in Secs. III D and III E we compare SKFT-derived\nformulas with widely used scattering theory of conven-\ntional scalar Gilbert damping [8, 61, 62] or spin-motive\nforce (SMF) theory [13, 19] of nonlocal damping, respec-\ntively. Finally, in Sec. III F, we discuss how to com-\nbine our SKFT-derived formulas to first-principles calcu-\nlations on realistic materials via density functional theory\n(DFT). We conclude in Sec. IV.\nII. SCHWINGER-KELDYSH FIELD THEORY\nFOR METALLIC FERROMAGNETS\nThe starting point of SKFT is the action [48] of metal-\nlic ferromagnet, S=SM+Se,\nSM=Z\nCdtX\nnh\n∂tMn(t)·An− H[Mn(t)]i\n,(3a)\nSe=Z\nCdtX\nnn′h\n¯ψn(t)\u0000\ni∂t−γnn′\u0001\nψn′(t) (3b)\n−δnn′JsdMn(t)·sn′(t)i\n,\nwhere SMis contribution from localized spins and Seis\ncontribution from conduction electrons. The integrationR\nCis along the Keldysh closed contour C[48]. Here the\nsubscript nlabels the site of a D-dimensional cubic lat-\ntice;∂tMn·Anis the Berry phase term [63, 64]; H[Mn]\nis the Hamiltonian of localized spins; ψn= (ψ↑\nn, ψ↓\nn)T\nis the Grassmann spinor [48] for an electron at site\nn;γnn′=−γis the nearest-neighbor (NN) hopping;\nsn=¯ψnσψnis the electronic spin density, where σis\nthe vector of the Pauli matrices; and Jsdis the magni-\ntude of sdexchange interaction between flowing spins of\nconduction electrons and localized spins. For simplicity,\nwe use ℏ= 1.\nThe Keldysh contour C, as well as all functions defined\non it, can be split into forward (+) and backward ( −)\nsegments [48]. These functions can, in turn, be rewritten\nasM±\nn=Mn,c±1\n2Mn,qfor the real-valued localized spins\nfield, and ψ±\nn=1√\n2(ψ1,n±ψ2,n) and ¯ψ±\nn=1√\n2(¯ψ2,n±\n¯ψ1,n) for the Grassmann-valued fermion fields ψnand¯ψn.\nThe subscripts candqrefer to the classical and quantum\ncomponents of time evolution. This rewriting yields thefollowing expressions for the two actions\nSM=Z\ndtX\nnMα\nnq\u0000\nϵαβγ∂tMβ\nn,cMγ\nnc+Bα\neff[Mn,c]\u0001\n,(4a)\nSe=Z\ndtdt′X\nnn′¯ψσ\nn\u0000ˇG−1\nnn′δσσ′−JsdˇMα\nnn′σα\nσσ′\u0001\nψσ′\nn′,(4b)\nwhere subscript σ=↑,↓is for spin; summation over\nrepeated Greek indices is implied; ψ≡(ψ1, ψ2)T;\nBeff=−δH/δMis the effective magnetic field; ϵαβγis\nthe Levi-Civita symbol; and ˇOare 2×2 matrices in the\nKeldysh space, such as\nˇGnn′=\u0012\nGRGK\n0GA\u0013\nnn′,ˇMα\nnn′=\u0012\nMcMq\n2Mq\n2Mc\u0013α\nnδnn′.\n(5)\nHere GR/A/K\nnn′(t, t′) are electronic re-\ntarded/advanced/Keldysh Green’s functions (GFs) [48]\nin the real-space representation of sites n.\nThe electrons can be integrated out [49] up to the sec-\nond order in Jsdcoupling, thereby yielding an effective\naction for localized spins only\nSeff\nM=Z\ndtX\nnMα\nn,qh\nϵαβγ∂tMβ\nn,cMγ\nn,c+Bα\neff[Mn,c]\n+Z\ndt′X\nn′Mα\nn′,c(t′)ηnn′(t, t′)i\n, (6)\nwhere\nηnn′(t, t′) = iJ2\nsd\u0010\nGR\nnn′(t, t′)GK\nnn′(t′, t)\n+GK\nnn′(t, t′)GA\nnn′(t′, t)\u0011\n, (7)\nis the non-Markovian time-retarded kernel. Note that\nterms that are second order in the quantum fluctuations\nMn,qare neglected [48] in order to write Eq. (6). The\nmagnetization damping can be explicitly extracted by\nanalyzing the kernel, as demonstrated for different ferro-\nmagnetic setups in Secs. III A and III B.\nIII. RESULTS AND DISCUSSION\nA. Nonlocality of Gilbert damping in metallic\nferromagnets in the absence of SOC\nSince ηnn′(t−t′) depends only on the difference t−t′, it\ncan be Fourier transformed to energy ε. Thus, the kernel\ncan be written down explicitly for low energies as\nηnn′(ε) =J2\nsdiε\n2πX\nk,qeik·(rn−rn′)eiq·(rn−rn′)Ak(µ)Aq(µ),\n(8)\nwhere Ak(µ)≡i[GR\nk(µ)−GA\nk(µ)] is the spectral func-\ntion [52] evaluated at chemical potential µ;kis a4\nwavevector; and rnandrn′are the position vectors of\nsites nandn′. Equation (8) remains finite in the clean\nlimit and for low temperatures, so it evades unphysical\ndivergences in the linear-response approaches [58–60]. By\ntransforming it back into the time domain, we minimize\nthe effective action in Eq. (6) with respect to the quan-\ntum fluctuations to obtain semiclassical equations of mo-\ntion for classical localized spins. This procedure is equiv-\nalent to the so-called large spin approximation [65, 66] or\na one loop truncation of the effective action. The higher\norder terms neglected in Eq. (6) contribute a stochas-\ntic noise that vanishes in the low temperature and large\nspin limit. Although the fluctuating effect of this noise\ncan modify the exact dynamics [54, 65], the determinis-\ntic regime suffices for a qualitative understanding and is\noften the main focus of interest [66, 67].\nThus, we arrive at the following extended LLG equa-\ntion\n∂tMn=−Mn×Beff,n+Mn×X\nn′λD\nnn′∂tMn′,(9)\nwhere the conventional αGMn×∂tMnGilbert term\nis replaced by the second term on the RHS exhibit-\ning nonlocal damping λD\nnn′instead of Gilbert damping\nscalar parameter αG. A closed expression for λD\nnn′can\nbe obtained for one-dimensional (1D), two-dimensional\n(2D) and three-dimensional (3D) metallic ferromagnets\nby considering quadratic energy-momentum dispersion of\ntheir conduction electrons\nλD\nnn′=\n\n2J2\nsd\nπv2\nFcos2(kF|rn−rn′|) 1D ,\nk2\nFJ2\nsd\n2πv2\nFJ2\n0(kF|rn−rn′|) 2D ,\nk2\nFJ2\nsd\n2πv2\nFsin2(kF|rn−rn′|)\n|rn−rn′|2 3D.(10)\nHere kFis the Fermi wavevector of electrons, vFis their\nFermi velocity, and J0(x) is the 0-th Bessel function of\nthe first kind.\nB. Nonlocality and anisotropy of Gilbert damping\nin metallic ferromagnets in the presence of SOC\nTaking into account that previous analytical calcu-\nlations [26–28] of conventional Gilbert damping scalar\nparameter always include SOC, often of the Rashba\ntype [68], in this section we show how to generalize\nEq. (8) and nonlocal damping extracted in the presence\nof SOC. For this purpose, we employ the Rashba Hamil-\ntonian in 1D, with its diagonal representation given by,\nˆH=P\nkσεkσˆc†\nkσˆckσ, where ˆ c†\nkσ/ˆckσcreates/annihilates\nan electron with wavenumber kand spin σoriented along\nthey-axis, εkσ=−2γcosk+ 2σγSOsinkis the Rashba\nspin-split energy-momentum dispersion, and γSOis the\nstrength of the Rashba SOC coupling. By switching\nfrom second-quantized operators ˆ c†\nkσ/ˆckσto Grassmann-\nvalued two-component fields [64] ¯cσ\nn/cσ\nn, where cσ\nn=\nFIG. 2. (a) Time evolution of two localized spins Mn, lo-\ncated at sites n= 1 and n′= 3 within a chain of 19 sites\nin the setup of Fig. 1(b), computed numerically by TD-\nNEGF+LLG scheme [40, 43, 56, 57]. The two spins are\ncollinear at t= 0 and point along the x-axis, while mag-\nnetic field is applied along the z-axis. (b) The same infor-\nmation as in panel (a), but for two noncollinear spins with\nangle ∈ {0,45,90,135,180}between them. (c) and (d) Ef-\nfective damping extracted from TDNEGF+LLG simulations\n(red dashed line) vs. the one from SKFT [black solid line plots\n1D case in Eq. (10)] as a function of the site n′of the second\nspin. The two spins are initially parallel in (c), or antiparallel\nin (d). The Fermi wavevector of conduction electrons is cho-\nsen as kF=π/2a, where ais the lattice spacing.\n(cσ\n1,n, cσ\n2,n)T, we obtain for the electronic action\nSe=Z\ndtdt′X\nnn′¯cσ\nn\u0002\n(ˇGσ\nnn′)−1δσσ′−JsdˇMα\nnn′σβ\nσσ′\u0003\ncσ′\nn′.\n(11)\nHere ˇGσ\nnn′is diagonal, but it depends on spin through\nεkσ. In addition, ˇMx,y,z\nnn′, as the matrix which couples to\nthe same σx,y,zPauli matrix in electronic action without\nSOC [Eq. (3b)], is coupled in Eq. (11) to a different Pauli\nmatrix σy,z,x.\nBy integrating electrons out up to the second order in\nJsd, and by repeating steps analogous to those of Sec. II\nwhile carefully differentiating the spin-split bands, we\nfind that nonlocal damping becomes anisotropic\nλ1D\nnn′=\nα⊥\nnn′0 0\n0α∥\nnn′0\n0 0 α⊥\nnn′.\n. (12)5\nwhere\nα⊥\nnn′=J2\nsd\nπ\u0012cos2(k↑\nF|rn−rn′|)\nv↑\nF2+cos2(k↓\nF|rn−rn′|)\nv↓\nF2\u0013\n,\n(13a)\nα∥\nnn′=J2\nsd\nπ|v↑\nFv↓\nF|\u0010\ncos\u0002\n(k↑\nF+k↓\nF)|rn−rn′|\u0003\n(13b)\n+ cos\u0002\n(k↑\nF−k↓\nF)|rn−rn′|\u0003\u0011\n,\nandk↑/↓\nFandv↑/↓\nFare the Fermi wavevectors and veloc-\nities, respectively, of the Rashba spin-split bands. This\nmeans that the damping term in Eq. (9) is now given by\nMn×P\nn′λ1D\nnn′·∂tMn′.\nWe note that previous experimental [69], numeri-\ncal [9, 70], and analytical [26–28] studies have also found\nSOC-induced anisotropy of Gilbert damping scalar pa-\nrameter. However, our results [Eqs. (12) and (13)] ex-\nhibit additional feature of nonlocality (i.e., damping at\nsitendepends on spin at site n′) and nonuniformity (i.e.,\ndependence on |rn−rn′|). As expected from Sec. III A,\nnonlocality persists for γSO= 0, i.e., k↑\nF=k↓\nF=kF,\nwith λ1D\nnn′properly reducing to contain αnn′three di-\nagonal elements. Additionally, the damping component\nα∥\nnn′given by Eq. (13b) can take negative values, re-\nvealing the driving capability of the conduction electrons\n(see Sec. III C). However, for realistic small values of γSO,\nthe driving contribution of nearby localized spins is like-\nwise small. Furthermore, the decay of nonlocal damping\nwith increasing distance observed in 2D and 3D, together\nwith the presence of intrinsic local damping from other\nsources, ensures that the system tends towards equilib-\nrium.\nC. Comparison of SKFT-derived formulas with\nnumerically exact TDNEGF+LLG simulations\nAn analytical solution to Eq. (9) can be obtained in\nfew special cases, such as for two exchange-uncoupled lo-\ncalized spins at sites n= 1 and n′̸= 1 within 1D wire\nplaced in an external magnetic field Bext=Bextez, on\nthe proviso that the two spins are collinear at t= 0.\nThe same system can be simulated by TDNEGF+LLG\nscheme, so that comparing analytical to such numeri-\ncally exact solution for trajectories Mn(t) makes it pos-\nsible to investigate accuracy of our derivation and ap-\nproximations involved in it, such as: truncation to J2\nsd\norder; keeping quantum fluctuations Mn,qto first order;\nand low-energy approximation used in Eq. (8). While\nsuch a toy model is employed to verify the SKFT-based\nderivation, we note that two uncoupled localized spins\ncan also be interpreted as macrospins of two distant ferro-\nmagnetic layers within a spin valve for which oscillatory\nGilbert damping as a function of distance between the\nlayers was observed experimentally [71]. Note that semi-\ninfinite NM leads from the setup in Fig. 1(b), always usedin TDNEGF+LLG simulations to ensure continuous en-\nergy spectrum of the whole system [40, 56], can also be\nincluded in SKFT-based derivation by using self-energy\nΣR/A\nk(ε) [52, 72] which modifies the GFs of the central\nmagnetic region in Fig. 1(b), GR/A\nk= (ε−εk−ΣR/A\nk)−1,\nwhere εk=−2γcosk.\nThe TDNEGF+LLG-computed trajectory M1(t) of lo-\ncalized spin at site n= 1 is shown in Figs. 2(a) and\n2(b) using two localized spins which are initially collinear\nor noncollinear, respectively. For the initially parallel\n[Fig. 2(a)] or antiparallel localized spins, we can ex-\ntract Gilbert damping from such trajectories because\nMz\n1(t) = tanh\u0000¯λ1D\nnn′Bextt/(1 + ( ¯λ1D\nnn′)2)\u0001\n[4, 40], where\nthe effective damping is given by ¯λ1D\nnn′=λ1D\n00±λ1D\nnn′\n(+ for parallel and −for antiparallel initial condition).\nThe nonlocality of such effective damping in Figs. 2(c)\nand 2(d) manifests as its oscillation with increasing sep-\naration of the two localized spins. The same result\nis predicted by the SKFT-derived formula [1D case in\nEq. (10)], which remarkably closely traces the numeri-\ncally extracted ¯λ1D\nnn′despite approximations involved in\nSKFT-based analytical derivation. Note also that the\ntwo localized spins remain collinear at all times t, but\ndamping remains nonlocal. The feature missed by the\nSKFT-based formula is the decay of ¯λ1D\nnn′with increasing\n|rn−rn′|, which is present in numerically-extracted effec-\ntive damping in Figs. 2(c) and 2(d). Note that effective\ndrastically reduced for antiparallel initial conditions, due\nto the driving capabilities of the conduction electrons, in\naddition to their dissipative nature. For noncollinear ini-\ntial conditions, TDNEGF+LLG-computed trajectories\nbecome more complicated [Fig. 2(b)], so that we can-\nnot extract the effective damping λ1D\nnn′akin to Figs. 2(c)\nand 2(d) for the collinear initial conditions.\nD. Comparison of SKFT-derived formulas with the\nscattering theory [8] of uniform local Gilbert\ndamping\nThe scattering theory of Gilbert damping αGwas\nformulated by studying a single domain ferromagnet\nin contact with a thermal bath [8]. In such a setup,\nenergy [8] and spin [61] pumped out of the system\nby time-dependent magnetization contain information\nabout spin-relaxation-induced bulk [8, 62] and interfa-\ncial [61] separable contributions to αG, expressible in\nterms of the scattering matrix of a ferromagnetic layer\nattached to two semi-infinite NM leads. For collinear lo-\ncalized spins of the ferromagnet, precessing together as\na macrospin, scattering theory-derived αGis a spatially-\nuniform scalar which can be anisotropic [62]. Its expres-\nsion is equivalent [62] to Kubo-type formulas [9, 34–36]\nin the linear response limit, while offering an efficient al-\ngorithm for numerical first-principles calculations [24, 25]\nthat can include disorder and SOC on an equal footing.\nOn the other hand, even if all localized spins are ini-\ntially collinear, SKFT-derived extended LLG Eq. (9) pre-6\nFIG. 3. (a) Comparison of trajectories of localized spins\nMz\nn(t), in the setup of Fig. 1(b) whose central region is\n1D metallic ferromagnet composed of 5 sites, using LLG\nEq. (9) with SKFT-derived nonlocal damping (solid red lines)\nvs. LLG equation with conventional spatially-independent\nαG= 0.016 (black dashed line). This value of αGis ob-\ntained by averaging nonlocal damping over the whole ferro-\nmagnet. The dynamics of Mn(t) is initiated by an external\nmagnetic field along the z-axis, while all five localized spins\npoint along the x-axis at t= 0. (b) Comparison of spin cur-\nrentISz\nR(t) pumped [56, 57, 61] by the dynamics of Mn(t) for\nthe two cases [i.e., nonuniform Mn(t) for nonlocal vs. uniform\nMn(t) for conventional damping] from panel (a). The Fermi\nwavevector of conduction electrons is chosen as kF=π/2a.\ndicts that due to nonlocal damping each localized spin\nwill acquire a distinct Mn(t) trajectory, as demonstrated\nby solid red lines in Fig. 3(a). By feeding these trajec-\ntories, which are affected by nonlocal damping [1D case\nin Eq. (10)] into TDNEGF+LLG simulations, we can\ncompute spin current ISz\nR(t) pumped [56, 57] into the\nright semi-infinite lead of the setup in Fig. 1(b) by the\ndynamics of Mn(t). A very similar result for pumped\nspin current is obtained [Fig. 3(b)] if we feed identical\nMn(t) trajectories [black dashed line in Fig. 3(a)] from\nconventional LLG equation with Gilbert damping scalar\nparameter, αG, whose value is obtained by averaging the\nSKFT-derived nonlocal damping over the whole ferro-\nmagnet. This means that scattering theory of Gilbert\ndamping [8], which in this example is purely due to inter-\nfacial spin pumping [61] because of lack of SOC and dis-\norder (i.e., absence of spin relaxation in the bulk), would\npredict a constant αGthat can only be viewed as the\nspatial average of SKFT-derived nonlocal and nonuni-\nform λ1D\nnn′. In other words, Fig. 3 reveals that different\ntypes of microscopic magnetization dynamics Mn(t) can\nyield the same total spin angular momentum loss into\nthe external circuit, which is, therefore, insufficient on\nits own to decipher details (i.e., the proper form of ex-\ntended LLG equation) of microscopic dynamics of local\nmagnetization.\n1.5 2.0 2.5 3.0\nw/a024vDW(aJ/¯ h)×10−2\nαG= 0.1\nEq.(9)\nRef.[13] withη= 0.05\nRef.[19] withη= 0.05\nEq.(1) withη= 0(a)\n0 25 50 75\nSite i−1.0−0.50.00.51.0Mα(t)t = 410 ¯ h/J\nα=x,y,z(b)\n0 25 50 75\nSite i−2−101(Mn×/summationtext\nn/primeλd\nnn/prime·∂tMn/prime)α×10−2\n(c)\n0 25 50 75\nSite i−2−101(M×D·∂tM)α×10−3\n(d)FIG. 4. (a) Comparison of magnetic DW velocity vDWvs.\nDW width wextracted from numerical simulations using: ex-\ntended LLG Eq. (9) with SKFT-derived nonlocal damping\n[Eq. (10), red line]; extended LLG Eq. (1) with SMF-derived\nin Ref. [13] nonlocal damping [Eq. (2), blue line] or SMF-\nderived nonlocal damping (green line) in Ref. [19] [with ad-\nditional term when compared to Ref. [13], see Eq. (14)]; and\nconventional LLG Eq. (1) with local Gilbert damping [i.e.,\nη= 0 in Eq. (2), black line]. (b) Spatial profile of DW within\nquasi-1D ferromagnetic wire at time t= 410 ℏ/J, where Jis\nexchange coupling between Mnat NN sites, as obtained from\nSKFT-derived extended LLG Eq. (9) with nonlocal damping\nλ2D\nnn′[Eq. (10)]. Panels (c) and (d) plot the corresponding spa-\ntial profile of nonlocal damping across the DW in (b) using\nSKFT-derived expression [Eqs. (9) and Eq. (10)] vs. SMF-\nderived [13] expression [second term on the RHS of Eq. (2)],\nrespectively.\nE. Comparison of SKFT-derived formulas with\nspin motive force theory [13] and [19] of nonlocal\ndamping\nThe dynamics of noncollinear and noncoplanar magne-\ntization textures, such as magnetic DWs and skyrmions,\nleads to pumping of charge and spin currents assumed\nto be captured by the spin motive force (SMF) the-\nory [16, 73, 74]. The excess angular momentum of dy-\nnamical localized spins carried away by pumped spin cur-\nrent of electrons appears then as backaction torque [57]\nexerted by nonequilibrium electrons onto localized spins\nor, equivalently, nonlocal damping [13, 17–19]. From this\nviewpoint, i.e., by using expressions for pumped spin cur-\nrent [13, 17–19], a particular form for nonlocal damp-\ning [second term on the RHS of Eq. (2)] was derived in\nRef. [13] from the SMF theory, as well as extended in\nRef. [19] with an additional term, while also invoking a\nnumber of intuitively-justified but uncontrolled approxi-\nmations.\nIn this Section, we employ an example of a magnetic\nfield-driven DW [Fig. 4(b)] of width wwithin a quasi-7\n1D ferromagnetic wire to compare its dynamics obtained\nby solving extended LLG Eq. (1), which includes non-\nlocal damping tensor [Eq. (2)] of Ref. [13], with the\ndynamics obtained by solving SKFT-derived extended\nLLG Eq. (9) whose nonlocal damping is different from\nRef. [13]. By neglecting nonlocal damping in Eq. (2),\nthe ferromagnetic domain wall (DW) velocity vDWis\nfound [75] to be directly proportional to Gilbert damping\nαG,vDW∝ −BextwαG, assuming high external magnetic\nfieldBextand sufficiently small αG. Thus, the value of αG\ncan be extracted by measuring the DW velocity. How-\never, experiments find that αGdetermined in this fashion\ncan be up to three times larger than αGextracted from\nferromagnetic resonance linewidth measurement scheme\napplied to the same material with uniform dynamical\nmagnetization [10]. This is considered as a strong evi-\ndence for the importance of nonlocal damping in systems\nhosting noncollinear magnetization textures.\nIn order to properly compare the effect of two different\nexpressions for the nonlocal damping, we use αG= 0.1\nin Eq. (1) and we add the same standard local Gilbert\ndamping term, αGMn×∂tMn, into SKFT-derived ex-\ntended LLG Eq. (9). In addition, we set λ2D\n00=ηin\nEq. (10), so that we can vary the same parameter ηin all\nversions of extended LLG Eqs. (1), and (9). Note that\nwe use λ2D\nnn′in order to include realistic decay of nonlo-\ncal damping with increasing distance |rn−rn′|, thereby\nassuming quasi-1D wire. By changing the width of the\nDW, the effective damping can be extracted from the DW\nvelocity [Fig. 4(a)]. Figure 4(a) shows that vDW∝wre-\ngardless of the specific version of nonlocal damping em-\nployed, and it increases in its presence—compare red,\nblue, and green data points with the black ones obtained\nin the absence of nonlocal damping. Nevertheless, the\nclear distinction between red, and blue or green data\npoints signifies that our SKFT-derived nonlocal damping\ncan be quite different from previously discussed SMF-\nderived nonlocal damping [13, 19], which are compara-\nble regardless of the inclusion of the nonadiabatic terms.\nFor example, the effective damping extracted from blue\nor green data points is D= 0.17 or D= 0.15, respec-\ntively, while λ2D\nnn′= 0.48. This distinction is further clar-\nified by comparing spatial profiles of SKFT-derived and\nSMF-derived nonlocal damping in Figs. 4(c) and 4(d),\nrespectively, at the instant of time used in Fig. 4(b). In\nparticular, the profiles differ substantially in the out-\nof-DW-plane or y-component, which is, together with\nthex-component, an order of magnitude greater in the\ncase of SKFT-derived nonlocal damping. In addition,\nthe SKFT-derived nonlocal damping is nonzero across\nthe whole wire , while the nonlocal damping in Eq. (2)\nis nonzero only within the DW width, where Mnvec-\ntors are noncollinear [as obvious from the presence of\nthe spatial derivative in the second term on the RHS\nof Eq. (2)]. Thus, the spatial profile of SKFT-derived\nnonlocal damping in Fig. 4(c) illustrates how its nonzero\nvalue in the region outside the DW width does not re-\nquire noncollinearity of Mnvectors.Since SKFT-derived formulas are independently con-\nfirmed via numerically exact TDNEGF+LLG simula-\ntions in Figs. 2(c) and 2(d), we conclude that previously\nderived [13] type of nonlocal damping [second term on\nthe RHS of Eq. (2)] does not fully capture backaction of\nnonequilibrium conduction electrons onto localized spins.\nThis could be due to nonadiabatic corrections [16, 19, 74]\nto spin current pumped by dynamical noncollinear mag-\nnetization textures, which are present even in the ab-\nsence of disorder and SOC [43]. One such correction was\nderived in Ref. [19], also from spin current pumping ap-\nproach, thereby adding a second nonlocal damping term\nηX\nβ′h\n(M·∂β′∂tM)M×∂β′M−M×∂2\nβ′∂tMi\n,(14)\ninto the extended LLG Eq. (1). However, combined us-\nage [green line in Fig. 4(a)] of both this term and the one\nin Eq. (2) as nonlocal damping still does not match the\neffect of SKFT-derived nonlocal damping [compare with\nred line in Fig. 4(a)] on magnetic DW. As it has been\ndemonstrated already in Fig. 3, the knowledge of total\nspin angular momentum loss carried away by pumped\nspin current [Fig. 3(b)], as the key input in the deriva-\ntions of Refs. [13, 19], is in general insufficient to decipher\ndetails of microscopic dynamics and dissipation exhibited\nby localized spins [Fig. 3(a)] that pump such current.\nF. Combining SKFT-derived nonlocal damping\nwith first-principles calculations\nObtaining the closed form expressions for the nonlocal\ndamping tensor λnn′in Secs. III A and III B was made\npossible by using simplistic model Hamiltonians and ge-\nometries. For realistic materials and more complicated\ngeometries, we provide in this Section general formulas\nwhich can be combined with DFT quantities and evalu-\nated numerically.\nNotably, the time-retarded dissipation kernel in\nEq. (7), from which λnn′is extracted, depends on the\nKeldysh GFs. The same GFs are also commonly used\nin first-principles calculations of conventional Gilbert\ndamping scalar parameter via Kubo-type formulas [29–\n33]. Specifically, the retarded/advanced GFs are ob-\ntained from first-principles Hamiltonians ˆHDFTDFT as\nˆGR/A(ε) =\u0002\nε−ˆHDFT+ˆΣR/A(ϵ)\u0003−1. Here, ˆΣR/A(ε) are\nthe retarded/advanced self-energies [52, 72] describing es-\ncape rate of electrons into NM leads, allowing for open-\nsystem setups akin to the scattering theory-derived for-\nmula for Gilbert damping [8, 62] and its computational\nimplementation with DFT Hamiltonians [24, 25]. Since\nescape rates are encoded by imaginary part of the self-\nenergy, such calculations do not require iηimaginary pa-\nrameter introduced by hand when using Kubo-type for-\nmulas [29–33] (where η→0 leads to unphysical divergent\nresults [58–60]). Therefore, ˆHDFTcan be used as an\ninput to compute the nonlocal damping tensor, via the8\ncalculation of the GFs ˆGR/A(ε) and the spectral function\nˆA(ε) =i\u0002ˆGR(ε)−ˆGA(ε)\u0003\n.\nFor these purposes, it is convenient to separate the\nnonlocal damping tensor into its symmetric and anti-\nsymmetric components, λαβ\nnn′=λ(αβ)\nnn′+λ[αβ]\nnn′, where the\nparenthesis (brackets) indicate that surrounded indices\nhave been (anti)symmetrized. They are given by\nλ(αβ)\nnn′=−J2\nsd\n2πZ\ndε∂f\n∂εTrspin\u0002\nσαAnn′σβAn′n\u0003\n, (15a)\nλ[αβ]\nnn′=−2J2\nsd\nπZ\ndε∂f\n∂εTrspin\u0002\nσαReˆGR\nnn′σβAn′n\n−σαAnn′σβReˆGR\nn′n\u0003\n+J2\nsd\n2πZ\ndε(1−2f)\n×Trspin\u0002\nσαReˆGR\nnn′σβ∂An′n\n∂ε−σα∂Ann′\n∂εσβReˆGR\nn′n\u0003\n,\n(15b)\nwhere f(ε) is the Fermi function, and the trace is taken\nin the spin space. The antisymmetric component either\nvanishes in the presence of inversion symmetry, or is of-\nten orders of magnitude smaller than the symmetric one.\nTherefore, it is absent in our results for simple models\non hypercubic lattices. As such, the nonlocal damping\ntensors in Eqs. (10) and (13), are fully symmetric and\nspecial case of Eq. (15a) when considering specific energy-\nmomentum dispersions and assuming zero temperature.\nIV. CONCLUSIONS AND OUTLOOK\nIn conclusion, we derived a novel formula, displayed\nas Eq. (15), for magnetization damping of a metallic fer-\nromagnet via unexploited for this purpose rigorous ap-\nproach offered by the Schwinger-Keldysh nonequilibrium\nfield theory [48]. Our formulas could open a new route for\ncalculations of Gilbert damping of realistic materials by\nemploying first-principles Hamiltonian ˆHDFTfrom den-\nsity functional theory (DFT) as an input, as discussed\nin Sec. III F. Although a thorough numerical exploration\nof a small two-spin system based on SKFT was recently\npursued in Ref. [54], our Eqs. (15) are not only applica-\nble for large systems of many localized spins, but are also\nrefined into readily computable expressions that depend\non accessible quantities.While traditional, Kubo linear-response [9, 30, 34–\n36] or scattering theory [8] based derivations produce\nspatially uniform scalar αG, SKFT-derived damping in\nEqs. (15) is intrinsically nonlocal and nonuniform as it\ndepends on the coordinates of local magnetization at two\npoints in space rnandrn′. In the cases of model Hamil-\ntonians in 1D–3D, we reduced Eqs. (15) to analytical ex-\npressions for magnetization damping [Eq. (10)], thereby\nmaking it possible to understand the consequences of\nsuch fundamental nonlocality and nonuniformity on lo-\ncal magnetization dynamics, such as: ( i) damping in\nEq. (10) osc illates with the distance between xandx′\nwhere the period of such oscillation is governed by the\nFermi wavevector kF[Figs. 1(c), 2(c), and 2(d)]; ( ii)\nit always leads to nonuniform local magnetization dy-\nnamics [Fig. 3(a)], even though spin pumping from it\ncan appear [Fig. 3(b)] as if it is driven by usually an-\nalyzed [8, 61] uniform local magnetization (or, equiv-\nalently, macrospin); ( iii) when applied to noncollinear\nmagnetic textures, such as DWs, it produces an order\nof magnitude larger damping and, therefore, DW wall\nvelocity, than predicted by previously derived [13] non-\nlocal damping [second term on the RHS of Eq. (2)].\nRemarkably, solutions of SKFT-based extended LLG\nEq. (9) are fully corroborated by numerically exact TD-\nNEGF+LLG simulations [40, 43, 56, 57] in 1D, despite\nthe fact that several approximations are employed in\nSKFT-based derivations. 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Uehling, Ferromagnetic resonance\nand spin diffusion in supermalloy, Phys. Rev. 164, 662\n(1967)." }, { "title": "2106.05182v1.Exact_solution_of_damped_harmonic_oscillator_with_a_magnetic_field_in_a_time_dependent_noncommutative_space.pdf", "content": "arXiv:2106.05182v1 [quant-ph] 14 May 2021Exact solution of damped harmonic oscillator with a\nmagnetic field in a time dependent noncommutative space\nManjari Duttaa∗,Shreemoyee Gangulyb†,Sunandan Gangopadhyayc‡\na,cDepartment of Theoretical Sciences ,\nS.N. Bose National Centre for Basic Sciences ,\nJD Block, Sector III, Salt Lake, Kolkata 700106, India\nbDepartment of Basic Science and Humanities,\nUniversity of Engineering and Management (UEM),\nB/5, Plot No.III, Action Area-III, Newtown, Kolkata 700156\nAbstract\nIn this paper we have obtained the exact eigenstates of a two d imensional damped harmonic oscillator in the\npresence of an external magnetic field varying with respect t o time in time dependent noncommutative space.\nIt has been observed that for some specific choices of the damp ing factor, the time dependent frequency of the\noscillator andthetimedependentexternalmagneticfield, t hereexistsinterestingsolutionsofthetimedependent\nnoncommutative parameters following from the solutions of the Ermakov-Pinney equation. Further, these\nsolutions enable us to get exact analytic forms for the phase which relates the eigenstates of the Hamiltonian\nwith the eigenstates of the Lewis invariant. Then we compute the expectation value of the Hamiltonian.\nThe expectation values of the energy are found to vary with ti me for different solutions of the Ermakov-Pinney\nequation corresponding to different choices of the dampingf actor, the time dependentfrequencyof theoscillator\nand the time dependent applied magnetic field. We also compar e our results with those in the absence of the\nmagnetic field obtained earlier.\n∗manjaridutta@boson.bose.res.in\n†ganguly.shreemoyee@gmail.com\n‡sunandan.gangopadhyay@bose.res.in, sunandan.gangopad hyay@gmail.com\n12\n1 Introduction\nTheLandauproblemofachargedparticlemovingin atwodimensionalp laneundertheinfluence ofamagneticfield\nacting perpendicular to the plane has been looked upon by physicists over the years, not only from the pedagogical\ninterest of formation of discrete energy levels known as Landau lev els, but also due to the multifaceted applications\nofthisproblem. In this context, an extremelyintriguingproblemisth at ofachargedoscillatorplacedin amagnetic\nfield, acting perpendicular to the plane in which it is oscillating. In additio n, one may also consider an electric\nfield lying along the plane of oscillation. The eigenfunction and the eigen values of a charged particle in a magnetic\nfield in the presence of a time-dependent background electric field w ith time-dependent mass and frequency have\nbeen determined in [1]. The problem becomes even more fascinating wh en one places the Landau oscillator in a\nnoncommutative phase space. This problem has been looked upon by an earlier study [2] in the presence of a time\ndependent magnetic field. The system there was considered in a non commutative space. The simplest setting of\nnoncommutative [NC] space is a two dimensional quantum mechanical space in which one replaces the standard\nset of commutation relations between the canonical coordinates b y NC commutation relations [ X,Y] =iθ, where\nθis a positive real constant. Study of quantum mechanical systems in such NC space has allured theoretical\nphysicists since the work by Synder [3]. Since then the necessity fo r NC spaces has been established to ensure the\nattainment of gravitational stability [4] in the present theories of quantum gravity, namely, string theory [5, 6]\nand loop quantum gravity [7]. This has triggered several studies on quantum mechanical systems in such spaces\nin the literature [8]-[18].\nHowever, the mentioned study [2] on charged oscillators in the pres ence of magnetic field, considers only\nnoncommutativity amongst position variables. So, we in our present communication extend the study to a space\nwhere noncommutativity exists not only amongst spatial variables b ut also amongst momentum variables. Also\nunlike the previous study we have considered the NC space to be time dependent. Moreover, our oscillator is\nconsidered to be damped by an explicit damping factor in order to mod el a realistic situation. A damped oscillator\nin two dimensional NC space has been studied by us in an earlier commun ication [19]. Before our communication\noneoftheveryfewworkswhichhadstudieddampedquantumharmo nicoscillatorsintwodimensionalspacewasthe\nwork by Lawson et.al.[20]. We extended their model to a two dimensional NC space. But at p resent our objective\nis to study how the interplay of damping and an external time depend ent magnetic field modulates the energetics\nof a charged oscillator in a time dependent NC space where spatial as well as momentum noncommutativity is\npresent.\nOur present study is one of the very first to study a damped quant um harmonic oscillator with time varying\nfrequency in two dimensional NC space in the presenceof a time-dep endent magnetic field. In our earlierstudy [19]\nwe had seen that the expectation value of energy of an oscillator de cays due to damping even in NC space. In\nthe present study we intend to investigate the change in energetic s of the damped quantum oscillator in NC space\nunder the influence of magnetic field having various kinds of time depe ndence. For this purpose we first set up\nthe Hamiltonian of the damped quantum harmonic oscillator in two dimen sional NC space under the presence of\na time varying magnetic field and then express it in terms of commutat ive variables. This is done in Section 2.\nAfter that we solve the Hamiltonian using the method of invariants [2 1, 22, 23] in Section 3. It must be noted that\nthe eigenfunction of the said Hamiltonian is a product of the eigenfun ction of the invariant and a phase factor.\nBoth the eigenfunction and phase factor are expressed in terms o f time dependent parameters which obey the\nnon-linear differential equation known as Ermakov-Pinney(EP) equ ation [24, 25]. Next in Section 4 we choose the\nparameters of the damped system in the presence of magnetic field in such a way that they satisfy all the equations\nrepresenting the system as well as provide us exact closed form so lutions for the system for various choices of time\nvariation of the applied field. In Section 5 we calculate the expectatio n value of energy of the damped oscillator3\nexplicitly and graphically explore how the time dependence of the magn etic field alters the time evolution of the\nenergetics of the damped quantum system having a time varying fre quency of oscillation in two dimensional NC\nspace.\n2 Model of the two-dimensional harmonic oscillator in magne tic field\nIn our study the system that we consider is a combination of two non -interacting damped harmonic oscillators\naffected by a time dependent magnetic field in two dimensional NC spac e. Both the oscillators have equal time\ndependent frequencies, coefficients of friction, equal mass and e qual charge in NC space. Such a model of two-\ndimensional Landau problem of harmonic oscillator in a time dependent magnetic field was considered in an earlier\ncommunication[2]in spatiallynoncommutativeconfigurationspace. I nthis work,however,weextendthe modelby\nconsidering the system in time dependent NC space. Also it must be no ted that the noncommutativity we consider\nis not restricted to spatial variables like the earlier studies but also e xtends to momentum noncommutativity.1\nThe Hamiltonian of the two dimensional oscillator in magnetic field has th e following form,\nH(t) =f(t)\n2M/bracketleftbig\n(P1−qA1)2+(P2−qA2)2/bracketrightbig\n+Mω2(t)\n2f(t)(X12+X22) ; (1)\nwhere the damping factor f(t) is given by,\nf(t) =e−/integraltextt\n0η(s)ds(2)\nwithη(s) being the coefficient of friction and Ai, the vector potential of a time dependent magnetic field B(t) is\nchosen in Coulomb gauge as,\nAi=−B(t)\n2ǫijXj; (3)\nwherei,j= 1,2 andǫij=−ǫjiwithǫ12= 1. Here ω(t) is the time dependent angular frequency of the\noscillators , Mandqare their mass and charge respectively. The position and momentum coordinates ( Xi,Pi)\nare noncommuting variables in NC space, that is, their commutators are [X1,X2]/negationslash= 0 and [P1,P2]/negationslash= 0. The\ncorresponding canonical variables ( xi,pi) in commutative space are such that the commutator [ xi,pj] =i/planckover2pi1δi,j,\n[xi,xj] = 0 = [pi,pj]; (i,j= 1,2).\nIn order to express the NC Hamiltonian in terms of the standard com mutative variables explicitly, we apply\nthe standard Bopp-shift relations [26] ( /planckover2pi1= 1):\nX1=x1−θ(t)\n2p2;X2=x2+θ(t)\n2p1 (4)\nP1=p1+Ω(t)\n2x2;P2=p2−Ω(t)\n2x1. (5)\nHereθ(t) and Ω(t) are the NC parameters for space and momentum respectively, su ch that [X1,X2] =iθ(t),\n[P1,P2] =iΩ(t) and [X1,P1] =i[1+θ(t)Ω(t)\n4] = [X2,P2]; (X1≡X,X2≡Y,P1≡Px,P2≡Py).\nThe Hamiltonian in terms of ( xi,pi) coordinates is therefore given by the following relation,\nH=a(t)\n2(p12+p22)+b(t)\n2(x12+x22)+c(t)(p1x2−p2x1). (6)\n1We shall be considering NC phase space in our work. However, w e shall be generically referring to this as NC space.4\nThe time dependent coefficients in the above Hamiltonian are given as,\na(t) =f(t)\nM+qB(t)f(t)θ(t)\n2M+1\n4/bracketleftbiggq2B2(t)f(t)\n4M+Mω2(t)\nf(t)/bracketrightbigg\nθ2(t) (7)\nb(t) =q2B2(t)f(t)\n4M+Mω2(t)\nf(t)+qB(t)f(t)Ω(t)\n2M+f(t)Ω2(t)\n4M(8)\nc(t) =1\n2/bracketleftbiggqB(t)f(t)\nM/parenleftbigg\n1+θ(t)Ω(t)\n4/parenrightbigg\n+Ω(t)f(t)\nM+/parenleftbiggq2B2(t)f(t)\n4M+Mω2(t)\nf(t)/parenrightbigg\nθ(t)/bracketrightbigg\n. (9)\nHere it must be noted that although our Hamiltonian given by Eqn.(6) h as the same form as that in [27] and\n[19] to study a system of a two dimensional harmonic oscillator and da mped harmonic oscillator in NC space, the\ntime dependent Hamiltonian coefficients (given by Eqn(s).(7),(8),(9) ) have a modified form. This is because our\npresent system of damped harmonic oscillator is studied in the prese nce of a time dependent magnetic field in\ntwo-dimensional NC space. Thus, both the damping factor f(t) and the magnetic field B(t) modulate and alter\nthe Hamiltonian coefficients from the form considered in earlier study [27] and [19] respectively. It is relevant to\nmention that those coefficients also differ from those obtained in [2] as the considered noncommutativity in that\nstudy is time independent and exists only in the configuration space.\n3 Solution of the model Hamiltonian\nIn order to find the solutions of the model Hamiltonian H(t) (Eqn.(6)) representing the two-dimensional damped\nharmonic oscillator with magnetic field in NC space, we follow the route s uggested by Lewis et.al.[21] in their\nwork. First we construct the time-dependent Hermitian invariant o peratorI(t) corresponding to our Hamiltonian\noperatorH(t) (given by Eqn.(6)). This is because if one can solve for the eigenfun ctions ofI(t),φ(x1,x2), such\nthat,\nI(t)φ(x1,x2) =ǫφ(x1,x2) (10)\nwhereǫis an eigenvalue of I(t) corresponding to eigenstate φ(x1,x2), one can obtain the eigenstates of H(t),\nψ(x1,x2,t), using the relation given by Lewis et. al.[21] which is as follows,\nψ(x1,x2,t) =eiΘ(t)φ(x1,x2) (11)\nwhere the real function Θ( t) which acts as the phase factor will be discussed in details later.\n3.1 The Time Dependent Invariant\nNext, followingthe approachtaken byLewis et.al.[21], weneed to constructthe operator I(t) which isan invariant\nwith respect to time, corresponding to the Hamiltonian H(t), as mentioned earlier, such that I(t) satisfies the\ncondition,\ndI\ndt=∂tI+1\ni[I,H] = 0. (12)\nThe procedure is to choose the Hermitian invariant I(t) to be of the same homogeneous quadratic form defined by\nLewiset. al.[21] for time-dependent harmonic oscillators. However, since we ar e dealing with a two-dimensional\nsystem in the present study, I(t) takes on the following form,\nI(t) =α(t)(p12+p22)+β(t)(x12+x22)+γ(t)(x1p1+p2x2). (13)5\nHere we will consider /planckover2pi1= 1 since we choose to work in natural units. Now, using the form of I(t) defined by\nEqn.(13) in Eqn.(12) and equating the coefficients of the canonical v ariables, we get the following relations,\n˙α(t) =−a(t)γ(t) (14)\n˙β(t) =b(t)γ(t) (15)\n˙γ(t) = 2[b(t)α(t)−β(t)a(t)] (16)\nwhere dot denotes derivative with respect to time t.\nTo express the above three time dependent parameters α,βandγin terms of a single time dependent parameter,\nwe parametrize α(t) =ρ2(t). Substituting this in Eqn(s).(14, 16), we get the other two param eters in terms of\nρ(t) as,\nγ(t) =−2ρ˙ρ\na(t), β(t) =1\na(t)/bracketleftbigg˙ρ2\na(t)+ρ2b+ρ¨ρ\na(t)−ρ˙ρ˙a\na2/bracketrightbigg\n. (17)\nNow, substituting the value of βin Eqn.(15), we get a non-linear equation in ρ(t) which has the form of the non-\nlinear Ermakov-Pinney (EP) equation with a dissipative term [27, 24, 25]. The form of the non-linear equation is\nas follows,\n¨ρ−˙a\na˙ρ+abρ=ξ2a2\nρ3(18)\nwhereξ2is a constant of integration. This equation has similar form to the EP e quation obtained in [27], which\nis expected since our H(t) has the same form as theirs. However, once again it should be ment ioned that the\nexplicit form of the time-dependent coefficients are different due to the presence of the external magnetic field as\nwell as the fact that the oscillator is damped.\nNow, using the EP equation we get a simpler form of βas,\nβ(t) =1\na(t)/bracketleftbigg˙ρ2\na(t)+ξ2a(t)\n��2/bracketrightbigg\n. (19)\nNext, substituting the expressions of α,βandγin Eqn.(13), we get the following expression for I(t),\nI(t) =ρ2(p12+p22)+/parenleftbigg˙ρ2\na2+ξ2\nρ2/parenrightbigg\n(x12+x22)−2ρ˙ρ\na(x1p1+p2x2). (20)\nThis form of the Lewis invariant in Cartesian coordinate is converted to polar coordinate using the same procedure\nas followed in our previous communication [19]. The invariant in polar coo rdinate takes the following form,\nI(t) =ξ2\nρ2r2+/parenleftbigg\nρpr−˙ρ\nar/parenrightbigg2\n+/parenleftBigρpθ\nr/parenrightBig2\n−/parenleftbiggρ/planckover2pi1\n2r/parenrightbigg2\n(21)\nwhere the canonical coordinates in polar representation takes th e following form,\npr=−i/parenleftbigg\n∂r+1\n2r/parenrightbigg\n, pθ=−i∂θ. (22)\nNow we note from Eqn.(21) that the invariant I(t) has the same form as that used in [27] to study the undamped\nharmonic oscillator in NC space. The time-dependent coefficients invo lved in the present study however differ due\nto the presence of external magnetic field and damping in our syste m. Thus, we can just borrow the expression of\neigenfunction and the phase factors from [27] for our present sy stem.6\n3.2 Eigenfunction and phase factor\nWe depict the set of eigenstates of the invariant operator I(t) as|n,l/angbracketright, following the convention in [27]. Here, n\nandlare integers such that n+l/greaterorequalslant0. So we have the condition l/greaterorequalslant−n. Thus, ifl=−n+m, thenmis a\npositive integer; and the corresponding eigenfunction in polar coor dinate system has the following form (restoring\n/planckover2pi1),\nφn,m−n(r,θ) =/angbracketleftr,θ|n,m−n/angbracketright (23)\n=λn(i√\n/planckover2pi1ρ)m\n√\nm!rn−mei(m−n)θ−a(t)−iρ˙ρ\n2a(t)/planckover2pi1ρ2r2\nU/parenleftbigg\n−m,1−m+n,r2\n/planckover2pi1ρ2/parenrightbigg\n(24)\nwhereλnis given by\nλ2\nn=1\nπn!(/planckover2pi1ρ2)1+n. (25)\nHere,U/parenleftbigg\n−m,1−m+n,r2\n/planckover2pi1ρ2/parenrightbigg\nis Tricomi’s confluent hypergeometric function [28, 29] and the eigen function\nφn,m−n(r,θ) satisfies the following orthonormality relation,\n/integraldisplay2π\n0dθ/integraldisplay∞\n0rdrφ∗\nn,m−n(r,θ)φn′,m′−n′(r,θ) =δnn′δmm′. (26)\nAgain following [27], the expression of the phase factor Θ( t) is given by,\nΘn,l(t) = (n+l)/integraldisplayt\n0/parenleftbigg\nc(T)−a(T)\nρ2(T)/parenrightbigg\ndT . (27)\nFor a given value of l=−n+m, it would be given by [27],\nΘn,m−n(t) =m/integraldisplayt\n0/parenleftbigg\nc(T)−a(T)\nρ2(T)/parenrightbigg\ndT . (28)\nWe shall use this expression to compute the phase explicitly as a func tion of time for various physical cases in the\nsubsequent discussion.\nThe eigenfunction of the Hamiltonian therefore reads (using Eqn(s ).(11, 24, 28))\nψn,m−n(r,θ,t) =eiΘn,m−n(t)φn,m−n(r,θ)\n=λn(i√\n/planckover2pi1ρ)m\n√\nm!exp/bracketleftbigg\nim/integraldisplayt\n0/parenleftbigg\nc(T)−a(T)\nρ2(T)/parenrightbigg\ndT/bracketrightbigg\n×rn−mei(m−n)θ−a(t)−iρ˙ρ\n2a(t)/planckover2pi1ρ2r2\nU/parenleftbigg\n−m,1−m+n,r2\n/planckover2pi1ρ2/parenrightbigg\n. (29)\n4 Solutions for the noncommutative damped oscillator\nin magnetic field\nIn this communication we are primarilyconcerned about the evolution of the solution due to the inclusion of a time\ndependent magnetic field in the system. For this purpose we want to find the eigenfunctions of the corresponding\nHamiltonian due to interplay of damping and magnetic field. The various kinds of damping in the presence of\nthe applied magnetic field are represented by various forms of the t ime dependent coefficients of the Hamiltonian,\nnamely,a(t),b(t) andc(t). However, the various forms must be constructed in such a way t hat they satisfy the7\nnon-linear EP equation given by Eqn.(18). The procedure of this con struction of exact analytical solutions is based\non the Chiellini integrability condition [30] and this formalism was followed in [27]. So, in this communication\nfor various forms of a(t) andb(t), we get the corresponding form of ρ(t) using the EP equation together with\nthe Chiellini integrability condition. In the subsequent discussion we s hall proceed to obtain solutions of the EP\nequation for the damped oscillator in a magnetic field considered in NC s pace.\n4.1 Solution Set-I for Ermakov-Pinney equation : Exponenti ally\ndecaying solutions\n4.1.1 The Solution Set\nThe simplest kind of solution set of the EP equation under damping is th e exponentially decaying set used in\n[27]. The solution set is given by the following relations,\na(t) =σe−ϑt, b(t) = ∆eϑt, ρ(t) =µe−ϑt/2(30)\nwhereσ,∆ andµareconstants.Here, ϑisanypositiverealnumber.Substitutingtheexpressionsfor a(t),b(t) andρ(t)\nin the EP equation, we can easily verify the relation between these co nstants to be as follows,\nµ4=4ξ2σ2\n4σ∆−ϑ2. (31)\n4.1.2 Study of the corresponding eigenfunctions\nWe now write down the eigenfunctions of the Hamiltonian for the chos en set of time-dependent coefficients. For\nthis purpose we need to choose explicit forms of the damping factor f(t) , angular frequency of the oscillator ω(t)\nand the applied magnetic field B(t). The eigenfunction of the invariant I(t) (which is given by Eqn.(24)) takes\non the following form for the solution Set-I:\nφn,m−n(r,θ) =λn(iµe−ϑt/2)m\n√\nm!rn−mei(m−n)θ−2σ+iµ2ϑ\n4σµ2e−ϑtr2\nU/parenleftbigg\n−m,1−m+n,r2eϑt\nµ2/parenrightbigg\n(32)\nwhereλnis given by\nλ2\nn=1\nπn![µ2exp(−ϑt)]1+n. (33)\nNext, we proceed to obtain explicit expressions of the phase facto rs for various forms of the damping factor f(t)\nand the angular frequency ω(t) of the oscillator. The value of the applied magnetic field B(t) is also tuned\naccordingly.\nAt first, we consider the most general form of damping factors an d the applied magnetic field which are given as\nfollows,\nf(t) =e−Γt;ω(t) =ω0e−δt/2;B(t) =B0eΛt; (34)\nwhere Γ and δare non-negative real constants and Λ is an arbitrary real const ant. Substituting these relations\nin Eqn(s).(7, 8), we get the most general form of the time depende nt NC parameters as,\nθ(t) =8Me−Γt\nq2B2\n0e2(Λ−Γ)t+4M2ω2\n0e−δt\n/radicalBigg\nq2B2\n0σe(2Λ−Γ−ϑ)t\n4M+ω2\n0e−δt/parenleftbig\nMσe(Γ−ϑ)t−1/parenrightbig\n−qB0e(Λ−Γ)t\n2M\n(35)\nΩ(t) =−qB0eΛt+2eΓt/radicalBig\nM∆e(ϑ−Γ)t−M2ω2\n0e−δt. (36)8\nIn order to get the exact analytical form of the phase factor we c hoose some suitable special forms of the constants\nϑ, Γ,δand Λ.\n/angbracketlefta/angbracketrightSet-I , Case I\nHere we set the constants\nϑ= Γ, δ= 0,Λ = 0. (37)\nSo, the parameters can be depicted by the following relations,\nf(t) =e−Γt;ω(t) =ω0;B(t) =B0. (38)\nTherefore, substituting Eqn.(38) in Eqn(s).(35,36), the reduced form of the NC parameters for this case are as\nfollows,\nθ(t) =8Me−Γt\nq2B2\n0e−2Γt+4M2ω2\n0/bracketleftBigg/radicalbigg\nq2B2\n0σe−2Γt\n4M+ω2\n0(Mσ−1)−qB0e−Γt\n2M/bracketrightBigg\nΩ(t) =−qB0+2eΓt/radicalBig\nM∆−M2ω2\n0. (39)\nIt can be checked that in the limit B→0, the expressions for θ(t) and Ω(t) reduce to those in [19]. Substituting\nthese relations in the expression for c(t) in Eqn.(9), we get,\nc(t) =1\nq2B2\n0e−2Γt+4M2ω2\n0/bracketleftBigg/parenleftbigg\n4M2ω2\n0+2qB0e−Γt/radicalBig\nM∆−M2ω2\n0/parenrightbigg/radicalbigg\nq2B2\n0σe−2Γt\n4M+ω2\n0(Mσ−1)\n−2qB0Mω2\n0e−Γt−q2B2\n0\nMe−2Γt/radicalBig\nM∆−M2ω2\n0/bracketrightbigg\n+/radicalbigg\n∆\nM−ω2\n0. (40)\nSubstituting the expressions of a(t),ρ(t) andc(t) in Eqn.(28),we can get an expression for the phase in a closed\nform as,\nΘn,l(t) = (n+l)/integraldisplayt\n0/bracketleftbigg\nc(t)−a\nρ2/bracketrightbigg\ndT= (n+l)/bracketleftBigg/radicalbigg\n∆\nM−ω2\n0−σ\nµ2/bracketrightBigg\nt\n+(n+l)ω0/radicalbig\n(Mσ−1)\nΓlogω0√\nMσ−1eΓt+/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)e2Γt\nω0√\nMσ−1+/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)\n+(n+l)ω0\nΓ\ntan−1 ω0eΓt\n/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)e2Γt−tan−1 ω0/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)\n\n+(n+l)/radicalbig\nM∆−M2ω2\n0\nMΓ\ntanh−12M/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)e2Γt\nqB0−tanh−12M/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)\nqB0\n\n−(n+l)/radicalbig\n(∆−Mω2\n0)σ\nΓ\ntanh−1/radicalBigg\nq2B2\n0σ+4Mω2\n0(Mσ−1)e2Γt\nq2B2\n0σ−tanh−1/radicalBigg\nq2B2\n0σ+4Mω2\n0(Mσ−1)\nq2B2\n0σ\n\n−(n+l)ω0\nΓ/bracketleftbigg\ntan−12Mω0eΓt\nqB0−tan−12Mω0\nqB0/bracketrightbigg\n+(n+l)/radicalbig\nM∆−M2ω2\n0\n2ΓMlogq2B2\n0e−2Γt+4M2ω2\n0\nq2B2\n0+4M2ω2\n0.(41)\n/angbracketleftb/angbracketrightSet-I , Case II9\nHere we set the constants\nϑ= Γ, δ= 0,Λ = Γ. (42)\nSo, the situation can be depicted by the following relations,\nf(t) =e−Γt;ω(t) =ω0;B(t) =B0eΓt. (43)\nTherefore, substituting Eqn.(42) in Eqn(s).(35,36), the reduced form of the NC parameters for this case are as\nfollows,\nθ(t) =8Me−Γt\nq2B2\n0+4M2ω2\n0/bracketleftBigg/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)−qB0\n2M/bracketrightBigg\n,Ω(t) =−qB0eΓt+2eΓt/radicalBig\nM∆−M2ω2\n0.(44)\nThe point that is to be noted is that the multiplication of the two time de pendent NC parameters obtained for\nthis case reduces to a constant value and later we observe that th e constant value is equal to the same found for\nanother case discussed in Eqn.(70). It can be checked that in the lim itB→0, the expressions for θ(t) and Ω(t)\nreduce to those obtained in [19]. Substituting these relations in the e xpression for c(t) in Eqn.(9), we get,\nc(t) =1\nq2B2\n0+4M2ω2\n0/bracketleftBigg/parenleftbigg\n4M2ω2\n0+2qB0/radicalBig\nM∆−M2ω2\n0/parenrightbigg/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)\n−2qB0Mω2\n0−q2B2\n0\nM/radicalBig\nM∆−M2ω2\n0/bracketrightbigg\n+/radicalbigg\n∆\nM−ω2\n0. (45)\nSubstituting the expressions of a(t),ρ(t) andc(t) in Eqn.(28),we can get an expression for the phase in a closed\nform in the following way.\nΘn,l(t) =(n+l)\nq2B2\n0+4M2ω2\n0/bracketleftBigg/parenleftbigg\n4M2ω2\n0+2qB0/radicalBig\nM∆−M2ω2\n0/parenrightbigg/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)\n−2qB0Mω2\n0−q2B2\n0\nM/radicalBig\nM∆−M2ω2\n0/bracketrightbigg\nt+(n+l)/bracketleftBigg/radicalbigg\n∆\nM−ω2\n0−σ\nµ2/bracketrightBigg\nt .(46)\nIn this case the phase is varying linearly with respect to time. In the lim itB0→0 , we can easily recover the\nsame form of the phase factor corresponding to the solution set I b obtained in [19].\n/angbracketleftc/angbracketrightSet-I , Case III\nHere we set the constants\nϑ= Γ, δ= 0,Λ =−Γ. (47)\nSo, the situation can be depicted by the following relations,\nf(t) =e−Γt, ω(t) =ω0, B(t) =B0e−Γt. (48)\nTherefore, substituting Eqn.(47) in Eqn(s).(35,36), the reduced form of the NC parameters for this case are as\nfollows,\nθ(t) =8Me−Γt\nq2B2\n0e−4Γt+4M2ω2\n0/bracketleftBigg/radicalbigg\nq2B2\n0σe−4Γt\n4M+ω2\n0(Mσ−1)−qB0e−2Γt\n2M/bracketrightBigg\nΩ(t) =−qB0e−Γt+2eΓt/radicalBig\nM∆−M2ω2\n0. (49)10\nSubstituting these relations in the expression for c(t) in Eqn.(9), we get,\nc(t) =1\nq2B2\n0e−4Γt+4M2ω2\n0/bracketleftBigg/parenleftbigg\n4M2ω2\n0+2qB0e−2Γt/radicalBig\nM∆−M2ω2\n0/parenrightbigg/radicalbigg\nq2B2\n0σe−4Γt\n4M+ω2\n0(Mσ−1)\n−2qB0Mω2\n0e−2Γt−q2B2\n0e−4Γt\nM/radicalBig\nM∆−M2ω2\n0/bracketrightbigg\n+/radicalbigg\n∆\nM−ω2\n0. (50)\nSubstituting the expressions of a(t),ρ(t) andc(t) in Eqn.(28),we can get an expression for the phase factor in\na closed form as ,\nΘn,l(t) = (n+l)/integraldisplayt\n0/bracketleftbigg\nc(t)−a\nρ2/bracketrightbigg\ndT= (n+l)/bracketleftBigg/radicalbigg\n∆\nM−ω2\n0−σ\nµ2/bracketrightBigg\nt\n+(n+l)ω0/radicalbig\n(Mσ−1)\n2Γlogω0√\nMσ−1e2Γt+/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)e4Γt\nω0√\nMσ−1+/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)\n+(n+l)ω0\n2Γ\ntan−1 ω0e2Γt\n/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)e4Γt−tan−1 ω0/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)\n\n+(n+l)/radicalbig\nM∆−M2ω2\n0\n2MΓ\ntanh−12M/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)e4Γt\nqB0−tanh−12M/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)\nqB0\n\n−(n+l)/radicalbig\n(∆−Mω2\n0)σ\n2Γ\ntanh−1/radicalBigg\nq2B2\n0σ+4Mω2\n0(Mσ−1)e4Γt\nq2B2\n0σ−tanh−1/radicalBigg\nq2B2\n0σ+4Mω2\n0(Mσ−1)\nq2B2\n0σ\n\n−(n+l)ω0\n2Γ/bracketleftbigg\ntan−12Mω0e2Γt\nqB0−tan−12Mω0\nqB0/bracketrightbigg\n+(n+l)/radicalbig\nM∆−M2ω2\n0\n4ΓMlogq2B2\n0e−4Γt+4M2ω2\n0\nq2B2\n0+4M2ω2\n0.\n(51)\n/angbracketleftd/angbracketrightSet-I , Case IV\nHere we set the constants\nϑ=δ= Λ = Γ. (52)\nSo, the situation can be depicted by the following relations,\nf(t) =e−Γt;ω(t) =ω0e−Γt/2;B(t) =B0eΓt. (53)\nSubstituting Eqn.(52) in Eqn(s).(35,36), the reduced form of the N C parameters for this case are as follows,\nθ(t) =8Me−Γt\nq2B2\n0+4M2ω2\n0e−Γt/bracketleftBigg/radicalbigg\nq2B2\n0σ\n4M+ω2\n0e−Γt(Mσ−1)−qB0\n2M/bracketrightBigg\n,Ω(t) =−qB0eΓt+2eΓt/radicalBig\nM∆−M2ω2\n0e−Γt.\n(54)\nSubstituting these relations in the expression for c(t) in Eqn.(9), we get,\nc(t) =1\nq2B2\n0+4M2ω2\n0e−Γt/bracketleftBigg/parenleftbigg\n4M2ω2\n0e−Γt+2qB0/radicalBig\nM∆−M2ω2\n0e−Γt/parenrightbigg/radicalbigg\nq2B2\n0σ\n4M+ω2\n0e−Γt(Mσ−1)\n−2qB0Mω2\n0e−Γt−q2B2\n0\nM/radicalBig\nM∆−M2ω2\n0e−Γt/bracketrightbigg\n+/radicalbigg\n∆\nM−ω2\n0e−Γt. (55)11\nWe are able to obtain the exact form of the phase factor and it has b een shown in the Appendix.\n4.2 Solution Set-II for Ermakov-Pinney equation: Rational ly decaying solutions\n4.2.1 The Solution Set\nWe now consider rationally decaying solutions of the EP equation similar to that used in [27] which is of the form,\na(t) =σ/parenleftbigg\n1+2\nk/parenrightbigg(k+2)/k\n(Γt+χ)(k+2)/k, b(t) =∆/parenleftbiggk\nk+2/parenrightbigg(2−k)/k\n(Γt+χ)(k−2)/k, ρ(t) =µ/parenleftbigg\n1+2\nk/parenrightbigg1/k\n(Γt+χ)1/k; (56)\nwhereσ, ∆,µ, Γ andχare constants such that (Γ t+χ)/negationslash= 0, andkis an integer. Substituting the expressions\nofa(t),b(t), andρ(t) in the EP equation, we can easily verify the relation between these c onstants to be as\nfollows,\nΓ2µ= (k+2)2(σ∆µ−ξ2σ2\nµ3). (57)\n4.2.2 Study of the corresponding eigenfunctions\nThe eigenfunction of the invariant operator I(t) [given by Eqn.(24)] for this solution Set-II is of the following\nform,\nφn,m−n(r,θ) =λn(iµ)m\n√\nm!/bracketleftbiggk+2\nk(Γt+χ)/bracketrightbiggm/k\nrn−mei(m−n)θ−[σ(k+2) +iµ2Γ] (Γt+χ)2/kk2/k\n2σ(k+2)(k+2)/kµ2r2\n×U/parenleftBigg\n−m,1−m+n,r2[k(Γt+χ)]2/k\nµ2(k+2)2/k/parenrightBigg\n(58)\nwhereλnis given by\nλ2\nn=1\nπn!µ2n+2/bracketleftbiggk(Γt+χ)\nk+2/bracketrightbigg2(1+n)/k\n. (59)\nIn order to get the eigenfunction of the Hamiltonian H(t), we need to calculate the associated phase factor. Once\nagain for this we need to fix up the forms of the damping factor f(t), angular frequency ω(t) of the oscillator and\nthe applied magnetic field B(t). In order to explore the solution of H(t) for rationally decaying coefficients, we\nchoose a rationally decaying form for ω(t),B(t) and setf(t) = 1. Thus, we have the following relations,\nη(t) = 0⇒f(t) = 1, ω(t) =ω0\n(Γt+χ), B(t) =B0\n(Γt+χ). (60)\n/angbracketlefte/angbracketrightSet-II , Case I\nAs we want to study how the nature of the rationally decaying solutio n gets altered when the system is placed in\na magnetic field, we set k= 2 in Eqn.(56). The system has been studied without applying any ext ernal field for\nthis particular kparameter in an earlier communication [19].\nWhen we set k= 2,the set a(t),b(t) andρ(t) takes the following simplified form,\na(t) =4σ\n(Γt+χ)2, b(t) = ∆, ρ(t) =/bracketleftbigg2µ2\nΓt+χ/bracketrightbigg1/2\n. (61)12\nSubstituting the expressions for a(t),b(t),ω(t) ,f(t) andB(t) in the Eqn(s).(7, 8), we get the time dependent\nNC parameters as,\nθ(t) =8M\nq2B2\n0+4M2ω2\n0/bracketleftBigg/radicalbigg\nq2B2\n0σ\nM+ω2\n04σM−ω2\n0(Γt+χ)2−qB0\n2M(Γt+χ)/bracketrightBigg\nΩ(t) =−qB0\n(Γt+χ)+2/radicalBigg\nM∆−M2ω2\n0\n(Γt+χ)2. (62)\nSubstituting these relations in the expression for c(t) in Eqn.(9) gives,\nc(t) =1\n4M2ω2\n0+q2B2\n0/bracketleftBigg/parenleftBigg\n4M2ω2\n0\n(Γt+χ)2+2qB0/radicalbig\nM∆(Γt+χ)2−M2ω2\n0\n(Γt+χ)2/parenrightBigg/radicalbigg\nq2B2\n0σ\nM−ω2\n0(Γt+χ)2+4ω2\n0σM\n−2qB0Mω2\n0\nΓt+χ−q2B2\n0/radicalbig\nM∆(Γt+χ)2−M2ω2\n0\nM(Γt+χ)/bracketrightBigg\n+/radicalBigg\n∆\nM−ω2\n0\n(Γt+χ)2. (63)\nThe additional terms that appear in the expression due to the pres ence of the magnetic field are mostly seen to be\ndecaying functions of time. Their contribution becomes more eviden t when we study the evolution of expectation\nvalue of energy with time in a later section. We are able to obtain the ex act form of the phase factor and it has\nbeen shown in the Appendix.\n/angbracketleftf/angbracketrightSet-II , Case II\nIt is observed from the solution set given by Eqn.(56), that the time dependent parameters a(t) andρ(t) vanish,\nwhileb(t) diverges if we set k=−2. In order to avoid this and study the system at this critical value o fk, we\nchoose the corresponding solution set to be,\na(t) =σ\n(Γt+χ)(k+2)/k, b(t) =∆\n(Γt+χ)(k−2)/k, ρ(t) =µ\n(Γt+χ)1/k. (64)\nWe now set k=−2 in Eqn. (64) to obtain the following set,\na=σ , b(t) =∆\n(Γt+χ)2, ρ=µ/radicalbig\nΓt+χ. (65)\nSubstituting these relations in the EP equation the constraint cond ition is found to be,\n−µ4Γ2+4σ∆µ4= 4ξ2σ2. (66)\nThe above relation matches with that found in Eqn.(31) while consider ingϑ= Γ.\nAs the Eqn.(58) vanish at k=−2, the eigenfunction of the invariant for this solution is needed to be calculated\nseparately and it is of the following form,\nφn,m−n(r,θ) =λn(iµ/radicalbig\n(Γt+χ)m\n√\nm!rn−mei(m−n)θ−2σ−iµ2Γ\n4σµ2(Γt+χ)r2\nU/parenleftbigg\n−m,1−m+n,r2\nµ2(Γt+χ)/parenrightbigg\n(67)\nwhereλnis given by\nλ2\nn=1\nπn![µ2(Γt+χ)]1+n. (68)\nAgain we consider the same explicit forms of the damping factor f(t), angular frequency ω(t) and applied\nmagnetic field B(t) as in Eqn.(60),\nf(t) = 1, ω(t) =ω0\n(Γt+χ), B(t) =B0\n(Γt+χ). (69)13\nSubstituting the expressions for a(t),b(t),ω(t) ,f(t) andB(t) in the Eqn(s).(7, 8), we get the time dependent\nNC parameters as,\nθ(t) =8M(Γt+χ)\nq2B2\n0+4M2ω2\n0/bracketleftBigg/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)−qB0\n2M/bracketrightBigg\n,Ω(t) = 2/radicalbig\nM∆−M2ω2\n0\n(Γt+χ)−qB0\n(Γt+χ).(70)\nOnce again it is interesting to note that a constant value is found aft er multiplication of the two time dependent\nNC parameters obtained above. Here we recall Eqn.(44) where we d iscussed that the multiplication of two time\ndependent NC parameters reduces to a constant value which is equ al to the same obtained for this case.\nSubstituting these relations in the expression for c(t) in Eqn.(9) gives,\nc(t) =1\n(4M2ω2\n0+q2B2\n0)(Γt+χ)/bracketleftBigg/parenleftbigg\n4M2ω2\n0+2qB0/radicalBig\nM∆−M2ω2\n0/parenrightbigg/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)\n−2qB0Mω2\n0−q2B2\n0/radicalbig\nM∆−M2ω2\n0\nM/bracketrightBigg\n+1\n(Γt+χ)/radicalbigg\n∆\nM−ω2\n0. (71)\nSubstituting these expressions for a(t),ρ(t) andc(t) in Eqn.(28), we get the following expression for the phase\nfactor in a closed form as,\nΘn,l(t) =(n+l)\n(4M2ω2\n0+q2B2\n0)Γ/bracketleftBigg/parenleftbigg\n4M2ω2\n0+2qB0/radicalBig\nM∆−M2ω2\n0/parenrightbigg/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)\n−2qB0Mω2\n0−q2B2\n0/radicalbig\nM∆−M2ω2\n0\nM/bracketrightBigg\nln(Γt+χ)\nχ+(n+l)\nΓ/bracketleftBigg/radicalbigg\n∆\nM−ω2\n0−σ\nµ2/bracketrightBigg\nln(Γt+χ)\nχ.\n(72)\n5 Analysis of the expectation value of energy\nIn this section, we intend to calculate the expectation value of ener gy. It is shown in [19] that the expectation\nvalue of energy /angbracketleftEn,m−n(t)/angbracketrightwith respect to energy eigenstate ψn,m−n(r,θ,t) can be expressed as,\n/angbracketleftEn,m−n(t)/angbracketright=1\n2(n+m+1)/bracketleftbigg\nb(t)ρ2(t)+a(t)\nρ2(t)+˙ρ2(t)\na(t)/bracketrightbigg\n+c(t)(n−m). (73)\nSubstituting the expression of c(t) in the above equation, the expectation value of energy for our mo del takes the\nfollowing form,\n/angbracketleftEn,m−n(t)/angbracketright=1\n2(n+m+1)/bracketleftbigg\nb(t)ρ2(t)+a(t)\nρ2(t)+˙ρ2(t)\na(t)/bracketrightbigg\n+(n−m)\n2/bracketleftbiggqB(t)f(t)\nM/parenleftbigg\n1+θ(t)Ω(t)\n4/parenrightbigg\n+Ω(t)f(t)\nM+/parenleftbiggq2B2(t)f(t)\n4M+Mω2(t)\nf(t)/parenrightbigg\nθ(t)/bracketrightbigg\n; (74)\nwhich reduces to the same obtained in [19] in the limit B→0.\nThe energy expression depends on the charge explicitly. It contain s both terms having linear and quadratic depen-\ndence on charge. So the energy does not remain invariant when the charge of the particle changes its sign. Another\nnotable point is that even when the frequency of oscillation ω→0, and the applied field B→0; the expectation\nvalue of energy is non-zero. This is because all the three paramete rs of the Hamiltonian a(t),b(t) andc(t) are\nfinite even as ω→0,B→0, as is clear from the Eqn(s).(7,8,9). Now we will proceed to study th e time-dependent\nbehaviour of /angbracketleftEn,m−n(t)/angbracketrightfor various types of damping and applied magnetic field.14\n5.1 Solution Set-I: Exponentially decaying solution\nFor the exponentially decaying solution given by Eqn.(30), the energ y expectation value takes the following form,\n/angbracketleftEn,m−n(t)/angbracketright= (n+m+1)µ2∆+c(t)(n−m) (75)\nwhere we have set the constant ξ2to unity and used the constraint relation given by Eqn.(31).\n/angbracketleftA/angbracketrightSet-I ,Case I\nHere we set f(t) =e−Γt,ω(t) =ω0andB(t) =B0. With this the energy expression for the ground state takes\nthe form,\n/angbracketleftEn,−n(t)/angbracketright= (n+1)µ2∆+n\nq2B2\n0e−2Γt+4M2ω2\n0/bracketleftbigg\n−2qB0Mω2\n0e−Γt−q2B2\n0\nMe−2Γt/radicalBig\nM∆−M2ω2\n0\n+/parenleftbigg\n4M2ω2\n0+2qB0e−Γt/radicalBig\nM∆−M2ω2\n0/parenrightbigg/radicalbigg\nq2B2\n0σe−2Γt\n4M+ω2\n0(Mσ−1)/bracketrightBigg\n+n/radicalbigg\n∆\nM−ω2\n0.(76)\nThe nature of this energy expectation value depends on the value o f the constants. Specifically, the sign of the\ncharge plays a crucial role for determining the nature of this expec tation value. It is noteworthy that in both the\nlimitst→ ∞andB→0 this energy expression gains the same constant value which is alrea dy found in [19]\nfor a damped oscillator in time dependent NC space. Apart from it, an inclusion of constant magnetic field in the\nsystem considered in [19] also makes the Hamiltonian non-hermitian af ter a certain limit of time beyond which\nthe energy becomes imaginary. The condition for getting the expec tation value of energy to be real is as follows,\nt≤1\n2Γlnq2B2\n0σ\n4Mω2\n0(1−Mσ). (77)\nIt is interesting to note that the upper bound of time below which the energy expectation value remains real does\nnot depend on the sign of charge of the oscillator, although the valu e of energy itself does. The nature of variation\nof energy expectation value with time is shown in Fig.1. Though the en ergy shows an initial decrease, eventually\nit tends to be a constant over time.\n/angbracketleftB/angbracketrightSet-I ,Case II\nHere we set f(t) =e−Γt,ω(t) =ω0andB(t) =B0eΓt. With this the energy expression for the ground state\ntakes the form,\n/angbracketleftEn,−n(t)/angbracketright= (n+1)µ2∆+n\nq2B2\n0+4M2ω2\n0/bracketleftBigg/parenleftbigg\n4M2ω2\n0+2qB0/radicalBig\nM∆−M2ω2\n0/parenrightbigg/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)\n−2qB0Mω2\n0−q2B2\n0\nM/radicalBig\nM∆−M2ω2\n0/bracketrightbigg\n+n/radicalbigg\n∆\nM−ω2\n0=constant . (78)\nAs expected we observe from Fig. 1, the energy is a constant over time. In the limit B→0, the constant value\nof energy reduces to the same obtained in [19] for a damped oscillato r in time dependent NC space.\n/angbracketleftC/angbracketrightSet-I ,Case III\nHere we set f(t) =e−Γt,ω(t) =ω0andB(t) =B0e−Γt. With this the energy expression for the ground state\ntakes the form,\n/angbracketleftEn,−n(t)/angbracketright= (n+1)µ2∆+n\nq2B2\n0e−4Γt+4M2ω2\n0/bracketleftbigg\n−2qB0Mω2\n0e−2Γt−q2B2\n0e−4Γt\nM/radicalBig\nM∆−M2ω2\n0\n+/parenleftbigg\n4M2ω2\n0+2qB0e−2Γt/radicalBig\nM∆−M2ω2\n0/parenrightbigg/radicalbigg\nq2B2\n0σe−4Γt\n4M+ω2\n0(Mσ−1)/bracketrightBigg\n+n/radicalbigg\n∆\nM−ω2\n0.(79)15\nThe nature of variation of the energy expectation value with time ha s almost the same characteristics as obtained\nin Case I. This is also observed graphically from Fig. 1. However, a clos er observation tells us that in Case III\nthe rate of decay is faster than in Case I for the same set of param eters. It must be because in Case III, unlike\nin Case I, the applied field is decaying as well with respect to time. The u pper bound of time beyond which the\nsystem becomes non-physical (since the energy ceases to be rea l after this time) is also half of that obtained in\nEqn.(77). Here the bound is found to be\nt≤1\n4Γlnq2B2\n0σ\n4Mω2\n0(1−Mσ). (80)\nAn important inference that can be drawn from the above study is t hat, in the presence of the external magnetic\nfield, the energy of a damped oscillatordecays with time if the field eith er decays with time or is atleast a constant.\nIf the field tends to grow as fast as the damping factor decays, th en the energy of the oscillator tends to be a\nconstant with time. In Fig. 1 we have also plotted the evolution of ene rgy with time in this situation if the applied\nfield is turned off. We see that if B=0, then the energy is a constant o ver time. This is expected from the energy\nexpectation value expressions given by Eqns. [76, 78 and 79]. Even a constant magnetic field is able to bring\nabout time variation in this energy value.\n/angbracketleftD/angbracketrightSet-I, Case IV\nHere we set f(t) =e−Γt,ω(t) =ω0e−Γt/2andB(t) =B0eΓt. With this the energy expression for the ground\nstate takes the form,\n/angbracketleftEn,−n(t)/angbracketright= (n+1)µ2∆+n\nq2B2\n0+4M2ω2\n0e−Γt/bracketleftbigg\n−2qB0Mω2\n0e−Γt−q2B2\n0\nM/radicalBig\nM∆−M2ω2\n0e−Γt\n+/parenleftbigg\n4M2ω2\n0e−Γt+2qB0/radicalBig\nM∆−M2ω2\n0e−Γt/parenrightbigg/radicalbigg\nq2B2\n0σ\n4M+ω2\n0e−Γt(Mσ−1)/bracketrightBigg\n+n/radicalbigg\n∆\nM−ω2\n0e−Γt.\n(81)\nIt can be verified that the above energy expectation value reduce s to a decaying form as obtained in [19] in the\nabsence of magnetic field. However, it is interesting to note that fo r certain choice of parameters (as shown in Fig.\n1) the energy may also exhibit an initial growth with time. Infact for t he given set of parameters in Fig. 1, it\ninitially decays and then increases. Eventually, as t→ ∞, the energy tends to be a constant. In this situation,\nif we turn off the applied magnetic field, then we see from Fig 1, the ene rgy decays off with time. However, due\nto the absence of the exponentially growing field, the energy of the oscillator is seen to be much lower. Also\nthe exponential growth in energy seen in the presence of the field is absent within the time range studied. The\nsystem also possess two lower bounds of time below which it becomes n on-physical due to the imaginary value of\nenergy. The conditions are as follows,\nt≥1\nΓln/bracketleftbiggMω2\n0\n∆/bracketrightbigg\n;t≥1\nΓln/bracketleftbigg4Mω2\n0(1−Mσ)\nq2B2\n0σ/bracketrightbigg\n. (82)\nThe greater of the two bounds serves as the actual lower bound.\n5.2 Solution Set-II: Rationally decaying solution\nIn previous section we considered two different solution set genera ted from Eqn.(56). The special form of the\nsolution shown in Eqn.(61) is directly produced by substituting k= 2 in the Eqn.(56) and the special form of the\nsolution shown in Eqn.(65) is obtained by substituting k=−2 in the modified form of Eqn.(56).16\n0 1 2 32000021000220002300024000 〈Ε〉/ω 0\nCase I\nCase II\nCase III\nCase IV\nCase I, B=0\nCase IV, B=0\nΓt\nFigure 1: A study of the variation of expectation value of energy, scal ed by1\nω0(/angbracketleftE/angbracketright\nω0) in order to make it dimen-\nsionless, as we vary Γt (again a dimensionless quantity). Here we consider mass M= 1, charge q=1, magnetic\nfieldB0=102,µ=1,∆=107,σ=107,ω0=103andΓ=1 in natural units. The constants n=1 and m=0. The\nexpectation value of energy /angbracketleftE/angbracketrightis calculated for exponentially decaying solution set when Case I:f(t) =e−Γt\n,ω(t) =ω0andB(t) =B0; Case II: f(t) =e−Γt,ω(t) =ω0andB(t) =B0eΓt; Case III: f(t) =e−Γt,\nω(t) =ω0andB(t) =B0e−Γtand Case IV: f(t) =e−Γt,ω(t) =ω0e−Γt/2andB(t) =B0eΓt. While for\nCase I and Case III the energy first decreases, then becomes co nstant with time, for Case II the energy remains\nconstant as we vary time. For Case IV the behaviour of energy w ith time is seen to be extremely interesting. It\nfirst decreases and then increases with time. Along with thes e, we have also plotted what happens in the absence of\nmagnetic field for comparison. When the angular frequency of oscillation is a constant (Case I, Case II and Case\nIII), if the magnetic field is set to zero, then the energy of th e oscillator is a constant with time. So, the magnetic\nfield, even when it is a constant brings about time variation i n energy for an exponentially damped oscillator having\na constant frequency. However, when the angular frequency i s decaying exponentially too (Case IV), then even\nwhen B=0, the energy decays with time. Nevertheless, the var iation of energy is remarkably different from that\nseen in the presence of the field for Case IV.\n5.2.1 Set-II, Case I\nFor the rationally decaying solution given by Eqn.(61), the energy ex pectation value takes the following form\n/angbracketleftEn,m−n(t)/angbracketright=(n+m+1)\n2(Γt+χ)/bracketleftbigg\n2/parenleftbiggσ\nµ2+∆µ2/parenrightbigg\n+µ2Γ2\n8σ/bracketrightbigg\n+(n−m)c(t). (83)\nwhere we have set the constant ξ2to unity and used the constraint relation given by Eqn.(57). Here we set\nf(t) = 1 ,ω(t) =ω0/(Γt+χ) andB(t) =B0/(Γt+χ). With this the energy expression for the ground state\ntakes the form,\n/angbracketleftEn,−n(t)/angbracketright=(n+1)\n2(Γt+χ)/bracketleftbigg\n2/parenleftbiggσ\nµ2+∆µ2/parenrightbigg\n+µ2Γ2\n8σ/bracketrightbigg\n+n\n4M2ω2\n0+q2B2\n0/bracketleftBigg\n−2qB0Mω2\n0\nΓt+χ−q2B2\n0/radicalbig\nM∆(Γt+χ)2−M2ω2\n0\nM(Γt+χ)\n+/parenleftBigg\n4M2ω2\n0\n(Γt+χ)2+2qB0/radicalbig\nM∆(Γt+χ)2−M2ω2\n0\n(Γt+χ)2/parenrightBigg/radicalbigg\nq2B2\n0σ\nM−ω2\n0(Γt+χ)2+4ω2\n0σM/bracketrightBigg\n+n/radicalBigg\n∆\nM−ω2\n0\n(Γt+χ)2;\n(84)17\n0 1 2 301000020000300004000050000〈Ε〉/ω0Case I\nCase I, B=0\nCase II\nCase II, B=0\nΓt\nFigure 2: A study of the variation of expectation value of energy, scal ed by1\nω0(/angbracketleftE/angbracketright\nω0) in order to make it di-\nmensionless, as we vary Γt (again a dimensionless quantity). Here we consider mass M= 1, charge q=1, mag-\nnetic fieldB0=1020,µ=1,∆=107,σ=107,ω0=103andΓ=1 in natural units. The constants n=1 and m=0.\nThe expectation value of energy /angbracketleftE/angbracketrightis calculated for rationally decaying solution set when /angbracketleftA/angbracketrightCase I:a(t) =\n4σ\n(Γt+χ)2, b(t) = ∆, ρ(t) =/bracketleftbigg2µ2\nΓt+χ/bracketrightbigg1/2\nand/angbracketleftB/angbracketrightCase II:a=σ , b(t) =∆\n(Γt+χ)2, ρ=µ√Γt+χ. For\nboth Case I and Case II the energy decays with time. However, t he rate of decay is higher for Case I as compared\nto Case II. This is expected since two of the Hamiltonian para metersa(t)andρ(t)are decaying with time for\nCase I, whereas only one parameter b(t)is decaying while another one ρ(t)is increasing with time for Case II.\nFor comparison, we have also plotted Case I and II, in the abse nce of the magnetic field in the same plot. It is\nnoteworthy, that although the energy of the oscillator deca ys even in the absence of the field, when we apply the\nfield the energy is enhanced. This is due to the presence of mag netic energy in the system in this situation.\nwhich is a decaying function of time and is seen to reduce to the same o btained in [19] in the limit B→0. The\ntime range beyond which the system becomes non-physical due to im aginaryenergy expectation value is as follows,\n1\nΓ/parenleftBigg\nω0/radicalbigg\nM\n∆−χ/parenrightBigg\n≤t≤1\nΓ/bracketleftBigg/radicalBigg\nq2B2\n0σ\nMω2\n0+4σM−χ/bracketrightBigg\n. (85)\nIn Fig. 2 we have done a comparative study of the time variation of en ergy in this situation, when the applied\nmagnetic field is present and when it is turned off. We see from the Figu re that in both these cases the energy\ndecays with time. However the energy of the oscillator is higher in the presence of the field due to the presence of\nthe magnetic energy in the system.\n5.2.2 Set-II, Case II\nFor the rationally decaying solution given by Eqn.(65), the energy ex pectation value takes the following form,\n/angbracketleftEn,m−n(t)/angbracketright=(m+n+1)µ2∆\nΓt+χ+c(t)(n−m). (86)\nwhere we have set the constant ξ2to unity and used the constraint relation given by Eqn.(66). Here we set\nf(t) = 1 ,ω(t) =ω0/(Γt+χ) andB(t) =B0/(Γt+χ). With this the energy expression for the ground state\ntakes the form,18\n/angbracketleftEn,−n(t)/angbracketright=(n+1)µ2∆\nΓt+χ+n\n(4M2ω2\n0+q2B2\n0)(Γt+χ)/bracketleftBigg\n−2qB0Mω2\n0−q2B2\n0/radicalbig\nM∆−M2ω2\n0\nM\n+/parenleftbigg\n4M2ω2\n0+2qB0/radicalBig\nM∆−M2ω2\n0/parenrightbigg/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)/bracketrightBigg\n+n\n(Γt+χ)/radicalbigg\n∆\nM−ω2\n0.(87)\nAs we see from Fig. 2 the corresponding energy expectation value d ecays off with time. Infact, it approaches zero\nat large time. However, the rate of decay is lower than Case I. This is expected since in Case II, unlike in Case\nI, while one of the Hamiltonian parameters b(t) is decreasing with time, another parameter ρ(t) is growing with\ntime. The resultant energy is decreasing with time since the rate at w hichb(t) is decaying (rate ∼t−2) is higher\nthan the rate at which ρ(t) is growing (rate ∼t1/2). When we turn off the field in this situation, we see from\nFig. 2, the energy still decays off due to the decaying angular frequ ency of the oscillator. But in this situation the\nenergy of the oscillator is reduced due to the absence of magnetic e nergy in the system.\n6 Conclusion\nIn conclusion, our primary objective through this study has been t o investigate the effect that an external time-\ndependent magnetic field has on a two dimensional damped harmonic o scillator in noncommutative space. The\nbehaviour of the system under the influence of this time varying field is seen to be dependent on the nature of the\nfield. For this purpose we have first set up the Hamiltonian for our sy stem in the presence of a general magnetic\nfield in noncommutative space. Then we map this Hamiltonian in terms of commutative variables by using a shift\nof variables connecting the noncommutative and commutative spac e, known in the literature as Bopp-shift. We\nhave then obtained the exact solution of this time dependent syste m in the presence of an applied time varying\nmagnetic field by using the well known Lewis invariant which in turn leads to a non-linear differential equation\nknown as the Ermakov-Pinney equation. Then we make various choic es of the parameters of the system and\nstudy the solutions depending on these choices as we tune the applie d magnetic field. In this study we have\nprimarily considered two different sets of parameters for our damp ed system, namely, exponentially decaying\nsolutions and rationally decaying solutions. Interestingly, the solut ions obtained make it possible to integrate the\nexpression of the phase factor exactly thereby giving an exact so lution for the eigenstates of the Hamiltonian.\nThen we compute the expectation value of the Hamiltonian. Expecte dly, the expectation value of the energy\nvaries with time. For the exponentially decaying system, the nature of the time dependent magnetic field crucially\ndetermines the nature of evolution of the energy with time. There is basically an interplay between the damping\nfactor, applied time varying magnetic field and time dependent angula r frequency of the harmonic oscillator in\ndetermining the time evolution. While an exponentially growing magnetic field is able to maintain the energy to\na constant value inspite of damping, a constant or an exponentially d ecaying field makes the energy fall off faster\nwith damping. Remarkably, the presence of an exponentially decayin g frequency along with the damping factor\nmakes the behaviour of the system under the influence of an expon entially growing field even more interesting.\nWhile initially the energy decays off with time due to the damping present in the system, later the energy starts\ngrowing under the influence of the growing field. For the rationally de caying situation, even when damping is not\npresent, a rationally decaying magnetic field in combination with a ratio nally decaying angular frequency is able\nto eventually damp out the energy of the oscillator. While the decayin g oscillation corresponding to the Case I of\nrational EP solution cannot remain physical at the zero value of ene rgy due to the existance of upper bound of\ntime, the same corresponding to the Case II of rational EP solution is physically damped out to be zero with time.\nHowever, we observe that at a given instant of time, the expectat ion value of energy is greater in the presence of19\nthe magnetic field than when the field is turned off. This is because mag netic energy is absent in the system in\nthe latter case.\nAppendix: Explicit forms of some phases\nWe discussed about the eigenfunctions corresponding to the expo nentially decaying EP solution set in section 4 .1.\nWe mentioned that the exact form of the phase factor to constru ct the eigenfunction of a damped oscillator having\nexponentially decaying frequency in the presence of a magnetic field increasing exponentially with respect to time\nin NC space can be found [Set I, Case IV]. The phase factor is as follow s,\nΘn,l(t) =2(n+l)\nΓ/bracketleftBigg/radicalbigg\n∆\nM−ω2\n0−/radicalbigg\n∆\nM−ω2\n0e−Γt/bracketrightBigg\n+2(n+l)\nΓ/braceleftBigg/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)−/radicalbigg\nq2B2\n0σ\n4M+ω2\n0(Mσ−1)e−Γt/bracerightBigg\n+(n+l)/bracketleftbigg\n−σ\nµ2/bracketrightbigg\nt+i(n+l)\nMΓ/radicalBig\nq2B2\n0+4M∆/bracketleftBigg\ntan−1/radicalbig\nq2B2\n0+4M∆\n2/radicalbig\nM2ω2\n0−M∆−tan−1/radicalbig\nq2B2\n0+4M∆\n2/radicalbig\nM2ω2\n0e−Γt−M∆/bracketrightBigg\n+qB0(n+l)\n2MΓlog(4M2ω2\n0+q2B2\n0)\n\n/parenleftBigg/radicalbigg\nω2\n0(Mσ−1)+q2B2\n0σ\n4MeΓt+qB0σ\n2eΓt/2/parenrightBigg2\n+ω2\n0(Mσ−1)2\n\n\n(4M2ω2\n0+q2B2\n0eΓt)\n\n/parenleftBigg/radicalbigg\nω2\n0(Mσ−1)+q2B2\n0σ\n4M+qB0σ\n2/parenrightBigg2\n+ω2\n0(Mσ−1)2\n\n\n+(n+l)/radicalbig\nq2B2\n0+4M∆\n2MΓlog\n(4M2ω2\n0+q2B2\n0)\n\nq2B2\n0σM2ω2\n0+4M3σ∆ω2\n0−2Mω2\n0q2B2\n0−4ω2\n0M2∆+eΓt\nq4B4\n0σ\n4+q2B2\n0∆\n\n+eΓtq2B2\n0σM∆−2qB0/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt(q2B2\n0+4M∆)\nω2\n0[Mσ−1]+q2B2\n0σ\n4MeΓt\n(M∆eΓt−M2ω2\n0)\n\n\n\n\n(4M2ω2\n0+q2B2\n0eΓt)\n\nq2B2\n0σM2ω2\n0+4M3σ∆ω2\n0−2Mω2\n0q2B2\n0−4ω2\n0M2∆+q4B4\n0σ\n4+q2B2\n0∆\n+q2B2\n0σM∆−2qB0/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt(q2B2\n0+4M∆)\nω2\n0[Mσ−1]+q2B2\n0σ\n4M\n(M∆−M2ω2\n0)\n\n\n\n+(n+l)√\nσ∆\nΓlog\nω2\n0M2σ∆−M∆ω2\n0−q2B2\n0σMω2\n0\n4+q2B2\n0σ∆\n2eΓt\n+/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtq2B2\n0σ∆(M∆eΓt−M2ω2\n0)\nω2\n0[Mσ−1]+q2B2\n0σ\n4MeΓt\n\n\n\nω2\n0M2σ∆−M∆ω2\n0−q2B2\n0σMω2\n0\n4+q2B2\n0σ∆\n2\n+/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtq2B2\n0σ∆(M∆−M2ω2\n0)\nω2\n0[Mσ−1]+q2B2\n0σ\n4M\n\n+(n+l)qB0\n2MΓlogq2B2\n0+4M2ω2\n0e−Γt\nq2B2\n0+4M2ω2\n0\n+(n+l)iqB0√\nMσ−1\n2MΓ×\n\nlogMσ∆−∆−q2B2\n0σ\n4−2Mω2\n0(Mσ−1)e−Γt−2i/radicalBigg\n(Mσ−1)/parenleftbigg\nω2\n0[Mσ−1]e−Γt+q2B2\n0σ\n4M/parenrightbigg\n(M∆−M2ω2\n0e−Γt)\nMσ∆−∆−q2B2\n0σ\n4−2Mω2\n0(Mσ−1)−2i/radicalBigg\n(Mσ−1)/parenleftbigg\nω2\n0[Mσ−1]+q2B2\n0σ\n4M/parenrightbigg\n(M∆−M2ω2\n0)\n\n(88)\nWe also discussed about the eigenfunctions corresponding to the r ationally decaying EP solution set in section\n4.2. We mentioned that the exact form of the phase factor to const ruct the eigenfunction of a damped oscillator20\nhaving rationally decaying frequency in the presence of a magnetic fi eld decaying rationally with respect to time\nin NC space [Set II, Case I] can be found. The phase factor is as follo ws,\nΘn,l(t) =−2(n+l)\nΓ/parenleftbiggσ\nµ2+qB0Mω2\n0\n4M2ω2\n0+q2B2\n0/parenrightbigg\nlogΓt+χ\nχ+4(n+l)M2ω2\n0\n(q2B2\n0+4M2ω2\n0)Γ/bracketleftBigg/radicalBigg/parenleftbiggq2B2\n0σ\nM+4ω2\n0σM/parenrightbigg1\nχ2−ω2\n0\n−/radicalBigg/parenleftbiggq2B2\n0σ\nM+4ω2\n0σM/parenrightbigg1\n(Γt+χ)2−ω2\n0+ω0/braceleftBigg\ntan−1√\nM ω0χ/radicalbig\n(q2B2\n0σ+4ω2\n0σM2)−ω2\n0χ2M\n−tan−1√\nM ω0(Γt+χ)/radicalbig\n(q2B2\n0σ+4ω2\n0σM2)−ω2\n0M(Γt+χ)2/bracerightBigg/bracketrightBigg\n+2(n+l)qB0\nq2B2\n0+4M2ω2\n0/bracketleftBigg\n1\nΓ/braceleftBigg/radicalBigg/parenleftbigg\n∆−Mω2\n0\nχ2/parenrightbigg/parenleftbigg4M2ω2\n0σ+\nq2B2\n0σ−Mω2\n0χ2/parenrightbigg\n−/radicalBigg/parenleftbigg\nM∆−M2ω2\n0\n(Γt+χ)2/parenrightbigg/parenleftbiggq2B2\n0σ\nM+4ω2\n0σM−ω2\n0(Γt+χ)2/parenrightbigg/bracerightBigg\n−2iMω2\n0\nΓ/braceleftBigg\nEllipticE/parenleftBigg\nisinh−1/bracketleftBigg\niω0√\nM(Γt+χ)/radicalbig\nq2B2\n0σ+4ω2\n0σM2/bracketrightBigg\n,q2B2\n0σ∆+4ω2\n0σM2∆\nM2ω4\n0/parenrightbigg\n−EllipticE/parenleftBigg\nisinh−1/bracketleftBigg\niω0√\nMχ/radicalbig\nq2B2\n0σ+4ω2\n0σM2/bracketrightBigg\n,q2B2\n0σ∆+4ω2\n0σM2∆\nM2ω4\n0/parenrightBigg/bracerightBigg\n−i(q2B2\n0σ∆+4ω2\n0σM2∆−M2ω4\n0)\nΓMω2\n0/braceleftBigg\nEllipticF/parenleftBigg\nisinh−1/bracketleftBigg\niω0√\nM(Γt+χ)/radicalbig\nq2B2\n0σ+4ω2\n0σM2/bracketrightBigg\n,q2B2\n0σ∆+4ω2\n0σM2∆\nM2ω4\n0/parenrightBigg\n−EllipticF\nisinh−1/bracketleftBigg\niω0√\nMχ/radicalbig\nq2B2\n0σ+4ω2\n0σM2/bracketrightBigg\n,q2B2\n0σ∆+\n4ω2\n0σM2∆\nM2ω4\n0\n\n\n\n+(n+l)\nΓ/bracketleftBigg/radicalbigg\n∆\nM(Γt+χ)2−ω2\n0−/radicalbigg\n∆\nMχ2−ω2\n0\n+ω0tan−1/radicalBigg\nMω2\n0\n∆(Γt+χ)2−Mω2\n0−ω0tan−1/radicalBigg\nMω2\n0\n∆χ2−Mω2\n0/bracketrightBigg\n−(n+l)q2B2\n0\nM(4M2ω2\n0+q2B2\n0)Γ/bracketleftbigg/radicalBig\nM∆(Γt+χ)2−M2ω2\n0\n−/radicalBig\nM∆χ2−M2ω2\n0+Mω0/parenleftBigg\ntan−1 Mω0/radicalbig\nM∆(Γt+χ)2−M2ω2\n0−tan−1Mω0/radicalbig\nM∆χ2−M2ω2\n0/parenrightBigg/bracketrightBigg\n. (89)\nHereEllipticF andEllipticE are the incomplete elliptic integrals of the first and second kinds resp ectively.\nReferences\n[1] L.M. Lawson , G.Y.H. Avossevou , J. Math. Phys. 59 (2018) 04210 9\n[2] G. Fiore and L. Gouba, J. Math. Phys. 52, 103509 (2011).\n[3] H.S. Synder, Phys. Rev. 71, 6871 (1947).\n[4] S. Doplicher, K. Fredenhagen, J.E. Roberts, Commun. Math. Ph ys. 172 (1995) 187.\n[5] D. Amati, M. Ciafaloni, G. Veneziano, Phys. Lett. B 216 (1989) 41 .\n[6] N. Seiberg, E. Witten, J. High Energy Phys. 09 (1999) 032.\n[7] C. Rovelli, Living Rev. Relativity 11, 5 (2008).\n[8] D. Bigatti, L. Susskind, Phys. Rev. D 62 (2000) 066004.\n[9] O.F. Dayi, A. Jellal, J. Math. Phys. 43 (2002) 4592.\n[10] B. Chakraborty, S. Gangopadhyay, A. Saha, Phys. Rev. D 70 (2004) 107707.\n[11] F.G. Scholtz, B. Chakraborty, Phys. Rev. D 71 (2005) 085005 .21\n[12] F.G. Scholtz, B. Chakraborty, S. Gangopadhyay, J. Govaert s, J. Phys. A 38 (2005) 9849.\n[13] B. Chakraborty, S. Gangopadhyay, A.G. Hazra, F.G. Scholtz, J. Phys. A 39 (2006) 9557.\n[14] R. Banerjee, S. Gangopadhyay, S.K. Modak, Phys. Lett. B 68 6 (2010) 181.\n[15] A. Saha, S. Gangopadhyay, S. Saha, Phys. Rev. D 83 (2011) 0 25004.\n[16] A. Saha, S. Gangopadhyay, S. Saha, Phys. Rev. D 97 (2018) 0 44015.\n[17] A. Smailagic, E. Spallucci, J. Phys. A 36 (2003) L467.\n[18] S. Gangopadhyay, F.G. Scholtz, Phys. Rev. Lett. 102 (2009) 241602.\n[19] M. Dutta, S. Ganguly, S. Gangopadhyay, Int. Journal. Theor . Phys. 59 (2020) 3852-3875.\n[20] L.M. Lawson , G.Y.H. Avossevou, L. Gouba , J. Math. Phys. 59 (2 018) 112101.\n[21] H.R. Lewis, Jr., W.B. Riesenfeld , J. Math. Phys. 10 (1969) 1458.\n[22] H.R. Lewis, Jr., J. Math. Phys. 9 (1968) 1976.\n[23] H.R. Lewis, Jr., Phys. Rev. Lett. 18 (1967) 510 ; Erratum Phys . Rev. Lett. 18 (1967) 636.\n[24] V. Ermakov, Univ. Izv. Kiev. 20 (1880) 1.\n[25] E. Pinney, Proc. Am. Math. Soc. 1 (1950) 681.\n[26] L. Mezincescu, “Star Operation in Quantum Mechanics”, [hep-t h/0007046].\n[27] S. Dey, A. Fring , Phys. Rev. D 90 (2014) 084005.\n[28] G.B. Arfken, H.J. Weber, “Mathematical Methods For Physicist s”, Academic Press, Inc.\n[29] A.F. Nikiforov, V.B. Uvarov, “Special Function of Mathematical Physics”, Birkh¨ auser, Basel, Switzerland,\n1988.\n[30] A. Chiellini, Bolletino dell’Unione Matematica Italiana 10, 301 (1931)." }, { "title": "2303.15837v1.Role_of_intersublattice_exchange_interaction_on_ultrafast_longitudinal_and_transverse_magnetization_dynamics_in_Permalloy.pdf", "content": "Role of intersublattice exchange interaction on ultrafast longitudinal and transverse\nmagnetization dynamics in Permalloy\nA. Maghraoui, F. Fras, M. Vomir, Y. Brelet, V. Halt\u0013 e, J. Y. Bigot, and M. Barthelemy\u0003\nUniversit\u0013 e de Strasbourg, CNRS, Institut de Physique et Chimie des\nMat\u0013 eriaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France\n(Dated: March 29, 2023)\nWe report about element speci\fc measurements of ultrafast demagnetization and magnetization\nprecession damping in Permalloy (Py) thin \flms. Magnetization dynamics induced by optical pump\nat 1:5eV is probed simultaneously at the M2;3edges of Ni and Fe with High order Harmonics for\nmoderate demagnetization rates (less than 50%). The role of the intersublattice exchange interaction\non both longitudinal and transverse dynamics is analyzed with a Landau Lifshitz Bloch description\nof ferromagnetically coupled Fe and Ni sublattices. It is shown that the intersublattice exchange\ninteraction governs the dissipation during demagnetization as well as precession damping of the\nmagnetization vector.\nPACS numbers: 71.20.Be, 75.40.Gb, 78.20.Ls, 78.47.+p\nI. INTRODUCTION\nUltrafast demagnetization of ferromagnets induced by\nfemtosecond laser pulses [1{3] promises novel applica-\ntions in data storage and processing technologies. Since\nits discovery, several microscopic mechanisms such as\nthe spin-orbit interaction [4{6], Elliott-Yafet scattering\ninduced spins-\rips [7], non-thermal excitations [8, 9],\nsuper-di\u000busive [10, 11] or ballistic spin-transport have\nbeen identi\fed to play a key role and their relative weight\ncan be element dependent [12]. Depending on magnetic\nanisotropies, such transient modi\fcation of the e\u000bective\nmagnetic \feld can trigger a coherent precession motion\nof the magnetization vector with a Gilbert damping [13]\nresulting from dissipation of energy to an external bath.\nThose longitudinal and transverse relaxation processes\nset a natural limit to optical manipulation of magneti-\nzation from femtosecond to nanosecond time scales. If\none aims to study the dynamics over such a large tempo-\nral scale, the Landau-Lifshitz Bloch (LLB) model [14] in\nwhich the e\u000bective \feld contains the essential microscopic\nmechanisms is well adapted. Among them, the exchange\ninteraction appears to be critical, but several aspects re-\nmain to be explored. In particular, in multi-compound\nmaterials, the intersublattice exchange interaction plays\na crucial role on the resulting global dynamics, acting\nas a spin momentum transfert between sublattices dur-\ning the demagnetization [15]. Over the last decade, it\nhas been investigated experimentally thanks to chemi-\ncal selectivity of XUV resonant probe of core levels of\ntransition metals (TM) and rare earths (RE). Time re-\nsolved Xray magnetic circular dichroism (XMCD) [16{20]\nand table-top high order harmonics (HH) probed time re-\nsolved magneto optical Kerr experiments (TMOKE)[21{\n29] have proven to o\u000ber a unique opportunity to study\nsublattice magnetization dynamics governed by dissipa-\n\u0003Corresponding author: barthelemy@unistra.frtion and momentum transfert mechanisms in all optical\nmagnetization switching in alloys [18{20]. In particular,\nthe demagnetization of each sublattice in a binary alloy\ncan be either accelerated or decelerated compared to the\npure element demagnetization [21, 22, 24]. This e\u000bect is\ndependent on the value of elemental magnetic momenta\nand on the ferro or antiferromagnetic nature of the ex-\nchange coupling [30, 31]. The case of Permalloy (Py) has\nattracted attention since various dynamical behaviors of\nsublattices magnetic momenta have been observed de-\npending on the photon energy range of the probe. On\none side, XMCD studies performed at L2;3-edges have\nshown a faster demagnetization of Ni momenta compared\nto Fe [31]. This observation is supported by the strong\ne\u000bective exchange coupling sustained by Fe momenta in\nPy so that Ni sublattice momenta are more submitted to\nthermal dissipation [32]. On the other side, HH TMOKE\nmeasurements show that during the early demagnetiza-\ntion of ferromagnetic Py, the momenta of Fe starts to\nrandomize before Ni momenta until a time of scale 10 fs\nafter which both sublattice relax together due to inter-\nsublattice exchange interaction (IEI) [24]. The origin of\na stronger coupling of Fe spins to the electronic system\ncompared to Ni remains unknown and deserves further\nexploration. In the present work, the magnetization dy-\nnamics of Fe and Ni sublattices of a 10 nm Permalloy thin\n\flm is studied with chemical selectivity over a wide tem-\nporal range as a function of excitation density. A table\ntop HH TMOKE con\fguration is used to measure both\ndemagnetization and precession at the M edges of Fe and\nNi. The role of strong intersublattice exchange interac-\ntion on longitudinal and transverse ultrafast magnetiza-\ntion dynamics is discussed for moderate demagnetization\namplitudes.\nII. EXPERIMENT\nIn our experiment, XUV sub 10 fs pulses are produced\nby HH generation in a Ne-\flled gas cell driven by 795arXiv:2303.15837v1 [cond-mat.mtrl-sci] 28 Mar 20232\nnm, 3 mJ, 1 kHz, 25 fs laser pulses. The resulting XUV\nprobe photons energies cover the 30 eV - 72 eV range and\nspan theM2;3-edges of Fe and Ni centered respectively at\n66 eV and 54 eV. Ultrafast demagnetization is induced\nby 795 nm, 25 fs pump with variable \ruence in a 10 nm\nthick Ni 80Fe20(Py) thin \flm with an in-plane anisotropy\ndeposited on a crystalline Al 2O3substrate by ion beam\nsputtering.\nTG sample \nIR pump HH probe \nz \nx y o H+ H- \nt \nFIG. 1: Principle of XUV probe IR pump transverse HH\nTMOKE experimental con\fguration with static magnetic\n\feld H along zaxis. TG: toroidal grating. CCD: Charge\nCoupled Device camera. Inset: example of re\rected spectra\non Py for the antiparallel orientations of the applied magnetic\n\feld +/-H (respectively blue solid line and red dotted line).\nFigure 1 illustrates the transverse time resolved\nmagneto-optical Kerr con\fguration used in this work.\nAn external static magnetic \feld ( H= 450 Oe) is ap-\nplied on the sample along the transverse axis, i:e:along\nz direction in \fgure 1 and perpendicularly to the plane\nof incidence xOy of the p-polarized IR pump and VUV\nprobe. The angle of incidence of the probe was set to\n45\u000ewith respect to the sample normal in order to maxi-\nmize the magnetic contrast obtained from spectrally re-\nsolved re\rectivity measurements [21]. In the inset of \fg-\nure 1, the re\rected XUV probe spectra Istat\nH+andIstat\nH\u0000is\nshown for two antiparallel orientations of the transverse\nmagnetic \feld H. The maximum intensity di\u000berence be-\ntween the two re\rected spectra is seen at the harmonics\nh45(centered at 66 eV) and h35(centered at 54 eV) corre-\nsponding to the M-edges of Ni and Fe respectively. Both\nspectra are further measured as a function of pump probe\ndelay by varying the optical path of the pump with a me-\nchanical delay line.\nIII. ULTRAFAST DEMAGNETIZATION IN\nPERMALLOY PROBED AT M-EDGES OF NI\nAND FE\nWe \frst consider the short time scale corresponding\nto demagnetization process in Permalloy. The elementalmagnetization dynamics of Ni and Fe elements m( q;\u001c)\nmeasured as a function of the pump-probe delay \u001cis\nthen integrated over each resonant qthharmonic:\n\u0001m\nm(q;\u001c) =Idyn\nH+(q;\u001c)\u0000Idyn\nH\u0000(q;\u001c)\nIstat\nH+(q)\u0000Istat\nH\u0000(q)(1)\nforq= 45 andq= 35 with Idyn\nH\u0006=Iwith\nH\u0006\u0000Istat\nH\u0006and\nIwith\nH\u0006being the intensity of signal with pump and Istat\nH\u0006\nwithout pump. Figure 2 shows demagnetization\u0001m\nm(q;\u001c)\n-0.4-0.20.0Δm/m\n8006004002000\n delay (fs)Ni\nFe\n-0.4-0.20.0Δm/m\n8006004002000\n delay (fs) Ni\n Fe\n-0.4-0.20.0Δm/m\n8006004002000\n delay (fs) Ni\n Fe(a)\n(b)\n(c)\nFIG. 2: Demagnetization dynamics in permalloy probed at\nthe Ni (full blue dots) and Fe (empty red dots) M-edges and\n\fts (grey lines: Fe, blue lines: Ni) for incident pump \ruences\nof a) 2:5 mJ/cm2. b) 3:8 mJ/cm2. c) 4:7 mJ/cm2.\nin Py at the M2;3edges of Fe (\u0001mFe\nmFe) and Ni (\u0001mNi\nmNi)\nintegrated over harmonics h35andh43respectively for\nthree increasing pump \ruences. Ni and Fe sublattices\nappear to demagnetize simultaneously. The demagneti-\nzation amplitude of both sublattices increases from 25\n% to 40 %. Contrary to reference [24], no reproducible\ndelay between the two sublattices demagnetizations is\nobserved with our pump duration of 25 fs. A possible\nexplanation could be a slight variation of intersublattice\nexchange interaction (sample dependent due to change\nof crystallinity or grating vs alloy) that may induce a\nchange of the temporal shift value. Moreover the di\u000ber-\nent conditions of HH generation could lead to a di\u000berent3\n-0.6-0.4-0.20.0|Δm/m|\n86420\npump fluence (mJ/cm2) Fe ; τmax\n Ni ; τmax\n Fe ; τ=5 ps\n Ni ; τ=5 ps\n \nFIG. 3: Demagnetization amplitudes measured at M-edges of\nNi and Fe in Py as a function of incident pump \ruence. For\nmaximum demagnetization (full symbols) and at a 5 ps pump\nprobe delay (empty symbols).\ntime duration of our probe resulting to a lower tempo-\nral resolution, or a delay between h35 and h45 due to a\npossible chirp.\nThe framework of analysis and data \ftting is described\nin the following, where details about linearized LLB are\npresented. This approach is based on the knowledge of\nthe laser induced temperature of the system. In order to\nde\fne such laser induced temperature, let us \frst explore\nthe demagnetization amplitude behavior with incident\npump \ruence.\nThe demagnetization amplitude of Ni and Fe in\npermalloy with respect to the pump \ruence is plotted\nin \fgure 3. For each point, two delays have been cho-\nsen. The \frst delay corresponds to maximal demagneti-\nzation level (the corresponding time delay range is 300\nfs - 500 fs from low to large pump \ruences), the second\ndelay is \fxed at 5 ps and corresponds to a thermal quasi-\nequilibrium between electrons and lattice baths (remag-\nnetization). Within the \ruence range of this experiment\nthe demagnetization amplitude is always found identical\nfor both sublattices. Moreover, up to 40% demagnetiza-\ntion given by the damage threshold of our sample, a lin-\near increase of demagnetization amplitude with the pump\n\ruence is observed, with the same slope for both delays.\nSuch linear behavior can be attributed to a coupling with\na thermalized bath[2], characterized by an energy of kBT,\nheated by the laser pulse. The vertical o\u000bset between\nthe two curves arises due to the magnetization recovery.\nIn LLB approach, it can be described as a temperature\nstep between maximal bath temperature (demagnetiza-\ntion) and cooling down of the bath due to coupling with\nlattice (remagnetization). One can notice that the linear\n\fts labelled \\maximum\" does not cross the zero variationat zero \ruence. At a delay of 5 ps a slight deviation from\nzero is also observed. This range of \ruence is either dif-\n\fcult to access experimentally or usually not considered.\nA hypothesis to explain such a behavior is a change of\nthe regime of interaction, at the origin of the demagneti-\nzation or remagnetization processes, with pump \ruence.\nThis could lead to a di\u000berent power dependent law in the\nrange of very low \ruences, but this aspect goes beyond\nthe scope of the present work. After thermalization of\nelectrons, the temperature dependent magnetization can\nbe approximated with a molecular \feld model as :\nm\u000f(T) = (1\u0000T\nTC)\u0014\u000f(2)\nwhereTCis the Curie temperature, \u0014\u000fis the critical ex-\nponent. In the following, the maximum amplitude of\ndemagnetization in our experiments is used to evaluate\nthe laser induced spin temperature Tthat is related to\nthe amplitude of \ructuations to which the spins are sub-\nmitted [34] in the 300-500 fs temporal range. As in ref\n[32], in our approach at short time scale, only the de-\nmagnetization process is considered, justi\fed by a slower\nrate of the re-magnetization process. The magnetic sys-\ntem can be considered initially in thermal equilibrium\nat a temperature of Ti= 300K, then for t=0 the bath\ntemperature is instantaneously changed to Tf. Thus, the\nmagnetization of the two sublattices will evolve towards\na new thermal equilibrium value given by Tf. Therefore,\nin this approach, the equivalent spin temperature is de-\n\fned at the maximum of demagnetization. By taking a\nCurie Weiss law ( \u0014= 1=2) andTC= 850 K in Ni 80Fe20,\nthe temperature range can be calibrated. In \fgure 3,\nthe evaluated spin temperature Tranges from 300 K to\n600 K, and the maximum demagnetization amplitudes of\n\fgure 2 correspond to T=TC= 0.35, 0.43 and 0.52.\nWe now analyze our experimental data in the frame of\nthe linearized LLB model. It considers an ensemble of\nrigid spins submitted to exchange interaction and cou-\npled to a thermal bath corresponding to either charges\nor phonons at the origin of dissipation [33{37]. It gives\na consistent approach that encompasses a broad tempo-\nral scale from femtoseconds to nanoseconds during which\nspin \rips as well as the magnetization precession take\nplace. By isolating the longitudinal contribution to mag-\nnetization dynamics at short time scales, it can be used\nto simulate ultrafast demagnetization in TM-RE com-\npounds. This method was \frst applied to ferrimagnets\nknown for their high potential for all optical switching\n[34], and more recently to better understand the role of\nintersublattice exchange interaction (IEI) in ferromag-\nnetic TM alloys [32]. In particular, this model allows\nto decipher quantitatively the role played by both the\nIEI and intrinsic dissipation of each sublattices magne-\ntization on the observed dynamics. Fe and Ni momenta\ndynamics in permalloy are described with the following4\n\frst order coupled rates equation:\n\u0012\n_mFe\n_mNi\u0013\n=Ak\u0012\nmFe\nmNi\u0013\n=\u0012\n\u0000\u0000FeFe \u0000FeNi\n\u0000NiFe\u0000\u0000NiNi\u0013\u0012\nmFe\nmNi\u0013\n(3)\nwhere the matrix Akdrives the dynamics. Its ele-\nments can be written as a function of micromagnetic\nparameters[32]:\n\u0000FeFe=1\n\u001cFe\nintra+\u001fNi\nk\n\u001fFe\nk1\n\u001cFeNi\nexch(4)\n\u0000FeNi=1\n\u001cFeNi\nexch(5)\n\u0000NiFe=1\n\u001cNiFe\nexch(6)\n\u0000NiNi=1\n\u001cNi\nintra+\u001fFe\nk\n\u001fNi\nk1\n\u001cNiFe\nexch(7)\nIn this basis, the elements of Akcontain two contribu-\ntions to longitudinal damping. The \frst one corresponds\nto intrasublattice demagnetization \u001cintra:\n1\n\u001cFe,Ni\nintra=\rFe,Ni\u000bFe,Ni\nk(T)\n\u001fFe,Ni\nk(T)(8)\nwhere\u000bFe,Ni\nkis longitudinal damping and \u001fFe,Ni\nkis the\ne\u000bective magnetic susceptibility, both being element and\ntemperature dependent quantities. \rFe,Nicorresponds to\nthe gyromagnetic ratio of each element. The second one\nis the intersublattice exchange mediated demagnetiza-\ntion:\n1\n\u001cFeNi,NiFe\nexch=\rFe,Ni\u000bFe,Ni\nk(T)JFeNi,NiFe\n\u0016Fe,Ni(9)\nwhereJFeNi=JNiFeis the IEI constant and \u0016Fe,Ni is the\natomic magnetic momentum.\nFinally, diagonal terms of Ak,\u0000\u0000FeFe and\u0000\u0000NiNi,\ncorrespond to a dissipation of magnetic momenta in each\nsublattice and via IEI with the second sublattice (eq.\n4). Non diagonal terms \u0000 NiFe and \u0000FeNi lead to an\nexchange of momentum between sublattices. It should\nbe underlined that, in this approach, the overall \row of\nmomentum, mediated by the IEI, is element dependent\nand weighted by the magnetic susceptibilities ratio.\nThe di\u000berential system (3) can be easily solved in the\neigen basis after diagonalization of Ak:\nAk=\u0012\n\u0000+0\n0 \u0000\u0000\u0013\n(10)\nwhere the two eigen values \u0000\u0006= 1=\u001c\u0006can be written as:\n\u0000\u0006=1\n2(\u0000FeFe+ \u0000NiNi\n\u0006p\n(\u0000FeFe\u0000\u0000NiNi)2+ 4\u0000 FeNi\u0000NiFe) (11)Finally the measured sublattices magnetization dynamics\ncan be expressed as a linear combination of the di\u000beren-\ntial system solutions:\n\u0001mFe\nmFe=AFeexp(\u0000t\n\u001c+) +BFeexp(\u0000t\n\u001c\u0000); (12)\n\u0001mNi\nmNi=ANiexp(\u0000t\n\u001c+) +BNiexp(\u0000t\n\u001c\u0000); (13)\nwhere the coe\u000ecients AFe,BFe,ANi,BNidepend on the\neigen vector components x\u0006= \u0000 FeNi=(\u0000FeFe\u00001=\u001c\u0006) as\nfollows:\nAFe= \u0001mFe\n0x+\n(x\u0000\u0000x+)(\u0001mNi\n0\n\u0001mFe\n0x\u0000\u00001) (14)\nBFe= \u0001mFe\n0x+\n(x\u0000\u0000x+)(1\u0000\u0001mNi\n0\n\u0001mFe\n0x+) (15)\nANi= \u0001mFe\n01\n(x\u0000\u0000x+)(\u0001mNi\n0\n\u0001mFe\n0x\u0000\u00001) (16)\nBNi= \u0001mFe\n01\n(x\u0000\u0000x+)(1\u0000\u0001mNi\n0\n\u0001mFe\n0x+) (17)\nwith \u0001mFe\n0and \u0001mNi\n0corresponding to the maximum\namplitude of demagnetization. It is important to notice\nthat\u001c\u0006only corresponds to \u001cFe;Niin the very low tem-\nperature range, when 1 =\u001cNiFe\nexch and 1=\u001cFeNi\nexch are negligible.\nIn the range of temperatures explored in our experiment,\ndue to IEI, the dynamics of each sublattice is a clear bi-\nexponential decay as shown in eq.12 and 13 where the\ndemagnetization time is a composition of \u001c\u0000and\u001c+.\nHaving in hands the T=TCvalues equivalent to pump\n\ruences from amplitudes of demagnetization of our ex-\nperiment, we can analyze our results in the frame of the\nlinearized LLB model. As shown earlier, the pump \ru-\nence range used in our experiments corresponds to an\nintermediate temperature range 0 :35\u0000FeNi). In-\ndeed, the IEI induced modi\fcations of both sublattice5\n0.1110 Γ (ps-1)\n0.80.60.40.20.0\n T/TcΓNiFe\nΓFeNi\nΓNiNi\nΓFeFe\n1.0\n0.9\n0.8\n0.7Δm/m\n5000\n τ (fs)Fe\nNi\n 150\n100\n50τ (fs)\n0.70.60.50.40.3\n T/Tcτ+\nτ−(a)\n(c)(b)\n120\n100\n80\n60\n40τintra (fs)\n0.70.60.50.40.3\n T/TcFe\nNi\nFIG. 4: Comparison of theoretical data from [32] (lines) and\nexperimental values (dots) of a) LLB matrix components for\npump energy densities of T=TC= 0.35, 0.42 and 0.52. Ex-\nperiment (markers) and theory (lines). Inset: example of M-\nedge demagnetization global \ftting (lines) giving rise to LLB\nmatrix element for 3.8 mJ/cm\u00002pump \ruence. b) charac-\nteristic times of M edges magnetization dynamics \u001c+(empty\nred dots),\u001c\u0000(full blue dots) and c) retrieved \u001cNi\nintra(full blue\ndots),\u001cFe\nintra(empty red dots), from experiment and compari-\nson with theoretical values as a function of temperature.\ndynamics are not the same, due to element dependence\nof \u0000FeNi;NiFe (via element dependence of \u000bFe;Ni\nk,\u0016Fe;Ni\nand\rFe;Ni). In order to discuss the intrasublattice dissi-\npation (without the contribution of IEI), we extract the\nintrasublattice demagnetization time from our measure-\nments, as shown in \fgure 4(c). It is deduced from the ex-\nperimentally retrieved values of Akmatrix elements and\nfrom equations 4 and 7 by taking a constant ratio of mag-\nnetic susceptibilities \u001fFe\nk=\u001fNi\nk= 2 (valid in the interme-\ndiate temperature range i.e T=TC<0.5 [32]). Up to T=TC\n= 0.52,\u001cNi\nintra\u0000\u001cFe\nintra<15 fs. Without IEI (\fgure 4(c)),\nFe sublattice undergoes a stronger dissipation. This dis-\nparity between intra sublattice dissipations is compen-\nsated by strong IEI leading to a common dynamics of\nboth sublattices (\fgure 4(b)). The above approach has\nthe advantage to explain the observed dynamics strongly\nin\ruenced by IEI. It shows that IEI mediated dissipation\ndoesn't have necessarily the same weight on each sub-\nlattice. Moreover, the rate of intrasublattice dissipation\nmediated by IEI is related to sublattices magnetic sus-\nceptibilities ratio, that is almost constant for moderate\nlaser induced temperature and diverges close to TC. Let\nus discuss this substantial di\u000berence with the de\fnition\nof a unique sub 10 fs exchange time observed in previous\nwork. Indeed, in the pioneer study proposed by Mathias\net al [24], an exchange interaction time is introduced as\na constant parameter that couples the two subsystems\nmagnetization dynamics. The key di\u000berences are based\non the following points. In ref [24], the process of IEI\nis considered in a conservative manner with equal ratesof magnetic momentum transfer between the two sub-\nlattices. The exchange interaction times are equal for\nboth sublattices and independent of \ruence. Finally the\ndata analysis, performed in this framework, imposes a\nstrong di\u000berence between \u001cFeand\u001cNi(See supplemen-\ntary informations of [24] ). The LLB based approach\nis fundamentally di\u000berent since it considers the e\u000bect of\nIEI as related to the conservative transfer of momentum\nbetween sublattices, but also to dissipation (eq. 4 - 7).\nSecondly, with LLB approach, these contributions are\nboth found element and temperature dependent (eq. 9)\ndue to longitudinal damping and magnetic susceptibili-\nties. The outcome analysis allows retrieving the intra-\nsublattice demagnetization times: between 80 and 100\nfs for both Fe and Ni, which is consistent with earlier\nobservations [7, 38].\nIV. ULTRAFAST MAGNETIZATION\nPRECESSION AND DAMPING IN PERMALLOY\nPROBED AT M-EDGES OF NI AND FE\nWe have shown the in\ruence of intersublattice ex-\nchange interaction on longitudinal magnetization dynam-\nics occuring on the hundreds of femtoseconds time scale.\nWe now address the question of how IEI does a\u000bect the\ndamping of transverse motion of magnetization vector,\nie precessional motion over hundreds of picoseconds. In\nthe following, the IR pump XUV probed TMOKE exper-\niments are performed on a 500 ps temporal range with\na tilt of the external magnetic \feld axis with a 10\u000ean-\ngle with respect to the sample plane. This con\fguration\nallows for transverse projection measurement of magneti-\nzation precession, simultaneously at M 2;3edges of Fe and\nNi. Both Kerr rotation signals \u0001 \u0012K=\u0012Kintegrated over\nh35 (Fe) and h43 (Ni) have been \ftted using the \ftting\nfunction:ANi,Fesin(2\u0019=TNi,Fe\nprt+\u001eNi,Fe)exp(\u0000t=TNi,Fe\nd) +\nBNi,Fe, whereANi,Fe,TNi,Fe\npr,\u001eNi,Fe,TNi,Fe\ndare respec-\ntively the precession amplitudes, periods, phases and\ndamping times of each sublattice. BNi,Feis an o\u000bset that\ncorresponds to long time delay magnetization recovery\ncompared to the temporal window of our measurements.\nAs seen in \fgure 5, the Ni and Fe momenta precession\nare measured selectively in Ni 80Fe20for three incident\npump \ruences corresponding to initial 25% to 35% of\nlaser induced demagnetization. The precession motion\nat Ni edge has a slightly higher amplitude. While the\nprecession signals are increased in amplitude with pump\n\ruence, Fe and Ni momenta still precess in phase. Within\nthe range of pump \ruences used, the precession period\nstays quasi constant TNi,Fe\npr\u0018150 ps. The damping time\nremains identical for both sublattices. It is found to be of\nabout 2 ns for the intermediate \ruence and is decreased\ndown to 300 ps at the highest \ruence. For the lowest \ru-\nence the damping time is di\u000ecult to extract within our\ntemporal window and lower signal to noise ratio. It is\nis estimated higher than 2 ns. Such increase of damp-\ning can be explained by the increasing of phonon medi-6\n-60-40-200ΔθΚ/θΚ (mrad)\n500 250 0\ndelay (ps) Ni\nFe\nFIG. 5: Precession measurements in permalloy probed with\nHigh order Harmonics. Fe M-edge (full red dots) and Ni M-\nedge (empty blue dots) magnetization dynamics of precession\nin Permalloy as a function of incident pump laser \ruence in-\ncreased from top to bottom: 1 :5 mJ/cm2; 2:5 mJ/cm2; 4\nmJ/cm2.\nated spin \rip rate with increasing \ruence. Indeed, the\ngenerated phonon density increases with pump \ruence.\nPhonons cause crystal \feld \ructuations that translates to\nthe magneto-crystalline anisotropy and leads to random\ntorques on the spins. We now analyze the Gilbert damp-\ning in the frame of the Landau Lifshitz Gilbert (LLG)\nequation. In the time scale of the transverse damping\n(three orders of magnitude longer compared to the short\ntime scale of ultrafast demagnetization), the precession\nmotion of two coupled sublattices \u000fand\u000emagnetization\ncan be written for m\u000f;\u000e(m\u000e\nsbeing the saturation magne-\ntization of the second sublattice \u000e) as [39]:\n_m\u000f\n\r\u000f=\u0000\u0000\nm\u000f\u0002H0\n\u000f\u0001\n\u0000\u000b\u000f\u0002\nm\u000f\u0002\u0000\nm\u000f\u0002H0\n\u000f\u0001\u0003\n+A\u000f\u000em\u000e\ns\b\u0000\nm\u000f\u0002m\u000e\u0001\n+\u000b\u000fA\u000f\u000em\u000e\ns\u0002\nm\u000f\u0002\u0000\nm\u000f\u0002m\u000e\u0001\u0003\t\n(18)\nThe \frst line of equation (18) corresponds to preces-\nsion of magnetization and damping related to e\u000bective\n\feldH0\n\u000f=H0+Hanis, where\r\u000fis the gyromagnetic\nratio,\u000b\u000fthe Gilbert damping, H0andHanisare the ap-\nplied and anisotropy \felds. The second line corresponds\nto the coupling of precession motion and damping via\nIEIJ\u000f\u000ebetween sublattices \u000fand\u000e. It corresponds to\na contribution to the e\u000bective \feld H00\n\u000f=\u0000A\u000f\u000em\u000e\nsn\u000eof\nthe second sublattice \u000eon the \frst sublattice \u000f. The ex-\nchange sti\u000bness parameter is de\fned by A\u000f\u000e=J\u000f\u000e=\u0016\u000e\u0016\u000f.\nNote that when A\u000f\u000eis lower than other elemental ef-\nfective \feld contributions, the two sublattices precess in-\ndependently. In our case, when A\u000f\u000edominates, the re-sulting motion corresponds to a single coupled precession\nmotion. The corresponding Gilbert damping can be eval-\nuated from the damping time TPy\ndas a single value for\neach \ruence. Considering a circular precession motion\nwith small angles, one has:\n\u000bPy= 1=(TPy\nd!Py) (19)\nwith!Py= 2\u0019=Tprbeing the precession pulsation. From\nour measurements in Py at M-edges of Ni and Fe (\fg-\nure 5) and by taking mPy\ns= 8.4 106A m\u00001, one has:\n\u000bNi,Py\u0018\u000bFe,Py\u00140:012 for the two \frst \ruences and\n\u000bNi,Py\u0018\u000bFe,Py= 0:079 for maximal \ruence. Moreover,\nin a strongly exchange coupled alloy, its Gilbert damp-\ning can be estimated as an e\u000bective damping from pure\nelements parameters [40]:\n\u000bPy\ne\u000b=mFe\ns\rNi\u000bFe+mNi\ns\rFe\u000bNi\nmFes\rNi+mNis\rFe(20)\nwhere\u000bi(i = Ni,Fe) is the pure element damping. To\ncompare the estimated value of damping \u000bPyfrom HHG\nexperiment to the one as a composition of pure elements\n\u000bPy\ne\u000b, we have performed precession measurements in\ntwo 10 nm thick \flms of pure Ni and pure Fe, using a\nTMOKE con\fguration with 25 fs, 800 nm pulses. The\nmeasured precession damping times TNi, pure\nd,TFe,pure\nd,\nat \fxed initial 20% demagnetization, allows retrieving\nthe corresponding Gilbert damping \u000bNi,pureand\u000bFe,pure\nby using equation (19). By taking mFe,pure\ns = 1.72 106A\nm\u00001; mNi,pure\ns = 4.85 106A m\u00001, the following Gilbert\ndamping values are obtained in pure thin \flms: \u000bNi,pure\n= 0.05 and \u000bFe,pure= 0.016. The e\u000bective damping\nin Py as a composition of pure elements damping\nobtained by equation (20): \u000bPy\ne\u000b= 0:041 (with\rFe=\n2.12 105m s\u00001A\u00001and\rNi= 2.03 105m s\u00001A\u00001) is\nin good agreement with values found from equation\n19. Finally, one can notice that the common Gilbert\ndamping value measured at both Fe and Ni M-edges in\nPy is close to the highest pure Ni value. It indicates\nthat the dissipation of precession is dominated by Ni\nsublattice contribution. This could be attributed to\nthe higher spin orbit coupling in Ni compared to Fe,\ngiving higher spin lattice dissipative contribution [12, 13].\nV. CONCLUSION\nIn this work, magnetization dynamics in Py induced\nby a 1.5 eV femtosecond pump pulse and probed by HH\nis investigated with chemical selectivity on Ni and Fe\nsublattices over a wide temporal scale. The role played\nby the IEI, in the sublattices damping and precession,\nhas been explored in the intermediate spin temperature\nrange. First, we show that demagnetization dynamics\nmeasured at M edge of each Fe and Ni sublattices of\npermalloy is well reproduced in the LLB framework. The7\npump \ruence dependent dynamics of each sublattice is\ncharacterized by double exponential decay with charac-\nteristics times \u001c+and\u001c\u0000both relying on elemental sus-\nceptibilities and longitudinal damping. This approach\nallows to distinguish two contributions to the demagne-\ntization time measured at M edges of each sublattice:\nthe \frst one corresponds to intrasublattice dissipation\ngoverned by longitudinal damping and magnetic suscep-\ntibilities, the second one is the IEI mediated dissipation\nresponsible of the strongly coupled response observed in\nthis study. An interesting prospective could be to study\nmagnetization dynamics beyond this range of excitation\ndensities, where both sublattices are expected to show\ndi\u000berent dynamics as predicted by LLB model.\nSecondly, we have shown that not only the longitudi-\nnal magnetization dynamics of each sublattice is dom-\ninated by IEI but also the magnetization vector orien-\ntation through precession and transverse damping. The\nstrong IEI drives the two sublattices to share a singleprecession mode of which the damping is a composition\nof pure elements damping. 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Melkov, Magnetization Oscilla-\ntions and Waves (1996)." }, { "title": "2307.00903v1.Magnetic_lump_motion_in_saturated_ferromagnetic_films.pdf", "content": "Magnetic lump motion in saturated ferromagnetic films\nXin-Wei Jin,1, 2Shi-Jie Shen,2Zhan-Ying Yang,1, 3and Ji Lin2,∗\n1School of Physics, Northwest University, Xi’an 710127, China\n2Department of Physics, Zhejiang Normal University, Jinhua 321004, China\n3Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China\n(Dated: July 4, 2023)\nIn this paper, we study in detail the nonlinear propagation of magnetic soliton in a ferromagnetic film. The\nsample is magnetized to saturation by an external field perpendicular to film plane. A new generalized (2+1)-\ndimensional short-wave asymptotic model is derived. The bilinear-like forms of this equation are constructed\nand exact magnetic line soliton solutions are exhibited. It is observed that a series of stable lumps can be\ngenerated by an unstable magnetic soliton under Gaussian disturbance. Such magnetic lumps are highly stable\nand can maintain their shapes and velocities during evolution or collision. The interaction between lump and\nmagnetic soliton, as well as interaction between two lumps, are numerically investigated. We further discuss\nthe nonlinear motion of lumps in ferrites with Gilbert-damping and inhomogeneous exchange effects. The\nresults show that the Gilbert-damping effects make the amplitude and velocity of the magnetic lump decay\nexponentially during propagation. And the shock waves are generated from a lump when quenching the strength\nof inhomogeneous exchange.\nI. INTRODUCTION\nThe propagation of electromagnetic wave in ordered\nmagnetic materials, especially in a ferromagnetic medium,\nplays a vital role in faster and higher density storage fields [1–\n3]. In particular, magnetic soliton(MS), which exists in both\nferro- and antiferro-magnets, is becoming a very promising\ninformation carrier because of its particle-like behavior and\nmaneuverability [4–9]. In the past few decades, a wide range\nof soliton-type propagation phenomena has been theoretically\npredicted [10–13], and some of them have been confirmed\nexperimentally [14, 15].\nIndeed, wave propagation in ferromagnetic media is well-\nknown as a highly nonlinear problem. A complete description\nof all types of nonlinear excitations is governed by the\nMaxwell equations coupled with Landau-Lifschitz equation.\nFor this moment, let us notice that a fully nonlinear theory has\nnot been developed. But the linear theory for sufficiently small\namplitudes was established and validated experimentally [16].\nIn order to obtain results valid in nonlinear regimes, or at\nleast weakly nonlinear, one has to resort to intermediate\nmodels (by introducing a small perturbative parameter related\nto the soliton wavelength) [17]. These models include long-\nwave model [18–20], modulational asymptotic model [21],\nand short-wave model [22–25]. Both long-wave model and\nmodulational asymptotic model are mainly used to explain\nand predict the behavior of large-scale phenomena owing to\ntheir long-wave-type approximate condition [26]. However,\nthis condition is not always applicable because the scale of\nmagnetic materials and devices are getting more refined and\nmore sophisticated. Moreover, the main practical interest of\nferrites is that they propagate microwaves [27, 28]. On the\ncontrary, from the viewpoint of applied physics, the short-\nwave-type approximation is much more relevant to available\nexperiments than the former one.\nSince Kraenkel et al. first proposed the short-wave model\n[29], quite a few related nonlinear evolution equations have\nbeen derived, which belong to the Kraenkel-Manna-Merle\n(KMM) system [22, 23, 30–32]. Some significant works\n∗Corresponding author: linji@zjnu.edu.cnhave been devoted to searching and explaining different\nexcitation patterns of ferromagnetic insulators. As for (1+1)-\ndimensional KMM system, the existence of multi-valued\nwaveguide channel solutions has been verified, and the\nnonlinear interaction properties were investigated between\nthe localized waves alongside the depiction of their energy\ndensities [22]. By applying the Hirota bilinear transformation\nmethod, the one- and two-soliton solutions were constructed\nwhile studying in details the solitons scattering properties\n[23]. This system is also solvable using the inverse scattering\nmethod [25]. It is noteworthy that this system possesses\nthe loop-soliton and spike-like soliton [33, 34], and the\nmagnetic loop-soliton dynamics have been extensively studied\n[35–37]. The propagation of electromagnetic waves in\nhigher-dimensional ideal ferromagnets has also been studied,\ncorresponding to the (2+1)-dimensional KMM system [26, 31,\n38, 39]. The analytical one-line-soliton solution as well as its\ntransverse stability have been reported [26]. It has been shown\nthat these structures were stable under certain conditions.\nOn the other hand, most previous studies have only focused\non the propagation of MS in ideal ferrites, which means\nsome important properties of the magnetic material were\nneglected. The main reason is that the nonlinear wave\nequation describing the propagation of electromagnetic waves\nin non-ideal ferromagnetic materials is no longer integrable.\nHowever, the Gilbert-damping and inhomogeneous exchange\neffects are essential features in a real ferromagnetic film, and\ntheir connection with MS motion is an important issue that has\nnot been explored so far. In this paper, we aim to investigate\ntheoretically and numerically the dynamics of the MS in a\nferromagnetic film including damping and the inhomogeneous\nexchange effect. The rest of this paper is organized as follows.\nIn Section 2, we review the physical background and derive\na new (2+1)-dimensional short-wave asymptotic model in\nferromagnetic media. In Section 3, the bilinear-like form\nof the reduced system is constructed and the analytical MS\nsolutions are acquired. In Section 4, the transmission stability\nof the magnetic soliton is numerically explored. The results\nshow that an unstable MS will split to some magnetic lumps\nby a small perturbation. The motions of these lumps under\nthe influence of damping and inhomogeneous exchange are\nanalysed in detail. We end this work in Section 5 with a brief\nconclusion and perspectives.arXiv:2307.00903v1 [nlin.PS] 3 Jul 20232\nII. PHYSICAL BACKGROUND\nA. Basic equations\nThe physical system under consideration is a saturated\nmagnetized ferrite film lying in the x−yplane, as shown in\nFig. 1. Different from Ref. [32], we consider the external\nfield H∞\n0perpendicular to the film, i.e., M0= (0,0,m). So the\ntransverse drift is avoided. The typical thickness of the film\nis about 0.5mm, and the width is approximately 10mm. We\nassume the propagation distance is large enough with regard\nto the wavelength, say more than 50cm. The evolution of the\nmagnetic field Hand the magnetization density Mis governed\nby the Maxwell equations coupled with Landau-Lifschitz-\nGilbert equation, which read as\n−∇(∇·H)+∆H=1\nc2∂2\n∂t2(H+M), (1a)\n∂\n∂tM=−γµ0M×Heff+σ\nMsM×∂\n∂tM, (1b)\nwhere c=1/p\nµ0˜εis the speed of light with the scalar\npermittivity ˜εof the medium, γis the gyromagnetic ratio,\nµ0being the magnetic permeability of the vacuum, σis the\ndamping constant, and Msis the saturation magnetization. The\neffective field Heffis given by [30]\nHeff=H−βn(n·M)+α∆M. (2)\nHere αandβare the constants of the inhomogeneous\nFigure 1. Ferrite film under consideration. The sample is magnetized\nto saturation by long strong magnetic field H∞\n0applied in the\nz-direction. The x-direction of the short wave propagation is\nperpendicular to the direction of static magnetization.\nexchange and the magnet anisotropy ( β>0 corresponds to\nthe easy-plane case), respectively. For a simple tractability, the\nunit vector nof the anisotropy axis is assumed to be along the\nzaxis (i.e., n≡ez). In order to transform the above systems to\ndimensionless equation, we rescale the quantities M,H, and t\nintoµ0γM/c,µ0γH/c, and ct. Thus, the constants µ0γ/cand\ncin Eqs.(2) and (3) are replaced by 1, Msbym=µ0γMs/c,\nandσby˜σ=σ/µ0γ[30].\nB. Linear analysis\nTo study the linear regime we look at a small perturbation of\na given solution. Equations (1) are linearized about the steady\nstate:\nM0= (0,0,m),H0=µM0. (3)where µis the strength of the internal magnetic field. Before\nproceeding further we assume that the ferromagnetic materials\nhave weak damping ¯σ∼ε˜σ. The exchange interaction\nparameter αand anisotropy parameter βare of order ε2and\nε3, respectively (i.e. ¯α=ε2α,¯β=ε3β). Let us seek for\nthe plane wave perturbation solution propagating along the x-\ndirection such as\nM=M0+εmexp[i(kx+ly−ωt)],\nH=H0+εhexp[i(kx+ly−ωt)],(4)\nwhere kandlare the wave numbers in the xandydirections, ω\nis the frequency. Vectors m= (mx,my,mz)andh= (hx,hy,hz)\nare arbitrary real scalar quantities.\nSubstituting Eq. (4) into (1) and (2) in the linear limit, it is\nreduced to\n\nω20 0 ω2−l2kl 0\n0 ω20 kl ω2−k20\n0 0 ω20 0 ω2−k2−l2\n−iωmµ 0 0 −m 0\n−mµ−iω 0 m 0 0\n0 0 −iω 0 0 0\n·\nmx\nmy\nmz\nhx\nhy\nhz\n=0\nThen we obtain the following dispersion relation\nm2(µ+1)\u0002\nµ(k2+l2−ω2)−ω2\u0003\n−ω2(k2+l2−ω2) =0\n(5)\nNote that we focus on studying the short-wave approximation\nk→∞[2]. It comes k0∼ε−1through a small parameter ε≪1\nlinked to the magnitude of the wavelength. Consequently, the\nfrequency expands accordingly as\nω=ω−1ε−1+ω1ε+ω3ε3+.... (6)\nThis assumption guarantees the phase velocity ω(k)/kand\nthe group velocity ∂ω/∂kare always bounded [3]. Now,\nreplacing Eq. (6) into the dispersion relation above, we obtain\na set of equations:\n•At order of ε−4:ω−1=±k0\n•At order of ε−2:ω1=\u0002\n(µ+1)m2+l2\u0003\n/2k0\n•higher order equations which determines ω3,ω5,...\nThe direction of the wave propagation is assumed to be\nclose to the xaxis, thus yvariable gives only account of a\nslow transverse deviation[40, 41]. Therefore lis assumed\nto be very small with respect to kand we write l=l0of\norder 0 with respect to ε. The phase up to order εis thus\n(x−t)/ε+l0y−εω1t,which motivates the introduction of\nnew variables:\nζ=1\nε(x−Vt),y=y,τ=εt. (7)\nThe variable ζdescribes the shape of the wave propagating at\nspeed V; it assumes a short wavelength about 1 /ε. The slow\ntime variable τaccounts for the propagation during very long\ntime on very large distances with regard to the wavelength.\nThe transverse variable yhas an intermediate scale, as in KP-\ntype expansions [26, 41]\nC. Multiple scale approach\nIn order to derive the nonlinear model, fields MandHare\nexpanded in power series of εas\nM=M0+εM1+ε2M2+ε3M3+...,\nH=H0+εH1+ε2H2+ε3H3+....(8)3\nwhere M0,H0,M1,H1,...are functions of (ζ,y,τ).\nWe consider the boundary conditions: lim\nζ→−∞M0=\n(0,0,m),lim\nζ→−∞Mj=lim\nζ→−∞Hj=0,(j̸=0). We derive\nthe following expressions by substituting Expansions (8) into\nequation (1):\n•At order ε−2:\nM0is a constant vector M0=(0,0,m),\n•At order ε−1:\nHx\n0=0,My\n1=0,Mz\n1=0,\n•At order ε0:\nMx\n1ζ=mHy\n0,\nMx\n2ζζ=−Hx\n2ζζ−Hy\n1ζτ\nMy\n2ζζ=−Hx\n1ζy+Hx\n0ζy\nMz\n2ζζ=Hz\n2ζτ+Hz\nyy\n•At order ε1:\nMx\n2ζ=−mHy\n1\nMy\n2ζ=m¯αMx\n1ζζ+¯σM1ζx−Mx\n1Hz\n0+mHx\n1\nMz\n2ζ=Mx\n1Hy\n0\nlet us introduce some independent variables XandTdefined\nasX=−mζ/2,Y=my,T=mτ.\nAfter eliminating H2andM2, we finally obtain the (2+1)-\ndimensional KMM equation:\nCXT=−BBX+CYY,\nBXT=BCX+BYY−sBX+ρBXX,(9)\nwhere observables B,Cand constants s,ρare defined by\nC=−X−ZX\n(Hz\n0/m)dX,B=Mx\n1/2m,\ns=−¯σ/2,ρ=¯αm2/4.(10)\nThis equation is new, which describes the evolution of\nmagnetization field Mand magnetic field Hwithin a ferrite\nfilm in presence of Gilbert-damping and inhomogeneous\nexchange. The quantities H0andM1refer to the zeroth and\nfirst-order expansion coefficients of the external magnetic field\nand the magnetization, respectively. For some simplicity,\nin the next, the independent variables X,YandTwill be\nrewritten as their lower cases x,yandt, respectively.\nIII. HIROTA’S BILINEARIZATION AND SOLITON\nSOLUTIONS OF THE (2+1)-DIMENSIONAL KMM\nEQUATION\nTo explore soliton solutions for the (2+1)-dimensional\nKMM equation (9), we consider a specific dependent variable\ntransformation\nB=G\nF,C=δx−2(lnF)t−2(lnF)y, (11)\nwhere δis an arbitrary constant. Consequently, the bilinear-\nlike forms of the (2+1)-dimensional KMM equation can be\nderived as follow\nF·(DxDt+sDx−D2\ny)G·F+G·(DxDy+D2\ny)F·F=δF2G\n(12a)\n∂x\u0014G2\n2F2−(DyDt+D2\nt)F·F\nF2\u0015\n+∂y\u0014(DyDt+D2\nt)F·F\nF2\u0015\n=0\n(12b)where G,Fare all differential functions of (x,y,t)to be\ndetermined. The symbols Dx,Dtrefer to the Hirota’s operators\nwith respect to the variable x,t, respectively. In order to\nconstruct the solitary wave solutions of Eq.(6), we expand\nGandFwith respect to a formal expansion parameter as\nG=εG1+ε3G3+ε5G5+...,F=1+ε2F2+ε4F4+ε6F6+...,\nin which εis a perturbation parameter and functions Gi,Fi,(i=\n1,2,3,...)are expansion coefficients of the above series. The\none-soliton solution could be constructed by truncating the\nperturbation expansion of GandFas follow\nG=eη1,F=1+k2A2\n16δ2e2η1. (13)\nSubstituting these expressions into Eq.(9) and solving the\nbilinear system recursively, in the absence of damping,\nthe analytical one-soliton solution of the (2+1)-dimensional\nKMM equation can be transformed into\nB=2δ\nksech(η1+η0),C=δx−2δ\nk[tanh(η1+η0)+1],\n(14)\nwhere η1=kx+ly+ [(l2−kl)/2k]t,η0=ln(k/4δ),kandl\nare arbitrary real constants. It should be noted that this soliton\nsolution exists only when the damping is neglected (s=0).\nSimilar to the procedure for constructing one-soliton solution,\nthe two-soliton solution can be given by treating the truncated\nperturbation expansions of GandFas\nG=A1eξ1+A2eξ2+C12eξ1+2ξ2+C21e2ξ1+ξ2, (15a)\nF=1+B11e2ξ1+B22e2ξ2+B12eξ1+ξ2+E12e2ξ1+2ξ2,(15b)\nwhere A1,A2,k1,k2are real constants, ξi=kix+liy+\u0002\n(l2\ni+δ)/ki\u0003\nt,(i=1,2), and the remaining parameters have\nthe following forms:\nBii=A2\nik2\ni\n16δ2,B12=A1A2\n2δ2k2\n1k2\n2\nk2+,k1l2=k2l1,\nCi j=AiA2\nj\n16δ2k2\njk2\n−\nk2+,E12=A2\n1A2\n2\n256δ4k2\n1k2\n2k4\n−\nk4+,(16)\nwhere k+=k1+k2,k−=k1−k2. Parameters Ai,Aj,ki,kj\nandli,(i=1,2,j=3−i)are arbitrary real constants.\nIV . NUMERICAL INVESTIGATION OF LINE-SOLITON\nAND MAGNETIC LUMPS\nA. Unstable MS splits into lumps\nWe now turn to the stability and interactions between MSs\nin a ferromagnetic film. The initial data is a MS perturbed\nby some position-dependent Gaussian wave packets with the\nfollowing expression:\nf=bexp\"\n−\u0012x−x0\nxr\u00132\n−\u0012y\nyr\u00132#\n, (17)\nwhere b,xrandyrcorrespond to the shape of the wave packet\nandx0is related to the perturbation position.\nThe time evolution results clearly show the instability of\nthe MS. For small bi, the MS will break up and eventually4\n(a)\n (b)\n(c)\n (d)\nFigure 2. Propagation of MS perturbed by a Gaussian disturbance.\n(a) Component Hz, (b) Component Hy, (c) and (d) are enlarged views\nof the indicated areas circled in red and black, respectively. The\nparameters are chosen as A1=A2=1,δ=−1,l1=l2=0,k1=\n1,k2=2,x0=−29,b=0.1,xr=1.5,yr=2.5 in (16) and (17).\nevolve into some stable two-dimensionally localized lumps , as\ndisplayed in Figs. 2(a) and 2(b). We observe that most of the\nenergy is always propagated as a lump, even if its speed may\ndiffer from the input. Such a magnetic lump is a solitary wave\npacket that maintains its shape and speed during propagation\nor collision.\nA complete single lump of magnetic field component Hz\n(component Hy) is circled in red (black) in Fig.2. The enlarged\nviews (see Figs.2(c) and 2(d)) provide a clear picture of the\nshape and contour map of the lump. It can be found that\ncomponent Hzis a dipole-mode lump, whereas component\nHyis a standard KP-lump. We also show the vector field\nof the magnetic lump in Fig.3(a). Note that magnetic field\ncomponent Hxis zero, the magnetic field is always in the y−z\nplane, hence the lump can be regarded as a 360◦domain wall\nlocalized in xandydirections. Fig.3(b) presents the magnetic\nfield along y=0. The blue and red arrows correspond to the\nmagnetic field intensity of component Hz,Hy, respectively.\nThe rest of this work is concerned with the propagation and\ninteraction behavior of these lumps in ferrite medium.\n(a)\n (b)\nFigure 3. (a) The vector field of the magnetic lump. (b) The magnetic\nlump along y=0. The blue and red arrows correspond to the\nmagnetic field intensity of components Hz,Hy, respectively.B. Lump motion in ferromagnets with damping or\ninhomogeneous exchange effects\nFigure 4. Three dimensional projections of lump at t=0,HandW\nrepresent the definitions of lump height and width, respectively.\nThe evolution behavior of the magnetic lump in the ideal\nferrite is quite simple and imaginable. Each lump maintains\nits shape while it travels at a constant speed. However, in most\nof real ferromagnetic materials, we have to take the Gilbert-\ndamping into account . For instance, the dimensionless\ndamping constant sranges from 0.048 to about 0.385 in\ngarnet ferrite films. Here we are going to study the dynamics\nof magnetic lump in a damped ferrite film. The typical\nferromagnetic film under consideration is a garnet ferrite film\nwith the dimensionless damping constant s=0.1. For a\nclearer view of the change in shape of the lump, we define\nHandWas the height and width of the lump, which\nare the vertical distance between the highest point and the\nlowest point and the horizontal distance along the propagation\ndirection, respectively. All of these are summarized in Fig.4.\nThe propagation of a lump on the garnet ferrite film\nis presented in Fig.5. As shown in Fig.5(a), the lump\ntravels forward a visible distance in the damped ferrite.\nBeyond that, comparing the profiles of lump between t=0\nand t=10, we evidently observe that the lump becomes\nsmaller and narrower. Fig.5(b) shows the lump height\nand width exhibit a tendency of exponential decay. The\nsolid blue line is the exponential fitting curve to H(t),\nwith the function expression being H(t) = A0e−st. We\nconfirm the above-mentioned amplitude attenuation law is\nuniversal by simulating the motion of lump in ferrites with\nvirous damping factors. Moreover, a definite relationship\nbetween the amplitude and the localization region of solitons\nis important for the soliton excitations. We analyze different\nsizes of numerical lumps and mark the width and height of\nlumps in the phase diagram (see Fig. 5(c)). The results show\nthat for a magnetic lump excitation, its width and height meet\na linear relationship within the error range ( W/H∼0.305).\nSo the lump excitation, upon decay, retains a soliton form.\nTherefore, in this system, the Gilbert-damping plays a role of\ndissipating energy during the motion of magnetic lumps and\nit is characterized by decreasing the amplitude and width of\nlump.\nThe inhomogeneities otherwise referred to as deformities is\ninevitable in real magnetic materials, and it can be caused by\neither external fields or the presence of defects, voids and gaps\nin the material. It has already been reported that the MS may\nbe deformed by the presence of inhomogeneities, in particular5\n(a)\n (b)\n (c)\nFigure 5. Evolution of a magnetic lump in a damped ferrite film with dimensionless damping constant s=0.1. (a) Comparison picture of lump\nwave at t=0 and t=10. (b) The variation of lump height H, lump width Wand velocity V . (c) Numerical relationship between the width and\nheight of magnetic lump.\nits structure and speed [35, 42]. In this present system,\nthe inhomogeneous exchange process is unignorable when\nthe wavelength of lump is comparable to the characteristic\nexchange length.\n(a)\n (b)\n(c)\n (d)\nFigure 6. Propagation of lump with and without the inhomogeneous\ninteraction, respectively.\nWe now move to study the lump motion in the presence of\ninhomogeneous exchange effect. The initial data is the stable\nmagnetic lump shown in Fig.5. As can be observed from Fig.\n6(a) and 6(b), in ferrite without exchange interaction, the lump\nsolution propagates at a constant speed and along the previous\npath. We then consider the non-equilibrium dynamics of lump\nby performing a sudden interaction quench. The pictures\nof component Hyat dimensionless times t=2 and t=4.5\nare shown in Fig. 6(c) and 6(d). As we see, for a quench\nfrom the non-interacting to strong inhomogeneous exchange\nferrite film, the lump oscillates rapidly and diffracts alongthe propagation direction. A two-dimensional shock wave\nis generated and propagates forward. The shock wave front\ncontinues to propagate in the negative direction along x-axis.\nFinally, the energy of lump will be dissipated into numberless\ntiny waves. Accordingly, considering that the lump would be\ndestroyed by the inhomogeneous exchange process, one has to\nconsider keeping its wavelength away from the characteristic\nexchange length in the lump-based microwave applications.\nC. Some examples of excitations and interactions\nThe evolution pattern given in Fig.2 reveals that the lump\nmoves at a larger velocity than the broken MS in the\npropagation. The reason is that the velocity of soliton solution\nis proportional to the soliton amplitude. During the formation\nof the lump, the original MS will be destroyed, and most of the\nenergy is concentrated in some certain centers, which causes\nthe amplitude (and velocity) of the lump to be greater than that\nof MS. These lumps with various speeds enable us to explore\nthe interaction between lump and soliton, as well as between\ntwo lumps.\nA typical example of lump-MS collision is shown in\nFig.7(a). The MS begins to break up around at t=4.\nSubsequently, the splitting lump is going to catch up and\ncollide with the front-MS. After the collision, the front-\nMS is destroyed and broken into several lumps with various\nsizes. It is remarkable that the lump keep its localized\nform before and after the collision almost unchanged. This\nphenomenon implies such two-component lumps are natural\nresults from this nonlinear propagation equations. Further\nsimulation shows these lump structures could be generated\nby a MS with random disturbance. Fig.7(b) depicts a\ncharacteristic inelastic collision between two lumps. We\ninitially generate two adjoining lumps. They are emitted by\nMS at dimensionless time t=6.5. The merging process can\nbe performed as follows. From t=7.5 tot=9.5, two lumps\nmerge simultaneously together and give birth to a new lump\nwhose amplitude is significantly greater than the amplitude of\nprevious lumps. Obviously there is a weak attraction between\ntwo lumps which results in their fusion. In addition to the\nfusion of the two lumps, we also observed an extraordinary\npeak at a specific moment (about t=9.5), which looks like a6\n(a)\n(b)\nFigure 7. (a) Collision between lump and MS. (b) Mergence of two lumps and the formation of a second-order rogue wave-like structure.\nsecond-order rogue wave. It appears to be the result of the\ninteraction between the ripples surrounding the two lumps.\nAfter the fusion, the rouge wave-like structure disappears and\nthe dynamics of the output is determined mainly by a single\nhigh-amplitude lump.\nV . CONCLUSION\nAs a conclusion, the nonlinear propagation of MS in a\nsaturation magnetized ferromagnetic thick film is studied in\ndetail. In the starting point, we derive the (2+1)-dimensional\nKMM system that governs the evolution of short MS waves\nin a saturated ferromagnetic film. The bilinear form of the\nKMM system is constructed and the MS solutions are obtained\nanalytically.\nAfter that, numerical simulations are performed to analyse\nthe evolution behaviours of MS. A significant observation\nis that the unstable MS can be destroyed by Gaussian\nperturbation and broken into some stable magnetic lumps.\nThese lumps exhibit high stability during the propagation.\nFurthermore, some examples are given to analyse the collision\nbehaviours between lump and MS, and the interaction between\ntwo lumps. It is found the lump keeps its shape and speed in\nthe collision with MS. The results confirm that the lump is astable propagation mode in this system and, more to the point,\nthe velocity of lump can be adjusted by its amplitude. Their\nrobustness and controllability provide the possibility for future\ninformation memory and logic devices. We also study the\npropagation of such a lump in ferrites subjected to influence\nof damping and inhomogeneous exchange effects. When the\nGilbert-damping of ferrite is considered, the lumps undergo\nthe following changes: the amplitude and the speed of lump\nare decreased, and the width of lump along the propagation\ndirection is getting narrow. It would cause a strong diffraction\nof the lump if we quench the interaction strength.\nWe hope our work will invoke follow-up experimental\nstudies of lump-based microwave applications. Addition-\nally, since only one- and two-line-soliton are obtained,\nthe integrability of the (2+1)-dimensional system Kraenkel-\nManna-Merle (KMM) remains an open issue. The existence\nof the higher-dimensional evolution system as well as the\nbulk polariton solution is an intriguing avenue for future\nexploration.\nACKNOWLEDGMENT\nThis work was supported by the National Natural Science\nFoundation of China under Great Nos. 11835011; 11675146;\n11875220;.\n[1] M Daniel, V Veerakumar, and R Amuda. 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Communications\nin Nonlinear Science and Numerical Simulation , 91:105437,\n2020." }, { "title": "1801.07408v1.The_dominancy_of_damping_like_torque_for_the_current_induced_magnetization_switching_in_Pt_Co_W_multilayers.pdf", "content": "1 \n The dominanc y of damping like torque for the current induced magnetization switching in \nPt/Co/ W multilayer s \nZelalem Abebe Bekele , Kangkang Meng *, Yong Wu, Jun Miao, Xiaoguang Xu, Yong Jiang * \nSchool of Materials Science and Engineering, University of Science and Technology Beijing, \nBeijing 100083, China \nAbstract \nTwo classes of spin -orbit coupling (SOC) mechanisms have been considered as candidate \nsources for the spin orbit torque (SOT): the spin Hall Effect (SHE) in heavy metals with strong \nSOC and the Rashba effect arising from broken inversion symmetry at material surfaces and \ninterfaces. In this work, we have investigated the SOT in perpendicularly magnetized Pt/Co/W \nfilms, which is compared with the results in Pt/Co/AlO x films. Theoretically , in the case of the \nasymmetric structure of trilayers with opposite sign of spin Hall angle, both damping like torque \nand field like torque due to the SHE and the Rashba effect will be enhanced . Using the harmonic \nmeasurements , we have chara cterized the effective fields corresponding to the damping like torque \nand the field like torque , but we have found the domi nancy of damping like torque in the Pt/Co/W \nfilms . It is much different from the results in the Pt/Co/AlO x films, in which both the damping like \ntorque and the field like torque are strong . \nKeywords: Spin-orbit coupling; Spin-orbit torque s; Perpendicular magnetic anisotropy ; Spin \npolarized transport in metals \n*Authors to whom correspondence should be addressed: \n*kkme ng@ustb.edu.cn \n*yjiang@ustb.edu.cn 2 \n 1. Introduction \nMagnetization reversal by an electric current is crucial issue for spintronic memory and \nlogic devices , such as spin transfer torque magnetic random access mem ory, in which \nmagnetization switching are occurred in a spin valve structure [1]. Recently, another type of \ntorques namely spin orbit torque (SOT) has attracted great interest as they can control the \nmagnetization in heavy metal (HM)/ ferromagnetic (FM) het erostructures with perpendicular \nmagnetic anisotropy (PMA) [2-13]. Typically, in HM/FM /oxide trilayers with PMA , the charge \ncurrent passing through the HM can produce pure spin current due to spin Hall effect (SHE) , \nwhich exert s a torque on the magne tization . Hence , the sign and magnitude of SOT are \nessentially determined by the spin Hall angle in the HM . On the other hand, the S OT can also \narise from the interfacial Rashba effect, for which the accumulated spins at the interface can \nforce the moments to change its direction by direct exchange coupling . These two mechanisms \nare all related to strong spin orbit coupling (SOC) [14-17]. Recent theoretical studies showed that \nboth the SHE and the Rashba effect can generate damping like torque and field like torque . \nCorrespondingly, the two kinds of torques yield to the damping like effective field HD~σ×m and \nthe field like effective field HF~σ, respectively , where m and σ are unit vectors of the \nmagnetization in the FM layer and non -equilibriu m spin polarization direction in the HMs. \nParticularly , the direction of σ depends on the sign of spin Hall angle SH of HMs. \nCurrently, most studies have focused on SOT in ultrathin FMs sandwiched between two \nHM layers , such as Pt/CoNiCo/Pt, Pt/Co/Ta , Pt/Co-Ni/W , and Pt/Co/Pt [18-23], in which both of \nthe HM layers have contributions to the spi n accumulation at the interface s. Because the \nefficiency of SOT relates directly to the magnitude of effective spin Hall angle (SH), it is \nconsidered that the overa ll SH will be enhanced if the sign of SH in the top and bottom HM layer 3 \n is opposite . In this work, we studied the SOT in Pt/Co/ W trilayers , in which the sign of SH is \npositive in Pt while the value is negative in W. Theoretically, in the case of asymmetric \nstructures with opposite sign of spin Hall angle, both the SHE and the Rashba effect will \nbe enhanced. Correspondingly, the two kinds of torques will be increased. However, we found \nlarge damping like torque in the Pt/Co/W multilayers as compared with the field like torque . \nThe result s were much different from the Pt/Co/AlO x films, in which both the damping like \ntorque and the field like torque are comparable. \n2. Results and discussion \n W(1.2)/Pt(6)/Co(1)/AlO x(3)/Pt(1) and W(1.2)/Pt(6) /Co(1)/W(2) (in nano meter from bottom \nto top) were prepared by magnetron sputtering system on Si/SiO 2 substrate as shown in Figure \n1(a). The two samples are referred to as sample I an d sample II. After the deposition process, we \nannealed both of the samples at 250 ° C for 40 min at the base pressure of 4.2 ×10-5 Pa to enhance \nthe PMA. The thin films were patterned into to 15 m×80 m Hall bar structure s by electron \nbeam lithography (EBL) . Figure 1(b) shows the schematic of the Hall bar along with the definition \nof the coordinate system. The magnetic hysteresis loops (M-H curves) as shown in Fi gure 1(c) \nindicate strong PMA in both of the two samples. On the other hand, t he Hall resistances (R H) \nas a function of the out -of-plane magnetic field (H) were measured at a source current of 0.1 mA \nas shown in Figure 1(d) . They also present strong PMA of the two samples. \nWe measured the SOT induced magnetization switching by applying a pulsed current with \nthe width of 50 s, and the resistance was measured after a 16 s delay under an external \nmagnetic field along the x-direction. Figure 2 (a) and ( b) showed the current induced \nmagnetizat ion switching with varying applied in-plane field for sample I and II, respectively . \nReductions of the critical switching currents were observed as the external field increase d for 4 \n both of the two samples. Furthermore, with applying appropriate field (~1000 Oe) the \nmagnetizations are fully switched. It can be proved by the changes of the Hal l resistance of the \ntwo samples, which are consistent with the results in R H-H measurements as shown in Figure \n1(d). Considering the resistivity of the W layer is much larger than that of the Pt layer and the AlO x \nlayer should be insulating, here we roughl y took the thickness of Co and Pt layers into account \nwhen determining the switching current density ( Jc) of the two samples. Therefore, with applying \nthe magnetic field of 3000 Oe (fully magnetization switching), the Jc in the Pt/Co/ AlO x and \nPt/Co/W films are (3.8± 0.051)× 1011 Am−2 and (2.04± 0.0044)× 1011 Am−2, respectively . \nThe harmonic measurements were determined by applying the sinusoidal AC current , for \nsample I the frequency is 365 Hz and the amplitude is ranged from 2.0 to 8.0 mA, for sample II the \nfrequency is 315 Hz a nd the amplitude is ranged from 2.0 to 2.6 mA. The first (V ) and second \n(V2) harmonic Hall voltage s are detected using two lock -in amplifier systems at the same time by \nsweeping the longitudinal ( Hx) and transverse ( Hy) fields . Before the harmonic measureme nts, we \nhave applied a large out -of-plane external field to the two samples, which remain saturated after \nthe field is turned off. Here, we will take the example of the harmonic measurements for sample I \nwith 5.0 mA and for sample II with 2.0 mA respectively as shown in Fig ure 3 , in which the results \nare measured with out -of-plane magnetization component M Z>0. Then, the values of the \ndamping like field ( HD) and field like effective field ( HF) of the two samples can be evaluated \nusing the following relation [24]: \n \n24122\n\nLT TL\nFDH HH\n (1) \nWhere \n defined as the ratio of planar Hall effect (PHE) resistance and anomalous Hall effect \n(AHE) resistance and ± sign is the direction of magnetization pointing ( ± M z) along out -of-palne 5 \n axis [13, 24 ]. The longi tudinal (HL) and transverse effective field (HT) can be calculated using \nthe following relation: \n\n2\n22XY\nLT\nXYVH\nHVH\n\n\n (2) \nTo measure the PHE resistance RPHE, a 3 0000 Oe strong in -plane magnetic field is applied to \nsaturate the magnetization of the sa mple s [9, 10, 13, 24] . Then, the R H was measured by rotating \nthe magnetic field in the plane of sample to rotate the magnetization along the \n direction. The \nplanar Hall measurement data can be fit ted using the equation \nH PHE 1 o 2R =R sin 2 α+φ+R sinα+φ+C\n to obtain the value of R PHE [9, 10, 13], \n1φand \n2φ are the \noffset angles in our measurement setup, C is the offset value in R H and the second term \noR sinα\n is the contribution of anomalous Hall effect due to small misalignment of external \nmagnetic field to the sample plane . The experimental data and fitt ed lines for R H- of the two \nsamples are shown in Figure 4(a) and (b) respectively . Then the values of \n in the two samples \ncan be determined to be 1.02 and 1.405 for sample I and II, respectively. According to the \nEquation (1), the HD and HF of the two samples can be finally determined and the results are \nshown in Figure 4 (c) and (d) . The effective fields vary linearly with the curr ent amp litude (I) , \nindicating that the effects of Joule heating are negligible in the measured current range. However, \nthe values of HF in the Pt/Co/W films are nearly ~88% smaller than HD, indicat ing that the current \ninduced magnetization switching of thi s sample has mostly depended on the damping like torque. \nFinally, the spin Hall angle SH were calculated using \n D s FM\nSH\nc2 H |e|M tθ=h|J |\n , where e is \ncharge of an electron, M s is saturation magnetization, t FM is thickness of ferromagnetic layer, J c 6 \n is applied current density , and \n is reduced Planck constant [24]. The effective SH up to 0.12 \nand 0.3004 were determined for sample I and sample II , respectively. Therefore , the effective \nSH will be enhanced when the Co layer is sandwich ed between two HM layers Pt and W with \nopposite signs of SH, and the damping like torque play a dominant role . \nIn the previous studies on SOT of HM/FM /oxide trilayers , the spin density generated by the \ninverse spin galvanic effect exerts a torque on the magnetization , which is attributed to \ninterfacial Rashba SOC [5-9]. A major difficulty to identify the physical origin of the SOT is that \nthe SHE also plays an important role in magnetic multilayers . Recently, Haney et al . have \ndeveloped semi -classical models for electron and spin transport in bilayer nanowires with a \nferromagnetic layer and a nonmagnetic layer with strong SOC [25]. They have proved that the \ndamping like torque is typically derived from the models describing the bulk SHE and the spin \ntransfer torque, and the field like torque is typically derived from a Rashba model describing \ninterfacial SOC. For the Rashba effect , an internal electric field gradient will arise at the interfa ces \nalong the direction of symmetry breaking. The directio n of Rashba effect induced spin \naccumulation as w ell as an effective magnetic field will be along E×P, where E is the internal \nelectric field, P is the electron momentum [26]. The direction of the E for a giv en interface can be \nobtained by considering the differences in the work function ( ) of the two materials at the \ninterface. The work functions of Pt, Co, W and AlO x are ~5.65 eV, 5.0 eV, 4.55 eV and 3.2 eV \n[27]. The smaller difference of work function in Pt/Co/W as compared with that in Pt/Co/AlO x \nfilms will decrease the spin accumulation fr om the Rashba effect . On the other hand , to obtain the \n-phase of W layer a dc sputtering power of 8 W was applied with the growth rate of 0.03nm/s , \nand the cubic lattice constant of -W is 0.5046 nm [28]. For the hexagonal lattice of Co , the in -\nplane lattice constant is about 0.2507nm, which is almost half of that in -W. Therefore, we 7 \n propose that the asymmetrical properties in Pt/Co/W are not as obvious as that in Pt/Co/AlO x \nfilms and the Rashba effect becomes smaller. Finally, as the resistivity of W is much smaller than \nthat of AlO x, the shunting effect is smaller in Pt/Co/W films . Thus, for a given current density, spin \naccumulation from the bulk SHE will be larger in Pt/Co/W films , for which the damping like \ntorque due to SHE will play a dominant role. \n3. Conclusion \nIn conclusion , we have investigated the SOT in perpendicularly magnetized Pt/Co/AlO x and \nPt/Co/W films. Using the harmonic measurements , we have characterized the effective fields \ncorresponding to the damping like torque and the field like torque and we have found obvious \ndamping like torque in the Pt/Co/W films while the field like torque was small . It is much different \nfrom the results in the Pt/Co/AlO x films, in which both the damping like torque and the field like \ntorque are strong . According to the weak asymmetrical properties in Pt/Co/W, we propose \nthe Rashba effect in this film is weak. \nACKNOWLEDGEMENT \nThis work was partially supported by the National Basic Research Program of China \n(2015CB921502), the National Science Foundation of China ( Grant Nos. 51731003, 61404125, \n51471029 , 51671019 , 11574027, 51501007, 51602022, 61674013, 51602025 ), and the \nFundamental Research Funds for the Central Universities (FRF -GF-17-B6). \nREFERENCES \n[1] Brataas A, Kent A D and Ohno H 2012 Nat. Mater . 11 372–381. \n[2] Liu L Q, Lee O J, Gudmundsen T J, Ralph D C and Buhrman R A 2012 Phys. Rev. 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Lett . 110 262401 . \n \n \n 10 \n Figure captions \n \nFigure 1 (a) Schematic structure of the multilayers. (b) SEM image of a Hall bar and \nmeasureme nt setup . (c) The M-H curves of the two samples. (d) RH-H curves of the two samples . 11 \n \n \nFigure 2 RH-I curves in sample I (a) and sample II (b) with varying in-plane magnetic field. 12 \n \n \nFigure 3 The first (V ) and second (V 2) harmonic Hall voltage s of the two samples by sweeping \nthe longitudinal ( Hx) and transverse ( Hy) fields . For sample I the frequency is 365 Hz and the \namplitude is 2.0 mA, and for sample II the frequency is 315 Hz and the amplitude is 2.0 mA. The \nresults were measured with out -of-plane magne tization component M Z>0. 13 \n \n \nFigure 4 (a) and (b) The R H- curves of the two samples. Black squares are experimental raw \ndata and the red lines are fitted curves. (c) and (d) T he damping like field ( HD) and field like \neffective field ( HF) versus applied current of the two samples . \n \n \n " }, { "title": "1803.10064v2.Dynamics_of_a_Magnetic_Needle_Magnetometer__Sensitivity_to_Landau_Lifshitz_Gilbert_Damping.pdf", "content": "arXiv:1803.10064v2 [physics.gen-ph] 19 Oct 2018Dynamics of a Magnetic Needle Magnetometer: Sensitivity to\nLandau–Lifshitz–Gilbert Damping\nY. B. Band1,2, Y. Avishai2,3,4, Alexander Shnirman3,5,6\n1Department of Chemistry, Department of Physics,\nDepartment of Electro-Optics, and the Ilse Katz Center for N ano-Science,\nBen-Gurion University, Beer-Sheva 84105, Israel\n2New York University and the NYU-ECNU Institute of Physics at NYU Shanghai,\n3663 Zhongshan Road North, Shanghai, 200062, China\n3Department of Physics, and the Ilse Katz Center for Nano-Sci ence,\nBen-Gurion University, Beer-Sheva 84105, Israel\n4Yukawa Institute for Theoretical Physics, Kyoto, Japan\n5Institut f¨ ur Theorie der Kondensierten Materie,\nKarlsruhe Institute of Technology, D-76128 Karlsruhe, Ger many\n6Institute of Nanotechnology, Karlsruhe Institute of Techn ology, D-76344 Eggenstein-Leopoldshafen, Germany\nAn analysis of a single-domain magnetic needle (MN) in the pr esence of an external magnetic\nfieldBis carried out with the aim of achieving a high precision magn etometer. We determine the\nuncertainty ∆ Bof such a device due to Gilbert dissipation and the associate d internal magnetic\nfield fluctuations that give rise to diffusion of the MN axis dir ectionnand the needle orbital angular\nmomentum. The levitation of the MN in a magnetic trap and its s tability are also analyzed.\nA rigid single-domain magnet with large total spin,\ne.g.,S≃1012/planckover2pi1, can be used as a magnetic needle magne-\ntometer (MNM). Recently Kimball, Sushkov and Budker\n[1] predicted that the sensitivity of a precessing MNM\ncan surpass that of present state-of-the-art magnetome-\nters by orders of magnitude. This prediction motivates\nour present study of MNM dynamics in the presence of\nan external magnetic field B. Such analysis requires in-\nclusion of dissipation of spin components perpendicular\nto the easy magnetization axis (Gilbert damping). It is\ndue to interactions of the spin with internal degrees of\nfreedom such as lattice vibrations (phonons), spin waves\n(magnons), thermal electric currents, etc. [2, 3]. Once\nthere is dissipation, fluctuations are also present [6], and\nresult in a source of uncertainty that can affect the ac-\ncuracy of the magnetometer. Here we determine the un-\ncertainty in the measurement of the magnetic field by a\nMNM. We also analyze a related problem concerning the\ndynamics of the needle’s levitation in an inhomogeneous\nmagnetic field, e.g., a Ioffe-Pritchard trap [8].\nThe Hamiltonian for a MN, treated asa symmetric top\nwith body-fixed moments of inertia IX=IY≡ I ��negationslash=IZ,\nsubject to a uniform magnetic field Bis,\nH=1\n2IˆL2+(1\n2IZ−1\n2I)ˆL2\nZ\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nHR−(ω0//planckover2pi1)(ˆS·ˆn)2\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nHA−ˆµ·B/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nHB,\n(1)\nwhere a hat denotes quantum operator. In the rotational\nHamiltonain HR,ˆLis the orbital angular momentum op-\nerator and ˆLZ=ˆL·ˆZis its component along the body-\nfixed symmetry axis. ˆSis the needle spin angular mo-\nmentum operator, and ˆnis the operator for nthat is the\nunit vector in the direction of the easy magnetization\naxis. The frequency appearing in the anisotropy Hamil-\ntonianHA[4] isω0= 2γ2KS/V, whereKis the strengthofthe anisotropy, Vis the needle volume, and γ=gµB//planckover2pi1\nis the gyromagnetic ratio, in which µBis the Bohr mag-\nnetron, and gis theg-factor (taken to be a scalar for\nsimplicity). In the expression for the Zeeman Hamilto-\nnianHB,ˆµ=gµBˆSis the magnetic moment operator.\nThe Heisenberg equations of motion are\n˙ˆS=−gµBB׈S+2ω0\n/planckover2pi1(ˆS׈n)(ˆS·ˆn),(2)\n˙ˆL=-2ω0\n/planckover2pi1(ˆS׈n)(S·ˆn), (3)\n˙ˆJ=−gµBB׈S, (4)\n˙ˆn=I−1\n/planckover2pi1[ˆL׈n+i/planckover2pi1ˆn], (5)\nwhereˆJ=ˆL+ˆSis the total angularmomentum operator\nandIis the moment of inertia tensor.\nThe dynamics of a MN can be treated semiclassically\nbecause Sis very large. A mean–field approximation\n[9–11] is obtained by taking quantum expectation values\nof the operator equations and assuming that for a given\noperator ˆA, the inequality/radicalBig\n∝angbracketleftˆA2∝angbracketright−∝angbracketleftˆA∝angbracketright2≪ |∝angbracketleftˆA∝angbracketright|holds,\n(an assumption warranted for large S). Hence, the ex-\npectation values of a product of operators on the RHS\nof Eqs. (2)-(5) can be replaced by a product of expecta-\ntion values. The semiclassical equations are equivalent\nto those obtained in a classical Lagrangian formulation.\nDissipation is accounted for by adding the Gilbert term\n[2, 4]−αS×(˙S//planckover2pi1−Ω×S//planckover2pi1) to the RHS of the expecta-\ntion value of Eq. (2) and subtracting it from the RHS of\nEq. (3). Here αis the dimensionless friction parameter,\nand the term Ω×Stransforms from body fixed to space\nfixed frames. Note that Gilbert damping is due to inter-\nnalforces, hence Jis not affected and Eq. (4) remains\nintact.2\nIt is useful to recast the semiclassical dynamical equa-\ntions of motion in reduced units by defining dimension-\nless vectors: the unit spin m≡S/S, the orbital angu-\nlar momentum ℓ≡L/S, the total angular momentum,\nj=m+ℓand the unit vector in the direction of the\nmagnetic field b=B/B:\n˙m=ωBm×b+ω0(m×n)(m·n)−αm×(˙m−Ω×m),(6)\n˙ℓ=−ω0(m×n)(m·n)+αm×(˙m−Ω×m),(7)\n˙n=Ω×n, (8)\n˙j=ωBm×b, (9)\nwhere the angular velocity vector Ωis given by\nΩ= (ω3−ω1)(ℓ·n)n+ω1ℓ\n= (ω3−ω1)[(j−m)·n]n+ω1(j−m).(10)\nHereωB=γ|B|is the Larmor frequency, ω1=S/IX,\nandω3=S/IZ. Similar equations were obtained in\nRef. [5], albeit assuming that the deviations of n(t) and\nm(t) frombare small. We show below that the dynam-\nics can be more complicated than simply precession of\nthe needle about the magnetic field, particularly at high\nmagnetic fields where nutation can be significant.\nFor the numerical solutions presented below we are\nguided by Ref. 1, which uses parameters for bulk cobalt,\nand take ω1= 100 s−1,ω3= 7000 s−1, anisotropy fre-\nquencyω0= 108s−1, Gilbert constant α= 0.01, tem-\nperature T= 300 K, and N=S//planckover2pi1= 1012. First, we elu-\ncidate the effects of Gilbert dissipation, and consider the\nshorttimebehaviorin aweakmagneticfield, ωB= 1s−1.\nThe initial spin direction is intentionally chosen notto be\nalong the easy magnetic axis; n(0) = (1 /2,1/√\n2,1/2),\nm(0) = (1 /√\n2,1/√\n2,0),ℓ(0) = (0 ,0,0). Figure 1(a)\nshows the fast spin dissipation as it aligns with the easy\naxis of the needle, i.e., m(t)→n(t) after a short time,\nand Fig. 1(b) shows relaxation of the oscillations in ℓ(t),\nwhileℓx(t) andℓy(t) approach finite values. Figure 1(c)\nshowsthe innerproduct m·n, which clearlytendstounity\nonthe timescaleofthe figure. Increasing αleadsto faster\ndissipation of m(t), but the short-time saturation values\nof bothm(t) andℓ(t) are almost independent of α.\nWe consider now the long time dynamics (still in\nthe weak field regime) and take the initial value of the\nspin to coincide with the easy magnetization axis, e.g.,\nm(0) =n(0) = (1 /√\n2,1/√\n2,0), with all other param-\neters unchanged. The spin versus time is plotted in\nFig. 2(a). The unit vectors m(t) andn(t) are almost\nidentical, andsincetheir z-componentisnearlyzero,they\nmove together in the x-yplane. In this weak field case,\nthe nutation is small, and the fast small-oscillations due\nto nutation are barely visible. The orbital angular mo-\nmentum dynamics is plotted in Fig. 2(b) [note the differ-\nenttimescalein (a)and(b)] andshowsthat ℓ(t) oscillates\nwith a frequency equal to that of the fast tiny-oscillation\u0001\u0001\u0001\u0002\u0001\u0003\u0001\n\u0001\u0002\u0001\u0003\u0001\u0002\u0004\u0001\u0001\u0002\u0004\u0003\u0001\u0002\u0005\u0001\u0001\u0002\u0005\u0003\u0001\u0002\u0006\u0001\u0004\u0001\u0001\u0002\n-\u0001\u0002\u0003\u0001\u0002\u0003\u0004\u0002\u0001{\u0001\u0001\u0007\u0001\u0002\u0007\u0001\u0003\u0007\u0001}\n\u0001\u0001\u0001\u0002\u0001\u0003\n\u0001\u0002\u0003\u0004\u0002\u0001\u0004\u0002\u0003\u0005\u0002\u0001\u0005\u0002\u0003\u0006\u0002\u0001\u0002\n-\u0001\u0002\u0001\u0005\u0001-\u0001\u0002\u0001\u0004\u0003-\u0001\u0002\u0001\u0004\u0001-\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0004\u0001{\u0001\u0001\u0007\u0001\u0002\u0007\u0001\u0003}\n\u0001\u0002\u0001\u0003\u0001\u0002\u0004\u0001\u0001\u0002\u0004\u0003\u0001\u0002\u0005\u0001\u0001\u0002\u0005\u0003\u0001\u0002\u0006\u0001\u0004\u0001\u0001\u0001\u0001\u0002\u0007\u0001\u0001\u0002\u0007\u0003\u0004\u0002\u0001\u0001{\u0002\u0002\u0003}\nFIG. 1: (color online) (a) The normalized spin vector mver-\nsus time for the low-field case at short times (5 orders of\nmagnitude shorter than in Fig. 2) when the initial spin is\nnot along the fast axis. (b) The reduced orbital angular mo-\nmentum vector ℓ(t). (c) The inner product m(t)·n(t) (the\nprojection of the spin on the fast magnetic axis of the needle .\nofm(t) [the oscillation amplitude is 0 .02|m(t)|]. Fig-\nure 2(c) shows a parametric plot of m(t) versus time.\nThe nutation is clearly very small; the dynamics of m(t)\nconsists almost entirely of precession at frequency ωB.\nFigure 3 shows the dynamics at high magnetic field\n(ωB= 105s−1) with all the other parametersunchanged.\nFigure 3(a) shows mversus time, and now the nutation\nis clearly significant. For the high magnetic field case,\nm(t) is also almost numerically equal to n(t).ℓ(t) is\nplotted in Fig. 3(b). Its amplitude is very large, ℓ(t)≈\n40m(t). However, its oscillation frequency is comparable\nwith that of m(t). In contrast with the results in Fig. 2,\nhere, in addition to precession of the needle, significant\nnutation is present, as shown clearly in the parametric\nplot of the needle spin vector m(t) in Fig. 3(c).\nWe now determine the uncertainty of the MNM due to\ninternal magnetic field fluctuations related to the Gilbert\ndamping. A stochastic force ξ(t), whose strength is de-\ntermined by the fluctuation–dissipation theorem [6], is3\n\u0001\u0001\u0001\u0002\u0001\u0003\u0001\n\u0001\u0002\u0003\u0004\u0005\u0006\u0002\n-\u0005\u0007\u0006-\u0006\u0007\b\u0006\u0007\b\u0005\u0007\u0006{\u0001\u0001\t\u0001\u0002\t\u0001\u0003\t\u0001}\n\u0001\u0001\u0001\u0002\u0001\u0003\n\u0001\u0002\u0003\u0004\u0002\u0001\u0004\u0002\u0003\u0005\u0002\u0001\u0005\u0002\u0003\u0006\u0002\u0001\u0002\n-\u0001\u0002\u0001\u0005\u0001-\u0001\u0002\u0001\u0004\u0003-\u0001\u0002\u0001\u0004\u0001-\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0004\u0001{\u0001\u0001\u0007\u0001\u0002\u0007\u0001\u0003}\nFIG. 2: (color online) Dynamics for thelow-field case ( ωB= 1\ns−1), over relatively long timescales relative to those in Fig. 1.\n(a)mversus time in units of seconds (note that nis indistin-\nguishable from mon the scale of the figure). (b) ℓ(t) (note\nthat it stays small compared to S). (c) Parametric plot of the\nneedle spin vector m(t) showing that nutation is almost im-\nperceptible for small fields [contrast this with the large fie ld\nresult in Fig. 3(c)]; only precession is important.\nadded to Eq. (6), in direct analogy with the treatment of\nBrownianmotionwherebothdissipationandastochastic\nforce are included [12]:\n˙m=m×(ωBb+ξ)+ω0(m×n)(m·n)\n−αm×(˙m−Ω×m). (11)\nξ(t) is internal to the needle and therefore it does not\naffect the total angular momentum jdirectly, i.e., ξ(t)\ndoes not appear in Eq. (9) [since the term −m×ξis also\nadded to the RHS of (7)]. However, as shown below, ξ(t)\naffectsℓas well as m, causing them to wobble stochas-\ntically. This, in turn, makes jstochastic as well via the\nZeeman torque [see Eq. (9)].\nThe fluctuation-dissipation theorem [6] implies\n∝angbracketleftξαξβ∝angbracketrightω≡/integraldisplay\ndt∝angbracketleftξα(t)ξβ(0)∝angbracketrighteiωt\n=δαβαωcoth(/planckover2pi1ω/2kBT)\nN≈δαβ2αkBT\n/planckover2pi1N,(12)\u0001\u0001\u0001\u0002\u0001\u0003\u0001\u0002\u0001\u0003\u0001\u0002\u0001\u0004\u0001\u0002\u0001\u0005\u0001\u0002\u0001\u0006\u0001\u0002\u0007\u0001\u0002\n-\u0007\u0002\u0001-\u0001\u0002\b\u0001\u0002\b\u0007\u0002\u0001{\u0001\u0001\t\u0001\u0002\t\u0001\u0003}\n\u0001\u0001\u0001\u0002\u0001\u0003\n\u0001\u0002\u0001\u0003\u0001\u0002\u0001\u0004\u0001\u0002\u0001\u0005\u0001\u0002\u0001\u0006\u0001\u0002\u0007\u0001\u0002\n-\u0004\u0001-\u0003\u0001\u0003\u0001\u0004\u0001{\u0001\u0001\n\b\u0001\u0002\n\u0000\u0001\u0003}\nFIG. 3: (color online) High-field case ( ωB= 105s−1). (a)\nm(t) [which is almost numerically equal to n(t)]. (b)ℓ(t)\n(note the ordinate axis scale is [ −40,40]). (c) Parametric plot\nof the needle spin vector m(t) showing that strong nutation\noccurs for large fields in addition to precession.\nwhereN=S//planckover2pi1, and the last approximation is ob-\ntained under the assumption that /planckover2pi1ω≪kBT. Note that\nEq. (11) should be solved together with Eqs. (8) and (9).\nThe presence of the anisotropy term in Eq. (11) makes\nnumerical solution difficult for large ω0. Hence, we con-\nsider a perturbative expansion in powers of λ≡ω1/ω0:\nm(t) =n0(t)+λδm(t)+...,n(t) =n0(t)+λδn(t)+...,\nj(t) =j0(t) +λδj(t) +.... Sinceω0is the largest fre-\nquency in the problem, the inequalities αω0≫ωB,ω1,ω3\nhold. Moreover, the Gilbert constant αis large enough\nto effectively pin m(t) ton(t) [hencej(t) =ℓ(t)+m(t)≈\nℓ(t)+n(t)]. Therefore, anadiabaticapproximationtothe\nset of dynamical stochastic equations can be obtained.\nThe zero order term in λreads:\n˙j0=ωBn0×b,˙n0=ω1j0×n0,(13)4\nwhereΩwas approximated by Ω0= (ω3−ω1)(j0·n0−\n1)n0+ω1(j0−n0) in Eqs. (8) and (10) in obtaining (13)\n[7]. The solution to Eqs. (13) [for times beyond which\nGilbert dissipation is significant so m(t)≈n(t)] is very\nclose to that obtained from Eqs. (6)-(8).\nExpanding Eq. (11) in powers of λand keeping only\nthe first order terms (the zeroth order term on the LHS\nvanishes since m0=n0), we get: ω1(δm−δn)×n0=\n˙n0−ωBn0×b+αn0×(˙n0−Ω0×n0)−n0×ξ. Taking\nEq. (13) into account and introducing the notation δη≡\nδm−δn, we obtain\nδη×n0=j0×n0−(ωB/ω1)n0×b−(1/ω1)n0×ξ,(14)\nand from Eqs. (8) and (9) we find\nd\ndtδj=ωB(δn+δη)×b, (15)\nd\ndtδn=ω1(j0−n0)×δn+ω1(δj−δn−δη)×n0\n=ω1j0×δn+ω1(δj−δη)×n0. (16)\nTo first order in λ,δn⊥n0(sincenmust be a unit\nvector), and δm⊥n0, henceδη⊥n0. Therefore, δη×\nb= [j0−(j0·n0)n0]×b+(ωB/ω1)[b−(b·n0)n0]×b+\nω−1\n1[ξ−(ξ·n0)n0]×bon the RHS of Eq. (15) and\nd\ndtδj=ωBδn×b+ωB[j0−(j0·n0)n0]×b\n−ω2\nB\nω1(b·n0)n0×b+ωB\nω1[ξ−(ξ·n0)n0]×b.(17)\nEquations (13), (16) and (17) form a closed system of\nstochastic differential equations [upon using Eq. (14) to\nsubstitute for δη×n0on the RHS of Eq. (16)]. With\nthe largest frequency ω0eliminated, a stable numerical\nsolution is obtained. Moreover, for small magnetic field\n(whereωBis the smallest frequency in the system), an\nanalytic solution of these equations is achievable. To ob-\ntain an analytic solution to Eqs. (13), let us transform\nto the frame rotating around Bwith frequency ωBto\nget equations of the formd\ndτv=d\ndtv+ωBb×v(which\ndefinesτ):\nd\ndτn0=−ω1n0×/parenleftbigg\nn0−j0+ωB\nω1b/parenrightbigg\n,(18)\nd\ndτj0=ωBb×/parenleftbigg\nn0−j0+ωB\nω1b/parenrightbigg\n.(19)\nIf the initial condition is n0(0)−j0(0)+(ωB/ω1)b= 0,\nthen, in the rotating frame j0(τ) andn0(τ) are constant\nvectors. Note that this initial condition is only slightly\ndifferent from the “ordinary” initial condition n0(0) =\nj0(0)since( ωB/ω1)≪1forsmallmagneticfields. Hence,\nin the rotating frame,\nd\ndτδn=ω1n0×(δn−δj+δη),(20)d\ndτδj=−ωBb×(δn−δj+δη).(21)\nWith the special initial conditionbeing satisfied, Eq. (14)\nbecomes δη×n0=−(1/ω1)n0×ξ, and Eqs. (20)-(21)\nbecome a set of first order differential equations with\ntime-independent coefficients. Their solution for initial\nconditions, δn(t= 0) = 0, δj(t= 0) = 0 is,\n/parenleftbiggδn(t)\nδj(t)/parenrightbigg\n=t/integraldisplay\n0dt1exp[C(t−t1)]C/parenleftbiggδη(t1)\n0/parenrightbigg\n,(22)\nwhere the constant matrix C=/parenleftbiggA−A\n−B B/parenrightbigg\nhas di-\nmension 6 ×6 and the 3 ×3 matrices AandBare given by\nAij=−ω1ǫijknk\n0,Bij=−ωBǫijkbk. Without loss of gen-\neralitywecanchoose n0=ˆzandb=ωB(cosθˆz+sinθˆx),\nwhereθis the angle between the easy magnetization\naxis and the magnetic field. In this basis, ∝angbracketleftδηxδηx∝angbracketrightω=\n∝angbracketleftδηyδηy∝angbracketrightω≈ω−2\n0∝angbracketleftξxξx∝angbracketrightω=ω−2\n0∝angbracketleftξyξy∝angbracketrightω=Sa(ω), and\n∝angbracketleftδηzδηz∝angbracketrightω= 0. Here ∝angbracketleftxx∝angbracketrightω≡/integraltext\ndteiωt∝angbracketleftx(t)x(0)∝angbracketrightand [see\nEq. (12)] Sa(ω) =αωcoth(/planckover2pi1ω/2kBT)\nω2\n0N≈2αkBT\nN/planckover2pi1ω2\n0.\nWe are particularly interested in the quantities\n∝angbracketleftδn2\ny(t)∝angbracketright ≡ ∝angbracketleftδny(t)δny(t)∝angbracketrightand∝angbracketleftδj2\ny(t)∝angbracketright ≡ ∝angbracketleftδjy(t)δjy(t)∝angbracketright\nbecause, in the basis chosen above, the y-axis is the di-\nrection of precession of n0aroundb. Using Eq. (22) we\nobtain∝angbracketleftδn2\ny(t)∝angbracketright ≈tω2\n1Sa(ω∼ω1). Assuming the pre-\ncession of nis measured, [or equivalently, the precession\nofm, since they differ only for short timescales of or-\nder (αω0)−1], the uncertainty in the precession angle is\n∝angbracketleft(∆ϕ)2∝angbracketright ≈tω2\n1Sa(ω∼ω1). We thus arrive at our central\nresult: the precision with which the precession frequency\ncan be measured is, ∆ ωB=√\n/angbracketleft(∆ϕ)2/angbracketright\nt≈ω1\nω0/radicalBig\n2αkBT\n/planckover2pi1N1√\nt.\nEquivalently, the magnetic field precision is,\n∆B=∆ωB\nγ≈/planckover2pi1\ngµBω1\nω0/radicalbigg\n2αkBT\n/planckover2pi1N1√\nt.(23)\nFor the parameters used in this paper we find ∆ B≈\n5×10−18√\nt[s]Tesla (independent of ωB). This result should\nbe compared with the scaling ∆ B∝t−3/2obtained in\nRef. 1. Therein, the initial uncertainty of the spin di-\nrection relative to the needle axis was estimated from\nthe fluctuation-dissipation relation and the deterministic\nprecession resulted in the t−3/2scaling of the precession\nangle uncertainty (in addition this angle was assumed to\nbe small). In contrast, we consider the uncertainty ac-\nquired due to Gilbert dissipation duringthe precession,\nallowing the precession angle to be large. Thus, the stan-\ndard1/√\ntdiffusion scalingis obtained and dominates for\ntimes that are even much longer than those considered\nin Ref. 1.\nIntheSupplementalMaterial[13]wediscussthreerele-\nvant related issues. (a) The time at which diffusion stops\nbecause equipartition is reached (we estimate the time5\nwhen the energy stored in stochastic orbital motion be-\ncomes of order kBT). (b) The uncertainty of the mag-\nnetic field for experiments in which the fast precession of\nnaroundjis averaged out in the measurement, and the\ndiffusion of jdetermines ∆ B. (c) We consider the related\nproblem of the dynamics and stability of a rotating MN\nin an inhomogeneous field (e.g., levitron dynamics in a\nIoffe-Pritchard trap [14, 15]).\nIn conclusion, we show that ∆ Bdue to Gilbert damp-\ning is very small; external noise sources, as discussed in\nRef. [1], will dominate over the Gilbert noise for weak\nmagnetic fields. A closed system of stochastic differen-\ntial equations, (13), (16) and (17), can be used to model\nthe dynamics and estimate ∆ Bfor large magnetic fields.\nA rotating MN in a magnetic trap can experience levi-\ntation, although the motion does not converge to a fixed\npoint or a limit cycle; an adiabatic–invariant stability\nanalysis confirms stability [13].\nThis work was supported in part by grants from the\nDFG through the DIP program (FO703/2-1). Useful\ndiscussions with Professor Dmitry Budker are gratefully\nacknowledged. A. S. was supported by DFG Research\nGrant No. SH 81/3-1.\n[1] D. F. J. Kimball, A. O. Sushkov, and D. Budker, Phys.\nRev. Lett. 116, 190801 (2016).\n[2] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443\n(2004)\n[3] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjet. 8153\n(1935). In L. D. Landau, Collected Papers. Ed. by D. ter\nHaar, (Gordon and Breach, New York, 1967), p. 101.\n[4] W. F. Brown Jr., Phys. Rev. 130, 1677 (1963).\n[5] H. Keshtgar, et al., Phys. Rev. B 95, 134447 (2017).\n[6] H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951).\n[7] We note in passing that Eqs. (13) are equivalent to the\nequations of motion of a symmetric top in a gravita-\ntional field when the top is anchored at a point a=an\non its axis a distance afrom the center of mass. The\nequations of motion are: dL/dt=T, where Land\nT=an×(−mgz) are taken with respect to the fixedpoint, and dn/dt=Ω×n. The angular velocity is given\nbyΩ=I−1\n1[L−(L·n)n] +I−1\n3(L·n)n, where the mo-\nments of inertia ( I1,I1,I3) are calculated relative to the\nfixed point. Introducing a characteristic scale L0so that\nL=L0j(jis not a unit vector and its length is not\nconserved) we obtain Eqs. (13) with ωB=mga/L 0and\nω1=L0/I1. Here, the analog of the magnetic field is the\ngravitational field and the analog of bisz.\n[8] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87,\n3989 (2000), and references therein.\n[9] O. Zobay and B. M. Garraway, Phys. Rev. A 61, 033603\n(2000); J. Liu, L. Fu, B.-Y. Ou, S.-G. Chen, D.-I. Choi,\nB. Wu, and Q. Niu, Phys. Rev. A 66, 023404 (2002).\n[10] Y. B. Band, I. Tikhonenkov, E. Pazyy, M. Fleischhauer,\nand A. Vardi, J. of Modern Optics 54, 697-706 (2007).\n[11] Y. B. Band, Phys. Rev. E 88, 022127 (2013); Y. B. Band\nand Y. Ben-Shimol, Phys. Rev. E 88, 042149 (2013).\n[12] H. P. Breuer and F. Petruccione, The Theory of\nOpen Quantum Systems (Oxford University, Cambridge,\n2002); M. Schlosshauer, Decoherence and the Quantum-\nto-Classical Transition (Springer, Berlin, 2007).\n[13] See Supplemental Material at\nhttp://link.aps.org/supplemental/10.1103/PhysRevLet t.121.160801\nwhich contains a discussion of the three issues enumer-\nated in the text, and which includes Refs. 16-20.\n[14] M. V. Berry, Proc. R. Soc. A 452, 1207 (1996).\n[15] A movie showing the dynamics of a Levitron can be seen\nathttps://www.youtube.com/watch?v=wyTAPW_dMfo .\n[16] Y. B. Band, Y. Avishai, A. Shnirman, “Dynamics of a\nMagnetic Needle Magnetometer: Sensitivity to Landau–\nLifshitz–Gilbert Damping”, Phys. Rev. Lett. (to be pub-\nlished).\n[17] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87,\n3989 (2000), and references therein; D. E. Pritchard,\nPhys. Rev. Lett. 51, 15 (1983).\n[18] C. C.Rusconi, V.P¨ ochhacker, K.Kustura, J.I.Ciracan d\nO. Romero-Isart, Phys. Rev. Lett. 119, 167202 (2017);\nC. C. Rusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac\nand O. Romero-Isart, Phys. Rev B 96, 134419 (2017); C.\nC. Rusconi and O. Romero-Isart, Phys. Rev B 93, 054427\n(2016).\n[19] S. Earnshaw, Trans. Camb. Phil. Soc. 7, 97-112 (1842).\n[20] D. R. Merkin, Introduction to the Theory of Stability ,\n(Springer–Verlag, New York, 1997); F. Verhulst, Non-\nlinear Differential Equations and Dynamical Systems ,\n(Springer–Verlag, Berlin, 1990).arXiv:1803.10064v2 [physics.gen-ph] 19 Oct 2018Supplemental Material for “Dynamics of a Magnetic Needle Ma gnetometer:\nSensitivity to Landau–Lifshitz–Gilbert Damping”\nY. B. Band1,2, Y. Avishai2,3,4, Alexander Shnirman3,5,6\n1Department of Chemistry, Department of Physics,\nDepartment of Electro-Optics, and the Ilse Katz Center for N ano-Science,\nBen-Gurion University, Beer-Sheva 84105, Israel\n2New York University and the NYU-ECNU Institute of Physics at NYU Shanghai,\n3663 Zhongshan Road North, Shanghai, 200062, China\n3Department of Physics, and the Ilse Katz Center for Nano-Sci ence,\nBen-Gurion University, Beer-Sheva 84105, Israel\n4Yukawa Institute for Theoretical Physics, Kyoto, Japan\n5Institut f¨ ur Theorie der Kondensierten Materie,\nKarlsruhe Institute of Technology, D-76128 Karlsruhe, Ger many\n6Institute of Nanotechnology, Karlsruhe Institute of Techn ology, D-76344 Eggenstein-Leopoldshafen, Germany\nIn this supplemental material we expand the discussion of the main t ext [1] and address the following three issues.\n(a) The time τeat which the diffusion of the magnetic needle axis direction nand the magnetic needle orbital\nangular momentum ℓstops because equipartition is reached, i.e., we estimate the time req uired for the energy stored\nin stochastic orbital motion to become of order kBT. (b) The uncertainty ∆ Bof the magnetic field for experiments\nin which the fast precession of naroundjis averaged out in the measurement process and the uncertainty ∆ Bis\ndetermined by the diffusion of j. (c) The dynamics of a magnetic needle in an inhomogeneous field, e.g., levitron\ndynamics of a rotating magnetic needle in a Ioffe-Pritchard trap [2], s ee Refs. [3–5].\n(a):τecan be estimated by noting that the diffusion determined in [1] stops once equipartition is reached. The\nenergy ∆ Estored in stochastic orbital motion is given by\n∆E∼/planckover2pi1ω1N/angbracketleftδℓ2/angbracketright, (1)\nwhere where N=S//planckover2pi1(note that δj−δn=δℓ). By requiring ∆ E∼kBTwe can estimate that the diffusion given\nby Eqs. (20-21) of [1] stops when τe∼ω2\n0/(αω3\n1) (this result can also be obtained by expanding Eq. (11) further in\npowers of λ≡ω1/ω0). For the parameters used in [1] this is an extremely long time ( τe∼1012s∼5 years). Hence,\nwe conclude that the diffusion of Eqs. (20-21) and the error estima tes given for ∆ Bin Ref. [1] are relevant for all\nreasonable times.\n(b): In [1] we calculate ∆ Bassuming the experimental measurement follows the temporal dyn amics of nandj.\nAn alternative assumption is that the precession of naroundjis averaged out by the measurement process and one\nmeasures the diffusion of j. For the latter we obtain the leading term\n/angbracketleftδj2\ny(t)/angbracketright ≈tω2\nBcos2θSa(ω∼ω1), (2)\nwhereSa(ω) is given in Eq. (23) of [1]. At θ=π/2 the leading contribution obtained in Eq. (2) vanishes and the\nremaining sub-leading term is\n/angbracketleftδn2\ny(t)/angbracketright ≈t2ω4\nB\nω2\n1Sa(ω∼ω1), (3)\nhence for θ/negationslash=π/2 we obtain\n∆B=∆ωB\nγ≈/planckover2pi1\ngµBωB\nω0cosθ/radicalbigg\n2αkBT\n/planckover2pi1N1√\nt, (4)\nwhereas at θ=π/2,\n∆B=∆ωB\nγ≈/planckover2pi1\ngµBω2\nB\nω0ω1/radicalbigg\n4αkBT\n/planckover2pi1N1√\nt. (5)\nTakingωB= 1s−1we obtain ∆ B≈cosθ×5×10−23√\nt[s]Tesla for θ/negationslash=π/2, and ∆ B≈7×10−25√\nt[s]Tesla for θ=π/2.2\n(c): A rotating magnet can be levitated in an inhomogeneous magnet ic field [3–5]. This is possible despite Earn-\nshaw’s theorem [6] from which one can conclude that levitation of a non-rotating ferromagnetin a static magnetic field\nis not possible. Two important factors regarding magnetic levitation are the forces on the magnet and its stability\n(ensuring that it does not spontaneously slide or flip into a configura tion without lift). The dynamics of a magnetic\nneedle in an inhomogeneous magnetic field can be modelled using Eqs. (6 ), (7) and (8) of [1] augmented by the\nequations of motion for the center of mass (CM) degrees of freed om of the needle,\n˙p=∇(µ·B(r)), (6)\n˙r=p/m , (7)\nwhererandpare the needle CM position and momentum vectors. Our numerical re sults show levitation of the\nmagnetic needle when the initial rotational angular momentum vecto r of the needle is sufficiently large and points\nin the direction of magnetic field at the center of the trap. We shall s ee that the dynamical variables do not evolve\nto a fixed point or a simple cyclic orbit. Moreover, a linear stability analy sis yields a 15 ×15 Jacobian matrix with\neigenvalues having a positive real part, so the system is unstable. However, a stability analysis of the system using\nthe adiabatic invariant |µ||B|[3] does yield a stable fixed point (contrary to the full numerical re sults which show a\nmore complicated levitation dynamics).\nFigure 1 shows the dynamics of the system over time in the trap. We u se the same magnetic needle parameters\nused in Fig. 2 of [1] and a Ioffe-Prichard magnetic field [2]\nB(r) =ex/parenleftbigg\nB′x−B′′\n2xz/parenrightbigg\n+ey/parenleftbigg\nB′y−B′′\n2zy/parenrightbigg\n+ez/parenleftbigg\nB0+B′′\n2(z2−x2+y2\n2)/parenrightbigg\n, (8)\nwith field bias B0, gradient B′, and curvature B′′parameters chosen so that the Zeeman energy and its variation ov er\nthe trajectory of the needle in the trap are substantial (as is clea r from the results shown in the figure). We start\nthe dynamics with initial conditions: r(0) = (0,0,0),p(0) = (0,0,0),m(0) = (0,0.0011/2,−(1−0.001)1/2) (almost\nalong the −zdirection), n(0) =m(0),ℓ(0) = (0 ,0,0.001) [this is large orbital angular momentum since ℓis the\norbital angular momentum divided by S]. Figure 1(a) shows the needle CM position r(t) versus time. Fast and slow\noscillations are seen in the xandymotion, whereas z(t) remains very close to zero. Figure 1(b) shows oscillations of\nthe CM momentum p(t) with time. px(t) andpy(t) oscillate with time, and pz(t) remains zero. Figure 1(c) plots the\nspinm(t) versus time. Initially, m(0) points almost in the −zdirection, and the tip of the needle n(t) =m(t) carries\nout nearly circular motion in the nx-nyplane. Figure 1(d) plots the orbital angular momentum ℓ(t). The components\nℓx(t) andℓy(t) undergo a complicated oscillatory motion in the ℓx(t)-ℓy(t) plane but ℓz(t)≈ℓz(0). Figure 1(e) is a\nparametric plot of m(t); the motion consists of almost concentric rings that are slightly dis placed one from the other.\nThe full dynamics show levitation but they do not converge to a fixed point or a limit cycle.\nQuite generally, for a system of dynamical equations, ˙ yi(t) =fi(y1,...,y n),i= 1,...n, a linear stability analysis\nrequires calculating the eigenvalues of the Jacobian matrix evaluate d at the equilibrium point y∗wheref(y∗) =0,\nJij=/parenleftBig\n∂fi\n∂yj/parenrightBig\ny∗[7]. The system is unstable against fluctuations if any of the eigenvalu es ofJijhave a positive real\npart. Equations (6), (7) and (8) of [1] together with Eqs. (6) and (7) above have a Jacobian matrix with eigenvalues\nwhose real part are positive, so the linear stability test fails. Howev er, if the Zeeman force −∇HZin Eq. (6) is\nreplaced by the gradient of the adiabatic invariant, µ·∇|B(r)|, none of the eigenvalues of the Jacobian matrix have\na positive real part and the system is linearly stable, i.e., the stability a nalysis using the adiabatic-invariant predicts\nstability. Note that substituting the adiabatic invariant for the Zee man energy in the full equations of motion yields\nr(t) andp(t) vectors that are constant with time and n(t),m(t) andℓ(t) are similar to the results obtained with\nthe full equations of motion (but the parametric plot of m(t) is a perfectly circular orbit). Thus, adiabatic–invariant\nstability analysis of a rotating magnetic needle in a magnetic trap confi rms stability of its levitation as obtained in\nthe numerical solution of the dynamical equations.3\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0001\u0005\u0001\u0001\u0004-\u0001\u0005\u0001\u0001\u0003\u0001\u0005\u0001\u0001\u0001\u0001\u0005\u0001\u0001\u0003\u0001\u0005\u0001\u0001\u0004\n\u0001{\u0001\u0001\u0002\u0001\u0003}\u0002\u0003\u0004\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0001\u0005\u0003\u0001-\u0001\u0005\u0001\u0002\u0001\u0005\u0001\u0001\u0001\u0005\u0001\u0002\u0001\u0005\u0003\u0001\n\u0001{\u0001\u0001\u0001\u0001\u0002\u0001\u0001\u0003}\u0002\u0001\u0002\u0002\u0002\u0003\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0003\u0005\u0001-\u0001\u0005\u0006-\u0001\u0005\u0007-\u0001\u0005\b-\u0001\u0005\u0004\u0001\u0005\u0001\n\u0001{\u0001\u0001\u0001\u0001\u0002\u0001\u0001\u0003}\u0002\u0001\u0002\u0002\u0002\u0003\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0001\u0005\u0001\u0001\u0001\u0006-\u0001\u0005\u0001\u0001\u0001\u0004\u0001\u0005\u0001\u0001\u0001\u0001\u0001\u0005\u0001\u0001\u0001\u0004\u0001\u0005\u0001\u0001\u0001\u0006\u0001\u0005\u0001\u0001\u0001\u0007\u0001\u0005\u0001\u0001\u0001\b\u0001\u0005\u0001\u0001\u0003\u0001\n\u0001{\u0001\u0001\u0001\u0001\u0002\u0001\u0001\u0003}\u0002\u0001\u0002\u0002\u0002\u0003\nFIG. 1: (color online) Dynamics of a needle in a Ioffe-Pritcha rd magnetic field. (a) rversus time, (b) pversus time, (c) m\nversus time (note that n(t) is indistinguishable from m(t) on the scale of the figure). (d) ℓversus time (note that |ℓ(t)|is small\ncompared to Sbut rotational angular momentum L(t) =Sℓ(t) is large since S= 1012). (e) Parametric plot of the needle spin\nvectorm(t) (nutation is very small for this case of small magnetic field ).4\n[1] Y. B. Band, Y. Avishai, A. Shnirman, “Dynamics of a Magnet ic Needle Magnetometer: Sensitivity to Landau–Lifshitz–\nGilbert Damping”, Phys. Rev. Lett. (to be published).\n[2] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87, 3989 (2000), and references therein; D. E. Pritchard,\nPhys. Rev. Lett. 51, 15 (1983).\n[3] M. V. Berry, Proc. R. Soc. A 452, 1207 (1996).\n[4] A movie a a Levitron can be seen at https://www.youtube.com/watch?v=wyTAPW_dMfo .\n[5] C. C. Rusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac and O. Romero-Isart, Phys. Rev. Lett. 119, 167202 (2017); C. C.\nRusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac and O. Romer o-Isart, Phys. Rev B 96, 134419 (2017); C. C. Rusconi and\nO. Romero-Isart, Phys. Rev B 93, 054427 (2016).\n[6] S. Earnshaw, Trans. Camb. Phil. Soc. 7, 97-112 (1842).\n[7] D. R. Merkin, Introduction to the Theory of Stability , (Springer–Verlag, New York, 1997); F. Verhulst, Nonlinear Differential\nEquations and Dynamical Systems , (Springer–Verlag, Berlin, 1990)." }, { "title": "1908.08629v2.Damping_enhancement_in_coherent_ferrite_insulating_paramagnet_bilayers.pdf", "content": "Damping enhancement in coherent ferrite/insulating-paramagnet bilayers\nJacob J. Wisser,1Alexander J. Grutter,2Dustin A. Gilbert,3Alpha T. N'Diaye,4\nChristoph Klewe,4Padraic Shafer,4Elke Arenholz,4, 5Yuri Suzuki,1and Satoru Emori6,\u0003\n1Department of Applied Physics, Stanford University, Stanford, CA, USA\n2NIST Center for Neutron Research, Gaithersburg, MD, USA\n3Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN, USA\n4Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA, USA\n5Cornell High Energy Synchrotron Source, Ithaca, NY, USA\n6Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA\n(Dated: October 29, 2019)\nHigh-quality epitaxial ferrites, such as low-damping MgAl-ferrite (MAFO), are promising\nnanoscale building blocks for all-oxide heterostructures driven by pure spin current. However, the\nimpact of oxide interfaces on spin dynamics in such heterostructures remains an open question. Here,\nwe investigate the spin dynamics and chemical and magnetic depth pro\fles of 15-nm-thick MAFO\ncoherently interfaced with an isostructural \u00191-8-nm-thick overlayer of paramagnetic CoCr 2O4\n(CCO) as an all-oxide model system. Compared to MAFO without an overlayer, e\u000bective Gilbert\ndamping in MAFO/CCO is enhanced by a factor of >3, irrespective of the CCO overlayer thickness.\nWe attribute this damping enhancement to spin scattering at the \u00181-nm-thick chemically disordered\nlayer at the MAFO/CCO interface, rather than spin pumping or proximity-induced magnetism. Our\nresults indicate that damping in ferrite-based heterostructures is strongly in\ruenced by interfacial\nchemical disorder, even if the thickness of the disordered layer is a small fraction of the ferrite\nthickness.\nI. INTRODUCTION\nEmerging spintronic device schemes leverage magnon\nspin currents in electrically insulating magnetic oxides\n(e.g., ferrites), unaccompanied by dissipative motion\nof electrons, for computing and communications\napplications1,2. Low-dissipation spintronic devices\nbecome particularly attractive if insulating ferrite thin\n\flms with low magnetic damping can serve as sources\nof magnon spin currents. Such low-damping ferrites\ninclude not only epitaxial garnet ferrites (e.g., YIG)3{11\nthat have been widely used in studies of insulating\nspintronics2{4,12{15, but also coherently strained epitaxial\nspinel ferrites16{18with crucial technical advantages over\ngarnets, such as lower thermal budget for crystallization,\nhigher magnon resonance frequencies, and potential to be\nintegrated coherently with other spinels and perovskites\nwith various functionalities19{22.\nIn general, low-damping ferrite thin \flms must be\ninterfaced with other materials to realize spintronic\ndevices. It is therefore essential to understand whether\nand how damping in the ferrite is impacted by the\nproximity to another material. For instance, to convert\nbetween electronic and magnonic signals through direct\nand inverse spin Hall or Rashba-Edelstein e\u000bects23,\nthe low-damping ferrite needs to be interfaced with\na nonmagnetic metal with strong spin-orbit coupling.\nSpin transport and enhanced damping through spin\npumping24in ferrite/spin-orbit-metal structures has\nalready been extensively studied3,4,12{15,25. Moreover,\nthe low-damping ferrite can be interfaced with an\ninsulating antiferromagnetic or paramagnetic oxide, in\nwhich signals can be transmitted as a pure magnon\nspin current26{40. While interfacing low-damping ferriteswith insulating anti/paramagnetic oxides has enabled\nprototypes of magnon spin valves37{39, the fundamental\nimpact of insulating oxide interfaces on spin dynamics\nhas remained mostly unexplored. In particular, it is an\nopen question whether or how damping of the ferrite is\nenhanced from spin dissipation within the bulk of the\nadjacent anti/paramagnetic oxide or from spin scattering\nat the oxide interface.\nHere, we investigate how room-temperature magnetic\ndamping in epitaxial ferrimagnetic spinel MgAl-ferrite\n(MgAl 1=2Fe3=2O4, MAFO) is impacted when interfaced\nwith an overlayer of insulating paramagnetic spinel\nCoCr 2O4(CCO)41,42. This epitaxial MAFO/CCO\nbilayer is an isostructural model system, possessing\na coherent interface with continuous crystal lattices\nbetween the spinel ferrite and paramagnet. We \fnd that\nthe presence of MAFO/CCO interface increases damping\nby more than a factor of >3 compared to MAFO without\nan overlayer. We attribute this damping enhancement {\nwhich is comparable to or greater than spin pumping\ne\u000bects reported for ferrite/spin-orbit-metal bilayers { to\nspin scattering by the ultrathin ( \u00181 nm) chemically\ndisordered layer at the MAFO/CCO interface. Our\n\fndings show that spin scattering at oxide interfaces\nhas a profound in\ruence on damping, even when the\nchemically disordered layer is a small fraction of the total\nmagnetic layer thickness.\nII. FILM GROWTH AND STRUCTURAL\nCHARACTERIZATION\nEpitaxial thin \flms of 15-nm-thick MAFO interfaced\nwith 1.3-8 nm of CCO overlayer were grown on as-arXiv:1908.08629v2 [cond-mat.mtrl-sci] 26 Oct 20192\n40 42 44 46MAO (004)\nMAFO/CCO\n (004)\nMAFO (004)log10(Intensity) (arb. units)\n2q (deg)CCO (004)\n-0.180 -0.175 -0.1700.570.580.590.600.610.620.63-\n \n (115)-(a) (c)\nqip(Å-1)qop(Å-1)\nCCO\n(25 nm)MAO(b)\n-0.2-0.1 0.00.10.2MAFOCCOIntensity (arb. units)\nDw004 (deg)MAFO/CCO\nFigure 1. (a) 2 \u0012-!scans of epitaxial MAFO(15 nm), CCO(25 nm), and MAFO(15 nm)/CCO(8 nm). The data are o\u000bset for\nclarity. (b) Rocking curve scans about the (004) \flm peak for the \flms shown in (a). (c) Reciprocal space map of epitaxial\nCCO(25 nm) coherently strained to the MAO substrate.\nreceived single-crystal MgAl 2O4(MAO) substrates via\npulsed laser deposition. A KrF 248 nm laser was\nincident on stoichiometric targets of MAFO and CCO\nwith \ruences of \u00191.5 J/cm2and\u00191.3 J/cm2,\nrespectively. Both \flms were grown in 10 mTorr (1.3\nPa) O 2and were cooled in 100 Torr (13 kPa) O 2.\nMAFO \flms were grown at 450\u000eC, whereas CCO \flms\nwere deposited at 300\u000eC in an attempt to minimize\nintermixing between the MAFO and CCO layers. These\ngrowth temperatures, much lower than >700\u000eC typically\nrequired for epitaxial garnets3{11, are su\u000ecient to fully\ncrystallize MAFO and CCO. The low crystallization\ntemperatures of the spinels o\u000ber an advantage over\nthe oft-studied garnets, with more opportunities for\nisostructural integration with coherent interfaces. The\nMAFO \flms exhibit a room-temperature saturation\nmagnetization of\u0019100 kA/m and a Curie temperature of\n\u0019400 K18. To obtain consistent ferromagnetic resonance\nresults, MAFO \flms were grown and subsequently\ncharacterized by ferromagnetic resonance (FMR) ex-situ;\nafter surface cleaning with ultrasonication in isopropanol,\nCCO overlayers were then deposited as described above.\nGrowth rates were calibrated via X-ray re\rectivity.\nOur structural characterization of MAFO and\nCCO shows high-quality, coherently strained \flms.\nIn symmetric 2 \u0012-!X-ray di\u000braction scans, only\npeaks corresponding to the (00 `) re\rections are\nobserved, indicating that the \flms are highly epitaxial.\nAdditionally, as seen in Fig. 1(a), Laue oscillations\naround the (004) Bragg re\rections in both single-layer\nMAFO and CCO layers as well as MAFO/CCO bilayers\ndenote smooth interfaces. Furthermore, MAFO, CCO,\nand MAFO/CCO samples all exhibit essentially the\nsame \flm-peak rocking curve widths (FWHM) of \u00190.06\u000e\n(Fig. 1(b)). Reciprocal space mapping of the ( \u00161\u001615)\nre\rection in 25-nm-thick single-layer CCO on MAO\n(Fig. 1(c)) reveals that the in-plane lattice parameter of\nthe \flm coincides with that of the substrate, indicating\nCCO is coherently strained to MAO. We note thatdespite the relatively large lattice mismatch between\nCCO and MAO of \u00193 %, coherently strained growth of\nCCO of up to 40 nm has been previously reported on\nMAO substrates41. For our CCO \flm, we calculate an\nout-of-plane lattice constant c\u00198:534\u0017A from the 2 \u0012-!\nscan; taking the in-plane lattice parameter a= 8:083\u0017A of\nthe MAO substrate, the resulting tetragonal distortion of\ncoherently strained CCO is c=a\u00191:055, similar to that\nfor coherently strained MAFO18.\nStructural characterization results underscore the\nquality of these epitaxial \flms grown as single layers and\nbilayers. Considering the comparable high crystalline\nquality for MAFO, CCO, and MAFO/CCO { as\nevidenced by the presence of Laue oscillations and narrow\n\flm-peak rocking curves { we conclude that MAFO/CCO\nbilayers (with the total thickness limited to \u001423 nm) are\ncoherently strained to the substrate. In these samples\nwhere the substrate and \flm layers are isostructural, we\nalso do not expect antiphase boundaries43{46. Indeed,\nwe \fnd no evidence for frustrated magnetism, i.e., high\nsaturation \feld and coercivity, that would arise from\nantiphase boundaries in spinel ferrites43{46; MAFO/CCO\nbilayers studied here instead exhibit soft magnetism, i.e.,\nsquare hysteresis loops with low coercivity <0.5 mT,\nsimilar to our previous report on epitaxial MAFO thin\n\flms18. Thus, MAFO/CCO is a high-quality all-oxide\nmodel system, which permits the evaluation of how spin\ndynamics are impacted by a structurally clean, coherent\ninterface.\nIII. FERROMAGNETIC RESONANCE\nCHARACTERIZATION OF DAMPING\nTo quantify e\u000bective damping in coherently strained\nMAFO(/CCO) thin \flms, we performed broadband\nFMR measurements at room temperature in a coplanar\nwaveguide setup using the same procedure as our prior\nwork16,18. We show FMR results with external bias3\nmagnetic \feld applied in the \flm plane along the [100]\ndirection of MAFO(/CCO); essentially identical damping\nresults were obtained with in-plane \feld applied along\n[110]47. Figure 2(a) shows the frequency fdependence of\nhalf-width-at-half-maximum (HWHM) linewidth \u0001 Hfor\na single-layer MAFO sample and a MAFO/CCO bilayer\nwith a CCO overlayer thickness of just 1.3 nm, i.e., less\nthan 2 unit cells. The linewidth is related to the e\u000bective\nGilbert damping parameter \u000beffvia the linear equation:\n\u0001H= \u0001H0+h\u000beff\ng\u00160\u0016Bf (1)\nwhere \u0001H0is the zero-frequency linewidth, his Planck's\nconstant,g\u00192:05 is the Land\u0013 e g-factor derived from the\nfrequency dependence of resonance \feld HFMR ,\u00160is the\npermeability of free space, and \u0016Bis the Bohr magneton.\nIt is easily seen from Fig. 2(a) that with the addition\nof ultrathin CCO, the damping parameter is drastically\nincreased, i.e., >3 times its value in bare MAFO.\nFigure 2(b) shows that the damping enhancement\nseen in MAFO/CCO is essentially independent of\nthe CCO thickness. This trend suggests that\nthe damping enhancement is purely due to the\nMAFO/CCO interface, rather than spin dissipation in\nthe bulk of CCO akin to the absorption of di\u000busive\nspin current reported in antiferromagnetic NiO26,35,48.\nWe note that other bulk magnetic properties of\nMAFO (e.g., e\u000bective magnetization, Land\u0013 e g-factor,\nmagnetocrystalline anisotropy) are not modi\fed by the\nCCO overlayer in a detectable way. We also rule\nout e\u000bects from solvent cleaning prior to CCO growth\nor thermal cycling in the deposition chamber up to\n300\u000eC, as subjecting bare MAFO to the same ex-\nsitu cleaning and in-situ heating/cooling processes as\ndescribed in Section II, but without CCO deposition,\nresults in no measurable change in damping. The\ndamping enhancement therefore evidently arises from the\nproximity of MAFO to the CCO overlayer.\nWe consider two possible mechanisms at the\nMAFO/CCO interface for the observed damping\nenhancement:\n(1) Spin current excited by FMR in MAFO\nmay be absorbed via spin transfer in an interfacial\nproximity-magnetized layer49of CCO, whose magnetic\nmoments may not be completely aligned with those of\nMAFO. While CCO by itself is paramagnetic at room\ntemperature, prior studies have shown that Co2+and\nCr3+cations in epitaxial CCO interfaced with a spinel\nferrite (e.g., Fe 3O4) can develop measurable magnetic\norder50. Such damping enhancement due to interfacial\nmagnetic layer is analogous to spin dephasing reported\nfor ferromagnets interfaced directly with proximity-\nmagnetized paramagnetic metal (e.g., Pt, Pd)49.\n(2) Even if CCO does not develop proximity-induced\nmagnetism, chemical disorder at the MAFO/CCO\ninterface may enhance spin scattering. For instance,\nchemical disorder may lead to an increase of Fe2+\n0 10 20 300246810HWHM Linewidth (mT)\nFrequency (GHz)(a)\n(b)MAFO/CCO\neff≈ 0.007\nMAFO\neff≈ 0.002\n0 2 4 6 80.0000.0020.0040.0060.0080.0100.012eff\nCCO thickness (nm)Figure 2. (a) HWHM FMR linewidth versus frequency\nfor MAFO(15 nm) and MAFO(15 nm)/CCO(1.3 nm). The\ne\u000bective Gilbert damping parameter \u000beffis derived from\nthe linear \ft. (b) \u000beffplotted against the CCO overlayer\nthickness. The dashed horizontal line indicates the average of\n\u000befffor MAFO without an overlayer.)\ncations at the MAFO surface, thereby increasing\nthe spin-orbit spin scattering contribution to Gilbert\ndamping in MAFO compared to its intrinsic composition\ndominated by Fe3+with weak spin-orbit coupling18,51.\nAnother possibility is that chemical disorder at the\nMAFO/CCO interface introduces magnetic roughness\nthat gives rise to additional spin scattering, perhaps\nsimilar to two-magnon scattering recently reported for\nferromagnet/spin-orbit-metal systems52.\nIn the following section, we directly examine interfacial\nproximity magnetism and chemical disorder to gain\ninsight into the physical origin of the observed damping\nenhancement in MAFO/CCO.\nIV. CHARACTERIZATION OF INTERFACE\nCHEMISTRY AND MAGNETISM\nTo evaluate the potential formation of a magnetized\nlayer in the interfacial CCO through the magnetic\nproximity e\u000bect, we performed depth-resolved\nand element-speci\fc magnetic characterization\nof MAFO/CCO bilayers using polarized neutron\nre\rectometry (PNR) and soft magnetic X-ray\nspectroscopy. PNR measurements were performed\nusing the PBR instrument at the NIST Center for\nNeutron Research on nominally 15-nm-thick MAFO\nlayers capped with either thick (5 nm) or thin (3 nm)4\nCCO overlayers. PNR measurements were performed in\nan in-plane applied \feld of 3 T at temperatures of 300\nK and 115 K, the latter case being slightly above the\nnominal 97 K Curie temperature of CCO41,42. Incident\nneutrons were spin-polarized parallel or anti-parallel to\nthe applied \feld both before and after scattering from\nthe sample, and the re\rected intensity was measured\nas a function of the perpendicular momentum transfer\nvector Q. The incident spin state of measured neutrons\nwere retained after scattering, corresponding to the\ntwo non-spin-\rip re\rectivity cross sections ( \"\"and##).\nSince all layers of the \flm are expected to saturate well\nbelow the applied \feld of 3 T, no spin-\rip re\rectivity is\nexpected and these cross sections were not measured.\nSince PNR is sensitive to the depth pro\fles of the\nnuclear and magnetic scattering length density (SLD),\nthe data can be \ftted to extract the chemical and\nmagnetic depth pro\fles of the heterostructure. In this\ncase, we used the Re\r1D software package for this\npurpose53. Figure 3(a,b) shows the 300 K re\rectivities\nand spin asymmetry curves of a nominal MAFO (15\nnm)/CCO (5 nm) sample alongside the depth pro\fle\n(Fig. 3(c)) used to generate the \fts shown. The\nbest \ft pro\fle (Fig. 3(c)) provides no evidence of a\nlayer with proximity-induced magnetization in the CCO.\nRather, we note that there appears to be a layer of\nmagnetization suppression near both the MAO/MAFO\nand MAFO/CCO interfaces. Further, the interfacial\nroughnesses of both the MAO/MAFO and MAFO/CCO,\n0.9(1) nm and 1.35(5) nm respectively, are signi\fcantly\nlarger than the CCO surface roughness of 0.27(3) nm\nand the bare MAFO surface roughness of <\u00180.5 nm54.\nThe interfacial roughnesses are signatures of chemical\nintermixing at the spinel-spinel interface leading to\ninterfacial suppression of the magnetization and/or Curie\ntemperature. Thus, we \fnd that the MAFO/CCO\ninterface, although structurally coherent, exhibits a\nchemically intermixed region on the order of one spinel\nunit cell thick on either side.\nTo obtain an upper limit of the proximity-induced\ninterfacial magnetization in CCO, we performed Markov-\nchain Monte-carlo simulations as implemented in the\nDREAM algorithm of the BUMPS python package.\nThese simulations suggest an upper limit (95% con\fdence\ninterval of) 7 emu/cc in the 1.5 nm of the CCO closest\nto the interface. In this case, the model evaluated the\nMAFO as a uniform structural slab but allowed for total\nor partial magnetization suppression at both interfaces,\nwhile the CCO layer was treated as a uniform slab with\nan allowed magnetization layer of variable thickness at\nthe interface.\nHowever, we note that equivalently good \fts are\nobtained using simpler models that \ft a single MAFO\nlayer with magnetically dead layers at the interfaces and\na completely nonmagnetic CCO layer. Equivalent results\nwere obtained for the thick CCO sample at 115 K and\nfor the thin CCO sample. We therefore conclude that the\nPNR results strongly favor a physical picture in which the\nFigure 3. (a) Spin-polarized neutron re\rectivity and (b)\nspin asymmetry of a MAFO (15 nm)/CCO (5 nm) bilayer\nalongside theoretical \fts. (c) Nuclear and magnetic scattering\n(scaled \u000210) length density pro\fle used to generate the \fts\nshown. Error bars represent \u00061 standard deviation.\nCCO is notmagnetized through the magnetic proximity\ne\u000bect.\nTo con\frm the PNR results and examine the e\u000bect\nof a CCO overlayer on the local environment of Fe\ncations in MAFO, we performed temperature-dependent\nX-ray absorption (XA) spectroscopy and X-ray magnetic\ncircular dichroism (XMCD) measurements at Beamline\n4.0.2 of the Advanced Light Source at Lawrence Berkeley\nNational Laboratory. We note that the detection\nmode (total electron yield) used here for XA/XMCD\nis sensitive to the top \u00195 nm of the sample, such that\nFe L edge signals from CCO-capped MAFO primarily\ncapture the cation chemistry near the MAFO/CCO\ninterface. Measurements were performed in an applied\n\feld of 400 mT along the circularly polarized X-ray beam,\nincident at 30\u000egrazing from the \flm plane. To minimize\ndrift e\u000bects during the measurement, multiple successive\nenergy scans were taken and averaged, switching both\napplied \feld direction and photon helicity so that all\nfour possible combinations of \feld direction and helicity\nwere captured at least once. XA and XMCD intensities\nwere normalized such that the pre-edge is zero and\nthe maximum value of the average of the (+) and\n(\u0000) intensities is unity. In the case of the Co L-\nedge, measurements were taken with energy sweeps\ncovering both Fe and Co edges, and for consistency\nboth edges were normalized to the highest XAS signal,\ncorresponding to the Fe L 3-edge.\nFigure 4(a) compares the XA of a bare MAFO \flm5\nFigure 4. (a) 300 K X-ray absorption spectra of MAFO and\nMAFO/CCO (3 nm) grown on MAO. (b) Photon helicity-\ndependent XA spectra and XMCD of the Fe L-edge for a\nMAFO/CCO (3 nm) bilayer at 300 K. (c) Co and (d) Cr\nL-edge XA and XMCD of the same bilayer.\nwith one capped by 3 nm of CCO. The two XA lineshapes\nare nearly identical, indicating the same average Fe\noxidation state and site-distribution in CCO-capped\nand uncapped MAFO \flms. It is therefore likely that\nthe reduced interfacial magnetization observed through\nPNR is a result of a defect-induced Curie temperature\nreduction, rather than preferential site-occupation of Co\nand Cr that might increase the Fe2+content in the\nintermixed interfacial region.\nWe further note that although a large XMCD signal\nis observed on the Fe-edge at 300 K (Fig. 4(b)), neither\nthe Co nor Cr L edges exhibit any signi\fcant magnetic\ndichroism, as shown in Figs. 4(c)-(d). Similar results\nare obtained on the Cr L edge at 120 K. Consistent\nwith the PNR results, we thus \fnd no evidence for\na net magnetization induced in the CCO through the\ninterfacial magnetic proximity e\u000bect.\nOur \fnding of suppressed interfacial magnetism\nin MAFO/CCO is reminiscent of earlier reports\nof magnetic dead layers in epitaxially-grown ferrite-\nbased heterostructures55{57. For example, prior\nPNR experiments have revealed magnetic dead layers\nat the interfaces of ferrimagnetic spinel Fe 3O4and\nantiferromagnetic rock-salt NiO or CoO, even when the\ninterfacial roughness is small (e.g., only 0.3 nm)55,56.\nA magnetic dead layer of 1 spinel unit cell has also\nbeen reported at the interface of Fe 3O4and diamagnetic\nrock-salt MgO grown by molecular beam epitaxy57.\nWe note that in these prior studies, the spinel ferrite\flms interfaced with the rock salts (NiO, CoO, MgO)\npossess antiphase boundaries. Suppressed magnetism\nis known to result from antiphase boundaries, as they\nfrustrate the long-range magnetic order and reduce\nthe net magnetization of the ferrite44. By contrast,\nthere is no evidence for antiphase boundaries in all-\nspinel MAFO/CCO grown on spinel MAO; therefore,\nthe suppressed magnetism at the MAFO/CCO interface\ncannot be attributed to antiphase-boundary-induced\nmagnetic frustration.\nAnother possible scenario is that magnetic dead layer\nformation is a fundamental consequence of the charge\nimbalance between di\u000berent lattice planes, as recently\nshown in a recent report of (polar) Fe 3O4undergoing\natomic reconstruction to avoid \\polar catastrophe\" when\ngrown on (nonpolar) MgO58. In our study on all-\nspinel heterostructures, there may also be some degree of\ncharge mismatch depending on the relative populations\nof cations on the tetrahedrally- and octahedrally-\ncoordinated sites at the MAFO/CCO interface, although\nthe charge mismatch is expected to be only \u0019\u00061, i.e.,\na factor of\u00195-6 smaller than that in MgO/Fe 3O458.\nThus, atomic reconstruction driven by charge imbalance\nappears unlikely as a dominant source of the magnetic\ndead layer in MAFO/CCO. We instead tentatively\nattribute the dead layer to atomic intermixing driven by\ndi\u000busion across the MAFO/CCO interface during CCO\noverlayer deposition.\nV. DISCUSSION\nOur PNR and XA/XMCD results (Section IV) indicate\nthat the damping enhancement observed in Section III\narises from chemical disorder, rather than proximity-\ninduced magnetism, at the MAFO/CCO interface.\nWe emphasize that this interfacial disordered layer\nis con\fned to within \u00192 spinel unit cells. We\nalso note that this interfacial disorder is due to\natomic intermixing, but not structural defects (e.g.,\ndislocations, antiphase boundaries), in this coherent\nbilayer system of MAFO/CCO. Nevertheless, this\nultrathin chemically disordered layer alone is evidently\nsu\u000ecient to signi\fcantly increase spin scattering.\nConsidering that the cation chemistry of Fe in MAFO\ndoes not change substantially (Fig. 4(a)), the interfacial\nspin scattering is likely driven by magnetic roughness,\nleading to a mechanism similar to two-magnon scattering\nthat accounts for a large fraction of e\u000bective damping in\nmetallic ferromagnet/Pt bilayers52.\nWe now put in context the magnitude of the damping\nenhancement \u0001 \u000beff, i.e., the di\u000berence in the e\u000bective\nGilbert damping parameter between CCO-capped and\nbare MAFO,\n\u0001\u000beff=\u000bbilayer\neff\u0000\u000bferrite\neff; (2)\nby comparing it with ferrite/spin-orbit-metal systems\nwhere spin pumping is often considered as the source6\n0.0000.0020.0040.0060.008\n MAFO/CCO\n [this study] MAFO/W\n [Riddiford] MAFO/Pt\n [Riddiford]YIG/Pt\n[Wang]Daeff\nYIG/Pt\n [Sun]\nFigure 5. Comparison of the enhancement of the e\u000bective\nGilbert damping parameter \u0001 \u000befffor MAFO/CCO and\nferrite/spin-orbit-metal bilayers. YIG/Pt [Sun], YIG/Pt\n[Wang], and MAFO/Pt(W) [Riddiford] are adapted from\nRefs.59,60, and61respectively. The values of \u0001 \u000befffrom the\nliterature are normalized for the saturation magnetization\nof 100 kA/m and magnetic thickness of 15 nm for direct\ncomparison with our MAFO/CCO result.\nof damping enhancement. Since damping enhancement\nfrom spin pumping or interfacial scattering scales\ninversely with the product of the saturation of\nmagnetization Msand the magnetic layer thickness tm,\nthe values of \u0001 \u000befftaken from the literature59{61are\nnormalized for direct comparison with the MAFO \flms\nstudied here with Ms= 100 kA/m and tm= 15 nm.\nAs summarized in Fig. 5, \u0001 \u000befffor MAFO/CCO\nis comparable to { or even greater than { \u0001 \u000beff\nfor ferrite/metal bilayers. This \fnding highlights that\nthe strength of increased spin scattering in a ferrite\ndue to interfacial chemical disorder can be on par\nwith spin dissipation due to spin pumping in metallic\nspin sinks. More generally, this \fnding suggests that\nspecial care may be required in directly relating \u0001 \u000beff\nto spin pumping across bilayer interfaces (i.e., spin-\nmixing conductance52), particularly when the FMR-\ndriven magnetic layer is directly interfaced with a spin\nscatterer.\nFurthermore, the strong interfacial spin scattering {\neven when the oxide interface is structurally coherent\nand the chemically disordered layer is kept to just <\u00182\nunit cells { poses a signi\fcant challenge for maintaining\nlow damping in ferrite/insulator heterostructures. This\nchallenge is partially analogous to the problem of reduced\nspin polarization in tunnel junctions consisting of spinelFe3O4and oxide barriers (e.g., MgO)62{65, which is also\nlikely due to interfacial chemical disorder and magnetic\ndead layers. However, we emphasize that the problems of\nantiphase boundaries43{46and charge-imbalance-driven\natomic reconstruction58, which have posed intrinsic\nchallenges for devices with MgO/Fe 3O4interfaces, are\nlikely not applicable to all-spinel MAFO/CCO. It is\ntherefore possible that deposition schemes that yield\nsharper interfaces, e.g., molecular beam epitaxy, can be\nemployed to reduce interfacial imperfections and hence\nspin scattering at MAFO/CCO for low-loss all-oxide\ndevice structures.\nVI. CONCLUSIONS\nWe have shown that e\u000bective damping in epitaxial\nspinel MgAl-ferrite (MAFO) increases more than\nthreefold when interfaced coherently with an insulating\nparamagnetic spinel of CoCr 2O4(CCO). This damping\nenhancement is not due to spin pumping into the\nbulk of CCO. Our depth-resolved characterization of\nMAFO/CCO bilayers also reveals no proximity-induced\nmagnetization in CCO or signi\fcant change in the\ncation chemistry of MAFO. We attribute the giant\ndamping enhancement to spin scattering in an ultrathin\nchemically disordered layer, con\fned to within 2 spinel\nunit cells across the MAFO/CCO interface. Our results\ndemonstrate that spin dynamics in ferrite thin \flms are\nstrongly impacted by interfacial disorder.\nAcknowledgements - This work was supported in\npart by the Vannevar Bush Faculty Fellowship program\nsponsored by the Basic Research O\u000ece of the Assistant\nSecretary of Defense for Research and Engineering and\nfunded by the O\u000ece of Naval Research through grant\nno. N00014-15-1-0045. 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Phys.\nLett.105, 102410 (2014)." }, { "title": "1812.10310v2.A_class_large_solution_of_the_2D_MHD_equations_with_velocity_and_magnetic_damping.pdf", "content": "arXiv:1812.10310v2 [math.AP] 6 Mar 2019A class large solution of the 2D MHD equations with velocity a nd\nmagnetic damping\nJinlu LiaMinghua Yangband Yanghai Yuc,∗\naSchool of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou 341000, China\nbDepartment of Mathematics, Jiangxi University of Finance a nd Economics, Nanchang, 330032, China\ncSchool of Mathematics and Statistics, Anhui Normal Univers ity, Wuhu, Anhui, 241002, China\nAbstract\nIn this paper, we construct a class global large solution to the two- dimensional MHD equations with\ndamp terms in the nonhomogeneous Sobolev framework.\nKeywords: 2D MHD equations; Large solutions; Damping; Besov space.\n2000 MSC: 35Q35, 35B35, 35B65, 76D03\n1. Introduction\nThis paper focuses on the following 2D incompressible magnetohydro dynamics (MHD) equations\n\n\n∂tu+u·∇u+µ(−∆)αu+∇p=b·∇b, x∈R2,t >0,\n∂tb+u·∇b+ν(−∆)βb=b·∇u, x ∈R2,t >0,\ndivu= divb= 0, x ∈R2,t≥0,\n(u,b)|t=0= (u0,b0), x ∈R2,(1.1)\nwhereu= (u1(t,x),u2(t,x))∈R2andb= (b1(t,x),b2(t,x))∈R2denote the divergence free velocity\nfield and magnetic field, respectively, p∈Ris the scalar pressure. µis the viscosity and νis the\nmagnetic diffusivity. The fractional power operator ( −∆)γwith 0< γ <1 is defined by Fourier\nmultiplier with symbol |ξ|2γ(see e.g. [ 7,12])\n(−∆)γu(x) =F−1|ξ|2γFu(ξ).\nWe make the convention that by γ= 0 we mean that ( −∆)γuis a damp term u. The magneto-\nhydrodynamic (MHD) equations which can be view as a coupling of incom pressible Navier–Stokes\nand Maxwell’s equations govern the motion of electrically conducting fl uids such as plasmas, liquid\nmetals and electrolytes, and play a fundamental role in geophysics, astrophysics, cosmology and\nengineering (see e.g. [ 10,4,9]). Due to the profound physical background and important math-\nematical significance, the MHD equations attracted quite a lot of at tention from many physicists\nand mathematicians in the past few years. Let us review some progr ess has been made about the\nMHD equations ( 1.1) which are more relatively with our problem. It is well known that the 2 D MHD\nequations ( 1.1) with−∆uand−∆b(namely, α=β= 1) have the global smooth solution([ 6]). In the\ncompletely inviscid case ( µ=ν= 0), the question of whether smooth solution of the MHD equations\n(1.1) with largeinitial data develops singularity infinite time remains complet ely open. Besides these\n∗Corresponding author.\nE-mail addresses: lijinlu@gnnu.cn (J. Li); ymh20062007@163.com (M . Yang); yuyanghai214@sina.com (Y. Yu);\nPreprint submitted to Elsevier March 7, 2019the two extreme cases, many intermediate cases, for example, th e 2D MHD equations with partial\ndissipation, has been studied by various authors. The issue of the g lobal regularity for the MHD\nequations ( 1.1) withµ >0,ν >0,α >0,β= 1 has been solved by Fan et al.[ 5]. Recently, Yuan and\nZhao [13] considered the MHD equations ( 1.1) with the dissipative operators weaker than any power\nof the fractional Laplacian and obtained the global regularity of th e corresponding system. On the\nother hand, Cao et al.[ 3], Jiu and Zhao [ 8] established the global regularity of smooth solutions to\nthe MHD equations ( 1.1) withµ= 0,ν >0,β >1 by different approach. Subsequently, Agelas [ 1]\nimproved this work with the diffusion ( −∆)βb(β >1) replaced by ( −∆)logκ(e−∆)b(κ >1).\nAs mentioned above, the global regularity for the completely inviscid MHD equations ( 1.1) with\nlarge initial data is still a challenging open problem. When α=β= 0, Wu et al [ 11] obtained that\nthe d-dimensional MHD equations ( 1.1) always possesses a unique global solution provided that the\ninitial datum is sufficiently small in the nonhomogeneous functional se ttingHswiths >1+d\n2. Our\nmain goal is to prove the global existence of solutions to ( 1.1) withα=β= 0 for a class of large\ninitial data.\nWe assume from now on that the damping coefficients µ=ν= 1, just for simplicity. Our main\nresult is stated as follows.\nTheorem 1.1. Letα=β= 0ands >2. Assume that the initial data fulfills divu0= divb0= 0and\nu0=U0+v0andb0=B0+c0\nwhere\nU0=/parenleftbigg\n∂2a0\n−∂1a0/parenrightbigg\nandB0=/parenleftbigg\n∂2m0\n−∂1m0/parenrightbigg\nwith\nsupp ˆa0(ξ),supp ˆm0(ξ)⊂ C:=/braceleftBig\nξ/vextendsingle/vextendsingle|ξ1−ξ2| ≤ε/bracerightBig\n. (1.2)\nThere exists a sufficiently small positive constant δ, and a universal constant Csuch that if\n/parenleftBig\n||v0||2\nHs+||c0||2\nHs+ε2(||a0||4\nHs+||m0||4\nHs)/parenrightBig\nexp/parenleftBig\nC(||a0||Hs+2+||m0||Hs+2)/parenrightBig\n≤δ,(1.3)\nthen the system (1.1)has a unique global solution.\nRemark 1.1. Letv0=c0= 0anda0=m0=ε−1\n2loglog1\nεχ, where the smooth function χsatisfying\nsuppˆχ∈/tildewideC,ˆχ(ξ)∈[0,1]andˆχ(ξ) = 1forξ∈1\n2/tildewideC,\nwhere\n/tildewideC:=/braceleftBig\nξ/vextendsingle/vextendsingle|ξ1−ξ2| ≤ε,1≤ξ2\n1+ξ2\n2≤2/bracerightBig\n.\nThen, direct calculations show that the left side of (1.3)becomes\nCε2/parenleftBig\nloglog1\nε/parenrightBig4\nexp/parenleftBig\nCloglog1\nε/parenrightBig\n.\nTherefore, choosing εsmall enough, we deduce that the system (1.1)has a global solution.\nMoreover, we also have\n||u0||L2/greaterorsimilarloglog1\nεand||b0||L2/greaterorsimilarloglog1\nε.\n2Remark 1.2. Considered the system (1.1)with0< α,β < 1, if the support condition (1.2)of the\nTheorem 1.1were replaced by\nsupp ˆa0(ξ),supp ˆm0(ξ)⊂ C:=/braceleftBig\nξ/vextendsingle/vextendsingle|ξ1−ξ2| ≤ε,1≤ξ2\n1+ξ2\n2/bracerightBig\n, (1.4)\nthe Theorem 1.1holds true.\nNotations : For the sake of simplicity, a/lessorsimilarbmeans that there is a uniform positive constant Csuch\nthata≤Cb. [A,B] stands for the commutator operator AB−BA, whereAandBare any pair of\noperators on some Banach space. In the paper, we will use the Bes ov space Bs\np,q, for more details,\nwe refer the readers to see the Chapter 2 in [ 2]. It is worth mentioning that the Besov space Bs\n2,2\ncoincides with the nonhomogeneous Sobolev spaces Hsfors >0, namely, Bs\n2,2(Rd) =Hs(Rd),where\nHs(Rd) :=/braceleftBig\nf∈ S′(Rd) :||f||Hs(Rd)<∞/bracerightBig\nwith the norm\n||f||Hs(Rd):=/parenleftBig/integraldisplay\nRd(1+|ξ|2)s|/hatwidef(ξ)|2dξ/parenrightBig1\n2.\n2. Reformulation of the System\nLet (a,m) be the solutions of the following system\n\n\n∂ta+a= 0,\n∂tm+m= 0,\n(a,m)|t=0= (a0,m0).(2.1)\nSetting\nU=/parenleftbigg∂2a\n−∂1a/parenrightbigg\nandB=/parenleftbigg∂2m\n−∂1m/parenrightbigg\n,\nwe can deduce from ( 2.1) that\n\n\n∂tU+U= 0,\n∂tB+B= 0,\ndivU= divB= 0,\n(U,B)|t=0= (U0,B0).(2.2)\nDenoting v=u−Uandc=b−B, the system ( 1.1) can be written as follows\n\n\n∂tv+u·∇v+v·∇U+u+∇p=b·∇c+c·∇B+f,\n∂tc+u·∇c+v·∇B+c=b·∇v+c·∇U+g,\ndivv= divc= 0,\n(v,c)|t=0= (v0,c0).(2.3)\nwhere\nf=−U·∇U+B·∇Bandg=−U·∇B+B·∇U.\n33. The Proof of Theorem 1.1\nBefore proceeding on, we present some estimates which will be used in the proof of Theorem 1.1.\nLemma 3.1. Fors >2, under the assumptions of Theorem 1.1, the following estimates hold\n||f||Hs+||g||Hs≤Ce−tε(||a0||2\nHs+2+||m0||2\nHs+2) (3.1)\nand\n||∇U||Hs+||∇B||Hs≤Ce−t(||a0||Hs+2+||m0||Hs+2). (3.2)\nProof of Lemma 3.1Notice that\nf1=−U·∇U1+B·∇B1\n=∂1a∂2∂2a−∂2a∂1∂2a−(∂1m∂2∂2m−∂2m∂1∂2m)\n= (∂1−∂2)a∂2∂2a+∂2a∂2(∂2−∂1)a+(∂2−∂1)m∂2∂2m+∂2m∂2(∂1−∂2)m\nand\nf2=−U·∇U2+B·∇B2\n=−∂1a∂2∂1a+∂2a∂1∂1a+∂1m∂2∂1m−∂2m∂1∂1m\n= (∂2−∂1)a∂1∂2a+∂2a∂1(∂1−∂2)a+(∂1−∂2)m∂1∂2m+∂2m∂1(∂2−∂1)m,\ndue to the fact that Hswiths >2 is a Banach algebra, then we have\n||f1||Hs/lessorsimilar||(∂1−∂2)a||Hs||a||Hs+2+||a||Hs+1||(∂2−∂1)a||Hs+1\n+||(∂1−∂2)m||Hs||m||Hs+2+||m||Hs+1||(∂2−∂1)m||Hs+1. (3.3)\nDirect calculations show that for τ≥0\n||a||Hτ+||m||Hτ≤e−t(||a0||Hτ+||m0||Hτ) (3.4)\nand\n||(∂1−∂2)a||Hτ+||(∂1−∂2)m||Hτ\n≤e−t(||(∂1−∂2)a0||Hτ+||(∂1−∂2)a0||Hτ)\n≤e−tε(||a0||Hτ+||m0||Hτ), (3.5)\nwhere we have used the conditions supp ˆ a0(ξ)⊂ Cand supp ˆ m0(ξ)⊂ C.\nIn view of the facts ( 3.4) and (3.5), we obtain from ( 3.3) that\n||f1||Hs/lessorsimilare−tε(||a0||2\nHs+2+||m0||2\nHs+2).\nSimilarly, we also have\n||f2||Hs/lessorsimilare−tε(||a0||2\nHs+2+||m0||2\nHs+2).\nThen, we get\n||f||Hs≤ ||f1||Hs+||f2||Hs/lessorsimilare−tε(||a0||2\nHs+2+||m0||2\nHs+2). (3.6)\n4An argument similar to that used above, we get\n||g||Hs/lessorsimilare−tε(||a0||2\nHs+2+||m0||2\nHs+2). (3.7)\nCombining ( 3.6) and (3.7) yields the desired result ( 3.1).\n(3.2) is just a consequence of ( 3.4). Thus, we complete the proof of Lemma 3.1./square\nProof of Theorem 1.1For notational simplicity, we set\nE(t) =/parenleftbig\n||v(t)||2\nHs+||c(t)||2\nHs/parenrightbig\n.\nApplying ∆ jto (2.3) and taking the L2inner product of the resulting equations with ∆ jvand ∆ jc,\nrespectively, we have\n1\n2d\ndt/parenleftBig\n||∆jv||2\nL2+||∆jc||2\nL2/parenrightBig\n+||∆jv||2\nL2+||∆jc||2\nL2=:5/summationdisplay\ni=1Ki, (3.8)\nwhere\nK1=−/integraldisplay\nR2[∆j,u·∇]v·∆jvdx−/integraldisplay\nR2[∆j,u·∇]c·∆jcdx,\nK2=/integraldisplay\nR2[∆j,b·∇]c·∆jvdx+/integraldisplay\nR2[∆j,b·∇]v·∆jcdx,\nK3=−/integraldisplay\nR2∆j(v·∇U)·∆jvdx−/integraldisplay\nR2∆j(v·∇B)·∆jcdx,\nK4=/integraldisplay\nR2∆j(c·∇B)·∆jvdx+/integraldisplay\nR2∆j(c·∇U)·∆jcdx,\nK5=/integraldisplay\nR2∆jf·∆jvdx+/integraldisplay\nR2∆jg·∆jcdx.\nMultiplying both sides of ( 3.8) by 22jsand summing up over j≥ −1 yields\n1\n2d\ndtE(t)+E(t) =5/summationdisplay\ni=1/summationdisplay\nj≥−122jsKi. (3.9)\nNext, we need to estimate the above terms involving Kifori= 1,···,5 as follows\n/summationdisplay\nj≥−122js|K1| ≤/summationdisplay\nj≥−122js||[∆j,u·∇]v||L2||∆jv||L2+/summationdisplay\nj≥−122js||[∆j,u·∇]c||L2||∆jc||L2\n/lessorsimilar||∇u||Hs−1/parenleftBig\n||v||2\nHs+||c||2\nHs/parenrightBig\n/lessorsimilar/parenleftBig\n||U||Hs+||v||Hs/parenrightBig\nE(t), (3.10)\nwhere we have used the commutator estimate (see Lemma 2.6 in [ 11])\n||[∆j,u·∇]f||Bs\n2,2≤C||∇u||Bs−1\n2,2||f||Bs\n2,2with div u= 0.\nSimilarly, we also have\n/summationdisplay\nj≥−122js|K2|/lessorsimilar/parenleftBig\n||B||Hs+||c||Hs/parenrightBig\nE(t). (3.11)\n5For the last three terms, by H¨ older’s inequality, we deduce\n/summationdisplay\nj≥−122js|K3|/lessorsimilar||∇U||Hs||v||2\nHs+||∇B||Hs||c||Hs||v||Hs, (3.12)\n/summationdisplay\nj≥−122js|K4|/lessorsimilar||∇U||Hs||c||2\nHs+||∇B||Hs||c||Hs||v||Hs, (3.13)\nand\n/summationdisplay\nj≥−122js|K5| ≤/summationdisplay\nj≥−122js||∆jf||L2||∆jv||L2+/summationdisplay\nj≥−122js||∆jg||L2||∆jc||L2\n≤C(||f||2\nHs+||g||2\nHs)+1\n2E(t). (3.14)\nInserting ( 3.10)–(3.14) into (3.9) yields that\nd\ndtE(t)+E(t)/lessorsimilarE3\n2(t)+/parenleftBig\n||∇U||Hs+||∇B||Hs/parenrightBig\nE(t)+||f||2\nHs+||g||2\nHs.(3.15)\nUtilizing the Lemma 3.1, we have from ( 3.15)\nd\ndtE(t)+E(t)/lessorsimilarE3\n2(t)+e−t/parenleftBig\n||a0||Hs+2+||m0||Hs+2/parenrightBig\nE(t)+e−tε2/parenleftBig\n||a0||4\nHs+2+||m0||4\nHs+2/parenrightBig\n.(3.16)\nNow, we define\nΓ := max {t∈[0,T∗) : sup\nτ∈[0,t]E(τ)≤η},\nwhereηis a small enough positive constant which will be determined later on.\nAssume that Γ < T∗. For all t∈[0,Γ], we obtain from ( 3.16) that\nd\ndtE(t)≤Ce−t/parenleftBig\n||a0||Hs+2+||m0||Hs+2/parenrightBig\nE(t)+Ce−tε2/parenleftBig\n||a0||4\nHs+2+||m0||4\nHs+2/parenrightBig\n,\nwhich follows from the assumption ( 1.3) that\nE(t)≤C/parenleftBig\nE0+ε2(||a0||4\nHs+||m0||4\nHs)/parenrightBig\nexp/parenleftBig\nC(||a0||Hs+2+||m0||Hs+2)/parenrightBig\n≤Cδ.\nChoosing η= 2Cδ, thus we can get\nE(t)≤η\n2fort≤Γ.\nSo if Γ< T∗, due to the continuity of the solutions, we can obtain there exists 0 < ǫ≪1 such\nthat\nE(t)≤η\n2fort≤Γ+ǫ < T∗,\nwhich is contradiction with the definition of Γ.\nThus, we can conclude Γ = T∗and\nE(t)≤C <∞for allt∈(0,T∗),\nwhich implies that T∗= +∞. This completes the proof of Theorem 1.1./square\n6Acknowledgments\nJ. Li was partially supported by NSFC (No.11801090). M. Yang was p artially supported by\nNSFC (No.11801236)\nReferences\nReferences\n[1] L. Agelas, Global regularity for logarithmically critical 2D MHD equa tions with zero viscosity.\nMonatsh. Math. 181 (2016), 245–266.\n[2] H. Bahouri, J.Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equa-\ntions, Grundlehren Math. Wiss., vol.343, Springer-Verlag, Berlin, He idelberg, 2011.\n[3] C. Cao, J. Wu, B. Yuan, The 2D incompressible magnetohydrodyn amics equations with only\nmagnetic diffusion. SIAM J. Math. Anal. 46 (2014), 588–602.\n[4] P.A. Davidson, An Introduction to Magnetohydrodynamics, Cam bridge University Press, Cam-\nbridge, England, 2001.\n[5] J. Fan, H. Malaikah, S. Monaquel, G. Nakamura, Y. Zhou, Global c auchy problem of 2D gen-\neralized MHD equations. Monatsh Math. 175 (2014), 127–131.\n[6] M. Sermange, R. Temam, Some mathematical questions related t o the MHD equations. Comm.\nPure Appl. Math. 36 (1983), 635–664.\n[7] N. Jacob, Pseudo Differential Operators and Markov Processe s. Vol. III: Markov Processes and\nApplications, Imperial College Press, 2005.\n[8] Q. Jiu, J. Zhao, Global regularity of 2D generalized MHD equations with magnetic diffusion. Z.\nAngew. Math. Phys. 66 (2015), 677–687.\n[9] J. Li, W. Tan, Z. Yin, Local existence and uniqueness for the non -resistive MHD equations in\nhomogeneous Besov spaces. Advances in Mathematics. 317 (2017 ) 786–798.\n[10] E. Priest, T. Forbes, Magnetic Reconnection, MHD Theory and Applications, Cambridge Uni-\nversity Press, Cambridge, 2000.\n[11] J. Wu , X. Xu, Z. Ye, Global smooth solutions to the n-dimensiona l damped models of incom-\npressible fluid mechanics with small initial datum. J. Nonlinear Sci. 25 (2 015), 157–192.\n[12] X. Wu, Y. Yu, Y. Tang, Global existence and asymptotic behavio r for the 3D generalized Hall-\nMHD system. Nonlinear Anal. 151 (2017) 41–50.\n[13] B. Yuan, J. Zhao, Global regularity of 2D almost resistive MHD eq uations. Nonlinear Anal.\nReal World Appl. 41 (2018), 53–65.\n7" }, { "title": "2107.00150v2.Origin_of_Nonlinear_Damping_due_to_Mode_Coupling_in_Auto_Oscillatory_Modes_Strongly_Driven_by_Spin_Orbit_Torque.pdf", "content": "Origin of Nonlinear Damping due to Mode Coupling in Auto-Oscillatory Modes\nStrongly Driven by Spin-Orbit Torque\nInhee Lee,1,\u0003Chi Zhang,1Simranjeet Singh,1,yBrendan McCullian,1and P. Chris Hammel1,z\n1Department of Physics, The Ohio State University, Columbus, OH 43210, USA\n(Dated: June 28, 2022)\nWe investigate the physical origin of nonlinear damping due to mode coupling between several\nauto-oscillatory modes driven by spin-orbit torque in constricted Py/Pt heterostructures by exam-\nining the dependence of auto-oscillation on temperature and applied \feld orientation. We observe a\ntransition in the nonlinear damping of the auto-oscillation modes extracted from the total oscillation\npower as a function of drive current, which coincides with the onset of power redistribution amongst\nseveral modes and the crossover from linewidth narrowing to linewidth broadening in all individual\nmodes. This indicates the activation of another relaxation process by nonlinear magnon-magnon\nscattering within the modes. We also \fnd that both nonlinear damping and threshold current in the\nmode-interaction damping regime at high drive current after transition are temperature independent,\nsuggesting that the mode coupling occurs dominantly through a non-thermal magnon scattering pro-\ncess via a dipole or exchange interaction rather than thermally excited magnon-mediated scattering.\nThis \fnding presents a promising pathway to overcome the current limitations of e\u000eciently con-\ntrolling the interaction between two highly nonlinear magnetic oscillators to prevent mode crosstalk\nor inter-mode energy transfer and deepens understanding of complex nonlinear spin dynamics in\nmultimode spin wave systems.\nSpin-orbit torque driven magnetic nano-oscillators\nhave recently emerged as potential charge current tun-\nable microwave sources for spintronics devices[1{13], as\nwell as fundamental elements for neuromorphic comput-\ning [14]. These oscillators use spin Hall e\u000bect to convert\nthe charge current into a pure spin current which is in-\njected into the ferromagnet, exerting an anti-damping\ntorque on the magnetization. Above the threshold cur-\nrent, coherent magnetic auto-oscillations (AO) are gen-\nerated at microwave frequencies.\nIn principle, these planar spin Hall nano-oscillators\n(SHNO) need not be limited in size and are expected\nto provide larger AO power because the Oersted \feld is\nrelatively small and uniform in the device unlike nano-\npillar spin-torque nano-oscillators (STNO). However, ad-\nditional damping channels arise in extended thin \flm\nstructures through nonlinear magnon scattering that pre-\nvents auto-oscillation [4]. These damping channels can be\nsuppressed by restricting the area of the AO by fabricat-\ning a nano-constriction [3, 10] as well as other methods of\nspatial mode con\fnement such as dimensional reduction\n[4, 15] and local dipolar \feld [16{19].\nNevertheless, in constriction-based SHNOs, mode\nsplittings are often observed [3, 10], leading to substan-\ntial linewidth broadening which degrades performance in\nterms of AO coherence and power. So far, little is known\nabout the origins of nonlinear damping as a consequence\nof multimode excitation in this system, and the mech-\nanism underlying mode coupling is still unknown. In\nnano-pillar STNOs, similar multimode behaviors such as\n\u0003lee.2338@osu.edu\nyPresent address: Department of Physics, Carnegie Mellon uni-\nversity, Pittsburgh, PA, 15213, USA\nzhammel@physics.osu.edumode hopping [20{26], mode coexistence [27{29] and 3-\nmagnon process [30, 31], have been reported along with\ntheoretical studies [20{26, 29, 32, 33] of mode coupling,\nincluding its description [21{26, 33], its e\u000bect on mode\ndecoherence [22, 24, 25] and its temperature dependence\n[22, 24, 25, 33], however, much remains to be understood.\nFor this study, we fabricate Pt(5nm)/Py(5nm) bilayer\ndevices in the form of a bow tie with an active center area\nof 600 nm\u00021\u0016m, as shown in Fig. 1(a). This sample\nstructure allows the generation of multiple AO modes by\napplying a high current density Jcthat is converted into\na spin current Jsthat, in turn, exerts a spin torque on the\nmagnetization of the Py (See Fig. 1(b)). By changing\nthe orientation of the magnetic \feld H0applied at angle\n\u0012with respect to the x-axis of the microstrip line, we tune\nthe eigenmodes of the spin waves de\fned by the spatial\npattern of the internal \feld in Py, which is signi\fcantly\nmodi\fed by its dipolar, or demagnetizing \feld.\nIndeed, when the orientation of the applied \feld\nchanges relative to the device edges causing mode con-\nstriction, spectral and spatial distributions of the result-\ning AO modes vary signi\fcantly. Fig. 2(a) shows the\nangle\u0012dependence of the AO spectrum measured with\nbias current IDC= 7 mA at temperature T= 77 K in\nan applied \feld H0= 570 Oe. As \u0012increases, more ex-\ncitation modes appear over a wider frequency range as a\nresult of stronger mode constriction. We perform micro-\nmagnetic simulations using MuMax3 [34](see Appendix\nB) to understand the complex spectra at various \u0012and to\nidentify the relevant AO modes. Fig. 2(b) shows the an-\ngle dependence of the AO spectra obtained from simula-\ntions conducted with the current density distribution and\nOersted \feld in Fig. 10 of Appendix. It well describes\nthe overall evolution of the experimental AO spectrum\nwith increasing angle in Fig. 2(a), although more accu-\nrate spatial information re\recting the high current den-arXiv:2107.00150v2 [cond-mat.mes-hall] 27 Jun 20222\nFIG. 1. (a) Optical image of the auto-oscillation device. (b)\nSchematic diagram of the 2D constricted Pt(5 nm)/Py(5 nm)\nbilayer structure of the bow tie shape in the dashed box of\n(a). Due to the strong spin-orbit coupling in Pt, the charge\ncurrent density Jc\rowing in the x direction along the axis\nof microstrip line is converted to pure spin current density Js\nwhich is injected into the ferromagnet Py with the appropri-\nate spin polarization required for anti-damping torque. \u0012is\nthe angle made by the direction of in-plane applied \feld H0\nrelative to the x-axis. (c) Four magnon scattering mediated\nby thermally excited magnons !th;iand!th;fin equilibrium\nwith thermal reservoir. !i+!th;i=!f+!th;f. (d) Four\nmagnon scattering by intermode interactions. d!is the fre-\nquency shift within the mode caused by dipole or exchange\ninteractions between !iand!j.\nsity nonlinearity and structural defects in the AO active\nregion seems necessary to fully account for the spectral\nshape details. Fig. 2(c)-(g) show the spatial pro\fles of\nthe spin wave eigenmodes corresponding to the spectral\npeaks in Fig. 2(b) at various angles \u0012. We \fnd that\nfor\u0012\u001465\u000e, the edge and bulk modes are combined,\nwhereas for \u0012\u001570\u000e, the edge and bulk modes are spa-\ntially separated due to the signi\fcant di\u000berence in the\ndemagnetizing \feld between the edge and center regions.\nThis separation also occurs spectrally (see Fig. 2(a) and\n2(b)) and the largest AO peak appears in one of the bulk\nmodes at frequencies higher than edge modes, which is\nin good agreement with previous AO results obtained on\nnanowires with \u0012= 85\u000e[4].\nThe spectral and spatial mode pro\fles for AO in the ac-\ntive mode shown in Fig. 2 di\u000ber signi\fcantly from those\nof the linear spin wave mode (see Fig. 6 in Appendix).\nThis demonstrates that the dynamics of the AO mode are\ndi\u000berent from those of the linear spin wave mode, aris-\ning from nonlinear e\u000bects such as complex bullet mode\ndynamics involved with mode size reduction[1, 3, 35] or\nmode size oscillation[5, 36]. In particular, this bullet\nmode e\u000bect appears to be more pronounced in bulk mode\n3a and 3b in Fig. 2(c) and bulk mode 3 in Fig. 2(d), away\nfrom the edge e\u000bect. The frequency jump as the signa-ture of the bullet mode is also discussed in Appendix D.\nIn the evolution of the auto-oscillation modes with in-\ncreasing bias current, we observe a transition in nonlinear\ndamping due to mode-mode coupling. As can be seen in\nFig. 3(a) obtained by measuring at \u0012= 65\u000e, as the cur-\nrentIDC{and hence the anti-damping torque{ increases.\nthe three main modes C1, C2, and C3 (labelled in Fig.\n3(e)) appear in the spectrum above each threshold cur-\nrent. We characterize these modes evolving at various\nIDCwith the resonance frequency f0, linewidth \u0001 f, and\npowerPobtained from Lorentzian \fts, which are shown\nin Fig. 3(b)-(d). Associated with the transition of non-\nlinear damping, we observe noticeable abrupt changes at\nthe current IMI= 7:1 mA, such as the turnover from\nthe linewidth narrowing to linewidth broadening (sign\nchange of \u0001 f=IDC) for all excitation modes (Fig. 3(c))\nand power saturation in C1 mode (Fig. 3(d)). This tran-\nsition marks the onset of another relaxation process with\nadditional damping. However, as IDCincreases in the\nhigh current regime ( IDC> IMI), we also observe i) a\nmonotonic redshift of the frequency for all three modes\nre\recting a monotonic reduction in saturation magnetiza-\ntion due to magnon excitation [37] (Fig. 3(b)), ii) a power\nincrease in the C2 and C3 modes despite their linewidth\nbroadening (Fig. 3(c) and 3(d)), and iii) an increase in\nthe total power (Fig. 3 (d)). All of these imply that\nthe magnon population is growing faster than excited\nmagnons can decay to other thermal reservoirs. There-\nfore, we conclude that nonlinear damping in the high cur-\nrent regime occurs through magnon redistribution from\nlow-frequency modes to high-frequency modes via nonlin-\near magnon-magnon scattering as shown schematically in\nFig. 1(c) or 1(d).\nThere are other possible causes for the nonlinear damp-\ning transition at IMI. The transition in the dependence of\npower on current shown in Fig. 3(d) may be described by\nthe selective excitation and selective saturation of each\nmode. However, C2 and C3 in multimode excitation can-\nnot be explained this way because their linewidth broad-\nening, along with the power increase, at IDC\u0015IMIis not\nconsistent with typical single-mode AO behavior which\nexhibits linewidth narrowing with increasing IDCas C1\ndoes forIDC< I MI. Power saturation of C1 may be\ncaused by Joule heating or spin pumping. However, Joule\nheating cannot saturate the power of C1 without simi-\nlarly a\u000becting C2 and C3. Crucially, Joule heating can\nbe observed in the spin torque FMR by applying a high\nnegative current ( IDC<0) corresponding to the posi-\ntive damping in which the spin current driven magnons\nare not generated. As the absolute value of IDC<0 in-\ncreases, the decreasing resonance \feld can change to an\nincreasing resonance \feld due to the decrease in satura-\ntion magnetization caused by Joule heating, similar to\nthe previous Brillouin light scattering results [37]. How-\never, this e\u000bect does not appear up to IDC= -8 mA in\nour data (see Fig. 13 in Appendix), showing that Joule\nheating has a small e\u000bect. The intensity of spin pumping\nis proportional to the power of the AO mode [38]. As3\nFIG. 2. (a) Angle dependence of the auto-oscillation spectrum measured at IDC= 7 mA,T= 77 K and H0= 570 Oe. We vary\nthe angle\u0012between H0andIDCin our experiments as shown in the diagram in the lower right corner of the \fgure. In the\nspectra for 65\u000eand 70\u000e, `C', `E' and `B' represent a combined edge-bulk mode, edge mode, and bulk mode, respectively. (b)\nSpectra for various \u0012obtained from micromagnetic simulations using Mumax3 performed with the current density distribution\nand Oersted \feld in Fig. 10 of Appendix. The corresponding IDCare 7.92 mA for \u0012= 55\u000e, 7.26 mA for \u0012= 60\u000e, 6.92 mA for\n\u0012= 65\u000e, 6.52 mA for \u0012= 70\u000e, and 6.36 mA for \u0012= 75\u000e, about 0.2 - 0.3 mA larger than the respective threshold current at\neach\u0012. In (a) and (b), each spectrum is o\u000bset. (c)-(g) Spatial eigenmode pro\fles corresponding to the spectral peaks indicated\nby the numbers in (b) at various \u0012. The dynamic magnetization amplitude mis normalized on the color scale of each image.\nAt\u0012= 70\u000eand 75\u000e, the spatial separation of edge (1-2) modes and bulk (3) modes occurs as shown in (c) and (d) as well as\ntheir spectral separation as shown in (b).\nshown in Fig. 3(c) and 3(d), the linewidths of C1, C2\nand C3 with di\u000berent powers start to increase together\natIMIwith increasing IDC, which cannot be explained by\nspin pumping. Therefore, we conclude that only mode-\nmode interaction can explain the coincidence of both the\npower transfer from C1 to the higher frequency modes\nC2 and C3 and the turnover from linewidth broadening\nto linewidth narrowing of all three modes at IMIfor in-\ncreasingIDC, eventually leading to an almost even power\ndistribution in the spectrum at 8 mA, as shown in Fig.\n3(e).\nIn Fig. 4 we show the temperature dependence of\nthe nonlinear damping of the AO system along with the\nthreshold current Ithand transition current IMI. Fig.\n4(a) shows the total power Ptotal summed for all exci-\ntation modes as a function of IDCobtained at various\ntemperatures for \u0012= 65\u000e. For each temperature, Ptotal\nhas two current regimes, `A' for IDC< IMIin which a\nsingle mode is dominant, and `B' for IDC\u0015IMIin which\nmode-mode interaction is active amongst multiple exci-\ntation modes, separated by the kink in Ptotalat whichP\nin the lowest frequency mode saturates. One of our re-\nmarkable \fndings is that total power Ptotal;Bfor all tem-\nperatures in regime B falls onto a single common curve\n(black line in Fig. 4(a)), whereas total power Ptotal;Ain\nregime A is temperature dependent. This suggests thatthe mode-mode coupling occurs through a non-thermal\nprocess (Fig. 1(d)) originating from exchange or dipole\ninteractions instead of the thermally excited magnon-\nmediated scattering (Fig. 1(c)). The rapid growth of\nPtotal;A(color lines in Fig. 4(a)) from each threshold cur-\nrentIthis eventually limited by Ptotal;B(black line in Fig.\n4(a)) at each corresponding transition current IMIfor all\ntemperatures. This indicates that the much faster relax-\nation process arising from the temperature-independent\nmode-mode interactions in regime B predominates over\nthe temperature dependent single mode relaxation pro-\ncess occurring in regime A. Note that magnon thermal-\nization through redistribution within the dynamic mag-\nnetic system is possible only if the relaxation via non-\nlinear magnon scattering arising from mode coupling is\nmuch faster than the relaxation to the external non-\nmagnetic systems in individual modes by intrinsic Gilbert\ndamping or spin pumping.\nIn order to quantify the nonlinear damping of the AO\nsystem, we discuss the parameter Qthat represents the\nchange in positive nonlinear damping \u0000 +\u0019\u0000G(1 +Qp)\nwith increasing AO power, where \u0000 Gis the Gilbert damp-\ning [39].Qas well asIthcan be obtained from the re-\nlationship between the normalized AO power pandIDC\n(see Appendix C). Fig. 4(b) and 4(c) show the temper-\nature dependence of the AO parameters in the regimes4\nFIG. 3. (a) Evolution of power spectral density (PSD) with\nincreasing bias current IDC, (b) resonance frequency f0vs.\nIDC, (c) linewidth \u0001 fvs.IDC, (d) power Pvs.IDCfor\nthe main modes C1, C2 and C3 for \u0012= 65\u000e. The FMR\nparameters in (b)-(d) are extracted by \ftting the data in (a)\nto Lorentzian functions. (e) Representative spectra at high\nIDCfor\u0012= 65\u000eshowing the redistribution of power among\nmodes at high IDC. (f) Evolution of PSD with increasing\nIDC, (g)f0vs.IDC, (h) \u0001fvs.IDC, (i)Pvs.IDCof the\nmain edge (E1, E2) and bulk (B1, B2, B3) modes for \u0012= 70\u000e.\nThe FMR parameters in (g)-(i) are extracted by \ftting the\ndata in (f) to Lorentzian functions. (j) Representative spectra\nat highIDCfor\u0012= 70\u000eshowing the redistribution of power\namong modes at high drive current. In (d) and (i), the black\ndiamond markers are the total power Ptotalsummed over all\nexcited modes, and IMIrepresents the current at which the\nnonlinear damping transition occurs. At very high IDC, the\nmain modes shown here are not clearly identi\fed due to the\nlarge linewidth broadening and the emergence of other excited\nmodes even though Ptotal can be obtained. Here H0= 570\nOe andT= 77 K for all measurements.\nFIG. 4. (a) Total power Ptotal of the AO as a function of\nIDCat various temperature T, (b) the nonlinear damping co-\ne\u000ecientQvs.T, (c) the threshold current Ithvs.Tand\nthe transition current IMIvs.Tfor\u0012= 65\u000e. Colored solid\ncurves are the linear \fts to Eq. (1) with Ith;0= 7:35 mA and\n\u0014th=\u00000:0117 mA=K and Eq. (2) with IMI;0= 8:34 mA and\n\u0014MI=\u00000:0147 mA=K forIthandIMI, respectively. Similarly,\n(d)Ptotalof the AO as a function of IDCat variousT, (e)Q\nvs.T, (f)Ithvs.TandIMIvs.Tfor\u0012= 70\u000e. Colored solid\ncurves are the linear \fts to Eq. (1) with Ith;0= 7:43 mA and\n\u0014th=\u00000:0124 mA=K and Eq. (2) with IMI;0= 8:12 mA and\n\u0014MI=\u00000:015 mA=K forIthandIMI, respectively. Colored\nsolid curves in Fig. 4(a) and 4(d) are the Ptotal calculated\nusing the theoretical equation (see Eq. (C4) in Appendix C)\nwithQAandIth;Ain 4(b)-(c) and 4(e)-(f), respectively. The\nblack curves in Fig. 4(a) and 4(d) are the Ptotal calculated\nusing the theoretical equation (see Eq. (C5) in Appendix C)\nwith the common values of QBandIth;Bin 4(b)-(c), 4(e)-(f),\nrespectively. Here `A' represents individual mode dominant\nregime (IDC< IMI) and `B' represents mode interaction ac-\ntivation regime ( IDC\u0015IMI). And the normalization factor\nN0= 9.5 pW and 8.7 pW, determined from \ftting, for \u0012= 65\u000e\nand\u0012= 70\u000e, respectively, (see Appendix C) and H0= 490\nOe. Temperature dependence of (g) minimum linewidth and\n(h) maximum power of C1 mode for \u0012= 65\u000e.5\nA and B discussed above: temperature-dependent QA,\nIth;A,IMI, and temperature-independent QB,Ith;B. The\nlarger value of QBrelative toQAobserved in Fig. 4(b),\nindicates that the mode couplings in regime B cause ad-\nditional nonlinear damping compared to regime A. In\nregime A where a single mode C1 is dominant, QAde-\npends on temperature such that it is almost constant at\nT\u0014125 K but increases for T > 125 K. This re\rects\nthe temperature dependence of the minimum linewidth\nand maximum power of C1 in Fig. 4(g) and 4(h) and\nthe existence of another temperature-independent relax-\nation mechanism below 125 K for an individual AO mode.\nAlso, asTincreases, both Ith;AandIMIdecrease linearly,\ndemonstrating that thermal \ructuation noise facilitates\nthe generation and stabilization of the AO mode and of\nmode-mode coupling with the smaller IDCsuch as\nIth(T) =Ith;0+\u0014thT (1)\nIMI(T) =IMI;0+\u0014MIT (2)\nwhereIth;0andIMI;0are the intrinsic threshold and tran-\nsition currents with thermal \ructuation excitation e\u000bects\nremoved, and \u0014thand\u0014MIare coe\u000ecients with respect\nto temperature change. The temperature dependence of\nthe intrinsic threshold, Ith;0(T), can be obtained from a\nseparately measured spin torque FMR, where Ith;0is es-\ntimated to be almost constant in the temperature range\nof 5 - 300 K (See Appendix E).\nThere are signi\fcant changes in the current evolution\nof the AO modes when \u0012changes from 65\u000eto 70\u000eas\nshown in Fig. 3(a) and 3(f). Edge-bulk combined modes\n(C1, C2, C3) at \u0012= 65\u000e(see Fig. 2(e) and Fig. 3(a)-(e))\nare spectrally and spatially separated into edge (E1, E2)\nand bulk (B1, B2, B3) modes at \u0012= 70\u000e(see Fig. 2(d)\nand Fig. 3(f)-(j)). As a result, for the AO modes existing\nat\u0012= 70\u000e(see Fig. 3(g)-(i)), the evolution of f0, \u0001f,\nandPof the AO modes with increasing IDCare more\ncomplex than for 65\u000e: the E1 and E2 modes split at 7\nmA, and there can be various mode-mode couplings with\ndi\u000berent strengths of dipolar and exchange interactions\ndepending on the spatial distance between the two inter-\nacting modes (e.g., edge-edge, edge-bulk and bulk-bulk).\nThis information can be valuable in developing strategies\nthat employ spectral and spatial mode separation to re-\nduce mode couplings and thus enhance the performance\nof auto-oscillators. We note that these mode couplings\nshould be distinguishable from the nonlinear bullet mode\ndynamics. The power sum of E1 and E2 monotonically\nincreases up to IMI= 7:2 mA until it saturates creating\na kink inPtotalas shown in Fig. 3(i). The temperature\ndependence of the AO parameters for \u0012= 70\u000eshown in\nFig. 4(e) and 4(f) is generally similar to that for \u0012= 65\u000e,\nexcept for a further increased QBvalue, perhaps due to\nthe increased number of intermode interaction routes al-\nlowed between a larger number of AO modes.\nIn conclusion, we observe the nonlinear damping tran-\nsition of the auto-oscillation modes through the total\npower as a function of drive current, which coincides with\nthe onset of power distribution amongst multiple modesand linewidth broadening of all individual modes. We\n\fnd the nonlinear damping due to mode-mode coupling\nto be independent of temperature, which suggests that\nthe mode coupling occurs through intermode interactions\nsuch as dipole and exchange interactions rather than\nthermally excited magnon mediated nonlinear scattering.\nThis study of nonlinear damping due to mode couplings\npresents a promising pathway to overcome the current\nlimitations of e\u000eciently controlling mode interactions in\nspin Hall nano-oscillators to prevent mode crosstalk or\ninter-mode energy transfer and deepens understanding\nof complex nonlinear spin dynamics in multimode spin\nwave systems. As one solution, we can generate well-\nde\fned AO modes locally excited by the dipole \feld from\na nano- or micron-scale permanent magnet [19], where\nthe number of modes and their frequency distribution\ncan be tuned by changing the local dipole \feld by mov-\ning a permanent magnet relative to the sample surface.\nFurthermore, the interaction between two spatially sep-\narated AOs can be controlled by adjusting the relative\nlateral distance in a scanned system [16{19].\nACKNOWLEDGMENTS\nWe thank Denis V. Pelekhov for helpful discussions.\nThis work was primarily supported by the Center for\nEmergent Materials, an NSF MRSEC, grant DMR-\n2011876.\nAppendix A: Angle Dependence of auto-oscillation\nspectrum over extended angle range\nFig. 5 shows the dependence of the auto-oscillation\nspectrum on angle, similar to Fig. 2(a), but over an\nextended angular range: \u0012= 50\u000e\u000085\u000e. At\u0012\u001465\u000e,\nthe lowest frequency mode has the largest amplitude and\nshifts slightly towards lower frequencies as \u0012increases.\nThe mode starts to split and its amplitude decreases\nabove 70\u000ewhile the frequency shifts monotonically lower\nwith increasing \u0012. On the other hand, the high frequency\nmodes hardly shift with increasing \u0012, and at\u0012\u001570\u000e, one\nof them has the largest amplitude among all excited AO\nmodes in the spectrum instead of the lowest frequency\nmode. Our micromagnetic simulations in the next sec-\ntion show that edge modes at low frequencies shift mono-\ntonically to lower frequencies, while bulk modes at high\nfrequencies hardly shift; this characteristic behavior al-\nlows edge and bulk modes to be di\u000berentiated for \u0012\u001570\u000e.\nAppendix B: Micromagnetic Simulations\nWe perform micromagnetic simulations using MuMax3\n[34] to understand complex AO spectra and identify their\nrelevant spatial mode pro\fles. In simulations, the com-\nputational dimension 2.6 \u0016m\u00029\u0016m\u00025 nm is subdi-6\nFIG. 5. Angle dependence of the auto-oscillation spectrum in\nFig. 2(a) of the main text extended to a wide angle range. It\nis measured at IDC= 7 mA,T= 77 K and H0= 570 Oe.\nvided into 5 nm\u000217.5 nm\u00025 nm cells. As magnetic\nparameters, we use the gyromagnetic ratio of \r=2\u0019=\n2.8 MHz/G and e\u000bective magnetization 4 \u0019Me\u000b= 6502\nG obtained from the ST-FMR data (see Appendix D)\nmeasured at IDC= 0 via Kittel equation.\nf0=\r\n2\u0019[H0(H0+ 4\u0019Me\u000b)]1=2(B1)\nThe standard values of exchange sti\u000bness Aex= 1:3\u0002\n10\u000011J/m and the Gilbert damping constant \u000b= 0.01\nfor permalloy are used.\n1. Linear Spin Wave Eigenmode\nFirst, we perform micromagnetic simulations in the\nlinear regime of magnetodynamics with no bias current.\nInitially, the magnetic system is excited by a sinc rf \feld\nwith an amplitude of 10 mT and a cuto\u000b frequency of\n40 GHz. Then Gilbert damping is turned o\u000b by making\n\u000b= 0, allowing the magnetic dynamic system to proceed\nfreely for 187 ns. We obtain the spectral and spatial pro-\n\fles of the modes by performing Fourier transform with\ndynamic motion after 62 ns to avoid initial transients [9].\nFig. 6(b)-(f) show the spatial pro\fle of the linear spin\nwave eigenmodes corresponding to each spectral peak in-\ndicated by the number in Fig. 6(a) for each angle \u0012. The\nedge and bulk modes are combined for \u0012\u001465\u000e, whereas\nthe edge and bulk modes are spatially and spectrally sep-\narated for \u0012\u001570\u000edue to the signi\fcant di\u000berence in\nthe demagnetizing \feld between the edge and center re-\ngions. Edge modes at low frequencies shift monotonicallyto lower frequencies, and bulk modes at high frequencies\nshift little.\n2. Self-Oscillatory Mode\nThe auto-oscillation mode of the system is obtained\nby solving the Landau-Lifshitz-Gilbert equation with an\nanti-damping spin torque applied to the active region of\nthe AO device. In the simulations, we try two di\u000berent\ncurrent density distributions shown in Fig. 7 and Fig. 10\nconsidered for a 1 mA bias current.\nAs an initial state in the simulation, the magnetic sys-\ntem is allowed to relax to a state close to the ground\nstate. The anti-damping torque proportional to the cur-\nrent density in Fig. 7 and 10 scaled by the current value\nis activated at 0 ns. If this anti-damping torque is smaller\nthan the Gilbert damping, the magnetization oscillations\ndecay, whereas if the anti-damping torque is larger than\nthe Gilbert damping, the magnetization oscillations grow\nuntil their amplitude saturates. We de\fne a threshold\ncurrent as that at which anti-damping is balanced with\nGilbert damping, where the magnetization oscillates with\na constant amplitude over time. The scale factor for the\ncurrent in the simulations is chosen so that the threshold\ncurrents in the simulations are as close as possible to the\nthreshold currents in the actual experimental data. The\nspectral and spatial pro\fles of the excitation modes are\nobtained by performing Fourier transforms of the time\ndependence of the magnetization dynamics.\na. Micromagnetic simulation using current density\ndistribution calculated in COMSOL\nFig. 7 shows the current density distribution of the AO\nsystem for a 1 mA current calculated using COMSOL [40]\nand the Oersted \feld produced by it. These determine\nthe anti-damping torque in micromagnetic simulations,\nand the corresponding results are shown in Fig. 8 and\nFig. 9.\nFigs. 8(a) and 8(b) show the evolution of power spec-\ntral density (PSD) with increasing bias current IDCfor\n\u0012= 65\u000eand\u0012= 70\u000e, respectively, and their represen-\ntative linecuts are shown in Fig. 8(c) and 8(d), respec-\ntively. These simulation data show the key features of\nthe experimental data in Fig. 3 of the main text, such as\nmonotonic redshift of the resonant frequency, mode am-\nplitude growing and mode amplitude saturation with in-\ncreasingIDC. In the simulation, the strong mode broad-\nening starts at a relatively lower IDCcompared to the\nexperimental data, so Fig. 8 shows the simulation result\napplicable only to the low current region of the exper-\nimental data Fig. 3. This indicates that the nonlinear\nmagnonic e\u000bect occurring at high currents, which can\ncause stronger self-excitation of the AO mode with high\ncoherence, seems to be lacking in the simulations. Since\nthermal e\u000bects play no role in the simulations, the mode7\nFIG. 6. (a) Spectra for various \u0012obtained from the micromagnetic simulation using Mumax3 with zero bias current, which\nre\rect the linear spin wave eigenmodes determined by the applied \feld H0and the constricted sample geometry. This is in\ncontrast to the case of nonlinear self-oscillatory modes with relatively larger amplitudes driven by the applied anti-damping\nspin torque. In (a), each spectrum is vertically o\u000bset. (b)-(f) Spatial eigenmode pro\fles corresponding to the spectral peaks\nindicated by the numbers in (a) at various \u0012. The dynamic magnetization amplitude mis normalized on the color scale of each\nimage. At\u0012= 70\u000eand 75\u000e, the spatial separation of edge (1-3) modes and bulk (4-5) modes occurs as shown in (b) and (c) as\nwell as their spectral separation as shown in (a).\nbroadening seen at high IDCarises purely from dipole\nor exchange spin-spin interactions of dynamic magne-\ntization occurring at relatively large cone angles. This\ndirectly demonstrates nonlinear damping due to mode\ncouplings that occur via dipole or exchange interactions,\nsupporting the main conclusion of the paper.\nFig. 9(a) shows the spectrum for various \u0012and Fig.\n9(b)-(f) show the spatial eigenmode pro\fles correspond-\ning to the spectral peaks indicated by the numbers in\nFig. 9(a). Most of the AO modes excited by the anti-\ndamping torque are edge modes that shift only to lower\nfrequencies with increasing \u0012, while the barely shifted\nbulk modes shown in the linear spin wave modes in Fig. 6\nare mostly suppressed. Compared with the spatial mode\npro\fles of the linear spin wave modes in Fig. 6, the edge\nmodes in this simulation seems to be selectively excited\nand evolved from the linear spin wave modes by the in-\nhomogeneous current density distribution.\nb. Micromagnetic simulation using current density\ndistribution of 2D Gaussian model\nIn order to test the e\u000bect of the inhomogeneous current\ndensity distribution on the excitation of the AO modes in\nthe simulation, we try a micromagnetic simulation using\nthe di\u000berent current density distribution in Fig. 10(a),\nwhere the current density Jxhas a 2D Gaussian distribu-\ntion with a maximum at the center and a broader distri-bution as function of position in the device compared to\nFig. 7(a), and Jy=Jz= 0. Indeed, the spectral shapes\nin this simulation are considerably di\u000berent as shown in\nFig. 2(b) of the main text and Fig. 11(c) and 11(d). In\ncontrast to Fig. 9, bulk modes excited around the cen-\ntral region are unsuppressed for \u0012= 70\u000eand 75\u000eand can\nhave larger mode amplitudes than edge modes, as shown\nin Fig. 2. These bulk modes exhibit improved agreement\nwith experimentally observed AO modes, in particular\nsmall frequency shift with increasing \u0012as shown in Fig.\n2(a) and Fig. 5. Also, the overall spectral shapes asso-\nciated with the number of the excited AO modes, their\nrelative frequencies and amplitudes, their frequency shift\nbehaviors for varying \u0012in Fig. 2(b) better match the\nexperimental data in Fig. 2(a) compared to the other\nAO simulations in Fig. 6(a) and Fig. 9(a). Therefore,\nwe conclude that the actual current density distribution\nis closer to the 2D Gaussian function in Fig. 10 than\nthat calculated using COMSOL in Fig. 7. We note that\nunidenti\fed structural defects around the edges of the\nAO active region of the sample can signi\fcantly alter the\nspectral shapes, especially for \u0012= 70\u000eand 75\u000e, where\nthe mode constriction e\u000bect is stronger.\nThis conclusion is further supported by the evolution\nof the power spectral density (PSD) with increasing bias\ncurrentIDCfor\u0012= 65\u000eand 70\u000eshown in Fig. 11(a) and\n11(b), respectively. In Fig. 11(a) and 11(c) for \u0012= 65\u000e,\nthe edge mode at the lowest frequency has the largest\namplitude at any IDCwhereas in Fig. 11(b) and 11(d)8\nFIG. 7. COMSOL [40] calculation of current densities in the\nAO structure for a 1 mA current: (a) x-component of current\ndensity distribution Jx, (b) y-component of current density\ndistribution Jy, (c) y-component of Oersted \feld Hy, and\n(d) z-component of Oersted \feld Hz.Jz= 0 andHx= 0.\nAnti-damping spin torque in the micromagnetic simulations\nis calculated based on these maps. The corresponding results\nare shown in Fig. 8 and Fig. 9.\nfor\u0012= 70\u000e, the bulk mode at \u00185.6 GHz grows with\nincreasingIDCand eventually has the larger amplitude\nthan any edge modes located at the lower frequencies at\nhighIDC, which agrees well with our experimental data\nin Fig. 3.\nAppendix C: Quantifying Nonlinear Damping with\nthe Coe\u000ecient Q\nThe nonlinear single-mode auto-oscillator can be de-\nscribed with a universal oscillator model, another form\nof the Landau-Lifshitz equation, derived by Slavin and\nTiberkevich [39],\ndc\ndt+i!(p)c+ \u0000+(p)c\u0000\u0000\u0000(p)c=fn(t) (C1)\nwherecis the complex dimensionless dynamic magneti-\nzation amplitude, p(=jcj2) is the oscillation power, !(p)\nis the power-dependent nonlinear frequency,\n\u0000+(p)\u0019\u0000G(1 +Qp) (C2)\n\u0000\u0000(p)\u0019\u001bI(1\u0000p) (C3)\nare the positive and negative damping constant, respec-\ntively,fn(t) is the thermal \ructuation noise, \u0000 G(=\u000b!)\nis the Gilbert damping, Qis the nonlinear damping coef-\n\fcient,\u001bis the spin-current e\u000eciency, and Iis the drive\ncurrent. On the left side of Eq. (C1), the second term\ndescribes precession, the third term describes damping,\nand the fourth term describes anti-damping. Note that\nFIG. 8. The evolution of power spectral density (PSD) with\nincreasing bias current IDCfor (a)\u0012= 65\u000eand (b)\u0012= 70\u000e\nobtained from micromagnetic simulations performed with the\ncurrent density distribution and Oersted \feld in Fig. 7. (c)\nRepresentative spectrum linecuts in (a) for various IDCat\n\u0012= 65\u000e. (d) Representative spectral linecuts in (b) for various\nIDCat\u0012= 70\u000e.\n!(p), \u0000+(p), and \u0000\u0000(p) are auto-oscillation power pde-\npendent.\nIn order to describe the nonlinear damping of the mul-\ntimode AO system in a simple and quantitative way, we\ncalculateQfrom the relationship with Ptotal =N0p,\nwhere forIDC0 in-\ncreases, the mode becomes narrower due to anti-damping\nand eventually splits into a couple of modes. In order to\nsee the overall shift of the ST-FMR peak, we normalize\nthe signalVmixto the maximum at each IDC, which is\nshown in Fig. 13 (b). With increasing IDC>0, the\nST-FMR peak shifts to the higher \feld because as the\namplitudes of self-localized AO modes grow, the static\nmagnetization decreases. On the other hand, as the ab-\nsolute value of IDC<0 increases up to -8 mA, the ST-\nFMR peak shifts monotonically to the lower \feld and\ndoes not show a shift to the higher \feld that might oc-\ncur due to the decrease in saturation magnetization by\nJoule heating. This means that even at IDC= 8 mA re-\ngardless of its sign, our device's magnetic system has not\nreached the current regime where the saturation magne-10\nFIG. 11. The evolution of power spectral density (PSD) with\nincreasing bias current IDCfor (a)\u0012= 65\u000eand (b)\u0012= 70\u000e\nobtained from micromagnetic simulations performed with the\ncurrent density distribution and Oersted \feld in Fig. 10. (c)\nRepresentative spectrum linecuts in (a) for various IDCat\u0012=\n65\u000e. (d) Representative spectrum linecuts in (b) for various\nIDCat\u0012= 70\u000e.\nFIG. 12. Noise level \u0011vs. temperature Tfor (a)\u0012= 65\u000eand\n(b)\u0012= 70\u000eobtained from the \ftting curves of solid colored\nlines in Fig. 4(a) and 4(d).\n.\nFIG. 13. (a) Spin torque FMR (ST-FMR) spectrum at var-\niousIDCranging from -8 mA to 8 mA. (b) The DC current\nevolution of ST-FMR where the signal Vmixis normalized to\nthe maximum at each IDCto see the shift of ST-FMR peak.\nObviously, the monotonic ST-FMR peak shift to the lower\n\feld with no change in the shift to the higher \feld for the\nintensity of IDC<0 increasing up to -8 mA indicates no re-\nduction in saturation magnetization that can be caused by\nJoule heating at large intensity of IDC.\nFIG. 14. Magnetic \feld dependence of resonance frequency\nobtained in (1) the main mode of ST-FMR measured at\nIDC= 0 mA (blue inverted triangle), (2) auto-oscillation (AO)\nmodes ofn= 1 (red circle), n= 2 (square orange), and n= 3\n(green triangle) measured at IDC= -7 mA, and (3) linear spin\nwave (SW) eigenmodes (red, orange, and green solid lines for\nn= 1;2, 3 respectively) obtained in the micromagnetic simu-\nlations using Mumax3 with IDC= 0 mA and 4 \u0019Ms= 6.5 kG.\nThe frequency discrepancy \u00180.5 GHz between the linear spin\nwave modes and the AO mode can be explained by the static\nmagnetization reduction and the frequency jump near the on-\nsetIththat has been observed as a feature of self-localization\nin the nonlinear \\bullet\"modes.\ntization decreases by Joule heating [37]. This provides\ncrucial evidence that the contribution of Joule heating\nto the nonlinear damping of the AO system described in\nthe main text is not signi\fcant at high IDCup to 8 mA.\nFig. 14 shows the resonance frequency f0of the auto-\noscillation as a function of the applied magnetic \feld H011\nFIG. 15. Temperature dependence of (a) Gilbert damping\nconstant\u000b, (b) saturation magnetization MS('Me\u000b), (c)\nspin Hall angle \u0012SH, obtained from ST-FMR. (d) Temperature\ndependence of the intrinsic threshold current Ith;0induced by\nthe spin Hall e\u000bect estimated by Eq. (E3) with parameters\nobtained from the ST-FMR in (a)-(c) in comparison with that\nestimated from the AO data in Fig. 4(c) of the main text.\ncompared to the linear spin wave eigenmode frequency\nand the resonance frequency of ST-FMR at IDC= 0 mA.\nThe AO mode appears below the corresponding linear\nspin wave mode with the discrepancy of \u00180.5 GHz due\nto two contributions: One by \u00180.2 GHz is the reduction\nof static magnetization due to increased cone angle shown\nin Fig. 3(b) and 3(g), and the other by \u00180.3 GHz is the\nfrequency jump of the AO mode from the linear spin wave\nmode frequency occurring near the onset Ith. The latter\nhas been observed as the signature of the bullet mode in\nself-localization [1, 3, 35].\nAppendix E: Temperature dependence of the\nintrinsic threshold current\nThe threshold current Ithwe obtain from measurement\nof the AO power has two contributions as expressed inEq. (1) of the main text: i) that due to thermal \ruc-\ntuation,\u0014thT, and ii) an intrinsic threshold current Ith;0\narising from the anti-damping torque due to the applied\nspin current. Here we estimate the temperature depen-\ndence of the intrinsic threshold current Ith;0(T) from the\nST-FMR data. According to the Slavin-Tiberkevich the-\nory described in Appendix C, the intrinsic Ith;0arising\nfrom the spin Hall e\u000bect is given by\nIth;0=\u000b!\n\u001b(E1)\nThe spin-current e\u000eciency \u001bis proportional to the\ntemperature-dependent \u0012SH=MSin our AO system [41].\n\u001b(T)/\u0012SH(T)\nMS(T)(E2)\nFrom Eq. (E1) and (E2), the intrinsic threshold is given\nas\nIth;0(T) =C\u000b(T)MS(T)\n\u0012SH(T)(E3)\nwhereCis a temperature independent constant. \u0012SHhas\nbeen found to be temperature independent with a value\nof 0.068 for 13 - 300 K [42], and \u000bfor Py \flm with same\nthickness 5 nm is also almost constant in the tempera-\nture range of 5 - 300 K regardless of capping materials\neven though there is a slight temperature dependence of\nsurface damping around 50 K [43]. Also, MScan be con-\nsidered to be almost constant at T\u001cTC. Therefore, Ith;0\ncan be taken to be almost constant for varying tempera-\nture. Based on this, we express Ith(T) determined from\nour AO power in Fig. 4 as a simple sum of the tempera-\nture independent Ith;0and additional thermal \ructuation\ncontribution linearly decreasing with temperature as Eq.\n(1) in the main text. 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Han, Experimental investigation\nof temperature-dependent gilbert damping in permalloy\nthin \flms, Scienti\fc Reports 6, 22890 (2016)." }, { "title": "1702.08408v2.Current_Induced_Damping_of_Nanosized_Quantum_Moments_in_the_Presence_of_Spin_Orbit_Interaction.pdf", "content": "Current Induced Damping of Nanosized Quantum Moments in the Presence of\nSpin-Orbit Interaction\nFarzad Mahfouzi\u0003and Nicholas Kioussisy\nDepartment of Physics and Astronomy, California State University, Northridge, CA, USA\n(Dated: November 10, 2021)\nMotivated by the need to understand current-induced magnetization dynamics at the nanoscale,\nwe have developed a formalism, within the framework of Keldysh Green function approach, to study\nthe current-induced dynamics of a ferromagnetic (FM) nanoisland overlayer on a spin-orbit-coupling\n(SOC) Rashba plane. In contrast to the commonly employed classical micromagnetic LLG simula-\ntions the magnetic moments of the FM are treated quantum mechanically . We obtain the density\nmatrix of the whole system consisting of conduction electrons entangled with the local magnetic\nmoments and calculate the e\u000bective damping rate of the FM. We investigate two opposite limiting\nregimes of FM dynamics: (1) The precessional regime where the magnetic anisotropy energy (MAE)\nand precessional frequency are smaller than the exchange interactions, and (2) The local spin-\rip\nregime where the MAE and precessional frequency are comparable to the exchange interactions. In\nthe former case, we show that due to the \fnite size of the FM domain, the \\Gilbert damping\"does\nnot diverge in the ballistic electron transport regime, in sharp contrast to Kambersky's breathing\nFermi surface theory for damping in metallic FMs. In the latter case, we show that above a critical\nbias the excited conduction electrons can switch the local spin moments resulting in demagnetization\nand reversal of the magnetization. Furthermore, our calculations show that the bias-induced anti-\ndamping e\u000eciency in the local spin-\rip regime is much higher than that in the rotational excitation\nregime.\nPACS numbers: 72.25.Mk, 75.70.Tj, 85.75.-d, 72.10.Bg\nI. INTRODUCTION\nUnderstanding the current-induced magnetization\nswitching (CIMS) at the nanoscale is mandatory for the\nscalability of non-volatile magnetic random access mem-\nory (MRAM) of the next-generation miniaturized spin-\ntronic devices. However, the local magnetic moments of a\nnanoisland require quantum mechanical treatment rather\nthan the classical treatment of magnetization commonly\nemployed in micromagnetic simulations, which is the cen-\ntral theme of this work.\nThe \frst approach of CIMS employs the spin transfer\ntorque (STT)1,2in magnetic tunnel junctions (MTJ) con-\nsisting of two ferromagnetic (FM) layers (i.e., a switch-\nable free layer and a \fxed layer) separated by an insulat-\ning layer, which involves spin-angular-momentum trans-\nfer from conduction electrons to local magnetization3,4.\nAlthough STT has proven very successful and brings the\nprecious bene\ft of improved scalability, it requires high\ncurrent densities ( \u00151010A/cm2) that are uncomfort-\nably high for the MTJ's involved and hence high power\nconsumption. The second approach involves an in-plane\ncurrent in a ferromagnet-heavy-metal bilayer where the\nmagnetization switching is through the so-called spin-\norbit torque (SOT) for both out-of-plane and in-plane\nmagnetized layers.5{8The most attractive feature of the\nSO-STT method is that the current does not \row through\nthe tunnel barrier, thus o\u000bering potentially faster and\nmore e\u000ecient magnetization switching compared to the\nMTJs counterparts.\nAs in the case of STT, the SO-STT has two compo-\nnents: a \feld-like and an antidamping component. Whilethe \feld-like component reorients the equilibrium direc-\ntion of the FM, the antidamping component provides the\nenergy necessary for the FM dynamics by either enhanc-\ning or decreasing the damping rate of the FM depending\non the direction of the current relative to the magneti-\nzation orientation as well as the structural asymmetry\nof the material. For su\u000eciently large bias the SOT can\novercome the intrinsic damping of the FM leading to ex-\ncitation of the magnetization precession.8The underlying\nmechanism of the SOT for both out-of-plane and in-plane\nmagnetized layers remains elusive and is still under de-\nbate. It results from either the bulk Spin Hall E\u000bect\n(SHE)9{12, or the interfacial Rashba-type spin-orbit cou-\npling,13{16or both17{19.\nMotivated by the necessity of scaling down the size\nof magnetic bits and increasing the switching speed, the\nobjective of this work is to develop a fully quantum me-\nchanical formalism, based on the Keldysh Green function\n(GF) approach, to study the current-induced local mo-\nment dynamics of a bilayer consisting of a FM overlayer\non a SOC Rashba plane, shown in Fig. 1.\nUnlike the commonly used approaches to investigate\nthe magnetization dynamics of quantum FMs, such as the\nmaster equation20, the scattering21or quasi-classical22\nmethods, our formalism allows the study of magnetiza-\ntion dynamics in the presence of nonequilibrium \row of\nelectrons.\nWe consider two di\u000berent regimes of FM dynamics: In\nthe \frst case, which we refer to as the single domain\ndynamics, the MAE and the precession frequency are\nsmaller than the exchange interactions, and the FM can\nbe described by a single quantum magnetic moment, of\na typically large spin, S, whose dynamics are governedarXiv:1702.08408v2 [cond-mat.mes-hall] 27 Apr 20172\nFIG. 1: (Color online) Schematic view of the FM/Rashba\nplane bilayer where the FM overlayer has length Lxand is\nin\fnite (\fnite) along the y-direction for the case of a single\ndomain (nano-island) discussed in Sec. III (IV). The magne-\ntization,~ m, of the FM precesses around the direction denoted\nby the unit vector, ~ nM, with frequency !and cone angle, \u0012.\nThe Rashba layer is attached to two normal (N) leads which\nare semi-in\fnite along the x-direction, across which an exter-\nnal bias voltage, V, is applied.\nmainly by the quantized rotational modes of the magne-\ntization. We show that the magnetic degrees of freedom\nentering the density matrix of the conduction electron-\nlocal moment entagled system simply shift the chemical\npotential of the Fermi-Dirac distribution function by the\nrotational excitations energies of the FM from its ground\nstate. We also demonstrate that the e\u000bective damping\nrate is simply the netcurrent along the the auxiliarym-\ndirection, where m= -S, -S+1, :::, +S, are the eigenval-\nues of the total Szof the FM. Our results for the change\nof the damping rate due to the presence of a bias volt-\nage are consistent with the anti-damping SOT of clas-\nsical magnetic moments,16,23, where due to the Rashba\nspin momentum locking, the anti-damping SOT, to low-\nest order in magnetic exchange coupling, is of the form,\n~ m\u0002(~ m\u0002^y), where ^yis an in-plane unit vector normal\nto the transport direction.\nIn the adiabatic and ballistic transport regimes due to\nthe \fnite S value of the nanosize ferromagnet our formal-\nism yields a \fnite \\Gilbert damping\", in sharp contrast to\nKambersky's breathing Fermi surface theory for damp-\ning in metallic FMs.24On the other hand, Costa and\nMuniz25and Edwards26demonstrated that the prob-\nlem of divergent Gilbert damping is removed by takinginto account the collective excitations. Furthermore, Ed-\nwards points out26the necessity of including the e\u000bect of\nlong-range Coulomb interaction in calculating damping\nfor large SOC.\nIn the second case, which corresponds to an indepen-\ndent local moment dynamics, the FM has a large MAE\nand hence the rotational excitation energy is compara-\nble to the local spin-\rip excitation (exchange energy).\nWe investigate the e\u000bect of bias on the damping rate of\nthe local spin moments. We show that above a criti-\ncal bias voltage the \rowing conduction electrons can ex-\ncite (switch) the local spin moments resulting in demag-\nnetization and reversal of the magnetization. Further-\nmore, we \fnd that, in sharp contrast to the single do-\nmain precessional dynamic, the current-induced damping\nis nonzero for in-plane and out-of-plane directions of the\nequilibrium magnetization. The bias-induced antidamp-\ning e\u000eciency in the local moment switching regime is\nmuch higher than that in the single domain precessional\ndynamics.\nThe paper is organized as follows. In Sec. II we present\nthe Keldysh formalism for the density matrix of the en-\ntagled quantum moment-conduction electron system and\nthe e\u000bective dampin/antdamping torque. In Sec. III we\npresent results for the current-induced damping rate in\nthe single domain regime. In Sec. IV we present results\nfor the current-induced damping rate in the independent\nlocal regime. We conclude in Sec. V.\nII. THEORETICAL FORMALISM\nFig. 1 shows a schematic view of the ferromagnetic\nheterostructure under investigation consisting of a 2D\nferromagnet-Rashba plane bilayer attached to two semi-\nin\fnite normal (N) leads whose chemical potentials are\nshifted by the external bias, Vbias. The magnetization\nof the FM precesses around the axis speci\fed by the\nunit vector, ~ nM, with frequency !and cone angle \u0012.\nThe FM has length, LFM\nx, along the transport direction.\nThe total Hamiltonian describing the coupled conduc-\ntion electron-localized spin moment system in the het-\nerostructure in Fig. 1 can be written as,\nHtot=X\nrr0;\u001b\u001b0Trfsdgh\u0010\n1s^H\u001b\u001b0\nrr0+\u000err0\u000e\u001b\u001b01s\u0016r+\u000err0Jsd~ \u001b\u001b\u001b0\u0001~sd(r) +\u000e\u001b\u001b0\u000err0HM\u0011\n \u0003\nfs0\ndgr0\u001b0 fsdgr\u001bi\n: (1)\nHere,~sd(r) is the local spin moment at atomic position\nr, the trace is over the di\u000berent con\fgurations of the lo-\ncal spin moments, fsdg, fsdgr\u001b=jfsdgi\n e\nr\u001bis the\nquasi-particle wave-function associated with the conduc-tion electron ( e) entangled to the FM states ( jfsdgi),\nJsdis thes\u0000dexchange interaction, 1sis the identity ma-\ntrix in spin con\fguration space, and ^ \u001bx;y;z are the Pauli\nmatrices. We use the convention that, except for r, bold3\nsymbols represent operators in the magnetic con\fgura-\ntion space and symbols with hat represent operators in\nthe single particle Hilbert space of the conduction elec-\ntrons. The magnetic Hamiltonian HMis given by\nHM=\u0000g\u0016BX\nr~Bext(r)\u0001~sd(r) (2)\n\u0000X\nhr;r0iJdd\nrr0\ns2\nd~sd(r0)\u0001~sd(r)\u0000X\nrJsd\nsd~sc(r)\u0001~sd(r);\nwhere, the \frst term is the Zeeman energy due to the\nexternal magnetic \feld, the second term is the magnetic\ncoupling between the local moments and the third term\nis the energy associated with the intrinsic magnetic \feld\nacting on the local moment, ~sd(r), induced by the local\nspin of the conduction electrons, ~sc(r).\nThe Rashba model of a two-dimensional electron gas\nwith spin orbit coupling interacting with a system of\nlocalized magnetic moments has been extensively em-\nployed14,27,28to describe the e\u000bect of enhanced spin-orbit\ncoupling solely at the interface on the current-induced\ntorques in ultrathin ferromagnetic (FM)/heavy metal\n(HM) bilayers. The e\u000bects of (i) the ferromagnet induc-\ning a moment in the HM and (ii) the HM with strong\nspin-orbit coupling inducing a large spin-orbit e\u000bect in\nthe ferromagnet (Rashba spin-orbit coupling) lead to a\nthin layer where the magnetism and the spin-orbit cou-\npling coexist.27\nThe single-electron tight-binding Hamiltonian29for\nthe conduction electrons of the 2D Rashba plane, H\u001b\u001b0\nrr0\nwhich is \fnite along the transport direction xand in\fnite\nalong theydirection is of the form,\n^H\u001b\u001b0\nxx0(kya) = [tcos(kya)\u000e\u001b\u001b0\u0000tsosin(kya)\u001bx\n\u001b\u001b0]\u000exx0(3)\n+t(\u000ex;x0+1+\u000ex+1;x0)\u000e\u001b\u001b0+itso(\u000ex;x0+1\u0000\u000ex+1;x0)\u001by\n\u001b\u001b0:\nHere,x;x0denote atomic coordinates along the trans-\nport direction, ais the in-plane lattice constant, and tso\nis the Rashba SOI strength. The values of the local e\u000bec-\ntive exchange interaction, Jsd= 1eV, and of the nearest-\nneighbor hopping matrix element, t=1 eV, represent a\nrealistic choice for simulating the exchange interaction of\n3dferromagnetic transition metals and their alloys (Fe,\nCo).30{32The Fermi energy, EF=3.1 eV, is about 1 eV\nbelow the upper band edge at 4 eV consistent with the\nab initio calculations of the (111) Pt surface33. Further-\nmore, we have used tso=0.5 eV which yields a Rashba\nparameter, \u000bR=tsoa\u00191.4 eV \u0017A (a=2.77 \u0017A is the in-\nplane lattice constant of the (111) Pt surface) consis-\ntent with the experimental value of about 1-1.5 eV \u0017A34\nand the ab initio value of 1 eV \u0017A28. However, because\nother experimental measurements for Pt/Co/Pt stacks\nreport35a Rashba parameter which is an order of mag-\nnitude smaller, in Fig.3 we show the damping rate for\ndi\u000berent values of the Rashba SOI.. For the results in\nSec. IV, we assume a real space tight binding for propa-\ngation along y-axis.The single particle propagator of the coupled electron-\nspin system is determined from the equation of motion\nof the retarded Green function,\n\u0012\nE\u0000i\u0011\u0000^\u0016\u0000^H\u0000HM\u0000Jsd\n2^~ \u001b\u0001^~sd\u0013\n^Gr(E) =^1;(4)\nwhere,\u0011is the broadening of the conduction electron\nstates due to inelastic scattering from defects and/or\nphonons, and for simplicity we ignore writing the identity\nmatrices ^1 and 1in the expression. The density matrix\nof the entire system consisting of the noninteracting elec-\ntrons (fermionic quasi-particles) and the local magnetic\nspins is determined (see Appendix A for details of the\nderivation for a single FM domain) from the expression,\n^\u001a=ZdE\n\u0019^Gr(E)\u0011f(E\u0000^\u0016\u0000HM)^Ga(E): (5)\nIt is important to emphasize that Eq. (5) is the central\nresult of this formalism which demonstrates that the ef-\nfect of the local magnetic degrees of freedom is to shift the\nchemical potential of the Fermi-Dirac distribution func-\ntion by the eigenvalues, \"m, ofHMjmi=\"mjmi, i.e.,\nthe excitation energies of the FM from its ground state.\nHere,jmiare the eigenstates of the Heisenberg model\ndescribing the FM. The density matrix can then be used\nto calculate the local spin density operator of the con-\nduction electrons, [ ~sc(r)]mm0=P\nss0\u001amm0\nss0;rr~ \u001bss0=2, which\nalong with Eqs. (2), (4), and (5) form a closed set\nof equations that can be solved self consistently. Since,\nthe objective of this work is the damping/anti-damping\n(transitional) behavior of the FM in the presence of bias\nvoltage, we only present results for the \frst iteration.\nEq. (5) shows that the underlying mechanism of the\ndamping phenomenon is the \row of conduction electrons\nfrom states of higher chemical potential to those of lower\none where the FM state relaxes to its ground state by\ntransferring energy to the conduction electrons. There-\nfore, the FM dynamical properties in this formalism is\ncompletely governed by its coupling to the conduction\nelectrons, where conservation of energy and angular mo-\nmentum dictates the excitations as well as the \ructua-\ntions of the FM sate through the Fermi distribution func-\ntion of the electrons coupled to the reservoirs. This is\ndi\u000berent from the conventional Boltzmann distribution\nfunction which is commonly used to investigate the ther-\nmal and quantum \ructuations of the magnetization.\nDue to the fact that the number of magnetic con\fgura-\ntions (i.e. size of the FM Hilbert space) grows exponen-\ntially with the dimension of the system it becomes pro-\nhibitively expensive to consider all possible eigenstates\nof theHMoperator. Thus, in the following sections we\nconsider two opposite limiting cases of magnetic con\fg-\nurations. In the \frst case we assume a single magnetic\nmoment for the whole FM which is valid for small FMs\nwith strong exchange coupling between local moments\nand small MAE. In this case the dynamics is mainly gov-\nerned by the FM rotational modes and local spin \rips can4\nFIG. 2: (Color online) Schematic representation of the quasi-\nparticles of the FM and conduction electron entangled states.\nThe horizontal planes denote the eigenstates, jS;miof the\ntotalSzof the FM with eigenvalues m=\u0000S;\u0000S+ 1;:::; +S\nalong the auxiliarym-direction. Excitation of magnetic state\ninduces a shift of the chemical potential of the Fermi-Dirac\ndistribution function leading to \row of quisiparticles along the\nm-direction which corresponds to the damping rate of the FM.\nThe FM damping involves two processes: (1) An intra-plane\nprocess involving spin reversal of the conduction electron via\nthe SOC; and (2) An inter-plane process involving quasiparti-\ncle \row of majority (minority) spin along the ascending (de-\nscending)m-direction due to conservation of total angular\nmomentum, where the interlayer hopping is accompanied by\na spin \rip of conduction electrons.\nbe ignored. In the second case we ignore the correlation\nbetween di\u000berent local moments and employ a mean \feld\napproximation such that at each step we focus on an indi-\nvidual atom by considering the local moment under con-\nsideration as a quantum mechanical object while the rest\nof the moments are treated classically. We should men-\ntion that a more accurate modeling of the system should\ncontain both single domain rotation of the FM as well\nas the local spin \ripping but also the e\u000bect of nonlocal\ncorrelations between the local moments and conduction\nelectrons, which are ignored in this work.\nIII. SINGLE DOMAIN ROTATIONAL\nSWITCHING\nIn the regime where the energy required for the excita-\ntion of a single local spin moment ( \u0019meV ) is much larger\nthan the MAE (\u0019\u0016eV) the low-energy excited states cor-\nrespond to rotation of the total angular momentum of the\nFM acting as a single domain and the e\u000bects of local spin\n\rips described as the second term in Eq 2, can be ignored.\nIn this regime all of the local moments behave collectively\nand the local moment operators can be replaced by the\naverage spin operator, ~sd(r) =P\nr0~sd(r0)=Nd=sd~S=S,\nwhereNdis the number of local moments and ~Sis the\ntotal angular momentum with amplitude S. The mag-\nnetic energy operator is given by HM=\u0000~B\u0001S, where,\n~B=g\u0016B~Bext+Jsd~ sc. Here, for simplicity we assume\n~ scto be scalar and independent of the FM state. The\neigenstates of HMoperator are then simply the eigen-states,jS;mi, of the total angular momentum Sz, with\neigenvalues m!=\u0000S!;:::; +S!, where!=Bzis the\nLarmor frequency. Thus, the wave function of the cou-\npled electron-spin con\fguration system, shown schemat-\nically in Fig. 2 is of the form, ms0r(t) =jS;mi\n s0r(t).\nOne can see that the magnetic degrees of freedom corre-\nsponding to the di\u000berent eigenstates of the Szoperator,\nenters as an additional auxiliary dimension for the elec-\ntronic system where the variation of the magnetic energy,\nhS;mjHMjS;mi=m!, shifts the chemical potentials of\nthe electrons along this dimension. The gradient of the\nchemical potential along the auxiliary direction, is the\nLarmor frequency ( \u0016eV\u0019GHz ) which appears as an\ne\u000bective \\electric \feld\"in that direction.\nSubstituting Eq (5) in Eq (A1)(b) and averaging over\none precession period we \fnd that the average rate of\nangular momentum loss/gain, which we refer to as the\ne\u000bective \\ damping rate \"per magnetic moment, can be\nwritten as\nTm=1\n2=(T\u0000\nm\u0000T+\nm); (6)\nwhere,\nT\u0006\nm=Jsd\n2SNdTrel[^\u001b\u0007S\u0006\nm^\u001am;m\u00061]: (7)\nis the current along the auxiliarym-direction in Fig. 2\nfrom them$m+ 1 (\u0006sign) state of the total Szof the\nFM. Here,Trel, is the trace over the conduction electron\ndegrees of freedom, and S\u0006\nm=p\nS(S+ 1)\u0000m(m\u00061)\nare the ladder operators. It is important to note that\nwithin this formalism the damping rate is simply the net\ncurrent across the mth-layer along the auxiliary direction\nassociated with the transition rate of the FM from state\nmto its nearest-neighbor states ( m\u00061).\nFig. 3 shows the damping rate as a function of the pre-\ncession cone angle, \u0012= cos\u00001(m\nS), for di\u000berent values of\nbias and for an in-plane e\u000bective magnetic \feld (a) along\nand (b) normal to the transport direction, and (c) an out-\nof-plane magnetic \feld. For cases (a) and (c) the damp-\ning rate is negative and relatively independent of bias for\nlow bias values. A negative damping rate implies that the\nFM relaxes towards the magnetic \feld by losing its angu-\nlar momentum, similar to the Gilbert damping rate term\nin the classical LLG equation, where its average value\nover the azimuthal precession angle, '=!t, is of the\nform,T=\u0000\u000bsdRd'\n2\u0019~ m\u0002(~ m\u0002~B)\u0001~ nM, which is nonzero\n(zero) when the unit vector ~ nMis along (perpendicular\nto) the e\u000bective magnetic \feld. The dependence of the\ndamping rate on the bias voltage when the e\u000bective mag-\nnetic \feld~Bis inplane and normal to the transport direc-\ntion can be understood by the spin-\rip re\rection mech-\nanism accompanied by Rashba spin-momentum locking\ndescribed in Ref.16. One can see that a large enough bias\ncan result in a sign reversal of the damping rate and hence\na magnetization reversal of the FM. It's worth mention-\ning that due to the zero-point quantum \ructuations of5\nthe magnetization, at \u0012= 0;\u0019(i.e.m=\u0006S) we have\nT 6=0 which is inversely proportional to the size of the\nmagnetic moment, S.\nIn Fig. 4(a) we present the e\u000bective damping rate ver-\nsus bias for di\u000berent values of the Rashba SOC. The re-\nsults show a linear response regime with respect to the\nbias voltage where both the zero-bias damping rate and\nthe slope,dT=dV increases with the Rashba SOC. This\nis consistent with Kambersky's mechanism of Gilbert\ndamping due to the SOC of itinerant electrons,24and the\nSOT mechanism16. Fig. 4(b) shows that in the absence\nof bias voltage the damping rate is proportional to t2\nsoand\nthe e\u000bect of the spin current pumped into the left and\nright reservoirs is negligible. This result of the t2\nsodepen-\ndence of the zero-bias damping rate is in agreement with\nrecent calculations of Costa and Muniz25and Edwards26\nwhich took into account the collective excitations. In the\npresence of an external bias, Tvaries linearly with the\nSOC, suggesting that to the lowest order it can be \ftted\nto\nT= sin2(\u0012)tso(c1tso~!+c2eVbias); (8)\nwherec1andc2are \ftting parameters.\nThe bias-induced e\u000eciency of the anti-damping SOT,\n\u0002\u0011~!(T(Vbias)\u0000T(0))=eVbiasT(0), describes how e\u000e-\ncient is the energy conversion between the magnetization\ndynamics and the conduction electrons. Accordingly, for\na given bias-induced e\u000eciency, \u0002, one needs to apply an\nexternal bias equal to ~!=e\u0002 to overcome the zero-bias\ndamping of the FM. Fig. 5 displays the anti-damping ef-\n\fciency versus the position of the Fermi energy of the FM\nfrom the bottom (-4 t=-4 eV) to the top (4 t=4eV) of the\nconduction electron band for the two-dimensional square\nlattice. The result is independent of the bias voltage and\nthe Larmor frequency in the linear response regime ( i.e.\nVbias;!\u001ct). We \fnd that the e\u000eciency peaks when the\nFermi level is in the vicinity of the bottom or top of the\nenergy band where the transport is driven by electron- or\nhole-like carriers and the Gilbert damping is minimum.\nThe sign reversal of the antidamping SOT is due to the\nelectron- or hole-like driven transport similar to the Hall\ne\u000bect.36\nClassical Regime of the Zero Bias Damping rate |\nIn the following we show that in the case of classical\nmagnetic moments ( S!1 ) and the adiabatic regime\n(!!0), the formalism developed in this paper leads to\nthe conventional expressions for the damping rate. In this\nlimit the system becomes locally periodic and one can\ncarry out a Fourier transformation from m\u0011Szspace\nto azimuthal angle of the magnetization orientation, ',\nspace. Conservation of the angular momentum suggests\nthat the majority- (minority-) spin electrons can propa-\ngate only along the ascending (descending) m-direction,\nwhere the hopping between two nearest-neighbor m-\nlayers is accompanied by a spin-\rip. As shown in Fig.\n2 the existence of spin-\rip hopping requires the presence\nof intralayer SOC-induced noncollinear spin terms which\nrotate the spin direction of the conduction electrons as\n04590135180−10−505B=[|B|,0,0]Damping Rate ( µeV)\n \n4590135180B=[0,|B|,0]\nCone Angle, θ (deg)4590135180B=[0,0,|B|]\nVbias=−5 mV\nVbias=0\nVbias=5 mV(a) (b) (c)\nStudent Version of MATLABFIG. 3: (Color online) E\u000bective damping rate for a single\nFM domain as a function of the precession cone angle, \u0012, for\nvarious bias values under an e\u000bective magnetic \feld which is\nin-plane (a) along and (b) normal to the transport direction\nand (c) out-of-plane. The length of the FM along the xdi-\nrection isLx= 25awhile it is assumed to be in\fnite in the\ny-direction, ~!= 10\u0016eV, the broadening parameter \u0011= 0,\nkBT= 10meV and the domain magnetic moment S= 200.\nThe results are robust with larger values of Sin either the\nballistic,\u0011\u001c~!, or dirty,\u0011\u001d~!, regimes.\n−2−1012−3−2−101\nBias Voltage (mV)Damping Rate ( µeV)\n \n−0.5 0 0.5−15−10−505\nSpin Orbit Coupling, tso (eV) \ntso=0\ntso=0.1 eV\ntso=0.2 eVVbias=−5 mV\nVbias=0\nVbias=5 mV(a) (b)\nStudent Version of MATLAB\nFIG. 4: (Color online) Damping torque versus (a) bias voltage\nand (b) spin-orbit coupling strength, for m= 0 corresponding\nto the precession cone angle of 90o. The precession axis of the\nFM is along the y-direction and the rest of the parameters are\nthe same as in Fig. 3. The zero-bias damping rate versus SOC\nshows at2\nsodependence while the damping rate under non-\nzero bias exhibits nearly linear SOC dependence.\nthey propagate in each m-layer. This is necessary for\nthe persistent \row of electrons along the 'auxiliary di-\nrection and therefore damping of the magnetization dy-\nnamics. Using the Drude expression of the longitudinal\nconductivity along the '-direction for the damping rate,\nwe \fnd that, within the relaxation time approximation,\n\u0011=!!1 , where the relaxation time of the excited con-\nduction electrons is much shorter than the time scale of6\n−4−3−2−101234−3\n−2\n−1\n0\n1\n2\n3\nFermi Energy (eV)Antidamping Efficiency (%)\n−4−2024−40−200Damping Rate ( µeV)\n \nVbias=5 mV\nVbias=0\nVbias=−5 mV\nStudent Version of MATLAB\nFIG. 5: (Color online) Bias-induced precessional anti-\ndamping e\u000eciency, \u0002 = ~!(T(Vbias)\u0000T(0))=eVbiasT(0), ver-\nsus the Fermi energy of the 2D Rashba plane in Fig.1, where\nthe energy band ranges from -4 eV to +4 eV. The magnetiza-\ntion precesses around the in-plane direction ( y\u0000axis) normal\nto the transport direction and the rest of the parameters of\nthe system are the same as in Fig. 3. Note, for magnetization\nprecession around the xandzaxis,T(Vbias) =T(0) for all\nprecession cone angles and hence \u0002=0. Inset shows the damp-\ning rate versus the Fermi energy for di\u000berent bias values used\nto calculate precessional anti-damping e\u000eciency.\nthe FM dynamics, Tis given by\nT=\u0000!\n\u0011X\nnZdkxdkyd'\n(2\u0019)3(v'\nn~k)2f0(\"n~k(')):(9)\nHere,v'\nn~k=@\"n~k(')=@' is the group velocity along the\n'-direction in Fig. 2, and \"n;~k=\"0(j~kj)\u0006j~h(~k)jfor the\n2D-Rashba plane, where \"0(j~kj) is the spin independent\ndispersion of the conduction electrons and ~h=atso^ez\u0002\n~k+1\n2Jsd~ m, is the spin texture of the electrons due to\nthe SOC and the s\u0000dexchange interaction. For small\nprecession cone angle, \u0012, the Gilbert damping constant\ncan be determined from \u000b=\u0000T=sd!sin2(\u0012), where the\nzero-temperature Tis evaluated by Eq. (9). We \fnd\nthat\n\u000b\u00191\n\u0011t2\nso\u0002\n(k+\nFa)2D+(EF) + (k\u0000\nFa)2D\u0000(EF)\u0003\n(1+cos2(\r));\n(10)\nwhereD+(\u0000)(E) is the density of states of the majority\n(minority) band, \ris the angle between the precession\naxis and the normal to the Rashba plane, and the Fermi\nwave-vectors ( k\u0006\nF) are obtained from, \"0(k\u0006\nF) =EF\u0007\nJsd=2. Eq. (10) shows that the Gilbert damping increases\nas the precession axis changes from in-plane ( \r=\u0019=2) to\nout of plane ( \r= 0),37which can also be seen in Fig. 3.\nIt is important to emphasize that in contrast to Eq. (9)\nthe results shown in Fig. 4 correspond to the ballistic\nregime with \u0011= 0 in the central region and the relaxation\nof the excited electrons occurs solely inside the metallic\nreservoirs. To clarify how the damping rate changes from\n10−810−610−410−2100−80−60−40−20020\nBroadening (eV)Damping Rate ( µeV)\n \nS=200, Vbias=0\nS=200, Vbias=3 mV\nS=300, Vbias=0\nS=300, Vbias=3 mV\nStudent Version of MATLABFIG. 6: (Color online) Precessional damping rate versus\nbroadening of the states in the presence (solid lines) and ab-\nsence (dashed lines) of bias voltage for two values of the do-\nmain sizeS= 200 and S= 300. In both ballistic, \u0011=!\u00190,\nand di\u000busive, \u0011=!\u001d1, regimes the precessional damping rate\nis independent of the domain size, while in the intermediate\ncase, the amplitude of the minimum of damping rate shows a\nlinear dependence versus S. Note that the value of the broad-\nening at which the damping rate is minimum varies inversely\nproportional to the domain size, S.\nthe ballistic to the di\u000busive regime we present in Fig.\n6 the damping rate versus the broadening, \u0011, of states\nin the presence (solid line) and absence (dashed line) of\nbias voltage. We \fnd that in both ballistic ( \u0011=!\u00190)\nand di\u000busive ( \u0011=!\u001d1) regimes the damping rate is\nindependent of the size of the FM domain, S. On the\nother hand, in the intermediate regime the FM dynamics\nbecome strongly dependent on the e\u000bective domain size\nwhere the minimum of the damping rate varies linearly\nwithS. This can be understood by the fact that the\ne\u000bective chemical potential di\u000berence between the \frst,\nm=\u0000Sand last,m=Slayers in Fig.3 is proportional\ntoSand for a coherent electron transport the conduc-\ntance is independent of the length of the system along\nthe transport direction. Therefore, in this case the FM\nmotion is driven by a coherent dynamics.\nIV. DEMAGNETIZATION MECHANISM OF\nSWITCHING\nIn Sec. III we considered the case of a single FM do-\nmain where its low-energy excitations, involving the pre-\ncession of the total angular momentum, can be described\nby the eigenstates jmiofSzand local spin \rip processes\nwere neglected. However, for ultrathin FM \flms or FM\nnanoclusters, where the MAE per atom ( \u0019meV ) is com-\nparable to the exchange energy between the local mo-\nments (Curie temperature), the low-energy excitations\ninvolve both magnetization rotation and local moments\nspin-\rips due to conduction electron scattering which can\nin turn change also S. In this case the switching is ac-7\nFIG. 7: (Color online) Spatial dependence of the local damp-\ning rate for the spin-1 =2 local moments of a FM island under\ndi\u000berent bias voltages ( \u00060:4V) and magnetization directions.\nFor the parameters we chose the size of the FM island to be\n25\u000225a2, the e\u000bective magnetic \feld, jBj= 20meV , the\nbroadening, \u0011= 0, andkBT= 10 meV.\ncompanied by the excitation of local collective modes that\ne\u000bectively lowers the amplitude of the magnetic ordering\nparameter. For simplicity we employ the mean \feld ap-\nproximation for the 2D FM nanocluster where the spin\nunder consideration at position ris treated quantum me-\nchanically interacting with all remaining spins through\nan e\u000bective magnetic \feld, ~B. The spatial matrix ele-\nments of the local spin operator are\n[^~sd;r]r1r2=~ sd(r1)\u000er1r2(1\u0000\u000er1r)1s+1\n2\u000er1r2\u000er1r~ \u001c;(11)\nwhere,~\u001cs are the Pauli matrices. The magnetic energy\ncan be expressed as, HM(r) =\u0000~B(r)\u0001~\u001c=2, where, the\ne\u000bective local magnetic \feld is given by,\n~B(r) =g\u0016B~Bext+ 4X\nr0Jdd\nrr0~ sd(r0) + 2Jsd~ sc(r):(12)\nThe equation of motion for the single particle propa-\ngator of the electronic wavefunction entangled with the\nlocal spin moment under consideration can then be ob-\ntained from,\n\u0012\nE\u0000^\u0016\u0000HM(r)\u0000^H\u0000Jsd\n2^~ \u001b\u0001^~sd;r\u0013\n^Gr\nr(E) =^1:(13)\nThe density matrix is determined from Eq. (5) which\ncan in turn be used to calculate the spin density of the\nconduction electrons, ~ sc(r) =Tr(^~ \u001b^\u001arr)=2, and the di-\nrection and amplitude of the local magnetic moments,\n~ sd(r) =Tr(~\u001c^\u001arr)=2.\nFig. 7 shows the spatial dependence of the spin-1\n2local\nmoment switching rate for a FM/Rashba bilayer (Fig.\n−101−30−20−100102030\n Damping Torque (meV)B=[|B|,0,0]\n−101\nBias Voltage (V)B=[0,|B|,0]\n−101B=[0,0,|B|]\n|B|=1 meV\n|B|=20 meV(c) (a) (b)\nStudent Version of MATLABFIG. 8: (Color online) Bias dependence of the average (over\nall sites) damping rate of the FM island for in-plane e\u000bective\nmagnetic \feld (or equilibrium magnetization) (a) along and\n(b) normal to the transport direction and (c) out-of-plane\nmagnetic \feld for two values of jBj.\n1) for two bias values ( Vbias=\u00060:4V) and for an in-\nplane e\u000bective magnetic \feld (a) along and (b) normal\nto the transport direction, and (c) an out-of-plane mag-\nnetic \feld. The size of the FM island is 25 a\u000225a, where\nais the lattice constant. Negative local moment switch-\ning rate (blue) denotes that, once excited, the local mo-\nment relaxes to its ground state pointing along the di-\nrection of the e\u000bective magnetic \feld; however positive\nlocal damping rate (red) denotes that the local moments\nremain in the excited state during the bias pulse dura-\ntion. Therefore, the damping rate of the local moments\nunder bias voltage can be either enhanced or reduced\nand even change sign depending on the sign of the bias\nvoltage and the direction of the magnetization. We \fnd\nthat the bias-induced change of the damping rate is high-\nest when the FM magnetization is in-plane and normal to\nthe transport directions similar to the single domain case.\nFurthermore, the voltage-induced damping rate is peaked\nclose to either the left or right edge of the FM (where the\nreservoirs are attached) depending on the sign of the bias.\nNote that there is also a \fnite voltage-induced damping\nrate when the magnetization is in-plane and and along\nthe transport direction ( x) or out-of-the-plane ( z).\nFig. 8 shows the bias dependence of the average (over\nall sites) damping rate for in- (a and b) and out-of-plane\n(c) directions of the e\u000bective magnetic \feld (direction\nof the equilibrium magnetization) and for two values of\njBj. This quantity describes the damping rate of the\namplitude of the magnetic order parameter. For an in-\nplane magnetization and normal to the transport direc-\ntion (Fig. 8) the bias behavior of the damping rate is lin-\near and \fnite in contrast to the single domain [Fig. 3(a)]\nwhere the damping rate was found to have a negligible\nresponse under bias. On the other hand, the bias behav-\nior of the current induced damping rate shows similar\nbehavior to the single domain case when the equilibrium8\n−4 −2 0 2 4−30−20−100102030\nFermi Energy (eV)Antidamping Efficiency (%)\n \nB=[20,0,0] meV\nB=[0,20,0] meV\nB=[0,0,20] meV\nStudent Version of MATLAB\nFIG. 9: (Color online) Bias-induced local anti-damping\ne\u000eciency due to local spin-\rip, \u0002 = jBj(T(Vbias)\u0000\nT(0))=eVbiasT(0), versus Fermi energy for di\u000berent equilib-\nrium magnetization orientations. For the calculation we chose\nVbias= 0:2 V and the rest of the Hamiltonian parameters are\nthe same as in Fig. 7.\nmagnetization direction is in-plane and normal to the\ntransport direction (Fig. 8(b)). For an out-of-plane ef-\nfective magnetic \feld [Fig. 8(c)] the damping torque has\nan even dependence on the voltage bias.\nIn order to quantify the e\u000eciency of the voltage in-\nduced excitations of the local moments, we calculate the\nrelative change of the average of the damping rate in the\npresence of a bias voltage and present the result versus\nthe Fermi energy for di\u000berent orientations of the magne-\ntization in Fig 9. We \fnd that the e\u000eciency is maximum\nfor an in-plane equilibrium magnetization normal to the\ntransport direction and it exhibits an electron-hole asym-\nmetry. The bias-induced antidamping e\u000eciency due to\nspin-\rip can reach a peak around 20% which is much\nhigher than the peak e\u000eciency of about 2% in the sin-\ngle domain precession mechanism in Fig. 5 for the same\nsystem parameters.\nFuture work will be aimed in determining the switch-\ning phase diagram16by calculating the local antidamping\nand \feld-like torques self consistently for di\u000berent FM\ncon\fgurations.\nV. CONCLUDING REMARKS\nIn conclusion, we have developed a formalism to in-\nvestigate the current-induced damping rate of nanoscale\nFM/SOC 2D Rashba plane bilayer in the quantum\nregime within the framework of the Kyldysh Green func-\ntion method. We considered two di\u000berent regimes of FM\ndynamics, namely, the single domain FM and indepen-\ndent local moments regimes. In the \frst regime we as-\nsume the rotation of the FM as the only degree of free-\ndom, while the second regime takes into account only\nthe local spin-\rip mechanism and ignores the rotation ofthe FM. When the magnetization (precession axis) is in-\nplane and normal to the transport direction, similar to\nthe conventional SOT for classical FMs, we show that the\nbias voltage can change the damping rate of the FM and\nfor large enough voltage it can lead to a sign reversal. In\nthe case of independent spin-1 =2 local moments we show\nthat the bias-induced damping rate of the local quantum\nmoments can lead to demagnetization of the FM and has\nstrong spatial dependence. Finally, in both regimes we\nhave calculated the bias-induced damping e\u000eciency as a\nfunction of the position of the Fermi energy of the 2D\nRashba plane.\nAppendix A: Derivation of Electronic Density\nMatrix\nUsing the Heisenberg equation of motion for the an-\ngular momentum operator, ~S(t), and the commutation\nrelations for the angular momentum, we obtain the fol-\nlowing Landau-Lifshitz equations of motion,\n\u0007i@\n@tS\u0006(t) =hzS\u0006(t)\u0000h\u0006(t)Sz(t) (A1a)\n\u0000i@\n@tSz(t) =1\n2\u0000\nh+(t)S\u0000(t)\u0000h\u0000(t)S+(t)\u0001\n(A1b)\n~hmm0(t) =1\n~X\nrJsd~smm0\nc(r) +g\u0016B\u000emm0~B(t);(A1c)\nwhere,S\u0006=Sx\u0006Sy(\u001b\u0006=\u001bx\u0006\u001by), is the angu-\nlar momentum (spin) ladder operators, ~smm0\nc(r) =\n1\n2P\n\u001b\u001b0~ \u001b\u001b\u001b0\u001amm0\n\u001b\u001b0;rris the local spin density of the con-\nduction electrons which is an operator in magnetic con-\n\fguration space. Here, \u001ais the density matrix of the\nsystem, and the subscripts, r;m;\u001b refer to the atomic\ncite index, magnetic state and spin of the conduction\nelectrons, respectively. In the following we assume a pre-\ncessing solution for Eq (A1)(a) with a \fxed cone angle\nand Larmor frequency !=hz. Extending the Hilbert\nspace of the electrons to include the angular momentum\ndegree of freedom we de\fne ms0i(t) =jS;mi\n s0i(t).\nThe equation of motion for the Green function (GF) is\nthen given\n\u0010\nE\u0000i\u0011\u0000^H(k) +n!\u0000n\n2SJsd(k)\u001bz\u0011\n^Gr\nnm(E;k) (A2)\n\u0000p\nS(S+ 1)\u0000n(n+ 1)\n2SJsd(k)\u001b\u0000^Gr\nn+1m(E;k)\n\u0000p\nS(S+ 1)\u0000n(n\u00001)\n2SJsd(k)\u001b+^Gr\nn\u00001m(E;k) =^1\u000enm\nwhere,n= (\u0000S;\u0000S+1;:::;S ) and the gauge transforma-\ntion n\u001bi(t)!ein!t n\u001bi(t) has been employed to remove\nthe time dependence. The density matrix of the system\nis of the form\n^\u001anm=e\u0000i(n\u0000m)!tSX\np=\u0000SZdE\n2\u0019^Gr\nnp2\u0011fp^\u0016^Ga\npm (A3)9\nwhere,fp^\u0016(E) =f(E\u0000p!\u0000^\u0016) is the equilibrium Fermi\ndistribution function of the electrons. Due to the fact\nthatp!are the eigenvalues of HM=\u0000g\u0016B~B\u0001S, one\ncan generalize this expression by transforming into a ba-\nsis where the magnetic energy is not diagonal which in\nturn leads to Eq (5) for the density matrix of the con-\nduction electron-local moment entagled system.\nAppendix B: Recursive Relation for GFs\nSince in this work we are interested in diagonal blocks\nof the GFs and in general for FMs at low temperaturewe haveS\u001d1, we need a recursive algorithm to be able\nto solve the system numerically. The surface Keldysh\nGFs corresponding to ascending ^ gu;r=<, and descending\n^gd;r=<, recursion scheme read,\n^gu;r\nn(E;k) =1\nE\u0000!n\u0000i\u0011n\u0000^H(k)\u0000^\u0006rn(E;k)\u0000n\n2SJsd(k)\u001bz\u0000(S\u0000\nn)2\n4S2Jsd(k)\u001b+^gu;r\nn\u00001(E;k)\u001b\u0000Jsd(k)(B1)\n^\u0006u;<\nn(E;k) =\u0000X\n\u000b\u0010\n2i\u0011n+^\u0006r\nn;\u000b(E;k)\u0000^\u0006a\nn;\u000b(E;k)\u0011\nfn\u000b+(S\u0000\nn)2\n4S2Jsd\u001b+^gu;r\nn\u00001^\u0006u;<\nn\u00001^gu;a\nn\u00001\u001b\u0000Jsd (B2)\n^gd;r\nn(E;k) =1\nE\u0000!n\u0000i\u0011n\u0000^\u0006rn(E;k)\u0000^H(k)\u0000n\n2SJsd(k)\u001bz\u0000(S+\nn)2\n4S2Jsd(k)\u001b\u0000^gu;r\nn+1(E;k)\u001b+Jsd(k)(B3)\n^\u0006d;<\nn(E;k) =\u0000X\n\u000b\u0010\n2i\u0011n+^\u0006r\nn;\u000b(E;k)\u0000^\u0006a\nn;\u000b(E;k)\u0011\nfn\u000b+(S+\nn)2\n4S2Jsd\u001b\u0000^gd;r\nn+1^\u0006d;<\nn+1^gd;a\nn+1\u001b+Jsd (B4)\nwhere, ^\u0006r\nn(E;k) =P\n\u000b^\u0006r\n\u000b(E\u0000!n;k) corresponds to the\nself energy of the leads, \u000b=L;R refers to the left and\nright leads in the two terminal device in Fig. 3 and S\u0006\nm=p\nS(S+ 1)\u0000m(m\u00061). Using the surface GFs we can\ncalculate the GFs as follows,\n^Gr\nn;m(E;k) =1\nE\u0000!n\u0000i\u0011n\u0000^H(k)\u0000^\u0006rn\u0000n\n2SJsd(k)\u001bz\u0000^\u0006r;u\nn\u0000^\u0006r;d\nn; n =m (B5)\n=S+\nn\n2S^gu;r\nn(E;k)Jsd(k)\u001b\u0000^Gr\nn+1;m(E;k); n6=m (B6)\n=S\u0000\nn\n2S^gd;r\nn(E;k)Jsd(k)\u001b+^Gr\nn\u00001;m(E;k); n6=m (B7)\nwhere the ascending and descending self energies are given by,\n^\u0006r;u\nn=(S\u0000\nn)2\n4S2Jsd(k)\u001b+^gu;r\nn\u00001(E;k)\u001b\u0000Jsd(k) (B8)\n^\u0006r;d\nn=(S+\nn)2\n4S2Jsd(k)\u001b\u0000^gd;r\nn+1(E;k)\u001b+Jsd(k) (B9)\nThe average rate of angular momentum loss/gain can be obtained from the real part of the loss of angular momentum\nin one period of precession,\nT0\nn=1\n2(T0\u0000\nn\u0000T0+\nn) =1\n2= X\nkTr[S\u0000\nn\n2S\u001b+Jsd(k)^\u001ann+1(k)\u0000S+\nn\n2S\u001b\u0000Jsd(k)^\u001ann\u00001(k)]!\n(B10)10\nwhich can be interpreted as the current \rowing across the layer n.\nT0\u0000=+\nn =X\nkZdE\n2\u0019iTrnh\n^\u0006d=u;r\nn(E;k)\u0000^\u0006d=u;a\nn(E;k)i\n^G<\nnn(E;k) +^\u0006d=u;<\nn (E)h\n^Gr\nnn(E;k)\u0000^Ga\nnn(E;k)io\n;(B11)\nAcknowledgments\nThe work at CSUN is supported by NSF-Partnership\nin Research and Education in Materials (PREM) GrantDMR-1205734, NSF Grant No. ERC-Translational Ap-\nplications of Nanoscale Multiferroic Systems (TANMS)-\n1160504, and US Army of Defense Grant No. W911NF-\n16-1-0487.\n\u0003Electronic address: Farzad.Mahfouzi@gmail.com\nyElectronic address: nick.kioussis@csun.edu\n1J. 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The terms of damping, whose weight associate d with them varies over time, are of the friction\ntype, and one of them has delay. This work will also address th e issue of existence and uniqueness of solution\nfor the model.\nMathematics Subject Classification (2010): Primary 35B40; Secondary 35Q74\nKeywords: Piezoelectric beam ·Energy decay ·magnetic effect ·time-varying delay ·time-varying weights\n1 Introduction\nIt is already known, since the 19th century that materials such as q uartz, Rochelle salt and barium titanate\nunder pressure produces electric charge/voltage, this phenome non is called the direct piezoelectric effect and\nwas discoveredby brothersPierreand JacquesCurie in 1880. This s ame materials, when subjected to an electric\nfield, produce proportional geometric tension. Such a phenomeno n is known as the converse piezoelectric effect\nand was discovered by Gabriel Lippmann in 1881 [47].\nCurrently, there is unanimity among researchers of the area, in ch aracterizing piezoelectric materials such\nas those that have the physical property of transforming mecha nical energy into electrical energy and vice versa\n[41, 45, 50], more precisely, they undergo mechanical deformation s when placed in an electric field and under\nmechanical loads they become electrically polarized. This type of spe cial property has great applicability in\nthe modern industry, therefore, these materials have been widely used in the production of electromechanical\ndevices, such as sensors for data collection, transducers for co nverting electric energy to mechanical energy or\nvice versa, resonators for timekeeping and telecommunication and actuators.\nThe ability of piezoelectric structures to generate deformations c ontrolled by electrical field applications\nand vice versa, attracts the attention of scientists from various areas in order to design mathematical and\ncomputational models capable of providing new knowledge and applica tions of these materials. In particular,\nMathematics can provideseveraltoolsfor studying the solution be haviorofpiezoelectric beam models, including\nNumerical Analysis, Dynamic Systems, Controllability and Stabilization .\nDue to the fact that magnetic energy has a relatively small effect on the general dynamics, magnetic effects\nare neglected in piezoelectric beam models. However, in closed loop, t he magnetic effect can cause oscillations\nin the output which results in the instability of the system, this tells us that the magnetic effect can cause a\nlimitation in the performance of the system [35, 52].\nIn many studies related to piezoelectric structures, the magnetic effect is neglected and only the mechanical\nand electrical effects are considered. In general the mechanical effects are modeled using Kirchhoff, Euler-\nBernoulli or Mindlin-Timoshenkoassumptions for small displacements [3, 14, 43, 50] and electricaland magnetic\neffects are added to the system generallyusing electrostatic, qua si-static and fully dynamic approaches[45]. The\nelectrostatic and quasi-static approaches (see for example [9, 14 , 16, 22, 41, 43, 45, 46]), despite being widely\nused, completely exclude the magnetic effect as well as its couplings w ith mechanical and electrical effects.\n∗carlos.mat.nonato@hotmail.com\n†jeremias@ufpa.br,∗Corresponding author\n‡raposo@ufsj.edu.br\n1Morris and ¨Ozer in [26, 27], proposed, a variational approach, a piezoelectric b eam model with a magnetic\neffect, based on the Euler-Bernoulli and Rayleigh beam theory for s mall displacement (the same equations for\nthe model are obtained if Mindlin-Timoshenko small displacement assu mptions are used [53]), they considered\nan elastic beam covered by a piezoelectric material on its upper and lo wer surfaces, isolated at the edges and\nconnected to a external electrical circuit to feed charge to the e lectrodes. As the voltage is prescribed at the\nelectrodes, the following Lagrangean was considered\nL=/integraldisplayT\n0[K−(P+E)+B+W]dt, (1.1)\nwhereK,P+E,BandBrepresent the (mechanical) kinetic energy, total stored energy , magnectic energy\n(electrical kinetic) of the beam and the work done by external for ces, respectively. For a beam of length\nLto thickness hand considering v=v(x,t),w=w(x,t) andp=p(x,t) as functions that represent the\nlongitudinal displacement of the center line, transverse displaceme nt of the beam and the total load of the\nelectric displacement along the transverse direction at each point x, respectively. So, one can assume that\nP+E=h\n2/integraldisplayL\n0/bracketleftbigg\nα/parenleftbigg\nv2\nx+h2\n12w2\nxx−2γβvxpx+βp2\nx/parenrightbigg/bracketrightbigg\ndx,B=µh\n2/integraldisplayL\n0p2\ntdx,\nK=ρh\n2/integraldisplayL\n0/parenleftbigg\nv2\nt+h2\n12w2\nt+w2\nt/bracketrightbigg\ndxandW=−/integraldisplayL\n0pxV(t)dx,(1.2)\nwhereV(t) is the voltage applied at the electrode. From Hamilton’s principle for a dmissible displacement\nvariations {v,w,p}ofLthe zero and observing that the only external force acting on the beam is the voltage\nat the electrodes (the bending equation is decoupled), they got th e system\nρvtt−αvxx+γβpxx= 0,\nµptt−βpxx+γβvxx= 0.(1.3)\nwhereρ,α,γ,µandβdenote the mass density, elastic stiffness, piezoelectric coefficient , magnetic permeability,\nwater resistance coefficient of the beam and the prescribed voltag e on electrodes of beam respectively, and in\naddition, the relationship is considered\nα=α1+γ2β, (1.4)\nthey assumed that the beam is fixed at x= 0 and free at x=L, and thus they got (from modeling) the following\nboundary conditions\nv(0,t) =αvx(L,t)−γβpx(L,t) = 0,\np(0,t) =βpx(L,t)−γβvx(L,t) =−V(t)\nh.(1.5)\nThen, the authors consider V(t) =kpt(L,t) (electrical feedback controller) in (1.5) and establish strong sta -\nbilization for almost all system parameters and exponential stability for system parameters in a null measure\nset.\nIt is worth mentioning that it is well known that piezoelectric beams wit hout the magnetic effect, in which\nthey are represented by a wave equation [27], are exactly observa ble [23] and exponentially stable [44].\nRamos et al. in [36] inserted a (mechanical) dissipative term δvtin (1.3)1, whereα >0 is a constant and\nconsidered the following boundary condition\nv(0,t) =αvx(L,t)−γβpx(L,t) = 0,\np(0,t) =βpx(L,t)−γβvx(L,t) = 0.(1.6)\nNote that, using (1.4), condition (1.6) is equivalent to Dirichlet-Neum ann condition\nv(0,t) =p(0,t) =vx(L,t) =px(L,t) = 0. (1.7)\nThe authors showed, by using energy method, that the system’s e nergy decays exponentially. This means that\nthe friction term and the magnetic effect work together in order to exponentially stabilizes the system.\nRamos et al. in [38] considered the piezoelectric beam with magnetic e ffect (1.3) with boundary conditions\ngiven by\nv(0,t) =αvx(L,t)−γβpx(L,t)+ξ1vt(L,t)\nh= 0,\np(0,t) =βpx(L,t)−γβvx(L,t)+ξ2pt(L,t)\nh= 0.(1.8)\n2They showed that the system is exponentially stable regardless of a ny relationship between system parameters\nand exponential stability is equivalent to exact observability at the b oundary.\nInparalleltothis, duetotechnologicaladvancesintheproduction anddesignofprecisioncontrolmechanisms\nsuch as sensors and actuators, there was an increasing need to s tudy the effects of information delay in order\nto improve the performance and control of these devices [15]. Dela y effects are present in almost every real\nmechanical system and in most situations such an effect is inevitable. Therefore, models that take into account\nthe effect of delay are more realistic [6].\nAn important exampleofthe delayeffect isthe active controlofcivil engineeringstructuresin which different\ncontrol systems are installed in tall buildings. The control process involvesseveral steps such as vibrational data\nmeasurement, filtering and conditioning of this data, computing con trol forces, transmission of data and signals\nto actuators, application of control forces necessary to the st ructure. If force applications are not synchronized\ndue to time delay, it can make the structure unstable [1].\nIn the context of models consisting of partial differential equation s, when we insert delay feedback terms\ninto models that were stable, they can become unstable [8, 7, 28]. Th erefore, for these types of models (formed\nby partial differential equations) we should be careful to analyze e ach case.\nNicaise et al. in [30], studied the following wave equation with boundary t ime-varying delay\nutt−∆u= 0 in Ω ×(0,∞),\nu= 0 in Γ D×(0,∞),\n∂u\n∂ν=−µ1ut−µ2ut(x,t−τ(t)) 0 in Γ N×(0,∞),\nu(x,0) =u0(x) andut(x,0) =u1(x) in Ω,\nut(x,t−τ(0)) =f0(x,t−τ(0)) in Γ N×(0,τ(0)),(1.9)\nwhere Ω ⊂Rnis bounded and smooth domain, µ1andµ2are positive constants, ν(x) represent the outer unit\nnormal vector to the point x∈Γ and∂u\n∂νis the normal derivative, Γ = Γ D∪ΓNis the boundary of Ω. At work,\nit was considered\nτ∈W2,∞([0,T]),∀T >0, (1.10)\n0<τ0≤τ(t)≤τ,∀t>0, (1.11)\nfor some constants τ0andτand there exists d>0 such that\nµ2<√\n1−dµ1 (1.12)\nwith\nτ′(t)≤d<1,∀t>0. (1.13)\nWith these assumptions, the authors showed that the system is ex ponentially stable.\nKirane et al. in [21], considered the following one-dimensional Timoshen ko beam model with variable delay\nτ(t) in the rotation angle equation\nρ1ϕtt−κ(ϕx+ψ)x= 0 in (0 ,1)×(0,∞),\nρ2ψtt−bψxx+κ(ϕx+ψ)+µ1ψt+µ2ψt(x,t−τ(t)) = 0 in (0 ,1)×(0,∞),(1.14)\nwhereρ1,ρ2,κandbare positive constants related to the beam’s physical properties, the delay function τ(t)\nsatisfies (1.10), (1.11) and (1.13). The authors showed that if (1.1 2) andρ1/κ=ρ2/bholds, then the system is\nexponentially stable.\nBenaissa et al. in [5] considered the following wave equation with delay a nd damping weights depending on\nthe time\nutt−∆u+µ1(t)ut+µ2(t)ut(x,t−τ) = 0 in Ω ×(0,∞),\nu= 0 in Γ ×(0,∞),\nu(x,0) =u0(x) andut(x,0) =u1(x) in Ω,\nut(x,t−τ(0)) =f0(x,t−τ(0)) in Ω ×(0,τ(0)),(1.15)\nwhere Ω ⊂Rnis a bouded domain with boundary Γ. Unlike previous works, the dampin gsµ1andµ2depend\non the time t, however the delay time τis constant. Under appropriate assumptions about the weights of the\ndampingµ1andµ2the authors obtained the exponential decay of the energy of the system.\nBarros et al. in [4] studied the problem (1.15) with Ω = (0 ,L)⊂Randτ=τ(t) a function of time t. Under\nappropriate assumptions for µ1(t) andµ2(t) and considering (1.10), (1.11) and (1.13) the authors showed tha t\nthe energy of the system decays exponentially.\n3Our intention in mentioning the last three works was to show situation s in which time-dependent delay\nfeedback appears τ=τ(t) as well as to show situations in which the weight of the damping may va ry, which\nmake the problem worse, undoubtedly more attractive and challeng ing.\nThere are numerous studies on exponential stability of linear syste ms considering the case where the delay is\nconstant [2, 12, 20, 28, 29, 39, 40, 42, 49]. Therearealso severa lstudies consideringnon-linearmodels with delay\nwhere the existence of attractors is investigated, among them, T imoshenko systems [11, 13, 37, 51], poroelastic\nsystems [10] and suspension bridge [31, 48].\nBased on the work mentioned above about piezoelectric beam and de lay feedback, we design and propose\nto study and question the exponential stability for the following sys tem\nρvtt−αvxx+γβpxx+µ1(t)vt+µ2(t)vt(x,t−τ(t)) = 0 in (0 ,L)×(0,∞),\nµptt−βpxx+γβvxx= 0 in (0 ,L)×(0,∞),(1.16)\nwith boundary conditions given by (1.6) and initial conditions\nv(x,0) =v0(x), vt(x,0) =v1(x), p(x,0) =p0(x), pt(x,0) =p1(x), x∈(0,L),\nvt(x,−sτ(0)) =v2(x,s),(x,s)∈(0,L)×(0,1),(1.17)\nwherev0,v1,v2,p0,p1are known functions belonging to appropriate functional spaces.\nIn (1.16) we are admitting that the delay is being considered in the long itudinal displacement of the beam,\nthis seems to us very natural since there are several studies on p iezoelectric structures considering the effect of\ndelay on the mechanical part of the system [19, 24, 33, 34].\nWe will use the standard multiplicative method to obtain the main result . The novelty of the work is found,\nbasically, in the application ofthis technique in a relativelynew model (p iezoelectricbeam with magneticeffect).\nThe article is organized as follows: in section 2, we will consider the ass umptions for the functions present\nin (1.16) as well as, through a change of variable, obtain a system eq uivalent to (1.16). In section 3, using the\nsemigroup theory of linear operators found in [18], the question of t he existence, uniqueness and regularity of\nthe solution will be addressed. In section 4, we will obtain the main res ult of this work, which is the proof of\nthe exponential decay for the system (1.1).\n2 Preliminaries and Assumptions\nIn this work we will consider the following assumptions:\n(A1)The delay function τ=τ(t), satisfies\nτ∈W2,∞([0,T]),∀T >0, (2.1)\nthere exist positive constants τ0,τ1andd, satisfying\n0<τ0≤τ(t)≤τ1,∀t>0 (2.2)\nand\nτ′(t)≤d<1,∀t>0; (2.3)\n(A2)µ1:R+→(0,+∞) is a non-increasing function of class C1(R+). In addition, there exists a constant\nM1>0, such that /vextendsingle/vextendsingle/vextendsingle/vextendsingleµ′\n1(t)\nµ1(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤M1,∀t≥0; (2.4)\n(A3)µ2:R+→Ris a function of class C1(R+),which is not necessarily positives or monotones. In addition,\nthere exist constants M2>0 andδ, with 0<δ<√\n1−d, such that\n|µ2(t)|≤δµ1(t), (2.5)\nand\n|µ′\n2(t)|≤M2µ1(t). (2.6)\nLet us now considerthe followingprocedure that can be found in [30], in orderto obtain a new (independent)\nvariable\nz(x,y,t) =vt(x,t−τ(t)y),(x,y,t)∈(0,L)×(0,1)×(0,∞). (2.7)\n4It is easily verified that the zsatisfies\nτ(t)zt(x,y,t)+(1−τ′(t)y)zy(x,y,t) = 0. (2.8)\nTherefore, by using (2.7) and (2.8) we can rewrite (1.16) as follows\nρvtt−αvxx+γβpxx+µ1(t)vt+µ2(t)z(x,1,t) = 0 in (0 ,L)×(0,∞),\nµptt−βpxx+γβvxx= 0 in (0 ,L)×(0,∞),\nτ(t)zt+(1−τ′(t)y)zy= 0 in (0 ,L)×(0,1)×(0,∞),(2.9)\nsubject to boundary conditions given in (1.6), that is,\nv(0,t) =p(0,t) =vx(L,t) =px(L,t) = 0 (2.10)\nand initial conditions\nv(x,0) =v0(x), vt(x,0) =v1(x), p(x,0) =p0(x), pt(x,0) =p1(x),in (0,L),\nz(x,y,0) =v2(x,y),in (0,L)×(0,1).(2.11)\n3 Well-posedness\nIn this section, using the theory of semigroups of linear operators found in [18], a result of existence, uniqueness\nand regularity will be obtained for the problem (2.9)-(2.11). Similar pr ocedures are found in [21, 25, 30].\nFirstly, consider the following spaces\nH∗(0,L) ={η∈H1(0,L);η(0) = 0} (3.1)\nand\nH=H∗(0,L)×L2(0,L)×H∗(0,L)×L2(0,L)×L2((0,L)×(0,1)). (3.2)\nWe define on Hthe following inner product\n/an}bracketle{tU,˜U/an}bracketri}htH=ρ/integraldisplayL\n0u/tildewideudx+µ/integraldisplayL\n0q/tildewideqdx+α1/integraldisplayL\n0vx˜vxdx\n+β/integraldisplayL\n0(γvx−px)(γ/tildewidevx−/tildewidepx)dx+/integraldisplayL\n0/integraldisplay1\n0z˜zdydx,(3.3)\nfor anyU= (v,u,p,q,z ),/tildewideU= (/tildewidev,/tildewideu,/tildewidep,/tildewideq,/tildewidez) inH.\nIntroducing U(t) = (v(t),vt(t),p(t),pt(t),z(t))TandU0= (v0,v1,p0,p1,v2)T, the system (2.9)-(2.11) can be\nwritten as the following abstract initial value problem in H\n/braceleftbiggUt(t) =A(t)U(t), t>0,\nU(0) =U0,(3.4)\nwhere the operator A(t) :D(A(t))⊂H→His given by\nA(t)\nv\nu\np\nq\nz\n=\nu\nρ−1(αvxx−γβpxx−µ1(t)u−µ2(t)z(·,1))\nq\nµ−1(βpxx−γβvxx)\n−1−τ′(t)y\nτ(t)zy\n, (3.5)\nwith\nD(A(t)) ={(v,u,p,q,z )∈H;v,p∈H2(0,L);u,q∈H∗(0,L);\nz∈L2(0,1;H1\n0(0,L)), z(·,0) =u}.(3.6)\nNote thatD(A(t)) is independent of t, that is,\nD(A(t)) =D(A(0)),∀t>0. (3.7)\nA general theory for not autonomous operators given by equatio ns of type (3.4) has been developed using\nsemigroup theory, see [17, 18, 32]. The simplest way to prove exist ence and uniqueness results is to show that\nthe triplet {(A,H,Y)}, withA={A(t);t∈[0,T]}, for some fixed T >0 andY=A(0), forms a CD-systems\n(Constant Domain system, see [17, 18]). More precisely, the follow ing theorem, which is due to Tosio Kato\n(Theorem 1.9 in [18]) gives the existence and uniqueness results:\n5Theorem 3.1. Assume that\n(i)Y=D(A(0))is dense in H;\n(ii)D(A(t)) =D(A(0)),∀t>0;\n(iii) for all t∈[0,T],A(t)generates a strongly continuous semigroup on Hand the family A={A(t);t∈\n[0,T]}is stable with stability constants Candmindependent of t(i.e., the semigroup (St(s))s≥0generated\nbyA(t)satisfies/bardblSt(s)W/bardblH≤Cems/bardblW/bardblH, for allW∈Hands≥0);\n(iv)∂tA(t)belongs to L∞\n∗([0,T],B(Y,H)), which is the space of equivalent classes of essentially bou nded,\nstrongly measurable functions from [0,T]into the set B(Y,H)of bounded operators from YintoH.\nThen, problem (3.4)has a unique solution\nU∈C([0,T);Y)∩C1([0,T);H), (3.8)\nfor any initial datum in Y.\nIn this way, we are ready to state and prove the main result of this s ection, which is\nTheorem 3.2 (Global solution). For anyU0∈D(A(0)), there exists a unique solution Uof(3.4)satisfying\nU∈C([0,+∞);D(A(0)))∩C1([0,+∞);H). (3.9)\nProof.We must show that A(t) meets the conditions of Theorem 3.1. In fact,\n(i)this condition can be proven using arguments analogous to those fo und in [21, 25, 29, 30].\n(ii)It has been observed in (3.7).\n(iii)In order to show that the operator A(t) generates a C0-semigroup on H, givent, we introduced the\ntime-dependent inner product on H(this internal product is equivalent to (3.3))\n/an}bracketle{tU,˜U/an}bracketri}htt=ρ/integraldisplayL\n0u/tildewideudx+µ/integraldisplayL\n0q/tildewideqdx+α1/integraldisplayL\n0vx˜vxdx\n+β/integraldisplayL\n0(γvx−px)(γ/tildewidevx−/tildewidepx)dx+ξ(t)τ(t)/integraldisplayL\n0/integraldisplay1\n0z˜zdydx,(3.10)\nfor anyU= (v,u,p,q,z ),/tildewideU= (/tildewidev,/tildewideu,/tildewidep,/tildewideq,/tildewidez) inH, where\nξ(t) =ξµ1(t) (3.11)\nandξis a positive constant such that\nδ√\n1−d<¯ξ<2−δ√\n1−d. (3.12)\nNote that\n/an}bracketle{tA(t)U,U/an}bracketri}htt=−µ1(t)/integraldisplayL\n0u2dx−µ2(t)/integraldisplayL\n0z(x,1)udx\n−ξ(t)\n2/integraldisplayL\n0/integraldisplay1\n0(1−τ′(t)y)∂\n∂yz2(x,y)dydx,(3.13)\nfor anyU= (v,u,p,q,z )T∈D(A(t)). Since\n(1−τ′(t)y)∂\n∂yz2=∂\n∂y/parenleftbig\n(1−τ′(t)y)z2/parenrightbig\n+τ′(t)z2, (3.14)\nfrom (3.13) and (3.14) we have\n/an}bracketle{tA(t)U,U/an}bracketri}htt=−µ1(t)/integraldisplayL\n0u2dx−µ2(t)/integraldisplayL\n0z(x,1)udx+ξ(t)\n2/integraldisplayL\n0u2dx\n−ξ(t)(1−τ′(t))\n2/integraldisplayL\n0z2(x,1)dx−ξ(t)τ′(t)\n2/integraldisplayL\n0/integraldisplay1\n0z2dydx.(3.15)\n6Now, applying Young’s inequality to the second term on the right side o f (3.15), we get\n/an}bracketle{tA(t)U,U/an}bracketri}htt≤−/parenleftbigg\nµ1(t)−ξ(t)\n2−|µ2(t)|\n2√\n1−d/parenrightbigg/integraldisplayL\n0u2dx\n−/parenleftbiggξ(t)\n2−ξ(t)τ′(t)\n2−|µ2(t)|√\n1−d\n2/parenrightbigg/integraldisplayL\n0z2(x,1)dx\n+ξ(t)|τ′(t)|\n2τ(t)τ(t)/integraldisplayL\n0/integraldisplay1\n0z2dydx.(3.16)\nFrom(A3)and (3.11), we obtain\n/an}bracketle{tA(t)U,U/an}bracketri}htt≤−µ1(t)/parenleftbigg\n1−ξ\n2−δ\n2√\n1−d/parenrightbigg/integraldisplayL\n0u2dx\n−µ1(t)/parenleftbiggξ(1−τ′(t))\n2−δ√\n1−d\n2/parenrightbigg/integraldisplayL\n0z2(x,1)dx\n+κ(t)/an}bracketle{tU,U/an}bracketri}htt,(3.17)\nwhere\nκ(t) =/radicalbig\n1+τ′(t)2\n2τ(t). (3.18)\nFrom (2.2), (2.3) and (3.12), we have\n1−ξ\n2−δ\n2√\n1−d>0 andξ(1−τ′(t))\n2−δ√\n1−d\n2>0. (3.19)\nTherefore we conclude that\n/an}bracketle{tA(t)U,U/an}bracketri}htt−κ(t)/an}bracketle{tU,U/an}bracketri}htt≤0, (3.20)\nwhich means that operator /tildewideA(t) =A(t)−κ(t)Iis dissipative (in the next steps we will use /tildewideAas a pivot to then\nrecover the intended properties of A).\nNow, we will prove the surjectivity of the operator λI−A(t), for fixed t >0. For this purpose, given\nF= (f1,f2,f3,f4,f5)T∈H, we seekU= (v,u,p,q,z )T∈D(A(t)) which is solution of\n(λI−A(t))U=F, (3.21)\nthat is, the entries of Usatisfy the system of equations\nλv−u=f1, (3.22)\nλρu−αvxx+γβpxx+µ1(t)u+µ2(t)z(x,1) =ρf2, (3.23)\nλp−q=f3, (3.24)\nλµq−βpxx+γβvxx=µf4, (3.25)\nλτ(t)z+(1−τ′(t)y)zy=τ(t)f5. (3.26)\nSuppose that we have found vandpwith the appropriated regularity. Therefore, from (3.22) and (3.2 4) we\nhave\nu=λv−f1, (3.27)\nq=λp−f3, (3.28)\nit is clear that u,q∈H∗(0,L). Furthermore, if τ′(t) = 0, then\nz(x,y) =u(x)e−λτ(t)y+τ(t)e−λτ(t)y/integraldisplayy\n0f5(x,s)eλτ(t)sds (3.29)\nis solution of (3.26) satisfying\nz(x,0) =u(x). (3.30)\n7Otherwise,\nz(x,y) =u(x)eσ(y,t)+τ(t)eσ(y,t)/integraldisplayy\n0f5(x,s)\n1−τ′(t)se−σ(s,t)ds, (3.31)\nwhere\nσ(y,t) =λτ(t)\nτ′(t)ln(1−τ′(t)y), (3.32)\nis solution of (3.26) satisfying (3.30). From now on, for practicality p urposes, we will consider τ′(t) = 0 (the\ncaseτ(t)/ne}ationslash= 0 is analogous), this way we have (taking into account (3.27))\nz(x,1) =ue−λτ(t)+τ(t)e−λτ(t)/integraldisplay1\n0f5(x,s)eλτ(t)sds\n= (λv−f1)e−λτ(t)+τ(t)e−λτ(t)/integraldisplay1\n0f5(x,s)eλτ(t)sds\n=λve−λτ(t)−f1e−λτ(t)+τ(t)e−λτ(t)/integraldisplay1\n0f5(x,s)eλτ(t)sds.(3.33)\nSubstituting (3.27) and (3.33) in (3.23), and (3.28) in (3.25), we obta in\nηv−αvxx+γβpxx=g1,\nλ2µp−βpxx+γβvxx=g2,(3.34)\nwhere\nη:=λ2ρ+λµ1(t)+λµ2(t)e−λτ(t),\ng1:=ρf2+λρf1+µ1(t)f1+µ2(t)f1e−λτ(t)−µ2(t)τ(t)e−λτ(t)/integraldisplay1\n0f5(x,s)eλτ(t)sds,\ng2:=µf4+λµf3.(3.35)\nIn orderto solve (3.34), we use a standard procedure, considerin gbilinear form Υ : (( H∗(0,L)×H∗(0,L))2→R,\ngiven by\nΥ((v,p),(/tildewidev,/tildewidep)) =η/integraldisplayL\n0v˜vdx+α/integraldisplayL\n0vx˜vxdx−γβ/integraldisplayL\n0px˜vxdx\n+λ2µ/integraldisplayL\n0p˜pdx+β/integraldisplayL\n0px˜pxdx−γβ/integraldisplayL\n0vx˜pxdx.(3.36)\nIt is not difficult to show that Υ is continuous and coercive, so by apply ing the Lax-Milgram’s Theorem, we\nobtain a solution for ( v,p)∈H∗(0,L)×H∗(0,L) for (3.34). In addition, it follows from (3.23) and (3.25) that\nv,p∈H2(0,L) and so (v,u,p,q,z )∈D(A(t)).\nTherefore, the operator λI−A(t) is surjective for all t>0. Sinceκ(t)>0, we have\nλI−/tildewideA(t) = (λ+κ(t))I−A(t) is surjective ∀t>0. (3.37)\nTo complete the proof of (iii), it’s suffices to prove that\n/bardblΦ/bardblt\n/bardblΦ/bardbls≤ec\n2τ0|t−s|,∀t,s∈[0,T], (3.38)\nwhere Φ = ( v,u,p,q,z )T,cis a positive constant and /bardbl·/bardbltis the norm associated to the inner product (3.10).\nFort,s∈[0,T], we have\n/bardblΦ/bardbl2\nt−/bardblΦ/bardbl2\nsec\nτ0|t−s|=/parenleftig\n1−ec\nτ0|t−s|/parenrightig/integraldisplayL\n0/bracketleftbig\nρu2+µq2+α1v2\nx+β(γvx−px)2/bracketrightbig\ndx\n+/parenleftig\nξ(t)τ(t)−ξ(s)τ(s)ec\nτ0|t−s|/parenrightig/integraldisplayL\n0/integraldisplay1\n0z2(x,y)dydx.(3.39)\nIt is clear that 1 −ec\nτ0|t−s|≤0. Now we will prove ξ(t)τ(t)−ξ(s)τ(s)ec\nτ0|t−s|≤0 for some c>0. In order to\ndo this, from (2.1) and MVT, we have\nτ(t) =τ(s)+τ′(r)(t−s), (3.40)\n8for somer∈(s,t). Sinceξis a non increasing function and ξ>0, we get\nξ(t)τ(t)≤ξ(s)τ(s) +ξ(s)τ′(r)(t−s), (3.41)\nwhich implies\nξ(t)τ(t)\nξ(s)τ(s)≤1+|τ′(r)|\nτ(s)|t−s|. (3.42)\nUsing (2.1) and that τ′is bounded, we deduce that\nξ(t)τ(t)\nξ(s)τ(s)≤1+c\nτ0|t−s|≤ec\nτ0|t−s|, (3.43)\nwhich proves (3.38) and therefore (iii)follows.\n(iv)Note that, from (A1), we have\nκ′(t) =τ′(t)τ′′(t)\n2τ(t)/radicalbig\n1+τ′(t)2−τ′(t)/radicalbig\n1+τ′(t)2\n2τ(t)2(3.44)\nis bounded on [0 ,T] for allT >0. Moreover\nd\ndtA(t)U=\n0\n−ρ−1[µ′\n1(t)u+µ′\n2(t)z(·,1)]\n0\n0\nτ′′(t)τ(t)y−τ′(t)(τ′(t)y−1)\nτ(t)2zy\n, (3.45)\nSinceτ′′(t)τ(t)ρ−τ′(t)(τ′(t)ρ−1)\nτ(t)2 is bounded on [0 ,T] by(A1), and considering (A2)and(A3), we have\nd\ndt˜A(t)∈L∞\n∗([0,T],B(D(A(0)),H)), (3.46)\nwhereL∞\n∗([0,T],B(D(A(0)),H))isthespaceofequivalenceclassesofessentiallybounded, stron glymeasurable\nfunctions from [0 ,T] intoB(D(A(0)),H).\nThen, (3.20), (3.37) and (3.38) imply that the family /tildewideA={/tildewideA(t) :t∈[0,T]}is a stable family of generators\ninHwith stability constants independent of t, by Proposition1 .1 from [18]. Therefore, the assumptions (i)-(iv)\nof Theorem 3.1 are verified. Thus, the problem\n/braceleftigg\n/tildewideUt=/tildewideA(t)/tildewideU,\n/tildewideU(0) =U0(3.47)\nhas a unique solution /tildewideU∈C([0,+∞),D(A(0)))∩C1([0,+∞),H) forU0∈D(A(0)). The requested solution of\n(3.4) is then given by\nU(t) =e/integraltextt\n0κ(s)ds/tildewideU(t), (3.48)\nbecause\nUt(t) =κ(t)e/integraltextt\n0κ(s)ds/tildewideU(t)+e/integraltextt\n0κ(s)ds/tildewideUt(t)\n=e/integraltextt\n0κ(s)ds(κ(t)+˜A(t))/tildewideU(t)\n=A(t)e/integraltextt\n0κ(s)ds/tildewideU(t)\n=A(t)U(t)(3.49)\nwhich concludes the proof.\n94 Exponential stability\nThis section is dedicated to study of the asymptotic behavior. We sh ow that the solution of problem (2.9)-(2.11)\nis exponentially stable using the multiplier technique.\nWe define the energy associated to the solution U(t) = (v(t),vt(t),p(t),pt(t),z(t)) of problem (2.9)-(2.11)\nby the following formula\nE(t) =1\n2/integraldisplayL\n0/bracketleftbig\nρv2\nt+µp2\nt+α1v2\nx+β(γvx−px)2/bracketrightbig\ndx+ξ(t)τ(t)\n2/integraldisplayL\n0/integraldisplay1\n0z2dydx. (4.1)\nOur effort consists in building a suitable Lyapunov functional by the e nergy method. The main goal in this\nsection is to prove the following stability result.\nTheorem 4.1. LetU(t) = (v(t),vt(t),p(t),pt(t),z(t))be the solution of (2.9)-(2.11)with initial data U0∈\nD(A(0))andE(t)the energy of U. Then there exist positive constants Mandγsuch that\nE(t)≤ME(0)e−γt,∀t≥0. (4.2)\nFor the proof of Theorem 4.1 we need several lemmas. Our first res ult states that the energy is a non-\nincreasing function and uniformly bounded above by E(0).\nLemma 4.2. LetU(t) = (v(t),vt(t),p(t),pt(t),z(t))be the solution of (2.9)-(2.11). Then the energy E(t)\nsatisfies\nd\ndtE(t)≤ −µ1(t)/parenleftbigg\n1−ξ\n2−δ\n2√\n1−d/parenrightbigg/integraldisplayL\n0v2\ntdx\n−µ1(t)/parenleftbiggξ(1−τ′(t))\n2−δ√\n1−d\n2/parenrightbigg/integraldisplayL\n0z2(x,1)dx≤0.(4.3)\nProof.Multiplying (2.9)1byvt, (2.9)2byptand integrating each of them by parts over [0 ,L], we get\n1\n2d\ndt/integraldisplayL\n0(ρv2\nt+α1v2\nx)dx+γβ/integraldisplayL\n0(γvx−px)vxtdx+µ1(t)/integraldisplayL\n0v2\ntdx+µ2(t)/integraldisplayL\n0z(x,1)vtdx= 0,(4.4)\n1\n2d\ndt/integraldisplayL\n0µp2\ntdx−β/integraldisplayL\n0(γvx−px)pxtdx= 0.(4.5)\nNow multiplying (2.9)3byξ(t)zand integrating over [0 ,L]×[0,1], we obtain\nτ(t)ξ(t)\n2/integraldisplayL\n0/integraldisplay1\n0d\ndtz2dydx+ξ(t)\n2/integraldisplayL\n0/integraldisplay1\n0(1−τ′(t)y)∂\n∂yz2dydx= 0, (4.6)\nwhich is equivalent to\nd\ndt/parenleftigg\nξ(t)τ(t)\n2/integraldisplayL\n0/integraldisplay1\n0z2dydx/parenrightigg\n=ξ(t)\n2/integraldisplayL\n0v2\ntdx−ξ(t)\n2/integraldisplayL\n0z2(x,1)dx+ξ(t)τ′(t)\n2/integraldisplayL\n0z2(x,1)dx\n+ξ′(t)τ(t)\n2/integraldisplayL\n0/integraldisplay1\n0z2dydx.(4.7)\nCombining (4.4), (4.5) and (4.7), we obtain\nd\ndtE(t) =−µ1(t)/integraldisplayL\n0v2\ntdx−µ2(t)/integraldisplayL\n0z(x,1)vtdx+ξ(t)\n2/integraldisplayL\n0v2\ntdx−ξ(t)\n2/integraldisplayL\n0z2(x,1)dx\n+ξ(t)τ′(t)\n2/integraldisplayL\n0z2(x,1)dx+ξ′(t)τ(t)\n2/integraldisplayL\n0/integraldisplay1\n0z2dydx.(4.8)\nApplying Young’s inequality and taking into account (3.12), (A2)(which results in ξ′(t)≤0), we have\nd\ndtE(t)≤ −/parenleftbigg\nµ1(t)−ξ(t)\n2−|µ2(t)|\n2√\n1−d/parenrightbigg/integraldisplayL\n0v2\ntdx\n−/parenleftbiggξ(t)\n2−ξ(t)τ′(t)\n2−|µ2(t)|√\n1−d\n2/parenrightbigg/integraldisplayL\n0z2(x,1)dx\n+ξ′(t)τ(t)\n2/integraldisplayL\n0/integraldisplay1\n0z2dydx\n≤ −µ1(t)/parenleftbigg\n1−¯ξ\n2−δ\n2√\n1−d/parenrightbigg/integraldisplayL\n0v2\ntdx\n−µ1(t)/parenleftbigg¯ξ(1−τ′(t))\n2−δ√\n1−d\n2/parenrightbigg/integraldisplayL\n0z2(x,1)dx≤0.(4.9)\n10Hence, the proof is complete.\nIn the previous result we observe that the energy functional res tores some energy terms with a negative sign.\nWe are interested in building a Lyapunov functional that restores t he full energy of the system with negative\nsign, and for this goal, we consider the following lemmas.\nLemma 4.3. IfU(t) = (v(t),vt(t),p(t),pt(t),z(t))is a solution of (2.9)-(2.11), then the functional I1, defined\nby\nI1(t) =ρ/integraldisplayL\n0vvtdx+γµ/integraldisplayL\n0vptdx (4.10)\nsatisfies the estimative\nd\ndtI1(t)≤ −α1\n2/integraldisplayL\n0v2\nxdx+ε1/integraldisplayL\n0p2\ntdx+c1/integraldisplayL\n0z2(x,1)dx+c1/parenleftbigg\n1+1\nε1/parenrightbigg/integraldisplayL\n0v2\ntdx, (4.11)\nfor any constants ε1>0andc1>0.\nProof.Taking derivative of I1(t), using (2.9) and integrating by parts, we arrive at\nd\ndtI1(t) =−α1/integraldisplayL\n0v2\nxdx+ρ/integraldisplayL\n0v2\ntdx+γµ/integraldisplayL\n0ptvtdx\n−µ1(t)/integraldisplayL\n0vtvdx−µ2(t)/integraldisplayL\n0z(x,1)vdx.(4.12)\nFrom(A2)and(A3), we have\nd\ndtI1(t) =−α1/integraldisplayL\n0v2\nxdx+ρ/integraldisplayL\n0v2\ntdx+γµ/integraldisplayL\n0ptvtdx\n+µ1(0)/integraldisplayL\n0|vtv|dx+δµ1(0)/integraldisplayL\n0|z(x,1)v|dx(4.13)\nEstimate (4.11) follows thanks to Young’s and Poincar´ e’s inequalities .\nLemma 4.4. IfU(t) = (v(t),vt(t),p(t),pt(t),z(t))is a solution of (2.9)-(2.11), then the functional I2, defined\nby\nI2(t) =ρ/integraldisplayL\n0vt(γv−p)dx+γµ/integraldisplayL\n0pt(γv−p)dx (4.14)\nsatisfies the estimative\nd\ndtI2(t)≤ −γµ\n2/integraldisplayL\n0p2\ntdx+3α1ε2/integraldisplayL\n0(γvx−px)2dx+c2\nε2/integraldisplayL\n0v2\nxdx\n+c2\nε2/integraldisplayL\n0z2(x,1)dx+c2/parenleftbigg\n1+1\nε2/parenrightbigg/integraldisplayL\n0v2\ntdx(4.15)\nfor any constants ε2>0andc2>0.\nProof.Taking derivative of I2(t), using (2.9) together with integration by parts, we obtain\nd\ndtI2(t) =−α1/integraldisplayL\n0vx(γvx−px)dx+ργ/integraldisplayL\n0v2\ntdx−ρ/integraldisplayL\n0vtptdx+γ2µ/integraldisplayL\n0ptvtdx\n−γµ/integraldisplayL\n0p2\ntdx−µ1(t)/integraldisplayL\n0vt(γv−p)dx−µ2(t)/integraldisplayL\n0z(x,1)(γv−p)dx.(4.16)\nFrom(A2)and(A3), we obtain\nd\ndtI2(t) =−α1/integraldisplayL\n0vx(γvx−px)dx+ργ/integraldisplayL\n0v2\ntdx−ρ/integraldisplayL\n0vtptdx+γ2µ/integraldisplayL\n0ptvtdx\n−γµ/integraldisplayL\n0p2\ntdx+µ1(0)/integraldisplayL\n0|vt(γv−p)|dx+δµ1(0)/integraldisplayL\n0|z(x,1)(γv−p)|dx.(4.17)\nWe then use Young’s and Poincar´ e’s inequalities to obtain (4.15).\n11Lemma 4.5. IfU(t) = (v(t),vt(t),p(t),pt(t),z(t))is a solution of (2.9)-(2.11), then the functional I3, defined\nby\nI3(t) =ρ/integraldisplayL\n0vtvdx+µ/integraldisplayL\n0ptpdx (4.18)\nsatisfies the estimative\nd\ndtI3(t)≤−α1\n2/integraldisplayL\n0v2\nxdx−β/integraldisplayL\n0(γvx−px)2dx+µ/integraldisplayL\n0p2\ntdx\n+c3/integraldisplayL\n0v2\ntdx+c3/integraldisplayL\n0z2(x,1)dx,(4.19)\nfor any constant c3>0.\nProof.Taking derivative of I3(t), use (2.9) and integrating by parts, yield\nd\ndtI3(t)≤ −α1/integraldisplayL\n0v2\nxdx−βγ2/integraldisplayL\n0v2\nxdx+2βγ/integraldisplayL\n0pxvxdx−β/integraldisplayL\n0p2\nxdx+ρ/integraldisplayL\n0v2\ntdx\n+µ/integraldisplayL\n0p2\ntdx−µ1(t)/integraldisplayL\n0vtvdx−µ2(t)/integraldisplayL\n0z(x,1)vdx.(4.20)\nFrom(A2)and(A3), and taking into account\n−β(γvx−px)2=−βγ2v2\nx+2βγvxpx−βp2\nx, (4.21)\nwe have\nd\ndtI3(t)≤ −α1/integraldisplayL\n0v2\nxdx−β/integraldisplayL\n0(γvx−px)2dx+ρ/integraldisplayL\n0v2\ntdx+µ/integraldisplayL\n0p2\ntdx\n+µ1(0)/integraldisplayL\n0|vtv|dx+δµ1(0)/integraldisplayL\n0|z(x,1)v|dx.(4.22)\nExploiting Young’s and Poincar´ e’s inequalities, we obtain the estimate s (4.19) and conclude the prove.\nAs in [21], taking into account the last lemma, we introduce the functio nal\nJ(t) =ξτ(t)/integraldisplayL\n0/integraldisplay1\n0e−2τ(t)yz2(x,y)dydx. (4.23)\nFor this functional we have the following estimate.\nLemma 4.6 ([21, Lemma 3.7]) .LetU(t) = (v(t),vt(t),p(t),pt(t),z(t))be solution of (2.9)-(2.11). Then the\nfunctionalJ(t)satisfies\nd\ndtJ(t)≤ −2J(t)+ξ/integraldisplayL\n0v2\ntdx. (4.24)\nNow we are in position to prove our principal result.\nProof of Theorem 4.1. WewilltoconstructasuitableLyapunovfunctional Lsatisfyingthefollowingequivalence\nrelation\nγ1E(t)≤L(t)≤γ2E(t),∀t≥0, (4.25)\nfor someγ1,γ2>0 and to prove that\nd\ndtL(t)≤ −λL(t),∀t≥0, (4.26)\nfor someλ>0, which implies\nL(t)≤L(0)e−λt,∀t≥0. (4.27)\nLet us define the Lyapunov functional\nL(t) =NE(t)(t)+3/summationdisplay\ni=1NiIi(t)+J(t), (4.28)\n12whereNi,i= 1,2,3 are positive real numbers which will be chosen later. By the Lemma 4 .2, there exists a\npositive constant Ksuch that\nd\ndtE(t)≤ −K/parenleftigg/integraldisplayL\n0v2\ntdx+/integraldisplayL\n0z2(x,1)dx/parenrightigg\n. (4.29)\nWe have that\n|L(t)−NE(t)|≤N1/parenleftigg\nρ/integraldisplayL\n0|vtv|dx+γµ/integraldisplayL\n0|ptv|dx/parenrightigg\n+N2/parenleftigg\nρ/integraldisplayL\n0|vt(γv−p)|dx+γµ/integraldisplayL\n0|pt(γv−p)|dx/parenrightigg\n+N3/parenleftigg\nρ/integraldisplayL\n0|vtv|dx+µ/integraldisplayL\n0|ptp|dx/parenrightigg\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξτ(t)/integraldisplayL\n0/integraldisplay1\n0e−2τ(t)yz2dydx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.(4.30)\nIt follows from (4.1), Young’s and Poincar´ e’s inequalities and from th e fact that τ(t)≤τ1for allt≥0 and\ne−2τ(t)y≤1 for ally∈(0,1) that\n|L(t)−NE(t)|≤γ3/integraldisplayL\n0/bracketleftbigg\nv2\nt+p2\nt+v2\nx+(γvx−px)2+/integraldisplay1\n0z2dy/bracketrightbigg\ndx≤γ3E(t) (4.31)\nfor some constant γ3>0. So, we can choose Nlarge enough that γ1:=N−γ3andγ2:=N+γ3, then\nγ1E(t)≤L(t)≤γ2E(t),∀t≥0 (4.32)\nholds.\nNow, taking derivative L(t), substitute the estimates (4.11), (4.15), (4.19), (4.23), (4.29) and setting\nN2=8\nγ, N3= 1, ε1=µ\nN1andε2=βγ\n48α1, (4.33)\nwe obtain that\nd\ndtL(t)≤ −/bracketleftbigg\nNK−c1/parenleftbigg\n1+N1\nµ/parenrightbigg\nN1−c2/parenleftbigg\n1+48α1\nβγ/parenrightbigg8\nγ−c3−ξ/bracketrightbigg/integraldisplayL\n0v2\ntdx\n−/parenleftbigg\nNK−c1N1−384α1c2\nβγ2−c3/parenrightbigg/integraldisplayL\n0z2(x,1)dx\n−/bracketleftbiggα1\n2N1−384α1c2\nβγ2+α1\n2/bracketrightbigg/integraldisplayL\n0v2\nxdx\n−2µ/integraldisplayL\n0p2\ntdx−β\n2/integraldisplayL\n0(γvx−px)2dx−2J(t).(4.34)\nFirst, let us choose N1large enough such that\nα1\n2N1−384α1c2\nβγ2+α1\n2>0. (4.35)\nNow, since ξ(t)τ(t) non-negativeand limited and choosing Nlarge enough that (4.34) is taken into the following\nestimate\nd\ndtL(t)≤−η/integraldisplayL\n0/bracketleftbigg\nv2\nt+p2\nt+v2\nx+(γvx−px)2+z2(x,1)+/integraldisplay1\n0z2dy/bracketrightbigg\ndx\n≤ −η/integraldisplayL\n0/bracketleftbigg\nv2\nt+p2\nt+v2\nx+(γvx−px)2+/integraldisplay1\n0z2dy/bracketrightbigg\ndx,(4.36)\nfor some positive constant η. Therefore, from (4.1), we have\nd\ndtL(t)≤ −ηE(t),∀t>0. (4.37)\n13In view of (4.32) and (4.37), we note that\nd\ndtL(t)≤ −λL(t),∀t>0, (4.38)\nwhich leads to\nL(t)≤L(0)e−λt,∀t>0. (4.39)\nThe desired result (4.2) follows by using estimates (4.32) and (4.39). Then, the proof of Theorem 4.1 is\ncomplete.\nReferences\n[1] A. K. Agrawaland J. N. Yang. Effect of fixed time delay on stability and performance of actively controlled\ncivil engineering structures. Earthquake Engineering & Structural Dynamics , 26(11):1169–1185, 1997.\n[2] T. A. Apalara. 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An integrated physical mode l that characterizes creep and hysteresis\nin piezoelectric actuators. Simulation Modelling Practice and Theory , 16(1):93–110, 2008.\n[53] A.¨Ozkan¨OzerandK.A.Morris. Modelinganelasticbeamwithpiezoelectricpatc hesbyincludingmagnetic\neffects. In 2014 American Control Conference , pages 1045–1050, 2014.\n16" }, { "title": "0805.3495v1.Intrinsic_and_non_local_Gilbert_damping_in_polycrystalline_nickel_studied_by_Ti_Sapphire_laser_fs_spectroscopy.pdf", "content": "Intrinsic and non-local Gilbert damping in\npolycrystalline nickel studied by Ti:Sapphire laser fs\nspectroscopy\nJ Walowski1, M Djordjevic Kaufmann1, B Lenk1, C Hamann2\nand J McCord2, M M unzenberg1\n1Universit at G ottingen, Friedirch-Hund-Platz 1, 37077 G ottingen, Germany\n2IFW Dresden, Helmholtzstra\u0019e 20, 01069 Dresden\nE-mail: walowski@ph4.physik.uni-goettingen.de\nAbstract. The use of femtosecond laser pulses generated by a Ti:Sapphire laser\nsystem allows us to gain an insight into the magnetization dynamics on time scales from\nsub-picosecond up to 1 ns directly in the time domain. This experimental technique is\nused to excite a polycrystalline nickel (Ni) \flm optically and probe the dynamics\nafterwards. Di\u000berent spin wave modes (the Kittel mode, perpendicular standing\nspin-wave modes (PSSW) and dipolar spin-wave modes (Damon-Eshbach modes)) are\nidenti\fed as the Ni thickness is increased. The Kittel mode allows determination of the\nGilbert damping parameter \u000bextracted from the magnetization relaxation time \u001c\u000b.\nThe non-local damping by spin currents emitted into a non-magnetic metallic layer\nof vanadium (V), palladium (Pd) and the rare earth dysprosium (Dy) are studied\nfor wedge-shaped Ni \flms 1 nm \u000030 nm. The damping parameter increases from\n\u000b= 0:045 intrinsic for nickel to \u000b > 0:10 for the heavy materials, such as Pd and\nDy, for the thinnest Ni \flms below 10 nm thickness. Also, for the thinnest reference Ni\n\flm thickness, an increased magnetic damping below 4 nm is observed. The origin of\nthis increase is discussed within the framework of line broadening by locally di\u000berent\nprecessional frequencies within the laser spot region.arXiv:0805.3495v1 [cond-mat.other] 22 May 2008Gilbert damping in Nickel thin \flms 2\n1. Introduction\nThe understanding of picosecond-pulsed excitation of spin packets, spin wave modes\nand spin currents is of importance in developing a controlled magnetic switching concept\nbeyond the hundred picosecond timescale and to test the speed of magnetic data storage\nmedia heading to the physical limits. Over the last years profound progress has been\nmade within that \feld by using femtosecond laser spectroscopy. The recent discoveries\nin ultrafast magnetization dynamics are heading to a new understanding [1{5] and\nnew all-optical switching concepts have been discovered [6]. In addition, the all-optical\nmethod has developed into a valuable tool to study the magnetization dynamics of\nthe magnetic precession and thereby access magnetocrystalline anisotropies and the\nmagnetic damping [7{11] or the dynamics of magnetic modes in nanometer sized arrays\nof magnetic structures [12, 13] and single magnetic nanostructures [14, 15]. Naturally,\none \fnds similarities and di\u000berences as compared to magnetic resonance techniques\nin frequency space (FMR) [16], optical techniques such as Brillouin light scattering\n(BLS) [17,18] and time-resolved techniques, for example pulsed inductive magnetometry\n(PIMM) [19]. Advantages and disadvantages of the di\u000berent techniques have already\nbeen compared in previous work [20{22]. The same concepts can be applied to the\nfemtosecond-laser-based all-optical spectroscopy techniques. Here we discuss their\nabilities, highlighting some aspects and peculiarities [11,23{27]:\ni. After excitation within the intense laser pulse, the nature of the magnetic relaxation\nmechanisms determine the magnetic modes observed on the larger time scale [5].\nFor a Ni wedge di\u000berent modes are found as the thickness is increased: coherent\nprecession (Kittel mode), standing spin waves (already found in [28]) and dipolar\nsurface spin waves (Damon-Eshbach modes) appear and can be identi\fed.\nii. Magnetic damping has been extracted by the use of fs spectroscopy experiments\nalready in various materials, epitaxial \flms, as a function of the applied \feld\nstrength, \feld orientation and laser excitation power [7{11]. Using the Kittel\nmode, we study the energy dissipation process caused by non-local damping by spin\ncurrents [29] in Ni by attaching a transition metal \flm (vanadium (V), palladium\n(Pd) and a rare earth \flm (dysprosium (Dy)) as a spin sink material and compare\nthem to a Ni reference sample. The present advantages and disadvantages of the\nmethod are discussed.\niii. A modi\fcation of the magnetic damping is found for the thinnest magnetic layers\nbelow 4 nm. The understanding of this e\u000bect is of high interest because of the\nincrease in methods used to study magnetic damping processes in the low \feld\nregion in the current literature. We present a simple model of line broadening\nknown from FMR [30{32] and adapted to the all-optical geometry that pictures\nthe e\u000bect of the increased intrinsic apparent damping observed. Therein a spread\nlocal magnetic property within the probe spot region is used to mimic the increased\napparent damping for the low \feld region.Gilbert damping in Nickel thin \flms 3\na)\nb)Side view:\nFigure 1. a) Schematics of the pump probe experiment to determine the change in\nKerr rotation as a function of the delay time \u001c. b) Experimental data on short and\nlong time scales. On top a schematic on the processes involved is given.\n2. Experimental Technique\nThe all-optical approach to measuring magnetization dynamics uses femtosecond laser\npulses in a pump-probe geometry. In our experimental setup a Ti:Sapphire oscillator\ngenerates the fs laser pulses which are then ampli\fed by a regenerative ampli\fer (RegA\n9050). This systems laser pulse characteristics are 815 nm central wave length, a\nrepetition rate of 250 kHz, a temporal length of 50 \u000080 fs and an energy of \u00181\u0016J\nper pulse. The beam is split into a strong pump beam (95% of the incoming power),\nwhich triggers the magnetization dynamics by depositing energy within the spot region,Gilbert damping in Nickel thin \flms 4\nand a weaker probe pulse (5% of the incoming power) to probe the magnetization\ndynamics via the magneto-optical Kerr e\u000bect delayed by the time \u001c, in the following\nabbreviated as time-resolved magneto-optic Kerr e\u000bect (TRMOKE). The schematic\nsetup and sample geometry is given in \fgure 1a). The spot diameters of the pump\nand probe beam are 60 \u0016m and 30\u0016m respectively. A double-modulation technique is\napplied to detect the measured signal adapted from [33]: the probe beam is modulated\nwith a photo-elastic modulator (PEM) at a frequency f1= 250 kHz and the pump\nbeam by a mechanic chopper at a frequency f2= 800 Hz. The sample is situated in\na variable magnetic \feld (0 \u0000150 mT), which can be rotated from 0\u000e(in-plane) to 90\u000e\n(out of plane) direction. The degree of demagnetization can be varied by the pump\n\ruence (10 mJ =cm2\u000060 mJ=cm2) to up to 20% for layer thicknesses around 30 nm and\nup to over 80% for layers thinner than 5 nm. The samples studied were all grown on\nSi(100) substrates by e-beam evaporation in a UHV chamber at a base pressure of\n\u00185\u000210\u000010mbar. For a variation of the thickness, the layers are grown as wedges with\na constant gradient on a total wedge length of 15 mm.\n3. Results and discussion\n3.1. Kittel mode, standing spin waves and Damon-Eshbach surface modes\nTo give an introduction to the TRMOKE signals \u0001 \u0012Kerr(\u001c) measured on the timescale\nfrom picoseconds to nanoseconds \frst, the ultrafast demagnetization on a characteristic\ntime scale\u001cMand the magnetic precessional motion damped on a time scale \u001c\u000bis shown\nfor a Ni \flm in \fgure 1b); the schematics of the processes involved on the di\u000berent time\nscales are given on the top. The change in Kerr rotation \u0001 \u0012Kerr(\u001c) shows a sudden drop\nat\u001c= 0 ps. This mirrors the demagnetization within a timescale of \u0018200 fs [34{36]. For\nthe short time scale the dynamics are dominated by electronic relaxation processes, as\ndescribed phenomenologically in the three temperature model [34] or by connecting the\nelectron-spin scattering channel with Elliot-Yafet processes, as done by Koopmans [36]\nand Chantrell [4] later. At that time scale the collective precessional motion lasting up\nto the nanosecond scale is initiated [28, 37]: the energy deposited by the pump pulse\nleads to a change in the magnetic anisotropy and magnetization, and thus the total\ne\u000bective \feld. Within \u001810 ps the total e\u000bective \feld has recovered to the old value\nand direction again. However, the magnetization, which followed the e\u000bective \feld,\nis still out of equilibrium and starts to relax by precessing around the e\u000bective \feld.\nThis mechanism can be imagined as a magnetic \feld pulse a few picoseconds long, and\nis therefore sometimes called an anisotropy \feld pulse. The resulting anisotropy \feld\npulse is signi\fcantly shorter than standard \feld pulses [38]. This makes the TRMOKE\nexperiment di\u000berent to other magnetization dynamics experiments.\nThe fact that the situation is not fully described by the model can be seen in the\nfollowing. Already van Kampen et al. [28] not only observed the coherent precessional\nmode, they also identi\fed another mode at a higher frequency than the coherentGilbert damping in Nickel thin \flms 5\nprecession mode, shifted by !k;n\u00182Ak2= 2An\u0019=t Ni2, the standing spin wave (PSSW)\nmode. It originates from the con\fnement of the \fnite layer thickness, where Ais the\nexchange coupling constant and nis a given order. Here we also present the \fnding of\ndipolar propagating spin waves. For all three, the frequency dependence as a function\nof the applied magnetic \feld will be discussed, a necessity for identifying them in the\nexperiments later on.\nFor the coherent precession the frequency dependence is described by the Kittel\nequation. It is derived by expressing the e\u000bective \feld in the Landau-Lifshitz-Gilbert\n(LLG equation) as a partial derivative of the free magnetic energy [39, 40]. Assuming\nnegligible in-plane anisotropy in case of the polycrystalline nickel (Ni) \flm and small\ntilting angles of the magnetization out of the sample plane (\feld is applied 35\u000eout of\nplane \fgure 1a)), it is solved as derived in [41]:\n!=\r\n\u00160s\n\u00160Hx\u0012\n\u00160Hx+\u00160Ms\u00002Kz\nMs\u0013\n; (1)\nFor the standing spin waves (PSSW) a similar equation is given. For the geometry\nwith the \feld applied 35\u000eout of plane (\fgure 1a) the frequencies !and!k;ndo not\nsimply add as in the \feld applied in plane geometry [41]:\n!=\r\n\u00160s\n(\u00160Hx+2Ak2\nMs)\u0012\n\u00160Hx+\u00160Ms\u00002Kz\nMs+2Ak2\nMs\u0013\n; (2)\nWhile the exchange energy dominates in the limit of small length scales, the\nmagnetic dipolar interaction becomes important at larger length scales. Damon and\nEshbach [42] derived by taking into account the dipolar interactions in the limit of\nnegligible exchange energy, the solution of the Damon-Eshbach (DE) surface waves\npropagating with a wave vector qalong the surface, decaying within the magnetic layer.\nThe wavelengths are found to be above the >\u0016m range for Ni [27].\n!=\r\n\u00160s\n\u00160Hx\u0012\n\u00160Hx+\u00160Ms\u00002Kz\nMs+M2\nS\n4[1\u0000exp(\u00002qtNi)]\u0013\n; (3)\nThe depth of the demagnetization by the femtosecond laser pulse is given by the\noptical penetration length \u0015opt\u001915 nm (\u0015= 800 nm). From the nature of the excitation\nprocess in the TRMOKE experiment one can derive that for di\u000berent thicknesses tNiit\nwill change from an excitation of the full \flm for a \u001810 nm \flm to a thin excitation\nlayer only for a few 100 nm thick \flm; thus the excitation will be highly asymmetric.\nThe model of the magnetic anisotropy \feld pulse fails to explain these e\u000bects since it is\nbased on a macrospin picture.\nAnother way to look at the excitation mechanism has been discussed by Djordjevic\net al. [5]. When the magnetic system is excited, on a length scale of the optical\npenetration depth short wavelength (high kvector) spin-wave excitations appear. As\ntime evolves, two processes appear: the modes with high frequency owning a fastGilbert damping in Nickel thin \flms 6\noscillation in space are damped very fast by giving part of the deposited energy to\nthe lattice. In addition, through multiple magnon interaction lower k-vector states are\npopulated, resulting in the highest occupation of the lowest energy modes at the end\n(e.g. the PSSW and DE modes here). As the Ni thickness is increased, the excitation\npro\fle becomes increasingly asymmetric, favoring inhomogeneous magnetic excitations,\nas the PSSW mode. The DE modes, due to their nature based on a dipolar interaction,\nare expected to be found only for higher thicknesses.\nFigure 2. Change in Kerr rotation after excitation on the long time scale for Cu 2nm =\nNi tNinm=Si(100) with tNi= 20 and b) their Fourier transform for di\u000berent applied\n\felds 0\u0000150 mT, (35\u000eout of plane (blue)). In c) the Fourier power spectra as color\nmaps for three Ni thicknesses tNi= 20, 40 and 220 nm are given. The data overlaid is\ndetermined form the peak positions. The straight lines are the analysis of the di\u000berent\nmodes and are identi\fed in the graph (Kittel model), perpendicular standing spin wave\n(PSSW) and dipolar surface spin wave (Damon Eshbach mode).Gilbert damping in Nickel thin \flms 7\nThe identi\fcation of the mode is important in determining a value for the magnetic\ndamping\u000b. Figure 2 pictures the identi\fcation of the di\u000berent modes and their\nappearance for di\u000berent Ni thicknesses. The data are handled as follows: for a\ntNi= 20 nm \flm on Si(100), covered with a 2 nm Cu protection layer, in a) the original\ndata after background subtraction and in b) its corresponding Fourier transform, shown\nfor increasing applied magnetic \feld. The evolution of the mode frequency and its\namplitude increase can be followed. An exponentially decaying incoherent background\nis subtracted from the data. This has to be done very carefully, to avoid a step-\nlike background which will be evident after Fourier transform as a sum of odd higher\nharmonics. The frequency resolution is limited by the scan range of 1 ns corresponding\nto \u0001!=2\u0019= 1 GHz. However, since the oscillation is damped within the scan range,\nthe datasets have been extended before Fourier transform to increase their grid points.\nA color map of the power spectrum is shown in \fgure 2c), where the peak positions are\nmarked by the data points overlaid. For the 20 nm thick \flm with tNi= 20 nm\u0018\u0015opt\nonly a single mode is observed. The mode is analyzed by 1 indicating the Kittel mode\nbeing present (data points and line in \fgure 2c), top) using Kz= 3:03\u0001104J=m3. With\nincreasing nickel thickness tNi= 40 nm> \u0015 opt, the perpendicular standing spin waves\n(PSSW) of \frst order are additionally excited and start to appear in the spectra (\fgure\n2c), middle). An exchange constant A= 9:5\u00011012J=m is extracted. In the limit of\ntNi= 220 nm\u001d\u0015opt(\fgure 1c)) the excitation involves the surface only. Hence, modes\nwith comparable amplitude pro\fle, e.g. with their amplitude decaying into the Ni layer,\nare preferred. Consequently DE surface waves are identi\fed as described by 3 and\ndominate the spectra up to critical \felds as high as \u00160Hcrit= 100 mT. For tNi= 220 nm\nthe wave factor is k= 2\u0016m (data points and line in \fgure 2c), bottom). For larger\n\felds than 100 mT the DE mode frequency branch merges into the Kittel mode [27].\nTo resume the previous \fndings for the \frst subsection, we have shown that in\nfact the DE modes, though they are propagating spin-wave modes, can be identi\fed\nin the spectra and play a very important role for Ni thicknesses above tNi= 80 nm.\nThey appear for thicknesses much thinner than the wavelength of the propagating\nmode. Perpendicular standing spin waves (PSSW) give an important contribution to\nthe spectra for Ni thicknesses above tNi= 20 nm. For thicknesses below tNi= 20 nm we\nobserve the homogeneously precessing Kittel mode only. This thickness range should\nbe used to determine the magnetic damping in TRMOKE experiments.\n3.2. Data analysis: determination of the magnetic damping\nFor the experiments carried out in the following with tNi<25 nm the observed dynamics\ncan be ascribed to the coherent precession of the magnetization (Kittel mode). The\nanalysis procedure is illustrated in the following using the data given in \fgure 3a). A\nPd layer is attached to a Ni \flm with the thicknesses (Ni 10 nm =Pd 5 nm=Si(100)) to\nstudy the non-local damping by spin currents absorbed by the Pd. The di\u000berent spectra\nwith varying the magnetic \feld strength from 0 mT \u0000150 mT are plotted from bottomGilbert damping in Nickel thin \flms 8\nto top (with the magnetic \feld tilted 35\u000eout of the sample plane).\n0 .0 4 5 0 .0 5 0 0 .0 5 5 \nα\n0 2 5 5 0 7 5 1 0 0 1 2 5 1 5 0 02468\nν [GHz]\nµ0 He x t [m T ]0 2 5 0 5 0 0 7 5 0 1 0 0 0 - 8 - 4 048\n \n∆θk [a.u.]\nτ [p s ]µ0He x t =\n1 5 0 m T \n1 4 0 m T \n1 3 0 m T \n1 2 0 m T \n1 1 0 m T \n1 0 0 m T \n 9 0 m T \n 8 0 m T \n 7 0 m T \n 6 0 m T \n 5 0 m T \n 4 0 m T \n 3 0 m T \n 0 m T a) b)5 nm Pd/ 10 n m Ni\n5 nm Pd/ 10 n m Ni\nFigure 3. a) Kerr rotation spectra for a Cu 2nm =Ni 10 nm=Pd 5 nm=Si(100) layer,\nmeasured for \felds applied from 0 \u0000150 mT (35\u000eout of plane, (blue)) and the \ftted\nfunctions (white, dashed). b) The magnetic damping \u000band precession frequencies\nextracted from the \fts to the measured spectra. The line is given by the Kittel mode\n(gray).\nThe data can be analyzed using the harmonic function with an exponential decay\nwithin\u001c\u000b:\n\u0001\u0012k\u0018exp\u0012\n\u0000\u001c\n\u001c\u000b\u0013\n\u0001sin(2\u0019(\u001c\u0000\u001c0)\u0017) +B(\u001c); (4)\nThe precession frequency \u0017=!=2\u0019and the exponential decay time \u001c\u000bof the\nprecession amplitude is extracted, where the function B(\u001c) stands for the background\narising from the uncorrelated magnetic and phonon excitations. To determine the\nGilbert damping parameter \u000bas given in the ansatz by Gilbert, the exponential decay\ntime\u001c\u000bhas to be related with \u000b. The LLG equation is solved under the same\npreconditions as for equation 1 using an exponential decay of the harmonic precession\nwithin\u001c\u000bfrom 4. Then the damping parameter \u000band can be expressed by the followingGilbert damping in Nickel thin \flms 9\nequation [41]:\n\u000b=1\n\u001c\u000b\r\u0010\nHx\u0000Kz\n\u00160Ms+Ms\n2\u0011: (5)\nIt is evident from 5 that in order to determine the Gilbert damping \u000bfrom the decay\nof the Kittel mode \u001c\u000b, the variables \r,MsandKzhave to be inserted, and therefore Kz\nhas to be determined beforehand.\nIn \fgure 3a), the background B(\u001c) is already subtracted. The \fts using 4 are\nplotted with the dashed lines on top of the measured spectra. The results are presented\nin b). The frequencies range from 3 GHz for 30 mT to 7 :5 GHz for the 150 mT applied\nmagnetic \feld. They increase linearly with the strength of the applied magnetic \feld for\nhigh \feld values. The extrapolated intersection with the ordinate is related to the square\nroot of the dipolar and anisotropy \feld. Using the Kittel equation (1), one determines\nthe out-of-plane anisotropy constant KzofKz= 6:8\u0001104J=m3. The calculated magnetic\ndamping\u000bas a function of the applied \feld is given in the graph below: this is mostly\nconstant but increases below 60 mT. Within the ansatz given by Gilbert, the damping\nconstant\u000bis assumed to be \feld-independent. We \fnd that this is ful\flled for most\nof the values: the average value of \u000b= 0:0453(4), consistent with earlier \fndings by\nBhagat and Lubitz from FMR experiments [43], is indicated by the line in the plot. The\n\u000bgiven in the following will always be averaged over a \feld region where the damping\nis Gilbert-like. A deviation from this value occurs for the small external \feld strengths.\nIt originates for two reasons: \ftting 5 with a few periods only does not determine a\nreliable value of the exponential precession decay time \u001c\u000band leads to a larger error.\nSecond, magnetic inhomogeneities mapping a spread in anisotropy energies within the\nprobe spot region can also be a source, and this becomes generally more important for\neven thicker \flms below 4 nm [31]. This will be discussed in more detail in the last\nsection of the manuscript.\n3.3. Intrinsic damping: nickel wedge\nFor our experiments Ni was chosen instead of Fe or Py as a ferromagnetic layer. The\nlatter would be preferable because of their lower intrinsic damping \u000bint, which make\nthe \flms more sensitive for detecting the non-local contribution to the damping. The\nreason for using Ni for our experiments is the larger signal excited in the TRMOKE\nexperiments. The magnetic damping \u000bintis used as a reference later on. The di\u000berent\nspectra with varying the Ni thickness tNiNixnm=Si(100) from 2 nm \u0014x\u001422 nm are\nplotted from bottom to top (with the constant magnetic \feld 150 mT and tilted 30\u000e\nout of plane) in \fgure 4a). The measurements were performed immediately after the\nsample preparation, in order to prevent oxidation on the nickel surface caused by the\nlack of a protection layer (omitted on purpose). The spectra show similar precession\nfrequency and initial excitation amplitude. However, the layers with tNi<10 nm show a\nfrequency shift visually recognized in the TRMOKE data. Furthermore, the precessionGilbert damping in Nickel thin \flms 10\namplitude decreases faster for the thinner layers. Figure 5 shows the frequencies and\nthe damping parameter extracted from the measured data in the intrinsic case for the\nnickel wedge sample (black squares). While the precession frequency given for 150 mT\nis almost constant above 8 nm Ni thickness, it starts to drop by about 25% for the\nthinnest layer. The magnetic damping \u000b(black squares) is found to increase to up to\n\u000b= 0:1, an indication that in addition to the intrinsic there are also extrinsic processes\ncontributing. It has to be noted that the change in \u000bis not correlated with the decrease\nof the precession frequency. The magnetic damping \u000bis found to increase below a\nthickness of 4 nm, while the frequency decrease is observed below a thickness of 10 nm.\nA priori\r,MsandKzcan be involved in the observed frequency shift, but they can\nnot be disentangled within a \ft of our \feld-dependent experiments. However, from our\nmagnetic characterization no evidence of a change of \randMsis found. A saturation\nmagnetization \u00160Ms= 0:659 T and g-factor of 2.21 for Ni are used throughout the\nmanuscriptzandKzis determined as a function of the Ni thickness, which shows a\n1=tNibehavior, as expected for a magnetic interface anisotropy term [44].\nThe knowledge of the intrinsic Gilbert damping \u000bintof the Ni \flm of a constant\nvalue for up to 3 nm thickness allows us to make a comparative study of the non-local\ndamping\u000b0, introduced by an adjacent layer of vanadium (V) and palladium (Pd) as\nrepresentatives for transition metals, and dysprosium (Dy) as a representative of the\nrare earths. Both damping contributions due to intrinsic \u000bintand non-local spin current\ndamping\u000b0are superimposed by:\n\u000b=\u000bint+\u000b0: (6)\nThey have to be disentangled by a study of the thickness dependence and compared\nto the theory of spin-current pumping, plus a careful comparison to the intrinsic value\n\u000binthas to be made.\n3.4. Non-local spin current damping: theory\nDynamic spin currents excited by a precessing moment in an adjacent nonmagnetic\nlayer (NM) are the consequence of the fact that static spin polarization at the interface\nfollows a dynamic movement of a collective magnetic excitation. The e\u000bect has already\nbeen proposed in the seventies [45,46] and later calculated within a spin reservoir model\nwith the spins pumped through the interfaces of the material by Tserkovnyak [29, 47].\nFor each precession, pumping of the spin current results in a corresponding loss in\nmagnetization, and thus in a loss of angular momentum. The spin information is lost\nand the backward di\u000busion damps the precession of the magnetic moment. In addition to\nthe \frst experiments using ferromagnetic resonance (FMR) [48{54] it has been observed\nin time- resolved experiments using magnetic \feld pulses for excitation [55,56]. In fact\nzAn altered g-factor by interface intermixing can not decrease its value below \u00182. Also, there is no\nevidence for a reduced Msfor lower thicknesses found in the Kerr rotation versus Ni thickness data.\nMore expected is a change in the magnetic anisotropy Kz. For the calculation of \u000blater on, the in\nboth cases (assuming a variation of Kzor an altered \r) the di\u000berences are negligible.Gilbert damping in Nickel thin \flms 11\na) b)\n0 250 500 750 10 00- 10 - 8 - 6 - 4 - 2 0\n2 2 n m 1 8 n m 1 0 n m 1 4 n m 8 n m 7 n m 6 n m 5 n m 4 n m 3 n m 2 n m \n \n∆θk [a.u.]\nτ [ p s ] 0 2 5 0 5 0 0 7 5 0 1 0 0 0 - 1 0 -8-6-4-20\n1 7 n m \n2 3 n m 1 4 n m 1 0 n m 8 n m 7 n m 6 n m 5 n m 4 n m 2 n m \n3 n m \n \n∆θk [a.u.]\nτ [p s]Ni r efer enc e x Ni/ 5 nm DydNi = dNi =\nFigure 4. a) Kerr rotation spectra for nickel layers from tNi= 2 nm\u000022 nm, measured\non the nickel wedge tNi= nm Ni=Si(100) and opposed in b) by a nickel wedge Al 2nm =\nDy 5nm= tNi= nm Ni=Si(100) with a 5 nm Dy spin-sink layer.\nthe non-local spin current damping is very closely related to the damping by spin-\rip\nscattering described within the s-d current model [57, 58] that uses the approximation\nof strongly localized d-states and delocalized s-states [59].\nA review describes the underlying circuit theory and dynamics of the spin currents\nat interfaces in detail [60]. The outcome of the theoretical understanding is that the\nadditional Gilbert damping is proportional to the angular momentum Ar;ltransmitted\nthrough the interface. Since each interface owns a characteristic re\rection and\ntransmission, the size of Ar;ldepends on the matching of the Fermi surfaces. The\nabsolute value is given by the total balance between transmitted angular momentum\nand the back \row. For the non-local damping \u000b0one \fnds:\n\u000b0=\r~G\"#\n4\u0019MstFM1\n1 +q\n\u001csf\n\u001celtanh\u0010\ntNM\n\u0015sd\u0011\u00001: (7)\nThe tanh function stems thereby from the di\u000busion pro\fle of the spin currents\ndetermined by the spin di\u000busion length \u0015sdwithin the non-magnetic material withGilbert damping in Nickel thin \flms 12\nthicknesstNM. Also, one \fnds from the analysis the ratio of the electron scattering\nrate\u001celversus the spin \rip rate \u001csf. The total amount of spin current through the\ninterfaces is determined by the interface spin mixing conductance G\"#. It is related\nto the magnetic volume. It is therefore that scales with the thickness of the magnetic\nlayertFM. The e\u000bective gyromagnetic ratio altered by the spin-current implies that in\naddition to an increased damping a small frequency shift will be observed. The non-local\nGilbert damping becomes important when it exceeds the intrinsic damping \u000bint.\n3.5. Non-local damping: vanadium, palladium and dysprosium\nDi\u000bering from other techniques, TRMOKE experiments require optical access for\nexcitation and detection, setting some restrictions to the layer stack assembly that can\nbe investigated with this method: a thick metallic layer on top of the magnetic layer is\nnot practical. Placing the damping layer below the magnetic layer is also unfavorable:\nby increasing the spin sink thickness the roughness of the metal \flm will increase with\nthe metals layer thickness and introduce a di\u000berent defects density, altering \u000bint. In the\nfollowing the nickel thickness will be varied and the spin sink thickness will be kept \fxed\nat 5 nm. To warrant that the nickel \flms magnetic properties are always comparable\nto the reference experiment ( Kz,\u000bint), they are always grown \frst on the Si(100). For\nthe Pd case the damping layer is below the Ni layer. Here the excitation mechanism\ndid not work and the oscillations were too weak in amplitude to analyze the damping\n\u000b, probably due to the high re\rectivity of Pd.\nThe results are presented in \fgure 4b) for the nickel wedge sample Ni xnm=Si(100)\nwith a 5 nm dysprosium (Dy) as a spin sink layer, covered by an aluminum protection\nlayer, as opposed to the nickel wedge sample data without this in a). The nickel layer\nthickness is varied from 2 nm \u0014x\u001422 nm. All spectra were measured in an external\nmagnetic \feld set to 150 mT and tilted 30\u000eout of plane. For the thinnest Ni thickness,\nthe amplitude of the precession is found to be smaller due to the absorption of the Dy\nlayer on the top. While the precession is equally damped for the Ni thicknesses ranging\nfrom 7 to 23 nm, an increased damping is found for smaller thicknesses below this. The\ndi\u000berence in damping of the oscillations is most evident for tNi= 4 and 5 nm.\nThe result of the analysis as described before is summarized in \fgure 5. In this\ngraph the data are shown for the samples with the 5 nm V, Pd, Dy spin-sink layer and\nthe Ni reference. While for the Ni reference, and Ni with adjacent V and Dy layer,\nthe frequency dependence is almost equal, indicating similar magnetic properties for\nthe di\u000berent wedge-like shaped samples, the frequency for Pd is found to be somewhat\nhigher and starts to drop faster than for the others. The most probable explanation is\nthat this di\u000berence is due to a slightly di\u000berent anisotropy for the Ni grown on top of\nPd in this case. Nevertheless, the magnetic damping found for larger thicknesses tNiis\ncomparable with the Ni reference. In the upper graph of \fgure 5 the Gilbert damping\nas a function of the Ni layer thickness is shown. While for the Pd and Dy as a spin\nsink material a additional increase below 10 nm contributing to the damping can beGilbert damping in Nickel thin \flms 13\nidenti\fed, for V no additional damping contribution is found.\n0 . 05 0 . 10 0 . 15 \n0 5 1 0 1 5 20 685 1 0 1 5 0 . 0 0 0 0 . 0 2 5 0 . 0 5 0 \ndN i [ nm] α\n \nν [GHz]\n x N i \nw i th : \n 5 n m V \n 5 n m P d \n 5 n m D y α−αint\ndN i [n m ]\nFigure 5. Gilbert damping parameters \u000band frequency \u0017as a function of the nickel\nlayer thickness for the intrinsic case and for di\u000berent damping materials of 5 nm V,\nPd, and Dy adjacent to the ferromagnet. \u000bis extracted from experiments over a large\n\feld region. The \fts are made using equation 5 and equation 7. In the inset the data\nis shown on a reciprocal scale. Below, the frequency is given (150 mT). The lines are\nguides for the eye.\nFor the adjacent V layer, since it is a transition metal with a low spin orbit-\nscattering (light material with low atomic number Z), with a low spin-\rip scattering rate\nand thus a spin di\u000busion length larger than the thickness tNM(d\u001c\u0015sd), no additional\ndamping will occur. For Pd and Dy the situation is di\u000berent: whereas the heavier Pd\nbelongs to the transition metals with a strong orbit-scattering (heavy material with\nhigh atomic number Z), Dy belongs to the rare earth materials. It owns a localized 4fGilbert damping in Nickel thin \flms 14\nmagnetic moment: therefore, both own a high spin-\rip scattering rate and we expect\nthe latter two to be in the region where ( t\u001d\u0015sd). In their cases the thickness of 5 nm\nof the spin-sink layer is chosen to be larger than the spin di\u000busion length ( tNM\u001d\u0015sd).\nIn this limit the spin current emitted from the magnetic layer through the interface is\ntotally absorbed within the non-magnetic layer. One can simplify 6 to:\n\u000b0(1) =\r~G\"#\n4\u0019Mst\u00001\nFM: (8)\nThis is called the limit of a perfect spin sink. The additional non-local spin current\ndamping is expected to behave inversely proportional with the nickel layer thickness\n\u0018t\u00001\nFM. The inset gives the analysis and the data point on a reciprocal scale. The slope\nshows a linear increase for thinner nickel layers, as expected for an inverse proportionality\nfor both the Pd and the Dy. Since the value for the intrinsic damping of the nickel \flm\nincreases below 4 nm this contribution has to be subtracted to reveal the spin-current\ncontribution. The value for \u000b0is then found to be 0 :07 for the 2 nm Ni =5 nm Pd \flm,\nwhich is in the order found by Mizukami by FMR for sputtered Permalloy \flms with\na Pd spin sink ( \u000b0= 0:04 for 2 nm Py =5 nm Pd) [49, 50]. A further analysis of the\nthickness dependence of \u000byields values for the prefactor in 7 for Pd (0 :33(3) nm) and Dy\n(0:32(3) nm) with the \ft given in the graph. From that value the real part of the interface\nspin mixing conductance in 7 can be calculated. It is found to be G\"#= 4:5(5)\u00011015\ncm1\nfor the Ni/Pd and Ni/Dy interface. The increase of the intrinsic damping \u000binthas\nbeen analyzed using an inverse thickness dependence (prefactor of 0 :1 nm). While it\ndescribes the data in the lower thickness range, it can be seen that it does not describe\nthe thickness dependence for the thicker range and thus, probably the increase does not\noriginate from an interface e\u000bect.\n3.6. Increased damping caused by anisotropy \ructuations: consequences for the\nall-optical approach\nIn this last part we want to focus on the deviation from the intrinsic damping \u000bintfor\nthe thin nickel layers itself ( tNi<4 nm). In the low \feld range (10 \u000050 mT) small\nmagnetization inhomogeneities can build up even when the magnetization appears to\nbe still saturated from the hysteresis curve (the saturation \felds are a few mT). For\nthese thin layers the magnetization does not align parallel in an externally applied \feld\nany more, but forms ripples. The in\ruence of the ripples on the damping is discussed\nin reference [32]. In the following we adopt this ansatz to the experimental situation\nof the TRMOKE experiment. We deduce a length scale on which the magnetization\nreversal appears for two di\u000berent Ni thicknesses and relate it to the diameter of our\nprobe spot. Lateral magnetic inhomogeneities were studied using Kerr microscopy at\ndi\u000berent applied magnetic \felds [44]. Magnetization reversal takes place at low \felds\nof a -0.5 to 2 mT. The resolution of the Kerr microscopy for this thin layer thickness\ndoes not allow us to see the extent of the ripple e\u000bect in the external \feld where the\nincrease of \u000band its strong \feld dependence is observed. However, the domains in theGilbert damping in Nickel thin \flms 15\ndemagnetized state also mirror local inhomogeneities. For our Ni xnm=Si(100) sample\nthis is shown in \fgure 6a) and b). The domains imaged using Kerr microscopy are\nshown for a 3 nm and a 15 nm nickel layer in the demagnetized state. The domains of\nthe 15 nm layer are larger than the probe spot diameter of 30 \u0016m, whereas the domains\nof the 3 nm layer are much smaller.\nd =15nmNi\nd =3nmNi\ndemagnetized\ndemagnetizeda)\nb)c)\nd)20µm\n20µm\nFigure 6. a) and b) Kerr microscopy images for the demagnetized state for 15 nm\nand 3 nm. c) and d) corresponding model representing the areas with slightly varying\nanisotropy\nFrom that observation, the model of local anisotropy \ructuations known from\nFMR [30, 31] is schematically depicted in \fgure 6c) and d). A similar idea was also\ngiven by McMichael [61] and studied using micromagnetic simulations. While for the\nthick \flm the laser spot probes a region of almost homogeneous magnetization state, for\nthe thin layer case the spot averages over many di\u000berent regions with slightly di\u000berent\nmagnetic properties and their magnetization slightly tilted from the main direction\naveraging over it. The TRMOKE signal determined mirrors an average over the probed\nregion. It shows an increased apparent damping \u000band a smaller \u001c\u000bresulting from the\nline broadening and di\u000berent phase in frequency space. While for the thick layer the\ntypical scale of the magnetic inhomogeneity is as large as the probe laser spot given\nand only 1-2 regions are averaged, for the thinner \flm of dNi= 3 nm many regions\nare averaged within a laser spot, as can be seen in 6b) and d). Because the magnetic\ninhomogeneity mapping local varying anisotropies becomes more important for smaller\n\felds, it also explains the strong \feld dependence of \u000bobserved within that region.\nFigure 7 shows data calculated based on the model, in which the upper curve (i)Gilbert damping in Nickel thin \flms 16\nis calculated from the values extracted from the experimental data for the 10 nm nickel\nlayer, curve (ii) is calculated by a superposition of spectra with up to 5% deviation\nfrom the central frequency at maximum and curve (iii) is calculated by a superposition\nof spectra of 7% deviation from the central frequency at maximum to mimic the line\nbroadening. The corresponding amplitudes of the superposed spectra related to di\u000berent\nKzvalues is plotted in the inset of the graph to the given frequencies. The apparent\ndamping is increased by 0.01 (for 5%) and reaches the value given in \fgure 3b) for\nthe 10 nm \flm determined for the lowest \feld values of 30 mT. These e\u000bects generally\nbecome more important for thinner \flms, since the anisotropy \ructuations arising from\nthickness variations are larger, as shown by the Kerr images varying on a smaller length\nscale. These \ructuations can be vice versa determined by the analysis.\n/s48 /s50/s53/s48 /s53/s48/s48 /s55/s53/s48 /s49/s48/s48/s48/s48/s50/s52\n/s55/s46/s48 /s55/s46/s53 /s56/s46/s48\n/s32/s32/s77/s32/s91/s97/s46/s117/s46/s93\n/s32/s91/s112/s115/s93/s40/s105/s105/s105/s41/s40/s105/s41\n/s40/s105/s105/s41/s40/s105/s105/s105/s41/s40/s105/s41/s65/s109/s112/s108/s105/s116/s117/s100/s101/s32/s65\n/s32/s91/s71/s72/s122/s93/s40/s105/s105/s41\nFigure 7. a) Datasets generated by superposing the spectra with the frequency spread\naccording to the inset: (i) is calculated from the values extracted from the experimental\ndata for the 10 nm Ni layer, (ii) by a superposition of spectra with up to 5% and (iii) is\ncalculated by a superposition of spectra owing 7% variation from the central frequency\nat maximum. The average precession amplitude declines faster if a higher spread of\nfrequencies (i.e. di\u000berent anisotropies) are involved.Gilbert damping in Nickel thin \flms 17\n4. Conclusion\nTo conclude, we have shown that all-optical pump-probe experiments are a powerful\ntool to explore magnetization dynamics. Although the optical access to the magnetic\nlayer allows an access to the surface only, magnetization dynamics can be explored\ndirectly in the time domain, resolving di\u000berent types of spin-wave modes (Kittel mode,\nperpendicular standing spin waves and Damon-Eshbach dipolar surface waves). This is\nin contrast to FMR experiments, where the measured data is a response of the whole\nsample. The obtained data can be similar to the \feld-pulsed magnetic excitations and\nthe Gilbert damping parameter \u000b, needed for the analysis of magnetization dynamics\nand the understanding of microscopic energy dissipation, can be determined from these\nexperiments. We have evaluated the contributions of non-local spin current damping\nfor V, Pd and Dy. 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Smith1, Anish Rai2,3, Youngmin Lim1, Timothy Hartnett4, Arjun Sapkota2,3, Abhishek \nSrivastava2,3, Claudia Mewes2,3, Zijian Jiang1, Michael Clavel5, Mantu K. Hudait5, Dwight D. \nViehland6, Jean J. Heremans1, Prasanna V. Balachandran4,7, Tim Mewes2,3, Satoru Emori1 \n1Department of Physics, Virginia Tech, Blacksburg, VA 24061, U.S.A. \n2Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, U.S.A. \n3Center for Materials for Information Technology (MINT), University of Alabama, Tuscaloosa, \nAL 35487, U.S.A . \n4Department of Material Science and Engineering, University of Virginia, \nCharlottesville, VA 22904, U.S.A. \n5Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA 24061, \nU.S.A. \n6Department of Materials Science and Engineering, Virginia Tech, Blacksburg, VA 24061, \nU.S.A. \n7Department of Mechanical and Aerospace Engineering, University of Virginia, \nCharlottesville, VA 22904, U.S.A. \n \n 2 \n To develop low -moment, low -damping metallic ferromagnets for power -efficient spintronic \ndevices, it is crucial to understand how magnetic relaxation is impacted by the addition of \nnonmagnetic elements. Here, we compare magnetic relaxation in epitaxial Fe films alloyed \nwith light nonmagnetic elements of V and Al. FeV alloys exhibit lower intrinsic damping \ncompared to pure Fe, reduced by nearly a factor of 2, whereas damping in FeAl alloys \nincreases with Al content . Our experimental and computat ional results indicate that \nreducing the density of states at the Fermi level , rather than the average atomic number, \nhas a more significant impact in lowering damping in Fe alloyed with light elements . \nMoreover, FeV is confirmed to exhibit an intrinsic Gi lbert damping parameter of ≃0.001, \namong the lowest ever reported for ferromagnetic metals. \n \nI. INTRODUCTION \n The relaxation of magnetization dynamics (e.g., via Gilbert damping) plays important \nroles in many spintronic applications, including those based on magnetic switching1,2, domain \nwall motion3,4, spin wave propagation5,6, and su perfluid -like spin transport7,8. For devices driven \nby spin -torque precessional dynamics1,9,10, the critical current density for switching is predicted \nto scale with the produ ct of the Gilbert damping parameter and the saturation magnetization 2,11. \nThus, it is desirable to engineer magnetic materials that possess both low damping and low \nmoment for energy -efficient operation . While some electrically insulating magnetic oxides have \nbeen considered for certain applications5,12,13, it is essential to engineer low -damping, low -\nmoment metallic ferromagnets for robust electrical readout via giant magnetoresistance and \ntunnel magnetoresistance. Fe is the elemental ferromagnet with the lowest intrinsic Gilbert \ndamping parameter ( ≃0.002)14,15, albeit with the highest saturation magnetization ( ≃2.0 T). 3 \n Recent experiments have reported that Gilbert damping can be further reduced by alloy ing Fe \nwith Co (also a ferromagnetic element), with Fe 75Co25 yielding an ultralow intrinsic Gilbert \ndamping parameter of ≃0.00116,17. However, Fe 75Co25 is close to the top of the Slater -Pauling \ncurve , such that its saturation magnetization is greater than that of Fe by approximately 20 %18. \nThere is thus an unmet need to engineer ferromagnetic alloys tha t simultaneously exhibit lower \ndamping and lower moment than Fe. \n A promising approach towards low -damping, low -moment ferromagnetic metals is to \nintroduce nonmagnetic elements into Fe . In addition to diluting the magnetic moment, \nnonmagnetic elements int roduced into Fe could influence the spin -orbit coupling strength ξ, \nwhich underlies spin relaxation via orbital and electronic degrees of freedom19–21. Simple atomic \nphysics suggests that ξ is related to the average atomic number of the alloy so that, \nconceivably, damping might be lowered by alloying Fe with lighter (lower -Z) elements. Indeed, \nmotivated by the premise of lowering damping through a reduced and presumably ξ, prior \nexperiments have explored Fe thin films alloyed with V20,22,23, Si24, and Al25. However, the \nexperimentally reported damping parameters for these alloys are often a factor of >2 higher22,23 ,25 \nthan the theoretically predicted intrinsic Gilbert damping parameter of ≃0.002 in Fe26 and do not \nexhibit a significant dependence on the alloy composition20,23,24. A possible issue is that the \nreported damping parameters – obtained from the frequency dependence of ferromagnetic \nresonance (FMR) linewidth with the film magnetized in -plane – may include contributions from \nnon-Gilbert relaxation induced by inhomogeneity and defects (e.g., two -magnon scattering)27–36, \nwhich can be affected by the alloying. Therefore, how Gilbert damping in Fe is impacted by \nalloying with low -Z elements remains an open question. 4 \n Here, we investigate the compositiona l dependence of magnetic relaxation at room \ntemperature in epitaxial thin films of ferromagnetic FeV and FeAl alloys. Both alloys are \ncrystalline bcc solid solutions and hence constitute excellent model systems. We employ two \nconfigurations of FMR measurem ents to gain complementary insights: (1) FMR with samples \nmagnetized in the film plane (similar to the prior experiments) to derive the “effective” Gilbert \ndamping parameter, 𝛼𝑒𝑓𝑓𝐼𝑃, which is found to include extrinsic magnetic relaxation due to two -\nmagnon scattering, and (2) FMR with samples magnetized perpendicular to the film plane to \nquantify the intrinsic Gilbert damping parameter, 𝛼𝑖𝑛𝑡, which is free of the two -magnon \nscattering contribution. \nSince Al ( Z = 13) is a much lighter element than V ( Z = 23), we might expect lower \nmagnetic relaxation in FeAl than FeV, if the smaller < Z> lowers intrinsic Gilbert damping via \nreduced ξ. Instead, we find a significant decrease in magnetic relaxation by alloying Fe w ith V – \ni.e., yielding an intrinsic Gilbert damping parameter of ≃0.001, on par with the lowest values \nreported for ferromagnetic metals – whereas damping in FeAl alloys increases with Al content . \nThese experimental results , combined with density functi onal theory calculations, point to the \ndensity of states at the Fermi level D(EF) as a plausible dominant factor for the lower (higher) \nGilbert damping in FeV (FeAl). We thus find that incorporating a low -Z element does not \ngenerally lower damping and that, rather, reducing D(EF) is an effective route for lower damping \nin Fe alloyed wi th a nonmagnetic element. Our findings confirm that FeV is an intrinsically \nultralow -damping alloy, as theoretically predicted by Mankovsky et al.26, which also possesses a \nlower saturation magnetization than Fe and FeCo. The combination of low damping and low \nmoment makes FeV a highly promising material for practical metal -based spintronic \napplications. 5 \n II. FILM DEPOSITION AND STRUCTURAL PROPERTIES \nEpitaxial Fe 100-xVx and Fe 100-xAlx thin films were grown using dc magnetron sputtering \non (001) -oriented MgO substrates. Prior to deposition, the substrates were annealed at 600 oC for \n2 hours37. The base pressure prior to deposition was < 5×10-8 Torr, and all film s were grown with \nan Ar pressure of 3 mTorr. Fe and V (Al) 2” targets were dc co -sputtered to deposit Fe 100-xVx \n(Fe 100-xAlx) films at a substrate temperature of 200 oC. By adjusting the deposition power, we \ntuned the deposition rate of each material (calibrated by X -ray reflectivity) to achieve the desired \natomic percentage x of V (Al). All FeV and FeAl films had a thickness of 25 nm, which is well \nabove the thickness regime where interfacial effects dominate31,38. The FeV (FeAl) films were \ncapped with 3 -nm-thick V (Al) deposited at room temperature to protect against oxidation, \nyielding a film structure of MgO/Fe 100-xVx(25nm)/V(3nm) or MgO/Fe 100-xAlx(25nm)/Al(3nm). \n We confirmed the epitaxial bcc structure of our thi n films using high resolution X -ray \ndiffraction. 2θ -ω scans show only the (002) peak of the film and the (002) and (004) peaks of the \nsubstrate, as shown in Fig ure 1. Rocking curve scans of the film peaks show similar full -width -\nat-half-maximum values of ≃ 1.3o irrespective of composition . The epitaxial relation between \nbcc Fe and MgO is well known16,39: the bcc film crystal is rotated 45o with respect to the \nsubstrate crystal , such that the [100] axis of the film lies parallel to the [110] axis of the \nsubstrate. The absence of the (001) film peak indicates that our epitaxial FeV and FeAl films are \nsolid sol utions rather than B2 -ordered compounds40. \n 6 \n III. MAGNETIC RELAXATION \n3.1. In -Plane Ferromagnetic Resonance \nMany spintronic devices driven by precessional magnetization dynamics are based on in -\nplane magnetized thin films. The equilibrium magnetization also lies in -plane for soft \nferromagnetic thin films dominated by shape anisotropy (i.e., negligible perpendicular magnetic \nanisotropy), as is the case for our epitaxial FeV and FeAl films. We therefore first discuss FMR \nresults w ith films magnetized in -plane. The in -plane FMR results further provide a basis for \ncomparison with previous studies20,22,23,25. \nSamples were placed with the film side facing a coplanar waveguid e (maximum \nfrequency 50 GHz) and magnetized by an e xternal field H (from a conventional electromagnet, \nmaximum field 1.1 T) along the in -plane [100] and [110] axes of the films. Here, unless \notherwise stated, we show results for H || [110] of the film. FMR spectra were acquired via field \nmodulation by sweeping H and fixing the microwave excitation frequency. \nExemplary spectra for Fe, Fe 80V20, and Fe 80Al20 are shown in Fig ure 2, where we \ncompare the peak -to-peak linewidths at a microwave excitation frequency of 20 GHz. We see \nthat the linewidth for Fe 80V20 shows a ≃ 25 % reduction compared to Fe. We further note that \nthe linewidth for the Fe 80V20 sample here is a factor of ≃ 2 narrower than that in previously \nreported FeV20; a possible origin of the narrow linewidth is discussed later . In contrast, Fe 80Al20 \nshows an enhancement in linewidth over Fe, which is contrar y to the expectation of lower \nmagnetic relaxation with a lower average atomic number. \nThe FMR linewidth is generally governed not only by magnetic relaxation, but also by \nbroadening contributions from magnetic inhomogeneities28,41,42. To disentangle the magnetic 7 \n relaxation and inhomogeneous broadening contributions to the linewidth, the typical prescription \nis to fit the frequency f dependence of linewidth ∆𝐻𝑝𝑝𝐼𝑃 with the linear relation41 \n∆𝐻𝑝𝑝𝐼𝑃=∆𝐻0𝐼𝑃+ℎ\n𝑔𝜇𝐵𝜇02\n√3𝛼𝑚𝑒𝑎𝑠𝐼𝑃𝑓, (1) \nwhere h is the Planck constant, 𝜇𝐵 is the Bohr magneton, 𝜇0 is the permeability of free space, \nand 𝑔 is the g-factor obtained from the frequency dependence of the resonance field (see Section \nIV and Supplementa l Material). In Eq. (1), the slope is attributed to viscous magnetic damping, \ncaptured by the measured damping parameter 𝛼𝑚𝑒𝑎𝑠𝐼𝑃, while t he zero -frequency linewidth ∆𝐻0𝐼𝑃 is \nattributed to inhomogeneo us broadening. The fitting with Eq. (1) was carried out for f 10 GHz, \nwhere H was sufficiently large to saturate the films. As is evident from the results in Fig ure 3, \nFe80V20 has lower linewidths across all frequencies and a slightly lower slope, i.e., 𝛼𝑚𝑒𝑎𝑠𝐼𝑃. On the \nother hand, Fe 80Al20 shows higher linewidths and a higher slope. \nThe measured viscous damping includes a small contribution from eddy currents, \nparameter ized by 𝛼𝑒𝑑𝑑𝑦 (Supplemental Material) , and a contribution due to radiative damping43, \ngiven by 𝛼𝑟𝑎𝑑 (Supplemental Material). Together these contributions make up ≃20 % of the total \n𝛼𝑚𝑒𝑎𝑠𝐼𝑃 for pure Fe and decrease in magnitude with increasing V or Al content . We subtract these \nto obtain the effective in -plane Gilbert damping parameter, \n 𝛼𝑒𝑓𝑓𝐼𝑃=𝛼𝑚𝑒𝑎𝑠𝐼𝑃−𝛼𝑒𝑑𝑑𝑦 − 𝛼𝑟𝑎𝑑. (2) \nAs shown in Fig ure 4a, 𝛼𝑒𝑓𝑓𝐼𝑃 remains either invariant or slightly decreases in Fe 100-xVx up to x = \n25, whereas we observe a monotonic enhancement of 𝛼𝑒𝑓𝑓𝐼𝑃 with Al content in Figure 4b . These \nresults point to lower (higher) damping in FeV (FeAl) and suggest a factor other than the average \natomic number governing magnetic relaxation in these alloys. However, such a conclusion \nassumes that 𝛼𝑒𝑓𝑓𝐼𝑃 is a reliable measure of intrinsic Gilbert damping . In reality, 𝛼𝑒𝑓𝑓𝐼𝑃 may include 8 \n a contribution from defect -induced two -magnon scattering27–31,35,36, a well -known non -Gilbert \nrelaxation mechanism in in -plane magnetized epitaxial films27,32 –34,44. We show in the next \nsubsection that substantial two -magnon scattering is indeed present in our FeV and FeAl alloy \nthin films. \n Although Eq. (1) is not necessarily the correct framework for quantifying Gilbert \ndamping in in -plane magnetized thin films, we can gain insight into the quality (homogeneity) of \nthe films from ∆𝐻0𝐼𝑃. For our samples, μ0∆𝐻0𝐼𝑃 is below ≈ 1 mT (see Fig ure 4c,d), which implies \nhigher film quality for our FeV samples than previously reported20. For example, Fe 73V27 in \nScheck et al. exhibits μ0∆𝐻0𝐼𝑃 ≃ 2.8 mT20, whereas Fe 75V25 in our study exhibits μ0∆𝐻0𝐼𝑃 ≃ 0.8 \nmT. Although 𝛼𝑒𝑓𝑓𝐼𝑃 is comparable between Scheck et al. and our study, the small ∆𝐻0𝐼𝑃 leads to \noverall much narrower linewidths in our FeV films (e.g., as shown in Figs. 2 and 3) . We \nspeculate that the annealing of the MgO substrate prior to film deposition37 – a common practice \nfor molecular beam epitaxy – facilitates high -quality epitaxial film growth and hence small ∆𝐻0𝐼𝑃 \neven by sputtering. \n \n3.2. Out -of-Plane Ferromagnetic Resonance \nTo quantify intrinsic Gilbert damping, we performed broadband FMR with the film \nmagnetized out -of-plane, which is the configuration that suppresses two -magnon scattering28–31. \nSamples were placed in side a W-band shorted -waveguide spectrometer (frequency range 70 -110 \nGHz) in a superconducting electromagnet that enabled measurements at fields > 4 T. This high \nfield range is well above the shape anisotropy field of ≤2 T for our films and hence sufficient to \ncompletely saturate the film out -of-plane. 9 \n The absence of two -magnon scattering in broadband out -of-plane FMR allows us to \nreliably obtain the measured viscous damping parameter 𝛼𝑚𝑒𝑎𝑠𝑂𝑃 by fitting the linear frequency \ndependence of the linewidth ∆𝐻𝑝𝑝𝑂𝑃, as shown in Figure 5, with \n∆𝐻𝑝𝑝𝑂𝑃=∆𝐻0𝑂𝑃+ℎ\n𝑔𝜇𝐵𝜇02\n√3𝛼𝑚𝑒𝑎𝑠𝑂𝑃𝑓. (3) \nWe note that the zero -frequency linewidth for the out -of-plane configuration ∆𝐻0𝑂𝑃 (Figure 6c,d) \nis systematically g reater than that for the in -plane configuration ∆𝐻0𝐼𝑃 (Figure 4c,d). Such a trend \nof ∆𝐻0𝑂𝑃>∆𝐻0𝐼𝑃, often seen in epitaxial films15,33,45, may be explained by the stronger \ncontribution of inhomogeneity to the FMR field when the magnetic precessional orbit is circular, \nas is the case for out -of-plane FMR, compared to the case of the highly elliptical precession in \nin-plane FMR41; however, the detailed mechanisms contributing to the zero -frequency linewidth \nremain the subject of future work . The larger ∆𝐻0𝑂𝑃 at high V and Al concentrations may be due \nto broader distributions o f anisotropy fields and saturation magnetization, or the presence of a \nsecondary crystal phase that is below the resolution of our X -ray diffraction results. \nThe absence of two -magnon scattering in out -of-plane FMR allows us to quantify the \nintrinsic Gilbert damping parameter, \n𝛼𝑖𝑛𝑡=𝛼𝑚𝑒𝑎𝑠𝑂𝑃−𝛼𝑒𝑑𝑑𝑦, (4) \nby again subtracting the eddy current contribution 𝛼𝑒𝑑𝑑𝑦. Since we utilize a shorted waveguide, \nthe contribution due to radiative damping does not apply. \nFrom the compositional dependence of 𝛼𝑖𝑛𝑡 as summarized in Figure 6a1, a reduction in \nintrinsic Gilbert damping is evidenced with V alloying. Our observation is in contrast to the \nprevious experiments on FeV alloys20,22,23 where the reported damping parameters remain >0.002 \n \n1 We were unable to carry out out -of-plane FMR measurements for FeV with x = 20 (Fig. 2(c,d )) as the sample had \nbeen severely damaged during transit. 10 \n and depend weakly on the V concentration. In particular, the observed minimum of 𝛼𝑖𝑛𝑡≃0.001 \nat x ≃ 25-30 is approximately half of the lowest Gilbert damping parameter previously reported \nfor FeV20 and that of pure Fe15. The low 𝛼𝑖𝑛𝑡 here is also comparable to the lowest damping \nparameters reported for ferromagnetic metals, such as Fe75Co2516,17 and Heusler compounds46–48. \nMoreover, t he reduced intrinsic damping by alloying Fe w ith V is qualitatively consistent with \nthe computational prediction by Mankov sky et al.26, as shown by the curve in Figure 6a. Our \nexperimental finding therefore confirms that FeV is indeed an intrinsically ultralow -damping \nferromagnet that possesses a smaller saturation magnetization than Fe. \nIn contrast to the reduction of 𝛼𝑖𝑛𝑡 observed in FeV alloys, FeAl shows an increase in \nintrinsic damping with increasing Al concentration, as seen in Figure 6b. Recalling that Al has an \natomic number of Z = 13 that is lower than Z = 23 for V, this trend clashes with the expectation \nthat lower < Z> red uces the intrinsic Gilbert damping through a reduction of the atomic spin -orbit \ncoupling. Thus, we are required to consider an alternative mechanism to explain the higher \n(lower) damping in FeAl (FeV), which we discuss further in Section V. \n \n3.3. Magnetic Relaxation: Practical Consideration s \nFor both FeV and FeAl alloys, 𝛼𝑖𝑛𝑡 derived from out -of-plane FMR (Figure 6a,b) is \nconsistently lower than 𝛼𝑒𝑓𝑓𝐼𝑃 derived from in -plane FMR (Fig ure 4a,b). Th is discrepancy \nbetween 𝛼𝑖𝑛𝑡 and 𝛼𝑒𝑓𝑓𝐼𝑃 implies a two-magnon scattering contribution to magnetic relaxation in \nthe in-plane configuration (Figure 4a,b). For many applications including spin -torque oscillators \nand magnonic devices , it is crucial to minimize magnetic relaxation in in-plane magnetized thin \nfilms. While the in -plane magnetic relaxation ( 𝛼𝑒𝑓𝑓𝐼𝑃≃0.002) is already quite low for the FeV \nalloys shown here, the low intrinsic Gilbert damping ( 𝛼𝑖𝑛𝑡≃0.001) points to the possibility of 11 \n even lower relaxation and narrow er FMR linewidths by minimizing two -magnon scattering and \ninhomogeneous linewidth broadening. Such ultralow magnetic relaxation in FeV alloy thin films \nmay be achieved by optimizing structural properties through growth conditions16 or seed layer \nengineering49. \nWhile ultralow intrinsic Gilbert damping values have been confirmed in high -quality \nepitaxial FeV, it would be desirable for device integration to understand how magnetic relaxation \nin FeV would be impacted by the presence of grain boundaries, i.e. in polycrystalline thin films. \nReports on polycrystalline FeCo49 suggest intrinsic damping values comparable to those seen in \nepitaxial FeCo16,17. While beyond the scope of this study, our future work will explore the \npossibility of low damping in polycrystalline FeV thin films. \n \nIV. SPECTROSCOPIC PARAMETERS \nThe results presented so far reveal that magnetic relaxation is reduced by alloying Fe with \nV, whereas it is increased by alloying Fe with Al. On the other hand, FeV and FeAl alloys \nexhibit similar compositional dependence of the spectroscopic parameters: effective \nmagnetization Meff (here, equivalent to saturation magnetization Ms), magnetocrystalline \nanisotropy field Hk, and the g-factor 𝑔 – all of which are quantified by fitting the frequency \ndependence of resonance field (Supplemental Material) . As shown in Fig ure 7a, there is a \nsystematic reduction in Meff with increasing concentration of V and Al. We also note in Fig ure 7b \na gradual reduction in magnitude of the in -plane cubic anisotropy. Both of these trends are \nexpected as magnetic Fe atoms are r eplaced with nonmagnetic atoms of V and Al. The reduction \nof Meff by ≃20% in the ultralow -damping Fe 100-xVx alloys with x = 25-30, compared to pure Fe, \nis of particular practical interest. The saturation magnetization of these FeV alloys is on par with 12 \n commonly used soft ferromagnetic alloys (e.g., Ni 80Fe2050, CoFeB51), but the damping parameter \nof FeV is several ti mes lower. Further, w hile FeV and FeCo in the optimal composition window \nshow similarly low intrinsic damping parameters, FeV provides the advantage of lower moment . \nWith the product 𝛼𝑖𝑛𝑡𝑀𝑒𝑓𝑓 approximately proportional to the critical current densi ty to excite \nprecessional dynamics by spin torque2,11, FeV is expected to be a superior material platform for \nlow-power spin tronic devices . \nThe g-factor 𝑔=2(1+𝜇𝐿/𝜇𝑆) is related to the orbital moment 𝜇𝐿 and spin moment 𝜇𝑆; \nthe deviation from the spin -only value of 𝑔= 2.00 provides insight into the strength of spin -orbit \ncoupling ξ52. As seen in Figure 7c, 𝑔 increases by 1-2% with both V and Al alloying, which \nsuggests that ξ increases slightly with the addition of these low -Z elements. This finding verifies \nthat < Z> is not necessarily a good predictor of ξ in a solid. Moreover, the higher 𝑔 for FeV is \ninconsistent with the scenario for lower damping linked to a reduced spin -orbit coupling. Thus, \nspin-orbit coupling alone cannot explain the observed behavior of Gilbert damping in Fe alloyed \nwith low -Z elements. \n \nV. DISCUSSION \nIn contrast to what has been suggested by prior experimental studies20,22 –25, we have \nshown that the reduction of average atomic number by alloying with a light element (e.g., Al in \nthis case) does not generally lower the intrinsic Gilbert dampin g of Fe. A possible source for the \nqualitatively distinct dependencies of damping on V and Al contents is the density of states at the \nFermi level, D(EF): it has been predicted theoretically that the intrinsic Gilbert damping \nparameter is reduced with decr easing D(EF), since D(EF) governs the availability of states for \nspin-polarized electrons to scatter into21,26,53 –55. Such a correlation between lower damping and 13 \n smaller D(EF) has been reported by recent experiments on FeCo alloys17,50, FeRh alloys40, CoNi \nalloys56, and Heusler compounds46,48,57. The similarity in the predicted composition dependence \nof the Gilbert damping parameter for FeCo and FeV26 suggests that the low damping of FeV may \nbe correlated with reduced D(EF). However, no prior experiment has corroborated this \ncorrelation for FeV or other alloys of Fe and light elements. \nWe therefore e xamine whether the lower (higher) damping in FeV (FeAl) compared to Fe \ncan be qualitatively explained by D(EF). Utilizing the Quantum ESPRESSO58 package to \nperform density functional theory calculations (details in Supplemental Material) , we calculated \nthe density of states for Fe, Fe 81.25V18.75, and Fe 81.25Al18.75. It should be recalled that although \nFeV and FeAl films measured experimentally her e are single -crystalline, they are solid solutions \nin which V or Al atoms replace Fe atoms at arbitrary bcc lattice sites. Therefore, f or each of the \nbinary alloys, we computed 6 distinct atomic configurations in a 2×2×2 supercell , as shown in \nFigure 8 . The spin -split density of states for each unique atomic configuration is indicated by a \ncurve in Figure 9. Here, D(EF) is the sum of the states for the spin -up and spin -down bands, \naveraged over results from the 6 distinct atomic configurations. \nAs summari zed in Fig ure 9 and Table 1, FeV has a smaller D(EF) than Fe, whereas FeAl \nhas a larger D(EF). These calculation results confirm a smaller (larger ) availability of states for \nspin-polarized electrons to scatter into in FeV (FeAl), qualitatively consistent with the lower \n(higher) intrinsic Gilbert damping in FeV (FeAl). \nWe remark that this correlation between damping and D(EF) is known to hold parti cularly \nwell in the limit of low electronic scattering rates 𝜏−1, where intra band scattering dominates21,54. \nGilmore et al. have pointed out that at sufficiently high electronic scattering rates, i.e., when \nℏ𝜏−1 is large enough that inter band scattering is substantial, the simple correlation between the 14 \n strength of Gilbert damping and D(EF) breaks down. It is unclear whether our FeV and FeAl \nalloy films at room temperature are in the intraband - or interband -dominated regime. Schoen et \nal. have argued that polycrystalline FeCo alloy films – with higher degree of structural disorder \nand likely higher electronic scattering rates than our epitaxial films – at room temperature are \nstill well within the intraband -dominated regime17. On the other hand, a recent temperature -\ndependent study on epitaxial Fe suggests coexistence of the intraband and interband \ncontributions at room temperature15. A consistent explanation for the observed room -temperature \nintrinsic damping in our alloy films is that the interband contribution depends weakly on alloy \ncomposition; it appears re asonable to conclude that D(EF), primarily through the intraband \ncontribution, governs the difference in intrinsic Gilbert damping among Fe, FeV, and FeAl . \n \nVI. SUMMARY \nWe have experimentally in vestigated magnetic relaxation in epitaxial thin films of Fe \nalloyed with low -atomic -number nonmagnetic elements V and Al . We observe a reduction in the \nintrinsic Gilbert damping parameter to 𝛼𝑖𝑛𝑡≃0.001 in FeV films , comparable to the lowest -\ndamping ferromagnetic metals reported to date. In contrast, an increase in damping is observed \nwith the addition of Al, demonstrating that a smaller average atomic number does not necessarily \nlower intrinsic damping in an alloy . Furthermore, our results on FeV and FeAl cannot be \nexplained by the change in spin -orbit coupling through alloying . Instead, we conclude that the \ndensity of states at the Fermi level plays a larger role in determining the magnitude of damping \nin Fe alloyed w ith lighter elements. Our work also confirms FeV alloys as promising ultra low-\ndamping , low-moment metallic materials for practical power -efficient spin -torque devices. \n 15 \n Acknowledgements: \nThis research was funded in part by 4 -VA, a collaborative partnership for advancing the \nCommonwealth of Virginia, as well as by the ICTAS Junior Faculty Program. D.A.S. \nacknowledges support of the Virginia Tech Graduate School Doctoral Assistantship. A. Sapkota \nand C. M . would like to acknowledge support by NSF -CAREER Award No. 1452670, A.R. and \nT.M. would like to acknowledge support by DARPA TEE Award No. D18AP00011, and A. \nSrivastava would like to acknowledge support by NASA Award No. CAN80NSSC18M0023. \n \nWe thank M.D. Stiles for helpful input regarding intrinsic damping mechanisms in alloys. \n \nThe data that support the findings of this study are available from the corresponding author upon \nreasonable request. \n \n Number of Spin -Up States (eV-1) \nat EF Number of Spin -Down States \n(eV-1) at EF \nFe 10.90 3.44 \nFe81.25V18.75 6.28 ± 1.80 4.61 ± 0.43 \nFe81.25Al18.75 6.81 ± 1.58 10.20 ± 3.03 \nTable 1: Number of spin -up and spin -down states at EF. For Fe81.25V18.75 and \nFe81.25Al18.75, the average and standard deviation of values for the 6 distinct atomic \nconfigurations (cf. 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Smogunov, P. Umari, and R.M. Wentzcovitch, J. Phys.Condens. \nMatter 21, 395502 (2009). \n 21 \n \n15 30 45 60 75 90\nMgO (004) MgO (004) MgO (004)MgO (002) MgO (002) \n MgO (002)BCC\nFe\n(002)\n Log(Intensity) (arb. units)BCC\nFe80V20\n(002)\n \n2q (deg)BCC\nFe80Al20\n(002) \nFigure 1: (a) 2θ-ω X-ray diffraction scans showing (00 2) and (004) substrate and (002) film \npeaks for bcc Fe, Fe 80V20, and Fe 80Al20. \n 22 \n \n-15 -10 -5 0 5 10 15 \n \nFe 2.70 mT\n FMR Signal (arb. units)Fe80V20 2.04 mT\n \nm0(H - HFMR) (mT)Fe80Al203.20 mT \nFigure 2: FMR spectra at f = 20 GHz with the magnetic field H applied in the film plane, fitted \nusing a Lorentzian derivative (solid curve ) for Fe, Fe 80V20 and Fe 80Al20. 23 \n \n0 10 20 30 40 5002468 Fe\n Fe80V20\n Fe80Al20\n Scheck et al.m0DHIP\nPP (mT)\nFrequency (GHz) \nFigure 3: FMR linewidths versus microwave frequency for the magnetic field applied within the \nplane of the film for three distinct alloys. The solid lines are linear fit s, described by Eq. (1), \nfrom which the effective damping parameter and zero frequency linewidth are determined. The \ndashed line represents the result for Fe 73V27 from Scheck et al.20 \n 24 \n \n0 10 20 30 40246\n0 10 20 302468\n0 10 20 30 40024\n0 5 10 15 20 25 3001 Fe100-xVx\n Scheck et al.aIP\neff x 103\naIP\neff x 103 Fe100-xAlxm0DHIP\n0 (mT)\nAlloy Composition, x (%)(a) (b)\n(c) (d)m0DHIP\n0 (mT)\nAlloy Composition, x (%) \nFigure 4: The effective damping parameter 𝛼𝑒𝑓𝑓𝐼𝑃 for (a) Fe 100-xVx and (b) Fe 100-xAlx and zero \nfrequency linewidth 𝜇0Δ𝐻0𝐼𝑃 for (c) Fe 100-xVx and (d) Fe 100-xAlx, obtained from in -plane FMR. \nThe solid symbols in (a) and (c) represent results reported by Scheck et al.20 \n 25 \n \n0 20 40 60 80 100 12001020304050\n Fe\n Fe70V30\n Fe70Al30m0DHOP\nPP (mT)\nFrequency (GHz) \nFigure 5: FMR linewidths versus applied microwave frequency for the magnetic field applied \nperpendicular to the plane of the film for three distinct alloys. The line is a linear fit, described \nby Eq. (3), from which the intrinsic Gilbert damping parameter and zero frequency linewidth are \ndetermined. \n 26 \n \n0 10 20 30 400123\n0 10 20 300246\n0 10 20 30 4001020\n0 10 20 3001020 Fe100-xVx\n Mankovsky et al.aint x 103 Fe100-xAlxaint x 103(a) (b)\n(c) (d)m0DHOP\n0 (mT)\nAlloy Composition, x (%)\nm0DHOP\n0 (mT)\nAlloy Composition, x (%) \nFigure 6: The intrinsic Gilbert damping parameter 𝛼𝑖𝑛𝑡 for (a) Fe 100-xVx and (b) Fe 100-xAlx and \nzero frequency linewidth 𝜇0Δ𝐻0𝑂𝑃 for (c) Fe 100-xVx and (d) Fe 100-xAlx, obtained from out -of-\nplane FMR. In (a), the dashed curve show s the predicted intrinsic damping parameter computed \nby Mankovsky et al.26 \n 27 \n \n0.81.21.62.02.4\n204060\n0 10 20 30 402.082.102.122.14 Fe\n Fe100-xVx\n Fe100-xAlx \n m0Meff (T)(a)\n |m0Hk| (mT)(b)\n g-factor\nAlloy Composition, x (%)(c) \nFigure 7: (a) Effective magnetization, (b) in -plane cubic anisotropy field, and (c) g-factor versus \nV and Al concentration. The solid (open) markers represent data from in -plane (out -of-plane) \nmeasurements . \n 28 \n \nFigure 8: The six unique atomic configurations from the supercell program for mimicking the \nFe81.25V18.75 or Fe81.25Al18.75 solid solution. \n29 \n \n-10010-10010\n-1.0 -0.5 0.0 0.5 1.0-10010 \n (a)\nFe81.25V18.75\n Density of States (eV-1)\n(b)Fe\n \nE - EF (eV)(c)Fe81.25Al18.75 \nFigure 9: Calculated spin-up (positive) and spin -down (negative) densit ies of states for (a) Fe, \n(b) Fe 81.25V18.75 and (c) Fe 81.25Al18.75. Results from the 6 distinct atomic configurations are shown \nin (b,c); the average densities of states at EF for Fe81.25V18.75 and Fe81.25Al18.75 are shown in \nTable 1. \n " }, { "title": "2309.11152v1.Evaluating_Gilbert_Damping_in_Magnetic_Insulators_from_First_Principles.pdf", "content": "Evaluating Gilbert Damping in Magnetic Insulators from First Principles\nLiangliang Hong,1, 2Changsong Xu,1, 2and Hongjun Xiang1, 2,∗\n1Key Laboratory of Computational Physical Sciences (Ministry of Education), Institute of Computational Physical Sciences,\nState Key Laboratory of Surface Physics, and Department of Physics, Fudan University, Shanghai 200433, China\n2Shanghai Qi Zhi Institute, Shanghai 200030, China\n(Dated: September 21, 2023)\nMagnetic damping has a significant impact on the performance of various magnetic and spin-\ntronic devices, making it a long-standing focus of research. The strength of magnetic damping is\nusually quantified by the Gilbert damping constant in the Landau-Lifshitz-Gilbert equation. Here\nwe propose a first-principles based approach to evaluate the Gilbert damping constant contributed\nby spin-lattice coupling in magnetic insulators. The approach involves effective Hamiltonian mod-\nels and spin-lattice dynamics simulations. As a case study, we applied our method to Y 3Fe5O12,\nMnFe 2O4and Cr 2O3. Their damping constants were calculated to be 0 .8×10−4, 0.2×10−4,\n2.2×10−4, respectively at a low temperature. The results for Y 3Fe5O12and Cr 2O3are in good\nagreement with experimental measurements, while the discrepancy in MnFe 2O4can be attributed\nto the inhomogeneity and small band gap in real samples. The stronger damping observed in Cr 2O3,\ncompared to Y 3Fe5O12, essentially results from its stronger spin-lattice coupling. In addition, we\nconfirmed a proportional relationship between damping constants and the temperature difference\nof subsystems, which had been reported in previous studies. These successful applications suggest\nthat our approach serves as a promising candidate for estimating the Gilbert damping constant in\nmagnetic insulators.\nI. INTRODUCTION\nRecent decades have witnessed rapid developments in\nmagnetics and spintronics [1–3]. A long-time pursuit in\nspintronics is to actively control and manipulate the spin\ndegrees of freedom in solid-state systems. Related fun-\ndamental studies involve spin transport, spin dynamics\nand spin relaxation [4]. Within these domains, magnetic\ndamping often plays a crucial role. Generally, stronger\ndamping enables a faster writing rate for magnetic mem-\nories, while lower damping leads to a longer propagation\ndistance of spin waves. Therefore, it is always essential\nto accurately evaluate the magnetic damping in different\nmaterials. For instance, yttrium iron garnet (YIG) is a\nhighly promising spintronic material due to its ultra-low\nmagnetic damping [5–7]. However, the intrinsic mecha-\nnism behind its unique property has yet to be fully eluci-\ndated, which partly motivates us to carry out this work.\nAt present, magnetic damping is typically represented\nby a phenomenological term in the well-known Landau-\nLifshitz-Gilbert (LLG) equation, which has been widely\nemployed to simulate magnetization dynamics [8, 9]. A\nbasic form of this equation can be written as,\n∂ ⃗ m\n∂t=−γ ⃗ m×⃗B+α\nm⃗ m×∂ ⃗ m\n∂t(1)\nwhere ⃗Brepresents the total magnetic field acting on the\nlocal dipole ⃗ m,mdenotes the norm of ⃗ m,γis the gyro-\nmagnetic ratio, and αis the Gilbert damping constant.\nThe second term on the right side, as we mentioned, leads\n∗hxiang@fudan.edu.cndirectly to the relaxation process, in which the rate of en-\nergy dissipation is determined by the damping constant.\nGiven the importance of αin magnetization dynamics,\nits origin has been extensively studied in the literature\n[10–13]. To the best of our knowledge, both intrinsic and\nextrinsic mechanisms contribute to the damping. Specif-\nically, the intrinsic factors include spin-lattice and spin-\nelectron couplings, while the extrinsic contributions pri-\nmarily involve lattice imperfections and eddy currents\n[14, 15].\nTwo types of first-principles based methods have been\ndeveloped to calculate the damping constants in the past.\nOne approach involves the breathing Fermi surface model\n[16, 17] and the torque correlation model [18, 19], while\nthe other is based on the scattering theory from linear\nresponse [20–22]. These methods have demonstrated re-\nmarkable success in studying the magnetic damping in\ntransition metals such as Fe, Co, and Ni. Despite be-\ning free from complicated experiments, which are mostly\nbased on ferromagnetic resonance, these theoretical ap-\nproaches still exhibit several limitations. Firstly, when\ndealing with complex systems, we often have to spend a\nsignificant amount of computing resources on the first-\nprinciples calculations. In addition, these methods are\nmore suitable for calculating the electronic contribution\nto Gilbert damping in metallic magnets, thus rarely tak-\ning the effect of spin-lattice coupling into consideration\n[14, 23].\nRecently, spin-lattice dynamics (SLD) simulations [24]\nhave been adopted as an alternative method to evaluate\nthe Gilbert damping parameters. In Ref. [23], the au-\nthors constructed an empirically parameterized Hamil-\ntonian model for a cobalt cluster. They coupled a pre-\nheated lattice with a fully ordered spin state, then per-\nformed SLD simulation. During the relaxation process,arXiv:2309.11152v1 [cond-mat.mtrl-sci] 20 Sep 20232\nthe energy of lattice and spin subsystems were recorded\nand fitted to the following logistic functions,\nUlat=Ulat\n0−∆U0\n1 + exp[ −η∆U0t−Θ](2)\nUmag=Umag\n0+∆U0\n1 + exp[ −η∆U0t−Θ](3)\nfrom which they extracted the relaxation rate Γ = η∆U0\nand calculated the damping constant α=ηµS/γ. Here,\nµSdenotes the magnitude of magnetic moments. In Ref.\n[25], the authors also built an empirical potential model\nfor a periodic bcc Fe system. They firstly applied an ex-\nternal magnetic field in the z-direction and thermalized\nthe system to a finite temperature. Then, the magnetiza-\ntion orientation of each atom was rotated artificially by\na same angle. Afterwards, the system would relax back\nto equilibrium, during which the averaged z component\nof atomic magnetization was recorded and fitted to the\nfollowing function,\nmz(t) = tanh\u0014α\n1 +α2γBext(t+t0)\u0015\n(4)\nwhere αwas exactly the Gilbert damping parameter to\nbe estimated. Since these works selected transition met-\nals as the research object, their results were both orders\nof magnitude smaller than the experimental values. In\naddition, the use of empirically parameterized models re-\nduced the accuracy of their simulated results.\nIn this work, we combine SLD simulations with first-\nprinciples based effective Hamiltonian models to evalu-\nate the damping constants in magnetic insulators, where\nthe dominant contribution results from spin-lattice cou-\nplings. Compared to the previous studies, our work has\nmade improvements mainly in two aspects. Firstly, the\nutilization of first-principles based Hamiltonian models\nin simulations enhances the reliability of our conclusions.\nBesides, the better choice of research objects allows for\ndemonstrating the superiority of SLD simulations. In\nparticular, the microscopic origin of low damping in YIG\nwill be investigated. The paper is organized as follows.\nIn Sec. II, we introduce our effective Hamiltonian model,\nparameterization methods, and a scheme for evaluating\nGilbert damping parameters. Then, both the validation\nand application of our method are presented in Sec. III.\nFinally, we summarize this work and give a brief outlook\nin Sec. IV.\nII. MODEL AND METHODS\nThis section is split into three parts. Firstly (in Sec.\nII A), we introduce a generic form of our effective Hamil-\ntonian model. Then, methods involving the calculation\nof model parameters are presented in Sec. II B. At the\nlast part (Sec. II C), we propose a novel scheme to de-\ntermine the Gilbert damping constant through dynamics\nsimulations.A. The Hamiltonian Model\nSince our purpose is to evaluate the contribution of\nspin-lattice coupling to magnetic damping, the effective\nHamiltonian model must incorporate both spin and lat-\ntice degrees of freedom. A concise and generic formula\nthat meets our basic requirements consists of the three\nterms as follows:\nH=HL({ui,α}) +HS({⃗ sj}) +HSLC({ui,α,⃗ sj}) (5)\nwhere αabbreviates three orthogonal axes, ui,αrepre-\nsents the displacement of atom i, and ⃗ sjis a unit vector\nthat represents the direction of spin j.\nThe first term HLin Hamiltonian model describes the\ndynamical behavior of individual phonons. Technically,\nwe take the atomic displacements as independent vari-\nables and expand the Hamiltonian to the second order\nwith Taylor series. Then, we have the form as,\nHL=1\n2X\nijX\nαβKij,αβui,αuj,β+1\n2X\ni,αMi˙ui,α˙ui,α(6)\nwhere Kij,αβ denotes the force constant tensor and Mi\nrepresents the mass of atom i.\nSimilarly, the second term HSdescribes the dynami-\ncal behavior of individual magnons. For simplicity but\nno loss of accuracy, we only considered the Heisenberg\nexchange interactions between neighbor magnetic atoms\nin this work, though more complex interactions could be\ntaken into account in principle. Therefore, this term can\nbe expressed as,\nHS=X\n⟨i,j⟩Jij⃗Si·⃗Sj (7)\nwhere Jijdenotes the isotropic magnetic interaction co-\nefficient.\nThe third term HSLCrepresents the coupling of spin\nand lattice subsystems, and is expected to describe the\nscattering process between phonons and magnons. As\nan approximation of the lowest order, this term can be\nwritten as,\nHSLC=X\n⟨i,j⟩X\nkα\u0012∂Jij\n∂uk,αuk,α\u0013\n⃗Si·⃗Sj (8)\nAccording to the theory of quantum mechanics, this\ncoupling term provides a fundamental description of the\nsingle-phonon scattering process, which is believed to be\ndominant among all scatterings in the low-temperature\nregion. This type of relaxation mechanism in ferromag-\nnetic resonance was systematically studied by Kasuya\nand LeCraw for the first time [26]. It’s worth noting that\na higher order of Taylor expansion could have been con-\nducted to improve the accuracy of Hamiltonian models\ndirectly. For instance, the scattering between individual\nphonons can be adequately described by the anharmonic\nterms. However, as one always has to make a trade-off3\nbetween the precision and complexity of models, in this\nwork we choose to neglect the high order terms since the\nanharmonic effects in current investigated systems are\nnot important.\nIn this study, we adopted the symmetry-adapted clus-\nter expansion method implemented in the Property Anal-\nysis and Simulation Package for Materials (PASP) [27]\nto build the Hamiltonian model presented above. This\npackage can identify the nonequivalent interactions and\nequivalent atom clusters in a crystal system by analyz-\ning its structural properties based on the group theory.\nA significant benefit of working with PASP is we are en-\nabled to describe the target system with the least number\nof parameters. In the next section, we will discuss how\nto calculate the model parameters for different materials.\nB. Calculation of Model Parameters\nFirstly, the Heisenberg exchange coefficients Jijand\nspin-lattice coupling constants ∂Jij/∂uk,αcan be calcu-\nlated with the four-state method [28, 29]. The basic flow\nis to construct four artificially designated spin states of\nthe target system, calculate the corresponding energies\nand forces based on the density functional theory (DFT),\nthen determine the parameters by proper combination of\nthose results. At the last step, the following formulas will\nbe used,\nJij=E↑↑+E↓↓−E↑↓−E↓↑\n4S2(9)\n∂Jij\n∂uk,α=F↑↑\nk,α+F↓↓\nk,α−F↑↓\nk,α−F↓↑\nk,α\n4S2(10)\nwhere Sis the spin quantum number of magnetic atoms,\nEis the total energy of system and Fk,αrefers to one\ncomponent of the force on atom k. The superscripts ( ↑↑,\n↓↓,↑↓,↓↑) specify the constrained spin states of system\nin the calculation. More technical information about the\nfour-state method can be found in the references [28, 29].\nCompared to other approaches, the four-state method of-\nfers an obvious advantage in that no additional DFT cal-\nculations are needed to determine the coupling constants\n∂Jij/∂uk,αonce the exchange coefficients Jijhave been\nobtained. This is because the energy and forces are typ-\nically provided simultaneously by one DFT calculation.\nSince atomic masses Mican be directly obtained from\nthe periodic table, more efforts are needed to deal with\nthe force constant tensor Kij,αβ. Currently, there are two\ncommonly adopted ways to calculate the force constant\ntensor: density functional perturbation theory (DFPT)\nand finite displacement method. Both of these methods\nare applicable to our task.\nHowever, we cannot directly take the force constant\ntensor obtained from first-principles calculations as the\nmodel parameter. This is because in dynamics simula-\ntions we usually expand crystal cells to reduce the un-\ndesired influence of thermal fluctuations, which leads toa conflict between the periodic boundary condition and\nthe locality (also known as nearsightedness [30, 31]) of\nmodels. To be more specific, when calculating the con-\ntribution of one atom or spin to the total energy, we tend\nto set a well designed cutoff radius and ignore the inter-\nactions beyond it. This step is essential when dealing\nwith a large-scale system, otherwise we will suffer from\nthe model complexity and the computational cost. Nev-\nertheless, if we set the elements of Kij,αβ that represent\nout-of-range interactions to be zero and leave the others\nunchanged, we may violate the so-called acoustic sum-\nmation rules:\nX\niKij,αβ = 0 for all j, α, β. (11)\nIt should be pointed out that a straightforward en-\nforcement of the acoustic summation rules, achieved by\nsubtracting errors uniformly from force constants, will\nbreak the inherent crystal symmetry inevitably, which is\nthe technique employed in phonopy [32]. To address the\nabove issues, we adopted a more appropriate method in\nthis work. Before a detailed introduction, it’s necessary\nto recall that not every element of the force constant ten-\nsor serves as an independent variable due to the crystal\nsymmetries. Taking the cubic cell of Y 3Fe5O12(contain-\ning 160 atoms) for example, there are 230400 elements in\nthe tensor. After symmetry analyses, we find that only\n597 independent variables {pn}are needed to adequately\ndetermine all the tensor elements {Kij,αβ({pn})}, where\nthe effect of locality is already considered. Afterwards,\nour method is to set a correction factor xnfor each vari-\nablepnand minimize the deviation of parameters under\nthe constraints of Eq. (11). A mathematical reformula-\ntion of this method can be written as,\nmin\n{xn}X\nn(xn−1)2,with\nX\niKij,αβ({xnpn}) = 0 for all j, α, β.(12)\nIn the case of Y 3Fe5O12, there are only 18 linearly inde-\npendent constraints, which allow the extremum problem\nto be solved rigorously. The modified force constant ten-\nsor restores positive definiteness and translational sym-\nmetry while maintaining the crystal symmetries. There-\nfore, the modified tensor meets the requirements for dy-\nnamics simulations. In Sec. III B, the effectiveness of this\napproximate method will be demonstrated through a spe-\ncific example.\nAll the first-principles calculations mentioned in this\nsection are carried out using the Vienna ab initial simu-\nlation package (VASP) [33–35]. The force constants and\nphonon spectra are obtained by phonopy [32]. The opti-\nmizations formulated in (12) are accomplished with the\nfunction optimize.minimize implemented in SciPy [36].4\nC. Evaluation of Damping Constants\nAfter the construction and parameterization of Hamil-\ntonian models, we are finally able to perform spin-lattice\ndynamics simulations. Before the evaluation of Gilbert\ndamping constants, we briefly introduce the framework\nof SLD to cover some relevant concepts. In practice, the\nmotion of magnetic moments follows the stochastic Lan-\ndau–Lifshitz–Gilbert (SLLG) equation [14],\nd⃗ mi\ndt=−γL⃗ mi×\u0010\n⃗Bi+⃗Bfl\ni\u0011\n−γLα⃗ mi\n|⃗ mi|×h\n⃗ mi×\u0010\n⃗Bi+⃗Bfl\ni\u0011i\n(13)\nwhere γLis the renormalized gyromagnetic ratio, ⃗Bi=\n−∂H/∂ ⃗ m iis the effective local magnetic field and ⃗Bfl\ni\nrefers to a stochastic field introduced by Langevin ther-\nmostat. At the same time, the motion of atoms obeys\nthe Newton’s equation,\nd˙ui,α\ndt=1\nMi\u0010\n⃗Fi,α+⃗Ffl\ni,α\u0011\n−ν˙ui,α (14)\nwhere νis the damping constant and ⃗Ffl\ni,αrefers to a\nstochastic force caused by thermal fluctuations. In this\nwork, ⃗Bfl\niand⃗Ffl\ni,αare modeled as normally distributed\nnoises with temperature-dependent variances,\nBfl\ni,β∼N\u0010\n0,p\n2αkBTS/γ|⃗ mi|δt\u0011\n(15)\nFfl\ni,β∼N\u0010\n0,p\n2νMikBTL/δt\u0011\n(16)\nwhere TSandTLrefer to the equilibrium temperature of\nspin and lattice subsystems respectively. During simula-\ntions, we can also measure the transient temperature of\neach subsystem with the following formulas [37],\nTS=P\ni|⃗ mi×⃗Bi|2\n2kBP\ni⃗ mi·⃗Bi, TL=1\n2kBNX\ni,αMi˙u2\ni,α (17)\nIn this work, the LLG equation is numerically solved\nwith the semi-implicit SIB method proposed by Mentink\net al. [38]. The Newton’s motion equation is integrated\nusing the Grønbech-Jensen-Farago Verlet-type method\n[39]. To ensure the stability of those algorithms, a step\nlength of 0 .5 or 0 .2 fs is adopted [40], where the shorter\none is used in energy-conserving simulations.\nBased on the combination of atomistic spin dynamics\n(ASD) and SLD simulations, a new scheme is proposed\nto evaluate the damping constant in magnetic materials.\nHere is the basic flow of this method and more details of\na specific application are presented in Sec. III B.\n1. Freeze the spin degree of freedom and thermalize\nthe lattice from 0 to TLin the simulation.\n2. Fix atomic positions and raise the temperature of\nspin to TS> TL. Compared to TL> TS, this type\nof nonequilibrium state is more common in actual\nscenarios.3. Perform an energy-conserving SLD simulation to\nrelax the system. Normally, the spin temperature\nwill decrease to the same as lattice and stay there\ntill the end.\n4. Conduct a series of ASD simulations with different\nGilbert damping constants. The initial states are\nthe same as in step 3 and the equilibrium temper-\natures are set to be TL.\n5. Compare the cooling rates ∂TS/∂tof spin system\nbetween SLD and ASD simulations to evaluate the\nequivalent Gilbert damping constant contributed\nby spin-lattice coupling.\nThe key point behind step 5 is that the cooling rates\nobserved in ASD simulations are related to the assigned\ndamping constants, while in SLD simulation the cooling\nrate is determined by the strength of spin-lattice cou-\npling. Note that the former relation can be viewed as a\nnatural deduction of the LLG equation,\n∂TS\n∂t=1\nCV∂Emag\n∂t∝ −1\nCV\u0012∂ ⃗ m\n∂t·⃗B\u0013\n∝ −1\nCV\u0014\u0012α\nm⃗ m×∂ ⃗ m\n∂t\u0013\n·⃗B\u0015\n∝α (18)\nwhere we have used Eq. (1) and simplified the formula of\nmagnetic energy as Emag∝ −⃗ m·⃗B.\nIII. RESULTS\nThis section is divided into four parts. In Sec. III A,\nseveral test results are presented to validate the accu-\nracy of SLD simulations, which are implemented in the\nPASP package. Subsequently, detailed calculations on\nthree magnetic materials, namely Y 3Fe5O12, MnFe 2O4\nand Cr 2O3, are discussed in the rest parts.\nA. Validations\nIn order to guarantee the reliability of our conclusions\nobtained from dynamics simulations, a series of pretests\nwere carried out. We select some representative results\nand present them in Fig. 1, where Cr 2O3is taken as the\nobject to be studied.\nFirstly, we set the ground state of Cr 2O3as the ini-\ntial state and performed a NVT simulation with Tset=\n150K. As shown in Fig. 1(a), the temperature of spin\nand lattice subsystems increased to 150 Kin less than 5\nps and stayed there till the end. Since we can approxi-\nmate Ek= 0.5ELandEp= 0.5EL+ES, Fig. 1(b) also\nindicates that the contribution of phonons and magnons\nto the excited state energy is around 87.5% and 12.5%\nrespectively. This result could be verified from another\nperspective. Note that there are totally 10 atoms in the5\nFIG. 1. NVT and NVE relaxations of a spin-lattice coupled system (Cr 2O3) within the framework of spin-lattice dynamics.\nThe top row plots the time evolution of temperatures and the bottom row shows the variation of potential, kinetic and total\nenergies. (a) & (b): NVT thermalization from TL=TS= 0KtoTL=TS= 150 K. (c) & (d): NVE relaxation with TL= 30K,\nTS= 175 Kinitially. (e) & (f): NVE relaxation with TL= 180 K,TS= 30Kinitially.\nunit cell of Cr 2O3, which contribute 30 kBto the heat ca-\npacity. Meanwhile, the 4 magnetic atoms will contribute\nanother 4 kBin the low temperature region. Therefore,\nwe can estimate that the contribution of magnons to the\ntotal heat capacity is close to 11.8%, which is consistent\nwith the result from dynamics simulations.\nIn Figs. 1(c) & 1(d), the initial state was set to be a\nnonequilibrium state with TL= 30KandTS= 175 K. As\nwe expected, the total energy was well conserved when\nthe system evolved to equilibrium. In addition, the final\ntemperature fell within the range of 48 K∼55K, which\nagrees with our previous analysis of the heat capacities.\nLastly, we simulated the relaxation process using an-\nother nonequilibrium excited state with TL= 180 Kand\nTS= 30Kas the initial state. As shown in Figs. 1(e) &\n1(f), the temperature of spin system increased gradually\nto equilibrium with the total energy conserved through-\nout the simulation. Also, the final temperature is around\n160K, which matches well with our analysis. It should be\npointed out that there exist two notable differences be-\ntween this case and the previous. Firstly, the subsystems\nultimately evolved to a same temperature in a finite time,which alleviated our concerns about the accuracy of SLD\nsimulations. Besides, the relaxation time ( τ2) was much\nlonger than that ( τ1) in Fig. 1(c). For this phenomenon,\na qualitative explanation is presented below.\nBased on the theory of second quantization, the Hamil-\ntonian model presented in Sec. II A can be expressed in\nthe following form [41, 42],\nHL=X\nqpℏωqp(b†\nqpbqp+ 1/2) (19)\nHS=X\nλϵλa†\nλaλ+Const. (20)\nHSLC=X\nλ,qpMλ,qpa†\nλ−qaλ\u0000\nb†\nqp−b−qp\u0001\n(21)\nwhere bqpdenotes the annihilation operator of phonons\nwith wave vector qin branch p, and aλrepresents the an-\nnihilation operator of magnons with wave vector λ. All\nthe parameters, namely ωqp,ϵλandMλ,qp, can be deter-\nmined from the effective Hamiltonian model in principle.\nAccording to the Fermi’s golden rule, we have\nW{nλ−q, nλ, Nqp→nλ−q+ 1, nλ−1, Nqp+ 1}=2π\nℏ|Mλ,qp|2(nλ−q+ 1)( nλ)(Nqp+ 1)δ(ϵλ−q−ϵλ+ℏωqp) (22)\nW{nλ−q, nλ, N−qp→nλ−q+ 1, nλ−1, N−qp−1}=2π\nℏ|Mλ,qp|2(nλ−q+ 1)( nλ)(N−qp)δ(ϵλ−q−ϵλ−ℏω−qp) (23)6\nFIG. 2. (a) The primitive cell of Y 3Fe5O12. The golden balls\nrepresent iron atoms, the cyan ball stand for yttrium atoms,\nand the red balls refer to oxygen atoms. (b) The magnetic\nground state of YIG. The arrows of different colors represent\nthe spin directions of Fe atoms. (c) The density of states ob-\ntained by DFT calculations. (d) The temperature dependence\nof average magnetization measured in MC and ASD simula-\ntions. For YIG, the phase transition point from ferrimagnetic\nto paramagnetic lies in 530 K approximately.\nwhere Wrepresents the probability of one-phonon emis-\nsion or absorption, nλdenotes the occupation number of\nmagnons and Nqpstands for phonons. Both nλandNqp\ncan be evaluated approximately using the Bose–Einstein\ndistribution. According to the above formulas, the scat-\ntering rate Wgrows linearly with Nand quadratically\nwith n. Compared to Fig. 1(c), there are more phonons\nbut fewer magnons in the case of Fig. 1(e), thus leading\nto a lower transition probability and a longer relaxation\ntime. More technical details about the second quantiza-\ntion of interactions between phonons and magnons can\nbe found in Ref. [41, 42].\nB. Damping constants in Y 3Fe5O12\nIn the field of spintronics, Y 3Fe5O12(yttrium iron gar-\nnet, YIG) has gained much attention due to its ultra-low\nmagnetic damping [5–7]. The unique property of this\nmaterial motivated us to investigate the intrinsic mecha-\nnism behind it. The crystal structure of YIG is presented\nin Fig. 3(a). There are totally 80 atoms in the primitive\ncell, of which 12 Fe ions are located in the center of oxy-\ngen tetrahedrons while the other 8 Fe ions are sited in\noxygen octahedrons. The magnetic ground state of YIG\nis illustrated in Fig. 3(b). The Fe ions situated in differ-\nent chemical environments contribute spins in opposite\ndirections, which makes YIG a typical ferrimagnetic ma-\nterial.TABLE I. The Heisenberg exchange coefficients J of YIG,\nwhere an effective spin S= 1 is adopted. For the FeO−FeO\npairs, the Greek letters ( α&β) refer to different chemical\nenvironments. All the results are calculated with the four-\nstate method.\nSpin Pair. Distance (Angst) J (meV)\n1NN FeT−FeO3.445 47.414\n1NN FeT−FeT3.774 2.399\n1NN FeO−FeO(α) 5.337 0.538\n1NN FeO−FeO(β) 5.337 5.055\n2NN FeT−FeO5.555 0.285\n2NN FeT−FeT5.765 3.437\nIn order to evaluate the Gilbert damping constants in\nYIG, our first step is to prepare an effective Hamilto-\nnian model. Considering the balance between precision\nand efficiency, the cutoff radius of interactions was set\nto be 11.0 Bohr for atomic pairs and 6.7 Bohr for 3-\nbody clusters. After symmetry analyses, we identified\n612 nonequivalent interactions in total, which included\n6 Heisenberg exchange terms and 9 spin-lattice coupling\nterms.\nTo determine the interaction parameters, we carried\nout a series of first-principles calculations, where a cu-\nbic cell was adopted to reduce the interference between\nadjacent cells caused by periodic boundary conditions.\nFollowing the settings in Ref. [43], we utilized the pro-\njector augmented-wave (PAW) method [44] and revised\nPerdew-Burke-Ernzerhof exchange-correlation functional\nfor solids (PBEsol) [45] in our calculations. Besides, the\nDFT+U method in its simplified form [46] was employed\nwhere the effective Hubbard U parameter was set to be\n4 eV for the 3 delectrons of Fe ions. In addition, a cutoff\nenergy of 520 eV for plane wave basis and a Γ-centered\n2×2×2 mesh of k-points were used in the DFT calcu-\nlations.\nIn Figure 2(c), we present the density of states (DOS)\nfor YIG. With a band gap of 1.863 eV, there is hardly\nany electric current occurring in the low temperature re-\ngion. Moreover, the Heisenberg exchange coefficients of\nYIG is listed in Table I. To verify the accuracy of these\nparameters, we conducted both Monte Carlo (MC) and\nASD simulations. The temperature dependence of aver-\nage magnetization is shown in Fig. 2(d), which reveals\nthe critical temperature of YIG to be 530 K. This result\nis slightly lower than the measured Curie temperature,\nTC= 560 K[5], but falls within our tolerance. The cal-\nculated results of coupling constants are provided in the\nsupplementary material.\nNext, we come to deal with the force constant tensor.\nIn order to demonstrate the impact of locality and val-\nidate the effectiveness of our optimization method, we\npresent some results pertaining to the tensor of YIG in\nTable II. Here we use “VASP” to tag the original tensor7\nTABLE II. The force constant tensor of YIG. The columns\nlabeled by A represent the sorted absolute values ofP\niKij,αβ\nand the columns labeled by B list the sorted eigenvalues of\nKij,αβ. For the cubic cell of YIG, we obtained the original\ntensor with the VASP package. Then, we eliminated the el-\nements that represent interactions beyond the cutoff radius.\nThis step was done by PASP. Finally, the tensor was modified\nto meet the requirement of translational symmetry through\nthe optimization formulated in (12).\nVASP PASP Modified\nNo. A B A B A B\n1 0.000 0.000 1.587 -0.102 0.000 0.000\n2 0.000 0.000 1.587 -0.102 0.000 0.000\n3 0.000 0.000 1.587 -0.102 0.000 0.000\n4 0.000 1.065 1.587 0.643 0.000 0.444\n5 0.000 1.065 1.587 0.643 0.000 0.444\n6 0.000 1.065 1.587 0.643 0.000 0.444\nobtained from DFT calculations, “PASP” to label the\nmodified tensor in which interactions beyond the cutoff\nradius are eliminated, and “Modified” to label the tensor\nafter optimization of independent variables. As shown in\nTable II, the “PASP” tensor violated the acoustic sum\nrule and was not positive semi-definite, whereas these is-\nsues were resolved for the “Modified” tensor. Although\nan obvious difference existed between the “PASP” and\n“Modified” tensor in terms of their eigenvalues, we still\nassumed the target system could be reasonably described\nby the “Modified” tensor and the validity of this assump-\ntion would be verified by the calculated results of damp-\ning constants. Additional details regarding the selection\nof tensor elements and the deviation of phonon spectra\nare provided in Fig. 3. According to figure 3(b) and 3(c),\nthe major deviation in phonon spectra resulted from the\nelimination of tensor elements, rather than the subse-\nquent modification.\nCompleting the preparation of Hamiltonian model, we\napplied the scheme proposed in Sec. II C to our first ob-\nject, Y 3Fe5O12. An instance is presented in Figure 4. We\nsetTL= 30K,TS= 180 Kfor the initial nonequilibrium\nstate and adopted an expanded supercell which contains\n12800 atoms in the simulation. Fig. 4(a) shows the time\nevolution of spin temperature in different types of simu-\nlations. By comparing the curves, we could roughly esti-\nmate that the equivalent damping constant in SLD simu-\nlation fell within the range of 10−3∼10−4. To make the\nestimation more precise, we calculated the initial cool-\ning rates ∂TS/∂t|t=0through polynomial (or exponen-\ntial) fittings and plotted them in Fig. 4(b). Afterwards,\na linear regression was performed to determine the quan-\ntitative relation between lg( −∂TS/∂t|t=0) and lg( α). As\nwe expected, the cooling rates in ASD simulations were\nproportional to the assigned damping constants. Then,\nwe combined the results of SLD and ASD simulations toevaluate the equivalent damping constant. This step was\naccomplished by identifying the intersection of red and\nblue lines in Figure 4(b). Finally, the damping constant\nwas determined to be αf= (2.87±0.13)×10−4in this\ncase. To verify our method and result, we present a com-\nparison between SLD and ASD (where we set α=αf)\nsimulations in Fig. 4(c). The curves agree well with each\nother in the initial stage but deviate in the second half.\nThis phenomenon is within our expectation, because in\nthe SLD simulation the lattice heats up as the spin cools\ndown, thereby slowing the energy transfer between two\nsubsystems.\nIn addition to the above case, we have measured the\nequivalent damping constants under different conditions\nto investigate the temperature dependence of magnetic\ndamping. The final results are summarized in Figure 5.\nDetails about the estimation of uncertainties are given in\nthe supplementary material. For Y 3Fe5O12, the damping\nconstants at different temperatures stay on the order of\n10−4, which is in good agreement with the experimental\nresults (3 .2×10−4[47], 2 .2×10−4[48], 1 .2–1.7×10−4\n[49]). For example, the damping constant in bulk YIG\nwas reported as 0 .4×10−4in Ref. [50]. Meanwhile, our\ncalculations yielded α= (2.8±0.3)×10−5at ∆T= 15\nK and α= (7.0±0.7)×10−5at ∆T= 30 K, where both\nTL= 0 K. Therefore, the experimental value corresponds\nroughly to the temperature region of ∆ T= 15∼30 K in\nour study. We believe such extent of thermal excitation\nis quite common in all kinds of spintronics experiments.\nMoreover, Fig. 5 indicates that αis approximately pro-\nportional to the temperature difference between subsys-\ntems. This outcome is also consistent with some com-\nputational works in the past [23, 25]. By comparing the\nsubfigures in Figure 5, we found that αhas little depen-\ndence on the lattice temperature, although here TLcould\nbe viewed to some extent as the ambient temperature of\nthe spin system.\nAs a supplement to Sec. III A, we further validate our\nsimulations by analyzing the measured cooling rates in\nFig. 5(a). By subtracting Eq. (23) from Eq. (22), the\ntransfer rate of energy between magnon and phonon sys-\ntems can be expressed as,\n˙Q=X\nqpℏωqp⟨˙Nqp⟩=X\nλ,qpTλ,qp (24)\nwhere Tλ,qpdenotes different transfer channels,\nTλ,qp∝(nλ−nλ−q)Nqp+nλ−qnλ+ 1 (25)\nAccording to the Bose–Einstein distribution, the number\nof magnons and phonons can be expressed as,\nnλ=1\neϵλ/kBTS−1, Nqp=1\neℏωqp/kBTL−1(26)\nWhen TSis high enough and TLis close to zero, we can\napproximate nλ=kBTS/ϵλ∝TSandNqpclose to zero.\nUnder these conditions, we have ˙Q∝T2\nS. This relation8\nFIG. 3. (a) The selection of force constant tensor elements for the cubic cell of YIG. An 160 ×160 zero-one matrix is used\nto show the result of selection, in which ’1’ denotes the interactions within cutoff radius and ’0’ represents the elements that\nare artificially eliminated. (b) The phonon spectrum calculated from the force constant tensor before and after the elimination\nof tensor elements. (c) The phonon spectrum calculated from the force constant tensor before and after the optimization of\nindependent variables.\nFIG. 4. (a) The time evolution of spin temperature in SLD and ASD simulations. The gray line represents the SLD simulation\nwhile the others refer to the ASD simulations with different damping constants. (b) The initial cooling rates ∂TS/∂t|t=0with\nrespect to the damping constants α, where the scaling of axis is set to be logarithm. The gray squares refer to the results of\nASD simulations and the blue line acts as the linear regression. The red circle is plotted by intersection of the blue line and\nthe horizontal red dash line, which represents the initial cooling rate in the SLD simulation. Then we can obtain the equivalent\ndamping constant from the abscissa of the red circle. (c) The comparison between ASD and SLD simulations. In the ASD\nsimulation, the Gilbert damping constant is set to be α= 2.87×10−4, which is exactly the result of our evaluation from the\nSLD simulation.\nFIG. 5. The temperature dependence of Gilbert damping constants for Y 3Fe5O12. The label of abscissa axis ∆ Trefers to\nTS−TLof the initial state in dynamical simulations. Measurements on the magnetic damping are performed under different\ninitial conditions of the lattice temperature: (a) TL= 0, (b) TL= 30K, (c)TL= 60K.9\nFIG. 6. The relation between damping constants αand spin-\nlattice coupling constants ∂Jij/∂uk,αin YIG. Through a lin-\near fitting, the slope is determined to be 2 .01, which agrees\nwell with our theoretical predictions.\nis well verified by linear regressions and the details are\nprovided in the supplementary material.\nFurthermore, the accuracy of our simulations can also\nbe proved from another perspective. According to Eqs.\n(22) and (23), the scattering rate Wgrows quadratically\nwith the coupling parameters Mλ,qp. Based on the theory\nof second quantization, Mλ,qpshall be proportional to\nthe coupling constants ∂Jij/∂uk,α. Therefore, under a\ndefinite condition of temperature, we have:\nα∝˙Q∝∆W∝M2\nλ,qp∝(∂Jij/∂uk,α)2(27)\nIn order to verify this relation, we adjusted the spin-\nlattice coupling constants of YIG coherently while keep-\ning the other model parameters unchanged. Then, SLD\nsimulations were carried out to evaluate the correspond-\ning damping constants. The result is plotted in Fig. 6,\nwhere the x-label “slcc” stands for the spin-lattice cou-\npling constants and the subscript “0” refers to the orig-\ninal situation. From a linear fitting, the slope is deter-\nmined to be 2 .01, which agrees well with our prediction.\nC. Damping constants in MnFe 2O4\nAfter the calculation on YIG, we applied our method\nto MnFe 2O4(MFO), which was reported to possess a\nlarge Gilbert damping constant in the literature [13, 51].\nAs shown in Fig. 7(a), MnFe 2O4has a typical structure\nof spinels, where A sites are surrounded by four oxygen\natoms and B sites are located in octahedrons. Generally,\nspinels can be classified into normal and inverse struc-\ntures according to the distribution of divalent and triva-\nlent cations between A/B sites. In experiments, MFO\nusually crystallizes into a mixed phase where the normal\nstructure occupies the major part (80% in bulk MFO\n[52]). Here, we only considered its normal structure in\nthis work. Also, the magnetic ground state of MFO is\nshown in Fig. 22(b), where the magnetic moments are\nantiparallel between A/B sites.\nFIG. 7. (a) The cubic cell of MnFe 2O4. The purple balls rep-\nresent manganese atoms, the golden balls refer to iron atoms,\nand the red balls stand for oxygen atoms. (b) The magnetic\nground state of MFO. The arrows of different colors repre-\nsent the spin directions of Mn and Fe atoms separately. (c)\nThe density of states obtained by DFT calculations. (d) The\ntemperature dependence of average magnetization measured\nin MC and ASD simulations. For MnFe 2O4, the phase tran-\nsition point from ferrimagnetic to paramagnetic lies in 730K\napproximately.\nFirstly, we started to construct an effective Hamilto-\nnian model for MFO. With the same cutoff settings for\nYIG, we found 105 nonequivalent interactions, including\n4 Heisenberg exchange terms and 10 spin-lattice coupling\nterms. Subsequently, DFT calculations were carried out\nto determine the interaction parameters. In these calcu-\nlations, we adopted a cubic cell containing 56 atoms and\na Γ-centered 4 ×4×4 grid mesh in the reciprocal space.\nBesides, UMn= 3.3 eV and UFe= 3.6 eV were used as the\neffective Hubbard parameters [52]. With the exception of\naforementioned settings, all the relevant first-principles\ncalculations were performed under the same conditions\nas in Sec. III B.\nThe DOS of MnFe 2O4is plotted in Fig. 7(c), yielding\na calculated band gap of 0.612 eV. This value does not\nmatch with the result of transport experiments, which re-\nported a much smaller band gap (0 .04–0.06 eV) [53]. In\naddition, MC and ASD simulations were performed using\nthe Heisenberg exchange coefficients listed in Table III.\nThe temperature dependence of average magnetization,\nshown in Fig. 7(d), suggests the critical temperature to\nbe around 730 K. This result is significantly higher than\nthe measured value of 573 K [54]. Both of the above dis-\ncrepancies may be attributed to the inevitable difference\nbetween the ideal normal spinel structure in calculations\nand the partially disordered samples in reality. Despite\nthis problem, we proceeded to describe the target system\nwith our Hamiltonian model and expected to see how far\nthe calculated results of damping constants would differ10\nTABLE III. The exchange coefficients J of MnFe 2O4, where\nan effective spin S= 1 is adopted.\nSpin Pair. Distance (Angst) J (meV)\n1NN Fe-Fe 3.003 6.835\n1NN Mn-Fe 3.521 33.224\n1NN Mn-Mn 3.667 3.956\n2NN Fe-Fe 5.201 0.929\nfrom experimental values.\nAfter the preparation of Hamiltonian model, we con-\nducted dynamics simulations to evaluate the equivalent\ndamping parameters in MFO at different temperatures.\nA supercell containing 13440 atoms was adopted in the\nsimulation, and the results are summarized in Fig. 10.\nThe average of calculated damping constants is around\n8×10−5, which is much smaller than the measured value,\n1.0×10−2[13, 51]. Two factors may account for this in-\nconsistency. Firstly, the inhomogeneity in real MnFe 2O4\nsamples greatly enhances the scattering of magnons and\nphonons, thereby increasing the damping constants. Ad-\nditionally, due to the narrow band gap observed in ex-\nperiments, eddy currents can arise at finite temperatures,\nwhich leads to a rapid loss of energy in the form of joule\nheat. As the result of these factors, we failed to obtain a\nreasonable estimation of Gilbert damping constants for\nMnFe 2O4with our methodology. On the other side, the\ncontribution of different relaxation mechanisms to FMR\nlinewidth has been studied comprehensively for MnFe 2O4\nin Ref. [53], which further confirms our analyses.\nD. Damping constants in Cr 2O3\nChromia (Cr 2O3) is a well-known collinear magneto-\nelectric antiferromagnet, which holds great prospects in\nthe field of spintronics [55–57]. As shown in Fig. 8(a),\nthe primitive cell of Cr 2O3contains 10 atoms, with each\nchromium atom bonded to the six oxygen atoms around\nit. Additionally, Fig. 8(b) displays the magnetic ground\nstate of Cr 2O3, where the spins of two nearest neighbor-\ning Cr atoms are oriented in opposite directions.\nAs a preliminary step in constructing the Hamiltonian\nmodel, we set the cutoff radius of interactions to be 11.0\nBohr for atomic pairs and 7.0 Bohr for 3-body clusters.\nThrough symmetry analyses, we identified 319 nonequiv-\nalent interactions, including 5 Heisenberg exchange terms\nand 21 spin-lattice coupling terms.\nAfterwards, a series of first-principles calculations were\nperformed to determine the model parameters. Following\nthe settings in Ref. [58], we adopted a hexagonal cell of\nCr2O3which contained a total of 90 atoms in the calcula-\ntions. Additionally, we used the LSDA+U method in its\nfull spherically symmetric form [59]. As to the Hubbard\nparameters, Jwas fixed at its recommended value of 0.6\nFIG. 8. (a) The primitive cell of Cr 2O3. The dark blue balls\nrepresent chromium atoms, and the red balls stand for oxygen\natoms. (b) The magnetic ground state. The arrows of differ-\nent colors represent the spin directions of Cr atoms. (c) The\ndensity of states obtained by DFT calculations. (d) The tem-\nperature dependence of sublattice magnetization measured in\nMC and ASD simulations. For Cr 2O3, the phase transition\npoint from ferrimagnetic to paramagnetic lies in 310K approx-\nimately.\nTABLE IV. The exchange coefficients J of Cr 2O3, in which\nan effective spin S= 1 is adopted.\nSpin Pair. Distance (Angst) J (meV)\n1NN Cr-Cr 2.640 44.778\n2NN Cr-Cr 2.873 29.269\n3NN Cr-Cr 3.411 -0.182\n4NN Cr-Cr 3.635 0.007\n5NN Cr-Cr 4.137 -0.500\neV, and Uwas adjusted to fit the N´ eel temperature ob-\nserved in experiments [60]. We found U= 2.0 eV was the\noptimal value for 3 delectrons of Cr ions. Except for the\nsettings specified above, all the DFT calculations were\nconducted under the same conditions as in Sec. III C.\nThe DOS of Cr 2O3is plotted in Fig. 8(c), which yields\na calculated band gap of 1.935 eV. This value indicates\nthat the energy dissipation of electric currents can be ne-\nglected in this system. Additionally, we list the Heisen-\nberg exchange coefficients of chromia in Table IV. Both\nMC and ASD simulations were performed to investigate\nthe temperature dependence of sublattice magnetization.\nAccording to Fig. 8(d), the critical point was determined\nto be 310 K approximately, which was quite consistent\nwith experimental observations. Also, the force constants\nof Cr 2O3went through the modification formulated in\nSec. II B, and the spin-lattice coupling parameters are\nprovided in the supplementary material.\nAfter the construction of Hamiltonian model, we con-\nducted a series of dynamics simulations to evaluate the11\nFIG. 9. (a) The 1NN FeT-FeOpair in Y 3Fe5O12. (b) The\n1NN Cr-Cr pair in Cr 2O3. The steel blue arrow stands for\nthe orientation of ∂J/∂u and the red number along with it\nrepresents the magnitude in unit of meV/Angst.\nequivalent damping parameters in Cr 2O3. An expanded\nhexagonal cell containing 14400 atoms was adopted for\nthe simulation, and the results are summarized in Fig. 11.\nAs two specific cases, our calculation yielded α= (1.31±\n0.14)×10−4at ∆T= 15 K and α= (2.7±0.3)×10−4\nat ∆T= 30 K, where both TL= 0 K. Therefore, the\ncalculated damping constants within ∆ T= 15∼30 K\nare quite close to 2 ×10−4, which is the estimated value\nreported in Ref. [61].\nFurthermore, the damping constants in Cr 2O3exhibit\na significant non-linear relation with the temperature dif-\nference of subsystems. Through logarithmic fittings, we\ncalculated the power exponents for Figures 11(a) to 11(c),\nand the results were 1.17, 1.62, 1.38. If we disregard the\ndifference between ∆ TandTfor the moment, these val-\nues are in good agreement with the theoretical prediction\nof Kasuya and LeCraw [26]. According to their study, the\nrelaxation rate varies as Tnwhere n= 1∼2 while n= 2\ncorresponds to a larger regime of temperatures.\nCompared to YIG, the greater magnetic damping ob-\nserved in chromia can be attributed to its significantly\nstronger spin-lattice coupling. As shown in Fig. 9, the\nmagnitude of principal spin-lattice coupling constant in\nCr2O3is two or three times larger than that in YIG. This\ncould be explained by the fact that direct exchange in-\nteraction between two magnetic atoms decreases rapidlywith their distance [62]. Therefore, owing to the shorter\ndistance of Cr-Cr pair, the direct exchange interaction\nbetween neighboring Cr atoms is believed to have a great\ncontribution to the spin-lattice coupling in Cr 2O3.\nIV. CONCLUSIONS\nIn summary, we propose a scheme to evaluate the con-\ntribution of spin-lattice coupling to the Gilbert damp-\ning in insulating magnetic materials. Our methodology\ninvolves first-principles based Hamiltonian models and\nspin-lattice dynamics simulations. Following a series of\nvalidations, we applied our method to three magnetic ma-\nterials, namely Y 3Fe5O12, MnFe 2O4and Cr 2O3. Their\ndamping constants were estimated separately, and the\nresults show that, in general, αis approximately propor-\ntional to the temperature difference between spin and\nlattice subsystems. Under the condition of ∆ T= 30\nK, the calculated damping constants are averaged to be\n0.8×10−4for YIG, 0 .2×10−4for MFO and 2 .2×10−4\nfor Cr 2O3. The results for YIG and Cr 2O3are in good\nagreement with experimental measurements, while the\ndiscrepancy for MFO can be attributed to the inhomo-\ngeneity and small band gap in real samples. 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Ebert\nDepartment of Chemistry/Phys. Chemistry, LMU Munich,\nButenandtstrasse 11, D-81377 Munich, Germany\n(Dated: May 30, 2018)\nThe modification of the magnetization dissipation or Gilber t damping caused by an inhomoge-\nneous magnetic structure and expressed in terms of a wave vec tor dependent tensor α(/vector q) is in-\nvestigated by means of linear response theory. A correspond ing expression for α(/vector q) in terms of\nthe electronic Green function has been developed giving in p articular the leading contributions to\nthe Gilbert damping linear and quadratic in q. Numerical results for realistic systems are pre-\nsented that have been obtained by implementing the scheme wi thin the framework of the fully\nrelativistic KKR (Korringa-Kohn-Rostoker) band structur e method. Using the multilayered system\n(Cu/Fe 1−xCox/Pt)nas an example for systems without inversion symmetry we demo nstrate the\noccurrence of non-vanishing linear contributions. For the alloy system bcc Fe 1−xCoxhaving inver-\nsion symmetry, on the other hand, only the quadratic contrib ution is non-zero. As it is shown, this\nquadratic contribution does not vanish even if the spin-orb it coupling is suppressed, i.e. it is a direct\nconsequence of the non-collinear spin configuration.\nPACS numbers: 71.15.-m,71.55.Ak, 75.30.Ds\nI. INTRODUCTION\nThe magnetization dissipation in magnetic materi-\nals is conventionally characterized by means of the\nGilbert damping (GD) tensor αthat enters the Landau-\nLifshitz-Gilbert (LLG) equation [1]. This positive-\ndefinite second-rank tensor depends in general on the\nmagnetization direction. It is well established that in\nthe case of spatially uniformly magnetized ferromagnetic\n(FM) metals two regimes of slow magnetization dynam-\nics can be distinguished, which are governed by differ-\nent mechanisms of dissipation [2–4]: a conductivity-like\nbehaviour occuring in the limiting case of ordered com-\npounds that may be connected to the Fermi breathing\nmechanism and a resistivity-likebehaviourshown by ma-\nterials with appreciable structural, chemical or tempera-\nture induced disorder and connected to a spin-flip scat-\nteringmechanism. Animportantissueisthatbothmech-\nanisms are determined by the spin-orbit coupling in the\nsystem (see e.g. [2, 4, 5]). During the last years, it was\ndemonstrated by variousauthors that first-principles cal-\nculationsforthe GD parameterforcollinearferromagntic\nmaterials allow to cover both regimes without use of any\nphenomenological parameters. In fact, in spite of the dif-\nferences concerning the formulation for the damping pa-\nrameter and the corresponding implementaion [6–8], the\nnumerical results are in generalin rathergood agreement\nwith each other as well as with experiment.\nIn the case of a pronounced non-collinear magnetic\ntexture, e.g. in the case of domain walls or topologi-\ncally nontrivial magnetic configurations like skyrmions,\nthe description of the magnetization dissipation assum-\ning a spatial-invariant tensor αis incomplete, and a non-\nlocal character of GD tensor in such systems has to be\ntaken into account [9–11]. This implies that the dissipa-\ntive torque on the magnetization should be representedby the expression of the following general form [12]:\nτGD= ˆm(/vector r,t)×/integraldisplay\nd3r′α(/vector r−/vector r′)∂\n∂tˆm(/vector r′,t).(1)\nIn the case of a magnetic texture varying slowly in space,\nhowever, an expansion of the damping parameter in\nterms of the magnetization density and its gradients [11]\nis nevertheless appropriate:\nαij=αij+αkl\nijmkml+αklp\nijmk∂\n∂rlmp(2)\n+αklpq\nij∂\n∂rkml∂\n∂rpmq+... ,\nwhere the first term αijstands for the conventional\nisotropic GD and the second term αkl\nijmkmlis associated\nwith the magneto-crystalline anisotropy (MCA). The\nthird so-called chiral term αklp\nijmk∂\n∂rlmpis non-vanishing\nin non-centrosymmetric systems. The important role of\nthis contribution to the damping was demonstrated ex-\nperimentally when investigating the field-driven domain\nwall(DW)motioninasymmetricPt/Co/Pttrilayers[13].\nAs an alternative to the expansion in Eq. (2) one can\ndiscuss the Fourier transform α(/vector q) of the damping pa-\nrametercharacterizinginhomogeneousmagneticsystems,\nwhich enter the spin dynamics equation\n∂\n∂t/vector m(/vector q) =−γ/vector m(/vector q)×/vectorH−/vector m(/vector q)×α(/vector q)∂\n∂t/vector m(/vector q).(3)\nIn this formulation the term linear in qis the first chiral\nterm appearing in the expansion of α(/vector q) in powers of q.\nFurthermore, it is important to note that it is directly\nconnected to the αklp\nijmk∂\n∂rlmpterm in Eq. (2).\nBy applying a gauge field theory, the origin of the\nnon-collinear corrections to the GD can be ascribed to\nthe emergent electromagnetic field created in the time-\ndependent magnetic texture [14, 15]. Such an emergent2\nelectromagneticfieldgivesrisetoaspincurrentwhosedi-\nvergence characterizes the change of the angular momen-\ntum in the system. This allows to discuss the impact of\nnon-collinearity on the GD via a spin-pumping formula-\ntion[9,14,16]. Somedetailsofthephysicsbehind thisef-\nfect depend on the specific propertiesofthe materialcon-\nsidered. Accordingly, different models for magnetisation\ndissipation were discussed in the literature [9, 12, 14, 17–\n19]. Non-centrosymmetric two-dimensional systems for\nwhich the Rashba-like spin-orbit coupling plays an im-\nportant role havereceived special interest in this context.\nThey have been discussed in particular by Akosa et al.\n[19], in order to explain the origin of chiral GD in the\npresence of a chiral magnetic structure.\nThe fourth term on the r.h.s. of Eq. (2) corresponds\nto a quadratic term of an expansion of α(/vector q) with re-\nspect to q. It was investigated for bulk systems with\nnon-magnetic [20] and magnetic [9] impurity atoms, for\nwhich the authors have shown on the basis of model con-\nsideration that it can give a significant correction to the\nhomogeneous GD in the case of weak metallic ferromag-\nnets. In striking contrast to the uniform part of the GD\nthis contribution does not require a non-vanishing spin-\norbit interaction.\nTo our knowledge, only very few ab-initio investiga-\ntions on the Gilbert damping in non-collinear magnetic\nsystems along the lines sketched above have been re-\nported so far in the literature. Yuan et al. [21] calcu-\nlated the in-plane and out-of-plane damping parameters\nin terms of the scattering matrix for permalloy in the\npresence of N´ eel and Bloch domain walls. Freimuth et\nal. [22], discuss the properties of a q-dependent Gilbert\ndamping α(/vector q) calculated for the one-dimensional Rashba\nmodelinthepresenceofthe N´ eel-typenon-collinearmag-\nnetic exchange field, demonstrating different GD for left-\nhanded and right-handed DWs. Here we extend the for-\nmalism developed before to deal with the GD in ferro-\nmagnets [6], to get access to non-collinear system. The\nformalism based on linear response theory allows to ex-\npand the GD parameters with respect to a modulation\nof the magnetization expressed in terms of a wave vector\n/vector q. Correspondingnumerical results will be presented and\ndiscussed.\nII. GILBERT DAMPING FOR\nNON-COLLINEAR MAGNETIZATION\nIn the following we focus on the intrinsic contribution\nto the Gilbert damping, excluding spin current induced\nmagnetizationdissipationwhich occursin the presenceof\nan external electric field. For the considerations on the\nmagnetization dissipation an adiabatic variation of the\nmagnetization in the time and space domain is assumed.\nMoreover, it is assumed that the magnitude of the local\nmagnetic moments is unchanged during a change of the\nmagnetization, i.e. the exchange field should be strong\nenough to separate transverse and longitudinal parts ofthe magnetic susceptibility. With these restrictions, the\nnon-local Gilbert damping can be determined in terms of\nthe spin susceptibility tensor\nχαβ(/vector q,ω) =i1\nV∞/integraldisplay\n0dt∝angbracketleftˆSα(/vector q,t)ˆSα(−/vector q,0)∝angbracketright0ei(ω−δ)t,(4)\nwhereˆSα(/vector q,t) is the /vector q- andt-dependent spin operator\nand reduced units havebeen used ( /planckover2pi1= 1). With this, the\nFourier transformationofthe real-spaceGilbert damping\ncan be represented by the expression [23, 24]\nααβ(/vector q) =γ\nM0Vlim\nω→0∂ℑ[χ−1]αβ(/vector q,ω)\n∂ω.(5)\nHereγ=gµBis the gyromagneticratio, M0=µtotµB/V\nis the equilibrium magnetization and Vis the volume of\nthe system. In order to avoid the calculation of the dy-\nnamical magnetic susceptibility tensor χ(/vector q,ω), which is\nthe Fourier transformed of the real space susceptibility\nχ(/vector r−/vector r′,ω), it is convenient to represent χ(/vector q,ω) in Eq.\n(5), in terms of a correlation function of time deriva-\ntives ofˆS. As˙ˆScorresponds to the torque /vectorT, that may\ninclude non-dissipative and dissipative parts, one may\nconsider instead the torque-torque correlation function\nπ(/vector q,ω) [24–27].\nAssuming the magnetization direction parallelto ˆ zone\nobtains the expression for the Gilbert damping α(/vector q)\nα(/vector q) =γ\nM0Vlim\nω→0∂ℑ[ǫ·π(/vector q,ω)·ǫ]\n∂ω. (6)\nwhereǫ=/bracketleftbigg\n0 1\n−1 0/bracketrightbigg\nis the transverse Levi-Civita tensor.\nThisimpliesthefollowingrelationshipofthe αtensorele-\nments with the elements of the torque-torque correlation\ntensorπ:αxx∼ −πyyandαyy∼ −πxx[24].\nUsing Kubo’s linear response theory in the Matsubara\nrepresentation and taking into account the translational\nsymmetry of a solid the torque-torque correlation func-\ntionπαβ(/vector q,ω) can be expressed by (see, e.g. [28]):\nπαβ(/vector q,iωn) =1\nβ/summationdisplay\npm∝angbracketleftTαG(/vectork+/vector q,iωn+ipm)\nTβG(/vectork,ipm)∝angbracketrightc,(7)\nwhereG(/vectork,ip) is the Matsubara Green function and ∝angbracketleft...∝angbracketrightc\nindicates a configurational average required in the pres-\nence of any disorder (chemical, structural or magnetic)\nin the system. Using a Lehman representation for the\nGreen function [28]\nG(/vectork,ipm) =/integraldisplay+∞\n−∞dE\nπℑG+(/vectork,E)\nipm−E(8)\nwithG+(/vectork,E) the retarded Green function and using the\nrelation\n1\nβ/summationdisplay\npm1\nipm+iωn−E11\nipm−E2=f(E2)−f(E1)\niωn+E2−E13\nfor the sum over the Matsubara poles in Eq. (7), the torque-torq ue correlation function is obtained as:\nπαβ(/vector q,iωn) =1\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE1\nπ+∞/integraldisplay\n−∞dE2\nπTr/angbracketleftbigg\nTαℑG(/vectork,E1)TβℑG(/vectork,E2)f(E2)−f(E1)\niωn+E2−E1/angbracketrightbigg\nc. (9)\nPerfoming finally the analytical continuation iωn→ω+iδone arrives at the expression\nΓαβ(/vector q,ω) =−π\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE1\nπ+∞/integraldisplay\n−∞dE2\nπTr/angbracketleftbigg\nTαℑG(/vectork+/vector q,E1)TβℑG(/vectork,E2)/angbracketrightbigg\nc(f(E2)−f(E1))δ(ω+E2−E1)\n=−π\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE\nπTr/angbracketleftbigg\nTαℑG(/vectork+/vector q,E)TβℑG(/vectork,E+ω)/angbracketrightbigg\nc(f(E)−f(E+ω)) (10)\nfor the imaginary part of the correlation function with Γ αβ(/vector q,ω) =−πℑπαβ(/vector q,ω). Accordingly one gets for the\ndiagonal elements of Gilbert damping tensor the expression\nααα(/vector q) =γ\nM0Vlim\nω→0∂[ǫ·Γ(/vector q,ω)·ǫ]\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nαα\n=γπ\nM0Vlim\nω→0∂\n∂ω1\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE\nπ2(f(E+ω)−f(E))Tr/angbracketleftbigg\nTβℑG(/vectork+/vector q,E)TβℑG(/vectork,E+ω)/angbracketrightbigg\nc\n=γ\nM0V1\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE\nπδ(E−EF)Tr/angbracketleftbigg\nTβℑG(/vectork+/vector q,E)TβℑG(/vectork,E)/angbracketrightbigg\nc\n=1\n4[ααα(/vector q,G+,G+)+ααα(/vector q,G−,G−)−ααα(/vector q,G+,G−)−ααα(/vector q,G−,G+)], (11)\nwhere the index βof the torque operator Tβis related to the index αaccording to Eq. 6, and the auxiliary functions\nααα(/vector q,G±,G±) =γ\nM0Vπ1\nΩBZ/integraldisplay\nd3kTr/angbracketleftbigg\nTβG±(/vectork+/vector q,EF)TβG±(/vectork,EF)/angbracketrightbigg\nc(12)\nexpressed in terms of the retarded and advanced Green function s,G+andG−, respectively.\nTo account properly for the impact of spin-orbit coupling when dealin g with Eqs. (11) and (12) a description of\nthe electronic structure based on the fully relativistic Dirac formalis m is used. Working within the framework of local\nspin density formalism (LSDA) this implies for the Hamiltonian the form [2 9]:\nˆHD=c/vectorα·/vector p+βmc2+V(/vector r)+β/vectorσ·ˆ/vector mBxc(/vector r). (13)\nHereαiandβare the standard Dirac matrices, /vectorσdenotes the vector of relativistic Pauli matrices, /vector pis the relativistic\nmomentum operator [30] and the functions V(/vector r) and/vectorBxc=/vectorσ·ˆ/vector mBxc(/vector r) are the spin-averaged and spin-dependent\nparts, respectively, of the LSDA potential [31] with ˆ/vector mgiving the orientation of the magnetisation.\nWith the Dirac Hamiltonian given by Eq. (13), the torque operator ma y be written as /vectorT=β[/vector σ׈/vector m]Bxc(/vector r).\nFurthermore, the Green functions entering Eqs. (11) and (12) a re determined using the spin-polarized relativistic\nversion of multiple scattering theory [29, 32] with the real space re presentation of the retarded Green function given\nby:\nG+(/vector r,/vector r′,E) =/summationdisplay\nΛΛ′Zn\nΛ(/vector r,E)τnm\nΛΛ′(E)Zm×\nΛ′(/vector r′,E)\n−δnm/summationdisplay\nΛ/bracketleftbig\nZn\nΛ(/vector r,E)Jn×\nΛ′(/vector r′,E)Θ(r′\nn−rn)\n+Jn\nΛ(/vector r,E)Zn×\nΛ′(/vector r′,E)Θ(rn−r′\nn)/bracketrightbig\n. (14)4\nHere/vector r,/vector r′refertoatomiccellscenteredatsites nandm, respectively,where Zn\nΛ(/vector r,E) =ZΛ(/vector rn,E) =ZΛ(/vector r−/vectorRn,E) isa\nfunction centered at the corresponding lattice vector /vectorRn. The four-component wave functions Zn\nΛ(/vector r,E) (Jn\nΛ(/vector r,E)) are\nregular (irregular) solutions to the single-site Dirac equation labeled by the combined quantum numbers Λ = ( κ,µ),\nwithκandµbeing the spin-orbit and magnetic quantum numbers [30]. Finally, τnm\nΛΛ′(E) is the so-called scattering\npath operator that transfers an electronic wave coming in at site minto a wave going out from site nwith all possible\nintermediate scattering events accounted for.\nUsing matrix notation with respect to Λ, this leads to the following exp ression for the auxilary damping parameters\nin Eq. (12):\nααα(/vector q,G±,G±) =γ\nM0Vπ1\nΩBZ/integraldisplay\nd3kTr/angbracketleftbigg\nTβτ(/vectork+/vector q,E±\nF)Tβτ(/vectork,E±\nF)/angbracketrightbigg\nc. (15)\nIn the case of a uniform magnetization, i.e. for q= 0 one obviously gets an expression for the Gilbert damping tensor\nas it was worked out before [7]. Assuming small wave vectors, the te rmτ(/vectork+/vector q,E±\nF) can be expanded w.r.t. /vector qleading\nto the series\nτ(/vectork+/vector q,EF) =τ(/vectork,E)+/summationdisplay\nµ∂τ(/vectork,E)\n∂kµqα+1\n2/summationdisplay\nµν∂τ(/vectork,E)\n∂kµ∂kνqµqν+... (16)\nthat results in a corresponding expansion for the Gilbert damping:\nα(/vector q) =α+/summationdisplay\nµαµqµ+1\n2/summationdisplay\nµναµνqµqν+... (17)\nwith the following expansion coefficients:\nα0±±\nαα=g\nπµtot1\nΩBZTrace/integraldisplay\nd3k/angbracketleftbigg\nTβτ(/vectork,E±\nF)Tβτ(/vectork,E±\nF)/angbracketrightbigg\nc(18)\nαµ±±\nαα=g\nπµtot1\nΩBZTrace/integraldisplay\nd3k/angbracketleftbigg\nTβ∂τ(/vectork,E±\nF)\n∂kµTβτ(/vectork,E±\nF)/angbracketrightbigg\nc(19)\nαµν±±\nαα=g\nπµtot1\n2ΩBZTrace/integraldisplay\nd3k/angbracketleftbigg\nTβ∂2τ(/vectork,E±\nF)\n∂kµ∂kνTβτ(/vectork,E±\nF)/angbracketrightbigg\nc, (20)\nand with the g-factor 2(1+ µorb/µspin) in terms of the spin and orbital moments, µspinandµorb, respectively, and the\ntotal magnetic moment µtot=µspin+µorb. The numerically cumbersome term in Eq. (20), that involves the sec ond\norder derivative of the matrix of /vectork-dependent scattering path operator τ(/vectork,E), can be reformulated by means of an\nintegration by parts:\n1\nΩBZ/integraldisplay\nd3kTβτ(/vectork,EF)Tβ∂2τ(/vectork,EF)\n∂kµ∂kν=/bracketleftBigg/integraldisplay /integraldisplay\ndkβdkγTi\nβτ(/vectork,E)Tj\nβ∂τ(/vectork,E)\n∂kβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleKα\n2\n−Kα\n2/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=0\n−/integraldisplay /integraldisplay /integraldisplay\ndkαdkβdkγTβ∂τ(/vectork,EF)\n∂kµTβ∂τ(/vectork,EF)\n∂kν/bracketrightBigg\n=−1\nΩBZ/integraldisplay\nd3kTβ∂τ(/vectork,EF)\n∂kµTβ∂τ(/vectork,EF)\n∂kν\nleading to the much more convenient expression:\nαµν±±\nαα=−g\n2πµtot/integraldisplay\nd3kTr/angbracketleftbigg\nTβ∂τ(/vectork,E±\nF)\n∂kµTβ∂τ(/vectork,E±\nF)\n∂kν/angbracketrightbigg\nc. (21)\nIII. RESULTS AND DISCUSSIONS\nThe scheme presented above to deal with the Gilbert\ndamping in non-collinear systems has been implementedwithin the SPR-KKR program package [33]. To exam-5\nine the importance of the chiral correction to the Gilbert\ndamping a first application of Eq. (19) has been made\nfor the multilayer system (Cu/Fe 1−xCox/Pt)nseen as a\nnon-centrosymmetricmodelsystem. Thecalculatedzero-\norder (uniform) GD parameter αxxand the correspond-\ning first-order (chiral) αx\nxxcorrection term for /vector q∝bardblˆxare\nplotted in Fig. 1 top and bottom, respectively, as a func-\ntion of the Fe concentration x. Both terms, αxxandαx\nxx,\n0 0.2 0.4 0.6 0.8 100.20.4αxx\n0 0.2 0.4 0.6 0.8 1xCo0123αxxx (a.u.)\nFIG. 1: The Gilbert damping parameters αxx(top) and\nαx\nxx(bottom) calculated for the model multilayer system\n(Cu/Fe 1−xCox/Pt)nusing Eqs. (18) and (19), respectively.\nincrease approaching the pure limits w.r.t. the Fe 1−xCox\nalloy subsystem. In the case of the uniform parame-\nterαxx, this increase is associated with the dominating\nbreathing Fermi-surface damping mechanism. This im-\nplies that the modification of the Fermi surface (FS) in-\nduced by the spin-orbit coupling (SOC) follows the mag-\nnetization direction that slowly varies with time. An ad-\nditional contribution to the GD, having a similar origin,\noccurs for the non-centrosymmertic systems with heli-\nmagnetic structure. In this case, the features of the elec-\ntronicstructure governedby the lackofinversionsymme-\ntry result in a FS modification dependent on the helicity\nof the magnetic structure. This implies a chiral contri-\nbution to the GD which can be associated with the term\nproportional to the gradient of the magnetization. Ob-\nviously, this additional modification of the FS and the\nassociated mechanism for the GD does not show up for\na uniform ferromagnet. As αis caused by the SOC one\ncan expect that it vanishes for vanishing SOC. This was\nindeed demonstrated before [5]. The same holds also for\nαxthat is cased by SOC as well.\nAnother system considered is the ferromagnetic alloy\nsystem bcc Fe 1−xCox. As this system has inversion sym-\nmetry the first-order term αµshould vanish. This expec-\ntation could also be confirmed by calculations that ac-count for the SOC. The next non-vanishing term of the\nexpansion of the GD is the term ∝q2. The correspond-\ning second-order term αxx\nxxis plotted in Fig. 2 (bottom)\ntogether with the zero-order term αxx(top). The bot-\n0 0.1 0.2 0.3 0.4 0.500.511.52αxx× 103\n0 0.1 0.2 0.3 0.4 0.5xCo012αxxxx ((a.u.)2)Fe1-xCox\nFIG. 2: The Gilbert damping terms αxx(top) and αxx\nxx(bot-\ntom) calculated for bcc Fe 1−xCox.\ntom panel shows in addition results for αxx\nxxthat have\nbeen obtained by calculations with the SOC suppressed.\nAs one notes the results for the full SOC and for SOC\nsuppressed are very close to each other. The small dif-\nference between the curves for that reason have to be as-\ncribed to the hybridization of the spin-up and spin-down\nsubsystems due to SOC. As discussed in the literature\n[9, 17, 20] a non-collinear magnetic texture has a corre-\nsponding consequence but a much stronger impact here.\nIn contrastto the GDin uniform FM systemswhereSOC\nisrequiredto breakthe totalspin conservationin the sys-\ntem,αxx\nxxis associated with the spin-pumping effect that\ncan be ascribed to an emergent electric field created in\nthe non-uniform magnetic system. In this case magnetic\ndissipation occurs due to the misalignment of the elec-\ntron spin following the dynamic magnetic profile and the\nmagnetization orientation at each atomic site, leading to\nthe dephasing of electron spins [16]\nIV. SUMMARY\nTo summarize, expressions for corrections to the GD\nofhomogeneoussystems werederived which areexpected\nto contribute in the case of non-collinear magnetic sys-\ntems. 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Ebert et al., The Munich\nSPR-KKR package , version 7.7,\nhttp://olymp.cup.uni-muenchen.de/ak/ebert/SPRKKR\n(2017), URL http://olymp.cup.uni-muenchen.de/ak/ebert/SPRKKR ." }, { "title": "1508.01497v1.On_the_spatial_scales_of_wave_heating_in_the_solar_chromosphere.pdf", "content": "arXiv:1508.01497v1 [astro-ph.SR] 6 Aug 2015Draftversion June4,2022\nPreprint typesetusingL ATEX styleemulateapjv. 5 /2/11\nON THESPATIALSCALES OF WAVEHEATINGINTHESOLARCHROMOSPHE RE\nRobertoSoler1,3, MarcCarbonell2,3, JoseLuisBallester1,3\n1Departament deF´ ısica, Universitat de les Illes Balears, E -07122, Palma deMallorca, Spain\n2Departament deMatem` atiques iInform` atica, Universitat deles Illes Balears, E-07122, Palma deMallorca, Spain and\n3Institute of Applied Computing &Community Code (IAC3),Universitat de les Illes Balears, E-07122, Palma deMallo rca, Spain\nDraftversion June 4,2022\nABSTRACT\nDissipationofmagnetohydrodynamic(MHD)waveenergyhasb eenproposedasaviableheatingmechanism\nin the solar chromosphericplasma. Here, we use a simplified o ne-dimensionalmodel of the chromosphereto\ntheoreticallyinvestigatethe physical processesand the s patial scales that are requiredfor the e fficient dissipa-\ntion of Alfv´ en waves and slow magnetoacoustic waves. We con sider the governing equations for a partially\nionized hydrogen-heliumplasma in the single-fluid MHD appr oximation and include realistic wave damping\nmechanismsthatmayoperateinthechromosphere,namelyOhm icandambipolarmagneticdi ffusion,viscosity,\nthermalconduction,andradiativelosses. Weperformanana lyticlocalstudyinthelimitofsmallamplitudesto\napproximatelyderivethe lengthscalesfor critical dampin gand efficient dissipation of MHD wave energy. We\nfind that the critical dissipation lengthscale for Alfv´ en w aves depends strongly on the magnetic field strength\nand ranges from 10 m to 1 km for realistic field strengths. The d amping of Alfv´ en waves is dominated by\nOhmic diffusion for weak magnetic field and low heights in the chromosph ere, and by ambipolar di ffusion\nfor strong magnetic field and medium /large heights in the chromosphere. Conversely, the damping of slow\nmagnetoacousticwaves is less e fficient, and spatial scales shorter than 10 m are requiredfor c ritical damping.\nThermal conduction and viscosity govern the damping of slow magnetoacoustic waves and play an equally\nimportant role at all heights. These results indicate that t he spatial scales at which strong wave heating may\nworkin thechromospherearecurrentlyunresolvedbyobserv ations.\nSubject headings: Sun: atmosphere — Sun: chromosphere — Sun: magnetic fields — S un: oscillations —\nwaves\n1.INTRODUCTION\nThe plasma heating of the upper solar atmosphere is one\nof the long-standingproblemsin solar physics. The physica l\nprocessesresponsibleforthetransportofenergyfromthe s o-\nlar interior and its dissipation in the atmospheric plasma a re\nunderintenseresearch(see,e.g.,Parnell&DeMoortel2012 ).\nOneof themechanismsthathasbeenproposedtoexplainthe\ntransport and dissipation of energy involves the propagati on\nand damping of magnetohydrodynamic (MHD) waves (see\nthe recent review by Arregui 2015, and references therein).\nIndeed,observationsindicatethatMHDwavesareubiquitou s\nin the solar atmosphere and can have the required energy to\nheat the upper layers (see, e.g., Cargill& deMoortel 2011;\nMcIntoshet al. 2011; Hahn& Savin 2014; Jesset al. 2015).\nWhile the overwhelming presence of the waves is demon-\nstratedbytheobservations,thephysicsbehindthedamping of\nthe waves and the deposition of wave energy into the plasma\nremainspoorlyknown.\nIn this paper, we perform a theoretical study on the condi-\ntionsforefficientdissipationofMHDwaveenergyinthesolar\nchromosphere. The realistic modelling of the physical pro-\ncesses in the chromospheric plasma is challenging. Because\nof the relatively low temperature, the chromosphericplasm a\nis only partially ionized and neutrals are predominant at lo w\nand medium altitudes. It has been shown that partial ioniza-\ntion effects have a strong impact on chromospheric dynam-\nics(see,e.g.,Mart´ ınez-Sykoraet al.2012;Leakeetal.20 14).\nIon-neutralcollisionsmayplaya crucialroleinthereleas eof\nmagnetic energy in the form of heat (Khomenko&Collados\n2012). Therefore, the consideration of partial ionization is a\nroberto.soler@uib.esunavoidablerequisiteforthe realistic descriptionof the chro-\nmosphericphysics.\nThe role of partial ionization on the damping of Alfv´ en\nwaves has been investigated in detail in the literature. Es-\ntimations of the damping rate due to ion-neutral collisions\nas a function of height in the chromosphere indicate that\nthe damping is most e fficient for waves with high fre-\nquencies closer to the local ion-neutral collision frequen cy\n(see, e.g., De Pontieuet al. 2001; Khodachenkoetal. 2004;\nLeakeet al. 2005; Soleret al. 2012a, among others). Re-\ncently, Soleret al. (2015) investigated in detail this phe-\nnomenon and showed that high-frequencyAlfv´ en waves can\nbecriticallydamped,i.e.,overdamped,becauseofion-neu tral\ncollisions. The energy carried by these overdamped waves\ncould be efficiently deposited in the plasma as a result of\nthe strong dissipation. In fact, previous computations of t he\nplasmaheatingrateobtainedfromnumericalsimulations(s ee\nGoodman 2011; Song& Vasyli¯ unas 2011; Tu &Song 2013;\nRussell &Fletcher 2013) showed that dissipation of Alfv´ en\nwave energy can provide a sustained heating over time that\nissufficienttocompensatethechromosphericradiativelosses\nat low altitudes. So, it is well established that partial ion iza-\ntioneffectsareimportantforthecorrectstudyofAlfv´ enwave\ndampingandassociatedheatinginthechromosphere.\nConcerning magnetoacoustic waves, Khodachenkoet al.\n(2004, 2006) obtained that thermal conduction and viscos-\nity can be important in their damping. Also, results from,\ne.g., Porteretal. (1994) and Carbonelletal. (2004) sugges t\nthat the effect of radiative losses should be taken into ac-\ncount depending on the plasma physical conditions. Con-\nversely, ion-neutral collisions may play a relatively less im-\nportant role in the direct damping of magnetoacoustic waves2\n(see Fortezaet al. 2007). However, partial ionization shou ld\nbeappropriatelyaccountedfor,since thepresenceofneutr als\nin the plasma modifies the e fficiency of thermal conduction\nand viscosity. Therefore,as discussed by Khodachenkoetal .\n(2006), the correct and complete description of the damping\nof the various types of MHD waves in the solar atmosphere\ningeneral,andinthechromosphereinparticular,requires the\nconsideration of all energy dissipation mechanisms, inclu d-\ningthermal,collisional,andfrictionale ffects. Weaddthatthe\ncorrectandcompletedescriptionofthedampingalsorequir es\ntheconsiderationofallthecomponentsoftheplasma,namel y\nthevariousionizedandneutralspecies.\nHereweareconcernedaboutthephysicalmechanismsthat\nare actually relevant in damping the waves and about the\nspatial scales that are required for the chromospheric damp -\ning to be efficient. We introduce the concept of ‘critical\ndissipation lengthscale’, which is related to the occurren ce\nofwave cutoffs. The existence of cuto ffs caused by the\nstrongdampingofthewavesiswelldocumentedinthelitera-\nture (see, e.g., Kulsrud&Pearce 1969; Mouschovias 1987;\nKamaya& Nishi 1998; Zaqarashvilietal. 2012; Soleretal.\n2013a,b;Vranjes&Kono2014;Soleret al.2015,amongoth-\ners). Thephysicalnatureofthewavecuto ffsresidesinthefact\nthat the damping is so strong that the wave restoring force is\neffectively suppressed (see a more extensive explanation in\nSoleret al.2013b). Asaresult,waveperturbationsdecayin a\ntimescale much shorter than the wave period, a phenomenon\ncalled critical damping or overdamping. For practical pur-\nposes, this means that the waves are unable to propagateand\nsotheycannottransportenergyawayfromthechromospheric\nmedium. All the energy initially stored in the wave pertur-\nbations is eventually deposited in the plasma in the form of\nheat. Therefore, overdamped waves are strong candidates to\nbe connected with e fficient plasma heating in the chromo-\nsphere(Soleret al.2015). Wavecuto ffsduetocriticaldamp-\ning are physicallyand conceptuallydi fferentfrom the cuto ffs\ndueto plasmastratificationstudiedin manypreviousworks.\nThe purposeof this paper is twofold. On the one hand, we\naimtoperformacomprehensivetheoreticalstudyofthephys -\nical mechanisms capable of producing the cuto ffof Alfv´ en\nandmagnetoacousticwavesunderchromosphericconditions .\nIt is common among theoretical works that investigate wave\ndamping to focus on the influence of one specific damping\nmechanismandignoreothere ffects. We areinterestedinpro-\nviding a consistent description of all relevant physical pr o-\ncesses that may be involvedin strong wave damping. On the\nother hand, we aim to obtain the critical dissipation length -\nscales for the various types of MHD waves. The value of\nthecriticaldissipationlengthscaleisrelevantbecausei tdeter-\nmines the spatial scales needed for wave heating to become\nefficient. Strong plasma heating produced by dissipation of\nMHDwaveenergywouldnecessarilyrequirespatialscalesof\ntheorderofthecriticaldissipationlengthscale.\nAnother important question from the theoretical point of\nview is the role of helium in the damping of the waves. The\nmajority of previous works that investigated wave damping\nignoredtheinfluenceofheliumandassumedachromospheric\nplasma composed of hydrogen alone. However, it has been\nshownthatthepresenceofneutralandionizedheliumcanen-\nhance the damping of Alfv´ en waves due to ion-neutral colli-\nsions(Zaqarashviliet al.2013). Asecondaryobjectiveoft his\nwork is to determine the impact of helium on the e fficiency\nofthedampingmechanismsand,consequently,onthecritica l\ndissipationlengthscales.This paper is organized as follows. Section 2 contains the\ndescriptionofthechromosphericmodel,thewavedissipati on\nmechanisms, and the basic equations governingwave propa-\ngation and damping. The critical dissipation lengthscales for\nAlfv´ en waves and slow magnetoacoustic waves in the chro-\nmosphericmodelareinvestigatedinSections3and4,respec -\ntively. Later, the results of this article are discussed in S ec-\ntion 5 and some concluding remarks are given in Section 6.\nFinally, the effect of dissipation on the nonlinear coupling\nof Alfv´ en waves and slow waves is briefly explored in Ap-\npendixA.\n2.BASIC EQUATIONS\n2.1.Partiallyionizedchromosphericmodel\nWe adopt a one-dimensional, static, gravitationally strat i-\nfiedmodelforthechromospherebasedonthesemi-empirical\nmodel F of Fontenlaet al. (1993), hereafter the FAL93-F\nmodel. The reason for choosing the FAL93-F model in-\nstead of more recent models is that the models tabulated in\nFontenlaetal. (1993) explicitly provide the variation of t he\nneutral and ionized helium densities with height. The same\nmodel was used in the previous work by Soleret al. (2015),\nbuttheinfluenceofheliumwasnotconsideredthere.\nWe treat the chromospheric medium as a partially ion-\nized hydrogen-helium plasma composed of electrons, pro-\ntons,neutralhydrogen,neutralhelium,andsinglyionized he-\nlium. Hereafter, subscripts e, p, H, He I, and He IIexplicitly\ndenote these species, respectively. The presence of doubly\nionized helium, He III, and that of heavier species is ignored\nbecause of their negligible abundance in the chromosphere.\nWedefinethefractionofspecies β=e,p,H,He I,orHeIIas\nξβ=ρβ\nρ, (1)\nwhereρβ=mβnβis the mass density of species β, withmβ\nandnβthe mass particle and number density, and ρ=/summationtext\nβρβ\nis the total mass density. We note that ξe≈0 owing to the\nvery small electron mass. We assume a strong thermal cou-\nplingandusethesametemperature, T,forallthespecies. Fig-\nure1(a)–(c)showsthedependenceonheight, h,ofthetemper-\nature,totaldensity,andfractionofthevariousspecies,r espec-\ntively. Thesharpchromosphere-to-coronatransitionregi onis\nlocatedat h≈2,000km,wherethetemperatureincreasesand\nthedensitydecreasesabruptly. Hydrogenismostlyneutral for\nh/lessorsimilar1,500 km, becomes to be ionized for h/greaterorsimilar1,500 km and\nis fully ionizedfor h/greaterorsimilar2,000 km. We note that there is a rel-\nativelylargeabundanceof neutralheliumat largealtitude sin\nthechromospherebecausethetemperatureisnothighenough\ntofullyionizehelium.\nThe variousspecies in the plasma exchangemomentumby\nmeans of particle collisions. The strength of the interacti on\ndepends on the so-called friction coe fficient,α. The friction\ncoefficient for collisions between two charged species q=e,\np, or He IIandq′=e, p, or He II, is (e.g., Spitzer 1962;\nBraginskii1965)\nαqq′=nqnq′e4lnΛqq′\n6π√\n2πǫ2\n0mqq′/parenleftig\nkBT/mqq′/parenrightig3/2, (2)\nwheremqq′=mqmq′//parenleftig\nmq+mq′/parenrightig\nis the reducedmass, eis the\nelectron charge, kBis Boltzmann’s constant, ǫ0is the permit-\ntivity of free space, and ln Λqq′is Coulomb’slogarithmgivenWave heatingscalesinthe chromosphere 3\nby(e.g.,Spitzer1962;Vranjes& Krstic2013)\nlnΛββ′=ln24πǫ3/2\n0k3/2\nBT3/2\ne3√nβ+nβ′. (3)\nIn turn, the friction coe fficient for collisions between a\ncharged or neutral species β=e, p, H, He I, or He II, and\na neutral species n =H or He Iis (e.g., Braginskii 1965;\nChapman&Cowling1970)\nαβn=nβnnmβn/bracketleftigg8kBT\nπmβn/bracketrightigg1/2\nσβn, (4)\nwhereσβnisthecollisioncrosssection. Recentestimationsof\nthisparametercanbefoundinVranjes&Krstic(2013)andan\nextensive discussion on the importance of its value for wave\ndampingisgivenin Soleretal. (2015).\nThe expressionsof the friction coe fficientsgivenaboveare\nalso valid for self-collisions, i.e., collisions between p articles\nofthe samespecies. Thetotal frictioncoe fficientofspeciesβ\nwiththeotherspeciesis\nαβ=/summationdisplay\nβ′/nequalβαββ′, (5)\nwhile the total friction coe fficient of speciesβincludingself-\ncollisionsisαβ,tot=αβ+αββ.\n2.2.Single-fluidMHD descriptionof plasmadynamics\nWe study the dynamics of the partially ionized chromo-\nsphericplasmawithintheframeworkofthesingle-fluidMHD\napproximation. The single-fluid MHD approximation as-\nsumes a strong coupling between ions, electrons, and neu-\ntralssothatallthespeciese ffectivelybehaveasonefluid(see\nBraginskii 1965). This approximation is valid in the chro-\nmosphere as long as the wave frequencies remain lower than\nthe ion-neutral collision frequency of the plasma and the\nwavelengths remain longer than the mean free path of par-\nticles between collisions (see, e.g., Zaqarashviliet al. 2 011,\n2013; Soleret al. 2013b). In this approximation, the basic\nMHD equations are written in terms of average quantities,\nwhile the effects of the interactions, i.e., collisions, between\nthe variousspecies remain in the formof nonidealterms(see\ndetails in, e.g., Fortezaet al. 2007; Meier&Shumlak 2012;\nKhomenkoetal. 2014). The basic single-fluid MHD equa-\ntionsusedinthisworkare\nDρ\nDt=−ρ∇·v, (6)\nρDv\nDt=−∇p−∇·ˆπ+ρg+1\nµ(∇×B)×B,(7)\n∂B\n∂t=∇×(v×B)−∇×(η∇×B)\n+∇×{ηA[(∇×B)×B]×B}, (8)\nDp\nDt=−γp∇·v+(γ−1)L, (9)\np=ρRT\n˜µ, (10)\nwith ˜µthe mean atomic weight of a hydrogen-heliumplasma\ngivenby\n˜µ=/parenleftigg\n2ξp+ξH+1\n4ξHei+1\n2ξHeii/parenrightigg−1\n.(11)Equations (6)–(10) are the continuity equation, momentum\nequation, induction equation, energy equation, and the equ a-\ntion of state, respectively. In these equations,D\nDt=∂\n∂t+v·∇\ndenotesthe material or total derivative, pis the thermal pres-\nsure,vis the velocity vector, Bis the magnetic field vec-\ntor,gis the acceleration of gravity, γis the adiabatic index,\nµthe magnetic permeability, ˆ πis the viscosity tensor, ηand\nηAare the coefficients of Ohmic and ambipolar di ffusion, re-\nspectively,Lrepresents the net e ffect of all the sources and\nsinks of energy,and Ris the ideal gas constant. We note that\nHall’s term is ignored in the induction equation due to its ir -\nrelevant role in damping the waves (Zaqarashviliet al. 2012 ;\nSoleretal. 2015).\n2.3.Dissipationmechanisms\nHere we discuss the importance of the various dissipation\nmechanismsthatappearinthe governingEquations(6)–(10) .\nEquation (8) contains two magnetic di ffusion terms. The\nfirsttermontheright-handsideofEquation(8)isOhmicdif-\nfusion,whichiscausedbyelectroncollisions. Thecoe fficient\nofOhmicdiffusionisgivenby\nη=αe\nµe2n2e. (12)\nEquation (12) corresponds to the classic Spitzer’s coe fficient\nofmagneticdiffusion(Spitzer1962). The secondtermonthe\nright-hand side of Equation (8) is ambipolar di ffusion. This\nterm contains the e ffect of ion-neutral collisions. The coe ffi-\ncient of ambipolar di ffusion in a hydrogen-helium plasma is\ngivenby(see Zaqarashviliet al. 2013)\nηA=ξ2\nHαHei+ξ2\nHeiαH+2ξHξHeiαHHei\nµ/parenleftig\nαHαHei−α2\nHHei/parenrightig.(13)\nFigure1(d)displaystheratio |B|2ηA/ηasafunctionofheight\nin the chromospheric model for three di fferent values of the\nmagnetic field strength. We see that ambipolar di ffusion is\ndominantthroughoutthechromosphereexceptatlowheights ,\nwhere Ohmic diffusion can be of importance for weak mag-\nnetic fields. Therefore, we shall consider both Ohmic and\nambipolardiffusioninourcomputations.\nViscosity in a partially ionized plasma is essentially dete r-\nmined by self-collisions of ions and neutrals. Electron vis -\ncosity can be safely neglected by virtue of the small electro n\nmass(see,e.g.,Braginskii1965;Meier& Shumlak2012). On\nthe one hand, we assume the neutral viscosity tensor, ˆ πn, to\nbe isotropic because of the absence of the direct e ffect of\nthe magnetic field on neutrals. Isotropy of neutral viscosit y\nmightbealteredbyion-neutralcollisions(seeVranjes201 4),\nbut this effect is usually neglected (Meier&Shumlak 2012).\nThus,theneutralviscositytensorusedhereis\nˆπn=−ζn/parenleftigg\n∇v+(∇v)⊤−ˆI2\n3∇·v/parenrightigg\n, (14)\nwhereˆIistheidentitytensorand ζnisthecoefficientofneutral\nviscositygivenby\nζn=mHn2\nH\nαH,tot+mHein2\nHei\nαHei,totkBT, (15)\nwherebothneutralhydrogenandneutralheliumareincluded .\nOn the other hand, the ion viscosity tensor, ˆ πi, has a compli-\ncated form in the presence of a magnetic field. It is usually4\nFigure1. Variation ofphysicalparameters withheightabovethephot osphereaccording tothechromospheric modelFofFontenla e t al.(1993): (a)temperature,\n(b) total density, (c) fraction of species, (d) ratio of ambi polar to Ohmic diffusivities for three di fferent magnetic field strengths, (e) ratio of neutral viscosi ty to\nion parallel viscosity, and (f) ratio of neutral thermal con ductivity to electron parallel thermal conductivity.\ndescribed as the sum of five components accounting for par-\nallel (or compressive) viscosity, perpendicular(or shear ) vis-\ncosity,andgyroviscosity(seethefullexpressioninBragi nskii\n1965). In the magnetizedchromosphericplasma parallel vis -\ncosityisseveralordersofmagnitudelargerthanbothperpe n-\ndicular viscosity and gyroviscosity. Then, we only conside r\ntheionparallelviscositycomponent,namely\nˆπi≈−3ζi,/ba∇dbl/parenleftigg\nbb−1\n3ˆI/parenrightigg/parenleftigg\nbb−1\n3ˆI/parenrightigg\n:∇v,(16)whereb=B/|B|istheunitvectorinthedirectionofthemag-\nnetic field andζi,/ba∇dblis the coefficient of ion parallel viscosity\ngivenby\nζi,/ba∇dbl=0.96mpn2\np\nαp,tot+mHeiin2\nHeii\nαHeii,totkBT,(17)\nwhere both protons and singly ionized helium are included.\nFigure1(e)displaysthe ratio ζn/ζi,/ba∇dblasa functionof heightin\nthe chromosphericmodel. We clearly see that neutral viscos -\nity is several orders of magnitude more important than ionWave heatingscalesinthe chromosphere 5\nviscosity throughout the chromosphere. Because of this re-\nsult, we are allowedto completelyneglection viscositycom -\nparedtoneutralviscosity. Weshallapproximate ˆ π≈ˆπninour\ncomputations.\nTheeffectofallthesourcesandsinksofenergyisincluded\ninthefunctionLintheright-handsideoftheenergyequation\n(Equation(9)). Theexpressionof Lis\nL=−∇·q−L(ρ,T)+j·E∗−ˆπ:∇v+H,(18)\nwhere the various terms on the right-hand side are: the di-\nvergence of heat flux due to thermal conduction q=−κ∇T,\nwithκthe thermal conductivity tensor; the radiative loss\nfunctionL(ρ,T); the generalized Joule heating j·E∗, with\nj=(∇×B)/µthe current density and E∗=E+v×Bthe\neffective electric field; the viscousheating ˆ π:∇v; and an ad-\nditionalandunspecifiedsourceofheating H.\nAs happens for viscosity, thermal conduction in a magne-\ntizedplasmaisstronglyanisotropic. Forconvenience,wes plit\nthe thermal conductivity tensor into its components parall el,\nκ/ba∇dbl, andperpendicular, κ⊥,to the magneticfield direction. Ina\nfullyionizedplasma κ/ba∇dblisessentiallydeterminedbyelectrons,\nwhileκ⊥ismainlyduetoionsandisnegligible. Inapartially\nionizedplasmatheisotropic thermalconductivityofneutr als\nhas to be added to both parallel and perpendicular compo-\nnents. Thus, the parallel and perpendicular components of\nthermal conductivity are approximated by κ/ba∇dbl≈κe,/ba∇dbl+κnand\nκ⊥≈κn, whereκe,/ba∇dblandκnare the parallel electron thermal\nconductivity and the total neutral thermal conductivity, r e-\nspectively,givenby\nκe,/ba∇dbl=3.2n2\nek2\nBT\nαe,tot, (19)\nκn=5\n3n2\nH\nαH,tot+n2\nHei\nαHe,totk2\nBT. (20)\nFigure1(e)displaystheratio κn/κe,/ba∇dblasafunctionofheightin\nthe chromospheric model. The neutral thermal conductivity\nis foundto be more importantthan the electron thermal con-\nductivity except at large altitudes, where both thermal con -\nductivitiesare of the same orderof magnitude. This is so be-\ncause hydrogenis largely ionized at those large altitudes, but\nhelium remains mostly neutral. Hence, we shall retain both\nneutral and electron thermal conductivities in the computa -\ntionsandexpressthedivergenceofthe heatfluxas\n∇·q=−B·∇/parenleftiggκe,/ba∇dbl\n|B|2B·∇T/parenrightigg\n−∇·(κn∇T).(21)\nThe radiative loss function, L(ρ,T), accounts for plasma\ncooling owing to radiative losses. Determining the chromo-\nspheric radiative losses as function of temperature and den -\nsity is a difficult task that requirescomplicatednumerical so-\nlutions of the radiative transfer problem. The radiative lo ss\nrate depends, e.g., on the completeness of the atomic model\nusedforthecalculation,ontheatomicprocessesincluded, on\nthe ionization equilibrium, and element abundance assumed ,\namong other factors. The full solution of the radiative tran s-\nfer problem is beyond the purpose and scope of the present\npaper. Here we follow a frequent alternative approach to ac-\ncount for the solar plasma radiative losses based on a semi-\nempirical parametrization of the radiative loss function ( see,\ne.g., Hildner 1974; Rosneretal. 1978; Klimchuk&Cargill\n2001;Schureetal.2009). Thismethodenablesustoincorpo-\nrate radiative losses in an approximate way without the needof solving the full radiative transfer problem. The inconve -\nnience of this approach is that the semi-empirical radiativ e\nloss function is obtained under the assumption of optically\nthinplasma,whilethecoolchromosphericplasmasofintere st\nhere do not completely satisfy this condition. Owing to fi-\nniteopticalthickness,theactualradiativelossesofthep lasma\nwould be probably reduced in some degree compared to the\noptically thin proxy. This fact should be taken into account\nwhen interpreting the e ffect of radiative losses on the results.\nTheexpressionoftheradiativelossfunctionwe useis\nL(ρ,T)=ρ2χ∗Tα, (22)\nwhereχ∗andαarepiecewiseconstantsdependingofthetem-\nperature. We usetheparametrizationof χ∗andαgiveninTa-\nble 1 of Soleretal. (2012b), which are obtained from up-to-\ndate computations of radiative losses derived from the CHI-\nANTI atomic database (Dereetal. 1997; Landiet al. 2012)\nassuming typical abundances in the solar atmosphere. The\nreader is referred to Parentiet al. (2006) and Parenti&Vial\n(2007) fordetails.\nThegeneralizedJouleterm j·E∗takesintoaccountplasma\nheating because of dissipation of electric currents. The ex -\npressionof j·E∗is\nj·E∗=µη/vextendsingle/vextendsingle/vextendsinglej/ba∇dbl/vextendsingle/vextendsingle/vextendsingle2+µηC|j⊥|2, (23)\nwhereηCistheso-calledCowling’sdi ffusivitygivenby\nηC=η+|B|2ηA, (24)\nandj/ba∇dblandj⊥arethecomponentsofthecurrentdensityparal-\nlel and perpendicular to the magnetic field, respectively, t hat\ncanbecomputedas\nj/ba∇dbl=1\nµ[(∇×B)·b]b, (25)\nj⊥=1\nµb×[(∇×B)×b]. (26)\nThus, Ohmic magnetic di ffusion is caused by the dissi-\npation of parallel currents, while Cowling’s magnetic dif-\nfusion, i.e., the joint e ffect of Ohmic and ambipolar di ffu-\nsion, is caused by the dissipation of perpendicular current s\n(Khomenko&Collados2012). Asexplainedbefore,weshall\nretain both Ohmic and ambipolar di ffusion in the following\ncomputations.\n2.4.Equationsforone-dimensionalwavepropagation\nWe reducetheset of basicequationsto the case oftheone-\ndimensional chromospheric model. We use a Cartesian co-\nordinate system and assume that the plasma properties vary\nalong the x-direction only, whereas yandzare ignorable di-\nrections. Therefore, the x-direction corresponds to the ver-\ntical direction, so that gravity is oriented in the negative x-\ndirection, namely g=(−g,0,0), withgthe acceleration of\ngravityatthesolarsurface. Wecanconvenientlyrotatethe co-\nordinatesystemfor vandBtolieonthe xy-planewithnoloss\nof generality. We redefine the xandydirectionsas the paral-\nlel,/ba∇dbl, and perpendicular, ⊥, directions, respectively. Hence,\nv=(v/ba∇dbl,v⊥,0) andB=(B/ba∇dbl,B⊥,0). The solenoidal condi-\ntion∇·B=0 imposes that∂B/ba∇dbl/∂x=0, while from the x-\ncomponentof Equation(8) we get ∂B/ba∇dbl/∂t=0. Therefore, B/ba∇dbl\nis a constant in both space and time. In this one-dimensional6\ncase,Equations(6)–(9)become\n∂ρ\n∂t=−∂\n∂x/parenleftbigρv/ba∇dbl/parenrightbig, (27)\n∂v/ba∇dbl\n∂t=−v/ba∇dbl∂v/ba∇dbl\n∂x−1\nρ∂p\n∂x+1\nρ∂\n∂x/parenleftigg4ζn\n3∂v/ba∇dbl\n∂x/parenrightigg\n−g\n−B⊥\nµρ∂B⊥\n∂x, (28)\n∂v⊥\n∂t=−v/ba∇dbl∂v⊥\n∂x+1\nρ∂\n∂x/parenleftigg\nζn∂v⊥\n∂x/parenrightigg\n+B/ba∇dbl\nµρ∂B⊥\n∂x, (29)\n∂B⊥\n∂t=B/ba∇dbl∂v⊥\n∂x−∂\n∂x/parenleftbigB⊥v/ba∇dbl/parenrightbig+∂\n∂x/parenleftigg\nηC∂B⊥\n∂x/parenrightigg\n,(30)\n∂p\n∂t=−v/ba∇dbl∂p\n∂x−γp∂v/ba∇dbl\n∂x+(γ−1)1\nµηC/parenleftigg∂B⊥\n∂x/parenrightigg2\n+∂\n∂xB2\n/ba∇dblκe,/ba∇dbl\nB2\n/ba∇dbl+B2\n⊥∂T\n∂x+∂\n∂x/parenleftigg\nκn∂T\n∂x/parenrightigg\n−L(ρ,T)\n+4ζn\n3/parenleftigg∂v/ba∇dbl\n∂x/parenrightigg2\n+ζn/parenleftigg∂v⊥\n∂x/parenrightigg2\n+H. (31)\nIn turn, the variations of temperature are related to those o f\npressureanddensityby\n1\nT∂T\n∂t=1\np∂p\n∂t−1\nρ∂ρ\n∂t. (32)\nLetusconsiderthestaticcasesothatweset v/ba∇dbl=v⊥=B⊥=\n0and∂/∂t=0. FromEquation(28)we get\n∂p\n∂x=−ρg. (33)\nThisistheconditionforagravitationallystratifiedplasm aand\nis satisfied by the chromosphericmodel. In turn, from Equa-\ntion (31) we find that the condition for the plasma to be in\nthermalequilibriumisthatthearbitraryheatingfunction must\nbe\nH=L(ρ,T)−∂\n∂x/bracketleftigg/parenleftbigκe,/ba∇dbl+κn/parenrightbig∂T\n∂x/bracketrightigg\n. (34)\nThe arbitrary heating term essentially balances the back-\ngroundradiativelosses since thecontributionofthermalc on-\nduction, i.e., the second term on the right-handside of Equa -\ntion (34), is almost negligible. We note that this heatingte rm\nis merely included to formally maintain the semi-empirical\nchromospheric model in thermal equilibrium. The presence\nof this arbitrary heating have no e ffect on the behavior and\ndampingofthewavessuperimposedonthebackgroundchro-\nmosphere.\nInspection of Equations (27)–(31) reveals that the velocit y\nlongitudinal component, v/ba∇dbl, and the thermal pressure, p, are\nnonlinearly coupled to the velocity and magnetic field trans -\nversecomponents, v⊥andB⊥. ThismeansthatAlfv´ enwaves,\nwhicharepolarizedinthetransversedirectiontothemagne tic\nfield,cannonlinearlydriveperturbationsassociatedwith lon-\ngitudinally polarized slow magnetoacoustic waves. On the\ncontrary, if v⊥andB⊥are initially zero, slow magnetoacous-\ntic wavescannotdriveAlfv´ enwavesinthe plasma.\n2.5.Approximatelocalanalysisof perturbations\nThe purpose of this paper is to obtain expressions for the\nlengthscalesthatgovernthedissipationofMHDwavesinthechromospheric plasma. To do so, we consider local pertur-\nbationssuperimposedonthebackgroundplasmaandperform\nan approximate study in the limit of small amplitudes. The\nuse of the local analysis is justified by the fact that, for the\nwaves to be dissipated in the chromosphere,the wavelengths\nand the associated dissipation lengthscales must be necess ar-\nily shorter than the chromospheric gravitational scale hei ght.\nThe pressure scale height in the chromosphere is ∼300 km,\nso that we must consider shorter lengthscales in this analy-\nsis. Consequently,the e ffect of gravity on the wave perturba-\ntions is ignored in the present local analysis, although gra vi-\ntational stratification is fully taken into account in the ba ck-\ngroundplasma.\nWe define the dimensionless parameter ǫ≡max|B⊥|/B/ba∇dbl\nandassumethatǫ≪1. Thenwewrite\nρ=ρ0+ǫ2ρ′, (35)\np=p0+ǫ2p′, (36)\nT=T0+ǫ2T′, (37)\nv/ba∇dbl=ǫ2v′\n/ba∇dbl, (38)\nv⊥=ǫv′\n⊥, (39)\nB/ba∇dbl=B0, (40)\nB⊥=ǫB′\n⊥, (41)\nwhere the subscript 0 denotes a backgroundquantity and the\nprime′denotes a perturbation. To separate the perturbations\nof Alfv´ en waves from those of slow waves, the perturbations\nof the perpendicular components of velocity and magnetic\nfield are assumed to be first-order in ǫ, while the perturba-\ntions of the remaining quantities are assumed to be second\norder inǫ. We substitute these quantities in Equations (27)–\n(31) and separate the various terms according to their order\nwithrespecttoǫ.\nThe conditions for static equilibrium (Equations (33) and\n(34)) are consistently recovered from the zeroth-order ter ms\ninǫ. The first-order equationsin ǫgovernthe behavior of v⊥\nandB⊥,andsotheydescribelinearAlfv´ enwaves,namely\n∂v′\n⊥\n∂t=˜ζn∂2v′\n⊥\n∂x2+B0\nµρ0∂B′\n⊥\n∂x, (42)\n∂B′\n⊥\n∂t=B0∂v′\n⊥\n∂x+ηC∂2B′\n⊥\n∂x2, (43)\nwhere˜ζn=ζn/ρ0. The second-order equations in ǫinvolve\nv′\n/ba∇dbl,p′,ρ′, andT′, so that they describe linear slow magne-\ntoacoustic waves and also the generation of slow magnetoa-\ncoustic waves due to the nonlinear coupling with the Alfv´ en\nwaves,namely\n∂ρ′\n∂t=−ρ0∂v′\n/ba∇dbl\n∂x, (44)\nρ0∂v′\n/ba∇dbl\n∂t=−∂p′\n∂x+4ζn\n3∂2v′\n/ba∇dbl\n∂x2−1\n2µ∂B′\n⊥2\n∂x, (45)\n∂p′\n∂t=−γp0∂v′\n/ba∇dbl\n∂x+(γ−1)/bracketleftigg/parenleftbigκe,/ba∇dbl+κn/parenrightbig∂2T′\n∂x2−ρ0Lρρ′\n−ρ0LTT′+1\nµηC/parenleftigg∂B′\n⊥\n∂x/parenrightigg2\n+ζn/parenleftigg∂v′\n⊥\n∂x/parenrightigg2,(46)\nwhereLρandLTarethepartialderivativesoftheradiativeloss\nfunctionwithrespecttodensityandtemperature,respecti vely,Wave heatingscalesinthe chromosphere 7\nnamely\nLρ≡/parenleftigg∂L\n∂ρ/parenrightigg\nρ0,T0,LT≡/parenleftigg∂L\n∂T/parenrightigg\nρ0,T0.(47)\nIn addition, p′,ρ′, andT′are related throughthe equation of\nstate asp′\np0=ρ′\nρ0+T′\nT0. (48)\nThe equations for higher orders in ǫrepresent nonlinear cor-\nrections on the Alfv´ en and slow waves. In this approximate\nstudy we consider su fficiently low amplitudes for the high-\norder corrections in ǫto be negligible. Therefore we restrict\nourselvestothefirst-orderandsecond-orderequationsin ǫ.\n3.CRITICAL DISSIPATIONLENGTHSCALEOF\nALFV´ENWAVES\nTo start with, we consider Alfv´ en waves, which are gov-\nerned by Equations (42) and (43). These two equations in-\nvolve the transverse components of vandBand can be ap-\npropriatelycombinedintoanequationfor B′\n⊥only,namely\n∂2B′\n⊥\n∂t2−c2\nA∂2B′\n⊥\n∂x2−/parenleftig\nηC+˜ζn/parenrightig∂3B′\n⊥\n∂x2∂t+ηC˜ζn∂4B′\n⊥\n∂x4=0,(49)\nwherec2\nA=B2\n0/µρ0is the Alfv´ en velocity squared. Equa-\ntion (49) governs linear Alfv´ en waves damped by Cowling’s\ndiffusion and viscosity. It can be shown that v′\n⊥satisfies the\nsameequation.\nLet us assume that a packet of Alfv´ en waves is generated\nimpulsivelybyaperturbationwithacertainspatialform. A ny\ndisturbanceintheplasmacanbeexpressedasasuperpositio n\nof Fourier modes. Each Fourier mode is characterized by a\nwavenumber, k, andafrequency,ω(k). Thus,wewrite\nB′\n⊥=/summationdisplay\nkBkexp[ikx−iω(k)t], (50)\nwhereBkistheamplitudeoftheFouriermodewithwavenum-\nberk. We substitute this expression into Equation (49) to\nfindthedispersionrelationforsmall-amplitude,linearAl fv´ en\nwavesin aviscous-resistiveplasma,namely\nω2(k)+ik2/parenleftig\nηC+˜ζn/parenrightig\nω(k)−k2c2\nA−k4ηC˜ζn=0.(51)\nEquation(51)waspreviouslyderivedby,e.g.,Chandrasekh ar\n(1961)andZaqarashviliet al.(2012).\nThe individual Fourier modes can be studied separately\nsince no interaction between di fferent Fourier modes of the\nwavepacket takes place in the small-amplitude,linear regi me\nconsidered here. We focus on one individual Fourier mode\nof the wavepacket, so we set kto a fixed value and ω(k) is\nobtained by solving Equation (51). The analytic solution to\nEquation(51)is\nω(k)=±kcA1−k2/parenleftig\nηC−˜ζn/parenrightig2\n4c2\nA1/2\n−ik2\n2/parenleftig\nηC+˜ζn/parenrightig\n.(52)\nThe first term on the right-hand side of Equation (52) is the\nrealpartofthefrequency,namely ωR(k),andthesecondterm\nis the imaginary part, namely ωI(k). The+and−signs in\nfront ofωR(k) correspond to waves propagating towards the\npositivex-direction (upwards) and towards the negative x-\ndirection (downwards), respectively. Hence, the initial p er-\nturbation naturally splits into two wavepackets propagati ngin opposite directions. The imaginary part of ω(k) defines a\ndamping time scale for the Fourier mode, so that its ampli-\ntude decreases in time following an exponential law, namely\nexp(−|ωI(k)|t). Ideal undamped Alfv´ en waves are recov-\nered when no dissipation is present ( ηC=˜ζn=0), so that\nωR(k)=±kcAandωI(k)=0 in that case. However, in\nthe presence of Cowling’s di ffusion and/or viscosity Alfv´ en\nwavesaredamped.\nThe quality factor, Q(k), is a dimensionless parameter that\ncharacterizeshow e fficientlydampeda specific Fouriermode\nis. Thequalityfactoriscommonlydefinedas\nQ(k)≡1\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleωR(k)\nωI(k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (53)\nThe quality factor is independent of the direction of wave\npropagation. The strength of wave damping depends on the\nvalue ofQ(k). According to the definition of Q(k) given in\nEquation(53),thewavesareweaklydampedor underdamped\nwhenQ(k)>1/2. In that case, most of the wave energy\nwould not dissipate in the chromosphere. The larger Q(k),\nthe less efficient damping, so that no wave energy dissipa-\ntion takes place if Q(k)→∞. Conversely, when Q(k)<1/2\nthewavesare overdamped and thewave energydissipationis\nverystrong: most of the wave energyis dissipated. The most\nextreme situation takes place when Q(k)=0, which corre-\nsponds to a wave cutoff. In a cutoffscenario, waves cannot\npropagateand all the energystored in the waves is deposited\nin the plasma. The expression of Q(k) computed from the\nAlfv´ enwavefrequency(Equation(52))is\nQ(k)=|ηC−˜ζn|\nηC+˜ζn√\nk2\nA−k2\n2kifk1 at the boundary, and\napproaches zero toward the center. The velocity becomes\nnonrelativistic, at the radius ¯ rN≈σ−1/2forσ≫1, and\nthe velocity is almost equal to coutside this radius. The\nwave traveling time in eq. (19) is almost determined by the\nslow region inside ¯ rN, and the system size may be regarded\nas being effectively reduced to σ−1/2L. We therefore have\nVAtosc/(σ−1/2L)≈2lnS, i.e,VAtosc/L≈2σ−1/2lnSfor\nthe large Sregime. This property can be seen from the\ncurves around log S≈50 in Fig. 1, except for σ= 104. A\nfactor of ( σ+ 1)1/2instead of σ1/2may provide a better\nextension to σ= 0, but a simple correction is used here.\nFigure 1 also shows that VAtosc/Lapproaches a con-\nstant in the small Sregime, for sufficiently large σ.\nAsymptotically the value of this constant as S→1 is em-\npirically VAtosc/L≈ctosc/L≈2.5, which is independent of\nσ, as far as σ≥102. In our model, the core size increases\nas ¯rc∝S−1/2, and hence the traveling time (19) becomes\nsmaller with decreasing S, but the lower bound is a few\ntimes the light crossing time for a region of size L.\nThe critical value Sc, which discriminates between con-\nstantVAtosc/Lfor smaller SandVAtosc/L∝lnSfor larger\nS, is given approximately by ln Sc∼σ1/2, or logSc∼\n0.4σ1/2. The transition is not very sharp but the relation\ndoes give the approximate boundary between two distinct\nbehaviors. Because log Sc∼1 forσ= 10 and log Sc∼40\nforσ= 104,whicharelocatedattheedgesofFig.1,thetwo\ndifferent behaviors are not clearly shown for these param-\neters. This critical value Scalso characterizes a transition\nin the decay time as will be discussed below.\nThe decay time is related to the imaginary part of ¯ ω,\ntdecay=L/((σ+1)1/2|Im(¯ω)|VA). Figure 2 shows the nor-\nmalized decay time VAtdecay/Las a function of Sfor sev-\neral values of σ. The time for σ= 0 scales as VAtdecay/L\n= 2(lnS)2/π2(Craig & McClymont (1991)). This scaling\nrelation is also broken by the inclusion of σ. The small\nand large Sregimes are different, as they are for the os-\ncillation time. A typical example is given by the curve for4 Y. Kojima et al.: Relativistic magnetic reconnection at X- type neutral points\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/MedSolidDiamond /MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle\n/Circle /Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/CircleΣ = 0\nΣ = 10\nΣ/Equal102\nΣ/Equal103\nΣ/Equal104Σ/Equal103.5\n12 51020 505102050100200\nlog SΤosc\nFig.1. Normalized oscillation time τosc≡VAtosc/Las a\nfunction of Lundquist number S, for magnetization param-\neter values σ= 0,10,102,103,103.5and 104.\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond/MedSolidDiamond\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle\n/SolidDownTriangle/SolidDownTriangle\n/SolidDownTriangle\n/SolidDownTriangle\n/SolidDownTriangle\n/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/SolidDownTriangle/Circle /Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/CircleΣ/Equal104\nΣ/Equal103.5\nΣ/Equal103\nΣ/Equal102Σ = 10Σ = 0\n12 51020 5010205010020050010002000\nlogSΤdecay\nFig.2. Normalized decay time τdecay≡VAtdecay/Las a\nfunction of Lundquist number S, for magnetization param-\neter values σ= 0,10,102,103,103.5and 104.\nσ= 103: the critical value is log Sc∼0.4σ1/2∼13 for this\ncase. Logarithmic dependence can be seen for log S >20,\nwhereas the curve becomes constant for log S <7. The\nrelation VAtdecay/L∝(lnS)2can be seen in the large S\nregime,S≫Sc, except for σ= 104, but the timescale is\nreduced to approximately VAtdecay/L≈2σ−1/2(lnS)2/π2\nforσ≫1. The factor σ−1/2can be interpreted as being\ndue to an effective reduction of the system’s size, as con-\nsidered for the oscillation time. The normalized decay time\nbecomes the minimum around Sc.\nIn the small Sregime,S≪Sc, normalized decay time\napproaches a constant value VAtdecay/L≈0.14σ. The nor-\nmalized decay time for fixed Sincreases with the magneti-\nzation parameter σ. The limit of σ→ ∞correspondsto the\nvacuum, in which there is no matter ( ρ= 0) and the dis-\nsipation time becomes infinite. This σ-dependence comes\nfrom taking account of the finiteness of cin the resistive\nlosses. (See eq. (15).) This effect can be neglected in the\nlargeSregime, where the approximation of instantaneous\ndissipation is good. However, the effect becomes evident in\nthe small Sregime.\nThe energy Eof perturbation decreases due to the\nOhmic dissipation\ndE\ndt=−η/integraldisplay\nj2dV. (20)The linearized form with Fourier component provides an\nexpression of the decay time as\nVAtdecay\nL= 2S/integraltext1\n0(δ¯εB+δ¯εE+δ¯εM)2π¯rd¯r\n/integraltext1\n0|δ¯j|22π¯rd¯r, (21)\nwhereδ¯εis dimensionless energy density of magnetic field,\nelectric field, kinetic energy of the fluid, and δ¯jis dimen-\nsionless current density. Their explicit forms are given by\nδ¯εB=1\n8π|δ¯B|2=1\n2|df\nd¯r|2, (22)\nδ¯εE=1\n8π|δ¯E|2=σ\n2|¯ωf|2, (23)\nδ¯εM=1\n2ρ0|δ¯v|2=¯r2\n2|¯ω|2|(1\n¯rd\nd¯r¯rd\nd¯r+ ¯ω2σ)f|2, (24)\nand\n|δ¯j|2=|(1\n¯rd\nd¯r¯rd\nd¯r+ ¯ω2σ)f|2. (25)\nSpatial distributions of these energy densities are displayed\nin Fig. 3 for S= 105,σ= 101and in Fig. 4 for S= 105,\nσ= 104. These functions are calculated by numerical so-\nlution outside ¯ rc, and by the analytic asymptotic form\neq. (18) inside it. Note that a sharp peak in δ¯εMand\nδ¯εBis located within ¯ rc. Both kinetic energy of matter\nand magnetic energy are accumulated from outer part to\nthe core( ∼¯rc), and are dissipated in the central region.\nHowever,distributionofelectricenergyisflat.Theseoverall\nfeatures are not so much different in Figs. 3 and 4, although\nthe sharp peak shifts by ¯ rc= (¯ω/((σ+1)1/2S))1/2.\nThe magnitude of δ¯εEis much smaller than that of δ¯εB\nin Fig. 3 ( σ= 101), whereas δ¯εEbecomes comparable to\nδ¯εBin Fig. 4 ( σ= 104). The electric energy is approxi-\nmately proportional to σ, as shown in eq.(23), and signif-\nicantly contributes to the sum of energy. Hence, the de-\ncay time becomes longer with the increase of σfor fixed\nS, since the total energy increases. (See eq.(21).) In the\nlargeSregime, however, the functions δ¯εBandδ¯εMare\nmuch larger than δ¯εE, so that the electric energy can be\nneglected. The decay time does not increase with σin this\nregime.\n4. Discussion and conclusions\nRelativistic MHD differs, in general, from the nonrelativis-\ntic case in at least three ways: (i) the Lorentz factor γ,\n(ii) the Coulomb force ρeE, and (iii) the displacement cur-\nrentc−1∂E/∂tin Maxwell’s equation. The Lorentz factor\nappears in the flow velocity and also in the resistivity of\nOhm’s law as a Lorentz contraction. The difference is of\norder (v/c)2in magnitude. Since we considered a linear\nperturbation from the static state, the inflow velocity is\nnot very large and the Lorentz factor may approximate to\nγ= 1. The magnitude of ρeEis of order ( v/c)2times the\nLorentz force j×B, and is hence neglected in nonrelativis-\ntic MHD. Moreover, the charge density is always zero due\nto the 2D X-point geometry considered here, so that the\nCoulomb force ρeEvanishes exactly. This leaves the dis-\nplacement current as a possible factor for the difference be-\ntween relativistic and nonrelativistic MHD. We have stud-\nied its effects, especially on the dynamics of the magneticY. Kojima et al.: Relativistic magnetic reconnection at X-t ype neutral points 5\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4\n-10 -8-6-4-2 0\nx2 x 103 δεE / δεBmax4 x10-25δεM / δεBmaxδεB / δεBmax\nFig.3.Normalized energy density δ¯εas a function of x=\nln¯rforS= 105andσ= 101. The function δ¯εMhas a sharp\npeak, and is shown with a reduction factor 4 ×10−25, while\nδ¯εEis magnified by 2 ×103.\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4\n-12-10-8-6-4-2 0\nx6 δεE / δεBmax\n6 x 10-14δεM / δεBmaxδεB / δεBmax\nFig.4.Normalized energy density δ¯εas a function of x=\nln¯rforS= 105andσ= 104. The function δ¯εMis shown\nwith a factor 6 ×10−14, whileδ¯εEis shown with a factor 6.\nreconnection using a simplified system based on linearized\nequations in the cold plasma limit. The magnetization pa-\nrameterσis incorporated in the basic equation through the\ndisplacement current and the oscillation and decay times\nfor the least-damped mode were calculated numerically for\nparameters S=10-1050andσ= 0-104.\nIn the system with σ= 0, for which the displacement\ncurrentcanbeneglected,theoscillationanddecaytimesare\nproportional to ln Sand (lnS)2, respectively. By including\nσ, these timescales are modified in different ways, in two\nregimes, which are characterized by S≫ScorS≪Scfor\nSc≈exp(σ1/2). For low resistivity, S≫Sc, a logarithmic\ndependence with Scan seen, but the timescales normalized\nby the boundary radius Land the Alfv´ en velocity VAbe-\ncome smaller with increasing σ. The smaller timescales can\nbe explained as being due to an effective reduction in the\nsize of the system, or the enlargement of the outer region\nwhere MHD waves propagate at almost the speed of light\nand the traveling time is negligible. On the other hand, for\nhigh resistivity, S≪Sc, a new feature appears in both\nthe oscillation and decay times, which do not depend on S.The oscillation time is a few times the light crossing time\nand does not depend on σ. The dissipation time becomes\nlonger in proportion to σand goes to infinity in the limit of\nσ→ ∞,thatis,nodissipationinthevacuum.Reconnection\nat the X point is thought to be “fast”, since the dissipation\ntime is scaled with (ln S)2. Actual time is of the order of\n10-103times crossing time with Alfv´ en velocity. The dis-\nplacement current significantly spoils the good property,\nand the timescale increases with σin high resistive region.\nThe increase of the decay time is related with deficiency of\nmatter, which is involved in the Ohmic dissipation.\nMagnetic reconnection is expected to be an important\nprocess of abrupt energy release in the solar and magne-\ntar flares. For example, the explosive tearing-mode recon-\nnection in the magnetar like the solar flares is discussed\n(Lyutikov (2006); Masada et al. (2010)). Dimensionless pa-\nrameters are however quite different in them: σ∼10−4\nandS∼1014in solar corona, whereas it is likely that\nσ≫1 andS≫1 in a magnetar magnetosphere. Present\nresult in an X-type collapse suggests the dissipation time\nt∼0.1σL/VA∼10−5σ(L/106cm) s under highly magne-\ntized environment. The spiky rise time ( <0.1s) or short\nduration ( <1s) of the magnetar flare may significantly\nconstrain σL. The energy of the flare ∆ E(∼1045erg)\nshould be a part of magnetic energy within the volume L3:\nB2\n0L3∼ρ0σL3>∆E. These two conditions provide an\nupper limit of σasσ <104.5(ρ0/(g/cm3))1/2. In such high\nenergy events, radiation and possibly pair creation may be\nimportant in the energy transfer. Further study is needed\nfor these effects. However, the results in this paper demon-\nstrate that the dynamics significantly depends on the mag-\nnetization parameter through the displacement current.\nAcknowledgements\nThis work was supported in part by a Grant-in-Aid\nfor Scientific Research (No.21540271) from the Japanese\nMinistry of Education, Culture, Sports, Science and\nTechnology.\nReferences\nBlackman, E. G. & Field, G. B. 1993, Physical Review Letters, 71,\n3481\nCraig, I. J. D. 2008, A&A, 487, 1155\nCraig, I. J. D., Litvinenko, Y. E., & Senanayake, T. 2005, A&A , 433,\n1139\nCraig, I. J. D. & McClymont, A. N. 1991, ApJ, 371, L41\nHassam, A. 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Kojima et al.: Relativistic magnetic reconnection at X- type neutral points\nZenitani, S., Hesse, M., & Klimas, A. 2009, ApJ, 696, 1385\nZenitani, S. & Hoshino, M. 2005, Physical Review Letters, 95 , 095001\nZenitani, S. & Hoshino, M. 2007, ApJ, 670, 702\nZenitani, S. & Hoshino, M. 2008, ApJ, 677, 530" }, { "title": "1907.00424v2.Non_linear_spin_torque__pumping_and_cooling_in_superconductor_ferromagnet_systems.pdf", "content": "Non-linear spin torque, pumping and cooling in superconductor/ferromagnet systems\nRisto Ojaj arvi,1,\u0003Juuso Manninen,2Tero T. Heikkil a,1,yand Pauli Virtanen1,z\n1University of Jyvaskyla, Department of Physics and Nanoscience Center,\nP.O. Box 35 (YFL), FI-40014 University of Jyv askyl a, Finland\n2Aalto University, Department of Applied Physics,\nLow Temperature Laboratory, P.O. Box 15100, FI-00076 AALTO, Finland\n(Dated: November 5, 2021)\nWe study the e\u000bects of the coupling between magnetization dynamics and the electronic degrees\nof freedom in a heterostructure of a metallic nanomagnet with dynamic magnetization coupled with\na superconductor containing a steady spin-splitting \feld. We predict how this system exhibits a\nnon-linear spin torque, which can be driven either with a temperature di\u000berence or a voltage across\nthe interface. We generalize this notion to arbitrary magnetization precession by deriving a Keldysh\naction for the interface, describing the coupled charge, heat and spin transport in the presence\nof a precessing magnetization. We characterize the e\u000bect of superconductivity on the precession\ndamping and the anti-damping torques. We also predict the full non-linear characteristic of the\nOnsager counterparts of the torque, showing up via pumped charge and heat currents. For the\nlatter, we predict a spin-pumping cooling e\u000bect, where the magnetization dynamics can cool either\nthe nanomagnet or the superconductor.\nI. INTRODUCTION\nThe intriguing possibility to control magnetization dy-\nnamics by spin torque suggested over two decades ago1\nand its reciprocal counterpart2,3of spin pumping4have\nbeen widely studied in magnetic systems. In such sys-\ntems charge and spin transport are closely linked and\nneed to be treated on the same footing. Recently there\nhas also been increased interest in coupling superconduc-\ntors to magnets and \fnding out how superconductivity\na\u000bects the magnetization dynamics5{19. On the other\nhand, recent work has shown that a combination of mag-\nnetic and superconducting systems results in giant ther-\nmoelectric e\u000bects20{24which couple charge and heat cur-\nrents. These works21,22also imply a coupling of spin and\nheat. However, a general description of the implications\nfor the magnetization dynamics, dynamical heat pump-\ning e\u000bects, and the behavior in the non-linear regime at\nenergies comparable to the superconductor gap \u0001, has\nbeen lacking.\nIn this work, we \fll this gap by constructing a theory\nwhich provides a combined description of pumped charge\nand heat currents, spin torques, magnetization damping,\nvoltage and thermal bias. We consider a metallic nano-\nmagnet F with a magnetization precessing at a rate \nwhich is determined by an external magnetic \feld, the\nshape of the magnet and the crystal anisotropy,26at a\nslowly varying angle \u0012to the precession axis [Fig. 1(a)].\nThe magnet is tunnel coupled to a superconducting elec-\ntrode S that also contains a constant spin-splitting (ex-\nchange or Zeeman) \feld25,27.\nMain features of the problem can be understood in a\ntunneling model, shown schematically in Fig. 1(b). Both\nthe spin splitting hand nonzero \n shift the spectrum,\nwhereas \n generates also e\u000bective spin-dependent chem-\nical potential shifts28providing a driving force which\npumps the currents across the interface. The inter-\nplay of the two enables a coupling between the magne-\nFIG. 1. (a) Schematic ferromagnetic island{superconductor\ntunnel junction (F/I/S) setup. The direction mof magneti-\nzation in F precesses at a rate \n at an angle \u0012around the\naxis (^z) of its e\u000bective \feld. Electron tunneling and intrin-\nsic damping produces torque \u001conm. The superconductor\nhas an internal spin splitting exchange \feld h, from exter-\nnal magnetic \feld, or a ferromagnetic insulator (FI) bilayer\nstructure25. We consider also thermal and electric biasing\n(\u000eT;V ). (b) \\Semiconductor picture\" for pumping, in the\nframe rotating with m(forhk^z). Grey solid line is the\nchemical potential when \n = 0. Increasing the precession fre-\nquency to \n6= 0 shifts both the spectrum and the chemical\npotentials (dashed lines) by \n cos \u0012in F and by \n in S. The\nexchange \feld honly shifts the spectrum in S.\ntization dynamics and the linear-response thermoelectric\ne\u000bect20,21,23originating from the spin-selective breaking\nof the electron-hole symmetry in the superconductor with\nrespect to the chemical potential. As a consequence, a\ntemperature di\u000berence between the two systems leads to\na thermal spin torque, which in a suitable parameter\nregime yields an anti-damping su\u000ecient to obtain \ripping\nor stable precession of the nanomagnet. The Onsager\ncounterpart of the thermal spin torque is a Peltier-type\ncooling (or heating) driven by the precessing magneti-\nzation. In the non-linear response, the precession also\npumps a charge current, as already shown in29. We dis-\ncuss the general picture for the spin-split superconductor,arXiv:1907.00424v2 [cond-mat.mes-hall] 17 Sep 20202\nand, in addition to the thermomagnetic e\u000bects, \fnd the\nKeldysh action [Eq. (20)] describing the stochastic prop-\nerties of the S/F junction. The action allows identify-\ning thermodynamical constraints, current noises, a spin-\ntronic \ructuation theorem and describes the probability\ndistribution of the magnetization direction and the spec-\ntrum of its oscillations.\nThe manuscript is structured as follows: We introduce\na simple tunneling model in Sec. II and discuss the tun-\nneling currents in Sec. III. Implications on magnetization\ndynamics are considered in Sec. IV, including thermal\ntransport associated with the ferromagnetic resonance\nand physics of spin torque oscillators driven by the ther-\nmal e\u000bects. In Sec. V we focus on studying the stochas-\ntic magnetization dynamics based on a Keldysh action\napproach to the tunneling model, and discuss probabil-\nity distributions and linewidths for the oscillators. We\nconclude in Sec. VI. Certain details of derivations are\npostponed to the Appendixes.\nII. TUNNELING MODEL\nThe main e\u000bects can be understood with a tunneling\nHamiltonian description (below ~=e=kB= 1),\nH=HS+^R(t)HF^R(t)y+X\njj0\u001bWjj0e\u0000iVtcy\nj\u001bdj0\u001b+ h:c:;\n(1)\nwherecj\u001banddj\u001bare the F and S conduction elec-\ntron operators and Wthe tunneling matrix elements\nfor spin/momentum states \u001b=\u0006,pj, andVis a bias\nvoltage. The Hamiltonian HSdescribes the spin-split\nsuperconductor23, andHFthe magnet with magnetiza-\ntion parallel to the ^ z-direction. The magnetization direc-\ntionm(t) = (cos\u001esin\u0012;sin\u001esin\u0012;cos\u0012) is speci\fed by a\nspin rotation matrix ^R(t)cj\u001b^R(t)y=P\n\u001b0R\u001b\u001b0(t)cj\u001b0. In\nthe frame rotating with R28,30, assumingm(t) varies adi-\nabatically so that an equilibrium electron distribution is\nmaintained, the Berry phase '(t) =Rtdt0_\u001e(1\u0000cos\u0012)\ncan be absorbed (c.f. Refs. 31, 32 and Appendix B) to\nthe spin rotation\nR=e\u0000i\u001e(t)\u001bz=2e\u0000i\u0012(t)\u001by=2ei\u001e(t)\u001bz=2e\u0000i'(t)\u001bz=2;(2)\nwhere\u001bx=y=z are the spin matrices. Varying m(t) re-\nsults to e\u000bective spin-dependent voltages30in the tun-\nneling part. For uniform precession, they are \n \u001b\u001b0=\n(\u001b\u0000\u001b0cos\u0012)\n=2 (see Fig. 1b). From the model, we can\ncompute in leading order in Wthe tunneling charge, en-\nergy, and spin currents ( Ic,_E,Is) via a standard Green\nfunction approach (see Ref. 33 and Appendix A). The\nassumption of local equilibrium implies that the rates of\ntunneling and other nonequilibrium-generating processes\non the magnetic island should be small compared to elec-\ntron relaxation.34{36\nConsider precession with frequency \n around the z-\naxis,\u001e(t) = \ntwithj_\u0012j \u001c \n. From the abovemodel, we \fnd the time-averaged currents and ~\u001cz=\n\u0000(m\u0002Is\u0002m)z,1,28thez-component of the time-\naveraged spin transfer torque:\nIc=GT\n2eZ1\n\u00001d\u000fX\n\u001b\u001b0h\u001bj\u001b0i2NS;\u001bNF;\u001b0[fF\u0000fS];(3)\n_ES=GT\n2e2Z1\n\u00001d\u000fX\n\u001b\u001b0\u000fh\u001bj\u001b0i2NS;\u001bNF;\u001b0[fF\u0000fS];(4)\n\u001cz=\u0000GTsin2\u0012\n8e2Z1\n\u00001d\u000fX\n\u001b\u001b0\u001bNS;\u001bNF;\u001b0[fF\u0000fS]:(5)\nHere,fF=f0(\u000f\u0000V\u0000\n\u001b\u001b0;TF),fS=f0(\u000f;TS) are\nthe Fermi distribution functions in F and S, h\u001bj\u001b0i2=\n(1 +\u001b\u001b0cos\u0012)=2 the spin overlap between mand the\nz-axis, andNS=F;\u001b =\u0006the densities of states (DOS) for\nup/down spins (quantization axis m(t) for F, and ^ zfor\nS) normalized by the Fermi level DOS per spin, and\nGTthe tunneling conductance. Of these, Eq. (3) was\npreviously discussed in Ref. 29 for h= 0. Using a\nbasic model for F and S, we have NF;\u001b= 1 +\u001bP\nandNS;\u001b=P\n\u00061\u0006\u001b^h\u0001^z\n2N0(\u000f\u0007h), whereP= (\u0017F;+\u0000\n\u0017F;\u0000)=(\u0017F;++\u0017F;\u0000) is the spin polarization in terms of\nthe majority/minority Fermi level DOS \u0017F;\u0006, andN0(\u000f)\nthe Bardeen-Cooper-Schrie\u000ber density of states37. The\ntunneling described by Eqs. (3{5) can be understood in a\nsemiconductor picture, as shown in Fig. 1b. The broken\nelectron-hole symmetry around the chemical potentials\nfor both spins in S and spin polarization in F results to\nthermally driven spin currents causing torques, and the\nrotation-induced potential shifts pump charge and heat\ncurrents.\nIII. TUNNELING CURRENTS\nExpanding for small voltage bias V, temperature dif-\nference\u000eT=TS\u0000TF, and the precession speed \n, the\ntime-averaged currents are described by a linear-response\nmatrix:\n0\n@Ic\n_ES\n\u001cz1\nA=0\n@G P\u000b cos\u0012 0\nP\u000bcos\u0012 G thT\u000b\n2sin2\u0012\n0\u0000\u000b\n2sin2\u0012\u0000G\n4sin2\u00121\nA0\n@V\n\u0000\u000eT=T\n\n1\nA;\n(6)\nwhereGandGthare the linear-response elec-\ntrical and thermal conductances. Here, \u000b=\n\u0000(GT=2)R1\n\u00001d\u000f\u000f[NS;+(\u000f)\u0000NS;\u0000(\u000f)]f0\n0(\u000f) is a thermo-\nelectric coe\u000ecient,20,21which originates from the ex-\nchange \feld hgenerating the electron-hole asymmetry\nin the superconductor. It is nonzero only when S is both\nsuperconducting and has a spin splitting h6= 0. The re-\nsponse matrix Lin Eq. (6) has the Onsager symmetry\nLij=Ltr\nji, where tr refers to time-reversal, \u000btr=\u0000\u000b,\nPtr=\u0000P.\nThe coe\u000ecient for charge pumping is here zero, un-\nlike in the ferromagnet{ferromagnet case,30because the3\nspin-(anti)symmetrized DOS of S is also (anti)symmetric\nin energy. This also suppresses linear-response contri-\nbutions to charge current from thermal magnetization\n\ructuations,31which are also related to the magnon spin{\nSeebeck e\u000bect3,18,31.\nImportantly, the spin splitting of the superconduc-\ntor enables the precession to pump energy current at\nlinear response, and as its Onsager counterpart, there\nis nonzero thermal spin torque (terms with \u000b6= 0).\nThis is made possible by the nonzero thermoelectric\ncoe\u000ecient20,21driving spin currents due to a tempera-\nture di\u000berence. This e\u000bect is (in metals) parametrically\nlarger by a factor \"F=\u0001\u001d1 than that from normal-state\nDOS asymmetry3,35,38in systems with Fermi energy \"F.\nA. Symmetries\nLet us now consider the joint probability Pof changes\n\u000ensand\u000eESin the electron number and energy of S, and\na change\u000emzin the magnetization of F, during a time in-\nterval of length t0. It satis\fes a \ructuation relation39,40:\nPt0(\u000en;\u000eES;\u000emz) =eT\u00001\nFV\u000en+(T\u00001\nS\u0000T\u00001\nF)\u000eES+T\u00001\nF\nS\u000emz\n\u0002Ptr\nt0(\u0000\u000en;\u0000\u000eES;\u000emz): (7)\nHere, we denote S=VMs=(~\r) as the e\u000bective\nmacrospin of the ferromagnetic island, Vand\rare the\nF volume and gyromagnetic ratio and Msthe magneti-\nzation. Moreover, Ptrcorresponds to reversed polariza-\ntions and precession ( NS=F;\u001b7!NS=F;\u0000\u001b, \n7!\u0000\n). The\nOnsager symmetry of Lijin Eq. (6) is a consequence of\n\ructuation relations41. The energy transfer \u000eEFinto the\nferromagnet (generally, \u000eEF6=\u000eES) is determined by\nenergy conservation \u000eEF+\u000eES=V\u000en + \nS\u000emz, which\nimplies _ES+_EF=IcV\u0000\n\u001cz. These results arise from\nthe symmetries of Eqs. (19, 20) below, for the case where\nthere is no external magnetic drive.\nB. Non-linear response\nThe pumped charge current is shown in Fig. 2(a), and\nthe energy current into S in Fig. 2(b). The charge pump-\ning is nonzero above the quasiparticle gap, j\nj&\u0001\u0006h.29\nThe heat current shows the presence of a region of cooling\nof either of the two leads, depending on the relative ori-\nentation ofhand \n^z. Nonzerohenables the N/S cooling\ne\u000bect to be present already at linear response, similarly\nas with voltage bias23,42.\nIV. MAGNETIZATION DYNAMICS\nThe Landau{Lifshitz{Gilbert{Slonczewski (LLG)\nequation for the tilt angle is\n\u0000S@tcos\u0012=\u001cz\u0000SA0\n sin2\u0012+\u0011; (8)\nFIG. 2. (a) Pumped di\u000berential current for TS=TF=\n0:1TCwhereTCis the critical temperature of the supercon-\nductor. Blue, yellow and red lines are for h=\u0000h^z;h^x;h^z,\nrespectively. (b) and (c) Energy current into the supercon-\nductor _ES(blue line) and into the magnet _EF(red line) for\n(b)h=\u0000h^zand for (c)h=h^z. F and S are at temperature\nT= 0:6TC. Dashed lines represent the linear response. In all\n\fgures,V= 0,\u0012=\u0019\n8,P= 1 andh= 0:3\u00010, where \u0001 0is the\nsuperconductor gap at zero temperature.\nFIG. 3. Electromagnetically driven FMR induced refrigera-\ntion forh=\u00000:3\u00010^z,P= 1, andA0= 0:1~GT=(e2S). (a)\nFor\u0015= 0, and (b) for \u0015= 1. Dynes broadening \u0000 = 10\u00003\u00010\nwas assumed43.\nwhere the spin transfer torque \u001czis given by Eq. (5).\nWe include the intrinsic Gilbert damping28phenomeno-\nlogically, and A0is the dimensionless damping constant.\nMoreover,\u0011is a Langevin term describing the torque\nnoise32,39,44,45with the correlation function h\u0011(t)\u0011(t0)i=\n2[D(\u0012) +SA0T] sin2(\u0012)\u000e(t\u0000t0); see below. Equilibrium\ntorques are here included in the LLG e\u000bective magnetic\n\feld \n^z(see Appendix A). We consider the limit of weak\ndamping, where it is su\u000ecient to consider only the equa-\ntion for the z-component.\nA. Heat balance in ferromagnetic resonance\nLet us consider a ferromagnetic resonance (FMR)26in\na thin magnetic layer on a spin-split S, driven by a reso-\nnant circularly polarized rf magnetic \feld (at frequency\n!= \n), and in the case of S acting as a reservoir at a\n\fxed temperature T. The electrical circuit is open, so4\nthat no charge \rows between F and S. The FMR driv-\ning acts as a power source. We assume that a fraction\n\u00152[0;1] of the power dissipated by the intrinsic Gilbert\ndamping heats the F electrons; the value of \u0015depends on\ninto which bath(s) its microscopic mechanism dissipates\nthe energy (see also Sec. V A below). In a steady state,\nthe total energy current into F, the overall torque, and\nthe charge current are zero:\n_EF;tot=_EF+\u0015PG= 0; (9)\n\u001cz+\u001cz;rf+\u001cz;G= 0; (10)\nIc= 0; (11)\nwhere\u001czandIcare the contributions related to the tun-\nneling between F and S, from Eqs. (3,5), and _EF=\nIcV\u0000\n\u001cz\u0000_ESis found from the tunneling model via\na similar calculation as in Eq. (4). Moreover, \u001cz;G=\n\u0000SA0\n sin2(\u0012) andPG=SA0\n2sin2(\u0012) are the torque\ndue to the intrinsic damping and the rate of work done\nby it. At resonance, the rf drive creates a torque \u001cz;rf=\n\rS(m\u0002hrf)z=\rShrfsin\u0012, wherehrfis the amplitude\nof the rf \feld. From the above it follows that the power\n_ES+_EF;tot=Prf\u0000(1\u0000\u0015)PG (12)\nis absorbed by the electron system, where Prf= \n\u001cz;rfis\nthe total rf power absorbed at resonance28.\nExpanding Eqs. (3{5) in the linear order in V,\u000eT=T\nand\u00122, but not in \n, we \fnd the charge and heat currents\n0\n@Ic\n_ES\n\u001cz1\nA=0\nB@G P\u000b P (G\u0000eG)\nP\u000b G thT \u000b +e\u000b+eG\n2\n0 0\u0000eG1\nCA0\n@V\n\u0000\u000eT=T\n\n4\u001221\nA:\n(13)\nUnlike the linear-response matrix in Eq. (6), the above\nmatrix is not symmetric, as there is no Onsager reci-\nprocity between \u001czand\u00122. The coe\u000ecients are\ne\u000b=GT\n2Z1\n\u00001d\u000fX\n\u001b\u0012\n\u000f\u0000\u001b\n2\u0013\nNS;\u001b(\u000f)f0(\u000f\u0000\u001b\n)\u0000f0(\u000f)\n\n(14)\neG=GT\n2Z1\n\u00001d\u000fX\n\u001b\u001bNS;\u001b(\u000f)f0(\u000f\u0000\u001b\n)\u0000f0(\u000f)\n\n(15)\nThese coe\u000ecients are de\fned so that lim \n!0eG=Gand\nlim\n!0e\u000b=\u000b, and they assume the values eGnormal =GT\nande\u000bnormal = 0 in the normal state.\nThe torque balance (10) determines the precession an-\ngle\u0012\u0019\rShrf=(SAe\u000b\n), whereSAe\u000b=SA0+eG\n4. To\nquadratic order in hrf,_EF=eG\n2\u00122=4\u0000_ES. Using this,\nand the conditions (9) and (11) for heat and charge cur-\nrents, we \fnd the FMR induced temperature di\u000berence\nFIG. 4. Maximum intrinsic damping, expressed as \n \u0002\n\u0015e2SA0=(GT\u00010), for which the system can be refrigerated,\nwithh=\u00000:3\u00010^z,P= 1 and Dynes broadening \u0000 =\n10\u00005\u00010. The maximum intrinsic damping is determined by\nsolvingA0from Eq. (16) with \u000eT= 0.\nand voltage\n\u0012V\n\u0000\u000eT\nT\u0013\n=\u0012\nG P\u000b\nP\u000b G thT\u0013\u00001\n\u0002\n \n\u0000P(G\u0000eG)\n4\n\u0000\u000b+e\u000b\n4\n +heG\n8+\u0015SA0i\n\n2!\n\u00122(16)\nThe coupling between _ESand\u00122is of the linear order in\n\n, whereas the coupling between Icand\u00122, the rf power,\nand the magnetic dissipation are of the quadratic order in\n\n. Thus, for \n\u001cTthe induced temperature di\u000berence\nand voltage are\nV'P\u000b\nGT\u000eT; \u000eT'\u000b\n2\u0010\nGth\u0000(P\u000b)2\nGT\u0011\n\u00122:(17)\nThe denominator eGth=Gth\u0000(P\u000b)2\nGTis always positive21.\nFor \n\u001cT, F is refrigerated when \u000b > 0, which cor-\nresponds to h\u0001^z < 0. Restoring the SI units, the\nmagnitude of the coe\u000ecient between \u000eTand \n\u00122is\nj~\u000b=(eGthe)j.~=kB.\nAt higher frequencies the magnetic dissipation, nonlin-\nearities ofe\u000bandeG, and the coupling between charge and\nprecession start to play a role and limit the attainable\ntemperature di\u000berence. For SA0=GT= 0:1, the magni-\ntude of the e\u000bect is illustrated in Fig. 3. The maximum\nvalue ofA0for which refrigeration is possible is shown in\nFig. 4 as a function of Tand \n. If\u0015= 1, the parame-\nter regime is similar to that where the spin-torque driven\noscillations occur (see Sec. IV B below). However, if the\nintrinsic damping dissipates the energy to systems di\u000ber-\nent from the F conduction electrons ( \u0015 < 1), refrigera-\ntion is easier to obtain than auto-oscillations. Therefore,\nmeasuring the temperature di\u000berence \u000eTvia the thermo-\nelectrically induced voltage Vallows for a direct study of\nthe energy dissipation mechanism of the intrinsic Gilbert\ndamping. Note that also in the absence of the spin split-\nting in S (and therefore \u000b= 0), it is possible to induce a5\nFIG. 5. (a) Torque vs. angle \u0012and voltage Vat \n = 0:3\u0001=~\nforTS=TF= 0:5TC,h=\u00000:3\u00010^zandP= 1. The arrows\nindicate where the torque drives the angle. The solid black\nline indicates the stable precession angle \u0012\u0003, and the dashed\nline the unstable one. At V= 0,\u0012\u0003= 0. (b) Torque vs. angle\nand temperature di\u000berence at \n = 0 :5\u0001=~forTS= 0:5TC,\nh= 0:3\u00010^z,P= 1, andV= 0. Moreover, SA0= 0.\nThe dashed green line indicates \u000eTo. (c) Magnetization dis-\ntribution normalized by its maximum value, for a thermally\ndriven spin oscillator with S= 100,TS= 0:5TC,h= 0:3\u00010^z,\nP= 1,V= 0 and \n = 0 :5\u0001=~. WhenTF\u00190:31TC(dashed\nline), the distribution is signi\fcantly bimodal. (d) Full width\nat half maximum (FWHM) of the dipole spectrum Sxx(!)\n(black line) and the average magnetization (red line) with\nGT=e2=~. The dashed line indicates \u0012\u0003and the dots corre-\nspond to panel (c).\nnon-zero voltage via FMR driving29. However, that gen-\nerally requires higher frequencies \n .\u0001 than the case\nanalyzed above.\nIf the thermoelectric coe\u000ecient is zero, F always heats\nup. In the normal state we have\n\u000eTnormal =\u0000GT+ 8\u0015SA0\n8Gth\n2\u00122<0; (18)\nwhich shows the combined heating e\u000bect from the dif-\nferent sources of dissipation. However, in that case the\ninduced voltage V= 0, and the temperature di\u000berence\nwould have to be measured via some other mechanism.\nB. Spin torques\nThe junction also exhibits a voltage-driven spin torque .\nWith an exchange \feld such that h\u0001^z<0 and \n .2h, the\ntorque due to tunneling becomes antidamping at large\nvoltages. When it exceeds the intrinsic damping, the\n\u0012= 0 equilibrium con\fguration is destabilized, and a\nnew stable steady-state con\fguration \u001cz;tot(\u0012\u0003) = 0 is\nestablished. An example of the signs of the torque and\nthe resulting con\fguration is shown in Fig. 5(a): The\nstable angle is \u0012\u0003= 0 at small voltages, after which there\nis a voltage range for which 0 < \u0012\u0003< \u0019. There, thesystem realizes a voltage-driven spin oscillator46,47. At\nlarge voltages the stable angle is \u0012\u0003=\u0019, corresponding\nto a torque-driven magnetization \rip.\nSimilarly, the thermal torque is shown in Fig. 5(b).\nDue to the nonzero linear-response coupling, it is anti-\nsymmetric in small \u000eT, in contrast to the voltage-driven\ntorque. Consequently, antidamping regions occur for\nboth signs of \n. In linear response [Eq. (6)], for tem-\nperature di\u000berences satisfying sgn( \u000b)\u000eT < \u000eT o= [1 +\ne2SA0=(~G)]P~\n=(2ejsj), the spin torque drives \u0012!0,\ndamping the precession. Here, s=\u0000P\u000b=(GT) is the\njunction thermopower, which can be jsj&kB=e.21Above\nthe critical temperature di\u000berence \u000eTo, the thermal spin\ntorque drives the system away from \u0012\u0003= 0 (or\u0012\u0003=\u0019for\n\n<0). The stable precession angle is shown in Fig. 5(b):\nthere is a range of \u000eTin which\u0012\u00036= 0;\u0019and the system\nexhibits thermally driven35spin oscillations.\nIn Fig. 5, we neglect the e\u000bect of the intrinsic damp-\ningA0on the magnetization oscillations. However, it is\nthe main obstacle in reaching auto-oscillations in FMR\ndevices, and we estimate its e\u000bect here. For the super-\nconducting systems, generally the e\u000bective bias jsj\u000eTcan\nbe at most \u0001. Considering the value \u000eTogiven above,\nthis results to a requirement for the resistance{area prod-\nuct of the S/F junction: RA.(RA)0=~\r\u0001\ne2A0MsdFj\nj\u0019\n10\u00004\n\u0016m2\u00021 T nm \u0001\n\u00160MsdFA0j~\nj, wheredFis the ferromag-\nnet thickness. Meeting the requirement is likely chal-\nlenging. Values RA\u00180:1 \n\u0016m2have been achieved in\n\u0018(100 nm)2lateral size magnetic junctions46,48. With\nsuchRAand\u00160MsdF= 5 T nm (e.g. Co layer46) and\nA0= 0:0128, the condition is satis\fed for f=j\nj=(2\u0019)<\n0:02\u0001=h\u00191 GHz (for Al as superconductor). The\nFMR refrigeration has a similar requirement but with\n\u00017!\u0001=\u0015, and hence may be easier to achieve, if the\nmicroscopic mechanism is such that \u0015<1.\nV. KELDYSH ACTION\nTo properly describe the metastable states in the mag-\nnetization precession, we need to extend the formalism.\nThe dynamics beyond average values can be described\nby an e\u000bective action S=S0+STfor the spin in-\ncluding the tunneling, derived32,34{36,39,45,49,50by retain-\ning the Keldysh structure51for the orientation of the\nmagnetization mean \feld. The action Sdescribes the\ngenerating function of the joint probability distribution\nPt0(\u000en;\u000eES;\u000eEF;\u000emz) [see Eq. (7)], with a source \feld\n\u001f,\u0018S,\u0018F,\u0010associated with each of the arguments. The\nfree part reads\nS0= 2SZ1\n\u00001dt\u0014\u0012\u0010\n2+\u001eq\u0013\n@t(cos\u0012)c\u0000(cos\u0012)q(_\u001ec\u0000\n)\u0015\n;\n(19)\nwherecandqdenote the symmetric/antisymmetric com-\nbinationsxc=q=x+\u0006x\u0000\n2of quantities on the two Keldysh6\nbranches (+ =\u0000), for example (cos \u0012)c=q=1\n2[cos(\u0012c+\n\u0012q)\u0006cos(\u0012c\u0000\u0012q)]. Concentrating on slow perturbations\naround the semiclassical ( S \u001d 1) precession trajectory\n\u001ec(t) = \nt, the tunnelling action can be expressed as\nST' \u0000iR1\n\u00001dtsTwith39\nsT=GT\n2Z1\n\u00001d\u000fX\n\u001b\u001b0=\u0006NF;\u001b0NS;\u001b\bcos\u0012q+\u001b\u001b0cos\u0012c\n2\n(20)\n\u0002[ei\u0011\u001b\u001b0fF(1\u0000fS) +e\u0000i\u0011\u001b\u001b0fS(1\u0000fF)]\n\u00001 +\u001b\u001b0(cos\u0012)c\n2[fF(1\u0000fS) +fS(1\u0000fF)]\t\n;\nwhere\u0011\u001b\u001b0=\u001f+\u000f\u0018S\u0000(\u000f\u0000V\u0000\n\u001b\u001b0)\u0018F\u00002\u001eq\n\u001b\u001b0\n\n. Here,\nwe have neglected terms that renormalize \n. For comput-\ning time averages, the source \felds are taken nonzero be-\ntweent= 0 andt=t0, e.g.\u001f(t) =\u001f\u0012(jt0j\u0000jtj)\u0012(tsgnt0).\nThe results (3{5) can be found as Ic=\u0000i@\u001fsTj0,\n_ES=\u0000i@\u0018SsTj0, and\u001cz=1\n2i@\u001eqsTj0, wherej0indicates\n\u001eq=\u0012q=\u001f=\u0018S=F= 0. Expansion around the saddle\npoint gives Eq. (8), and the correlator characterizing the\nspin torque noise is D=\u00001\n8@2\n\u001eqsTj0csc2\u0012=\u00001\n8@2\n\u0012qsTj0.\nA. Intrinsic damping\nWe can include the phenomenological Gilbert damp-\ning termA0m\u0002_mof the LLG equation into a corre-\nsponding term in the action, iSG=R1\n\u00001dtsG(t). With\nthe weak-damping assumptions _\u001ec'\n,j_\u0012cj \u001c j _\u001ej,\nthe leading term in the torque is produced by sG'\n\u00002iSA0\n sin2(\u0012c)\u001eq.\nFurther reasoning is required for thermodynamic con-\nsistency. Let us \frst assume that the Gilbert damping\nis caused by a coupling that ultimately dissipates energy\ninto the bath of conduction electrons in F ( \u0015= 1). We\ncan express the conservation of energy in conversion of\nmagnetic energy to energy of conduction electrons as the\nsymmetrysG[\u0018F+x;\u001eq+ \nx=2] =sG[\u0018F;\u001eq] for allx.\nIn addition, to preserve the thermodynamic \ructuation\nrelations and the second law at equilibrium, the \ructu-\nation symmetry sG[\u0018F;\u001eq] =sG[iT\u00001\nF\u0000\u0018F;\u0000\u001eq] should\nbe ful\flled.39The above \fxes the series expansion in \u0018F,\n\u001eq,T\u00001\nFto have the form\nsG[\u0018F;\u001eq]'\u00002A0Ssin2(\u0012c)\"\ni\n\u0012\n\u001eq\u0000\n2\u0018F\u0013\n(21)\n+ 2TF(\u001eq\u0000\n2\u0018F)2#\n+::: :\nIf the Gilbert damping dissipates energy directly to mul-\ntiple baths (e.g. magnons, phonons), more terms of\nthis form appear, where \u0018FandTFshould be replaced\nby the corresponding bath variables, and only a frac-\ntion 0\u0014\u0015\u00141 of the total A0comes from conduc-\ntion electrons. Including Eq. (21) in the total actionS=S0+ST+SGthen produces e.g. the correlation func-\ntion of the Langevin noise terms in Eq. (8), and the addi-\ntional term in the heat balance equation Eq. (9). These\nare of course possible to \fnd also directly, by assuming\nthe \ructuation-dissipation theorem, and reasoning about\nmagnetic work done by the damping.\nFor the external rf drive, we similarly have a term\nsrf= 2iSmq\u0001\rhrf'2i\rhrfSsin(\u0012c)\u001eq, at resonance.\nIt does not obey the above energy conservation symme-\ntry, as power is externally provided and the mechanism\ngenerating hrfis not included in the model. As a conse-\nquence, as noted in Eq. (12) _ES;tot+_EF;tot6= 0, and the\n\ructuation relation (7) is modi\fed.\nB. Spin oscillator\nThe probability distribution of the magnetization angle\n\u0012can be obtained from Eqs. (19,20)39,44, within a semi-\nclassical method applied to ~ sT=sTj\u0012q=\u001f=\u0018j=039,51. In\nthis approach, at equilibrium, the \ructuation symmetry\n~sT(\u001eq=\u0000i\n=2T) = 0 results to the Boltzmann distri-\nbutionP(cos\u0012) =NeScos(\u0012)\n=T. In the nonequilibrium\ndriven state ( V6= 0,\u000eT6= 0), the distribution deviates\nfrom this.\nThe probability distribution is shown in Fig. 5(c) for\nthe thermally driven oscillator. The \fgure shows the spin\ntorque-driven transition from the magnetization pointing\nin the direction of the magnetic \feld (cos \u0012= 1) for high\nTF, to the opposite direction of the \feld (cos \u0012=\u00001)\nat lowTF. In the intermediate range TF\u00190:25{0:3Tc,\nthe probability distribution becomes bimodal, re\recting\nthe two locally stable con\fgurations in Fig. 5(b): one of\nthese corresponds to the oscillating state.\nC. Emission spectrum\nA driven spin oscillator produces electromagnetic emis-\nsion which can be detected.46,47This can be character-\nized with the classical correlator of the magnetic dipole,\nwhose spectrum is approximately a Lorentzian centered\nat frequency \n. The classical spectrum of the magnetic\ndipole correlator can be written as\nSxx(!) =S2Z1\n\u00001dt0ei!t0hmx(t0)mx(0)i; (22)\nwheremx= cos\u001esin\u0012, and the average is over\nthe driven steady state of the system. To evalu-\nate it, the average over \u001ecan be taken \frst, not-\ning thathcos\u001e(t0) cos\u001e(0)i\u001e=1\n2Rehei\u001e(t0)\u0000i\u001e(0)i\u001e=\n1\n2ReR\nD[\u001ec;\u0012q]eiSei\u001ec(t0)\u0000i\u001ec(0)=1\n2ReR\nD[\u001ec;\u0012q]eiS0,\nwhere the exponential factor is removed by a shift\n(cos\u0012)q7!(cos\u0012)q+ sgn(t0)\u0012(jt0j\u0000jtj)\u0012(tsgnt0)=(2S).\nForS \u001d 1, this results to S0\u0000S'\nt0+\nijt0jS\u00002Dcsc2\u0012c=: (t0) so thathmx(t0)mx(0)i\u001e'7\n1\n2sin2\u0012Reei (t0). Evaluating the Fourier transform, we\nget\nSxx(!)'1\n2X\n\u0006hD=[(!\u0006\n)2+ (S\u00002Dcsc2\u0012c)2]i\u0012:\n(23)\nA similar calculation is done in Ref. 44, via Langevin\nand Fokker{Planck approaches. The remaining average\nis over the steady state distribution P(cos\u0012).\nThe linewidth of the spectrum [black line in Fig. 5(d)]\nin this nonequilibrium system is a non-trivial function of\nthe system parameters. For TF\u00190:31TCprecession at \u0012\u0003\nbecomes possible, and as a result the linewidth ( /csc2\u0012)\nnarrows rapidly, becoming signi\fcantly smaller than the\nnear-equilibrium \ructuations at \u0012\u00180;\u0019.\nVI. DISCUSSION\nIn this work, we explain how the thermomagnetoelec-\ntric e\u000bect of a spin-split superconductor couples the mag-netization in a magnetic tunnel junction to the temper-\nature di\u000berence across it. The thermoelectric coe\u000ecient\nin the superconducting state is generally large, and en-\nables a magnetic Peltier e\u000bect and thermal spin torque,\nwith prospects for generating thermally driven oscilla-\ntions detectable via spectroscopy. Superconductivity also\no\u000bers possibilities to characterize and control the thermal\nphysics via both the electric and magnetic responses or\nexternal \feld coupling of the magnetization.\nACKNOWLEDGMENTS\nWe thank A. Di Bernardo for discussions. 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Nussinov, A. Shnirman, and A. V. Balatsky,\nPhys. Rev. Lett. 92, 107001 (2004).\n51A. Kamenev, Field theory of non-equilibrium systems\n(Cambridge University Press, 2011).\n52G. Eilenberger, Z. Phys 214, 195 (1968).\nAppendix A: Tunneling currents\nCalculation of the tunneling currents from the\nmodel (1) in the main text can be done with standard\nGreen function approaches.33Assuming a spin and mo-\nmentum independent matrix element ( Wjj0=W), the\nk-spin component of the spin current to S reads:\nIk\ns=GT\n32Z1\n\u00001d\u000ftr\u001bk\n2[(R\u0014gFRy)+\u0014gS\u0000\u0014gS(R\u0014gFRy)\u0000]K;\n(A1)\nwhere the superscript Krefers to the Keldysh com-\nponent and GT=\u0019\u0017F\u0017SjWj2is the normal state\ntunneling conductance. The charge and energy cur-\nrents can be obtained by replacing \u001bk=27!^\u001c3and\n\u001bk=27!\u000fin Eq. (A1), respectively. Here, \u001bj\nand ^\u001cjare Pauli matrices in the spin and Nambuspaces, with the basis ( \"; #;\u0000 y\n#; y\n\"), andX+(\u000f;t) =R\ndt0ei\u000f(t\u0000t0)X(t;t0),X\u0000(\u000f;t) =R\ndt0ei\u000f(t0\u0000t)X(t0;t).\nMoreover, \u0014gF=S(\u000f) =2i\n\u0019\u0017F=S^\u001c3P\nj\u0014GF=S(\u000f;pj) are state-\nsummed Keldysh Green's functions, normalized by the\ntotal density of states (DOS) \u0017F=Sat Fermi level, of the\nferromagnet and the spin-split superconductor. The ro-\ntation matrix\nR=e\u0000i\u001e\u001bz=2e\u0000i\u0012\u001by=2ei\u001e\u001bz=2\n\u0002e\u0000iRtdt_\u001e(1\u0000cos\u0012)\u001bz=2e\u0000iV^\u001c3t(A2)\ncontains the Euler angles of the time-dependent magne-\ntization direction vector ( m\u0001\u001b=R\u001bzRy), a Berry phase\nfactor, and voltage bias V. The Berry phase appears\nfrom the Green function31,32of the conduction electrons\nin F following adiabatically the changing magnetization.\nFor a metallic ferromagnet, ^ gR\nF\u0000^gA\nF'2P\n\u0006(^\u001c3\u0006\u001bz)\u0017F;\u0006\n\u0017F\nand ^gK= [^gR\u0000^gA](1\u00002f0(\u000f)), where\u0017F;\"=#:=\u0017F;\u0006are\nthe densities of states of majority/minority spins at the\nFermi level and f0(\u000f) = (1 +e\u000f=T)\u00001is the Fermi distri-\nbution function.\nEvaluating Eq. (A1) for the di\u000berent currents pro-\nduces Eqs. (3{5) in the main text, with NS=F;\u001b =\u0006=\n1\n2tr[1+^\u001c3\n21+\u001b\u001bz\n2(^gR\nS=F\u0000^gA\nS=F)].\nBeyond linear response (6), we \fnd the second-order\ncontributions to the current and torque:\n\u000e(2)Ic=\u0000\u000b\u0000\n2;0\n2\u0014\nsin2(\u0012)\u0010\nV\n\u0000Pcos\u0012\n4\n2\u0011\n(A3)\n+Pcos(\u0012)V2\u0015\n\u0000Pcos(\u0012)A\n2\u0012\u000eT\nT\u00132\n\u0000B\u000eT\nTV ;\n\u000e(2)\u001cz\nsin2(\u0012)=\u000b\u0000\n2;0\n4h\nV2\u0000Pcos(\u0012)V\n +3 + cos(\u0012)\n8\n2i\n(A4)\n+A\n4\u0012\u000eT\nT\u00132\n+B\n4\u000eT\nT\n;\nwhere\u000b\u0007\ni;j =\u0000(GT=2)R1\n\u00001d\u000f\u000fj[NS;+(\u000f)\u0007\nNS;\u0000(\u000f)]f(i)\n0(\u000f), andA= 2\u000b\u0000\n1;1+\u000b\u0000\n2;2,B=\u000b+\n1;0+\u000b+\n2;1.\nFor \n\u001c\u0001, the onset of the voltage driven spin oscil-\nlations [Fig. 5(a)] can be determined from Eqs. (6) and\n(A4) to occur at Vo=\u00064q\ne2SAe\u000b\n=\u000b\u0000\n2;0.\nIn addition to the spin transfer torque (STT) discussed\nin the main text, the electron transfer between F and\nthe spin-split S generates also other torque components\nacting onF. This e\u000bect can be found from Eq. (A1), and\nappears in the torque components \u001cx=yperpendicular to\nthe equilibrium magnetization ^ z.\nIn the main text, we neglect these torques, because\nany equilibrium torques can be absorbed to a renormal-\nization of the e\u000bective magnetic \feld, and moreover, in\nthe limit of weak damping and torques the components\nperpendicular to ^ zsuch that\u001cx=y\u001cS\n have little e\u000bect\non the dynamics. In contrast, the component in the main\ntext has a signi\fcant e\u000bect already at \u001cz\u0018A0S\n\u001cS\n.9\nFor completeness, we write here the expressions for all\ntorques, as obtained from Eq. (A1). Equation (5) in the\nmain text gives the dissipative contribution to \u001cz. Similar\ncontributions can be found for \u001cx=y:\n\u001cx=y=\u0000GT\n8Z1\n\u00001d\u000fX\n\u001b\u001b0(1 +\u001b\u001b0cos\u0012)2\n2NS;x=y (A5)\n\u0002[fF(\u000f\u0000\n\u001b\u001b0\u0000V)\u0000fS(\u000f)];\nwhereNS;0=x=y=z =1\n2tr1+\u001c3\n2\u001b0=x=y=z\n2(^gR\nS\u0000^gA\nS).\nIn addition, there are two remaining contributions, the\nequilibrium spin torque, and a Kramers{Kronig counter-\npart to the density of states term. Terms of the latter\ntype commonly appear in calculations of time-dependent\nresponse. To \fnd it, we need ^ gR+A= ^gR+ ^gA. We can\nevaluate them e.g. in a model with a parabolic spec-\ntrum in 3D, \u0018k=k2=(2m)\u0000\u0016. In the superconductor,\nh;\u0001\u001c\u0016Sand in the magnet, \u0001 = 0. Evaluating the\nmomentum sum yields\n^gR+A\nS\u0016S!1'^gR\nS;qcl+ ^gA\nS;qcl+ ^gR+A\nFjhF7!h;\u0016F7!\u0016S;(A6)\n^gR+A\nF = 2iaRep\n\u0000[(\u000f\u0000hF\u001bz)^\u001c3+\u0016F]=j\u0016Fj+C:\nHere ^gR=A\nS;qcl, are quasiclassical low-energy Green\nfunctions52, 1=a =P\n\u0006p\n1\u0006hF=\u0016F, and\nhF=\u00172\nF#\u0000\u00172\nF\"\n\u00172\nF#+\u00172\nF\"\u0016Fthe internal exchange \feld in F\nin the model. Moreover, Care scalars independent of\n\u000f,h, and \u0001, and drop out from expressions for the\nobservables here.\nNeglecting terms of order \u0001 =\u0016;T=\u0016; \n=\u0016, we \fnd the\nremaining terms in the spin current,\nI00\nS=I00\nS;eq+\u000eI00\nS (A7)\n\u000eI00\nS=\u0000GT\n64Z1\n\u00001d\u000fX\n\u001b\u001b0tanh\u000f\u0000\n\u001b\u001b0\u0000V\n2TF(A8)\n\u0002(\u001b^z+\u001b0m(t))\u0002P(\u000f)NF;\u001b0;whereP(\u000f) =1\n2itr1+\u001c3\n2\u001b[^gR\nS;qcl(\u000f) + ^gA\nS;qcl(\u000f)]. It has the\nsymmetryP(\u0000\u000f) =P(\u000f). For a BCS superconductor,\nthe integrand is nonzero only inside the gap, j\u000f\u0006hj<\u0001.\nThe equilibrium spin current I00\nS;eqis related to the\nexchange coupling between F and FI mediated by the\nelectrons in the superconductor. It can be absorbed to a\nsmall renormalization of the e\u000bective magnetic \feld act-\ning on F. While its value can be calculated in the above\ntunneling model, the model is not su\u000ecient for describ-\ning this non-Fermi surface term in the realistic situation.\nThe superconducting correction \u000eI00\nSvanishes at equilib-\nrium, but may contribute to nonequilibrium response.\nThis torque however has \u001c00z= 0 and can be neglected\nsimilarly as in Eq. (A5).\nAppendix B: Adiabatic Green function\nIn the tunneling calculation of Eq. A1, an expres-\nsion for the adiabatic Green function of the elec-\ntrons on the ferromagnet with dynamic magnetiza-\ntion appears. For completeness, we discuss its mean-\ning here. The nonequilibrium Green function for free\nelectrons in a time-dependent exchange \feld, H(t) =P\nn\u001b\u001b0cy\nn\u001b[Hn(t)]\u001b\u001b0cn\u001b0,Hn(t) =\u000fn+h(t)\u0001\u001b, with a\nthermal initial state at t= 0 isG>\nn(t;t0) =\u0000iUn(t;0)(1\u0000\n\u001an)Un(0;t0)y, wherei@tUn(t;t0) = [\u000fn\u0000h(t)\u0001\u001b]Un(t;t0),\nU(t;t) = 1, and \u001an= [1 +eHn(0)=T]\u00001. In an\nadiabatic approximation for j_hj \u001ch2,Un(t;t0)'\ne\u0000i(t\u0000t0)\u000fnR(t)ei'n(t;t0)\u001bz=2R(t0)y, whereR(t)\u001bzR(t)y=\nh(t)\u0001\u001band'n(t;t0) =iRt\nt0dt00tr\u001bzR(t00)y@t00R(t00). In\nterms of Euler angles h= (cos\u001esin\u0012;sin\u001esin\u0012;cos\u0012) we\nwriteR=e\u0000i\u001e\u001bz=2e\u0000i\u0012\u001by=2ei\u001e\u001bz=2e\u0000i\u001f\u001bz=2. The function\n\u001f(t) is arbitrary, but Undoes not depend on it. For\nsimplicity, we choose \u001f=Rtdt0_\u001e(1\u0000cos\u0012), which gives\n'n= 0. With this choice, the adiabatic Green function\nbecomes\nG>\nn(t;t0) =R(t)G>\nn;0(t\u0000t0)R(t0)y; (B1)\nand the electron Berry phase appears only in the rotation\nmatrix. This is equivalent to the \\rotating frame\" picture\nused in the main text and other works28,30." }, { "title": "1602.07325v1.Experimental_Investigation_of_Temperature_Dependent_Gilbert_Damping_in_Permalloy_Thin_Films.pdf", "content": "1 Experimental Investigation of Temperature-Dependent Gilbert \nDamping in Permalloy Thin Films \nYuelei Zhao1,2†, Qi Song1,2†, See-Hun Yang3, Tang Su1,2, Wei Yuan1,2, Stuart S. P. Parkin3,4, Jing \nShi5*, and Wei Han1,2* \n1International Center for Quantum Materials, Peking University, Beijing, 100871, P. R. China \n2Collaborative Innovation Center of Quantum Matter, Beijing 100871, P. R. China \n3IBM Almaden Research Center, San Jose, California 95120, USA \n4Max Planck Institute for Microstructu re Physics, 06120 Halle (Saale), Germany \n5Department of Physics and Astronomy, Univers ity of California, Riverside, California 92521, \nUSA \n†These authors contributed equally to the work \n*Correspondence to be addressed to: jing.shi @ucr.edu (J.S.) and weihan@pku.edu.cn (W.H.) \n \n \nAbstract \nThe Gilbert damping of ferromagnetic materials is arguably the most important but least \nunderstood phenomenological parameter that dictates real-time magnetization dynamics. \nUnderstanding the physical origin of the Gilbert damping is highly relevant to developing future \nfast switching spintronics devices such as magnetic sensors and magnetic random access memory. Here, we report an experimental stud y of temperature-dependent Gilbert damping in \npermalloy (Py) thin films of varying thicknesses by ferromagnetic resonance. From the thickness \ndependence, two independent cont ributions to the Gilbert damping are identified, namely bulk \ndamping and surface damping. Of particular inte rest, bulk damping decreases monotonically as \nthe temperature decreases, while surface da mping shows an enhancement peak at the 2 temperature of ~50 K. These results provide an important insight to the physical origin of the \nGilbert damping in ultr athin magnetic films. \n \nIntroduction \nIt is well known that the magnetization dynamics is described by the Landau-Lifshitz-Gilbert \nequation with a phenomenological parameter called the Gilbert damping ( α),1,2: \n eff\nSdM dMMH Mdt M dtαγ=− × + × \n (1) \nwhere M\nis the magnetization vector, γis the gyromagne tic ratio, and SM M=\n is the saturation \nmagnetization. Despite intense theore tical and experimental efforts3-15, the microscopic origin of \nthe damping in ferromagnetic (FM) metallic ma terials is still not well understood. Using FM \nmetals as an example, vanadium doping decreases the Gilbert damping of Fe3 while many other \nrare-earth metals doping increase s the damping of permalloy (Py)4-6,16. Theoretically, several \nmodels have been developed to explain some key characteristics. For example, spin-orbit \ncoupling is proposed to be the intrinsic or igin for homogenous time-varying magnetization9. The \ns-d exchange scattering model assumes that damp ing results from scattering of the conducting \nspin polarized electrons with the magnetization10. Besides, there is the Fermi surface breathing \nmodel taking account of the spin scattering with the lattice defects ba sed on the Fermi golden \nrule11,12. Furthermore, other damping mechanisms in clude electron-electron scattering, electron-\nimpurity scattering13 and spin pumping into the adjacent nonmagnetic layers14, as well as the two \nmagnon scattering model, which refers to that pa irs of magnon are scatte red by defects, and the \nferromagnetic resonance (FMR) mode moves into short wavelength spin waves, leading to a 3 dephasing contribution to the linewidth15. In magnetic nanostructu res, the magnetization \ndynamics is dictated by the Gilbert damping of the FM materials which can be simulated by \nmicromagnetics given the boundaries and dimens ions of the nanostructures. Therefore, \nunderstanding the Gilbert damping in FM materials is particularly important for characterizing \nand controlling ultrafast responses in magnetic nanostructures that ar e highly relevant to \nspintronic applications such as magne tic sensors and magnetic random access memory17. \nIn this letter, we report an expe rimental investigation of the G ilbert damping in Py thin films \nvia variable temperature FMR in a modified multi-functional insert of physical property \nmeasurement system with a coplanar waveguide (see methods for details). We choose Py thin \nfilms since it is an interesting FM metallic material for spintronics due to its high permeability, nearly zero magnetostriction, low coercivity, a nd very large anisotropi c magnetoresistance. In \nour study, Py thin films are gr own on top of ~25 nm SiO\n2/Si substrates with a thickness ( d) range \nof 3-50 nm by magnetron sputtering (see methods for details). A capping layer of TaN or Al 2O3 \nis used to prevent oxidation of the Py during m easurement. Interestingly, we observe that the \nGilbert damping of the thin Py films ( d <= 10 nm) shows an enhanced peak at ~ 50 K, while \nthicker films ( d >= 20 nm) decreases monotonically as the temperature decreases. The distinct \nlow-temperature behavior in the Gilbert dampi ng in different thickness regimes indicates a \npronounced surface contribution in the thin limit. In fact, from the linear relationship of the \nGilbert damping as a function of the 1/ d, we identify two contribu tions, namely bulk damping \nand surface damping. Interestingl y, these two contributions show very different temperature \ndependent behaviors, in whic h the bulk damping decreases m onotonically as the temperature \ndecreases, while the surface damping indicates an enhancement peak at ~ 50 K. We also notice \nthat the effective magnetization sh ows an increase at the same temperature of ~50 K for 3 and 5 4 nm Py films. These observations could be all related to the magnetization reorientation on the \nPy surface at a certain temperatur e. Our results are important for theoretical investigation of the \nphysical origins of Gilbert damping and also us eful for the purpose of designing fast switching \nspintronics devices. \nResults and Discussion \nFigure 1a shows five representative curves of the forward amplitude of the complex \ntransmission coefficients (S 21) vs. in plane magnetic field meas ured on the 30 nm Py film with \nTaN capping at the frequencies of 4, 6, 8, 10 an d 12 GHz and at 300 K after renormalization by \nsubtracting a constant background. These experiment al results could be fitted using the Lorentz \nequation18: \n 2\n21 0 22()\n() ( )resHSSHH HΔ∝Δ+ − (2) \nwhere S0 is the constant describing the coefficient for the transmitted microwave power, H is the \nexternal magnetic field, Hres is the magnetic field under the resonance condition, and ΔH is the \nhalf linewidth. The extracted ΔH vs. the excitation frequency ( f) is summarized in Figures 1b and \n1c for the temperature of 300 K and 5 K respect ively. The Gilbert damp ing could be obtained \nfrom the linearly fitted curves (red lin es), based on the following equation: \n 02()H fHπαγΔ= + Δ (3) \nin which γ is the geomagnetic ratio and ΔH0 is related to the inhom ogeneous properties of the \nPy films. The Gilbert damping at 300 K and 5 K is calculated to be 0.0064 ± 0.0001 and 0.0055 \n± 0.0001 respectively. 5 The temperature dependence of the Gilbert damp ing for 3-50 nm Py films with TaN capping \nlayer is summarized in Figure 2a. As d decreases, the Gilbert damping increases, indicative of \nthe increasing importance of the film surfaces. Interestingly, fo r thicker Py films (e.g. 30 nm), \nthe damping decreases monotonically as the temper ature decreases, which is expected for bulk \nmaterials due to suppressed sca ttering at low temperature. As d decreases down to 10 nm, an \nenhanced peak of the damping is obser ved at the temperature of ~ 50 K. As d decreases further, \nthe peak of the damping becomes more pronounce d. For the 3 nm Py film, the damping shows a \nslight decrease first from 0.0126 ± 0.0001 at 3 00 K to 0.0121 ± 0.0001 at 175 K, and a giant \nenhancement up to 0.0142 ± 0.0001 at 50 K, and then a sharp decrease back down to 0.0114 ± \n0.0003 at 5 K. \nThe Gilbert damping as a function of the Py th icknesses at each temperature is also studied. \nFigure 2b shows the thickness dependence of the Py damping at 300 K. As d increases, the \nGilbert damping decreases, which indicates a surface/interface enhanced damping for thin Py \nfilms19. To separate the damping due to the bul k and the surface/interface contribution, the \ndamping is plotted as a function of 1/ d, as shown in Figure 2c, and it follows this equation as \nsuggested by theories19-21. \n 1()BSdαα α=+ (4) \nin which the Bα and Sα represent the bulk and surface da mping, respectively. From these \nlinearly fitted curves, we are able to separate the bulk damping term and the surface damping \nterm out. In Figure 2b, the best fitted parameters for Bα and Sα are 0.0055 ± 0.0003 and 0.020 ± \n0.002 nm. To be noted, there are two insulating mate rials adjacent to the Py films in our studies. 6 This is very different from previous studies on Py/Pt bilayer systems, where the spin pumping \ninto Pt leads to an enhanced magnetic dampi ng in Py. Hence, the enhanced damping in our \nstudies is very unlikely resulti ng from spin pumping into SiO 2 or TaN. To our knowledge, this \nsurface damping could be related to interfacial spin f lip scattering at the interface between Py \nand the insulating layers, which ha s been included in a generalized spin-pumping theory reported \nrecently21. \nThe temperature dependence of the bulk damp ing and the surface damping are summarized \nin Figures 3a and 3b. The bulk damping of Py is ~0.0055 at 300 K. As the temperature decreases, \nit shows a monotonic decrea se and is down to ~0.0049 at 5 K. Th ese values are consistent with \ntheoretical first principle calculations21-23 and the experimental valu es (0.004-0.008) reported for \nPy films with d ≥ 30 nm24-27. The temperature dependence of the bulk damping could be \nattributed to the magnetization rela xation due to the spin-lattice scattering in the Py films, which \ndecreases as the temperature decreases. \nOf particular interest, the surface damping sh ows a completely different characteristic, \nindicating a totally different mechanism from th e bulk damping. A strong enhancement peak is \nobserved at ~ 50 K for the surface damping. Could this enhancement of this surface/interface \ndamping be due to the strong spin-orbit coupli ng in atomic Ta of Ta N capping layer? To \ninvestigate this, we measure the damping of the 5 nm and 30 nm Py films with Al 2O3 capping \nlayer, which is expected to exhibit much lo wer spin-orbit coupling compared to TaN. The \ntemperature dependence of the Py damping is su mmarized in Figures 4a and 4b. Interestingly, \nthe similar enhancement of the damping at ~ 50 K is observed for 5 nm Py film with either Al 2O3 \ncapping layer or TaN layer, whic h excludes that the origin of the feature of the enhanced 7 damping at ~50 K results from th e strong spin-orbit coupling in TaN layer. These results also \nindicate that the mechanism of this feature is most likely related to the common properties of Py \nwith TaN and Al 2O3 capping layers, such as the crysta lline grain boundary and roughness of the \nPy films, etc. \nOne possible mechanism for the observed peak of the damping at ~50 K could be related to a \nthermally induced spin reorientation transition on the Py surface at that temperature. For \nexample, it has been show n that the spin reorientation of Py in magnetic tunnel junction structure \nhappens due to the competition of different magne tic anisotropies, which c ould give rise to the \npeak of the FMR linewidth around the temperature of ~60 K28. Furthermore, we measure the \neffective magnetization ( Meff) as a function of temperature. Meff is obtained from the resonance \nfrequencies ( fres) vs. the external magnetic field via the Kittel formula29: \n 12() [ ( 4 ) ]2res res res efffH H Mγππ=+ (4) \nin which Hres is the magnetic field at the resonance condition, and Meff is the effective \nmagnetization which contains the saturation ma gnetization and other anisotropy contributions. \nAs shown in Figures 5a and 5b, the 4π*M eff for 30 nm Py films w ith TaN capping layer are \nobtained to be ~10.4 and ~10.9 kG at 300 K and 5 K respectively. The temperature dependences \nof the 4π*M eff for 3nm, 5 nm, and 30 nm Py films are s hown in Figures 6a-6c. Around ~50 K, an \nanomaly in the effective magnetization for thin Py films (3 and 5 nm) is observed. Since we do \nnot expect any steep change in Py’s saturation magnetization at this temperature, the anomaly in \n4π*M eff should be caused by an anisot ropy change which coul d be related to a sp in reorientation. \nHowever, to fully understand the underlying mechan isms of the peak of the surface damping at ~ \n50 K, further theoretical and e xperimental studies are needed. 8 Conclusion \nIn summary, the thickness and temperature dependences of the Gilbert damping in Py thin \nfilms are investigated, from which the contributio n due to the bulk damping and surface damping \nare clearly identified. Of particular interest, the bulk damping decreases monotonically as the \ntemperature decreases, while the surface damping develops an enhancement peak at ~ 50 K, \nwhich could be related to a thermally induced spin reorientation for the surface magnetization of \nthe Py thin films. This model is also consistent with the observation of an enhancement of the \neffective magnetization below ~50 K. Our expe rimental results will contribute to the \nunderstanding of the intrinsic and ex trinsic mechanisms of the Gilber t damping in FM thin films. \n \nMethods \nMaterials growth. The Py thin films are deposited on ~25 nm SiO 2/Si substrates at room \ntemperature in 3×10- 3 Torr argon in a magnetron sputtering sy stem with a base pressure of ~ \n1×10-8 Torr. The growth rate of the Py is ~ 1 Å/s. To prevent ex situ oxidation of the Py film \nduring the measurement, a ~ 20 Å TaN or Al 2O3 capping layer is grown in situ environment. The \nTaN layer is grown by reactive sputtering of a Ta target in an argon-nitrogen gas mixture (ratio: \n90/10). For Al 2O3 capping layer, a thin Al (3 Å) layer is deposited first, and the Al 2O3 is \ndeposited by reactive spu ttering of an Al target in an ar gon-oxygen gas mixture (ratio: 93/7). \nFMR measurement. The FMR is measured using the vector network analyzer (VNA, Agilent \nE5071C) connected with a coplanar wave guide30 in the variable temperature insert of a \nQuantum Design Physical Properties Measuremen t System (PPMS) in the temperature range \nfrom 300 to 2 K. The Py sample is cut to be 1 × 0.4 cm and attached to the coplanar wave guide 9 with insulating silicon paste. For each temper ature from 300 K to 2 K, the forward complex \ntransmission coefficients (S 21) for the frequencies between 1 - 15 GHz are recorded as a function \nof the magnetic field sweeping from ~2500 Oe to 0 Oe. \n \nContributions \nJ.S. and W.H. proposed and supervised the studies. Y.Z. and Q.S. performed the FMR \nmeasurement and analyzed the data. T.S. and W.Y. helped the measurement. S.H.Y. and S.S.P.P. \ngrew the films. Y.Z., J.S. and W.H. wrote the manuscript. All authors commented on the \nmanuscript and contributed to its final version. \n \nAcknowledgements \nWe acknowledge the fruitful discussions with Ryuichi Shindou, Ke Xia, Ziqiang Qiu, Qian \nNiu, Xincheng Xie and Ji Feng and the support of National Basic Research Programs of China \n(973 Grants 2013CB921903, 2014CB920902 and 2015 CB921104). Wei Han also acknowledges \nthe support by the 1000 Talents Program for Young Scientists of China. \n \nCompeting financial interests \nThe authors declare no compe ting financial interests. \n \n \nReferences: \n \n1 Landau, L. & Lifshitz, E. On the theory of the dispersion of magnetic permeability in \nferromagnetic bodies. Phys. Z. Sowjetunion 8, 153 (1935). \n2 Gilbert, T. L. A phenomenological theory of damping in ferromagnetic materials. \nMagnetics, IEEE Transactions on 40, 3443-3449, doi:10.1109/TMAG.2004.836740 \n(2004). 10 3 Scheck, C., Cheng, L., Barsukov, I., Frait, Z. & Bailey, W. E. 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B 78, 020404 (2008). \n14 Brataas, A., Tserkovnyak, Y. & Bauer, G. E. W. Scattering Theory of Gilbert Damping. \nPhys. Rev. Lett. 101, 037207 (2008). \n15 Arias, R. & Mills, D. L. Extrinsic contri butions to the ferromagne tic resonance response \nof ultrathin films. Phys. Rev. B 60, 7395-7409 (1999). \n16 Walowski, J., Müller, G., Djordjevic, M., M ünzenberg, M., Kläui, M., Vaz, C. A. F. & \nBland, J. A. C. Energy Equilibration Pro cesses of Electrons, Magnons, and Phonons at \nthe Femtosecond Time Scale. Phys. Rev. Lett. 101, 237401 (2008). \n17 Stiles, M. D. & Miltat, J. in Spin Dynamics in Confined Magnetic Structures III Vol. 101 \nTopics in Applied Physics (eds Burkard Hillebrands & André Thiaville) Ch. 7, 225-308 \n(Springer Berlin Heidelberg, 2006). \n18 Celinski, Z., Urquhart, K. B. & Heinrich, B. Using ferromagnetic resonance to measure \nthe magnetic moments of ultrathin films. J. Magn. Magn. Mater. 166, 6-26 (1997). \n19 Barati, E., Cinal, M., Edwards, D. M. & Umerski, A. Gilbert damping in magnetic \nlayered systems. Phys. Rev. B 90, 014420 (2014). \n20 Tserkovnyak, Y., Brataas, A., Bauer, G. E. W. & Halperin, B. I. Nonlocal magnetization \ndynamics in ferromagnetic heterostructures. Rev. Mod. Phys. 77, 1375-1421 (2005). \n21 Liu, Y., Yuan, Z., Wesselink, R. J. H., St arikov, A. A. & Kelly, P. J. Interface \nEnhancement of Gilbert Damping from First Principles. Phys. Rev. Lett. 113, 207202 \n(2014). 11 22 Starikov, A. A., Kelly, P. J ., Brataas, A., Tserkovnyak, Y. & Bauer, G. E. W. Unified \nFirst-Principles Study of Gilbert Dampi ng, Spin-Flip Diffusion, and Resistivity in \nTransition Metal Alloys. Phys. Rev. Lett. 105, 236601 (2010). \n23 Mankovsky, S., Ködderitzsch, D., Woltersdo rf, G. & Ebert, H. First-principles \ncalculation of the Gilbert damping paramete r via the linear response formalism with \napplication to magnetic transition metals and alloys. Physical Review B 87, 014430 \n(2013). \n24 Bailey, W., Kabos, P., Mancoff, F. & Russe k, S. Control of magnetization dynamics in \nNi81Fe19 thin films through the us e of rare-earth dopants. Magnetics, IEEE Transactions \non 37, 1749-1754 (2001). \n25 Rantschler, J. O., Maranville, B. B., Malle tt, J. J., Chen, P., McMichael, R. D. & \nEgelhoff, W. F. Damping at no rmal metal/permalloy interfaces. Magnetics, IEEE \nTransactions on 41, 3523-3525 (2005). \n26 Luo, C., Feng, Z., Fu, Y., Zha ng, W., Wong, P. K. J., Kou, Z. X., Zhai, Y., Ding, H. F., \nFarle, M., Du, J. & Zhai, H. R. Enhancem ent of magnetization damping coefficient of \npermalloy thin films with dilute Nd dopants. Phys. Rev. B 89, 184412 (2014). \n27 Ghosh, A., Sierra, J. F., Auffret, S., Ebels, U. & Bailey, W. E. Dependence of nonlocal \nGilbert damping on the ferromagnetic layer t ype in ferromagnet/Cu/Pt heterostructures. \nAppl. Phys. Lett. 98, 052508 (2011). \n28 Sierra, J. F., Pryadun, V. V., Russek, S. E ., García-Hernández, M., Mompean, F., Rozada, \nR., Chubykalo-Fesenko, O., Snoeck, E., Miao, G. X., Moodera, J. S. & Aliev, F. G. \nInterface and Temperature Dependent Magnetic Properties in Permalloy Thin Films and \nTunnel Junction Structures. Journal of Nanoscience and Nanotechnology 11, 7653-7664 \n(2011). \n29 Kittel, C. On the Theory of Ferromagnetic Resonance Absorption. Phys. Rev. 73, 155 \n(1948). \n30 Kalarickal, S. S., Krivosik, P., Wu, M., Patt on, C. E., Schneider, M. L., Kabos, P., Silva, \nT. J. & Nibarger, J. P. Ferromagnetic reso nance linewidth in metallic thin films: \nComparison of measurement methods. J. Appl. Phys. 99, 093909 (2006). \n 12 \nFigure Captions \n \nFigure 1. Measurement of Gilbert damping in Py thin films via ferromagnetic resonance \n(Py thickness = 30 nm). a, Ferromagnetic resonance spectra of the absorption for 30 nm Py thin \nfilms with TaN capping layer at gigahertz frequencies of 4, 6, 8, 10 and 12 GHz at 300 K after \nnormalization by background subtraction. b, c, The half linewidths as a function of the resonance \nfrequencies at 300 K and 5 K respectively. The red solid lines indicate the fitted lines based on \nequation (3), where the Gilbert damp ing constants could be obtained. \n \nFigure 2. Temperature dependence of the Gilber t damping of Py thin films with TaN \ncapping. a, The temperature dependence of the Gilbert damping fo r 3, 5, 10, 15, 20, 30, and 50 \nnm Py films. b, The Gilbert damping as a function of the Py thickness, d, measured at 300 K. c, \nThe Gilbert damping as a function of 1/ d measured at 300 K. The linear fitting corresponds to \nequation (4), in which the slope and the intercep t are related to the surf ace contribution and bulk \ncontribution to the total Gilber t damping. Error bars correspond to one standard deviation. \n Figure 3. Bulk and surface damping of Py thin films with TaN capping layer. a, b, The \ntemperature dependence of the bulk damping an d surface damping, respectively. The inset table \nsummarizes the experimental values reported in early studies. Error bars correspond to one \nstandard deviation. \nFigure 4. Comparison of the Gilbert damping of Py films with different capping layers. a, \nb, Temperature dependence of the Gilbert dampi ng of Py thin films with TaN capping layer 13 (blue) and Al 2O3 capping layer (green) for 5 nm Py a nd 30 nm Py, respectively. Error bars \ncorrespond to one standard deviation. \nFigure 5. Measurement of effective magnetizat ion in Py thin films via ferromagnetic \nresonance (Py thickness = 30 nm). a, b, The resonance frequencies vs. the resonance magnetic \nfield at 300 K and 5 K, respectively. The fitted li nes (red curves) are obtained using the Kittel \nformula. \nFigure 6. Effective magnetization of Py fi lms as a function of the temperature. a, b, c, \nTemperature dependence of the effective magnetizati on of Py thin films of a thickness of 3 nm, \n5 nm and 30 nm Py respectively. In b, c, the blue/green symbols correspond to the Py with \nTaN/Al\n2O3 capping layer. \n \n 0\n500\n1000\n1500\n2000\n-0.3\n-0.2\n-0.1\n0.0\n0.1\n 4 \n 6 \n 8\n 10 \n 12 \n \nS\n21\n (dB) \n \nH (Oe)\nT=300 K\nf\n (GHz)\n0\n4\n8\n12\n16\n0\n10\n20\n30\n \n\nH (Oe)\n \nf (GHz)\nT=300 K\n0\n4\n8\n12\n16\n0\n10\n20\n30\n \n\nH (Oe)\n \nf (GHz)\nT=5 K\nb\nc\na\nFigure 10\n50\n100\n150\n200\n250\n300\n0.006\n0.008\n0.010\n0.012\n0.014\nd\n (nm)\n 3 \n 15 \n 5 \n 20\n 10 \n 30 \n \n 50 \n \na\n \nTemperature (K)\n0.0\n0.1\n0.2\n0.3\n0.004\n0.006\n0.008\n0.010\n0.012\n0.014\n \na\n \n \n1/\nd\n (nm\n-1\n)\n0\n10\n20\n30\n0.006\n0.008\n0.010\n0.012\n0.014\n \n \nd\n (nm)\n \na\na\nb\nc\nFigure \n20\n50\n100\n150\n200\n250\n300\n0.0040\n0.0045\n0.0050\n0.0055\n0.0060\nTheory\n Ref. 21, 22\n Ref. 23\n Temperature (K)\n \na\nB\n \na\na\nExp.\n0.006\nRef. 24\n0.004\n-\n0.008\nRef.\n25\n0.007\nRef.\n26\n0.0067\nRef. 27\n0\n50\n100\n150\n200\n250\n300\n0.016\n0.018\n0.020\n0.022\n0.024\n0.026\n0.028\n0.030\n Temperature (K)\na\nS\n (nm)\n \nb\nFigure \n30\n50\n100\n150\n200\n250\n300\n0.004\n0.006\n0.008\n0.010\n 5 nm Py/TaN\n 5 nm Py/Al\n2\nO\n3\n Temperature (K)\na\n \n \n0\n50\n100\n150\n200\n250\n300\n0.004\n0.005\n0.006\n0.007\n 30 nm Py/TaN\n 30 nm Py/Al\n2\nO\n3\n Temperature (K)\na\n \n \na\nb\nFigure \n4a\nb\n0\n500\n1000\n1500\n2000\n0\n4\n8\n12\n16\n \nf\n (GHz)\n \nH\n (Oe)\nT=300 K\n0\n500\n1000\n1500\n2000\n0\n4\n8\n12\n16\n \nf\n (GHz)\n \nH (Oe)\nT=5 K\nFigure \n58.6\n8.8\n9.0\n9.2\n9.4\n9.6\n4\n\nM\neff\n (kG) \n 5 nm Py/TaN\n 5 nm Py/Al\n2\nO\n3\n \n6.2\n6.3\n6.4\n6.5\n6.6\n6.7\n6.8\n6.9\n4\n\nM\neff\n (kG) \n 3 nm Py/TaN\n \n0\n50\n100\n150\n10.6\n10.7\n10.8\n10.9\n11.0\n 30 nm Py/TaN\n 30 nm Py/Al\n2\nO\n3\n4\n\nM\neff\n (kG) \n \na\nb\nc\nTemperature (K) \nFigure \n6" }, { "title": "1901.01941v1.Giant_anisotropy_of_Gilbert_damping_in_epitaxial_CoFe_films.pdf", "content": "Giant anisotropy of Gilbert damping in epitaxial CoFe \flms\nYi Li,1, 2Fanlong Zeng,3Steven S.-L. Zhang,2Hyeondeok Shin,4Hilal Saglam,2, 5Vedat Karakas,2, 6Ozhan\nOzatay,2, 6John E. Pearson,2Olle G. Heinonen,2Yizheng Wu,3, 7,\u0003Axel Ho\u000bmann,2,yand Wei Zhang1, 2,z\n1Department of Physics, Oakland University, Rochester, MI 48309, USA\n2Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA\n3State Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200433, China\n4Computational Sciences Division, Argonne National Laboratory, Argonne, IL 60439, USA\n5Department of Physics, Illinois Institute of Technology, Chicago IL 60616, USA\n6Department of Physics, Bogazici University, Bebek 34342, Istanbul, Turkey\n7Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China\n(Dated: January 8, 2019)\nTailoring Gilbert damping of metallic ferromagnetic thin \flms is one of the central interests in\nspintronics applications. Here we report a giant Gilbert damping anisotropy in epitaxial Co 50Fe50\nthin \flm with a maximum-minimum damping ratio of 400 %, determined by broadband spin-torque\nas well as inductive ferromagnetic resonance. We conclude that the origin of this damping anisotropy\nis the variation of the spin orbit coupling for di\u000berent magnetization orientations in the cubic lattice,\nwhich is further corroborate from the magnitude of the anisotropic magnetoresistance in Co 50Fe50.\nIn magnetization dynamics the energy relaxation rate\nis quanti\fed by the phenomenological Gilbert damping\nin the Landau-Lifshits-Gilbert equation [1], which is a\nkey parameter for emerging spintronics applications [2{\n6]. Being able to design and control the Gilbert damp-\ning on demand is crucial for versatile spintronic device\nengineering and optimization. For example, lower damp-\ning enables more energy-e\u000ecient excitations, while larger\ndamping allows faster relaxation to equilibrium and more\nfavorable latency. Nevertheless, despite abundant ap-\nproaches including interfacial damping enhancement [7{\n9], size e\u000bect [10, 11] and materials engineering [12{14],\nthere hasn't been much progress on how to manipulate\ndamping within the same magnetic device. The only\nwell-studied damping manipulation is by spin torque [15{\n18], which can even fully compensate the intrinsic damp-\ning [19, 20]. However the requirement of large current\ndensity narrows its applied potential.\nAn alternative approach is to explore the intrinsic\nGilbert damping anisotropy associated with the crys-\ntalline symmetry, where the damping can be continu-\nously tuned via rotating the magnetization orientation.\nAlthough there are many theoretical predictions [21{25],\nmost early studies of damping anisotropy are disguised\nby two-magnon scattering and linewidth broadening due\nto \feld-magnetization misalignment [26{29]. In addition,\nthose reported e\u000bects are usually too weak to be consid-\nered in practical applications [30, 31].\nIn this work, we show that a metallic ferromagnet can\nexhibit a giant Gilbert damping variation by a factor\nof four along with low minimum damping. We inves-\ntigated epitaxial cobalt-iron alloys, which have demon-\nstrated new potentials in spintronics due to their ultralow\ndampings [32, 33]. Using spin-torque-driven and induc-\ntive ferromagnetic resonance (FMR), we obtain a four-\nfold (cubic) damping anisotropy of 400% in Co 50Fe50thin\n\flms between their easy and hard axes. For each angle,the full-range frequency dependence of FMR linewidths\ncan be well reproduced by a single damping parame-\nter\u000b. Furthermore, from \frst-principle calculations and\ntemperature-dependent measurements, we argue that\nthis giant damping anisotropy in Co 50Fe50is due to the\nvariation of the spin-orbit coupling (SOC) in the cu-\nbic lattice, which di\u000bers from the anisotropic density of\nstate found in ultrathin Fe \flm [30]. We support our\nconclusion by comparing the Gilbert damping with the\nanisotropic magnetoresistance (AMR) signals. Our re-\nsults reveal the key mechanism to engineer the Gilbert\ndamping and may open a new pathway to develop novel\nfunctionality in spintronic devices.\nCo50Fe50(CoFe) \flms were deposited on MgO(100)\nsubstrates by molecular beam epitaxy at room temper-\nature, under a base pressure of 2 \u000210\u000010Torr [34]. For\nspin-torque FMR measurements, i) CoFe(10 nm)/Pt(6\nnm) and ii) CoFe(10 nm) samples were prepared. They\nwere fabricated into 10 \u0016m\u000240\u0016m bars by photolithog-\nraphy and ion milling. Coplanar waveguides with 100-\nnm thick Au were subsequently fabricated [18, 35]. For\neach layer structure, 14 devices with di\u000berent orienta-\ntions were fabricated, as shown in Fig. 1(a). The geom-\netry de\fnes the orientation of the microwave current, \u0012I,\nand the orientation of the biasing \feld, \u0012H, with respect\nto the MgO [100] axis (CoFe [1 10]).\u0012Iranges from 0\u000e\nto 180\u000ewith a step of 15\u000e(D1 to D14, with D7 and D8\npointing to the same direction). For each device we \fx\n\u0012H=\u0012I+ 45\u000efor maximal recti\fcation signals. In addi-\ntion, we also prepared iii) CoFe(20 nm) 40 \u0016m\u0002200\u0016m\nbars along di\u000berent orientations with transmission copla-\nnar waveguides fabricated on top for vector network an-\nalyzer (VNA) measurements. See the Supplemental Ma-\nterials for details [36].\nFig. 1(b) shows the angular-dependent spin-torque\nFMR lineshapes of CoFe(10 nm)/Pt devices from dif-\nferent samples (D1 to D4, hard axis to easy axis) atarXiv:1901.01941v1 [cond-mat.mtrl-sci] 7 Jan 20192\nFIG. 1. (a) Upper: crystalline structure, axes of bcc Co 50Fe50\n\flm on MgO(100) substrate and de\fnition of \u0012Hand\u0012I.\nLower: device orientation with respect to the CoFe crystal\naxis. (b) Spin-torque FMR lineshapes of i) CoFe(10 nm)/Pt\ndevices D1 to D4 measured. (c) Resonances of D1 and D4\nfrom (b) for \u00160Hres<0. (d) Resonances of iii) CoFe(20\nnm) for\u0012H= 45\u000eand 90\u000emeasured by VNA FMR. In (b-d)\n!=2\u0019= 20 GHz and o\u000bset applies.\n!=2\u0019= 20 GHz. A strong magnetocrystalline anisotropy\nas well as a variation of resonance signals are observed.\nMoreover, the linewidth increases signi\fcantly from easy\naxis to hard axis, which is shown in Fig. 1(c). We have\nalso conducted rotating-\feld measurements on a sec-\nond CoFe(10 nm)/Pt device from a di\u000berent deposition\nand the observations can be reproduced. This linewidth\nanisotropy is even more pronounced for the CoFe(20 nm)\ndevices without Pt, measured by VNA FMR (Fig. 1d).\nFor the CoFe(10 nm) devices, due to the absence of the\nPt spin injector the spin-torque FMR signals are much\nweaker than CoFe/Pt and completely vanish when the\nmicrowave current is along the easy axes.\nFigs. 2(a-b) show the angular and frequency de-\npendence of the resonance \feld Hres. In Fig. 2(a), the\nHresfor all four sample series match with each other,\nwhich demonstrates that the magnetocrystalline proper-\nties of CoFe(10 nm) samples are reproducible. A slightly\nsmallerHresfor CoFe(20 nm) is caused by a greater e\u000bec-\ntive magnetization when the thickness increases. A clear\nfourfold symmetry is observed, which is indicative of the\ncubic lattice due to the body-center-cubic (bcc) texture\nof Co 50Fe50on MgO. We note that the directions of the\nhard axes has switched from [100] and [010] in iron-rich\nalloys [33] to [110] and [1 10] in Co 50Fe50, which is con-\nω/2πμ0Hres (T) μ0Hres (T) [110]\n[110][100][010](a) (b) CoFe(10 nm)/Pt \nω/2π=2045o90 o135o\n135o180o 225oCoFe(10 nm)/Pt \nCoFe(10 nm) CoFe(20 nm) θH:\n[100]\n[110][010]FIG. 2. (a) Resonance \feld \u00160Hresas a function of \u0012Hat\n!=2\u0019= 20 GHz for di\u000berent samples. Diamonds denote the\nrotating-\feld measurement from the second CoFe(10 nm)/Pt\ndevice. The black curve denotes the theoretical prediction.\n(b)\u00160Hresas a function of frequency for the CoFe(10 nm)/Pt\ndevices. Solid curves denote the \fts to the Kittel equation.\nsistent with previous reports [37, 38].\nThe magnetocrystalline anisotropy can be quanti-\n\fed from the frequency dependence of \u00160Hres. Fig.\n2(b) shows the results of CoFe(10 nm)/Pt when HB\nis aligned to the easy and hard axes. A small uniax-\nial anisotropy is found between [1 10] (0\u000eand 180\u000e) and\n[110] (90\u000e) axes. By \ftting the data to the Kittel equa-\ntion!2=\r2=\u00162\n0(Hres\u0000Hk)(Hres\u0000Hk+Ms), where\n\r= 2\u0019(geff=2)\u000128 GHz/T, we obtain geff= 2:16,\n\u00160Ms= 2:47 T,\u00160H[100]\nk= 40 mT,\u00160H[010]\nk= 65 mT\nand\u00160H[110]\nk=\u00160H[110]\nk=\u000043 mT. Taking the disper-\nsion functions from cubic magnetocrystalline anisotropy\n[39, 40], we obtain an in-plane cubic anisotropy \feld\n\u00160H4jj= 48 mT and a uniaxial anisotropy \feld \u00160H2jj=\n12 mT. Fig. 2(a) shows the theoretical predictions from\nH4jjandH2jjin black curve, which aligns well with all\n10-nm CoFe samples.\nWith good magnetocrystalline properties, we now turn\nto the energy relaxation rate. Fig. 3(a) shows the full-\nwidth-half-maximum linewidths \u00160\u0001H1=2of the spin-\ntorque FMR signals at !=2\u0019= 20 GHz. Again, a fourfold\nsymmetry is observed for CoFe(10 nm)/Pt and CoFe(10\nnm), with the minimal (maximal) linewidth measured\nwhen the \feld lies along the easy (hard) axes. For\nCoFe(10 nm) devices, we did not measure any spin-torque\nFMR signal for HBalong the hard axes ( \u0012H= 45\u000e, 135\u000e\nand 225\u000e). This is due to the absence of the Pt spin\ninjector as well as the near-zero AMR ratio when the rf\ncurrent \rows along the easy axes, which will be discussed\nlater. For all other measurements, the linewidths of CoFe\ndevices are smaller than for CoFe/Pt by the same con-\nstant, independent of orientation (upper diagram of Fig.\n3a). This constant linewidth di\u000berence is due to the spin\npumping contribution to damping from the additional Pt\nlayer [41, 42]. Thus we can deduce the intrinsic damp-\ning anisotropy from CoFe(10 nm)/Pt devices, with the3\nω/2π 105, 195 deg 75, 165 deg 120, 210 deg 135, 225 deg(HA) 45, 135 deg (HA) \n60, 150 deg \n90, 180 deg(EA) θHCoFe(10 nm)/Pt \n(b) = -\n=-\n[100] [110] [110] [010](a) ω/2π=20 \nω/2π θH\n0, 90 deg 15, 75 deg 22.5, 67.5 deg 30, 60 deg 42.5, 50 deg \n40, 52.5 deg \n37.5, 55 deg CoFe(20 nm) (VNA) \n(c)CoFe(10 nm)/Pt CoFe(10 nm) \n90 deg (EA) \nfor CoFe \nFIG. 3. (a) \u00160\u0001H1=2as a function of \u0012Hat!=2\u0019= 20 GHz\nfor the CoFe(10 nm) series in Fig. 2(a). Top: Addtional\nlinewidth due to spin pumping of Pt. The green region de-\nnotes the additional linewidth as 4 :5\u00060:7 mT. (b-c) \u00160\u0001H1=2\nas a function of frequency for (b) CoFe(10 nm)/Pt and (c)\nCoFe(20 nm) samples. Solid lines and curves are the \fts to\nthe data.\ndamping shifted from CoFe(10 nm) devices by a constant\nand is much easier to measure.\nIn Fig. 3(b-c) we show the frequency dependence of\n\u00160\u0001H1=2of CoFe(10 nm)/Pt devices from spin-torque\nFMR and CoFe(20 nm) devices from VNA FMR. For\nboth the easy and hard axes, linear relations are ob-\ntained, and the Gilbert damping \u000bcan be extracted\nfrom\u00160\u0001H1=2=\u00160\u0001H0+ 2\u000b!=\r with the \fts shown\nas solid lines. Here \u00160\u0001H0is the inhomogeneous broad-\nening due to the disorders in lattice structures. In Fig.\n3(b) we also show the linewidths of the CoFe(10 nm)\ndevice along the easy axis ( \u0012H= 90\u000e), which has a\nsigni\fcant lower linewidth slope than the easy axis of\nCoFe(10 nm)/Pt. Their di\u000berences yield a spin pump-\ning damping contribution of \u0001 \u000bsp= 0:0024. By using\n\u0001\u000bsp=\r\u0016hg\"#=(4\u0019MstM), we obtain a spin mixing con-\nductance of g\"#(CoFe/Pt) = 25 nm\u00002, which is compa-\nrable to similar interfaces such as NiFe/Pt [43, 44]. For\n\u0012Hbetween the easy and hard axes, the low-frequency\nlinewidth broadenings are caused by the deviation of\nmagnetization from the biasing \feld direction, whereas\nat high frequencies the \feld is su\u000ecient to saturate the\nmagnetization. In order to \fnd the damping anisotropy,\nwe \ft the linewidths to the angular model developed bySuhl [45, 46], using a single \ft parameter of \u000band the\nextractedH2jjandH4jjfrom Fig. 2. The solid \ftting\ncurves in Fig. 3(b) nicely reproduce the experimental\npoints.\nThe obtained damping anisotropy for all the samples\nare summarized in Fig. 4, which is the main result of\nthe paper. For CoFe(10 nm)/Pt samples, \u000bvaries from\n0.0056 along the easy axis to 0.0146 along the hard axis.\nBy subtracting the spin pumping \u0001 \u000bspfrom both values,\nwe derive a damping anisotropy of 380%. For CoFe(20\nnm) samples measured by VNA FMR, \u000bvaries from\n0.0054 to 0.0240, which yields an anisotropy of 440% and\nreproduces the large anisotropy from spin-torque FMR.\nThis giant damping anisotropy implies, technologically,\nnearly four times smaller critical current to switch the\nmagnetization in a spin-torque magnetic random access\nmemory, or to excite auto-oscillation in a spin-torque os-\ncillator, by simply changing the magnetization orienta-\ntion from the hard axis to the easy axis within the same\ndevice. In addition, we emphasize that our reported\ndamping anisotropy is not subject to a dominant two-\nmagnon scattering contribution, which would be mani-\nfested as a nonlinear linewidth softening at high frequen-\ncies [28, 31]. For this purpose we have extended the fre-\nquency of spin-torque FMR on CoFe(10 nm)/Pt up to 39\nGHz, see the Supplemental Materials for details [36]. We\nchoose CoFe(10 nm)/Pt samples because they provide\nthe best signals at high frequencies and the additional Pt\nlayer signi\fcantly helps to excite the dynamics. Linear\nfrequency dependence of linewidth persists throughout\nthe frequency range and \u0001 H0is unchanged for the two\naxes, with which we can exclude extrinsic e\u000bects to the\nlinewidths. We also note that our result is substantially\ndi\u000berent from the recent report on damping anisotropy\nin Fe/GaAs [30], which is due to the interfacial SOC and\ndisappears quickly as Fe becomes thicker. In compari-\nson, the Gilbert damping anisotropy in Co 50Fe50is the\nintrinsic property of the material, is bonded to its bulk\ncrystalline structure, and thus holds for di\u000berent thick-\nnesses in our experiments.\nIn order to investigate the dominant mechanism for\nsuch a large Gilbert damping anisotropy, we perform\ntemperature-dependent measurements of \u000band the re-\nsistivity\u001a. Fig. 5(a) plots \u000bas a function of 1 =\u001afor\nthe CoFe(10 nm)/Pt and CoFe(20 nm) samples and for\nHBalong the easy and hard axes. The dominant lin-\near dependence reveals a major role of conductivitylike\ndamping behavior. This is described by the breathing\nFermi surface model for transition-metal ferromagnets,\nin which\u000bcan be expressed as [23, 24, 47{49]:\n\u000b\u0018N(EF)j\u0000\u0000j2\u001c (1)\nwhereN(EF) is the density of state at the Fermi level, \u001c\nis the electron relaxation time and \u0000\u0000=h[\u001b\u0000;Hso]iE=EF\nis the matrix for spin-\rip scatterings induced by the SOC\nHamiltonian Hsonear the Fermi surface [48, 49]. Here \u001c4\n(b) CoFe(10 nm) CoFe(20 nm) CoFe(20 nm) CoFe(10 nm)/Pt - ∆α sp \nCoFe(10 nm ) 400 %\n100 %\nFIG. 4. Renormalized damping and its anisotropy for\nCoFe(10 nm) and CoFe(20 nm), measured from spin-torque\nFMR and VNA FMR, respectively. For CoFe(20 nm)/Pt sam-\nples, \u0001\u000bsphas been subtracted from the measured damping.\nis proportional to the conductivity (1 =\u001a) from the Drude\nmodel, with which Eq. (1) gives rise to the behaviors\nshown in Fig. 5(a).\nFor the origin of damping anisotropy, we \frst check\nthe role of N(EF) by ab-initio calculations for di\u000berent\nordered cubic supercells, which is shown in the Supple-\nmental Materials [36]. However, a negligible anisotropy\ninN(EF) is found for di\u000berent magnetization orienta-\ntions. This is consistent with the calculated anisotropy\nin Ref. [30], where less than 0.4% change of N(EF) was\nobtained in ultrathin Fe \flms. The role of \u001ccan also be\nexcluded from the fact that the resistivity di\u000berence be-\ntween the easy and hard axes is less than 2% [36]. Thus\nwe deduce that the giant damping anisotropy of 400% is\ndue to the change of j\u0000\u0000j2, or the SOC, at di\u000berent crys-\ntalline directions. In particular, unlike the single element\nFe, disordered bcc Fe-Co alloy can possess atomic short-\nrange order, which gives rise to local tetragonal crystal\ndistortions due to the di\u000berent lattice constants of Fe and\nCo [2{4]. Such local tetragonal distortions will preserve\nglobal cubic symmetry but can have large e\u000bects on the\nSOC. We emphasize that our CoFe samples, which did\nnot experience annealing, preserve the random disorder.\nOur \frst principle calculations also con\frm the role of lo-\ncal tetragonal distortions and its enhancement on SOC,\nsee the Supplemental Materials for details [36].\nThe anisotropy of the SOC in Co 50Fe50can be re\rected\nby its AMR variation along di\u000berent crystalline orienta-\ntions. The AMR ratio can be de\fned as:\nAMR(\u0012I) =\u001ak(\u0012I)\n\u001a?(\u0012I)\u00001 (2)\nwhere\u001ak(\u0012I) and\u001a?(\u0012I) are measured for the biasing\n\feld parallel and perpendicular to the current direction,\nrespectively. The main contribution of AMR is the asym-\nmetrics-delectron scatterings where the s-orbitals are\nmixed with magnetization-containing d-orbitals due toSOC [53, 54]. Since both the damping and AMR origi-\nnate from SOC and, more precisely, are proportional to\nthe second order of SOC, a large damping anisotropy is\nexpected to be accompanied by a large AMR anisotropy\nand vice versa. Furthermore, due to the fourfold sym-\nmetry, the AMR should be invariant when the current\ndirection is rotated by 90 degrees, as the AMR is a func-\ntion of\u0012Ias de\fned in Eq. (1). Thus the damping and\nAMR should exhibit similar angular dependence on \u0012H\nand\u0012I, respectively.\nIn Fig. 5(b) we compare renormalized \u000b(\u0012H) with\nCoFe(20 nm) CoFe(10 nm)/Pt : (a)\n300 K 8 K F(θI)/F max (b) \n,10 nm \n20 nm 10 nm \n20 nm \nFIG. 5. (a) \u000b(T) as a function of 1 =\u001a(T).T= 8 K, 30 K, 70\nK, 150 K and 300 K for CoFe(10 nm)/Pt and T= 8 K and\n300 K for CoFe(20 nm). Dashed and dotted lines are guides\nto eyes. (b) Renormalized \u000b(\u0012H) and AMR( \u0012I) andF(\u0012I) for\nCoFe(10 nm)/Pt and CoFe(20 nm). Circles, crosses and +\ndenote\u000b, AMR and F, respectively.\nAMR(\u0012I) for 10-nm and 20-nm CoFe samples, where the\nAMR values are measured from Hall bars with di\u000berent\n\u0012I. The AMR ratio is maximized along h100iaxes and\nminimized alongh110iaxes, with a large anisotropy by a\nfactor of 10. This anisotropy is also shown by the inte-\ngrated spin-torque FMR intensity for CoFe(10 nm)/Pt,\nde\fned asF(\u0012I) = \u0001H1=2Vmax\ndc [17, 18] and plotted in\nFig. 5(b). The large AMR anisotropy and its symme-\ntry clearly coincide with the damping anisotropy mea-\nsured in the same samples, which con\frms our hypoth-\nesis of strong SOC anisotropy in CoFe. Thus we con-\nclude that the damping anisotropy is dominated by the\nvariation of SOC term in Eq. (1). In parallel, we also\ncompare\u000b(\u0012H) and AMR( \u0012I) for epitaxial Fe(10 nm)\nsamples grown on GaAs substrates [36]. Experimentally\nwe \fnd the anisotropy less is than 30% for both damping\nand AMR, which helps to explain the presence of weak\ndamping anisotropy in epitaxial Fe [30].5\nWe compare our results with prior theoretical works on\ndamping anisotropy [23, 24]. First, despite their propor-\ntional relationship in Fig. 5(a), the giant anisotropy in\n\u000bis not re\rected in 1 =\u001a. This is because the s-dscatter-\ning, which dominates in the anisotropic AMR, only con-\ntributes a small portion to the total resistivity. Second,\nneither the anisotropy of damping nor AMR are sensitive\nto temperature. This is likely because the thermal excita-\ntions at room temperature ( \u00180:025 eV) are much smaller\nthan the spin-orbit coupling ( \u00180:1 eV [47]). Third, the\ndamping tensor has been expressed as a function of M\nanddM=dt[24]. However in a fourfold-symmetry lat-\ntice and considering the large precession ellipticity, these\ntwo vectors are mostly perpendicular to each other, point\ntowards equivalent crystalline directions, and contribute\nequivalently to the symmetry of damping anisotropy.\nIn summary, we have experimentally demonstrated\nvery large Gilbert damping anisotropy up to 400% in\nepitaxial Co 50Fe50thin \flms which is due to their bulk,\ncubic crystalline anisotropy. We show that the damping\nanisotropy can be explained by the change of spin-orbit\ncoupling within the breathing Fermi surface model, which\ncan be probed by the corresponding AMR change. Our\nresults provide new insights to the damping mechanism\nin metallic ferromagnets, which are important for opti-\nmizing dynamic properties of future magnetic devices.\nWe are grateful for fruitful discussions with Bret Hein-\nrich. W.Z. acknowledges supports from the U.S. Na-\ntional Science Foundation under Grants DMR-1808892,\nMichigan Space Grant Consortium and DOE Visit-\ning Faculty Program. Work at Argonne, including\ntransport measurements and theoretical modeling, was\nsupported by the U.S. Department of Energy, Of-\n\fce of Science, Materials Science and Engineering Di-\nvision. Work at Fudan, including thin \flm growth\nand fabrication, was supported by the Nat'l Key Ba-\nsic Research Program (2015CB921401), Nat'l Key Re-\nsearch and Development Program (2016YFA0300703),\nNSFC (11734006,11474066,11434003), and the Program\nof Shanghai Academic Research Leader (17XD1400400)\nof China. O.O. and V.K. acknowledge supports\nfrom Bogazici University Research Fund (17B03D3),\nTUBITAK 2214/A and U.S. Department of State Ful-\nbright Visiting Scholar Program.\n\u0003wuyizheng@fudan.edu.cn\nyho\u000bmann@anl.gov\nzweizhang@oakland.edu\n[1] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443\n(2004).\n[2] S. 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B 10, 4626 (1974).7\nSupplemental Materials:Giant anisotropy of Gilbert damping in epi-\ntaxial CoFe \flms\nbyYi Li, Fanlong Zeng, Steven S.-L. Zhang, Hyeondeok Shin, Hilal Saglam, Vedat Karakas, Ozhan Ozatay, John E.\nPearson, Olle G. Heinonen, Yizheng Wu, Axel Ho\u000bmann and Wei Zhang\nCrystallographic quality of Co 50Fe50\flms\nFIG. S-1. Crystallographic characterization results of CoFe \flms. (a) RHEED pattern of the CoFe(10 nm) \flm. (b) XRD\nof the CoFe(10 nm) and (20 nm) \flms. (c) X-ray re\rectometry measured for the CoFe(20 nm) \flm. (d) AFM scans of the\nCoFe(20 nm) \flm. (e) Rocking curves of the CoFe(20 nm) \flm for [100] and [110] rotating axes.\nFig. S-1 shows the crystallographic characterization for the epitaxial CoFe samples. Re\rection high-energy electron\ndi\u000braction (RHEED) shows very clear and sharp patterns which shows high quality of the epitaxal \flms. X-ray\ndi\u000braction (XRD) yields clear CoFe(002) peaks at 2 \u0012= 66:5\u000e. X-ray re\rectometry scan of the CoFe (20 nm) \flm\nshows a good periodic pattern and the \ft gives a total thickness of 19.84 nm. Atomic-force microscopy (AFM) scans\nfor 10\u0016m\u000210\u0016m and 100 nm\u0002100 nm scales show smooth surface with a roughness of 0.1 nm. Lastly XRD rocking\ncurves for [100] and [110] rotating axes show a consistent linewidth of 1.45\u000e, which indicates isotropic mosaicity of\nthe CoFe \flms.\nAs a result of the crystallographic characterizations, we believe our MBE-grown CoFe samples are epitaxial, have\nsmooth surfaces and exhibit excellent crystalline quality. Moreover, we can exclude the source of inhomogeneity\nfrom misorientation of crystallities (mosaicity) due to isotropic rocking curves. This means the inhomogeneous FMR\nlinewidth broadening is isotropic, as is consistent with the experiments.\nDevice geometries for Spin-torque FMR and VNA FMR measurements.\nFig. S-2 shows the device geometry for Spin-torque FMR and VNA FMR measurements. For spin-torque FMR,\nwe have prepared CoFe(10 nm)/Pt, CoFe(10 nm) and Fe(10 nm) devices. A second CoFe(10 nm)/Pt sample is also\nprepared for rotating-\feld measurements. For VNA FMR, we have prepared CoFe(20 nm) samples. All the CoFe\n\flms are grown on MgO(100) substrates; the Fe \flm is grown on a GaAs(100) substrate. Au (100 nm) coplanar\nwaveguides are subsequently fabricated on top of all devices. For VNA FMR samples, an additional SiO 2(100 nm) is8\nFIG. S-2. (a) Spin-torque FMR devices of CoFe(10 nm)/Pt samples. (b) Illustration of the Spin-torque FMR circuit. (c) Front\nand (d) back view of the VNA FMR devices for CoFe(20 nm) samples.\ndeposited between CoFe and Au for electric isolation. The CoFe(20 nm) bars is only visible from the back view in\nFig. S-2(d).\nSpin-torque FMR lineshapes\nFigure S-3 shows the full lineshapes of (a) CoFe(10 nm)/Pt(6 nm), (b) CoFe(10 nm) and (c) Fe(10 nm) devices\nmeasured at !=2\u0019= 20 GHz. The Fe \flms were deposited on GaAs substrates by MBE growth. (a) and (b) are used\nto extract the resonance \felds and linewidths in Figs. 2(a) and 3(a) of the main text. (c) is used to examine the\ncorrelation between damping anisotropy and AMR anisotropy.\nSpin-torque FMR linewidths as a function of frequency for CoFe(10 nm) devices.\nFigure S-4(a) shows the spin-torque FMR linewidths for CoFe(10 nm) devices. Because there is no spin torque\ninjection from Pt layer, the FMR signals are much weaker than CoFe(10 nm)/Pt and the extracted linewidths are\nmore noisy. The excitation of the dynamics is due to the magnon charge pumping e\u000bect [1] or inhomogeneities of the\nOersted \felds. No signal is measured for the rf current \rowing along the easy axis (magnetic \feld along the hard\naxis, see Fig. S-3b), because of the negligible AMR ratio.\nFigure S-4(b) shows the angular dependence of the extracted Gilbert damping for CoFe(10 nm)/Pt and CoFe(10\nnm). The former is extracted from Fig. 3(b) of the main text. The latter is extracted from Fig. S-4(a). The blue\ndata points for CoFe(10 nm)/Pt are obtained from the resonances at negative biasing \felds. Those data are used in\nFig. 4 of the main text.9\nFIG. S-3. Spin-torque FMR lineshapes of (a) CoFe(10 nm)/Pt, (b) CoFe(10 nm) and (c) Fe(10 nm) devices measured at\n!=2\u0019= 20 GHz. \u0012H\u0000\u0012Iis \fxed to 45\u000e.\nFIG. S-4. (a) \u00160\u0001H1=2as a function of frequency for CoFe(10 nm) devices. Solid lines and curves are the \fts to the experiments.\n\u0012H\u0000\u0012Iis \fxed to 45\u000e. (b)\u000bas a function of \u0012Hfor CoFe(10 nm)/Pt and CoFe(10 nm) devices.\nSpin-torque FMR for CoFe(10 nm)/Pt up to 39 GHz.\nFig. S-5 shows the spin-torque FMR lineshapes and linewidths up to 39 GHz for CoFe(10 nm)/Pt devices along\nthe easy and hard axes ( \u0012H= 90\u000eand45\u000e). At!=2\u0019= 32:1 GHz (Fig. S-5a), the spin-torque FMR amplitude is\n0.1\u0016V for the easy axis and 0.02 \u0016V for the hard axis. 10 seconds of time constant is used to obtained the signals.\nThroughout the frequency range, linewidths demonstrate good linear dependence on frequency as shown Fig. S-5(b).\nFor the hard axis the signal has reached the noise bottom limit at 32.1 GHz. For the easy axis the noise bottom limit\nis reached at 39 GHz. The two linear \fts yield \u000b= 0:0063 and\u00160\u0001H0= 1:8 mT for the easy axis and \u000b= 0:00153\nand\u00160\u0001H0= 1:5 mT for the hard axis. The two damping parameters are close to the values obtained below 20 GHz\nin the main text. Also the inhomogeneous linewidth \u00160\u0001H0nicely match between easy and hard axes.10\nFIG. S-5. High-frequency ST-FMR measurement of i) CoFe(10 nm)/Pt for the biasing \feld along the easy axis ( \u0012H= 90\u000e) and\nhard axis (\u0012H= 45\u000e). Left: lineshapes of ST-FMR at !=2\u0019= 32:1 GHz. Right: linewidth as a function of frequency. Lines\nare linear \fts to the data by setting both \u000band \u0001H0as free parameters.\nLow-temperature FMR linewidths and dampings for CoFe(10 nm)/Pt and CoFe(20 nm).\nFIG. S-6. (a-b) \u00160\u0001H1=2as a function of frequency for CoFe(10 nm)/Pt devices at di\u000berent temperatures. (c) Extracted\ndamping at di\u000berent temperatures, same as in Fig. 4 of the main text.\nFigure S-6 shows the frequency dependence of linewidths for extracting temperature-dependent Gilbert damping\nin Fig. 5(a) of the main text.\nFor CoFe(10 nm)/Pt samples, we plot both \u000band resistivity \u001ameasured at di\u000berent temperatures in Fig. S-6(c).\nThe measurements of \u001awere conducted with a biasing magnetic \feld of 1 Tesla parallel to the current direction, so\nthat the AMR in\ruence is excluded. Also the resistivity variation between the easy and hard axes is very small, about\n1%, which is much smaller than the damping anisotropy.\nWe have also conducted the low-temperature VNA FMR of the new CoFe(20 nm) samples at 8 K, in addition to\nthe room-temperature measurements. The linewidths data are shown in Fig. S-6(d) for both easy and hard axes.\nThe extracted damping are: \u000b= 0:0054 (EA, 300 K), 0.0061 (EA, 8 K), 0.0240 (HA, 300 K) and 0.0329 (HA, 8 K).\nThose values are used in Fig. 4(b) and Fig. 5(a) of the main text.\nFor CoFe(10 nm) the damping anisotropy decreases from 380 % at 300 K to 273 % at 30 K by taking out the spin\npumping damping enhancement (an unexpected reduction of alpha happens at 8 K for the hard axis). For CoFe(2011\nnm) the damping anisotropy increases from 440 % at 300 K to 540 % at 8 K. Thus a clear variation trend of damping\nanisotropy in CoFe \flms remains to be explored.\nFirst-principle calculation of N(EF)anisotropy for Co 50Fe50\nFIG. S-7. Density of states as a function of energy. EFis the Fermi level.\nFirst-principle calculations were done using QUANTUM ESPRESSO for a cubic lattice of Co 50Fe50of CsCl, Zintl\nand random alloy structures. Supercells consisting of 4 \u00024\u00024 unit cells were considered with a total of 128 atoms (64\ncobalt and 64 iron atoms). The calculations were done using plane-wave basis set with a 180 Ry kinetic energy cut-o\u000b\nand 1440 Ry density cut-o\u000b. For both Co and Fe atoms, fully relativistic PAW pseudopotentials were used. Figure\nS-7 shows the density of states (DOS) of the CsCl form for di\u000berent magnetization orientations \u0012in thexy-plane.\nClearly, DOS exhibits no anisotropy ( <0:1% variation at E=EF). No anisotropy was found in the Zintl form, either.\nThus, we conclude that the Gilbert damping anisotropy in Co 50Fe50cannot be caused by a variation of N(EF) with\nrespect to magnetization direction in ideal ordered structures.\nSOC induced by atomic short-range order (ASRO)\nIn our experiment, because the Co 50Fe50\flms were grown by MBE at low temperatures, they do not form the\nordered bcc B2 structure but instead exhibit compositional disorder. Transition metal alloys such as CoPt, NiFe, and\nCoFe tend to exhibit ASRO [2{4]. The ASRO in CoFe is likely to give rise to local tetragonal distortions because of the\ndi\u000berent lattice constants of bcc Fe and (metastable) bcc Co at 2.856 \u0017A and 2.82 \u0017A, respectively. Such local tetragonal\ndistortions will preserve global cubic (or four-fold in-plane) symmetry, but can have large e\u000bects on the SOC, with\nconcomitant e\u000bect on spin-orbit induced magnetization damping. For example, \frst-principle calculations using the\ncoherent-potential approximation for the substitutionally disordered system shows that a tetragonal distortion of 10%\nin the ratio of the tetragonal axes aandcgives rise to an magnetocrystalline anisotropy energy (MAE) density [2, 3]\nof about 1 MJ/m3. These results are consistent with our observed MAE in Co 50Fe50.\nTo con\frm this mechanism, we performed DFT-LDA calculations on 50:50 CoFe supercells consisting of a total\nof 16 atoms for CsCl, zintl, and random alloy structures; in the random alloy supercell, Co or Fe atoms randomly\noccupied the atomic positions in the supercell. Note that all CoFe geometries are fully relaxed, including supercell\nlattice vectors.\n1. Structural relaxation including spin-orbit coupling (SOC) shows local tetragonal distortions for random alloy\nsupercell. Among the three di\u000berent CoFe phases, tetragonal c/a ratio for the supercell in optimized geometry\nis largest (1.003) in the random alloy supercell with SOC, which means local tetragonal distortions are more12\nFIG. S-8. Density of states (DOS) for (a) CsCl, (b) Zintl, and (c) alloy form of CoFe with SOC (black solid) and without SOC\n(red solid).\nTABLE I. Relaxed atomic positions (including SOC) of the alloy structure. In the ideal CsCl or Zintl structures, the atomic\npositions are all multiples of 0.25 in units of the lattice vector components.\nAtom x-position y-position z-position\nCo 0.003783083 0.000000000 0.000000000\nFe -0.001339230 0.000000000 0.500000000\nFe -0.002327721 0.500000000 0.000000000\nFe 0.002079922 0.500000000 0.500000000\nFe 0.502327721 0.000000000 0.000000000\nFe 0.497920078 0.000000000 0.500000000\nCo 0.496216917 0.500000000 0.000000000\nFe 0.501339230 0.500000000 0.500000000\nCo 0.250000000 0.250000000 0.254117992\nFe 0.250000000 0.250000000 0.752628048\nFe 0.250000000 0.750000000 0.247371952\nCo 0.250000000 0.750000000 0.745882008\nCo 0.750000000 0.250000000 0.250415490\nCo 0.750000000 0.250000000 0.746688258\nCo 0.750000000 0.750000000 0.253311742\nCo 0.750000000 0.750000000 0.749584510\ndominant in random alloy compared to CsCl and Zintl structures. [c/a values : CsCl (0.999), Zintl (0.999),\nAlloy (1.003)]. In addition, the alloy system exhibited local distortions of Co and Fe position relative to their\nideal positions. In contrast, in CsCl and Zintl structures the Co and Fe atoms exhibited almost imperceptible\ndistortions. Table shows the relaxed atomic positions in the alloys structure in units of the lattice vectors. In\nthe ideal (unrelaxed) system, the positions are all at multiples of 0.25; the relaxed CsCl and Zintl structures no\ndeviations from these positions larger than 1 part in 106\n2. SOC changes the density of states (DOS) at the Fermi energy, notably for the random alloy but notfor the CsCl\nand Zintl structures. Figure S-8 shows DOS for (a) CsCl, (b) Zintl, and (c) random alloy structure with SOC\n(black lines) and without it red lines). We can see signi\fcant DOS di\u000berence for the random alloy supercell\nwith SOC where tetragonal distortions occurred, while almost no changes are observed in the CsCl and Zintl\nstructures.\n3. The local distortions in the alloy structure furthermore gave rise to an energy anisotropy with respect to the\nmagnetization direction. The energy (including SOC) of the relaxed alloy structure for di\u000berent directions of\nthe magnetization is shown in Fig. S-9. While the supercell was rather small, because of the computational\nexpense in relaxing the structure with SOC, so that no self-averaging can be inferred, the \fgure demonstrates\nan induced magnetic anisotropy that arises from the SOC and local distortions. No magnetic anisotropy was\ndiscernible in the CsCl and Zintl structures.\nAs a result from the DFT calculation, we attribute the large SOC e\u000bect in damping anisotropy of Co 50Fe50to local\ntetragonal distortions in disordered Co and Fe alloys. These distortions give rise to SOC-induced changes of DOS at\nthe Fermi level, as well as magnetic anisotropy energy with respect to the crystallographic axes.13\nFIG. S-9. Change in total energy (per supercell) of the alloy structure as function of the magnetization direction.\n\u0003wuyizheng@fudan.edu.cn\nyho\u000bmann@anl.gov\nzweizhang@oakland.edu\n[1] C. Ciccarelli, K. M. D. Hals, A. Irvine, V. Novak, Y. Tserkovnyak, H. Kurebayashi, A. Brataas and A. Ferguson, Nature\nNano. 10, 50 (2015)\n[2] S. Razee, J. Staunton, B. Ginatempo, E. Bruno, and F. Pinski, Phys. Rev. B 64, 014411 (2001).\n[3] Y. Kota and A. Sakuma, Appl. Phys. Express 5, 113002 (2012).\n[4] I. Turek, J. Kudrnovsk\u0013 y, and K. Carva, Phys. Rev. B 86, 174430 (2012)." }, { "title": "1709.00187v3.Scaling_of_the_Rashba_spin_orbit_torque_in_magnetic_domain_walls.pdf", "content": "arXiv:1709.00187v3 [cond-mat.mes-hall] 21 Jul 2019Scaling of the Rashba spin-orbit torque in magnetic\ndomain walls\nD. Wang\nCollege of Engineering Physics, Shenzhen Technology University, Sh enzhen 518118,\nGuangdong, China\nYan Zhou\nSchool of Science and Engineering, The Chinese University of Hong K ong, Shenzhen,\nShenzhen 518172, Guangdong, China\nE-mail:wangdaowei@sztu.edu.cn ,zhouyan@cuhk.edu.cn\nAbstract. Spin-orbit torque in magnetic domain walls was investigated by solving t he\nPauli-Schr¨ odinger equation for the itinerant electrons. The Rash ba interaction considered\nis derived from the violation of inversion symmetry at interfaces bet ween ferromagnets\nand heavy metals. In equilibrium, the Rashba spin-orbit interaction g ives rise to a torque\ncorresponding to the Dzyaloshinskii-Moriya interaction. When ther e is a current flowing,\nthe spin-orbit torque experienced by the itinerant electrons in sho rt domain walls has\nboth field-like and damping-like components. However, when the dom ain wall width is\nincreased, the damping-like component, which is the counterpart o f the non-adiabatic\nspin transfer torque, decreases rapidly at the domain wall center . In contrast to the non-\nadiabatic spin transfer torque, the damping-like spin-orbit torque does not approach to\nzero far away from the domain wall center, even in the adiabatic limit. The scattering\nof spin-up and spin-down wave functions, which is caused by the Ras hba spin-orbit\ninteraction and the spatial variation of magnetization profile in the d omain wall, gives\nrise to the finite damping-like spin-orbit torque.\nKeywords : spin-orbit torque, Rashba spin-orbit interaction, magnetic doma in wall2\n1. Introduction\nEver since its discovery, the spin degree of freedom of electrons p lays an important role in\nmodern physics, and the spin quantum number is established as an int rinsic property of\nfundamentalparticles. Duetothesimultaneouspresence ofthes pinandorbitalmotion, the\ninterplay between the spin and orbital degrees of freedom contrib utes a small correction to\nthe Hamiltonian for isolated atoms, which can be observed as the fine splitting of spectral\nlines. Although it is a small correction to the total energy, the spin- orbit interaction (SOI),\nwhich is a relativistic effect, can play a crucial role in magnetically order ed systems with\ncompeting interactions. It is well known that the magnetic anisotro py is determined by\nboth the crystal field and the SOI [1, 2]. The magneto-optical Far aday and Kerr effects of\nlight propagating through a ferromagnet also derive from the SOI [3 ].\nAt the interface between a metallic ferromagnet (FM) and a nonmag netic heavy\nmetal (HM), due to the reduced coordination number, hence the c orrespondingly lowered\nsymmetry, and the strong spin-orbit coupling (SOC) provided by th e HM, perpendicular\nmagnetic anisotropy (PMA) [4, 5] can result in, bringing the magnetiz ation vector to a\ndirection perpendicular to the interface, instead of lying in the inter face, which is required\nby the demagnetization energy. Since the spin and orbital degrees are coupled to give rise\ntothePMAandtheitinerantandlocalisedelectronsare s-dexchangecoupled, ifanelectric\ncurrent is flowing in the HM, the orbital motion of electrons will inevita bly influence the\nlocal spin dynamics. This action of electric current on magnetization can be viewed as\na modification of the PMA by the current [6]. An equivalent effective Ra shba field was\nactually first proposed by several groups [6, 7, 8, 9] for the FM/H M bilayer system without\ninversion symmetry. In a system without inversion symmetry, ther e can exist electric field\nalongthesymmetry violationdirection. Inthestaticcoordinatefra meofamoving electron,\nthe electric field is transformed into an effective Rashba field acting o n the electron. This\nis the physical origin of the Rashba field experienced by the local mag netization [10].\nFollowing works showed that, in addition to the field-like torque corre sponding to\nthe Rashba field, there is another damping-like contribution to the t orque, dubbed spin-\norbit torque (SOT), experienced by the local magnetization [11]. Ph ysically, those two\ntorque components can be described by a gauge field if the SOI is not too large [12].\nWhen the diffusive motion of electrons is considered, the SOT exhibits complex angular\ndependence [13, 14]. Adding to the complexity of the form of the SOT , a quantum kinetic\ntheory treatment found more terms other than the field-like and d amping-like ones both\nin systems with homogeneous [15] and textured [16] magnetization d istribution. Density\nfunctional theory calculation [17] showed that, while the field-like te rm comes primarily\nfrom the Rashba field, the damping-like term can be attributed to a s pin current caused\nby the spin Hall effect. Due to its origin from the SOC provided by the in terfacial heavy\natoms, the field-like term is very sensitive to the interface quality an d thickness of the\nferromagnetic layer [18]. A recent fully relativistic investigation of t he same problem gave\nqualitatively similar results [19].\nAlmost all the previous works deal with the SOT in a ferromagnet with uniformly\ndistributed magnetization. The effect of magnetization textures o n the SOT is largely left\ninoblivion. Investigations onthespintransfertorque[20,21,22,2 3](STT) indomainwalls\n(DWs)demonstratedthatthereisatransitionfromadiabatictono n-adiabaticregimes[24].3\nIn this paper, we would like to investigate the evolution of the SOT in DW s, which are\nthe most common example of magnetization textures. What we cons ider is the dynamics\nof electrons in a N´ eel magnetic DW, subject to the effective Rashb a magnetic field caused\nby the breaking of inversion symmetry. It is well known that, due to the Rashba field, in\nequilibrium there exists the Dzyaloshinskii-Moriya [25, 26] (DM) torq ue [27, 28, 29],\nτDM∝ˆm×/parenleftBigg\nˆx∂mz\n∂x−ˆz∂mx\n∂x/parenrightBigg\n, (1)\nwhich favours a nonuniform arrangement of magnetization. ˆ mis the normalized\nmagnetization vector, ˆ xand ˆzare unit vectors pointing to the xandzdirections, and\nthe DW profile varies in the xdirection. Electrons are confined in the xyplane. Under\nthe influence of an electric current flowing along the xdirection, the SOT arises due to the\nsame Rashba field, which possesses both field-like and damping-like co mponents [11, 12],\nτSO=αˆm׈y+βˆm×(ˆm׈y) (2)\nwith decomposition coefficients αandβ. ˆyis a unit vector directed along the ydirection.\nThe first term is the field-like Rashba torque, with the effective Rash ba field directed along\nˆy, and the second is the corresponding damping-like torque. For a ge neral current flowing\nalong direction specified by a unit vector for current density, ˆj, the field direction ˆ yshould\nbe replaced by ˆ z׈j, showing that the effective Rashba field is perpendicular to both the\nsymmetry breaking directions (ˆ zandˆj).\nThe main task of this paper is to verify the Rashba origin of the DM tor que in\nequilibrium and the non-equilibrium SOT, Eqs. (1) and (2), and study t he scaling of\nthe SOT in DWs with respect to the DW width. Using a minimal Hamiltonian w ith\nSOC, we can show that the DM torque and the SOT are actually derive d from the same\nRashba SOI. In addition, it can be found that, similar to the scaling of the non-adiabatic\nSTT inside a DW [24], the damping-like component at the DW center deca ys rapidly with\nthe increase of the DW width. However, the damping-like SOT far awa y from the DW\ncenter approaches to a constant, sizable value, in contrast to th e non-adiabatic STT which\napproaches asymptotically to zero. This finite damping-like SOT in DWs in the adiabatic\nlimit is derived from the scattering of spin-up and spin-down wave fun ctions caused by\nthe Rashba SOC and spatial variation of magnetization, as shown by our perturbation\nanalysis.\nExperimentally, the existence and magnitude of the SOT at HM/FM int erfaces are\nstill far from reaching a consensus. Originally, the fast current-d riven DW motion observed\nin Pt/Co/AlO xthin films was postulated to be caused by the Rashba field [30, 31]. Us ing\nthe same Rashba field, the current-induced magnetization switchin g was explained [32].\nHowever, it was shown later that the spin current generated in the HM due to the spin\nHall effect could also explain the experimentally observed switching [33 ]. Our results show\nthat, in the adiabatic limit, which is relevant to most experimental mea surements, the\nfield-like torque becomes the dominant torque, but there still exist s a damping-like torque\nwith comparable magnitude in magnetization textures. Hence the re sultant magnetization\ndynamics and switching is dramatically different from that driven by a p ure damping-like\nspin Hall torque. This could help discriminate the driving force behind t he experimental\nobservations, for particular the DW motion in HM/FM systems.\nThe organization of this paper is as follows. First we will formulate in Se c. 2 our\ncalculation of the equilibrium and non-equilibrium SOTs using the equatio n of motion for4\nthe spin density, which is derived from the Pauli-Schr¨ odinger for itin erant electrons. Then\nwe will use the results of Sec. 2 to discuss the case of a uniform magn etization distribution\nin Sec. 3, which can be viewed as the extreme adiabatic limit. The numer ical calculation\nof the SOT in a DW is given in Sec. 4, and a perturbation analysis of the o btained scaling\nof the SOT is given in Sec. 5. Finally, Sec. 6 gives a summary of our work .\n2. Outline of Theory\nThe staring point of our discussion is the Hamiltonian for electrons mo ving in a\nmagnetization texture [6, 7, 9],\nH=p2\n2me+µBσ·M+αR\n¯hσ·(p׈z), (3)\nwhereme, ¯h,p=−i¯h∇andµBare the electron mass, the reduced Planck’s constant,\nthe momentum operator and the Bohr magneton, respectively. αRis the Rashba constant,\nwhich characterizes the broken inversion symmetry [34], and the R ashba term can be\nincorporated into the kinetic energy term, by forming a covariant d erivative operator [12].\nPrevious density functional theory investigation found that the S OT [17] and DM torque\n[35] are primarily confined to the interface, hence we need only to co nsider the motion\nof the electrons in the interface, which is a 2D plane. ˆ zis the unit normal vector of the\ninterface. Due to the broken inversion symmetry, electrons expe rience an effective in-plane\nmagnetic field which is perpendicular to the 2D linear momentum, as cha racterized by\nthe third term in the Hamiltonian H.σ= ˆxσx+ ˆyσy+ ˆzσzis the vector Pauli matrix,\nandσx,σyandσzare the Pauli matrices. The magnetization texture is described by\nM=M(ˆxsinθcosφ+ ˆysinθsinφ+ ˆzcosθ). Physically, the Hamiltonian Hdescribes the\nenergy of conduction electrons in a solid, interacting through the s-dexchange interaction\nwith the localized electrons. In our simple treatment, we will only cons ider the itinerant\nHamiltonian as given in Eq. (3), while the local magnetic moments are as sumed to be\nstatic, asdescribed by M. SincetheCoulombinteractionbetween electronsisnotexplicitly\nincluded in our model Hamiltonian, the exchange interaction respons ible for the long range\nferromagnetic order is not present. The magnetization texture is used to simulate the\nexchange interaction between the conduction electrons.\nDue to the insufficient consideration of the local magnetization dyna mics, what we can\ncalculate using the Hamiltonian His actually the torques acting on the itinerant electron\nmagnetization. However, suppose that the itinerant and localized s ubsystems are only\ncoupled through the s-dexchange term in the Hamiltonian H,σ·M, the torques acting\non the itinerant magnetization will be retro-acted on the localized ma gnetization. To\nthe first order of the Rashba coupling constant, the itinerant and localized magnetization\nis parallel to each other. Given this fact, the form of the torques s hould be identical,\nregardless of whether the torques are acting on the localized magn etization or not, if higher\norder corrections to the torques experienced by the local magne tization are neglected.\nThis argument is consistent with the conservation of the total ang ular momentum of the\nwhole system, comprised of the localized and itinerant electron subs ystems. Hence the\ntorques calculated for the itinerant magnetization can also be viewe d as acting on the local\nmagnetization, at least in the case of equilibrium or in the presence of a steady current.\nWe will only present the results for the itinerant electrons in the follo wing.5\nFor a general discussion about the spin dynamics corresponding to the Hamiltonian\nH, we consider the time dependent Pauli-Schr¨ odinger equation i¯h∂ψ/∂t=Hψfor the\ndetermination of the spinor wave function ψ. The resulting conservative charge current is\n2me\n¯hj=i(ψ†∇ψ−∇ψ†ψ)−kαψ†(ˆz×σ)ψ. (4)\nkα= 2meαR/¯h2is an effective wave number. The term proportional to the Rashba\ncoupling is identical in form to the contribution of a vector potential to the current. This\nis not surprising, since the Rashba term can actually be absorbed int o the kinetic energy,\nadding an effective vector potential to the momentum operator [12 ]. Multiplying the Pauli-\nSchr¨ odinger equation with the vector Pauli matrix, instead of the unit matrix in the spinor\nspace, we can get the equation of motion for the spin density, which reads as\n2me\n¯h∂s\n∂t=∇·Q+2k2\nBˆM×s+2kα\n¯hψ†/vector σ×(ˆz×p)ψ, (5)\nwheres=ψ†σψis the magnetic moment density of conduction electrons, and the\ncorresponding spin current density is defined by\nQ=i(ψ†∇σψ−∇ψ†σψ)−kαψ†(ˆz×σ)σψ. (6)\nWhat we defined here as the magnetic moment density is only proport ional to the spin\noperator, different from the usual definition using the spin angular momentum operator\n¯hσ/2. As we are only interested in the magnetization originating from the electrons and\nthe torque experienced by the electrons, and the spin angular mom entum is proportional\nto the magnetic moment, we do not distinguish between them hereaf ter. The wave number\nkBis defined by the Zeeman energy ¯ h2k2\nB/2me=µBM, andˆMis the unit direction vector\nfor the local magnetization, ˆM=M/M. Although the spin-current density is Hermitian\nin the conventional sense, i.e. integrated over the whole space, it is not a real quantity\nlocally. This non-Hermitian character arises from the term proport ional to the Rashba\ncoupling constant. By retaining only the Hermitian part of this term, a Hermitian spin\ncurrent density can be constructed as\nQ=i(ψ†∇σψ−∇ψ†σψ)+kαǫij3ˆiˆjψ†ψ, (7)\nwhereǫijkis the antisymmetric Levi-Civita symbol and a summation over repeat ed indices\nis implied. We have also used numbers 1, 2 and 3 to denote the x,yandzdirection unit\nvectors, respectively. The anti-Hermitian part in the original spin c urrent density cancels\nwith the anti-Hermitian part of the precessional term caused by th e Rashba field, and the\nremaining precessional term is\nτ= 2kαℑ(ˆzψ†σ·∇ψ−ψ†σz∇ψ). (8)\nNow the spin current density and the precessional term are both H ermitian. This is\nwhat can be expected from the start, since the Hermitian spin dens ity requires that each\ncontributing term be Hermitian. The net effect of the Rashba field is t o modify the spin-\ncurrent density by adding a term proportional to the Rashba cons tant. Eqs. (7) and (8)\nwill be used to calculate the STT, DM torque and SOT, when averaged over the whole\nFermi sphere (equilibrium) or surface (with current flowing).\nThe final equation for the magnetization is\n2me\n¯h∂s\n∂t=∇·Q+2k2\nBˆM×s+τ, (9)6\nwhich is identical in form to the equation of motion for the spin density obtained in [36],\nexcept that the orbital angular momentum is not included here. The physical meaning\nof those terms in the right hand side is quite obvious. The first term is the spin current\ntorque acting on the magnetization vector, which is resulted from t he spatial variation of\nthe spin density. In the ground state, this term gives rise to the ex change torque in a\nmagnetization texture, which is proportional to ˆ m× ∇2ˆm. The second term is the s-d\nexchange torque, acting on the itinerant magnetization due to the presence of the static\nlocal magnetization. The last term is the SOT derived from the Rashb a term in the\nHamiltonian H. In equilibrium, the SOT amounts to the usual DM torque. When ther e\nis a current flowing, the exchange torque reduces to the convent ional STT, and the SOT\nhas the form of a sum of the field-like and damping-like torques. Due t o torque balance in\nthe steady state, the torque corresponding to the spin accumula tion, which is given by the\nsecond term, includes both the STT and SOT contributions.\nThe formula given above describe only single Bloch states in the recipr ocal space.\nTo obtain the actual physical properties, especially the equilibrium a nd non-equilibrium\nmagnetization and various torques, integration in the momentum sp ace has to be\nperformed. Specifically, this means that the equilibrium magnetizatio n is given by the\nintegral\nm(ρ) =/integraldisplayd2k\n(2π)2s(k,ρ)fD(ǫk), (10)\nwhereρis a position vector in the 2D electron gas plane, kis the Bloch wave vector in the\nmomentum space, fDis the Fermi-Dirac distribution function, and ǫkis the energy of the\nBloch state. As we are only interested in the zero-temperature be haviour, the integration\nover the whole k-space reduces to the integration over the 2D Fermi sphere. Oth er\nequilibrium quantities can be given similarly. Using the relaxation time app roximation,\nthe non-equilibrium spin accumulation induced in the presence of an ele ctric fieldEalong\nthexdirection is expressed as\nδm(ρ) =/integraldisplayd2k\n(2π)2s(k,ρ)/parenleftbigg\nfD/parenleftbigg\nk−eEτ0\n¯hˆx/parenrightbigg\n−fD(k)/parenrightbigg\n=−eEτ0\n(2π)2¯h/contintegraldisplay\ndϕkxs(k,ρ),(11)\nwhereτ0is the relaxation time constant, eis the electron charge, and ϕthe angle of the\nwave vector relative to the x-axis. Since the temperature is zero Kelvin, the integration is\nconfined to the Fermi surface, which is a circle in the 2D case conside red here. The same\nexpression holds for other non-equilibrium quantities, such as the S TT and SOT.\n3. Solution for a uniform magnetization distribution\nThe analytical solution to the Pauli-Schr¨ odinger equation is genera lly difficult to find.\nIn the case of a uniform magnetization distribution, the correspon ding Pauli-Schr¨ odinger\nequation is easy to solve [37]. Although the situation for a uniform mag netization\ndistribution is simple, insights still can be gained by a thorough analysis of the relationship\nbetween equilibrium and non-equilibrium quantities. In addition, throu gh the examination\nof this simple and well known situation, our approach to the calculatio n of the torques will\nbe demonstrated.\nThe magnetizationisuniformlymagnetized alongthe zdirection, so ˆM= ˆz. Withthis\nmagnetization distribution, the Hamiltonian Hcommutes with the momentum operator7\n∇. The Hamiltonian Hcan then be diagonalized by a rotation in the spinor space, and the\nsolution has the form ψ±= exp(ik·ρ)Uη±, where the spinors η±are the eigenvectors of\nthe Pauli matrix σz,σzη±=±η±.U= exp(−iϕ\n2σz)exp(−iϑ\n2σx) is a rotation matrix in the\nspinor space, with ϑgiven by tan ϑ=kαk/k2\nB.k=/radicalBig\nk2x+k2yis themodulus of the2D wave\nvectork. The rotation corresponding to Uis first a rotation around the x-axis byϑ, then\na rotation around the z-axis byϕ. Using an effective wave number kǫ,k2\nǫ=k2±k2\nBsecϑ,\nthe energy of an electron with momentum kis given by ǫk= ¯h2k2\nǫ/2me. As there is no\nmagnetization variation in space, the spin-up and spin-down wave fu nctions correspond to\nthe±branches of the dispersion relation. For a magnetization texture, this one-to-one\ncorrespondence does not exist. In cases where there is no confu sion arising, we still use\nthe spin-up and spin-down terminology to refer to the ±dispersion branches. From the\ndispersion relation, the Fermi wave vectors are given by\nk±\nF=\nk2\nF+k2\nα\n2∓/radicalBigg\nk4\nB+k2\nFk2α+k4\nα\n4\n1/2\n,\nwherekFis the Fermi wave number corresponding to the case without the ex change\nsplitting caused by the local magnetization and the Rashba SOI.\nThe corresponding momentum specific equilibrium magnetization can b e calculated\nas\nm±(k) =ψ†\n±σψ±= (ˆzcosϑ+ ˆxsinϑsinϕ−ˆysinϑcosϕ)η†\n±σzη±.(12)\nObviously, the momentum specific magnetization is not parallel to the uniform background\nmagnetization, which is parallel to ˆ z. The modification to the wave function due to the\nRashba interaction gives rise to a transverse component of the ma gnetization, making the\nmagnetization vector parallel to the total effective field, which is co mprised of the exchange\nfield due to the local magnetization and the Rashba field. The total m agnetization is an\nintegral over the Fermi sphere,\nm±=/integraldisplayk±\nF\n0kdk\n(2π)2/integraldisplay2π\n0dϕm±(k) =±ˆz\n2π/integraldisplayk±\nF\n0kdkcosϑ, (13)\nwhich is parallel to ˆ z. The current-induced non-equilibrium magnetization (or the spin\naccumulation) is given by an integral over the Fermi surface,\nδm±=−eEτ0\n(2π)2¯h/contintegraldisplay\ndϕkxm±(k) =±ˆyeEτ0\n4π¯hk±\nFsinϑ±, (14)\nwhich is proportional to the Rashba constant (through sin ϑ±) and perpendicular to the\nequilibrium magnetization. ϑ±is the angle ϑevaluated with the Fermi wavenumber k±\nF.\nWe will nowturn to thecalculation of thevarious torques acting onth e magnetization.\nAccording to Eq. (7), the spin-current density Qis a constant. Hence, the divergence of\nthe spin-current density is zero, ∇ ·Q= 0. This indicates that, in this case, both the\nequilibrium exchange torque and the non-equilibrium STT are zero, sin ce there are no\nmagnetization gradients. In the equation of motion for the spin den sity, there is another\ntorque (Eq. (8)) contribution arising from the Rashba term,\nτ±(k) =−kαkη†\n±σzη±cosϑ. (15)8\nThis contribution amounts to the DM torque, which is zero in equilibrium for a uniform\ndistribution of magnetization (cf. Eq. (1)). When there is a curren t flowing, this\ncontribution becomes ��nite,\nτ±=−eEτ0\n(2π)2¯h/contintegraldisplay\ndϕkxτ±(k) =±ˆxeEτ0\n4π¯hk2\nBk±\nFsinϑ±. (16)\nτ±is the spin-resolved SOT. It is easy to check that the non-equilibrium torqueτ±exactly\ncancels thetorque resulting fromthespin accumulation, which ispro portionalto the vector\nproduct ˆz×δm±, to guarantee that the time dependence of the total spin density is zero,\nds/dt∝k2\nBˆz×δm±+τ±= 0. The total SOT is a sum of the spin-up and spin-down\ncontributions, τ=τ++τ−. In this case of a uniform magnetization distribution, the\nSOTτhas only the field-like contribution, and the corresponding Rashba fi eld is along the\ny-axis, which is consistent with previous investigations [6, 7, 8, 9].\n4. Dzyaloshinskii-Moriya and spin-orbit torques\nThe case of a uniform magnetization distribution discussed in the pre vious section can\nbe viewed as to give the SOT in the extreme adiabatic limit. Since the gra dient of\nthe magnetization is zero, the only remaining torque is the Rashba fie ld-like torque, or\nthe adiabatic part of the SOT. The DM torque, whose manifestation requires the spatial\nvariation of the magnetization vector, is absent in this extremely ad iabatic limit. To see\nthe effects of the DM torque and SOT, we need to consider a magnet ization texture. The\nmagnetization texture we consider for the study of the DM torque and SOT is a Walker\nDW profile, φ= 0 and cos θ=−tanhx/δ, whereδ=/radicalBig\nA/Kis the DW width. Ais\nthe exchange constant of the material, and Kis the anisotropy constant. With this N´ eel\nmagnetization texture [39], the Pauli-Schr¨ odinger equation is\n/bracketleftBig\n−∇2+k2\nBσ·ˆM−ikα(σx∂y−σy∂x)/bracketrightBig\nψ=k2\nǫψ, (17)\nwhere the abbreviation ∂stands for the partial derivative operator, ∂xψ=∂ψ/∂xand\n∂yψ=∂ψ/∂y. Since the magnetization profile is a function of xonly, the wave function\ncan be assumed to have the form ψ= exp(ikyy)χ(x), simplifying the equation to\n/bracketleftBig\n−∂2\nx+k2\nBσ·ˆM+kα(σxky+iσy∂x)/bracketrightBig\nχ=/parenleftBig\nk2\nǫ−k2\ny/parenrightBig\nχ. (18)\nThe physics behind this equation will become more transparent if a tr ansformation to\nthe coordinate of the local magnetization is made. This transforma tion amounts to a\nunitary rotation, U= exp(−iθσy/2). After this unitary rotation, the Hamiltonian His\ntransformed into ˜H=U†HU∝ −D2\nx+ikασyDx+k2\nBσz+kαky(σxcosθ+σzsinθ). In the\nlocalcoordinate, themomentumindependent partofthemagnetic fieldisdiagonalized, and\nthe ordinary derivative is replaced by the covariant derivative, Dx=∂x−iθ′σy/2. To make\nthe expression more compact, we have used a prime to denote the s patial derivative along\nthexdirection,θ′=∂xθ. The appearance of a vector potential in the covariant derivative ,\nwhich is proportional to the spatial variation of the magnetization, is of great importance\nforthedynamicsofmagnetization. TheDMtorqueismediatedbythe intrinsicspincurrent\nassociated with the vector potential [40]. The second derivative of the angleθ′,θ′′, which\nis proportional to the commutator between ∂xandDx, determines the non-adiabaticity of\nthe torque.9\n-50 -25 0 25 50-0.4-0.200.20.4Mag. & Exchange field (a. u.)Mx\nMy\nMz\nmx\nmy\nmz\nFigure 1. Equilibrium magnetization ( m) distribution over the DW region, with the DW\nwidthkFδ= 5. The components of the s-dexchange field ( M) are also displayed. The\nmisalignment between the exchange field Mand the itinerant magnetization mis mainly\ncaused by the Rashba interaction. The itinerant magnetization lies in thexzplane.\nSeeking an analytic solution to the Pauli-Schr¨ odinger equation (18) describing\nelectrons moving in a nonuniform magnetization texture is difficult. In the absence of the\nRashba term, an exact solution exists for spin spirals [37]. The existe nce of such an exact\nsolution can be traced back to the vanishing of the second derivativ e of the magnetization\nangle, with respect to the spatial coordinate. Since this derivative is not zero for a DW,\nan exact solution is still missing now. In the presence of the Rashba t erm, there are even\nno exact solutions for spin spirals. This fact can be readily understo od, based on the fact\nthat the effective exchange field has a contribution from the σxRashba term in Eq. (18).\nThe corresponding additional momentum dependent xfield makes the spatial dependence\nof the direction of the effective exchange field more complicated tha n that of a simple\nmagnetization distribution. This complexity caused by the Rashba te rm renders the task\nof finding an exact solution harder.\nmxmymz∂Qx∂Qy∂Qzτxτyτz\n+−−+−−+−−\n−++−++−++\nTable 1. Parity for the magnetization ( m), divergence of the spin current density ( ∂Q)\nand the SOT ( τ). The symbols + and −denote the even and odd parities, respectively. In\nthe table, the first row is for the equilibrium quantities, while the seco nd row corresponds\nto the case with a current flowing. Due to the additional kxfactor for the Fermi surface\nintegration, the current induced quantities have parity opposite t o that of the equilibrium\nones.\nGiven the above consideration, we adopt a scattering matrix metho d to numerically\nsolve the Pauli-Schr¨ odinger equation. The eigenfunction can be co nstructed by injecting\nplane waves at a large distance from the DW center, evolving accord ing to the Pauli-\nSchr¨ odinger equation, Eq. (18), and then requiring that the linea r combination of the\nfunctions at the DW center to satisfy the continuity condition. The same method was10\n-50 -25 0 25 50-0.0100.01DM torque (a. u.)xDM\nyDM\nzDM\nFigure 2. DM torque acting on the magnetization of the itinerant electrons, w ith\nthe DW width kFδ= 5. The DM torque has only y-component, which corresponds\nto an effective field lying in the xzplane. This in-plane effective field is described by\nHDM=HDM(ˆxˆm′\nz−ˆzˆm′\nx), with the prime symbol (′) denotes the spatial derivative\nalong the xdirection. The constant HDMis proportional to the Rashba constant αR.\nused for a similar discussion without the Rahsba term [24]. In the nume rical calculation,\na particle-hole symmetry of the Hamiltonian Hof Eq. (18), H=σxPTHTPσx, can be\nemployed to reduce the number of the wave functions to be comput ed.PandTare the\nxinversion and time reversal operators, respectively. A similar part icle-hole symmetry\nwas found for the Hamiltonian for magnons inside DWs [38]. Employing th e particle-hole\nsymmetry and identifying the pair of wave functions ψandσxPψas waves with opposite\nmomenta, the parity of various physical quantities, such as the ma gnetization and the\nSOT, can be determined as given in Table 1. When the physical quantit ies in which we\nare interested are computed using Eqs. (7) and (8) with the numer ically obtained wave\nfunctions, it can be verified that the parity relations given in Table 1 a re actually obeyed.\nThe equilibrium itinerant magnetization distribution for kB= 0.4kFandkα= 0.1kFis\nshown in Figure 1. It is obvious that the conduction electron magnet ization distribution is\nnot everywhere parallel to the local magnetization vector. As the total torque is zero, the\ns-dexchange torque due to the local magnetization has to balance the itinerant exchange\ntorque, which is given by the divergence of the spin density current and proportional\nto ˆm׈m′′, and the torque caused by the Rashba interaction, the DM torque which is\nproportional to ˆ m×HDM(cf. Eq. (1)). The DM field HDMis proportional to ˆ xˆm′\nz−ˆzˆm′\nx\n[27, 28, 29]. This balance between torques inevitably introduces a de viation of the itinerant\nmagnetization from the local magnetization. As the exchange field is a second order\nderivative, the term proportional to the exchange field gives a sma ller contribution to\nthe deviation, as compared to the DM term, except for shorter DW s, where the quantum\nconfinement effect is prominent. The equilibrium DM torque is shown in F igure 2. It is\nseen immediately that the DM torque agrees well with the form τDM= ˆm×HDM, having\nonly aycomponent.\nTurning to the case where there is a current following along the xdirection. The SOT\nfor the DW width kFδ= 5.0 with kB= 0.4kFandkα= 0.1kFis shown in Fig. 3. We\ncan see that for such a short DW width, the field-like and damping-like torques are both11\n-50 -25 0 25 50-202Spin orbit torque (a. u.)xSO\nySO\nzSO\n-50 -25 0 25 50123-500 -250 0 250 50001234\nFigure 3. Spatial variation of SOT with DW width kFδ= 5. For this small DW width,\nboth the field-like ( xandz) and the damping-like ( y) components are comparable in\nmagnitude. The decomposition of the total SOT to both torques is g iven by τSO=\nαˆm׈y+βˆm×( ˆm׈y). The corresponding decomposition coefficients αandβare plotted\nin the lower inset. The upper inset gives the coefficients αandβfor a long DW with\nwidthkFδ= 50. For the long DW, the damping-like torque is negligibly small at the D W\ncenter, while it is not infinitesimally small far away from the DW center.\npresent. The field-like torque corresponds to the effective Rashb a field, which is given by\nˆm׈y, while the damping-like torque has the form ˆ m×(ˆm׈y). [7] For the magnetization\nlying in the xzplane, the field-like torque has both xandzcomponents, but the damping-\nlike torque has only a ycomponent. Hence the total SOT can be written as that already\ngiven by Eq. (2), τSO=αˆm׈y+βˆm×(ˆm׈y). The decomposition coefficients αand\nβare displayed in the insets to Figure 3. The observable spatial variat ion of the SOT far\naway from the DW center is caused by quantum interference, and it will decay away as the\nDW width is increased (cf. the upper inset to Figure 3).\nThe critical length for the transition from non-adiabatic to adiabat ic behaviour is\ndefined asδc=kF/k2\nB,[24] which is kFδc= 6.25 with our parameters. As the DW width is\nincreased above δc, the magnitude of the damping-like SOT should decrease exponentia lly,\nbasedonasimilarinvestigationonthenon-adiabaticSTT[24]. Numerica lly, itisdifficultto\nverify this stipulation, since the spin-down wave function diverges e xponentially when the\nenergy is within the exchange energy gap, which is brought about by the instability of the\nPauli-Schr¨ odinger equationtherein. WeusedtheGNUmultipleprecis ionarithmeticlibrary\n[41] to circumvent this problem, by retaining more significant digits in t he computation for\nlonger DWs.\nThe scaling of the field-like (adiabatic) and damping-like (non-adiabat ic) components\nof the SOT at the DW center ( x= 0) and infinity ( x=±∞) is shown in Figure 4. The\nfield-like component approaches asymptotically to a constant value with the increase of the\nDW width for both x= 0 andx=±∞, while the damping-like component decays rapidly\natx= 0 andlevels off toa finite value at x=±∞. This behaviour of reaching a finitevalue\nfor the damping-like torque far away from the DW center is in stark c ontrast to that of the\nnon-adiabatic STT: In the adiabatic limit, the non-adiabatic STT deca ys exponentially to\nzero [24]. Furthermore, the decay of the damping-like SOT at x= 0 cannot be described12\n0 10 20 30 40 50 60 7010-410-310-2Coefs. (a. u.)\n0.010.020.03Coefs. (a. u.)\n(0)\n(0)\n()\n()\nFigure 4. Dependence of the field-like (adiabatic, α) and damping-like (non-adiabatic,\nβ) SOT coefficients on the DW width, with kB/kF= 0.4 and kα/kF= 0.1. The plotted\ncoefficients correspond to the values of αandβat the DW center ( x= 0) and far away\nfrom the DW center ( |x|=∞), where the magnetization variation is maximized and\nminimized respectively. The logarithmic y-axis scale is different for coefficients at x= 0\n(lefty-axis) and |x|=∞(righty-aixs). The solid and dash-dotted lines are guides to the\neye. The small, but discernable, oscillation in α(∞) is caused by the quantum confinement\neffect shown in Fig. 3\nby a single exponential form, which can be seen obviously from Figure 4. A better measure\nof the non-adiabaticity of the SOT can be given by the ratio β/α, which is shown in Figure\n5 for bothx= 0 andx=±∞. The decay of the non-adiabaticity is still not exponential,\nwhich is more prominent for the x=±∞case.\nBefore turning to a perturbation treatment of the scaling of the n on-adiabaticity of\nSOT, it is worth emphasizing that we could not consider the spin Hall cu rrent in our model\ncalculation. Hence the damping-like torque given here is not originate d from the spin Hall\neffect as in previous investigations [11, 17, 19], but derives from th e pure presence of a\nDW. The appearance of the damping-like SOT looks similar to the anti-d amping SOT in\n(Ga,Mn)As [42], although the physical origin is actually quite different. In (Ga,Mn)As the\nanti-damping SOT originates from the intrinsic Berry curvature dur ing the acceleration of\nelectrons, while the damping-like SOT in our case is a steady-state pr operty. Furthermore,\nthe non-equilibrium anti-damping SOT is proportional to the Rashba c onstant, our steady-\nstate damping-like SOT is of second order in the Rashba constant (c f. inset to Figure 5\nand Eq. (29)).\n5. Perturbation analysis\nTo obtain insight into the scaling behaviour of the field-like and damping -like SOT\ncomponents, we resort to a perturbation analysis of the SOT. The Hamiltonian H=\n−∂2\nx+k2\nBσ·ˆM+kα(σxky+iσy∂x)canbetransformedintoadifferentform, ˜H=−[∂x−i(kα+\nα′)σy/2]2+λzσz+k2\nα/4, throughaunitarytransformation ˜H=U†\nyHUy. Theunitarymatrix\nUy= exp(−iασy/2) depends on the x-coordinate due to the position-dependent angle α,\nwhich is defined by tan α= tanθ+kαkysecθ/k2\nB. The positive local effective exchange field\nnow is both position and momentum dependent, λ2\nz=k4\nB+k2\nαk2\ny+ 2kαkyk2\nBsinθ. From13\nthis expression, it is obvious that the difficulty in solving the Hamiltonian analytically\ncan be traced back to the position variation of the effective exchan ge fieldλzwith finite\nRashba coupling kα. In the case of zero kαorky, there exists analytical solutions at least\nfor a harmonic exchange field ( α′=θ′= constant). The transformed Hamiltonian can be\ndecomposed into two parts, ˜H=H0+V, and hence allows a perturbative analysis. The\nunperturbed Hamiltonian is given by H0=−∂2\nx+λ0σz+ikασy∂x, and the perturbation is\nV=kαα′/2+α′2/4+iσy(α′∂x+α′′/2)+(λz−λ0)σz. Now the constant but momentum\ndependent exchange field is λ0=/radicalBig\nk4\nB+k2αk2y. In terms of the wave function ξof˜H,\nthe wave function ψofHcan be readily obtained through a unitary transformation with\noperatorUy,ψ=Uyξ.\nThe solution ξ0=Uxη±exp(ikxx) toH0gives the zeroth order approximation to the\nexact wave function of ˜H. The unitary operator Uxis given by Ux= exp(−iβσx/2) with\ntanβ=kαkx/λ0. With this approximate wave function, the magnetization vector ca n be\nreadily calculated as\nm±(k) = ((ˆzcosα+ ˆxsinα)cosβ−ˆysinβ)η†\n±σzη±. (19)\nIt is interesting to note that, although there is a magnetization com ponent along the y\ndirection, the magnetization in the xzplane is parallel to the local effective exchange\nfield. Given this everywhere alignment of the in-plane itinerant magne tization vector and\nthe local effective exchange field, we can take the zeroth approxim ation as giving the\nproper definition of the adiabatic limit for the motion of electrons insid e a magnetization\ntexture. However, this alignment between the in-plane magnetizat ion component and the\nlocal exchange field does not guarantee the everywhere alignment between the equilibrium\nand the local magnetization vectors, since the effective exchange field depends on both\nmomentum and position, while the local magnetization depends on pos ition only.\nDue to the linear dependence on kxof theycomponent of the magnetization vector,\nthe momentum averaged magnetization in equilibrium lies entirely in the xzplane. But for\na finite Rashba coupling constant, the equilibrium magnetization is not exactly parallel to\nthe local magnetization, as stated above. Nevertheless, the dev iation is only of the second\norder inkα, and to the first order of kα, the itinerant magnetization is parallel to the local\nmagnetization,\nm±∝ ±ˆM. (20)\nThe current-induced spin accumulation is given by an average over t he Fermi surface, with\nnon-zero contribution from the ycomponent only\nδm±∝ ±kαˆy. (21)\nSimilar to the case of a uniform magnetization distribution discussed in Sec. 3, the spin\naccumulation is perpendicular to the equilibrium magnetization and pro portional to the\nRashba constant.\nThe momentum specific SOT (Eq. (8)) follows straightforwardly fro m the adiabatic\nwave function, too. It has the form\nτ±(k) =−4kα(kcosαcosβ+ ˆz(kysinβ−kxcosβsinα))η†\n±σzη±.(22)\nThe equilibrium SOT has only a ycomponent,\nτ±∝ˆyk2\nαsin2θη†\n±σzη±, (23)14\n10 20 30 40 50 60 7010-210-1100\n0.40.71\n00 0.05 0.100.51\nFigure 5. Non-adibaticity ( β/α) atx= 0 (left y-axis) and |x|=∞(righty-axis) as a\nfunction of the DW width, with kB/kF= 0.4 and kα/kF= 0.1. The data are normalized\nto unity at the smallest DW width. The solid curves are fitted to the da ta points using the\nexpression Eq. (29) given in the main text. It is obvious that a single e xponential decay\nof the non-adiabaticity cannot fit the data. The inset shows the de pendence of α(∞)\nandβ(∞) on the Rashba coupling constant kαwithkFδ= 10. The linear and parabolic\ndependence of αandβonkαrespectively is approximately obeyed. The solid lines in the\ninset are guides to the eye.\nwhilethecurrentinducedSOThastheformofthatofauniformmagn etizationdistribution,\nδτ±∝ −kαˆy×m±. (24)\nThe appearance of the adiabatic DM torque, Eq. (23), is accompan ied by the non-zero\nvariation of the magnitude of the magnetization, although the spin d ensity has a constant\nmagnitude for a single Bloch state. The variation of the magnetizatio n magnitude appears\nafter thek-space average.\nThe first order correction ξ1to the wave function ξis determined by ( H0−ǫ0)ξ1=\n−Vξ0. With the Green’s function\nG(k) = (H0−ǫ0)−1=k2−ǫ0−λkUxσzU†\nx\n(k2−ǫ0)2−λ2\nk(25)\nink-space, the first order wave function ξ1can be simply expressed as\nξ1(k) =−2πG(k)V(k,ki)Ux(ki)η±, (26)\nwhere 2πV(k,ki) =/integraltextdxexp(−ikx)V(x)exp(ikix) is thek-space representation of the\npotential. The expression for ξ1has a simple interpretation: the incoming wave with\nmomentum kiis scattered by the potential V(k,ki) into the final state with momentum k,\nwhile the Green’s function in momentum space gives the propagation f actor. To the first\norder ofkα, the explicit form of V(k,ki) is\nV(k,ki) =∆k\n8cschπδ∆k\n2−kα\n8/parenleftBiggky\nk2\nB1+δ2(∆k)2\nδ−2/parenrightBigg\nsechπδ∆k\n2\n−k+ki\n4/parenleftBigg\nsechπδ∆k\n2−kαδ\nk2\nBky∆kcschπδ∆k\n2/parenrightBigg\nσy+kαδ\n2k2\nBkyλ0σzsechπδ∆k\n2,(27)\nwhich brings about the exponential decay of the physical quantitie s on the DW width\nthrough the hyperbolic secant and cosecant functions with finite m omentum transfer15\n∆k=ki−k[43]. When the momentum transfer is zero, the k-space potential has a\nsimple form,\nV(ki,ki) =kα\n4/parenleftBigg\n1−ky\n2k2\nBδ/parenrightBigg∆θ\nπ−∆cosθ\n8πδ−ki/parenleftBiggkαky\nk2\nB∆cosθ\n2π+∆θ\n2π/parenrightBigg\nσy+kαkyδ∆θ\n2πσz,(28)\nwhere ∆θ=θ(∞)−θ(−∞) =πand ∆cosθ= cosθ(∞)−cosθ(−∞) =−2. We\nintentionally retain ∆ θand ∆cosθin Eq. (28) to highlight the topological origin of each\nterm. It isV(ki,ki), the potential with zero momentum transfer, that introduces d eviation\nfrom the exponential decay of observables as required by Eq. (27 ).\nAtx= 0 andx=±∞, the damping-like ( y) SOT component can be calculated using\nthe first order wave function and has the form as given in Eq. (2) wit h the coefficient\nβ∝k2\nα/parenleftbigg\nc+a\nδ2+be−γδ/parenrightbigg\n(29)\ntothelowest orderin kα, wherea,b,candγareall constants. Thelast, exponential termof\n��comesfromthescattering oftheoriginalwave withafinitemomentu m transfer(∆ k∝negationslash= 0),\nwhile the first two terms with zero momentum transfer (∆ k= 0). The interference terms\nthatresultfromtheinnerproductofwaveswithdifferentwavevec torsmakenocontribution\nto the damping-like SOT component at infinity, since they are averag ed to zero after an\nintegration over the Fermi surface. The constant cis of the order of unity at x=±∞,\nhence as the DW width approaches asymptotically to a very large valu e, the damping-like\nSOT approaches to a constant value at ±∞. However, due to the cancellation between\nthe contributions from the spin-up and spin-down wave functions, the constant term at\nthe DW center x= 0 has a smaller value as compared to its value at ±∞. The parabolic\ndependence on kαand the scaling behaviour of the non-adiabaticity are described by t he\nform ofβgiven by Eq. (29), as can be seen from Figure 5.\nIt would be a surprise to know that there exists a sizable damping-like SOT far away\nfrom the DW center, even in the case of a large DW width. One would ex pect a zero\ndamping-like SOT at x=±∞, since the magnetization variation far away from the DW\ncenterx= 0 is negligibly small, and the deviation from the case of a uniform magne tization\ndistribution should be very small, too. To resolve this contradiction, the non-local nature\nof the damping-like SOT, or all the physical quantities in quantum mec hanics, should be\nemphasized. The wave function in our case considered here is exten ded, instead of being\nlocalized to a region with finite extension, through the whole space, h ence the presence of\ntheDWwillinfluencethewholewavefunction, ratherthanonlylocallya tregionswherethe\nDW is present. Specifically, the asymptotic damping-like SOT at x=±∞is determined\nby the Fourier transformation of the localized potential Vwith zero momentum transfer,\nwhich is an integration over the whole space and gives rise to the non- local character of\nthe damping-like SOT. Hence, even at x=±∞where the magnetization variation of the\nDW is infinitesimal, due to the pure existence of the DW, the damping-lik e SOT is still\nnot zero, in contrast to what can be expected for a uniform magne tization distribution.\nIn a more mathematical point of view, the different behaviour of dam ping-like SOT is a\nreflection of the fact that a DW cannot be continuously deformed t o a single domain state\n[44]. Accordingly in our consideration of the SOT in DWs, we can vary th e DW width,\nbut we cannot take the limit of letting the DW width approaching infinity ,δ→ ∞. The\nreason is simple: if δ→ ∞, there would actually be no variation of magnetization in space,16\nhence no DWs. In addition, the boundary condition used to determin e the Walker DW\nprofile,dθ/dx= 0 atx=±∞, is meaningless in the limit δ→ ∞.\n6. Conclusion\nTo summarize, we have studied the magnetization dynamics of itinera nt electrons confined\nto the interface of ferromagnets/heavy metals. Due to the violat ion of the inversion\nsymmetry, which is the direct consequence of the combination of diff erent materials around\nthe interface, electrons moving in the interface are affected by an effective Rashba field. By\nnumerically solving the Pauli-Schr¨ odinger equation for electrons mo ving inside a N´ eel DW,\nwe found that in equilibrium the Rashba field reduces to the DM field. Wit h a current\nflowing, the Rashba field is transformed into the Rashba SOT acting o n the itinerant\nmagnetization. The SOT has both field-like and damping-like componen ts. In the non-\nadiabatic limit, the field-like and damping-like components are compara ble in magnitude.\nWhen the DW width is increased, hence bringing the system into the ad iabatic limit, the\nfield-like component becomes the dominant one. However, far away from the DW center,\nthe magnitude of the damping-like torque is still comparable to that o f the field-like torque,\nas far as the Rashba coupling constant is sizable. This finite contribu tion to the damping-\nlike torque can be traced back to the scattering of the spin-up and spin-down components\ninduced by the Rashba SOI and the continuous variation of magnetiz ation in the DW.\nFurther studies to include the effects of impurity scattering and re alistic electronic band\nstructures could test whether the same scaling behaviour of the d amping-like SOT will\npersist.\nAcknowledgments\nWe would like to express our gratitude to Prof. Jiang Xiao for his valua ble comments and\ndiscussions, especially for bringing us to the topic of SOT in magnetic D Ws and sharing\nhis code on STT simulation. This work was supported by the President ’s Fund of The\nChinese University of Hong Kong, Shenzhen, the National Natural Science Foundation of\nChina (Grant No. 11574137), and the Shenzhen Fundamental Res earch Fund (Grant Nos.\nJCYJ20160331164412545 and JCYJ20170410171958839).\nReferences\n[1] R. C. 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Das Sarma\nCondensed Matter Theory Center and Joint Quantum Institute,\nDepartment of Physics, University of Maryland, College Park, MD 20742-4111, USA\n(Dated: March 9, 2022)\nWe study theoretically quantum dynamics of interacting bosons in arti\fcial magnetic \felds as\nengineered in recent ultracold atomic experiments, where quantum cyclotron orbital motion has\nbeen observed. With exact numerical simulations and perturbative analyses, we \fnd that inter-\nactions induce damping in the cyclotron motion. The damping time is found to be dependent on\ninteraction and tunneling strengths monotonically, while its dependence on magnetic \rux is non-\nmonotonic. Su\u000eciently strong interactions would render bosons dynamically localized inhibiting\nthe cyclotron motion. The damping predicted by us can be construed as an interaction-induced\nquantum decoherence of the cyclotron motion.\nI. INTRODUCTION\nCyclotron orbits and Landau levels formed by elec-\ntrons moving in magnetic \felds play an essential role in\nthe emergence of several novel phenomena in solid state\nsystems. Semiclassical cyclotron orbital motion in two\ndimensional electron gas gives rise to the Hall conduc-\ntance and then eventually to quantized Hall conductance\nin high enough magnetic \felds. In quantum Hall insu-\nlators, chiral edge states mediating dissipationless edge\ncurrent [1, 2] can be understood as quantum cyclotron\norbits bounded by the edges in a strip-like geometry. In\nthe current work, we study the cyclotron motion of in-\nteracting bosonic neutral atoms in an optical lattice sub-\njected to arti\fcial gauge \felds which act like e\u000bective\nexternal magnetic \felds in the lattice leading to novel\nphysics [3].\nUltracold atomic gases con\fned in optical lattices, be-\ncause of their unprecedented controllability, allow for\nquantum simulations of various lattice Hamiltonians,\ne.g., Bose-Hubbard models [4, 5], where both equilibrium\nmany-body physics [6, 7] and non-equilibrium dynamics\nhave been extensively studied theoretically [8] and ex-\nperimentally [9, 10]. Recent experiments created a two\ndimensional square lattice pierced by magnetic \rux (\\ar-\nti\fcial gauge \felds\") by engineering laser assisted tunnel-\ning [11{13]. Due to non-trivial Berry curvatures in such\na system, charge neutral atoms, e.g.87Rb, loaded into\nthis \rux lattice behave like \\charged\" bosons experienc-\ning strong magnetic \felds, and the consequent e\u000bective\nLorentz force results in atomic cyclotron motion, which\nhas been experimentally observed [12]. While these inter-\nesting experimental developments are largely motivated\nby considerations of observing fascinating new equilib-\nrium many-body phases such as atomic quantum spin\nHall insulators [14{20], the observed non-equilibrium cy-\nclotron dynamics of bosons itself [12] is extremely inter-\nesting and requires theoretical understanding. In partic-\nular, the interaction e\u000bect on the dynamical cyclotron\nmotion is obviously of great interest, and is the main\ntopic of study in the current work.\nIn this article, we study cyclotron dynamics of inter-\nacting bosons with both exact numerical simulations andperturbative analyses. Weak interactions are found to\ninduce damping e\u000bects (i.e., quantum decoherence) in\nthe dynamics. We \fnd that while the damping time (or\nthe decoherence time) monotonically decreases with in-\ncreasing tunneling and interaction, it has non-monotonic\nbehavior with varying magnetic \rux. With the pertur-\nbative analyses, the damping e\u000bect is attributed to spe-\nci\fc scattering processes, and such physics is established\nto be generic for interacting bosons in arti\fcial gauge\n\felds, i.e., not relying on the model Hamiltonian used in\nour numerical simulations. With su\u000eciently strong in-\nteractions, cyclotron dynamics is completely suppressed\nand the bosons form a dynamically localized state, anal-\nogous to self-trapping e\u000bects observed in Bose-Einstein\ncondensates in double-well potentials [21{24]. Our \fnd-\ning of the cyclotron damping e\u000bect suggests importance\nof interactions and many-body physics in quantum trans-\nport of bosons in arti\fcial gauge \felds, which is of great\ninterest in recent atomic gases [18, 25, 26]. Quantum\nsimulations of this damping e\u000bect in such controllable\nsystems would help understand relaxations in Hall trans-\nport experiments in complex electronic materials where\nthe decoherence could be attributed to various origins.\nFIG. 1. An optical lattice plaquette with magnetic \rux \b\nand its spectra. (a) The plaquette system where the sites 1,\n2, 3 and 4 are located at (0 ;0), (0;\u00001), (1;\u00001), and (1;0) in\nour coordinate choice. (b) The single-particle energy spectra\nwith varying \b.arXiv:1312.5747v2 [cond-mat.quant-gas] 24 Jun 20142\nII. SYSTEM AND MODEL HAMILTONIAN\nIn the experimental setup to observe the cyclotron mo-\ntion [12], a plaquette of four lattice sites is isolated in\na two dimensional optical lattice by suppressing inter-\nplaquette tunnelings with superlattice techniques. To\nstudy the cyclotron motion, we look at one isolated pla-\nquette threaded by magnetic \rux as illustrated in Fig. 1.\nThe model Hamiltonian describing bosons loaded into\nthis plaquette is ( ~= 1 throughout)\nH=H0+V\nH0=\u0000Kh\nei\b=2(by\n2b1+by\n4b3) +h:c:i\n\u0000Jh\nby\n3b2+by\n4b3+h:c:i\nV=U\n2X\njby\njby\njbjbj; (1)\nwherebjis a bosonic annihilation operator for the\njth site. This engineered Hamiltonian connects to\ncharged bosons in magnetic \felds through Peierls substi-\ntution [27, 28]. The tunneling strength Jis \fxed in our\ncalculation to be 0 :5\u00022\u0019kHz following the experimental\nsituation. The free part of the Hamiltonian can be writ-\nten asH0=P\njj0H(0)\njj0by\njbj0, withH(0)the single-particle\nHamiltonian matrix. We study the quantum dynamics\nassuming an initial state\nj\ti=1p\nN!\u0002\n y\u0003Nj0i; (2)\nwith y=1p\n2\u0010\nby\n3+by\n4\u0011\n, which describes Nbosons pre-\npared in a superposed state of sites 3 and 4. The physics\ndescribed here is otherwise robust against the choice of\n as long as it is not \fne-tuned.\nTo characterize the cyclotron motion, the time depen-\ndent occupation numbers njand an average position vec-\ntor~X(t) = (x(t);y(t)) are de\fned as\nnj(t) =1\nNh\t(t)jby\njbjj\t(t)i; (3)\n~X(t) =X\nj~Rjnj(t); (4)\nwherej\t(t)iis the time evolved many-body state, and\n~Rjis the position of the j-th site (see Fig. 1(a)). The\ninitial state is not an eigenstate of the Hamiltonian and is\nthus not stationary. For non-interacting bosons, we have\nj\t(t)i=1p\nN!\u0002\n y(t)\u0003Nj0i;with y(t) =e\u0000iH0t yeiH0t=\nP\nj j(t)by\nj;where the coe\u000ecients j(t) are determined\nby the single-particle Schr odinger equation i@t j(t) =P\nj0H(0)\njj0 j0(t). In this state, bosons actually rotate in\nthe plaquette when the engineered magnetic \rux is non-\nzero (see Fig. 2), which is a quantum analogue of classical\ncharged particles moving in a magnetic \feld, and this\nquantum cyclotron motion is undamped. The densityinhomogeneity among the four sites oscillates without\nany relaxation. One useful quantity in this dynamical\nprocess is the occupation fraction P (t) =N (t)=Nwith\nN (t) =h\t(t)j y(t) (t)j\t(t)i;\nwhereN (t) can be thought as the occupation num-\nber of the initially occupied single-particle mode (t).\nAlthough the quantum state is fully dynamical and in-\nvolves fast oscillations on the tunneling time scale, J\u00001,\n(around one millisecond), non-interacting bosons remain\nin the single-particle state (t), and the occupation frac-\ntionP (t) remains unity, indicating a perfectly coherent\nbosonic cyclotron motion in the non-interacting optical\nlattice.\nFIG. 2. Cyclotron motion of non-interacting bosons with var-\nious magnetic \rux. Bosons are circulating in the plaquette\nwith \fnite magnetic \rux ( \u0019=2 and\u0019=4 in this plot), and the\ncircular dynamics is a quantum analogue of cyclotron motion.\nFor magnetic \rux \b = 0, the dynamics cannot be identi\fed\nas cyclotron motion.\nIII. WEAKLY INTERACTING BOSONS\nA. Numerical simulations\nWe \frst simulate the dynamics (Figs. 3,4) with an ex-\nact treatment of the many-body Schr odinger equation\ni@tj\t(t)i=Hj\t(t)i, where the Hamiltonian Hand the\ntime evolved state j\t(t)iare represented in a complete\nbasis\njM1M2M3M4i=Y\nj1p\nMj!\u0010\nby\nj\u0011Mj\nj0i:\nIn our numerical simulations, the total particle number\nis \fxed to be 8, i.e., the mean \flling is two particles\nper site. In Fig. 4, the cyclotron motion illustrated by\noscillations in the average position ~X(t) = (x(t);y(t))\nshows damping in the presence of repulsive interactions.\nAfter several (quasi-)periods of cyclotron motion, ~X(t)3\nFIG. 3. Cyclotron motion with various interaction strengths.\nThe average position ( x(t);y(t)) (see the main text) illustrates\nthe rotation of bosons in the plaquette. (a) The cyclotron\nmotion of non-interacting bosons, where damping does not\noccur. (b), (c) and (d) show the interacting case with varying\ninteraction strength U. Interaction e\u000bects make ( x(t);y(t))\ncollapse into the center of the plaquette after several periods\nof rotation. The periods it costs for the rotation to collapse\ndecrease with increasing interaction strength. Here we use\nthe parameters K= 0:25\u00022\u0019kHz, and \b = 0 :735\u0002\u0019=2 as\nrealized in the experiment [12].\ncollapses to the regime around the center of the plaque-\ntte (Fig. 3). In this case, the -mode occupation frac-\ntionP (t) no longer remains unity, nonetheless it still\nremains quasi-static, namely, does not exhibit fast oscil-\nlations. The damping of oscillation amplitudes in ~X(t)\nis found to coincide with the decrease in P (t) (Fig. 4).\nThe damping of cyclotron motion is thus well captured\nbyP (t). Physically, the decrease in P (t) is caused by\ninteraction processes where bosons are scattered out of\ntheir originally occupied mode (this physical picture is\nborne out by our perturbative analysis presented below).\nIts coincidence with the cyclotron motion damping im-\nplies that the scattered bosons do not contribute to the\ncyclotron motion coherently, thus contributing to quan-\ntum decoherence.\nThe strength of damping can be quanti\fed by a damp-\ning time (decoherence time) \u001cdamp which we de\fne to be\nthe time it takes for half of the bosons in the mode\nto be scattered into other single-particle states, namely\nthe time when P (t=\u001cdamp) reaches 1=2. The damping\ntime\u001cdamp is found to be inversely proportional to the\ninteraction strength when it is su\u000eciently weak. For the\nparameters used in experiments [12]|\b \u00190:735\u0002\u0019=2,\nandK\u00190:25\u00022\u0019kHz, the damping time is around 10ms\nfor an interaction strength of U= 0:05\u00022\u0019kHz (Fig. 5).\nThus, our predicted interaction-induced cyclotron de-\ncoherence should be observable within the experimen-\nFIG. 4. Damping of cyclotron motion and decay of the oc-\ncupation fraction P (t). Top panel shows P (t) obtained by\n2nd order perturbation theory and by exact numerical simula-\ntions. The 2nd order perturbation result agrees with numer-\nics at short time as expected. In the intermediate regime\nwheret < \u001c damp,P (t) is well described by an empirical\n\ftP\ft(t) (Eq. (5)) and the \ftting error is negligible. Bot-\ntom panel shows the average position ( x(t);y(t)). Compar-\ning two panels, the damping in x(t) andy(t) coincides with\ndecrease of P (t). In this plot we use U= 0:02\u00022\u0019kHz,\nK= 0:25\u00022\u0019kHz and \b = 0 :735\u0002\u0019=2.\ntal time scales for moderate values of on-site interaction\nstrength. In the intermediate regime t < \u001c damp, we \fnd\nthat the time dependence of P (t) can be empirically\ndescribed (see Fig. 4) by a two-parameter \ftting formula\nP\ft(t) =1\n1 +\r\u0000\re\u0000(t=\u001c)2; (5)\nwhere\u001cand\rare the \ftting parameters. This \ftting for-\nmula is proposed from extending our perturbative results\n(to present below) to longer time. After the cyclotron\nmotion relaxes, i.e., t>\u001c damp and~X(t) collapses to the\nplaquette center, P\ft(t) no longer captures the dynam-\nics ofP (t) (see Fig. 6). We note that the decoherence\nprocess in Eq. (5) is not a simple temporal exponential\nrelaxation phenomenon.\nWe have also studied the dependence of \u001cdamp on the\n(complex) tunneling strength K and the applied magnetic\n\rux \b. We \fnd that \u001cdamp decreases with increasing\nK. The dependence of \u001cdamp on magnetic \rux exhibits a\nnon-monotonic behavior, having a minimum around \u0019=2\n(Fig. 5). When \b reaches \u0019, the spectra of H(0)be-\ncome degenerate (Fig. 1(b)) and the cyclotron dynamics\nchanges dramatically. Actually even with the \rux value\nclose to\u0019, the -mode occupation fraction P (t), as well\nas the oscillation amplitudes of ~X(t), yield long-time os-\ncillations, which we can attribute to the small energy\nscale in the single-particle spectrum near the degeneracy\npoint. Also P (t) is then no longer well-described by the4\nempirical \ft P\ft(t) (Eq. (5)).\nFIG. 5. Damping time with di\u000berent tunnelings, interactions\nand magnetic \rux. (a) The dependence of damping time\n\u001cdamp onKand \b, where Uis \fxed to be 0 :02\u00022\u0019kHz.\nThe lobe structure in (a) implies that \u001cdamp decreases mono-\ntonically with increasing Kand that it has non-monotonic\nbehavior with increasing \b. The minima of \u001cdamp locates\naround \b = \u0019=2. (b) shows its dependence on U, where we\nchooseK= 0:25\u00022\u0019kHz, and \b = 0 :735\u0002\u0019=2.\nFIG. 6. Long time behavior of the cyclotron decoherence. In\nthis plot we use U= 0:02\u00022\u0019kHz,K= 0:25\u00022\u0019kHz and\n\b = 0:735\u0002\u0019=2.\nB. Perturbative analysis\nTo better understand the cyclotron damping found in\nthe numerics, we carry out a perturbative analysis with\nthe standard time-dependent perturbation theory (see\nthe Appendix). Here it is useful to introduce single parti-\ncle modes\u001fy\nl=1;2;3(t)j0i, which are orthogonal to y(t)j0i.\nThese modesf y(t);\u001fy\nl(t)gform an instantaneous com-\nplete basis for the single-particle states. Similar to y(t),\nwe have\u001fy\nl(t) =e\u0000iH0t\u001fy\nl(0)eiH0t. The operators by\njare\nthen expanded as by\nj= \u0003\nj(t) y(t) +\u001f\u0003\nlj(t)\u001fy\nl(t):The oc-cupation fraction of (t) is obtained as\nP\t(t) =\n1\u0000U2(N\u00001)2X\nljIlj2\u0000U2(N\u00001)X\nl1l2jIl1l2j2;(6)\nwith\nIl=Zt\nt0dt0X\njj j(t0)j2 j(t0)\u001f\u0003\nlj(t0);\nIl1l2=Zt\nt0dt0X\nj j(t)2\u001f\u0003\nl1j(t0)\u001f\u0003\nl2j(t0): (7)\nExpanding the single-particle wavefunctions j(t) and\n\u001flj(t) in the eigen-basis of H(0)as\n j(t) =X\n\u000b'\u000b\u0015\u000b\nje\u0000i\u000f\u000bt;\n\u001flj(t) =X\n\u000b\u0014l\u000b\u0015\u000b\nje\u0000i\u000f\u000bt;\n[\u0015\u000b\njis the\u000bth eigenstate ofH(0)with energy \u000f\u000b], we get\nIl=C(1)\nlt+O(t0), andIl1l2=C(2)\nl1l2t+O(t0), with\nC(1)\nl= 2X\nj\u000b\u000b0j\u0015\u000b\njj2j\u0015\u000b0\njj2j'\u000bj2'\u000b0\u0014\u0003\nl\u000b0\nC(2)\nl1l2= 2X\nj\u000b\u000b0j\u0015\u000b\njj2j\u0015\u000b0\njj2'\u000b'\u000b0\u0014\u0003\nl1\u000b\u0014\u0003\nl2\u000b0; (8)\nprovided that there is no \fne-tuned degeneracy in the\nspectrum ofH(0). ThenP (t) simpli\fes to\nP (t)\u00191\u0000U2t2\n\u0002\"\n(N\u00001)2X\nljC(1)\nlj2+ (N\u00001)X\nl1l2jC(2)\nl1l2j2#\n:(9)\nThis 2nd order perturbative result is checked against ex-\nact numerics (see Fig. 4). The \ftting formula P\ft(t)\n(Eq. (5)) can be thought as an empirical extension of this\nperturbative result to longer time. The physical picture\nthat emerges isjC(1)\nlj2describes one-particle loss rate and\njC(2)\nl1l2j2two-particle loss rate (Fig. 7). The damping time\nis estimated from our perturbative analysis to be\n\u001cdamp/U\u00001\nq\n(N\u00001)2P\nljC(1)\nlj2+ (N\u00001)P\nl1l2jC(2)\nl1l2j2:\n(10)\nCarrying out the summations in Eq. (8) numerically, we\n\fnd that two particle processes dominate over single par-\nticle ones, when the particle number is not too large, say\nN < 10. With bosons scattered into the \u001fmodes, the\ndepletion of N causes the damping of cyclotron motion.\nThe dependence of \u001cdamp on tunneling, interaction, and\nmagnetic \rux found in numerical simulations is repro-\nduced in the perturbative analysis, and in particular, the5\nnon-monotonic dependence on the magnetic \rux is re-\nproduced. The long-time oscillations in P (t) show up\nnaturally in the integrals of Eq. (7) near \u0019-\rux, where\nthe spectral degeneracy actually invalidates Eq. (8).\nGiven the perturbative analysis, the damping phenom-\nena in cyclotron motion are expected to be generic for\ninteracting bosons in arti\fcial magnetic \felds. Despite\nthe used speci\fc model Hamiltonian (Eq. (1)) in numer-\nical simulations, the described damping physics is rather\nmodel independent .\nFIG. 7. Schematic diagrams of particle-loss from the initially\noccupied single-particle mode . Bosons are scattered into\nthe modes \u001flin interaction processes. (a) and (b) illustrate\nthe single- and two-particle loss, respectively.\nIV. STRONG INTERACTIONS AND\nDYNAMICAL LOCALIZATION\nWe further look at stronger interactions, which are po-\ntentially accessible in experiments, for example by imple-\nmenting deep lattices. The perturbative analysis would\nno longer be reliable in the strongly interacting limit.\nOur numerical simulations show that bosons tend to lo-\ncalize for strong interaction, suppressing cyclotron mo-\ntion completely. The quantity characterizing the local-\nization phenomenon is the number imbalance among the\nfour sites\n\u0001n(t) = (n3(t) +n4(t))\u0000(n1(t) +n2(t));(11)\nwhose time average\n\u0001n=1\nTZT\n0dt\u0001n(t)\ndistinguishes localized and delocalized states. In our sim-\nulations, we choose Tto be 2 seconds and convergence\nis checked for longer time. As shown in Fig. 8, in a de-\nlocalized state with weak interactions, the number im-\nbalance \u0001noscillates fast (at tunneling time scale) in\ntime and the time average \u0001nvanishes. In a localized\nstate with strong interactions, \u0001 nstill oscillates in time\nbut is otherwise always positive, and thus \u0001nis \fnite,\nmeaning that bosons are localized on sites 3 and 4. The\nparticle transfer from sites 3 and 4 to other two sites\nis suppressed. An intuitive picture to understand thislocalization is that the tunneling probability, with large\nrepulsion, is greatly suppressed because bosons have to\ntunnel all together in order to preserve energy. In the in-\ntermediate/crossover regime, the dynamics in \u0001 nyields\n\ructuations at very long time scale, which makes it chal-\nlenging to determine a precise transition point in numer-\nics. Another property of the localized state is that the\n -mode occupation fraction P (t) yields fast oscillations,\ni.e., is no longer quasi-static. This peculiar dynamical lo-\ncalization of strongly repulsive bosons is a generalization\nof self-trapping in double-wells to the plaquette system,\nand is an important testable prediction of our theory.\nFIG. 8. Dynamical localization at strong interaction. (a),\nThe number imbalance dynamics \u0001 n(t) (Eq. (11)) for delo-\ncalized, localized and intermediate states, where the interac-\ntion strengths are chosen to be U=~= 0:2;5;1 (2\u0019kHz), re-\nspectively. (b), Time averaged number imbalance \u0001nvarying\ninteraction strengths. In this plot we use K= 0:25\u00022\u0019kHz,\nand \b = 0:735\u0002\u0019=2.\nV. DISCUSSION AND CONCLUSION\nAlthough this work focused on a speci\fc model Hamil-\ntonian as motivated by the recent experiments [12, 13],\nthe studied interaction induced damping in atomic cy-\nclotron motion is expected to be a generic phenomenon.\nIn particular, the damping mechanism as shown in Fig. 7\nand the derived damping time in Eq. (10) are actu-\nally model-independent and directly applicable to more\ngeneric magnetic Hamiltonians as well. For example, the\nneglected trap e\u000bects as in the experimental setup [12, 13]\ncould be easily included within our developed framework.\nThe presence of the shallow trap in principle generates\nweak potential di\u000berence among the four sites (Fig. 1(a))\nand further modi\fes the tunneling amplitudes, and such\ne\u000bects are captured by our analytic formula (Eq. (10)).\nWith a reasonable assumption that the induced poten-\ntial di\u000berence and the modi\fed tunneling amplitudes are\nsmaller than 10% of J, we \fnd that the physics presented\nin this work is robust. One relevant question in this con-\ntext is whether the damping discovered by us is really a\n`quantum collapse' phenomenon (e.g. Jaynes-Cummings\nmodel [29]) with the revival of the cyclotron motion at\na very long time. It is perhaps possible, in principle, for\nthe system to revive at a very long time, but the fact6\nthat our analytical theory agrees with our direct numer-\nical simulations and that we see no revival in the simu-\nlation indicates that such a revival, even if it happens,\nwill occur at an unphysically long time of little interest\nto laboratory experiments.\nOur predictions of interaction induced damping, deco-\nherence, and dynamical localization (i.e. complete sup-\npression) of the recently reported bosonic cyclotron mo-\ntion [12] in optical lattices in the presence of an arti\f-\ncial magnetic \rux should be directly experimentally ob-\nservable since all our results presented in this work use\nreasonable parameters easily achieved in the laboratory.\nThe observation of our predicted novel dynamical phe-\nnomena will be a direct manifestation of interaction ef-\nfects on the quantum dynamics of Bose-Hubbard model\nin an e\u000bective magnetic \feld.\nVI. ACKNOWLEDGMENTS\nWe would like to thank Jay Deep Sau, Kai Sun and\nAnatoli Polkovnikov for helpful discussions. This work is\nsupported by JQI-NSF-PFC, ARO-Atomtronics-MURI,\nand AFOSR-JQI-MURI.\nAppendix A: Details of perturbative analysis for\ncyclotron damping\nThe details of perturbative analysis of the cyclotron\ndamping dynamics are given here. With standard per-\nturbation theory, the time-dependent quantum state in\nthe interaction picture reads\nj\tI(t)i=A(t)j\t(0)\nIi+j\t(1)\nI(t)i+j\t(2)\nIi+O(U3);(A1)\nwith the leading part j\t(0)\nIi=j\t(t= 0)i, the renormal-\nization factor\nA(t) = 1\u0000iZt\nt0dt0h\t(0)\nIjVI(t0)j\t(0)\nIi\n\u0000Zt\nt0dt0Zt0\nt0dt00h\t(0)\nIjVI(t0)VI(t00)j\t(0)\nIi;(A2)the \frst order correction\nj\t(1)\nI(t)i=\u0000iZt\nt0dt0PVI(t0)j\t(0)\nIi;\nand the second order correction\nj\t(2)\nI(t)i=\u0000Zt\nt0dt0Zt0\nt0dt00PVI(t0)VI(t00)j\t(0)\nIi;\nwhere the projection operator is P= 1\u0000j\t(0)\nIih\t(0)\nIj\nand the interaction term VI(t) =eiH0tVe\u0000iH0t. Then\nthe occupation number of the (t) mode,N (t), is given\nby\nN (t) =NjA(t)j2+h\t(1)\nI(t)j y j\t(1)\nIi+O(U3):(A3)\nThe statej\t(2)\nIidoes not contribute to this order because\nh\t(0)\nIj y j\t(2)\nIi= 0.\nIt is useful to introduce single particle modes\n\u001fy\nl=1;2;3(t)j0i, which are orthogonal to y(t)j0i. These\nmodesf y(t);\u001fy\nl(t)gform an instantaneous complete ba-\nsis for the single-particle states. Similar to y(t), we have\n\u001fy\nl(t) =e\u0000iH0t\u001fy\nl(0)eiH0t, and\u001fl(0) will be shortened as\n\u001flin the following. The operators by\njare then expanded\nas\nby\nj= \u0003\nj(t) y(t) +\u001f\u0003\nlj(t)\u001fy\nl(t):\nThe renormalization factor A(t) is given by\n1\u0000jA(t)j2=U2N(N\u00001)2X\nljIlj2\n+1\n2U2N(N\u00001)X\nl1l2jIl1l2j2;(A4)\nwithIlandIl1l2given in Eq. (7). The perturbed state\nj\t(1)\nI(t)iis\nj\t(1)\nI(t)i=\u0000i(N\u00001)p\nNU\"X\nlIl\u001fy\nl#\n yN\u00001\np\n(N\u00001)!j0i\n\u0000i\n2p\nN(N\u00001)U\"X\nl1l2Il1l2\u001fy\nl1\u001fy\nl2#\n yN\u00002\np\n(N\u00002)!j0i:(A5)\nThe obtained occupation fraction of the (t) mode is\ngiven in Eq. (9).\n[1] R. B. Laughlin, Phys. Rev. B 23, 5632 (1981).\n[2] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and\nM. den Nijs, Phys. Rev. Lett. 49, 405 (1982).\n[3] V. Galitski and I. B. 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Jaynes and F. W. Cummings, Proceedings of the\nIEEE 51, 89 (1963)." }, { "title": "1204.5342v1.Nonlocal_feedback_in_ferromagnetic_resonance.pdf", "content": "Nonlocal feedback in ferromagnetic resonance\nThomas Bose and Steffen Trimper\nInstitute of Physics, Martin-Luther-University, D-06099 Halle, Germany\u0003\n(Dated: April 27, 2022)\nAbstract\nFerromagnetic resonance in thin films is analyzed under the influence of spatiotemporal feedback\neffects. The equation of motion for the magnetization dynamics is nonlocal in both space and time\nandincludesisotropic, anisotropicanddipolarenergycontributionsaswellastheconservedGilbert-\nand the non-conserved Bloch-damping. We derive an analytical expression for the peak-to-peak\nlinewidth. It consists of four separate parts originated by Gilbert damping, Bloch-damping, a mixed\nGilbert-Bloch component and a contribution arising from retardation. In an intermediate frequency\nregimetheresultsarecomparablewiththecommonlyusedLandau-Lifshitz-Gilberttheorycombined\nwith two-magnon processes. Retardation effects together with Gilbert damping lead to a linewidth\nthe frequency dependence of which becomes strongly nonlinear. The relevance and the applicability\nof our approach to ferromagnetic resonance experiments is discussed.\nPACS numbers: 76.50.+g; 76.60.Es; 75.70.Ak; 75.40.Gb\n\u0003thomas.bose@physik.uni-halle.de; steffen.trimper@physik.uni-halle.de\n1arXiv:1204.5342v1 [cond-mat.mes-hall] 24 Apr 2012I. INTRODUCTION\nFerromagnetic resonance enables the investigation of spin wave damping in thin or ul-\ntrathin ferromagnetic films. The relevant information is contained in the linewidth of the\nresonance signal [1–3]. Whereas the intrinsic damping included in the Gilbert or Landau-\nLifshitz-Gilbert equation [4, 5], respectively, predicts a linear frequency dependence of the\nlinewidth [6], the extrinsic contributions associated with two-magnon scattering processes\nshow a nonlinear behavior. Theoretically two-magnon scattering was analyzed for the case\nthat the static external field lies in the film plane [7, 8]. The theory was quantitatively\nvalidated by experimental investigations with regard to the film thickness [9]. Later the\napproach was extended to the case of arbitrary angles between the external field and the\nfilm surface [10]. The angular dependence of the linewidth is often modeled by a sum of\ncontributions including angular spreads and internal field inhomogeneities [11]. Among oth-\ners, two-magnon mechanisms were used to explain the experimental observations [12–17]\nwhereas the influence of the size of the inhomogeneity was studied in [18]. As discussed in\n[3, 14] the two-magnon contribution to the linewidth disappears for tipping angles between\nmagnetization and film plane exceeding a critical one \bcrit\nM=\u0019=4. Recently, deviations from\nthis condition were observed comparing experimental data and numerical simulations [17].\nSpin pumping can also contribute to the linewidth as studied theoretically in [19]. How-\never, a superposition of both the Gilbert damping and the two-magnon contribution turned\nout to be in agreement very well with experimental data illustrating the dependence of the\nlinewidth on the frequency [16, 20–23]. Based on these findings it was put into question\nwhether the Landau-Lifshitz-Gilbert equation is an appropriate description for ferromag-\nnetic thin films. The pure Gilbert damping is not able to explain the nonlinear frequency\ndependence of the linewidth when two-magnon scattering processes are operative [3, 24].\nAssuming that damping mechanisms can also lead to a non-conserved spin length a way\nout might be the inclusion of the Bloch equations [25, 26] or the the Landau-Lifshitz-Bloch\nequation [27, 28] into the concept of ferromagnetic resonance.\nAnother aspect is the recent observation [29] that a periodic scattering potential can alter\nthe frequency dependence of the linewidth. The experimental results are not in agreement\nwith those based upon a combination of Gilbert damping and two-magnon scattering. It\nwas found that the linewidth as function of the frequency exhibits a non monotonous be-\n2havior. The authors [29] suggest to reconsider the approach with regard to spin relaxations.\nMoreover, it would be an advantage to derive an expression for the linewidth as a measure\nfor spin damping solely from the equation of motion for the magnetization.\nTaking all those arguments into account it is the aim of this paper to propose a gener-\nalized equation of motion for the magnetization dynamics including both Gilbert damping\nand Bloch terms. The dynamical model allows immediately to get the magnetic susceptibil-\nity as well as the ferromagnetic resonance linewidth which are appropriate for the analysis\nof experimental observations. A further generalization is the implementation of nonlocal\neffects in both space and time. This is achieved by introducing a retardation kernel which\ntakes into account temporal retardation within a characteristic time \u001cand a spatial one\nwith a characteristic scale \u0018. The last one simulates an additional mutual interaction of\nthe magnetic moments in different areas of the film within the retardation length \u0018. Re-\ncently such nonlocal effects were discussed in a complete different context [30]. Notice that\nretardation effects were already investigated for simpler models by means of the Landau-\nLifshitz-Gilbert equation. Here the existence of spin wave solutions were in the focus of the\nconsideration [31]. The expressions obtained for the frequency/damping parameters were\nconverted into linewidths according to the Gilbert contribution which is a linear function\nof the frequency [31, 32]. In the present approach we follow another line. The propagating\npart of the varying magnetization is supplemented by the two damping terms due to Gilbert\nand Bloch, compare Eq. (9). Based on this equation we derive analytical expressions for the\nmagnetic susceptibility, the resonance condition and the ferromagnetic resonance linewidth.\nDue to the superposition of damping and retardation effects the linewidth exhibits a non-\nlinear behavior as function of the frequency. The model is also extended by considering\nthe general case of arbitrary angles between the static external field and the film surface.\nMoreover the model includes several energy contributions as Zeeman and exchange energy\nas well as anisotropy and dipolar interaction. The consequences for ferromagnetic resonance\nexperiments are discussed.\nII. DERIVATION OF THE EQUATION OF MOTION\nIn order to define the geometry considered in the following we adopt the idea presented\nin [10], i.e. we employ two coordinate systems, the xyz-system referring to the film surface\n3ΘMey\nex,eX\nezMS\neZeY\nΘHH0\n/Bullet\n/Bullet\n/BulletξM(z1)\nM(z2)\nM(z3)hrf\nd\nlxlzFIG. 1. (Color online) The geometry referring to the film and the magnetization. Further descrip-\ntion in the text.\nand the XYZ-system which is canted by an angle \u0002Mwith respect to the film plane. The\nsituation for a film of thickness dis sketched in Fig. 1. The angle \u0002Mdescribing the direction\nof the saturation magnetization, aligned with the Z-axis, originates from the static external\nfieldH0which impinges upon the film surface under an angle \u0002H. Therefore, it is more\nconvenient to use the XYZ-system for the magnetization dynamics. As excitation source\nwe consider the radio-frequency (rf) magnetic field hrfpointing into the x= X-direction. It\nshould fulfill the condition hrf\u001cH0. To get the evolution equation of the magnetization\nM(r;t),r= (x;y;z )we have to define the energy of the system. This issue is well described\nin Ref. [10], so we just quote the most important results given there and refer to the cited\nliterature for details. Since we consider the thin film limit one can perform the average along\nthe direction perpendicular to the film, i.e.\nM(rk;t) =1\ndZd=2\n\u0000d=2dyM(r;t); (1)\nwhere rk= (x;0;z)lies in the film plane. In other words the spatial variation of the\nmagnetization across the film thickness dis neglected. The components of the magnetization\npoint into the directions of the XYZ-system and can be written as [33]\nM(rk;t) =MX(rk)eX+MY(rk)eY+\u0012\nMS\u0000M2\nX(rk) +M2\nY(rk)\n2MS\u0013\neZ:(2)\n4Typically the transverse components MX;Yare assumed to be much smaller than the satu-\nration magnetization MS. Remark that terms quadratic in MX;Yin the energy will lead to\nlinear terms in the equation of motion. The total energy of the system can now be expressed\nin terms of the averaged magnetization from Eq. (1) and reads\nH=Hz+Hex+Ha+Hd: (3)\nThe different contributions are the Zeeman energy\nHz=\u0000Z\nd3rH0sin (\u0002 H\u0000\u0002M)MY(rk)\n\u0000Z\nd3rH0cos (\u0002 H\u0000\u0002M)\u0012\nMS\u0000MX(rk)2+MY(rk)2\n2MS\u0013\n;(4)\nthe exchange energy\nHex=D\n2MSZ\nd3r\u0002\nrMX(rk)\u00032+\u0002\nrMY(rk)\u00032; (5)\nthe surface anisotropy energy\nHa=HSMSV\n2sin2(\u0002M) +HS\n2sin(2\u0002 M)Z\nd3rM Y(rk)\n+HS\n2MScos(2\u0002 M)Z\nd3rM Y(rk)2\u0000sin2(\u0002M)Z\nd3rM X(rk)2;(6)\nand the dipolar energy\nHd=2\u0019M2\nSVsin2(\u0002M) +\u0019Z\nd3r\u001a\n2MSsin(2\u0002 M)MY(rk)\n+\u0012dk2\nz\nkksin2(\u0002M)\u0000(dkk\u00002) cos2(\u0002M)\u00002 sin2(\u0002M)\u0013\nMY(rk)2\n+\u0012dk2\nx\nkk\u00002 sin2(\u0002M)\u0013\nMX(rk)2\u00002dkxkz\nkksin(\u0002 M)MX(rk)MY(rk)\u001b\n:(7)\nIn these expressions V=lxlzdis the volume of the film, Ddesignates the exchange stiffness\nandHS/d\u00001represents the uniaxial out-of-plane anisotropy field. If HS<0the easy axis\nis perpendicular to the film surface. The in-plane anisotropy contribution to the energy is\nneglected but it should be appropriate for polycrystalline samples [16]. Moreover kk=jkkj\nis introduced where kk=kxex+kzezis the wave vector of the spin waves parallel to the\nfilm surface. Eqs. (3)-(7) are valid in the thin film limit kkd\u001c1. In order to derive Hdin\nEq. (7) one defines a scalar magnetic potential and has to solve the corresponding boundary\n5value problem inside and outside of the film [34]. As result [10] one gets the expressions in\nEq. (7).\nIn general if the static magnetic field is applied under an arbitrary angle \u0002Hthe mag-\nnetization does not align in parallel, i.e. \u0002M6= \u0002 H. The angle \u0002Mcan be derived from\nthe equilibrium energy Heq=H(MX= 0;MY= 0). Defining the equilibrium free energy\ndensity asfeq(\u0002M) =Heq=Vaccording to Eqs. (3)-(7) one finds the well-known condition\nsin(\u0002 H\u0000\u0002M) =4\u0019M S+HS\n2H0sin(2 \u0002 M) (8)\nby minimizing feqwith respect to \u0002M. We further note that all terms linear in MYin\nEqs. (3)-(7) cancel mutually by applying Eq. (8) as already pointed out in Ref. [10].\nThe energy contributions in Eqs. (3) and the geometric aspects determine the dynamical\nequation for the magnetization. The following generalized form is proposed\n@\n@tM(rk;t) =ZZ\ndr0\nkdt0\u0000(rk\u0000r0\nk;t\u0000t0)(\n\r\u0002\nHeff(r0\nk;t0)\u0002M(r0\nk;t0)\u0003\n+\u000b\u0014\nM(r0\nk;t0)\u0002@\n@t0M(r0\nk;t0)\u0015\n\u00001\nT2M?(r0\nk;t0))\n;(9)\nwhere\r=g\u0016B=~is the absolute value of the gyromagnetic ratio, T2is the transverse\nrelaxation time of the components M?=MXeX+MYeYand\u000bdenotes the dimensionless\nGilbertdampingparameter. Thelatterisoftentransformedinto G=\u000b\rM Srepresentingthe\ncorresponding damping constant in unit s\u00001. The effective magnetic field Heffis related to\nthe energy in Eqs. (3)-(7) by means of variational principles [35], i.e. Heff=\u0000\u000eH=\u000eM+hrf.\nHere the external rf-field hrf(t)is added which drives the system out of equilibrium.\nRegarding the equation of motion presented in Eq. (9) we note that a similar type was\napplied in [12] for the evaluation of ferromagnetic resonance experiments. In this paper\nthe authors made use of a superposition of the Landau-Lifshitz equation and Bloch-like\nrelaxation. Here we have chosen the part which conserves the spin length in the Gilbert form\nand added the non-conserving Bloch term in the same manner. That the combination of\nthesetwodistinctdampingmechanismsissuitablefortheinvestigationofultrathinmagnetic\nfilms was also suggested in [24]. Since the projection of the magnetization onto the Z-axis is\nnot affected by T2this relaxation time characterizes the transfer of energy into the transverse\ncomponents of the magnetization. This damping type is supposed to account for spin-spin\nrelaxation processes such as magnon-magnon scattering [33, 36]. In our ansatz we introduce\n6another possible source of damping by means of the feedback kernel \u0000(rk\u0000r0\nk;t\u0000t0). The\nintroduction of this quantity reflects the assumption that the magnetization M(rk;t2)is\nnot independent of its previous value M(rk;t1)providedt2\u0000t1< \u001c. Here\u001cis a time\nscale where the temporal memory is relevant. In the same manner the spatial feedback\ncontrols the magnetization dynamics significantly on a characteristic length scale \u0018, called\nretardation length. Physically, it seems to be reasonable that the retardation length differs\nnoticeably from zero only in z-direction which is shown in Fig. 1. As illustrated in the figure\nM(x;z1;t)is affected by M(x;z2;t)while M(x;z3;t)is thought to have negligible influence\nonM(x;z1;t)sincejz3\u0000z1j>\u0018. Therefore we choose the following combination of a local\nand a nonlocal part as feedback kernel\n\u0000(rk\u0000r0\nk;t\u0000t0) =\u0000 0\u000e(rk\u0000r0\nk)\u000e(t\u0000t0)\n+\u00000\n4\u0018\u001c\u000e(x\u0000x0) exp\u0014\u0000jz\u0000z0j\n\u0018\u0015\nexp\u0014\u0000(t\u0000t0)\n\u001c\u0015\n; t>t0:(10)\nThe intensity of the spatiotemporal feedback is controlled by the dimensionless retardation\nstrength \u00000. The explicit form in Eq. (10) is chosen in such a manner that the Fourier-\ntransform \u0000(kk;!)!\u00000for\u0018!0and\u001c!0, and in case \u00000= 1the ordinary equation\nof motion for the magnetization is recovered. Further,R\ndrkdt\u0000(rk;t) = \u0000 0<1, i.e. the\nintegral remains finite.\nIII. SUSCEPTIBILITY AND FMR-LINEWIDTH\nIf the rf-driving field, likewise averaged over the film thickness, is applied in X-direction,\ni.e.hrf(rk;t) =hX(rk;t)eX, the Fourier transform of Eq. (9) is written as\n\u0014i!\n\r\u0000(kk;!)+1\n\rT2+H21(kk)\u0015\nMX(kk;!) =\u0000\u0014\nH1(kk) +i\u000b!\n\r\u0015\nMY(kk;!);\n\u0014i!\n\r\u0000(kk;!)+1\n\rT2+H12(kk)\u0015\nMY(kk;!) =\u0014\nH2(kk) +i\u000b!\n\r\u0015\nMX(kk;!)\u0000MShX(kk;!):\n(11)\n7The effective magnetic fields are expressed by\nH1(kk) =H0cos(\u0002 H\u0000\u0002M) + (4\u0019M S+HS) cos(2 \u0002 M)\n+ 2\u0019dkkMS \nk2\nz\nk2\nksin2(\u0002M)\u0000cos2(\u0002M)!\n+Dk2\nk\nH2(kk) =H0cos(\u0002 H\u0000\u0002M)\u0000(4\u0019M S+HS) sin2(\u0002M)\n+ 2\u0019dM Sk2\nx\nkk+Dk2\nk;(12)\nand\nH12(kk) = 2\u0019dM Skxkz\nkksin(\u0002 M) =\u0000H21(kk): (13)\nThe Fourier transform of the kernel yields\n\u0000(kk;!) =\u00000(1 + i!\u001c) + \u0000 1\n2 (1 + i!\u001c)(!2\u001c2\u001c1)'\u00000+ \u0000 1\n2\u0000i\n2\u00001!\u001c;\n\u00001=\u00000\n1 +\f2; \f =\u0018kz;(14)\nwhere the factor 1=2arises from the condition t > t0when performing the Fourier trans-\nformation from time into frequency domain. In Eq. (14) we discarded terms !2\u001c2\u001c1.\nThis condition is fulfilled in experimental realizations. So, it will be turned out later the\nretardation time \u001c\u001810 fs. Because the ferromagnetic resonance frequencies are of the order\n10:::100 GHz one finds!2\u001c2\u001810\u00008:::10\u00006. The retardation parameter \f=\u0018kz, introduced\nin Eq. (14), will be of importance in analyzing the linewidth of the resonance signal. With\nregard to the denominator in \u00001, compare Eq. (14), the parameter \fmay evolve ponderable\ninfluence on the spin wave damping if this quantity cannot be neglected compared to 1.\nAs known from two-magnon scattering the spin wave modes can be degenerated with the\nuniform resonance mode possessing wave vectors kk\u0018105cm\u00001. The retardation length \u0018\nmay be estimated by the size of inhomogeneities or the distance of defects on the film sur-\nface, respectively. Both length scales can be of the order \u001810:::1000 nm, see Refs. [18, 29].\nConsequently the retardation parameter \fcould reach or maybe even exceed the order of 1.\nLet us stress that in case \f= 0,\u001c= 0,\u00000= 1and neglecting the Gilbert damping,\ni.e.\u000b= 0, the spin wave dispersion relation is simply \rp\nH1(kk)H2(kk)\u0000H2\n12(kk). This\nexpression coincides with those ones given in Refs. [7] and [10].\nProceeding the analysis of Eq. (11) by defining the magnetic susceptibility \u001fas\nM\u000b(kk;!) =X\n\f\u001f\u000b\f(kk;!)h\f(kk;!);f\u000b;\fg=fX;Yg;(15)\n8whereh\fplays the role of a small perturbation and the susceptibility \u001f\u000b\fexhibits the\nresponse of the system. Eq. (15) reflects that there appears no dependence on the direction\nofkk.\nSince the rf-driving field is applied along the eX-direction it is sufficient to focus the\nfollowing discussion to the element \u001fXXof the susceptibility tensor. From Eq. (11) we\nconclude\n\u001fXX(kk;!) =MSh\nH1(kk;!) +i\u000b!\n\ri\nh\nH1(kk;!) +i\u000b!\n\rih\nH2(kk;!) +i\u000b!\n\ri\n+h\ni!\n\r\u0000(kk;!)+1\n\rT2i2:(16)\nBecause at ferromagnetic resonance a uniform mode is excited let us set kk= 0in Eqs. (12)-\n(13). Considering the resonance condition we can assume \f=\u0018kz= 0. For reasons men-\ntioned above we have to take \f=\u0018kz6= 0when the linewidth as a measure for spin damping\nis investigated. Physically we suppose that spin waves with non zero waves vectors are not\nexcited at the moment of the ferromagnetic resonance. However such excitations will evolve\nduring the relaxation process. In finding the resonance condition from Eq. (16) it seems to\nbe a reasonable approximation to disregard terms including the retardation time \u001c. Such\nterms give rise to higher order corrections. In the same manner all the contributions orig-\ninated from the damping, characterized by \u000bandT2, are negligible. Let us justify those\napproximation by quantitative estimations. The fields H1,H2and!=\rare supposed to\nrange in a comparable order of magnitude. On the other hand one finds \u000b\u001810\u00003:::10\u00002,\n!T2\u001810\u00002and!\u001c\u001810\u00004. Under these approximations the resonance condition reads\n\u0012!r\n\r\u00132\n= \u00002\n0H1(kk= 0)H2(kk= 0): (17)\nThisresultiswellknownforthecasewithoutretardationwith \u00000= 1. Althoughtheretarda-\ntion time\u001cand the retardation length \u0018are not incorporated in the resonance condition, the\nstrength of the feedback may be important as visible in Eq. (17). Now the consequences for\nthe experimental realization will be discussed. To address this issue the resonance condition\nEq. (17) is rewritten in terms of the resonance field Hr=H0(!=!r)leading to\nHr=1\n2 cos(\u0002 H\u0000\u0002M)8\n<\n:s\n(4\u0019M S+HS)2cos4(\u0002M) +\u00121\n\u000002!r\n\r\u00132\n\u0000(4\u0019M S+HS)(1\u00003 sin2(\u0002M))9\n=\n;:(18)\n9ΘM[deg]\nΘH[deg]Γ0= 0 .7\nΓ0= 1 .0\nΓ0= 1 .3FIG. 2. (Color online) Dependence of the magnetization angle \u0002Mon the angle \u0002Hunder which the\nstatic external field is applied for !r=(2\u0019) = 10 GHz . The parameters are taken from [16]: 4\u0019MS=\n16980 G,HS=\u00003400 G;\r= 0:019 GHz=G.\nThe result is arranged in the in the same manner as done in [16]. The difference is the\noccurrence of the parameter \u00000in the denominator. In [16] the gyromagnetic ratio \rand\nthe sum (4\u0019M S+HS)were obtained from \u0002H-dependent measurements and a fit of the\ndata according to Eq. (18) with \u00000= 1under the inclusion of Eq. (8). If the saturation\nmagnetization can be obtained from other experiments [16] the uniaxial anisotropy field HS\nresults. Thus, assuming \u000006= 1the angular dependence \u0002M(\u0002H)and the fitting parameters\nas well would change. In Fig. 2 we illustrate the angle \u0002M(\u0002H)for different values of \u00000and\na fixed resonance frequency. If \u00000<1the curve is shifted to larger \u0002Mand for \u00000>1to\nsmaller magnetization angles. To produce Fig. 2 we utilized quantitative results presented\nin [16]. They found for Co films grown on GaAs the parameters 4\u0019M S= 16980 G ,HS=\n\u00003400 Gand\r= 0:019 GHz=G. As next example we consider the influence of HSand denote\nH(0)\nS=\u00003400 Gthe anisotropy field for \u00000= 1andH(R)\nSthe anisotropy field for \u000006= 1. The\nabsolute value of their ratio jH(R)\nS=H(0)\nSj, derived from Hr(H(0)\nS;\u00000= 1) =Hr(H(R)\nS;\u000006= 1),\nisdepictedinFig.3forvariousfrequencies. Inthisgraphweassumedthatallotherquantities\nremain fixed. The effect of a varying retardation strength on the anisotropy field can clearly\nbeseen. Thechangeinthesignoftheslopeindicatesthattheanisotropyfield H(R)\nSmayeven\nchange its sign. From here we conclude that the directions of the easy axis and hard axis\nare interchanged. For the frequencies 4 GHzand10 GHzthis result is not observed in the\nrange chosen for \u00000. Moreover, the effects become more pronounced for higher frequencies.\n10/vextendsingle/vextendsingle/vextendsingleH(R)\nS/H(0)\nS/vextendsingle/vextendsingle/vextendsingle\nΓ04 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHzFIG. 3. (Color online) Effect of varying retardation strength on the uniaxial anisotropy field for\nvarious frequencies and \u0002M=\u0019=3.4\u0019MS= 16980 G ,HS=\u00003400 G;\r= 0:019 GHz=G, see [16].\nIn Fig. 3 we consider only a possible alteration of the anisotropy field. Other parameters like\nthe experimentally obtained gyromagnetic ration were unaffected. In general this parameter\nmay also experiences a quantitative change simultaneously with HS.\nLet us proceed by analyzing the susceptibility obtained in Eq. (16). Because the following\ndiscussion is referred to the energy absorption in the film, we investigate the imaginary part\nofthesusceptibility \u001f00\nXX. SinceexperimentallyoftenaLorentziancurvedescribessufficiently\nthe resonance signal we intend to arrange \u001f00\nXXin the form A0=(1 +u2), whereA0is the\nabsolute value of the amplitude and uis a small parameter around zero. The mapping to a\nLorentzian is possible under some assumptions. Because the discussion is concentrated on\nthe vicinity of the resonance we introduce \u000eH=H0\u0000Hr, whereHris the static external\nfield when resonance occurs. Consequently, the fields in Eq. (12) have to be replaced by\nH1;2!H(r)\n1;2+\u000eHcos(\u0002 H\u0000\u0002M). Additionally, we take into account only terms of the order\np\n\u000f\u0015in the final result for the linewidth where f\u000f;\u0015g/f!=\r[\u000b+!\u001c] + 1=(\rT2)g. After a\nlengthy but straightforward calculation we get for \u000eH=H(r)\n1;2\u001c1and using the resonance\ncondition in Eq. (17)\n\u001f00\nXX(!) =A0\n1 +h\nH0\u0000Hr\n\u0001Ti2; A0=MS\n(1 +\u0014) cos(\u0002 H\u0000\u0002M) \u0001T; \u0014=H(r)\n2\nH(r)\n1:(19)\nHere we have introduced the total half-width at half-maximum (HWHM) \u0001Twhich can be\n11brought in the form\n\u0001T=1\ncos(\u0002 H\u0000\u0002M)q\n\u00012\nG+ \u00012\nB+ \u00012\nGB+ \u00012\nR: (20)\nThe HWHM is a superposition of the Gilbert contribution \u0001G, the Bloch contribution \u0001B,\na joint contribution \u0001GBarising from the combination of the Gilbert and Bloch damping\nparts in the equation of motion and the contribution \u0001Rwhich has its origin purely in the\nfeedback mechanisms introduced into the system. The explicit expressions are\n\u0001G=!\n\rs\n\u000b\u0014\n\u000b\u000016p\u0014\n(1 +\u0014)\u00000\u00001!\u001c\n(\u00000+ \u0000 1)3\u0015\n; (21a)\n\u0001B=4 \u00000\n(\u00000+ \u0000 1)p\u0014\n(1 +\u0014)s\n1\n(\rT2)2\u00004 \u00001\n(\u00000+ \u0000 1)2!\n\r!\u001c\n\rT2; (21b)\n\u0001GB=s\n8\u00000\n(\u00000+ \u0000 1)p\u0014\n(1 +\u0014)\u000b!\n\r2T2; (21c)\n\u0001R=8p\u0014\n(1 +\u0014)!\n\r\u00000\u00001!\u001c\n(\u00000+ \u0000 1)3: (21d)\nThe parameter \u00001is defined in Eq. (14). If the expressions under the roots in Eqs. (21a)\nand (21b) are negative we assume that the corresponding process is deactivated and does\nnot contribute to the linewidth \u0001HT. Typically, experiments are evaluated in terms of the\npeak-to-peak linewidth of the derivative d\u001f00\nXX=dH0, denoted as \u0001H\u0011. One gets\n\u0001H\u0011=2p\n3\u0001\u0011; (22)\nwhere the index \u0011stands for G(Gilbert contribution), B(Bloch contribution), GB(joint\nGilbert-Bloch contribution), R(pure retardation contribution) or Tdesignating the total\nlinewidth according to Eq. (20) and Eqs. (21a)-(21d). Obviously these equations reveal a\nstrong nonlinear frequency dependence, which will be discussed in the subsequent section.\nIV. DISCUSSION\nAs indicated in Eqs. (20) - (22) the quantity \u0001H\u0011consists of well separated distinct\ncontributions. Thebehaviorof \u0001H\u0011isshowninFigs.4-6asfunctionofthethreeretardation\nparameters, the strength \u00000, the spatial range \fand the time scale \u001c. In all figures the\nfrequencyf=!=(2\u0019)is used. In Fig. 4 the dependence on the retardation strength \u00000is\n12∆HT[G]4 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHz∆Hη[G]\nΓ0∆HG\n∆HB\n∆HGB\n∆HR\n∆HTf= 70 GHzFIG. 4. (Color online) Influence of the retardation strength \u00000on the peak-to-peak linewidth \u0001HT\nfor various frequencies (top graph) and on the single contributions \u0001H\u0011forf= 70 GHz (bottom\ngraph). \u0001B= 0is this frequency region. The parameters are: \u0002H= \u0002 M= 0,\f= 0:5,\u000b= 0:01,\nT2= 5\u000210\u00008s;\u001c= 1:7\u000210\u000014s. The other parameters are 4\u0019MS= 16980 G ,HS=\u00003400 G;\r=\n0:019 GHz=G, compare [16].\nshown. As already observed in Figs. 2 and 3 a small change of \u00000may lead to remarkable\neffects. Hence we vary this parameter in a moderate range 0:5\u0014\u00000\u00142. The peak-to-peak\nlinewidth \u0001HTas function of \u00000remains nearly constant for f= 4 GHz andf= 10 GHz ,\nwhereas for f= 35 GHz a monotonous growth-up is observed. Increasing the frequency\nfurther tof= 50 GHz and70 GHzthe curves offers a pronounced kink. The subsequent\nenhancement is mainly due to the Gilbert damping. In the region of negative slope we\nset\u0001HG(\u00000) = 0, while in that one with a positive slope \u0001HG(\u00000)>0grows and tends\nto2\u000b!=(p\n3\r)for\u00000!1. The other significant contribution \u0001HR, arising from the\nretardation decay, offers likewise a monotonous increase for growing values of the retardation\nparameter \u00000. This behavior is depicted in Fig. 4 for f= 70 GHz . Now let us analyze the\ndependence on the dimensionless retardation length \f=\u0018kz. Because\fis only nonzero if\n13∆HT[G]4 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHz∆Hη[G]\nβ∆HG\n∆HB\n∆HGB\n∆HR\n∆HTf= 70 GHzFIG. 5. (Color online) Influence of the dimensionless retardation length \f=\u0018kzon the total\npeak-to-peak linewidth \u0001HTfor various frequencies (top graph) and on the single contributions\n\u0001H\u0011forf= 70 GHz (bottom graph); \u0001B= 0in this range. The parameters are: \u0002H= \u0002 M= 0,\n\u00000= 1:1,\u000b= 0:01,T2= 5\u000210\u00008s;\u001c= 1:7\u000210\u000014s. The other parameters: 4\u0019MS= 16980 G ,\nHS=\u00003400 Gand\r= 0:019 GHz=Gare taken from [16].\nkz6= 0this parameter \u0018accounts the influence of excitations with nonzero wave vector. We\nargue that both nonzero wave vector excitations, those arising from two-magnon scattering\nand those originated from feedback mechanisms, may coincide. Based on the estimation\nin the previous section we consider the relevant interval 10\u00002\u0014\f\u001410. The results are\nshown in Fig.5. Within the range of \fone recognizes that the total peak-to-peak linewidths\n\u0001HTforf= 4 GHz andf= 10 GHz offer no alteration when \fis changed. The plotted\nlinewidths are characterized by a minimum followed by an increase which occurs when \f\nexceeds approximately 1. This behavior is the more accentuated the larger the frequencies\nare. The shape of the curve can be explained by considering the single contributions as\nis visible in the lower part in Fig. 5. While both quantities \u0001HG(\f)and\u0001HR(\f)remain\nconstant for small \f,\u0001HG(\f)tends to a minimum and increases after that. The quantity\n14∆HT[G]4 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHz∆Hη[G]\nτ[fs]∆HG\n∆HB\n∆HGB\n∆HR\n∆HTf= 70 GHzFIG. 6. (Color online) Influence of the retardation time \u001con the total peak-to-peak linewidth\n\u0001HTfor various frequencies (top graph) and on the single contributions \u0001H\u0011forf= 70 GHz\n(bottom graph). \u0001B= 0in this region. The parameters are \u0002H= \u0002 M= 0,\f= 0:5,\u000b= 0:01,\nT2= 5\u000210\u00008s;\u00000= 1:1; the other parameters are taken from [16]: 4\u0019MS= 16980 G ,HS=\n\u00003400 G;\r= 0:019 GHz=G.\n\u0001HR(\f)develops a maximum around \f\u00191. Thus, both contributions show nearly opposite\nbehavior. The impact of the characteristic feedback time \u001con the linewidth is illustrated\nin Fig. 6. In this figure a linear time scale is appropriate since there are no significant\neffects in the range 1 fs\u0015\u001c\u00150. The total linewidth \u0001HT(\u001c)is again nearly constant\nforf= 4 GHz andf= 10 GHz . In contrast \u0001HT(\u001c)reveals for higher frequencies two\nregions with differing behavior. The total linewidth decreases until \u0001HG(\u001c)becomes zero.\nAfter that one observes a positive linear slope which is due to the retardation part \u0001HR(\u001c).\nThis linear dependency is recognizable in Eq. (21d), too. Below we will present arguments\nwhy the feedback time \u001cis supposed to be in the interval 0< \u001c < 100 fs. Before let us\nstudy the frequency dependence of the linewidth in more detail. The general shape of the\ntotal linewidth \u0001HT(!)is depicted in Fig. 7. Here both the single contribution to the\n15∆Hη[G]\nf[GHz]∆HG\n∆HB\n∆HGB\n∆HR\n∆HTFIG. 7. (Color online) Frequency dependence of all contributions to the peak-to-peak linewidth for\n\u0002H= \u0002 M= 0,\f= 0:5,\u000b= 0:01,T2= 5\u000210\u00008s,\u001c= 1:7\u000210\u000014sand\u00000= 1:2. Parameters taken\nfrom Ref. [16]: 4\u0019MS= 16980 G ,HS=\u00003400 Gand\r= 0:019 GHz=G. The Bloch contribution\n\u0001HBis shown in the inset.\nlinewidth and the total linewidth are shown. Notice that the total linewidth is not simply\nthe sum of the individual contributions but has to be calculated according to Eq. (20). One\nrealizes that the Bloch contribution \u0001HBis only nonzero for frequencies f\u00146 GHzin the\nexamples shown. Accordingly \u0001HB= 0in Figs. 4-6 (lower parts) since these plots refer to\nf= 70 GHz . The behavior of the Gilbert contribution deviates strongly from the typically\napplied linear frequency dependence. Moreover, the Gilbert contribution will develop a\nmaximum value and eventually it disappears at a certain frequency where the discriminant\nin Eq. (21a) becomes negative. Nevertheless, the total linewidth is a nearly monotonous\nincreasing function of the frequency albeit, as mentioned before, for some combinations of\nthe model parameters there might exist a very small frequency region where \u0001HGreaches\nzero and the slope of \u0001HTbecomes slightly negative. The loss due to the declining Gilbert\npart is nearly compensated or overcompensated by the additional line broadening originated\nbytheretardationpartandthecombinedGilbert-Blochterm. Thelatteroneis \u0001HGB/pf\nand\u0001HR/f2, see Eqs. (21c)-(21d). In the frequency region where \u0001HG= 0only \u0001HGB\nand\u0001HRcontribute to the total linewidth, the shape of the linewidth is mainly dominated\nby\u0001HR. Thispredictionisanewresult. Thebehavior \u0001HR/f2, obtainedinourmodelfor\nhigh frequencies, is in contrast to conventional ferromagnetic resonance including only the\nsum of a Gilbert part linear in frequency and a two-magnon contribution which is saturated\n16at high frequencies. So far, experimentally the frequency ranges from 1 GHzto225 GHz,\nsee [21]. Let us point out that the results presented in Fig. 7 can be adjusted in such a\nmanner that the Gilbert contribution will be inoperative at much higher frequencies by the\nappropriate choice of the model parameters. Due to this fact we suggest an experimental\nverification in more extended frequency ranges. Another aspect is the observation that\nexcitations with a nonzero wave vector might represent one possible retardation mechanism.\nRegarding Eqs. (21a)-(21d) retardation can also influence the linewidth in case kz= 0\n(i.e.\f= 0and\u00001= \u0000 0). Only if\u001c= 0the retardation effects disappear. Therefore let us\nconsider the time domain of retardation and its relation to the Gilbert damping. The Gilbert\ndamping and the attenuation due to retardation can be considered as competing processes.\nSo temporal feedback can cause that the Gilbert contribution disappears. In the same\nsense the Bloch contribution is a further competing damping effect. In this regard temporal\nfeedback has the ability to reverse the dephasing process of spin waves based on Gilbert and\nBloch damping. On the other hand the retardation part \u0001Rin Eq. (21d) is always positive\nfor\u001c > 0. Thus, the retardation itself leads to linewidth broadening in ferromagnetic\nresonance and consequently to spin damping. Whether the magnitude of retardation is able\nto exceed the Gilbert damping depends strongly on the frequency. With other words, the\nfrequency of the magnetic excitation ’decides’ to which damping mechanisms the excitation\nenergyistransferred. Ourcalculationsuggeststhatforsufficienthighfrequenciesretardation\neffects dominate the intrinsic damping behavior. Thus the orientation and the value of the\nmagnetization within the retardation time \u001cplays a major role for the total damping.\nGenerally, experimental data should be fit according to the frequency dependence of the\nlinewidth in terms of Eqs. (20)-(22). To underline this statement we present Fig. 8. In this\ngraph we reproduce some results presented in [7] for the case \u0002H= \u0002 M= 0. To be more\nspecific, we have used Eq. (94) in [7] which accounts for the two-magnon scattering and\nthe parameters given there. As result we find a copy of Fig. 4 in [7] except of the factor\n2=p\n3. Further, we have summed up the conventional Gilbert linewidth /fwith the Gilbert\ndamping parameter \u000b1= 0:003. This superposition yields to the dotted line in Fig. 8. The\nresult is compared with the total linewidth resulting from our retardation model plotted as\nsolid line. To obtain the depicted shape we set the Gilbert damping parameter according\nto the retardation model \u000b2= 0:0075, i.e. to get a similar behavior in the same order of\nmagnitude of \u0001HTwithin both approaches we have to assume that \u000b2is more than twice\n17∆HT[G]\nf[GHz]retardation model\nGilbert+2-magnonFIG. 8. (Color online) Comparison with the two-magnon model. Frequency dependence of the total\npeak-to-peak linewidth \u0001HTfor\u0002H= \u0002 M= 0,\f= 0:5,\u000b1= 0:003,\u000b2= 0:0075,T2= 5\u000210\u00008s,\n\u001c= 1:22\u000210\u000014sand\u00000= 1:2. Parameters taken from [7]: 4\u0019MS= 21000 G ,HS=\u000015000 Gand\nfrom [37]:\r= 0:018 GHz=G(derived from g= 2:09for bulk Fe). The dotted line is a superposition\nof Fig. 4 in [7] reflecting the two-magnon contribution and the Gilbert contribution (denoted as\n\u000b1in the text) linear in the frequency.\nas large compared to \u000b1.\nFinally we discuss briefly the \u0002H-dependence of the linewidth which is shown in Fig. 9.\nIn the upper part of the figure one observes that \u0001HT(\u0002H)exhibits a maximum which is\nshifted towards lower field angles as well as less pronounced for increasing frequencies. The\nlower part of Fig. 9, referring to f= 10 GHz , displays that the main contribution to the total\nlinewidth arises from the Gilbert part \u0001HG. This result for f= 10 GHz is in accordance\nwith the results discussed previously, compare Fig. 7. For higher frequencies the retardation\ncontribution \u0001HRmay exceed the Gilbert part.\nV. CONCLUSIONS\nA detailed study of spatiotemporal feedback effects and intrinsic damping terms offers\nthat both mechanisms become relevant in ferromagnetic resonance. Due to the superposi-\ntion of both effects it results a nonlinear dependence of the total linewidth on the frequency\nwhich is in accordance with experiments. In getting the results the conventional model in-\ncluding Landau-Lifshitz-Gilbert damping is extended by considering additional spatial and\n18linewidth ∆ HT[G]\n4 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHzlinewidth ∆ Hη[G]\nΘH[deg]∆HB\n∆HR\n∆HGB\n∆HG\n∆HTf= 10 GHzFIG. 9. (Color online) Angular dependence of the total peak-to-peak linewidth \u0001HTfor various\nfrequencies (top graph) and all contributions \u0001H\u0011forf= 10 GHz (bottom graph) with \f= 0:5,\n\u000b= 0:01,T2= 5\u000210\u00008s,\u001c= 1:7\u000210\u000014sand\u00000= 1:1. The parameters are taken from\n[16]: 4\u0019MS= 16980 G ,HS=\u00003400 Gand\r= 0:019 GHz=G.\ntemporal retardation and non-conserved Bloch damping terms. Our analytical approach\nenables us to derive explicit expressions for the resonance condition and the peak-to-peak\nlinewidth. We were able to link our results to such ones well-known from the literature.\nThe resonance condition is affected by the feedback strength \u00000. The spin wave damping is\nlikewise influenced by \u00000but moreover by the characteristic memory time \u001cand the retar-\ndation length \u0018. As expected the retardation gives rise to an additional damping process.\nFurthermore, the complete linewidth offers a nonlinear dependence on the frequency which\nis also triggered by the Gilbert damping. From here we conclude that for sufficient high\nfrequencies the linewidth is dominated by retardation effects. Generally, the contribution of\nthedifferentdampingmechanismstothelinewidthiscomprisedofwellseparatedrateswhich\nare presented in Eqs. (20)-(22). Since each contribution to the linewidth is characterized\nby adjustable parameters it would be very useful to verify our predictions experimentally.\n19Notice that the contributions to the linewidth in Eqs. (20)-(22) depend on the shape of\nthe retardation kernel which is therefore reasonable not only for the theoretical approach\nbut for the experimental verification, too. One cannot exclude that other mechanisms as\nmore-magnon scattering effects, nonlinear interactions, spin-lattice coupling etc. are likewise\nrelevant. Otherwise, we hope that our work stimulates further experimental investigations\nin ferromagnetic resonance.\nWe benefit from valuable discussions about the experimental background with Dr. Khali\nZakeri from the Max-Planck-Institute of Microstructure Physics. 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Lett. 49, 658\n(2000)\n22" }, { "title": "1402.6996v1.N_2_supersymmetric_radiation_damping_problem_on_a_noncommutative_plane.pdf", "content": "arXiv:1402.6996v1 [hep-th] 24 Feb 2014N=2 supersymmetric radiation damping problem\non a noncommutative plane\nEverton M. C. Abreua,b,∗Albert C. R. 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 29, 2021\nDedicated to the memory of Prof. Wilson Oliveira\nAbstract\nIt is well known that a direct Lagrangian description of radi ation damping is still missing. In\nthis paper a specific approach of this problem was used, which is the standard way to treat the\nradiation damping problem. A N= 2 supersymmetric extension for the model describing the ra di-\nation damping on the noncommutative plane with electric and magnetic interactions was obtained.\nThe entire supercharge algebra and the total Hamiltonian fo r the system were analyzed. Finally,\nnoncommutativity features were introduced and its consequ ences were explored..\nPACS numbers: 11.15.-q; 11.10.Ef; 11.10.-z; 41.60.-m\nKeywords: noncommutativity, supersymmetry, radiation damping\n∗Electronic address: evertonabreu@ufrrj.br\n†Electronic address: albert@fisica.ufjf.br\n1N=2 supersymmetric radiation damping problem.... 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 relative to the self-intera ction of the particle with\nits own electromagnetic field which creates the radiation damping [1– 3].\nThe analysis of dissipative systems in quantum theory has a great int erest and relevance\neither because of fundamental reasons [4] or because of its applic ations in our real world [5–\n9]. The explicit time dependence of the Lagrangian and Hamiltonian ope rators introduces\na major difficulty in this study since the canonical commutation relatio ns are not preserved\nin time. Different approaches have been used in order to apply the ca nonical quantization\nscheme to dissipative systems [10–15].\nAnother way to handle with the problem of quantum dissipative syste ms is to enlarge\nthe target’s phase-space in a way that we will have to deal with an eff ective isolated system\ncomposed by the original system plus its time-reversed copy [16–18 ]. The new degrees\nof freedom thus introduced may be represented by a single equivale nt (collective) degree\nof freedom for the bath, which absorbs the energy dissipated by t he system. In order to\nimplement a canonical quantization formalism, we must first double th e dimension of the\ntarget phase-space. The objective of this procedure is to obey t he canonical quantization\nscheme, which requires an effective isolated system.\nTo study the quantum dynamics of an accelerated charge, we have to use indirect repre-\nsentations since the energy, the linear momentum and the angular m omentum that are all\ncarried by the radiation field are lost. The consequences concernin g the motion of the charge\nare known as radiation 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 indicates 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.\nA complete description of radiation damping is still missing. Hence, in th is paper we\nhave discussed some aspects of RD framework concerning algebra ic noncommutativity and\nsupersymmetry, as well as the relative resulting physics, of cours e. Notice that to talk about\nthese issues in a RD system is very difficult because we have to deal wit h two systems (the\nparticle and the reservoir) and consequently both mathematical a nd physical features are\nnot trivial, as we will see.\nIn this work we have introduced a N= 2 supersymmetric extension for the RD model\nin addition to the N= 1 supersymmetric version introduced in [19, 20]. We have used\nthe nonrelativistic (2 + 1)-dimensional pseudo-Euclidean space mod el describing the RD\n(representedbytheequation(2)below)onthenoncommutative( NC)planewhichintroduced\nan interaction term into the free model through the N= 2 superfield technique.\nAs we said just above, it is important to notice that in fact there are two phase-spaces\nconsidered here. The first one is where the RD occurs and the seco nd one is the doubled\nphase-space where the details and the relevance will be described in section II and in the\nreferences quoted there. It is in this doubled phase-space that w e have carried out the\nconsiderations described in this work, i. e., noncommutativity and N= 2 supersymmetry.N=2 supersymmetric radiation damping problem..... 3\nFor example, concerning only the noncommutativity issue, it can be s hown easily that the\noriginal space is commutative whereas in this doubled phase-space w e will show precisely in\nthis paper that it is NC. We hope that our work can bring some light on t he understanding\nof this extended space.\nThe organization of this paper is: in section 2 we will carry out a very b rief review of the\nmechanical model with a Chern-Simons term developed in [21] and its G alilean-symmetric\nversion, i.e., the LSZ model. In section 3 we will present a symplectic st ructure for the\nmodel in order to introduce the noncommutativity through the var iables used in [22, 23].\nIn section 4 we will show the supersymmetric extension of the model, 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\nfor the free particle model which is quasi-invariant under D= 2 Galilei symmetry as\nLLSZ=1\n2m˙x2\ni−κεij˙xi¨xj, i,j= 1,2, (1)\nwhereκhas dimensions of mass ×time. It can be shown [21] (by following the methods\nof ref. [24]) that this Lagrangian is quasi-invariant. The model (1) d oes not have a usual\nGalilei symmetry. We can describe it by the exotic, two-fold centrally extended Galilei\nsymmetry with non-commutating boosts. It was analyzed carefully in [21] and later in [25].\nThe authors in [21] have demonstrated that the model describes t he superposition of a free\nmotion in an NC external and an oscillatory motion in an NC internal spa ce. AN= 2\nsupersymmetric extension of (1) was accomplished in [26] which analy zed particles in the\nNC space with electric and magnetic interactions. A supersymmetriz ation of (1) was firstly\nobtained in [27]. Other considerations can be found in [28].\nThe radiation damping model. In [20, 29] another point of view concerning the study\nof RD [1, 30, 31] was presented, where it was introduced a Lagrang ian formalism in two\ndimensions given by\nLRD=1\n2mgij˙xi˙xj−γ\n2εij˙xi¨xj, i,j= 1,2, (2)\nwhereεijis the Levi-Civita antisymmetric tensor and gijis the metric for the pseudo-\nEuclidean plane [32] which is given by\ngij=gij=diag(1,−1). (3)\nWe are using the Einstein sum convention for repeated indices. The m odel (2) was shown to\nhave (1+1)-Galilean symmetry. The dynamical group structure as sociated with the system\nisSU(1,1) [29]. The supersymmetrization N= 1 of (3) was studied in [33].\nThe Lagrangian (2) 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\nheat bath coupled to the 1-system [19, 20]. It shows that the dissip ative term, as a matter\nof fact, acts as a coupling term between both the 1-system and th e 2-system in this space.N=2 supersymmetric radiation damping problem..... 4\nSpecifically, wehaveasystem formedbythechargeanditstime-rev ersed image, thatglobally\nbehaves like a closed system described by equation (2).\nNotice that the Lagrangian (2) is similar to the one discussed in [21] (a ction (1)), which\nis a special nonrelativistic limit of the particle with torsion [34]. However , in this case we\nhave a pseudo-Euclidean metric and the RD constant ( γ) which acts as a coupling constant\nof a Chern-Simons-like term. The RD constant γplays the same mathematical role of the\n“exotic” parameter κin (2) [21, 26]. 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 [27] is that, besides the metric, the physical sy stems are different. The\nRD constant γis not a simple coupling constant. It depends on the physical proper ties of\nthe charged particle, like the charge eand massmwhich are related to the objects in its\nequation of motion. This last one depicts an interaction between the charge and its own\nradiation field.\nIII. NONCOMMUTATIVITY\nThe analysis of NC geometry and its applications in physics requires a g reat amount of\ntime since it has many implications in several subjects such as quantu m mechanics, high\nenergy physics and condensed matter [35]. As an example we can brie fly describe the planar\nsystems of condensed matter that deal with a perpendicular magn etic field and becomes\nitself NC in the lowest Landau level [35]. Besides, the NC parameter c an be identified with\nthe inverse of the magnetic field. The study of NC theories has rece ived a special attention\nthrough the last years thanks to the possibility that noncommutat ivity can explain the\nphysics of the Early Universe. It has been used in many areas of the oretical physics [36],\nclassical mechanics [37], cosmology [38] and Lorentz invariance [39].\nLet us introduce a Lagrangian multiplier which connects ˙ xtozand after that we will sub-\nstitute all differentiated x-variables in the Lagrangian (2) by z-variables, one can construct\na first-order Lagrangian\nL(0)=gijpi(˙xj−zj)+m\n2gijzizj−γ\n2εijzi˙zj, (4)\nwhere the equations of motion can be written through the symplect ic structure [40], as\nωij˙ξj=∂H(ξ)\n∂ξi(5)\nand the symplectic two form is\n(ω) =\n0 g 0\n−g 0 0\n0 0−γε\n (6)\nwhere\nε=/parenleftbigg\n0 1\n−1 0/parenrightbigg\n, (7)N=2 supersymmetric radiation damping problem..... 5\nandgwas given in Eq. (3) and 0denotes the 2 ×2 null matrix. H(ξl) is the Hamiltonian\nandξiare the symplectic variables.\nLet us use a modified version of the variables introduced in [22, 23] a s\nQi=γgij(mzj−pj),\nXi=xi+εijQj,\nPi=pi. (8)\nWe can consider that our Lagrangian can be divided in two disconnect ed parts in order to\ndescribe the “external” and “internal” degrees of freedom. So,\nL(0)=L(0)\next+L(0)\nint (9)\nwhere\nL(0)\next=gijPi˙Xj+γ\n2εijPi˙Pj−1\n2mgijPiPj, (10)\nand\nL(0)\nint=1\n2γεijQi˙Qj+1\n2mγ2gijQiQj. (11)\nThe internal coordinates, /vectorQ, and the external ones, /vectorX, do not depend on each other [22]\nand we can see that they do not commute. The respective nonvanis hing Poisson brackets\nare\n{Xi,Xj}=γεij,{Xi,Pj}=gij,\n{Qi,Qj}=γεij, (12)\nwhere we can see clearly that the RD constant acts as the NC param eter that appears in the\ncanonical NC approach. Hence, we can conclude the presence of R D constant in any result\nfrom now on is a consequence of the NC effect.\nNowwewill introduceaninteractiontermintothe“external” sector (equation(10))which\ndoes not modify the internal sector and it is represented by a pote ntial energy term U(X)\ninvolving NC variables, as follows\nLext=gijPi˙Xj+γ\n2εijPi˙Pj−1\n2mgijPiPj−U(X). (13)\nThis result leads us to a deformation of the constraint algebra since the constraint now\ninvolves a derivative of the potential [29].\nNotice that the Lagrangian in Eqs. (10), (11) and (13) are formed by objects that show a\nNCalgebra described inEq. (12). The standardNCprocedure istor ecover thecommutative\nalgebra in (12) using the so-called Bopp shift\nxi=ˆXi+1\n2γǫijpj, (14)\nwhere the hat defines a NC variable. Substituting (14) into (12) we w ill have that {xi,xj}=\n0. The same can be made with Qiso that\nQi=ˆQi+1\n2γǫijPj, (15)N=2 supersymmetric radiation damping problem..... 6\nwhere ˆpi=piandPi=ˆPiand consequently {pi,pj}={Pi,Pj}= 0. Substituting (14)\nand (15) into the Lagrangians (10), (11) and (13) results in a Lagr angian defined in NC\nphase-space.\nTheNCeffect of(8)istoseparatethesectors, i.e., theexternal a ndinternal ones. Namely,\ntheRDNCeffectwastopromote, from D= 2phase-spaceRDscenario, theanalysisof D= 1\ndissipative dynamics.\nIV. THE N= 2SUPERSYMMETRIC MODEL\nIn [41] it was constructed the supersymmetric extension of the LS Z model in Eq. (2)\nin the NC plane. An entire SUSY investigation was carried out and its N= 2 extension\nwas provided in [26] where interesting physical results were obtaine d. The objective of this\nsection is to construct the N= 2 framework for RD in a NC plane since its N= 1 was\naccomplished in [33]. In this way our starting point is the Lagrangian in E q. (13).\nTo obtain the supersymmetric extension of the model described by the Lagrangian (13),\nwe will use a Grassmannian variable that will be connected with each co mmuting space\ncoordinate which represents the system degrees of freedom. We are considering only N= 2\nSUSY for a non-relativistic particle, which is described by the introdu ction of two real\nGrassmannian variables Θ and ¯Θ (the Hermitian conjugate of Θ) in the configuration space,\nbut all the dynamics is parametrized by the time t[42, 43].\nLet us carry out the Taylor expansion for the real scalar superco ordinate as\nXi→ Xi(t,Θ,¯Θ) =Xi(t)+iψi(t)Θ+i¯Θ¯ψi(t)+¯ΘΘFi(t) (16)\nand their canonical supermomenta\nPi(t)→ Pi(t,Θ,¯Θ) =iηi(t)−iΘ(Pi(t)+ifi(t))−¯ΘΘ˙ηi(t), (17)\nwhich under the infinitesimal supersymmetry transformation laws\nδt=i¯ǫθ+i¯ǫΘ, δΘ =ǫandδ¯Θ = ¯ǫ, (18)\nwhereǫis a complex Grassmannian parameter. We can also write that\nδXi= (ǫ¯Q+¯ǫQ)Xi (19)\nandδPi= (ǫ¯Q+¯ǫQ)Pi, (20)\nwhere both Qand¯Qare the two SUSY generators\nQ=∂\n∂¯Θ+iΘ∂\n∂t,¯Q=−∂\n∂Θ−i¯Θ∂\n∂t. (21)\nIntermsof( Xi(t),Pi(t),Fi,fi), thebosonic(even) components and( ψi(t),¯ψi(t),ηi(t)), the\nfermionic (odd) components, we can 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, (22)N=2 supersymmetric radiation damping problem..... 7\nand\nδηi=ǫ(Pi+ifi);δPi= 0;δfi= 2¯ǫ˙ηi. (23)\nThe super-Lagrangian for the super point particle with N= 2, which is invariant under\nthe transformations (22) and (23), can be written as the following integral (we have used for\nsimplicity that m= 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)] (24)\nwhereDis the covariant derivative ( D=∂Θ−i¯Θ∂t) and¯Dis its Hermitian conjugate. The\nsuperpotential U[X] is a polynomial function of the supercoordinate\nLet us expand the superpotential U[X] in Taylor series and if we maintain Θ ¯Θ (because\nonly these terms survive after integrations over Grassmannian va riables Θ and ¯Θ), we have\nthat\nU[X] =Xi∂U[X(t)]\n∂Xi+XiX∗\nj\n2∂2U[X(t)]\n∂Xi∂Xj+... (25)\n=Fi¯ΘΘ∂iU[X(t)]+¯ΘΘψi¯ψj∂i∂jU[X(t)]+...\nwhere the derivatives ∂i=∂\n∂Xiare such that Θ = 0 = ¯Θ, which are functions only of the\nX(t) even coordinate. Substituting equation (25) into equation (24), we can write, after\nintegrations that\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)], (26)\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 that\nfi(t) =gij∂jU[X(t)], (27)\nFi(t) =fi+γgilεlj˙fj\n=gij∂jU[X(t)]−γεij∂j∂kU[X]˙Xk(t), (28)\nwhere wehave to eliminatethevariable fiaswell asitsderivative in Fi. Now, if wesubstitute\n(27) and (28) into (26) the auxiliary variables can be completely elimina ted, 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 . (29)N=2 supersymmetric radiation damping problem..... 8\nNote that, as in [26], we can rewrite equation (29) 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, (30)\nwhich is invariant under standard gauge transformations Aµ→A′\nµ=Aµ+∂µΛ, where\nA0(X,t) =−1\n2gij∂iU∂jU (31)\nand\nAk(X,t) =γ\n2εij∂iU∂j∂kU , (32)\nwhere both can be identified in [26] with the scalar potential A0(in this case we have a\npseudo-Euclidean metric) and the vector potential Ak. Notice that both potentials above\nare not independent. The vector potential introduces a magnetic fieldB=εij∂iAjgiven by\nB(X) =γ\n2εilεjk(∂i∂lU)(∂j∂kU), (33)\nwhere we can see that the noncommutativity introduced by the par ameterγgenerates both\na constant magnetic field [26] and an electric field given by Ei=∂iA0which can be written\nas\nEi(X) =−gjk∂i∂jU∂kU . (34)\nThe Euler-Lagrange equations, in this case, are\nm∗˙Xi=Pi−meγεijEj+mγεijψl¯ψk∂l∂k∂jU, (35a)\n˙Pi=egijεjl˙XlB+begijEj−gijψl¯ψk∂l∂k∂jU, (35b)\nwhereEiandBare the electric and magnetic field, respectively, and\nm∗=m(1−eγB)\nis an effective mass. Notice that noncommutativity introduces a cor rection term in order to\nobtain an effective mass for the system.\nHowever, this way of introducing electromagnetic interaction modifi es the symplectic\nstructure of the system which determines the NC phase-space ge ometry, for the bosonic\nsector, equation (12), we have\n{Xi,Xj}=m\nm∗γεij,{Xi,Pj}=m\nm∗gij,\n{Pi,Pj}=m\nm∗bεij, (36)\nwhere we have the value eγB/ne}ationslash= 1 in order to avoid a singularity [44, 45]. Notice that the\nalgebrain(36) isdifferent fromtheonein(12)where themomenta co mmute. Inother words,\nthe noncommutativity introduces a new NC algebra to the system be sides the modification\nin the old one.N=2 supersymmetric radiation damping problem..... 9\nConcerning the fermionic sector, the Euler-Lagrange equations a re\niγεij¨¯ηj+bigij˙¯ηj−i˙ψi= 0,\niγεij¨ηj+bigij˙ηj−i˙¯iψ= 0, (37)\nfor the fermionic variables ( η,¯η). For the fermionic variables ( ψi,¯ψi) the Euler-Lagrange\nequations are\ni˙ηi+gik¯ψj∂k∂jU= 0\ni˙¯ηi−gikψj∂j∂kU= 0. (38)\nwhere the fermionic variables ( ψi,¯ψi) do not have dynamics.\nSo, analogously to [26] we have here that noncommutativity have int roduced electric\nand magnetic fields into the system. In the case of RD, studied here , described in an NC\nhyperbolic phase-space, the movement of a charged particle have an extra electromagnetic\nenergy that did not appear in an N= 1 SUSY analysis [33]. This result agrees with the fact\nthat noncommutativity does not change the physics of the system . However, we understand\nthat this electromagnetic energy is an extra one due to the NC feat ure of the phase-space.\nThis result is also different, as it should be expected, from the one ob tained in [26] where\nonly a magnetic interaction appear.\nA. The harmonic oscillator solutions\nIn order to obtain an interesting solution of equations (35) let us co nsider a specific form\nfor the superpotential\nU(X) =ω\n2gijXiXj, (39)\nwhich has clearly an harmonic-like form.\nIt is easy to see that in both equations (35a) and (35b) the last ter m that have three\nderivatives disappear and so we have two new equations that show t he separation of both\nthe fermionic and bosonic sectors, so that,\nm∗˙Xi=Pi−eγǫijEj (40a)\nand\n˙Pi=egjlǫij˙XlB+egijEj. (40b)\nComputing a second time derivative of equation (40a) and using (40b ) we have that\nm∗¨Xi=egijǫjl˙XlB+egijEj−eγǫij˙Ej (41)\nand this differential equation will disclose a very well known result.\nFrom (31), (32) and (39), we can write that,\nA0(X,t) =−ω2\n2gijXiXj, (42a)\nAk(X,t) =γ\n2ω2ǫkjXj. (42b)N=2 supersymmetric radiation damping problem..... 10\nSubstituting these equations into (33) and (34) we have that\nB=γω2(43)\nand\nEi=−ω2gijXj, (44)\nand finally, substituting these both equations into (41) it is easy to s how that\n¨Xi−γeω2\n1−γ2ω2e(gijǫjk+gjkǫij)˙Xk+eω2\n1−γ2ω2eXi= 0, (45)\nwhich is the equation of a damped harmonic oscillator. We can see clear ly that the second-\nterm of (45) represents a dissipative force proportional to the v elocity and in the last term\nof (45), we have that\nω2\n0=eω2\n1−γ2ω2e\ncan be seen as the natural frequency of this oscillator, ω0. Notice that the RD constant is\nresponsible for the dissipative force and it also affects the frequen cy. The instantaneous rate\nof energy of the oscillator in (45) can be written as\ndE\ndt=m∗γeω2\n1−γ2ω2e˙Xi, (46)\nso that the RD constant also affects the energy rate. Notice that whenγ= 0 we have the\nstandard and well known results. We have also that γ2ω2e/ne}ationslash= 1, obviously. If we have the\nmetric in(63) andwe use thepseudo-Euclidean planegiven in(4), we c an see that the second\nterm in (45) disappears and we have that\n¨Xi+eω2\n1−γ2ω2eXi= 0\n=⇒¨Xi+ω2\n0Xi= 0,\nwhich is the equation for the standard harmonic oscillator that has t he standard solutions.\nHowever, the difference is the NC contribution.\nFrom (45) 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 (22) and (23), and the La-\ngrangian (30), we can compute the supercharge algebra, throug h the Noether’s theorem.\nThe results for the charge operator are given by\nQ=igij(Pi−iWi)ψjand¯Q=igij(Pi+iWi)¯ψj, (47)\nwhereWi(X) =∂iU(X).N=2 supersymmetric radiation damping problem..... 11\nThe supercharge algebra is\n{Q,Q}={¯Q,¯Q}= 0, (48)\nand\n{Q,¯Q}=−2iH. (49)\nMoreover, we can easily carry out a canonical calculation of the Ham iltonian and we obtain\nthat\nH=Hb+Hf, (50)\nwhere the bosonic Hamiltonian Hbis given by\nHb=1\n2gij(PiPj+WiWj), (51)\nand the fermionic part Hfcan be written as\nHf=m\nm∗/bracketleftBig\nieB(X)εij¯ψiψj+gik∂jWk(X)¯ψiψj/bracketrightBig\n. (52)\nNote that the second term in Hbis proportional to the scalar potential, equation (31), 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 keeps the symplectic structure of equation (12). In [26] this transformation has been\nconsidered andtheauthorshaveobtainedthesameexpression fo rthemagneticfieldequation\n(33).\nV. REMARKS AND CONCLUSIONS\nA fundamental property of all charged particles is that the electr omagnetic energy is\nradiatedwhenever theyareaccelerated. Therecoil momentum of thephotonsemitted during\nthisprocessisequivalent toareactionforcecorresponding tothe particleself-interactionwith\nits own electromagnetic field, which originates the RD.\nHere the supersymmetric model was separated in two parts, name ly, “external” and “in-\nternal” degrees of freedom of the supersymmetric model in terms of the new variables, where\nthe RD constant introduced noncommutativity into the coordinate sector. We have shown\na way to introduce an electromagnetic coupling.\nWe have calculated the supersymmetric N= 2 extension of the RD model. We have\ndemonstrated that the noncommutativity introduced by the para meter generated a constant\nmagnetic field. With this result, combined with the electric field we have obtained a general\nexpression for the damped harmonic oscillator which results in the st andard harmonic oscil-\nlator in our pseudo-Euclidean space. We have seen that in the NC spa ce, the non-physical\nsolutions, namely the pre-acceleration solutions, have disappeare d. After that, we have com-\nputed the supercharges algebra and the total Hamiltonian of the s ystem, which was divided\ninto two parts: bosonic and fermionic.\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 analyze the Euler-Lagrange equ ations (35), (37) and (38).N=2 supersymmetric radiation damping problem..... 12\nVI. 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Lett. 60, 1692 (19 88).\n[41] L. Lapointe, H. Ujino and L. Vinet, Ann. Phys. 314 (2004) 464.\n[42] C. A. P. Galv˜ ao and C. Teitelboim, J. Math. Phys. 21, 186 3 (1980).\n[43] G. Junker and S. Mattiensen, J. Phys. A 27, L751 (1995).\n[44] C. Duval and P. A. Horv´ athy, J. Phys. A 34, 10097 (2001).\n[45] P. A. Horv´ athy and M. S. Plyushchay, Nucl. Phys. B714, 2 69 (2005)." }, { "title": "1607.02042v1.Damping_of_Alfven_waves_by_Turbulence_and_its_Consequences__from_Cosmic_Rays_Streaming_to_Launching_Winds.pdf", "content": "arXiv:1607.02042v1 [astro-ph.HE] 6 Jul 2016Draft version July 8, 2016\nPreprint typeset using L ATEX style emulateapj v. 5/2/11\nDAMPING OF ALFVEN WAVES BY TURBULENCE AND ITS CONSEQUENCES: FROM COSMIC-RAYS\nSTREAMING TO LAUNCHING WINDS\nA. Lazarian\nDepartment of Astronomy, University of Wisconsin-Madison , 475 Charter St., Madison, WI 53705\nDraft version July 8, 2016\nABSTRACT\nThis paper considers turbulent damping of Alfven waves in magnetize d plasmas. We identify two\ncases of damping, one related to damping of cosmic rays streaming in stability, the other related to\ndamping of Alfven waves emitted by a macroscopic wave source, e.g. stellar atmosphere. The phys-\nical difference between the two cases is that in the former case the generated waves are emitted in\nrespect to the local direction of magnetic field, in the latter in respe ct to the mean field. The scaling\nof damping is different in the two cases. We the regimes of turbulence ranging from subAlfvenic to\nsuperAlfvenic we obtain analytical expressions for the damping rat es and define the ranges of appli-\ncability of these expressions. Describing the damping of the stream ing instability, we find that for\nsubAlfvenic turbulence the range of cosmic ray energies influenced by weak turbulence is unpropor-\ntionally large compared to the range of scales that the weak turbule nce is present. On the contrary,\nthe range of cosmic ray energies affected by strong Alfvenic turbu lence is rather limited. A number\nof astrophysical applications of the process ranging from launchin g of stellar and galactic winds to\npropagation of cosmic rays in galaxies and clusters of galaxies is cons idered. In particular, we discuss\nhow to reconcile the process of turbulent damping with the observe disotropy of the Milky Way cosmic\nrays.\nSubject headings: cosmic rays, MHD, turbulence, radiation mechanisms: non-therma l-turbulence-\ngalaxies, clusters: radio continuum\n1.INTRODUCTION\nAstrophysical plasmas are magnetized and turbulent\n(see McKee & Ostriker 2007, Sharma et al. 2009, Bran-\ndenburg & Lazarian 2013). The propagation of Alfvenic\nwaves in MHD turbulence is a problem of great astro-\nphysical significance with applications to key astrophys-\nical processes (see Uhlig et al. 2012, Wiener, Oh & Guo\n2013, van der Holst et al. 2014, Lynch et al. 2014).\nWaves in astrophysical environments can arise from in-\nstabilities, e.g. from the cosmic ray (CR) instability\ndue to streaming (Lerche 1967, Kulsrud & Pearce 1969,\nWentzel 1969, Skilling 1971). They can also be induced\nin the environment by macroscopic sources, e.g. they\ncan be the result of vibrations of the stellar surface or\narise from magnetic reconnection (see Konigl 2009 and\nref. therein, Suzuki 2013). The most well-known con-\nsequences of wave propagation and damping range from\nheating of coronal gas in Solar atmosphere (e.g. Arber,\nBrady & Shelyag 2016, Reep & Russell 2016), acceler-\nation and scattering of cosmic rays (e.g. Jokipii 1966,\nSchlickeiser 2002, 2003), and launching of solar and stel-\nlar winds (e.g. Suzuki & Inutsuka 2005, van Ballegooi-\njen, & Asgari-Targhi 2016). The damping of waves in-\nduced by CR streaming is important for the confinement\nand acceleration of CRs in interstellar, intergalactic and\ninterplanetary environments (see Ensslin et al. 2011,\nBadruddin, & Kumar, A. 2016, Kulsrud 2005). In more\ngeneral terms, the damping must be understoond in or-\nder to answer the long-standing question of the relative\nimportanceofwavesandturbulenceinvariousastrophys-\nical settings (see Petrosian 2015).\nInitial studies of Alfven wave damping in turbulent\nplasmas were done in Silimon & Sudan (1989) in a modelof isotropic turbulence which does not correspond to the\npresent understanding of magnetized turbulence (see §2\nfor more discussion of MHD turbulence). More recently,\nturbulent damping of Alfven waves was suggested as a\nprocess for suppressing of CR streaming instability in\nYan & Lazarian (2002, henceforth YL02). This process\nwas quantified in an important study by Farmer & Gol-\ndreich (2004, henceforth FG04) for the original Goldre-\nich & Sridhar (1995, henceforth GS95) model of Alfvenic\nturbulence and was tested numerically in Beresnyak &\nLazarian (2008).\nFG04 employs the original GS95 model of strong\nAlfvenic turbulence that assumes that turbulence is in-\njectedisotropically with velocity uLthat is equal to the\nAlfven velocity VA. Although important, this regime\ndoesnotcoverthefullvarietyofastrophysicalconditions.\nMoreover,theworkdealsonlywiththeAlfvenwavesgen-\nerated by CR streaming instability and does not cover\nAlfven waves from the external macroscopic source. At\nthe same time, the propagationofAlfven wavesin turbu-\nlent plasmas is important for understanding of heating\nof stellar atmospheres, galactic halos, intracluster media\n(see Zhuravleva et al. 2014). The momentum deposited\nby Alfven waves in plasmas is important for launching\nstellar and galactic winds.\nFirst of all, as we discuss further in the paper, the\ndamping of Alfven waves generated by streaming in-\nstability is different from the damping of Alfven wave\ngenerated by a microscopic source. Moreover, strong\nisotropicallyinjected MHDturbulence discussedin FG04\nis one of the types of MHD turbulence that is relevant\nto astrophysical settings. Depending on the injection\nscale, injection velocity and media magnetization, mag-\nnetized turbulent motions can demonstrate regimes of2 Lazarian\nisotropicsuperAlfvenicturbulence, extremelyanisotropic\nweakturbulence, and strongsubAlfvenic turbulence with\nanisotropic injection. Turbulent motions at the injection\nscale can also damp Alfven waves.\nThe quantitative studies of the damping of the Alfven\nwaves injected with respect to (a) local magnetic fields\nand (b) global mean field by MHD turbulence in the\naforementioned different regimes, as well as with the\nturbulence at scales larger than the turbulence injection\nscale, is the goal of this paper.\nIn particular, in the present paper we address the\ndamping of Alfven waves by turbulence taking into ac-\ncountboth the spatialstructureofthe frontofthe Alfven\nwave and what are the actual properties of MHD turbu-\nlence with which these wavesareinteracting. The former\ndependsonhowtheAlfvenwavesweregenerated,thelat-\nter depends on the properties of MHD turbulence in dif-\nferent regimes. As we mentioned earlier, the FG04 study\naddresses the damping of Alfven waves generated by the\nstreaming instability of cosmic rays (see Kulsrud 2005,\nLongair2011), which correspondsto the wavesgenerated\nwith respect to the local direction of the magnetic field\nthat is sampled by the streaming particles. The distinc-\ntion between the local and global systems of reference\n(Lazarian & Vishniac, 1999, henceforth LV99, Cho &\nVishniac 2000, Maron & Goldreich 2001, Cho, Lazarian\n& Vishniac 2002) plays a crucial role in the present day\ntheory of MHD turbulence and, as we will show later,\nthe damping of Alfven waves differs in the case of Alfven\nwaves launched with respect to the local magnetic field\nand with respect to the mean magnetic field. This im-\nportant distinction has not been addressed in the ear-\nlier studies. Similarly, we are not aware of quantitative\nstudies of Alfven wave damping by MHD turbulence in\ndifferent regimes.\nWe believe that providingthe study of the Alfven wave\ndampingforawiderangeofpossibleturbulentastrophys-\nical settings is very important. Indeed, magnetic turbu-\nlence in stellar and pulsar atmospheres, in halos of spi-\nral galaxies MHD is subAlfvenic with the Alfven Mach\nnumber MA=uL/VA<1. In spiral arms, different\nparts of the interstellar media exhibit different degrees\nof magnetization (Draine 2011). For instance, molecular\nclouds may be superAlfvenic (see Luntilla et al. 2008),\nalthough this is not a universally accepted opinion (see\nLi & Henning 2011). Turbulence in clusters of galaxies is\ngenerally accepted to be superAlfvenic (see Brunetti &\nLazarian 2007, Brunetti & Jones 2014, Brunetti 2016).\nIn what follows, in §2 we derive the relations for dif-\nferent regimes of the MHD turbulence that we employ in\nthis paper. By providing this derivation we clarify the\npointsthatareessentialforourfurtherexplanationofthe\nnatureofAlfven wavedamping. Then weprovideaphys-\nical picture of damping in §3. Historically, the damping\nof streaming instability by turbulence was the first to be\naddressed (YL02, FG04). This type of damping is asso-\nciated with Alfven waves generated with respect to the\nlocal magnetic field. Therefore, in §4 we quantify the\ndamping of waves generated in the local system of refer-\nence for both subAlfvenic and superAlfvenic turbulence.\nThere we also define the ranges of wavelength for which\nthe damping by weak and strong subAlfvenic turbulence\nare applicable as well as define the range of applicabil-\nity of damping by superAlfvenic turbulence. We laterconsider the second case of damping of Alfven waves,\ni.e. the case of damping of waves generated by a macro-\nscopic source. Therefore, for the aforementioned vari-\nety of MHD turbulence regimes, we consider the case of\ndamping of Alfven waves that are injected with respect\nto the mean field in §5. We comparethe turbulent damp-\ning of Alfven waves with the non-linear Landau damping\nof Alfven waves in §6 and briefly outline some of the as-\ntrophysical implications of our study in §7. We compare\nour results with those in earlier works in §8. Our dis-\ncussion is provided in §9 and our summary is given in\n§10.\n2.REGIMES OF ALFVENIC TURBULENCE\nMHD theory is applicable to astrophysical plasmas at\nsufficiently large scales and for many astrophysical sit-\nuations the requirement for Alfvenic turbulence to be\napplicable is that the turbulence is studied at scales sub-\nstantially larger than the ion gyroradius ρi(see more in\nKulsrud 2005, Eyink et al. 2011). In what follows we\nconsider this criterium satisfied. For Alfvenic turbulence\nvelocity and magnetic field fluctuations have the same\nscaling and thus below we focus on velocity scaling.\nMHD turbulence can be decomposed into the cascades\nof Alfven, slow and fast modes (GS95, Lithwick & Gol-\ndreich 2001) the concept that has been elaborated and\nproved numerically (Cho & Lazarian 2002, 2002, Kowal\n& Lazarian 2010, Takamoto & Lazarian 2016). For non-\nrelativistic turbulence the Alfvenic cascade is marginally\naffected by two other fundamental MHD modes (see Cho\n& Lazarian 2002, Takamoto & Lazarian 2016, cf. Stone\net al. 1998) and therefore we focus on Alfvenic tur-\nbulence that is responsible for Alfven wave non-linear\ndamping.\nThe the pioneering studies of Alfvenic turbulence were\ndone by Iroshnikov (1964) and Kraichnan (1965) for a\nhypothetical model of isotropic MHD turbulence. Later\nstudies (see Montgomery & Turner 1981, Matthaeus et\nal,1983, Shebalin et al 1983, Higdon 1984) discovered the\nanisotropic nature of the energy cascade and paved the\nway for further advancements in the field. The break-\nthrough work by GS95 provided the theory of trans-\nAlfvenic turbulence, i.e. turbulence corresponding to in-\njectionvelocity uLattheinjectionscale LequaltoAlfven\nvelocityVAwhich corresponds to the Alfven Mach num-\nber\nMA=uL\nVA, (1)\nequal to unity. The generalizationof the GS95 theory for\nMA<1 andMA>1 was obtained later (LV99, Lazar-\nian 2006). Note that the original GS95 theory was also\naugmented by the concept of local systems of reference\n(LV99, Cho & Vishniac 2000, Maron & Goldreich 2001,\nCho, Lazarian & Vishniac 2002) that specifies that the\nturbulent motions should be viewed not in the system\nof reference of the mean magnetic field, but in the lo-\ncal system of reference of the turbulent eddies. This is\nquite a natural concept, which, however, was missed by\nthe earlier studies. Indeed, for the small scale turbulent\nmotions the only magnetic field that matters is the mag-\nnetic field in their vicinity. Thus this local field, rather\nthan the mean field, should be considered. Keeping this\nin mind, in what follows, we describe turbulent motions\nusing not parallel and perpendicular wave numbers, butTurbulent Damping of Alfven waves 3\nparallel to local magnetic field size of the eddy l/bardbland\nperpendicular scale l⊥. The distinction between the lo-\ncal system of reference related to the local field and the\nglobal system of reference related to the mean field is\nalso important for the two cases of Alfven wave damping\nthat we consider in this work.\nIn Table 1 (see §9) we present various regimes of\nAlfvenic turbulence and the ranges for which those are\napplicable. Below we consider first subAlfvenic tur-\nbulence corresponding corresponding the Alfven Mach\nnumberMA<1. For Alfvenic perturbations the relative\nperturbationsofvelocitiesandmagneticfieldsarerelated\nin the following way:\nδBl\nB=δBl\nBLBL\nB=ul\nuLMA=ul\nVA, (2)\nwhereBlis the perturbation of the magnetic field Bat\nthescale l,BListheperturbationofthe magneticfieldat\nthe injection scale L, whileulis the velocity fluctuation\nat the scale lin the turbulent flow with energy injected\nwith the velocity uL.\nTo understand the nature of non-linear damping of\nAlfven waves, it is useful to get an insight into a de-\ntailedpictureofcascadingbyAlfventurbulence(seeCho,\nLazarian & Vishniac 2003 for more details). Consider\ncolliding Alfvenic wave packets with parallel scales lland\nperpendicular scales l⊥. The change of energy per colli-\nsion is\n∆E∼(du2\nl/dt)∆t, (3)\nwhere the first term represents the change of the energy\nof a packet in time as it interacts with the oppositely\nmoving wave packet. Naturally, the time of the interac-\ntion is the time of the passage of the given wave packet\nthrough the oppositely moving wavepacket of the size l/bardbl,\nthus the interaction time ∆ t∼l/bardbl/VA. To estimate the\ncharacteristic rate of cascading one should accept that\nthe cascading of our wave packet is happening due to\nthechangeofthestructureoftheoppositelymovingwave\npacket which is happening with the rate ul/l⊥. Indeed,\nmore structures are being created in the passing package\nas the background field affected by the opposite package\nevolves. Therefore Eq. (3) becomes\n∆E∼ul·˙ul∆t∼(u3\nl/l⊥)(l/bardbl/VA), (4)\nThe fractional energy change per collision is the ratio\nof ∆EtoE,\nf≡∆E\nu2\nl∼ull/bardbl\nVAl⊥, (5)\nwhich is the measure of the strength of the nonlinear\ninteraction. fis the ratio of the rate of shearing of the\nwave packet ul/l⊥to the rate of the propagation of the\nwave packet VA/l/bardbl. If the shearing rate is much smaller\nthan the propagation rate, for f≪1 the cascading is a\nrandom walk process which means that\nℵ=f−2, (6)\nsteps are required for the cascading, i.e. the cascading\ntime is\ntcas∼ ℵ∆t. (7)\nForℵ>1, the turbulence is called weakand ifℵ ≈1 the\nturbulence is called strong.TheAlfvenic3-waveresonantinteractionsaregoverned\nby relations for wavevectors that reflect momentum con-\nservation k1+k2=k3, and frequencies reflecting energy\nconservation ω1+ω2=ω3. With Alfven wavepackets in-\nteracting with oppositely moving kand have the disper-\nsion relation ω=VA|k/bardbl|, wherek/bardbl∼l−1\n/bardblis the compo-\nnent of wavevector parallel to the background magnetic\nfield the increase of k⊥∼l−1\n⊥occurs. The decrease of\nl⊥withl/bardblbeing fixed signifies the increase of the energy\nchange per collision. This eventually makes ℵof the or-\nder of unity and the approximation of the weak Alfvenic\nturbulence breaks down. For ℵ ≈1 the GS95 critical\nbalanced condition\null−1\n⊥≈VAl−1\n/bardbl, (8)\nis satisfied with the cascading equal to the wave period\n∼∆t. The value of ℵcannot decrease further and the\nturbulence evolves as strong Alfvenic turbulence . There-\nfore the further decrease of l⊥entails the corresponding\ndecrease of l/bardblto keep the critical balance satisfied. The\nability to change l/bardblmeans that the frequencies of in-\nteracting waves increase, which is possible as cascading\nintroduces the uncertainty in wave frequency ωof the\norder of 1 /tcas.\nThe cascadingturbulent energy flux for incompressible\nfluid is (Batchelor 1953):\nǫ≈u2\nl/tcas=const, (9)\nwhich in the hydrodynamic case provides\nǫhydro≈u3\nl/l≈u3\nL/L=const, (10)\nwhere the relation tcas≈l/ulis used.\nFor the weak cascade ℵ ≫1 provides (LV99)\nǫw≈u4\nl\nV2\nA∆t(l⊥/l/bardbl)2≈u4\nL\nVAL, (11)\nwhere Eqs. (9) and (7) are used. The second relation in\nEq. (11) follows from the assumed isotropic injection of\nturbulence at the scale L.\nTaking into account that l/bardblis constant, it is easy to\nsee that Eq. (11) provides\nul∼uL(l⊥/L)1/2, (12)\nwhich is different from the Kolmogorov ∼l1/3scaling.1\nFor the accepted model of isotropically injected turbu-\nlence at scale L, the initial l/bardbl=Land the transition to\nℵ ≈1, i.e. to strong turbulence, occurs (LV99)\nltrans∼L(uL/VA)2≡LM2\nA. (13)\nThus, weak turbulence has a limited, i.e. [ L,LM2\nA], iner-\ntial range and at scalesless than LM2\nAit transits into the\nregime of strong turbulence. The velocity corresponding\nto the transition follows from ℵ ≈1 condition given by\nEqs. (6) and (5):\nutrans≈VAltrans\nL≈VAM2\nA. (14)\n1Using the relation kE(k)∼u2\nkit is easy to see that the spec-\ntrum of weak turbulence is Ek,weak∼k−2\n⊥(LV99, Galtier et al.\n2000).4 Lazarian\nThe relations for the strong turbulence in the sub-\nAlfvenic regime obtained in LV99 can be easily derived\nas follows. Indeed, the turbulence becomes strong and\ncascades over one wave period, which according to Eq.\n(8) is equal to l⊥/ul. Substituting the latter in Eq. (9)\none gets\nǫs≈u3\ntrans\nltrans≈u3\nl\nl=const, (15)\nwhichisanalogoustothehydrodynamicKolmogorovcas-\ncade in the direction perpendicular to the local direction\nof the magnetic field. This cascade starts at ltransand\nhastheinjectionvelocitygivenbyEq. (14). Thus(LV99)\nul≈VA/parenleftbiggl⊥\nL/parenrightbigg1/3\nM4/3\nA. (16)\nIn terms of the injection velocity uLEq. (16) can be\nrewritten as\nδul≈uL/parenleftbiggl⊥\nL/parenrightbigg1/3\nM1/3\nA. (17)\nSubstituting the latter expression in Eq. (8) one gets the\nrelation between the parallel and perpendicular scales of\nthe eddies (LV99):\nl/bardbl≈L/parenleftbiggl⊥\nL/parenrightbigg2/3\nM−4/3\nA. (18)\nThe relations given by Eq. (18) and (16) reduce to the\nwell-known GS95 scaling for transAlfvenic turbulence if\nMA≡1.\nThesuperAlfvenic turbulence corresponds to uL> VA,\nwhich is equivalent to MA>1. ForMA≫1 the turbu-\nlence at scales close to the injection scale is essentially\nhydrodynamic as the influence of magnetic forces is of\nmarginal importance, i.e. the velocity is Kolmogorov\nul=uL(l/L)1/3. (19)\nThe cascading nature changes at the scale\nlA=LM−3\nA, (20)\nat which the turbulent velocity becomes equal to the\nAlfven velocity (Lazarian 2006). The rate of cascading\nforl < lAcan be written as:\nǫsuperA≈u3\nl/l≈M3\nAV3\nA/L=const. (21)\nThis cascading can be related to the GS95 cascade, if\nthe scale given by Eq. (21) is taken as the injection scale\nfor the transAlfvenic turbulence and the corresponding\nscaling follows from Eq. (22) and (16) for the injection\nofVA. In other words:\nl/bardbl≈L/parenleftbiggl⊥\nL/parenrightbigg2/3\nM−4/3\nA, (22)\nul≈uL/parenleftbiggl⊥\nL/parenrightbigg1/3\nM1/3\nA. (23)\nThe relations above are used in the discussion of the\nAlfven wave damping that follows.3.GENERAL REMARKS ON TURBULENT DAMPING OF\nALFVEN WAVES\nLinear Alfven waves propagate without inducing irre-\nversibledistortions on each other. The situation changes\nwhen Alfven waves interact with Alfvenic turbulence.\nThere the structure of magnetic field lines changes sig-\nnificantly over the time of propagation of the wave and\nthis causes the non-linear distortion. As a result, the\nwavesundergocascadinganddissipatetogetherwith tur-\nbulence. The difference between this process and the\nother wave damping processes that turbulent damping\ndoes not depend on plasma microphysics.\nWe provide calculations further in this paper, but here\nweprovidesomesimpleargumentsexplainingthe physics\nof the two regimes of damping that we consider further\nalong. The dynamically important magnetic field of the\nturbulent fluid is aligned better locally on the scale of\nthe small eddies. Therefore the Alfven waves emitted\nparallel to the local magnetic field, as this is the case of\nAlfven waves emitted by the streaming instability, will\nexperience the least distortions from the oppositely mov-\ning eddies. Formally, the wave moving exactly parallel\nto the magnetic field corresponds to a wave packet with\nl⊥=∞. Therefore the least distorted Alfven waves can\nbe described as the Alfven packages having the largest\nvalue of l⊥. The wave packages are most efficiently dis-\ntorted by the oppositely moving wave packages with the\nsamel⊥. The larger l⊥the longer time for the evolution\nof the corresponding packages, e.g. for strong GS95 tur-\nbulence it corresponds to l⊥/vl∼l2/3\n⊥. We will show be-\nlowthat for the given Alfven wavewavelengththe largest\nl⊥corresponds exactly to the case of emission parallel to\nthe local direction of the magnetic field. This is the case\nof Alfven wave damping relevant to the streaming insta-\nbility. The other case is the Alfven wave emission by a\nmacroscopic source. If Alfven waves are emitted at an\nangleθto the magnetic field, the value of l⊥is smaller\nand therefore the turbulent damping of Alfven waves is\nfaster. This is also the case of, for instance, if Alfven\nwaves are emitted parallel to the mean field. There the\ndispersion of the magnetic field direction in respect to\nthe mean field acts as the angle θ. Naturally, the damp-\ning of Alfven waves depends on the regime of turbulence\nthat the waves interact with. For instance, if the turbu-\nlent perturbations are weakly non-linear, as in the case\nof weak turbulence, these perturbations should induce\ndampingwhichisslowercomparedwiththestronglynon-\nlinear perturbations of the strong turbulence.\nIn what followswe assume that the propagatingAlfven\nwaves are weak enough so that they do not distort the\nAlfvenic turbulence. In the opposite case we expect the\nturbulence to get more features of the turbulence with\nnon-zero cross helicity which is frequently called ”imbal-\nanced turbulence”. A tested model of such turbulence\nin Beresnyak & Lazarian (2008b) predicts a significant\ndecrease of cascading for the strong wave packages. We\nmay expect that the perturbations significantly stronger\nthan the background turbulence can disturb the back-\nground turbulence inducing its imbalance. Thus the\nwaves can potentially propagate over longer distances\ncompared with the estimates that we are going to ob-\ntain below.Turbulent Damping of Alfven waves 5\n4.DAMPING OF THE STREAMING INSTABILITY\nIn this section we consider the damping of Alfvenic\nwaves that are generated with respect to the local mag-\nnetic field direction. This sort of damping is associated\nwith the damping of the streaming instability of cosmic\nrays (YL02, FG04). The damping of Alfven waves in\nthe global system of reference relevant to the damping of\nAlfven waves emitted by macroscopic sources is consid-\nered in§5.\n4.1.Streaming instability and local system of reference\nAlfven waves can be generated by different astrophysi-\ncalsources. ThestreaminginstabilityofCRsisanimpor-\ntant process that generates Alfven waves. The genera-\ntionhappens asparticlesinteractwith the localmagnetic\nfield and the sampling scale for the magnetic field is the\nLarmor radius of the energetic particle. This is a typi-\ncal situation when one must consider the local system of\nreference related to the local direction of the wondering\nmagnetic field (LV99, Cho & Vishniac 2000, Maron &\nGoldreich 2001, Cho, Lazarian & Vishniac 2002).\nThe growth rate of the streaming instability for the\ndirection parallel to the local direction of the magnetic\nfield is given by the expression (see Kulsrud & Pearce\n1969):\nΓcr≈ΩBncr(> γ)\nni/parenleftbiggvstream\nVA−1/parenrightbigg\n,(24)\nwhere Ω B=eB/mcis the particle gyrofrequency, ncris\nthe number density of CRs with gyroradius r > λ=\nγmc2/eB. The streaming velocity vstreamenters Eq.\n(24) as a ratio with VA.\nFor the instability to operate, the growth rate given by\nEq. (24) shouldexceedthe rateofthe turbulentdamping\nthat we quantify below.\n4.2.Damping by SubAlfvenic strong turbulence\nWe consider first the case of strong subAlfvenic turbu-\nlence. Treating subAlfvenic strong turbulence as a test\ncase, we consider different ways of deriving the result.\nThe first approach that we present is based on calcu-\nlating the distortion of the wave by evolving turbulent\nfluctuations as the waves propagate along magnetic field\nlines. The distortion of the wavefront arises from the\nmagnetic field lines wondering over angle θx. This angle\ndepends on the fluctuations of the magnetic field δBx\ninduced by turbulence with perpendicular scale x. Sim-\nple geometric considerations suggest that the distortion\ninduced by a wave propagating along magnetic field for\nthe time tis\nδx≈VAtsin2θx≈VAt/parenleftbiggδBx\nB/parenrightbigg2\nt,(25)\nwhere the perturbation induced by turbulence evolves as\n/parenleftbiggδBx\nB/parenrightbigg\nt≈/parenleftbiggux\nVA/parenrightbigg/parenleftbiggt\nx/ux/parenrightbigg\n, (26)\nwhereuxis the velocity corresponding to the magnetic\nfield fluctuation δBx. The time tin Eq. (26) is less\nthan the eddy turnover time x/uxand the ratio reflects\nthe partial sampling of the magnetic perturbation by thewave. Substituting the scaling of strong subAlfvenic tur-\nbulence for uxin Eq. (26) one can rewrite Eq. (25) as\nδx≈V3\nAM16/3\nAt3\nx2/3L4/3. (27)\nThe damping of the wave with the wavelength λcorre-\nsponds to the ”resonance condition” δx=λand substi-\ntuting this in Eq. (27) one can express the perpendicular\nscaleofthe ”resonance”magnetic perturbationsthat dis-\ntort the wave:\nx≈V9/2\nAt9/2M8\nA\nλ3/2L2. (28)\nThe required time for the damping is equal to the\nturnover of the resonant eddy:\nt≈x\nul≈V2\nAt3M4\nA\nλL, (29)\nwhich gives the rate of turbulent damping of Alfven\nwaves\nΓsubA≈t−1, (30)\nor\nΓsubA,s≈VAM2\nA\nλ1/2L1/2. (31)\nFor transAlfvenic turbulence, i.e. MA= 1 this result\ntransfers to the one in FG04. We point out, however, the\nsquare of the Alfven Mach number dependence, which\nmeans a significant change of the damping rate for sub-\nAlfvenic turbulence. We also note that in FG04 the in-\njection scale for turbulence was defined not as the ac-\ntual injection scale, but the scale at which the turbulent\nvelocity becomes equal to the Alfven one. We discuss\nthe implications of this in §8 where we compare our ap-\nproach/results with those in FG04.\nFor isotropic injection of turbulence the maximal per-\npendicular scale of strong subAlfvenic motions is given\nbyxmax=LM2\nA. Therefore, if one substitute this in Eq.\n(28) and simultaneously uses Eq. (30) and Eq. (31) to\nexpresst, one gets\nλmax,s≈LM4\nA. (32)\nFor the streaming instability the particles emit Alfven\nwaves of the order of the particle gyroradius rL. There-\nfore the range of rLis limited to\nrL< LM4\nA, (33)\nwhich can be a serious limitation if MAis sufficiently\nsmall. The larger energy particles the interactions hap-\npen with weak turbulence. We discuss this regime of\ndamping in §4.3, while below we provide another deriva-\ntion of Eq. (31).\nBecause of the significance of wave damping it is also\nuseful to present a more intuitive derivation of the same\nresult that is based on the notion of propagating wave\npackets that we employed obtaining Eq. (4). Consider\ntwo oppositely moving packets with the perpendicular\nscalex′∼k′−1\n⊥. Each packet induces magnetic field dis-\ntortionθ′\nxof the oppositely moving waves. Consider a\nlocally emitted Alfven wave moving parallel to the local\ndirection of magnetic field with wavenumber k−1\n/bardbl∼λ. It\nis easy to see that the wave gets distorted by interacting6 Lazarian\nwith turbulence with the perpendicular k⊥∼k/bardblsinθ′\nx.\nThe interactions of a wave with k⊥and the oppositely\nmoving packages will be most efficient if it is ”resonant”\ni.e.k′\n⊥=k⊥.2This suggests the relation k/bardblsinθx=k⊥,\nwhich determines the perpendicular scale of the wave\npackage which will cascade the wave\nλ≈xsinθx≈xδBx\nB. (34)\nInserting the scaling given by Eqs. (26) and (16) it is\npossible to get the expression for the ”resonant” perpen-\ndicular scale x:\nx=L1/4λ3/4M−1\nA, (35)\nwhich can then be used to find the rate of damping de-\nfined as Γ subA,s≈ux/x, which reproduces the earlier\nresult given by Eq. (31). Within this approach the max-\nimalwavelengthoftheAlfvenicwavethatcanbedamped\nby strong subAlfvenic turbulence can be obtained from\nEq. (34) if the scale ltransis used instead of x, i.e.\nλmax,s≈/parenleftbiggutrans\nVA/parenrightbigg\nltrans≈LM4\nA,(36)\nwhich coinsides with the result given by Eq. (32). The\nminimal scale of waves that are being damped depend\non the perpendicular scale ofthe smallest Alfvenic eddies\nlmin. UsingEq. (34) and the scalingofstrongturbulence\ngiven by Eq. (16) one can get the range of rLaffected\nby turbulent damping arising from strong subAlfvenic\nturbulence:\nl4/3\nmin\nL1/3M4/3\nA< rL< LM4\nA, (37)\nwhich indicates that the waves much smaller than lmin\ncan be damped. The value of lmincan be large in par-\ntially ionized gas (see Xu et al. 2015). Due to the differ-\nences of rLfor protons and electrons Eq. (37) presents\na situation when the streaming instability of electrons is\ndamped, while it is damped for protons.\nThe damping of streaming instability for rL<\nl4/3\nmin\nL1/3M4/3\nAis present, but significantly reduced. An esti-\nmate of it can be obtained by considering the distortion\nδx≪λgiven by Eq. (27) for the time period of the wave\nλ/VA, which is significantly less than the period of the\neddy at the scale lmin,teddy≈l2/3\nminL1/3/(VAM4/3\nA). The\ndistortions accumulate as a random walk with the time\nstep given by teddy. The damping requires λ/δxsteps,\nwhich results in the damping rate\nΓsub,s,r L≪lmin≈M12\nAVAr4\nL\nl2\nminL3, (38)\nwhich also illustrates inefficiency of damping by turbu-\nlence with the perpendicular scale larger than the ”reso-\nnant” scale.\n4.3.Damping by subAlfvenic weak turbulence\n2It is possible to show that the interactions with smaller and\nlarger turbulent scales is subdominant compared with the in terac-\ntion with the ”resonant” scale.For waves longer than λmax,sthe wave is cascaded\nthrough weak interactions together with the correspond-\ning wavepackets, the perpendicular wave scales for which\nare given by Eq. (34). The difference here, however, is\nthat the scaling of weak turbulence given by Eq. (12)\nshould be used. This gives the relation between the\nAlfven wave wavelength and the perpendicular scale of\nthe ”resonant” weak mode l⊥\nλ=l⊥/parenleftbiggl⊥\nL/parenrightbigg1/2\nMA, (39)\nwhich provides the weak eddy perpendicular scale\nl⊥≈λ2/3L1/3M−2/3\nA. (40)\nUnlike strong turbulence, the weak wave packets are\ncascading ℵtimes slower (see Eqs (7), (6)), with\nℵ ≈/parenleftbiggVAl⊥\nulL/parenrightbigg2\n, (41)\nwhere it is taken into account that the parallel scale of\nweak turbulence wavepackets is equal to the injection\nscaleL. The rate of turbulent damping of the Alfven\nwave is therefore\nΓsubA,w≈(ℵ∆t)−1=ℵ−1VA\nL, (42)\nwhich gives\nΓsubA,w≈VAM8/3\nA\nλ2/3L1/3, (43)\nwhich compared to the case of the earlier discussed\ndamping given by Eq. (31) shows even stronger depen-\ndence on MAas well as a different dependence of the\nwavelength λ. Being applicable to weak turbulence, this\nresult does not transfer for MA= 1 to that in FG04 and\ntherefore it is essential to define the range of its applica-\nbility in terms of wavelength λ.\nThe maximal wavelength of the Alfven waves that can\nbe cascaded by the weak cascade can be obtained by\nsubstituting l⊥=L, i.e. using the energy injection scale,\nin Eq. (39). This gives:\nλmax,w≈LMA. (44)\nThus the particles emitting Alfven waves of the order of\ntheir gyroradius should have the range of gyroradii\nLM4\nA< rL< LM A (45)\nin order to interact with weak turbulence. This is pro-\nvided that LM4\nAis larger than the damping scale of tur-\nbulent motions. Otherwise the lower boundary in Eq.\n(45) is given by lmin.\nWaveswith λ > λmax,wwillinteractwithturbulenceat\nthe injection scale L. Such waves cascade by the largest\nwave packets whose cascading rate is ℵ−1VA\nL, i.e.\nΓouter≈ ℵ−1VA\nL≈M2\nAVA\nL, (46)\nwhich is valid for λ < L. In the case of λ≫Lthe result\nin Eq. (56) is being reduced by another random walk\nfactor (L/λ)2, i.e.\nΓouter,extreme ≈ ℵ−1VA\nLL2\nλ2≈M2\nAVA\nLL2\nλ2.(47)Turbulent Damping of Alfven waves 7\nwhich can be important for the damping of Alfvenic\nwaves by turbulence injected at small scales.\n4.4.Damping by SuperAlfvenic turbulence\nThe case of superAlfvenic turbulence for scales less\nthan the scale of the transfer to MHD regime, i.e. lA=\nLM−3\nA, can be obtained from our earlier results through\nthe following considerations. At lAthe turbulence be-\ncomes Alfvenic and this scale can be considered as the\nturbulence injection scale. The injection velocity at this\nscale isVAand therefore the resulting damping rate can\nbe obtained by substituting lAas the injection scale L\nandVL=VAin Eq. (31). As a result:\nΓsuper≈VA\nl1/2\nAλ1/2=VAM3/2\nA\nL1/2λ1/2. (48)\nIn a sense this is a case of transAlfvenic turbulence if lA\nis associated with the turbulence injection scale. This\ncase corresponds to the FG04 where the turbulence in-\njection scale was definedto be the scale LMHDat which\nthe injection velocity becomes equal to VA. Thus for\nsuperAlfvenic turbulence LMHD=lA.\nTreating lAas the effective injection scale one can eas-\nily get from Eq. (32) the maximal wavelength up to\nwhich the above treatment of the non-linear damping is\napplicable:\nλmax,super ≈lA=LM−3\nA. (49)\nFor the streaming instability we associate λwith the gy-\nroscalerLand therefore define the corresponding range\ngyroscales as\nl4/3\nmin\nL1/3MA< rL< LM−3\nA, (50)\nprovided that lmin< LM−3\nA. In the opposite case of\nlmin> lAthe turbulence is does not get Alfvenic over\neven at the smallest scales.\nFor wavelengths larger than those given by Eq. (49)\nand therefore for rL> LM−3\nAthe damping is induced\nby Kolmogorov-type isotropic hydrodynamic turbulence\nwhich folds magnetic fields over the scale of eddies. The\ncharacteristic damping rate in this case is expected to\ncoincide with the turnover time of the corresponding ed-\ndies, i.e.\nΓhydro≈uλ\nλ≈VAMA\nL1/3λ2/3, (51)\nwhere we used Eq. (19).\n4.5.Other forms of presenting our results\nEmissionofAlfven wavesby energeticparticlesmoving\nalong magnetic field lines presents the most important\ncase of the emission of Alfven waves in the local system\nof reference. The resonant emission along local magnetic\nfield direction corresponds to the condition\nλ=rL, (52)\nwhererL=γmc2/eBis the Larmor radius of the reso-\nnant particle with a relativistic factor γ. We shall use\nEq. (52) in expressions below.Expressing wave damping through the cascading rate\nis another way of presenting our results. Cascading of\nturbulent energy is a source of media heating. This can\nprovideupper limits on the level of turbulence in astro-\nphysical environments, which is valuable when the scales\nof the turbulent motions and injection rates are difficult\nto estimate.3The cascading rate of the weak turbulence\ngiven by Eq. (11) can be rewritten as\nǫw≈V3\nAM4\nA\nL, (53)\nwhich shows a decrease of the energy dissipation by a\nfactorM4\nAcompared with the case of transAlfvenic tur-\nbulence. For rL< LM4\nAthe damping rate for waves\ncan be obtained by expressing MAfrom Eq. (53) and\nsubstituting it in Eq. (31):\nΓsubA,s≈ǫ1/2\nw\nV1/2\nAr1/2\nL, (54)\nwhich differs from the expression in FG04 by the use of\nthe cascading rate for weak turbulence ǫwinstead of the\ncascading rate for strong turbulence. Thus the obtained\ndampingrateforsubAlfvenic turbulence is M2\nAtimes less\nthanin thecaseoftrans-Alfvenicturbulence(seealsoEq.\n(31)).4\nForLM4\nA< λ < LM Awe get the expression which\nis significantly different in its form from that in FG04.\nIndeed, expressing MAfrom Eq. (53) and substituting\nit in Eq. (43) we can get\nΓsubA,w≈ǫ1/3\nwL1/3\nVAr2/3\nL≈ǫ1/3\nwM4/3\nA\nr2/3\nL.(55)\nThe expression given by Eq. (55) has a slower depen-\ndence on the dissipation rate compared to Eq. (54).\nThe suppression of damping rate by the factor M8/3\nA(see\nEq. (43)) is important and for MA≪1 it explains the\nsmooth transition to the regime of insignificant Alfven\nwave damping that is present for marginally perturbed\nmagnetic fields.\nDealing with the damping of Alfven waves emitted by\nparticles with larger rL, one can obtain the expression\nfor the damping for LMA< rL< L(see Eq. (39))\nthat corresponds to the damping by the outer scale of\nturbulent motions:\nΓouter≈ǫ1/2\nw\nL1/2V1/2\nA. (56)\nFor superAlfvenic strong MHD turbulence if one ex-\npressesMAfrom Eq. (21) and substitutes it in Eq. (48)\n3The situation is changing with the development of new tech-\nniques that obtain the injection scale and injection veloci ty from\nobservations (see Chepurnov et al. 2010, 2015, Lazarian &\nPogosyan 2012).\n4It is interesting to note the special property of damping by\nstrong turbulence. The damping depends only on the turbulen t\nenergy dissipation rate and not on the scale of the energy inj ection.\nNote, that the turbulent damping in other regimes is very diff erent\nand does not show this remarkable universality.8 Lazarian\nit is easy to get\nΓsuper≈ǫ1/2\nsuper\nV1/2\nAr1/2\nL, (57)\nwhich has formally the same form as the expression for\nthe damping for subAlfvenic strong turbulence given by\nEq. (54). The cardinal difference between the two ex-\npressions, assuming that the injection scale Lis the\nsame, stems from the differences in the cascading rates\nin superAlfvenic and subAlfvenic turbulence. The sub-\nAlfvenic turbulence induces the significant reduction of\nthe cascading rate compared to the transAlfvenic tur-\nbulence, the superAlfvenic strong MHD turbulence in-\nduces a significant increase of dissipation compared to\nthe transAlfvenic case. Thus, for the same L, the damp-\ning of Alfven waves by superAlfvenic turbulence is more\nefficient than by the subAlfvenic one. The damping rate\nforrL> λmax,super where the latter is given by Eq. (49)\nis produced by hydrodynamic turbulence and therefore\nis\nΓhydro≈ǫ1/3\nhydro\nr2/3\nL. (58)\nIn view of the astrophysical importance of damping in\nsubAlfvenic turbulence, it is useful to rewrite the expres-\nsions given by Eq. (31) and (43) in terms of λmax,sgiven\nby Eq. (36), namely\nΓsubA,s≈VA\nL/parenleftbiggλmax,s\nrL/parenrightbigg1/2\n, rL< λmax,s,(59)\nand\nΓsubA,w≈VA\nL/parenleftbiggλmax,s\nrL/parenrightbigg2/3\n, rL> λmax,s.(60)\nExpressed in this form Eq. (59) explicitly shows that the\ndamping by strong turbulence Γ sub,sis faster than the\nAlfven crossing rate of the injection scale eddies, while\nin the case of weak turbulence Eq. (60) shows that Γ w\nis slower that the aforementioned rate.\nForLMA< rL< L, the damping rate can be written\nas (see Eq. (56) and Eq. (36))\nΓouter≈ΓsubA,srL\nL, (61)\nwhichpresentsanotherformfortheAlfvenwavedamping\nby turbulence at the outer scale.\n5.DAMPING OF ALFVEN WAVES GENERATED IN THE\nGLOBAL SYSTEM OF REFERENCE\nBelow we consider the damping of Alfven waves gen-\nerated by an outside source which is not related to the\nmagnetic field structure. It is important to realize that\nsuch waves should be viewed as being in the global sys-\ntem of reference and therefore our earlier treatment of\nthe damping is not applicable. This is a separate case of\ndamping relevant to many astrophysical settings, e.g. to\nthe emission waves by stellar surface activity (see §6.5).\n5.1.Case of Strong SubAlfvenic turbulence\nConsider first an Alfven wave moving at an angle θ≫\nδB/Bwith respect to the meanmagnetic field. In thissituation one can disregard the dispersion of propagation\nangles that arises from turbulent magnetic wandering.\nFor this purpose we use sin θinstead of sin θxin Eq. (34)\nand get for the perpendicular scale of eddies:\nx≈λ\nsinθ. (62)\nThe rest goes along the same line of reasoning that we\nemployed in §3.1. Indeed, the rate of the wave damping\nis equalto the turnoverrate ofstrongsubAlfvenic eddies.\nTherefore using Eq. (62) it is easy to get\nΓsubA,s,θ≈VAM4/3\nAsin2/3θ\nλ2/3L1/3, (63)\nwhich provides the non-linear damping rate of an Alfven\nwave moving at the angle θwith respect to the mean\nfield.\nIn terms of the cascading rate of weak turbulence ǫw\n(see Eq. (11)), the above damping rate for the wave can\nbe rewritten as:\nΓsubA,s,θ≈ǫ1/3\nwsin2/3θ\nλ2/3. (64)\nThe turbulent damping given by Eq.(64) is applicable\nto\nlminsinθ < λ < LM2\nAsinθ, (65)\nwherelminis the minimal scale, i.e. the perpendicular\ndamping scale, and LM2\nA=ltransis the maximal scale\nfor the extent of the turbulent cascade.\nNaturally, for this expression our approximation θ≫\nδB/Bfails if the wave is launched parallel to the mean\nmagnetic field. The directions of the local magnetic field\nexperience dispersion and this makes the actual θ0not\nzero. In the global system of reference the dispersion is\ndetermined by the magnetic field variations at the injec-\ntion scale (see Cho et al. 2002). Therefore\nθ0≈BL\nB≈MA. (66)\nSubstituting this into Eq. (63) we get\nΓsubA,s, 0≈ǫ1/3\nwM2/3\nA\nλ2/3, (67)\nwhich is different from our expression for the damping\nof Alfvenic waves moving along the local direction of the\nmagneticfield (see Eqs. (31), (54)). The difference stems\nfrom the difference in Alfven waves generated in respect\nto the local system of reference and in global system of\nreference. The rate given by Eq. (67) is applicable to\nthe range\nlminMA< λ < LM3\nA, (68)\nwhich trivially follows from Eqs.(65) and (66).\n5.2.The case of Weak SubAlfvenic turbulence\nForweaksubAlfvenicturbulenceinthecase θ≫δB/B\nwe shall use Eq. (62) to relate the wavelength λto the\nscale of perpendicular motions that the wave interacts\nwith while cascadingaswell as Eq. (42) to get the damp-\ning rate corresponding to such motions. As a result,\nΓweak,global,θ ≈VAsinθM2\nA\nλ≈ǫ1/2L1/2sinθ\nV3/2\nAλ,(69)Turbulent Damping of Alfven waves 9\nwhere in the damping is expressed through the weak cas-\ncading rate ǫw.\nThe applicability of this type of damping is applicable\nto\nLM2\nAsinθ < λ < LM Asinθ, (70)\nwhere the last inequality is obtained by substituting the\nmaximal perpendicular scale of eddies LMAforxin\nEq.(62).\nFor the propagation along the mean magnetic field one\nshould take into account Eq. (66) which results in\nΓweak,global, 0≈VAM3\nA\nλ≈VAǫ3/4L3/4\nλV5/4\nA.(71)\nThe range of the applicability of this damping rate is\nLM3\nA< λ < LM2\nA, (72)\nwhere Eq.(66) and (70) were used.\n5.3.Other cases\nFor strong superAlfvenic turbulence, i.e. for damping\nby turbulent motions at scales less than lAone can still\nuse our approach above and consider damping of Alfven\nwaves with λ < l A(see Eq. (49)). The damping by\neddies less than lAhappens by one eddy turnover time.\nIf the wave is at an angle θto the magnetic field within a\nmagnetic eddy < lAthen the damping happens over one\nturnovertime for the motions ofthe size xdefined by Eq.\n(62). The procedures analogous to those we employed\nabove provide\nΓsuper,global,θ ≈VAMAsin2/3θ\nλ2/3L1/3, (73)\nwhere in superAlfvenic turbulence angle θchanges from\none strong turbulence eddy of size lAto another. There-\nfore an averaging over such changing directions should\nbe performed which for the random distribution of direc-\ntions provides the damping rate of /angbracketleftsin2/3θ/angbracketright= 3/5.\nFor Alfven waves from a macroscopic source ≫lAthe\nturbulentvolumecanbeconsideredasconsistingofMHD\ncells with the regular MHD turbulence but with the in-\njection of transAlfvenic turbulence at the scale lA. The\nwave damping will differ depending on the angle θbe-\ntween the magnetic field in a given cell and the wave\npropagation direction. The rate of damping can be ob-\ntained by substituting in Eq. (63) the actual angle θas\nwell asMA= 1 and L=lA. The minimal wavelength in\nthis case depends on the lmin∼θ.\nAt scales larger than lAthe turbulence is essentially\nhydrodynamic. Therefore, for turbulent damping by su-\nperAlfvenic eddies of size larger than lAas well as for\ndamping by outer-scale eddies there is no difference be-\ntween local and global frames. Therefore our earlier re-\nsults are applicable.\n5.4.Finite-sized macroscopic emitter\nOur considerations obtained for an infinitely extended\nmicroscopic emitter can be generalized for the finite size\nemitter. If the size of the emitter yand the wave is\nemitted attheangle θ≫δB∗/B, then ourconsiderations\nin§5.1-5.2 stay the same. Note, however, that B∗in this\ncase ismin[BL,Bdamp], where Bdampis the magneticfield deviation at the scale of wave damping, i.e. ldamp≈\nΓ−1\nglobal,θVA, where Γ global,θare, for instance, defined for\nweak and strong subAlfvenic turbulence in §5.1 and§5.2.\nIf the wave is emitted parallel to the local magnetic\nfield at the scale y, then we are have to deal with the\nintermediate case having features of Alfven wave damp-\ning in local and global systems of reference. Indeed, the\nvariations of the magnetic field directions should be cal-\nculatedatthescaleof yandcomparedwiththevariations\nof magnetic field at the ”resonant”scale. For strong sub-\nAlfvenic turbulence this scale is given by Eq. (28) and\nfor weak subAlfvenic turbulence by Eq. (40). The latter\ntwo scales depend on λ. Therefore we expect to see the\nscaling corresponding to Alfven wave damping if the y\nis smaller than the values given by the aforementioned\nequations. Note that the damping of the waves emitting\nwith respect to the mean magnetic field will be happen-\ning inhomogeneously with patches where the local mag-\nnetic field happens to be parallel to the wavefronthaving\nthe ability to support the Alfven wave propagation for a\nlonger period of time.\n6.COMPARISON WITH NON-LINEAR LANDAU DAMPING\nIt is important to compare the turbulent damping that\nwestudy in this paperwith the non-linearLandaudamp-\ning process that can also damp Alfven waves (see Kul-\nsrud 2005). The latter damping is inversely proportional\nto the square root of CR scaleheight Lz, so we may ex-\npect that this process is subdominant for weak gradients\nof the cosmic ray distribution. Indeed, the ratio of the\nrate of turbulent subAlfvenic damping and the rate non-\nlinear Landau damping GNLcan be evaluated to give:\nΓsubA,s\nΓNL≈B3/2\nµGn1/4\ni,−3L1/2\nz,100M2\nA\nL1/2\n100T1/4\n4keVn1/2\nCR,−10γn/2−2\n100,(74)\nwhereT4keV= (T/4keV),BµG= (B/1µG),Lz,100=\n(Lz/100kpc), ni\n−3= (ni/10−3cm−3),nCR\n−10=nCR(γ >\n1)/10−10cm−3),γ100=γ/100, and is scaled to n= 4.6.\nNote that nCR(> γ) = 10−10γ−1.6cm−3of the order a\nCR energy density in equipartition with a ∼µG B-field.\nTherefore, if the CR profile falls ( nCR→0) and flat-\ntens (Lz→ ∞) the non-linear Landau damping becomes\nsubdominant.\nOnemaywonderwhetherforveryweaklevelsofturbu-\nlenceMA→0 the non-linear Landau damping may be-\ncome important. The latter, however, is a self-regulated\nprocess as the suppression of the streaming instability is\nboundtoallowtheCRstospreadfast, decreasingthe CR\ngradient. Potentiallyresonancescatteringcould mitigate\nthis spreading. However, this depends on the presence\nof fast modes that, in the absence of streaming insta-\nbilities, were identified in Yan & Lazarian (2002) as a\nmajor factor of cosmic ray scattering in the interstel-\nlar plasmas..5In many instancies, e.g. galactic halos,\nthese modes are subject to significant collisionless damp-\ning and therefore their efficiency of controlling of the CR\n5The importance of fast waves is easy to understand. One\nshould recall that that due to the extreme anisotropy of the t en-\nsor that describe the Alfven turbulence at small scales (Cho et al.\n2002), the scattering by Alfvenic modes of the MHD cascade in i-\ntiated at the large injection scale Lis very small (Chandran 2000,\nYan & Lazarian 2002).10 Lazarian\nspreading is limited. On the contrary, there is no such\na self-regulation for turbulent damping of CR streaming\nwhich makes the process dominant in most astrophysical\nsettings. In many instances when the turbulent damping\nfails, the non-linear Landau damping is unlikely to damp\nthe CR streaming either. For instance, we argue in the\nnext section that the turbulent damping ofthe streaming\ninstability is not important for the Milky Way halo due\nto the low level of turbulence there. There we do not\nexpect non-linear damping to be important there due to\nthe self-regulation which entails the increase of Lz.\n7.ASTROPHYSICAL IMPLICATIONS\nIn what follows we discuss in detail the problem of\ndamping of streaming instability in our Galaxy and\nsketch some other astrophysical implications of the im-\nproved understanding of Alfven wave damping that we\nhave obtained in this paper. The detailed treatment of\nthese implications will be provided elsewhere.\n7.1.Streaming of CRs in galaxies\nOne of the simplest models of the galactic CR prop-\nagation is the so-called ”leaky box model” (see Longair\n2011). Within this model CRs propagate freely within\nthe galactic disk, while they experience streaming insta-\nbility as they enter a fully ionized halo surrounding the\ngalaxy. Freezoomingthroughthegalacticdiskispossible\nas, in the leaky box model, the galactic disk is assumed\nto be partially ionized and therefore the streaming in-\nstability is being suppressed by ion-neutral damping (see\nKulsrund & Pearce 1968). This model is surely naive,\nas the galactic disk is definitely not fully filled with par-\ntially ionized gas. In fact, a significant fraction of the the\ngalactic disk is filled with hot ionized gas (McKee & Os-\ntriker 1977, see Draine 2011). Moreover, the leaky model\ndoes not account for turbulent damping of streaming in-\nstability.\nThe first treatment of CR propagation that took the\nstreaming instability damping into account was done by\nFG04. This study came to a paradoxical conclusion,\nnamely, that turbulence suppresses streaming instabil-\nity for most of the CR energies and therefore it is really\ndifficulttounderstandtheobservedhighisotropyofCRs.\nBelow we subject the the problem to scrutiny and come\nto the conclusions that are different in FG04. In partic-\nular, we will show that (a) damping is produced by weak\nrather than strong Alfvenic turbulence and therefore is\nreduced, (b) the turbulent dissipation rate assumed in\nFG04 to be equal to the plasma cooling rate is overesti-\nmate of the actual dissipation rate, as this way of esti-\nmating disregards other important heating mechanisms.\nOur study above shows that turbulent damping of the\nstreaming instability changes significantly with whether\nthe damping is performed by strong or weak Alfvenic\nturbulence. Note, that it is natural to assume from the\nvery beginning that turbulence in the halo is subAlfvenic\nrathertransAflvenicorsuperAlfvenic. Thisfactiseasyto\nunderstand. Indeed, theISMisturbulent(seeArmstrong\net al. 1995, Elmegreen & Scalo 2004, McKee & Ostriker\n2007, Chepurnov & Lazarian 2010) and the sources of\nturbulence driving, whether they are related to super-\nnovae (see MacLow 2004, Draine 2011) or magnetoro-\ntational instability (see MacLow & Klessen 2004), are\nwithin the galactic disk. The magnetic field in the halois expected to become more and more quiescent with the\ngreater distance from the disk as turbulence decays dif-\nfusing from the disk. Our quantitative estimates based\non the observational data that we provide below support\nthis intuitive notion.\nFor subAlfvenic turbulence it is possible to express\nthe streaming rate using the textbook approach to the\nstreaming instability (see Kulsrud 2005), but equating\nthe turbulent damping rate to the streaming damping\nrate in Eq. (24) (see FG04):\nvstream≈VA/parenleftbigg\n1+ΓnirL\nγcncr/parenrightbigg\n, (75)\nwhere we used the relation rL=γcΩ−1. Note, that\nthe rates of damping Γ are different for weak and strong\nturbulence. In particular, the damping is by strong tur-\nbulence if Γ = Γ sub,s, i.e. for rL< LM4\nA, and by weak\nturbulence if Γ = Γ w, i.e. for rL> LM4\nA. For the Milky\nWay galaxy the quantities that enter Eq. (75) can be\nestimated for the hot coronal gas of the halo, i.e. plasma\nwith density ni≈10−3cm−3temperature T≈106K\nand magnetic field B≈3µG. An upper limit of the\nAlfven Mach number MAmay be obtained for the galac-\ntic halo assuming that the turbulent velocity dispersion\nin the halo is the same as in the disk, i.e. 106cm/s.\nIndeed, as the sources of the turbulence are localized\nin the galactic disk and the turbulence decays quickly\n(see Stone et al. 1998, Cho & Lazarian 2002), this value\nis a substantial overestimate of the turbulent velocities.\nWith the Alfven velocity for the parameters above being\nVA≈2×107cm/s, one gets MA<1/20. Therefore\nthe critical value of rLisLM4\nA≈5×1014cm, where\nwe assumed the injection scale Lequal to 100 pc. The\nrelativistic proton gyroradius is rL≈1012γcm, which\nmeans that the streaming that is controlled by strong\nturbulence is applicable up to γ <500. This conclusion\ncontradicts to the use of damping by strong turbulence\nassumed in FG04. In fact, we believe that the expression\nof streaming affected by strong turbulence, namely,\nvstream,s≈VA/parenleftBigg\n1+/bracketleftbiggǫw\n700erg s−1g−1/bracketrightbigg1/2\nγ1.1/parenrightBigg\n,(76)\nis applicable to the damping of turbulence in the galactic\ndisk rather to any parts of the Milky Way halo. Note,\nthat Eq. (76) in its form coincides with the expression\nfor the streaming velocity in FG04. This coincidence is\nthe result of the remarkable universality of the damping\nby strong Alfvenic turbulence that we discussed earlier\nin§4.5. However, the significant difference of Eq. (76)\nexpression is the use of the weak turbulence cascading\nrateǫw. This rate differs from the one for transAlfvenic\nturbulence by a factor M4\nA.\nMost of the streaming cosmic rays in the Milky Way\nhalo are expected to have rL> LM4\nA. For them, using\nthe expression for damping by weak turbulence, i.e. Eq.\n(55), it easy to obtain\nvstream,w ≈VA/parenleftBigg\n1+ǫ1/3\nwnir1/3\nLM4/3\nA\nγcncr/parenrightBigg\n.(77)\nIt is evident that Eq. (77) is very different from Eq.\n(76). The most significant difference stems from the factTurbulent Damping of Alfven waves 11\nthat the streaming velocity in Eq. (77) does depend not\nonly on the dissipation rate ǫw, but also on the Alfven\nMachnumber MA. It isalsoimportant that the damping\nrate in Eq. (77) depend on the ǫ1/3\nwrather than ǫ1/2\nwas\nin Eq. (76). Indeed, both factors above help avoiding\n”streaming catastrophe” outlined in FG04. Additional,\nbut less important factor that helps us is that the second\nterm of Eq. (77) scales as γ0.94compared to γ1.1in Eq.\n(76).\nTo find the streaming velocity using Eq. (77) one\nshould know both the dissipation of weak turbulence ǫw\nand the Alfven Mach number MA. Two different esti-\nmates ofthe cascadingratewerepresentedin FG04. One\nwas based on the cooling rate for the hot gas, the other\nwas based on the supernovae energy injection rate. The\nlatter is readily available. Indeed, it is accepted that the\nsupernovae are releasing 1051ergs of mechanical energy\nin the gas once every one million years in the disk area of\n100 pc2(see Draine 2011). Assuming that the resulting\nturbulence is transAlfvenic and therefore decays in one\nAlfven crossingtime the FG04 obtained the estimates for\nthe turbulent dissipation rate of ≈25 erg s−1g−1. We\nbelieve that this is an estimate that has relevance to the\ngalactic disk, rather than to the galactic halo. Such a\nsignificant rate of turbulent dissipation according to Eq.\n(76) should ensure that within the media of the galactic\ndiskAlfvenicturbulencesuppressesstreaminginstability,\nwhich corresponds to the disk part of the ”leaky box”\nmodel.\nThe situation is very different for the hot plasmas in\nthe Milky Way halo. There the second estimate in FG04\nbased on the gas cooling might be relevant. Indeed,\nthe turbulent cascading rate determines hot gas heating\nand this cannot be larger that the radiative cooling rate,\nwhich is about 0 .06 erg s−1g−1(see Binney & Tremaine\n1987). Infact, thisprovidesthe upper limit fortheturbu-\nlent cascading and the actual rate, as we discuss further,\nmay be substantially lower.6Indeed, turbulent heating\nis not the only way of heating the halo plasmas. For\ninstance, we may consider heating that comes from CR\nstreaming (see Wiener et al. 2013b). The irreversible\nenergy transfer from streaming CRs to gas provides the\nvolumetric heating rate (see Kulsrud 2005):\nΓheat≈ −VA▽Pcr. (78)\nTo get the heating per unit of mass one has to divide the\nheating rate given by Eq. (78) by the plasma density.\nTaking as a rough estimate the energy density of CRs to\nbe 1 eV per cm−3and the characteristic scale of the CR\nchange to be Lcr≈5 kpc, one gets heating ∼0.06 erg\ns−1g−1, which coincides with the cooling rate in Binney\n& Tremaine (1987). This may indicate that the galactic\nhalo is heated by cosmic ray streaming that does take\nplacein the halo environment. As a result, the cascad-\ning rate adopted in FG04 significantly overestimates the\nactual turbulence cascading in the halo gas of the Milky\nWay7. We feel that the most important is to establish\n6The rate of turbulent heating by supernovae above and the\nupper limit of turbulent heating at hand are so different both due\nto the decrease of turbulent velocities in the galactic halo compared\nto the disk and to the decrease of Alfven Mach number MA. The\nlatter makes turbulence less dissipative in proportion to MA.\n7We note parenthetically that the adopted cascading of ∼0.06whether streaming instability really fails in the realisti-\ncally turbulent Milky Way halo. Therefore, for the rest\nof our discussion, we concentrate on showing that the\nconclusions about the ”streaming catastrophe” reached\nin FG04 are not obtained on the self-consistent basis.\nIt is well known that the anisotropy is less than 0 .1%\nfor the CRs with γ <106(see Longair 2011). As the\nAlfven velocity in the hot plasmas is ∼0.1% ofcFG04\nassumed that the the second term in bracketsof Eq. (76)\nis not larger than unity.8This provided Vstreaming ≈\nVA(1+0.01γ1.1) for the cascading of ∼0.06 erg s−1g−1.\nOn the basis of this estimate, FG04 concluded that to\navoid the contradiction with the observational data for\nγ∼106oneshouldassumethattheratefortheturbulent\ndissipation is less than 4 ×10−11erg s−1g−1, which is\nvery different from the assumed ∼0.06 erg s−1g−1rate.\nOn the basis of this FG06 came to the conclusion that\nstreaming instability is not feasible as the solution for\nsolving the problem of explaining the observed isotropy\nof cosmic rays.\nAs we pointed above, for realistic magnetization of the\ngalactic halo MA≪1 and Eq. (77) rather than Eq.\n(76) should be used to determine the streaming veloci-\nties.9It is safe to say that in the situation the turbulent\ndamping in galactic halo not being constrained observa-\ntionallyit isprematuretobe alarmedabout the failureof\nstreaming instability to explain the cosmic ray isotropy\nare premature. In fact, we expect the turbulent veloc-\nity to decrease fast with the distance from the galactic\nplane. In addition, due to the drop ofthe plasmadensity,\nwe also expect the exponential increase of VAwith the\ndistance from the galactic plane. Therefore in Eq. (77)\nbothǫwandMAare likely to decrease exponentially, i.e.\n∼exp(−H/h), where His the halo size ∼5 kpc, and\nh∼Lis the scale height of the galactic disk, which is\none or two hundred parsecs. Therefore it is likely that\natsomedistance from the disk ≫Lthe second term in\nbrackets in Eq. (77) becomes small. This is what the\nonly thing that is required for the streaming instability\nto isotropize CRs.\nExpressing the streaming velocity through the turbu-\nlence dissipation rate is advantageousonly when this dis-\nsipation rate is readily available from observations. In\nthe situation of galactic halo when the turbulent heating\nmay not be the dominant process it seems advantageous\nto use the other expressions for the turbulent damping\nrate, e.g. for the damping by weak turbulence to substi-\ntute in Eq. (75) the expression for damping given by Eq.\n(43). This way we get:\nvstream,w ≈VA/parenleftBigg\n1+VAnir1/3\nLM8/3\nA\nL1/3γcncr/parenrightBigg\n,(79)\nwhere the turbulence injection scale Lcan be obtained\nfrom observations with statistical techniques using spec-\nerg s−1g−1corresponds to MA≈0.2 if the injection scale L= 100\npc isadopted. Thissuggests that even with this cascading ra te that\nwe argue to be an overestimate, the turbulence is subAlfveni c.\n8The difference in terms of strong turbulence cascading assum ed\ninFG04 and the weak that isemployed inEq. (76)isnot importa nt\nfor the argument as we discussed earlier.\n9In fact, for MA<0.003 the CR with γ= 106interact with\nthe turbulence in the outer scale, which further reduces tur bulent\ndamping (see Eq. (56).12 Lazarian\ntral lines (see Chepurnov et al. 2010, 2015) or syn-\nchrotronemission (see Lazarian& Pogosyan2012, 2016),\nwhile the Alfven Mach number MAcan be obtained us-\ning anisotropy studies with spectral lines (see Esquivel &\nLazarian 2005, Burkhart et al. 2014, Esquivel, Lazarian\n& Pogosyan2015, Kandel, Lazarian& Pogosyan2016ab)\nor synchrotron studies (see Lazarian & Pogosyan 2012,\n2016, Herron et al. 2016). In particular the variations\nofLandMAwith the distance from the observer can\nbe obtained using multifrequency polarization studies as\nexplained in Lazarian & Pogosyan (2016). We believe\nthat this is a promising future direction of research.\nIn fact, in view of our study, the ”leaky box” model\ncan be reformulated. Instead of suppression of stream-\ning instability in the disk by ion-neutral collisions, the\ninstability is likely to be efficiently suppressed by turbu-\nlence there. At the same time, the streaming instability\ncan be present in the Milky Way halo, returning and\nisotropizing CRs.\nNaturally, apart from the streaming instability, there\nare also other ways to isotropize CRs. We also note that\nadditional sources of CR isotropization come from the\nscattering of CRs as well as from magnetic field wander-\ning10. Due to the Richardson dispersion (see Lazarian &\nYan 2014) CRs following magnetic field lines spread su-\nperbalisticallyin the directionperpendicularto themean\nmagneticfield, modifyinganddecreasingtheanisotropies\n(seeLopez-Barqueroetal. 2015). Inaddition, fastmodes\nthat were identified as the major source of scattering in\nthegalacticenvironmentsinYan&Lazarian(2002,2004)\ncan provide significant isotropization of CR. These pos-\nsibilities were not considered in FG04. On the contrary,\nthe idea that is mentioned there, i.e. of confinement us-\ning magnetic mirror arising from dense molecular clouds\n(Chandran 2000) looks problematic. Indeed, in view of\nthe lowfilling factorofdensecloudsit looksunrealisticto\nthink that CRs have to encounter many magnetic bottles\ncreated this way prior to their leaving the galaxy. In ad-\ndition, with the new data that shows that the strength of\nmagnetic fields stay in a significant fraction of molecular\nclouds on the level close to the value of the field in diffuse\ninterstellar medium (Crutcher et al. 2010), the confine-\nment efficiency of magnetic bottles created by molecular\ncloudsis veryquestionable11. Moreover,the formationof\nmagnetic bottles does not ensure particle anisotropy, as\nthe magnetic bottles formed by molecular clouds are sta-\ntionary and therefore they do not change the adiabatic\ninvariant of the particles confined by the bottles.\n7.2.Acceleration of particles in shocks and reconnection\nlayers, impact on reconnection\nWe also want to stress that the issues related to CR\nstreaming are not limited to the observed CR isotropy.\nWe believe that the CR streaming instability can be\npresent also in the galactic disk but at places of signif-\n10Magnetic field wandering for Alfvenic turbulence was first\ndescribed in LV99 and later employed in solving different pro b-\nlems from thermal conduction of magnetized plasmas (Naraya n &\nMedvedev 2002, Lazarian 2006) to shock acceleration (Lazar ian &\nYan 2014).\n11The effect of poor correlation of density and magnetic field\nwas explained in Lazarian, Esquivel & Crutcher (2012) as the con-\nsequence of process of turbulent reconnection or ”reconnec tion dif-\nfusion” (LV99, Lazarian 2005, Santos-Lima et al. 2010).icantly higher than average CR flux, e.g. near places\nof CR acceleration, e.g. shocks (see Bell 1978, Schlick-\neiser 2002) or reconnection sites (de Gouveia dal Pino &\nLazarain 2005, Lazarian 2005, Drake et al. 2006, Lazar-\nian & Opher 2008).\nThe acceleration of CRs in shocks is an accepted pro-\ncessforexplainingthepopulationofgalacticCRs(Krym-\nski et al. 1978, Bell 1978, Armstrong & Decker 1979).\nTo be efficient, the process of returning of CRs back to\nthe shock must also be efficient. Potentially, the stream-\ning instability should be important for returning parti-\ncles back (see Longair 2011). Turbulence, however, is\nlikely to complicate the process. In fact, apart from the\npre-existing turbulence, there is turbulence that is gen-\nerated both in the precursor (Beresnyak et al. 2009, del\nVale et al. 2016) and the postshock media (Giacalone\n& Jokipii 2007). This superAlfvenic small-scale turbu-\nlence is expected to efficiently damp the streaming. At\nthe same time, the same turbulence also generates a tur-\nbulent magnetic field, which can act as a magnetic mir-\nror that returns the CRs back to the shock. Therefore,\nit is likely that the CR acceleration in shocks proceeds\nwithout the important contribution from the streaming\ninstability.\nStreaming instability can also return particles acceler-\nated by magnetic reconnection to the reconnection site\nenhancing the First order Fermi acceleration that arises\nfromreconnection(deGouveiadalPino&Lazarian2005,\nDrake et al. 2006). The corresponding reconnection can\nproceed both when large scale magnetic field reconnects\nreleasing its free energy and within multiple reconnec-\ntion regions in the steady state turbulence. The latter\nprocess was recently considered in Brunetti & Lazarian\n(2016). The role of the streaming instability depends on\nthe level of turbulence in the system. Generically, we\nexpect the level of turbulence to increase in the recon-\nnection regions as magnetic reconnection progresses (see\nLazarian et al. 2016) and therefore the role of streaming\ninstability to decrease. However, the study of the pa-\nrameter space for which the streaming is important both\nfor magnetic reconnection and shock CR acceleration is\nbeyond the scope of the present study.\nWe also note that magnetic reconnection can be a\nsource of Alfvenic waves (see Kigure et al. 2010). As\nthe processof reconnectionhappens genericallyin turbu-\nlent fluids, it is natural that the generated Alfven waves\nshould experience turbulent damping. Eventually, as\nwe discussed, this should contribute to generating more\nturbulence in the reconnection region. Turbulence was\nshown in LV99 to change the nature of magnetic recon-\nnection making it independent of plasma resistivity (see\nmore in Kowal et al. 2009, 2012, Eyink, Lazarian, Vish-\nniac 2011, Eyink et al. 2013, Eyink 2015, Lalescu et al.\n2015). Turbulence is being generated by reconnection\nthus inducing fast reconnection in the case when the ini-\ntial state of magnetized plasmas is not turbulent (Beres-\nnyak 2013, Oishi et al. 2015, Lazarian et al. 2015). In\nhighly magnetized plasmas with magnetic energy signifi-\ncantly exceeding thermal energy, the transition to turbu-\nlent reconnection has an explosive character with higher\nlevelofturbulenceincreasingtherateofreconnectionand\nthe higher reconnection increasingthe level of turbulence\n(LV99,Lazarian&Vishniac2009). Ourstudyshowsthat\nthe transition to turbulence is inevitable even if initiallyTurbulent Damping of Alfven waves 13\na significant part of energy leaves the reconnection zone\nin the form of Alfven waves.\n7.3.Implications for galaxy clusters\nIn WOG (see also Ensslin et al. 2011, Pinzke et\nal. 2015) streaming instability suppression was invoked\nto explain the bimodality of the cluster radio emission,\nnamely, the fact that the majority of clusters are radio-\nquiet (Brunetti et al. 2007, 2009, Brown et al. 2011,\nBrunetti & Jones 2014 and ref. therein), and it is only\nthe clusters associated with merger activity that demon-\nstrate radio halos. The authors above suggested a way\nto account for this property by assuming that the CRs\nescape at superAlfvenic speeds and this fast escape turns\noff the radio galaxies (see Ensslin et al. 2011).\nIt was shown in WOG that non-linear Landau damp-\ning (e.g. Felice & Kulsrud 2001) is too weak to inhibit\nwavegrowth,whileturbulentdamping(YL02,FG04)can\nsuppress the instability. This conclusion agrees with our\nanalysis in §6. Moreover, our present study allows us to\nexpressthe resultsin WOGin termsofthe actualparam-\neters of the turbulence in galaxy clusters, e.g. their mag-\nnetization and the turbulence injection scale. This tur-\nbulence is accepted to be superAlfvenic (see Brunetti &\nLazarian 2007, Miniatti & Beresnyak 2015). The Alfven\nMach number of the intracluster medium is expected to\nvary depending on the level of turbulence. In Brunetti\n& Lazarian (2016) the range of MAwas estimated to be\nfrom 3 to 9. The value of lAthus may range from ap-\nproximately 10 pc to 0.3 pc if we assume the injection\nscale of 102pc. Our study dictates that these values of\nlAshould be used in WOG for LMHDthat they employ\nin their study while dealing with the streaming instabil-\nity by strong MHD turbulence. Adopting lA=rL= 1\npc one gets that CR with γ <106interact with strong\nturbulence, as it is assumed in WOG. At the same time\nthe streaming induced by CR with higher γis affected\nby the damping induced by superAlfvenic turbulence in\nhydrodynamic regime.\nOur quantitative insight strengthen the conclusion in\nWOG that the CRs can stream rapidly in the presence\nof superAlfvenic turbulence. However, the consequences\nof this effect for the dynamics of CRs on large scales are\nnot easyto evaluate. Indeed, the escape ofCRs islimited\nnot only by streaming but also by turbulence scattering\nas well as the diffusion of magnetic field lines. The latter\nin superAlfvenic turbulence are entangled on the scale lA\n(e.g. Brunetti&Lazarian2007),whichproducestheran-\ndom walk with the scale of lA. This entails the increase\nof the escape time by a factor ( D/lA)2, whereDis the\nlength of order of Mpc that the particles should cover,\nwhile our estimate of lAis of the order of 1pc. These\nare the complications that should be considered in the\nfuture quantitative models. The process of turning off\nand on can also be explained by merger-induced scenar-\nios of turbulent reacceleration as discussed in detail by\nBrunetti & Jones (2014 and ref. therein). A synthesis\nof the approaches above will be presented in a future\npublication.\n7.4.Streaming of CRs and ionization of molecular\nclouds\nStreaming of CRs into molecular clouds is an interest-\ning process that requires further studies. For instance,in a recent paper by Schlickeiser et al. (2016) stream-\ning instability arising as the CRs penetrate molecular\nclouds was described. This study, however, does not ac-\ncount for a possible suppression of streaming instability\nby ambient turbulence. For superAlfvenic turbulence,\nthe processes of penetration of CR inside the clouds are\ngoing to be modified. As a result one can imagine a situ-\nation in which the coefficient for the ”along the magnetic\nfield” diffusion is larger in the outer turbulent parts of\nthe molecular cloud and smaller at the inner part of the\nmolecular cloud where the streaming instability operates\nand creates waves scattering CRs. In this situation the\ndensity of CRs may potentially be higher in the inte-\nrior of molecular clouds than in the ambient interstellar\nmedium. This can also be relevant to explaining obser-\nvations (see McCall et al. 2003, Le Petit et al. 2004)\nwhich suggest significant variations of the CR density\nin molecular gas. In realistic inhomogeneous interstellar\ngas one can expect regions where streaming instability is\nsuppressed and regions where it operates, creating sig-\nnificant variations of CR diffusion and CR density.\n7.5.Heating of plasmas and launching of winds\nWhile the damping of Alfven waves by turbulence has\nbecome a well accepted process in the field of CR re-\nsearch, in other fields the studies of Alfven damping fre-\nquently ignore the turbulent nature of the magnetized\nplasmas and focus instead of wave steepening and pure\nplasma effects. Therefore we would like to point out that\nour results are applicable to heating of stellar corona by\nAlfven waves and launching of stellar winds by damping\nof Alfven waves (see Suzuki & Inutsuka 2005, Verdini et\nal. 2005, Evans et al. 2009, Vidotto & Jatenco-Pereira\n2010, Verdini et al. 2010, Suzuki 2015). The cascading\nthat we consider results in efficient dissipation of Alfven\nwaves and this dissipation is very robust, i.e. it does not\ndepend on the microphysics of plasma processes. Our\nresults show that in highly magnetized regions of solar\natmosphere with low Alfven Mach number MAAlfven\nwavescanpropagatelargerdistancesthaninregionswith\nlowerMA. This should be accounted in the quantiative\nmodelingofwindlaunchingandplasmaheating. Wenote\nthat the turbulent damping scenario does not require ef-\nficientcouplingbetweenAlfvenandfastmodeturbulence\nthat is assumed in some of the studies (see Cramer et al.\n2014), it does not require having non-linear Alfven waves\noflargeamplitudeeither(cf. Airapetianetal. 2010). For\ninstance, we believe that the turbulent damping can be\nrelevant to explaining the observed ”unexpected” damp-\ning of Alfven waves in the regions above the Sun’s polar\ncoronal holes (Hahn et al. 2012). These and other issues\nshouldbeclarifiedbythefurtherresearchwhichaccounts\nfor the turbulent damping of Alfven waves.\nHeating by waves emitted by various sources can be\nan important source of heating of turbulent plasmas in\ngalaxy clusters. Our study provides a way to quantify\nthe distribution of heating as a function of the distance\nfrom the source.\nHeating by Alfven waves emitted by processes on the\nstellar surface and the processes of launching of stellar\nwinds are intrinsically connected. Alfven waves emitted\nby stars carry momentum. This momentum is deposited\nwith the plasmas as Alfven waves dissipate and this can\nbetheprocessthatlaunchesthewinditselforcontributes14 Lazarian\nto the process of the wind launching, e.g. together with\nthe radiation force (see Suzuki 2011, 2015). The efficient\nturbulent damping of Alfven waves that we have demon-\nstrated in this paper makes this process efficient.\nInterestingly enough, a sufficiently strong flux of\nAlfven waves can induce an instability, resulting in the\nformation of the area of enhanced turbulence damping.\nConsider a train of Alfven waves subject to turbulent\ndamping in magnetized plasmas where a particular re-\ngion has an enhanced level of turbulence. This area\nwill induce stronger damping of Alfven waves. Those,\nas they cascade, will decrease their perpendicular scale\nuntil their parallel and perpendicular scales eventually\nsatisfy Eq. (8) corresponding to the turbulence critical\nbalance. So the cascading Alfven waves will create more\nturbulence which through inverse cascading can produce\nmotions that can cascade the Alfven waves more effi-\nciently.12The turbulence initially gets imbalanced, but\nin realistic turbulent media with density inhomogeneities\nas well as in the presence of parametric instabilities (del\nZanna et al. 2001) the scattered waves become a part of\nthe balanced MHD turbulence.\nThe dissipation of Alfven waves that we discussed\nabovehappensintheglobalsystemofreference. However\nthis does not exhaust all the possibilities for launching\nthe winds. For instance, CR can launch winds getting\ncoupled with the magnetized plasmas through stream-\ning instability (see Recchia et al. 2016). In the latter\ncase, turbulent damping of Alfven waves happens in the\nlocal system of reference. Our work shows that quan-\ntitative models of CR-driven winds (see Ruskowski et\nal. 2016) should account for the spatial change of tur-\nbulent damping arising from the change of MA. We ex-\npect to see near the galactic disk the superAlfvenic CR\nstreamingthat wasinvokedbyRuskowskiet al. (2016)in\ntheir modeling. This, however,should changeto Alfvenic\nstreaming in the galactic halo as the turbulent damping\nis expected to get less efficient there. The consequences\nof this change are difficult to evaluate without detailed\ncalculations.\n8.COMPARISON WITH EARLIER WORKS\nSilimon & Sudan (1989) for their studies of Alfven\nwaves damping used models of MHD turbulence that\nwere not supported by further research. Yan & Lazarian\n(2002) suggested that the streaming instability can be\nsuppressed by turbulence, but did not provide a quan-\ntitative study of the processes. In this situation the\nclosest study to the present one is the quantitative pi-\noneering study of the streaming instability damping by\nstrong Alfvenic turbulence in FG04. In view of the the-\nory provided in this paper this is a particular case of\ndamping. As we identified in our paper, this is the case\nof the Alfven waves that are emitted in the local sys-\n12This turbulence can scatter fluctuations of smaller wavelen gth\nthan the original wavelength of the train. It can also transf er some\npart of the energy to large scales through the inverse cascad ing\nprocess and thus increase the turbulent damping of the origi nal\ntrain of Alfven waves. The difference between the scales at wh ich\nthe damping of waves occurs depends on the angle θbetween the\ndirection of Alfven waves and the mean magnetic field as well a s\nAlfven Mach number of the original turbulence. For sufficient ly\nlargeθor/and sufficiently large MAthe scales of turbulent damp-\ning and the transfer of of the energy into the energy of turbul ent\nmotions can be close making the instability efficient.tem of reference. In the present paper we also identified\nthe other regime of damping, i.e. when the damping of\nAlfven waves emitted by a macroscopic source. The lat-\nter damping happens with respect to the mean magnetic\nfield, i.e. in the global system of reference. The scalings\nof the damping are different in two cases (see Table 1).\nAs for the streaming instability damping, our treat-\nment is different from FG04 and we explain the differ-\nences below. The model of turbulence adopted in FG04\nis based on the assumption that the turbulent energy is\ninjected at the scale LMHDwith velocity VA. This is\na case of transAlfenic turbulence with the caveat that\nFG04 does not associate LMHDdirectly with the injec-\ntion scale, but definesthis scale as the scale at which\nturbulence becomes transAlfvenic . Thus defined, LMHD\ncan be associated with the scale lAfor the transition\nto the MHD regime of transAlfvenic turbulence that we\nquantified in this paper. The extension of the FG04 ap-\nproach to strong subAlvenic turbulence is problematic,\nhowever. No quantitative expressions of this LMHDare\ngiven in FG04 but the paper contains a footnote ”Turbu-\nlence can also be injected at smaller velocities on smaller\nscales, in which case LMHDshould be considered an ex-\ntrapolationbeyondtheactualouterscaleofthecascade.”\nThis extrapolation has not been elaborated and it faces\nconceptual difficulties. Indeed, as we discussed in §2, the\nsubAlfvenicturbulence hastworegimes,weakturbulence\nand strong turbulence. The regime of strongsubAlfvenic\nturbulenceisverydifferent fromthe caseoftransAlfvenic\nturbulence. In transition to strong subAlfvenic turbu-\nlence happens at the scale ttransand the energy is be-\ning injected anisotropically at this scale, which is in con-\ntrast to the isotropic injection for the transAflvenic tur-\nbulence. At scales larger that ltransthe turbulence is not\nany more strong, but follows a weak turbulence cascade\nwith a very different scaling (see Table 1). Thus there is\nno physically justified way ofdefining LMHDfor the sub-\nAlfvenic injection of energy in the system. Nevertheless,\nwhen expressed in terms of the energy dissipation13our\nresults look for strong subAlfvenic turbulence similar to\nthose in FG04, with the difference that the weak cascad-\ning rather than strong cascading rate enter the formulae.\nThis coincidence stems from the fact that in this partic-\nular regime the damping does depend on the dissipation\nrate only and not on the injection scale. This makes\nthe case of strong turbulence special, as in other regimes\nboth the turbulence dissipation rate and the injection\nscale influence the streaming instability damping.\nWe have quantified the streaming instability damping\nfor a variety of different regimes of turbulence, including\n(a)hydro-likesuperAlfvenic, (b) magneticsuperAlfvenic,\n(c) weak turbulence subAlfvenic, (d) strong turbulence\nsubAlfvenic. The case(b) coincideswith theonein FG04\nif we identify LMHDthere with lAin this paper. Damp-\ningofstreaminginstabilityindifferentturbulentregimes\nare important for different astrophysical environments.\nFor instance, we identified weak turbulence as the major\nagent for streaming instability damping in subAlfvenic\n13We find this way of presenting results may sometimes be con-\nfusing, as in many cases the turbulent dissipation is not dir ectly\nmeasurable in view of multiple sources of media heating. On t he\ncontrary, the scale of the turbulence Land the magnetic Mach\nnumber MAcan be observationally measured as we discuss in this\npaper.Turbulent Damping of Alfven waves 15\nturbulence in the Milky Way halo.\nIn terms ofastrophysicalconsequences, we believe that\nthere are no reasonsto claim ofthe catastrophicsuppres-\nsion ofthe streaming instability by turbulence in galactic\nenvironmentsand thereforedonot agreewith the conclu-\nsion in FG04 related to the crisis in explaining observed\ndegree of isotropy of the Milky Way CRs. Indeed, for the\nparameters expected for the turbulence in the galactic\nhalo, we found that the turbulent damping should arise\nfrom the interactionofCRs with weakturbulence, rather\nthan with strong turbulence as it is assumed in FG04.\nThis reduces the damping. We provided arguments sug-\ngesting that the estimate of the turbulence dissipation\nrate in FG04 that is based on the cooling of the hot gas\nis, in fact, an upper limit, which does not constrain the\nactual turbulence dissipation rate. Therefore we do not\nbelieve that the streaminginstability must be suppressed\nby turbulence in the Galactic halo. We also pointed out\nto the self-regulating nature of the competing non-linear\nLandau damping of the CR streaming instability. This\ndamping shuts out as soon as freely streaming particles\nspread into space decreasing the gradients in CR distri-\nbution. Thus we conclude that there is no evidence to\nclaim that the streaming instability is suppressed in the\nMilky Way and therefore it cannot isotropize CRs.\nThe astrophysical implications of our study is not lim-\nitedbyCR isotropization,however. Theexpressionsthat\nwe obtained for the damping of Alfven waves emitted\nby macroscopic sources describe new ways for launching\nstellar/galactic winds and heating cosmic plasmas.\n9.DISCUSSION OF RESULTS\nIn this paper we have presented the calculations of\nAlfven wave damping arising from Alfvenic turbulence.\nWe have dealt both with the case of Alfven waves gener-\nated in the local system of reference, as this is the case\nof Alfvenic waves generated e.g. by streaming instabil-\nity, and with the case of Alfven waves generated by an\nexternal source, e.g. by magnetic perturbations in stellar\natmospheres. We haveprovidedthe study foravarietyof\npossible astrophysicalconditions from superAlfvenic tur-\nbulence, i.e. for MA>1 to subAlfvenic turbulence, i.e.\nforMA<1. We have shown significant changes of wave\ndamping depending on MAand point out the difference\nin Alfven wave damping for waves generated in the lo-\ncal system of reference and launched with respect to the\nmean magnetic field. We have demonstrated that some\nof the paradoxes noted in the literature disappear when\nthe variations of the turbulence magnetization are taken\ninto account. In particular, we have demonstrated that\nthe streaming instability can be present in the galactic\nhalo, allowing isotropization of CRs in the Milky Way.\nThe different regimes of damping that we have consid-\nered in the paper are applicable to various astrophysical\nsettings and should be accounted for within the detailed\nmodeling.\nSome of our results are presented in a concise form in\nTable 1. This table describes both the regimes of turbu-\nlence and the damping rates for Alfven waves that this\nturbulence entails. We see, that, compared to the earlier\nstudy in FG04, a variety of different scalings are present.\nThe Table also describes the ranges of applicability of\ndifferent regimes of turbulent damping. Both the damp-\ning of waves in the local system of reference, correspond-ing to the waves generated by streaming instability and\ndamping of waves emitted by external sources parallel to\nthe mean magnetic field are presented. In particular, Ta-\nble 1 illustrates that the scalings of damping in the two\nsituationsand the rangesof the wavesfor which damping\nis applicable are different (see the two last columns, the\nfirst provides the damping of the streaming instability\nand the range of the CR Larmor radii rLfor which this\ndamping works, the second column is for the damping of\nthe waves launched by the external source parallel to the\nmean magnetic field and the range of the wavelengthsfor\nwhich the damping acts). Other cases, e.g. Alfven waves\nemitted at an arbitrary angle, as well as damping of the\nAlfvenwavesbyouter-scaleturbulencearealsopresented\nin the current paper. We would like to stress the impor-\ntant role of weak turbulence for the suppression of the\nstreaming instability at low MA. While the weak turbu-\nlence has a limited inertial range [ LMA,L], it can affect\nCR streaming for rLin the range [ LM4\nA,LMA]. For in-\nstance, for a moderate MA= 0.1, the weak turbulence\nthatispresentoveronedecaderangeofscalescancontrol\nthe propagation of CRs over 3 decades of energy scales.\nThe range of energies of cosmic rays whose streaming is\naffected by strong subAlfvenic turbulence is significantly\nreduced. Thus, as we discussed in §7.1, for the Milky\nWay galactic halo we expect most of the CRs streaming\nto interact with weak rather than strong turbulence.\nOur study employs a number of simplifying assump-\ntions the importance of which we would like to dis-\ncuss. The first of them is that we can consider Alfvenic\nturbulence separately from the turbulence induced by\nother modes. This issue has been studied theoretically\nand quantified numerically (GS95, Lithwick & Goldreich\n2001, Cho & Lazarian 2002, 2003). The rates of trans-\nfer of energy from Alfven to compressible (fast and slow)\nmodes did not exceed the 10% to 15% in the study of\nCho & Lazarian (2002). This is a reasonable degree of\naccuracy for the approximation we employed here.\nA more serious point is related to the model of turbu-\nlence that is chosen. Our study makes use of the theory\nof balanced MHD turbulence,14i.e. when the flow of\nenergy in the opposite direction is the same, while local-\nized astrophysical sources and sinks of turbulent energy\nmay make Alfvenic turbulence imbalanced, i.e. with the\nflow of energy in one direction exceeding the flow in the\nopposite direction. Solar wind up to 1AU presents an\nexample of such imbalanced turbulence. A few theories\nhave been suggested to account for imbalance (e.g. Lith-\nwick & Goldreich 2007, Beresnyak & Lazarain 2008b,\nChandran 2008, Perez & Boldyrev 2009). Among these\n14While there are still debates regarding the nature of this tu r-\nbulence, we do not believe that there is any evidence in favor of\nthe modifications of GS95 theory that have been suggested so f ar\n(Boldyrev 2005, 2006, Beresnyak & Lazarian 2006, Gogoberid ze\n2007). These attempts were taken in order to explain low reso -\nlution numerical simulations that were getting a power spec trum\nof Alfvenic turbulence that was systematically more shallo w than\nthek−5/3spectrum suggested by GS95. It was shown, however,\nin Beresnyak & Lazarian (2010) that MHD turbulence is less lo cal\nthan hydro and therefore higher resolution numerical simul ations\nare required in order to reveal the actual spectrum of MHD tur bu-\nlence. Later simulations by Beresnyak (2014) supported thi s. In\nview of this we do not find it useful to present our results in te rms\nof modified MHD theories, although it would be very easy to do\nso.16 Lazarian\nTABLE 1\nRegimes of MHD turbulence and turbulent Alfven wave damping\nType Injection Range Spectrum Instability damping rate Wav e damping rate\nof MHD turbulence velocity of scales E(k) and rLrange and wavelength range\nWeak VL< VA[ltrans,L]k−2\n⊥VAM8/3\nA\nr2/3\nLL1/3,LM4\nA< rL< LM AVAM3\nA\nλ,LM3\nA< λ < LM2\nA\nStrong\nsubAlfvenic VL< VA[lmin,ltrans]k−5/3\n⊥VAM2\nA\nr1/2\nLL1/2,l4/3\nmin\nL1/3< rL< LM4\nAVAM2\nA\nλ2/3L1/3,lminMA< λ < LM3\nA\nHydro-like\nsuperAlfvenic VL> VA[lA,L] k−5/3 VAMA\nr2/3\nLL1/3,lA< rL< LVAMA\nλ2/3L1/3,lA< λ < L\nStrong\nsuperAlfvenic VL> VA[lmin,lA]k−5/3\n⊥VAM3/2\nA\nr1/2\nLL1/2,l4/3\nmin\nL1/3MA< rL< lAVAMAsin2/3θ\nλ2/3L1/3,lminsinθ < λ < l A\nLandlminare the injection and perpendicular dissipation scales, re spectively. MA≡δB/B,ltrans=LM2\nAforMA<1 andlA=LM−3\nA.\nforMA<1. For weak Alfvenic turbulence ℓ/bardbldoes not change. The waves are sent parallel to the mean field, θvaries as discussed in §5.3.\ntheories, the one by Beresnyak & Lazarian (2008b) was\nshown to correspond to numerical simulations in Beres-\nnyak & Lazarian (2009). For small imbalances, this the-\nory smoothly transfers to GS95, while large imbalances\nare difficult to create in astrophysical media because of\nreflection of Alfvenic perturbations in realistically com-\npressible and inhomogeneous media. Therefore we be-\nlieve that our present study can provide a reasonable\nestimate for such situations.15\nOur treatment was presented for a single scale of en-\nergyinjection. Inrealastrophysicalsituationssmallscale\nenergyinjection takesplace alongwith the largescaleen-\nergy cascade. The local energy injection may dominate\nthe dynamics of small scale eddies. For instance, locally,\nturbulence in the precursor of supernovae shocks defi-\nnitely dominates the turbulence of the large scale Galac-\ntic cascade. Our treatment can be generalized for such\nsituations.\nThis study has important astrophysical implications.\nNaturally, astrophysical fluids exhibit a variety of turbu-\nlent regimes. The damping of the waves also depends on\nhow the Alfven waves are launched. We have quantified\nthe whole variety of the regimes of Alfven wave damping\nby strong and weak subAlfvenic turbulence, turbulence\nat the injection scale and at the dissipation scale. We\nidentified the difference in damping for the waves emit-\nted by streaming particles and macroscopic astrophysi-\ncal sources. The latter are essential e.g. for launching\nstellar and galactic winds, or for the heating of intra-\ncluster media. Our exploration of the damping of the\nstreaming by superAlfvenic turbulence is the closest to\nthat described in the earlier studies. For a number of\nimplications we have just sketched the possible physics\nand do not get into the quantitative details. This is nat-\nural, as the Alfven wave damping in turbulent media is\nwidely spread in astrophysical settings and this paper is\nfocused on quantifying different regimes of the process\nrather than its numerous astrophysical consequences.\n15When the imbalance of turbulence is important, our treat-\nment can be generalized using the anisotropies of the compon ent\nof Alfvenic turbulence propagating in the opposite directi on to\nwave propagation. For instance, Beresnyak & Lazarian (2008 b)\npredicted different anisotropies for stronger and weaker op positely\npropagating components.We would like to stress that our paper is focused on\nturbulent damping of Alfven waves. We deal with the\nnon-linear Landau damping (see Kulsrud 2005) only to\nthe extent that is required for the purpose of the com-\nparison of the importance of two mechanisms for the\ndamping of the streaming instability. We argue that the\nnon-linear Landau damping acts in self-regulating fash-\nion and therefore it can be in many instances subdom-\ninant compared to the damping by turbulence. Indeed,\nif the streaming instability is suppressed by the afore-\nmentioned mechanism, it allows the spread of the CRs,\nincreasing the scale hight of their distribution. This, in\nits turn, suppresses the non-linear Landau damping. On\nthe basis of this reasoning, we believe that in most astro-\nphysical situations turbulent damping is more important\nthat the non-linear Landau damping of Alfven waves.\n10.SUMMARY\nOur study of Alfven wave damping in MHD turbu-\nlence revealed a variety of damping regimes with impor-\ntant astrophysical consequences. We express our results\nthrough the magnetization of the media, which is given\nby the Alfven Mach number MAand the turbulence in-\njection scale L. Both quantities can be obtained through\nobservations. We quantified the wave damping for differ-\nent regimes of subAlfvenic, superAlfvenic turbulence as\nwell as for damping of Alfven waves by the turbulence at\nthe injection scale. In every case we obtained the range\nof wavelengths for which the damping in the particular\nregime is applicable. This work opens ways for studies\nof the consequences of Alfven wave damping in various\nastrophysicalsettings. Those include launching of stellar\nand galactic winds, heating of the media, control of the\nCR streaming instability, etc. Our results can be briefly\nsummarized as follows:\n•The damping is different if Alfven waves are gen-\nerated in the local system of reference and in the\nglobal system of reference when Alfven waves are\nlaunched with respect to the mean field. The for-\nmer case takes place when e.g. when particles\nsubject to the streaming instability generate waves\nwith respect to the magnetic field that they inter-\nactwith, whilethe lattercasetakesplacee.g. whenTurbulent Damping of Alfven waves 17\nAlfven wavesareinjected intoturbulent mediaby a\nexternal macroscopic source. The structure of the\nAlfven wavefrontsandtheir interactionwith turbu-\nlence in two cases is different and this entails the\ndifference in turbulent damping.\n•ForAlfvenwaveslaunchedbystreaminginstability,\ntheirdampinginsubAlfvenicturbulenceisdifferent\nfor weak and strong regimes of turbulence. In both\ncases, the damping is significantly slowercompared\nto the case of the superAlfvenic turbulence. The\nweak turbulence, while being present over a lim-\nited range of scales, can affects CR streaming over\na significant range of energies. On the contrary,\nthe range of the damping by strong subAlfvenic\nturbulence is significantly reduced. This results,\nfor instance, that the CR streaming in low MAen-\nvironments of galactic halos is mostly affected by\nweak turbulence.\n•The damping of Alfven waves launched with re-\nspecttothemeanmagneticfielddependsonthean-\ngle between the mean field and the direction of the\nwave propagation. In the limiting case of Alfven\nwaves propagating along the mean magnetic field,\nthe damping is different compared to that present\nfor the wavessend along the local direction of mag-netic field by the streaming instability.\n•The study suggests many important astrophysical\nconsequences, many of which are still to be elabo-\nrated. For instance, the efficient damping of Alfven\nwavesgeneratedbyastrophysicalsources,e.g. stars\nand galaxies, provides heating of the media and\nlaunching of winds as Alfven waves deposit both\ntheir energy and momentum into the ambient tur-\nbulent magnetized plasmas. At the same time, we\ndo not expect the damping of streaming instability\nby turbulence in the Milky Way galactic halo to be\nto be strong enough to affect the observed level of\ngalactic CR isotropy.\nI acknowledge the NSF grant AST 1212096, NASA\ngrant NNX14AJ53G and the NSF Center for Magnetic\nSelf Organization (CMSO) as well as a distinguished vis-\nitor PVE/CAPES appointment at the Physics Gradu-\nate Program of the Federal University of Rio Grande\ndo Norte, the INCT INEspao and Physics Graduate\nProgram/UFRN.Productiveand stimulating discussions\nwith Jungyeon Cho and Gianfranco Brunetti as well as\nhelpful comments of the anonymous referee are acknowl-\nedged.\nREFERENCES\nAirapetian, V., Carpenter, K. G., & Ofman, L. 2010, ApJ, 723,\n1210\nArber, T. D., Brady, C. S., & Shelyag, S. 2016, ApJ, 817, 94\nArmstrong, T. P., & Decker, R. B. 1979, Particle Acceleratio n\nMechanisms in Astrophysics, 56, 101\nArmstrong, J. W., Rickett, B. J., & Spangler, S. 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Miyazaki1\n1)WPI Advanced Institute for Materials Research, Tohoku Univ ersity,\nSendai 980-8577, Japan\n2)Department of Applied Physics, Tohoku University, Sendai 9 80-8579,\nJapan\n(Dated: 23 July 2018)\nLaser-inducedmagnetizationprecessional dynamicswasinvestiga tedinepitaxialfilms\nof Mn 3Ge, which is a tetragonal Heusler-like nearly compensated ferrimag net. The\nferromagnetic resonance (FMR) mode was observed, the preces sion frequency for\nwhich exceeded 0.5 THz and originated from the large magnetic anisot ropy field of\napproximately 200 kOe for this ferrimagnet. The effective damping c onstant was\napproximately 0.03. The corresponding effective Landau-Lifshitz c onstant of approx-\nimately 60 Mrad/s and is comparable to those of the similar Mn-Ga mate rials. The\nphysical mechanisms for the Gilbert damping and for the laser-induc ed excitation of\nthe FMR mode were also discussed in terms of the spin-orbit-induced damping and\nthe laser-induced ultrafast modulation of the magnetic anisotropy , respectively.\na)Electronic mail: mizukami@wpi-aimr.tohoku.ac.jp\n1Among the various types of magnetization dynamics, coherent mag netization precession,\ni.e.,ferromagneticresonance(FMR),isthemostfundamentaltype, andplaysamajorrolein\nrf spintronics applications based on spin pumping1–5and the spin-transfer-torque (STT).6,7\nSpin pumping is a phenomenon through which magnetization precessio n generates dc and rf\nspin currents in conductors that are in contact with magnetic films. The spin current can be\nconverted into anelectric voltage throughthe inverse spin-Hall eff ect.8The magnitude of the\nspin current generatedvia spinpumping is proportionaltothe FMRf requency fFMR;4,5thus,\nthe output electric voltage is enhanced with increased fFMR. In the case of STT oscillators\nand diodes, the fFMRvalue for the free layer of a given magnetoresistive devices primarily\ndetermines the frequency range for those devices.9,10An STT oscillator and diode detector\nat a frequency of approximately 40 GHz have already been demonst rated;11–13therefore, one\nof the issues for consideration as regards practical applications is the possibility of increasing\nfFMRto hundreds of GHz or to the THz wave range (0.1-3 THz).11,14\nOnesimple methodthroughwhich fFMRcanbeincreased utilizes magneticmaterials with\nlarge perpendicular magnetic anisotropy fields Heff\nkand small Gilbert damping constants\nα.13,15,16This is because fFMRis proportional to Heff\nkand, also, because the FMR quality\nfactor and critical current of an STT-oscillator are inversely and d irectly proportional to α,\nrespectively. The Heff\nkvalue is determined by the relation Heff\nk= 2Ku/Ms−4πMsfor thin\nfilms, where KuandMsare the perpendicular magnetic anisotropy constant and saturat ion\nmagnetization, respectively. Thus, materials with a small Ms, largeKu, and low αare\nvery favorable; these characteristics are similar to those of mate rials used in the free layers\nof magnetic tunnel junctions integrated in gigabit STT memory applic ation.17We have\npreviously reported that the Mn-Ga metallic compound satisfies the above requirements,\nand that magnetization precession at fFMRof up to 0.28 THz was observed in this case.18\nA couple of research groups have studied magnetization precessio n dynamics in the THz\nwave range for the FePt films with a large Heff\nk, and reported an αvalue that is a factor of\nabout 10 larger than that of Mn-Ga.19–21Thus, it is important to examine whether there are\nmaterialsexhibiting properties similartothoseofMn-Gaexist, inorde r tobetter understand\nthe physics behind this behavior.\nIn this letter, we report on observed magnetization precession at fFMRof more than 0.5\nTHz for an epitaxial film of a Mn 3Ge metallic compound. Also, we discuss the relatively\nsmall observed Gilbert damping. Such THz-wave-range dynamics ca n be investigated by\n2means of a THz wave22or pulse laser. Here, we use the all-optical technique proposed\npreviously;23therefore, the mechanism of laser-induced magnetization preces sion is also dis-\ncussed, because this is not very clearly understood.\nMn3Ge has a tetragonal D0 22structure, and the lattice constants are a= 3.816 and\nc= 7.261˚A in bulk materials [Fig. 1(a)].24,25The Mn atoms occupy at two non-equivalent\nsites in the unit-cell. The magnetic moment of Mn I(∼3.0µB) is anti-parallel to that of\nMnII(∼1.9µB), because of anti-ferromagnetic exchange coupling, and the net magnetic\nmoment is ∼0.8µB/f.u. In other words, this material is a nearly compensated ferrima gnet\nwith a Curie temperature Tcover 800 K.26The tetragonal structure induces a uniaxial\nmagnetic anisotropy, where the c-axis is the easy axis.24The D0 22structure is identical to\nthat of tetragonally-distorted D0 3, which is a class similar to the L2 1Heusler structure;\nthus, D0 22Mn3Ge is also known as a tetragonal Heusler-like compound, as is Mn 3Ga.27\nThe growth of epitaxial films of D0 22Mn3Ge has been reported quite recently, with these\nfilms exhibiting a large Kuand small Ms, similar to Mn-Ga.28–30Note that Mn 3Ge films\nwith a single D0 22phase can be grown for near stoichiometric compositions.29,30Further, an\nextremely large tunnel magnetoresistance is expected in the magn etic tunnel junction with\nMn3Ge electrodes, owing to the fully spin-polarized energy band with ∆ 1symmetry and the\nBloch wave vector parallel to the c-axis at the Fermi level.29,31These properties constitute\nthe qualitative differences between the Mn 3Ge and Mn 3Ga compounds from the material\nperspective.\nAll-optical measurement for the time-resolved magneto-optical K err effect was employed\nusing a standard optical pump-probe setup with a Ti: sapphire laser and a regenerative\namplifier. The wavelength and duration of the laser pulse were appro ximately 800 nm and\n150 fs, respectively, while the pulse repetition rate was 1 kHz. The p ulse laser beam was\ndivided into an intense pump beam and a weaker probe beam; both bea ms weres-polarized.\nThe pump beam was almost perpendicularly incident to the film surface , whereas the angle\nof incidence of the probe beam was ∼6◦with respect to the film normal [Fig. 1(b)].\nBoth laser beams were focused on the film surface and the beam spo ts were overlapped\nspatially. The probe and pump beams had spot sizes with 0.6 and 1.3 mm, respectively.\nThe Kerr rotation angle of the probe beam reflected at the film surf ace was analyzed using\na Wollaston prism and balanced photodiodes. The pump beam intensity was modulated\nby a mechanical chopper at a frequency of 360 Hz. Then, the volta ge output from the\n3FIG. 1. (a) Illustration of D0 22crystal structure unit cell for Mn 3Ge. (b) Diagram showing\ncoordinate system used for optical measurement and ferroma gnetic resonance mode of magnetiza-\ntion precession. The net magnetization (= MII−MI) precesses about the equilibrium angle of\nmagnetization θ, whereMI(MII) is the magnetization vector for the Mn I(MnII) sub-lattice. (b)\nOut-of-plane normalized hysteresis loop of the Kerr rotati on angle φkmeasured for the sample.\nphotodiodes was detected using a lock-in amplifier, as a function of d elay time of the pump-\nprobe laser pulses. The pump pulse fluence was ∼0.6 mJ/cm2. Note that the weakest\npossible fluence was used in order to reduce the temperature incre ase while maintaining the\nsignal-to-noise ratio. A magnetic field Hof 1.95 T with variable direction θHwas applied\nusing an electromagnet [Fig. 1 (b)].\nThec-axis-oriented Mn 3Ge epitaxial films were grown on a single-crystalline (001) MgO\nsubstrate with a Cr seed layer, and were capped with thin MgO/Al lay ers at room tempera-\nture using a sputtering method with a base pressure below 1 ×10−7Pa. The characteristics\nof a 130-nm-thick film with slightly off-stoichiometric composition (74 a t% Mn) deposited\nat 500◦C are reported here, because this sample showed the smallest coer civity (less than\n1 T) and the largest saturation magnetization (117 emu/cm3) of a number of films grown\nwith various thicknesses, compositions, and temperatures. Thes e properties are important\nto obtaining the data of time-resolved Kerr rotation angle φkwith a higher signal-to-noise\nratio, because, as noted above, Mn 3Ge films have a large perpendicular magnetic anisotropy\n4field and a small Kerr rotation angle.30Figure 1(c) displays an out-of-plane hysteresis loop\nofφkobtained for a sample without pump-beam irradiation. The loop is norm alized by the\nsaturation value φk,sat 1.95 T. The light skin depth is considered to be about 30 nm for the\nemployed laser wavelength, so that the φkvalue measured using the setup described above\nwas almost proportional to the out-of-plane component of the ma gnetization Mzwithin the\nlight skin depth depth. The loop shape is consistent with that measur ed using a vibrating\nsample magnetometer, indicating that the film is magnetically homogen eous along the film\nthickness and that value of φk/φk,sapproximates to the Mz/Msvalue.\nFigure 2(a) shows the pump-pulse-induced change in the normalized Kerr rotation angle\n∆φk/φk,s(∆φk=φk−φk,s) as a function of the pump-probe delay time ∆ twith an applied\nmagnetic field Hperpendicular to the film plane. ∆ φk/φk,sdecreases quickly immediately\nafter the pump-laser pulse irradiation, but it rapidly recovers within ∼2.0 ps. This change\nis attributed to the ultrafast reduction and ps restoration of Mswithin the light skin depth\nregion, and is involved in the process of thermal equilibration among t he internal degrees of\nfreedom, i.e., the electron, spin, and lattice systems.32. After the electron system absorbs\nlight energy, the spin temperature increases in the sub-ps timesca le because of the heat\nflow from the electron system, which corresponds to a reduction in Ms. Subsequently, the\nelectron and spin systems are cooled by the dissipation of heat into t he lattices, which have\na high heat capacity. Then, all of the systems reach thermal equilib rium. This process is\nreflected in the ps restoration of Ms. Even after thermal equilibrium among these systems is\nreached, the heat energy remains within the light skin depth region a nd the temperature is\nslightly higher than the initial value. However, this region gradually co ols via the diffusion\nof this heat deeper into the film and substrate over a longer timesca le. Thus, the remaining\nheat causing the increased temperature corresponds to the sma ll reduction of ∆ φk/φk,safter\n∼2.0 ps.\nWith increasing θHfrom out-of-plane to in-plane, a damped oscillation becomes visible\nin the ∆φk/φk,sdata in the 2-12 ps range [Fig. 2(b)]. Additionally, a fast Fourier tran sform\nof this data clearly indicates a single spectrum at a frequency of 0.5- 0.6 THz [Fig. 2(c)].\nThese damped oscillations are attributed to the temporal oscillation ofMz, which reflects\nthe damped magnetization precession,23because the zcomponent of the magnetization\nprecession vector increases with increasing θH. Further, the single spectrum apparent in\nFig. 2(c) indicates that there are no excited standing spin-waves ( such as those observed in\n5thick Ni films), even though the film is thicker than the optical skin de pth.23\nFerrimagnets generally have two magnetization precession modes, i.e., the FMR and\nexchange modes, because of the presence of sub-lattices.33In the FMR mode, sub-lattice\nmagnetization vectors precess while maintaining an anti-parallel dire ction, as illustrated in\nFig. 1(b), such that their frequency is independent of the exchan ge coupling energy between\nthe sub-lattice magnetizations. On the other hand, the sub-lattic e magnetization vectors\nare canted in the exchange mode; therefore, the precession fre quency is proportional to the\nexchange coupling energy between them and is much higher than tha t of the FMR mode.\nAs observed in the case of amorphous ferrimagnets, the FMR mode is preferentially excited\nwhen the pump laser intensity is so weak that the increase in tempera ture is lower than the\nferrimagnet compensation temperature.34No compensation temperature is observed in the\nbulk Mn 3Ge.25,26Also, the temperature increase in this experiment is significantly sma ller\nthanTcbecause the reduction of Msis up to 4 %, as can be seen in Fig. 2(a). Therefore,\nthe observed magnetization precession is attributed to the FMR mo de. Further, as the\nmode excitation is limited to the light skin depth, the amplitude, freque ncy, and etc., for\nthe excited mode are dependent on the film thickness with respect t o the light skin depth.\nThis is because the locally excited magnetization precession propaga tes more deeply into\nthe film as a spin wave in cases where fFMRis in the GHz range.23Note that it is reasonably\nassumed that such a non-local effect is negligible in this study, becau se the timescale of the\ndamped precession discussed here ( ∼1-10 ps) is significantly shorter than that relevant to a\nspin wave with wavelength comparable to the light skin depth ( ∼100 ps).\nThe FMR mode in the THz-wave range is quantitatively examined below. When the ex-\nchangecouplingbetween thesub-latticemagnetizationsissufficient ly strongandthetemper-\nature is well below both Tcand the compensation temperature, the magnetization dynamics\nfor a ferrimagnet can be described using the effective Landau-Lifs hitz-Gilbert equation35\ndm\ndt=−γeffm×/bracketleftbig\nH+Heff\nk(m·z)z/bracketrightbig\n+αeffm×dm\ndt, (1)\nwheremis the unit vector of the net magnetization parallel (anti-parallel) to the magnetiza-\ntion vector MII(MI) for the Mn II(MnI) sub-lattice [Fig. 1(b)]. Here, the spatial change of\nmis negligible, as mentioned above. Heff\nkis the effective value of the perpendicular magnetic\nanisotropyfieldincluding thedemagnetizationfield, even thoughthe demagnetizationfieldis\nnegligibly small for thisferrimagnet (4 πMs= 1.5 kOe). Further, γeffandαeffaretheeffective\n6FIG. 2. Change in Kerr rotation angle ∆ φknormalized by the saturation value φk,sas a function\nof pump-probe delay time ∆ t: (a) for a short time-frame at θH= 0◦and (b) for a relatively long\ntime-frame and different values of θH. The solid curves in (a) and (b) are a visual guide and values\nfitted to the data, respectively. The data in (b) are plotted w ith offsets for clarity. (c) Power\nspectral density as a function of frequency fand magnetic field angle θH.\n7values of the gyromagnetic ratio and the damping constant, respe ctively, which are defined\nasγeff= (MII−MI)/(MII/γII−MI/γI) andαeff= (αIIMII/γII−αIMI/γI)/(MII/γII−MI/γI),\nrespectively, using the gyromagnetic ratio γI(II)and damping constant αI(II)for the sub-\nlattice magnetization of Mn I(II). In the case of Heff\nk≫H,fFMRand the relaxation time of\nthe FMR mode τFMRare derived from Eq. (1) as\nfFMR=γeff/2π/parenleftbig\nHeff\nk+Hz/parenrightbig\n, (2)\n1/τFMR= 2παefffFMR. (3)\nHere,Hzis the normal component of H. Figure 3(a) shows the Hzdependence of the\nprecession frequency fp. This is obtained using the experimental data on the oscillatory\npart of the change in ∆ φk/φk,svia least-square fitting to the damped sinusoidal func-\ntion, ∆φk,p/φk,sexp(−t/τp)sin(2πfp+φp), with an offset approximating the slow change\nof ∆φk/φk,s[solid curves, Fig. 2(b)]. Here, ∆ φk,p/φk,s,τp, andφpare the normalized am-\nplitude, relaxation time, and phase for the oscillatory part of ∆ φk/φk,s, respectively. The\nleast-square fitting of Eq. (2) to the fpvs.Hzdata yields γeff/2π= 2.83 GHz/kOe and\nHeff\nk= 183 kOe [solid line, Fig. 3(a)]. The γeffvalue is close to 2.80 GHz/kOe for the free\nelectron. The value of Heff\nkis equal to the value determined via static measurement (198\nkOe)30within the accepted range of experimental error. Thus, the analy sis confirms that\nthe THz-wave range FMR mode primarily results from the large magne tic anisotropy field in\nthe Mn 3Ge material. The αeffvalues, which are estimated using the relation αeff= 1/2πfpτp\nfollowing Eq. (3), are also plotted in Fig. 3(a). The experimental αeffvalues are indepen-\ndent ofHzwithin the accepted range of experimental error, being in accorda nce with Eq.\n(3); the mean value is 0.03. This value of αefffor D0 22Mn3Ge is slightly larger than the\npreviously reported values for for D0 22Mn2.12Ga (∼0.015) and L1 0Mn1.54Ga (∼0.008).18\nIn the case of metallic magnets, the Gilbert damping at ambient tempe rature is primarily\ncaused by phonon and atomic-disorder scattering for electrons a t the Fermi level in the\nBloch states that are perturbed by the spin-orbit interaction. Th is mechanism, the so-\ncalled Kambersky mechanism,36,37predicts α∝M−1\ns, so that it is more preferable to use\nthe Landau-Lifshitz constants λ(≡αγMs) for discussion of the experimental values of α\nfor different materials. Interestingly, λeff(≡αeffγeffMs) for Mn 3Ge was estimated to be 61\nMrad/s, which is almost identical to the values for D0 22Mn2.12Ga (∼81 Mrad/s) and L1 0\nMn1.54Ga(∼66Mrad/s). The λfortheKamberkymechanism isapproximatelyproportional\n8FIG. 3. (a) Normal component of magnetic field Hdependence on precession frequency fpand\neffective dampingconstant αeffforMn 3Gefilm. (b)Oscillation amplitudeoftheKerrrotation angle\n∆φk,p/φk,scorresponding to the magnetization precession as a functio n of the in-plane component\nofH. The solid line and curve are fit to the data. The dashed line de notes the mean value of αeff.\ntoλ2\nSOD(EF), whereλSOisthespin-orbitinteractionconstant and D(EF)isthetotaldensity\nof states at the Fermi level.37The theoretical values of D(EF) for the above materials are\nroughly identical, because of the similar crystal structures and co nstituent elements, even\nthough the band structures around at the Fermi level differ slight ly, as mentioned at the\nbeginning.18,29Furthermore, the spin-orbit interactions for Ga or Ge, depending on the\natomic number, may not differ significantly. Thus, the difference in αefffor these materials\ncan be understood qualitatively in terms of the Kambersky mechanis m. Further discussion\nbased on additional experiments is required in order to obtain more p recise values for αeff\nand to examine whether other relaxation mechanisms, such as extr insic mechanisms (related\nto the magnetic inhomogeneities), must also be considered.\nFinally, the excitation mechanism of magnetization precession in this s tudy is discussed\nbelow, in the context of a previously proposed scenario for laser-in duced magnetization\n9precession in Ni films.23The initial equilibrium direction of magnetization θis determined\nby thebalance between HandHeff\nk[Fig. 1(b)]. Duringtheperiodinwhich thethree internal\nsystems are not in thermal equilibrium for ∆ t <∼2.0 ps after the pump-laser irradiation\n[Fig. 2(a)], not only the value of Ms, but also the value of the uniaxial magnetic anisotropy,\ni.e.,Heff\nk, is altered. Thus, the equilibrium direction deviates slightly from θand is restored,\nwhich causes magnetization precession. This mechanism may be exam ined by considering\nthe angular dependence of the magnetization precession amplitude . Because the precession\namplitudemaybeproportionaltoanimpulsive torquegeneratedfro mthemodulationof Heff\nk\nin Eq. (1), the torque has the angular dependence |m0×(m0·z)z|, wherem0is the initial\ndirection of the magnetization. Consequently, the z-component of the precession amplitude,\ni.e., ∆φk,p/φk,s, is expressed as ∆ φk,p/φk,s=ζcosθsin2θ∼ζ/parenleftbig\nHx/Heff\nk/parenrightbig2, whereζis the\nproportionalityconstant and Hxisthe in-plane component of H. The experimental values of\n∆φk,p/φk,sare plotted as a function of Hxin Fig. 3(b). The measured data match the above\nrelation, which supports the above-described scenario. Although ζcould be determined via\nthe magnitude and the period of modulation of Heff\nk, it is necessary to consider the ultrafast\ndynamics of the electron, spin, and lattice in the non-equilibrium stat e in order to obtain a\nmore quantitative evaluation;38,39this is beyond the scope of this report.\nIn summary, magnetization precessional dynamics was studied in a D 022Mn3Ge epitaxial\nfilm using an all-optical pump-probe technique. The FMR mode at fFMRup to 0.56 THz\nwas observed, which was caused by the extremely large Heff\nk. A relatively small damping\nconstant of approximately 0.03 was also obtained, and the corresp onding Landau-Lifshitz\nconstant for Mn 3Ge were shown to be almost identical to that for Mn-Ga, being in quali-\ntatively accordance with the prediction of the Kambersky spin-orb it mechanism. 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Bulanov4, 5, 6\n1Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08544, USA\n2National Research Nuclear University MEPhI, Kashirskoe sh. 31, 115409, Moscow, Russia\n3Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Moscow Region 141700, Russia\n4National Institutes for Quantum and Radiological Sciences and Technology,\n8-1-7 Umemidai, Kizugawa, Kyoto 619-0215, Japan\n5Institute of Physics of the Czech Academy of Sciences v.v.i. (FZU),\nNa Slovance 1999/2, 18221, Prague, Czech Republic\n6A. M. Prokhorov Institute of General Physics of the Russian Academy of Sciences, Vavilov Street 38, Moscow, 119991, Russia.\n(Dated: September 10, 2018)\nIn contrast to hydrodynamic vortices, vortices in plasma contain an electric current circulating\naround the center of the vortex, which generates a magnetic \feld localized inside. Using computer\nsimulations, we demonstrate that the magnetic \feld associated with the vortex gives rise to a\nmechanism of dissipation of the vortex pair in a collisionless plasma, leading to fast annihilation\nof the magnetic \feld with its energy transforming into the energy of fast electrons, secondary\nvortices, and plasma waves. Two major contributors to the energy damping of double vortex system,\nnamely, magnetic \feld annihilation and secondary vortex formation, are regulated by the size of the\nvortex with respect to the electron skin depth, which scales with the electron gamma-factor, \re,\nasR=de/\r1=2\ne. Magnetic \feld annihilation appears to be dominant in mildly relativistic vortices,\nwhile for the ultrarelativistic case, secondary vortex formation is the main channel for damping of\nthe initial double vortex system.\nI. INTRODUCTION\nFormation and evolution of localized nonlinear struc-\ntures such as vortices and solitons play a crucial role\nin the physics of continuous media [1, 2]. For instance,\ndrift wave dynamics in tokamak plasmas can be described\nwithin the framework of the Hasegawa-Mima (HM) equa-\ntion [3], which has a well-known point vortex solution.\nThe vortices may a\u000bect energy and particle transport sig-\nni\fcantly [4, 5]. The formation of \fnite-radius relativis-\ntic electron vortex structures associated with quasistatic\nmagnetic \feld generation provides one of the pathways\nfor the electromagnetic \feld energy depletion in laser\nplasmas [6]. The late stage of the vortex evolution re-\nsulting in strong plasma density modulations has been\nrevealed in the experiments [7] using proton radiogra-\nphy. Electron vortex pairs are also observed in simula-\ntions of relativistic shocks, being responsible for electron\nenergization in the upstream region [8]. Understanding\nthe dynamics of vortex structures in plasmas is impor-\ntant for developing the theory of relativistic plasma tur-\nbulence [9]. Relativistic electron vortex dynamics may\nalso be a signi\fcant factor in the late stages of relativis-\ntic Weibel-like instability, which can arise in superstrong\nlaser-plasma interaction [10], as well as in colliding astro-\nphysical \rows of electron-positron plasmas [11].\nIn contrast to hydrodynamical vortices, which are sus-\ntained by \ruids comprised of neutral particles, vortices\nin plasmas are sustained by the rotational motion of\ncharged particles, leading to nonzero circular electric cur-\nrent, which forms a magnetic \feld inside the vortex [12].\nIn the case of small radius vortices, which correspond to\nthe point-vortex solution of the HM equation, the vor-\ntex internal energy is conserved during the interactionprocess. However, in the case of \fnite radius vortices,\nwe expect the \fnite-radius and electromagnetic interac-\ntion e\u000bects to become prominent, leading to a fast vortex\nenergy dissipation with its transformation into the en-\nergy of fast particles. Below, using two dimensional (2D)\nParticle-In-Cell (PIC) simulations with the code REMP\n[13], we demonstrate how pairs of vortices interact be-\nyond the point vortex approximation. We reveal the ef-\nfect of relativistic annihilation of the binary electron vor-\ntices magnetic \feld that leads to vortex pair dampening.\nII. SIMULATION SETUP\nThe simulation parameters are as follows. For clarity,\nwe describe the simulation setup in terms of an arbi-\ntrary spatial scale parameter, \u0015, and then immediately\nrescale the model to the physically relevant units. We\nset a slab of electron plasma (assuming immobile ions)\nwith a constant density gradient along the xaxis, so\nthe electron plasma density equals ne=nmax = 0:1 at\nx= 55\u0015andne=nmax = 1 atx= 95\u0015, with width\n40\u0015and zero temperatures for electrons. We measure\nspatial parameters in \u0015, temporal { in 2 \u0019=! 0=\u0015=c,\ndensities { in n0=me!2\n0=4\u0019e2, electromagnetic \felds\n{ inE0=me!0c=e, wheremeis electron mass, eis\nthe absolute value of electron charge, cis the speed of\nlight in vacuum. For the sake of simplicity, we intro-\nduce circularly symmetric electron vortices. They are\ninitiated by accumulating the localized magnetic \feld\nduring a number of timesteps at the beginning of the\nsimulation [12, 14]. For the simulations presented, elec-\ntron vortices are formed with various maximum mag-\nnetic \felds: Bmax= 0:5;1;2;4;6:5;35 in plasma witharXiv:1708.07803v2 [physics.plasm-ph] 4 Aug 20182\nnmax= 0:16;0:36;0:64;1;4;16, respectively. Hereafter,\nwe will refer to the simulation parameters by the mag-\nnetic \feld amplitude Bmax. The vortex centres are lo-\ncated around points x= 75\u0015andy=\u00004\u0015;4\u0015. We\nchoose our parameters in such a way that the condition\n!2\npe\u001c!2\nBholds [15], so the electrons can be considered\nmagnetized. Here !2\npe= 4\u0019nee2=meis the plasma fre-\nquency and !B=eB=m ecis the electron gyrofrequency.\nThe computational grid is 150 \u0015\u0002120\u0015with 32 nodes\nper\u0015, boundary conditions are periodic. We have also\nqualitively veri\fed the results of our simulations with a\nlarger domain resolution (64 and 128 nodes per \u0015). The\ninitial particle-in-cell number corresponding to the max-\nimum electron density is equal to 100. The total number\nof particles is about 108. The integration timestep is\n0.0155. The total time of the simulations is 500 time\nunits.\nFor the sake of clarity, we further rescale our numeri-\ncal model to physically relevant units appearing from the\nsimple electron vortex model. It can be formulated as fol-\nlows. Let us assume that the electron moves in a circular\norbit around the uniformly distributed immobile and pos-\nitively charged ions. Then, the electric \feld experienced\nby the electron is E= 2\u0019enR , whereRis the radius of\nthe electron vortex and nis the ion density. Assuming\nthe electron to have a speed ve\u0019c, we obtain the mag-\nnetic \feld to be B= 2\u0019enR . Radial force balance for the\nelectron can be written as vepe=R=\u0000eE, which gives\nan expression connecting electron vortex radius and elec-\ntron momentum, R= (pec=2\u0019ne2)1=2\u0019dep2\re. Thus,\nwe \fx\u0015= (4\u00192nmax=n0\u0001mec=pe)1=2R, normalizing all\nspatial quantities to R, temporal frequencies to crossing\nfrequency!cr=c=R, \felds toE0\n0=me!crc=e, densities\n{ ton0\n0=me!2\ncr=4\u0019e2.\nIII. PIC SIMULATION RESULTS AND\nTHEORETICAL ESTIMATES\nIn our simulations, we expect to observe the following\nscenario: \frst, when two vortices are far away from each\nother (>5R), they would be stationary unless we were to\ntake into account the e\u000bects of a \fnite vortex radius. In\nthe latter case, we can expect that the vortices will move\nperpendicularly to the density gradient (parallel to the\ny-axis), due to the conservation of the Ertel's invariant\nI= \n=n, where \n is the vorticity and nis the electron\ndensity [16]. The velocity of such motion is estimated\nas \nR2jrn=nj, which is /c=80 and has turned out to\nbe fairly consistent with the simulation results presented\nbelow. Then, when the vortex interaction becomes signif-\nicant (it scales as K0(j\u0001y=dej) with the vortex separation\n\u0001y,K0is the modi\fed Bessel function of second kind,\nsee, e.g., [4]), we expect the binary vortex to start mov-\ning along the xaxis and possibly follow one of the com-\nplicated trajectories discussed in Ref. [4]. The typical\nvelocities of such motion are Vbin\u00190:2\u00000:5c. Eventu-\nally, the vortex binary tightening until \u0018Rwill lead to\nFIG. 1. Sketch of the binary vortex evolution - z compo-\nnent of the magnetic \feld: approaching each other (t=330),\nformation of the dipole vortex structure (t=414), radiation\nof electromagnetic waves and formation of a dipole magnetic\n\feld structure in the wake of the dipole vortex (t=467), de-\ncay of the dipole vortex into smaller electron vortices, which\nform von Karman vortex rows (t=488), and the magnetic \feld\nannihilation, leading to electron heating (t=501). 2 dewidth\nscale, tightly connected to the annihilation process, is demon-\nstrated.\nthe \fnite-radius e\u000bects coming into play, which are be-\nyond the scope of applicability of the point vortex theory\ndescribed in Ref. [4]. To reveal the \fnite vortex radius\ne\u000bects and the e\u000bects of magnetic interaction we perform\nthe PIC simulations.\nFigure 1 illustrates typical evolution of the Bzcom-\nponent of the magnetic \feld observed during the sim-\nulation (for Bmax= 2). When the binary vortex sys-\ntem is tight enough (i.e. distance between the closest\npoints of the vortices is \u0018de, wherede=c=!peis the\nelectron skin depth, Fig. 1, t=330), the point vortex\napproximation breaks down. The electron currents of\nthe two vortices, both directed along the xaxis in the\nclosest point of approach, attract each other and form\na magnetic-dipole vortex structure (Fig. 1, t=414) [17].\nThe structure observed has an analogue in hydrodynam-3\nics, which is known as the Larichev-Reznik dipole vortex\nsolution [18]. This type of structure is believed to be\nstable in the hydrodynamic case [19]. However, in our\ncase, the magnetic structure moves along the + xdirec-\ntion, losing the majority of its magnetic energy by turn-\ning it into electromagnetic waves (Fig. 1, t=467; Fig.\n3b), accelerated electrons and forming of von Karman-\nlike streets of secondary vortices (Fig. 1, t=488, 501;\nFig. 3b, Fig. 3c), though, secondary vortex formation\ndoes not decrease the total magnetic energy of the system\nsigni\fcantly. The direction of the binary vortex motion\nmay be de\rected from the straight propagation along the\nxaxis, as the binary components disintegrate unequally\non the secondary vortices, and the resulting binary vor-\ntex with unequal components de\rects in the direction of\nthe larger vortex component, in agreement with [4]. The\nrapidly accelerated electrons are a sign of the relativis-\ntic magnetic \feld annihilation. The annihilation of the\nmagnetic \feld was observed in PIC simulations previ-\nously in a di\u000berent geometry [20] between the azimuthal\nmagnetic \felds formed by two parallel laser pulses prop-\nagating in a nonuniform underdense plasma and leads\nto electron heating. Though the overall physics of the\nAmpere's law is the same in both cases, as well as the\nsignature of rapid electron energization, in [20] the dis-\nplacement current arose as a result of the magnetic \felds\nexpanding towards each other due to the negative density\ngradient along the propagation axis of the laser pulses.\nIn our case, the two vortices are pushed towards each\nother by the \fnite-radius e\u000bect of the vortex drift mo-\ntion. Still, in both cases the dynamics of the magnetic\n\felds is guided by the conservation of the Ertel's invari-\nant. The process of secondary vortex formation may be\ncaused by vortex boundary bending, observed in simula-\ntions previously [12]. Secondary vortices are not subject\nto the vortex \flm instability [6], as the \fnite vortex ra-\ndius e\u000bects dominate the motion of the vortices which\nare separated by a few de. The role of the relativistic\ne\u000bects is demonstrated using auxiliary simulations with\nnmax= 0:36 with a large range of Bmaxfrom 0.1 to 2.\nIt was demonstrated that the magnetic \feld damping in\nthe nonrelativistic case is at least three times longer, and\nthe electric \felds coming from the displacement current\nterm in Ampere's law are negligible, see [21].\nA simple model of the magnetic \feld annihilation of\nelectron vortices may be written as follows. The radius\nof a vortex is connected to the electron momentum by\nrelationR=de= (2pe=mec)1=2. Thus, the nonrelativis-\ntic vortices have radius R\u0014deand the ultrarelativis-\ntic vortices have R\u001dde. Ampere's law is generally\nstated asr\u0002B= 4\u0019=c\u0001J+ 1=c\u0001@E=@t. It may\nbe rewritten as an order-of-magnitude estimate, using\njr\u0002Bj\u0019j@B=@yj\u0018jB=dj, wheredis the typical spa-\ntial gradient scale length, jJj\u0019enecfor the limit when\nve\u0018c,j@E=@tj\u0018E=\u001c, where\u001cis the typical temporal\nscale. Finally, it yields d=de=B=(1 +E=! pe\u001c) (B and\nE are dimensionless). Thus, it is clear from this equation\nthat reaching descale (d\u0014de) is necessary for the mag-netic \feld annihilation through the displacement current\nterm (see, e.g., [20, 21]). Thus, the more relativistic the\nvortex is (in terms of pe=mec\u0019\rparameter), the harder\nit is to squeeze the dipole vortex down to a descale. That\nbeing said, large vortices (in terms of descale) are harder\nto damp via the magnetic \feld annihilation.\nLet us compare two types of simulations with the same\nparameters except for the signs of the magnetic \felds in\nthe vortices. Thus, in one case the vortices move to-\nwards each other and interact (Figure 2, blue line), in\nthe other case they move away from each other and do\nnot decay on the timescale of the simulations (Figure\n2, dashed black line). Figure 2 shows the rate of mag-\nnetic energy dissipation in both simulations. Here, we\ncan distinguish at least two mechanisms of vortex dis-\nsipation - slow (dashed lines, dissipation time is larger\nthan 103!\u00001\ncr) and fast (solid lines, typically less or much\nless than 103!\u00001\ncr). The \frst mechanism can probably\nbe attributed to the formation of spiral density waves in\nthe electron plasma, which are seen in the early stages\nof simulations (e.g., see spiral perturbations of electron\ndensity in Fig. 3a and Fig. 4a). In our simulations, this\nmechanism gives us the rate of dissipation which dissi-\npates no more than 20% of the magnetic energy during\nthe simulation time, so it will not impact the character-\nistic lifetime of the electron vortex, or at least will make\na contribution on a longer timescale than the fast dis-\nsipation, which will be discussed below. In turn, fast\nvortex dissipation can destroy the vortex pair on a much\nshorter timescale. Synchrotron losses, in comparison to\nelectromagnetic solitons, are also negligible in the elec-\ntron vortex case [22].\nAs the result of the magnetic energy dissipation, we\nobserve a bunch of electrons being accelerated approx-\nimately in + xdirection, adding up to \u001860mecto the\nelectron momentum in comparison to the maximum elec-\ntron momentum of the stationary electron vortices in the\ncase ofBmax= 35. Figure 4 demonstrates the e\u000bect of\nthe electron acceleration. The energy of electrons is large\nenough for the bunch to escape the plasma region. Ac-\ncording to Figure 2, we see that the more relativistic\nvortices, with larger \r-factors, are harder to annihilate,\nin agreement with our theoretical model. Secondary vor-\ntices, which are more prominent in the simulations with\nhigher\rfactors of the initial vortices, are also more sta-\nble against the magnetic \feld annihilation, which results\nin the saturation of the magnetic \feld energy in the sys-\ntem (see Figure 2, aqua and purple lines).\nIt is also important to note that the immobile ion ap-\nproach is justi\fed only if !pi=!pe\u001c1 and 2\u0019=! piis\ngreater than the total simulation time. Besides, the bi-\nnary vortex motion should be fast enough so we could\nignore the ion motion: Vbin=R\u001d!pi, whereRis the\ntypical radius of the vortex. Otherwise, the binary sys-\ntem of vortices does not move according to the HM equa-\ntion, but they evolve independently [12] until two vortex\nboundaries collide. The e\u000bects of ion inertia on the bi-\nnary vortex system will be considered in a separate paper.4\nFIG. 2. Normalized magnetic \feld energy evolution over time\nfor various cases - drifting single vortices (dashed black line),\nmerging vortices (dashed brown line), dissipating vortices for\nBmax= 1:0 (blue),Bmax= 2:0 (green),Bmax= 4:0 (red),\nBmax= 6:5 (aqua), and Bmax= 35 (purple). In the case of\nsmaller vortices (in terms of R=de) the magnetic \feld anni-\nhilation dominates the vortex damping, in the case of larger\ndensity values - secondary vortex formation mitigates the to-\ntal magnetic energy dissipation.\nThe simulation setup used in the problem, such as a\nplasma density gradient, is implemented in order to con-\nsider the adiabatic switching-on of the vortex interaction\ne\u000bects. Thus, we may observe the same e\u000bect of vor-\ntex damping in homogeneous plasmas when forming tight\nbinary systems of vortices using our numerical scheme.\nHowever, in order to exclude the e\u000bect of the initial gener-\nation process, which inevitably will cause strong coupling\nbetween the vortex pair, and demonstrate the stability of\nsingle electron vortices, we decided to form vortices far\naway from each other, making sure that the vortex gen-\neration process does not impact their interaction and the\nmagnetic \feld energy is almost constant over the sim-\nulation time (for non-interacting vortices). The dashed\nblack line in Figure 2 demonstrates the evolution of the\nmagnetic energy in the single vortex drift case. In gen-\neral, the lifetime of the electron vortex binaries in the\nhomogeneous plasma appears to be longer than in the\nnonzero density gradient case.\nIt is also natural to discuss a system of binary vor-\ntices with the same polarization of magnetic \feld. In\nthe point vortex approximation, they will simply rotate\naround each other in the case of homogeneous plasma [4].\nHowever, it turns out that the \fnite radius vortices are\nsubject to a merger process, which may also lead to mi-\nnor electromagnetic energy dissipation (Figure 2, dashed\nbrown line) via the spiral density wave formation by the\nresulting ellipsoidal vortex [23], which turned out to be\nin principle agreement with the results of the hydrody-\nnamical simulations of the 2D vortex merger process [24].a)\nb)\nc)\nFIG. 3. a) Electron density distribution for t=32 (simulation\nwithBmax= 0:5). Spiral density waves, which possibly cor-\nrespond to the electromagnetic energy dissipation mechanism\non early stages of vortex evolution, are seen; b) Bzcomponent\nof the magnetic \feld for t=592 (simulation with Bmax= 6:5).\nAroundx= 193 and y=\u00008 we observe the emission of the\nelectromagnetic wave. c) Bzcomponent of the magnetic \feld\nfor t=798 (simulation with Bmax= 35:0). The von Karman-\nlike street of secondary vortices is observed in the wake region\nof the dipole vortex.\nIV. CONCLUSIONS\nIn conclusion, we presented the computer simulation\nresults on the interaction of electron vortex binaries.\nThese structures are often seen in 2D PIC simulations\nof various laser-plasma con\fgurations and are crucial for\nunderstanding the superstrong magnetic \feld evolution\nand turbulence in relativistic plasmas. If the binary vor-5\na)\nb)\nFIG. 4. Electron density distribution at a)t=112 and b)t=167\n(Bmax=4 simulation). The annihilation of the magnetic \feld\nleads to the formation of an electron bunch with an energy\nallowing to escape the plasma into vacuum.tex system is tight enough, the point vortex approxima-\ntion breaks down, and the binary vortex is subject to the\nfast annihilation. The vortex annihilation leads to accel-\neration of the electron bunches, which in its turn leads\nto propagating electrostatic waves. In the case of larger\n\rfactor of the initial vortices (i.e., for simulations with\nBmax= 4 and more), we also observe formation of the\nvon Karman-streets of secondary vortices, the motion of\nwhich is stabilized by the drift motion due to the \fnite-\nradius e\u000bects. Mildly relativistic electron vortex pairs\ndamp mainly through the annihilation of the magnetic\n\feld, while ultrarelativistic electron vortex pairs decay\nvia the secondary vortex formation. 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Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: November 24, 2022)\nThe dynamics of a magnetic moment or spin are of high interest to applications in technology.\nDissipation in these systems is therefore of importance for improvement of efficiency of devices,\nsuch as the ones proposed in spintronics. A large spin in a magnetic field is widely assumed to\nbe described by the Landau-Lifshitz-Gilbert (LLG) equation, which includes a phenomenological\nGilbert damping. Here, we couple a large spin to a bath and derive a generic (non-)Ohmic damping\nterm for the low-frequency range using a Caldeira-Leggett model. This leads to a fractional LLG\nequation, where the first-order derivative Gilbert damping is replaced by a fractional derivative of\norders≥0. We show that the parameter scan be determined from a ferromagnetic resonance\nexperiment, where the resonance frequency and linewidth no longer scale linearly with the effective\nfield strength.\nIntroduction. — The magnetization dynamics of mate-\nrials has attracted much interest because of its techno-\nlogical applications in spintronics, such as data storage\nor signal transfer [1–3]. The right-hand rule of magnetic\nforces implies that the basic motion of a magnetic mo-\nment or macrospin Sin a magnetic field Bis periodic\nprecession. However, coupling to its surrounding (e.g.,\nelectrons, phonons, magnons, and impurities) will lead\nto dissipation, which will align SwithB.\nSpintronics-based devices use spin waves to carry sig-\nnals between components [4]. Contrary to electronics,\nwhich use the flow of electrons, the electrons (or holes)\nin spintronics remain stationary and their spin degrees\nof freedom are used for transport. This provides a sig-\nnificant advantage in efficiency, since the resistance of\nmoving particles is potentially much larger than the dis-\nsipation of energy through spins. The spin waves con-\nsist of spins precessing around a magnetic field and they\nare commonly described by the Landau-Lifshitz-Gilbert\n(LLG) equation [5]. This phenomenological description\nalso includes Gilbert damping, which is a term that\nslowly realigns the spins with the magnetic field. Much\neffort is being done to improve the control of spins for\npractical applications [6]. Since efficiency is one of the\nmain motivations to research spintronics, it is important\nto understand exactly what is the dissipation mechanism\nof these spins.\nAlthough the LLG equation was first introduced phe-\nnomenologically, since then it has also been derived from\nmicroscopic quantum models [7, 8]. Quantum dissipation\nis a topic of long debate, since normal Hamiltonians will\nalways have conservation of energy. It can be described,\nfor instance, with a Caldeira-Leggett type model [9–13],\nwhere the Hamiltonian of the system is coupled to a bath\nof harmonic oscillators. These describe not only bosons,\nbut any degree of freedom of an environment in equilib-\nrium. These oscillators can be integrated out, leading toan effective action of the system that is non-local and ac-\ncounts for dissipation. The statistics of the bath is cap-\ntured by the spectral function J(ω), which determines\nthe type of dissipation. For a linear spectral function\n(Ohmic bath), the first-order derivative Gilbert damping\nis retrieved.\nThe spectral function is usually very difficult to calcu-\nlate or measure, so it is often assumed for simplicity that\nthe bath is Ohmic. However, J(ω) can have any contin-\nuous shape. Hence, a high frequency cutoff is commonly\nput in place, which sometimes justifies a linear expan-\nsion. However, a general expansion is that of an sorder\npower-law, where scould be any positive real number.\nA spectral function with such a power-law is called non-\nOhmic, and we refer to sas the “Ohmicness” of the bath.\nIt is known that non-Ohmic baths exist [14–23] and that\nthey can lead to equations of motion that include frac-\ntional derivatives [24–28]. Because fractional derivatives\nare non-local, these systems show non-Markovian dynam-\nics which can be useful to various applications [29–31].\nHere, we show that a macroscopic spin in contact\nwith a non-Ohmic environment leads to a fractional LLG\nequation, where the first derivative Gilbert damping gets\nreplaced by a fractional Liouville derivative. Then, we ex-\nplain how experiments can use ferromagnetic resonance\n(FMR) to determine the Ohmicness of their environ-\nment from resonance frequency and/or linewidth. This\nwill allow experiments to stop using the Ohmic assump-\ntion, and use equations based on measured quantities\ninstead. The same FMR measurements can also be done\nwith anisotropic systems. Aligning anisotropy with the\nmagnetic field may even aid the realization of measure-\nments, as this can help reach the required effective field\nstrengths. In practice, the determination of the type of\nenvironment is challenging, since one needs to measure\nthe coupling strength with everything around the spins.\nHowever, with the experiment proposed here, one canarXiv:2211.12889v1 [cond-mat.mes-hall] 23 Nov 20222\nessentially measure the environment through the spin it-\nself. Therefore, the tools that measure spins can now also\nbe used to determine the environment. This information\nabout the dissipation may lead to improved efficiency,\nstability, and control of applications in technology.\nDerivation of a generalized LLG equation. — We con-\nsider a small ferromagnet that is exposed to an external\nmagnetic field. Our goal is to derive an effective equa-\ntion of motion for the magnetization. For simplicity, we\nmodel the magnetization as one large spin (macrospin)\nˆS. Its Hamiltonian (note that we set /planckover2pi1andkBto\none) reads ˆHs=B·ˆS−KˆS2\nz, where the first term\n(Zeeman) describes the coupling to the external mag-\nnetic fieldB, and the second term accounts for (axial)\nanisotropy of the magnet. However, since a magnet con-\nsists of more than just a magnetization, the macrospin\nwill be in contact with some environment. Following\nthe idea of the Caldeira-Leggett approach [9–13, 32], we\nmodel the environment as a bath of harmonic oscillators,\nˆHb=/summationtext\nαˆp2\nα/2mα+mαω2\nαˆx2\nα/2, where ˆxαand ˆpαare\nposition and momentum operators of the α-th bath oscil-\nlator with mass mαand eigenfrequency ωα>0. Further-\nmore, we assume the coupling between the macrospin and\nthe bath modes to be linear, ˆHc=/summationtext\nαγαˆS·ˆxα, where\nγαis the coupling strength between macrospin and the\nα-th oscillator. Thus, the full Hamiltonian of macrospin\nand environment is given by ˆH=ˆHs+ˆHc+ˆHb.\nNext, we use the Keldysh formalism in its path-integral\nversion [33, 34], which allows us to derive an effective ac-\ntion and, by variation, an effective quasi-classical equa-\ntion of motion for the macrospin. For the path-integral\nrepresentation of the macrospin, we use spin coherent\nstates [34]|g/angbracketright= exp(−iφˆSz) exp(−iθˆSy) exp(−iψˆSz)|↑/angbracketright,\nwhereφ,θ, andψare Euler angles and |↑/angbracketrightis the eigen-\nstate of ˆSzwith the maximal eigenvalue S. Spin co-\nherent states provide an intuitive way to think about\nthe macrospin as a simple vector S=/angbracketleftg|ˆS|g/angbracketright=\nS(sinθcosφ,sinθsinφ,cosθ) with constant length Sand\nthe usual angles for spherical coordinates θandφ. For\nspins, the third Euler angle ψpresents a gauge freedom,\nwhich we fix as in Ref. [35] for the same reasons explained\nthere.\nAfter integrating out the bath degrees of freedom, see\nSup. Mat. [36] for details, we obtain the Keldysh partition\nfunctionZ=/integraltext\nDgexp[iS], with the Keldysh action\nS=/contintegraldisplay\ndt/bracketleftbig\nS˙φ(1−cosθ)−Beff(Sz)·S/bracketrightbig\n−/contintegraldisplay\ndt/contintegraldisplay\ndt/primeS(t)α(t−t/prime)S(t/prime). (1)\nThe first term, called Berry connection, takes the role\nof a kinetic energy for the macrospin; it arises from\nthe time derivative acting on the spin coherent states\n(−i∂t/angbracketleftg|)|g/angbracketright=S˙φ(1−cosθ). The second term is the po-\ntential energy of the macrospin, where we introduced aneffective magnetic field, Beff(Sz) =B−KSzez, given by\nthe external magnetic field and the anisotropy. The third\nterm arises from integrating out the bath and accounts\nfor the effect of the environment onto the macrospin;\nthat is, the kernel function α(t−t/prime) contains informa-\ntion about dissipation and fluctuations. Dissipation is\ndescribed by the retarded and advanced components\nαR/A(ω) =/summationtext\nα(γ2\nα/2mαω2\nα)ω2/[(ω±i0)2−ω2\nα], whereas\nthe effect of fluctuations is included in the Keldysh com-\nponent,αK(ω) = coth(ω/2T) [αR(ω)−αA(ω)]. This is\ndetermined by the fluctuation-dissipation theorem, as we\nassume the bath to be in a high-temperature equilibrium\nstate [33, 34, 37].\nFrom the Keldysh action, Eq. (1), we can now de-\nrive an equation of motion for the macrospin by taking\na variation. More precisely, we can derive quasi-classical\nequations of motion for the classical components of the\nanglesθandφby taking the variation with respect to\ntheir quantum components [38]. The resulting equations\nof motion can be recast into a vector form and lead to a\ngeneralized LLG equation\n˙S(t) =S(t)×/bracketleftbigg\n−Beff[Sz(t)] +/integraldisplayt\n−∞dt/primeα(t−t/prime)S(t/prime) +ξ(t)/bracketrightbigg\n,\n(2)\nwith the dissipation kernel [39] given by\nα(ω) =/integraldisplay∞\n−∞dε\nπεJ(ε)\n(ω+i0)2−ε2, (3)\nwhere we introduced the bath spectral density J(ω) =/summationtext\nα(πγ2\nα/2mαωα)δ(ω−ωα) [33, 36]. The last term in\nEq. (2) contains a stochastic field ξ(t), which describes\nfluctuations (noise) caused by the coupling to the bath;\nthe noise correlator for the components of ξ(t) is given\nby/angbracketleftξm(t)ξn(t/prime)/angbracketright=−2iδmnαK(t−t/prime). Next, to get a\nbetter understanding of the generalized LLG equation,\nwe consider some examples of bath spectral densities.\nFractional Landau-Lifshitz-Gilbert equation. — For the\ngeneralized LLG equation (2), it is natural to ask: In\nwhich case do we recover the standard LLG equation?\nWe can recover it for a specific choice of the bath spectral\ndensityJ(ω), which we introduced in Eq. (3). Roughly\nspeaking,J(ω) describes two things: first, in the delta\nfunctionδ(ω−ωα), it describes at which energies ωα\nthe macrospin can interact with the bath; second, in\nthe prefactor πγ2\nα/2mαωα, it describes how strongly the\nmacrospin can exchange energy with the bath at the fre-\nquencyωα. In our simple model, the bath spectral den-\nsity is a sum over δ-peaks because we assumed excitations\nof the bath oscillators to have an infinite life time. How-\never, also the bath oscillators will have some dissipation\nof their own, such that the δ-peaks will be broadened. If,\nfurthermore, the positions of the bath-oscillator frequen-\nciesωαis dense on the scale of their peak broadening, the\nbath spectral density becomes a continuous function in-\nstead of a collection of δ-peaks. In the following, we focus3\non cases where the bath spectral density is continuous.\nSince the bath only has positive frequencies, we have\nJ(ω≤0) = 0. Even though J(ω) can have any pos-\nitive continuous shape, one might assume that it is an\napproximately linear function at low frequencies; that is,\nJ(ω) =α1ωΘ(ω)Θ(Ωc−ω), (4)\nwhere Θ(ω) = 1 forω > 0 and Θ(ω) = 0 forω < 0 and\nΩcis some large cutoff frequency of the bath such that we\nhaveωsystem/lessmuchT/lessmuchΩc. Reservoirs with such a linear\nspectral density are also known as Ohmic baths. Insert-\ning the Ohmic bath spectral density back into Eq. (3),\nwhile sending Ω c→∞ , we recover the standard LLG\nequation,\n˙S(t) =S(t)×/bracketleftBig\n−Beff[Sz(t)] +α1˙S(t) +ξ(t)/bracketrightBig\n,(5)\nwhere the first term describes the macrospin’s precession\naround the effective magnetic field, the second term—\nknown as Gilbert damping—describes the dissipation of\nthe macrospin’s energy and angular momentum into the\nenvironment, and the third term describes the fluctu-\nations with/angbracketleftξm(t)ξn(t/prime)/angbracketright= 4α1Tδmnδ(t−t/prime), which\nare related to the Gilbert damping by the fluctuation-\ndissipation theorem. Note that the same results can\nbe obtained without a cutoff frequency by introducing\na counter term, which effectively only changes the zero-\nenergy level of the bath, see Sup. Mat. [36] for details.\nThe assumption of an Ohmic bath can sometimes be\njustified, but is often chosen out of convenience, as it is\nusually the simplest bath type to consider. To our knowl-\nedge, there has been little to no experimental verification\nwhether the typical baths of magnetizations in ferromag-\nnets are Ohmic or not. To distinguish between Ohmic\nand non-Ohmic baths, we need to know how the mag-\nnetization dynamics depends on that difference. Hence,\ninstead of the previous assumption of a linear bath spec-\ntral density (Ohmic bath), we now assume that the bath\nspectral density has a power-law behavior at low frequen-\ncies,\nJ(ω) = ˜αsωsΘ(ω)Θ(Ωc−ω), (6)\nwhere we refer to sas Ohmicness parameter [40]. It is\nconvenient to define αs= ˜αs/sin(πs/2) and we should\nnote that the dimension of αsdepends on s. Fors= 1\nwe recover the Ohmic bath. Correspondingly, baths with\ns < 1 are called sub-Ohmic and baths with s > 1 are\ncalled super-Ohmic. For 0 1, the friction decreases\nwith time, so longer movements will be more likely once\nset in motion.\nWithin the fractional LLG equation, we thus see two\nimportant new regimes. For s < 1 (sub-Ohmic), the\nfriction is more likely to relax (localize) the spin (e.g.\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.10.51510\n[ωd-(B 0-KS)]/[α sS(B 0-KS)s][αsS(B 0-KS)s]2sin2θ/Ω2s0.20.40.60.81.1.21.41.61.8\nFIG. 1. A lin-log plot of the amplitude sin2θas a function of\ndriving frequency ωdplotted in dimensionless units for several\nvalues ofs. The resonance peaks change, depending on s.\nThe resonance frequency ωresand linewidth ∆ H/2have been\noverlayed with crosses. The red dashed crosses have been\ncalculated numerically, whereas the black solid crosses are the\nderived results from Eqs. (10) to (12).4\nsub-diffusion) towards the B-field direction. For small\nmovements, the friction could be very small, whereas it\nwould greatly increase for bigger movements. This could\ndescribe a low dissipation stable configuration. For s >\n1 (super-Ohmic), the friction could reduce as the spin\nmoves further, which in other systems is known to cause\nL´ evy-flights or super-diffusion [24, 25, 42]. This might\nlead the system to be less stable, but can potentially also\ngreatly reduce the amount of dissipation for strong signal\ntransfer: In a similar way to the design of fighter-jets,\nunstable systems can be easily changed by small inputs,\nwhich leads to more efficient signal transfer.\nFerromagnetic Resonance. — FMR is the phenomenon\nwhere the spin will follow a constant precession in a ro-\ntating external magnetic field. The angle θfrom thez-\naxis at which it will do so in the steady state will vary\naccording to the driving frequency ωdof the magnetic\nfield. Close to the natural frequency of the precession,\none generally finds a resonance peak [43]. We assume a\nmagnetic field of the form\nBeff(t) =\nΩ cos(ωdt)\nΩ sin(ωdt)\nB0−KSz\n, (9)\nwhere Ω is the strength of the rotating component, and\nwe will neglect thermal noise. We search for a steady\nstate solution of S(t) in the rotating frame where Beff(t)\nis constant. We will assume a small θapproximation\nwhere the ground state is in the positive zdirection, i.e.\n0<Ω/lessmuchB0−KSandαsS/lessmuch(B0−KS)1��s. Then (see\nSup. Mat. [36] for details of the calculations), we find\nthat the resonance occurs at a driving frequency\nωres≈(B0−KS) + (B0−KS)sαsScos/parenleftBigπs\n2/parenrightBig\n.(10)\nIt should be noted that this is different from what was to\nbe expected from any scaling arguments, since the cosine\nterm is completely new compared to previous results [43],\nand it vanishes precisely when s= 1. However, this new\nnon-linear term scales as ( B0−KS)s, which is an easily\ncontrollable parameter. In the limit where B0−KSis\nsmall (resp. large), the linear term will vanish and the\ns-power scaling can be measured for the sub(resp. super)-\nOhmic case. The amplitude at resonance is found to be\nsin2θres≈Ω2\n/bracketleftbig\nαsS(B0−KS)ssin/parenleftbigπs\n2/parenrightbig/bracketrightbig2, (11)\nand the Full Width at Half Maximum (FWHM) linewidth\nis given by\n∆H/2≈2αsS(B0−KS)ssin/parenleftBigπs\n2/parenrightBig\n. (12)\nDepending on the experimental setup, it might be eas-\nier to measure either the resonance location or the width\n0.0 0.5 1.0 1.5 2.001234\nωres/(B0-KS)ΔH/2/[αsS(B0-KS)s]\n0.01 0.10 1 10 1000.0110s0.20.40.60.81.1.21.41.61.8FIG. 2. A plot of the linewidth in Eq. (13) as a function of res-\nonance frequency for several values of s. The inset shows the\nsame plot in a log-log scale, where the slope of the linewidth\nis precisely the Ohmicness sof the bath.\nof the peak. Nevertheless, both will give the opportu-\nnity to see the sscaling inB0−KS. The presence\nof the anisotropy provides a good opportunity to reach\nweak or strong field limits. In fact, the orientation of the\nanisotropy can help to add or subtract from the magnetic\nfield, which should make the required field strengths more\nreachable for experiments. Some setups are more suit-\nable for measuring the width as a function of resonance\nfrequency. When s= 1, this relation can be directly\nderived from Eqs. (10) and (12). However, when s/negationslash= 1,\nthe relation can only be approximated for strong or weak\ndamping. For small αsS, we see that\n∆H/2≈2αsS(ωres)ssin/parenleftBigπs\n2/parenrightBig\n. (13)\nThe resonance peaks have been calculated numerically\nin FIG. 1 in dimensionless values. The red dashed lines\nshow the location of the numerically calculated peak and\nthe FWHM line width. The black solid lines show the\nlocation of the analytically approximated result for the\npeak location and FWHM line width [Eqs. (10) and (12)].\nFor smallαsSand Ω, we see a good agreement be-\ntween the analytical results and the numerical ones, al-\nthough sub-Ohmic seems to match more closely than\nsuper-Ohmic. This could be due to the greater sta-\nbility of sub-Ohmic systems, since the approximations\nmight affect less a stable system. As one might expect\nfrom the thought experiment presented earlier, we can\nsee in FIG. 1 that sub-Ohmic systems require higher,\nmore energetic, driving frequencies to resonate, whereas\nsuper-Ohmic systems already resonate at lower, less en-\nergetic, driving frequencies. In FIG. 2, we provide a plot\nof Eq. (13) to facilitate further comparison with experi-\nments. If the assumption of Gilbert damping was correct,\nall that one would see is a slope of one in the log-log inset.\nConclusion. — By relaxing the Ohmic Gilbert damp-\ning assumption, we have shown that the low-frequency\nregime of magnetization dynamics can be modeled by a5\nfractional LLG equation. This was done by coupling the\nmacrospin to a bath of harmonic oscillators in the frame-\nwork of a Caldeira-Leggett model. The Keldysh formal-\nism was used to compute the out-of-equilibrium dynam-\nics of the spin system. By analyzing an FMR setup, we\nfound ans-power scaling law in the resonance frequency\nand linewidth of the spin, which allows for a new way to\nmeasure the value of s. This means that experiments in\nmagnetization dynamics and spintronics can now avoid\nthe assumption of Gilbert damping and instead measure\nthe Ohmicness of the environment. This could aid in a\nbetter understanding of how to improve efficiency, stabil-\nity, and control of such systems for practical applications.\nAcknowledgments. — This work was supported by the\nNetherlands Organization for Scientific Research (NWO,\nGrant No. 680.92.18.05, C.M.S. and R.C.V.) and (partly)\n(NWO, Grant No. 182.069, T.L. and R.A.D.).\n[1] I. D. Mayergoyz, G. Bertotti, and C. 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Dalir and M. Bashour, Applied Mathematical Sci-\nences 4, 1021 (2010).\n[31] C. Gardiner and P. Zoller, Quantum noise: a handbook of\nMarkovian and non-Markovian quantum stochastic meth-\nods with applications to quantum optics (Springer Science\n& Business Media, 2004).\n[32] A. O. Caldeira and A. J. Leggett, Phys. Rev. A 31, 1059\n(1985).\n[33] A. Kamenev, Field theory of non-equilibrium systems\n(Cambridge University Press, 2011).\n[34] A. Altland and B. D. Simons, Condensed Matter Field\nTheory (Cambridge University Press, 2010), 2nd ed.\n[35] A. Shnirman, Y. Gefen, A. Saha, I. S. Burmistrov, M. N.\nKiselev, and A. Altland, Physical Review Letters 114,\n176806 (2015).\n[36] R. C. Verstraten, T. Ludwig, R. A. Duine, and\nC. Morais Smith, Supplementary material .\n[37] R. Kubo, Reports on Progress in Physics 29, 255 (1966).\n[38] In a straightforward variation with respect to quantum\ncomponents, we would only obtain a noiseless quasi-\nclassical equation of motion because the information\nabout noise (fluctuations) is included in the Keldysh\npart ofα(t−t/prime), which appears in the action only with\neven powers of quantum components. However, there is a\nway [44] that allows us to retain information about noise\nin the quasi-classical equation of motion; see also [33, 34].\nNamely, we perform a Hubbard-Stratonovich transforma-\ntion to linearize the contribution quadratic in quantum\ncomponents. This linearization comes at the cost of in-\ntroducing a new field, which takes the role of noise.\n[39] The dissipation kernel αis closely related to the retarded\nαRand advanced αAcomponents. Namely, it is given by\nα(ω) =−αR(ω)−αA(−ω).\n[40] From a mathematical perspective, any continuous but\nnot smooth function can still be expanded in a power-law\nfor small enough parameters. Hence, the only assumption\nthat we make in this model is that the frequencies in the\nsystem are very small. Then, there will always exist an\n0< s∈Rsuch that this expansion holds. In contrast,6\nthe Ohmic expansion can only be made for smooth con-\ntinuous functions.\n[41] This definition does not have any boundary conditions,\nas they would have to be at −∞ and would dissipate\nbefore reaching a finite time. One can, however, enforce\nboundary conditions by applying a very strong magnetic\nfield for some time such that the spin aligns itself, and\nthen quickly change to the desired field at t= 0.\n[42] A. A. Dubkov, B. Spagnolo, and V. V. Uchaikin, Interna-\ntional Journal of Bifurcation and Chaos 18, 2649 (2008).\n[43] T. Ludwig, I. S. Burmistrov, Y. Gefen, and A. Shnirman,\nPhysical Review Research 2, 023221 (2020).\n[44] A. Schmid, Journal of Low Temperature Physics 49, 609\n(1982).The fractional Landau-Lifshitz-Gilbert equation\nSupplementary Material\nR.C. Verstraten1, T. Ludwig1, R.A. Duine1,2, C. Morais Smith1\n1Institute for Theoretical Physics, Utrecht University,\nPrincetonplein 5, 3584CC Utrecht, The Netherlands\n2Department of Applied Physics, Eindhoven University of Technology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: November 24, 2022)\nCONTENTS\nI. Keldysh microscopic model 1\nA. Hamiltonian 1\nB. Keldysh partition function 2\nC. Quasi-classical equation of motion 4\nD. Generalized Landau-Lifshitz-Gilbert equation 8\nII. Fractional derivative from non-Ohmic spectral function 9\nA. Calculating the effective Greens functions 9\nB. Ohmic spectral function 11\nC. Sub-Ohmic spectral function 13\nD. Super-Ohmic spectral function 15\nE. Comparison Ohmic versus non-Ohmic 17\nIII. FMR powerlaw derivation 18\nA. Ferromagnetic Resonance 18\nB. Resonance frequency and amplitude 20\nC. Calculating the FWHM linewidth 21\nIV. Dimensional analysis 22\nReferences 22\nI. KELDYSH MICROSCOPIC MODEL\nFor pedagogical reasons we start with a microscopic derivation of the usual LLG equation before going into the\nfractional one. In this section, we combine spin coherent states with the Keldysh formalism [1, 2] to derive a stochastic\nLangevin-like equation of motion of a (macro) spin [3].\nA. Hamiltonian\nIn the main text, we introduced a spectral function J(ω) with a cutoff frequency Ω c. This was originally done from\nthe perspective that any spectral function could be expanded to linear order; hence, the model would only be valid\nup to some highest frequency. However, the cutoff is also important for the model to be realistic, since any physical\nspectral function should vanish as ω→∞ . In the main text, we stated that the same results can be obtained by\nintroducing a constant counter term in the Hamiltonian. This is a term which exactly completes the square of the\ncoupling term and the harmonic potential of the bath and can be seen as a normalization of the zero-energy level. If\nwe instead start the model with this counter term and drop the cutoff, we will get a Greens function αct(ω), which is\nprecisely such that the original Greens function can be written as α(ω) =α(0) +αct(ω), i.e., the counter term in the\nHamiltonian removes the zero frequency contribution of the Greens function. This α(ω= 0) generates a term in the\nequation of motion that goes as/integraltext∞\n0d/epsilon1J(/epsilon1)\nπ/epsilon1[S(t)×S(t)]. Since the integral is finite, with a frequency cutoff in J(ω),\nthe entire term is zero due to the cross product. This means that the equation of motion will be identical if we startarXiv:2211.12889v1 [cond-mat.mes-hall] 23 Nov 20222\neither from the regular Hamiltonian with a frequency cutoff, or with a counter term and no cutoff. Here, we choose\nto show the method that includes a counter term, because then we do not need to calculate terms which would have\ncanceled either way.\nThe microscopic system that we describe is a large spin in an external magnetic field, where the spin is linearly\ncoupled to a bath of harmonic oscillators in the same way as in Refs. [4–9]. Therefore, our Hamiltonian has the form\nof a system, coupling, bath, and counter term; H(t) =Hs+Hc+Hb+Hct, where\nHs=B·ˆS−KS2\nz,\nHc=/summationdisplay\nαγαˆS·ˆxα,\nHb=/summationdisplay\nαˆp2\nα\n2mα+mαω2\nα\n2ˆx2\nα,\nHct=/summationdisplay\nαγ2\nα\n2mαω2αˆS2. (1)\nHere,Bis the (effective) magnetic field, ˆSis the spin, Kis thez-axis anisotropy, γαis the coupling strength, and\nαis the index over all harmonic oscillators which have position ˆxα, momentum ˆpα, massmαand natural frequency\nωα. Notice that the counter term is constant, since S2is a conserved quantity, and that we have indeed completed\nthe square, such that\nH(t) =B·ˆS−KS2\nz+/summationdisplay\nαˆp2\nα\n2mα+/bracketleftBigg/radicalbigg\nmαω2α\n2ˆxα+/radicalBigg\nγ2α\n2mαω2αˆS/bracketrightBigg2\n. (2)\nB. Keldysh partition function\nWe will use the Keldysh formalism to derive a quasi-classical equation of motion. Since this is an out-of-equilibrium\nsystem, a common choice would be to use the Lindblad formalism with a master equation [10]. However, Lindblad can\nonly describe Markovian systems, which will not be the case when we introduce a non-Ohmic bath. In the Keldysh\nformalism, one starts with an equilibrium density matrix in the far past (effectively infinite on the relevant time scale).\nThis then gets evolved with the time evolution operator as usual. However, in contrast to ordinary path integrals,\nonce the present has been reached, one evolves back to the infinite past. Since there is infinite time for evolution,\nwe can reach out-of-equilibrium states adiabatically. The benefit of integrating back to the infinite past is that we\nbegin and end with the same in-equilibrium system, which means equilibrium techniques can be used, at the cost of\nhaving both the forward ( O+) and backward ( O−) quantities to take care of. To reach useful results, one can apply\na Keldysh rotation to the classical ( Oc= (O++O−)/2) and quantum ( Oq=O+−O−) components with the added\nnotation/vectorO=/parenleftbigg\nOc\nOq/parenrightbigg\n. To derive a quasi-classical equation of motion, the action can be expanded in all the quantum\ncomponents, after which the Euler-Lagrange equation for the quantum components provides the equation of motion\nin terms of the classical components.\nFIG. 1. Figure extracted from Ref. [2]. The Keldysh contour starts at t=−∞, evolves forward to some time t, and then\nevolves backwards in time to t=−∞.\nTo begin, we write down the Keldysh partition function\nZ= Tr/braceleftbigg\nTKexp/bracketleftbigg\n−i/contintegraldisplay\nKdtH(t)/bracketrightbigg\nρ0/bracerightbigg\n, (3)3\nwhereTKis the Keldysh time ordering, ρ0is the density matrix at t=−∞, and the integral runs over the Keldysh\ncontour, as shown in FIG. 1. After discretizing the Keldysh time integral in the way of FIG. 1, we can rewrite the\ntrace as path-integrals over the spin coherent state |g/angbracketrightand the oscillators |ˆxα/angbracketrightand|ˆpα/angbracketright. This yields\nZ=/integraldisplay\nDg/productdisplay\nα/integraldisplay\nDˆxα/integraldisplay\nDˆpαeiS[g,{ˆxα},{ˆpα}], (4)\nwith the Keldysh action\nS[g,{ˆxα},{ˆpα}] =/contintegraldisplay\nKdt/bracketleftBigg\n(−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2\nz,g\n+/summationdisplay\nα/parenleftBigg\n−γαSg·ˆxα+ˆpα·˙ˆxα−γ2\nαS2\ng\n2mαω2α−ˆp2\nα\n2mα−mαω2\nα\n2ˆx2\nα/parenrightBigg/bracketrightBigg\n, (5)\nwhere we defined Sg=/angbracketleftg|ˆS|g/angbracketright.\nThe continuous path-integral seems to miss the boundary term /angbracketleftˆx1,α,g1|ρ0|ˆx2N,α,g2N/angbracketright/angbracketleftˆp2N,α|ˆx2N,α/angbracketright, but it is\nincluded in the Keldysh contour, as it connects the beginning and final contour time at t=−∞; see Ref. [2].\nNow, we will integrate out the bath degrees of freedom, beginning by completing the square and performing the\nGaussian integral over ˆpα. The Gaussian contribution in ˆpαwill act as a constant prefactor, so it will drop out of\nany calculation of an observable due to the normalization. Hence, we can effectively set it to one to find\n/integraldisplay\nDˆpαexp/bracketleftbigg\n−i/contintegraldisplay\nKdt/parenleftbiggˆp2\nα\n2mα−ˆpα·˙ˆxα/parenrightbigg/bracketrightbigg\n= exp/bracketleftbigg\ni/contintegraldisplay\nKdt/parenleftBig\n−mα\n2ˆxα∂2\ntˆxα/parenrightBig/bracketrightbigg\n, (6)\nwhere we also did a partial integration in ˆxα. Next we will perform a similar approach for the positions, but it is\nuseful to apply the Keldysh rotation first. Note that we can directly rewrite the integral over the Keldysh contour as\na regular time integral over the quantum components. However, one must still rewrite the contents of the integral in\nterms of the quantum and classical parts of the variables, since the Keldysh rotation does not immediately work for\nproducts. The action can first be written as\niS[g,{ˆxα}] =i/integraldisplay\ndt/parenleftBig\n[(−i∂t/angbracketleftg|)|g/angbracketright]q−[B·Sg]q+K/bracketleftbig\nS2\nz,g/bracketrightbigq\n−/summationdisplay\nα/braceleftBigg\n[γαSg·ˆxα]q+γ2\nα[S2\ng]q\n2mαω2α+/bracketleftBigmα\n2ˆxα/parenleftbig\n∂2\nt+ω2\nα/parenrightbigˆxα/bracketrightBigq/bracerightBigg/parenrightBigg\n. (7)\nWe can then derive that\n−[γαSg·ˆxα]q=−γα/bracketleftbig\nS+\ng·ˆx+\nα−S−\ng·ˆx−\nα/bracketrightbig\n=−γα/bracketleftbigg/parenleftbigg\nSc\ng+1\n2Sq\ng/parenrightbigg\n·/parenleftbigg\nˆxc\nα+1\n2ˆxq\nα/parenrightbigg\n−/parenleftbigg\nSc\ng−1\n2Sq\ng/parenrightbigg\n·/parenleftbigg\nˆxc\nα−1\n2ˆxq\nα/parenrightbigg/bracketrightbigg\n=−γα/bracketleftbig\nSc\ngˆxq\nα+Sq\ngˆxc\nα/bracketrightbig\n=−γα/bracketleftbigg/parenleftbigSc\ngSq\ng/parenrightbig\nτx/parenleftbigg\nˆxc\nα\nˆxq\nα/parenrightbigg/bracketrightbigg\n, (8)\nwhere we introduced τx=/parenleftbigg\n0 1\n1 0/parenrightbigg\nin the Keldysh (classical, quantum) space represented by an upper index candq\nrespectively. Next, we want to derive a similar form for the part of the action that is quadratic in ˆxα. Since these are\nharmonic oscillators in equilibrium, we can refer the reader to Ref. [2], noting that a unit mass was used there, and\nconclude that\n/bracketleftBig\n−mα\n2ˆxα/parenleftbig\n∂2\nt+ω2\nα/parenrightbigˆxα/bracketrightBigq\n=/parenleftbigˆxc\nαˆxq\nα/parenrightbig/parenleftBigg\n0/bracketleftbig\nG−1\nα/bracketrightbigA\n/bracketleftbig\nG−1\nα/bracketrightbigR/bracketleftbig\nG−1\nα/bracketrightbigK/parenrightBigg/parenleftbigg\nˆxc\nα\nˆxq\nα/parenrightbigg\n, (9)\nwhere the retarded and advanced Greens functions read\n[G−1\nα]R/A(t−t/prime) =δ(t−t/prime)mα\n2[(i∂t±i0)2−ω2\nα]. (10)\nThe±i0 is introduced because we need an infinitesimal amount of dissipation on the bath for it to remain in equilibrium\nand the sign is tied to causality. This is because there is also an infinitesimal amount of energy transfer from the4\nmacroscopic spin to each of the oscillators. This results in an extra first-order derivative term, which is found by\nmultiplying out the square with i0. One might want to set these terms to zero immediately, but as it turns out,\nthese are very important limits, which shift away poles from integrals that we need to compute later. Once that is\ndone, the limits are no longer important for the final result, and they may finally be put to zero. Since the bath is in\nequilibrium, we can use the fluctuation dissipation theorem to compute the Keldysh component using\nGK\nα(ω) =/bracketleftbig\nGR\nα(ω)−GA\nα(ω)/bracketrightbig\ncoth/parenleftBigω\n2T/parenrightBig\n.\nThe ˆxdependent part of the action is now given by\niSX=i/integraldisplay\ndt/bracketleftbigg\n−γα/parenleftbigSc\ngSq\ng/parenrightbig\nτx/parenleftbigg\nˆxc\nα\nˆxq\nα/parenrightbigg\n+/parenleftbigˆxc\nαˆxq\nα/parenrightbig\nG−1\nα/parenleftbigg\nˆxc\nα\nˆxq\nα/parenrightbigg/bracketrightbigg\n, (11)\nwhich we can compute by completing the square to find\niSX=i/integraldisplay\ndt/bracketleftbigg\n−γ2\nα\n4/vectorST\ng/parenleftbigg\n0GA\nα\nGR\nαGK\nα/parenrightbigg\n/vectorSg/bracketrightbigg\n. (12)\nBefore we write down the final effective action, we also have to rewrite the quadratic part in Sin a similar vector\nform, which is\n−γ2\nα[S2\ng]q\n2mαω2α=−γ2\nα\n2mαω2α/parenleftbigSc\ngSq\ng/parenrightbig\nτx/parenleftbiggSc\ng\nSq\ng/parenrightbigg\n. (13)\nCombining everything together, we find that the partition function of the system is given by Z=/integraltext\nDg eiS[g], with\nthe effective action\niS[g] =i/integraldisplay\ndt/braceleftBigg\n/bracketleftbig\n(−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2\nz,g/bracketrightbigq−/integraldisplay\ndt/prime/vectorST\ng(t)/parenleftbigg\n0αA\nαRαK/parenrightbigg\n(t−t/prime)/vectorSg(t/prime)/bracerightBigg\n, (14)\nwhereαA/R(t−t/prime) =/summationtext\nα/parenleftBig\nγ2\nα\n4GA/R\nα(t−t/prime) +γ2\nα\n2mαω2αδ(t−t/prime)/parenrightBig\nandαK(t−t/prime) =/summationtext\nαγ2\nα\n4GK\nα(t−t/prime).\nC. Quasi-classical equation of motion\nIn the quasi-classical regime, we are interested in solutions where the quantum components ( q) are small compared\nto the classical components ( c). We can thus neglect terms of O[(q)3], but we must be careful with ( q)2. We can\nuse a Hubbard-Stratonovich transformation to convert ( q)2terms into an expression with just ( q), but with a new\nfieldξadded to the path integral [3]. The action will then contain only terms of linear order in ( q), which means the\npartition function has the form Z∼/integraltext\nDcDq exp[if(c)q] =/integraltext\nDc1\n2πδ[f(c)]. Hence, only solutions that satisfy f(c) = 0\ncontribute to the path integral. Within that subset, we want to minimize the action.\nIn order to derive the equation of motion of the system, we must understand the relation between |g/angbracketrightandSg=\n/angbracketleftg|S|g/angbracketright. Using the Euler angle representation [1], we can describe |g/angbracketrightas\n|g/angbracketright=g|↑/angbracketright=e−iφSze−iθSye−iψSz|↑/angbracketright=e−iφSze−iθSy|↑/angbracketrighte−iψS(15)\nand similarly\n/angbracketleftg|=eiψS/angbracketleft↑|eiθSyeiφSz. (16)\nNote that the ψangle is now independent of the quantum state |↑/angbracketright, since this angle is describing the rotation of the\nvector pointing in the spin direction, which is symmetric. Hence, this will yield a gauge symmetry.\nUsing the Euler angle representation in the first terms of Eq. (14), we see that\n(−i∂t/angbracketleftg|)|g/angbracketright=/parenleftBig\n˙ψSeiψS/angbracketleft↑|eiθSyeiφSz+eiψS/angbracketleft↑|˙θSyeiθSyeiφSz+eiψS/angbracketleft↑|eiθSy˙φSzeiφSz/parenrightBig\ne−iφSze−iθSy|↑/angbracketrighte−iψS\n=˙ψS+˙θ/angbracketleft↑|Sy|↑/angbracketright+˙φ/angbracketleft↑|eiθSySze−iθSy|↑/angbracketright. (17)5\nWe note that/angbracketleft↑|Sy|↑/angbracketright= 0, while the last term includes a rotation of the spin up state by θdegrees in the ydirection\nand then measures the Szcomponent of that state, which is Scosθ. Hence,\n(−i∂t/angbracketleftg|)|g/angbracketright=˙ψS+˙φScosθ. (18)\nWe now define a new variable χsuch thatψ=χ−φ, which results in\n(−i∂t/angbracketleftg|)|g/angbracketright= ˙χS−˙φ(1−cosθ)S. (19)\nMaking use of the Euler angle representation, we also see that\nSg=S\nsinθcosφ\nsinθsinφ\ncosθ\n. (20)\nWe see thatB·Sg=S[Bxsinθcosφ+Bysinθsinφ+Bzcosθ]. Similarly, KS2\nz,g=KS2cos2θ. Now, we still have to\ncompute the quantum parts of these quantities. We first note that\nSq\ng,x/S= [sinθcosφ]q= 2 cosθcsinθq\n2cosφccosφq\n2−2 sinθccosθq\n2sinφcsinφq\n2;\nSq\ng,y/S= [sinθsinφ]q= 2 sinθccosθq\n2cosφcsinφq\n2+ 2 cosθcsinθq\n2sinφccosφq\n2;\nSq\ng,z/S= [cosθ]q=−2 sinθcsinθq\n2;\n[cos2θ]q=−2 sinθccosθcsinθq. (21)\nNext, we will choose a gauge for χas in Ref. [11], which is\n˙χc=˙φc(1−cosθc)\nχq=φq(1−cosθc). (22)\nDefiningp= 1−cosθ, we see that [(−i∂t/angbracketleftg|)|g/angbracketright]q=/bracketleftBig\n˙χS−˙φpS/bracketrightBig\nq=S/bracketleftBig\nφq˙pc−˙φcpq/bracketrightBig\n. Now,pq= 2 sinθcsinθq\n2and\n˙pc=˙θcsinθccosθq\n2+˙θq\n2cosθcsinθq\n2,which leads to\n[(−i∂t/angbracketleftg|)|g/angbracketright]q=S/bracketleftBig\nφq˙pc−˙φcpq/bracketrightBig\n=S/bracketleftBigg\nφq˙θcsinθccosθq\n2+φq˙θq\n2cosθcsinθq\n2−2˙φcsinθcsinθq\n2/bracketrightBigg\n. (23)\nNext, we want to express B·Sq\ngin terms of Euler angles. We see that\nB·Sq\ng=S[Bxsinθcosφ+Bysinθsinφ+Bzcosθ]q\n= 2S/bracketleftBig\nBx/parenleftbigg\ncosθcsinθq\n2cosφccosφq\n2−sinθccosθq\n2sinφcsinφq\n2/parenrightbigg\n+By/parenleftbigg\nsinθccosθq\n2cosφcsinφq\n2+ cosθcsinθq\n2sinφccosφq\n2/parenrightbigg\n−Bzsinθcsinθq\n2/bracketrightBig\n, (24)\nwhere we used the results from Eq. (21). Similarly, we have\nK/bracketleftbig\nS2\nz,g/bracketrightbigq=KS2[cos2θ]q=−2KS2sinθccosθcsinθq. (25)\nCombining these results, we conclude that\n/bracketleftbig\n(−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2\nz,g/bracketrightbigq=S/bracketleftBigg\nφq˙θcsinθccosθq\n2+φq˙θq\n2cosθcsinθq\n2−2(−Bz+KScosθc+˙φc) sinθcsinθq\n2\n−2Bx/parenleftbigg\ncosθcsinθq\n2cosφccosφq\n2−sinθccosθq\n2sinφcsinφq\n2/parenrightbigg\n−By/parenleftbigg\nsinθccosθq\n2cosφcsinφq\n2+ cosθcsinθq\n2sinφccosφq\n2/parenrightbigg/bracketrightBigg\n. (26)6\nRemark that this expression only contains odd powers of ( q), so that we can neglect all higher-order terms to get\n/bracketleftBig\n(−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2\nz,g/bracketrightBigq\n=S/bracketleftBig\n−θqsinθc(−Bz+KScosθc+˙φc)\n−θqcosθc(Bxcosφc+Bysinφc) +φqsinθc(˙θc+Bxsinφc−Bycosφc)/bracketrightBig\n. (27)\nNow, we focus on the part of the action in Eq. (14) that comes from the bath, given by\niSb[g] =−i/integraldisplay\ndt/integraldisplay\ndt/prime/vectorST\ng(t)/parenleftbigg\n0αA\nαRαK/parenrightbigg\n(t−t/prime)/vectorSg(t/prime). (28)\nLet us first consider what Sq\ngandSc\ngare in terms of φandθ. By performing some trigonometric operations on each\nof the components, we find that\nSc\ng=S\nsinθccosθq\n2cosφccosφq\n2−cosθcsinθq\n2sinφcsinφq\n2\nsinθccosθq\n2sinφccosφq\n2+ cosθcsinθq\n2cosφcsinφq\n2\ncosθccosθq\n2\n (29)\nand\nSq\ng= 2S\ncosθcsinθq\n2cosφccosφq\n2−sinθccosθq\n2sinφcsinφq\n2\nsinθccosθq\n2cosφcsinφq\n2+ cosθcsinθq\n2sinφccosφq\n2\n−sinθcsinθq\n2\n. (30)\nBy expanding in the quantum components of Sc\ngandSq\ng, we see that\nSc\ng= (q)0+O/parenleftbig\n(q)2/parenrightbig\n,\nSq\ng= (q)1+O/parenleftbig\n(q)3/parenrightbig\n.\nSince the action only contains terms with at least one Sq\ng, we know that the only way to obtain a term of order ( q)2\nis from (Sq\ng)2. Hence, we may neglect all terms beyond linear ( q) inS(c/q)\ng in the quasi-classical regime. This results\nin\nSc\ng=S\nsinθccosφc\nsinθcsinφc\ncosθc\n, (31)\nSq\ng=S\nθqcosθccosφc−φqsinθcsinφc\nφqsinθccosφc+θqcosθcsinφc\n−θqsinθc\n. (32)\nA useful remark for later is that this shows that\nSq\ng=θq∂\n∂θcSc\ng+φq∂\n∂φcSc\ng. (33)\nGoing back to iSb[g], we can rewrite this as a convolution, in the sense that\niSb[g] =−i/integraldisplay\ndt/bracketleftbig\nSc\ng(t)·/parenleftbig\nαA∗Sq\ng/parenrightbig\n(t) +Sq\ng(t)·/parenleftbig\nαR∗Sc\ng/parenrightbig\n(t) +Sq\ng(t)·/parenleftbig\nαK∗Sq\ng/parenrightbig\n(t)/bracketrightbig\n, (34)\nwhere (f∗g)(t) =/integraltext∞\n−∞dt/primef(t−t/prime)g(t/prime). We see that the first two terms contain precisely one quantum component,\nbut the last term has two quantum components. When writing down the Euler-Lagrange equation of motion, it is\nimportant to realize that the convolution operation will act as if it is a simple multiplication, since the convolution\nobeys\nd\ndx(f(x)∗g)(t) =/parenleftbiggdf\ndx∗g/parenrightbigg\n(t). (35)7\nWe now concentrate on the ( q)2part of this action, for which we would like to use a Hubbard-Stratonovich transfor-\nmation in order to reduce this to linear in ( q). Recall that a Hubbard-Stratonovich transformation is given by\nexp/bracketleftBig\n−a\n2x2/bracketrightBig\n=/radicalbigg\n1\n2πa/integraldisplay\nDξexp/bracketleftbigg\n−ξ2\n2a−ixξ/bracketrightbigg\n. (36)\nHowever, we see that our action does not contain any purely quadratic terms, but rather a Greens functional shape\nasSq\ng(t)αK(t−t/prime)Sq\ng(t/prime). Hence, to use a Hubbard-Stratonovich like transformation, we must derive it from a Greens\nfunction exponential, similarly to Ref. [3]. Assuming that this is renormalizable and that αKcan be rewritten into a\ndistribution, we have\n1 =/integraldisplay\nDξexp/bracketleftbigg\n−1\n2/integraldisplay\ndt/integraldisplay\ndt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)/bracketrightbigg\n=/integraldisplay\nDξexp/bracketleftbigg\n−1\n2/integraldisplay\ndt/integraldisplay\ndt/prime/parenleftbigg\nξ(t)−2/integraldisplay\ndt/prime/primeSq\ng(t/prime/prime)αK(t/prime/prime−t)/parenrightbigg\n[−2iαK]−1(t−t/prime)/parenleftbigg\nξ(t/prime)−2/integraldisplay\ndt/prime/prime/primeαK(t/prime−t/prime/prime/prime)Sq\ng(t/prime/prime/prime)/parenrightbigg/bracketrightbigg\n=/integraldisplay\nDξexp/bracketleftbigg\n−1\n2/integraldisplay\ndt/integraldisplay\ndt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)\n−iSq\ng(t)δ(t−t/prime)ξ(t/prime)−iξ(t)δ(t−t/prime)Sq\ng(t/prime)−2iSq\ng(t)αK(t−t/prime)Sq\ng(t/prime)/bracketrightbig\n=/integraldisplay\nDξexp/bracketleftbigg\n−1\n2/integraldisplay\ndt/integraldisplay\ndt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)−2iSq\ng(t)δ(t−t/prime)ξ(t/prime)−2iSq\ng(t)αK(t−t/prime)Sq\ng(t/prime)/bracketrightbigg\n,\nwhere we used that/integraltext\ndt/primeαK(t−t/prime)[αK]−1(t/prime−t/prime/prime) =δ(t−t/prime/prime) and that 2 iαKis positive real. Therefore, we find that\nexp/bracketleftbigg\n−i/integraldisplay\ndt/integraldisplay\ndt/primeSq\ng(t)αK(t−t/prime)Sq\ng(t/prime)/bracketrightbigg\n=/integraldisplay\nDξexp/bracketleftbigg\n−1\n2/integraldisplay\ndt/integraldisplay\ndt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)/bracketrightbigg\n·exp/bracketleftbigg\ni/integraldisplay\ndtSq\ng(t)ξ(t)/bracketrightbigg\n. (37)\nThe double integral in the first exponential signifies the statistical properties of ξ. For instance, if αKis delta-like,\nthenξwould have Gaussian statistics (e.g. white noise), but in general we will have time correlated noise defined by\nαK[3], such that\n/angbracketleftξ(t)ξ(t/prime)/angbracketright=−2iαK(t−t/prime). (38)\nSince there is no gdependence in the double ξexponential, we will leave it out of S[g] and only remember these\nstatistics. Our partition function is then given by\nZ=/integraldisplay\nDξexp (iSn[ξ])/integraldisplay\nDgexp (iSsc[g,ξ]), (39)\nwhere the noise action is given by\niSn[ξ] =−1\n2/integraldisplay\ndt/integraldisplay\ndt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime) (40)\nand the semi-classical action is given by\niSsc[g,ξ] =i/integraldisplay\ndtS/bracketleftBig\n−θqsinθc(−Bz+KScosθc+˙φc)−θqcosθc(Bxcosφc+Bysinφc)\n+φqsinθc(˙θc+Bxsinφc−Bycosφc)/bracketrightBig\n+i/integraldisplay\ndt/bracketleftbig\nξ(t)Sq\ng(t)/bracketrightbig\n−i/integraldisplay\ndt/bracketleftbig\nSc\ng(t)·/parenleftbig\nαA∗Sq\ng/parenrightbig\n(t) +Sq\ng(t)·/parenleftbig\nαR∗Sc\ng/parenrightbig\n(t)/bracketrightbig\n, (41)\nwhereSc\ng(t) andSq\ng(t) include only up to first-order corrections in quantum components. Assuming that αA/Rcan\nbe written in terms of distributions, we can define the distribution αdiss(t) =−αR(t)−αA(−t) and rewrite the\nsemi-classical action as\niSsc[g,ξ] =i/integraldisplay\ndtS/bracketleftBig\n−θqsinθc(−Bz+KScosθc+˙φc)−θqcosθc(Bxcosφc+Bysinφc)\n+φqsinθc(˙θc+Bxsinφc−Bycosφc)/bracketrightBig\n+i/integraldisplay\ndt/bracketleftbig/parenleftbig\nαdiss∗Sc\ng/parenrightbig\n(t) +ξ(t)/bracketrightbig\nSq\ng(t). (42)8\nRecall that, using the Euler angles, we have/integraltext\nDg=/integraltext\nDθDφ sin(θ). Technically, the factor of sin( θ) would end up\nin the action. However, since one could define ρ= cos(θ) as a new variable in order to avoid this, we know that this\nterm is not relevant to the physics. Hence, we can disregard it.\nSince all terms in iSsc[g,ξ] are either linear in θqorφq, we find two Euler-Lagrange equations of the form\nδLsc\nδθq= 0 andδLsc\nδφq= 0. (43)\nRemembering Eq. (33), we see thatδSq\ng(t)\nδθq=δSc\ng(t)\nδθcandδSq\ng(t)\nδφq=δSc\ng(t)\nδφc. Hence, the e.o.m. can be rearranged to yield\n˙φc=1\nSsinθc/bracketleftbig\n−B(Sc\nz) +/parenleftbig\nαdiss∗Sc\ng/parenrightbig\n(t) +ξ(t)/bracketrightbig\n·δSc\ng(t)\nδθc(44)\nand\n˙θc=−1\nSsinθc/bracketleftbig\n−B(Sc\nz) +/parenleftbig\nαdiss∗Sc\ng/parenrightbig\n(t) +ξ(t)/bracketrightbig\n·δSc\ng(t)\nδφc, (45)\nwhereB(Sc\nz) =\nBx\nBy\nBz−KSc\nz\n.\nD. Generalized Landau-Lifshitz-Gilbert equation\nWe want to show that the equations found by the microscopic model are in fact precisely of the LLG form. For\nthis, we will have to start from the LLG equation, and introduce the same two Euler angles θandφfor the spin, and\nshow that this gives rise to the same set of equations as previously deduced.\nWe begin with the generalized LLG equation\n˙S(t) =S(t)×[−B(Sz) + (αdiss∗S) (t) +ξ(t)], (46)\nwhereαdiss(t) =−αR(t)−αA(−t),/angbracketleftξ(t)ξ(t/prime)/angbracketright=−2iαK(t−t/prime) andB(Sz) = (Bx,By,Bz−KSz)T. Since the velocity\nofSis always perpendicular to S, we know that the magnitude of Sis constant. Hence, we can go to spherical\ncoordinates, such that\nS=S\nsinθcosφ\nsinθsinφ\ncosθ\n. (47)\nInserting this into the LLG equation, we firstly see that\n˙S=˙θ∂S\n∂θ+˙φ∂S\n∂φ=˙θS\ncosθcosφ\ncosθsinφ\n−sinθ\n+˙φS\n−sinθsinφ\nsinθcosφ\n0\n.\nNow, we notice that the RHS of the LLG equation can, without loss of generality, be written as S(t)×rwith\nr= (x,y,z )T. Working this out explicitly, we find that the LLG equation ˙S=S×rbecomes\nS\n˙θcosθcosφ−˙φsinθsinφ\n˙θcosθsinφ+˙φsinθcosφ\n−˙θsinθ\n=S\nzsinθsinφ−ycosθ\nxcosθ−zsinθcosφ\nysinθcosφ−xsinθsinφ\n. (48)\nWe note that the equation corresponding to the zcomponent can be written as\n˙θ=−1\nsinθr·\n−sinθsinφ\nsinθcosφ\n0\n=−1\nSsinθr·∂S\n∂φ. (49)9\nNow, we add up the ˆ xand ˆyequations, such that the ˙θcancels (i.e.−ˆxsinφ+ ˆycosφ). This yields\n˙φsinθ(sin2φ+ cos2φ) =−zsinθ(sin2φ+ cos2φ) +ycosθsinφ+xcosθcosφ,\nwhich simplifies to\n˙φ=1\nsinθr·\ncosθcosφ\ncosθsinφ\n−sinθ\n=1\nSsinθr·∂S\n∂θ. (50)\nBy inserting r=−B(Sz) + (αdiss∗S) (t) +ξ(t), we see that this is identical to the equations derived from the\nmicroscopic model\n˙φc=1\nSsinθc/bracketleftbig\n−B(Sz) +/parenleftbig\nαdiss∗Sc\ng/parenrightbig\n(t) +ξ(t)/bracketrightbig\n·δSc\ng(t)\nδθc; (51)\n˙θc=−1\nSsinθc/bracketleftbig\n−B(Sz) +/parenleftbig\nαdiss∗Sc\ng/parenrightbig\n(t) +ξ(t)/bracketrightbig\n·δSc\ng(t)\nδφc. (52)\nTherefore, we may conclude that our microscopic model is described by the generalized LLG equation.\nFor the fractional LLG equation, we are in particular interested in the case where αdiss∗S=αsDs\ntS, whereDs\ntis\na fractional derivative. For instance, assuming 0 0, however, we see that the exponential goes to zero when ω→−i∞.\nHence, we can close the ωintegration along the bottom. Thus, using the residue theorem (reversing the integration\ndirection), we find\n(αdiss∗S)(t) =−αssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n−∞dt/prime/integraldisplay∞\n0dε2πiΘ(t−t/prime)\n/bracketleftBig\ne−i(ε−i0)(t−t/prime)(ε−i0)2εs−1\n(ε−i0 +i0 +ε)+e−i(−ε−i0)(t−t/prime)(−ε−i0)2εs−1\n(−ε−i0 +i0−ε)/bracketrightBig\nS(t/prime)\n=−iαssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplayt\n−∞dt/prime/integraldisplay∞\n0dε/bracketleftbigg\ne−iε(t−t/prime)εs+1\nε+eiε(t−t/prime)εs+1\n−ε/bracketrightbigg\nS(t/prime)\n=−iαssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplayt\n−∞dt/prime/integraldisplay∞\n0dε/bracketleftBig\ne−iε(t−t/prime)−eiε(t−t/prime)/bracketrightBig\nεsS(t/prime)\n=−2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplayt\n−∞dt/prime/integraldisplay∞\n0dεsin[ε(t−t/prime)]εsS(t/prime)\n=−2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/braceleftbigg/bracketleftbig\nεs−1cos[ε(t−t/prime)]S(t/prime)/bracketrightbigt/prime=t\nt/prime=t0−/integraldisplayt\nt0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg\n=−2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/braceleftbigg\nεs−1S(t)−εs−1cos[ε(t−t0)]S(t0)−/integraldisplayt\nt0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg\n.(80)\nThe first term vanishes because of the cross product with S(t) in the LLG equation. The second term is where we\nhad to be careful. Here, we should realize that the −∞ is physically only indicating that it is a time very far in the\npast. So, to avoid unphysical infinities, we introduced a finite initial time t0and we will take t0→−∞ later. For\nthis, we need to introduce some fractional derivative notation. We define the Riemann-Liouville (RL) and Caputo (C)14\nderivatives of order s, with an integer nsuch thatn≤s 0).\nThis results in\nsin2θ=Ω2\n(B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftbigπs\n2/parenrightbig\n≈Ω2\n(B0−KS)2y2+ (αsS)2(B0−KS)2s(1 + 2sy)−2αsS(B0−KS)s+1(1 +sy)ycos/parenleftbigπs\n2/parenrightbig. (117)\nNow, we put the derivative with respect to yequal to zero, to get\n(B0−KS)2y+s(αsS)2(B0−KS)2s−αsS(B0−KS)s+1(1 + 2sy) cos/parenleftBigπs\n2/parenrightBig\n= 0. (118)\nHence, we find that\ny=−s(αsS)2(B0−KS)2s+αsS(B0−KS)s+1cos/parenleftbigπs\n2/parenrightbig\n(B0−KS)2−2sαsS(B0−KS)s+1cos/parenleftbigπs\n2/parenrightbig\n=αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig\n+O(αsS)2, (119)\nwhich results in\nωres≈(B0−KS)/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBig\n= (B0−KS) +αsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig\n. (120)\nWe see that the resonance frequency gets shifted by a small amount, depending on s, which scales non-linearly.\nInserting this result into Eq. (116), we can now also find an approximation for the amplitude at resonance:\nsin2θres=Ω2\n(B0−KS−ωres)2+ (αsS|ωres|s)2+ 2αsS|ωres|s(B0−KS−ωres) cos/parenleftbigπs\n2/parenrightbig\n≈Ω2/braceleftBig/bracketleftBig\nαsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig/bracketrightBig2\n+/parenleftBig\nαsS/vextendsingle/vextendsingle/vextendsingle(B0−KS) +αsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingles/parenrightBig2\n−2αsS/vextendsingle/vextendsingle/vextendsingle(B0−KS) +αsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingles\nαsS(B0−KS)scos2/parenleftBigπs\n2/parenrightBig/bracerightBig−1\n=Ω2\n[αsS(B0−KS)s]2/bracketleftBig\ncos2/parenleftBigπs\n2/parenrightBig\n+/vextendsingle/vextendsingle/vextendsingle1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingle2s\n−2/vextendsingle/vextendsingle/vextendsingle1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingles\ncos2/parenleftBigπs\n2/parenrightBig/bracketrightBig−1\n=Ω2\n[αsS(B0−KS)s]2/bracketleftBig\n1−cos2/parenleftBigπs\n2/parenrightBig\n+O(αsS)/bracketrightBig−1\n≈Ω2\n/bracketleftbig\nαsS(B0−KS)ssin/parenleftbigπs\n2/parenrightbig/bracketrightbig2. (121)\nSince the sine function decreases as smoves away from one, we see that the amplitude actually increases for non-Ohmic\nenvironments.21\nC. Calculating the FWHM linewidth\nNext, we are interested not only in the location of the resonance, but also how sensitive the resonance is to the\ndriving frequency. One way to describe this is by using the Full Width at Half Maximum measure. This provides a\nwell-defined line width independently of the shape of the peak. It is found by measuring the width of the peak at\nhalf the height of its maximum. This can be measured in the laboratories, but it can also be computed. Since our\nfunction of interest is of the form sin2θ(ωd) = Ω2/g(ωd), it makes sense to approximate the inverse function instead\nof the regular one. To this end, we will translate the FWHM measurement to the inverse function, and then Taylor\nexpandg(ωd) near resonance as a parabola to solve for the new condition of this inverse function. Notice that from\nEq. (116), we have\ng(ωd) = (B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftBigπs\n2/parenrightBig\n. (122)\nThe FWHM condition is\nΩ2\ng(ωd)= sin2θ(ωd) =sin2θ(ωres)\n2=Ω2\n2g(ωres), (123)\nhence we must solve for 2 g(ωres) =g(ωd). To this end, let us assume that ωd=ωres+yand expand g(ωd) iny. We\nwill use that\n|a+y|n≈an+nan−1y+1\n2n(n−1)an−2y2\nfor smallyanda>0. Then,\ng(ωres+y)\n= (B0−KS−ωres−y)2+ (αsS|ωres+y|s)2+ 2αsS|ωres+y|s(B0−KS−ωres−y) cos/parenleftBigπs\n2/parenrightBig\n≈(B0−KS−ωres)2+ (αsSωs\nres)2+ 2αsSωs\nres(B0−KS−ωres) cos/parenleftBigπs\n2/parenrightBig\n+y/parenleftBig\n−2(B0−KS−ωres) + 2s(αsS)2ω2s−1\nres−2αsScos/parenleftBigπs\n2/parenrightBig/braceleftbig\nωs\nres+sωs−1\nres[ωres−(B0−KS)]/bracerightbig/parenrightBig\n+y2/bracketleftBig\n1 +s(2s−1)(αsS)2ω2s−2\nres−2sαsSωs−1\nrescos/parenleftBigπs\n2/parenrightBig\n+s(s−1)αsSωs−2\nres(B0−KS−ωres) cos/parenleftBigπs\n2/parenrightBig/bracketrightBig\n=g(ωres) +y/parenleftBigg\n2αsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig\n+ 2s(αsS)2(B0−KS)2s−1/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBig2s−1\n−2αsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig/braceleftBigg/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBigs\n+sαsScos/parenleftBigπs\n2/parenrightBig\n(B0−KS)s−1/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBigs−1/bracerightBigg/parenrightBigg\n+y2/braceleftBigg\n1 +s(2s−1)(αsS)2(B0−KS)2s−2/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBig2s−2\n−2sαsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBigs−1\n−s(s−1)(αsS)2(B0−KS)2s−2cos2/parenleftBigπs\n2/parenrightBig/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBigs−2/bracerightBigg\n≈g(ωres) +y/braceleftBig\n2s(αsS)2(B0−KS)2s−1/bracketleftBig\n1−2 cos2/parenleftBigπs\n2/parenrightBig/bracketrightBig/bracerightBig\n+y2/braceleftBig\n1−2s(αsS)(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig\n+ (αsS)2(B0−KS)2s−2/bracketleftBig\ns(2s−1)−3s(s−1) cos2/parenleftBigπs\n2/parenrightBig/bracketrightBig/bracerightBig\n+O(αsS)3.\n(124)\nNow, we set 2 g(ωres) =g(ωres+y) =g(ωres) +by+ay2, and remark that g(ωres) = (αsS)2(B0−KS)2ssin2/parenleftbigπs\n2/parenrightbig\n, in\norder to find that\ny=−b±/radicalbig\nb2+ 4ag(ωres)\n2a=⇒∆FWHM =/radicalbig\nb2+ 4ag(ωres)\na. (125)22\nHence, we find that the lowest-order contribution to the linewidth is given by\n∆FWHM≈/radicalBig\n4(αsS)2(B0−KS)2ssin2/parenleftbigπs\n2/parenrightbig\n+O(αsS)3\n1 +O(αsS)\n= 2(αsS)(B0−KS)ssin/parenleftBigπs\n2/parenrightBig\n+O(αsS)2. (126)\nIV. DIMENSIONAL ANALYSIS\nThe fractional derivative in the LLG equation has an impact on the dimensions of quantities. We can firstly see\nthat in the chosen units, we have [ B0−KS] = [ωd] = time−1. Assuming Sto be dimensionless, then [ ωd] = [αsDs\ntS] =\n[αs][ωd]s, hence [αs] = [ωd]1−s. We can now start to understand what we mean when we say that certain quantities\nare small, since this has to be relative to something else. For instance, when we say αsSis small, we understand this\nasαsS/lessmuch(B0−KS)1−s. For Ω it is simpler, since there is no fractional derivative acting with it. Hence, for Ω small\nwe simply mean Ω /lessmuchB0−KS. We can now also define some dimensionless variables, such as α/prime\ns=αsS(B0−KS)s−1\nand Ω/prime= Ω/(B0−KS). We have used these variables in the figures to show the general behavior of the quantities.\n[1] A. Altland and B. D. Simons, Condensed Matter Field Theory (Cambridge University Press, 2010), 2nd ed.\n[2] A. Kamenev, Field theory of non-equilibrium systems (Cambridge University Press, 2011).\n[3] A. Schmid, Journal of Low Temperature Physics 49, 609 (1982).\n[4] A. O. Caldeira and A. J. Leggett, Physical Review Letters 46, 211 (1981).\n[5] A. O. Caldeira and A. J. Leggett, Physica A: Statistical Mechanics and its Applications 121, 587 (1983).\n[6] A. O. Caldeira and A. J. Leggett, Annals of Physics 149, 374 (1983).\n[7] A. O. Caldeira, An introduction to macroscopic quantum phenomena and quantum dissipation , vol. 9780521113755 (Cam-\nbridge University Press, 2012).\n[8] A. O. Caldeira and A. J. Leggett, Phys. Rev. A 31, 1059 (1985).\n[9] U. Weiss, Quantum dissipative systems (World scientific, 2012).\n[10] C. Gardiner and P. Zoller, Quantum noise: a handbook of Markovian and non-Markovian quantum stochastic methods with\napplications to quantum optics (Springer Science & Business Media, 2004).\n[11] A. Shnirman, Y. Gefen, A. Saha, I. S. Burmistrov, M. N. Kiselev, and A. Altland, Physical Review Letters 114, 176806\n(2015).\n[12] R. C. Verstraten, R. F. Ozela, and C. M. Smith, Physical Review B 103, L180301 (2021).\n[13] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic press, 2014)." }, { "title": "2203.06693v2.Continuum_damping_of_topologically_protected_edge_modes_at_the_boundary_of_a_magnetized_plasma.pdf", "content": "Continuum damping of topologically-protected edge modes at the boundary of\nmagnetized plasmas\nRoopendra Singh Rajawat∗and G. Shvets\nSchool of Applied and Engineering Physics, Cornell University, Ithaca, NY 14850\nV. Khudik\nDepartment of Physics and Institute for Fusion studies, The University of Texas at Austin, TX 78712\n(Dated: February 23, 2024)\nRecent extension of the topological ideas to continuous systems with broken time-reversal\nsymmetry, such as magnetized plasmas, provides new insights into the nature of scattering-free\ntopologically-protected surface plasma waves (TSPWs). We demonstrate a unique characteristic of\nTSPWs propagating above the electron cyclotron frequency: their collisionless damping via coupling\nto the continuum of resonant modes localized inside a smooth plasma-vacuum interface. Damped\nTSPWs retain their unidirectional nature and robustness against backscattering. When sheared\nmagnetic field creates a boundary between damped and undamped TSPWs, the two refract into\neach other without reflections.\nIntroduction.- Regardless of their state of matter –\nsolid [1], gaseous and fluid [2], or plasma [3] – a wide\nrange of materials is found to exhibit non-trivial topolog-\nical properties that fundamentally impact wave propaga-\ntion at domain walls between topologically-distinct bulk\nmaterials. Specifically, bulk materials possessing propa-\ngation bandgaps (i.e., acting as “insulators” for wave-like\npolaritonic perturbations of certain energies) and lack-\ning the time-reversal (TR) symmetry can be character-\nized by an integer invariant, known as the Chern num-\nber [4], assigned to every bandgap. The bulk-boundary\ncorrespondence (BBC) principle [5] – originally estab-\nlished in condensed matter physics [1, 6, 7] and later ex-\ntended to photonics and metamaterials [8, 9], cold atomic\ngases[10], and classical fluids [11] – predicts the existence\nand number of gapless unidirectional edge states that are\nspectrally-located within a common bandgap of the two\nbulk materials sharing a domain wall.\nThere are several remarkable properties of topologi-\ncal edge states. (i) In real space, they are robust to\nbackscattering with respect to various domain wall per-\nturbations, e.g., changes along (e.g., sharp turns) and\nacross (e.g., interface smoothness) its perimeter. (ii) In\nreciprocal space, their dispersion curves connect the two\ncontinua of the bulk modes (CBM) by spanning the cor-\nresponding topological bandgap. While properties (i)\nand (ii) are always present in condensed matter systems,\nwhere the band structure arises from the lattice period-\nicity, the situation can be quite different for continuous\nsystems, such as moving/rotating fluids or magnetized\nplasmas [3, 12, 13], where intrinsic material resonances\n(e.g., electron cyclotron resonance) are responsible for\nthe creation of multiple continuous propagation bands.\nThe key distinctive characteristic of a continuous topo-\nlogical medium with a smooth domain wall is that a new\ncontinuous spectrum of modes can arise in addition to\nthe continuum modes of the bulks.\nIn the specific case of magnetized plasmas comprising\n(c)(d)HPISPI𝑥10𝑛𝑛!(a)𝑦𝑥𝑧01.0𝑛/𝑛!Ramp Vacuum\n𝑩𝟎Plateau(b)\nExponential Decay (Theory)FIG. 1. (a) Surface wave excitation at the interface between\na magnetized plasma slab (green) and vacuum by a dipole\nsource (blue disk) oscillating with ωdr= 0.9ωp0. Generated\nTSPW (red wavy arrow) propagates unidirectionally along\nthe interface. (b) Plasma density profiles for hard (dashed\nline) and smooth (solid line) plasma interfaces. (c,d) Snap-\nshots of time-averaged E2\nzin az= const-plane for (c) HPI,\nand (d) SPI. Parameters: ωc/ωp0= 0.75,kz/kp= 1.0, and\nkpδl= 0.2 for SPI.\nmobile electrons and immobile neutralizing ions – the\nsubject of this Letter – the new continuum of modes cor-\nresponds to local upper hybrid resonances: ω2\nUH(x) =\nω2\nc+ω2\np(x), where ωc=eB0/mc is the cyclotron fre-\nquency of an electron with mass/charge m/e rotating\nin a uniform magnetic field B0=B0ˆz, and ωp(x) =p\n4πe2n(x)/mis the local plasma frequency inside the\ndomain wall determined by the local unperturbed plasma\ndensity n(x). In the simplest case of a semi-infinite vol-\nume of cold magnetized plasma with uniform density n0\ninterfaced with a vacuum region through a smooth do-\nmain wall, the continuum of localized modes (CLM) oc-\ncupies the spectral region ωc< ω < ΩUH, where Ω UHis\nthe upper-hybrid frequency of the bulk.arXiv:2203.06693v2 [physics.plasm-ph] 22 Feb 20242\nThe presence of the CLM inside the topological\nbandgap raises a number of fundamental questions that\ndo not arise in the case of a periodic topological\nmedium. For example, it is unclear if topologically ro-\nbust bandgap-spanning modes – possibly damped – can\nexist inside the CLM spectral band [3, 14], as one would\nexpect based on the BBC principle. While the BBC prin-\nciple can be validated for continuous models with sharp\ninterfaces [15], it is not known if it still holds for smooth\ninterfaces. An even more important open question is\nhow property (i) is affected, i.e., whether the presence\nof the CLM band affects the robustness of edge state\nto backscattering when a domain wall makes a sharp\nturn. It also remains to be understood how the pro-\nfile of the domain wall affects the damping of the edge\nstates when their frequency overlaps with the continuum\nof local modes. In this Letter, we use particle-in-cell\n(PIC) simulations and analytic modeling to demonstrate\nthe existence of a new class of backscattering-robust uni-\ndirectional quasi-modes spanning the entire CLM band,\nand develop an analytic theory predicting their propaga-\ntion properties.\nAs a model for investigating topological surface plasma\nwaves (TSPWs), we use a physical configuration shown\nin Fig. 1(a): a planar slab of magnetized plasma with\ndensity n(x)≡n(x) (where n(x)→n0forx≪Lint) in-\nterfacing with a vacuum region at x≈Lint. Two types of\ninterfaces shown in Fig. 1(a) are utilized: a hard plasma\ninterface (HPI: solid line) and a smooth plasma interface\n(SPI: dashed line). To excite TSPWs with a fixed ax-\nial wavenumber kz, we use a periodic chain of oscillating\npoint sources with dipole moments p≡ezpz(blue dot in\nFig. 1(a)) oscillating as pz∝cos (kz·z−ωt) spaced by\nPz≪2π/kzin the z−direction placed near the plasma-\nvacuum interface.\nThe excitation of TSPW’s with kz=ωp0/cpropagat-\ning along with the interface ( ±yz−plane) was modeled\nby a first-principles 3D-PIC simulation code VLPL[16]\nfor cold collisionless plasmas with magnetic field corre-\nsponding to ωc= 0.75ωp0: see [17] for the details of\nthe PIC simulation setup. For these parameters, the\nupper (lower) edge of the first (second) bulk propaga-\ntion band with the Chern number C1=−1 (C2= 1)\nis at ω−≈0.4ωp0(ω+≈ωp0), resulting in a complete\nbandgap for ω−< ω < ω +. For the smooth plasma inter-\nface, our driven PIC simulations reveal the existence of\nundamped TSPWs in the lower portion of the bandgap:\nω−< ω < ω c, as was recently demonstrated using eigen-\nvalue simulations [3, 14]. The dispersion relation ω(ky)\nfor these modes is plotted as a green line in Fig. 2 for an\nSPI represented by a linear density ramp with the length\nδl= 0.2k−1\np(where k−1\np≡c/ωp0is the collisionless skin\ndepth of the bulk plasma). Eigenvalue simulations do\nnot reveal any edge states – damped or undamped – in\nthe rest of the bulk bandgap overlapping with the CLM\nband extending from ωcto Ω UH> ω +.\n(𝑏)\n𝑥!𝑥\"#𝑥$%𝑥&𝑥!'𝐂(𝑐)𝜔\"#𝜔!'𝜔(Ω$%x𝜔)*(𝑒)\n(𝑎)\n𝐶+=−1𝐶,=1\n(𝑑)FIG. 2. (a) Propagation band structure for a semi-infinite\nmagnetized plasma slab with a smooth plasma/vacuum in-\nterface. Yellow curves: propagating bulk and ramp-localized\ncontinuum modes. Undamped surface waves: HPI-TSPW\n(black solid) and SPI-TSPW (green solid). Damped SPI-\nTSPWs: spatial propagation ( ky: red solid line) and decay\n(γ: blue dashed line) constants. Chern numbers are indicated\nfor the bulk bands. Stars at ω=ωdr: driven modes from\nFig. 1(c,d). (b,c) Integration contours in the complex (b) ω-\nand (c) x- planes used for calculating the dispersion relation.\n(d,e) Spatial profiles of (d) Ex, and (e) Bzoscillation am-\nplitudes extracted from the PIC simulation. Shaded region:\nlinear density ramp of the SPI. Red dashed line: upper-hybrid\nresonance at xUH. Plasma parameters: same as in Fig. 1.\nHowever, driven PIC simulations at ωdr= 0.9ωp0\nfor the same SPI (Fig. 1(d)) and for the HPI (Fig.\n1(c): δl= 0) present a different picture. The red-\ncolored images of the time-averaged E2\nzreveal that the\nedge states are indeed launched inside the CLM band\nfor both interface types, and illustrate several of their\nproperties. First, SPI and HPI support unidirectional\nedge states propagating in the positive y−direction. Sec-\nond, while the HPI-TSPW mode propagates undamped\nwith ky≈0.8kp, the SPI-TSPW appears to be exponen-\ntially decaying, with the corresponding complex-valued\nwavenumber ˜ky≈(0.8−0.075i)kp. The lack of damping\nin the HPI case is not surprising because the CLM band\ncollapses into a single point at ω= Ω UHoutside of the\nbulk bandgap. However, the presence of weakly-damped\nmodes in the SPI case is puzzling because no such modes\nare revealed by eigenvalue simulations. The collision-\nless nature of our PIC simulations raises the question of\nhow the energy of TSPWs is dissipated in the absence of\nplasma viscosity.3\nNevertheless, several conjectures – validated below –\ncan be made and utilized for interpreting the physics of\nthe collisionlessly-damped unidirectional TSPWs. First,\napart from finite damping, TSPW propagation along rel-\natively sharp ( δl≪k−\np1) interfaces appears similar to\ntheir propagation along the HPI: ˜ky≈ky. Therefore, a\nperturbed solution for SPI-TSPWs can be obtained us-\ning the field profile of HPI-TSPWs as the unperturbed\nsolution [18], and kpδlas a small parameter. Second,\nthe unidirectional nature of the SPI-TSPWs indicates\nthat they are robust to backscattering by abrupt (i.e.,\non a scale shorter than ∆ y∼k−1\ny) perturbations of\nthe domain wall. Third, collisionless damping of SPI-\nTSPWs appears to be a kinetic effect that results from\nstrong plasma heating occurring at the specific loca-\ntionx=xrsuch that the unperturbed HPI-TSPW fre-\nquency ωresonates with highly-localized upper-hybrid\nresonances: ωUH(xr) =ω. Resonant excitation of local-\nized upper-hybrid waves can produce energetic electrons\n[19].\nFinally, the decaying mode SPI-TSPW is highly\nreminiscent of weakly-damped surface waves in un-\nmagnetized plasmas that were interpreted as quasi-\nmodes generated by phase mixing of multiple local res-\nonances [20]. The key distinction between the true\nexponentially-decaying eigenmodes (e.g., due to finite\nfluid viscosity) and the quasi-modes is that the latter\nexhibit short-term exponential and long-term power-law\ndecay that can be viewed as a form of Landau damp-\ning [21] in physical space [13, 22]. While continuum\ndamping of electromagnetic waves is not unusual in\nplasma physics [23, 24], to our knowledge, it has never\nbeen studied in the context of topologically-robust sur-\nface waves in magnetized plasmas.\nTo construct a cold fluid description of the edge states,\nwe assume that their electric ( E) and magnetic ( B)\nfields are harmonic, i.e. proportional to eik·x−iωt, where\nk=eyky+ezkzis the in-plane propagation wavenum-\nber. Temporal or spatial damping rates of TSPWs\nare included in their complex-valued frequency ( ω) or\nwavenumber ( ky), respectively [25]. After space-time\nFourier transformation of Maxwell equations [26], we ob-\ntain:\n∂ψ\n∂x=−i\nϵt(x)Mψ, ψ = (Bx, By, Bz, Ez)T(1)\nwhere ψis a four-component vector comprising the field\ncomponents remaining continuous across the discontinu-\nous HPI. The 4 ×4 matrix M(x,k, ω) is given by\nM=\n0 kyϵtkzϵt 0\n−kyϵt 0 0 k0ϵaϵt\nk2\n0ϵ2\ntg−ϵtk2\nz\nkz−ikzϵgikyϵg−k0ky\nkzϵ2\ntg\nik0ϵgϵ2\ntk2\n0−k2\nz\nk0k0ky\nkz−ikyϵg\n,(2)\nwhere k0=ω/c,ϵt=ω2−ω2\nUH(x)\nω2−ω2c,ϵg=ωc\nωω2\np(x)\nω2c−ω2, andϵa=ω2−ω2\np(x)\nω2 are the x-dependent components of the\ncold plasma dielectric tensor [27], and ϵ2\ntg≡ϵ2\nt−ϵ2\ng. The\nremaining two components of the electric field are ex-\npressed in terms of ψcomponents:\nϵtEx=ikyk0ϵgEz−kykzBz\nk0kz+kz\nk0By−ik0\nkzϵgBx,(3)\nEy=\u0012ky\nkz\u0013\nEz−\u0012k0\nkz\u0013\nBx. (4)\nTherefore, by integrating Eq.(1) between x=xin→\n−∞andx=xout→+∞endpoints (deep inside and out-\nside the plasma) and assuming the vanishing of ψ→0 at\nthe endpoints, a TSPW dispersion relation in the form\nofD(ω, ky) = 0 can be obtained [20, 28] for any SPI.\nFor the TSPWs in the ω−< ω < ω cfrequency range,\nthex-integration of Eq.(1) can be carried out along the\nreal axis because no singularities are encountered for any\nreal-valued xin< x < x outfor any plasma density pro-\nfilen0(x). The resulting solution of the dispersion rela-\ntion yields a real-valued (damping-free) dispersion rela-\ntionω(ky) [14] plotted as a green solid line in Fig. 2(a).\nThe situation changes dramatically for the real-valued\nTSPW frequencies inside the CLM band ( ω > ω c) be-\ncause the 1 /ϵt(x) factor in Eq.(1) acquires a pole sin-\ngularity at the local upper-hybrid resonance location\nx=xUH(ω) defined by ω2=ω2\nc+ω2\np(xUH). The am-\nbiguity of integrating Eq.(1) across the singularity is re-\nsolved by analytically continuing the dispersion function\n(D→D∗) into the lower-half of the complex ω-plane\nto preserve causality [13, 20, 22]. If D∗(ω, ky) possesses\na pole ˜ ωQM≡ωre\nQM+iωim\nQM(where ωim<0), then its\nanalyticity requires that the integration between xinand\nxoutbe carried out along the path Cin the complex- x\nplane passing below the complex-valued point xrdefined\nasωUH(xr) = ˜ωQM[13, 20]: see Fig. 2(c).\nIterative application of such integration for different\ntrial frequencies ˜ ωtryields the quasi-mode dispersion re-\nlation ˜ ωQM(ky) satisfying the condition of D∗(˜ωQM, ky) =\n0 for a given SPI density profile n0(x): see [17] for the\nmathematical details of calculating D∗(˜ωQM, ky) = 0.\nIn Fig. 2(a), we present an example of the complex\nwavenumber ˜ky(ω)≡ky+iγcalculated for the real-\nvalued frequencies ω > ω cassuming the SPI in the form\nof a linearly-varying plasma density ramp extending from\nxin=−0 to xout=δl+ 0: n(x) =n0(1−x/δl) and\nδl= 0.2k−1\np. Note that the resonant point xr≡xUH\n(where xUH/δl=\u0000\nω2\nUH−ω2\u0001\n/ω2\np0) is also real-valued,\nso the the integration contour Cincludes a semi-circle\nabove x=xUH. The numerically calculated oscillation\n(decay) constants ky(γ) are plotted in Fig. 2(a) as solid-\nred (dashed-blue) lines, respectively.\nNote that the decay constant γis normalized to k2\npδl\nin Fig. 2(a): a result that holds rather accurately for\nsharp plasma-vacuum interfaces with δl≪k−1\np. To\nmake further analytic progress in understanding this scal-4\ning with δl, as well as the properties of the TPSW\nquasi-modes, we assume that the electromagnetic fields\nψare nearly-uniform inside the SPI. Then, their incre-\nment ∆ ψ≡ψv−ψp(where p(v) superscripts label\nthe fields in the plasma (vacuum) regions) can be es-\ntimated as ∆ ψ≈iMψ\u0010Rδl\n0dx/ϵ t(x)\u0011\n. After expand-\ningϵt(x) in the vicinity of x=xUHasϵt≈˜x/˜δl,\nwhere ˜δl=δl−xUHis the distance of the local upper-\nhybrid resonance point from the edge of the plasma and\n˜x=x−xUH, the integral over the pole is calculated\nto be approximately equal to −iπ˜δl. To leading order\ninδl, we find that BxandByare continuous ( Bp\nx,y−\nBv\nx,y≈0) while BzandEzchange across the interface:\nkz(Bp\nz−Bv\nz)≈iπ˜δlωcR/ω andk0(Ep\nz−Ev\nz)≈ −π˜δlR.\nHere R=ky(kzBz+iωcEz/c)−k2\nzBy−ik0ωcBx/cis\ncalculated on the vacuum side of the interface.\nUsing these boundary conditions, a dispersion relation\nD∗(ω,˜ky) = 0 for the edge states can be expressed as\nfollows (see [17] for more details):\n(κ1+κ0+Q1) (P2−ky) = (κ2+κ0+Q2) (P1−ky) (5)\nwhere κ0andκ1,2are the inverse decay lengths in vac-\nuum and plasma, respectively [18], and the remaining\nreal-valued coefficients are defined in [17].\n(𝑎)\n𝑦𝑧𝑥\n𝐵!(𝑦)̂𝑧source\n(𝑑)\n𝑦𝑧𝑥\n𝐵!̂𝑧source\n(𝑐)\n(𝑏)\nobstacle\nFIG. 3. Reflectionless surface wave propagation in a sheared\nmagnetic field ⃗B=B0(y)ˆ z (a,b) and around an obstacle (c,d).\n(b) Refraction of a propagating TSPW ( y <100k−1\np) with fre-\nquency ωdr= 0.67ωp0into a damped TSPW ( y >100k−1\np):\nEzat the plasma-vacuum interface (color-coded) and the cy-\nclotron frequency ωc(y) (magenta line, scale on the right) lin-\nearly sheared on the Lshear\ny = 2k−1\npscale. Plasma parameters\nin (a,b): same as in Fig 1. (d) A snapshot of time-averaged\nE2\nz(yellow color) of a damped TSPW with ωdr= 0.7ωp0.\nPlasma parameters in (c,d): kz/kp= 0.8,ωc/ωp0= 0.5, and\nkpδl≈0.6. Inset: the TSPW spatially decays inside each of\nthe propagation arms.\nFrom Eq.(5), it has been demonstrated [18] that in the\nHPI case ( δl= 0) there always exists a real-valued kyroot\nfor the following spectral range: ω1< ω < ω 2, where\nω2,1=\u0010q\n2ω2\np0+ω2c±ωc\u0011\n/2. For a fixed kz/kp= 1,the resulting dispersion curve (black solid line) for a\nHPI-TSPW is plotted in Fig. 2(a). In the SPI case, it\nfollows from Eq.(5) that as long as xUHis located in-\nside the ramp, the propagation wavenumber ˜kymust be\ncomplex, with its imaginary part γproportional to the\nfields discontinuity – just as our plots of kyandγin\nFig. 2(a) indeed demonstrate for a specific density ramp\nwith δl= 0.2k−\np1.\nRemarkably, while the ky(ω) curves for the HPI- and\nSPI-TSPW overlap for small values of ky, continuum\ndamping removes the unphysical flattening of the HPI-\nTSPW dispersion curve for large propagation wavenum-\nbers|ky| ≫kp, thereby removing the divergences of the\nlocal density of states and of the thermal electromagnetic\nenergy density [25, 29]. As a result, the combination of\nthe lossless TSPW ( ω−< ω < ω c) and the lossy quasi-\nmode ( ωc< ω < ω +) produces a propagation mode span-\nning the entire topological bandgap bracketed between\nthe propagation bands of the bulk continua. Importantly,\nthe group velocity ∂ω/∂k y>0 of the mode is positive for\nall frequencies inside the topological bandgap, thus indi-\ncating unidirectional propagation that is expected to be\nprotected against backscattering. The normalized spa-\ntial damping rate γ/k2\npδlof the topological quasi-mode\nis observed to be at its minimum for the long-wavelength\nmodes with positive ky< k p, and to increase for the\nslowly-propagating short-wavelength modes.\nHow “real” are the continuum-damped quasi-modes,\ngiven that they represent only one part of the plasma\nresponse, in addition to the continuum of modes from\ntheωc< ω < ΩUHrange [12, 13, 23]? In fact, the\nlong-term behavior of the fields inside the SPI region is\nexpected to be non-exponential [20, 22]. However, their\nearly- and medium-term behavior is accurately described\nas exponential decay in time or space, depending on the\ninitial/boundary conditions. This is confirmed by our\nPIC simulations presented in Fig. 1(d) that show nearly-\nexponentially decaying surface waves with the damping\nconstant γPIC≈0.08kpand the propagation number\nkyPIC≈0.8kp. Both are in close agreement with the cor-\nresponding complex-valued propagation constant of the\nsurface quasi-mode: ˜ky= (0.8−0.075i)kp.\nAnother sign of the quasi-modes physical significance\nis that PIC simulations indeed confirm their unusual field\ndistribution predicted by the analytic theory. For exam-\nple, it follows from Eq.(4) that the largest field compo-\nnentEx∝1/ϵt(x) is expected to peak and change its sign\nat the upper-hybrid location xUHinside the linear SPI\ndensity ramp (6 < kpx <6.2) as observed in Figs. 2(e,f),\nwhere it can be seen that |Ex| ≫ | Bz|inside the ramp.\nMoreover, the Bz(x) profile is nearly-discontinuous at\nx=xUH. This is consistent with the jump condition\nforψthat was used to derive the quasi-mode dispersion\nrelation given by Eq.(5).\nNext, we demonstrate the reflectionless conversion of\na collisionless TSPW into a weakly-damped quasi-mode.5\nThe setup is shown in Fig. 3(a), where the boundary\nbetween the two (left and right) plasmas with identical\ndensity profiles comprises a linear magnetic shear region\nwith the length Ly= 2k−1\np, and the radiation source on\nthe left side of the shear region launching a TPSW with\nωdr= 0.67ωp0. The corresponding cyclotron frequencies\non the two sides of the shear region are ω(l)\nc=ωp0and\nω(r)\nc= 0.5ωp0, respectively, and the two bulk plasmas\nshare a common topological bandgap that includes the\nsource frequency: ω(l,r)\n−< ω dr< ω(l,r)\n+, where ω(l)\n−≈0.45,\nω(l)\n+≈1,ω(r)\n−≈0.25 and ω(r)\n+≈1 (see Fig.S1 and the\ndetails of bulk band structures in [17]).\nBecause the source frequency also satisfies the ω(r)\nc<\nωdr< ω(l)\nccondition, the launched mode is an undamped\n(damped) TPSW on the left (right) sides of the magnetic\nshear region centered at y= 100 k−1\np. The correspond-\ning propagation wavenumbers kyon the opposite sides of\nthe shear region have opposite signs across the magnetic\nshear layer: kl,r\ny≈ ∓0.2kp, respectively. Therefore, while\nthe group velocities v(gr)\ny=∂ω/∂k y>0 of the TSPWs\nare positive on the two sides of the boundary, their phase\nvelocities v(ph)\ny=ω/kychange sign. Therefore, the elec-\ntric field profiles shown in Fig. 3(b) reveal reflectionless\nnegative refraction [30] of the incident undamped TPSW\ninto the damped one.\nFinally, we demonstrate the robustness of damped SPI-\nTSPW when encountering a sharp rectangular obstacle\n(see Fig. 3(c)) in its path. The dimensions of the obstacle\nare chosen as Lx×Ly×Lz= 4.7k−1\np×12.6k−1\np×7.9k−1\np.\nAs shown in Fig. 3(d), the launched TSPW seamlessly\npropagates around the obstacle without any reflection or\nradiation into the bulk plasma, thus manifesting topolog-\nical protection despite their damping [31]. The collision-\nless damping observed before and around the obstacle\n(see inset of Fig. 3(d)) is a manifestation of the absorp-\ntion by localized upper-hybrid resonances.\nIn summary, we have established the existence of\ncollisionlessly-damped, yet topologically-protected sur-\nface quasi-modes in a magnetized plasma. Using PIC\nsimulations and theoretical calculations, we have iden-\ntified the damping mechanism through the coupling to\na continuum of upper-hybrid resonances localized in-\nside the plasma-vacuum domain wall. While the cal-\nculations presented here assume mobile electrons in the\nbackground of immobile, future work will address the fi-\nnite ion mass and the possibility of topological effects in\nmulti-species plasmas.\nThis work is supported by Air Force of Scientific\nResearch (AFOSR) through Stanford University under\nMURI Award no. FA9550-21-1-0244. The authors\nwould like to thank Texas Advanced Computing Center\n(TACC) for providing HPC resources.∗rupn999@gmail.com\n[1] M. Z. Hasan and C. L. Kane, Colloquium: Topological\ninsulators, Rev. Mod. Phys. 82, 3045 (2010).\n[2] P. Delplace, J. B. Marston, and A. Venaille, Topological\norigin of equatorial waves, Science 358, 1075.\n[3] J. B. Parker, J. B. Marston, S. M. Tobias, and\nZ. 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Lett. 111,\n257401 (2013)." }, { "title": "2401.14708v1.Efficient_Control_of_Magnetization_Dynamics_Via_W_CuO___text_x___Interface.pdf", "content": "Efficient Control of Magnetization Dynamics Via W/CuO XInterface\nAntarjami Sahoo,1Haifeng Ding,2Antonio Azevedo,3and Subhankar Bedanta1\n1)Laboratory for Nanomagnetism and Magnetic Materials (LNMM), School of Physical Sciences,\nNational Institute of Science Education and Research (NISER), an OCC of Homi Bhabha National Institute (HBNI), Jatni 752050,\nOdisha, India\n2)National Laboratory of Solid State Microstructures, Department of Physics, Nanjing\nUniversity and Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093,\nPeople’s Republic of China\n3)Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901,\nBrazil\n(*Electronic mail: sbedanta@niser.ac.in)\n(Dated: 29 January 2024)\nMagnetization dynamics, which determine the speed of magnetization switching and spin information propagation, play\na central role in modern spintronics. Gaining its control will satisfy the different needs of various spintronic devices.\nIn this work, we demonstrate that the surface oxidized Cu (CuO X) can be employed for the tunability of magnetization\ndynamics of ferromagnet (FM)/heavy metal (HM) bilayer system. The capping CuO Xlayer in CoFeB/W/CuO Xtrilayer\nreduces the magnetic damping value in comparison with the CoFeB/W bilayer. The magnetic damping even becomes\nlower than that of the CoFeB/CuO Xby∼16% inferring the stabilization of anti-damping phenomena. Further, the\nreduction in damping is accompanied by a very small reduction in the spin pumping-induced output DC voltage in\nthe CoFeB/W/CuO Xtrilayer. The simultaneous observation of anti-damping and spin-to-charge conversion can be\nattributed to the orbital Rashba effect observed at the HM/CuO Xinterface. Our experimental findings illustrate that the\ncost-effective CuO Xcan be employed as an integral part of modern spintronics devices owing to its rich underneath\nspin-orbital physics.\nI. INTRODUCTION\nIntensive research on spintronics such as magnetic ran-\ndom access memory (MRAM), has been carried out in the\nlast decade, as it shows potential for lower power consump-\ntion and instant-on capability1. Spintronics takes the advan-\ntage of magnetic materials which helps in information stor-\nage and non-volatile operation. At the same time, the en-\ndurance of spintronics devices is far superior to that of re-\nsistive and phase-change memory technologies1. Spintron-\nics also unravels a rich array of fundamental physics arising\ndue to the bulk, interface, and external perturbation induced\nphenomena2. The efficient electrical control of magnetiza-\ntion in spin-orbit-torque (SOT) based devices is important for\nfuture spintronics applications. The magnetic damping is a\ncritical parameter that determines the required energy and op-\nerational speed of SOT-MRAM devices. The scattering of the\nmagnons by the conduction electrons limits the opportunity of\nreducing the magnetic damping of ferromagnet (FM) metals3.\nSeveral valiant efforts have been put forward to minimize\nthe damping of FMs for future spin-orbitronics applications3.\nFerrimagnetic insulators, like YIG and Heusler alloys, are\nfound experimentally to exhibit low damping. However, the\ninsulating ferrimagnets cannot be a suitable component as\nsome spin-orbitronics applications require the flow of charge\ncurrent through the magnetic layer. Meantime, the fabrication\nprotocols like high-temperature annealing hinder the integra-\ntion of Heusler alloys into the CMOS technology3. Hence, it\nis highly required to find alternative methodologies for reduc-\ning the magnetic damping of FM materials that are compatible\nwith current information technology.\nRecently, the damping of some FM/Nonmagnet (NM) bi-layers have been interestingly found to be less compared to\nthat of the parent FMs4,5. The reduction in damping, also\nknown as anti-damping, in the Py/Pt bilayer has been at-\ntributed to the formation of Rashba-like states at the FM/NM\ninterface. The non-equilibrium spin accumulation at such\nan interface is believed to generate an anti-damping torque\nvia the inverse Rashba Edelstein effect (IREE). Though here\nthe spin Rashba effect (SRE) plays a vital role, anti-damping\nphenomena can also be realized due to orbital Rashba effect\n(ORE). Traditionally, the orbital angular momentum (OAM)\nwas believed not to play any significant role in spintronics ap-\nplications due to its quenching by crystal field in solids. But\nin recent times, it has been observed that the OAM mediated\norbital torque (OT) can induce magnetization dynamics in the\nFM layer6,7. For example, a large current induced SOT is\nevident in the surface oxidized Cu/Py bilayer even without\nthe presence of any large SOC material6,7. This has been\nattributed to the OAM based OT in CuO X, which does not\nrequire any large SOC materials for its origin. Further, the\nevolution of ORE/OT can be mainly interfacial, as the resis-\ntivity of CuO Xis expected to be very high. Hence, the ORE,\nsimilar to the SRE can be thought of playing a vital role in\ncontrolling the magnetization dynamics of adjacent Py layer.\nThe OAM polarization arising due to the orbital hybridization\ninteracts with the electric field generated at the interfaces with\nstructural asymmetry. This leads to the formation of OAM\ntexture of electronic states in the k-space, which is known\nas the ORE8,9. Though the ORE has a similarity with the\nSRE, a high SOC at the interface is not required, unlike SRE\nfor evolution ORE. Upon the application of an external elec-\ntric field, a non-equilibrium OAM accumulates at the bound-\naries, and this process is known as orbital Rashba-Edelstein\neffect (OREE). It is the orbital counterpart of spin Rashba-arXiv:2401.14708v1 [cond-mat.mtrl-sci] 26 Jan 20242\nTABLE I. Stacking of different heterostructures with their corresponding nomenclatures.\nStacking Nomenclature\nSi/SiO 2(300 nm)/CFB (7 nm)/CuO X(3 nm) SF\nSi/SiO 2(300 nm)/CFB (7 nm)/ W (5 nm) SA\nSi/SiO 2(300 nm)/CFB (7 nm)/W (5 nm)/CuO X(3 nm) SA1\nSi/SiO 2(300 nm)/CFB (7 nm)/Cu (10 nm)/W (5 nm) SB\nSi/SiO 2(300 nm)/CFB (7 nm)/Cu (10 nm)/W (5 nm)/CuO X(3 nm) SB1\nEdelstein effect and onsager reciprocal effect is known as the\ninverse orbital Rashba-Edelstein effect (IOREE). The OAM\naccumulation can modulate the magnetization dynamics of\nadjacent FM layer with finite SOC via orbital torque. This\ntype of electrically generated torque helps in the development\nof modern magnetic nanodevices. For example, a large torque\nhas been reported in CoFe/Cu/Al 2O3heterostructures with-\nout the requirement of heavy metals (HMs)10. The extremely\nlarge Hall conductivity could not be explained by simple spin\ntorque mechanism and the OAM mediated orbital Edelstein\neffect at the Cu/Al 2O3interface was attributed to this type\nof exotic magnetization dynamics phenomena. It has also\nbeen found that the CuO Xcapping significantly enhances the\nSOT efficiency in the thulium iron garnet (TmIG)/Pt bilayer11.\nThough the CuO Xhas weak SOC, the inversion symmetry\nbreaking at the Pt/CuO Xinterface leads to the orbital cur-\nrent via OREE and hence, the nonlocal generation of SOTs\nin TmIG/Pt/CuO Xtrilayer11. The spin current injected into\nthe Pt/CuO Xinterface in YIG/Pt/CuO Xtrilayer also gets con-\nverted to the charge current with higher efficiency compared\nto the YIG/Pt via IOREE process12. In both of these ex-\namples, the use of heavy metal Pt helps in harnessing OAM\nvia the orbital-to-spin conversion mechanism due to its high\nSOC. Hence, the combination of HM and surface oxidized\nCu can play a vital role in determining the magnetization\ndynamics of adjacent FMs, which can be detected by either\nOREE or IOREE. In this manuscript, we report the interest-\ning magnetization dynamics in Co 20Fe60B20(CFB)/W/CuO X\nvia ferromagnetic resonance study. The CuO Xcapping re-\nduces the magnetic damping of the trilayer to a value below\nthe damping of CFB without the application of any exter-\nnal DC current, while the spin-to-charge current conversion\nin CFB/W/CuO Xtrilayer still remains significant. The anti-\ndamping phenomenon can be attributed to the IOREE effect\nat the W/CuO Xinterface, and it paves an alternative path for\nlowering the magnetic damping of FMs, and consequently, the\nfabrication of power-efficient spintronics devices.\nII. EXPERIMENTAL METHODS\nFour different types of heterostructures with CFB (7\nnm)/Cu (0, 10 nm)/W (5 nm)/CuO X(0, 3 nm) and CFB (7\nnm)/CuO X(3 nm) stacking have been fabricated on Si/SiO 2\n(300 nm) substrates for the investigation of magnetization dy-\nnamics and spin pumping phenomena. In addition, the CFB\n(7 nm)/W (10 nm)/CuO X(3 nm) heterostructure was also fab-\nricated to reaffirm the stabilization of βphase of W. The het-erostructures can be read as shown in TABLE I. The CFB,\nβ-W, and Cu layers were grown by DC magnetron sputtering\n(Manufactured by EXCEL Instruments, India) at room tem-\nperature. The top thin Cu layer oxidizes naturally to form\nthe CuO Xcapping in SF, SA1 and SB1 heterostructures. We\nhave also fabricated a 10 nm Cu thin film on Si/SiO 2to in-\nvestigate the formation of the natural oxidation of Cu. Be-\nfore the fabrication of heterostructures, thin films of CFB,\nW, and Cu were prepared for thickness calibration and study\nof magnetic and electrical properties. The base pressure of\nthe sputtering chamber was usually maintained at ∼4×10−8\nmbar prior to the deposition. The structural characterizations\nof individual thin films and heterostructures were performed\nby x-ray diffraction (XRD) and x-ray reflectivity (XRR) mea-\nsurements. Before the fabrication of heterostructures, the\nthickness of individual layers was calibrated via XRR, and\nthe growth of different materials of the heterostructures was\nmonitored using a quartz crystal monitor. The supercon-\nducting quantum interference device based vibrating sample\nmagnetometer (SQUID-VSM) was employed for the static\nmagnetization characterization. The magnetization dynamics\nwere investigated by a lock-in based ferromagnetic resonance\n(FMR) spectrometer manufactured by NanOsc, Sweden. The\nheterostructures were kept in a flip-chip manner on the co-\nplanner waveguide (CPW) and the FMR spectra in 4-17 GHz\nrange were recorded for all the samples. The FMR spectrom-\neter set-up is also equipped with an additional nano-voltmeter\nusing which spin-to-charge conversion phenomena of all the\ndevices are measured via Inverse Spin Hall Effect (ISHE) with\n15 dBm RF power. The contacts were given at the two oppo-\nsite ends of 3 mm × 2 mm devices using silver paste to mea-\nsure the spin pumping induced voltage difference across the\nsamples13. The schematics illustrating the CFB/W/CuO Xhet-\nerostructure and spin-to-charge conversion process owing to\nthe spin injection from CFB are shown in Fig. 1 (a).\nIII. RESULTS AND DISCUSSION\nThe grazing incidence x-ray diffraction (GIXRD) was per-\nformed for all the heterostructures. The XRD patterns of CFB\n(7 nm)/W (5, 10 nm)/CuO X(3 nm) heterostructures are shown\nin Fig. 1 (b). The presence of (210) Bragg’s peaks of W at ∼\n39.8◦indicates the stabilization of βphase of W (A-15 crys-\ntal structure)14,15. The Bragg’s peaks are more prominent for\nheterostructures with 10 nm thick W layers as diffraction in-\ntensity increases with increasing W thickness. The XRD pat-\nterns for SA, SA1, SB, and SB1 are similar indicating W is of3\nFIG. 1. (a) Schematic of Si/SiO 2/CFB/W/CuOx heterostructure illustrating the spin to charge conversion phenomena, (b) GIXRD patterns of\nSi/SiO 2/CFB (7 nm)/W (5, 10 nm)/CuOx (3 nm) heterostructures\ntheβ-phase in all these heterostructures. Here, we have not\nused reactive gases like O 2and N 2for the growth of β-W un-\nlike some previous reports16. We don’t observe other Bragg’s\npeaks of W in the XRD patterns, inferring an oriented growth\nofβ-W. We do not also observe the (110), (200), and (210)\nBragg’s peaks for the bcc α-W in the XRD patterns15. Fur-\nther, the fullwidth half maximum of (210) Bragg’s peaks of W\ndecreases as we increase the thickness of W (Fig. S1). The\nstabilization of this type of oriented βphase of W is quite im-\nportant for future SOT device fabrication, as βphase of W\nyields high spin-orbit coupling.\nThe Cu capping layer was allowed to oxidize naturally in\nambient environment at laboratory. After that, the formation\nof the top CuO Xlayer in SA1 and SB1 heterostructures was\nconfirmed by GIXRD and I-V measurements. The 10 nm\nthick Cu layer without any capping displays a Bragg’s peak\nat 82.2◦, whereas the same peak is absent for the SB het-\nerostructure, where the 10 nm Cu is capped with 5 nm W (Fig.\n2 (a)). The Bragg’s peak at ∼82.2◦has also been previouslyreported for Cu film17, inferring the formation of surface ox-\nidized Cu in the SA1 and SB1 heterostructures. Further, the\nI-V measurement was also performed for the bare 10 nm thick\nCu film. The I-V plot (Fig. 2 (b)) shows a non-linear behav-\nior, and it is quite similar to the I-V properties of the semicon-\nducting copper oxide films18,19. These experiments reaffirm\nthe natural oxidation of Cu if not capped by any other layer.\nIn fact, a similar natural oxidation of Cu has also been re-\nported previously and has been attributed to the evolution of\nIOREE12.\nNext, we study the effect of the top CuO Xon the magnetiza-\ntion dynamics of heterostructures via FMR. The heterostruc-\ntures are placed in a flip-chip manner on CPW. Fig. S2 shows\nthe typical FMR spectra of SA1 measured in the 4-17 GHz\nrange. Similar types of FMR spectra were also recorded for\nother heterostructures. All the FMR spectra were fitted to the\nderivative of symmetric and antisymmetric Lorentzian func-\ntion to evaluate the resonance field ( Hres) and full-width-at-\nhalf-maximum of magnetic field swept-absorption ( ∆H)13,20:\nFMR Signal =K14(∆H)(H−Hres)\n[(∆H)2+4(H−Hres)2]2−K2(∆H)2−4(H−Hres)2\n[(∆H)2+4(H−Hres)2]2+Offset , (1)\nwhere K1andK2are the antisymmetric and symmetric ab-\nsorption coefficients, respectively. The Hresand∆Hextracted\nfor various resonance frequencies from the Lorentzian fit of\nthe field dependent FMR absorption are shown in Fig. 3. The\nHresdependent fof different heterostructures are plotted in\nFig. 3 (a). The fvsHresplots are fitted using the Kittel’s\nequation13,20:\nf=γ\n2πq\n(HK+Hres)(HK+Hres+4πMe f f), (2)where\n4πMe f f=4πMS+2KS\nMStFM\nandHK,KS, and tFMare the anisotropy field, perpendicular\nsurface anisotropy constant, and the thickness of FM, respec-\ntively. Further, γis the gyromagnetic ratio and 4 πMe f frepre-\nsents the effective magnetization. The 4 πMe f fextracted from\nthe fitting gives similar values as compared with the satura-4\nFIG. 2. (a) GIXRD patterns of Si/SiO 2/Cu (10 nm) and Si/SiO 2/CFB (7 nm)/Cu (10 nm)/W (5 nm) heterostructures, (b) I-V plot of Si/SiO 2/Cu\n(10 nm) heterostructure\nFIG. 3. (a) Frequency ( f) versus resonance field ( Hres) and (b) linewidth ( ∆H) versus frequency ( f) behaviour for all the heterostructures\nlisted in TABLE I. The solid lines in (a) and (b) are the best fits to equations 2 and 3, respectively.\ntion magnetization value (4 πMS) calculated from the SQUID\nVSM measurements. Further, the effective Gilbert damping\nconstant and hence, the magnetization relaxation mechanism\nare studied from the resonance frequency dependent FMR\nlinewidth behavior. The frequency domain measurement of\nferromagnetic resonance linewidth ∆Hallows the separation\nof intrinsic and extrinsic contributions to the magnetic damp-\ning by the following equation,\n∆H=∆H0+4παe f f\nγf, (3)\nwhere the ∆H0is known as the inhomogeneous linewidth and\nrepresents the extrinsic contribution to the damping of the pre-\ncessing magnetization. The value of αe f frepresents the in-\ntrinsic contribution to the damping. The ∆Hvsfplots of all\nthe heterostructures are shown in Fig. 3 (b). All the resonance\nfrequency dependent ∆Hplots are fitted by the equation 3 toevaluate the effective Gilbert damping ( αe f f) constant20. The\nlinear dependency of ∆Honfindicates the magnetic damp-\ning is mainly governed by intrinsic mechanism via electron-\nmagnon scattering rather than the extrinsic two magnon scat-\ntering. The energy during the magnetization precession can\nalso be transferred between the uniform and non-uniform pre-\ncession modes via two-magnon scattering, resulting an addi-\ntional contribution to the intrinsic damping in αe f f. This usu-\nally leads to non-linear frequency-dependent ∆Hbehavior. As\nwe do not observe the non-linearity in the ∆Hvsfplots, the\ncontribution of the two-magnon scattering can be neglected.\nThe values of αe f fof different heterostructures are shown\nin TABLE II. The αe f fof CFB/W is found to be larger\ncompared to that with CFB/CuO X, inferring possible spin-\npumping due to the presence of a high SOC β-W layer. Inter-\nestingly, the magnetic damping of CFB/W/CuO Xdecreases\nand even becomes less than that of CFB/CuO X. Such type5\nFIG. 4. (a) The magnetic field swept FMR spectra and ISHE pattern of SA1 (Si/SiO 2/CFB (7 nm)/W (5 nm)/CuOx (3 nm)) heterostructure;\n(b) FMR Spectra, VMEAS , Lorentzian Fit to VMEAS andVSYM versus applied magnetic field of SA1 (Si/SiO 2/CFB (7 nm)/W (5 nm)/CuOx (3\nnm)) heterostructure.\nTABLE II. Effective magnetic damping of various heterostructures.\nSF SA SA1 SB SB1\nαe f f(±0.0001) 0.0060 0.0070 0.0053 0.0065 0.0051\nof decrease can be termed as anti-damping, which has been\nestablished in the heterostructures without flowing any DC\ncurrent. To further confirm the anti-damping behaviour, the\ndynamics of another pair of heterostructures, SB and SB1,\nwhere a thick Cu layer is inserted between CFB and W, have\nalso been investigated. The SB heterostructure, where there\nis no top CuO Xcapping, shows an enhancement in damp-\ning similar to SA (TABLE II). While the αe f fin SB1, where\nthe CFB/Cu/W is capped with CuO Xagain becomes lower\ncompared to that of the CFB/CuO X. Hence, we can con-\nclude that the W/CuO Xinterface plays a vital role in control-\nling the magnetization precession. The naturally oxidized Cu\nlayer has proven to exhibit OREE, both experimentally and\ntheoretically8,9,10,11. Thus, we can expect the evolution of or-\nbital Rashba state at the W/CuO Xinterface. The high SOC\nof W can facilitate the spin-to-orbital conversion of angular\nmomentum, which can lead to the accumulation of OAM at\nthe W/CuO Xinterface. The accumulated OAM can be con-\nverted to the charge current via IOREE, which can induce\nanti-damping like torque similar to the anti-damping torque\nexperienced at Py/Pt interface due to the non-equilibrium spin\naccumulation4,15. As the spin diffusion length of Cu is quite\nlarge, the 10 nm-thick Cu spacer layer efficiently transports\nthe spin current from CFB to W and vice-versa. Hence, theSB1 heterostructure also exhibits the anti-damping behaviour\nsimilar to SA1 arising due to W/CuO Xinterface. The lower-\ning of magnetic damping value by the HM/CuO Xoverlayer\ncan be used as an effective tool for achieving low damping\nmagnetic materials, which is important for power-efficient\nspintronics applications.\nAs the W/CuO Xinterface induces anti-damping, it is also\nimportant to investigate the spin-to-charge conversion in these\ntypes of heterostructures. The spin pumping induced charge\ncurrent measurements were performed for all the heterostruc-\ntures. Fig. 4 (a) shows the typical field-dependent DC voltage\n(Vdc) measured across the SA1 heterostructure under FMR\nconditions. The measured DC voltage reverses its polarity\nwhen the magnetic field reverses its direction. This confirms\nthe presence of spin pumping-induced spin-to-charge conver-\nsion in our heterostructures. A similar type of ISHE has also\nbeen observed in SA. As the presence of thick Cu layer sig-\nnificantly reduces the resistance of the devices, accurate spin\npumping-induced charge current measurement of SB and SB1\nheterostructures was not possible. The lower resistances of the\nSB and SB1 samples produce voltages in the tens of nV and\nhence, the noise level plays a vital role in the measured data.\nIn order to separate the symmetric (V SYM) and asymmetric\n(VASYM ) components, the V MEAS vsHplots were fitted with\nthe following Lorentzian function:\nVdc=VSYM(∆H)2\n(∆H)2+(H−Hres)2+VASYM(∆H)(H−Hres)\n(∆H)2+(H−Hres)2(4)\nThe V MEAS , its Lorentzian fit, and V SYMobtained around the resonance condition of SA1 are shown in Fig. 4 (b).6\nFIG. 5. Angel-dependent V SYM and their fits for (a) CFB (7 nm)/W (5 nm) [SA] and (b) CFB (7 nm)/W (5 nm)/CuO X(3 nm) [SA1]\nheterostructures\nThe angle-dependent spin-to-charge conversion measure-\nments were performed for both the SA and SA1 heterostruc-\ntures. The V SYMvsφdata (Fig. 5) were fitted with the fol-\nlowing equation to exclude the spin rectification effects and\nevaluate the spin pumping voltage (V SP)13:\nVSYM=VSPCos3(φ)+VAHECos(φ)Cos(θ)\n+VAMR⊥\nSYM Cos(2φ)Cos(φ)+VAMR∥\nSYMSin(2φ)Cos(φ)(5)\nThe V SPfor SA and SA1 heterostructures was found to be\n∼5.6 µV and ∼4.9 µV , respectively. The small reduction\nof V SPof CFB/W/CuO Xheterostructure compared to CFB/W\nbilayer can be due to IOREE induced spin-to-charge conver-\nsion current at the W/CuO Xinterface, which is in opposite\ndirection to the already present ISHE-induced charge current\nin CFB/W. Here, We cannot also completely neglect the ef-\nfect of slight variation in W thickness, which can induce a\nsmall change in V SP. However, the anti-damping phenomena,\nwhich mainly corroborate the spin accumulation at W/CuO X\ninterface, infer the IOREE mechanism to be the most promi-\nnent one.The magnetic damping in FM/HM bilayer usually\nincreases when the spin pumping-induced spin-to-charge con-\nversion is present in the system. This can also be seen in\nour CFB/W bilayer. Whereas, interestingly, the presence of\nthe top CuO Xcapping layer reduces the magnetic damping\nwithout much affecting the spin-to-charge conversion current.\nThus, this type of system, where the reduction in damping\nand efficient spin-to-charge interconversion are observed si-\nmultaneously, can bring about a paradigm shift in spintronics\ndevice applications. It can certainly pave the path for the de-\nvelopment of low-operating-power spintronics devices.\nIV. CONCLUSION\nIn conclusion, we have presented the control of mag-\nnetic damping of FM/HM bilayer when capped with sur-face oxidized Cu. The CuO Xlayer lowers the damping\nvalue of CoFeB/W/CuO Xtrilayer even below that for the\nCoFeB/CuO X. The strong anti-damping via CuO Xcapping is\naccompanied by a very small reduction in the spin-to-charge\ninterconversion phenomena as evident in the spin pumping-\nFMR experiment. The rich spin-orbital physics leading to the\ninverse orbital Rashba effect at HM/CuO Xinterface can at-\ntribute to the simultaneous observation of anti-damping and\nsizable spin-pumping induced output voltage.\nACKNOWLEDGMENTS\nWe acknowledge the Department of Atomic Energy (DAE),\nthe Department of Science and Technology (DST) of the Gov-\nernment of India, SERB project CRG/2021/001245. A.S.\nacknowledges the DST-National Postdoctoral Fellowship in\nNano Science and Technology.\nREFERENCE\n1S. Bhatti, R. Sbiaa, A. Hirohata, H. Ohno, S. Fukami, and S. Pira-\nmanayagam, “Spintronics based random access memory: a review,” Ma-\nterials Today 20, 530–548 (2017).\n2Q. Shao, P. Li, L. Liu, H. Yang, S. 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Tiusan, “Ferromagnetic-resonance-induced spin pumping\nin Co 20Fe60B20/Pt systems: Damping investigation,” Journal of Physics D:\nApplied Physics 51, 045002 (2018)." }, { "title": "1705.07489v2.Dynamical_depinning_of_chiral_domain_walls.pdf", "content": "Dynamical depinning of chiral domain walls\nSimone Moretti,\u0003Michele Voto, and Eduardo Martinez\nDepartment of Applied Physics, University of Salamanca, Plaza de los Caidos, Salamanca 37008, Spain.\nThe domain wall depinning \feld represents the minimum magnetic \feld needed to move a domain\nwall, typically pinned by samples' disorder or patterned constrictions. Conventionally, such \feld\nis considered independent on the Gilbert damping since it is assumed to be the \feld at which the\nZeeman energy equals the pinning energy barrier (both damping independent). Here, we analyse\nnumerically the domain wall depinning \feld as function of the Gilbert damping in a system with per-\npendicular magnetic anisotropy and Dzyaloshinskii-Moriya interaction. Contrary to expectations,\nwe \fnd that the depinning \feld depends on the Gilbert damping and that it strongly decreases for\nsmall damping parameters. We explain this dependence with a simple one-dimensional model and\nwe show that the reduction of the depinning \feld is related to the \fnite size of the pinning barriers\nand to the domain wall internal dynamics, connected to the Dzyaloshinskii-Moriya interaction and\nthe shape anisotropy.\nI. INTRODUCTION\nMagnetic domain wall (DW) motion along ferromag-\nnetic (FM) nanostructures has been the subject of in-\ntense research over the last decade owing to its po-\ntential for new promising technological applications1,2\nand for the very rich physics involved. A consider-\nable e\u000bort is now focused on DW dynamics in systems\nwith perpendicular magnetic anisotropy (PMA) which\npresent narrower DWs and a better scalability. Typ-\nical PMA systems consist of ultrathin multi-layers of\nheavy metal/FM/metal oxide (or heavy metal), such as\nPt=Co=Pt3,4or Pt=Co=AlOx5{7, where the FM layer has\na thickness of typically 0 :6\u00001 nm. In these systems,\nPMA arises mainly from interfacial interactions between\nthe FM layer and the neighbouring layers (see Ref.8and\nreferences therein). Another important interfacial ef-\nfect is the Dzyaloshinskii-Moriya interaction (DMI)9,10,\npresent in systems with broken inversion symmetry such\nas Pt/Co/AlOx. This e\u000bect gives rise to an internal in-\nplane \feld that \fxes the DW chirality (the magnetization\nrotates always in the same direction when passing from\nup to down and from down to up domains) and it can\nlead to a considerably faster domain wall motion10and to\nnew magnetic patterns such as skyrmions11or helices12.\nNormally, DWs are pinned by samples' intrinsic disorder\nand a minimum propagation \feld is needed in order to\novercome such pinning energy barrier and move the DW.\nSuch \feld is the DW depinning \feld ( Hdep) and it repre-\nsents an important parameter from a technological point\nof view since a low depinning \feld implies less energy\nrequired to move the DW and, therefore, a energetically\ncheaper device.\nFrom a theoretical point of view, DW motion can be\ndescribed by the Landau-Lifshitz-Gilbert (LLG) equa-\ntion13which predicts, for a perfect sample without dis-\norder, the velocity vs\feld curve depicted in Fig. 1 and\nlabelled as Perfect . In a disordered system, experi-\nments have shown that a DW moves as a general one-\ndimensional (1D) elastic interface in a two-dimensional\ndisordered medium3,4and that it follows a theoreticalvelocityvsdriving force curve, predicted for such inter-\nfaces14,15(also shown in Fig. 1 for T= 0 andT= 300K).\nMoreover, this behaviour can be reproduced by including\ndisorder in the LLG equation16{18. At zero temperature\n(T= 0) the DW does not move as long as the applied\n\feld is lower than Hdep, while, at T6= 0, thermal ac-\ntivation leads to DW motion even if H < H dep(the so\ncalled creep regime). For high \felds ( H >> H dep) the\nDW moves as predicted by the LLG equation in a per-\nfect system. Within the creep theory, the DW is con-\nsidered as a simple elastic interface and all its internal\ndynamics are neglected. Conventionally, Hdepis consid-\nered independent of the Gilbert damping because it is as-\nsumed to be the \feld at which the Zeeman energy equals\nthe pinning energy barrier19,20(both damping indepen-\ndent). Such assumption, consistently with the creep the-\nory, neglects any e\u000bects related to the internal DW dy-\nnamics such as DW spins precession or vertical Bloch\nlines (VBL) formation21. The damping parameter, for\nits part, represents another important parameter, which\ncontrols the energy dissipation and a\u000bects the DW veloc-\nity and Walker Breakdown22. It can be modi\fed by dop-\ning the sample23or by a proper interface choice as a con-\nsequence of spin-pumping mechanism24. Modi\fcations of\nthe DW depinning \feld related to changes in the damping\nparameter were already observed in in-plane systems23,25\nand attributed to a non-rigid DW motion23,25. Oscilla-\ntions of the DW depinning \feld due to the internal DW\ndynamics were also experimentally observed in in-plane\nsimilar systems26. Additional dynamical e\u000bects in soft\nsamples, such as DW boosts in current induced motion,\nwere numerically predicted and explained in terms of DW\ninternal dynamics and DW transformations27,28.\nHere, we numerically analyse the DW depinning \feld\nin a system with PMA and DMI as function of the Gilbert\ndamping. We observe a reduction of Hdepfor low damp-\ning and we explain this behaviour by adopting a simple\n1D model. We show that the e\u000bect is due to the \fnite\nsize of pinning barriers and to the DW internal dynam-\nics, related to the DMI and shape anisotropy \felds. This\narticle is structured as follows: in Section II we present\nthe simulations method, the disorder implementation andarXiv:1705.07489v2 [cond-mat.mes-hall] 25 Aug 20172\ntheHdepcalculations. The main results are outlined and\ndiscussed in Section III, where we also present the 1D\nmodel. Finally, the main conclusions of our work are\nsummarized in Section IV.\n●●●●\n●\n●\n●\n●●●●●●●●●●�=��\n�=����\n●�������\n������� ������� ��������\n����★\nFIG. 1. DW velocity vsapplied \feld as predicted by the LLG\nequation in a perfect system and by the creep law atT= 0\nandT= 300K.\nII. MICROMAGNETIC SIMULATIONS\nWe consider a sample of dimensions\n(1024\u00021024\u00020:6) nm3with periodic bound-\nary conditions along the ydirection, in order to simulate\nan extended thin \flm. Magnetization dynamics is\nanalysed by means of the LLG equation13:\ndm\ndt=\u0000\r0\n1 +\u000b2(m\u0002He\u000b)\u0000\r0\u000b\n1 +\u000b2[m\u0002(m\u0002He\u000b)];\n(1)\nwhere m(r;t) =M(r;t)=Msis the normalized magneti-\nzation vector, with Msbeing the saturation magnetiza-\ntion.\r0is the gyromagnetic ratio and \u000bis the Gilbert\ndamping. He\u000b=Hexch+HDMI+Han+Hdmg+Hz^uz\nis the e\u000bective \feld, including the exchange, DMI, uni-\naxial anisotropy, demagnetizing and external \feld con-\ntributions13respectively. Typical PMA samples param-\neters are considered: A= 17\u000210\u000012J=m,Ms= 1:03\u0002\n106A=m,Ku= 1:3\u0002106J=m3andD= 0:9 mJ=m2,\nwhereAis the exchange constant, Dis the DMI constant\nandKuis the uniaxial anisotropy constant. Disorder is\ntaken into account by dividing the sample into grains\nby Voronoi tessellation29,30, as shown in Fig. 2(a). In\neach grain the micromagnetic parameters fMs;Dc;Kug\nchange in a correlated way in order to mimic a normally\ndistributed thickness31:\ntG=N(t0;\u000e)!8\n<\n:MG= (MstG)=t0\nKG= (Kut0)=tG\nDG= (Dct0)=tG; (2)\nwhere the subscript Gstands for grain, t0is the aver-\nage thickness ( t0= 0:6nm) and\u000eis the standard devi-\nation of the thickness normal distribution. The sample\nis discretized in cells of dimensions (2 \u00022\u00020:6)nm3,smaller than the exchange length lex\u00185nm. Grain size\nis GS=15 nm, reasonable for these materials, while the\nthickness \ructuation is \u000e= 7%. Eq. (1) is solved by the\n\fnite di\u000berence solver MuMax 3.9.329.\nA DW is placed and relaxed at the center of the sample\nas depicted in Fig. 2(b). Hdepis calculated by applying\na sequence of \felds and running the simulation, for each\n\feld, until the DW is expelled from the sample, or until\nthe system has reached an equilibrium state (i.e. the DW\nremains pinned): \u001cmax<\u000f(\u000b).\u001cmaxindicates the maxi-\nmum torque, which rapidly decreases when the system is\nat equilibrium. It only depends on the system parame-\nters and damping. For each value of \u000b, we choose a spe-\nci\fc threshold, \u000f(\u000b), in order to be sure that we reached\nan equilibrium state (see Supplementary Material32for\nmore details). The simulations are repeated for 20 dif-\nferent disorder realizations. Within this approach, Hdep\ncorresponds to the minimum \feld needed to let the DW\npropagate freely through the whole sample. In order to\navoid boundaries e\u000bects, the threshold for complete de-\npinning is set tohmzi>0:8, wherehmziis averaged over\nall the realizations, i.e. hmzi=PN\ni=1hmzii=N, where\nN= 20 is the number of realizations. We checked that,\nin our case, this de\fnition of Hdepcoincides with tak-\ningHdep= MaxfHi\ndepg, withHi\ndepbeing the depinning\n\feld of the single realization. In other words, Hdepcor-\nresponds to the minimum \feld needed to depin the DW\nfrom any possible pinning site considered in the 20 real-\nizations33.\nFollowing this strategy, the DW depinning \feld is nu-\nmerically computed with two di\u000berent approaches:\n(1) by Static simulations, which neglect any precessional\ndynamics by solving\ndm\ndt=\u0000\r0\u000b\n1 +\u000b2[m\u0002(m\u0002He\u000b)]: (3)\nThis is commonly done when one looks for a minimum\nof the system energy and it corresponds to the picture\nin whichHdepsimply depends on the balance between\nZeeman and pinning energies.34\n(2) by Dynamic simulations, which include precessional\ndynamics by solving the full Eq. (1). This latter method\ncorresponds to the most realistic case. Another way to\nestimate the depinning \feld is to calculate the DW veloc-\nityvs\feld curve at T= 0 and look for minimum \feld at\nwhich the DW velocity is di\u000berent from zero. For these\nsimulations we use a moving computational region and\nwe run the simulations for t= 80ns (checking that longer\nsimulations do not change the DW velocity, meaning that\nwe reached a stationary state). This second setup re-\nquires more time and the calculations are repeated for\nonly 3 disorder realizations.\nUsing these methods, the depinning \feld Hdepis cal-\nculated for di\u000berent damping parameters \u000b.3\n(a) (b)\nxy\n(c)\nFIG. 2. (a) Grains structure obtained by Voronoi tassellation.\n(b) Initial DW state. (c) Sketch of the internal DW angle \u001e.\nIII. RESULTS AND DISCUSSION\nA. Granular system\nOur \frst result is shown in Fig. 3(a)-(b), which depicts\nthe \fnal average magnetization hmzias function of the\napplied \feld for di\u000berent damping parameters. In the\nStatic simulations (Fig. 3(a)) Hdepdoes not depend on\ndamping, so that a static depinning \feld can be de\fned.\nConversely, in the Dynamic simulations (Fig. 3(b)), Hdep\ndecreases for low damping parameters. The depinning\n\feld is indicated by a star in each plot and the static\ndepinning \feld is labelled as Hs. The same result is ob-\ntained by calculating Hdepfrom the DW velocity vsap-\nplied \feld plot, shown in Fig. 3(c). The stars in Fig. 3(c)\ncorrespond to the depinning \felds calculated in the pre-\nvious simulations and they are in good agreement with\nthe values predicted by the velocity vs\feld curve. The\ndynamical depinning \feld \u00160Hd, normalized to the static\ndepinning \feld \u00160Hs= (87\u00061)mT, with \u00160being the\nvacuum permeability, is shown in Fig. 3(d) as function of\nthe damping parameter \u000b.Hdsaturates for high damp-\ning (in this case \u000b\u00150:5) while it decreases for low damp-\ning untilHd=Hs\u00180:4 at\u000b= 0:02. This reduction must\nbe related to the precessional term, neglected in the static\nsimulations. The same behaviour is observed with di\u000ber-\nent grain sizes (GS=5 and 30 nm) and with a di\u000berent\ndisorder model, consisting of a simple variation of the Ku\nmodule in di\u000berent grains. This means that the e\u000bect is\nnot related to the grains size or to the particular disorder\nmodel we used.\nAdditionally, Fig. 4 represents the DW energy35as\nfunction of DW position and damping parameter for\n\u00160Hz= 70 mT. At high damping, the average DW en-\nergy density converges to \u001b1\u001810 mJ=m2, in good agree-\nment with the analytical value \u001b0= 4pAK0\u0000\u0019D=\n10:4 mJ=m2, whereK0is the e\u000bective anisotropy K0=\nKu\u0000\u00160M2\ns=2. On the contrary, for low damping, the\nDW energy increases up to \u001b(0:02)\u001814 mJ=m2. This\nincrease, related to DW precessional dynamics, reduces\nthe e\u000bective energy barrier and helps the DW to over-\n●●●●●●●●●●●●●●●●●\n○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■\n●α=����\n○α=���\n■α=���\n���������������������<��> ★(�) ������\n��\n●●●●●●●●●●●●●●●●●\n○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■\n�� �� �� ��������������������������\n������� �����(��)<��>★★★(�) �������\n●●●●●●●●●○○○○○○○○○■■■■■■■■■●α=����\n○α=���\n■α=���\n�� �� �� ��������������\n������� �����(��)�� ��������(�/�)★★★(�)\n●●●●● ● ●\n������������������������������������\n�������α��/��(�)FIG. 3. Average hmzias function of applied \feld for dif-\nferent damping parameters for the (a) Static simulations and\n(b)Dynamic simulations. (c) DW velocity vs applied \feld for\ndi\u000berent damping. (d) Dynamical depinning \feld, normalized\ntoHs, as function of damping.\ncome the pinning barriers. Fig. 4(c) shows the total en-\nergy of the system (including Zeeman). As expected36,\nthe energy decreases as the DW moves.\nFinally, Fig. 5 shows the DW motion as function of\ntime for\u000b= 0:02 and\u000b= 0:5, along the same grain\npattern (and therefore along the same pinning barriers).\nThe applied \feld is \u00160Hz= 70mT, which satis\fes\nHd(0:02) (��/��)(�)\n●●●●●●●●●●●●\n●●●●●●●●●●●●○○○○○○○○○○○○○○○○○○○○○○○■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■\n���������������-��-�������\n�� ��������(��)����� ������ �������(��/��)\n(�)\nFIG. 4. (a) DW energy density as function of DW posi-\ntion for di\u000berent damping. The \fnal drop corresponds to\nthe expulsion of the DW. (b) Average DW density as funci-\nton of damping. Dashed line represents the analytical value\n\u001b1\u001810 mJ=m2. (c) Total energy density of the system as\nfunction of DW position for di\u000berent damping parameters.\nof VBL and their complex dynamics21. Further insights\nabout this mechanism are given by analysing the DW\ndepinning at a single energy barrier as described in the\nnext subsection.\nB. Single barrier\nIn order to understand how the DW precessional dy-\nnamics reduces Hdep, we micromagnetically analysed the\nDW depinning from a single barrier as sketched in Fig. 6.\nWe considered a strip of dimensions (1024 \u0002256\u00020:6)nm3\nand we divided the strip into two regions, R1andR2,\nwhich are assumed to have a thickness of t1= 0:58 and\nt2= 0:62 nm respectively. Their parameters vary ac-\ncordingly (see Sec. II), generating the DW energy bar-\nrier (\u000e\u001b) shown in Fig. 6(b). A DW is placed and re-\nlaxed just before the barrier. The \fnite size of the DW\n(\u0019\u0001DW\u001815 nm, with \u0001 DWbeing the DW width pa-\nrameter) smooths the abrupt energy step and, in fact,\nthe energy pro\fle can be successfully \ftted by using theBloch pro\fle22\n\u001bDW=\u001b0+\n+\u0012\u000e\u001b\n2\u0013\u001a\n1 + cos\u0012\n2 arctan\u0014\nexp\u0012x0\u0000x\n\u0001DW\u0013\u0015\u0013\u001b\n;\n(4)\nwherex0= 20 nm is the step position, while \u001b0and\n\u001b1are the DW energies at the left and right side of the\nbarrier as represented in Fig. 6(b). This means that\nthe pinning energy barrier has a spatial extension which\nis comparable to the DW width. By performing the\nsame static and dynamic simulations, we obtain a static\ndepinning \feld of \u00160Hs= 120 mT and, when decreasing\nthe damping parameter, we observe the same reduction\nof the depinning \feld as in the granular system (see\nFig. 6(c)). In this case the DW behaves like a rigid\nobject whose spins precess coherently and no VBL\nnucleation is observed. Hence, Hdepreduction does not\ndepend directly on the presence of VBL but on the more\ngeneral mechanism of spins' precession already present\nin this simpli\fed case.\nNevertheless, an important characteristic of these single\nbarrier simulations is that the barrier is localized and it\nhas a \fnite size which is of the order of the DW width.\nNote that the same holds for the granular system:\ndespite a more complex barrier structure, the dimension\nof the single barrier between two grains has the size of\nthe DW width.\nThus, in order to understand the interplay between the\nDW precessional dynamics and the \fnite size of the bar-\nrier, we considered a 1D collective-coordinate model with\na localized barrier. The 1D model equations, describing\nthe dynamics of the DW position qand the internal angle\n\u001e(sketched in Fig. 2(c)), are given by16\n(1 +\u000b2)_\u001e=\r0[(Hz+Hp(q))\n\u0000\u000b\u0012\nHKsin 2\u001e\n2\u0000\u0019\n2HDMIsin\u001e\u0013\n|{z }\nHint(\u001e)];(5)\n(1 +\u000b2)_q\n\u0001DW=\r0[\u000b(Hz+Hp(q))\n+\u0012\nHKsin 2\u001e\n2\u0000\u0019\n2HDMIsin\u001e\u0013\u0015\n;(6)\nwhereHK=MsNxis the shape anisotropy \feld, favour-\ning Bloch walls, with Nx=t0log 2=(\u0019\u0001DW)37being the\nDW demagnetizing factor along the xaxis.HDMI =\nD=(\u00160Ms\u0001DW) is the DMI \feld. Hint(\u001e) represents\nthe internal DW \feld, which includes DMI and shape\nanisotropy. Hintfavours Bloch ( \u001e=\u0006\u0019=2) or N\u0013 eel wall\n(\u001e= 0 or\u001e=\u0019) depending on the relative strength\nofHKandHDMI. In our system, the DMI dominates\nover shape anisotropy since \u00160HDMI\u0018170 mT while\n\u00160HK\u001830 mT. Hence, the DW equilibrium angle is5\nOut[64]=\nOut[60]=\n… (a) 𝜶=𝟎.𝟎𝟐time 0 0.1 ns 0.2 ns 0.3 ns 4 ns\ntime 0 0.1 ns 0.2 ns 0.3 ns 4 ns(b) 𝜶=𝟎.𝟓\n… \nOut[395]=mx\nOut[395]=mx\nOut[62]=\nOut[65]=\nFIG. 5. (a) Snapshots of the magnetization dynamics at subsequent instants under \u00160Hz= 70mT, for two di\u000berent damping:\n(a)\u000b= 0:02 and (b) \u000b= 0:5. The grains pattern, and therefore the energy barrier, is the same for both cases. In order to let\nthe DW move across more pinning sites, these simulations were performed on a larger sample with Lx= 2048 nm.\n\u001e=\u0019(\u001e= 0 or\u001e=\u0019additionally depends on the sign\nof the DMI). Hp(q) is the DW pinning \feld, obtained\nfrom the DW energy pro\fle (Eq. (4)) as follows: the max-\nimum pinning \feld is taken from the static simulations\nwhile the shape of the barrier is taken as the normalized\nDW energy gradient (see Supplementary Material32for\nmore details),\nHp(q) =Hs\u0012@\u001bDW(x)\n@x\u0013\nN=\n= 2Hsexp\u0010\nx0\u0000q\n\u0001DW\u0011\nsinh\n2 arctan\u0010\nexp\u0010\nx0\u0000q\n\u0001DW\u0011\u0011i\n1 + exp\u0010\n2(x0\u0000q)\n\u0001DW\u0011 :(7)\nThe corresponding pinning \feld is plotted in Fig. 7(a).38\nThe results for the dynamical Hdep, obtained with this\nmodi\fed 1D model, are plotted in Fig. 6(c) and they\nshow a remarkable agreement with the single barrier mi-\ncromagnetic simulations. This indicates that the main\nfactors responsible for the reduction of Hdepare already\nincluded in this simple 1D model. Therefore, additional\ninsights might come from analysing the DW dynamics\nwithin this 1D model. Fig. 7(b) and (c) represents the\nDW internal angle \u001eand the DW position qas function\nof time for di\u000berent damping. The plots are calculated\nwith\u00160Hz= 55 mT which satis\fes Hdep(0:02)< Hz<\nHdep(0:1)< H dep(0:5). As shown in Fig. 7(b) and (c),\nbelow the depinning \feld ( \u000b= 0:1,\u000b= 0:5), both the\ninternal angle and the DW position oscillate before reach-\ning the same \fnal equilibrium state. However, the am-plitude of these oscillations (the maximum displacement)\ndepends on the damping parameter. Fig. 7(d) shows the\n\fnal equilibrium position as function of the applied \feld\nfor di\u000berent damping. The equilibrium position is the\nsame for all damping and it coincides with the position\nat whichHz=Hp(q). Conversely, the maximum dis-\nplacement, shown in Fig. 7(e), strongly increases for low\ndamping parameters. For applied \feld slightly smaller\nthan the depinning \feld, the DW reaches the boundary\nof the pinning barrier, meaning that a further increase\nof the \feld is enough to have a maximum displacement\nhigher than the barrier size and depin the DW. In other\nwords, the decrease of the depinning \feld, observed in\nthe single barrier simulations, is due to DW oscillations\nthat depend on \u000band that can be larger than the bar-\nrier size, leading to DW depinning for lower \feld. The\nDW dynamics and the depinning mechanism are further\nclari\fed in Fig. 7(f) and Fig. 7(g). Fig. 7(f) represents\nthe DW coordinates fq;\u001egfor\u00160Hz= 55 mT and dif-\nferent damping. Before reaching the common equilib-\nrium state, the DW moves in orbits (in the fq;\u001egspace)\nwhose radius depends on the damping parameter. For\n\u000b= 0:5 (black line) the DW rapidly collapse into the \f-\nnal equilibrium state. Conversely, for \u000b= 0:1 (red open\ncircles), the DW orbits around the equilibrium state be-\nfore reaching it. If the radius of the orbit is larger than\nthe barrier size the DW gets depinned, as in the case\nof\u000b= 0:02 (blue full circles). This mechanism is also\nrepresented in Fig. 7(g), where the DW orbits are placed\nin the energy landscape. The energy is calculated as6\n●●●●●●●●●●●●●●●●●μ� ����������� ��●�� ��������������������������������������������\n�������α��/��○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○�� �������������(�)-���������������������������������������(��)σ��(��/��)��δσσ�σ�(b)\n(c)\nR1R2yx(a)\nFIG. 6. (a) Sketch of the two regions implemented for the\nsingle barrier (SB) micromagnetic simulations. (b) DW en-\nergy as function of DW position along the strip. Blue solid\nline represents the analytical value, red points the DW con-\nvoluted energy (due to the \fnite size of the DW) while black\ndashed line a \ft using Eq. 4. (c) Dynamical depinning \feld,\nnormalized to the static depinning \feld, for the single bar-\nrier simulations as function of damping, obtained from full\nmicromagnetic simulations and the 1D model.\n\u001b(q;\u001e) =\u001bDW(q;\u001e)\u00002\u00160MsHzq, where\u001bDWis given by\nEq. (4). Fig. 7(g) shows that the equilibrium state cor-\nresponds to the new minimum of the energy landscape.\nFurthermore, it con\frms that the applied \feld is below\nthe static depinning \feld, at which the pinning barrier\nwould have been completely lifted. Nevertheless, while\nreaching the equilibrium state, the DW moves inside the\nenergy potential and, if the radius of the orbit is larger\nthan the barrier size, the DW can overcome the pinning\nbarrier, as shown for \u000b= 0:02 in Fig. 7(g).\nAt this point we need to understand why the amplitude\nof the DW oscillations depends on damping. By solving\nEq. (5) and Eq.(6) for the equilibrium state ( _ q= 0, _\u001e=\n0) we obtain\n_q= 0)jHp(q)j=Hz+Hint(\u001e)\n\u000b\n\u0019Hz\u0000\u0019\n2HDMI\n\u000bsin\u001e; (8)\n_\u001e= 0)jHp(q)j=Hz\u0000\u000bHint(\u001e)\n\u0019Hz+\u000b\u0019\n2HDMIsin\u001e; (9)\nsince\u00160HDMI\u001d\u00160HKand, therefore, Hint\u0019\n\u0000(\u0019=2)HDMIsin\u001e. These equations have a single com-\nmon solution which corresponds to jHp(q)j=Hzand\n\u001e=\u001e0=\u0019(at whichHint(\u0019) = 0). However, at t= 0,the DW starts precessing under the e\u000bect of the applied\n\feld and, if \u001e6=\u0019whenjHp(q)j=Hz, the DW does not\nstop at the \fnal equilibrium position but it continues its\nmotion, as imposed by Eq. (8) and (9). In other words,\nthe DW oscillations in Fig. 7(b) are given by oscillations\nof the DW internal angle \u001e, around its equilibrium value\n\u001e0=\u0019. These oscillations lead to a modi\fcation of the\nDW equilibrium position due to the DW internal \feld\n(Hint(\u001e)), which exerts an additional torque on the DW\nin order to restore the equilibrium angle. As previously\ncommented, if the amplitude of these oscillations is large\nenough, the DW gets depinned. From Eq. (8) we see\nthat the new equilibrium position (and therefore the am-\nplitude of the oscillations) depends on the DMI \feld, the\nvalue of the DW angle \u001eand the damping parameter.\nIn particular, damping has a twofold in\ruence on this\ndynamics: one the one hand, it appears directly in\nEq. (8), dividing the internal \feld, meaning that for the\nsame deviation of \u001efrom equilibrium, we have a stronger\ninternal \feld for smaller damping. On the other hand,\nthe second in\ruence of damping is on the DW internal\nangle: once the DW angle has deviated from equilibrium,\nthe restoring torque due to DMI is proportional to the\ndamping parameter (see Eq. (9)). Hence, a lower damp-\ning leads to lower restoring torque and a larger deviation\nof\u001efrom equilibrium. The maximum deviation of \u001efrom\nequilibrium ( \u000e\u001e=\u001emax\u0000\u001e0) is plotted in Fig. 8(b) as\nfunction of damping for \u00160Hz= 40 mT. As expected, a\nlower damping leads to a larger deviation \u000e\u001e.\nIn this latter section, the DW was set at rest close to\nthe barrier and, therefore, the initial DW velocity is zero.\nNevertheless, one might wonder what happens when the\nDW reaches the barrier with a \fnite velocity. We simu-\nlated this case by placing the DW at an initial distance\nd1= 200 nm from the barrier. The depinning is further\nreduced in this case (see Supplementary Material32for\nmore details). However, in the static simulations, the de-\npinning \feld remains constant, independently from the\nvelocity at which the DW reaches the barrier, meaning\nthat the reduction of Hdepis again related to the DW\nprecession. When the DW starts from d1it reaches the\nbarrier precessing, thus with a higher deviation from its\nequilibrium angle, leading to a higher e\u000bect of the inter-\nnal \feld.\nC. Di\u000berent DMI and pinning barriers\nFinally, by using the 1D model it is possible to ex-\nplore the dependence of Hdepon the pinning potential\namplitudeHs(related to the disorder strength) and on\nthe DMI constant D. The depinning \feld as function of\ndamping for di\u000berent values of Hsis plotted in Fig. 9(a).\nThe reduction of Hdepis enhanced for larger values of\nHs(strong disorder). This is consistent with our expla-\nnation, since for strong disorder we need to apply larger\n\felds that lead to larger oscillations of \u001e.\nFig. 9(b) represents the dynamical Hdepas function of7\n●●●●●●●●●●●●●●●●●●●●●●●●\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼�����������������������������������(��)�� ��������(��)●●●●●●●●■■■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆◆◆▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼○○○○○○○○○○○○○○○○○○○○○○●α=����■α=���◆α=���▲α=����▼α=���○α=�������������������������������������������������\n������� �����μ���(��)���������������(��)������������(��)●●●●●●●■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆◆▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼○○○○○○○○○○○○○○○○○○○○○●α=����■α=���◆α=���▲α=����▼α=���○α=�������������������������������������������������\n������� �����μ���(��)�����������(��)������������(��)-�������������-���-��-�������������(��)μ���(��)\nMax Displacement\nEq. Position(a)(b)(c)(d)\n(e)(f)\n(g)●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ 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▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼●α=����○α=���α=������������������������������������\n�(��)ϕ(°)\nFIG. 7. (a) Pinning \feld obtained from Eq. (7) as function of DW position. DW position internal angle \u001eas function of\ntime for di\u000berent damping parameter and \u00160Hz= 55 mT. (c) DW position qas function of time for di\u000berent damping and\n\u00160Hz= 55 mT. (d) Equilibrium position as function of applied \feld for di\u000berent damping. (e) Maximum DW displacement as\nfunction of the applied \feld for di\u000berent damping. (f) DW coordinates fq;\u001egfor\u00160Hz= 55 mT and di\u000berent damping. (g)\nDW coordinatesfq;\u001eginside the energy landscape: \u001b=\u001bDW(q;\u001e)\u00002\u00160MsHzq.\n������������������������������\n�������αδϕ(°)\nFIG. 8. Maximum deviation of \u001efrom its equilibrium posi-\ntion as function of damping.\ndamping for \u00160Hs= 120 mT and di\u000berent DMI con-\nstants (expressed in term of the critical DMI constant\nDc= 4pAK0=\u0019= 3:9 mJ=m2)39. In this case, the reduc-\ntion ofHdepis enhanced for low DMI, until D= 0:05Dc,\nbut a negligible reduction is observed for D= 0. This\nnon-monotonic behaviour can be explained by looking at\nthe dependence of \u000e\u001eandHinton the DMI constant.\nFig. 10(a) shows the maximum \ructuation \u000e\u001eas func-\ntion of DMI for \u00160Hz= 30 mT. \u000e\u001eincreases for low\nDMI and it has a maximum at \u0019HDMI =HK, which\nin our case corresponds to D= 0:014Dc. The increase\nof\u000e\u001efor small values of Dis due to the smaller restor-\ning torque in Eq. (9). This holds until \u0019HDMI =HK,\nwhere shape anisotropy and DMI are comparable and\nthey both a\u000bect the DW equilibrium con\fguration. As a\nconsequence, the reduction of Hdepis enhanced by de-creasingDuntilD\u00180:014Dc, while it is reduced if\n0< D < 0:014Dc. Another contribution is given by\nthe amplitude of the internal \feld, Hint. Fig. 10(b) de-\npicts\u00160Hintas function of \u000e\u001eandD. The maximum\n\u000e\u001e, obtained at \u00160Hz= 30 mT, is additionally marked\nin the plot. The internal \feld decreases with the DMI\nbut this reduction is compensated by an increase in \u000e\u001e,\nwhich leads to an overall increase of \u00160Hint, as discussed\nin the previous part. However, at very low DMI, the in-\nternal \feld is dominated by shape anisotropy and, inde-\npendently on the DW angle displacement, it is too small\nto have an e\u000bect on the depinning mechanism. Note,\nhowever, that the amplitude of Hintshould be compared\nwith the amplitude of the pinning barrier Hs. Fig. 9(b)\nis calculated with \u00160Hs= 120 mT and the internal \feld,\ngiven by shape anisotropy ( HK=2\u001815 mT), has indeed\na negligible e\u000bect. However, larger e\u000bects are observed,\nin the caseD= 0, for smaller Hs, with reduction of Hdep\nup toHd=Hs\u00180:6, as shown in Fig. 9(c), which is calcu-\nlated with\u00160Hs= 30 mT. In other words, the reduction\nof the depinning \feld depends on the ratio between the\npinning barrier and the internal DW \feld.\nFinally, it is interesting to see what happens for\nweaker disorder and di\u000berent DMI in the system with\ngrains. Fig. 11 shows the dynamical Hdep, for di\u000berent\npinning potential and di\u000berent DMI, obtained in the\ngranular system. The results are in good agreement with\nwhat predicted by the 1D model for di\u000berent disorder\nstrengths. However, we observe a smaller dependence\non the DMI parameter. This is due to two reasons:8\n●●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ●\n■■■■■■■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■\n◆◆◆◆◆◆◆◆◆◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲\n●μ���=�� ��\n■μ���=�� ��\n◆μ���=�� ��\n▲μ���=��� ��\n������������������������������������\n�������α��/��(�)\n●●●●●●●●●●●●●●●●●●●●\n■■■■■■■■■■■■■■■■■■■■\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼\n○○○○○○○○○○○○○○○○○○○○\n□□□□□□□□□□□□□□□□□□□□\n●�=���� � �■�=��� � �\n◆�=��� � �▲�=��� � �\n▼�=��� � �○�=��� � � □�=���\n���������������������������������������\n�������α��/��(�) μ���=��� ��\n●●●●●●●●●●●●●●●●●●●●\n■■■■■■■■■■■■■■■■■■■■\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼\n○○○○○○○○○○○○○○○○○○○○\n□□□□□□□□□□□□□□\n□□□□□□\n●�=���� � �■�=���� � �\n◆�=���� � �▲�=���� � �\n▼�=���� � �○�=���� � � □�=���\n���������������������������������������\n�������α��/��(�) μ���=�� ��\nFIG. 9. (a) Dynamical Hdepas function of damping for di\u000ber-\nentHs(disorder strength). (b) Dynamical Hdepas function\nof damping for di\u000berent DMI constant and \u00160Hs= 120 mT.\n(c) Dynamical Hdepas function of damping for di\u000berent DMI\nconstant and \u00160Hs= 30 mT.\n(1) in the system with grains the static pinning barrier\nis\u00160Hs= 87 mT and the dependence of the depinning\n\feld with DMI is smaller for smaller barriers, as shown\nin Fig. 9(c). (2) The DW motion in the granular\nsystem presents the formation of VBL which might also\ncontribute to the reduction of the depinning \feld. The\nmechanism is the same: a VBL is a non-equilibrium\ncon\fguration for the DW (as a deviation of \u001efrom\nequilibrium) that generates additional torques on the\nDW, which contribute to the DW depinning.\n����� ���� ��� �������������\n�/��δϕ(°)(�)\nπ����=��\nμ�����(��)\n� ��� ��� ��� ���\n������������������������������\n�/��δϕ(°)(�)FIG. 10. (a) Max DW angle \ructuation \u000e\u001e=\u001emax\u0000\u001eeq\nas function of DMI for \u00160Hz= 30 mT. (b) Internal DW\n\feld\u00160Hintas function of DMI and \u000e\u001e. The green points\ncorrespond the max \ructuation plotted in (a). Note that the\nscale is logarithmic in (a).\n●●●●● ● ●\n□□□□□ □\n●μ���=�� ��\n□μ���=�� ����������������/��(�)\n●●●●● ● ●\n◇◇◇◇◇ ◇\n○○○○○ ○\n●�=��� ��/��~���� �\n◇�=��� ��/��~���� �\n○�=�\n������������������������������\n�������α��/��(�)\nFIG. 11. (a) Dynamical Hdepas function of damping for dif-\nferentHs(disorder strength). (b) Dynamical Hdepas function\nof damping for di\u000berent DMI constants.9\nIV. CONCLUSIONS\nTo summarize, we have analysed the DW depinning\n\feld in a PMA sample with DMI and we found that Hdep\ndecreases with the damping parameter with reductions\nup to 50%. This decrease is related to the DW inter-\nnal dynamics and the \fnite size of the barrier: due to\nDW precession, the DW internal angle ( \u001e) deviates from\nequilibrium and triggers the internal DW \feld (DMI and\nshape anisotropy) which tries to restore its original value.\nAt the same time, the internal \feld pushes the DW above\nits equilibrium position within the energy barrier. This\nmechanism leads to DW oscillations and, if the ampli-\ntude of the oscillations is higher than the barrier size,\nthe DW gets depinned for a lower \feld. Deviations of \u001e\nfrom equilibrium and DW oscillations are both damping\ndependent and they are enhanced at low damping.\nIn the system with grains the mechanism is the same\nbut deviations from the internal DW equilibrium include\nthe formation of VBL with more complex dynamics.\nThe e\u000bect is enhanced for low DMI (providing that\u0019HDMI> H K) and for stronger disorder since we need\nto apply larger external \felds, which lead to larger DW\noscillations. These results are relevant both from a tech-\nnological and theoretical point of view, since they \frstly\nsuggest that a low damping parameter can lead to a\nlowerHdep. Furthermore, they show that micromagnetic\ncalculations of the depinning \feld, neglecting the DW\nprecessional dynamics can provide only an upper limit\nforHdep, which could actually be lower due to the DW\nprecessional dynamics.\nV. ACKNOWLEDGEMENT\nS.M. would like to thank K. Shahbazi, C.H. Mar-\nrows and J. Leliaert for helpful discussions. This work\nwas supported by Project WALL, FP7- PEOPLE-2013-\nITN 608031 from the European Commission, Project No.\nMAT2014-52477-C5-4-P from the Spanish government,\nand Project No. SA282U14 and SA090U16 from the\nJunta de Castilla y Leon.\n\u0003Corresponding author: simone.moretti@usal.es\n1D. Allwood, Science 309, 1688 (2005).\n2S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008).\n3P. J. Metaxas, J. P. Jamet, A. Mougin, M. Cormier,\nJ. Ferr\u0013 e, V. Baltz, B. Rodmacq, B. Dieny, and R. L.\nStamps, Physical Review Letters 99, 217208 (2007).\n4J. Gorchon, S. Bustingorry, J. Ferr\u0013 e, V. Jeudy, A. B.\nKolton, and T. Giamarchi, Physical Review Letters 113,\n027205 (2014), arXiv:1407.7781.\n5T. A. Moore, I. M. Miron, G. Gaudin, G. Serret, S. Auf-\nfret, B. Rodmacq, A. Schuhl, S. Pizzini, J. Vogel, and\nM. Bon\fm, Applied Physics Letters 93, 262504 (2008),\narXiv:0812.1515.\n6I. M. Miron, T. Moore, H. Szambolics, L. D. Buda-\nPrejbeanu, S. Au\u000bret, B. 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In fact, by in-creasing the sample dimension along the xdirection, we\nincrease the probability of \fnding the highest possible hj\nin the single realization and the average of Hi\ndepwill tend\nto the maximum.\n34This is solved by the Relax solver of MuMax with the as-\nsumption\u000b=(1 +\u000b2) = 1.\n35The DW energy is calculated as the energy of the system\nwith the DW minus the energy of the system without the\nDW (uniform state). The pro\fle is obtained by moving the\nDW with an external applied \feld and then subtracting the\nZeeman energy.\n36X. Wang, P. Yan, J. Lu, and C. He, Annals of Physics\n324, 1815 (2009), arXiv:0809.4311.\n37S. Tarasenko, a. Stankiewicz, V. Tarasenko, and J. Ferr\u0013 e,\nJournal of Magnetism and Magnetic Materials 189, 19\n(1998).\n38The same results are obtained with a Gaussian barrier,\nmeaning that the key point is the \fnite size of the barrier\nrather than its shape.\n39ForD >D c, DW have negative energies and the systems\nspontaneously breaks into non-uniform spin textures.\nAppendix A: Maximum torque and equilibrium state\nIn this section we show in more detail how the maximum torque represents an indicator of the equilibrium state.\nMaximu torque is de\fned as\n\u001cmax\n\r0= Maxf\u00001\n1 +\u000b2mi\u0002He\u000b;i\u0000\u000b\n1 +\u000b2mi\u0002(mi\u0002He\u000b;i)g=1\n\r0Max\u0012dmi\ndt\u0013\n; (A1)\nover all cells with label i=f1;:::;N =Nx\u0001Nyg. MuMax3.9.329can provide this output automatically if selected.\nWe perform the same simulations as indicated in the main text, without any stopping condition, but simply running\nfort= 20 ns. Fig. 12(a) shows the average mzcomponent for \u000b= 0:2 andBz= 10 mT, while Fig. 12(b) depicts the\ncorresponding maximum torque. We can see that, once the system has reached equilibrium, the maximum torque has\ndropped to a minimum value. The same results is obtained for di\u000berent damping but the \fnal maximum torque is\ndi\u000berent. Numerically this value is never zero since it is limited by the code numerical precision and by the system\nparameters, in particular by damping.\nFig. 12(c) represents the maximum torque as function of applied \feld for di\u000berent damping. The maximum torque\nis clearly independent on the applied \feld but depends on the damping value. Finally, Fig. 12(d) shows the max\ntorque as function of damping. The maximum torque decreases with damping and it saturates for \u000b\u00150:5 since we\nhave reached the minimum numerical precision of the code29. For higher damping the maximum torque oscillates\naround this minimum sensibility value, as shown in the inset of Fig. 12(d). The value obtained with these preliminary\nsimulations is used to set a threshold \u000f(\u000b) for the depinning \feld simulations in order to identify when the system has\nreached an equilibrium. Furthermore, additional tests were performed, without putting any max torque condition,\nbut simply running the simulations for a longer time ( t= 80;160 ns) and calculating the depinning \feld in order\nto ensure that the results obtained with these two method were consistent, i.e., that we have actually reached an\nequilibrium state with the maximum torque condition.11\n������������������-���-���-���\n����(��)<��>⨯��-�α=���(�)\n���������������������������\n����(��)��� ������/γ �(��)α=���(�)\n○ ○ ○ ○ ○ ○ ○ ○\n□ □ □ □ □ □ □ □\n◇◇◇◇◇◇◇◇\n○α=����□α=���◇α=���\n� �� �� �� ��������������������\nμ���(��)��� ������/γ �(��)(�)\n●\n●●\n��������������������-�������������������\n�������α��� ������/γ �(��)(�)\n���������������������\n����(��)τ���/γ�(��)α=���\nFIG. 12. (a) average mzas function of time. (b) Max torque/ \r0(\u001cmax) as function of time. \u001cmaxrapidly decreases when the\nsystem is at equilibrium. (c) Max torque as function of applied \feld for di\u000berent damping. (d) Max torque at equilibrium as\nfunction of damping. The inset shows the max torque as function of time for \u000b= 0:5.\nAppendix B: 1D energy barrier\nAs commented in the main text, the pinning \feld implemented in the 1D model simulations is obtained by using the\nshape of the DW energy pro\fle derivative @\u001b(x)=@x(beingxthe DW position) and the amplitude of the depinning\n\feld obtained in the full micromagnetic simulations Hsfor the single barrier case. Namely\nHdep=Hs\u0012@\u001b(x)\n@x\u0013\nN; (B1)\nwhere we recall that Nstands for the normalized value. This choice might sound unusual and needs to be justi\fed.\nIn fact, having the DW energy pro\fle, the depinning \feld could be simply calculated as20\nHdep=1\n2\u00160Ms@\u001b(x)\n@x: (B2)\nThis expression is derived by imposing that the derivative of the total DW energy E(x) = 2\u00160MsHzx+\u001b(x) (Zeeman\n+ internal energy) must be always negative. However, in our case also Ms(x) depends on the DW position and the\nresults obtained with Eq. B2 is di\u000berent from the depinning \feld measured in the static single barrier simulations.\nFor this reason we use Eq. B1 which keep the correct barrier shape and has the measured static value.\nFinally, we recall that equivalent results are obtained by using a simple Gaussian shape for the pinning \feld, meaning\nthat the key point is the localized shape of the barrier, rather than its exact form.\nAppendix C: Dynamical depinning for a moving Domain Wall\nIn this section we show the results for the dynamical depinning \feld when the DW is placed at an initial distance of\nd1= 200 nm from the barrier. In this way the DW hits the pinning with an initial velocity. The d0case corresponds to\nthe DW at rest relaxed just before the barrier and extensively analysed in the main text. Also for this con\fguration\nwe performed static and dynamic simulations, neglecting or including the DW precessional dynamics respectively.\nThe depinning \feld for the d1case is further reduces at small damping, reaching Hd=Hs\u00180:08 (Hd= 9 mT and\nHs= 120 mT) at \u000b= 0:02. Nevertheless, the depinning \feld remains constant in the static simulations independently\non the velocity at which the DW hits the barrier. This suggests that, rather than related to the DW velocity, the\nreduction is again related to the DW precession. When the DW starts from d1it reaches the barrier precessing, thus\nwith a higher displacement from its equilibrium angle, leading to a higher e\u000bect of the internal \feld.12\n●●●●●●\n○○○○○○ ◆◆◆◆◆ ◆ □□□□□ □\n●��(�������)\n○��(�������)◆��(������)\n□��(������)\n��� ��� ��� ��� ������������������������\n�������α��/��\nFIG. 13. Dynamical depinning \feld as function of damping for static and dynamic simulations for the d0andd1cases." }, { "title": "2310.06447v1.Emerging_Spin_Orbit_Torques_in_Low_Dimensional_Dirac_Materials.pdf", "content": "Emerging Spin-Orbit Torques in Low Dimensional Dirac Materials\nJoaqu´ ın Medina Due˜ nas,1, 2Jos´ e H. Garc´ ıa,1and Stephan Roche1, 3\n1ICN2 — Catalan Institute of Nanoscience and Nanotechnology,\nCSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain\n2Department of Physics, Universitat Aut` onoma de Barcelona (UAB),\nCampus UAB, Bellaterra, 08193 Barcelona, Spain\n3ICREA — Instituci´ o Catalana de Recerca i Estudis Avan¸ cats, 08010 Barcelona, Spain\n(Dated: October 11, 2023)\nWe report a theoretical description of novel spin-orbit torque components emerging in two-\ndimensional Dirac materials with broken inversion symmetry. In contrast to usual metallic inter-\nfaces where field-like and damping-like torque components are competing, we find that an intrinsic\ndamping-like torque which derives from all Fermi-sea electrons can be simultaneously enhanced\nalong with the field-like component. Additionally, hitherto overlooked torque components unique to\nDirac materials, emerge from the coupling between spin and pseudospin degrees of freedom. These\ntorques are found to be resilient to disorder and could enhance the magnetic switching performance\nof nearby magnets.\nSpin-orbit torque (SOT) nonvolatile magnetic memo-\nries represent an emerging technology that leverages the\nintrinsic spin-orbit coupling (SOC) within metals to con-\nvert charge current into a spin source, denoted as S,\nfurther harnessed to manipulate a magnetic state [1, 2].\nWhen oriented along unit vector ˆm, this magnetic state\nis subjected to a torque Tproportional to ˆm×Sdriv-\ning its magnetization dynamics and eventually achieving\nmagnetization reversal [3]. This mechanism offers new\npossibilities regarding energy efficiency and miniaturiza-\ntion of devices, surpassing traditional multi-ferromagnet\nsetups [4].\nSOT is typically broken down into two main contribu-\ntions: the field-like (FL) and damping-like (DL) torques.\nThe FL torque induces precession of the magnetization,\nwhile the DL torque aligns it along an effective spin-orbit\nfield [3, 5, 6]. For efficient magnetic switching, both\nstrong FL and DL torques are necessary; however, in\nstandard metal/magnet interfaces they have competing\norigins. The FL torque usually stems from the Rashba-\nEdelstein effect enabled by the reduced symmetry at the\ninterface, whereas the DL torque arises from the injec-\ntion of angular momentum from the metallic bulk states\ntowards the magnet via the spin Hall effect [7, 8]. In this\ncontext, Van der Waals heterostructures offer a novelty\nto diversify and simultaneously enhance the efficiency\nof both torque components. Indeed, interfacial effects\ndominate the torque physics in such heterostructures, so\nthe proper design of crystal symmetries can enable novel\ntorque responses with tailor-made properties for driving\nthe magnetization dynamics of nearby magnetic materi-\nals [9]. Surprisingly, despite the suppression of the per-\npendicular spin Hall effect, DL torque has been reported\nand tentatively explained in terms of Berry curvature\neffects [10, 11]. Additionally, skew-scattering has been\ntheoretically proposed as a potential source of extrinsic\ninterfacial DL torques, but the lack of microscopic infor-\nmation concerning interface quality does not facilitate aconvincing explanation [12, 13].\nDirac materials present favorable conditions for effi-\ncient FL torque due to their unique spin-momentum lock-\ning [14]. Specifically, transport conveyed by a single spin-\nhelical band represents an optimal regime for the Rashba-\nEdelstein effect. Such phenomenon has been manifested\nby the edge states of 3D topological insulators [15] and in\ngraphene/transition metal dichalcogenide (TMD) bilay-\ners [16–18]. Although magnetic exchange is detrimental\nto the Rashba-Edelstein effect, new torque components\nbeyond the FL contribution can be created by reducing\nthe system’s symmetry [19]. For instance, recent mea-\nsurements with WTe 2/Py [20] and CuPt/CoPt [21] in-\nterfaces have reported the emergence of unconventional\ntorques, leading to the strongly desired field-free magne-\ntization switching .\nIn this Letter, we develop a general theory for SOT\nmechanisms in 2D-based materials, based on symmetry\nanalysis, semi-classical modeling, and quantum simula-\ntions in realistic electronic models. We first employ group\ntheory to determine the minimal set of torque contribu-\ntions common to all 2D systems, beyond the standard\nFL contribution. We then use Boltzmann transport the-\nory to elucidate the nature of the emerging torques. We\nfinally quantify the SOT response for a few illustrative\ncases using Kubo quantum transport simulations. We re-\nveal that the origin of a DL torque in 2D-based systems\nderives from the spin-momentum locking of accelerating\nelectronic states and manifests through the entire Fermi\nsea. Furthermore, we find that Dirac materials present\nhitherto overlooked torque contributions which can in-\nduce non-trivial magnetization dynamics.\nBeyond field-like torque in 2D.— We begin by identi-\nfying which torque components are allowed by symme-\ntries, beyond the standard FL contribution. We exam-\nine non-magnetic point group C∞v, which represents the\nhighest symmetry group that supports Rashba SOC, ex-\nhibiting in-plane axial symmetry and an infinite set ofarXiv:2310.06447v1 [cond-mat.mes-hall] 10 Oct 20232\nmirror planes perpendicular to the 2D plane, while lack-\ning mirror symmetry parallel to it. Notably, C∞vencom-\npasses the minimal set of potential torques in 2D systems,\nas any additional torque can only be enabled by reduc-\ning the symmetry group. To determine the minimal set\nof torques, the non-equilibrium spin density is expressed\nas a function of the orientation of the magnetization ˆm\ncharacterized by a polar angle θ, and an azimuthal angle\nφrelative to the applied electric field E(see Fig. 1-(a),\ninset). We retain only the components compatible with\nthe system’s symmetries and expand them up to second\norder in terms of their angular dependence [3, 22]. This\nprocedure enables us to separate the non-equilibrium spin\ndensity in two contributions, S=SI+SII, with\nSI=χFL(1−ξFLsin2θ)HSOC−χDLˆm×HSOC, (1)\ncorresponding to the standard FL and DL terms, causing\nthe magnetization to precess and align along the effective\nSOC field HSOC≡ˆz×E. The adimensional parameter\nξFLrepresents a second order term that modulates the\nmagnetization precession as it approaches the plane. The\nsecond contribution reads:\nSII=−E(χ∥cosφ+χ⊥cosθsinφ) sinθˆz, (2)\nwhere χ∥generates an anisotropic damping of the in-\nplane magnetization with respect to the out-of-plane\ncomponent, while χ⊥competes with the aforementioned\nstabilization along HSOC, favoring an alignment parallel\nor antiparallel to the current depending on the direction\nofmz. These torques could be ascertained by observ-\ning non-trivial angular dependence in magnetoresistive\nexperiments.\nNote that this procedure does not constitute a pertur-\nbative expansion . Therefore, these torques, as well as\nthose of higher order, can contribute on equal footing,\nand their existence and strength will depend on the un-\nderlying symmetries and competing fields. To discern\ntheir relevance, we develop a dual approach. Using the\nKubo quantum transport framework, we first determine\nthe non-equilibrium spin density at Fermi level εFusing\nthe Kubo-Bastin formula [23, 24],\nS(εF) =−2ℏZ\ndεf(ε)Im Trh\nδ(ε−ˆH)ˆs∂εG+(ˆj·E)i\n,\n(3)\nwhere ˆH,ˆsandˆjare the Hamiltonian, spin and current\ndensity operators respectively, fis the Fermi-Dirac dis-\ntribution, and G+= lim η→0[ˆH−ε+iη]−1is the retarded\nGreen’s function. Moreover, we distinguish Fermi-sea\nand Fermi-surface contributions adopting the decompo-\nsition proposed in Ref. [25]. We numerically compute\nEq. (3) employing a kernel polynomial method (KPM)\nexpansion which includes the choice of a finite broaden-\ningη[26]. We simulate disordered systems via real-space\nlinear scaling numerical methods, reaching a precision of\n(a) \n(b) Fermi surface\n(c) Fermi sea\nelectric field\nFIG. 1. Physical origin of the FL and DL torques. (a)Spin\ntexture of a 2DEG with Rashba SOC and out-of-plane mag-\nnetization ( θ= 0). Inset: scheme of the magnetization ori-\nentation. When applying an electric field E:(b)the Fermi\nsurface drifts overpopulating states with k·E>0, generat-\ning a non-equilibrium spin density along HSOC=ˆz×E.(c)\nThe acceleration of the carriers generates a forcing ∼d\ndtBk\n(gray). Interaction between the equilibrium spins (blue) and\nthe forcing originates a current-induced spin texture (red),\nresponsible for the DL torque.\nη≈15 meV in a system with >106orbitals [27–29]. For\npristine systems, we develop a k-space KPM calculation\nof Eq. (3), reaching η∼1 meV precision in systems as\nlarge as >109orbitals [29]. To compute Sas a function\nof the magnetization, we select an optimal set of magne-\ntization configurations, each requiring a separate Kubo\ncalculation, resulting in 14 magnetization configurations\nfor each system [29].\nSimultaneously, we develop a semi-classical approach\nbased on Boltzmann transport theory to understand the\nunderlying microscopic mechanisms of these torques [30].\nWithin this theory, the non-equilibrium spin density is\nS(εF) =X\nµZd2k\n(2π)2h\nδfµ,ksµ,k+f(Eµ,k)δsµ,ki\n, (4)\nwith Eµ,kand|Eµ,k⟩the eigenvalue and eigenstate re-\nspectively of an electron with crystal momentum kand\nband index µ, and sµ,k≡ ⟨Eµ,k|ˆs|Eµ,k⟩the mean value\nof the spin operator ˆs, commonly referred to as the spin\ntexture. The first term in Eq. (4) represents the stan-\ndard Boltzmann transport contribution stemming from\nthe current-induced variation of the carrier occupation\nδfµ,k. The origin of the FL torque has been extensively\nstudied and arises from the current-induced drift of the\nFermi surface, which combined with the helical spin tex-\nture of Rashba systems produces an in-plane spin den-\nsity perpendicular to the current [31, 32], as illustrated\nin Fig. 1-(b).\nThe second term in Eq. (4) remains less understood.\nIt originates from the adiabatic transport of Bloch states\nunder an electric field and serves as a quantum mechan-\nical correction to the semi-classical result [33]. We here\noffer an intuitive interpretation of this aspect: as the\ncarriers accelerate in the momentum-dependent Rashba3\nfield, a dynamic magnetic field develops within their rest\nframe. This field acts as a driving force that nudges\nthe spin away from its equilibrium, producing a current-\ninduced spin texture throughout the entire Fermi sea, as\ndepicted in Fig. 1-(c).\nTo illustrate this point, let us consider a Bloch Hamil-\ntonian that incorporates a spinless component H0,k, to-\ngether with an effective magnetic field Bk, which encap-\nsulates both the exchange and SOC fields. The Hamil-\ntonian reads ˆHk=H0,k−1\n2Bk·ˆs(we take the gyro-\nmagnetic ratio equals to unity). In equilibrium, the spin\nstates align along Bk. When applying an electric field E,\nthe electrons accelerate according to ℏdk\ndt=−eE, with e\nthe elementary charge. The spin dynamics emerging from\nthis non-equilibrium state is described by the Ehrenfest\ntheorem, yieldingd\ndtsµ,k+ℏ−1Bk×sµ,k= 0. We decom-\npose the spin texture into a component aligned with the\ninstantaneous effective field and a perturbation induced\nby the current: sµ,k=µˆBk+δsµ,k, with µ=±the\nspin majority/minority band index. Within the linear\nresponse regime, the non-equilibrium spin texture reads\nδsµ,k=−µeℏ\nB3\nkBk×(E· ∇k)Bk. (5)\nThis result reveals a crucial point: the interplay between\nthe equilibrium and dynamic magnetic fields results in a\ncurrent-induced shift of the spin texture. Notably, this\nshift is the main contributor to the DL torque observed\nin 2D magnetic Rashba systems.\nSemi-classical torque mechanisms in a 2D electron\ngas.— To highlight our theory’s capabilities, we begin\nby showcasing the presence of a DL torque in an s-wave\n2D electron gas (2DEG) with C∞vsymmetry. The mag-\nnetization is characterized by an exchange splitting Jex\nalong the magnetization direction ˆm, while the Rashba\nSOC field is helical with an isotropic amplitude Λ R,k.\nThe Hamiltonian is ˆHk=H0,k−1\n2ΛR,kˆφ·ˆs−1\n2Jexˆm·ˆs,\nwith H0,kthe kinetic term, and ˆφ=ˆz׈k. We deter-\nmine the non-equilibrium spin density employing Boltz-\nmann transport assuming an isotropic momentum relax-\nation time τin the weak disorder regime. A perturba-\ntive expansion in terms of the magnetization direction is\nappropriate only if the effective magnetic field is domi-\nnated either by the exchange or Rashba term. We be-\ngin by defining the leading order effective magnetic field,\nB0,k= (J2\nex+ Λ2\nR,k)1/2, isotropic in momentum space.\nFor a dominant exchange splitting, which is usually the\nexperimental condition, B0≈Jexis momentum indepen-\ndent, while for a dominant Rashba splitting B0,k≈ΛR,k.\nThe conventional torques, represented in SI, show a\nsimilar behavior in both regimes. The FL torque is\nessentially determined by the spin helicity, sµ,k·ˆφ≈\nµΛR,k/B0,k. It derives from the Fermi-surface contribu-\ntion to Eq. (4), and is proportional to the density of statesand the electron mobility following:\nχFL=µeτkµ\n4πℏΛR,kµ\nB0,kµ, (6)\nwhere µ=±is the band index, the electron mobility is\ncontained within τ, and kµis the isotropic part of the\nFermi momentum, which encodes the density of states,\ndefined such that εF=H0,kµ−µ\n2B0,kµ.\nThe DL torque on the other hand, is determined by\nthe Fermi-sea integral of the current-induced spin tex-\ntureδsµ,k.δsµ,kemerges from the interaction between\nthe equilibrium spin texture, and the variation of the\neffective magnetic field in the accelerating electron rest\nframe via spin-momentum locking, as given by Eq. (5).\nOnly the Rashba field Λ R,kˆφcontributes to the latter,\nyieldingR\ndφ(E· ∇k)Bk=π\nk∂k(kΛR,k)HSOC, whose in-\nteraction with the spin texture component parallel to the\nmagnetization ˆmgenerates the DL torque\nχDL=µeJex\n4πZkµ\n0dk∂k(kΛR,k)\nB3\n0,k. (7)\nThis expression reveals that the DL torque is generated\nby the interplay between spin-momentum locking and ex-\nchange splitting throughout the entire Fermi sea. In the\nparticular case of a dominant exchange splitting the effec-\ntive field is momentum-independent, and the DL torque\nbecomes χDL=ℏ\nτJexχFL, thus offering a way to deter-\nmine the momentum relaxation time via the χFL/χDL\nratio. Additionally, the interaction between the dynamic\nmagnetic field and the equilibrium Rashba field produces\nan out-of-plane current-induced spin density inducing an\nunconventional torque χ∥, which is only non-zero in the\nRashba dominated regime [29].\nWe finally find that the second order torques are van-\nishing in parabolic systems, χ⊥=ξFL= 0. In the ex-\nchange dominated regime, the spin texture presents a\ndominant component aligned with ˆm, while Rashba SOC\nintroduces a helical component as well as an anisotropic\nmodulation to the dominating one. Hence, the drift of the\nFermi surface, in addition to the FL torque, also yields\na non-equilibrium spin density parallel to the magneti-\nzation, which stems from the spin texture component\nparallel to ˆmand cannot exert any torque, but could\nbe relevant for charge-to-spin conversion [29]. Further-\nmore, our theory fully matches quantum simulations for\nthe description of DL torque in a s-wave 2DEG [29].\nUnconventional torques in Dirac Matter.— We now\nuse our theory to reveal unconventional torques in Dirac\nsystems enabled by the additional angular momentum\nprovided by the pseudospin degree of freedom. As a min-\nimal model, we consider a C∞vDirac Hamiltonian doted\nwith exchange splitting, given by ˆHk=ℏvk·ˆσ+ ∆σz−\n1\n2λR(ˆs׈σ)·ˆz−1\n2Jexˆm·ˆs, with vthe velocity of mass-\nless Dirac electrons, ∆ their effective mass, and λRthe4\nDirac point\nsemi-gap(b)\n0 30 60 -30 -600\n-11\n20 40-0.80.0\n0 10 -100\n-22(a)\n0\n-30\n-603060\nDirac point semi-\ngap\nspin susceptibility\nFIG. 2. SOT in Dirac matter. (a)Band structure of Dirac system with Rashba SOC and exchange splitting [ θ= 0]. (b)The\nconventional SOTs, χFLandχDL, are driven by semi-classical effects, evinced by the agreement between semi-classical (dotted\ncurves) and Kubo-Bastin (solid curves) frameworks. The insets show torque responses, computed by the Kubo framework,\nwhich are not captured by semi-classical effects. Left inset: All unvconventional torques show a strong enhancement near the\nband inversion at charge neutrality. Right inset: χDLandχ∥shift from semi-classical to quantum-driven mechanisms at the\nDirac point, as the semi-gap is dominated by the effective mass (∆ ̸= 0, solid curves) or Rashba SOC (∆ = 0, dashed curves)\nrespectively. [ v∼106m/s,λR= 8 meV, Jex= 60 meV, ∆ = 8 meV, η= 2 meV at the band center and τ=ℏ/η, except if\nindicated otherwise]\nRashba SOC parameter, while the pseudospin is repre-\nsented by Pauli vector ˆσ. Such system may be realized by\nan insulating ferromagnet/graphene/TMD trilayer [13],\nwhere proximity effects induce SOC and exchange split-\nting [34–36]. Rashba splitting in graphene/TMD inter-\nfaces is typically of order ∼1 meV [37–40], while proxim-\nity with a ferromagnet can reach exchange splittings as\nlarge as ∼100 meV [41, 42]. We thus focus on the regime\nwith a dominant exchange splitting.\nSpin-momentum locking in Dirac electrons is medi-\nated by the pseudospin. The 2DEG paradigm may\nbe recovered in absence of spin-pseudospin correlations,\ni.e.⟨σisj⟩=⟨σi⟩⟨sj⟩, allowing us to define an effec-\ntive Rashba field for each pseudospin polarized set of\nbands. This is the case far from band crossings, where\nRashba SOC perturbatively imprints a helical component\nto the spin texture. Drastically different is the situa-\ntion near band crossings, where Rashba SOC acts non-\nperturbatively by hybridizing bands of opposite pseu-\ndospin polarizations, while also inducing non-negligible\nspin-pseudospin correlations. These two regimes must\nbe analyzed separately.\nIn the former regime, the pair of bands with oppo-\nsite pseudospin polarization, or equivalently, opposite\nvelocities, are decoupled. We thus recover the 2DEG\nparadigm by adequately adjusting for the corresponding\nkinetic Hamiltonian and Rashba field, allowing us to un-\nderstand the FL and DL torques from simple band struc-\nture properties, shown in Fig. 2-(a): At energies larger\nthan the magnetic splitting the spectrum presents two\nFermi contours with positive velocity which compete due\nto their opposite helicities. Such competition is energyindependent due to the linear dispersion. For energies\nlower than the magnetic splitting, the two Fermi con-\ntours present the same helicity, but with opposite veloc-\nity. The inner Fermi contour vanishes when approaching\nthe spin-split Dirac point, where ∆ opens a semi-gap.\nWithin the semi-gapped energy window a single spin-\nhelical band remains, representing an optimal condition\nfor maximizing the SOT efficiency. The semi-classical\nresults shown by dotted curves in Fig. 2-(b), which are\nin very good agreement with Kubo-Bastin calculations\n(shown in solid lines). Additionally, we recover the pre-\nviously obtained relation for the momentum relaxation\ntime and the torque ratio, τ= (ℏ/Jex)χFL/χDL.\nNear the band inversion at charge neutrality the hy-\nbridization between bands of opposite velocity breaks\nthe 2DEG paradigm. To elucidate the torques in this\nregime, the full quantum response provided by the Kubo-\nBastin formula is required. The results are shown in\nFig. 2-(b) left inset, where strong unconventional torques\nare seen near the Rashba gap, vanishing in the 2DEG\ncase. The origin of these torques is two-fold. Despite\nthe dominant exchange splitting, Rashba SOC acts non-\nperturbatively and induces strong anisotropies in the\ndispersion. Furthermore, non-negligible spin-pseudospin\ncorrelations quench the spin texture [43, 44] and mod-\nify the coupled spin-pseudospin dynamics [29]. These\ntwo features are only possible due to the additional pseu-\ndospin degree of freedom.\nAt the Dirac point, the SOT physics may shift from\nsemi-classical to quantum driven effects. While the effec-\ntive mass ∆ acts separately on each spin-split Dirac cone,\nRashba SOC couples them favoring spin-pseudospin en-5\n(a) (b)\n0 50 -500.00.1\n-0.1\n0 1 2100\n90\n80\n70\nFIG. 3. The FL and DL torques are robust against Anderson\ndisorder (of strength W).(a)SOT for pristine (solid) and\nW= 2.7 eV disordered (dashed) systems. (b)Peak torque\nvalues relative to the pristine system. [ v∼106m/s,λR=\n10 meV, Jex= 40 meV, ∆ = 5 meV and η= 15 meV at the\nband center.]\ntanglement [43, 44]. At the spin-split Dirac point the\nspectrum is semi-gapped, thus transport should be dom-\ninated by the band of the opposite cone, whose physics\nis mainly determined by the kinetic term of the Hamilto-\nnian, insensitive to ∆. This is indeed the case for massive\nDirac electrons, as shown in Fig. 2-(b) right inset (solid\ncurves), where |χDL|increases across the semi-gap while\nχ∥remains negligible. However, the torques change dra-\nmatically for massless electrons, as |χDL|is minimal and\n|χ∥|peaks within the semi-gap, shown in Fig. 2-(b) right\ninset (dashed curves). This qualitative change is gen-\nerated due to spin-pseudospin entanglement dominating\nthe gap, indicating a large torque originated from a van-\nishing Fermi contour.\nWe finally compute the Kubo-Bastin response via lin-\near scaling numerical methods [27–29] considering real-\nspace Anderson disorder of strength W. The results are\nshown in Fig. 3, focusing on the FL and DL torques. For\nWas high as 2 .7 eV, the χFLandχDLpeaks are only\nreduced to 68% and 79% of their respective pristine val-\nues. Dictated by a Fermi-sea contribution, χDLshows a\nbetter resilience than χFL.\nIn conclusion, we have developed a general theory of\nSOT for interfaces involving low dimensional (Dirac) ma-\nterials with broken inversion symmetry. Novel types of\nSOT components, resilient to disorder effects, have been\nfound to emerge and superimpose, enhancing the torque\ncapability of these interfaces for more efficient magneti-\nzation switching, enabling further exploration of ultralow\nenergy memory and spintronic applications in ultracom-\npact devices.\nThe authors thank D. Garc´ ıa Ovalle and A. Man-\nchon for fruitful discussions. The authors ac-\nknowledge funding from Ministerio de Ciencia e In-\nnovacion (MCIN) under grant PID2022-138283NB-\nI00/MCIN/AEI/10.13039/501100011033 and the Eu-\nropean Regional Development Fund. J.M.D. ac-\nknowledges support from MCIN (grant FPI PRE2021-\n097031). J.H.G. acknowledge funding from the Eu-\nropean Union (ERC, AI4SPIN, 101078370). S.R andJ.H.G, acknowledge grant PCI2021-122035-2A-2 funded\nby MCIN/AEI/10.13039/501100011033 and European\nUnion “NextGenerationEU/PRTR”, funding from the\nEuropean Union’s Horizon 2020 research and inno-\nvation programme under grant No 881603, and the\nsupport from Departament de Recerca i Universitats\nde la Generalitat de Catalunya. ICN2 is funded\nby the CERCA Programme/Generalitat de Catalunya\nand supported by the Severo Ochoa Centres of Excel-\nlence programme, Grant CEX2021-001214-S, funded by\nMCIN/AEI/10.13039.501100011033.\n[1] K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda,\nS. Maekawa, and E. Saitoh, Electric Manipulation of Spin\nRelaxation Using the Spin Hall Effect, Phys. Rev. Lett.\n101, 036601 (2008), publisher: American Physical Soci-\nety.\n[2] I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq,\nA. Schuhl, S. 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Rev.\nB102, 041403(R) (2020), publisher: American Physical\nSociety.8\nSUPPLEMENTAL NOTES ON KUBO CALCULATIONS\nWe present additional details for the Kubo-Bastin calculations of SOT. In order to distill the macroscopic torques\naccording to their dependence on the magnetization direction, we select an evenly spaced sample of 2M+ 1θpoints\nand 2 M φ points, with M∈Nthe order of the Sexpansion.\nCalculations of disordered systems, as presented in Fig. 3 of the main text, are performed by real-space simulations\nvia linear scaling numerical methods [27, 28]. We consider a finite system of 1024 ×1024 unit cells including random\nreal-space disorder, and use a stochastic approximation of the trace for each disorder configuration. We compute the\nnon-equilibrium spin density for a band center broadening of 15 meV, corresponding to 1867 Chebyshev moments,\nand average over 11 disorder configurations.\nIn order to reach higher energy resolution, we develop a k-space representation of the KPM expansion of the\nKubo-Bastin formula for pristine systems, as presented in Fig. 2 of the main text. We consider a finite system\nof 14691 ×14691 unit cells, which in the pristine limit corresponds to a 14691 ×14691 k-point grid sampling the\nBrillouin zone. Aiming to describe the response of the system near the Dirac points, we keep only the kpoints with\neigenvalues within a ±300 meV energy range from charge neutrality, reducing the computational cost in more than\n99%. Furthermore, focusing on a finite energy window about charge neutrality allows us to reach lower broadening\nwith a smaller Chebyshev expansion. We compute the non-equilibrium spin density for a band center broadening of\n2 meV, corresponding to 581 Chebyshev moments.\nSEMI-CLASSICAL CALCULATIONS OF SOT IN A 2D ELECTRON GAS\nIn this section we present the complete calculations for the semi-classical spin density in a 2D electron gas. We analyze\nas well the specific case of the 2D electron gas expanded to lowest order in momentum, including a comparison between\nboth semi-classical and Kubo-Bastin methods.\nWe begin expanding the relevant physical magnitudes in series in terms of the magnetization direction. Note that\nthe expansion must be physically sustained by a weak interplay between the exchange and Rashba fields, which we\nincorporate at the end of the calculation. We define B0,k= (J2\nex+ Λ2\nR,k)1/2the leading order term of the magnetic\nfield. The dispersion of band µis\nEµ,k=H0,k−µ\n2B0,k−µ\n2JexΛR,k\nB0,k(ˆm·ˆφ) . (8)\nFor the Fermi momentum of band µwe assume an ansatz of the form kF,µ(φ) =kµ+δkµ(ˆm·ˆφ), with kµthe leading\ncontribution isotropic in k-space and independent of the magnetization direction. Intersection with Fermi level εF\nyields the dominant contribution kµsuch that εF=H0,kµ−µ\n2B0,kµ, and the first order correction\nδkµ=µ\n2JexΛR,kµ\nB0,kµ∂k(H0,kµ−µ\n2B0,kµ). (9)\nCaution must be taken near the band edges, as if the band edge occurs at a finite momentum the Fermi surface is\nstrongly anisotropic, thus escaping the validity regime of the present calculations.\nWe calculate the non-equilibrium spin density within the Boltzmann semi-classical framework, as given by Eq. (4) of\nthe main text. At zero temperature fµ,k= Θ( εF−Eµ,k), with Θ the Heavyside step function. Assuming a momentum\nrelaxation time τ, the current-induced variation of the carrier occupation is δfµ,k=eτ\nℏδ(εF−Eµ,k)(E·∇k)Eµ,k. The\nFermi-surface and Fermi-sea contributions to the non-equilibrium spin density respectively are\nSsurf\nµ=eτ\n4π2ℏZ2π\n0dφ k(E· ∇k)Eµ,k\n|∇kEµ,k|sµ,k\f\f\f\f\f\nk=kF,µ\n=µeτkµ\n4πℏΛR,kµ\nB0,kµ \nˆz×E−J2\nex\nB2\n0,kµ(ˆz×E·ˆm)ˆm!\n,(10a)9\nSsea\nµ=1\n4π2Z2π\n0dφZ\ndk[Θ(kµ−k) +δ(kµ−k)δkµ(ˆm·ˆφ)]k δsµ,k\n=−µJex\n4π\"\nˆm×(ˆz×E)Zkµ\n0dk∂k(kΛR,k)\nB3\n0,k+ (ˆm·E)ˆz \nΛ3\nR,kµ\nB4\n0,kµ∂k(2µH0,kµ−B0,kµ)−Zkµ\n0dk3Λ3\nR,k\nB5\n0,k!#\n.\n(10b)\nThe first term in Eqs. (10a) and (10b) yield the final results for the field-like and damping-like torques, presented in\nEqs. (6) and (7) of the main text respectively. The unconventional torque χ∥stems from the second term in Eq. (10b),\nwhich is only non-negligible in the Rashba-dominated regime. Finally, we note that the Fermi-Surface contribution\nyields a spin density of the form ∝(ˆz×E·ˆm)ˆm; which, though is allowed by symmetries, does not represent a SOT\nas it is parallel to the magnetization, and is thus excluded from Eqs. (1) and (2) of the main text.\nThe semi-classical non-equilibrium spin density is calculated in a weak disorder limit as τ→ ∞ . In the Kubo-Bastin\nformalism this regime is represented by a vanishing broadening η→0. Under these conditions, the Fermi surface\ncontribution to the Kubo-Bastin formula, as indicated by the Bonbien & Manchon decomposition [25], reduces to\nthat obtained in Boltzmann transport. On the other hand, the Kubo-Bastin Fermi sea contribution reads\nSsea→ −ℏX\nµ,µ′Zd2k\n(2π)2f(Eµ,k)−f(Eµ′,k)\n(Eµ,k−Eµ′,k)2Im⟨Eµ,k|ˆj·E|Eµ′,k⟩⟨Eµ′,k|ˆs|Eµ,k⟩\n=−ℏX\nµ=±Zd2k\n(2π)22f(Eµ,k)\nB2\nkIm⟨Eµ,k|ˆj·E|E−µ,k⟩⟨E−µ,k|ˆs|Eµ,k⟩\n=X\nµ=±Zd2k\n(2π)2f(Eµ,k)δsµ,k(11)\nA straight-forward calculation reveals that Im ⟨Eµ,k|ˆj·E|E−µ,k⟩⟨E−µ,k|ˆs|Eµ,k⟩=e\n2sk×(E· ∇k)Bk, proportional\nto the current-induced spin texture, presented in Eq. (5) of the main text. Thus, the Fermi sea contribution in the\nKubo-Bastin formalism corresponds exactly to that of the Boltzmann framework.\n2DEG: Parabolic dispersion\nWe analyze the particular case of the 2DEG hamiltonian with a parabolic kinetic term, characterized by the effective\nmass meff, and a linear Rashba field Λ R,k=λRk. The Hamiltonian is\nˆHk=k2\n2meff−λRk\n2ˆφ·ˆs−Jex\n2ˆm·ˆs, (12)\nwhere we use ℏ= 1. The isotropic part of the Fermi momentum of band µ, at Fermi level εFis\nk2\nµ= 2meffεF+m2\neffλ2\nR\n2+µmeff\u0012\nJ2\nex+ 2meffλ2\nRεF+m2\neffλ4\nR\n4\u00131/2\n. (13)\nThe non-equilibrium spin density of band µis given by\nχFL=µeτλRk2\nµ\n4πℏ(J2ex+λ2\nRk2µ)1/2, (14a)\nχDL=µe\n2πλR\"\n1−Jex\n(J2ex+λ2\nRk2µ)1/2#\n, (14b)\nχ∥=−µe\n2πλR\"\n1−Jex\n(J2ex+λ2\nRk2µ)1/2−Jexλ2\nRk2\nµ\n2(J2ex+λ2\nRk2µ)3/2 \n1 +meffλ2\nR\n2µ(J2ex+λ2\nRk2µ)1/2−meffλ2\nR!#\n, (14c)10\nKB\nsc\nKB\nsc\nKB\nscKB\nKB\nKB\nsc\nKB\nsc\nKB\nscKB\nKB(a) Dominant exchange field(b) Dominant Rashba field\n(a1)\n(a2)(a3)\n(a4)(b1)\n(b2)(b3)\n(b4)\nFIG. 4. Non-equilibrium spin density in 2DEG, comparing semi-classical [sc] and Kubo-Bastin [KB] results. The Rashba\npseudogap is delimited by Eµ,k=0=µ\n2Jex(dashed blue lines). (a)Dominant exchange field [ Jex= 40meffλ2\nR, and 1 .5meffλ2\nR\nKubo broadening at the band center, corresponding to 187 Chebyshev moments]. (b)Dominant Rashba field [ Jex= 0.1meffλ2\nR,\nand 0 .015meffλ2\nRKubo broadening at the band center, corresponding to 340 Chebyshev moments]. The yellow shaded region\nshows the energy range in which the Fermi momentum is not defined in all φdirections. Panels (a3), (b3) only show Kubo-\nBastin torques, as the corresponding semi-classical torques are zero. All calculations performed fixing meff, with the energy\nand distance scales determined by meffλ2\nRand ( meffλR)−1respectively\nχtorque-less =µeτJ2\nexλRk2\nµ\n4πℏ(J2ex+λ2\nRk2µ)3/2, (14d)\nwith χtorque-less representing the spin density contribution Storque-less =−χtorque-less (ˆm·ˆz×E)ˆmwhich cannot exert\ntorque, while χ⊥=ξFL= 0.\nWe compare the results obtained analytically via semi-classical calculations with those obtained from Kubo-Bastin\nsimulations. For a dominant exchange field the results are shown in Fig. 4-(a), revealing an excellent agreement\nbetween both methods. All non-zero contributions, except for χ∥, are maximal at the high energy edge of the Rashba\npseudogap as a single spin-helical band takes part in transport. χ∥on the other hand is supressed by the exchange\nfield and increases at higher energies along with the Rashba field magnitude. These results show that all the SOT\nmechanisms are captured by the semi-classical theory.\nThe analysis is more complicated for a dominant Rashba field, whose results are shown in Fig. 4-(b), which cannot\ndominate throughout the entire spectrum as it is proportional to the momentum. Furthermore, the dispersion shifts\nfrom Rashba to kinetic dominance yielding a band edge at a finite momentum. To leading order the band edge is\nεedge=−1\n8meffλ2\nR−1\n2m−1\neffλ−2\nRJ2\nex, which occurs at momentum kedge=1\n2(m2\neffλ4\nR−4J2\nex)1/2for band µ= +. Within\nan energy window of width JexλRkedge(1−m2\nz)(Jex+λ2\nRk2\nedge)−1/2about εedge, highlighted in yellow in Fig. 4-(b)\nthe Fermi momentum is not defined for all φdirections. Due to the breakdown of the approximations, we do not\nexpect accurate results for the unconventional torques near εedgeas they involve a perturbative treatment of the\ndispersion anisotropies. Note that this limitation does not lie in the semi-classical framework itself, which was proven\nto yield equivalent results to the Kubo-Bastin framework, but lie in the approximations performed in order to isolate\nthe separate contributions to the spin density. Fig. 4-(b) reveals that the conventional FL and DL torques are fully\ncaptured by the semi-classical theory. The semi-classical results for the unconventional torques are valid outside the\nyellow shaded region, showing good agreement with Kubo-Bastin calculations, specially for energies higher than the\nRashba pseudo-gap.\nSPIN DYNAMICS AND SEMI-CLASSICAL SPIN-ORBIT TORQUE IN DIRAC SYSTEM\nIn this section we present the calculations for the semi-classical torques in a Dirac system.\nThe spin-pseudospin dynamics are given by coupled equations, which not only involve the spin and pseudospin\ntextures, but also spin-pseudospin correlations. The dynamic equations are\nd\ndt⟨ˆσ⟩ −2(ℏvk+ ∆ˆz)× ⟨ˆσ⟩+λR⟨(ˆz׈s)׈σ⟩= 0 , (15a)11\nd\ndt⟨ˆs⟩+Jexˆm× ⟨ˆs⟩ −λR⟨(ˆz׈σ)׈s⟩= 0 . (15b)\nWithin a perturbative treatment of Rashba SOC, the spin and pseudospin textures are, to leading order, independently\ndefined by the exchange and bare Dirac Hamiltonians respectively. The states present a strong spin and pseudospin\npolarization |s|=|σ| ≈1 +O(λR), and the spin-pseudospin correlations are disentangled as λR⟨ˆσiˆsj⟩=λR⟨ˆσi⟩⟨ˆsj⟩.\nWe may now define the effective magnetic field as a classical variable, namely, Bk=B0−λRˆz×σk, which presents\na spin-pseudospin coupling term proportional to λR, and a SOC-independent term B0=Jexˆm. Equivalently, we\ndefine a pseudomagnetic field βk=β0,k+λRˆz×sk, with β0,k=−2(ℏvk+ ∆ˆz). Note that a separate magnetic\nfield is defined for each pseudospin polarized set of bands. The spin and pseudospin textures are obtained solving the\nequations in equilibrium (d\ndtσk=d\ndtsk= 0), yielding to first order\nsµν,k=µ\u0014\nˆB0−νλR\nB0ˆB0×\u0010\nˆB0×\u0010\nˆz׈β0,k\u0011\u0011\u0015\n, (16a)\nσµν,k=−ν\u0014\nˆβ0,k−µλR\nβ0,kˆβ0,k×\u0010\nˆβ0,k×\u0010\nˆz׈B0\u0011\u0011\u0015\n, (16b)\nwith ν, µ=±the particle/hole and spin majority/minority band indexes respectively. The strong pseudospin polariza-\ntion of the bands allows us to treat it as a classical variable. We rewrite the full Hamiltonian as a collection of two sep-\narate Hamiltonians, one for each pseudospin polarization, each with spin as its only internal degree of freedom, namely\nˆHk=L\nν=±ˆHν,k. The pseudospin-polarized Hamiltonians ˆHν,kpresent the same structure as the 2DEG Hamilto-\nnian as given in Eq. (7) of the main text, with H0,ν,k=νp\n(ℏvk)2+ ∆2and Λ R,ν,k=−νℏvkλR[(ℏvk)2+ ∆2]−1/2.\nFrom this point the results obtained for the 2DEG case may be used, taking care of performing the sum over all\npseudospin polarizations. Note that the performed approach, though consistent with a perturbative treatment of\nRashba SOC to first order, does not accurately capture the behaviour of the system near band crossings where SOC\nacts non-perturbatively, as evinced by the lack of a gap at the band crossing between both spin-split Dirac cones." }, { "title": "2401.12022v1.Damping_Enhanced_Magnon_Transmission.pdf", "content": "Damping-Enhanced Magnon Transmission\nXiyin Ye,1Ke Xia,2Gerrit E. W. Bauer,3, 4and Tao Yu1,∗\n1School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China\n2School of Physics, Southeast University, Jiangsu 211189, China\n3WPI-AIMR and Institute for Materials Research and CSRN, Tohoku University, Sendai 980-8577, Japan\n4Kavli Institute for Theoretical Sciences, University of the Chinese Academy of Sciences, Beijing 100190, China\n(Dated: January 23, 2024)\nThe inevitable Gilbert damping in magnetization dynamics is usually regarded as detrimental\nto spin transport. Here we demonstrate in a ferromagnetic-insulator–normal-metal heterostructure\nthat the strong momentum dependence and chirality of the eddy-current-induced damping causes\nalso beneficial scattering properties. Here we show that a potential barrier that reflects magnon wave\npackets becomes transparent in the presence of a metallic cap layer, but only in one direction. We\nformulate the unidirectional transmission in terms of a generalized group velocity with an imaginary\ncomponent and the magnon skin effect. This trick to turn presumably harmful dissipation into useful\nfunctionalities should be useful for future quantum magnonic devices.\nIntroduction .—Magnonic devices save power by ex-\nploiting the collective excitations of the magnetic or-\nder, i.e., spin waves or their quanta, magnons, for non-\nreciprocal communication, reprogrammable logics, and\nnon-volatile memory functionalities [1–10]. The possibil-\nity to modulate magnon states and their transport in fer-\nromagnets by normal metals or superconductors brings\nfunctionalities to spintronics [11–14], quantum informa-\ntion [15–21], and topological materials [22, 23]. The pre-\ndiction of inductive magnon frequency shifts by supercon-\nducting gates on magnetic insulators [24–30] have been\nexperimentally confirmed [31]. Normal metals are not\nequally efficient in gating magnons [32–35], but the stray\nfields of magnetically driven “eddy currents” [36–43] sig-\nnificantly brake the magnetization dynamics [36].\nThe intrinsic Gilbert damping seems to be detrimental\nto transport since it suppresses the magnon propagation\nlength. However, in high-quality magnets such as yt-\ntrium iron garnet (YIG) films, this is not such an issue\nsince the magnon mobility is often limited by other scat-\ntering processes such as two-magnon scattering by disor-\nder, and measurements can be carried out in far smaller\nlength scales.\nNatural and artificial potential barriers are impor-\ntant instruments in electronics and magnonics by confin-\ning and controlling the information carriers. They may\nguide magnon transport [31, 44], act as magnonic logic\ngate [45], induce magnon entanglement [18, 46], and help\ndetecting exotic magnon properties [47–50]. In the lin-\near transport regime, the transmission of electrons and\nmagnons through an obstacle has always been assumed\nto be symmetric, i.e., the same for a wave or particle\ncoming from either side.\nIn this Letter, we address the counter-intuitive ef-\nfect that the strong momentum-dependent eddy-current-\ninduced damping by a normal metal overlayer as shown\nin Fig. 1 may help surmount obstacles such as mag-\nnetic inhomogeneities [51], artificial potential barriers\nformed by surface scratches [52], or dc-current carryingwires [46]. Here we focus on the band edges of magnetic\nfilms that are much thinner than the extinction length of\nthe Damon-Eshbach surface states in thick slabs and are\ntherefore not chiral. Instead, the effect therefore origi-\nnates from the Oersted fields generated by the eddy cur-\nrents in the overlayer that act in only half of the recip-\nrocal space [7] and causes magnon accumulations at the\nsample edges or magnon skin effect [8, 9]. The trans-\nmission through a barrier that is small and symmetric\nfor magnons with opposite wave numbers in an uncov-\nered sample becomes unidirectional with the assistance\nof dissipative eddy currents.\nFIG. 1. Ferromagnetic insulator-normal metal heterostruc-\nture. An in-plane external magnetic field H0orients the\nmagnetization at an angle θwith the ˆz-direction. The yellow\nsheet between the normal metal and ferromagnetic insulator\nindicates suppression of the exchange interaction and conven-\ntional spin pumping.\nModel and non-perturbation theory .—We consider the\nferromagnetic insulator (FI)-normal metal (NM) het-\nerostructure with thickness 2 dFanddMand an in-plane\nmagnetic field H0in Fig. 1. The saturated equilib-\nrium magnetization Msmakes an angle θwith the ˆz-\ndirection such that the torques exerted by the external\nand anisotropy fields cancel. For convenience, we set\nθ= 0 in the following discussion and defer results forarXiv:2401.12022v1 [cond-mat.mes-hall] 22 Jan 20242\nfinite θto the Supplemental Material (SM) [53]. We gen-\neralize a previous adiabatic theory [7, 36] to the full elec-\ntrodynamics of the system by self-consistently solving the\nMaxwell equations coupled with the linearized Landau-\nLifshitz (LL) equations and Ohm’s Law. This treatment\nbecomes exact in the limit of an instantaneous response\nof the metal electrons and high-quality ultrathin mag-\nnetic films.\nThe driving force is an externally generated spatiotem-\nporal magnetization dynamics M(r, t) =M(r, ω)e−iωtat\nfrequency ω. According to Maxwell’s theory, the electric\nfieldEobeys the wave equation ∇2E(r, ω)+k2\n0E(r, ω) =\n−iωµ0JM, where the wave number k0=ω√µ0ε0,µ0(ε0)\nis the vacuum permeability (permittivity), and JM=\n∇×Mis the “magnetization current” [54]. Disregarding\nthe intrinsic Gilbert damping, the LL equation\niωM=−µ0γM×Heff[M] (1)\ngoverns the magnetization dynamics in the FI, where γ\nis the gyromagnetic ratio. The effective magnetic field\nHeff[M] =−δF[M]/δM(r), where the free energy Fis\na functional of the magnetization. It includes the static\nfieldH0, the dipolar field Hd, and (in the FI) the ex-\nchange field Hex=αex∇2Mthat depends on the spin-\nwave stiffness αex. In the presence of the NM layer,\nHeff[M] also contains the Oersted magnetic fields gen-\nerated by the “eddy” currents J=σE, where the elec-\ntrical conductivity σis real. This defines a closed self-\nconsistency problem that we solve numerically.\nWe consider a thin FI film with constant Ms=\n(0,0, Ms). The transverse fluctuations M(r, ω) =\n(Mx(k, ω), My(k, ω),0)eik·rwith in-plane wave vectors\nk= (0, ky, kz) are small precessions with iMx(k, ω) =\nakMy(k, ω), where the complex ellipticity akbecomes\nunity for circular motion.\nThe electric-field modes outside the magnet are plane\nwaves with wave numbers km=p\nω2µ0ε0+iωµ0σ,\nwhere σ= 0 in the absence of an NM layer. The continu-\nity of electric and magnetic fields provides the interface\nboundary conditions. The field in the FI\nEη={x,y,z}(−dF⩽x⩽dF)\n=E(0)\nη(−dF⩽x⩽dF) +RkE(0)\nη(x=dF)e−iAk(x−dF)\nis now modified by the reflection coefficient\nRk=\u0000\nA2\nk−B2\nk\u0001\neiBkdM−\u0000\nA2\nk−B2\nk\u0001\ne−iBkdM\n(Ak−Bk)2eiBkdM−(Ak+Bk)2e−iBkdM,(2)\nwhere E(0)is the solution of Eq. (1) inside the FI without\nthe NM cap [53], Ak=p\nk2\n0−k2, and Bk=p\nk2m−k2.\nThe reflection is isotropic and strongly depends on the\nwave vector. Naturally, Rk= 0 when dM= 0. On the\nother hand, when |k|= 0, the electric field cannot escape\nthe FI, since the reflection is total with Rk=−1.A corollary of Maxwell’s equation—Faraday’s Law—\nreads in frequency space iωµ0[Hd(r, t) +M(r, t)] =∇ ×\nE(r, t). When the magnetization of sufficiently thin mag-\nnetic films is uniform, the Zeeman interaction is propor-\ntional to the spatial average Hdover the film thickness.\nReferring to SM for details [53], we find\nHd,x=\u0014\n−Rk\n4A2\nkdFak(e2iAkdF−1)2(−iAkak+ky)\n+i\n2AkdF(e2iAkdF−1)\u0015\nMx≡ζx(k)Mx,\nHd,y=\"\n−Rk\n4iAkdF(e2iAkdF−1)2 \n−ky\niAkak+k2\ny\nA2\nk+ 2!\n+k2\ny\nA2\nk−k2\ny\nA2\nk1\n2iAkdF(e2iAkdF−1)#\nMy≡ζy(k)My.\nBy substitution into the LL equation (1), the spin wave\neigenfrequencies and ellipticities become\nω(k) =µ0γq\n(˜H0−ζx(k)Ms)(˜H0−ζy(k)Ms),(3a)\nak=q\n(˜H0−ζy(k)Ms)/(˜H0−ζx(k)Ms), (3b)\nwhere ˜H0=H0+αexk2Ms. Imω(k)̸= 0 because of the\nJoule heating due to the eddy currents in the cap layer.\nChiral damping and frequency shifts .—The stray elec-\ntric fields of spin waves propagating perpendicular to the\nmagnetization are chiral, i.e., they depend on their prop-\nagation direction by a hand rule. When kz= 0,E=Ezˆz\nis along the equilibrium magnetization and Ez∝My\nis complex only for positive ky. We illustrate the re-\nsults of the self-consistent calculations for dF= 100 nm,\ndM= 500 nm, conductivity σ= 6.0×107(Ω·m)−1\nfor copper at room temperature [55], applied magnetic\nfield µ0H0= 0.02 T, µ0Ms= 0.178 T, the exchange\nstiffness αex= 3×10−16m2for YIG [56], and γ=\n1.77×1011(s·T)−1. The presence of the NM cap lay-\ners shifts the relative phases between the stray electric\nfields and that of the generating spin waves. We focus\nhere on the wave numbers ky=±1µm−1in Fig. 2(a)\n[Fig. 2(b)] at which the electric field is in-phase (out-\nof-phase) with the transverse magnetization Myˆy. The\nresponse to an in-phase (out-of-phase) electric field is dis-\nsipative (reactive). Both components decay in the FI and\nthe vacuum as ∝1/|k|. In the NM, the in-phase compo-\nnent is screened only in the metal region on the scale of\na skin depth λ=p\n2/(ωµ0σ)∼1.5µm at ω= 11 GHz.\nThe out-of-phase electric field, on the other hand, cre-\nates only a reactive response and is therefore symmetric\nabove and below the metallic film. Also in this case the\ndamping is modulated for constant Gilbert damping by\nthe associated spin wave frequency shift in Fig. 2(b), an\neffect that cannot be captured by the adiabatic approxi-\nmation [7, 36].3\nFIG. 2. The system responds strongly to a phase difference\nbetween the spin waves and their wave vector-dependent ac\nelectric stray fields E. ReEcauses damping [(a)] and Im E\na frequency shift [(b)]. Im Ezgoverns the spin wave vector\ndependence of the chiral damping [(c)]. (d) illustrates the\nstrong ky-dependence of the damping of the lowest standing\nspin wave for Cu thicknesses dM={50,100,200,500}nm. (e)\nshows the real and imaginary parts of the reflection coefficient\nRkthat causes the frequency shifts plotted in (f).\nThe chirality of the radiated electric field controls the\nbackaction of the NM layer that modifies the magnon\ndispersion in a chiral fashion. Figure 2(c) illustrates\nthe strong wave vector-dependent damping coefficient\nαeff(k) =|Imωk|/Reωk. Spin waves propagating in the\npositive ˆy-direction decay much faster than those along\nthe negative direction, while the damping for positive\nand negative kzis the same. According to Fig. 2(d), the\ncalculated damping for kz= 0 in Fig. 2(c) increases (de-\ncreases) with the thickness of the Cu (YIG) film. The\nenhancement of the damping saturates for NM thick-\nnesses dN>1/p\nk2+ 1/λ2, depending on the skin depth\n(λ∼1.5µm) and the wave number 1 /kof the electric\nfield. Moreover, the Kittel mode at k= 0 in Fig. 2(e)\nis not affected by the metal at all because the reflection\ncoefficient Rk=−1, which implies that the dynamics\nof the FI and metal fully decouple. Indeed, recent ex-\nperiments do not find a frequency shift of the FMR by a\nsuperconducting overlayer [57, 58]. The additional damp-\ning by eddy currents reported by Ref. [39] is caused bythe width of the exciting coplanar waveguide, a finite-size\neffect that we do not address here.\nThe real part of Rkin Fig. 2(e) causes an in-phase\nOersted magnetic field that chirally shifts the spin wave\nfrequencies by as much as ∼1 GHz, see Fig. 2(f). Refer-\nence [59] indeed reports a frequency shift of perpendicular\nstanding spin wave modes in Bi-YIG films in the presence\nof thin metallic overlayers.\nThe predicted effects differ strongly from those caused\nby spin pumping due to the interface exchange coupling\nαsp= (ℏγ/M sdF)Reg↑↓, where g↑↓is the interfacial spin\nmixing conductance [60]. αspdoes not depend on the\nthickness of the metal and vanishes like 1 /dF. The fre-\nquency shift scales like Im g↑↓/dFand is very small even\nfor very thin magnetic layers. In contrast, the eddy\ncurrent-induced damping is non-monotonic, scaling like\n∝dFwhen 2 kdF≪1, vanishing for much thicker mag-\nnetic layers, and reaching a maximum at dF∼2λ.\nUnidirectional transmission of wave packets through a\npotential barrier .—The transmission of a wave packet im-\npinging from the left or right at a conventional potential\nbarrier is the same [61]. In the presence of a metal cap,\nthis does not hold for magnons in thin magnetic films.\nBefore turning to the potential scattering in this\nmodel, we have to address the effect of the edges. When\nmagnons propagate in the negative direction without\ndamping but decay quickly when propagating in the op-\nposite one, those reflected at the left boundary of the\nsample accumulate, which is a non-Hermitian skin ef-\nfect [62–65]. We substantiate this conclusion by nu-\nmerical calculations for a two-dimensional square lat-\ntice model with ˆ mi= (1 /√\nN)P\nkˆmkeik·ri, where ˆ mk\nis the annihilation operator of magnons with frequency\nωkfrom Eq. (3a) and ilabels the sites and Nis the\nnumber of sites. The Hamiltonian in the real space\nˆH0=P\nijtjiˆm†\njˆmi, where tji= (1/N)P\nkℏωkeik·(rj−ri)is\na hopping amplitude between possibly distant sites iand\njand the summation is over the first Brillouin zone. With\na coarse-grained lattice constant of ay=az= 0.1µm the\nreciprocal lattice vector 2 π/ay,zis much larger than the\nmagnon modes of interest (refer to the SM [53] for de-\ntails). When the frequencies ωkare complex, the Hamil-\ntonian is non-Hermitian, i.e.,tji̸=t∗\nij.\nFigure 3(a) shows the winding path of the real and\nimaginary eigenfrequencies with wave number. In the\ninterval ky= [−25,25]µm−1and an applied magnetic\nfield parallel to the boundary with θ= 0, the complex\ncomponent is hysteretic, indicating localization of modes\nat opposite boundaries. Figure 3(b)-(c) show the average\nspatial distributions W(r) = (1 /Nm)PNm\nl=1|ϕl(r)|2ofNm\nlowest-frequency eigenstates ϕl(r) for ky∈[−1,1]µm−1\nandkz∈[−1,1]µm−1. When the static magnetic field\naligns with the sample boundary z-axis, i.e.θ= 0 in\nFig. 3(b), the magnons tend to accumulate at the left4\nedge. In the antiparallel configurations θ=π[Fig. 3(c)],\nthe magnons aggregate at the right. In the noncollinear\nconfiguration with θ=π/4 [Fig. 3(d)], the maxima shifts\nto the upper-left corner. While Wis an average, we\nalso illustrate the localization of individual low-frequency\nmodes in SM [53].\nFIG. 3. The magnon skin effect caused by chiral damping.\n(a) Complex spectral winding under periodic boundary con-\nditions when kyevolves from −25 to 25 µm−1forθ= 0. (b)-\n(d) corresponds to the edge or corner aggregations of magnon\neigenstates for other magnetic configurations θ∈ {0, π, π/ 4}.\nWe now illustrate the effect of square potential barri-\ners of width dand height u0,ˆV(y) =u0[Θ(y+d/2)−\nΘ(y−d/2)], where Θ( x) is the Heaviside step function,\non the magnon transmission along ˆy(⊥Ms). With in-\ncoming ⟨y|k0⟩=eik0y, the scattered states |ψs⟩obey the\nLippmann-Schwinger formula [66]\n|ψs⟩=|k0⟩+1\niℏ∂t−ˆH0+i0+ˆV|ψs⟩. (4)\nwhere ˆH0=P\nkℏωkˆm†\nkˆmkis the magnon Hamiltonian for\nan extended film. The transmitted waves read\n⟨y|ψs⟩=\u001aT+(k0)eik0y,{y, k0}>0\nT−(k0)eik0y,{y, k0}<0. (5)\nIn the weak scattering limit |u0d| ≪ | ℏvk0|,\nT±(k0) = 1±\u0012iℏvk0\nu0d−vk0\n2|vk0|\u0013−1\n≈1∓iu0d\nℏvk0,(6)\nwhere vk0=∂ωk/∂k|k=k0ˆyis a generalized group ve-\nlocity that dissipation renders complex. The imaginary\npart of the group velocity and transmission amplitudes\ndepend on the direction of the incoming wave:\nD±(k0) =|T±(k0)|2≈1±2Im\u0012u0d\nℏvk0\u0013\n. (7)For example, with u0/ℏ= 30.5 GHz, d= 0.1µm,k0=\n±0.8µm−1,vk0>0= (2.32 + 0 .52i) km/s and vk0<0=\n−(2.64 + 0 .16i) km/s lead to T+(k0>0)≈0.6 while\nT−(k0<0)≈0.9, so even in the weak scattering limit the\nNM cap layer significantly and asymmetrically reduces\nthe transmission probability.\nWe can assess the strong scattering regime with |u0d|≳\n|ℏvk0|by numerical calculations but find dramatic ef-\nfects on the time evolution of a real-space spin-wave\npacket as launched, e.g., by a current pulse in a mi-\ncrowave stripline. We adopt a Gaussian shape Ψ( r,0) =\ne−(r−r0)2/(2η2)eiq0·rcentered at r0with a width η≫ay,z\nthat envelopes a plane wave with wave vector q0and\nˆV(r) = u0f(r) with either f(|y−˜y0|< d) = 1 or\nf(|z−˜z0|< d) = 1, where ˜ y0and ˜z0are the center of the\nbarriers. According to Schr¨ odinger’s equation Ψ( r, t) =\neiˆHt/ℏΨ(r, t= 0) with ˆH=ˆH0+ˆV(r). Numerical results\nin Fig. 4(a) and (b) u0d≪ |ℏvk0|agree with perturbation\ntheory (7) in the weak scattering regime. However, when\n|ℏvk0|≲u0dand|Im(v−k0)| ≪ | Im(vk0)|≲|Re(v±k0)|\nthe transmission and unidirectionality becomes almost\nperfect. Figure 4(c) and (d) show a nearly unidirectional\ntransmission of the wave packet through the potential\nbarrier for the Damon-Eshbach configuration q0⊥Ms;\nit is transparent for spin waves impinging from the left,\nbut opaque for those from the right. In the calculations,\nq0=q(0)\nyˆywith q(0)\ny=±5µm−1andη= 3µm≫d.\nThe potential barrier is peaked with d=ay,z= 0.1µm\nand its height u0/ℏ= 15 GHz is relatively weak (the\nregular on-site energy ∼13 GHz). Also, dM= 50 nm\nanddF= 20 nm. The results are insensitive to the de-\ntailed parameter values (see SM [53]). The red and blue\ncurves are the incident and reflected wave packets, re-\nspectively. When q(0)\ny<0, the barrier does not affect the\nwave packet that propagates freely through the poten-\ntial barrier and accumulates on the left edge [Fig. 4(c)].\nWhen q(0)\ny>0, as shown in Fig. 4(d), the barrier reflects\nthe wave packet nearly completely, which we associate\nagain with the skin effect since these magnons cannot ac-\ncumulate on the right side. The unidirectional transmis-\nsion is therefore a non-local phase-coherent phenomenon\nthat involves the wave function of the entire sample.\nSince we find the skin effect to be crucial, its absence\nin waves propagating in the ˆz-direction must affect the\ntransport over the barrier. Indeed, our calculations in\nFig. 4(e) and (f) find strong reflection for both propaga-\ntion directions, even when reducing the barrier height by\nan order of magnitude to u0= 1.5 GHz (see SM [53]).\nDiscussion and conclusion .—In conclusion, we cal-\nculate the chiral damping, chiral frequency shift, and\nanomalous transport of magnonic modes in ferromag-\nnetic films with NM cap layers beyond the adiabatic ap-\nproximations. We predict anomalous unidirectional spin\ntransport over potential barriers. This effect is rooted\nin the non-Hermitian magnon skin effect and reflects the5\nFIG. 4. Calculated transmissions [(a) and (b)] and time evolu-\ntion of spin-wave packets in the presence of a potential barrier\nat the origin when q0⊥Ms[(c) and (d)] and q0∥Ms[(e)\nand (f)], where Msand the applied magnetic field are parallel\nto the sample edge with θ= 0. The red and blue curves rep-\nresent, respectively, the incident and scattered wave packets\nwith propagation directions indicated by arrows.\nglobal response of the entire system to a local perturba-\ntion. 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Howes\nDepartment of Physics and Astronomy, University of Iowa, Io wa City, IA, 52242\nand\nEliot Quataert\nDepartment of Astronomy, University ofCalifornia, Berkel ey, CA, 94720\nDraft version July 5, 2018\nABSTRACT\nMeasurements of small-scale turbulent fluctuations in the solar wind find a non-zero right-handed\nmagnetic helicity. This has been interpreted as evidence for ion cyclo tron damping. However, theoret-\nical and empirical evidence suggests that the majority of the ener gy in solar wind turbulence resides in\nlow frequency anisotropic kinetic Alfv´ en wave fluctuations that ar e not subject to ion cyclotron damp-\ning. We demonstrate that a dissipation range comprised of kinetic Alf v´ en waves also produces a net\nright-handed fluctuating magnetic helicity signature consistent wit h observations. Thus, the observed\nmagnetic helicity signature does not necessarily imply that ion cyclotr on damping is energetically\nimportant in the solar wind.\nSubject headings: turbulence — solar wind\n1.INTRODUCTION\nThe identification of the physical mechanisms respon-\nsible for the dissipation of turbulence in the solar wind,\nandforthe resultingheatingofthesolarwindplasma, re-\nmainsanimportantandunsolvedproblemofheliospheric\nphysics. An important clue to this problem is the ob-\nserved non-zero fluctuating magnetic helicity signature\nat scales corresponding to the dissipation range of solar\nwind turbulence.\nMatthaeus et al. (1982) first proposed the “fluctuat-\ning” magnetic helicity as a diagnostic of solar wind tur-\nbulence, defining the “reduced fluctuating” magnetic he-\nlicity spectrum derivable from observational data (see\n§3 below). A subsequent study, corresponding to scales\nwithin the inertial range, found values that fluctuated\nrandomly in sign, and suggested an interpretation that\n“a substantial degree of helicity or circular polarization\nexists throughout the wavenumber spectrum, but the\nsenseofpolarizationorhandednessalternatesrandomly”\n(Matthaeus & Goldstein 1982). Based on a study of the\nfluctuating magnetic helicity of solutions to the linear\nVlasov-Maxwell dispersion relation, Gary (1986) sug-\ngested instead that, at inertial range scales, all eigen-\nmodes have a very small intrinsic normalized fluctuating\nmagnetic helicity, eliminating the need to invoke an en-\nsemble of waves with both left- and right-handed helicity\nto explain the observations.\nSubsequent higher time resolution measurements, cor-\nresponding to scales in the dissipation range, exhibited a\nnon-zero net reduced fluctuating magnetic helicity signa-\nture, with the sign apparently correlated with the direc-\ntion of the magnetic sector (Goldstein et al. 1994). As-\nsuming dominantly anti-sunward propagating waves, the\nstudyconcludedthatthesefluctuationshadright-handed\nhelicity. The proposed interpretation was that left-hand\npolarized Alfv´ en/ion cyclotron waves were preferentially\ndampedbycyclotronresonancewiththeions, leavingun-\ndamped right-hand polarized fast/whistler waves as thedominant wave mode in the dissipation range, producing\nthe measured net reduced fluctuating magnetic helicity.\nWe refer to this as the cyclotron damping interpretation .\nA subsequent analysis of more solar wind inter-\nvals confirmed these findings for the dissipation range\n(Leamon et al. 1998b). Leamon et al. (1998a) argued\nthat a comparison of the normalized cross-helicity in the\ninertial range (as a proxy for the dominant wave prop-\nagation direction in the dissipation range) to the mea-\nsured normalized reduced fluctuating magnetic helicity\nprovides evidence for the importance of ion cyclotron\ndamping, which would selectively remove the left-hand\npolarized waves from the turbulence; using a simple rate\nbalance calculation, they concluded that the ratio of\ndamping by cyclotron resonant to non-cyclotron reso-\nnant dissipation mechanisms was of order unity. A re-\ncentstudyperformingthesameanalysisonamuchlarger\ndata set concurred with this conclusion (Hamilton et al.\n2008).\nIn this Letter, we demonstrate that a dissipation range\ncomprised of kinetic Alfv´ en waves produces a reduced\nfluctuating magnetic helicity signature consistent with\nobservations. A dissipation range of this nature re-\nsults from an anisotropic cascade to high perpendicu-\nlar wavenumber with k⊥≫k/bardbl; such a cascade is con-\nsistent with existing theories for low-frequency plasma\nturbulence (Goldreich & Sridhar 1995; Boldyrev 2006;\nHowes et al. 2008b; Schekochihin et al. 2009), numerical\nsimulations (Cho & Vishniac 2000; Howes et al. 2008a),\nand observations in the solar wind (Horbury et al. 2008;\nPodesta2009). Ourresultsimplythatnoconclusionscan\nbe drawnabout the importanceofion cyclotrondamping\nin the solarwind based on the observedmagnetic helicity\nsignature alone.\n2.FLUCTUATING MAGNETIC HELICITY\nThemagnetichelicityis definedasthe integraloverthe\nplasma volume Hm≡/integraltext\nd3rA·B, whereAis the vector\npotentialwhichdefinesthemagneticfieldvia B=∇×A.2 Howes and Quataert\nThis integral is an invariant of ideal Magnetohydrody-\nnamics (MHD) in the absence of a mean magnetic field\n(Woltjer 1958a,b). Matthaeus & Goldstein (1982) chose\nto set aside the complications associated with the pres-\nence of a mean magnetic field, defining the fluctuating\nmagnetic helicity byH′\nm=/integraltext\ndrδA·δB, where the fluc-\ntuating quantities denoted by δdo not include contribu-\ntions from the mean field.\nModeling the turbulent magnetic field1by\nB(r,t) =B0ˆz+/summationdisplay\nkB(k)ei(k·r−ωt)(1)\nin a periodic cube of plasma with volume L3, we ob-\ntainH′\nm=L3/summationtext\nkH′\nm(k), where the fluctuating mag-\nnetic helicity density for each wave vector kis defined\nbyH′\nm(k)≡A(k)·B∗(k). Here B(−k) =B∗(k) and\nω(−k) =−ω∗(k) are reality conditions and B∗(k) is the\ncomplex conjugate of the Fourier coefficient. Specifying\nthe Coulomb gauge ∇·A= 0, we obtain\nH′\nm(k)=iByB∗\nz−B∗\nyBz\nkx=iBzB∗\nx−B∗\nzBx\nky\n=iBxB∗\ny−B∗\nxBy\nkz, (2)\nwhere the components Bj(k) arise from the eigenfunc-\ntions of the linear wave mode. It is easily shown that\nthis result is invariant to rotation of the wave vector k,\nalong with its corresponding linear eigenfunction, about\nthe direction of the mean magnetic field. The normalized\nfluctuating magnetic helicity density is defined by\nσm(k)≡kH′\nm(k)/|B(k)|2, (3)\nwherek=|k|. This normalized measure has values\nwithin the range [ −1,+1], where negative values de-\nnoteleft-handedhelicityandpositivevaluesdenoteright-\nhanded helicity.\nWe numerically calculate σm(k) over the k⊥–k/bardblplane\nfor the eigenmodes of the linear Vlasov-Maxwell dis-\npersion relation (Stix 1992) for a proton and electron\nplasma with an isotropic Maxwellian equilibrium dis-\ntribution function for each species and no drift veloc-\nities (see Howes et al. 2006, for a description of the\ncode). The dispersion relation depends on five param-\netersω=ωVM(k⊥ρi,k/bardblρi,βi,Ti/Te,vthi/c), for ion Lar-\nmorradius ρi, ionplasmabeta βi, iontoelectrontemper-\nature ratio Ti/Te, and ion thermal velocity to the speed\nof lightvthi/c.\nWe specify plasma parameters characteristic of the so-\nlar wind at 1 AU: βi= 1,Ti/Te= 1, and vthi/c= 10−4.\nFigure 1 is a contour plot of σm(k) obtained by solving\nfor the Alfv´ en wave root over the k⊥–k/bardblplane, then us-\ning the complex eigenfunctions to determine σm(k). The\nMHD regime corresponds to the lower left corner of the\nplot,k/bardblρi≪1 andk⊥ρi≪1; here, the Alfv´ en wave with\nk⊥∼k/bardblis linearly polarized with σm≃0. As one moves\nup vertically on the plot to the regime k/bardbl≫k⊥, the so-\nlution becomes left-handed with values of σm→ −1. In\n1We assume that turbulent fluctuations are reasonably modele d\nas a collection of linear wave modes. Nonlinear interaction s, ne-\nglected here, will serve to replenish energy lost from wave m odes,\nso we neglect the linear wave damping and take only the real fr e-\nquency.Fig. 1.— Normalized fluctuating magnetic helicity density σm(k)\n[eq. 3] for linear Alfv´ en waves over the k⊥–k/bardblplane. The MHD\nAlfv´ en wave (MHD Alfven), ion cyclotron wave (ICW), and kin etic\nAlfv´ en wave (KAW) regimes are identified. Plasma parameter s are\nrepresentative of the near-Earth solar wind.\nthis regime of nearly parallel wave vectors, the solution\nrepresents Alfv´ en waves in the limit k/bardblρi≪√βiand ion\ncyclotronwavesin the limit k/bardblρi/greaterorsimilar√βi. The linear wave\nmode becomes strongly damped via the ion cyclotron\nresonance at a value of k/bardblρi/greaterorsimilar√βi(Gary & Borovsky\n2004). This is precisely the behavior supporting the cy-\nclotrondampinginterpretationofthemeasuredmagnetic\nhelicity in the solar wind.\nBut the Alfv´ en wave solution does not always produce\nleft-handed magnetic helicity. If one moves instead from\nthe MHD regime horizontally to the right, the solution\nbecomes right-handed with σm→+1 ask⊥ρi→1, a be-\nhaviorpreviouslyfoundbyGary(1986). Inthisregimeof\nnearly perpendicular wave vectors with k⊥≫k/bardbl, the so-\nlution represents Alfv´ en waves in the limit k⊥ρi≪1 and\nkinetic Alfv´ en waves in the limit k⊥ρi/greaterorsimilar1. Thus, if the\ndissipation range is comprised of kinetic Alfv´ en waves,\nas suggested by theories for critically balanced, low-\nfrequency plasma turbulence (Schekochihin et al. 2009;\nHowes et al. 2008a), one would expect to observe a pos-\nitive normalized fluctuating magnetic helicity signature\nin that regime.\n3.REDUCED FLUCTUATING MAGNETIC HELICITY\nUnfortunately, due to the limitations of single-point\nsatellite measurements, equations (2) and (3) cannot be\nused directly to calculate the fluctuating magnetic he-\nlicity from observations; approximations must be intro-\nduced to define a related measurable quantity. In this\nsection, we calculate the reduced fluctuating magnetic\nhelicity density, as defined by Matthaeus et al. (1982)\nand used by subsequent authors, for the magnetic field\ndefinedbyequation(1), but withoutassumingthe Taylor\nhypothesis.\nThe two-point, two-time magnetic field correlation\nfunction is\nRij(r,t) =/angb∇acketleftδBi(x,τ)δBj(x+r,τ+t)/angb∇acket∇ight,(4)\nwhere the angle brackets specify an ensemble average,\ndefined here by /angb∇acketlefta(r,t)/angb∇acket∇ight=L−3/integraltext\nd3xa(x,r,t). We find\nRij(r,t) =/summationdisplay\nkB∗\ni(k)Bj(k)ei(k·r−ωt)(5)Solar Wind Magnetic Helicity 3\nwhere the reality conditions ensure that this quantity is\nreal.\nWechoosetosamplethiscorrelationfunctionatamov-\ning probe with position given by r=−vt; this corre-\nspondstosatellitemeasurementsofthesolarwind, where\nthe probe is stationary and the solar wind is stream-\ning past the probe at velocity v. Thus, we may de-\ntermine the reduced magnetic field correlation function,\nRr\nij(t) =Rij(r,t)|r=−vt, obtaining the form\nRr\nij(t) =/summationdisplay\nkB∗\ni(k)Bj(k)e−i(k·v+ω)t.(6)\nThe reduced frequency spectrum, defined by Sr\nij(ω′) =\n(1/2π)/integraltext\ndt′Rr\nij(t′)eiω′t′, is then given by\nSr\nij(ω′) =/summationdisplay\nkB∗\ni(k)Bj(k)δ[ω′−(k·v+ω)].(7)\nThis demonstrates that the frequency ω′of the fluctua-\ntions sampledby the movingprobeis the Dopplershifted\nfrequency ω′=k·v+ω. Note that adopting the Tay-\nlor hypothesis (Taylor 1938), as often done in studies\nof solar wind turbulence, corresponds to dropping ωin\nequation (7).\nThereduced fluctuating magnetic helicity density is de-\nfined by\nH′r\nm(k1) = 2Im[Sr\n23(k1)]/k1. (8)\nwhere the effective wavenumber is calculated from the\nmeasured frequency using k1=ω′/v, assuming the\nTaylor hypothesis is satisfied (Matthaeus et al. 1982;\nMatthaeus & Goldstein 1982), and we have chosen an\northonormal basis with direction 1 along the direction\nof sampling ˆv=v/|v|and directions 2 and 3 in the\nplane perpendicular to ˆv. Thenormalized reduced fluc-\ntuating magnetic helicity density is given by σr\nm(k1) =\nk1H′r\nm(k1)/|B(k1)|2, where|B(k1)|2is the trace power.\nThe relation between the reduced fluctuating magnetic\nhelicity density H′r\nm(k1) and the fluctuating magnetic he-\nlicity density H′\nm(k) can be seen by writingthe spectrum\nin terms of the Doppler-shifted frequency ω′instead of\nk1,H′r\nm(ω′)≡2Im[S23(ω′)]/(ω′/v). Using equation (7)\nand2Im[ a∗b] =i(ab∗−a∗b), thereducedfluctuatingmag-\nnetic helicity density can be written as\nH′r\nm(ω′)=/summationdisplay\nk/parenleftbiggi[B2(k)B∗\n3(k)−B∗\n2(k)B3(k)]\nω′/v/parenrightbigg\n×δ[ω′−(k·v+ω)] (9)\nEquation(9), the experimentallyaccessiblequantity, is\nin terms of the magnetic field measurements in a frame\ndefined by the solar wind velocity v. To write this in\nterms of the theoretically calculable H′\nm(k) (eq. 2), we\nexpress the magnetic field components B2andB3in the\nx,y,zcoordinate system. To do so, define the probe ve-\nlocity in spherical coordinates about the direction of the\nmean magnetic field: v=vsinθcosφˆx+vsinθsinφˆy+\nvcosθˆz. The orthonormal basis specified with respect toˆvcan be written as\nˆe1=ˆv= sinθcosφˆx+sinθsin��ˆy+cosθˆz\nˆe2=ˆz׈v/|ˆz׈v|=−sinφˆx+cosφˆy\nˆe3=ˆe1׈e2=−cosθcosφˆx−cosθsinφˆy+sinθˆz,\n(10)\nFinally, we exploit the fact that the solutions of the\nVlasov-Maxwell dispersion relation depend only on the\nperpendicular and parallel components of the wave vec-\ntork⊥andk/bardblwith respect to the mean magnetic field,\nand not on the angle about the field; thus the eigenfunc-\ntionforawavevector k=k⊥ˆx+k/bardblˆzcanberotatedbyan\nangleαabout the mean magnetic field to yield the solu-\ntion for any wavevector k′=k⊥cosαˆx+k⊥sinαˆy+k/bardblˆz.\nUsing the above, the reduced fluctuating magnetic helic-\nity density Hr\nm(ω′) in equation (9) becomes\nH′r\nm(ω′)=/summationdisplay\nkH′\nm(k)k⊥sinθcosα+k/bardblcosθ\nk⊥sinθcosα+k/bardblcosθ+ω/v\n×δ[ω′−(k′·v+ω)], (11)\nwhere we have specified the azimuthal angle of the probe\nvelocityφ= 0 without loss of generality. It is clear from\nthis equation that all possible wave vectors k′that give\nthe same Doppler shifted frequency ω′will contribute\nto the sum for the reduced fluctuating magnetic helicity\ndensity at the frequency ω′.\n4.DISCUSSION\nPredicting the values of H′r\nm(ω′) for solar wind tur-\nbulence based on equation (11) requires understanding\nthreeissues: the scalingofthe magneticfluctuation spec-\ntrum with wavenumber, the imbalance of Alfv´ en wave\nenergy fluxes in opposite directions along the mean mag-\nnetic field, and the variation of the angle θbetween the\nsolar wind velocity vand the mean magnetic field.\nThe 1-D magnetic energy spectrum in the solar wind\ntypically scales as k−5/3\n1in the inertial range and kp\n1in\nthe dissipation range, where −2≤p≤ −4 (Smith et al.\n2006) and the effective wavenumber is k1=ω′/v. It is\nclearfromequation(11)that, whentheplasmaframefre-\nquencyωis negligible, the Doppler-shifted observed fre-\nquencyalwaysresultsin an effective wavenumber k1≤k,\nwith equality occurring only when the velocity vis\naligned with the wave vector k. We assume that, for\nhomogeneous turbulence at the dissipation range scales,\nturbulent energy at fixed k⊥andk/bardblis uniformly spread\nover wave vectors with all possible angles αabout the\nmean magnetic field. Because the fluctuation amplitude\ndeceases for larger effective wavenumbers, the contribu-\ntion toH′r\nm(ω′) is maximum at angle α= 0; for angles α\nyielding a Doppler shift to lower effective wavenumbers\nk1<(k2\n⊥+k2\n/bardbl)1/2, the higher amplitude fluctuations at\nthose lower wavenumbers will contribute more strongly\ntoH′r\nm(ω′). An accurate calculation of the magnetic he-\nlicity signature based on equation (11) must take into\naccount the scaling of the magnetic energy spectrum.\nTo compare to σr\nm(k1) derived from observations (for\nexample, see Figure 1 of Leamon et al. (1998b)), we con-\nstruct the normalized quantity\nˆσr\nm(k1) =/summationtext\nkH′\nm(k)k′·v\nk′·v+ωδ[ω′−(k′·v+ω)]/summationtext\nk[|B(k)|2/k]δ[ω′−(k′·v+ω)].(12)4 Howes and Quataert\nFig. 2.— Normalized reduced fluctuating magnetic helicity\nˆσr\nm(k1) vs. effective wavenumber k1due to a turbulent spectrum\nof kinetic Alfv´ en waves with θ= 60◦. The solid line corresponds to\nthe model 1-D energy spectrum while the dashed line correspo nds\nto ak−1spectrum.\nIn evaluating equation (12), we assume a model 1-D en-\nergy spectrum2that scales as k−5/3forkρi≪1 and\nk−7/3forkρi≫1, consistent with theories for crit-\nically balanced turbulence (Goldreich & Sridhar 1995;\nHowes et al. 2008a; Schekochihin et al. 2009) and solar\nwind observations (Smith et al. 2006). In Figure 2, we\nplot ˆσr\nm(k1) vs. effective wavenumber k1=ω′/vfor a\nturbulent spectrum filling the MHD Alfv´ en and kinetic\nAlfv´ en wave regimes ( k⊥> k/bardblandk/bardblρi<1) forβi= 1,\nTi/Te= 1,vthi/c= 10−4,θ= 60◦, andv/vA= 10. The\ncontributions to ˆ σr\nm(k1) for all angles αof each wave\nvector are collected in 120 logarithmically spaced bins in\nDoppler-shifted frequency. The results are rather insen-\nsitive to the scaling of the 1-D magnetic energy spectrum\nover the range from k−1tok−4. The solid line in Fig-\nure 2 correspondsto the model spectrum assumedabove,\nwhile the dashed line corresponds to a k−1energy spec-\ntrum. Figure 2 demonstrates that turbulence consisting\nof Alfv´ en and kinetic Alfv´ en waves produces a positive\n(right-handed) magnetic helicity signature in the dissi-\npation range at k1ρi/greaterorsimilar1.\nTheanalysispresentedinFigure2considersonlywaves\nwithk/bardbl>0, so all of the waves in the summation in\nequation (11) are traveling in the same direction. If\nthere were an equal Alfv´ en wave energy flux in the oppo-\nsite direction—a case of balanced energy fluxes, or zero\ncross helicity—the net ˆ σr\nm(k1) would be zero due to the\nodd symmetry of H′\nm(k) ink/bardbl. It is often observed, at\nscalescorrespondingtotheinertialrange,thattheenergy\nflux in the anti-sunward direction dominates, leading to\na large normalized cross helicity (Leamon et al. 1998a).\nIf this imbalance of energy fluxes persists to the smaller\nscales associated with the dissipation range, a non-zero\nvalue of ˆ σr\nm(k1) is expected. However, theories of im-\nbalanced MHD turbulence (Chandran 2008, and refer-\nences therein) predict that the turbulence is “pinned” to\nequal values of the oppositely directed energy fluxes at\nthe dissipation scale. This implies that, at sufficiently\nhigh wavenumber k1, the value of ˆ σr\nm(k1) should asymp-\ntote to zero. Thus, ˆ σr\nm(k1) in Figure 2 would likely dropto zero more rapidly than shown, leaving a smaller pos-\nitive net ˆ σr\nm(k1) around k1ρi∼1, consistent with ob-\nservations (Goldstein et al. 1994; Leamon et al. 1998b;\nHamilton et al. 2008). We defer a detailed calculation of\nthe effects of imbalance to a future paper.\nThe angle θbetween B0andvis likely to vary during\na measurement; this angle does not typically sample its\nfull range0 ≤θ≤π, but hassomedistribution about the\nParker spiral value. Calculations of ˆ σr\nm(k1) over 0≤θ≤\nπyield results that are qualitatively similar to Figure 2,\nso this averagingwill not significantly changeour results.\nTaken together, we have demonstrated that a solar\nwind dissipation range comprised of kinetic Alfv´ en waves\nproduces a magnetic helicity signature consistent with\nobservations, as presented in Figure 2. The under-\nlying assumption of the cyclotron damping interpreta-\ntionofmagnetichelicitymeasurements,aninterpretation\nthatdominates the solarwind literature(Goldstein et al.\n1994; Leamon et al. 1998a,b; Hamilton et al. 2008), is\nthe slab model, k=k/bardblˆzandk⊥= 0, i.e., purely par-\nallel wave vectors. As shown in Figure 1, only in the\nlimitk/bardbl≫k⊥does the Alfv´ en wave root generate a left-\nhanded helicity σm→ −1 ask/bardblρi→√βi; in the same\nlimit, the fast/whistler root generates a right-handed\nhelicityσm→+1 in a quantitatively similar manner\n(see Figure 9 of Gary (1986)). Strong ion cyclotron\ndamping of the Alfv´ en/ion cyclotron waves as k/bardblρi→1\n(Gary & Borovsky 2004) would leave a remaining spec-\ntrumofright-handedfast/whistlerwaves, asproposedby\ncyclotron damping interpretation. However, only if the\nmajorityoftheturbulentfluctuationshave k/bardbl/greaterorsimilark⊥isthe\nslablimitapplicable,andonlyifsignificantenergyresides\nin slab-like fluctuations are the conclusions drawn about\nthe importance of cyclotron damping valid. There is, on\nthe other hand, strong theoretical and empirical support\nfor the hypothesis that the majority of the energy in so-\nlar wind turbulence has k⊥≫k/bardbl(see Howes et al. 2008a\nand references therein). In this case, there is a transi-\ntion to kinetic Alfv´ en wave fluctuations at the scale of\nthe ion Larmor radius. This Letter demonstrates that a\ndissipation range comprised of kinetic Alfv´ en waves pro-\nduces a reduced fluctuating magnetic helicity signature\nconsistent with observations.\nG. G. H. thanks Ben Chandran for useful discussions.\nG. G. H. was supported by the DOE Center for Multi-\nscale Plasma Dynamics, Fusion Science Center Coopera-\ntive Agreement ER54785. E. Q. and G. G. H. were sup-\nported in part by the David and Lucille Packard Foun-\ndation. E. Q. was also supported in part by NSF-DOE\nGrant PHY-0812811 and NSF ATM-0752503.\n2On 150 ×150 logarithmic gridpoints over k⊥ρi,k/bardblρi∈\n[10−3,102], the modelweights B2asafunction of k= (k2\n⊥+k2\n/bardbl)1/2usingB2(k) =B2\n0{[(kρi)−1/3+(kρi)4/3]/[1+(kρi)2]}2.\nREFERENCES\nBoldyrev, S. 2006, Phys. Rev. Lett., 96, 115002\nChandran, B. D. G. 2008, Astrophys. J., 685, 646\nCho, J., & Vishniac, E. T. 2000, Astrophys. J., 539, 273\nGary, S. P. 1986, J. Plasma Phys., 35, 431\nGary, S. P., & Borovsky, J. E. 2004, J. Geophys. Res., 109, 610 5Goldreich, P., & Sridhar, S. 1995, Astrophys. J., 438, 763\nGoldstein, M. L., Roberts, D. A., & Fitch, C. A. 1994,\nJ. Geophys. Res., 99, 11519\nHamilton, K., Smith, C. W., Vasquez, B. J., & Leamon, R. J.\n2008, J. Geophys. Res., 113, A01106Solar Wind Magnetic Helicity 5\nHorbury, T. S., Forman, M., & Oughton, S. 2008,\nPhys. Rev. Lett., 101, 175005\nHowes, G. G., Cowley, S. C., Dorland, W., Hammett, G. W.,\nQuataert, E., & Schekochihin, A. A. 2006, Astrophys. J., 651 ,\n590\n—. 2008a, J. Geophys. Res., 113, A05103\nHowes, G. G., Dorland, W., Cowley, S. C., Hammett, G. W.,\nQuataert, E., Schekochihin, A. A., & Tatsuno, T. 2008b,\nPhys. Rev. Lett., 100, 065004\nLeamon, R. J., Matthaeus, W. H., Smith, C. W., & Wong, H. K.\n1998a, Astrophys. J., 507, L181\nLeamon, R. J., Smith, C. W., Ness, N. F., Matthaeus, W. H., &\nWong, H. K. 1998b, J. Geophys. Res., 103, 4775\nMatthaeus, W. H., & Goldstein, M. L. 1982, J. Geophys. Res., 8 7,\n6011Matthaeus, W. H., Goldstein, M. L., & Smith, C. 1982,\nPhys. Rev. Lett., 48, 1256\nPodesta, J. J. 2009, Astrophys. J., 698, 986\nSchekochihin, A. A., Cowley, S. C., Dorland, W., Hammett,\nG. W., Howes, G. G., Quataert, E., & Tatsuno, T. 2009,\nAstrophys. J. Supp., 182, 310\nSmith, C. W., Hamilton, K., Vasquez, B. J., & Leamon, R. J.\n2006, Astrophys. J. Lett., 645, L85\nStix, T. H. 1992, Waves in Plasmas (New York: American\nInstitute of Physics)\nTaylor, G. I. 1938, Proc. Roy. Soc. A, 164, 476\nWoltjer, L. 1958a, Proc. Nat. Acad. Sci., 44, 489\n—. 1958b, Proc. Nat. Acad. Sci., 44, 833" }, { "title": "2006.16510v1.Negative_Gilbert_damping_in_cavity_optomagnonics.pdf", "content": "Negative Gilbert damping in cavity optomagnonics\nYunshan Cao\u0003and Peng Yany\nSchool of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Films and Integrated Devices,\nUniversity of Electronic Science and Technology of China, Chengdu 610054, China\nExceptional point (EP) associated with the parity-time ( PT) symmetry breaking is receiving considerable\nrecent attention by the broad physics community. By introducing balanced gain and loss, it has been realized in\nphotonic, acoustic, and electronic structures. However, the observation of magnonic EP remains elusive. The\nmajor challenge is to experimentally generate the negative Gilbert damping, which was thought to be highly\nunlikely but is demanded by the PT symmetry. In this work, we study the magneto-optical interaction of\ncircularly-polarized lasers with a submicron magnet placed in an optical cavity. We show that the o \u000b-resonant\ncoupling between the driving laser and cavity photon in the far-blue detuning can induce the magnetic gain (or\nnegative damping) exactly of the Gilbert type. A hyperbolic-tangent function ansatz is found to well describe\nthe time-resolved spin switching as the intrinsic magnetization dissipation is overcome. When the optically\npumped magnet interacts with a purely lossy one, we observe a phase transition from the imbalanced to passive\nPTsymmetries by varying the detuning coe \u000ecient. Our findings provide a feasible way to manipulate the sign\nof the magnetic damping parameter and to realize the EP in cavity optomagnonics.\nIntroduction. —One of the most fundamental principles in\nquantum mechanics is that a physical observable should be\ndescribed by a Hermitian operator to guarantee real eigenval-\nues [1]. However, Bender and Boettcher [2] reported a class\nof non-Hermitian Hamiltonians that allow entirely real spec-\ntrum as long as the combined parity ( P) and time (T)-reversal\nsymmetries are respected. By tuning system parameters, both\nthe eigenvalues and eigenstates of the PT-symmetric Hamil-\ntonian simultaneously coalesce [3, 4], giving rise to a non-\nHermitian degeneracy called exceptional point (EP). The na-\nture around the EP that is accompanied by a phase transi-\ntion can trigger many intriguing phenomena, such as unidi-\nrectional invisibility [5, 6], loss-induced laser suppression and\nrevival [7] and optical transparency [8], laser mode selection\n[9], and EP enhanced sensing [10–13]. Over the past decades,\nthe experimental observation of EPs has been realized in a\nbroad field of photonics [14–17], acoustics [18, 19], and elec-\ntronics [20–22]. Very recently, the concept of PTsymmetry\nis attracting significant attention in spintronics and magnonics\n[23–31]. The simplest way to obtain a PT-symmetric system\nconsists in coupling two identical subsystems, one with gain\nand the other with equal amount of loss. The composite sys-\ntem isPT symmetric because space reflection interchanges\nthe subsystems, and time reversal interchanges gain and loss.\nIndeed, aPT-symmetric magnetic structure composed of two\nidentical ferromagnets with balanced gain and loss was first\nproposed by Lee et al. [23] and subsequently investigated by\nYang et al. [27]. One recent breakthrough was made by Liu\net al. [32] who reported EP in passive PT-symmetric devices\nin the form of a trilayer structure with two magnetic layers\nof di\u000berent (positive) Gilbert damping. However, the exper-\nimental observation of genuine PT symmetry for magnons\n(the quanta of spin waves)—as elementary excitations in or-\ndered magnets—is still elusive. The di \u000eculty lies in that the\nGilbert damping can hardly be tuned to be negative [33, 34].\nThe past ten years have witnessed the development and\napplication of spin cavitronics, allowing cavity photons res-\nonantly coupled to magnons with the same microwave fre-quency [35–46]. One recent trend beyond microwaves is the\nrealization of the parametric coupling between optical lasers\nand magnons, that would generate interesting new opportuni-\nties. Tantalizing physics indeed has been demonstrated, such\nas nonreciprocal Brillouin light scattering [47], microwave-\nto-optical converting [48, 49], optical cooling of magnons\n[50], etc. In these studies, considerable interests have been\ndrawn to the scalar properties of magnons, e.g., magnon num-\nber (population), temperature, and chemical potential, which\nis successful to describe the small-angle spin precession. In\ncontrast, their vectorial behavior, i.e., the full time-evolution\nof the magnetic moment driven by optical lasers, remains\nlargely unexplored, with few exceptions [51]. It has been\nshown that a ferromagnetic-to-antiferromangetic phase tran-\nsition may emerge in the vicinity of the magnonic EP [27]. In\nsuch case, the magnetic moment would significantly deviate\nfrom its equilibrium direction, and a vectorial field descrip-\ntion becomes more relevant than a scalar one.\nS\nz\nx y\nCircularly polarized \nlaser beamOptical cavity(a) (b)\n(c)ωlas > ωcav\nωlas‘ωm\nRed-detuningBlue-detuning\nωlas < ωcav\nωlas‘\nωm\nFIG. 1: (a) Schematic illustration of a macrospin Sinteracting\nwith three orthogonally propagating circularly-polarized lasers (red\nbeams) in an optical cavity. O \u000b-resonant coupling between the driv-\ning laser (!las) and the cavity photon ( !cav) mediated by magnons\n(!m\u001c!cav) in the blue (b) and red (c) detuning regimes.\nIn this Letter, we propose to realize the negative GilbertarXiv:2006.16510v1 [cond-mat.mtrl-sci] 30 Jun 20202\ndamping by considering the optomagnonic interaction be-\ntween three orthogonally propagating circularly-polarized\nlasers and a submicron magnet placed in an optical cavity [see\nFig. 1(a)]. By solving the coupled equations of motion and\nintegrating the photon’s degree of freedom, we derive the an-\nalytical formula of the optical torque acting on the macrospin.\nIn the far-blue detuning, we find that the optical torque exactly\ntakes the Gilbert form \u0000\u000bopt\nS˙S\u0002Swith\u000bopt>0 (see below).\nThe total Gilbert damping becomes negative when the intrin-\nsic dissipation is overcome. In such case, a hyperbolic-tangent\nfunction ansatz is found to well describe the time-resolved\nspin switching. We further study the optically pumped spin\ninteracting with a purely lossy one, and observe a phase transi-\ntion from the imbalanced to passive PTsymmetries by vary-\ning the detuning parameter.\nModel. —The proposed setup is schematically plotted in\nFig. 1(a). Three circularly-polarized laser beams propagat-\ning respectively along x;y;zdirections drive the parametric\ncoupling with a macrospin S=(ˆSx;ˆSy;ˆSz) inside the optical\ncavity. The Hamiltonian reads\nH=\u0000~!0ˆSz\u0000~X\nj=x;y;z\u0010\n\u0001j\u0000gjˆSj\u0011\nˆcy\njˆcj+Hdr; (1)\nwhere!0=\rB0is the Larmor frequency around the exter-\nnal magnetic field B0pointing to the negative z-direction with\n\rbeing the gyromagnetic ratio, \u0001j=!las;j\u0000!cavis the de-\ntuning between the laser frequency !las;jand the cavity reso-\nnant frequency !cav, and ˆ cy\nj(ˆcj) is the creation (annihilation)\noperator of the optical cavity photons, with j=x;y;z. The\ncoupling strength gjbetween the spin and optical photon orig-\ninates from the Faraday-induced modification of the electro-\nmagnetic energy in ferromagnets [52]. The last term describes\nthe interaction between the driving laser and the cavity pho-\ntonHdr=i~P\nj(Ajˆcy\nj\u0000h:c:), where Aj=(2\u0014jPj=~!las;j)1=2is\nthe field amplitude, with \u0014jthe laser loss rate and Pjbeing the\ndriving power.\nThe Heisenberg-Langevin equations of motion for coupled\nphotons and spins are expressed as ( o\u0011hˆoi),\n˙cj=(i\u0001j\u0000\u0014j)cj\u0000igjSjcj+Aj; (2a)\n˙Sx=!0Sy+gynySz\u0000gznzSy; (2b)\n˙Sy=\u0000!0Sx\u0000gxnxSz+gznzSx; (2c)\n˙Sz=\u0000gynySx+gxnxSy; (2d)\nwhere nj=hˆcy\njˆcjiis the average photon number in the cav-\nity. Because the spin dynamics usually is much slower than\noptical photons, one can expand the cavity photon operator as\ncj(t)\u0019cj0(t)+cj1(t)+\u0001\u0001\u0001, in orders of ˙Sj. Equation (2a) then\ncan be recast in series\n0=(i\u0001j\u0000\u0014j)cj0\u0000igjSjcj0+Aj; (3a)\n˙cj0=(i\u0001j\u0000\u0014j)(cj0+cj1)\u0000igjSj(cj0+cj1)+Aj;(3b)\nby keeping up to the first-order terms. We can therefore derivethe formula of photon number in the cavity\nnj(t)\u0019 jcj0j2+2Re[ c\u0003\nj0cj1]\n=A2\nj\n(\u0001j\u0000gjSj)2+\u00142\nj\u00004\u0014jA2\njgj(\u0001j\u0000gjSj)\nh\n(\u0001j\u0000gjSj)2+\u00142\nji3˙Sj:(4)\nSubstituting (4) into Eqs. (2b)-(2d), we obtain\n˙S=\u0000\rS\u0002Be\u000b+\u000b\nS(˙S\u0002S)\u0000\fopt\u0002S; (5)\nwhere the e \u000bective magnetic field Be\u000b=\u0000B0ez+Boptin-\ncludes both the external magnetic field and the optically in-\nduced magnetic field\nBopt=X\nj\r\u00001gjA2\nj\n(\u0001j\u0000gjSj)2+\u00142\njej; (6)\nwhich is the zeroth-order of ˙Sj. The second term in the right\nhand side of (5) is the intrinsic Gilbert damping torque, with\nS=jSjthe total spin number and \u000b > 0 being the intrinsic\nGilbert damping constant. The last term in (5) represents the\noptical torque with the anisotropic e \u000bective field\n\fopt=X\nj4\u0014jA2\njg2\nj(\u0001j\u0000gjSj)\nh\n(\u0001j\u0000gjSj)2+\u00142\nji3˙Sjej; (7)\nwhich is linear with the first-order time-derivative of Sj. Be-\nlow, we show that the anisotropic nature of (7) can be smeared\nout under proper conditions.\nNegative Gilbert damping. —To obtain the optical torque of\nexactly the Gilbert form, we make two assumptions: (i) the\nthree laser beams are identical, i.e., Aj=A;gj=g;\u0014j=\u0014;\nand\u0001j= \u0001; (ii) the optomagnonic coupling works in the far\ndetuning regime, i.e., j\u0011j\u001d1 with\u0011= \u0001=(gS), which allows\nus to drop the gjSjterms in Eq. (7). The optically induced\ne\u000bective fields then take the simple form\nBopt=\r\u00001gA2\n\u00012+\u00142X\njej; (8)\nand\n\fopt=\u000bopt\nS˙S;with\u000bopt=4\u0014A2g2S\u0001\n(\u00012+\u00142)3(9)\nbeing the laser-induced magnetic gain or loss that depends the\nsign of the detuning \u0001. Based on the above results, we finally\nobtain the optically modulated spin dynamics\n˙S=\u0000\rS\u0002Be\u000b+\u000be\u000b\nS(˙S\u0002S); (10)\nwith\u000be\u000b=\u0000\u000bopt+\u000b. One can observe that a negative ef-\nfective Gilbert constant ( \u000be\u000b<0) emerges in the far-blue de-\ntuning regime, i.e., 1 < \u0011 < \u0011 c. In case of the red detun-\ning (\u0011 < 0), we have \u000bopt<0, which indicates the enhance-\nment of the magnetic attenuation. In the deep-blue detuning3\nηηηPTηC\nηC=7.11η P (W)P (μWĎ\nr (m)αopt /α Bopt (μT)αeff =0\nBopt =333 μT(a)\n(b)(c)\n(d)\nηPTηCBopt =440 μTαeff =-α\n×\nFIG. 2: Optically induced magnetic gain (a) and magnetic field (b)\nvs. the optical detuning parameter \u0011. (c)\u0011PT(orange) and \u0011C(green)\nas a function of the driving laser power. (d) Radius dependence of\nthe laser power at the compensation point \u0011C=7:11.\nregime (\u0011>\u0011 c), driving lasers can still generate the magnetic\ngain (\u000bopt>0) but cannot compensate the intrinsic dissipa-\ntion, i.e., 0 < \u000b opt< \u000b. Here\u0011cis the critical detuning pa-\nrameter at which the e \u000bective Gilbert damping vanishes. The\nphysics can be understood from the diagram plotted in Figs.\n1(b) and 1(c): In the blue detuning regime ( !las> ! cav), mi-\ncrowave magnons are emitted in the non-resonant interaction\nbetween the driving laser and the cavity photon, representing\na magnetic gain. On the contrary, they are absorbed in the red\ndetuning (!las< ! cav), manifesting a magnon absorption or\ncooling. Below we discuss practical materials and parameters\nto realize this proposal.\nMaterials realizations. —For a ferromagnetic insulator like\nyttrium ion garnet (YIG), the intrinsic Gilbert constant \u000btyp-\nically ranges 10\u00003\u001810\u00005[53–55]. We take \u000b=10\u00004\nin the following calculations. The magneto-optical coupling\nstrength is determined by the Faraday rotation coe \u000ecient\u0012F\nof the materials gS'c\u0012F=p\u000fr, with cthe speed of light and\n\u000frthe relative permittivity (for YIG, we choose \u000fr=15 [56]\nand\u0012F=188\u000e=cm [57]). We thus have gS=2\u0019\u00191 GHz. The\noptical cavity is set at the resonant frequency !cav=2\u0019=100\nTHz with the loss rate \u0014=2\u0019=1 GHz. For a YIG sphere of\nradius r=10 nm and spin density \u001as\u00191028m\u00003, we esti-\nmate the total spin number S=\u001asr3\u0019104and the coupling\nstrength g=2\u0019\u00190:1 MHz. Materials parameters are summa-\nrized in Table I. Because g\u001c\u0014, all interesting physics occurs\nin the weak coupling regime. A negative \u000be\u000bis demanded for\nrealizing thePTsymmetry in magnetic system. Considering\nthe driving laser with a fixed power P=1\u0016W, the e \u000bective\nTABLE I: Parameters for optical cavity and YIG.\n!cav=2\u0019 \u0014= 2\u0019 ! 0=2\u0019 gS=2\u0019 r\u000b\n100 THz 1 GHz 10 GHz 1 GHz 10 nm 10\u00004Gilbert-type magnetic gain is \u000be\u000b=\u0000\u000bat\u0011PT'6:16, and\nthe critical gain-loss point \u000be\u000b=0 occurs at\u0011C'7:11, indeed\nsatisfying the large-detuning condition j\u0011j\u001d1 in deriving (9).\nFigure 2(a) shows the monotonically decreasing dependence\nof the optically induced magnetic gain \u000bopton the detuning pa-\nrameter\u0011. The\u0011-dependence of the optical field is plotted in\nFig. 2(b), showing that it monotonically decreases with the in-\ncreasing of the detuning, too. Enhancing the laser power will\npush the two critical points \u0011Cand\u0011PTinto the deep detuning\nregion, as demonstrated in Fig. 2(c). For a magnetic sphere\nof larger volume (1 \u0016m)3\u0018(1 mm)3that contains a total spin\nnumber S=1010\u00181019with the reduced magneto-optical\ncoupling strength g=2\u0019=10\u00001\u001810\u000010Hz, the required laser\npower then should be 6 \u001815 orders of magnitude higher than\nthe nm-scale sphere case, as shown in Fig. 2(d).\nTime-resolved spin flipping. —To justify the approximation\nadopted in deriving the Gilbert-type magnetic gain, we di-\nrectly simulate the time evolution of the unit spin components\n(sj\u0011Sj=S) based on both Eq. (5) and Eq. (10). Numer-\nical results are, respectively, plotted in Figs. 3(a) and 3(b)\nfor the same detuning parameter \u0011=1:8 (corresponding to\nan e\u000bective magnetic gain \u000be\u000b=\u00000:0453) and!0=2\u0019=10\nGHz. Both figures show that the very presence of the negative\nGilbert damping can flip the spin in a precessional manner,\nwith similar switching curves. The fast Fourier transforma-\ntion (FFT) analysis of the spatiotemporal oscillation of sxalso\nconfirms this point (see the insets). Although the analytical\nform of sz(t) by solving (5) generally is unknown [58, 59], we\nfind an ansatz that can well describe the time-resolved spin\nswitching\nsj sj\nszη=1.8 (αeff =-0.0453 )\n(a) (c)\n(d) (b)szsysx\nττ τ\nηEq. (5)\nEq. (10)τ0=98.9 (a) fitting\n(b) fitting\nTheoryτ0=107.8’’\nτ0=100.4’τp=22.1τp=22.4’’τp=14.9’ tanh(- )τ-τ0τp\nτ0’\nτ0’’\nTheory\nτp’\nτp’’\nTheory46810121401020 9.779.71\nFrequency (GHz)Frequency (GHz)FFT of s x FFT of s x46810121401020\nFIG. 3: Time evolution of unit spin components ( sx;sy;sz) at de-\ntuning\u0011=1:8 based on Eq. (5) (a) and Eq. (10) (b). Insets\nshow the FFT spectrum of sx. (c) Theoretical fittings of szusing\nthe hyperbolic-tangent ansatz (11) (dashed curves). The solid green\ncurve is the analytical formula without any fitting. (d) Numerical re-\nsults of the \u0011-dependence of the two characteristic times \u001c0and\u001cp,\ncomparing with formula (12) (solid curves).4\nsz(\u001c)'tanh \n\u0000\u001c\u0000\u001c0\n\u001cp!\n; (11)\nwhich is reminiscent of the Walker solution for modeling the\nprofile of 180\u000emagnetic domain wall [60] by replacing the\ntime coordinate \u001cwith the space coordinate x. Here\u001c0is\nthe switching time, \u001cprepresents the life-time of uniform\nmagnons, and \u001c=!0t. From perturbation theory, we derive\nthe analytical form of these two parameters\n\u001cp=\u00001+\u000b2\ne\u000b\n\u000be\u000b;and\u001c0=\u001cptanh\u00001vt\n1\u00004B2\nopt\nB2\ne\u000b:(12)\nFigure 3(c) shows the time evolution of sz. Symbols repre-\nsent the numerical results, dashed curves label the theoretical\nfittings of ansatz (11), and the solid curve is the analytical for-\nmula without fitting. The fitted switching time \u001c0\n0=100:4\n(\u001c00\n0=107:8) and magnon life-time \u001c0\np=14:9 (\u001c00\np=22:4)\nfrom from Eq. (5) [Eq. (10)] compare well with the analyt-\nical formula (12) which gives \u001c0=98:9 and\u001cp=22:1. We\nfurther show that the analytical ansatz agrees excellently with\nnumerical results in a broad range of detuning parameters, as\nplotted in Fig. 3(d).\nPhase transition in spin dimers. —We have shown that un-\nder proper conditions, one can realize the Gilbert-type mag-\nnetic gain which is essential for observing PT-symmetry in\npurely magnetic structures. Next, we consider the optically\npumped spin Sinteracting with a lossy one S0, as shown in\nFig. 4(a). The coupled spin dynamics is described by the\nLandau-Lifshitz-Gilbert equation\n˙s=\u0000\rs\u0002Be\u000b+!exs\u0002s0+\u000be\u000b˙s\u0002s; (13a)\n˙s0=\u0000\rs0\u0002B0\ne\u000b+!exs0\u0002s+\u000b˙s0\u0002s0; (13b)\nwhere s(0)\u0011S(0)=Sis the unit spin vector. Since the optically\ninduced magnetic field is the same order of magnitude with\nthe geomagnetic field (much smaller than B0), it can be safely\nignored. Spin s0is exchange coupled to the optically pumped\nspins, and su \u000bers an intrinsic Gilbert damping. If \u000be\u000b=\u0000\u000b,\nthe two-spin system satisfies the PT-symmetry: Eqs. (13) are\ninvariant in the combined operation of the parity P(s$s0\nandBe\u000b$B0\ne\u000b) and the time reversal T(t!\u0000 t,s!\u0000s,\ns0!\u0000s0,Be\u000b!\u0000Be\u000b, and B0\ne\u000b!\u0000B0\ne\u000b).\nAssuming a harmonic time-dependence for the small-angle\nspin precession sx;y(t)=sx;yei!twithjsx;yj \u001c 1, one can\nsolve the eigenspectrum of Eqs. (13). By tuning the spin-\nspin coupling strength !ex, we observe a transition from exact\nPT phase to the broken PT phase, separated by the EP at\n!c\nex=2\u0019=1 MHz for\u0011=\u0011PT=6:16, as shown in Figs. 4(b)\nand 4(c). Interestingly, the unequal gain and loss, i.e., \u000be\u000b<0\nand\u000be\u000b,\u0000\u000b, leads to an imbalanced parity-time ( IPT )-\nsymmetry. In this region ( \u0011>\u0011 IPT=5:66), the eigenfrequen-\ncies have di \u000berent real parts but share the identical imaginary\none, as plotted in Fig. 4(d). A passive parity-time ( PPT )-\nsymmetry is further identified when \u000be\u000b>0. In such case\nRe[(ω-ω0)/2π] (MHz)(b)(a)\n(d)\n(c)\nωex /2π (MHz)Im[(ω-ω0)/2π] (MHz)\nηη=ηPT\nηPTηPPT ηIPTωex \n2πc\n=1 MHz(e)\nωexs s’\nωex \n2π=1.5 MHzFIG. 4: (a) Spin dimmer consisting of an optically pumped spin sand\na purely lossy one s0. Evolution of eigenfrequencies vs. the exchange\ncoupling (b,c) at the detuning \u0011PT=6:16, and vs. the detuning pa-\nrameter (d,e) at the exchange coupling !ex=2\u0019=1:5 MHz.\n(\u0011 > \u0011 PPT=7:11), the imaginary part of both branches is\nsmaller than their intrinsic damping [see Fig. 4(e)].\nDiscussion. —In the above derivation, we focus on the case\nthat the intrinsic Gilbert damping is isotropic. Our approach\ncan also be generalized to treat the case when the intrinsic\ndamping is anisotrpic [61, 62]. The three propagating lasers\nthen should be accordingly adjusted to match the tensor form\nof the intrinsic magnetic damping, by modulating the driving\npower or the frequency of each beam, for instance. The red-\ndetuning region is appealing to cool magnons to the subtle\nquantum domain. Inspired by PT-symmetric optics [19], we\nenvision a giant enhancement of the magnonic gain and an\nultralow-threshold magnon lasing in a two-cavity system with\nbalanced optical gain and loss, which is an open question for\nfuture study. While the magnonic passive PTsymmetry has\nbeen observed by Liu et al. [32], the exact and imbalanced\nPTphases are still waiting for the experimental discovery.\nConclusion. —To summarize, we have proposed an opto-\nmagnonic method to generate the negative Gilbert damp-\ning in ferromagnets, by studying the parametric dynamics\nof a macrospin coupled with three orthogonally propagating\ncircularly-polarized lasers in an optical cavity. We analyti-\ncally derived the formula of the optical torque on the spin\nand identified the condition for the magnetic gain exactly in\nthe Gilbert form. 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Adams4\n1,2,3University at Bu\u000balo, The State University of New York\n4Lawrence Berkeley National Laboratory\nMarch 23, 2023\nAbstract\nWe present a study of the standard plasma physics test, Landau damping, using the Particle-In-Cell\n(PIC) algorithm. The Landau damping phenomenon consists of the damping of small oscillations in\nplasmas without collisions. In the PIC method, a hybrid discretization is constructed with a grid of\n\fnitely supported basis functions to represent the electric, magnetic and/or gravitational \felds, and a\ndistribution of delta functions to represent the particle \feld. Approximations to the dispersion relation\nare found to be inadequate in accurately calculating values for the electric \feld frequency and damping\nrate when parameters of the physical system, such as the plasma frequency or thermal velocity, are\nvaried. We present a full derivation and numerical solution for the dispersion relation, and verify the\nPETSC-PIC numerical solutions to the Vlasov-Poisson for a large range of wave numbers and charge\ndensities.\nKey words| Simulation, Particle-In-Cell, PETSc, Plasma, Landau damping\n1 Introduction\nIn 1936, Lev Landau \frst formulated a simple kinetic model, now referred to as the Fokker-Plank equation\nin Landau form or simply just the Landau equation, for the description of charged particles in a plasma\nperforming Coulomb collisions [24]. Ten years later, Landau furthered this discovery by predicting the\ndamping of non-relativistic, collisionless plasma oscillations, or Langmuir waves, for the \frst time [25]. The\nbasic concept proposed in that paper, that a conservative phenomenon exhibits irreversible behaviors, has\nsince in\ruenced hundreds of papers and become one of the foundational problems in plasma physics. Thus,\nthe phenomenon is now referred to as Landau damping . In his seminal paper, Landau used the solution to\nthe Cauchy problem for the linearized Vlasov-Poisson equation around a spatially homogeneous Maxwellian\nequilibrium. Landau solved the equation analytically using Fourier and Laplace transforms and concluded\nthat the electric \feld damps exponentially and that the decay is a function of the wavenumber, k, of the\nperturbation. In [5], Bohm and Gross provide a simple explanation for the damping in plasmas. In essence,\nplasmas exhibit a tendency to remain approximately \feld free. Therefore, if electric \felds are introduced,\neither by external disturbance or by an incomplete space charge neutralization, the newly introduced \felds\nwill be forced out by a reaction from the free charges.\nThrough the years, numerous others have extensively examined Landau damping in literature [12, 19, 35].\nIn 2009, a rigorous solution to the nonlinear Vlasov-Poisson equation was given by Villani and Mouhot in [30].\nIn their paper, the damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic\nand spatial variables, rather than exchanges of energy, with phase mixing being the driving mechanism.\nDeveloped in parallel to the theory behind Landau damping, numerical methods for approximating\nsolutions to the kinetic plasma system were pioneered by Vlasov [36]. The Particle-In-Cell (PIC) method has\nbeen a popular choice for numerically simulating plasmas since its inception [17, 18], as it can considerably\nreduce the complexity of the system in comparison to a direct N-body methods. The PIC method is a\nhybrid discretization algorithm comprised of two separate sets of bases for evaluation of di\u000berent aspects of\nthe problem. These bases are the particle basis, where the particle is represented by some (usually radially\n1arXiv:2303.12620v1 [physics.plasm-ph] 22 Mar 2023symmetric) shape function, and the mesh basis, where a mean \feld approach may be taken to computing\ndi\u000berent \feld quantities from external and self consistent forces. Typically, the continuum \feld solve is\nhandled by employing the \fnite element method, although other formulations have used splines [10], \fnite\ndi\u000berence methods [4], etc.\nIn this paper, we present a Particle-In-Cell (PIC) method for solving the Vlasov-Poisson system using the\nPortable Extensible Toolkit for Scienti\fc Computing (PETSc) [2, 3]. PETSc-PIC uses symplectic integration\nschemes [1] for particle pushing while conducting \feld solves with a \fnite element method [21, 26]. The goal\nof PETSc is to provide composable pieces from which optimal simulations can be constructed. PETSc user\nlevel APIs allow applications to delay implementation choices, such as solver details, until runtime using\ndynamic con\fguration [6]. PETSc-PIC solvers fully conserve the moments, mass, momentum and energy,\nat each time step while also preserving entropy monotonicity. Recent advances in the PETSc-PIC code [32]\nalso include conservative projections between the \fnite element and particle basis, a key step towards hybrid\nFEM-particle algorithms.\n2 Problem Formulation\nConsider the Vlasov-Poisson system, a common variation of the more general Vlasov-Maxwell-Landau\nsystem of equations in the non-relativistic case where the magnetic and collisional e\u000bects are neglected. It\ncan be an e\u000bective model for strongly non-Maxwellian plasmas. The Vlasov equation,\n@f\n@t+v\u0001@f\n@x\u0000qe\nmE\u0001@f\n@v= 0; (1)\ndescribes the evolution of the phase space distribution, f(x;v;t), de\fned over the domain ( x;v)2RD\u0002RD\nwhereDis the spatial dimension. The electric \feld is obtained using Poisson's equation,\n\u0001\u001e(x;t) =\u0000\u001a\n\u000f0; (2)\nwhere\u001eis the electric potential, \u001ais the charge density and E=\u0000r\u001e. The charge density contains a\nneutralizing background term, \u001b, such that,\n\u001a(x;v;t) =\u001b\u0000qeZ\nRDf(x;v;t)dv: (3)\nThis neutralizing background simulates the e\u000bect of ions on the electrons in the domain. The use of a\nstationary, uniform background charge is based on the assumption that the ions are much heavier than the\nelectrons and thus feel little in\ruence from them.\nIn order to study the linear Landau damping phenomenon, we consider the initial particle distribution,\nf(x;v;t = 0) =1p\n2\u0019v2\nthe\u0000v2=2v2\nth(1 +\u000bcos(kx)) (4)\n(x;v) = [0;2\u0019=k]\u0002[\u0000vmax;vmax]\nwherevth=p\nKTe=m,\u000b= 0:01,k= 0:5,vmax= 10 and the boundaries are periodic. An important piece\nof PIC methods for the Vlasov-Poisson system is the reduction of noise. A primary source of noise in PIC\nmethods can be traced to the initial discrete distribution of particles in phase space. We mimic a \\quiet\nstart\" [7, 13, 14] continuum initialization in this work by placing particles at the center of the spatial and\nvelocity cells and weighting them based on the initial distribution function f(x;v;t = 0). This method is also\nused in [31], where particles are further remapped back to the cell centers every few steps. The remapping\nstep provides enough noise reduction to accurately observe nonlinear e\u000bects in damping, however, as we are,\nfor now, concerned only with the linear case of Landau damping, we will ignore the remapping phase.\n2.1 Linear Landau Damping\nWe seek to \frst derive a set of equations to understand the damping of plasma oscillations in our\nsystem and to calculate expected values for the damping rate and electric \feld oscillation frequency. These\n2expressions are found by \frst deriving the dispersion relation for a plasma. The derivation shown follows\nfrom [9]. Consider a uniform plasma with an initial distribution f0(v) with zero initial electric and magnetic\n\felds, E0=B0= 0. To \frst order, the perturbation in f(x;v;t ) is denoted by f1(x;v;t ) such that,\nf(x;v;t ) =f0(v) +f1(x;v;t ): (5)\nPlugging (5) in to (1) gives,\n@f1\n@t+v\u0001@f1\n@x\u0000qe\nmeE1\u0001@f0\n@v= 0: (6)\nAssuming that the ions are massive and \fxed and that the waves are one-dimensional plane waves\nf1/ei(kx\u0000!t), (6) becomes,\nf1=iqeEx\nme@f0=@vx\n!\u0000kvx: (7)\nRecall the Poisson equation (2), with the potential \u001ereplaced by the divergence of the electric \feld,\nr\u0001E=r\u0001E1=\u0000\u001a\n\u000f0(8)\n=\u00001\n\u000f0\u0012\n\u001b\u0000qeZ\n(f0(v) +f1(x;v;t ))dv\u0013\n:\nWith zero initial electric \feld, the electric \feld vector is replaced by the electric perturbation, E1, which\ntakes the form E1=Exei(kx\u0000!t)^ x. Furthermore, at equilibrium, the neutralizing background is equal to the\ntotal weight of the electron distribution, \u001b=qeR\nf0dv, leaving only the perturbation term f1in the Poisson\nequation. Thus we are left with,\nik\u000f0Ex=\u0000qeZ\nf1dv: (9)\nSubstituting (7) into (9) and dividing by ik\u000f0Ex, we have,\n1 =q2\ne\nkme\u000f0Z@f0=@v\n!\u0000kvdv: (10)\nThe integral in (10) is a three-dimensional integral, however for the Maxwellian or other factorable\ndistribution, integration in the 2nd and 3rd dimension is simple. Evaluating the (10) integral in the 2nd and\n3rd dimension, and substituting in the plasma frequency, !p=\u0000\nneq2\ne=m\u000f 0\u00011=2, leaves the dispersion relation,\n1 =!2\np\nk2Z1\n\u00001@f0=@vx\nvx\u0000(!=k)dvx: (11)\nLandau showed that this problem can be solved rigorously by means of the Laplace transform method.\nImportantly, it is necessary to go around the singularity in the integrand in (11) in the complex plane. The\nsolution to (11) takes the form,\n!=!r+i\r; (12)\nwhere!rrepresents the real oscillations of the plasma and \rthe imaginary, which Landau showed to be the\npart of the solution driving the damping of the oscillations. Following Landau's method [9], an approximation\nfor the oscillation and damping terms can be derived, given by,\n!r= 1 +3\n2^k2; (13)\n\r=\u0000r\u0019\n81\n^k3exp\u0014\n\u00001\n2^k2\u0015\n:\nA normalized form of the wavenumber khas been introduced to simplify the equations going forward. The\nnormalized wavenumber, ^kis given by,\n^k=kvth\n!p(14)\n3wherevth=p\nKT=m is the thermal velocity. For all examples, we non-dimensionalize so that vth= 1. The\nreal part of the solution to (11) was similarly derived by Vlasov in [36], however Vlasov did not account for\nthe imaginary damping term.\nThese approximations are valid for the case where ^k\u001c1 but their accuracy degrades considerably as ^k\napproaches 1 and higher. Even when ^k= 0:5, the calculated values for !rand\rdi\u000ber from the numerical\nresults by at least 5%. In [29], McKinstrie draws similar conclusions, electing to derive more accurate forms\nof (13) by expanding !rin powers of ^k,\n!r= 1 +3\n2^k2+15\n8^k4+147\n16^k6; (15)\n\r=\u0000r\u0019\n8\u00121\n^k3\u00006^k\u0013\nexp\u0014\n\u00001\n2^k2\u00003\n2\u00003^k2\u000012^k4\u0015\n: (16)\nThese new expressions are more accurate for ^kup to 0:4 but still diverge from the correct values as ^kincreases\nfurther. Shalaby et. al. provided further re\fnements to these equations in [33], using a numerical \ftting\nformula, taking the form,\n!=1 +3\n2^k2+15\n8^k4+147\n16^k6+ 736:437^k8\u000014729:3^k10(17)\n+ 105429 ^k12\u0000370151 ^k14+ 645538 ^k16\u0000448190 ^k18;\n\r=\u0000r\u0019\n8\u00121\n^k3\u00006^k\u000040:7173^k3+ 3900:23^k5\u00002462:25^k7\u0000274:99^k9\u0013\nexp\u0014\n\u00001\n2^k2\u00003\n2\u00003^k2\u000012^k4\u0000575:516^k6+ 3790:16^k8\n\u00008827:54^k10+ 7266:87^k12i\n:\nThese equations give good estimates for !rand\rin the case where ^k= 0:5, which is of particular interest\nin this paper. In fact, the values obtained from (17) in the case where ^k= 0:5 and all other parameters\n(!p,vth,qe,etc.) are assumed to be 1 :0 match those commonly listed as \\analytic solutions\" [9, 31, 38]. That\nbeing said, the accuracy of the numerical \ft still decreases considerably for ^k>0:6.\nAn alternate, and as we will show, more accurate way to calculate !rand\rfor given values of ^kis to\n\fnd them by computing the zeros of (11). This was done by Canosa in [8] for values of kranging from 0 :25\nto 2:0 in increments of 0 :05 (see Table 1 for a selection of values). A comparison of the approximations by\nLandau, McKinstrie and Shalaby to the zero \fnding results from Canosa is shown in Section 4.\n^k!r\r\n0.25 1.1056 -0.0021693\n0.5 1.4156 -0.15336\n0.75 1.7371 -0.46192\n1.0 2.0459 -0.85134\n1.5 2.6323 -1.7757\n2.0 3.1891 -2.8272\nFigure 1: Values for!rand\rfor given values of ^kfrom [8].\n3 PETSc-PIC\nPETSc, the Portable, Extensible Toolkit for Scienti\fc Computation, is a well-known library for numerical\nmethods. It provides parallel data management, structured and unstructured meshes, linear and nonlinear\nalgebraic solvers and preconditioners, optimization algorithms, time integrators and many more functions.\nThe PETSc-PIC algorithm relies on two modules to handle the particle and mesh solves simultaneously. The\n4\frst, DMPlex [22, 23, 26], is a PETSc module for generic unstructured mesh creation, manipulation, and\nI/O [16]. It decouples user applications from the implementation details of common mesh and discretization\ntasks. The other important module for this work, DMSwarm [28], provides a fully parallel solution for pure\nparticle methods (e.g. DEM, SPH, EFG) and for particle-mesh methods (e.g. PIC, FLIP, MPM, GIMP).\nWe start with discussion of the particle methods in the PETSc-PIC algorithm. A method must \frst be\nchosen to represent the particle space, and for interpolation between the mesh and particle representations.\nThere are numerous choices in shape functions for this purpose, however in our case a simple delta function\nrepresentation of particles is chosen. Thus the approximation of the distribution function is de\fned in the\nparticle space as,\nfp=X\np\u0000 !!p\u000e(x\u0000xp); (18)\nwhere\u0000 !!pis the vector of weights, xare the con\fguration space variables and xprepresents the particle\nposition and velocity, respectively. The \fnite element representation, using a function space V, is given by\nthe weighted sum of basis functions,\nfFE=X\nifi i(x); (19)\nwhere i2Vdenotes the basis functions and fithe associated \fnite element coe\u000ecient.\nThe Vlasov equation is a linear hyperbolic equation which may be written in a simpler form,\n@f\n@t+z\u0001rqf= 0; (20)\nwhere q= (x;v) is the phase space variable and z= (v;\u0000qeE=m) is the combined force. The force term\n\u0000qeE=mis independent of velocity, and therefore (20) may be written in the conservative form,\n@f\n@t+rq\u0001(zf) = 0: (21)\nGiven this new advective form of the Vlasov equation, we can rewrite the equation for the characteristics\nQ= (X;V),\ndQ\ndt=z; (22)\nwhich reexpressed with the original phase-space variables gives,\ndX\ndt=V; (23)\ndV\ndt=\u0000qe\nmE:\nSince particles follow characteristics, the Vlasov equation in the particle basis becomes\ndxp\ndt=vp; (24)\ndvp\ndt=\u0000qe\nmE:\nThe equations of motion are stepped forward in time using structure-preserving symplectic integrators\nwhich have been well studied [15]. The electric \feld is solved concurrently at each step using a \fnite element\nsolver, discussed in the next section.\n3.1 PETSc-FEM\nAt each step in the simulation, the Poisson equation is solved using the \fnite element method. The\ngradient of the potential, i.e. the electric \feld, is then interpolated across each cell at the particle locations.\nThe interpolated electric \feld is then applied to the particles in the form of the Coulomb force.\nThe PETSc-FEM method is abstractly formalized by the Ciarlet triple [11, 20], such that a \fnite element\nis a triple (T;V;V0), where,\n5•the domainTis a bounded, closed subset of Rd(ford= 1;2;3;:::) with nonempty interior and piecewise\nsmooth boundary;\n•the spaceV=V(\n) is a \fnite-dimensional function space on Tof dimension n;\n•the set of degrees of freedom (nodes) V0=fl1;l2;:::;l ngis a basis for the dual space, that is, the space\nof bounded linear functionals on V.\nThe cellTtogether with the local function space Vand the set of rules for describing the functions in Vis the\n\fnite element . The discretization in PETSc is handled by the PETScFE object, which contains a PetscSpace\n(V), PetscDualSpace ( V0), and DMPlex (T). PETScFE supports simplicial elements, tensor cells, and some\nspecial cells such as pyramids.\nIn general, the \fnite element solve for the Poisson equation can be accomplished using the standard H1\nfunction space. In the H1space, the weak form of the Poisson equation is,\nZ\n\nr i\u0001r\u001e=Z\n\n i; (25)\nwhere 2VandVis the set of basis functions on the cell. The elements are then constructed such that\nthe basis functions are continuous across the cell boundaries.\n3.2 Conservative Projections\nTo preserve the conservation laws in a PIC simulation, a method must be constructed to conservatively\nproject between the particle and grid representations. Weak equality of the representations,\nZ\n\n ifFE=Z\n\n ifP (26)\nis enforced on the representations to achieve this [27, 32]. Restricting this equivalence to the \fnite-\ndimensional analogues gives the matrix-vector form,\nMfFE=Mpfp; (27)\nwhere M is the \fnite element mass matrix,\nM=Z\n\n i j; (28)\nMpis the particle mass matrix,\nMp=Z\n! i\u000e(x\u0000xp); (29)\nfFEis a vector containing the \fnite element coe\u000ecients and fpis the vector of particle weights. The\nentries ofMpcontain evaluations of the \fnite element basis functions at particle locations with rows being\ndetermined by the basis function index, and columns being determined by the particle indices. Moving from\nthe particle basis to the mesh, we must invert the \fnite element mass matrix, which is easily accomplished\nwith CG/Jacobi [37]. In the other direction, we must invert a rectangular particle mass matrix, usually with\nLSQR [32].\n4 Numerical Results\nIn this section, the results of this numerical study are presented. We consider the one-dimensional (1X-\n1V) case of the Vlasov-Poisson system. According to (15), derived in Section 2.1, and the zero \fnding\ndata from Canosa [8], the damping rate should be \r=\u00000:153 and the frequency of oscillations should be\n!r= 1:416. All runs were conducted on a single 2.4 [ GHz ] 8-Core Intel Core i9 processor with 64 [ GB] of\nmemory. The example code and packages/options required to run it are provided in Appendix A.\n6To begin, we show results from the densest run of the PETSc-PIC simulation with 160 spatial cells and\n8;000 particles per cell and a PIC timestep of dt= 0:3. Figure 2 shows the maximum value of the electric\n\feld,Emax= max \njEj, over time. The values for \rand!rwere measured by \ftting the peaks of the given\ndata. The frequency of oscillations describes the frequency of the electric \feld completing one full oscillation.\nSince each oscillation of Emaxis the equivalent of one half of the electric \feld period, we count two Emax\noscillations for each plasma oscillation. Values achieved by the PETSc-PIC algorithm, \r=\u00000:1531 and\n!r= 1:4124 agree within 1% of the analytic values from Canosa and Shalaby et. al., which are assumed to\nbe the most accurate for the case k= 0:5.\nFigure 2: The maximum value of the electric \feld as a function of time for the one-dimensional linear\nLandau damping problem.\nThe total error in the moments, shown in Figure 3, was shown to be stable over the entire runtime. At\nearly times, the error in momentum and energy \ructuate but each converges by t= 20. This convergence\ncomes from the use of basic symplectic integrator in PETSc-PIC which guarantees the error does not grow\nover time. We also note that the error in the particle solve and the \fnite element solve is exactly equal,\napart from an increased level of noise in the particle solve. This con\frms the e\u000bectiveness of the conservative\nprojector used in PETSc-PIC. More detailed tests of the conservative projector can be found in [32].\nConvergence studies were conducted in which either the mesh or the number of particles per cell were\nincreased while the other was held constant. In the case of mesh convergence, the number of particles per\ncell was held at Nv= 8;000 while in the particle number convergence tests, the number of mesh cells was\nNx= 100. We naively expect Monte Carlo convergence in particle number, O\u0010\n1=p\nN\u0011\n, and we indeed\nachieve this for !rin the upper left of Figure 4. Since we have such a regular initial particle distribution,\n7Figure 3: The total mass, momentum and energy error for the particle and \fnite element solve. The\nmoment errors all converge to zero given a long enough time.\nwe might hope to see Quasi-Monte Carlo convergence, O(1=N), and we do see this superconvergence in \rin\nthe upper right of Figure 4. We expect O\u0000\nh2\u0001\nconvergence in the mesh resolution hsince this controls the\nerror in the electric \feld, and we see this in gamma in the lower right of Figure 4. However, this convergence\nshould quickly saturate as particle error begins to dominate, which we see in !rin the lower left of Figure 4.\n4.1 Variations in Wavenumber and Charge Density\nWe have thus far shown that the PETSc-PIC algorithm is an accurate and structure-preserving method for\nmodeling plasma systems. We next present results from tests in which the wavenumber k, and consequently\nthe domain size, and the charge density were varied. Varying either of these values impacts the value of the\nnon-dimensional wavenumber ^k. In the case of the wavenumber k, the calculated values for !rand\rwere\ncompared to the values obtained with the approximation equations (13) and (15), the numerical \ft (17)\nand the zero \fnding results from Table 1. The results from PETSc-PIC, shown in Figure 5, clearly show\na strong deviation of the approximation equations and the numerical \ft for ^k >0:5 while closely matching\nthe zero \fnding data. This demonstrates that these approximations quickly break down outside of the small\nparameter range typically chosen in numerical studies of Landau damping. When considering real plasma\nsystems in which values for k,!p, etc. are more dynamic, it is far more e\u000bective to use zero \fnding methods\nto calculate expected values for !rand\r.\nIt may be naively assumed that data from numerical tests with varying charge densities will match the\napproximation equations (13), (15) and (17) or even the zero \fnding data from Canosa, however, these\nanalytic results are based on an assumption of unchanging charge density. More speci\fcally, these results\nare based on charge densities such that the plasma frequency, !p, is always unity. Therefore, to accurately\ncompare analytic results to our data we must resolve the dispersion relation for varying charge densities. A\nzero \fnding algorithm, using Newton's method [34], was employed to calculate new analytic values for !r\nand\rwith charge densities ranging from 0 :1 to 2:0. The zero \fnding algorithm calculates multiple values for\n!rand\rhowever we select the solution containing the largest \rwhich corresponds to the smallest !r. Other\n8Figure 4: (Top) Particle per cell number convergence plots for Nx= 100 and (bottom) mesh convergence\nplots forNv= 8;000.\nsolutions found by the algorithm represent less dominant modes which can be ignored for the purposes of this\nstudy. Figure 6 contains the results from the new zero \fnding algorithm along with data from numerical tests\nwhich agree perfectly. We observe that when the charge density is increased, the frequency of oscillations\nalso increases. This matches the expected physical behavior of an electrically charged plasma.\nWe have extended our zero \fnding algorithm to the case where the charge density approaches zero\n(^k!1 ) to make note of an interesting phenomenon. At a charge density of zero, the dispersion relation\nhas no solution. We capture this in Figure 6, where we observe that both !rand\rare asymptotic atR\nf0=V= 0. This can similarly be observed in the numerical results from our PETSc-PIC algorithm. As\nthe charge density is decreased, the rate at which the electric \feld oscillations becomes too large to resolve\nnumerically. In the case ofR\nf0=V= 0:25, shown in Figure 7, we can only observe two full oscillations of the\nelectric \feld before the simulation becomes too noisy. Theoretically, the charge density could be decreased\nasymptotically in our simulations to observe the damping rate and frequency trends but in practice there is\ntoo much noise to resolve any real processes in the plasma.\n9Figure 5: A comparison of various approximations for !rand\rto root \fnding results and numerical results\nfrom PETSc-PIC. Plots on the right are zoomed in on the region 0 :0\u0014^k\u00140:75 to show the accuracy of\neach approximation before they diverge from the data.\n5 Conclusions and Future Work\nWe have presented PETSc-PIC, a structure-preserving Particle-in-Cell algorithm for solving the\nelectrostatic Vlasov-Poisson systems. The accuracy of our algorithm has been demonstrated by comparing\nthe frequency of electric \feld oscillations and the damping rate of the oscillations to analytic values. We\nhave also shown that the approximations for the frequency and damping rate break down outside of narrow\nranges for the wavenumber and charge density. These approximations are often cited in numerical Landau\ndamping studies without further context or reference to the equations used to compute the parameters,\nwhich can lead to complications in reproducing results. We have sought to provide a complete picture of\nLandau damping and the numerical methods we have used to simulate this phenomenon.\nFuture work with the PETSc-PIC algorithm will fall into two primary categories: improvements to\nthe algorithm and the extension of the Landau damping test to multi-dimensional and nonlinear cases.\nImprovements to the algorithm will focus on reformulation using a mixed form of the Poisson equation and\nH(div) \fnite elements. We expect that C0electric \felds will decrease the noise in our particle representation\nover time. While we have not observed any major negative impacts from using H1\fnite elements in the test\nproblem chosen for this paper, Landau damping, using a mixed form makes a notable di\u000berence in the case of\nTwo-Stream Instability. PETSc currently includes support for the H(div) conforming \fnite elements Brezzi-\nDouglas-Marini (BDM) and Raviart-Thomas (RT) on simplicial grids, however RT elements are currently\n10Figure 6: Numerical results for varying charge densities compared to zero \fnding data. The charge density\nis represented on the x-axis as the integral of the initial distribution over the domain volume.\nFigure 7: A comparison of electric \feld oscillations given di\u000berent charge densities.\nthe only element type supported on tensor cells. We will also replicate our tests in parallel, allowing us to\nincrease the number of particles per cell by several orders of magnitude, reducing the largest source of error\nin the code.\nNonlinear Landau damping is a more complex in that non-damping phenomenon, such as plasma echo,\nare present. Vitally, the linearization of the Vlasov equation, used as the fundamental approximation in\n11the study of linear Landau damping, does not guarantee that the asymptotic behavior of the linear Vlasov\nequation is an approximation of the asymptotic behavior of the nonlinear Vlasov equation [30]. There are\nreasons to doubt that the study of the linearized equations gives any hint on the long-time behavior of the\nnonlinear equations. Therefore if an algorithm is desired that can accurately capture the long-time behavior\nof a plasma, the nonlinear case of Landau damping must also be considered.\nA Appendix A\nThe data presented in this paper can be recreated with PETSc using the DMSwarm example\nex9 ($PETSCDIR=src=dm=impls=swarm=tests=ex 9:c). Exact runtimes may vary depending on the\narchitecture and compiler. The DMSwarm example can be run using the following options:\n./ex9 -dm_plex_dim 2 -dm_plex_simplex 0 -dm_plex_box_bd periodic,none\n-dm_plex_box_faces 10,1 -dm_view\n-dm_plex_box_lower 0,-0.5 -dm_plex_box_upper 12.5664,0.5\n-dm_swarm_num_species 1 -dm_swarm_num_particles 50\n-vdm_plex_dim 1 -vdm_plex_simplex 0 -vdm_plex_box_faces 7500\n-vdm_plex_box_lower -10 -vdm_plex_box_upper 10\n-petscspace_degree 1 -em_type primal\n-em_pc_type svd -em_snes_atol 1.e-12\n-ts_type basicsymplectic -ts_basicsymplectic_type 1\n-ts_max_time 500 -ts_max_steps 1000 -ts_dt 0.03\n-fake_1D -cosine_coefficients 0.01,0.5 -charges -1.0,1.0\n-perturbed_weights -periodic\nThis example uses a 100 square cell mesh on the domain ( x;v)2[0;4\u0019]\u0002[\u000010;10], with 8000 particles\nper cell. A \frst-order basic symplectic integrator is chosen as the time integration method and H1\fnite\nelements are chosen for the \feld solves.\nReferences\n[1] Shrirang Abhyankar et al. PETSc/TS: A Modern Scalable DAE/ODE Solver Library . Preprint\nANL/MCS-P5061-0114. ANL, Jan. 2014.\n[2] Satish Balay et. al. PETSc/TAO Users Manual . English. Version Revision 3.18. Argonne National\nLaboratory. 2022. 310 pp.\n[3] Satish Balay et al. PETSc Web page .https://petsc.org/ . 2022. url:https://petsc.org/ .\n[4] Je\u000brey W. Banks et al. \\High-Order Accurate Conservative Finite Di\u000berence Methods for Vlasov\nEquations in 2D+2V\". In: SIAM Journal on Scienti\fc Computing 41.5 (2019), B953{B982. doi:\n10.1137/19M1238551 . eprint: https://doi.org/10.1137/19M1238551 .url:https://doi.org/10.\n1137/19M1238551 .\n[5] D. Bohm and E. P. Gross. \\Theory of Plasma Oscillations. A. Origin of Medium-Like Behavior\". In:\nPhys. Rev. 75 (12 1949), pp. 1851{1864. doi:10.1103/PhysRev.75.1851 .url:https://link.aps.\norg/doi/10.1103/PhysRev.75.1851 .\n[6] Jed Brown, Matthew G. Knepley, and Barry Smith. \\Run-time extensibility and librarization of\nsimulation software\". In: IEEE Computing in Science and Engineering 17.1 (Jan. 2015), pp. 38{45.\ndoi:10.1109/MCSE.2014.95 .\n[7] J.A. Byers. \\Noise Suppression Techniques in Macroparticle Models of Collisionless Plasmas\". In:\nProceedings of the Fourth Conference of Numerical Simulation of Plasmas . Ed. by NTIS. NRL,\nWashington, D.C., 1970.\n[8] Jos\u0013 e Canosa. \\Numerical solution of Landau's dispersion equation\". In: Journal of Computational\nPhysics 13.1 (1973), pp. 158{160. issn: 0021-9991. doi:https://doi.org/10.1016/0021-9991(73)\n90131-9 .url:https://www.sciencedirect.com/science/article/pii/0021999173901319 .\n12[9] F.F. Chen. Introduction to Plasma Physics and Controlled Fusion . 2nd ed. Vol. 1. Plenum Press, New\nYork, 1984.\n[10] C.Z Cheng and Georg Knorr. \\The Integration of the Vlasov Equation in Con\fguration Space\". In:\nJournal of Computational Physics 22.3 (1976), pp. 330{351. issn: 0021-9991. doi:https://doi.org/\n10.1016/0021-9991(76)90053-X .url:https://www.sciencedirect.com/science/article/pii/\n002199917690053X .\n[11] Philippe G. Ciarlet. Numerical Analysis of the Finite Element 0)is the hopping parameter, and J(>0)\naccounts for s-dexchange interaction [Eq. (3)]. The (\u00001)i\nfactor in the exchange term reflects the two-sublattice nature of\nthe AFM. The conduction subsystem unit cell consists of two\nbasis atoms, similar to the magnetic subsystem. As a result,\nthere are four distinct electron bands: two due to there being\ntwo basis atoms per unit cell, and twice this due to the two\npossible spin polarizations per electron. Disregarding applied\nfields, these bands constitute two spin-degenerate bands. We\nlabel these bands 1 and 2, where the latter is higher in energy.The itinerant electron Hamiltonian [Eq. (2)] is diagonalized\ninto an eigenbasis (c1k\u001b;c2k\u001b)with eigenenergies \u000f1k=\u0000\u000fk\nand\u000f2k= +\u000fk, where\u000fk=p\nJ2S2+t2j\rkj2, where\rk=P\nh\u000eie\u0000ik\u0001\u000e. The itinerant electron dispersion is depicted in Fig.\n2.\nThe magnetic and conduction subsystems interact through\ns-dexchange interaction, parametrized by J:\nHI=\u0000JX\niSi\u0001si; (3)\nwhere si=P\n\u001b\u001b0cy\ni\u001b\u001b\u001b\u001b0ci\u001b0is the spin of the itinerant elec-\ntrons at site i, where \u001bis the vector of Pauli matrices. The term\nwhich is zeroth order in the magnon operators, and thus ac-\ncounts for the static magnetic texture, is already included in He\n[Eq. (2)]. To first order in magnon operators, the interaction\nHamiltonian can be compactly written as\nHe\u0000m=X\n\u0015\u001aX\nkk0qcy\n\u0015k\"c\u001ak0#\u0010\nWA;\u0015\u001a\nkk0qay\n\u0000q+WB;\u0015\u001a\nkk0qbq\u0011\n+h.c.;(4)\nwhere\u0015and\u001aare summed over the electron band indices. As\ndetailed in the Supplemental material, WA;\u0015\u001a\nkk0qandWB;\u0015\u001a\nkk0q, both\nlinear in J, are coe \u000ecients determining the amplitudes for\nscattering between the itinerant electrons and the aqandbq\nmagnons, respectively. Specifically, when considering plane\nwave states, WA=B;\u0015\u001a\nkk0qbecomes a delta function, thereby enforc-\ning the conservation of crystal momentum in a translationally\ninvariant lattice. Inclusion of disorder or other many-body\ne\u000bects results in deviation of the eigenstates from ideal plane\nwaves causing a wave vector spread around its mean value [ 2].\nThe delta function, associated with an exact crystal momentum\nconservation, is thus transformed to a peaked function with\nfinite width (\u0001k). The\u0015\u001acombinations 11and22describe\nintraband electron scattering, while 12and21describe in-\nterband scattering. Intraband scattering is illustrated in Fig.\n2. Interband scattering is prohibited within our model due to\nenergy conservation, since the uniform q=0magnon energy\nis much smaller than the band gap.\nThe scattering described by He\u0000m[Eq. (4)] transfers spin\nangular momentum between the magnetic and conduction sub-\nsystems. The itinerant electrons are assumed to maintain a\nthermal distribution thereby acting as a perfect spin sink. This\nis consistent with a strong conduction electron spin relaxation\nobserved in metallic AFMs [ 70,71]. As a result, the magnetic\nsubsystem spin is e \u000bectively damped through the s-dexchange\ninteraction.\nGilbert damping.— In the Landau-Lifshitz-Gilbert (LLG)\nphenomenology for two-sublattice AFMs, dissipation is ac-\ncounted via a 2\u00022 Gilbert damping matrix [ 72]. Our goal here\nis to determine the elements of this matrix in terms of the\nparameters and physical observables within our microscopic\nmodel. To this end, we evaluate the spin current “pumped”\nby the magnetic subsystem into the sconduction electrons,\nwhich dissipate it immediately within our model. The angu-\nlar momentum thus lost by the magnetic subsystem appears\nas Gilbert damping in its dynamical equations [ 72,73]. The3\n-\n/2 -\n /4 0\n /2\n /4\ne\ne\nkF,1a/epsilon1=µ1\nm\ne\ne\nm\n/epsilon1=µ2\nkF,2a\nFIG. 2: The selectron dispersion in metallic AFM model,\nwhere the red and blue dispersions depict electron bands 1 and\n2, respectively. Illustrations of intraband electron-magnon\nscattering at two di \u000berent Fermi levels, \u00161and\u00162, are added.\nThe depicted momentum transfer is exaggerated for clarity.\nsecond essential ingredient in identifying the Gilbert damping\nmatrix from our microscopic theory is the idea of coherent\nstates [ 74,75]. The classical LLG description of the magne-\ntization is necessarily equivalent to our quantum formalism,\nwhen the magnetic eigenmode is in a coherent state [ 74–76].\nDriving the magnetization dynamics via a microwave field,\nsuch as in the case of ferromagnetic resonance experiments,\nachieves such a coherent magnetization dynamics [73, 77].\nThe spin current pumped by a two-sublattice magnetic sys-\ntem into an electronic bath may be expressed as [78]\nIz=Gmm(m\u0002˙m)z+Gnn(n\u0002˙n)z\n+Gmn\u0002(m\u0002˙n)z+(n\u0002˙m)z\u0003;(5)\nwhere mand nare the magnetization and Néel field nor-\nmalized by the sublattice magnetization, respectively. Here,\nGi j=\u000bi j\u0002(M=j\rj), where\u000bi jare the Gilbert damping co-\ne\u000ecients,\ris the gyromagnetic ratio of the delectrons\nandMis the sublattice magnetization. Considering the uni-\nform magnetization mode, Izis the spin current operator\nIz=i[He\u0000m;Sz][79], where Sz=P\niSz\ni. We get\nIz=iX\n\u0015\u001aX\nkk0qcy\n\u0015k\"c\u001ak0#\u0010\nWA;\u0015\u001a\nkk0qay\n\u0000q+WB;\u0015\u001a\nkk0qbq\u0011\n\u0000h.c.:(6)\nThe expectation value of this operator assuming the uniform\nmagnetization mode to be in a coherent state corresponds to\nthe spin pumping current [Eq. (5)].\nIn order to evaluate the spin pumping current from Eq. (6),\nwe follow the method employed to calculate interfacial spin\npumping current into normal metals in Refs. [ 73,77,78], and\nthe procedure is described in detail therein. Briefly, this method\nentails assuming the magnetic and conduction subsystems to\nbe independent and in equilibrium at t=\u00001, when the mu-\ntual interaction [Eq. (4)] is turned on. The subsequent timeevolution of the coupled system allows evaluating its physical\nobservables in steady state. The resulting coherent spin-current\ncorresponds to the classical spin current Izthat can be related\nto the motion of the magnetization and the Néel field [Eq. (5)].\nAs a last step, we identify expressions for (m\u0002˙m)z,(m\u0002˙n)z\nand(n\u0002˙n)zin terms of coherent magnon states, which enables\nus to identify the Gilbert damping coe \u000ecients\u000bmm,\u000bnnand\n\u000bmn.\nResults.— Relegating the detailed evaluation to Supplemen-\ntal Material, we now present the analytic expression obtained\nfor the various coe \u000ecients [Eq. (5)]. A key assumption that\nallows these simple expressions is that the electronic density of\nstates in the conduction subsystem does not vary significantly\nover the magnon energy scale. Furthermore, we account for a\nweak disorder phenomenologically via a finite scattering length\nlassociated with the conduction electrons. This results in an\ne\u000bective broadening of the electron wavevectors determined by\nthe inverse electron scattering length, (\u0001k)=2\u0019=l. As a result,\nthe crystal momentum conservation in the system is enforced\nonly within the wavevector broadening. By weak disorder we\nmean that the electron scattering length is much larger than\nthe lattice parameter a. Ifkandk0are the wave vectors of the\nincoming and outgoing electrons, respectively, we then have\n(k\u0000k0)a=(\u0001k)a\u001c1. This justifies an expansion in the wave\nvector broadening (\u0001k)a. The Gilbert damping coe \u000ecients\nstemming from intraband electron scattering are found to be\n\u000bmm=\u000b0(\u0018J)\u0000\u000b0(\u0018J)\n40BBBBBBBB@1+\u00182\nJ\u0010\n\u00182\nJ+8\u00004 cos2(kFa)\u0011\n\u0010\n\u00182\nJ+4 cos2(kFa)\u001121CCCCCCCCA[(\u0001k)a]2;\n\u000bnn=\u000b0(\u0018J)\n40BBBBBB@1+sin2(kFa)\ncos2(kFa)\u00182\nJ\u0010\n\u00182\nJ+4 cos2(kFa)\u00111CCCCCCA[(\u0001k)a]2:\n(7)\nwhere\u0018J=JS=t,kFis the Fermi momentum and ais the lattice\nparameter, and where\n\u000b0(\u0018J)=\u0019v2J2\n8g2(\u0016)j˜Vj24 cos2(kFa)\n\u00182\nJ+4 cos2(kFa): (8)\nHere, vis the unit cell volume, g(\u000f)is the density of states\nper unit volume, \u0016is the Fermi level, and !0is the energy of\ntheq=0magnon mode. ˜Vis a dimensionless and generally\ncomplex function introduced to account for the momentum\nbroadening dependency of the scattering amplitudes. It satisfies\n˜V(0)=1and0\u0014j˜V(\u0001k)j\u00141within our model. These analytic\nexpressions for the Gilbert damping parameters constitute one\nof the main results of this letter.\nDiscussion.– We straightaway note that \u000bnn=\u000bmm\u0018\n[(\u0001k)a]2\u001c1.\u000bnnis strictly dependent upon (\u0001k)a, and is non-\nzero only if there is disorder and a finite electron momentum\nbroadening. \u000bmmis large even when considering a perfectly\nordered crystal. This latter result is in good accordance with\nrecent first-principles calculations in metallic AFMs [ 56,66].\nWe moreover observe that both \u000bmmand\u000bnnare quadratic\ninJandg(\u0016). This result is shared by Gilbert damping ow-\ning to spin-pumping in insulating ferrimagnet |normal metal4\ne\n e\nm\nkFa/epsilon1=µ\nFIG. 3: A schematic depiction of magnon-induced interband\nscattering in a band crossing at the Fermi level.\n(NM) and AFM |NM bilayers with interfacial exchange cou-\npling [ 78]. Metallic AFMs bear a close resemblance to these\nbilayer structures. There are however two main di \u000berences:\nThes-dexchange coupling exists in the bulk of metallic AFMs,\nwhereas it is localized at the interface in the bilayer structures.\nAdditionally, the itinerant electron wave functions are qual-\nitatively di \u000berent in metallic AFMs and NMs, owing to the\nmagnetic unit cell of the AFM. Indeed, these di \u000berences turn\nout to leave prominent signatures in the Gilbert damping in\nmetallic AFMs.\nThe uniform mode magnon energy is much smaller than the\nelectron band gap within our simple model. Interband scat-\ntering is thus prohibited by energy conservation. However,\nin real AFM metals, the electron band structure is more com-\nplex. There may for instance exist band crossings [ 80–82].\nIn such materials, magnon-induced interband electron scatter-\ning should also contribute to Gilbert damping, depending on\nthe position of the Fermi surface. Motivated by this, we now\nconsider Gilbert damping stemming from interband scattering,\nwhile disregarding the energy conservation for the moment,\nlabeling the coe \u000ecients\u000bI\nmmand\u000bI\nnn. We then find the same\nexpressions as in Eq. (7) with the roles of \u000bI\nmm;nninterchanged\nwith respect to \u000bmm;nn. This implies that \u000bI\nnnis large and inde-\npendent of electron momentum broadening, whereas \u000bI\nmmis\nproportional to the electron momentum broadening squared.\nAlthough arriving at this result required disregarding the en-\nergy conservation constraint, the qualitative e \u000bect in itself is\nnot an artifact of this assumption. In materials with a band\ncrossing, as depicted in Fig. 3, \u000bI\nnn=\u000bI\nmm> \u000b nn=\u000bmmis a gen-\neral result. This generic principle derived within our simple\nmodel provides valuable guidance for designing materials with\nan engineered Gilbert damping matrix.\nWe now provide a rough intuitive picture for the damping\ndependencies obtained above followed by a more mathemati-\ncal discussion. Consider a conventional di \u000braction experiment\nwhere an incident probing wave is able to resolve the two\nslits only when the wavelength is comparable to the physical\nseparation between the two slits. In the case at hand, the wave-\nfunctions of electrons and magnon participating in a scatteringprocess combine in a way that the wavenumber by which the\nconservation of crystal momentum is violated becomes the\nprobing wavenumber within a di \u000braction picture. Therefore,\nthe processes conserving crystal momentum have vanishing\nprobing wavenumber and are not able to resolve the opposite\nspins localized at adjacent lattice sites. Therefore, only the aver-\nage magnetization is damped leaving the Néel field una \u000bected.\nWith disorder, the probing wavenumber becomes non-zero and\nthus also couples to the Néel field. The interband scattering,\non the other hand, is reminiscent of Umklapp scattering in a\nsingle-sublattice model and the probing wavenumber matches\nwith the inverse lattice spacing. Therefore, the coupling with\nthe Néel field is strong.\nThe Gilbert damping in metallic AFMs here considered is\ncaused by spin pumping from the magnetic subsystem into\nthesband. This spin pumping induces electron transitions\nbetween spin\"/#states among the selectrons. The Gilbert\ndamping coe \u000ecients depend thus on transition amplitudes pro-\nportional to products of itinerant electron wave functions such\nas y\n\u0015k\"(x) \u001ak0#(x). The damping e \u000bect on sublattice A depends\non this transition amplitude evaluated on the A sublattice, and\nequivalently for the B sublattice. Assuming without loss of gen-\nerality that site i=0belongs to sublattice A, we find in the one-\ndimensional model that the damping on sublattice A is a func-\ntion ofP\njcos2\u0010\u0019xj\n2a\u0011\n y\n\u0015k\"(xj) \u001ak0#(xj), whereas the damping\non sublattice B is a function ofP\njsin2\u0010\u0019xj\n2a\u0011\n y\n\u0015k\"(xj) \u001ak0#(xj).\nEquivalently, by straightforwardly using that m=(mA+mB)=2\nandn=(mA\u0000mB)=2, this analysis predicts that \u000bmmis a\nfunction ofP\nj y\n\u0015k\"(xj) \u001ak0#(x), whereas\u000bnnis a function of\nP\njcos\u0010\u0019xj\na\u0011\n y\n\u0015k\"(xj) \u001ak0#(x). Assuming plane wave solutions\nof the electron wave functions, and if we consider intraband\nscattering only, we more concretely find that \u000bmmis a function\nof(1\u0000i(\u0001k)a), where iis the imaginary unit, whereas \u000bnnis a\nfunction of ( \u0001k)a. This coincides well with Eq. (7).\nAbove, we presented a discussion of interband scattering in\nthe minimal model where the band gap artificially was set to\nzero. In this limit, the upper electron band is a continuation\nof the lower band with a \u0006\u0019=amomentum shift. We may then\nwrite 2k\u001b= 1;k+\u0019=a;\u001b. Under the assumption of a disappear-\ning band gap, momentum-conserving interband scattering at\nmomentum kis therefore equivalent to intraband scattering be-\ntween kandk\u0006\u0019=a. This is the exact phase shift which results\nin a small\u000bmmand a large \u000bnnconsistent with the discussion\nabove. In real metallic AFMs with complex band structures,\nthe exact wave function relations unveiled above do not apply.\nHowever, interband transition amplitudes will undoubtedly\ncarry a position dependent phase. This position dependence\nresults in a dephasing of transition amplitudes at neighboring\nlattice sites, which gives rise to a non-negligible \u000bnn. The pre-\ncise damping coe \u000ecients in real metallic AFMs depend on the\ndetailed electron wave functions. We may however generally\nconclude that \u000bI\nnn=\u000bI\nmm>\u000b nn=\u000bmm.\nConclusion.— We have provided a microscopic derivation\nof Gilbert damping resulting from magnon decay through s-d\nexchange interaction in metallic antiferromagnets. Analytic5\nexpressions for Gilbert damping coe \u000ecients resulting from in-\ntraband electron scattering are presented, while Gilbert damp-\ning resulting from interband electron scattering is discussed on\na conceptual level. We find that intraband electron scattering\ngives rise to a large magnetization damping and a negligible\nNéel field damping. The intraband Néel field damping is pro-\nportional to the inverse electron scattering length squared, and\ndisappears exactly if there is no crystal disorder. 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Andrieu1* \n \n1 Institut Jean Lamour, UMR CNRS 7198, Université de Lorraine, 54506 Vandoeuvre lès Nancy, France \n2 Synchrotron SOLEIL -CNRS, L’Orme des Merisiers, Saint -Aubin, BP48, 91192 Gif -sur-Yvette, France \n3 Laboratoire Léon Brillouin, IRAMIS, CEA Saclay, 91191 Gif sur Yvette, France \n \n*Email c orresponding author: stephane.andrieu@univ- lorraine.fr \n \n \nAbstract \nThe prediction of ultra -low magnetic damping in Co 2MnZ Heusler half -metal thin -film \nmagnets is explored in this study and the damping response is shown to be linked to the \nunderlying electronic properties. By substituting the Z elements in high crystalline quality \nfilms (Co2MnZ with Z=Si, Ge, Sn, Al , Ga, Sb) , electronic properties such as the minority spin \nband gap, Fermi energy position in the gap and spin polarization can be tuned and the consequence on magnetization dynamics analyzed. The experimental results allow us to \ndirectly explore the interplay of spin polarization, spin gap, Fermi energy position and the \nmagnetic damping obtained in these films, together with ab initio calcul ation predictions. The \nultra-low magnetic damping coefficients measured in the range 4.1 10\n-4 – 9 10-4 for Co 2MnSi, \nGe, Sn, Sb are the lowest values obtained on a conductive layer and offers a clear \nexperimental demonstration of theoretical predict ions on Half -Metal Magnetic Heus ler \ncompounds and a pathway for future materials design. \n Guillemard et al, Accepted for publication in Physical Review Applied (2019) \n \n \n \n2 \n I - INTRODUCTION \nAs a result of many theoreti cal and experimental advances , spintronic , electronics that \nuse both the charge and spin of the electron, is progressing . Prediction s of many phenomena \nlike high magnetoresistance MgO -based magnetic tunnel junction [1,2], magnetization \nreversal by s pin-transfer torque s (STT) [3,4], magnetization reversal by spin-orbit torque s \n(SOT) [5] or all-optical switching (AOS) by direct laser excitation [6] offer possibilities to \ndesign magnetoresistive random access memories, magnetic sensors and novel logic devices . \nEven more the search for systems with high conversion efficiency of charge to spin or spin to \ncharge current conversi on [7] are being explored for fundamental understanding of spin-\ntransport and for applications in low -energy -consumption devices. Most spintronic device \nconsists of thin- film heterostructures where interesting physics emerges at the interfaces [ 8]. \nFor continued progress , magnetic materials with specific and dedicated properties are needed, \nsuch as a high Curie temperature and an appropriate magnetic anisotropy for thermal stability \n[9], a high spin polarization at the Fermi energy to obtain fully spin- polarized current s, and a \nsmall magnetic damping to easily generate magnetization precession . All of these properties \nare desirable for STT, SOT and AOS based devices [ 9] but also in spin- wave- based devices, \nan emergent research field called magnonics [10]. \nHowever it is increasing challenging to achieve low magnetic damping in metallic \nmagnetic materials. The magnetic damping reflects the ability of the magnetiza tion to precess \naround an effective magnetic field. D issipation occurs due to interactions with the \nenvironment , the precession amplitude decreases and the oscillating magnetization return s to \nalign with the effective field. This damping process is characterized by the phenomenological \nGilbert damping coefficient α within the Landau -Lifshitz -Gilbert (LLG) formalism [ 11-14]. \nFor many emerging spintro nic and magnonic application, t his is particularly important in low -\npower applications that exploit magnetic dynamics such as STT switching where the \nswitching current is directly proportional to α [15]. This is also true for SOT devices where a \nprecessing magnetization generates a charge current in a metal and vice et versa [9,16]. \nWhile low damping parameters are often obtained in ferrimagnetic insulating oxides \nsuch as yttrium -iron garnet (YIG) where α=7.35 10-5 can be observed in the bulk [17], \nmagnetic metals typically have much higher damping where Fe -V alloys having a damping \naround 2 10-3 [18, 19] was considered state of the art for a thin film. Recently there was \nprogress in getting damping as low as α =2.1 10-3 in Fe -Co alloys [20]. However, a broader \nclass of materials where ultralow magnetic damping in metallic magnets emerges as half-\nmetal magnetic (HMM) behavior . Such mater ials emerged in 1983 by De Groot et al . [21] Guillemard et al, Accepted for publication in Physical Review Applied (2019) \n \n \n \n3 \n who reported predictions of peculiar el ectronic property of the NiMnSb half Heusler \ncompound. Its electronic band structure was predicted to be in between a metal and an \ninsulator. For the majority spin, responsible for the macroscopic magnetization, this material \nis a metal since electronic states are available around the Fermi energy. However, for minority \nspins, there is a gap around the Fermi energy. This peculiar property was called half -metal lic \nmagnetic (HMM) behavior. The NiMnSb material is thus a metal for majority spins whereas \nit is an insulator (at 0K) for minority spins. Such properties lead to a full spin -polarization at \nthe Fermi energy, making this material an excellent candidate for spin current generation. \nFurthermore, additional theoretical studies performed on HMM materials highlighted another \nphysical property of major importance in spintronic : their magnetic damping coefficients \nwere predicted to be extremely low compared to other conductive materials (a factor 100 \nbelow in the 10-5 to 10-4 range ) [22-24]. \nIn HMM materials, extremely low magn etic damping coefficients are predicted due to \nthe following reason. T he electronic band structure imposes no dens ity of states for minority \nspin. This spin channels exchange is thus forbidden and leads to continuous precession [ 22]. \nIn practice, other dissipation processes are pos sible leading to non -zero damping coefficient, \nbut even with taking them into account damping coefficient as low as 10-5 are predicted \n[22,24]. The precession damping is thus much smaller in a HMM than in a regular \nferromagnetic material (with non -zero den sity of states at the Fermi energy for both spin \nchannels). The HMM materials are thus very promising materials for applications. \nAfter the paper of De Groot [ 21], HMM properties were theoretically predicted for many \nHeusler compounds [ 25-27]. However, it took a significant effort for experimentalists to obtain \na direct verification . Indeed, the direct evidence of a spin gap in the minority spin channel was \nreported only very recently in Co 2MnSi [28, 29]. On the other hand, as small magnetic \ndamping coefficients were reported for several Heusler compounds , the measured values still \nremained in the 10-3 range [30-35] which is at least 10 times larger than prediction and \ncomparable with epitaxial FeV alloys [ 19] which are not HMM. In 2016, we measured for the \nfirst time a damping coefficient in the 10-4 range [29] (7.10-4 in Co 2MnSi, which was \nconfirmed by another group in 2018 [36]). However, the fact that the magnetic damping \nvalues reported in the literature dedicated to HMM are often higher than 10-3 is puzzling . In \nfact, a recent theoretical study reports that the magnetic damping can strongly varies \naccording to chemical disorder in the unit cell [24]. Thus, one big challenge when growing \nX2YZ full-Heusler thin films is to be as close as possible to the exact stoichiometry and \nachieving the chemical ly-ordered L21 phase as the outstanding properties of Co- based Guillemard et al, Accepted for publication in Physical Review Applied (2019) \n \n \n \n4 \n Heusler compounds are most often predicted for this L21 phase . However , Heusler alloys can \ncrystall ize in several phases with a low er chemical ordering [ 37] witho ut changing the atomic \nsites in the lattice . The most encountered disordered phase is the B2 one where Y and Z atoms \nare randomly arranged each other , leading to a primitive unit cell instead of the FCC cell \n(𝐹𝐹3�𝐹→𝑃𝐹3�𝐹). Up to now , the extent to which the chemical disorder affects the physical \nproperties is not clear . On one hand, ab initio calculations [24,38,39 ] and experiments [ 39,40 ] \nshow that the physical properties ( Curie temperature , cell parameter, magnetic moment, \nmagnetic damping constant and spin polarization at E F) are not drastically different between \nthe L2 1 and B2 phase s and the half -metallic spin gap should be conserved. O n the other hand, \nthe correlation between the degree of chemical ordering and the physical properties suffers a \nlack of experimental evidence since it is utterly complex to manipulate the degree of chemical \nordering without adding other parameters to take into account (like introducing doping or \nreducing the crystallographic quality of the layer). \nIn th is work, we present a systematic study of Co 2MnZ Heusler thin films epitaxially \ngrown by m olecular beam epitaxy (MBE) with Z=Al, Si, Ga, Ge, Sn, Sb. The chemical \nordering inside the Heusler FCC lattice is examined by using in situ Reflect ion High Energy \nElectron Diffraction (RHEED) and ex situ Transmission Electron Microscopy (TEM) . The \nspin polarization at the Fermi energy was determined by spin -resolved photoemission \nspectroscopy (SR -PES) experiments performed on the CASSIOPEE beamline at SOLEIL \nsynchrotron source. Finally , the damping coefficients were measured by using FerroMagnetic \nResonance (FMR) in a perpendicular geometry (dc magnetic field applied perpendicular to \nthe film plane) . Ultra-low damp ing (in the range 4 . 10-4 to 9 10-4) is observed here for at least \n4 of these 6 materials and is discussed according to the spin polarization determined \nexperimentally and compared to theoretical predictions . \n \nII – EXPERIMENTAL RESULTS \nSamples growth and structural characterization - All the films were grown by using \nMolecular Beam Epitaxy. The Heusler films stoichiometry was accurately controlled by using quartz microbalances (see appendix ). In situ Electron diffraction ( RHEED) performe d all \nalong the growth process allows us to control the epitaxial process and the chemical ordering in the Heusler unit cell. Indeed, if chemical ordering occurs, additional RHEED streaks \nshould be observed along the Heusler [110] azimuth compared to the A 2 or B2 phase [\n41]. In \nfigure 1 are shown the typical RHEED patterns obtained after the growth process as well as \nthe corresponding S TEM -HAADF images . All the films shows additional streaks along [110] Guillemard et al, Accepted for publication in Physical Review Applied (2019) \n \n \n \n5 \n RHEED patterns and STEM -HAADF images show the correct positioning of the Z element \nmeaning that the films structure is compatible with the L2 1 structure, except C o2MnAl for \nwhich a B2 structure takes place, as reported in the literature [ 30]. \n \n \nFigure 1: a) Scheme of the epitaxial rela tionship between Co 2MnZ films and MgO substrate, b) \nCo2MnZ (001) reciprocal lattice explored by electron diffraction (RHEED), c) Atomic columns along \nthe [110] azimuth for the L 21 ordering, d) RHEED patterns along [110] for the Co 2MnZ series, e) \nSTEM -HAADF micrographs along [110] zone axis for the Co 2MnSi, Co 2MnAl and Co 2MnSn films \n(zoom in inset). Note the small weight of Si (14electrons) compared to Sn (50 electrons). The Z atoms \nare at the correct position in the cell as confirmed by RHEED and TEM except for Co 2MnAl where a \nmixing of Mn and Si are observed leading to a B2 structure (no half streaks on the RHEED pattern and no contrast between the Mn and Z columns by microscopy). \n \nSpin polarization – Since SR-PES is a surface technique, the films were grow n in a MBE \nchamber coupled to the synchrotron beamline (see appendix and ref. [29, 42] ). The PES for \neach spin channel and the resulting Spin Polarization (SP) are shown in figure 2 for the \nGuillemard et al, Accepted for publication in Physical Review Applied (2019) \n \n \n \n6 \n Co2MnZ series studied here. The results are discussed as a fun ction of the Z element with the \nsemiconductor terminology type III, IV and V and for simplicity we note the different Heusler \ncompounds as C o2MnZtype. \n For the Co2MnZIV compounds (Z=Si/Ge/Sn ) the SR -PES spectra show a similar PES \nshape which is expected regarding the constant number of valence electrons in each \ncompound. Importantly, one should note that they manifest the same transition s depicted in \nour previous paper on Co2MnSi [29], where two features are explained . First, the loss of spin \npolarization at E F is caused by the presence of polarized surface states split by exchange \ncoupling (noted S1 for majority spin and S2 for minority spin in figure 2). Indeed, these S 1 and \nS2 transitions completely disappear when covering the surface by an atomic layer and the high \nspin polarization is thus observed up to the Fermi energy (we tested MgO, Mn and MnSi in \n[29]). It should be noted that these surface states are resonant with photon energy [29] so that \nthere are observed only in the photon energy range 25- 45 eV depending on the compounds \n(and are not observed in ref.28). We obs erved exactly the same behavior for Ge and Sn. \nHowever, there is a significant difference between Co 2MnSi when compared to Co 2MnGe and \nCo2MnSn. The minority spin gap is large in Co 2MnSi, whereas it decreases in Co 2MnGe and \nCo2MnSn. So the beginning of the spin gap is observed in Co 2MnSi (around - 0.4 eV ) despite \nthe surface state contribution around E F, but thi s is no longer the case for Ge and Sn. In other \nwords, the PES experiments including the surface states do not allow one to obtain the bulk \nspin polarization. To access it, one can change the photon polarization from P to S. We \npreviously showed that the spin gap can thus be observed up to E F in Co2MnSi [29]. We \nconsequently performed the same experiment on Co 2MnGe and Co 2MnSn. Results are shown \nin figure 2. Like in Co 2MnSi using a S photon polarization, the suppression of the transitions \ncoming from the s urface states allows us to see the spin gap in Co 2MnGe but more \ninterestingly, a full spin polarization is now observed confirming Co2MnGe is HMM. The \n(pseudo) spin gap is also observed for Co 2MnSn. Even though it is much larger than in figure \n2 using P polarization this compound is not fully spin polarization but has a SP of 80%. \n For Co2MnZIII compounds , the number of valence electrons is lower than in type IV \nso EF should decrease. This is actually observed and the former surface states are above EF \nand not occupied so the SP is maximum at E F. Therefore, the maximum of spin polarization \nvisible on figure 2 for the Co 2MnZIII compounds nearly corresponds to the beginning of the \nminority spin gap. The SP was observed to be close to 60% for Co 2MnAl whereas a full SP \nwas reached for Co 2MnGa. Guillemard et al, Accepted for publication in Physical Review Applied (2019) \n \n \n \n7 \n For the Co 2MnZV = Sb compound, a quite small SP is observed (around 35 %) but one \nimportant feature has to be taking into account. Sb is well known to strongly segregate during \nMBE growth [43]. This segregation was also observed in Sb- based Heusler compounds like \nNiMnSb [44] or Co2TiSb [45]. We directly observed this segregation using Auger \nspectroscopy. Since PES is a surface technique, the electronic properties of such a Sb \nenriched surface are included with the underneath Co 2MnSb PES contribution a nd may \nsignificantly affect the measured SP. \n \n \nFigure 2: Spin -Resolved photoemission spectra measured for the Co 2MnZ series. An incoming photon \nenergy of 37 eV is used for Z =Si, Al, Ga, Sb, 30 eV for Z=Ge and 27 eV for Z=Sn. We changed the \nincoming photon energy because of surface states, resonant in photon energy, which depends on the \nZIV element in the compound. T he left graphs were obtained using the P photon polarization. Surface \ntransitions are observed and noted S 1 (majority spin channel) and S 2 (minority spin channel). S 1 and \nS2 are responsible for the loss of spin polarization at E F (see text). Besides of tha t, spin gaps are \nobserved for Co 2MnGa and Co 2MnSi. The right graphs were obtained using the S polarization of the \nphotons on Co 2MnSi(001), Co 2MnGe(001) and Co 2MnSn(001). The S 1 and S 2 transitions coming from \nsurface states are now forbidden leading to a cl ear observation of the minority spin gap. \n \nGuillemard et al, Accepted for publication in Physical Review Applied (2019) \n \n \n \n8 \n Magnetic damping measurements - The magnetic damping properties of these films were \nextracted by using FMR measurements performed on the samples grown on the CASSIOPPE \nbeamline (after capping them with Au) but also on samples grown in our MBE at Institut Jean \nLamour. FMR experiment allows studying the precession dynamic of the electronic magnetic \nmoment [11-14,46,47 ] in ferromagnetic materials (see appendix for details) . Typical VNA-\nFMR measurement s performed on Co 2MnGe (left column) and Co 2MnSi (right column) are \nshown in figure 3. Figure 3 -a corresponds to the S 11 reflection coefficient [ 48] for 1.9 T and \n2.38 T of magnetic flux density, respectively. For each field value, the real and imaginary \nparts of the dynamic susceptibility are fitted simultaneously (i.e. using the same parameters to \nenhance the fitting reliability) to extract the position of the resonance frequency peak (figure \n3-b) and its Half W idth at Half M aximum (HWHM) (figure 3 -c). At f = 0, the curve crosses \nthe axis at 𝐻= 𝑀𝑒𝑒𝑒≈𝑀𝑆 for small magnetic anisotropy , which is the case in these layers \n[49,50 ]. The measured effective magnetic moments per formula unit are reported in table I for \nthe whole Co 2MnZ series and are compared to theoretical values. Our experimental results \ntend to follow the 𝑀𝑆=Λ−24 μB/f.u Slater -Pauling rules [ 51] where Λ is the total number \nof valence electron in the Heusler structure. Finally, the linewidth slopes in figure 3 -c give \ndamping values equal to 5.3 10-4 for Co2MnGe and 4.6 10-4 for Co2MnSi at 290K (6.1 10-4 \nand 4.1 10-4 at 8K) , the lowest of the Co 2MnZ series (Figure 3 -d). Moreover, inhomogeneous \nlinewidth values Δ𝑓0 in Co2MnGe and Co 2MnSi (respectively 23.6 and 14.5 MHz) are very \nsmall, almost comparable with bulk magnetic insulators such as YIG. Such small values \nimply excellent homogeneity of the magnetic properties (hence a high cryst al quality) in our \nfilms. Finally, o ne should note that this kind of FMR measurement leads to an effective value \nof the damping larger than the intrinsic value for the material [20,52,53 ]. Indeed, the effective \ndamping measured here gathers extrinsic contributions such as the radiative damping, the \neddy current damping and the spin- pumping contributions. Thus the magnetic damping values \nreported in table I inclu de these extrinsic contributions and correspond to a n upper limit of the \ntrue values . Recently, Shaw et al . [35] demonstrated a method to remove the extrinsic \ndamping contributions in FMR measurements and get a better estimation of the true magnetic damping in Co\n2MnGe (measured damping 1.5 10-3 which becomes smaller than 10-3 after \ncorrections). Thus the theoretical values are likely found to be lower than experimental ones \nwithout correction as reported here . Even with these extrinsic contributions, the measured \ndamping values of the whole Co 2MnZ series are in the ultra- low damping range (four out of \nsix in the 10-4 range and two below 2.10-3) in agreement with theoretical predictions. \n Guillemard et al, Accepted for publication in Physical Review Applied (2019) \n \n \n \n9 \n \nFigure 3: Perpendicular FMR on Co 2MnGe (left) and Co 2MnSi. Left -a) S 11 parameter of the \nscattering matrix, b) dependence of the FMR with field, c) evolution of the linewidth versus FMR. A \nreview of the measured damping for the Co 2MnZ series is also presented (right). The best magnetic \ndamping obtained up to now in conductive FeV alloys thin films and bulk insulating YIG using our set -\nup are shown as a comparison. \n \nIII - DISCUSSION \n All the experimental results extracted from this work are reported in table I together \nwith theoretical predictions found in the literature. Looking first the PES results obtained on \nCo2MnZ IV compounds, the observed full spin polarization for Si and Ge is consistent with \ntheoretical calculations regarding the calculated width of the spin gap and the position of the \nFermi energy in the middle of the gap. For Co2MnSi band structure calculations , the Fermi \nenergy is located right in the middle of the spin gap [24, 54] and a large gap is obtained ( 0.8 \neV [54], 0.41 eV [24,55 ]). Our results are consistent with these theoretical predictions since \nthe measured (half -) gap between the minority spin valence band to E F is around 0.4 eV \n(figure 2 ). In the case of Co 2MnGe, the calculated gap is smaller (0.58 eV [56]) and E F is \ncloser to the minority valence band, again consistent with our observations (half -gap around \n0.25 eV – see figure 2 ). Finally, regarding Co2MnSn bands calculations, Kandpa l et al. [56] \nfound EF in the gap and close to the valence band ( so HMM) whereas Ishida et al. [55] found \nGuillemard et al, Accepted for publication in Physical Review Applied (2019) \n \n \n \n10 \n it 0.06 eV below the top valence band (so not HMM ). Our results do not allow us to conclude \nabout this point (HMM or not) since our measurements are broadened by temperature (300K). \nHowever, both scenarios are consistent with the limited SP reported here . To summarize, the \nSR-PES results on these type IV Heusler compounds are well described by ab initio \ncalculations. The presence of this minority spin gap at the Fermi energy should reduce the \nmagnetic damping due to the removal of one conduction channel res ponsible for scattering \nprocesses involving spin flip (in other words, it should decrease the energy dissipation \nthrough spin relaxation in the system). Our magnetic damping measurements are in agreement \nwith this mechanism. Indeed, the larger the spin gap the smaller the magnetic damping. These \nlarge spin gaps in Co 2MnSi and Co 2MnGe also explain why the magnetic damping is rather \nindependent on the temperature for these two compounds (thermal fluctuation energy lower \n \n Our experiments (290 K) Theory (0 K) \nZ a (Å) Meff \n(μB / f.u) α Max \nof SP MS \n(μB/f.u) α SP ΔGap \n(eV) EF - EVB \n(eV) Type III Al 5.76 4.4 1.1 10-3 \n(290 K) \n1.15 10-3 \n(30 K) 60 % 4,3 \n[38] - 0.68 \n[56] 0.66 \n[56] 0 \n[55] \nGa 5.77 5.4 2 10-3 \n(290 K) \n1.8 10-3 \n(30 K) 100 % 4.09 \n[57] - 0.67 \n[56] 0.3 \n[56] 0 \n[55] Type IV Si 5.65 5.1 4.6 10-4 \n(290 K) \n4.1 10-4 \n(8 K) 100 % 4,94 \n[51] 6 10-5 \n[22] 100 \n[56] \n[24] \n[54] 0.41 \n[24] \n0.81 \n[54] ~ 0.2 \n[24] \n0.33 \n[54] \nGe 5.75 5 5.3 10-4 \n(290 K) \n6.1 10-4 \n(8 K) 100 % 4,94 \n[51] 1.9 10-4 \n[21] 100 \n[53] [55] 0.58 \n[55] \n0.54 \n[53] 0.07 \n[60] \n0.03 \n[53] \nSn 6.00 5.6 9. 10-4 \n(290 K) \n10-3 \n(8 K) 81 % 4,98 \n[51] 7 10-4 \n[59] 0.77 \n[56] 0.41 \n[56] \n0.17 \n[55] 0 \n[56] \n-0.06 \n[55] Type V \nSb 5.96 5.1 9.6 10-4 \n(290 K) 36 % 6 \n[58] - 100 \n[57] 0.65 \n[58] \n0.33 \n[55] 0.56 \n[58] \n0.39 \n[55] \nTable I : review of the experimental results obtained by FMR and SR -PES and comparison with \navailable theoretical results. \n Guillemard et al, Accepted for publication in Physical Review Applied (2019) \n \n \n \n11 \n than the spin gap even at room temperat ure). But our FMR result allows us t o go further in the \ncase of Co 2MnSn, since its very low magnetic damping strongly suggests that Co2MnSn is a \ntrue HMM as predicted by [ 56]. To conclude, our results highlight a clear correlation between \nthe theoretical gap width and Fermi energy position with the experimental SP and magnetic \ndamping values. \n Regarding bands calculations on Co2MnZIII = Al or Ga considering the L21 phase , the \ntheoretical SP at E F are respectively equal to 68 and 67% respectively [56]. Full SP is not \nreached because a small minority spin DOS remains at the Fermi energy . Strictly speaking, \nthese compounds are not predicted to be true HMM . However, this former DOS is so small \nthat a strong SP is still calculated. To account for this peculi ar behav ior, theoreticians use the \nterm “pseudo gap” [ 57]. The SR -PES results shown in figure 2 for the B2 ordered Co 2MnAl \nare in good agreement with this theoretical prediction since a SP of 60 % is obtained at E F. It \nis also consistent with the theoretical work of B. Pradines et al . [24], explaining that the spin \npolarization at E F should stay unchanged between the L2 1 and B2 chemical phase s. However, \nthe situation is different for Co 2MnGa. We observed a full SP in clear disagreement with \ntheory. Our experimental results show that E F is located in a true band gap, but very close to \nthe minority spin valence band. As a consequence, this leads to a low magnetic damping (compared to regular ferromagnetic layers) but h igher than type IV compounds due to the \nproximity of E\nF with the valence band. \n For the Co 2MnSb compound, theoretical calculation s lead to a large spin gap ( 0.65 \neV) with E F very close to the empty minority spin conduction band (0.09 eV below) in [ 58] or \neven in the conduction band in [ 55]. The experimental results are puzzling. As our PES results \nare not consistent with a HMM behavior, a very low magnetic damping is however observed \nconsistent with a HMM behavior. In fact, the low SP can be explained by considering Sb \nsegregation at the surface that does not affect the bulk SP . We conclude that Co 2MnSb is \nlikely HMM in the bulk, but its magnetic damping is higher than for type IV materials \nbecause of the E F proximity with the minorit y conduction band. These observations finally \npoint out that our SR-PES analysis is pertinent to get bulk properties for (001) Co2MnSi, Ge, \nSn, Ga, and Al but not for (001) Co 2MnSb. \n \nIV - CONCLUSIONS \n This work is a clear experimental demonstration that the unique electronic band \nstructure of Half -Metal Magnetic materials leads to ultra -low magnetic damping. T his study \nallows us to determine several key -points to achieve ultra-low damping. The most important Guillemard et al, Accepted for publication in Physical Review Applied (2019) \n \n \n \n12 \n ingredients are a large minority spin band gap, and a suitable position of the Fermi energy in \nthe bandgap, that is not too close to minority spin val ence and conduction bands. To achieve \nthis behavior requires growing high- quality crystalline films with control of the stoichiometry. \nThe measured magne tic damping values of the series match qualitatively with the spin \npolarization at E F predicted by the theoretical calculations and the SP measured by SR -PES \nwithout surface states. First, Co2MnSi and Co 2MnGe have Fermi energy inside a large half -\nmetallic gap and present ultra -low magnetic damping values below 6.10-4 (although the \nintrinsic damping will be even lower). Second , Co2MnSn and Co 2MnSb, with Fermi energy \nvery close to a minority spin band (valence band for the former, conduction band for the \nlatter) have magnetic damping values in between (6.10-4 ≤ α ≤ 10-3). And third, Co 2MnZIII (Al, \nGa) are not predicted to be pure HMM and their damping value s are obtained above the rest \nof the series with 10-3 ≤ α ≤ 2.10-3. Nevertheless, the case of Co 2MnGa is puzzling since we \nobserved a full spin polarization whereas the magnetic damping coefficient is the highest in \nthe series. Finally, t he ultra -low magnetic damping we obtained in Co 2MnSi (4.1 10-4) and \nCo2MnGe (5.3 10-4) have never been observed in a conductive material . One should note that \nin devices usually grown by sputtering the crystalline quality is not as good as in epitaxial \nfilms. However, the spin gap is predicted in the whole Brillouin Zone so there is no \nfundamental reason for not getting ultra -small damping in sputtered Heusler -based devices. \nSuch ability to get easy precession of the magnetization offers extraordinary opportunities for \ngetting more efficient spintronic devices, like Spin -Torque FMR, Spin Pumping FMR, \nOptical switching or magnonic based device s. \n \nACKNOWLEDGMENT \nThis work was supported partly by the french PIA project “Lorraine Université \nd’Excellence”, reference ANR -15-IDEX -04-LUE. Some UHV e xperiments were performed \nusing equipment from the TUBE —Da νm funded by FEDER (EU), ANR, the Region Lorra ine \nand Grand Nancy. We ack nowledge Eric E. Fullerton from the Center for Memory and \nRecording Research (University of California San Diego -USA) for his critical reading of the \nmanuscript. \n Guillemard et al, Accepted for publication in Physical Review Applied (2019) \n \n \n \n13 \n APPENDIX \n Samples growth and structural characterization - The substrates used to get single -\ncrystalline Heusler films are MgO(001) due to the small misifit between MgO and Co2MnZ \nlayers considered here . All the films were grown by using Molecular Beam Epitaxy. Co, Si \nand Ge were evaporated using electron guns, whereas Mn, Al, Ga, Sn and Sb by using \nKnudsen cells. Epitaxial films were obtained directly on MgO for Z=Al, S i, Ga, Ge . As this \nprocess was not successful with Z=Sb, Co 2MnSb(001) films were obtained on 10nm V(001) \nbuffer layer grown on MgO. The Heusler fil ms were all 20 nm thick . The starting temperature \nfor the growth was fixed to 450°C (measured by a thermocouple beside the sample holder) \nand the Heusler films were thus heated up to 750°C after the growth to improve the chemical \nordering in the lattice an d the surface quality. Auger spectroscopy is systematically performed \nafter the growth process and allowed us to verify that no surface contamination occurs (no O and C detected). The stoichiometry is a crucial point to get the best electronic and magnetic \nproperties of these compounds. For this purpose, the fluxes of each element were calibrated by using a quartz microbalance located at the sample’s location . The fluxes variations during \nthe process are observed to be below 2%. To achieve the good stoichiometry, the Co flux is \nfixed to 2. 10\n14 at/cm2/s and the Mn and Z elements fluxes to 1014 at/cm2/s. Considering these \nfluxes and a sticking coefficient on the quartz equal to 1 for all these materials , the time to \ncomplete a layer is expected to be equal to 3.0 seconds (for Co 2MnSi). We measured exactly \nthis completion time by using RHEED intensity oscillations performed during the growth. \nMoreover, the desired total thickness of the films (fixed by using the fluxes and the growth \nduration) is in perfect agreement with the final thickness determined by XRD reflectivity \nmeasurements . RHEED performed all along the growth process also allows us to verify that \nthe Heusler lattice grows with the epitaxial relationship MgO [100] (001) // Co 2MnZ [110] \n(001), meaning that the Heusler single -crystal is turned by 45° compared to the MgO lattice as \nshown in figure 1. As RHEED only gives information about the surface, θ -2θ XRD \nexperiments were performed ex situ. All the films shows (111) peaks by XRD meaning that \nthe films structure is compatible with the L2 1 structure, except C o2MnAl for which a no (111) \npeak was observed which is compatible with the B2 structure , as reported in the literature \n[30]. Finally, STEM -HAADF investigations were carried out using a JEM - ARM 200F Cold FEG \nTEM/STEM operating at 200 kV and equipped with a spherical aberration (Cs) probe and image \ncorrectors (point resolution 0.12 nm in TEM mode and 0.078 nm in STEM mode). \n Spin- resolved photo- emission spectroscopy (SR -PES) : SR -PES experiments were \nperformed on the CASSIOPEE beamline at SOLEIL synchrotron. The set -up and analysis Guillemard et al, Accepted for publication in Physical Review Applied (2019) \n \n \n \n14 \n process are detailed in [42] (beamline description) and [ 29] (study of Co 2MnSi). The spin-\nresol ution is obtained using a Mott detector with a n overall energy resolution of 150 meV in \nthe photon energy range 30- 40 eV used in this study. As discussed in ref. [ 29], our \nexperimental conditions allow us to explore around 80% of the Brillouin Zone according to \nthe photon energy range used in this study. As the films magnetization is always in -plane in \nour samples, the films were first magnetized in situ before the photoemission measurement by \napplying an in- plane magnetic field equal to 200 0e sufficient to saturate the magnetization \n(the coercive fields in our samples are alwa ys below 100 Oe). As the SR -PES measurement \nhas to be performed without any magnetic field, the spin polarization extracted from raw PES \nspectra is obtained at remanence. The remanence is systematically measured on the same \nfilms ex situ after capping the samples with 5 nm thick gold. The remanence was observed to \nvary from 80 to 100% depending on the Co 2MnZ compound. The true spin polarizations \nshown in this paper are thus obtained by correcting the SR -PES spectra from remanence. \n \n Magnetic damping measurements : Like in NMR, the precession occurs when an \noscillating magnetic field of small amplitude is generated perpendicular to the magnetic moment equilibrium which corresponds, in a ferromagnet , to the magnetization direction. \nThis equilibrium posit ion is imposed by the effective field derived from magnetic free energy \ndensity that includes exchange, dipolar interactions and magnetocrystalline anisotropy \nenergies. The magnetic damping is, by definition, inversely related to the lifetime of the \npreces sion. The damping value is thus extracted by looking at the linewidth of the resonance \npeak in frequency. The larger the linewidth is, the higher the damping and the shorter the \nprecession motion. FMR experiments were performed in the perpendicular geometr y where \nthe static magnetic field is applied out of the plane of the film in order to avoid extrinsic broadening of the linewidth due to the 2- magnons scattering [\n46,47 ]. The RF magnetic field is \ngenerated , thanks to a Vector Network Analyser (VNA -FMR) , in a coplanar waveguide \n(ground- signal -ground geometry) on top of which the sample is placed face down. \nMeasurements are performed in reflection geometry. The physical parameter extracted from these experiments is the S\n11 coefficient of the scattering matrix of the line , from which the \ndynamic susceptibility of the magnetic layer is extracted [47]. In the case of a perpendicular \ngeometry, the Kittel law [ 11] (equation 1) becomes linear and so does the evolution of the \npeak’s linewidth versus its own resonance frequency. The slope of this curve is directly the \nmagnetic damping value (equation 2) : \n𝑓𝑟=𝛾0(𝐻−𝑀𝑆+𝐻𝑘⊥)=𝛾0�𝐻−𝑀𝑒𝑒𝑒� (1) Guillemard et al, Accepted for publication in Physical Review Applied (2019) \n \n \n \n15 \n ∆𝑓=2𝛼𝛾0�𝐻−𝑀𝑒𝑒𝑒�+2∆𝑓0=2𝛼𝑓𝑟+2∆𝑓0 (2) \nwhere 𝑓𝑟 refers to the resonance frequency, 𝛾0 the gyromagnetic ratio of the electron, 𝐻 the \nmagnetic field strength, 𝑀𝑆 the magnetization, 𝐻𝑘⊥ the perpendicular magnetic anisotropy, \nthe effective magnetization 𝑀𝑒𝑒𝑒=𝑀𝑆−𝐻𝑘⊥, Δ𝑓 the full width at half maximum, ∆𝑓0 the \ninhomogeneous half linewidth and 𝛼 the Gilbert magnetic damping. The shift in frequency of \nthe resonance peak versus field is imposed by the gyromagnetic ratio 𝛾 0 of the electron, which \nshould be equal to 28 GHz/T in the case of a of a pure delocalized ferromagnetic model ( free \nelectron model) . However, in a solid, this ratio can be different mainly due to spin- orbit \ncoupling since 𝛾0 is proportional to the Landé 𝑔-factor. 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Pradines, R. Arras, and L. Calmels, Effects of partial B2, D03 and A2 disorders on the \nmagnetic properties of the Heusler alloys Co\n2FeAl, Co 2MnSn and Co 2MnAl for spintronic \napplications , in preparation . \n \n[60] S. Ouardi, Electronic Structure and Physical Properties of Heusler Compounds for Thermoelectric \nand Spintronic Applications , Ph.D Thesis , Johannes Gutenberg- University at Mainz, Germany \n(2012) . " }, { "title": "1807.11808v3.Comparative_study_of_methodologies_to_compute_the_intrinsic_Gilbert_damping__interrelations__validity_and_physical_consequences.pdf", "content": "Comparative study of methodologies to compute the intrinsic Gilbert damping:\ninterrelations, validity and physical consequences\nFilipe S. M. Guimar~ aes,\u0003J. R. Suckert, Jonathan Chico, Juba Bouaziz, Manuel dos Santos Dias, and Samir Lounis\nPeter Gr unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J ulich & JARA, 52425 J ulich, Germany\n(Dated: December 20, 2018)\nRelaxation e\u000bects are of primary importance in the description of magnetic excitations, leading\nto a myriad of methods addressing the phenomenological damping parameters. In this work, we\nconsider several well-established forms of calculating the intrinsic Gilbert damping within a uni\fed\ntheoretical framework, mapping out their connections and the approximations required to derive\neach formula. This scheme enables a direct comparison of the di\u000berent methods on the same footing\nand a consistent evaluation of their range of validity. Most methods lead to very similar results for\nthe bulk ferromagnets Fe, Co and Ni, due to the low spin-orbit interaction strength and the absence\nof the spin pumping mechanism. The e\u000bects of inhomogeneities, temperature and other sources of\n\fnite electronic lifetime are often accounted for by an empirical broadening of the electronic energy\nlevels. We show that the contribution to the damping introduced by this broadening is additive, and\nso can be extracted by comparing the results of the calculations performed with and without spin-\norbit interaction. Starting from simulated ferromagnetic resonance spectra based on the underlying\nelectronic structure, we unambiguously demonstrate that the damping parameter obtained within\nthe constant broadening approximation diverges for three-dimensional bulk magnets in the clean\nlimit, while it remains \fnite for monolayers. Our work puts into perspective the several methods\navailable to describe and compute the Gilbert damping, building a solid foundation for future\ninvestigations of magnetic relaxation e\u000bects in any kind of material.\nI. INTRODUCTION\nDynamical processes lie at the core of magnetic manip-\nulation. From the torques acting on the magnetic mo-\nments to how fast they relax back to their equilibrium\norientations, a material-speci\fc time-dependent theory\nis essential to describe and predict their behavior. In\nmost cases, the description of the time evolution of the\nmagnetization is done via micromagnetics1or atomistic\nspin dynamics (ASD)2,3approaches, in which the mag-\nnetization is considered either as a classical continuous\nvector \feld or as individual 3D vectors on a discrete\nlattice, respectively. They have been successfully used\nto describe a plethora of magnetic phenomena, ranging\nfrom spin waves in low dimensional magnets4, domain\nwalls5and skyrmion6dynamics to thermal stability of\nmagnetic textures7. These approaches model the mag-\nnetization dynamics via a phenomenological equation of\nmotion that contains both precessional and relaxation\nterms.\nA \frst attempt to address these processes was per-\nformed by Landau and Lifshitz (LL), by considering\na Larmor-like precessional torque and adding to it a\n(weaker) damping term of relativistic origin8. Since its\nphenomenological inception in 1935, the precise nature\nof the relaxation processes has been a source of intense\ndebate. In particular, the original LL formulation was\nfound to not properly describe situations in which the\ndamping was large. This problem was addressed by\nGilbert, who introduced a Rayleigh-like dissipation term\ninto the magnetic Lagrangian, thus obtaining the now-ubiquitous Landau-Lifshitz-Gilbert (LLG) equation9,\ndM\ndt=\u0000\rM\u0002B+\u000b\nMM\u0002dM\ndt\n=\u0000e\rM\u0002B\u0000\u000be\r\nMM\u0002(M\u0002B):(1)\nwhere\r >0 is the gyromagnetic factor, Mis the (spin)\nmagnetic moment, Bis the time-dependent e\u000bective\nmagnetic \feld acting on M, and\u000bis the scalar damping\nparameter named after Gilbert. The upper form of the\nLLG equation is due to Gilbert, and the lower one shows\nthat it is equivalent to a LL equation with a renormalized\ngyromagnetic factor, e\r=\r=(1 +\u000b2). The \frst term in\nthe right-hand side of Eq. (1) describes the precession of\nthe magnetic moments around the e\u000bective \feld, while\nthe second term is the Gilbert damping one, that de-\nscribes the relaxation of the magnetic moments towards\nB. This equation corrects the previously mentioned issue\nfor large values of \u000b, for which the original LL equation\nis expected to fail10,11.\nThe ferromagnetic resonance (FMR) technique is one\nof the most common procedures to probe magnetiza-\ntion dynamics12, in which the damping parameter is re-\nlated to the linewidth of the obtained spectra13. Al-\nthough many measurements have been carried out in bulk\nmaterials12,14{18, their description at low temperatures is\nstill controversial19{22. This can be attributed to the dif-\nferent intrinsic and extrinsic mechanisms that can con-\ntribute to the relaxation processes23{36. When varying\nthe temperature, two distinct regimes could be identi-\n\fed in the measured relaxation parameters37. For high\ntemperatures, a proportionality between the linewidth\nand the temperature was observed in most of the exper-arXiv:1807.11808v3 [cond-mat.mes-hall] 19 Dec 20182\niments. It was called resistivity-like, due to the simi-\nlarity with the temperature dependence of this quantity.\nA conductivity-like regime (linewidth inversely propor-\ntional to the temperature) was identi\fed at low temper-\natures for certain materials such as Ni15,17, but not for\nFe18,38. It was also seen that di\u000berent concentrations\nof impurities a\u000bected this low-temperature regime, even\nsuppressing it altogether16.\nFrom the theoretical point-of-view, the calculation of\nthe Gilbert parameter is a challenging problem due to\nthe many di\u000berent mechanisms that might be at play for\na given material39,40. Perhaps this is why most of the\ntheoretical approaches have focused on contributions to\nthe damping from electronic origin. The ultimate goal\nthen becomes the development of a predictive theory of\nthe Gilbert damping parameter, based on the knowledge\nof a realistic electronic structure of the target magnetic\nmaterial. The ongoing e\u000borts to complete this quest\nhave resulted in the development of a myriad of tech-\nniques21,22,37,41{43. Comparisons between a few of these\napproaches are available44,45, including experimental val-\nidation of some methods24,46, but a complete picture is\nstill lacking.\nWe clarify this subject by addressing most of the well-\nestablished methods to calculate the Gilbert damping\nfrom \frst principles. First, we connect the many dif-\nferent formulas, highlighting the approximations made\nin each step of their derivations, determining what con-\ntributions to the damping they contain, and establish-\ning their range of validity. These are schematically illus-\ntrated in Fig. 1. Second, we select a few approaches and\nevaluate the Gilbert damping within a uni\fed and con-\nsistent framework, making use of a multi-orbital tight-\nbinding theory based on \frst-principles electronic struc-\nture calculations. FMR simulations and the mapping of\nthe slope of the inverse susceptibility are used to bench-\nmark the torque correlation methods based on the ex-\nchange and spin-orbit torques. We apply these di\u000berent\ntechniques to bulk and monolayers of transition metals\n(Fe, Ni and Co), for which the spin pumping mecha-\nnism is not present and only the spin-orbit interaction\n(SOI) contributes to the relaxation. Disorder and tem-\nperature e\u000bects are included by an empirical broadening\nof the electronic energy levels37,43,47,48. Third, we engage\na longstanding question regarding the behavior of the\ndamping in the low-temperature and low-disorder limits:\nshould the intrinsic contribution to the Gilbert damping\ndiverge for clean systems? Our results using the con-\nstant broadening model demonstrate that the divergence\nis present in the clean limit of 3D systems but not of\nthe 2D ones49, which we attest by eliminating the pos-\nsibility of them being caused by numerical convergence\nissues or di\u000berent anisotropy \felds. Our results also in-\ndicate that the limit !!0 is not responsible for the\ndivergence of the intrinsic damping, as it is commonly\nattributed19,37,43,50. Finally, we propose a new way to\nobtain the spin-orbit contribution that excludes the \fc-\ntitious temperature/disorder contribution caused by thearti\fcial broadening51,52: they can be discounted by sub-\ntracting the values of damping calculated without SOI.\nFor bulk systems, this yields the total damping, while in\nlayered materials this method should also discount part\nof the spin pumping contribution. In Ref. 20, where tem-\nperature and disorder are included via a CPA analogy, a\nsimilar arti\fcial increase of \u000bfor high temperatures was\nremoved by including vertex corrections.\nThis work is organized as follows. We start, in Sec. II,\nwith a brief overview of the di\u000berent methods proposed\nin the literature. In Sec. III, we explain the theory used\nto calculate the response functions. We then turn to\nthe distinct theoretical forms of calculating the damping:\nIn Sec. IV, we analyze the di\u000berent approaches related\nto the spin-spin responses, while in Sec. V, the torque\nmethods are explored. We then discuss the obtained re-\nsults and conclude in Sec. VII. The Hamiltonian used in\nthe microscopic theory is given in Appendix A, while the\nanisotropy \felds for the 3D and 2D systems together with\nthe transverse dynamical magnetic susceptibility given\nby the LLG equation are given in Appendix B.\nII. OVERVIEW OF METHODS ADDRESSING\nINTRINSIC GILBERT DAMPING\nWe now focus on the di\u000berent methods to describe\nthe microscopic contributions to the Gilbert parameter,\nwhich encompasses e\u000bects that transfers energy and an-\ngular momentum out of the magnetic system. Within\nthese mechanisms, the relativistic SOI comes to the fore.\nThis is often referred to as the intrinsic contribution to\nthe damping, and was \frst identi\fed by Landau and Lif-\nshitz8. The origin of this damping mechanism lies in\nthe non-hermiticity of the relativistic corrections to the\nspin Hamiltonian when the magnetization precesses26,27.\nThe elementary magnetic excitations, called magnons,\ncan also be damped via Stoner excitations (electron-hole\npairs with opposite spins)33,34,53. Alternatively, the con-\nduction electrons can carry spin angular momentum even\nin absence of the SOI. This leads to damping via the spin-\npumping mechanism32,54{56.\nEarly models proposed to describe these processes al-\nready argued that the interaction between the magnetic\nmoments and the conduction electrons is a key ingre-\ndient57. This led to the so-called breathing Fermi sur-\nface model, where the shape of the Fermi surface de-\npends on the orientation of the magnetization through\nthe SOI41. This approach, however, could only capture\nthe conductivity-like regime, which diverges at low tem-\nperatures. The decay of magnons into Stoner excitations\nwas also considered early on39, describing well the exper-\nimental behavior of Ni but also missing the increase at\nlarger temperatures of other materials.\nAn important progress was made by Kambersky us-\ning the spin-orbit torque correlation function to calculate\nthe damping parameter37. This approach captures both\nconductivity- and resistivity-like behaviors, which were3\n↵\u0000↵noSOI\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\nInversion\nInversion for low frequencies + sum ruleDyson equation of susceptibilityEquation of motion of susceptibility\nSpectral representation at T=0KSpectral representation at T=0KLow SOI\nLow spin pumpingNo spin pumpingComputational costsFull SOI Spin pumpingFMR linewidthSlope of inverse susceptibility\nSlope of inverse mean-field susceptibility\nExchange torque correlation at Fermi surfaceSlope of mean-field SO-torque susceptibility with SOIProduct of spectral functions of opposite spinsSlope of mean-field susceptibilitySpin correlation at Fermi surfaceSO-torque correlation at Fermi surface with SOISpin response methodsTorque response methods\nEquation of motion of susceptibility + perturbation theory\nDyson equation for Green function + Orbital quenchingSlope of SO-torque susceptibility without SOI\nSpectral representation at T=0KSlope of mean-field SO-torque susceptibility without SOISO-torque correlation at Fermi surface without SOILarge broadening\nFigure 1. Diagram exhibiting the di\u000berent methods investigated in this work, their connections and range of validity. Two\ngroups are identi\fed: one related to the spin susceptibility (spin response methods), including the ferromagnetic resonance\nand the slope of the inverse susceptibility that involves a direct mapping of this quantity to the LLG equation; and the other\nassociated with torque responses, for which approximations need to be taken. The steps indicated by solid lines represent\nexact connections, while dashed arrows involve some kind of approximation. The arrow on the left points from the methods\nthat require less computational power (lower part) to the more demanding ones (upper part). Boxes are hyper-linked with the\nrespective equations and sections.\nshown to originate from the intra- and interband transi-\ntions, respectively58. Recently, this so-called torque cor-\nrelation method was re-obtained using a di\u000berent per-\nturbative approach19, spurring discussions about the va-\nlidity of the obtained results, specially the divergence\ncaused by the intraband transitions22. A similar method\nalso based on torque correlation functions was developed\nusing a scattering theory approach42involving the ex-\nchange torque operator instead of the spin-orbit torque\none. Results obtained in this way also present diverg-\ning behavior in the clean limit of 3D structures20. A\nsimilar scattering framework was used to explain the\nenhancement of the Gilbert damping due to the spin\npumping in thin \flms32. Yet another method relating\nthe Gilbert damping to the spin-spin response was pro-\nposed and related to the existing spin-orbit torque cor-\nrelation method43. It also presented diverging intraband\ncontributions when the parameter used to broaden the\ndelta functions (which mimics the e\u000bect of disorder or\ntemperature) was taken to zero59. The vertex correc-\ntions proposed in Ref. 59 did not remove this diver-\ngence. More recently, Costa and Muniz21showed that\nthe damping parameters of layered structures remain \f-nite in the zero broadening limit, when extracted directly\nfrom the linewidth of the dynamical magnetic suscepti-\nbility (within the random phase approximation).\nSeveral of these methods have been implemented\nfor material-speci\fc calculations20,47,49,58,60{62, and some\napproaches were compared and related43{45,63. In this\nwork, we start our analysis with the uniform frequency-\ndependent spin-spin susceptibility, which is measured ex-\nperimentally in FMR setups, to derive the other expres-\nsions for the damping parameter based on the spin- and\ntorque-correlation methods.\nIII. MICROSCOPIC THEORY\nWe begin by setting the grounds of the theory we use\nto evaluate the di\u000berent formulas of the Gilbert damping\non equal footing. The electronic structure of the system\nis described by the mean-\feld Hamiltonian\n^H=^H0+^Hxc+^HSOI+^Hext: (2)\nThe paramagnetic band structure is described by ^H0\nwithin a multi-orbital tight-binding parametrization. An4\ne\u000bective local electron-electron interaction within the\nmean-\feld approximation is included in ^Hxc, which is re-\nsponsible for ferromagnetism. We also account for spin-\norbit interaction through ^HSOI, and the interaction with\nexternal static magnetic \felds via ^Hext. The explicit\nforms of all the terms are given in Appendix A.\nIn this work, we investigate the di\u000berent methods to\ncompute the intrinsic Gilbert damping utilizing the pro-\ntotypical bulk magnets Fe (bcc), Co (fcc) and Ni (fcc),\nand also square lattices corresponding to the (001) planes\nof those materials, with the same nearest-neighbor dis-\ntances as in its bulk forms.\nFor simplicity, we consider the spin-orbit interaction\nand the local e\u000bective Coulomb interaction only on the\ndorbitals, with U= 1 eV64{66for all systems, and the\nspin-orbit strengths \u0015Fe\nSOI= 54 meV67,\u0015Co\nSOI= 70 meV68,\nand\u0015Ni\nSOI= 133 meV68. The magnetic ground state is\nfound by self-consistently enforcing charge neutrality for\nthe bulk materials69. For the monolayer cases, the total\nnumber of electrons in the atomic plane is decreased to\nn= 7:3 (Fe),n= 8:1 (Co) and n= 9:0 (Ni), as the re-\nmaining charge spills into the vacuum (which we are not\nexplicitly taking into account within the model). The\nground-state properties (spin moment M, orbital mo-\nmentM`and magnetic anisotropy energy K) obtained\nwithin this framework are listed in Table I. The easy axis\nfor all the bulk systems and the monolayers were found\nto be along the (001) direction. We emphasize that our\ngoal is not to achieve the most realistic description of the\nelectronic structure of these materials, but rather to de-\n\fne a concrete set of cases that allow us to compare the\ndi\u000berent methods to compute the Gilbert damping.\nThe magnetic excitations are described using linear re-\nsponse theory, where the transverse magnetic response\n\u000eM(t) due to an oscillatory magnetic \feld \u000eB(t) is given\nby70\n\u000eM\u000b(t) =Z\ndt0\u001f\u000b\f(t\u0000t0)\u000eB\f(t0); (3)\nwhere the convention to sum over repeated indices of\nthe components \f=fx;y;zgis used. This approach\ncaptures the orbitally-averaged part of the response. The\nbulk monolayer\nbcc Fe fcc Co fcc Ni Fe Co Ni\nM(\u0016B) 2.32 1.48 0.43 2.90 1.90 0.96\nM`(\u0016B) 0.072 0.079 0.055 0.28 0.22 0.20\nK(meV) 0.19 0.26 0.084 1.7 1.8 1.9\nTable I. Ground state properties of the investigated systems.\nMandM`denotes the spin and orbital magnetic moments,\nrespectively. Values obtained for \u0011= 1:36 meV. The mag-\nnetic anisotropy constant Kis obtained from the anisotropy\n\felds given by Eq. (B3).magnetic susceptibility is given by\n\u001f\u000b\f(t\u0000t0) =\u00004\n\n^S\u000b(t);^S\f(t0)\u000b\u000b\n= 4i\n\u0002^S\u000b(t);^S\f(t0)\u0003\u000b; (4)\nin atomic units. ^S\u000b(t) is the\u000b-component of the spin op-\nerator. In the \frst line of the equation above, we reprise\nthe double-bracket notation of Zubarev for the spin-spin\nretarded Green function71. This notation is convenient\nfor the derivations of Sec. V.\nFor the crystal symmetries of the systems we are in-\nterested in, it is convenient to work in the circular ba-\nsis^S\u0006=^Sx\u0006i^Sy, which diagonalizes the susceptibil-\nity matrix with components \u001f\u0000+(t) and\u001f+\u0000(t). The\nfrequency- and wave vector-dependent transverse suscep-\ntibility\u001f\u0000+(q;!) is obtained within the random phase\napproximation (RPA), which captures the collective spin\nwave modes21,72, as well as the possible decay into\nparticle-hole excitations (Stoner modes) described by the\nsingle-particle response function \u001f\u0000+\n0(!). Considering\nmatrices that take into account the orbital dependency,\nthe two susceptibilities are related by\n[\u001f\u0000+]\u00001= [\u001f\u0000+\n0]\u00001\u00001\n4U: (5)\nHere,U\u0016\u0017=U\u000e\u0016\u0017is a matrix with the e\u000bective lo-\ncal Coulomb interaction strength within the dorbitals.\nIt plays a similar role to the exchange-correlation ker-\nnel in the adiabatic local-density approximation of time-\ndependent DFT calculations73. We de\fne the transverse\nmagnetic response of the system by summing the suscep-\ntibility matrix over all the dorbitals.\nThe uniform single particle transverse susceptibility\n\u001f\u0000+\n0(!) =\u001f\u0000+\n0(q= 0;!), obtained within the mean-\n\feld approximation, is expressed in terms of the single-\nparticle Green functions as\n\u001f\u0000+\n0;\u0016\u0017(!) =1\n\u0019NX\nkZ\u000fF\nd\"\b\nG\"\"\n\u0016\u0017(k;\"+!) ImG##\n\u0017\u0016(k;\")\n+ ImG\"\"\n\u0016\u0017(k;\")\u0002\nG##\n\u0016\u0017(k;\"\u0000!)\u0003\u0003o\n:\n(6)\nThe sum is over the wave vectors in the \frst Brillouin\nzone, with Ntheir number. The indices \u0016;\u0017represent\norbitals and \u000fFis the Fermi level.\nIn the spirit of many preceding works37,43,47,48, the\ne\u000bect of temperature and disorder is modeled by in-\ntroducing a constant band broadening \u0011on the en-\nergy levels, such that G(!)!G(!+ i\u0011). The imag-\ninary part of the Green function is then de\fned as\nImG\u0016\u0017(!) =1\n2ifG\u0016\u0017(!+ i\u0011)\u0000G\u0016\u0017(!\u0000i\u0011)g. This ap-\nproach attempts to capture all the intrinsic e\u000bects origi-\nnated from the electronic structure of the system.\nThe imaginary part of the susceptibility is related\nto the energy dissipation of the system74, encoding\nthe relaxation mechanism of the magnetization towards\nequilibrium. The damping parameter is then obtained5\nby mapping the transverse magnetic susceptibility ob-\ntained from the quantum mechanical calculation de-\nscribed above to the phenomenological form provided by\nthe LLG, Eq. (1). On the following sections, we present\ndi\u000berent mapping procedures involving several approx-\nimations and explore their range of validity when the\nbroadening \u0011is taken to zero (clean limit).\nIV. SPIN RESPONSE METHODS\nA. Ferromagnetic resonance\nMagnetic excitations can be investigated by applying\ntime-dependent perturbations. This is done in FMR ex-\nperiments where the magnetic sample is subjected to a\nstatic magnetic \feld and an oscillatory radio-frequency\none. By varying either the strength of the static compo-\nnent or the frequency of the oscillatory \feld, the system\ncan be driven through magnetic resonance. This setup\nyields the uniform mode of the transverse magnetic sus-\nceptibility. As the Gilbert parameter describes the relax-\nation mechanisms of the magnetization, it is related to\nthe linewidth of the resonance peak21,75.\nWe simulate this kind of experiments by calculating\nthe transverse magnetic response relying on the linear\nresponse theory discussed in Sec. III, and mapping the\nimaginary part of the susceptibility into the result ob-\ntained from the LLG equation (see Appendix B),\nIm\u001f\u0000+(!) =\u00002\u000b\r!M\n[!\u0000\r(Bext+Ban)]2+ (\u000b!)2:(7)\nWhen \fxing the frequency and varying Bext;z, this func-\ntion presents a resonance at Bres= (!\u0000\rBan;z)=\r\nwith linewidth given by the full width at half maxi-\nmum \u0001B= 2\u000b!=\r . On the other hand, when the\n\feld is kept \fxed and the frequency is varied, the res-\nonance is located at !res=\r(Bext;z+Ban;z)=p\n1 +\u000b2\nwith full width at half maximum approximately given by\n\u0001!\u00192\u000b\rjBext;z+Ban;zj, in the limit \u000b\u001c175.\nThe Gilbert parameter can then be obtained either by\n\ftting Eq. (7) or through the ratio between the linewidth\nand the resonance position. In this sense, a divergence of\nthe damping when \u0011!0 seems counter-intuitive, since\nthis would imply that either the resonance position ( Bres\nor!res) goes to zero or that the corresponding linewidth\nincreases drastically. In the presence of SOI, the SU(2)\nrotational symmetry is broken and the anisotropy \feld\nBan;zshifts the resonance position to a \fnite value | it\ncosts a \fnite amount of energy to set the magnetization\ninto precession76. Therefore, the divergence of the damp-\ning parameter can only happen if the linewidth increases\nand goes to in\fnity.\nTo verify this claim, we simulate FMR experiments\nin fcc Co bulk by calculating the imaginary part of the\ntransverse magnetic susceptibility as a function of the\nfrequency!, in the presence of the spin-orbit interac-\ntion. In Fig. 2a, we present the obtained spectra fordi\u000berent values of the broadening \u0011. When a relatively\nlarge value of the broadening is used, \u0011= 13:6 meV (solid\ncurve), the spectra displays a broad resonance peak,\nwhich can be characterized by a value \u000b= 1:3\u000210\u00002,\nobtained by \ftting the linear response data with Eq. (7).\nWhen the broadening of the energy levels is decreased\nto\u0011= 4:1 meV (dashed curve), the peak shifts and be-\ncomes sharper ( \u000b= 3:8\u000210\u00003), as one intuitively ex-\npects when disorder and/or temperature decreases. No-\ntice that most of the change in \u000bis due to the change\nin the peak width, while the resonance shift is relatively\nsmall. This can be viewed as a consequence of the smaller\nenergy overlap between the bands, which decrease possi-\nble interband transitions58. Surprisingly, by further de-\ncreasing the broadening to \u0011= 0:41 meV (dotted curve),\nthe peak becomes broader when compared to the pre-\nvious case, with \u000b= 5:6\u000210\u00003. This counter-intuitive\nresult represents a shorter lifetime of the magnetic excita-\ntion when the electronic lifetime (mean time between two\nsuccessive scattering events) \u001c=\u0011\u00001becomes longer.\nObtaining the damping from the FMR curves is com-\nputationally demanding, though. The response function\nmust be calculated for many frequencies (or magnetic\n\felds) to resolve the peak. For the case of low broaden-\nings that require many k-points in the Brillouin zone for\na converged result, this task becomes prohibitive. In the\nnext section, we provide alternative methods to obtain\nthe Gilbert parameter based on the static limit of the\nsusceptibility, and compare their outcomes with the ones\nobtained using the resonance approach.\nB. Inverse Susceptibility Method\nWe proceed now to investigate a di\u000berent mapping\nof the microscopic transverse susceptibility to the LLG\nequation and possible approximations to simplify the cal-\nculation of the Gilbert damping. From Eq. (B4), one can\nsee that\u000bde\fnes the slope of the imaginary part of the\ninverse susceptibility43, i.e.,\n\u000b= 2\rMlim\n!!0Im[\u001f\u0000+(!)]\u00001\n!: (8)\nWe will refer to this as the inverse susceptibility method\n(ISM). The mapping to the LLG model of the slope at\nsmall frequencies has a great advantage over the FMR\none since it only requires a single frequency-point calcula-\ntion, instead of a full sweep over frequencies or magnetic\n\felds for the \ftting procedure.\nIn Fig. 2b, we display the damping parameter for bcc\nFe, fcc Co and fcc Ni bulk systems calculated as a func-\ntion of the electronic energy broadening. We also include\nthe results obtained from the FMR approach (solid sym-\nbols), which compare well with the ISM given in Eq. (8).\nNote that although Eq. (8) has an explicit linear depen-\ndence on the spin moment M, the susceptibility implic-\nitly depends on its value. The obtained curves are in-\nversely related to M: highest for Ni ( M\u00180:45\u0016B), low-6\n0.5 0.55 0.6 0.6502468·105\nFrequencyω(meV)−Imχ−+(ω) (states/eV)η1= 13.6 meV→α= 1.3·10−2\nη2= 4.1 meV→α= 3.8·10−3\nη3= 0.41 meV→α= 5.6·10−3\n1 10 10010−210−1100101102\nBroadening η(meV)Gilbert damping αXC-TCM ISM\nFe\nFe 5·λSOI\nFe 10·λSOI\n1 10 10010−310−210−1100101\nBroadening η(meV)Gilbert damping αISM Fe\nISM Co\nISM Co no SOI\nFMR Co\nISM Ni\n1 10 10010−310−210−1100101\nBroadening η(meV)Gilbert damping αFe monolayer\nCo monolayer\nNi monolayer10 100 1,000Temperature (K)\n10 100 1,000Temperature (K)\n10 100 1,000Temperature (K)(a)\n(b)(c)\n(d)\nFigure 2. Characteristics of the Gilbert damping in 3D and 2D systems in presence and absence of SOI. (a) Ferromagnetic\nresonance spectra for fcc Co, in presence of spin-orbit interaction and no external \feld, calculated for three di\u000berent decreasing\nbroadenings \u00111= 13:6 meV (solid), \u00112= 4:1 meV (dashed) and \u00113= 0:41 meV (dotted). The values of the Gilbert damping\ngiven in the legend box, obtained by \ftting to Eq. (7), decrease from the \frst case to the second, but increases again when \u0011is\nfurther decreased. (b) Gilbert damping in presence of spin-orbit interaction for bcc Fe (blue triangles), fcc Co (red circles, solid\nline) and fcc Ni (green squares) as a function of the broadening, obtained from the slope of the inverse susceptibility, Eq. (8).\nAll values were computed with 108k-points in the full Brillouin zone. Solid red circles are the values obtained from the FMR\nspectra in (a), while the open red circles connected by dashed lines represent the damping parameter for fcc Co when SOI is\nnot included in the calculations. (c) Damping parameter for bcc Fe for di\u000berent SOI strenghts: \u0015SOI= 54:4 meV, 5 \u0002\u0015SOI,\nand 10 \u0002\u0015SOI. (d) Gilbert damping of Fe, Co and Ni monolayers in the presence of SOI. No increase in the Gilbert damping\nis seen when the broadening \u0011is decreased.\nest for Fe (M\u00182:3\u0016B) and Co in-between ( M\u00181:5\u0016B).\nThis trend is con\frmed by setting the SOI strength \u0015SOI\nto the same values for all the elements (not shown). The\nposition of the minimum value of \u000bis connected with\n\u0015SOI, which determines when the intraband or interband\ntransitions become more important58. To substantiate\nthis claim, we employed the technique of arti\fcially scal-\ning the\u0015SOI, as previously done in connection to the\nmagnetic anisotropy energy77. The results are shown in\nFig. 2c, where the SOI strength \u0015SOIof Fe bulk is mag-\nni\fed by factors of 5 and 10. Indeed, the minimum can\nclearly be seen to shift to larger values of \u0011.An important aspect to be considered is the conver-\ngence of Eq. (6) | failing to achieve numerical precision\nmay give rise to spurious results49,78. This can be partly\nsolved using sophisticated schemes to perform those cal-\nculations79,80. When the broadening is lowered, the con-\nvergence of the wave vector summation is a\u000bected by\nthe increasingly dominant role of the poles of the Green\nfunctions in the vicinity of the Fermi energy. For that\nreason, to capture the intricacies of the electronic states\n| in particular, the important contributions from the\nsmall gaps opened by the weak SOI |, we calculated\nthe slope of the response function using a very \fne in-7\ntegration mesh on the Brillouin zone reaching up to 109\nk-points. The results in Fig. 2c also demonstrate that the\ndivergence is not an issue of numerical convergence, since\nthis behavior is shifted to larger values of broadenings,\nfor which the convergence is more easily achieved.\nNevertheless, such diverging e\u000bect only occurs in the\npresence of spin-orbit interaction. In Fig. 2b we also dis-\nplay the values of \u000bfor Co fcc obtained using the ISM\nwhen the SOI is not included in the calculations (cir-\ncles connected by dashed lines). In this case, \u000bnoSOI lin-\nearly goes to zero when the broadening is decreased21.\nThe non-vanishing damping when SOI is not present\ncan be interpreted as originating from the \fnite elec-\ntronic lifetimes introduced by the constant broadening\nparameter. As it stands, \u0011represents the coupling to a\n\fctitious reservoir51,52providing dissipation mechanisms\nthat physically should originate from disorder or temper-\nature, for example.\nObtaining the damping from the FMR spectra when\nSOI is not present requires an applied magnetic \feld,\nsuch that the resonance frequency becomes \fnite and\navoiding an in\fnite response at zero frequency (repre-\nsenting no cost of energy due to the rotational symmetry,\ni.e., the Goldstone mode). Nevertheless, the results pre-\nsented in Fig. 2d were obtained using the ISM without\nany applied \feld. Calculations with an applied magnetic\n\feld shifting the peak to the original anisotropy energy\nwere indistinguishable from those values (with variations\nsmaller than 3%). This is accordance to the phenomeno-\nlogical expectations expressed through Eq. (B4), where\nthe slope is independent of the magnetic \feld.\nOne can put our results for bulk ferromagnets into\nperspective by comparing with low dimensional systems.\nWe investigated this case within our linear response ap-\nproach, using monolayers of Fe, Co and Ni. The calcu-\nlations follow the same procedure, except that the sum\noverkvectors in Eq. (6) is restricted to the 2D Brillouin\nzone. The results are presented as triangles (Fe), cir-\ncles (Co) and squares (Ni) connected by dotted lines in\nFig. 2d, and once again exhibit a monotonous decay with\nthe decrease of \u0011. We note that previous calculations of\nthe damping parameter in thin \flms also did not \fnd it\nto increase rapidly for decreasing broadening21,49.\nBesides the dimensionality, another main di\u000berence\nfrom the bulk to the layered case is the larger anisotropy\n\felds of the latter (see Table I). Nevertheless, this can-\nnot explain the non-diverging behavior in the monolay-\ners. We have already shown that by arti\fcially increas-\ning the SOI strength of the bulk | and, consequently, its\nanisotropy \feld |, the conductivity-like behavior of the\ndamping occurs at even larger broadenings (see Fig. 2c).\nOn the other hand, to rule out a possible divergence hap-\npening at lower broadenings ( \u0011<0:1 meV, not reachable\nin our calculations), we have also scaled up \u0015SOIof the\nmonolayers by one order of magnitude. This resulted in\nlarger dampings, nonetheless, the same decreasing be-\nhaviour with \u0011!0 was observed (not shown). There-\nfore, the divergence can only be attributed to the three-dimensionality of the ferromagnet.\nC. Approximate static limit methods\nWe now look back to Fig. 1 and proceed to perform\napproximations on Eq. (8) in order to simplify the calcu-\nlations of the damping parameter. Here we follow Ref. 43.\nFirst, we use Eq. (5) that relates the RPA susceptibility\nmatrix to the mean-\feld response matrix \u001f0, such that\nIm\u001f\u00001\u0019Im\u001f\u00001\n0. Although Uis a real matrix, the sum\nover orbitals ( \u001f=P\n\u0016\u0017\u001f\u0016\u0017) ends up mixing the real\nand imaginary parts of the matrix elements. Only when\nRe\u001f\u00001\n0=U=4 the relation above becomes an equality.\nThis means that, within our model with Uacting only on\nthedorbitals,\u001fmust also be de\fned by summing over\nthose orbitals only. Under the previous assumption, we\nobtain\n\u000b\u00192\rMlim\n!!0Im[\u001f\u0000+\n0(!)]\u00001\n!: (9)\nThis relation is only valid when \u001f\u0000+\n0is decoupled from\nthe other types of susceptibilities (transverse and longi-\ntudinal), as in the systems we investigate in this work.\nThe damping parameter can therefore be obtained from\nthe single-particle transverse susceptibility \u001f0.\nFor frequencies !in the meV range (where the col-\nlective spin excitations are located), \u001f\u0000+\n0has a simple\n!-dependence81:\n\u001f\u0000+\n0(!)\u0019Re\u001f0(0) + i!Im\u001f0\n0(0): (10)\nwhere\u001f0\n0(0) =d\u001f\u0000+\n0\nd!\f\f\f\n!=0. These results are valid also in\nthe presence of spin-orbit coupling. Using Eq. (10), the\nGilbert damping can be written as\n\u000b\u0019\u00002\rM\u0002\nRe\u001f\u0000+\n0(0)\u0003\u00002lim\n!!0Im\u001f\u0000+\n0(!)\n!:(11)\nAlthough the expansion of the susceptibility for low fre-\nquencies was used, no extra approximation is employed,\nsince Eq. (9) is calculated in the limit !!0. Re\u001f\u0000+\n0(0)\ncan be obtained using the sum rule that relates the\nstatic susceptibility with the magnetic moments76. For\n3dtransition metals, the external and the spin-orbit\n\felds are three orders of magnitude smaller than U, and\nso the static susceptibility of the bulk systems reads\nRe\u001f\u0000+\n0(0)\u00194=U. Thus,\n\u000b\u0019\u0000\rMU2\n8lim\n!!0Im\u001f\u0000+\n0(!)\n!: (12)\nFinally, from Eq. (6) it is possible to show that Eq. (12)8\nsimpli\fes as\n\u000b\u0019\rMU2\n2\u0019NX\nk;\u0016\u0017TrfImG\u0017\u0016(k;\u000fF)^S\u0000ImG\u0016\u0017(k;\u000fF)^S+g\n=\r\n2M\u0019NX\nk;\u0016\u0017TrfImG\u0017\u0016(k;\u000fF)^T\u0000\nxcImG\u0016\u0017(k;\u000fF)^T+\nxcg\n=\rMU2\u0019\n8NX\nk;\u0016\u0017n#\n\u0017\u0016(k;\u000fF)n\"\n\u0016\u0017(k;\u000fF):\n(13)\nwheren\u001b\n\u0016\u0017(k;\u000fF) =\u00001\n\u0019ImG\u001b\u001b\n\u0016\u0017(k;\u000fF) is the matrix el-\nement of the spectral function of spin \u001bcalculated at\nkand\u000fF. The second equation is written in terms\nof the \\exchange-correlation torque operator\", T\u0006\nxc=\n\u0000i\u0002^S\u0006;^Hxc\u0003\n=\u0007iUM^S\u0006. This result is equivalent\nto the one obtained in Ref. 42, which we reference as\ntheexchange torque correlation method (XC-TCM) |\nalthough, in reality, it relates \u000bwith the spin-spin re-\nsponse. The last step in Eq. (13) connects the damping\nwith the product of spectral functions of opposite spins\nat the Fermi level, as shown theoretically in Ref. 81 and\ncon\frmed experimentally in Ref. 46.\nIn Fig. 2c, we compare the results obtained with this\napproximated method with the ISM described before, for\nthe di\u000berent values of SOI scalings. For the bulk tran-\nsition metals we investigate, the approximation is very\ngood, since the SOI is relatively small. In fact, even\nwhen the SOI is scaled one order of magnitude higher,\nthe results of the XC-TCM are still very good.\nThe formulas in Eq. (13) show that we have arrived\nat the bottom of the triangle in Fig. 1. These forms\ndo not involve an integral over energy, which simpli\fes\nsubstantially the calculation of \u000b. For that reason, they\nare suitable for \frst-principles approaches (e.g., Refs. 20\nand 62). This concludes our investigations of the spin\nresponse methods. In the next section, we take a di\u000berent\npath to calculate the Gilbert damping.\nV. TORQUE RESPONSE METHODS\nDespite the simplicity of the methods based on the spin\nsusceptibility discussed in the previous section, seminal\nwork was based on a di\u000berent type of response function.\nThe main idea, \frst proposed by Kambersky37, is to di-\nrectly relate \u000bto the spin-orbit interaction. Here, our\naim is twofold. First, we connect the spin susceptibility\nwith the spin-orbit torque response via the equation of\nmotion, clarifying the damping mechanisms captured by\nthis formalism. Second, we compare the results obtained\nwith both types of methods.\nWe start with the equation of motion for the spin-spin\nsusceptibility. Its time-Fourier transform can be written\nas19\n!\n\n^S\u0000;^S+\u000b\u000b\n!=M+\n\n\u0002^S\u0000;^H\u0003\n;^S+\u000b\u000b\n!; (14)whereM=\u00002\n^Sz\u000b\n. From the Hamiltonian given in\nEq. (2), the commutator [ ^S\u0000;^H\u0003\nhas four contributions:\nkinetic (spin currents, from ^H0), exchange torque (from\n^Hxc), external torque (from ^Hext) and spin-orbit torque\n(from ^HSOI). In presence of SOI, the total spin magnetic\nmoment is not a conserved quantity and spin angular\nmomentum can be transferred to the orbital degrees of\nfreedom. For bulk systems subjected to static external\n\felds and in the present approximation for the electron-\nelectron interaction, the only two non-vanishing torques\nare due to the external \feld and the spin-orbit interac-\ntion. It also follows from these assumptions that the\nmechanisms that contribute to the relaxation arises then\nfrom the spin-orbit torques ^T\u0006\nSOI=\u0000i\u0002^S\u0006;^HSOI\u0003\nand\nfrom the broadening of the energy levels \u0011.\nIt can be shown19that the inverse of the susceptibility\n\u001f\u0000+(!) =\n\n^S\u0000;^S+\u000b\u000b\n!is given by\n\u0002\n\u001f\u0000+(!)\u0003\u00001=\u0002\n\u001f\u0000+\nnoSOI (!)\u0003\u00001\u0002\n1 +\u001f\u0000+\nnoSOI (!) \u0000(!)\u0003\u00001\n\u0019\u0002\n\u001f\u0000+\nnoSOI (!)\u0003\u00001\u0000\u0000(!):\n(15)\nHere,\u001f\u0000+\nnoSOI (!) is the susceptibility calculated excluding\nthe SOI contribution to the Hamiltonian. The connection\nbetween the two susceptibilities in Eq. (15) is provided\nby the quantity\nM2\u0000(!) = i\n\u0002^T\u0000\nSOI;^S+\u0003\u000b\n+\n\n^T\u0000\nSOI;^T+\nSOI\u000b\u000b\n!:(16)\nUsing Eq. (8), and noticing that the \frst term on the\nright-hand side of the equation above does not contribute\nto the imaginary part, we \fnd\n\u000b=\u000bnoSOI\u00002\r\nMlim\n!!0Im\n\n^T\u0000\nSOI;^T+\nSOI\u000b\u000b\n!\n!: (17)\n\u000bnoSOI is the contribution obtained by inputting\n\u001f\u0000+\nnoSOI (!) into Eq. (8), which is \fnite due to the broad-\nening\u0011.\nKambersky37\frst obtained this same result following a\ndi\u000berent approach. In our framework, this would involve\nstarting from Eq. (5) and exploiting the consequences of\nthe fact that the collective spin excitations ( !\u0018meV)\nhave low frequencies when compared to the exchange en-\nergy (U\u0018eV). On the other hand, Hankiewicz et al.19\ndescribed the same expansion for low SOI, and justi\fed\nits use for !.\rBext. Finally, Edwards22shows that\nthis formula is equivalent to a perturbation theory cor-\nrect to\u00152\nSOI(compared to \rBext\u0000!). For that rea-\nson, he suggests that the states used in the calculation\nof\n\n^T\u0000\nSOI;^T+\nSOI\u000b\u000b\n!should not include SOI, since the op-\nerator ^T\u0000\nSOI/\u0015SOI. Due to the orbital quenching in the\nstates without SOI, this leads to the absence of intra-\nband contributions and, consequently, of the divergent\nbehavior for \u0011!082.\nIn this approach, temperature and disorder e\u000bects are\nincluded in \u000bnoSOI (shown in Fig. 2d for Co), while the9\nspin-orbit intrinsic broadening is calculated by the sec-\nond term in Eq. (17), which can also be obtained as\n\u000b\u0000\u000bnoSOI . An extra advantage of calculating the damp-\ning as the aforementioned di\u000berence is that one explic-\nitly subtracts the contributions introduced by \u0011, provid-\ning similar results to those obtained with vertex correc-\ntions20. Considering the torque-torque response within\nthe mean-\feld approximation (an exact result in the per-\nturbative approach22), we obtain, similarly to Eq. (13),\n\u000b\u0000\u000bnoSOI =\n2\r\nM\u0019NX\nkTrfImG(k;\u000fF)^T\u0000\nSOIImG(k;\u000fF)^T+\nSOIg:\n(18)\nIn this formula, the involved quantities are matrices in\nspin and orbital indices and the trace runs over both.\nThis is known as Kambersy's formula, commonly used in\nthe literature43,44,47,49,58, which we refer to as spin-orbit\ntorque correlation method (SO-TCM). As in Eq. (13), it\nrelates the damping to Fermi level quantities only. When\nthe SOI is not included in the calculation of the Green\nfunctionsG(k;\u000fF) and enters only through the torque\noperators, we name it perturbative SO-TCM22. These\nmethods are placed at the bottom right of Fig. 1, with\nthe main approximations required indicated by the long\ndashed arrows.\nWe now proceed to compare these approaches with the\nISM explained in Sec. IV B. Fig. 3 presents the calcula-\ntions of the SOI contribution to the damping parameter\nof bulk Fe (a), Co (b) and Ni (c) using the SO-TCM ob-\ntained in Eq. (18), when no external \feld is applied. Both\napproaches, including SOI (red curve with squares) in\nthe Green functions or not (green curve with triangles),\nare shown. For a meaningful comparison, we compute\n\u000b\u0000\u000bnoSOI within the ISM.\nWe \frst note that the perturbative approach suggested\nby Edwards22describes reasonably well the large broad-\nening range (i.e., mostly given by the interband transi-\ntions), but deviates from the other approaches for low \u0011.\nThis is an expected behaviour since it does not include\nthe intraband transitions that display the \u0011\u00001behav-\nior within the constant broadening model. In the clean\nlimit, the Gilbert damping computed from the pertur-\nbative SO-TCM approaches zero for all elements, in a\nvery monotonic way for Co and Ni, but not for Fe. This\nmethod is thus found to be in agreement with the other\nones only when \u0015SOI\u001c\u0011. The SO-TCM formula in-\ncluding the SOI in the states (i.e., Kambersky's formula)\nmatches very well \u000bobtained within ISM in the whole\nrange of broadenings.\nFinally, after demonstrating that the SO-TCM pro-\nvides very similar results to the ISM, we can use it to\nresolve the wave-vector-dependent contributions to the\nGilbert parameter by planes in the reciprocal space, as\n\u000b(kmax\nz) =kmax\nzX\njkzj\u000b(kz); (19)\n1 10 10010−310−2Gilbert damping α(a)Fe\nISM ( α−αnoSOI )\nSO-TCM\nperturbative SO-TCM10 100 1,000Temperature [K]\n1 10 10010−410−310−2Gilbert damping α(b)Co\n10 100 1,000\n1 10 10010−410−310−210−1100\nBroadening η[meV]Gilbert damping α(c)Ni\n10 100 1,000Figure 3. Comparison between \u000b\u0000\u000bnoSOI for (a) Fe bcc, (b)\nCo fcc and (c) Ni fcc, obtained using the inverse susceptibility\nmethod (ISM) with the spin-orbit-torque correlation method\n(SO-TCM) with and without SOI in the states (perturbative\nSO-TCM). All the points were computed with 108k-points in\nthe full Brillouin zone.\nwhere\u000b(kz) is given by the right-hand side of Eq. (18)\nsummed over kx;ky. The result, displayed in Fig. 4, uses\n100 million k-points for all curves and shows the expected\ndivergence in presence of SOI and a decrease with \u0011when\nthis interaction is absent. In every case, most of the con-\ntribution arises from the \frst half ( kmax\nz<0:4). Note\nthat when the broadening of the energy levels is low, the\nintegrated alpha without SOI (Fig. 4b) displays step-\nlike contributions, while when SOI is present, they are\nsmoother. This is a consequence of the damping being10\n01234Gilbert damping α(·10−2)(a) With SOI\nη(meV):\n0.14\n0.41\n0.68\n1.1\n1.4\n4.1\n6.8\n10.9\n13.6\n40.8\n0 0.2 0.4 0.6 0.8 100.20.40.60.81\nkmax\nzGilbert damping α(·10−3)(b) Without SOIDecreasingη\nDecreasingη\nFigure 4. Integrated Gilbert damping for fcc Co as a function\nofkzplotted against the maximum value, kmax\nz(see Eq. (19)),\nwith SOI (a) and without SOI (b). The curves were obtained\nusing the SO-TCM given in Eq. (18). Colors represent di\u000ber-\nent values of the broadening \u0011(in units of meV). The value of\n\u000bforkmax\nz= 0 (i.e., a single value of kzin the sum) represents\na two-dimensional system, whilst for kmax\nz= 1 the sum covers\nthe whole 3D Brillouin zone. In the latter case, the damping\ndecreases when \u0011is decreased without SOI, while it increases\ndrastically when SOI is present. For 2D systems, the\ncaused by interband transitions in the former and intra-\nband in the latter.\nThe convergence of the previous results for the smallest\n\u0011including SOI were tested with respect to the total\nnumber of k-points in the Brillouin zone in Fig. 5. By\ngoing from 10 million to 10 billion k-points, the results\nvary\u001820%. However, compared with the result shown\nin Fig. 4a, the damping gets even larger, corroborating\nonce more the divergent results.\nVI. DISCUSSIONS\nIn this section, we make a few \fnal remarks on the pre-\nviously obtained results and we go beyond bulk systems\nto comment on the approximations taken and additional\nphysical mechanisms that may come into play in other\n0 0.2 0.4 0.6 0.8 1024\nkmax\nzGilbert damping α(·10−2)\nk-points:\n107\n108\n109\n1010Figure 5. Integrated Gilbert damping for fcc Co as a func-\ntion ofkzplotted against the maximum value, kmax\nz, for\n\u0011= 0:14 meV and di\u000berent amount of k-points (up to 10\nbillion) in the Brillouin zone.\nmaterials. We also provide a new analytical explanation\nfor the divergence of the damping parameter within the\nconstant broadening model.\nOur \frst comment regards the application of static\nmagnetic \felds B. As described in Refs. 19 and 22, the\napproximations done in Eq. (15) to derive an expression\nfor\u000binvolves comparisons between the excitation en-\nergy andB. However, all the results we have presented\nhere were obtained in absence of static \felds. We also\nperformed calculations including external magnetic \felds\nup toB\u00187 T, and the computed damping parameter is\nweakly in\ruenced by their presence. We conclude that\nthe validity of the SO-TCM formula given in Eq. (18)\ndoes not hinge on having a magnetic \feld, supporting\nthe arguments already given in Ref. 19.\nA further remark concerns the approximations made\nto obtain the mean-\feld result in Eq. (12). We assumed\nthat SOI is weak when using the magnetic sum rule.\nThis approximation may break down when this is not\nthe case. The spin pumping also a\u000bects the magnetic\nsum rule, which may worsen the agreement with the ISM\nresults. Although this contribution is not present in the\ninvestigated (bulk-like) systems, it plays an important\nrole in magnetic multilayers. This e\u000bect enhances the\ndamping factor32,54,55. Furthermore, the SO-TCM ex-\nplicitly excludes spin pumping, as this is described by\n^I\u0000\nS=\u0000i\u0002^S\u0000;^H0\u0003\n, dropped from the equation of mo-\ntion. These validity conditions are indicated in Fig. 1 by\nthe large blue rectangle (low SOI), red triangle (low spin\npumping) and green rectangle (no spin pumping).\nAnother mechanism that opens new spin relaxation\nchannels is the coupling between transverse and longi-\ntudinal excitations induced by the SOI. This was one of\nthe reasons raised in Ref. 21 to explain the divergence of\nthe damping parameter. However, this is absent not only11\nwhen the system has full spin rotational symmetry83, but\nalso when rotational symmetry is broken by the SOI in\n2D and 3D systems for the symmetries and materials we\ninvestigated. Even though the damping is \fnite in the\n\frst two cases (as shown in Fig. 2d), the divergence is\nstill present in the latter (Fig. 2b).\nWe can also recognize that the mathematical expres-\nsion for\u000bin terms of the mean-\feld susceptibility given\nin Eq. (12) is similar to the conductivity one (i.e., the\nslope of a response function)84| which leads to the same\nissues when approaching the clean limit ( \u0011!0). How-\never, the physical meaning is the exact opposite: While\nthe divergence of the conductivity represents an in\fnite\nacceleration of an ideal clean system, in\fnite damping\ndenotes a magnetic moment that is instantly relaxed in\nwhichever direction it points (as d M=dt!0 for\u000b!1 )\n| i.e., no dynamics10,11. This means that a clean 3D\nspin system is in\fnitely viscous. Within the constant\nbroadening model, the divergence of the Gilbert damp-\ning can also be seen analytically by comparing Eq. (12)\nwith the calculations of the torkance done in Ref. 48.\nBy replacing the torque operator and the current density\nby the spin lowering and raising operators, respectively,\nthe even contribution (in the magnetization) to the re-\nsponse function vanishes and only the odd one remains.\nIn this approximation, it is also seen that only the Fermi\nsurface quantities are left, while the Fermi sea does not\ncontribute85. In the limit of low broadenings, this con-\ntribution is shown to diverge as \u0011\u00001. This divergence\narises from intraband transitions which are still present\nin the clean limit, and originate from the \fnite electronic\nlifetimes introduced by the constant broadening approx-\nimation.\nThe static limit ( !!0) is another reason that many\nauthors considered to be behind the divergent damping\nbehavior19,37,43,50. This limit is taken in Eq. (8) in or-\nder to eliminate the contribution of terms nonlinear in\nfrequency from the inverse susceptibility (e.g., inertia ef-\nfects68,86). They can be present in the full microscopic\ncalculation of the susceptibility but are not included in\nthe phenomenological model discussed in Appendix B.\nAdding the quadratic term in frequency leads to an in-\nverse susceptibility given by\nIm[\u001f\u0000+(!)]\u00001=\u0000!\n2\rM(\u000b\u0000!I)\nwhereIis the o\u000b-diagonal element of the moment of iner-\ntia tensor86. The \ft to the expression linear in frequency\nthen yields an e\u000bective \u000be\u000b(!). In the vicinity of the res-\nonance frequency, \u000be\u000b(!res) =\u000b\u0000!resI, which is clearly\nreduced in comparison to the one obtained in the static\nlimit,\u000be\u000b(0) =\u000b. According to Ref. 68, I\u0018\u000b=\u0011, which\nexplains the discrepancy between the FMR and the ISM\nseen in Fig. 2b as \u0011!0. We can then conclude that the\nstatic limit is not the culprit behind the divergence of \u000b\nin the clean limit.VII. CONCLUSIONS\nIn this work, we presented a study of di\u000berent meth-\nods to calculate the intrinsic Gilbert damping \u000b, o\u000bering\na panorama of how the approaches are related and their\nrange of validity (see Fig. 1). They can be grouped into\nthree main categories: the methods that directly employ\nthe results of full microscopic calculations of the dynam-\nical magnetic susceptibility \u001f(!) (FMR and ISM); the\nexchange-torque method (XC-TCM), which is also based\non\u001f(!) but making use of the mean-\feld approximation;\nand the spin-orbit torque-correlation method (SO-TCM),\nobtained from the (spin-orbit) torque-torque response via\nan equation of motion for \u001f(!). While the FMR, ISM\nand XC-TCM include all the contributions to the mag-\nnetic relaxation, the SO-TCM provides only the intrinsic\ncontribution due to the angular momentum transfer to\nthe orbital degrees of freedom (not including, for exam-\nple, the spin pumping mechanism). The XC- and SO-\nTCM, given by Eqs. (13) and (18), are predominant in\nthe literature due to their simplicity in obtaining \u000bin\nterms of Fermi level quantities. It is important to note,\nhowever, that they rely on approximations that may not\nalways be full\flled21.\nIn order to implement and compare the di\u000berent meth-\nods, we constructed a uni\fed underlying framework\nbased on a multi-orbital tight-binding Hamiltonian using\nas case studies the prototypical bulk 3D systems: bcc Fe,\nfcc Co and fcc Ni. For this set of materials, the di\u000berent\nmethods lead to similar results for \u000b, showing that the\ncorresponding approximations are well-founded. Even\nwhen the SOI strength is scaled up by one order of mag-\nnitude, this excellent agreement remains, as we explic-\nitly veri\fed for bcc Fe. We found one method that falls\nout-of-line with the others in the clean limit, namely the\nperturbative form of the SO-TCM formula22,82. In this\ncase, although the equation is identical to the well-known\nKambersky formula, Eq. (18), the electronic states used\nto evaluate it do not include SOI. By comparison with\nthe other methods, we conclude that the results obtained\nby the perturbative SO-TCM are only valid in the large\nbroadening regime (compared to the SOI strength). Cen-\ntral to our analysis was a careful study of the convergence\nof our results with respect to the number of k-points,\nreaching up to 1010k-points in the full Brillouin zone.\nThe behavior of \u000bis intimately connected with the con-\nstant broadening approximation for the electronic life-\ntimes. For high temperatures, the Gilbert damping in-\ncreases with increasing temperature ( \u000b\u0018\u0011), while for\nlow temperatures it diverges for 3D ferromagnets ( \u000b\u0018\n1=\u0011), but not for 2D (ferromagnetic monolayers). Our\ncalculations revealed that the high temperature values\nof\u000barise mostly from the broadening of the electronic\nstates. In Ref. 20, the strongly increasing behaviour of\n\u000bfor high temperatures was found to be spurious, and\ncured employing a more realistic treatment of disorder\nand temperature, and the so-called vertex corrections.\nWe found that the contribution of the intrinsic SOI to \u000b12\nis additive to the one arising from the broadening, and\ncan be easily extracted by performing a calculation of\n\u000bwithout SOI and subtracting this result from the SOI\none,\u000b\u0000\u000bnoSOI . Combined with the ISM, this provides\na relatively simple and accurate way to obtain the in-\ntrinsic damping, which discounts contributions from the\nadditional broadening \u0011. This establishes an alternative\nway of accessing the high temperature regime of \u000b.\nThe low-temperature divergence of \u000bwhen approach-\ning the clean limit for 3D ferromagnets has also been the\nsubject of much discussion. The \frst di\u000eculty is in es-\ntablishing numerically whether this quantity actually di-\nverges or not. Our results consistently show an increase\nof\u000bwith decreasing \u0011, down to the smallest achievable\nvalue of\u0011= 0:14 meV (Fig. 5), with no hints of a plateau\nbeing reached, but only when accounting for SOI. This\ndivergence arises from the intraband contributions to \u000b,\nas discussed in Ref. 58. Refs. 22 and 82 used pertur-\nbation theory arguments to claim that such intraband\ncontributions should be excluded. However, as we dis-\ncussed in Sec. IV B, adapting the formalism of Ref. 48 to\nthe calculation of \u000bshows that these intraband terms are\nenabled by the constant broadening approximation, and\nso should be included in the calculations. Contrary to\nthe high temperature regime, works that employ a more\nrealistic treatment of disorder and temperature still \fnd\nthe diverging behavior of \u000b20,52.\nIn real experiments, any kind of material disturbance\nsuch as disorder or temperature e\u000bects leads to a \fnite\nvalue of the damping. Besides that, a non-uniform com-\nponent of the oscillatory magnetic \feld (either from the\napparatus itself or due to its limited penetration into the\nsample) induces excitations with \fnite wave vectors and\n\fnite linewidths39,87. A di\u000berent way to determine the\ndamping parameter is using the time-resolved Magneto-\nOptic Kerr E\u000bect (TR-MOKE)40,88. It has the advan-\ntage that, as it accesses a smaller length scale ( \u00181µm)\nthan FMR experiments (which probe the whole magnetic\nvolume), the measured magnetic properties are more ho-\nmogeneous and thus the e\u000bect of linewidth broadening\nmay be weaker. The magnetic excitations in nanomag-\nnets can also be probed by recent re\fnements of FMR\nexperimental setups89,90.\nAlthough the methods we described here are gen-\neral, we did not explicitly addressed non-local sources\nof damping such as the spin-pumping32. As a future\nproject, we plan to ascertain whether our conclusions\nhave to be modi\fed for systems where this mechanism\nis present. Systems that combine strong magnetic el-\nements with heavy ones possessing strong SOI are ex-\npected to have anisotropic properties, as well-known for\nthe magnetic interactions91. It is then natural to explore\nwhen the Gilbert damping can also display signi\fcant\nanisotropy, becoming a tensor instead of a scalar quan-\ntity47,78. Indeed, this has been observed experimentally\nin magnetic thin \flms92,93. As the SOI, magnetic non-\ncollinearity can also lead to other forms of damping in do-\nmain walls and skyrmions50,94{98. From the microscopicpoint of view, the potential coupling between transverse\nand longitudinal degrees of freedom allowed by the non-\ncollinear alignment should also be considered. Lastly,\nhigher order terms in frequency, such as the moment of\ninertia68,86,99{101, might also become important in the\ndynamical magnetic susceptibility for large frequencies\nor for antiferromagnets, for instance.\nThe description of magnetization dynamics of real ma-\nterials helps to design new spintronic devices able to con-\ntrol the \row of information. Our work sheds light on fun-\ndamental questions about the main relaxation descrip-\ntions used in the literature and sets ground for future\ntheoretical predictions.\nAppendix A: Ground-state Hamiltonian\nIn this Appendix, we give the explicit forms of the\nterms in the Hamiltonian written in Eq. 2. As the inves-\ntigated systems only have one atom in the unit cell, the\nsite indices are omitted.\nThe electronic hoppings in the lattice are described by\n^H0=1\nNX\nk\u001bX\n\u0016\u0017t\u0016\u0017(k)cy\n\u0016\u001b(k)c\u0017\u001b(k); (A1)\nwithcy\n\u0016\u001b(k) andc\u0017\u001b(k) being the creation and annihila-\ntion operators of electrons with spin \u001band wave vector\nkin the orbitals \u0016and\u0017, respectively. The tight-binding\nparameters t\u0016\u0017(k) were obtained by \ftting paramagnetic\nband structures from \frst-principles calculations up to\nsecond nearest neighbors102, within the two-center ap-\nproximation103.\nThe electron-electron interaction is characterized by\na local Hubbard-like104interaction within the Lowde-\nWindsor approximation105, resulting in the mean-\feld\nexchange-correlation term\n^Hxc=\u0000X\n\u00162d\n\u001bU\n2(\nM\u000b\u001b\u000b\n\u001b\u001b0+X\n\u00172d\u000en\u0017(2\u000e\u001b\u001b0\u000e\u0016\u0017\u0000\u000e\u001b\u001b0))\ncy\n\u0016\u001b(k)c\u0016\u001b0(k):\n(A2)\nHere,Uis the local e\u000bective Coulomb interaction, M\u000b\nand\u001b\u000bare the\u000b-component of the magnetic moment\nvector (summed over the dorbitals) and of the Pauli\nmatrix, respectively. \u000en\u0016is the change in the occupation\nof orbital\u0016compared to the DFT calculations included\nin Eq. A1. M\u000band\u000en\u0016are determined self-consistently.\nThe atomic SOI is described by\n^HSOI=\u0015X\n\u0016\u0017\n\u001b\u001b0^L\u000b\n\u0016\u0017^S\u000b\n\u001b\u001b0cy\n\u0016\u001b(k)c\u0017\u001b0(k);(A3)\nwhereL\u000bandS\u000bare the\u000bcomponents of the orbital and\nspin vector operators, respectively. The strength of the\nSOI,\u0015, is also obtained from \frst-principles calculations.13\nThe interaction with a static magnetic \feld Bextis\ndescribed by\n^Hext=B\u000b\nextX\n\u0016\u0017\n\u001b\u001b0(^L\u000b\n\u0016\u0017\u000e\u001b\u001b0+\u001b\u001b\u001b0\u000e\u0016\u0017)cy\n\u0016\u001b(k)c\u0016\u001b0(k);\n(A4)\nwhere\u0016Bis absorbed to B\u000b\nextand we used gL= 1 and\ngS= 2 as the Land\u0013 e factors for the orbital and spin\nangular momentum.\nAppendix B: Phenomenology of FMR\nThe semi-classical description of the magnetization is\nobtained using the Landau-Lifshitz-Gilbert (LLG) equa-\ntion (1)9. The e\u000bective \feld acting on the magnetic mo-\nment is obtained from the energy functional of the system\nasBe\u000b(t) =\u0000@E=@M. For the symmetries we investi-\ngate, the model energy106for the 3D cubic cases77can\nbe written as\nE3D(M) =K4\nM4(M2\nxM2\ny+M2\nyM2\nz+M2\nxM2\nz)\u0000M\u0001Bext;\n(B1)\nwhile for 2D systems,\nE2D(M) =\u0000K2\nM2M2\nz\u0000M\u0001Bext: (B2)\nPositive values of K4andK2yield easy magnetization\ndirection along the (001) direction.\nWe consider magnetic moments pointing along the easy\naxis, which de\fnes the ^ zdirection. Static magnetic \felds\nare applied along the same orientation. The magnetic\nmoment is set into small angle precession, M=M^ z+\u000eMx(t)^ x+\u000eMy(t)^ y, by an oscillatory \feld in the trans-\nverse plane, i.e., Bext(t) =Bext^ z+\u000eBext(t). In this form,\nthe e\u000bective \feld (linear in the transverse components of\nthe magnetization) is given by Be\u000b(t) =Ban(t)+Bext(t),\nwith\nB3D\nan(t) =\u00002K4\nM2(\u000eMx^ x+\u000eMy^ y) , and B2D\nan=2K2\nM^ z\n(B3)\nbeing the anisotropy \felds for 3D and 2D systems, respec-\ntively. In the following expressions, K4andK2appear\nin the same way, so they are denoted by K.\nThe Fourier transform of the linearized equation of mo-\ntion can be written using the circular components \u000eM\u0006=\n\u000eMx\u0006i\u000eMy. Within this convention, \u000eM\u0000=\u000eB\u0000=\n\u001f\u0000+=2 and\n\u001f\u0000+(!) =\u00002\rM\n[!\u0000\r(Bext+Ban)]\u0000i\u000b!; (B4)\nwhereBan= 2K=M .\nACKNOWLEDGMENTS\nWe are very grateful to R. B. Muniz, A. T. Costa\nand D. M. Edwards for fruitful discussions. 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Perpignan Via Domitia, Laboratoire de Mathématiques et Physique, F -66860, Perpignan, France \n2School of Physics, University College Dublin, Belfield, Dublin 4, Ireland \n3Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland \n4Kotel’nikov Institute of Radio Engineering and Electronics of the Russia n Academy of Sciences, Vvedenskii Square 1, \nFryazino, Moscow Region, 141120, Russia \n5Laboratoire des Solides Irradiés, Ecole Polytechnique, 91128 Palaiseau Cedex, France \nThermal fluctuations of nanomagnets driven by spin -polarized currents are treated via the Landau -Lifshitz -Gilbert equation as \ngeneralized to include both the random thermal noise field and Slonczewski spin -transfer torque (STT) term s. The magnetization \nreversal time of such a nanomagnet is then evaluated for wide ranges of damping by using a method which generalizes the solution of \nthe so -called Kramers turnover problem for mechanical Brownian particles thereby bridging the very low damping (VLD) and \nintermediate damping (ID) Kramers escape rates , to the analogous magnetic turnover problem. The reversal time is then evaluated for \na nanomagnet with the free energy density given in the standard form of superimposed easy -plane and in -plane easy -axis anisotropies \nwith the dc bias field along the easy axis. \n \nIndex Terms — Escape rate, Nanomagnets, Reversal time of the magnetization, Spin -transfer t orque . \n \nI. INTRODUCTION \nue to the spin-transfer torque (STT) effect [1 -6], the \nmagnetization of a nanoscale ferromagnet may be altered \nby spin -polarized currents . This phenomenon occurs because \nan electri c current with spin polarization in a ferromagnet has \nan associated flow of angular momentum [3,7] ther eby \nexerting a macroscopic spin torque. The phenomenon is the \norigin of the novel subject of spintronics [7,8], i.e., current -\ninduced control over magnet ic nanostructures . Common \napplications are very high-speed current -induced \nmagnetization switching by (a) reversing the orien tation of \nmagnetic bits [3,9 ] and (b) using spin polarized currents to \ncontrol steady state microwave oscillations [9 ]. This is \naccomplished via the steady state magnetization precession \ndue to STT representing the conversion of DC input into an \nAC output voltage [3]. Unfortunately , thermal fluctuations \ncannot now be ignored due to the nanometric size of STT \ndevices, e.g., leading to mainly noise -induced switching at \ncurrents far less than the critical switching current without \nnoise [10] as corroborated by experiments (e.g., [11]) \ndemonstrating that STT near room temperature significantly \nalters thermally activated switching processes . These now \nexhibit a pronounced dependence on both material and \ngeometrical parameters. Consequently, an accurate account of \nSTT switching effects at finite temperatures is necessary in \norder to achieve further improvements in the design and \ninterpretatio n of experiments, in view of the manifold practical applications in spintronics, random access memory \ntechnology, and so on. \nDuring the last decade, various analytical and numerical \napproaches to the study of STT effects in the thermally \nassisted magnetiza tion reversal (or switching) time in \nnanoscale ferromagnets have been developed [6,7,12 -26]. \nTheir objective being to generalize methods originally \ndeveloped for zero STT [12,27 -32] such as stochastic \ndynamics simulations (e. g., Refs. [21 -25]) and extensio ns to \nspin Hamiltonians of the mean first passage time (MFPT) \nmethod (e.g., Refs. [16] and [17] ) in the Kramers escape rate \ntheory [33,34]. However, unlike zero STT substantial progress \nin escape rate theory including STT effects has so far been \nachieved o nly in the limit of very low damping (VLD), \ncorresponding to vanishingly small values of the damping \nparameter \n in the Landau -Lifshitz -Gilbert -Slonczewski \nequation (see Eq. (5) below). Here the pronounced time \nseparation between fast precessional and slow energy changes \nin lightly damped closed phase space trajectories (ca lled \nStoner -Wohlfarth orbits) has been exploited in Refs. \n[7,14, 16,17] to formulate a one -dimensional Fokker -Planck \nequation for the energy distribution function which may be \nsolved by quadratures. This equation is essentially similar to \nthat derived by Kramers [ 33] in treating the VLD noise -\nactivated escape rate of a point Brownian particle from a \npotential well although the Hamiltonian of the magnetic \nproblem is no longer separable and additive and the barrier \nheight is now STT depend ent. The Stoner -Wohlfarth orbits \nand steady precession along such an orbit of constan t energy \noccur if the spin -torque is strong enough to cancel out the \ndissipative torque. The origin of the orbits arises from the \nbistable (or, indeed, in general multistable) structure of the \nanisotropy potential. This structure allows one to define a \nnonconservative “effective” potential with damping - and D \nManuscript received April 6, 2017; revised June 27, 2017; accepted July \n24, 2017. Date of publication July 27, 2017; date of current ver -sion \nSeptember 18, 2017. Correspondin g author: Y. P. Kalmykov (e -mail: \nkalmykov@univ -perp.fr). \nColor versions of one or more of the figures in this paper are available \nonline at http://ieeexplore.ieee.org . \nDigital Object Identifier: 10.1109/TMAG.2017. 2732944 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017 \n 2 \ncurrent -dependent potential barrier s between stationary self -\noscillatory states of the magnetization, thereby permitting one \nto estimate the reversal (switching) time between these states . \nThe magnetizat ion reversal time in the VLD limit is then \nevaluated [16,17,35 ] both for zero and nonzero STT. In \nparticular, for nonzero STT , the VLD reversal time has been \nevaluated analytically in Refs. [16,17 ]. Here it has been shown \nthat in the high barrier limit, an asymptotic equation for the \nVLD magnetization reversal time from a single well in the \npresence of the STT is given by \n \nVLD\nTST1\nCES . (1) \nIn Eq. (1), \n is the damping parameter arising from the \nsurroundings , \nTST\nAE\nEfe is the escape rate render ed by \ntransition state theory (TST) which ignores effects due to the \nloss of spins at the barrier [34], \nAEf is the well precession \nfrequency, \nE is the damping and spin -polarized -current \ndependent effective en ergy barrier, and \nCES is the \ndimensionless action at the saddle point C (the action is given \nby Eq. (13) below). \nThe most essential fea ture of the results obtained in Refs. \n[16,17,35 ] and how they pertain to this paper is that they apply \nat VLD only where the inequality \n1\nCES holds meaning \nthat the energy loss per cycle of the almost periodic motion at \nthe critical en ergy is much less than the thermal energy . \nUnfortunately for typical values of the material parameters \nCES\n may be very high (\n310 ), meaning that this inequality \ncan be fulfilled only for \n0.001 . In addition, both \nexperimental and theoretical estimates suggest higher values \nof of the order of 0.001 -0.1 ( see, e.g., Refs. [6,36 -38]), \nimplying that the VLD asymptotic results are no longer valid \nas they will now differ substantially from the true value of the \nreversal time . These considerations suggest that the \nasymptotic calculations for STT should be extended to include \nboth the VLD and intermediate damping (ID) regions. This is \nour primar y objective here . Now like point Brownian particles \nwhich are governed by a separable and additive Hamiltonian , \nin the escape rate problem as it pertains to magnetic moments \nof nanoparticles, three regimes of damping appear [ 12,33,34]. \nThese are (i) very low damping \n( 1)\nCES , (ii) intermediate -\nto-high damping (IHD) \n( 1)\nCES , and (iii) a more or less \ncritically damped turnover regime \n( ~ 1)\nCES . Also , Kramers \n[33] obtained his now -famous VLD and IHD escape rate \nformulas for point Brownian particles by assuming in both \ncases that the energy barrier is much greater than the thermal \nenergy so that the concept of an escape rate applies. He \nmentioned, however, that he could not find a general method \nof attack in order to obtain an escape rate formula valid for \nany damp ing regime. This problem, namely the Kramers \nturnover, was initially solved by Mel’nikov and Meshkov \n[39]. They obtained an escape rate that is valid for all values \nof the damping by a semi heuristic argument, thus constituting a solution of the Kramers tu rnover problem for point particles. \nLater, Grabert [40] and Pollak et al . [41] have presented by \nusing a coupled oscillator model of the thermal bath , a \ncomplete solution of the Kramers turnover problem and have \nshown that the turnover escape rate formula can be obtained \nwithout the ad hoc interp olation between the VLD and IHD \nregimes as used by Mel’nikov and Meshkov . Finally, Coffey \net al. [42,43 ] have shown for classical spins that at zero STT , \nthe magnetization reversal time for values of damping up to \nintermediate values, \n1, can also be evaluated via the \nturnover formula for the escape rate bridging the VLD and ID \nescape rates, namely, \n \nTST1\n()\nCE AS , (2) \nwhere \n()Az is the so-called depopulation factor, namely [39-\n42] \n \n 2\n2\n0ln 1 exp[ ( 1/4)]1\n1/4()z\nd\nA z e\n \n\n . (3) \nNow the ID reversal time (or the lower bound of the reversal \ntime) may always be evaluated via TST as [32,34] \n \nID\nTST1 . (4) \nTherefore b ecause \n()\nCCEE A S S is the energy loss per \ncycle at the critical energy \n0\nCES [39] (i.e. , in the VLD \nlimit) , Eq. (2) transparently reduces to the VLD Kramers \nresult, Eq. (1). Moreover in the ID range, where \n( ) 1\nCE AS , \nEq. (2) reduces to the TST Eq. (4). Nevertheless in the high \nbarrier limit \n1,\nCES \n given by Eq. (2) can substantially \ndeviate in the damping range \n0.001 1 both from \nID , \nEq. (4), and \nVLD , Eq. (1). Now, the approach of Coffey et al. \n[42,43 ] generalizing the Kramers turnover results to classical \nspins (nanomagnets) was developed for zero STT, \nnevertheless, it can also be used to account for STT effects. \nHere we shall extend th e zero STT results of Refs. \n[14,16,17,39 -42] treating the damping dependence of STT \neffects in the magnetization reversal of nano scaled \nferro magnets via escape rate theory in the most important \nrange of damping comprising the VLD and ID ranges , \n1. \nII. MODEL \nThe object of our study is the role played by STT effects in the \nthermally assist ed magnetization reversal using an adaptation \nof the theory of thermal fluctuations in nanomagnets \ndeveloped in the seminal work s of Néel [27] and Brown \n[28,29]. The Néel -Brown theory i s effect ively an adaptation of \nthe Kramers theory [ 33,34 ] originally given for point \nBrownian particles to magnetization relaxa tion governed by a \ngyromagnetic -like equation which is taken as the Langevin \nequation of the pro cess. Hence, the verification of that theory \nin the pure (i.e., without STT) nanomagnet context nicely \nillustrates the Kramers conception of a thermal relaxation \nprocess as escape over a potential barrier arising from the IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017 \n 3 \nshuttling action of the Brownian m otion. However, it should \nbe recalled throughout that unlike nanomagnets at zero STT \n(where the giant spin escape rate theory may be effectively \nregarded as fully developed), devices based on STT , due to the \ninjection of the spin -polarized current, invaria bly represent an \nopen system in an out-of-equilibrium steady state. This is in \nmarked contrast to the conventional steady state of \nnanostructures characterized by the Boltzmann equilibrium \ndistribution that arises when STT is omitted . Hence both the \ngover ning Fokker -Planck and Langevin equations and the \nescape rate theory based on these must be modified . \nTo facilitate our di scussion, we first describe a schematic \nmodel of the STT effect. The archetypal model (Fig. 1 (a)) of \na STT device is a nanostructure compr ising two magnetic \nstrata label ed the free and fixed layers and a nonmagnetic \nconducting spacer. The fixed layer is much more strongly \npinned along its orientation than the free one. If an electric \ncurrent is passsed through the fixed layer it become s spin -\npolarized . Thus , the current , as it encounters the free layer, \ninduces a STT . Hence, the magnetization \nM of the free layer \nis altered . Both ferromagnetic layers are assumed to be \nuniformly magnetized [3,6]. Although th is gia nt coherent spin \napproximation cannot explain all observations of the \nmagnetization dynamics in spin -torque systems, nevertheless \nmany qualitative features needed to interpret experimental \ndata are satisfactorily reproduced. Indeed, the current -induced \nmagnetization dynamics in the free layer may be described by \nthe Landau -Lifshitz -Gilbert -Slonczewski equation including thermal fluctuations , i.e., the usual Landau -Lifshitz -Gilbert \nequation [ 44] incl uding STT, however augmented by a \nrandom magnetic field \n()tη which is regarded as white noise. \nHence it now becomes a magnetic Langevin equation \n[3,6,7,12 ], viz., \n \nS u u H η u u u u I\n . (5) \nHere \n/SMuM is the unit vector directed along \nM , \nSM is \nthe saturation magnetization, and is the gyromagnetic -type \nconstant . The effective magnetic field \nH comprising the \nanisotropy and external applied fields is defined as \n \n0SkT E\nvMHu . (6) \nHere E is the normalized free energy density of the free layer \nconstituting a conservative potential, \nv is the free layer \nvolume , \n7 2 1\n04 10 JA m in SI units, and \nkT is the \nthermal energy. For purposes of illustration , we sh all take \n,)(E\n in the standard form of superimposed easy -plane and \nin-plane easy -axis anisotropies plus the Zeeman term due to \nthe applied magnetic field \n0H [45] (in our notation): \n \n22 2, ) sin cos sin cos )( ( 2 cos h E . (7) \nIn Eq. (7) and are the polar and azimuthal angles in the \nusual spherical polar coordinate system , \n0S/ (2 ) h H M D\n \nand \n2\n0S / ( ) v M D kT\n are the external field and anisotropy \nparameters, \n/1DD \n is the biaxiality parameter \ncharacterized by \nD\n and \nD thereby encompassing both \ndemagnetizing and magnet ocrystalline anisotropy effects \n(since \n and \n are determined by both the volume an d the \nthickness of the free layer, th eir numerical values may vary \nthrough a very large range, in particular, they can be very \nlarge , > 100 [45]). The form of Eq. (7) implies that both the \napplied field \n0H and the unit vector \nPe identifying the \nmagnetization direction in the fixed layer are directed along \nthe easy X-axis (see Fig. 1(a)) . In general, \n,()E as \nrendered by Eq. (7) has two equivalent saddle points C and \ntwo nonequivalent wells at \nA and \nA (see Fig.1(b) ). Finally , \nthe STT induced field \nSI is given by \n \n0S\nSkT\nvMIu , (8) \nwhere \n is the normalized non conservative potential due to \nthe spin -polarized current, which in its simplest form i s \n \n ( , )PJ eu . (9) \nIn Eq. (9), \n()P J b I e kT\n is the dimensionless STT \nparameter , I is the spin -polarized current regarded as positive \nif electrons flow from the free into the fixed layer, e is the \nelectronic charge, \n is Planck’s reduced constant , and \nPb is a \nparameter determined by the spin polarization factor \nP [1]. \nAccompanying the magnetic Langevin equation (5) (i.e., the \nstochastic differential equation of the random magnetization \nprocess) , one has the Fokker -Planck equation for th e evolution \nof the associated probability density function \n( , , )Wt of \norientations of \nM on the unit sphere, viz., [ 6,12,16 ] \n \nX e u Z \nY M \n \neasy axis H0 \nfixed layer free layer I eP \n(a) \n \n \n (b) \nFig. 1. (a) Geometry of the problem: A STT device consists of two \nferromagnetic strata labelled the free and fixed layers, respectively, and a \nnormal conducting spacer all sandwiched on a pillar between two ohmic \ncontact s [3,6]. Here I is the spin -polarized current, M is the magnetization of \nthe free layer, H0 is the dc bias magnetic field. The magnetization of the \nfixed layer is directed along the unit vector eP. (b) Free energy potential of \nthe free layer presented in the standard form of superimposed easy -plane and \nin-plane easy -axis anisotropies, Eq. (7), at = 20 and h = 0.2 . IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017 \n 4 \n \nFPLWWt , (10) \nwhere \nFPL is the Fokker -Planck operator in phase space \n( , )\n defined via [6,12,26] \n \n1\nN\n1\n11FP1 ( )L sin2 sin\n1 ( ) 1\nsin sin\n( ) ( )sinWEWW\nEW\nEEW \n \n\n\n\n\n\n \n \n \n\n \n \n (11) \nand \n0 N1\nS( ) / (2 ) v M kT is the free diffus ion time \nof the magnetic moment. If \n0= (zero STT), Eq. (10) \nbecomes the original Fokker -Planck equation derived by \nBrown [33] for magnetic nanoparticles . \nIII. ESCAPE RATES AND REVE RSAL TIME IN THE DAM PING \nRANGE \n1 \nThe magnetization reversal tim e can be calculated exactly by \nevaluating the smallest nonvanishing eigenvalue \n1 of the \nFokker -Planck operator L FP in Eq. (10) [32,34 ,42]. Thus \n1 is \nthe inverse of the longest relaxation time of the magnetization \n11/\n, which is usually associated with the reversal time . \nIn the manner of zero STT [42,43], the calculation of \n1 can \nbe approximately accomplished using the Mel’nikov -Meshkov \nformalism [39]. This relies on the fact that in the high barrier \nand underda mped limit s, one may rewrite the Fokker -Planck \nequation, Eq. (10), as an energy -action diffusion equation. \nThis in turn is very similar to that for translating point \nBrownian particles moving along the x-axis in an external \npotential V(x) [7,17,42] . In the under damp ed case, which is the \nrange of interest, for the escape of spins from a single \npotential well with a minimum at a point A of the \nmagnetocristalline anisotropy over a single saddle point C, the \nenergy distribution function \n()WE for magnetic moments \nprecessing in the potential well can then be found via an \nintegral equation [42], which can be solved for \n()WE by the \nWiener –Hopf method. Then, the flux -over-population method \n[33,34] yields the decay ( escape ) rate as \n1/CAJN . Here \nconstCJ\n is the probability current density over the sadd le \npoint and \n()C\nAE\nAEN W E dE is the well population while the \nescape rate is rendered as the product of the depopulation \nfactor \n( ),\nCE AS Eq. (3), and the TST escape rate \nTST\nAE\nEfe\n. In the preceding equation \nE is the effective \nspin-polari zed current dependent energy barrier given by \n \n1\nAC\nCE\nE EAEVdE E E ES , (12) \nwhere \nAE is the energy at the bottom of the potential well, \nCE\n is the energy at the saddle point, and the dimensionless \naction \nES and the dimensionless work \nEV done by the STT are defined as [7,17] \n \nEEd SE \nuuu\n , (13) \n \nEE d Vuuu\n , (14) \nrespectively. T he contour integrals in Eqs. (13) and (14) are \ntaken along the energy trajectory \nconstE and are to be \nevaluated in the vanishing damping sense. \nFor the bistable potential, Eq. (7), having two nonequivalent \nwells \nA and \nA with minima \n( 1 2 ) Eh\n at \n0A \nand \nA , respectively, and two equivalent saddle points C \nwith \n2\nCEh at \ncosC h (see Fig. 1(b)) we see that two \nwells and two escape routes over two saddle points are \ninvolved in the relaxation process . Thus, a finite probability \nfor the magnetic dipole to return to the initi al well having \nalready visited the second one exists. This possibility cannot \nbe ignored in the underdamped regime because then the \nmagnetic dipole having entered the second well loses its \nenergy so slowly that even after several precessions, thermal \nfluctuations may still reverse it back over the potential barrier. \nIn such a situation, on applying the Mel’nikov -Meshkov \nformalism [39] to the free energy potential, Eq. (7), and the \nnonconservative potential, Eq. (9), the energy distribution \nfunction s \n()WE and \n()WE for magne tic moments \nprecessing in the two potential well s can then be found by \nsolving two coupled integral equations for \n()WE and \n()WE\n. These then yield the depopulation factor \n, ()\nCCEE A S S\n via the Mel’nik ov-Meshkov formula for two \nwells, viz., [39] \n \n( ) ( )\n((), )CC\nCC\nCCEE\nEE\nEEA S A S\nA S SA S S\n\n\n . \nHere \n()Az is the depopulation factor for a single well \nintroduced in accordance with Eq. (3) above while \nCES are the \ndimensionless action s at the energy saddle point s for two \nwells. These are to be calculated via Eq. (13) by integrating \nalong the energy trajectories \nC EE between two saddle \npoints and are explicitly given by \n \n2\n3/2\n12\n21\n12(1 )\n(1(1 2 a4\n(1\nrct)\n)1an )(1 )\n)1 (1CCEEh\nhhhES\nhd\nh\nh \n\n\n\n\n\n \n \n \n\nuuu\n (15) \n(at zero dc bias field, h = 0, these simplify to \nCCEESS \n4\n). Furthermore, the overall TST escape rate \nTST for \na bistable potential, Eq. (7), is estimated via the individual \nescape rates \nTST\n from each of the two wells as \n \n TST TSTTST2.EEffee \n (16) \nIn Eq. (16), the factor 2 occurs because two magnetization \nescape r outes from each well over the two saddle points exist, \nwhile \nE are the effective spin -polarized current dependent IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017 \n 5 \nbarrier heights for two wells (explicit equations for \nE are \nderived in Appendix A). In addit ion \n \n01(1 )(1 )2f h h (17) \nare the corresponding well precession frequencies, where \n1\n0S 2MD\n is a precession time constant . Thus, the \ndecay rate \n1 becomes \n2\n2(1 ) ( , )1\n0\n(1 ) ( , )(1 )(1 )\n(( ) ( )\n()\n, 1 )(1 )CC\nCCJh F hEE\nEE\nJh F hA S A S\nAh h e\nhhS\neS\n\n\n\n\n\n\n\n\n \n\n \n\n \n\n(18) \nwhere both the functions \n( , )Fh occurring in each \nexponential are given by the analytical formula: \n \n22\n2\n1\n12\n2 1 1 2(1 ) (1( , ) 12 2 2 (1\n(1 21 arctan\n(1)(1 )\n)\n)1\n)(1 ) 1 (1 )hhFhhh h\nhh h\nh h h \n\n\n\n\n \n \n\n\n\n (19) \nand 0.38 is a numerical par ameter (see Eq. (A.6), etc. in \nAppendix A ). For zero STT, J = 0, Eq. (18) reduces to the \nknown results of the Néel -Brown theory [32,43] for classical \nmagnetic moments with superimposed easy -plane and in -plane \neasy-axis anisotropies plus the Zeeman term due to the applied \nmagnetic field. In contrast to zero STT, for normalized spin \ncurrents J 0, depends on \n not only through the \ndepopulation factors \n()\nCE AS but also through the spin-\npolarized current dependent effective barrier heights \nE . \nThis i s so because part s of the arguments of the exponentials \nin Eq. (18) , namely Eq. (19), are markedly dependent on the \nratio \n/J and the dc bias field parameter. The turnover Eq. \n(18) also yields a n asymptotic estimate for the inverse of the \nsmallest nonvanishing eigenvalue of the Fokker -Planck \noperator \nFPL in Eq. (10). In additio n, one may estimate two \nindividual reversal times, namely, \n from the deeper well \naround the energy minimum at \n0A and \n from the \nshallow well around the energy minimum at \nA (see Fig. \n1(b)) as \n \n2(1 ) ( , )\n02\n( ) (1 )(1 )\nCJh F h\nEe\nA S h h\n\n\n \n . (20) \nThe individual times are in general unequal, i.e., \n . In \nderiving Eqs. (18) and (20), all terms of order \n22, , ,JJ etc. \nare neglected. This hypothesis is true only for the \nunderdamped regime , α < 1, and weak spin-polarized currents, \nJ<<1. ( Despite these restrictions as we will see below Eqs. \n(18) and (20) still yield accurate estimates for \n for much \nhigher values of J). Now, \n can also be calculated \nnumerically via the method of statistical moments developed \nin Ref. [26] whereby t he solution of the Fokker -Planck \nequation (10) in configuration space is reduced to the task of solving an infinite hierarchy of differential -recurrence \nequations for the averaged spherical harmonics \n( , ) ( )lmYt \ngoverning the magnetization relaxation . (The \n( , )lmY are the \nspherical harmonics [46 ], and the angular brackets denote the \nstatistical aver aging ). Thus one can evaluate \n numerically \nvia \n1 of the Fokker -Planck operator L FP in Eq. (10) by using \nmatrix continued fr actions as described in Ref. [47 ]. We \nremark that the r anges of applicability of the escape rate \ntheory and the matrix continued -fraction method are in a sense \ncomplementary because escape rate theory cannot be used for \nlow potential barrie rs, \n3E , while the matrix continued -\nfraction method encounters substantial computation al \ndifficulties for very high potential barriers \n25E in the \nVLD range, \n410 . Thus , in the foregoing se nse, numerical \nmethods and escape rate theory are very useful for the \ndetermination of τ for low and very high potential barriers, \nrespectively. Nevertheless , in certain (wide) ranges of model \nparameters both methods yield accurate results for the reversal \ntime ( here these ranges are \n5 30, 3, and \n410 ). \nThen the numerically exact benchmark solution provided by \nthe matrix continued fraction method allows one to test the \naccuracy of the analytical es cape rate equations given above. \nIV. RESULTS AND DISCUSSIO N \nThroughout the calculations, the anisotropy and spin -\npolarization parameters will be taken as \n0.034 D\n , \n20 , \nand \n0.3P (\n0.3 0.4P are typical of ferromagnetic \nmetals) just as in Ref. 6. Thus for \n5 1 1mA s . 10 , 22 \n300T\nK\n, \n24~10v\n3m , and a current density of the order \nof \n7~ 10\n2A cm in a 3 nm thick layer of cobalt with \n61\nS 1 1. Am 04 M\n, we have the following estimates for the \nanisotropy (or inverse temperature ) parameter \n20.2 , \ncharacteristic time \n1\n0S2()MD\n0.48 ps, and spin -\npolarized current parameter \n( ) ~1P J b I e kT\n . In Figs. 2 \nand 3, we compare from the asymptotic escape rate Eq . (18) \nwith \n1\n1 of the Fokker –Planck operator as calculated \nnumerically via matrix continued fraction s [26]. Apparently, \nas rendered by the turnover equation (18) and \n1\n1 both lie \nvery close to each other in the high barrier limit, where the \nasymptotic Eq. (18) provides an accurate approximation \nto\n1\n1. In Fig. 2, is plotte d as a function of \n for various J. \nAs far as STT effects are concerned they are governed by the \nratio \n/J so that by altering \n/J the ensuing variation of \nmay exceed several or ders of magnitude (Fig. 2) . Invariably \nfor J << 1, which is a condition of applicability of the escape \nrate equations (1) and (18), STT effects on the magnetization \nrelaxation are pronounced only at very low damping, << 1 . \nFor \n1 , i.e. high damping, STT influences the reversal \nprocess very weak ly because the STT term in Eq. (5) is then \nsmall compared to the damping and random field terms . \nFurthermore, may greatly exceed or, on the other hand, be \nvery much less than the value for zero STT , i.e., J = 0 (see Fig. \n2). For example, as J decreases from positive values, \n IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017 \n 6 \nexponentially increases attaining a maximum at a critical \nvalue of the spin -polarized current and then smoothly switches \nover to exponential decrease as \nJ is further increased \nthrough negative values of J [26]. Now, t he temperature , \nexternal d.c. bias field, and dam ping dependence of can \nreadily be understood in terms of the effective potential \nbarriers \nE in Eq. (18). For example, for \n5, the \ntemperature dependence of has the customary Arrhenius \nbehavior \n~,Ee where \nE , Eq. (19), is markedly \ndependent on \n/J (see Fig. 3 a). Furthermore, the slope of \n1()T\n significantly decreases as the dc bias field parameter h increases due to lowering of the barrier height \nE owing to \nthe action of the external field (see Fig. 3b) . Now, although \nthe range of applicability of Eqs. (18) and (20) is ostensibly \nconfined to weak spin -polarized currents, J << 1, they can still \nyield accurate estimates for the reversal time for much higher \nvalues of J far exceeding this condition (see Fig. 3 a). \nThus , the turnover formula for , Eqs. (18) and (20), \nbridgi ng the Kramers VLD and ID escape rates as a function \nof the damping parameter for point particles [35,39 -41] as \nextended by Coffey et al. [42,43] to the magnetization \nrelaxation in nanoscale ferromagnets allows us (via the further \nextension to include STT embodied in Eq. (18)) to accurately \nevaluate STT effects in the magnetization reversal time of a \nnanomagnet driven by spin -polarized current in the highly \nrelevant ID to VLD damping range. This (underdamped) range \nis characterized by \n1 and the asymptotic escape rates are \nin complete agreement with independent numerical results \n[17]. Two particular merits of the escape rate equations for the \nreversal time are that (i) they are relatively simple ( i.e., \nexpressed via elementary functions) and (ii) that they can be \nused in those parameter ranges, where numerical methods \n(such as matrix continued fractions [17]) may be no longer \napplicable , e.g., for very high barriers , \n25E . Hence , one \nmay conclude that the damping dependence of the \nmagnetization reversal time is very marked in the \nunderdamped regime \n1 , a fact which may be very \nsignificant in int erpreting many STT experiments. \nV. APPENDIX A: CALCULATION OF \n( , )Fh IN EQ. (19) \nFor the bistable potential given by Eq. (7), and the \nnonconservative potential, Eq. (9), the spin -polarized current \ndependent effective barrier heights \nE for each of the two \nwells are given by (cf. Eq. (12)) \n \n21(1 ) ( , )h J F E h \n , (A.1) \nwhere \n \n( , )C\nAVFhSd\n\n \n\n\n , (A.2) \nwith \n/E , \n/ 1 2AAEh \n , \n2/CCEh . The \ndimensionless action \nS and the dimensionless work done by \nthe STT \nV for the deeper well can be calculated analytically \nvia elliptic integrals as described in detail in Ref. [17] yielding \n \n2 2\n0\n2\n22(1 )\n1\n2 ( )11 ( ) ( )\n)(2\n1\n(1 )( )\n() (142),(1 )( ( ( ) ) 1)p Ehd hpf\nEm hqq q m K m\nq h q mhpq q mS\nm\nKm\n\n\n\n\n \n \n\n\n\n \n\n\n\n \n \n \n \n\n \n\n \n uuu\n (A.3) \n\n54321: J = 0.2\n2: J = 0.1\n3: J = 0\n4: J = 0.1\n5: J = 0.2/ \nh =0.15\n =20\n = 201 \nFig. 2. Reversal time \n0/ vs the damping parameter \n for various values \nof the spin-polarized current parameter J. Solid lines : numerical calculations \nof the inverse of t he smallest nonvanishing eigenvalue \n1\n01() of the \nFokker –Planck operator , Eq. (11). Asterisks: the turnover formula, Eq. (18). \n \n54\n/ 3211: J = 1\n2: J = \n3: J = \n4: J = \n5: J = \nh = 0.1\n = 0.01\n = 20\n(a)\n \n (b)\n4\n/ 321 1: h = 0.0\n2: h = 0.1\n3: h = 0.2\n4: h = 0.3\n = 0.01\n = 20\nJ = \n\n \nFig. 3 . Reversal time \n0/ vs. the anisotropy (inverse temperature) \nparameter for various spin-polarized currents J (a) and dc bias field \nparameters h (b). Solid lines: numerical solution for the inverse of the \nsmallest nonvanishing eigenvalue \n1\n01() of the Fokker –Planck operator , \nEq. (11). Asterisks: Eq. (18). IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017 \n 7 \n \n\n0\n2 23\n2\n2( 1)\n1 ( 1)\n( | )1\n2\n21 2 1()\n( ) (1\n(|)121 ( ) (,))phV\nm\nm\nqhdhf\nq hpK\nhp q E m m qqq q m K mm\n\n\n\n\n \n\n\n\n \n\n \n\n \n \n\n \n \n\nueu\n (A.4) \nwhere \n \n2\n2\n21( 1)ph\n\n , \n1\n1eqe\n , \n \n\n1 (1 )\n(1 ) 1eemee\n , \n2\n( 1)hepph\n\n , \n()Km\n, \n()Em , and \n( | )am are the complete elliptic integrals \nof the fir st, seco nd, and third kinds, respectively [48], and \nf \nis the precession frequency in the deeper well at a given \nenergy, namely, \n \n0( 1)(1\n())(1 )\n8p e efKm\n\n . (A.5) \nThe quantities \nS , \nV , and \nf for the shallower well are \nobtained simply by replacing the dc bias field parameter \nh by \nh\n in all the equations for \nS , \nV , and \nf . We remark that \nS\n and \nV in Eqs. (A.3) and (A.4) differ by a factor 2 from \nthose given in Ref. [17]. This is so because \nS and \nV are \nnow calculated between the saddle points and not over the \nprecession period . When \n( , )C , \nS in Eqs. (A.3) \nreduces to \nCES , Eq. (15). \nIn the parameter ranges \n01h and \n1 , the integral in \nEq. (A.2) can be accurately evaluated analytically using an \ninterpolation function for \n/VS between t he two limiting \nvalues \n/\nAAVS and \n/\nCCVS at \n1A h\n and \n2\nCh , \nnamely \n \n11\nC AA\nA C AA\nCAV VV V\nS S S S\n \n \n \n , (A.6) \nwhere 0.38 is an interpolation parameter yielding the best \nfit of \n/VS in the interval \n.AC These limiting \nvalues can be calculated from Eqs. (A.3) and (A.4) yielding \nafter tedious algebra: \n \n1\n22A\nAV\nh S\n\n (A.7) \nand \n2\n2\n1\n12\n2 1 1 2)(1 ) 1 (112 (1\n(1 21 arct)\n)1an\n(1 (1 )(1 ) ) 1C\nCV h\nh Sh\nhh h\nh h h\n\n\n\n\n\n\n\n\n\n\n \n \n. (A.8) \nHence with Eqs. 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IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017 \n 9 \n " }, { "title": "1709.10365v1.Non_local_Gilbert_damping_tensor_within_the_torque_torque_correlation_model.pdf", "content": "Non-local Gilbert damping tensor within the torque-torque correlation model\nDanny Thonig,1,\u0003Yaroslav Kvashnin,1Olle Eriksson,1, 2and Manuel Pereiro1\n1Department of Physics and Astronomy, Material Theory, Uppsala University, SE-75120 Uppsala, Sweden\n2School of Science and Technology, Orebro University, SE-701 82 Orebro, Sweden\n(Dated: July 19, 2018)\nAn essential property of magnetic devices is the relaxation rate in magnetic switching which\ndepends strongly on the damping in the magnetisation dynamics. It was recently measured that\ndamping depends on the magnetic texture and, consequently, is a non-local quantity. The damping\nenters the Landau-Lifshitz-Gilbert equation as the phenomenological Gilbert damping parameter\n\u000b, that does not, in a straight forward formulation, account for non-locality. E\u000borts were spent\nrecently to obtain Gilbert damping from \frst principles for magnons of wave vector q. However,\nto the best of our knowledge, there is no report about real space non-local Gilbert damping \u000bij.\nHere, a torque-torque correlation model based on a tight binding approach is applied to the bulk\nelemental itinerant magnets and it predicts signi\fcant o\u000b-site Gilbert damping contributions, that\ncould be also negative. Supported by atomistic magnetisation dynamics simulations we reveal the\nimportance of the non-local Gilbert damping in atomistic magnetisation dynamics. This study gives\na deeper understanding of the dynamics of the magnetic moments and dissipation processes in real\nmagnetic materials. Ways of manipulating non-local damping are explored, either by temperature,\nmaterials doping or strain.\nPACS numbers: 75.10.Hk,75.40.Mg,75.78.-n\nE\u000ecient spintronics applications call for magnetic ma-\nterials with low energy dissipation when moving magnetic\ntextures, e.g. in race track memories1, skyrmion logics2,3,\nspin logics4, spin-torque nano-oscillator for neural net-\nwork applications5or, more recently, soliton devices6. In\nparticular, the dynamics of such magnetic textures |\nmagnetic domain walls, magnetic Skyrmions, or magnetic\nsolitons | is well described in terms of precession and\ndamping of the magnetic moment mias it is formulated\nin the atomistic Landau-Lifshitz-Gilbert (LLG) equation\nfor sitei\n@mi\n@t=mi\u0002\u0012\n\u0000\rBeff\ni+\u000b\nms@mi\n@t\u0013\n; (1)\nwhere\randmsare the gyromagnetic ratio and the\nmagnetic moment length, respectively. The precession\n\feldBeff\niis of quantum mechanical origin and is ob-\ntained either from e\u000bective spin-Hamilton models7or\nfrom \frst-principles8. In turn, energy dissipation is\ndominated by the ad-hoc motivated viscous damping in\nthe equation of motion scaled by the Gilbert damping\ntensor\u000b. Commonly, the Gilbert damping is used as\na scalar parameter in magnetization dynamics simula-\ntions based on the LLG equation. Strong e\u000borts were\nspend in the last decade to put the Gilbert damping\nto a \frst-principles ground derived for collinear mag-\nnetization con\fgurations. Di\u000berent methods were pro-\nposed: e.g. the breathing Fermi surface9{11, the torque-\ntorque correlation12, spin-pumping13or a linear response\nmodel14,15. Within a certain accuracy, the theoretical\nmodels allow to interpret16and reproduce experimental\ntrends17{20.\nDepending on the model, deep insight into the fun-\ndamental electronic-structure mechanism of the Gilbertdamping\u000bis provided: Damping is a Fermi-surface ef-\nfect and depending on e.g. scattering rate, damping\noccurs due to spin-\rip but also spin-conservative tran-\nsition within a degenerated (intraband, but also inter-\nband transitions) and between non-degenerated (inter-\nband transitions) electron bands. As a consequence of\nthese considerations, the Gilbert damping is proportional\nto the density of states, but it also scales with spin-orbit\ncoupling21,22. The scattering rate \u0000 for the spin-\rip tran-\nsitions is allocated to thermal, but also correlation ef-\nfects, making the Gilbert damping strongly temperature\ndependent which must be a consideration when applying\na three-temperature model for the thermal baths, say\nphonon14, electron, and spin temperature23. In particu-\nlar, damping is often related to the dynamics of a collec-\ntive precession mode (macrospin approach) driven from\nan external perturbation \feld, as it is used in ferromag-\nnetic resonance experiments (FMR)24. It is also estab-\nlished that the Gilbert damping depends on the orien-\ntation of the macrospin25and is, in addition, frequency\ndependent26.\nMore recently, the role of non-collective modes to the\nGilbert damping has been debated. F ahnle et al.27\nsuggested to consider damping in a tensorial and non-\nisotropic form via \u000bithat di\u000bers for di\u000berent sites i\nand depends on the whole magnetic con\fguration of the\nsystem. As a result, the experimentally and theoret-\nically assumed local Gilbert equation is replaced by a\nnon-local equation via non-local Gilbert damping \u000bijac-\ncounting for the most general form of Rayleigh's dissi-\npation function28. The proof of principles was given for\nmagnetic domain walls29,30, linking explicitly the Gilbert\ndamping to the gradients in the magnetic spin texture\nrm. Such spatial non-locality, in particular, for discrete\natomistic models, allows further to motivate energy dis-arXiv:1709.10365v1 [cond-mat.mtrl-sci] 29 Sep 20172\nij\nαij\nq\nFIG. 1: Schematic illustration of non-local energy dissipation\n\u000bijbetween site iandj(red balls) represented by a power\ncord in a system with spin wave (gray arrows) propagation q.\nsipation between two magnetic moments at sites iand\nj, and is represented by \u000bij, as schematically illustrated\nin Fig. 1. An analytical expression for \u000bijwas already\nproposed by various authors14,31,32, however, not much\nwork has been done on a material speci\fc, \frst-principle\ndescription of the atomistic non-local Gilbert damping\n\u000bij. An exception is the work by Gilmore et al.32who\nstudied\u000b(q) in the reciprocal space as a function of the\nmagnon wave vector qand concluded that the non-local\ndamping is negligible. Yan et al.29and Hals et al.33, on\nthe other hand, applied scattering theory according to\nBrataas et al.34to simulate non-collinearity in Gilbert\ndamping, only in reciprocal space or continuous meso-\nscopic scale. Here we come up with a technical descrip-\ntion of non-locality of the damping parameter \u000bij, in\nreal space, and provide numerical examples for elemental,\nitinerant magnets, which might be of high importance in\nthe context of ultrafast demagnetization35.\nThe paper is organized as follows: In Section I, we\nintroduce our \frst-principles model formalism based on\nthe torque-torque correlation model to study non-local\ndamping. This is applied to bulk itinerant magnets bcc\nFe, fcc Co, and fcc Ni in both reciprocal and real space\nand it is analysed in details in Section II. Here, we will\nalso apply atomistic magnetisation dynamics to outline\nthe importance in the evolution of magnetic systems. Fi-\nnally, in the last section, we conclude the paper by giving\nan outlook of our work.\nI. METHODS\nWe consider the torque-torque correlation model in-\ntroduced by Kambersk\u0013 y10and further elaborated on by\nGilmore et al.12. Here, \fnite magnetic moment rotations\ncouple to the Bloch eigenenergies \"n;kand eigenstates\njnki, characterised by the band index nat wave vec-tork, due to spin-orbit coupling. This generates a non-\nequilibrium population state (a particle-hole pair), where\nthe excited states relax towards the equilibrium distribu-\ntion (Fermi-Dirac statistics) within the time \u001cn;k=1=\u0000,\nwhich we assume is independent of nandk. In the adi-\nabatic limit, this perturbation is described by the Kubo-\nGreenwood perturbation theory and reads12,36in a non-\nlocal formulation\n\u000b\u0016\u0017(q) =g\u0019\nmsZ\n\nX\nnmT\u0016\nnk;mk+q\u0000\nT\u0017\nnk;mk+q\u0001\u0003Wnk;mk+qdk:\n(2)\nHere the integral runs over the whole Brillouin zone\nvolume \n. A frozen magnon of wave vector qis consid-\nered that is ascribed to the non-locality of \u000b. The scat-\ntering events depend on the spectral overlap Wnk;mk+q=R\n\u0011(\")Ank(\";\u0000)Amk+q(\";\u0000) d\"between two bands \"n;k\nand\"m;k+q, where the spectral width of the electronic\nbandsAnkis approximated by a Lorentzian of width \u0000.\nNote that \u0000 is a parameter in our model and can be spin-\ndependent as proposed in Ref. [37]. In other studies, this\nparameter is allocated to the self-energy of the system\nand is obtained by introducing disorder, e.g., in an al-\nloy or alloy analogy model using the coherent potential\napproximation14(CPA) or via the inclusion of electron\ncorrelation38. Thus, a principle study of the non-local\ndamping versus \u0000 can be also seen as e.g. a temperature\ndependent study of the non-local damping. \u0011=@f=@\"is\nthe derivative of the Fermi-Dirac distribution fwith re-\nspect to the energy. T\u0016\nnk;mk+q=hnkj^T\u0016jmk+qi, where\n\u0016=x;y;z , are the matrix elements of the torque oper-\nator ^T= [\u001b;Hso] obtained from variation of the mag-\nnetic moment around certain rotation axis e.\u001band\nHsoare the Pauli matrices and the spin-orbit hamilto-\nnian, respectively. In the collinear ferromagnetic limit,\ne=ezand variations occur in xandy, only, which al-\nlows to consider just one component of the torque, i.e.\n^T\u0000=^Tx\u0000i^Ty. Using Lehmann representation39, we\nrewrite the Bloch eigenstates by Green's function G, and\nde\fne the spectral function ^A= i\u0000\nGR\u0000GA\u0001\nwith the\nretarded (R) and advanced (A) Green's function,\n\u000b\u0016\u0017(q) =g\nm\u0019Z Z\n\n\u0011(\")^T\u0016^Ak\u0010\n^T\u0017\u0011y^Ak+qdkd\":(3)\nThe Fourier transformation of the Green's function G\n\fnally is used to obtain the non-local Gilbert damping\ntensor23between site iat positionriand sitejat position\nrj,\n\u000b\u0016\u0017\nij=g\nm\u0019Z\n\u0011(\")^T\u0016\ni^Aij\u0010\n^T\u0017\nj\u0011y^Ajid\": (4)\nNote that ^Aij= i\u0000\nGR\nij\u0000GA\nji\u0001\n. This result is consis-\ntent with the formulation given in Ref. [31] and Ref. [14].\nHence, the de\fnition of non-local damping in real space3\nand reciprocal space translate into each other by a\nFourier transformation,\n\u000bij=Z\n\u000b(q) e\u0000i(rj\u0000ri)\u0001qdq: (5)\nNote the obvious advantage of using Eq. (4), since it\nallows for a direct calculation of \u000bij, as opposed to tak-\ning the inverse Fourier transform of Eq. (5). For \frst-\nprinciples studies, the Green's function is obtained from\na tight binding (TB) model based on the Slater-Koster\nparameterization40. The Hamiltonian consists of on-site\npotentials, hopping terms, Zeeman energy, and spin-orbit\ncoupling (See Appendix A). The TB parameters, includ-\ning the spin-orbit coupling strength, are obtained by \ft-\nting the TB band structures to ab initio band structures\nas reported elsewhere23.\nBeyond our model study, we simulate material spe-\nci\fc non-local damping with the help of the full-potential\nlinear mu\u000en-tin orbitals (FP-LMTO) code \\RSPt\"41,42.\nFurther numerical details are provided in Appendix A.\nWith the aim to emphasize the importance of non-\nlocal Gilbert damping in the evolution of atomistic\nmagnetic moments, we performed atomistic magnetiza-\ntion dynamics by numerical solving the Landau-Lifshitz\nGilbert (LLG) equation, explicitly incorporating non-\nlocal damping23,34,43\n@mi\n@t=mi\u00020\n@\u0000\rBeff\ni+X\nj\u000bij\nmj\ns@mj\n@t1\nA:(6)\nHere, the e\u000bective \feld Beff\ni =\u0000@^H=@miis allo-\ncated to the spin Hamiltonian entails Heisenberg-like ex-\nchange coupling\u0000P\nijJijmi\u0001mjand uniaxial magneto-\ncrystalline anisotropyP\niKi(mi\u0001ei)2with the easy axis\nalongei.JijandKiare the Heisenberg exchange cou-\npling and the magneto-crystalline anisotropy constant,\nrespectively, and were obtained from \frst principles44,45.\nFurther details are provided in Appendix A.\nII. RESULTS AND DISCUSSION\nThis section is divided in three parts. In the \frst part,\nwe discuss non-local damping in reciprocal space q. The\nsecond part deals with the real space de\fnition of the\nGilbert damping \u000bij. Atomistic magnetization dynam-\nics including non-local Gilbert damping is studied in the\nthird part.\nA. Non-local damping in reciprocal space\nThe formalism derived by Kambersk\u0013 y10and Gilmore12\nin Eq. (2) represents the non-local contributions to the\nenergy dissipation in the LLG equation by the magnonwave vector q. In particular, Gilmore et al.32con-\ncluded that for transition metals at room temperature\nthe single-mode damping rate is essentially independent\nof the magnon wave vector for qbetween 0 and 1% of\nthe Brillouin zone edge. However, for very small scat-\ntering rates \u0000, Gilmore and Stiles12observed for bcc Fe,\nhcp Co and fcc Ni a strong decay of \u000bwithq, caused by\nthe weighting function Wnm(k;k+q) without any sig-\nni\fcant changes of the torque matrix elements. Within\nour model systems, we observed the same trend for bcc\nFe, fcc Co and fcc Ni. To understand the decay of the\nGilbert damping with magnon-wave vector qin more de-\ntail, we study selected paths of both the magnon qand\nelectron momentum kin the Brillouin zone at the Fermi\nenergy\"Ffor bcc Fe (q;k2\u0000!Handq;k2H!N),\nfcc Co and fcc Ni ( q;k2\u0000!Xandq;k2X!L) (see\nFig. 2, where the integrand of Eq. (2) is plotted). For\nexample, in Fe, a usually two-fold degenerated dband\n(approximately in the middle of \u0000H, marked by ( i)) gives\na signi\fcant contribution to the intraband damping for\nsmall scattering rates. There are two other contributions\nto the damping (marked by ( ii)), that are caused purely\nby interband transitions. With increasing, but small q\nthe intensities of the peaks decrease and interband tran-\nsitions become more likely. With larger q, however, more\nand more interband transitions appear which leads to an\nincrease of the peak intensity, signi\fcantly in the peaks\nmarked with ( ii). This increase could be the same or-\nder of magnitude as the pure intraband transition peak.\nSimilar trends also occur in Co as well as Ni and are\nalso observed for Fe along the path HN. Larger spectral\nwidth \u0000 increases the interband spin-\rip transitions even\nfurther (data not shown). Note that the torque-torque\ncorrelation model might fail for large values of q, since\nthe magnetic moments change so rapidly in space that\nthe adiababtic limit is violated46and electrons are not\nstationary equilibrated. The electrons do not align ac-\ncording the magnetic moment and the non-equilibrium\nelectron distribution in Eq. (2) will not fully relax. In\nparticular, the magnetic force theorem used to derive\nEq. (3) may not be valid.\nThe integration of the contributions in electron mo-\nmentum space kover the whole Brillouin zone is pre-\nsented in Fig. 3, where both `Loretzian' method given\nby Eq. (2) and Green's function method represented\nby Eq. (3) are applied. Both methods give the same\ntrend, however, di\u000ber slightly in the intraband region,\nwhich was already observed previously by the authors\nof Ref. [23]. In the `Lorentzian' approach, Eq. (2), the\nelectronic structure itself is una\u000bected by the scattering\nrate \u0000, only the width of the Lorentian used to approx-\nimateAnkis a\u000bected. In the Green function approach,\nhowever, \u0000 enters as the imaginary part of the energy\nat which the Green functions is evaluated and, conse-\nquently, broadens and shifts maxima in the spectral func-\ntion. This o\u000bset from the real energy axis provides a more\naccurate description with respect to the ab initio results\nthan the Lorentzian approach.4\nΓHq(a−1\n0)\nΓ H\nk(a−1\n0)\nFe\nΓX\nΓ X\nk(a−1\n0)\n Co\nΓX\nΓ X\nk(a−1\n0)\n Ni\n(i) (ii) (ii)\nFIG. 2: Electronic state resolved non-local Gilbert damping obtained from the integrand of Eq. (3) along selected paths in the\nBrillouin zone for bcc Fe, fcc Co and fcc Ni. The scattering rate used is \u0000 = 0 :01 eV. The abscissa (both top and bottom in\neach panels) shows the momentum path of the electron k, where the ordinate (left and right in each panel) shows the magnon\npropagation vector q. The two `triangle' in each panel should be viewed separately where the magnon momentum changes\naccordingly (along the same path) to the electron momentum.\nWithin the limits of our simpli\fed electronic structure\ntight binding method, we obtained qualitatively similar\ntrends as observed by Gilmore et al.32: a dramatic de-\ncrease in the damping at low scattering rates \u0000 (intra-\nband region). This trend is common for all here ob-\nserved itinerant magnets typically in a narrow region\n00:02a\u00001\n0the damping\ncould again increase (not shown here). The decay of \u000b\nis only observable below a certain threshold scattering\nrate \u0000, typically where intra- and interband contribu-\ntion equally contributing to the Gilbert damping. As\nalready found by Gilmore et al.32and Thonig et al.23,\nthis point is materials speci\fc. In the interband regime,\nhowever, damping is independent of the magnon propa-\ngator, caused by already allowed transition between the\nelectron bands due to band broadening. Marginal vari-\nations in the decay with respect to the direction of q\n(Inset of Fig. 3) are revealed, which was not reported be-\nfore. Such behaviour is caused by the break of the space\ngroup symmetry due to spin-orbit coupling and a selected\nglobal spin-quantization axis along z-direction, but also\ndue to the non-cubic symmetry of Gkfork6= 0. As a re-\nsult, e.g., in Ni the non-local damping decays faster along\n\u0000Kthan in \u0000X. This will be discussed more in detail in\nthe next section.\nWe also investigated the scaling of the non-local\nGilbert damping with respect to the spin-orbit coupling\nstrength\u0018dof the d-states (see Appendix B). We observe\nan e\u000bect that previously has not been discussed, namely\nthat the non-local damping has a di\u000berent exponential\nscaling with respect to the spin-orbit coupling constant\nfor di\u000berentjqj. In the case where qis close to the Bril-\nlouin zone center (in particular q= 0),\u000b/\u00183\ndwhereas\nfor wave vectors jqj>0:02a\u00001\n0,\u000b/\u00182\nd. For largeq,\ntypically interband transitions dominate the scatteringmechanism, as we show above and which is known to\nscale proportional to \u00182. Here in particular, the \u00182will\nbe caused only by the torque operator in Eq. (2). On the\nother hand, this indicates that spin-mixing transitions\nbecome less important because there is not contribution\nin\u0018from the spectral function entering to the damping\n\u000b(q).\nThe validity of the Kambserk\u0013 y model becomes ar-\nguable for\u00183scaling, as it was already proved by Costa\net al.47and Edwards48, since it causes the unphysical\nand strong diverging intraband contribution at very low\ntemperature (small \u0000). Note that there is no experi-\nmental evidence of such a trend, most likely due to that\nsample impurities also in\ruence \u0000. Furthermore, various\nother methods postulate that the Gilbert damping for\nq= 0 scales like \u00182 9,15,22. Hence, the current applied\ntheory, Eq. (3), seems to be valid only in the long-wave\nlimit, where we found \u00182-scaling. On the other hand,\nEdwards48proved that the long-wave length limit ( \u00182-\nscaling) hold also in the short-range limit if one account\nonly for transition that conserve the spin (`pure' spin\nstates), as we show for Co in Fig. 11 of Appendix C. The\ntrends\u000bversusjqjas described above changes drastically\nfor the `corrected' Kambersk\u0013 y formula: the interband re-\ngion is not a\u000bected by these corrections. In the intraband\nregion, however, the divergent behaviour of \u000bdisappears\nand the Gilbert damping monotonically increases with\nlarger magnon wave vector and over the whole Brillouin\nzone. This trend is in good agreement with Ref. [29].\nFor the case, where q= 0, we even reproduced the re-\nsults reported in Ref. [21]; in the limit of small scattering\nrates the damping is constant, which was also reported\nbefore in experiment49,50. Furthermore, the anisotropy\nof\u000b(q) with respect to the direction of q(as discussed\nfor the insets of Fig. 3) increases by accounting only for\npure-spin states (not shown here). Both agreement with5\n510−22Fe\n0.000\n0.025\n0.050\n0.075\n0.100\nq: Γ→H\n2510−2α(q)Co\nq: Γ→X\n510−225\n10−310−210−110+0\nΓ (eV)Ni\nq: Γ→X\nFIG. 3: (Color online) Non-local Gilbert damping as a func-\ntion of the spectral width \u0000 for di\u000berent reciprocal wave vector\nq(indicated by di\u000berent colors and in units a\u00001\n0). Note that q\nprovided here are in direct coordinates and only the direction\ndi\u000bers between the di\u000berent elementals, itinerant magnets.\nThe non-local damping is shown for bcc Fe (top panel) along\n\u0000!H, for fcc Co (middle panel) along \u0000 !X, and for fcc Ni\n(bottom panel) along \u0000 !X. It is obtained from `Lorentzian'\n(Eq. (2), circles) and Green's function (Eq. (3), triangles)\nmethod. The directional dependence of \u000bfor \u0000 = 0:01 eV is\nshown in the inset.\nexperiment and previous theory motivate to consider \u00182-\nscaling for all \u0000.\nB. Non-local damping in real space\nAtomistic spin-dynamics, as stated in Section I (see\nEq. (6)), that includes non-local damping requires\nGilbert damping in real-space, e.g. in the form \u000bij. This\npoint is addressed in this section. Such non-local con-\ntributions are not excluded in the Rayleigh dissipation\nfunctional, applied by Gilbert to derive the dissipation\ncontribution in the equation of motion51(see Fig. 4).\nDissipation is dominated by the on-site contribution\n-101 Fe\nαii= 3.552·10−3\n˜αii= 3.559·10−3\n-101αij·10−4Co\nαii= 3.593·10−3\n˜αii= 3.662·10−3\n-10\n1 2 3 4 5 6\nrij/a0Ni\nαii= 2.164·10−2\n˜αii= 2.319·10−2FIG. 4: (Color online) Real-space Gilbert damping \u000bijas\na function of the distance rijbetween two sites iandjfor\nbcc Fe, fcc Co, and fcc Ni. Both the `corrected' Kambersk\u0013 y\n(red circles) and the Kambersk\u0013 y (blue squares) approach is\nconsidered. The distance is normalised to the lattice constant\na0. The on-site damping \u000biiis shown in the \fgure label. The\ngrey dotted line indicates the zero line. The spectral width is\n\u0000 = 0:005 eV.\n\u000biiin the itinerant magnets investigated here. For both\nFe (\u000bii= 3:55\u000110\u00003) and Co ( \u000bii= 3:59\u000110\u00003) the\non-site damping contribution is similar, whereas for Ni\n\u000biiis one order of magnitude higher. O\u000b-site contri-\nbutionsi6=jare one-order of magnitude smaller than\nthe on-site part and can be even negative. Such neg-\native damping is discernible also in Ref. [52], however,\nit was not further addressed by the authors. Due to\nthe presence of the spin-orbit coupling and a preferred\nglobal spin-quantization axis (in z-direction), the cubic\nsymmetry of the considered itinerant magnets is broken\nand, thus, the Gilbert damping is anisotropic with re-\nspect to the sites j(see also Fig. 5 left panel). For ex-\nample, in Co, four of the in-plane nearest neighbours\n(NN) are\u000bNN\u0019\u00004:3\u000110\u00005, while the other eight are\n\u000bNN\u0019\u00002:5\u000110\u00005. However, in Ni the trend is opposite:\nthe out-of-plane damping ( \u000bNN\u0019\u00001:6\u000110\u00003) is smaller\nthan the in-plane damping ( \u000bNN\u0019 \u00001:2\u000110\u00003). In-\nvolving more neighbours, the magnitude of the non-local6\ndamping is found to decay as 1=r2and, consequently, it\nis di\u000berent than the Heisenberg exchange parameter that\nasymptotically decays in RKKY-fashion as Jij/1=r353.\nFor the Heisenberg exchange, the two Green's functions\nas well as the energy integration in the Lichtenstein-\nKatsnelson-Antropov-Gubanov formula54scales liker\u00001\nij,\nG\u001b\nij/ei(k\u001b\u0001rij+\b\u001b)\njrijj(7)\nwhereas for simplicity we consider here a single-band\nmodel but the results can be generalized also to the multi-\nband case and where \b\u001bdenotes a phase factor for spin\n\u001b=\";#. For the non-local damping the energy integra-\ntion is omitted due to the properties of \u0011in Eq. (4) and,\nthus,\n\u000bij/sin\u0002\nk\"\u0001rij+ \b\"\u0003\nsin\u0002\nk#\u0001rij+ \b#\u0003\njrijj2:(8)\nThis spatial dependency of \u000bijsuperimposed with\nRuderman-Kittel-Kasuya-Yosida (RKKY) oscillations\nwas also found in Ref. [52] for a model system.\nFor Ni, dissipation is very much short range, whereas in\nFe and Co `damping peaks' also occur at larger distances\n(e.g. for Fe at rij= 5:1a0and for Co at rij= 3:4a0).\nThe `long-rangeness' depends strongly on the parameter\n\u0000 (not shown here). As it was already observed for the\nHeisenberg exchange interaction Jij44, stronger thermal\ne\u000bects represented by \u0000 will reduce the correlation length\nbetween two magnetic moments at site iandj. The same\ntrend is observed for damping: larger \u0000 causes smaller\ndissipation correlation length and, thus, a faster decay\nof non-local damping in space rij. Di\u000berent from the\nHeisenberg exchange, the absolute value of the non-local\ndamping typically decreases with \u0000 as it is demonstrated\nin Fig. 5.\nNote that the change of the magnetic moment length\nis not considered in the results discussed so far. The\nanisotropy with respect to the sites iandjof the non-\nlocal Gilbert damping continues in the whole range of the\nscattering rate \u0000 and is controlled by it. For instance, the\nsecond nearest neighbours damping in Co and Ni become\ndegenerated at \u0000 = 0 :5 eV, where the anisotropy between\n\frst-nearest neighbour sites increase. Our results show\nalso that the sign of \u000bijis a\u000bected by \u0000 (as shown in\nFig. 5 left panel). Controlling the broadening of Bloch\nspectral functions \u0000 is in principal possible to evaluate\nfrom theory, but more importantly it is accessible from\nexperimental probes such as angular resolved photoelec-\ntron spectroscopy and two-photon electron spectroscopy.\nThe importance of non-locality in the Gilbert damping\ndepend strongly on the material (as shown in Fig. 5 right\npanel). It is important to note that the total | de\fned as\n\u000btot=P\nj\u000bijfor arbitrary i|, but also the local ( i=j)\nand the non-local ( i6=j) part of the Gilbert damping do\nnot violate the thermodynamic principles by gaining an-\ngular momentum (negative total damping). For Fe, the\n-101\n1. NN.\n2. NN.Fe\n34567αii\nαtot=/summationtext\njαijαq=0.1a−1\n0αq=0\n-10αij·10−4Co\n123456\nαij·10−3\n-15-10-50\n10−210−1\nΓ (eV)Ni\n5101520\n10−210−1\nΓ (eV)FIG. 5: (Color online) First (circles) and second nearest\nneighbour (triangles) Gilbert damping (left panel) as well as\non-site (circles) and total Gilbert (right panel) as a function of\nthe spectral width \u0000 for the itinerant magnets Fe, Co, and Ni.\nIn particular for Co, the results obtained from tight binding\nare compared with \frst-principles density functional theory\nresults (gray open circles). Solid lines (right panel) shows the\nGilbert damping obtained for the magnon wave vectors q= 0\n(blue line) and q= 0:1a\u00001\n0(red line). Dotted lines are added\nto guide the eye. Note that since cubic symmetry is broken\n(see text), there are two sets of nearest neighbor parameters\nand two sets of next nearest neighbor parameters (left panel)\nfor any choice of \u0000.\nlocal and total damping are of the same order for all\n\u0000, where in Co and Ni the local and non-local damp-\ning are equally important. The trends coming from our\ntight binding electron structure were also reproduced by\nour all-electron \frst-principles simulation, for both de-\npendency on the spectral broadening \u0000 (Fig. 5 gray open\ncircles) but also site resolved non-local damping in the\nintraband region (see Appendix A), in particular for fcc\nCo.\nWe compare also the non-local damping obtain from\nthe real and reciprocal space. For this, we used Eq. (3)\nby simulating Nq= 15\u000215\u000215 points in the \frst magnon\nBrillouin zone qand Fourier-transformed it (Fig. 6). For7\n-1.0-0.50.00.51.0αij·10−4\n5 10 15 20 25 30\nrij/a0FFT(α(q));αii= 0.003481\nFFT(G(k));αii= 0.003855\nFIG. 6: (Color online) Comparing non-local Gilbert damping\nobtained by Eq. (5) (red symbols) and Eq. (4) (blue symbols)\nin fcc Co for \u0000 = 0 :005 eV. The dotted line indicates zero\nvalue.\nboth approaches, we obtain good agreement, corroborat-\ning our methodology and possible applications in both\nspaces. The non-local damping for the \frst three nearest\nneighbour shells turn out to converge rapidly with Nq,\nwhile it does not converge so quickly for larger distances\nrij. The critical region around the \u0000-point in the Bril-\nlouin zone is suppressed in the integration over q. On\nthe other hand, the relation \u000btot=P\nj\u000bij=\u000b(q= 0)\nfor arbitrary ishould be valid, which is however violated\nin the intraband region as shown in Fig. 5 (compare tri-\nangles and blue line in Fig. 5): The real space damping\nis constant for small \u0000 and follows the long-wavelength\nlimit (compare triangles and red line in Fig. 5) rather\nthan the divergent ferromagnetic mode ( q= 0). Two\nexplanations are possible: i)convergence with respect to\nthe real space summation and ii)a di\u000berent scaling in\nboth models with respect to the spin-orbit coupling. For\ni), we carefully checked the convergence with the summa-\ntion cut-o\u000b (see Appendix D) and found even a lowering\nof the total damping for larger cut-o\u000b. However, the non-\nlocal damping is very long-range and, consequently, con-\nvergence will be achieved only at a cut-o\u000b radius >>9a0.\nForii), we checked the scaling of the real space Gilbert\ndamping with the spin-orbit coupling of the d-states\n(see Appendix B). Opposite to the `non-corrected' Kam-\nbersk\u0013 y formula in reciprocal space, which scales like\n\u00183\nd, we \fnd\u00182\ndfor the real space damping. This indi-\ncates that the spin-\rip scattering hosted in the real-space\nGreen's function is suppressed. To corroborate this state-\nment further, we applied the corrections proposed by\nEdwards48to our real space formula Eq. (4), which by\ndefault assumes \u00182(Fig. 4, red dots). Both methods, cor-\nrected and non-corrected Eq. (4), agree quite well. The\nsmall discrepancies are due to increased hybridisations\nand band inversion between p and d- states due to spin-\norbit coupling in the `non-corrected' case.\nFinally, we address other ways than temperature (here\nrepresented by \u0000), to manipulate the non-local damping.\nIt is well established in literature already for Heisenberg\nexchange and the magneto crystalline anisotropy that\n-0.40.00.40.81.2αij·10−4\n1 2 3 4 5 6 7\nrij/a0αii= 3.49·10−3αii= 3.43·10−3FIG. 7: (Color online) Non-local Gilbert damping as a func-\ntion of the normalized distancerij=a0for a tetragonal dis-\ntorted bcc Fe crystal structure. Here,c=a= 1:025 (red circles)\nandc=a= 1:05 (blue circles) is considered. \u0000 is put to 0 :01 eV.\nThe zero value is indicated by dotted lines.\ncompressive or tensial strain can be used to tune the mag-\nnetic phase stability and to design multiferroic materials.\nIn an analogous way, also non-local damping depends on\ndistortions in the crystal (see Fig. 7).\nHere, we applied non-volume conserved tetragonal\nstrain along the caxis. The local damping \u000biiis marginal\nbiased. Relative to the values of the undistorted case,\na stronger e\u000bect is observed for the non-local part, in\nparticular for the \frst few neighbours. Since we do a\nnon-volume conserved distortion, the in-plane second NN\ncomponent of the non-local damping is constant. The\ndamping is in general decreasing with increasing distor-\ntion, however, a change in the sign of the damping can\nalso occur (e.g. for the third NN). The rate of change\nin damping is not linear. In particular, the nearest-\nneighbour rate is about \u000e\u000b\u00190:4\u000110\u00005for 2:5% dis-\ntortion, and 2 :9\u000110\u00005for 5% from the undistorted case.\nFor the second nearest neighbour, the rate is even big-\nger (3:0\u000110\u00005for 2:5%, 6:9\u000110\u00005for 5%). For neigh-\nbours larger than rij= 3a0, the change is less signi\fcant\n(\u00000:6\u000110\u00005for 2:5%,\u00000:7\u000110\u00005for 5%). The strongly\nstrain dependent damping motivates even higher-order\ncoupled damping contributions obtained from Taylor ex-\npanding the damping contribution around the equilib-\nrium position \u000b0\nij:\u000bij=\u000b0\nij+@\u000bij=@uk\u0001uk+:::. Note that\nthis is in analogy to the magnetic exchange interaction55\n(exchange striction) and a natural name for it would\nbe `dissipation striction'. This opens new ways to dis-\nsipatively couple spin and lattice reservoir in combined\ndynamics55, to the best of our knowledge not considered\nin todays ab-initio modelling of atomistic magnetisation\ndynamics.\nC. Atomistic magnetisation dynamics\nThe question about the importance of non-local damp-\ning in atomistic magnetization dynamics (ASD) remains.8\n0.40.50.60.70.80.91.0M\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nt(ps)0.5\n0.1\n0.05\n0.01αtot\nαij\n0.5 1.0 1.5 2.0 2.5 3.0\nt(ps)Fe\nCo\nFIG. 8: (Color online) Evolution of the average magnetic mo-\nmentMduring remagnetization in bcc Fe (left panel) and\nfcc Co (right panel) for di\u000berent damping strength according\nto the spectral width \u0000 (di\u000berent colors) and both, full non-\nlocal\u000bij(solid line) and total, purely local \u000btot(dashed line)\nGilbert damping.\nFor this purpose, we performed zero-temperature ASD\nfor bcc Fe and fcc Co bulk and analysed changes in the\naverage magnetization during relaxation from a totally\nrandom magnetic con\fguration, for which the total mo-\nment was zero (Fig. 8)\nRelated to the spectral width, the velocity for remag-\nnetisation changes and is higher, the bigger the e\u000bective\nGilbert damping is. For comparison, we performed also\nASD simulations based on Eq. (2) with a scalar, purely\nlocal damping \u000btot(dotted lines). For Fe, it turned out\nthat accounting for the non-local damping causes a slight\ndecrease in the remagnetization time, however, is overall\nnot important for relaxation processes. This is under-\nstandable by comparing the particular damping values\nin Fig. 5, right panel, in which the non-local part ap-\npear negligible. On the other hand, for Co the e\u000bect\non the relaxation process is much more signi\fcant, since\nthe non-local Gilbert damping reduces the local contribu-\ntion drastically (see Fig. 5, right panel). This `negative'\nnon-local part ( i6=j) in\u000bijdecelerates the relaxation\nprocess and the relaxation time is drastically increased\nby a factor of 10. Note that a `positive' non-local part\nwill accelerate the relaxation, which is of high interest for\nultrafast switching processes.\nIII. CONCLUDING REMARKS\nIn conclusion, we have evaluated the non-locality of\nthe Gilbert damping parameter in both reciprocal and\nreal space for elemental, itinerant magnets bcc Fe, fcc\nCo and fcc Ni. In particular in the reciprocal space,\nour results are in good agreement with values given in\nthe literature32. The here studied real space damping\nwas considered on an atomistic level and it motivates\nto account for the full, non-local Gilbert damping in\nmagnetization dynamic, e.g. at surfaces56or for nano-\nstructures57. We revealed that non-local damping canbe negative, has a spatial anisotropy, quadratically scales\nwith spin-orbit coupling, and decays in space as r\u00002\nij.\nDetailed comparison between real and reciprocal states\nidenti\fed the importance of the corrections proposed by\nEdwards48and, consequently, overcome the limits of the\nKambersk\u0013 y formula showing an unphysical and experi-\nmental not proved divergent behaviour at low tempera-\nture. We further promote ways of manipulating non-local\nGilbert damping, either by temperature, materials dop-\ning or strain, and motivating `dissipation striction' terms,\nthat opens a fundamental new root in the coupling be-\ntween spin and lattice reservoirs.\nOur studies are the starting point for even further in-\nvestigations: Although we mimic temperature by the\nspectral broadening \u0000, a precise mapping of \u0000 to spin\nand phonon temperature is still missing, according to\nRefs. [14,23]. Even at zero temperature, we revealed a\nsigni\fcant e\u000bect of the non-local Gilbert damping to the\nmagnetization dynamics, but the in\ruence of non-local\ndamping to \fnite temperature analysis or even to low-\ndimensional structures has to be demonstrated.\nIV. ACKNOWLEDGEMENTS\nThe authors thank Lars Bergqvist, Lars Nordstr om,\nJustin Shaw, and Jonas Fransson for fruitful discus-\nsions. O.E. acknowledges the support from Swedish Re-\nsearch Council (VR), eSSENCE, and the KAW Founda-\ntion (Grants No. 2012.0031 and No. 2013.0020).\nAppendix A: Numerical details\nWe performkintegration with up to 1 :25\u0001106mesh\npoints (500\u0002500\u0002500) in the \frst Brillouin zone for bulk.\nThe energy integration is evaluated at the Fermi level\nonly. For our principles studies, we performed a Slater-\nKoster parameterised40tight binding (TB) calculations58\nof the torque-torque correlation model as well as for the\nGreen's function model. Here, the TB parameters have\nbeen obtained by \ftting the electronic structures to those\nof a \frst-principles fully relativistic multiple scattering\nKorringa-Kohn-Rostoker (KKR) method using a genetic\nalgorithm. The details of the \ftting and the tight binding\nparameters are listed elsewhere23,59. This puts our model\non a \frm, \frst-principles ground.\nThe tight binding Hamiltonian60H=H0+Hmag+\nHsoccontains on-site energies and hopping elements H0,\nthe spin-orbit coupling Hsoc=\u0010S\u0001Land the Zeeman\ntermHmag=1=2B\u0001\u001b. The Green's function is obtained\nbyG= (\"+ i\u0000\u0000H)\u00001, allows in principle to consider\ndisorder in terms of spin and phonon as well as alloys23.\nThe bulk Greenian Gijin real space between site iandj\nis obtained by Fourier transformation. Despite the fact\nthat the tight binding approach is limited in accuracy, it\nproduces good agreement with \frst principle band struc-\nture calculations for energies smaller than \"F+ 5 eV.9\n-1.5-1.0-0.50.00.51.01.5\n5 10 15 20 25 30\nrij(Bohr radii)Γ≈0.01eVTB\nTBe\nDFT\nαDFT\nii= 3.9846·10−3\nαTB\nii= 3.6018·10−3-1.5-1.0-0.50.00.51.01.5\nΓ≈0.005eV\nαDFT\nii= 3.965·10−3\nαTB\nii= 3.5469·10−3αij·10−4\nFIG. 9: (Colour online) Comparison of non-local damping ob-\ntained from the Tight Binding method (TB) (red \flled sym-\nbols), Tight Binding with Edwards correction (TBe) (blue\n\flled symbols) and the linear mu\u000en tin orbital method (DFT)\n(open symbols) for fcc Co. Two di\u000berent spectral broadenings\nare chosen.\nEquation (4) was also evaluated within the DFT and\nlinear mu\u000en-tin orbital method (LMTO) based code\nRSPt. The calculations were done for a k-point mesh\nof 1283k-points. We used three types of basis func-\ntions, characterised by di\u000berent kinetic energies with\n\u00142= 0:1;\u00000:8;\u00001:7 Ry to describe 4 s, 4pand 3dstates.\nThe damping constants were calculated between the 3 d\norbitals, obtained using using mu\u000en-tin head projection\nscheme61. Both the \frst principles and tight binding im-\nplementation of the non-local Gilbert damping agree well\n(see Fig. 9).\nNote that due to numerical reasons, the values of\n\u0000 used for the comparisons are slightly di\u000berent in\nboth electronic structure methods. Furthermore, in the\nLMTO method the orbitals are projected to d-orbitals\nonly, which lead to small discrepancies in the damping.\nThe atomistic magnetization dynamics is also per-\nformed within the Cahmd simulation package58. To\nreproduce bulk properties, periodic boundary condi-\ntions and a su\u000eciently large cluster (10 \u000210\u000210)\nare employed. The numerical time step is \u0001 t=\n0:1 fs. The exchange coupling constants Jijare\nobtained from the Liechtenstein-Kastnelson-Antropov-\nGubanovski (LKAG) formula implemented in the \frst-\nprinciples fully relativistic multiple scattering Korringa-\nKohn-Rostoker (KKR) method39. On the other hand,\nthe magneto-crystalline anisotropy is used as a \fxed pa-\nrameter with K= 50\u0016eV.\n012345678α·10−3\n0.0 0.02 0.04 0.06 0.08 0.1\nξd(eV)2.02.22.42.62.83.03.2γ\n0.0 0.1 0.2 0.3 0.4\nq(a−1\n0)-12-10-8-6-4-20αnn·10−5\n01234567\nαos·10−3 1.945\n1.797\n1.848\n1.950\n1.848\n1.797\n1.950FIG. 10: (Color online) Gilbert damping \u000bas a function of\nthe spin-orbit coupling for the d-states in fcc Co. Lower panel\nshows the Gilbert damping in reciprocal space for di\u000berent\nq=jqjvalues (di\u000berent gray colours) along the \u0000 !Xpath.\nThe upper panel exhibits the on-site \u000bos(red dotes and lines)\nand nearest-neighbour \u000bnn(gray dots and lines) damping.\nThe solid line is the exponential \ft of the data point. The\ninset shows the \ftted exponents \rwith respect wave vector\nq. The colour of the dots is adjusted to the particular branch\nin the main \fgure. The spectral width is \u0000 = 0 :005 eV.\nAppendix B: Spin-orbit coupling scaling in real and\nreciprocal space\nKambersk\u0013 y's formula is valid only for quadratic spin-\norbit coupling scaling21,47, which implies only scattering\nbetween states that preserve the spin. This mechanism\nwas explicitly accounted by Edwards48by neglecting the\nspin-orbit coupling contribution in the `host' Green's\nfunction. It is predicted for the coherent mode ( q= 0)21\nthat this overcomes the unphysical and not experimen-\ntally veri\fed divergent Gilbert damping for low tem-\nperature. Thus, the methodology requires to prove the\nfunctional dependency of the (non-local) Gilbert damp-\ning with respect to the spin-orbit coupling constant \u0018\n(Fig. 10). Since damping is a Fermi-surface e\u000bects, it\nis su\u000ecient to consider only the spin-orbit coupling of\nthe d-states. The real space Gilbert damping \u000bij/\u0018\r\nscales for both on-site and nearest-neighbour sites with\n\r\u00192. For the reciprocal space, however, the scaling is\nmore complex and \rdepends on the magnon wave vec-\ntorq(inset in Fig. 10). In the long-wavelength limit,\nthe Kambersk\u0013 y formula is valid, where for the ferromag-\nnetic magnon mode with \r\u00193 the Kambersk\u0013 y formula\nis inde\fnite according to Edwards48.10\n10−32510−2α(q)\n10−310−210−110+0\nΓ (eV)0.000\n0.025\n0.050\n0.075\n0.100\nq: Γ→XCo\nFIG. 11: (Colour online) Comparison of reciprocal non-local\ndamping with (squares) or without (circles) corrections pro-\nposed by Costa et al.47and Edwards48for Co and di\u000berent\nspectral broadening \u0000. Di\u000berent colours represent di\u000berent\nmagnon propagation vectors q.\nAppendix C: Intraband corrections\nFrom the same reason as discussed in Section B, the\nrole of the correction proposed by Edwards48for magnon\npropagations di\u000berent than zero is unclear and need to\nbe studied. Hence, we included the correction of Ed-\nward also to Eq. (3) (Fig. 11). The exclusion of the spin-\norbit coupling (SOC) in the `host' clearly makes a major\nqualitative and quantitative change: Although the in-\nterband transitions are una\u000bected, interband transitions\nare mainly suppressed, as it was already discussed by\nBarati et al.21. However, the intraband contributions are\nnot totally removed for small \u0000. For very small scat-\ntering rates, the damping is constant. Opposite to the\n`non-corrected' Kambersk\u0013 y formula, the increase of the\nmagnon wave number qgives an increase in the non-\nlocal damping which is in agreement to the observation\nmade by Yuan et al.29, but also with the analytical modelproposed in Ref. [52] for small q. This behaviour was ob-\nserved for all itinerant magnets studied here.\nAppendix D: Comparison real and reciprocal\nGilbert damping\nThe non-local damping scales like r\u00002\nijwith the dis-\ntance between the sites iandj, and is, thus, very long\nrange. In order to compare \u000btot=P\nj2Rcut\u000bijfor arbi-\ntraryiwith\u000b(q= 0), we have to specify the cut-o\u000b ra-\ndius of the summation in real space (Fig. 12). The inter-\nband transitions (\u0000 >0:05 eV) are already converged for\nsmall cut-o\u000b radii Rcut= 3a0. Intraband transitions, on\nthe other hand, converge weakly with Rcutto the recipro-\ncal space value \u000b(q= 0). Note that \u000b(q= 0) is obtained\nfrom the corrected formalism. Even with Rcut= 9a0\nwhich is proportional to \u001930000 atoms, we have not\n0.81.21.62.0αtot·10−3\n4 5 6 7 8 9\nRcut/a00.005\n0.1\nFIG. 12: Total Gilbert damping \u000btotfor fcc Co as a function\nof summation cut-o\u000b radius for two spectral width \u0000, one in\nintraband (\u0000 = 0 :005 eV, red dottes and lines) and one in the\ninterband (\u0000 = 0 :1 eV, blue dottes and lines) region. The\ndotted and solid lines indicates the reciprocal value \u000b(q= 0)\nwith and without SOC corrections, respectively.\nobtain convergence.\n\u0003Electronic address: danny.thonig@physics.uu.se\n1S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008), URL http://www.sciencemag.org/cgi/doi/\n10.1126/science.1145799 .\n2J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nature\nNanotech 8, 742 (2013), URL http://www.nature.com/\ndoifinder/10.1038/nnano.2013.176 .\n3A. Fert, V. Cros, and J. 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B\n76, 035107 (2007), URL https://link.aps.org/doi/10.\n1103/PhysRevB.76.035107 ." }, { "title": "2102.12420v1.Finding_the_mechanism_of_wave_energy_flux_damping_in_solar_pores_using_numerical_simulations.pdf", "content": "Astronomy &Astrophysics manuscript no. main ©ESO 2021\nFebruary 25, 2021\nFinding the mechanism of wave energy flux damping in solar pores\nusing numerical simulations\nJ.M. Riedl1, C.A. Gilchrist-Millar2, T. Van Doorsselaere1, D.B. Jess2, and S.D.T. Grant2\n1Centre for mathematical Plasma Astrophysics (CmPA), KU Leuven\nCelestijnenlaan 200B bus 2400, 3001 Leuven, Belgium\ne-mail: juliamaria.riedl@kuleuven.be\n2Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University Belfast\nBelfast, BT7 1NN, U.K.\nReceived; accepted\nABSTRACT\nContext. Solar magnetic pores are, due to their concentrated magnetic fields, suitable guides for magnetoacoustic waves. Recent\nobservations have shown that propagating energy flux in pores is subject to strong damping with height; however, the reason is still\nunclear.\nAims. We investigate possible damping mechanisms numerically to explain the observations.\nMethods. We performed 2D numerical magnetohydrodynamic (MHD) simulations, starting from an equilibrium model of a single\npore inspired by the observed properties. Energy was inserted into the bottom of the domain via di \u000berent vertical drivers with a period\nof 30s. Simulations were performed with both ideal MHD and non-ideal e \u000bects.\nResults. While the analysis of the energy flux for ideal and non-ideal MHD simulations with a plane driver cannot reproduce the\nobserved damping, the numerically predicted damping for a localized driver closely corresponds with the observations. The strong\ndamping in simulations with localized driver was caused by two geometric e \u000bects, geometric spreading due to diverging field lines\nand lateral wave leakage.\nKey words. Waves – Methods: numerical – Sun: photosphere – Sun: oscillations – sunspots – Magnetohydrodynamics (MHD)\n1. Introduction\nSolar pores are macroscopic features resembling small sunspots\nlacking a penumbra, but can also occur as a precursor or remnant\nof sunspots (Garcia de La Rosa 1987; Sobotka 2003; Thomas\n& Weiss 2004). Given their nearly circular symmetry and high\nmagnetic field concentrations, pores act as e \u000ecient wave guides\nfor magnetoacoustic waves, allowing wave flux to enter higher\nregions of the solar atmosphere (Jess et al. 2015) where the en-\nergy can then be dissipated (Grant et al. 2018).\nThe observational evidence of waves in solar pores is vast.\nPhotospheric sausage modes in pores were identified by, e.g.,\nFujimura & Tsuneta (2009) (sausage and /or kink waves), Mor-\nton et al. (2011) (being fast waves according to Moreels et al.\n2013), Dorotovi ˇc et al. (2014) (standing slow and fast waves),\nGrant et al. (2015) (propagating slow surface waves), Keys et al.\n(2018) (surface and body waves), and Gilchrist-Millar et al.\n(2021) (propagating slow surface and body waves, hereafter re-\nferred to as GM21). These authors all found evidence of wave\nperiods of around 3 and /or 5 minutes, indicating the likely role\nof photospheric p-modes as a driver for the waves.\nThe propagation to the chromosphere was studied by\nBalthasar et al. (2000), who confirmed, by using the Vacuum\nTower Telescope (VTT) on Tenerife, the presence of five-minute\noscillations for the magnetic field in the deep photosphere, as\nseen in other observations. Using the Transition Region and\nCoronal Explorer (TRACE) observations, they found a peak at\na period of three minutes in the chromosphere. Stangalini et al.\n(2011) reported longitudinal acoustic waves reaching the chro-mosphere in both three- and five-minute bands. They underline\nthe strong connection between wave transmission and magnetic\nfield geometry, which suggests that for pore models special at-\ntention should be paid to the definition of the magnetic field, as\nalso suggested by Jess et al. (2013).\nHowever, how far waves in solar pores propagate into higher\nlayers of the solar atmosphere is still unclear. Khomenko & Col-\nlados (2006) conducted numerical simulations of waves in a\nsmall sunspot; they used a localized wave source to study wave\npropagation, refraction, and mode conversion. They found that\ndue to the vertical and horizontal stratification of the Alfvén\nspeed, (low \f) fast waves are refracted in the chromosphere back\ndown to the photosphere, while slow modes continue propagat-\ning up. Recent simulations of a chromospheric resonance layer\nabove a sunspot done by Felipe et al. (2020) show that actual\nwave propagation only takes place between the photosphere and\nchromosphere. A chromospheric resonance layer was previously\nalso simulated by Botha et al. (2011) and observed by Jess et al.\n(2020). On the other hand, Riedl et al. (2019) showed in concen-\ntrated flux tubes that plane waves are converted to tube (sausage\nand kink) waves that are able to propagate to the corona since\nthe tube structure greatly a \u000bects the wave propagation (Cally &\nKhomenko 2019; Khomenko & Cally 2019).\nGrant et al. (2015), and more recently GM21, measured wave\nenergy throughout the lower atmosphere of solar pores, and in-\ndeed report significant energy flux damping as a function of\nheight. Analytic calculations of Yu et al. (2017) show that the\nobserved damping could at least be partly explained by reso-\nnant damping of slow sausage waves. Although they find that\nArticle number, page 1 of 12arXiv:2102.12420v1 [astro-ph.SR] 24 Feb 2021A&A proofs: manuscript no. main\nthis damping mechanism is stronger than previously expected,\nthe numerical studies of Chen et al. (2018), validated by ana-\nlytic calculations of Geeraerts et al. (2020), show that damping\ndue to electrical resisitvity is much more potent than that due\nto resonant absorption. However, this alone is not enough to ac-\ncount for the damping. Flux could also be lost due to leaky tube\nwaves (Cally 1986). Leaking waves had already been observed\nby Stangalini et al. (2011) and Morton et al. (2012). Grant et al.\n(2015) mentioned that part of the waves in their observations\nare reflected, which fits the simulations of Khomenko & Colla-\ndos (2006), and that mode conversion (Cally 2001; Bogdan et al.\n2003) might play a role. Frequency dependent damping for slow\nmagnetoacousic waves in sunspot umbrae is discussed by Kr-\nishna Prasad et al. (2017), who find that higher frequencies are\ndamped more strongly. The authors suspect this behavior occurs\ndue to radiative and /or conductive losses.\nAnother important factor to consider is the cuto \u000bfrequency\npresent in stratified media (Lamb 1909). Acoustic waves with\nlower frequencies than the cuto \u000bfrequency cannot propagate,\nbut are evanescent standing waves. This e \u000bect can be used to\ndetermine the cuto \u000bfrequency of the solar atmosphere (Felipe\net al. 2018), which indicates that five-minute waves like those\nobserved by GM21 should be below the cuto \u000b. However, the\nphase lag and propagation speed between di \u000berent heights sug-\ngest that the observed waves in GM21 are indeed propagat-\ning as evanescent waves should not show any phase di \u000berences\n(Carlsson & Stein 1997). On the other hand, the picture is not\ncompletely black and white. Centeno et al. (2006) summarized\nthe e \u000bects of the cuto \u000bfrequency; when radiative losses are\ntaken into account, they find that there is no clear value for\nthe cuto \u000bfrequency, but a transition between mainly evanescent\nand mainly propagating waves. Therefore, it is possible that the\nwaves in GM21 are partly subject to the cuto \u000b, which could ac-\ncount for at least part of the observed damping. For the sake of\nthis study, however, we assume the waves to be 100% propagat-\ning.\nIn this paper we aim to expand our understanding of the\ndamping mechanisms in solar pores by explaining the observed\ndamping with simple two-dimensional (2D) numerical simula-\ntions, using a model inspired by the observational parameters\nobtained by GM21 for their pore 3. We insert propagating waves\nat the bottom of the domain with a vertical velocity driver with\na frequency above the cuto \u000bfrequency, and study the result-\ning wave energy flux with height in comparison with the data\nfrom GM21 for di \u000berent setups. In Section 2 we briefly reiter-\nate the most important points of GM21 before introducing the\nmodel, the numerical setup, and the approach for calculating the\nwave energy flux. The results, distinguished by driver location,\nare presented in Section 3 and thoroughly discussed in Section\n4. A short discussion about the case of a driver with frequency\nlower than the cuto \u000bfrequency is presented in Appendix A.\n2. Methods\n2.1. Observations\nThe model developed in this work is inspired by the observations\ndetailed in GM21, who utilized data obtained by the Facility In-\nfrared Spectropolarimeter (FIRS; Jaeggli et al. 2010) based at\nthe National Science Foundation’s Dunn Solar Telescope (DST),\nSacramento Peak, New Mexico. The FIRS data consist of sit-\nand-stare slit-based spectropolarimetric observations of the de-\ncaying active region NOAA 12564, which was captured between\n14:09 - 15:59 UT on 2016 July 12 in the Si I 10827 Å spectral\nFig. 1. Energy flux across all five observed pores as a function of height.\nThe color scale is logarithmic. Pore boundaries are shown as white\ndashed lines. The green solid line shows the inclination angle of the\nmagnetic field. From GM21.\nline. The observations acquired contain a set of five solar pores\nthat were positioned along a unique straight-line configuration.\nTo cover all pores in a single FIRS exposure, the DST coude ta-\nble was rotated so the spectrograph slit passed through the center\nof each photospheric pore boundary.\nAn examination of the photospheric Si I 10827 Å line bisec-\ntor velocities showed periods on the order of five minutes across\nall pore structures. Through spectropolarimetric inversions us-\ning the Stokes Inversion based on Response functions (SIR; Ruiz\nCobo & del Toro Iniesta 1992) code, the local plasma densities,\nmagnetic field strengths, and temperatures were deduced as a\nfunction of atmospheric height spanning the range 0 – 500 km.\nThe central pore (pore 3 in GM21) exhibited the best signal-to-\nnoise ratio, and so was selected for comparison with the present\ntheoretical work.\nFor pore 3 documented by GM21, the magnetic fields were\nfound to be close to vertical toward the pore center, with field\nstrengths of 2400 G and 1000 G at atmospheric heights of 0\nkm and 500 km, respectively. Temperatures ranged from 5000\nK to 3500 K and densities spanned from 8.5x10\u00004kg m\u00003to\n9.8x10\u00006kg m\u00003across the same height range. Parameters de-\nrived from the inversions were combined with mean square ve-\nlocities to calculate energy flux estimates as a function of at-\nmospheric height (Equation 12). The energy flux across all five\npores as a function of height is displayed in Figure 1. Pore 3 was\nfound to exhibit considerable energy damping with an average\nenergy flux on the order of 25 kW m\u00002at an atmospheric height\nof 100 km, dropping to 1.5 kW m\u00002at 500 km. The damping\nmechanisms producing this drop in energy flux remain elusive.\nIn addition, an increase in energy flux toward the boundaries of\npore 3 indicated the presence of surface mode waves.\n2.2. Model\nIn order to investigate the wave damping in solar pores with\nnumerical simulations, we first need to create a 2D gravita-\ntionally stratified magnetohydrostatic (MHS) equilibrium atmo-\nsphere that resembles the observational data. For a MHS equlib-\nArticle number, page 2 of 12J.M. Riedl et al.: Finding the mechanism of wave energy flux damping in solar pores using numerical simulations\nrium the following condition must be fulfilled\nrp\u00001\n\u00160(r\u0002B)\u0002B\u0000\u001ag=0; (1)\nwhere pis the gas pressure, \u001ais the density, Bis the magnetic\nfield,\u00160is the magnetic permeability, and gis the gravitational\nacceleration.\nWe start by choosing the magnetic field in the z-direction\nBz(x;z)=a(z)\"\narctan x+r(z)\ns(z)!\n\u0000arctan x\u0000r(z)\ns(z)!#\n+b(z);(2)\nwhich results in a bundle of strong vertical magnetic field of a(z)\ninside the pore above the background field b(z) outside the pore,\nresembling the observations. The parameter r(z) describes the ra-\ndius of the pore, while s(z) is the smoothness parameter, which\ndefines the thickness of the transition between pore and back-\nground. For the sake of simplicity, the written dependence on\nthe vertical coordinate ( z) is omitted from now on for these four\nparameters.\nThe parameters defining Equation 2 are\na=0:33Gaxis [T];r=2\u0002105\npGaxis[m];\nb=0:05Gside [T];s= 0:1r[m]; (3)\nwith exponential functions approximating the observed magnetic\nfield strength of pore 3 from GM21 at the axis of the pore Gaxis=\n0:1 exp(\u0000z=300000) +0:07 [T] and the side of the pore Gside=\n0:1 exp(\u0000z=300000) +0:02 [T].\nBecause div B=0, we know that in 2D\n@Bx\n@x=\u0000@Bz\n@z: (4)\nTherefore,\nBx(x;z)=\u0000Z@Bz\n@zdx+h(z)\n=\u0000da\ndz\"s\n2ln (x\u0000r)2+s2\n(x+r)2+s2!\n+(x+r) arctan\u0012x+r\ns\u0013\n\u0000(x\u0000r) arctan\u0012x\u0000r\ns\u0013\u0015\n\u0000a\"1\n2ds\ndzln (x\u0000r)2+s2\n(x+r)2+s2!\n+dr\ndzarctan\u0012x+r\ns\u0013\n+dr\ndzarctan\u0012x\u0000r\ns\u0013#\n\u0000db\ndzx;(5)\nwhere we assume that h(z)=0 because then the solution is anti-\nsymmetric around x=0.\nIn order to get a solution that fulfills both components of\nEquation 1,\n@2p\n@x@z=@2p\n@z@x(6)\nmust be true. By di \u000berentiating the x-component of Equation\n1 with respect to zand the z-component with respect to x, and\ncombining the resulting derivatives with Equation 6, we find a\nconstraint for the density,\n@\u001a\n@x=1\n\u00160g\"@Bx\n@x @Bz\n@x\u0000@Bx\n@z!\n+Bx @2Bz\n@x2\u0000@2Bx\n@z@x!\n+@Bz\n@z @Bz\n@x\u0000@Bx\n@z!\n+Bz @2Bz\n@x@z\u0000@2Bx\n@z2!#\n;(7)assuming g=[0;\u0000g] with g=274 m /s. The density can then be\nobtained with\n\u001a(x;z)=Z@\u001a\n@xdx+f(z): (8)\nThe function f(z) is of great importance here as it defines the\ngravitational stratification of the density. We therefore set f(z) to\nbe equal to the average density obtained from the observations\nof GM21 for their pore 3. Since @\u001a=@ xalso has a dependence on\nz, we add a small constant to \u001ato ensure its non-negativity. Due\nto the complexity of Equation 8 it is solved numerically.\nFrom the second component of Equation 1, the pressure can\nbe calculated with\np(x;z)=Z@p\n@zdz+j(x): (9)\nAs long as the pressure is symmetric around the pore axis at\nx=0, there is no need to add a function j(x). However, we add\na constant to ensure a positive pressure. This equation is also\nsolved numerically.\nTheoretically, the described model is in MHS equilibrium.\nHowever, numerical calculations as used in the solution of the\nmodel and in the simulation code itself are imperfect, often re-\nsulting in somewhat unstable behavior, especially when gravity\nis involved. Therefore, using the boundary conditions described\nin Section 2.3, we simulate the model without driver for a phys-\nical time of 1300 seconds to let it settle down. After this time,\nthere are no significant changes to density, magnetic field, or\npressure on a timescale compared to a few driver periods. This\nslightly relaxed atmosphere is then used as the initial condition\nfor our simulations. We note, however, that even after the slight\nrelaxation there are still significant velocities within the domain,\nmeaning that the resulting model atmosphere has not completely\nsettled to a MHS equilibrium.\nThe top panel of Figure 2 shows the vertical magnetic field\ncomponent of the initial atmosphere, with field lines shown in\norange. Due to the symmetry of the problem, only half of the\npore is included in our model, with the pore axis being located at\nx=0. The pore itself is located on the left side of the plot, where\nthe magnetic field is strong and mainly vertical. The deviation of\nthe horizontal profile from the arctan-shape of Equation 2 oc-\ncurs because of the equilibration process. The comparison of the\nmodel with the observations of GM21 (Figure 2 bottom) shows\ngreat similarity. It should be noted, however, that the horizontal\nextent of our model pore (FWHM radius \u00190:44 Mm) is smaller\nthan the pores in the observations (radius \u00192:5 Mm). Even so,\nwhen comparing the magnetic field inclination of the model at-\nmosphere with the field inclination of pore 3 from GM21 in the\ndirection perpendicular to the slit (thus perpendicular to the line\nof five pores), while taking the di \u000berent radii into account, the\nfield inclinations also coincide quite well (see Figure 3). The\nplasma-\fin our model is higher than unity everywhere, with val-\nues ranging from 2 to 6.5 inside the pore and higher values up to\n40 and higher outside.\nSimilarly, Figure 4 shows the density of the settled model\nand the comparison to observations, where the density struc-\ntures seen appear during the equilibration process. It is imme-\ndiately apparent, that the model density is much less stratified\nwith height than the observations, even though we added the ob-\nservational density as stratification in Equation 8. This is caused\nby the e \u000bect of@\u001a=@ xcalculated by Equation 7 already having\na dependence on z, which in total decreases the stratification. In\naddition, it is also slightly decreased when the atmosphere is al-\nlowed to settle down. However, it should be noted that for the\nArticle number, page 3 of 12A&A proofs: manuscript no. main\nFig. 2. Top: Vertical component of the magnetic field of the settled\nmodel atmosphere. Orange lines depict the magnetic magnetic field\nlines. Bottom: Comparison of the model atmosphere with the observa-\ntions from GM21 of pore 3. The maximum observational value within\nthe pore is shown by obs. peak , while the horizontal average across the\nwhole pore is shown by obs. average . The green lines show the model\nvalues for the indicated lines in the top figure (line 1 at pore axis).\nobservations in GM21, the density is not a direct output of the\ninversions, but is instead determined through solving equations\nof state using inferred inversion outputs, under the assumption\nof hydrostatic (HS) equilibrium. This simplifying assumption is\nproblematic in strong magnetic fields as it ignores the Lorentz\nforce, thus providing notable uncertainties on the densities input\ninto the model, of up to an order of magnitude (Borrero et al.\n2019).\nNonetheless, the density values from GM21 are still consis-\ntent with those from semi-empirical models like that of Maltby\net al. (1986), who considered a magnetized atmosphere at the\ncenter of a sunspot umbra. They also assumed a HS equilibrium;\nhowever, this assumption is valid for the center of an axially\nsymmetric sunspot as the magnetic terms in Equation 1 vanish.\nTherefore, we have to assume that the observational values of\nthe density are more reliable than the model values.\nThe smaller pore radius and less stratified density in our\nmodel compared to the observations are due to compromises be-\ning made when solving Equation 1. Once a non-force-free mag-\nnetic field is chosen, the density or pressure cannot be freely\nchosen, but only manipulated through the addition of integration\nconstants, as can be seen in Equations 8 and 9. Therefore, in\norder to obtain a stable model for our simulations, certain con-\ncessions have to be made. In addition, due to the same reasons,\nour model also results in a plasma- \f >1 inside the pore, as op-\nposed to a low \ffound by GM21 within the pores. The impact\nof the di \u000berences between observations and theoretical model on\nour results is discussed in Section 4.1.\nFig. 3. Comparison of the magnetic field inclination between model\n(red) and observations for pore 3 of GM21 (black) as a function of hor-\nizontal distance normalized to the pore radius. The pore radius for the\nmodel was assumed to be 0.44 Mm, while the radius of the observed\npore is 2.5 Mm. The observational values are taken along the line per-\npendicular to the slit. Model values are mirrored around x=0 and are\nshown for the bottom (solid line), middle (dashed line), and top part\n(dotted line) of the model. The vertical dashed gray lines show the bor-\nder of the pores at x=1.\nFig. 4. Top: Logarithm of the density of the settled model atmosphere.\nOrange lines depict the magnetic magnetic field lines. Bottom: Compar-\nison of the model atmosphere with the observations from GM21 of pore\n3. The maximum observational value within the pore is shown by obs.\npeak , while the horizontal average across the whole pore is shown by\nobs. average . The green lines show the model values for the indicated\nlines in the top figure (line 1 at pore axis).\n2.3. Numerical setup\nAll our simulations are conducted using the PLUTO code\n(Mignone et al. 2007), which solves the magnetohydrodynamic\n(MHD) equations when using the respective module. Fluxes are\nArticle number, page 4 of 12J.M. Riedl et al.: Finding the mechanism of wave energy flux damping in solar pores using numerical simulations\ncomputed using a linearized Roe Riemann solver, while the time\nstep is advanced using an unsplit second-order accurate charac-\nteristic tracing method, which is less dissipative than multi-step\nalgorithms. To deal with the inevitable occurrence of div Bwe\nuse the mixed hyperbolic /parabolic divergence cleaning tech-\nnique of Dedner et al. (2002), which is further discussed in\nMignone et al. (2010). Gravity is added using a body force with\nconstant acceleration toward the negative z-direction.\nKeeping a model atmosphere stable when gravity is included\ncan often prove di \u000ecult and is highly contingent on the bound-\nary conditions at boundaries perpendicular to the gravitational\nacceleration. In our case it was not possible to set fully open\nboundary conditions at the top boundary. We therefore expand\nthe model atmosphere at the top to add a thick high-viscosity\nlayer to absorb all outgoing waves, e \u000bectively having an open\nboundary. We use the same boundary condition for the right\nboundary. The viscosity is treated with an explicit time integra-\ntion. Including the viscous layers the domain ranges from 0 to\n3 Mm in the x-direction and from -0.095 to 0.795 Mm in the z-\ndirection with 1000 \u0002297 cells, leading to a spatial resolution of\n3 km in both directions. Excluding the viscous layers, a physical\ndomain remains ranging from 0 to 2.68 Mm with 894 cells in the\nx-direction and from -0.095 to 0.475 Mm with 191 cells in the z-\ndirection. Only this physical domain is used for the analysis and\nfigures. The height of 0 Mm is defined as the bottom of the pho-\ntosphere. After settling the calculated model from Section 2.2\nfor 1300 s (defined as t=0 s in the plots), the simulations are\nrun for an additional 200 s.\nDue to the symmetry of the system our model only includes\nhalf of a solar pore, with the pore axis being located at the left\nboundary. Thus, the boundary conditions there are set to be re-\nflective. At the bottom boundary we set pressure, density, and\nmagnetic field to fixed values that fulfill the equations presented\nin Section 2.2 for the initial model before the equilibration. The\nhorizontal velocity is set to 0. For simulations without driver, the\nvertical velocity is set to 0 as well. When a driver is included the\nvertical velocity is set according to\nvz;driver=Asin 2\u0019\nTt!\n; (10)\nwith the amplitude A=160 m /s and the period T=30 s. Since\nthe driver purely perturbs the velocity, some of the driver energy\nimmediately flows into pressure and density perturbations. Be-\ntween the ghost cells (additional cells outside the computation\ndomain to enable numerical integration) including the driver and\nthe first cell of the domain, the root mean square of the velocity\nperturbation is therefore reduced to levels observed by GM21 at\nthe bottom of the pores of about 50 m /s. This short period for the\ndriver was chosen because a typical p-mode period of 300 s is\nclose to the cuto \u000bperiod in our model, leading to the formation\nof standing waves due to reflections. However, we want to in-\nvestigate propagating waves and their damping. In addition, for\nlonger periods the wavelength would increase accordingly, caus-\ning the resulting waves to not fit into the domain. For the sake\nof completeness, we also did simulations with a low-frequency\ndriver below the cuto \u000bfrequency, and we show a crude analysis\nin Appendix A.\nFor some of our simulations we include non-ideal e \u000bects like\nviscosity, resistivity, and thermal conduction. Those e \u000bects were\nadded using explicit time integration, and for expected values in\nthe photosphere ( ReandRmtaken from Ossendrijver 2003). In\nthe case of the simulations with viscosity, where viscosity was\nalso present in the physical domain, simulations were done with\nexaggerated values for the viscous shear coe \u000ecient.2.4. Wave energy flux\nThe energy flux can be calculated as (e.g., Goedbloed & Poedts\n2004)\nE=\u00001\n\u00160(v\u0002B)\u0002B+ \u001av2\n2+\u001a\b +\r\n\r\u00001p!\nv; (11)\nwhere \b =\u0000gz+const:is the gravitational potential. The left\nterm of Equation 11 is the Poynting flux, which is the magnetic\ncomponent of the energy flux, whereas the other terms describe\nthe hydrodynamic component. Since in our model \f >1 every-\nwhere and the driver mainly excites acoustic waves, the hydro-\ndynamic component is dominant in our simulations.\nThere are still velocities up to nearly 2 km /s within the whole\nphysical domain or up to \u0018350 m /s within the pore after settling\nthe atmosphere for 1300 s. These velocities are higher than the\ndriver amplitude. Thus, in addition to simulations with a driver,\nwe also conduct simulations without a driver, allowing us to ex-\ntract the e \u000bects caused by the input waves alone. This is done by\nsubtracting all the variables of the simulations without a driver\nfrom the variables of the simulations with a driver, e \u000bectively\ngiving us the perturbed variables\n\u001a0=\u001adriver\u0000\u001anodriver; p0=pdriver\u0000pnodriver;\nv0=vdriver\u0000vnodriver; B0=Bdriver\u0000Bnodriver:\nTo obtain the wave energy flux, these perturbed variables are put\ninto Equation 11, in a process that is similar to linearization.\nIn GM21, on the other hand, the wave energy flux was cal-\nculated as\nE=\u001avghv2i; (12)\nwith vgbeing the group speed and hv2ibeing the mean square\nvelocity. For our simulations, Equations 11 and 12 yield simi-\nlar trends with absolute values in the same order of magnitude.\nUsing Equation 11 facilitates a more detailed analysis, which is\npossible due to the much more detailed knowledge of the data in\nsimulations compared to observations.\n3. Results\nWe conducted a range of simulations, including and removing\nnon-ideal e \u000bects, and applying di \u000bering drivers. Depending on\nthe driver location, the results can be divided into two distinct\ngroups, which are discussed in the following.\n3.1. Driver located at whole bottom boundary\nWe applied the velocity driver described in Equation 10 on the\nwhole bottom boundary, resulting in plane fast waves propagat-\ning upward at approximately the sound speed. A single snapshot\nof the vertical velocity perturbation after two driver periods is\nshown in Figure 5. The amplitude of the waves increases with\nheight, as is expected due to the conservation of energy in a\nstratified plasma. The wave fronts are not completely horizon-\ntal, but have a jagged form at the pore location. This happens\ndue to di \u000bering wave speeds at di \u000berent locations. The vertical\nwave ridges visible at x\u00190:6 Mm and the right boundary, and\nmore pronounced at later times, as seen in the movie of the time\nsequence, are wave fronts of slow waves.\nFigure 6 shows the time-averaged wave energy flux as a func-\ntion of height relative to the first measure point obtained from\nthe observations of GM21 for their pore 3. Both simulation and\nArticle number, page 5 of 12A&A proofs: manuscript no. main\nFig. 5. Snapshot of the vertical velocity perturbation after two periods\nfor the full driver and ideal MHD. The gray lines show magnetic field\nlines. The blue line highlights the field line considered for the analysis\nin Figure 6. The full time sequence is available as a movie online.\nFig. 6. Relative wave energy flux parallel to the magnetic field averaged\nover time as a function of height for the full driver. The solid green line\nshows the energy flux along the pore axis, whereas the dashed purple\nline shows the average flux across the pore up to the field line high-\nlighted in Figure 5. The observational data (black line with symbols)\nare from pore 3 of GM21. All fluxes are normalized to the first obser-\nvational data point.\nobservational data were normalized to the data point at z=0:1\nMm. The time average of the simulation data was taken over the\nfirst period of the propagating wave. The figure shows the en-\nergy flux at the pore axis, where the magnetic field line is vertical\n(green line), and the flux averaged from the pore axis to the loca-\ntion of the field line highlighted in Figure 5 (purple dashed line).\nThe observational data were also averaged in time and across the\npore.\nIt is evident from Figure 6 that there is no indication of wave\ndamping with height in our simulations; instead, the energy flux\neven increases with height, which could be explained by waves\nbeing refracted into the pore, as discussed in Section 4.1. The\nlack of damping is not only the case for ideal MHD, but also\nwhen resistivity, viscosity, or thermal conduction is included.\nEven for exaggerated viscosity no damping is achieved. We as-\nsume this is the case because we are studying a very narrow slab\nof atmosphere of a few hundred kilometers, leaving little time\nfor non-ideal e \u000bects to a \u000bect the waves. Therefore, we fail to re-\nFig. 7. Snapshot of the vertical velocity perturbation after two periods\nfor the localized driver for ideal MHD. The gray lines show magnetic\nfield lines. The red bar below the x-axis indicates the driver location.\nThe blue line highlights the field line considered for the analysis in Fig-\nure 8. The full time sequence is available as a movie online.\nproduce the observed damping with a plain driver located at the\nwhole bottom boundary.\n3.2. Localized driver\nSolar pores are magnetic structures that do not form in the pho-\ntosphere but are already present below the solar surface. As so-\nlar pores are good wave guides, it is valid to assume that only\nthe pore itself may be driven. Numerical simulations (Cameron\net al. 2007) supported by observations (Cho et al. 2013) sug-\ngest that rapid cooling within pores could lead to downflows that\ncollide with the plasma of lower layers to produce rebounding\nupflows, which further motivates the assumption of a localized\ndriver. Moreover, previous simulations (Kato et al. 2016) show\nthat photospheric bu \u000beting by turbulent motions lead to the e \u000e-\ncient excitation of waves. We therefore alter our driver to a step-\nfunction driver that is only present in the inner part of the pore\nvz;driver=8>><>>:Asin\u00102\u0019\nTt\u0011\nx\u00140:2Mm\n0 x>0:2Mm:\nFigure 7 shows a snapshot of the vertical velocity perturba-\ntion after two periods for the step-function driver. In contrast to\nthe respective figure for the full driver, the wave fronts are not\nhorizontal and the maximum amplitude is lower because the at-\nmosphere is only driven at one location (indicated by the red\nbar below the x-axis). The blue line highlights a field line rooted\nslightly outside the driver location at x=0:22 Mm. There are\nclearly waves present beyond this field line, suggesting that the\nwaves do not purely propagate along the magnetic field.\nIf we now study the wave energy flux as a function of height\nfor the simulation with localized driver, as shown in Figure 8\n(left), it is immediately apparent that the energy flux is now\nstrongly damped, in stark contrast to the simulations with the full\ndriver. This sudden drop in wave energy flux with height by just\nchanging the driver location can be explained by two geometric\nmechanisms: geometric spreading and lateral wave leakage.\n3.2.1. Geometric spreading\nThe magnetic field lines in our model diverge with height. There-\nfore, if the waves were perfectly propagating along the field\nArticle number, page 6 of 12J.M. Riedl et al.: Finding the mechanism of wave energy flux damping in solar pores using numerical simulations\nFig. 8. Left: Relative wave energy flux parallel to the magnetic field av-\neraged over time as a function of height for the localized driver. The\nsolid green line shows the energy flux along the pore axis, whereas the\ndashed purple line shows the average flux across the pore up to the field\nline highlighted in Figure 7. The observational data (black line with\nsymbols) are from pore 3 of GM21. All fluxes are normalized to the\nfirst observational data point. Right: Comparison of flux damping in the\nsimulation with localized driver with the e \u000bects of geometric damping.\nThe solid green line shows the same data as in the left plot for compar-\nison. The other lines are explained in the text.\nlines, the flux along a single field line, as well as the average flux\nat each height within the pore, would be expected to drop due\nto the flux being distributed across a wider area with increasing\nheights. The decrease in flux with height due to this mechanism\nis proportional to 1 =Rin 2D geometry, where Ris the distance\nbetween the pore axis and a specific field line. Such a curve is\nshown in Figure 8 (right, dotted red line) for the field line high-\nlighted in Figure 7. Since this curve drops substantially less with\nheight than the wave flux, there must be another mechanism with\napproximately equal significance.\nIn addition, if only geometric spreading caused the damp-\ning, the wave flux parallel to the magnetic field integrated across\nthe pore should be constant with height because the same total\namount of flux would be contained inside the pore at all heights.\nThis is not the case, which can be seen with the dash-dotted or-\nange line in Figure 8. Therefore, flux must escape from the pore\nthrough its edges.\n3.2.2. Lateral leakage\nIn our simulations with a localized driver, we observe waves\npropagating out of the solar pore, which decreases the flux in-\nside the pore. This is the case because magnetoacoustic waves\ncan propagate at an inclined angle with respect to the magnetic\nfield. In a homogeneous plasma, the phase speed of fast and slow\nmagnetoacoustic waves is (e.g., Goedbloed et al. 2019)\nvfa=sl(\u0012)=q\nv2s+v2\nAp\n226666641\u00060BBBB@1\u00004v2\nccos2\u0012\nv2s+v2\nA1CCCCA1=237777751=2\n; (13)\nwhere vsis the sound speed, vAthe Alfvén speed, vc=vAvs=(v2\nA+\nv2\ns)1=2the cusp speed, and \u0012the angle between the propaga-\ntion direction and the magnetic field. The positive (negative)\nsign is for the calculation of the phase speed of the fast (slow)\nwave. In a plasma where vs>vA(approximately \f > 1), thephase speed of fast waves takes the shape of a flattened quasi-\ncircle with vfa(\u0012=0;\u0019)=vsin the magnetic field direction\nandvfa(\u0012=\u0019=2;3\u0019=2)=(v2\nA+v2\ns)1=2perpendicular to it. On\nthe other hand, slow waves take the shape of double quasicir-\ncles with vsl(\u0012=0;\u0019)=vAin the magnetic field direction\nandvsl(\u0012=\u0019=2;3\u0019=2)=0 perpendicular to the magnetic field.\nTherefore, also for slow waves there is still a non-zero phase\nspeed for all directions except exactly perpendicular to the mag-\nnetic field. This e \u000bect was previously used by Nakariakov & Zi-\nmovets (2011) to explain flare ribbon propagation.\nAssuming local homogeneity and utilizing Equation 13, we\ncan apply the Huygens-Fresnel principle to theoretically predict\nthe locations of fast and slow wave fronts. In order to do this we\nassume that the wave originates from a point source. In this point\nthe phase speed in all directions is calculated, supplying us with\ninformation of the wave front location in the next snapshot. For\nall subsequent snapshots we calculate the phase speed in each\npoint of the previous wave front. The next fast (slow) wave front\nis then the outer edge of all fast-wave quasicircles (slow-wave\ndouble quasicircles).\nWe let our theoretical wave fronts for both fast and slow\nwaves propagate from two point sources at the bottom of the do-\nmain: one at the pore axis at ( x;z)=(0;\u00000:095) Mm and one at\nthe edge of the driver at ( x;z)=(0:2;\u00000:095) Mm, starting from\nt=0 s. Figure 9 shows the saturated wave energy flux parallel\n(left) and perpendicular (right) to the magnetic field at t=124\ns. The theoretical fast wave fronts (solid lines) and slow wave\nfronts (dashed lines) are overplotted in green. By examining the\ntime sequence, which is available as movies online, it is evident\nthat there are many waves propagating with the exact same shape\nand speed as the theoretical fast wave fronts outside the pore. We\ntherefore identify those waves as fast waves. They can be seen\nmost clearly in Figure 9 between x\u00190:7 Mm and x\u00191:7 Mm.\nThere are also waves propagating out of the pore with the\nsame shape and speed as the theoretical slow wave fronts. We\ntherefore identify these waves as slow waves. In Figure 9 they\ncan be predominantly seen in the perpendicular flux compo-\nnent between the two dashed lines. When observing the time\nsequence for the parallel component and focusing on that region\nan interaction between fast and slow waves can be seen. How-\never, at this point the flux has already exited the pore, and we\nthus do not discuss this further.\nWhen observing the full time sequence of the movies of Fig-\nure 9, the theoretical wave fronts eventually develop a dip close\nto the border of the pore (e.g., snapshot 35). This is especially\nprominent for the fast waves, and is also seen in the simulation\ndata. The reason for this dip is the density structure at that lo-\ncation, which can be seen in Figure 4. The di \u000berence in density\nleads to a di \u000berence in phase speed.\nAlthough there are clearly waves leaking out of the pore,\nmost of the flux is contained within the pore, following the\nmagnetic field lines. To estimate the e \u000bect of lateral leakage on\nthe damping of energy flux with height, we compare the time-\nintegrated total flux present along the field line highlighted in\nblue in Figures 7 and 9 (which is the total flux lost laterally)\nwith the time-integrated total flux inside the pore at the bottom\nof the domain (which is the total incoming flux). The time inte-\ngration of the flux is calculated for the first wave front over one\nperiod Tfor all locations\nEt(x;z)=Zt2(z)\nt1(z)E(x;z;t)dt; (14)\nArticle number, page 7 of 12A&A proofs: manuscript no. main\nFig. 9. Snapshot of the wave energy flux parallel (left) and perpendicular (right) with respect to the magnetic field. The color range is saturated.\nThe solid (dashed) green lines show the first theoretical wave fronts of the fast (slow) waves; the gray lines show the magnetic field lines. The red\nbars below the x-axis indicate the driver location. The blue lines highlight the field line considered for the analysis in Figure 8. Movies of the full\ntime sequence are available online.\nwhere Eis the wave energy flux according to Equation 11 and\nt1(z) and t2(z) is the time of the beginning and end, respectively,\nof the first wave front at height z, with t2(z)=t1(z)+T.\nFor the calculation of the escaped flux we chose a field line\nrooted slightly outside the driver region in order to be sure that\nall flux at that location has exited the pore. We then integrate\nthe time-integrated flux components along this field line from\nthe root of the field line at the bottom of the domain until height\nz, before calculating the time-integrated total flux. The total es-\ncaped flux is then\nEt;esc(z)=0BBBBB@\"Zl(z)\n0Et;k(x;z)dl#2\n+\"Zl(z)\n0Et;?(x;z)dl#21CCCCCA1=2\n;(15)\nwhere l(z) is the length of the field line at height zand the in-\ntegrals of the fluxes are taken along the field line. Here Et;kand\nEt;?are the parallel and perpendicular components of the time-\nintegrated energy flux (Equation 14) with respect to the magnetic\nfield (and therefore the field line), with Et;k?Et;?. The integra-\ntion is done before calculating the absolute value to allow flux\nwith opposing signs to cancel out.\nSimilarly, the total flux contained in the pore at the bottom\nof the domain is calculated by\nEt;bot=0BBBBB@\"Zxl\n0Et;x(x;z=zbot)dx#2\n+\"Zxl\n0Et;z(x;z=zbot)dx#21CCCCCA1=2\n;\n(16)\nwhere xl=0:22 Mm is the x-position of the field line root, zbot=\n\u00000:095 Mm is the z-location of the bottom of the domain, and\nthe integrals are taken horizontally across the pore at the bottom\nof the domain. Here Et;xandEt;zare the x- and z-components of\nthe time-integrated energy flux, with Et;x?Et;z.\nThe e \u000bect of wave leakage on the damping is then estimated\nby\nedamp(z)=1\u0000Et;esc(z)\nEt;bot: (17)\nThe result of Equation 17 is shown in Figure 8 (right) as the\nblue dashed line. There is a significant di \u000berence between thisline and the line showing missing flux when only considering\ngeometric spreading (orange dash-dotted line). Both methods are\nestimates, and we expect the actual e \u000bect of lateral wave leakage\nto lie between these lines.\n4. Conclusions and discussion\nWe created a MHS model close to equilibrium, which was in-\nspired by observational data of a solar pore (GM21) and investi-\ngated possible damping mechanisms by driving the model with\na vertical velocity perturbation at the bottom of the domain. We\nfound that, even if viscosity, resistivity, or thermal conduction\nare included, the strong damping from the observations could\nnot be reproduced at all by using a driver that covers the whole\nbottom boundary. When switching to a localized driver, however,\nthe results show strong damping in our simulations. This damp-\ning occurs because of a) geometric spreading, where the flux is\nspread over a wider area due to diverging field lines and b) lat-\neral wave leakage, where waves leave the pore. Therefore, even\nif only considering classic wave e \u000bects, significant damping can\nbe achieved. Wave leakage at the edge of a solar pore was indeed\nalready observed by Stangalini et al. (2011).\n4.1. Effects of differences between observed pore and model\n& comparison of simulations to observations\nIt was mentioned in Section 2.2 there are di \u000berences between\nour model and the observational data example pore (GM21, pore\n3). The di \u000berences in density and pressure profiles mainly lead\nto di\u000berences in characteristic wave speeds. This does not a \u000bect\nthe damping due to geometric spreading, as this damping mech-\nanism is only dependent on the magnetic field structure, which is\nsimilar to the observations, with nearly vertical inclination inside\nthe pore and nearly horizontal field lines outside.\nAn important point we have to note, however, is the sound\nspeed profile, as shown in Figure 10. In our model, the sound\nspeed generally increases with height, whereas it is the oppo-\nsite for the observations. In addition, there is a strong horizontal\nstructuring, with lower speeds at the center and the border of the\npore. From applying Snell’s refraction law, as also discussed in\nArticle number, page 8 of 12J.M. Riedl et al.: Finding the mechanism of wave energy flux damping in solar pores using numerical simulations\nFig. 10. Sound speed of the initial atmosphere. Gray lines show mag-\nnetic field lines. The red bar below the x-axis indicates the driver loca-\ntion for simulations with localized driver. The blue line highlights the\nfield line considered for the analysis in Figure 8. Contours for the sound\nspeed are shown in thick black lines.\nthe context of sunspots by Khomenko & Collados (2006), we\nknow that waves travelling into a medium with higher phase\nspeed refract away from the line perpendicular to the constant-\nphase-speed-line. If in our simulations the fast (acoustic) waves\nare propagating along the diverging field lines, they are refracted\naway from the pore. Therefore, should the fast lateral waves\nin our simulations exclusively occur because of refraction, we\nwould not expect acoustic waves escaping laterally for the ob-\nservations of pores like in GM21. The e \u000bect of lateral leaking\nfor magnetic waves should be the same, however, as the Alfvén\nspeed profile in our simulations is similar to the observations.\nEvidence of at least some fast wave refraction occurring in\nour simulations is seen in the amplitude of the wave energy flux,\nwhere the amplitude is increased at the center and the bound-\nary of the pore compared to the region in between. Those re-\ngions coincide with regions of lower sound speeds and waves\nare therefore refracted toward those regions. The increased am-\nplitude in the pore boundary therefore does not occur because\nof sausage surface waves. Since the sound speed is higher at the\nlocation just outside the pore, waves that are located outside the\npore would be refracted into the pore. This could be one of the\nreasons why the energy flux profile increases with height for the\nfull driver (Figure 6), as there are ample waves present outside\nthe pore to be refracted. In addition, fast wave energy flux that\nescaped from the pore was eventually refracted down toward the\nbottom of the domain in the simulations with localized driver.\nThis can be seen in Figure 9 (right), where the perpendicular flux\ncomponent for the fast waves outside the pore is mainly positive\nand therefore directed downward, considering the nearly hori-\nzontal field lines. This refraction of fast waves is similar to what\nwas found by Khomenko & Collados (2006).\nThe observations of pore 3 GM21 also show higher energy\nflux concentrations at the pore boundaries. Contrary to the events\nin our simulations, it was found that these flux concentrations\nare due to surface sausage modes. This could possibly promote\nadditional lateral wave leakage as flux already present at the edge\nof the pore could more easily escape.\nA crucial di \u000berence between observations and simulations\nis that due to the cadence of the instruments, GM21 were only\nable to investigate slow waves, whereas in this paper we have acombination of slow and fast waves. By splitting the energy flux\ninto magnetic (Poynting) and hydrostatic contributions, slow and\nfast waves could have been studied separately. However, most\nof the slow waves in our simulations with localized driver stem\nfrom the sharp edge of the step-function, causing most of the\nslow waves being concentrated just inside and atop the consid-\nered field line marking the boundary of the pore in our analysis\n(blue highlighted field line in e.g. Figure 9), with little slow wave\nflux inside the rest of the pore. We therefore only considered the\ntotal flux for our analysis, as our estimate for the influence of\ndamping due to lateral wave leakage (Equation 17) would not\nhave worked for slow waves alone. On the other hand, there was\nno need to exclude slow waves from the same analysis as the\nmagnitude of the Poynting flux is about three orders of magni-\ntude smaller than the hydrostatic component.\nIn our model \f > 1 everywhere, whereas \f < 1 is ex-\npected inside the pores according to the observations. This ba-\nsically means that the fast waves inside the pore in this paper\ncorrespond to the slow waves observed in GM21 as they both\nhave predominantly acoustic properties. While slow waves are\nallowed to propagate in all directions except directly perpendic-\nular to the magnetic field (see Equation 13), their phase speed\nas a function of angle to the magnetic field has a di \u000berent shape\nthan for fast waves. While, according to our results, slow waves\nalso leave the pore, it is possible that due to this di \u000berent shape\nfewer low\fslow acoustic waves (observations) would leave the\npore than fast acoustic waves in our simulations. However, the\nslow acoustic waves in observations are still comparable to the\nfast acoustic waves simulated here. It is therefore reasonable to\nassume that within the pore the slow wave energy flux would\nbe dominant over the fast wave energy flux if our model atmo-\nsphere had \f < 1 in that region. Applying this assumption to\nthe real world highlights one of the di \u000eculties in observing fast\nmodes: the fast wave flux would be overshadowed by the slow\nwave flux. In addition, having a low plasma- \finside the pore\ninevitably leads to a \f=1 (or vs=vA) layer at the border of\nthe pore with high \foutside. In these layers waves are strongly\nsubjected to mode conversion (Cally 2005, 2006; Schunker &\nCally 2006; Hansen et al. 2015). Whether these mode conver-\nsions increase the amount of energy flux escaping from the pore\nor have a channeling e \u000bect in the pore will have to be determined\nin future work.\n4.2. On other limitations of the current study\nIn this work, we did not account for any radiative losses. Ac-\ncording to Carlsson & Stein (2002), acoustic waves in the pho-\ntosphere are much more damped at higher frequencies, meaning\nthat the impact of this damping mechanism in our simulations\nwould be larger than for the observations of GM21, who observe\nlonger periods.\nOur simulations were done on a 2D Cartesian grid. In 2D, the\n“area” inside the pore at each height is just a 1D line. Therefore,\nwe estimated the damping due to geometric spreading to be pro-\nportional to 1 =R(z) with R(z) the distance between the pore axis\nand a field line. In 3D, however, we expect the wave energy flux\ndue to this e \u000bect to decrease with 1 =R(z)2. Estimating the change\nin e\u000bect from 2D to 3D for wave leakage is more di \u000ecult. We\nassume that it is dependent on the ratio of the area inside the pore\nto the area that has been available for flux to escape, which is the\nmantle of the pore up to a specific height. This ratio is R(z)=l(z) in\n2D and R(z)2\u0019=(2R(z)\u0019l(z)) in 3D, with l(z) describing the length\nof the considered field line from the root up to a certain height z.\nTherefore, the dependence R(z)=l(z) can also be assumed for 3D.\nArticle number, page 9 of 12A&A proofs: manuscript no. main\nThe increase in e \u000eciency of geometric spreading for 3D could\naccount for the di \u000berence between the damping in our simula-\ntions and the observed damping. We note that by assuming a 2D\ngeometry in our simulations we have excluded the possibility of\nAlfvén waves.\n4.3. Concluding remarks and future work\nAs discussed above, there are both slow and fast waves present\nin our simulations. The slow waves are predominantly excited\nat the edge of the step-function driver. Simulations using a\nGaussian-shaped driver instead show that slow waves are excited\nat the flank of the Gaussian, mostly at the steepest location. This\nleads to the conclusion that any kind of localized vertical driver\nwould excite both slow and fast waves. Therefore, we also expect\nboth kinds of waves to be present in the photosphere at all times.\nWhile slow modes have been observed in the photosphere many\ntimes, temporal resolution has so far limited similar studies for\nfast waves. However, future instruments on the Daniel K. Inouye\nSolar Telescope (DKIST), European Solar Telescope (EST), and\nNational Large Solar Telescope (NLST) might provide the ca-\ndence needed to observe fast waves propagating at an inclined\nangle with respect to the magnetic field.\nObserving the leaking waves as seen in our simulations\nmight be challenging as the magnitude of the vertical (line-of-\nsight) velocity perturbations is roughly a factor of ten lower than\nthe perturbations inside the pore. However, since the wave fronts\nof the leaking waves are inclined from the vertical (as seen in\nFigure 9), an observer from above would see the integrated ef-\nfects of waves in di \u000berent phases (i.e., positive and negative ve-\nlocities within the same pixel). This would lead to spectral line\nbroadening. The possibility to observe the leaking waves us-\ning this e \u000bect can be investigated using forward modeling tech-\nniques, such as the FoMo code developed by Van Doorsselaere\net al. (2016).\nAcknowledgements. The authors thank the referee for their constructive com-\nments. JMR and TVD have received funding from the European Research Coun-\ncil (ERC) under the European Union’s Horizon 2020 research and innovation\nprogramme (grant agreement No. 724326). CAG-M, DBJ, and SDTG are grate-\nful to Invest NI and Randox Laboratories Ltd. for the award of a Research & De-\nvelopment Grant (059RDEN-1) that allowed the research framework employed\nto be developed. DBJ and SDTG also wish to acknowledge the UK Science and\nTechnology Facilities Council (STFC) for funding under the Consolidated Grant\nST/T00021X /1.\nReferences\nBalthasar, H., Collados, M., & Muglach, K. 2000, Astronomische Nachrichten,\n321, 121\nBogdan, T. J., Carlsson, M., Hansteen, V . H., et al. 2003, ApJ, 599, 626\nBorrero, J. M., Pastor Yabar, A., Rempel, M., & Ruiz Cobo, B. 2019, A&A, 632,\nA111\nBotha, G. J. J., Arber, T. D., Nakariakov, V . M., & Zhugzhda, Y . D. 2011, ApJ,\n728, 84\nCally, P. S. 1986, Sol. 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M. 2011, A&A, 534,\nA65\nThomas, J. H. & Weiss, N. O. 2004, ARA&A, 42, 517\nVan Doorsselaere, T., Antolin, P., Yuan, D., Reznikova, V ., & Magyar, N. 2016,\nFrontiers in Astronomy and Space Sciences, 3, 4\nYu, D. J., Van Doorsselaere, T., & Goossens, M. 2017, A&A, 602, A108\nArticle number, page 10 of 12J.M. Riedl et al.: Finding the mechanism of wave energy flux damping in solar pores using numerical simulations\nAppendix A: Simulations with driver below the\ncutoff frequency\nIn order to focus on propagating waves all simulations in this\npaper have so far been conducted with a driver frequency well\nabove the expected cuto \u000bfrequency of the model atmosphere,\nand therefore also with a period much smaller than the five-\nminute waves observed by GM21. This means that for the cur-\nrent study, all e \u000bects of the cuto \u000bfrequency have been ignored.\nHowever, as mentioned in Section 1, even if the waves of GM21\ndefinitely have a propagating character, they might be altered to\npartly evanescent waves by the existence of the cuto \u000bfrequency\n(Centeno et al. 2006). In this section we explore the possibility\nof damping due to evanescent waves by conducting the same two\nexperiments as before, namely simulations with full driver and\nlocalized driver, but with a lower driver frequency.\nAppendix A.1: Cutoff frequency and new driver period\nIt is commonly accepted that acoustic waves with frequencies\nbelow the cuto \u000bfrequency are not allowed to propagate, but are\nstanding and evanescent. However, it is di \u000ecult to define an ex-\nact value for the cuto \u000bfrequency, and numerous di \u000berent def-\ninitions exist. Centeno et al. (2006) show that when radiative\nlosses are involved there is no clear cuto \u000bfrequency that dis-\ntinguishes between fully propagating or fully evanescent waves.\nFelipe et al. (2018) compared analytical definitions for the cut-\no\u000bfrequency suitable for sunspot umbrae from Lamb (1909),\nSchmitz & Fleck (1998), and Roberts (2006) to the observed\ncuto\u000b. The results generally agree. Using the same analytical\nexpressions as discussed in Felipe et al. (2018) on the observa-\ntional data obtained by GM21 for pore 3 shows that waves with\na period of five minutes indeed have a lower frequency than the\ncuto\u000bfrequency for at least most of the observed domain.\nAccording to the same equations, a driver period of five min-\nutes would still result in a frequency above the cuto \u000bfrequency\nfor our model atmosphere. To mimic the conditions of the obser-\nvations, we choose a longer driver period of T=7 minutes for\nthe following simulations. To include at least one full period of\nthe driver the simulations are run for 500 seconds.\nAppendix A.2: Results\nFigure A.1 shows the height–time graph of the wave energy flux\nparallel to the magnetic field at the axis of the pore for the sim-\nulation with localized driver. The characteristic speeds (starting\nfrom steepest: fast speed vfa(\u0012=\u0019=2)=(v2\nA+v2\ns)1=2, sound speed,\nAlfvén speed, cusp speed) are plotted as black lines, while the\ncontour at value zero is shown in red. The initial part of the first\nwave (i.e., the initial disturbance where the flux is above zero for\nthe first time) propagates with the sound speed (black dashed line\noverplotted on first red line) as it did for the propagating waves in\nSection 3. Then, however, the waves get altered by the e \u000bects of\nthe cuto \u000bfrequency to approximately standing waves within less\nthan half a driver period, as can be seen from the nearly vertical\nfeatures in the figure. This is not what was observed in GM21,\nwho found clear evidence of propagating waves. The di \u000berence\nmight be accounted for by the neglect of radiative losses in our\nsimulations.\nWe performed the same study for the wave energy flux damp-\ning as in Sections 3.1 and 3.2, but for the low-frequency driver.\nFigure A.2 shows the results for the full driver, while Figure A.3\nshows the results for the localized driver. It is immediately appar-\nent that the energy flux for the full driver is now heavily damped\nFig. A.1. Parallel wave energy flux as a function of height and time\nat the pore axis for the simulation with localized driver with a period\nof 7 minutes. The black lines show (from steepest to flattest) the fast\nspeed (dash-dotted line), sound speed (dashed line), Alfvén speed (dot-\nted line), and cusp speed (solid line). The red lines show the contours\nfor zero flux. The frequency of the energy flux is approximately dou-\nbled compared to the driver period because of phase di \u000berence between\np0andv0(see Equation 11).\nFig. A.2. Same as Figure 6, but for a driver period of 7 minutes.\nas well, about the same amount as the energy flux for the local-\nized high-frequency driver (Figure 8). The energy flux for the lo-\ncalized low-frequency driver (Figure A.3) is damped even more,\nprobably because the damping with height is not decreased by\ninward refracted waves as for the full driver.\nAppendix A.3: Discussion\nIt is obvious that the choice of driver frequency strongly a \u000bects\nthe damping in our simulations. However, whether this is purely\ndue to evanescent waves is not fully clear.\nOn the one hand, the dash-dotted orange curve in Figure A.3,\nwhich shows the damping without e \u000bect of geometric spreading,\nstrongly follows the solid green line, which is the full damping\nin our simulation with the localized low-frequency driver. This\nhints that geometric spreading has little to no e \u000bect in this case.\nAt the same time the dashed blue line, which is an estimate for\nthe influence of lateral leakage, is nearly constant, meaning that\nthis e \u000bect is also not very strong. Therefore, a crucial damping\nmechanism is missing, which is likely the reflection of waves\ndue to the cuto \u000bfrequency.\nArticle number, page 11 of 12A&A proofs: manuscript no. main\nFig. A.3. Same as Figure 8, but for a driver period of 7 minutes.\nOn the other hand, these new simulations and their analysis\nare subject to some limiting factors. First of all, due to the low\nfrequency, the wavelengths of the resulting waves are signifi-\ncantly longer than the size of the computational domain. This\ncould lead to strange boundary e \u000bects influencing the results.\nSince the ratio of the wavelength to the size of the pore (which is\nsmaller in our model than in the observations) also changes dras-\ntically, this could account for the decreased e \u000bects of geometric\ndamping and lateral wave leakage. In addition, due to the waves\nstarting at some final time t0, there are no waves present in the\ndomain before the first waves reach a certain height (i.e., left of\nthe first red line in Figure A.1). Therefore, when integrating the\nwave energy flux over time, the lower integration boundary t1(z)\nwas chosen by using a relative threshold to determine the onset\nof the first wave at every height. This line basically coincides\nwith the sound speed line (dashed) in Figure A.1. The upper in-\ntegration boundary was then determined by t2(z)=t1(z)+T, with\nTbeing the driver period. E \u000bectively, the time integration for the\nsimulations with high-frequency driver was done over the first\nperiod of the wave, as a translation of t1(z) byT=30 s resulted\nin at2(z) being located right in front of the next wave train. This\nis not the case for the low-frequency waves because they change\nfrom propagating to standing waves within the first wave period,\nmeaning that their steepness changes in Figure A.1. Therefore,\nit is not clear over which time period the integration should be\nperformed, and the choice might a \u000bect the shape of the damping\ncurves in Figures A.2 and A.3.\nMoreover, even if the limitations listed above have little to no\ne\u000bect, there are still no propagating waves in our low-frequency\nsimulations, as opposed to the observations of GM21. Therefore,\nthe damping in the low-frequency simulations due to evanescent\nwaves is expected to be much stronger than for the observations,\nwhere the waves were at least partly propagating. This validates\nthe study of the other damping mechanisms presented in this pa-\nper.\nArticle number, page 12 of 12" }, { "title": "2210.13764v1.Microscopic_structure_of_electromagnetic_whistler_wave_damping_by_kinetic_mechanisms_in_hot_magnetized_Vlasov_plasmas.pdf", "content": "Microscopic structure of electromagnetic whistler wave damping by kinetic\nmechanisms in hot magnetized Vlasov plasmas\nAnjan Paul1, 2and Devendra Sharma1, 2\n1Institute for Plasma Research, Bhat, Gandhinagar, India, 382428\n2Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai 400094, India\n(Dated: October 26, 2022)\nThe kinetic damping mechanism of low frequency transverse perturbations propagating parallel\nto the magnetic \feld in a magnetized warm electron plasma is simulated by means of electromag-\nnetic (EM) Vlasov simulations. The short-time-scale damping of the electron magnetohydrodynamic\nwhistler perturbations and underlying physics of \fnite electron temperature e\u000bect on its real fre-\nquency are recovered rather deterministically, and analyzed. The damping arises from an interplay\nbetween a global (prevailing over entire phase-space) and the more familiar resonant-electron-speci\fc\nkinetic damping mechanisms, both of which preserve entropy but operate distinctly by leaving their\ncharacteristic signatures on an initially coherent \fnite amplitude modi\fcation of the warm electron\nequilibrium distribution. The net damping results from a deterministic thermalization, or phase-\nmixing process, largely supplementing the resonant acceleration of electrons at shorter time scales,\nrelevant to short-lived turbulent EM \ructuations. A kinetic model for the evolving initial transverse\nEM perturbation is presented and applied to signatures of the whistler wave phase-mixing process\nin simulations.\nI. INTRODUCTION\nThe electromagnetic turbulence prevails in collective\nexcitations of charged matter interacting with, and by\nmeans of, the electromagnetic \feld over a vast range of\nspatiotemporal scales, usually terminated by dissipation\nat the \fner scales. The solar-wind spectrum, for exam-\nple, shows that beyond a frequency breakpoint a deviation\nexists from the inertial rage characterized by exponent -\n5/3 of power law [1, 2]. In one of the plousible scenarios,\nthe whistler \ructuations can be the fundamental mode\nand central means of dissipation in this weak turbulence\nregime [3]. A steepening present in the spectrum lead-\ning to considerably high spectral exponents ( \u00182-3) sug-\ngests presence of considerable damping alongside to the\nintra-spectral energy transfer [3]. Besides by conversion\nto electrostatic modes [4], damping by kinetic transverse\nwave-particle interaction must operate on the short lived\nexcitations [5{8] initiated by sponteneous \feld \ructua-\ntions. Fresh perturbations, so triggered, excite warm\nplasma eigenmodes by leaving long lasting remnants of\ntheir initially enforced phase-space structure in the mem-\nory of nonthermal kinetic distributions [9]. The asymp-\ntotic long-time solutions of the collisionless kinetic for-\nmulation [10, 11] applied to them therefore have large\nscope of sophistication by admitting a strong determin-\nistic thermalization , or phase-mixing [9], alongside the\ndamping evaluated in usual time asymptotic, t!0 limit.\nThe general kinetic evolution produced collisionless\ndamping of electromagnetic \ructuations [12] involves a\nrather complicated phase-space dynamics and is most\naccessible by deterministic Vlasov simulations [13, 14].\nOnly a limited number of studies have rather determin-\nistically simulated the dynamics of the transverse elec-\ntromagnetic excitations and their damping/stability in a\nhot collisionless magnetized plasma [15, 16], even as the\nprocess remains critical for determining the operationalstate of turbulence and the transport associated with it\nboth in space plasmas [3, 17] as well as in modern mag-\nnetic con\fnement fusion experiments [18].\nIn the collisionless limit, the modi\fcations made to\ntemperature, or width, of an initially equilibrium warm\nelectron velocity distribution produce a higher order cor-\nrection to the resonant wave particle interaction process.\nThe analytical model predicts a related downward shift\nin the whistler wave frequency in collisionless plasmas\nwith hotter electrons [8, 19, 20]. The recovered strength\nof damping due to wave particle interaction however re-\nmains underestimated in comparison with that produced\nby the computer simulations implemented with reason-\nably low collisionality.\nThis paper addresses above aspects of kinetic whistler\ndamping mechanism, subsequent to the recovery of gen-\neral electromagnetic modes of a magnetized plasma\nand their dispersive characterization in our \rux-balance\nbased [13, 21] Vlasov simulations. This is accompanied\nby illustration of its detailed phase-spatiotemporal evolu-\ntion. The interaction of electromagnetic modes, propa-\ngating parallel to the magnetic \feld with the resonant\nparticles is studied, particularly recovering the damp-\ning of the whistler waves via full kinetic mechanism\nand comparison of the simulation results with those an-\nalytically prescribed in the linear Landau theory limit\n[9, 10]. Presented simulations and analysis enter the\n\fner regime of kinetic phase-mixing of the electromag-\nnetic mode uniquely achievable by Vlasov simulations.\nWe qualitatively recover the phase-mixing e\u000bects show-\ning the dominance of frequency v\u0001kof the ballistic term\n(/exp(ikvt)) [9] accounting for short time response, in\naddition to time asymptotic Landau damping results that\nare routinely applied to turbulent electromagnetic \ruc-\ntuations of su\u000eciently short life time. First quantitative\nanalysis of the phase-spatiotemporal evolution of a 4D\nelectron phase-space distribution perturbation associatedarXiv:2210.13764v1 [physics.plasm-ph] 25 Oct 20222\nwith the transverse electromagnetic whistler mode simu-\nlated in a hot magnetized Vlasov plasma is presented.\nThe present paper is organized as follows. In Sec. II\nthe electromagnetic Vlasov-Maxwell model considered\nfor the simulation is presented. Vlasov simulation im-\nplementation and its characterization by dispersion of\ntransverse EM perturbation is presented in Sec. III. In\nSec. IV the simulations of Landau damping of transverse\nwhistler wave perturbations in a warm electron plasma\nis presented followed by their comparison with analyt-\nical results. Based on the evolution of simulated per-\nturbations, the signatures of phase-mixing supplemented\nlandau damping of the initial perturbation are presented\nin Sec. V. The kinetic model for initial transverse EM\nperturbations is presented and solved in Sec. V B. Con-\nclusions are presented in Sec. VI.\nII. THE ELECTROMAGNETIC VLASOV\nPLASMA MODEL\nThe electromagnetic magnetized plasma modes simu-\nlated in this paper follow pure kinetic formulation and\nare well represented by the solutions of collisionless, fully\nnonlinear Vlasov equation for species \u000b,\n@f\u000b\n@t+v\u0001@f\u000b\n@x+q\u000b\nm\u000b\u0012\nE+v\u0002B\nc\u0013@f\u000b\n@v= 0:(1)\nThe evolution of electric \feld Eand magnetic \feld B\nfor the electromagnetic processes follows the Maxwell's\nequations,\nr\u0002B=1\nc@E\n@t+4\u0019\ncJ; (2)\nr\u0002E=\u00001\nc@B\n@t; (3)\nr\u0001B= 0 (4)\nwhere the current density Jis described by\nJ=X\n\u000bZ1\n\u00001dvq\u000bvf\u000b (5)\nConsistent with the electron magnetohydrodynamic\nregime excitations, we use an externally applied constant\nmagnetic \feld B0and in\fnitely massive ions. The sub-\nscript for species \u000btherefore only assumes value, e, repre-\nsenting electrons (and henceforth omitted), which are the\nonly mobile species and contributing to the perturbation.\nThe full nonlinear kinetic model (1)-(5) is implemented\nfor the case of waves propagating parallel to an applied\nmagnetic \feld ( B0kk) where principle modes are high\nand low frequency right and left handed circularly po-\nlarized modes. This set up includes the whistler waves\nthat are right handed polarized and propagate in the fre-\nquency range \n ci< ! < \nce, where \n c\u000bis the gyrofre-\nquency of the species \u000b.III. KINETIC SIMULATIONS OF\nELECTROMAGNETIC WAVES IN\nMAGNETIZED WARM VLASOV PLASMA\nThe simulations presented in this analysis progress\nby numerically evolving the magnetized plasma (elec-\ntron) velocity distribution function faccording to the\ndynamics of the phase-\ruid \row [9] which is governed\nby the collisionless Vlasov equation (1), and associated\nMaxwell's equations (2)-(4). The 4-dimensional (4D \u0011\n1x-3v) phase-space simulations are performed using an\nadvanced \rux balance technique [13, 21, 22] generalized\nto simulate the electromagnetic plasma modes in a large\nrange of magnetization of plasma species. The results of\nsimulations are characterized \frst against the analytical\ndispersion relation of left and right handed circularly po-\nlarized low and high frequency modes and then against\nboth analytic and numerical evaluation of the linear Lan-\ndau damping descriptions [8, 9, 23].\nA. The simulation set-up\nIn order to simulate the waves propagating parallel to\nan ambient magnetic \feld B0in a wide range of frequency\n!and wave vector k, we have assumed a setup where\nboth the B0andkare aligned to z-axis and the periodic\nboundary condition is used at both the boundaries of the\none-dimensional simulation zone located between z= 0\nandL. The setup therefore assumes symmetry along\nboth ^xand ^ydirections with \fnite spread of electron\nvelocities along these dimensions, besides the direction ^ z.\nThe three dimensional equilibrium velocity distribution\nfor the electron species is considered to be a Maxwellian,\nf(z;v) =\u00121\n2\u0019v2\nth\u00133=2Y\nj=x;y;zexp\u001a\n\u0000(vj\u0000hvji)2\n2v2\nth\u001b\n;(6)\nwherevth= (Te=me)1=2is electron thermal velocity, Teis\nelectron temperature in energy units and meis electron\nmass.\nAt timet= 0, the equilibrium distribution is per-\nturbed with sinusoidal perturbations (having variation\nalong ^z) with initial amplitudes of hvi1,E1andB1\n(/exp(ikz)) consistent with transverse electromagnetic\n(right or left handed circularly polarized) linear magne-\ntized plasma modes propagating with a desired k[5, 24].\nB. Dispersion characterization of electromagnetic\nplasma modes\nWe present the dispersion relations recovered for the\ncase of very small electron thermal velocity ( vth= 0:001\nc) simulated and its comparison with the corresponding\nanalytical cold plasma dispersion relation [24, 25]. Spe-\nci\fc to simulation cases presented in this section, we have3\n0 2 4 6 8 10\nck/pe\n0.00.20.40.60.81.01.2/pe\nAnalytical R mode\nAnalytical L mode\nce\n1\npe\n2\n=ck\nSimulation\nFIG. 1. Dispersion of simulated frequency (dots) compared\nto the Right and Left handed polarized branches of the ana-\nlytical dispersion relation (solid line).\nused a 3-dimensional velocity space grid of rather mod-\nerate size having 32 \u000232\u000232 grid points, in combination\nwith spatial grid size also of 32 grid points. In Fig. 1,\nthe comparison is presented of the simulated values of\nfrequency!of the perturbation plotted as function of\nk, with the right and left handed circularly polarized\n(RHCP and LHCP) branches of the analytical dispersion\nrelation in the limit of in\fnitely massive ions ( !pe\u001d!pi),\nk2\nR;L=!2\nc2\"\n1\u0000!2\npe\n!(!\u0007\nce)#\n: (7)\nConsidering the parameters, vth= 0:001 c, the ratio\nof electron cyclotron frequency and plasma frequency\n\nce=!pe= 0:1 and su\u000eciently small initial perturba-\ntion exclusively in electron average velocity amplitude\n\u0001V=hvi1max\u0000hvi1min = 2\u000210\u00003c, the high fre-\nquency (! > \nce) RHCP and LHCP mode phase ve-\nlocities are su\u000eciently high for these waves to stay out\nof resonance with the cold electrons chosen for this case.\nMoreover, for the low frequency whistler waves excited\n(dots on the red curve in the region 0 < ! < \ncein\nFig. 1), the electron thermal velocity vth= 0:001ccho-\nsen is still su\u000eciently low for no signi\fcant resonant elec-\ntron population to be available at the resonant velocity,\nvz=vres= (!\u0000\nce)=k. In rest of this analysis we exclu-\nsively characterize this low frequency whistler branch of\nthe perturbation for relatively warmer electrons in order\nto analyze its resonant damping.\nIV. LANDAU DAMPING OF THE\nTRANSVERSE WHISTLER MODE\nAn advanced simulation set up, with grid size of\n64\u000264\u000264\u000264 is implemented in the following sets\nof simulations, by adopting the electron velocity range\n[-0.26, 0.26] c. The velocity distribution of electrons as\nfunction of the parallel velocity vk\u0011vzis plotted in\nFig. 2 atvx=vy= 0 for a range of electron thermal ve-\nlocityvth= 0:001c(inner most pro\fle) to 0 :026c(outer\n0.2\n 0.0 0.2\nvz/c0.00.20.40.60.81.01.2f/fe\nvres\nvpFIG. 2. The electron distribution function (normalized to its\nmaximum value) plotted as function of parallel velocity vk\u0011\nvzatvx=vy= 0. The thermal velocity of electrons for the\ncurves with increasing width ranges from vth= 0:001c(inner\nmost pro\fle) to 0 :026c(outer most pro\fle) , respectively. The\nvertical dotted lines indicate resonant velocity vresand phase\nvelocityvpof the low frequency whistler mode at k= 1:0!p=c.\nmost pro\fle) explored in the cases presented in this sec-\ntion. The resonant velocity vres= (!\u0000\nce)=kand phase\nvelocityvpof the low frequency whistler mode for pa-\nrametersk= 1:0!p=cand \nce=!p= 0:1, are indicated by\nvertical dotted line. The pro\fles for smaller values of vth\nshow the population of resonant electrons drops to neg-\nligible values such that no resonant damping is present\nfor these cases.\nIn Fig. 3(a), the time evolution of the amplitude of\nthe velocity perturbation \u0001 Vis plotted in the simu-\nlations done for the range of electron thermal velocity\nvth= 0:001cto 0:026cfor the value of k= 0:8!pe=c.\nTwo additional set of simulations done for the values of\nk= 0:897 andk= 1:0!pe=care presented in Fig. 3(b)\nand Fig. 3(c), respectively. The time evolution of \u0001 Vin\nall the cases above cover the initial evolution of the wave\namplitude only for the time duration \u0001 t\u00183!\u00001\np(i.e.,\nabout a fraction of one complete cycle of the whistler\nwave with frequency !<\ncewhile \nce=!p= 0:1) which\nis the time duration in which the linear Landau damp-\ning rate remains reasonable estimate for the wave damp-\ning. The short time evolution in Fig. 3 has su\u000eciently\nlow numerical widening of ffor resolving the e\u000bect of\nTevariation, which is varied with relatively much larger\nincrements in this study.\nA. Comparison of simulations with analytic\nwhistler damping rates\nThe damping rates of whistler velocity perturbation\namplitude can be compared with the available analytic\napproximations of the Landau damping rate of whistlers.\nFor the purpose of comparison we have used the analytic\nresults from kinetic formulation in certain limiting cases.\nThe numerical evaluation of analytical expression is done4\n0123\ntpe\n1.00\n1.01\n1.02\n1.03\n1.04\n1.05V/c \n×103\n(c)1.00\n1.01\n1.02\n1.03V/c \n×104\n(b)1.000\n1.005\n1.010\n1.015\n1.020V/c \n×103\n(a)\nFIG. 3. The decay in the velocity perturbation \u0001 vforvth=\n0:001c(lowest damping) to 0 :026c(highest damping) for (a)\nk= 0:8!pe=c, (b)k= 0:897!pe=cand (c)k= 1:0!pe=c. The\nmodulation seen in some cases are residual RHCP and LHCP\nexcitations ( !R;L>\ncesatisfying (18)).\nin the limits where such approximations become unavail-\nable. We begin by using the kinetic dispersion relation\nfor the electromagnetic waves ([26] and Sec. V B) given\nby,\nc2k2\u0000!2=8\u00192q2\ne\nm1Z\n\u000011Z\n0h\n(!\u0000kvz)@fe0\n@v?+kv?@fe0\n@vzi\n(!\u0000kvz\u0006\nce)\n\u0002v2\n?dv?dvz: (8)\nAssuming isotropy in the velocity space,\n@f0\n@vk=@f0\n@v?; (9)\nadditionally considering the whistler wave frequency\nlimit \nci\u001c!\u001c\nceand for wavelengths larger than\nelectron gyroradius, \u0015>\u001ace, we recover [9],\n\r\u0019\u0000!2\npe\njkkjvth1\n1 +k2c2=!2rexp \n\u0000\n2\nce\nk2\nkv2\nth!\n:(10)However, for the cases explored in the present simulations\nhave!<\u0018\nceand the expression (10) underestimates the\ndamping rate \r, yielding negligible values in comparison\nto what recovered in the simulations. For example, for\nk= 1:0!pe=c,vth= 0:026c, \nce= 0:1!pe,\r\u001810\u00008!pe,\nalthough a considerably higher damping ( \r\u001810\u00003!pe) is\nrecovered in the simulations. An estimate from (8) in the\nlimit!<\u0018\ncemore relevant to simulations is prescribed\nby Gary [23] in the form of a general expression for \r,\n\r\u0019\u0000\u0019\n2kX\n\u000b!2\np\u000bp\n2\u0019vth\u000bexp \n\u0000(!\u0006\nc\u000b)2\n2k2\nkv2\nth\u000b!\n(11)\nwhich can be used as an alternate analytic estimate to\ncompare with theory the whistler damping rate recov-\nered in the simulations. For an even improved analytic\nestimate of damping rates we have also used the exact\nexpression (8) and evaluated \rnumerically, by the fol-\nlowing procedure.\nAs in the simulation where ions are in\fnitely mas-\nsive, considering electron equilibrium distribution to be\nMaxwellian,\nf0=\u00121\n2\u0019\u00133\n2 1\nvthkev2\nth?eexp \n\u0000v2\n?\n2v2\nth?e\u0000v2\nk\n2v2\nthke!\n;\n(12)\nand substituting it in the dispersion (8), one obtains [8],\n!2+!2\npeI\u0000c2k2= 0; (13)\nwhere,\nI=v2\nth?e\nv2\nthke\u00001 +\"\nv2\nth?e\nv2\nthke!\u0006\nce\n\u0007\nce+ 1#\n\u0007\ncep\n2kvthkeZ(\u0010);\n(14)\nandZ(\u0010) is the plasma dispersion function [27],\nZ(\u0010) =1p\u00191Z\n\u00001e\u0000t2\nt\u0000\u0010dt; (15)\nwith the argument\n\u0010=!\u0007\ncep\n2kvthke: (16)\nThe upper and lower sign in Eq. (14) are for RHCP and\nLHCP waves, respectively. Applying dispersion (13) to\nwhistlers which are right handed polarized, and consid-\nering isotropy of the distribution ( vthk=vth?=vth) we\nget (13) in the form,\n!2+!!2\npep\n2kvthZ(\u0010)\u0000c2k2= 0: (17)\nNote that under cold plasma approximation where imagi-\nnary part of the function Z(\u0010) =ip\u0019e\u0000\u00102\u00001\n\u0010is negligible5\n012\nVth/c×102\n4\n2\n0i/pe\n×103\nSimulation\napprox. 1\napprox. 2\napprox. 3\nFIG. 4. Comparison of simulated damping rate \rwith\nthat obtained from analytical approximations, approx-1 (10),\napprox-2 (19) and approx-3 (11) done for k= 1:0!p=cor sim-\nulation box length L= 2\u0019c=! p.\nandZ(\u0010)\u0018\u00001=\u0010, such that with \u0010= (!\u0000\nce)=p\n2kvth\none readily recovers from (17), the cold plasma dispersion\nrelation for the RHCP waves,\n!2\u0000!!2\npe\n(!\u0000\nce)\u0000c2k2= 0; (18)\nwhich is the whistler dispersion relation in the limit \n ci<\n!<\nce. Considering !=!r+i!i, in the warm plasma\nRHCP dispersion (17) we obtain for \r=!i\u001c!r[8, 20],\n\r=\u0000!rp\u0019e\u0000\u00102p\n2kvth\n[!rZ0(\u0010) +p\n2kvthZ(\u0010) + 2!rk2v2\nth]; (19)\nwhereZ0(\u0010) is the derivative of Z(\u0010) with respect to \u0010.\nComparison of the damping rate recovered in simula-\ntions with the analytical approximations (10), (11) and\n(19) is presented in Fig. 4 for the above range of vthvalues\nfor which simulations are performed using k= 0:897!p=c\nor simulation box length L= 7c=!p. Note that for the\nnearly cold electron case, vth!0, the simulations (blue\ndots) duly recover a whistler propagation free of any\ndamping (or growth), con\frming that the deviation from\nthis undamped propagation recovered at large electron\ntemperature values (larger vth>0) represents pure ki-\nnetic e\u000bects in the simulations. The simulations however\nshow increasing damping from \fnite vthvalues while all\nthe other analytical approximations prescribe compara-\nble damping only beyond vth>1:0\u000210\u00002c, as discussed\nfurther below.\nSuitability of approximation (10) for the present range\nis ruled out by the negligible resonant damping pre-\nscribed by (10) as plotted in Fig. 4 using green line in\ncomparison to other estimates of \r, namely, (11) and\n(19) plotted with red and black lines, respectively. The\nanalytical approximation (11) very closely estimates the\nnumerically evaluated values of the exact expression (19).\nWe note that despite no clear approximations used in\nthem (at least, in the numerically computed \rfrom (19))\nbesides their linear origin, they still show \fnite discrep-\nancy from the simulated damping rates. In general,\n0.1 0.4 0.7 1.0\nck/pe\n1\n012345r/pe\n×102\n1\n012345\ni/pe\n×102\nFIG. 5. Simulated frequency and damping rate \ras function\nofkand a range in electron thermal velocity values vth=\n0:001 to 0:026. The vertical dashed lines indicate kvalues for\nwhich sets of simulations are performed by varying vth.\nthe simulations recover stronger damping rates for the\nwhistlers than what prescribed by the analytical damp-\ning rate approximations duly obtained from the kinetic\ndispersion relation. We explore the origin of this dis-\ncrepancy for whistlers excitations in Sec. V based on the\ndeterministic evolution of the electron distribution func-\ntion in essential 4D phase-space set up available from the\nsimulations. The missing physics is located in the small\ntime evolution of the initial perturbations as asymptotic\ncontributions are duly accounting for in estimating the\nkinetic Landau damping rate. For turbulent situations,\nwhere the excited \ructuations are short lived, the short\ntime contributions need to be accounted for. Moreover,\nthey might dominate the asymptotic mechanisms in cer-\ntain relevant limits.\nIn Fig. 5 the analytically obtained whistler frequency\n!and more accurate numerically computed damping\nrate from (19) are plotted as function of kfor several\n(>10) di\u000berent values of electron thermal velocity in\nrangevth= 0:001 to 0:026. In Fig 6, the !rand\rval-\nues are separately compared with analytical values where\nthe markers represent simulated data and variation is\ndone in both kandvthvalues. As in Fig 6, simulations\ncon\frm the analytical prediction that the !rslightly re-\nduces at larger electron temperature, though the analyt-\nical!roverestimates this reduction for larger vth. The\ncomparison of \rpresented in Fig. 6 shows that while at\nk= 0:8!p=cthe damping rate in simulation is recovered\nto be stronger than the corresponding analytic values, at\nlargerkvalues, 0:897!p=cand 1!p=c, the overlap in the\nanalytic and simulated \rvalues is better, especially at\nlargervthvalues. It can also be noted that at smaller k,\nanalytical expressions underestimate of the damping and\nthe spread in kinetic \rvalues is larger than the analytical\nvalue spread. At larger k, on the other hand, the ana-\nlytical approximation overestimates \rand the kinetically\nobtained\rvalues have a narrower spread.6\n0.00.20.40.60.81.0r/pe\n×101\n(a)\n0.991.001.014.254.504.755.00×102\n0.20.40.60.81.0\nck/pe\n1.0\n0.8\n0.6\n0.4\n0.2\n0.0i/pe\n×102\n(b)\n0.981.001.024\n2\n0×103\nFIG. 6. Comparison of simulated frequencies (a) !rand (b)\n!rwith those obtained from analytical approximation (19)\nand corresponding !ras functions of kandvth. Markers\nat variouskvalues represent simulation data such that the\nmarkers and curves from top to bottom correspond to vth=\n0:001 to 0:026.\nV. PHASE-MIXING SUPPLEMENTED\nDAMPING OF THE EMHD PERTURBATIONS\nRelatively larger damping of the whistler perturbations\nat smaller kin comparison to the analytical estimates\n(10), (11) and (19) is interpreted based on the deter-\nministic evolution of the electron distribution function\ngoverned by the Vlasov equation as simulated and an-\nalyzed in this section. In warm electron plasmas the\ninitial damping is found to have considerable contribu-\ntion from the corresponding deterministic phase-mixing\nprocess of the fresh (initial) perturbations made to the\nplasma, con\frming to a propagating whistler wave. Note\nthat the term phase-mixing used here refers to the varied\nresponse of the electrons of di\u000berent velocities (from a\nwider spread about mean) to the propagating wave per-\nturbation. The deterministic deformations of the veloc-\nity distribution function that evolve antisymmetrically\nabout a reference velocity (distinct for electrostatic and\nelectromagnetic waves) in the phase-space however in-\ntroduce incoherence between the substructures of observ-\nable macroscopic spatio-temporal variations, or collective\nwave perturbations. The resultant weakening of the per-turbations translates in to an additional damping (than\nwhat caused by the resonant electrons) detectable in the\ndeterministic Vlasov simulation data. At shorter time\nscales this contribution supplements, and modi\fes, the\ndamping of the (transverse) wave, producing enhanced\ndamping rates in comparison to the pure resonant damp-\ning essentially estimated considering the long time limit\n[9]. Clear signatures of both resonant and non resonant\nprocesses are visible in the simulated 3-dimensional elec-\ntron velocity distribution function evolving in the phase-\nspace.\nA. Phase-space evolution of whistler wave\nperturbation in the Vlasov simulations\nThe plots of electron distribution function perturba-\ntionf1=f\u0000f0, as available from the simulations done\nwith denser grid size, 64 \u0002160\u0002160\u0002160, are ana-\nlyzed in order to understand the kinetics of the evolution\nof whistler wave perturbation in the plasma. Since the\nsystem is periodic and transverse, analyzing the fdata\nfrom any chosen spatial location is equivalent. The lo-\ncationz=L=2, or the center of the simulation domain,\nis therefore chosen here as the representative location in\nthe simulation domain. For added accuracy, both fand\nf0are evolved under equivalent Vlasov simulation op-\neration in order to obtain f1(t) presented below. From\nthe full 3-dimensional velocity distribution function avail-\nable, the evolution of f1(z=L=2;vz;vx;vy) at various\nstages of time is presented in Fig. 7 from the cold elec-\ntron case simulation, performed with su\u000eciently small Te\nsuch that thejvresj\u001dvth= 0:001c. Because of the small\nelectron temperature, both the resonant and phase veloc-\nity of the whistler are far removed from the distribution\nwidth (along vzkk) and thef1plotted in three columns\nof frames from left to right correspond to velocity values\nvz=\u0000vth, 0 andvth, respectively. The same data is plot-\nted in Fig. 8 from simulation of warmer electrons case,\njvresj \u0014vth= 0:001c, such that su\u000ecient population\nof resonant electrons is present with vz=vres, produc-\ning dominant and analyzable kinetic e\u000bects. The rows of\nframes from top to bottom in both plots correspond to\ntime fromt= 0 to 20!\u00001\npe(\u00112\n\u00001\nce).\nIn both cold and warm electron cases plotted in top\nrows of Fig. 7 and Fig. 8, respectively, the f1plotted\natt= 0, consistent with a whistler phase-space per-\nturbation in the velocity space, has a nonzero velocity\nperturbation purely aligned to ^ vy. The following rows\ncover evolution over a period of 20 !\u00001\npe, or one complete\nwhistler cycle. We \frst discuss evolution of perturbation\nin the cold electron case presented in Fig. 7. Consider\nthe evolution of the perturbation component located at\nvz\u00180 plotted in the second column of the Fig. 7. Be-\ncause of being drawn from vanishing vz, this component\noff1does not have electrons that leave to, or arrive from,\nthe adjacent zlocations as the time progresses (i.e., from\ntop frame to bottom frame). Therefore, the participating7\n5\n05vy/c×103\n(a)\n6\n3\n036×105\n5\n05vy/c×103\n(b)\n6\n3\n036×105\n5\n05vy/c×103\n(c)\n5.0\n2.5\n0.02.55.0×105\n5\n05vy/c×103\n(d)\n6\n3\n036×105\n5\n05vy/c×103\n(e)\n5.0\n2.5\n0.02.55.0×105\n5\n05\n×103\n5\n05vy/c×103\nvx/c(f)\n6\n4\n2\n024×105(g)\n1.6\n0.8\n0.00.81.6×107\n(h)\n1.6\n0.8\n0.00.81.6×107\n(i)\n1.6\n0.8\n0.00.8×107\n(j)\n1.6\n0.8\n0.00.8×107\n(k)\n1.2\n0.6\n0.00.61.2×107\n5\n05\n×103\nvx/c(l)\n1.2\n0.6\n0.00.61.2×107(m)\n8\n4\n048×105\n(n)\n8\n4\n048×105\n(o)\n6\n3\n036×105\n(p)\n6\n3\n036×105\n(q)\n6\n3\n036×105\n5\n05\n×103\n(r)\nvx/c6\n3\n036×105\nFIG. 7. Evolution of the electrons velocity distribution func-\ntion perturbation in the transverse velocity-space plane vx-vy\nfor the cold electrons case, with vth= 0:001c. Frames from\ntop to bottom correspond to equal intervals over one whistler\ncycle (\u001820!\u00001\npe) fromt= 0, while those from left to right are\nforvz=\u0000vth, 0, andvth, respectively. Red and black arrows\nindicate, B1andhvi1, respectively.\nelectrons are simply displaced to newer angular locations\nwith (angular) velocity \n ceon the same transverse veloc-\nity planevx-vyin the subsequent frames, with respect to\ntheir original locations in the t= 0 frame. However the\npolarization of the perturbation in this case rotates with\nangular velocity of the wave !, dissimilar to \n ce(> !,\nthe reason for which is well encoded in the linear the-\nory of waves in magnetized plasmas). Drawn from the\nsimulation data for hvi1andB, the black and red ar-\nrows represent phases of right handed polarized velocity\nand magnetic \feld vectors, respectively, for the reference.\nThe rotation in perturbation polarization consistent with\nwave frequency con\frms that the initial perturbation ef-\n\fciently excited a whistler eigenmode in the simulated\n0.1\n0.00.1vy/c(a)\n4\n2\n024\n0.1\n0.00.1vy/c(b)\n5.0\n2.5\n0.02.55.0\n0.1\n0.00.1vy/c(c)\n4\n048\n0.1\n0.00.1vy/c(d)\n8\n4\n048\n0.1\n0.00.1vy/c(e)\n8\n4\n048\n0.1\n 0.0 0.1\nvx/c0.1\n0.00.1vy/c(f)\n8\n4\n048(g)\n1.0\n0.5\n0.00.51.0×102\n(h)\n1.0\n0.5\n0.00.51.0×102\n(i)\n8\n4\n048×101\n(j)\n6\n3\n036×101\n(k)\n6\n3\n036×101\n0.1\n 0.0 0.1\nvx/c(l)\n8\n4\n04×101(m)\n5.0\n2.5\n0.02.55.0\n(n)\n1.5\n0.01.53.0\n(o)\n1\n0123\n(p)\n1.5\n0.01.53.0\n(q)\n0.00.61.21.82.4\n0.1\n 0.0 0.1\nvx/c(r)\n1\n01234FIG. 8. Evolution of the electrons velocity distribution func-\ntion perturbation in the transverse velocity-space plane vx-vy\nfor the warm electrons case, with vth= 0:02c. Frames from\ntop to bottom correspond to equal intervals over one whistler\ncycle (\u001820!\u00001\npe) fromt= 0, while those from left to right\nare forvz=vres, 0, andvphase, respectively. Red and black\narrows indicate, B1andhvi1, respectively.\nmagnetized plasma right from t= 0.\nThe kinetic e\u000bects because of resonant electrons are\naddressed further below in warm electron case since for\ncold electrons the velocity distribution drops to negligible\nvalues at the resonant velocity. The same is evident from\nthe identical evolution of the perturbation component at\nvz=j\u0006vthj\u001cjvresj, plotted in the columns 1 and 3,\nrespectively, which show rotation of perturbation polar-\nization with same frequency and a coherent response of\nelectrons of all vzvalues due to the distribution function\nbeing nearly a delta function in velocities both along and\nacross k. Therefore, the \ruid theory results ( \r!0,!\ngiven by (18)) remain suitably recoverable for this cold\nelectron case as clear from the data point corresponding8\ntovth\u00180:001cin Fig. 4.\nWe now show that an identical initial perturbation,\nthat excited a nearly ideal whistler eigenmode in the\nabove cold electron case, evolves in a warm electron mag-\nnetized plasma exhibiting a far richer kinetic response.\nThis initial evolution, covering 20 !\u00001\nperemains applicable,\nfor example, to the short lived transverse whistler mode\n\ructuations characteristically running much of weak elec-\ntromagnetic plasma turbulence [3]. There evolution\ntherefore necessitates invoking initial phase-mixing de-\nscription of analytical kinetic theory results in addition\nto the routine estimates of damping by pure resonant\nelectrons.\nConsidering the evolution for the perturbation compo-\nnent atvz\u00180;z=L=2 plotted in the middle column\nof the frames in Fig. 8 we note that its polarization ro-\ntates nearly with the wave frequency !, however with a\nvisible shortfall in comparison to the cold electron case.\nThe electrons being su\u000eciently warm in this case, it is\nnow possible to examine the f1evolution at the resonant\nvelocity for this set of data, which is noted to be much\ndistinct as plotted in the \frst column of the Fig. 8.\nNote that individual electrons streaming with resonant\nvelocityvzwould see the wave frequency Doppler-shifted\nto \nceand therefore will be perpetually accelerated by\nthe wave E1in transverse the direction. The resulting\nproliferation of electrons to higher v?is visible with time\nin frames from (a)-(f). It is important to notice, though,\nthat, unlike the coherent response of the full distribution\nin cold electron case, the polarization of the perturbation\nin the resonant electrons has rotated with a considerably\nlower frequency, thereby covering a lesser angle as com-\npared to the bulk, or vz\u00180, electrons.\nQuantitative estimation of this shift in frequency is\npossible based on the fact that because of \n ce> !, in\ntime\u000etthe gyrating electrons cover a far larger angle\nin the transverse plane than the rotating polarization of\nthe perturbation. Consequently, as the time progresses\nthevz=vreselectrons bring imprint of an increasingly\nolder phase of the wave, distinct from the instant phase\nof the bulk electron component polarization (seen in the\ncorresponding middle column frame plotted for vz= 0).\nThis is despite of resonant electrons arriving (because of\ntheir negative streaming velocity vres<0) from a forward\nlocationz=\u000ez >L= 2 having a slightly advanced phase\nof the wave.\nThe convected phase \u001e(z=L=2;t=\u000et) of the po-\nlarization for resonant perturbation component (pertur-\nbation component with vz=vres) at timet=\u000etcan\nbe calculated by using a transformation between present\nand older phase of the polarization of the perturbation,\n\u001e(L=2;\u000et) =\u001e(L=2 +\u000ez;0) =\u001e0+\u000e\u001ez+\u000e\u001e\u0012;(20)\nwhere\u001e0is instant phase of the polarization of the bulk\nperturbation at z=L=2 (as visible in vz= 0,t=\u000et\nframe),\u001e(z;t) is the general phase of the polarization\n(ideally known from the cold electron limit), \u000e\u001ezand\n\u000e\u001e\u0012are the phase di\u000berence arising from change in zand\u0012, respectively. For the resonant component of the per-\nturbation, we have,\n\u000e\u001ez=\u000ezk=vres(\u0000\u000et)k;\u000e\u001e\u0012= (\u0000\u000et) \nce:(21)\nSincevres<0 and \nce> ! = \nce\u0000jvresjk, we have\n\u000e\u001ez>0,\u000e\u001e\u0012<0 andj\u000e\u001e\u0012j> \u000e\u001ez, meaning that the\nphase of resonant component of the perturbation relative\nto instant bulk electron phase will be negative. These\nphase delays are in excellent agreement with the phase\ndi\u000berence in the frames presenting the resonant compo-\nnent of the perturbation in the \frst column of Fig. 8.\nAn even distinct evolution is witnessed in the pertur-\nbation component at the positive values of parallel ve-\nlocity,vz>0, as potted in third column of Fig. 8 for\nvz=vphase. Before explanation of this behavior, which\nstrongly relates to phase-mixing process of the whistler\nperturbation, it can be clearly noted that the strength of\nmacroscopic wave variables (especially J1) is going to be\nweakened with time if velocity integration is performed\nat advanced times using f1(vx;vy) that has its peaks in\nvx-vyplane initially overlapping for all vz, but dispersed,\nat future times, over a wider range of angles for di\u000berent\nvz.\nIn thef1evolution plotted at vz=vphase in third col-\numn of Fig. 8, the streaming (and simultaneously gy-\nrating) electrons experience the wave \felds purely non-\nrotating. At each \u000etthey import, from their original\nlocations to z=L=2, the phase as determined by the\ntransformation (20). In this case, however, with each gy-\nration completed, the electrons additionally drift orthog-\nonal to the phase of the wave electric \feld experienced by\nthem. Consequently, the imported f1is additionally dis-\nplaced away from v?= 0. With time progressing, the f1\nsteadily sampled at z=L=2 witnesses newer electrons\narriving from backward locations. An increasing drift\nis therefore witnessed in f1, rotated at an incremented\nangle invx-vyplane. This results in periodically sepa-\nrating red and blue patches, absent initially. Note that\nthe initial transverse velocity perturbation at t= 0 is in-\ntroduced with no such nonuniformity with respect to vz.\nThis nonuniformity clearly develops over the time-scale\nof an electron gyroperiod.\nB. Kinetic model for initial-value transverse\nelectromagnetic perturbations\nOut of the several kinetic modi\fcations of f1witnessed\nabove, those attributable to the wave \felds and those to\nballistic e\u000bects can be distinguished and analyzed based\non the linearized Laplace-Fourier transformed Vlasov-\nMaxwell system of equations,\n(p+ik\u0001v)fkp1(v) =fk1(t= 0)\n+e\nme\u0012v\u0002B0\nc\u0013\n\u0001rvfkp1(v)\n+e\nme\u0012\nEkp1+v\u0002Bkp1\nc\u0013\n\u0001rvf0(v) (22)9\n0\n20\n2\n40\n60\n/ f(vx,vy)max\n80\n100\nt +ce\n-0.2\n 1\n120\nvz/c\n0\n0\n0.2\n2\nt +ce\n0.2\n 1\nvz/c\n0\n0\n-0.2\nFIG. 9. Time evolution of the maximum amplitude, f1max, of\nelectrons velocity distribution function perturbation f1(vx,vy)\nplotted in Figs. 7 and 8 in full range of vz, for the hot electrons\ncase (vth= 0:02c). Plots (a) and (b) present relatively rotated\nviews of the same surface for clearer exposure of the both sides\nabout the surface maximum.\nBkp1=Bk1(t= 0) +c\nip(k\u0002Ekp1): (23)\nand\nEkp1=Ek1(t= 0)\u0000c\nip\u0012\nk\u0002Bkp1\u00004\u0019\ncjkp1\u0013\n;(24)\nwherep=\u0000i!r+!iandEkp1andBkp1are the trans-\nformed wave electric and magnetic \felds, respectively.\nUsing the well known identity for conjugate compo-\nnents of the velocity variable, vx=v?cos\u001eandvy=\nv?sin\u001e,\n@fpk1(v)\n@\u001e=\u0012\n\u0000vy@\n@vx+vx@\n@vy\u0013\n=\u0000(v\u0002^ z)\u0001rvfpk1(v);\nEq. (22) can be rewritten as,\n(p+ik\u0001v)\n\ncefkp1(v) +@fpk1(v)\n@\u001e=fk1(t= 0)\n\nce\n+e\nme\nce\u0012\nEkp1+v\u0002Bkp1\nc\u0013\n\u0001rvf0(v):(25)\nNote that the wave \feld generated vzspeci\fc modi\fca-\ntions, including (i) the acceleration of resonant electrons\nand (ii) the slowly building electron drift for vz\u0018vphase,\ncan be resolved, approximately, from the fundamentally\nkinetic ballistic modi\fcations in the f1at shorter times\n(\n\u00001\nce< t < !\u00001) when the latter dominate the former.\nIn order to do this, we \frst write the right handed polar-\nized component of the Vlasov equation [26] applicable to\nthe whistler mode,\nd\nd\u001e\u001a\nf1+(v) exp\u0014(p+ikvz)\n\nce\u001e\u0015\u001b\n=\nfk1+(t= 0)\n\nceexp\u0014(p+ikvz)\n\nce\u001e+i\u001e\u0015\n+F+(vz;v?) exp\u0014(p+ikvz)\n\nce\u001e+i\u001e\u0015\n(26)\nwherev?= (v2\nx+v2\ny)1\n2,fk1+is right handed circularly\npolarized part of the initial perturbation. The quantity\n0\n2\n5\n/ fmax/ / fmax(t=0)\nt +ce\n-0.2\n 1\n10\nvz/c\n0\n00.2FIG. 10. Surface in the \fgure normalized to individual initial\nvalues at each vz.\nF+(vz;v?) is obtained by ignoring Bk(t= 0) in Eq. (23),\nas,\nF+(vz;v?) =e\nme\nce\u0014\u0012\n1\u0000kvz\n!\u0013@f0\n@v?+kvz\n!@f0\n@v?\u0015E+p\n2\n(27)\nwhereE+=Ex\u0000iEy. Both sides of Eq. (26) can be\nintegrated with respect to \u001efrom\u001e=\u00001 to instant\u001e\nby assuming very small imaginary part in !, yielding the\ntime asymptotic solution,\nfpk1+(v) =fk1+(t= 0)ei\u001e\n(p+ikvz+i\nce)\n+\nce\n(p+ikvz+i\nce)F+(vz;v?)ei\u001e:(28)\nThis means that all the right handed circularly polar-\nized transverse modes are obtainable by accounting for\nthe poles of the second term, which represents the con-\ntribution of the electric and magnetic \felds of the wave.\nFor example, the Whistler mode corresponds to E+ob-\ntained in terms of B+andJ+using (22)-(24) in the limit\n\npi\nce) RHCP and LHCP waves (see Fig. 3) and\nshould not be confused with the discussed low frequency\nhigh amplitude oscillations in the small vzregion.This additional frequency in the surface plot contours\nfor largervz(>0) results from the streaming electrons\nimporting information of polarization from adjacent lo-\ncations based on the transformation Eq. (20). Note that\nbecause of the periodic set up, for a larger velocity, say\nvz\u0018n(k\u000et)\u00001, the information being imported would\ncontainncycles of modulation, corresponding to ncon-\nsecutive spatial periods sampled by fast streaming elec-\ntrons in the time interval \u000etpastt= 0. The frequency\nof temporal modulation in f1max(vz;t) therefore must in-\ncrease with vzvalue being examined at a \fxed location\nz=L=2. This dependence on vzis indeed con\frmed by\nplotting, in Fig. 10, the amplitude f1max(vz;t) normal-\nized to its t= 0 reference value for each vz, such that\nthe modulations become clearly identi\fable even at the\nlargervzwhere the distribution function f0as well as the\namplitude of f1drop to signi\fcantly small values.\nThe perturbation information at z=L=2 arriving\nloaded on electrons streaming with phase velocity is\nplayed faster than the rotation of bulk polarization (as\n! < \nce) but imports an increasingly delayed phase\n(causing the rotation appearing to stop or reversed over\ncertain intervals). Additionally, after certain time inter-\nval, it begins to display the signatures of freshly devel-\nopedE1\u0002B0drift.\nFinally, since electrons with vz> v resare e\u000bectively\nmagnetized to a higher degree on the wave time scale\nexperienced by them, ( !\u0000kvz)\u00001\u001d\n\u00001\nce, they partici-\npate rather e\u000eciently in the collective wave propagation.\nHowever, a Doppler down-shifted frequency experienced,\nand responded to, by this majority of electrons causes the\nwave collective !rto reduce in accordance with disper-\nsion (17). This reduction in !ris duly captured by the\npresent Vlasov simulation results, as clear by copmparing\nFig. 7 with Fig. 8.\nVI. SUMMARY AND CONCLUSIONS\nThe kinetic damping mechanism of low frequency\ntransverse perturbation propagating parallel to the mag-\nnetic \feld in a magnetized warm plasma is simulated\nby means of Vlasov simulations. The impact of \fnite\ntemperature on the wave frequency and the underly-\ning physics of the analytical approximations prescrib-\ning reduction in the real frequency of the wave in a\nwarm plasma is addressed and resolved by full 4D phase-\nspatiotemporal evolution simulated of the electron distri-\nbution function.\nThe analysis of the Vlasov simulation output illustrates\nand estimates the evolution of initially coherent pertur-\nbations that undergo damping by mutually separable ini-\ntial kinetic evolution operating over the global and local\nregions of the of electron velocity space. The net kinetic\ne\u000bects contributed reduction in the strength of pertur-\nbations is caused by both, (i) a phase-mixing process\npresent throughout the velocity space and (ii) the reso-\nnant damping which is dominant at the resonant velocity,11\naccelerating only the resonant electrons. The quantita-\ntive analysis of the amplitude oscillation produced by the\nformer is done and their interplay, at short times, with\nthe modi\fcations of the initial perturbations by the reso-\nnant electrons is shown to result in enhanced decay of the\nperturbation amplitude as compared to typical long time\nestimates of the damping. It is emphasized that while\nboth these mechanisms in a collisionless warm plasma\nconserve the entropy, it is only the resonant e\u000bects that\ncontributes to the routinely estimated damping rate \r.\nThe analytically predicted reduction in the real wave fre-\nquency!ris duly recovered by the simulations. This is\nshown to follow from a greater e\u000bective magnetization\nof dominant vz> vreselectron population, which, how-ever, experiences frequency of the wave Doppler-shifted\nto smaller values.\nWe conclude by discussing that access to rather self-\nconsistent analytical initial value solutions of the problem\n(22)-(24) remain of high relevance. They should enable\nthe output from the present fully kinetic electromagnetic\ntransverse wave Vlasov simulations to more carefully de-\ntermine, for example, the spectral boundaries of the elec-\ntromagnetic collisionless plasma turbulence. The associ-\nated particle and energy transport in the space and fusion\nplasma, in turn, can also be approached more determin-\nistically.\nAcknowlegement: The simulations are performed on the\nIPR HPC supercomputing cluster ANTYA.\n[1] J. E. Borovsky, Journal of Geophysical Research 117,\n5104 (2012).\n[2] J. J. Podesta, D. A. Roberts, and M. L. Goldstein, The\nAstrophysical Journal 664, 543 (2007).\n[3] S. P. Gary, S. Saito, and Y. Narita, The Astrophysical\nJournal 716, 1332 (2010).\n[4] X. Xu, L. Chen, C. Zhou, X. Liu, Z. Xia, J. J. Simpson,\nand Y. Zhang, Journal of Geophysical Research: Space\nPhysics 125, e2019JA027750 (2017).\n[5] R. A. Helliwell, Whistlers and related ionospheric phe-\nnomena (Press, Stanford, Calif: Stanford Univ., 1965).\n[6] F. Xiao, S. Liu, X. Tao, Z. Su, Q. Zhou, C. Yang, Z. He,\nY. He, Z. Gao, D. N. Baker, H. E. Spence, G. D. Reeves,\nH. O. Funsten, and J. B. Blake, Journal of Geophysical\nResearch: Space Physics 122, 3201{3211 (2017).\n[7] R. M. Thorne, Geophys. Res. Lett. 37, L22107 (2010).\n[8] L. Chen, R. M. Thorne, Y. Shprits, and B. Ni, Journal of\nGeophysical Research: Space Physics 118, 2185 (2013).\n[9] N. A. Krall and A. W. Trivelpiece, Principles of Plasma\nPhysics (San Francisco Press Inc., San Francisco, 1986).\n[10] L. D. Landau, C. R. Acad. Sci. U. R. S. S. 44, 311 (1944).\n[11] L. D. Landau, J. Phys. U.S.S.R. 10, 25 (1946).\n[12] R. Lutomirski, The Physics of Fluids 13, 149 (1970).\n[13] E. Fijalkow, Computer physics communications 116, 329\n(1999).\n[14] A. Mangeney, F. Califano, C. Cavazzonic, and\nP.Travnicek, Journal of Computational Physics 179, 495\n(2002).\n[15] L. Palodhi, F. Califano, and F. Pegoraro, Plasma Physics\nand Controlled Fusion 52, 095007 (2010).[16] F. Valentini, P. Travnicek, F. Califano, P. Hellinger, and\nA. Mangeney, Journal of Computational Physics 225,\n753 (2007).\n[17] J. R. Shuster, N. Bessho, S. Wang, and J. Ng, Physics\nof Plasmas 28, 122902 (2021).\n[18] T. F ul op, H. Smith, and G. Pokol, Physics of Plasmas\n16, 022502 (2009).\n[19] C. Schreiner, P. Kilian, and F. Spanier, Communications\nin Computational Physics 21, 947 (2017).\n[20] C. Schreiner, Numerical modelling of the microphysical\nfoundation of astrophysical particle acceleration , Ph.D.\nthesis, North-West University (South Africa), Potchef-\nstroom Campus (2016).\n[21] D. Mandal and D. Sharma, Journal of Physics: Confer-\nence Series 759, 012068 (2016).\n[22] D. Mandal, D. Sharma, and H. Schamel, Phys. Plasmas\n27, 022102 (2020).\n[23] S. P. Gary, Theory of space plasma microinstabilities , 7\n(Cambridge university press, 1993).\n[24] F. F. Chen, Introduction to Plasma Physics (Springer\nScience & Business Media, 2012).\n[25] T. H. Stix, Waves in plasmas (Springer Science & Busi-\nness Media, 1992).\n[26] J. A. Bittencourt, Fundamentals of plasma physics\n(Springer Science & Business Media, 2013).\n[27] M. Abramowitz and I. A. Stegun, Handbook of Mathe-\nmatical Functions with Formulas, Graphs, and Mathe-\nmatical Tables , ninth dover printing, tenth gpo printing\ned. (Dover, New York, 1964)." }, { "title": "1603.07977v1.Large_spin_pumping_effect_in_antisymmetric_precession_of_Ni___79__Fe___21___Ru_Ni___79__Fe___21__.pdf", "content": "Large spin pumping e\u000bect in antisymmetric precession of\nNi79Fe21/Ru/Ni 79Fe21\nH. Yang,1Y. Li,1and W.E. Bailey1,a)\nMaterials Science and Engineering, Dept. of Applied Physics and Applied Mathematics, Columbia University,\nNew York NY 10027\n(Dated: 16 September 2021)\nIn magnetic trilayer structures, a contribution to the Gilbert damping of ferromagnetic resonance arises from\nspin currents pumped from one layer to another. This contribution has been demonstrated for layers with\nweakly coupled, separated resonances, where magnetization dynamics are excited predominantly in one layer\nand the other layer acts as a spin sink. Here we show that trilayer structures in which magnetizations are\nexcited simultaneously, antisymmetrically, show a spin-pumping e\u000bect roughly twice as large. The antisym-\nmetric (optical) mode of antiferromagnetically coupled Ni 79Fe21(8nm)/Ru/Ni 79Fe21(8nm) trilayers shows a\nGilbert damping constant greater than that of the symmetric (acoustic) mode by an amount as large as\nthe intrinsic damping of Py (\u0001 \u000b'0.006). The e\u000bect is shown equally in \feld-normal and \feld-parallel to\n\flm plane geometries over 3-25 GHz. The results con\frm a prediction of the spin pumping model and have\nimplications for the use of synthetic antiferromagnets (SAF)-structures in GHz devices.\nPumped spin currents1,2are widely understood to in-\n\ruence the magnetization dynamics of ultrathin \flms\nand heterostructures. These spin currents increase the\nGilbert damping or decrease the relaxation time for thin\nferromagnets at GHz frequencies. The size of the e\u000bect\nhas been parametrized through the e\u000bective spin mixing\nconductance g\"#\nr, which relates the spin current pumped\nout of the ferromagnet, transverse to its static (time-\naveraged) magnetization, to its precessional amplitude\nand frequency. The spin mixing conductance is inter-\nesting also because it determines the transport of pure\nspin current across interfaces in quasistatic spin trans-\nport, manifested in e.g. the spin Hall e\u000bect.\nIn the spin pumping e\u000bect, spin current is pumped\naway from a ferromagnet / normal metal (F 1/N) in-\nterface, through precession of F1, and is absorbed else-\nwhere in the structure, causing angular momentum loss\nand damping of F1. The spin current can be absorbed\nthrough di\u000berent processes in di\u000berent materials. When\ninjected into paramagnetic metals (Pt, Pd, Ru, and oth-\ners), the spin current relaxes exponentially with para-\nmagnetic layer thickness3{5. The relaxation process has\nbeen likened to spin-\rip scattering as measured in CPP-\nGMR, where spin-\rip events are localized to heavy-metal\nimpurities6and the measurement reveals the spin di\u000bu-\nsion length \u0015SD. When injected into other ferromagnets\n(F2in F 1/N/F 2), the spin current is absorbed through\nits torque on magnetization5,7. A similar process appears\nto be relevant for antiferromagnets as well8.\nIn F 1/N/F 2structures, only half of the total possible\nspin pumping e\u000bect has been detected up until now. For\nwell-separated resonances of F1andF2, only one layer\nwill precess with large amplitude at a given frequency\n!, and spin current is pumped from a precessing F1into\na staticF2. If both layers precess symmetrically, with\na)Electronic mail: Contact author. web54@columbia.eduthe same amplitude and phase, equal and opposite spin\ncurrents are pumped into and out of each layer, causing\nno net e\u000bect on damping. The di\u000berence between the\nsymmetric mode and the uncoupled mode, increased by\na spin pumping damping \u000bspwas detected \frst in epi-\ntaxial Fe/Au/Fe structures9. However, if the magnetiza-\ntions can be excited with antisymmetric precession, the\ncoupled mode should be damped by twice that amount,\n2\u000bsp. Takahashi10has published an explicit prediction of\nthis \"giant spin pumping e\u000bect\" very recently, including\nan estimate of a fourfold enhanced spin accumulation in\nthe central layer.\nIn this paper, we show that a very large spin pump-\ning e\u000bect can be realized in antisymmetric precession of\nPy(8 nm)/Ru(0.70-1.25 nm)/Py(8 nm) synthetic antifer-\nromagnets (SAF, Py=Ni 79Fe21). The e\u000bect is roughly\ntwice that measured in Py trilayers with uncoupled\nprecession. Variable-frequency ferromagnetic resonance\n(FMR) measurements show, for structures with magne-\ntization saturated in the \flm plane or normal to the \flm\nplane, that symmetric (acoustic mode) precession of the\ntrilayer has almost no additional damping, but the an-\ntisymmetric (optical mode) precession has an additional\nGilbert damping of \u00180.006, compared with an uncou-\npled Py(8nm) layer in a F 1/N/F 2structure of\u00180.003.\nThe interaction stabilizes the antiparallel magnetization\nstate of SAF structures, used widely in di\u000berent elements\nof high-speed magnetic information storage, at GHz fre-\nquencies.\nMethod: Ta(5 nm)/Cu(3 nm)/Ni 79Fe21(8\nnm)/Ru(tRu)/Ni 79Fe21(8 nm)/Cu(3 nm)/SiO 2(5 nm),\ntRu= 0.7 - 1.2 nm heterostructures were deposited by\nultrahigh vacuum (UHV) sputtering at a base pressure\nof 5\u000210\u00009Torr on thermally oxidized Si substrates.\nThe Ru thckness range was centered about the second\nantiferromagnetic maximum of interlayer exchange\ncoupling (IEC) for Py/Ru/Py superlattices, 8-12 \u0017A,\nestablished \frst by Brillouin light scattering (BLS)\nmeasurement11. Oscillatory IEC in this system, as aarXiv:1603.07977v1 [cond-mat.mtrl-sci] 25 Mar 20162\nfunction of tRu, is identical to that in the more widely\nstudied Co/Ru( tRu)/Co superlattices12, 11.5 \u0017A, but is\nroughly antiphase to it. An in-plane magnetic \feld bias\nof 200 G, rotating in phase with the sample, was applied\nduring deposition as described in13.\nThe \flms were characterized using variable fre-\nquency, swept-\feld, magnetic-\feld modulated ferromag-\nnetic resonance (FMR). Transmission measurements\nwere recorded through a coplanar waveguide (CPW) with\ncenter conductor width of 300 \u0016m, with the \flms placed\ndirectly over the center conductor, using a microwave\ndiode signal locked in to magnetic \feld bias modulation.\nFMR measurements were recorded for magnetic \feld bias\nHBapplied both in the \flm plane (parallel condition, pc)\nand perpendicular to the plane (normal condition, nc.)\nAn azimuthal alignment step was important for the nc\nmeasurements. For these, the sample was rotated on twoaxes to maximize the \feld for resonance at 3 GHz.\nFor all FMR measurements, the sample magnetization\nwas saturated along the applied \feld direction, simplify-\ning extraction of Gilbert damping \u000b. The measurements\ndi\u000ber in this sense from low-frequency measurements of\nsimilar Py/Ru/Py trilayer structures by Belmenguenai et\nal14, or broadband measurements of (sti\u000ber) [Co/Cu] \u000210\nmultilayers by Tanaka et al15. In these studies, e\u000bects\non\u000bcould not be distinguished from those on inhomo-\ngeneous broadening.\nModel: In the measurements, we compare the mag-\nnitude of the damping, estimated by variable-frequency\nFMR linewidth through \u0001 H1=2= \u0001H0+ 2\u000b!=j\rj, and\nthe interlayer exchange coupling (IEC) measured through\nthe splitting of the resonances. Coupling terms between\nlayersiandjare introduced into the Landau-Lifshitz-\nGilbert equations for magnetization dynamics through\n_mi=\u0000mi\u0002(\riHe\u000b+!ex;imj) +\u000b0mi\u0002_mi+\u000bsp;i(mi\u0002_ mi\u0000mj\u0002_ mj) (1)\nincgsunits, where we de\fned magnetization unit vec-\ntors as m1=M1=Ms;1,m2=M2=Ms;2withMs;ithe\nsaturation moments of layer i. The coupling constants\nare, for the IEC term, !ex;i\u0011\riAex=(Ms;iti), where\nthe energy per unit area of the system can be written\nuA=\u0000Aexmi\u0001mj, andtiis the thickness of layer i. Pos-\nitive values of Aexcorrespond to ferromagnetic coupling,\nnegative values to antiferromagnetic coupling. The spin\npumping damping term is \u000bsp;i\u0011\r\u0016h~gFNF\n\"#=(4\u0019Ms;iti),\nwhere ~gFNF\n\"# is the spin mixing conductance of the tri-\nlayer in units of nm\u00002.\u000b0is the bulk damping for the\nlayer.\nThe collective modes of 1 ;2 are found from small-\namplitude solutions of Equations 1 for i= 1;2. General\nsolutions for resonance frequencies with arbitrary mag-\nnetization alignment, not cognizant of any spin pump-\ning damping or dynamic coupling, were developed by\nZhang et al12. In our experiment, to the extent possi-\nble, layers 1 ;2 are identical in deposited thickness, mag-\nnetization, and interface anisotropy (each with Cu the\nopposite side from Ru). Therefore if !irepresents the\nFMR frequency (dependent on bias \feld HB) of each\nlayeri, the two layers have !1=!2=!0. In this\nlimit, there are two collective modes: a perfectly sym-\nmetric mode Sand a perfectly antisymmetric mode A\nwith complex frequencies f!S= (1\u0000i\u000b0)!0andf!A=\n(1\u0000i\u000b0\u00002i\u000bsp) (!0+ 2!ex). The Gilbert damping for\nthe modes, \u000bk=\u0000Im(f!k)=Re(!k\nf), wherek= (S;A),\nand the resonance \felds Hk\nBsatisfy\nHA\nB=HS\nB+ 2HexHex=\u0000Aex=(MstF) (2)\n\u000bA=\u000bS+ 2\u000bsp\u000bsp=\r\u0016h~gFNF\n\"#=(4\u0019Ms;iti) (3)\nand!ex=\rHex. Note that there is no relationshipin this limit between the strength of the exchange cou-\nplingAexand the spin-pumping damping 2 \u000bspexpressed\nin the antisymmetric mode. The spin pumping damping\nand the interlayer exchange coupling can be read sim-\nply from the di\u000berences in the the Gilbert damping \u000b\nand resonance \felds between the antisymmetric and sym-\nmetric modes. The asymmetric mode will have a higher\ndamping by 2 \u000bspfor anyAexand a higher resonance\n\feld forAex<0, i.e. for antiferromagnetic IEC: because\nthe ground state of the magnetization is antiparallel at\nzero applied \feld, antisymmetric excitation rotates mag-\nnetizations towards the ground state and is lower in fre-\nquency than symmetric excitation.\nResults: Samplepc\u0000andnc\u0000FMR data are shown in\nFigure 1. Raw data traces (lock-in voltage) as a func-\ntion of applied bias \feld HBat 10 GHz are shown in the\ninset. We observe an intense resonance at low \feld and\nresonance weaker by a factor of 20-100 at higher \feld. On\nthe basis of the intensities, as well as supporting MOKE\nmeasurements, we assign the lower-\feld resonance to the\nsymmetric, or \"acoustic\" mode and the higher-\feld res-\nonance to the antisymmetric, or \"optical\" mode. Similar\nbehavior is seen in the nc- andpc\u0000FMR measurements.\nIn Figure 1a) and c), which summarizes the \felds-\nfor-resonance !(HB), there is a rigid shift of the\nantisymmetric-mode resonances to higher bias \felds HB,\nas predicted by the theory. The lines show \fts to\nthe Kittel resonance, !pc=\rr\nHeff\u0010\nHeff+ 4\u0019Meff\ns\u0011\n,\n!nc=\r\u0000\nHeff\u00004\u0019Meff\ns\u0001\nwith an additional e\u000bective\n\feld along the magnetization direction for the antisym-\nmetric mode: Heff;S =HB, andHeff;A =HB\u0000\n8\u0019Aex=(4\u0019MstF).\nIn Figure 2, we show coupling parameters, as a func-3\nHBHeffm(t)\nHBHeff m(t)(a)\n(c)(b)\n(d)10 GHz10 GHz\n1/21/2\nFIG. 1. FMR measurement of Ni 79Fe21(8\nnm)/Ru(tRu)/Ni 79Fe21(8 nm) trilayers; example shown\nfortRu= 1.2 nm. Inset : lock-in signal, transmitted power\nat 10 GHz, as a function of bias \feld HB, for a) pc-FMR\nand c) nc-FMR. A strong resonance is observed at lower HB\nand a weaker one at higher HB, attributed to the symmetric\n(S) and antisymmetric (A) modes, respectively. a), c): Field\nfor resonance !(HB) for the two resonances. Lines are \fts\nto the Kittel resonance expression, assuming an additional,\nconstant, positive \feld shift for !A,Hex=\u00002Aex=(MstF)\ndue to antiferromagnetic interlayer coupling Aex<0. b)\npc-FMR and d) nc-FMR linewidth as a function of frequency\n\u0001Hpp(!) for \fts to Gilbert damping \u000b.\ntion of Ru thickness, extracted from the FMR measure-\nments illustrated in Figure 1. Coupling \felds are mea-\nsured directly from the di\u000berence between the symmet-\nric and antisymmetric mode positions and plotted in\nFigure 2a. We convert the \feld shift to antiferromag-\nnetic IEC constant Aex<0 through Equation 2, us-\ning the thickness tF= 8 nm and bulk magnetization\n4\u0019Ms= 10.7 kG4. The extracted exchange coupling\nstrength in pc-FMR has a maximum antiferromagnetic\nvalue ofAex=\u00000.2 erg/cm2, which agrees to 5% with\nthat measured by Fassbender et al11for [Py/Ru] Nsu-\nperlattices.\nThe central result of the paper is shown in Figure 2 b).\nWe compare the damping \u000bof the symmetric ( S) and an-\ntisymmetric ( A) modes, measured both through pc-FMR\nandnc-FMR. The values measured in the two FMR ge-\nometries agree closely for the symmetric modes, for which\nsignals are larger and resolution is higher. They agree\nroughly within experimental error for the antisymmetric\nmodes, with no systematic di\u000berence. The antisymmetric\nmodes clearly have a higher damping than the symmetric\nmodes. Averaged over all thickness points, the enhanced\ndamping is roughly \u000bA\u0000\u000bS= 0.006.\nDiscussion: The damping enhancement of the anti-\nsymmetric ( A\u0000) mode over the symmetric ( S\u0000) mode,\nshown in Figure 2b), is a large e\u000bect. The value is close\nto the intrinsic bulk damping \u000b0\u00180.007 for Ni 79Fe21.\n0.7 0.8 0.9 1.0 1.1 1.2 1.301002003004005006002Hex (Oe)a)\nnc,Hexpc,HexMOKE,Hex\n0.7 0.8 0.9 1.0 1.1 1.2 1.3\ntRu (nm)0.0060.0080.0100.0120.0140.0160.018α\nα0α0+αspα0+2αspb)pc, S\nnc, Spc, A\nnc, A0.000.050.100.150.20\n−Aex (erg/cm2)FIG. 2. Coupling parameters for Py/Ru/Py trilayers. a):\nInterlayer (static) coupling from resonance \feld shift of an-\ntisymmetric mode; see Fig 1 a),c). The antiferromagnetic\nexchange parameter Aexis extracted through Eq 2, in agree-\nment with values found in Ref11. The line is a guide to the\neye. b) Spin pumping (dynamic) coupling from damping of\nthe symmetric (S) and antisymmetric (A) modes; see Fig 1\nb), d). The spin pumping damping for uncoupled layer pre-\ncession in Py/Ru/Py, \u000bspis shown for comparsion. Dotted\nlines show the possible e\u000bect of \u0018100 Oe detuning for the two\nPy layers. See text for details.\nWe compare the value with the value 2 \u000bspexpected from\ntheory for the antisymmetric mode and written in Eq 3.\nThe interfacial spin mixing conductance for Ni 79Fe21/Ru,\nwas found in Ref.16to be ~gFN\n\"#= 24 nm\u00002. For a F/N/F\nstructure, in the limit of ballistic transport with no spin\nrelaxation through N, the e\u000bective spin mixing conduc-\ntance is ~gFN\n\"#=2: spin current must cross two interfaces\nto relax in the opposite Flayer, and the conductance re-\n\rects two series resistances17. This yields \u000bsp= 0:0027.\nThe observed enhancement matches well with, and per-\nhaps exceeds slightly, the \"giant\" spin pumping e\u000bect of\n2\u000bsp, as shown.\nLittle dependence of the Gilbert damping enhancement\n\u000bA\u0000\u000bSon the resonance \feld shift HA\u0000HScan be ob-\nserved in Figure 2 a,b. We believe that this independence\nre\rects close tuning of the resonance frequencies for Py\nlayers 1 and 2, as designed in the depositions. For \f-\nnite detuning \u0001 !de\fned through !2=!0+ \u0001!and\n!1=!0\u0000\u0001!, the modes change. Symmetric and anti-\nsymmetric modes become hybridized as S0andA0, and4\nthe di\u000berence in damping is reduced. De\fning g\u0001!2=\n(1\u0000i\u000b0\n) (1\u0000i\u000b0\n\u00002i\u000bsp\n) \u0001!2, it is straightfor-\nward to show that for the nc-case, the mode frequen-\ncies are!S0;A0= (f!S+f!A)=2\u0006q\n(f!S\u0000f!A)2=4 +g\u0001!2.\nThe relevant parameter is the frequency detuning nor-\nmalized to the exchange (coupling) frequency, z\u0011\n\u0001!=(2!ex); ifz\u001d1, the layers have well-separated\nmodes, and each recovers the uncoupled damping en-\nhancement of \u000bsp,\u000bS0;A0=\u000b0+\u000bspidenti\fed in Refs5,9.\nThe possibility of \fnite detuning, assuming ( !2\u0000\n!1)=\r= 100 Oe, is shown in Figure 2b), with the dot-\nted lines. The small \u0000zlimit for detuning \fnds sym-\nmetric e\u000bects on damping of the S0andA0modes, with\n\u000bS0=\u000b0+ 2\u000bspz2and\u000bA0=\u000b0+ 2\u000bsp(1\u0000z2), respec-\ntively, recovering perfect symmetric and antisymmetric\nmode values for z= 0. We assume that the \feld split-\nting shown in Figure 2 a) gives an accurate measure of\n2!ex=\r, as supported by the MOKE results. This value\ngoes into the denominator of z. We \fnd a reasonable \ft\nto the dependence of SandAdamping on Ru thickness,\nimplicit in the coupling. For the highest coupling pionts,\nthe damping values closely reach the low- zlimit, and we\nbelieve that the \"giant\" spin pumping result of 2 \u000bspis\nevident here.\nWe would like to point out next that it was not a-\npriori obvious that the Py/Ru/Py SAF would exhibit\nthe observed damping. Ru could behave in two limits in\nthe context of spin pumping: either as a passive spin-\nsink layer, or as a ballistic transmission layer supporting\ntransverse spin-current transmission from one Py layer to\nthe other. Our results show that Ru behaves as the latter\nin this thickness range. The symmetric-mode damping of\nthe SAF structure, extrapolated back to zero Ru thick-\nness, is identical within experimental resolution ( \u001810\u00004)\nto that of a single Py \flm 16 nm thick measured in nc-\nFMR (\"\u000b0\" line in Fig 2b.) If Ru, or the Py/Ru interface,\ndepolarized pumped spin current very strongly over this\nthickness range as has been proposed for Pt18, we would\nexpect an immediate increase in damping of the acous-\ntic mode by the amount of \u0018\u000bsp. Instead, the volume-\ndependent Ru depolarization in spin pumping has an (ex-\nponential) characteristic length of \u0015SD\u001810 nm5, and\nattenuation over the range explored of \u00181 nm is negli-\ngible.\nPerspectives: Finally, we would like to highlight some\nimplications of the study. First, as the study con\frms\nthe prediction of a \"giant\" spin pumping e\u000bect as pro-\nposed by Takahashi10, it is plausible that the greatly en-\nhanced values of spin accumulation predicted there may\nbe supported by Ru in Py/Ru/Py synthetic antiferro-\nmagnets (SAFs). These spin accumulations would di\u000ber\nstrongly in the excitation of symmetric and antisymmet-\nric modes, and may then provide a clear signature intime-resolved x-ray magnetooptical techniques19, similar\nto the observation of static spin accumulation in Cu re-\nported recently20.\nSecond, in most device applications of synthetic an-\ntiferromagnets, it is not desirable to excite the antisym-\nmetric (optical) mode. SAFs are used in the pinned layer\nof MTJ/spin valve structures to increase exchange bias\nand in the free layer to decrease (magnetostatic) stray\n\felds. Both of these functions are degraded if the opti-\ncal, or asymmetric mode of the SAF is excited. Accord-\ning to our results, at GHz frequencies near FMR, the\nsusceptibility of the antisymmetric mode is reduced sub-\nstantially, here by a factor of two (from 1/ \u000b) for nc-FMR ,\ndue to spin pumping. This reduction of \u001fon resonance\nwill scale inversely with layer thickness. The damping,\nand susceptibility, of the desired symmetric (acoustic)\nmode is unchanged, on the other hand, implying that\nspin pumping favors the excitation the symmetric mode\nfor thin Ru, the typical operating point.\nWe acknowledge NSF-DMR-1411160 for support.\n1Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev.\nLett. 88, 117601 (2002).\n2Y. Tserkovnyak, A. Brataas, G. Bauer, and B. Halperin, Reviews\nin Modern Physics 77, 1375 (2005).\n3S. Mizukami, Y. Ando, and T. Miyazaki, Journal of Magnetism\nand Magnetic Materials 239, 42 (2002).\n4A. Ghosh, J. F. Sierra, S. Au\u000bret, U. Ebels, and W. E. Bailey,\nApplied Physics Letters 98, (2011).\n5A. Ghosh, S. Au\u000bret, U. Ebels, and W. E. Bailey, Phys. Rev.\nLett. 109, 127202 (2012).\n6J. Bass and W. Pratt, Journal of Physics: Condensed Matter 19,\n41 pp. (2007).\n7G. Woltersdorf, O. Mosendz, B. Heinrich, and C. H. Back, Phys-\nical Review Letters 99, 246603 (2007).\n8P. Merodio, A. Ghosh, C. Lemonias, E. Gautier, U. Ebels,\nV. Baltz, and W. Bailey, Applied Physics Letters 104(2014).\n9B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Ur-\nban, and G. E. W. Bauer, Phys. Rev. Lett. 90, 187601 (2003).\n10S. Takahashi, Applied Physics Letters 104(2014).\n11J. Fassbender, F. Nortemann, R. Stamps, R. Camley, B. Hille-\nbrands, G. Guntherodt, and S. Parkin, Journal of Magnetism\nand Magnetic Materials 121, 270 (1993).\n12Z. Zhang, L. Zhou, P. E. Wigen, and K. Ounadjela, Phys. Rev.\nB50, 6094 (1994).\n13C. Cheng, N. Sturcken, K. Shepard, and W. Bailey, Review of\nScienti\fc Instruments 83, 063903 (2012).\n14M. Belmeguenai, T. Martin, G. Woltersdorf, M. Maier, and\nG. Bayreuther, Physical Review B 76(2007).\n15K. Tanaka, T. Moriyama, M. Nagata, T. Seki, K. Takanashi,\nS. Takahashi, and T. Ono, Applied Physics Express 7(2014).\n16N. Behera, M. S. Singh, S. Chaudhary, D. K. Pandya, and P. K.\nMuduli, Journal of Applied Physics 117(2015).\n17See eqs. 31, 74, 81 in Ref2.\n18J.-C. Rojas-S\u0013 anchez, N. Reyren, P. Laczkowski, W. Savero, J.-\nP. Attan\u0013 e, C. Deranlot, M. Jamet, J.-M. George, L. Vila, and\nH. Ja\u000br\u0012 es, Phys. Rev. Lett. 112, 106602 (2014).\n19W. Bailey, C. Cheng, R. Knut, O. Karis, S. Au\u000bret, S. Zohar,\nD. Keavney, P. Warnicke, J.-S. Lee, and D. Arena, Nature Com-\nmunications 4, 2025 (2013).\n20R. Kukreja, S. Bonetti, Z. Chen, D. Backes, Y. Acremann, J. A.\nKatine, A. D. Kent, H. A. D urr, H. Ohldag, and J. St ohr, Phys.\nRev. Lett. 115, 096601 (2015)." }, { "title": "0905.4699v2.Resonant_Nonlinear_Damping_of_Quantized_Spin_Waves_in_Ferromagnetic_Nanowires.pdf", "content": "Boone et al. Page 1 of 10 October 15, 2009 Resonant Nonlinear Damping of Quantized Spin Waves in \nFerromagnetic Nanowires: A Spin-Torque Ferromagneti c \nResonance Study \n \nC. T. Boone 1, J. A. Katine 2, J. R. Childress 2, V. Tiberkevich 3, A. Slavin 3, J. Zhu 1, X. Cheng 1, \nI. N. Krivorotov 1 \n1. Department of Physics and Astronomy, University of California, Irvine, California 92697 \n2. Hitachi Global Storage Technologies, San Jose, Cali fornia 95135 \n3. Department of Physics, Oakland University, Rocheste r, Michigan 48309 \n \nAbstract \n \nWe use spin torque ferromagnetic resonance to measu re the spectral properties of dipole-\nexchange spin waves in permalloy nanowires. Our mea surements reveal that geometric \nconfinement has a profound effect on the damping of spin waves in the nanowire geometry. \nThe damping parameter of the lowest-energy quantize d spin wave mode depends on \napplied magnetic field in a resonant way and exhibi ts a maximum at a field that increases \nwith decreasing nanowire width. This enhancement of damping originates from a \nnonlinear resonant three-magnon confluence process allowed at a particular bias field \nvalue determined by quantization of the spin wave s pectrum in the nanowire geometry.Boone et al. Page 2 of 10 October 15, 2009 Spin transfer torque (STT) [1-4] is emerging as a n ew tool for studies of magnetization \ndynamics in nanostructures [5-13]. In this Letter, we use STT to measure properties of dipole-\nexchange spin waves in permalloy (Py = Ni 86Fe 14) nanowires. Our measurements reveal that \ngeometric confinement of spin waves in nanowires ha s a profound effect on the spin wave \ndamping [14]. We find a resonant dependence of the damping parameter of the lowest-energy \nspin wave mode on external magnetic field − a maximum of damping is observed at a field that \nincreases with decreasing nanowire width. We explai n this resonant enhancement of damping by \na nonlinear three-magnon confluence process in whic h two quanta of the first ( n=1) spin wave \nmode merge into a single quantum of the third ( n =3) mode. \nBrillouin light scattering [15,16], ferromagnetic r esonance (FMR) [17-20] and magneto-\noptical techniques [21,22] have been applied to stu dies of spin waves in ferromagnetic wires. \nHere we use spin torque FMR (st-FMR) [5,6], to make measurements of the spectral properties \nof dipole-exchange spin waves in Py nanowires. In t his technique, a ferromagnetic nanocontact \nis used to apply an ac spin-polarized current, Iac , to a metallic nanowire on a nonmagnetic \nmetallic substrate as illustrated in Fig. 1. When t he frequency of the current flowing \nperpendicular to the nanowire into the substrate is equal to the eigen-frequency of a spin wave \nmode, this mode is resonantly excited by the ac STT from the current. The excited mode induces \nresistance oscillations at the nanocontact due to t he giant magnetoresistance effect and generates \na rectified voltage, Vdc , proportional to the mode amplitude [5,6]. As the frequency of the ac \ncurrent, f, is swept through a spin wave eigen-frequency, a r esonance peak or a trough appears in \nthe plot of Vdc versus f. The peaks and troughs in the st-FRM spectrum, Vdc (f), give information \nabout spin wave frequencies, amplitudes and damping parameters. Boone et al. Page 3 of 10 October 15, 2009 \n \nFig. 1. (color online): (a) Scanning electron micro scopy (SEM) image \nof a Py nanowire device with five current-polarizin g SAF nanocontacts \nand Au/Ta top leads. (b) Schematic side view of the sample. We make st-FMR measurements for a set of eight 6-nm thick and 100-250 nm wide Py \nnanowires lithographically defined on a Cu/Ta under layer that serves as the bottom electrical \nlead. Spin-polarized current is injected into the n anowire through a Co 50 Fe 50 / Ru/ Co 50 Fe 50 / \nIr 20 Cr 3Mn 77 synthetic antiferromagnet (SAF) nanopillar pattern ed on top of the Py nanowire and \nseparated from the nanowire \nby an 8-nm thick Cu spacer, as \nshown in Fig. 1. The top Au/Ta \nlead is connected to the SAF \nfixed layer and current is \napplied between the top and \nthe bottom leads. Since the Cu spacer layer above t he Py layer is only partially milled away to \nprotect the Py nanowire surface, some lateral curre nt spreading in the spacer takes place. Five \nspin current injector nanocontacts are attached to the Py nanowire. In this Letter, we report \nsingle-contact st-FMR measurements. The device in F ig. 1(a) is made in a multi-step \nnanofabrication process starting from a Ta(5)/ Cu(3 0)/ Ta(3)/ Cu(30)/ Ta(5)/ Cu(3)/ Py(6)/ \nCu(8)/ Co 50 Fe 50 (3.5)/ Ru(0.8)/ Co 50 Fe 50 (3.5)/ Ir 20 Cr 3Mn 77 (7)/ Cu(10)/ Ru(5)/ Ta(2.5) multilayer \n(thicknesses are in nanometers). The values of satu ration magnetization of the Py ( M = 580 \nemu/cm 3) and Co 50 Fe 50 (1480 emu/cm 3) films are measured by vibrating sample magnetomet ry. \nThe nanowire axis is perpendicular to the SAF excha nge bias direction in order to maximize the \nmagnitude of STT [6]. For st-FMR measurements, we apply modulated microwave current to \nthe nanocontact and measure the resulting rectified voltage, Vdc , using lock-in detection [6]. The \nfrequency of the current is swept in the 1–15 GHz r ange and, after subtraction of a background \nvoltage due to Ohmic heating [13], the resulting Vdc (f) gives the st-FMR spectrum. Fig. 2(a) Boone et al. Page 4 of 10 October 15, 2009 \n \nFig. 2. (color online): (a) Evolution of st-FMR spe ctra with increasing \nmagnetic field applied parallel to a 240 nm wide na nowire. The curves \nare vertically offset for clarity. (b) st-FMR spect rum showing three \nspin wave resonances. (c), (d): Mode frequency vers us field applied \nparallel to the nanowire for a 140 nm wide wire and a 240 nm wide \nwire. Symbols are data and lines are Eq. (1) fits w ith the nanowire \nwidth w and the pinning parameter δ as fitting parameters. Inset in (c) \nis the dependence of the damping parameters of thre e spin wave \nmodes in the 240 nm wide nanowire on magnetic field . Inset in (d) \nshows the dispersion relation calculated from Eq. ( 1) for a 140 nm \nwire at 400 Oe. The dashed arrows indicate the two -magnon \nscattering channels and the solid arrow shows the kx=0 three-magnon \nconfluence process active only at the field for whi ch the frequency of \nthe n=1 mode is half the frequency of the n=3 mode. shows a series of st-FMR spectra measured at severa l fields applied along the nanowire axis of a \n240 nm wide Py nanowire at T = 4.2 K. These spectra exhibit three resonances (peaks and \ntroughs) corresponding to the lowest three spin wav e modes in the nanowire, as shown in Fig. \n2(b). All three modes shift to higher frequencies, and the frequency spacing among the modes \ndecreases with increasing magnetic field [15]. The observed resonances are due to spin waves in \nthe Py nanowire since the lowest-frequency SAF acou stic mode at H = 0 is expected at f >13 \nGHz − well above the \nfrequencies of the resonances in \nFig. 2(b) [23]. The line shape of \neach mode can be fit to a sum of \na symmetric and an \nantisymmetric Lorentzians, \nindicating that not only STT but \nalso magnetic field from the top \nlead contribute to excitation of \nspin waves [13]. From the \namplitude Vdc (f1) of the lowest-\nenergy mode we estimate the \nprecession cone angle of this \nmode to be 4.6 °/mA. In our \nsample geometry, spin waves \nwith wave vectors ranging from 0 to max \nxkcan be excited, where kx is the wave vector along the \nnanowire axis, and the value of max \nxk is determined by the extent of current spreading i n the Cu Boone et al. Page 5 of 10 October 15, 2009 spacer. Therefore, the line shape of each mode in t he st-FMR spectra is a convolution of line \nshapes of spin waves with kx < max \nxk. For our samples this inhomogeneous line broadenin g does \nnot exceed line broadening due to intrinsic spin wa ve damping as discussed later in this Letter. \nThis allows us to identify positions of minima and maxima of the resonances in st-FMR spectra \nwith the frequencies of spin wave modes at k x = 0. \nTo quantify the effect of confinement on spin waves in nanowires, we compare the \nmeasured spin wave frequencies to a theoretical exp ression for the frequency of quantized \ndipole-exchange spin wave width modes in thin-film strips of rectangular cross section [24,25]: \n( ) ( )\n\n − −+ +\n\n\n\n\n\n\n\n − −−+ + =dd\nMA\nMH k\ndd\nMA\nMHM f\nnn\nn\nnyn \nnn\nn nκκκ\nπ π κ κκκ\nπ πππγ exp 1\n2 4exp 11\n2 4422\n2 22\n2\n2 (1) \nIn this equation, fn is the frequency of n-th mode, γ /2 π = 2.95 MHz/Oe is the gyromagnetic \nratio, d is the nanowire thickness, H is the magnetic field along the strip axis, A = 10 -6 erg/cm is \nthe exchange constant, and 2\nnκ= 2\nxk + 2\nyn k where kyn is the quantized spin-wave wave vector \nalong the width of the strip. In our measurements, both even and odd modes are excited due to \ninhomogeneous current injection over the nanowire w idth arising from the current injector being \nmisaligned with the nanowire center as shown in Fig . 1(a). The quantized wave vector, kyn, \ngenerally can be expressed as kyn =( )\nwnnπδ−, where w is the nanowire width and 0 ≤ δn ≤ 1 is a \npinning parameter determined by the boundary condit ions for dynamic magnetization at the \nnanowire edges ( δn = 0 corresponds to complete pinning, whereas δn = 1 describes free spins). \nWe fit Eq. (1) to the fn(H) data using δn and w as fitting parameters, assuming that δn is \nindependent of n and is identical for all studied nanowires, δn = δ. We use w as a fitting \nparameter because we find ~ 20% spread in the spin w ave frequencies for nanowires of Boone et al. Page 6 of 10 October 15, 2009 nominally identical width. This fitting procedure g ives high-quality fits shown in Fig. 2(c) and \n2(d) for fn(H) with δ ≈ 0.5 and the nanowire width w similar to the nanowire width measured \nwith SEM (135 nm and 225 nm with ~ 10% standard devi ation for two groups of the nanowires \nstudied). A recent theory of boundary conditions fo r dipole-exchange spin waves in thin-film \nelements [25] predicts values of δ smaller ( δ ≈ 0.2, corresponding to stronger edge pinning) than \nδ = 0.5 found in our experiment. We expect some edge pinning reduction for nanowires with \nthicknesses comparable to or smaller than the excha nge length of Py (~ 5 nm) (see Fig. 2 in \n[25]). The edge pinning in our system can be also r educed because the magnetic properties of Py \nnear the nanowire edges are modified by ion milling [26, 27], and, therefore, the assumptions of \nspatially homogeneous magnetization and exchange us ed in [25] are not satisfied for our system. \nFitting a mode spectral line \nshape to a linear combination of \nsymmetric and antisymmetric \nLorentzians [13] gives the mode \nhalf width at half maximum, ∆fn, \nwhich carries information about \nthe mode damping parameter, αn. \nDue to inhomogeneous line \nbroadening caused by excitation of \nspin waves with kx in the (0 − kxmax ) \nrange, ∆fn sets an upper bound on \nαn: αn ≤( ) M Hfn\nπ γπ\n22\n+∆≡nα~[28]. \n \nFig. 3 (color online): (a), (b) Damping parameter of the first mode \n1~α versus field for 140 nm and 125 nm nanowires, show ing a \nmaximum of damping at a field Hmax (425 Oe and 550 Oe, \nrespectively). (c) Damping versus ac drive current for the 125 nm \nwire at H = Hmax (550 Oe) and H < Hmax (375 Oe). (d) H max versus \nfrequency of the first mode at H = Hmax . Squares are data, line shows \nthe field at which frequency of the third mode is e qual to twice the \nfrequency of the first mode as calculated from Eq. (1), data labels \nshow the nanowire width obtained from Eq. (1) fit o f the fn(H) data. Boone et al. Page 7 of 10 October 15, 2009 The inset in Fig. 2(c) shows nα~ for the first three modes of a nanowire with w = 240 nm as a \nfunction of field. The high-field ( H > 250 Oe) value of 1~α=0.013 ± 0.002 does not significantly \nexceed the value of Gilbert damping parameter α ≈ 0.01 for thin Py films [28]. Using the \ndispersion relation of Eq. (1) and assuming intrins ic damping α =0.008 [12], inhomogeneous line \nbroadening gives 1~α=0.013 for kxmax = 2 ×10 5 cm -1. The width of the region where spin torque is \napplied to the nanowire, π/kxmax =160 nm, is approximately twice as large as the phy sical width of \nthe current injector, which can be explained by cur rent spreading in the Cu spacer. We observe a \nlarge increase of nα~for H < 250 Oe for all modes. A low-field increase of da mping of similar \nmagnitude was previously observed in thin Py films [28,29]. This damping enhancement can be \nattributed to enhanced inhomogeneous broadening [30 ] and to incomplete saturation of \nmagnetization at low fields [27]. \nThe inset in Fig. 2(c) also demonstrates that the d amping parameters of the n ≥ 2 modes \nare enhanced compared to damping of the n=1 mode. This increase of damping can be explained \nby two-magnon scattering of spin waves [31,32]. Hig her order ( n ≥ 2) width modes can scatter \ninto lower order modes with the same frequency and kx ≠0 via a two-magnon process leading to \nthe linewidth broadening. A likely mechanism for th e observed large two-magnon scattering is a \nrandom magnetic potential induced by the nanowire e dge roughness or by magnetization pinning \ncenters. In contrast, two-magnon scattering of the n=1 mode to other modes is prohibited by \nconservation of energy, and the high-field value of the damping parameter for this mode is close \nto the Py thin-film value. \nFor nanowires with w < 220 nm, we observe a non-monotonic dependence of damping of \nthe n=1 mode on magnetic field as shown in Fig. 3(a) and 3(b). The damping as a function of Boone et al. Page 8 of 10 October 15, 2009 field exhibits a maximum at a field Hmax that increases with decreasing w. This damping \nmaximum is a nonlinear effect as demonstrated in Fi g. 3(c) that shows 1~α versus Iac measured at \nHmax =550 Oe and at H = 375 Oe < Hmax for a 125-nm wide nanowire. The damping increases \nlinearly with Iac (and thus with the mode amplitude) for all field v alues, but the rate of increase is \nsignificantly larger at H = Hmax , indicating that an additional nonlinear damping-en hancing \nprocess is active at H = Hmax . Fig. 3(d) shows the dependence of Hmax on frequency of the first \nmode at this field, max \n1f, for nanowires of different widths. We also note t hat at Hmax , the \nfrequency of the lowest-energy mode, f1, is equal to half the frequency of the third mode, 2 f1 = f3. \nThe solid line in Fig. 3(d) shows the field at whic h 2 f1 = f3 as a function of f1 calculated from Eq. \n(1). This theoretical curve is in excellent agreeme nt with the observed dependence of Hmax \non max \n1f. Therefore, we conclude that the observed enhancem ent of damping at Hmax is due to a \nthree-magnon confluence process in which two spin w aves of the n= 1 mode merge into a single \nspin wave of the n= 3 mode [33,34]. Due to quantization of the spin wav e spectrum in nanowires, \nthis energy- and momentum-conserving scattering cha nnel has a resonant character and exists \nonly at a particular magnetic field, Hmax . \nIn summary, we use st-FMR to study spin waves in Py nanowires and find that two-\nmagnon scattering enhances the damping of higher or der quantized spin wave width modes \ncompared to the damping of the lowest-energy mode. We also observe that the damping of the \nlowest-energy width mode resonantly depends on the bias magnetic field and exhibits a \nmaximum at a particular field value. We attribute t his maximum to a nonlinear three-magnon \nconfluence process in which two magnons of the firs t width mode merge into a single magnon of \nthe third width mode. This resonant dependence of t he spin wave damping parameter on \nmagnetic field originates from quantization of the spin wave spectrum in the nanowire geometry. Boone et al. Page 9 of 10 October 15, 2009 Our work demonstrates that the nonlinear three-magn on confluence process creates an additional \nresonant damping channel for the lowest-energy spin wave mode of a ferromagnetic nanowire \nand gives rise to an unusual dependence of spin wav e damping on external magnetic field. This \nwork was supported by the NSF, and by the NRI throu gh the Western Institute of \nNanoelectronics. Boone et al. Page 10 of 10 October 15, 2009 References \n \n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159 , L1 (1996) \n[2] L. Berger, Phys. Rev. B 54 , 9353 (1996) \n[3] D. C. Ralph, M. D, Stiles, J. Magn. Magn. Mater ., 320 , 1190 (2008) \n[4] J. A. Katine et al., Phys. Rev. Lett. 84 , 3149 (2000) \n[5] A. A. Tulapurkar et al., Nature 438 , 339 (2005) \n[6] J. C. Sankey et al., Phys. Rev. Lett. 96 , 227601 (2006) \n[7] S. I. Kiselev et al., Nature 425 , 380 (2003) \n[8] W. H. Rippard et al., Phys. Rev. Lett. 92 , 027201 (2004) \n[9] Q. Mistral et al., Appl. Phys. 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Slavin, Phys. Rev. B 72 , 014463 (2005) \n[26] M. P. Kostylev et al., Phys. Rev. B 76 , 054422 (2007) \n[27] R. P. Cowburn et al., J. Appl. Phys. 87 , 7067 (2000) \n[28] J. P. Nibarger, R. Lopusnik, T. J. Silva, Appl . Phys. Lett. 82 , 2112 (2003) \n[29] T. J. Silva et al., J. Appl. Phys. 85 , 7849 (1999) \n[30] G. Counil et al. , J. Appl. Phys. 95 , 5646 (2004) \n[31] R. Arias, D. L. Mills, Phys. Rev. B 60 , 7395 (1999) \n[32] R. D. McMichael, D. J. Twisselmann, A. Kunz, P hys. Rev. Lett. 90 , 227601 (2003) \n[33] M. Sparks, “Ferromagnetic Relaxation Theory”, McGraw-Hill, p. 95 (1964) \n[34] R. N. Costa Filho, M. G. Cottam, G. A. Farias, Phys. Rev. B 62 , 6545 (2000) " }, { "title": "1812.01334v3.Spin_transport_in_a_magnetic_insulator_with_zero_effective_damping.pdf", "content": "Spin transport in a magnetic insulator with zero e\u000bective damping\nT. Wimmer,1, 2,\u0003M. Althammer,1, 2,yL. Liensberger,1, 2N. Vlietstra,1\nS. Gepr ags,1M. Weiler,1, 2R. Gross,1, 2, 3, 4and H. Huebl1, 2, 3, 4,z\n1Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany\n2Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany\n3Nanosystems Initiative Munich (NIM), Schellingstra\u0019e 4, 80799 M unchen, Germany\n4Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M unchen, Germany\n(Dated: November 19, 2019)\nApplications based on spin currents strongly pro\ft from the control and reduction of their ef-\nfective damping and their transport properties. We here experimentally observe magnon mediated\ntransport of spin (angular) momentum through a 13 :4 nm thin yttrium iron garnet \flm with full\ncontrol of the magnetic damping via spin-orbit torque. Above a critical spin-orbit torque, the fully\ncompensated damping manifests itself as an increase of magnon conductivity by almost two orders of\nmagnitude. We compare our results to theoretical expectations based on recently predicted current\ninduced magnon condensates and discuss other possible origins of the observed critical behaviour.\nIntroduction - There is broad interest in using the spin\ndegree of freedom for information transport. This makes\nthe e\u000ecient manipulation of spin currents an important\nbut also challenging task [1{4]. Magnons, the quantized\nexcitations of the spin system in a magnetically ordered\nmaterial, are one of the most promising candidates for\nthe transport of spin information. However, in contrast\nto the number of charge carriers in an electronic conduc-\ntor, the magnon number in a spin conductor is not con-\nserved. Inevitably, magnon mediated spin currents only\nprevail on a characteristic length scale, which is mainly\ndetermined by the magnetic Gilbert damping of the ma-\nterial. Therefore, e\u000ecient ways of reducing and tuning\nthe magnetic damping represent an important step for\nspin transport devices.\nOne possible way to manipulate spin currents is\nto employ spin orbit torques (SOTs) in heavy metal\n(HM)/ferromagnetic insulator (FMI) bilayers [5{8].\nDriving a charge current through the HM in contact with\nthe FMI, an antidamping-like spin torque can be exerted\non the magnetization of the FMI. Above a critical cur-\nrent, the magnetic damping is completely compensated\nvia the SOT. For nano-structured devices, this damping\ncompensation manifests itself in the emergence of auto-\noscillations of the magnetization [5, 6, 9, 10]. Previous\nexperiments [7] demonstrated a 10-fold increase of the\npropagation length of coherent spin-waves in a HM/FMI\nwaveguide upon application of a large charge current to\nthe HM. Cornelissen et al [11] reported that also the dif-\nfusive transport of incoherently generated magnons can\nbe controlled by charge currents in HM/FMI nanostruc-\ntures.\nIn this Letter, we demonstrate the full compensation of\nthe magnetic damping in a nanometer-thick yttrium iron\ngarnet (YIG) \flm via SOT caused by a charge current in\nan adjacent HM layer. Above a threshold current den-\nsity in the HM, we observe a highly non-linear increase of\nmagnon conductivity by almost two orders of magnitude,\nindicating vanishing magnon decay. Our experimentalobservations can be rationalized by a SOT induced damp-\ning compensation of the magnetization dynamics. In this\ncontext, we will discuss two possible scenarios leading to\nthe damping compensation: (i) a strong overpopulation\nof modes by incoherent magnons and (ii) the formation\nof a coherent auto-oscillation state [6] equivalent to a\nswasing state [12]. In addition to (ii), Bender et al. pre-\ndict the formation of a magnon Bose-Einstein condensate\n(BEC) for SOT levels below the swasing phase. The on-\nset and smooth transition of the observed change in the\nmagnon conductance might be indicative for this BEC\nphase [12, 13].\nObservation - The principle of our magnon conduc-\ntance measurement is inspired by recent DC magneto-\ntransport experiments that infer magnon transport prop-\nerties in YIG [14{22]. As shown in Fig. 1 (a), magnons\nare injected from a Pt strip (injector) into a 13 :4 nm thick\nYIG \flm by the spin Hall e\u000bect (SHE) [23, 24] using a\nlow-frequency (13 Hz) charge current Iac= 50 µA in the\nPt strip. The di\u000busive transport of these magnons is\nquanti\fed by electrically measuring the magnon density\nbelow a second Pt strip (detector) as the \frst harmonic\nvoltage signal Vacvia lock-in detection, exploiting the in-\nverse SHE (we plot one quadrature containing the entire\nlock-in signal). Cornelissen et al. [11] demonstrated that\nthe magnon transport in such an arrangement can be\ncontrolled by a DC charge current Idcapplied to a third\n(modulator) strip placed in between injector and detector\n(c.f. Fig. 1 (a)). The modulator current causes a \fnite\nspin chemical potential \u0016sat the Pt/YIG interface, lead-\ning to an enhanced magnon density in YIG. Since the\nmagnon chemical potential \u0016mis expected to grow with\n\u0016s, we can tune \u0016mby varying Idc.\nIn contrast to Ref. [11], we here focus on the non-\nlinear regime of this magnon transport. Our physical\npicture of the magnon transport is condensed in Fig. 1\n(b) and (c). For the sake of simplicity, we only consider\nmagnon transport beneath the modulator and therefore\ndisregard the magnon decay on either side of the modu-arXiv:1812.01334v3 [cond-mat.mtrl-sci] 18 Nov 20192\nH\nφyz\nx\nVac+\n-\nMIacIdc\ninjector\ndetectormodulator(a)\nw\nwd\ndw\nPt+\n-\n+\n-\nt\nYIG\n zero effective dampingy\nIdc = Icrit(c)YIG\nIdc = 0(b)\n/uni03BCs = 0 Pt\nMnmac\nynmac\nIs Pt\nFigure 1. (a) Schematic depiction of the device, electrical\nconnection scheme and the coordinate system with the in-\nplane rotation angle 'of the applied magnetic \feld \u00160H.\n(b), (c) Illustrations of the magnon transport from injector to\ndetector. We here only consider magnon transport directly\nbelow the modulator. (b) For Idc= 0, magnons (blue wiggly\narrows) generated by the injector di\u000buse from left to right.\nMagnon decay events, indicated by red crosses, result in a\n\fnite lifetime and a corresponding characteristic spin di\u000bu-\nsion length depicted as a exponential decay of the magnon\ndensitynac\nm(orange solid line). The modulator only stati-\ncally a\u000bects the transport properties via magnon absorption.\n(c) ForIdc=Icrit, the modulator current is large enough to\ncompensate the magnetic damping of the YIG, resulting in\ne\u000bectively vanishing magnon decay beneath the modulator.\nThe damping compensation is illustrated by a large magnon\naccumulation beneath the modulator.\nlator. When Idc= 0 (panel (b)), the magnon density nac\nm\nfrom the injector decays exponentially (orange solid line).\nForIdc=Icrit(panel (c)), the threshold current for the\ndamping compensation is reached, the magnon lifetime\ndiverges and spin transport with an e\u000bectively vanishing\nmagnon decay ensues. This corresponds to a zero e\u000bec-\ntive damping state and is illustrated by the large magnon\naccumulation beneath the modulator.\nTo investigate the magnon propagation in the thin\nYIG layer for di\u000berent modulator currents Idc, we mea-\nsureVacas a function of the magnetic \feld orientation\n+Idc-Idc (a) (b)\nA(+/uni03BC0H) A(-/uni03BC0H) A(+/uni03BC0H) A(-/uni03BC0H)Figure 2. Detector signal Vacplotted versus the rotation angle\n'of the in-plane \feld at \u00160H= 50 mT for (a) positive and\n(b) negative DC bias currents Idcin the modulator. (a) The\nmagnon transport signal for Idc>0 is signi\fcantly increased\nat'=\u0006180\u000eand mostly una\u000bected at '= 0\u000e. (b) For\nIdc<0, we observe a 180\u000eshifted behavior, where the signal\nincrease is evident at '= 0\u000e, while unchanged for '=\u0006180\u000e.\n'(c.f. Fig. 1 (a)) with a \fxed magnetic \feld strength\nof\u00160H= 50 mT at T= 280 K. The result is shown\nin Fig. 2, where the black data points show the charac-\nteristic (cos2') modulation expected for magnon trans-\nport between injector and detector for Idc= 0. This\nresults from the variation of the magnon injection with\n', with maxima expected for Hperpendicular to Iac\n('=\u0000180\u000e;0\u000e;180\u000e) [14, 15]. Note that we observe\na \fnite o\u000bset signal even at '=\u000690\u000e. Since this o\u000b-\nset signal is found to be non-reproducible in di\u000berent\nmeasurement setups, we attribute this to a spurious ex-\nperimental artifact. The rather triangular shape of the\nangle dependent measurement for Idc= 0 is due to the\ncubic magnetocrystalline anisotropy of the YIG \flm (see\nRef. [25]), which results in non-collinear orientations of\nthe magnetization Mand the external \feld H. Most\nimportantly, however, a signi\fcant enhancement of the\nmagnon transport signal is observed at '=\u0006180\u000ein\nFig. 2 (a) for Idc>0. This can be understood by a\nmagnon accumulation underneath the modulator, which\nincreases the magnon conductivity and results in a larger\nVac. In the same way, a decrease of Vacis expected for\n'= 0\u000edue to the magnon depletion obtained in this\ncon\fguration. This, however, is counterbalanced by ther-\nmally injected magnons present due to Joule heating of\nthe modulator strip. Figure 2 (b) shows the measurement\nfor the inverted DC current direction ( Idc<0). Here, we\nobserve the expected 180\u000eshifted case: an enhancement\nfor'= 0\u000eand no signi\fcant change for \u0006180\u000e. This be-\nhaviour is consistent with an accumulation of magnons3\nfor the given current and magnetic \feld direction.\nFor a quantitative analysis of the data presented in\nFig. 2, we extract the signal amplitudes A(+\u00160H) and\nA(\u0000\u00160H) as a function of Idcfor various magnetic \feld\namplitudes H(see Fig. 3). In the low bias regime\n(jIdcj<0:4 mA ), the A(Idc) curves can be modeled by\na superposition of a linear and quadratic dependence as\nalready reported by Cornelissen et al. [11]. However, we\nobserve a two orders of magnitude improved control of\nthe magnon conductivity compared to Ref. [11]. This is\nin agreement with the predicted magnetic layer thickness\ndependence of the modulation e\u000eciency [11]. A quanti-\ntative comparison to the model of Ref. [11] is shown in\nthe Supplemental Material (SI) [25]. In addition, and\nmost importantly, we see a pronounced deviation from\nthe linear transport modulation [11] for large Idc. This\nmanifests itself by a shoulder in the A(Idc) curves for\nIdc>0:5 mA (marked by black triangles in Fig. 3 for\npositiveIdc).\nA(+/uni03BC0H) A(-/uni03BC0H)\n100 mT60 mT30 mT\n-100 mT-60 mT-30 mT150 mT\n-150 mTcurrent density (1011 A/m2)\nFigure 3. Extracted amplitudes A(+\u00160H) andA(\u0000\u00160H) (as\nindicated in Fig. 2) of the magnon transport signal for dif-\nferent external magnetic \felds plotted versus the DC current\nIdcin the modulator. The transition into the damping com-\npensation state for positive Idcis indicated by black triangles\n(maximum slope of the curves). The transition shifts to larger\nDC currents with increasing external magnetic \felds.\nWe now focus on the magnon transport properties,\nwhich we express by an e\u000bective magnon resistance Rs\nYIG.\nTo this end, we evaluate Rs\nYIGmeasured between in-\njector and detector as a function of the modulator cur-\nrent. The magnon resistance in YIG can be directly de-\nduced from the magnon transport amplitudes Aplotted\nin Fig. 3 (see SI [25]). However, Acontains contribu-\ntions from thermal (quadratic in Idc) as well as SHE\ninduced magnon injection e\u000bects (linear in Idc). We\ncorrect for both of those contributions, leading to the\nRs\nYIG(Idc) dependence shown in Fig. 4 (a) (for details\nsee Ref. [25]). Thus, Rs\nYIG(Idc) enables us to determine\nthe impact on magnon transport stemming solely from\nnon-linear and non-quadratic modulations of the magnon\ntransport, i.e. from the damping compensation regime.\nForIdc<0:4 mA, we observe a constant Rs\nYIG. We de\fne\n(a)\n(b)sIcritIonFigure 4. (a) Magnon resistance Rs\nYIGof the YIG chan-\nnel between injector and detector for a magnetic \feld of\n\u00160H= 50 mT.Rs\nYIGis corrected for e\u000bects associated with\n(linear) SHE and (quadratic) thermal magnon injection ef-\nfects. A very steep decrease of Rs\nYIGforIon< I dc< I crit\nis evident. The reduction of Rs\nYIGby 0:13 \n is compatible\nwith a vanishing magnon resistivity underneath the modu-\nlator strip. (b) Critical currents Ion=critversus applied \feld\n\u00160H. The right y-axis shows the critical chemical potentials\n\u0016c=2=c=1from Eq. (1) (solid red and blue lines).\na characteristic onset current Ion, at which the magnon\nresistanceRs\nYIGstarts to drop rapidly by 0 :13 \n and sat-\nurates at a \fnite value above the second characteristic\ncurrentIcrit. Here, we de\fne Ionas the current at which\nRs\nYIGdrops by 10 % compared to the constant resistance\nobserved for small Idc.Icritis taken at the current level\nwhereRs\nYIGreaches its minimum value. The magnon\nresistance data also allows us to roughly estimate the\nresistance within the damping compensated region be-\nneath the modulator. Assuming a serial resistor network\nmodel [18] (see also the SI [25]), and zero magnon resis-\ntance underneath the modulator strip when the damping\nis compensated, we expect Rs\nYIG= 0:19 \n forIdc>Icrit.\nThis in good agreement with our data shown in Fig. 4\n(a). We can further roughly estimate the magnon resis-\ntivity\u001as\nYIGforIdc>Icritand obtain 8 :16 n\n m, which is\nalmost two orders of magnitude smaller than the magnon\nresistivity for Idc< Ion(0:54µ\n m) [25]. Thus, the ob-\nserved magnon resistance shows similarities to the sud-\nden electrical resistance drop of a superconductor at the\nsuperconducting phase transition.\nInterpretation - Our magnon transport measurements\nshow that the magnon conductance is strongly enhanced4\nfor largeIdc, i.e. when the damping is compensated un-\nderneath the modulator strip. The strong enhancement\nsuggests a vanishing magnon resistivity, which could be\ninterpreted as spin super\ruidity [26{29]. As the damp-\ning compensation may also lead to coherent magnetiza-\ntion dynamics, this warrants the question how a coherent\nmagnetization state created by ferromagnetic resonance\n(FMR) a\u000bects the transport properties. In stark con-\ntrast to the reduction of the magnon resistance due to\nthe damping compensation, we \fnd an increase of the\nmagnon resistance when coherently driving the YIG mag-\nnetization by a microwave magnetic \feld [25] [30]. This\ndemonstrates that the e\u000bective compensation of mag-\nnetic damping is responsible for the formation of the\nultra-low magnon resistance state - and not the coher-\nence of the magnetization precession. In particular, we\nwant to emphasize that damping compensation not nec-\nessarily results in a coherent precession of the magneti-\nzation, but also a broad frequency spectrum of excited\nmodes is a possible scenario. Thus, taking the damp-\ning compensation as the bottom line of our experimental\nobservations, we provide two possible scenarios explain-\ning our \fndings: (i) a strong overpopulation of magnons\nin a broad frequency spectrum leads to the compensa-\ntion of the magnon damping, but no coherent magne-\ntization precession is achieved. (ii) Similar to a spin-\ntorque-oscillator [6], the compensation of the magnetic\ndamping leads to a coherent auto-oscillation state of the\nmagnetization, equivalent to a swasing phase as discussed\nbelow. Here, the terminology of swasing is adopted from\nRef. [12] describing the spin wave analogon of lasing [31].\nIn the following, we will compare our data to Ref. [12],\nwhich theoretically predicts magnon condensation and\nswasing under DC pumping [12]. We want to emphasize\nthat this swasing instability is identical to the threshold\nfor auto-oscillations in spin Hall oscillators, as observed\nin Refs. [5, 6, 9, 10] (see [25] for a thorough derivation of\nthis equivalence). Note, that this threshold condition is\nindependent of the scenario and thus also holds for the in-\ncoherent case (i), since damping compensation is given by\nthe equality of the magnon relaxation and pumping rate\nand hence assumes no coherence of the excited modes.\nThis corresponds to the case c= 1 in the subsequent dis-\ncussion (see Eq. (1)). In addition, the model by Bender\net al. [12] also discusses the formation of a magnon BEC\n(c= 2 in Eq. (1)), which is de\fned by a \fnite population\nof magnons in the ground state. We rewrite their model\nto conform to our in-plane magnetized case [25] and \fnd\nthe spin chemical potentials \u0016c=2, corresponding to the\nformation of a magnon BEC, as well as \u0016c=1, i.e. the so-\ncalled swasing instability in the magnon BEC phase, to\nbe given by\n\u0016c=2=c=1=\u0012\n1 +\u000be\u000b\nc\u0001\u000bsp\u0013 \u0014\n~\r\u00160\u0012\nH+Ms\n2\u0013\u0015\n:(1)Here,\u0016s=\u0016c=2corresponds to the critical spin chemi-\ncal potential for the formation of a magnon BEC, while\n\u0016s=\u0016c=1corresponds to the swasing instability, which\nis equivalent to the full damping compensation [25]. Fur-\nthermore,\u000be\u000bis an e\u000bective damping parameter [25] ,\n\u000bspis the spin pumping induced damping enhancement\nof the FMI, ~is the reduced Planck constant, \ris the\ngyromagnetic ratio and \u00160is the vacuum permeability.\nThe criteria further depend on the external magnetic \feld\nmagnitude Hand the saturation magnetization Ms. The\nspin chemical potential is related to the applied current\nby\u0016s= [e\u0012SHIdctanh (\u0011)]=[w\u001be\u0011] [18, 32, 33], where e\nis the elementary charge, wdenotes the width of the Pt\nstrip,\u001beand\u0012SHare the electrical conductivity and the\nspin Hall angle of the Pt. Moreover, \u0011=tHM=(2ls) is the\nratio of the thickness of the Pt strip tHMand its spin dif-\nfusion length ls. For a comparison of our data to Eq. (1),\nwe plot the experimentally determined critical currents\nIonandIcritas a function of the applied magnetic \feld\nin Fig. 4 (b). For Ion(Icrit), we observe a characteris-\ntic current around 0 :45 mA (0:6 mA) for\u00160H < 50 mT\nand both critical currents increase with the applied mag-\nnetic \feld strength for \u00160H > 50 mT. We can solve the\ncondition\u0016s=\u0016c=2(\u0016s=\u0016c=1) forIdcand identify\nthe result with the aforementioned characteristic current\nIon(Icrit). Hence, we can quantitatively corroborate the\n\feld dependence of the critical currents observed in Fig. 4\n(b). Using the values \u001be= 1:74\u00021061=\nm,\u0012SH= 0:11,\nls= 1:5 nm,w= 500 nm and tPt= 3:5 nm to calculate\n\u0016s, we \fnd good quantitative agreement of model and\nexperimental data for both Icrit(spheres and red line)\nandIon(stars and blue line). The characteristic param-\neters\u000be\u000band\u000bspentering Eq. (1) are determined in-\ndependently using ferromagnetic resonance experiments\npresented in the SI [25]. The strong increase of the the-\noretically predicted threshold currents at small magnetic\n\felds is not properly re\rected by the experimental data.\nAs discussed in Ref. [25], however, this may be caused by\nan in-plane magnetocrystalline anisotropy \feld, e.g. due\nto the cubic anisotropy of our YIG \flm.\nFor an intuitive understanding of the BEC and swas-\ning scenario excited using spin Hall physics, we refer to\nRef. [12]. Here, the threshold of the BEC is determined\nby the presence of a \fnite population of magnons in the\nground state, corresponding to a phase transition of sec-\nond order. In contrast, the swasing threshold is associ-\nated with the full compensation of the intrinsic damping\nand can be identi\fed with a coherent magnetization pre-\ncession. The di\u000berence between those threshold values\noriginates from the fact that magnons are an excitation\nwith a \fnite lifetime and hence a non-conserved quantity.\nThe observation of a smooth transition of Rs\nYIGin Fig. 4\n(a) thus might be indicative of this second order phase\ntransition, where magnons are condensing continuously\ninto a steady state BEC.\nSummary - We \fnd ultra-low magnon resistance in-5\ndicating an e\u000bectively vanishing magnon decay in a\nHM/FMI bilayer under the application of a large cur-\nrent density to the HM. The damping compensation is\nachieved by employing spin-orbit torque mediated spin\ncurrent injection in a YIG/Pt heterostructure. We dis-\ncuss our data by comparing it to the theoretically pre-\ndicted threshold conditions for the transition into a DC\ncharge current pumped magnon BEC [12]. 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Rev.\nB94, 174437 (2016).\n[53] B. Flebus, S. A. Bender, Y. Tserkovnyak, and R. A.\nDuine, Phys. Rev. Lett. 116, 117201 (2016)." }, { "title": "1706.08525v2.High__β__Effects_on_Cosmic_Ray_Streaming_in_Galaxy_Clusters.pdf", "content": "MNRAS 000, 000{000 (0000) Preprint 27 May 2022 Compiled using MNRAS L ATEX style \fle v3.0\nHigh Beta E\u000bects on Cosmic Ray Streaming in Galaxy\nClusters\nJoshua Wiener1;2, Ellen G. Zweibel2;3, & S. Peng Oh1\n1Department of Physics; University of California; Santa Barbara, CA 93106, USA.\n2Department of Astronomy, University of Wisconsin-Madison, Madison, WI 53706, USA.\n3Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA.\n27 May 2022\nABSTRACT\nDi\u000buse, extended radio emission in galaxy clusters, commonly referred to as radio\nhalos, indicate the presence of high energy cosmic ray (CR) electrons and cluster-\nwide magnetic \felds. We can predict from theory the expected surface brightness of a\nradio halo, given magnetic \feld and CR density pro\fles. Previous studies have shown\nthat the nature of CR transport can radically e\u000bect the expected radio halo emission\nfrom clusters (Wiener et al. 2013). Reasonable levels of magnetohydrodynamic (MHD)\nwave damping can lead to signi\fcant CR streaming speeds. But a careful treatment of\nMHD waves in a high \fplasma, as expected in cluster environments, reveals damping\nrates may be enhanced by a factor of \f1=2. This leads to faster CR streaming and\nlower surface brightnesses than without this e\u000bect. In this work we re-examine the\nsimpli\fed, 1D Coma cluster simulations (with radial magnetic \felds) of Wiener et al.\n(2013) and discuss observable consequences of this high \fdamping. Future work is\nrequired to study this e\u000bect in more realistic simulations.\n1 INTRODUCTION\nGalaxy clusters are the largest gravitationally bound objects\nin the universe, and are host to processes described by ev-\nery area of physics. For some clusters, this includes particle\nacceleration to very high energies, generating cosmic rays\n(CRs). Cosmic ray electrons (CRe) at these energies will\nemit synchrotron emission in the presence of magnetic \felds\nwhich is observable as radio emission.\nGalaxy clusters can exhibit radio emission of a few mor-\nphological types. We focus here on radio haloes, which refers\nto extended, di\u000buse radio emission. Studies have shown that\nwhere radio haloes are detected, the total radio luminosity\nof the halo correlates well with total X-ray luminosity of\nthe host cluster. But many clusters do not have radio haloes\nat all. Thus there is a peculiar bimodality in the presence\nof radio haloes in galaxy clusters that must be explained\n(Brunetti et al. (2007), En\u0019lin et al. (2011), Basu (2012);\nSommer & Basu (2014)).\nRelated to this problem is the source of the CRe which\nproduce the emission. CRe at the required energies ( \u00181-\n10 GeV) have short cooling times ( \u0018<100 Myr) for typi-\ncal cluster magnetic \feld strengths ( \u00183\u0016G) and densities\n(\u001810\u00003cm\u00003). The presence of a radio halo in a given clus-\nter therefore indicates relatively recent injection of CRe. The\n`hadronic model' claims that this is provided by hadronic\ncollisions of CR protons (CRp) with ambient thermal nuclei\nthat result in pions, which then decay into secondary CRe.\nA rival model, `turbulent reacceleration', hypothesizes that\nlow-energy CRe undergo Fermi-II reacceleration by gas mo-\ntions during mergers (Brunetti & Lazarian 2007). However,the abundance and distribution of CRp is still an important\ningredient in this model, as secondary CRe produced during\nhadronic collisions provide seeds for reacceleration (Brunetti\n& Lazarian 2011; Pinzke et al. 2017).1An excellent summary\nof the state of the \feld can be found in Brunetti & Jones\n(2014).\nAn issue with the hadronic model is that high energy\nCRp have much longer ( \u001810-100 Gyr) cooling times. If a\ncluster undergoes some merger or structure formation pro-\ncess that may be responsible for accelerating CRs, then we\nmay naively expect the resulting radio halo to last for as\nlong as or longer than the age of the universe. If this were\nthe case, we would struggle to explain clusters that don't ex-\nhibit radio halo emission, since every cluster is expected to\naccelerate CRp at some point in its history through such a\nmechanism. A lack of hadronic emission implies either that\nthe assumed acceleration e\u000eciencies are too high or that\nthe CRp are quickly diluted by transport e\u000bects. We inves-\ntigate this second possibility. Note that CRp are frequently\ninvoked in turbulent reacceleration models as well, to gener-\nate secondary seed electrons for reacceleration. Observations\nare best \ft by a \rat CRp pro\fle, which can be explained by\nCR streaming (Pinzke et al. 2017).\nBuilding on the work of En\u0019lin et al. (2011), Wiener\net al. (2013) attempted to resolve this issue by suggest-\ning that, for favorable magnetic \feld orientations, CRp can\ntravel at high speeds away from cluster centers, quickly re-\n1Acceleration from the thermal pool is precluded by strong\nCoulomb losses (Petrosian & East 2008).\nc\r0000 The AuthorsarXiv:1706.08525v2 [astro-ph.HE] 6 Oct 20172Wiener, Zweibel, & Oh\nducing the CRp density and thus the expected radio lu-\nminosity on appropriate time scales. Bulk CR transport\nis limited by the streaming instability, which ampli\fes hy-\ndromagnetic waves traveling in the same direction as the\nmean cosmic ray velocity if the cosmic rays are anisotropic\nin the frame of the wave (Kulsrud & Pearce 1969). How-\never, if other plasma processes damp the wave then a larger\nstreaming speed is required to excite the instability, result-\ning in faster cosmic ray transport. In Wiener et al. (2013)\nit was shown that under galaxy cluster plasma conditions\nthe strongest damping mechanism is the turbulent damp-\ning process proposed by Yan & Lazarian (2002); Farmer &\nGoldreich (2004).\nGalaxy cluster plasmas are characterized by large ra-\ntios of thermal to magnetic pressure, a parameter usually\ndenoted as \f. In this work we show that, in high \fenvi-\nronments, wave damping is actually stronger than the rate\nquoted in Wiener et al. (2013). Therefore, CRs can travel\neven faster than previously thought and radio halos turn\no\u000b faster than we previously estimated for the same den-\nsity and magnetic \feld pro\fle2. While this may explain the\nbimodality of radio haloes in the hadronic model, it can-\nnot reconcile other issues with the hadronic model. Among\nothers, gamma-ray non-detections in the Coma cluster put\nstrict limits on the amount of CRp present (Brunetti et al.\n(2012); Zandanel & Ando (2014); Ackermann et al. (2016);\nBrunetti et al. (2017)). As such, this work should not be seen\nas an attempt to strengthen the hadronic model of radio\nhaloes so much as a more accurate theory of CR transport\nin clusters in general, which is still an important ingredient\nin the reacceleration model (Pinzke et al. (2017)).\nAlso, although the time evolution of radio halos is the\nmain focus of the paper, we note that faster streaming may\nalso help to explain the puzzling lack of di\u000buse \r-rays from\ngalaxy cluster cores (Pfrommer (2008); Pinzke & Pfrommer\n(2010); En\u0019lin et al. (2011); Ahnen et al. (2016)). It may\nalso be germane to understanding radio mini-halo luminosi-\nties (Jacob & Pfrommer 2017). While the hadronic model is\nproblematic for understanding giant radio halos, it remains\nthe chief contender for understanding kinematically quies-\ncent radio mini-halos (Zandanel et al. 2014), where the CRp\ncould be sourced by a central AGN. Note that modulo loss\nprocesses, the contribution of the AGN jet to CRe could be\nnon-negligible as well.\nInx2 we review the theory of MHD wave damping in\ncollisionless, high \fplasmas and show how it modi\fes previ-\nously estimated turbulent damping rates. An important part\nof this calculation is an estimate of the e\u000bective collisionality\nof the plasma under the assumption that microturbulence is\npresent due to pressure anisotropies (Schekochihin & Cow-\nley 2007); we argue that the level of microturbulence is low\nenough that Alfv\u0013 en waves at the characteristic wavelengths\nexcited by cosmic ray steaming instabilities are nevertheless\ncollisionlessly damped. We also discuss the possible e\u000bects\nof the magnetic \feld topology, which is not taken into ac-\ncount in our simulations. In x3 we describe the simulation\n2As we discuss inx2.3, we use a very simple 1D halo model with\npurely radial magnetic \felds for comparison with Wiener et al.\n(2013). More sophisticated simulations are needed in future work\nto examine the e\u000bects of this damping in real clusters.setup, which is similar to the setup we used previously for\nthe Coma cluster (Wiener et al. 2013). Sections x4 andx5\ngives the results and conclusion, respectively.\n2 MHD WAVES IN TURBULENT, HIGH BETA\nPLASMA\n2.1 Wave damping\nAs mentioned in x1, the streaming speed as a function of\nenergy is determined by balancing the growth rate of the\nstreaming instability (Kulsrud & Cesarsky 1971)\n\u0000cr(kk) =!cpnCR(>\rR)\nni\u0012vD(\rR)\nvA\u00001\u0013\n(1)\nagainst the mechanism(s) that damp the wave. In (1), !cp\nis the proton gyrofrequency (for proton cosmic rays), \rR\u0011\n!cp=ckkis the minimum Lorentz factor of a resonant cosmic\nray, andvDis the energy dependent streaming speed.\nEquation (1) is computed assuming a perfectly straight\nand uniform background magnetic \feld B0. Under this as-\nsumption, the fastest growing waves propagate parallel to\nB0, with waves of any given wavenumber kkbeing driven\nprimarily by cosmic rays with gyroradii rL\u0018k\u00001\nk.\nIfB0has curvature or perpendicular structure, strict\nparallel propagation is impossible: there is a minimum an-\ngle\u0012minwhich depends on the background \feld structure.\nFarmer & Goldreich (2004) evaluated \u0012minfor Alfv\u0013 en waves\nexcited by cosmic ray streaming in a background \feld upon\nwhich an anisotropic MHD turbulent cascade is imposed,\nand found it to be\ntan\u0012min=\u0012\u0015k\n\u0015?\u0013\n\u0018\u0012rL\nLMHD\u00131=4\n\u0018\u0012rL\u000f\nv3\nA\u00131=4\n;(2)\nwhereLMHD is the length scale at which the characteristic\nbulk speed is the Alfv\u0013 en speed vA, and the turbulent energy\ncascade rate is \u000f\u0011v3\nA=LMHD. The perpendicular lengthscale\n\u0015?and turbulent velocity v\u0015?corresponding to \u0012minare\n\u0015?\u0018r3=4\nLL1=4\nMHD\u0018r3=4\nL(v3\nA=\u000f)1=4; (3)\nand\nv\u0015?\u0018vA\u0012\u0015?\nLMHD\u00131=3\n\u0018vA\u0012rL\nLMHD\u00131=4\n: (4)\nAccording to the Farmer & Goldreich (2004) scenario,\nMHD waves are progressively sheared by the turbulent cas-\ncade, eventually transferring their power to the dissipation\nscale. The corresponding \\turbulent damping rate\" is there-\nfore on the order of the eddy turnover rate at the perpen-\ndicular scale\n\u0000damp\u0018v\u0015?\n\u0015?\u0018\u000f1=3\n\u00152=3\n?\u0018\u0012\u000f\nrLvA\u00131=2\n; (5)\nwhere in the last equality we have used eqns. (3) and (4).\nThis was the damping rate used in the ZEUS simulations in\nWiener et al. (2013). The Farmer & Goldreich (2004) calculations\nhave recently been generalized by Lazarian (2016) to a variety of\nscenarios.\nA key assumption in deriving eqn. (5) is that the dissipation\nscale is much shorter than the wavelength of the wave. If this is\nnot the case, the dissipation rate can exceed the shearing rate.\nIn collisionless, high \fplasmas, MHD waves of even small\nMNRAS 000, 000{000 (0000)3\nobliquity are subject to strong ion Landau damping (Foote &\nKulsrud 1979), a process whereby ions with parallel velocity vk\nwhich satis\fes the resonance condition vk=!=kkabsorb energy\nfrom the wave due to acceleration by its parallel electric \feld3.\nWe brie\ry review the Foote & Kulsrud calculation here.\nThey de\fned\n\f=Pg\nPB=\u001akBT=\u0016mp\nB2=8\u0019=v2\ni\nv2\nA; (6)\nand framed their analysis in terms of the rescaled wave frequency\n!and wave number k:\n\u0017\u0011!\nk0vA; l\u0011kcos\u0012\nk0; k 0\u0011!civA\nv2\ni;\nwhere!ciis the thermal ion cyclotron frequency. They also in-\ntroduce a parameter \u000b=\u00191=2\f1=2l2tan2\u0012.\nFoote & Kulsrud (1979) show that in the small llimit, which\napplies to most of the waves ampli\fed by streaming cosmic rays,\nthe dispersion relation can be approximated asymptotically by\n\u0017=l\u0000i\u000b\n4l\u00061\n4l\u0012\nl6\u0000\u000b2+4i\u000b3\nl2\u00131=2\n(7)\nExpanding eqn. (7) for small \u000b, the damping rate is given by\n\u0000damp =\u0000k0vAIm(\u0017)\u0019k0vA\u000b\n4l(8)\nUsing (2) for \u0012and restoring dimensional units, we arrive at the\nresult\n\u0000damp\u0019p\u0019\n4k0vA\f1=2ltan2\u0012\u0019p\u0019\n4\f1=2r\u00001\nLvA \nrL\u000f\nv3\nA!1=2\n) \u0000damp\u0019p\u0019\n4\f1=2\u0012\u000f\nrLvA\u00131=2\n; (9)\nwhere we have used lk0=kcos\u0012=r\u00001\nL.\nComparison with (5) shows that with this treatment the\nwave damping rate is enhanced by a factor of \f1=2. In the cluster\nenvironments simulated in Wiener et al. (2013), where \fcan be\nof the order of 100 in the cluster centers, this factor can lead to\nsigni\fcantly higher CR streaming speeds than those predicted by\n(5).\nThe factor of \f1=2can be heuristically understood as follows.\nThe Landau damping rate is given by the rate at which resonant\nions absorb energy from oblique waves; \u0000Landau\ndamp\u0018kvitan2\u0012. The\nturbulent damping rate is given by the rate at which a pair of\ninteracting Alfv\u0013 en waves cascade, \u0000turb\ndamp\u0018kvAtan2\u0012. The ge-\nometrical factor of tan2\u0012is the same for these pairwise inter-\nactions. Thus, the Landau damping rate is larger by a factor\n\u0018vi=vA\u0018\f1=2.\n2.2 Collisionality\nThe analysis by Foote & Kulsrud (1979) assumes a collisionless\nplasma. We must verify that the plasmas in galaxy cluster en-\nvironments we will be simulating are su\u000eciently collisionless. In\nparticular, we must determine whether microinstabilities driven\nby pressure anisotropy can render the thermal ions essentially col-\nlisional. The relevant comparison to make here is the ion mean\nfree path to the typical wavelength of the Alfv\u0013 en waves in ques-\ntion.\n3This process is not to be confused with nonlinear Landau\ndamping, in which the thermal ions resonate with the low fre-\nquency beat wave generated by the interaction of two higher fre-\nquency waves (Kulsrud 2005). NLLD was \frst invoked for galaxy\ncluster plasmas by Loewenstein et al. (1991) and was argued to\nbe subdominant in turbulent galaxy cluster plasmas in Wiener\net al. (2013).The long mean free paths, relatively weak magnetic \felds,\nand pervasive large scale turbulence in galaxy cluster plas-\nmas make them attractive candidates for shear driven pressure\nanisotropy (Schekochihin & Cowley 2006) through distortion of\nthe magnetic \feld and preservation of the particles' adiabatic\ninvariants. We follow the notation of Kunz et al. (2011) in the\nfollowing discussion.\nShear in a \ruid with velocity \feld uand magnetic \feld direc-\ntionbwill drive pressure anisotropy while collisions will oppose\nit. Balancing the two yields an equilibrium anisotropy given by\nBraginskii (1965)\n\u0001i\u0011P?;i\u0000Pk;i\nPi=2:9\n\u0017ii\u0012\nbb:ru\u00001\n3r\u0001u\u0013\n: (10)\nIn the above, the isubscript indicates we are referring to ions and\n\u0017iiindicates the frequency of Coulomb collisions. These collisions\nserve to isotropize the pressure.\nA complication arises if collisions are so infrequent that the\npressure anisotropy exceeds the threshold for microinstabilities:\n\u0001i>1\n\fi;mirror instability\n\u0001i<\u00002\n\fi;\frehose instability\nIf Coulomb collisions alone are not su\u000ecient to prevent these\ninstabilities, we may expect two possible responses of the \ruid.\nIn one scenario, the shear S\u0011bb:ru\u00001=3r\u0001uwill adjust\nitself to reduce the pressure anisotropy until marginal stability is\nachieved. In this case the collision frequency does not change - it\nis simply the Coulomb collision frequency and the mean free path\nis the Coulomb mean free path. This is the working hypothesis in\nKunz et al. (2011).\nIn the other scenario, the shear Sremains unchanged and\ninstead the resulting magnetic \ructuations driven by the insta-\nbilities will themselves scatter the ions. This increases the e\u000bective\ntotal scattering frequency \u0017i, which now includes these magnetic\nscatterings in addition to Coulomb scattering, until marginal sta-\nbility is achieved:\n\u0001i=2:9\n\u0017iS=2\u0018\n\fi!\u0017i=1:45\fi\n\u0018S; (11)\nwhere\u0018is -1 for the \frehose instability, and 1/2 for the mirror\ninstability. In this case, the total collision frequency is enhanced,\nand so the mean free path is shorter than the Coulomb mean free\npath.\nThe above marginal stability criterion therefore provides a\nlower limit on the ion scattering frequency \u0017i, and thus an upper\nlimit on the ion mean free path \u0015i=vi=\u0017i. This limit depends\non the shear forcing S.\nLet us approximate this forcing as S\u0018U=L for some char-\nacteristic speed Uand length scale L. If we assume a turbulent\ncascade with Kolmogorov scaling, U3=L= const., we can relate\nanyUandLto the outer scales U0andL0. The marginal stability\ncriterion then becomes\n\u0017i\u00181:45\fi\n\u0018U\nL=1:45\fi\n\u0018U0\nL0\u0012L0\nL\u00132=3\n(12)\nThe highest collision frequency will therefore be dictated by the\nsmallest scale of turbulence, the dissipation scale Ld. This scale is\ndetermined by balancing the cascade rate Ud=Ldwith the viscous\ndissipation rate, which itself depends on the collision frequency,\nand is of order v2\ni=(\u0017iL2\nd). We obtain\nUd\nLd=U0\nL0\u0012L0\nLd\u00132=3\n\u0018v2\ni\n\u0017iL2\nd\n)\u0012Ld\nL0\u00134=3\n\u0018v2\ni\n\u0017iU0L0\nMNRAS 000, 000{000 (0000)4Wiener, Zweibel, & Oh\nLd\nL0=\u0012v2\ni\n\u0017iU0L0\u00133=4\n(13)\nCombining (12) and (13) we arrive at an expression for the ion\ncollision rate at marginal stability\n\u0017i\u00181:45\fi\n\u0018U0\nL0\u0012\u0017iU0L0\nv2\ni\u00131=2\n)\u0017i\u0018\u00121:45\fi\n\u0018\u00132U3\n0\nL0v2\ni(14)\nindicating an ion mean free path of\n\u0015i\u00180:48\u00182L0\n\f2\ni\u0012vi\nU0\u00133\n\u00180:48L0\n\f2\niM3\n0(15)\nwhere we have de\fned the Mach number at the driving scale\nM0\u0011U0=vi.\nWe can plug this solution for \u0017iback in to (13) to \fnd the\ndissipation length:\nLd\u0018L0\u0012v2\ni\n\f2\niU4\n0=v2\ni\u00133=4\n\u0018L0M\u00003\n0\f\u00003=2\ni(16)\nNote that the turbulent velocity at this scale is\nUd=U0\u0012Ld\nL0\u00131=3\n\u0018U0M\u00001\n0\f\u00001=2\ni=vA:\nThat is, under the assumption of marginal stability, the charac-\nteristic turbulent velocity at the dissipation scale is about the\nsame as the Alfv\u0013 en speed. This means the dissipation scale Ldis\nabout the same as the length scale LMHD de\fned in Wiener et al.\n(2013)'s streaming simulations as a measure of the wave damping\nrate, where it is assumed to be of order 100 kpc.\nNow we can make an estimate of the mean free path as a\nfunction of LMHD :\n\u0015i\u0018L0\f\u00002\niM\u00003\n0\u0018Ld\f\u00001=2\ni\u001814 kpc\u0012LMHD\n100 kpc\u0013\u0012\fi\n50\u0013\u00001=2\n(17)\nThis is much longer than the several AU gyroradii of the \u0018100\nGeV CR protons that generate the secondary e\u0006which produce\nthe observed radio emission. The assumption of collisionless wave\ndamping is therefore sound if we assume marginal stability to\nMHD microinstabilities.\nWe can also compare the above result to the Coulomb mean\nfree path\u0015ii. If\u0015iiis signi\fcantly shorter, then Coulomb colli-\nsions alone are enough to keep the pressure anisotropy (11) low\nenough to avoid MHD microinstabilities. We have\n\u0015ii=viti=s\nkT\nmppmp(kT)3=2\n4p\u0019ln \u0003e4ni\u00195\u00021020cmT2\n7\nni;\u00003(18)\nwhere we approximate ln \u0003 \u001930 and we adopt the subscript\nnotationQx=Q=10xin cgs units.\nIn the central regions of our simulated Coma cluster (see\nx3),T7= 9:5 andni;\u00003= 3:4, so\u0015ii\u00194:4 kpc. This is less\nthan the mean free path derived from marginal stability above,\nimplying Coulomb collisions su\u000eciently isotropize the pressure to\navoid instabilities, at least in the cluster center. But this is still\nwell above the AU scale gyroradii, implying we are safely in the\ncollisionless damping regime.\nThe fact that Ld\u0018LMHD when the plasma is marginally\nstable to microinstabilities raises potentially serious issues. It\ncould imply that there is no MHD inertial range, since the\nReynolds number at the Alfv\u0013 en scale is of order unity, and parallel\ntrans-Alfv\u0013 enic motions simply dissipate. Indeed, recent analytic\nand numerical work \fnds an upper limit on shear Alfv\u0013 en \ructua-\ntions of\u000eB?=B0\u0018\f\u00001=2(Squire et al. 2016, 2017), above which\nthe perturbation is rapidly quenched by the \frehose instability.\nWhile CR-streaming driven turbulence lies below this limit, the\n101102103\nR (kpc)101102103104105SB @ 1.4 GHz (Jy/sr)Quad 315\nQuad 225\nQuad 135\nQuad 45\nSimulation\nDeiss et al 1997Figure 1. 1.4 GHz surface brightness observations of the Coma\ncluster. Azimuthally averaged data from Deiss et al. (1997)\n(points) is compared with data from di\u000berent quadrants of Coma\nfrom Brown & Rudnick (2011) (thin lines - data provided by Larry\nRudnick, personal communication). Overlayed in the thick line is\nthe initial surface brightness of our model cluster.\nbackground turbulence does not, and it is unclear whether equa-\ntion (2) (which relies on canonical Goldreich-Sridhar theory) still\napplies. Such issues lie well beyond the scope of this paper, but\nthey raise important caveats to keep in mind.\n2.3 Magnetic Topology and Other Unmodeled\nFactors\nAs will be explained in x3, our numerical simulations are 1D\nspherically symmetric. This is primarily for two reasons - \frst,\nthis simpli\fes the physics for ease of computation. Second, we\nwant an apples-to-apples comparison with the simulations from\nWiener et al. (2013), which also employed these simpli\fcations\nfor the \frst reason. We discuss here the possible implications of\nthese simpli\fcations.\nReal galaxy clusters are, of course, not spherically symmet-\nric. The Coma cluster, which we use as our characteristic cluster,\nhas noticeable azimuthal dependence in the radio surface bright-\nness, as reported by Brown & Rudnick (2011). We compare their\n1.4 GHz observations of di\u000berent quadrants of the Coma clus-\nter using the Green Bank Telescope (provided by Larry Rudnick,\npersonal communication) with the azimuthally averaged observa-\ntions of Deiss et al. (1997) as well as the initial surface brightness\nin our model cluster in \fgure 1. There is clearly non-symmetric\nstructure in the radio signal. The comparison between the two\nsets of data is further complicated by subtraction of the fore-\nground signal from our own Galaxy, which is handled di\u000berently\nin each paper. As far as they relate to the construction of our\ninitial CR pro\fle, these di\u000berences in the data are of little conse-\nquence - streaming times in our simulations are not sensitive to\nsmall changes in the initial CR pro\fle.\nHowever, the departure from spherical symmetry has other,\nmore signi\fcant implications for the evolution of out model CRs.\nThe most important of these is the structure of the cluster's mag-\nnetic \feld. Our 1D simulations necessarily have perfectly radial\nmagnetic \felds, which is a best-case scenario for streaming. In\nthe limit of small cross-\feld di\u000busion, CRs can only stream along\nmagnetic \feld lines. They can thus leave the cluster most quickly\nif the \feld lines are radial.\nMNRAS 000, 000{000 (0000)5\nBut in a real cluster the magnetic \feld may be signi\fcantly\ntangled. Indeed, the very basis of our streaming model is that\nthe presence of MHD turbulence provides a wave damping mech-\nanism. This same turbulence will also tangle the magnetic \feld\non some length scale, increasing the escape time of CRs in our\nstreaming model in some potentially complicated way.\nFortunately we can make reasonable estimates of how the\nescape time is a\u000bected by such tangling. Suppose we want to\nknow how long it takes for a CR to travel a distance Daway\nfrom the center of our cluster. In the 1D symmetric case with\nperfectly radial magnetic \feld lines, this time is just\ntst\u0018D=vst;\nwhere we simplify to a case where the streaming speed vstis\nroughly constant in space and time. If instead, the \feld is tangled\non some length scale L, we can treat the transport of CRs as\na random walk with steps of length L, travel at speed vst. CR\ntransport is then e\u000bectively di\u000busive with di\u000busion coe\u000ecient\n\u0014\u0018Lvst:\nThe time for CRs to di\u000buse out to a distance Dis then\ntdi\u000b\u0018D2\n\u0014\u0018D\nLD\nvst\u0018D\nLtst:\nIn other words, tangling of the \feld on scale Lincreases the escape\ntime of CRs by roughly D=L.\nIs this a large factor? We can approximate the tangling\nlength scale consistently by \fnding the scale where the kinetic\nenergy density in turbulence equals the magnetic energy density.\nBelow this scale, turbulent motions are too weak to bend the \feld\nlines. This occurs when\n1\n2\u001av2\nt=B2\n8\u0019=1\n2\u001av2\nA\nUsing our de\fnition of LMHD as the length scale where the tur-\nbulent speed vtis equal to the Alfv\u0013 en speed vA(seex2.2), we\nsee that the tangling length scale described above is just of order\nLMHD . Our \fducial value is 100 kpc, and our simulated cluster\nhas radius 1 Mpc. We may then expect the e\u000bective escape speed\nto exceed the streaming times in our simulation by about a factor\nof 10.\nThere are also some observational constraints on the mag-\nnetic \feld structure in galaxy clusters which rely on Faraday ro-\ntation measure (RM) observations of radio sources in or behind\nthe clusters. The theoretical background for this technique is de-\nscribed in detail by many authors (see Murgia et al. (2004) and\nFeretti et al. (2012) for just two examples). In this framework, the\nstructure of the magnetic \feld is described by a simple power law\nin Fourier space, jBkj2/k\u0000n, between two length scales \u0003 min\nand \u0003 max.\nBonafede et al. (2010) use RM observations in this way to\nconstrain the magnetic \feld of the Coma cluster in particular.\nThey claim that the \feld which best \fts the RM data is tangled\non scales ranging from \u0003 min\u00182 kpc to \u0003 max\u001834 kpc, with a\nsteep Kolmogorov-like spectrum of n= 11=3. This suggests that\nmost of the power is on large scales, i.e. most of the \\steps\" in\nthe \feld line random walk are of length \u0003 max\u001834 kpc. This\nis reasonably close to our above estimate of LMHD . We note,\nhowever, that in general the magnetic \feld structure of galaxy\nclusters is not well constrained by observations. So while it is\nhard to say with any certainty, we may reasonably expect our\nsimpli\fed model to underestimate streaming times by a factor of\n10 - 30.\nIn the context of our question about radio halo turno\u000b times,\nthis is not a cripplingly large factor. It extends the turno\u000b time\nfrom hundreds of Myr to a few Gyr, still in the range of reason-\nable turno\u000b times for the hadronic model. More importantly for\nthe context of this work, this factor is independent of the high\nbeta e\u000bects considered here. If our older simulations from Wiener\net al. (2013) are o\u000b by a factor of 10, then the new simulationspresented here are also o\u000b by this same factor, and our main point\nis unchanged.\nOf course there are non-linear e\u000bects which complicate this -\nas the CR density drops the streaming speed increases. If escape\nspeeds are initially slower by a factor of 10, the overall shuto\u000b\ntime may be changed by an entirely di\u000berent factor. Determining\nthe e\u000bects of this non-linearity would require non-1D simulations\nwhich are beyond the scope of this work. However, we are encour-\naged that in the apples-to-apples comparison we make here, the\nhigh-beta e\u000bect causes an increase in the initial streaming speeds\nby a factor of \f1=2\u00188 in the central regions of the cluster. So dif-\nferences brought about by these high beta e\u000bects should at least\nbe comparable to the e\u000bects of \feld tangling.\nStill, it is di\u000ecult to assess the e\u000bects of \feld tangling and\nnon-linear evolution other than the above speculation. If we have\noverestimated the \feld tangling scale by a factor of about 10,\nwhich is possible considering the spectrum in Bonafede et al.\n(2010) extends down to 2 kpc, then our escape speed becomes\nof the order of the age of the universe even without accounting\nfor the non-linearity of the evolution. Future observational con-\nstraints on magnetic \feld structure in galaxy clusters may there-\nfore prove critical to the question of long range CR streaming.\nThere is another point to be made about the magnetic \feld\ntopology which may be important. We have talked about the\ne\u000bects of a tangled \feld, but what if the \feld lines never leave\nthe cluster? Consider a single \feld line which extends out to some\nradiusRin the cluster before folding back on itself and returning\ntowards the center. Then along this \feld line, streaming will even\nout the CR density out to R, but CRs will be unable to leak out\npastR. If all the \feld lines worked this way, no CRs would be able\nto escape, although they could spread out evenly within radius R.\nHowever, if some percentage of \feld lines leave the cluster, then\nCRs will always be able to leak out along these lines. The halo\ndropo\u000b time would then be primarily determined by two factors -\nthe percentage of \feld lines which exit the cluster, and the rate of\ncross \feld di\u000busion which allows CRs on trapped lines to migrate\nto neighboring lines which escape.\n3 SIMULATION SETUP\nWe reproduce here the spherically symmetric ZEUS hydrody-\nnamic simulations of the Coma cluster from Wiener et al. (2013),\nbut with the damping rate enhanced by the factor of \f1=2from\nthe analysis inx2. As mentioned in x2.3, spherical symmetry is\nclearly a drastic simpli\fcation (L. Rudnick; personal communica-\ntion) but we assume it here to keep the problem tractable and to\nfocus on the di\u000berence between the present treatment and Wiener\net al. (2013). As in Wiener et al. (2013), we model the Coma clus-\nter with density and temperature pro\fles given by\nne\n10\u00003cm\u00003= 3:4\"\n1 +\u0012r\n294 kpc\u00132#\u00001:125\n(19)\nT= 8:25 keV\"\n1 +\u0012r\n460 kpc\u00132#\u00000:32\n(20)\nThese pro\fles are taken from Pinzke & Pfrommer (2010), with\nthe density inferred from X-ray observations (Briel et al. (1992)).\nThe magnetic \feld is assumed to scale with gas density as\nB=B0\u0012ne(r)\nne(0)\u0013\u000bB\n; (21)\nwithB0= 5\u0016G and\u000bB= 0:3, as suggested by constraints\nfrom Bonafede et al. (2010) (Wiener et al. (2013) also considered\n\u000b= 0:5 which is more in line with Bonafede et al. (2010); the\nchoice of\u000b= 0:3 here is to properly compare with the results\nfrom Wiener et al. (2013). It is conservative in the sense that it\nMNRAS 000, 000{000 (0000)6Wiener, Zweibel, & Oh\nimplies a lower \fand therefore a weaker Landau damping e\u000bect).\nThe above density, temperature, and magnetic \feld pro\fles are\n\fxed for these simulations - only the CR distribution is evolved\nin time.\nThe initial CR distribution is of the form used in Pinzke\n& Pfrommer (2010), motivated by cosmological hydrodynamic\nsimulations of galaxy clusters where cosmic rays are accelerated\nvia di\u000busive shock acceleration:\nfp(r;pp) =C(r)X\ni\u0001ip\u0000\u000bip (22)\n\u0001= (0:767;0:143;0:0975)\u000b= (2:55;2:3;2:15): (23)\nHere and in the equations to follow, the momenta pare expressed\nin units ofmcfor the appropriate m. Namely, for actual momen-\ntumPwe will work in terms of pp=P=mpcandpe=P=mec. In\nthis framework we have to be careful how we de\fne our distribu-\ntion functions fpandfe. To be unambiguous, let us de\fne them\nas such:\ndnp(r;pp) =fp(r;pp)dpp\ndne(r;pe) =fe(r;pe)dpe\nwhere dnp(r;pp) is the di\u000berential number density of CRp at ra-\ndiusrand unitless momentum pp, and similarly for the electrons.\nWe choose the normalization Cto be of the form4\nC(r) =(Cvir\u0000Ccenter )\n1 +\u0010\nr\nrtrans\u0011\u0000\fC+Ccenter: (24)\nIn Pinzke & Pfrommer (2010), the parameters Cvir,Ccenter ,\nrtrans, and\fCare determined from scaling relations. We instead\nchoose values that roughly reproduce the observed synchrotron\nradiation. For Coma, these are Ccenter = 6\u000210\u000011cm\u00003,Cvir=\n5:2\u000210\u000011cm\u00003,rtrans = 55 kpc,\fC= 1:09.\nWith the above initial distribution we evolve the CRp dis-\ntribution function forward in time according to\n@fp\n@t+ (u+vA)\u0001rfp=1\n3p@fp\n@pr\u0001(u+vA)\n+1\np3r\u0001\u0012\u0000dampB2n\n4\u00193mp\n0vAn\u0001rfp\njn\u0001rfpj\u0013 (25)\nas in Skilling (1971). In the above, uandvAare the gas and\nAlfv\u0013 en velocities, \n 0is the non-relativistic gyrofrequency, and n\nis a unit vector which points along the magnetic \feld. The last\nterm represents di\u000busion with respect to the Alfv\u0013 en wave frame\ndue to wave damping, and is directly a\u000bected by the discussion in\nx2. More details of this equation and its numerical evolution can\nbe found in Wiener et al. (2013). We set u\u00110 in the simulations\ndiscussed here, but include it in eqn. (25) for completeness.\nAs the CRp distribution function evolves, we derive from\nit at every time step a steady-state secondary CRe distribution\nfunction according to\nfe(r;pe) =1\nj_pejZ1\npedp0\nese(r;p0\ne); (26)\nwhich balances the source function from the CRp hadronic colli-\nsions\nse(r;pe) =4\n316me\nmpcnN(r)\u001bppfp\u0012\nr;pp=16me\nmppe\u0013\n(27)\n4Although this resembles the normalization in Pinzke & Pfrom-\nmer (2010), please note the di\u000berence between our C(r) (which\nhave dimensions of number density) used here and the ~C(r) =\nC(r)\u001a(r)=mp(unitless) used by Pinzke & Pfrommer (2010). We\nutilize this formula as a convenience, but our CR density pro\fle\nis actually much \ratter than in their simulation.with the losses from synchrotron emission and inverse Compton\n(IC) scattering\n_pe(r;pe) =4\n3\u001bTp2\ne\nmec(\"B(r) +\"cmb): (28)\nIn (27),nNis the number density of target nucleons and we\nassume we are far above the pion production threshold.\nThis steady state model includes two implicit assumptions.\nFirst, the energy losses of the secondary CRe are dominated by\nsynchrotron and IC losses. Namely this means we assume the\nCRe don't stream on time scales faster than their loss times (\u0018<\n100 Myr), a condition we will have to check a posteriori . This\nassumption will turn out to be well satis\fed. Second, we assume\nthat the source function seis not changing signi\fcantly on these\nsame time scales. This will turn out not to be very well satis\fed,\nas we discuss below in x4.\nOnce we have the secondary CRe distribution function at\neach time step, we can then determine the synchrotron emissivity\nat frequency \u0017. From Rybicki & Lightman (1979),\nj\u0017(r) = 0:333p\n3\n2\u0019e3B(r)\nmec2Z1\n0dpefe(r;pe)F\u0012\u0017\n\u0017c\u0013\n: (29)\nIn the above, \u0017c= 3eBp2\ne=4\u0019mecand the function Fis an integral\nof a modi\fed Bessel function, F(x) =xR1\nxK5=3(x0)dx0. With\nthe emissivity in hand, we determine surface brightness and total\nluminosity from simple spatial integrals5.\n4 RESULTS\nThe total 1.4 GHz luminosity of our simulated Coma cluster as a\nfunction of time is shown in Figure 2. Three CR transport models\nare compared: in the \frst, there is no wave damping and CRs\nstream at the Alfv\u0013 en speed. In the second, there is wave damping\naccording to (5). In the third, there is wave damping with the\nhigh-\fcorrection factor, (9). We see that the increased factor of\n\f1=2in the damping rate allows CRs to stream out even faster\nthan in our original simulation, causing the radio luminosity to\nalso drop on faster time scales. The streaming speeds for the\nrelevant 100 GeV CRp are shown in \fgure 3. This \fgure shows\nthe streaming speed at a \fxed radius of 300 kpc as a function\nof time. Initially the streaming speeds are larger with the high- \f\ncorrection in comparison to without, and this enhancement grows\nwith time due to the non-linear evolution of the streaming speeds.\nWe may wonder if these enhanced streaming speeds inter-\nfere with any of our assumptions, namely our assumption that\nCRe secondary losses are dominated by synchrotron and inverse\nCompton losses. CRe will stream at the same speeds as CRp of\nthe same energy, and if the CRe stream out on time scales compa-\nrable to their loss times, our steady state treatment is incorrect.\nLooking at Figure 3 we may suspect that this is the case, as the\n\u0018104km/s streaming speeds imply streaming times of \u0018100 Myr\nacross our 1-2 Mpc cluster.\nHowever, for the electrons, the relevant energies for 1.4 GHz\nemission are 1-10 GeV. As shown in Figure 4, CRp at these en-\nergies stream out much more slowly, even in our high- \fmodel6.\n5We reiterate that the hadronic model for giant radio haloes\nis disfavored by gamma-ray non-detections. We include a radio\nsurface brightness prediction for direct comparison with Wiener\net al. (2013). The dimming of the model radio halo can also be\nthought of as a simple proxy for the evolution of CRp.\n6Note that most CR energy resides at \u0018GeV energies, and\nthat even with the high \fcorrection these CRs remain more\nor less locked to the wave frame. Thus, CR heating in cluster\ncores (Loewenstein et al. 1991; Guo & Oh 2008; Pfrommer 2013;\nRuszkowski et al. 2017) remains viable.\nMNRAS 000, 000{000 (0000)7\n0 100 200 300 400 500\nt (Myr)102910301031L1.4GHz (erg s−1 Hz−1)\nNo Damping\nLMHD= 100 kpc\nLMHD= 100 kpc, High Beta Correction\nFigure 2. Total 1.4 GHz luminosity as a function of time for the\nComa cluster simulation. The quicker streaming speeds in the\nnew simulation result in a quicker turn-o\u000b of the rxhibited radio\nhalo.\n0 100 200 300 400 500\nt (Myr)102103104105vs (km s−1) (96.5 GeV, 299 kpc)Alfven speed\nLMHD= 100 kpc\nLMHD= 100 kpc, High Beta Correction\nFigure 3. Streaming speeds as a function of time for 100 GeV CR\nprotons at a radius of 300 kpc in the Coma cluster simulation. The\nenhanced damping due to high \fe\u000bects signi\fcantly increases the\nstreaming speed.\nWe can expect the streaming times of our CRe secondaries to be\n\u00181-2 Gyr, much longer than their loss times. Note that since the\nstreaming instability growth rate is proportional to nCR(equation\n(1)), andnCR;p\u001dnCR;e, the CRe actually scatter o\u000b the wave\n\feld created by the much more abundant CRp. If they manage\nto achieve spatial separation, the CRe could in principle stream\nmuch faster.\nStill, the streaming speeds of the 100 GeV CRp which source\nthese electrons are high, implying a short crossing time. Our\nsteady state model assumes that the CRp distribution changes\non time scales much longer than the CRe loss times, such that\nthe production of secondaries `quickly' reaches an equilibrium be-\nfore the CRp density changes too much. This assumption is no\nlonger satis\fed, so a more complete treatment of secondary CRe\nproduction and transport may be necessary. However, we have\n0 100 200 300 400 500\nt (Myr)102103104105vs (km s−1) (5.0 GeV, 299 kpc)Alfven speed\nLMHD= 100 kpc\nLMHD= 100 kpc, High Beta CorrectionFigure 4. Streaming speeds as a function of time for 5 GeV CR\nprotons at a radius of 300 kpc in the Coma cluster simulation.\nAt these lower energies, the streaming speeds are not signi\fcantly\nsuper-Alfv\u0013 enic.\nused somewhat arti\fcial initial conditions which do not re\rect\nthe cosmological build up of CRp. As long as the injection time is\nlonger than the streaming time (likely to be true for CRs sourced\nby structure formation (giant radio halos), less so for CRs sourced\nby AGN activity (radio mini-halos)), CRs will have a \rat distri-\nbution, which slowly increases in normalization.\n5 CONCLUSION\nDespite signi\fcant advances in observations and modeling, the\norigin and transport of cosmic rays in galaxy clusters remains\nincompletely understood. As long as this is the case, radiative\ndiagnostics based on cosmic rays will be provisional, and a full\nassessment of the role of cosmic rays in the dynamics and energy\nbalance of clusters will elude us.\nIn this paper, we have added a new ingredient to the prop-\nagation problem: the role of ion Landau damping in suppress-\ning the growth of Alfv\u0013 en waves excited by cosmic ray streaming.\nAlthough waves which propagate exactly along the background\nmagnetic \feld B0are undamped, inhomogeneity in B0precludes\nperfectly parallel propagation. At the large \f\u00118\u0019Pg=B2val-\nues characteristic of galaxy clusters, oblique waves are subject to\nstrong Landau damping due to their parallel electric \felds.\nWhen the propagation angle is calculated assuming B0is\na uniform \feld with a superimposed anisotropic MHD turbulent\ncascade, the resulting damping rate (9) is of the same form as\nthe damping rate estimated from the turbulent shearing rate (5),\nwhich was taken as the dominant damping mechanism in earlier\nwork (Wiener et al. 2013), but exceeds the turbulent damping\nrate by a factor of \f1=2. It is worth noting that the assumption of\nan anisotropic turbulent cascade is key to this result: if the scale\nof variation of B0were taken to be a global scale, the resulting\ndamping rate would be very slow (Zweibel 2003).\nSince Landau damping is a collisionless process, it is impor-\ntant to check whether the e\u000bective ion mean free path is much\nlonger than the wavelengths of the Alfven waves. Estimating the\nmean free path \u0015iito Coulomb scattering is straightforward and\nthe result (18) shows that the waves are indeed collisionless with\nrespect to Coulomb interactions. However, we also estimated for\nthe \frst time the mean free path \u0015idue to scattering from mi-\nMNRAS 000, 000{000 (0000)8Wiener, Zweibel, & Oh\ncroscale instabilities driven by pressure anisotropy (17), and found\nthat it too is much larger than the wavelengths of the Alfv\u0013 en\nwaves in question, fully validating the role of Landau damping\nin suppressing the Alfv\u0013 en waves that scatter cosmic rays in tur-\nbulent galaxy cluster plasmas. Note that there are potentially\nimportant caveats about the nature of MHD turbulence in a high\n\fplasma (Squire et al. 2016, 2017), which are beyond the scope\nof this paper.\nWe then reconsidered the evolution of an idealized version of\nthe radio halo of the Coma cluster following Wiener et al. (2013),\nbut including Landau damping. In this model, the halo is pro-\nduced by secondary electrons and positrons created in hadronic\ninteractions between cosmic ray protons and thermal cluster gas.\nAs expected, transport of the cosmic ray protons is much faster,\nquickly depleting them and turning o\u000b the halo on even faster\ntimescales than predicted in the original model. This strengthens\nthe case for rapid evolution of radio halos, at least in this idealized\n(1D, radial magnetic \feld) model cluster, unless the cosmic ray\nprimary proton source is continuously replenished. This setup is\nthe most optimistic case there is for evolution via streaming, an\nimportant caveat of this work. Future simulations with higher di-\nmensionality and more complicated \feld topologies are necessary\nto study this e\u000bect in real clusters.\nThe problem of cosmic ray electron transport on galaxy clus-\nters was \frst raised by Ja\u000be (1977), who pointed out that the\nradiative loss time of cosmic ray electrons is much shorter than\ntheir transport time at the Alfv\u0013 en speed from the core of the clus-\nter. If the primary CRe could stream out to large distances in less\nthan one cooling time, they could supply the radio emission in the\noutskirts without being reaccelerated. However, even in our ideal\nscenario this does not seem to be the case. Although we have not\nundertaken a detailed comparison of Ja\u000be's model with ours, the\ntransport speeds of 5 GeV electrons shown in Figure 4 are well\nbelow the\u00182000 km/s transport speed that Ja\u000be estimated was\nnecessary to form the halo with primary electrons.\nThe rapid transport speed of cosmic ray protons due to Lan-\ndau damping of Alfv\u0013 en waves could help to explain the stringent\nupper limits on di\u000buse \r-ray emission from galaxy cluster cores re-\nported by Pinzke & Pfrommer (2010); En\u0019lin et al. (2011); Ahnen\net al. (2016). It also has important consequences for hadronic pro-\nduction of CRe `seeds' for turbulent reacceleration - rapid stream-\ning produces a \rat CRp pro\fle, giving a seed population which\nbetter \fts observational constraints (Pinzke et al. 2017). It is also\nrelevant to the tension between observed radio halo luminosities\nand models of CR heating of cluster cores (Jacob & Pfrommer\n2017), since high energy CRs responsible for the former stream\nat much higher velocities than the lower energy CRs relevant for\nthe latter. These issues are beyond the scope of the current paper,\nbut a topic for future work.\nAcknowledgements:\nWe are happy to acknowledge discussions with Matt Kunz,\nEliot Quataert, and Larry Rudnick. JW and EGZ acknowledge\nsupport by NSF Grant AST-1616037, the WARF Foundation, and\nthe Vilas Trust. SPO acknowledges support from NASA grant\nNNX15AK81G.\nREFERENCES\nAckermann M., et al., 2016, ApJ, 819, 149\nAhnen M. L., et al., 2016, A&A, 589, A33\nBasu K., 2012, MNRAS, 421, L112\nBonafede A., Feretti L., Murgia M., Govoni F., Giovannini G.,\nDallacasa D., Dolag K., Taylor G. B., 2010, A&A, 513, A30\nBraginskii S. I., 1965, Reviews of Plasma Physics, 1, 205\nBriel U. G., Henry J. 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G., 2003, ApJ, 587, 625\nMNRAS 000, 000{000 (0000)" }, { "title": "1912.02500v1.Steering_magnonic_dynamics_and_permeability_at_exceptional_points_in_a_parity_time_symmetric_waveguide.pdf", "content": "Steering magnonic dynamics and permeability\nat exceptional points in a parity-time\nsymmetric waveguide\nXi-guang Wang,y,zGuang-hua Guo,yand Jamal Berakdar\u0003,z\nySchool of Physics and Electronics, Central South University, Changsha 410083, China\nzInstitut für Physik, Martin-Luther Universität Halle-Wittenberg, D-06120 Halle/Saale,\nGermany\nE-mail: jamal.berakdar@physik.uni-halle.de\nAbstract\nTuning the low-energy magnetic dynamics is a key element in designing novel mag-\nnetic metamaterials, spintronic devices and magnonic logic circuits. This study uncov-\ners a new, highly effective way of controlling the magnetic permeability via shaping\nthe magnonic properties in coupled magnetic waveguides separated by current carrying\nspacer with strong spin-orbit coupling. The spin-orbit torques exerted on the waveg-\nuidesleadstoanexternallytunableenhancementofmagneticdampinginonewaveguide\nandadecreaseddampingintheother, constitutingsoamagneticparity-time(PT)sym-\nmetric system with emergent magnetic properties at the verge of the exceptional point\nwhere magnetic gains/losses are balanced. In addition to controlling the magnetic per-\nmeability, phenomena inherent to PT-symmetric systems are identified, including the\ncontrolonmagnonpoweroscillations, nonreciprocalmagnonpropagation, magnontrap-\nping and enhancement as well as the increased sensitivity to magnetic perturbation and\nabrupt spin reversal. These predictions are demonstrated analytically and confirmed\n1arXiv:1912.02500v1 [cond-mat.mes-hall] 5 Dec 2019by full numerical simulations under experimentally feasible conditions. The position of\nthe exceptional points and the strength of the spontaneous PT symmetry breaking can\nbe tuned by external electric and/or magnetic fields. The roles of the intrinsic magnetic\ndamping, and the possibility of an electric control via Dzyaloshinskii-Moriya interac-\ntion are exposed and utilized for mode dispersion shaping and magnon amplification\nand trapping. The results point to a new route to designing optomagnonic waveguides,\ntraps, sensors, and circuits.\nKeywords\nMagnonic circuits, PT-symmetry breaking, spin orbit torque, non-Hermitian dynamics, Op-\ntomagnonics, magnetic switching\nIntroduction\nNanomagnetism is the backbone of spin-based memories, data processing and sensorics. In a\ngeneric magnet, the permeability, meaning the magnetic response to a weak external pertur-\nbation is governed by the behavior of the spin waves which are collective transverse oscilla-\ntions (with their quantum termed magnon) around the ground state. Miniaturized magnonic\nlogiccircuits1–6andwaveguidesoperatedatlowenergycostwithnegligibleOhmiclosseswere\ndemonstrated. Furthermore, geometric confinements, nanostructuring, and material design\nallow a precise spectral shaping and guiding of magnons, which is reflected respectively in a\nmodified magnetic response. Here we point out an approach based on a magnonic gain-loss\nmechanism in two waveguides with a normal-metal spacer. The two magnetic waveguides\nare coupled via the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. Driving charge\ncurrent in a spacer layer with a strong spin-orbit coupling (cf. 1(a)), spin orbit torques\n(SOTs) are exerted on the magnetizations of the waveguides. In effect, SOT adds to the\nintrinsic magnetic damping, as evident from Eq. (1). It is thus possible to achieve a case\n2where SOT-induced magnetic losses in one waveguide are balanced by antidamping in the\nother waveguide. This is a typical case of a PT-symmetric system as realized for instance\nin optical systems.7–12A hallmark of PT-symmetric systems is that, even if the underly-\ning Hamiltonian is non-Hermitian, the eigenvalues may be real,13–15and turn complex when\ncrossing the \"exceptional point\" and entering the PT-symmetry broken phase upon varying a\nparametric dependence in the Hamiltonian. Examples were demonstrated in optics and pho-\ntonics,8–12,16–21optomechanics,22,23acoustics24,25and electronics.26–29Also, PT-symmetric\ncavity magnon-polaritons were discussed involving phonon dissipation or electromagnetic\nradiation as well as parametric driving or SOT effects.30–35\nOur main goal is the design and demonstration of PT-symmetric magnonic waveguides\nwhich are controllable by feasible external means that serve as a knob to tune the system\nacross the exceptional point. In addition to the documented advantages of magnons, this\nwould bring about new functionalities that can be integrated in optomagnonic, spintronic,\nand magnonic circuits.\nSpin-torque driven PT-symmetric waveguides\nOur magnon signal propagates along two magnetic waveguides (which define the ~ xdirection)\ncoupled via RKKY exchange interaction (cf. Fig, 1(a)). A charge current flowing in a spacer\nwith a large spin Hall angle (such as Pt) applies a SOT ~T1k~ yon first waveguide enhancing\nthe effective damping, and a SOT ~T2k\u0000~ yon second waveguide weakening the effective\ndamping. The polarization directions of the spin Hall effect induced transverse spin currents\n~T1=~ z\u0002~jPtin WG1 and ~T2= (\u0000~ z)\u0002~jPtin WG2, are related to the charge current density\n~jPt. In a generic ferromagnet and for the long wavelength spin excitations of interest here,\nto describe the magnetic dynamics it is sufficient to adopt a classical continuous approach\nand solve for the equations of motion of the magnetization vector fields ~Mp(~ r;t)(p= 1;2\n3a \nFigure 1: (a) Two magnetic waveguides (labeled as WG1 and WG2, as an example we\nuse YIG in the numerical simulations) are coupled via RKKY interaction with metallic\nspacer that has a large spin Hall angle (here Pt). Driving a charge current ~jPtalong the\nspace (xdirection) results in spin-Hall torques acting on the magnetic waveguides. The\ntorques damp or antidamp the magnetic dynamics in WG1 and WG2 resulting so in PT-\nsymmetric structure with a new (PT) symmetry behavior of the magnetic permeability.\nMagnon wave packets are launched locally at one end of WG1 or WG2 (left side in the figure)\nand the propagation characteristics of the magnonic signal is steered, amplified or suppressed\nby external fields that drive the waveguides from the PT-symmetric to the PT-symmetry\nbroken phase through the exceptional point where magnetic losses in WG1 balance magnetic\nantidamping in WG2. This can be achieved for instance by changing the ratio between\nthe intrinsic coupling strength between the waveguides \u0014and the strength of the spin-Hall\ntorques!J. (b) Real and (c) imaginary parts of two eigenmode frequency f=!=(2\u0019)as\nwe scan!J=\u0014at the wave vector kx= 0:1nm\u00001. (d-f) Spatial profiles of propagating spin\nwave amplitude for different loss/gain balance (different !J), when the spin waves are locally\nexcited either in WG1 or WG2. The color change from blue to red corresponds to a linear\namplitude change ranging from 0 to the maximum of input signal. The local microwave field\nexcites spin waves at the left side of the waveguide and has a frequency of 20 GHz. The\nlength (along xaxis) of waveguides in (d-f) is 580 nm. 4enumerates the two waveguides), which amounts to propagating the Landau-Lifshitz-Gilbert\n(LLG) equation,36–39\n@~Mp\n@t=\u0000\r~Mp\u0002~He\u000b;p+~Mp\nMs\u0002\"\n\u000b@~Mp\n@t\u0000\rcJ~Tp\u0002~Mp#\n: (1)\nThe waveguides are located at z= +z0andz=\u0000z0. We are interested in small transversal\nexcitations and hence it is useful to use the unit vector field ~ mp=~Mp=MswhereMsis the\nsaturation magnetization and \ris the gyromagnetic ratio. \u000bis the conventional Gilbert\ndamping inherent to magnetic loses in each of the waveguides. The effective field ~He\u000b;p=\n2Aex\n\u00160Msr2~ mp+JRKKY\n2\u00160Mstp~ mp0+H0~ yconsists of the internal exchange field, the interlayer RKKY\ncoupling field, and the external magnetic field applied along the yaxis, where p;p0= 1;2,\nandp06=p.Aexis the exchange constant, JRKKYis the interlayer RKKY exchange coupling\nstrength,tpis the thickness of the pth layer, and \u00160is the vacuum permeability. Of key\nimportance to this study is the strength cJ=T\u0012SH~Je\n2\u00160etpMsof SOT which is proportional\nto charge-current density Jeand the spin Hall angle \u0012SHin the spacer layer, for instance,\nat the exceptional point defined in the following study, cJ= 1\u0002105A/m corresponds to a\ncharge current density of Je= 9\u0002108A/cm2in Pt.40Tis the transparency at the interface,\nandeis the electron charge. Our proposal applies to a variety of settings, in particular\nsynthetic antiferromagnets41offer a good range of tunability. To be specific, we present\nhere numerical simulations for Pt interfaced with a Yttrium-Iron-Garnet (YIG) waveguides\nas experimentally realized for instance in Ref. [ 40] corresponding to the following values\nMs= 1:4\u0002105A/m,Aex= 3\u000210\u000012J/m (technical details of the numerical realization are\nin the supplementary materials). For the Gilbert damping we use \u000b= 0:004but note that\ndepending on the quality of the waveguides \u000bcan be two order of magnitude smaller. The\ninterlayer exchange constant JRKKY = 9\u000210\u00005J/m2, which is in the typical range.42For\nthe waveguide thickness we used t1;2= 4nm. A large enough magnetic field H0= 2\u0002105\nA/m is applied along +ydirection to bring the WGs to a remnant state.\n5Magnonic coupled wave-guide equations with spin-orbit\ntorque\nFor a deeper understanding of the full-fledge numerical simulations presented below, it\nis instructive to formulate an analytical model by considering small deviations of ~ ms;p=\n(\u000emx;p;0;\u000emz;p)away from the initial equilibrium ~ m0;p=~ y. Introducing p=\u000emx;p+i\u000emz;p\nwe deduce from linearizing Eq. (1) the coupled waveguide equations\ni@ 1\n@t\u0000[(!0\u0000\u000b!J)\u0000i(!J+\u000b!0)] 1+q 2= 0;\ni@ 2\n@t\u0000[(!0+\u000b!J) +i(!J\u0000\u000b!0)] 2+q 1= 0:(2)\nFor convenience, we introduce in addition to the coupling strength q=\rJRKKY\n(1+i\u000b)\u00160Mstp, the SOT\ncoupling at zero intrinsic damping \u0014=\rJRKKY=(2\u00160Mstp) =qj\u000b!0. The intrinsic frequency\nof the waveguides is given by !0=\r\n1+\u000b2(H0+2Aex\n\u00160Msk2\nx+JRKKY\n2\u00160Mstp)which is for the material\nstudied here is in the GHz. Essential for the behavior akin to PT-symmetric systems is\nthe SOT-driven gain-loss term !J=\rcJ\n1+\u000b2. The wavevector along xdirection is kx. Eq. (2)\nadmits a clear interpretation: The magnonic guided modes in the first waveguide ( WG1) are\nsubject to the confining complex potential V(z) =VR(z)+iVi(z)withVR(z0) =!0\u0000\u000b!Jand\nVi(z0) =\u0000!J\u0000\u000b!0. In WG2 the potential is VR(\u0000z0) =!0+\u000b!JandVi(\u0000z0) =!J\u0000\u000b!0.\nThe mode coupling is mediated by qwhich determines the periodic magnon power exchange\nbetween WG1 and WG2 in absence of SOT.\nFor a PT symmetric system the condition VR(z0) =VR(\u0000z0)andVi(z0) =\u0000Vi(\u0000z0)must\napply, which is obviously fulfilled if the intrinsic damping is very small ( \u000b!0). Comparing\nthecurrentandthephotoniccase, inthelattercasethesignoftheimaginarypartoftheWGs\nrefractive index is tuned. Here we control with SOT the imaginary part of the permeability\nwhich we explicitly prove by deriving and analyzing of the magnetic susceptibility (cf. Supp.\nMaterials). ThisfindingpointstoanewroutefordesigningPT-symmetricmagneto-photonic\nstructures via permeability engineering. We note, for a finite magnetic damping \u000ba PT-\n6behavior is still viable as confirmed by the full numerical simulations that we discuss below.\nMagnon dynamics across the spontaneous PT-symmetry\nbreaking transition\nThe dispersion !(kx)of the modes governed by Eq. (2) reads\n/s45/s50/s120/s49/s48/s56\n/s45/s49/s120/s49/s48/s56\n/s48\n/s49/s120/s49/s48/s56\n/s50/s120/s49/s48/s56/s48/s50/s48/s52/s48/s54/s48\n/s32/s32/s102/s32/s32/s40/s71/s72/s122/s41\n/s107\n/s120/s32/s40/s109 /s41\nFigure 2: Merging of the acoustic ( !J= 0, solid squares) and optical magnon ( !J= 0,\nsolid dots) modes dispersion Re[!](kx)when approaching the loss/gain-balanced exceptional\npoint!J=\u0014(open dots).\n!= (1\u0000i\u000b)!0\u0006q\nq2\u0000!2\nJ+ 2i\u000b!2\nJ+\u000b2!2\nJ (3)\nwhich describes both the acoustic and optical magnon modes43and depends parametrically\non!Jandq. For\u000b!0(in which case q\u0011\u0014) the eigenvalues are always real in the\nPT-symmetric regime below the gain/loss-balanced threshold !J=\u0014< 1. At the exceptional\npoint!J=\u0014= 1, the two eigenvalues and eigenmodes become identical. For !J=\u0014 > 1(by\nincreasing the current density for instance) we enter the PT-symmetry broken phase, and the\neigenvalues turn complex, as typical for PT-symmetric systems.21,44The splitting between\nthe two imaginary parts is determined by 2\u0014[(!J=\u0014)2\u00001]1=2and is tunable by external\nfields. This fact is useful when exploiting the enhanced waveguides sensitivity to magnetic\nperturbations round the exceptional point. Allowing for a small damping \u000bdoes not alter\nthe modes behavior, as demonstrated by the full numerical results in a Fig. 1(b-c). The\n7full magnon dispersions ( Re[!]versuskxcurves ) for !J=\u0014< 1and!J=\u0014= 1are shown in\nFig. 2. The symmetry of our waveguide brings in a special behavior of the magnon signal\ntransmission, meaning the propagation of a superposition of eigenmodes: Without charge\ncurrent in the spacer ( !J= 0), a signal injected at one end in one waveguide oscillates\nbetween WG1 and WG2 (due to the coupling \u0014) in a manner that is well-established in\ncoupled wave guide theory (cf. Fig. 1(d) ). Switching on the charge current, !J=\u0014becomes\nfinite and the beating of the magnon power between WG1 and WG2 increases (cf. Fig. 1(e)),\nas deducible from Eqs. (2), and also encountered in optical wave guides.44Eqs. (2) also\nindicatethatneartheexceptionalpoint, amagnonicwavepacketinjectedinonewaveguideno\nlonger oscillates between the two waveguides but travels simultaneously in both waveguides,\nas confirmed in Fig. 1(f) by full numerical simulations. This behavior resembles the optics\ncase.10We note that in our waveguides, this limit is simply achieved by tuning the external\nelectric and magnetic fields that then change the ratio !J=\u0014. We also found in line with\nRef. [ 10] a non-reciprocal propagation below the exceptional point. Passing the exceptional\npoint (!J=\u0014> 1) the magnonic signal always propagates in the guide with gain and is quickly\ndamped in the guide with loss.\nEnhanced sensing at PT-symmetry breaking transition\nTo assess the susceptibility of our setup to external magnetic perturbations we apply an\nexternal microwave field ~hpwhich adds to effective field in the LLG equation. In frequency\nspace we deduced that e p=P\np0\u001fpp0\rehm;p0(tilde stands for Fourier transform), with hm;p=\nhx;p+ihz;p, and\u001fpp0is the dynamic magnetic susceptibility which has the matrix form\n\u001f=1\n(!k\u0000i\u000b!\u0000!)2+!2\nc\u0000\u001420\n@(!k\u0000i\u000b!) + (i!c\u0000!) \u0014\n\u0014 (!k\u0000i\u000b!)\u0000(i!c\u0000!)1\nA;(4)\nwith!c=\rcJand!k=\r(H0+2Aexk2\nx\n\u00160Ms+JRKKY\n2\u00160Mstp).\n8/s49/s53\n/s51/s48/s48/s46/s48/s49/s46/s48/s120/s49/s48/s45/s56\n/s48/s49\n/s49/s53\n/s51/s48/s48/s46/s48/s51/s46/s48/s120/s49/s48/s45/s56\n/s48/s49\n/s48 /s53/s48/s48/s48/s55\n/s48 /s53/s48/s48/s48/s49/s48\n/s48 /s52/s48/s48/s48/s48/s56/s48\n/s48 /s52/s48/s48/s48/s48/s54/s48\n/s50/s48 /s52/s48/s45/s50/s46/s48/s120/s49/s48/s52/s48/s46/s48/s50/s46/s48/s120/s49/s48/s52\n/s48 /s49/s48/s48/s45/s49/s46/s52/s120/s49/s48/s53/s48/s46/s48/s49/s46/s52/s120/s49/s48/s53\n/s73/s109\n/s91\n/s93\n/s74/s32/s47/s32\n/s102/s32/s40/s71/s72/s122/s41/s32/s74/s32/s47/s32/s73/s109\n/s91\n/s93\n/s102\n/s32/s40/s71/s72/s122/s41/s98/s65/s109/s112/s32/s40/s65/s47/s109/s41\n/s32/s32/s65/s109/s112/s32/s40/s65/s47/s109/s41\n/s120 /s32/s40/s110/s109/s41/s97\n/s99\n/s74\n/s65/s109/s112/s32/s40/s65/s47/s109/s41\n/s32/s32/s65/s109/s112/s32/s40/s65/s47/s109/s41\n/s120 /s32/s40/s110/s109/s41/s100\n/s74\n/s74\n/s32/s32/s32\n/s120 /s32/s40/s110/s109/s41/s101/s74\n/s32/s32/s32\n/s120 /s32/s40/s110/s109/s41/s102/s32/s32/s77\n/s120/s32/s40/s65/s47/s109/s41\n/s116/s32/s40/s110/s115/s41/s103\n/s32/s32/s77\n/s121/s32/s40/s65/s47/s109/s41\n/s116/s32/s40/s110/s115/s41/s104Figure 3: Magnetic susceptibility Im[\u001f11](a) and Im[\u001f12](b) as functions of fand!J.\nPeaks in Im[\u001f11]andIm[\u001f12]are found at the exceptional point !J=\u0014. (c-f) Exciting spin\nwaves of frequency 20 GHz at x= 0in WG1, spatial profiles of spin wave amplitude for\ndifferent!J. Black solid line and red dashed line represents the amplitudes in WG1 and\nWG2, respectively. Near !=\u0014, a slight variation in !Jcauses marked changes in the spin\nwave amplitudes (e-f), while the change is negligible near != 0:7\u0014(c-d). (g-h) At !=\u0014,\nincreasing the external magnetic field ( H0= 2\u0002105A/m) in WG1 by 100 A/m, the time\ndependence of Mx(g) andMy(h) atx= 2000nm in WG2.\n9Near the exceptional point the system becomes strongly sensitive, for instance to changes\nin the charge current term !c, as testified by the behavior of the susceptibility which is\ndemonstrated for the imaginary parts of \u001f11and\u001f12in Fig. 3. The high sensitivity of the\nexcited spin waves on !cnear the exception point (see Fig. 3(c-f)) is exploitable to detect\nslight changes in charge current density cJ.\nFurthermore, near the exceptional point, our setup is strongly sensitive to changes in\nthe magnetic environment. As an example, at !J=\u0014, if the magnetic field H0(or local\nmagnetization) is increased by 100 A/m in WG1, large amplitude spin-wave oscillations\nare generated in WG2, as evidenced by the time dependence of Mx(x= 2000nm) in WG2\n(Fig. 3(g)). The spin wave amplification leads eventually to a reversal of Myin WG2 (Fig.\n3(h)). Away from the PT-breaking transition, e.g. for !J= 0:7\u0014, whenH0is reduced by\nthe same amount in WG1, virtually no changes in propagating spin waves are observed (not\nshown). Obviously, this magnon amplification may serve as a tunable sensor for the magnetic\nenvironment.\nCurrent-induced switching in magnetic PT-symmetric junc-\ntions\nA special feature of magnetic systems is the possibility of current-induced switching (de-\nscribed by Eq. (1) but not by Eqs. (2)).45In fact, for large current densities we are well\nabove the exceptional point. In this case the magnetic system becomes unstable towards\nswitching. We find with further increasing the charge current density (enhancing !J), the\nlocal magnetization in guide 2 is indeed switched to \u0000y. Magnon dynamics above the excep-\ntional point is still possible however by tuning the spacer material properties or its thickness\nto obtain a smaller \u0014, for instance with JRKKY = 9\u000210\u00007J/m2and\u000b= 0:01. In this case\nthe condition !J\u001d\u000b!0is not satisfied anymore, and the influence of intrinsic magnetic\nlosses (\u000b) in both wave guides is important. Nonetheless, even without reaching the strict\n10a\nc\ne\nf\nb\nd\nωJ = 2κ ωJ = 2κ\nωJ = 3κ\nωJ = 3κFigure 4: (a-b) Real and imaginary parts of the eigenmodes !as varying the loss/gain\nbalance by scanning !J(meaning, the SOT strength). The wave vector is kx= 0:03nm\u00001\nand the intrinsic coupling between WG1 and WG2 \u0014is lowered, as compared to Fig. 1 (by\nchoosingJRKKY = 9\u000210\u00007J/m2). (c-d) Spatial profiles of magnon wave amplitudes (as\nnormalized to their maxima) for !J= 2\u0014, and ( Re[!] = 2\u0019\u000220GHz). Black dashed lines\ncorresponds to WG1 and red solid line to WG2. (e-f) Time dependence (at the location\nx= 2000nm) and the spatial profiles (at t= 40ns) of thexcomponent of the magnetization\nMxfor!J= 3\u0014. The color variation from blue to red corresponds to a Mxchange from the\nnegative maximum to the positive maximum.\n11PT-symmetric condition, we still observe that the real parts of the two eigenvalues merge at\nthe same point !J=\u0014, and the two imaginary parts become different when !J>\u0014, as shown\nby Fig. 4(a-b). When !J= 2\u0014, the two imaginary parts are both negative, meaning that\nboth modes are evanescent. The propagation of magnonic signal launched in one waveguide\nend is shown in Fig. 4 (c-d) evidencing that the spin waves in the two waveguides decay\ndifferently. An input signal in the waveguide with enhanced damping leads to an evanescent\nspin wave in WG1. Injecting the signal in WG2, the attenuation of spin wave is weaker, and\nits amplitude is always larger. When !J= 3\u0014andIm[!]of the optical magnon mode turns\npositive, we observe that SOT induces spin wave amplification with time (Fig. 4(e-f)). This\nfinding is interesting for cavity optomagnonics.46\nFor input signal in WG1 or WG2, the spin wave amplitude is always larger in WG2 with a\nnegative effective damping. Also, the excited spin wave amplitude is much larger when the\ninput is in the WG2. Thus, no matter from which waveguide we start, the output signal is\nalways distributed at the end of WG2, a fact that can be employed for constructing magnonic\nlogic gates.\nDzyaloshinskii-Moriya interaction in electrically controlled\nPT-symmetric waveguides\nInmagneticlayersandattheirinterfacesanantisymmetricexchange,alsocalledDzyaloshinskii-\nMoriya (DM), interaction47,48may exist. In our context it is particularly interesting that\nthe DM interaction may allow for a coupling to an external electric field ~Eand voltage\ngates. The contribution to the system free energy density in the presence of DM and ~E\nisEelec=\u0000~E\u0001~P, with the spin-driven polarization ~P=cE[(~ m\u0001r)~ m\u0000~ m(r\u0001~ m)].49,50\nThis alters the magnon dynamics through the additional term ~Helec=\u00001\n\u00160Ms\u000eEelec\n\u000e~ min the\neffective field ~He\u000b. To uncover the role of DM interaction on the magnon dynamics in PT\nsymmetric waveguides we consider three cases: (i) The two waveguides experience the same\n12-0.15 0.00 0.15030 60 \n-0.15 0.00 0.15030 60 \n0 1 2-5 05\n0 1 2912 kx (nm-1 ) kx (nm-1 )f (GHz) f (GHz) \nf (GHz) f (GHz) Ez = 2 MV/cm Ez = −2 MV/cm\nRe[ω/2 π]Im[ω/2 π]\nWG1 WG1 \nWG2 WG2 ωJ / κ ωJ / κaa bb\nddccFigure 5: Control of coupled magnonic waveguide characteristics by an external electric\nfield in presence of Dzyaloshinskii-Moriya interaction. (a-b) Applying a static electric field\n~E1;2= (0;0;Ez)withEz=\u00062MV/cm to both waveguides modify the magnon dispersion\nRe[!](kx)curves for!J= 0(solid dots) and !J=\u0014(open dots). (c) Real and imaginary\nparts of two eigenmodes !as functions of !Jand in the presence of two static electric\nfields (or voltages) applied with opposite polarity to the two waveguides ( ~E1= (0;0;Ez)\nand~E2= (0;0;\u0000Ez), andEz= 2MV/cm) at kx= 0:1nm\u00001. (d) Spatial profiles of the\npropagating spinwave amplitudes when applying electric field in WG1 ( ~E1= (0;0;Ez)with\nEz= 2MV/cm, and ~E2= (0;0;0)). Color scale from blue to red corresponds to amplitude\nchange from 0 to its maximum.\n13static electric field ~E1;2= (0;0;Ez).\n(ii) The electric fields in the two waveguides are opposite to each other, i.e. ~E1= (0;0;Ez)\nand~E2= (0;0;\u0000Ez).\n(iii) The electric field is applied only to waveguide 1. These situations can be achieved by\nelectric gating.\nFor the case (i) with ~E1;2= (0;0;Ez),!0=\r\n1+\u000b2(H0\u00002cEEzkx\n\u00160Ms+2Aex\n\u00160Msk2\nx+JRKKY\n2\u00160Mstp)in Eq. 2,\nand the condition for PT-symmetry still holds. Applying an electric field along the zaxis\ncauses an asymmetry in the magnon dispersion. As shown by Fig. 5, the positive Ezshifts\nthe dispersion towards positive kxwhile a negative Ezshifts it in the opposite direction.\nWith increasing !J, the changes of Re[ !] and Im[!] (not shown) are similar to these in Fig.\n1(b-c).\nAs for the case ~E1= (0;0;Ez)and~E2= (0;0;\u0000Ez), in the two equations (2) !0is\ndifferent. Explicitly: !0=\r\n1+\u000b2(H0\u00072cEEzkx\n\u00160Ms+2Aex\n\u00160Msk2\nx+JRKKY\n2\u00160Mstp)where the\u0000sign applies\nfor WG1 and the +sign corresponds to WG2. Hence, under an asymmetric electric field\nthe potential VRis not even ( VR(z0)6=VR(\u0000z0)), and the PT-symmetry condition can not\nbe satisfied. The !Jdependence of Re[ !] and Im[!] are shown in Fig 5, and no exceptional\npoint can be strictly identified in this case.\nFor case (iii), we set ~E1= (0;0;Ez)and~E2= (0;0;0). The PT-symmetry condition is\nnot satisfied. When the electric field is applied only to a single guide, it shifts selectively the\nmagnon dispersion relation in this guide. Therefore, the magnon wave in the lower frequency\nrange propagates solely in the guide with the electric field. As shown in Fig. 5(d), we excite\nthe magnonic wavepacket with a frequency in the WG1 or WG2, the magnonic wave always\npropagates in the waveguide 1 which amounts to a magnon channeled by the electric field,\nwhile the propagation in the other guide is suppressed. This example illustrates yet another\nhandle to steer magnonic waves swiftly and at low energy consumption by pulsed electric\ngating.\n14Conclusions\nMagnonic waveguides based on magnetic junctions that exhibit a transition from a PT-\nsymmetric to a PT-symmetry broken phase may act near the transition (exceptional) point\nas effective sensor for changes in external fields and in the magnetic environments and also\nserve as magnonic amplifier or magnetic switch. The particular behavior of the waveguides\nmagnetic susceptibility is also reflected in the permeability (cf. supplementary materials)\npointing to a new route to PT-symmetric magneto-photonics. Magnonic propagation is\nhighly controllable by external electric and magnetic fields that can derive the system across\nthe exceptional point and lead to controlled power distribution in the waveguides as well as\nnon-reciprocal or amplified magnon waves. DM interaction allows for dispersion engineering\nvia external electric fields, and for PT-symmetry based large-amplitude spin excitations.\nThese observations underline the potential of PT-symmetric magnonics as the basis for\nadditional functionalities of magnetophotonic, spintronics and cavity magnonic devices that\nare highly controllable by external parameters.\nAcknowledgement\nThis research is financially supported by the DFG through SFB 762 and SFB TRR227,\nNational Natural Science Foundation of China (No. 11704415, 11674400, 11374373), and\nthe Natural Science Foundation of Hunan Province of China (No. 2018JJ3629).\nAuthor contributions\nXGW performed all numerical simulations and analytical modeling. JB conceived and su-\npervised the project. XGW and JB wrote the paper . XGW , JB, and GHG discussed,\ninterpret and agreed on the content.\nCompeting interests:\nThe authors declare no competing\n15financial and non-financial interests.\nFull data availability statements: All technical details for producing the figures are\nenclosed in the supplementary materials. Data are available from the authors upon request.\nSupporting Information Available\nThe following files are available free of charge.\n\u000fpt-symmetry-supp.pdf: Numerical simulation details, dynamic magnetic permeability\nin separated waveguides, and the influence of intrinsic damping.\nReferences\n(1) Chumak, A. V.; Vasyuchka, V. I.; Serga, A. A.; Hillebrands, B. Magnon spintronics.\nNat. Phys. 2015,11, 453–461.\n(2) Wang, Q.; Pirro, P.; Verba, R.; Slavin, A.; Hillebrands, B.; Chumak, A. V. Reconfig-\nurable nanoscale spin-wave directional coupler. Sci. Adv. 2018,4, e1701517.\n(3) Kruglyak, V. V.; Demokritov, S. O.; Grundler, D. Magnonics. J. Phys. D: Appl. 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Lett. 2014,113, 037202.\n21" }, { "title": "2105.06472v1.On_Inhibition_of_Rayleigh__Taylor_Instability_by_Horizontal_Magnetic_Field_in_an_Inviscid_MHD_Fluid_with_Velocity_Damping.pdf", "content": "arXiv:2105.06472v1 [math.AP] 13 May 2021On Inhibition of Rayleigh–Taylor Instability by Horizonta l\nMagnetic Field in an Inviscid MHD Fluid with Velocity Dampin g\nFei Jianga, Song Jiangb, Youyi Zhaoa,∗\naCollege of Mathematics and Computer Science, Fuzhou Univer sity, Fuzhou, 350108, China.\nbInstitute of Applied Physics and Computational Mathematic s, Beijing, 100088, China.\nAbstract\nItisstillanopenproblemwhethertheinhibitionphenomenonofRayleig h–Taylor(RT)instability\nby horizontal magnetic field can be mathematically proved in a non-re sistive magnetohydrody-\nnamic (MHD) fluid in a two-dimensional (2D) horizontal slab domain, sin ce it had been roughly\nverified by a 2D linearized motion equations in 2012 [43]. In this paper, we find that this in-\nhibition phenomenon can be rigorously verified in the inhomogeneous, incompressible, inviscid\ncase with velocity damping. More precisely, there exists a critical nu mbermCsuch that if the\nstrength |m|of horizontal magnetic field is bigger than mC, then the small perturbation solution\naround the magnetic RT equilibrium state is exponentially stable in time. Our result is also the\nfirst mathematical one based on the nonlinear motion equations for the proof of inhibition of\nflow instabilities by a horizontal magnetic field in a horizontal slab doma in. In addition, we also\nprovide a nonlinear instability result for the case |m| ∈[0,mC). Our instability result presents\nthat horizontal magnetic field can not inhibit the RT instability, if it’s st rength is to small.\nKeywords: non-resistive MHD fluds; inviscid fluid; damping; Rayleigh–Taylor insta bility;\nexponential stability.\n1. Introduction\nThe equilibrium of the heavier fluid on top of the lighter one and both su bject to the gravity\nis unstable. In this case, the equilibrium state is unstable to sustain s mall disturbances, and this\nunstable disturbance will grow and lead to the release of potential e nergy, as the heavier fluid\nmoves down under the gravity force, and the lighter one is displaced upwards. This phenomenon\nwas first studied by Rayleigh [37] and then Taylor [40], is called therefo re the RT instability. In\nthe last decades, this phenomenon has been extensively investigat ed from mathematical, physical\nand numerical aspects, see [3, 41] for examples. It has been also w idely investigated how other\nphysical factors, such as elasticity [29, 31], rotation [2, 3], interna l surface tension [14, 19, 45],\nmagnetic field [21, 22, 27, 28, 43, 44] and so on, influence the dynam ics of RT instability.\nIn this paper, we are interested in the phenomenon of inhibition of RT instability by magnetic\nfield. This topic goes back to the theoretical work of Kruskal and S chwarzchild [32]. They\nanalyzed the effect of an impressed horizontal magnetic field on the growth of RT instability in\nthe horizontally periodic motion of a completely ionized plasma (with zer o resistance) in three\ndimensions in 1954, and pointed out that the curvature of the magn etic lines can influence the\n∗Corresponding author.\nEmail addresses: jiangfei0951@163.com (Fei Jiang), jiang@iapcm.ac.cn (Song Jiang),\nzhaoyouyi957@163.com (Youyi Zhao)\nPreprint submitted to Elsevier May 17, 2021development of instability, but can not inhibit the growth of RT instab ility. The inhibition\nof RT instability by an impressed vertical magnetic field was first verifi ed for inhomogeneous,\nincompressible, non-resistive magnetohydrodynamic (MHD) fluids in three dimensions by Hide\n[3, 16]. In 2012, Wang noticed that an impressed horizontal magnet ic field can inhibit RT\ninstability in a non-resistive MHD fluid in two dimensions [43]. Later Jiang –Jiang further found\nthat magnetic field always inhibit RT instability, if the condition is satisfie d that the non-slip\nvelocity boundary-value condition is imposed in the direction of impres sed magnetic field. In this\npaper, we call such condition the “fixed condition” for the sake of s implicity. It should be noted\nthat all the results stated previously are based on the linearized no n-resistive MHD equations.\nFor reader’s convenience, we also summarize the known linear result s as follows:\nWhether an impressed horizontal/vertical magnetic field can inhibit R T\ninstability in a slab domain with a non-slip boundary-value condition of ve locity?\nhorizontal vertical\n2DYes Yes\n3D No Yes\nRecently, Jiang–Jiang further established a so-called magnetic inhib ition theory in viscous\nnon-resistive MHD fluids, which reveals the physical effect of fixed c ondition in magnetic inhibi-\ntion phenomena [25]. Roughly specking, let us consider an element line a long an impressed field\nin the rest state of a MHD fluid, then the element line can be regarded as an elastic string. Thus,\nthe bent element line will restore to its initial location under the magne tic tension, the fixed\ncondition, as well as viscosity. By the magnetic inhibition mechanism of a non-resistive MHD\nfluid, the assertions in the table above seem to be obvious. However , rigorous mathematical\nproofs are not easy.\nThanks to the multi-layers method developed in the proof of well-pos ed problem of surface\nwaves [15], recently the inhibition phenomenon of RT instability by a mag netic field had been\nrigorously proved based on the (nonlinear) non-resistive MHD equa tions under a fixed condition,\nfor example, Wang verified the inhibition phenomenon by an impressed non-horizontal magnetic\nfield in the stratified incompressible viscous MHD fluid in a 3D slab domain [4 4]; moreover, he\nalso proved that the impressed horizontal magnetic field can not inh ibit the RT instability for\nthe horizontally periodic motion. Similar results can be also found in oth er magnetic inhibition\nphenomena, see [23] for Parker instability and [26] for thermal inst ability.\nHowever at present it is still an open problem whether the phenomen on of inhibition of RT\ninstability by a horizontal magnetic field can be rigorously proved in a n on-resistive MHD fluid\nin a 2D slab domain. To our knowledge, there are also not any available m athematical proof\nfor the inhibition of other flow instabilities by a horizontal magnetic fie ld in a horizontal slab\ndomain in the both 2D and 3D cases. The purpose of this paper is to mo ve a step in this\ndirection. Fortunately we find that this inhibition phenomenon can be mathematically verified\nin the inhomogeneous, incompressible, inviscid, non-resistive MHD flu id with velocity damping\nin two-dimensions. More precisely, there exists a critical number mCsuch that if the strength\n|m|of a horizontal magnetic field is bigger than mC, then the small perturbation solution around\nthe magnetic RT equilibrium state is exponentially stable in time, i.e., RT ins tability can be\ninhibited by a horizontal magnetic field in a 2D slab domain. Next we math ematically formulate\nour result.\n1.1. Mathematical formulation for the magnetic RT problem\nThe system of motion equations of an inhomogeneous, incompressib le, inviscid, non-resistive\nMHD fluid with velocity damping in the presence of a gravitational field in a two-dimensional\n2domain Ω reads as follows:\n\n\nρt+v·∇ρ= 0,\nρvt+ρv·∇v+∇(P+λ|M|2/2)+aρv=λM·∇M−ρge2,\nMt+v·∇M=M·∇v,\ndivv= divM= 0.(1.1)\nNext we shall explain the mathematical notations in system (1.1).\nThe unknowns ρ:=ρ(x,t),v:=v(x,t),M:=M(x,t) andP:=P(x,t) denote the density,\nvelocity, magnetic field andkinetic pressure ofincompressible MHD flu ids, resp..x∈Ω⊂R2and\nt>0 are spacial variables and time variables, resp.. The constants λ>0,g>0 anda/greaterorequalslant0 stand\nfor the permeability of vacuum, the gravitational constant and th e velocity damping coefficient,\nresp..e2= (0,1)Trepresents the vertical unit vector, and −ρge2the gravitational force, where\nthe superscript T denotes the transposition.\nSince we consider the horizontally periodic motionsolution of (1.1), we define thehorizontally\nperiodic domain\nΩ := 2πT×(0,h), (1.2)\nwhereT=R/Z. For the horizontally periodic domain Ω, the 1D periodic domain 2 πT×{0,h},\ndenoted by ∂Ω, customarily is regarded as the boundary of Ω. For the well-posed ness of the\nsystem (1.1), we shall pose the following initial-boundary value condit ions:\n(ρ,v,M)|t=0= (ρ0,v0,M0), (1.3)\nv|∂Ω·/vectorn= 0, (1.4)\nwhere/vectorndenotes the outward unit normal vector on ∂Ω. Here and in what follows, we always\nuse the superscript 0 to emphasize the initial data.\nNow we choose a RT density profile ¯ ρ:= ¯ρ(x2), which is independent of x1and satisfies\n¯ρ∈C4(Ω),inf\nx∈Ω¯ρ>0, (1.5)\n¯ρ′|x2=y2>0 for somey2∈ {x2|(x1,x2)T∈Ω}, (1.6)\nwhere ¯ρ′:= d¯ρ/dx2andΩ :=R×[0,h]. We remark that the second condition in (1.5) prevents\nus from treating vacuum, while the condition in (1.6) is called RT conditio n, which assures that\nthere is at least a region in which the density is larger with increasing he ightx2and leads to the\nclassical RT instability, see [20, Theorem 1.2].\nWith RT density profile in hand, we further define a magnetic RT equilibr iarM:= (¯ρ,0,¯M),\nwhere¯M= (m,0)Tandmis a constant. We often call ¯Man impressive horizontal magnetic\nfield, while the pressure profile ¯Punder the equilibrium state is determined by the relation\n∇¯P=−¯ρge2in Ω. (1.7)\nDenoting the perturbation around the magnetic RT equilibria by\n̺=ρ−¯ρ, v=v−0, N=M−¯M,\n3and then using the relation (1.7), we obtain the system of perturba tion equations from (1.1):\n\n\n̺t+v·∇(̺+ ¯ρ) = 0,\n(̺+ ¯ρ)vt+(̺+ ¯ρ)v·∇v+∇β+aρv\n=λ(N+¯M)·∇N−̺ge2,\nNt+v·∇N= (N+¯M)·∇v,\ndivv= divN= 0,(1.8)\nwhereβ:=P−¯P+λ(|M|2− |¯M|2)/2, and we call βthe total perturbation pressure. The\ncorresponding initial-boundary value conditions read as follows:\n(̺,v,N)|t=0= (̺0,v0,N0), (1.9)\nv|∂Ω·/vectorn= 0. (1.10)\nWe call the initial-boundary value problem (1.8)–(1.10) magnetic RT pr oblem for the sake of\nsimplicity. Obviously, to mathematically prove the inhibition of RT instab ility by a horizontal\nmagnetic field in a 2D slab domain, it suffices to verity the stability in time o f magnetic RT\nproblem with some non-trivial initial datum .\nWemention thatthewell-posedness problemofinviscid fluids withveloc ity damping hadbeen\nwidely investigated, see [17, 33, 35, 38, 39, 42, 47, 48] for example s. Recently, some authors had\nfurther studied the well-posedness problem of the motion equation s of incompressible inviscid,\nnon-resistive MHD fluids, i.e., taking ρ= 1 andg= 0 in (1.1). For examples, Wu–Wu–Xu first\ngiven the existence of unique global(-in-time) solutions with algebraic decay-in-time for the 2D\nCauchy problem with small initial perturbation [46], and Du–Yang–Zho u obtained the existence\nof unique global solutions with exponential decay-in-time for the init ial-boundary value problem\nin a 2D slab domain with small initial perturbation around some non-triv ial equilibria [6]. It\nshould benoted that the mathematical methods adopted in[6, 46] f or thewell-posedness problem\nnot applied to our stability problem, therefore next we shall reform ulate magnetic RT problem\nin Lagrangian coordinates as in [23, 26, 44].\n1.2. Reformulation in Lagrangian coordinates\nLet the flow map ζbe the solution to the initial-value problem\n/braceleftBigg\n∂tζ(y,t) =v(ζ(y,t),t) in Ω,\nζ(y,0) =ζ0(y) in Ω ,(1.11)\nwhere the invertible mapping ζ0:=ζ0(y) maps Ω to Ω, and satisfies\nJ0:= det∇ζ0= 1 in Ω, (1.12)\nζ0·/vectorn=y·/vectornon∂Ω. (1.13)\nHere and in what follows, “det” denotes a determinant of matrix.\nWe denote the Eulerian coordinates by ( x,t) withx=ζ(y,t) and the Lagrangian coordinates\nby (y,t)∈Ω×R+\n0, whereR+\n0:= [0,∞). We further assume that, for each fixed t>0,\nζ|y2=i:R→Ris aC2(R)-diffeomorphism mapping for i= 0, h, (1.14)\nζ:Ω→Ω is aC2(Ω)-diffeomorphism mapping . (1.15)\n4Sincevsatisfies the divergence-free condition, and non-slip boundary-v alue condition (1.10),\nwe can deduce from (1.11)–(1.13) that\nJ:= det∇ζ= 1 in Ω,\nζ·/vectorn=y·/vectornon∂Ω.\nWe define the matrix A:= (Aij)2×2via\nAT= (∇ζ)−1:= (∂jζi)−1\n2×2.\nThen we further define the differential operators ∇A, divAand curl Aas follows: for a scalar\nfunctionfand a vector function X:= (X1,X2)T,\n∇Af:= (A1k∂kf,A2k∂kf)T,divA(X1,X2)T:=Alk∂kXl,curlAX:=A1k∂kX2−A2k∂kX1,\nwhere we have used the Einstein convention of summation over repe ated indices, and ∂k:=∂yk.\nInparticular, curl X:= curlIX, whereIrepresents anidentity matrix. In addition, we will denote\n(curlAX1,...,curlAXn)TbycurlA(X1,...Xn)Tfor simplicity, where Xi= (Xi\n1,Xi\n2)Tis a vector\nfunction for 1/lessorequalslanti/lessorequalslantn.\nDefining the Lagrangian unknowns\n(ϑ,u,Q,B )(y,t) = (ρ,v,P+λ|M|2/2,M)(ζ(y,t),t) for (y,t)∈Ω×R+\n0,\nthen in Lagrangian coordinates, the initial-boundary value problem o f (1.1), (1.3) and (1.4) is\nrewritten as follows: \n\nζt=u, ϑt= 0,divAu= 0,\nϑut+∇AQ+aϑu=λB·∇AB−ϑge2,\nBt=B·∇Au,divAB= 0,\n(ζ,ϑ,u,B )|t=0= (ζ0,ϑ0,u0,B0),\n(ζ−y,u)|∂Ω·/vectorn= 0,(1.16)\nwhere (ϑ0,v0,B0) = (ρ0(ζ0),v0(ζ0),M0(ζ0)). In addition, the relation (1.7) in Lagrangian coor-\ndinates reads as follows:\n∇A¯P(ζ2) =−¯ρ(ζ2)ge2. (1.17)\nLetη=ζ−y,η0=ζ0−y,q=Q−¯P(ζ2)−λ|¯M|2/2,A= (∇η+I)Tand\nGη:= ¯ρ(η2(y,t)+y2)−¯ρ(y2). (1.18)\nIfζ0,ϑ0andB0satisfy\nB0=m∂1ζ0andϑ0= ¯ρ(y2),\nthen the initial-boundary value problem (1.16), together with the re lation (1.17), implies that\n\n\nηt=u,\n¯ρut+∇Aq+a¯ρu=λm2∂2\n1η+gGηe2,\ndivAu= 0,\n(η,u)|t=0= (η0,u0),\n(η,u)|∂Ω·/vectorn= 0(1.19)\n5and\nϑ= ¯ρ(y2), B=m∂1ζ, (1.20)\nplease refer to [22] for the derivation.\nIt should be noted that (1.19), together with (1.20), also implies (1.1 6). In addition, noting\nthatqis the sum of the perturbation pressure and perturbation magnet ic pressure in Lagrangian\ncoordinates, however we still call qthe perturbation pressure for the sake of simplicity. From\nnow on, we call the initial-boundary value problem (1.19) the transfo rmed MRT problem. Ob-\nviously the stability problem of magnetic RT problem reduces to invest igate the stability of the\ntransformed MRT problem.\n1.3. Notations\nBefore stating our main results on the transformed MRT problem, w e shall introduce simpli-\nfied notations throughout this paper.\n(1) Simplified basic notations: Ia:= (0,a) denotes a time interval, in particular, I∞=R+.S\ndenotes the closure of the set S⊂Rnwithn/greaterorequalslant1, in particular, IT:= [0,T]./integraltext\n:=/integraltext\nΩ=/integraltext\n(0,2π)×(0,h). (u)Ωdenotes the mean value of uin a periodic box. a/lessorsimilarbmeans that a/lessorequalslantcb.\nIf not stated explicitly, the positive constant cmay depend on g,a,λ,m, ¯ρand Ω in the\ntransformed MRT problem, and may vary from one place to other pla ce. Sometimes we\nusecifori/greaterorequalslant1 to replace c, and to emphasize that ciis fixed value. αalways denotes the\nmultiindex with respect to the variable y,|α|=α1+α2is called the order of multiindex,\n∂α:=∂α1\n1∂α2\n2, [∂α,φ]ϕ:=∂α(φϕ)−φ∂αϕand\n[∂αcurl∂j\n1A,φ]χ:=∂αcurl∂j\n1A(φχ)−φ∂αcurl∂j\n1Aχforj= 0,1.\n“∇if∈X” represents that ∂αf∈Xfor any multiindex αsatisfying |α|=i, whereXdenotes\nsome set of functions.\n(2) Simplified Banach spaces, norms and semi-norms:\nLp:=Lp(Ω) =W0,p(Ω), Hi:=Wi,2(Ω), Hi:={w∈Hi|(w)Ω= 0},\nH1,i:={w∈Hi|∂1w∈Hi}, Hj\ns:={w∈Hj|w|∂Ω·/vectorn= 0},\nHj\nσ:={w∈Hj\ns|divw= 0}, H1,j\ns:=Hj\ns∩H1,j, Lp\nTHi:=Lp(IT,Hi),\n/ba∇dbl·/ba∇dbli:=/ba∇dbl·/ba∇dblHi,/ba∇dbl·/ba∇dblk,i:=/ba∇dbl∂k\n1·/ba∇dbli,/ba∇dbl·/ba∇dblk,i:=/radicalBig/summationtext\n0/lessorequalslantl/lessorequalslantk/ba∇dbl·/ba∇dbl2\nl,i,\nwhere 1/lessorequalslantp/lessorequalslant∞, andi,k/greaterorequalslant0,j/greaterorequalslant1.\nIn addition, for simplicity, we denote/radicalBig/summationtext\n1/lessorequalslantk/lessorequalslantj/ba∇dblfk/ba∇dbl2\nXby/ba∇dbl(f1,...,fj)/ba∇dblX, where/ba∇dbl·/ba∇dblXrep-\nresents a norm or a semi-norm, and fkmay be a scalar function, a vector or a matrix for\n1/lessorequalslantk/lessorequalslantj.\n6(3) simplified function classes: for integer j/greaterorequalslant1,\nHj\n1:={w∈Hj|det(∇w+I) = 1}, H1,j\n1,s:=H1,j\ns∩Hj\n1,\nC0\nB,weak(IT,L2) :=L∞\nTL2∩C0(IT,L2\nweak),\nHj\n∗:={ξ∈Hj|ξ(y)+ysatisfies the diffeomorphism\nproperties as ζin (1.14) and (1.15) },\nC0(IT,H1,j\ns) :={η∈C(IT,Hj\ns)| ∇j∂1η∈C0\nB,weak(IT,L2)},\nH1,4\n1,∗,T:={η∈C0(IT,H1,4\ns)|η(t)∈H4\n∗for eacht∈IT},\nU4\nT:={u∈C0(IT,H3\ns)| ∇4u∈C0\nB,weak(IT,L2),\nut∈C0(IT,H2\ns), ut∈L∞\nTH3},\nQ4\nT:={q∈C(IT,H3)∩L∞\nTH4|qt∈L∞\nTH3}.\n(4) Energy integral: for any given w∈H1,\nE(w) :=/integraldisplay\ng¯ρ′w2\n2dy−λ/ba∇dblm∂1w/ba∇dbl2\n0.\n(5) Energy and dissipation functionals:\nE:=/ba∇dbl(η,∂1η,u)/ba∇dbl2\n4+/ba∇dbl(ut,∇q)/ba∇dbl2\n3,Ep:=/ba∇dbl(η,∂1η,u)/ba∇dbl2\n1,3+/ba∇dbl(u,ut,∇q)/ba∇dbl2\n3,\nD:=/ba∇dbl(u,∂1η)/ba∇dbl2\n4+/ba∇dbl(ut,∇q)/ba∇dbl2\n3,Dp:=/ba∇dbl(u,∂1η)/ba∇dbl2\n1,3+/ba∇dbl(u,ut,∇q)/ba∇dbl2\n3.\nWe callE, resp.Dthe total energy, resp. dissipation functionals, and Ep, resp.Dpthe\npartial energy, resp. dissipation functionals. In addition, we use t he following notation for\nsimplicity\nI0:=/ba∇dbl(η0,∂1η0,u0)/ba∇dbl2\n4. (1.21)\n1.4. Main results\nNow we introduce the stability result for the transformed MRT prob lem.\nTheorem 1.1 (Stability) .Assume¯ρsatisfies (1.5)–(1.6)and\n|m|>mC:=/radicalBigg\nsup\nw∈H1σg/integraltext\n¯ρ′w2\n2dy\nλ/ba∇dbl∂1w/ba∇dbl2\n0. (1.22)\nWe further assume a>0,(η0,u0)∈(H1,4\n1,s∩H4\n∗)×H4\nsanddivA0u0= 0, whereA0:= (∇η0+I)−T.\nThen there exist a sufficiently small constant δ>0such that, for any (η0,u0)satisfyingI0/lessorequalslantδ2,\nthe transformed MRT problem (1.19)admits a unique global classical solution (η,u,q)in the\nfunction class H1,4\n1,∗,∞×U4\n∞×Q4\n∞. Moreover, the solution enjoys the estimate (2.14)and the\nfollowing properties:\n(1)the energy inequality in differential form: for a.e. t>0,\nd\ndt˜E+D/lessorequalslant0 (1.23)\nfor some functional ˜E, which belongs to W1,∞(R+)1and is equivalent to Ea.e.t>0.\n1Since˜E ∈W1,∞(R+), there exists a function ˜E∈W1,∞(R+)∩AC(R+\n0) such that ˜E=˜Eand˜E′=˜E′for a.e.\nt∈R+.\n7(2)the stability estimate of total energy: for a.e. t>0,\nE(t)+/integraldisplayt\n0D(τ)dτ/lessorsimilarI0. (1.24)\n(3)the exponential stability estimates: for a.e. t>0,\nec1t(/ba∇dblη2(t)/ba∇dbl2\n3+Ep(t))+/integraldisplayt\n0ec1τDp(τ)dt/lessorsimilarI0, (1.25)\nec1t/ba∇dblη1(t)−η∞\n1/ba∇dbl2/lessorsimilar√\nI0 (1.26)\nfor some positive constant c1, whereη∞\n1∈H2only depends on y2.\nRemark 1.1. If the assumptions of (1.6) and (1.22) are replaced by\n¯ρ′/lessorequalslant0 in Ω and |m|>0,\nthen the conclusions in Theorem 1.1 also hold. In addition, Theorem 1.1 also holds for the case\ng= 0.\nRemark 1.2. By the assumptions of ¯ ρ, it is easy to check that\n00,\ndet(∇η+I) = 1 in Ω ×IT, (1.27)\nsup\nt∈IT/ba∇dbl(η,∂1η,u)(t)/ba∇dbl4/lessorequalslantδ∈(0,1]. (1.28)\n8Forsufficientlysmall δ,wefirstderivethehorizontal-typeenergyinequality, andthenth ecur-type\nenergy inequality, see (2.44) and (2.58). Summing up the both energ y inequalities of horizontal-\ntype and cur-type, we can arrive at the total energy inequality\nd\ndt˜E+D/lessorsimilar√\nED (1.29)\nfor some (total) energy functional ˜E, which is equivalent to Eunderthe stability condition (1.22).\nIn particular, (1.29) further implies (1.23), which yields the prioristability estimate (1.24).\nThanks to the prioriestimate (1.24) and the unique local(-in-time) solvability of the trans formed\nMRT problem (1.19) in Proposition 2.2, we immediately get the unique glob al solvability of the\nproblem (1.19).\nIn addition, similarly to (1.23), we can also establish the partial energ y inequality\nd\ndt˜Ep+Dp/lessorequalslant0, (1.30)\nwhere˜Epis equivalent to Ep. Thanks to the observation that /ba∇dblη/ba∇dbl1,3/lessorsimilar/ba∇dblη/ba∇dbl2,3by the horizontal\nperiodicity, we immediately see that\nEpis equivalent to Dp.\nThus the partial energy inequality (1.30), together with the above equivalence, immediately\nimplies the exponential stability of partial energy (1.25). Finally, (1.2 6) can be easily deduced\nfrom (1.25) by an asymptotic analysis method.\nRecently Pan–Zhou–Zhu investigated the well-posdeness of the eq uations of a viscous MHD\nfluid in a 3D periodic domain [36] under the assumption of that the initial data satisfies some\nodevity condition. Motivated by Pan–Zhou–Zhu’s result, we have th e following exponential\nstability of total energy.\nCorollary 1.1. If additionally the initial data (η0,u0)in Theorem 1.1 satisfies the odevity con-\nditions\n(η0\n1,u0\n1)(y1,y2) =−(η0\n1,u0\n1)(−y1,y2), (1.31)\n(η0\n2,u0\n2)(y1,y2) = (η0\n2,u0\n2)(−y1,y2). (1.32)\nthen the solution (η,u,q)in Theorem 1.1 also satisfies the odevity conditions\n(η1,u1)(y1,y2,t) =−(η1,u1)(−y1,y2,t),(η2,u2)(y1,y2,t) = (η2,u2)(−y1,y2,t),(1.33)\nand enjoys the following exponential stability of total ene rgy: for a.e. t>0,\nec2tE(t)+/integraldisplayt\n0ec2τD(τ)dτ/lessorsimilarI0(1.34)\nfor some positive constant c2.\nThe key idea to prove Corollary 1.1 is that Eis equivalent to Dunder the odevity conditions,\nand thus we immediately get the exponential stability (1.34). The det ailed derivation will be\nprovided in Section 3.\nWe can not expect the stability result for transformed MRT problem under the condition\n|m| ∈(0,mC). In fact, this condition results in RT instability.\n9Theorem 1.2 (Instability) .Let¯ρsatisfy(1.5)–(1.6)anda/greaterorequalslant0. If|m| ∈(0,mC), the equilibria\n(¯ρ,0,¯M)is unstable in the Hadamard sense, that is, there are positiv e constants m0,ǫ,δ0, and\n((˜η0,ηr),(˜u0,ur))∈H4\nssuch that for any δ∈(0,δ0]and the initial data\n(η0,u0) :=δ(˜η0,˜u0)+δ2(ηr,ur)∈(H5\ns∩H5\n1∩H5\n∗)×H5\ns,\nthere is a unique solution (η,u,q)to the transformed MRT problem (1.19), where(η,u,q)∈\nH1,4\n1,∗,τ×U4\nτ×Q4\nτfor anyτ∈ITmax,Tmaxdenotes the maximal time of existence of the solution,\nanddivA0u0= 0withA0:= (∇η0+I)−T. However, for 1/lessorequalslanti,j/lessorequalslant2, andk= 0,1,\n/ba∇dbl∂k\njχi(Tδ)/ba∇dblL1/greaterorequalslantǫ\nfor some escape time Tδ:=1\nΛln2ǫ\nm0δ∈ITmax, whereχcan be taken by ηoru.\nThe proof of Theorem 1.2 is based on a so-called bootstrapinstability method. The bootstrap\ninstability method has its origin in [12, 13]. Later, various versions of t he bootstrap approach\nwere presented by many authors, see [8, 11, 30] for examples. In particular, recently Jiang–Jiang–\nZhan proved the existence of the RT instability solution under L1-norm for the stratified viscous,\nnon-resistive MHD fluids [30] . In this paper, we adapt the version of the bootstrap instability\nmethod in [30] to prove Theorem 1.2. It should be noted that the aut hors in [30] considered\nthe RT instability in viscous fluids. However our problem is the inviscid ca se, thus there exist\nsome details, which are different to the Ref. [30], in the proof of Theo rem 1.2. In particular,\nthe absence of the strong continuity of highest order of spacial d erivatives of ( η,u) results in\nsome troubles. Fortunately, these troubles can be overcome by m aking use of the stability of\nlocal(-in-time) solutions (2.80) and the weak continuity, i.e. ∇4(∂1η,u)∈C0\nB,weak(Iτ,L2) for any\nτ∈ITmax.\nWe mention that Jang–Guo ever proved the RT instability of inviscid flu ids in a 2D periodic\ndomain [18]. It is not clear to authors that whether Jang–Guo’s resu lt’s can be extended to the\nslab domain. In other word, it is not clear that whether Theorem 1.2 also holds for the cas e\nm= 0, such case will be further investigated in future .\nThe rest of this paper is organized as follows. In Sections 2–4, we pr ovide the proof of\nTheorem 1.1, Corollary 1.1 and Theorem 1.2 in sequence. In Section 5, we will establish the\nlocal well-posdedness result for the transformed MRT problem (1.1 9). Finally, in Appendix A,\nwe list some well-known mathematical results, which will be used in Sect ions 2–5.\n2. Proof of Theorem 1.1\nThis section is devoted to the proof of Theorem 1.1. The key step is t o derive the total energy\ninequality (1.23)andthepartialenergyinequality (1.30)forthetra nsformedMRTproblem(1.19)\nbya prioriestimates. Tothisend, let( η,u,q)beasolutionto(1.19)andsatisfy (1.27)and(1.28),\nwhereδis sufficiently small, and the smallness of δdepends on g,a,λ,m, ¯ρand Ω. It should be\nnoted that a,mand ¯ρsatisfy the assumptions in Theorem 1.1. Next we proceed with a priori\nestimates.\n2.1. Preliminary estimates\nTo being with, we shall establish some preliminary estimates involving ( η,u).\nLemma 2.1. For any given t∈IT, we have\n10(1)the estimates for AandAt:\n/ba∇dblA/ba∇dblC0(Ω)+/ba∇dblA/ba∇dbl3/lessorsimilar1, (2.1)\n/ba∇dblAt/ba∇dbli,j/lessorsimilar/ba∇dblu/ba∇dbli,j+1for0/lessorequalslanti+j/lessorequalslant3. (2.2)\n(2)the estimates ˜A:\n/ba∇dbl˜A/ba∇dbli/lessorsimilar/ba∇dblη/ba∇dbli+1for0/lessorequalslanti/lessorequalslant3, (2.3)\n/ba∇dbl˜A/ba∇dbli,j/lessorsimilar/ba∇dblη/ba∇dbli,j+1for1/lessorequalslanti+j/lessorequalslant4andi/greaterorequalslant1. (2.4)\nHere and in what follows ˜A:=A−I.\n(3)the estimate of divu: for0/lessorequalslanti/lessorequalslant3,\n/ba∇dbldivu/ba∇dbli/lessorsimilar/ba∇dbl(η,u)/ba∇dbl4/ba∇dbl(η,u)/ba∇dbl1,i. (2.5)\n(4)the estimates involving gravity term: for sufficiently small ,\n/ba∇dblGη/ba∇dbl3/lessorsimilar/ba∇dblη2/ba∇dbl3, (2.6)\n/ba∇dblG/ba∇dblk,0/lessorsimilar/braceleftBigg\n/ba∇dblη2/ba∇dbl2/ba∇dblη2/ba∇dbl0 fork= 0;\n/ba∇dblη2/ba∇dbl3/ba∇dbl∂1η2/ba∇dblk−1,0for1/lessorequalslantk/lessorequalslant4,(2.7)\nwhereG:=g(Gη−¯ρ′η2).\nProof. (1)–(3) Recalling (1.27) and the definitions of Aand˜A, we can compute out that\nA=/parenleftbigg∂2η2+1−∂1η2\n−∂2η1∂1η1+1,/parenrightbigg\nand thus\n˜A=/parenleftbigg∂2η2−∂1η2\n−∂2η1∂1η1/parenrightbigg\n. (2.8)\nMaking use of (1.19) 1, (1.28), theembedding inequality (A.1) andthe product estimate (A .3),\nwe can easily deduce (2.1)–(2.5) from (1.19) 3and the expressions of Aand˜A.\n(4) We turn to verifying (2.6) and (2.7). By (1.28) and Lemma A.8, ζ:=η+ysatisfies the\ndiffeomorphism properties (1.14) and (1.15) for sufficiently small δ. Thus ¯ρ(l)(y2+η2) for any\ny∈Ω makes sense, and\n¯ρ(l)(y2+η2)−¯ρ(l)(y2) =/integraldisplayη2\n0¯ρ(l+1)(y2+z)dz (2.9)\nfor 0/lessorequalslantl/lessorequalslant3. Moreover, for any given t∈IT,\nsup\ny∈Ω/vextendsingle/vextendsingle/vextendsingle¯ρ(l+1)/vextendsingle/vextendsingle\ny2=y2+η2/vextendsingle/vextendsingle/vextendsingle/lessorsimilar1, (2.10)\nsup\ny∈Ωsup\nz∈Ψ/vextendsingle/vextendsingle¯ρ(l+1)(y2+z)/vextendsingle/vextendsingle/lessorsimilar1, (2.11)\n11where Ψ := {τ|0/lessorequalslantτ/lessorequalslantη2}forη2/greaterorequalslant0 and := ( η2,0] forη2<0. Making use of (1.28),\n(2.9)–(2.11) and the embedding inequality (A.1), we easily deduce (2.6 ) from the definition of\nGη.\nSince\n¯ρ(m)(y2+η2)−¯ρ(m)(y2) = ¯ρ(m+1)(y2)η2+/integraldisplayη2\n0(η2(y,t)−z) ¯ρ(m+2)(y2+z)dz\nfor 0/lessorequalslantm/lessorequalslant2, it is easy to see that\n/ba∇dblG/ba∇dbl0/lessorsimilar/ba∇dblη2/ba∇dbl2/ba∇dblη2/ba∇dbl0\nand\n/vextenddouble/vextenddouble∂k\n1G/vextenddouble/vextenddouble\n0=/vextenddouble/vextenddouble∂k−1\n1/parenleftbig\n¯ρ′(y2+η2)∂1η2/parenrightbig\n−¯ρ′∂k\n1η2/vextenddouble/vextenddouble\n0\n/lessorsimilar/vextenddouble/vextenddouble/parenleftbig\n¯ρ′(y2+η2)−¯ρ′/parenrightbig\n∂k\n1η2/vextenddouble/vextenddouble\n0+/vextenddouble/vextenddouble[∂k−1\n1,¯ρ′(y2+η2)]∂1η2/vextenddouble/vextenddouble\n0\n/lessorsimilar/ba∇dblη2/ba∇dbl3/ba∇dbl∂1η2/ba∇dblk−1,0for 1/lessorequalslantk/lessorequalslant4.\nPutting the above two estimates together yields (2.7). /square\nLemma 2.2. We have\n(1)the estimate of divη: for0/lessorequalslanti/lessorequalslant3,\n/ba∇dbldivη/ba∇dblj,i/lessorsimilar/braceleftBigg\n/ba∇dblη/ba∇dbl3/ba∇dblη/ba∇dbl1,iforj= 0;\n/ba∇dblη/ba∇dbl3/ba∇dblη/ba∇dbl1,i+1forj= 1,(2.12)\n/ba∇dbldivη/ba∇dblj,i/lessorsimilar/ba∇dblη/ba∇dbl3/ba∇dblη/ba∇dbl2,2fori+j= 4andj/greaterorequalslant2. (2.13)\n(2)the estimate of η2:\n/ba∇dblη2/ba∇dblj/lessorsimilar/braceleftBigg\n/ba∇dblη/ba∇dbl1,0forj= 0,1;\n/ba∇dblη/ba∇dbl1,j−1for2/lessorequalslantj/lessorequalslant4.(2.14)\nProof. (1) Recalling (1.27), we can compute out that\ndivη=∂1η2∂2η1−∂1η1∂2η2. (2.15)\nExploitingtheproductestimate(A.3), wecaneasilydeduce(2.12)–( 2.13)fromtherelationabove.\n(2) Noting that\nη2|∂Ω= 0 and∂2η2= divη−∂1η1, (2.16)\nthus, using (2.16) and (A.5), we get\n/ba∇dblη2/ba∇dbl0/lessorsimilar/ba∇dbl(∂1η1,divη)/ba∇dbl0,\n/ba∇dblη2/ba∇dblj/lessorsimilar/ba∇dblη2/ba∇dbl0+/ba∇dbl∇η2/ba∇dblj−1/lessorsimilar/ba∇dbl(∂1η,divη)/ba∇dblj−1for 1/lessorequalslantj/lessorequalslant4.\nWe immediately get (2.14) from the two estimates above and (2.12). /square\n12Lemma 2.3. Let the multiindex αsatisfy|α|/lessorequalslant3and\nWα=∂α(curlAt(¯ρu)−λm2curl∂1A∂1η). (2.17)\n(1)Then we have\n/ba∇dblWα/ba∇dbl0/lessorsimilar/ba∇dbl(∂1η,u)/ba∇dbl2\n4, (2.18)\n/ba∇dbl∂α(curlA(¯ρχ)−curl(¯ρχ))/ba∇dbl0/lessorsimilar/ba∇dblη/ba∇dbl4/ba∇dblχ/ba∇dbl4, (2.19)\nwhereχ=η,∂1ηandu.\n(2)Forα2/\\e}atio\\slash= 2, we have\n/ba∇dblWα/ba∇dbl0/lessorsimilar/ba∇dbl(∂1η,u)/ba∇dbl4/ba∇dbl(∂1η,u)/ba∇dbl1,3, (2.20)\n/ba∇dbl∂α(curlA(¯ρu)−curl(¯ρu))/ba∇dbl0/lessorsimilar/ba∇dbl(η,u)/ba∇dbl4(/ba∇dblη/ba∇dbl2,2+/ba∇dblu/ba∇dbl1,3), (2.21)\n/ba∇dbl∂α(curlA(¯ρ∂i\n1η)−curl(¯ρ∂i\n1η))/ba∇dbl0/lessorsimilar/ba∇dblη/ba∇dbl4/ba∇dblη/ba∇dbl2,3fori= 0,1. (2.22)\nProof. The estimates (2.18) and (2.20) can be easily derived by using (2.2), ( 2.4) and the\nproduct estimate (A.3). Noting that\ncurlAw−curlw=˜A1k∂kw2−˜A2k∂kw1,\nthen (2.19) and (2.21)–(2.22) can be easily derived by using (2.3), (2 .8), (A.3) and (A.7). /square\n2.2. Horizontal-type energy inequalities\nThis section is devoted to establishing the total/partial horizontal- type energy inequalities.\nLet 0/lessorequalslantk/lessorequalslant4, then we apply ∂k\n1to (1.19) to get that\n\n\n∂k\n1ηt=∂k\n1u,\n¯ρ∂k\n1ut+∂k\n1∇Aq+a¯ρ∂k\n1u=λm2∂k+2\n1η+g¯ρ′∂k\n1η2e2+∂k\n1Ge2,\n∂k\n1divAu= 0,\n(∂k\n1η,∂k\n1u)|∂Ω·/vectorn= 0.(2.23)\nThus we can derive from (2.23) the following horizontal spatial estim ates of (η,u).\nLemma 2.4. For0/lessorequalslantk/lessorequalslant4,\nd\ndt/parenleftbigg/integraldisplay\n¯ρ∂k\n1η·∂k\n1udy+a\n2/ba∇dbl√¯ρ∂k\n1η/ba∇dbl2\n0/parenrightbigg\n−E(∂k\n1η)/lessorsimilar/ba∇dbl√¯ρ∂k\n1u/ba∇dbl2\n0+√\nEDp,(2.24)\nd\ndt/parenleftbig\n/ba∇dbl√¯ρu/ba∇dbl2\nk,0−E(∂k\n1η)/parenrightbig\n+c/ba∇dbl√¯ρu/ba∇dbl2\nk,0/lessorsimilar√\nEDp. (2.25)\nProof. Multiplying (2.23) 2by∂k\n1η, resp.∂k\n1uinL2, then, using the integrating by parts and\n(2.23)1, we have that\nd\ndt/parenleftbigg/integraldisplay\n¯ρ∂k\n1η·∂k\n1udy+a\n2/ba∇dbl√¯ρ∂k\n1η/ba∇dbl2\n0/parenrightbigg\n−E(∂k\n1η)\n=/ba∇dbl√¯ρ∂k\n1u/ba∇dbl2\n0+/integraldisplay\n∂k\n1G∂k\n1η2dy−/integraldisplay\n∂k\n1∇Aq·∂k\n1ηdy, (2.26)\n13resp.\n1\n2d\ndt/parenleftbig\n/ba∇dbl√¯ρu/ba∇dbl2\nk,0−E(∂k\n1η)/parenrightbig\n+a/ba∇dbl√¯ρu/ba∇dbl2\nk,0\n=/integraldisplay\n∂k\n1G∂k\n1u2dy−/integraldisplay\n∂k\n1∇Aq·∂k\n1udy. (2.27)\nBy (2.7), (2.14) and (A.7), we can estimate that\n/integraldisplay\n∂k\n1G∂k\n1η2dy/lessorequalslant/ba∇dblG/ba∇dblk,0/ba∇dblη2/ba∇dblk,0/lessorsimilar√\nEDp, (2.28)\n/integraldisplay\n∂k\n1G∂k\n1u2dy/lessorequalslant/ba∇dblG/ba∇dblk,0/ba∇dblu2/ba∇dblk,0/lessorsimilar√\nEDp. (2.29)\nNext we estimate for the integrals involving the pressure in (2.26)–( 2.27) by two cases.\n(1) We first consider the case k= 0.\nBy the integration by parts and the boundary-value condition (2.23 )4withk= 0,\n−/integraldisplay\n∇Aq·ηdy=/integraldisplay\nqdivηdy−/integraldisplay\n∇˜Aq·ηdy.\nNext we estimate for the two integrals on the right hand of the abov e identity.\nLetK:=η1(−∂2η2,∂1η2)T. Then the identity (2.15) can be rewritten as follows\ndivη= divK. (2.30)\nBy (2.14), the embedding inequality (A.1) and (A.7), we have\n/ba∇dblK/ba∇dbl0/lessorsimilar/ba∇dblη/ba∇dbl2/ba∇dblη2/ba∇dbl1/lessorsimilar/ba∇dblη/ba∇dbl2/ba∇dblη/ba∇dbl2,1, (2.31)\nExploiting (2.30) and (2.31), we have\n/integraldisplay\nqdivηdy=−/integraldisplay\nK·∇qdy/lessorequalslant/ba∇dblK/ba∇dbl0/ba∇dbl∇q/ba∇dbl0/lessorsimilar√\nEDp. (2.32)\nNoting that\n∇˜Aq·η=η1(∂2η2∂1q−∂1η2∂2q)+η2(∂1η1∂2q−∂2η1∂1q),\nthus, using (2.14), (A.3) and (A.7), we deduce that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n∇˜Aq·ηdy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar/ba∇dblη/ba∇dbl2/ba∇dblη/ba∇dbl1,1/ba∇dbl∇q/ba∇dbl0/lessorsimilar√\nEDp. (2.33)\nCombining with (2.32) and (2.33), we get\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n∇Aq·ηdy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilar√\nEDp. (2.34)\nPutting (2.28) with k= 0 and (2.34) into (2.26) yields (2.24) with k= 0.\nIn addition, by the integration by parts, (2.23) 3and (2.23) 4withk= 0, we have\n−/integraldisplay\n∇Aq·udy=/integraldisplay\nqdivAudy= 0. (2.35)\n14Thus putting (2.29) with k= 0 and (2.35) into (2.27) with k= 0 yields (2.25) with k= 0.\n(2) Now we further consider the case k/\\e}atio\\slash= 0.\nMaking use of (2.12), (2.13) and (A.7), we can estimate that\n−/integraldisplay\n∂k\n1∇Aq·∂k\n1ηdy=/integraldisplay\n∂k−1\n1∇˜Aq·∂k+1\n1ηdy+/integraldisplay\n∂k\n1qdiv∂k\n1ηdy\n/lessorequalslant/ba∇dbl∂k−1\n1∇˜Aq/ba∇dbl0/ba∇dbl∂k+1\n1η/ba∇dbl0+/ba∇dbl∂k\n1q/ba∇dbl0/ba∇dbl∂k\n1divη/ba∇dbl0\n/lessorsimilar√\nEDp. (2.36)\nPutting (2.28) and (2.36) into (2.26) yields (2.24) for k/\\e}atio\\slash= 0.\nExploiting (2.23) 3, we have\n−/integraldisplay\n∂k\n1∇Aq·∂k\n1udy=/integraldisplay\n∂k\n1qdivA∂k\n1udy−/integraldisplay\n[∂k\n1,A]∇q·∂k\n1udy\n=/integraldisplay\n∂k\n1q∂k\n1divAudy−/integraldisplay\n∂k\n1q[∂k\n1,AT] :∇udy−/integraldisplay\n[∂k\n1,A]∇q·∂k\n1udy\n=−/integraldisplay\n∂k\n1q[∂k\n1,AT] :∇udy−/integraldisplay\n[∂k\n1,A]∇q·∂k\n1udy/lessorsimilar√\nEDp. (2.37)\nConsequently, putting (2.29) and (2.37) into (2.27) yields (2.25) with k/\\e}atio\\slash= 0. The proof is\ncompleted. /square\nNow we shall establish stabilizing estimates for E(∂k\n1η) appearing in (2.24) and (2.25).\nLemma 2.5. It holds that\n/ba∇dblη/ba∇dbl2\ni+1,0/lessorsimilar−E(∂i\n1η)+/ba∇dblη/ba∇dbl4/ba∇dblη/ba∇dbl2\n2,3for any0/lessorequalslanti/lessorequalslant4. (2.38)\nProof. By the definition of mC, it is easy to see that\n−/integraldisplay\ng¯ρ′w2\n2dy/greaterorequalslant−λm2\nC/ba∇dbl∂1w/ba∇dbl2\n0for anyw∈H1\nσ,\nwhich, together with the stability condition |m|>mC, implies that\n/ba∇dblw/ba∇dbl2\n1,0/lessorsimilarλ(m2−m2\nC)/ba∇dbl∂1w/ba∇dbl2\n0/lessorsimilar−E(w) for anyw∈H1\nσ. (2.39)\nLet us consider the Stokes problem\n/braceleftBigg\n−∆˜η+∇̟= 0,div˜η= divηin Ω,\n˜η= 0 on ∂Ω.(2.40)\nBy the existence theory of Stokes problem, there exist a unique so lution (˜η,̟)∈H4×H3to\n(2.40). Moreover, ∂j\n1(˜η,̟) is also the solution of Stokes problem for 1 /lessorequalslantj/lessorequalslant3 and\n/ba∇dbl˜η/ba∇dblj,2/lessorsimilar/ba∇dbldivη/ba∇dblj,1/lessorsimilar/ba∇dblη/ba∇dbl4/ba∇dblη/ba∇dbl2,3, (2.41)\nwhere we have used (2.12), (2.13) and (A.7) in the last inequality abov e.\n15Now we use ∂i\n1(η−˜η) to rewrite E(∂i\n1η) as follows:\nE(∂i\n1η) =E(∂i\n1(η−˜η))+E(∂i\n1˜η)−Ii, (2.42)\nwhere\nIi:= 2λm2/integraldisplay\n∂i+1\n1η·∂i+1\n1˜ηdy−2g/integraldisplay\n¯ρ′∂i\n1η2∂i\n1˜η2dy.\nNote that∂i\n1(η−˜η)∈H1\nσ, thus, we use (2.39) to get\n/ba∇dblη−˜η/ba∇dbl2\ni+1,0/lessorsimilar−E(∂i\n1(η−˜η)),\nwhich, together with (2.42) and Young’s inequality, yields\n/ba∇dblη/ba∇dbl2\ni+1,0/lessorsimilarE(∂i\n1˜η)−E(∂i\n1η)−Ii+/ba∇dbl˜η/ba∇dbl2\ni+1,0. (2.43)\nMaking use of (2.14), (2.41) and (A.7), we can estimate that\nE(∂i\n1˜η)−Ii+/ba∇dbl˜η/ba∇dbl2\ni+1,0/lessorsimilar/ba∇dblη/ba∇dbl4/ba∇dblη/ba∇dbl2\n2,3for 0/lessorequalslanti/lessorequalslant4.\nFinally, putting the above estimate into (2.43) yields (2.38). This comp letes the proof. /square\nThanks to Lemmas 2.4–2.5 and (2.14), we easily get the total/partial horizontal-type energy\ninequalities.\nLemma 2.6. There exist two functionals EandEof(η,u)such that\nd\ndtE+c/ba∇dbl(u,∂1η)/ba∇dbl2\n4,0/lessorsimilar√\nEDp, (2.44)\nd\ndtE+c(/ba∇dblu/ba∇dbl2\n0+/ba∇dbl∂1(u,∂1η)/ba∇dbl2\n3,0)/lessorsimilar√\nEDp, (2.45)\n/ba∇dbl(η,∂1η,u)/ba∇dbl2\n4,0−c/ba∇dblη/ba∇dbl3/ba∇dblη/ba∇dbl2\n2,3/lessorsimilarE/lessorsimilar/ba∇dbl(η,∂1η,u)/ba∇dbl2\n4,0, (2.46)\n/ba∇dblu/ba∇dbl2\n0+/ba∇dbl∂1(η,∂1η,u)/ba∇dbl2\n3,0−c/ba∇dblη/ba∇dbl4/ba∇dblη/ba∇dbl2\n2,3/lessorsimilarE/lessorsimilar/ba∇dblu/ba∇dbl2\n0+/ba∇dbl∂1(η,∂1η,u)/ba∇dbl2\n3,0.(2.47)\n2.3. Curl-type energy inequality\nThis section is devoted to establishing curl-type energy inequalities. By (1.17), we have\ncurlA(gGηe2) = curl A(−g¯ρe2) =A1j∂j(−g¯ρ) =g¯ρ′∂1η2.\nThus applying curl Ato (1.19) 2yields that\n∂tcurlA(¯ρu)+acurlA(¯ρu) =λm2curlA/parenleftbig\n¯ρ−1∂2\n1(¯ρη)/parenrightbig\n+g¯ρ′∂1η2+curl At(¯ρu).(2.48)\nLet the multiindex αsatisfy|α|/lessorequalslant2. Recalling the definition of Wαin (2.17), thus applying\n∂αto (2.48) yields\n∂t∂αcurlA(¯ρu)+a∂αcurlA(¯ρu)\n=∂1(λm2(¯ρ−1∂αcurlA(¯ρ∂1η)+[∂αcurlA,¯ρ−1](¯ρ∂1η))+∂α(g¯ρ′η2))+Wα. (2.49)\nNow we derive the following energy estimates for curl uand curl∂1η.\n16Lemma 2.7. For multiindex αsatisfying |α|/lessorequalslant3, we have\nd\ndt/integraldisplay/parenleftbig\n2∂αcurlA(¯ρη)∂αcurlA(¯ρu)+a|∂αcurlA(¯ρη)|2/parenrightbig\ndy+c/ba∇dbl∂αcurlA(¯ρ∂1η)/ba∇dbl2\n0\n/lessorsimilar/ba∇dbl∂αcurlA(¯ρu)/ba∇dbl2\n0+/ba∇dblλm2[∂αcurlA,¯ρ−1](¯ρ∂1η)+∂α(g¯ρ′η2)/ba∇dbl2\n0\n+/braceleftBigg√\nED;√\nEDpforα1/greaterorequalslant1,(2.50)\nd\ndt/parenleftBig\n/ba∇dbl∂αcurlA(¯ρu)/ba∇dbl2\n0+λ/ba∇dblm/radicalbig\n¯ρ−1∂αcurlA(¯ρ∂1η)/ba∇dbl2\n0/parenrightBig\n+c/ba∇dbl∂αcurlA(¯ρu)/ba∇dbl2\n0\n/lessorsimilar/ba∇dblλm2[∂αcurlA,¯ρ−1]/parenleftbig\n¯ρ∂2\n1η/parenrightbig\n+∂α(g¯ρ′∂1η2)/ba∇dbl2\n0+/braceleftBigg√\nED;√\nEDpfor|α|/lessorequalslant1orα1/greaterorequalslant1.(2.51)\nProof. (1) Multiplying (2.49) by ∂αcurlA(¯ρη) inL2yields that\nd\ndt/integraldisplay/parenleftBig\n∂αcurlA(¯ρη)∂αcurlA(¯ρu)+a\n2|∂αcurlA(¯ρη)|2/parenrightBig\ndy+λm2/ba∇dbl√¯ρ−1∂αcurlA(¯ρ∂1η)/ba∇dbl2\n0\n=Iα\n1+Iα\n2+/ba∇dbl∂αcurlA(¯ρu)/ba∇dbl2\n0\n−/integraldisplay/parenleftbig\nλm2[∂αcurlA,¯ρ−1](¯ρ∂1η)+∂α(g¯ρ′η2)/parenrightbig\n∂αcurlA(¯ρ∂1η)dy, (2.52)\nwhere\nIα\n1:=/integraldisplay/parenleftbig\n∂αcurlAt(¯ρη)∂αcurlA(¯ρu)+Wα∂αcurlA(¯ρη)/parenrightbig\ndy\n−/integraldisplay/parenleftbig\n∂α(g¯ρ′η2)+λm2∂αcurlA∂1η/parenrightbig\n∂αcurl∂1A(¯ρη)dy,\nIα\n2:=a/integraldisplay\n∂αcurlAt(¯ρη)∂αcurlA(¯ρη)dy.\nMaking use of (2.1), (2.2), (2.14), (2.18), (2.20), (A.3) and (A.7), w e can estimate that\nIα\n1/lessorsimilar/braceleftBigg√\nED;√\nEDpforα1/greaterorequalslant1(2.53)\nand\nIα\n2/lessorsimilar√\nEDpforα1/greaterorequalslant1. (2.54)\nIn addition, noting the structure\ncurlAt(¯ρη) =∂t˜A1j∂j(¯ρη2)−∂t˜A2j∂j(¯ρη1)\nthus, similarly to (2.53) with further using integral formula by parts , (1.19) 1and (2.8), we easily\nestimate that\nIα\n2/lessorsimilar√\nED. (2.55)\nFinally, putting (2.53)–(2.55) into (2.52), and then using Young’s ineq uality and the lower-\nbound condition infy∈Ω¯ρ>0, we get (2.50) immediately.\n17(2) Multiplying (2.49) by ∂αcurlA(¯ρu) inL2yields that\n1\n2d\ndt(/ba∇dbl∂αcurlA(¯ρu)/ba∇dbl2\n0+λ/ba∇dblm/radicalbig\n¯ρ−1∂αcurlA(¯ρ∂1η)/ba∇dbl2\n0)+a/ba∇dbl∂αcurlA(¯ρu)/ba∇dbl2\n0\n=/integraldisplay/parenleftbig\nλm2/bracketleftbig\n∂αcurlA,¯ρ−1]/parenleftbig\n¯ρ∂2\n1η/parenrightbig\n+∂α(g¯ρ′∂1η2)/parenrightbig\n∂αcurlA(¯ρu)dy+Iα\n3, (2.56)\nwhere we have defined that\nIα\n3:=λm2/integraldisplay\n¯ρ−1∂αcurlA(¯ρ∂1η)∂α(curlAt(¯ρ∂1η)−curl∂1A(¯ρu))dy\n+/integraldisplay\n(λm2[∂αcurl∂1A,¯ρ−1](¯ρ∂1η)+Wα)∂αcurlA(¯ρu)dy.\nSimilarly to (2.53), it is easy to estimate that\nIα\n3/lessorsimilar/braceleftBigg√\nED;√\nEDpfor|α|/lessorequalslant1 orα1/greaterorequalslant1.(2.57)\nPlugging (2.57) into (2.56) and then using Young’s inequalities, we obta in (2.51). This completes\nthe proof. /square\nNow we use Lemmas 2.7 to further derive the total/partial vorticity -type energy inequalities.\nProposition 2.1. There exist two functionals Ecul\npandEculof(η,u)such that\nd\ndtEcul+c/ba∇dbl(u,∂1η)/ba∇dbl2\n4/lessorsimilar/ba∇dbl(u,∂3\n1u,∂5\n1η)/ba∇dbl2\n0+√\nED, (2.58)\nd\ndtEcul\np+c(/ba∇dbl(u,∂1η)/ba∇dbl2\n1,3+/ba∇dblu/ba∇dbl2\n3)/lessorsimilar/ba∇dbl(u,∂3\n1u,∂5\n1η)/ba∇dbl2\n0+√\nEDp, (2.59)\n/ba∇dblcurl(¯ρη,¯ρ∂1η,¯ρu)/ba∇dbl2\n3−c/ba∇dblη/ba∇dbl4/ba∇dbl(η,∂1η,u)/ba∇dbl2\n4/lessorsimilarEcul/lessorsimilar/ba∇dbl(η,∂1η,u)/ba∇dbl2\n4, (2.60)\n/ba∇dblcurl(¯ρu)/ba∇dbl2\n2+/ba∇dblcurl(¯ρu,¯ρ∂1η)/ba∇dbl2\n1,2−c/ba∇dbl(η,u)/ba∇dbl4/ba∇dbl(u,∂1η)/ba∇dbl2\n1,3\n/lessorsimilarEcul\np/lessorsimilar/ba∇dbl(u,∂1(η,∂1η,u))/ba∇dbl2\n3. (2.61)\nProof. Let 1/lessorequalslanti/lessorequalslant3 andf=∂1ηor∂2\n1η. Then, for any multiindex βsatisfying |β|= 3−i, it\nis easy to check\n/ba∇dbl[∂i\n1∂βcurlA,¯ρ−1](¯ρf)/ba∇dbl0/lessorsimilar/ba∇dblf/ba∇dbli,2−i+/braceleftBigg\n0 for i= 1;\n/ba∇dblη/ba∇dbl3,2/ba∇dblf/ba∇dbl1fori= 2,3.\nIn addition, for any multiindex αsatisfying |α|/lessorequalslant3, it obviously holds that\n/ba∇dbl[∂αcurlA,¯ρ−1](¯ρf)/ba∇dbl0/lessorsimilar/ba∇dblf/ba∇dbl|α|.\nExploiting (A.7) and the two estimates above, we can derive from (2.5 0) forα1/greaterorequalslant1 and (2.51)\nwith the both cases of α1/greaterorequalslant1 and|α|/lessorequalslant2 that\nd\ndtEcul\np,i+c/ba∇dblcurlA(¯ρu,¯ρ∂1η)/ba∇dbl2\ni,3−i/lessorsimilar/ba∇dblη/ba∇dbl2\ni+2,3−i+√\nEDp, (2.62)\nd\ndtEcul\np,4+c/ba∇dblcurlA(¯ρu)/ba∇dbl2\n2/lessorsimilar/ba∇dblη/ba∇dbl2\n2,2+√\nEDp (2.63)\n18forsomefunctionals Ecul\np,iandEcul\np,4,whichareequivalentto /ba∇dblcurlA(¯ρη,¯ρ∂1η,¯ρu)/ba∇dbl2\ni,2−iand/ba∇dblcurlA(¯ρu,\n¯ρ∂1η)/ba∇dbl2\n2, resp..\nSimilarly, we can easily derive from (2.14), (2.50) and (2.51) that\nd\ndt˜Ecul+c/ba∇dblcurlA(¯ρu,¯ρ∂1η)/ba∇dbl2\n3/lessorsimilar/ba∇dblη/ba∇dbl2\n2,3+√\nED, (2.64)\nfor some functional ˜Ecul, which is equivalent to /ba∇dblcurlA(¯ρη,¯ρ∂1η,¯ρu)/ba∇dbl2\n3.\nMaking use of (2.5), (2.12), (2.13) and Hodge-type elliptic estimate ( A.9), we can deduce\nthat, for 0 /lessorequalslantj/lessorequalslant3,\n/ba∇dbl(u,∂1η)/ba∇dblj,4−j/lessorsimilar/ba∇dbl(u,∂1η)/ba∇dblj,0+/ba∇dbl(divu,curlu,div∂1η,curl∂1η)/ba∇dblj,3−j\n/lessorsimilar/ba∇dbl(u,∂1η,curl(¯ρu),curl(¯ρ∂1η))/ba∇dblj,2−j\n+/ba∇dbl(η,u)/ba∇dbl4/ba∇dbl(η,u)/ba∇dbl1,3+/braceleftBigg\n/ba∇dblη/ba∇dbl3/ba∇dblη/ba∇dbl1,2forj= 0,\n/ba∇dblη/ba∇dbl3/ba∇dblη/ba∇dbl2,2for 1/lessorequalslantj/lessorequalslant3(2.65)\nand\n/ba∇dblu/ba∇dbl2\n3/lessorsimilar/ba∇dbl(u,curl(¯ρu))/ba∇dbl2+/ba∇dbl(η,u)/ba∇dbl4/ba∇dbl(η,u)/ba∇dbl1,2. (2.66)\nMaking use of (2.21), (2.22), (2.65), (2.66), the interpolation inequ ality (A.2) and (A.7), we\nfurther derive from (2.62)–(2.64) that\nd\ndtEcul\np,i+c/ba∇dbl(u,∂1η)/ba∇dbl2\ni,4−i/lessorsimilar/ba∇dbl(u,∂2\n1η)/ba∇dbl2\ni,3−i+√\nEDpfor 1/lessorequalslanti/lessorequalslant3,\nd\ndtEcul\np,4+c/ba∇dblu/ba∇dbl2\n3/lessorsimilar/ba∇dblu/ba∇dbl2\n0+/ba∇dblη/ba∇dbl2\n2,2+√\nEDp\nd\ndt˜Ecul+c/ba∇dbl(u,∂1η)/ba∇dbl2\n4/lessorsimilar/ba∇dbl(u,∂2\n1η)/ba∇dbl2\n3+√\nED.\nConsequently, we immediately get (2.58) and (2.59) from the three e stimates above by using\n(A.2) and (A.7), where Ecul:=˜Ecul+cEcul\npandEcul\np=Ecul\np,1+c/summationtext4\nj=2Ecul\np,jfor some constants c;\nmoreover, exploiting (2.19), (2.21), (2.22) and (A.3), we easily see t hat (2.60) and (2.61) hold.\nThis completes the proof. /square\n2.4. Equivalent estimates\nThis section is devoted to establishing the following equivalent estimat es.\nLemma 2.8. For sufficiently small δ, we have\nEis equivalent to /ba∇dbl(η,∂1η,u)/ba∇dbl2\n4, (2.67)\nDis equivalent to /ba∇dbl(u,∂1η)/ba∇dbl2\n4, (2.68)\nEp,Dpand/ba∇dbl(u,∂1(u,∂1η))/ba∇dbl2\n3are equivalent , (2.69)\nwhere the equivalent coefficients in (2.67)–(2.69)are independent of δ.\nProof. To obtain (2.67)–(2.69), the key step is to establish the estimates o f∇qandut. By\n(1.19)2, we have/braceleftBigg\n−div(∇q/¯ρ) =f1in Ω,\n∇q/¯ρ·/vectorn=f2·/vectornon∂Ω,\n19where\nf1:=div/parenleftbig\nut+au−λm2¯ρ−1∂2\n1η+ ¯ρ−1(∇˜Aq−gGηe2)/parenrightbig\n,\nf2:=−∇˜Aq/¯ρ.\nNote that\n/integraldisplay\nf1dy+/integraldisplay\n∂Ωf2·/vectorndy1= 0,\nthus applying the elliptic estimate (A.22) yields\n/ba∇dbl∇q/ba∇dbl3/lessorsimilar/ba∇dblf1/ba∇dbl2+/ba∇dblf2/ba∇dbl3\n/lessorsimilar/ba∇dbl(divAtu,div˜Aut,div˜Au)/ba∇dbl2+/ba∇dbl(∂2\n1η,∇˜Aq,Gη)/ba∇dbl3.\nMaking use of (2.2), (2.3), (2.6) and the product estimate (A.3), we further derive that, for\nsufficiently small δ,\n/ba∇dbl∇q/ba∇dbl3/lessorsimilar/ba∇dbl(η2,∂2\n1η)/ba∇dbl3+/ba∇dbl(η,u)/ba∇dbl4/ba∇dbl(u,ut)/ba∇dbl3. (2.70)\nSimilarly, dividing (1.19) 2by ¯ρ, and then applying /ba∇dbl·/ba∇dbl3to the resulting identity, we get\n/ba∇dblut/ba∇dbl3=/ba∇dbl(λm2∂2\n1η+gGηe2−∇Aq−a¯ρu)/¯ρ/ba∇dbl3\n/lessorsimilar/ba∇dbl(η2,∂2\n1η,u,∇q)/ba∇dbl3,\nwhich, together with (2.70), implies, for sufficiently small δ,\n/ba∇dbl(ut,∇q)/ba∇dbl3/lessorsimilar/ba∇dbl(η2,∂2\n1η,u)/ba∇dbl3. (2.71)\nThanks to (2.14), (2.71) and (A.7), we easily see that (2.67)–(2.69) hold. /square\n2.5.A priori stability estimate\nNow we are in a position to establish the a priori stability estimates under the a priori\nassumptions (1.27) and (1.28).\nSimilarly to (2.65), we have, for sufficiently small δ,\n/ba∇dblη/ba∇dbl4/lessorsimilar/ba∇dbl(η,curl(¯ρη))/ba∇dbl3. (2.72)\nMaking use of (2.46), (2.60), (2.65) with i= 0, (2.68), (2.72) and the interpolation inequality\n(A.2), we can derive from (2.44) and (2.58) that, for sufficiently sma llδ,\nd\ndt˜E+cD/lessorsimilar√\nED, (2.73)\nwhere˜E:=Ecul+cEfor some constant cand\n˜Eis equivalent to /ba∇dbl(η,∂1η,u)/ba∇dbl2\n4. (2.74)\nExploiting (1.28) and (2.67), we further deduce from (2.73) that th ere exists a positive constant\nδ1, such that, for any δ∈(0,δ1],\nd\ndt˜E+cD/lessorequalslant0. (2.75)\n20In particular, by (2.74), we easily get from (2.75) that, for some c3/greaterorequalslant1,\nE(t)+/integraldisplayt\n0D(τ)dτ/lessorequalslantc3/ba∇dbl(η0,∂1η,u0)/ba∇dbl2\n4. (2.76)\nSimilarly to (2.75), making use of (2.47), (2.61), (2.65), (2.66), (2.69 ), the interpolation\ninequality (A.2) and (A.7), we derive from (2.45) and (2.59) that, for sufficiently small δ,\nd\ndt˜Ep+cDp/lessorequalslant√\nEDp, (2.77)\nwhere˜Ep:=Ecul\np+cEfor some constant cand\n˜Ep,Ep,Dpand/ba∇dbl(u,∂1(η,∂1η,u))/ba∇dbl2\n3are equivalent . (2.78)\nThus, exploiting (2.67) and (2.78), we further deduce from (2.77) t hat\nd\ndt˜Ep+2c1˜Ep/lessorequalslant0,\nwhich, together with (2.14) and (2.78), further implies that\nec1t(/ba∇dblη2(t)/ba∇dbl2\n4+Ep(t))+/integraldisplayt\n0ec1τDp(τ)dτ/lessorsimilar/ba∇dbl(u0,∂1(η0,∂1η0,u0))/ba∇dbl2\n1,3. (2.79)\nThis completes the derivation of the a prioristability estimates (1.24) and (1.25).\n2.6. Proof of Theorem 1.1\nNow we state the local well-posedness result for the transformed MRT problem (1.19).\nProposition 2.2. Letb>0,a/greaterorequalslant0be constants and ι>0be the constant in Lemma A.8. We\nassume that ¯ρsatisfy(1.5),(η0,u0)∈(H1,4\ns∩H4\n∗)×H4\ns,/ba∇dbl(u0,∂1η0)/ba∇dbl4/lessorequalslantbanddivA0u0= 0,\nwhereA0:= (∇ζ0)−Tandζ0=η0+y. Then there exist a sufficiently small constant δ2/lessorequalslantι/2,\nsuch that if η0satisfies\n/ba∇dblη0/ba∇dbl4/lessorequalslantδ2,\nthe transformed MRT problem (1.19)admits a unique local-in-time classical solution (η,u,q)∈\nC0(IT1,H1,3\ns)×U3\nT×Q3\nTfor some local existence time T >0. Moreover, ηsatisfies\nsupt∈IT/ba∇dblη/ba∇dbl4/lessorequalslant2δ22\nand\nsupt∈IT/ba∇dbl(η,∂1η,u)/ba∇dbl4/lessorequalslantc4√\nI0 (2.80)\nfor some positive constant c4/greaterorequalslant1. It should be noted that δ2andTmay depend on g,a,λ,m,¯ρ\nandΩ; moreover Tfurther depends on b.\n2Sincesupt∈IT/ba∇dblη/ba∇dbl4/lessorequalslantι, we have, by Lemma A.8,\ninf(y,t)∈R2×ITdet(∇η+I)/greaterorequalslant1/4.\n21Proof. The proof of Proposition 2.2 will be provided in Section 5. /square\nThanks to the prioriestimate (2.76) and Proposition 2.2, we can easily establish Theorem\n1.1. Next we briefly give the proof.\nLet (η0,u0) satisfies the assumptions in Theorem 1.1, and\nI0/lessorequalslantδ2,whereδ:= min{δ1,δ2}/√c3c2\n4,\nwherec3andc4are the constants in (2.76) and (2.80). By Proposition 2.2 and Lemma A.8, there\nexists a unique local solution ( η,u,q) to the transformed MRT problem (1.19) with a maximal\nexistence time Tmax, which satisfies that\n•for anyτ∈ITmax, the solution ( η,u,q) belongs to H1,4\n1,∗,τ×U4\nτ×Q4\nτand\nsupt∈Iτ/ba∇dblη/ba∇dbl4/lessorequalslant2δ2;\n•limsupt→Tmax/ba∇dblη(t)/ba∇dbl4>δ2or limsupt→Tmax/ba∇dbl(u,∂1η)(t)/ba∇dbl4=∞, ifTmax<∞.\nLet\nT∗:= sup/braceleftbig\nτ∈ITmax/vextendsingle/vextendsingle/ba∇dbl(η,∂1η,u)(t)/ba∇dbl4/lessorequalslant√c3c2\n4δfor anyt/lessorequalslantτ/bracerightbig\n.\nWe easily see that the definition of T∗>0 makes sense and T∗>0. Thus, to obtain the\nexistence of global solutions, next it suffices to verify T∗=∞. Next we shall prove this fact by\ncontradiction.\nWe assume that T∗<∞. Then, for any given T∗∗∈IT∗,\nsupIT∗∗/ba∇dbl(η,∂1η,u)(t)/ba∇dbl4/lessorequalslant√c3c2\n4δ/lessorequalslantδ1. (2.81)\nThanks to (2.81), we can follow the argument of (2.73) by further u sing a regularity method as\nin the derivation of (5.71) to verify that\nd\ndt˜E+cD/lessorequalslant0 for a.e.t∈IT∗,where˜E ∈W1,∞(IT∗). (2.82)\nReferring to (2.76), we can further derive from (2.80) and (2.82) t hat\n/ba∇dblE(t)/ba∇dblL∞(IT∗)+/integraldisplayt\n0D(τ)dτ/lessorequalslantc3lim\nt→0supτ∈It/ba∇dbl(η,∂1η,u)(τ)/ba∇dbl2\n4/lessorequalslantc3c2\n4δ2.(2.83)\nBy the continuity of ( η,u) and the fact\nsupt∈Iτ/ba∇dblf/ba∇dbl0=/ba∇dblf/ba∇dblL∞τL2for anyf∈C0\nB,weak(Iτ,L2) withτ >0,\nwe further derive from (2.83) that\nsupIT∗/ba∇dbl(η,∂1η,u)(t)/ba∇dbl4/lessorequalslant√c3c4δ. (2.84)\nWe take (η(T∗∗),u(T∗∗)) as a initial data. Noting that, by (2.84) and the definition of δ,\n/ba∇dbl(η,∂1η,u)(T∗∗)/ba∇dbl4/lessorequalslantB:=√c3c4δand/ba∇dblη(T∗∗)/ba∇dbl4/lessorequalslantδ2,\n22then, by Proposition 2.2, there exists a unique local-in-time classical solution, denoted by ( η∗,u∗,\nq∗), to the transformed MRT problem (1.19) with ( η(T∗∗),u(T∗∗)) in place of ( η0,u0); moreover\nsupt∈[T∗∗,T]/ba∇dbl(η∗,∂1η∗,u∗)/ba∇dbl4/lessorequalslantc4B/lessorequalslant√c3c2\n4δand supt∈[T∗∗,T]/ba∇dblη∗/ba∇dbl4/lessorequalslant2δ2,\nwhere the local existence time T >0 depends on b,g,a,λ,m, ¯ρand Ω.\nIn view of the existence result of ( η∗,u∗,q∗), the uniqueness conclusion in Proposition 2.2 and\nthe fact that Tmaxdenotes the maximal existence time, we immediately see that Tmax>T∗+T/2\nand supt∈[0,T∗+T/2]/ba∇dbl(η,∂1η,u)/ba∇dbl4/lessorequalslant√c3c2\n4δ. This contradicts with the definition of T∗. Hence\nT∗=∞and thusTmax=∞. This completes the proof of the existence of global solutions.\nThe uniqueness of global solution is obvious due to the uniqueness re sult of local solutions in\nProposition 2.2 and the fact supt/greaterorequalslant0/ba∇dblη/ba∇dbl4/lessorequalslant2δ2. To complete the proof of Theorem 1.1, we shall\nverify that the solution ( η,u,q) satisfies the properties (1.23)–(1.26).\nReferring to (2.82) and (2.83), we see that the global solution ( η,u) enjoys (1.23) and (1.24).\nSimilarly, we also verify that the global solution also satisfies (1.25) by referring to the derivation\nof (2.79) and (2.83).\nFinally, we shall derive (1.26). By (1.25), we easily see that\n∂1η(t)→0 inH2ast→ ∞ (2.85)\nand\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0udτ/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n3/lessorsimilar/integraldisplayt\n0/ba∇dblu/ba∇dbl3dτ/lessorsimilar/radicalbig\nI0for anyt>0 (2.86)\nThanks to (2.86), there exist a subsequence {tn}∞\nn=1and some function η∞\n1∈H3such that\n/integraldisplaytn\n0u1dτ→η∞\n1−η0\n1weakly inH3.\nExploiting (1.19)1, (1.25) and weakly lower semi-continuity, we have\n/ba∇dblη1(t)−η∞\n1/ba∇dbl3/lessorequalslantliminf\ntn→∞/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplaytn\ntu1dτ/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n3/lessorsimilar√\nI0liminf\ntn→∞/integraldisplaytn\nte−c1τdτ/lessorsimilar√\nI0e−c1t,\nwhich, together with (2.85), yields that (1.26) holds and η∞\n1only depends on y2. This completes\nthe proof of Theorem 1.1.\n3. Proof of Corollary 1.1\nThis section is devoted to the proof of Corollary 1.1. Let ( η,u,q) be the classical solution\nconstructed by Theorem 1.1 with initial data ( η0,u0) further satisfying the odevity conditions\n(1.31) and (1.32). Next we first verify the solution preserves the o devity conditions in (1.33).\nLetψ= (−η1,η2)(−y1,y2,t),w= (−u1,u2)(−y1,y2,t) andp= (q1,q2)(−y1,y2,t). Since the\nclassical solution of the transformed MRT problem (1.19) is unique, t hus, to get the relation\n(1.33), it obviously suffices to verify that ( ψ,w,p) is also the classical solution of (1.19). It is\nobvious that ( ψ,w) satisfies (1.19) 1. Next we shall verify that ( ψ,w) also satisfies (1.19) 2and\n(1.19)3.\nDefining\nA=/parenleftbigg∂2η2+1−∂1η2\n−∂2η1∂1η1+1/parenrightbigg\nandB=/parenleftbigg∂2ψ2+1−∂1ψ2\n−∂2ψ1∂1ψ1+1/parenrightbigg\n,\n23then\n(B11,B22) = (A11,A22)|y1=−y1and (B12,B21) =−(A12,A21)|y1=−y1. (3.1)\nBy (3.1),\n∇Bp= (−A1i,A2i)T∂iq|y1=−y1 (3.2)\nand\ndivBw= divAu|y1=−y1= 0. (3.3)\nBy (3.3), we see that ( ψ,w) satisfies (1.19) 3as (η,u).\nThanks to (3.2), we have\n¯ρ∂twi+∇Bp+a¯ρwi=/braceleftBigg\n−(¯ρ∂tu1+A1i∂iq+ ¯ρu1)|y1=−y1fori= 1;\n(¯ρ∂tu2+A2i∂iq+a¯ρu2)|y1=−y1fori= 2,\n∂2\n1(ψ1,ψ2) =∂2\n1(−η1,η2)|y1=−y1andgGψe2=gGηe2/vextendsingle/vextendsingle\ny1=−y1,\nwhereGψis defined in (1.18) with φ2in place ofη2. Hence we see that ( ψ,w) also satisfies (1.19) 2\nby the three identity above. This completes the proof of the verific ation of preserving odevity of\nsolutions.\nThanks to (1.33) and (A.8), we easily get, for 0 /lessorequalslanti/lessorequalslant3,\n/ba∇dbl∂i\n2η1/ba∇dbl0/lessorsimilar/ba∇dbl∂1∂i\n2η1/ba∇dbl0,\nwhich, together the estimate (2.14) satisfied by η, implies that\nEis equivalent to D.\nThus we can further derive (1.34) from (1.23) and (1.24). This comp letes the proof of Corollary\n1.1.\n4. Proof of Theorem 1.2\nThis section is devoted to the proof of instability of transformed MR T problem in Theorem\n1.2. We will complete the proof by five subsections. In what follows, t he fixed positive constant\ncI\nifori/greaterorequalslant1 may depend on g,a,λ,m, ¯ρand Ω.\n4.1. Linear instability\nTo begin with, we exploit modified variational method of ODE as in [14, 29 ] to prove the\nexistence of unstable solutions of the following linearized MRT problem under the instability\ncondition |m| ∈(0,mC):\n\n\nηt=u,\n¯ρut+∇q+a¯ρu=λm2∂2\n1η+g¯ρ′η2e2,\ndivu= 0,\n(η,u)|∂Ω·/vectorn= 0.(4.1)\n24Proposition 4.1. Leta/greaterorequalslant0and¯ρsatisfy(1.5)and(1.6). If|m| ∈[0,mC), then the zero solution\nis unstable to the linearized MRT problem. That is, there is a n unstable solution (η,u,q) :=\neΥt(w/Υ,w,β)to the above problem, where\n(w,β)∈H5\nσ×H4\nsolves the boundary-value problem\n\n\nΥ2¯ρw+Υ∇β+aΥ¯ρw=m2∂2\n1w+g¯ρ′w2e2inΩ,\ndivw= 0 inΩ,\nw·/vectorn= 0 on∂Ω.(4.2)\nwith some growth rate Υ∈(2Λ/3,Λ], whereΛsatisfies\nE(v)/lessorequalslant(Λ2+aΛ)/ba∇dbl√¯ρv/ba∇dbl2\n0for anyv∈H1\nσ. (4.3)\nIn addition,\n/ba∇dblwi/ba∇dbl0/ba∇dbl∂1wi/ba∇dbl0/ba∇dbl∂2wi/ba∇dbl0/\\e}atio\\slash= 0fori= 1,2. (4.4)\nProof. Similarly to [14, 29], next we divide the proof of Proposition 4.1 into five s teps.\n(1) Let\n˜E(ψ,ξ) =/integraldisplayh\n0/parenleftBig\ng¯ρ′|ψ|2−λm2/parenleftBig\nξ2|ψ|2+|ψ′|2/parenrightBig/parenrightBig\ndy2\nand\nF:={ξ∈Z/{0} |˜E(ψ,ξ)>0 for someψ∈H1\n0(0,h)}.\nNext we prove the first assertion that the set of instability f requencies Fis not empty.\nRecalling the definition of mCand the condition |m| ∈[0,mC), we see that there exists a\nfunctionω:= (ω1,ω2)T∈H1\nσ, such that\n/integraldisplay\ng¯ρ′ω2\n2dy−λm2/ba∇dbl∂1ω/ba∇dbl2\n0>0. (4.5)\nLet ˆω(ξ,y2) be the Fourier coefficient of ω(y1,y2) for fixedy2, i.e.,\nˆω(ξ,y2) =/integraldisplay2π\n0ω(y1,y2)e−iξy1dy1.\nWe define the functions ϕandψby the following relations\nˆω1(ξ,y2) =iϕ(ξ,y2) and ˆω2(ξ,y2) =−ψ(ξ,y2),\nwhere (ˆω1,ˆω2)T= ˆωandξ∈Z. Obviously,\n/hatwidest∂1ω1=ξϕand/hatwidest∂1ω2=−iξψ. (4.6)\nSinceω2|∂Ω= 0, thenψ∈H1\n0(0,h). By divω= 0, we have\nξϕ+ψ′= 0,\n25which, together with the fact ψ|y2=0, h= 0, implies that\nψ(0,y2) = 0 forξ= 0. (4.7)\nExploiting (4.6), (4.7), and Fubini and Parseval Theorems, we have\nE(ω) =1\n2π/integraldisplayh\n0/summationdisplay\nξ∈Z/{0}/parenleftBig\ng¯ρ′|ψ|2−λm2/parenleftBig\n|ξψ|2+|ψ′|2/parenrightBig/parenrightBig\ndy2.\nThe above identity together with (4.5) imply that there is a ξ∈Z/{0}andψ∈H1\n0(0,h), such\nthat\n˜E(ψ,ξ)>0.\nHence the set of instability frequencies Fis not empty.\n(2) We define that\nJ(ψ,ξ) :=/integraldisplayh\n0¯ρ/parenleftBig\n|ξψ|2+|ψ′|2/parenrightBig\ndy2,H:=/braceleftbig\nψ∈H1\n0(0,h)|J(ψ,ξ) = 1/bracerightbig\n,\nΞ(ψ,ξ,s) :=ξ2˜E(ψ,ξ)−asJ(ψ,ξ).\nFor any given ξ∈F, we define that\nCξ:= sup{s∈R|Ξ(ψ,ξ,s)>0 for someψ∈H1\n0(0,h)}.\nRecalling the definition of F, we easily see that\n00.\nNext we prove the second assertion that, for any given (ξ,s)∈F×[0,Cξ), there exist α>0\nand a classical solution ψ0∈H1\n0(0,h)∩H4(0,h), which satisfies the following modified boundary-\nvalue problem:\n/braceleftBigg\n(α+as)(¯ρξ2ψ0−(¯ρψ′\n0)′) =gξ2¯ρ′ψ0−λm2ξ2(ξ2ψ0−ψ′′\n0)in(0,h),\nψ0(0) =ψ0(h) = 0.(4.8)\nMoreover, supψ∈HΞ(ψ,ξ,s) = Ξ(ψ0,ξ,s).\nTo being with, we shall consider the following variational problem:\nα:= sup\nψ∈HΞ(ψ,ξ,s), (4.9)\nwhereξ∈Fands∈R+\n0. Sinceαdepends on s, sometimes we denote αbyα(s). Obviously,\nα(s2)−α(s1) =a(s1−s2) for anys1, s2/greaterorequalslant0, (4.10)\ng/ba∇dbl¯ρ′/¯ρ/ba∇dblL∞/greaterorequalslantα>0 for any 0 /lessorequalslants0. (4.12)\nSince 0/lessorequalslantα <∞, there exists a maximizing sequence {ψn}∞\nn=1⊂ Hsuch thatψn→ψ0\nweakly inH1\n0(0,h) and strongly in L2(0,h). By the convergence results of ψnand the weakly\nlower semi-continuity, we easily get\nJ(ψ0,ξ)/lessorequalslantlim\nn→∞/ba∇dbl√¯ρψn/ba∇dbl2\nL2(0,h)+liminf\nn→∞/ba∇dblψ′\nn/ba∇dbl2\nL2(0,h)/lessorequalslantliminf\nn→∞J(ψn,ξ) = 1 (4.13)\n26and\nα= lim\nn→∞Ξ(ψn,ξ,s)/lessorequalslantglim\nn→∞/integraldisplayh\n0¯ρ′|ψn|2dy2−λm2lim\nn→∞/integraldisplayh\n0ξ2|ψn|2dy2\n−λm2liminf\nn→∞/integraldisplayh\n0|ψ′\nn|2y2−as/lessorequalslantΞ(ψ0,ξ). (4.14)\nFrom now on, we consider the case s∈[0,Cξ).Now we prove ψ0/\\e}atio\\slash= 0 by contradiction. We\nassume that ψ0= 0, then there exists a subsequence {ψn}∞\nn=1(still denoted by ψnfor simplicity),\nsuch that /ba∇dblψ′\nn/ba∇dblL2(0,h)→b>0 and/ba∇dbl√¯ρψ′\nn/ba∇dblL2(0,h)→1. Thus we immediately see that\nΞ(ψn,ξ,s)→α=−(λbm2ξ2+as)/lessorequalslant0 fors∈[0,Cξ) anda/greaterorequalslant0,\nwhich contradicts with (4.11). Thus we get ψ0/\\e}atio\\slash= 0.\nSinceψ0/\\e}atio\\slash= 0, we see from (4.13) that\n00and\nψ∈H1\n0(0,h)∩H4(0,h), such that\n/braceleftBigg\nγ2(¯ρξ2ψ−(¯ρψ′)′) =gξ2¯ρ′ψ−λm2ξ2(ξ2ψ−ψ′′)−aγ(¯ρξ2ψ−(¯ρψ′)′)in(0,h),\nψ(0) =ψ(h) = 0.(4.17)\nMoreover,\nsup\nχ∈HΞ(χ,ξ,γ) = Ξ(ψ,ξ,γ) =γ >0. (4.18)\nThe above assertion obviously holds for a= 0 by the second assertion, thus it suffices to consider\nthe casea>0.\nThanks to (4.10)–(4.12), it is easy to see that there exists a fixed p ointγsatisfying (4.18).\nMoreover, by the second assertion, there exists ψ∈H1\n0(0,h)∩H5(0,h) satisfying (4.17) with γ.\nThis completes the proof of the third assertion.\n(4)Now we are in the position to the proof of existence of a soluti on(w,β)to the boundary-\nvalue problem (4.2)with some Υ>0\nLetψandγbe constructed in the third assertion for any given ξ∈F. From now on, we\ndenote (ψ,γ) by (ψξ,γξ) to emphasize the dependence of ξ. We define Λ := supξ∈Fγξ. It is easy\nto see that\n−ξ∈F,(ψξ,γξ) = (ψ−ξ,γ−ξ) for anyξ∈Fand 0<Λ<∞.\nDenoting that\nϕξ:=−ξ−1ψ′\nξandϑξ:= (γ2¯ρϕξ+aγ¯ρϕξ+λm2ξ2ϕξ)/γξξ,\nthen it is easy to check that\n\n\nγ2\nξ¯ρϕξ−γξξϑξ+aγξ¯ρϕξ=−λm2ξ2ϕξ in (0,h),\nγ2\nξ¯ρψξ+γξϑ′\nξ+aγξ¯ρψξ=−λm2ξ2ψξ+g¯ρ′ψξin (0,h),\nξϕξ+ψ′\nξ= 0 in (0 ,h),\nψξ(0) =ψξ(h) = 0.(4.19)\nLet\nw1(y) =−iϕξ(y2)(eiy1ξ−e−iy1ξ) and (w2,θ)(y) = (ψξ(y2),ϑξ(y2))(eiy1ξ+e−iy1ξ),\nthenw1,w2andθare real-value functions. Recalling (4.18) and (4.19) 4, we easily see that ψξ,\nϕξ,ψ′\nξandϕ′\nξare non-zero function. Thus we have /ba∇dblwi/ba∇dbl0/ba∇dbl∂1wi/ba∇dbl0/ba∇dbl∂2wi/ba∇dbl0/\\e}atio\\slash= 0 fori= 1 and 2.\nLetw= (w1,w2)Tandβ:=θ−(θ)Ω, then (w,β)∈H5\nσ×H4. By (4.19), we easily see that\n(w,β) satisfies (4.2) with γξin place of Υ. By the definition of Λ, there exists an ξ0∈Fsuch\nthatγξ0∈(2Λ/3,Λ]. Now we take ξ=ξ0and Υ =γξ0. Consequently, we see that the solution\n(w,β,Υ) satisfies (4.4) and the boundary-value problem (4.2) with Υ ∈(2Λ/3,Λ]. In addition,\nit is easy to see that ( η,u,q) :=eΥt(w/Υ,w,β) is a solution of (4.1).\n(5)To complete the proof of Proposition 4.1, we shall verifies (4.3).\nLetξ∈Z, and the two real-valued functions ˜ ϕ,˜ψ∈H1\n0(0,h) satisfy\nξ˜ϕ+˜ψ′= 0. (4.20)\n28In particular, we have\n˜ψ=˜ψ′= 0 ifξ= 0. (4.21)\nThanks to (4.18) and the fact γ/lessorequalslantΛ, we see that\n˜E(˜ψ,ξ)/lessorequalslantξ−2/parenleftbig\nΛ2+aΛ/parenrightbig\nJ(˜ψ,ξ),ifξ∈F. (4.22)\nIn addition, using the definition of Fand (4.21), we have\n˜E(˜ψ,ξ)/lessorequalslant0 ifξ∈Z/F. (4.23)\nExploiting (4.22) and (4.23), we obtainthat, forany ξ∈Z, andfor any ˜ ϕ,˜ψ∈H1\n0(0,h)satisfying\n(4.20),\n˜E(˜ψ,ξ)/lessorequalslant(Λ2+aΛ)/integraldisplayh\n0¯ρ/parenleftBig\n|˜ϕ|2+|˜ψ|2/parenrightBig\ndy2. (4.24)\nLetv∈H1\nσ, ˆv(ξ,y2) be the Fourier coefficient of v(y1,y2) for fixedy2,ϕ(ξ,y2) = iˆv1(ξ,y2)\nandψ(ξ,y2) = ˆv2(ξ,y2). Then (ϕ,ψ) satisfiesξϕ+ψ′= 0. Consequently, making use of (4.24),\nand the Fubini’s and Parseval’s theorems, we have\nE(v) =1\n2π/summationdisplay\nξ∈Z˜E(ψ,ξ)/lessorequalslant1\n2π(Λ2+aΛ)/summationdisplay\nξ∈Z/integraldisplayh\n0¯ρ/parenleftbig\n|ϕ|2+|ψ|2/parenrightbig\ndy2\n/lessorequalslant(Λ2+aΛ)/ba∇dbl√¯ρv/ba∇dbl2\n0,\nwhich yields (4.3). This completes the proof. /square\n4.2. Nonlinear energy estimates\nThis section is devoted to establishing the following Gronwall-type ene rgy inequality for the\nsolutions of the transformed MRT problem.\nProposition 4.2. LetΥ>0be provided by Proposition 4.1 and (η,u,q)be the local solution\nconstructed by Proposition 2.2. There exist δI\n1andcI\n1>0such that, if /ba∇dbl(η,∂1η,u)/ba∇dbl4/lessorequalslantδI\n1in\nsome time interval I˜T⊂IT, whereITis the existence time interval of (η,u,q), then there exists\na functional E(t)of(η,u,q)satisfies the Gronwall-type energy inequality\nE(t)/lessorequalslantcI\n1/parenleftbigg\nI0+/integraldisplayt\n0/ba∇dblη2(τ)/ba∇dbl2\n0dτ/parenrightbigg\n+Υ/integraldisplayt\n0E(τ)dτ (4.25)\nfor a.e.t∈I˜T, where the constants δI\n1may depend on g,a,λ,m,¯ρandΩ, andE(t)∈W1,∞(I˜T)\nsatisfies, for a.e. t∈I˜T,\nE,Eand/ba∇dbl(η,∂1η,u)/ba∇dbl2\n4are equivalent . (4.26)\nProof. Let (η,u,q) be the local solution constructed by Proposition 2.2. We further a ssume\nthat\nE(t)/lessorequalslantδ∈(0,1] for anyt∈I˜T⊂IT. (4.27)\n29Similarly to Lemma 2.4, for sufficiently small δ, we can easily derive that, for a.e. t∈IT\nd\ndt/parenleftbigg/integraldisplay\n¯ρ∂k\n1η·∂k\n1udy+a\n2/ba∇dbl√¯ρη/ba∇dbl2\nk,0/parenrightbigg\n+c/ba∇dblη/ba∇dbl2\nk+1,0/lessorsimilar/ba∇dbl(η2,u)/ba∇dbl2\nk,0+√\nED,(4.28)\nd\ndt/ba∇dbl(√¯ρu,√\nλm∂1η)/ba∇dbl2\nk,0+c/ba∇dblu/ba∇dbl2\nk,0/lessorsimilar/ba∇dblη2/ba∇dbl2\nk,0+√\nED, (4.29)\nwhere 0/lessorequalslantk/lessorequalslant4.\nWe also verify that ( η,u) satisfies (2.58) for a.e. t∈I˜T. Thus we derive from (2.58) satisfied\nby (η,u), (4.28) and (4.29) that, for some sufficiently small ς∈(0,1],\nd\ndtE+cς/ba∇dbl(u,∂1η)/ba∇dbl2\n3/lessorsimilar(1+ς−1)/ba∇dblη2/ba∇dbl2\n3,0+(1+ς+ς−1)√\nED, (4.30)\nwhere\nE:=ςEcul+ς−1/ba∇dbl(√¯ρu,√\nλm∂1η)/ba∇dbl2\n3,0+a\n2/ba∇dbl√¯ρη/ba∇dbl2\n3,0+3/summationdisplay\nk=0/integraldisplay\n¯ρ∂k\n1η·∂k\n1udy∈W1,∞(I˜T).\nSimilarly to (2.67), (2.68) and (2.74), for sufficiently small δ, we have, for a.e. t∈I˜T,\nE,Eand/ba∇dbl(η,∂1η,u)/ba∇dbl2\n4are equivalent , (4.31)\nDis equivalent to /ba∇dbl(u,∂1η)/ba∇dbl2\n4, (4.32)\nwhere the equivalent coefficients in (4.31) are independent of δ.\nExploiting the interpolation inequality (A.2) yields, for any ε∈(0,1],\n/ba∇dblη2/ba∇dblk,0/lessorsimilar/braceleftBigg\nε−1/ba∇dblη2/ba∇dbl0+ε/ba∇dblη2/ba∇dbl2 fork= 1;\nε−(k−1)/(4−k)/ba∇dblη2/ba∇dbl1,0+ε/ba∇dbl∂1η2/ba∇dbl3for 2/lessorequalslantk/lessorequalslant4.(4.33)\nConsequently, making use of (4.31)–(4.33), we easily derive (4.25) f rom (2.80) and (4.30) for\nsufficiently small δ. /square\n4.3. Construction of nonlinear solutions\nFor any given δ>0, let\n(ηa,ua,qa) =δeΥt(˜η0,˜u0,˜q0), (4.34)\nwhere (˜η0,˜u0,˜q0) := (w/Υ,w,β) and (w,β,Υ) is provided by Proposition 4.1. Then ( ηa,ua,qa)\nis also a solution to the linearized MRT problem (4.1), and enjoys the es timate, for any i/greaterorequalslant0,\n/ba∇dbl∂i\nt(ηa,ua)/ba∇dbl5+/ba∇dbl∂i\ntqa/ba∇dbl4= ΥiδeΥt(/ba∇dbl(˜η0,˜u0)/ba∇dbl5+/ba∇dbl˜q0/ba∇dbl4)/lessorsimilarΥiδeΥt. (4.35)\nIn addition, we have by (4.4) that\n/ba∇dblχj/ba∇dbl0/ba∇dbl∂1χj/ba∇dbl0/ba∇dbl∂2χj/ba∇dbl0>0, (4.36)\nwhereχ= ˜η0or ˜u0, andj= 1, 2.\nSince the initial data of linear solution ( ηa,ua,qa) does not satisfy the necessary compatibility\nconditions in the transformed MRT problem in general. Therefore, w e shall modify the initial\ndata of the linear solution.\n30Proposition 4.3. Let(˜η0,˜u0) := (w/Υ,w)be provided by (4.34), then there exists a constant\nδI\n2∈(0,1], such that for any δ∈(0,δI\n2], there exists (ηr,ur)enjoying the following properties:\n(1) The modified initial data\n(ηδ\n0,uδ\n0) :=δ(˜η0,˜u0)+δ2(ηr,ur)\nbelongs to (H5\n1∩H5\ns)×H5\nsand satisfies the compatibility condition\ndivAδ\n0uδ\n0= 0inΩ,\nwhereAδ\n0is defined as Awithηδ\n0in place of η.\n(2) Uniform estimate:\n/ba∇dbl(ηr,ur)/ba∇dbl5/lessorequalslantcI\n2, (4.37)\nwhere the positive constant cI\n2is independent of δ.\nProof. Please refer to [25, Lemma 4.2] or [30, Proposition 5.1]. /square\nNow we define that\ncI\n3:=/ba∇dbl(˜η0,∂1˜η0,˜u0)/ba∇dbl4+cI\n2>0, (4.38)\nδ0:= min/braceleftbiggδ2\n2cI\n3c4,δI\n2,δI\n1\n2cI\n3c2\n4/bracerightbigg\n/lessorequalslant1, (4.39)\nwherec4/greaterorequalslant1 is the constant in (2.80).\nLetδ/lessorequalslantδ0. Sinceδ/lessorequalslantδI\n2, we can use Proposition 4.3 to construct ( ηδ\n0,uδ\n0), which satisfies\n/ba∇dbl(ηδ\n0,∂1ηδ\n0,uδ\n0)/ba∇dbl4/lessorequalslantcI\n3δ/lessorequalslantδ2.\nBy Proposition 2.2, there exists a (nonlinear) unique solution ( η,u,q) of the transformed MRT\nproblem (1.19) with initial value ( ηδ\n0,uδ\n0) in place of ( η0,u0), where (η,u,q)∈H1,4\n1,∗,τ×U4\nτ×Q4\nτ\nfor anyτ∈ITmaxandTmaxdenotes the maximal time of existence.\nLetǫ0∈(0,1] be a constant, which will be given in (4.65). We define\nTδ:= Υ−1ln(ǫ0/δ)>0,i.e.,δeΥTδ=ǫ0, (4.40)\nT∗:= sup/braceleftbig\nt∈ITmax/vextendsingle/vextendsinglesupτ∈[0,t)/ba∇dbl(η,∂1η,u)(τ)/ba∇dbl4/lessorequalslant2cI\n3c4δ0/bracerightbig\n, (4.41)\nT∗∗:= sup/braceleftbig\nt∈ITmax/vextendsingle/vextendsinglesupτ∈[0,t)/ba∇dblη(τ)/ba∇dbl0/lessorequalslant2cI\n3δeΥτ/bracerightbig\n. (4.42)\nSince (η,u) satisfies (2.80) with ( ηδ\n0,uδ\n0) in place of ( η0,u0) for someT∈ITmax,\nc4/ba∇dbl(η,∂1η,u)(t)/ba∇dbl2\n4/vextendsingle/vextendsingle\nt=0=c4/ba∇dbl(ηδ\n0,∂1ηδ\n0,uδ\n0)/ba∇dbl4/lessorequalslantcI\n3c4δ <2cI\n3c4δ, (4.43)\nand\n/ba∇dblη(t)/ba∇dbl3|t=0=/ba∇dblηδ\n0/ba∇dbl3/lessorequalslantcI\n3δ <2cI\n3δ,\nthusT∗>0,T∗∗>0 and\n/ba∇dblη(T∗∗)/ba∇dbl0= 2cI\n3δeΥT∗∗,ifT∗∗0 by (4.36).\nNow we define that\nǫ0= min/braceleftBigg/parenleftbiggcI\n3\n2cI\n5/parenrightbigg2\n,cI\n3c4δ0\ncI\n4,m2\n0\n4|cI\n5|2/bracerightBigg\n>0. (4.65)\nWe claim that\nTδ=Tmin= min/braceleftbig\nTδ,T∗,T∗∗/bracerightbig\n/\\e}atio\\slash=T∗orT∗∗, (4.66)\nwhich can be showed by contradiction as follows:\n35(1) IfTmin=T∗∗, thenT∗∗< T∗/lessorequalslantTmaxorT∗∗=T∗<+∞. Noting that√ǫ0/lessorequalslantcI\n3/2cI\n5, then\nby (4.34), (4.38), (4.40) and (4.48) that\n/ba∇dblη(T∗∗)/ba∇dbl0/lessorequalslant/ba∇dblηa(T∗∗)/ba∇dbl0+/ba∇dblηd(T∗∗)/ba∇dbl0\n/lessorequalslantδeΥT∗∗(cI\n3+cI\n5√\nδeΥT∗∗)/lessorequalslantδeΥT∗∗(cI\n3+cI\n5√ǫ0)\n/lessorequalslant3cI\n3δeΥT∗∗/2<2cI\n3δeΥT∗∗,\nwhich contradicts to (4.44).\n(2) IfTmin=T∗, thenT∗0. The notations P(x1,...,xn) and˙P(x1,...,xn) represent the generic\npolynomials with respect to the parameters x1,...,xn, where all the coefficients in Pand˙P\nare equal one, and ˙Pfurther satisfies ˙P(0,...,0) = 0. It should be noted that P(x1,...,xn),\n˙P(x1,...,xn) andcκmay vary from line to line. a/lessorsimilarκbmeans that a/lessorequalslantcκb, wherecκdenotes a\ngeneric positive constant, which may depend on κ,a,g,λ,m, ¯ρand Ω.\nWe always use the notations B, resp.Jto represent ( ∇ς+I)−T, resp. det( ∇ς+I), whereς\nat least satisfies\nς∈C0(IT,H4) and inf (y,t)∈ΩTdet(∇ς+I)/greaterorequalslant1/4 for someT >0. (5.4)\nIn addition, we define that\nA4,1/4\nT,ι:={ψ∈C0(IT,H4\ns)| /ba∇dblψ/ba∇dbl3/lessorequalslantι} (5.5)\nand\nST:={(ξ,w,β)∈C(IT,H4\ns)×C0(IT,H4\ns)×(C0(IT,H3)∩L∞\nTH4)|\n∂2\n1ξ∈L2\nT1H4,∇4∂1ξ∈C0\nB,weak(IT,L2),∇Bβ∈L2\nTH4,\n(∇Bβ/¯ρ)|∂Ω·/vectorn=−ge2·/vectorn}, (5.6)\nfor some constant T >0,ιis the positive constant provided in Lemma A.8, ¯ ρsatisfies (1.5)\nandgis the gravity constant. It should be noted that the function ς, which belongs to A4,1/4\nT,ι,\nautomatically satisfies (5.4)by Lemma A.8.\nFinally, some preliminary estimates for BandJare collected as follows.\n37Lemma 5.1. Letςsatisfy(5.4)withT >0. Then\n/ba∇dblB−I/ba∇dbl3/lessorsimilar˙P(/ba∇dblς/ba∇dbl4), (5.7)\n/ba∇dblB/ba∇dblC0(Ω)+/ba∇dblJ/ba∇dblC0(Ω)+/ba∇dbl(J,J−1)/ba∇dbl3/lessorsimilarP(/ba∇dblς/ba∇dbl4)for anyt∈IT. (5.8)\n(1) If additionally ςfurther satisfies (∂1ς,ςt)∈L∞\nTH4, then\n/ba∇dbl∂t(B,J)/ba∇dbl3/lessorsimilar˙P(/ba∇dbl(ς,ςt)/ba∇dbl4), (5.9)\n/ba∇dbl∂1B/ba∇dbl3/lessorsimilar˙P(/ba∇dblς/ba∇dbl1,4)for a.e.t∈IT. (5.10)\n(2) If additionally ςfurther satisfies ςt∈L∞\nTH3andςtt∈L∞\nTH2, then\n/ba∇dbl∂tt(B,J)/ba∇dbl1/lessorsimilar˙P(/ba∇dblς/ba∇dbl4,/ba∇dblςt/ba∇dbl3,/ba∇dblςtt/ba∇dbl2)for a.e.t∈IT. (5.11)\nProof. Since the derivations of the estimates (5.7)–(5.11) are very elemen tary, we omit it. /square\n5.1. Solvability of the linear κ-approximate problem (5.2)\nIn this section, we construct the unique local solution of the the line arκ-approximate problem\n(5.2). To this purpose, we shall rewrite the linear κ-approximate problem (5.2) as the following\nequivalent problem (in the sense of classical solutions):\n\n\nηt−κ∂2\n1η=u,\n¯ρut+∇BQ+a¯ρu=λm2∂2\n1η−g¯ρe2,\n−divB(∇BQ/¯ρ) =K1,\n(η,u)|t=0= (η0,u0),\n(η,u)|∂Ω·/vectorn= 0,∇BQ|∂Ω·/vectorn=−g¯ρe2·/vectorn,(5.12)\nwhereς|t=0=η0,η0satisfies\ndivA0u0= 0,A0= (∇η0+I)−T,infy∈Ω(detA0)>0,\nand\nK1:=adivBu−divBtu−JtdivBu/J −λm2divB/parenleftbig\n∂2\n1η/¯ρ/parenrightbig\n. (5.13)\nThen the solvability of the linear κ-approximate problem reduces to the solvability of the initial-\nboundary value problem (5.12).\nWe want to establish the local well-posdeness result for (5.12) by an iteration method. To\nbegin with, we shall investigate the solvability of the linear problem\n\n\nηt−κ∂2\n1η=w,\nut+au=K2,\n−divB(∇BQ/¯ρ) =K1,\n(η,u)|t=0= (η0,u0),\n(η,u)|∂Ω·/vectorn= 0,∇BQ·/vectorn|∂Ω=−g¯ρe2·/vectorn,(5.14)\n38where (ς,w,θ) is given, and\nK2:= (λm2∂2\n1ξ−∇Bθ)/¯ρ−ge2.\nIt is easy see that the above linear problem is equivalent to the followin g three sub-problems:\nthe initial-boundary value problem of partly dissipative equation for η\n\n\nηt−κ∂2\n1η=w,\nη|t=0=η0,\nη|∂Ω·/vectorn= 0,(5.15)\nthe initial-value problem of ODE for u\n\n\nut+au=K2,\nu|t=0=u0,\nu|∂Ω·/vectorn= 0(5.16)\nand the Neumann boundary-value problem of elliptic equation for q\n/braceleftBigg\n−divB(∇BQ/¯ρ) =K1in Ω,\n∇BQ/¯ρ·/vectorn=−ge2·/vectornon∂Ω.(5.17)\nThus the solvability of the linear problem (5.14) reduces to the solvab ility of the three sub-\nproblems above. Next we establish the global well-posedness result s for the above three sub-\nproblems in sequence.\nProposition 5.1. LetT >0,w∈L2\nTH4\nsandη0∈H1,4\ns. Then the initial-boundary value problem\n(5.15)admits a unique solution η∈C(IT,H4\ns), which satisfies ∇4∂1η∈C0\nB,weak(IT,L2)and the\nestimates\nsupt∈IT(/ba∇dblη/ba∇dbl4+√κ/ba∇dbl∂1η/ba∇dbl4)+κ/ba∇dbl∂2\n1η/ba∇dblL2\nTH4\n/lessorsimilar/ba∇dblη0/ba∇dbl4+√κ/ba∇dblη0/ba∇dbl1,4+(1+√\nT)/ba∇dblw/ba∇dblL2\nTH4. (5.18)\nProof. We define the difference quotient with respect variable y1as follows:\nDτ\n1f(y1,y2) = (f(y1+τ,y2)−f(y1,y2))/τforτsatisfyin |τ| ∈(0,1).\nLetε∈(0,1),χbe a 1D standard mollifier (see [34, pp. 38] for the definition), and χε(s) :=\nχ(s/ε)/ε. Let ˜w=win ΩTand ˜w= 0 outside Ω ×(R\\IT). We define the mollification of ˜ wwith\nrespect tot:\nSt\nε(˜w) :=χε∗˜w. (5.19)\nThenSt\nε(˜w)∈C∞(R,H4\ns). We can check that\n/ba∇dblSt\nε(˜w)/ba∇dblL2\nTH4/lessorsimilar/ba∇dblw/ba∇dblL2\nTH4, (5.20)\nSt\nε(˜w)→wstrongly in L2\nTH4.\n39Now we consider the τ-approximate problem for (5.15):\n\n\nητ\nt=L(ητ)+St\nε(˜w) in ΩT,\nητ|t=0=η0in Ω,\nητ·/vectorn= 0 on ∂Ω,(5.21)\nwhereL:H4\ns→H4\nsby the rule L(f) =κD−τ\n1Dτ\n1fforf∈H4\ns.\nIt is easy to see that L(̟)∈C0(IT,H4\ns) for̟∈C0(IT,H4\ns) andL∈ L(H4\ns), where L(H4\ns)\nis a set of all linear bounded operators of H4\ns. In particular, L∈C0(IT,L(H4\ns)). By existence\ntheory of the initial-value problem of a abstract ODE equation (see [3 4, Proposition 2.17]), there\nexists a unique solution ητ∈C0(IT,H4\ns)∩C1(IT,H4\ns) to (5.21). Obviously ητ,ητ\ntautomatically\nbelong toL2\nTH4\nsby the second conclusion in Lemma A.10 and the separability of H4\ns.\nLetαsatisfy 0 /lessorequalslant|α|/lessorequalslant4. Applying ∂αto (5.21) 1, and then multiplying the resulting identity\n∂αηεinL2, we have, for a.e. t∈IT,\nd\ndt/integraldisplay\n|∂αητ|2dy+κ/integraldisplay\n|Dh\n1∂αητ|2dy=/integraldisplay\nSt\nε(∂α˜w)·∂αητdy. (5.22)\nMaking use of (5.20), we easily deduce from (5.22) that,\n/ba∇dblητ/ba∇dblC0(IT,H4)/lessorsimilar/ba∇dblη0/ba∇dbl4+√\nT/ba∇dblw/ba∇dblL2\nTH4. (5.23)\nIn addition, we easily deduce from (5.21) 1that\nκ\n2d\ndt/ba∇dblDτ\n1∂αητ/ba∇dbl2\n0+/integraldisplay\n|∂αητ\nt|2dy=/integraldisplay\n∂α(St\nε(w))·∂αητ\ntdy. (5.24)\nNoting that\n/ba∇dblDτ\n1∂αητ|t=0/ba∇dbl0/lessorsimilar/ba∇dbl∂αη0/ba∇dbl1,0, (5.25)\nthus, making use of Young’s inequality, (5.20) and (5.21) 1, we deduce from (5.24) and (5.25) that\nκ/ba∇dblDτ\n1ητ/ba∇dbl2\nC0(IT,H4)+/ba∇dbl(L(ητ),ητ\nt)/ba∇dbl2\nL2\nTH4/lessorsimilarκ/ba∇dblη0/ba∇dbl2\n1,4+/ba∇dblw/ba∇dbl2\nL2\nTH4, (5.26)\nThanks to the regularity of ητ, we can we easily derive (5.21) 1that, for any ϕ∈H1,\n/integraldisplay\nDτ\n1∂α(ητ(y,t)−ητ(y,s))·ϕdy\n=−/integraldisplayt\ns/integraldisplay\n(∂αSt\nε(˜w)+L(∂αητ))·D−τ\n1ϕdydτ. (5.27)\nExploiting (5.20), the uniform estimate of L(ητ) in (5.26) and the fact\n/ba∇dblD−τ\n1ϕ/ba∇dbl0/lessorsimilar/ba∇dblϕ/ba∇dbl1,0,\nwe easily derive from the identity (5.27) that\nDτ\n1∇4ητis uniformly continuous in H−1. (5.28)\nHere and in what follows, H−1denotes the dual space of H1\n0:={w∈H1|w|∂Ω= 0}.\n40Making use of (5.23), (5.26), (5.28) and (A.42)–(A.44), there exist s a subsequence (still de-\nnoted byητ) of{ητ}|τ|∈(0,1)such that, for τ→0,\nητ\nt⇀ηε\ntweakly inL2\nTH4\ns, ητ→ηεstrongly in C0(IT,H3),\nητ⇀ηεweakly-* in L∞\nTH4\ns, Dτ\n1∇4ητ→ ∇4∂1ηεinC0(IT,L2\nweak),\nDτ\n1ητ⇀∂1ηεweakly-* in L∞\nTH4andD−τ\n1Dτ\n1(ητ)⇀∂2\n1ηεweakly inL2\nTH4.\nMoreover, the limit function ηεis just the unique solution to the problem\n\n\nηε\nt=κ∂2\n1ηε+St\nε(˜w) in ΩT,\nηε|t=0=η0in Ω,\nηε·/vectorn= 0 on ∂Ω\nand satisfies\nsupt∈IT(/ba∇dblηε/ba∇dbl4+√κ/ba∇dbl∂1ηε/ba∇dbl4)+κ/ba∇dbl∂2\n1ηε/ba∇dblL2\nTH4\n/lessorequalslant/ba∇dblη0/ba∇dbl4+√κ/ba∇dblη0/ba∇dbl1,4+(1+√\nT)/ba∇dblw/ba∇dblL2\nTH4.\nThus we further have ηε∈C0(IT,H4\ns).\nNoting that ηεenjoys the uniform estimate above, thus we again get a limit function η, which\nsatisfies the desired conclusion in Proposition 5.1, by taking limit of som e sequence of {ηε}ε>0.\nWe omit such limit process, since it is very similar to the argument of obt ainingηε. /square\nProposition 5.2. Leta∈RandK2∈L1\nTH4\nsandu0∈H4\ns, then the initial-boundary value\nproblem(5.16)admits a unique solution u∈C0(IT,H4\ns), which satisfies ut∈L1\nTH4\nsand\n/ba∇dblu(t)/ba∇dbl4/lessorsimilar/ba∇dblu0/ba∇dbl4+/ba∇dblK2/ba∇dblL1\nTH4. (5.29)\nProof. Let\nu=u0e−at+/integraldisplayt\n0K2(τ)e−a(t−τ)dτ. (5.30)\nSinceK2∈L1\nTH4\nsandu0∈H4\ns, it is easy to check that ugiven by (5.30) is the unique solution\nof (5.16) 1–(5.16) 2; moreover, ubelongs toC0(IT,H4\ns) and satisfies (5.29). /square\nProposition 5.3. LetT >0,ιbe the positive constant provided in Lemma A.8, ¯ρsatisfy(1.5)\nandς∈A4,1/4\nT,ιdefined by (5.5). IfK1∈C0(IT,H1)∩L∞\nTH2and satisfies/integraltext\nK1Jdy= 0for each\nt∈IT, then there exists a unique solution Q∈C0(IT,H3), which satisfies the boundary-value\nproblem(5.17)for eacht∈ITand enjoys the estimate\n/ba∇dblQ/ba∇dblL∞\nTH4/lessorsimilarP(/ba∇dblς/ba∇dblL∞\nTH4)/parenleftbig\n/ba∇dblg/ba∇dbl3+/ba∇dblK1/ba∇dblL∞\nTH2/parenrightbig\n. (5.31)\n(1) If we additionally assume that K1∈L2\nTH3, then\n/ba∇dbl∇BQ/ba∇dblL2\nTH4/lessorsimilarP(/ba∇dblς/ba∇dblL∞\nTH4)/parenleftBig√\nT/ba∇dblg/ba∇dbl4+/ba∇dblK1/ba∇dblL2\nTH3/parenrightBig\n. (5.32)\n(2) If we additionally assume that ςt∈L∞\nTH3andK1\nt∈L∞\nTH1, then\n/ba∇dblQt/ba∇dblL∞\nTH3/lessorsimilarP(/ba∇dblς/ba∇dblL∞\nTH4,/ba∇dblςt/ba∇dblL∞\nTH3)(/ba∇dblg/ba∇dbl3+/ba∇dblK1/ba∇dblL∞\nTH2+/ba∇dblK1\nt/ba∇dblL∞\nTH1).(5.33)\n41Proof. Letϕ=ς+y, thenϕis aC1-diffeomorphism mapping by Lemma A.8. We denote\nϕ−1the inverse mapping of ϕwith respect to variable y, and then define ˜K1:=K1(ϕ−1,t). By\n(A.46),˜K1∈C0(IT,H1)∩L∞\nTH2.\nNow we consider the following Neumann boundary-value problem of ellip tic equation\n/braceleftBigg\n−div(∇p/˜ρ) =˜K1in Ω,\n∇p/˜ρ·/vectorn=−ge2·/vectornon∂Ω,(5.34)\nwhere ˜ρ= ¯ρ(ϕ−1\n2). It is easy to see that\n/integraldisplay\n˜K1dx=/integraldisplay\nK1Jdy= 0 =−/integraldisplay\n∂Ωge2·/vectorndy1.\nBy Lemma A.7, the second assertion in Lemma A.10, (A.41), (A.48) and the separability of H3,\nthere exists a unique solution p∈C0(IT,H3)∩L∞\nTH4of the boundary-value problem (5.34) such\nthat\n/ba∇dblp/ba∇dblL∞\nTH4/lessorsimilarP(/ba∇dblς/ba∇dblL∞\nTH4)(/ba∇dblg/ba∇dbl3+/ba∇dblK1/ba∇dblL∞\nTH2). (5.35)\nLet˜Q=p(ϕ), then˜Q∈C0(IT,H3)∩L∞\nTH4by (A.45). Wefurther define that Q:=˜Q−(˜Q)Ω,\nthenQ∈C0(IT,H3)∩L∞\nTH4and satisfies (5.31) by (A.47). Since psatisfies (5.34), we have\n/braceleftBigg\n−divB(∇BQ/¯ρ) =K1in Ω,\n∇BQ/¯ρ·/vectorn=−ge2·/vectornon∂Ω.\nIn addition, the uniqueness of Qis obvious in the class C0(IT,H3)∩L∞\nTH4.\n(1) If we additionally assume that K1∈L2\nTH2, then˜K1∈L2\nTH2. By Lemma A.10, we have\np∈L2\nTH4and\n/ba∇dblp/ba∇dblL2\nTH4/lessorsimilarP(/ba∇dblς/ba∇dblL∞\nTH3)(√\nT/ba∇dblg/ba∇dbl3+/ba∇dbl˜K1/ba∇dblL2\nTH2). (5.36)\nBy (5.36), (A.47) and (A.48),\n/ba∇dbl∇BQ/ba∇dblL2\nTH4=/ba∇dbl∇p(x,t)|x=ϕ(y,t)/ba∇dblL2\nTH4\n/lessorsimilarP(/ba∇dblς/ba∇dblL∞\nTH4)/ba∇dbl∇p/ba∇dblL2\nTH4/lessorsimilarP(/ba∇dblς/ba∇dblL∞\nTH4)/parenleftBig√\nT/ba∇dblg/ba∇dbl4+/ba∇dblK1/ba∇dblL2\nTH3/parenrightBig\n,\nwhich yields (5.32).\n(2) By (A.50), it is easy to see that\n˜K1\nt∈L∞\nTH1and∂t(˜ρ,(1/˜ρ))∈L∞\nTH3. (5.37)\nThanks to Lemma A.7, it is easy to verify that there exists a unique so lutionχ∈L∞\nTH3to\n/braceleftBigg\n−div(∇χ/˜ρ) =˜K1\nt+div(∂t(1/˜ρ)∇p) in Ω,\n∇χ/˜ρ·/vectorn=∂t(1/˜ρ)∇p·/vectorn on∂Ω.(5.38)\nMoreover, by (5.35), (A.22), (A.48) and (A.50),\n/ba∇dblp/ba∇dblL∞\nTH4+/ba∇dblχ/ba∇dblL∞\nTH3\n/lessorsimilarP(/ba∇dblς/ba∇dblL∞\nTH4,/ba∇dblςt/ba∇dblL∞\nTH3)(/ba∇dblg/ba∇dbl3+/ba∇dblK1/ba∇dblL∞\nTH2+/ba∇dblKt/ba∇dblL∞\nTH1). (5.39)\n42Lett∈ITandDsϑ=/parenleftbig\nϑ(y,t+s)−ϑ(y,t)/parenrightbig\n/swheret+s∈IT. Sincepsatisfies (5.34), thus\n/braceleftBigg\n−div(∇Dsp/˜ρ(x,t+s)) =Ds˜K1+div(Ds(1/˜ρ)∇p) in Ω,\n∇Dsp/˜ρ(x,t+s)·/vectorn=−Ds(1/˜ρ)∇p·/vectorn on∂Ω.(5.40)\nSubtracting (5.38) from (5.40) yields that\n\n\n−div(∇(χ−Dsp)/˜ρ(x,t+s))\n=˜K1\nt−Ds˜K1+div((∂t(1/˜ρ)−Ds(1/˜ρ))∇p\n+(1/˜ρ(x,t)−1/˜ρ(x,t+s))∇χ) in Ω ,\n(∇(χ−Dsp)/˜ρ(x,t+s))·/vectorn\n= ((∂t(1/˜ρ)−Ds(1/˜ρ))∇p(x,t)\n+(1/˜ρ(x,t+s))−1/˜ρ(x,s))∇χ)·/vectorn on∂Ω.(5.41)\nApplying the estimate (A.22) to (5.41), we have, for a.e. t∈IT,\n/ba∇dblχ−Dsp/ba∇dbl3/lessorsimilar/ba∇dbl˜K1\nt−Ds˜K1/ba∇dbl1\n+/ba∇dbl((∂t(1/˜ρ)−Ds(1/˜ρ))∇p,(1/˜ρ(x,t+s)−1/˜ρ(x,s))∇χ)/ba∇dbl2.(5.42)\nNoting that the generalized derivative with respect to tis automatically strong derivative, we\neasilyseethatthetwotermsontherighthandoftheinequality (5.42 )convergeto0fora.e. t∈IT\nby (5.37). So, /ba∇dbl(Dsp−χ)/ba∇dbl2\n3→0 ass→0 for a.e.t∈IT. This means that the strong derivative\nofpwith respect to tis equal to that of χ. In addition, it is easy to check that p∈AC0(IT,H3),\nthuspt=χ, whereptdenotes the generalized derivative of p. Hence,pt∈L∞(ITH3) satisfies\n(5.39) with ptin place ofχ. Thanks to (A.47) and (A.49), we immediately get (5.33) from (5.39).\nThis completes the proof of Proposition 5.3. /square\nWith Propositions 5.1–5.3 in hand, next we will use an iteration method t o establish an\nexistence result of a unique local solution to the linearized κ-problem (5.2).\nProposition 5.4. LetA4,1/4\nα,ιandSαare defined by (5.5)and(5.6)with some positive constant\nαin place ofT, resp.. We assume that a/greaterorequalslant0,¯ρsatisfy(1.5),(η0,u0)∈H1,4\ns×H4\nsandς∈A4,1/4\nα,ι\nsatisfiesςt∈C0(Iα,H2\ns)∩L∞\nαH4, then theκ-approximate problem (5.2)defined on Ωαadmits a\nunique solution, denoted by (ηκ,uκ,Qκ), which belongs to Sα.\nProof. LetT/lessorequalslantα. Thanks to the regularity of ( ς,ςt) and the relation\n∂j(JBij) = 0 fori= 1,2, (5.43)\nit is easy to check that, for any ( ξ,w,β)∈ST,\nK1(ξ,w)∈C0(IT,H1)∩L∞\nTH1∩L2\nTH3and/integraldisplay\nK1(ξ,w)Jdy= 0 for each t∈IT,\nwhereK1(ξ,w) is defined by (5.13) with ( ξ,w) in place of ( η,u).\nBy Proposition 5.3, there exists a function Q1∈C0(IT,H3)∩L∞\nTH4such that ∇BQ1∈L2\nTH4\nand\n/braceleftBigg\n−divB(∇BQ1/¯ρ) = 0 in Ω ,\n∇BQ1/¯ρ·/vectorn=−ge2·/vectornon∂Ω.\n43In view of Propositions 5.1–5.3 and the facts above, we easily see tha t there exist a solu-\ntion sequence {(ηn,un,Qn)}∞\nn=1defined on ITand the solutions ( ηn,un,Qn) enjoy the following\nproperties:\n(1) (η1,u1) = 0, andQ1satisfies∇BQ1|∂Ω·/vectorn=−ge2·/vectorn.\n(2) (ηn,un,Qn)∈STforn/greaterorequalslant1.\n(3) forn/greaterorequalslant2, (ηn,un,Qn) satisfies the following relations:\n\n\nηn\nt−κ∂2\n1ηn=un−1,\nun\nt+aun= (λm2∂2\n1¯ηn−1−∇BQn−1)/¯ρ−ge2=:K2,n,\n−divB(∇BQn/¯ρ) =K1(ηn,un),\n(ηn,un)|t=0= (η0,u0),\n(ηn,un)|∂Ω·/vectorn= 0,∇BQn|∂Ω·/vectorn=−g¯ρe2·/vectorn,\nwhereK1(ηn,un) is defined by (5.13) with ( ηn,un) in place of ( η,u),K1,n∈C0(IT,H1)∩\nL∞\nTH2∩L2\nTH3,K2,n∈L2\nTH4\ns, and\n/integraldisplay\nK1,nJdy= 0 for each t∈IT.\nWe further define (¯ ηn,¯un,¯Qn) := (ηn−ηn−1,un−un−1,Qn−Qn−1) forn/greaterorequalslant3. Then\n\n\n¯ηn\nt−κ∂2\n1¯ηn= ¯un−1,\n¯un\nt+a¯un= (λm2∂2\n1¯ηn−1−∇B¯Qn−1)/¯ρ,\n−divB/parenleftbig\n∇B¯Qn/¯ρ/parenrightbig\n=adivB¯un−divBt¯un−JtdivB¯un/J −λm2divB(∂2\n1¯ηn/¯ρ),\n(¯ηn,¯un)|t=0= (0,0),\n(¯ηn,¯un)|∂Ω·/vectorn= 0,∇B¯Qn|∂Ω·/vectorn= 0.\nThanks to Propositions 5.1–5.3, (5.8) and (5.9), we easily estimate th at, forn/greaterorequalslant3,\n/ba∇dbl¯ηn/ba∇dblC0(IT,H4s)+/ba∇dbl∂1¯ηn/ba∇dblL∞\nTH4+/ba∇dbl∂2\n1¯ηn/ba∇dblL2\nTH4\n/lessorsimilarκ(1+√\nT)/ba∇dbl¯un−1/ba∇dblL2\nTH4/lessorsimilarκT(1+√\nT)/ba∇dbl¯un−1/ba∇dblC0(IT,H4s),\n/ba∇dbl¯un/ba∇dblC0(IT,H4s)/lessorsimilar√\nT/ba∇dbl(∇B¯Qn−1,∂2\n1¯ηn−1)/ba∇dblL2\nTH4,\n/ba∇dbl¯Qn/ba∇dblL∞\nTH4/lessorsimilarP(/ba∇dbl(ς,ςt)/ba∇dblL∞αH4)/parenleftBig\n/ba∇dbl¯un/ba∇dblC0(IT,H4s)+/ba∇dbl∂1¯ηn/ba∇dblL∞\nTH4/parenrightBig\n,\n/ba∇dbl∇B¯Qn/ba∇dblL2\nTH4/lessorsimilarP(/ba∇dbl(ς,ςt)/ba∇dblL∞αH4)/parenleftBig√\nT/ba∇dbl¯un/ba∇dblC0(IT,H4s)+/ba∇dbl∂2\n1¯ηn/ba∇dblL2\nTH4/parenrightBig\n.\nWe immediately see fromthe above four estimates that there exists a sufficiently small T1(the\nsmallness depends on κ,g,λ,m, ¯ρ, Ω and the norm /ba∇dbl(ς,ςt)/ba∇dblL∞αH4) such that, for T= min{T1,α},\n{(ηn,∂1ηn,∂2\n1ηn,un,Qn,∇BQn)}∞\nn=1\nis a Cauchy sequence in C0(IT,H4\ns)×L∞\nTH4×L2\nTH4×C0(IT,H4\ns)×L∞\nTH4×L2\nTH4. Thus we can\nget one limit function ( η,u,Q), which is the unique local solution of the initial-boundary value\n44problem (5.12) and also the unique local solution of the linear κ-approximate problem (5.2). In\naddition, it is easy to see that ( η,u,Q)∈STby further using Propositions 5.1, 5.3 and trace\ntheorem.\nNoting that the local time T1is independent of the initial data and the local solution con-\nstructed above satisfies ( η,u)|t=T∈H1,4\ns×H4\ns, thus, ifT < α, we can further extend the local\nsolution to be a global solution defined on Iαby finite steps; moreover the obtained global solu-\ntion is the unique solution of of the linearized κ-approximate problem (5.2) and belongs to Sα.\nThis completes the proof of Proposition 5.4. /square\n5.2. Solvability of linearized problem (5.3)\nTo investigate the solvability of linearized problem (5.3), we shall first deriveκ-independent\nestimates of the solutions of the linear problem (5.2).\nLemma 5.2. Under the assumptions of Proposition 5.4, we further assume thatκ∈(0,1],\n/ba∇dblη0/ba∇dbl4/lessorequalslantδ∈R+andςsatisfies\n∂1ς∈L∞\nαH4, ς|t=0=η0.\nThen there exist polynomials ˙P(/ba∇dblς/ba∇dblL∞αH4),P(/ba∇dbl(ς,∂1ς,ςt)/ba∇dblL∞αH4),P(I0,/ba∇dbl(ς,∂1ς,ςt)/ba∇dblL∞αH4)and\npositive constants c,c5/greaterorequalslant1,δ3(may depending on g,a,λ,m,¯ρandΩ), such that, for any\nδ/lessorequalslantδ3, the local solution (ηκ,uκ,Qκ)provided by Proposition 5.4 enjoys the following estimates :\nsupt∈IT1Eκ(t)/lessorequalslant2c5(I0+T˙P(/ba∇dblς/ba∇dblL∞\nT1H4)), (5.44)\nsupt∈IT1Eκ(t)+κ/ba∇dbl∂2\n1ηκ/ba∇dbl2\nL2\nT1H4/lessorequalslant2c5I0+1, (5.45)\n/ba∇dblQκ/ba∇dblL∞\nT1H4+/ba∇dbluκ\nt/ba∇dblL∞\nT1H3/lessorsimilarP/parenleftBig\nI0,/ba∇dbl(ς,ςt)/ba∇dblL∞\nT1H4/parenrightBig\n, (5.46)\nwhereEκ(t) :=/ba∇dbl(ηκ,∂1ηκ,uκ)(t)/ba∇dbl2\n4,I0is defined by (1.21), and\nT1:= min{1/3c5P(/ba∇dbl(ς,∂1ς,ςt)/ba∇dblL∞αH4),α,1}. (5.47)\nMoreover, for (ηκ,uκ)restricted in IT1,\n∇4∂1ηκand∇4uκare uniformly continuous in H−1. (5.48)\nIf additionally ςtt∈L∞\nT1H2, then\n/ba∇dblQκ\nt/ba∇dblL∞\nT1H3/lessorsimilarP(I0,/ba∇dbl(ς,ςt)/ba∇dblL∞\nT1H4,/ba∇dblςtt/ba∇dblL∞\nT1H2). (5.49)\nProof. Frow now on we denote ( ηκ,uκ,Qκ) by (η,u,Q), and letT/lessorequalslantmin{α,1}and\n/ba∇dblη0/ba∇dbl4/lessorequalslantδ∈(0,1].\nNext we establish the desired uniform estimates for ( η,u,Q) by eight steps.\n(1)Estimate of η.\nRecalling (5.18), we immediately get\nsupt∈IT/ba∇dblη(t)/ba∇dbl2\n4+κ2/ba∇dbl∂2\n1η/ba∇dbl2\nL2\nTH4/lessorsimilar/ba∇dblη0/ba∇dbl2\n1,4+/ba∇dblu/ba∇dbl2\nL2\nTH4/lessorsimilarI0+Tsupt∈ITEκ(t).(5.50)\n(2)Estimate of Q.\n45Noting that Qsatisfies/braceleftBigg\n−divB(∇BQ/¯ρ) =K1in Ω,\n∇BQ/¯ρ·/vectorn=−ge2·/vectornon∂Ω,(5.51)\nwhereK1is defined by (5.13). By (5.8), (5.9), (5.31) and (5.50), we have\n/ba∇dblQ/ba∇dblL∞\nTH4/lessorsimilarP(/ba∇dblς/ba∇dblL∞\nTH4)/parenleftbig\n/ba∇dblg/ba∇dbl3+/ba∇dblK1/ba∇dblL∞\nTH2/parenrightbig\n/lessorsimilarP/parenleftbig\n/ba∇dbl(ς,ςt)/ba∇dblL∞\nTH4/parenrightbig/parenleftBig\n1+/radicalbig\nEκ(t)/parenrightBig\n. (5.52)\n(3)L2-norm energy estimate of u.\nLetqς=Q−¯P(ς2+y2), andGς:= ¯ρ(ς2(y,t)+y2)−¯ρ(y2). Then (5.2) 2can be rewritten as\nfollows:\n¯ρut+∇Bqς+a¯ρu=λm2∂2\n1η+gGςe2. (5.53)\nMultiplying (5.53) with JuinL2then yields that\n1\n2d\ndt/integraldisplay\n¯ρ|u|2Jdy+a/integraldisplay\n¯ρ|u|2Jdy\n=1\n2/integraldisplay\n¯ρ|u|2Jtdy+λm2/integraldisplay\n∂2\n1η·uJdy+/integraldisplay\n(gGς−∇Bqς)·uJdy. (5.54)\nUsing the boundary-value condition of ( ς,u), the integration by parts, (5.2) 3, (5.8) and (5.43),\nwe have\n/integraldisplay\n(gGς−∇Bqς)·uJdy=g/integraldisplay\nGςu2Jdy/lessorsimilar˙P(/ba∇dblς/ba∇dbl3)/ba∇dblu2/ba∇dbl0. (5.55)\nIn addition, making use of (5.2) 1, (5.8), (5.9) and the integration by parts, we get\nλm2/integraldisplay\n∂2\n1η·uJdy\n=−λm2\n2d\ndt/integraldisplay\n|∂1η|2Jdy+λm2/integraldisplay\n(|∂1η|2Jt/2−∂1η·η∂1J −κ|∂2\n1η|2J)dy\n/lessorequalslant−λm2\n2d\ndt/integraldisplay\n|∂1η|2Jdy−κλm2/integraldisplay\n|∂2\n1η|2Jdy+cP(/ba∇dbl(ς,ςt)/ba∇dbl3)/ba∇dblη/ba∇dbl2\n2. (5.56)\nNoting that 1 /4/lessorequalslantJ/lessorsimilar1, thus, plugging (5.55)–(5.56) into (5.54) and then integrating the\nresulting inequality over (0 ,t), we immediately get, for any t∈IT,\n/ba∇dbl(u,∂1η)(t)/ba∇dbl2\n0+κ/integraldisplayt\n0/ba∇dbl∂2\n1η(τ)/ba∇dbl2\n0dτ\n/lessorsimilarI0+T˙P(/ba∇dblς/ba∇dblL∞\nTH3)+TP/parenleftbig\n/ba∇dbl(ς,ςt)/ba∇dblL∞\nTH3/parenrightbig\nsupt∈IT/ba∇dbl(η,u)(t)/ba∇dbl2\n2. (5.57)\n(4)Curl estimate of (η,u).\nApplying curl Bto (5.2) 2yields that\n∂tcurlB(¯ρu)+acurlB(¯ρu) =λm2curlB/parenleftbig\n¯ρ−1∂2\n1(¯ρη)/parenrightbig\n+curl Bt(¯ρu)−g¯ρ′B12. (5.58)\nLet the multiindex αsatisfy|α|/lessorequalslant2. Applying ∂αto (5.58) yields\n∂t∂αcurlB(¯ρu)+a∂αcurlB(¯ρu) =λm2¯ρ−1∂1∂αcurlB(¯ρ∂1η)+Kα\n3+Kα\n4, (5.59)\n46where we have defined that\nK3\nα:=λm2∂1/parenleftbig\n[∂αcurlB,¯ρ−1](¯ρ∂1η)/parenrightbig\n−∂α(g¯ρ′B12),\nK4\nα:=∂αcurlBt(¯ρu)−λm2∂αcurl∂1B(∂1η).\nMultiplying (5.59) by ∂αcurlB(¯ρu) inL2, we obtain, for a.e. t∈IT,\n1\n2d\ndt/integraldisplay\n|∂αcurlB(¯ρu)|2dy+a/integraldisplay\n|∂αcurlB(¯ρu)|2dy\n−λm2/integraldisplay\n¯ρ−1∂1∂αcurlB(¯ρ∂1η)∂αcurlB(¯ρu)dy\n=/integraldisplay\nK3\nα∂αcurlB(¯ρu)dy+/integraldisplay\nK4\nα∂αcurlB(¯ρu)dy=:I6+I7. (5.60)\nUsing the integral by parts and (5.2) 1again, we obtain\n−λm2/integraldisplay\n¯ρ−1∂1∂αcurlB(¯ρ∂1η)∂αcurlB(¯ρu)dy\n=λ\n2d\ndt/integraldisplay\n¯ρ−1|m∂αcurlB(¯ρ∂1η)|2dy+κλm2/integraldisplay\n¯ρ−1|∂αcurl/parenleftbig\n¯ρ∂2\n1η/parenrightbig\n|2dy+I8+I9,(5.61)\nwhere we have defined that\nI8:=λm2/integraldisplay\n¯ρ−1∂αcurlB(¯ρ∂1η)∂α(curl∂1B(¯ρu)−curl∂tB(¯ρ∂1η))dy,\nI9:=κλm2/integraldisplay/parenleftbigg\n¯ρ−1∂��curl∂1B(¯ρ∂1η)∂αcurlB(¯ρ∂2\n1η)+|∂αcurlB−I/parenleftbig\n¯ρ∂2\n1η/parenrightbig\n|2\n+2∂αcurlB−I/parenleftbig\n¯ρ∂2\n1η/parenrightbig\n∂αcurl/parenleftbig\n¯ρ∂2\n1η/parenrightbig/parenrightbigg\ndy.\nThanks to the estimates (5.7)–(5.10), we can estimate that\nI6+I7+I8/lessorsimilar˙P(/ba∇dblς/ba∇dbl4)+P(/ba∇dbl(ς,∂1ς,ςt)/ba∇dbl4)/ba∇dbl(u,∂1η)/ba∇dbl2\n4,\nI9/lessorsimilarκ(P(/ba∇dbl(ς,∂1ς)/ba∇dbl4)/ba∇dblη/ba∇dbl1,4(/ba∇dblcurl(¯ρη)/ba∇dbl2,3+/ba∇dblη/ba∇dbl1,4)\n+/ba∇dblB−I/ba∇dbl3(/ba∇dblB−I/ba∇dbl3/ba∇dblη/ba∇dbl2\n2,4+/ba∇dblη/ba∇dbl2,4/ba∇dblcurl(¯ρη)/ba∇dbl2,3).\nNoting that ς|t=0=η0, by Newton–Leibniz formula and (5.7) with t= 0, we have\n/ba∇dblB(t)−I/ba∇dbl3/lessorequalslant/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftbigg\nB0−I,/integraldisplayt\n0Bτ(τ)dτ/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n3/lessorsimilar/ba∇dblη0/ba∇dbl4+TP(/ba∇dbl(ς,ςt)/ba∇dblL∞\nTH5),(5.62)\nwhereB0:=B|t=0. Making use of the above three estimates, we deduce from (5.60)– (5.61) that\nd\ndt/vextenddouble/vextenddouble/vextenddouble/parenleftBig\ncurlB(¯ρu),/radicalbig\nλ/¯ρmcurlB(¯ρ∂1η)/parenrightBig/vextenddouble/vextenddouble/vextenddouble2\n3+cκ/ba∇dblcurl(¯ρη)/ba∇dbl2\n2,3\n/lessorsimilar˙P(/ba∇dblς/ba∇dbl4)+P(/ba∇dbl(ς,∂1ς,ςt)/ba∇dbl4)/ba∇dbl(u,∂1η)/ba∇dbl2\n4+κ(/ba∇dblη0/ba∇dbl4+TP(/ba∇dbl(ς,ςt)/ba∇dblL∞\nTH4))/ba∇dblη/ba∇dbl2\n2,4.(5.63)\nIn addition, similarly to (5.62), we have\n/ba∇dblcurlf(t)/ba∇dbl3/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/parenleftBig\ncurlB(0)−If(t),curlBf(t),curl/integraltextt\n0Bτdτf(t)/parenrightBig/vextenddouble/vextenddouble/vextenddouble\n3(5.64)\n/lessorsimilar/ba∇dblη0/ba∇dbl4/ba∇dblf(t)/ba∇dbl4+TP(/ba∇dbl(ς,ςt)/ba∇dbl4)/ba∇dblf(t)/ba∇dbl4+/ba∇dblcurlBf(t)/ba∇dbl3.\n47Integrating (5.63) over (0 ,t), and then using H¨ older’s inequality and the above estimate and\n(5.62), we conclude that\n/ba∇dblcurl(u,∂1η)(t)/ba∇dbl2\n3+κ/integraldisplayt\n0/ba∇dblcurl(¯ρη)(τ)/ba∇dbl2\n2,3dτ\n/lessorsimilarI0+T˙P(/ba∇dblς/ba∇dblL∞\nTH4)+(/ba∇dblη0/ba∇dbl4+TP(/ba∇dbl(ς,∂1ς,ςt)/ba∇dblL∞\nTH4)supt∈IT/ba∇dbl(u,∂1η)/ba∇dbl2\n4\n+κ/parenleftbig\n/ba∇dblη0/ba∇dbl4+TP(/ba∇dbl(ς,ςt)/ba∇dblL∞\nTH4)/parenrightbig/integraldisplayt\n0/ba∇dblη(τ)/ba∇dbl2\n2,4dτ+/ba∇dbl(u,∂1η)(t)/ba∇dbl2\n3. (5.65)\n(5)Divergence estimate of (η,u).\nSimilarly to (5.64), we have\n/ba∇dbldivf(t)/ba∇dbl3/lessorequalslant/vextenddouble/vextenddouble/vextenddouble/parenleftBig\ndivB0−If(t),divBf(t),div/integraltextt\n0Bτdτf(t)/parenrightBig/vextenddouble/vextenddouble/vextenddouble\n3.\nNoting that div Bu= 0 for any t∈IT, takingf=uin the above estimate yields\n/ba∇dbldivu/ba∇dbl3/lessorsimilar(/ba∇dblη0/ba∇dbl4+TP(/ba∇dbl(ς,ςt)/ba∇dbl4))/ba∇dblu/ba∇dbl4. (5.66)\nApplying div B∂1to (5.2) 1and then using (5.2) 3, we have\n∂t(divB∂1η)−κ∂1divB∂2\n1η= divBt∂1η−div∂1Bu−κdiv∂1B∂2\n1η=:K5.(5.67)\nWe define the mollification of f∈L2\nTL2with respect to y1as follows:\nS1\nε(f) :=χε∗η. (5.68)\nIt is easy to check that\nS1\nε(f)→finL2\nTL2. (5.69)\nThen we can derive from (5.67) that\n∂tS1\nε(∂αdivB∂1η)−κ∂1S1\nε(∂αdivB∂2\n1η) =S1\nε(∂αK5), (5.70)\nwhere the multiindex αsatisfies|α|/lessorequalslant3.\nForφ∈C∞\n0(IT), we multiply the above identity by S1\nε(∂αdivB∂1η)φinL2(ΩT) to get that\n/integraldisplayT\n0/parenleftbigg1\n2/ba∇dblS1\nε(∂αdivB∂1η)/ba∇dbl2\n0φτ+κ/ba∇dblS1\nε(∂αdivB∂2\n1η)/ba∇dbl2\n2φ/parenrightbigg\ndτ\n=/integraldisplayT\n0/integraldisplay\n(S1\nε(∂αK5)S1\nε(∂αdivB∂1η)−κS1\nε(∂αdivB∂2\n1η)S1\nε(∂αdiv∂1B∂1η))dyφdτ.\nThanks to (5.69), we can take limits by ε→0 in the above identity to get that\n/integraldisplayT\n0/parenleftbigg1\n2/ba∇dbl∂αdivB∂1η/ba∇dbl2\n0φt+κ/ba∇dbl∂αdivB∂2\n1η/ba∇dbl2\n2φ/parenrightbigg\ndτ\n=/integraldisplayT\n0/integraldisplay\n(∂αK5∂αdivB∂1η−κ∂αdivB∂2\n1η∂αdiv∂1B∂1η)dyφdτ.\n48In particular, we have, for a.e. t∈IT,\n1\n2d\ndt/ba∇dbl∂αdivB∂1η/ba∇dbl2\n0+κ/ba∇dbl∂αdivB∂2\n1η/ba∇dbl2\n2\n=/integraldisplay\n(∂αK5∂αdivB∂1η−κ∂αdivB∂2\n1η∂αdiv∂1B∂1η)dy. (5.71)\nFollow the argument of (5.65), we can derive from (5.71) that\n/ba∇dbldiv∂1η(t)/ba∇dbl2\n3+κ/integraldisplayt\n0/ba∇dbldivη(τ)/ba∇dbl2\n2,3dτ\n/lessorsimilarI0+(/ba∇dblη0/ba∇dbl2\n4+TP(/ba∇dbl(ς,∂1ς,ςt)/ba∇dblL∞\nTH4))supt∈IT/ba∇dbl(u,∂1η)/ba∇dbl2\n4\n+κ/parenleftbig\n/ba∇dblη0/ba∇dbl4+TP(/ba∇dbl(ς,ςt)/ba∇dblL∞\nTH4)/parenrightbig/integraldisplayt\n0/ba∇dblη(τ)/ba∇dbl2\n2,4dτ\n+κP(/ba∇dblς/ba∇dblL∞\nTH1,4)/integraldisplayt\n0/ba∇dblη(τ)/ba∇dbl2,4/ba∇dbl∂1η(τ)/ba∇dbl4dτ. (5.72)\nConsequently, we immediately deduce from (5.65), (5.66) and (5.72) that\n/ba∇dbl(divu,curlu,div∂1η,curl∂1η)(t)/ba∇dbl2\n3+κ/integraldisplayt\n0/ba∇dbl(divη,curl(¯ρη))(τ)/ba∇dbl2\n2,3dτ\n/lessorsimilarI0+T˙P(/ba∇dblς/ba∇dblL∞\nTH4)+(/ba∇dblη0/ba∇dbl4+TP(/ba∇dbl(ς,∂1ς,ςt)/ba∇dblL∞\nTH4)supt∈IT/ba∇dbl(u,∂1η)(t)/ba∇dbl2\n4\n+κ/parenleftbig\n/ba∇dblη0/ba∇dbl4+TP(/ba∇dbl(ς,ςt)/ba∇dblL∞\nTH4)/parenrightbig/integraldisplayt\n0/ba∇dblη(τ)/ba∇dbl2\n2,4dτ\n+κP(/ba∇dblς/ba∇dblL∞\nTH1,4)/integraldisplayt\n0/ba∇dblη(τ)/ba∇dbl2,4/ba∇dbl∂1η(τ)/ba∇dbl4dτ+/ba∇dbl(u,∂1η)(t)/ba∇dbl2\n3 (5.73)\n(6)Summing up the estimates (η,u,Q).\nThanks to the estimates (5.50), (5.57), (5.73), the interpolation in equality (A.2) and the\nHodge-type elliptic estimate (A.9), we have, for sufficiently small δ,\nsupt∈ITEκ(t)+κ/ba∇dbl∂2\n1η/ba∇dbl2\nL2\nTH4\n/lessorequalslantc5(I0+T˙P(/ba∇dblς/ba∇dblL∞\nTH4))\n+c5TP(/ba∇dbl(ς,∂1ς,ςt)/ba∇dblL∞\nTH4)/parenleftBig\nsupt∈ITEκ(t)+κ/ba∇dbl∂2\n1η/ba∇dbl2\nL2\nTH4/parenrightBig\n(5.74)\nand\nsupt∈ITEκ(t)+κ/ba∇dbl∂2\n1η/ba∇dbl2\nL2\nTH4\n/lessorequalslantc5I0+c5TP(/ba∇dbl(ς,∂1ς,ςt)/ba∇dblL∞\nTH4)/parenleftBig\n1+supt∈ITEκ(t)+κ/ba∇dbl∂2\n1η/ba∇dbl2\nL2\nTH4/parenrightBig\n.(5.75)\nIt should be noted that the two polynomials above are same.\nNow we use c5andPin (5.75) to define T1by (5.47), and thus get (5.44), resp. (5.45) from\n(5.74), resp. (5.75) by taking T=T1. Finally, making use of (5.2) 2satisfied by ( η,u,Q), (5.45)\nand (5.52), we easily get (5.46).\n(7) Let the multiindex βsatisfy|β|= 4. Obviously, there exists isuch that 1 /lessorequalslanti/lessorequalslant4 and\nβi/\\e}atio\\slash= 0. Letβ−satisfyβ−\ni=βi−1 andβ−\nj=βjforj/\\e}atio\\slash=i, andβ+satisfyβ−\ni= 1 andβ+\nj= 0\n49forj/\\e}atio\\slash=i. Similarly to (5.27), we can deduce from (5.2) 1and (5.2) 2that, for any ϕ∈H1\n0and for\nanys,t∈IT,\n/integraldisplay\n∂β∂1(η(t)−η(s))·ϕdy=−/integraldisplayt\ns/integraldisplay\n∂β(u+κ∂2\n1η)·∂1ϕdydτ, (5.76)\n/integraldisplay\n∂β(¯ρu(t)−¯ρu(s))·ϕdy\n=/integraldisplayt\ns/integraldisplay\n(∂β−∇BQ·∂β+ϕ−λm2∂β∂1η·∂1ϕ−∂β(g¯ρe2+a¯ρu)·ϕ)dydτ.(5.77)\nMaking use of the uniform estimates (5.45) and (5.46), we easily dedu ce the assertion in (5.48)\nfrom the two identities above.\n(8) If additionally ςtt∈L∞\nT1H2, we can apply the second conclusion in Proposition 5.3 to\n(5.51). Then, by further using the estimates (5.11), (5.33), (5.45 ) and (5.46), we can easily get\n(5.49). This completes the proof. /square\nThanks to Lemma 5.2, we establish the unique local solvability of the κ-approximate problem\n(5.3) by a compactness argument.\nProposition 5.5. Let the assumptions of Lemma 5.2 be satisfied, δ0,T1be provided by Lemma\n5.2 andςtt∈L∞\nT1H2, then, for any δ/lessorequalslantδ3, the linearized problem (5.3)defined on ΩT1admits a\nunique solution (ηL,uL,QL)∈C0(IT1,H1,4\ns)×U4\nT1×Q4\nT1; moreover the solution satisfies\nsupt∈IT1/ba∇dblη(t)/ba∇dbl4/lessorequalslant/ba∇dblη0/ba∇dbl4+/radicalbig\nT(2c5I0+1), (5.78)\nsupt∈IT1EL(t)/lessorequalslant2c5(I0+T˙P(/ba∇dblς/ba∇dblL∞\nTH4)), (5.79)\nsupt∈IT1EL(t)/lessorequalslant2c5I0+1, (5.80)\n/ba∇dblQL/ba∇dblL∞\nT1H4+/ba∇dbluL\nt/ba∇dblL∞\nT1H3/lessorsimilarP/parenleftBig\nI0,/ba∇dbl(ς,ςt)/ba∇dblL∞\nTαH4/parenrightBig\n, (5.81)\n/ba∇dblQL\nt/ba∇dblL∞\nT1H3/lessorsimilarP/parenleftBig\nI0,/ba∇dbl(ς,ςt)/ba∇dblL∞\nTαH4,/ba∇dblςtt/ba∇dblL∞\nT1H2/parenrightBig\n, (5.82)\nwhereEL(t) := supt∈IT1/ba∇dbl(ηL,∂1ηL,uL)(t)/ba∇dbl2\n4.\nProof. Let (ηκ,uκ,Qκ)∈STbe the solution of the linearized problem (5.2) stated as in Lemma\n5.2. Thanks to the κ-independent estimates (5.44)–(5.46), (5.48) and (5.49), we can e asily fol-\nlow the compactness argument as in the proof of Proposition 5.1 to o btain a limit function\n(ηL,uL,QL), which is the solution of the linearized problem (5.3), satisfies (5.79) –(5.82) and\nbelongs to C0(IT1,H1,4\ns)×U4\nT1×Q4\nT1.\nThe estimate (5.78) is obvious by (5.3) 1satisfied by ( η,u) and (5.80). In addition, the\nuniqueness can be easily verified by a standard energy method. The proof of Proposition 5.5 is\ncomplete. /square\n5.3. Proof of Proposition 2.2\nLet (η0,u0) satisfy the assumptions in Proposition 2.2, /ba∇dblη0/ba∇dbl4/lessorequalslantδ/lessorequalslantmax{δ3,ι}/2, and\nT2:= min{1/3c5P(c2\n4(b+δ3)2),δ2/(2c5(b+δ3)+1),1/4c5}<1,\n50where the positive constant c5/greaterorequalslant1 and the polynomial are provided by (5.47) and δ3is the\nconstant in Lemma 5.2 with α=T2. By Lemma 5.2, Proposition 5.4 and Lemma A.8, for any\nT/lessorequalslantT2, we can construct a solution sequence\n{(ηn,un,Qn)}∞\nn=1⊂C0(IT,H1,4\ns)×U4\nT×Q4\nT\nsuch that\n(1) (η1,u1,Q1) = (η0,u0,0).\n(2) (ηn+1,un+1,Qn+1) satisfies\n\n\nηn+1\nt=un+1,\n¯ρun+1\nt+∇AnQn+1+a¯ρun+1=λm2∂2\n1ηn+1−g¯ρe2,\ndivAnun+1= 0,\n(ηn+1,un+1)|t=0= (η0,u0),\n(ηn+1,un+1)·/vectorn= 0,∇AnQn+1·/vectorn=−ge2¯ρ·/vectornon∂Ω(5.83)\nfor anyn/greaterorequalslant1, where An= (∇(ηn+I))−T.\n(3) forn/greaterorequalslant2, (ηn,un,Qn) enjoys the following estimates:\nsupt∈IT/ba∇dblηn(t)/ba∇dbl4/lessorequalslant/ba∇dblη0/ba∇dbl4+/radicalbig\nT(2c5I0+1)/lessorequalslant2δ/lessorequalslantmin{δ3,ι}/lessorequalslant1,(5.84)\nsupt∈ITEn(t)/lessorequalslantc2\n4I0, (5.85)\nsupt∈ITEn(t)+/ba∇dblQn/ba∇dblL∞\nTH4+/ba∇dbl∂t(un,Qn)/ba∇dblL∞\nTH3/lessorsimilarP(I0), (5.86)\nwhereEn(t) :=/ba∇dbl(ηn,∂1ηn,un)(t)/ba∇dbl2\n3.\nSimilarly to (5.8)–(5.9), by using (5.84) and (5.86), we have, for any n/greaterorequalslant1,\nsupt∈IT/ba∇dbl(An,(Jn)−1,An\nt,Jn\nt)/ba∇dbl3/lessorsimilarP(I0), (5.87)\nwhereJn:= det(∇ηn+I). In addition, similarly to (5.76) and (5.77), by a regularity method,\nwe can easily derive from (5.83) 1and (5.83) 2that, for any ϕ∈H1\n0, (ηn,un) satisfies\n/integraldisplay\n∂β∂1(ηn(t)−ηn(s))·ϕdy=−/integraldisplayt\ns/integraldisplay\n∂βun·∂1ϕdydτ,\n/integraldisplay\n∂β(¯ρun(t)−¯ρun(s))·ϕdy\n=/integraldisplayt\ns/integraldisplay\n(∂β−∇An−1Qn·∂β+ϕ−λm2∂β∂1ηn·∂1ϕ−∂β(g¯ρe2+a¯ρun)·ϕ)dydτ.\nThus we further derive from the above two identities, (5.86) and (5 .87) that\n∇3∂1ηnand∇3unare uniformly continuous in H−1, (5.88)\nwheren/greaterorequalslant2. Next we further prove that {(ηn,un,Qn)}∞\nn=1is a Cauchy sequence.\nFrom now on, we always assume n/greaterorequalslant2. We define that\n¯ηn+1:=ηn+1−ηn,¯un+1:=un+1−un,¯Qn+1:=Qn+1−Qnand¯An:=An−An−1.\n51Then it follows from (5.83) that\n\n\n¯ηn+1\nt= ¯un+1,\n¯ρ¯un+1\nt+∇An¯Qn+1+a¯ρ¯un+1=λm2∂2\n1¯ηn+1−∇¯AnQn,\ndivAn¯un+1=−div¯Anun,\n(¯ηn+1,¯un+1)|t=0= (0,0),\n(¯ηn+1,¯un+1)|∂Ω·/vectorn= 0.(5.89)\nIt follows from (5.89) 1and (5.89) 4that, for 0 /lessorequalslanti/lessorequalslant4,\nsupt∈IT/ba∇dbl¯ηn+1(t)/ba∇dbli+√κ/ba∇dbl∂1¯ηn+1/ba∇dblL2\nTHi/lessorsimilarT/ba∇dbl¯un+1/ba∇dblL∞\nTHi. (5.90)\nBy (5.86), (5.87), (5.89) 1and (5.90), we can estimate that, for 0 /lessorequalslanti/lessorequalslant3,\nsupt∈IT/ba∇dbl¯An/ba∇dbli/lessorsimilarTP(I0)/ba∇dbl¯un/ba∇dblL∞\nTHi+1, (5.91)\nsupt∈IT/ba∇dbl¯An\nt/ba∇dbli/lessorsimilarP(I0)/ba∇dbl¯un/ba∇dblL∞\nTHi+1. (5.92)\nLetζn=ηn+y. (ζn)−1denotes the inverse function of ζnwith respect to the variable y. We\ndefine that\nK5:= (λm2∂2\n1¯ηn+1−∇¯AnQn)/¯ρ, K6:=K5−¯un+1\nt−a¯un+1,\n(β,̺,˜K5,˜K6) := (¯Qn+1,¯ρ,K5,K6)|y=(ζn)−1(x,t),\nthen, by (5.89) 2,/braceleftBigg\ndiv(∇β/̺) = div˜K6in Ω,\n(∇β/̺)·/vectorn=˜K5·/vectornon∂Ω.\nApplying the elliptic estimate (A.22) to the above boundary-value pro blem and then making use\nof (A.47) and (A.48), we have\n/ba∇dbl¯Qn+1/ba∇dbl2/lessorsimilarP(/ba∇dblηn/ba∇dbl3)/ba∇dblβ/ba∇dbl2/lessorsimilarP(/ba∇dblηn/ba∇dbl3)(/ba∇dbl˜K5/ba∇dbl1+/ba∇dbldiv˜K6/ba∇dbl1)\n/lessorsimilarP(/ba∇dblηn/ba∇dbl3)(/ba∇dblK5/ba∇dbl1+/ba∇dbldivAK6/ba∇dbl0). (5.93)\nNoting that\ndivAnK6= divA((λm2∂2\n1¯ηn+1−∇¯AnQn)/¯ρ)+div ¯Anun+divAn\nt¯un+1+∂tdiv¯Anun,\nthus, by using (5.86), (5.87), (5.91) and (5.92), we easily estimate t hat\n/ba∇dblK5/ba∇dbl1+/ba∇dbldivAK6/ba∇dbl0/lessorsimilarP(I0)/ba∇dbl/parenleftbig\n¯un,¯un+1,∂1¯ηn+1/parenrightbig\n/ba∇dblL∞\nTH2.\nPutting the above estimate into (5.93) yields\n/ba∇dbl¯Qn+1/ba∇dblC0(IT,H2)/lessorsimilarP(b)/ba∇dbl/parenleftbig\n¯un,¯un+1,∂1¯ηn+1/parenrightbig\n/ba∇dblL∞\nTH2. (5.94)\nSimilarly to the derivation of (5.57), we have that\nsupt∈IT/ba∇dbl(¯un+1,∂1¯ηn+1)(t)/ba∇dbl2\n0/lessorsimilarTP(I0)/ba∇dbl/parenleftbig\n¯un,¯un+1,∂1¯ηn+1/parenrightbig\n/ba∇dbl2\nL∞\nTH2. (5.95)\n52By (5.89) 1–(5.89) 3, we have\n∂t/parenleftbig\ndivAn∂1¯ηn+1/parenrightbig\n= divAn\nt∂1¯ηn+1−∂1(div¯Anun)−div∂1An¯un+1, (5.96)\n∂t/parenleftbig\ncurlAn/parenleftbig\n¯ρ¯un+1/parenrightbig/parenrightbig\n+acurlAn/parenleftbig\n¯ρ¯un+1/parenrightbig\n=λm2(¯ρ−1∂1/parenleftbig\ncurlAn/parenleftbig\n¯ρ∂1¯ηn+1/parenrightbig/parenrightbig\n+∂1/bracketleftbig\ncurlAn,¯ρ−1/bracketrightbig\n(¯ρ∂1¯ηn+1)\n−curl∂1An∂1¯ηn+1)+curl An\nt/parenleftbig\n¯ρ¯un+1/parenrightbig\n+cur ¯An(∇An−1Qn). (5.97)\nMaking use of (5.86), (5.87) and (5.91), we can follow the argument o f (5.73) to derive from\n(5.89)3, (5.96) and (5.97) that, for a.e. t∈IT,\n/ba∇dbl(div¯un+1,curl¯un+1,div∂1¯ηn+1,curl∂1¯ηn+1)(t)/ba∇dbl2\n1\n/lessorsimilar(/ba∇dblη0/ba∇dbl3+TP(I0))/ba∇dbl/parenleftbig\n¯un,∂1¯ηn,¯un+1,∂1¯ηn+1/parenrightbig\n/ba∇dbl2\nL∞\nTH2+/ba∇dbl/parenleftbig\n¯un+1,∂1¯ηn+1/parenrightbig\n/ba∇dbl2\n1. (5.98)\nConsequently, summing up the estimates (5.90), (5.95) and (5.98) a nd then using the inter-\npolation inequality (A.2), Hodge-type elliptic estimate (A.9) and Young ’s inequality, we obtain,\nfor sufficiently small δ,\nsupt∈IT/ba∇dbl/parenleftbig\n¯ηn+1,∂1¯ηn+1,¯un+1/parenrightbig\n/ba∇dbl2\n2/lessorsimilarTP(b)/ba∇dbl/parenleftbig\n¯un,∂1¯ηn,¯un+1,∂1¯ηn+1/parenrightbig\n/ba∇dblL2\nTH2. (5.99)\nIn addition, by (5.86), (5.87) and (5.89) 2, we get that\n/ba∇dbl¯un+1\nt/ba∇dblC0(IT,H1)/lessorsimilarP(b)/parenleftBig\n/ba∇dbl/parenleftbig\n¯un+1,∂1¯ηn+1/parenrightbig\n/ba∇dblC0(IT,H2)+/ba∇dbl¯Qn+1/ba∇dblC0(IT,H2)/parenrightBig\n. (5.100)\nBy (5.94), (5.99) and (5.100), we immediately see that the sequence {(ηn,un,un\nt,Qn)}∞\nn=1is\na Cauchy sequence in C0(IT,H1,2×H2×H1×H2) for sufficiently small T∈(0,T2], where the\nsmallness depends on b,g,a,λ,m, ¯ρand Ω. Hence\n(ηn,un,un\nt,Qn)→(η,u,ut,Q) strongly in C0(IT,H1,2×H2×H1×H2).\nThanks to the strong convergence of ( ηn,un,un\nt,Qn), we can take the limit in (5.83), and then get\nalocal classical solution ( η,u,Q)to theproblem (5.1). Let q=Q−¯P(ζ2)−λ|¯M|2/2, then(η,u,q)\nis also a local classical solution to the transformed MRT problem (1.19 ) by (1.17) and (1.18). In\naddition, thanks to (5.85), (5.86) and (5.88), we easily follow the com pactness argument as in\nthe proof of Proposition 5.5 to further derive that\n(η,u,q)∈C0(IT,H1,4\ns)×U4\nT×Q4\nT,\nsupt∈IT/ba∇dblηn(t)/ba∇dbl2\n4/lessorequalslant2δ/lessorequalslantι, (5.101)\nsupt∈IT/ba∇dbl(η,∂1η,u)/ba∇dbl2\n4/lessorequalslantc4I0.\nIn addition, it is easy to check that ( η,u,q) is the unique solution of the transformed MRT\nproblem (1.19) in C0(IT,H1,4\ns)×U4\nT×Q4\nTdue to (5.101). This completes the proof of Proposition\n2.2.\nAppendix A. Analytic tools\nThis appendix is devoted to providing some mathematical results, wh ich have been use in\nprevious sections. It should be noted that Ω appearing in what follow s is still defined by (1.2)\nand we will also use the simplified notations appearing in Section 1.3. In a ddition, Ω T:= Ω×IT\nanda/lessorsimilarbstill denotes a/lessorequalslantcb, however the positive constant cdepends on the parameters and\ndomain in the lemma, in which cappears.\n53Lemma A.1. (1)Embedding inequality (see [1, 4.12 Theorem]): Let D⊂R2be a domain\nsatisfying the cone condition, then\n/ba∇dblf/ba∇dblC0(D)=/ba∇dblf/ba∇dblL∞(D)/lessorsimilar/ba∇dblf/ba∇dblH2(D). (A.1)\n(2)Interpolation inequality in Hj(see [1, 5.2 Theorem]): Let Dbe a domain in R2satisfying\nthe cone condition, then, for any given 0/lessorequalslantj 0. Then\n/ba∇dblw/ba∇dblW1,p(D)/lessorsimilar/ba∇dbl∇w/ba∇dblLp(D) (A.4)\nfor anyw∈W1,p(D)withu/vextendsingle/vextendsingle\nΓ= 0in the sense of trace.\nRemark A.1. By Lemma A.2, we easily deduce that\n/ba∇dblw/ba∇dblW1,p(0,a)/lessorsimilar/ba∇dblw′/ba∇dblLp(0,a)\nfor anyw∈W1,p(0,a) withw(0) = 0 orw(a) = 0. Thus we further have\n/ba∇dbl̟/ba∇dbl0/lessorsimilar/ba∇dbl∂2̟/ba∇dbl0for any̟∈H1\n0:={υ∈H1|υ|∂Ω= 0}. (A.5)\nLemma A.3. Poincar´ e inequality (see [34, Lemma 1.43]): Let 1/lessorequalslantp<∞, andDbe a bounded\nLipchitz domain in Rnforn/greaterorequalslant2or a finite interval in R. Then for any w∈W1,p(D),\n/ba∇dblw/ba∇dblLp(D)/lessorsimilar/ba∇dbl∇w/ba∇dblp\nLp(D)+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nDwdy/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\n. (A.6)\nRemark A.2. By Poincar´ e inequality, we have, for any given i/greaterorequalslant0,\n/ba∇dblw/ba∇dbl1,i/lessorsimilar/ba∇dblw/ba∇dbl2,ifor anywsatisfying∂1w, ∂2\n1w∈Hi. (A.7)\nRemark A.3. By Poincar´ e inequality, we also have\n/ba∇dbl̟/ba∇dbl0/lessorsimilar/ba∇dbl∂1̟/ba∇dbl0for any̟∈H1satisfying̟(y1,y2) =−̟(−y1,y2). (A.8)\nLemma A.4. Hodge-type elliptic estimates: If w∈Hi\nswithi/greaterorequalslant1, then\n/ba∇dbl∇w/ba∇dbli−1/lessorsimilar/ba∇dbl(divw,curlw)/ba∇dbli−1. (A.9)\n54Proof. By a regularity method, we can verify that, for /lessorequalslantj/lessorequalslanti−1,\n/ba∇dbl∂j\n1∇w/ba∇dbl2\n0=/ba∇dbl∂j\n1divw/ba∇dbl2\n0+/ba∇dbl∂j\n1curlw/ba∇dbl2\n0, (A.10)\nwhich yields (A.9) for i= 1.\nNext we further consider the case i/greaterorequalslant2. Since\n∆w=∇divw+∇⊥curlw, (A.11)\nwhere∇⊥:= (−∂2,∂1)T, we derive from (A.10) and (A.11) that, for any 1 /lessorequalslantl+k/lessorequalslanti−1, 1/lessorequalslantk,\n/ba∇dbl∂k+1\n2∂l\n1w/ba∇dbl0=/ba∇dbl∂k−1\n2∂l\n1/parenleftbig\n∆w−∂2\n1w/parenrightbig\n/ba∇dbl0\n/lessorequalslant/ba∇dbl∂k−1\n2∂l\n1(∇divw,∇⊥curlw)/ba∇dbl0+/ba∇dbl∂k−1\n2∂l+2\n1w/ba∇dbl0,\nwhich further yields that\n/ba∇dbl∂k+1\n2∂l\n1w/ba∇dbl0/lessorsimilar/ba∇dbl(divw,curlw)/ba∇dbli−1+/ba∇dbl∂k−1\n2∂l+2\n1w/ba∇dbl0. (A.12)\nBy an induction method, we easily derive from (A.10) and (A.12) that\n/ba∇dbl∂k+1\n2∂l\n1w/ba∇dbl0/lessorsimilar/ba∇dbl(divw,curlw)/ba∇dbli−1,\nwhich, together with (A.10), yields\n/ba∇dbl∇w/ba∇dbli−1/lessorsimilar/ba∇dbl(divw,curlw)/ba∇dbli−1.\nThis completes the proof of Lemma A.4. /square\nLemma A.5. Extension theorem: Let i/greaterorequalslant0,h>0,δ=h/(i+1), andf∈Hi, then there exists\na extension operator Ei\nδsuch that Ei\nδ:f∈Hi→Hi(T×R)such that\nEi\nδ(f) = 0fory2<−δ/2, h+δ/20and for any given s∈IT, there exists a δ>0such that for any t∈ITsatisfying |t−s|<δ,\n/ba∇dblf(ϕ(y,t),s)−f(ϕ(y,s),s)/ba∇dbl0/lessorequalslantε.\nProof. It is well-know that, for any χ∈L2(Rn) withn/greaterorequalslant1,\nlim\nh→0/integraldisplay\nRn|χ(x+h)−χ(x)|dx= 0,\nsee the theorem of continuity in the mean of integral in [49, Theorem 4.21].\nRecalling the proof of the above assertion, we easily see that, for a nyχ∈L2(Rn)∩L∞(Rn)\nwithn/greaterorequalslant1,\nlim\nh→0/integraldisplay\nRn|χ(x+h)−χ(x)|2dx= 0. (A.39)\nThen following the argument of (A.39) with further using the assump tions ofςand the\nembedding inequality (A.1), we easily get desired conclusion in Lemma A.9 . /square\n60Lemma A.10. Some results for functions with values in Banach spaces: Let T >0, integersi,\nj/greaterorequalslant0be given and 1/lessorequalslantp/lessorequalslant∞.\n(1) Assume f∈Lp\nTHi,∂k\n1∇if∈Lp\nTL2for any1/lessorequalslantk/lessorequalslantj, thenf∈Lp\nTHj,i, whereHj,i:=\n{w∈Hi|∂k\n1w∈Hifor any1/lessorequalslantk/lessorequalslantj}.\n(2) LetXbe a separable Banach space and T >0. Ifw∈C0(IT,X), thenw:IT→Xis a\nstrongly measurable function and\n/ba∇dblw/ba∇dblX∈C0(IT). (A.40)\n(3) Letj/greaterorequalslanti+1andcbe a constant. If f∈Lp(IT,Hi)with1

/braceleftBigg/integraltextT\n0/ba∇dblf(t)/ba∇dblp\nHj,idtforp∈[1,∞);\nesssupt∈IT/ba∇dblf/ba∇dblHj,iforp=∞,\nwe immediately get f∈Lp\nTHj,i.\n(2) The conclusion is obvious by Pettis theorem (see Theorem 7 in APP ENDIX E in [7]) and\nthe separability of X. In addition, (A.40) is obvious due to the triangle inequality of norm.\n(3) By Lemma A.5, there exists a function ˜f∈Hi(T×R) such that\n˜f|Ω=f,/ba∇dbl˜f/ba∇dblHi(T×R)/lessorsimilar/ba∇dblf/ba∇dbli.\n62Letε∈(0,1). Then we denote Sε(˜f) the mollifications of ˜fwith respect to the 2D variable\n(y1,y2). Obviously, Sε(˜f)∈Lp\nTHj\nSε(˜f)→fstrongly in Lp\nTHiforp>1, (A.54)\n/ba∇dblSε(˜f)/ba∇dbli/lessorsimilar/ba∇dblf/ba∇dbljfor anyt∈IT,\nwhich implies that\n/ba∇dblSε(˜f)/ba∇dblLp\nTHj/lessorsimilar/ba∇dblg/ba∇dblLp\nTHj.\nThus there exists χ∈Lp\nTHjsuch that\nSε(˜f)⇀fweakly inLp\nTHjforp>1, (A.55)\nwhich, together with (A.54), yields (A.41) for p>1.\nThanks to the above result for p>1, we easily see that (A.41) also holds for p=∞.\n(4)–(a) Since {Dh\n1f}h∈(0,1)is uniformly bounded in L∞\nTHj,i, then, for any multindex αsatis-\nfying|α|=i,\nDτ\n1∂αf ⇀ωweakly-* in L∞\nTL2(a subsequence) .\nThis mean that, for any χ∈H1and for any φ∈C∞\n0(IT),\n−/integraldisplayT\n0/integraldisplay\n∂αf∂1χdyφdt=−lim\nτ→0/integraldisplayT\n0/integraldisplay\n∂αfDτ\n1χdyφdt\n= lim\nτ→0/integraldisplayT\n0/integraldisplay\nDτ\n1∂αfχdyφdt=/integraldisplayT\n0/integraldisplay\nωχdyφdt(a subsequence) . (A.56)\nSinceH1is a separable space, thus we further derive from (A.56) that ω=∂1∂αf=∂α∂1f.\nThus, by the first assertion in Lemma A.10, we get f∈L∞\nTH1,i.\n(4)–(b) If additionally {D−τ\n1Dτ\n1f}|τ|∈(0,1)further is uniformly bounded in L2\nTHiwith respect\ntoτ∈(0,1), then, for any multiindex αsatisfying |α|=i,\nD−τ\n1Dτ\n1∂αf→ψαinL2\nTL2andDτ\n1∇if→ ∇i∂1finL2\nTL2(a subsequence) .(A.57)\nBy(A.57), foranymultiindex αsatisfying |α|=i, foranyχ∈C2(Ω), andforany φ∈C∞\n0(IT),\n−/integraldisplayT\n0/integraldisplay\n∂α∂1f∂1χdyφdt=−lim\nh→0/integraldisplayT\n0/integraldisplay\nDh\n1∂αfDh\n1χdyφdt\n= lim\nh→0/integraldisplayT\n0/integraldisplay\nD−h\n1Dh\n1∂αfχdyφdt=/integraldisplayT\n0/integraldisplay\nψαχdyφdt(a subsequence) .\nBy a density argument, we further derive from the above identity t hat, for any χ∈H1and\nfor anyφ∈C∞\n0(IT),\n−/integraldisplayT\n0/integraldisplay\n∂α∂1f∂1χdyφdt=/integraldisplayT\n0/integraldisplay\nψαχdyφdt,\nwhich implies that ∂2\n1∇if=∇i∂2\n1f∈L2\nTL2. Moreover f∈L2\nTH2,i.\n63(4)–(c) If additionally Dτ\n1∂αfεinC0(IT,L2\nweak) andDτ\n1∂αfεis uniformly continuous in H−1,\nthen we have\nDτ\n1∂αfε→ϑinC0(IT,L2\nweak) (a subsequence) ,\nwhich implies that, for any χ∈L2and for any φ∈C∞\n0(IT),\nlim\nτ→0/integraldisplayT\n0/integraldisplay\nDτ\n1∂αfεχdyφdt=/integraldisplayT\n0/integraldisplay\nϑχdyφdt(a subsequence) (A.58)\nThus we get ϑ=∂α∂1ffrom (A.56) and (A.58).\n(5)–(a) Let us first consider the case f∈C0(IT,Hi). Lets∈ITbe any given. Thanks to\n(A.9), it is easy to check that for any ε>0, there exists a δsuch that for any t∈ITsatisfying\n|t−s|<δ,\n/ba∇dblF(y,t)−F(y,s)/ba∇dbli=/ba∇dblf(ϕ(y,t),t)−f(ϕ(y,s),s)/ba∇dbli/lessorequalslantε.\nThus\nF:=f(ϕ,t)∈C0(IT,Hi). (A.59)\nNow we further consider the case f∈Lp\nTHi. Let˜f=fin ΩTand˜f= 0 outside Ω ×(R\\IT).\nLetSt\nε(˜f) be defined as in (5.19). We can check that St\nε(˜f)∈C0(R,Hi) and\nSt\nε(˜f(x,t))|x=ϕ→F(y,s) strongly in Lp\nTHiforp>1. (A.60)\nIn addition, it is easy to verify that, for a.e. t∈IT,\n/ba∇dblF/ba∇dbli/lessorsimilarP(/ba∇dblς/ba∇dbl3)/ba∇dblf/ba∇dbli. (A.61)\nThus we easily see from (A.59), (A.60) and (A.61) that (A.45) and (A.4 7) hold. Next we turn\nto deriving (A.46) and (A.48).\nThanks to the regularity ς∈C0(IT,H3), we have (after possibly being redefined on a set of\nmeasure zero)\n˜ϕ(˜y) :ΩT→˜ϕ(ΩT) is a homeomorphism mapping , (A.62)\nwhere ˜ϕ(y,t) := (ϕ(y,t),t), please refer to (8.12) and (8.13) in [30]. In particular, for given t,\nϕ−1(y,t)∈C0(Ω) andϕ−1(y,t) : Ω→Ω is a homeomorphism mapping (A.63)\nwhereϕ−1(y,t) denotes the inverse mapping of ϕwith respect to y.\nIt is easy to check that\n∇(ϕ−1) = (∇ϕ)−T|y=ϕ−1. (A.64)\nThanks to Lemma A.9, (A.62), (A.63) and (A.64), similarly to (A.59), we can verify that\nϕ−1∈C0(IT,H3) (A.65)\nThus (A.46) obviously holds by following the argument of (A.59) again.\n64(5)–(b) Let φ∈C∞\n0(IT) andψ∈C∞\n0(Ω). LetSt\nδ, resp.Sεdenote the 1D, resp. 2D mollifiers\nwith respect to variables t, resp. (y1,y2). LetSt\nνis defined as St\nδwithνin place of δ. Then we\ncan compute out that, for sufficiently small δ,ε, andν,\n−/integraldisplayT\n0/integraldisplay\nSt\nδ(Sδ(f(x,t)))|x=y+Stν(ς))ψφtdydt\n=/integraldisplayT\n0/integraldisplay\n(St\nδ(Sε(ft(x,t)))|x=y+Stν(ς)+St\nν(ςt)·St\nδ(Sε(∇f(x,t)))|x=y+Stν(ς))ψφdydt,\nwhich implies that\n−/integraldisplayT\n0/integraldisplay\nFψφtdydt=/integraldisplayT\n0/integraldisplay\n(ft+ςt·∇f)|x=ϕψφdydt.\nNoting that C∞\n0(Ω) is density in L2, thus we have, for any ϕ∈L2and for any φ∈C∞\n0(IT),\n−/integraldisplayT\n0/integraldisplay\nFϕφtdydt=/integraldisplayT\n0/integraldisplay\n(ft+ςt·∇f)|x=ϕϕφdydt,\nwhich immediately implies that Ft= (ft(x,t) +ςt·∇f(x,t))|x=ϕ∈Lp\nTL2. Exploiting the third\nassertion in Lemma A.10 and (A.47), we further have Ft∈Lp\nTHi.\nThanks to the regularity ( ς,ςt), we have (after possibly being redefined on a set of measure\nzero)\n˜ϕ(˜y) :ΩT→˜ϕ(ΩT) is a homeomorphism mapping ,\n˜ϕ(˜y) : ΩT→˜ϕ(ΩT) is aC1-diffeomorphic mapping ,\nwhere ˜ϕ(y,t) := (ϕ(y,t),t), please refer to (8.12) and (8.13) in [30]. Moreover,\n˜ϕ−1(˜x) = (ϕ−1(y,t),t),∇˜x˜ϕ−1= (∇˜y˜ϕ)−1|˜y=˜ϕ−1,\nwhere ˜x= (x,t). In particular, we compute out that\n∂tϕ−1=−((∇ϕ)−1ςt)|y=ϕ−1, (A.66)\nThus we immediately get (A.50) by (A.65) and (A.66). This completes th e proof. /square\nAcknowledgements. 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Zeng, Almost global solutions to the three-dimensio nal isentropic inviscid flows with damping\nin a physical vacuum around barenlatt solutions, Arch. Rati on. Mech. Anal. 239 (2021) 553–597.\n[49] M.Q. Zou, Real function theory (in chinese), Beijing Un iversity Publisher, Beijing, 2004.\n67" }, { "title": "1701.08771v1.Torsional_Alfvén_resonances_as_an_efficient_damping_mechanism_for_non_radial_oscillations_in_red_giant_stars.pdf", "content": "arXiv:1701.08771v1 [astro-ph.SR] 30 Jan 2017Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 7 November 2018 (MN L ATEX style file v2.2)\nTorsional Alfv´ en resonances as an efficient damping\nmechanism for non-radial oscillations in red giant stars\nShyeh Tjing Loi⋆and John C. B. Papaloizou †\nDepartment of Applied Mathematics and Theoretical Physics , University of Cambridge, Centre for Mathematical Science s,\nWilberforce Road, Cambridge CB3 0WA, UK\nLast compiled: 7 November 2018\nABSTRACT\nStars are self-gravitating fluids in which pressure, buoyancy, rot ation and magnetic\nfields provide the restoring forces for global modes of oscillation. P ressure and buoy-\nancy energetically dominate, while rotation and magnetism are gener ally assumed to\nbe weak perturbations and often ignored. However, observation s of anomalously weak\ndipole mode amplitudes in red giant stars suggest that a substantial fraction of these\nare subject to an additional source of damping localised to their cor e region, with in-\ndirect evidence pointing to the role of a deeply buried magnetic field. I t is also known\nthat in many instances the gravity-mode character of affected mo des is preserved, but\nso far no effective damping mechanism has been proposed that acco mmodates this\naspect. Here we present such a mechanism, which damps the oscillat ions of stars har-\nbouringmagnetisedcoresviaresonantinteractionswith standingA lfv´ enmodesofhigh\nharmonic index. The damping rates produced by this mechanism are q uantitatively\non par with those associated with turbulent convection, and in the r ange required\nto explain observations, for realistic stellar models and magnetic field strengths. Our\nresults suggest that magnetic fields can provide an efficient means o f damping stel-\nlar oscillations without needing to disrupt the internal structure of the modes, and\nlay the groundwork for an extension of the theory of global stellar oscillations that\nincorporates these effects.\nKey words: stars: oscillations — stars: magnetic field — stars: interiors — metho ds:\nanalytical — MHD\n1 INTRODUCTION\nSurface convection in many stars stochastically excites\nglobal oscillations (normal modes), which can be de-\ntected through the intensity fluctuations associated with\ntemperature variations induced at the stellar surface\n(Houdek & Dupret 2015). These normal modes can be re-\ngarded as standing superpositions of waves associated with\nrestoringforces producedbypressure,buoyancy,theCorio lis\nforce (in the presence of rotation) and the Lorentz force (if\nthe star harbours a magnetic field). Pressure and buoyancy\neffects dominate energetically over those produced by rota-\ntion and magnetic fields, and so to a first approximation one\nidentifies in the asymptotic limit of high and low frequen-\ncies two types of modes: p-modes, restored mainly by pres-\nsure and associated with large surface displacements; and\ng-modes, restored mainly by buoyancy and associated with\n⋆E-mail: stl36@cam.ac.uk\n†E-mail: jcbp2@damtp.cam.ac.uklarge interior displacements(Deubner & Gough1984).Inre-\nality, and particularly in the case of evolved stars, modes a re\nnot purely one type or the other but have mixed character,\nexhibiting large fluid displacements both near the surface\nand in the deep interior (Osaki 1975).\nThe fluid displacement field ξ(r,t) at the point with po-\nsition vector rassociated with a normal mode of oscillation\ncan be described by a spatial amplitude function modulated\nby a time-harmonic component exp( −iωt). If rotation and\nmagnetic fields are weak, which is the case for the vast ma-\njority of stars, then there is negligible departure of the st el-\nlar background from spherical symmetry. This allows one\nto expand the spatial part in terms of vectorial spherical\nharmonics, i.e. the overall fluid displacement can be writte n\nξ(r,t) = [ξrYm\nℓˆr+ξh∇Ym\nℓ+ξTˆr×∇Ym\nℓ]exp(−iωt),(1)\nwhere we adopt spherical polar coordinates ( r,θ,φ) and\nthere is an implicit summation over spherical harmonics\nYm\nℓ(θ,φ). Radial dependencies are captured solely by the\nscalar functions ξr(r),ξh(r) andξT(r), which describe dis-\nplacements in three mutually orthogonal directions for giv en\nc/circlecop†rt0000 RAS2S. T. Loi and J. C. B. Papaloizou\nℓandm. The first two terms on the RHS of Eq. (1) are col-\nlectively referred to as the spheroidal component, while th e\nthird (involving ξT) is the torsional component. Mathemat-\nically, spheroidal motions are those for which ( ∇×ξ)r= 0,\nthe subscript rdenoting the radial component. Physically,\nthese are motions that involve deformation but no twist.\nTorsional motions have ∇ ·ξ= 0, and correspond to mo-\ntions that involve twist but no deformation.\nIn the absence of rotation and magnetic fields, one can\nshow from the fluid equations of motion that for ω∝negationslash= 0,\nξT= 0, implying that pressure and buoyancy are only capa-\nble of restoring spheroidal motions. However, it is possibl e to\naccess thethirdspatial degreeoffreedom(torsional motio ns)\nin the presence of rotation and/or magnetic fields. In this\nwork we ignore rotation. Our aim is to investigate the dy-\nnamical consequences of interactions between torsional mo -\ntions restored by the Lorentz force with the usual spheroida l\n(i.e., p- and g-) modes. We deal with the limit where the\nmagnetic field is sufficiently weak that it does not disrupt\nthe structure of the spheroidal modes. We find that reso-\nnant interactions between the two types of modes can pro-\nvide an efficient energy sink for spheroidal motions. This\nsource of damping may be potentially important for explain-\ning the anomalously low amplitudes of non-radial (particu-\nlarly dipole) modes observed in some evolved stars.\nThe existence of red giant stars exhibiting low ampli-\ntudes of their dipole ( ℓ= 1) modes was first reported by\nMosser et al. (2012a), accounting for roughly 20% of their\nsample. For the remainder, the higher amplitudes of their\nℓ= 1modes are consistent with theprimary source ofdamp-\ning being convection alone. Given that the red giant popu-\nlation appears to be divided into those with either high or\nlowℓ= 1 mode amplitudes, with relatively few intermediate\ncases, we shall refer to this as the dipole dichotomy problem.\nWhile the frequencies of the low-amplitude ℓ= 1 modes are\nclose to those predicted by the usual asymptotic relation\n(Tassoul 1980; Gough 1986) obeyed by the remainder of the\nsample (Mosser et al. 2011, 2012b), their widths are consid-\nerably larger (Garc´ ıa et al. 2014), suggesting that the ℓ= 1\nmodes of these stars are subject to an additional source of\ndamping. Follow-up analyses by Stello et al. (2016) estab-\nlished that a dichotomy also exists for the ℓ= 2 modes,\nbut to a lesser extent than ℓ= 1. Radial ( ℓ= 0) modes\nappear to be unaffected. As argued by Garc´ ıa et al. (2014),\nsources of damping such as turbulent viscosity localised to\nthe convective envelope should affect all low-degree modes\nto a similar extent, and so to selectively affect non-radial\nmodes the extra source of damping needs to be localised to\nthe core. A further piece of evidence is that the behaviour\nis mass-dependent (Mosser et al. 2012a), being restricted t o\nstars more massive than 1.1M ⊙(Stello et al. 2016). This is\nroughly the threshold mass above which stars on the main\nsequence possess convective rather than radiative cores.\nConvective regions of stars are strongly associated with\ndynamo action and therefore the existence of a magnetic\nfield (Proctor & Gilbert 1994; Charbonneau & MacGregor\n2001). Numerical simulations suggest that convective core\ndynamos in massive stars may generate field strengths of\n10–100kG or more (Brun et al. 2005; Featherstone et al.\n2009). The long timescales of magnetic diffusion in stel-\nlar cores, which greatly exceed nuclear timescales, sug-\ngest that after the dynamo ceases at the end of the star’smain sequence life the field should relax into a long-\nlived equilibrium state if sufficiently large-scale magneti c\nstructure can be retained during the cessation phase. Al-\nthough there have been many previous works investigat-\ning the effects of magnetic fields on stellar oscillations\n(e.g., Campbell & Papaloizou 1986; Cunha & Gough 2000;\nRincon & Rieutord 2003; Reese et al. 2004; Lee 2007), these\nare not directly applicable here as they have mainly been\nconcerned with cases where the magnetic field of interest\nextends beyond the star and is dynamically significant only\nin a thin layer near the surface. Prior to the discovery of\nthedipole dichotomyproblem, the existence of core-confine d\nfields, though not generally disputed, was not considered to\ngive rise to observable consequences. Until recently, very lit-\ntle attention has been paid to their possible influence on\nstellar oscillations.\nThe link to main-sequence dynamo action led to sug-\ngestions that the mechanism behind the dipole dichotomy\nmight involve a deeply buried magnetic field (Garc´ ıa et al.\n2014). Follow-up theoretical work by Fuller et al. (2015) an d\nLecoanet et al. (2016) has established that if the magnetic\nfield strength exceeds a critical threshold, complete con-\nversion of gravity waves to magnetoacoustic waves occurs,\nwhich then dissipate within the core (damping processes as-\nsociated with conversion between different wave modes have\nbeen previously been investigated mainly in the context of\nthe solar atmosphere, e.g. Spruit & Bogdan (1992)). This\nacts to selectively damp non-radial modes, while also imply -\ningthat modes having the character of g-modes could not be\nconstructed (only pure p-modes could exist). An additional\nprediction is that affected p-modes should be magnetically\nsplit toan extent comparable to g-mode period spacings and\nrotational splittings (Cantiello et al. 2016). However, mo re\ndetailed analyses of the observational data performed by\nMosser et al. (2016) indicate that (i) additional splitting of\nthis sort is not seen, (ii) the measured mode amplitudes are\ninconsistent with total energy conversion, and (iii) in man y\ninstances the mixed character of the modes is retained. The\ncurrent consensus is that an independent mechanism is re-\nquired to explain the existence of stars possessing mixed\nmodes with weak amplitudes, and where the amplitude de-\npression is only partial.\nIn this work we present a new mechanism for damp-\ning spheroidal modes involving resonant interactions with\ntorsional Alfv´ en modes localised to the magnetised core (a\nsimilar idea was briefly speculated on by Reese et al. (2004)\nin the context of roAp stars, but this has not been pursued\nfurther in any context). No critical field strength is neces-\nsary: our mechanism is capable of operating in the regime\nbelow the threshold required by Fuller et al. (2015). More-\nover, no disruption to the structure of the spheroidal modes\nis required, implying that the mixed character of modes can\nbe retained. Damping rates are a function of several param-\neters including the field strength, so in general one expects\nonly partial energy loss. In Section 2 we describe the back-\nground stellar model and magnetic field configuration used.\nIn Section 3 we explain the details of the damping mech-\nanism, and present quantitative results for the applicatio n\nof this to a 2M ⊙red giant model in Section 4. We discuss\nobservational consequences and limitations in Section 5. W e\nconclude in Section 6.\nc/circlecop†rt0000 RAS, MNRAS 000, 000–000Magnetic damping of stellar oscillations 3\n0 5\nr (m)×10901234m (kg)×1030\n0 1 2\nr (m)×1080246T (K)×107\n0 1 2\nr (m)×10801234p (Pa)×1019\n0 1 2\nr (m)×10802468ρ (kg/m3)×107\n0 5\nr (m)×10911.21.41.61.8Γ1\n0 1 2\nr (m)×10800.020.040.060.080.1N (Hz)\n0 5\nr (m)×109024681012c (m/s)×105\n0 1 2\nr (m)×10800.020.040.060.080.1Sl (Hz)l = 0\nl = 1\nl = 2\nl = 30 5\n×10705×1029\nFigure 1. From left to right, top to bottom: plots of the enclosed mass, temperature, pressure, density, adiabatic index, buoyanc y\nfrequency, soundspeed and Lamb frequency profiles,fora2 M⊙redgiantstellarmodel(generated byCESAM).Note thatthe t emperature,\npressure, density and buoyancy/Lamb frequency plots are on ly shown up to 5% of the stellar radius. In the plot of enclosed mass (top\nleft), an inset plot zooming in to 1% of the stellar radius has been included to illustrate the high central mass concentra tion.\n2 MODELS\n2.1 Red giant stellar model\nWe will illustrate our mechanism in the context of a 2M ⊙\nred giant model whose background profiles were generated\nby the CESAM (Code d’Evolution Stellaire Adaptatif et\nModulaire) stellar evolutionary code (Morel 1997). We ob-\ntained the parameter grids from an online source1. The var-\nious background quantities are shown plotted in Fig. 1. The\nage ofthemodelis 963Myr,whenthestaris atanintermedi-\nate position in its ascent along the red giant branch (RGB).\nAt this stage the radius of the star is 7.7R ⊙and the dynam-\nical timescale (given by/radicalbig\nR3∗/GM∗, whereR∗andM∗are\nthe stellar radius and mass) is 6.8hr. Although our proposed\nmechanism is quite general, we have chosen to demonstrate\nit using a reasonably realistic stellar model to obtain mean -\ningful estimates of the damping rates for comparison with\nobservation.\n2.2 Magnetic field configuration\nFollowing cessation of convective fluid motions and there-\nfore the dynamo at the end of the main sequence, the\nmagnetic field is expected to relax rapidly into an equi-\nlibrium state (note that the Lorentz force during dynamo\n1https://www.astro.up.pt/helas/stars/cesam/A/data/operation is strong enough to influence the velocity field)\n(Braithwaite & Spruit 2015). The timescale of relaxation is\nthe Alfv´ en travel time across the core, which can be as short\nas 1yr (for a core diameter of Rc∼100Mm,B∼10kG and\nρ∼100gcm−3). Where they exist in nature, fields in non-\nconvective regions should therefore obey the force-balanc e\ncondition\n∇p+ρ∇Φ =1\nµ0(∇×B)×B, (2)\nwherep,ρ, Φ andBare the pressure, density, gravitational\npotential and magnetic field, respectively. If that were not\nthe case then they would evolve towards such a state on the\nAlfv´ en timescale.\nTo model a physically realistic equilibrium field that\nmight be found in a red giant core, one seeks a solu-\ntion to Eq. (2) that (i) is spatially confined, (ii) is fi-\nnite throughout, (iii) is continuous at the boundary in-\nterior to which the field is confined, and (iv) is stable.\nThe third condition is needed to avoid infinite current\nsheets on the boundary. It has been shown that purely\npoloidal and purely toroidal fields are unstable (Tayler 197 3;\nMarkey & Tayler 1973; Flowers & Ruderman 1977), and so\nfor stability, magnetic equilibria necessarily involve a m ix-\nture of poloidal and toroidal components. Numerical stud-\nies, which find that random initial fields tend to settle into\nmixed poloidal-toroidal configurations with roughly compa -\nrable strengths of the two components, support this no-\nc/circlecop†rt0000 RAS, MNRAS 000, 000–0004S. T. Loi and J. C. B. Papaloizou\ntion (Braithwaite & Spruit 2004; Braithwaite & Nordlund\n2006). Notably, this means that a simple dipole field is in-\nadequate for our purposes, since it violates all four criter ia.\nMore generally, it can be shown that magnetic fields which\nare force free throughout, spatially confined and continu-\nous on the boundary must vanish identically (Roberts 1967;\nBraithwaite & Spruit 2015), and so we require a non-force\nfree configuration.\nEarly analytic work by Prendergast (1956) successfully\nobtained an axisymmetric, mixed poloidal-toroidal soluti on\nsatisfying the first three criteria. Though originally deri ved\nfor incompressible stars, its extension tocompressibilit y pro-\nducesaqualitativelysimilar result(Braithwaite & Nordlu nd\n2006; Duez & Mathis 2010). Although we do not check for\nstability of this configuration for the red giant model con-\nsidered here, its stability for n= 3 polytropes has previ-\nously been verified numerically (Duez et al. 2010). In addi-\ntion we expectthatconfinedfieldswith poloidal andtoroidal\ncomponents of comparable strength should in general be\nadequately characterised by the configuration we employ.\nTo obtain this field solution, one begins by introducing the\npoloidal flux function ψ(r,θ), in terms of which an axisym-\nmetric magnetic field B= (BR,Bφ,Bz) may be written\nB=1\nR∇ψ׈φ+Bφˆφ. (3)\nNote that ( R,φ,z) will be used to refer to cylindrical polar\ncoordinates, while ( r,θ,φ) are spherical polar coordinates\n(theφcoordinate is the same for each and refers to the\nazimuthal direction). Physically, ψis a flux surface label,\nmeaning that poloidal projections of the field lines are the\nlevel surfaces of ψ. Substituting Eq. (3) into Eq. (2) and\napplyingtheappropriatevectoridentities, onecanshowth at\nthe quantity F≡RBφis invariant on flux surfaces, i.e. F=\nF(ψ). If in addition we assume a barotropic configuration,\nwe arrive at a nonlinear PDE known as the Grad-Shafranov\nequation:\n∆∗ψ+FdF\ndψ=−µ0ρR2G, (4)\nwhere\n∆∗≡∂2\n∂R2+∂2\n∂z2−1\nR∂\n∂R\n≡∂2\n∂r2+(1−µ2)∂2\n∂µ2, (5)\nµ≡cosθ, andG=G(ψ) is another flux surface invari-\nant. Neither the functional forms of ForGare constrained\nwithin ideal MHD (different choices correspond to different\nequilibria), and so following Prendergast (1956), a simpli fy-\ning choice is to set F(��) =λψandG(ψ) =−β/µ0, whereλ\nandβare constants. Separating ψ(r,θ) = Ψ(r)sin2θturns\nEq. (4) into\nΨ′′−/parenleftbigg2\nr2−λ2/parenrightbigg\nΨ =βρr2, (6)\nwhich is an inhomogeneous, second-order ODE that can be\nsolved using the method of Green’s functions. Applying the\nboundary conditions Ψ(0) = 0, Ψ( r1) = 0 and Ψ′(r1) = 0,\nwherer1is the boundary of the field region, this producesthe result\nΨ(r) =βλr\nj1(λr1)/bracketleftbigg\nf(r,r1;λ)/integraldisplayr\n0ρ(ξ)ξ3j1(λξ)dξ\n+j1(λr)/integraldisplayr1\nrρ(ξ)ξ3f(ξ,r1;λ)dξ/bracketrightbigg\n,(7)\nwhere\nf(ξ1,ξ2;λ)≡j1(λξ2)y1(λξ1)−j1(λξ1)y1(λξ2),(8)\nj1andy1are spherical Bessel functions of the first and sec-\nond kind, and λis a root of\n/integraldisplayr1\n0ρ(ξ)ξ3j1(λξ)dξ= 0. (9)\nThe oscillatory nature of j1means that more than one pos-\nsible value of λmay satisfy Eq. (9); we chose to use the\nsmallest one. We have set r1= 0.005R∗(the inferred bound-\nary of what used to be the convective core) on the basis of\ninspection of the stellar profiles. This corresponds to a mas s\ncoordinate of 0.11 M∗. For theρ(r) profile of the red giant,\nwe obtainλ= 2643.2R−1\n∗, yielding comparable poloidal and\ntoroidal fieldstrengths (the maximumvalues of each ofthese\ndiffer by less than a per cent).\nThe components of Bin terms of Ψ are given in spher-\nical polar coordinates by\nBr(r,θ) =2\nr2Ψ(r)cosθ,\nBθ(r,θ) =−1\nrΨ′(r)sinθ, (10)\nBφ(r,θ) =−λ\nrΨ(r)sinθ.\nThe corresponding field configuration, which we will refer to\nasPrendergast’s solution , is displayed in Fig. 2.\nPrendergast’s solution qualitatively resembles a dipole\nin its angular dependence, but unlike a dipole field possesse s\nno singularity, has a mixed poloidal-toroidal topology, an d\nvanishes smoothly at the spherical boundary r=r1in all\nthree components of B. Beyond this radius we set B=0,\ni.e. we neglect the envelope field under the assumption that\nthis is much weaker than the core field. The overall strength\nof the Prendergast field is controlled through the parameter\nβ, which sets the amplitude of the field but not its shape.\nInspection of the CESAM grid of models for this star shows\nthat the core contracts by roughly a factor of 10 in radius\nfrom 403 Myr (on the main sequence) to 963 Myr. Under\nconservation of magnetic flux, central field strengths shoul d\nincrease by a factor of about 100. If magnetic field strengths\non themain sequence were 10–100 kG,then one expectsfield\nstrengths in the red giant core to be of order several MG.\nWe have set βsuch that the central field strength is 4 MG.\n3 ALFV ´EN RESONANCE DAMPING\nMECHANISM\nIn this section we present a mechanism for damping\nspheroidal modes through interaction with an embedded\nmagnetic field, illustrating this for a red giant containing\na Prendergast field in the core. This section is structured\nas follows. First, we isolate eigenmodes of oscillation tha t\nin the limit of a weak magnetic field are purely torsional\nc/circlecop†rt0000 RAS, MNRAS 000, 000–000Magnetic damping of stellar oscillations 5\nR (R*)z (R*)\n \n01234\nx 10−3−4−3−2−101234x 10−3\n00.005 0.010.015 0.02012345\nr (R*)N2r/g\n00.005 0.010.015 0.020510x 106\nr (R*)ρ/〈ρ〉\nFigure 2. The Prendergast magnetic field solution (top) calcu-\nlated over the assumed core region (0.005 of the stellar radi us).\nThis region is shown shaded in the bottom two panels, which pl ot\nthe dimensionless squared buoyancy frequency and mass dens ity\nprofiles near the centre of the star. In the top panel, a select ion\nof magnetic flux surfaces (poloidal field loops) are shown as b lack\nlines, while the underlying colour represents the strength of the\ntoroidal component. The absolute scaling of the field at this stage\nisarbitrary;illustrated here isjustthe overall geometry . Note that\nthe solution is axisymmetric and so only a meridional half-p lane\nneeds to be shown.in nature (e.g. Mestel 2012) and correspond to standing\nAlfv´ en waves localised to the field region (Section 3.1). We\nshow that these couple to the spheroidal modes through the\nLorentz force (Section 3.2). We then incorporate viscous an d\nOhmic dissipation andshow that this produces adampingof\nthe torsional modes, implying that the torsional problem is\none of a driven-damped mechanical oscillator (Section 3.3) .\nUnder resonant conditions, the rates of driving and dissi-\npation for such a system exactly balance. In the context of\nthe stellar problem, this means that where resonances be-\ntween spheroidal and torsional modes exist, the energy diss i-\npated equals the work done against the Lorentz force by the\nspheroidal motions. Integrating this over the star, we arri ve\nat an analytical expression for the overall damping rate γof\na spheroidal mode (Section 3.4). Throughout this work we\nassume linearity of the fluid motions, and specialise to the\ncase of axisymmetric modes.\n3.1 Torsional Alfv´ enic oscillations\nThe equation of motion of a driven oscillator can be written\n∂2ξ\n∂t2+L[ξ] =S(r,t), (11)\nwhereξdenotes mechanical displacement, Lis a linear op-\nerator containing the spatial derivatives of ξand the source\ntermSrepresents the external driving/forcing. Here Sis re-\ngarded as being external because it does not depend on ξ,\nalthough it may be a function of position rand timet. The\nnormal modes of the oscillator are the solutions of Eq. (11)\nwithS=0(the homogeneous problem). Imposing a time-\nharmonic dependence ξ∝exp(−iωt), this corresponds to\nthe eigenproblem L[ξ] =ω2ξ, satisfied for only special val-\nues ofω2=ω2\n0. These are the natural frequencies of the\noscillator, and the associated forms of ξare the eigenfunc-\ntions of the system.\nThe fluid equation of motion in the absence of rotation\ncan be written\nρ/parenleftbigg∂u\n∂t+u·∇u/parenrightbigg\n=−∇p−ρ∇Φ−1\n2∇B2+(B·∇)B,\n(12)\nwhereu=Dξ/Dtis the fluid velocity and we have absorbed\nthe usualµ0factors into the definition of B. The last two\nterms on the RHS of Eq. (12) correspond to the magnetic\npressure and tension, respectively. We here adopt the Cowl-\ning approximation under which the gravitational potential\nis fixed and depends only on r. Then upon linearising and\ntaking the curl of Eq. (12) the r-components of the first\nthree terms on the RHS vanish, leaving magnetic tension as\nthe only force capable of restoring torsional motions. In th e\naxisymmetric case, which we focus on here, the torsional di-\nrection corresponds to the φ(azimuthal) direction (we com-\nmenton thenon-axisymmetric case inSection 5.2). Consider\nnow the torsional component of Eq. (12), which linearises to\ngive\nρ0∂2ξφ\n∂t2=B0\nR·∇(RB′\nφ)+B′\nR·∇(RB0φ).(13)\nSubscript 0’s denote static background quantities, while\nprimes denote (small) time-dependent perturbations about\nthe background. Using the linearised induction equation\nc/circlecop†rt0000 RAS, MNRAS 000, 000–0006S. T. Loi and J. C. B. Papaloizou\nB′=∇×(ξ×B0), this allows us toexpress Eq. (13) in terms\nof justξand background quantities. This can be written in\nthe form\n∂2ξφ\n∂t2+LT[ξφ] =fTS\nρ0, (14)\nwhere\nLT[ξφ] =−B0\nρ0R·∇/bracketleftbigg\nR2B0·∇/parenleftbiggξφ\nR/parenrightbigg/bracketrightbigg\n, (15)\nfTS=−B0\nR·∇/bracketleftbigg\nR2ξ·/parenleftbiggB0φ\nR/parenrightbigg\n+RB0φ(∇·ξ)/bracketrightbigg\n+1\nR[∇×(ξ×B0)]·∇(RB0φ).(16)\nSee thatfTSdepends only on the spheroidal displacement\nξS≡(ξR,0,ξz), notξφ, and can thus be regarded as a forc-\ning term in Eq. (14) provided that the spheroidal displace-\nment is assumed known, which will effectively be the case\nwhen the magnetic field is weak. In that case the spheroidal\nmodes will be relatively unperturbed.\nLet us examine the operator LTmore closely. Although\nat first glance this appears to depend on three spatial di-\nmensions, notice that B0· ∇=Bp∂/∂s, whereBpis the\nmagnitude of the poloidal component of B, andsis arc\nlength (i.e. physical distance) along the poloidal project ions\nof the field lines. Hence the problem is intrinsically one-\ndimensional. Rescaling to a new distance coordinate σobey-\ning dσ/ds= 1/(R2Bp), the eigenproblem reduces to\n∂2ηφ\n∂t2=v2\nA∂2ηφ\n∂σ2, (17)\nwhereηφ≡ξφ/Ris a new scaled fluid displacement\nandv2\nA≡1/(ρ0R4). One recognises Eq. (17) as the one-\ndimensional wave equation with spatially varying advectio n\nspeedvA=vA(σ), which can be identified as the Alfv´ en\nspeed with respect to the new coordinates. From here it be-\ncomes more convenient to work in terms of ηφrather than\nξφ. The particular form of Eq. (17) allows the solutions to\nbe understood intuitively as standing waves on stretched\n1D loops. These are quantised vibrations whose frequencies\nincrease as the spatial scale decreases.\nWe solved Eq. (17) as a matrix eigenvalue problem on a\ndiscrete 1D grid for each flux surface, with periodic bound-\nary conditions and spatial derivatives approximated by cen -\ntred differences. We calculated the eigenmodes Xj(σ,ψ) and\neigenfrequencies ω2\n0,j(ψ) on 1000 evenly-spaced (in ψ) flux\nsurfaces with 5000 uniformly-spaced (in σ) points on each\nsurface. Here j∈Z+is the harmonic index. Although the\ntotal number of eigenfunctions obtainable by this method\nequals the number of grid points, the accuracy of the so-\nlutions is expected to degrade for larger ω2\n0,jwhere spatial\nscales of the associated eigenfunctions become too small to\nbe adequately resolved. On each flux surface we restricted\nthe eigenfunctions used in further analysis to the 1000 hav-\ning the lowest eigenfrequencies.\nThe parameter βwas chosen so as to produce the distri-\nbution of the Alfv´ en speed, vA, shown in the upper panel of\nFig. 3. A selected eigenmode is illustrated in the lower pane l\nof Fig. 3. Since the problem is axisymmetric, the spatial am-\nplitude function need only be displayed on a poloidal field\nloop, corresponding to a longitudinal slice of the flux sur-\nface. The full solution is obtained by sweeping each loop in\nR (R*)z (R*)\n \n01234\nx 10−3−4−3−2−101234x 10−3\n0.511.522.5x 10−4\n-2\n2-1\n40z (R*)×10-3\n12\nηφ×10-3×10-3\nR (R*)0 2\n-20\nFigure 3. Spatial distribution of the Alfv´ en speed overlaid with\nan arbitrary field loop (top), and the amplitude function for the\nj= 7 eigenmode on that loop (bottom). Colour bar units are in\nterms of the dynamical speed/radicalbig\nGM∗/R∗. In the bottom panel,\nthe equilibrium position of the field line is shown in black an d\nthe displaced position in red. Arrows are an aid to visualisi ng the\ndirection of the displacement.\na circle about the axis of symmetry (here the z-axis). The\nmotion can be envisaged as segments of each flux surface\n(which are tori in 3D) twisting with respect to others. For\na given wave speed vA, one expects ω0,jfor fixedjto in-\ncrease as the length of the field loop shrinks. This is indeed\nobserved in our model: the spatial distribution of ω0,jfor\nj= 300 is shown in Fig. 4.\nFor agivenfluxsurface, one alsoexpects ω0,jtoincrease\nwithj. Only an even number of nodes is allowed for vibra-\ntions on a loop, so j= 1 has zero nodes, j= 2,3 have two,\nj= 4,5havefour, andsoon. Hence jis roughlyproportional\nto the number of wavelengths around the loop, but there is\na paired structure to the spectrum. This can be seen in the\nc/circlecop†rt0000 RAS, MNRAS 000, 000–000Magnetic damping of stellar oscillations 7\nFigure 4. Spatial distribution of the j= 300 eigenfrequen-\ncies. Colour bar units are in terms of the dynamical frequenc y/radicalbig\nGM∗/R3∗. Since eigenmodes are localised to individual flux sur-\nfaces,ω0,jis constant on any flux surface for given j. Smaller flux\nsurfaces tend to have higher ω0,jfor fixedj.\ninset toFig. 5, which plots ω0,jversusjfor aselection of flux\nsurfaces. Note that despite having equal numbers of nodes,\neachpair still correspond todistinct eigenmodes, thesebe ing\nodd and even versions of one another (e.g. for a constant vA\nthey would be the sine and cosine solutions), with slightly\ndifferenteigenfrequencies. Notethatsimilar behaviour of the\nperiodic solutions of the Mathieu equation occurs. The over -\nall slope of ω0,jversusjshould also be proportional to the\nfundamental frequency (larger for smaller loops, in line wi th\nthe picture of a vibrating string). As can be seen in Fig. 5,\nthis is indeed the case.\n3.2 Coupling with spheroidal motions\nThe perturbation to the Lorentz force can be subdivided\ninto terms that depend on the spheroidal displacement ξS\nbut not the torsional displacement ξφ, and the terms that\ndepend onξφbut notξS(this is possible because only terms\nlinear in ξare retained). Writing this out in components, we\ncan express this separation as\nfS(ξ) =fSS(ξS)+fST(ξφ),\nfT(ξ) =fTS(ξS)+fTT(ξφ), (18)\nwherefSandfTand the spheroidal and torsional compo-\nnents to the Lorentz force perturbation, fSSare the terms in\nfSthat depend only on ξS,fSTare those that depend only\nonξφ, etc. This allows us to observe the following coupled0 200 400 600 800 1000051015202530\njω0,j1902002107.27.47.67.888.28.4\nFigure 5. The torsional spectrum calculated for 10 evenly-spaced\n(inψ) flux surfaces. Each track corresponds to one flux surface,\nand flux surfaces of lower tracks enclose those of higher ones .\nAlthough apparently continuous, the tracks are in fact made up\nof discrete points, since j∈Z+. The discreteness can be seen in\nthe inset plot, which zooms in to a small portion of the overal l\nspectrum. The paired structure reflects approximately dege nerate\nmodes, which have equal numbers of nodes but are odd and even\nversions of one another. Frequencies are given in units of th e\ndynamical frequency,/radicalbig\nGM∗/R3∗.\nstructure of the equations of motion:\n∂2ξS\n∂t2+LS[ξS] =fST(ξφ)\nρ0(19)\n∂2ξφ\n∂t2+LT[ξφ] =fTS(ξS)\nρ0, (20)\nwhere\nLS[ξS] =1\nρ0/bracketleftbig\n∇p′+ρ′∇Φ0−fSS(ξS)/bracketrightbig\n,(21)\nLT[ξφ] =−fTT(ξφ)\nρ0. (22)\nWe see that the coupling between spheroidal and tor-\nsional motions is provided by the Lorentz force. To first or-\nder, given that the Lorentz force is much smaller than the\nforces of pressure and buoyancy, the fSSterm in Eq. (21)\ncan be neglected. This is akin to assuming that the mag-\nnetic field has negligible effect on the spheroidal eigensolu -\ntion (i.e. we still get the usual p- and g-modes). An impor-\ntant term not included explicitly in Eq. (19) is the forcing\nassociated with (purely spheroidal) convective motions, t he\nsource of energy for the whole system. Given that the cou-\npling from spheroidal motions into torsional motions and\nback into spheroidal motions is a second-order process, the\ndirect contribution of convection should dominate over fST\nin the spheroidal equation of motion when magnetic fields\nare weak (the coupling strength scales like the magnetic\npressureB2, which is far smaller than the gas pressure). We\nshall thusneglect all magnetic terms in Eq. (19). Incontras t,\nthe Lorentz force has first-order significance in providing\nboth the driving and the restoration of torsional motions,\nsince there are no pressure or buoyancy forces to compete\nwith, and cannot be neglected in Eq. (20).\nc/circlecop†rt0000 RAS, MNRAS 000, 000–0008S. T. Loi and J. C. B. Papaloizou\nWith magnetic terms neglected, finding the spheroidal\neigenmodes reduces to the standard hydrodynamic problem\noflinearadiabaticstellar oscillations. IntheCowlingap prox-\nimation (i.e. neglecting the perturbation to the gravitati onal\npotential), this involves solving the following second-or der\nsystem of ODEs:\ndξr\ndr=−/parenleftbigg2\nr−1\nΓ1Hp/parenrightbigg\nξr+1\nρ0c2s/parenleftbiggS2\nℓ\nω2−1/parenrightbigg\np′\ndp′\ndr=ρ0(ω2−N2)ξr−1\nΓ1Hpp′, (23)\nwhich can be achieved by standard numerical techniques for\nODE eigenvalue problems. We did this by first interpolating\nthe CESAM profiles onto a finer grid (to capture the small\nspatial scales of g-mode oscillations) and then solving Eqs\n(23) using the shooting method. The quantities Γ 1,Hp,cs,\nSℓandNcharacterise thestellarbackgroundandcorrespond\nto the adiabatic index, pressure scale height, sound speed,\nLamb frequency and buoyancy frequency, respectively. The\nhorizontal component ξhof the fluid displacement is related\ntop′through\nξh=p′\nrω2ρ0. (24)\nWe conducted a near-exhaustive search for all\nspheroidal eigenmodes between ω= 1 and 20 (expressed as\na multiple of the dynamical frequency/radicalbig\nGM∗/R3∗), refin-\ning this near the locations of p-dominated modes (of which\none exists per radial order), for spherical harmonic degree s\nℓ= 0, 1, 2 and 3. The objective was to selectively extract\nthe modes that are observable experimentally, which are re-\nstricted to those with low ℓ(due to geometric cancellation\neffects),strongp-modecharacter (associated withlarger s ur-\nface motions and therefore intensity variations) and locat ed\nwithin several radial orders of ω= 10 (typical frequency\nof maximum excitation for solar-like oscillators). Figure 6\nshows several ℓ= 0 modes and one ℓ= 1 mode found by\nthe search. The ℓ= 0 modes (top left) are oscillatory only\nnear the surface, while the ℓ= 1 mode (top right and bot-\ntom right) has significant mixed character and is oscillator y\nboth near the surface and near the centre. Mixing occurs\nalso forℓ= 2 and 3, but due to the weaker coupling for\nhigherℓ, the modes have purer p- or g-like character com-\npared toℓ= 1. Radial orders n(bottom left) were computed\nusing the Eckart scheme (Eckart 1960; Scuflaire 1974; Osaki\n1975), where the convention is to count p-type (g-type) ra-\ndial crossings positively (negatively). The large negativ e val-\nues for the ℓ >0 modes indicate that these have a large\nnumber (hundreds) of oscillations in the g-mode cavity.\nRecall that from the point of view of the torsional equa-\ntion of motion, spheroidal fluid motions act as a forcing\nfunction through their associated Lorentz force. In genera l,\nif the forcing applied to a mechanical oscillator contains\none or more frequencies that match its natural frequencies,\nthen resonant excitation occurs. The strength of the excita -\ntion depends on the geometric similarity between the forcin g\nfunction at the resonant frequencies and the corresponding\neigenmodes, which can be quantified as the coefficients of\nthe eigenfunction expansion of the forcing function. Figur e\n7 shows the spatial distribution of fTSnear the core for the\nmode shown on the right of Fig. 6. Overlaid in black is the\nsame field loop shown in Fig. 3. The fine-scale oscillations of0 0.5 1−2−1012x 10−6\nr (R*)\n0 0.5 1\nx 10−3−2−1012x 10−6\nr (R*)0 0.5 1−3−2−1012\nr (R*)ξr\n \nn = 1\nn = 2\nn = 3\nn = 4\n−1000 −500 005101520\nnωS\n \nl = 0\nl = 1\nl = 2\nl = 3\nFigure 6. A selection of eigenmodes and eigenfrequencies for the\nstellar model examined here, showing the four lowest-order radial\nmodes (top left), and an ℓ= 1 mixed mode near ω= 10 (top\nright) with the central regions shown enlarged on the bottom\nright so that the g-type oscillations can be seen. Red and bla ck\nin the two righthand plots correspond to horizontal and radi al\nfluid displacements ξhandξr, respectively. Note that the scaling\nofξis arbitrary. The frequencies (expressed as a multiple of th e\ndynamical frequency) of the first four lowest spherical degr ees are\nplotted versus radial order on the bottom left.\nξSproduce corresponding fine-scale oscillations of fTS(ξS).\nThis illustrates how the excitation of high harmonics by low -\ndegree modes can occur: in general the field lines cut across\nmany radial shells, enabling large nto map to large j.\nThe torsional equation of motion in terms of the scaled\nfluid displacement ηφcan be written\n∂2ηφ\n∂t2−1\nρ0R4∂2ηφ\n∂σ2=Fφ, (25)\nwhereFφ=fTS/ρ0R. We now wish to derive an expression\nfor the expansion coefficients of Fφ(the spheroidal forcing)\nwith respect to the torsional eigenmodes Xjidentified in\nSection 3.1, i.e. the quantities ajin\nFφ(σ,ψ) =/summationdisplay\njaj(ψ)Xj(σ,ψ). (26)\nFirst, we need to establish the orthogonality relation for\nXj. Substituting the eigensolution ω2\n0,j,Xjinto the homo-\ngeneous form of Eq. (25), we have\n−ρ0R4ω2\n0,jXj=∂2Xj\n∂σ2. (27)\nIntegrating twice by parts and applying periodic boundary\nconditions, one can show that\n/contintegraldisplay\nX∗\nk∂2Xj\n∂σ2dσ=/contintegraldisplay\nXj∂2X∗\nk\n∂σ2dσ, (28)\nwhere the integral is around a closed field loop. Multiplying\nEq. (27) by X∗\nkand integrating, and using Eq. (28), we find\nc/circlecop†rt0000 RAS, MNRAS 000, 000–000Magnetic damping of stellar oscillations 9\nFigure 7. Spatial distribution of fTS(the torsional component\nof the Lorentz force associated with spheroidal motions) fo r the\nmixed mode shown on the right of Fig. 6, overlaid with the flux\nsurface shown in Fig. 3. Only the region near the centre is sho wn.\nFluid displacements have been normalised so that the total e n-\nergy of the mode is unity. One sees that there is a cross-cut of the\nfield loop across many radialnodes (sign changes) of fTS, suggest-\ning that this should preferentially excite high-index harm onics on\nthat loop.\nthat\n/parenleftbig\nω2\n0,j−ω2\n0,k/parenrightbig/contintegraldisplay\nρ0R4XjX∗\nkdσ= 0, (29)\nwhich implies that unless ω2\n0,j=ω2\n0,k, it must be that/contintegraltext\nρ0R4XjX∗\nkdσ= 0. It is possible to normalise the Xjso\nthat/contintegraltext\nρ0R4XjX∗\njdσ= 1. Doing so, we arrive at the desired\nexpression\naj=/contintegraldisplay\nR3fTSX∗\njdσ. (30)\nThe values of ajfor the spheroidal mode shown in Fig. 7\nand each of the torsional modes are plotted in Fig. 8. The\nspheroidal displacements have been normalised such that\nthe total energy E(Unno et al. 1989) equals unity, i.e.\nE=ω2/integraldisplay\nρ0r2/bracketleftbig\nξ2\nr(r)+ℓ(ℓ+1)ξ2\nh(r)/bracketrightbig\ndr= 1.(31)\nAs a comment, the values obtained for ajfor the current\nmodel are substantially lower than the maximum physically\nallowed values, which would occur under conditions of com-\nplete geometric overlap (perfect constructive interferen ce).\nAs an order-of-magnitude estimate, the upper bound on aj\nis given by R1/2\ncB3/2ξL−2, whereRcis the size of the core,\nξis the characteristic fluid displacement and Lis the char-\nacteristic length scale of variation in ξ. For our red giant\nmodel, substituting appropriate values for these paramete rs\nFigure 8. The torsional spectrum computed for500 flux surfaces.\nPoints are coloured according to the strength of coupling wi th\nthe spheroidal mode whose Lorentz force distribution is sho wn\nin Fig. 7. The coupling strength is quantified as |aj|, the abso-\nlute value of the coefficient of the eigenfunction expansion ( see\nEq. (30)).\nyields a physical upper bound on ajof the order 104. As can\nbe seen from Fig. 8, the actual values obtained are aj∼100.\nAlso apparent from Fig. 8is that the torsional spectrum\nis very dense. In fact, for every value of j∈Z+there is a\ncontinuumof ω0,jvalues, reflecting the existence of a contin-\nuum of flux surfaces. However, the number of resonances is\nfinite due to the discrete nature of j, and at a given ωequals\nthejrange intersected by a horizontal line at that value of\nω. For the spheroidal mode shown in Fig. 7, which is near\nω= 10, we identify around 900 resonances, i.e. there exist\nthis number of magnetic flux surfaces which have a torsional\nmode with this eigenfrequency.\n3.3 Dissipative effects\nWe shall now incorporate dissipative effects, with the goal\nof eluciating their role in contributing to the damping of th e\ntorsional oscillations. In the following derivation it wil l be\nassumed that the damping coefficients in units of the dy-\nnamical frequency are much less than unity (highly under-\ndamped oscillator), so that the eigenfrequencies and eigen -\nmodes derived above remain valid. At the end of this section\nwe evaluate the expression for the damping coefficient ob-\ntained and verify that it is indeed small.\nWith dissipative terms included, the momentum and\ninduction equations are\n∂2ξ\n∂t2+L[ξ]−ν∇2∂ξ\n∂t= 0 (32)\n∂B′\n∂t−νm∇2B′=∇×/parenleftbigg∂ξ\n∂t×B0/parenrightbigg\n,(33)\nwhereνandνmare the viscous and Ohmic dissipation co-\nefficients, and L[ξ] refers to the linearised form of the RHS\nof Eq. (12). Anticipating small scales we have retained only\nthe highest order derivatives in the viscous force and rate\nof Ohmic diffusion. Invoking a time-harmonic separation,\nEq. (33) becomes an inhomogeneous Helmholtz equation\nc/circlecop†rt0000 RAS, MNRAS 000, 000–00010S. T. Loi and J. C. B. Papaloizou\nwhich can be solved by means of an integration kernel to\nyieldB′(r)≈ ∇×(¯ξ×B0), where\n¯ξ(r) =/integraldisplay/integraldisplay/integraldisplay\nK(r−r′)ξ(r′)d3r′, (34)\nK(r) =iω\n4πνm|r|exp/bracketleftbigg\n−(1−i)/radicalbiggω\n2νm|r|/bracketrightbigg\n.(35)\nHere we have assumed that ξvaries much more rapidly over\nspace than B0, which itself varies on a scale much larger\nthan/radicalbig\nνm/ω. As far as the Lorentz force is concerned the\neffect of dissipation is thus to replace ξby¯ξ, which are re-\nlated through ξ=¯ξ−(νm/iω)∇2¯ξ. Substituting this into\nEq. (32), neglecting products of νandνm(given that both\nare small), defining ¯ ηφ≡¯ξφ/Rand retaining only diffu-\nsive terms involving second order spatial derivatives, the\ntorsional component can be written\n∂2¯ηφ\n∂t2−1\nρ0R4∂2¯ηφ\n∂σ2−νtot∂2\n∂n2∂¯ηφ\n∂t=Fφ,(36)\nwhereνtot=ν+νmis the total dissipation coefficient and\nnis the direction normal to the flux surfaces. The reason\nfor retaining only this spatial part of the Laplacian is that\nthe finest-scale structure is likely to develop in the direc-\ntion perpendicular rather than parallel to the flux surfaces ,\nas a result of phase mixing. This refers to the decorrelation\nof oscillations on adjacent surfaces having slightly differ ent\neigenfrequencies, which occurs on a timescale correspond-\ning to the inverse of their frequency difference (cf. the beat\nphenomenon). In the case of our model, the decorrelation\ntimescale associated withthespatialscale alongfluxsurfa ces\n(∼10−5R∗) is only ∼10 dynamical times, muchshorter than\nthe dissipation timescale associated with the same length\nscale (∼106dynamical times). Decorrelation will therefore\nproceed down to much smaller scales before its development\nis halted by viscous/resistive effects.\nWeseek asolution tothecoefficients bjoftheeigenfunc-\ntion expansion ¯ ηφ(t,σ,ψ) =/summationtext\njbj(ψ)Xj(σ,ψ)e−iωt. Noting\nthat∂/∂n=RBp∂/∂ψ, we get\n/bracketleftbig\nω2\n0,j−ω2/bracketrightbig\nbj(ψ)+ iνtotωR2B2\np∂2bj(ψ)\n∂ψ2=aj(ψ).(37)\nLet us focus on a small region in ψnear a resonant sur-\nfaceψ0whosej-th harmonic is of frequency ω0,j(ψ0) =ω.\nLocally, we adopt the Taylor expansion ω2\n0,j(ψ)≈ω2+\n2ωω′\n0,j(ψ0)[ψ−ψ0]. Consider the change of variable x=\nC[ψ−ψ0] whereC= (2ω′\n0,j(ψ0)/R2B2\np)1/3, which turns\nEq. (37) into\nbjx+ iνtot∂2bj\n∂x2=Caj(ψ0)\n2ωω′\n0,j(ψ0). (38)\nThe RHS, which we will call A, can be regarded as roughly\nconstantovertheresonantlayer.Equation(38)canbesolve d\nusing Fourier transforms ( x→kandbj(x)→˜bj(k)), yield-\ning\n˜bj(k) =/braceleftBigg\n0 k>0\niAexp[k3νtot/3]k<0(39)\nIn the limit νtot→0,˜bj/iAtends to 1 −H(k) whereH(k)\nis the Heaviside step function. This satisfies i F[1−H(k)] =\n1/(x−i0). Here Fdenotes a Fourier transform and i0 is\nan infinitesimal imaginary component that we identify withthe damping contribution iΓ ω(cf. the Landau prescription\nfrom plasma physics). The final solution for bjis then\nbj(ψ) =aj(ψ0)\nω2\n0,j(ψ)−ω2−iΓω. (40)\nSince the RHS of Eq. (38) is approximately constant, we in-\nfer the characteristic scale to be x∼ν1/3\ntot. The local damp-\ning rate can be estimated from the first-order term of the\nTaylor expansion of ω2\n0,j(ψ), and has the approximate ex-\npression Γ ∼[2ω′\n0,j(ψ0)RBp]2/3ν1/3\ntot. Up to a factor of order\nunity, Γ turns out to be the inverse of the timescale required\nfor decorrelation to occur over the width of the resonant\nlayer. Further inspection reveals that this width is precis ely\nthat for which the timescales of decorrelation and dissipa-\ntion are equal, providing the physical interpretation for t he\nassociated loss process as being closely linked to phase mix -\ning. Given the small magnetic Prandtl numbers in stellar\ninteriors,νtotis dominated by the Ohmic dissipation coeffi-\ncientνm≈109T−3/2m2s−1(Spitzer 1962). For our model\n(T∼107K) we find that Γ ∼10−3inverse dynamical times,\nand that the widths of the resonant layers are ∼10−7R∗.\n3.4 Overall damping rates\nA major objective of this work is to estimate the overall\ndamping rate γof spheroidal modes due to the resonant\ncoupling with torsional modes. We will now combine results\nfrom preceding sections to arrive at an expression for γ.\nThe total rate of work done by the torsional component\nof the Lorentz force associated with spheroidal motions is\ndE\ndt=/integraldisplay/integraldisplay/integraldisplay/parenleftbigg\nfTS∂ξ∗\nφ\n∂t+f∗\nTS∂ξφ\n∂t/parenrightbigg\nd3r\n= 4πRe/bracketleftbigg/integraldisplay/integraldisplay\nρ0R4Fφ∂η∗\nφ\n∂tdσdψ/bracketrightbigg\n.(41)\nInvoking the eigenfunction expansions for Fφandηφ, elim-\ninatingbjin favour of ajusing Eq. (40), making use of the\northonormality relation for Xjand averaging over one oscil-\nlation period, we obtain the time-averaged rate of work\n/angbracketleftbiggdE\ndt/angbracketrightbigg\n≈4π/summationdisplay\nj|aj(ψ0)|2/integraldisplayΓ\nh(ψ)+Γ2dψ, (42)\nwhere\nh(ψ)≡/parenleftbiggω2\n0,j(ψ)\nω−ω/parenrightbigg2\n. (43)\nThis assumes that ajvaries slowly over the width of the\nresonant region and can be approximated by its value at the\nresonant surface ψ=ψ0whereω0,j(ψ0) =ω. In the limit of\nsmallΓtheresonantregion isspatially narrow,andsowecan\napproximate h(ψ) by the first term of its Taylor expansion\naboutψ0. This allows us to straightforwardly evaluate the\nψ-integral in Eq. (42). We arrive at the final expression\n/angbracketleftbiggdE\ndt/angbracketrightbigg\n= 2π2/summationdisplay\nj|aj(ψ0)|2/parenleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingledω0,j\ndψ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nψ0/parenrightBigg−1\n,(44)\nwhich we see is independent of the local damping coefficient\nΓ. This reflects a basic property of driven-damped oscilla-\ntors that near a resonance, in the limit of weak dissipation\nc/circlecop†rt0000 RAS, MNRAS 000, 000–000Magnetic damping of stellar oscillations 11\n(regardless of what this may be or how it physically arises),\nthe system always adjusts itself so that the rate of driving\nand dissipation are in balance. The global damping rate of\na spheroidal mode is then γ=∝angb∇acketleftdE/dt∝angb∇acket∇ight/E, whereE, given\nin Eq. (31), is the total energy of the mode. If we were to\nnormalise the fluid displacements such that E= 1, thenγ\nis simply given by Eq. (44).\n4 RESULTS\n4.1 Application: damping red giant oscillations\nThe summation in Eq. (44) for any given spheroidal mode\nhaving frequency ωis over all torsional modes (indexed by\nj) whose frequencies equal this. Our approach to evaluat-\ning this was to scan over the 1000 ×1000 torsional modes\ncomputed and saved for the ones whose ω0,jlay closest to\nthe target frequency. If the difference fell below a certain\nthreshold (taken to be 0.1 frequency units), this was classe d\nas a resonance and the contribution of the mode was added\nto the sum. The typical ωspacing between the torsional\nmodes in our set was ∼0.01 frequency units, so variations\nabout the threshold of 0.1 units affected the results very lit -\ntle. The purpose of the threshold was to restrict the j-range\nof modes resonant at a given ωto be as close to the true\nrange as possible, without losing modes comfortably within\nthe true range. A further point to be made is that every j\nvalue was scanned independently, which restricts the resul t-\ning set of modes to one per jbut not one per flux surface.\nAt the low resolution of the grid (1000 flux surfaces) com-\npared to the j-range of the spectrum (of order this value,\nas can be seen from Fig. 8), there are instances where a\nflux surface contributes more than one mode to the sum.\nThough not strictly realistic, it is inevitable with this di s-\ncretised approach that the true resonant surface for given\njis approximated by one close by, and this may be shared\nbetween more than one j. Given the slow variation in the\nstructure of the torsional modes with space, however, this i s\nunlikely to give rise to systematic errors. Our quantitativ e\nresults are impacted more heavily by the fact that only a\nfinite number of modes were generated, since this excludes\nsome number of potentially resonant modes (implications\nare discussed further in Section 5.2).\nFigure 9 plots the damping times tdamp≡γ−1calcu-\nlated for the p-dominated modes of the red giant model de-\nscribed in Section 2. The field strength is scaled such that\nthecentralvalueis 4MG, inline with expectationsfrom sim-\nulations of main sequence core dynamos (Brun et al. 2005)\nand magnetic flux conservation during core contraction into\nthe red giant stage. We find that damping times for the\nℓ= 0 modes are typically in excess of 1023days, many or-\nders of magnitude longer than for the ℓ >0 modes, which\nare damped on timescales of months through this mecha-\nnism. The distinguishing feature between radial and non-\nradial modes which is likely to explain this is the structure\nof the spatial amplitude function within the core. For ra-\ndial modes, spatial oscillations occur only near the surfac e;\nnear the core the amplitude function follows a very smooth\nexponential decay. This results in very tiny ajvalues for\nthe torsional modes (which are localised to the core) having\nfrequencies near ω= 10. In contrast, the spatial amplitude567891011121314151022102310241025tdamp (d)\n \n56789101112131415101102103104105106107\nωStdamp (d)\n \nl = 1\nl = 2\nl = 3l = 0\nFigure 9. Damping times for the most p-dominated modes of the\nfour lowest-degree spherical harmonics. Radial modes are s hown\non a separate plot (top) since their damping times differ grea tly\nfrom the non-radial modes (bottom). For non-radial modes, a t\nfrequencies below a certain threshold (here ω≈11), one sees\nthat lower degrees experience stronger damping. Note the lo ga-\nrithmic scale on the vertical axis. The unit of ωSis the dynamical\nfrequency, while tdampis expressed in days.\nfunctions of non-radial modes near ω= 10 have hundreds\nof finely-spaced oscillations within the core, and so are abl e\nto strongly overlap with torsional modes having jvalues of\nseveral hundred.Atthefieldstrengths expectedfor red gian t\ncores, it turns out that the eigenfrequencies associated wi th\ntorsional modes having j∼100 lie in the vicinity of ω= 10,\nmaking resonances between spheroidal and torsional modes\npossible. (For ρ∼105gcm−3,B∼1MG andRc∼10Mm\nwe havevA∼10ms−1, implying a fundamental Alfv´ en fre-\nquency of ∼1µHz. Spheroidal modes in red giants are ex-\ncited near ∼100µHz.)\n4.2 Dependence on spherical harmonic degree\nFor the three non-radial degrees examined we find system-\natic differences in the characteristic damping times, these\ndiffering byroughly an order of magnitude between adjacent\nℓ. At the field strength considered, damping times below a\ncertain threshold frequency (occurring e.g. near ω= 11 for\nℓ= 1; this “step” feature is commented on more below) are\n101–102days forℓ= 1, 102–103days forℓ= 2, and 103–105\ndays forℓ= 3. Particularly for ℓ= 1 these damping times\nare comparable to those associated with turbulent convec-\ntion (10–30 days), which suggests that dampingthrough res-\nonant interactions with Alfv´ en modes should impact mode\namplitudes at an observable level. These would be most pro-\nnounced for ℓ= 1, followed by ℓ= 2 and then ℓ= 3.\nThe physical reason for the dependence of damping\nrates onℓis likely to be the variation in the strength of\ncoupling between the p- and g-mode cavities, which deter-\nc/circlecop†rt0000 RAS, MNRAS 000, 000–00012S. T. Loi and J. C. B. Papaloizou\n6789101112131410−1410−1210−1010−8\nωSMl\nFigure 10. Mode inertia as a function of frequency for the\nspheroidal modes of the red giant model. Black, red, green an d\nblue correspond to ℓ= 0, 1, 2 and 3, respectively. Though the\nspectrum is discrete, individual points (one for each mode) have\nbeen joined by straight lines to aid visualisation of the pat tern.\nAsterisks mark the most p-dominated modes, identified as loc al\nminima in the red, green and blue curves. These are the ones fo r\nwhich damping times have been computed and plotted in Fig. 9.\nmines the extent of mode mixing. For a fixed frequency the\ng-mode cavity is of the same size regardless of ℓ, but the\np-mode cavity is larger for smaller ℓ. Consequently mode\nmixing is most effective for ℓ= 1, followed by ℓ= 2, then\nℓ= 3, and so on for higher multipoles. One measure of the\ng-like character of a mode is the mode inertia\nMℓ=/integraltext\nρ0r2/bracketleftbig\nξ2\nr(r)+ℓ(ℓ+1)ξ2\nh(r)/bracketrightbig\ndr\nξ2r(R∗)+ℓ(ℓ+1)ξ2\nh(R∗),(45)\nwhich gives an indication of the amount of mass displaced\nby the mode. Modes with smaller (larger) inertia are more\np-like (g-like), preferentially localised to the envelope (core)\nwhere densities are lower (higher). The plot of Mℓversus\nmode frequency (see Fig. 10) exhibits periodic modulations\nonce per radial order (one unit of ω), where the modes hav-\ning minimum Mℓare the most p-dominated ones. Obser-\nvations favour the detection of p-dominated modes because\nthese give rise to larger fluid motions at the surface, and so\nwe selectively consider these modes here. These are marked\nwith asterisks in Fig. 10. One can see that the ℓ= 1 p-\ndominated modes (red asterisks) have the largest inertia\nand thus the most g-like character compared to other ℓ.\nThe larger core fluid displacements associated with the p-\ndominated ℓ= 1 modes enhances their rate of damping due\nto interactions with the torsional Alfv´ en modes, compared\nwith the p-dominated modes of higher ℓ(green and blue\nasterisks).\n4.3 Scaling with field strength and core size\nFor a fixed core size Rc, damping rates associated with this\nmechanism are expected to be smaller for weaker magnetic\nfields. We have fTS∝B2,Xj∝B1/2and/contintegraltext\ndσ∝B−1, so\nif we were to ignore changes in ajassociated with details\nof the mode geometries, then from Eq. (30) we infer that\naj∝B3/2. In addition the number of available resonancesat givenωis inversely proportional to B, and|dω0,j/dψ|\nis independent of B, so from Eq. (44) this implies that\nγ∝B2. The increase in γwithB, which appears to be\ndriventhrough the ajdependence, arises physically from the\nincreased couplingbetween spheroidal andtorsional motio ns\nwhen the Lorentz force is stronger. While the above scaling\nargument is fairly simplistic, it does appear to be the case\nnumerically that γincreases with increasing B(doubling the\nfield strength decreases the characteristic damping times b y\na factor of several).\nAlso of interest is the predicted variation of γwithRc.\nAs a star ascends the red giant branch its core contracts and\nenvelope expands. One expects the damping rate of modes\nnear the frequency of maximum excitation νmax(which is it-\nself subject to variation) to change accordingly. To estima te\nthis we will assume conservation of mass and magnetic flux,\nsothatρ0∝R−3\ncandB∝R−2\nc.Inthissituation fTS∝R−5\nc,/contintegraltext\ndσ∝RcandXj∝R−1\nc, soaj∝R−2\nc. We also need to\naccount for changes in the torsional eigenfrequencies: the se\ngo asω0,j∼vA/RcwherevA∼B/√ρ0∝R−1/2\nc, so both\nω0,jand|dω0,j/dψ| ∝R−3/2\nc.\nTo predict the dependence of γonRc, it remains to\ndetermine the effect of changing the frequency of excita-\ntion, which enters into γthrough/summationtext\nj, the summation over\nresonant modes. As can be seen from Fig. 8, the torsional\neigenspectrum fills a wedge in ( ω0,j,j)-space. The numberof\nresonances at frequency ωequals the range in jintersected\nby a horizontal line placed at that frequency. If all ω0,jare\nboosted by a given factor, then the wedge is stretched up-\nward leaving a proportionally smaller j-intersection range\nat the location of a fixed horizontal line. Likewise, if the li ne\nrepresenting ωis shifted downwards by some factor, then\nthej-range intersected decreases proportionally. Inspection\nof the CESAM models for this star shows that between 963\nand 1021Myr, R∗roughly quadruples while Rchalves, sug-\ngesting that empirically R∗∝R−2\nc. We know that νmax∝\nR−3/2\n∗, so putting all this together,/summationtext\nj∝νmax/ω0,j∝R9/2\nc.\nTogether with the |aj|2|dω0,j/dψ|−1∝R−5/2\ncdependence\nof the contribution from each mode, this leads to γ∝R2\ncas\nthe predicted scaling dependence for spheroidal modes near\nνmax. We thus predict that dampingrates should decrease as\nthe core contracts. The dominating influence here is the re-\nduction in the numberof available resonances, which falls o ff\nmore quickly than the strength of spheroidal-torsional cou -\npling grows (acting alone, the latter would tend to drive up\nγthrough the increase in Band therefore ajasRcshrinks).\n5 DISCUSSION\n5.1 Comparison with observations\n5.1.1 Impact on mode visibilities\nMode visibilities (i.e. amplitudes) vℓ, which are a measure\nof the area under the peaks in the power spectra associated\nwith a given ℓ, are usually expressed in a form where they\nare normalised with respect to some other quantity. If this\nis with respect to the area under ℓ= 0 peaks, then for ex-\namplev1≈1.54 (Ballot et al. 2011). In the context of the\ndipole dichotomy problem, following Mosser et al. (2016), i t\nis more convenient to normalise with respect to the area un-\nc/circlecop†rt0000 RAS, MNRAS 000, 000–000Magnetic damping of stellar oscillations 13\nder theℓ= 1 peaks of stars which fall in the high-amplitude\n(“normal”) group. Normal stars thus have v1≈1. For a\nfixed rate of excitation, the visibility of a mode decreases\nproportionally with the overall damping rate, and so the\nlatter definition of v1allows it to be expressed in terms of\nthe envelope- and core-associated damping rates γeandγ\nas\nv1=γe\nγe+γ=tdamp\nte+tdamp, (46)\nwherete≡γ−1\ne. In practice, tecan be measured from the\nlinewidths of the radial modes and is characteristically 15 ±\n5days (see Mosser et al. 2016, fig. 3b)\nAs can be seen from fig. 3a of Mosser et al. (2016), the\nlow-amplitude group of red giants have v1values close to 0.1\nfor stars with dynamical frequencies ∆ νnear 12µHz, andv1\nvalues around 0.7 for stars with ∆ νnear 4µHz, with some\nscatter about this trend. Let us take teto be 15days. From\nFig. 9 it appears that tdamplies between 30 and 50 days\nforℓ= 1, producing v1of 0.67–0.77. Increasing the field\nstrength increases the damping and lowers the visibilities :\nat triple the field strength, tdampvalues would be roughly\nnine-fold lower producing v1of 0.18–0.27. The full range of\nobserved visibilities for stars with depressed ℓ= 1 modes\ncan thus be accounted for through modest variations of the\nfield strength. For ℓ= 2,tdampvalues are about an order\nof magnitude greater than ℓ= 1, placing v2(defined in an\nanalogous manner) at around 0.93–0.97. At triple the field\nstrength,v2would be in the range 0.60–0.88. The damping\ntimes computed for ℓ= 3 are about an order of magni-\ntude larger still, which would give v3values of 0.98–0.99\n(this would be much more difficult, if not impossible, to de-\ntect compared to ℓ= 1 and 2). Hence the ℓ-dependence\nof the mode amplitude depression observed experimentally\n(Stello et al. 2016) is reproduced by our mechanism.\n5.1.2 Explaining the dichotomy\nGiven that γdepends on B, one possible explanation for\nthe dichotomy in the red giant population is that this re-\nflects a dichotomy in the field strengths. This could be re-\nlated to the stability of magnetic equilibria, where the ini tial\ndynamo-generated field relaxes into one of two or more pos-\nsible states characterised by different equilibrium streng ths\n(e.g. Braithwaite 2008). The helicity (aproximately con-\nserved during relaxation), which in turn depends on the\ndynamo mechanism, may play a role. However, further in-\nvestigation along this line is beyond the scope of this work.\nAnother possibility is that this relates to a property\nof the results not yet discussed in much detail, and that is\nthe dramatic step-like feature present in the ajdistribution.\nThis refers to the broad “shoulder” tracing out a roughly\nhorizontal linepastabout j/greaterorsimilar400nearω= 11inFig. 8.Just\nbelow this line ajvalues are large, but step down by about\ntwoorders ofmagnitude aboveit. This does notappear tobe\nassociated with any particular flux surface; rather, it seem s\nas though each fluxsurface has a jvalue above which ajsud-\ndenly becomes small, and the associated ω0,jis roughly the\nsame for all flux surfaces. We notice that the position of this\nstep migrates downward for spheroidal modes of increasing\nfrequencyωS, and so we suggest that it may be related to\nthe spatial scale Lof the g-mode oscillations, which givena characteristic vAare associated with a certain frequency\nvA/L. AsωSincreases, so does L, which would qualitatively\nexplain the direction of its migration. One could conceive o f\nthis as being related to the condition ωS=vA/Ldescribing\namatch of both the frequencyand spatial scale of spheroidal\nand torsional modes, which gives rise to optimal coupling.\nTorsional modes with smaller spatial scales rapidly become\ndifficult to excite. The step feature can be clearly seen in\nFig. 9 as the strong jump in tdampnearω= 11 forℓ= 1,\nandω= 14 forℓ= 2 (forℓ= 3 this lies slightly off the edge\nof the plot, near ω= 16). Hence there is also a dependence\nonℓof the step location, this being at higher frequencies for\nlargerℓ.\nRegardless of the origin of this feature, if it turns out to\nbe ubiquitous among different stellar models then this could\nexplain the dichotomy as being created by νmaxlying above\nor below the step. If ωS(the frequency of a given spheroidal\nmode)lies belowthestep, ajandhenceγvalues will behigh.\nAsωSincreases, there will come a point where it meets the\nfrequency of the step and ajvalues strongly drop. Modes of\nhigherωSwould be subject to much weaker damping. The\nlocation of the step depends also on the field strength: for\nstronger fields (larger vA) this occurs at higher frequencies.\nThough there may be other factors involved, this means that\nan intrinsic spread of field strengths among the red giant\npopulation could produce the dichotomy. It is worth noting\nthat this picture implies the possibility of stars for which\nthe step lies near νmax, about which modes of several radial\norders are usually detectable. For such stars, one expects t o\nsee low visibilities at frequencies below the step, and high\n(“normal”) visibilities above. Interestingly, such stars have\nindeed been reported in the literature: at least three are\nknown (Garc´ ıa et al. 2014; Mosser et al. 2016).\n5.1.3 Dependence on evolutionary stage\nAs mentioned earlier in this section, observed values of v1\nare noted to increase as the dynamical frequency ∆ νde-\ncreases. Variations in ∆ νare closely tied to evolutionary\nstage, since expansion of the envelope as the star ascends\nthe RGB causes ∆ νto drop. Accompanying the evolution\nis a contraction of the core, which we previously argued in\nSection 4.3 is associated with a drop in γ. The predicted\nscaling is roughly γ∝R2\nc, so that a star with v1≈0.45 at\n∆ν= 12µHz would end up with v1≈0.6 at ∆ν= 4µHz.\nAlthough somewhat shallower than the trend seen observa-\ntionally, a large number of approximations have been used,\nsome of which may be questionable. For example, if it turns\nout that contraction of the core allows the field configura-\ntion to relax further (so that ψis not in fact conserved),\nthis would weaken the dependence of ajonRcand steepen\nthe rise inv1towards smaller ∆ ν. However, it is encourag-\ning that the simple scaling dependencies predicted by our\nmechanism qualitatively reproduce this aspect of the obser -\nvations.\n5.2 Limitations\nThe exact quantitative values presented here are clearly se n-\nsitive to the background model, and although we have only\nconsidered one stellar model and field configuration, we have\nc/circlecop†rt0000 RAS, MNRAS 000, 000–00014S. T. Loi and J. C. B. Papaloizou\nendeavoured to use ones that are as realistic as possible. Th e\ncurrent work serves mainly to illustrate the viability of ou r\nmechanism for producing damping rates that are compa-\nrable to other known sources of damping (e.g. convection),\ngivenreasonable fieldstrengths.Moredetailedinvestigat ions\nof parameter space and the examination of different stellar\nmodels would be the subject of future work.\nThroughout we have considered only axisymmetric\nspheroidal and torsional modes, where the axis of symme-\ntry matches that of the background field. This has been\nconvenient for the purposes of the analytic treatment here.\nWe do not expect the generalisation to non-axisymmetry to\nadversely impact our results. The quality of the geometric\noverlap between the non-radial spheroidal modes and the\ntorsional Alfv´ en modes owes to the cross-cut of field lines\nacross a large number of fine-scale spatial oscillations, wh ich\nfor low-degree spheroidal modes is relatively unaffected by\nangular orientation. The description of torsional modes in\nthe non-axisymmetric case is complicated by the involve-\nment of motions in the poloidal as well as the toroidal di-\nrection. However, in the limit of small poloidal scales and\nsmallm(azimuthal order), it can be shown that the non-\naxisymmetric torsional eigenfunctions closely resemble t hose\nfound for the axisymmetric case in that they are dominated\nbyφrather than θ-displacements. We therefore expect them\nto physically interact with the non-axisymmetric spheroid al\nmodes in a similar way to the axisymmetric case presented\nhere, although the mathematical treatment would be less\ntrivial. A more detailed investigation of non-axisymmetri c\neffects will be deferred to future work.\nWe have assumed linearity of the fluid motions, even\nthough the smallness of the damping coefficient Γ suggests\nthat large limiting amplitudes of torsional motion may be\nattained. If reaching nonlinear amplitudes, one might ex-\npect wave breaking to occur. Further work to investigate\nthe complications of nonlinearity has yet to be performed.\nWe used a particular magnetic field configuration (the\nPrendergast solution) which was convenient to implement\nsince it can be written down in closed form. We acknowledge\nthat it is only one possible solution to the Grad-Shafranov\nequation; other equilibria may be permitted in reality. How -\never the geometry of the field is not particularly impor-\ntant to our mechanism, because for the same characteristic\nAlfv´ en speed, the frequencies of potentially resonant tor -\nsional modes are set by the spatial scale of the g-mode os-\ncillations. This has no bearing on the length or the shape\nof the field loops. Notice that the typical conditions (field\nstrengths, densities etc.) in red giant cores are such that t he\nratio of the Alfv´ en speed to the g-mode wavelength fortu-\nitously coincides with the frequency of maximum excitation\nνmaxof thespheroidal modes. Resonant interactions proceed\neffectively for this reason (an alternate way of understand-\ning this is the requirement that gravity wave phase speeds\nmatch the Alfv´ en speed). This should also be true of differ-\nent field configurations, including ones for which the spatia l\nscale of the field loops may be much smaller, as long as core\nfield strengths are similar across the red giant population.\nThe only other source of damping considered here, be-\nsides that of our proposed mechanism, is that arising from\nconvection. In reality, radiative diffusion also contribut es to\nthe damping of g-mode oscillations. We have done some\nrough estimates of the expected rate of damping from radia-tive diffusion and find that it is several orders of magnitude\nsmaller than that associated with our proposed mechanism\n(forℓ= 1 and 2; they may be on par for ℓ= 3). This is in\nline with previous works, which have determined that radia-\ntive damping is small compared to convective damping for\np-dominated modes and have difficulty accounting for the\nlow dipole mode amplitudes seen red giants (Dupret et al.\n2009; Garc´ ıa et al. 2014); it also offers no explanation for\nthe dichotomy.\nThe primary source of systematic error in our quantita-\ntive estimates of γarises from the finiteness of the grid. This\nimpacts the calculation in two ways. Firstly, the torsional\nmodes considered here were limited to those with j/lessorequalslant1000,\nwhich means that any resonant modes with j >1000 are\nlost to the sum in Eq. (44). Importantly, this means that\nthe damping rates presented here are likely to be system-\naticunderestimates , perhaps by a few tens of per cent. A\nsecond way in which γmay be affected is by inaccuracies in\nthe shapes of the torsional modes when jis comparable to\nthe number of grid points. Tests comparing the eigenspectra\ngenerated under different grid resolutions indicate that in -\naccuracies become substantial for jvalues in excess of about\nhalf the number of grid points (discrepancies in ω0,jof or-\nder unity are reached, when compared to a five-fold increase\nin the number of grid points). If an eigenfunction is not re-\nsolved properly its ajvalues will not be accurate, and it is\nnotstraightforward topredict whethersystematically hig her\nor lower values would be obtained. For the sake of caution,\nwe restricted the largest j-value considered to be one-fifth of\nthe number of grid points. The overall error in our damping\nrates is thus likely to be dominated by the former effect.\nFinally, a key assumption is that the structure of the\nspheroidal modes is unaffectedbythemagnetic field. Clearly\nif they were to be modified significantly in the region of the\ncore, then this would impact our quantitative results. We\nstress that our mechanism, as presented, is designed to op-\nerate in the weak-field regime where this assumption holds.\nThe work of Fuller et al. (2015) and Lecoanet et al. (2016)\nhas suggested that drastic alteration to spheroidal mode\nstructure is to be expected if the field exceeds a critical\nstrength, and so we expect that in this regime our mech-\nanism would break down, or require modification. We there-\nfore address a complementary regime to that of the above\nauthors. On this note, the significance of this work is that\nit demonstrates a new mechanism for damping stellar oscil-\nlations that does not need to disrupt the structure of the\nmodes in any part of the star. This is precisely what is re-\nquired to account for the existence of the weak-amplitude\nmixed modes reported by Mosser et al. (2016).\n6 SUMMARY AND FUTURE WORK\nWe have presented a mechanism for damping spheroidal\nmodes of red giant stars via resonant interactions with tor-\nsional Alfv´ en modes localised to a magnetised core. Quan-\ntitative estimates of the associated damping rates indicat e\nthat these can be comparable to those of envelope-based\nsources.Thismaybeaviableanswertothedipoledichotomy\nproblem in the case of weak core fields, where the structure\nof the modes is expected to be preserved. To our knowledge,\nc/circlecop†rt0000 RAS, MNRAS 000, 000–000Magnetic damping of stellar oscillations 15\nthere is no other mechanism which has been proposed that\nachieves this for the weak-field regime.\nOur mechanism can be summarised as follows. Turbu-\nlent convection in the envelope excites the usual spheroida l\nmodes restored by pressure and buoyancy. If there is a mag-\nnetic field present inside the core, then this allows in addi-\ntion for the existence of torsional oscillations restored b y\nmagnetic tension. These torsional modes can be thought\nof as quantised vibrations on closed field loops (standing\nAlfv´ en waves). Coupling to spheroidal motions proceeds vi a\nthe Lorentz force, since in general the Lorentz force associ -\nated with spheroidal fluid displacements has a component in\nthe torsional direction (and vice versa). Though the funda-\nmental frequencies of torsional modes are small for realist ic\nfield strengths, resonances with spheroidal modes are pos-\nsible through the excitation of high loop harmonics. The\nstrength of the interaction is determined by the quality of\nthe geometric overlap between the two types of modes. In\nred giants, spheroidal eigenmodes have extremely fine-scal e\nradial structure in the core, providing the ability for effici ent\noverlap with high loop harmonics through the cross-cut of\nfield lines across many radial oscillations. Under conditio ns\nof weak but non-zero dissipation, the energy lost to exci-\ntation of torsional resonances equals the work done against\nthe Lorentz force by the spheroidal motions. The singular\nnature of the interaction acts as an energy sink, giving rise\nto a global damping of the spheroidal modes that is larger\nwhen more resonances are available. The observability of\nthis effect relies on the existence of modes that have large\namplitudes both in the core, so that overlaps with torsional\nresonances are significant, and at the surface, so that the\nmode can be observed. Evolved stars meet these conditions\nthrough a strong coupling of their p- and g-mode cavities,\nwhich forms modes of mixed character.\nAs already discussed in Section 5.2, the effects of non-\nlinearity and non-axisymmetry have yet to be dealt with.\nA number of numerical issues also limit the accuracy of\nour quantitative results, which would need to be addressed\nbefore a detailed exploration of parameter space (e.g. ex-\namining mass and age dependencies) is attempted. Also of\ninterest is to investigate the consequences for our mecha-\nnism of modification to the spheroidal mode structure by a\nstrong core field. The recent work of Lecoanet et al. (2016)\nhas presented local calculations of the process, but furthe r\nwork (particularly the formulation of a global description )\nwill be necessary to develop a general theory for the mag-\nnetic damping of stellar oscillations that encompasses bot h\nstrong- and weak-field regimes.\nACKNOWLEDGMENTS\nWe thank Mike Proctor and Henrik Latter for helpful com-\nments and discussions. STL is supported by funding from\nthe Cambridge Australia Trust.\nREFERENCES\nBallot J., Barban C., Van’t Veer-Menneret C., 2011, A&A,\n531, 124\nBraithwaite J., 2008, MNRAS, 386, 1947Braithwaite J., Nordlund A., 2006, A&A, 450, 1077\nBraithwaite J., Spruit H. C., 2004, Nature, 431, 819\nBraithwaite J., SpruitH. C., 2015, Living Reviews, pp 1–57\nBrun A. S., Browning M. K., Toomre J., 2005, ApJ, 629,\n461\nCampbell C. G., Papaloizou J. C. B., 1986, MNRAS, 220,\n577\nCantiello M., Fuller J., Bildsten L., 2016, ApJ, 824, 14\nCharbonneau P., MacGregor K. B., 2001, ApJ, 559, 1094\nCunha M. 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University\nof Tokyo Press\nc/circlecop†rt0000 RAS, MNRAS 000, 000–000" }, { "title": "1607.05878v1.Electromagnon_in_the_Z_type_hexaferrite_____rm_Ba___x___rm_Sr___1_x___3_rm_Co_2Fe__24_O__41__.pdf", "content": "arXiv:1607.05878v1 [cond-mat.mtrl-sci] 20 Jul 2016Electromagnon in the Z-type hexaferrite (BaxSr1−x)3Co2Fe24O41\nFilip Kadlec,1Christelle Kadlec,1Jakub V´ ıt,1,2Fedir Borodavka,1Martin Kempa,1\nJan Prokleˇ ska,3Josef Burˇ s´ ık,4R´ obert Uhreck´ y,4St´ ephane Rols,5Yi Sheng Chai,6\nKun Zhai,6Young Sun,6Jan Drahokoupil,1Veronica Goian,1and Stanislav Kamba1,∗\n1Institute of Physics, Czech Academy of Sciences,\nNa Slovance 2, 182 21 Prague 8, Czech Republic\n2Faculty of Nuclear Science and Physical Engineering,\nCzech Technical University, Bˇ rehov´ a 7, 115 19 Prague 1, Cz ech Republic\n3Department of Condensed Matter Physics, Faculty of Mathema tics and Physics,\nCharles University, Ke Karlovu 5, 121 16 Prague 2, Czech Repu blic\n4Institute of Inorganic Chemistry, Czech Academy of Science s, 250 68 ˇReˇ z, Czech Republic\n5Institut Laue-Langevin, Boˆ ıte Postale 156, 38042 Grenobl e Cedex 9, France\n6Institute of Physics, Chinese Academy of Sciences, Beijing , P. R. China\n(Dated: July 21, 2016)\nWe studied experimentally the high-temperature magnetoel ectric (Ba xSr1−x)3Co2Fe24O41pre-\npared as ceramics ( x= 0, 0.2) and a single crystal ( x= 0.5) using inelastic neutron scattering, THz\ntime-domain, Raman and far-infrared spectroscopies. The s pectra, measured with varying temper-\nature and magnetic field, reveal rich information about the c ollective spin and lattice excitations.\nIn the ceramics, we observed an infrared-active magnon whic h is absent in Eω⊥zpolarized THz\nspectra of the crystal, and we assume that it is an electromag non active in Eω/bardblzpolarized spectra.\nOn heating from 7 to 250K, the frequency of this electromagno n drops from 36 to 25cm−1and its\ndamping gradually increases, so it becomes overdamped at ro om temperature. Applying external\nmagnetic field has a similar effect on the damping and frequenc y of the electromagnon, and the\nmode is no more observable in the THz spectra above 2T, as the t ransverse-conical magnetic struc-\nture transforms into a collinear one. Raman spectra reveal a nother spin excitation with a slightly\ndifferent frequency and much higher damping. Upon applying m agnetic field higher than 3T, in\nthe low-frequency part of the THz spectra, a narrow excitati on appears whose frequency linearly\nincreases with magnetic field. We interpret this feature as t he ferromagnetic resonance.\nI. INTRODUCTION\nThe current research on multiferroic materials is moti-\nvated by both an incomplete understanding of their fun-\ndamental physical properties and their potential in re-\nalizing novel devices which would make use of static or\ndynamic magnetoelectric (ME) coupling in non-volatile\nmemories, spintronics, magnonics and microwave de-\nvices. For practical applications, a strong ME coupling\nnear room temperature is required. BiFeO 3, the most in-\ntensively studied multiferroic, exhibits a large ferroelec-\ntric (FE) polarization and multiferroicity up to 643K.\nHowever, its spiral magnetic structure persists up to an\nexternal magnetic field of 19T and, therefore, the ME\ncoupling is rather low up to 18T.1Much larger ME cou-\npling was observed in spin-order-induced ferroelectrics.2\nUnfortunately, most of these materials exhibit multifer-\nroic properties only below ca. 50–100K. Nevertheless,\nsome ferrites with hexagonal symmetry, called hexafer-\nrites, exhibit magnetic-field-induced electric polarization\nclose to or even far above room temperature and their\nME coupling can be very high.3–5\nBased on their chemical formulas and crystal struc-\ntures, hexaferrites can be classified into several types:\nM-type, such as (Ba ,Sr)CoxTixFe12−2xO19,Y-type\n(Ba,Sr)2Me2Fe12O22,W-type (Ba ,Sr)Me 2Fe16O27,Z-\ntype(Ba ,Sr)3Me2Fe24O41,X-type(Ba ,Sr)2Me2Fe28O46,\nandU-type (Ba ,Sr)4Me2Fe36O60, where Me is a biva-lent metal ion (e.g. Co, Mg, Zn).5,6The structures of\nhexaferrites can be described as sequences of three basic\nbuildingblocks(usuallydenoted by S, R,andT) periodi-\ncallystackedalongthe zaxes. Since the hexagonalstruc-\ntures of hexaferrites are associated with centrosymmet-\nricP63/mmcorR¯3mspace groups, no FE polarization\nshould exist in these materials. Nevertheless, their ferri-\nmagnetic structures are spiralor heliconical and they can\nbeeasilychangedbyexternalmagneticfieldtotransverse\nconical, wherean electric polarizationofthe orderoftens\nofµC/m2can appear due to the inverse Dzyaloshinskii-\nMoriyainteraction( ∝Si×Sj) betweennon-parallelspins\nSiandSj. In this case the centrosymmetric structure\nshould be broken and the crystal symmetry lowered. Im-\nportantly, the magnetic field needed to induce a polariza-\ntion can be very low (of the order of millitesla) and the\neffect can be remanent.4,7By contrast, the polarization\ndisappears in higher magnetic fields (usually above 2T)\nwhen the magnetic structure changes.5\nMEpropertiesoftheZ-typehexaferriteSr 3Co2Fe24O41\nwere reported first time by Kitagawa et al.8They discov-\neredthat this highly resistivematerial exhibits magnetic-\nfield induced electric polarization at least up to 300K.\nME and magnetodielectric effects in Sr 3Co2Fe24O41were\nconfirmed later by other authors.9–14Chunet al.3in-\nvestigated (Ba xSr1−x)3Co2Fe24O41and discovered that\nthe ME effect is the highest in (Ba 0.52Sr2.48)Co2Fe24O41.\nMoreover, for this composition, the magnetic structure2\nchangesfromtransverseconicaltocollinearferrimagnetic\nstructure at a temperature as high as 400K, so the ME\neffect can be measured well above room temperature.3,9\nIn general, depending on the crystal and magnetic\nstructures, the ME coupling may be due to one of three\ndifferent mechanisms: exchange striction (magnetostric-\ntion∝Si·Sj), inverse Dzyaloshinskii-Moriya interaction\nor spin-dependent hybridization of the panddorbitals.2\nThe same mechanisms can be also responsible for the\ndynamic ME coupling, which activates magnons in the\nTHz or far-infrared dielectric permittivity spectra and\ntherefore they are called electromagnons .2,15,16By con-\ntrast, common magnons impact only upon the magnetic\nsusceptibility spectra in the microwave or THz ranges.\nThey are also called ferromagnetic (FMR) and antifer-\nromagnetic resonances, and their frequencies correspond\ntoacoustic-likeandoptic-likemagnons,respectively,with\nwavevectors qfrom the Brillouin-zone center ( q= 0). As\nforelectromagnons,theyhavefrequentlywavevectorsout\nof the Brillouin-zone center ( q∝negationslash= 0) and they can be ex-\ncited by THz photons (with a wavevector q∼0) only\nif the magnetic structure is modulated, which is true in\npractically all spin-order-induced multiferroics. As re-\ngards the hexaferrites, an electromagnon was reported\nonly in the Y-type compound Ba 2Mg2Fe12O22.17–19In-\nterestingly, it was observed not only in the spin-induced\nFE phase below 50K, but also in the paraelectric one at\n90K, if an external magnetic field (0 .4T≤µ0H≤1.6T)\nwasapplied ; in that casethe magneticstructurechanged\nfrom proper screw to longitudinal (for H∝bardblz) or trans-\nverse conical (for H⊥z).18The activation of the elec-\ntromagnon in Eω∝bardblzpolarized spectra was explained by\nthe exchange striction, although the static electric po-\nlarization P⊥zin hexaferrites comes from the inverse\nDzyaloshinskii-Moriya interaction.18\nSince the Z-type hexaferrites (Ba xSr1−x)3Co2Fe24O41\nexhibit ferrimagnetic phase transitions at temperatures\nas high as TN= 700K and ME coupling up to nearly\n400K, one can expect electromagnons to be activated in\ntheir THz spectra at much higher temperatures than in\nY-type hexaferrites. For that reason we have performed\ndetailed THz time-domain, infrared (IR), Raman and\ninelastic neutron scattering (INS) spectroscopic studies\nfrom 5 to 900K on (Ba xSr1−x)3Co2Fe24O41ceramics\nwithx= 0 and 0.2 and a single crystal with x= 0.5.\nThree spin excitations including one electromagnon were\ndiscovered.\nII. SAMPLES AND EXPERIMENTAL DETAILS\nPowders of hexagonal ferrite with a composition\n(BaxSr1−x)3Co2Fe24O41(x= 0 and 0.2) were pre-\npared by the Pechini type in-situ polymerizable com-\nplex method relying on immobilization of metalloorganic\nprecursor complexes in a rigid organic polymer network,\nthus ensuring the compositionalhomogeneityof the com-\nplex oxide. First, calculated amounts of strontium car-bonate (SrCO 3), barium carbonate (BaCO 3), cobalt ni-\ntrate (Co(NO 3)2·6H2O), and iron nitrate (Fe(NO 3)3·\n9H2O; all chemicals from Sigma-Aldrich) were decom-\nposed in a 0.1 mol/l solution of nitric acid in distilled\nwater. After their complete dissolution, a calculated\namount of a polymer gel formed by reaction of citric\nacid (HOOCCH 2C(OH)-(COOH)CH 2COOH) with ethy-\nlene glycol (HOCH 2CH2OH) in water was added to this\nsolution, mixed and heated to 130◦C. With contin-\nued heating over several hours the clear solution became\nhighly viscous, gradually gelled and finally polymerized\ninto avoluminous resin. After breakingthe resin, its dry-\ning (at 150◦C) and charring (at 350◦C) for 24 hrs, the\nresulting powder was heat-treated in an oxygen atmo-\nsphere at 1200◦C for 12 hrs. A powder X-ray diffraction\nmeasurement proved a single-phase composition of the\nproduct. Cold isostatic pressing (pressure 300MPa) and\nsubsequent sintering at 1200◦C in oxygen atmosphere\nwere used to obtain dense ceramics of the Z-phase ferrite.\nSingle-and double-sidepolished pellets with diametersof\n6mm and thicknesses of2 and 0.48mm were preparedfor\nthe IR and THz studies, respectively.\nA (Ba 0.5Sr0.5)3Co2Fe24O41single crystal with a natu-\nral hexagonal plane was grown by the flux method.20It\nhad a diameter of 4–5mm and a thickness of 1.8mm.\nLow-temperature IR reflectivity measurements in the\nfrequencyrange30–670cm−1(or, equivalently, 1–20THz)\nwere performed using a Bruker IFS-113v Fourier-\ntransform IR spectrometer equipped with a liquid-He-\ncooled Si bolometer (1.6K) serving as a detector. Room-\ntemperature mid-IR spectra up to 5000cm−1were ob-\ntained using a pyroelectric deuterated triglycine sulfate\ndetector.\nTHz complex transmittance from 3 to 50cm−1(with\nthe upper limit due to sample opacity) was measured\nusing a custom-made time-domain spectrometer. For\nthe low-temperature IR reflectivity and THz complex\ntransmittance spectroscopy, Oxford Instruments Optis-\ntat cryostats with mylar and polyethylene windows, re-\nspectively, were used. THz spectroscopy with magnetic\nfield was performed using a custom-made time-domain\nspectrometercomprisinganOxfordInstrumentsSpectro-\nmag cryostat with a superconducting magnet, allowing\nus to apply an external magnetic field of up to 7T; the\nFaraday geometry (wavevector parallel to the magnetic\nfield) was used.\nINS was measured on a powder sample (9.75g) using\nthe neutron time-of-flight spectrometer IN4 in the Insti-\ntut Laue-Langevin in Grenoble, France.\nFor Raman studies, a Renishaw RM1000 Micro-\nRaman spectrometer equipped with a CCD detector and\nBragg filters was used. The experiments were performed\nin the backscattering geometry within the 10–800cm−1\nrange using an Ar laser with the wavelength of 514nm\nand an Oxford Instruments Optistat optical continu-\nous He-flow cryostat. Further, using a Quantum design\nPPMS 9T instrument, we carried out measurements of\nthe magnetic susceptibility, magnetization, and of the3\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48\n/s40/s66/s97\n/s48/s46/s50/s83/s114\n/s48/s46/s56/s41\n/s51/s67/s111\n/s50/s70/s101\n/s50/s52/s79\n/s52/s49\n/s84\n/s67/s51/s63\n/s84\n/s67/s50/s32/s66/s61/s50/s32/s84\n/s32/s66/s61/s49/s32/s84\n/s32/s66/s61/s48/s46/s53/s32/s84\n/s32/s66/s61/s48/s46/s48/s53/s32/s84\n/s32/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s84\n/s67/s49\nFIG. 1: Temperature dependence of the magnetization\nfor polycrystalline (Ba 0.2Sr0.8)3Co2Fe24O41taken at various\nmagnetic fields. The temperatures of magnetic phase transi-\ntions are marked by arrows.\nME effect, in a temperature interval from 5 to 1000K\nwith a magnetic field of up to 9T.\nIII. RESULTS AND DISCUSSION\nA. Magnetic and magnetoelectric properties\nFig. 1 shows the magnetization Mof the\n(Ba0.2Sr0.8)3Co2Fe24O41ceramics as a function of\ntemperature for several values of magnetic field. These\ndependences are similar to the previously published\ndata obtained with Sr 3Co2Fe24O41ceramics8and\na Ba0.52Sr2.48Co2Fe24O41single crystal.3Thus, the\nmagnetic phase diagram of (Ba xSr1−x)3Co2Fe24O41\nis probably relatively independent of xat least for\n0≤x≤0.2. The collinear ferrimagnetic structure with\nspins parallel to the zaxis appears below TC1≈700K.\nAtTC2≈500K, the spins start to rotate towards the\nxy-plane and at Tcon≈400K a transverse conical struc-\nture is stabilized.3Near 260–300K, we observed shallow\nminima in the M(T) curves, similarly to Refs. 3,9.\nThis feature has not been satisfactorily explained yet,\nand we suppose it is due to some further changes in\nthe magnetic structure. Subsequently, we measured\nmagnetization curves M(H) at various temperatures\n(see Fig. 1 in Suppl. Materials21) and found some\nfeatures due to magnetic phase transitions in the range\nfrom 0.1 to 1T. The magnetization reversal processes\nduring these transitions are explained in Ref. 22. ME\nmeasurements of P(H) at 10K (see Fig. 2 in Suppl.\nMaterials21) revealed changes in electric polarization\ninduced by magnetic field greater than 0.01T, attaining\na maximum at 0.3T and vanishing near 2T. A similar\nbehavior was reported for a single crystal3, where the\nmaximum polarization was, two orders of magnitude\nhigher than in our ceramics. For temperatures higherthan 50K, a strong leakage conductivity prevented us\nfrom acquiring meaningful ME data.\nB. Phonon spectra\nWe performed the factor-group analysis of lattice vi-\nbrations in (Ba,Sr) 3Co2Fe24O41for the centrosymmetric\nhexagonal space group P63/mmc(D4\n6h), taking into ac-\ncount the site symmetries published in Refs. 23,24 with\nthe same crystallographic sites shared between Fe/Co\nand Ba/Sr atoms. The unit cell contains 2 ×70 atoms,\nand the analysis yields:\nΓD4\n6h= 29A2u(z)+37E1u(x,y)+26A1g(x2+y2,z2)+\n+33E1g(xz,yz)+36E2g(x2−y2,xy)+27B2u+\n+34E2u+28B1g+7A1u+8A2g+8B1u+7B2g\n(1)\nwherex,y, andzmarkelectric polarizationsofthe IR ra-\ndiation for which the phonons are IR active, whereas the\nrest of the symbols in parentheses are components of the\nRaman tensor. After subtracting two acoustic phonons,\n64 IR-active and 95 Raman-active phonons are predicted\nin the spectra. Additional 119 phonons are silent, i.e.\ninactive in the IR or Raman spectra.\nAssumingthatthe crystalstructureundergoesanequi-\ntranslational spin-induced FE phase transition, the soft\nmode in the paraelectric phase should have, according to\nthetablesinRef.25, the A2usymmetry, andtheresulting\nacentric space group will be P6mm(C1\n6v). The factor-\ngroup analysis of phonons from the Brillouin-zone center\nreads\nΓC1\n6v= 55A1(z,x2+y2,z2)+70E1(x,y,xz,yz )+55B2\n+70E2(x2−y2,xy)+15A2+15B1.(2)\nThus, in the FE phase, one can expect 123 IR-active\nmodes, 193 modes can be theoretically observed in Ra-\nman spectra and 85 modes remain silent. Nevertheless,\nthe intensities of the newly activated modes are expected\nto be very low, because the polar distortion (propor-\ntional to the polarization) in spin-order-induced ferro-\nelectrics and probably also in (Ba xSr1−x)3Co2Fe24O41is\nfourordersofmagnitudesmallerthaninthecanonicalFE\nBaTiO 3. We note that, due to this fact, no new phonons\nbelowTCwere reported also in other spin-induced fer-\nroelectrics, only small shifts of phonon frequencies were\nobserved due to the spin-phonon coupling.26,27Anyway,\nthe factor-group analysis is useful for the discussion of\nelectromagnon activity in both IR and Raman spectra\nbelow.\nFig.2 shows the IR reflectivity spectra of the\n(Ba0.2Sr0.8)3Co2Fe24O41ceramics and of the\n(Ba0.5Sr0.5)3Co2Fe24O41single crystal at selected\ntemperatures. The single crystal was grown and pol-\nished with the surface normal [0001], therefore only E1u\nmodes are active in its Eω∝bardblx,ypolarized spectra ( Eω4\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s99/s101/s114/s97/s109/s105/s99/s115\n/s32/s32/s82/s101/s102/s108/s101/s99/s116/s105/s118/s105/s116/s121\n/s87 /s97/s118/s101/s110/s117/s109/s98/s101/s114/s32/s40/s99/s109/s45/s49\n/s41/s99/s114/s121/s115/s116/s97/s108/s44/s32/s99/s101/s114/s97/s109/s105/s99/s115\n/s32 /s32/s32 /s32/s32/s51/s48/s48/s32/s75\n/s32 /s32/s32 /s32/s32/s50/s48/s48/s32/s75\n/s32 /s32/s32 /s32/s49/s48/s48/s32/s75\n/s32 /s32/s32 /s32/s32/s32/s32/s32/s55/s32/s75/s115/s105/s110/s103/s108/s101/s32/s99/s114/s121/s115/s116/s97/s108\n/s69/s32 /s122\nFIG. 2: Temperature dependence of IR reflectivity\nspectra of (Ba 0.2Sr0.8)3Co2Fe24O41ceramics and a\n(Ba0.5Sr0.5)3Co2Fe24O41single crystal. The latter spec-\ntrum was taken in polarization Eω⊥zandHω⊥z, i.e. only\nE1usymmetry phonons are seen.\ndenotes the electric vector of the incident radiation). In\nthe spectra of ceramics, both A2uandE1umodes are IR\nactive, but their intensities are reduced. We identified\nonly 21E1uphonons in the single crystal and 22 phonons\nin the ceramics, which is much less than expected from\nthe factor-group analysis. The discrepancy is apparently\ncaused by small intensities and/or overlapping of some\nmodes. It is worth noting that no new modes appear\nin the IR spectra on cooling, i.e. no signature of any\nstructural phase transition is seen below 300K. The\nintensities of the reflection bands only increase on\ncooling due to reduced phonon damping with lowering\ntemperature. The phonon parameters obtained from fits\nof spectra taken at 10K are listed in Tab. I in Suppl.\nMaterials.21\nC. THz studies\nFig. 3 shows complex refractive index spectra of\nthe (Ba 0.2Sr0.8)3Co2Fe24O41ceramics obtained by time-\ndomain THz spectroscopy between 7 and 800K. The\nlow-frequency increase in n(ω) andκ(ω) occurring above\n300K is due to the sample conductivity arising in the\nmicrowave region; furthermore, two clear resonances are\nseen in the spectra. The higher-frequency one (near\n∼45cm−1) is present at all temperatures and it exhibits\nonly a small softening on heating. The lower-frequency\noneappearsat250Knear25cm−1andmarkedlyhardens\nand sharpens on cooling (its frequency reaches 35cm−1\nat 7K).\nUpon applying magnetic field, the lower-frequency res-\nonance broadens, shifts towards lower frequencies and fi-\nnally disappears at magnetic field values above 2T. This/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s32/s56/s48/s48/s32/s75\n/s32/s54/s48/s48/s32/s75\n/s32/s51/s48/s48/s32/s75/s32/s107/s32/s50/s48/s48/s32/s75\n/s32/s49/s53/s48/s32/s75\n/s32/s49/s48/s48/s32/s75\n/s32/s32/s32/s32/s32/s55/s32/s75\n/s32/s32\n/s87 /s97/s118/s101/s110/s117/s109/s98/s101/s114/s32/s40/s99/s109/s45/s49\n/s41/s40/s98/s41/s52/s46/s52/s52/s46/s54/s52/s46/s56\n/s32/s32\n/s110/s40/s97/s41\n/s66\n/s101/s120/s116/s32/s61/s32/s48/s32/s84\nFIG. 3: Spectra of the complex refractive index of the\n(Ba0.2Sr0.8)3Co2Fe24O41ceramics determined by THz spec-\ntroscopy at various temperatures.\nbehavior is shown in Fig. 4 for 50K; qualitatively similar\nmagnetic-field dependences were observed at tempera-\ntures up to 250K (see Fig. 3 in Supplement21). This is\na signature of a magnetic excitation; we assume that it\ndisappears from the spectra due to the magnetic phase\ntransitionfromthetransverseconicalto acollinearstruc-\nture. We expect this mode to remain active up to Tcon∼=\n400K when the magnetic structure changes. Neverthe-\nless, its damping dramatically increases with tempera-\nture, so the mode gradually becomes overdamped. Con-\nsequently, above ∼250K, it is seen only as a broad fea-\ntureless background in the κ(ω) spectra, which overlaps\nwith the conductivity appearing above 300K (see Fig. 3\nin Supplement21). Note that the magnetic mode disap-\npears from the THz spectra near the temperature TC3\nmarked in Fig. 1. The mode seen near 45cm−1is appar-\nently a phonon, because its shape does not change with\nmagnetic field and it remains in the spectra up to the\nparamagnetic phase.\nAs the magnetic field is further increased, another nar-\nrow excitation appears in the low-frequency part of the\nTHz spectra. Its resonance frequency linearly increases\nwith the magnetic field as ωFMR=γHwith the propor-\ntionality constant γ= 0.032THz/T (see inset of Fig. 4),\nwhich roughly corresponds to the gyromagnetic ratio of\na free electron ( γ= 0.028THz/T). Such behavior is typ-5\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55/s48/s50/s52/s54/s56/s53/s48/s32/s75\n/s110\n/s32/s107/s32/s32/s32/s32/s32/s32/s32/s32/s48/s32/s84\n/s32/s48/s46/s54/s50/s53/s32/s84\n/s32/s32/s32/s32/s32/s32/s32/s32/s49/s32/s84\n/s32/s32/s32/s32/s32/s32/s32/s32/s50/s32/s84\n/s32/s32/s32/s32/s32/s32/s32/s32/s52/s32/s84\n/s32/s32/s32/s32/s32/s32/s32/s32/s53/s32/s84\n/s32/s32/s32/s32/s32/s32/s32/s32/s55/s32/s84\n/s32/s32\n/s87 /s97/s118/s101/s110/s117/s109/s98/s101/s114/s32/s40/s99/s109/s45/s49\n/s41/s40/s98/s41/s51/s46/s53/s52/s46/s48/s52/s46/s53\n/s32/s32\n/s40/s97/s41\n/s32/s32/s53/s48/s32/s75\n/s50/s53/s48/s32/s75/s40/s99/s41\n/s32/s32/s70/s77/s82/s32/s40/s99/s109/s45/s49\n/s41\n/s48/s72/s32/s40/s84/s41/s70/s77/s82/s61/s32 /s46/s72\nFIG. 4: Magnetic-field dependence of a) index of refraction\nand b) extinction coefficient in the THz region measured at\n50K. c) Magnetic field dependence of the ferrimagnetic res-\nonance frequency ωFMRin (Ba 0.2Sr0.8)3Co2Fe24O41ceramics\nat 50 and 250K.\nical of FMRs.28The same resonance is seen up to 250K\n(the highest value attainable in the magnetic cryostat)\nand at 7T, for all temperatures, it reaches a frequency\nof≈7.5cm−1. (see Fig. 4c and Fig.3 in Supplement21).\nWithout magnetic field, the FMR can be observed in\nthe microwave range. These observations are beyond the\nscope of the present article and will be presented sepa-\nrately.\nIn multiferroics, simultaneously magnetically and elec-\ntrically active spin excitation are called electromagnons.\nThese can be distinguished from magnons by comparing\nthe polarized IR spectra of crystals taken in all possible\npolarizations. Z-hexaferrite single crystals can be easily\ngrown only in the hexagonalplane, therefore we disposed\nmerely of a (0001) single crystal plate. In Fig. 5, po-\nlarized THz spectra of the (Ba 0.5Sr0.5)3Co2Fe24O41sin-\ngle crystal with Eω⊥z⊥Hω(Hωdenoting the mag-\nnetic vector of the incident beam) are compared with the\nspectra of ceramics. In the single crystal, we observed\na very weak and narrow excitation at 31cm−1. Since\nthis mode is independent of temperature and external\nmagnetic field (not shown) and its frequency is clearly\nlower than that of the magnetic excitation in ceramics,\nwe interpret it as a weak phonon with the E1usymme-/s50/s48 /s52/s48 /s54/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s72 /s99/s69 /s99\n/s99/s101/s114/s97/s109/s105/s99/s115/s44/s32/s32/s99/s114/s121/s115/s116/s97/s108\n/s32/s32/s32 /s32/s32/s32/s32/s32/s55/s75\n/s32/s32/s32 /s32/s49/s53/s48/s75\n/s32/s32/s32 /s32/s51/s48/s48/s75/s32/s107\n/s32/s32\n/s87 /s97/s118/s101/s110/s117/s109/s98/s101/s114/s32/s40/s99/s109/s45/s49\n/s41/s40/s98/s41/s52/s46/s51/s52/s46/s52/s52/s46/s53/s52/s46/s54/s52/s46/s55/s52/s46/s56\n/s32/s32\n/s110\n/s40/s97/s41\nFIG. 5: Comparison of THz spectra of a) index of refrac-\ntion and b) extinction coefficient in (Ba 0.2Sr0.8)3Co2Fe24O41\nceramics and (Ba 0.5Sr0.5)3Co2Fe24O41single crystal at three\ntemperatures. The spectra of the single crystal are polariz ed\nEω⊥z⊥Hω, but the [0001] crystal plane was tilted by\n6◦from the sample surface, therefore the narrow weak mode\nnear 44cm−1is probably a leakage phonon mode from the\nEω/bardblzpolarized spectra. The oscillations in the spectra of\nthe single crystal are artifacts due to echoes from multiple\nreflections in the sample. They can be avoided by time win-\ndowing, but then the spectral resolution is lower and the pea k\nnear 32cm−1is not resolved.\ntry. As the spin excitation near 36cm−1is not active in\nthe (Eω⊥z;Hω∝bardblz) polarized THz spectra29, it must\nbe an electromagnon active for Eω∝bardblz. The existence of\nthis electromagnon was reported already in 2014 at the\nMarch APS meeting by Chun29but never published.\nThe two excitations near 44 and 57cm−1are phonons.\nThe former one is much weaker than the corresponding\nphonon in ceramics, so it is probably due to a leakage of\nanA2u(z) mode.\nTHz spectra of the (Ba 0.5Sr0.5)3Co2Fe24O41single\ncrystal measured in external magnetic field (not shown)\nrevealed a FMR below 10cm−1with a frequency identi-\ncal with that of the FMR in ceramics (see Fig. 4). This\nprovides an evidence that the FMR has a magnon-like\ncharacter and that it is active in the Eω⊥z⊥Hωpo-\nlarized THz spectra.\nTHz spectra of Sr 3Co2Fe24O41ceramics revealed the\nsame excitations like in (Ba 0.2Sr0.8)3Co2Fe24O41, i.e.,6\na phonon near 45cm−1and an electromagnon near\n35cm−1, (see Fig. 4 in Supplement21) which vanishes\nfrom the spectraat 250K. Thisprovidesanevidence that\nthe crystalline and magnetic structures of both samples\nare almost identical. This was also confirmed by very\nrecent magnetic studies of (Ba xSr1−x)3Co2Fe24O41with\nxranging from 0 to 1.22\nD. Raman scattering\nIn the non-centrosymmetric FE phases, the electro-\nmagnons have to be both IR and Raman active, similarly\nto the case of BiFeO 3.30In our case, the electromagnon\nis active in the Eω∝bardblz-polarized THz spectra, so if the\nstructure is FE (space group P6mm), according to the\nselection rules in Eq. (2), the electromagnon has the A1\nsymmetry and it should be also Raman active in the z2-\npolarized spectra.\nWe measured temperature-dependent Raman spectra\nof the single crystal (Fig. 6) and ceramics (see Fig. 5\nin Supplement21) on cooling down to 4K. Indeed, be-\nlow 250K, a distinctive excitation appears in the low-\nfrequency part of the spectra. The inset of Fig. 6 com-\npares the temperature dependences of frequencies of the\nRaman-activeexcitationsinthe singlecrystaland ceram-\nicswith thoseofthe IRactiveelectromagnonin ceramics.\nThe frequencies of all three excitations are very similar\nand all of them decrease on heating towards Tcon. Nev-\nertheless, a detailed analysis reveals some differences: (i)\nThe Raman-active mode seen in the ceramics has a fre-\nquency systematically higher by 7–10cm−1than the IR-\nactive one observed in the same sample. (ii) Upon heat-\ning, the Raman-active mode detected in the single crys-\ntal softens faster than those in ceramics. (iii) In both\nsamples, the Raman-active spin excitation has a damp-\ning systematically 2–3 times higher than the IR-active\nmode.\nThe first difference can be possibly a consequence of\nan angular dependence of the mode frequency (oblique\nmode)relevanttothesingle-crystallineceramicgrains(in\nfact, this Raman mode was observed only in few grains,\nand we suppose that their zcrystal axes were oriented\nalmost parallel to the sample surface). The second obser-\nvation could be theoretically owing to somewhat differ-\nent Ba concentrations in the ceramics ( x= 0.2) and the\nsingle crystal ( x= 0.5), but since we know that the mag-\nnetic structure is identical for both compositions, this\npossibility is not likely. Instead, we note that owing to\nthe sintering, there may be a varying mechanical stress,\nwhich may influence the temperature dependences of the\nmode frequencies in the individual grains. Finally, the\ndifferentdampingsofthemodesinRamanandTHz spec-\ntra do not support the idea of identical modes either.\nConsequently, we are not sure that the spin excitations\nseen in both THz and Raman spectra are the same. In\ncase the Raman mode is not identical with the IR-active\none, the Raman mode could be due to an antiferromag-/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48/s50/s48/s51/s48/s52/s48\n/s73/s82/s45/s97/s99/s116/s105/s118/s101/s32/s32/s82/s97/s109/s97/s110/s45/s97/s99/s116/s105/s118/s101/s32/s109/s111/s100/s101/s32\n/s32/s32/s32/s32/s105/s110/s32/s99/s101/s114/s97/s109/s105/s99/s115\n/s32/s32/s99/s101/s114/s97/s109/s46\n/s32/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s32/s83/s112/s105/s110/s32/s101/s120/s99/s105/s116/s97/s116/s105/s111/s110/s32/s102/s114/s101/s113/s46/s32/s40/s99/s109/s45/s49\n/s41\n/s82/s97/s109/s97/s110/s45/s97/s99/s116/s105/s118/s101/s32/s109/s111/s100/s101/s32\n/s32/s32/s105/s110/s32/s120/s40/s122/s122/s41/s120 /s32/s115/s112/s101/s99/s116/s114/s117/s109/s40/s66/s97\n/s48/s46/s53/s83/s114\n/s48/s46/s53/s41\n/s51/s67/s111\n/s50/s70/s101\n/s50/s52/s79\n/s52/s49/s82/s101/s100/s117/s99/s101/s100/s32/s82/s97/s109/s97/s110/s32/s105/s110/s116/s101/s110/s99/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s82/s97/s109/s97/s110/s32/s115/s104/s105/s102/s116/s32/s40/s99/s109/s45/s49\n/s41/s32/s32/s32/s32/s32/s52/s75\n/s32/s32/s32/s53/s48/s75\n/s32/s49/s48/s48/s75\n/s32/s49/s53/s48/s75\n/s32/s50/s48/s48/s75\n/s32/s50/s53/s48/s75\n/s32/s51/s48/s48/s75/s120/s40/s122/s122/s41/s120\nFIG. 6: Temperature dependence of x(zz)xRaman spectra of\nthe (Ba 0.5Sr0.5)3Co2Fe24O41single crystal. The inset shows\nthe temperature dependenceof spin excitation frequencies ob-\ntained from these spectra and from unpolarized Raman and\nTHz spectra of the (Ba 0.2Sr0.8)3Co2Fe24O41ceramics.\nnetic resonance or more probably multi-magnon scatter-\ning involving magnons with the highest magnon density\nof states (DOS) .\nIn the opposite case, if the Raman and IR-active ex-\ncitations corresponded to the same vibrational mode, it\nwould be an argument supporting the FE P6mmcrystal\nsymmetry of the Z-hexaferrite. Then, the FE polariza-\ntionPwould be oriented along the zaxis, which is in\ncontradiction with Ref. 3, where magnetic-field induced\nchanges in polarization in the hexagonal plane were ob-\nserved. One might also hypothetically assume that at\nzero magnetic field, P∝bardblz, and that Ptilts away from\nthezaxis in external magnetic field; in such a case, the\ncrystal structure would change to monoclinic. However,\nin the FE phase, a much higher number of phonons than\nobserved should be present in both IR and Raman spec-\ntra. For example, at 4K, we resolved 20 modes in z2-\npolarized Raman spectra and 21 IR-active phonons in\nEω⊥zspectra. These numbers are in better agreement\nwith the 26 A1g(z2) and 36E1u(x,y) modes predicted for\nthe paraelectric P63/mmcphasethan with the 54 A1(z2)\nand 70E1(x,y) modes expected for the FE P6mmsym-\nmetry.\nWe can conclude that our spectra do not support any\nFEdistortion in zeromagneticfield. This isin agreement\nwith the known literature; up to now, in the structural\nstudies of Z-hexaferrite (Ba xSr1−x)3Co2Fe24O41, no po-\nlar space group was resolved23,24and the electric polar-\nizationP⊥zwas observed only in an external magnetic\nfield.3,8Nevertheless, for final proving or disproving of\nthe polar crystal structure, we propose further comple-\nmentary experiments, such as second-harmonic genera-\ntion or high-resolution electron diffraction.7\nE. Inelastic neutron scattering\nSome electromagnons activated in the THz dielectric\nspectra by exchange coupling have wavevectors from the\nBrillouin-zoneboundary,31–33wherethemagnonDOSat-\ntainsamaximum. Suchmagnonsshouldberecognizedby\nthe corresponding maxima of intensity in the INS spec-\ntra. Using the powder, we have performed INS experi-\nments with various energy resolutions. Fig. 7a shows a\nmap representing the orientation-averaged scattering in-\ntensity at T= 5K. The high INS intensity seen as a\ncolumn at Q≈4˚A−1and for all Qat energies above\n15 meV corresponds to the phonon DOS. Near the neu-\ntron momentum transfer value of Q= 1.3˚A−1, a mag-\nnetically active branch (marked by the dashed line) ex-\ntends in energy up to at least 20meV. The magnon DOS\n(proportional to the scattering intensity integrated over\nthe interval 1 .2≤Q≤1.4˚A) exhibits a small maxi-\nmum near 8meV ≈64cm−1(see Fig. 7b). Near this\nenergy we see only a weak excitation in the Eω⊥z\npolarized IR reflectivity spectra (see Fig. 2 and Tab. I\nin Supplement21); based solely on our data, we cannot\ndecide whether this is an electromagnon or a phonon.\nTheelectromagnonenergy,accordingtoourTHzspectra,\namounts to 4–5.5 meV, but at the low temperatures used\nin INS experiments, at this energy transfer value, a min-\nimum in magnon DOS is seen (Fig. 7b). Subsequently,\nthe wavevector of the electromagnon is most probably\nnot from the Brillouin zone center or boundary. This is\nrather surprising because recent polarized INS studies of\na Y-type hexaferrite claimed that the observed electro-\nmagnon with similar features as the one in Z-hexaferrite\nwas a zone-center mode.19.\nIn the transverse-conical structure, the electromagnon\ncan induce an oscillating electric polarization along the\nzaxis. Therefore we suggest that it is activated by the\nexchange-striction mechanism. Note that in both Z- and\nY-type hexaferrites, the static polarization Pappears\ndue to the inverse Dzyaloshinskii-Moriya interaction2,8\nperpendicularly to the zaxis, if one applies an exter-\nnal magnetic field H⊥zwhereby the transverse conical\nstructure is stabilized.3,19However, the magnetic struc-\nture of the Y-type hexaferrite is longitudinally conical\natµ0H= 0T,17,18whereas that of the Z-hexaferrite at\nzero and small magnetic fields is transverse conical. The\nelectromagnons in both materials share several features:\n(i) they are active in Eω∝bardblzeven in zero magnetic field,\n(ii) their frequencies shift with external magnetic field\nand, finally, (iii) they disappear from the spectra above\nsome threshold magnetic field, when the transverse or\nlongitudinal conical magnetic structures disappear.\nIV. CONCLUSION\nInconclusion, byusingseveralcomplementaryspectro-\nscopic techniques, we have obtained a comprehensive set\nof information on the spin and lattice dynamics of the0 2 4 60510152025\nMomentum transfer Q ( ˚A−1)Energy (meV)\n \n(a)\n0500100015002000\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48/s54/s48/s48\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41/s32/s45/s32/s81/s99/s117/s116/s32/s64/s32/s40/s49/s46/s51/s32/s43/s45/s32/s48/s46/s49/s41/s32/s197/s45/s49\n/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41/s32 /s32/s61/s32/s49/s46/s53/s32/s197\n/s32 /s32/s61/s32/s49/s46/s55/s32/s197/s32\n/s32 /s32/s61/s32/s50/s46/s50/s32/s197\n/s32 /s32/s61/s32/s51/s46/s48/s32/s197\n/s40/s98/s41\nFIG. 7: (a) INS intensity as a function of momentum and\nenergy transfers, measured on (Ba 0.2Sr0.8)3Co2Fe24O41at\n5K. The dashed line marks the signal corresponding to the\nmagnon branch. The remaining signal comes from phonon\nDOS. The energy resolution was ∆ E= 1.5meV, the wave-\nlength of the incident neutrons was 1.5 ˚A. (b) Magnon DOS\ncalculated from the INS spectra for 1 .2≤Q≤1.4˚A taken\nwith various wavelengths λof the incident neutrons.\nZ-type hexaferrite compounds (Ba xSr1−x)3Co2Fe24O41,\nin broad frequency and temperature ranges.\nIn the low-temperature THz spectra of\n(BaxSr1−x)3Co2Fe24O41ceramics, a soft spin exci-\ntation near 35cm−1was discovered, whose frequency\nsoftens on heating towards Tcon≈400K and its damping\nincreases. An external magnetic field exceeding 2T\ninduces a change of magnetic structure and the spin\nexcitation vanishes from the THz spectra. THz spectra\nobtained on a single crystal revealed the same magnon\nin theEω∝bardblzpolarized spectra, therefore we claim that\nit is an electromagnon. Since the excitation is observed\nin the transverse conical magnetic structure, we propose\nthat it is activated by the exchange striction mechanism.8\nA spin excitation with a similar frequency was discov-\nered in Raman spectra. Should it be the same electro-\nmagnon, the sample would be FE with a polarization\nP∝bardblzandtheP6mmspacegroup. Thisisratherunlikely,\nbecause up to now, only P⊥zoriented polarization was\nobserved and the numbers of phonon modes observed in\nthe IR and Raman spectra are much lower than those al-\nlowed in the FE phase. Nevertheless, further structural,\nmagnetoelectric and second-harmonic-generation exper-\niments appear necessary to clearly prove or disprove a\npolar phase in (Ba xSr1−x)3Co2Fe24O41.\nUpon applying magnetic field higher than 3T, in the\nlow-frequency part of the THz spectra, a narrow exci-\ntation appears whose frequency linearly increases with\nmagnetic field. Its behavior is independent on tempera-\nture (investigated up to 250K) and since the proportion-alityconstantoftheresonancefrequencyonthe magnetic\nfield corresponds to the gyromagnetic ratio of a free elec-\ntron, we interpret this excitation as the ferromagnetic\nresonance.\nAcknowledgments\nThis work was supported by the Czech Science Foun-\ndation projects15-08389Sand 14-18392S,the programof\nCzech Research Infrastructures, project LM2011025 and\nMˇSMTprojectLD15014. TheexperimentsinILLGreno-\nble were carriedout within the project LG14037financed\nby the Ministry of Education of the Czech Republic.\n∗Authors to whom correspondence should be addressed; e-\nmail: kamba@fzu.cz; kadlecf@fzu.cz\n1M. Tokunaga, M. Azuma, andY.Shimakawa, J. Phys. Soc.\nJpn.79, 064713 (2010).\n2Y. Tokura, S. Seki, and N. Nagaosa, Rep. Prog. Phys. 77,\n076501 (2014).\n3S. H. Chun, Y. S. Chai, B.-G. Jeon, H. J. Kim, Y. S. Oh,\nI. Kim, H. Kim, B. J. Jeon, S. Y. Haam, J.-Y. Park, et al.,\nPhys. Rev. 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ELECTROMAGNON IN Z-TYPE\nHEXAFERRITE (BaxSr1−x)3Co2Fe24O41-\nSUPPLEMENTARY MATERIALS\n/s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48/s48/s49/s48/s50/s48/s51/s48/s40/s66/s97\n/s48/s46/s50/s83/s114\n/s48/s46/s56/s41\n/s51/s67/s111\n/s50/s70/s101\n/s50/s52/s79\n/s52/s49\n/s49/s48/s48/s32/s75/s54/s32/s75\n/s50/s48/s48/s32/s75\n/s52/s53/s48/s32/s75\n/s53/s48/s48/s32/s75\n/s32/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s54/s48/s48/s32/s75\nFIG. 1: Magnetization curves of polycrystalline\n(Ba0.2Sr0.8)3Co2Fe24O41taken at various temperatures.\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s32/s32/s32/s32/s32/s32/s32/s32/s50/s32/s84\n/s32/s32/s32/s32/s32/s32/s32/s32/s51/s32/s84\n/s32/s32/s32/s32/s32/s32/s32/s32/s52/s32/s84\n/s32/s32/s32/s32/s32/s32/s32/s32/s53/s32/s84\n/s32/s32/s32/s32/s32/s32/s32/s32/s54/s32/s84\n/s32/s32/s32/s32/s32/s32/s32/s32/s55/s32/s84/s50/s53/s48/s32/s75/s110\n/s32/s107/s32/s32/s32/s32/s32/s32/s32/s32/s48/s32/s84\n/s32/s32/s32/s32/s32/s48/s46/s53/s32/s84\n/s32/s32/s32/s48/s46/s55/s53/s32/s84\n/s32/s32/s32/s32/s32/s32/s32/s32/s49/s32/s84\n/s32/s32/s32/s32/s32/s49/s46/s53/s32/s84\n/s32/s32\n/s87 /s97/s118/s101/s110/s117/s109/s98/s101/s114/s32/s40/s99/s109/s45/s49\n/s41/s40/s98/s41/s52/s46/s50/s52/s46/s52/s52/s46/s54/s52/s46/s56\n/s32/s32\n/s40/s97/s41\nFIG. 3: Magnetic-field dependence of a) index of refraction\nand (b) extinction coefficient of (Ba 0.2Sr0.8)3Co2Fe24O41\nceramics measured at 250K./s49/s48/s45/s51\n/s49/s48/s45/s50\n/s49/s48/s45/s49\n/s49/s48/s48/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s80/s32/s40 /s67/s47/s109/s50\n/s41/s49/s48/s32/s75\n/s32/s32\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41\nFIG. 2: Magnetic-field dependence of polarization changes\natT= 10K for the (Ba 0.2Sr0.8)3Co2Fe24O41ceramics. The\nblue and red curves show changes upon decreasing and\nincreasing magnetic field, respectively.\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s32/s87 /s97/s118/s101/s110/s117/s109/s98/s101/s114/s32/s40/s99/s109/s45/s49\n/s41/s110\n/s32/s107\n/s32/s32\n/s40/s98/s41/s52/s46/s48/s52/s46/s50/s52/s46/s52/s52/s46/s54/s52/s46/s56\n/s32/s50/s57/s52/s32/s75\n/s32/s50/s53/s48/s32/s75\n/s32/s50/s48/s48/s32/s75\n/s32/s49/s53/s48/s32/s75\n/s32/s49/s48/s48/s32/s75\n/s32/s32/s32/s53/s48/s32/s75\n/s32/s32/s32/s32/s32/s55/s32/s75/s32/s32\n/s40/s97/s41/s83/s114\n/s51/s67/s111\n/s50/s70/s101\n/s50/s52/s79\n/s52/s49\nFIG. 4: Temperature dependence of THz spectra of the\ncomplex refractive index of Sr 3Co2Fe24O41ceramics10\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48 /s55/s48/s48 /s56/s48/s48/s40/s66/s97\n/s48/s46/s50/s83/s114\n/s48/s46/s56/s41\n/s51/s67/s111\n/s50/s70/s101\n/s50/s52/s79\n/s52/s49\n/s51/s48/s48/s49/s48/s48/s53/s48/s52/s82/s101/s100/s117/s99/s101/s100/s32/s82/s97/s109/s97/s110/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s44/s32/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115\n/s82/s97/s109/s97/s110/s32/s115/s104/s105/s102/s116/s32/s40/s99/s109/s45/s49\n/s41/s84/s40/s75/s41\n/s50/s53/s48/s50/s48/s48/s49/s53/s48/s99/s101/s114/s97/s109/s105/s99/s115\nFIG. 5: Temperature dependence of Raman spectra of the (Ba 0.2Sr0.8)3Co2Fe24O41ceramics measured with the polarizer and\nanalyzer set parallel. It is important to stress that the ele ctromagnon seen below 50cm−1was observed only in few ceramic\ngrains. This is consistent with its activity only in the z2Raman spectra. The grains are randomly oriented and only few of\nthem have their zaxes lying in the sample plane.11\nTABLE I: Comparison of phonon fitting parameters describing IR reflectivity and THz transmission spectra of\n(Ba0.2Sr0.8)3Co2Fe24O41ceramics and those of the (Ba 0.5Sr0.5)3Co2Fe24O41single crystal ( Eω⊥zandHω⊥z) obtained\natT= 10K. The high-frequency contributions to permittivity we reε∞= 4.92 in the ceramics and ε∞= 4.79 in the single\ncrystal. The phonons seen in the IR reflectivity spectra of th e single crystal are assumed to have the E2usymmetry. The two\nphonons in THz transmission spectra below 60cm−1are much weaker in the crystal than in the ceramics, therefor e we assume\nthey have the A2usymmetry and they appear as a leakage from the Eω/bardblzspectra (the zcrystal axis was tilted by 6◦from\nthe normal crystal plane). The oscillators present only in t he ceramics are assumed to have the A2usymmetry.\nCeramics Single crystal\nSymmetry ωTO(cm−1) γTO(cm−1) ∆ ε ω TO(cm−1) γTO(cm−1) ∆ ε\nA2u? 45.26 1.69 0.12 43.78 0.24 0.02\nA2u? 56.91 1.92 0.04\nE2u 66.33 9.19 0.24\n87.09 29.80 4.03\nE2u 98.43 19.29 1.80\nE2u 109.56 10.83 0.82\nA2u 123.04 8.29 0.33\nA2u 138.23 8.65 0.15\nE2u 160.06 15.62 0.18 166.91 8.79 0.24\nE2u 172.04 11.37 0.17 176.32 9.32 0.21\nA2u 190.02 17.04 0.36\nE2u 211.84 14.00 0.46 209.70 28.11 0.68\nA2u 222.54 7.28 0.14\nE2u 240.52 12.85 0.54 239.23 11.88 1.70\nA2u 252.93 11.33 0.30\n267.91 29.74 0.89\nE2u 285.88 13.01 0.77 285.02 8.46 2.01\nE2u 298.72 16.58 1.31 297.01 11.34 0.89\nE2u 309.85 6.11 0.10 309.85 12.21 1.22\nE2u 320.97 17.47 1.30 323.54 4.62 1.01\nE2u 359.49 6.46 0.05 358.83 11.09 0.22\nE2u373.18 42.74 1.53374.90 15.91 0.59\nE2u 385.81 23.89 0.50\nE2u 445.08 39.37 0.59 431.25 9.52 1.04\nA2u 503.29 20.03 0.08\nE2u 540.95 12.12 0.02 539.23 27.63 0.79\nE2u 566.62 59.22 0.81 560.49 21.00 0.64\nE2u 595.73 36.54 0.33 590.59 14.14 0.38" }, { "title": "1203.2319v2.Magnetic_damping_of_a_carbon_nanotube_NEMS_resonator.pdf", "content": "Magnetic damping of a carbon nanotube NEMS resonator\nD R Schmid, P L Stiller, Ch Strunk and A K H ¨uttel\nInstitute for Experimental and Applied Physics, University of Regensburg,\n93040 Regensburg, Germany\nE-mail: andreas.huettel@physik.uni-regensburg.de\nAbstract. A suspended, doubly clamped single wall carbon nanotube is characterized at\ncryogenic temperatures. We observe specific switching effects in dc-current spectroscopy of\nthe embedded quantum dot. These have been identified previously as nano-electromechanical\nself-excitation of the system, where positive feedback from single electron tunneling drives\nmechanical motion. A magnetic field suppresses this effect, by providing an additional\ndamping mechanism. This is modeled by eddy current damping, and confirmed by measuring\nthe resonance quality factor of the rf-driven nano-electromechanical resonator in an increasing\nmagnetic field.\nPACS numbers: 63.22.Gh, 62.25.Jk, 73.63.KvarXiv:1203.2319v2 [cond-mat.mes-hall] 14 Jun 2012Magnetic damping of a carbon nanotube NEMS resonator 2\n/c7e/c56/c67\n/c49/c64/c63/c2f/c6c/c66/c56/c73/c64\n/c72/c66/c20/c67/c65/c6e/c2e/c63/c72/c79/c6f/c73/c74/c61/c74\n/c28/c61/c29\n/c30/c2e/c35/cb5/c6d\n/c31/cb5/c6d/c28/c62/c29\n/c32/c30/c30/c6e/c6d/c37/c30/c30/c6e/c6d/c36/c30/c30/c6e/c6d/c35/c30/c30/c6e/c6d\n/c20/c32/c30/c30/c20/c34/c30/c30/c20/c36/c30/c30/c20/c31/c30/c30/c30\n/c2d/c31/c30 /c2d/c35 /c20/c30 /c20/c35 /c20/c31/c30 /c56/c20/c28/c56/c29/c67/c66\n/c28/c4d/c48/c7a/c29/c28/c64/c29\n/c41/c42/c41/cb4/c42/cb4\n/c20/c30/c2d/c31 /c20/c30/c20/c32/c39/c30/c6d/c4b\n/c20/c31 /c20/c33/c49/c20/c28/c41/c29\n/c56/c20/c28/c56/c29/c67/c2d/c31/c32/c20/c31/c30/c2d/c31/c30/c20/c31/c30/c2d/c38/c20/c31/c30 /c28/c63/c29\nFigure 1. Measurement setup, chip geometry and basic sample characterization. (a) Scanning\nelectron microscope (SEM) image of a typical suspended carbon nanotube, combined with a\nsimplified sketch of the measurement electronics. dc gate and bias voltages are applied and the\nlow-frequency current signal measured via a preamplifier; in addition a radio frequency signal\ncan be applied via an antenna hanging several centimeters above the chip. (b) SEM image of\nthe electrode geometry, displaying concentric, ring segment shaped contact electrodes around\na nanotube growth catalyst island. All transport measurements presented here are measured\nacross the same 700nm wide trench. (c) Measurement of the dc current Idc(Vg)through the\nnanotube device as function of back gate voltage Vgfor an applied bias Vsd=0:2mV. (d)\nObserved rf-driven mechanical resonance frequencies f(Vg)as function of back gate voltage\nVg, see text for details.\nNano-electromechanical resonator systems provide an intriguing field of research, where\nboth technical applications and fundamental insights into the limits of mechanical motion are\npossible. Among these systems, carbon nanotubes provide the ultimate electromechanical\nbeam resonator [1, 2, 3], because of their stiffness, low mass, and high aspect ration. At\nthe same time, they are an outstanding material for transport spectroscopy of quantum dots at\ncryogenic temperatures [4, 5]. Chemical vapour deposition (CVD) has been shown to produce\non chip defect-free single wall carbon nanotubes [6]. By performing this growth process as\nlast chip fabrication step, suspended defect- and contamination-free macromolecules can be\nintegrated into electrode structures and characterized. On the electronic side, this has led\nto many valuable insights into, e.g., the physics of spatially confined few-carrier systems\n[7, 8, 9]. In terms of nano-electromechanical systems, these ultra-clean nanotubes have shown\nexceedingly high mechanical quality factors at cryogenic temperatures [10]. This has allowed\nfor the observation of direct interaction between single electron tunneling and mechanical\nmotion [11, 12, 13].\nIn this article, we report on low temperature transport spectroscopy measurements on\na suspended, doubly clamped carbon nanotube, as displayed in figure 1(a). The carbon\nnanotube acts as an ultra-clean quantum dot as well as a nano-electromechanical transversalMagnetic damping of a carbon nanotube NEMS resonator 3\nresonator. Figure 1(b) shows a typical chip electrode structure including dimensions. On a\nhighly p+ doped Si substrate with \u0018300nm thermally grown SiO 2on top, contact patterns\nare defined via electron beam lithography and evaporation of 40nm rhenium. This metal layer\ndirectly serves as etch mask for subsequent anisotropic dry etching of the oxide, generating\ndeep trenches between the electrodes. As last fabrication step, CVD growth catalyst is\nlocally deposited at the center of each contact electrode structure and the nanotube growth\nis performed [6].\nElectronic transport measurements were conducted in a3He evaporation cooling system\natT3He=290mK, and in a dilution refrigerator at Tmc,base=25mK. The electronic\nmeasurement setup, as sketched in figure 1(a), closely follows Refs. [10, 11]. A gate voltage\nVgis applied to the substrate as back gate, a bias voltage Vsdacross the device. The\nresulting dc current through the device is measured via a preamplifier, as required for Coulomb\nblockade transport spectroscopy [14]. An antenna suspended close to the chip provides means\nto apply a radio-frequency signal contact-free.\nAt first, we characterize the basic electronic and electromechanical properties of the\ndevice. As can be seen from the dc current measurement in figure 1(c), our device exhibits\nthe typical electronical behavior of a very clean and regular small band gap nanotube. The\nmeasurement displays the dc current Idcas function of the applied gate voltage Vg, for a low\nconstant dc bias voltage Vsd=0:2mV. For Vg<0:5V, highly transparent contacts in hole\nconduction lead to a rapid transition from Coulomb blockade to the Fabry-Perot interference\nregime [15]. Around Vg'0:75V, current is suppressed as the electrochemical potential is\nlocated within the semiconducting band gap. For Vg>1V, electron conduction becomes\nvisible through sharp, well-defined Coulomb blockade oscillations with the characteristic\nfour-fold pattern of the carbon nanotube level structure [16]. Regular Kondo conductance\nenhancement [17] emerges for Ne\u0000>15, again confirming the presence of a defect-free single\nwall carbon nanotube.\nWhen a radiofrequency signal is applied at mechanical resonance, the nanotube vibrates,\nleading to a change in detected, time-averaged dc current [10]. This signal can be identified\nvia its characteristic dependence on the back gate voltage Vg: electrostatical forces on the\ninfluenced charge on the nanotube lead to mechanical tension, and thereby an increase in\nresonance frequency of the transversal vibration mode. Figure 1(d) shows a map of such\nresonance positions, displaying the resonance frequency as function of back gate voltage Vg.\nIt thus characterizes the basic electromechanical properties of our device. Among several\nother weaker features, four clear structures, plotted in figure 1(d) and labelled A, B, A’, and\nB’, can be seen in the observed frequency range, with the overall gate voltage dependence\ntypical for the mechanical response of a tensioned carbon nanotube resonator [1, 2, 10].\nTraces A’ and B’ coincide over a wide range with double the frequency of traces A\nand B (plotted in figure 1(d) as thin black lines). It appears unlikely that these represent\nhigher mechanical modes, since in the low tension limit an exact frequency doubling is not\nexpected [18]. Instead, A’ and B’ can represent different driving mechanisms for the modes\nof A and B. In literature, e.g., parametric resonance has been demonstrated in measurements\non nanotube resonators [19, 20]. The observation of the two modes A and B is consistentMagnetic damping of a carbon nanotube NEMS resonator 4\n/c30/c30/c2e/c30/c20/c54/c56/c20/c28/c6d/c56/c29/c73/c64\n/c2d/c31\n/c33/c2e/c39/c32 /c33/c2e/c39/c37 /c56/c20/c28/c56/c29/c67/c61/c74/c65 /c33/c2e/c39/c32 /c33/c2e/c39/c37 /c56/c20/c28/c56/c29/c67/c61/c74/c65 /c33/c2e/c39/c32 /c33/c2e/c39/c37 /c56/c20/c28/c56/c29/c67/c61/c74/c65 /c33/c2e/c39/c32 /c33/c2e/c39/c37 /c56/c20/c28/c56/c29/c67/c61/c74/c65 /c33/c2e/c39/c32 /c33/c2e/c39/c37 /c56/c20/c28/c56/c29/c67/c61/c74/c65/c64/c49/c2f/c64/c56\n/c32/c28/c65/c2f/c68/c29/c56/c20/c28/c56/c29/c67/c61/c74/c65/c56/c20/c28/c6d/c56/c29/c73/c64/c31\n/c2d/c31/c30/c32\n/c31\n/c30/c64/c49/c2f/c64/c56\n/c32/c28/c65/c2f/c68/c29\n/c33/c2e/c37/c35 /c33/c2e/c38 /c33/c2e/c38/c35 /c33/c2e/c39 /c33/c2e/c39/c35 /c34/c2e/c30\n/c30/c2e/c32/c20/c54 /c30/c2e/c34/c20/c54 /c30/c2e/c36/c20/c54 /c30/c2e/c38/c20/c54\n/c2d/c34/c30\n/c2d/c32/c30\n/c20/c30/c20/c33/c2e/c39/c32 /c20/c33/c2e/c39/c34 /c20/c33/c2e/c39/c36/c49/c20/c28/c6e/c41/c29/c64/c63\n/c56/c20/c28/c56/c29/c67/c30/c2e/c30/c54\n/c30/c2e/c32/c54\n/c30/c2e/c34/c54\n/c30/c2e/c36/c54\n/c30/c2e/c38/c54\n/c31/c2e/c30/c54/c31\n/c30/c32\n/c4e/c3d/c34/c30/c65/c6c /c34/c31 /c34/c32 /c33/c39 /c33/c38\n/c28/c61/c29\n/c30/c2e/c30/c20/c54\n/c28/c62/c29\n/c28/c63/c29 /c64/c49/c2f/c64/c56\n/c32/c28/c65/c2f/c68/c29\n/c20/c30/c20/c30/c2e/c32/c20/c30/c2e/c34/c20/c30/c2e/c36/c20/c30/c2e/c38\n/c20/c33/c2e/c39/c32 /c20/c33/c2e/c39/c34 /c20/c33/c2e/c39/c36 /c56/c20/c28/c56/c29/c67/c28/c64/c29\nFigure 2. Feedback effects in a non-driven resonator (all measurements at Tmc,base). (a)\nDifferential conductance d I=dVsd(Vg;Vsd)(no rf signal applied; lock-in measurement with\nan excitation of Vsd,ac=5mV RMS at 137Hz) of the carbon nanotube quantum dot at zero\nmagnetic field, displaying four-fold shell filling combined with Kondo effect and traces of\nsuperconductivity in the metallic leads (see text). At finite bias, strong switching effects\nattributable to mechanical self-excitation become visible, indicated by white arrows [11, 21].\n(b) Detail of (a) for increasing magnetic field, this time plotting the numerical derivative\nof the simultaneously measured dc current Idc(Vg;Vsd). Already at B=0:8T, the self-\ndriving effects are completely suppressed. (c) dc current Idc(Vg)along the trace of constant\nVsd=\u00001:15mV, as marked in (b) with a dotted line, and (d) corresponding differential\nconductance d I=dVsd(Vg).\nwith mechanical motion of two adjacent suspended nanotube segments of different length, as\nvisible in the chip geometry of figure 1(b). Assuming the minimum resonance frequency close\nto charge neutrality to be the case of vanishing mechanical tension, the ratio of the minimum\nfrequencies of A and B, fmin,A=fmin,B=253MHz =182MHz'1:39 agrees very well with\nthe expectation from the different trench widths (`B=`A)2= (700nm =600nm )2'1:36. The\ndetailed mechanism leading to the signal contribution of the second nanotube segment next to\nthe contacted 700nm gap is still under investigation.\nIn the following we return to measurements without any applied rf driving signal.\nFigure 2(a) displays a lock-in measurement of the differential conductance of the suspended\ncarbon nanotube quantum dot, as function of gate voltage Vgand bias voltage Vsd. A positive\ngate voltage is used to tune the quantum dot into the regime with an electron number of\n38\u0014Ne\u0000\u001442, where it is strongly tunnel-coupled to the contact electrodes. Several featuresMagnetic damping of a carbon nanotube NEMS resonator 5\non the plot can immediately be identified and are well understood. The narrow, approximately\ngate voltage independent conductance minimum around Vsd=0 in figure 2(a) is caused\nby superconductivity of the metallic rhenium leads; two Kondo ridges of enhanced low-\nbias conductance become clearly visible around Vg=3:84V ( Ne\u0000=39) and Vg=4:0V\n(Ne\u0000=41).\nIn addition, the differential conductance signal from figure 2(a) exhibits sharply\ndelineated regions of modified signal level, often accompanied by switching behavior, see\nwhite arrows in the figure. This has already been observed previously in clean suspended\ncarbon nanotube quantum dots [11]. As predicted in Refs. [21, 22] and confirmed in\nRef. [11], in these parameter regions single electron tunneling from dc current alone suffices\nto coherently drive the mechanical motion via a positive feedback mechanism. In turn, this\nbecomes visible in the recorded current or conductance signal as well.\nThe panels of figure 2(b) display a detail enlargement of the parameter region of\nfigure 2(a), this time plotting as differential conductance the numerical derivative of the dc\ncurrent recorded simultaneously with the lock-in signal. Although this value is affected by\na larger noise level, it reproduces more faithfully one-time switching events while sweeping\nthe bias voltage, which delimit the feedback regions. A clear substructure emerges inside\nthe feedback region, which so far has not found any equivalent in theoretical considerations.\nIn addition, when increasing an externally applied magnetic field perpendicular to the chip\nsurface, the parameter regions of positive feedback shrink. As can be seen from the panels\nof figure 2(b), applying a magnetic field of B=0:8T already completely suppresses the self-\ndriving effect within the observed region.\nThis is further illustrated by the line traces of figure 2(c) and (d), displaying the dc\ncurrent (c) and the differential conductance (d) as function of gate voltage Vgat constant\nVsd=\u00001:15mV across the parameter regions of figure 2(b). While no significant changes\ntake place outside the positive feedback region, the discontinuous behavior at zero field\nbecomes smooth, and only slight fluctuations remain at B=1T. In particular the current\nagrees very well with the prediction of Refs. [21, 22] in the cases of present and suppressed\nfeedback.\nWhile it has been shown in Ref. [11] that large electronic tunnel rates are an\nimportant prerequisite for self-excitation, here the conductance remains unchanged outside\nthe feedback-dominated regions. The magnetic field does not significantly influence the\nelectronic tunnel rates, excluding such a mechanism for the suppression of the self-\nexcitation. A second prerequisite is a high mechanical quality factor [21, 22], since the\nfeedback mechanism has to compensate and overcome damping of the mechanical oscillation.\nConsequently, the suppression of self-driving indicates a magnetic field induced additional\ndamping mechanism.\nTo verify this conclusion from the pure dc measurements, we measure the frequency\ndependence of the radio frequency-driven resonator response. An additional damping\nmechanism in a magnetic field should here become visible as a resonance peak broadening, i.e.\na decrease in effective quality factor Q. A constant positive gate voltage Vg=3:91V is used\nto tune the quantum dot into the Coulomb blockade region with electron number Ne\u0000=40.Magnetic damping of a carbon nanotube NEMS resonator 6\n/c28/c62/c29 /c28/c64/c29 /c28/c61/c29\n/c31/c54\n/c33/c54\n/c2d/c37/c35 /c2d/c32/c35 /c30 /c44/c66/c20/c28/c6b/c48/c7a/c29 /c2d/c35/c30/c30/c2e/c30/c30/c2e/c35/c31/c2e/c30/c6e/c6f/c72/c6d/c61/c6c/c69/c7a/c65/c64/c20 /c64/c49\n/c34/c39/c31/c2e/c34 /c34/c39/c31/c2e/c37 /c66/c20/c28/c4d/c48/c7a/c29/c2d/c30/c2e/c32/ca6/c20/c28/c72/c61/c64/c29/c30/c2e/c30/c64/c49/c20/c28/c70/c41/c29\n/c30/c2e/c30/c30/c2e/c35/c31/c2e/c30/c31/c2e/c35\n/c30/c2e/c32/c58\n/c59/c52\n/c43/c4e/c54/c28/c63/c29/c43\n/c31/c35/c30/c30/c30\n/c42/c20/c28/c54/c29/c51\n/c31/c32/c30/c30/c30/c30/c32/c35/c30/c30/c30\n/c30 /c32\nFigure 3. Broadening of the driven mechanical resonance in a magnetic field (all\nmeasurements at Tmc,base). (a) Observed mechanical resonance for Vg=3:91V and Vsd=\n0:3mV (nominal maximal driving signal after attenuators \u000042dBm). The mechanical driving\nsignal is amplitude-modulated at low frequency fam=137Hz; the plot displays the lock-\nin response (upper panel: x,y, lower panel: f) at that frequency of the averaged current\nthrough the carbon nanotube quantum dot [10]. (b) Selected mechanical resonance curves\n(ylock-in signal) for B=1T and B=3T (nominal maximal driving signal \u000041dBm). At\nthe higher magnetic field, a slightly broader resonance can be observed. (c) Circuit model\nof electromechanical damping by Ohmic dissipation, see text. (d) Quality factor Qof the\nresonance extracted from multiple traces as in (b), as function of magnetic field. The solid line\nprovides a model fit, assuming both a magnetic field induced and a magnetic field independent\ndamping contribution (see text).\nBecause of the transparent tunnel barriers to the leads, significant cotunneling conductance\non the order of Gcot'0:4e2=hcan still be observed in this parameter region, enabling the\ndetection of the mechanical resonance in dc current. Extending the mechanical resonance\ndetection setup of Refs. [10, 11] to increase sensitivity, the applied radio frequency signal is\namplitude-modulated at a low frequency fam=137Hz, such that the period 1 =fam'7ms\nis larger than the oscillation decay timescale \u0018Q=fexpected from literature [10]. The\ncorresponding low-frequency modulation of the current signal is recorded by a lock-in\namplifier. In addition, we drive at double frequency (A’ in figure 1(d)) as this results in a\nstronger resonance signal.\nFigure 3(a) displays a typical resulting in-phase ( x), out-of-phase ( y), and phase angle ( f)\namplitude modulation response signal as function of the driving frequency f. In the in-phase\n(x) response, a multi-peak structure emerges. Indications of this multi-peak shape (see arrows\nin figure 3(a)) remain visible even at lowermost driving power and suggest a more complex\ncoupling of the radiofrequency driving signal into the electromechanical system than only\nactuation via electrostatical force [10, 20, 19]. As can be seen from both the yresponse and\nthe phase angle, in spite of the low amplitude modulation frequency, a distinct phase shift of\nthe response on resonance is still visible. The shift of approximately Df=0:2rad corresponds\nto a delay time of Dt=0:35ms, on the order of 105mechanical oscillation cycles. Given that\npreviously observed nanotube resonators [10, 20] have exhibited quality factors on that orderMagnetic damping of a carbon nanotube NEMS resonator 7\nof magnitude, this is consistent with mechanical storage of vibration energy and later release\nwithin one amplitude modulation cycle, leading to a delayed driving response.\nTo avoid fitting of the multi-peak structure in the in-phase ( x) signal, we focus in the\nfollowing on the out-of-phase ( y) signal induced by the phase shift. Figure 3(b) shows selected\nfrequency response traces of the mechanical resonance, recorded at an external magnetic field\nofB=1T and B=3T, respectively, and pointing towards a slight broadening of the peak\nstructure at higher magnetic field. Evaluation of many similar curves, including repeated\nmeasurements at the same magnetic field value and over a large driving power range, leads\nto the plot of figure 3(d). Here, the width of the resonance peaks is plotted in terms of an\nexperimentally observed quality factor Qas function of the magnetic field B. Indeed, the\nmeasured peak width increases (and Qdecreases) significantly above B=1T.\nA straightforward circuit model sketched in figure 3(c) can be used to describe the\nmagnetic field dependence. A vibration component perpendicular to the magnetic field leads\nto an induced ac voltage across the resonator. We assume the carbon nanotube resonator to\nbe partially electrically shortened in the rf signal frequency range via an Ohmic resistance R\nand a large parasitic capacitance. For simplicity, we do not take into account the deflection\nshape but assume a uniform beam deflection along the entire nanotube of length Land mass\nmto estimate the magnetic flux modulation. As a result, eddy currents lead to a damping of\nthe mechanical motion corresponding to\nQm(B) =q\nB2q=2pfRm\nL22p\n2; (1)\nas both induced voltage and resulting eddy current are proportional to B. Assuming an\nadditional magnetic field independent resonator damping, which determines the zero external\nfield quality factor Q0, we obtain the expression\nQ(B) =Q0Qm(B)\nQ0+Qm(B)(2)\nThe solid line in figure 3(d) provides a best fit of this function to the data, using Q0andqas\nfree parameters and resulting in the values q=5:381\u0002105T2andQ0=25020. As visible\nin figure 3 this model describes our measurement results well. This thereby confirms the\npresence of a magnetic-field induced damping mechanism. Using the resonance frequency f\nand estimating L'700nm and m'1:3\u000110\u000021kg, we obtain a value for the Ohmic resistance\nofR'200kWin the replacement circuit of figure 3(c).\nAs a last remark, using the fit function of figure 3(d) one obtains Q(0:8T)=Q(0T)'0:97,\ni.e. only a very small decrease of the effective quality factor within the magnetic field range\ncovered in figure 2. A likely reason for this is that the resonance peak widths evaluated in\nfigure 3(d) do not solely correspond to the device quality factor entering the self-excitation,\nbut are broadened by additional mechanisms, leading to an underestimation of the low-field\nquality factor Q0.\nSummarizing, we characterize a quantum dot in a suspended ultra-clean single wall\ncarbon nanotube, which also acts as nano-electromecanical resonator. We observe how\nfeedback and self-driving effects, where only dc current is sufficient to drive resonator motion,\nare suppressed in a finite magnetic field. The conclusion that the magnetic field inducesMagnetic damping of a carbon nanotube NEMS resonator 8\nan additional damping mechanism is confirmed by tracing the driven resonator response\nas a function of magnetic field. We model the decrease of the mechanical quality factor\nsuccessfully using eddy current damping.\nAcknowledgments\nThe authors would like to thank the Deutsche Forschungsgemeinschaft (Emmy Noether grant\nHu 1808/1, SFB 631 TP A11, GRK 1570) and the Studienstiftung des deutschen V olkes for\nfinancial support.\nReferences\n[1] Sazonova V , Yaish Y , ¨Ust¨unel H, Roundy D, Arias T A and McEuen P L 2004 Nature 431284\n[2] Witkamp B, Poot M and van der Zant H 2006 Nano Letters 62904\n[3] Reulet B, Kasumov A Y , Kociak M, Deblock R, Khodos I I, Gorbatov Y B, V olkov V T, Journet C and\nBouchiat H 2000 Phys. Rev. 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(b) 2472974\n[13] Lassagne B, Tarakanov Y , Kinaret J, Garcia-Sanchez D and Bachtold A 2009 Science 281107\n[14] Kouwenhoven L P, Marcus C M, McEuen P L, Tarucha S, Westervelt R M and Wingreen N S 1997 Electron\ntransport in quantum dots (Kluwer)\n[15] Liang W, Bockrath M, Bozovic D, Hafner J H, Tinkham M and Park H 2001 Nature 411665–669\n[16] Oreg Y , Byczuk K and Halperin B I 2000 Phys. Rev. Lett. 85365–368\n[17] Goldhaber-Gordon D, Shtrikman H, Mahalu D, Abusch-Magder D, Meirav U and Kastner M A 1998\nNature 391156–159\n[18] Cleland A N 2003 Foundations of Nanomechanics (Springer Verlag)\n[19] Eichler A, Chaste J, Moser J and Bachtold A 2011 Nano Letters 112699\n[20] Laird E A, Pei F, Tang W, Steele G A and Kouwenhoven L P 2012 Nano Letters 12193–197\n[21] Usmani O, Blanter Y M and Nazarov Y V 2007 Phys. Rev. B 75195312\n[22] Usmani O 2006 Strong Feedback in Nanoelectromechanical Systems Ph.D. thesis Technische Universiteit\nDelft" }, { "title": "0908.3146v1.Surface_Alfven_Wave_Damping_in_a_3D_Simulation_of_the_Solar_Wind.pdf", "content": "arXiv:0908.3146v1 [astro-ph.SR] 21 Aug 2009Surface Alfv´ en Wave Damping in a 3D Simulation of the Solar W ind\nR. M. Evans1and M. Opher1\nGeorge Mason University, 4400 University Drive, MSN 3F3, Fa irfax, VA 22030\nV. Jatenco-Pereira2\nUniversidade de S˜ ao Paulo, Inst Astronomico e Geofisico, Rua do Mat˜ ao 1226, Cidade\nUniversit´ aria, BR Sao Paulo, SP 05508-900, Brazil\nand\nT. I. Gombosi3\nCenter for Space Environment Modeling, University of Michig an, 2455 Hayward Street, Ann\nArbor, MI 48109\nrevansa@gmu.edu\nABSTRACT\nHere we investigate the contribution of surface Alfv´ en wav e damping to the heating\nof the solar wind in minima conditions. These waves are prese nt in regions of strong\ninhomogeneities in density or magnetic field (e. g., the bord er between open and closed\nmagnetic field lines). Using a 3-dimensional Magnetohydrod ynamics (MHD) model, we\ncalculate the surface Alfv´ en wave damping contribution be tween 1-4 R⊙(solar radii),\nthe region of interest for both acceleration and coronal hea ting. We consider waves with\nfrequencies lower than those that are damped in the chromosp here and on the order of\nthose dominating the heliosphere: 3 ×10−6−10−1Hz. In the region between open and\nclosed field lines, within a few R⊙of the surface, no other major source of damping has\nbeen suggested for the low frequency waves we consider here. This work is the first to\nstudy surface Alfv´ en waves in a 3D environment without assu ming a priori a geometry\nof field lines or magnetic and density profiles. We demonstrat e that projection effects\nfrom the plane of the sky to 3D are significant in the calculati on of field line expansion.\nWe determine that waves with frequencies >2.8×10−4Hz are damped between 1-4 R⊙.\nIn quiet sun regions, surface Alfv´ en waves are damped at fur ther distances compared\nto active regions, thus carrying additional wave energy int o the corona. We compare\nthe surface Alfv´ en wave contribution to the heating by a var iable polytropic index and\nfind that it an order of magnitude larger than needed for quiet sun regions. For active\nregions the contribution to the heating is tweny percent. As it has been argued that\na variable gamma acts as turbulence, our results indicate th at surface Alfv´ en wave\ndamping is comparable to turbulence in the lower corona. Thi s damping mechanism– 2 –\nshould be included self consistently as an energy driver for the wind in global MHD\nmodels.\nSubject headings: Sun: corona, Sun: magnetic fields, Sun: solar wind, waves\n1. Introduction\nThe physical mechanisms behind the heating of the solar coro na and the acceleration of the\nfast solar wind are two major unresolved issues in solar phys ics. Thermal heating alone is not\nsufficient to bring models into agreement with observations o f the lower corona and at Earth\n(Usmanov & Goldstein 2003). The acceleration of the solar wi nd occurs predominantly within a\nfew solar radii of the surface (Hartmann & MacGregor 1980; Gr all et al. 1996). Additionally, Solar\nand Heliospheric Observatory (SOHO) observations have sho wn that ions are heated below 4 R⊙\n(Kohl et al. 1998; Esser et al. 1999). Grall et al. (1996) sugg ested that because the locations for\nthe heating of the corona and the acceleration of the solar wi nd are the same, it is possible that\nthe same mechanism could contribute to both.\nOnesourceofheatingismagneticreconnectionassociatedw ithflaresandnanoflaresatthesolar\nsurface. Using extreme ultraviolet observations, Patsour akos & Klimchuk (2009) have suggested\nthe heating of active regions is impulsive, and therefore co uld be associated with nanoflares. In the\nhigh speed solar wind, reconnection events current sheets a nd filaments have been suggested as a\nheating mechanism (Mattheaus et al. 2003).\nIn the lower corona, the region of interest in this work, damp ing of Alfv´ en waves is known\nto produce nonthermal acceleration and to bring models into agreement with observations both\nnear the Sun and at large distances (Parker 1965). The possib le damping mechanisms for Alfv´ en\nwaves in the photosphere to lower corona are numerous and inc lude nonlinear damping (Wentzel\n1989; Ofman & Davila 1997), turbulent cascade (Hollweg 1986 ; Mattheaus et al. 1999), phase mix-\ning (Heyvaerts & Priest 1983; Parker 1991), Landau damping ( Hollweg 1971), neutral collisional\n(De Pontieu, Martens & Hudson2001;Leake et al.2005),ion-c yclotron damping(Isenberg, Lee & Hollweg\n2001), and surface Alfv´ en wave damping (Ionson 1978) (and r eferences herein).\nGlobal magnetohydrodynamics (MHD) models that do not inclu de waves employ different\nmethodssuchasempiricalheatingfunctionsorvariedpolyt ropicindexdistributiontodriveandheat\nthe solar wind and are able to match well with Ulysses, Yohkoh , Helios and Advanced Composition\nExplorer data (Mikic et al. 1999; Groth et al. 2000; Roussev e t al. 2003). To include Alfv´ en wave\ndamping in a realistic model of the solar environment, physi cal damping mechanisms must be\nspecified. Some MHD models use Alfv´ en waves without specify ing a damping mechanism. Global\n(Usmanov & Goldstein 2006) and local (Cranmer, van Ballegoo ijen & Edgar 2007) models include\nadditional equations for the waves and prescribe an empiric al damping length. These studies are\nbenchmarked with Ulysses and Helios observations. Alterna tively, one can include the wave energy– 3 –\nand momentum without damping (Lionello, Linker & Mikic 2009 ), and match SOHO Extreme\nUltraviolet and Yohkoh soft X-ray observations.\nOf the numerous possible mechanisms for damping Alfv´ en wav es, those which we expect to be\nimportant for low frequency waves are nonlinear turbulent d amping, phase mixing, surface Alfv´ en\nwaves. In this paper, we study the contribution of surface Al fv´ en wave damping by utilizing a 3D\nglobal MHD simulation of a thermally-driven solar wind (Coh en et al. 2007). In this model, the\nlower corona is the inner boundary (chromosphere and transi tion region are not resolved.) Waves\nare not explicitly included in the model, nor can low frequen cy waves be resolved due to spatial\nand temporal resolution limits (to be discussed further in S ection 2.2).\nWe consider waves with frequencies lower than those that are completely damped in the chro-\nmosphere and on the order of those dominating the heliospher e: 3×10−6−10−1Hz (periods 3\nseconds to 3 days). Alfv´ en waves have been detected in the lo wer layers of the solar atmosphere\nusing both ground based observations (Jess et al. 2009) and t he Hinode spacecraft (Okamoto et al.\n2007; Cirtain et al. 2007; De Pontieu et al. 2007) with period s 2-4 minutes (4 .2−8.3×10−3Hz).\nGround based observations indicate the presence of Alfv´ en waves in the corona with periods of five\nminutes (3 .3×10−3Hz) (Tomczyk et al. 2007; Tomczyk & McIntosh 2009). At 1 AU, th e dominant\nwave power is in waves with periods of 1-3 hours (9 .2×10−5to 2.8×10−4Hz) (Belcher & Davis\n1971).\nInthisstudyweareinterestedinadampingmechanismthatac tsinthelowercorona(Grall et al.\n1996). In the chromosphere, Alfv´ en waves with frequencies above 0.6 Hz are damped by ion-neutral\ncollisional dampingandfrequenciesbelow10−2Hzwereunaffected(De Pontieu, Martens & Hudson\n2001; Leake et al. 2005). Cranmer & van Ballegooijen (2005) f ound that waves below 10−2Hz were\nnot damped by any mechanism in the chromosphere. Cranmer & va n Ballegooijen (2005) also\nfound that nonlinear damping occurred over the extended cor ona. Verdini & Velli (2007) found\nwaves 10−6−10−4Hz were reflected by a gradient in the background Alfv´ en profi le, and dissipated\nnot in the lower layers of the atmosphere, but over distance o f a few solar radii. In the corona,\nthe ion-cyclotron frequency is 104−6Hz, so cyclotron resonance damping is not relevant for low\nfrequency waves. Phase mixing of outgoing and reflected inco ming Alfv´ en waves has also been\nstudied Suzuki & Inutsuka (2005, 2006).\nUsing a combination of three damping mechanisms (nonlinear damping, surface Alfv´ en wave\ndamping, and phase mixing), Jatenco-Pereira & Opher (1989) were able to match observations of\nmass loss rates and terminal velocities for cool, giant star s. They applied their model to the Sun\nand were able to obtain coronal heating and match wind veloci ty and Alfv´ en wave power density\nobservations in a 1D simulation (Jatenco-Pereira, Opher & Y amamoto 1994). In this paper, we\nextend their work on surface Alfv´ en wave damping in a 3D simu lation of the solar corona.\nSurface Alfv´ en waves form on a magnetic interface - a finite t hickness boundary separating\ntwo regions of plasma with a strong inhomogeneity in magneti c field and/or density. The Alfv´ en\nwave in each region can interact, damp and transfer energy in to the resonant layer separating the– 4 –\ntwo plasmas (resonant absorption). Ionson (1978) first util ized surface Alfv´ en waves and resonant\nabsorption as a mechanism to heat coronal loops. The transfe r of MHD wave energy by resonant\nabsorption was also studied in Hollweg (1987) and Wentzel (1 979). An alternative dissipation\nmechanism for surface Alfv´ en waves is nonlinear wave steep ening (Ruderman 1992). These and\nother efforts, e.g. (Lee & Roberts 1986) have resulted in dampi ng lengths which depend on the\nfrequency of the waves, the nature of the magnetic interface , and the local plasma parameters\n(density, magnetic field and velocity).\nUtilizing these relations, the profile of the damping length in the wind has been estimated\n(Jatenco-Pereira & Opher 1989; Narain & Sharma 1998). All pr evious studies made assumptions\nabout the wind. For example, (Narain & Sharma 1998) calculat ed nonlinear viscous damping of\nsurface Alfv´ en waves in polar coronal holes. They assumed t wo values of the superradial expansion\nof the magnetic field lines, profiles for density (based on obs ervations), and a single frequency (0.01\nHz). They obtained one profile, and concluded that the nonlin ear damping of the surface Alfv´ en\nwaves in region of strong magnetic field expansion should con tribute significantly to the heating in\nthe solar wind.\nThe surface Alfv´ en wave damping length depends on the profil e of the background Alfv´ en\nspeed. As was shown in a survey of Alfv´ en speed profiles from s everal MHD models (Evans et al.\n2008), theAlfv´ en profilesforVerdini & Velli (2007) andCra nmer, van Ballegooijen & Edgar(2007)\nare almost identical below 10 R⊙, and the profiles were different from MHD models using empirica l\nheating functions. The profile for Usmanov & Goldstein (2006 ) is similar to these two, but differs\nvery low in the corona. Evans et al. (2008) concluded that the inclusion of Alfv´ en waves with\nempirical damping brought MHD models better in alignment wi th local heating studies that had\nthe best agreement with observations.\nIn the present study, we quantify the surface Alfv´ en wave da mping length for use in an MHD\nwave-driven model. We expect to find surface Alfv´ en waves in the border between the fast and slow\nsolar wind for two reasons: a) the gradient in density and b) t he superradial expansion of the open\nmagnetic field required to fill the space over closed streamer s. We will focus on the superradial\nexpansion of the field lines and compare our 3D MHD model with t he observational study of\nDobrzycka et al. (1999). Our calculations show that waves wi th periods less than 1 hour (frequency\ngreater than 2.8 ×10−4Hz) are damped in the region between 1-4 R⊙, the region of interest for\nboth solar wind acceleration and coronal heating (Grall et a l. 1996). No other major source of\ndamping has been suggested for these waves in this region. We demonstrate the importance of\nthe 3D geometry in our results. We show that the contribution of the damping of surface Alfv´ en\nwaves to the wind is on the order of magnitude as (or in some cas es larger than) turbulence. It is\nimportant to note that this study of waves in a solar wind solu tion was not self consistent - we did\nnot consider any back effects on the waves from the plasma.\nThe paper is organized as follows: in section 2 we describe th e theory and numerical simulation\nbackground. In section 3 we calculate the location of corona l hole boundaries, and the damping– 5 –\nlength profile along those lines. In section 4 we estimate the energy surface Alfv´ en waves will\ndeposit below 4 R⊙. We also calculate the heating invoked in a semi-empirical t hermodynamical\nmodel, compare with the wave flux and discuss the results. Fin ally, conclusions can be found in\nsection 5.\n2. Methods and Data\n2.1. Theory\nThe inhomogeneity in density and/or magnetic field that give s rise to the surface Alfv´ en waves\ncan be described analytically as either a discontinuity or fi nite layer (Hasegawa & Uberoi 1982),\nsuch as a flux tube. For the case of a rapidly expanding flux tube of width a, (where ais much\nsmaller than the radius of the flux tube), surface Alfv´ en wav es form on the inner and outer surfaces.\nThese waves can interact, damp and deposit energy into the su rrounding plasma with a damping\nrate (Lee & Roberts 1986),\nΓSW=π(¯ka)/parenleftbiggω2\n2−ω2\n1\n8ω/parenrightbigg\n(1)\nwhere¯kis the average wave number, ωis the frequency and ω1andω2are the Alfv´ en wave\nfrequency on either side of the flux tube (1 representing insi de and 2 outside.) We assume the\nwidth of the flux tube to be much smaller than the radius. This a llows us to take ¯ka= 0.1, as in\nJatenco-Pereira & Opher (1989). If the frequency on the outs ide is much larger than the frequency\ninside (i.e., a strong inhomogeneity), then the damping rat e is\nΓSW=πω(¯ka)\n4√\n2. (2)\nThe surface Alfv´ en wave damping length can be written as the Alfv´ en speed vA=/radicalig\nB2\n4πρ\ndivided by the damping rate,\nLSW=vA\nΓSW=vA4√\n2\nωπ(¯ka). (3)\nThe initial damping length L0can be written as\nL0=vA04√\n2\nωπ(¯ka), (4)\nwhich, by taking ¯ka= 0.1, can be simplified to\nL0= 18vA0\nω. (5)\nUtilizing the relation that a∝A(r)1\n2∝rS\n2(whereSis the superradial expansion factor of the\nfield line), and fixing the frequency of the waves to be constan t with height, the damping length in– 6 –\nthe inertial frame is\nLSW=L0/parenleftigr0\nr/parenrightigS\n2/parenleftbiggvA\nvA0/parenrightbigg2\n(1+MA) (6)\nwhereMA=uSW\nvAis Alfv´ en Mach Number, uSWis the solar wind speed, Bis the magnetic field\nstrength, ρis the mass density. The subscript 0 indicates the variable i s to be evaluated at the\nreference height.\nIn the present study the model does not treat the chromospher e or the transition region; the\nlower corona is the inner boundary. The cell size at the inner boundary is3\n128R⊙, and so we chose\nr0=1.04R⊙as our reference height to assure that calculations would no t include the solar surface\ninner boundary cells.\nWe quantify the expansion of open field lines by S, given by.\nAcs(r) =Acs(r0)/parenleftbiggr\nr0/parenrightbiggS(r)\n(7)\nwhereAcs(r) is the cross sectional area of the flux tube at distance r. A value of 2 for Sindicates\npure radial expansion. The lines which border closed field li nes must open faster than radial to fill\nthespaceabovetheclosed loops. Instudieswhere Sisnotafunctionof r, typical values chosenwere\n2-6 for open field regions from 1-10 R⊙(Moore et al. 1991; Jatenco-Pereira, Opher & Yamamoto\n1994; Narain & Sharma 1998). Here we determine Sexplicitly as a function of height from 1.04-10\nR⊙.\nA similar parameter for the expansion of a field line is the sup erradial diverging factor or\nsuperradial enhancement factor f, as in\nAcs(r) =Acs(r0)/parenleftbiggr\nr0/parenrightbigg2\nf(r). (8)\nWe will use a 3D MHD model as a laboratory to estimate the contr ibution of the surface waves to\nthe wind from 1.04-10 R⊙.\n2.2. Generation of Steady State\nWe obtain steady state solar wind solutions by using the Sola r Corona component of the\nSpaceWeather ModelingFramework(SWMF), developedbytheU niversityofMichigan (Toth et al.\n2005). This 3D global magnetohydrodynamics (MHD) model inc orporates Michelson Doppler Im-\nager (MDI) magnetograms to generate an initial magnetic fiel d configuration with the Potential\nField Source Surface model (see Cohen et al. (2007) for detai ls). An initial density is assumed on\nthe solar surface (3 .4×108cm−3), and the MHD equations are evolved in local time steps (12,0 00\niterations) and time accurate calculations (for ten minute s) to achieve steady state solutions for\nsolar minima conditions in a 24 ×24×24R⊙domain.– 7 –\nThe steady state was generated with Carrington Rotation (CR ) 1912. Solar wind solutions\nfrom SWMF were validated in Cohen et al. (2008) from CR1916-1 929 by comparing with Advanced\nComposition Explorer and Wind satellite data (near 1 AU). CR 1912 was chosen to allow for com-\nparisons to an observational study of the expansion of open fi eld lines (Dobrzycka et al. 1999).\nWaves occur naturally as a perturbation to the MHD equations , and so their presence may be\nexpected when solving the MHD equations in space and time. Ho wever, in global simulations waves\nhavetobeincludedexplicitly (Usmanov & Goldstein 2003)du etotimeandspatial limitations. The\ntime step of this simulation (0.2 seconds) is less than the sm allest period considered in this analysis\n(3 seconds). Additionally, the grid resolution is not enoug h to spatially resolve the waves.\n3. Coronal Hole Boundary Analysis\n3.1. Location and Expansion Factor\nDobrzycka et al. (1999) (herein referred to as DO99) charact erized the large-scale solar mag-\nnetic topology during solar minima conditions (August 1996 ) using data from the Solar and Helio-\nspheric Observatory’s (SOHO) Ultraviolet Coronagraph Spe ctrometer instrument. They analyzed\nthe latitudinal dependence of two line emission intensitie s and found the values were constant\nwithin the large polar coronal holes but suddenly increased at the border of the holes and equa-\ntorial streamers. DO99 used this increase in intensity to id entify the colatitude of coronal hole\nboundary (CHB), i.e. the border between open and closed magn etic field lines. In the present\nstudy, we identify the CHB locations as the open field line wit h the largest colatitude (the angle\nmeasured from the pole to the equator) in the steady state des cribed in the previous section. We\ncompare our calculation of the field expansion with the obser ved values from DO99.\nFigure 1 shows the 3D coronal hole boundary (CHB) field lines o btained from the model. The\ndifferent colors refer to: red as northeast (NE); blue as south east (SE); green as southwest (SW);\nand purple as northwest (NW). The solar surface is shown colo red by the radial component of the\nmagnetic field. On 1996 August 17, there was an active region n ear the center of the disk, and\nmostly quiet sun at the intersection of the plane of the sky wi th the photosphere.\nThe first two rows of Table 1 provide CHB colatitudes (angle me asured from the pole to the\nCHB footpoint) found in DO99 and this study. We find that 3 of th e 4 simulated CHB are higher in\nlatitude (i.e., smaller colatitude) compared to DO99. As we discuss below, these differences could\nbe due to projection effects.\nThe superradial expansion factor Sand superradial enhancement factor fwere calculated as\na function of height for each CHB field line. In Figure 2 we show ffor the CHB lines in this study.\nAdditionally, we have included the fprofiles which correspond to the minimum and maximum\nasymptotic values of ffor CHBs determined in DO99. Rows 3 and 4 in Table 1 provide the\nasymptotic superradial enhancement factor value for each l ine. We find that both Sandfcover– 8 –\na larger range of values compared to DOB99. Only the SW line fr om our simulation falls in the\nrange from DO99 and only from 3-6 R⊙. In general, we find that 3 of the 4 boundary lines (all\nexcept SW) have lower values in our simulation compared to DO 99.\nThe best agreement for the location of the CHB between the stu dies is the SW line. The\nsouthern hemisphere from the simulation is more similar to t he results deduced from observations\nintheplaneoftheskyinDO99thanthenorthernhemisphere. O verall, wefindthatDobrzycka et al.\n(1999) had larger values of f, and a small range of values, for the same field lines. As we sho w\nbelow, this result is due to 3D vs. 2D projection effects.\n3.2. Projection Effects\nIn order to quantify the significance of 2D projection effects i n the context of the expansion of\nfield lines, we calculated ffor four field lines shown in Figure 3a. The projection of the fi eld lines\non the plane of the sky is shown in Figure 3b. Contours of solar wind speed (inkm\ns) are shown in\nFigures 3a and b, and the solar surface is shown as white spher e. Figure 3c provides the superradial\nenhancement factor f, calculated according to Equation 8 for each field line (labe led A, B, C and\nD as in Figure 3b). The calculation of the 3D line is shown as so lid lines, and the 2D projection\nline calculation is shown as dashed lines.\nFigure 3c demonstrates that a 2D projection of a 3D field line o n the plane of the sky can\noverestimate the divergence of the line. The 3D lines from A- D all approach values of f 4 while\nthe 2D estimates vary between 4 and 13. This projection effect e xplains why the values for ffrom\nour simulation are smaller than those determined in the obse rvational study of DO99. It is crucial,\ntherefore, to do studies of surface Alfv´ en wave damping in a 3D simulation in order to capture the\ntrue divergence of the field lines.\n3.3. Damping Length\nIn Figure 4, we plot LSW(see Eq. 6) which was calculated using parameters ρ,Bandusw\nfrom the steady state solution. Figure 4a presents LSWfor the coronal hole boundary field lines\nin Figure 1 with frequency 4 .17×10−3Hz, normalized to the initial damping length L0of the SW\nline (chosen because of the agreement with DO99). This norma lization allows for comparison of\nthe profile features from different source regions as a functio n of height. In Figure 4b we feature\nonly the SW line and present LSWfor several frequencies, from 3 .3×10−1−3.8×10−6Hz. It\ncan be seen in Figure 4b that frequencies above 2 .8×10−4(short dashed line) will be appreciably\ndamped within a few solar radii of the surface.\nFigure 4a shows distinctly different profiles from the souther n and northern CHB lines. We\nexamined the source region of each footpoint and found that t he SE and SW lines originated near– 9 –\nsmall active regions in which the radial component of the mag netic field Br≈50 G. Both northern\nhemisphere lines originated from quiet sun regions ( Br≈1 G). For SWMF and other MHD models,\nEvans et al. (2008) showed that the Alfv´ en profile will conta in a maximum, or hump, if the source\nregion is quiet sun. The profile from an active region in globa l models begins at a maximum value,\nand drops to less than a few hundredkm\nswithin one solar radius from the surface.\nThe profile of LSWis controlled by the Alfv´ en speed profile. The normalized pr ofiles in Fig.\n4a show that the position corresponding to LSW= 1R⊙is closest to the Sun for active regions.\nThe profiles from quiet Sun source regions have a plateau, pus hingLSW= 1R⊙further from the\nSun. The implication of this result can be seen in the equatio n relating the Alfv´ en wave energy\ndensity,\nǫSW=/parenleftbiggMA0\nMA/parenrightbigg/parenleftbigg1+MA0\n1+MA/parenrightbigg2\nexp/parenleftbigg\n−r\nLSW/parenrightbigg\n. (9)\nIf the damping length is 1 R⊙or less, then the waves will be damped close to the Sun. Theref ore,\nthe presence of the hump means the energy of the surface Alfv´ en wave can travel further into the\ncorona before substantial damping occurs. This means that t he quiet sun region will damp more\nsurface waves at further distances, so it is more efficient in c arrying the wave momentum out into\nthe corona. Active regions will damp closest to the Sun.\n4. Dissipation of Wave Energy and Heating\n4.1. Wave Energy\nIn the previous section we considered how surface Alfv´ en wa ves at the solar surface with the\nfrequency range 3 .8×10−6to 3.3×10−1Hz (Cranmer & van Ballegooijen 2005) would be damped\nin our background solar wind environment. We found that wave s with frequency above 2 .8×10−4\nHz were appreciably damped below 4 R⊙. We now consider the contribution of their wave flux to\nthe energy of the wind. There will be a contribution to the mom entum of the wind as well, but in\nthis analysis we ignore this contribution. We assume that th e wave damping will contribute solely\nto heating the wind; therefore we derive here an upper limit o n their contribution to the heating of\nthe plasma. The spectra of the surface Alfv´ en waves (Jatenc o-Pereira, Opher & Yamamoto 1994)\nis:\nφAW(ω) =φ0/parenleftigω\n¯ω/parenrightig−αerg\ncm2sHz(10)\nwhereφ0= 1.3×105erg\ncm2sHz, ¯ωis the mean frequency in the observed range and the power inde x\ncorresponding to the low frequency waves we are considering isα=0.6 (Tu et al. 1989).\nWe assume that this flux of Surface Alfv´ en waves is propagati ng along open field lines during\nsolar minima. This flux will decrease with distance as\nφ(ω,r) =φ0/parenleftigω\n¯ω/parenrightig−α\nexp/parenleftbigg/integraldisplay\n−dr\nLSW(ω,r)/parenrightbigg\n(11)– 10 –\nwhereLSWis the damping length, for each frequency. With the damping l engths calculated in\nthe previous section, we calculate how much flux is lost betwe enR= 1.04 (our self-imposed\nreference height) and 4 R⊙. The height of 4 R⊙was chosen because the contribution to solar wind\nacceleration and coronal heating must be deposited within a few solar radii of the Sun (Kohl et al.\n1998; Esser et al. 1999).\nThe contribution to the solar wind at each frequency is\nφlost(ω) =φ0/parenleftigω\n¯ω/parenrightig−α\n1−exp\n/integraldisplayr2\nr1−dr\nL0(ω)/parenleftbigr0\nr/parenrightbigS\n2/parenleftig\nvA\nvA0/parenrightig2\n(1+MA)\n\n (12)\nwhere the limits are r1= 1.04R⊙andr2= 4R⊙, and the definition of Lswfrom Equation 6 has\nbeen included ( vA,S,MA, are all functions of r.)\nWe replace L0in Eq. 12 with Eq. 5. The total flux lost is found using\nφlost,total=/integraldisplayω2\nω1φ0/parenleftigω\n¯ω/parenrightig−α\n1−exp\n/integraldisplayr2\nr1−ωrS\n2vA0\n18rS\n2\n0v2\nA(1+MA)dr\n\ndω (13)\nwhere the limits are ω1= 2.8×10−4Hz andω2= 0.3 Hz.\nNext we compare the potential contribution of the wave flux to the heating in the model (Eq.\n22). It should be stressed that we are not doing a self-consis tent calculation: we are estimating\nwave flux from a model that does not include waves, and we are no t considering any feedback of\nthe waves on the plasma.\n4.2. Heating of the Corona\nThe simulation analyzed in this paper is that of a thermally d riven solar wind. The polytropic\nindex Γ in themodel varies in space by utilizing theWang-She eley-Arge model andthe conservation\nof energy along a solar wind field line using the Bernoulli equ ation. This serves to artificially heat\nthe wind (Cohen et al. 2007) in a manner mimicking turbulence Roussev et al. (2003). Figure 5\nshows the distribution of Γ in the plane of the sky on 1996 Augu st 17. In this section we quantify\nthe heating due to this variable gamma, and do a comparison wi th the energy deposited by damped\nsurface Alfv´ en waves, as in Eq. 13.\nThe first law of thermodynamics attributes changes in the int ernal energy Uof a gas to work\ndone on or by the gas W, and heat added to or removed from the gas Q. In the case of an ideal\ngas, the change in internal energy can be written as dU=cvdT, wherecvis the specific heat at\nconstant volume. The work is expressed as dW=−pdV, wherepis the pressure and Vis the\nvolume. The first law can therefore be written as\ndQ=cvdT+pdV. (14)– 11 –\nBy introducing the ideal gas equation of state and assuming t hat the ratios of specific heats are\nconstants, one can derive a polytropic equation,\np\nρcp−c\ncv−c=p\nρα=const. (15)\nwhereαis referred to as the polytropic index. The notation stems fr om Parker (1963) to clarify\nthat this index can (but need not) be the ratio of specific heat s, and that we are not necessarily\nconsidering an adiabatic process. The symbol γis typically used for the ratio of specific heats,\nand in the case of an adiabatic expansion (no heating enters o r leaves the system), α=5\n3. An\nisothermal wind expansion would be characterized by α= 1. Observations of the solar wind have\nindicated that α=1.46-1.58 in the heliosphere (Totten, Freeman & Arya 1995) . A value closer to\nunity is adopted in some global MHD models in the region near t he Sun in order to generate fast\nsolar wind and match temperature observations in the helios phere (Usmanov & Goldstein 2003).\nAll previous discussion had the underlyingassumption that αwas constant with height. If that\ncondition is not met, then the polytropic index is referred t o as an effective (or local) polytropic\nindex and written as Γ (Totten, Freeman & Arya 1995). The poly tropic equation (Eq. 15) is\nmodified todlnP\ndr= Γdlnρ\ndr+lnρdΓ\ndr(16)\nsuchthat therelationshipbetween densityandpressureisn otsimple. ThevariationofΓwithheight\nhas been utilized to heat the solar wind used in this paper. We will characterize the additional\nheating provided by the prescribed distribution of Γ in our s olar wind simulation, and argue that\nsurface Alfv´ en waves damped near the Sun could replace this artificial heating and move the model\ntowards a more physical treatment of the solar environment. For a solar wind with a constant ratio\nof specific heats γ, the conservation of energy can be written as (Manchester et al. 2004):\n∂ε\n∂t+∇·/bracketleftbigg\nu/parenleftbigg\nε+p+B2\n8π/parenrightbigg\n−(u·B)B\n4π/bracketrightbigg\n=ρg·u+q (17)\nwhere p is the thermal pressure, q is the additional heating f unction, and the energy density is\nε=ρu2\n2+p\nγ−1+B2\n8π. (18)\nRecent global MHD studies adopt an exponential function for the form of q with several free\nparameters in order to benchmark the model with observation s during solar minima conditions\n(Groth et al. 2000; Manchester et al. 2004). Substituting Eq . 18 into Eq. 17, and setting time\nderivatives to zero for a steady solar wind, we find:\n∇·/bracketleftbigg\nu/parenleftbiggγp\nγ−1+ρu2\n2+B2\n4π/parenrightbigg\n−(u·B)B\n4π/bracketrightbigg\n=ρg·u+q. (19)\nAs discussed at the beginning of this section, there is no hea ting function q in the simulation used\nin this paper, and the ratio of specific heats γis replaced by the effective gamma Γ,\n∇·/bracketleftbigg\nu/parenleftbiggΓp\nΓ−1+ρu2\n2+B2\n4π/parenrightbigg\n−(u·B)B\n4π/bracketrightbigg\n=ρg·u. (20)– 12 –\nAlthough Γ has both latitudinal and azimuthal dependence be low 4R⊙, we consider only at the\nradial variation, and so we replace ∇byd\ndr. We assume that we have exactly the same solar\nwind solution in the two cases we are considering: that of a va riable gamma and of an additional\nvolumetric heating function with γ=5\n3. In order to quantify the amount of heating in the model\nwith variable gamma, we subtract equation 20 from 19:\nq=−d\ndr/bracketleftbigg\nur/parenleftbiggΓp\nΓ−1−γp\nγ−1/parenrightbigg/bracketrightbigg\n. (21)\nEquation 21 can be written as:\nq=−/bracketleftbiggd(urp)\ndr/parenleftbiggΓ\nΓ−1−γ\nγ−1/parenrightbigg\n−/parenleftbiggdΓ\ndrurp\n(Γ−1)2/parenrightbigg/bracketrightbigg\n. (22)\nKnowing how Γ, pandurvary along any radial line, and setting γ=5\n3we can integrate\nequation 22 between r1=1.04R⊙andr2=4R⊙to find the heat input along any field line:\nQ=/integraldisplayr2\nr1qdrerg\ncm2s. (23)\nThis equation gives the heat deposited into the system betwe en the two heights. We compare Q\nwith the flux of damped surface Alfv´ en waves (Equation 13.)\nTable 1 provides φlostandQfor the CHB field lines. The expansion fof the NW line from the\nsimulation has the best match to the observations (5.55 comp ared to 6.0). The surface Alfv´ en wave\nflux along this line, and along the NE lin, is larger than the he ating Q by an order of magnitude.\nThe geometrical properties of the SW line also match well wit h observations, however the wave flux\nfor it (and also the SE line) account for 20 % of the required he ating. The distinction between the\nsouthern and northern lines is the source region: they come f rom a stronger magnetic field region.\nNear an active region, the second term in Eq. 22 (which includ es the pressure and radial velocity)\nis larger than a quiet sun region. Therefore, we expect Q to be larger than the surface Alfv´ en wave\nflux along lines from active regions.\nAs we assumed all of the wave flux goes to heating, this procedu re gives an upper limit on\nthe contribution of the damping of surface Alfv´ en waves alo ng an open magnetic field line to the\nheating along that line. A random sampling of 7 open field line s in the northern hemisphere with\nfootpoints in the plane of the sky (see Figure 3) yielded φlostthat were on the order, or an order\nof magnitude larger than the Q.\n5. Conclusions\nThis work is the first study to look at surface Alfv´ en waves in a 3D environment without\nassuming a priori a geometry of the field lines or magnetic and density profiles and strengths. We– 13 –\nshowed the calculation of the expansion of field lines must be done in a 3D environment. Our\ncalculations show that waves with periods less than 1 hour (f requency greater than 2.8 ×10−4Hz)\nare damped in the region between 1-4 R⊙, the region of interest for both solar wind acceleration\nand coronal heating (Grall et al. 1996). We showed that the qu iet sun region will damp surface\nwaves at further distances, so it is more efficient in carrying the wave momentum out into the\ncorona. Surface waves formed on flux tubes with footpoints in an active region will damp closer to\nthe Sun. The required heating from an active region was found to be larger than the damping of\nsurface Alfv´ en wave flux, therefore another mechanism (suc h as turbulence) may be the dominant\nheating in these regions, with surface Alfv´ en waves contri buting approximately 20% of the heating.\nWe estimated damping of surface Alfv´ en waves in the border b etween open and closed field\nlines at heights 1.04-4 R⊙due to the superradial expansion of the field lines. As some of the wave\nflux would go to the momentum of the wind, we provide an upper li mit on the contribution of\nsurface Alfv´ en waves to the heating of the solar wind. In the region between open and closed field\nlines, within a few solar radii of the surface, no other major source of damping has been suggested\nfor the low frequency waves we consider here.\nOur results demonstrate that it is not necessary to have turb ulence in order to heat the solar\nwind - and that it is imperative to include the physics of surf ace Alfv´ en wave damping in solar\nwind models in order to more physically model the heating. Su rface Alfv´ en waves could also be\npresent in the solar wind, in the flux tube structures said to fi ll interplanetary space (Borovsky\n2008). Another environment which could support these waves are Corotating Interaction Regions\n(CIRs), due to the inhomogeneity in density present in these structures (Tsubouchi 2009). Both of\nthese topics will be addressed in future works.\nIt is important to note that this study of waves in a solar wind solution was not self consistent\n- we did not consider any back effects on the waves from the plasm a. We simply tried to estimate\nif the waves could produce the heating required to create the solar wind solution from the model.\nIn the future we will pursue other damping mechanisms with th e goal of incorporating the key\nmechanisms of wave damping in self-consistently in global M HD models to improve the lower\ncorona.\nWe would like to thank NASA Ames for the use of the Columbia sup ercomputer. This research\nwas supported by the NSF CAREER Grant ATM 0747654 and LWS NNGO 6GB53G .– 14 –\nREFERENCES\nBelcher, J. W. & Davis, L., Jr. 1971, J. Geophys. Res., 76, 16, 3534\nBorovsky, J. E. 2008, J. Geophys. Res., 113, A08110\nCirtain, J. W. et al. 2007, Science, 318, 1580\nCohen, O, et al. 2007, ApJ, 654, L163\nCohen, O., Sokolv, I. V., Roussev, I. I. & Gombosi, T. 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R. 1983, A&A, 117, 220\nHollweb, J. V. 1971, Phys. Rev. Lett., 27, 20\nHollweg, J. V. 1986, J. Geophys. Res., 91, A4, 4111\nHollweg, J. V. 1987, ApJ, 312, 880\nIonson, J. A. 1978, ApJ, 226, 650.\nIsenberg, P. A., Lee, M. A. & Hollweg, J. V. 2001, J. Geophys. R es., 106, A4, 5649– 15 –\nFig. 1.— 3D coronal hole boundary field lines obtained from th e simulation for 1996 August 17.\nThe different colors refer to: red as northeast line (NE); blue as southeast line (SE); green as\nsouthwest line (SW);and purple as northwest line (NW). The b lack lines show closed equatorial\nstreamers, and the arrows give the direction of the magnetic field. The solar surface is shown\ncolored by the radial component of the magnetic field.– 16 –\nR [Rs]f\n2 345678910123456789\nLowerLimit(DO99)\nUpperLimit(DO99)\nNE\nSE\nNW\nSW\nFig. 2.— The superradial expansion factor f(see Eq. 8) for the lines in Fig. 1 (colors corre-\nspond) from 1.04-10 solar radii (Rs). We also show the profile s which correspond to the minimum\n(dashed line) and maximum (dash-dot line) values of fdetermined in the observational study by\nDobrzycka et al. (1999). We find that fcovers a larger range of values compared to the observa-\ntional study.– 17 –\nR [Rs]f\n1 2 3 4551015\nA(3D)\nB(3D)\nC (3D)\nD (3D)\nA(2D)\nB(2D)\nC (2D)\nD (2D)(c)\nFig. 3.— Four field lines (A, B, C and D) shown (a) in 3D and (b) as their projections on the\n2D plane of the sky. Contours of solar wind speed (inkm\ns) are shown on the plane, and the solar\nsurface is shown as a white sphere. The expansion factor calc ulated from the 3D field line and 2D\nprojections for each line is shown in (c). The 2D projection c an overestimate the expansion factor\nas compared to the 3D calculation.– 18 –\nR [Rs]Lsw[Rs]\n2 34567891010-210-1100101102\nNE\nSE\nNW\nSW(a)\nX [Rs]Lsw[Rs]\n2 34567891010-310-210-1100101102103104\n(b)\nFig. 4.— a) Surface Alfv´ en damping length ( Lsw) profiles for the coronal hole boundaries in Fig.\n1 (colors correspond), normalized to the L0of the SW line (green) from 1.04-10 solar radii (Rs).\nNote the plateau in the profiles for lines whose source region on the sun is quiet sun, differing from\nfield lines with footpoints near active regions, whose profil es drop quickly. b) Profiles for the SW\ncoronal hole boundary field line corresponding to different fr equencies. From the bottom up to top\nprofile: 3 .3×10−1Hz (dash-dot-dot line); 1 ×10−2Hz (dash-dot line); 4 .17×10−3Hz (dot line);\n1.67×10−4Hz (long dash line); 2 .8×10−4Hz (short dash line); 1 .5×10−5Hz (thin solid line);\n3.8×10−6Hz (thick solid line). Waves with frequencies above 2 .8×10−4are appreciably damped\nbelow 4Rs.– 19 –\nFig. 5.— The distribution of Γ, the effective polytropic index , in the plane of the sky on 1996\nAugust 17. Γ is specified on the solar surface (shown as white s phere) using the Bernoulli integral,\nand is interpolated to 1.1 at 2.5 Rs (inner black circle). Abo ve 2.5Rs, Γ varies linearly until 12.5Rs\n(outer black circle), above which it has the value 1.5. The wh ite lines show the boundaries of grid\nrefinement.– 20 –\nJatenco-Pereira, V. & Opher, R. 1989, A&A, 209, 327\nJatenco-Pereira, V., Opher, R. & Yamamoto, L. C. 1994, ApJ, 4 32, 409\nJess, D. B, Mathioudakis, M., Erd´ elyi, R., Crockett, P. J., Keenan, F. P. & Christian, D. J. 2009,\nScience, 323, 1582\nKohl, J. L. et al. 1998, ApJ, 501, L127\nLeake, J. E., Arber, T. D. & Khodachenko, M. L. 2005, A&A, 442, 1091\nLee, M. A. & Roberts, B. 1986, ApJ, 301, 430\nLionello, R., Linker, J. A. & Mikic, Z. 2009, ApJ, 690, 902\nManchester, W. 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K. & Inutsuka, S. 2005, ApJ, 632, L49\nSuzuki, T. K. & Inutsuka, S. 2006, J. Geophys. Res., 111, A061 01\nTomczyk, S. et al. 2007, Science, 317, 1192– 21 –\nTomczyk, S. & McIntosh, S. W. 2009, ApJ, in press\nToth, G. et al. 2005, J. Geophys. Res., 110, A12226\nTotten, T. L., Freeman, J. W. & Arya, S. 1995, J. Geophys. Res. , 100, A1, 13\nTsubouchi, K. 2009, J. Geophys. Res., 114, A02101\nTu, C. Y., Marsch, E. & Thieme, K. M. 1989, Geophys. Res. Lett. 17, 283\nUsmanov, A. V. & Goldstein, M. L. 2003, J. Geophys. Res., 108, A9, 1354\nUsmanov, A. V. & Goldstein, M. L. 2006, J. Geophys. Res., 111, A07101\nVerdini, A. & Velli, M. 2007, ApJ, 662, 669\nWentzel, D. G. 1979, ApJ, 227, 319\nWentzel, D. G. 1989, ApJ, 336, 1073\nThis preprint was prepared with the AAS L ATEX macros v5.2.– 22 –\nTable 1: Properties of Coronal Hole Boundary Field Lines\nNE NW SE SW\nθ0a(Dobrzycka et al. 1999) 29.0◦31.0◦23.3◦28.0◦\nθ0This Study 37.2◦22.3◦19.2◦25.2◦\nf10R⊙b(Dobrzycka et al. 1999) 6.56 6.00 7.30 6.5\nf10R⊙This Study 2.56 5.55 4.71 7.62\nQc(erg\ncm2s) 8 .9×1038.0×1032.7×1053.1×105\nφlostd(erg\ncm2s) 6 .1×1046.2×1046.4×1046.4×104\naColatitude of the line - angle measured from the pole to the fo otpoint on the solar surface.\nbValue for the superradial enhancement factor at R=10 R⊙\ncHeating calculated along the field line (see section 4.2)\ndAlfv´ en wave flux deposited into the wind (see section 4.1)" }, { "title": "2301.04156v2.Cosmic_Ray_Drag_and_Damping_of_Compressive_Turbulence.pdf", "content": "Draft version August 15, 2023\nTypeset using L ATEXtwocolumn style in AASTeX631\nCosmic Ray Drag and Damping of Compressive Turbulence\nChad Bustard1and S. Peng Oh2\n1Kavli Institute for Theoretical Physics, University of California - Santa Barbara, Kohn Hall, Santa Barbara, CA 93107, USA\n2Department of Physics, University of California - Santa Barbara, Broida Hall, Santa Barbara, CA 93106, USA\nSubmitted to ApJ\nABSTRACT\nWhile it is well-known that cosmic rays (CRs) can gain energy from turbulence via second order\nFermi acceleration, how this energy transfer affects the turbulent cascade remains largely unexplored.\nHere, we show that damping and steepening of the compressive turbulent power spectrum are expected\nonce the damping time tdamp∼ρv2/˙ECR∝E−1\nCRbecomes comparable to the turbulent cascade time.\nMagnetohydrodynamic (MHD) simulations of stirred compressive turbulence in a gas-CR fluid with\ndiffusive CR transport show clear imprints of CR-induced damping, saturating at ˙ECR∼˜ϵ, where ˜ ϵis\nthe turbulent energy input rate. In that case, almost all the energy in large scale motions is absorbed by\nCRs and does not cascade down to grid scale. Through a Hodge-Helmholtz decomposition, we confirm\nthat purely compressive forcing can generate significant solenoidal motions, and we find preferential CR\ndamping of the compressive component in simulations with diffusion and streaming, rendering small-\nscale turbulence largely solenoidal, with implications for thermal instability and proposed resonant\nscattering of E∼>300 GeV CRs by fast modes. When CR transport is streaming dominated, CRs also\ndamp large scale motions, with kinetic energy reduced by up to to an order of magnitude in realistic\nECR∼Egscenarios, but turbulence (with a reduced amplitude) still cascades down to small scales with\nthe same power spectrum. Such large scale damping implies that turbulent velocities obtained from the\nobserved velocity dispersion may significantly underestimate turbulent forcing rates, i.e. ˜ ϵ≫ρv3/L.\n1.INTRODUCTION\nCosmic rays (CRs) and magnetized turbulence are\nboth ubiquitous in the Universe, and their interplay has\nlong been a fascinating topic of research. Fluctuations\nat the small-scale end of a turbulent cascade, on scales of\norder the CR gyroscale, are frequently invoked to scat-\nter individual CRs, creating the high degree of observed\nCR isotropy and the long residence times of CRs in the\nMilky Way disk and its surrounding halo relative to the\nlight crossing time (Amato & Blasi 2018; Becker Tjus &\nMerten 2020). In such a scenario, dubbed the “extrin-\nsic turbulence” model (Zweibel 2017), the resulting bulk\nCR transport is magnetic field-aligned diffusion, with an\nenergy-dependent spatial diffusion coefficient κ||and CR\nfluxFCR∝κ||∇PCR. CRs in this picture can also gain\nenergy from repeated scattering off gyroscale fluctua-\nCorresponding author: Chad Bustard\nbustard@ucsb.edutions, a second order Fermi mechanism called “resonant\nreacceleration.”\nPhenomenological models of CR propagation fit to di-\nrect and indirect CR observables (Hanasz et al. 2021)\nhave traditionally assumed a Kolmogorov scaling for tur-\nbulence, appropriate for hydrodynamic turbulence; how-\never, our understanding of CR scattering by turbulence\nhas been refined over time with new insights into magne-\ntohydrodynamic (MHD) turbulence. Most profoundly,\nMHD turbulence differs from hydrodynamic turbulence\nin that MHD forces and hence turbulence are no longer\nisotropic. The resulting anisotropy of slow and Alfv´ en\nmodes (Goldreich & Sridhar 1995) makes them ineffi-\ncient CR scatterers, as CRs interact with multiple un-\ncorrelated eddies during one gyro-orbit, essentially can-\nceling out gyroresonant contributions from each eddy\n(Chandran 2000).\nCompressible fast modes, whose velocities are inde-\npendent of magnetic field direction, are more isotropic\n(Cho & Lazarian 2003) and therefore considered the best\ncandidate for CR scattering (Yan & Lazarian 2004); al-arXiv:2301.04156v2 [astro-ph.HE] 14 Aug 20232 Bustard & Oh\nthough, the degree of isotropy decreases with decreasing\nscale due to strong collisionless and viscous damping,\nhence the efficacy of CR scattering decreases with de-\ncreasing CR energy (Kempski & Quataert 2022). Fast\nmode scattering, then, is most plausible for higher en-\nergy CRs ( E >300 GeV).\nForE < 300 GeV, where most of the CR energy\nresides, CRs can largely create scattering perturba-\ntions themselves through a resonant streaming insta-\nbility (Wentzel 1968; Kulsrud & Pearce 1969). The\nresulting transport is no longer purely diffusive; in-\nstead, CRs “stream” down their field-aligned pressure\ngradient at the local Alfv´ en speed vA=B/√4πρwith\nFCR∝vAPCR, and additional, energy-dependent CR\ndiffusivity ( FCR∝ ∇PCR) is introduced by wave damp-\ning1, e.g. ion-neutral damping, nonlinear Landau damp-\ning, and turbulent damping (Skilling 1971; Farmer &\nGoldreich 2004; Blasi et al. 2012; Wiener et al. 2013;\nZweibel 2017; Bustard & Zweibel 2021). There is also\nan important difference regarding energy transfer be-\ntween CRs and hydromagnetic waves: whereas extrinsic\nturbulence is generated externally, in self-confinement,\nthe free energy to generate waves comes from the CRs\nthemselves, and this energy is subsequently dissipated\ninto the thermal gas via wave damping at a rate H=\n−dECR/dt=vA· ∇PCR. We refer to this collisionless\nenergy transfer as streaming energy loss / gas heating.\nWhile considerable effort has been put towards ex-\nploring resonant-scale interactions between CRs and ei-\nther self-generated (e.g. Skilling 1975; Felice & Kulsrud\n2001; Bai et al. 2019; Holcomb & Spitkovsky 2019) or\nexternally driven (e.g. Giacalone & Jokipii 1999; Yan &\nLazarian 2002; Reichherzer et al. 2020) waves, somewhat\nless focus has been given to the interplay between CRs\nand turbulence on scales much larger than a CR gyrora-\ndius (less than an AU for a GeV CR proton in a ∼µG\nfield). In particular, we will focus on scales larger than\nthe CR mean free path due to pitch angle scattering2,\nwhere the collective CR population is well-described as a\nfluid that experiences compressions and rarefactions in\nthe turbulent flow, leading to energy transfer between\nthe CRs and turbulence. To distinguish this from its\nresonant-scale counterpart, the flow of energy from tur-\nbulence to the bulk CR fluid is called non-resonant reac-\n1Note that, depending on the functional form of the damping rate,\nthe “diffusive” term may not be truly diffusive (see e.g. Skilling\n1971 or Appendix B3 of Hopkins et al. 2021 for examples)\n2This is usually around a pc both in phenomenological models of\nMilky Way CR propagation motivated by extrinsic turbulence\nand in self-confinement models.celeration (Ptuskin 1988), and its efficiency depends on\nCR transport model.\nFor purely diffusive CR transport, non-resonant reac-\nceleration is maximally efficient when CRs are well-\ntrapped in the turbulent flow ( κ < v phL0, where vph\nis the phase speed of compressive fluctuations and L0\nis the outer eddy scale). When streaming is taken into\naccount, the interaction between perturbed CR and gas\nvariables is fundamentally altered. While CR diffusion\nintroduces a π/2 phase shift between CR and density\nperturbations, leading to a CR force that damps fluctu-\nations much like a damped harmonic oscillator, both\nthe change in flux ( FCR∝PCRinstead of FCR∝\n∇PCR) and the associated energy loss that accompany\nstreaming transport modify the CR force (Tsung et al.\n2022). As we showed in Bustard & Oh (2022) (from\nnow on referred to as Paper I), CR reacceleration /\nturbulent damping rates become dependent on plasma\nβ=Pg/PB; they remain largely unchanged in high- β\nplasmas like the intracluster medium (ICM) where reac-\nceleration is a leading explanation for radio halos (e.g.\nBrunetti & Lazarian 2011; Brunetti & Jones 2014), but\nthey are stunted significantly in low- βplasmas.\nDespite non-resonant reacceleration being a fairly in-\nefficient process compared to diffusive shock acceleration\n(a first order Fermi mechanism), with minimum growth\ntimes lengthened even further by streaming transport,\nit was pointed out by Thornbury & Drury (2014); Drury\n& Strong (2017) that a significant fraction of total CR\npower in galaxies could come from reacceleration, conse-\nquently creating a large sink for turbulent energy. In this\npaper, we present analytical estimates and CR+MHD\nsimulations suggesting that CRs in very plausible astro-\nphysical environments can divert significant amounts of\nturbulent energy, essentially acting as an unsual form\nof viscosity. The outcome is a CR-modified route to gas\nheating, rather than the typical conversion to heat at the\ndissipation scale, and a damped turbulent energy spec-\ntrum with decreased small-scale, compressive power.\nThese changes are, of course, strongest in environ-\nments where CRs are dynamically important such as the\nISM (where CR energy densities are roughly in equipar-\ntition with turbulent and magnetic energy densities;\nBoulares & Cox 1990) and the Milky Way circumgalac-\ntic medium (which may be energetically dominated by\nCRs; e.g. Ji et al. 2020), but they would affect any pro-\ncess that relies on compressive motions. For instance,\ncompressions seed thermal instability (Field 1965; Mc-\nCourt et al. 2012; Mohapatra et al. 2022), which is fre-\nquently invoked, for instance, to explain the existence\nof cold CGM clouds (Putman et al. 2012). Fluctuations\nthat scatter CRs are not immune to these modificationsCosmic Ray Effects on Turbulence 3\neither. Low-energy, self-confined CRs could sap energy\nfrom the turbulent fast mode cascade at large scales, de-\ncreasing the available small-scale power needed to scat-\nterhigh energy CRs.\nThis paper is outlined as follows. In §2, we discuss our\nsimulation method and setup. In §3, we analytically es-\ntimate and then quantify in simulations the fractions of\nturbulent driving and gas heating that are channeled\nthrough CRs. We then analytically derive how CR-\ninduced damping should affect MHD turbulence spectra\n(§4.1) and the conditions under which damping rates\ncan exceed cascade rates ( §4.2). In §4.3, we present\nexploratory simulations strongly suggestive of these an-\nalytic estimates and show sensitivities to streaming vs\ndiffusive CR transport. We discuss regimes of applica-\nbility and implications in §5 and conclude in §6.\n2.SIMULATION SETUP\nWe begin by briefly describing the simulation method-\nology and setup, which is described in more detail in\nPaper I. Using the Athena++ MHD code (Stone et al.\n2020) coupled with an additional CR module that mod-\nels CR diffusive and streaming transport in a fluid ap-\nproximation using a two-moment method originally de-\nveloped for radiation transport (Jiang & Oh 2018), we\nnumerically solve the ideal MHD equations plus two ad-\nditional equations for the CR energy and energy flux.\nAll simulations begin with a flat background (no gra-\ndients) consisting of CRs, gas, and magnetic fields,\nwith a constant net (straight) magnetic field in the ˆ x\ndirection. We stir turbulence following an Ornstein-\nUhlenbeck random process (Uhlenbeck & Ornstein 1930;\nEswaran & Pope 1988), randomly generating velocity\nperturbations between modes k = 1 and 3 in a cubic\nbox of width 2L. For driving, we set the autocorrelation\ntimescale to be tcorr=L/csand drive fluctuations every\ntdrive= 2×10−3(L/cs). For the parameter scans in §3,\nwe use grids of size 1283and 2563. We simulate fluids\nwith either an isothermal equation of state, where the\nthermal energy is fixed, or an adiabatic equation of state.\nThe latter results in a gradual rise in the gas pressure\ndue to a combination of CR heating and grid-scale dis-\nsipation of the cascade, which we decompose and quan-\ntify. These simulations all use purely compressive forc-\ning, with two turbulent driving rates ˜ ϵ=dE/dt , result-\ning in approximately Ms∼0.15 and Ms∼0.5 turbu-\nlence with a weak dependence on plasma βsince MHD\nforces counteract motions. We avoid solenoidal driving\nto avoid turbulent amplification of magnetic fields, so\nthat we can evolve simulations at approximately fixed\nplasma β. To a good approximation, solenoidal drivingonly amplifies magnetic fields, while compressive driving\nenergizes CRs.\nAt our parameter scan resolution of 2L/256, the cas-\ncade exhibits only a short inertial range, and in test-\ning we find that the spectral slope in pure MHD runs\n(no CRs) is intermediate between E(k)∼k−2and\nE(k)∼k−3/2– a shallower slope is expected for com-\npressive fast modes, but the exact exponent has been\nhighly debated. In our analytic estimates ( §4.1), we\nwill explore CR-induced deviations to different initial\nspectra, but we particularly note significant changes to\nKraichnan turbulence where E(k)∼k−3/2initially. For\n§4.3, where we want to test deviations from this spec-\ntrum due to CR drag, we increase the resolution to\n2L/512, though we find that the main trends are well-\nrecovered even with a resolution of 2L/256 (see Ap-\npendix). Higher resolution simulations giving a larger\ninertial range would be preferable, but to ensure an ac-\ncurate treatment of CR propagation and influence, the\ntwo-moment method has an effective, maximum speed\nof light parameter vmthat must be much larger than\nother propagation speeds in the system and that sets\nthe Courant-limited timestep. In Paper I, we found\nthat vm∼50csgives seemingly converged CR heating\nrates and reacceleration rates. With this choice, our\nMHD+CR simulations are about a factor of 8 more ex-\npensive than pure hydro turbulence sims, prohibiting us\nfrom going to much higher resolution.\n3.COSMIC RAY DIVERSION OF TURBULENT\nENERGY\nWe’ll begin with a short review of non-resonant reac-\nceleration (see e.g. Ptuskin 1988; Chandran & Maron\n2004; Lynn et al. 2012 and §2 of Paper I for greater\ndetail) and its relation to the turbulent damping rate.\nVariables used in our discussion are summarized in Ta-\nble 1. As discussed in Paper I, “drag” against CRs pro-\nvides a frictional force on compressive motions known\nas Ptuskin damping (Ptuskin 1981). It is similar to ra-\ndiative damping of sound waves, which famously leads\nto Silk damping of acoustic waves in the early universe\n(Silk 1968). In general, since Ek/tdamp∼PCR/tgrow, we\nhave3:\ntdamp∼ρv2max\u0012tgrow\nPCR,1\n˜ϵ\u0013\n∼max\u0000\nM2\nctgrow,tinject\u0001\n(1)\n3In this paper, we use the notation ˜ ϵto denote the turbulent\ndriving rate in units of turbulent energy density per unit time,\nand we use ϵto denote the driving rate in units of v2(velocity\nsquared) per unit time. In hydrodynamic turbulence, ˜ ϵ≡ρv3/L\nandϵ≡v3/L, but these equivalences don’t hold in CR-modified\nturbulence.4 Bustard & Oh\nTable 1. Simulation parameters, CR module settings, and other variable definitions\nParameter Definition / Setting / Equation Additional Notes\nL Half box size k = 2 mode\nL0 Outer eddy scale k = 3 mode\ntdrive 2×10−3(L/cs) Turbulence driven every tdrive\ntcorr L/cs Autocorrelation time\n˜ϵ,ϵ Input turbulent energy rate, dE/dt ρv3/L,v3/Lin hydro turbulence\nvm 50cs Effective maximum speed of light\nκ CR diffusion coefficient Assumed to be field-aligned only ( κ=κ||)\nβ Pg/PB Plasma beta\ncsp\nγPg/ρ Gas sound speed\nvphp\n(γPg+γCRPCR+PB)/ρ Compressive wave phase speed\nvA B/√4πρ Alfv´ en speed\nccp\nγCRPCR/ρ Effective CR sound speed\nMs,Mph,MA,Mcv/cs,v/vph,v/vA,v/cc Mach numbers\nH vA· ∇PCR “Collisionless” CR loss rate / gas heating rate\nfCR,fth,fCR,heating˙ECR/˜ϵ,˙Eth/˜ϵ,< H > /˜ϵ Fraction of ˜ ϵ→CRs, thermal gas, CR heating\nE(k) Kinetic energy spectrum ∝k−5/3(Kolmogorov), k−2(Burgers), k−3/2(Kraichnan)\ntinject ρv2/˜ϵ Energy injection time\ntcascade kE(k)/F(k) Cascade time (see Equation 11)\ntgrow p2/Dpp CR reacceleration time ( §3 and Paper I)\ntdamp ∼ρv2max\u0010\ntgrow\nPCR,1\n˜ϵ\u0011\n∼max\u0000\nM2\nctgrow,tinject\u0001\nTurbulent damping time (Equation 1)\nwhere Mc≡v/ccis the Mach number in units of the CR\neffective sound speed, cc∼p\nPCR/ρ, and tinject≡ρv2/˜ϵ.\nEquation 1 is a general expression for the damping time,\nfor which one can plug in the appropriate tgrow, the CR\nreacceleration (or growth) time.\nWorking in the limit of purely diffusive spatial CR\ntransport with isotropic diffusion coefficient κ, the reac-\nceleration time can be derived in two limits depending\non the ratio of diffusion time tdiff=l2/κto compressive\nwave crossing time tsc=l/vphacross an eddy of length\nlin a medium with compressive phase velocity vph∼\n(Ptot/ρ)1/2∼[Pg+PB+PCR)/ρ]1/2. In the fast diffusion\nlimit ( tdiff≪tsc, or equivalently, κ≫vphl), deriving\nthe CR momentum diffusion coefficient Dppfollows the\ntextbook argument for second order Fermi acceleration:\nDpp∼(∆p)2/τscatter ∼p2v2/(c2τscatter )∼p2v2/κ. The\nenergy growth time, defined as p2/Dppis\ntgrow∼κ\nv2;κ >> v phl (2)\nIn the opposite limit of slow diffusion ( tdiff≫tsc,\nor equivalently, κ≪vphl),Dpp∼(δp)2/τdiff∼\n(p2v2/v2\nph)(κ/l2), and the growth time is\ntgrow∼p2\nDpp∼v2\nphl2\nv2κ;κ << v phl (3)Joining the two regimes in the middle, the minimum\ngrowth time is tgrow∼(vphl/v2) when κ∼vphl.\nStrictly speaking, these scalings are appropriate if CR\ndiffusion is isotropic, if streaming is negligible, and if\nall reacceleration comes from eddies of a single scale l.\nRelaxing these assumptions introduces further modifi-\ncations. In the fast diffusion limit ( κ≫vphl), there are\nalso correction factors that decrease the growth time\nif anisotropic rather than isotropic spatial diffusion is\naccounted for (Chandran & Maron 2004). Additional\nstreaming transport, widely applicable for CRs with en-\nergy E⪅300 GeV, introduces a correction factor that\ndecreases reacceleration rates by fcorr= 1−p\n2/βand\nfcorr= (1−p\n2/β)1/2in the slow and fast diffusion\nregimes, respectively (Paper I); and in the slow diffusion\nlimit ( κ≪vphl), multiple eddies contribute to reaccel-\neration, with relative contributions dependent upon the\nshape of the turbulent power spectrum (see Equation 4\nin Paper I for a more general expression). If the wave\nspectrum is Burgers-like ( E(k)∼k−2), roughly consis-\ntent with our simulations, eddies at each logarithmic\ninterval in the inertial range contribute equally to reac-\nceleration, and tgrowhas a broad minimum of tgrow∼\n(vphl/v2) throughout the entire range of κ||< vphl.\nIf we work in the limit of a single outer-scale eddy\n(i.e., we only consider eddies of size L0), in the fastCosmic Ray Effects on Turbulence 5\ndiffusion ( κ≫vphL0) regime, where tgrow∼κ/v2then\nEquation 1 gives tdamp∼κ/c2\nc, in agreement with the\nclassic (much more detailed) calculation of this effect by\nPtuskin (1981). Working instead in the broad regime of\nmaximal reacceleration, where CRs are well-trapped in\nthe turbulent flow (when κ < v phL0), the characteristic\ngrowth time is tgrow∼(vphL0/v2), which gives:\ntdamp∼max\u0012vphL0\nc2c,tinject\u0013\n(4)\nNote that tdamp is velocity independent.\nWith these reacceleration times in mind, we can now\nestimate the fraction of turbulent energy forcing ˜ ϵthat\ngoes toward CRs. It is given by\nfCR∼˙ECR\n˜ϵ∼ECR\n˜ϵtgrow(5)\nFor example, for Kolmogorov turbulence, where ˜ ϵ∼\nρv3/L, and for the characteristic growth time tgrow∼\n9/2vphL/v2this gives:\nfCR∼ECR/tgrow\nρv3/L∼max\u00122\n3MphPCR\nρv2,1\u0013\n(6)\nNote that Equation 6 is approximate and assumes\n˜ϵ∼ρv3/L, which is only true in the limit where\nCRs do not back-react on the flow. In general, ˜ ϵ−\nρv3/L−ECR/tgrow∼0, and fCR∼ECR/tgrow/˜ϵ∼\n(ECR/tgrow)(ρv3/L)−1(1−fCR). This gives\nfCR∼ECR\ntgrow(ρv3/L)\u0012\n1 +ECR\ntgrow(ρv3/L)\u0013−1\n∼\u00122\n3MphPCR\nρv2\u0013\n/\u0012\n1 +2\n3MphPCR\nρv2\u0013 (7)\nwhich agrees with Equation 6 in the appropriate limits.\nIn both Equations 6 and 7, the maximum value of 1\nreflects energy conservation: CRs cannot gain more en-\nergy than is injected by turbulent forcing, hence fCR∼\n2\n3MphPCR\nρv2is only valid for ˙ECR<˜ϵ. Within this regime,\nthe fraction of kinetic energy deposited into CRs is small\nifPCR≪ρv2, in which case most energy is deposited in\nthe thermal gas; however, for higher PCR, the fraction\nincreases and can become quite substantial at close to\nequipartition values.\nFigure 1 compares this expectation to simulations and\nis one of the key results of this paper. The y-axis\nshows the partitioning of the input energy rate into\nCRs ( fCR=˙ECR/˜ϵ) and thermal energy ( fth=˙Eth/˜ϵ)\nfor varying PCR/Pg, keeping ˜ ϵfixed, for purely diffusive\nCRs. Unlike our previous simulations, which all used an\n0.00.20.40.60.81.0fiCRs back-react on\nflow, fCR saturates\n101\n100101102\nPCR/Pg0.00.20.40.60.81.0fi\nfCR>0 due to\n numerical dissipation2\n3phPCR\nv2\n2\n3phPCR\nv2\n(1+2\n3phPCR\nv2)\nfCR fthFigure 1. The average CR energy gain rate and ther-\nmal energy gain rate relative to the turbulent driving rate\n(fCR=˙ECR/˜ϵandfg=˙Eg/˜ϵ, respectively) for simulations\nwithout streaming, as a function of PCR/Pg. These all are\nadiabatic, Ms∼0.5 simulations on a 1283grid, with β∼1.\nTop: κ||∼0.15L0vph, where CR energy gain is maximized.\nThe dashed black curve is the analytic expectation from\nEquation 6, showing good agreement when PCR/Pg<1, and\nthe dash-dotted curve shows Equation 7, which accounts for\nCR back-reaction on the flow and subsequent saturation of\nfCR.Bottom: κ||∼0. For PCR≫Pg, even κ∼0 leads\nto significant fractions of turbulent energy converted to CR\nenergy, but this CR reacceleration is due to numerical diffu-\nsion caused by finite resolution.\nisothermal equation of state, these simulations have an\nadiabatic equation of state, which makes it easier to con-\nfirm energy conservation. Together, the contributions to\n˙ECRand ˙Ethsum to ∼80−90% of the driving rate, with\nthe rest going towards small magnetic and kinetic energy\nincreases. The top and bottom panels show simulations\neach without streaming and with κ= 0.15L0vphand\nκ= 0, respectively. For PCR/Pg<1,fCRfollows the\nexpectation from Equation 6 (shown as a black dashed\nline) quite well, an indication that turbulent reacceler-\nation is diverting the driving energy to CRs at the ex-6 Bustard & Oh\nDiffusion OnlyStreaming OnlyDiff + StreamAdiab. Stream OnlyAdiab. Diff + StreamDiffusion OnlyStreaming OnlyDiff + StreamAdiab. Stream OnlyAdiab. Diff + StreamDiffusion OnlyStreaming OnlyDiff + StreamAdiab. Stream OnlyAdiab. Diff + Stream 00.20.40.60.81\nFigure 2. Partitioning of input turbulent energy rate ˜ ϵinto three different channels: CR reacceleration fCR, dissipation via CR\ncollisionless heating fCR,heating (i.e. streaming energy loss), and grid-scale heating fth−fCR,heating . Without CRs, this choice\nof ˜ϵproduces Ms∼0.15 turbulence. Each simulation here starts with PCR∼Pgbut with varying CR transport treatments,\neither with diffusion only (all with κ∼0.15vphL0) or diffusion plus additional streaming. For each β, the first three simulations\nuse an isothermal equation of state, so there is no gas heating. The last two, denoted by “Adiab.”, use an adiabatic equation\nof state, in which case the total thermal gas heating rate is the sum of CR heating and grid-scale heating. With diffusion\nonly, reacceleration is very efficient: most turbulent energy is soaked up by CRs. With streaming, both gas heating and CR\nenergization are relatively inefficient in the low- βregime, but for β∼10, 100, CR heating is the dominant energy channel.\nInstead of turbulent energy cascading to small scales and eventually dissipating into thermal energy at the grid scale, CRs\nintercept this energy transfer at large scales; astoundingly, even in these subsonic flows very high fractions of turbulent energy\nare channelled through CRs when PCR∼>Pg.\npense of thermal gas heating. Similar simulations with4\nκ= 0 show far lower fCR, again revealing the depen-\ndence of reacceleration on diffusion coefficient. Note\nthat while we previously only tested analytic expecta-\ntions for the growth time tgrow (on which Equation 6\ndepends) when the gas is isothermal in Paper I, they\ncontinue to hold when the gas is adiabatic.\nAsPCR/Pgincreases, fCRdeviates from the analytic\nexpression in Equation 6; fCRincreases more slowly to-\nwards the asymptotic bound fCR∼1 than in our ansatz.\n4In practice, κhas a non-zero value because of numerical diffusion,\nbut here this has little impact up until PCR≫Pg.Nonetheless, for PCR/Pg∼>1, what immediately stands\nout is the large fraction of energy diverted to CRs, with\nfCRas large as 0.8 when PCR/Pg>1. These large val-\nues of fCRclearly come at the expense of thermal heat-\ning5, with fthdecreasing from fth≈1 when PCR≪Pg\ntofth<0.2 when PCR> Pg.\nIn the above, purely diffusive case, turbulent energy\ndirectly accelerates CRs. When streaming is included,\nenergy is also lost to collisionless heating at a rate\nH=vA·∇PCR. In Fig 2, we quantify the partitioning of\n5Since we enforce purely compressive driving, magnetic field am-\nplification is very weak, and fCR+fth≈1 for an adiabatic setup.Cosmic Ray Effects on Turbulence 7\nturbulent kinetic energy into direct acceleration of CRs\n(fCR) and gas heating ( fth) in simulations with fixed\n˜ϵproducing undamped Ms∼0.15. We distinguish be-\ntween collisionless heating by CRs fCR,heating (red bars),\nand heating due to turbulence which cascades down to\nthe grid scale and dissipates fth−fCR,heating (orange\nbars). Note that, in all cases (see e.g. the adiabatic\nβ= 10, 100 simulations), the sum of fCR,fCR,heating ,\nandfth−fCR,heating can be slightly greater or slightly\nlower than 1; we average each dE/dt over the final 1/4\nof the simulation snapshots, and during this time inter-\nval of fully developed turbulence, kinetic and magnetic\nenergy can, on average, be slightly decreasing or slightly\nincreasing. For that reason, the sum of all bars shown\nfor each simulation in Figure 2 lands between 0.95 and\n1.05 of the input driving rate.\nWhen streaming is included, fCRis a small and weakly\nincreasing function of β, consistent with Paper I and ev-\nident in Figure 2. Here, we fix the initial state to have\nPCR∼Pgfor each simulation and quantify the CR en-\nergy gain rate as we did in Figure 1. Despite the fact\nthat a negligible fraction of energy fCRends up in CRs,\nthe latter nonetheless have a strong impact on the turbu-\nlent cascade. In MHD simulations, turbulence cascades\nto grid scales where numerical diffusion dominates6and\nsubsequently dissipates, heating the gas. Thus, fthis a\ngood barometer of how much kinetic energy flux makes\nit to the dissipation scale; however, that is not the case\nwith turbulence modified by streaming CRs. In the adi-\nabatic streaming simulations quantified in Figure 2, the\ntotal amount of gas heating is a weak function of β, but\nactually much of that heating is done by CR streaming\nenergy loss instead of classical small-scale dissipation.\nOnly∼60% (for β∼1) to <10% (for β∼10, 100) of\nthe driving energy makes it to the grid scale, with the\nremaining energy channeled through CRs.\nNote that in our estimate of tdamp (Equation 4), we\nhave notincluded the effects of CR streaming on tgrow.\nIf we did, tdamp would be substantially longer in low β\nenvironments. However, as we have seen, this is incor-\nrect. When CR streaming is present, the kinetic energy\nof compressive motions is still absorbed by CRs at large\nscales. This energy is subsequently returned to the gas\nin the form of heat via CR streaming, and so streaming\nimpedes the secular growth of CR energy, resulting in\nthe lower growth times explored in Paper I. However,\ndiversion of kinetic energy away from the turbulent cas-\ncade and damping of compressive motions still happens\nat a similar rate, even at low β(Figure 2). CR streaming\n6In high resolution simulations with explicit viscosity, it would\ninstead cascade to the viscous scale.provides an avenue for gas motions to quickly dissipate\nin the form of heat without going through the turbu-\nlent cascade. In this case, CRs can be thought of as\nproviding an unusual form of viscosity.\nTo summarize: once Pc/Pg∼>1, and for β∼>10,\nour simulations show that the energy input in turbu-\nlent driving appears to be almost completely diverted to\nCRs, with only ∼10% remaining which cascades down\nto grid scales. This is irrespective of whether streaming\nis absent (in which case CRs store the energy) or present\n(in which case CRs thermalize a significant fraction via\ncollisionless heating). This is astonishing efficiency, con-\nsidering that strong shocks convert at best ∼10−30%\nof kinetic energy to CRs. For β∼1, the fraction of en-\nergy routed through CRs is slightly lower, ∼80% in the\ndiffusion only case, and ∼50% with both CR streaming\nand diffusion. We now turn to some implications of this\nfinding.\n4.COSMIC RAY IMPRINTS ON KINETIC\nENERGY SPECTRA\nIn certain regimes, CRs are clearly an important en-\nergy sink for fluid motions. When turbulent energy is\ndiverted to the CR population, it either\n1. Directly accelerates CRs through non-resonant\nreacceleration\n2. (If CR streaming is significant) Heats the gas at\nscales lCR≫ldissthrough collisionless energy\ntransfer by self-confined CRs (streaming energy\nloss), where ldissis the Kolmogorov dissipation\nscale.\nIn either case, energy that originally would have cas-\ncaded to small scales is siphoned out of the turbulent\ncascade, and it is interesting to ask what imprint this\nmight have on the kinetic energy spectrum. In this sec-\ntion, we first focus on the effects of purely diffusive CRs,\nleaving an initial exploration of streaming CR transport,\nthe effects of which are less straightforward and deserve\nfuture follow-up, to §4.4. We will first explore CR mod-\nifications to Kolmogorov and Kraichnan spectra analyt-\nically and discuss astrophysical regimes where spectra\ncould be heavily modified. Of the compressible MHD\nmodes, it is thought that slow modes have a Kolmogorov\nspectrum ( E(k)∝k−5/3) and fast modes have a Kraich-\nnan spectrum ( E(k)∝k−3/2) (Cho & Lazarian 2003),\nthough this is still debated. In our simulations, compres-\nsive forcing gives rise to something intermediate between\nKraichnan and Burgers turbulence ( E(k)∝k−2), and\nwe will see that CR damping also has noticeable effects\nin this regime.8 Bustard & Oh\n4.1. Analytic Theory\nWe can solve for the turbulent power spectrum by\nsolving the dynamic equation (Landau & Lifshitz 1987).\nIf we consider a turbulent energy injection rate ϵinjected\nat some outer scale L=k−1\nL(where ϵ∼v2\nl/tcascade ∼\nconst in the absence of damping, and tcascade depends on\nthe form of turbulence), then in steady state the com-\nbined effects of the cascade to smaller scales and damp-\ning must balance injection:\nϵ δ(k−kL) =∂\n∂kF(k) + Γ( k)E(k) (8)\nwhere E(k) is the power spectrum of turbulence, F(k) is\nthe turbulent cascade flux in k-space, and Γ( k)∼t−1\ndamp\nis the damping rate. While Equation 8 makes no as-\nsumption on the turbulent spectrum or the damping\nrate, we now must adopt choices for each. First, Equa-\ntion 4 in Paper I describes the CR reacceleration rate\nt−1\ngrow from an ensemble of waves across many scales;\nhowever, to assess the impact of CRs on turbulence at\na given scale, we need to consider just the amount of\nenergy that CRs sap from individual eddies of scale l.\nAssuming we are in the fast transport regime ( κ > v phl),\ntgrow∼κ/v2\nl, hence, tdamp∼ρv2\nltgrow/PCR∼κ/c2\nc.\nΓ(k) = t−1\ndampis then scale-independent. In the slow\ntransport regime, tgrow∼v2\nphl2/(v2\nlκ) and tdamp ∼\nρv2\nphl2/(PCRκ). Because the latter is scale-dependent,\nwe’ll make the simplifying assumption that diffusion is\nfast, such that tdamp is scale-independent. This is not\nunreasonable, especially at small scales, because for a\ngiven κ||, transport across smaller and smaller scales is\nincreasingly in the fast regime. As we’ll see, our simula-\ntions display similar behavior to our following analytics\nthat assume fast diffusion.\nWhile the above Γ( k) is scale-independent and there-\nfore makes no assumption on cascade physics, the cas-\ncade flux F(k) depends on the type of turbulence: for\nKolmogorov turbulence, F(k)∼[E(k)]3/2k5/2, while for\nisotropic Kraichnan turbulence, F(k)∼k3[E(k)]2/vph.\nIn the absence of damping (Γ( k) = 0), integrating both\nsides of Equation 8 with respect to kgives E(k)∼\nϵ2/3k−5/3andE(k)∼(ϵvph)1/2k−3/2, the power spectra\nfor Kolmogorov and Kraichnan turbulence respectively.\nThe first and second terms on the right hand side of\nEquation 8 have units of v2/k×(t−1\ncascade, t−1\ndamp) respec-\ntively. In Fig. 3, we solve Equation 8 for various values\noftdamp/tcascade . It is easy to understand the asymptotic\nbehavior. When tcascade ≪tdamp, the first term on the\nRHS dominates: injected energy cascades before it can\ndamp, and we obtain the usual Kolmogorov/Kraichnan\npower spectra. On the other hand, if tdamp≪tcascade ,\nthen the second term on the RHS dominates, which givesϵ∼ΓR\nE(k)dk∼Γv2, or\nv2∼ϵtdamp∼v2\n0\u0012tdamp\ntcascade\u0013\n(9)\nwhere v2\n0andtcascade are the velocity and cascade time\nat the outer scale in the absence of damping; for a given\nenergy forcing ϵ, the velocity at the outer scale is re-\nduced. However, since tdamp∼>tinject, the damping time\ncannot be made arbitrarily small. We discuss this fur-\nther in §4.2.\nTo understand the behavior at smaller scales, note\nthat the cascade time is scale-dependent, while for non-\nresonant CR acceleration, tdamp is independent of scale.\nWe are accustomed to thinking of the cascade time de-\ncreasing towards small scales (for instance, tcascade ∝\nl2/3, l1/2for undamped Kolmogorov, Kraichnan tur-\nbulence respectively). However, damping changes the\nscale dependence of velocity, further reducing velocities\nat small scales, and thus increasing cascade times at\nthese scales. If tcascade /tdamp still decreases towards\nsmall scales, then the cascade eventually takes over\nand the spectrum rebounds from damping. However,\niftcascade /tdamp instead increases towards small scales,\nthen damping becomes increasingly dominant and the\nspectrum will cut off precipitously. Since tdamp is inde-\npendent of k, what matters is the scale dependence of\ntcascade .\nFrom Equation 8, the cascade time can be written as:\ntcascade ∼kE(k)\nF(k)∼1\n[k3E(k)]1/2(Kolmogorov) (10)\n∼vph\nk2E(k)(Kraichnan) (11)\nwhere we have used F(k)∼[E(k)]3/2k5/2,F(k)∼\nk3[E(k)]2/vphfor Kolmogorov and Kraichnan turbu-\nlence respectively. When damping operates, E(k) will\nsteepen from standard Kolmogorov/Kraichnan spectra.\nFrom Equation 11, we see that for a power spectrum\nE(k)∝k−α,tcascade increases with kforα∼>3\n(Kolmogorov), α∼>2 (Kraichnan). The steepening\nof the power spectrum slope is controlled by the rel-\native strength of damping, i.e. tcascade /tdamp at large\nscales. If this is sufficiently large, it produces a power\nspectrum with a slope steeper than the critical value,\nand we have a runaway: tcascade /tdamp continually in-\ncreases towards small scales, producing a rapid cutoff\nin the velocity power spectrum. However, if the initial\nvalue of tcascade /tdamp produces a power spectrum with\nan index shallower than the critical slope, then damp-\ning initially ‘takes a bite’ out of the turbulent cascade,\nbuttcascade /tdamp decreases towards small scales, untilCosmic Ray Effects on Turbulence 9\n1 10 100 1000k 0.0010.010.11E(k)k53Kolmogorov\n1 10 100 1000k0.0010.010.11E(k)k32Kraichnan\n1 10 100 1000k0.0010.010.11E(k)k2Burgers\ntcasc\ntdamp=0tcasc\ntdamp=1\n2tcasc\ntdamp=1tcasc\ntdamp=1.25tcasc\ntdamp=1.5tcasc\ntdamp=2.1\nk21\nk3\nFigure 3. Modified kinetic energy spectra for a Kolmogorov (left), Kraichnan (middle), and Burgers (right) cascade with\nvarying levels of CR damping, all with vph= 2v0. E(k) is in units of the outer-scale, undamped kinetic energy, where k denotes\nthe wavenumber. Different line colors denote different ratios of the cascade time to the damping time, showing that if damping\nbecomes competitive, the outer scale velocity decreases, and the slope of the spectrum steepens. Dashed lines show E(k) =k−2\nandE(k) =k−3for comparison. For tcascade /tdamp⪆1.5, 1 for Kolmogorov and Kraichnan, respectively, the cascade sharply\ncuts off at progressively smaller k. For smaller tcascade /tdamp, CRs damp fluctuations, but the cascade returns to its normal\nscaling at large k. For Burgers turbulence, which is not a genuine cascade, there can be an appreciable decrease in power at\nsmall k, but at high k, the spectrum recovers a k−2slope.\ndamping becomes negligible, the original cascade domi-\nnates and the spectrum recovers its original undamped\npower law slope.\nWe clearly see confirmation of this bifurcation in\nsmall scale damping in Fig. 3. We see that we re-\nquire tcascade /tdamp∼>1.5 at the outer scale for crit-\nical damping in a Kolmogorov cascade (so that the\npower spectrum steepens beyond E(k)∝k−3), or\ntcascade /tdamp∼>1 at the outer scale for critical damp-\ning in a Kraichnan cascade (so that the power spectrum\nsteepens beyond E(k)∝k−2). Indeed, tcascade /tdamp∼\n1 causes a perfect transformation of the Kraichnan spec-\ntrum from a E(k)∝k−3/2spectrum to a Burgers-like\nE(k)∝k−2spectrum.\nThis bifurcation in the existence of small scale turbu-\nlence is important, so we restate it in simpler terms.\nDamping can change the slope of the velocity power\nspectrum E(k)∝k−α, and hence the scale dependence\nof velocity v(k)∝k(1−α)/2(using v2∼kE(k)), but it\ndoes not change the physics of the turbulent cascade.\nThe latter can be encapsulated in the form of cascade\ntimes tcascade ∼l/vl(Kolmogorov), tcascade ∼lvph/v2\nl\n(Kraichnan). Using v(k)∝k(1−α)/2, these relations im-\nplytcascade ∝k(α−3)/2(Kolmogorov), and tcascade ∝\nkα−2(Kraichnan), which gives critical slopes α= 3,2\nrespectively, in line with our previous arguments. The\nscale dependence of tcascade determines if turbulence is\ncompletely damped at small scales, or recovers with the\noriginal (undamped) power-law scaling.\nThe right panel of Figure 3 shows modified “Burg-\ners” spectra where we’ve solved Equation 8 with F(k)∼\nk2E(k). In this case, even when tcascade /tdamp >1, themodified kinetic energy spectra never show cut-offs, in-\nstead always converging to a k−2spectrum at high k,\nbut there is a substantial decrease in small-scale power\ncompared to the undamped case. We briefly note that\nEquation 8 does not really apply to Burgers turbulence\nE(k)∝k−2, which is not a genuine turbulent cascade,\nbut rather an instantaneous jump from large to small\nscales via shocks which arise from non-linear steepen-\ning. However, Ptuskin damping creates friction which\ncan balance non-linear steepening and prevent shock for-\nmation. We can see this by examining Burgers’ equation\nin the presence of Ptuskin damping:\n∂v\n∂t+v· ∇v=−Γv (12)\nFor Γ >∇v, the damping term exceeds the non-linear\nterm, so that damping exceeds non-linear steepening\nwhen the nonlinear time tNL∼L/v > t damp. The out-\ncome of this is uncertain. Figure 3 suggests that wave\namplitudes will be most significantly damped at low k,\nafter which steepening still occurs but with reduced am-\nplitude. In any case, tNL/tdamp potentially plays a sim-\nilar role to tcascade /tdamp, and as such, we will use tNL\nas a proxy for tcascade in our simulation analysis ( §4.3).\n4.2. What is tcascade /tdamp?\nThe results of the previous section show that the ra-\ntiotcascade /tdamp is the critical parameter determining\nthe efficacy of small scale damping, and that there is\na critical value ( tcascade /tdamp∼>1.5,1 for Kolmogorov\nand Kraichnan turbulence, respectively) such that the\nturbulence spectrum will show a cutoff. Here, we inves-10 Bustard & Oh\ntigate the conditions under which these thresholds may\nbe crossed.\nWe have previously argued from energy conservation\nthat ˙ECR∼<˜ϵin steady state, hence tdamp≥tinject∼\nρv2/˜ϵ∼L/v, the timescale on which kinetic energy is in-\njected. In Appendix A, we confirm this expectation and\nalso show how various scalings, such as δρ/ρ, δv/v , can\nbe understood as a function of PCR/Pg, orv/cs, v/v ph.\nWhen does tdamp reach the minimal value of tinject∼\nL/v, so that almost all of the injected kinetic energy is\ndirectly dissipated in cosmic rays? Equating the first\nand second terms in brackets in Equation 4, tdamp∼\ntinject when:\nMph∼<\u0012Pc\nPtot\u0013\n(13)\nEquation 13 is only an order of magnitude estimate; the\nexact threshold must come from numerical simulations.\nNonetheless, it illustrates the relevant physics: damping\nsaturates when the turbulent Mach number is small and\nthe CR energy density is high.\nIftdamp reaches its minimal value of tinject∼L/v,\nthen:\ntcascade\ntdamp∼1 (Kolmogorov)\n∼1\nMph(Kraichnan)(14)\nFrom Fig 3, we see that it is unclear whether\ndamping will be strong enough to enforce a small\nscale cutoff in a Kolmogorov cascade (which requires\ntcascade /tdamp∼>1.5), but any subsonic turbulence in a\nKraichnan cascade which satisfies Equation 13 will au-\ntomatically have tcascade /tdamp∼>1), the threshold for\ncritical damping there. The increase in tcascade /tdamp\nis not due to a decrease in the damping time (which\nhas a floor at tinject), but rather the increased cascade\ntime in MHD turbulence. Longer cascade times are as-\nsociated with wave turbulence, where wave-wave inter-\nactions produce non-linearities which eventually cause\nturbulence to cascade (Nazarenko 2011). Other forms\nof wave turbulence can be present, for instance, in sys-\ntems with strong stratification (Wang et al. 2022) or\nrotation.\nNote that even if the threshold for critical damping\n(i.e. exponential suppression of small-scale power) is not\nmet, Figure 3 shows that the damping of gas motions\ncan still be significant.\n4.3. Simulations\nThe results of §4.1, 4.2 are useful for guiding expec-\ntations and driving intuition. Nonetheless, given the\ncomplex non-linearities, they require validation by nu-\nmerical simulation – a difficult task, given the limitedinertial range of standard resolution simulations. We\nnow present a set of simulations which, to our knowl-\nedge, are the first CR hydrodynamics simulations specif-\nically probing CR influence on turbulent kinetic energy\nspectra. While a more complete set of simulations with\ndifferent driving modes and higher resolution awaits, we\nalready see that CRs suppress small-scale fluctuations.\nWe focus first on the case where Ptuskin damping is\nmaximized, running a series of diffusion-only simulations\nnear the CR energy gain ‘sweet spot’ κ∼0.15L0vph,\nwhere v2\nph∼(Pc+Pg+PB)/ρ. We vary the input driving\nrate ˜ϵby an order of magnitude to create turbulence with\nundamped Ms∼0.5 andMs∼0.15 , where Ms=v/cs\nandcs∼p\nPg/ρis the gas sound speed (thus, Mph=\nv/vphdecreases as Pc/Pgincreases). Plasma beta, β=\nPg/PB, are denoted in each figure and represent rough\nvalues for the presented suite of simulations; while each\nsimulation starts with the same β, the saturated value\nofβchanges by a small amount depending on whether\nCRs are present, what CR transport model is assumed,\netc.\nFigure 4 shows simulation kinetic energy spectra for\nbothMs∼0.5 and Ms∼0.15 simulation sets, each\nnormalized by the k = 3 mode power for the Ms∼0.5\nMHD-only simulation. Different colors denote different\ninitial PCR/Pg, ranging from 0 to 1 .5. Points denote\naverage kinetic energies, and the shaded regions denote\nthe minimum and maximum kinetic energies taken over\n10 snapshots at late times when we see converged spec-\ntra, typically between 8 and 10 eddy turnover times af-\nter the simulation starts (see Appendix for more about\ntime convergence). Importantly, we note that the iner-\ntial ranges in our MHD-only simulations display some-\nthing between a Kraichnan ( k−3/2) and a Burgers-like\n(k−2) spectrum, with significant power at high k due\nto the generation of solenoidal modes rather than fast\nmodes despite our purely compressive forcing (see Sec-\ntion 4.5). The k−2compressive component we find\nis frequently seen in hydrodynamic simulations with\ncompressive driving, due to non-linear steepening (e.g.,\nMiniati 2015). Thus, the analytic models of §4.1,4.2\nwhere we assume a Kraichnan spectrum do not exactly\napply. Nonetheless, we can look for qualitative agree-\nment.\nWe can use the non-linear steepening time as a proxy\nfor the cascade time: tcasc∼tNL∼L0/vL, where vL\nis the outer-scale velocity. Ratios of cascade time to\ndamping time, calculated with tdamp =< ρv2/˙PCR>,\nare noted in the legend. The trend agrees at least\nqualitatively with Figure 3. Power both at large and\nsmall scales is decreased when PCR≥Pg, consistent\nwith mild Ptuskin damping when tcascade ∼tdamp. AsCosmic Ray Effects on Turbulence 11\n100101102\nk103\n102\n101\n100k2E(k)dkk2\n 1\nCR-Modified Kinetic Energy Spectra\ns0.5\nMHD\n(PCR\nPg, tNL\ntdamp) (0.3, 0.12)\n(PCR\nPg, tNL\ntdamp) (1.5, 0.37)\ns0.15\nMHD\n(PCR\nPg, tNL\ntdamp) (0.3, 0.25)\n(PCR\nPg, tNL\ntdamp) (1.5, 0.62)\nFigure 4. The turbulent kinetic energy spectrum, multiplied by k2for a set of Ms∼0.5 and Ms∼0.15 diffusion-only\nsimulations, keeping κ= 0.15L0vphandβ∼1, as we vary PCR/Pg. Spectra are normalized to the k=3 mode for the Ms∼0.5\nMHD run. Ratios of the outer scale nonlinear time to damping time, calculated with tdamp =< ρv2/˙PCR>andtNL=L0/v, are\nalso denoted. Points show the energy in each k-bin averaged over 10 outputs at late times, when turbulence is fully developed,\nwhile shaded regions show the minima and maxima during those time periods. Each simulation was run on a 5123grid. While\nMHD runs produce overall spectra shallower than k−2, CRs damp fluctuations, slightly decreasing the power in low k modes\nwhile steepening the spectra at high k.\ntcascade /tdamp increases, the spectrum deviates further\nand further from the MHD case. For example, the\ntcascade /tdamp∼0.62,Ms∼0.15 simulation has be-\ntween 10 and 100 times less power in high-k modes than\nthe MHD run. Projections of density, kinetic energy,\nand magnetic energy for these Ms∼0.15,β∼10\nsimulations vary quite obviously, as seen in Figure 5,\nwith fluctuations clearly damped in the PCR∼Pgcase\n(bottom row) compared to the MHD case (top row).\nHigher Mach number simulations appear to show damp-\ning, as well, but the effect is less obvious. This is in\nline with expectations from our previous discussion that\ntcascade /tdamp is maximized for smaller values of stirring\nvelocity.\nWhile our analytic predictions and preliminary simu-\nlations suggest that Ptuskin damping could play a role in\nsuppressing the compressible turbulent cascade at smallscales, it may appear hazardous to draw conclusions\nbased on moderate resolution simulations with limited\ninertial range. We therefore refer the reader to Figure\n6, which shows kinetic energy spectra for simulations on\na 5123grid, each with initial β∼10 but diffusion co-\nefficients varying between κ||∼(0−15)L0cs. Clearly,\nthe strongest damping effect occurs when κ||is near the\nsweet-spot ( κ||∼(0.15−1.5)L0cs), and the effect dimin-\nishes as κ||increases. Maybe most importantly, signifi-\ncant spectral changes do not occur in the absence of CR\ntransport ( κ||∼0), suggesting that numerical diffusion\nplays a negligible role.\nWe also refer the reader back to §3 and Figure 1,\nwhere we presented a separate, more robust diagnostic\nof the suppression of the turbulent cascade by Ptuskin\ndamping: via the heating of adiabatic gas. In hydro-\ndynamic simulations of adiabatic gas, we have found12 Bustard & Oh\nMHD\n PCR∼Pgκ||∼0.15vphL0Density\nKinetic Energy\nMagnetic Energy\nFigure 5. Projections perpendicular to the initial magnetic field direction of density (left column), kinetic energy (middle\ncolumn), and magnetic energy (right column) after ∼10 eddy turnover times, normalized by their average values in the MHD\nonly case. Top: MHD-only simulations with β∼10 and Ms∼0.15.Bottom: Simulations with the same βand forcing rate,\nbut with PCR∼Pgand diffusive CR transport. Density, velocity, and magnetic fluctuations are all suppressed compared to the\nMHD case.\nthat ˙Egas→˜ϵ, as it should. However, in adiabatic\nsimulations with CRs, we have found ˙Egas→0, while\n˙ECR→˜ϵ, i.e. almost all of the turbulent energy is ab-\nsorbed by the CRs (see Figure 1). Furthermore, all of\nthis energy is absorbed at large scales, which are well re-\nsolved. The shift to CRs receiving almost all the energy\nof the turbulent cascade is genuine turbulent accelera-\ntion, not due to numerical diffusion in the CR module.\nWe infer this from numerical convergence in our CR ac-\nceleration rates, as well as the close conformance to ana-\nlytic expectations. Nonetheless, we have tested this ex-\nplicitly by checking energy absorption for the two-fluid\ncase when κ= 0 (bottom panel of Figure 1); in this case\n˙Egas/˜ϵ→0.8 when PCR/Pg∼1, i.e. gas heating is once\nagain large.\nIf Ptuskin damping does not allow gas motions to\ncascade the ∼2 decades to grid scale in our simula-\ntions to enable dissipation, this strongly suggests that\nreal turbulence should not be able to cascade down\nthe many more decades to e.g. the gyroscale of CRs,\nwhere fast modes are frequently invoked to scatter CRs\nwith E⪆300 GeV. Of course, it is still imperative to\ntest these ideas in much higher resolution simulations,preferably with a spectral code that can better resolve\nan MHD Kraichnan cascade.\nWhile we’ve focused on the kinetic energy spectra so\nfar, we have yet to show that the magnetic energy spec-\ntra, which is most important for CR scattering, shows\nthe same damping trends. Figure 7 shows the kinetic\nenergy spectra (left panel), magnetic energy spectra\n(middle), and CR energy spectra (right) for our set of\n5123,β∼10 simulations with varying CR diffusivi-\nties. The kinetic energy spectra are identical to that\nin Figure 6, and they show considerable damping when\nκ∼0.15L0cs, i.e. at the sweet-spot diffusivity where\ndamping is most efficient. Similarly, for that same simu-\nlation, the magnetic energy spectrum is clearly damped,\nbut as κvaries off the sweet-spot, more small-scale power\nremains. The CR energy spectra are quite different: the\namplitude of small-scale CR pressure fluctuations mono-\ntonically decreases with increasing κ, because strong dif-\nfusion damps small-scale CR perturbations.\n4.4. Streaming vs Diffusion\nIn the pure diffusion limit, Γ( k) is well known, and\nas we’ve shown analytically and numerically, the result-Cosmic Ray Effects on Turbulence 13\n100101102\nk102\n101\n100k2E(k)dk10\n100101102\nk102\n101\n100E(k)/E(k)MHDMHD ||0\n ||0.15L0cs\n ||1.5L0cs\n ||15L0cs\nFigure 6. Kinetic energy spectra of 5123simulations when PCR∼Pgbut varying the diffusion coefficient from κ||∼0 (where\nthe only diffusivity is numerical) to the most efficient reacceleration regime ( κ||∼0.15vphL0andκ||∼1.5vphL0) to the fast\ndiffusion regime ( κ||∼15vphL0). Note how the power spectrum is somewhat different for the two fluid system even in the\nabsence of CR transport, presumably because of changes to the phase velocity and other adiabatic properties, but deviations\nfrom the MHD spectrum are mild compared to simulations with added diffusion. Left: k2E(k)dknormalized by the k= 3 MHD\nvalue. Right: Ratio of each spectrum to the MHD spectrum. Note how diffusion introduces a characteristic scale lCRwhere the\nkinetic energy is reduced: in the fast diffusion limit, lCR> L 0, and the outer-scale kinetic energy drops significantly while the\nrest of the spectrum retains the same shape as the MHD case. Going to smaller κ||, overall changes are more drastic because\nreacceleration is more efficient but also the scale where the spectrum cuts off most dramatically shifts to lCR< L 0.\n100101102\nk102\n101\n100k2EK(k)dk10\n100101102\nk102\n101\n100k2EB(k)dk\n100101102\nk103\n102\n101\n100k2ECR(k)dkMHD ||0\n ||0.15L0cs\n ||1.5L0cs\n ||15L0cs\nFigure 7. Kinetic energy spectra (left), magnetic energy spectra (middle), and CR energy spectra (right) multiplied by k2for\n5123,β∼10 simulations with varying κ. Note that the left panel is the same as in Figure 6 and that the y-axis of the right\npanel extends down to 10−3rather than 10−2for the other panels. Overall, magnetic energy spectra follow the same trend as\nkinetic energy spectra, showing damped small-scale power when the diffusivity is near the sweet-spot κ∼0.15L0cs. CR energy\nspectra instead show an approximately monotonic decrease in small-scale power with increasing diffusivity, as strong diffusion\ndamps small-scale CR perturbations.\ning CR drag damps turbulence at large scales, chang-\ning kinetic energy spectral slopes and even introducing\ncut-offs. The functional form for Γ( k) is more uncertain\nwhen streaming transport is introduced. Since we found\nin Paper I that streaming stunts reacceleration rates due\nto fundamental changes to CR-turbulent interactions,\nit’s tempting to append the plasma β-dependent correc-\ntion factors from Paper I to Γ( k). If this were true, weak\nCR reacceleration should imply very weak changes to\nthe kinetic energy spectrum; however, we’ve run a num-ber of simulations with CR streaming, including some\nwith no diffusive transport where reacceleration is abso-\nlutely negligible, that clearly modify the kinetic energy\nspectra. We present some simple scalings which match\nour simulations, but defer a detailed study to future\nwork.\nAll simulations in this section start with PCR∼Pg\nand assume an isothermal equation of state. Figure 8\nshows the kinetic energy spectra for 2563simulations of\nvarying β∼1, 10, 100, each with different CR transport14 Bustard & Oh\n102\n101\n1001\nk2E(k)dk\n102\n101\n100E(k)/E(k)MHD\n102\n101\n10010\n102\n101\n100\n100101102\nk102\n101\n100100\n100101102\nk102\n101\n100\nMHD ||0.15vphL0\n ||0 + Streaming\n ||0.15vphL0 + Streaming\nFigure 8. Kinetic energy spectra for Ms∼0.15, 2563simulations each with PCR/Pg∼1 but varying CR transport and varying\nthe initial plasma βfrom 1 (top) to 10 (middle) to 100 (bottom). The magenta-colored lines show the resulting MHD (no CR)\nspectra as a reference. The left column shows k2E(k)dknormalized by the MHD value at k= 3, while, to more clearly show\nthe changes in spectral shape, the right column shows the ratio of each spectrum to the MHD spectrum. For diffusion only,\nefficient reacceleration damps the kinetic energy spectrum, resulting in less power at small scales compared to the MHD case.\nHowever, with streaming included, both reacceleration rates and field-aligned CR pressure gradients depend on β. At low β\n(low Alfv´ en Mach number MA=v/vA), streaming negates reacceleration, and the kinetic energy spectra revert to the MHD\ncase. For larger β, however, reacceleration becomes somewhat more efficient, causing damping, and a more significant fraction\nof turbulent energy is channeled through CRs and lost via streaming energy transfer. This latter effect, most clearly evident\nin the streaming only simulations (red curves), decreases the overall kinetic energy in the system but doesn’t appear to induce\ncut-offs like the diffusion-only runs.Cosmic Ray Effects on Turbulence 15\nmodels but the same turbulent driving rate, which for\nsimulations without CRs (MHD only) give a sonic Mach\nnumber Ms∼0.15. A partial version of Figure 8, using\na 5123domain, is included in the Appendix and shows\nsimilar behavior. The left column shows each spectrum\nmultiplied by k2, normalized to the peak value of the\nMHD spectrum at k=3. The right column, in order to\nmore clearly show differences in the spectral shape and\noverall kinetic energy, shows each spectrum divided by\nthe MHD spectrum. Note the similarity of the pure\nstreaming power spectra to the streaming + diffusion\npower spectra; in this parameter range, streaming dom-\ninates over diffusion. We seek to answer two main ques-\ntions about these results:\nHow does streaming vs diffusive transport affect the\noverall kinetic energy in the gas?\nThe top panel of Figure 9 quantifies the partitioning of\nturbulent forcing that ends up in CRs ( fCR=˙ECR/˜ϵ),\nas well as fE= (ρv3/L)/˜ϵvs the steady-state plasma\nbeta, βf. Filled circles denote simulations with stream-\ning and diffusion, while empty circles have just stream-\ning. The streaming plus diffusion results quantify what\nwe see by eye in the kinetic energy spectra: increasing β\nleads to smaller turbulent velocities; in each case, CRs\ntake only a very small amount of the total energy forc-\ning, with most energy input instead removed from the\nsystem by streaming energy loss.\nThe bottom panel of Figure 9 shows the same sim-\nulations but with the y-axis showing the damped ki-\nnetic energy vs the undamped case. Overplotted is a\nline showing a β−1/2scaling, which appears to fit the\ndata quite well. At face value, it is counterintuitive\nthat in the streaming dominated case, CR damping is\nstronger at higher β, i.e. when vAis smaller. To in-\nterpret this, it’s important to note that Alfv´ en Mach\nnumbers for each run saturate at MA<1, meaning\nthat Alfv´ en crossing times are faster than eddy turnover\ntimes; hence, streaming transport is relatively fast. Fast\nstreaming transport leads to small field-aligned CR pres-\nsure gradients / large field-aligned CR scale lengths\nlCR=PCR/(ˆb· ∇PCR). Compared to CRs with slow\ndiffusive transport, streaming CRs have comparatively\nsmall pressure gradients and absorb less energy (via the\nv· ∇PCRterm) in sub-Alfv´ enic flows. This may par-\ntially explain the behavior seen in Figure 8, where, for\ninstance, β∼1 leads to fast streaming transport, hence\nsmall CR pressure gradients, and little to no change in\nthe kinetic energy spectrum.\nAt the same time, it is important to realize that CR\ntransport timescales are notsimply ∼L/v A, since CR\npressure gradients and magnetic fields are often mis-\n101102\nf\n102\n101\n100fE=(v3/L)/\n102\n101\n100\nfCR=(ECR)/\n101102\nf\n101\n100v2/v2\n0\n1/2\ns0.15 (if undamped)\ns0.5 (if undamped)\ns0.75 (if undamped)\n101\n100\nfCR=(ECR)/\n||0.15L0vph + streaming\n||=0 + streaming\nFigure 9. Isothermal, 2563(circle symbols) and 5123(dia-\nmond symbols, bottom panel only) simulations with stream-\ning and PCR∼Pg. The MHD (undamped) version of these\nsimulations give Ms=v0/cs∼0.15 (black), 0.5 (green), and\n0.75 (cyan). The top panel shows only the Ms∼0.15 points\nand shows the partitioning of turbulent forcing that ends up\nin CRs ( fCR=˙ECR/˜ϵ; green points and right y-axis), as\nwell as fE= (ρv3/L)/˜ϵ(black points and left y-axis) vs the\nsteady-state plasma beta βf. While for diffusion there was a\nclear correlation between fCRandfE, now, fCRis small, and\nfEcorrelates inversely with β, at least in this sub-Alfv´ enic\nregime studied. The bottom panel shows the turbulent ki-\nnetic energy relative to the undamped case, where ρv3\n0/L∼˜ϵ.\nWith CRs, even with streaming only transport where there\nis no reacceleration, v2/v2\n0∝β−1/2, at least roughly, in this\nsub-Alfv´ enic or “fast transport” regime. There is also a weak\ntrend towards larger overall damping with increasing driving\nrate (larger Mshas smaller v2/v2\n0), at fixed β.16 Bustard & Oh\naligned. Thus, for instance, CR heating rates (which\nnaively scale as ∼vA/L) somewhat counter-intuitively\ndecrease as magnetic field strengths and hence vAin-\ncrease. This is because increased magnetic tension in\nsub-Alfvenic turbulence results in poorer alignment be-\ntween magnetic fields and CR pressure gradients, reduc-\ningvA·∇Pc(see Figure 4 in Paper I). This qualitatively\nfits with the bottom panel of Figure 9, assuming colli-\nsionless energy loss drives the damping.\nWhile we do not have a rigorous argument for the\nv2/v2\n0∝β−1/2scaling, which we present as an outcome\nof our simulations, we can give the following heuristic ar-\ngument: v2∝ϵtdamp (from equation 9), where naively\ntdamp∝L/vA. However, since CR heating rates (and\nhence turbulent damping rates) scale as vA· ∇Pc, we\nknow that tdamp also depends on Pc, PB, where in sub-\nAlfvenic turbulence PBcontrols the relative alignment\nbetween vAand∇Pcvia magnetic tension. From dimen-\nsional analysis, we must have tdamp∼L/vA(PB/Pc)α,\nwhere α= 1 since tdamp ∝theat∝P−1\nc. If so,\nv2∝tdamp∝PB/vA∝vA∝β−1/2. Future work will\nhave to test more carefully the scalings in the ansatz\ntdamp ∝L/v A(PB/Pc)∼vAL/c2\ncfor the streaming\ndominated case, which closely resembles the expression\ntdamp∼vphL/c2\ncin the sweet spot for the diffusion dom-\ninated case. What is striking in our simulations is that\nCR ‘drag’ in the streaming dominated case consistently\nseems to render undamped super-Alfvenic turbulence\nsub-Alfvenic, even though the rise in magnetic energy\ndensity (and hence rise in vA) is very mild; most of the\nchange in MAis due to reduced gas velocities.\nDoes streaming change the shape of kinetic energy\nspectra, as diffusion does?\nStreaming CRs, which don’t themselves take an appre-\nciable amount of turbulent energy input, still nonethe-\nless sap kinetic energy from the system. How the kinetic\nenergy spectra change, however, is fundamentally differ-\nent between streaming and diffusive transport. Chang-\ningβ(changing MA) in streaming-dominated simula-\ntions effectively changes the ratio of transport time to\neddy turnover time. To glean further insight, it’s inter-\nesting to compare to simulations with purely diffusive\ntransport but varying diffusion coefficients.\nFigure 6 shows kinetic energy spectra for simulations\non a 5123grid, each with initial β∼10 but diffusion co-\nefficients varying between κ||∼(0.15−15)vphL0. Our\nfiducial case of κ||∼0.15vphL0shows that damping, in\nthe slow diffusion regime, exerts meaningful drag on an\nentire hierarchy of scales, beginning at the outer scale;\nin other words, damping and cascade rates are competi-\ntive over a large range of k. Moving to the fast diffusionregime ( κ||∼15vphL0), this is clearly not the case: the\ndiffusion length scale is larger than the outer eddy scale,\nand the damping rate is only competitive with the cas-\ncade time at large scales, leaving the cascade to operate\nuninterrupted after CRs have reduced the outer-scale\nkinetic energy.\nFollowing similar logic, we infer that, for streaming-\ndominated transport in sub-Alfv´ enic turbulence, Γ( k)\nmust be weighted heavily towards small k, causing an\ninitial reduction in outer-scale kinetic energy but an\nunimpeded cascade at larger k. Thus, we see that the\npower spectrum when streaming is included has the\nsame shape over the effective inertial range of the sim-\nulations k∼<30, albeit with a lower normalization (in\ntheβ∼10,100 cases, when damping is effective). In\nthe dissipation range, k∼>30, there is additional steep-\nening compared to the MHD case, though whether this\nis numerical or physical is as yet unclear.\n4.5. Compressive vs Solenoidal Components\nWhile we intend to follow this manuscript with a\nlarger simulation suite and more detailed analysis of\nCR-modified turbulence, we include a preliminary anal-\nysis here of compressive vs solenoidal motions to display\nsome characteristics we anticipate from an expanded\nsimulation suite. Our arguments so far have focused on\nCR damping of compressive fluctuations, but our kinetic\nenergy spectra contain both compressive and solenoidal\nmotions despite being seeded with purely compressive\nforcing. In hydrodynamic turbulence, compressive mo-\ntions completely dominate in subsonic turbulence driven\nwith purely compressive forcing (Federrath et al. 2010),\nbut in MHD turbulence, magnetic fields affect this bal-\nance. Namely, for the sub-Alfvenic, β∼10,Ms∼0.15\nsimulations we’ve focused on, we expect from previous\nwork (Lim et al. 2020) that the combination of com-\npressive fluctuations and magnetic tension will generate\nsolenoidal power, even at a level comparable to the com-\npressive power. This holds true in our simulations.\nWe use a standard Hodge-Helmholtz decomposition\nto separate compressive and solenoidal components as a\nfunction of scale, and we plot their power spectra, multi-\nplied by k2and normalized by the corresponding power\nof the MHD simulation’s k= 3 mode, for a subset of\nourβ∼10 and β∼100 simulations with and without\nCRs in Figure 10. For the MHD simulations, the inte-\ngrated fractions of solenoidal power to total power are\nEsol/Etot∼0.42 for β∼10 and Esol/Etot∼0.11 for\nβ∼100. These values are in-line with those in Lim\net al. (2020), with magnetic tension playing a small role\nin solenoidal generation at higher β. Interestingly, the\nsolenoidal component is comparable to the compressiveCosmic Ray Effects on Turbulence 17\n100101102\nk104\n103\n102\n101\n100k2EK(k)dk10\n100101102\nk100\nCompressive\nSolenoidalMHD ||0.15L0cs\n ||0.15L0cs + Streaming\n100101102\nk0.10.20.30.40.50.60.70.80.91.0v2\nx(k)\nv2\ntot(k)dk10\n100101102\nk100\nFigure 10. Top row: Kinetic energy spectra, multiplied by k2and normalized to the k= 3 MHD value, decomposed into\ncompressive (solid lines) and solenoidal (dashed lines) components. Each simulation was run on a 5123grid. The left panel\nshows β∼10 simulations, where solenoidal power is a significant fraction of the total power and dominates at small scales,\nleading to a shallower than k−2spectrum. The right panel shows β∼100 simulations, where compressive modes are dominant\nat almost all scales in the MHD case. CRs considerably damp the compressive fluctuations, though, which in turn decreases the\npower in solenoidal motions that are generated by a combination of compressions and magnetic tension. In both the β∼10 and\nβ∼100 cases with streaming, compressive damping leads to an increased ratio of solenoidal to compressive power. However,\nthis “divergence cleaning” is not pronounced in the pure diffusion run, where solenoidal and compressive power decrease by\nabout the same amount. Bottom row: Power in velocity fluctuations vxalong the initial mean magnetic field divided by the\npower in total velocity fluctuations. This quick measure of anisotropy roughly tracks the compressive vs solenoidal motions seen\nabove: with CR streaming present, CR damping leads to more anisotropy (higher fraction of vxpower), consistent with a larger\nfraction of anisotropic solenoidal modes rather than isotropic compressive modes. The diffusion only case, which shows a kink\nat high-k, is an outlier whose analysis we leave for future work.\ncomponent or even dominates at small scales. That our\nkinetic energy spectra are shallower than k−2at large\nk, then, seems to be due to Alfven modes rather than\nfast modes, consistent with recent literature suggesting\nthat, even with primarily compressive driving, signifi-\ncant turbulent energy lies instead in Alfven modes (see\ne.g. Figure 2 in Makwana & Yan 2020, or Gan et al.\n2022 for a full spatio-temporal decomposition of fast,\nslow, and Alfven modes).When CRs are present, compressive (and in some\ncases, solenoidal power) decreases. For the β∼10 case\nwith CR diffusion, we measure Esol/Etot= 0.36, very\ncomparable to the MHD case with Esol/Etot= 0.42.\nWhen streaming is present, we measure Esol/Etot= 0.67\nfor the β∼10 simulation (compare to Esol/Etot= 0.42\nfor MHD), and we measure Esol/Etot= 0.35 for the\nβ∼100 simulation (compare to Esol/Etot= 0.11 for\nMHD).18 Bustard & Oh\nOur interpretation, barring further work that we save\nfor a future paper, is that CRs preferentially damp com-\npressive motions consistent with the analytic deriva-\ntions of this paper, but since compressive motions com-\nbine with magnetic tension to drive solenoidal mo-\ntions in sub-Alfvenic turbulence, both compressive and\nsolenoidal components are suppressed. That CRs pref-\nerentially damp compressive rather than solenoidal mo-\ntions is evidenced by our two simulations with CR\nstreaming, which show a “divergence cleaning” effect\nwhere the ratio of solenoidal to compressive power in-\ncreases. However, this divergence cleaning is less appar-\nent in the diffusion only run. We defer a fuller discus-\nsion of the difference between diffusion and streaming\neffects to later work where we drive both compressive\nand solenoidal modes, rather than relying on solenoidal\nmotions generated by compressive forcing, since in this\ncase damping of compressive motions can easily damp\nsolenoidal power as well.\nFor brevity, we defer most other analyses of the cas-\ncade to future work, but we do point out one additional\noutcome of damping: CRs can change the anisotropy of\nthe cascade. In the bottom row of Figure 10, we plot\nthe power in velocity fluctuations along the initial mag-\nnetic field direction (ˆ x) over the power in all directions.\nThis quick diagnostic of anisotropy follows our intuition\nfrom above: In the simulations with CR streaming, as\nCRs damp isotropic compressive motions, and turbu-\nlence is dominated by anisotropic solenoidal motions,\neddies become more elongated along the mean field di-\nrection, with the fractional power in vxfluctuations in-\ncreasing from the MHD values, especially in k= 5−20\nrange. There is no such increase in anisotropy in the dif-\nfusion only run, consistent with the lack of divergence\ncleaning. Indeed, there is an apparent downturn toward\nisotropy at high k, albeit in a range where results may\nnot be numerically reliable.\n5.DISCUSSION\n5.1. Regimes of CR Modified Turbulence\nIn§4, we showed both analytically and numerically\nthat CRs can have a significant impact on the power\nspectrum of turbulence. In particular, we showed that\nas the damping time decreases relative to the turbu-\nlent cascade time, the turbulent power spectra will be\nsteepen and then cut off abruptly at small scales (for\ntcascade /tdamp∼>1). These results should eventually be\ncarefully checked by higher resolution numerical simu-\nlations. Nonetheless, we clearly already have seen in\n§3 and Fig 1 a situation where CR damping of mo-\ntions is stronger than the rate at which energy cascades\nto smaller scales, so that little energy reaches the gridscale. Our analytic estimates can guide expectations as\nto which environments these effects might be important.\nIntra-cluster medium (ICM) —In the ICM, although sonic\nMach numbers are typically low ( Ms∼0.1−0.3), the\nabsence of hadronic γ-ray emission gives an upper bound\nonPc/Ptot<<1 (typically less than a few percent; Ack-\nermann et al. 2014), so that Equation 13 is not satisfied\nthere. The CR energy density is too small to appreciably\naffect gas motions, and it is unlikely that CR reacceler-\nation appreciably damps the turbulent cascade.\nInterstellar medium (ISM) —In the ISM, CR damping\ncould be potentially important: Pc/Ptot∼ O(1) is rel-\natively large. In the diffusion only case, the main un-\ncertainty lies in the CR acceleration rate. The most\nefficient reacceleration occurs for diffusivities in the\nrange κ||< v phL0∼3×1026cm2/s for ISM-like pa-\nrameters (see Table 2 in Paper I). Canonical values of\nκ∼1028−1029cm2/s used in galactic propagation mod-\nels are much larger, i.e. we are sufficiently far away from\nthe ‘sweet spot’ that acceleration and hence damping\ntimes could be long. On the other hand, if CR stream-\ning dominates transport, then since the ISM has β∼1,\ndamping is small, as we have seen.\nCircumgalactic medium (CGM) —Finally, the galactic\nhalo and CGM are strong candidates for significant CR\ndamping. For the diffusion only case, these regions oc-\ncupy a sweet-spot where κ∼vphL0,Mph<1, and if,\nas suggested by simulations of Milky Way mass galaxies\n(e.g. Butsky & Quinn 2018; Ji et al. 2020), Pc/Pg∼>1,\nthen Pc/Ptotis order unity. For these conditions, Equa-\ntion 13 is satisfied, so that tdamp∼tinject. Thus, for\ninstance, from Equation 14, tcascade /tdamp∼ M−1\nph∼2\nfor a compressive Kraichnan cascade with Mph∼0.5:\nthe compressive cascade will be steepened beyond the\ncritical threshold of E(k)∝k−2and abruptly cut off, so\nthere is no small scale turbulence. We see hints of this\nin Figure 4 for the Ms∼0.5 case, but given our limited\ndynamic range, spectral changes are more obvious for\nMs∼0.15, when tNL/tdamp is even larger.\nOnce streaming is included, we have also seen that\nthere can be considerable damping in the β∼10−100\ncases, with a weak trend towards larger damping with in-\ncreasing driving rate at fixed β(see how v2/v2\n0is smaller\nfor larger Ms), potentially because CRs are more effi-\nciently trapped in turbulent eddies as the Alfven Mach\nnumber approaches unity. This differs from the diffu-\nsion only case, where stronger turbulence implies smaller\ntNL/tdamp and weaker damping. For a fixed driving\nrate that produces transonic MHD turbulence in a β∼\n10 environment (reasonable CGM parameters), adding\nstreaming-dominated CRs up to equipartition PCR∼PgCosmic Ray Effects on Turbulence 19\ndamps turbulent kinetic energy by a factor of ∼5 or\ngreater (bottom panel of Figure 9).\nFor each of these regimes, there is also a question of\nthe turbulent driving scale relative to the CR mean free\npath, i.e. whether our fluid assumption of CR transport\nis valid. For both self-confinement and extrinsic tur-\nbulence models of CR scattering, the typical mean free\npath for a GeV CR in the ISM is ∼1 pc, which is not too\nfar below the typical driving scale of turbulence ( ∼100\npc). However, if self-confinement is stronger, then the\nmean free path is shorter, and the separation between\ndriving scale and mean free path is larger. Similarly, in\nthe CGM and ICM, the driving scale is much larger, so\nthis scale separation is not an issue.\nFinally, given the possibilities for CR-modified turbu-\nlence in ISM and CGM environments described above,\nhow do these results compare to observations of elec-\ntron density fluctuations measured through interstellar\nscintillation (Armstrong et al. 1995), which show den-\nsity fluctuations on a wide range of scales, i.e. the\n“Big Power Law” in the sky? We believe our results\nare consistent with these observations for two reasons:\n(i) The observed spectrum ∝k−5/3is consistent with\nKolmogorov turbulence and is, therefore, unlikely to be\ngenerated by a purely compressive fast mode cascade.\nFrom our findings, CRs preferentially damp compres-\nsive fluctuations ( §4.5), allowing solenoidal motions to\nextend over a wide range of scales consistent with the\n“Big Power Law”. (ii) In any case, in the β∼1 ISM, if\nstreaming is dominant, CRs do not significantly modify\nthe power spectrum (Figure 8). Signatures of small scale\ndamping are more likely to be seen in the β >1 CGM,\nif the CR energy density is significant (as is suggested\nby simulations; Ji et al. 2020).\n5.2. Implications of CR-Modified Turbulence\nThe implications of such CR-modified spectra are pos-\nsibly quite intriguing. For instance, a CR-induced cut-\noff could significantly affect the spatial scale of thermal\ninstability, since there are no small scale compressive\nmotions, unless there is direct driving at those scales.\nAlso, since Ptuskin damping only affects compressive\nmotions, not solenoidal motions, Ptuskin damping can\npotentially make turbulence less Burgers-like and more\nKolmogorov-like. It would be interesting to explore this\n‘divergence-cleaning’ effect in simulations with a mix-\nture of driving modes.\nPerhaps the most interesting consequence of CR\ndamping of turbulence is its implication for scattering\nof high-energy CRs by fast modes in an extrinsically\ndriven turbulent cascade. This is frequently invoked to\nexplain the scattering of CRs with E∼>300GeV (Yan& Lazarian 2004), since self-confinement is too weak\nto explain observed isotropy and confinement times.\nHowever, the resonant scattering invoked (transit time\ndamping) requires the turbulence to cascade many or-\nders of magnitude, to the ∼300AU gyroscale of such\nCRs. Fig 3 shows that for tcasc/tdamp = 1, a Kraich-\nnan ( E(k)∝k−3/2) fast mode spectrum will steepen to\na Burgers ( E(k)∝k−2) spectrum, which already has\ntoo little small scale power to efficiently scatter CRs\nvia transit time damping (Miniati 2015; Pinzke et al.\n2017), and even higher values of tcasc/tdamp∼ M−1\nphwill\ncompletely eliminate turbulence at small scales. While\nthis needs further study, low-energy CRs, by damping\nturbulent fluctuations at large scales, could divert tur-\nbulent energy that would otherwise scatter high-energy\nCRs. This potentially adds to the long list of problems\nwith ‘standard’ theories of CR scattering in the Milky\nWay which have been recently pointed out (Kempski &\nQuataert 2022; Hopkins et al. 2022).\nRegardless of whether CR drag introduces a cut-off to\nkinetic energy spectra, it is clear from our simulations\nthat CRs in both diffusion-dominated and streaming-\ndominated transport regimes can sap a significant frac-\ntion of the turbulent forcing rate. This breaks the usual\ncorrespondence between turbulent velocity and turbu-\nlent driving rate, i.e. for hydrodynamic turbulence,\nρv3\n0/L∼˜ϵ. Now, ρv3/L∼fE˜ϵ, where the new cor-\nrection factor fEcan be ≪1. As derived in Equa-\ntion 9, v2/v2\n0∝tdamp/tcascade , which for streaming-\ndominated transport in sub-Alfv´ enic turbulence gives\nv2/v2\n0∝β−1/2(Figure 9). In the CGM, where we ex-\npect damping to be most significant, turbulent velocities\nobtained from the observed velocity dispersion may sig-\nnificantly underestimate the turbulent forcing rate, i.e.\n˜ϵ≫ρv3/L.\n6.CONCLUSIONS\nIn this paper, we present analytical estimates and ac-\ncompanying MHD+CR simulations probing CR effects\non turbulence, namely the damping of turbulence by\nlarge-scale, CR-induced drag on compressive gas mo-\ntions. Our main findings are as follows:\n•Despite long CR reacceleration times, the damping\ntime due to CR reacceleration can be very competi-\ntive with the turbulent cascade time.\ntdamp∼ρv2max\u0012tgrow\nPCR,1\n˜ϵ\u0013\n∼max\u0012\nM2\nctgrow,ρv2\n˜ϵ\u0013\n(15)\nwhere Mc=v/ccis the Mach number with respect\nto the CR sound speed cc∼p\nPc/ρ, and ˜ ϵ=ρv3/L\nis the turbulent energy injection rate. Our key fig-\nures are Figures 1 and 2, where we confirm that CRs20 Bustard & Oh\ncan divert a significant fraction of turbulent energy\nthat would otherwise dissipate as heat at small scales.\nConditions for strong damping are met under quite\nreasonable conditions (Equation 13); the CGM is an\nespecially strong candidate for this damping.\n•If CR diffusion dominates transport, and if the ratio\nof the damping time to the cascade time is sufficiently\nshort, small scale compressive turbulence should be\nexponentially suppressed (see Figure 3). This sup-\npression of small scale turbulence has abundant im-\nplications for e.g. thermal instability, “divergence-\ncleaning” of turbulence spectra (e.g. Figure 10), and\nsuppression of fast modes at small scales, which have\nbeen invoked to scatter high-energy CRs (see §4.1).\nWe see compelling signatures of damping in our sim-\nulation spectra ( §4.3; Figure 4), but these effects de-\nserve future study with higher resolution simulations\nthat capture a larger turbulent inertial range.\n•The effects of streaming transport are more com-\nplex and deserve follow-up. Importantly, tgrow in\nEquation 15 does notinclude the suppression of CR\nreacceleration by streaming (the βdependent fac-\ntors identified in Bustard & Oh 2022), which would\nsubstantially increase damping times. Instead, from\nFigure 2, diversion of turbulent energy through CRs\nremains strong even in the presence of CR stream-\ning, for our simulations where MA∼<1. Instead\nof introducing spectral cut-offs, streaming uniformly\ndecreases the normalization of the turbulent power\nspectrum, but not its shape, with the turbulent ki-\nnetic energy scaling as v2∝vA∝β−1/2(Fig 8, Fig9). This is possibly because damping operates pre-\ndominantly at the largest scales in the ‘fast trans-\nport’ regime (here, the sub-Alfv´ enic regime). Such\nlarge scale damping implies energetic input and tur-\nbulent heating rates (much of which gets channeled\ninto CR collisionless heating) can be much larger\nthan standard estimates for Kolmogorov turbulence,\n˜ϵ≫ρv3/L.\nACKNOWLEDGMENTS\nThe authors gratefully acknowledge Navin Tsung,\nMax Gronke, Yan-Fei Jiang, Christoph Federrath, Hui\nLi, and Ellen Zweibel, as well as the organizers and par-\nticipants of the KITP “Fundamentals of Gaseous Ha-\nlos” workshop. We also thank our anonymous referee\nfor an extremely detailed and perceptive report that\nsignificantly improved our paper. CB was supported\nby the National Science Foundation under Grant No.\nNSF PHY-1748958 and by the Gordon and Betty Moore\nFoundation through Grant No. GBMF7392. SPO was\nsupported by NSF grant AST-1911198, and NASA grant\n19-ATP19-0205.\nComputations were performed on the Stampede2 and\nPSC-Bridges2 supercomputers under allocation TG-\nPHY210004 provided by the Extreme Science and Engi-\nneering Discovery Environment (XSEDE), which is sup-\nported by National Science Foundation grant number\nACI-1548562 (Towns et al. 2014).\nSoftware: Athena++ (Stone et al. 2020), yt (Turk\net al. 2011), Matplotlib (Hunter 2007), Mathematica\n(Wolfram Research, Inc. 2021)\nAPPENDIX\nA.TURBULENT PROPERTIES AND DAMPING IN A COSMIC RAY DOMINATED MEDIUM\nCRs can influence velocity and density perturbations in a turbulent medium, but the extent depends on the relative\npartition of CR vs thermal energy, as well as the CR diffusivity / transport speed. Commer¸ con et al. (2019) simulate\nCRs in a turbulent box with purely diffusive transport and a bi-stable ISM (with radiative cooling). They found\nthat trapped CRs modify the gas flow, change the density PDF, and provide support against thermal instability,\nmaintaining the gas in an intermediate temperature state that is classically thermally unstable. It remains to be seen\nhow these simulations would change when streaming is included. The perturbative heating term from CR streaming\naffects thermal instability (Kempski & Quataert 2020), and in low- βplasmas where this heating is most significant, CR\nstreaming can also drive acoustic waves unstable, generating a “stair-case” cosmic ray pressure profile and additional\nmultiphase gas (Tsung et al. 2022; Quataert et al. 2022).\nOur simulation setup is quite different from that of Commer¸ con et al. (2019), most notably because we don’t include\nradiative cooling, so we don’t attempt a detailed comparison, but we do find some qualitatively similar behavior.\nFigure 11 shows δρ/ρ,δv/v, and δPCR/PCRfor diffusion-only simulations with varying PCR/Pgand either κ= 0\norκ= 0.15L0vph(where vphdepends on PCR/Pg). When CRs are dynamically unimportant ( PCR/Pg≪1), we\nrecover the MHD expectation that δρ/ρ∼δv/v∼ M s= 0.5. Deviations from this relation start when PCR/Pg⪆1.\nInterestingly, we find that δρ/ρ isindependent ofPc/Pg, while δPc/Pc∝1/Pc(i.e., δPc∼const is independent ofCosmic Ray Effects on Turbulence 21\n102\n101\n100101102\nPCR/Pg102\n101\n/ \nv/v\nPCR/PCR\nv/cs=s\nv/vph=ph\nv/v2\nph1/PCR\n0.15L0vph\n=0\nFigure 11. Fluctuating density (blue symbols), velocity (black symbols), and CR pressure (green symbols) as a function of\nPCR/Pg. Open circles denote simulations with non-zero diffusion coefficient κ||∼0.15L0vph, and open diamonds denote purely\nadvective CR transport ( κ= 0). None of these simulations include additional streaming transport.\nPc/Pg). At the same time, we find that the velocity divergence ∇ ·v∝P−1/2\nc (not shown), i.e. it does depend on Pc.\nThis might appear puzzling, since one expects density fluctuations and velocity divergence to be directly related, yet\nthe former is independent of Pc, while the latter shows dependence.\nA key to understanding these results is to realize that the ‘sweet spot’ κ∼Lvphis really still in the ‘fast diffusion\nregime’. The ratio tdiffuse /tsc∼lvph/κis only unity at the outer scale l∼L; at smaller scales, tdiffuse /tsc<1 and\ndiffusion dominates. In this diffusion dominated regime, CRs diffuse out of eddies before they contribute significantly\nto resisting compression – i.e., they do not provide a significant restoring force (instead, they provide drag). In\nparticular, they do not contribute to the phase velocity vph. Thus, δρ/ρ∼δPg/Pg∼v/cs, where csis the gas sound\nspeed, independent of PCR. This is roughly consistent with Commer¸ con et al. (2019) (see their Figures 5 and 6), which\nfinds a similar dependence on κand a clear decrease in δPCR/PCRasPCR/Pgincreases.\nUsing this information, we can better interpret the lower bound on damping time that we infer from our simulations.\nImportantly, as Ptuskin damping saturates ( PCR/Pg→ ∞ ,fCR→1), the maximum rms CR pressure perturbation\nis⟨∆PCR⟩rms∼ρv2. This is a strict upper bound, since the free energy to create CR pressure perturbations is\nderived from kinetic energy (similarly, ∆ PCR,∆Pgat a shock cannot exceed the ram pressure ρv2). In this limit,\n∆PCR/PCR∼ρv2/PCR∝1/PCR. Finally, in the diffusion dominated limit, CR compression is balanced by diffusion,\nPCR,0(∇ ·v)∼ −∇ · (κ∇PCR,1)∼κρv2/L2, which implies that\n∇ ·v∝κ\nPCR,0, (A1)\nwhere PCR,0, PCR,1refer to the unperturbed and perturbed CR pressure, respectively. Thus, in the regime where we fix\nthe sweet-spot diffusion coefficient κ∼vphL0andPCR/Pg∼>1 (so that vph∝P1/2\nCR), then κ∝P1/2\nCR, and∇·v∝P−1/2\nCR.\nWe have also verified in our simulations that ∇ ·v∝κfor constant PCR, and∇ ·v∝1/PCRfor constant κ.\nThese results thus indicate that the damping time cannot become arbitrarily small. If drag forces are given by:\n˙v∼1\nρ∇PCR,1∼L/v∼teddy. Thus, δPCR∼<ρv2implies that\ntdamp∼>teddy. This is equivalent to the statement that the work done by CR forces in opposing gas motions cannot22 Bustard & Oh\n100101102\nk101\n100k2E(k)dk10\n MHD\n||0.15vphL0\nTime Interval\n6.1 - 7.4 teddy\n8.6 - 9.9 teddy\n11.1 - 12.3 teddy\n100101102\nk101\n100k2E(k)dk10\n ||0 + Streaming\n||0.15vphL0 + Streaming\nTime Interval\n6.1 - 7.4 teddy\n8.6 - 9.9 teddy\n11.1 - 12.3 teddy\nFigure 12. Time convergence of select spectra, each run on a 2563grid. The diffusion-only simulations, which show the most\ndamping, converge very early. 5123simulations (not shown) are similarly converged with respect to time.\nexceed the energy input rate: v· ∇PCR,1∼<˜ϵ∼ρv3/L, which implies ∇PCR,1∼<ρv2/L, consistent with Equation\nA2. Thus, for Kolmogorov turbulence, we expect tcascade /tdamp∼1 for maximally efficient Ptuskin damping. In\nthe Kraichnan case, however, tcascade /tdamp can be greater than 1 if the Mach number, relative to the velocity of\ncompressible fluctuations, is small. This is not due to any decrease in the damping time; instead, it is due to cascade\ntimes being lengthened when vphis large.\nB.TIME CONVERGENCE\nFigure 12 shows the average kinetic energy spectra for 2563simulations with varying CR transport model, measured\nat different time intervals. Most importantly, the diffusion-only simulations show converged, clearly damped spectra\neven at early times. Spectra for simulations with CR streaming are also well-converged but at somewhat later times.\nNote that these time intervals over which we pull out kinetic energy spectra are much later than the saturation of bulk\nturbulent quantities (e.g. kinetic energy, magnetic energy, etc.), which occurs after only a few eddy turnover times.\nC.RESOLUTION CONVERGENCE\nA good test of how inherently diffusive our CR module is, and whether that accounts for some observed spectral\nchanges, is to run simulations with no explicit CR diffusion at various resolutions. Figure 13 compares spectra for\nourβ= 10 MHD simulations to simulations with PCR∼Pgand purely advective CR transport (no streaming and\nκ||= 0). For grid sizes of 2563and 5123, we see in both cases that, in the inertial range up until k∼20, there is\nno appreciable damping due to the presence of CRs, confirming again that CR transport is the cause for clear and\nobvious damping seen in Figures 4, 8, 6 beginning at small k.\nFigure 14 shows kinetic energy spectra for 5123simulations when transport is included. These simulations only\ncomprise part of those on a 2563grid (compare to Figure 8 in §4.4) because computer resource limits prohibit us\nfrom running the streaming only ( κ∼0 + streaming) simulations. In any case, the streaming only simulations and\nthe streaming + diffusion simulations are both streaming dominated in this sub-Alfv´ enic regime, so we expect their\nspectra to look very similar, as we saw in Figure 8.\nThe MHD and diffusion only spectra look qualitatively similar to those on a 2563grid. 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G. 2017, Physics of Plasmas, 24, 055402" }, { "title": "1806.03172v1.Brownian_motion_of_magnetic_domain_walls_and_skyrmions__and_their_diffusion_constants.pdf", "content": "Brownian motion of magnetic domain walls and skyrmions,\nand their di\u000busion constants\nJacques Miltat,\u0003Stanislas Rohart, and Andr\u0013 e Thiaville\nLaboratoire de Physique des Solides, Universit\u0013 e Paris-Sud,\nUniversit\u0013 e Paris-Saclay, CNRS, UMR 8502, F-91405 Orsay Cedex, France\n(Dated: October 8, 2018)\nExtended numerical simulations enable to ascertain the di\u000busive behavior at \fnite temperatures\nof chiral walls and skyrmions in ultra-thin model Co layers exhibiting symmetric - Heisenberg - as\nwell as antisymmetric - Dzyaloshinskii-Moriya - exchange interactions. The Brownian motion of\nwalls and skyrmions is shown to obey markedly di\u000berent di\u000busion laws as a function of the damping\nparameter. Topology related skyrmion di\u000busion suppression with vanishing damping parameter,\nalbeit already documented, is shown to be restricted to ultra-small skyrmion sizes or, equivalently,\nto ultra-low damping coe\u000ecients, possibly hampering observation.\nI. INTRODUCTION\nThe prospect of ultra-small stable information bits in\nmagnetic layers in presence of the Dzyaloshinskii-Moriya\n(DM) interaction [1] combined to the expectation of their\nminute current propagation [2], notably under spin-orbit\ntorques [3], builds up a new paradigm in information\ntechnology. In stacks associating a metal with strong\nspin-orbit interactions e.g. Pt and a ferromagnetic metal\nsuch as Co, that may host isolated skyrmions, large do-\nmain wall velocities have also been forecast [4] and ob-\nserved [5]. The DM interaction induces chiral magnetiza-\ntion textures, walls or skyrmions, that prove little prone\nto transformations of their internal structure, hence their\nextended stability and mobility.\nIn order, however, to achieve low propagation cur-\nrents, steps will need to be taken towards a reduction\nof wall- or skyrmion-pinning. Recent experimental stud-\nies indicate that skyrmions fail to propagate for cur-\nrents below a threshold roughly equal to 2 1011Am\u00002for\n[Pt/Co/Ta]nand [Pt/CoFeB/MgO]nmultilayers [6], or\n2:5 1011Am\u00002for [Pt/(Ni/Co/Ni)/Au/(Ni/Co/Ni)/Pt]\nsymmetrical bilayers [7]. Only in one seldom instance did\nthe threshold current fall down to about 2 :5 1010Am\u00002\nfor a [Ta/CoFeB/TaO] stack, still probably, however, one\norder of magnitude higher than currents referred to in\nsimulation work applying to perfect samples [8].\nIn a wall within a Co stripe 50 nm wide, 3 nm thick, the\nnumber of spins remains large, typically 216for a 5 nm\nwide wall. A skyrmion within a Co monolayer (ML) over\nPt or Ir, on the other hand, contains a mere 250 spins,\nsay 28. Assuming that a sizeable reduction of pinning\nmight somehow be achieved, then a tiny structure such\nas a skyrmion is anticipated to become sensitive, if not\nextremely sensitive, to thermal \ructuations.\nIn this work, we show, on the basis of extended nu-\nmerical simulations, that both chiral walls and skyrmions\nwithin ferromagnets obey a di\u000busion law in their Brow-\nnian motion at \fnite temperature [9, 10]. The di\u000busion\n\u0003jacques.miltat@u-psud.fr\nx z q \nwS tS L \na) b) q FIG. 1. a) Wall within a narrow stripe: wSis the stripe width,\ntSits thickness. The stripe element length Lis solely de\fned\nfor computational purposes. qis the wall displacement; b)\nsnapshot of the magnetization distribution: color coding after\nmx. The wall region mx\u00191 appears red. Thermal \ructua-\ntions are visible within domains: T= 25 K,wS= 100 nm,\ntS= 0:6 nm,\u000b= 0:5.\nlaw is shown to be valid over a broad range of damp-\ning parameter values. The thermal di\u000busion of domain\nwalls seems to have attracted very little attention, ex-\ncept for walls in 1D, double potential, structurally un-\nstable, lattices [11], a source of direct inspiration for\nthe title of this contribution. Chiral magnetic domain\nwalls are found below to behave classically with a mobil-\nity inversely proportional to the damping parameter. As\nshown earlier [12, 13], such is not the case for skyrmions,\na behavior shared by magnetic vortices [14]. Vortices and\nskyrmions in ferromagnetic materials are both charac-\nterized by a de\fnite topological signature. In contradis-\ntinction, skyrmions in antiferromagnetic compounds are\ncharacterized by opposite sign spin textures on each sub-\nlattice, with, as a result, a classical, wall-like, dependence\nof their di\u000busion constant [15]. Lastly, ferrimagnets do\ndisplay reduced skyrmion Hall angles [16], most likely\nconducive to modi\fed di\u000busion properties.arXiv:1806.03172v1 [cond-mat.mes-hall] 8 Jun 20182\nII. DOMAIN WALL DIFFUSION\nWe examine here, within the micromagnetic frame-\nwork, the Langevin dynamics of an isolated domain wall\nwithin a ferromagnetic stripe with thickness tS, width\nwSand \fnite length L(see Fig. 1). The wall is located\nat mid-position along the stripe at time t= 0. Thermal\nnoise is introduced via a stochastic \feld ~HRduncorrelated\nin space, time and component-wise, with zero mean and\nvariance\u0011proportional to the Gilbert damping parame-\nter\u000band temperature T[17] :\nh~HRdi=~0\nhHi\nRd(~ r;t)Hj\nRd(~r0;t0)i=\u0011\u000eij\u000e(~ r\u0000~r0)\u000e(t\u0000t0)\n\u0011=2kBT\n\r0\u00160MS\u000b(1)\nwhere,kBis Boltzmann constant, \u00160and\r0are the vac-\nuum permability and gyromagnetic ratio, respectively,\nMSthe saturation magnetization. Written as such, the\nfunctions\u000e(~ r\u0000~r0) and\u000e(t\u0000t0) have the dimension of\nreciprocal volume and time, respectively. Applied to nu-\nmerical simulations, the variance of the stochastic \feld\nbecomes\u0011=2kBT\n\r0\u00160MSVdt\u000b, whereVis the computation\ncell volume and dtthe integration time step.\nA. Simulation results\nThe full set of numerical simulations has been per-\nformed by means of an in-house code ported to graph-\nical processing units (GPU's). Double precision has\nbeen used throughout and the GPU-speci\fc version of\nthe \"Mersenne twister\" [18] served as a source of long-\nsequence pseudo-random numbers generator.\nMaterial parameters have been chosen such as to mimic\na 3-ML Co layer (thickness tS= 0:6 nm) on top of Pt\nwith an exchange constant equal to A= 10\u000011J/m, a\nMs= 1:09 106A/m saturation magnetization, a Ku=\n1:25 106J/m3uniaxial anisotropy constant allowing for\na perpendicular easy magnetization axis within domains,\nand a moderate-to-high DM interaction (DMI) constant\nDDM= 2 mJ/m2. In order to temper the neglect of short\nwavelength excitations [19], the cell size has been kept\ndown toLx=Ly= 1 nm, whilst Lz=tS= 0:6 nm.\nThe stripe length has been kept \fxed at L= 1\u0016m,\na value compatible with wall excursions within the ex-\nplored temperature range. The latter has, for reasons\nto be made clear later, been restricted to \u00191=3 of the\npresumed Curie temperature for this model Co layer. Fi-\nnally, the integration time constant, also the \ructuating\n\feld refresh time constant, has been set to dt= 25 fs.\nAs shown by the snapshot displayed in Fig. 1b, the wall\nmay acquire some (moderate) curvature and/or slanting\nduring its Brownian motion. Because wall di\u000busion is\ntreated here as a 1D problem, the wall position qis de-\n0510152025\n487488489Time [ns]Wall Position [nm]ΔtqFIG. 2. Excerpt of a wall trace displaying wall position \ructu-\nations vstime:T= 77 K,\u000b= 0:5,wS= 100 nm,tS= 0:6 nm.\nqis the wall displacement during time interval \u0001 t.\n\fned as the average position owing to :\nq=L\nNxNyPNx\ni=1PNy\nj=1mz(i;j)\n[hmziL\u0000hmziR](2)\nwhere,iandjare the computation cell indices, Nxand\nNythe number of cells along the length and the width\nof the stripe, respectively, hmziLis the \ructuations aver-\naged value of the zmagnetization component far left of\nthe domain wall,hmziRthe average value of mzfar right.\nRegardless of sign, hmziRandhmziLare expected to be\nequal in the absence of any Hz\feld.\nFig. 2 displays the position as a function of time of a\nwall within a wS= 100 nm wide stripe immersed in a\nT= 77 K temperature bath. A 2 ns physical time win-\ndow has been extracted from a simulation set to run for\n1:5\u0016s. The \fgure shows short term wall position \ruc-\ntuations superimposed onto longer time di\u000busion. Ac-\ncording to Einstein's theory of Brownian motion [9], the\nprobability P(x;t) of \fnding a particle at position xat\ntimetobeys the classical di\u000busion equation @tP(x;t) =\nD@2\nx2P(x;t) with, as a solution, a normal (gaussian) dis-\ntributionP(x;t) = 1=p\n4\u0019Dtexp(\u0000x2=4Dt), whereDis\nthe di\u000busion constant.\nSo does the raw probability of \fnding a (sti\u000b) wall in\na narrow stripe at position qafter a time interval \u0001 t, as\nshown in Fig. 3 (see Fig. 2 for variable de\fnition). It\nought to be mentioned that the average wall displace-\nmenthq(\u0001t)iis always equal to 0, with an excellent ac-\ncuracy, provided the overall computation time is large\nenough. The \ft to a normal distribution proves rather\nsatisfactory, with, however, as seen in Fig. 3, a slightly\nincreasing skewness in the distributions as a function of\nincreasing \u0001 t. Skewness, however, 1) remains moderate3\n05 1031 1041.5 1042 104\n-30-20-100102030Δt = 0.2 nsN\nq - [nm]05 1031 1041.5 1042 104\n-30-20-100102030Δt = 0.5 ns\nq - [nm]N\n05 1031 1041.5 1042 104\n-30-20-100102030Δt = 1.0 ns\nq - [nm]N\n05 1031 1041.5 1042 104\n-30-20-100102030q - [nm]Δt = 2.0 nsN\nFIG. 3. Wall within stripe: event statistics with time interval\n\u0001tas a parameter; \u000b= 0:5,wS= 100 nm, tS= 0:6 nm,\nT= 25K. The continuous blue lines are \fts to a gaussian\ndistribution, the variance of which increases with \u0001 t.\nup to \u0001tvalues typically equal to 5 \u000010 ns, 2) is seen to\nreverse sign with time interval (compare Fig. 3b and c),\nexcluding intrinsic biasing. The distributions standard\ndeviation is clearly seen to increase with increasing \u0001 t.\nAlternatively, one may represent the variance hq2i\n(hqi= 0) as a function of the time interval \u0001 t: if di\u000bu-\nsion applies, then a linear dependence is expected, with\na 2Dslope for a one-dimensional di\u000busion. Fig.4a shows,\nfor various temperatures, that a linear law is indeed ob-\nserved. Lastly, as shown in Fig.4b, the di\u000busion constant\nincreases linearly with increasing temperature. The er-\nror bars measuring the departure from strict linearity in\nFig.4a remain limited in extent. For the stripe width\nand damping parameter considered here ( wS= 100 nm,\n\u000b= 0:5), the ratio of di\u000busion constant to temperature\nis found to amount to D=T= 0:187 nm2ns\u00001K\u00001.\nB. Wall di\u000busion constant (analytical)\nThiele's equation [20] states that a magnetic texture\nmoves at constant velocity ~ vprovided the equilibrium of\n3 forces be satis\fed:\n~G\u0002~ v+\u000bD~ v=~F (3)\nwhere,~Fis the applied force, ~FG=~G\u0002~ vis the gyrotropic\nforce,~Gthe gyrovector, ~FD=\u000bD~ vthe dissipation force,\nDthe dissipation dyadic.\nFor the DMI hardened N\u0013 eel wall considered here : ~G=\n050100150200250300\n012345Δt [ns]< q2 > [nm2]\n25° K50° K77° K120° K150° K\na)0102030\n050100150T [K]D [nm2 ns-1]\nb)FIG. 4. a) Variance hq2i(nm2) of the wall displacement vs\ntime interval \u0001 twith temperature Tas a parameter. Thick\nlines represent a linear \ft to data; b) Di\u000busion constant D\nas a function of temperature (square full symbols). Dis pro-\nportional to the slope of the hq2ivs\u0001tcurves in Fig.4a (see\ntext for details). The error bars are deduced from the slopes\nof straight lines through the origin that encompass all data\npoints in Fig.4a for a given temperature and the \ft time\nbracket, 1\u00005 ns. For the sake of legibility, the error bars\nhave been moved-up by 2 :5 units. Continuous line: linear\n\ft through the origin. The dashed line is the analytical ex-\npectation in the \"low\" noise limit. \u000b= 0:5,wS= 100 nm,\ntS= 0:6 nm.\n~0. For a 1D wall, the Thiele equation simply reads :\n\u000bDxxvx=Fx (4)\nwhere,Dxx=\u00160MS\n\r0R\nV(@~ m\n@x)2d3r.\nThe calculation proceeds in two steps, \frst evaluate\nthe force, hence, according to Eqn.4, the velocity auto-\ncorrelation functions, then integrate vstime in order to\nderivehq2i. The force, per de\fnition, is equal to minus\nthe partial derivative of the energy Ew.r.t. the displace-\nmentq, namelyFx=\u0000@E\n@q=\u0000\u00160MSR\nV@~ m\n@x\u0001~H d3r.\nFormally,\nhFx(t)Fx(t0)i= (\u00160MS)2\u0002 (5)*Z\nV@~ m(~ r;t)\n@x\u0001~H(~ r;t)d3rZ\nV@~ m(~r0;t0)\n@x\u0001~H(~r0;t0)d3r0+\nAs noticed earlier [14], since the random \feld noise is\n\"multiplicative\" [17], moving the magnetization vector\nout of the average brackets is, strictly speaking, not al-\nlowed, unless considering the magnetization vector to\nonly marginally di\u000ber from its orientation and modulus\nin the absence of \ructuations (the so-called \"low\" noise\nlimit [14]):\nhFx(t)Fx(t0)i= (\u00160MS)2\u0002 (6)\nZ\nVX\ni;j\"\n@mi(~ r;t)\n@x@mj(~r0;t0)\n@xD\nHi(~ r;t)Hj(~r0;t0)E#\nd3r d3r0\nIf due account is being taken of the fully uncorrelated4\ncharacter of the thermal \feld (Eqn.1), the force auto-\ncorrelation function becomes:\nhFx(t)Fx(t0)i= 2\u000bkBTDxx\u000e(t\u0000t0) (7)\nThe velocity auto-correlation function follows from\nEqn.4. Lastly, time integration ( q(t) =Rt\n0vx(t0)dt0)\nyields :\nhq2(t)i= 2Dt;D=kBT\n\u000bDxx(8)\nIn order to relate the di\u000busion constant to a more directly\nrecognizable wall mobility, Dxxmay be expanded as :\nDxx=\u00160MS\n\r02wStS\n\u0001T(9)\nwhere, \u0001Thas been called the Thiele wall width (implic-\nitly de\fned in [21]). Dmay thus be expressed as :\nD=kBT\n2\u00160MS1\nwStS\r0\u0001T\n\u000b(10)\nthus, proportional to the wall mobility \r0\u0001T=\u000b.\nA directly comparable result may be obtained after\nconstructing a full Langevin equation from the ( q;\u001e)\nequations of domain wall motion (Slonczewski's equa-\ntions [22]), where \u001eis the azimuthal magnetization angle\nin the wall mid-plane. In this context, the wall mobility\nis\u0016W=\r0\u0001=\u000b, where \u0001 is the usual wall width, inci-\ndentally equal to the Thiele wall width in the case of a\npure Bloch wall. The Langevin equation [10] here reads:\nmD\n2wStSd2\nq2\u000b\ndt2+1\n22\u00160Ms\n\u0016WwStSd\nq2\u000b\ndt=kBT(11)\nwhere,mDis D oring's wall mass density (kg =m2):\nmD=\u0000\n1 +\u000b2\u0001\u0012\r0\n2\u00160Ms\u0013\u000021\n\u0019jDDMj(12)\nan expression valid in the limit jDDMj\u001dKE\u000b=\nKu\u00001\n2\u00160M2\ns. Note that the DMI constant DDMex-\nplicitly enters the expression of the wall mass, as a con-\nsequence of the wall structure sti\u000bening by DMI. In the\nstationary regime, hq2iis proportional to time tand the\nwall di\u000busion constant exactly matches Eqn.10, after sub-\nstitution of \u0001 Tby \u0001. Finally, the characteristic time for\nthe establishment of stationary motion is:\nt0=mD1\n2\u00160Ms\r0\u0001\n\u000b(13)\nFor the parameters of our model 3-ML Co layer on top\nof Pt, D oring's mass density is equal to \u00183 10\u00008kg=m2\nfor\u000b= 0:5, and the characteristic time amounts to\nt0'25 ps. Still for \u000b= 0:5,wS= 100 nm and\ntS= 0:6 nm,D=Tamounts to 0 :153 nm2ns\u00001K\u00001for\n\u0001T= 4:13 nm, i.e. the value computed from a properly\nconverged wall pro\fle at T= 0. The relative di\u000berence\n0255075\n050100150T [K]a)wS = 25 nmwS = 50 nmwS = 100 nmD [nm2 ns-1]\n00.250.50.751\n00.010.020.030.040.051/wS [nm-1]D /T [nm2 ns-1 K-1]\nb)FIG. 5. a) Di\u000busion constant Das a function of temperature\nwith the stripe width wSas a parameter (full symbols); b)\nD=Tas a function of the inverse of the stripe width. \u000b= 0:5,\ntS= 0:6 nm, throughout. Solid blue lines: linear \ft through\nthe origin, dashed line: analytical expectation.\nbetween simulation and theoretical values is found to be\nof the order of\u001920%.\nOwing to Eqn.10, Dis expected to prove inversely pro-\nportional to both the stripe width wSand the Gilbert\ndamping parameter \u000b, a behavior con\frmed by simula-\ntions. Fig.5a displays the computed values of the dif-\nfusion coe\u000ecient as a function of temperature with the\nstripe width as a parameter, whilst Fig.5b states the lin-\near behavior ofDvswS\u00001. The slope proves, however,\nsome 13:5% higher than anticipated from Eqn.10. Lastly,\nthe 1=\u000bdependence is veri\fed in Fig.6 showing the com-\nputed variation of Dvstemperature with \u000bas a param-\neter for a narrow stripe ( wS= 25 nm) as well as the\ncorresponding \u000bdependence ofD=T. The dotted line\nrepresents Eqn.10 without any adjusting parameter. The\nrelative di\u000berence between simulation data and theoret-\nical expectation is beyond, say \u000b= 0:25, seen to grow\nwith increasing \u000bbut also appears to be smaller for a\nnarrow stripe as compared to wider tracks.\nAltogether, simulation results only moderately depart\nfrom theoretical predictions. The Brownian motion of\na DMI-sti\u000bened wall in a track clearly proves di\u000busive.\nThe di\u000busion constant is classically proportional to the\nwall mobility and inversely proportional to the damping\nparameter. Unsurprisingly, the smaller the track width,\nthe larger the di\u000busion constant. In order to provide an\norder of magnitude, the di\u000busion induced displacement\nexpectation,p\n2D\u0001t, for a wall sitting in a 100 nm-wide,\npinning-free, track for 25 ns at T= 300 K proves essen-\ntially equal to\u0006the stripe width.\nIII. SKYRMION DIFFUSION\nOutstanding observations, by means of Spin Polarized\nScanning Tunneling Microscopy, have revealed the exis-\ntence of isolated, nanometer size, skyrmions in ultra-thin5\n0255075100125150175200\n020406080100α = 0.125α = 0.25α = 0.5α = 0.075α = 0.05\nα = 0.8a)T [K]D [nm2 ns-1]\n0246810\n-20-1001020\n00.20.40.60.81D /T [nm2 ns-1 K-1](%)\nαb)wS = 25 nmwS = 50 nmwS = 100 nm\nFIG. 6. a) Di\u000busion constant Das a function of temperature\nwith the damping constant \u000bas a parameter ( wS= 25 nm,\ntS= 0:6 nm). Solid blue lines: linear \ft through the ori-\ngin; b)D=T(large semi-open symbols) as a function of \u000bfor\nwS= 25 nm and tS= 0:6 nm; dotted blue curve: analyt-\nical expectation. Full symbols: relative di\u000berence between\ncomputational and analytical results (%).\nFIG. 7. a) Snapshot of a skyrmion immersed in a 12.5 K\ntemperature bath ( \u000b= 0:5), together with the underlying\nlattice. Red cells: sz\u0019+1, blue cells: sz\u0019\u00001. The white\ncross indicates the barycenter of lattice site positions satisfy-\ningsz\u00150:5.\n\flms such as a PdFe bilayer on an Ir(1111) single crystal\nsubstrate [23] [24]. We analyse below the thermal motion\nof skyrmions in a model system made of a Co ML on top\nof Pt(111). We deal with skyrmions with a diameter of\nabout 2:5 nm containing at T= 0 about 250 spins.\nA. Simulation results\nIn order to monitor the Brownian motion of an iso-\nlated skyrmion, rather than micromagnetics, it is pre-\nferred to simulate the thermal agitation of classical\nspins,~ s(jsj= 1), on a triangular lattice. Lat-\ntice e\u000bects and frequency cuto\u000bs in thermal excitations\nare thus avoided. Such simulations have already been\nused e.g. for the determination of the barrier to col-\nlapse of an isolated skyrmion [25, 26]. The parame-\nFIG. 8. Example of skyrmion trajectory. Distances in atomic\nunits (1 at:u:= 2:51\u0017A). The trajectory started at the origin\nof coordinates at time t= 0 and stopped at the cross location\nat physical time t\u0019100 ns.T= 25 K,\u000b= 1.\nters are: lattice constant a= 2:51\u0017A, magnetic mo-\nment\u0016At= 2:1\u0016B/atom, Heisenberg exchange nearest\nneighbor constant J= 29 meV/bond, Dzyaloshinskii-\nMoriya exchange d=\u00001:5 meV/bond, magnetocrys-\ntalline anisotropy 0 :4 meV/atom. The stochastic \feld\nis still de\fned by Eqn.1 after substitution of the prod-\nuctMSVby the magnetic moment per atom. The code\nfeatures full magnetostatic (dipole-dipole) interactions.\nFast Fourier Transforms implementation ensues from the\ndecomposition of the triangular lattice into two rectangu-\nlar sublattices, at the expense of a multiplication of the\nnumber of dipole-dipole interaction coe\u000ecients. Lastly,\nthe base time step, also the stochastic \feld refresh time,\nhas been given a low value in view of the small atomic\nvolume, namely dt= 2:5 fs for\u000b\u00150:1,dt= 1 fs below.\nTime steps that small may be deemed little compatible\nwith the white thermal noise hypothesis [17]. They are in\nfact dictated by the requirement for numerical stability,\nprimarily w.r.t. exchange interactions.\nFig.7 is a snapshot of an isolated skyrmion in the\nmodel Co ML with a temperature raised to 12 :5 K. The\nskyrmion is at the center of a 200 at. u.- i.e. \u001950 nm-size\nsquare computation window, that contains 46400 spins\nand is allowed to move with the di\u000busing skyrmion. Do-\ning so alleviates the computation load without restricting\nthe path followed by the skyrmion. Free boundary con-\nditions (BC's) apply. The window, however, proves su\u000e-\nciently large to render the con\fning potential created by\nBC's ine\u000bective. The skyrmion position as a function of\ntime is de\fned simply as the (iso)barycenter of the con-\ntiguous lattice site positions x(k),y(k), wheresz\u00150:5:\nqSk\nx=1\nKKX\nk=1x(k) ;qSk\ny=1\nKKX\nk=1y(k) (14)6\n01 1042 1043 1044 1045 1046 1047 1048 104\n01 1042 1043 1044 1045 1046 1047 104\n-50050NΔt = 0.2 ns\nqx,y - [at. u.]01 1042 1043 1044 1045 1046 1047 1048 104\n01 1042 1043 1044 1045 1046 1047 104\n-50050Δt = 0.5 nsN\nqx,y - [at. u.]\n01 1042 1043 1044 1045 1046 1047 1048 104\n01 1042 1043 1044 1045 1046 1047 104\n-50050NΔt = 1.0 ns\nqx,y - [at. u.]01 1042 1043 1044 1045 1046 1047 1048 104\n01 1042 1043 1044 1045 1046 1047 104\n-50050N\nqx,y - [at. u.]Δt = 2.0 ns\nFIG. 9. Skyrmion: event statistics with time interval \u0001 t\nas a parameter for the displacement components qx(black\nfull symbols) and qy(red open symbols), labeled qx;yin the\n\fgures. In each panel, the curves have been o\u000bset vertically\nfor legibility. Solid lines: \ft to a gaussian distribution. \u000b=\n0:25,T= 25 K\nwhere,kis the lattice site index, Kthe number of lattice\nsites satisfying the above condition. Such a de\fnition\nproves robust vsthermal disorder such as displayed in\nFig. 7. Similarly to the case of wall di\u000busion, we analyze\n\frst the distributions of the displacement components\nqx;qy. The event statistics for each value of the time\ninterval is clearly gaussian (see Fig.9). However, the noise\nin the distributions appears larger when compared to the\nwall case. It also increases faster with \u0001 t. On the other\nhand, the raw probabilities for hq2\nxiandhq2\nyibarely di\u000ber\nas anticipated from a random process. The behavior of\nhq2i(q2=q2\nx+q2\ny)vs\u0001tis displayed in Fig.10a.\nThe range of accessible temperatures is governed by\nthe thermal stability of the tiny skyrmion within a Co\nML: with a lifetime of '1\u0016s at 77 K [25{28], tem-\nperatures have been con\fned to a \u001450 K range. When\ncompared to the wall case (Fig.4a), the linear dependence\nofhq2iwith respect to \u0001 tappears less satisfactory, al-\nthough, over all cases examined, the curves do not display\na single curvature, but rather meander gently around a\nstraight line. The slope is de\fned as the slope of the\nlinear regression either for time intervals between 0 :25\nand 2:5 ns (thick line segments in Fig.10a) or for the full\nrange 0 to 5 ns (dashed lines). Then, the ratio of the\ndi\u000busion constant to temperature, D=T, for an isolated\nskyrmion within the model Co ML considered here is\nequal to 0:250 and 0:249 nm2ns\u00001K\u00001, respectively, for\n\u000b= 0:5 (see Fig.10b). The di\u000berence proves marginal.\nLastly, error bars appear even narrower than in the wall\n01000200030004000\n01234550° K25° K12.5° K4.2° K< q2 > [at.u.2]\nΔt [ns]a)01020\n0255075T [K]b)D [nm2 ns-1]FIG. 10. a) Variance (at :u:2) of the skyrmion displacement\nhq2ivstime interval \u0001 twith temperature Tas a parameter.\nThick and dashed lines represent a linear \ft to data with\ndi\u000berent time coverage, namely [0 :25\u00002:5 ns] and [0\u00005 ns];\nb) Di\u000busion constant Das a function of temperature for a\n[0:25\u00002:5 ns]- (open symbols) and [0 \u00005 ns]- (full symbols)\nlinear \ft. Solid blue line: linear \ft through the origin. Dashed\nline: analytical expectation in the \"low\" noise limit. In order\nto ensure legibility, the error bars as de\fned in the caption\nof Fig.4 and pertaining to the [0 :25\u00002:5 ns] \ft time bracket\nhave been moved-up by one unit. \u000b= 0:5.\ncase.\nB. Skyrmion di\u000busion constant (analytical)\nThe gyrovector ~Gin Thiele's equation (Eqn.3) has in\nthe case of a skyrmion or a vortex, and in many other\ninstances such as lines within walls, a single non-zero\ncomponent, here Gz. Thiele's equation, in components\nform, reads:\n\u0000Gzvy+\u000b[Dxxvx+Dxyvy] =Fx\n+Gzvx+\u000b[Dyxvx+Dyyvy] =Fy(15)\nBecause of the revolution symmetry of a skyrmion at rest,\nDxyorDyxmay safely be neglected and Dyy=Dxx.\nAccordingly, the velocities may be expressed as:\nvx=\u000bDFx+GFy\nG2+ (\u000bD)2;vy=\u000bDFy\u0000GFx\nG2+ (\u000bD)2(16)\nwhere,G=Gz,D=Dxx=Dyy.\nSimilarly to the stochastic \feld, the force components\nare necessarily uncorrelated. The velocity autocorrela-\ntion functions may now be obtained following the same\nlines as in the wall case, yielding, in the low noise ap-\nproximation:\nhvx(t)vx(t0)i=hvy(t)vy(t0)i= 2kBT\u000bD\nG2+ (\u000bD)2\u000e(t\u0000t0)\n(17)7\n00.050.10.150.20.250.30.35\n-20-1001020304050\n0246810α(%)D /T [nm2 ns-1 K-1]\nFIG. 11. Computed values of D=Tvs\u000b(large open symbols);\nblack line: guide to the eye; blue (resp. red) solid curves: an-\nalytical values with [ \r0SAt=\u00160\u0016At]D= 4\u0019(resp. 14:5). The\nblue curve thus corresponds to the Belavin-Polyakov pro\fle\nlimit. The relative di\u000berence between simulation and theory\nis indicated by small full symbols (% : right scale).\nThe average values of the displacements squared, hq2\nxi\nandhq2\nyifollow from time integration:\n\nq2\nx(t)\u000b\n=\nq2\ny(t)\u000b\n= 2kBT\u000bD\nG2+ (\u000bD)2t (18)\nAs shown previously [12, 13], the di\u000busion constant for a\nskyrmion thus reads:\nD=kBT\u000bD\nG2+ (\u000bD)2(19)\nThe following relations do apply:\n\nq2\nx(t)\u000b\n=\nq2\ny(t)\u000b\n= 2Dt\n\nq2(t)\u000b\n=\nq2\nx(t) +q2\ny(t)\u000b\n= 4Dt(20)\nRelation (19) implies a peculiar damping constant de-\npendence with, assuming for the time being DandGto\nhave comparable values, a gradual drop to zero of the\ndi\u000busion constant with decreasing \u000b(\u000b\u00141), termed\n\"di\u000busion suppression by G\" by C. Sch utte et al. [12].\nDi\u000busion suppression is actually not a complete surprise\nsince, for electrons in a magnetic \feld, a similar e\u000bect is\nleading to the classical magnetoresistance. A similar de-\npendenceD(\u000b) is also expected for a vortex. Boundary\nconditions, however, add complexity to vortex di\u000busion.\nWhat nevertheless remains, is a linear dependence of D\nvs\u000b[14], namely, di\u000busion suppression.\nThe classical expressions for GzandDxxvalid for a\nmagnetization continuum need to be adapted when deal-\ning with discrete spins. We obtain:\nGz=\u00160\u0016At\n\r0X\nk[~ s(k)\u0001[@x~ s(k)\u0002@y~ s(k)]]\nDxx=\u00160\u0016At\n\r0X\nkh\n[@x~ s(k)]2i (21)where,\u0016Atis the moment per atom.\nThe dimensionless product\r0SAt\n\u00160\u0016AtGz(Eqn.21), where\nSAtis the surface per atom, amounts to 4 \u0019, irrespec-\ntive of the skyrmion size in a perfect material at T= 0.\nStated otherwise, the skyrmion number is 1 [29]. In\nthe Belavin-Polyakov pro\fle limit [30], the dimention-\nless product\r0SAt\n\u00160\u0016AtDxx(Eqn.21) also amounts to 4 \u0019. In\nthis limit,Dis proportional to \u000b=(1+\u000b2).Dxxincreases\nwith skyrmion radius beyond the Belavin-Polyakov pro-\n\fle limit (see supplementary material in [7]). For a\nskyrmion at rest in the model Co ML considered here,\nD=Dxx\u001914:5\u00160\u0016At=(\r0SAt). For that value of\nDxx, and for the parameters used in the simulations,\nD=T, the ratio of the theoretical skyrmion di\u000busion con-\nstant to temperature, is equal 0 :234 nm2ns\u00001K\u00001, for\n\u000b= 0:5 (SAt=a2p\n3=2), to be compared to the 0 :250\nvalue extracted from simulations. More generally, Fig.11\ncompares numerical D=Tvalues calculated for a broad\nspectrum of \u000bvalues with theoretical expectations for\nD= 14:5\u00160\u0016At=(\r0SAt) and in the Belavin-Poliakov\nlimit. The average di\u000berence between analytical and sim-\nulation results is, in the \u000b= (0;1) interval, seen to be of\nthe order of'15%.\nIV. DISCUSSION\nIn the present study of thermal di\u000busion characteris-\ntics, satisfactory agreement between simulations and the-\nory has been attained for DMI sti\u000bened magnetic tex-\ntures, be it walls in narrow tracks or skyrmions. The\n\u000bdependence of the di\u000busion constants has been thor-\noughly investigated, with, as a result, a con\frmation of\nBrownian motion suppression in the presence of a non-\nzero gyrovector or, equivalently, a topological signature.\nThe theory starts with the Thiele relation applying to\na texture moving under rigid translation at constant ve-\nlocity. Furthermore, the chosen values of the components\nof the dissipation dyadic, are those valid for textures at\nrest, atT= 0. The\u000bdependence of the di\u000busion con-\nstants clearly survives these approximations. And, yet, a\nwall within a narrow stripe or a skyrmion in an ultra-thin\nmagnetic layer are deformable textures, as obvious from\nFigs.1,7. Simulations, on the other hand, rely on the\npioneering analysis of Brownian motion, here meaning\nmagnetization/spin orientation \ructuations [17], within\na particle small enough to prove uniformly magnetized\nand then extend the analysis to ultra-small computation\ncell volumes down to the single spin. Both approaches\nrely on the hypothesis of a white -uncorrelated- noise at\n\fnite temperature.\nThe discussion of results is organized in two parts. In\nthe \frst, results are analyzed in terms of a sole action of\nstructure plasticity on the diagonal elements of the dis-\nsipation dyadic. In the second, we envisage, without fur-\nther justi\fcation, how the present results are amended if,\nin the di\u000busion constants of walls and skyrmions (Eqns.8\nand 19), the gyrotropic and dissipation terms are re-8\n01 10-102 10-10\n20406080100f (GHz)S\n 12.5 K 25 K T = 50 K \na)0123\n010203040506070T(K)< rEq > [nm]\nb)\nFIG. 12. a) Power spectrum Sof the time series rEq(t) for\nthree temperatures. The hatched area corresponds to the fre-\nquency range where a signature of the fundamental skyrmion\nbreathing mode is anticipated to be observed ( \u001939:3 GHz,\nin the present case); b) Equivalent skyrmion radius hrEqias\na function of temperature. Error bars correspond to \u00061\u001b\nof the gaussian distribution, itself a function of temperature.\n\u000b= 0:5, throughout.\nplaced by their time average as deduced from simulations.\nA. Size e\u000bects\nThe integral de\fnition of wall position adopted in this\nwork (Eqn.2) allows for a 1D treatment of wall di\u000busion,\nthus ignoring any di\u000busion characteristics potentially as-\nsociated with wall swelling, tilting, curving or meander-\ning. Additional information is, however, available in the\ncase of skyrmions. We concentrate here on the number,\nn, of spins within the skyrmion satisfying the condition\nsz\u00150:5, and its \ructuations as a function of time. The\nsurface of the skyrmion is nSAtand its equivalent radius,\nrEq, is de\fned by r2\nEq=nSAt=\u0019. The skyrmion radius\nrEqis found to \ructuate with time around its average\nvalue, according to a gaussian distribution that depends\non temperature, but becomes independent of the autocor-\nrelation time interval beyond \u001925 ps. The power spec-\ntrum of the time series rEq(t), shown in Fig.12a, excludes\nthe existence of a signi\fcant power surge around the\nfundamental breathing mode frequency of the skyrmion\n(\u001939:3 GHz for the present model Co ML) [31]. The\nskyrmion radius as de\fned from the discrete ndistribu-\ntion is thus subject to white noise. The average radius\nhrEqi, on the other hand, varies signi\fcantly with tem-\nperature, increasing from \u00191:6 nm to 2:4 nm when the\ntemperature is increased from 4 :2 K to 50 K (Fig.12b)\nand the diagonal element of the dissipation dyadic is ex-\npected to increase with increasing skyrmion radius [3, 7].\nOwing to relations (19,21), the maximum of D(\u000b) is\nfound for\u000b=Gz=Dxx=G=D . For\u000b < G=D , resp.\n\u000b > G=D ,Dincreases, resp. decreases, with D, hence\nthe relative positions of the blue and black continuous\ncurves in Fig.11. At maximum, Dis independent of D\nand amounts to kBT\r0SAt\n\u00160\u0016At1\n2G=kBT\r0SAt\n\u00160\u0016At1\n8\u0019. It ensues\nz\t\r \nf\t\r α\t\n\r\nR/Δ\t\n\ra) b) 00.20.40.60.81\n01020304050R /!\"#D / #\" < 0#D / #\" > 0FIG. 13. Di\u000busion suppression: a) general shape of function\nf(\u000b;R= \u0001) with 0 < \u000b < 1, 1< R= \u0001<50; b) crest line\nseparating the region of di\u000busion suppression ( @D=@\u000b > 0)\nfrom region @D=@\u000b< 0.\nthat the discrepancy between numerical and analytical\nDvalues around \u000b= 1 may not be relaxed by a sole\nvariation of D. On the other hand, allowing Dto increase\nwith skyrmion radius, itself a function of temperature,\nleads to an increase (decrease) of the di\u000busion coe\u000ecient\nfor\u000bG=D ).\nLikely more important is the reduction, as a function\nof skyrmion size, of the \u000bwindow where di\u000busion sup-\npression is expected. If including the ( R=\u0001 + \u0001=R) de-\npendence of Dxx(see supplementary material in [7]; \u0001 is\nthe wall width and Rthe skyrmion radius), the skyrmion\ndi\u000busion constant may be expressed as:\nD=kBT\r0SAt\n\u00160\u0016At1\n8\u0019f\u0012\n\u000b;R\n\u0001\u0013\n\u0011=R\n\u0001;\u0018=1\n2\u00121 +\u00112\n\u0011\u0013\n;f(\u000b;\u0011) =2\u000b\u0018\n1 + (\u000b\u0018)2(22)\nThe general shape of function f(\u000b;R= \u0001) is shown in\nFig.13a. The maximum of f(\u000b;R= \u0001) is equal to 1 for\nall values of \u000bandR=\u0001. The crest line R\u000b= \u0001 is\nseen to divide the parameter space into two regions (see\nFig.13b), a region close to the axes where @D=@\u000b > 0,\ni.e. the region of di\u000busion suppression, from the much\nwider region where @D=@\u000b < 0, that is, the region of\nwall-like behavior for skyrmion di\u000busion. Clearly, the \u000b\nwindow for di\u000busion suppression decreases dramatically\nwith increasing skyrmion size R=\u0001. A \frst observation\nof skyrmion Brownian motion at a video recording time\nscale (25 ms) may be found in the Supplementary Ma-\nterial of Ref.[32]. Skyrmions are here unusually large\nand most likely escape the di\u000busion suppression window\n(\u000b<0:02 forR=\u0001 = 50). Combining skyrmion thermal\nstability with general observability and damping parame-\nter tailoring may, as a matter of fact, well prove extremely\nchallenging for the observation of topology related di\u000bu-\nsion suppression.9\n0.750.80.850.90.951\n101214161820\n020406080100120140160T (K)< DVF >< mz / mz(T=0) >< sz / sz(T=0) >1 ML3 ML : 0.6 nm\nFIG. 14. Average reduced zmagnetization or spin component\nas a function of temperature (left scale) and time averaged\nvalue of the sole vector function, hDVFi, within the diagonal\nelement of the dissipation tensor in the skyrmion case (right\nscale). These results prove independent of the damping pa-\nrameter provided the time step in the integration of the LLG\nequation be suitably chosen.\nB. Time averaging\nOne certainly expects from the simulation model a fair\nprediction of the average magnetization hMziorhSzivs\ntemperature T, at least for temperatures substantially\nlower than the Curie temperature TC. Fig.14 shows the\nvariation ofhMzi=Mz(T= 0) orhSzi=Sz(T= 0) with\ntemperature for the two model magnetic layers of this\nwork. Although simulation results do not compare unfa-\nvorably with published experimental data [33{35], where,\ntypically, the Curie temperature amounts to \u0019150Kfor\n1 ML, and proves larger than 300 Kfor thicknesses above\n2 ML, a more detailed analysis, potentially including dis-\norder, ought to be performed.\nhGzi=\u00160\u0016Athszi\n\r0hX\nk[~ s(k)\u0001[@x~ s(k)\u0002@y~ s(k)]]i\n=\u00160\u0016Athszi\n\r0SAthGVF\nzi\nhDxxi=\u00160\u0016Athszi\n\r0hX\nk[@x~ s(k)]2i\n=\u00160\u0016Athszi\n\r0SAthDVF\nxxi(23)\nLet us now, without further justi\fcation, substitute in\nthe expression of the skyrmion di\u000busion coe\u000ecient time\naveraged values of GandD, owing to relations (23).\nKeeping in mind the geometrical meaning of GVF\nz, the\ndimensionless vector function in G,hGziis anticipated\nto be a sole function of hszi. Inversely, DVF\nxx, the (di-\nmensionless) vector function in hDxxi, a de\fnite posi-\ntive quantity, steadily increases with thermal disorder.\nIt is even found to be proportional to temperature (notshown). Its time averaged value for the sole skyrmion\nmay only be obtained by subtraction of values computed\nin the presence and absence of the skyrmion.\nFor the skyrmion in our model Co monolayer, hDVF\nxxi\nis found to increase moderately with temperature (see\nFig.14), a result also anticipated from an increase with\ntemperature of the skyrmion radius. Besides, both hGzi\nandhDxxiare expected to decrease with temperature\ndue to their proportionality to hszi.hDxxiis thus sub-\nject to two competing e\u000bects of temperature T. Present\nevidence, however, points at a dominating in\ruence of\nhsz(T)i.\nV. SUMMARY AND OUTLOOK\nSummarizing, it has been shown that the Brownian\nmotion of chiral walls and skyrmions in DMI materials\nobeys di\u000busion equations with markedly di\u000berent damp-\ning parameter ( \u000b) dependence. Although not a new re-\nsult, skyrmions Brownian motion suppression with de-\ncreasing\u000b(\u000bT N) to\nP60\n3cm0(T 0andDz>0parametrise the antiferro-\nmagnetic intra-plane exchange interaction and the easy-plane anisotropy, respectively. The inter-plane exchange is\ntwo orders of magnitude weaker than the intra-plane exchange; thus, we neglect it24.Dy<0stabilises the ground\nstate asP60\n3c0m(Mithen points along equivalent crystallographic yaxes), whereas Dy>0stabilisesP60\n3cm0(with\nMialong equivalent crystallographic xaxes). \u0001\u00150is a weak fourth-order in-plane anisotropy, which becomes\nrelevant at the SRT, where Dycrosses zero. The three-sublattice antiferromagnet exhibits three optically excitable\nspin precessions called X,Y, andZmode18,25, which relate to an oscillating net magnetisation along the x,y, orz\naxes, respectively. We investigate the damping-like torque in relation to spin excitations via the IFE, which leaves us\nwith theZ-mode excitation illustrated in Fig. 1c.\nFigure 2a shows an exemplary measurement of the time-resolved Faraday rotation in YMnO 3after optical excita-\ntion with a circularly polarised pump pulse at 30 K. Two regimes can be distinguished: (i) a pronounced peak around\n0 psreflecting the direct interaction between pump and probe pulse and (ii) a damped sinusoidal modulation of the\nFaraday rotation that marks the ensuing spin precession and relaxation.\nFrom a fit of the latter, we extract the relaxation time \u001c, frequency !and initial phase \u001e0of the precession (see\nMethods). Figure 2b shows a magnified view of the region of the temporal overlap along with an extrapolation of\nthe sinusoidal fit. Most strikingly, we find a finite magnetisation at 0 ps. This is unexpected because the conventional\nfield-like torque of the IFE only perturbs the antiferromagnetic order, but does not induce a net magnetisation at the\nmoment of the excitation14,26–28. We will show that the observed immediate onset of a finite net magnetisation is a4\n05 01 001 502 00Faraday rotation (arb. units)P\nump-probe delay (ps)T = 30 K05 0100150200cD\nelay (ps)/s116/s119-\n10 1 bD\nelay (ps)initial phase /s1020a\nFIG. 2.Z-mode precession in YMnO 3. atime-resolved Faraday rotation following the impulsive optical excitation at time t= 0.\nbMagnified view of the region around t= 0.cOffset-corrected oscillation (see Methods). The red line is an exponentially\ndamped sine-fit with the dashed line as its envelope. The extrapolation of the fit towards t= 0is shown in band reveals a finite\ndeflection at t= 0. This initial deflection yields a temporal shift of the spin precession by approximately 280 fs , which amounts\nto an initial phase of 0.1 for a precession with a period of 18:2 ps.\nconsequence of spin damping during the impulsive spin excitation, which has been neglected so far.\nOriginally, YMnO 3is in the magnetic ground state with Mi(t <0) =M0^ xi, whereM0is the sublattice saturation\nmagnetisation. The optomagnetic field pulse of the IFE can be modelled as HIFE(t) =H\u0012\u000e(t)^ z, whereHis the\neffective magnetic field strength, \u0012is the laser pulse duration, and \u000e(t)denotes the Dirac delta function29. Analytically\nintegrating Equation (1) from t=\u00001to+0withHi=HIFEyields:\nMi(+0) =M00\nBBB@1\n\r\u00160H\u0012\n\u000b\r\u0016 0H\u00121\nCCCA: (3)\nThe canting along the zaxis is typically significantly smaller than the ycanting because \u000b\u001c1. However, any\nspin canting along the magnetic hard axis ( z) will be enhanced during the spin precession as a combined effect of\nantiferromagnetic exchange and magneto-crystalline anisotropy. This so-called exchange enhancement1,14,17,29,30is\nan effect that is intrinsic to AFMs, but is absent in ferromagnets. Thus, whereas damping effects during the spin\nexcitation may be neglected for ferromagnets, the neglecting is not generally justified in AFMs. In fact, we will show\nthat despite \u000b\u001c1, the damping-like contribution to the total spin excitation may even dominate over the field-like\ncontribution.\nThe threefold rotational symmetry, which is preserved during the Z-mode precession, allows us to describe the an-5\ntiferromagnetic dynamics by considering only one spin precessing in effective out-of-plane and in-plane anisotropies\nJ=3\n2\u0015+Dz+jDyjandD= \u0001M2\n0+jDyj, respectively (Supplementary Note 1). For the zcomponent of the\noscillating net magnetisation M, this yields a damped sinusoidal oscillation with\n!=!0p\n1\u0000A2\u000b2=4; (4)\n\u001c=2\nA\u000b! 0; (5)\ntan\u001e0=A\u000bp\n1\u0000A2\u000b2=4\n1\u0000A2\u000b2=2: (6)\nHere, ~!0= 2g\u0016BM0pJDis the precession frequency for the undamped case with the Land ´e factorg= 2and the\nBohr magneton \u0016B.A=p\nJD\u00001\u001d1is the exchange enhancement factor1,29. Geometrically, Acorresponds to the\naspect ratio of the elliptical spin precession14,17. Note that the initial phase \u001e0depends on the damping during the\nimpulsive spin excitation in direct relation to the initial phase in Fig. 2 and the ultrafast damping-like torque TDL.\nUltrafast spin damping in HoMnO 3\nWe further investigate the role of the damping-like torque by performing temperature-dependent measurements of\nthe optically induced spin dynamics. Temperature affects both the Gilbert-damping parameter \u000b(and therefore the\nmagnitude ofTDL) and the magneto-crystalline anisotropy (and therefore the exchange enhancement A). In partic-\nular, we trace the Zmode through the first-order SRT in HoMnO 3and towards the second-order antiferromagnetic-\nparamagnetic phase transition in YMnO 3(see Supplementary Fig. 2 for exemplary time-domain data). Conceptually,\nwe thus investigate an order–order and order–disorder phase transition, thereby showcasing the general importance\nof spin damping during ultrafast antiferromagnetic spin excitations.\nWe first consider the case of HoMnO 3. Based on measurements as in Fig. 2, we extract !0,\u001cand\u001e0as depicted in\nFigs. 3a–c, respectively. The SRT causes a dip of !0around 33 K. The fact that the frequency does not drop to zero\nat the SRT highlights the importance of the fourth-order anisotropy term in Equation (2). All anisotropy parameters\ncan be extracted from a fit of the temperature dependence of !0with a minimal model (see Methods). (The deviations\nbelow 25 K are caused by the incipient ordering of the Ho(4b) moments24,31,32, see Supplementary Fig. 3).\nMoreover, using Eqs. (4) and (5), we determine the exchange-enhanced damping A\u000band, in combination with\nthe extracted anisotropy constants, its constituents Aand\u000b(Figs. 3d–f). We find a significant increase in A\u000b\nbetween 30 K and45 K. Together with Equation (6) we can then calculate the temperature dependence of the initial\nphase and compare it with the direct measurement (Fig. 3c). The agreement around the SRT and towards higher\ntemperatures is impressive and demonstrates the importance of the previously unrecognised damping-like torque\nTDL. More strikingly, combining Eqs. (2) and (3) yields the energy density transferred into the magnetic lattice during\nthe excitation as6\nRelaxation timea\nHo(4b)\norderingTSR\nb\ncd\ne\nf\nFIG. 3. Temperature-dependent Z-mode characterisation in HoMnO 3. aFrequency, brelaxation time and ctangent of the\ninitial phase of the optically induced spin precession as function of temperature. Red: model calculations (see Methods). In\nthe blue-shaded region .25 K , incipient ordering of the Ho(4b) moments leads to discrepancies between data and the model\n(see Supplementary Fig. 3). Grey line: exponential decay as guide to the eye. dExchange-enhanced damping parameter A\u000b\nextracted from Z-mode frequency and relaxation time (Eqs. (4) and (5)). eTemperature-dependent aspect ratio Aof the spin-\nprecession ellipses. fResulting Gilbert damping parameter \u000b. Yellow line: function proportional to Bose-Einstein distribution\nwith quasiparticle energy of 4:3 meV .\n\u0001F=F(+0)\u0000F(\u00001)\u00193(A2\u000b2+ 1)DM2\ny; (7)\nwhere the first term originates from the ultrafast damping-like torque along the zaxis, whereas the second term is re-\nlated to the field-like in-plane torque of the IFE. Thus, as A\u000bexceeds one, spin excitation via the damping-like torque\nof the IFE even becomes the dominant excitation mechanism. The enhancement of Aaround the SRT is based on\nchanges of the magneto-crystalline anisotropy, while the anomalous increase of \u000bmatches the observations of a\ntemporary small-domain state in passing the SRT23. Such a small-domain state exhibits a high density of domain\nwalls which will locally affect the spin precession frequency. Thus, we attribute the enhanced Gilbert damping to\ndephasing due to magnon scattering at the walls33.7\ntan �0 Frequency (meV) Lifetime (ns)TN\nCorrected model\nUncorrected model\nFIG. 4. Temperature-dependent Z-mode characterisation in YMnO 3. aFrequency, brelaxation time, and ctangent of the initial\nphase of the optically induced spin precession as function of temperature. Green and red: model calculations with and without\ncorrection for non-damping-based excitation mechanisms, respectively. Grey line: fitted exponential decay.\nUltrafast spin damping in YMnO 3\nThe general trend of \u000bis captured well by an offset Bose-Einstein distribution for a quasiparticle with an energy of\n\u00184:3 meV , suggesting that scattering with the crystal-field excitations of the Ho 4felectrons (\u00193:5 meV24) contributes\nsignificantly to the spin damping. This is confirmed by measurements on YMnO 3(Fig. 4), yielding relaxation times\nthat are\u001820 times longer than those for HoMnO 3in line with the unoccupied 4forbital. Accordingly, the extracted\nvalues for the temperature-dependent Gilbert-damping parameter \u000bin YMnO 3(Fig. 5) are an order of magnitude\nlower than those in HoMnO 3. At low temperature, \u000bcan be approximated by a Bose-Einstein distribution with a\nquasiparticle energy of 2:6 meV , which corresponds to a hybrid spin-lattice mode in YMnO 334. Assuming that the\nfrequency of this mode is, just like the Zmode frequency, proportional to the magnitude of the antiferromagnetic\norder parameter, we can consistently describe the temperature dependence up to the N ´eel temperature.\nNote, however, that despite good qualitative agreement, Fig. 4c reveals a difference between the measured and\nmodelled value of the initial phase \u001e0. This indicates secondary, non-damping-based excitation mechanisms, such\nas an optical spin transfer torque35or optical orientation36, which could become relevant because of the smallness\nof\u000bin YMnO 3as compared with that in HoMnO 3. These mechanisms involve an angular momentum transfer from8\n0123G\nilbert damping parameter /s97x10-3T N1\n0203040506070800.000.050.100.150.200.25T\nemperature (K)Exchange-enhanced damping A/s97\nFIG. 5. Temperature-dependent damping of the Zmode in YMnO 3.Gilbert-damping parameter \u000band exchange-enhanced\ndampingA\u000bare shown in the same plot with A= 69:3(see Methods, Table 1). Full line: offset Bose-Einstein distribution with\nconstant quasiparticle energy E=2:6 meV . Dashed line: offset Bose-Einstein distribution with E/!0andE(0 K) =2:6 meV .\nthe circularly polarised pump pulse to the material. As the angular momentum of the circular laser pulse is directed\nalong thezaxis, we heuristically include such a contribution in our model by setting Mi;z(+0) =\u000b\r\u0016 0H\u0012M 0+\u000ein\nEquation (3), which changes the initial phase to\ntan\u001e0\u0019A\u0012\n\u000b+\u000e\n\r\u00160H\u0012M 0\u0013\n: (8)\nWith this, we restore the quantitative agreement between the measured and modelled initial phase in Fig. 4c.\nDISCUSSION\nSummarising, we thus demonstrated the indispensable role of the previously neglected ultrafast damping-like\ntorque during the optical excitation of AFMs, both experimentally and theoretically. The torque impulsively generates\na magnetisation along the minor axis of the intrinsically highly elliptical spin precession in AFMs. The extreme ellip-\nticity converts the small damping-related magnetisation into a large precessional canting. Because of this leverage\neffect, this damping-based mechanism can even facilitate the dominant spin excitation mechanism in striking contrast\nto its complete neglect thus far. In addition to studying the spin dynamics deep in the antiferromagnetic phase, we\nalso traced the associated Zmode in the vicinity of both first- and second-order phase transitions. This enabled us\nto verify the role of the ultrafast damping-like torque in distinctly different scenarios. We find that the Gilbert-damping\nparameter\u000bremains almost constant throughout the SRT in HoMnO 3, while the changes of the magneto-crystalline\nanisotropy driving the SRT lead to a two-fold increase in the exchange enhancement factor A. In contrast, as we\napproach the N ´eel temperature TN(here in YMnO 3), the magneto-crystalline anisotropy remains unchanged, but \u000b\ndiverges. Although fundamentally different, we find an enhanced ultrafast damping-like torque in both cases, which\nagain highlights the general importance of this hitherto neglected excitation mechanism.9\nMore specifically, in terms of damping mechanisms we identified scattering with the 4fcrystal-field excitations\nof HoMnO 3as the leading contribution to the spin damping. Therefore, the choice of the A-site ion enables us\nto tune the strength of the damping across several orders of magnitude. This is, on one hand, an useful degree\nof freedom for tuning material properties to specific spintronics applications. On the other hand, it allowed us to\nidentify the presence of additional non-damping-based mechanisms inducing a net magnetisation. With respect to\nthis, we note that any ultrafast mechanism that induces a net magnetisation in an AFM will benefit from the observed\nexchange enhancement. While field-like excitations via the IFE or using THz pulses typically induce spin-cantings of\nthe order of 1\u000e15,37, there are various possibilities beyond increasing the pump fluence that may allow us to exceed\na90\u000espin canting and thus induce antiferromagnetic switching. Exploiting different excitation mechanisms such as\nthermally and optically induced ultrafast spin-transfer torques35,38,39may drastically increase the impulsively induced\nnet magnetisation. Increasing the exchange enhancement, e.g. by tuning the system close to a SRT, will increase\nthe ellipticity of the spin precession. Therefore, the impulsively induced net magnetisation translates into a larger\nmodulation of the antiferromagnetic order parameter thereby reducing the switching threshold. Ultimately, our results\nshow that both field-like and damping-like torques — a combination that has proved to be powerful for the electrical\ncontrol of magnetic order7,40— are available optically, too, and can be utilized to act on the magnetic order. Thus,\noptically generated torques might provide the long sought-after tool enabling the efficient realisation of ultrafast\ncoherent precessional switching of AFMs.\nMETHODS\nSetup\nBoth samples are flux-grown and polished, (0001)-oriented platelets of approximately 55µmthickness. Our setup\nis schematically shown in Fig. 1b. The fundamental light source is a regeneratively amplified laser system providing\n130 fs pulses at 1:55 eV photon energy and 1 kHz repetition rate. We use the output of an optical parametric amplifier\ncombined with a quarter-wave plate to generate circularly polarised pump pulses at 0:95 eV photon energy, which\ncreate an effective magnetic field in the sample via the IFE. The pump fluence on the sample is approximately\n60 mJ cm\u00002. By pumping the material far below the nearest absorption resonance at \u00191:6 eV41, we minimise parasitic\nheating effects42. We then probe the ensuing dynamics by measuring the time-resolved Faraday rotation \b(t)of a\nlinearly polarised probe pulse at 1:55 eV with balanced photodetectors. The probe fluence is <0:1 mJ cm\u00002. Both\nthe pump and probe beams are arranged in a quasi-collinear geometry with the beam propagation direction along\nthe hexagonal zaxis, which is the optic axis. Thus, the incident light experiences optical isotropy and complications\narising from birefringence or other anisotropic optical effects are avoided. The sample is mounted in a cryostat for\nvarying the temperature between \u00195 Kand300 K .\nThe oscillatory part of the signal (Fig. 2a) is fitted by\n\b(t) =A0+A1e\u0000t=t1+A2e\u0000t=\u001csin (!t+\u001e0): (9)10\nWhile the first two terms model parasitic contributions to the signal arising from thermal electron dynamics, we extract\nthe frequency !(T), relaxation time \u001c(T)and initial phase \u001e0(T)from the oscillatory third term. Error bars in Figs. 3,\n4 and 5 reflect the standard error of that fit or its propagation through Eqs. (4)-(6).\nTemperature dependence of !0\nAs stated in the main text, the analytic solution of the Landau-Lifshitz-Gilbert equation yields (Supplementary\nNote 1): ~!0= 2g\u0016BM0pJD. According to mean-field theory, the temperature-dependent sublattice magnetisation\nM0(T)can be described as a paramagnet aligned by the exchange field of the neighbouring sublattices, i.e. M0(T) =\nNg\u0016 BSBS=2(M0(T)TNT\u00001), whereBS=2is the Brillouin function of an S= 2 Mn spin16,43,44andN= 6=Vwith\nV=0:374 nm3, being the unit cell volume, is the number density of Mn atoms. We model the SRT in HoMnO 3\nby introducing a linear dependence of the anisotropy parameter Dyon temperature. The SRT temperature TSR\nis defined as the point where Dyvanishes24, i.e.Dy(T) =\u0014(T\u0000TSR). For YMnO 3, assuming a temperature-\nindependent value for Dyyields a reasonable fit. This model is fitted to the measured temperature dependence of\n!0. The results are summarised in Table I.\nParameter Unit HoMnO 3 YMnO 3\nTN [K] 69.2(7) 74.7(1)\nTSR [K] 33.5(2) —\n\u0015 [T2cm3J\u00001] 70.84571.125\nDz [T2cm3J\u00001] 11.04513.925\nDy(T) [T2cm3J\u00001]\u0014(TSR\u0000T) 0.0251(1)\n\u0014 [T2cm3J\u00001K\u00001] 0.0093(6) 0\n\u0001\u0001(g\u0016B)2[T2cm3J\u00001] 0.0354(9) 0\nJ [T2cm3J\u00001] 117.2 +jDyj 120.6\nD [T2cm3J\u00001] \u0001M2\n0+jDyj 0.0251(1)\nA=p\nJ=D — Fig. 3e 69.3\nTABLE I. Fit parameters. Values for\u0015=J\nN(g\u0016B)2andDz=Kz\nN(g\u0016B)2were calculated from the exchange interaction Jand the\nout-of-plane anisotropy constant Kzgiven in the respective references.\nDATA AVAILABILITY\nThe data that support the findings of this study are available from the corresponding author on reasonable request.\nSource data are provided with this paper.\n\u0003ctzschaschel@fas.harvard.edu\nCurrent address : Department of Chemistry and Chemical Biology, Harvard University, Cambridge 02138, USA11\n[1] O. Gomonay, V. Baltz, A. 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Kiryukhin, P . Jain, and M. R. Fitzsimmons, Magnetic structures and dynamics of multiferroic systems\nobtained with neutron scattering, npj quantum mater. 1, 16003 (2016).\nACKNOWLEDGMENT\nThe authors thank B. A. Ivanov and A. V. Kimel for valuable discussions and R. Gm ¨under for experimental assis-\ntance. T.S. was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI (Nos. JP19H01828,13\nJP19H05618, JP19K21854 and JP26103004) and the JSPS Core-to-Core Program (A. Advanced Research Net-\nworks). C.T. acknowledges support by the SNSF under fellowship P2EZP2-191801. C.T. and M.F . acknowledge\nsupport from the SNSF project 200021/147080 and by FAST, a division of the SNSF NCCR MUST.\nAUTHOR CONTRIBUTIONS\nC.T. and T.S. conceived the project. C.T. designed and evaluated the experiments. T.S. and M.F . supervised the\nproject. All authors discussed the results and contributed to writing the manuscript.\nCORRESPONDING AUTHOR\nCorrespondence to Christian Tzschaschel.\nCOMPETING INTERESTS\nThe authors declare no competing interests." }, { "title": "2002.02686v1.Engineering_Co__2_MnAl__x_Si___1_x___Heusler_compounds_as_a_model_system_to_correlate_spin_polarization__intrinsic_Gilbert_damping_and_ultrafast_demagnetization.pdf", "content": "1 \n Engineering Co 2MnAl xSi1-x Heusler compounds as a model system to \ncorrelate spin polarization, intrinsic Gilbert damping and ultrafast \ndemagnetization \nC. Guillemard1,2, W. Zhang1*, G. Malinowski1, C. de Melo1, J. Gorchon1, S. Petit -\nWatelot1, J. Ghan baja1, S. Mangin1, P. Le Fèvre2, F. Bertran2, S. Andrieu1* \n1 Institut Jean Lamour, UMR CNRS 7198, Université de Lorraine, 54500 Nancy France \n2 Synchrotron SOLEIL -CNRS, Saint -Aubin, 91192 Gif -sur-Yvette, France \nAbstract: \nEngineering of magnetic materials f or developing better spintronic applications relies on \nthe control of two key parameters: the spin polarization and the Gilbert damping \nresponsible for the spin angular momentum dissipation. Both of them are expected to \naffect the ultrafast magnetization dyna mics occurring on the femtosecond time scale. \nHere, we use engineered Co2MnAl xSi1-x Heusler compounds to adjust the degree of spin \npolarization P from 60 to 100% and investigate how it correlates with the damping. We \ndemonstrate experimentally that the damping decreases when increasing the spin \npolarization from 1.1 10-3 for Co 2MnAl with 63% spin polarization to an ultra -low value \nof 4.10-4 for the half -metal magnet Co 2MnSi. This allows us investigating the relation \nbetween these two parameters and the ultrafast demagnetization time characterizing the \nloss of magnetization occurring after femtosecond laser pulse excitation. The \ndemagnetization time is observed to be inversely proportional to 1 -P and as a consequence \nto the magnetic damping, which can be attributed to the similarity of the spin angular \nmomentum dissipation processes responsible for these two effects. Altogether, our high \nquality Heusler compounds allow controlling the band structure and therefore the channel \nfor spin angular momentum dissipation. \n \n * corresponding authors : \nwei.z hang @univ -lorraine.fr \nstephane.andrieu@univ -lorraine.fr 2 \n I - INTRODUCTION \nDuring the last decades, extensive magnetic materials research has strived to \nengineer denser, faster and more energy efficient processing and data storage devices. On \nthe one hand, a high spin polarization has been one of the most important ingredients th at \nhave been seek [1]. For example, the spin polarization is responsible for a high readout \nsignal in magnetic tunnel junction based devices [2,3] . Additionally, a high spin \npolarization results in a decrease of the threshold current for magnetization reve rsal by \nspin torques [4] required for the development of spin-transfer -torque magnetic random \naccess memory devices [5] , for gyrotropic dynamics in spin -torque nano -oscillators [6] \nand for magnetic domain wall motion [7]. On the other hand, the intrinsic m agnetic \nenergy dissipation during magnetization dynamics, which is determined by the Gilbert \ndamping constant, needs to be low in order to build an energy efficient device. Fortunately, \nspin polarization and damping are usually closely related in magnetic materials. \nNowadays, manipulation of the magnetization on the femtosecond timescale has \nbecome an outstanding challenge since the demonstration of subpicosecond \nmagnetization quenching [8] and magnetization reversal on the picosecond timescale [9]. \nDespite the theoretical and experimental work that has been reported up to now , the \nrelationship between the polarization at the Fermi level or the magnetic damping and the \nultrafast demagnetization excited by femtosecond lasers, remains unclear [10-15]. Indeed, \nnumerous mech anisms have been proposed but no consensus has yet been reached. In \nparticular, efforts have been undertaken to unify the magnetization dynamics on the \nnanosecond timescale and the ultrafast demagnetization considering that the sp in-flip \nmechanisms involved in both phenomena could be the same [10-11,16] . Regarding the \ninfluence of the damping on the demagnetization time, different predictions have been \nreported both experimentally and theoretically . In this situation, the need for engineered \nsamples in which the spin -polarization and magnetic damping are well controlled is of \nutmost importance to unveil their role on the ultrafast magnetization dynamics. \nHeusler compounds are a notable class of magnetic materials allowing for tunabl e \nspin-polarization and magnetic damping [ 17]. The absence of available electronic states \nin the minority band at the Fermi level leads to very high spin polarization and ultra -low \ndamping due to a strong reduction of spin scattering [ 18-23]. Recently, ultra-low damping 3 \n coefficient associated with full spin polarization at the Fermi energy was reported in \nCo2Mn-based Heusler compounds , [22-23]. Among those alloys, Co 2MnSi has the \nsmallest damping down to 4.1 x 10-4 with 100% spin -polarization while Co 2MnAl , which \nis not predicted to be a half -meta llic magnet, has a damping of 1.1 x 10-3 and a spin -\npolarization of 60 %. \nIn the present work, we used Co 2MnAl xSi1-x quaternary Heusler compounds \ngrown by Molecular Beam Epitaxy (MBE) to tune the spin -polarization at the Fermi \nenergy . Controlling the amount of Al within the alloys allows tuni ng the spin -polarization \nfrom 60 to 100 % as measured by spin resolved photoemission. We show that the \nmagnetic damping parameter for these alloys is among the lowest reported in the literature \nand decreases when the spin -polarization increases. Ultrafast magnetization dynamics \nexperiments were thus performed on these prototype samples. This complete \nexperimental characterization allows us to directly correlat e the ultrafast magnetization \ndynamics to these parameters and comparing our results to the different theory discussed \nabove. \nThe Co 2MnSi compound grows in the L2 1 structure whereas the Co 2MnAl com pound \ngrows in the B2 phase as shown by STEM -HAADF analysis [22]. Such different \nstructures are directly observable during the growth by Reflexion High Energy Electron \nDiffraction (RHEED ) since the surface lattice is different for bot h compounds. Ind eed, \nhalf streaks are observed along Co 2MnSi [110] azimuth due to the L2 1 chemical ordering \n[24] which is not the case for Co2MnAl [22]. The RHEED analysis on Co2MnAl xSi1-x \nfilms with x= 0, ¼ ,½ , ¾ ,1 reveals a regular decrease of these half -streaks intensity with \nx (Figure 1 a). This information that concerns only the surface is confirmed in the entire \nthickness of the films by using x -ray diffraction. Indeed, the (111) peak typical of the \nchemical ordering in the L2 1 structure clearly decreases and disappears with x ( Figure \n1b). 4 \n \nFigure 1 : a) RHEED patterns along [110] showing the progressive vanishing of the half -streaks \n(observed on Co 2MnSi, x=0) at the surface with x. b) Confirmation of the transition from L2 1 to \nB2 chemical ordering in the entire film by the vanishing of the (111) peak and displacement of \n(220) peak with x as shown by x -ray diffraction. c) Spatial distribution of both chemical ordering \nin the films deduced from STEM -HAADF experiments: as the L 21 structure is observed in the \nentire Co 2MnSi film (x=0 ), and the B2 one i n Co 2MnAl (x=1 ), a mixing of both structure is clearly \nobserved for x=0.5 . \n \nIn addition, the displacement of the (220) peak with x allows us to extract a linear \nvariation of the lattice constant (Figure 1b ), as observed in the case of a solid solution. \nThis is an indication that the L2 1 chemical ordering progressively vanishes when \nincreasing the Al substitution rate 𝑥. However, the chemical disorder distribution in the \nfilms cannot be easily determined by using the electron and x -ray diffraction analyses. To \naddress this point, a STEM HAADF analysis has been carried on the Co 2MnAl ½Si½ films \nwith a comparison with Co 2MnSi and Co 2MnAl. A clear mixing of both structures is \n5 \n observ ed for x=½ where around 50% is L2 1 chemically ordered and 50%, B2, with typical \ndomains size around 10nm along the growth axis (001) and a few nm in the plane of the \nfilm ( Figure 1c). \nThe electronic properties of the Co 2MnAl xSi1-x(001) series were studied using spin -\nresolved photoemission (SR -PES) and ferromagnetic resonance (FMR). The SR -PES \nspectra were obtained by using the largest slit acceptance of the detector (+/ - 8°) at an \nangle of 8° of the normal axis of the surface. Such geometry allows us to analyze all the \nreciprocal space as confirmed by similar experiments but performed on similar \npolycrystalline films [23]. Getting the spin-polarization dependence with x using raw SR -\nPES spectra is however not obvious due to the existence of surface states systematically \nobserved on Co 2MnSi but also on other Co 2Mn-based Heusler compounds [19, 22-23]. \nTo get the bulk spin polarization, we thus used the S polarization of the photon beam. \nIndeed , we have shown that the surface states are no more detected due to their symmetry \n[19] without any loss of information on the bulk band structure [ 23]. The corresponding \nSR-PES spectra are shown in figure 2 . As expec ted, we thus obtain a tunable spin \npolarization at EF from 100% to 63% by substituting Si by Al, as shown in figure 3 . \n \nFigure 2 : spin -resolved photoemission spectra using P photon polarization (left), S photon \npolarization (middle) and resulting spin polarization curves (right) for the Co 2MnAl xSi1-x series, \n6 \n The radiofrequency magnetic dynamics of the films were thus studied using \nferromagnetic resonance (FMR) . The magnetic damping coefficient , the effective \nmagnetic moment Ms (close to the true moment in our films due to very small anisotropy \n– see [ 22]), and the inhomogeneous linewidth f0 were thus extracted from the \nmeasurements performed on the Co 2MnAl xSi1-x(001) series. The results obtained on the \nsame series used for photoemission experiments are shown in table I . As shown in figure \n3, a clear correlation is observed between the spin polarization at EF and the magnetic \ndamping coefficient , as theoretically expected. An ultra -low value was obtained for \nCo2MnSi (x= 0) due to the large spin gap [ 22]. By substituting Al by Si, the magnetic \ndamping increase is explained by the decrease of the spin polarization. \nCo2MnAl xSi1-x Spin polarization \n(%) Ms \n(µB/f.u.) \n(x 10-3) f0 \n(MHz) g factor \n(0.01) \nx = 0 973 5.08 0.460.05 14.3 2.01 \nx = 0.25 903 4.85 0.730.15 21.7 1.99 \nx = 0.5 833 4.85 0.680.15 9 2.01 \nx = 0.75 703 4.8 1.000.05 81.5 2.00 \nx = 1 633 4.32 1.100.05 22 2.01 \nTable 1: data extracted from spin -resolved photoemission and ferromagnetic resonance \nexperiments performed on the Co 2MnAl xSi1-x series. \n \nFigure 3: -top- spin polarization and magnetic damping dependence with Al content for the \nCo2MnAl xSi1-x series and –bottom - magnetic damping versus spin polarization . The lines are \nguide to the eyes. \n7 \n In addition, t he magnetization is also observed to decrease with x in agreement with the \nSlater -Pauling description of the valence band electrons in Heusler compounds [25]. \nIndeed, as a 5 µ B magnetic moment per cell is expected for Co2MnSi (type IV valence \nelectrons), it should decrease to 4 when replacing Si by Al (type III) as actually observed \n(Table I ). Finally, the FMR susceptibilities reach extremely small inhomogeneous \nlinewidth f0, a proof of the excellent homogeneity of the magnetic properties (hence a \nhigh crystal quality) in our films. \nFigure 4 (a) shows the ultrafast demagnetization curves measured on the same \nCo2MnAl xSi1-x series with a maximum magnetization quenching ~1 5%. The temporal \nchanges of the Kerr signals ∆𝜃𝑘(𝑡) were normalized by the saturation value 𝜃𝑘 just before \nthe pump laser excitation. The time evolution of magnetization on sub -picosecond \ntimescales c an be fitted according to Eq. (2 ) in terms of the three -Temperature M odel \n(3TM) [26], which describes the energy distribution among electrons, phonons, and spins \nafter laser excitation. \n−∆𝑀(𝑡)\n𝑀={[𝐴1\n(𝑡𝜏0+1 ⁄ )0.5−𝐴2𝜏𝐸−𝐴1𝜏𝑀\n𝜏𝐸−𝜏𝑀𝑒−𝑡\n𝜏𝑀−𝜏𝐸(𝐴1−𝐴2)\n𝜏𝐸−𝜏𝑀𝑒−𝑡\n𝜏𝐸]Θ(𝑡)}∗𝐺(𝑡,𝜏𝐺) (2) \nwhere 𝐺(𝑡,𝜏𝐺) represents the convolution product with the Gaussian laser pulse profile, \nG\n is the full width at half maximum (FWHM) of the laser pulses. Θ(𝑡) is the Heavyside \nfunction . The constant A1 represents the amplitude of demagnetization obtained after \nequilibrium between the electrons, spins, and phonons is reestablished while A 2 is \nproportional to the initial electron temperature raise . The two critical time parameters \n𝜏𝑀,𝜏𝐸 are the ultrafast demagnetization time and magnetization recovery time, \nrespectively. In the low fluence regime, which corresponds to our measurements, 𝜏𝐸 \nbecomes close to the electron -phonon relaxation time . A unique value of 𝜏𝐸=550 ±\n20 𝑓𝑠 was used for fitt ing the demagnetization curves for all samples. T he ultrafast \ndemagnetization time 𝜏𝑀 decrease s from 380 ±10 fs for Co 2MnSi to 165 ±10 fs for \nCo2MnAl (Figure 4b). The evolution of the demagnetization time with both spin \npolarization P and Gilbert damping 𝛼 is presented in figure 4c and 4d . A clear linear \nvariation between 1𝜏𝑀⁄ and 1−𝑃 is observed in this series . As the magnetic damping 𝛼 \nis proportional to P here, this means that 1𝜏𝑀⁄ is proportional to 𝛼 too. A similar relation 8 \n between these two par ameters was proposed by Koopmans et al. [10]. However, they also \npredicted an influence of the Curie temperature . As the Curie temperature in Heusler \ncompounds changes with the number of valence electron s and because the Co 2MnAl xSi1-\nx behave as solid solutions as indicated by the lattice spacing variation ( Figure 1b), we \nthus consider a linear decrease of 𝑇𝑐 with x going from 985 K to 697 K as exper imentally \nmeasured for x =0 and x=1, respectively. To test this possi ble influence of the Curie \ntemperature on the ultrafast magnetization dynamics , we plot in figure 4d first the product \n𝜏𝑀.𝛼 and second the product 𝜏𝑀.𝛼.𝑇𝑐(𝑥)𝑇𝑐(𝐶𝑜2𝑀𝑛𝑆𝑖 ) ⁄ . These results demonstrate that \nthe Curie temperature does not influence the ultrafas t demagnetization in our samples . \n \nFigure 4 : (a) Ultrafast demagnetization curves obtained for different Al concentration x . The \ncurves have been shifted vertically for sake of clarity. The solid lines represent fitted curves \nobtained using Eq. ( 2). (b) Ultrafast demagnetization time as a function of Al content x, (c) the \ninverse of 𝜏𝑀 as a function of 1-P, P being the spin polarization at E F, and d) test of Koopmans \nmodel with and without taking into account the Curie temperature of the films (see text). \n9 \n One can now compare our experimental results with existing theoretica l models. \nWe first discuss the dependence between the magnetic damping and the spin polarization. \nUltra -low magnetic damping values are predicted in Half -Metal Magnet (HMM) Heusler \ncompounds and explained by the lack of density of state at the Fermi energy for minority \nspin, or in other words by the full spin polarization [18,27,28] . Consequently, the \nmagnetic damping is expected to increase when creating some states in the m inority band \nstructure around the Fermi energy that is when decreasing the spin polarization [28]. If \nwe confirmed in previous experimental works that ultra -low magnetic damping \ncoefficients are actually observed especially on HMM Co2MnSi and Co 2MnGe [19,2 2-\n23], we could not state any quantitative dependence between the damping values and the \nspin polarization. As prospected, the Co 2MnAl xSi1-x alloys are shown here to be ideal \ncandidates to address this point . This allows us getting a clear experimental demonstration \nof these theoretical expectations. Furthermore, a linear dependence between the magnetic \ndamping and the spin polarization is obtained. This behavior may be explained by the \nmixing of both L2 1 and B2 phases in the films. To the best of our knowledge, this \nexperimental result is the first quantitative demonstration of the link between the \nmagnetic damping and spin polarization. \n Second , the dependence between the magnetic damping and the demagnetization \ntime observed here is a clear opportunity to test the different theoretical explanations \nproposed in the literature to explain ultrafast dynamics . In the last 15 years, t he influence \nof the damping on the ultrafast dynamics has been explored, both theoreticall y and \nexperimentally. The first type of prediction we want to address is the link between the \ndemagnetization time and the electronic structure via the spin polarization P. Using a \nbasic approach considering the Fermi golden rule, several groups [12,13] proposed that \nthe demagnetization process is linked to the population of minority and majority spin \nstates at E F, leading to a dependence of the spin-scattering rate proportional to 1 -P [13]. \nAs this spin scattering rate is linked to the inverse of the dem agnetization delay time , the \n𝜏𝑀~(1−𝑃)−1 law was proposed . This law is clearly verified i n our samples series. One \nshould note that this is a strong experimental demonstration since we compare samples \ngrown in the same conditions , so with the same control of the stoichiometry and structural \nproperties . 10 \n However, one point is still not clear since much larger demagnetization times in \nthe picosecond timescale would be expected for large band gap and full spin -polarization. \nIn the case of small band ga p of the order of 0.1 eV, Mann et al [13] showed that thermal \neffects from the heated electron system lead to a decrease of 𝜏𝑀. They calculated a \nreduction of the spin -flip suppression factor from 104 for a gap of 1 eV to 40 for a gap of \n0.3 eV. However, the band gap of our Co 2MnSi was calculated to be around 0.8 eV with \na Fermi energy in the middle of the gap [27,28] . This was corroborated by direct \nmeasurement using SR -PES [19, 22 ]. Therefore, according to their model, we should \nexpect a much longer demagnetization time for Co 2MnSi. However, the largest values \nreported by several groups [13, 29] all on HMM materials are of the same order of \nmagnitude, i.e. around 350 to 400fs . This probably means that a limitation exists due to \nanother physical reason . One hypothesis should be to consider the 1.5eV photon energy \nwhich is much larger than the spin gap. During the excitation, the electrons occupying the \ntop minority spin valence band can be directly excited into the conduction band. In a \nsimilar way, maj ority spin electrons are excited at energies higher than the spin band gap. \nBoth of these effects may allow for spin flips scattering and only the majority electrons \nexcited within the spin band gap energy range cannot flip their spins. Even if such photon \nenergy influence is not considered based on the argument that the timescale for photon \nabsorption followed by electronic relaxation is very fast compared to the magnetic \nrelaxation process [16 ], performing experiments by changing the excitation wavelength \nto energies below the spin band gap would be very interesting to better understand \nultrafast magnetization dynamics. \n Concerning the dependence between the demag netization time and the magnetic \ndamping , different theoretical models have been proposed and two opposite trends were \nobtained; 𝛼 and 𝜏𝑀 being either directly [15] or inversely [10 ] proportional . From the \nexperimental side, the inverse proportionality between 𝜏𝑀 and 𝛼 proposed by Koopmans \net al. [10] could not be reproduced by doping a thin Permalloy film with rare -earth atoms \n[14]. However, the introduction of these rare -earth elements strongly modifies the \nmagnetic relaxation properties and could induce different relaxation channels for 𝜏𝑀 and \n𝛼 [30]. Zhang et al. performed a similar st udy using thin Co/Ni multilayers and observed \na direct proportionality between 𝜏𝑀 and 𝛼 [15]. However, the damping extracted in their 11 \n study should be strongly influenced by the heavy metal Pt capping and seed layers which \nmay induce strong spin pumping effect during the magnetization precession [30]. \nFurthermore, they did not take into account the influence of the Curie temperature. \nTherefore, in these studies, extrinsic effects might influence the magnetization dynamics \nin a different way on both time scales which makes more complex the comparison \nbetween theory and experiments. Therefore, o ur results offer a nice opportunity to \ndisentangle the se different effects. According to different studies , the ultrafast \ndemagnetization slows down when approaching the Curie temperature [ 10,16, 32,33]. In \nother words, a larger difference between the initial temperature and 𝑇𝑐 would lead to a \nfaster demag netization . In our samples, 𝑇𝑐 goes up from Co2MnAl to Co2MnSi, whereas \nthe demagnetization process becomes slower . Therefore, we conclude that, in the present \ncase, the Curie temperatures of our samples are too high to affect 𝜏𝑀 which only depends \non the intrinsic propertie s of the films, i.e. Gilbert damping and spin polarization. This \nalso clarifies some points reported by Müller et al. work [ 12]. In their paper, they first \nreported a very fast demagnetization process in Co 2MnSi(110) and second a slow one in \nCrO 2 and LaSrMnO 3 films with 𝑇𝑐 values close to room temperature (390 K 360 K \nrespectively). Therefore, it is not possible to state whether the very slow demag netization \nprocess in these compounds is due to a low 𝑇𝑐 or a large spin polarization. Furthermore, \nrecent experimental results demonstrated a large decrease in the spin polarization at the \nFermi level in CrO 2 as function of the temperature, resulting in less than 50% at 300 K \n[34]. In our samples we disentangle these two effects and the longest demagnetization \ntime is found for Co 2MnSi (𝜏𝑀=380 𝑓𝑠), a true half -metal magnet with a 0.8 eV spin \ngap and a large 𝑇𝑐. \n In summary, we first demonstrate experimentally that substituting Si by Al in \nCo2MnAl xSi1-x Heusler compounds allows us to get a tunable spin polarization at E F from \n60% in Co 2MnAl to 100% in Co 2MnSi, indicati ng the transition from metallic to half \nmetallic behaviors. Second, a strong correlation between the spin polarization and the \nGilbert magnetic damping is established in these films . This confirms the theoretical \njustification of ultra -low magnetic damping in Ha lf-Metal -Magnet s as a consequence of \nthe spin gap. Third , the ultrafast spin dynamics results also nicely confirm that the spin \ngap is at the origin of the increase of the relaxation time. Our experiments allow us to go 12 \n further by establishing clear relati onships between the spin polarization, the magnetic \ndamping and the demagnetization time. A n inverse relationship between demagnetization \ntime and Gil bert damping is established in these alloys , which agrees well with the model \nproposed by Mann et al. [13] and with Koopmans et al. [10] but without considering any \ninfluence of Curie temperature much larger than room temperature in these films . \nExperimental section \n Co2MnAl xSi1-x(001) quaternary Heusler compounds are grown by Molecular \nBeam Epitaxy using an MBE machine e quipped with 24 materials. The s toichiometry is \naccurately controlled during the growth by calibration of the Co, Mn, Si and Al atomic \nfluxes using a quartz microbalance located at the pl ace of the sample. The error on each \nelem ent concentration is less than 1 % [23]. The films are grown directly on MgO(001) \nsubstrates, with the epitaxial relationship [100] (001) MgO // [110] (001) Heusler \ncompound. The thickness is fixed to 20nm. \n The phot oemission experiments were done at the CASSIOPEE beamline at \nSOLEIL synchrotron source. The films were grown in a MBE connected to the beamline \n(see [19,22,35 ] for details). The SR -PES spectra were obtained by using the largest slit \nacceptance of the detec tor (+/ - 8°) at an angle of 8° of the normal axis of the surface. Such \ngeometry allows us to analyze all the reciprocal space on similar polycrystalline films \n[23]. \n The radiofrequency magnetic dynamics of the films were thus studied using \nferromagnetic resonance (FMR). A Vectorial Network Analyzor FMR set -up was used in \nthe perpendicular geometry (see [ 22] for experimental details) where the static magnetic \nfield is applied out of the plane of the film in order to avoid extrinsic bro adening of the \nlinewidth due to the 2 -magnons scattering [ 36,37]. \n Ultrafast magnetization dynamics were investigat ed using polar time -resolved \nmagneto -optical Kerr (TR -MOKE) experiments. An amplified Ti -sapphire laser \nproducing 35 fs pulses at 800 nm with a repetition rate of 5 KHz is used . The pump beam \nis kept at the fundamental mode and is focused down to spot size of ~260 𝜇𝑚 while the \nprobe is frequency doubled to 400 nm and focused to a spot size of ~60 𝜇𝑚. 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Jánossy, \n Phys. Rev. B 73, 144424 (2006) \n[37] Kh. Zakeri, J. Lindner, I. Barsukov, R. Meckenstock, M.Farle, U. von Hörsten, H.Wende, \nW. Keune, J. Rocker, S. S. Kalarickal, K. Lenz, W. Kuch, K. Baberschke, Z. Frait, \n Phys. Rev. B 76, 104416 (2007) " }, { "title": "2011.10437v1.The_effect_of_magnetic_field_on_the_damping_of_slow_waves_in_the_solar_corona.pdf", "content": "Astronomy &Astrophysics manuscript no. finitebeta_aa ©ESO 2020\nNovember 23, 2020\nThe effect of magnetic field on the damping of slow waves in the\nsolar corona\nT. J. Duckenfield1, D. Y . Kolotkov1;2, and V . M. Nakariakov1;3\n1Centre for Fusion, Space and Astrophysics, Department of Physics, University of Warwick, CV4 7AL, UK\ne-mail: T.Duckenfield@warwick.ac.uk\n2Institute of Solar-Terrestrial Physics SB Russian Academy of Sciences, Irkutsk 664033, Russia\n3St Petersburg Branch, Special Astrophysical Observatory, Russian Academy of Sciences, St Petersburg, 196140, Russia\nReceived 2 November 2020 /Accepted 20 November 2020\nABSTRACT\nContext. Slow magnetoacoustic waves are routinely observed in astrophysical plasma systems such as the solar corona, and are usually\nseen to damp rapidly. As a slow wave propagates through a plasma, it modifies the equilibrium quantities of density, temperature,\nand magnetic field. In the corona and other plasma systems, the thermal equilibrium is comprised of a balance between continuous\nheating and cooling processes, the magnitudes of which vary with density, temperature and magnetic field. Thus the wave may induce\na misbalance between these competing processes. Its back reaction on the wave has been shown to lead to dispersion, and amplification\nor damping, of the wave.\nAims. This e \u000bect of heating /cooling misbalance has previously been studied in the infinite magnetic field approximation, in a plasma\nwhose thermal equilibrium comprises of optically thin radiative losses and field-aligned thermal conduction, balanced by an (un-\nspecified) heating process. In this work we extend this analysis by considering a non-zero \fplasma. The importance of the e \u000bect of\nmagnetic field in the rapid damping of slow waves in the solar corona is evaluated, and compared to the e \u000bects of thermal conduction.\nMethods. A linear perturbation under the thin flux tube approximation is considered, and a dispersion relation describing the slow\nmagnetoacoustic modes is found. The dispersion relation’s limits of strong non-adiabaticity and weak non-adiabaticity are studied.\nThe characteristic timescales are calculated for plasma systems with a range of typical coronal densities, temperatures and magnetic\nfield strengths.\nResults. The number of timescales characterising the e \u000bect of misbalance is found to remain at two, as with the infinite magnetic\nfield case. In the non-zero \fcase, these two timescales correspond to the partial derivatives of the combined heating /cooling function\nwith respect to constant gaspressure and with respect to constant magnetic pressure. The predicted damping times of slow waves\nfrom thermal misbalance in the solar corona are found to be of the order of 10–100 minutes, coinciding with the wave periods and\ndamping times observed. Moreover the slow wave damping by thermal misbalance is found to be comparable to the damping by\nfield-aligned thermal conduction. The change in damping with plasma- \fis complex and depends on the coronal heating function’s\ndependence upon the magnetic field in particular. Nonetheless we show that in the infinite field limit, the wave dynamics is insensitive\nto the dependence of the heating function on the magnetic field, and this approximation is found to be valid in the corona so long as\nthe magnetic field strength is greater than approximately 10 G for quiescent loops and plumes, and 100 G for hot and dense loops.\nConclusions. Thermal misbalance may damp slow magnetoacoustic waves rapidly in much of the corona, and its inclusion in our\nunderstanding of slow mode damping may resolve discrepancies between observations and theory relying on compressive viscosity\nand thermal conduction alone.\nKey words. Magnetohydrodynamics (MHD) - Waves – Sun: oscillations – Radiation mechanisms: thermal - Sun: corona\n1. Introduction\nSlow magnetoacoustic waves are a common feature of many\nplasma systems, and the study of their properties allows one to\nprobe the local plasma conditions which otherwise may be dif-\nficult to measure. Often these plasma systems are maintained\nat thermal equilibrium by a delicate balance between continu-\nous heating and cooling mechanisms - one example being the\nsolar corona, which is cooled by radiative losses and heated\nby some as-yet undetermined heating process(es). The e \u000bect of\nthese heating and cooling mechanisms vary with the plasma pa-\nrameters. As a slow wave propagates through the plasma, the\nwave perturbs both the plasma’s mechanical and thermal equi-\nlibria, through modifications in the local density, temperature\nand magnetic field strength. Thus if the plasma is steadily be-\ning heated and cooled at thermal equilibrium, the wave inducesa misbalance between these competing processes. This leads to\nthe transfer of energy between the wave and the plasma referred\nto as a heating /cooling misbalance .\nPrevious studies of the e \u000bects of this wave-induced thermal\nmisbalance under the infinite magnetic field approximation have\nshown that the plasma may act as a dissipative or active medium,\ndamping the wave or growing its amplitude (Nakariakov et al.\n2000; Kumar et al. 2016). The presence of characteristic times\nassociated with the thermal misbalance may also cause disper-\nsion, such that any broadband pulse is dispersed by the medium\ninto a quasi-periodic slow wave train (Zavershinskii et al. 2019).\nSuch observable e \u000bects upon the wave by the heating /cooling\nmisbalance are related to the properties of the heating and cool-\ning processes themselves, specifically their derivatives with re-\nspect to the thermodynamic parameters of the plasma: density\n\u001a, temperature Tand potentially magnetic field strength B. In\nArticle number, page 1 of 11arXiv:2011.10437v1 [astro-ph.SR] 20 Nov 2020A&A proofs: manuscript no. finitebeta_aa\nKolotkov et al. (2019) the damping by thermal misbalance of hot\ncoronal loops observed by SUMER was considered, and it was\nfound that in the regime of enhanced damping, the theoretically\nobtained damping rates coincide with those seen in observations.\nRecently, Kolotkov et al. (2020) demonstrated the potential for\nconstraining the unknown coronal heating function, using obser-\nvations of the solar corona such as the observed rapid damping\nof slow modes, and coronal slow waves’ thermal instability and\nacoustic overstability.\nThese previous works rest on the assumption of infinitely\nstrong magnetic field and consider perturbations to the local\nplasma density and temperature, following from the seminal\nwork of Field (1965) analysing perturbations to an infinite ho-\nmogeneous plasma. Yet to fully understand the e \u000bects of the\nheating /cooling misbalance, the influence of non-zero \fmust be\nstudied, since any non-zero magnetic field fluctuations will in-\nteract with the density, temperature and velocity perturbations\nand therefore a \u000bect the wave evolution and propagation speed\n(Afanasyev & Nakariakov 2015; Nakariakov et al. 2017). Some\nmagnetic field measurements of coronal structures have found\nmagnetic field strengths can drop to 10 G and below (e.g. a value\nof 4 G reported in Lin et al. 2004), implying that the magnetic\npressure may not totally dominate over gas pressure everywhere.\nMoreover, the source of energy for the solar coronal heating is\nknown to be the magnetic field, and so it is natural to allow a\ndependence of the heating /cooling misbalance upon the mag-\nnetic field strength. It is therefore important to investigate the\nrole magnetic e \u000bects have upon the dispersion and damping by\nheating /cooling misbalance.\nThe rapid damping of slow magnetoacoustic modes observed\nin the solar corona is well documented, yet ambiguity remains\nregarding its origin (e.g. De Moortel 2009). Thermal conduc-\ntion and compressive viscosity are invoked as damping mecha-\nnisms, however we contend that the e \u000bect of wave-induced heat-\ning/cooling misbalance can be of equal importance. The inclu-\nsion of thermal misbalance as an additional damping mechanism\ncould resolve discrepancies seen in the frequency dependencies\nof observed slow mode damping, such as for the standing slow\nmodes of hot loops reported in Mariska (2006), the propagating\nslow modes in coronal holes detailed in Gupta (2014), and the\npropagating slow modes in the warm corona analysed in Krishna\nPrasad et al. (2014). The phase shifts between density and tem-\nperature measured in e.g. Krishna Prasad et al. (2018) disagree\nwith those predicted from theory (Owen et al. 2009), and simi-\nlarly the growth in polytropic index (estimated via phase shift)\nwith temperature observed in Van Doorsselaere et al. (2011); Kr-\nishna Prasad et al. (2019) are also a mystery. The series of pa-\npers culminating in Wang & Ofman (2019) try to rectify these\nand similar discrepancies between 1D slow mode damping the-\nory and observations through anomalous thermal conduction and\nviscosity coe \u000ecients – the inclusion of thermal misbalance pro-\nvides an alternative, perhaps more physically motivated, expla-\nnation.\nIn this work we extend the results of Kolotkov et al. (2019)\nto investigate the e \u000bects of thermal misbalance in non-zero \f\nplasma upon a slow wave using the thin flux tube approximation.\nA non-adiabatic linear dispersion relation is derived, and its lim-\nits of weak and strong non-adiabaticity are explored in Section 2.\nEstimates of the damping time of slow waves in the solar corona\nby thermal misbalance, its sensitivity to the dependence of the\nheating /cooling function on B, and comparisons with other dis-\nsipation mechanisms are the topics of Section 3. Discussion and\nconcluding remarks are made in Section 4.2. Dispersion relation\n2.1. Derivation\nIn this work we use the first order thin flux tube approxima-\ntion, which formally corresponds to the first order of the Tay-\nlor expansion of the MHD variables with respect to the radial\ncoordinate, derived by Roberts & Webb (1978) and Zhugzhda\n(1996). The governing equations are the same as for Nakariakov\net al. (2017), neglecting the viscous dissipation in the momen-\ntum equation and slightly adjusting the definition of the thermal\nheating /cooling function Qto have the units of W kg-1(matching\nthe definition in Field 1965; Kolotkov et al. 2019):\ndp\ndt\u0000\rp\n\u001ad\u001a\ndt=(\r\u00001) \n\u0014k@2T\n@z2\u0000\u001aQ(\u001a;T;B)!\n; (1)\n\u001adu\ndt+@p\n@z=0; (2)\np+B2\n2\u00160=pext\ntotal; (3)\n@B\n@t+u@B\n@z+2Bvr=0 (4)\n@\u001a\n@t+2\u001avr+@\n@z(\u001au)=0; (5)\np=kB\nm\u001aT (6)\nAs usual, pis the plasma pressure, \u001ais the plasma density, T\nis the temperature, kBis the Boltzmann constant, \u00160is the mag-\nnetic permeability of free space, mis the mean particle mass,\nand\ris the polytropic index. Also uis the wave-induced flow\nspeed along the tube (i.e. in zdirection), Bis the component of\nmagnetic field strength along the tube, pext\ntotalis the total external\npressure and vris the radial derivative of the radial component of\nplasma velocity. All of these quantities are measured at the axis\nof the (infinitesimally thin) flux tube. The right hand side of the\nenergy equation (1) represent thermodynamic processes ongoing\ninside the plasma. The first term is the (field-aligned) thermal\nconduction, for which we use the standard estimation of coe \u000e-\ncient\u0014k\u001910\u000011T5=2Wm-1K-1. The second term is the combi-\nnation of some unspecified heating H(\u001a;T;B) and optically thin\nradiative coolingL(\u001a;T), combined in the net heat /loss function\nQ(\u001a;T;B)=L\u0000H . We note Qdepends on Bonly if the heating\ntermHis a function of B, since the radiative losses Lare known\nto be independent of B.\nThus, in addition to the perturbation of the mechanical equi-\nlibrium provided by the force balance, in this work we consider\na wave-induced perturbation of the thermal equilibrium of the\ncorona. It is important to remark explicitly that, following from\nthe previous works on thermal misbalance, we allow both the\nheating and cooling functions to be perturbed. This is is con-\ntrast to several previous works in which the heating term is held\nconstant, which is to say remains unperturbed by the wave, such\nas Claes & Keppens (2019); Kaneko & Yokoyama (2017); De\nMoortel & Hood (2003) do when setting up their simulations.\nWe consider linear perturbations of a mechanical equilib-\nrium, characterised by the constant quantities denoted p0,\u001a0,\nB0,T0, and pext\ntotal, and without steady flows. In addition we con-\nsider Q0=0 in the equilibrium, motivated by the continued ex-\nistence of the corona. The parallel thermal conduction does not\ncontribute to this equilibrium because the plasma temperature is\nuniform. Let the perturbations of the equilibrium quantities be\nArticle number, page 2 of 11T. J. Duckenfield et al.: The e \u000bect of magnetic field on the damping of slow waves in the solar corona\nsmall,\np=p0+p1; \u001a=\u001a0+\u001a1;T=T0+T1;\nB=B0+B1; vr=v1;u=u1;\nwhere the subscript 1 denotes small perturbations. In the follow-\ning, exceptions are made for vr;usince these are small quantities\nabout zero anyway and so we leave their subscripts alone. We\nsubstitute these quantities into Equations (1) - (6) and keep only\nthe linear terms of the small quantities to find:\n@\n@tp1\u0000C2\nS@\n@t\u001a1=\n(\r\u00001) \n\u0014k@2\n@z2T1\u0000\u001a0h\nQ\u001a\u001a1+QTT1+QBB1i!\n; (7)\n\u001a0@\n@tu+@\n@zp1=0; (8)\np1+B0B1\n\u00160=0; (9)\n@\n@tB1+2B0vr=0; (10)\n@\n@t\u001a1+2\u001a0vr+\u001a0@\n@zu=0; (11)\np1\u0000kB\nm(\u001a0T1+T0\u001a1)=0: (12)\nThe parameter C2\nS=\rp0=\u001a0is the sound speed at equilibrium,\nandQT;Q\u001a;QBare the partial derivatives of the combined heat-\ning/cooling function Q(Qx=@Q=@x), evaluated at the equilib-\nrium.\nBy using these equations, several assumptions have been\nmade which are worth mentioning. Firstly, since in a slow wave\nin a low-beta plasma, any change in the external pressure pext\ntotalto\nthe flux tube is neglected (e.g. see Edwin & Roberts 1983), we\nconcentrate on waves propagating inside the flux tube, taking\nthat the slow waves are always in the trapped regime. Secondly,\nthe obliqueness of the wavefronts are accounted for through the\nuse ofvr– this is valid when the wavelength of the perturbations\n(parallel to the field) is much longer than the transverse spatial\nscale, determined by the width of the waveguiding plasma non-\nuniformity. This is the applicability condition of the thin flux\ntube approximation, and is a key di \u000berence to the plane acoustic\nwave case used elsewhere. Finally it should be noted that, for all\nnon-adiabatic processes in general (whose assorted characteris-\ntic timescales we call \u001ci), the assumption \r=CP=CVis only\nvalid when !\u001d\u001c\u00001\ni, i.e. when the wave is of su \u000eciently high\nfrequency that it is adiabatic or weakly non-adiabatic (for exam-\nple see the discussion in Van Doorsselaere et al. 2011). In general\nnon-adiabatic scenarios (Zavershinskii et al. 2019) or non-zero \f\nplasmas (Nisticò et al. 2017), the ratio of specific heats can vary.\nIn our estimations, we use \r=5=3 without loss of generality.\nSince we have no flows and we have uniformity in the zdi-\nrection, we take a Fourier transform by making the ansatz of\nplane waves, that is to say we assume a harmonic dependence\nupon the time and spatial coordinates for all perturbed variables\n/exp (\u0000i!t+ikz) where!is the frequency and kis the parallelwavenumber. The resulting linearised set of equations yield the\nfollowing dispersion relation:\n!3+A(k)!2+B(k)!+C(k)=0 (13)\nwhere the coe \u000ecients are\nA=iC2\nT\nC2\nS(QT\nCV+\u0014k\n\u001a0CVk2+(\r\u00001)\nC2\nAh\nT0QT\u0000B0QB\u0000\u001a0Q\u001a\n+T0\n\u001a0\u0014kk2i)\n;\nB=\u0000C2\nTk2;\nC=\u0000i(\r\u00001)C2\nT\nC2\nS\u0010\nT0QT\u0000\u001a0Q\u001a+T0\n\u001a0\u0014kk2\u0011\nk2=0:\nThe term CAis the standard Alfvén speed defined through C2\nA=\nB2\n0=\u00160\u001a0, and the term CTis the tube speed defined through\nC\u00002\nT=C\u00002\nS+C\u00002\nA. Equation (13) is cubic in !yet quartic in\nk, that is to say asymmetric with respect to space and time. This\ndispersion relation describes two oppositely-directed propagat-\ning slow waves and an entropy mode, made into a thermal mode\nby the non-adiabatic e \u000bects (e.g. De Moortel & Hood 2003). The\ntube speed appears in the coe \u000ecient of the !term, thus in the\nadiabatic limit the equation reduces to the wave equation with\nC2\nTas the speed, as expected for so-called tube waves. This ex-\npression also agrees exactly with the infinite magnetic field case\n(Eq. 7, Kolotkov et al. 2019) in the limit B!1 .\nRegarding the thermal conduction terms, we see the term\n(C2\nT=CS)2\u0014kk2=\u001a0CVin the!2coe\u000ecient A, which is propor-\ntional to the term in the infinite magnetic field case (e.g. Eq. 8 in\nKolotkov et al. 2019) but modified by the ratio of tube to sound\nspeed squared. Thus there is a non-zero \fmodification to the ef-\nfect by thermal conduction on the waves, which is qualitatively\nconsistent with the result in Afanasyev & Nakariakov (2015).\nIt is convenient to re-express the non-adiabatic terms us-\ning characteristic timescales, which for the thermal misbalance\nterms are fully determined by the equilibrium parameters and\npartial derivatives Q\u001a;QB;QT. Note that these timescales are\nnotdetermined by the heating and cooling processes separately.\nRather, these characteristic timescales are determined by how\nquickly the perturbation returns to, or destroys, the equilibrium.\nIn the case with infinite magnetic field, the characteristic\ntimescales were written in terms of QT[p]and QT[\u001a]=QT,\nwhere QT[p]means the partial derivative taken with respect to\ntemperature assuming constant gas pressure (Zavershinskii et al.\n2019). The introduction of a finite magnetic field means there\nis a separate, magnetic pressure term B2=2\u00160. Thus we con-\nsider separately the derivative with respect to constant gas pres-\nsure QT[gas p], and with respect to constant magnetic pressure\nQT[mag p]. To write the additional terms in the derivatives of Q\nby the magnetic field, the relevant equations are the radial pres-\nsure balance (Eq. 9) and the ideal gas law (Eq. 12), finding:\n@\u001a\n@T=\u0000m\nkBp0\nT2\n0=\u0000\u001a0\nT0;\n@p\n@T=kB\nm\u001a0;\n@B\n@T=\u0000\u00160\nB0@p\n@T=\u0000\f\n2B0\nT09>>>>>>>>>>>>=>>>>>>>>>>>>;=)QT[gas p]=QT\u0000\u001a0\nT0Q\u001a\nQT[mag p]=QT\u0000\f\n2B0\nT0QB:\nGathering terms in the dispersion relation (Eq. 13) we can define\ntwo characteristic timescales,\n\u001c1=CP\nQT[gas p]=\rCV\nQT[gas p]; \u001c 2=CV\nQT[mag p]: (14)\nArticle number, page 3 of 11A&A proofs: manuscript no. finitebeta_aa\nIt is striking that despite there being three di \u000berent dependencies\ninQ(\u001a;T;B), the e \u000bects of heating /cooling misbalance can be\nexpressed in terms of these two timescales evaluated at constant\ngas and magnetic pressures. Comparing these timescales with\nthe infinite magnetic field case (Kolotkov et al. 2019; Zavershin-\nskii et al. 2019) we see that \u001c1is identical, whilst \u001c2is di\u000berent\nonly by a magnetic correction term in QT[mag p], which goes to\nQTas the plasma- \fgoes to zero. We also use the characteristic\ntimescale for thermal conduction (in the infinite magnetic field\nlimit) - for wavelength \u0015=2\u0019=k- as given in Kolotkov et al.\n(2019), namely \u001ccond(k)=\u001a0CV\u00152=\u0014k. Pulling these definitions\ntogether, we write the dispersion relation as\n!3+i2\n2+\r\f(4\u00192\n\u001ccond(k)\u0012\n1+\f\n2\u0013\n+1\n\u001c2+\r\f\n21\n\u001c1)\n!2\n\u0000C2\nTk2!\u0000iC2\nT(1\n\r4\u00192\n\u001ccond(k)+1\n\u001c1)\nk2=0: (15)\nIt may be seen that Equation (15) is a \u000bected by the magnetic\nfield in several ways: the phase speed CS!CT, the terms\nwith plasma- \f, and also implicitly through the timescale \u001c2(i.e.\nvia the product \fQB=2). Interestingly, only the last of these is\na\u000bected by the dependence of Qupon magnetic field B. This\nmeans that even if the heating model is independent of magnetic\nfield, the properties of the wave are still a \u000bected by the magnetic\nfield strength. The reverse is also true, if \fgoes to zero the wave\ndynamics is not a \u000bected by the magnetic field even if the heating\nfunction has some dependence on it.\nThe non-zero \fe\u000bects on the real part of !(and hence phase\nspeed) are well known, and it has been demonstrated they may\nbe important for slow waves in some wave guides such as hot\nflaring loops (Afanasyev & Nakariakov 2015), as well as the\ndetermination of cut-o \u000bfrequency in the solar atmosphere. By\naccounting for the obliqueness of the waves, the wave speed is\nmade to depend on the absolute value of the magnetic field via\nCT, which is sub-sonic and sub-Alfvénic.\n2.2. Limit of weak non-adiabaticity\nSimilar to the previous works on damping of magnetoacous-\ntic waves by thermal conduction, the upper and lower limits\nof non-adiabaticity are now derived. We begin with the limit of\nweak non-adiabaticity, in which the wave is only mildly a \u000bected\nby the transfer of energy with the active medium. In this limit\n!\u001d1=\u001c1;2;cond, thus we rearrange the dispersion relation (15)\nassuming!,0 into\n!2=C2\nTk2(\n1\u0000i\"1\nC2\nT!2\nk2 1\n!\u001c2+\r\f\n21\n!\u001c1!C2\nT\nC2\nS\u00001\n!\u001c1\n+4\u00192\n!\u001ccond(k)0BBBB@1\nC2\nT!2\nk2\u0014\n1+\f\n2\u0015C2\nT\nC2\nS\u00001\n\r1CCCCA#)\n; (16)\n(consistent with Eq. (21) Nakariakov et al. 2017, which deals\nwith the same limiting case). Taking the limit of 1 =!\u001c condand\n1=!\u001c 1;2as small parameters, the Taylor expansion of the disper-\nsion relation reduces to\n!2\u0019C2\nTk2(\n1\u0000i!\u000012\n2+\r\f\" \r\u00001\n\r!4\u00192\n\u001ccond+1\n\u001c2\u00001\n\u001c1#)\n:(17)\nTo evaluate the !\u00001in the imaginary component of Equa-\ntion (17), perturbation theory is used. In the zeroth order !\u0019CTk, and using this yields the following solution to the weakly\nnon-adiabatic dispersion relation:\n!R\u0019CTk; (18)\n!I\u0019\u00001\n2 2\n2+\r\f!\"\r\u00001\n\r4\u00192\n\u001ccond+1\n\u001c2\u00001\n\u001c1#\n: (19)\nThe phase speed of the weakly non-adiabatic wave is the tube\nspeed CTas expected, so the phase speed is reduced as \fin-\ncreases. In the limit of \f!0 this equation coincides with the\nresults in Kolotkov et al. (2019).\nFrom Equation (19) we are motivated to form the single,\ncombined timescale TM(\u001c1;\u001c2) which can be referred to as a\ncharacteristic damping time of the heating /cooling misbalance\nin the weakly non-adiabatic regime:\n1\nTM=C2\nT\nC2\nS 1\n\u001c2\u00001\n\u001c1!\n;\n= 2\n2+\r\f! 1\n\u001c2\u00001\n\u001c1!\n;\n=2\n2+\r\f(\r\u00001\n\rQT\nCV+1\n\r\u001a0\nT0Q\u001a\nCV\u0000\f\n2B0\nT0QB\nCV)\n:(20)\nFrom a physical standpoint, this equation implies that in the con-\nsidered limit of weak non-adiabaticity, the perturbations of T,\u001a,\nBall a\u000bect the heat /loss function Qindependently (i.e. the ef-\nfects of QT,Q\u001aandQBare additive in Eqs. 19 and 20). In the ab-\nsence of thermal conduction, the e-folding time of the slow wave\namplitude by thermal misbalance is accordingly Tdamp=2TM,\nconsistent with the relationship between damping time and char-\nacteristic times defined in De Moortel & Hood (2003). In a simi-\nlar fashion, we form a timescale TcondB (k;\f) characteristic of the\nwave damping time by thermal conduction in the limit of weak\nnon-adiabaticity, related to \u001ccond(k) by\nTcondB =\u0012\n1+\r\f\n2\u0013\r\n\r\u00001\u001ccond(k)\n4\u00192; (21)\n=)!I\u0019\u00001\n2 1\nTcondB+1\nTM!\n: (22)\nIf the limit of weak non-adiabaticity applies, there is no thermal\nconduction, andTM>0, then we may interpretTMas the damp-\ning time over which the thermal misbalance is attenuating the\nslow wave . Similarly, if the limit of weak non-adiabaticity ap-\nplies andTM<0, energy is supplied from the medium into the\nwave, amplifying the slow wave over the characteristic timescale\nTM. Regarding the e \u000bect of non-zero \f, the phase speed is re-\nduced as\fincreases through CT, and the e \u000bect on the damping\ndepends on the sign of QB– note if there is no magnetic depen-\ndence ( QB=0), then the damping e \u000bect is always lessened with\nincreasing\f.\nThe e \u000bect of thermal conduction is always to damp ( TcondB is\nstrictly positive). From Equation (21) it is clear that in the limit of\nweak non-adiabaticity, the damping e \u000bect of thermal conduction\nis reduced as the plasma- \fincreases. In the weak non-adiabatic\nlimit, the e \u000bect of thermal conduction and the e \u000bect of thermal\nmisbalance upon the wave increment !Iare additive, shown in\nEquation (22).\nIn the presence of both thermal conduction and thermal mis-\nbalance, whilst remaining in the weakly non-adiabatic limit, the\nArticle number, page 4 of 11T. J. Duckenfield et al.: The e \u000bect of magnetic field on the damping of slow waves in the solar corona\ndamping time of the wave is Tdamp=2=(T\u00001\ncondB+T\u00001\nM), equiva-\nlently\nTdamp=2+\r\f \r\u00001\n\r4\u00192\u0014k\n\u001a0CV!\nk2+\u001c1\u0000\u001c2\n\u001c1\u001c2: (23)\n2.3. Limit of strong non-adiabaticity\nThe limit of strong non-adiabaticity describes slow magnetoa-\ncoustic waves for which !\u001c1=\u001c1;2;cond, which is to say these\nwaves are highly a \u000bected by the exchange of energy with the ac-\ntive medium. The dispersion relation (Eq. 15) may be expressed\nin the following way (where we have divided through by !,0):\n!2=C2\nTk21+i(1\n\r4\u00192\n!\u001ccond+1\n!\u001c1)\n1+i(2\n2+\r\f (1+\f=2)4\u00192\n!\u001ccond+1\n!\u001c2+\r\f\n21\n!\u001c1!)\n(24)\nAfter Taylor expansion we find the strong limit to be\n!2\u0019C2\nSk28>>>>>>><>>>>>>>: 1\n\r4\u00192\n\u001ccond+1\n\u001c1!\n4\u00192\n\u001ccond+1\n\u001c2+\r\f\n2 1\n\r4\u00192\n\u001ccond+1\n\u001c1!\n\u0000i! \r\u00001\n\r!4\u00192\n\u001ccond+1\n\u001c2\u00001\n\u001c1\n 4\u00192\n\u001ccond+1\n\u001c2+\r\f\n2 1\n\r4\u00192\n\u001ccond+1\n\u001c1!!29>>>>>>>>=>>>>>>>>;: (25)\nThe change from C2\nTtoC2\nSis caused by pulling out a factor of 1 +\n\r\f=2. Equation (25) agrees with strong limit in the infinite field\nlimit as\f!0 as it should (Zavershinskii et al. 2019). In order\nto deal with the !in the imaginary component, we again apply\nthe perturbation approach. In the zeroth order !is approximated\nby!Ras seen in Equation (25), yielding the following solution\nto the highly non-adiabatic dispersion relation:\n!R\u00191p\rCSk 4\u00192\n\u001ccond+\r\n\u001c1!1=2\n\"4\u00192\n\u001ccond+1\n\u001c2+\f\n2 4\u00192\n\u001ccond+\r\n\u001c1!#1=2; (26)\n!I\u0019\u00001\n2C2\nSk2\u0012\n1+\r\f\n2\u0013 \r\u00001\n\r4\u00192\n\u001ccond+1\n\u001c2\u00001\n\u001c1!\n\"4\u00192\n\u001ccond+1\n\u001c2+\f\n2 4\u00192\n\u001ccond+\r\n\u001c1!#2:(27)\nWe must make it clear that di \u000berent combinations of signs of\n\u001c1;\u001c2may lead to very di \u000berent behaviour, e.g. complex phase\nspeeds (that is, even non-propagating modes) or the develop-\nment of thermal instabilities of a non-acoustic nature (see Field\n1965). Restricting our attention to the case of a stable propa-\ngating slow wave ( \u001c1;\u001c2>0), in the absence of thermal con-\nduction the non-adiabatic wave propagates at the speed !R=kwith!R=CSk(\u001c2=(\u001c1+(\r\f=2)\u001c2))1=2, which in the infinite\nfield case is simply CSp\u001c2=\u001c1.In non-zero \fplasma, the phase\nspeed of the highly non-adiabatic wave is reduced compared to\nthe infinite magnetic field case. The wave will damp if Equa-\ntion (27) is negative. The e \u000bect of non-zero \fupon this wave\nincrement /decrement (that is, growth or damping) is governed\nby Equation (27), and is di \u000berent for di \u000berent plasma conditions\n(equivalently its impact depends on the relative magnitudes of\n\u001ccond;\u001c1;\u001c2). Unlike the weakly non-adiabatic case, the e \u000bects\nof the di \u000berent non-adiabatic mechanisms upon the wave incre-\nment/decrement!Iare not additive.\nConsidering only the thermal conduction terms ( \u001c1;2\u001d\n\u001ccond), it is found that !R=k\u0019CS(\r(1+\f=2))\u00001=2. This tends\nto the isothermal sound speed CS=p\ras\f!0 consistent with\nDe Moortel & Hood (2003); as with the case for no thermal con-\nduction, for increasing \fthis phase speed is reduced. The ef-\nfect of thermal conduction on the wave decrement is always to\ndamp (!I<0). However, as the wave approaches the isothermal\nregime (without misbalance) !Ibecomes proportional to !\u001ccond\nwhich is a small parameter in the strong limit. In other words, al-\nthough the e \u000bect of thermal conduction is always to damp, in the\nisothermal regime the damping ceases. The e \u000bect of increasing\n\fis to reduce !Iand hence increase damping times (equivalent\nto lessening the rate of damping).\n3. Damping of slow waves in the corona\n3.1. Estimation of damping time for non-zero plasma-beta\nWe now focus on the damping e \u000bect of the thermal misbalance\nupon slow magnetoacoustic waves in the corona. The charac-\nteristic timescales of wave-induced thermal misbalance (Eq. 14)\nvary with T0,\u001a0,B0, as well as the parameters dictating the heat-\ning and cooling rates HandL. Up to this point, all our results\nhave been expressed in terms of a generic heating /cooling func-\ntion Q, whose derivatives with respect to thermal equilibrium\nare treated as free parameters, and applicable to any plasma con-\nditions for which the governing equations may be satisfied. In\norder to fully explore our results in the coronal context how-\never, we now pin down a functional form of Qand pick some\nplasma parameter ranges to evaluate. We consider temperatures\nranging from 0 :5 MK to 20 MK and electron number densities\nranging from 1\u0002108cm\u00003to 5\u00021012cm\u00003, since many typical\ncoronal structures have been detected at these temperatures and\ndensities, such as plumes and coronal loops (De Moortel 2009).\nSome coronal structures do exist outside of these ranges, such as\nprominences, however for such conditions the e \u000bects of partial\nionisation, non-LTE conditions and optical thickness can not be\nneglected.\nWe parameterise the coronal heating /cooling function as\nQ=L(\u001a;T)\u0000H(\u001a;T;B)(=8><>:L(\u001a;T) from CHIANTI ;\nH(\u001a;T;B)=h0\u001aaTbBc:(28)\nwhere the coe \u000ecient h0is determined from the initial thermal\nequilibrium condition, Q0=0,=) h0=L0=\u001aa\n0Tb\n0Bc\n0, and\nthe power indices a,bandcare treated as free parameters. We\nsynthesise the coronal optically thin radiation function L(\u001a;T)\nfrom CHIANTI atomic database v. 9.0.1 (Dere et al. 1997, 2019)\nfor the densities and temperatures from those intervals.\nWe do not know the values of a;b;c, as this is essentially the\ncoronal heating problem. As discussed in Kolotkov et al. (2020),\nmany previous authors such as Ibanez S. & Escalona T. (1993);\nCarbonell et al. (2006) have considered five models for a;borigi-\nnating from appendix B of Rosner et al. (1978). However all five\nArticle number, page 5 of 11A&A proofs: manuscript no. finitebeta_aa\nof these are incompatible with the observations of widespread\ncoronal thermal stability and the rapid damping of slow (acous-\ntic) waves. Following Kolotkov et al. (2020) we consider the val-\nues of a=1=2,b=\u00007=2, for which both thermal stability and\nacoustic stability are always satisfied in coronal conditions, in\naccordance with observations. We study the change in damping\nwith the parameter c, and the change in damping with plasma- \f.\nWe now estimate the absolute values of the characteristic\nthermal misbalance damping time TMin the limit of weak non-\nadiabaticity (recall the connection to the wave damping time\nTdamp=2TM, in the absence of damping by thermal conduc-\ntion). The e \u000bect of a weaker magnetic field (i.e. higher plasma- \f)\nonTMmay be seen in Figure 1, for two heating models with dif-\nferent magnetic dependencies (chosen for illustration purposes\nonly). Looking only at the e \u000bect of reduced magnetic field (scan-\nning downwards in Fig. 1) across the range of typical coro-\nnal magnetic field strengths, the damping time decreases due to\nnon-zero\fe\u000bects, particularly temperatures over \u00182 MK; the\nchange in magnetic field strength has the most pronounced e \u000bect\non hotter, denser plasma (i.e. where plasma- \fis greater) whereas\nthe cooler loops and plumes remain largely una \u000bected.\nThere is a distinction between the impact of the finite mag-\nnetic field (non-zero plasma- \f) upon the wave-induced ther-\nmal misbalance, and the e \u000bect of any dependence of the heat-\ning/cooling function upon magnetic field strength (non-zero QB).\nTo demonstrate the latter of these, consider the di \u000berence intro-\nduced by the change in dependence of Hupon B(scanning left\nto right in Fig. 1). For these heating functions the magnetic heat-\ning power-law index ( c=0!c=1) has made the damping\ntime vary lesswith magnetic field strength. In other words, the\nheating scenario with c=1 has a stabilising e \u000bect on the wave\ndynamics. This may not be the case for other values of c(e.g.\nc=\u00001), this is not discussed in this work. The di \u000berence is ap-\nparent only for lower magnetic field strengths. At infinite mag-\nnetic field, there is no di \u000berence between damping times for the\ntwo heating models – see panels (a) and (b) in Figure 1, which\nare almost identical. This implies that above a certain magnetic\nfield strength, the infinite magnetic field approximation is valid\nregardless of the heating functional dependence upon magnetic\nfield.\n3.2. Sensitivity of the wave damping to the dependence of\nheating function upon magnetic field\nIn order to estimate the magnetic field strength above which the\ninfinite magnetic field approximation is appropriate, we consider\nthe combined damping e \u000bect of thermal misbalance and thermal\nconduction. This is necessary because the thermal conduction\ndamping term is also a \u000bected by non-zero plasma- \f(see Eq. 21),\nso its exclusion would not allow for delineating a complete pic-\nture. For simplicity, we only consider the weakly non-adiabatic\nlimit, such that the angular frequency of the slow wave is ap-\nproximated by Equations (18), (19) and (22). We specify three\npertinent examples of damped slow waves seen in the corona: the\n3-minute upwardly propagating slow waves above a sunspot ob-\nserved in the SDO /AIA 171 Å bandpass which peaks at 0 :63 MK\n(De Moortel 2009), the slow waves seen in a plume observed\nwith the 193 Å bandpass (which peaks at 1 :3 MK) (e.g. Krishna\nPrasad et al. 2014), and the standing oscillations in hot loops ob-\nserved by SUMER, with a formation temperature of Fe XIX of\n6:3 MK) (e.g. recently reviewed in Nakariakov et al. 2019). As\nbefore we use the illustrative choice of heating scenario takenfrom Kolotkov et al. (2020), H /\u001a1=2T\u00007=2Bc, and let cvary\nfrom -1, 0, 1.\nFigure 2 shows that for su \u000eciently strong magnetic field\nstrengths, slow waves in the presence of all of the heating models\nshown have converged to the quality factor calculated for the in-\nfinite magnetic field case. This is regardless of the dependence of\nheating model upon magnetic field, in our case controlled by the\npower-law index c. For the warm quiescent corona, as demon-\nstrated by the top two panels, a magnetic field strength greater\nthan around 10 G is su \u000ecient for the infinite magnetic field ap-\nproximation to be appropriate. For the conditions typical of hot\nloops seen by SUMER, which are often post-flare, the non-zero\n\fe\u000bects are still important at high magnetic field strengths, and\nso a magnetic field strength of approximately one order of mag-\nnitude higher (\u0018100 G) is required for the damping to be inde-\npendent of the heating /cooling function’s dependence upon B.\nThis di \u000berence in behaviour may also be seen in Figure 1 since\nit is the hot, dense plasmas (upper right quadrants of those plots)\nfor which the change with plasma- \fand with the change in mag-\nnetic dependence cis greatest and visible. Returning to Figure 2,\nit may also be seen that if QB=0 (black lines) or the plasma- \fis\nsu\u000eciently small to neglect the e \u000bects of QB, then the damping\nis always diminished with increasing \f(stronger magnetic fields\nmean more damping) – consistent with Subsection 2.2.\nWhen the plasma- \fgreatly exceeds 1, that is to say when\nthe plasma is pressure dominated as opposed to magnetically\ndominated, the timescales for both the thermal misbalance and\nthe thermal conduction depart greatly from the infinite magnetic\nfield values. This is apparent in the bottom panel of Figure 2,\nwhere the quality factors have diverged greatly in the \f >1 re-\ngion. Also, the e \u000bect of plasma- \fupon the wave damping can\nnow be to decrease or increase the damping, depending on the\nsign of QB. We do not consider the regime \f>1 since it is more\napplicable to chromospheric plasma and below, necessitating the\naddition of further physical e \u000bect such as optically thick radia-\ntion, partial ionisation and non-LTE conditions.\n3.3. Comparison with damping by thermal conduction\nAs Figure 1 demonstrates, the damping timescale for the ther-\nmal misbalance in the infinite magnetic field case is of the same\norder as the observed periodicity of slow waves in many typi-\ncal coronal conditions, and the same order again as the observed\ndamping times (some tens of minutes). To check this holds true\nwhen accounting for non-zero \fplasma, as well as compare the\ndamping by thermal misbalance with the (conventionally domi-\nnant) damping by thermal conduction, we calculate the charac-\nteristic damping times due to thermal misbalance TMandTcondB\nfor typical combinations of coronal densities, temperatures and\nmagnetic field strengths. The results are shown in Table 1.\nWe stress that the variation of thermal conduction damp-\ning timeTcondB with both\fand\u0015means that its relevance\nis extremely broad, and the thermal misbalance timescale TM\ndepends on the exact parameterisation of the as-yet unknown\ncoronal heating function. Even so, for typical coronal situations\nTable 1 leads us to conclude that when comparing the damp-\ning of slow waves by thermal misbalance with the damping by\nfield-aligned thermal conduction, we find the e \u000bect of the heat-\ning/cooling misbalance could be of equal or greater importance .\nAs a specific example, consider the propagating 3 minute oscilla-\ntions seen in 171 Å with a wavelength \u0015\u001922 Mm. BothTMand\nTcondB are of the same order as the wave period, and the quality\nfactor calculated for this combination of parameters is \u00182 (top\nArticle number, page 6 of 11T. J. Duckenfield et al.: The e \u000bect of magnetic field on the damping of slow waves in the solar corona\n(a) Plot ofTMwith c=0 at 100 G.\n (b) Plot ofTMwith c=1 at 100 G.\n(c) Plot ofTMwith c=0 at 12 G.\n (d) Plot ofTMwith c=1 at 12 G.\n(e) Plot ofTMwith c=0 at 4 G.\n (f) Plot ofTMwith c=1 at 4 G.\nFig. 1: Variation of the characteristic thermal misbalance damping timescale TMwith magnetic field, and with di \u000berent power-law\nindex cwhereH/\u001a1=2T\u00007=2Bc. Scanning down the column shows how the damping time changes as magnetic field B0decreases:\npanels (a), (b) at 100 G; panels (c), (d) at 12 G; panels (e), (f) at 4 G. Comparing left-to-right shows the e \u000bect of a di \u000berent\npower-law index c(left side c=0, right side c=1), whilst all other parameters are held the same. Symbols correspond to specific\nplasma conditions (see Table 1). Note that panels (a) and (b) are practically identical, since the plasma- \feverywhere in these plots\nis su\u000eciently close to zero for the infinite magnetic field approximation to apply, which is independent of @Q=@B(see Subsec. 3.2).\nArticle number, page 7 of 11A&A proofs: manuscript no. finitebeta_aa\nTypical value Loop in 171 Å Plume in 193 Å Hot loop in SUMER\n(Symbol on plots) (square) (star) (circle)\nTemperature, T0 0.63 MK 1.3 MK 6.3 MK\nNumber density, ne 1:5\u0002109cm\u000030:5\u0002109cm\u000031\u00021010cm\u00003\nPeriod, P \u00193\u000010 min\u00198\u000018 min\u001910\u000020 min\nTM,B=1 3.2 min 23 min 23 min\nTM,B=34G3.2 min\n(\f=0.01)23 min\n(\f=0.00)30 min\n(\f=0.38)\nTM,B=12G3.3 min\n(\f=0.05)24 min\n(\f=0.03)81 min\n(\f=3.0)\nTM,B=4G4.4 min\n(\f=0.43)29 min\n(\f=0.28)548 min'9 hr\n(\f=27)\nTcondB ,B=1 10 min 11 min 27 min\nTcondB ,B=34G10 min\n(\f=0.01)11 min\n(\f=0.00)36 min\n(\f=0.38)\nTcondB ,B=12G11 min\n(\f=0.03)12 min\n(\f=0.03)97 min\n(\f=3.0)\nTcondB ,B=4G14 min\n(\f=0.43)14 min\n(\f=0.28)650 min'11 hr\n(\f=27)\n\u001crad 9.7 min 60 min 70 min\nTable 1: Table comparing the characteristic timescales calculated for the typical values of three coronal plasma non-uniformities in\nwhich rapidly decaying slow modes have been observed: warm quiescent loops seen in 171 Å (De Moortel 2009), coronal plumes\nseen in 193 Å (e.g. Krishna Prasad et al. 2014), and hot dense loops observed in the Fe XIX channel by SUMER (Nakariakov et al.\n2019). The three points ( T;\u001a) are marked on the plots in Fig. 1, and are the same parameters used for the three plots in Fig. 2. The\ncharacteristic timescale TMis calculated using Eq. (20) for heating model H=\u001a1=2T\u00007=2. The thermal conduction damping time\nTcondB is calculated from Eq. (21) using wavelengths \u0015=22 Mm, 100 Mm and 250 Mm respectively. A range of magnetic field\nstrengths (hence \f) are presented. The characteristic radiative timescale \u001cradis calculated from Eq. (29)\npanel of Fig. 2). This is consistent with the observations of the\nrapid damping of these propagating slow waves.\nThe evaluation of the e \u000bect of the heating and cooling upon\nthe slow wave may easily be confused with the cooling timescale\nof the host plasma, often defined as (see e.g. Eq. (6) in De Moor-\ntel & Hood 2004)\n\u001crad=\rCVT0\nL0(\u001a0;T0): (29)\nAlthough the values of timescales \u001cradin Table 1 look similar to\ntheir counterparts the misbalance damping timescales TM, from\na physical point of view they are independent and describe fun-\ndamentally di \u000berent processes. The quantity \u001cradis associated\nwith the host plasma (not the wave), it depends on the magni-\ntude of the radiative losses, and neglects the influence of coronal\nheating which indisputably exists. The cooling with the char-\nacteristic time \u001cradoccurs when the heating of the plasma is\nsuddenly switched o \u000b. In contrast, the characteristic timescales\n\u001c1;\u001c2and damping time TMare determined by the derivatives of\nthe complete heating /cooling function and the plasma parame-\nters, and characterise the e \u000bect of the wave-induced misbalance\nupon the wave when both cooling and heating processes are still\noperating. Thus, the radiative timescale does not reflect if the ef-\nfect of misbalance between heating and cooling is important for\nthe slow magnetoacoustic wave: the heating /cooling misbalance\nmay have a great e \u000bect even if\u001cradis far from the wave period\n!.\nThe slow waves considered in Table 1 all lie comfortably in\nthe weakly non-adiabatic regime, !\u0002f\u001ccond;\u001c1;\u001c2g\u001d 1. This\nmay not necessarily be true for all slow waves in the corona.Supposing the thermal conduction were strong enough to be in\nthe strongly non-adiabatic regime !\u001ccond\u001c1, the damping time\nshould be calculated using Equation (27) and tends to no damp-\ning by thermal conduction in the isothermal limit (De Moortel &\nHood 2003). In contrast, thermal misbalance may cause strong\ndamping even in the isothermal regime in which the conductive\ndamping is very weak, via the wave’s perturbations to density\nand magnetic field, not temperature. This makes the damping by\nthermal misbalance a viable mechanism for damping in isother-\nmal regimes.\n4. Discussion and Conclusions\nThe importance of non-adiabatic e \u000bects for the damping of slow\nmodes has been shown in many previous works, however in\nsome cases the importance of the presence of steadily operating\ncoronal heating for slow modes has not been realised because\nthe heating term is considered a constant (that is, unperturbed )\n(e.g. De Moortel & Hood 2004). As we have shown, if the coro-\nnal heating mechanism is acting during the oscillation, then the\ndamping e \u000bect of wave-induced misbalance between the heat-\ning and cooling mechanisms can be significant and should not\nbe neglected. We must stress that in our study the energy for\nheating does not come from the slow wave, and is supplied by\nsome other mechanism.\nThe potential for the inclusion of damping by thermal mis-\nbalance to explain the various discrepancies between observed\nslow mode damping and theory is clearly enormous. In the\nweakly non-adiabatic limit, the damping (or amplification) by\nheating /cooling misbalance does not change with wavenumber\nArticle number, page 8 of 11T. J. Duckenfield et al.: The e \u000bect of magnetic field on the damping of slow waves in the solar corona\nFig. 2: Plots of damping time over period (quality factor) against\nmagnetic field strength, for 3 heating models with di \u000bering de-\npendencies on magnetic field strength, which are each coloured\n(Eqs. 23, 18). [ Top] Quality factors calculated for plasma param-\neters corresponding to an upwardly propagating slow wave in a\ncoronal fan loop above a sunspot, T=0:63 MK, electron den-\nsity of ne=1:5\u0002109cm\u00003, and periodicity set to 3.0 minutes\ncorresponding to a wavelength of \u0015=22 Mm. [ Middle ] Qual-\nity factors calculated for a slow wave propagating in a coronal\nplume, T=1:3 MK, ne=5\u0002108cm\u00003, and wavelength \u0015set\nto 100 Mm corresponding to a periodicity of 9 minutes. [ Bot-\ntom] Quality factors calculated for a standing slow wave in a hot\nloop observed by SUMER, T=6:3 MK, ne=1\u00021010cm\u00003,\u0015\n=250 Mm yielding a periodicity of 11 minutes. Grey shading\nmarks the\f > 1 region, and the dotted line in the bottom plot\nmarks a region in which TM<0 and so the e \u000bect of thermal\nmisbalance is destabilising the plasma.\nk, meaning its e \u000bect is universal for di \u000berent length structures. Inthe general non-adiabatic case, there is some dependence upon\nk. The damping by thermal conduction always varies with length\nscale. Thus if the slow wave is damped by both thermal conduc-\ntion and thermal misbalance (e.g. Eq. 23), the dependence of\ndamping time upon frequency would be more complicated than\na straight line of gradient 2 on a log-log plot (as was previously\nexpected since \u001ccond/k\u00002). The inclusion of thermal misbalance\nas a damping mechanism may therefore explain the unexpected\nfrequency dependencies found in Krishna Prasad et al. (2014),\nsince the gradients of best fit on period vs damping length plots\ncan take a range of values depending on the relative contribu-\ntions ofTMandTcondB . Moreover, the di \u000berence seen between\nthe damping of slow waves observed in plumes and those seen\nin sunspots (Krishna Prasad et al. 2014); the unexpected depen-\ndence of damping length upon temperature reported in Krishna\nPrasad et al. (2019); and the change of the damping’s frequency\ndependence with height in Gupta (2014) are naturally explained\nby the variation of \u001cdamp upon plasma- \fand/or the variation\nofTMwith density, temperature and heating function. Further-\nmore, the thermal misbalance will likely introduce a phase shift\nbetween density and temperature in the same manner as ther-\nmal conduction does (see Sec 3.1.2, Owen et al. 2009) which\nmay explain the results in e.g. Krishna Prasad et al. (2018). The\nvariation of this phase shift with the e \u000bectiveness of the ther-\nmal misbalance would have the consequence of making mea-\nsurements of the coronal polytropic index, measured using den-\nsity/temperature phase shifts, also vary with the e \u000bectiveness of\nthe thermal misbalance. As we have demonstrated in this work\n(see also Kolotkov et al. 2020) the e \u000bect of thermal misbalance\nvaries with temperature in the corona, and so one may expect\nthat the polytropic index would vary with temperature as was the\ncase in Krishna Prasad et al. (2019), though further validation is\nwarranted.\nThe damping and dispersion of slow waves in the corona by\nwave-induced thermal misbalance are subject to non-zero \fef-\nfects, some of which are irrespective of the heating /cooling func-\ntion whilst further e \u000bects may occur if there is any dependence\nofQupon magnetic field. Regarding purely non-zero \fe\u000bects,\nwe have found that a wave propagating through a non-zero \f\nplasma will always have a reduced phase speed compared to\na wave in the infinite magnetic field case , whilst its e \u000bect on\nthe wave attenuation depends on the exact plasma conditions. In\nthe case of a damped wave in weakly non-adiabatic plasma, any\nreduction in the magnetic field strength (increase in plasma- \f)\nwill increase the damping time . The e \u000bect of thermal conduc-\ntion is diminished (compared to the infinite field case) as the\nplasma-\fgrows, such that both the isothermal phase speed and\nthe damping rate from thermal conduction are reduced. Regard-\ning the e \u000bects of any dependence of Qupon of magnetic field\nstrength, one important e \u000bect on slow waves which may be im-\nportant even for low- \fplasma, is the stability of the plasma to\nthe isentropic instability (and potentially the thermal instability\nas well). In this work we have focussed on the damping e \u000bect\nof thermal misbalance upon slow waves, enforced by our choice\nof heating model informed by Kolotkov et al. (2020), since such\nheating model(s) is chosen such that slow modes are damped ev-\nerywhere in the corona. This may not be the case everywhere,\nsince this assumption renders the phenomenon of coronal rain\nfrom thermal instability an impossibility, which is evidently not\ntrue. However the topic of instability in a non-zero \fplasma with\nmagnetically dependent heating will be the subject of its own\ndedicated work in the future.\nOne major result of this work is that the infinite magnetic\nfield approximation is good for the quiescent corona when the\nArticle number, page 9 of 11A&A proofs: manuscript no. finitebeta_aa\nmagnetic field strength is above \u001810 G. The magnetic field\nstrengths in coronal structures are di \u000ecult to observe directly,\noften relying on seismological inference. Typical values in trans-\nversely oscillating coronal loops lie in the tens of Gauss (e.g. see\nthe inferences in tables B.1 and B.2 in Arregui et al. 2019), but\nvalues of several kilogauss have been reported at the base of the\ncorona above exceptionally strong sunspots (e.g. Anfinogentov\net al. 2019). Thus it may be concluded that, for the majority of\nthe quiescent corona, the e \u000bects of the dependency of the heat-\ning model upon magnetic field strength may be safely neglected,\nand the infinite magnetic field approximation used instead. The\nsituation may be di \u000berent in particularly hot loops such as af-\nter a flare, where the plasma- \ftends to be higher, and it is in\nthese non-zero \fregions in which any dependence of the heat-\ning/cooling function upon magnetic field strength may be probed\nin a manner analogous to Kolotkov et al. (2020). A natural gen-\neralisation of our study would be the consideration of the heating\nscenario which depend also upon the height above the bottom of\nthe corona.\nThe key results of this paper may be summarised into the\nfollowing:\n1. The dispersion relation governing slow magnetoacoustic\nwaves along an infinitely thin cylinder with non-zero \fwas\nderived. Crucially, two timescales ( \u001c1and\u001c2) that charac-\nterise the e \u000bect of wave-induced thermal misbalance are\ngeneralised for the non-zero \fcase (these timescales were\nfound in the infinite magnetic field case in e.g. Kolotkov\net al. 2019). These are inversely proportional to the combined\nheating /cooling functions’ derivatives with respect to tem-\nperature at constant gaspressure, and with respect to tem-\nperature at constant magnetic pressure respectively (Eq. 14).\n2. The e \u000bect of heating /cooling misbalance in the limit of weak\nnon-adiabaticity was found, applicable for waves in which\nthe exchange of energy with the medium is only mild. Such\nwaves propagate at the tube speed CT, and their amplitude\ndamping may be calculated through Equation (22). In this\nlimit the two characteristic timescales for thermal misbal-\nance may be combined into a single damping time TM,\nwhose e \u000bect on the wave decrement is additive to that from\nthermal conduction (Eqs. (17)–(19)). The sign of TMmay\nbe positive (enhanced damping) or negative (reduced damp-\ning or over-stability). A change in magnetic field strength\n(plasma-\f) will change the damping rate depending on the\nsign of QB. IfQB=0 (or the magnetic field is su \u000eciently\nstrong for the infinite field approximation to be valid), then\na decrease in magnetic field strength (increase in plasma- \f)\nwill always lessen the damping rate. Thermal conduction al-\nways acts to damp the wave, and its e \u000bect is also reduced as\n\fincreases.\n3. The e \u000bect of thermal misbalance in the limit of strong non-\nadiabaticity was found, applicable for waves in which the\nexchange of energy with their medium is extreme (Eqs. (25)–\n(27)). In this limit, the e \u000bects upon the wave decrement\nby parallel thermal conduction and by thermal misbalance\nare not additive. The limited isothermal phase speed in this\nregime is reduced for greater \f.\n4. The damping e \u000bect of wave-induced thermal misbalance\nupon slow magnetoacoustic waves is important for a wide\nrange of coronal conditions, demonstrated through Table 1\nby reason of the heating /cooling misbalance’s damping\ntimescaleTMcoinciding with typical observed coronal slow\nwave periods and damping times. The damping by thermal\nmisbalance is of comparable importance to the damping ef-\nfect by thermal conduction. The di \u000berent physical origins(and therefore di \u000berent parametric dependencies) of these\nomnipresent damping mechanisms may explain the discrep-\nancies reported between observations of slow mode damping\nin the corona and theory.\n5. The quality factors for 3 minute slow mode oscillations\nabove sunspots, slow modes in plumes, and in hot (post-flare)\nloops are estimated, considering both damping both thermal\nconduction and damping by wave-induced thermal misbal-\nance. For su \u000eciently large \fplasma, the damping of slow\nwaves is independent of the heating /cooling functional de-\npendence upon magnetic field. As a rule of thumb, the in-\nfinite magnetic field approximation is valid for studying the\ne\u000bect of thermal misbalance in the quiescent corona for mag-\nnetic field strengths greater than \u001910 G.\nAcknowledgements. The work was supported by the STFC consolidated grant\nST/T000252 /1. D.Y .K. acknowledges support from the budgetary funding of Ba-\nsic Research program No. II.16. V .M.N. acknowledges the Russian Foundation\nfor Basic Research grant No. 18-29-21016. CHIANTI is a collaborative project\ninvolving George Mason University, the University of Michigan (USA), and the\nUniversity of Cambridge (UK).\nReferences\nAfanasyev, A. N., & Nakariakov, V . M. 2015, Astron. Astrophys., 573, A32,\ndoi:10.1051/0004-6361/201424516\nAnfinogentov, S. A., Stupishin, A. G., Mysh’yakov, I. I., & Fleishman, G. D.\n2019, Astrophys. J., 880, L29, doi: 10.3847/2041-8213/ab3042\nArregui, I., Montes-Solís, M., & Asensio Ramos, A. 2019, Astron. Astrophys.,\n625, A35, doi: 10.1051/0004-6361/201834324\nCarbonell, M., Terradas, J., Oliver, R., & Ballester, J. L. 2006, Astron. Astro-\nphys., 460, 573, doi: 10.1051/0004-6361:20065528\nClaes, N., & Keppens, R. 2019, Astron. Astrophys., 624, 1, doi: 10.1051/\n0004-6361/201834699\nDe Moortel, I. 2009, Space Sci. 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J., 886, 2, doi: 10.3847/1538-4357/\nab478f\nZavershinskii, D. I., Kolotkov, D. Y ., Nakariakov, V . M., Molevich, N. E., &\nRyashchikov, D. S. 2019, Phys. Plasmas, 26, 082113, doi: 10.1063/1.\n5115224\nZhugzhda, Y . D. 1996, Phys. Plasmas, 3, 10, doi: 10.1063/1.871836\nArticle number, page 11 of 11" }, { "title": "1312.0300v1.Critical_Field_of_Spin_Torque_Oscillator_with_Perpendicularly_Magnetized_Free_Layer.pdf", "content": "arXiv:1312.0300v1 [cond-mat.mes-hall] 2 Dec 2013Critical Field of Spin Torque Oscillator with Perpendicula rly Magnetized Free Layer\nTomohiro Taniguchi, Hiroko Arai, Sumito Tsunegi, Shingo Tamaru, Hito shi Kubota, and Hiroshi Imamura\nNational Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center,\n1-1-1 Umezono, Tsukuba 305-8568, Japan\nThe oscillation properties of a spin torque oscillator cons isting of a perpendicularly magnetized\nfree layer and an in-plane magnetized pinned layer are studi ed based on an analysis of the energy\nbalance between spin torque and damping. The critical value of an external magnetic field applied\nnormal to the film plane is found, below which the controllabl e range of the oscillation frequency\nby the current is suppressed. The value of the critical field d epends on the magnetic anisotropy, the\nsaturation magnetization, and the spin torque parameter.\nThe self-oscillation of the magnetization in a spin\ntorque oscillator (STO) has been studied extensively be-\ncause of its potential application to spintronics devices\nsuch as microwave generators and recording heads of\nhigh-density hard disk drives [1–10]. The self-oscillation\nof the magnetization is induced when the energy supply\nfrom the spin torque balances with the energy dissipa-\ntion due to the damping [11]. Recently, it was found\nthat the STO consisting of a magnetic tunnel junction\n(MTJ) with a perpendicularly magnetized free layer and\nan in-plane magnetized pinned layer [12–14] showed a\nlarge power ( ∼0.5µW) and a narrow linewidth ( ∼50\nMHz)[15], makingagreatadvancetowardtherealization\nof STO devices.\nPrecise control of the oscillation frequency by the cur-\nrent is necessary for STO application. To this end, it is\nimportant to clarify the oscillation properties of STOs.\nDepending on the magnetization directions of the free\nand pinned layers, STO can be classified into four types.\nThe self-oscillation of the STO was first observed in the\nin-plane magnetized system in 2003 [1]. An MTJ with\nan in-plane magnetized free layer and a perpendicularly\nmagnetized pinned layer was also developed because a\nsignificant reduction in the switching current was ex-\npected [3, 16, 17]. An STO in which both the free\nand pinned layers were perpendicularly magnetized was\ntheoretically studied because its axial symmetry made\nthe analyses easy [18]. Contrary to these three types,\nthe oscillation properties of STO with a perpendicularly\nmagnetized free layer and an in-plane magnetized pinned\nlayer remain unclear.\nIn this letter, we theoretically study the oscillation\nproperties of an STO with a perpendicularly magnetized\nfree layer and an in-plane magnetized free layer based on\nthe analyses of the energy balance between spin torque\nand damping. We find that an external magnetic field\napplied normal to the film plane plays a key role in the\nself-oscillation of the magnetization. A critical value of\ntheappliedfieldexistsbelowwhichthecontrollablerange\nof the oscillation frequency by the current is suppressed.\nThe value of the critical field depends on the perpendic-\nular magnetic anisotropy, the saturation magnetization,\nand the spin torque parameter.pmelectron (I>0)\nspin torquedamping damping spin torquez\nθ\nxHappl\nFIG. 1: Schematic view of the system, where mandpare\nthe unit vectors pointing in the magnetization directions o f\nthe free and pinned layers, respectively. The tilted angle o f\nthe magnetization mfrom the z-axis is denoted as θ. The\narrows indicate the directions of spin torque and damping.\nThe applied field is denoted as Happl.\nThe system we consider is schematically shown in Fig.\n1, where the unit vectors pointing in the magnetization\ndirections of the free and pinned layers are denoted as m\nandp, respectively. The z-axis is normal to the film\nplane while the x-axis is parallel to p. The applied\nfield,Happl, is parallel to the z-axis. The current is de-\nnoted as I, where the positive current corresponds to\nthe electrons flowing from the free layer to the pinned\nlayer. The magnetic energy density of the free layer,\nE=−MHapplmz−[M(HK−4πM)/2]m2\nz, consists of\nthe Zeeman energy and the uniaxial anisotropy energy,\nwhereMandHKare the magnetization and the crys-\ntalline anisotropy along the z-axis, respectively. Because\nwe are interested in the perpendicularly magnetized free\nlayer,HKshouldbe largerthan the demagnetizationfield\n4πM. The energy density has two minima at m=±ez.\nThroughout this letter, the initial state is assumed to be\nm=ez. It should be noted that a trajectory with a con-\nstantmz= cosθcorresponds to the constant energy line\nof this system. The magnetization dynamics is described\nby the Landau-Lifshitz-Gilbert (LLG) equation [19–23],\ndm\ndt=−γm×H−γHsm×(p×m)+αm×dm\ndt,(1)\nwhereγandαare the gyromagnetic ratio and Gilbert2\ndamping constant, respectively. The magnetic field is\ndefined as H=−∂E/(∂Mm). The strength of the spin\ntorque is\nHs=/planckover2pi1ηI\n2eMV(1+λm·p), (2)\nwhereVis the volume of the free layer. Two dimension-\nless parameters, ηandλ, whose ranges are 0 < η <1\nand−1< λ < 1, determine the magnitude of the\nspin polarization and the dependence of the spin torque\nstrength on the relative angle between the magnetiza-\ntions(cos−1m·p), respectively. The relationshipsamong\nη,λ, and other material parameters depend on the the-\noretical model: for example, in the ballistic transport\ntheory in MTJs [21, 23], ηis proportional to the spin\npolarization of the density of state of the free layer and\nλ=η2. The form of eq. (2) is common for spin torque in\nnot only MTJs but also giant magnetoresistive (GMR)\nsystems [22, 24]. In particular, λplays a key role in the\nmagnetization dynamics of this system.\nIn the self-oscillation state, the energy supply by the\nspin torque balances with the energy dissipation due to\nthe damping, and therefore, the magnetization precesses\non the constant energy line. From eq. (1), the energy\nchange due to the spin torque and the damping is de-\nscribed as dE/dt=−MH·(dm/dt) =Ws+Wα, where\nWs=γMHs\n1+α2[p·H−(m·p)(m·H)−αp·(m×H)],\n(3)\nWα=−αγM\n1+α2/bracketleftbig\nH2−(m·H)2/bracketrightbig\n, (4)\nare the work done by the spin torque and the dissipation\ndue to the damping, respectively [25]. By assuming that\nthe magnetization tilts from the z-axis with an angle θ=\ncos−1mzandaveraging dE/dtoveroneprecessionperiod\nτ, we found that the current I(θ) satisfying dE/dt=\n(1/τ)/contintegraltext\ndt(dE/dt) = 0 is [26]\nI(θ) =2αeλMV\n/planckover2pi1ηcosθ/parenleftBigg\n1/radicalbig\n1−λ2sin2θ−1/parenrightBigg−1\n×[Happl+(HK−4πM)cosθ]sin2θ.(5)\nThe oscillation frequency at the angle θisf(θ) = 1/τ=\nγ[Happl+(HK−4πM)cosθ]/(2π). The angle θincreases\nwith increasing the current, which results red-shift of the\noscillation frequency [27]. The critical current for preces-\nsion is defined as Ic= limθ→0I(θ), and is given by\nIc=4αeMV\n/planckover2pi1ηλ(Happl+HK−4πM).(6)\nThe critical current Icdiverges in the limit of λ→0\nbecause when Hsis independent of the relative angle of\nthe magnetization, and when the equilibrium direction\nof the magnetization is perpendicular to p, the energy0.25current, I(θ) (mA) \nangle, θ (deg)0 30 60 90 (a) Happl=0 \n0.260.27current, I(θ) (mA) \nangle, θ (deg)(b) Happl=10 (Oe)current, I(θ) (mA) \nangle, θ (deg)0 30 60 90 (c) Happl=3 (kOe)\n40 \n080 1200 20 60 80 40 0.2650.280.285\n0.275\n0.27Ic\nI>Ic\nFIG. 2: Dependences of I(θ) [eq. (5)] on the tilted angle of\nthe magnetization θfor (a)Happl= 0, (b) 10, and (c) 3 ×103\nOe, respectively, where lim θ→0I(θ) =Ic. The ranges of θin\n(a) and (c) are 0 ≤θ≤90◦while that in (b) is 0 ≤θ≤80◦\nto emphasize the local minimum of I(θ).\nsupply from the spin torque over one precession period is\nzero, making it impossible for the spin torque to induce\nthe magnetization dynamics.\nFigures 2(a)-2(c) show the dependences of I(θ), eq.\n(5), on the tilted angle of the magnetization θwith\nHappl= 0, 10, and 3 ×103Oe, respectively. The values\nof the other parameters are HK= 18.6 kOe, 4πM= 18.2\nkOe,V=π×60×60×2 nm3,η= 0.54,λ=η2,\nγ= 17.32 MHz/Oe, and α= 0.005, which are estimated\nfrom the experiments [15, 28]. Depending on the value\nof the applied field, the dependence of I(θ) onθis dras-\ntically changed, from which the following three distin-\nguishable current dependences shown in Figs. 2(a)-2(c)\nare expected.\nFirst, in theabsenceoftheappliedfield, I(θ) monoton-\nically decreases as the angle θincreases, and remains fi-\nnite in the limit of θ→π/2, as shownin Fig. 2(a). These\nindicate that, once the current magnitude reaches the\ncritical current Ic, the magnetization immediately moves\ntothefilmplane( xy-plane)because I(0< θ < π/ 2)< Ic.\nFigure 3 shows the time evolutions of the component of\nmforI= 0.3 mA> Ic≃0.27 mA. The magnetization\nreachesθ=π/2, and the dynamics stops at m=−ex\nbecause both the field torque and the spin torque, which\nare the first and second terms on the right-hand side of\neq. (1), are zero at this point. Therefore, self-oscillation3\nmz\nmy\nmx\ntime (μs)magnetization 1\n-1 0\n0 1 2\nFIG. 3: Magnetization dynamics in the absence of the ap-\nplied field, where the red, blue, and black lines correspond\ntomx,my, andmz, respectively. The red line ( mx) below 1\nµs overlaps the blue line. The current magnitude is 0.3 mA\n(> Ic≃0.27 mA).\ncannot be realized in the absence of the applied field. It\nshould be noted that since the spin torque prefers the\nanti-parallel alignment of the magnetizations for I >0,\nmstops at −ex, although all torques are also zero at\n+ex.\nSecond, when the magnetic field is smaller than a cer-\ntain value Hc, i.e., 0 < Happl< Hc,I(θ) shows a lo-\ncal minimum, as shown in Fig. 2(b). The theoreti-\ncal formula and the value of Hcare derived below. In\nthis intermediate region, when the current magnitude\nreachesIc, the magnetization moves to a certain angle\nθ0, which satisfies I(θ0) =Ic. For example, in Fig. 2(b),\nθ0≃74◦forHappl= 10 Oe. By increasing the current\nmagnitude from Ic, the tilted angle θcontinuously in-\ncreases from θ0. The self-oscillation can be realized with\nthe frequency f(θ). It should be noted that, below Ic,\nthe power spectrum of the STO peaks at the ferromag-\nnetic resonance (FMR) frequency fFMR=f(θ= 0) =\nγ(Happl+HK−4πM)/(2π) due to the mag-noise effect\n[15]. Since the tilted angle θdiscontinuously changes\natIc, the discontinuity of the oscillation frequency as a\nfunction of the current is expected, as shown in Fig. 4\n(a).\nThird,I(θ) monotonically increases as the current in-\ncreasesfortheappliedfieldsatisfying Hc< Happl. Inthis\ncase, the tilted angle of the magnetization continuously\nincreases as the current increases from Ic. Therefore, the\noscillation frequency changes from fFMRfor the current\naboveIc, as shown in Fig. 4 (b).\nAn important assumption used above is that the tilted\nangle of the magnetization, θ, is constant during the pre-\ncession. Based on this assumption, eq. (5) predicts that\nlimθ→π/2I(θ) =∞forHappl>0, which means that\nthe magnetization cannot reach the film plane. Then,\nthe oscillation frequency saturates to f(θ=π/2) =\nγHappl/(2π) in the large current limit. Also, the control-\nlable range of the oscillation frequency for Hc< Happl\nisf(θ= 0)−f(θ=π/2) =γ(HK−4πM)/(2π), which\ndepends on the perpendicular anisotropy only. Strictly\nspeaking, however, the angle θis not a constant during\nthe precession because the directions of the spin torque1.5\n0.5\n01.0frequency (GHz) (a)\n00 andmx<0 are opposite, as shown in Fig.\n1. When θbecomes close to π/2 by a large current, the\nmagnetization can reach the film plane and stops its dy-\nnamics because the spin torque for mx<0 moves the\nmagnetization closer to the film plane. Therefore, in re-\nality, the solid lines in Figs. 4(a) and 4(b) break at a\ncertain current above which the self-oscillation cannot be\nrealized. The magnitude of such current depends on the\napplied field magnitude. However, the investigation of\nsuch current or field magnitude requires a breakthrough\nof the constant θassumption, and is beyond the scope of\nthis letter.\nThe reason why the existence of the applied field de-\ntermines whether the magnetization can reach θ=π/2\nor not is as follows. The tilted angle of the magneti-\nzationθis determined by the balance between the spin\ntorque and the damping. In the absence of the applied\nfield, the energy dissipation due to the damping, eq. (4),\nrapidly decreases as the angle θincreases, compared with\nthe work done by spin torque, eq. (3), because Wαis\non the second order of the field H= (HK−4πM)mzez\nwhileWsis on the first order of H. Then, once the spin\ntorque overcomes the damping at θ= 0, the energy sup-\nply from the spin torque is always larger than the energy\ndissipation due to the damping during 0 < θ < π/ 2.\nTherefore, the magnetization can reach θ=π/2. Be-\ncause both the energy supply from the spin torque and\nthe energy dissipation due to the damping are zero at4\nθ=π/2, i.e.,dE(θ=π/2)/dt= 0, the magnetization\ndynamics stops at θ=π/2. However, in the pres-\nence of the applied field, the damping can balance with\nthe spin torque at 0 < θ < π/ 2 because of the pres-\nence of the constant term Happlin the magnetic field\nH= [Happl+ (HK−4πM)mz]ez. Therefore, the self-\noscillation of the magnetization with the angle θcan be\nrealized. At θ=π/2, the direction of the spin torque\nis parallel to the film plane, which means that the work\ndone by spin torque is zero. On the other hand, the en-\nergy dissipation due to the damping remains finite, i.e.,\ndE(θ=π/2)/dt=−αγMH2\nappl/(1+α2). Therefore, the\nmagnetization cannot reach θ=π/2.\nThe value of the critical field Hccan be determined by\nthe condition in which dI(θ)/dθ >0 nearθ/greaterorsimilar0, and is\ngiven by\nHc=3λ2\n2−3λ2(HK−4πM), (7)\nwhich is 59 Oe for the above parameters. We empha-\nsize that Hcdepends on the magnetic anisotropy and\nthe spin torque parameter λonly. As discussed above,\nwhen the applied field magnitude is larger than eq. (7),\nthe controllable range of the oscillation frequency by the\ncurrent is γ(HK−4πM)/(2π). On the other hand, when\nHappl< Hc, the controllable range is suppressed because\nofthediscontinuouschangeofthetiltedangleofthemag-\nnetization.\nLet us briefly discuss the relationbetween our previous\nwork [15] and this work. Reference [15] experimentally\ninvestigatedthe currentdependence ofthe oscillationfre-\nquencyofSTO.Becausetheapplied fieldmagnitudeused\nin Ref. [15] (typically, 2 kOe) is much larger than Hc,\nthe continuous change of the oscillation frequency was\nobserved, as shown in Fig. 3 (c) of Ref. [15].\nAt the end of this letter, let us mention that the sit-uation considered here is similar to the switching of the\nperpendicularly magnetized free layer by an in-plane po-\nlarized spin current injected by the spin-Hall effect [29].\nSimilar to the above discussion, in the spin-Hall system,\nthe magnetizationcannot crossoverthe film plane by the\nspin torque only, in principle. Therefore, to assist or pre-\nvent the switching, a magnetic field which has a compo-\nnent along the film plane was applied [29]. In the case of\nSTO discussed in this letter, the field-like torque [30–32],\nwhich is neglected in the above calculation, plays a role\nof a torque due to a magnetic field along the film plane.\nThen, the field-like torque may changes the magnetiza-\ntion dynamics, especially in the zero-field limit. The in-\nvestigation of the effect of the field-like torque will be an\nimportant work in future.\nIn conclusion, we studied the oscillation properties of\nthe STO consisting of a perpendicularly magnetized free\nlayer and an in-plane magnetized pinned layer by analyz-\ning the energy balance between the spin torque and the\ndamping. We found the existence of the critical value\nof the external magnetic field applied normal to the film\nplane,Hc. When the applied field is below Hc, the tilted\nangleofthemagnetizationdiscontinuouslychangesabove\nthe critical current. The controllable range of the oscil-\nlation frequency by the current is suppressed due to the\ndiscontinuity. Above Hc, the controllable range of the\noscillation frequency is γ(HK−4πM)/(2π). The value of\nthe critical field depends on the perpendicular magnetic\nanisotropy, the saturation magnetization, and the spin\ntorque parameter λ.\nThe authors would like to acknowledge H. Naganuma,\nT. Yorozu, H. Maehara, A. 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Phys. 4(2008)\n67." }, { "title": "2102.11117v1.Robust_formation_of_nanoscale_magnetic_skyrmions_in_easy_plane_thin_film_multilayers_with_low_damping.pdf", "content": "Robust formation of nanoscale magnetic skyrmions in easy-plane thin \flm multilayers\nwith low damping\nLuis Flacke,1, 2, aValentin Ahrens,3Simon Mendisch,3Lukas K orber,4, 5Tobias B ottcher,6Elisabeth Meidinger,1, 2\nMisbah Yaqoob,1, 2Manuel M uller,1, 2Lukas Liensberger,1, 2Attila K\u0013 akay,4Markus Becherer,3Philipp Pirro,6\nMatthias Althammer,1, 2Stephan Gepr ags,1Hans Huebl,1, 2, 7Rudolf Gross,1, 2, 7and Mathias Weiler1, 2, 6, b\n1Walther-Mei\u0019ner Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany\n2Physics Department, Technical University of Munich, 85748 Garching, Germany\n3Department of Electrical and Computer Engineering,\nTechnical University of Munich, 80333 Munich, Germany\n4Helmholtz-Zentrum Dresden-Rossendorf e.V., Institute of Ion Beam Physics and Materials Research, 01328 Dresden, Germany\n5Fakult at Physik, Technische Universit at Dresden, 01062 Dresden, Germany\n6Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit at Kaiserslautern, 67663 Kaiserslautern, Germany\n7Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany\n(Dated: February 2021; Revised February 2021)\nWe experimentally demonstrate the formation of room-temperature skyrmions with radii of about\n25 nm in easy-plane anisotropy multilayers with interfacial Dzyaloshinskii-Moriya interaction (DMI).\nWe detect the formation of individual magnetic skyrmions by magnetic force microscopy and \fnd\nthat the skyrmions are stable in out-of-plane \felds up to about 200 mT. We determine the interlayer\nexchange coupling as well as the strength of the interfacial DMI. Additionally, we investigate the\ndynamic microwave spin excitations by broadband magnetic resonance spectroscopy. From the\nuniform Kittel mode we determine the magnetic anisotropy and low damping \u000bG<0:04. We\nalso \fnd clear magnetic resonance signatures in the non-uniform (skyrmion) state. Our \fndings\ndemonstrate that skyrmions in easy-plane multilayers are promising for spin-dynamical applications.\nMagnetic skyrmions [1] attract increasing interest for\ntheir potential use in novel devices for information stor-\nage and processing. Atomic-scale skyrmion lattices [2]\nand low threshold current densities for spin transfer\ntorque movement of skyrmions were reported [3]. The\nroom-temperature stabilization of skyrmions in entirely\nmetallic, thin-\flm heterostructures [4, 5] has been iden-\nti\fed as an important step towards real-world applica-\ntions in information technologies. However, uniting all\nproperties required for applications in a single material\nsystem is challenging. Metallic multilayer systems can\nbe fabricated by sputter deposition techniques and allow\nto stabilize skyrmions at room temperature by the in-\nterface Dzyaloshinskii-Moriya interaction (iDMI) [4, 6].\nThis makes them promising candidates to resolve the\nchallenges imposed by material design. For thin-\flm sys-\ntems, an uniaxial anisotropy has been suggested to be a\nrequirement for skyrmion formation in bulk-DMI [7] and\niDMI [8] systems, which translates to perpendicular mag-\nnetic anisotropy (PMA) in thin-\flm heterostructures.\nHowever, both the large spin-orbit coupling materials re-\nquired to induce iDMI and the very small thickness of the\nferromagnet necessary for the PMA, increase damping of\nthe magnetization dynamics due to spin pumping [9]. In\nturn, the high damping leads to smaller skyrmion veloci-\nties or, alternatively, requires higher current densities for\nfast skyrmion motion [10, 11]. In addition, the PMA mul-\ntilayer skyrmion systems usually is associated with a low\n\feld stability of skyrmions, as the spin textures shrink\nwith increasing magnetic \feld and are annihilated byrather small external \felds of 10 mT - 100 mT [4{6]. The\ntwo challenges, high magnetic damping and low \feld sta-\nbility can be overcome as discussed recently by Banerjee\net al. [12]. They show that switching from uniaxial to a\ntwo-dimensional (2D) easy plane anisotropy allows to sig-\nni\fcantly enhance the \feld robustness of the skyrmions.\nWith this lifted PMA restriction, it is thus possible to\nuse thicker ferromagnetic (FM) layers which still support\nsu\u000eciently iDMI to create skyrmions but result in lower\ndamping due to spin pumping.\nIn this work, we experimentally explore the use of a\n2D easy plane anisotropy in magnetic multilayers for the\nstabilization of skyrmions. The multilayer system based\non Pt, CoFe and Ir also establishes the required iDMI\nand hosts skyrmions with diameter of less than 100 nm\nwhile showing comparatively low damping of their mag-\nnetization dynamics.\nSmall skyrmions in thin-\flm multilayers can be stabi-\nlized by the iDMI, which favors perpendicularly aligned\nneighboring spins S1;2:\nHDMI=\u0000D12\u0001(S1\u0002S2)\nThe sign and magnitude of the iDMI vector D12is deter-\nmined by the involved materials and their stack sequence,\nallowing for additive DMI contributions from the two op-\nposite interfaces of a magnetic thin \flm [4{6, 13, 14].\nThin-\flm multilayer systems employing iDMI for\nskyrmion formation typically rely on PMA. This is re-\n\rected in the condition for the e\u000bective anisotropy Ke\u000b=\nKu\u0000\u00160M2\ns=2>0, which omits higher order spin orbitarXiv:2102.11117v1 [cond-mat.mtrl-sci] 22 Feb 20212\nFIG. 1. a) Sketch of the multilayer system, with a sixfold repetition of the Pt/CoFe/Ir trilayer (numbers give the \flm thickness\nin nm). b) MFM images recorded with OOP magnetic \felds, \feld values are indicated at the top of each frame. With decreasing\n\feld, the magnetic texture shows transitions from a saturated ferromagnetic state (1), to isolated skyrmions (2 ;3), to a dense,\nskyrmion arrangement (4 \u00006) and eventually forms a maze state at zero \feld (7). Individual skyrmions between the spin-spirals\ncan still be found at zero \feld. The three highlighted skyrmions (marked with three colored circles in (7)) are analyzed further\nin Fig. 2 a). Scalebar in all images corresponds to 500 nm.\ncoupling terms and dipolar contributions [7, 8]. Here,\nKu>0 represents an anisotropy with easy axis along the\n\flm normal (out-of-plane, OOP), and the second term\ndescribes the shape anisotropy of the thin \flm \feld with\nthe vacuum permeability \u00160and the saturation magneti-\nzationMs. Recent theoretical calculations however also\npredict the existence of skyrmions with increased \feld\nstability within the Ke\u000b<0 regime [12, 15{17]. Not\nrequiring PMA allows to use thicker ferromagnetic (FM)\nlayers with reduced spin pumping damping contributions,\nwhile iDMI is still su\u000eciently large to enable skyrmion\nformation.\nWe use sputter deposited heterostructues composed\nof [Pt(0:6)=Co25Fe75(1:1)=Ir(0:7)]6multilayers (numbers\ngive the layer thickness in nanometers), as sketched in\nFig. 1 a) to reveal the abovementioned properties and\nbene\fts. Additionally, we fabricated a sample series with\nvarying Ir thickness in order to determine the interlayer\nRKKY coupling. In the following, we denote the spe-\nci\fc Co 25Fe75alloy simply as CoFe. The details of the\nsample fabrications are discussed in the supplementary\ninformation (SI) [18].\nIn order to observe of the formation of skyrmions\nin our multilayers, we use magnetic force microscopy\n(MFM). Fig. 1 b) presents 2 \u0016m\u00022\u0016m MFM phase\ncontrast images recorded with external magnetic \felds\n0 mT\u0014\u00160Hext\u0014309 mT applied in the OOP direc-\ntion. In the following, we discuss the individual MFM\nimages shown in Fig. 1 b). MFM image (1) shows van-ishing contrast, as the sample is magnetically saturated,\nmeaning that all magnetic moments are aligned parallel\nto the applied magnetic \feld. By reducing \u00160Hextbe-\nlow the saturation magnetic \feld, small individual dots\nwith an apparent diameter below 100 nm start to emerge\n(2), which we attribute to skyrmion formation. A \fxed\nchirality of the skyrmions is expected, as we determine\na high interface DMI value of Dint\u00191:86 mJ=m2with\nBLS measurements [19] on reference samples (see SI [18]).\nThis value agrees well with DMI strengths reported for\nsuperlattices with similar materials [13]. The density\nof skyrmions increases further on reducing the magnetic\n\feld magnitude until a dense, unordered arrangement is\nformed (3\u00006). Even lower magnetic \felds eventually\nlead to the formation of a labyrinthine state at rema-\nnence, shown in image (7). However, obviously individ-\nual skyrmions maintain stabilized even at Hext= 0, as\nindicated by the three circles. The multilayer thus hosts\nskyrmions for 0\u0014\u00160Hext\u0014190 mT. This \feld range for\nskyrmion formation is roughly twice the range reported\nfor toKe\u000b\u00150 systems [4, 6, 13, 20, 21].\nIn Fig. 2 we take a closer look at the magnetic \feld\ndependent skyrmion size distribution to elucidate the\nqualitatively di\u000berent behavior compared to PMA sys-\ntems [4, 5, 22]. Fig. 2 a) depicts the azimuthally averaged\nradial pro\fle of the magnetic contrast of the three high-\nlighted skyrmions in image (7) of Fig. 1 b) vs. the radial\ndistance from their center position. Using a Gaussian \ft,\nwe extract the skyrmion diameter das the full-width-half-3\nFIG. 2. a) Azimuthally averaged MFM phase contrast of the\nskyrmions highlighted in Fig. 1 (o\u000bset is used for clarity).\nThe radial pro\fle is \ftted with a Gaussian function, from\nwhich the radius r= (25:4\u00061:6) nm is found at Hext=\n0. b) Apparent radius vs. applied magnetic \feld in OOP\ndirection. The black squares indicate the average value taken\nfrom 28 - 115 skyrmions from images (2 \u00006) of Fig. 1 b) for\na \feld down sweep starting at saturation, whereas, the blue\ncircles indicate skyrmion sizes determined from an up sweep\nstarting at (66\u00066) mT. The y-error represents the standard\ndeviation of the size distribution. The x-error bar indicates\nthe uncertainty of the \feld calibration. The red dotted line is\na guide to the eye.\nmaximum of the MFM signal. We de\fne the apparent\nradius asr=d=2. At\u00160Hext= 0 mT, the three extracted\nradii are about 25 nm. For larger \u00160Hextwe extracted the\nradial pro\fle of 28 - 115 skyrmions from graphs (2 \u00006)\nand plot the average radius and its standard deviation\nvs. the applied \feld. For \u00160Hext<150 mT,rre-\nmains unchanged within experimental uncertainty, while\nrdiverges at the transition between the skyrmion state\nand the ferromagnetic alignment. This behavior is in\ncontrast to the typically observed reduction of rwith\nHextobserved for skyrmions in PMA-based multilay-\ners [4, 5, 13]. We additionally veri\fed experimentally\nthat the skyrmion formation shown in Fig. 1 is non-\nhysteretic within experimental uncertainty, i.e., the same\nskyrmion size is obtained for given \u00160Hext, regardless of\n\feld history (see SI [18]). This indicates a strong in-\n\ruence of the iDMI on the skyrmion formation in our\nsamples.\nWe note that a qualitatively similar radius increase\ncan be found in antiferromagnetically coupled skyrmion\nbilayer systems [23]. However, no antiferromagnetic\n(AFM) coupling was found in our sample. By employ-\ning anomalous Hall e\u000bect (AHE) measurements to ex-\ntract the saturation \feld \u00160HS(see SI [18]) and varying\nthe Ir spacer thickness as shown in Fig. 3 a) the AFM\nand FM RKKY coupling regions can be separated. Ir\nhas been shown to induce strong AFM coupling between\nmagnetic layers in the ultra-thin limit [24, 25]. As the\ncoupling strength varies signi\fcantly with Ir thickness,\nso does the required \feld to overcome the AFM orderand obtain saturation. Within the FM regime, \u00160HS\nstays constant [26]. The sample studied by MFM has\nan Ir thickness tIr= 7\u0017A and is thus FM coupled. This\nrules out AFM coupling as the origin of the observed in-\ndependence of skyrmion size on external magnetic \feld\nmagnitude.\nFIG. 3. a) Saturation \felds \u00160HSof samples with varying Ir\nthicknesstIrare determined by anomalous Hall e\u000bect mea-\nsurements. In the AFM RKKY coupling regime, HScorre-\nlates with the strength of exchange coupling. Within the FM\nregime,HSis independent of coupling strength, and small\npockets in the hysteresis curves allow for a qualitative sep-\naration of the regimes with AFM and FM coupling. Cor-\nresponding samples show a characteristic \feld \u00160HB<\u0016 0HS\nbelow which saturation starts to break down. b) The hystere-\nsis curves from the sample discussed in Fig. 1 are recorded\nby SQUID magnetometry in IP and OOP direction. The\ninsets depict MFM images at the indicated \felds (vertical\ndotted lines) along the descending OOP branch. The IP\ngraph exhibits an easy-plane switching loop with a coerciv-\nity of less than 4 mT. The decrease of magnetic moment for\nj\u00160Hextj<200 mT is attributed to domain formation due to\niDMI. Scalebar in all images corresponds to 500 nm.\nOur observation is in accordance with the theoretical\nwork by Banerjee et al. who predicted a stable skyrmion\nsize for a large \feld range for easy-plane anisotropy thin\n\flms. The contribution of the easy-plane \"compass\"\nanisotropy to the free energy stemming from higher or-\nder spin-orbit coupling terms [12] can possibly explain\nwhy we observe an almost magnetic \feld independent\nskyrmion size. This is further supported by the observa-\ntion of a qualitatively di\u000berent skyrmion size evolution\nin our micromagnetic simulations that do not include\nanisotropies caused by higher order spin-orbit coupling\nterms (see SI [18])\nFor our system, the easy plane anisotropy is veri\fed by\nsuperconducting quantum interference device (SQUID)\nmagnetometry. We show the in-plane (IP) and the\nOOP magnetization curves in Fig. 3 b). The shape of\nthe OOPMvs.Hextloop is characteristic for samples\nwith a maze-state formation, where skyrmions are often\nfound [6, 13, 24]. Due to domain formation, also PMA\nsystems involving labyrinthine arrangement can exhibit4\nFIG. 4. The colorplot shows the real part of the background\ncorrected FMR signal. For \u00160Hext>240 mT applied along\nthe OOP direction we see the characteristic FM response of\nthe CoFe-multilayer. The black squares indicate the micro-\nmagnetic simulation results (see SI [18]). Below the criti-\ncal \feld we see an additional dynamic response, which we\nattribute to the skyrmion background. Dashed lines and\nthe gray area indicate the \feld and its uncertainty, respec-\ntively, at which the correspondingly numbered MFM graphs\nin Fig. 1 were recorded. The Gilbert damping parameter is\n\u000bG= (37\u00061)\u000210\u00003(see SI [18]).\na similar hysteresis curve [13, 24]. We verify the easy\nplane anisotropy by performing IP SQUID magnetome-\ntry measurements. The IP behavior qualitatively di\u000bers\nfrom the OOP curve and also from Ke\u000b\u00150 systems,\nwhere the IP curve shows hard-axis behavior [13, 20]. We\nobserve an easy-plane switching loop with a coercivity of\nless than 4 mT. The gradual decrease of magnetization\nfor\u00160Hext\u0014200 mT is attributed to domain formation.\nThe formation of a maze state at remanence despite easy-\nplane anisotropy is attributed to the strong iDMI [27].\nWe have so far demonstrated that stable skyrmions\ncan form in easy-plane anisotropy thin \flms with DMI.\nWe now turn to the dynamic magnetic properties of such\nthin \flm multilayers. To this end, we determine the dy-\nnamic microwave response of the sample by placing it\non a coplanar waveguide (CPW) and recording the mi-\ncrowave transmission S21through the CPW with a vector\nnetwork analyzer (VNA) [28, 29]. The frequency depen-\ndent background is removed by calculating the normal-\nized \feld-derivative [30] @DS21=@H and Re(@DS21=@H)\nis shown in Fig. 4.\nFor\u00160Hext'240 mT, we observe the ferromagnetic\nresonance (FMR) of the fully aligned CoFe magnetiza-\ntion, which follows the known OOP Kittel equation for\nthin \flms [31]:\n2\u0019fres=\u00160\r(Hext\u0000Me\u000b): (1)\nHere,\ris the gyromagnetic ratio, Hextis the applied\nmagnetic \feld, and Me\u000b=Ms\u0000Hkwith the perpendic-\nular anisotropy \feld Hk= 2Ku=(\u00160MS). We \ft the datain Fig. 4 to extract resonance \felds and linewidths as de-\ntailed in the SI [18]. We determine \u00160Me\u000b= (201:8\u0006\n0:1) mT, again con\frming the easy plane anisotropy. We\nused the determined parameters, as input variables for\nmicromagnetic simulations and were able to qualitatively\nreproduce the hysteresis curves as well as the formation\nof chiral spin textures within the \u00160Me\u000b>0 system (see\nSI [18]). From \ftting the linewidth of the signal for a\nbroad frequency range we extract the Gilbert damping\nparameter and obtain \u000bG= 0:037 (see SI [18]). This\nvalue is almost an order of magnitude lower than those re-\nported on skyrmion host multilayers of Pt/CoFeB/MgO\n(\u000b= 0:5) [32] and Ir/Fe/Co/Pt ( \u000b= 0:1) [13]. Another\ncommon FM material used for skyrmion multilayers is\nCo, where \u000b\u00190:3 is determined for layer thicknesses\nt<1 nm sandwiched between Pt layers [33]. As pointed\nout by Fert et al. , such a decrease in magnetic damp-\ning is expected to result in substantial improvement for\nskyrmion motion [11].\nFor\u00160Hext<240 mT,fresvs.Hextshows a qualita-\ntively di\u000berent behavior with fresincreasing for decreas-\ningjHextj. This is attributed to a rotation of Mtowards\nthe \flm plane and the formation of a non-uniform mag-\nnetic texture in accordance with our MFM and SQUID\nmagnetometry data. We attribute the resonance ob-\nserved for \u00160Hext<240 mT to the dynamic preces-\nsion of the magnetic moments in the quasi-uniform back-\nground (see Fig. 1) and not to spin dynamics within the\nskyrmions. This is in accordance with the vanishing am-\nplitude of the resonance towards Hext= 0, where no\nuniform background remains. Our observation is qual-\nitatively di\u000berent to that made by Montoya et al. [34],\nwhere various resonances of dipolar skyrmions in DMI-\nless Fe/Gd superlattices were observed. Even though the\nmagnetic resonance data look similar, Montoya et al. ob-\nserved a phase transition of a \feld polarized state di-\nrectly to a phase with coexisting stripes and skyrmions.\nOur MFM data, however, reveal a smooth transition\nfrom parallel alignment, over individual skyrmions, to\na dense skyrmion arrangement and eventually a maze\nstate formation. Our results are also qualitatively di\u000ber-\nent from a study of PMA iDMI thin-\flms with magnetic\nskyrmions [35], where magnetic resonance is also ob-\nserved in a phase with coexisting stripes and skyrmions.\nWe explicitly con\frm a larger magnetic \feld range\nwith skyrmion existence than comparable PMA sys-\ntems [4, 6, 13, 20, 21] by MFM. Our measurements show a\nconstant skyrmion size over a wide range of external mag-\nnetic \felds (0 mT to \u0018150 mT) as predicted by Baner-\njeeet al. [12]. This size evolution opposes the typical\ndecrease of skyrmion size with increasing magnetic \feld\ndue to Zeeman energy contributions [4]. We explicitly\nrule out antiferromagnetic coupling between layers as an\nexplanation for the unusual size dependence by varying\nthe spacer thickness. The easy-plane anisotropy in our\n\flms is veri\fed by (SQUID) magnetometry and broad-5\nband magnetic resonance. Microwave spectroscopy ex-\nperiments reveal a happily low damping within the sys-\ntem, making it promising for more detailed studies of\nspin dynamics. Compared to other experimentally real-\nized easy-plane skyrmion systems like polar magnets [36],\nbulk- and interface DMI thin \flms [37, 38], as well\nas DMI-less Fe/Gd superlattices [39], the investigated\nmultilayer system uniquely provides room-temperature\nnanoscale skyrmions in a low damping host material for\na large \feld range. The signi\fcant reduction of damping\ncan result in faster skyrmion motion [11]. The high sta-\nbility of the skyrmion size over a broad range of external\nmagnetic \felds may be also bene\fcial for devices.\nAcknowledgments. { We acknowledge \fnancial support\nby the Deutsche Forschungsgemeinschaft (DFG, Ger-\nman Research Foundation) via WE5386/4-1, WE5386/5-\n1 and Germany's Excellence Strategy EXC-2111-\n390814868.\naluis.\racke@wmi.badw.de\nbweiler@physik.uni-kl.de\n[1] S. Muhlbauer, B. Binz, F. 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A major goal of the br oader heliophysics\ncommunity is to identify the physical mechanisms responsible for the dissipation of the\nturbulence andto quantifythe consequentrateofplasmaheating .One ofthe mechanisms\nproposed to damp turbulent fluctuations in weakly collisional space a nd astrophysical\nplasmas is electron Landau damping. The velocity-space signature o f electron energiza-\ntion by Landau damping can be identified using the recently developed field-particle\ncorrelation technique. Here, we perform a suite of gyrokinetic tur bulence simulations\nwith ion plasma beta values βi= 0.01,0.1,1,and 10 and use the field-particle correlation\ntechnique to characterize the features of the velocity-space sig natures of electron Lan-\ndau damping in turbulent plasma conditions consistent with those obs erved in the solar\nwind and planetary magnetospheres. We identify the key features of the velocity-space\nsignatures of electron Landau damping as a function of varying plas maβito provide a\ncritical framework for interpreting the results of field-particle co rrelation analysis of in\nsituspacecraft observations of plasma turbulence.\n1. Introduction\nPlasma turbulence plays an important role in the transport of energ y throughout\nthe heliosphere, governing the conversion of the energy of large- scale magnetic fields and\nplasma flows into heat of the plasma species. Under the hot and diffus e conditions typical\nof heliospheric plasmas, collisionless interactions between the electr omagnetic fields and\nthe individual plasma particles govern the rate of damping of the tur bulent fluctuations\nand the resulting particle energization. Landau damping (Landau 19 46) is one of several\ncollisionless mechanisms that may transfer electromagnetic energy to particle energy in\na weakly collisional plasma, and it has been proposed to be of consider able importance\nin dissipating the turbulent energy in the solar wind (Leamon et al.1998; Leamon et al.\n1999; Quataert 1998; Howes et al.2008a; Schekochihin et al.2009; Howes et al.2011a;\nHowes 2011; TenBarge & Howes 2013; Howes 2015; Li et al.2016). Recent analyses of\nobservations in the Earth’s turbulent magnetosheath plasma from theMagnetospheric\nMultiscale (MMS) mission (Burch et al.2016) have provided the first in situevidence of\nelectron Landau damping acting to dissipate turbulence in any space plasma (Chen et al.\n2019; Afshari et al.2021). Understandinghow energyis transferredto plasmapartic lesin\nturbulent dissipation is a major goal in heliophysics, needed to explain the long-standing\nproblem of how the solar corona is heated to temperatures above a million Kelvin (Edl´ en\n1943, 1945) in which turbulence is thought to play a crucial role (With broe & Noyes\n†Email address for correspondence: sarah-horvath@uiowa.e du2\n1977; Heyvaerts & Priest 1983; Parker 1988; Klimchuk 2006; Cran mer 2009; Chandran\n2010), or to predict the observed non-adiabatic temperature pr ofile of the expanding\nsolar wind (Richardson & Smith 2003). A variety of mechanisms are like ly to be involved\nin the dissipation process; descriptions of these can be found in a re cent literature review\n(Verscharen et al.2019).\nIn the solar wind, observations show that the incompressible (Alfv´ enic) component\ndominatesturbulentfluctuationswithintheinertialrange(Bruno & Carbone2005;Alexandrova et al.\n2008). As the turbulent cascade of Alfv´ enic fluctuations reache s the ion kinetic length\nscales, finite Larmor radius effects lead to the development of a non zero component of\nthe electric field parallel to the local mean magnetic field. Particles wit h a parallel ve-\nlocity near the parallel phase velocity of an Alfv´ en wave can resona ntly interact with\nthe parallel electric field of the wave and thereby gain or lose energy . This results in\nnet energization of the particles if, in the region of the resonance, the initial slope of\nthe distribution function is negative (positive) for particles with v/bardbl>0 (v/bardbl<0). This\ncollisionless transfer of energyfrom the electric field to the particle s is entropy conserving\nand thus reversible. However, in practice arbitrarily weak Coulomb c ollisions acting on\nsmall velocity scales in the particle distribution function serve to the rmalize the energy,\nrealizing thermodynamic heating of that plasma species (Howes et al.2006; Howes 2008;\nSchekochihin et al.2009). Landau damping is capable of energizing ions at perpendicular\nscalesnearthe ion Larmorradius, k⊥ρi∼1, and electronsat smallerperpendicularscales\nk⊥ρi/greaterorsimilar1 (Leamon et al.1999; Quataert 1998; Howes et al.2008a; Schekochihin et al.\n2009; Howes et al.2011a; Howes 2011; TenBarge & Howes 2013; Howes 2015; Told et al.\n2015; Kiyani et al.2015; Li et al.2016).\nRecent analyses of burst-mode measurements from the Magnetospheric Multiscale\n(MMS)mission(Burch et al.2016)usingthefield-particlecorrelationtechnique(Klein & Howes\n2016; Howes et al.2017; Klein et al.2017) haveprovidedthe first direct observationalev-\nidence of electron Landau damping acting to damp turbulence in Eart h’s magnetosheath\nplasma (Chen et al.2019). Following this groundbreaking study, a survey of the field-\nparticle correlation signatures in 20 MMSintervals was conducted, revealing evidence\nfor electron Landau damping in 19 of the intervals. In one-third of t hose intervals, the\nstudy indicated that this mechanism may be responsible for a dominan t fraction of the\ndamping required to terminate the solar wind turbulent cascade (Af shariet al.2021).\nMotivatedbytheseresults,wepreviouslyusedahigh-resolution,g yrokineticsimulation\nto re-createthe field-particle correlation signature of electron L andau damping in plasma\nconditions matching the first MMSinterval that yielded an in situobservation of elec-\ntron Landau damping (Chen et al.2019). The study demonstrated that electron Landau\ndamping is observable in gyrokinetic turbulence simulations and confir med the complex\nfeatures of Landau damping in the regime of dispersive kinetic Alfv´ e n waves (KAWs)\n(Horvath et al.2020). In particular, we showed how the bipolar signatures of Land au\ndamping of KAWs at different phase velocities may appear together in velocity-space to\nproduce field-particle correlations with multiple bipolar signatures or with superimposed\nsignatures that create a single, broadened structure. Further more, we demonstrated that\nthe broadband frequency content in velocity-space fluctuations increases the difficulty of\nobserving signatures in broadband turbulence due to the inability to choose the corre-\nlation interval τsuch that it averages over an integer multiple of all of the wave mode s\nrepresented (Klein & Howes 2016; Klein et al.2017; Howes et al.2017; Horvath et al.\n2020)\nIn this paper, it is our goal to follow up on our previous work by applyin g the field-\nparticle correlation technique to a suite of gyrokinetic simulations. T hrough this, we will\nconstruct a framework for identifying the presence of electron L andau damping in situCharacterizing the Signature of Electron Landau Damping 3\nthroughout the inner heliosphere. Though the fiducial field-partic le correlation signature\nof ion Landau damping is well understood (Klein & Howes 2016; Klein et al.2017), our\nfirst study showed that electron Landau damping produces more c omplicated signatures\nduetothedispersivenatureofKAWsatspatialscalessmallerthant heionLarmorradius,\nk⊥ρi/greaterorsimilar1 (Horvath et al.2020). Further, the range of wavenumbers and frequencies ove r\nwhich KAWs are dispersive changes with the ion plasma beta βi= 8πniTi/B2—the ratio\nof the ion thermal to magnetic pressure—and with the ion-to-elect ron temperature ratio\nTi/Te.Bothofthesequantitieschangethroughouttheheliosphere.In ordertounderstand\nhow the signature of electron Landau damping will appear at differen t locations in the\ninner heliosphere, therefore, we create four high-resolution gyr okinetic simulations at\ndifferent ion plasma beta βivalues and analyze these simulations with the field-particle\ncorrelation technique, described in Sec. 2.1. In addition, we create and analyze a set of\nsingle kinetic Alfv´ en wave simulations to aid in interpreting the field-pa rticle correlation\nsignatures in turbulence. Both of these simulation types are descr ibed in more detail in\nSec. 2.2. The insights gained from the single-wave runs are discusse d in Sec. 3, and we\npresent results from the turbulent simulations in Sec. 4, and discus s the implications for\nobserving these signatures in situin Sec. 5.\n2. Methods\n2.1.The Field Particle Correlation Technique\nThefield-particlecorrelationtechnique(Klein & Howes2016;Howes et al.2017;Klein et al.\n2017) uses single-point measurements of the velocity distribution f unction and electro-\nmagnetic fields to measure the rate of change of the particle phase-space energy density\nws(x,v,t) =1\n2msv2fs(x,v,t) and to identify particle energization mechanisms in weakly\ncollisional plasmas. Energization mechanisms interact with particles in specific regions\nof velocity-space, thereby creating characteristic patterns as net energy is transferred\nbetween fields and particles. Correlating field and particle data toge ther generates these\ncharacteristic velocity-space signatures and, in doing so, makes p ossible the identifica-\ntion of the physical mechanism at work and determines the net chan ge in phase-space\nenergy density. The field-particle correlation technique has been s uccessfully applied to\nunderstand particle energization in a wide range of fundamental pla sma physics pro-\ncesses: (i) collisionless damping of electrostatic (Klein & Howes 2016; Howeset al.2017)\nand electromagnetic (Klein et al.2017; Howes 2017; Howes et al.2018) plasma waves;\n(ii) kinetic instabilities (Klein 2017); (iii) damping of electromagnetic plas ma turbulence\nthrough ion Landau damping (Klein et al.2017; Howes et al.2018; Klein et al.2020),\nion transit-time damping (Arzamasskiy et al.2019; Cerri et al.2021), electron Landau\ndamping(Chen et al.2019;Horvath et al.2020;Afshari et al.2021;Horvath et al.2022),\nion cyclotron damping (Klein et al.2020; Afshari et al.2023), electron magnetic pump-\ning (Montag & Howes2022), ion stochastic heating (Arzamasskiy et al.2019; Cerri et al.\n2021); (iv) electron energization in collisionless magnetic reconnect ion (McCubbin et al.\n2022); and(v) ionand electronaccelerationin collisionlessshocks(J unoet al.2021,2023;\nBrownet al.2023).\nThe field-particle correlation is derived by multiplying the Vlasov equat ion for species\nsbymsv2/2 to obtain an equation for the rate of change of phase-space ene rgy density\n∂ws(x,v,t)/∂t. Time-averaging the electric field term in the resulting equation over a\ncorrelation interval τyields the form of the field-particle correlation. The contribution\nfrom the component of the parallel electric field yields the net energ y transfer from Lan-\ndau damping at agiven spatial point r0, and is called the parallel field-particle correlation4\n(Howeset al.2017; Klein et al.2017):\nCE/bardbl(r0,v,t;τ) =1\nτ/integraldisplayt+τ/2\nt−τ/2−qsv2\n/bardbl\n2∂fs(r0,v,t′)\n∂v/bardblE/bardbl(r0,t′)dt′. (2.1)\nTaking a cylindrical coordinate system in velocity-spacethat is aligne d with the direction\nof the local mean magnetic field, ( v/bardbl,v⊥,θ), we can integrate the parallel correlation\nover the gyrophase θto obtain the parallel field-particle correlation in gyrotropic phase\nspace,CE/bardbl(v/bardbl,v⊥,t), with the correlation (time-averaging) interval τcentered at time\ntand taken at a specified spatial location r0(e.g., see contour plots in Fig. 7). Below\nwe suppress the r0argument, with the implication that each field-particle correlation is\ncomputed using data from a single spatial point. The resulting energ y transfer between\ntheparallelelectricfieldandtheparticlesincludestwocontributions inthecaseofLandau\ndamping: (i) the conservative, oscillating energy transfer betwee n the waves and the\nparticles that is associated with undamped waves; and (ii) the secula r energy transfer\nfromthe parallelelectricfieldtothe particlesthat isassociatedwith collisionlessdamping\nviatheLandauresonance.Inthecaseofplasmaturbulence,theo scillatoryenergytransfer\nis often larger in amplitude than the secular energy transfer. Apply ing a time average\nover a suitably chosen correlation interval cancels out the large-a mplitude oscillatory\nenergy transfer to reveal the smaller amplitude signature of the c ollisionless damping.\nForturbulencesimulations,onemaychooseacorrelationinterval τthatislongerthanthe\nperiod of waves at the simulation domain scale (Klein & Howes 2016; How eset al.2017;\nKleinet al.2017). For spacecraft observations, in which there is a broadban d frequency\nspectrum, including wave periods much longer than the MMSburst-mode intervals used\nfor the analysis, one may employ a high-pass filter to the electric field to eliminate\nthe large-amplitude, low frequency contribution to undamped wave motion, isolating the\nwaveperiodsatkineticscaleswherecollisionlessdampingisexpectedt oarise(Chen et al.\n2019; Afshari et al.2021).\nThe velocity-space signatures of the particle energization can be p resented in a variety\nof useful ways. To determine if the signature is coherent over time ,CE/bardbl(v/bardbl,v⊥,t) may be\nintegrated over the perpendicular velocity v⊥to obtain the reduced parallel correlation\nCE/bardbl(v/bardbl,t). This rate of energization vs. v/bardblcan be plotted as a line plot at a single time\nt(e.g., see lower panels of Fig. 7), or the v/bardbl-dependence at each time can be stacked to\ngenerate a timestack plot ofCE/bardbl(v/bardbl,t) (e.g., see Fig. 8, central contour plots), revealing\nthepersistenceoftheenergizationmechanismovertime.Todeter minetheneteffectofthe\nenergization mechanism as a function of v/bardbl, integrating the reduced parallel correlation\nover time yields the time-integrated, reduced parallel correlation CE/bardbl(v/bardbl) (e.g., lower\npanels of Fig. 8), providing the full time-averaged change in the pha se-space energy\ndensitywsas a function of v/bardbl. Alternatively, one may instead integrate the reduced\nparallel correlation over v/bardblto obtain the total rate of change of spatial energy density at\nposition r0due to the parallel electric field, ( ∂Ws/∂t)E/bardbl(e.g., left-hand panels of Fig. 8).\nNote that ( ∂Ws/∂t)E/bardbl=je,/bardblE/bardblis simply the rate of electromagnetic work done by the\nparallel electric field on the electrons.\n2.2.AstroGK simulations\nThe Astrophysical Gyrokinetics Code, AstoGK(Numata et al.2010), was used to gener-\nate a suite of four plasma turbulence simulations spanning expected conditions in the\nsolar wind of the inner heliosphere. These four simulations use a realis tic mass ratio,\nmi/me= 1836,in orderto properlyseparatethe ion andelectron scaleswit hin the turbu-\nlent cascade, have a unity temperature ratio Ti/Te= 1, and sweep through four different\nplasma beta values, βi= 0.01,0.1,1,10. Each AstroGK simulation evolves the comple-Characterizing the Signature of Electron Landau Damping 5\n100101102\nk⊥ρi10−310−1101103EB⊥∝k−3.5\n⊥\n∝k−3.2\n⊥\n∝k−2.9\n⊥\n∝k−2.8\n⊥βi= 0.01\nβi= 0.1\nβi= 1\nβi= 10\nFigure 1. Perpendicular magnetic energy spectra ( EB⊥) of the four simulations, as a function\nofk⊥ρi, time-averaged over a representative portion of the full si mulations. The vertical dotted\nlines mark the driving scale ( k⊥ρi= 2) and the smallest fully resolved scale ( k⊥ρi= 42). By-eye\npower law fits are shown as references for each spectrum.\nmentary, perturbed gyrokinetic distribution function, gs(v/bardbl,v⊥,t), where gs(v/bardbl,v⊥) =\nhs(v/bardbl,v⊥)−qsF0s\nT0s∝angbracketleftφ−v⊥·A⊥∝angbracketrightRs,hsis the gyrokinetic distribution function, and ∝angbracketleft∝angbracketrightRs\ndenotes a gyroaverageabout a fixed gyrocenter coordinate Rs(Numata et al.2010). We\nchoosegs(v/bardbl,v⊥,t) at a fixed probe position r0as the distribution function fs(r0,v,t)\nin our computation of the parallel field-particle correlation by (2.1).\nWe choose the physical ( x,y,z) and velocity ( λ,ε) space resolution in each simulation\nto be (nx,ny,nz,nλ,nε,ns) = (64,64,32,128,32,2), where nsis the number of plasma\nspecies and velocity-space is partitioned into a gyrotropic grid by pit ch angle λ=v2\n⊥/v2\nand energy ε=v2. Velocity vis normalized to the thermal velocity, vts=/radicalbig\n2Ts/ms, for\neach species. Each simulation domain is an elongated 3D Eulerian slab wit h dimensions\nL/bardbl×L2\n⊥, given by L/bardbl= 2πa0andL⊥=πρi, where the ion thermal Larmor radius is\nρi=vti/Ωi, and the ion cyclotron frequency is given by Ω i=qiB0/mic. The elongation\nof the simulation domain is characterized by the small gyrokinetic exp ansion parameter,\nwhereǫ∼ρi/a0≪1 (Howes et al.2006, 2008 b, 2011b). For these parameters, the fully\nresolved range of perpendicular scales is given by 2 /lessorequalslantk⊥ρi/lessorequalslant42, or 0.05/lessorsimilark⊥ρe/lessorsimilar1,\ncapturing a broad range of dispersive kinetic Alfv´ en wave frequen cies (see Figure 2).\nNote that, for each timeslice, the fields are output at each point on the high-resolution\nspatial grid but the electron velocity distribution function is output only at 24 individual\n“probes” spread throughout the simulation box. Sixteen are in the mid-plane ( z= 0) and\ntheremainingeightalongthe z-axis(x=y= 0),asindicatedinFigure2ofHorvath et al.\n(2020).\nAn oscillating Langevin antenna (TenBarge et al.2014) drives counter-propagating,\nperpendicularly polarized Alfv´ en waves with ( kxρi,kyρi,kza0) = (2,0,±1) and (0 ,2,±1)\nto launch a turbulent cascade for each simulation. The driving frequ enciesω0and decor-\nrelation rates γ0for each driven mode are listed in Table 1, where ω≡ω/k/bardbl0vAand\nk/bardbl0= 2π/L/bardbl. The amplitude of the driving is chosen in accordance with critically bal-\nanced kinetic Alfv´ en wave (KAW) turbulence (Howes et al.2008a, 2011a) in order to\nself-consistently produce a strong turbulent cascade that begin s at the driving scale of\nk⊥ρi= 2. The turbulent cascadetransfersenergyto eversmallerlengt h scales,ultimately\nreaching k⊥ρi= 42where the resolvedelectron collisionlessdamping is sufficiently str ong\nto terminate the cascade, enabling a turbulent steady state to be achieved. To prevent\na build-up of fine-scale structure in velocity space that is unresolve d by our finite grid,6\nβiTi/Temi/me−γ0/ω0ω0ωsdωmaxT0Tmin∆t/T0νe,fνi,f\n0.01 1 1836 0.17 2.044 - 5.57 3.074 1.128 68.9 0.4 0.1\n0.1 1 1836 0.063 2.082 2.845 14.9 3.018 0.422 13.6 0.6 0.3\n1 1 1836 0.042 1.576 3.856 38.56 3.987 0.163 2.63 1.2 1.2\n10 1 1836 0.19 0.6832 0.989 13.55 9.197 0.464 1.01 1.0 2.0\nTable 1. Parameters of the four turbulent simulations.\nwe set non-zero collisionalities for the ions and electrons. This allows t he energy that is\ntransferred to the particles via collisionless Landau damping to ther malize. These values\nmay be changed somewhat during the course of a simulation to ensur e good conservation\nof energy. In Table 1, we display the final values of the electron and ion collisionalities\nin normalized units ( νe,f≡νe,f/k/bardbl0vA;νi,f≡νi,f/k/bardbl0vA) for each simulation. These\nvalues may be compared with the maximum damping rate ( −γmax, not listed) and the\ndriving-scale wave frequency ( ω0) to ensure that νsis large enough to dissipate fine-\nscale structure in velocity space but small enough such that the dr iving scale remains\ncollisionless.\nWe present the average spectra of the perpendicular magnetic fie ld energy for the four\nsimulations in steady state in Fig. 1. The spectra have been separat ed in amplitude in\norder to highlight the individual power law slopes, and by-eye fits (da shed black lines)\nare presented alongside the data. At βi= 10 and βi= 1, the slope of the perpendicular\nmagnetic energy spectra agree well with dissipation-range spectr a that have been ob-\nserved in βi/greaterorsimilar1 plasmas near 1 AU, EB⊥∝k−2.8\n⊥(Sahraoui et al.2013). The simulation\nspectrasteepen asbetadecreases,up to EB⊥∝k−3.5\n⊥atβi= 0.01.Suchlowbetaplasmas\nare rarely observed near Earth (Wilson et al.2018), and dissipation-range simulations\nof turbulence at this beta are uncommon, though one recent work studying the low- β\nlimit of plasma turbulence also finds steep magnetic energy spectra ( Zhouet al.2023).\nAdditionally, our steepest spectra is within the range of early dissipa tion-range observa-\ntions (Leamon et al.1998) and the trend we observe of the spectra steepening as bet a\ndecreases is consistent with the electrons transferring more ene rgy from the turbulence\nwithin the dissipation range in the case of more significant damping (To ldet al.2015).\nThe linear phase velocity normalized to the electron thermal velocity ,ω/k/bardblvte, and\nthe normalized damping rate −γ/ωfor the KAWs over the parameter range of each of\nthese four simulations are computed using the linear Vlasov-Maxwell dispersion rela-\ntion solver PLUME (Klein & Howes 2015) and are presented in Fig. 2. The normalized\nfrequencies ω0≡ω0/(k/bardblvA) of the domain scale KAWs with wavevector components\nk⊥0ρi= 2πρi/L⊥= 2 and k/bardbl0a0= 2πa0/L/bardbl= 1 are listed in Table 1. Note that the\nAlfv´ en wave and electron thermal velocity normalizations of the fr equency are related by\nω/k/bardblvte=ωβ−1/2\ni(Ti/Te)1/2(me/mi)1/2. Also, the normalized Alfv´ en wave frequencies\nhave departed from the MHD limit of ω= 1, since we are modeling kinetic Alfv´ en waves\nwithk⊥ρi>1 (Fig. 2(a)). The maximum frequency KAW found in each simulation ca n\nbe determined from Fig. 2 (listed in Table 1 as ωmax), along with the frequency corre-\nsponding the onset of strong damping at −γ/ω∼0.1 (Fig. 2(b)), which is listed as ωsd\nin Table 1 if this frequency is above that of the domain scale.\nThe post-saturation length of each of the four simulations, ∆ t/T0, is listed in Table 1\nin terms of time normalized to the period of the domain-scale kinetic Alf v´ en waves,\nT0= 2π/ω0. Note that the timescale separation between the faster electron thermalCharacterizing the Signature of Electron Landau Damping 7\n(a) (b)\nFigure 2. Linear Vlasov-Maxwell dispersion relation for plasmas wit hTi/Te= 1,\nmi/me= 1836, and βi= 0.01 (cyan), 0 .1 (green), 1 (blue), and 10 (red). (a): Parallel phase ve-\nlocity normalized to the electron thermal velocity ω/k/bardblvte. (b): Normalized damping rate −γ/ω,\nwith the onset of strong damping ( −γ/ω/greaterorsimilar0.1) marked by a horizontal dashed line.\nvelocity and the slower Alfv´ en velocity, vte/vA=β1/2\ni(Te/Ti)1/2(mi/me)1/2, decreases\nfor lower values of βi. Since the maximum timestep in AstroGK is determined by the\nelectron thermal velocity, lower βisimulations are substantially less computationally\ncostly to run.\nIn addition to the nonlinear turbulence simulations described above, we also use\nAstroGK to model a total of20 individually-driven,damped kinetic Alfv´ en wav esat a sin-\ngle value of k⊥ρi. Foreachvalue of βi= 0.01,0.1,1,10we use the oscillatingLangevinan-\ntenna to separately drive linear modes at k⊥ρi= 2,4,8,16,32, where ( kxρi,kyρi,kza0) =\n(k⊥ρi,0,1) in each case. The parallel spatial grid and velocity-space resolut ions for each\nwave are ( nz,nλ,nε,ns) = (32,128,32,2), where the variables and normalization are the\nsame as described for the turbulence runs. Each driven, damped lin ear wave was evolved\nfor a minimum of 10 wave periods.\n3. Velocity-Space Signatures of Single Kinetic Alfv´ en Waves\nTo develop a framework for interpreting the velocity-space signat ures of electron Lan-\ndau damping in turbulence simulations, we first present the results o f applying field-\nparticle correlation analysis to the suite of single KAWs described in Se c. 2. The analysis\nof single, damped waves eliminates the complications which arise from o verlapping bipo-\nlar signatures of Landau damping in broadband turbulence, as illustr ated in Figure 1\nof Horvath et al.(2020). Thus, we are able to more easily identify trends in the behav -\nior of the field-particle correlation signatures of this mechanism as t he ion plasma βiis\nchanged. Later, these clearly identified features will aid our interp retation of signatures\nin the turbulence simulations, which we present in Sec. 4.\nFig.3shows(a)agyrotropicsignature CE/bardbl(v/bardbl,v⊥,t)and(b)atimestackplot CE/bardbl(v/bardbl,t)\nof the field-particle correlation for an individual KAW simulation with βi= 0.1 and\nk⊥ρi= 8. In panel (a), the characteristic bipolar form of the field-part icle correlation\nsignature of Landau damping (Klein & Howes 2016; Klein et al.2017; Howes et al.2017;\nChenet al.2019; Horvath et al.2020) is clearly visible in both the colormap plot and in\nthe line plot of the time-integrated, reduced parallel correlation CE/bardbl(v/bardbl) shown below. In8\nFigure 3. (a) Gyrotropic field-particle correlation signature CE/bardbl(v/bardbl,v⊥) att/T0= 6.75 and (b)\ntimestack plot CE/bardbl(v/bardbl,t) of a driven, damped wave simulation with βi= 0.1,k⊥ρi= 8, and\na correlation interval of τ/T0= 1. In the time-integrated lower panel in (b), the full-widt h at\nhalf-maximum of the positive portion of the bipolar signatu re is indicated.\nthis example, the correlation interval τis equal to a full wave period ( τ/T0= 1) and the\ngyrotropiccorrelationin (a) is showncenteredattime t/T0= 6.75.The significantenergy\ntransfer shown in the bipolar velocity-space signature is localized in v/bardblaround the paral-\nlel phase velocity of the damped wave (vertical dashed line, vph,/bardbl/vte=ω/k/bardblvte= 0.51),\nwith a loss of phase-spaceenergy density (blue) at v/bardbl< vph,/bardbland a gain of of phase-space\nenergy density (red) at v/bardbl> vph,/bardbl. The dominance of the positive region indicates a net\ntransfer of energy from the parallel electric field to the resonant electrons. As expected\nfor Landau damping, the bipolar structure does not vary in the per pendicular direction\napart from an exponential decrease at high perpendicular velocitie s due to the equilib-\nrium velocity distribution. Therefore, we may integrate over all of v⊥to yield the total\nchange in wsas a function of v/bardblat the spatial point being considered (lower panel). This\nlower panel is then calculated for all times and combined to create (b ) atimestack plot of\nthe correlation CE/bardbl(v/bardbl,t). The panel beneath the timestack plot is the time-integrated,\nreduced parallel correlation CE/bardbl,e(v/bardbl). The panel to the left of the timestack is the rate of\nchangeoftheelectronspatialenergydensityduetotheparallele lectricfield,( ∂We/∂t)E/bardbl,\nwhich is calculated by integrating the correlation over v/bardbland is equal to je,/bardblE/bardbl.\n3.1.Velocity-Space Signature Location and Wave Phase Velocity\nA key feature of a bipolar signature of Landau damping is its concent ration in velocity-\nspace about the parallel phase velocity of the damped wave (Klein & H owes 2016). The\nform of the signature is consistent with flattening of the distributio n function around the\nresonant velocity, as predicted by quasilinear theory of Landau da mping, and indicates\nthat a resonant mechanism is producing the energization (Howes et al.2017). Through\nthe sweep of βiandk⊥ρiparameter space provided by our 20 single-wavesimulations, we\nshow here quantitatively that the location of the zero crossing in th e bipolar signature of\nLandau damping (the velocity v/bardbl/vtewhereCE/bardbl(v/bardbl) = 0) is indeed very well correlated\nwith the parallel phase velocity of the damped wave. These two quan tities are plotted\nagainst each other in Fig. 4(a), and the results lie almost entirely on t he line of v/bardbl=\nvph. The linear Vlasov-Maxwell dispersion relation (Stix 1992; Klein & Howe s 2015) was\nused to find vphfor each wavemode, and the zeros are found from gyrotropic plot s with\ncorrelation interval τ/T0= 1. Note that the exception to the close agreement between\nv/bardblandvph,/bardbloccurs for the signatures with βi= 10 (circles). This error likely arises due\nto a combination of the small KAW phase velocity at high βiand the finite v/bardbl-resolution\n∆v/bardbl/vte= 0.133 of the simulations. The slight deviation of the other signatures f rom the\nv/bardbl=vphline appears to fall within the error due to finite v/bardbl-resolution.Characterizing the Signature of Electron Landau Damping 9\nFigure 4. (a) Velocity-space signature location v/bardbl/vtevs. parallel wave phase velocity vph,/bardbl/vte\nfor the linear runs. (b) Full-width, half maximum (FWHM) val ue of the bipolar signatures\nvs. the damping rate −γ/ωof the wave. Plasma βiandk⊥ρiare differentiated using marker\nstyles and color, respectively.\n3.2.Velocity-Space Signature Width and Wave Damping Rate\nThe second observation we are able to make using the single-wave sim ulations is that\nthe width of the field-particle correlation signatures changes with b othβiandk⊥ρi. The\nwidth tendstovarydirectlywith k⊥ρi(forinstance,seethefivesingle-wavesignaturesfor\nβi= 1 shown in Appendix A) and indirectly with βi. Similarly, the damping rate of the\nKAWs increaseswith k⊥ρiand decreaseswith βi, suggestingapossible relationship.Note\nthat this connection with k⊥ρiis true specifically for dispersive KAWs, where vphgener-\nally increases monotonically with k⊥ρi. Indeed, when the signature width—quantified by\nthe full-width, half maximum (FWHM) of the positive portion of the bipo lar structure\n(e.g., see the lower panel of Fig. 3)—is plotted against the correspondin g normalized\ndamping rate of the KAW in the simulation, −γ/ω, a logarithmic dependence emerges,\nroughly described by FWHM ∝1\n6ln(−γ/ω). This relationship is shown in Fig. 4(b).\nThoughlogarithmicdependencies areweak,broadeningofthe signa turesdueto increased\ndamping rates has plausible theoretical support. Specifically, it may be analogous to the\nfrequency-space broadening of the Fourier transform of a damp ed sine wave. A pure sine\nwave yields a Dirac delta function in frequency space under a Fourier transform, while a\ndamped wave yields a peak of finite width due to its decrease in amplitud e over time.\n3.3.The Effect of Weighting by v2\n/bardblin the Parallel Field-Particle Correlation\nAnother feature that becomes clear through the suite of single-w ave signatures is the\neffect of v2\n/bardblin the definition of the field-particle correlation, Eq. (2.1). Since par ticle ve-\nlocities are normalized by the species thermal velocity, this factor e nhances the velocity-\nspacesignaturefor v/bardbl/vte>1andsuppressesthe signaturefor v/bardbl/vte<1.†This behavior\nis due to the mathematical form of the field-particle correlation, wh ich arises from the\nderivation of the rate of change of phase-space energy density. Therefore, the enhance-\nment or suppression of the energy transfer rate due to v2\n/bardblis a physical effect that can\nhinder the ability to recognize bipolar velocity-space signatures. Sp ecifically, for damped\n†Note that the inflection point of vte= 1 is peculiar to our normalization choice. However,\na similar effect would be observed for any choice of normaliza tion.10\n0.0 0.5 1.0 1.5 2.0\nv∝bardbl/vte010 Amplitude\nions(a)\nv2\n∝bardbl\n2\nqsE∝bardbl∂f\n∂v∝bardbl\nCE∝bardbl\n0.0 0.5 1.0 1.5 2.0\nv∝bardbl/vte010 Amplitude\nelectrons(b)\nFigure 5. Diagram of the effect of v2\n/bardbl-weighting on bipolar signatures of Landau damping.\nwaves that are resonant with particles at low parallel velocities with v/bardbl/vte<1, the\npreferential suppression of signal at small v/bardblcan mask the characteristic bipolar struc-\nture that identifies the energization as being due to Landau damping . Fig. 5 presents\ntwo sketches of this effect, showing the separate magnitudes of t he weighting by v2\n/bardbl/2\n(cyan dashed) and the remaining parallel electric field correlation qsE/bardbl∂f/∂v /bardbl(blue dot-\ndashed) along with their product (red solid). The relative magnitude s of the negative\nand positive regions of the bipolar signature of CE/bardblwhen (a) vph/vte>1 is contrasted\nwith the relative magnitudes when (b) vph/vte<1. For the Landau damping of ions, the\nwave resonance is typically vph,/bardbl/vti≃1, as in Fig. 5(a), where the bipolar structure of\nthe signature of Landau damping is easily observed. For electron La ndau damping, how-\never, the resonances are typically less than the electron thermal velocityvph,/bardbl/vte<1,\nand asymptotically reach vph/vte→1 at the smallest spatial scales ( k⊥ρe→1), as seen\nin Fig. 2(a). Additionally, as βiincreases, the parallel phase velocity ω/k/bardblvteacross all\ndissipation range scales becomes an even smaller fraction of vte. Therefore, for electron\nLandau damping signatures in high βiplasmas, the resonant velocities are often deep\nin the center of the electron distribution function, so the negative portion of the signa-\nture can be significantly suppressed and become nearly unobserva ble, as in Fig. 5(b).\nFor our twenty single-wave simulations binned in parallel velocity to a r esolution of\n∆v/bardbl= 0.133vte, thev2\n/bardbl-weighting made the negative portion of the bipolar signatures\ndifficult or impossible to observefor resonantwaveswith vph/lessorequalslant0.3vte: the casefor eleven\nout of the twenty signatures. ‡\n3.4.Non-resonant feature\nFinally, we use the single-wave simulations to address a prominent, no n-resonant feature\nthat begins to appear in the field-particle correlations for βi/greaterorsimilar1. Specifically, we see a\nlarge-amplitude feature across all parallel velocities that is antisym metric about v/bardbl= 0.\nFig. 6 shows the non-resonant feature for two individual wave simu lations at perpen-\ndicular wavenumber k⊥ρi= 8 and (a) βi= 1 and (b) 10. At βi= 1, the non-resonant\nfeature is visible but subdominant to the resonant, bipolar signatur e which is centered\naroundv/bardbl/vte= 0.133. Atβi= 10, however, the amplitude of the non-resonant feature\nhas completely eclipsed the amplitude of the resonant feature, whic h is centered close to\nzero atv/bardbl/vte= 0.018.\nOur explanation of the source of this non-resonant feature are e lectron density oscilla-\n‡When the factor of v2\n/bardblis removed from the correlation in such cases, the bipolar st ructure\nbecomes clear.Characterizing the Signature of Electron Landau Damping 11\nFigure 6. Timestack field-particle correlation of driven, damped kin etic Alfv´ en waves at\nk⊥ρi= 8 with (a) βi= 1 and (b) β= 10 and τ/T0= 1 for both. The bipolar signature of\nLandau damping appears in (a) along with an antisymmetric, n on-resonant feature. In (b), the\nnon-resonant feature is dominant.\ntions,δne, that arise naturally in the turbulent plasma at βi/greaterorsimilar1. A local spatial density\nperturbation has a Maxwellian δf, which is even in vand therefore contributes an odd\nvariationin v/bardblto thecorrelationdue tothe velocityderivativeofthe distribution f unction\nin Eq. (2.1). The effect of a density perturbation on the parallel field -particle correlation\nis illustrated in a diagram shown in Figure 5(b) of McCubbin et al.(2022).AstroGK\nevolves the perturbed electron distribution function (Numata et al.2010), which in our\nkinetic Alfv´ en wavesimulations containsdisturbances due to reson antfield-particle inter-\nactions and due to non-resonant density oscillations. A density per turbation may appear\nin the field-particle correlation at high βidue to a combination of two factors. First,\nasβiincreases, the relative magnitude of the compressive density pert urbation to the\nresonant perturbations grows as the KAWs become more compres sible (TenBarge et al.\n2012; Matteini et al.2020) and as the phase velocity of the damped KAW decreases\nand moves deeper into the core of the electron distribution functio n. Second, in order\nfor the compressive signature to appear in the field-particle corre lation, it must have an\nappropriate phase-offset from the parallel electric field fluctuatio n. At low βi,E/bardbland\n(∂fe/∂v/bardbl)δneare approximately out of phase, φ≃π/2; asβiincreases, the phase offset\nshifts and the product of E/bardbland (∂fe/∂v/bardbl)δnecontributes more significantly to the field-\nparticle correlation. Appendix B discusses the density perturbatio n and how it appears\nin the field-particle correlation in greater detail. In Section 5.5.1, we p resent a technique\nfor removing this feature from the data.\n4. Turbulence analysis\nNext, we present the results of the field-particle correlation analy sis of the four full\nturbulence simulations. The features of the bipolar signatures of e lectron Landau damp-\ning that were isolated in the suite of damped individual waves — including variations\nin signature width, suppression of the negative portion of the bipola r signatures at low\nresonant velocity, and the presence of a non-resonant feature arising in βi= 1 and 10\n— are also present. To illustrate some of the commonly observed fea tures, we present\nexamples of the gyrotropic parallel field-particle correlation CE/bardbl(v/bardbl,v⊥,t) in Fig. 7 and\nof the timestack plot of CE/bardbl(v/bardbl,t) in Fig. 8.\n4.1.Gyrotropic Field-Particle Correlations\nFig. 7 shows six example gyrotropic field-particle correlations, two f rom each of the\nturbulence simulations at βi= 0.01,0.1 and 1. We exclude βi= 10, since signatures are\nnot easily observed in this case, given that the v/bardbl-resolution of our simulations is binned\nat ∆v/bardbl/vte= 0.133 while the largest fully resolved wave mode in the βi= 10 simulation12\nhas a phase velocity around vph/vte∼0.1. Though the positive portion of the damping\nsignature can been seen in some cases, the negative portion is unre solved (and would be\nstrongly suppressed due to the factor of v2\n/bardbl). Further, the large relative magnitude of the\nnon-resonant feature dominates the field-particle correlations a tβi= 10.\nEach example plot in Fig. 7 shows a contour map of CE/bardbl,e(v/bardbl,v⊥), the gyrotropic\ncorrelation, that has been averaged over a correlation interval o f length τ, included in\nthe caption in terms of both the largest ( T0) and smallest ( Tmin) KAW periods resolved\nwithin each simulation, and centered at some time t/T0(displayed in the upper left-hand\ncorner). The range of phase velocities at which we expect to see sig natures of Landau\ndamping are marked with vertical dotted lines. The inner vertical line s correspond to\nthe magnitude of the phase velocity of the lowest frequency waves in this range: having\na period of either T0(the driving-scale wave) or Tmax(the wave at the onset of strong\ndamping) for βi/greaterorequalslant0.1. This minimum damped frequency occurs when the damping rate\nreaches a threshold of −γ/ω= 0.1, which we take to be the onset of strong damping.\nFig. 2(b) shows that for βi= 0.1 and 1, this wave is above the driving-scale wave of\nk⊥ρi= 2. For βi= 10, the initial peak above −γ/ω= 0.1 is due to ion Landau damping;\ntherefore, we take the onset of strong electron damping to be ar oundk⊥ρi∼3. The\nvertical dotted lines at higher velocities correspond to the highest frequency wave that\nwe expect to undergo Landau damping (with period Tmin), found simply by the taking\nthe peak of the curve in Figure 2(a). The phase velocities of these t wo limiting cases\nare marked for both upward ( v/bardbl/vte>0) and downward ( v/bardbl/vte<0) wave propagation.\nEach line plot in the lower panels of Fig. 7 displays the reduced correlat ion integrated\nover allv⊥,CE/bardbl,e(v/bardbl).\nThe top row of Fig 7 shows gyrotropic correlations from the βi= 0.01 turbulent\nsimulation. Panel (a) is from a probe in the z= 0 plane of the simulation box, calculated\nusing a correlation interval of length τ≃17.2T0(τ≃47.0Tmin) and shown centered at\ntimet/T0= 43.52.This example containsa pair of bipolar signatures:one at positive and\nthe other at negative parallel velocity, indicating damping of KAWs pr opagating in both\ndirections relative to the background magnetic field. As shown in Fig. 4(a), the location\nof the zero crossing in the integrated correlation CE/bardbl(v/bardbl) in the lower panel corresponds\nto the phase velocity of the damped wave. Thus, the upward-prop agating damped wave\nhas a lower phase velocity than the downward propagating damped w ave. However, the\nsignature at v/bardbl/vte>0 is significantly wider than the one at v/bardbl/vte<0. This seems to\ncontradict the conclusion of Fig. 4(b), which predicts a direct relat ionship between phase\nvelocity and signature width for constant βi. However, the v/bardbl/vte>0 signature has a\nsubtle, divided positive peak. This may indicate that the signature is a superposition of\nthe damping signatures of two waves closely spaced in parallel phase velocity such that\ntheir individual bipolar signatures cannot be resolved. Instead, th e two signatures merge\ninto a single, broadened peak (Horvath et al.2020). This interpretation is supported by\ngyrotropic plots centered at earlier times and by the timestack plot of the correlation at\nthis probe point. The plot in panel (b) is an example of an imbalanced sig nature, which\noccurs when damping is present for only upward or downward propa gating waves at a\ngiven time. In this case, an upward-propagating wave produces a b ipolar signature but\nno clear resonant damping signature is visible for v/bardbl/vte<0. This correlationis averaged\nover the same interval τ, but is taken at a probe point along the z-axis (z∝negationslash= 0) and is\ncentered at an earlier time.\nThe second rowofFig. 7 containscorrelationsfrom the βi= 0.1 simulation, taken from\ntwo different probe points along the z-axis (z∝negationslash= 0). Each is averaged over a correlation\ninterval of τ= 2.47T0(17.7Tmin) and is centered at t/T0= 10.49. Note that the\nvelocity range marked by the pairs of vertical dotted lines has shift ed toward smallerCharacterizing the Signature of Electron Landau Damping 13\nFigure 7. Gyrotropic parallel field-particle correlations CE/bardbl,e(v/bardbl,v⊥) from different probes in\nthe (a,b) βi= 0.01 (τ/T0≃17.2, τ/T min≃47.0), (c,d) βi= 0.1 (τ/T0≃2.47, τ/T min≃17.7)\nand (e,f) βi= 1 (τ/T0≃2.43, τ/T min≃35.0) simulations. The features of the velocity-space\nsignatures are described in the text.\nv/bardbl, reflecting the shift toward smaller parallel phase velocities in the dis persion relation\nasβiincreases, as shown in Fig. 2(a). In (c), the example correlation sh ows signatures\nof damped KAWs propagating in both directions along the backgroun d magnetic field.\nForv/bardbl/vte>0, there is evidence of two bipolar signatures, spaced far enough a part to\nbe resolved in v/bardblspace. The higher-frequency signature is centered around the u pper\nvertical dotted line, indicating that the damped wave is at the maximu m phase velocity\nthat we expect in this simulation. This signature is also wider than the s ignature at\nsmallerv/bardbl, consistent with our theoretical expectations illustrated in Fig. 4( b). Note\nalso that the lower resonant-velocity has a significantly suppresse d negative portion of\nits bipolar signature that is difficult to observe at this color scale, con sistent with the\neffect ofv2\n/bardblweighting as illustrated in Fig. 5. For v/bardbl/vte<0, one signature is visible just14\nbelow the dotted line marking the magnitude of the most highly damped phase-velocity.\nThe relative width of this signature is between that of the two upwar d-propagating wave\nsignatures, as expected. Panel (d) shows an example of another imbalanced signature\nwith a clear velocity-space signature of electron Landau damping fo r only downward-\npropagatingwaves.In this case,there aretwobipolarstructure s:one centeredat v/bardbl/vte≃\n−1 and another at v/bardbl/vte≃ −0.25. Again, the low-velocity signature is strongly affected\nby thev2\n/bardblweighting; however, this fainter signature is persistent througho ut the whole\nsimulation. The structures in the positive velocity region of the plot d o not show a clear\nbipolar signature indicative of electron Landau damping of upward pr opagating kinetic\nAlfv´ en waves, but rather may simply be the remnants of oscillatory fluctuations that are\npoorly cancelled out by the average over our chosen correlation int ervalτ.†\nThe third row of Fig. 7 shows gyrotropic correlations from the βi= 1 simulation,\nfrom two different probe points in the midplane ( z= 0). These correlations are averaged\nover an interval τ= 1.43T0(τ= 35.0Tmin). As before, the vertical dotted lines mark\nout a range of resonant parallel phase velocities that are yet near er tov/bardbl= 0. The\nlargest expected velocity is now at v/bardbl/vte<1, meaning that the v2\n/bardbl-weighting will act\nto suppress all of the signatures relative to any incomplete cancella tion that appears at\nv/bardbl/vte>1. These examples show how the non-resonant feature begins to d ominate the\ncorrelations at βi/greaterorsimilar1, and this feature is particularly prominent in panel (e). Despite\nthis, a faint bipolar signature of an upward-propagating wave can s till be seen in the\ncolor map of this example at the shown scale, though it has become ind istinguishable in\nthe reduced correlation due to the overwhelming magnitude of the n on-resonant feature.\nAs discussed for the damped individual waves, this feature arises d ue to the increased\ncompressibilityofKAWsat βi/greaterorsimilar1,anddue tothe changingphaseoffset φ∝negationslash=π/2between\nE/bardbland the compressive density fluctuation in ∂f/∂v /bardbl. In many cases, this non-resonant\nfeature hinders the observation of bipolar signatures at βi/greaterorsimilar1, though at times it can\nbe removed so that the signatures may be observed via a folding tec hnique discussed\nin Section 5.5.1. At the time of the correlation in panel (f), the non-r esonant feature is\nsmaller in amplitude relative to the resonant signatures, and a bipolar structure is visible\n(with a somewhat attenuated negative portion) at v/bardbl/vte≃0.3.\nIn summary, this section presents one of the key results of this pa per, the gyrotropic\nvelocity-space signature of electron Landau damping CE/bardbl(v/bardbl,v⊥), as exemplified for βi=\n0.01 in Fig. 7(a) at v/bardbl/vte≃ −1.2 and (b) at v/bardbl/vte≃+0.75, forβi= 0.1 in Fig. 7(c) at\nv/bardbl/vte≃ −0.8 and +1 .1 and (d) at v/bardbl/vte≃ −1.0, and for βi= 1 in Fig. 7(f) at v/bardbl/vte≃\n+0.4. These qualitative and quantitative features of the velocity-spa ce signatures as a\nfunction of the ion plasma βiprovide an essential framework for identifying electron\nLandau damping in both kinetic turbulence simulations and spacecraf t observations.\n4.2.Timestack Plots of the Field-Particle Correlation\nTo observe how the gyrotropic velocity-space signatures of elect ron energization in Fig. 7\nevolve over time, we take advantage of the fact that these gyrot ropic signatures do not\nstrongly depend on v⊥(other than the exponential decrease with increasing v⊥, a con-\nsequence of the underlying Maxwellian equilibrium velocity distribution) to integrate\n†Note that it is impossible to completely cancel out oscillat ory signal from field-particle cor-\nrelations in the case of broadband turbulence, since only os cillations from constant amplitude\nwaves with periods of τ(or integer multiples of τ) will exactly cancel. Furthermore, turbu-\nlence is intermittent in space and time, meaning that the mag nitude of the remnant, oscillatory\nsignal that does not cancel out over the correlation interva l in broadband turbulence is expect-\ned—and generally observed—to be a highly variable value. Th is imperfect cancellation leads to\nan effective amplitude threshold for the identification of el ectron Landau damping.Characterizing the Signature of Electron Landau Damping 15\nFigure 8. Timestacks of the reduced field-particle correlations CE/bardbl,e(v/bardbl) from the simulations\nwith plasma (a) βi= 0.01 (τ/T0≃17.2, τ/Tmin≃47.0), (b) 0.1 (τ/T0≃4.94, τ/Tmin≃35.3),\n(c) 1 (τ/T0≃1.43, τ/Tmin≃35.0), and (d) 10 ( τ/T0≃0.06, τ/Tmin≃1.21). The correlations\nin panels (a) and (b) are calculated from the same probe point s as those in Figure 7(b) and (c),\nrespectively.\noverv⊥and generate a timestack plot of reduced parallel correlation CE/bardbl(v/bardbl,t). These\ntimestack plots are invaluable for determining the persistence of a s ignature over time\nand for establishing an overall view of the net energy transfer at a given location. In\nFig. 8, we present one timestack plot from a single probe in each of th e four simulations\nat (a)βi= 0.01, (b) 0 .1, (c) 1, and (d) 10. In each, we integrate over the full duration\nof the simulation to obtain the time-integrated, reduced parallel co rrelation CE/bardbl(v/bardbl) at\nthe probe position (lower panels); we also integrate over v/bardblto obtain the rate of change\nof the spatial energy density ( ∂Ws/∂t)E/bardbl=je,/bardblE/bardbldue to the parallel electric field.\nIn Fig. 8(a), we show the timestack from the βi= 0.01 simulation at the same probe\npoint as the gyrotropic correlation in Fig. 7(b) and averaged over t he same correlation\ninterval τ/T0≃17.2, τ/Tmin≃47.0. Note that the v/bardbl/vte>0 signature observed\natt/T0= 26.18 in the gyrotropic plot was present at the beginning of the averag ing\ninterval and persisted for over 30 domain-scale wave periods. Oth er signatures (of both\nupward and downward propagating waves) appear later in the simula tion. The proximity\nof these signatures in velocity-space can make it difficult to distinguis h their bipolar\nform due to overlap with signatures at other parallel phase velocitie s in addition to the\nv2\n/bardblsuppression of the bipolar form. The timestack from the βi= 0.1 simulation in (b) is\ntaken from the same probe point as the gyrotropic plot in Fig. 7(c) b ut averaged over\na longer correlation interval, τ/T0≃4.94, τ/Tmin≃35.3. Att/T0= 10.49—the central16\ntime of the correlation interval for the gyrotropic plot—the three signatures seen in the\ngyrotropic plot are clearly visible. The timestack shows that each of these signatures are\npresent through the whole simulation, but two of the three shift so mewhat in velocity,\nleading to the overlap of signatures at slightly different resonant pa rallel velocities when\nintegrated over the full simulation (lower panel).\nIn Fig. 8(c), we plot a timestack from the βi= 1 simulation, taken at a midplane\n(z= 0) probe that is not represented in Fig. 7, but averaged over the same correlation\nintervalτ/T0≃1.43, τ/Tmin≃35.0. At this value of βi, the broad non-resonant feature\nhas become prominent in the correlation,though the loweramplitude resonantsignatures\ncan still be seen at small parallel velocities. For instance, at v/bardbl/vte≃ −0.2 a signature\nis visible in the range 1 .2< t/T 0<2, which creates an obvious positive region in the\nlower, time-integrated panel. Note that the negative portion of th is bipolar signature is\nnot visible in the timestack at this color scale and does not appear in th e time-integrated\ncorrelation.At many probe points, such evidence of a damping signa ture at lower parallel\nvelocities than the non-resonant feature is much more difficult to ob serve. When βiis\nincreased further, these difficulties are exacerbated, as illustrat ed in Fig. 8(d) for the\nβi= 10 simulation. This correlation is taken from a midplane probe, and av eraged over\na mereτ/T0≃0.06, orτ/Tmin≃1.21. The correlation interval is a very small fraction\nof the the driving scale wave period T0, so we would not expect to see bipolar signatures\nin the timestack due to these waves. The full interval of this simulat ion is long enough\nto giveτ/T0≃1.01 (shown in the lower panel), so in order to display a timestack at this\nplasma beta we have chosen a low value of τ/T0for the contour plot. As in the case of\nβi= 1, at some probe points in the βi= 10 simulation evidence of energization due to\nLandau damping can be found by observing positive bumps near the r ange of expected\nresonant velocities in the time-integrated correlation. However, t he increased amplitude\nof the non-resonant feature with respect to the resonant damp ing signatures, coupled\nwith the finite resolution of our parallel velocity grid, make damping sig natures nearly\nimpossible to observe clearly in correlations for βi= 10. It is worthwhile noting here\nthat, at higher ion plasma beta values βi/greaterorsimilar1, ion Landau and transit-time damping are\nexpected toplayasignificantrolein thedissipationofturbulent ener gyatscales k⊥ρi∼1\n(Kleinet al.2017), so electron Landau damping of turbulent energy that reac hes the\nkinetic Alfv´ en wave cascade at k⊥ρi≫1 is expected to be less significant in the overall\ndissipation of the turbulent cascade.\n5. Discussion: Features of the Velocity-Space Signatures of Electron\nLandau Damping as a Function of Plasma βi\nIn the previous sections, we have demonstrated general qualitat ive and quantitative\nfeatures that are present in the velocity-space signatures of ele ctron Landau damping\nas a function of the ion plasma βifor linear kinetic Alfv´ en waves. These indicate the\ngeneral trends that should be visible in the velocity-space signatur es generated by turbu-\nlence dissipation measured at different locations throughout the inn er heliosphere. Next,\nwe presented example velocity-space signatures from our suite of four kinetic plasma\nturbulence simulations to illustrate how these general trends appe ar in the context of\nbroadband turbulent energy dissipation via the mechanism of electr on Landau damping.\nHere, we recap these features and discuss implications for making in situobservations of\nelectron Landau damping.Characterizing the Signature of Electron Landau Damping 17\n5.1.Imbalanced Velocity-Space Signatures\nSince the earliest studies of turbulence in magnetohydrodynamic (M HD) plasmas, it\nwas recognized that the turbulent cascade is mediated by nonlinear interactions be-\ntween Alfv´ en wavepackets propagating up and down the magnetic field (Iroshnikov\n1963; Kraichnan1965; Sridhar & Goldreich1994;Goldreich & Sridhar 1995). These “col-\nlisions” between counterpropagatingAlfv´ en wavepacketshave b een invoked as the funda-\nmentalbuildingblockofastrophysicalplasmaturbulence,withtheo retical(Howes & Nielson\n2013),numerical(Nielson et al.2013),andexperimental(Howes et al.2012,2013;Drake et al.\n2013) confirmation of the underlying physics. Further work has illus trated the role of the\nAlfv´ en wave collisions in the self-consistent development of curren t sheets in kinetic\nplasma turbulence (Howes 2016; Verniero et al.2018; Verniero & Howes 2018) and the\nrole of Landau damping in the dissipation of those current sheets (H oweset al.2018).\nWhen the turbulent fluctuations havemoreAlfv´ enicenergyflux pr opagatingin onedirec-\ntion along the local magnetic field than the other direction—known as imbalanced tur-\nbulence(Lithwick & Goldreich 2003; Lithwick et al.2007; Beresnyak & Lazarian 2008;\nChandran2008;Markovskii & Vasquez2013),orturbulencewith n onzerocrosshelicity—\nit is predicted to alter the dynamics of the turbulent cascade (Meyr andet al.2021;\nSquireet al.2022), and therefore will likely also impact the dominant mechanisms o f its\ndissipation.\nEven if the turbulence is balanced on longer timescales, during short periods of time\nlocal regions of imbalanced turbulence may develop. Averaged over space and time, such\nimbalanced regions are not thought to effect the overall energy sp ectra of the turbulent\ncascade (Perez & Boldyrev 2009). Further, in this work we find tha t one may measure\na local dominance of wave damping in one direction or another even if t he turbulence\nremains relatively balanced locally. The field-particle correlation analy sis of our suite of\nturbulence simulations frequently finds asymmetric signatures of e lectron Landau damp-\ning,e.g., Fig. 7(b). An inspection of the timestack plots in Fig. 8 clearly shows that the\ndamping of the turbulent fluctuations variessignificantly in time over the evolution ofthe\nsimulations, with upward or downward propagating fluctuations dom inating the energy\ntransfer to the electrons over intervals of time from a few to tens of characteristic wave\nperiods. This is consistent with the results of our previous simulation s (Horvath et al.\n2020) and with in situobservations in the magnetosheath (Afshari et al.2021). Using\nMMSdata, Afshari et al.(2021) connected the imbalance in the wave damping with the\ndominant direction of the Poynting flux in the data interval. In our su ite of simulations\npresented here, however, we find that the overall flux of electro magnetic energy remains\nroughly balanced at all times. This is not surprising since the turbulen ce is driven by ap-\nproximately equal-amplitude, oppositely-directed Alfv´ en waves. N onetheless, on shorter\ntimescalesand atindividual probepositions inthe simulations,local imb alancesofdamp-\ning arise that lead to the asymmetric signatures that are frequent ly observed in these\nsimulations and in MMSobservations (Afshari et al.2021).\n5.2.Temporal Intermittency and the Overlap of Velocity-Space S ignatures\nAs was found in previous field-particle correlation studies of electro n Landau damping\nin simulations (Horvath et al.2020, 2022), we find that electron damping signatures in\nbroadband, dispersive KAW turbulence contain more structure th an the simple bipolar\nsignature of a single, damped wave. This was illustrated in Fig. 7(c), w here the time-\nintegrated signature at v/bardbl>0 is composed of at least two overlapping bipolar signa-\ntures, creating a broadened, double-peaked structure. In the timestack plot of Fig. 8(b),\nclosely-spaced signatures overlap in the time-average to create t he appearance of a sin-\ngle signature that ‘shifts’ in v/bardbl. Note that, if the resonant velocities were spaced more18\ncloselyandwereunresolvedalong v/bardbl,thiswouldresultinasmoothlybroadenedsignature.\nThe superposition of multiple bipolar signatures creating a complicate d final signature\nwas most commonly observed in our lowest beta simulation ( βi= 0.01), which has the\nlargestrangeofexpectedresonantparallelphasevelocities.Add itionally, AstroGK evolves\nlow-beta plasmas significantly faster than high-beta plasmas, resu lting in a longer time\ninterval overwhich temporally intermittent wavedamping can be obs erved(see the simu-\nlation lengths, ∆ t/T0, in Table 1). If we continued running the higher beta simulations to\ncreate longer time intervals, we would expect to see the effects of s ignature superposition\nubiquitously (with the exception of the βi= 10 simulation, in which the v/bardblresolution is\ninsufficient to resolve multiple signatures in the region of expected da mping). In obser-\nvations, obtaining long time intervals—in terms of the underlying kinet ic Alfv´ en wave\nperiods—is not a problem. On the other hand, finite instrumental re solution in parallel\nvelocity (Verniero et al.2021a,b) is a significant limitation that must be taken into ac-\ncount when interpreting the velocity-space signatures from the fi eld-particle correlation\nanalysis of spacecraft observations.\n5.3.Velocity-Space Resolution and Weighting by v2\n/bardbl\nThe effects of v2\n/bardblweighting on the velocity-space signatures are visible in the various\nexamples from the turbulence simulations in Figures 7 and 8. In partic ular, we have\nshown that this weighting may obscure the negative portion of a bipo lar signature of\nelectron Landau damping when the parallel resonant phase velocity falls at a sufficiently\nsmall value of v/bardbl/vte. From our individual wave simulations in Sec. 3, we found that our\nbinned parallel velocity resolution of ∆ v/bardbl/vte= 0.133, coupled with the v2\n/bardblweighting,\nresulted in difficulty observing the full bipolar structure of the veloc ity-space signatures\nof electron Landau damping at parallel velocities v/bardbl/vte/lessorequalslant0.3. The entire dispersive\nrange of parallel phase velocities is above this threshold for the βi= 0.01 simulations,\nbut asβiis increased, more and more of the dispersive range of phase velocit ies falls\nbelow this threshold. For in situsignatures, clear bipolar signatures may therefore be\nmore easily observed for a given finite resolution in parallel velocity clo ser to the Sun,\nwhereβimay be smaller due to increased local magnetic energy density relativ e to the\nplasma thermal energy density. For plasma conditions and velocity r esolutions where\nv2\n/bardbldoes result in obscuring the negative portion of the characteristic bipolar structure,\nelectron Landau damping may still be identified through the timestac k correlations. If\nenergization is persistent and localized at a given parallel velocity in a r egime where the\nnegative portion is not observed, this still indicates the action of a r esonant energization\nmechanism that may be identified as electron Landau damping.\n5.4.Variations in Velocity-Space Signature Width\nAs discussed above, in both individual kinetic Alfv´ en wave and turbu lence simulations,\nwe observe that the velocity-space signature of electron Landau damping varies in width\nas the perpendicular wavenumber and plasma parameters change. We explain this rela-\ntionship by showing that the increased width (quantified by the full-w idth half maximum\nvalue of the positive portion of the bipolar signature) is associated w ith an increased nor-\nmalized damping rate of KAWs, as shown in Figure 4(b). Our simulations indicate that\nwe should expect to observe widths of the velocity-space signatur es of electron Landau\ndamping that depend on the local plasma βiand the perpendicular scale k⊥ρiof the\nwave being damped, which is generally a monotonically increasing funct ion of the par-\nallel phase velocity, as shown in dispersion relation plotted in Figure 2( a). As observed\nin the gyrotropic plot of Fig. 7(a) at v/bardbl/vte>0, if an unexpectedly wide velocity-\nspace signature is observed, it may indicate the damping of multiple dis persive kineticCharacterizing the Signature of Electron Landau Damping 19\nAlfv´ en wavepackets with different characteristic perpendicular w avenumbers k⊥ρi(and\ntherefore multiple resonant parallel phase velocities ω/k/bardblvte). In this case, the overlap of\nmultiple bipolar signatures is expected to lead to a broadened velocity -space signature,\nas illustrated in Figure 1 of Horvath et al.(2020).\n5.5.Non-resonant feature\nFinally, here we show that, as the plasma parameters are varied, a n on-resonant fea-\nture may arise in the field-particle correlation that obscures the sig natures of secular\nparticle energization in velocity-space. Our results show, for a con stant ion-to-electron\ntemperature ratio, that a non-resonant feature due to electro n density oscillations be-\ncomes problematic for βi/greaterorequalslant1. This feature is antisymmetric in v/bardbl, so it cancels out when\nthe correlation is integrated over parallel velocity, yielding zero net change to the total\nelectron spatial energy density We(r,t). However, it is the velocity-space signatures of\nthe rate of change of the phase-space energy density we(r,v,t) that are used to distin-\nguish the mechanisms of energy transfer using the field-particle co rrelation technique.\nTherefore, this non-resonant feature may hinder the identificat ion of energization mech-\nanisms in high- βi, KAW turbulence which has non-negligible levels of compressibility\nand in which the phase offset between E/bardbland the compressive fluctuation of ∂fe/∂v/bardbl\nhas shifted away from φ≃π/2.\n5.5.1.Eliminating the Non-Resonant Feature by ‘Folding’ Paralle l Velocity-Space\nAt the expense of losing the distinction between signatures of upwa rd (v/bardbl>0) and\ndownward ( v/bardbl<0) propagating waves, the non-resonant feature can be remove d by\ntaking advantage of its antisymmetric nature. ‘Folding’ the reduce d parallel correlation\nacrossv/bardbl= 0, obtaining CE/bardbl(|v/bardbl|,t) =CE/bardbl(v/bardbl>0,t)+CE/bardbl(v/bardbl<0,t), causes the large-\namplitude, non-resonant feature to cancel out, often revealing an underlying bipolar\nvelocity-space signature, as shown in Fig. 9. In (a), we plot the time stack correlation\nCE/bardbl(v/bardbl,t) from the βi= 1 simulation, which clearly shows an antisymmetric feature\nthat obscures a portion of the expected velocity-range for Land au damping. When the\ndata in the central and lower panels are folded over v/bardbl= 0—like closing a book with\nv/bardbl= 0 as the spine—a bipolar structure is revealed that is persistent th roughout the\nentire simulated interval. The amplitude of the resulting signature of electron Landau\ndamping has a peak amplitude about one-quarter of the amplitude of the broad non-\nresonant feature. Though by folding the correlation we have lost in sight into whether the\nsignature is due to the damping of upward or downward propagating waves, it reveals\nthe structure in velocity-spacethat is necessary for identifying t he mechanism of electron\nLandau damping. This method of ‘seeing beneath’ the non-resonan t feature was effective\nfor ourβi= 1 simulations, but we note that in our βi= 10 simulations the Landau\nresonances with the electrons are so close to v/bardbl= 0 that the velocity-space resolution\nof the simulation is insufficient for revealing signatures with certainty , even after folding\nthe data.\nAlternatively, high-pass filtering could be an effective way of removin g a low-frequency\nnon-resonant feature while retaining the distinction between the d amping of forward\nand backward propagating waves. We tested this with some succes s in our simulations,\nbut since the timestep in AstroGK simulations varies dynamically in order to satisfy\nthe numerical Courant–Friedrichs–Lewy stability condition, it is unc lear if the results\nof the filtered AstroGK datasets are reliable. We report that applying a fast-Fourier\ntransform, backward and forward high-pass Butterworth filter s, and an inverse fast-\nFourier transform did remove the non-resonant feature in tests performed on the βi= 120\nFigure 9. (a): Reduced parallel field-particle correlation CE/bardbl(v/bardbl,t) timestack plot in the\nβi= 1 turbulence simulation using τ= 2.6Tmax. (b): The same timestack plot in (a) folded\nacrossv/bardbl= 0 to obtain CE/bardbl(|v/bardbl|,t), showing a clear signature of electron Landau damping\natv/bardbl/vte∼0.1 (albeit with the negative region suppressed by the v2\n/bardblweighting) that persists\nthroughout thesimulation, with apeakamplitude about one q uarterof thenon-resonantfeature.\nsimulation. Due to the variable timestep, however, we did not pursue this method of\nremoving the non-resonant feature in depth.\n6. Summary and Conclusions\nApplication of the field-particle correlation technique to measureme nts of weakly col-\nlisional plasma turbulence in space and astrophysical plasma environ ments generates\nvelocity-space signatures of particle energization. These signatu res can be used to iden-\ntify the specific physical mechanisms that dominate the dissipation o f the turbulence.\nRecent field-particle correlation analyses of MMSobservations of turbulence in Earth’s\nmagnetosheath plasma have provided the first direct evidence of e lectron Landau damp-\ning as a significant mechanism for the dissipation ofspace plasma turb ulence (Chen et al.\n2019; Afshari et al.2021). Here, we employ linear kinetic Alfv´ en wave (KAW) simula-\ntions and a suite of four gyrokinetic simulations of turbulence with βi= 0.01,0.1,1,10\nandTi/Te= 1 to characterize the qualitative and quantitative features of th e velocity-\nspace signature of electron Landau damping in a turbulent space pla sma. Linear KAW\nsimulations have demonstrated that the zero-crossing of the bipo lar velocity-space sig-\nnature of electron Landau damping falls at the resonant parallel ph ase velocity of the\nwave, and that the width of the signature increases with increasing normalized damping\nrate. Furthermore, unlike in the case ofion Landaudamping whereb oth the negative and\npositive regionsof the bipolar velocity-spacesignatureare clearly o bservable(Klein et al.\n2017; Howes et al.2018; Klein et al.2020), forthe electronLandaudamping ofdispersive\nKAWs, the negativeregionofthe bipolarsignaturecan be suppress edby the v2\n/bardblweighting\nin the field-particle correlation for waves with low parallel phase veloc ities compared to\nthe electron thermal velocity, vph,/bardbl/vte/lessorequalslant0.3.\nAkeyresultofthis paperisthecharacterizationofthetypicalgyr otropicvelocity-space\nsignatures CE/bardbl(v/bardbl,v⊥) of electron Landau damping, with typical examples presented in\nFigure 7 for βi= 0.01,0.1,and 1. Similarly, timestack plots of the reduced parallel field-\nparticle correlation CE/bardbl(v/bardbl,t), shown in Figure 8, provide a clear means to determine\nthe electron energization rates as a function of time, showing both temporal intermit-\ntency and the overlap of bipolar velocity-space signatures of KAW w avepackets withCharacterizing the Signature of Electron Landau Damping 21\ndifferent resonant parallel phase velocities. Furthermore, even in balanced turbulence—\ncharacterized by approximately equal wave energy fluxes up and d own the local mean\nmagnetic field—over periods of a few to tens of KAW periods, one fre quently observes\nan imbalance of the velocity-space signatures of electron Landau d amping of upward and\ndownwardpropagatingKAWs,consistentwithprevious MMSobservations(Afshari et al.\n2021). In addition, a significant non-resonant feature that is asy mmetric in v/bardblappears\ndue to the plasma density fluctuations associated with KAWs at highe r values of plasma\nbeta,βi/greaterorsimilar1. This large-amplitude, non-resonant feature can make it difficult t o identify\nthe velocity-space signatures of electron Landau damping, but a m ethod of folding over\nthe correlationsin v/bardblcan effectively eliminate the feature, revealing the generally smaller\namplitude velocity-space signatures of resonant damping. All of th ese factors should be\ntaken into consideration when analyzing the field-particle correlatio n signatures of elec-\ntron Landau damping. It is our hope that these results can provide insight for interpret-\ning future field-particle correlation analysis of in situspacecraft observations: helping to\nidentify the signatures of electron Landau damping throughout th e inner heliosphere and\nadvancing the wider goal of determining the contribution of this mec hanism to electron\nenergization in the solar wind and other weakly collisional plasma enviro nments.\nFunding\nThis workwassupportedby NASAFINESST Grant80NSSC20K1509a swellasNASA\ngrants 80NSSC18K1217 and 80NSSC18K1371. Numerical simulation s were performed\nusing the Extreme Science and Engineering Discovery Environment ( XSEDE), which is\nsupported by National Science Foundation grant number ACI-154 8562, through alloca-\ntion TG-PHY090084.\nDeclaration of interests\nThe authors report no conflicts of interest.\nAppendix A. Velocity-Space Signatures of Electron Landau Damping\nof Single Kinetic Alfv´ en Waves\nHere we reproduce timestack plots of the field-particle correlation sCE/bardbl(v/bardbl,t) in steady\nstate for five of the twenty individual kinetic Alfv´ en wave (KAW) sim ulations. For a βi=\n1 plasma, we plot the cases of a KAW with k⊥ρi= 2,4,8,16,and 32. Each correlation\nshows clear electron energization due to Landau damping of the wav e by the resonant\nelectrons, where the resonant parallel phase velocity for each wa ve is indicated by the\nvertical dotted lines. At the parallel velocity-space resolution of t heAstroGK simulations,\nthenegativeportionofthebipolardampingsignatureisonlyvisibleinth etime-integrated\npanels for the two smallest-scale waves (those at the largest reso nant velocities), but all\nhave a clear, localized, and persistent energization signature due t o the action of Landau\ndamping. The non-resonant feature, antisymmetric in v/bardbl, is visible in these correlations.\nThough the compressibility of the KAWs increases with k⊥ρi(Matteini et al.2020), the\nnon-resonant feature is most clearly visible at lower k⊥ρiin these plots, likely due to the\nlower damping rate and the resonance’s increased proximity to v/bardbl= 0 for these larger\nperpendicular wavelength waves.22\nFigure 10. Field-particle correlations of a set of driven, damped kine tic Alfv´ en waves with\nβi= 1 and (a) k⊥ρi= 2, (b) 4, (c) 8, (d) 16, and (e) 32. The zero-crossing of the bi polar\nsignatures correspond to the parallel wave phase velocity ( vertical dotted lines), and the width\nof the signature increases with increasing damping rate as k⊥ρiincreases.\nAppendix B. Non-Resonant Feature Analysis\nHere we explore more deeply the antisymmetric, non-resonant fea ture that appears in\nthe field-particle correlation, which we attribute to the electron de nsity perturbations,\nδne, that constitute a dynamic part of the kinetic Alfv´ en wave.This no n-resonantfeature\narises in the field-particle correlation due to (1) an increase in the am plitude of δne\nrelative to the resonant perturbations and (2) the relative phase between E/bardbland∂f/∂v /bardbl\nasafunctionof v/bardbl.Thecompressivenatureofthenon-resonantfeature,itsamplit ude,and\nits dependence onthe phaseoffset φareseenclearlywhen the factorsin the mathematical\nform of the correlation are divided into separate plots, as shown in F ig. 11.\nThe reduced parallel field-particle correlation CE/bardbl(v/bardbl,t) is shown in Fig. 11(a) for the\nindividualwavesimulationat k⊥ρi= 8,βi= 0.1.Thereduced,complementaryperturbed\ndistribution function ge(v/bardbl,t) (Numata et al.2010) is plotted in (b). The location of the\nresonant KAW phase velocity is marked by the vertical dashed line at v/bardbl/vte= 0.51, and\nthe resonant perturbations in ge(v/bardbl,t) peak broadly around this resonant parallel veloc-\nity. In the background, at lower amplitude, there is a broad oscillato ry signature that is\nan even function in v/bardbland that spans all velocities (fading to zero with the distribution\nfunction at large v/bardbl). This non-resonant, even perturbation in ge(v/bardbl,t) represents com-\npression in the electron density, δne, when integrated over v/bardbl. In (d), we show the parallel\nvelocity derivative of the reduced perturbed distribution function ,∂ge/∂v/bardbl. Here, it canCharacterizing the Signature of Electron Landau Damping 23\nFigure 11. Detail of the density perturbations giving rise to the non-r esonant feature in the\nE/bardblfield-particle correlations of high- βiplasmas. The parallel electric field-particle correlation\nCE/bardbl,e(v/bardbl) (a); the perturbed distribution function ge(b); the parallel electric field weighted by\nparallel velocity factor E/bardblv2\n/bardbl(c); and the parallel velocity derivativeof the perturbedd istribution\nfunction ∂ge/∂v/bardbl(d).\nbe seen that the even function in ge, yielding δne∝negationslash= 0, turns into an odd function when\nthe derivative with respect to v/bardblis computed. Finally, (c) contains the parallel electric\nfield multiplied by v2\n/bardbl; note that E/bardblitself is constant over v/bardbl. Upon close inspection, it\ncan be observedthat the oscillation in E/bardbland the backgroundoscillationin ∂ge/∂v/bardblhave\nthe same period but are offset in phase. This phase offset φis ultimately what causes\nthe density oscillation to lead to the odd, non-resonant signature t hat appears in the\nfield-particle correlation in panel (a).\nIn the main body of this paper, we have explained how the relative amp litudes of\nthe resonant and density perturbations to gechange due to the combined effects of\nincreasing compressibility of KAWs and to the decreasing phase-velo city of KAWs as βi\nincreases. However, we did not discuss the origin of the phase offse t between the density\nperturbation and the parallel electric field from φ≃π/2 at low βitoφ∝negationslash=π/2 at higher\nβi. If the electron response is approximately instantaneous, the de nsity perturbation will\nbe exactly π/2 out of phase with the finite E/bardblthat drives it. In this case, the average\nacceleration of the particles by E/bardblis zero. This is approximately the background phase\nrelationship we observe in the low βi(0.01 and 0 .1) single-wave runs.\nThe phase offset is approximately φ≃π/2 at all parallel velocities except those that\nresonantly interact with the E/bardblwave around vph=ω/k/bardbl, as shown in Fig. 12. For small\nperpendicular wavenumbers at large βi(1 and 10) the phase offset is also φ≃π/2. As\nk⊥ρiincreases, however, the phase offset deviates from π/2 and moves toward πor 0,\nresulting in detectable non-resonant features in the field-particle correlation signatures.\nAsβiandk⊥ρiincrease, the compressive wave begins to lag E/bardbl, shifting the relative\nphase��and thereby enabling E/bardblto do work on the particles, which results in a local24\nFigure 12. The phase offset φbetween E/bardbland the density oscillation in ∂ge/∂v/bardblas a function\nofv/bardbl/vte. Horizontal sections, separated by black lines, correspon d to different βi. 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Sci. 120(23), e2220927120." }, { "title": "1512.03610v1.Ultra_low_magnetic_damping_of_a_metallic_ferromagnet.pdf", "content": "Ultra -low magnetic damping of a metallic ferromagnet \nMartin A. W. Schoen,1, 2 Danny Thonig,3 Michael L. Schneider,1 T. J. Silva,1 Hans T. Nembach,1 Olle \nEriksson,3 Olof Karis,3 and Justin M. Shaw1* \n \n1Quantum E lectromagnetics Division, National Institute of Standards and Technology, Boulder, CO, 80305 , USA \n2Institute of Experimental and Applied Physics, University of Regens burg, 93053 Regensburg, Germany \n3Department of Physics and Astronomy, University Uppsala S -75120 Uppsala, Sweden \nDated: 12/11/2015 \nPACS numbers: 71.70.Ej,76.50.+g, 75.78.n,71.20.Be \n*Corresponding author: justin.shaw@nist.gov \n \n \nThe phenomenology of magnetic damping is of critical importance for devices that seek to exploit \nthe electronic spin degree of freedom since damping strong ly affects the energy required and speed \nat which a device can operate. However, theory has struggled to quantitatively predict the \ndamping, even in common ferromagnetic materials1–3. This presents a challenge for a broa d range \nof applications in spin tronics4 and spin -orbitron ics that depend on materials and structures with \nultra -low damping. Such systems enable many experimental investigations that further our \ntheoretical understanding of numerous magnetic phenomena such as damping and spin -\ntransport5 mediated by chirality6 and the Rashba effect. Despite this requirement, it is believed \nthat achieving ultra -low damping in metallic ferromagnets is limited due to the scattering of \nmagnons by the conduction electrons. However, we report on a binary alloy of Co and Fe that \novercomes this obstacle and exhibits a damping parameter approaching 10-4, which is comparable \nto values reported only for ferrimagnetic insulators7,8. We explain this phenomenon by a unique \nfeature of the bandstructure in this system: T he density of states exhibits a sharp minimum at the \nFermi level at the same alloy concentration at whic h the minimum in the magnetic damping is \nfound. This discovery provides both a significant fundamental understanding of damping \nmechanisms as well as a test of the theoretical predictions put forth by Mankovsky et al.3. \n \nIn recent decades, several theoretical appr oaches have attempt ed to quantitatively predict \nmagnetic damping in metallic systems. One of the early promising theories was that of Kambersky , \nwho introduced the so -called breathing Fermi surface model .9–11 More recently, Gilmore and Stiles2 as \nwell as Thonig et al.12 demonstra ted a generalized torque correlation model that include both intraband \n(conductivity -like) and interband (resistivity -like) transitions. The use of scattering theory to describe \ndamping was later applied by Brataas et al.13 and Liu et al.14 to describe damping in transition metals . \nA numerical realization of a linear response damping model was implemented by Mankovsky3 for Ni -\nCo, Ni -Fe, Fe -V and Co -Fe alloys. For the Co -Fe alloy , these calculations predict a minimum intrinsic \ndamping of αint ≈ 0.000 5 at a Co -concentration of 1 0 % to 20 %, but was not experimentally \nobserved.15 \nUnderlying this theoretical work is the goal of achieving new systems with ultra -low damping \nthat are required in many magnonic and spin-orbitronics applications .7,8 Ferrimagnetic insulators such \nas yttrium -iron-garnett (YIG) have long been the workhorse for these investigations since YIG films \nas thin as 25 nm have experimental damping parameters as low as 0.9×10−4.16 Such ultra-low damping \ncan be achieved in insulating ferrimagnets in part due to the absence of conduction electrons and \ntherefore the suppression of magnon -electron scattering. H owever, insulators cannot be used in most \nspintronic and spin -orbitronic applications where a charge current through the magnetic material is \nrequired nor is the requirement of growth on gadolinium g allium g arnet templates compatible with \nspintronics and c omplementary metal -oxide semiconductor (CMOS) fabrication processes. One \nproposed alternative class of materials are Heusler alloys, some of which are theoretically predicted to \nhave damping parameters as low as 10-4.17 While such values have yet to be realized, damping \nparameters as low as 0.001 have been reported for Co 2FeAl18 and NiMnSb19. However, Heusler alloys have non -trivial fabrication constraints, such as high -temperature annealing , which are also \nincompatible with spintronic and CMOS device fabrication constraints . \nIn contrast, metallic ferromagnets such a s 3d transition metals are ideal candidate materials for \nthese applications since high quality ma terials can be produced at room temperature (RT) without the \nrequirement of annea ling. However, ultra -low damping is thought to be unachievable in metallic \nsystems since damping in conductors is dominated by magnon -electron scattering in the conduction \nband resulting in a damping parameter over an order of magnitude higher than those found in high -\nquality YIG. \n Inspired by Mankovsky’s theoretical prediction of ultra -low damping in the CoxFe1−x alloy system3, \nwe systematically studied the compositional dependence of the damping parameter in Co xFe1−x alloy s, \ninclu ding careful evaluation of spin -pumping and radiative damping contributions. Polycrystalline \nCoxFe1−x alloy films , 10 nm thick, were sputter -deposited at RT with Cu/Ta seed and capping layers . \nX-ray diffraction ( XRD) reveals that the CoFe alloys display a body -centered -cubic (bcc) phase over a \nCo concentration of 0 % to 60 %, a face -centered -cubic (fcc) phase above 80 % Co, and a mixed phase \nbetween 60 % and 80 % Co, in good agreement with the bulk phase diagram of this system. The \ndamping parameter is determined from broadband ferromagnetic resonance (FMR) spectroscopy \nwhich measures the susceptibility over frequencies spanning 5 GHz to 40 GHz. An example of 𝑆21(𝐻) \nvector -network -analyzer transmission data is shown in Figure 1a and b, toge ther with fits to the \ncomplex susceptibility for the real and imaginary parts, respectively. The total damping parameter αtot \nis determined from the frequency depen dence of the linewidth obtained from these su sceptibility fits, \naccording to equation (1), \n \n∆𝐻=2ℎ𝛼tot\n𝑔μ0μB 𝑓+∆𝐻0 (1) \n \nwhere µ0 is the vacuum permeability, µB is the Bohr magneton, h is Planck’s constant, g is the Landé \ng-factor , and ΔH0 is the inhomogeneous linewidth . \n \n \nFigure 1: Ferromagnetic resonance spectra, measured via FMR and the resulting linewidth as a \nfunction of frequency . a and b, respectively , show the real and imaginary part s of the S21(H) \ntransmission parameter (black circles) with the complex susceptibility fit (red li nes). c, the line -widths \n(symbols) are plotted versus the frequency for Co, Fe, Co 20Fe80 and Co 25Fe75. The uncertainties in the \nline-widths were ob tained by means of the standard method for the determination of confidence limits \non estimated parameters for nonlinea r models under the assu mption of Gaussian white noise . The lines \nare error -weighted fits to equation (1), which are used to determine both the total damping αtot and the \ninhomogeneous linewidth broadening for each alloy. \n \nThe measured total damping αtot versus alloy composition for 10 nm films is plotted in Figure 2. αtot \nshows a clear minimum of (2.1 ± 0.15) × 10−3 at a Co concentration of 25 %. However, as a result of the \nutilized measurement geometry and the structure of the sample, there are several extrinsi c contributions \nto αtot that are independent of αint. \nThe first contribution —the result of the inductive coupling of the precessing magnetization and the \ncoplanar waveguide (CPW)—is radiative damping αrad.20 The FMR system is designed and optimized \nto couple microwaves into the ferromagnet, and therefore, by virtue of reciprocity, the system is efficient \nat coupling mi crowaves out of the ferromagnet. For very thin films or films with low saturation \nmagnetization , αrad is typically not a significant contribution to the total damping and can be ignored. \nHowever, in the present case, αrad must be accounted for in the anal ysis due to the combination of a very \nhigh saturation magnetization and the exceptionally small value of αint. As described in the supplemental \ninformation section (SI), we calculate and experimentally validate the contribution of αrad to the total \ndamping, which is plotted in Figure 2. \n \n \n \n \n \nFigure 2: The total measured damping with radiative and interfacial contributions. The total \ndamping αtot (red crosses with lines), spin - pumping αsp (gray line) and radiative αrad (green line) \ncontributions to the damping, and the intrinsic damping αint are plotted against the respective Co \nconcentration. The errors are propagated from the line -width fits. The crystal structure of the alloys, \nobtained from XRD, signified by the color regions in the plot . \n \nThe second non -negligible contribution to the total damping is the damping enhancement due to \nspin-pumping into the adjacent Cu/Ta layers. The spin-pumping contribution αsp can be determined from \nthe thickness dependence of ( αtot - αrad) since it behaves as an interfacial damping term21. Indeed, we \nmeasured the thickness dependence of ( αtot - αrad) for many alloy samples in order to quantify and account \nfor αsp (see SI), which is displayed in Figure 2. \nContributions from eddy -current damping22 are estimat ed to be smaller than 5% and are neglected. \nFinally, two -magnon scattering is minimized in the perpendicular geometry used in this investigation \nand its contribution is disregarded23. \n \n \n \nFigure 3: Calculated electron density of states (DOS ) and it s comparison to the intrinsic damping. \na Electronic structure of bulk Co xFe1-x. The DOS is shown for different Co concentrations, as indicated. \nEF is the Fermi energy . b The extracted intrinsic damping (black squares, left axis) is compared to the \ntheory in Mankovsky et al.3 , for a temperature of 0 K (blue line) and for a temperature of 300 K for \npure Fe (blue cross). The calculated DOS at the Fermi energy n(EF) is plotted on the right axis (red line) . \nThe y-offset of n(EF) is chosen deliberately in order to demonstrate that the concentration -dependent \nfeatures o f the damping directly correlate to features of n(EF). We cannot exclude concentration -\nindependent contributions to the damping, which are accounted for by the 0.4 eV-1 y-offset. \nThe total measured damping becomes , αtot = αint + (αrad /2) + αsp, allowing the intrinsic damping αint \nto be determined, which is presented in Figure 2. For many values of αint, the contributions of αsp and \nαrad are of similar magnitude, showing the importance of accounting for these contributions . For 25 % \nCo αint now di splays a sharp minimum in damping of (5 ± 1.5) × 10−4, which is astonishing for a \nconductor. Indeed, αint < 0.001 have been measured only in ferrimagnetic insulators .24 \nThese results raise the question why αint can be so low in the presence of conduction electrons. To \ngain a deeper understanding, we performed electronic structure calculations for Co xFe1-x within a full -\nrelativistic , multiple -scattering approach (Korringa -Kohn -Rostoker method25, KKR) using the coherent \npotential approximation (CPA)26,27 over the entire range of compositions (see SI). Several representative \nexamples are given in Fig ure 3a. \nThe d−states (peak in the DOS below EF) for pure Fe are not fully occupied . Consistent with the rigid \nband model28, the d−states shift to lower energies when the concentration of Co increases , and become \nfully oc cupied at 25 % Co, coinciding with the minimum in n(EF) displayed in Figure 3a , which \noriginates from the hybridization between majority Fe eg and minority Co t2g states. \nEbert et al.1 suggested that αint is proportional to n(EF) in the breathing Fermi -surface model (i.e., \nintraband transitions) in the cases of a minimally varying spin -orbit coupling (SOC ) (as is the case for \nthe Co xFe1-x system) and small electron -phonon coupling2,29. Alternatively, i nterband transitions become \nsignificant only if bands have a finite overlap due to band broadening, caused for example by coupling \nto the phonon s. As a result, interband transitions suppressed at low -temperature and energy dissipatio n \nbecomes dominated by intraband transitions. Our RT measurements of Co xFe1-x satisfy this “low -\ntemperature” condition since the electron -phonon coupling is < 20 meV for pure bcc Fe, and < 30 meV \nfor pure hcp Co . Band broadening due to disorder is about 15 meV for the , at EF dominating, eg states \n(50 meV for the t 2g states) in Co25Fe75 and varies up to 55 meV for the e g states (150 meV for the t 2g \nstates) over the whole range of composition. T hese calculations show that the band broadening effect at \nRT is too small to provide significant interband damping, consistent with the almost perfect \nproportionality between n(EF) for all alloy compositions in the bcc phase (0 % to 60 % Co). Such a \nproportionality requires an offset of 0.4 eV-1, which originates f rom the fact the n(EF) is a superposition \nof all states, some of which do not contribute significantly to the damping. \nThe calculations of αint by Mankovsky3 (included in Fig 3b) show a minimum value of αint ≈ 0.0005 \nbetween 10 % and 20 % Co, which differs from the sharp minimum we find at 25 % Co in both the \nexperimental data and the calculated values of n(EF). Remarkably, with the exception of pure Fe, the \ncalculated values of αint (at 0 K) agree with our r esults within a factor of ≈ 2. Furthermore, the agreement \nis greatly improved for pure Fe, when a finite temperature of 300 K is included. While not perfect, the \nagreement between those calculations and our results is compelling and provides the critical f eedback \nneeded for further refinement of theory. \nWe therefore demonstrate and conclude that αint is largely determined by n(EF) in the limit of \nintraband scattering. Secondly, our work shows that a theoretical understanding of damping requires an \naccurate account of all contributions to the damping parameter. Furthermore, if the theoretical \nexplanation put forth here to explain low damping holds in general, it is natural to utilize data -mining \nalgorithms to screen larger groups of materials in o rder to iden tify additional low -damping systems. \nExamples of such studies to identify new materials for use , for example, in scintillators have been \npublished30, and the generalization to applications in mag netization dynamics is straight forward . \n \nAuthor responsibility \nMAWS and DT wrote the manuscript, JS and HN conceived of the experiment, MAWS deposited the \nsamples, and performed the SQUID measurements and analysis, MAWS, MLS and HN performed the \nFMR measurements and analysis, JS performed the XRD measurements and anal ysis, DT and OE \nperformed the first -principle s DFT calculations. All authors contributed to the interpretation of the \nresults. \n \nAcknowledgements \n \nThe authors are grateful to Mark Stiles and Eric Edwards for valuable discussions. \n \nReferences. \n1. Ebert, H., Mankovsky, S., Ködderitzsch, D. & Kelly, P. J. Ab Initio Calculation of the Gilbert Damping \nParameter via the Linear Response Formalism. Phys. Rev. Lett. 107, 66603 –66607 (2011). \n2. Gilmore, K., Idzerda, Y. U. & Stiles, M. D. Identification of the Dominant Precession -Damping Mechanism \nin Fe, Co, and Ni by First -Principles Calculations. Phys. Rev. Lett. 99, 027204 (2007). \n3. Mankovsky, S., Ködderitzsch, D., Woltersdorf, G. & Ebert, H. First -principles calculation o f the Gilbert \ndamping parameter via the linear response formalism with application to magnetic transition metals and \nalloys. Phys. Rev. B 87, 014430 (2013). \n4. Žutic, I., Fabian, J. & Das Sarma, S. Spintronics: Fundamentals and applications. Rev. Mod. Phys . 76, 323 –\n410 (2004). \n5. Emori, S., Bauer, U., Ahn, S. -M., Martinez, E. & Beach, G. S. D. Current -driven dynamics of chiral \nferromagnetic domain walls. Nat Mater 12, 611 –616 (2013). \n6. Jué, E. et al. Chiral damping of magnetic domain walls. accepted in Nat. Materials (2015). \n7. Allivy Kelly, O. et al. Inverse spin Hall effect in nanometer -thick yttrium iron garnet/Pt system. Applied \nPhysics Letters 103, (2013). \n8. Onbasli, M. C. et al. Pulsed laser deposition of epitaxial yttrium iron garnet films with l ow Gilbert damping \nand bulk -like magnetization. APL Materials 2, (2014). \n9. Kamberský, V. FMR linewidth and disorder in metals. Czechoslovak Journal of Physics B 34, 1111 –1124 \n(1984). \n10. Kamberský, V. On ferromagnetic resonance damping in metals. Czechosl ovak Journal of Physics B 26, \n1366 –1383 (1976). \n11. Kambersky, V. & Patton, C. E. Spin -wave relaxation and phenomenological damping in ferromagnetic \nresonance. Phys. Rev. B 11, 2668 –2672 (1975). \n12. Thonig, D. & Henk, J. Gilbert damping tensor within the b reathing Fermi surface model: anisotropy and non -\nlocality. New J. Phys. 16, 013032 (2014). \n13. Brataas, A., Tserkovnyak, Y. & Bauer, G. E. W. Scattering Theory of Gilbert Damping. Phys. Rev. Lett. 101, \n037207 (2008). 14. Liu, Y., Starikov, A. A., Yuan, Z. & Kelly, P. J. First -principles calculations of magnetization relaxation in \npure Fe, Co, and Ni with frozen thermal lattice disorder. Phys. Rev. B 84, 014412 (2011). \n15. Oogane, M. et al. Magnetic Damping in Ferromagnetic Thin Films. Japanese Journal of Applied Physics 45, \n3889 –3891 (2006). \n16. Chang, H. et al. Nanometer -Thick Yttrium Iron Garnet Films With Extremely Low Damping. Magnetics \nLetters, IEEE 5, 1–4 (2014). \n17. Liu, C., Mewes, C. K. A., Chshiev, M., Mewes, T. & Butler, W. H. Origin of low Gilbe rt damping in half \nmetals. Applied Physics Letters 95, (2009). \n18. Mizukami, S. et al. Low damping constant for Co2FeAl Heusler alloy films and its correlation with density \nof states. Journal of Applied Physics 105, (2009). \n19. Dürrenfeld, P. et al. Tunabl e damping, saturation magnetization, and exchange stiffness of half -Heusler \nNiMnSb thin films. ArXiv e -prints (2015). \n20. Schoen, M. A. W., Shaw, J. M., Nembach, H. T., Weiler, M. & Silva, T. J. Radiative damping in waveguide -\nbased ferromagnetic resonance measured via analysis of perpendicular standing spin waves in sputtered \npermalloy films. Phys. Rev. B 92, 184417 (2015). \n21. Tserkovnyak, Y., Brataas, A. & Bauer, G. E. W. Enhanced Gilbert Damping in Thin Ferromagnetic Films. \nPhys. Rev. Lett. 88, 117601 (2 002). \n22. Lock, J. M. Eddy current damping in thin metallic ferromagnetic films. British Journal of Applied Physics \n17, 1645 (1966). \n23. Hurben, M. J. & Patton, C. E. Theory of two magnon scattering microwave relaxation and ferromagnetic \nresonance linewidth in magnetic thin films. Journal of Applied Physics 83, 4344 –4365 (1998). \n24. Sun, Y. et al. Damping in Yttrium Iron Garnet Nanoscale Films Capped by P latinum. Phys. Rev. Lett. 111, \n106601 (2013). \n25. Zabloudil, J., Hammerling, R., Szunyogh, L. & Weinberger, P. Electron Scattering in Solid Matter . \n(Springer: Berlin, 2005). \n26. Durham, P. J., Gyorffy, B. L. & Pindor, A. J. On the fundamental equations of the Korringa -Kohn -Rostoker \n(KKR) version of the coherent potential approximation (CPA). J. Phys. F 10, 661 (1980). \n27. Faulkner, J. S. & Stocks, G. M. Calculating properties with the coherent -potential approxi mation. PRB 21, \n3222 (1980). \n28. Stern, E. A. Rigid -Band Model of Alloys. Phys. Rev. 157, 544 (1967). \n29. Kamberský, V. On the Landau -Lifshitz relaxation in ferromagnetic metals. Can. J. Phys. 48, 2906 (1970). \n30. Ortiz, C., Eriksson, O. & Klintenberg, M. Data mining and accelerated electronic structure theory as a tool \nin the search for new functional materials. Computational Materials Science 44, 1042 –1049 (2009). \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Methods: Ultra -low magnetic damping of a metallic \nferromagnet \nMartin A. W. Schoen,1, 2 Danny Thonig,3 Michael Schneider,1 Thomas J. Silva,1 Hans T. Nembach,1 Olle \nEriksson,3 Olof Karis ,3 and Justin M. Shaw1* \n \n1Quantum Electromagnetics Division, National Institute of Standards and Technology, Boulder, CO, 80305 \n2Institute of Experimental and Applied Physics, University of Regensburg, 93053 Regensburg, Germany \n3Department of Physics and Astronomy, University Uppsala S -75120 Uppsala, Sweden \n \nDated: 12/11/2015 \n PACS numbers: 71.70.Ej,76.50.+g, 75.78.n,71.20.Be \n*Corresponding author: justin.shaw@nist.gov \n \n \nSample preparation \nThe sample s were deposited by DC magnetron sputtering at an Ar pressure of approximately 0.67 Pa ( 5 \n× 10-3 Torr) in a chamber with a base -pressure of less than 5 × 10-6 Pa (4 × 10-8 Torr). The alloys were \ndeposited by co -sputtering from two targets with the deposition rates calibrated by X -ray reflectometry \n(XRR). The repeatability of the deposition rates was found to be better than 3% variation over the course \nof this study. For all deposited alloys, the combined deposition rate was kept at approximately 0.25 \nnm/s, to ensure similar growth conditions. Furthermore the Co 20Fe80 and the Co 25Fe75 samples were \nreplicated by depositing from single stoichiometric targets, to prove the reproducibility of the results. \nSamples with a thickness of 10 nm were fabricated over the full alloy composition range and additional \nthickness series (7 nm, 4nm, 3nm and 2 nm) were fabricated for the pure elements and select \nintermediate alloy concentr ations (20% Co, 25% Co, 50% Co and 85% Co). \n \nX-ray diffraction measurement. \nThe crystal structure of the alloys was determined by X -ray diffraction XRD using an in -plane geometry \nwith parallel beam optics and a Cu K α X-ray source. The in -plane geometry all ows the signal from the \nCo-Fe alloys to be isolated from the high -intensity signal coming from the silicon substrate. These \nmeasurements yield both the in -plane lattice constant s and the crystal structure, as shown in the \nsupplemental material, section 1. The deposition rates were calibrated using XRR using the same system \nconfigured for the out -of-plane geometry. \n \nSQUID measurement. \nWe measured the in -plane hysteresis curves at 300 K to determine the magnetic moment of the sample. \nSample were first diced with a precision diamond saw such that an accurate value of the volume of the \nsample could be calculated. The s aturation magnetization MS for all alloy samples is then determined \nby normalizing the measured moment to the volume of the CoFe in the sample . These values are shown \nin the supplemental material. \n \n \nVNA -FMR measurement \nThe FMR measurement utilized a room -temperature -bore superconducting magnet, capable of applying \nstatic magnetic fields as high as 3 T. An approximately 150 nm p oly(methyl methacryl ate) (PMMA) \ncoat was first applied to the samples to prevent electrical shorting of the co -planar waveguide (CPW) \nand to protect the sample surface. Sample were placed face down on the CPW and m icrowave fields \nwere applied in the plane of the sample, at a frequenc ies that ranged from 10 GHz to 40 GHz. A vector -network -analyzer ( VNA ) was connected to both sides of the CPW and the complex S21(H) transmission \nparameter was determined. The iterative susceptibility fit of S21(H) was done with the method describe d \nby Nembach et al.1. All fits were constrained to a field window that was 3 times the linewidth around \nthe resonance field in order to minimize the fit residual. We verified that this does not change the results , \nbut reduces the influence of measurement noise on the error bars of the fitted values. \n \nCalculation of the DOS \nElectronic ground state calc ulations have been performed by a full -relativistic multiple -scattering \nGreen’s function method ( Korringa -Kohn -Rostoker method2, KKR ) that relies on the local s pin-\ndensity approximation (LDA) to density functional theory. We utilize d Perdew –Wang exc hange \ncorrelation functionals3–6. \n \nIn our multiple -scattering theory , the electronic structure is described by scattering path operators τij \n(i,j lattice site indices) 2, where , in the spin -angular -mome ntum representation , we consider angular \nmomenta up to lmax = 3 and up to 60 x 60 x 60 points in reciprocal space. The substitutional alloys are \ntreated within the coherent potential approximation (CPA)7,8. Co impurities in the Fe host lattice are \ncreated in the effective CPA medium by defect matrices. The CPA medium is described by scattering \npath operators. The site -dependent potentials are con sidered in the atomic sphere approximation \n(potentials are spherically symmetric within muffin -tin spheres and constant in the interstitials). \nThe DOS is obtained from the integrated spectral density2 \n \n𝑛(𝐸)= 1\n𝜋∫ 𝐼𝑚[tr(𝐺(𝐸+i𝜂,𝒌))]d𝒌\nΩBZ \n \nwith the small positive energy η. The integration in reciprocal space k runs over the first Brillouin \nzone ΩBZ. \n \n \n \nReferences \n \n1. Nembach, H. T. et al. Perpendicular ferromagnetic resonance measurements of damping and Lande g -factor \nin sputtered (Co_2Mn)_1 -xGe_x films. Phys. Rev. B 84, 054424 (2011). \n2. Zabloudil, J., Hammerling, R., Szunyogh, L. & Weinberger, P. Electron Scattering in Solid Matter . (Springer: \nBerlin, 2005). \n3. Becke, A. D. Density -functional exchange -energy approximation with correct asymptotic behavior. Phys. Rev. \nA 38, 3098 –3100 (1988). \n4. Langreth, D. C. & Mehl, M. J. Beyond the local -density approximation in calculations of gro und-state \nelectronic properties. Phys. Rev. B 28, 1809 –1834 (1983). \n5. Perdew, J. P. et al. Erratum: Atoms, molecules, solids, and surfaces: Applications of the generalized gradient \napproximation for exchange and correlation. Phys. Rev. B 48, 4978 –4978 (1993). \n6. Perdew, J. P. & Wang, Y. Accurate and simple analytic representation of the electron -gas correlation energy. \nPhys. Rev. B 45, 13244 –13249 (1992). \n7. Asada, T. & Terakura, K. Generalized -gradient -approximation study of the magnetic and cohesive p roperties \nof bcc, fcc, and hcp Mn. Phys. Rev. B 47, 15992 –15995 (1993). \n8. Paxton, A. T., Methfessel, M. & Polatoglou, H. M. Structural energy -volume relations in first -row transition \nmetals. Phys. Rev. B 41, 8127 –8138 (1990). \n \n \n \n \n \n \n Supplemental Inform ation: Ultra -low magnetic damping of \na metallic ferromagnet \nMartin A. W. Schoen,1, 2 Danny Thonig,3 Michael Schneider,1 Thomas J. Silva,1 Hans T. Nembach,1 Olle \nEriksson,3 Olof Karis,3 and Justin M. Shaw1 \n \n1Quantum Electromagnetics Division, National Institute of Standards and Technology, Boulder, CO, 80305 \n2Institute of Experimental and Applied Physics, University of Regensburg, 93053 Regensburg, Germany* \n3Department of Physics and Astronomy , University Uppsala S -75120 Uppsala, Sweden \n \nDated: 12/11/2015 \n \nX-ray diffraction measurement \n \nFigure S1 shows several in -plane geometry X -ray diffraction (XRD) spectra for a series of select alloys. \nFrom 100 % to 80 % Co, there is no evidence of a body -centered cubic peak and we can conclude that \nthe phase of the CoFe is purely face-centered cubic (fcc). At a concentration of 75% (not shown), a bcc \npeak emerges in the spectrum. An fcc (220) peak is always visible in the spectra due to the signal from \nthe Cu seed and capping layers. The presence of the signal from the C u layers complicates the analysis \nsince the point at which the crystalline phase transitions from a mixed fcc and bcc to a pure bcc phase \nfor the Co xFe1-x alloys cannot be exactly determined. However , there is strong evidence that the phase \ntransition occurs at a Co concentration in the vicinity of 60 % to 70 % Co. This can be observed in Fig \nS2b which shows a plot of the lattice constants as a function of the concentration. For samples with a \nCo conce ntration smaller than 70 %, the lattice constant for the fcc phase becomes constant at a value \nthat is expected for pure Cu. This is in contrast to the rapidly changing value of the lattice constant \nobserved in alloys with a concentration of Co exceeding 7 0 %. To put these results in context , the bulk \nphase diagram for this system predicts a fcc to bcc transition at approximately 75 % Co1. Thus, our \nresults shows that our thin films initiate a bcc phase at a slightly higher value of Co concentration. \n \nFigure S1: XRD spectra. The spectra are shown for select alloy concentrations. The Cu fcc peak is \nalways visible and is overlaid with the fcc Co pe ak. The Co bcc peak is visible only for Co concentrations \nbelow 70%. \n \n \nIt is important to point out that we do not measure the expected hexagonal close -packed (hcp) crystal \nstructure for the pure Co samples .1 This suppression of the hcp structure was previously found in growth \nof Co on Cu and explained by the strained growth of the Co on Cu suppressing the hcp phase.2–4 Indeed, \nwe confirm this result by comparing pure Co films grown with a Ta/Cu seed and capping layer and one \ngrown with only a Ta seed and capping layer . Figure S2a shows an XRD plot in the vicinity of the \nhcp(010) peak for the 10 nm pure Co samples with a Ta/Cu seed and a Ta seed. The sample with Ta \nseed layer exhibits a clear hcp(010) peak, indicating an hcp structure. In contrast, the sample that \ninclude s Cu in the seed layer shows no evidence of the hcp(010) peak. . \n \n \n \nFigure S2: XRD characterization. In a the XRD spectrum of pure Co samples with a Ta and a Ta/Cu \nseed layer. The via XRD determined lattice constant is pl otted against the respective Co concentration \nin b. The crystalline structure is denoted and coded in color. \n \nSaturation and effective magnetization. \n \nThe magnetization of all of our samples were measured and verified using two approaches. The \nfirst is the direct measurement of saturation magnetization Ms by superconducting quantum interference \ndevice (SQUID) magnetometry. These measured values of Ms are displayed in Figure S3 as a function \nof the concentration of Co in the alloy. For comparison we also include the Slater -Pauling curve1,5,6 in \nthe plot. In the bcc phase , MS displays almost perf ect agreement with the Slater -Pauling curve indicative \nof the high quality of the samples used in this study. Howe ver, below 80 % Co, the Slater -Pauling curve \nunderestimates Ms slightly when the crystalline phase of the CoFe alloy becomes exclusively fcc. \n Ferromagnetic resonance (FMR) also provides a measurement of the effective magnetization \nMeff, which is equal to Ms−Hk, where Hk is the perpendicular anisotropy of the material. As a result of \nthe very small thickness of our samples, we expect a non -negligible value of Hk to result from the \ninterfaces of the material with the seed and capping layers7,8. Indeed, Meff in Figure S3 are smaller than \nMs, indicative of the presence of a perpendicular interface anisotropy. If the difference between Meff and \nMs is interfacial in origin, a thickness dependence of the Meff should produce the condition of Ms=Meff \nas the thickness t→ ∞. For a select subset of alloys, we varied the thickness of the alloy and measured \nMeff. From an extrapolation of thicknes s t→ ∞ we determined Ms. These values are included in Figure \nS3 and show reasonable agreement with the values of Ms determined from SQUID magnetometry, \ndemonstrating the equivalency of both measurement methods. \nThe minimum of the DOS curve at EF, shown in Figure 3, is coupled to the Slater -Pauling maximum of \nthe magnetization curve, which appears in the present work at Co concentration of approximately 25 %. \nFor this concentration the large peak in the DOS, immediately above the Fermi level, is entirely of spin-\ndown character, and alloys with higher Co concentration unavoidably populate this peak, which reduces \nthe magnetization. In the present work, and in several other theoretical9–11 and experimental12,13 works \nthe maximum of the magnetization (and the minimum in the density of states) appears between 20 % \nand 30 % Co, while in the work of Ebert14,15 and, thus, also in the calculations of Mankovsky16 it appears \nat lower Co concentrations of approximately 15 %. This may explain why in the work of Mankovsky16 \nthe theory predicts a minimum of the damping at this concentration. \n \n \nFigure S3: The variation of the saturation magnetization and effective magnetization with \nconcentration. Meff (black squares) determined from FMR and Ms (blue triangles) determined from \nSQUID magnetometry are plotted versus alloy concentration. For comparison , Ms is also determined \nby extrapolating the linear regression of Meff vs. 1/ t, as described in the text. The extrapolated values \nfor Ms are included in the plot (red crosses) and they match Ms determined by SQUID at those alloy \nconcentrations reasonably well . This shows that SQUID and FMR meas urement are consistent with \neach other. For comparison , the Slater -Pauling curve1 is also shown (gray dotted line). The crystalline \nphase of th e alloys are also indicated in the figure. \n \n \nCalculation and direct measurement of the radiative damping contribution . \n \nRadiative damping in the perpendicular FMR geometry, used in our study, arises from the inductive \ncoupling of the dynamic in -plane components of the magnetization to the wave -guide. By Faraday’s \nlaw, only the magnetization component perpendicular to the direction of the wave -guide can couple \neffectively, therefore the radiative damping is anisotropic and has to be calculated according ly. Thus, \nassuming a uniform magnetization profile and excitation field in the sample, the radiative damping αrad \nin our measurement is calculated via17 \n \n𝛼rad= 𝛾 𝜇02 𝑀𝑠𝑡𝑙\n8 𝑍0𝑤𝑐𝑐, (S1) \n \n \n \nwhere γ=gμB/ħ is the gyromagnetic ratio, Z0 =50 Ω the impedance, wcc is the width of the wave guide, \nand t and l are the thickness and length of the sample on the waveguide. This calculated value of αrad can \nalso be measured directly, by placing a spacer between sample and waveguide. Following the argument \nin Schoen et al.17, a 100 µm glass spa cer decreases αrad by approximately a factor of ten, making it , \nwithin errors , undetectable in our measurement. Indeed, when the spacer is inserted , the damping for \nboth the Co 20Fe80 and Co 25Fe75 decreases significantly, shown in Table S1, and the damping decrease \nmatches the calculated αrad = 0.00065 for both samples reasonably well . \n \n Sample α without spacer α with spacer αrad \nCo20Fe80 0.00230±0.00003 0.0016±0.00015 0.0007±0.00015 \nCo25Fe75 0.0021±0.00015 0.0018±0.0002 0.0003±0.00025 \n \nTable S1: Direct measurement of the radiative damping contribution . The damping of the \nCo20Fe80 and the Co 25Fe75 sample measured with and without a 100 µm spacer between sample \nand waveguide. The radiative damp ing αrad is determined from the difference in damping with \nand without spacer. \n \nDetermination of the interfacial damping enhancement \n \nThe flow of spin-angular momentum into adjacent layers (in our sample geometry the Cu/Ta cap and \nseed layers) further enhances the measured damping . This non -local damping or spin -pumping \ncontribution αsp is purely interfacial and can therefore be determined from the thickness dependence of \nthe damping for a given alloy concentration18. We measured the damping of thickness series of the pure \nelements and select alloy concentrations (20 % Co, 25 % Co, 50 % Co and 85 % Co). In order to correctly \naccount for the inte rfacial contribution αsp we first remove the radiative contribution αrad from αtot and in \nFigure S4a plot the dependence of (αtot - αrad ) on the inverse thickness 1 /t for select alloy concentrations. \nαsp is described by \n 𝛼sp= 2𝑔𝑒𝑓𝑓↑↓𝜇𝐵𝑔\n4𝜋𝑀𝑆𝑡. (S2) \nThe factor of 2 accounts for the two nominally equal interfaces of the alloy to the seed and cap layers. \nThe effective spin mixing conductance 𝑔eff↑↓ then is determined from the slope of the linear fits to the \n(αtot - αrad) vs. 1/ t data and plott ed in Fig ure S4b. All values of 𝑔eff↑↓ are in the range of expected values \nbased on previous reports19,20. Using interpola ted values of 𝑔eff↑↓ for all alloy concentrations (gray line in \nFigure S4b), the non -local interface effects for all alloy compositions are straightforwardly determined \nwith equation S2. \n \nFigure S4: Determination of the interfacial damping contribution . The (αtot - αrad) vs. 1/t dependence \n(symbols) for the pure elements and select alloy concentrations (20% Co, 25% Co, 50% Co and 85% \nCo) with fits to Equation 2 (lines) is plotted in a. From these fits the effective spin mixing conductance \ngeff↑↓ is calculated and plotted in b. The gray line is an interpolation of the calculated values. \n \n \nComparison of αint to other theories \n \nGilmore et al.21 calculated the damping as a funct ion of the electron -phonon self -energy Γ. They found \na minimum in damping for the pure elements bcc Fe (𝛼int=0.0013 ), hcp Co (𝛼int=0.0011 ) and \nfcc Ni (𝛼int=0.017). A comparison of these values to the extracted 𝛼𝑖𝑛𝑡 for bcc Fe 𝛼int=0.0024 ) and \nfcc Co (𝛼int=0.0026 ) shows that these calculated values are well below what we estimate to be the \nintrinsic damping. However, an adjustment of the electron -phonon self-energies Γ to 3 meV for Fe and \n2 meV for Co covers our findings. Nevertheless, we acknowledge that our sputtered films contain some \ndisorder and strain. Thus, it is conceivable that epitaxially grown, single crystal CoFe alloys may exhibit \neven lower values of damping. \n \n \n \n \n1. Bozorth, R. M. Ferromagnetism . (IEEE Press: 2003). \n2. Kief, M. T. & Egelhoff, W. F. Growth and structure of Fe and Co thin films on Cu(111), Cu(100), \nand Cu(110): A comprehensive study of metastable film growth. Phys. Rev. B 47, 10785 –10814 \n(1993). \n3. Pelzl, J. et al. Spin-orbit -coupling effects on g -value and damping factor of the ferromagnetic \nresonance in Co and Fe films. Journal of Physics: Condensed Matter 15, S451 (2003). \n4. Tischer, M. et al. Enhancement of Orbital Magnetism at Surfaces: Co on Cu(100). Phys. Rev. Lett. \n75, 1602 –1605 (1995). \n5. Pauling, L. The Nature of the Interatomic Forces in Metals. Phys. Rev. 54, 899 –904 (1938). \n6. Slater, J. C. Electronic Structure of Alloys. Journal of A pplied Physics 8, 385 –390 (1937). \n7. Farle, M., Mirwald -Schulz, B., Anisimov, A. N., Platow, W. & Baberschke, K. Higher -order \nmagnetic anisotropies and the nature of the spin -reorientation transition in face -centered -tetragonal \nNi(001)/Cu(001). Phys. Rev. B 55, 3708 –3715 (1997). \n8. Shaw, J. M., Nembach, H. T. & Silva, T. J. Measurement of orbital asymmetry and strain in \nCo_90Fe_10/Ni multilayers and alloys: Origins of perpendicular anisotropy. Phys. Rev. B 87, \n054416 (2013). \n9. James, P., Eriksson, O., Johansson, B. & Abrikosov, I. A. Calculated magnetic properties of binary \nalloys between Fe, Co, Ni, and Cu. Phys. Rev. B 59, 419 –430 (1999). \n10. Richter, R. & Eschrig, H. Spin -polarised electronic structure of disordered BCC FeCo alloys from \nLCAO -CPA. Journal of Physics F: Metal Physics 18, 1813 (1988). \n11. Wu, D., Zhang, Q., Liu, J. P., Yuan, D. & Wu, R. First -principles prediction of enhanced magnetic \nanisotropy in FeCo alloys. Applied Physics Letters 92, (2008). \n12. Paduani, C. & Krause, J. C. Electronic structure and magnetization of Fe -Co alloys and multilayers. \nJournal of Applied Physics 86, 578 –583 (1999). \n13. Stearns, M. B. Magnetic Properties of 3d, 4d and 5d Elements, Alloys and Compounds . (Springer \nBerlin: 1984). \n14. Chadov, S. et al. Orbital magnetism in transition metal systems: The role of local correlation \neffects. EPL 82, 37001 (2008). \n15. Ebert, H. & Battocletti, M. Spin and orbital polarized relativistic multiple scattering theory - With \napplications to Fe, Co, Ni and Fe(x)Co(1 -x). Solid State Communications 98, 785 –789 (1996). \n16. Mankovsky, S., Ködderitzsch, D., Woltersdorf, G. & Ebert, H. First -principles calculation of the \nGilbert damping parameter via the linear response formalism with application to magneti c \ntransition metals and alloys. Phys. Rev. B 87, 014430 (2013). \n17. Schoen, M. A. W., Shaw, J. M., Nembach, H. T., Weiler, M. & Silva, T. J. Radiative damping in \nwaveguide -based ferromagnetic resonance measured via analysis of perpendicular standing spin \nwaves in sputtered permalloy films. Phys. Rev. B 92, 184417 (2015). \n18. Tserkovnyak, Y., Brataas, A. & Bauer, G. E. W. Enhanced Gilbert Damping in Thin Ferromagnetic \nFilms. Phys. Rev. Lett. 88, 117601 (2002). \n19. Czeschka, F. D. et al. Scaling Behavior of t he Spin Pumping Effect in Ferromagnet -Platinum \nBilayers. Phys. Rev. Lett. 107, 046601 (2011). \n20. Weiler, M., Shaw, J. M., Nembach, H. T. & Silva, T. J. Detection of the DC Inverse Spin Hall \nEffect Due to Spin Pumping in a Novel Meander -Stripline Geometry. Magnetics Letters, IEEE 5, \n1–4 (2014). \n21. Gilmore, K., Idzerda, Y. U. & Stiles, M. D. Identification of the Dominant Precession -Damping \nMechanism in Fe, Co, and Ni by First -Principles Calculations. Phys. Rev. Lett. 99, 027204 (2007). \n \n " }, { "title": "1311.1778v1.Spin_Orbit_Torques_and_Anisotropic_Magnetization_Damping_in_Skyrmion_Crystals.pdf", "content": "arXiv:1311.1778v1 [cond-mat.mes-hall] 7 Nov 2013Spin-Orbit Torques and Anisotropic Magnetization Damping in Skyrmion Crystals\nKjetil M. D. Hals and Arne Brataas\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway\nThe length scale of the magnetization gradients in chiral ma gnets is determined by the relativistic\nDzyaloshinskii-Moriya interaction. Thus, even conventio nal spin-transfer torques are controlled by\nthe relativistic spin-orbit coupling in these systems, and additional relativistic corrections to the\ncurrent-induced torques and magnetization damping become important for a complete understand-\ning of the current-driven magnetization dynamics. We theor etically study the effects of reactive and\ndissipative homogeneous spin-orbit torques and anisotrop ic damping on the current-driven skyrmion\ndynamics in cubic chiral magnets. Our results demonstrate t hat spin-orbit torques play a significant\nrole in the current-inducedskyrmion velocity. The dissipa tive spin-orbit torque generates a relativis-\ntic Magnus force on the skyrmions, whereas the reactive spin -orbit torque yields a correction to both\nthe drift velocity along the current direction and the trans verse velocity associated with the Magnus\nforce. The spin-orbit torque corrections to the velocity sc ale linearly with the skyrmion size, which\nis inversely proportional to the spin-orbit coupling. Cons equently, the reactive spin-orbit torque\ncorrection can be the same order of magnitude as the non-rela tivistic contribution. More impor-\ntantly, the dissipative spin-orbit torque can be the domina nt force that causes a deflected motion of\nthe skyrmions if the torque exhibits a linear or quadratic re lationship with the spin-orbit coupling.\nIn addition, we demonstrate that the skyrmion velocity is de termined by anisotropic magnetization\ndamping parameters governed by the skyrmion size.\nI. INTRODUCTION\nThe manipulation of submicron-scale magnetic el-\nements via electric currents has paved the way for\na promising new class of magnetoelectronic devices\nwith improved scalability, faster execution time, and\nlower power consumption.1,2The dominant mechanism\nof current-driven magnetic excitations in conventional\nmetallic ferromagnets is the spin-transfer-torque (STT)\neffect,3,4in which spin angular momentum is transferred\nfrom spin-polarized currents to the magnetization, gen-\nerating the torque τ(r,t) on the magnetization:1,2\nτ(r,t) =−(1−βm×)(vs·∇)m. (1)\nHere, the vector vsis proportional to the out-of-\nequilibrium current density Jand the spin polarization\nof the current. The first term in Eq. (1) describes the\nreactive STT, whereas the second term, which is propor-\ntional to β, describes the dissipative STT.\nIn systems that lack spatial-inversion symmetry, an\nalternative manner of generating current-induced mag-\nnetization torques is via relativistic intrinsic spin-\norbit coupling (SOC). Orbital momentum is (via SOC)\ntransferred to the spins, causing a so-called spin-\norbit torque (SOT) on the magnetization.5–21Recently,\nSOTs have been observed to lead to remarkably effi-\ncient current-driven magnetization dynamics in ultra-\nthin magnetic films and strained ferromagnetic semicon-\nductors.7–13,18–20,22,23Similar to the STT in Eq. (1),\nsuch SOTs also have reactive and dissipative contribu-\ntions.21,24Although the reactive homogeneous SOT is\nknown to scale linearly with the SOC to the lowest order,\n6,7there are only a few theoretical works regarding the\ndissipative SOT that typically predict it to be smaller.24\nHowever, a recent experiment and theory demonstrate\nthatthedissipativehomogeneousSOTcanbeofthesameorder of magnitude as the reactive part.21SOTs can en-\nable the design of significantly simpler devices because\nthe torques originate from a direct conversion of orbital\nangularmomentum intospinexcitationsandappeareven\nin homogenous ferromagnets with no external sources of\nspin-polarized currents.\nMagnetic skyrmions are vortex-like spin configura-\ntions that cannot be continuously deformed into a ho-\nmogeneous magnetic state.25Experimentally, magnetic\nskyrmion phases were first reported in bulk chiral mag-\nnets,26–32and more recently, they also have been ob-\nserved in magnetic thin films.33–38Under the applica-\ntion of a weak external magnetic field, a chiral mag-\nnetic system crystallizes into a two-dimensional lattice\nof skyrmions in which the magnetic moment at the\ncore of each vortex is antiparallel to the applied field,\nwhereas the peripheral magnetic moments are parallel.\nIn ultra-thin magnetic films, the skyrmion phase be-\ncomes more energetically favorable with decreasing film\nthickness,34–36,38,39and for a single atomic layer of Fe, a\nskyrmion structure at the atomic length scale has been\nobserved even in the absence of an external magnetic\nfield.34\nIn the context of spintronic applications, promising\ncharacteristics of skyrmions are the extremely low depin-\nningcurrents31,32thatarerequiredtomovethem andthe\nfact that they avoid pinning centers.40A proposed expla-\nnation of the latter feature is that the Magnus force acts\non the skyrmions, leading to a deflection of their mo-\ntion.40,41This force is closely linked to the topological\nHall effect and can be viewed as the reaction of the fic-\ntitious Lorentz force experienced by the itinerant quasi-\nparticles as their spins adiabatically align with the local\nmagnetization direction.\nThe underlying physical mechanism that gives rise to\nthe skyrmion phase is the Dzyaloshinskii-Moriya inter-2\naction (DMI).42,43The DMI has the same relativistic\norigin as the homogeneous SOTs observed in ferromag-\nnetic heterostructures and strained ferromagnetic semi-\nconductors. They both arise from the combined effect\nof spin-orbit coupling and broken spatial-inversion sym-\nmetry. Neglected in earlier works concerning current-\ninduced skyrmion motion, a reactive SOT in chiral mag-\nnets was predicted and studied for systems in the helical\nphase in Ref. 44. However, the effect of reactive and dis-\nsipative SOTs on the skyrmion dynamics in bulk chiral\nmagnets remains unknown. Such studies are important\nbecauseSOTs canbe equallyasimportantas the conven-\ntional STTs in Eq. (1). This equal importance is because\nthe typical length scale of the spatial variations of the\nmagnetization is determined by the DMI, i.e., the mag-\nnetization gradients ∂imjscale as ∂imj∼D/J, where\nDis the DMI parameter and Jis the spin stiffness. Be-\ncauseDis linear in the SOC, even the non-relativistic\nSTT in Eq. (1) is proportional to the SOC to the lowest\norder in the relativistic corrections. Thus, the SOTs in\nchiral magnets can be of the same order of magnitude as\nthe conventional STT. Thus, a complete understanding\nof the current-drivendynamics of chiral magnets requires\na correct treatment of SOC effects on both the current-\ninduced torques and magnetization damping.\nThe magnetization dynamics driven by currents or ex-\nternal fields strongly depends on dissipation. In isotropic\nsystems, the damping can typically be assumed to be de-\ncoupled fromthe magnetizationdirectionand itstexture.\nHowever, this model does not necessarily extend to chi-\nral magnets. First, the broken spatial-inversion symme-\ntry allows for terms that are linear, not only quadratic,\nin the magnetization gradients. Second, chiral magnets\nhave a preferred direction, and thus, the dissipation is\nlikely not isotropic and independent of the texture struc-\nture.\nIn this paper, we present a theoretical study of the\ncurrent-induceddynamicsofskyrmionsthat correctlyac-\ncounts for the effects of SOC on both current-induced\ntorques and magnetization damping. Our results demon-\nstrate that SOC generates reactive and dissipative ho-\nmogeneous SOTs that lead to important corrections to\nthe drift velocity along the current direction and to the\nMagnus force. Another essential consequence of the SOC\nis that the skyrmions experience effective damping and\ntorque parameters that depend on the current direction\nrelative to the crystallographic axes.\nThis paper is organized as follows. Section II intro-\nduces the phenomenology of SOTs that was presented in\nRef. 45 and performs a similar phenomenological expan-\nsion for the Gilbert damping tensor. In Section III, we\napplythe phenomenologytothe studyofcurrent-induced\nskyrmion dynamics in cubic chiral magnets and derive a\ncollective coordinate description for the skyrmion veloc-\nity. Our results are summarized in Section IV.II. PHENOMENOLOGICAL EXPANSION\nWe include the SOC effects phenomenologically by de-\nriving an equation for the magnetization dynamics that\nsatisfies the symmetry of the underlying crystal struc-\nture. In deducing expressions for the torques and dis-\nsipation, we perform a phenomenological expansion of\nthe magnetization-damping tensor and current-induced\ntorques in terms of the magnetization and its gradients.\nThe magnetic system is assumed to satisfy the local ap-\nproximation, in which the magnetization dynamics de-\npends only on the local properties of the system. In\nthis approximation, the magnetization dynamics can be\nphenomenologically expressed in terms of the Landau-\nLifshitz-Gilbert-Slonczewski (LLGS) equation:\n˙m=−γm×Heff+m×α˙m+τ. (2)\nHere,m(r,t) is the unit vector of the magnetization\nM(r,t) =Msm(r,t),γis (minus) the gyromagnetic ra-\ntio, and Heff(r,t) =−δF[M]/δMis the effective field\ndetermined by the functional derivative of the magnetic\nfree-energy functional F[M] =/integraltext\ndrF.F(r,t) is the\nfree-energy density. The second term on the right side\nof Eq. (2) describes the magnetization damping, where\nα(r,t) is a symmetric, positive-definite second-rank ten-\nsor that depends on the local magnetization direction\nand local magnetization gradients. The torque τ(r,t) in\nEq. (2) represents the current-induced torques.\nThe free-energy density of a magnetic system is, to the\nsecond order in the magnetization gradients,\nF=Jij∂iM·∂jM+DijkMi∂jMk+K(1)\nijMiMj+\nK(2)\nijklMiMjMkMl−M·B. (3)\nThe tensor Jijis the spin stiffness, Dijkdescribes the\nDMI, and Bis an external magnetic field. In an\ninversion-symmetric system, Dijk= 0. The two terms\nproportional to K(1)\nijandK(2)\nijklrepresent the two first\nharmonics in the phenomenological expansion of the\nmagnetocrystalline anisotropy energy. Jij,Dijk,K(1)\nij,\nandK(2)\nijklare polar tensors that are invariant under the\npoint group of the system.46In the above equation and\nin what follows, we assume summation over repeated in-\ndices, and ∂iis a shorthand notation for ∂/∂ri.\nA similar phenomenological expansion can be per-\nformed for the Gilbert damping tensor α(r,t). To the\nsecond order in the spatial magnetization gradients, we\nobtain\nαij=α(0)\nij+α(1)\nijklmkml+α(2)\nijklpmk∂lmp+\nα(3)\nijklpq∂kml∂pmq+α(4)\nijklpqmk∂l∂pmq.(4)\nAgain, the tensors α(0)\nij,α(1)\nijkl,α(2)\nijklp, andα(3,4)\nijklpqare in-\nvariant under the point group of the system, and the\ntensorα(2)\nijklponlyappearsinsystemswith brokenspatial-\ninversion symmetry, such as chiral magnets, the focus of3\nthis study. Terms that are odd under time reversal do\nnot appear in the expansion because such terms do not\nrepresent dissipative processes.\nRecently, in Ref. 45, we carried out a phenomenologi-\ncal expansion of the current-induced torque τ(r,t). The\nstarting point of the derivation is to write down the most\ngeneral form of the torque in the linear-response regime\nand in the local approximation:\nτ(r,t) =m×ηJ. (5)\nThe second-rank tensor ( η[m,∇m])ij, which we refer to\nas thefieldance tensor , depends on the local magnetiza-\ntion direction and local gradients of the magnetization.\nThe fieldance tensor contains all information concerning\nthe local torque. The general form of the fieldance ten-\nsor is not known in systems with strong SOC, but recent\nexperiments indicate that it has a complicated struc-\nture.18–21Therefore, we expand the fieldance tensor in\npowers of miand∂jmito find simplified expressions for\nthe allowed torques:\nηij= Λ(r)\nij+Λ(d)\nijkmk+βijkl∂kml+Pijklnmk∂lmn+...(6)\nThe first and second terms in Eq. (6) represent the re-\nactive and dissipative homogeneous SOTs, respectively,\nthat recently have been observed experimentally.7,8,21\nThese terms are only present in systems with broken\nspatial-inversionsymmetry. ThelasttwotermsinEq.(6)\nrepresent a generalization of the torque in Eq. (1). In the\nnon-relativisticlimit, in which the systemis invariantun-\nder separate rotations of the spin space and coordinate\nspace, the βijklterm and Pijklnterm are equal to the\ndissipative and reactive STTs in Eq. (1), respectively.45\nWhen SOC is included, the forms of the tensors in\nEq. (6) are, as for the free energy and magnetization\ndamping, determined by the crystal symmetry: Λ(r)\nijand\nPijklnbecome invariant axial tensors of the point group,\nwhereas Λ(d)\nijkandβijklbecome invariant polar tensors of\nthe point group. Higher-order terms in the expansion of\nthe fieldancetensorrepresenttorqueswith higherdegrees\nof anisotropy. However, the leading-order terms explic-\nitly written in Eq. (6) provide a sufficient description of\nthe current-driven dynamics for many materials.\nIII. CHIRAL MAGNETS AND SKYRMION\nDYNAMICS\nWe now focus on bulk magnets with cubic B20-type\ncrystal structures. Examples of such magnets include\nMnSi, FeGe, and (Fe,Co)Si. The symmetry of these sys-\ntems is described by the tetrahedral point group T (in\nthe Sch¨ onflies notation).\nWe begin by writing down the specific form of the in-\nvariant tensors in systems described by the point group\nT.47,48The invariant tensors lead to phenomenological\nexpressions for the free energy, Gilbert damping tensor,and current-induced torque given in Sec. II. A collec-\ntive coordinate description is then applied to model the\ncurrent-driven dynamics of the magnetic system.\nA. Invariant tensors\nThe point group T is the smallest of the five cubic\npoint groups and consists of 12 proper rotation opera-\ntors: the identity operator; three two-fold rotation op-\nerators about the x, y, and z axes; and eight three-fold\nrotation operators about the body diagonals of the cube.\nBecause the group only consists of proper rotations, the\ninvariant axial and polar tensors have the same form.46\nIn the present study, expressions are required for all ten-\nsors written explicitly in Section II, with the exception of\nα(3)andα(4)in Eq. (4) because we expand the Gilbert\ndamping tensor to the lowest order in the magnetization\ngradients. Thus, we will consider invariant tensors up to\nthe fifth rank.\nInvariant second-rank tensors Tijare described by one\nindependent tensor coefficient:\nTij=Tδij. (7)\nHere,δijis the Kronecker delta. Invariant third-rank\ntensors are described by two independent tensor coeffi-\ncients, and their non-vanishing elements satisfy the two\nsymmetry relations\nTxyz=Tyzx=Tzxy, (8)\nTxzy=Tyxz=Tzyx. (9)\nInvariant fourth-rank tensors are described by seven in-\ndependent coefficients that satisfy the relations\nTxxxx=Tyyyy=Tzzzz, (10)\nTxxyy=Tyyzz=Tzzxx(p= 3), (11)\nTxxzz=Tyyxx=Tzzyy(p= 3), (12)\nwherep= 3 indicates the three different symmetry\nrelations obtained from the expression by holding the\nfirst index constant and permuting the last three in-\ndices. For example, Eq. (11) also yields the relations\nTxyyx=Tyzzy=TzxxzandTxyxy=Tyzyz=Tzxzx.\nInvariant fifth-rank tensors are determined by 20 inde-\npendent tensorcoefficients whose non-vanishingelements\nsatisfy the symmetry relations\nTxxxyx=Tyyyzx=Tzzzxy(p= 20).(13)\nHere,p= 20 refers to the 20 independent symmetry re-\nlations obtained from Eq. (13) via permutations of the\nfive indices.\nThere are 68 independent tensor coefficients that de-\ntermine the required free energy, torques, and damping\nto describe current-induced skyrmion motion. Whereas\nthe number of parameters might appear overwhelming,\nwe demonstrate below that only certain combinations of\nthese parameters appear in the final results, which are\nmore transparent.4\nB. Collective coordinate description\nWe apply a collective coordinate description to model\nthe current-driven magnetization dynamics.49The mag-\nnetic state is assumed to be parameterized by a set of\ntime-dependent collective coordinates {ai(t)|i= 1,2,...}\nsuch that m(r,t) =m(r,{ai(t)}). The equations of mo-\ntion for the collective coordinates are then given by\n(Γij−Gij) ˙aj=γFi+Li. (14)\nHere, the matrices Γ ijandGijareGij=/integraltext\nm·\n[(∂m/∂ai)×(∂m/∂aj)]dV and Γ ij=/integraltext\n(∂m/∂ai)·\nα(∂m/∂aj)dV, and Fiis the force attributable to the ef-\nfective field, Fi=/integraltext\nHeff·(∂m/∂ai)dV. The force on the\ncollectivecoordinatesattributable tothe current-induced\ntorque is represented by Li=/integraltext\n(∂m/∂ai)·[m×τ]dV.\nWe compute the above matrices and vectors, which are\ngoverned by the 68 independent tensor coefficients dis-\ncussed in the previous section, and determine the rate of\nchange of the collective coordinates.\nC. Equations of motion\nA current density J= [Jx,Jy,Jz] is applied to the\nsystem, and an external magnetic field is applied along\nthezaxis such that a lattice of skyrmions forms in the\nxyplane. The skyrmion lattice is assumed to be undis-\ntorted during the current-driven dynamics. In this ap-\nproximation, we can disregard rotations of the magnetic\ntexture50, and the skyrmions can be considered a lattice\nof non-interacting particles, where each skyrmion is de-\nscribed by\nmx=2qR(y−ry)\n(x−rx)2+(y−ry)2+R2,(15)\nmy=−2qR(x−rx)\n(x−rx)2+(y−ry)2+R2,(16)\nmz=−q(x−rx)2+(y−ry)2−R2\n(x−rx)2+(y−ry)2+R2.(17)\nThe two-dimensional center-of-mass coordinates rxand\nryarethecollectivecoordinatesthatdescribethedynam-\nical evolution of each skyrmion. Distortions introduce an\nadditional collective coordinate, which describes the ro-\ntational motion of the skyrmions.50The parameters R\nandq∈ {1,−1}are the size and topological charge of\nthe skyrmion, respectively.\nDisregarding terms that are of second order in the\nGilbert damping parameters, the collective coordinate\nformulas presented in Section IIIB generate the veloc-ity of the center of mass:\n/parenleftbigg\n˙rx\n˙ry/parenrightbigg\n=−/parenleftbigg/parenleftbig\nPeff\ny+RΛeff\nr/parenrightbig\nJx /parenleftbig\nPeff\nx+RΛeff\nr/parenrightbig\nJy/parenrightbigg\n+ (18)\nq/parenleftbigg−/parenleftbig\nPeff\nyβeff\ny−Peff\nxαeff\ny/parenrightbig\nJy /parenleftbig\nPeff\nxβeff\nx−Peff\nyαeff\nx/parenrightbig\nJx/parenrightbigg\n+\nqR\n−/parenleftBig\nΛeff\nd,y−Λeff\nrαeff\ny/parenrightBig\nJy/parenleftBig\nΛeff\nd,x−Λeff\nrαeff\nx/parenrightBig\nJx\n.\nHere,Peff\nxandPeff\ny(Peff\nxβeff\nxandPeff\nyβeff\ny) areeffective re-\nactive (dissipative) STT parameters that are linear com-\nbinationsofthe tensorcoefficients Pijkln(βijkln) andαeff\nx\nandαeff\nyare effective Gilbert damping parameters that\nare linear combinations of the tensor coefficients α(0)\nij,\nα(1)\nijkl, andα(2)\nijklp. Theparameters βeff\nxandβeff\nyaredimen-\nsionless and determined by the ratio between the dissipa-\ntive and reactive STT. Their magnitude is proportional\nto the spin-flip rate, which is of second order in the SOC.\nThe effective Gilbert damping parameters depend on the\nskyrmion size Rin combination with the α(2)\nijklptensor,\ni.e., the parameters can be decomposed into two terms,\nαeff\nx,y=α′\nx,y+R−1α′′\nx,ywhereα′′\nx,ydepends on α(2)\nijklp. The\neffective reactive and dissipative homogeneous SOT pa-\nrameters Λeff\nr, Λeff\nd,x, and Λeff\nd,yare proportional to the ten-\nsorcoefficientsofΛ(r)\nijandΛ(d)\nijk. The effective parameters\nin Eq. (18) determine the current-driven magnetization\ndynamics and can be extracted from experiments. Their\nexplicit forms and relations to the invariant tensors are\ngiven in Appendix A.\nTo revealthe effects ofthe SOC, let us comparethe ex-\npression for the velocity of the center of mass in Eq. (18)\nwith the conventional expression for the velocity in the\nnon-relativistic limit, in which the homogeneous SOTs\nvanish, and the STT and the Gilbert damping parame-\nters satisfy the symmetry relations Peff≡Peff\nx=Peff\ny,\nβeff≡βeff\nx=βeff\ny, andαeff≡αeff\nx=αeff\ny:\n˙r=−PeffJ+qPeff(βeff−αeff)ˆz×J.(19)\nComparingEqs.(18)andEq.(19)indicatesthattheSOC\nintroduces several important effects on the skyrmion dy-\nnamics.\nFirst, the equations of motion in Eq. (18) are no longer\nrotationally symmetric about the zaxis. Clearly, the ef-\nfective torque and damping parameters that govern the\nmotion along the xandyaxes differ because there are no\nsymmetry operations that relate the two axes. Thus, dif-\nferent current-driven velocities can be observed for cur-\nrents applied along the two directions, and a measure-\nment of this velocity anisotropy provides a simple test\nfor investigating the importance of the SOC. A current\nalong the zaxis does not influence the velocity in the\nlinear-response regime.\nSecond, the SOTs strongly affect the skyrmion mo-\ntion along the current direction. The reactive homoge-\nneousSOTleadstoarenormalizationofthe drift velocity5\nalong the current direction that scales linearly with the\nskyrmion size R. The reason for this linear dependency\nis that the homogeneous SOTs do not depend on the\nmagnetization gradients. Thus, the homogeneous SOTs\ncouple more strongly to largerskyrmions, whose textures\nare distributed over a larger spatial region. Because Λeff\nr\nis linear in the SOC, whereas Rscales as the inverse of\nthe SOC (and thus the product RΛeff\nris independent of\nthe SOC), the reactive SOT contribution to the drift ve-\nlocity can be of the same orderof magnitude as the terms\nthat are induced by the reactive STT, i.e., the terms pro-\nportional to Peff\nxandPeff\ny.\nThird, SOTs also strongly influence the Magnus force.\nThe terms proportional to qin Eqs. (18)-(19) repre-\nsent the transverse drift velocity induced by the Mag-\nnus force. Both the reactive and dissipative homogenous\nSOTs produce corrections to the Magnus-force-induced\nmotion that scale linearly with R. Using the same ar-\nguments as above, the reactive SOT yields a transverse\ndrift velocity ∼RΛeff\nrαeff\nithat can be of the same or-\nder of magnitude as the non-relativistic terms ∼Peff\niβeff\ni\nand∼Peff\niαeff\ni(i∈ {x,y}) ifβeff\ni<<1. The most in-\nteresting observation from Eq. (18) is the contribution\nof the dissipative homogeneous SOT to the transverse\nvelocity. In contrast to the terms that arise from the\nSTTs and reactive SOT, the dissipative SOT produces\na transverse velocity that depends neither on damping\nparameters nor the dimensionless effective βparameters.\nThe velocity is solely determined by the values of RΛeff\nd,x\nandRΛeff\nd,y. There is little knowledge regarding the mag-\nnitude of the dissipative homogeneous SOT and how it\ndepends on the SOC in chiral magnets. However, a re-\ncent experiment concerning (Ga,Mn)As indicated that\nthe dissipative part can be comparable in magnitude to\nthe reactive part. If the same result is applicable to chi-\nral magnets, the dissipative homogeneous SOT provides\nthe largest contribution to the transverse drift velocity\nand is the dominant driving force that causes deflected\nmotion of the skyrmions. This relativistic Magnus force\nis not linked to the fictitious magnetic field generated by\nthe spin texture but instead arises from the dissipative\npart of the out-of-equilibrium spin density generated by\nthe SOC combined with an applied electric field.\nThe three independent tensor coefficients that de-\nscribe the reactive and dissipative homogeneous SOTs\ncan be extractedfrom spin-orbitferromagneticresonance\n(FMR) measurements.11,21An external magnetic field is\nused to align the magnetization along different directions\nrelative to the bar direction, and an alternating current\nis applied to produce microwave SOTs within the sample\nthat resonantly drive the magnetization. The reflected\ndirect current contains information regarding the magni-\ntudeoftheSOTs. Webelievethatsuchameasurementof\nthe SOTs will be one of most interesting tasks for future\nexperimental work concerning chiral magnets.\nJust prior to the submission of our paper, a relatedtheoreticalworkconcerningSOTsandskyrmionsin mag-\nnetic thin films waspresented.51However,that workcon-\nsiders a different symmetry class that is intended for the\ndescription of ultra-thin ferromagnetic heterostructures,\nin which there is complete rotational symmetry and bro-\nken spatial-inversion symmetry along a transverse direc-\ntion. Both reactive and dissipative Rashba SOTs are\nconsidered in their study, and they demonstrate that the\nSOTs also play a significant role in the skyrmion dy-\nnamics of these systems. However, only isotropic and\nspatially independent damping is considered.\nIV. SUMMARY\nIn summary, we studied the effects of SOC on the\ncurrent-driven dynamics of skyrmions in cubic chiral\nmagnets. We performed a phenomenological expan-\nsion of the Gilbert damping tensor and current-induced\ntorques that accounts for the relativistic SOC effects. A\ncollective-coordinate description was applied to model\nthe current-induced motion of the skyrmions. Our re-\nsults demonstrated that the skyrmion velocity depends\non the direction of the applied current relative to the\ncrystallographic axes and that the SOTs contribute sig-\nnificantly to the current-induced velocity. The reactive\nSOT induces a correction to both the parallel and trans-\nverse drift velocities of the skyrmions that is of the same\norder of magnitude as the non-relativistic contributions.\nIf the dissipative SOT exhibits a linear or quadratic rela-\ntionship with the SOC, it produces a relativistic Magnus-\nforce motion that is larger than the transverse drift ve-\nlocity induced by conventional STTs. The SOTs cannot\nbe neglected in the modeling of current-driven skyrmion\ndynamics because they do not depend on the gradients\nof the magnetization and couple more strongly to larger\nskyrmions.\nV. ACKNOWLEDGMENTS\nK.M.D.H. wouldliketothankRembertDuine forstim-\nulating discussions of SOTs in chiral magnets. K.M.D.H.\nand A.B. would like to thank Jairo Sinova for notifying\nus of Ref. 5 and its important contribution to the theory\nof SOTs.\nAppendix A: Effective Parameters\nBelow, we provide the expressions for the effective\ntorque and damping parameters in terms of the tensor\ncoefficients of the invariant tensors given in Section II.\nThe effective damping parameters are6\nαeff\nx=1\n60(60a(0)+12a(1)\nxxxx+19a(1)\nxxyy+29a(1)\nxxzz−10a(1)\nxyxy−10a(1)\nxyyx−2a(1)\nxzxz−2a(1)\nxzzx)−\n1\n30R(3a(2)\nxxxyz−5a(2)\nxxxzy−9a(2)\nxxyxz+10a(2)\nxxyzx+7a(2)\nxxzxy−6a(2)\nxxzyx+8a(2)\nxyxxz−6a(2)\nxyxzx−\n2a(2)\nxyyyz+4a(2)\nxyyzy−2a(2)\nxyzxx+2a(2)\nxyzzz+2a(2)\nxzxxy−2a(2)\nxzxyx−2a(2)\nxzyxx+2a(2)\nxzyyy), (A1)\nαeff\ny=1\n60(60a(0)+12a(1)\nxxxx+29a(1)\nxxyy+19a(1)\nxxzz−6a(1)\nxyxy−6a(1)\nxyyx−6a(1)\nxzxz−6a(1)\nxzzx)−\n1\n30R(5a(2)\nxxxyz−3a(2)\nxxxzy−7a(2)\nxxyxz+6a(2)\nxxyzx+9a(2)\nxxzxy−10a(2)\nxxzyx+2a(2)\nxyxzx−2a(2)\nxyyyz+\n2a(2)\nxyzxx−2a(2)\nxyzzz−6a(2)\nxzxxy+6a(2)\nxzxyx+2a(2)\nxzyxx−2a(2)\nxzyyy−4a(2)\nxzzyz). 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A. Tretiakov, D. Clarke, Gia-Wei Chern, Ya. B. Baza-\nliy, and O. Tchernyshyov, Phys. Rev. Lett. 100, 127204\n(2008).\n50K. Everschor, M. Garst, R. A. Duine, and A. Rosch, Phys.\nRev. B84, 064401 (2011).\n51M. E. Knoester, J. Sinova, and R. A. Duine,\narXiv:1310.2850." }, { "title": "2105.03576v1.A_second_order_numerical_method_for_Landau_Lifshitz_Gilbert_equation_with_large_damping_parameters.pdf", "content": "A SECOND-ORDER NUMERICAL METHOD FOR\nLANDAU-LIFSHITZ-GILBERT EQUATION WITH LARGE\nDAMPING PARAMETERS\nYONGYONG CAI, JINGRUN CHEN, CHENG WANG, AND CHANGJIAN XIE\nAbstract. A second order accurate numerical scheme is proposed and imple-\nmented for the Landau-Lifshitz-Gilbert equation, which models magnetization\ndynamics in ferromagnetic materials, with large damping parameters. The\nmain advantages of this method are associated with the following features:\n(1) It only solves linear systems of equations with constant coe\u000ecients where\nfast solvers are available, so that the numerical e\u000eciency has been greatly im-\nproved, in comparison with the existing Gauss-Seidel project method. (2) The\nsecond-order accuracy in time is achieved, and it is unconditionally stable for\nlarge damping parameters. Moreover, both the second-order accuracy and the\ngreat e\u000eciency improvement will be veri\fed by several numerical examples\nin the 1D and 3D simulations. In the presence of large damping parameters,\nit is observed that this method is unconditionally stable and \fnds physically\nreasonable structures while many existing methods have failed. For the do-\nmain wall dynamics, the linear dependence of wall velocity with respect to the\ndamping parameter and the external magnetic \feld will be obtained through\nthe reported simulations.\n1.Introduction\nFerromagnetic materials are widely used for data storage due to the bi-stable\nstates of the intrinsic magnetic order or magnetization. The dynamics of magneti-\nzation has been modeled by the Landau-Lifshitz-Gilbert (LLG) equation [9,13]. In\nparticular, two terms are involved in the dynamics of the LLG equation: the gyro-\nmagnetic term, which is energetically conservative, and the damping term, which\nis energetically dissipative.\nThe damping term is important since it strongly a\u000bects the energy required and\nthe speed at which a magnetic device operates. A recent experiment on a magnetic-\nsemiconductor heterostructure [25] has indicated that the Gilbert damping constant\ncan be adjusted. At the microscopic level, the electron scattering, the itinerant\nelectron relaxation [11], and the phonon-magnon coupling [16, 17] are responsible\nto the damping, which can be obtained from electronic structure calculations [19].\nFor the application purpose, tuning the damping parameter allows one to optimize\nthe magneto-dynamic properties in the material, such as lowering the switching\ncurrent and increasing the writing speed of magnetic memory devices [23].\nWhile most experiments have been devoted to small damping parameters [4,14,\n22], large damping e\u000bects are observed in [10,18]. The magnetization switching time\nDate : May 11, 2021.\n2010 Mathematics Subject Classi\fcation. 35K61, 65N06, 65N12.\nKey words and phrases. Micromagnetics simulations, Landau-Lifshitz-Gilbert equation,\nsecond-order method, large damping parameter.\n1arXiv:2105.03576v1 [physics.comp-ph] 8 May 20212 Y. CAI, J. CHEN, C. WANG, AND C. XIE\ntends to be shorter in the presence of the large damping constant [18]. Extremely\nlarge damping parameters ( \u00189) are presented in [10].\nThe LLG equation is a vectorial and nonlinear system with the \fxed length of\nmagnetization in a point-wise sense. Signi\fcant e\u000borts have been devoted to design\ne\u000ecient and stable numerical methods for micromagnetics simulations; see [6, 12]\nfor reviews and references therein. Among the existing numerical works, semi-\nimplicit schemes have been very popular since they avoid a complicated nonlinear\nsolver while preserving the numerical stability; see [2, 7, 24], etc. In particular,\nthe second-order accurate backward di\u000berentiation formula (BDF) scheme is con-\nstructed in [24], with a one-sided interpolation. In turn, a three-dimensional lin-\near system needs to be solved at each time step, with non-constant coe\u000ecients.\nMoreover, a theoretical analysis of the second order convergence estimate has been\nestablished in [5] for such a BDF2 method. As another approach, a linearly implicit\nmethod in [2] introduces the tangent space to deal with the length constraint of\nmagnetization, with the \frst-order temporal accuracy. As a further extension, high-\norder BDF schemes have been constructed and analyzed in a more recent work [1].\nAn unconditionally unique solvability of the semi-implicit schemes has been proved\nin [1,5], while the convergence analysis has required a condition that the temporal\nstep-size is proportional to the spatial grid-size. However, an obvious disadvantage\nhas been observed for these semi-implicit schemes: the vectorial structure of the\nLLG equation leads to a non-symmetric linear system at each time step, which\ncannot be implemented by an FFT-based fast solver. In fact, the GMRES is often\nused, while its e\u000eciency depends heavily on the temporal step-size and the spatial\ngrid-size, and extensive numerical experiments have indicated much more expensive\ncomputational costs than standard Poisson solvers [24].\nThe Gauss-Seidel projection method (GSPM) is another popular set of numerical\nalgorithms since only linear systems with constant coe\u000ecients need to be solved at\neach time step [8,15,21]. This method is based on a combination of a Gauss-Seidel\nupdate of an implicit solver for the gyromagnetic term, the heat \row of the harmonic\nmap, and a projection step to overcome the sti\u000bness and the nonlinearity associated\nto the LLG equation. In this numerical approach, the implicit discretization is only\napplied to the scalar heat equation implicitly several times; therefore, the FFT-\nbased fast solvers become available, due to the symmetric, positive de\fnite (SPD)\nstructures of the linear system. The original GSPM method [20] turns out to be\nunstable for small damping parameters, while this issue has been resolved in [8] with\nmore updates of the stray \feld. Its numerical e\u000eciency has been further improved\nby reducing the number of linear systems per time step [15]. One little de\fciency\nof GSPM is its \frst-order accuracy in time.\nMeanwhile, in spite of these improvements, the GSPM method is computation-\nally more expensive than the standard Poisson solver, because of the Gauss-Seidel\niteration involved in the algorithm. An additional de\fciency of the GSPM is its\n\frst-order accuracy in time. Moreover, most of the above-mentioned methods have\nbeen mainly focused on small damping parameters with the only exception in a\ntheoretical work [1]. In other words, there has been no numerical method designed\nspeci\fcally for real micromagnetics simulations with large damping parameters. In\nthis paper, we propose a second-order accurate numerical method to solve the LLG\nequation with large damping parameters, whose complexity is also comparable toA SECOND-ORDER METHOD FOR LLG EQUATION 3\nsolving the scalar heat equation. To achieve this goal, the LLG system is refor-\nmulated, in which the damping term is rewritten as a harmonic mapping \row. In\nturn, the constant-coe\u000ecient Laplacian part is treated by a standard BDF2 tem-\nporal discretization, and the associated dissipation will form the foundation of the\nnumerical stability. Meanwhile, all the nonlinear parts, including both the gyro-\nmagnetic term and the remaining nonlinear expansions in the damping term, are\ncomputed by a fully explicit approximation, which is accomplished by a second\norder extrapolation formula. Because of this fully explicit treatment for the nonlin-\near parts, the resulting numerical scheme only requires a standard Poisson solver at\neach time step. This fact will greatly facilitate the computational e\u000borts, since the\nFFT-based fast solver could be e\u000eciently applied, due to the SPD structure of the\nlinear system involved at each time step. In addition, the numerical stability has\nbeen demonstrated by extensive computational experiments, and these experiments\nhas veri\fed the idea that the dissipation property of the heat equation part would\nbe able to ensure the numerical stability of the nonlinear parts, with large damping\nparameters.\nThe rest of this paper is organized as follows. In section 2, the micromagnetics\nmodel is reviewed, and the numerical method is proposed, as well as its comparison\nwith the GSPM and the semi-implicit projection method (SIPM). Subsequently,\nthe numerical results are presented in section 3, including the temporal and spa-\ntial accuracy check in both the 1D and 3D computations, the numerical e\u000eciency\ninvestigation (in comparison with the GSPM and SIPM algorithms), the stability\nstudy with respect to the damping parameter, and the dependence of domain wall\nvelocity on the damping parameter and the external magnetic \feld. Finally, some\nconcluding remarks are made in section 4.\n2.The physical model and the numerical method\n2.1.Landau-Lifshitz-Gilbert equation. The LLG equation describes the dy-\nnamics of magnetization which consists of the gyromagnetic term and the damping\nterm [3,13]. In the nondimensionalized form, this equation reads as\nmt=\u0000m\u0002he\u000b\u0000\u000bm\u0002(m\u0002he\u000b) (2.1)\nwith the homogeneous Neumann boundary condition\n(2.2)@m\n@\u0017\f\f\f\n@\n= 0;\nwhere \n is a bounded domain occupied by the ferromagnetic material and \u0017is unit\noutward normal vector along @\n.\nIn more details, the magnetization m: \n\u001aRd!R3;d= 1;2;3 is a three-\ndimensional vector \feld with a pointwise constraint jmj= 1. The \frst term on the\nright-hand side in (2.1) is the gyromagnetic term and the second term stands for\nthe damping term, with \u000b>0 being the dimensionless damping coe\u000ecient.\nThe e\u000bective \feld he\u000bis obtained by taking the variation of the Gibbs free energy\nof the magnetic body with respect to m. The free energy includes the exchange\nenergy, the anisotropy energy, the magnetostatic energy, and the Zeeman energy:\n(2.3)F[m] =\u00160M2\ns\n2\u001aZ\n\n\u0000\n\u000fjrmj2+q\u0000\nm2\n2+m2\n3\u0001\n\u00002he\u0001m\u0000hs\u0001m\u0001\ndx\u001b\n:4 Y. CAI, J. CHEN, C. WANG, AND C. XIE\nTherefore, the e\u000bective \feld includes the exchange \feld, the anisotropy \feld, the\nstray \feldhs, and the external \feld he. For a uniaxial material, it is clear that\nhe\u000b=\u000f\u0001m\u0000q(m2e2+m3e3) +hs+he; (2.4)\nwhere the dimensionless parameters become \u000f=Cex=(\u00160M2\nsL2) andq=Ku=(\u00160M2\ns)\nwithLthe diameter of the ferromagnetic body and \u00160the permeability of vacuum.\nThe unit vectors are given by e2= (0;1;0),e3= (0;0;1), and \u0001 denotes the\nstandard Laplacian operator. For the Permalloy, an alloy of Nickel (80%) and\nIron (20%), typical values of the physical parameters are given by: the exchange\nconstantCex= 1:3\u000210\u000011J/m, the anisotropy constant Ku= 100 J/m3, the sat-\nuration magnetization constant Ms= 8:0\u0002105A/m. The stray \feld takes the\nform\nhs=1\n4\u0019rZ\n\nr\u00121\njx\u0000yj\u0013\n\u0001m(y)dy: (2.5)\nIf \n is a rectangular domain, the evaluation of (2.5) can be e\u000eciently done by the\nFast Fourier Transform (FFT) [20].\nFor brevity, the following source term is de\fned\nf=\u0000Q(m2e2+m3e3) +hs+he: (2.6)\nand the original PDE system (2.1) could be rewritten as\nmt=\u0000m\u0002(\u000f\u0001m+f)\u0000\u000bm\u0002m\u0002(\u000f\u0001m+f): (2.7)\nThanks to point-wise identity jmj= 1, we obtain an equivalent form:\n(2.8)mt=\u000b(\u000f\u0001m+f) +\u000b\u0000\n\u000fjrmj2\u0000m\u0001f\u0001\nm\u0000m\u0002(\u000f\u0001m+f):\nIn particular, it is noticed that the damping term is rewritten as a harmonic map-\nping \row, which contains a constant-coe\u000ecient Laplacian di\u000busion term. This fact\nwill greatly improve the numerical stability of the proposed scheme.\nFor the numerical description, we \frst introduce some notations for discretization\nand numerical approximation. Denote the temporal step-size by k, andtn=nk,\nn\u0014\u0004T\nk\u0005\nwithTthe \fnal time. The spatial mesh-size is given by hx=hy=hz=\nh= 1=N, andmn\ni;j;`stands for the magnetization at time step tn, evaluated at the\nspatial location ( xi\u00001\n2;yj\u00001\n2;z`\u00001\n2) withxi\u00001\n2=\u0000\ni\u00001\n2\u0001\nhx,yj\u00001\n2=\u0000\nj\u00001\n2\u0001\nhyand\nz`\u00001\n2=\u0000\n`\u00001\n2\u0001\nhz(0\u0014i;j;`\u0014N+ 1). In addition, a third order extrapolation\nformula is used to approximate the homogeneous Neumann boundary condition.\nFor example, such a formula near the boundary along the zdirection is given by\nmi;j;1=mi;j;0;mi;j;N +1=mi;j;N:\nThe boundary extrapolation along other boundary sections can be similarly made.\nThe standard second-order centered di\u000berence applied to \u0001 mresults in\n\u0001hmi;j;k=mi+1;j;k\u00002mi;j;k+mi\u00001;j;k\nh2x\n+mi;j+1;k\u00002mi;j;k+mi;j\u00001;k\nh2y\n+mi;j;k+1\u00002mi;j;k+mi;j;k\u00001\nh2z;A SECOND-ORDER METHOD FOR LLG EQUATION 5\nand the discrete gradient operator rhmwithm= (u;v;w )Treads as\nrhmi;j;k=2\n64ui+1;j;k\u0000ui\u00001;j;k\nhxvi+1;j;k\u0000vi\u00001;j;k\nhxwi+1;j;k\u0000wi\u00001;j;k\nhxui;j+1;k\u0000ui;j\u00001;k\nhyvi;j+1;k\u0000vi;j\u00001;k\nhywi;j+1;k\u0000wi;j\u00001;k\nhyui;j;k +1\u0000ui;j;k\u00001\nhzvi;j;k +1\u0000vi;j;k\u00001\nhzwi;j;k +1\u0000wi;j;k\u00001\nhz3\n75:\nSubsequently, the GSPM and the SIPM numerical methods need to be reviewed,\nwhich could be used for the later comparison.\n2.2.The Gauss-Seidel projection method. The GSPM is based on a combi-\nnation of a Gauss-Seidel update of an implicit solver for the gyromagnetic term,\nthe heat \row of the harmonic map, and a projection step. It only requires a series\nof heat equation solvers with constant coe\u000ecients; as a result, the FFT-based fast\nsolvers could be easily applied. This method is \frst-order in time and second-order\nin space. Below is the detailed outline of the GSPM method in [8].\nStep 1. Implicit Gauss-Seidel:\ngn\ni= (I\u0000\u000f\u0001t\u0001h)\u00001(mn\ni+ \u0001tfn\ni); i= 2;3;\ng\u0003\ni= (I\u0000\u000f\u0001t\u0001h)\u00001(m\u0003\ni+ \u0001tf\u0003\ni); i= 1;2; (2.9)\n(2.10)0\n@m\u0003\n1\nm\u0003\n2\nm\u0003\n31\nA=0\n@mn\n1+ (gn\n2mn\n3\u0000gn\n3mn\n2)\nmn\n2+ (gn\n3m\u0003\n1\u0000g\u0003\n1mn\n3)\nmn\n3+ (g\u0003\n1m\u0003\n2\u0000g\u0003\n2m\u0003\n1)1\nA:\nStep 2. Heat \row without constraints:\n(2.11) f\u0003=\u0000Q(m\u0003\n2e2+m\u0003\n3e3) +h\u0003\ns+he;\n(2.12)0\n@m\u0003\u0003\n1\nm\u0003\u0003\n2\nm\u0003\u0003\n31\nA=0\n@m\u0003\n1+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n1+f\u0003\n1)\nm\u0003\n2+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n2+f\u0003\n2)\nm\u0003\n3+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n3+f\u0003\n3)1\nA:\nStep 3. Projection onto S2:\n(2.13)0\n@mn+1\n1\nmn+1\n2\nmn+1\n31\nA=1\njm\u0003\u0003j0\n@m\u0003\u0003\n1\nm\u0003\u0003\n2\nm\u0003\u0003\n31\nA:\nHerem\u0003denotes the intermediate values of m, and stray \felds hn\nsandh\u0003\nsare\nevaluated at mnandm\u0003, respectively.\nRemark 2.1. Two improved versions of the GSPM have been studied in [15], which\nturn out to be more e\u000ecient than the original GSPM. Meanwhile, it is found that\nboth improved versions become unstable when \u000b > 1, while the original GSPM\n(outlined above) is stable even when \u000b\u001410. Therefore, we shall use the original\nGSPM in [8]for the numerical comparison in this work.6 Y. CAI, J. CHEN, C. WANG, AND C. XIE\n2.3.Semi-implicit projection method. The SIPM has been outlined in [5,24].\nThis method is based on the second-order BDF temporal discretization, combined\nwith an explicit extrapolation. It is found that SIPM is unconditionally stable and\nis second-order accurate in both space and time. The algorithmic details are given\nas follows.\n(2.14)8\n>>>>>><\n>>>>>>:3\n2~mn+2\nh\u00002mn+1\nh+1\n2mn\nh\nk=\u0000^mn+2\nh\u0002\u0000\n\u000f\u0001h~mn+2\nh+^fn+2\nh\u0001\n\u0000\u000b^mn+2\nh\u0002\u0010\n^mn+2\nh\u0002(\u000f\u0001h~mn+2\nh+^fn+2\nh)\u0011\n;\nmn+2\nh=~mn+2\nh\nj~mn+2\nhj;\nwhere ~mn+2\nhis an intermediate magnetization, and ^mn+2\nh,^fn+2\nhare given by the\nfollowing extrapolation formula:\n^mn+2\nh= 2mn+1\nh\u0000mn\nh;\n^fn+2\nh= 2fn+1\nh\u0000fn\nh;\nwithfn\nh=\u0000Q(mn\n2e2+mn\n3e3) +hn\ns+hn\ne. The presence of cross product in the\nSIPM yields a linear system of equations with non-symmetric structure and vari-\nable coe\u000ecients. In turn, the GMRES solver has to be applied to implement this\nnumerical system. The numerical evidence has revealed that, the convergence of\nGMRES solver becomes slower for larger temporal step-size kor smaller spatial\ngrid-sizeh, which makes the computation more challenging.\n2.4.The proposed numerical method. The SIPM in (2.14) treats both the\ngyromagentic and the damping terms in a semi-implicit way, i.e., \u0001 mis computed\nimplicitly, while the coe\u000ecient functions are updated by a second order accurate,\nexplicit extrapolation formula. The strength of the gyromagnetic term is controlled\nby \u0001m+fsince the length of mis always 1. Meanwhile, the strength of the\ndamping term is controlled by the product of \u0001 m+fand the damping parameter\n\u000b. For small \u000b, say\u000b\u00141, it is reasonable to treat both the gyromagentic and\nthe damping terms semi-implicitly. However, for large \u000b, an alternate approach\nwould be more reasonable, in which the whole gyromagentic term is computed by\nan explicit extrapolation, while the nonlinear parts in the damping term is also\nupdated by an explicit formula, and only the constant-coe\u000ecient \u0001 mpart in the\ndamping term is implicitly updated. This idea leads to the proposed numerical\nmethod. To further simplify the presentation, we start with (2.8), and the numerical\nalgorithm is proposed as follows.\n(2.15)8\n>>>>>>>>>><\n>>>>>>>>>>:3\n2~mn+2\nh\u00002mn+1\nh+1\n2mn\nh\nk=\u0000^mn+2\nh\u0002\u0010\n\u000f\u0001h^mn+2\nh+^fn+2\nh\u0011\n+\u000b\u0010\n\u000f\u0001h~mn+2\nh+^fn+2\nh\u0011\n+\u000b\u0010\n\u000fjrh^mn+2\nhj2\u0000^mn+2\nh\u0001^fn+2\nh\u0011\n^mn+2\nh;\nmn+2\nh=~mn+2\nh\nj~mn+2\nhj;A SECOND-ORDER METHOD FOR LLG EQUATION 7\nwhere\n^mn+2\nh= 2mn+1\nh\u0000mn\nh;\n^fn+2\nh= 2fn+1\nh\u0000fn\nh:\nTable 1 compares the proposed method, the GSPM and the SIPM in terms\nof number of unknowns, dimensional size, symmetry pattern, and availability of\nFFT-based fast solver of linear systems of equations, and the number of stray \feld\nupdates. At the formal level, the proposed method is clearly superior to both the\nGSPM and the SIPM algorithms. In more details, this scheme will greatly improve\nthe computational e\u000eciency, since only three Poisson solvers are needed at each\ntime step. Moreover, this numerical method preserves a second-order accuracy in\nboth space and time. The numerical results in section 3 will demonstrate that the\nproposed scheme provides a reliable and robust approach for micromagnetics simu-\nlations with high accuracy and e\u000eciency in the regime of large damping parameters.\nTable 1. Comparison of the proposed method, the Gauss-Seidel\nprojection method, and the semi-implicit projection method.\nProperty or number Proposed method GSPM SIPM\nLinear systems 3 7 1\nSize N3N33N3\nSymmetry Yes Yes No\nFast Solver Yes Yes No\nAccuracy O(k2+h2)O(k+h2)O(k2+h2)\nStray \feld updates 1 4 1\nRemark 2.2. To kick start the proposed method, one can apply a \frst-order al-\ngorithm, such as the \frst-order BDF method, in the \frst time step. An overall\nsecond-order accuracy is preserved in this approach.\n3.Numerical experiments\nIn this section, we present a few numerical experiments with a sequence of damp-\ning parameters for the proposed method, the GSPM [8] and the SIPM [24], with\nthe accuracy, e\u000eciency, and stability examined in details. Domain wall dynamics\nis studied and its velocity is recorded in terms of the damping parameter and the\nexternal magnetic \feld.\n3.1.Accuracy and e\u000eciency tests. We set\u000f= 1 andf= 0 in (2.8) for conve-\nnience. The 1D exact solution is given by\nme= (cos(X) sint;sin(X) sint;cost)T;\nand the corresponding exact solution in 3D becomes\nme= (cos(XYZ ) sint;sin(XYZ ) sint;cost)T;\nwhereX=x2(1\u0000x)2,Y=y2(1\u0000y)2,Z=z2(1\u0000z)2. In fact, the above exact\nsolutions satisfy (2.8) with the forcing term g=@tme\u0000\u000b\u0001me\u0000\u000bjrmej2+me\u0002\n\u0001me, as well as the homogeneous Neumann boundary condition.8 Y. CAI, J. CHEN, C. WANG, AND C. XIE\nFor the temporal accuracy test in the 1D case, we \fx the spatial resolution\nash= 5D\u00004, so that the spatial approximation error becomes negligible. The\ndamping parameter is taken as \u000b= 10, and the \fnal time is set as T= 1. In the 3D\ntest for the temporal accuracy, due to the limitation of spatial resolution, we take\na sequence of spatial and temporal mesh sizes: k=h2\nx=h2\ny=h2\nz=h2= 1=N0\nfor the \frst-order method and k=hx=hy=hz=h= 1=N0for the second-\norder method, with the variation of N0indicated below. Similarly, the damping\nparameter is given by \u000b= 10, while the \fnal time Tis indicated below. In turn,\nthe numerical errors are recorded in term of the temporal step-size kin Table 2. It\nis clear that the temporal accuracy orders of the proposed numerical method, the\nGSPM, and the SIPM are given by 2, 1, and 2, respectively, in both the 1D and\n3D computations.\nThe spatial accuracy order is tested by \fxing k= 1D\u00005,\u000b= 10,T= 1 in 1D\nandk= 1D\u00003,\u000b= 10,T= 1 in 3D. The numerical error is recorded in term of\nthe spatial grid-size hin Table 3. Similarly, the presented results have indicated\nthe second order spatial accuracy of all the numerical algorithms, including the\nproposed method, the GSPM, and the SIPM, respectively, in both the 1D and 3D\ncomputations.\nTo make a comparison in terms of the numerical e\u000eciency, we plot the CPU time\n(in seconds) vs. the error norm kmh\u0000mek1. In details, the CPU time is recorded\nas a function of the approximation error in Figure 1a in 1D and in Figure 1b in\n3D, with a variation of kand a \fxed value of h. Similar plots are also displayed in\nFigure 1c in 1D and Figure 1d in 3D, with a variation of hand a \fxed value of k. In\nthe case of a \fxed spatial resolution h, the proposed method is signi\fcantly more\ne\u000ecient than the GSPM and the SIPM in both the 1D and 3D computations. The\nSIPM is slightly more e\u000ecient than the GSPM, while such an advantage depends\non the performance of GMRES, which may vary for di\u000berent values of kandh. In\nthe case of a \fxed time step size k, the proposed method is slightly more e\u000ecient\nthan the GSPM, in both the 1D and 3D computations, and the GSPM is more\ne\u000ecient than the SIPM.\n3.2.Stability test with large damping parameters. To check the numerical\nstability of these three methods in the practical simulations of micromagnetics with\nlarge damping parameters, we consider a thin \flm of size 480 \u0002480\u000220 nm3with\ngrid points 100\u0002100\u00024. The temporal step-size is taken as k= 1 ps. A uniform\nstate along the xdirection is set to be the initial magnetization and the external\nmagnetic \feld is set to be 0. Three di\u000berent damping parameters, \u000b= 0:01;10;40,\nare tested with stable magnetization pro\fles shown in Figure 2. In particular, the\nfollowing observations are made.\n\u000fThe proposed method is the only one that is stable for very large damping\nparameters;\n\u000fAll three methods are stable for moderately large \u000b;\n\u000fThe proposed method is the only one that is unstable for small \u000b.\nIn fact, a preliminary theoretical analysis reveals that, an optimal rate convergence\nestimate of the proposed method could be theoretically justi\fed for \u000b>3. Mean-\nwhile, extensive numerical experiments have implied that \u000b > 1 is su\u000ecient to\nensure the numerical stability in the practical computations.A SECOND-ORDER METHOD FOR LLG EQUATION 9\nTable 2. The numerical errors for the proposed method, the\nGSPM and the SIPM with \u000b= 10 andT= 1. Left: 1D with\nh= 5D\u00004; Right: 3D with k=h2\nx=h2\ny=h2\nz=h2= 1=N0\nfor GSPM and k=hx=hy=hz=h= 1=N0for the proposed\nmethod and SIPM, with N0speci\fed in the table.\n1D 3D\nkk\u0001k1k\u0001k 2k\u0001kH1k=hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 4.459D-4 5.226D-4 5.588D-4 1/20 6.171D-4 4.240D-4 4.246D-4\n2.0D-2 1.147D-4 1.345D-4 1.436D-4 1/24 4.381D-4 3.010D-4 3.014D-4\n1.0D-2 2.899D-5 3.402D-5 3.631D-5 1/28 3.268D-4 2.245D-4 2.248D-4\n5.0D-3 7.192D-6 8.529D-6 9.119D-6 1/32 2.531D-4 1.739D-4 1.741D-4\n2.5D-3 1.699D-6 2.321D-6 2.518D-6 1/36 2.017D-4 1.386D-4 1.387D-4\norder 2.007 1.961 1.957 { 1.902 1.903 1.903\n(a)Proposed method\n1D 3D\nkk\u0001k1k\u0001k 2k\u0001kH1k=h2k\u0001k1k\u0001k 2k\u0001kH1\n2.5D-3 2.796D-4 2.264D-4 1.445D-3 1/36 4.194D-4 2.683D-4 2.815D-4\n1.25D-3 1.425D-4 1.174D-4 7.720D-4 1/64 2.388D-4 1.399D-4 1.500D-4\n6.25D-4 7.170D-5 5.940D-5 4.026D-4 1/144 1.069D-4 6.106D-5 6.736D-5\n3.125D-4 3.591D-5 2.971D-5 2.069D-4 1/256 6.021D-5 3.442D-5 3.860D-5\n1.5625D-4 1.799D-5 1.488D-5 1.054D-4 1/400 3.855D-5 2.208D-5 2.501D-5\norder 0.991 0.984 0.945 { 0.992 1.032 1.000\n(b)GSPM\n1D 3D\nkk\u0001k1k\u0001k 2k\u0001kH1k=hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 4.315D-4 5.111D-4 8.774D-4 1/20 6.170D-4 4.240D-4 4.249D-4\n2.0D-2 1.128D-4 1.334D-4 2.255D-4 1/24 4.380D-4 3.010D-4 3.016D-4\n1.0D-2 2.872D-5 3.399D-5 5.706D-5 1/28 3.268D-4 2.245D-4 2.251D-4\n5.0D-3 7.174D-6 8.552D-6 1.433D-5 1/32 2.531D-4 1.739D-4 1.743D-4\n2.5D-3 1.721D-6 2.333D-6 3.784D-6 1/36 2.017D-4 1.386D-4 1.389D-4\norder 1.991 1.951 1.969 { 1.902 1.903 1.902\n(c)SIPM\nUnder the same setup outlined above, we investigate the energy dissipation of\nthe proposed method, the GSPM, and the SIPM. The stable state is attainable at\nt= 2 ns, while the total energy is computed by (2.3). The energy evolution curves\nof di\u000berent numerical methods with di\u000berent damping parameters, \u000b= 2;5;8;10,\nare displayed in Figure 3. One common feature is that the energy dissipation rate\nturns out to be faster for larger \u000b, in all three schemes. Meanwhile, a theoretical\nderivation also reveals that the energy dissipation rate in the LLG equation (2.1)\ndepends on \u000b, and a larger \u000bleads to a faster energy dissipation rate. Therefore,\nthe numerical results generated by all these three numerical methods have made a\nnice agreement with the theoretical derivation.10 Y. CAI, J. CHEN, C. WANG, AND C. XIE\nTable 3. The numerical errors of the proposed method, the\nGSPM and the SIPM with \u000b= 10 andT= 1. Left: 1D with\nk= 1D\u00005; Right: 3D with k= 1D\u00003.\n1D 3D\nhk\u0001k1k\u0001k 2k\u0001kH1hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 7.388D-3 7.392D-3 8.243D-3 1/2 4.261D-3 2.472D-3 2.472D-3\n2.0D-2 1.848D-3 1.848D-3 2.061D-3 1/4 9.822D-4 5.595D-4 5.753D-4\n1.0D-2 4.621D-4 4.621D-4 5.153D-4 1/8 2.453D-4 1.390D-4 1.424D-4\n5.0D-3 1.155D-4 1.155D-4 1.288D-4 1/16 6.137D-5 3.471D-5 3.554D-5\norder 2.000 2.000 2.000 { 2.035 2.047 2.037\n(a)Proposed method\n1D 3D\nhk\u0001k1k\u0001k 2k\u0001kH1hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 7.388D-3 7.392D-3 8.244D-3 1/2 4.256D-3 2.470D-3 2.470D-3\n2.0D-2 1.848D-3 1.848D-3 2.061D-3 1/4 9.810D-4 5.589D-4 5.744D-4\n1.0D-2 4.619D-4 4.622D-4 5.158D-4 1/8 2.447D-4 1.388D-4 1.423D-4\n5.0D-3 1.153D-4 1.156D-4 1.302D-4 1/16 6.103D-5 3.468D-5 3.613D-5\norder 2.000 2.000 1.995 { 2.037 2.047 2.030\n(b)GSPM\n1D 3D\nhk\u0001k1k\u0001k 2k\u0001kH1hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 7.388D-3 7.392D-3 8.243D-3 1/2 4.261D-3 2.472D-3 2.472D-3\n2.0D-2 1.848D-3 1.848D-3 2.061D-3 1/4 9.822D-4 5.595D-4 5.753D-4\n1.0D-2 4.621D-4 4.621D-4 5.153D-4 1/8 2.453D-4 1.390D-4 1.424D-4\n5.0D-3 1.155D-4 1.155D-4 1.288D-4 1/16 6.137D-5 3.471D-5 3.554D-5\norder 2.000 2.000 2.000 { 2.035 2.047 2.037\n(c)SIPM\nMeanwhile, we choose the same sequence of values for \u000b, and display the energy\nevolution curves in terms of time up to T= 2 ns in Figure 4. It is found that the\nproposed method have almost the same energy dissipation pattern with the other\ntwo methods for moderately large damping parameters \u000b= 2;5;8. In the case of\n\u000b= 10, the SIPM has a slightly di\u000berent energy dissipation pattern from the other\ntwo numerical methods.\n3.3.Domain wall motion. A Ne\u0013 el wall is initialized in a nanostrip of size 800 \u0002\n100\u00024 nm3with grid points 128 \u000264\u00024. An external magnetic \feld of he= 5 mT\nis then applied along the positive xdirection and the domain wall dynamics is\nsimulated up to 2 ns with \u000b= 2;5;8. The corresponding magnetization pro\fles are\nvisualized in Figure 5. Qualitatively, the domain wall moves faster as the value of\n\u000bincreases. Quantitatively, the corresponding dependence is found to be linear;\nsee Figure 6. The slopes \ftted by the least-squares method in terms of \u000bandhe\nare recorded in Table 4.A SECOND-ORDER METHOD FOR LLG EQUATION 11\n10-610-510-410-3100101102\nProposed method\nGSPM\nSIPM\n(a)Varyingkin 1D up to\nT= 1\n1.8 2 2.2 2.4 2.6 2.8 3 3.2\n10-7101102103\nProposed method\nGSPM\nSIPM(b)Varyingkin 3D up to T=\n0:1\n10-510-410-310-2101102103\nProposed method\nGSPM\nSIPM\n(c)Varyinghin 1D up to\nT= 1\n10-410-310-210-1100101102103\nProposed method\nGSPM\nSIPM(d)Varyinghin 3D up to\nT= 1\nFigure 1. CPU time needed to achieve the desired numerical ac-\ncuracy, for the proposed method, the GSPM and the SIPM, in\nboth the 1D and 3D computations. The CPU time is recorded as\na function of the approximation error by varying korhindepen-\ndently. CPU time with varying k: proposed method >1, the second term on the right-hand side\ncan be neglected. In a similar manner to Eq. 10, we can relate the mag netic noise in the\nrotating frame to that of the lab frame and SPxcan be simplified to\nSPx(ω) =1\n8P2\nzγ2\nKT2\n2\n(ω−ωL)2T2\n2+1[SBx(ω)+SBy(ω)], (13)\nwhere we take as our convention a one-sided power spectral dens ity [35].\nWith the addition of spin-damping the transfer function changes to h(ω) =iγK\niω+1\nT2dand\nthe noise power spectral density is the same as in Eq. 13, but with T2replaced with T2d.\nFrom Eq. 9 and 13 the signal to noise ratio of the absorptive signal, u nder steady-state\nconditions and for long acquisition time T, is\nSNR(ω) =SNR0\n/radicalbig\n1+(ω−ωL)2T2\n2d, (14)\nwhereSNR0=/parenleftBig\nB1√\nT/parenrightBig/radicalBig\n1\nSBis the resonant SNR and SB=1\n2/bracketleftbig\nSBx(ω)+SBy(ω)/bracketrightbig\nrepre-\nsents the average magnetic noise in any given direction. From this ex pression it is easy to\nsee that the resonant SNR does not depend on damping and that th e FWHM line width is\n∆ω=√\n12\nT2d=(1+DF)√\n12\nT2. (15)\nTherefore the bandwidth of the sensitivity for an absorptive signa l increases as the return\ndifference, without loss of SNR, as long as the only noise is magnetic no ise.\n9B. Spin-projection noise\nWeconsider, at first, onlyasingle potassium atominthecell, but leave offthe superscript\nifor notational simplicity. As described by Ref. [36], the spin project ion noise associated\nwith measurement of Sxcan be calculated by\nSSx(ω) = 2×/integraldisplay0\n∞RSx(t)/parenleftbig\ne−iωt+eiωt/parenrightbig\ndt, (16)\nwhere the symmetrized spin-spin autocorrelation function RSxis given by\nRSx(t) =1\n2Tr/braceleftbig\nρ(0)/bracketleftbig\nSH\nx(t)SH\nx(0)+SH\nx(0)SH\nx(t)/bracketrightbig/bracerightbig\n. (17)\nIn the above expression, ρ(0) is the density matrix at time t= 0 and SH\nxis the operator\nSxin the Heisenberg representation. In the absence of magnetic nois e and in the limit of\nhigh polarization, the solution to the Bloch equation in the Larmor rot ating frame and with\ndamping gives\n/an}bracketle{tSx/an}bracketri}ht= Tr/braceleftbigg\nρ(0)[SxcosωLt−SysinωLt]e−t\nT′\n2/bracerightbigg\n(18)\n= Tr/braceleftbig\nρ(0)SH\nx(t)/bracerightbig\n.\nEquation 18 implies that SH\nx(t) can be replaced by ( SxcosωLt−SysinωLt)e−t\nT′\n2in Eq. 17,\nin which case the spin-spin autocorrelation function becomes\nRSx(t) = Tr/braceleftbigg\nρ(0)/bracketleftbigg\nS2\nxcosωLτ−1\n2(SySx+SxSy)sinωLτ/bracketrightbigg\ne−τ\nT′\n2/bracerightbigg\n. (19)\nTherefore the average power spectral density per atom is\nSSx(ω) =/planckover2pi12\n2T2d\n1+T2\n2d(ω−ωL)2, (20)\nwhere we have taken the limit that ωis close to ωLandωLT2>>1. This agrees with\nquantum mechanical expression of Ref. [36] derived for a spin-1/2 particle. While it is clear\nfrom Eq. 20 that the resonant noise density is reduced with spin-da mping the net power is\nnot. Therefore this reduction would not be considered spin-squee zing, rather it represents\nthe broadening of the spectrum; nevertheless the ability to easily v ary resonant noise and\nwidth may be of use in quantum control.\nThe noise power spectral density for the net magnetic moment alon g the probe direction\nis related to SFx(ω) of Eq. 20 by\nSPx= 4SSx(ω)\n/planckover2pi12N. (21)\n10Therefore the SNR under spin-projection noise is determined by Eq s. 9 and 21:\nSNR=SNR0\n/radicalbig\n1+(ω−ωL)2T2\n2d/radicalbigg\nT2d\nT2, (22)\nwhere the resonant undamped SNR0is\nSNR0=B1√\nT/bracketleftBigg\nPzγK/radicalbigg\nNT2\n8/bracketrightBigg\n. (23)\nNote the inverse of the square bracketed expression in Eq. 23 rep resents the undamped\nresonant field sensitivity, or the spin-projection noise expressed in terms of field. From\nEq. 22, the SNR bandwidth increases as ∆ ω=(1+DF)√\n12\nT2, as in the case of magnetic noise.\nHowever unlike the case of magnetic noise, this broadening comes at a cost to SNR; the\nresonant SNR decreases in proportion to√\n1+DF.\nC. Photon shot noise\nThrough interaction with the K atoms, the polarization angle of the p robe beam after\nthe magnetometer φis shifted from its original phase φ0byφ=φ0+αφPx. During feedback,\nusing the optical Bloch equations, and in the limit that ωis close to ωLandωLT2>>1, the\nFourier transform of φis equal to the transform of φ0times the transfer function h(ω) =\n1\nT2+i(ω−ωL)\n1\nT2d+i(ω−ωL). Therefore the power spectral density of φis\nSφ(ω) =/bracketleftbigg1+(ω−ωL)2T2\n2\n1+(ω−ωL)2T2\n2d/bracketrightbiggT2\n2d\nT2\n2Spsn(ω), (24)\nwhereSpsnis the standard white photon shot noise.\nTherefore the SNR from photon shot noise alone can be expressed as\nSNR=SNR0/radicalBigg\n1\n[1+(ω−ωL)2T2\n2][1+(ω−ωL)2T2\n2d], (25)\nwhere the resonant SNR under no damping is given by\nSNR0=B1√\nT/bracketleftBigg\nPzγKT2αφ/2/radicalbig\nSpsn/bracketrightBigg\n. (26)\nFrom Eq. 25 it easy to see that the resonant SNR does not change w ith damping, but the\nFWHMlinewidthofthisSNRmodestly increases from2\nT2withnodampingto√\n12\nT2forinfinite\ndamping, with most of the increase occurring for damping factors u nder 10.\n11D. Total noise and bandwidth\nThe measurement of the noise under spin-damping in principle permits the identification\nof the separate contributions of spin-projection noise SS=α2SPxfrom Eq. 21, photon\nshot noise SP=α2\nGSφof Eq. 24, and magnetic noise SB=α2SPxof Eq. 13. The total\nmagnetometer noise power spectral density can be expressed as\nSV(ω)≡ SS(ω)+SP(ω)+SB(ω)\n=A2\nn+(ω−ωL)2T2\n2dB2\nn\n1+(ω−ωL)2T2\n2d, (27)\nwhere in the second expression the functional dependence on ωhas been made explicit such\nthatA2\nnrepresents the amplitude on resonance and B2\nnthe base noise at large off-resonance\nvalues. The two parameters A2\nnandB2\nncan be expressed in terms of the resonant noise\nspectral densities with no damping applied, denoted in the following by a zero superscript,\nB2\nn=S0\nP (28)\nA2\nn=ax2+bx= (S0\nP+S0\nB)x2+S0\nSx, (29)\nwherex≡T2d\nT2=1\n(1+DF). If in addition, to these noise sources, there is an out-of-loop nois e\nsource, say for instance from the spectrometer itself, both bas e noise power B2\nnand the\namplitude noise A2\nnwould be increased by this constant noise.\nThe SNR under the combined noise can be found using Eq. 27. Both th e loss of SNR\nand the broadening of the SNR with spin damping depend on the relativ e amounts of the\ndifferent types of noise. In our experimental case where magnetic noise and photon shot\nnoise dominate over spin-projection noise, we find that broadening with little loss of SNR\ncan occur for damping factors on the order of 10 or less.\nE. Measuring noise\nFor a finite acquisition time Tof the noise signal V(t), the ensemble average of the\nperiodogram [ PT(ω)] can be taken as a measurement of the frequency distribution of the\nnoise [37]\n[PT(ω)]≡1\nT/bracketleftbig\n|F{V(t)}|2/bracketrightbig\n(30)\n=/integraldisplayT\n−T/parenleftbigg\n1−|τ|\nT/parenrightbigg\nR(τ)e−iωτdτ, (31)\n12whereF{V(t)}is the Fourier transform and R(τ) is the autocorrelation function of V(t). In\nthe limit that Tis much larger than the characteristic decay time of R(τ) withτ, [PT(ω)]\napproaches half the power spectral density,1\n2S(ω). More generally, the integral on the right\nhand side of Eq. 30 can be viewed as the Fourier transform of the au tocorrelation func-\ntion multiplied by a triangular function, or the power spectral densit y convoluted with the\nfunction Tsinc2(ωT/(2π)). For finite acquisitions times, the features of the power spectr al\ndensity are broadened on the order of1\nTto give [PT(ω)].\nIn this paper we focus on the absorptive signal part of the signal a s measured through\nquadrature detection, as is typically used in magnetic resonance te chniques. Such phase-\nsensitive detection is needed for an optimal signal to noise ratio and to distinguish the true\nsignal from interfering signals. The noise spectra for absorptive s ignals is/radicalBig\n1\n2[PT(ω)] and\ntherefore for long Tis equivalent to1\n2/radicalbig\nS(ω). In the next section, however, the presented\nnoise data is scaled so as to represent/radicalbig\nS(ω) for ease of comparison with derived expressions\nfor noise spectral density.\nIV. RESULTS\nA. Spin-damping at long times\nWhen spin-damping is applied to the magnetometer, both the signal a nd noise are sup-\npressed when resonant with the Larmor frequency of the magnet ometer and their effective\nwidths are broadened, as shown in Fig. 2 for cell 1. For the absorpt ive signal the resonant\namplitude Asis inversely proportional to the return difference (1+ DF), while the FWHM\nwidth Γ s=1\nπT2dis proportional to the return difference. This is clearly demonstrat ed in\nFig. 3, where Asand Γ sare determined from fits of the signal-versus-frequency data to a\nLorentzian function, the formexpected from Eq. 9. For clarity, t heparameters in Fig. 3 have\nbeen normalized with respect to their undamped counterparts A0\nsand Γ0\ns. Representative\nabsorptive signals and fits for select damping factors are shown in t he inset of Fig. 2.\nAs evident in Fig. 2, the noise spectra have a very different function al form from the\nsignal. Fits of these spectra to the square root of Eq. 27 are depic ted as solid lines. Good\nagreement between the data and fits are observed, except for t he highest damping factors,\nwhere the fit slightly underestimates the on-resonance amplitude a nd deviates from the high\n13/s45/s52/s48 /s45/s50/s48 /s48 /s50/s48 /s52/s48/s49/s48/s49/s48/s48\n/s48/s46/s57/s55/s57/s46/s54/s53\n/s45/s52 /s45/s50 /s48 /s50 /s52/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48\n/s48/s46/s48/s48/s49/s46/s57/s51/s51/s46/s56/s54/s53/s46/s55/s57/s55/s46/s55/s50/s57/s46/s54/s53/s49/s49/s46/s53/s56\n/s40/s110/s114/s97/s100/s47/s72/s122/s49/s47/s50\n/s41/s32/s78/s111/s105/s115/s101/s32/s65/s109/s112/s108/s105/s116/s117/s100/s101/s32/s40/s110/s86/s47/s72/s122/s49/s47/s50\n/s41\n/s102/s32/s45/s32 /s102\n/s76/s32/s40/s107/s72/s122/s41/s32/s68/s70 /s32/s61/s32/s48\n/s32/s68/s70 /s32/s61/s32/s48/s46/s51/s50\n/s32/s68/s70 /s32/s61/s32/s48/s46/s57/s51\n/s32/s68/s70 /s32/s61/s32/s52/s46/s50\n/s32/s68/s70 /s32/s61/s32/s57/s46/s48\n/s32/s68/s70 /s32/s61/s32/s49/s55/s46/s53\n/s32/s68/s70 /s32/s61/s32/s52/s51\n/s32/s68/s70 /s32/s61/s32/s56/s49\n/s32/s77/s101/s97/s115/s117/s114/s101/s100/s32/s110/s111/s32/s112/s117/s109/s112/s32/s110/s111/s105/s115/s101\n/s32/s67/s97/s108/s99/s117/s108/s97/s116/s101/s100/s32/s115/s104/s111/s116/s32/s110/s111/s105/s115/s101/s32/s68/s70 /s32/s61/s32/s48\n/s32/s68/s70 /s32/s61/s32/s48/s46/s57/s51\n/s32/s68/s70 /s32/s61/s32/s57/s46/s48\n/s32/s68/s70 /s32/s61/s32/s56/s49/s32\n/s40 /s114/s97/s100/s41/s70/s84/s32/s111/s102/s32/s49/s54/s32/s109/s115/s32/s83/s105/s103/s110/s97/s108/s32/s40 /s86/s41\n/s102/s32/s45/s32 /s102\n/s76/s32/s40/s107/s72/s122/s41\nFIG. 2. Spectra of measured magnetometer noise (dotted line s) for various damping factors,\nshowing suppression of on-resonance noise amplitude Anto well below photon shot noise at the\nhigher damping factors. The noise spectra for each damping f actor is fitted (solid lines) to Eq. 27.\nAlso plotted are the measured noise (red) with no pump beam an d the expected photon shot\nnoise (black dashed line). The inset shows similar suppress ion of the magnetometer output to a\nreference RF magnetic signal of 209 fT (points), fitted to a Lo rentzian function. Acquisition time\nwas 16.4 ms, more than an order of magnitude larger than K T2, and this data is expressed both\nin volts output from the polarimeter, left axis, and rotatio n angle of the probe polarization, right\naxis.\noff-resonance frequency behavior. The width of the noise peak/d ip is predicted by Eq. 27\nto be equal to Γ n, which is equivalent to Γ s, and therefore should increase as the return\ndifference. The fit parameter Γ ndemonstrates this predicted behavior in Fig. 3. In contrast,\nthe resonant noise amplitude Anis suppressed as the signal for low damping factors, but\nis suppressed less than the signal at higher damping factors. As ex plored more below, this\nbehavior is expected from Eq. 29. The third parameter, Bn, predicts the far off-resonant\namplitude of the noise corresponding to the photon shot noise. The slight increase in the\nmeasured shot noise ( ≃25 nV/Hz1/2) is due to additional observed noise from the balanced\npolarimeter as is measured in the absence of both probe and pump ligh t.\n14/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/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\n/s87/s105/s100/s116/s104/s32/s69/s120/s112/s97/s110/s115/s115/s105/s111/s110/s32/s65/s109/s112/s108/s105/s116/s117/s100/s101/s32/s83/s117/s112/s112/s114/s101/s115/s115/s105/s111/s110\n/s68/s97/s109/s112/s105/s110/s103/s32/s70/s97/s99/s116/s111/s114/s32/s40 /s68/s70 /s41/s32/s40/s49/s32/s43/s32 /s68/s70 /s41\n/s32/s83/s105/s103/s110/s97/s108/s32/s97/s109/s112/s108/s105/s116/s117/s100/s101/s32/s115/s117/s112/s112/s114/s101/s115/s115/s105/s111/s110/s32/s40 /s65\n/s115/s48\n/s32/s47/s32 /s65\n/s115/s41\n/s32/s83/s105/s103/s110/s97/s108/s32/s119/s105/s100/s116/s104/s32/s101/s120/s112/s97/s110/s115/s105/s111/s110/s32/s40\n/s115/s32/s47/s32\n/s115/s48\n/s41\n/s32/s78/s111/s105/s115/s101/s32/s97/s109/s112/s108/s105/s116/s117/s100/s101/s32/s115/s117/s112/s112/s114/s101/s115/s115/s105/s111/s110/s32/s40 /s65\n/s110/s48\n/s32/s47/s32 /s65\n/s110/s41\n/s32/s78/s111/s105/s115/s101/s32/s119/s105/s100/s116/s104/s32/s101/s120/s112/s97/s110/s115/s105/s111/s110/s32/s40\n/s110/s32/s47/s32\n/s110/s48\n/s41\nFIG. 3. The magnetometer signal amplitude suppression, defi ned asA0\ns/As(solid blue), and\nlinewidth expansion, defined as Γ s/Γ0\ns(open blue), scale as (1 + DF) (black solid line). The on-\nresonance noise amplitude suppression, A0\nn/An(solid red), is linear at low damping factors ( <20)\nbut clearly reaches a noise limit at higher DF, while the expansion of noise width, Γ n/Γ0\nn(open\nred), is linear for the range of damping factors measured.\nFor comparison, a set of noise measurements are made with a secon d K cell, cell 2, which\noperated with a higher level of environmental noise. For both cells, the resonant noise power\nA2\nnis plotted in Fig. 4 as a function of x=1\n1+DF. The measured noise power is fitted to\na quadratic polynomial of the form ax2+bx+c, corresponding to Eq. 29. From the fit\nwe extract the noise contributions, with acorresponding to S0\nP+S0\nB, andb, to the spin-\nprojection noise. Parameter crepresents the limit of noise suppression, and may be due to\nexternal noise added outside of the feedback loop or noise folded b ack into the spectrum\nfrom aliasing effects and limitations in the spectrometer’s filtering. Th is noise power is more\nthan an order of magnitude larger than the noise floor of the spect rometer.\n15/s49/s48/s45/s51\n/s49/s48/s45/s50\n/s49/s48/s45/s49\n/s49/s48/s48/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51/s49/s48/s52/s49/s48/s53\n/s57/s46/s51/s120/s49/s48/s45/s51/s57/s46/s51/s120/s49/s48/s45/s50/s57/s46/s51/s120/s49/s48/s45/s49/s57/s46/s51/s120/s49/s48/s48/s57/s46/s51/s120/s49/s48/s49/s57/s46/s51/s120/s49/s48/s50\n/s32/s77/s101/s97/s115/s117/s114/s101/s100/s32/s110/s111/s105/s115/s101/s32/s45/s32/s67/s101/s108/s108/s32/s49\n/s32/s70/s105/s116/s32/s116/s111/s32/s113/s117/s97/s100/s114/s97/s116/s105/s99/s32/s102/s117/s110/s99/s116/s105/s111/s110\n/s32/s77/s101/s97/s115/s117/s114/s101/s100/s32/s110/s111/s105/s115/s101/s32/s45/s32/s67/s101/s108/s108/s32/s50\n/s32/s70/s105/s116/s32/s116/s111/s32/s113/s117/s97/s100/s114/s97/s116/s105/s99/s32/s102/s117/s110/s99/s116/s105/s111/s110/s67/s97/s108/s99/s117/s108/s97/s116/s101/s100/s32/s115/s112/s105/s110/s45/s112/s114/s111/s106/s101/s99/s116/s105/s111/s110/s32/s110/s111/s105/s115/s101/s67/s97/s108/s99/s117/s108/s97/s116/s101/s100/s32/s112/s104/s111/s116/s111/s110/s32/s115/s104/s111/s116/s32/s110/s111/s105/s115/s101\n/s40/s110/s114/s97/s100/s50\n/s47/s72/s122/s41/s32/s82/s101/s115/s111/s110/s97/s110/s116/s32/s78/s111/s105/s115/s101/s32/s80/s111/s119/s101/s114/s32/s40/s110/s86/s50\n/s47/s72/s122/s41/s32\n/s49/s32/s47/s32/s40/s49/s32/s43/s32 /s68/s70 /s41\nFIG. 4. For cell 1 (open blue) and cell 2 (solid red), the plot o f resonant noise power A2\nnas a\nfunction of suppression, fitted to a quadratic polynomial (l ines). The values of all parameters for\nthe two cells are given in Table I.\nThe values of the fit parameters for both data sets are given in Tab le I. The measured\nspin-projection noise is similar in magnitude and agrees fairly well with t he predicted values.\nWhile the calculated noise takes into consideration the reduced polar izations of 83% and\n78%forcells 1and2 respectively, thederivationofEq. 23relies heav ily ontheatomicsystem\nbeing in the high polarization limit; this may be responsible for the obser ved trend that the\npredicted noise is higher than measured noise, particularly for the lo wer polarization cell.\nThe quoted errors for the calculated spin-projection noise are du e to the uncertainty in the\nparameters nK,Pz,T2, and the volume of the cell. Fig. 4 clearly shows that at high damping\nfactors, we are able to suppress the total magnetometer noise p ower by about three order\nof magnitude below photon shot noise and two orders of magnitude b elow the undamped\nspin-projection noise.\nThe SNR of themagnetometer is simply calculated by taking the ratio o f the fit equations\n16TABLE I. Using the fit parameters from the data in Fig. 4, the wi ngs of the noise curve Bn,\nand the magnetometer responsivity, we find the resonant nois e contributions. The measured and\npredicted values for spin-projection noise are in reasonab le agreement. For the magnetic noise,\nwhich is predominantly environmental, we give as a predicte d lower bound the calculated light\nshift noise. The measured shot noise is close to the predicte d value. For the prediction of the\nout-of-loop noise we only give a lower bound corresponding t o the base noise of the spectrometer\nitself; aliasing effects may account for the observed noise.\nFit ParametersMagnetic Noise Photon Shot Spin-projection Out-of-loop\n/radicalbig\na−B2nnoise,Bnnoise,√\nbnoise,√c\n(aT/√\nHz) (aT/√\nHz) (aT/√\nHz) (aT/√\nHz)\nCell 1Measured 248±19 107±7 35±5 3±1\nPredicted >2±1 100±2 45±9 >0.4\nCell 2Measured 361±28 118±8 31±5 4±1\nPredicted >2±1 105±2 47±9 >1.0\ncorresponding to the measured signal and the noise for each DF. Figure 5, shows that as\nthe damping is increased the SNR bandwidth, or sensitivity bandwidth , increases, but at the\nsame time the resonant SNR decreases. For damping factors of ab out 20 or less, however,\nthis loss of signal is quite small, so that broadening of the bandwidth in this regime comes\nwith little cost. For cell 1, a damping factor of 17.5 increases the det ection bandwidth\nof the magnetometer by a factor of 2.8 over 0.70 kHz with ∼10% loss in on-resonance\nsensitivity, while cell 2 shows a bandwidth increase of 3 .7×over 0.74 kHz with almost no\nloss in sensitivity for DF= 20. The difference between the two data sets can be mostly\nattributedto thehigher level of environmental noise experienced by cell 2compared to cell 1.\nThe increase in bandwidth in an atomic magnetometer can significantly reduce the detec-\ntion time when the frequency of the signal to be detected is not well known. For example,\nin NQR detection the resonant frequency of the material is temper ature dependent; our test\nsubstance has a temperature coefficient of 100 Hz /◦C. Therefore, a factor of 3 increase in\nbandwidth without loss in sensitivity is equivalent to a factor of 3 incre ase in the acceptable\ntemperature variation of the substance under detection.\n17/s49 /s49/s48 /s49/s48/s48/s48/s53/s49/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s45/s49/s48 /s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s83/s78/s82\n/s40/s110/s111/s114/s109/s97/s108/s105/s122/s101/s100/s41/s32\n/s32/s83/s78/s82/s32/s66/s97/s110/s100/s119/s105/s100/s116/s104\n/s40/s110/s111/s114/s109/s97/s108/s105/s122/s101/s100/s41\n/s49/s32/s43/s32 /s68/s70\n/s32/s32/s83/s78/s82/s32/s40/s110/s111/s114/s109/s97/s108/s105/s122/s101/s100/s41\n/s102/s32/s45/s32 /s102\n/s76/s32/s40/s107/s72/s122/s41/s32/s68/s70/s32 /s61/s32/s48\n/s32/s68/s70/s32 /s61/s32/s50/s48\n/s32/s68/s70 /s32/s61/s32/s53/s48\nFIG. 5. Plot of the measured magnetometer SNR(open points) and bandwidth (solid points), for\nboth cell 1 (black square) and cell 2 (red circles), shows goo d agreement to the predicted (lines)\nvalues. Theinset isthemagnetometer SNRforthreedampingfactors asafunctionof frequencyfor\ncell 2, and shows that sensitivity bandwidth can be broadene d with little loss of SNR for DF≤20.\nB. Spin-damping at short times and in the presence of ringing\nAny net magnetization transverse to B0has a ring down with the time constant T2. If\nsuch a component exists at the beginning of a measurement the ass ociated ringing can dwarf\nthe signal of interest, as demonstrated in Fig. 6. This is particularly detrimental for short\ndata acquisition times or short-lived signals, as shown in Fig. 7. To illust rate the potentially\ncatastrophic effects of ringing in a high Qatomic magnetometer we apply a long perturbing\npulse ending at time t= 0, the beginning of the data acquisition windows of Fig. 6. During\nthe first millisecond in Fig. 6(a), the ringing clearly masks the desired s ignal, in this case a\nthree times smaller radio-frequency pulse applied at t= 60µs.\nThe application of spin damping in the first 60 µs permits for the quick damping of\n18/s48 /s49 /s50 /s51/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48\n/s48 /s49 /s50 /s51/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48\n/s48 /s49 /s50/s48/s49/s48/s50/s48/s54/s48/s32/s117/s115/s101/s99/s46/s32/s102/s101/s101/s100/s98/s97/s99/s107/s32/s119/s105/s110/s100/s111/s119\n/s32/s32/s77/s97/s103/s110/s101/s116/s111/s109/s101/s116/s101/s114/s32/s79/s117/s116/s112/s117/s116/s32/s40 /s86/s41/s32/s82/s105/s110/s103/s105/s110/s103/s32/s43/s32/s97/s112/s112/s108/s105/s101/s100/s32/s82/s70/s32/s102/s105/s101/s108/s100\n/s32/s68/s97/s109/s112/s101/s100/s32/s114/s105/s110/s103/s105/s110/s103/s32/s43/s32/s97/s112/s112/s108/s105/s101/s100/s32/s82/s70/s32/s102/s105/s101/s108/s100\n/s32/s65/s112/s112/s108/s105/s101/s100/s32/s82/s70/s32/s102/s105/s101/s108/s100\n/s98/s32\n/s32/s84/s105/s109/s101/s32/s40/s109/s115/s41\n/s32/s32\n/s32/s82/s105/s110/s103/s105/s110/s103\n/s32/s68/s97/s109/s112/s101/s100/s32/s114/s105/s110/s103/s105/s110/s103\n/s32/s66/s97/s115/s101/s32/s110/s111/s105/s115/s101\n/s70/s101/s101/s100/s98/s97/s99/s107/s32/s104/s117/s109/s112/s97\n/s32\n/s32/s32\nFIG. 6. Plots (a) and (b) demonstrate the application of spin -damping with the magnetometer\ninitially in a perturbed state, a state created by a RF pulse o f amplitude 1 .13 pT applied for t≤0.\n(a) The magnetometer response to a 0 .37 pT RF signal, applied at t= 60µs, is obscured by\nthe transient ringing from the initial perturbed state (dot ted line). Application of spin-damping\nduring a short window quickly eliminates the transient and p ermits the clear observation of the\nsignal (solid line) as compared to when there is no initial pe rturbation (dashed line). (b) The\nringing transient naturally decays with a time constant of T2= 0.7 ms (dotted line), but under\ndamping decays in less than 60 µs (solid line). However, a small feedback hump, arises after the\ndamping field is turned off, due to inhomogeneity in B0across the K cell.\nthe ringing and clear detection of the desired signal, shown as a solid lin e in Fig. 6(a).\nFigure 6(b) shows that the ringing decay constant is reduced by ap proximately a factor of\n50 under the effects of damping. In both figures, the negative fee dback starts at a high\ndamping factor of ∼150 for approximately the first 20 µs and is smoothly ramped down to\nDF= 0 over the following 40 µs, so as to avoid the creation of undesirable transients from\nthe turn-off of damping.\nThere is, however, a small rise in the magnetometer signal following t he application of\nfeedback; the arrow in Fig. 6(b) indicates the emergence of this “f eedback hump.” Through\n19modeling, it is determined that this small rise is due to the inhomogeneit y inB0across the\nK cell. The applied feedback field forces the net magnetic moment of t he cell to zero. Some\nisochromats across the cell become 180◦out of phase from one another and once damping is\noff, individual isochromats with different Larmor frequencies partia lly rephase and a small\nmagnetic moment re-emerges. For measurements in which the phas e of the signal can be\ncontrolled separately from the perturbation, as is common for ech o experiments in magnetic\nresonance, flipping or cycling the phase of the signal can be used to cancel the effects of this\nfeedback hump. Such phase cycling is commonly used to suppress th e effect of the transients\ncreated by the refocusing pulse. The ameliorating impact of phase c ycling is shown in Fig. 7,\nthroughcomparing theSNRdataofcolumns(5)and(7)tocolumns ( 6)and(8), respectively.\nThe combination of spin-damping and phase-cycling together leads t o a strong and rapid\nsuppression of the transients, at the same time helping to avoid sat uration and a potentially\nlong recovery time of the spectrometer. Furthermore, the use o f an atomicmagnetometer for\ndetection, permits the use of a low- Qprobe for excitation thus preventing long-time ringing\nof the excitation coil.\nIn addition to the coherent transient added by the feedback hump , the turning on and\noff of spin-damping adds noise to the magnetic field detection, even w hen the magnetometer\nbegins in an aligned state. This noise can be greatly reduced, but not eliminated, by shaping\nthe spin-damping to turn-off gently as was done for the data in Fig. 6 . By comparing the\nSNR of a signal acquired without damping, column (1) of Fig. 7, to SNR with damping\napplied before data acquisition, column (2), we can see that the loss of SNR is particularly\nevident for data acquisition over short times. Note the shorter win dow associated with\ncolumn (1) has a SNR that is nearly a factor of 5 smaller than the large r window, a result\nconsistent with theoretical predictions.\nOne way to avoid the noise associated with switching damping off is to lea ve damping\non during data acquisition. As discussed in the previous section, this can be done for low\ndamping factors without loss of signal and with an increase in sensitiv ity bandwidth. The\nbenefits to SNR can be clearly observed in Fig. 7, by comparing column s (2) and (5) where\nthere is no damping in the window, to columns (3) and (7) where dampin g,DF= 10, is\nleft on during the window. Combining both phase-cycling and damping d uring acquisition,\npermitsustoretainthesensitivity ofthemagnetometereveninthe presenceofringing, Fig.7\ncolumn (1) to column (8). Therefore, and particularly for short win dows as is necessary in\n20/s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56/s48/s50/s48/s52/s48/s54/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48\n/s32/s116/s111/s32\n/s32/s122/s101/s114/s111/s32\n/s112/s104/s97/s115/s101\n/s99/s121/s99/s108/s105/s110/s103/s32/s116/s111/s32\n/s68/s70/s61/s49/s48\n/s112/s104/s97/s115/s101\n/s99/s121/s99/s108/s105/s110/s103/s32/s116/s111/s32\n/s68/s70/s61/s49/s48/s32/s116/s111/s32\n/s32/s122/s101/s114/s111/s32/s110/s111/s110/s101\n/s32/s116/s111/s32\n/s68/s70/s61/s49/s48/s32/s116/s111/s32\n/s32/s122/s101/s114/s111/s32/s110/s111/s110/s101/s87/s105/s116/s104/s32/s82/s105/s110/s103/s105/s110/s103/s78/s111/s32/s82/s105/s110/s103/s105/s110/s103/s32/s77/s97/s103/s110/s101/s116/s111/s109/s101/s116/s101/s114/s32/s83/s105/s103/s110/s97/s108/s45/s116/s111/s45/s78/s111/s105/s115/s101/s32/s82/s97/s116/s105/s111\n/s32/s32\n/s68/s97/s109/s112/s105/s110/s103/s32/s50/s52/s32/s109/s115/s32/s119/s105/s110/s100/s111/s119/s32 /s32/s49/s46/s53/s32/s109/s115/s32/s119/s105/s110/s100/s111/s119\nFIG. 7. Results showing that application of spin-damping ca n reduce the negative effects of per-\nturbation noise and recover the magnetometer sensitivity. Two forms of damping are tested. Both\nstart with DF= 150 for the first20 µs of the 60 µsfeedback window, butin one theDF is smoothly\nreduced to zero, while in the other the DF is reduce to 10 and ke pt at this value throughout the\nacquisition window. SNR is measured for a 24 ms (sparse hatch ing) and a 1 .5 ms (dense hatch-\ning) window in the absence, columns (1)-(3), and in the prese nce, (4)-(8) of ringing created by a\nperturbing pulse three times larger than the detected signa l. As shown in (1)-(3), the switching\noff of damping adds noise, but with damping retained during ac quisition the SNR is regained.\nMeasurement 4 shows the significant loss of SNR due to noise fr om an initial perturbation of the\nK spins. The SNR is partially regained with damping (5). The a ddition of phase cycling (6) or\ndamping during the window (7) further increases the SNR, and with the combination of the two\ntechniques (8) the SNR for both window sizes is in agreement w ith the SNR when ringing is not\npresent (1).\nmagneticresonance echo trains, itisimportanttohave bothcontin uance ofdamping into the\n21windowtoavoidswitching noiseandtheuseofphasecycling tominimizet hefeedback hump.\nArmed with both these tools, spin-damping promises to be quite usef ul in the reduction of\nunwanted delay, or dead-time, before data acquisition.\nV. CONCLUSION\nIn this work, we have demonstrated that negative magnetic feedb ack can effectively be\nused to rapidly damp the ringing of the K spins from some unwanted init ial perturbation.\nUnder spin-damping the effective T2can be reduced by more than an order of magnitude,\ntherefore permitting the clear observation of short-lived signals, which otherwise would be\nobscured by the use of a high Qatomic magnetometer.\nFurthermorewefindthatthemagnetometer suppresses notonly coherent signals, butalso\nnoise. Damping effects the spectrum of the noise, both amplitude an d shape, according to\nthe type of noise, so that we are able to separately measure magne tic, photon shot, and spin-\nprojection noise. While the net power in the magnetic and photon sho t noise are reduced\nunder damping, the power in spin-projection noise remains the same even as its spectrum\nis broadened. The magnetic noise spectrum also broadens, with the effective T2simply\nreplacing the undamped T2in the spectral shape. The photon shot noise, however, becomes\ncolored under the presence of negative feedback, giving the noise spectrum an inverted\nappearance. In total we observe a resonant noise an order of ma gnitude lower than the\nundamped photon shot noise, implying the closed-loop production of polarization-squeezed\nlight.\nFor phase-sensitive detection, the signal and noise are broadene d under damping so as\nto increase the bandwidth of the magnetometer. For magnetic and photon shot noise, this\nincrease is not accompanied by loss of SNR, while for spin-projection noise the resonant\nSNR decreases as the square root of the effective T2. Therefore in our system, which is dom-\ninated by magnetic and photon shot noise, we observed a three time s increase in detection\nbandwidth with little degradation to the sub-femtoTesla sensitivity o f the magnetometer.\n22ACKNOWLEDGMENTS\nWe would like to acknowledge Philip Naudus for his modeling of the spin sys tem under\ndamping and field inhomogeneity. This work was supported by NSF gra nts #0730473 and\n#054798.\n[1] D. I. Hoult, Review of Scientific Instruments 50, 193 (1979).\n[2] E. Baudin, K. Safiullin, S. W. Morgan, and P. J. Nacher, Jou rnal of Physics: Conference\nSeries294, 012009 (2011).\n[3] I. M. Savukov, S. J. Seltzer, and M. V. Romalis, Journal of Magnetic Resonance 185, 214\n(2007).\n[4] I. M. Savukov and M. V. Romalis, Phys. Rev. Lett. 94, 123001 (2005).\n[5] M. P. Ledbetter, I. M. Savukov, D. Budker, V. Shah, S. Knap pe, J. Kitching, D. J. Michalak,\nS. Xu, and A. Pines, Proceedings of the National Academy of Sc iences105, 2286 (2008).\n[6] S. Xu, C. W. 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Walden, Spectral Analysis for Physical Applications (Cambridge\nUniversity Press, Cambridge, 1993).\n[36] M. Braun and J. K¨ onig, Phys. Rev. B 75, 085310 (2007).\n[37] R. G. Brown and P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman\nFiltering , 3rd ed. (Wiley, New York, 1997).\n24" }, { "title": "2101.02794v2.Mechanisms_behind_large_Gilbert_damping_anisotropies.pdf", "content": "Mechanisms behind large Gilbert damping anisotropies\nI. P. Miranda1, A. B. Klautau2,∗A. Bergman3, D. Thonig3,4, H. M. Petrilli1, and O. Eriksson3,4\n1Universidade de São Paulo, Instituto de Física,\nRua do Matão, 1371, 05508-090, São Paulo, SP, Brazil\n2Faculdade de Física, Universidade Federal do Pará, Belém, PA, Brazil\n3Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden and\n4School of Science and Technology, Örebro University, Fakultetsgatan 1, SE-701 82 Örebro, Sweden\n(Dated: November 22, 2021)\nA method with which to calculate the Gilbert damping parameter from a real-space electronic\nstructure method is reported here. The anisotropy of the Gilbert damping with respect to the\nmagnetic moment direction and local chemical environment is calculated for bulk and surfaces\nof Fe 50Co50alloys from first principles electronic structure in a real space formulation. The size\nof the damping anisotropy for Fe 50Co50alloys is demonstrated to be significant. Depending on\ndetails of the simulations, it reaches a maximum-minimum damping ratio as high as 200%. Several\nmicroscopic origins of the strongly enhanced Gilbert damping anisotropy have been examined, where\nin particular interface/surface effects stand out, as do local distortions of the crystal structure.\nAlthough theory does not reproduce the experimentally reported high ratio of 400% [Phys. Rev.\nLett. 122, 117203 (2019)], it nevertheless identifies microscopic mechanisms that can lead to huge\ndamping anisotropies.\nIntroduction: Magnetic damping has a critical impor-\ntanceindeterminingthelifetime,diffusion,transportand\nstability of domain walls, magnetic vortices, skyrmions,\nand any nano-scale complex magnetic configurations [1].\nGiven its high scientific interest, a possibility to obtain\nthis quantity by means of first-principles theory [2] opens\nnew perspectives of finding and optimizing materials for\nspintronic and magnonic devices [3–8]. Among the more\npromising ferromagnets to be used in spintronics devices,\ncobalt-iron alloys demonstrate high potentials due to the\ncombination of ultralow damping with metallic conduc-\ntivity [4, 9].\nRecently, Li et al.[10] reported an observed, gi-\nant anisotropy of the Gilbert damping ( α) in epitaxial\nFe50Co50thin films (with thickness 10 −20nm) reach-\ning maximum-minimum damping ratio values as high as\n400%. TheauthorsofRef. [10]claimedthattheobserved\neffect is likely due to changes in the spin-orbit coupling\n(SOC) influence for different crystalline directions caused\nby short-range orderings that lead to local structural dis-\ntortions. This behaviour differs distinctly from, for ex-\nample, pure bcc Fe [11]. In order to quantitatively pre-\ndict the Gilbert damping, Kambersky’s breathing Fermi\nsurface (BFS) [12] and torque-correlation (TC) [13] mod-\nels are frequently used. These methods have been ex-\nplored for elements and alloys, in bulk form or at sur-\nfaces, mostly via reciprocal-space ab-initio approaches,\nin a collinear or (more recently) in a noncollinear con-\nfiguration [14]. However, considering heterogeneous ma-\nterials, such as alloys with short-range order, and the\npossibility to investigate element specific, non-local con-\ntributions to the damping parameter, there are, to the\nbest of our knowledge, no reports in the literature that\nrely on a real space method.\nIn this Letter, we report on an implementation of ab\ninitiodamping calculations in a real-space linear muffin-tin orbital method, within the atomic sphere approxi-\nmation (RS-LMTO-ASA) [15, 16], with the local spin\ndensity approximation (LSDA) [17] for the exchange-\ncorrelation energy. The implementation is based on the\nBFSandTCmodels, andthemethod(SupplementalMa-\nterial - SM, for details) is applied to investigate the re-\nported, huge damping anisotropy of Fe50Co50(100)/MgO\nfilms [10]. A main result here is the identification of\na microscopic origin of the enhanced Gilbert damping\nanisotropy of Fe50Co50(100) films, and the intrinsic rela-\ntionships to the local geometry of the alloy. Most signifi-\ncantly, wedemonstratethatasurfaceproducesextremely\nlarge damping anisotropies that can be orders of magni-\ntude larger than that of the bulk. We call the attention\nto the fact that this is the first time, as far as we know,\nthat damping values are theoretically obtained in such a\nlocal way.\nResults: We calculated: i)ordered Fe50Co50in theB2\nstructure (hereafter refereed to as B2-FeCo) ii)random\nFe50Co50alloysinbccorbctstructures, wherethevirtual\ncrystal approximation (VCA) was applied; iii)Fe50Co50\nalloys simulated as embedded clusters in a VCA matrix\n(host). In all cases VCA was simulated with an elec-\ntronic concentration corresponding to Fe50Co50. The ii)\nandiii)alloys were considered as in bulk as well as in\nthe (001) surface, with bcc and bct structures (here-\nafter correspondingly refereed as VCA Fe50Co50bcc,\nVCA Fe50Co50bct, VCA Fe50Co50(001) bcc and VCA\nFe50Co50(001) bct). The effect of local tetragonal distor-\ntions was considered with a localc\na= 1.09ratio (SM for\ndetails). All data for cluster based results, were obtained\nfrom an average of several different configurations. The\ntotal damping for a given site iin real-space ( αt, Eqs. S6\nand S7 from SM) can be decomposed in non-local, αij\n(i/negationslash=j), and local (onsite), αonsite(orαii,i=j) contri-\nbutions, each of them described by the tensor elements2\nανµ\nij=g\nmiπ/integraldisplay\nη(/epsilon1)Tr/parenleftBig\nˆTν\niImˆGijˆTµ\njImˆGji/parenrightBig\nd/epsilon1,(1)\nwheremiis the total magnetic moment localized in the\nreference atomic site i,µ,ν={x,y,z},ˆTis the torque\noperator, and η(/epsilon1) =∂f(/epsilon1)\n∂/epsilon1the derivative of the Fermi\ndistribution. The scalar αijparameter is defined in the\ncollinear regime as αij=1\n2(αxx\nij+αyy\nij).\nTo validate our methodology, the here obtained total\ndamping for several systems (such as bcc Fe, fcc Ni, hcp\nand fcc Co and B2-FeCo) were compared with estab-\nlished values available in the literature (Table S1, SM),\nwhere an overall good agreement can be seen.\nFig. 1 shows the non-local contributions to the damp-\ning for bcc Fe and B2-FeCo. Although the onsite contri-\nbutions are around one order of magnitude larger than\nthe non-local, there are many αijto be added and total\nnet values can become comparable. Bcc Fe and B2-FeCo\nhave very different non-local damping contributions. El-\nement resolved αij, reveal that the summed Fe-Fe in-\nteractions dominate over Co-Co, for distances until 2a\ninB2-FeCo. We observe that αijis quite extended in\nspace for both bcc Fe and B2-FeCo. The different con-\ntributions to the non-local damping, from atoms at equal\ndistance arises from the reduced number of operations in\nthe crystal point group due to the inclusion of SOC in\ncombination with time-reversal symmetry breaking. The\nB2-FeCo arises from replacing every second Fe atom in\nthe bcc structure by a Co atom. It is interesting that this\nreplacement (i.e. the presence of Co in the environment)\nsignificantly changes the non-local contributions for Fe-\nFe pairs , what can more clearly be seen from the Insetin\nFig. 1, where the non-local damping summed over atoms\nat the same relative distance for Fe-Fe pairs in bcc Fe\nandB2-FeCo are shown; the non-local damping of Fe-Fe\npairs are distinctly different for short ranges, while long\nranged (further than ∼2.25 Å) contributions are smaller\nand more isotropic.\nThe damping anisotropy, i.e. the damping change,\nwhen the magnetization is changed from the easy axis\nto a new direction is1\n∆αt=/parenleftBigg\nα[110]\nt\nα[010]\nt−1/parenrightBigg\n×100%, (2)\nwhereα[110]\ntandα[010]\ntare the total damping obtained\nfor magnetization directions along [110]and [010], re-\nspectively. Analogousdefinitionalsoappliesfor ∆αonsite.\n1We note that this definition is different to the maximum-\nminimum damping ratio, defined asα[110]\nt\nα[010]\nt×100%, from Ref.\n[10].We investigated this anisotropy in surfaces and in bulk\nsystems with (and without) tetragonal structural distor-\ntions. Our calculations for VCA Fe50Co50bcc show a\ndamping increase of ∼13%, when changing the magne-\ntization direction from [010]to[110](Table S2 in the\nSM). The smallest damping is found for the easy magne-\ntization axis, [010], which holds the largest orbital mo-\nment (morb) [18]. For VCA Fe50Co50bcc we obtained\na small variation of ∼2%for the onsite contribution\n(α[010]\nonsite = 8.94×10−4andα[110]\nonsite = 8.76×10−4),\nwhat implies that the anisotropy comes mostly from\nthe non-local contributions, particularly from the next-\nnearest neighbours. For comparison, ∆αt∼3%(with\n∆αonsite∼0.4%) in the case of bcc Fe, what corrobo-\nrates the reported [11] small bcc Fe anisotropy at room\ntemperature, andwiththebulkdampinganisotropyrates\n[19].\nWe also inspected the chemical inhomogeneity influ-\nence on the anisotropy, considering the B2-FeCo alloy,\nwhere the weighted average damping (Eq. S7 of SM)\nwas used instead. The B2-FeCo bcc (∼7%) and VCA\nFe50Co50bcc (∼13%) anisotropies are of similar magni-\ntudes. Both B2structure and VCA calculations lead to\ndamping anisotropies which are significantly lower than\nwhatwasobservedintheexperiments, anditseemslikely\nthatthepresenceofdisorderincompositionand/orstruc-\ntural properties of the Fe/Co alloy would be important\nto produce large anisotropy effects on the damping.\n\nα\nij\n\t\n×\n\t\n10\n-4\n−3\n−2\n−1\n0\n1\n2\n3\n4\n\nNormalized\n\t\ndistance\n1.0\n1.5\n2.0\n2.5\n3.0\n\n\t\nB2\n\t\nFe-Co\n\t\nB2\n\t\nFe-Fe\n\t\nB2\n\t\nCo-Co\n\t\nbcc\n\t\nFe-Fe\n\t\n−10\n0\n10\n20\n\n\t\n1.0\n1.5\n2.0\n2.5\n3.0\nFigure 1. (Color online) Non-local damping contributions,\nαij, in (Fe-centered) bulk B2-FeCo and bcc Fe, as a function\nof the normalized distance in lattice constant units a.Inset:\nNon-local contributions from only Fe-Fe pairs summed, for\neach distance, in bcc Fe bulk (empty blue dots) and in the\nB2-FeCo (full red dots). The onsite damping for Fe (Co) in\nB2-FeCo isαFe\nonsite = 1.1×10−3(αCo\nonsite = 0.8×10−3) and for\nbcc Fe it is αFe\nonsite = 1.6×10−3. The magnetization direction\nisz([001]). Lines are guides for the eyes.\nWeanalyzedtheroleoflocaldistortionsbyconsidering3\na hypothetical case of a large, 15%(c\na= 1.15), distortion\non thez-axis of ordered B2-FeCo. We found the largest\ndamping anisotropy ( ∼24%) when comparing the results\nwith magnetization in the [001](α[001]\nt= 10.21×10−3)\nand in the [010](α[010]\nt= 7.76×10−3) directions. This\nconfirms that, indeed, bct-like distortions act in favour of\nthe∆αtenhancement (and therefore, of the maximum-\nminimum damping ratio), but the theoretical data are\nnot large enough to explain the giant value reported ex-\nperimentally [10].\nNevertheless, in the case of an alloy, the local lattice\ndistortions suggested in Ref. [10] are most to likely occur\nin an heterogeneous way [20], with different distortions\nfor different local environments. To inspect this type\nof influence on the theoretical results, we investigated\n(Table S3, SM) clusters containing different atomic con-\nfigurations embedded in a VCA Fe50Co50matrix (with\nFe bulk lattice parameter); distortions were also consid-\nered such that, locally in the clusters,c\na= 1.15(Ta-\nble S4, in the SM). Moreover, in both cases, two types\nof clusters have to be considered: Co-centered and Fe-\ncentered. The αtwas then computed as the sum of the\nlocal and non-local contributions for clusters with a spe-\ncific central (Fe or Co) atom, and the average of Fe-\nand Co-centered clusters was taken. Fe-centered clus-\ntershaveshownlargeranisotropies, onaverage ∼33%for\nthe undistorted (∼74%for the distorted) compared with\n∼8%fortheundistortedCo-centeredclusters( ∼36%for\nthe distorted). Although these results demonstrate the\nimportance of both, local distortions as well as non-local\ncontributions to the damping anisotropy, they are not\nstill able to reproduce the huge observed [10] maximum-\nminimum damping ratio.\nWe further proceed our search for ingredients that\ncould lead to a huge ∆αtby inspecting interface effects,\nwhich are present in thin films, grain boundaries, stack-\ning faults and materials in general. Such interfaces may\ninfluence observed properties, and in order to examine\nif they are relevant also for the reported alloys of Ref.\n[10], we considered these effects explicitly in the calcu-\nlations. As a model interface, we considered a surface,\nwhat is, possibly, the most extreme case. Hence, we per-\nformed a set of αtcalculations for the Fe50Co50(001),\nfirst on the VCA level. Analogous to the respective bulk\nsystems, we found that the onsite contributions to the\ndamping anisotropy are distinct, but they are not the\nmain cause ( ∆αonsite∼18%). However, the lack of in-\nversion symmetry in this case gives a surprisingly large\nenhancement of ∆αt, thus having its major contribution\ncoming from the non-local damping terms, in particular\nfrom the next-nearest neighbours. Interestingly, negative\nnon-local contributions appear when αtis calculated in\nthe[010]direction. These diminish the total damping\n(the onsite contribution being always positive) and gives\nrise to a larger anisotropy, as can be seen by comparisonof the results shown in Table I and Table S5 (in the Sup-\nplemental Material). In this case, the total anisotropy\nwas found to be more than ∼100%(corresponding to a\nmaximum-minimum damping ratio larger than 200%).\nA compilation of the most relevant theoretical results\nobtained here is shown in Fig. 2, together with the ex-\nperimental data and the local density of states (LDOS)\natEFfor each magnetization direction of a typical atom\nin the outermost layer (data shown in yellow). As shown\nin Fig. 2, the angular variation of αthas a fourfold ( C4v)\nsymmetry, with the smallest Gilbert damping occurring\nat 90◦from the reference axis ( [100],θH= 0◦), for both\nsurface and bulk calculations. This pattern, also found\nexperimentally in [10], matches the in-plane bcc crys-\ntallographic symmetry and coincides with other mani-\nfestations of SOC, such as the anisotropic magnetoresis-\ntance [10, 21]. Following the simplified Kambersky’s for-\nmula [13, 22], in which (see SM) α∝n(EF)and, there-\nfore, ∆α∝∆n(EF), we can ascribe part of the large\nanisotropy of the FeCo alloys to the enhanced LDOS dif-\nferences at the Fermi level, evidenced by the close corre-\nlation between ∆n(EF)and∆αtdemonstrated in Fig. 2.\nThus, as a manifestation of interfacial SOC (the so-called\nproximity effect [23]), the existence of ∆αtcan be under-\nstoodintermsofRashba-likeSOC,whichhasbeenshown\nto play an important role on damping anisotropy [24, 25].\nAnalogous to the bulk case, the higher morboccurs where\nthe system presents the smallest αt, and the orbital\nmoment anisotropy matches the ∆αtfourfold symme-\ntry with a 90◦rotation phase (see Fig. S3, SM). Note\nthat a lower damping anisotropy than Co50Fe50(001) is\nfound for a pure Fe(001) bcc surface, where it is ∼49%\n(Table S2, SM), in accordance with Refs. [7, 26], with\na dominant contribution from the onsite damping val-\nues (conductivity-like character on the reciprocal-space\n[19, 27]).\nThe VCA surface calculations on real-space allows to\ninvestigate the layer-by-layer contributions (intra-layer\ndamping calculation), as shown in Table I. We find that\nthemajorcontributiontothedampingsurfaceanisotropy\ncomes from the outermost layer, mainly from the differ-\nence in the minority 3dstates around EF. The deeper\nlayers exhibit an almost oscillatory ∆αtbehavior, simi-\nlar to the oscillation mentioned in Ref. [28] and to the\nFriedel oscillations obtained for magnetic moments. The\ndamping contributions from deeper layers are much less\ninfluenced by the inversion symmetry breaking (at the\nsurface), as expected, and eventually approaches the typ-\nicalbulklimit. Therefore, changesintheelectronicstruc-\ntureconsiderednotonlytheLDOSoftheoutermostlayer\nbut a summation of the LDOS of all layers (including the\ndeeper ones), which produces an almost vanishing differ-\nence between θH= 0◦andθH= 45◦(also approaching\nthe bulk limit). The damping anisotropy arising as a sur-\nface effect agrees with what was observed in the case of\nFe [7] and CoFeB [29] on GaAs(001), where the damping4\n0o45o90o\n135o\n180o\n225o\n270o315o[100][110][010]\n[−110]\n0.10.20.3\nΔn(EF)\n(st./Ry−at.)\n0.0050.0100.015\nαt\nθH\nFigure 2. (Color online) Total damping and LDOS difference\natEF,∆n(EF), as a function of θH, the angle between the\nmagnetizationdirectionandthe [100]-axis. Squares: (redfull)\nVCAFe 50Co50(001)bcc. Triagles: (greenfull)averageover32\nclusters (16 Fe-centered and 16 Co-centered), with bcc struc-\nture at the surface layers (SM) embedded in a VCA medium;\n(gray open) similar calculations, but with a local lattice dis-\ntortion. Circles: (yellow open) ∆n(EF)betweenθH= 0◦and\nthe current angle for a typical atom in the outermost layer\nof VCA Fe 50Co50(001) bcc; (blue full) experimental data [10]\nfor a 10-nm Fe 50Co50/Pt thin film; (purple full) average bulk\nVCA Fe 50Co50bcc; and (brown full) the B2-FeCo bulk. Lines\nare guides for the eyes.\nanisotropy diminishes as the film thickness increases.\nTable I. Total intra-layer damping ( αt×10−3) and anisotropy,\n∆αt(Eq. 2), of a typical (VCA) atom in each Fe 50Co50(001)\nbcc surface layer for magnetization along [010]and[110]di-\nrections. In each line, the sum of all αijin the same layer is\nconsidered. Outermost (layer 1) and deeper layers (2-5).\nLayerαt[010]αt[110] ∆αt\n1 7.00 14.17 +102.4%\n2 1.28 1.16 −9.4%\n3 2.83 3.30 +16.6%\n4 2.18 1.99 −8.7%\n5 2.54 2.53 −0.4%\nWe also studied the impact of bct-like distortions in\nthesurface, initiallybyconsideringtheVCAmodel. Sim-\nilartothebulkcase,tetragonaldistortionsmaybeimpor-\ntant for the damping anisotropy at the surface, e.g. when\nlocal structural defects are present. Therefore, localized\nbct-like distortions of the VCA medium in the surface,\nparticularly involving the most external layer were inves-\ntigated. The structural model was similar to what was\nused for the Fe50Co50bulk, consideringc\na= 1.09(see\nSM). Our calculations show that tetragonal relaxations\naround a typical site in the surface induce a ∆αt∼75%,\nfromα[010]\nt= 8.94×10−3toα[110]\nt= 15.68×10−3. Themain effect of these distortions is an enhancement of the\nabsolute damping values in each direction with respect to\nthe pristine (bcc) system. This is due to an increase on\nαonsite, fromα[010]\nonsite = 7.4×10−3toα[010]\nonsite = 9.5×10−3,\nand fromα[110]\nonsite = 8.7×10−3toα[110]\nonsite = 11.7×10−3;\nthe resulting non-local contributions remains similar to\ntheundistortedcase. Theinfluenceofbct-likedistortions\non the large damping value in the Fe50Co50surface is in\nline with results of Mandal et al.[30], and is related to\nthe transition of minority spin electrons around EF.\nWe then considered explicit 10-atom Fe50Co50clusters\nembedded in a VCA FeCo surface matrix. The results\nfrom these calculations were obtained as an average over\n16 Fe-centered and 16 Co-centered clusters. We con-\nsidered clusters with undistorted bcc crystal structure\n(Fig. 2, yellow open circles) as well as clusters with lo-\ncal tetragonal distortions (Fig. 2, black open circles). As\nshown in Fig. 2 the explicit local tetragonal distortion\ninfluences the damping values ( α[010]\nt= 10.03×10−3and\nα[110]\nt= 14.86×10−3)andtheanisotropy, butnotenough\ntoreproducethehugevaluesreportedintheexperiments.\nA summary of the results obtained for each undis-\ntorted FeCo cluster at the surface is shown in Fig. 3:\nCo-centered clusters in Fig. 3(a) and Fe-centered clusters\nin Fig. 3(b). A large variation of αtvalues is seen from\nclustertocluster, dependingonthespatialdistributionof\natomic species. It is clear that, αtis larger when there is\na larger number of Fe atoms in the surface layer that sur-\nroundsthecentral, referenceclustersite. Thiscorrelation\ncan be seen by the numbers in parenthesis on top of the\nblue symbols (total damping for each of the 16 clusters\nthat were considered) in Fig. 3. We also notice from the\nfigure that the damping in Fe-centered clusters are lower\nthan in Co-centered, and that the [010]magnetization di-\nrection exhibit always lower values. In the Insetof Fig. 3\nthe onsite contributions to the damping, αonsite, and the\nLDOS atEFin the central site of each cluster are shown:\na correlation, where both trends are the same, can be ob-\nserved. The results in Fig. 3 shows that the neighbour-\nhood influences not only the local electronic structure at\nthe reference site (changing n(EF)andαonsite), but also\nmodifies the non-local damping αij, leading to the cal-\nculatedαt. In other words, the local spatial distribution\naffects how the total damping is manifested, something\nwhich is expressed differently among different clusters.\nThis may open up for materials engineering of local and\nnon-local contributions to the damping.\nConclusions: We demonstrate here that real-space\nelectronic structure, based on density functional theory,\nyield a large Gilbert damping anisotropy in Fe50Co50al-\nloys. Theory leads to a large damping anisotropy, when\nthe magnetization changes from the [010]to the [110]di-\nrection, which can be as high as ∼100%(or200%in the\nminimum-maximum damping ratio) when surface calcu-\nlations are considered. This is in particular found for5\n\u0001\u0001\u0002\u0003\u0004\u0005\u0006\u0007\n\u0005\u0004\u0005\b\u0005\u0007\u0005\t\u0005\n\u0001\n\u0001\u0001\n\u000b\f\u0001\r\u000e\u0006\u000f\u0010\u0011\u0002\u0010\u0012\u0010\u0013\u0001\u0002\u0001\u0014\u0005\u0004\u0005\u0015\u0001\u0002\u0001\u0014\u0004\u0004\u0005\u0015\u0001\u0001\u0016\u0016\u0003\u0004\u0005\u0006\u0007\u0005\u0004\u0005\b\u0005\u0007\u0005\n\u0001\u0002\u0003\u0004\u0005\b\u0005\u0007\u0005\t\u0005\n\u0001\b\t\u0017\u0018\u0004\u0005\u0004\b\u0004\t\u0004\u0017\u0001\u0006\u0007\u0007\b\t\n\u000b\n\f\b\u0001\u0002\u0003\u0004\u0005\b\t\n\u000b\n\f\u0006\u0007\u0007\b\t\u000b\u000b\n\f\b\u0001\u0002\u0003\u0004\u0005\b\t\u000b\u000b\n\f\n\u0001\u0001\u0002\u0003\u0001\u0004\u0005\u0006\u0007\n\u0005\b\u0004\u0005\u0004\b\t\u0005\n\u0001\n\u000b\f\r\u0002\u000e\u000f\u0001\u0010\f\u0011\u0012\u000e\u000f\t\u0013\u0014\u0015\u0004\u0005\u0004\t\u0004\u0013\u0004\u0014\u0001\u0016\u0012\u0017\u0001\u0018\u000e\u0006\u0019\u000e\u0010\u0002\u000e\u000f\u000e\u001a\u0001\u0001\u001b\u001b\u0003\u0001\u0004\u0005\u0006\u0007\u0005\b\u0004\u0005\u0004\b\t\u0005\n\u0001\u0002\u0003\u0004\u0005\u0004\u0014\u0004\u0015\t\u0005\t\t\n\u0001\t\u0013\u0014\u0015\u0004\u0005\u0004\t\u0004\u0013\u0004\u0014\u0001(0)(1)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(3)(3)(4)(4)\n(1)(1)(2)(2)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(4)(5)\nFigure 3. Damping for the [010](open circles) and [110](full\ncircles) magnetization directions for distinct types of 10-atom\nFe50Co50bcc clusters, embedded in VCA Fe 50Co50(001) bcc\nand without any distortion around the reference atom (for\nwhichαtandαonsiteare shown). (a) Co-centered and (b)\nFe-centered clusters. The quantity of Fe atoms in the surface\nlayers (near vacuum) are indicated by the numbers in paren-\nthesis and the results have been ordered such that larger val-\nues are to the left in the plots. Insets:αonsitefor the [010]\n(red open circles) and [110](blue filled circles) magnetization\ndirections, and corresponding local density of states, n(EF),\nat the Fermi level (green filled and unfilled triangles) at the\ncentral atom (placed in the outermost layer) for both types\nof clusters. Lines are guides for eyes.\ncontributions from surface atoms in the outermost layer.\nHence the results presented here represents one more ex-\nample, in addition to the well known enhanced surface\norbital moment [31], of the so-called interfacial spin-orbit\ncoupling. This damping anisotropy, which holds a bcc-\nlike fourfold ( C4v) symmetry, has a close relation to the\nLDOS difference of the most external layer at EF(ma-\njorly contributed by the minority dstates), as well as\nto the orbital moment anisotropy with a 90◦phase. As\na distinct example of an interface, we consider explicitly\nthe Fe50Co50cluster description of the alloy. In this case,\nbesides an onsite contribution, we find that the damp-\ning anisotropy is mostly influenced by non-local next-\nnearest-neighbours interactions.\nSeveral Gilbert damping anisotropy origins are also\ndemonstrated here, primarily related to the presence of\ninterfaces, alloy composition and local structural distor-\ntions (as summarized in Table S6, in the SM [32]). Pri-\nmarily we find that: ( i) the presence of Co introduces anenhanced spin-orbit interaction and can locally modify\nthe non-local damping terms; ( ii) the randomness of Co\nin the material, can modestly increase ∆αtas a total ef-\nfect by creating Co-concentrated clusters with enhanced\ndamping; ( iii) at the surface, the spatial distribution of\nFe/Co, increases the damping when more Fe atoms are\npresent in the outermost layer; and ( iv) the existence\nof local, tetragonal distortions, which act in favour (via\nSOC) of the absolute damping enhancement, by modify-\ning theαonsiteof the reference atom, and could locally\nchange the spin relaxation time. Furthermore, in rela-\ntionship to the work in Ref. [10], we show here that bulk\nlike tetragonal distortions, that in Ref. [10] were sug-\ngested to be the key reason behind the observed huge\nanisotropy of the damping, can in fact not explain the\nexperimental data. Such distortions were explicitly con-\nsidered here, using state-of-the-art theory, and we clearly\ndemonstrate that this alone can not account for the ob-\nservations.\nAlthough having a similar trend as the experimen-\ntal results of Ref. [10], we do not reproduce the most\nextreme maximum-minimum ratio reported in the ex-\nperiment,∼400%(or∆αt∼300%). The measured\ndamping does however include effects beyond the intrin-\nsic damping that is calculated from our electronic struc-\nturemethodology. Other mechanismsare knownto influ-\nence the damping parameter, such as contributions from\neddy currents, spin-pumping, and magnon scattering, to\nname a few. Thus it is possible that a significant part\nof the measured anisotropy is caused by other, extrin-\nsic, mechanisms. Despite reasons for differences between\nobservation and experiment on films of Fe50Co50alloys,\nthe advancements presented here provide new insights on\nthe intrinsic damping anisotropy mechanisms, something\nwhich is relevant for the design of new magnetic devices.\nAcknowledgements: H.M.P. and A.B.K. acknowledge\nfinancial support from CAPES, CNPq and FAPESP,\nBrazil. The calculations were performed at the computa-\ntional facilities of the HPC-USP/CENAPAD-UNICAMP\n(Brazil), at the National Laboratory for Scientific Com-\nputing (LNCC/MCTI, Brazil), and at the Swedish Na-\ntional Infrastructure for Computing (SNIC). I.M. ac-\nknowledge financial support from CAPES, Finance Code\n001, process n◦88882.332894/2018-01, and in the Insti-\ntutional Program of Overseas Sandwich Doctorate, pro-\ncess n◦88881.187258/2018-01. 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Bergman3, D. Thonig3,4, H. M. Petrilli1, and O. Eriksson3,4\n1Universidade de São Paulo, Instituto de Física,\nRua do Matão, 1371, 05508-090, São Paulo, SP, Brazil\n2Faculdade de Física, Universidade Federal do Pará, Belém, PA, Brazil\n3Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden and\n4School of Science and Technology, Örebro University, Fakultetsgatan 1, SE-701 82 Örebro, Sweden\n(Dated: November 22, 2021)\nI. Theory\nThe torque-correlation model, first introduced by\nKamberský [1], and later elaborated by Gilmore et al.\n[2], can be considered as both a generalization and an\nextended version of the breathing Fermi surface model,\nwhich relates the damping of the electronic spin orienta-\ntion, with the variation in the Fermi surface when the\nlocal magnetic moment is changed. In this scenario,\nand considering the collinear limit of the magnetic or-\ndering, due to the spin-orbit coupling (SOC), the tilting\nin magnetization ˆmby a small change δˆmgenerates a\nnon-equilibrium population state which relaxes within a\ntimeτtowards the equilibrium. We use an angle θ, to\nrepresent the rotation of the magnetization direction δˆm.\nIftheBlochstatesofthesystemsarecharacterizedbythe\ngenericbandindex natwavevector k(withenergies /epsilon1k,n),\nit is possible to define a tensorfor the damping, that has\nmatrix elements (adopting the isotropic relaxation time\napproximation)\nανµ=gπ\nm/summationdisplay\nn,mdk\n(2π)3η(/epsilon1k,n)/parenleftbigg∂/epsilon1k,n\n∂θ/parenrightbigg\nν/parenleftbigg∂/epsilon1k,m\n∂θ/parenrightbigg\nµτ\n~\n(S1)\nwhich accounts for both intraband ( n=m, conductivity-\nlike) and interband ( n/negationslash=m, resistivity-like) contribu-\ntions [2]. Here µ,νare Cartesian coordinate indices,\nthat will be described in more detail in the discussion\nbelow, while η(/epsilon1k,n) =∂f(/epsilon1)\n∂/epsilon1/vextendsingle/vextendsingle/vextendsingle\n/epsilon1k,nis the derivative of\nthe Fermi distribution, f, with respect to the energy\n/epsilon1, andn,mare band indices. Therefore, the torque-\ncorrelation model correlates the spin damping to vari-\nations of the energy of single-particle states with respect\nto the variation of the spin direction θ, i.e.∂/epsilon1k,n\n∂θ. Us-\ning the Hellmann-Feynmann theorem, which states that\n∂/epsilon1k,n\n∂θ=/angbracketleftψk,n|∂H\n∂θ|ψk,n/angbracketright, and the fact that only the spin-\norbit Hamiltonian Hsochanges with the magnetization\ndirection, the spin-orbit energy variation is given by\n∂/epsilon1k,n(θ)\n∂θ=/angbracketleftψk,n|∂\n∂θ/parenleftbig\neiσ·ˆnθHsoe−iσ·ˆnθ/parenrightbig\n|ψk,n/angbracketright(S2)\nin which σrepresents the Pauli matrices vector, and\nˆnis the direction around which the local moment hasbeen rotated. The expression in Eq. S2 can be eas-\nily transformed into∂/epsilon1k,n(θ)\n∂θ=i/angbracketleftψk,n|[σ·ˆn,Hso]|ψk,n/angbracketright\nand we call ˆT= [σ·ˆn,Hso]thetorqueoperator. In\nview of this, it is straightforward that, in the collinear\ncase in which all spins are aligned to the zdirection,\nσ·ˆn=σµ(µ=x,y,z), originating the simplest {x,y,z}-\ndependent torque operator ˆTµ. Putting together the in-\nformation on Eqs. S1 and S2, and using the fact that\nthe imaginary part of the Greens’ functions can be ex-\npressed, in Lehmann representation, as Im ˆG(/epsilon1±iΛ) =\n−1\nπ/summationtext\nnΛ\n(/epsilon1−/epsilon1n)2+Λ2|n/angbracketright/angbracketleftn|, then it is possible to write in\nreciprocal-space [3]:\nανµ=g\nmπ/integraldisplay /integraldisplay\nη(/epsilon1)Tr/parenleftBig\nˆTνImˆGˆTµImˆG/parenrightBig\nd/epsilon1dk\n(2π)3.(S3)\nIn a real-space formalism, the Fourier transformation\nof the Green’s function is used to find a very similar ex-\npression emerges for the damping element ανµ\nijrelative to\ntwo atomic sites iandj(at positions riandrj, respec-\ntively) in the material:\nανµ\nij=g\nmiπ/integraldisplay\nη(/epsilon1)Tr/parenleftBig\nˆTν\niImˆGijˆTµ\njImˆGji/parenrightBig\nd/epsilon1,(S4)\nwhere we defined mi= (morb+mspin)as the total mag-\nnetic moment localized in the reference atomic site iin\nthe pair{i,j}. The electron temperature that enters into\nη(ε)is zero and, consequently, the energy integral is per-\nformed only at the Fermi energy. In this formalism, then,\nthe intraband and interband terms are replaced by onsite\n(i=j) and non-local ( i/negationslash=j) terms. After calculation of\nall components of Eq. S4 in a collinear magnetic back-\nground, we get a tensor of the form\nαij=\nαxxαxyαxz\nαyxαyyαyz\nαzxαzyαzz\n, (S5)\nwhich can be used in the generalized atomistic Landau-\nLifshitz-Gilbert (LLG) equation for the spin-dynamics\nof magnetic moment on site i[4]:∂mi\n∂t=mi×/parenleftBig\n−γBeff\ni+/summationtext\njαij\nmj·∂mj\n∂t/parenrightBig\n. Supposing that all spins are\nparallel to the local zdirection, we can define the scalar2\nαvalue as the average between components αxxandαyy,\nthat is:α=1\n2(αxx+αyy).\nOnce one has calculated the onsite ( αonsite) and the\nnon-local (αij) damping parameters with respect to the\nsite of interest i, the total value, αt, can be defined as\nthe sum of all these α’s:\nαt=/summationdisplay\n{i,j}αij. (S6)\nIn order to obtain the total damping in an heteroge-\nneous atomic system (more than one element type), such\nas Fe50Co50(with explicit Fe/Co atoms), we consider the\nweighted average between the different total local damp-\ning values ( αi\nt), namely:\nαt=1\nMeff/summationdisplay\nimiαi\nt, (S7)\nwheremiis the local magnetic moment at site i, and\nMeff=/summationtext\nimiis the summed total effective magnetiza-\ntion. This equation is based on the fact that, in FMR ex-\nperiments, the magnetic moments are excited in a zone-\ncentered, collective mode (Kittel mode). In the results\npresented here, Eq. S7 was used to calculate αtofB2-\nFeCo, both in bcc and bct structures.\nII. Details of calculations\nThe real-space linear muffin-tin orbital on the atomic-\nsphere approximation (RS-LMTO-ASA) [5] is a well-\nestablished method in the framework of the DFT to de-\nscribetheelectronicstructureofmetallicbulks[6,7], sur-\nfaces [8, 9] and particularly embedded [10] or absorbed\n[11–14] finite cluster systems. The RS-LMTO-ASA is\nbased on the LMTO-ASA formalism [15], and uses the\nrecursion method [16] to solve the eigenvalue problem\ndirectly in real-space. This feature makes the method\nsuitable for the calculation of local properties, since it\ndoes not depend on translational symmetry.\nThe calculations performed here are fully self-\nconsistent, and the spin densities were treated within\nthe local spin-density approximation (LSDA) [17]. In all\ncases, we considered the spin-orbit coupling as a l·sterm\nincluded in each variational step [18–20]. The spin-orbit\nis strictly necessary for the damping calculations due to\nits strong dependence on the torque operators, ˆT. In\nthe recursion method, the continued fractions have been\ntruncated with the Beer-Pettifor terminator [21] after 22\nrecursion levels ( LL= 22). The imaginary part that\ncomes from the terminator was considered as a natural\nchoice for the broadening Λto build the Green’s func-\ntions ˆG(/epsilon1+iΛ), which led to reliable αparameters in\ncomparison with previous results (see Table S1).To account for the Co randomness in the experimental\nFe50Co50films [22], some systems were modeled in terms\nof the virtual crystal approximation (VCA) medium of\nFe50Co50, considering the bcc (or the bct) matrix to have\nthe same number of valence electrons as Fe50Co50(8.5\ne−). However, we also investigated the role of the Co\npresence, as well as the influence of its randomness (or\nordering), by simulating the B2(CsCl) FeCo structure\n(a=aFe). The VCA Fe50Co50andB2-FeCo bulks were\nsimulated by a large matrix containing 8393 atoms in\nreal-space, the first generated by using the Fe bcc lat-\ntice parameter ( aFe= 2.87Å) and the latter using the\noptimized lattice parameter ( a= 2.84Å). Thisachoice\nin VCA Fe50Co50was based on the fact that it is eas-\nier to compare damping results for Fe50Co50alloy and\npure Fe bcc bulk if the lattice parameters are the same,\nand the use of the aFehas shown to produce trustwor-\nthyαtvalues. On the other hand, bct bulk structures\nwithc\na= 1.15(B2-FeCo bct and VCA Fe50Co50bct) are\nbased on even larger matrices containing 49412 atoms.\nThe respective surfaces were simulated by semi-matrices\nof the same kind (4488 and 19700 atoms, respectively),\nconsidering one layer of empty spheres above the outer-\nmost Fe50Co50(or pure Fe) layer, in order to provide a\nbasis for the wave functions in the vacuum and to treat\nthe charge transfers correctly.\nWe notice that the investigations presented here are\nbased on a (001)-oriented Fe50Co50film, in which only a\nsmall lattice relaxation normal to the surface is expected\nto occur (∼0.1%[23]).\nDamping parameters of Fe-centered and Co-centered\nclusters, embedded in an Fe50Co50VCA medium, have\nbeen calculated (explicitly) site by site. In all cases,\nthese defects are treated self-consistently, and the po-\ntential parameters of the remaining sites were fixed at\nbulk/pristine VCA surface values, according to its envi-\nronment. When inside the bulk, we placed the central\n(reference) atom of the cell in a typical site far away\nfrom the faces of the real-space matrix, avoiding any un-\nwanted surface effects. We considered as impurities the\nnearest 14 atoms (first and second nearest neighbours,\nup to 1a) from the central atom, treating also this sites\nself-consistently, in a total of 15 atoms. We calculated\n10 cases with Fe and Co atoms randomly positioned: 5\nwith Fe as the central atom (Fe-centered) and 5 with Co\nas the central atom (Co-centered). An example (namely\ncluster #1 of Tables S3 and S4), of one of these clusters\nembedded in bulk, is represented in Fig. S1(a). As the\nself-consistent clusters have always a total of 15 atoms,\nthe Fe (Co) concentration is about 47% (53%) or vice-\nversa. On the other hand, when inside the surface, we\nplaced the central (reference) atom of the cluster in a\ntypical site of the most external layer (near vacuum),\nsince this has shown to be the layer where the damping\nanisotropy is larger. Therefore, we considered as impu-\nrities the reference atom itself and the nearest 9 atoms3\n(up to 1a), in a total of 10 atoms (and giving a perfect\n50% (50%) concentration). An example of one of these\nclusters embedded in a surface is shown in Fig. S1(b).\n(a)\n(b)\nFigure S1. (Color online) Schematic representation of an ex-\nampleof: (a)Fe-centered15-atomclusterembeddedinaVCA\nFe50Co50bcc bulk medium; (b) Co-centered 10-atom cluster\nembedded in a VCA Fe 50Co50(001) bcc surface medium. Yel-\nlow and blue spheres represent Fe and Co atoms, respectively,\nwhile gray atoms represent the VCA Fe 50Co50sites (8.5 va-\nlence e−). The Fe(Co) concentration in the clusters are: (a)\n53% (47%) and (b) 50% (50%) . The total number of atoms\nincluding the surrounding VCA sites are: (a) 339 and (b) 293.\nThey were all accounted in the sum to obtain αtat the central\n(reference) Fe (a) and Co (b) site.\nTo simulate a bct-like bulk distortion, the 8 first neigh-\nbours of the central atom were stretched in the cdi-\nrection, resulting in ac\na= 1.15ratio. On the other\nhand, when embedded on the Fe50Co50(001) bcc surface,\nthe central (reference) atom is placed in the outermost\nlayer (near vacuum), and we simulate a bct distortion\nby stretching the 4 nearest-neighbours (on the second\nlayer) to reproduce ac\na= 1.09ratio (the maximum per-\ncentage that the atoms, in these conditions, could be\nmoved to form a bct-like defect). In this case, a total\nof 10 atoms (the nearest 9 atoms from the central one\n– up to 1a– and the reference atom) were treated self-\nconsistently, analogous to as shown in Fig. S1(b). As in\nthe case of the pristine bcc Fe50Co50clusters embedded\nin the VCA surface, we considered a total of 32 10-atom\nclusters with different Fe/Co spatial distributions, being\n16 Fe-centered, and 16 Co-centered.III. Comparison with previous results\nTheab-initio calculation of the Gilbert damping, in\nthe collinear limit, is not a new feature in the literature.\nMainly, the reported theoretical damping results are for\nbulk systems [2, 4, 24–28], but, some of them even stud-\nied free surfaces [29]. Therefore, in order to demonstrate\nthe reliability of the on-site and total damping calcula-\ntions implemented here in real-space, a comparison of\nthe presently obtained with previous (experimental and\ntheoretical) results, are shown in Table S1. As can be\nseen, our results show a good agreement with previously\nobtainedαvalues, including some important trends al-\nreadypredictedbefore. Forexample, thereducedGilbert\ndamping of Co hcp with respect to the Co fcc due to\nthe reduction of the density of states at the Fermi level\n[24, 28], (∼10.92states/Ry-atom in the hcp case and\n∼16.14states/Ry-atom in the fcc case).\nIV. Details of the calculated damping values\nThe damping values obtained for the systems studied\nhere are shown in Tables (S2-S5). These data can be use-\nful for the full understanding of the results presented in\nthe main text. For easy reference, in Table S2 the αtof a\ntypical atom in each system (bulk or surface) for different\nspin quantization axes are shown. These data are plot-\nted in Fig. 2 of the main text. The obtained values show\nthat, indeed, for bulk systems the damping anisotropies\nare not so pronounced as in the case of Fe50Co50(001)\nbcc surface.\nAs observed in Table S2, the increase in αtwhen\nchanging from the bcc Fe50Co50(c\na= 1) to the bct\nFe50Co50bulk structure (c\na= 1.15) is qualitatively con-\nsistent to what was obtained by Mandal et al.[33] (from\nαt= 6.6×10−3in the bcc to αt= 17.8×10−3in the\nbct, withc\na= 1.33[33]).\nTables S3 and S4 refer to the damping anisotropies\n(∆αt) for all Fe-centered and Co-centered clusters stud-\nied here, with different approaches: ( i) bcc clusters em-\nbedded in the VCA medium (Table S3) and ( ii) bct-like\nclusters embedded in the VCA medium (Table S4).\nIn comparison with bct-like clusters, we found larger\nabsoluteαtvalues but lower damping anisotropies. In\nall cases, Fe-centered clusters present higher ∆αtper-\ncentages.\nIn Table S5 the onsite damping anisotropies ( ∆αonsite)\nfor each layer of the Fe50Co50(001) bcc surface (\"1\" repre-\nsents the layer closest to vacuum) are shown. In compar-\nison with the total damping anisotropies (Table I of the\nmain text), much lower percentages are found, demon-\nstrating that the damping anisotropy effect comes ma-\njorly from the non-local damping contributions.\nThe most important results concerning the largest\ndampinganisotropiesaresummarizedinTableS6, below.4\nTable S1. Total damping values ( ×10−3) calculated for some bulk and surface systems, and the comparison with previous\nliterature results. The onsite contributions are indicated between parentheses, while the total damping, αt, are indicated\nwithout any symbols. All values were obtained considering the [001]magnetization axis. The VCA was adopted for alloys,\nexcept for the Fe 50Co50bcc in theB2structure (see Eq. S7). Also shown the broadening Λvalue considered in the calculations.\nBulks a(Å) This work Theoretical Experimental Λ(eV)\nFe bcc 2.87 4.2(1.6) 1 .3[2]a/(3.6)[4] 1.9[30]/2.2[31]\nFe70Co30bcc 2.87 2.5(0.7) − 3−5[32]d\nFe50Co50bcc 2.87 3.7(1.0)[VCA]/ 2.3(1.0)[B2]1.0[25]c[VCA]/ 6.6[33] [B2] 2.3[27]\nNi fcc 3.52 27.8(57.7) 23 .7[34]/( 21.6[4])b26.0[31]/24.0[35]\nNi80Fe20(Py) fcc 3.52 9.8(12.1) 3 .9[25]c8.0[30]/5.0[35]\nCo fcc 3.61 [3] 3.2(5.3) 5 .7[28]/(3.9[4])b11.0[30]∼5×10−2\nCo hcp 2.48/4.04 [28] 2.1(6.2) 3 .0[28] 3.7[31]\nCo85Mn15bcc [36] 2.87 [28] 6.2(4.2) 6 .6[28] −\nCo90Fe10fcc 3.56 [37] 3.6(4.2) − 3.0[35]/4.8[37]\nSurfaces a(Å) This work Theoretical Experimental\nFe(001) bcc [110] 2.87 5.8(5.4)e− 7.2[38]h/6.5[39]i\nFe(001) bcc [100] 2.87 3.9(4.4)f∼4[29]g4.2[40]j\nNi(001) fcc 3.52 80.0(129.6)∼10[29]g/12.7[41]m22.1[42]l\nPdFe/Ir(111) [43] fcc 3.84 3.9(2.7)n− −\nPdCo/Ir(111) [44] fcc 3.84 3.2(14.7)o− −\naWith Λ∼2×10−2eV.\nbWith Λ = 5×10−3eV.\ncWith Λ∼1.4×10−4eV.\ndFor a 28%Co concentration, but the results do not significantly change for a 30%Co concentration. Range including results before and\nafter annealing.\neOf a typical atom in the more external surface layer (in contact with vacuum), in the [110] magnetization direction.\nfOf a typical atom in the more external surface layer (in contact with vacuum), in the [100] magnetization direction.\ngFor a (001) bcc surface with thickness of N= 8ML (the same number of slabs as in our calculations), and Λ = 10−2eV.\nhAnisotropic damping obtained for a 0.9 nm Fe/GaAs(001) thin film (sample S2 in Ref. [38]) in the [110] magnetization direction.\niAnisotropic damping obtained for a 1.14 nm Fe/InAs(001) thin film in the in-plane [110] hard magnetization axis.\njFor a 25-nm-thick Fe films grown on MgO(001).\nkFor epitaxial Fe(001) films grown on GaAs(001) and covered by Au, Pd, and Cr capping layers.\nlIntrinsic Gilbert damping for a free 4×[Co(0.2 nm)/Ni(0.6 nm)](111) multilayer. Not the same system as Ni(001), but the nearest system\nfound in literature.\nmFor a Co | Ni multilayer with Ni thickness of 4 ML (fcc stacking).\nnOf a typical atom in the Fe layer.\noOf a typical atom in the Co layer.\nThe alloys with short-range orders (SRO) are described\nas FeCo clusters (with explicit Fe and Co atoms) embed-\nded in the Fe50Co50VCA medium – with and without\nthe bct-like distortion. In this case, the damping is cal-\nculated as a weighted average (Eq. S7). As discussed in\nthe main text, it can be seen from Table S6 that distor-\ntions and disorder can increase the anisotropy but the\nmajor effect comes from the surface. We notice that the\nnumber of clusters considered is limited in the statistical\naverage.\nIV. Kambersky’s simplified formula\nInordertoconnecttheanisotropyoftheGilbertdamp-\ning to features in the electronic structure, we consider in\nthe following Kambersky’s simplified formula for Gilbert\ndamping [47, 48]α=1\nγMs/parenleftBigγ\n2/parenrightBig2\nn(EF)ξ2(g−2)2\nτ.(S8)\nHere,γis the gyromagnetic ratio, n(EF)represents the\nLDOS at the Fermi level, ξis the SOC strength, τis the\nelectron scattering time, Msis the spin magnetic mo-\nment, andgis the spectroscopic g-factor [35, 49]. Note\nthat Eq. S8 demonstrates the direct relation between\nαandn(EF), often discussed in the literature, e.g., in\nRef. [27]. Our first principles calculations have shown\nno significant change in ξ, upon variation of the mag-\nnetization axis, for the FeCo systems ( ξCo= 71.02meV\nandξFe= 53.47meV). Hence, we can soundly relate the\ndamping anisotropy ∆αtto∆n(EF).\nFigure S2 shows how the LDOS difference (per atom)\n∆n(E)between the [010]and [110]magnetization di-\nrections is developed in pure Fe(001) bcc and in VCA\nFe50Co50(001) bcc surfaces, respectively. In both cases,5\nTableS2. Totaldamping( αt×10−3)ofatypicalatomin\neach system for the spin quantization axes [010](θH=\n90◦) and [110](θH= 45◦); also shown for the [001]and\n[111]. Bulk and surface bct systems are simulated with\nc\na= 1.15.\nBulks\nBulk αt[010]αt[110] ∆αt\nFe bcc 4.18 4.31 +3.1%a\nB2-FeCo bcc 2.28 2.44 +7.2%\nB2-FeCo bct 7.76 8.85 +12.4%\nVCA Fe 50Co50bcc 3.70 4.18 +13.0%\nVCA Fe 50Co50bct 4.69 5.10 +8.7%\nαt[010]αt[001] ∆αt\nB2-FeCo bct 7.76 10.21 +24.1%\nVCA Fe 50Co50bct 4.69 5.75 +22.6%\nαt[010]αt[111] ∆αt\nFe bcc 4.18 4.56 +9.1%b\nSurfaces\nSurface αt[010]αt[110] ∆αt\nFe(001) bcc 3.85 5.75 +49.4%\nFe/GaAs(001) bcc [38] 4.7(7) 7.2(7) +53(27)%c\nFe/MgO(001) bcc [45] 3.20(25) 6.15(20) +92(14)%d\nVCA Fe 50Co50(001) bcc 7.00 14.17 +102.4%\nVCA Fe 50Co50(001) bct 15.20 14.80−2.6%\nαt[010]αt[001] ∆αt\nVCA Fe 50Co50(001) bct 15.20 15.56 +2.4%\nVCA Fe 50Co50(001) bcc 7.00 9.85 +40.7%\naMankovsky et al.[24] find a damping anisotropy of ∼12%\nfor bulk Fe bcc at low temperatures ( ∼50K) between\n[010] and [011] magnetization directions. For this result,\nthe definition α=1\n2(αxx+αyy)was used.\nbThis result agrees with Gilmore et al.[46], which find\nthat the total damping of pure Fe bcc presents its higher\nvalue in the [111] crystallographic orientation and the\nlower value in the [001] direction, except for high scatter-\ning rates. Also agrees with Mankovsky et al.[24] results.\ncAnisotropic damping obtained for a 0.9 nm Fe/GaAs(001)\nthin film (sample S2 in Ref. [38]) in the [010] and [110]\nmagnetization directions.\ndFor a Fe(15 nm)/MgO(001) film at T= 4.5K in the high-\nest applied magnetic field, in which only intrinsic contri-\nbutions to the anisotropic damping are left.\nthe chosen layer,denoted as first, is the most external\none (near vacuum). the VCA Fe50Co50(001) bcc we also\ncalculated ∆n(E)for all layers summed (total DOS dif-\nference).\nAs can be seen, although in all cases the quantity\n∆n(E)exhibits some oscillations, differently from what\nwe observe forthe pureFe(001) surface case, at the Fermi\nenergy, there is a non-negligible difference in the minor-\nity spin channel ( 3dstates) for the VCA Fe50Co50(001).\nConsidering the results presented in Table I (main text)\nthe larger contribution to the damping anisotropy comes\nfrom the most external layer. The results by Li et al.\n[22] indicate a small difference (for two magnetization\ndirections) of the total density of states at the FermiTable S3. Total damping anisotropy ( ×10−3) of all stud-\nied Co-centered and Fe-centered bcc clusters for the spin-\nquantization axis [010]and [110], considering the 15-atom\nFeCo cluster together with the VCA medium in the summa-\ntion for total damping.\nCo-centered\nCluster # αt[010]αt[110] ∆αt\n1 10.11 9.65 4.8%\n2 8.09 6.96 16.2%\n3 7.81 7.02 11.3%\n4 7.11 7.02 1.3%\n5 7.48 6.88 8.7%\nAverage 8.12 7.51 8.1%\nFe-centered\nCluster # αt[010]αt[110] ∆αt\n1 2.68 2.03 32.0%\n2 2.49 2.05 21.5%\n3 2.56 1.86 37.6%\n4 2.45 1.79 36.9%\n5 2.76 2.01 37.3%\nAverage 2.59 1.95 32.8%\nTable S4. Total damping anisotropy ( ×10−3) of all stud-\nied Co-centered and Fe-centered bcc clusters for the spin\nquantization axis [010]and [110], with bct-like distortions/parenleftbigc\na= 1.15/parenrightbig\n, considering the 15-atom FeCo cluster together\nwith the VCA medium in the summation for total damping.\nCo-centered\nCluster # αt[010]αt[110] ∆αt\n1 5.85 4.37 33.9%\n2 5.95 4.21 41.3%\n3 5.88 4.35 35.2%\n4 5.90 4.41 33.8%\n5 5.86 4.34 35.0%\nAverage 5.89 4.34 35.7%\nFe-centered\nCluster # αt[010]αt[110] ∆αt\n1 2.36 1.39 69.8%\n2 2.27 1.32 72.0%\n3 2.22 1.26 76.2%\n4 2.25 1.26 78.6%\n5 2.42 1.38 75.4%\nAverage 2.30 1.32 74.2%\nTable S5. Onsite damping ( αonsite×10−3) of a typical atom\nin each layer of the VCA Fe 50Co50(001) bcc for the spin quan-\ntization axis [010]and[110].\nLayerαonsite[010]αonsite[110] ∆αonsite\n1 7.36 8.70 +18.2%\n2 0.63 0.69 +9.5%\n3 1.41 1.44 +2.1%\n4 0.87 0.86−1.1%\n5 0.99 0.97−2.0%6\nTableS6. SummaryofthemainFe 50Co50dampinganisotropy\nresults for: pure ordered ( B2) alloy; pure random (VCA) bulk\nalloy; bcc bulk together with short-range order (SRO) clus-\nters (see Table S3); bulk together with bct-like distorted clus-\nters inside (see Table S4); surface calculations, in the pristine\nmode and with explicit bct-like clusters embedded (surface +\ndistortion). The maximum-minimum ratio according to Ref.\n[22] isα[110]\nt\nα[010]\nt×100%.\nStructure ∆αtMax-min ratio\nOrdered alloy bcc 7.2% 107.2%\nOrdered alloy bct 24.1% 124.1%\nRandom alloy bcc 13% 113%\nRandom alloy bct 22.6% 122.6%\nRandom alloy + SRO 14.9% 114.9%\nRandom alloy + SRO + Distortion 47.2% 147.2%\nSurface (external layer) 102.4% 202.4%\nSurface (ext. layer) + Distortion 75.4% 175.4%\n10-nm Co 50Fe50/Pt [22] (exp.) 281.3% 381.3%\n−2−1 0 1 2\n−0.02 −0.01 0 0.01 0.02Δn(E)[010]−[110] (states/Ry−atom)\nEnergy (E−EF) (Ry)Fe(001) bcc (first)\nVCA Fe50Co50(001) bcc (first)\nVCA Fe50Co50(001) bcc (all)\nFigure S2. LDOS difference (per atom), ∆n(E), between the\n[010]and[110]magnetization directions, for both spin chan-\nnels (full lines for majority spin and dashed lines for minority\nspin states), in the outermost layer in pure Fe(001) bcc (in\nblack); outermost layer in VCA Fe 50Co50(001) bcc (in blue);\nand all layers summed in VCA Fe 50Co50(001) bcc (in red).\nlevel,N(EF), what the authors claim that could not ex-\nplain the giant maximum-minimum damping ratio ob-\nserved. So, in order to clarify this effect in the VCA\nFe50Co50(001) bcc, ∆n(E)was also calculated for the all\nlayers summed, what is shown in Fig. S2 (in red). This\ndifference is in fact smaller if we consider the DOS of\nthe whole system, with all layers summed. However, if\nwe consider only the most external layer, then the LDOS\nvariation is enhanced. This is consistent with our theo-\nretical conclusions. As we mention in the main text, this\ndo not rule out a role also played by local (tetragonal-\nlike) distortions and other bulk-like factors in the damp-\ning anisotropy.For the outermost layer of Fe(001) bcc, the calculated\nLDOSatEFis∼20.42states/Ry-atominthe[110]direc-\ntion and∼20.48states/Ry-atom in the [010] direction,\nwhich represents a difference of ∼0.3%and agrees with\nthe calculations performed by Chen et al.[38].\nV. Correlation with anisotropic orbital moment\nBesides the close relation exhibited between ∆αt\nand∆n(EF), we also demonstrate the existence of an\nanisotropic orbital moment in the outermost layer, in\nwhich the fourfold symmetry ( C4v) matches the damp-\ning anisotropy with a 90◦phase. Fig. S3 shows this\ncorrelation between ∆αtand∆morbfor two situations:\n(i) for a typical atom in the outermost layer of VCA\nFe50Co50(001) bcc (blue open dots); and ( ii) for a typi-\ncal atom in the VCA Fe50Co50bcc bulk, considering the\nsame ∆morbscale. For case ( i) we find orbital moments\ndifferencesmorethanoneorderofmagnitudehigherthan\ncase (ii).\n0o45o90o\n135o\n180o\n225o\n270o315o[100][110][010]\n[−110]\n0.51.52.5\nΔmorb\n(µB/atom × 10−3)\n0.0050.0100.015\nαt\nθH\nFigure S3. (Color online) Total damping and orbital moment\ndifference, ∆morbas a function of θH, the angle between the\nmagnetizationdirectionandthe [100]-axis. Squares: (redfull)\nVCA Fe 50Co50(001) bcc. Circles: (blue open) morbdifference\nbetweenθH= 90◦and the current angle for a typical atom in\nthe outermost layer of VCA Fe 50Co50(001) bcc; and (yellow\nfull) samemorbdifference but for a typical atom in the VCA\nFe50Co50bcc bulk (in the same scale). Lines are guides for\nthe eyes.\nVI. Contribution from next-nearest-neighbours\nFinally, we show in Fig. S4 the summation of all non-\nlocal damping contributions, αij, for a given normalized\ndistance in the outermost layer of VCA Fe50Co50(001)7\nbcc. As we can see, the next-nearest-neighbours from a\nreference site (normalized distanced\na= 1) have very dis-\ntinctαijcontributions to αtfor the two different mag-\nnetization directions ( [010]and[110]), playing an impor-\ntant role on the final damping anisotropy. We must note,\nhowever, that these neighbours in a (001)-oriented bcc\nsurface are localized in the same layer as the reference\nsite, most affected by the interfacial SOC. Same trend is\nobserved ford\na= 2, however less intense. This is con-\nsistent with our conclusions, about the relevance of the\noutermost layer on ∆αt.\n−3−2−1 0 1 2\n 0.5 1 1.5 2 2.5 3 3.5 4∑αij × 10−3\nNormalized distance[110]\n[010]\nFigure S4. 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Rev. 76, 743 (1949)." }, { "title": "1704.03326v1.CoFeAlB_alloy_with_low_damping_and_low_magnetization_for_spin_transfer_torque_switching.pdf", "content": "arXiv:1704.03326v1 [cond-mat.mtrl-sci] 11 Apr 2017CoFeAlB alloy with low damping and low magnetization for spi n transfer torque\nswitching\nA. Conca,1,∗T. Nakano,2T. Meyer,1Y. Ando,2and B. Hillebrands1\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\n2Department of Applied Physics, Tohoku University, Japan\n(Dated: June 29, 2021)\nWe investigate the effect of Al doping on the magnetic propert ies of the alloy CoFeB. Comparative\nmeasurements of the saturation magnetization, the Gilbert damping parameter αand the exchange\nconstantasafunctionoftheannealingtemperature forCoFe B andCoFeAlBthinfilmsare presented.\nOur results reveal a strong reduction of the magnetization f or CoFeAlB in comparison to CoFeB.\nIf the prepared CoFeAlB films are amorphous, the damping para meterαis unaffected by the Al\ndoping in comparison to the CoFeB alloy. In contrast, in the c ase of a crystalline CoFeAlB film, α\nis found to be reduced. Furthermore, the x-ray characteriza tion and the evolution of the exchange\nconstant with the annealing temperature indicate a similar crystallization process in both alloys.\nThe data proves the suitability of CoFeAlB for spin torque sw itching properties where a reduction\nof the switching current in comparison with CoFeB is expecte d.\nThe alloy CoFeB is widely used in magnetic tunnel-\ning junctions in combination with MgO barriers due to\nthe large magnetoresistance effect originating in the spin\nfiltering effect [1–4]. For the application in magnetic ran-\ndom accessmemories, the switching ofthe magnetization\nof the free layer via spin transfer torque (STT) with spin\npolarised currents is a key technology. However, the re-\nquired currents for the switching process are still large\nand hinder the applicability of this technique. The criti-\ncal switching current density for an in-plane magnetized\nsystem is given by [5]\nJc0=2eαMStf(HK+Hext+2πMS)\n/planckover2pi1η(1)\nwhereeis the electron charge, αis the Gilbert damping\nparameter, MSis the saturation magnetization, tfis the\nthickness of the free layer, Hextis the external field, HK\nis the effective anisotropy field and ηis the spin transfer\nefficiency. Fromtheexpressionitisclearthat, concerning\nmaterial parameters, Jc0is ruled by the product αM2\nS.\nFor out-of-plane oriented layers, the term 2 πMSvanishes\nand then JC0is proportional to αMS[6]. Even in the\ncase of using pure spin currents created by the Spin Hall\neffect, the required currents are proportional to factors\nof the form αnMSwithn= 1,1/2 [7]. A proper strat-\negy to reduce the critical switching currents is then de-\nfined by reducing the saturation magnetization. This can\nbe achieved by the development of new materials or the\nmodification of known materials with promising prop-\nerties. Since the compatibility with a MgO tunneling\nbarrier and the spin filtering effect must be guaranteed\ntogether with industrial applicability, the second option\nis clearly an advantage by reducing MSin the CoFeB al-\nloy. In this case, a critical point is that this reduction\nmust not be associated with an increase of the damping\nparameter α.In the last years, several reports on doped CoFeB al-\nloys have proven the potential of this approach. The\nintroduction of Cr results in a strong reduction of MS\n[8–10], however, it is sometimes also causing an increase\nof the damping parameter [8]. The reduction of MSby\ndoping CoFeB with Ni is smaller compared to a doping\nwith Cr but it additionally leads to a reduction of α[8].\nFIG. 1. (Color online) θ/2θ-scans for 40 nm thick films of\nCo40Fe40B20(top) and Co 36Fe36Al18B10(bottom) showing\nthe evolution of crystallization with the annealing temper a-\nture.2\nFIG. 2. (Color online) Evolution of the saturation magnetiz a-\ntion for CoFeB and CoFeAlB with the annealing temperature\nTann.\nIn constrast, the reduction of magnetization with V is\ncomparable to Cr [9] but to our knowledge no values for\nαhave been published. In the case of doping of CoFeB\nby Cr or by V, a reduction of the switching current has\nbeen shown [8, 9].\nIn this Letter, we report on results on Al doped CoFeB\nalloy thin films characterized by ferromagnetic resonance\nspectroscopy. The dependence of MS, the Gilbert damp-\ning parameter αand the exchange constant on the an-\nnealing temperature is discussed together with the crys-\ntalline structure of the films and the suitability for STT\nswitching devices.\nThe samples are grown on Si/SiO 2substrates us-\ning DC (for metals) and RF (for MgO) sput-\ntering techniques. The layer stack of the sam-\nples is Si/SiO 2/Ta(5)/MgO(2)/FM(40)/MgO(2)/Ta(5)\nwhere FM = Co 40Fe40B20(CoFeB) or Co 36Fe36Al18B10\n(CoFeAlB). Here, the values in brackets denote the layer\nthicknesses in nm. In particular, the FM/MgO interface\nis chosen since it is widely used for STT devices based\non MTJs. This interface is also required to promote the\ncorrect crystallizationof the CoFeB layerupon annealing\nsince the MgO layer acts as a template for a CoFe bcc\n(100)-oriented structure [1–3] with consequent B migra-\ntion.\nThe dynamic properties and material parameters were\nstudied by measuring the ferromagnetic resonance using\na strip-line vector network analyzer (VNA-FMR). For\nthis, the samples were placed face down and the S 12\ntransmission parameter was recorded. A more detailed\ndescription of the FMR measurement and analysis pro-\ncedure is shown in previous work [11, 12]. Brillouin light\nspectroscopy (BLS) was additionally used for the mea-\nsurement of the exchange constant. The crystalline bulk\nproperties of the films were studied by X-ray diffractom-\netry (XRD) using the Cu-K αline.\nFigure 1 shows the θ/2θ-scans for CoFeB (top) andFIG. 3. (Color online) Linewidth at a fixed frequency of 18\nGHz (a) and Gilbert dampingparameter αdependenceon the\nannealing temperature T ann(b). The αvalue for Tann= 500◦\nis only a rough estimation since the large linewidth value do es\nnot allow for a proper estimation. The inset shows the linear\ndependence of the linewidth on the frequency exemplarily fo r\nCoFeAlB annealed at 350◦C and 400◦C. The red lines are a\nlinear fit.\nCoFeAlB (bottom) samples annealed at different tem-\nperatures T ann. The appearance of the CoFe diffractions\npeaks, as shown by the arrowsin Fig. 1 indicate the start\nof crystallization at high annealing temperatures of more\nthan 400◦C. In the case of lower annealing temperatures\nor the as-deposited samples, the FM layer is in an amor-\nphous state. The first appearance of the (200) diffraction\npeak occurs at the same point for both alloys showing a\nverysimilarthermalevolution. Thissimplifiesasubstitu-\ntion ofCoFeB by the Al alloyin tunneling junctions since\nthe same annealing recipes can be applied. This is criti-\ncal since the used values must be also optimized for the\nquality of the tunneling barrier itself or the perpendicu-\nlar anisotropy induced by the FM/MgO interface. The\n(110) CoFe peak is also present for both material compo-\nsitions owing to a partial texturing of the film. However,\nthe larger intensity of the (200) peak is not compatible\nwith a random crystallite orientation but with a domi-\nnant (100) oriented film [13, 14]. This is needed since the\nspin filtering effect responsible for the large magnetore-\nsistance effect in MgO-based junctions requires a (100)3\nFIG.4. (Color online)Dependenceoftheproduct αM2\nSonthe\nannealing temperature T annfor CoFeB and CoFeAlB. This\nquantityisrulingtheswitchingcurrentinin-planemagnet ized\nSTT devices as shown in Eq. 1.\norientation.\nThe dependence of the FMR frequency on the external\nmagnetic field is described by Kittel’s formula [15]. The\nvalue ofMeffextracted from the Kittel fit is related with\nthe saturation magnetization of the sample and the in-\nterfacial properties by Meff=MS−2K⊥\nS/µ0MSdwhere\nK⊥\nSis the interface perpendicular anisotropy constant.\nFor the thickness used in this work (40 nm) and physi-\ncally reasonable K⊥\nSvalues, the influence of the interface\nis negligible and therefore Meff≈MS. For details about\nthe estimation of Meffthe reader is referred to [12].\nFigure 2 shows the obtained values for MSfor all sam-\nples. AstrongreductionforCoFeAlBin comparisonwith\nstandard CoFeB is observed and the relative difference is\nmaintained for all T ann. The evolution with annealing is\nvery similar for both alloys. Significantly, the increase in\nMSstartsforvaluesofT annlowerthan expectedfromthe\nappearance of the characteristic CoFe diffraction peaks\nin the XRD data (see Fig. 1). This shows that the mea-\nsurement of MSis the more sensitive method to probe\nthe change of the crystalline structure.\nFor CoFeB a saturationvalue around MS≈1500kA/m\nis reached at T ann= 450◦C. This is compatible with val-\nues reported for CoFe (1350-1700 kA/m) [16, 17] and\nCoFeB (1350-1500 kA/m) [17, 18]. On the contrary, for\nCoFeAlB the introduction of Al reduces the magnetiza-\ntion of the samples and the annealing does not recover\nto CoFe-like values.\nFigure 3(a) shows the dependence of the magnetic\nfield linewidth on T annmeasured at a fixed frequency\nof 18 GHz. From the linear dependence of this linewidth\non the FMR frequency, the Gilbert damping parameter\nis extracted (as exemplarily shown for the CoFeAlB al-\nloy in the inset in Fig. 3(b)) and the results are shown\nin Fig. 3(b). For T annvalues up to 350◦C, where theFIG. 5. (Color online) Dependence of the exchange con-\nstantAexon the annealing temperature T annfor CoFeB and\nCoFeAlB. The top panels show typical BLS spectra for ma-\nterials (see text).\namorphousphaseisstill dominating, almostnodifference\nbetween both alloys is observed. With increasing tem-\nperature the damping increases for both alloys but the\nevolution is different. For CoFeAlB the increase starts\nalmost abruptly at T ann= 400◦C, reaches a maximum\naroundα= 0.02 and then decreases again to α= 0.012\nfor Tann= 500◦C. In contrast, the increase for CoFeB\nis more smoothly with T annand increases stadily with\nhigher T ann. In fact, due to the large linewidths reached\nfor Tann= 500◦C, the value of αcannot be properly\nestimated and only a lower limit of 0.03-0.04 can be\ngiven. This situation is represented by the dashed line\nin Fig. 3(b). It is important to note here that when the\ncrystallization process is fulfilled (i.e. for T ann= 500◦C)\nαis much lower for the Al doped alloy. This is rele-\nvant for the application in tunneling junctions where a\nfull crystallization is required for the presence of the spin\nfiltering effect originating large magnetoresistance values\nin combination with MgO barriers [4].\nFor further comparison of both alloys, the quantity\nαM2\nShasbeen calculatedand plotted in Fig. 4. As shown\nin Eq. 1, this value is ruling the critical switching current\nin in-plane magnetized systems. We observe for the al-\nloys showing a mostly amorphous phase (T ann<400◦C)\na slight improvement for CoFeAlB in comparison with\nCoFeB due to the lower MS. However, for fully crys-\ntalline films (T ann= 500◦C), the CoFeAlB shows a much\nsmaller value for αM2\nS. Since a full crystalline phase is\nneeded for any application of this alloy in MTJ-based de-\nvices, this denotes a major advantage of this compound\ncompared to standard.\nThe exchange constant Aexis a critical parameter that\nis strongly influenced by the introduction of Al. Its esti-\nmationinrequiredformodelingthespintorqueswitching\nbehaviorofthe alloys. The accessto the constantisgiven4\nby the dependence of the frequency of the perpendicular\nstanding spin-wave (PSSW) modes on the external static\nmagnetic field [19]. As shown in previous works [12, 20],\nitispossibletoobservethePSSWmodesinmetallicfilms\nwith a standard VNA-FMR setup. However, the signal\nis strongly reduced compared to the FMR peak. For the\nsamples presented in this paper, the PSSW peak could\nnot be observed for T ann>400◦C since the increased\ndamping leads to a broadening and lowering of the peak\nwhich prevents the estimation of Aex. For this reason,\nBLS spectroscopy is used for the measurement of the fre-\nquency position of the PSSW modes. This technique has\nalargersensitivityforthePSSWmodesthanVNA-FMR.\nFigure 5(c) shows the evolution of Aexupon annealing\nfor both alloys. For the films dominated by the amor-\nphous phase the value is much lower for CoFeAlB which\nis also compatible with the lower magnetization. How-\never, asthe crystallizationevolves,theexchangeconstant\nincreases stronger than for CoFeB and the same value is\nobtained for the fully crystallized films. This fact points\nto a similar role of Al and B during the crystallization\nprocess: when the CoFe crystallitesform, the light atoms\nare expelled forming a Al-B-rich matrix embedding the\nmagnetic crystallites. This explains also the similar evo-\nlution observed in the XRD data shown in Fig. 1. The\nlower maximal magnetization obtained for the CoFeAlB\ncan be explained by the reduced CoFe content but also\na certain number of residual Al and B atoms in the crys-\ntallites, which may differ for both alloys.\nTheAexvalues for as-deposited CoFeB films are very\nsimilar to previous reports [12, 20, 21]. Concerning the\nvalues for the crystallized samples, since the properties\nare strongly dependent on the B content and of the ra-\ntio between Co and Fe as well as on the exact annealing\nconditions, a comparison with literature has to be made\ncarefully. Nevertheless, the maximal value and the evo-\nlution with T annfor CoFeB is similar to the one reported\nby some of the authors [12]. Also results for alloys with\nthe same B content arecompatiblewith ourdata [22, 23].\nCoFeB films with reduced B content show larger values\n[17], the same is true for CoFe alloys with values between\n3.84-2.61 ×1011J/m depending on the exact stoichiome-\ntry [16, 17]. This may again be a hint that a rest of Al\nor B is present in the CoFe crystallites.\nIn summary, the presented experimental results show\nthat CoFeAlB is a good candidate as alternative to\nCoFeB for spin torque switching devices due to the re-\nduction of the factor αM2\nSwhich dominates the critical\nswitching current. This reduction was found to originate\nfrom a strong reduction of the saturation magnetization\nandadecreaseddampingparameter αforfullycrystalline\nCoFeAlB films. Furthermore, the results reveal a larger\nthermal stability of the damping properties in CoFeAlB\ncompared to CoFeB. The absolute values of MSand the\nexchange constant Aexfor crystalline films point to a for-\nmation of CoFe crystallines with a non-vanishing contentof the lights atoms embedded in a B or Al matrix.\nFinancial support by M-era.Net through the\nHEUMEM project, the DFG in the framework of\nthe research unit TRR 173 Spin+X and by the JSPS\nCore-to-Core Program is gratefully acknowledged.\n∗conca@physik.uni-kl.de\n[1] S.YuasaandD.D.Djayaprawira, J.Phys.D:Appl.Phys.\n40, R337-R354 (2007).\n[2] S. Yuasa,Y. Suzuki, T. Katayama, and K. Ando,\nAppl. Phys. Lett. 87, 242503 (2005).\n[3] Y. S. Choi, K. Tsunekawa, Y. Nagamine, and\nD. Djayaprawira, J. Appl. Phys. 101, 013907 (2007).\n[4] X.-G. Zhang and W. H. Butler, J. Phys.: Condens. Mat-\nter15R1603, (2003).\n[5] Z. Diao, Z. Li, S. Wang, Y. Ding, A. Panchula, E. Chen,\nL.-C. Wang, and Y. Huai, J. Phys. D: Appl. Phys. 19,\n165209 (2007).\n[6] K. L. Wang, J. G. Alzate, and P.K. Amiri, J. Phys. D:\nAppl. Phys. 46, 074003 (2013).\n[7] T. 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Lett., 3,\n3000204 (2012)." }, { "title": "2201.07842v1.Active_tuning_of_plasmon_damping_via_light_induced_magnetism.pdf", "content": "Title: Active tuning of plasmon damping via light induced magnetism . \n \nAuthor list : \nOscar Hsu -Cheng Cheng1*, Boqin Zhao1*, Zachary Brawley3, Dong Hee Son1,2, Matthew \nSheldon1,3 \n1Department of Chemistry, Texas A&M University, College Station, TX, USA. \n2 Center for Nanomedicine, Institute for Basic Science and Graduate Program of Nano \nBiomedical Engineering, Advanced Science Institute, Yonsei University, Seoul, Republic of Korea. \n3Department of Material Science and Engineering, Texas A&M University, College Station, \nTX, USA. \n*These authors contributed equally to this work. \ne-mail: dhson@chem.tamu.edu; sheldonm@tamu.edu \n Keywords: plasmonic, inverse Faraday effect, Raman thermometry, magnetoplasmon, \n Abstract \nCircularly polarized optical excitation of plasmonic nanostructures causes \ncoherent circulating motion of their electrons, which in turn, gives rise to strong optically \ninduced magnetization — a phenomenon known as the inverse Faraday effect (IFE). In \nthis study we report how the IFE also significantly decreases plasmon damping. By \nmodulating the optical polarization state incident on achiral plasmonic nanostructures \nfrom linear to circular , we observe reversible increases of reflectance by 78% as well as \nsimultaneous increases of optical field concentration by 35.7% under 10\n9 W/m2 \ncontinuous wave (CW) optical excitation. These signatures of decreased plasmon \ndamping were also monitored in the presence of an externally applied magnetic field (0.2 \nT). The combined interactions allow an estimate of the light -induced magnetization , which \ncorresponds to an effective magnetic field of ~1.3 T during circularly polarized CW \nexcitation (109 W/m2). We rationalize the observed decreases in plasmon damping in \nterms of the Lorentz forces acting on the circulating electron trajectories . Our results \noutline strategies for actively modulating intrinsic losses in the metal , and thereby , the \noptical mode quality and field concentration via opto- magnetic effects encoded in the \npolarization state of incident light. \n Introduction \nThe r eversible modulation of plasmonic resonances in metal nanostructures using \nexternal stimul i – i.e., “ active plasmonics” — is currently of great interest for potential \napplications in sensing, optoelectronic devices, and light-based information processing\n1. \nCommonly explored strategies modulate plasmon resonance frequenc ies by altering the \nsurrounding dielectric environment of nanostructures, for example, with \nthermoresponsive materials2-4. Modification of plasmonic modes based on distance -\ndependent optical coupling between nanostructures in compliant media under stress and \nstrain has also been demonstrated5-7. Alternatively, t he optical properties of the metal \ncomprising a nanostructure can be reversibly modulated. Within the D rude model, the \ncomplex dielectric function of a metal at angular frequency, 𝜔𝜔, is well described in terms \nof the electrical carrier density, 𝑛𝑛, and the damping constant, 𝛾𝛾, for the carrier oscillations : 𝜀𝜀(𝜔𝜔) = 1−𝜔𝜔𝑝𝑝2\n𝜔𝜔2+𝑖𝑖𝜔𝜔𝑖𝑖 (1) \nwhere 𝜔𝜔𝑝𝑝=�𝑛𝑛𝑒𝑒2\n𝜀𝜀𝑜𝑜𝑚𝑚𝑒𝑒 is the bulk plasma frequency, also depending on the electron charge, \n𝑒𝑒, effective mass, 𝑚𝑚𝑒𝑒, and vacuum permittivity, 𝜀𝜀𝑜𝑜8. Researchers have shown that \ncapacitive surface charging of metals when they are integrated into electrochemical cells \nresult s in reversible shifts of their plasmon resonance frequency through the dependence \non 𝑛𝑛9-11. In the time domain, pulsed laser excitation can similarly perturb electronic carrier \npopulations giving rise to transient modulation of plasmonic behavior12-17. \n In comparison, the possibility of actively tuning plasmon damping in the steady \nstate, and the opportunities for manipulating plasmonic behavior, has been studied much \nless18-21. In equation (1) , 𝛾𝛾 reflects several different microscopic processes connected \nwith the conductivity and mean free path of electrical carriers in the metal such as \nelectron- electron scattering , electron- phonon scattering , surface scattering22, and \nchemical interface damping with surface adsorbates23, in addition to other loss pathways \nsuch as radiation damping24 and Landau damping25,26. Usually, 𝛾𝛾 is considered to be an \nintrinsic property that is determined by the chemical identity of the metal27,28, surface \nchemistry and morphology, such as the crystal grain size29-32, or the modal behavior at a \nparticular frequency19,33, e.g. near field localization versus far field out -coupling . However , \nchanges in plasmon damping have a profound impact on the ability of a nanostructure to \nconcentrate optical power in a particular mode , known as the quality factor or “ Q” factor. \nDecreasing 𝛾𝛾 lowers the imaginary part of the permittivity , decreasing ohmic losses from \ncarrier motion at the optical frequency and increasing overall optical scattering or \nreflectance. L ower damping also provides greater field enhancement at sub-wavelength \n“hot spots” , improving efficiency for localized sensing, photochemistry, or heating via \nphotothermalization. \nIn a series of recent studies, the Vuong laboratory reported anomalously large \nmagneto- optical (MO) responses from colloidal Au nanoparticles under small magnetic \nfields (~1 mT) and low intensity circularly polarized (CP) excitation (<1 W/ cm2)34,35. The \nlarge MO response was attributed to the interaction between ext ernal magnetic fields and \ncirculating drift current s in the metal that were resonantly excited during CP excitation \n(Fig. 1a). The generation of drift current s from coherent charge density waves that \ncirculate in metals during CP optical cycle s has been studied extensively theoretically36-\n38, and is understood to contribute to optically induced static (DC) magnetism . This \nbehavior is also known as the Inverse Faraday Effect (IFE)36-41. Our recent experimental \nstudy of 100 nm Au nanoparticle colloids showed ultrafast modulation of effective \nmagnetic fields up to 0.038 T under a peak intensity of 9.1×1013 W/m2, confirming MO \nactivity and optically -induced magnetism many orders of magnitude great er than for bulk \nAu42. Notably, Vuong et al. also reported an apparent increase of the volume averaged \nelectrical conductivity when an external magnetic field was aligned with the light -induced \nmagnetic field. Considering several recent theoretical and experiment al magnetic circular \ndichroism (MCD) studies43-45, these results can be interpreted as a decrease in plasmonic \ndamping when the microscopic electron motion in a plasmon resonance contributes to \nDC magnetization. \nIn this study we show that plasmon damping is indeed strongly modified by \nmagnetic interactions , whether magnetism is induced externally using an applied magnetic field or created internally with light via the IFE. We observe that the normal \nincidence backscattering , i.e. the reflectance, of array s of non-chiral Au nanostructures is \nincreased by 78% when controlling the ellipticity of incident light from linearly polarized \n(LP) to CP during 109 W/m2 continuous wave (CW) excitation. Further, we query the \noptical field concentration at hot spots in the metal by taking advantage of recently refined \nelectronic Raman thermometry techniques46-53. Localized photothermal heating induces \ntemperatures ~2 3 K greater during CP excitation than during LP excitation under an \nexcitation intensity of ~109 W/m2, suggesting active modulation of optical field \nenhancement by 35.7% at hot spots. The local heating is further modulated by the \npresence of an externally applied magnetic field (~0.2 T) , supporting the underlying \nmagnetic origin of these phenomena and allowing estimation of the light -induced effective \nmagnetic fields at hot spots (~1.3 T). Taken together, our results indicate reversible \nmodulation of the plasmon damping that can exceed ~ 50%, solely by controlling the \nellipticity of the incident radiation. \n \n \n \nFig. 1 Sample overview . (a) Relationship between incident light helicity, induced \nelectronic motion, induced magnetization from the IFE (MIFE), and induced magnetization \n(Mind) from an external static magnetic field (Bapp) in a plasmonic nanostructure. When \nincident light has left -handed circular polarization (LHCP), M IFE and Mind are parallel. (b) \nSchematic of the Au nanodisk array sample (not to scale). ( c) Top view and ( d) side view \nof the local field enhancement during 532 nm excitation. Width and height not to scale in (d). (e) Optical image of the array on an Au film (f ) SEM image. Scale bar s: (c) 100 nm \n(e) 40 µm and ( f) 1 µm. \n \nResult and Discussion \nSamples consisting of 100 µm x 100 µm arrays of 400 nm diameter by 100 nm \nheight disk-shaped gold nanostructures in a square lattice pattern (700 nm pitch) were \ndeposited on 38 nm thick Al 2O3 layer on top of 100 nm thick gold films using electron-\nbeam lithography ( Fig. 1b, e, f , see Methods section). The nanodisk shape supports \ncirculating electronic currents during CP excitation (Fig. 1a). Periodic arrays provide high \nabsorpt ivity across the visi ble spectrum (Fig. 1e, Fig. 2a), aiding photothermal heating for \nthe Raman thermometry studies detailed below. The overall sample geometry is achiral, \nhighly symmetric, and exhibits no polarization or ellipticity dependence for absorption or scattering (neglecting non- linear effects), as confirmed by full wave optical simulations \n(FDTD method, see S upporting Information). \n \n \nFig. 2 Spectroscop y of gold nanodisk array s. (a) Absorption spectrum. The dashed \nline indicates 532 nm. (b) Electronic Raman spectrum during 532 nm CW excitation. Different spectral regions provide information analyzed in this study. Orange box: anti -\nStokes signal used for temperature fitting. Purple box: 532 nm backscattering (Rayleigh \nline, filtered here) used to quantify ellipticity -dependent reflection. Green box: the broad \nenergy distribution of the Stokes region is fit for the plasmon damping, 𝛾𝛾. \n \nFig. 3 Confocal spectral mapping under different helicit ies. (a, d) Confocal Raman \nand (b, e) backscattering intensity map of the gold nanodisk array under 532 nm CW \nexcitation with (a, b) CP or (d, e) LP. (c, f) Line scans of the backscattering efficiency \nalong the region between (blue) or over (red) nanodisk s with LP (dashed trace) or CP \n(solid trace) excitation. Inset: optical image of the sample array. The green box indicates \nthe region of the confocal map. Scale bar: 4 µm. \n \n \nSamples were mounted onto a piezo- driven microscope stage, and confocal maps \nof the electronic Raman (eR) spectrum (Fig . 3a, d ) or the backscattering intensity at 532 \nnm (Fig. 3b, e) were collected as a function of position over the edge of an array during linearly polarized (LP) or circularly polarized (CP) excitation. A representative eR spectrum is shown in Fig . 2b. In comparison with typical Raman signals that result from \ninelas tic scattering with vibrational modes in a sample, the eR signal is due to inelastic \ninteractions with the electron gas at the metal surface. The broad eR signal therefore \nprovides information about the energetic distribution of electrons, such as their \ntemperature. As clearly seen by comparison of F ig. 3a, d , and Fig. 1 c, d, the eR signal is \nstrongest at “hot spots” at the edge of nanodisks where field enhancement is greatest . In \ncontrast, the sample backscattering, or reflectance, (Fig. 3b, e) is lower ov er the array \ndue to pronounced plasmonic absorption, ultimately giving rise to localized photothermal \nheating. \nThe backscattering efficiency of the nanodisks is strongly modulated based on the \npolarization state (ellipticity) of the incident light. As displayed in F ig. 3c, f, we compared \nthe backscattered light intensity from different locations over the nanodisk array and the \nadjacent Au film during LP (dashed trace) or CP (solid trace) excitation. The signal \nintensity was converted to backscattering effic iency by referencing the Au film region to \na smooth Ag mirror, in order to rule out any polarization- dependent instrument response. \n(see Methods section for details). In all locations over the array the backscattering \nefficiency is generally larger for CP versus LP, with a maximum relative increase of \nbackscattering up to 78 % at the edge of individual nano disks during CP excitation. \nWe rule out the possibility that this trend is due to inherent differences in the \nellipticity -dependent scattering efficiency based on sample geometry, because the \nsample is not chiral. Neglecting optically induced magnetization or other non- linear effects, \nthe total absorption and scattering of the nanostructure array is expected to show no dependence on beam ellipticity. Indeed, we have performed linear, full wave optical simulations (FDTD method, see Supporting information) tha t confirm no difference in the \nabsorption or scattering efficiency based on LP or CP excitation. We hypothesize that the large difference in backscatteri ng efficiency observed experimentally results from \nellipticity -dependent modulation of the plasmon damping. This interpretation is \nqualitatively supported by additional optical simulations ( see Supporting information) that \nshow a comparable increase of bac kscattering , depending on sample location, when \ndamping is decreased by 50% or greater . \nDecreased plasmon damping also results in more concentrated optical fields at hot \nspots, which can lead to more pronounced local photothermal heating. In terms of \nequat ion (1) , as the damping constant 𝛾𝛾 decreases the imaginary part of the dielectric \nfunction 𝐼𝐼𝑚𝑚(𝜀𝜀(𝜔𝜔)) also decreases, giving rise to a larger Q -factor and greater local field \nenhancement\n54. The local power density for heating, 𝑞𝑞(𝑥𝑥,𝑦𝑦,𝑧𝑧), depends on field \nenhancement as55,56: \n𝑞𝑞(𝑥𝑥,𝑦𝑦,𝑧𝑧) = 1\n2𝐼𝐼𝑚𝑚[𝜀𝜀(𝑥𝑥,𝑦𝑦,𝑧𝑧)]𝜀𝜀0|𝐸𝐸(𝑥𝑥,𝑦𝑦,𝑧𝑧)|2 (2) \nwhere 𝐸𝐸(𝑥𝑥,𝑦𝑦,𝑧𝑧) is the local electric field. Thus, lower plasmon damping provides a net \nincrease of heating power and correspondingly larger temperatures at locations with \nstrong field enhancement. Intuitively, lower damping increases the cross section that \nfunnels light energy into a plasmonic hot spot. See simulations of this effect for the \nnanodisk array geometry in Supporting information. \nRaman signal intensity also depends on local field enhancement, scaling as |𝐸𝐸|4 \n57. Therefore Raman- based thermometry techniques primarily probe the nanostructure \ntemperature at hot spots. We measured the sample temperature at hot spots by adapting \nan anti -Stokes (aS) Raman thermometry method developed by Xie et al46. Experimentally, \nit has been shown that the spectral intensity of the aS eR signal, 𝑆𝑆(∆𝜔𝜔), is thermally \nactivated according to a Bose- Einstein distribution. The aS spectrum collected from a \nsample at an unknown temper ature, 𝑇𝑇𝑙𝑙, can be normalized by a spectrum collected at a \nknown temperature, 𝑇𝑇0, according to \n𝑆𝑆(∆𝜔𝜔)𝑇𝑇𝑙𝑙\n𝑆𝑆(∆𝜔𝜔)𝑇𝑇0 = 𝑒𝑒𝑒𝑒𝑝𝑝 �−ℎ𝑐𝑐∆𝜔𝜔\n𝑘𝑘𝐵𝐵𝑇𝑇0� −1\n𝑒𝑒𝑒𝑒𝑝𝑝 �−ℎ𝑐𝑐∆𝜔𝜔\n𝑘𝑘𝐵𝐵𝑇𝑇𝑙𝑙� −1 (3) \nwhere ℎ is Plank constant, 𝑐𝑐 is the speed of light, ∆𝜔𝜔 is the wavenumber (negat ive for \nanti-Stokes), and 𝑘𝑘𝐵𝐵 is Boltzmann constant. Spectral features that do not change with \nthermal activation, such as the frequency -dependent signal enhancement factor, cancel \nout, so that 𝑇𝑇𝑙𝑙 is the only unknown fitting parameter. \nWith fixed linear polarization, we measured the aS spectra of samples as a function \nof excitation power to induce variable amounts of photothermal heating , and we fit for 𝑇𝑇 𝑙𝑙. \nThe spectra were normalized by a spectrum collected at the lowest possible power that \npreserved good signal -to-noise ( I = 7.2 × 106 W/m2), with the goal of inducing minimal heating above room temperature, i.e. 𝑇𝑇0 ≈ 298 K. We observe that 𝑇𝑇𝑙𝑙 increases linearly \nwith LP excitation intensity (Fi g. 4, blue line), in good agreement wit h many other reports \nof gold and copper nanostructures46,50,58. The linear fit to the temperature trend shows a \ny-intercept near room temperature, at the limit of zero incident power, further confirming \nthe accuracy of the thermometry technique. The fitted slope of the trend ( 4.6×10-8 K·m2/W) \ndescribes the “heating efficiency” of the sample under LP excitation. \nWe determined changes in the sample temperature, and hence modification of the \nplasmon damping, by measuring the eR response while varying the ellipticity of the \nexcitation beam. However, given the complex spectral dependence of the eR signal, \nsimilar procedures as those described for the backscattering study could not be used to correct for the ellipti city-dependent instrument response. Instead, we devised a dual \nbeam configuration (see M ethods section for details). In summary, two separate CW 532 \nnm laser beams were coincident on the sample. A low power “Beam 1” was maintained with linear polarization. A second, higher power “Beam 2” was used to induce variable \nmagnetization and damping in the sample by controlling excitation ellipticity. The eR \nspectrum resultant from Beam 1 was isolated and fit for 𝑇𝑇\n𝑙𝑙 by collecting a spectrum with \nboth Beam 1 and Beam 2 incident, and then subtracting the spectrum collected with only \nBeam 2 incident. This procedure allowed us to probe the sample temperature using a \nbeam that had non- changing incident power and linear polarization (Beam 1) while the \nsample was excited with variable ellipticity and power. The accuracy of this dual -beam \nmethod is confirmed in Fig . 4 (red trace). The fitted 𝑇𝑇𝑙𝑙 are reported as a function of the \ntotal incident power of Beam 1 and Beam 2, with the power of Beam 1 held constant and both bea ms maintained in linear polarization. A similar y -intercept and heating efficiency \nis observed using either thermometry technique. \n \n \nFig. 4 Photothermal heating with a single - or dual-beam geometr y. (a) Fitted \nnanostructure temperature, 𝑇𝑇\n𝑙𝑙 as function of excitation intensity, 𝐼𝐼, when only Beam 1 \n(blue) or when both Beam 1 and Beam 2 (red) were incident . Blue linear fit: 𝑇𝑇𝑙𝑙 (𝐾𝐾) =\n 4.6×10−8𝐼𝐼 + 299. For the dual -beam study the intensity of Beam 1 was kept at 4.8×108 \n(W/m2), and the intensi ty of Beam 2 varied between 6.6 – 9.5×108 (W/m2). Red linear fit: \n𝑇𝑇𝑙𝑙 (𝐾𝐾) = 4.6×10−8𝐼𝐼+299. \n \nFig. 5. Ellipticity dependent photothermal heating. Fitted nanostructure temperature \nas a function of Beam 2 ellipticity . The intensity of Beam 1 was kept at 4.8×108 W/m2. The \nintensity of Beam 2 was kept at 9.5×108 W/m2. The dashed line is a guide for the eye. \nSee Fig . S8 for a detailed explanation of ellipticity and ellipticity angle. \n \n \nWe then examined the dependence on the ellipticity of Beam 2 while the power of \nboth beams was held constant (Fig . 5). For both right -handed or left -handed circular \npolarization (RHCP or LHCP), 𝑇𝑇𝑙𝑙 increases with increasing ellipticity. For the same \nmagnitude ellipticity but opposite helicity, the increase of 𝑇𝑇𝑙𝑙 is simila r. This trend indicates \nenhanced field concentration at hot spots when the ellipticity -dependent magnetization in \nthe sample is increased. The maximum increase of temperature observed for CP compared with LP is ~23 K. Based on the heating efficiency determ ined under LP \nexcitation (Fig . 4a, red line), this same temperature increase would be expected if the \nsample received 5 × 10\n8 W/m2 more incident power. Since the total excitation power was \nkept constant at 1.4 × 109 W/m2, we conclude that the switch from LP to CP excitation \nequivalently increases the heating power at hot spots by 35.7%. We emphasize that field enhancement is not expected to depend on excitation ellipticity, neglecting non- linear \neffects, because the sampl e is not chiral. \nWe also studied sample behavior under an externally applied magnetic field. For \nthis experiment the ellipticity of Beam 2 was kept at either 0 (LP), +0.67 (LHCP), or - 0.67 \n(RHCP). Note that the ellipticity for RHCP was limited to this ran ge based on our \nexperimental geometry (see Methods section). An external magnetic field B\napp = 0.2 T \nwas applied parallel to the direction of light propagation (Fig. 1a), and 𝑇𝑇𝑙𝑙 was measured \nas a function of total incident power. As summarized in Fig. 6, the fitted 𝑇𝑇𝑙𝑙 are larger under \nLHCP excitation compared to RHCP, while both polarizations cause greater heating than LP. This effect can be rationalized in terms of the direction between the optically induced \nmagnetization , M\nIFE, and the applied magneti c field , Bapp. When MIFE and B app are anti -\nparallel, as for RHCP, the optically induced circular electron motion is opposite the direction favored by the Lorentz forces from the external magnetic field, resulting in an increase of damping and lower optical field enhancement. In support of this picture, Gu. \net al. theoretically analyzed the behavior of a free electron gas in a nanoparticle under \nCP excitation and predicted that the optically induced magnetic moment is enhanced \n(suppressed) when an external m agnetic field is aligned (anti -aligned), due to the Lorentz \nforces on individual electrons that perturb their circular movement59. Theoretical studies \nof magnetoplasmons also predict decreases in damping when rotating surface charge \ndensity waves provide magnetization parallel with externally applied magnetic fields43. \nThe difference in heating efficiency during LHCP and RHCP excitation allows an \nestimate of the strength of the optically induced magnetization, MIFE, at hot spots in terms \nof the magnetization, Mind, that results from Bapp (see full calculations in Supporting \nInformation) . Assuming that the temperature increase compared to LP excitation is \nlinearly proportional to the net magnetization Mind + MIFE we determine an “effective” \nmagnetic field, B eff, at hot spots to be 1.3T for the highest incident power of 1.45× 109 \nW/m2 and 0.67 ellipticity . Note that B eff is not the magnetic field produced by optically \nexciting the nanostructure, but rather , corresponds to the field strength of a hypothetical \nexternal magnet that would produce the same magnetization in the dark as observed \nduring CP optical excitation with no B app. This estimate also assumes that M ind and MIFE \nare either aligned or anti -aligned, though their orientation may be more complex \nmicroscopically40. When n ormalized for optical power density, the observed \nmagnetization is in good agreement with our previous time- resolved studies of ensembles \nof Au colloids42 (Table S1) . \nFinally, we comment that the plasmon damping can als o be estimated directly by \nfitting to the Stokes side eR spectrum (green box, Fig. 2b). As discussed in detail in a \nrecent report from our laboratory53, the eR spectrum reflects the approximately Lorentzian \ndistribution of non- thermal electron -hole pairs that have been generated during the \nplasmon damping process, i.e., the natural linewidth of the excited plasmon. The fitted \ndamping observed under LP ( 1.42×109 W/m2) was 34.1 meV and the lowest damping \nobserved under CP ( 1.42× 109 W/m2, ellipticity of 0.94) was 31.6 meV. These values can \nequivalently be reported as a plasmon dephasing time of 19.3 fs (LP) or 20.8 fs (CP) and \nare comparable to values commonly reported in ultrafast transient absorption studies of Au nanostructures\n21. While this fitted estimate of the ellipticity -dependent change in \ndamping is somewhat smaller c ompared to the estimate based on computational \nmodeling of the sample backscattering study discussed above, both measures consistently indicate a significant decrease of plasmon damping during CP excitation. \nFig. 6. Photothermal heating under an external magnetic field and different \nhelicit ies. Fitted nanostructure temperature (𝑇𝑇𝑙𝑙) as a function of total intensity (𝐼𝐼) with \nBapp = 0.2 T and variable excitation helicity . The magnitude of ellipticity is 0.67 for b oth \nLHCP and RHCP. The linear fits to each temperature trend are LHCP: 𝑇𝑇𝑙𝑙 (𝐾𝐾) =\n 8.1×10−8𝐼𝐼+293; RHCP: 𝑇𝑇 𝑙𝑙 (𝐾𝐾) = 7.3×10−8𝐼𝐼+292; LP: 𝑇𝑇𝑙𝑙 (𝐾𝐾) = 4.8×10−8𝐼𝐼+305. \n \n \nConclusion \nWe have demonstrated the ability to modulate plasmon damping in achiral \nplasmonic gold nanodisk arrays by controlling incident light ellipticity. Confocal mapping \nrevealed that CP excitation leads to enhanced efficiency for backscattering, consistent \nwith an overall decrease of damping. A dual -beam Raman thermometry technique \nquantified localized heating in samples. We observe more efficient photothermal heating \nwhen the ellipticity of incident light increases, regardless of helicity (RHCP or LHCP ), \nindicat ing greater field enhancement at hot spots. The simultaneous increase of \nscattering and absorption is a telltale signature of decreased damping in plasmonic \nabsorbers. In comparison, under an external magnetic field, RHCP and LHCP excitation \nprovide differ ent amounts of heating. This behavior suggests that the microscopic origin \nof decreased damping is the interaction between the optically driven coherent electron \nmotion and Lorentz forces from DC magnetic fields, whether magnetic fields are optically \ninduc ed or externally applied. Our results provide further insight into electron dynamics \ninside plasmonic nanostructures during CP excitation and suggest multiple new strategies \nfor controllably modulating heating, magnetization, reflectance, damping, and rela ted \nphotophysical effects. \n \n \n \n \n \n \nMethods \n \nNanostructure Fabrication \nPrior to fabrication, a silicon substrate was cleaned using a combination of base \npiranha and UV -ozone. A 100 nm gold mirror was then thermally evaporated (Lesker PVD \nelectron- beam evaporator) onto the silicon substrate. A 38 nm thick Al 2O3 dielectric layer \nwas then deposited on the gold mirror by RF sputtering (Lesker PVD RF sputterer) and \nthe thickness was determined using ellipsometry. Next, 950 PMMA A4 was spin- coated \nonto the Al 2O3 as the electron beam resist layer. Electron- beam lithography (TESCAN \nMIRA3 EBL) was performed to pattern the 100 µm × 100 µm nanodisk array into the e-\nbeam resist. After development, a 5 nm chrome adhesion layer was thermally deposit ed \non the surface of the exposed PMMA, followed by a 100 nm layer of gold. Finally, liftoff was performed in acetone using a combination of pipet pumping and sonication, leaving only the nanodisk array on the surface of the substrate. The morphology of gol d nanodisks \narray is confirmed by SEM (Fig . 1f). \n Raman Spectroscopy \nRaman spectra were taken using a confocal microscope system (Witec RA300) \nand spectrometer (UHTS300, grating = 300 g/mm). For Raman spectral mapping, the excitation source was 532 nm CW Nd:YAG laser and the Raman spectra were collected by a 100x objective (Zeiss EC Epiplan Neofluar, NA = 0.9, WD = 0.31mm). A pair \nconsisting of a holographic 532 nm notch filter (RayShield Coupler, Witec) and a 532 nm \nnotch filter (NF533- 17, Thorlab) were added to prevent saturation of the spectrometer. \nThe obtained Raman spectra were corrected by the transmission spectra of the notch filter. The resolution of the Raman map was 50 nm in lateral dimensions (both x and y \ndirection) and 100 nm in the z direction. The optimal z height was determined by \nmaximizing the Raman Stokes signal. The 532 nm backscattering efficiency under \ndifferent elli pticity was obtained by referencing to a silver mirror. \nFor the dual-beam Raman spectroscopy experiment, two 532 nm CW laser were \nused. Beam 2 was coupled through free space, and was a 532 CW diode laser with a spot size of 2 um\n2 on the sample. The ellipt icity of Beam 2 was controlled by a half \nwaveplate and a quarter waveplate optically in series . The function of Beam 2 was to \ngenerate circular current s in the gold nanostructures and actively tune the damping \nconstant. The difference in the highest achiev able ellipticity for LHCP and RHCP was a \nresult of the limitation of the beam splitter. Beam 1 was coupled through a fiber coupler \n(Rayshield coupler) and was sourced by a 532 nm CW Nd:YAG laser , with a spot size of \n0.55 µm2 when focused. Beam 1 always had lower intensity than Beam 2 and was linearly \npolarized. Both beams were focused by a 100x objective (Zeiss EC Epiplan Neofluar, NA = 0.9, WD = 0.31mm) on the gold nanostructures. Below the sample, there was a slot for inserting a magnet, which has magneti c field parallel to the incident light (point ing \ndownward) with magnetic field strength = 0.2 T near the surface of gold nanostructure. \n Acknowledgement s \nWe thank Prof. Luat Vuong for helpful discussions . This work was funded in part by the \nNational Science Foundation (Grant DMR -2004810). M.S. also acknowledges support \nfrom the Welch Foundation (A -1886). \n \n \nAuthor contributions \nO.H.- C.C. and B.Z. carried out the measurements and analyzed the data. O.H.-C.C. \ndrafted the manuscript. B.Z. and performed the simulations and drafted the SI . 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B \n27 (2010). \n Supporting information for \nActive tuning of plasmon damping via light induced magnetism \n \nAuthor list: \nOscar Hsu -Cheng Cheng1*, Boqin Zhao1*, Zachary Brawley3, Dong Hee Son1,2, \nMatthew Sheldon1,3 \n1Department of Chemistry, Texas A&M University, College Station, TX, USA. \n2 Center for Nanomedicine, Institute for Basic Science and Graduate Program \nof Nano Biomedical Engineering, Advanced Science Institute, Yonsei University, Seoul, Republic of Korea \n3Department of Material Science and Engineering, Texas A&M University, \nCollege Station, TX, USA. \n*These authors contributed equally to this work. \ne-mail: dhson@chem.tamu.edu; sheldonm@tamu.edu \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 1. Full wave electromagnetic simulations \n1.1 Simulations Setup \nThree -dimensional full wave electromagnetic simulations were performed using \nfinite difference time domain (FDTD) methods (Lumerical, Ansys Inc.). A 100 \nnm height and 2 18 nm radius (determined by the SEM image) Au nanodisk on \na 40 nm thick Al 2O3 layer (with a 5 nm Cr adhension layer in between) on top \nof Au film defined the simulation geometry . Periodic boundary conditions were \napplied on all sides perpendicular to the substrate with a simulation region of \n700 nm x 700 nm, representing the periodicity of the nanostructure. A plane \nwave source was injected from above, normal to the substrate. The \nbackscattered radiation was collected by a 2- D monitor placed above the plane \nwave source. \n \nThe refractive index of Al 2O3 was calculated as a linear combination of the \nexperimental index values from Palik1 for Al 2O3 and Al, so that the imaginary \npart at 532 nm is close to 0. 2, to account for excess Al in the layer during \nnanostructure fabrication as determined in control studies . The permittivity \nvalues for Au was modeled with the analytical function based on Drude- Lorentz \nmodel2: \n𝜀𝜀(𝜔𝜔)=1−𝑓𝑓0𝜔𝜔p,02\n𝜔𝜔2+𝑖𝑖𝛤𝛤0𝜔𝜔+∑𝑓𝑓j𝜔𝜔p,j2\n𝜔𝜔j2−𝜔𝜔2−𝑖𝑖𝛤𝛤j𝜔𝜔𝑗𝑗max\n𝑗𝑗=1 (S1) \nwhere 𝜀𝜀(𝜔𝜔) is the relative permittivity, 𝛤𝛤0 is the intraband damping constant, \n𝛺𝛺p=�𝑓𝑓0𝜔𝜔p,0 is the plasma frequency associated with intraband transition. \n𝑗𝑗max is the number of Lorentz oscillators (in our case, 5) with strength 𝑓𝑓j , \nfrequency 𝜔𝜔j and damping constant 𝛤𝛤j for each individual oscillator. \n In order to systematically study the effect of modulation of damping on optical \nproperties observed in the experiment, we studies the dependence on the \nintraband damping term 𝛤𝛤\n0, and all the interband damping terms 𝛤𝛤j in equation \n(S1). Bulk Au values were used for simulations represent ing LP excitation in the \nexperiment. These damping terms were reduced up to 50% of their original \nvalues to generate new permittivity values to represent the CP excitation in the \nexperiment, applied to the Au film and the entire Au nanodisk. \n 1.2 Backscattering simulations \nThe confocal backscattering map in the experiment ( Fig. 3) show s the spatial \ndependence of backscattering on the nanodisk array. In order to reproduce this \nspatial dependence behavior in the simulation, we mapped out the distribution \nof Poynting vector along z direction (Pz) of the scattered field on a x -y plane \nslightly above the top surface of the nanodisk . We then averaged around each point with a Gaussian -like decay which approximates the effect of Gauss ian \nbeam excitation in the experiment ( Fig. S1a ). The backscattering intensity is \nthe largest near the center of the nanodisk and lowest between the gaps. When \ncomparing the backscattering spatial distribution at different damping ( Fig. S1b ), \nit is clear that decreasing damping increases the backscattering intensity along \nthe black dash line in Fig. S1a , which is consistent with experimental results \n(Fig. 3c, f ). \n However, with 50% damping modulation, the magnitude of backscattering \nchange pr edicted by the simulation was not as large as that observed in the \nexperiment at all locations . We hypothesize that the damping modulation is \nhighly spatially non- uniform in the experimental system with maxima near hot \nspots, unlike the assumption of uniform damping modulation across the \nnanostructure in the modeling. A spatial dependent change in damping in \nexperiments is likely a key factor that is not accounted for in these simulations, limiting the quantitative accuracy of the result. Nonetheless, the modeling \ncorroborates that the observed experimental trend is consistent with decreased damping in the metal. \n \n \n \nFig. S1 a) Simulated backscattering map on the x -y plane above the nanodisk \ntop surface with Gaussian- like weighting. The r ed line circles region of the \nnanodisk. b) Backscattering intensity plotted along the black dashed line in a), \nat bulk damping and 50% damping, as well as the backscattering from an Au \nfilm for reference. \n \n \nIn addition to the backscattering map, the overal l backscattering intensity from \nthe nanostructure, essentially the average of the spatial dependent backscattering intensity, was also simulated ( Fig. S2 ). The overall value also \nincreases by around 0.5% with damping values decreased by half , further \nhighlighting the increased backscattering efficiency with decreasing damping. \n \nIf we label the overall backscattered percentage of power as R, the absor bed \npercentage of power would be 1- R, due to the opacity of the nanostructure. \nTherefore, the absorption spectrum of the nanostructure can be computed \naccordingly . Fig. S3 shows the computed absorption spectrum of the \nnanostructure . Although i t does not exactly match the experimental absorption \nspectrum in Fig . 2a, but the general shape is very similar. \n \nWe emphasize that simulation does not model any nonlinear effects . If we only \nchange the ellipticity of the excitation source in the simulations , without \nmodulating Au permittivity values , the absorption and the backscattering \nintensity values of the nanostructure remain constant . This suggests that the \ndifference in absorption and scattering seen in the simulation must come from some nonlinear effects , such as modulation of damping . \n \n \nFig. S2 Overall backscattering efficiency of gold nanodisk array s with damping \nof 100% and 50% of bulk damping. \n \nFig. S3 Simulated absorption spectrum of gold nanodisk array s with the bulk \ndamping constant. \n \n \n1.3 Plasmonic heating simulations \nWe mentioned in the main article that the local power density for heating, \n𝑞𝑞(𝑥𝑥,𝑦𝑦,𝑧𝑧), depends on field enhancement as : \n𝑞𝑞(𝑥𝑥,𝑦𝑦,𝑧𝑧) = 1\n2𝐼𝐼𝐼𝐼[𝜀𝜀(𝑥𝑥,𝑦𝑦,𝑧𝑧)]𝜀𝜀0|𝐸𝐸(𝑥𝑥,𝑦𝑦,𝑧𝑧)|2 (S3) \nwhere 𝐸𝐸(𝑥𝑥,𝑦𝑦,𝑧𝑧) is the local electric field. We simulated the local electric field \ndistribution 𝐸𝐸(𝑥𝑥,𝑦𝑦,𝑧𝑧) on the nanodisk with damping modulation to verify the \nexperimentally observed increase in photothermal heating with decreasing \ndamping. Here, we mapped the electric field intensity (|E|2) distribution on th e \ntop surface of the nanodisk , multiplied by the imaginary permittivity, which is \nproportional to the local heating power. Fig. S 4 shows the calculated results for \nboth bulk damping and 50% damping. At 50% damping, the local heating at hot \nspots is significantly higher than the bulk damping case. \n \n \nFig. S4 Heating map on the top surface of the nanodisk a) with bulk damping, \nb) with 50% damping and c) the difference between 50% damping and bulk damping. \n2. Raman fitting methodology \nThe method used in the manuscript to extract the temperature of the plasmonic \nnanostructure was adopted from Xie et al3 and from many of our previous \nreports4-7. Basically, the anti -Stokes electronic Raman scattering signal from the \nplasmonic nanostructure is associated with the thermalized electron distribution \nthat is approximately thermally equilibrated with the metal lattice. Therefore, the \nspectral intensity of the anti -stokes spectrum 𝑆𝑆(∆𝜔𝜔) can be characterized with \na Bose- Einstein distribution: \n𝑆𝑆(∆𝜔𝜔)∝𝐶𝐶×𝐷𝐷(∆𝜔𝜔)×1\nexp�−ℎ𝑐𝑐∆𝜔𝜔\n𝑘𝑘B𝑇𝑇�−1 (S3) \nwhere ℎ is Plank constant, 𝑐𝑐 is the speed of light, ∆𝜔𝜔 is the wavenumber \n(negative for anti -Stokes), 𝑘𝑘B is Boltzmann constant , and 𝑇𝑇 is the \ntemperature of the sample. 𝐶𝐶 is a scaling factor. 𝐷𝐷(∆𝜔𝜔) is a correction factor \nproportional to the photonic density of states (PDOS) of the nanostructure. In \npractice, the PDOS is approximated with reflection4-6, absorption, or dark -field \nscattering8 spectrum. However, there is not a direct experimental measurement \nof the PDOS . Therefore, we follow the procedure of Xie et al3 and have \nnormalized the Raman spectrum at an unknown temperature 𝑇𝑇𝑙𝑙 to a reference \nRaman spectrum at a known temperature 𝑇𝑇0, to remove the dependence of \n𝐷𝐷(∆𝜔𝜔): \n𝑆𝑆(∆𝜔𝜔)𝑇𝑇𝑙𝑙\n𝑆𝑆(∆𝜔𝜔)𝑇𝑇0 = 𝐶𝐶1×𝐷𝐷(∆𝜔𝜔)×(𝑒𝑒𝑒𝑒𝑒𝑒 �−ℎ𝑐𝑐∆𝜔𝜔\n𝑘𝑘B𝑇𝑇0� −1)\n𝐶𝐶2×𝐷𝐷(∆𝜔𝜔)×(𝑒𝑒𝑒𝑒𝑒𝑒 �−ℎ𝑐𝑐∆𝜔𝜔\n𝑘𝑘B𝑇𝑇l� −1)=𝐶𝐶×𝑒𝑒𝑒𝑒𝑒𝑒 �−ℎ𝑐𝑐∆𝜔𝜔\n𝑘𝑘B𝑇𝑇0� −1\n𝑒𝑒𝑒𝑒𝑒𝑒 �−ℎ𝑐𝑐∆𝜔𝜔\n𝑘𝑘B𝑇𝑇l� −1 (S4) \nThe reference spectrum is usually chosen to be the spectrum at lowest \nillumination intensity, assuming no laser heating, and therefore the sample \ntemperature is assumed to be at room temperature. \n By correctly normalizing the spectral counts to the illumination power and integration time, 𝐶𝐶 is theoretically a value of 1. Therefore, the only parameter \nto solve for is the unknown temperature 𝑇𝑇\nl . In practice, fitting accuracy is \nimproved if 𝐶𝐶 is a free fit parameter (usually very close to 1) to account for \nsmall changes in the collection efficiency during the course of the measurement. \n 3. The accuracy of Raman -fitted temperature s \n \nFig. S5 Fitted temperature of the nanostructure compared to the thermal \nstage temperature. Blue asterisk with error bar: average fitted temperature \nand standard deviation. Red dotted line: reference line with T stage = T fit. \n \n We have shown that the sample temperature extracted from the Raman spectrum with equation ( 3) is physically accurate. The same nanostructure \nsample used in the manuscript was heated to a pre- set temperature on a \nthermal stage ( TS1500, Linkam). Raman spectra were taken at different \nthermal stage temperatures with only beam 1 present. The intensity of beam 1 \nwas set to be very low for negligible laser heating. The Raman spectrum \ncollected at 298 K thermal stage temperature was chosen for the reference spectrum and 𝑇𝑇\n0 was set to be 298 K in the fitting. \n The fitted temperatures were compared to the corresponding thermal stage \ntemperatures, as shown in Fig. S 5. Standard deviation (the blue bar around the \ndata points) was calculated based on the fitted temperatures for s everal spectra \nat the same thermal stage temperature, which is around 1 K, showing the high \nprecision of the measurements. The data points lie close to the red reference line with T\nstage = T fit, confirming our ability to accurately report the nanostructure \ntemperatures with anti-Stokes Raman thermometry. \n 4. Determination of optically -induced magnetism \n𝑀𝑀IFE at hot spots \nAlthough bulk Au has been reported to display diamagnetism, there have been \nnumerous reports pointing to observed paramagnetic, or even ferromagnetic \nbehavior in various nanoscale Au systems9-12. According to the theory of the \nIFE13, 14 and by studying the direction of magnetization in this report, we \nconcluded that the Au nanodisk arrays behave paramagnetically (under \ncircularly polarized optical pumping) . However, the (volume) susceptibility is not \nknown. Here, we may label it symbolically with 𝜒𝜒V and carry it through our \nsubsequent calculations . In addition, our previous experimental study15 also \nindicates a paramagnetic behavior in nanoscale Au. Therefore, we model the \nmagnetization of Au under an external magnetic field (in the dark) as follows : \n𝐵𝐵app=𝜇𝜇0�1\n𝜒𝜒V�𝑀𝑀ind (S5) \nwhere 𝜇𝜇0 is the vacuum permeability, 𝐵𝐵app is the external applied magnetic \nfield and 𝑀𝑀ind is the induced magnetization. \n \nThe induced magnetization in response to the external magnetic field 𝑀𝑀ind is \na separate contribution to the sample magnetism in addition to the light -induced \nmagnetism 𝑀𝑀IFE (Fig. 1a) , and both effects are expected to contribute to the \ndamping experienced by coherently driven, circulating electrons during optical pumping . Therefore, the temperature increase measured in the experiment (Fig. \n6, Fig. S6 ) is assumed to scale linearly with the total induced magnetism 𝑀𝑀\nind+\n𝑀𝑀IFE. \n First, we calculate the strength of 𝑀𝑀\nind under a 0.2 T applied external magnetic \nfield. By applying equation (S5), with 𝜇𝜇0=1.257 ×10−6 H/m, we can derive \nthat 𝑀𝑀ind=1.6×105𝜒𝜒V A/m. We determine 𝑀𝑀IFE in the experiment based the \nfitted temperature data at the highest incident intensity in Fig. 6 (also Fig. S6 ). \nNote that the temperature without an external magnetic field , when 𝑀𝑀ind=0, \nwould have been 405 K for both LHCP and RHCP. This means the samples \nexperience a temperature increase of ~ 33 K during CP compared to LP when \nonly the 𝑀𝑀IFE alters the damping in the sample. The additional 5 K increase or \ndecrease of temperature (see Fig. S6) during LHCP or RHCP , respectively, is \ndue to the interaction with 𝑀𝑀ind . Therefore, 𝑀𝑀IFE is calculated to be \napproximately 6.6x larger than 𝑀𝑀ind, or 1.05×106𝜒𝜒V A/m, which corresponds \nto induced magnetic flux density of 1.3𝜒𝜒V T. According to equation (S5), t his is \nequivalent to 𝑀𝑀ind that would be produced under a 1.3 T external applied \nmagnetic field, which we label as B eff. \n Lastly, we can calculate the magnetic moment per Au atom m\nAu based on the \nrelationship \n𝑀𝑀IFE=𝑁𝑁Au\n𝑉𝑉𝐼𝐼Au (S6) \nIn an Au crystal lattice, t he number density of Au atoms 𝑁𝑁Au\n𝑉𝑉 is 58.9 nm-3. \nTherefore, 𝐼𝐼Au=1.9𝜒𝜒V𝜇𝜇B. \n We compare our results with our previous experimental study on Au \nnanocolloids, as well as two experiment measurements on Au film, summarized in Table. S1 , assuming that the IFE magnetism is linearly proportional to the \nincident optical power (as suggested by theoretical studies on the IFE13, 14, 16). \nOur analysis in this study only reveal s the magnetic behavior at hot spots, \nbecause most of the Raman signal comes from hot spots on the nanodisk. In \nRef. [1 5], the magnetic moment per Au atom was averaged over the entire \nnanoparticle, which could be the reason for a slightly smaller value for induced \nmagnetic moment per atom compared to this study. Both nanoscale Au system \npossess excitation intensity normalized magnetic moment per atom ( 𝐼𝐼Au/𝐼𝐼) of \nalmost 4 orders of magnitude larger than Au film, indicating an enhancement of \nthe IFE phenomenon in nanoscale compared to bulk . \n \nFor reference, i f 𝜒𝜒V is on the order of 10−5 (a typical value for bulk Au) , the \nIFE induced magnetic field at hot spots during in the study is estimated to be \non the order of 10−5 T. \n \n Table. S1 Comparison between induced magnetic moment per atom due to the IFE in different reports. \n \n System Excitation \nIntensity \n𝐼𝐼 (W/m2) 𝑀𝑀IFE (A/m) 𝐼𝐼Au (𝜇𝜇B) 𝐼𝐼Au/𝐼𝐼 \n(10−10𝜇𝜇B/(W\nm2)) \nThis work Au nanodisk \narray ~109 (CW) 1.05×106𝜒𝜒V 1.9𝜒𝜒V b19𝜒𝜒V \nRef. [15]15 Au \nnanopartic le \ncolloid ~9×1013 \n(pulsed) 3.0×104𝜒𝜒V 2.8×104𝜒𝜒V 3.1𝜒𝜒V \nRef. [1 7]17 Au film ~13×1013 \n(pulsed) a4.5×106𝜒𝜒V 8.2𝜒𝜒V 6.3×10−4𝜒𝜒V \nRef. [1 8]18 Au film ~13×1013 \n(pulsed) a1.1×107𝜒𝜒V 20𝜒𝜒V 1.5×10−3𝜒𝜒V \na Several assumptions were made to calculate this value. The path length in Au film is estimated by the \nskin depth (1/absorption coefficient). The verdet constant of Au film is extracted from Ref. [17]19. The \ninduced magnetization was also normalized to the excitation frequency. 𝜒𝜒𝑉𝑉 of Au f ilm was used as bulk \nAu value. \nb At hot spots. \nFig. S6 Diagram to aid induced magnetic field calculation. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n5. Optical setup \n \nFig. S7. Optical setup for dual -beam Raman spectroscopy. CW: continuous \nwave; LP: linear polarizer; HWP: half waveplate; QWP: quarter waveplate; BS: \nbeam splitter. \n 6. Definition of ellipticity \n \nFig. S8. Ellipticity explanation. This figure shows a left -handed elliptically \npolarized light. φ is the ellipticity angle. Ellipticity is defined as the intensity \nratio between the short and long axis. Ellipticity angle = 0, - 45\no, +45o or \nEllipticity = 0, 1, - 1 represent LP, LHCP, RHCP , respectively. \n \nReferences \n1. Palik, E. D., Handbook of optical constants of solids. Academic Press Handbook Series 1985. \n2. Rakic, A. D.; Djurisic, A. B.; Elazar, J. M.; Majewski, M. L., Optical properties of metallic films \nfor vertical -cavity optoelectronic devices. Appl. Opt. 1998 . 37, (22), 5271- 5283. \n3. Xie, X.; Cahill, D. G., Thermometry of plasmonic nanostructures by anti -Stokes electronic Raman \nscattering. Appl. Phys. Lett. 2016. 109, (18), \n4. Hogan, N.; Wu, S.; Sheldon, M., Photothermalization and Hot Electron Dynamics in the Steady \nState. J. Phys. Chem. C 2019. 124, (9), 4931- 4945. \n5. Wu, S. X.; Hogan, N.; Sheldon, M., Hot Electron Emission in Plasmonic Thermionic Converters. \nACS Energy Lett. 2019. 4, (10), 2508- 2513. \n6. Hogan, N.; Sheldon, M., Comparing steady state photothe rmalization dynamics in copper and gold \nnanostructures. J. Chem. Phys. 2020. 152, (6), 061101. \n7. Wu, S. X.; Cheng, O. H. C.; Zhao, B.; Hogan, N. Q.; Lee, A.; Son, D. H.; Sheldon, M., The \nconnection between plasmon decay dynamics and the surface enhanced R aman spectroscopy background: \nInelastic scattering from non -thermal and hot carriers. J. Appl. Phys. 2021. 129, (17), \n8. Cai, Y . Y .; Sung, E.; Zhang, R.; Tauzin, L. J.; Liu, J. G.; Ostovar, B.; Zhang, Y .; Chang, W. S.; \nNordlander, P.; Link, S., Anti -Stoke s Emission from Hot Carriers in Gold Nanorods. Nano Lett. 2019. \n19, (2), 1067- 1073. \n9. Ulloa, J. A.; Lorusso, G.; Evangelisti, M.; Camon, A.; Barbera, J.; Serrano, J. L., Magnetism of \nDendrimer -Coated Gold Nanoparticles: A Size and Functionalization Study. J. Phys. Chem. C 2021. 125, \n(37), 20482- 20487. \n10. Nealon, G. L.; Donnio, B.; Greget, R.; Kappler, J. P.; Terazzi, E.; Gallani, J. L., Magnetism in gold \nnanoparticles. Nanoscale 2012. 4, (17), 5244- 5258. \n11. Tuboltsev, V .; Savin, A.; Pirojenko, A.; Raisan en, J., Magnetism in nanocrystalline gold. ACS Nano \n2013. 7, (8), 6691- 6699. \n12. Trudel, S., Unexpected magnetism in gold nanostructures: making gold even more attractive. Gold \nBull. 2011 . 44, (1), 3 -13. \n13. Hertel, R., Theory of the inverse Faraday effect in metals. J. Magn. Magn. Mater. 2006. 303, (1), \nL1-L4. \n14. Hertel, R.; Fahnle, M., Macroscopic drift current in the inverse Faraday effect. Phys. Rev. B 2015. \n91, (2), \n15. Cheng, O. H. C.; Son, D. H.; Sheldon, M., Light -induced magnetism in plasmonic gold \nnanoparticles. Nat. Photonics 2020. 14, (6), 365- +. \n16. Sinha -Roy, R.; Hurst, J.; Manfredi, G.; Hervieux, P. A., Driving Orbital Magnetism in Metallic \nNanoparticles through Circularly Polarized Light: A Real -Time TDDFT Study. ACS Photonics 2020 . 7, \n(9), 2429- 2439. \n17. Kruglyak, V . V .; Hicken, R. J.; Ali, M.; Hickey, B. J.; Pym, A. T. G.; Tanner, B. K., Measurement \nof hot electron momentum relaxation times in metals by femtos econd ellipsometry. Phys. Rev. B 2005 . \n71, (23), \n18. Kruglyak, V . V .; Hicken, R. J.; Ali, M.; Hickey, B. J.; Pym, A. T. G.; Tanner, B. K., Ultrafast third -\norder optical nonlinearity of noble and transition metal thin films. Journal of Optics a- Pure and Ap plied \nOptics 2005. 7, (2), S235 -S240. \n19. Mcgroddy, J. C.; Mcaliste.Aj; Stern, E. A., Polar Reflection Faraday Effect in Silver and Gold. \nPhysical Review 1965. 139, (6a), 1844- &. " }, { "title": "1507.07762v1.Phenomenology_of_chiral_damping_in_noncentrosymmetric_magnets.pdf", "content": "arXiv:1507.07762v1 [cond-mat.mes-hall] 28 Jul 2015Phenomenology of chiral damping in noncentrosymmetric mag nets\nCollins Ashu Akosa1, Ioan Mihai Miron2,3,4, Gilles Gaudin2,3,4, and Aur´ elien Manchon1∗\n1King Abdullah University of Science and Technology,\nPhysical Science and Engineering Division, Thuwal 23955, S audi Arabia\n2Univ. Grenoble Alpes INAC-SPINTEC F-38000 Grenoble France\n3CNRS INAC-SPINTEC F-38000 Grenoble France and\n4CEA INAC-SPINTEC F-38000 Grenoble France\n(Dated: June 18, 2021)\nA phenomenology of magnetic chiral damping is proposed in th e context of magnetic materi-\nals lacking inversion symmetry breaking. We show that the ma gnetic damping tensor adopts a\ngeneral form that accounts for a component linear in magneti zation gradient in the form of Lif-\nshitz invariants. We propose different microscopic mechani sms that can produce such a damping in\nferromagnetic metals, among which spin pumping in the prese nce of anomalous Hall effect and an ef-\nfective ”s-d” Dzyaloshinskii-Moriya antisymmetric exchange. The impl ication of this chiral damping\nin terms of domain wall motion is investigated in the flow and c reep regimes. These predictions have\nmajor importance in the context of field- and current-driven texture motion in noncentrosymmetric\n(ferro-, ferri-, antiferro-)magnets, not limited to metal s.\nPACS numbers: 75.40.Gb,76.60.Es,75.60.Ch\nUnderstanding energy relaxation processes of fast dis-\nsipative systems at the nanoscale is of paramount im-\nportance for the smart design and operation of ultra-\nfast nano devices. In this respect, magnetic heterostruc-\ntures have drawn increasing enthusiasm in the past ten\nyears with the optical control of magnetic order at the\nsub-picosecond scale [ 1,2] and the promises of ultra-\nfast domain wall motion in asymmetrically grown metal-\nlic multilayers [ 3]. In fact, the observation of ultrahigh\ncurrent-driven velocity in ultrathin multilayers with po-\ntentially strong disorder has come as a surprise and trig-\ngeredintenseinvestigationson the physicsemergingfrom\nsymmetry breaking in strong spin-orbit coupled magnets\n[4,5]. These studies have unravelled the essential role\nplayedbyDzyaloshinskii-Moriyainteraction[ 6,7](DMI),\nan antisymmetric exchange interaction that emerges in\nmagnets lacking spatial inversion symmetry. This in-\nteraction forces neighboring spin to align perpendicular\nto each other and competes with the ferromagnetic ex-\nchange resulting in distorted textures such as spin spirals\nor skyrmions, as observed in bulk inversion asymmetric\nmagnets [ 8] (B20, ZnS or pyrochlores three-dimensional\ncrystals) as well as at the interface of transition met-\nals [9,10] (Mn/W, Fe/W, Pt/Co etc.). In perpendic-\nularly magnetized domain walls, this interaction favors\nN´ eel over Bloch configuration [ 11–13], a key ingredient\nexplaining most of the thought-provoking observations\nreported to date [ 14]. The dynamics of spin waves can\nalso be modified by DMI, which distorts the energy dis-\npersion [ 15] and results in a relaxation that depends on\nthe propagation direction [ 16,17].\nAnother crucial aspect of fast dynamical processes is\nthe natureofthe energyrelaxation. In the hydrodynamic\nlimit of magnetic systems, this dissipation is written in\ntheformofanon-localtensor[seeEq. ( 1)]whosecomplex\nphysics is associated with a wide variety of mechanismssuch as many-magnon scattering [ 18] and itinerant elec-\ntronspin relaxation[ 19]. Sincethe energyrelaxationrate\nof spin waves depends on their wave vector (the higher\nthe spin wave energy, the stronger its dissipation), the\nmagnetic damping of smooth magnetic textures (i.e. in\nthe longwavelengthlimit q) depends onthe inverseofthe\nexchange length, q∼1/∆. In inversion symmetric sys-\ntems, thisresultsinacorrectiontothemagneticdamping\nof the order of 1 /∆2[20–22]. However, in magnetic sys-\ntems lacking inversion symmetry such as the systems in\nwhich DMI is observed (i.e. B20 and ZnS crystals and\ntransition metal interfaces), one can reasonably expect\nthat the energydissipationbecomeschiral, namely that a\ncomponent linear in the magnetization gradient emerges\nthereby fulfilling Neumann’s principle stating that ”any\nphysical properties of a system possesses the symmetry\nof that system”.\nIn the present work, we phenomenologically explore\nthe nature of the magnetic damping in noncentrosym-\nmetric magnets and reveal that spatial inversion sym-\nmetry breaking results in the emergence of such a chiral\ndamping that vanishes when the symmetry ofinversionis\nrecovered. The inclusion of such a chiral damping in the\nequation of motion of magnetic textures opens appealing\navenues to solve recent puzzling observations that can\nnot be fully accounted for with DMI only.\nSymmetry considerations - The equation of motion\ngoverning the dynamics of continuous magnetic textures\nis given by the extended Landau-Lifschitz-Gilbert (LLG)\nequation\n∂tm=−γm×Heff+m(r)×/integraldisplay\ndr′α(r,r′)∂tm(r′),(1)\nwherem(r,t) =M(r,t)/Msis a unit vector in the direc-\ntion of the magnetization M(r,t),γis the gyromagnetic\nratio,Heffis the effective field incorporating the external2\napplied, anisotropy, exchange, DMI and demagnetizing\nfields, and α(r,r′) is the magnetic damping expressed as\na non-local second-rank tensor. In the limit of smooth\ntextures, the tensorial components of the damping is a\nfunction of the magnetization direction and of its spatial\ngradients, αij=αij(m,∇m). Performing an expansion\nup to the first order in magnetization gradient, one ob-\ntains (see also [ 23])\nαij=αij\n0+/summationdisplay\nlmKij\nlmmlmm+/summationdisplay\nklmLij\nklmmk∂lmm.(2)\nThe first term is the isotropic damping, the second term\n(∼Kij\nlm) amounts for the anisotropy arising from the\ncrystalline environment and the third term is the chiral\ndamping. It should be noted that only terms bilinear in\nmagnetization direction mi, i.e. even under time rever-\nsal symmetry, are retained in the expansion. Since the\nfocus of this study is on the chiral nature of magnetic\ndissipation, we ignore the anisotropy term ( ∝Kij\nlm) at\nthis stage. Spatial inversion symmetry breaking imposes\nthe third term of Eq. ( 2) to reduce to a sum of Lif-\nshitz antisymmetric invariants, ∝mk∂lmm−mm∂lmk\n(i.e.Lij\nklm=−Lij\nmlk). To illustrate the general form of\nthis chiral damping, we consider two prototypical sys-\ntems. In the case of a cubicthree-dimensional system\nwith bulk spatial inversion symmetry breaking (e.g. B20\nor ZnS crystals), all the three directions ( x,yandz) are\nequivalent, and the chiral damping adopts the general\nform\nαij=αij\n0+αij\n3d∆m·[∇×m], (3)\nwhere ∆ defines the characteristic exchange length. In a\ntwo-dimensionalsystemwithinterfacialsymmetrybreak-\ning along z, i.e. invariant under C∞zrotation symmetry,\nthe damping takes the form\nαij=αij\n0+αij\nz∆m·[(z×∇)×m].(4)\nAs dictated by Neumann’s principle, the chiral damp-\ning possesses the same symmetry as DMI given by Na-\ngaosa et. al. [ 8] and Thiaville et. al. [ 14], respectively.\nIn addition, Onsager reciprocity imposes that αij=αji.\nTherefore, using simple symmetry arguments, one can\nconstruct a chiral damping up to linear order in mag-\nnetization gradient. These considerations suggest that\nsuch a damping is present in noncentrosymmetric (ferro-\n, ferri- and antiferro-)magnets in general, not limited to\nmetals. However, this phenomenology does not provide\ninformation regarding the strength of the chiral damping\nitself, the relative values of the off-diagonal tensor ele-\nments (∼αi/negationslash=j) compared to the diagonal ones ( ∼αii)\nnor does it indicate the underlying physical mechanisms\nresponsible for it. Let us now turn our attention towards\nthe microscopic origin of such chiral damping.\nMicroscopic origin of the chiral damping -In this work,\nwe focus our attention on magnetic textures in metalswhere the magnetic damping is driven by the spin relax-\nation of itinerant electrons [ 19]. In ferromagnets with in-\nterfacialRashba spin-orbitcoupling, the magnetic damp-\ning adopts the form of a tensor linear in both magneti-\nzation gradient and Rashba strength, as derived by Kim\net al. [24] and Wang et al. [ 25]. In this model, the\nmagnetic dissipation is mediated by the same spin-orbit\ncoupled itinerant electrons which mediate the DMI on\nthe local magnetic moment (see e.g. Ref [ 26]). Besides\nthis effect, we here propose two additional mechanisms\nthat can contribute to the chiral damping.\nThe first mechanism arises from the interplay between\nspin motive force and anomalous Hall effect. It has been\nrecently shown that time-dependent spin textures gen-\nerate a local spin current [ 27],Js\ni=gµB¯hG0\n4e2∂tm×∂im,\nflowing along the direction of the texture ei, polarized\nalong∂tm×∂im(G0is the electrical conductivity),\nand that induces a magnetic damping at the second or-\nder of spatial gradient [ 20]. When anomalous Hall ef-\nfect is present in the ferromagnet, this primary spin\ncurrentJs\niis converted into a secondary spin current\nJs\nj=θHP[Js\ni·(ei×ej)]ei×ej, flowing along ejand\npolarized along ei×ej. Here,θHis the spin Hall angle\nandPis the spin polarization in the ferromagnet. This\nsecondary spin current can be injected in an adjacent\nspin sink with strong spin relaxation, thereby inducing a\ndamping torque on the magnetization, similarto the spin\npumping mechanism [ 28]. Consideringa one-dimensional\ndomain wall along xdeposited on a heavy metal with an\ninterface normal to z, we obtain a damping torque on the\nform [29]\nτ=θHAgµB¯hPG0\n4e2[(∂tm×∂im)·y]m×(y×m),(5)\nAbeing a renormalization factor arising from the spin\ncurrent backflow. This damping torque is proportional\nto∼sin2ϕ,ϕbeing the azimuthal angle of the magne-\ntization, and vanishes when the wall is either in Bloch\n(ϕ= 0) or N´ eel configuration ( ϕ=π/2).\nThe second mechanism is directly related to DMI. In\ntransition metal ferromagnets, the orbital characters of\nthe delocalized ( spdhybridized) and localized electrons\n(pdhybridized) are mixed so that both types of electrons\ncontribute to the magnetic exchange. This is particu-\nlarly true in the case of DMI: ab initio calculations indi-\ncate that interactions beyond the next-nearest neighbor\ncontribute significantly to the total DMI [ 30], proving\nthat delocalized orbitals are crucial in determining the\noverall strength of DMI. Therefore, by parsing the to-\ntal spinSiinto localized ( d-dominated) and delocalized\n(s-dominated) contributions, Si=Sd\ni+ˆss\ni, the DMI be-\ntween sites iandjcan be phenomenologically rewritten\nDij·Si×Sj=Ddd\nij·Sd\ni׈Sd\nj+Dsd\nij·Sd\ni׈ss\nj. The\nfirst term only involves orbital overlap between localized\nstateswhilethesecondtermdescribesthechiralexchange\nbetween the local spin and the itinerant spin. In the con-3\ntinuous limit, the Hamiltonian of the itinerant electron\ncan be then written\nˆHsd=ˆp2\n2m+Jexm·ˆσ\n+D\n¯h[(z׈p)×m]·ˆσ+αR\n¯h(z׈p)·ˆσ,(6)\nwhereˆσis the vector of Pauli spin matrices, Jex,D\nandαRare the strength of the exchange, s-dDM and\nRashba spin-orbit interactions respectively. In other\nwords, because ofthe magnetic texture the itinerant elec-\ntron spin experiences an additional effective field of the\nform∼(z׈p)×m. This introduces a chirality in the\nmagnetic damping mediated by itinerant electrons.\nFrom this point, deriving the effective magnetic damp-\ning follows the standard procedure. One can extract the\nequationofmotionoftheitinerantspinsfromEq. ( 6)and\ndefine the spin current induced by the moving magnetic\ntexture (see Supplementary Materials [ 29]). Following\nthis method, we obtain a semi-classical Bloch equation\nfor the itinerant electron spin density s=∝angb∇acketleftˆσ∝angb∇acket∇ightas\n∂ts+∇·J=−1\nτexs×m−∆\nτD(∇z×m)×s−∆\nτR∇z×s−Γre,\n(7)\nwhere∇z=z×∇, Γrerepresentsthe spin relaxationand\ndephasing, ∆ is the exchange length, τex= ¯h/2Jexis the\nspin precession time, τD= ¯h∆/DandτR= ¯h∆/αRare\nthe characteristic time scales for the DMI and Rashba\ninteraction respectively. J=−D∇⊗sis the spin cur-\nrent density tensor, Dbeing the diffusion constant. Let\nus now write the spin density in the form s=nsm+δs,\nwherens(δs) is the (non-)equilibrium spin density, and\nassume the relaxation time approximation such that\nΓre(s) =1\nτsfδs+1\nτϕm×(δs×m), accounting for the spin-\nflip relaxation ( ∼τsf) and the spin dephasing ( ∼τϕ).\nAfter some algebra [ 29], one obtains the torque τgener-\nated by a precessing magnetization ∼∂tm\nτ/˜ns≈(1+χξ−βm×)[−∂tm (8)\n+λD[((z×∇)×m)×(m×∂tm+ξ∂tm)]⊥\n+λR[(z×∇)×(m×∂tm+ξ∂tm)]⊥].\nIn this expression, β=τex/τsf,χ=τex/τϕ,ξ=χ+β,\nand ˜ns=ns/(1+ξ2) and the subscript ⊥indicates that\nthe torque is defined perpendicular to the magnetization\nm. The first term has been derived previously [ 31], the\nsecond term ( λD= ∆τex/τD) arises from the s-dDMI\nexchange, and the third term ( λR=τex/τR) arises from\nRashba spin-orbit coupling [ 24,25]. The total torque τ\ncontributes both to the renormalization of the gyromatic\nratio (terms that preserve time-reversal symmetry) and\nto dissipation (terms that break time-reversal symme-\ntry).\nDomain wall motion - To illustrate the effect of\nthis chiral damping on the dynamics of magnetic tex-\ntures in noncentrosymmetric metals, we first derivethe equation of motion of a one-dimensional perpen-\ndicular domain wall, such as the ones commonly ob-\nserved in heavy metal/ferromagnet asymmetric mul-\ntilayers [ 3–5]. The magnetization is defined m=\n(cosϕsinθ,sinϕsinθ,cosθ), where ϕ=ϕ(t) is the az-\nimuthal angle and θ(x) = 2tan−1(exp[s(x−X)/∆]),X\nbeing the domain wall centre, and s=±1 defining the\ndomain wall chirality ( ↑↓or↓↑, respectively). We con-\nsider a magnetic domain wall submitted to a magnetic\nanisotropyfield Hk=Hksinθsinϕy, favoringBlochcon-\nfiguration and an applied magnetic field H=Hxxfavor-\ning N´ eel configuration. The damping torque is given by\nEq. (8)(for the sake of simplicity, we do not consider\nthe influence of the torque arising from anomalous Hall\neffect, Eq. ( 5), in this work). The rigid domain wall\ndynamics is described by the coupled equations\ns∂τx\n∆=π\n2(−Hxsinϕ+Hk\n2sin2ϕ)+(α−sνcosϕ)Hz,\n∂τϕ=−sπ\n2(α−sµcosϕ)(−Hxsinϕ+Hk\n2sin2ϕ)+Hz,\n(9)\nwhere we defined τ=γt,µ= (λD−ξλS)(π˜ns/4∆), and\nν= (βλD+λS)(π˜ns/4∆) (α,ν,µ≪1). In Eq. ( 9), we\nneglectedthecomponentsofthetorque τthatareevenin\nmagnetization (i.e. that renormalizes the gyromagnetic\nratio) and only consider the dissipative components. The\ndamping due to s-dDMI and Rashba SOC both produce\na contribution proportional to scosϕ, i.e. it depends on\nthe domain wall chirality sas well as on the direction of\nthe azimuthal angle ϕ. Notice that the DM field does not\nexplicitly enters these equations since it can be simply\nmodeled by a chiral in-plane field ∼sHx[14].\nLet us now investigate the influence of this damping\non the field-driven motion of a domain wall submitted\nto both perpendicular ( Hz) and in-plane ( Hx) magnetic\nfields. In this example, we chose µ=ν=αcfor sim-\nplicity. Figure 1(a) shows the (time-averaged) velocity\nof the domain wall as a function of Hzfor difference\nchiral damping strengths and Hx= 0. These veloc-\nity curves display the usual Walker breakdown around\nHz≈12−19 mT, with an additional kink at nega-\ntiveHzdirectly attributed to the chiral damping. Be-\nlow Walker breakdown and in the absence of in-plane\nfieldHx, the domain wall azimuthal angle ϕobeys\nHz=sπ\n4Hk(α−sαccosϕ)sin2ϕ, which produces a\nkink around Hz≈sαcHWB/αwhereHWB=απHk/4,\nwhich is associated with a jump in the effective damp-\ning,αeff=α−αccosϕ[see Fig. 1(c)]. When applying\na large in-plane field Hx, the azimuthal angle is ϕ≈0\n(N´ eel configuration) and the damping does not depend\nonHz. Hence, the kink disappears as shownin Fig. 1(b).\nMore interestingly, when the domain wall is tuned from\nBloch to N´ eel configuration by an in-plane field [see Figs.\n1(d)-(f)], the effective damping is strongly modified [Fig.4\n−50 0 5000.511.5Φ/π\nµ0Hz = 5 mT (e)\n αc = 0 αc = 0.25α αc = 0.5α αc = 0.75α\n−50 0 500.20.40.60.8\nµ0Hx(mT) αeff \nµ0Hz = 5 mT (f)−50 0 50102030Velocity (m/s)µ0Hz = 5 mT (d)\n−40−20 020400.40.60.8\nµ0Hz(mT) αeff (c)\nµ0Hx = 0 mT−40−20 02040−20020Velocity (m/s)(a)\nµ0Hx = 0 mT\n−40−20 02040−50050Velocity (m/s)(b)\nµ0Hx = 25 mT\nFIG. 1. (Color online) (a,b) Domain wall velocity as a func-\ntion of perpendicular magnetic field for different chiral dam p-\ning strengths, at Hx=0 mT and Hx=25 mT, respectively;\n(c) corresponding effective damping. (d) Domain wall veloc-\nity as a function of in-plane magnetic field for different chir al\ndamping strengths, and for Hz=5 mT; (e,f) corresponding\nazimuthal angle and effective damping, respectively. The pa -\nrameters are α= 0.5 ,HWB= 12.5 mT.\n1(f)] and domain wall velocity becomes strongly asym-\nmetric as reported on Fig. 1(d). Notice that the kink is\nstill observable at small negative in-plane field.\nWe now turn our attention towards the creep regime,\nwhich is of most importance for field-driven domain wall\nmotion in ultrathin disordered multilayers. In this case,\nthe creep law predicts [ 32]\nvcreep=v0exp/parenleftBig\n−(Tc/T)(Hc/Hz)1/4/parenrightBig\n(10)\nwhereTcis the critical temperature (close to the depin-\nning temperature in fact), Tis the sample temperature,\nHzis the applied field and Hcis the critical field needed\nto overcome the disorder pinning potential. This expres-\nsion assumes Hz≪Hc, as well as Tc≫T. The ex-\nponent gathers only terms accounting for the disordered\nenergy landscape of the system and does not include any\ndissipative contributions in principle. In contrast, the\ncoefficient v0depends on both the energy landscape as\nwell as on the viscosity of the elastic wall. In the limit\nof anoverdamped membrane (i.e., when α∂tm≫∂tm),\none can show that v0≈γ∆Hc/α[32]. It is mostly prob-\nable that even for strongly disordered ferromagnets, the\ncreeping domain wall is not in the overdamped regime,\nalthough very large damping ( α≈0.5) have been re-\nported in Pt/Co/AlOx and Pt/Co/Pt systems [ 3]. How-\never, since the theoretical description of the creep motion\nof intermediate damped systems is not fully understood,\nwe propose to investigate the impact of chiral damping\nin this limit.\nToevaluatetheimpactofthechiraldamping, wefollow\nthe procedure proposed by Je et al. [ 33] and rewrite thecreep law\nvcreep(H) =η0σDW(Hx)β\nαeff(Hx)exp/parenleftBigg\n−χ0σ1/4\nDW(Hx)\nσ1/4\nDW(0)H−1\n4z/parenrightBigg\n.\n(11)\nHere,σDWisthe (chiral)energyofthe domainwalland β\nis a coefficient that describes the scaling law between the\ncritical force Hcand the domain wall energy and η0and\nχ0are normalization factors which can be chosen to fit\nexperimental data of Ref. [ 33]. In the phenomenological\nmodel adopted here, the domain wall energy reads\nσDW=σ0+π∆µ0Ms/bracketleftBig\n1\n2Hkcosϕ−Hx/bracketrightBig\ncosϕ.(12)\nIn the right-handside of Eq. ( 12), the first term accounts\nfor theϕ-independent contribution to the magnetic en-\nergy, the second term is the in-plane magnetic anisotropy\nfavoring Bloch configuration and the last term is the in-\nplane longitudinal magnetic field favoring N´ eel configu-\nration. The energy minimization ∂ϕσDW= 0 gives [ 33]\ncosϕ=/braceleftbigg\nHx/Hk:|Hx| ≤Hk,\nsign(Hx/Hk) :|Hx|> Hk.(13)\nand the corresponding domain wall energy\nσDW\nσ0=/braceleftbigg\n1−(Hx)2/HDWHk:|Hx| ≤Hk\n1+(Hk−2|Hx|)/HDW:|Hx|> Hk(14)\nwhereHDW=π\n2∆µ0Ms/σ0. The magnetic damping αeff\nis written in the simplest form αeff=α+sαccosϕ. To\ninvestigate the impact of chiral damping on the creep\nmotion, we chose HDW= 1T, andHk= 50mT. The\nnormalized velocity v(Hx)/v(0) of a domain wall as a\nfunction of the in-plane field and for different strengths\nof chiral damping αcis represented on Fig. 2. It shows\nthree distinct regions: a smooth variation of the velocity\nin the intermediate field region, |Hx|< Hk, where the\ndomain wall is changed from one N´ eel chirality to an-\nother, as well as two external region, |Hx|> Hk, where\nthe domain wall remains in the N´ eel configuration. In\nthis case, the velocity increases following the exponential\nlaw given above. Notice that in this regime and for such\na one-dimensional domain wall, the DMI results in an\neffective in-plane magnetic field whose sign depends on\nthe chirality of the wall, i.e. Hx→Hx−sHDMI(see e.g.\nRef. [33]). Thus, including DMI in the calculation only\nresults in a horizontal shift of the velocity curve in Fig.\n2.\nWe acknowledge that the present phenomenology re-\nmains essentially qualitative and although the orders of\nmagnitude discussed in this work are globally consistent\nwith the experimental observations, a microscopic the-\nory of chiral damping using, for instance, density func-\ntion theory techniques, as well as a more comprehensive5\nFIG. 2. (Color online) Normalized domain wall velocity as\na function of in-plane magnetic field Hxin the presence of a\ndriving field Hz= 20mT. In-plane magnetic field pushes the\ndomain from a Bloch to a N´ eel configuration and hence in-\ncreases the azimuthal angle ϕthereby modifying the damping\nof the wall. A kink in the velocity is observed at Hx=Hk,\nwhen the domain wall saturates in the N´ eel configuration\nwhen the damping becomes independent on the in-plane field\nHx.\nmodel of the creep motion of magnetic domain walls are\nhighly desirable to quantitatively confront with experi-\nments. Nonetheless, the symmetry principles discussed\ninthisLetterarequitegeneralandensuretheexistenceof\nsuch a chiral damping in any magnetic structures (ferro-\nmagnets, antiferromagnets, chiralmagnets, but alsomet-\nals and insulators etc.) presenting spatial inversion sym-\nmetrybreaking. Thephysicalmechanismsresponsiblefor\nthis chiral damping can be spin-orbit coupling but also\ndipolar coupling or magnetic frustrations, as in the case\nof DMI. 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B 88, 214401 (2013)." }, { "title": "2308.05955v2.Dynamical_Majorana_Ising_spin_response_in_a_topological_superconductor_magnet_hybrid_by_microwave_irradiation.pdf", "content": "Dynamical Majorana Ising spin response in a topological superconductor-magnet\nhybrid by microwave irradiation\nYuya Ominato,1, 2Ai Yamakage,3and Mamoru Matsuo1, 4, 5, 6\n1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.\n2Waseda Institute for Advanced Study, Waseda University, Shinjuku, Tokyo 169-8050, Japan.\n3Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n4CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan\n6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: March 20, 2024)\nWe study a dynamical spin response of surface Majorana modes in a topological superconductor-\nmagnet hybrid under microwave irradiation. We find a method to toggle between dissipative and\nnon-dissipative Majorana Ising spin dynamics by adjusting the external magnetic field angle and\nthe microwave frequency. This reflects the topological nature of the Majorana modes, enhancing\nthe Gilbert damping of the magnet, thereby, providing a detection method for the Majorana Ising\nspins. Our findings illuminate a magnetic probe for Majorana modes, paving the path to innovative\nspin devices.\nIntroduction.— The quest for Majoranas within matter\nstands as one of the principal challenges in the study of\ncondensed matter physics, more so in the field of quan-\ntum many-body systems [1]. The self-conjugate nature\nof Majoranas leads to peculiar electrical characteristics\nthat have been the subject of intensive research, both\ntheoretical and experimental [2]. In contrast, the focus of\nthis paper lies on the magnetic properties of Majoranas,\nspecifically the Majorana Ising spin [3–8]. A distinctive\ncharacteristic of Majorana modes, appearing as a surface\nstate in topological superconductors (TSC), is its exceed-\ningly strong anisotropy, which makes it behave as an Ising\nspin. In particular, this paper proposes a method to ex-\nplore the dynamical response of the Majorana Ising spin\nthrough the exchange interaction at the magnetic inter-\nface, achieved by coupling the TSC to a ferromagnet with\nferromagnetic resonance (FMR) (as shown in Fig.1 (a)).\nFMR modulation in a magnetic hybrid system has at-\ntracted much attention as a method to analyze spin ex-\ncitations in thin-film materials attached to magnetic ma-\nterials [9, 10]. Irradiating a magnetic material with mi-\ncrowaves induces dynamics of localized spin in magnetic\nmaterials, which can excite spins in adjacent thin-film\nmaterials via the magnetic proximity effect. This setup\nis called spin pumping, and has been studied intensively\nin the field of spintronics as a method of injecting spins\nthrough interfaces [11, 12]. Recent studies have theoret-\nically proposed that spin excitation can be characterized\nby FMR in hybrid systems of superconducting thin films\nand magnetic materials [13–18]. Therefore, it is expected\nto be possible to analyze the dynamics of surface Majo-\nrana Ising spins using FMR in hybrid systems.\nIn this work, we consider a TSC-ferromagnetic insula-\ntor (FI) hybrid system as shown in Fig. 1 (a). The FMR\nis induced by microwave irradiation on the FI. At the\ninterface between the TSC and the FI, the surface Ma-\n(b)\n(c)(a)\nFI~~\n~~Microwave\nϑS\nY, yX\nxZhdcHex\nTSC\n(d)\nhdchdc+δhα+δα\nHz\nFIG. 1. (a) The TSC-FI hybrid schematic reveals how,\nunder resonance frequency microwave irradiation, localized\nspins commence precessional motion, consequently initiating\nthe dynamical Majorana Ising spin response at the TSC inter-\nface. (b) In the TSC context, the liaison between a spin-up\nelectron and a spin-down hole with the surrounding sea of\nspin-triplet Cooper pairs drastically modulate their proper-\nties; notably, a spin-down hole can engage with a spin-triplet\nCooper pair, thereby inheriting a negative charge. (c) No-\ntably, spin-triplet Cooper pairs amass around holes and scat-\nter around electrons, thereby eroding the rigid distinction be-\ntween the two. (d) The interplay between the Majorana mode\nand the localized spin manipulates the FMR spectrum, trig-\ngering a frequency shift and linewidth broadening.\njorana modes interact with the localized spins in the FI.\nAs a result, the localized spin dynamics leads to the dy-\nnamical Majorana Ising spin response (DMISR), which\nmeans the Majorana Ising spin density is dynamically in-\nduced, and it is possible to toggle between dissipative and\nnon-dissipative Majorana Ising spin dynamics by adjust-\ning the external magnetic field angle and the microwave\nfrequency. Furthermore, the modulation of the localizedarXiv:2308.05955v2 [cond-mat.mes-hall] 19 Mar 20242\nspin dynamics due to the interface interaction leads to a\nfrequency shift and a linewidth broadening, which reflect\nthe properties of the Majorana Ising spin dynamics. This\nwork proposes a setup for detecting Majorana modes and\npaves the way for the development of quantum comput-\ning and spin devices using Majoranas.\nModel.— We introduce a model Hamiltonian Hconsist-\ning of three terms\nH=HM+HFI+Hex. (1)\nThe first, second, and third terms respectively describe\nthe surface Majorana modes on the TSC surface, the bulk\nFI, and the proximity-induced exchange coupling. Our\nfocus is on energy regions significantly smaller than the\nbulk superconducting gap. This focus allows the spin ex-\ncitation in the TSC to be well described using the surface\nMajorana modes. The subsequent paragraphs provide\ndetailed explanations of each of these three terms.\nThe first terms HMdescribes the surface Majorana\nmodes,\nHM=1\n2Z\ndrψT(r)\u0010\nℏvˆkyσx−ℏvˆkxσy\u0011\nψ(r),(2)\nwhere r= (x, y),ˆk= (−i∂x,−i∂y),vis a constant\nvelocity, and σ= (σx, σy, σz) are the Pauli matrices.\nThe two component Majorana field operator is given by\nψ(r) = ( ψ→(r), ψ←(r))T, with the spin quantization\naxis along the xaxis. The Majorana field operators sat-\nisfy the Majorana condition ψσ(r) =ψ†\nσ(r) and the an-\nticommutation relation {ψσ(r), ψσ′(r)}=δσσ′δ(r−r′)\nwhere σ, σ′=→,←. We can derive HMby using surface-\nlocalized solutions of the BdG equation based on the bulk\nTSC Hamiltonian. The details of the derivation of HM\nare provided in the Supplemental Material [19].\nA notable feature of the surface Majorana modes is\nthat the spin density is Ising like, which we call the Majo-\nrana Ising spin [3–8]. The feature follows naturally from\nthe Majorana condition and the anticommutation rela-\ntion. The Majorana Ising spin density operator is given\nbys(r) := ψT(r)(σ/2)ψ(r) = (0 ,0,−iψ→(r)ψ←(r))\n(See the Supplemental Material for details [19]). The\nanisotropy of the Majorana Ising spin is the hallmark of\nthe surface Majorana modes on the TSC surface.\nThe second term HFIdescries the bulk FI and is given\nby the ferromagnetic Heisenberg model,\nHFI=− JX\n⟨n,m⟩Sn·Sm−ℏγhdcX\nnSZ\nn, (3)\nwhere J>0 is the exchange coupling constant, Snis the\nlocalized spin at site n,⟨n, m⟩means summation for near-\nest neighbors, γis the electron gyromagnetic ratio, and\nhdcis the static external magnetic field. We consider the\nspin dynamics of the localized spin under microwave irra-\ndiation, applying the spin-wave approximation. This al-\nlows the spin excitation to be described by a free bosonic\noperator, known as a magnon [20].The third term Hexrepresents the proximity exchange\ncoupling at the interface between the TSC and the FI,\nHex=−Z\ndrX\nnJ(r,rn)s(r)·Sn=HZ+HT,(4)\nHZ=−cosϑZ\ndrX\nnJ(r,rn)sz(r)SZ\nn, (5)\nHT=−sinϑZ\ndrX\nnJ(r,rn)sz(r)SX\nn, (6)\nwhere the angle ϑis shown in Fig. 1 (a). HZis the\ncoupling along the precession axis and HTis the coupling\nperpendicular to the precession axis. In our setup, HZ\nleads to gap opening of the energy spectrum of the surface\nMajorana modes and HTgives the DMISR under the\nmicrowave irradiation.\nDynamical Majorana Ising spin response.— We con-\nsider the microwave irradiation on the FI. The coupling\nbetween the localized spins and the microwave is given\nby\nV(t) =−ℏγhacX\nn\u0000\nSX\nncosωt−SY\nnsinωt\u0001\n,(7)\nwhere hacis the microwave amplitude, and ωis the mi-\ncrowave frequency. The microwave irradiation leads to\nthe precessional motion of the localized spin. When the\nfrequency of the precessional motion and the microwave\ncoincide, the FMR occurs. The FMR leads to the DMISR\nvia the exchange interaction. The DMISR is character-\nized by the dynamic spin susceptibility of the Majorana\nmodes, ˜ χzz(q, ω), defined as\n˜χzz(q, ω) :=Z\ndre−iq·rZ\ndtei(ω+i0)tχzz(r, t),(8)\nwhere χzz(r, t) := −(L2/iℏ)θ(t)⟨[sz(r, t), sz(0,0)]⟩\nwith the interface area L2and the spin den-\nsity operator in the interaction picture, sz(r, t) =\nei(HM+HZ)t/ℏsz(r)e−i(HM+HZ)t/ℏ. For the exchange cou-\npling, we consider configuration average and assume\n⟨P\nnJ(r,rn)⟩ave=J1, which means that HZis treated\nas a uniform Zeeman like interaction and the interface\nis specular [21]. Using eigenstates of Eq. (2) and after a\nstraightforward calculation, the uniform spin susceptibil-\nity is given by\n˜χzz(0, ω)\n=−X\nk,λ|⟨k, λ|σz|k,−λ⟩|2f(Ek,λ)−f(Ek,−λ)\n2Ek,λ+ℏω+i0,\n→ −Z\ndED (E)E2−M2\n2E2f(E)−f(−E)\n2E+ℏω+i0, (9)\nwhere |k, λ⟩is an eigenstate of HMwith eigenenergy\nEk,λ=λp\n(ℏvk)2+M2, (λ=±).M=J1Scosϑis\nthe Majorana gap, f(E) = 1 /(eE/kBT+ 1) is the Fermi3\ndistribution function, and D(E) is the density of states\ngiven by\nD(E) =L2\n2π(ℏv)2|E|θ(|E| − |M|), (10)\nwith the Heaviside step function θ(x). It is important to\nnote that the behavior of the uniform spin susceptibil-\nity is determined by the interband contribution, which is\nproportional to the Fermi distribution function, i.e., the\ncontribution of the occupied states. This mechanism is\nsimilar to the Van Vleck paramagnetism [22]. The con-\ntribution of the occupied states often plays a crucial role\nin topological responses [23].\nReplacing the localized spin operators with their statis-\ntical average values, we find the induced Majorana Ising\nspin density, to the first order of J1S, is given by\nZ\ndr⟨sz(r, t)⟩= ˜χzz\n0(0,0)J1Scosϑ\n+ Re[˜ χzz\n0(0, ω)]hac\nαhdcJ1Ssinϑsinωt, (11)\nwhere ˜ χzz\n0(0,0) is the spin susceptibility for M= 0. The\nfirst term originates from HZand gives a static spin den-\nsity, while the second term originates from HTand gives\na dynamic spin density. Figure 2 shows the induced Ising\nspin density as a function of time at several angles. As\nshown in Eq. (11), the Ising spin density consists of the\nstatic and dynamic components. The dynamic compo-\nnent is induced by the precessional motion of the local-\nized spin, which means one can induce the DMISR using\nthe dynamics of the localized spin.\nThe inset in Fig. 2 shows Im˜ χzz(0, ω) as a function of\nϑat a fixed frequency. When the frequency ℏωis smaller\nthan the Majorana gap, Im˜ χzz(0, ω) is zero. Once the\nfrequency overcomes the Majorana gap, Im˜ χzz(0, ω) be-\ncomes finite. The implications of these behaviors are that\nif the magnon energy is smaller than the Majorana gap,\nthere is no energy dissipation due to the DMISR. How-\never, once the magnon energy exceeds the Majorana gap,\nfinite energy dissipation associated with the DMISR oc-\ncurs at the surface of the TSC. Therefore, one can toggle\nbetween dissipative and non-dissipative Majorana Ising\nspin dynamics by adjusting the precession axis angle and\nthe microwave frequency.\nFMR modulation.— The retarded component of the\nmagnon Green’s function is given by GR(rn, t) =\n−(i/ℏ)θ(t)⟨[S+\nn(t), S−\n0(0)]⟩with the interaction picture\nS±\nn(t) =eiHFIt/ℏS±\nne−iHFIt/ℏ. The FMR signal is char-\nacterized by the spectral function defined as\nA(q, ω) :=−1\nπIm\"X\nne−iq·rnZ\ndtei(ω+i0)tGR(rn, t)#\n.\n(12)\nSSImχzz(0, ω) ˜⟨s z⟩\n2\n1ωtϑ\nFInon-dissipativenon-dissipativedissipativedissipativeTSC\nFITSC000.00.51.0\nπ/4\nπ/2\n0 π/4 π/20\nϑ2π\nπFIG. 2. The induced Ising spin density, with a unit\n˜χzz\n0(0,0)J1S, is presented as a function of ωtandϑ. The\nfrequency and temperature are set to ℏω/J1S= 1.5 and\nkBT/J 1S= 0.1, respectively. The coefficient, hac/αhdc, is\nset to 0 .3. The static Majorana Ising spin density arises\nfrom HZ. When the precession axis deviates from the di-\nrection perpendicular to the interface, the precessional mo-\ntion of the localized spins results in the dynamical Majorana\nIsing spin response (DMISR). Energy dissipation due to the\nDMISR is zero for small angles ϑas the Majorana gap ex-\nceeds the magnon energy. However, once the magnon energy\novercomes the Majorana gap, the energy dissipation becomes\nfinite. Therefore, one can toggle between dissipative and non-\ndissipative DMISR by adjusting ϑ.\nFor uniform external force, the spectral function is given\nby\nA(0, ω) =2S\nℏ1\nπ(α+δα)ω\n[ω−γ(hdc+δh)]2+ [(α+δα)ω]2.\n(13)\nThe peak position and width of the FMR signal is given\nbyhdc+δhandα+δα, respectively. hdcandαcorre-\nspond to the peak position and the linewidth of the FMR\nsignal of the FI alone. δhandδαare the FMR modu-\nlations due to the exchange interaction HT. We treat\nHM+HFI+HZas an unperturbed Hamiltonian and HT\nas a perturbation. In this work, we assume the specular\ninterface, where the coupling J(r,rn) is approximated\nasDP\nn,n′J(r,rn)J(r′,rn′)E\nave=J2\n1. The dynamics\nof the localized spins in the FI is modulated due to the\ninteraction between the localized spins and the Majo-\nrana Ising spins. In our setup, the peak position and the\nlinewidth of the FMR signal are modulated and the FMR4\nmodulation is given by\nδh= sin2ϑSJ2\n1\n2NγℏRe˜χzz(0, ω), (14)\nδα= sin2ϑSJ2\n1\n2NℏωIm˜χzz(0, ω), (15)\nwhere Nis the total number of sites in the FI. These for-\nmulas were derived in the study of the FMR in magnetic\nmultilayer systems including superconductors. One can\nextract the spin property of the Majorana mode from the\ndata on δhandδα. Because of the Ising spin anisotropy,\nthe FMR modulation exhibits strong anisotropy, where\nthe FMR modulation is proportional to sin2ϑ.\nFigure 3 shows the FMR modulations (a) δαand (b)\nδh. The FMR modulation at a fixed frequency increases\nwith angle ϑand reaches a maximum at π/2, as can be\nread from Eqs. (14) and (15). When the angle ϑis fixed\nand the frequency ωis increased, δαbecomes finite above\na certain frequency at which the energy of the magnon\ncoincides with the Majorana gap. When ϑ < π/ 2 and\nℏω≈2M,δαlinearly increases as a function of ωjust\nabove the Majorana gap. The localized spin damping is\nenhanced when the magnon energy exceeds the Majorana\ngap. At ϑ=π/2 and ω≈0, the Majorana gap vanishes\nandδαis proportional to ω/T. In the high frequency\nregion ℏω/J 1S≫1,δαconverges to its upper threshold.\nThe frequency shift δhis almost independent of ωand\nhas a finite value even in the Majorana gap. This behav-\nior is analogous to the interband contribution to the spin\nsusceptibility in strongly spin-orbit coupled band insula-\ntors, and is due to the fact that the effective Hamiltonian\nof the Majorana modes includes spin operators. It is im-\nportant to emphasize that although the Majorana modes\nhave spin degrees of freedom, only the zcomponent of the\nspin density operator is well defined. This is a hallmark\nof Majorana modes, which differs significantly from elec-\ntrons in ordinary solids. Note that δhis proportional to\nthe energy cutoff, which is introduced to converge energy\nintegral for Re˜ χzz(0, ω). The energy cutoff corresponds\nto the bulk superconducting gap, which is estimated as\n∆∼0.1[meV] ( ∼1[K]). Therefore, our results are ap-\nplicable in the frequency region below ℏω∼0.1[meV]\n(∼30[GHz]). In addition, we assume that Majorana gap\nis estimated to be J1S∼0.01[meV] ( ∼0.1[K]).\nDiscussion.— Comparing the present results with spin\npumping (SP) in a conventional metal-ferromagnet hy-\nbrid, the qualitative behaviors are quite different. In con-\nventional metals, spin accumulation occurs due to FMR.\nIn contrast, in the present system, no corresponding spin\naccumulation occurs due to the Ising anisotropy. Also, in\nthe present calculations, the proximity-induced exchange\ncoupling is assumed to be an isotropic Heisenberg-like\ncoupling. However, in general, the interface interaction\ncan also be anisotropic. Even in such a case, it is no qual-\nitative change in the case of ordinary metals, although a\n0.00.5\n(a) (b)\nϑℏω/J1S 0\nπ/4\nπ/2024\nϑℏω/J1S 0\nπ/4\nπ/2024δ α δ h10\n0FIG. 3. The temperature is set to kBT/J 1S= 0.1. (a)\nThe damping modulation δαonly becomes finite when the\nmagnon energy exceeds the Majorana gap; otherwise, it van-\nishes. This behavior corresponds to the energy dissipation of\nthe Majorana Ising spin. (b) The peak shift is finite, except\nforϑ= 0, and is almost independent of ω. This behavior\nresembles the spin response observed in strongly spin-orbit\ncoupled band insulators, where the interband contribution to\nspin susceptibility results in a finite spin response, even within\nthe energy gap.\ncorrection term due to anisotropy is added [24]. There-\nfore, the Ising anisotropy discussed in the present work\nis a property unique to the Majorana modes and can\ncharacterize the Majorana excitations.\nLet us comment on the universal nature of the toggling\nbetween non-dissipative and dissipative dynamical spin\nresponses observed in our study. Indeed, such toggling\nbecomes universally feasible when the microwave fre-\nquency and the energy gap are comparable, and when the\nHamiltonian and spin operators are non-commutative,\nindicating that spin is not a conserved quantity. The\nnon-commutativity can be attributed to the presence of\nspin-orbit couplings [25–27], and spin-triplet pair corre-\nlations [28].\nMicrowave irradiation leads to heating within the FI,\nso that thermally excited magnons due to the heating\ncould influence the DMISR. Phenomena resulting from\nthe heating, which can affect interface spin dynamics, in-\nclude the spin Seebeck effect (SSE) [29], where a spin\ncurrent is generated at the interface due to a tempera-\nture difference. In hybrid systems of normal metal and\nFI, methods to separate the inverse spin Hall voltage due\nto SP from other signals caused by heating have been\nwell studied [30]. Especially, it has been theoretically\nproposed that SP and SSE signals can be separated us-\ning a spin current noise measurement [24]. Moreover, SP\ncoherently excites specific modes, which qualitatively dif-\nfers from SSE induced by thermally excited magnons [14].\nTherefore, even if heating occurs in the FI in our setup,\nthe properties of Majorana Ising spins are expected to\nbe captured. Details of the heating effect on the DMISR\nwill be examined in the near future.\nWe also mention the experimental feasibility of our the-\noretical proposals. As we have already explained, the\nFMR modulation is a very sensitive spin probe. Indeed,\nthe FMR modulation by surface states of 3D topological5\ninsulators [31] and graphene [32–36] has been reported\nexperimentally. Therefore, we expect that the enhanced\nGilbert damping due to Majorana Ising spin can be ob-\nservable in our setup when the thickness of the ferromag-\nnetic insulator is sufficiently thin.\nFinally, it is pertinent to mention the potential candi-\ndate materials where surface Majorana Ising spins could\nbe detectable. Notably, UTe 2[37], Cu xBi2Se3[38, 39],\nSrxBi2Se3and Nb xBi2Se3[40] are reported to be in a p-\nwave superconducting state and theoretically can host\nsurface Majorana Ising spins. Recent NMR measure-\nments indicate that UTe 2could be a bulk p-wave su-\nperconductor in the Balian-Werthamer state [41], which\nhosts the surface Majorana Ising spins with the per-\npendicular Ising anisotropy, as considered in this work.\nAxBi2Se3(A= Cu, Sr, Nb) is considered to possess in-\nplane Ising anisotropy [8], differing from the perpendic-\nular Ising anisotropy explored in this work. Therefore,\nwe expect that it exhibits anisotropy different from that\ndemonstrated in this work.\nConclusion.— We present herein a study of the spin\ndynamics in a topological superconductor (TSC)-magnet\nhybrid. Ferromagnetic resonance under microwave irra-\ndiation leads to the dynamically induced Majorana Ising\nspin density on the TSC surface. One can toggle between\ndissipative and non-dissipative Majorana Ising spin dy-\nnamics by adjusting the external magnetic field angle and\nthe microwave frequency. Therefore, our setup provides\na platform to detect and control Majorana excitations.\nWe expect that our results provide insights toward the\ndevelopment of future quantum computing and spintron-\nics devices using Majorana excitations.\nAcknowledgments.— The authors are grateful to R.\nShindou for valuable discussions. This work is partially\nsupported by the Priority Program of Chinese Academy\nof Sciences, Grant No. XDB28000000. We acknowl-\nedge JSPS KAKENHI for Grants (Nos. JP20K03835,\nJP21H01800, JP21H04565, and JP23H01839).\nSUPPLEMENTAL MATERIAL\nSurface Majorana modes\nIn this section, we describe the procedure for deriv-\ning the effective Hamiltonian of the surface Majorana\nmodes. We start with the bulk Hamiltonian of a three-\ndimensional topological superconductor. Based on the\nbulk Hamiltonian, we solve the BdG equation to demon-\nstrate the existence of a surface-localized solution. Us-\ning this solution, we expand the field operator and show\nthat it satisfies the Majorana condition when the bulk\nexcitations are neglected. As a result, on energy scales\nmuch smaller than the bulk superconducting gap, the\nlow-energy excitations are described by surface-localized\nMajorana modes. The above procedure is explained inmore detail in the following. Note that we use rfor three-\ndimensional coordinates and r∥for two-dimensional ones\nin the Supplemental Material.\nWe start with the mean-field Hamiltonian given by\nHSC=1\n2Z\ndrΨ†\nBdG(r)HBdGΨBdG(r), (16)\nwithr= (x, y, z ). We consider the Balian-Werthamer\n(BW) state, in which the pair potential is given by\n∆ˆk=∆\nkF\u0010\nˆk·σ\u0011\niσywith the bulk superconducting gap\n∆. Here, we do not discuss the microscopic origin of the\npair correlation leading to the BW state. As a result, the\nBdG Hamiltonian HBdGis given by\nHBdG=\nεˆk−EF 0 −∆\nkFˆk−∆\nkFˆkx\n0 εˆk−EF∆\nkFˆkx∆\nkFˆk+\n−∆\nkFˆk+∆\nkFˆkx−εˆk+EF 0\n∆\nkFˆkx∆\nkFˆk− 0 −εˆk+EF\n,\n(17)\nwith ˆk±=ˆky±iˆkz,ˆk=−i∇, and εˆk=ℏ2ˆk2\n2m. The four\ncomponent Nambu spinor ΨBdG(r) is given by\nΨBdG(r) :=\nΨ→(r)\nΨ←(r)\nΨ†\n→(r)\nΨ†\n←(r)\n, (18)\nwith the spin quantization axis along the xaxis. The\nmatrices of the spin operators are represented as\nσx=\u00121 0\n0−1\u0013\n, (19)\nσy=\u0012\n0 1\n1 0\u0013\n, (20)\nσz=\u00120−i\ni0\u0013\n. (21)\nThe fermion field operators satisfy the anticommutation\nrelations\n{Ψσ(r),Ψσ′(r′)}= 0, (22)\n{Ψσ(r),Ψ†\nσ′(r′)}=δσσ′δ(r−r′), (23)\nwith the spin indices σ, σ′=→,←.\nTo diagonalize the BdG Hamiltonian, we solve the BdG\nequation given by\nHBdGΦ(r) =EΦ(r). (24)\nWe assume that a solution is written as\nΦ(r) =eik∥·r∥f(z)\nu→\nu←\nv→\nv←\n, (25)6\nwithk∥= (kx, ky) and r∥= (x, y). If we set the four\ncomponents vector to satisfy the following equation (Ma-\njorana condition)\n\n0 0 1 0\n0 0 0 1\n1 0 0 0\n0 1 0 0\n\nu→\nu←\nv→\nv←\n=±\nu→\nu←\nv→\nv←\n, (26)\nwe can obtain a surface-localized solution. If we take a\npositive (negative) sign, we obtain a solution localized\non the top surface (bottom surface). As we will consider\nsolutions localized on the bottom surface below, we take\na negative sign. Finally, we obtain the normalized eigen-\nvectors of the BdG equation given by\nΦλ,k∥(r) =eik∥·r∥\n√\nL2fk∥(z)uλ,k∥, (27)\nwith\nfk∥(z) =Nk∥sin(k⊥z)e−κz, (28)\nNk∥=s\n4κ(k2\n⊥+κ2)\nk2\n⊥, (29)\nκ=m∆\nℏ2kF, (30)\nk⊥=q\nk2\nF−k2\n∥−κ2, (31)\nand\nu+,k∥=\nu+,→k∥\nu+,←k∥\nv+,→k∥\nv+,←k∥\n=1√\n2\nsinϕk∥+π/2\n2\n−cosϕk∥+π/2\n2\n−sinϕk∥+π/2\n2\ncosϕk∥+π/2\n2\n,(32)\nu−,k∥=\nu−,→k∥\nu−,←k∥\nv−,→k∥\nv−,←k∥\n=1√\n2\n−cosϕk∥+π/2\n2\n−sinϕk∥+π/2\n2\ncosϕk∥+π/2\n2\nsinϕk∥+π/2\n2\n.(33)\nThe eigenenergy is given by Eλ,k∥=λ∆k∥/kF. We can\nshow that the eigenvectors satisfy\nu−,−k∥=u+,k∥. (34)\nConsequently, the field operator is expanded as\nΨBdG(r) =X\nk∥\u0012\nγk∥eik∥·r∥\n√\nL2+γ†\nk∥e−ik∥·r∥\n√\nL2\u0013\n×fk∥(z)u+,k∥+ (bulk modes) ,(35)\nwhere γk∥(γ†\nk∥) is the quasiparticle creation (annihila-\ntion) operator with the eigenenergy E+,k∥. Substitutingthe above expression into Eq. (16) with omission of bulk\nmodes and performing the integration in the z-direction,\nwe obtain the effective Hamiltonian for the surface states\nHM=1\n2Z\ndr∥ψT(r∥)\u0010\nℏvˆkyσx−ℏvˆkxσy\u0011\nψ(r∥),(36)\nwhere v= ∆/ℏkFand we introduced the two component\nMajorana field operator\nψ(r∥) =\u0012ψ→(r∥)\nψ←(r∥)\u0013\n, (37)\nsatisfying the Majorana condition\nψσ(r∥) =ψ†\nσ(r∥), (38)\nand the anticommutation relation\nn\nψσ(r∥), ψσ′(r′\n∥)o\n=δσσ′δ(r∥−r′\n∥). (39)\nThe spin density operator of the Majorana mode is\ngiven by\ns(r∥) =ψ†(r∥)σ\n2ψ(r∥). (40)\nThexcomponent is given by\nsx(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00121/2 0\n0−1/2\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=1\n2\u0002\nψ†\n→(r∥)ψ→(r∥)−ψ†\n←(r∥)ψ←(r∥)\u0003\n=1\n2\u0002\nψ2\n→(r∥)−ψ2\n←(r∥)\u0003\n= 0. (41)\nIn a similar manner, the yandzcomponents are given\nby\nsy(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00120 1/2\n1/2 0\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=1\n2\u0002\nψ†\n→(r∥)ψ←(r∥) +ψ†\n←(r∥)ψ→(r∥)\u0003\n=1\n2\b\nψ→(r∥), ψ←(r∥)\t\n= 0, (42)\nand\nsz(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00120−i/2\ni/2 0\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=−i\n2\u0000\nψ†\n→(r∥)ψ←(r∥)−ψ†\n←(r∥)ψ→(r∥)\u0001\n=−iψ→(r∥)ψ←(r∥), (43)\nrespectively. As a result, the spin density operator is\ngiven by\ns(r∥) =\u0000\n0,0,−iψ→(r∥)ψ←(r∥)\u0001\n. (44)\nOne can see that the spin density of the Majorana mode\nis Ising like.7\nMajorana Ising spin dynamics\nIn this section, we calculate the Ising spin density in-\nduced on the TSC surface by the proximity coupling Hex.\nHexconsists of two terms, HZandHT.HZleads to the\nstatic spin density and HTleads to the dynamic spin\ndensity. First, we calculate the static spin density. Next,\nwe calculate the dynamic spin density.\nThe total spin density operator is given by\nsz\ntot=Z\ndr∥sz(r∥). (45)\nThe statistical average of the static spin density is calcu-\nlated as\n⟨sz\ntot⟩=−X\nk∥M\n2Ek∥\u0002\nf(Ek∥)−f(−Ek∥)\u0003\n→ −\u0012L\n2πℏv\u00132Z∆\nMEdEZ2π\n0dϕM\n2E[f(E)−f(−E)]\n=−Z∆\n0dED (E)f(E)−f(−E)\n2EM. (46)\nAt the zero temperature limit T→0, the static spin\ndensity is given by\n⟨sz\ntot⟩=1\n2L2\n2π(ℏv)2(∆−M)M≈˜χzz\n0(0,0)M, (47)\nwhere ˜ χzz\n0(0,0) = D(∆)/2 and we used ∆ ≫M.\nThe dynamic spin density is given by the perturbative\nforce\nHT(t) =Z\ndr∥sz(r∥)F(r∥, t), (48)\nwhere F(r∥, t) is given by\nF(r∥, t) =−sinϑX\nnJ(r∥,rn)\nSX\nn(t)\u000b\n≈ −sinϑJ1Sγhacp\n(ω−γhdc)2+α2ω2cosωt\n=:Fcosωt. (49)\nThe time dependent statistical average of the Ising spin\ndensity, to the first order of J1S, is given by\nZ\ndr∥\nsz(r∥, t)\u000b\n=Z\ndr∥Z\ndr′\n∥Z\ndt′χzz(r∥−r′\n∥, t′)F(r′\n∥, t−t′)\n= Re\u0002\n˜χzz(0, ω)Fe−iωt\u0003\n≈Re[˜χzz\n0(0, ω)]Fcosωt, (50)\nwhere we used Re˜ χzz\n0(0, ω)≫Im˜χzz\n0(0, ω). The real part\nof ˜χzz(0, ω) is given by\nRe˜χzz(0, ω) =−PZ\ndED (E)E2−M2\n2E2f(E)−f(−E)\n2E+ℏω,\n(51)where Pmeans the principal value. When the integrand\nis expanded with respect to ω, the lowest order correc-\ntion term becomes quadratic in ω. In the frequency range\nconsidered in this work, this correction term is signifi-\ncantly smaller compared to the static spin susceptibility\nRe˜χzz(0,0). Therefore, the spin susceptibility exhibits\nalmost no frequency dependence and remains constant\nas a function of ω. The imaginary part of ˜ χzz(0, ω) is\ngiven by\nIm˜χzz(0, ω)\n=πD(ℏω/2)(ℏω/2)2−M2\n2(ℏω/2)2[f(−ℏω/2)−f(ℏω/2)].\n(52)\nFMR modulation due to the proximity exchange\ncoupling\nIn this section, we provide a brief explanation for the\nderivation of the FMR modulations δhandδα. The FMR\nmodulations can be determined from the retarded com-\nponent of the magnon Green’s function, which is given\nby\n˜GR(k, ω) =2S/ℏ\nω−ωk+iαω−(2S/ℏ)ΣR(k, ω),(53)\nwhere we introduce the Gilbert damping constant αphe-\nnomenologically. In the second-order perturbation calcu-\nlation with respect to HT, the self-energy is given by\nΣR(k, ω) =−\u0012sinϑ\n2\u00132X\nq∥|˜J(q∥,k)|2˜χzz(q∥, ω),(54)\nwhere ˜J(q∥,0) is given by\n˜J(q∥,k) =1\nL2√\nNZ\ndr∥X\nnJ(r∥,rn)ei(q∥·r∥+k·rn)\n(55)\nThe pole of ˜GR(k, ω) signifies the FMR modulations,\nincluding both the frequency shift and the enhanced\nGilbert damping. These are given by\nδh=2S\nγℏReΣR(0, ω), δα =−2S\nℏωImΣR(0, ω).(56)\nFrom the above equations and Eq. (54), it is apparent\nthat FMR modulations provide information regarding\nboth the properties of the interface coupling and the dy-\nnamic spin susceptibility of the Majorana modes.\nThe form of matrix element ˜J(q∥,0) depends on the\ndetails of the interface. In this work, we assume the\nspecular interface. |˜J(q∥,0)|2is given by\n|˜J(q∥,0)|2=J2\n1\nNδq∥,0. (57)8\nUsing Eq. (57), the self-energy for the uniform magnon\nmode is given by\nΣR(0, ω) =−\u0012sinϑ\n2\u00132J2\n1\nN˜χzz(0, ω). (58)\n[1] F. Wilczek, Nat. 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Jpn. 92, 063701 (2023)." }, { "title": "1001.2845v1.Resonance_Damping_in_Ferromagnets_and_Ferroelectrics.pdf", "content": "arXiv:1001.2845v1 [cond-mat.other] 16 Jan 2010Resonance Damping in Ferromagnets and Ferroelectrics\nA. Widom\nPhysics Department, Northeastern University, Boston, MA U SA\nS. Sivasubramanian\nNSF Nanoscale Science & Engineering Center for High-rate Na nomanufacturing,\nNortheastern University, Boston MA USA\nC. Vittoria and S. Yoon\nDepartment of Electrical and Computer Engineering, Northe astern University, Boston, MA USA\nY.N. Srivastava\nPhysics Department and & INFN, University of Perugia, Perug ia IT\nThe phenomenological equations of motion for the relaxatio n of ordered phases of magnetized\nand polarized crystal phases can be developed in close analo gy with one another. For the case of\nmagnetized systems, thedrivingmagnetic fieldintensityto ward relaxation was developedbyGilbert.\nFor the case of polarized systems, the driving electric field intensity toward relaxation was developed\nby Khalatnikov. The transport times for relaxation into the rmal equilibrium can be attributed to\nviscous sound wave damping via magnetostriction for the mag netic case and electrostriction for the\npolarization case.\nPACS numbers: 76.50.+g, 75.30.Sg\nI. INTRODUCTION\nIt has long been of interest to understand the close\nanalogiesbetween orderedelectric polarized systems, e.g.\nferroelectricity , and ordered magnetic systems, e.g. fer-\nromagnetism . Atthe microscopiclevel, the sourceofsuch\nordering must depend on the nature of the electronic en-\nergy spectra. The relaxation mechanism into thermal\nequilibrium state must be described by local electric field\nfluctuationsforthe electricpolarizationcaseandbymag-\nnetic intensity fluctuations for the magnetization case;\nSpecifically, the field fluctuations for each case\nGpol\nij(r,r′,t) =1\n¯h/integraldisplayβ\n0/an}bracketle{t∆Ej(r′,−iλ)∆Ei(r,t)/an}bracketri}htdλ,\nGmag\nij(r,r′,t) =1\n¯h/integraldisplayβ\n0/an}bracketle{t∆Hj(r′,−iλ)∆Hi(r,t)/an}bracketri}htdλ,\nwherein β=¯h\nkBT,(1)\ndetermine the relaxation time tensor for both cases via\nthe fluctuation-dissipation formula1–4\nτij=/integraldisplay∞\n0lim\nV→∞/bracketleftbigg1\nV/integraldisplay\nV/integraldisplay\nVGij(r,r′,t)d3rd3r′/bracketrightbigg\ndt.(2)\nWe have unified the theories of relaxation in ordered po-\nlarized systems and ordered magnetized systems via the\nKubo transport time tensor in Eqs.(1) and (2).\nThe transport describing the relaxation of or-\ndered magnetization is the Landau-Lifshitz-Gilbert\nequation5–7. This equation has been of considerable\nrecent interest8–10in describing ordered magnetic reso-\nnancephenomena11–14. The equationdescribingthe elec-tric relaxation of an ordered polarization is the Landau-\nKhalatinikov-Tani equation15–17. This equation can be\nsimplymodeled18–21witheffectiveelectricalcircuits22–25.\nInformation memory applications26–29of such polarized\nsystem are of considerable recent interest30–32.\nThe unification of the magnetic Gilbert-Landau-\nLifshitz equations and the electric Landau-Khalatnikov-\nTani equations via the relaxation time tensor depends\non the notion of a nonequilibrium driving field . For the\nmagnetic case, the driving magnetic intensity Hddeter-\nmines the relaxation of the magnetization via the torque\nequation\n˙M=γM×Hd, (3)\nwhereinγisthegyromagneticratio. Fortheelectriccase,\nthe driving electric field Eddetermines the relaxation of\nthe polarization via the equation of motion for an ion of\nchargeze\nm¨r=zeEd. (4)\nThe unification of both forms of relaxation lies in the\nclose analogy between the magnetic driving intensity Hd\nand the electric driving field Ed.\nIn Sec.II the thermodynamics of ordered magnetized\nand polarized systems is reviewed. The notions of mag-\nnetostrictionand electrostrictionaregiven aprecise ther-\nmodynamic definition. In Sec.III, the phenomenology of\nthe relaxation equations are presented. The magnetic\ndriving intensity Hdand the electric driving field Edare\ndefined in terms of the relaxation time tensor Eq.(2). In\nSec.IV, we introduce the crystal viscosity tensor. From a\nKubo formula viewpoint, the stress fluctuation correlax-2\nation\nFijkl(r,r′,t) =1\n¯h/integraldisplayβ\n0/an}bracketle{t∆σkl(r′,−iλ)∆σij(r,t)/an}bracketri}htdλ,(5)\ndetermines the crystal viscosity\nηijkl=/integraldisplay∞\n0lim\nV→∞/bracketleftbigg1\nV/integraldisplay\nV/integraldisplay\nVFijkl(r,r′,t)d3rd3r′/bracketrightbigg\ndt.(6)\nFor models of magnetic relaxation wherein acoustic heat-\ning dominates via magnetostriction33and for models of\nelectric relaxation wherein acoustic heating dominates\nvia electrostriction, the relaxation time tensor in Eq.(2)\ncan be related to the viscosity tensor Eq.(6). An inde-\npendent microscopic derivation of viscosity induced re-\nlaxationisgiveninAppendixA. IntheconcludingSec.V,\nthe sound wave absorption physics of the viscous damp-\ning mechanism will be noted.\nII. THERMODYNAMICS\nOur purpose is to review the thermodynamic proper-\nties of both magnetically ordered crystals and polariza-\ntion ordered crystals. The former is characterized by\na remnant magnetization Mfor vanishing applied mag-\nnetic intensity H→0 while the latter is characterized by\na remnant polarization Pfor vanishing applied electric\nfieldE→0.\nA. Magnetically Ordered Crystals\nLetwbe the enthalpy per unit volume. The funda-\nmental thermodynamic law determining the equations of\nstate for magnetically ordered crystals is given by\ndw=Tds+H·dM−e:dσ, (7)\nwhereinsistheentropyperunitvolume, Tisthetemper-\nature,eis the crystal strain and σis the crystal stress.\nThe magnetic adiabatic susceptibility is defined by\nχ=/parenleftbigg∂M\n∂H/parenrightbigg\ns,σ. (8)\nIf\nN=M\nM⇒N·N= 1 (9)\ndenotes a unit vector in the direction of the magnetiza-\ntion, then the tensor Λ ijkldescribing adiabatic magne-\ntostriction coefficients may be defined as34\n2ΛijklNl=M/parenleftbigg∂eij\n∂Mk/parenrightbigg\ns,σ=−M/parenleftbigg∂Hk\n∂σij/parenrightbigg\ns,M.(10)When the system is out of thermal equilibrium, the driv-\ning magnetic intensity is\nHd=H−/parenleftbigg∂w\n∂M/parenrightbigg\ns,σ−τ·/parenleftbigg∂M\n∂t/parenrightbigg\n,(11)\nwherein τare the relaxation time tensor transport co-\nefficients which determine the relaxation of the ordered\nmagnetic system into a state of thermal equilibrium.\nB. Ordered Polarized Crystals\nThe fundamental thermodynamic law determining the\nequations of state for ordered polarized crystals is given\nby\ndw=Tds+E·dP−e:dσ, (12)\nwherein wis the enthalpy per unit volume, sis the en-\ntropy per unit volume, Tis the temperature, eis the\ncrystal strain and σis the crystal stress. The electric\nadiabatic susceptibility is defined by\nχ=/parenleftbigg∂P\n∂E/parenrightbigg\ns,σ. (13)\nThe tensor βijkdescribing adiabatic electrostriction co-\nefficients may be defined as34\nβijk=/parenleftbigg∂eij\n∂Pk/parenrightbigg\ns,σ=−/parenleftbigg∂Ek\n∂σij/parenrightbigg\ns,P.(14)\nThe piezoelectric tensor is closely related to the elec-\ntrostriction tensor via\nγijk=/parenleftbigg∂eij\n∂Ek/parenrightbigg\ns,σ=/parenleftbigg∂Pk\n∂σij/parenrightbigg\ns,E=βijmχmk.(15)\nWhen the system is out of thermal equilibrium, the driv-\ning electric field is\nEd=E−/parenleftbigg∂w\n∂P/parenrightbigg\ns,σ−τ·/parenleftbigg∂P\n∂t/parenrightbigg\n,(16)\nwherein τis the relaxation time tensor transport coef-\nficients which determine the relaxation of the ordered\npolarized system into a state of thermal equilibrium.\nIII. RESONANCE DYNAMICS\nHere we shall show how the magnetic intensity Hd\ndrives the magnetic resonance equations of motion in\nmagnetically ordered systems. Similarly, we shall show\nhowtheelectricfield Eddrivesthe polarizationresonance\nequations of motion for polarized ordered systems.3\nA. Gilbert-Landau-Lifshitz Equations\nThe driving magnetic intensity determines the torque\non the magnetic moments according to\n∂M\n∂t=γM×Hd. (17)\nEmploying Eqs.(11) and (17), one finds the equations for\nmagnetic resonance in the Gilbert form\n∂M\n∂t=γM×/bracketleftBigg\nH−/parenleftbigg∂w\n∂M/parenrightbigg\ns,σ−/parenleftbiggα\nγM/parenrightbigg\n·∂M\n∂t/bracketrightBigg\n,(18)\nwherein the Gilbert dimensionless damping tensor αis\ndefined as\nα= (γM)τ. (19)\nOnemaydirectlysolvetheGilbert equationsforthe driv-\ning magnetic intensity according to\nHd+α·/parenleftbig\nN×Hd/parenrightbig\n=H−/parenleftbigg∂w\n∂M/parenrightbigg\ns,σ.(20)\nEqs.(17) and (20) expressthe magneticresonancemotion\nin the Landau-Lifshitz form.\nB. Landau-Khalatnikov-Tani Equations\nThe driving electric field gives rise to a polarization\nresponse according to\n∂2P\n∂t2=/parenleftBigg\nω2\np\n4π/parenrightBigg\nEd, (21)\nwhereinωpis the plasma frequency. A simple derivation\nof Eq.(21) may be formulated as follows. In a large vol-\numeV, the polarization due to charges {zje}is given\nby\nP=/parenleftbigg/summationtext\njzjerj\nV/parenrightbigg\n. (22)\nIf the drivingelectric field acceleratesthe chargesaccord-\ning to\nmj¨rj=zjeEd, (23)\nthen Eq.(21) holds true with the plasma frequency\nω2\np= 4πe2lim\nV→∞/bracketleftBigg/summationtext\nj(z2\nj/mj)\nV/bracketrightBigg\n= 4πe2/summationdisplay\nanaz2\na\nma,(24)\nwhereinnais the density of charged particles of type a.\nThe polarization resonance equation of motion follows\nfrom Eqs.(16) and (21) as17\n/parenleftbigg4π\nω2p/parenrightbigg∂2P\n∂t2+τ·∂P\n∂t+∂w(P,s,σ)\n∂P=E.(25)The electric field Einduces the polarization Pat reso-\nnant frequencies which are eigenvalues of the tensor Ω\nfor which\nΩ2=ω2\npχ−1\n4π≡ω2\np(ǫ−1)−1. (26)\nThedecayratesforthepolarizationoscillationsareeigen-\nvalues of the tensor Γfor which\nΓ=ω2\npτ\n4π. (27)\nIfthedecayratesarelargeonthescaleofthetheresonant\nfrequencies, then the equation of motion is over damped\nso that\nmin\njΓj≫max\niΩiimplies\nτ·∂P\n∂t+∂w(P,s,σ)\n∂P=E. (28)\nEq.(28) represents the Landau-Khalatnikov equation for\npolarized systems.\nIV. HEATING RATE PER UNIT VOLUME\nLet us here consider the heating rate implicit in relax-\nation processes. Independently of the details of the mi-\ncroscopic mechanism for generating such heat, the rates\nof energy dissipation are entirely determined byτ. Ex-\nplicitly, the heating rates per unit volume for magnetiza-\ntion and polarization are given, respectively, by\n˙qM=∂M\n∂t·τ·∂M\n∂t, (29)\nand\n˙qP=∂P\n∂t·τ·∂P\n∂t. (30)\nFinally, the notion of crystal viscosity ηijklis introduced\ninto elasticity theory35via the heating rate per unit vol-\nume from rates of change in the strain ∂e/∂t; It is\n˙qe=∂eij\n∂tηijkl∂ekl\n∂t. (31)\nCrystal viscosity is employed to describe, among other\nthings, sound wave attenuation. Our purpose is to de-\nscribe how heating rates in Eqs.(29) and (30) can be re-\nlated to the heating rate in Eq.((31)). This allows us to\nexpressthetransportcoefficients τintermsofthecrystal\nviscosity.\nA. Relaxation via Magnetostriction\nFrom the magnetostriction Eq.(10), it follows that\nmagnetic relaxation gives rise to a strain\n∂eij\n∂t=2\nMΛijklNk∂Ml\n∂t, (32)4\nand thereby to the heating rate,\n˙q=4\nM2∂Mi\n∂t(ΛmnqiNq)ηmnrs(ΛrskjNk)∂Mj\n∂t,(33)\nin virtue of Eq.(31). Employing Eqs.(29) and (33), we\nfind that the magnetic relaxation transport coefficient in\nthe magnetostriction model\nτij=4\nM2(ΛmnqiNq)ηmnrs(ΛrskjNk).(34)\nThe Gilbert damping tensor follows from Eqs.(19) and\n(34) as\nαij=4γ\nM(ΛmnqiNq)ηmnrs(ΛrskjNk).(35)\nThe central relaxation tensor Eq.(35) describes the mag-\nnetic relaxation in terms of the magnetostriction coeffi-\ncients and the crystal viscosity.\nB. Relaxation via Electrostriction\nFromtheelectrostrictionEq.(14), it followsthatatime\nvarying polarization gives rise to a time varying strain\n∂eij\n∂t=βijk∂Pk\n∂t, (36)\nand thereby to the heating rate,\n˙q=∂Pi\n∂tβkliηklmnβmnj∂Pj\n∂t, (37)\nin virtue of Eq.(31). Employing Eqs.(30) and (37), we\nfind that the electric relaxation transport coefficient in\nthe electroostriction model\nτij=βkliηklmnβmnj. (38)\nThecentralrelaxationtensorEq.(38) describesthe polar-\nization relaxation time tensor coefficients in terms of the\nelectrostriction coefficients and the crystal viscosity. The\nimplications ofthe electrostrictionmodel forthe Landau-\nKhalatnikov equation is to the authors knowledge a new\nresult.\nV. CONCLUSIONS\nFor ordered polarized and magnetized systems, we\nhave developed phenomenological equations of motion inclose analogywith one another. For the magnetized case,\nthe relaxation is driven by the magnetic intensity Hd\nyielding the Gilbert equation of motion7. For the polar-\nized case, the relaxation is driven by the electric field Ed\nyielding the Tani equation of motion17. In both cases,\nthe relaxation time tensor τis determined by the crystal\nviscosity as derived in the Appendix A; i.e. in Eqs.(A3)\nand (A6). The viscosity can be measured independently\nfrom the magnetic or electrical relaxation by employing\nsound absorption techniques36.\nAppendix A: Kubo formulae\nFrom the thermodynamic Eq.(10), the fluctuations in\nthe magnetic intensity are given by magnetostriction, i.e.\n∆Hk(r,t) =−/parenleftbigg2ΛijklNl\nM/parenrightbigg\n∆σij(r,t).(A1)\nEqs.(A1), (1) and (5) imply\nGmag\nij(r,r′,t) =\n4\nM2(ΛmnqiNq)Fmnrs(r,r′,t)(ΛrskjNk).(A2)\nEmploying Eqs.(A2), (2) and (6), one finds the central\nresult for the magnetic relaxation time tensor; It is\nτmag\nij=4\nM2(ΛmnqiNq)ηmnrs(ΛrskjNk) =αij\nγM.(A3)\nFrom the thermodynamic Eq.(14), the fluctuations in\nthe electric intensity are given by electrostriction, i.e.\n∆Ek(r,t) =−βijk∆σij(r,t). 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Bhatia, Ultrasonic Absorption , Oxford University\nPress, Oxford (1967)." }, { "title": "1706.04670v2.Temperature_dependent_Gilbert_damping_of_Co2FeAl_thin_films_with_different_degree_of_atomic_order.pdf", "content": "1 \n Temperature -dependent Gilbert damping of Co 2FeAl thin films with different degree of \natomic order \nAnkit Kumar1*, Fan Pan2,3, Sajid Husain4, Serkan Akansel1, Rimantas Brucas1, Lars \nBergqvist2,3, Sujeet Chaudhary4, and Peter Svedlindh1# \n \n1Department of Engineering Sciences, Uppsala University, Box 534, SE -751 21 Uppsala, \nSweden \n2Department of Applied Physics, School of Engineering Sciences, KTH Royal Institute of \nTechnology, Electrum 229, SE -16440 Kista, Sweden \n3Swedish e -Science Research Center, KTH Roy al Institute of Technology, SE -10044 \nStockholm, Sweden \n4Department of Physics, Indian Institute of Technology Delhi, New Delhi -110016, India \n \nABSTRACT \nHalf-metallicity and low magnetic damping are perpetually sought for in spintronics materials \nand full He usler alloys in this respect provide outstanding properties . However, it is \nchallenging to obtain the well -ordered half-metallic phase in as -deposited full Heusler alloys \nthin films and theory has struggled to establish a fundamentals understanding of the \ntemperature dependent Gilbert damping in these systems. Here we present a study of the \ntemperature dependent Gilbert damping of differently ordered as -deposited Co 2FeAl full \nHeusler alloy thin films. The sum of inter - and intraband electron scattering in conjunction \nwith the finite electron lifetime in Bloch states govern the Gilbert damping for the well -\nordered phase in contrast to the damping of partially -ordered and disordered phases which is \ngoverned by interband electronic scat tering alone. These results, especially the ultralow room \ntemperature intrinsic damping observed for the well -ordered phase provide new fundamental \ninsight s to the physical origin of the Gilbert damping in full Heusler alloy thin films. \n 2 \n INTRODUCTION \nThe Co-based full Heusler alloys have gained massive attention over the last decade due to \ntheir high Curie temperature and half-metallicity; 100% spin polarization of the density of \nstates at the Fermi level [1 -2]. The room temperature half- metallicity and lo w Gilbert \ndamping make them ideal candidates for magnetoresistive and thermoelectric spintronic \ndevices [3]. Co2FeAl (CFA), which is one of the most studied Co-based Heusler alloys , \nbelongs to the 𝐹𝐹𝐹𝐹 3𝐹𝐹 space group, exhibits half-metallicity and a high C urie temperature \n(1000 K) [2, 4]. In CFA, half-metallicity is the result of hybridization between the d orbitals \nof Co and Fe. The d orbitals of Co hybridize resulting in bonding (2e g and 3t 2g) and non-\nbonding hybrids (2e u and 3t 1u). The bonding hybrids of Co further hybridise with the d \norbitals of Fe yielding bonding and anti -bonding hybrids. However, the non-bonding hybrids \nof Co cannot hybridise with the d orbitals of Fe. The half-metallic gap arises from the \nseparation of non-bonding states, i.e. the conduction band of e u hybrids and the valence band \nof t 1u hybrids [5, 6]. However, chemical or atomic disorder modifies the band hybridization \nand results in a reduc ed half-metallicity in CFA. The ordered phase of CFA is the L2 1 phase, \nwhich is half -metallic [7]. The partially ordered B2 phase forms when the Fe and Al atoms \nrandomly share their sites, while the disordered phase forms when Co, Fe, and Al atoms \nrandomly share all the sites [5-8]. These chemical disorders strongly influence the physical \nproperties and result in additional states at the Fermi level therefore reducing the half-\nmetallicity or spin polarization [7, 8]. It is challenging to obtain the ordered L2 1 phase of \nHeusler alloys in as-deposited films, which is expecte d to possess the lowest Gilbert damping \nas compared to the other phases [4, 9-11]. Therefore, in the last decade several attempts have \nbeen made to grow the ordered phase of CFA thin films employing different methods [4, 9-\n13]. The most successful attempts used post -deposition annealing to reduce the anti -site \ndisorder by a thermal activation process [4]. The observed value of the Gilbert damping for \nordered thin films was found to lie in the range of 0.001-0.004 [7-13]. However, the \nrequirement of post -deposition annealing might not be compatible with the process constraints \nof spintronics and CMOS devices. The annealing treatment requirement for the formation of \nthe ordered phase can be circumvented by employing energy enhanced growth mechanisms \nsuch as io n beam sputtering where the sputtered species carry substantially larger energy, ~20 \neV , compared to other deposition techniques [14, 15]. This higher energy of the sputtered \nspecies enhances the ad -atom mobility during coalescence of nuclei in the initial stage of the \nthin film growth, therefore enabling the formation of the ordered phase. Recently we have 3 \n reported growth of the ordered CFA phase on potentially advantageous Si substrate using ion \nbeam sputtering. The samples deposited in the range of 300°C to 500°C substrate temperature \nexhibited nearly equivalent I(002)/I(004) Bragg diffraction intensity peak ratio, which \nconfirms at least B2 order ed phase as it is difficult to identify the formation of the L2 1 phase \nonly by X -ray diffraction analysis [16] . \nDifferent theoretical approaches have been employed to calculate the Gilbert damping in Co -\nbased full Heusler alloys, including first principle calculations on the ba sis of (i) the torque \ncorrelation model [17], (ii) the fully relativistic Korringa -Kohn-Rostoker model in \nconjunction with the coherent potential approximation and the linear response formalism [8] , \nand (iii) an approach considering different exchange correlation effects using both the local \nspin density approximation including the Hubbard U and the local spin density approximation \nplus the dynamical mean field theory approximation [7]. However, very little is known about \nthe temperature dependence of the Gilbert damping in differently ordered Co-based Heusler \nalloys and a unifying conse nsus between theoretical and experimental results is still lacking. \nIn this study we report the growth of differently ordered phases, varying from disordered to \nwell-ordered phases, of as -deposited CFA thin films grown on Si employing ion beam \nsputtering a nd subsequently the detailed temperature dependent measurements of the Gilbert \ndamping. The observed increase in intrinsic Gilbert damping with decreasing temperature in \nthe well -ordered sample is in contrast to the continuous decrease in intrinsic Gilbert damping \nwith decreasing temperature observed for partially ordered and disordered phases. These \nresults are satisfactorily explained by employing spin polarized relativistic Korringa -Kohn-\nRostoker band structure calculations in combination with the local spin density \napproximation. \nSAMPLES & METHODS \nThin films of CFA were deposited on Si substrates at various growth temperatures using ion \nbeam sputtering system operating at 75W RF ion-source power ( 𝑃𝑃𝑖𝑖𝑖𝑖𝑖𝑖). Details of the \ndeposition process as well as st ructural and magnetic properties of the films have been \nreported elsewhere [16]. In the present work to study the temperature dependent Gilbert \ndamping of differently ordered phases (L2 1 and B2) we have chosen CFA thin films deposited \nat 573K, 673K and 773K substrate temperature ( 𝑇𝑇𝑆𝑆) and the corresponding samples are named \nas LP573K, LP673K and LP773K, respectively. The sample thickness was kept constant at 50 \nnm and the samples were capped with a 4 nm thick Al layer. The capping layer protects the \nfilms by forming a 1.5 nm thin protective layer of Al 2O3. To obtain the A2 disordered CFA 4 \n phase, the thin film was deposited at 300K on Si employing 100W ion-source power, this \nsample is referred to as HP300K. Structural and magnetic properties of this film are presented \nin Ref. [18]. The absence of the (200) diffraction peak in the HP300K sample [18] reveals that \nthis sample exhibits the A2 disordered structure. The appearance of the (200) pea k in the LP \nseries samples clearly indicates at least formation of B2 order [16]. Employing the Webster model along with the analysis approach developed by Takakura et al. [19] we have calculated \nthe degree of B2 ordering in the samples, S\nB2= �I200 I220⁄\nI200full orderI220full order�� , where \nI200 I220⁄ is the experimentally obtained intensity ratio of the (200) and (220) diffractions and \nI200full orderI220full order⁄ is the theoretically calculated intensity ratio for fully ordered B2 structure \nin polycrystalline films [20]. The estimated values of SB2 for the LP573, LP673, and LP773 \nsamples are found to be ∼ 90 %, 90% and 100%, respectively , as presented in Ref. [20]. The \nI200 I400⁄ ratio of the (200) and (400) diffraction peaks for all LP series samples is ∼ 30 %, \nwhich compares well with the theoretical value for perfect B2 order [21, 22]. Here it is \nimportant to note that the L21 ordering parameter, SL21, will take different values depending \non the degree of B2 ordering. S L21 can be calculated from the I111 I220⁄ peak ratio in \nconjunction with the SB2 ordering parameter [19]. However, in the recorded grazing incident \nXRD spectra on the polycrystalline LP samples (see Fig. 1 of Ref. [16]) we did not observe \nthe (111) peak. This could be attributed to the fact t hat theoretical intensity of this peak is only \naround two percent of the (220) principal peak. The appearance of this peak is typically \nobserved in textured/columnar thicker films [19, 23 ]. Therefore, here using the experimental \nresults of the Gilbert damping, Curie temperature and saturation magnetization, in particular employing the temperature dependence of the Gilbert damping that is very sensitive to the \namount of site disorder in CFA films, and comparing with corresponding results obtained \nfrom first principle calculations, we provide a novel method for determining the type of \ncrystallographic ordering in full Heusler alloy thin films. \nThe observed values of the saturation magnetization ( µ0MS) and coercivity ( µ0Hci), taken \nfrom Refs. [16, 18] are presented in Table I. The temperature dependence of the magnetization \nwas recorded in the high temperature region (300–1000K) using a vibrating sample \nmagnetometer i n an external magnetic field of 𝜇𝜇 0𝐻𝐻=20 mT. An ELEXSYS EPR \nspectrometer from Bruker equipped with an X -band resonant cavity was used for angle \ndependent in-plane ferromagnetic resonance (FMR) measurements . For studying the 5 \n temperature dependent spin dynamics in the magnetic thin films, an in-house built out -of-\nplane FMR setup was used. The set up, using a Quantum Design Physical Properties \nMeasurement System covers the temperature range 4 – 350 K and the magnetic field range \n±9T. The system employs an Agilent N5227A PNA network analyser covering the frequency \nrange 1 – 67 GHz and an in-house made coplanar waveguide. The layout of the system is \nshown in Fig. 1. The complex transmission coefficient ( 𝑆𝑆21) was recorded as a function of \nmagnetic field for different frequencies in the range 9-20 GHz and different temperatures in \nthe range 50-300 K. All FMR measurements were recorded keeping constant 5 dB power. \nTo calculate the Gilbert damping, we have the used the torque –torque correlation model [7, \n24], which includes both intra - and interband transitions. The electronic structure was \nobtained from the spin polarized relativistic Korringa -Kohn-Rostoker (SPR- KKR) band \nstructure method [24, 25] and the local spin density approximation (LSDA) [26] was used for \nthe exchange correlation potential. Relativistic effects were taken into account by solving the Dirac equation for the electronic states, and the atomic sphere approximation (ASA) was employed for the shape of potentials. The experimental bulk value of the lattice constant [27] \nwas used. The angular momentum cut -off of 𝑙𝑙\n𝑚𝑚𝑚𝑚𝑚𝑚 =4 was used in the mu ltiple -scattering \nexpansion. A k-point grid consisting of ~1600 points in the irreducible Brillouin zone was \nemploye d in the self -consistent calculation while a substantially more dense grid of ~60000 \npoints was employe d for the Gilbert damping calculation. The exchange parameters 𝐽𝐽 𝑖𝑖𝑖𝑖 \nbetween the atomic magnetic moments were calculated using the magnetic force theorem \nimplemented in the Liechtenstein -Katsnelson -Antropov-Gubanov (LKAG) formalism [28, 29] \nin order to construct a parametrized mod el Hamiltonian. For the B2 and L2 1 structures, the \ndominating exchange interactions were found to be between the Co and Fe atoms, while in A2 the Co-Fe and Fe -Fe interactions are of similar size. Finite temperature properties such as the \ntemperature dependent magnetization was obtained by performing Metropolis Monte Carlo \n(MC) simulations [30] as implemented in the UppASD software [31, 32] using the \nparametrized Hamiltoni an. The coherent potential approximation (CPA) [33, 34] was ap plied \nnot only for the treatment of the chemical disorder of the system, but also used to include the \neffects of quasi -static lattice displacement and spin fluctuations in the calculation of the \ntemperature dependent Gilbert damping [35–37] on the basis of linear response theory [38]. \nRESULTS & DISCUSSION \nA. Magnetization vs. temperature measurements 6 \n Magnetization measurements were performed with the ambition to extract values for the \nCurie temperature ( 𝑇𝑇𝐶𝐶) of CFA films with different degree of atomic order; the results a re \nshown in Fig. 2(a). Defining 𝑇𝑇𝐶𝐶 as the inflection point in the magnetization vs. temperature \ncurve, the observed values are found to be 810 K, 890 K and 900 K for the LP573K, LP773K \nand LP673K samples, respectively. The 𝑇𝑇𝐶𝐶 value for the HP300K sample is similar to the \nvalue obtained for LP573. Using the theoretically calculated exchange interactions, 𝑇𝑇𝐶𝐶 for \ndifferent degree of atomic order in CFA varying from B2 to L2 1 can be calculated using MC \nsimul ations. The volume was kept fixed as the degree of order varied between B2 and L2 1 \nand the data presented here represent the effects of differently ordered CFA phases. To obtain \n𝑇𝑇𝐶𝐶 for the different phases, the occupancy of Fe atoms on the Heusler alloy 4a sites was varied \nfrom 50% to 100%, corresponding to changing the structure from B2 to L2 1. The estimated \n𝑇𝑇𝐶𝐶 values , cf. Fig. 2 (b), monotonously increases from 𝑇𝑇 𝐶𝐶=810 K (B2) to 𝑇𝑇𝐶𝐶=950 K \n(L2 1). A direct comparison between experimental and calculated 𝑇𝑇𝐶𝐶 values is hampered by the \nhigh temperature (beyond 800K) induced structural transition from well -ordered to partially -\nordered CFA phase which interferes with the magnetic transition [39, 40]. The irreversible \nnature of the recorded magnetization vs . temperature curve indicates a distortion of structure \nfor the ordered phase during measurement , even though interface alloying at elevated \ntemperature cannot be ruled out . The experimentally observe d 𝑇𝑇𝐶𝐶 values are presented in \nTable I. \nB. In-plane angle dependent FMR measurements \nIn-plane angle dependent FMR measurements were performed at 9.8 GHz frequency for all \nsamples; the resonance field 𝐻𝐻𝑟𝑟 vs. in -plane angle 𝜙𝜙𝐻𝐻 of the applied magnetic field is plotted \nin Fig. 3. The experimental results have been fitted using the expression [41], \n𝑓𝑓=\n𝑔𝑔∥𝜇𝜇𝐵𝐵𝜇𝜇0\nℎ��𝐻𝐻𝑟𝑟cos(𝜙𝜙𝐻𝐻−𝜙𝜙𝑀𝑀)+2𝐾𝐾𝑐𝑐\n𝜇𝜇0𝑀𝑀𝑠𝑠cos4(𝜙𝜙𝑀𝑀−𝜙𝜙𝑐𝑐)+2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠cos2(𝜙𝜙𝑀𝑀−𝜙𝜙𝑢𝑢)��𝐻𝐻𝑟𝑟cos(𝜙𝜙𝐻𝐻−\n𝜙𝜙𝑀𝑀)+ 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒+𝐾𝐾𝑐𝑐\n2𝜇𝜇0𝑀𝑀𝑠𝑠(3+cos4(𝜙𝜙𝑀𝑀−𝜙𝜙𝑐𝑐)+2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠cos2 (𝜙𝜙𝑀𝑀−𝜙𝜙𝑢𝑢)��12�\n, (1) \nwhere 𝑓𝑓 is resonance frequency , 𝜇𝜇𝐵𝐵 is the Bohr magneton and ℎ is Planck constant . 𝜙𝜙𝑀𝑀, 𝜙𝜙𝑢𝑢 \nand 𝜙𝜙𝑐𝑐 are the in -plane directions of the magnetization, uniaxial anisotropy and cubic \nanisotropy, respectively , with respect to the [100] direction of the Si substrate . 𝐻𝐻𝑢𝑢=2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠 and \n𝐻𝐻𝑐𝑐=2𝐾𝐾𝑐𝑐\n𝜇𝜇0𝑀𝑀𝑠𝑠 are the in-plane uniaxial and cubic anisotropy fields , respectively, and 𝐾𝐾𝑢𝑢 and 𝐾𝐾𝑐𝑐 7 \n are the uniaxial and cubic magnetic anisotrop y constant s, respectively, 𝑀𝑀𝑠𝑠 is the saturation \nmagnetization and 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 is the effective magnetization . By considering 𝜙𝜙 𝐻𝐻 ∼ 𝜙𝜙𝑀𝑀, 𝐻𝐻𝑢𝑢 and 𝐻𝐻𝑐𝑐 \n<<𝐻𝐻𝑟𝑟<< 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒, equation (1) can be simplified as: \n𝐻𝐻𝑟𝑟=�ℎ𝑒𝑒\n𝜇𝜇0𝑔𝑔∥𝜇𝜇𝐵𝐵�21\n𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒−2𝐾𝐾𝑐𝑐\n𝜇𝜇0𝑀𝑀𝑠𝑠cos4(𝜙𝜙𝐻𝐻−𝜙𝜙𝑐𝑐)−2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠cos2(𝜙𝜙𝐻𝐻−𝜙𝜙𝑢𝑢) . (2) \nThe extracted cubic anisotropy fields µ 0Hc ≤ 0.22mT are negligible for all the samples. The \nextracted in -plane Landé splitting factors g∥ and the uniaxial anisotropy fields µ0Hu are \npresented in T able I. The purpose of the angle dependent FMR measurements was only to \ninvestigate the symmetry of the in -plane magnetic anisotropy. Therefore, care was not taken \nto have the same in -plane orientation of the samples during angle dependent FMR \nmeasurements, which explains why the maxima appear at diffe rent angles for the different \nsamples. \nC. Out-of-plane FMR measurements \nField -sweep out -of-plane FMR measurements were performed at different constant \ntemperatures in the range 50K – 300K and at different constant frequencies in the range of 9-\n20 GHz. Figure 1(b) shows the amplitude of the complex transmission coefficient 𝑆𝑆21(10 \nGHz) vs. field measured for the LP673K thin film at different temperatures. The recorded \nFMR spectra were fitted using the equation [42], \n𝑆𝑆21=𝑆𝑆�∆𝐻𝐻2��2\n(𝐻𝐻−𝐻𝐻𝑟𝑟)2+�∆𝐻𝐻2��2+𝐴𝐴�∆𝐻𝐻2��(𝐻𝐻−𝐻𝐻𝑟𝑟)\n(𝐻𝐻−𝐻𝐻𝑟𝑟)2+�∆𝐻𝐻2��2+𝐷𝐷∙𝑡𝑡, (3) \nwhere 𝑆𝑆 represents the coefficient describing the transmitted microwave power, 𝐴𝐴 is used to \ndescribe a waveguide induced phase shift contribution which is, however, minute , 𝐻𝐻 is \napplied magnetic field, ∆𝐻𝐻 is the full-width of half maxim um, and 𝐷𝐷∙𝑡𝑡 describes the linear \ndrift in time (𝑡𝑡) of the recorded signal. The extracted ∆ 𝐻𝐻 vs. frequency at different constant \ntemperatures are shown in Fig. 4 for all the samples. For brevity only data at a few \ntemperatures are plotted. The Gilbert damping was estimated using the equation [42 ], \n∆𝐻𝐻=∆𝐻𝐻0+2ℎ𝛼𝛼𝑒𝑒\n𝑔𝑔⊥𝜇𝜇𝐵𝐵𝜇𝜇0 (4) \nwhere ∆𝐻𝐻0 is the inhomogeneous line -width broadening, 𝛼𝛼 is the experimental Gilbert \ndamping constant , and 𝑔𝑔⊥ is the Landé splitting factor measured employing out -of-plane \nFMR. The insets in the figures show the temperature dependence of 𝛼𝛼. The effective 8 \n magnetization ( 𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒) was estimated from the 𝑓𝑓 vs. 𝐻𝐻𝑟𝑟 curves using out -of-plane Kittel’s \nequation [43], \n𝑓𝑓=𝑔𝑔⊥𝜇𝜇0𝜇𝜇𝐵𝐵\nℎ�𝐻𝐻𝑟𝑟−𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒�, (5) \nas shown in Fig. 5 . The temperature dependence of 𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 and 𝜇𝜇0∆𝐻𝐻0 are shown as insets in \neach figure . The observed room temperature values of 𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 are closely equal to the 𝜇𝜇0𝑀𝑀𝑠𝑠 \nvalues obtained from static magnetizat ion measurements, presented in T able I. The extracted \nvalues of g⊥ at different temperatures are within error limits constant for all samples. \nHowever , the difference between estimated values of g ∥ and g⊥ is ≤ 3%. This difference c ould \nstem from the limited frequency range used since these values are quite sensitive to the value \nof 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒, and even a minute uncertainty in this quantity can result in the observed small \ndifference between the g∥ and g⊥ values. \nTo obtain the intrinsic Gilbert damping (𝛼𝛼 𝑖𝑖𝑖𝑖𝑖𝑖) all extrinsic contributions to the experimental 𝛼𝛼 \nvalue need to be subtracted. In metallic ferromagnets , the intrinsic Gilbert damping is mostly \ncaused by electron magnon scattering, but several other extrinsic co ntributions can also \ncontribute to the experimental value of the damping constant. One contribution is two -\nmagnon scattering which is however minimized for the perpendicular geometry used in this \nstudy and therefore this contribution is disregarded [44]. Another contribution is spin-\npumping into the capping layer as the LP573K, LP673K and LP773K samples are capped \nwith 4 nm of Al that naturally forms a thin top layer consisting of Al2O3. Since spin pumping \nin low spin-orbit coupling materials with thickness less than the spin-diffusion length is quite \nsmall this contribution is also disregarded in all samples. However, the HP300K sample is \ncapped with Ta and therefore a spin-pumping contribution have been subtracted from the \nexperimental 𝛼𝛼 value ; 𝛼𝛼𝑠𝑠𝑠𝑠= 𝛼𝛼𝐻𝐻𝐻𝐻300𝐾𝐾(with Ta capping )−𝛼𝛼𝐻𝐻𝐻𝐻300𝐾𝐾(without capping )≈\n1×10−3. The third contribution arises from the inductive coupling between the precessi ng \nmagnetization and the CPW , a reciprocal phenomenon of FMR, known as radiative damping \n𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 [45]. This damping is directly proportional to the magnetization and thickness of the thin \nfilms samples and therefore usually dominates in thicker and/or high magnetization samples. \nThe l ast contribution is eddy current damping ( αeddy) caused by eddy current s in metallic \nferromagnetic thin films [ 45, 46]. As per Faraday’s law the time varying magnetic flux density \ngenerate s an AC voltage in the metallic ferromagnetic layer and therefore result s in the eddy 9 \n current damping . Thi s damping is directly proportional to the square of the film thickness and \ninversely proportional to the resistivity of the sample [ 45]. \nIn contrast to eddy -current damping, 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 is independent of the conductivity of the \nferromagnetic layer, hence this damping mechanism is also operati ve in ferromagnetic \ninsulators. Assuming a uniform magnetization of the sample the radiative damping can be \nexpressed as [45], \n𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟= 𝜂𝜂𝜂𝜂𝜇𝜇02𝑀𝑀𝑆𝑆𝛿𝛿𝛿𝛿\n2 𝑍𝑍0𝑤𝑤 , (6) \nwhere 𝛾𝛾=𝑔𝑔𝜇𝜇𝐵𝐵ℏ� is the gyromagnetic ratio, 𝑍𝑍0 = 50 Ω is the waveguide impedance, 𝑤𝑤 = 240 \nµm is the width of the waveguide, 𝜂𝜂 is a dimensionless parameter which accounts for the \nFMR mode profile and depends on boundary conditions, and 𝛿𝛿 and 𝑙𝑙 are the thickness and \nlength of the sample on the waveguide, respectively. The strength of this inductive coupling \ndepends on the inductance of the FMR mode which is determined by the waveguide width, \nsample length over waveguide, sample saturation magnetization and sample thickness. The \ndimensions of the LP573K, LP673K and LP773K samples were 6.3×6.3 mm2, while the \ndimensions of the HP300K sample were 4×4 mm2. Th e 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 damping was estimated \nexperimentally as explained by Schoen et al. [45] by placing a 200 µm thick glass spacer \nbetween the waveguide and the sample , which decreases the radiative damping by more than \none order magnitude as shown in Fig. 6(a). The measured radiative damping by placing the \nspacer between the waveguide and the LP773 sample, \n𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟=𝛼𝛼𝑤𝑤𝑖𝑖𝑖𝑖ℎ𝑖𝑖𝑢𝑢𝑖𝑖 𝑠𝑠𝑠𝑠𝑚𝑚𝑐𝑐𝑒𝑒𝑟𝑟 −𝛼𝛼𝑤𝑤𝑖𝑖𝑖𝑖ℎ 𝑠𝑠𝑠𝑠𝑚𝑚𝑐𝑐𝑒𝑒𝑟𝑟≈ (2.36 ±0.10×10−3) − (1.57 ±0.20×10−3)=\n0.79±0.22×10−3. The estimated value matches well with the calculated value using Eq. \n(6); 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟= 0.78 ×10−3. Our results are also analogous to previously reported results on \nradiative damping [45]. The estimated temperature dependent radiative damping values for all \nsamples are shown in Fig. 6(b). \nSpin wave precession in ferromagnetic layers induces an AC current in the conducting \nferromagnetic layer which results in eddy current damping. It can be expressed as [45, 46], \n𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 = 𝐶𝐶𝜂𝜂𝜇𝜇02𝑀𝑀𝑆𝑆𝛿𝛿2\n16 𝜌𝜌 , (7) \nwhere 𝜌𝜌 is the resistivity of the sample and 𝐶𝐶 accounts for the eddy current distribution in the \nsample ; the smaller the value of 𝐶𝐶 the larger is the localization of eddy currents in the sample. \nThe measured resistivity values between 300 K to 50 K temperature range fall in the ranges \n1.175 – 1.145 µΩ-m, 1 .055 – 1.034 µΩ -m, 1 .035 – 1.00 µΩ -m, and 1. 45 – 1.41 µΩ -m for the \nLP573K, LP673 , LP773 and HP300K samples, respect ively. The parameter 𝐶𝐶 was obtained 10 \n from thickness dependent experimental Gilbert damping constants measured for B2 ordered \nfilms, by line ar fitting of 𝛼𝛼−𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟≈𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 vs. 𝛿𝛿2 keeping other parameters constant (cf. Fig. \n6(c)). The fit to the data yield ed 𝐶𝐶 ≈ 0.5±0.1. These results are concurrent to those \nobtained for permalloy thin films [45]. Since the variations of the resistivity and \nmagnetization for the samples are small , we have used the same 𝐶𝐶 value for the estimation of \nthe eddy current damping in all the samples. The estimated temperature dependent values of \nthe eddy current damping are presented in Fig. 6(d). \nAll these contributions have been subtracted from the experimentally observed values of 𝛼𝛼. \nThe estimated intrinsic Gilbert damping 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 values so obtained are plotted in Fig. 7(a) for all \nsamples. \nD. Theoretical results: first principle calculations \nThe calculated temperature dependent intrinsic Gilbert damping for Co 2FeAl phases with \ndifferent degree of atomic order are shown in Fig. 7(b). The temperature dependent Gilbert \ndamping indicates that the lattice displacements and spin fluctuations contribute differently in \nthe A2, B2 and L2 1 phases. The torque correlation model [47, 48] describes qualitatively two \ncontributions to the Gilbert damping. The first one is the intraband scattering where the band \nindex is always conserved. Since it has a linear dependence on the electron lifetime, in the \nlow temperature regime this term increases rapid ly, it is also known as the conductivity like \nscattering. The second mechanism is due to interband transitions where the scattering occurs \nbetween bands with different indices. Opposite to the intraband scattering, the resistivity like \ninterband scattering with an inverse depe ndence on the electron lifetime increases with \nincreas ing temperature. The sum of the intra - and interband electron scattering contributions \ngives rise to a non-monotonic dependence of the Gilbert damping on temperature for the L2 1 \nstructure. In contrast to the case for L2 1, only interband scattering is present in the A2 and B2 \nphases, which results in a monotonic increase of the intrinsic Gi lbert damping with increas ing \ntemperature. This fact is also supported by a previous study [37 ] which showed that even a \nminute chemical disorder can inhibit the intraband scattering of the system. Our theoretical results manifest that the L2\n1 phase has the lowest Gilbert damping around 4.6 × 10−4 at 300 \nK, and that the value for the B2 phase is only slightly larger at room temperature. According \nto the torque correlation model, the two main contributions to damping are the spin orbit \ncoupling and the density of states (DOS) at the Fermi level [47 , 48]. Since the spin orbit \nstrength is the same for the different phases it is enough to focus the discussion on the DOS 11 \n that provide s a qualitative explanation why damping is found lower in B2 and L2 1 structure s \ncompare d to A2 structure. The DOS at the Fermi level of the B2 phase (24.1 states/Ry/f.u; f.u \n= formula unit ) is only slightly larger to that of the L2 1 phase (20.2 states/Ry/f.u.) , but both \nare significantly smaller than for the A2 phase (59.6 states/Ry/f.u.) as shown in Fig. 8. The \ngap in the mi nority spin channel of the DOS for the B2 and L2 1 phases indicate half-\nmeta llicity, while the A2 phase is metallic. The atomically resolved spin polarized DOS \nindica tes that the Fermi -level states mostly have contr ibutions from Co and Fe atoms. For \ntransition elements such as Fe and Ni, it has been reported that the intrinsic Gilbert damping \nincreases significantly below 100K with decreas ing temperature [37]. The present electronic \nstructure calculations were performed using Green’s functions, which do rely on a \nphenomenological relaxation time parameter, on the expense that the different contributions to \ndamping cannot be separated eas ily. The reported results in Ref. [37] are by some means \nsimilar to our findings of the temperature dependent Gilbert damping in full Heusler alloy films with different degr ee of atomic order. The intermediate states of B2 and L2\n1 are more \nclose to the trend of B2 than L2 1, which indicates that even a tiny atomic orde r induced by the \nFe and Al site disorder will inhibit the conductivity -like channel in the low temperature \nregion. The theoretically calculated Gilbert damping constants are matching qualitatively with \nthe experimentally observed 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 values as shown in Fig. 7. However, the theoretically \ncalculated 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 for the L2 1 phase increases rapidly below 100K, in co ntrast to the \nexperimental results for the well -ordered CFA thin film (LP673K ) indicating that \n𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 saturates at low temperature. This discrepancy between the theoretical and experimental \nresults can be understood taking into account the low temperature behaviour of the life time τ \nof Bloch states. The present theoretical model assum ed that the Gilbert damping has a linear \ndependence on the electron lifetime in intraband transitions which is however correct only in \nthe limit of small lifetime, i.e., 𝑞𝑞𝑣𝑣𝐹𝐹𝜏𝜏≪1, where q is the magnon wave vector and 𝑣𝑣𝐹𝐹 is the \nelectron Fermi velocity. However, in the low temperature limit the lifetime 𝜏𝜏 increases and as \na result of the anomalous skin effect the intrinsic Gilbert damping saturates \n𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖∝ tan−1𝑞𝑞𝑣𝑣𝐹𝐹𝜏𝜏𝑞𝑞𝑣𝑣𝐹𝐹� at low temperature [37], which is evident from our experimental \nresults. \nRemaining discrepancies between theoretical and experimental values of the intrinsic Gilbert \ndamping might stem from the fact that the samples used in the present study are 12 \n polycrystalline and because of sample imperfections these fil ms exhibit significant \ninhomogenous line -width broadening due to superposition of local resonance fields. \nCONCLUSION \nIn summary , we report temperature dependent FMR measurements on as -deposited Co 2FeAl \nthin films with different degree of atomic order. The degree of atomic ordering is established \nby comparing experimental and theoretical results for the temperature dependent intrinsic \nGilbert damping constant. It is evidenced that the experimentally observed intrinsic Gilbert \ndamping in samples with atomic disorder (A2 and B2 phase samples) decreases with \ndecreasing temperature. In contrast, the atomically well -ordered sample, which we identify at \nleast partial L21 phase, exhibits an intrinsic Gilbert damping constant that increases with \ndecreasing temperat ure. These temperature dependent results are explained employing the \ntorque correction model including interband transitions and both interband as well as \nintraband transitions for samples with atomic disorder and atomically ordered phases, \nrespectively. \nACKNOWLEDGEMENT \nThis work is supported by the Knut and Alice Wallenberg (KAW) Foundation, Grant No. \nKAW 2012.0031 and from Göran Gustafssons Foundation (GGS), Grant No. GGS1403A. The \ncomputations were performed on resources provided by SNIC (Swedish National \nInfrastructure for Computing) at NSC (National Supercomputer Centre) in Linköping, \nSweden. S. 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Mills, Phys. Rev. B 77, 214405 (2008). \n45. M. A. W. Schoen, J. M. Shaw, H. T. Nembach, M. Weiler, and T. J. Silva, Phys. Rev. \nB 92, 184417 (2015). \n46. Y . Li and W. E. Bailey, Phys. Rev. Lett. 116 , 117602 (2016). \n47. V. Kambersk´y, Czech. J. Phys. B 26, 1366 (1976). \n48. K. Gilmore, Y . U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99 , 027204 (2007) . \n 15 \n Table I P arameters describing magnetic properties of the different CFA samples. \nSample 𝜇𝜇0𝑀𝑀𝑆𝑆(𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒) \n(T) 𝜇𝜇0𝐻𝐻𝑐𝑐𝑖𝑖 \n(mT) 𝑔𝑔∥(𝑔𝑔⊥) \n 𝜇𝜇0𝐻𝐻𝑢𝑢 \n(mT) 𝑇𝑇𝐶𝐶 \n(K) 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 \n(× 10-3) \nLP573K 1.2±0.1 (1.09 1±0.003) 0.75 2.06 (2.0) 1.56 810 2.56 \nLP673K 1.2±0.1 (1.11 0±0.002) 0.57 2.05 (2.0) 1.97 >900 0.76 \nLP773K 1.2±0.1 (1.081 ±0.002) 0.46 2.05 (2.0) 1.78 890 1.46 \nHP300K 0.9±0.1 (1.066 ±0.002) 1.32 2.01 (2.0) 3.12 -- 3.22 \n 16 \n Figure 1 \nFig. 1. (a) Layout of the in -house made VNA-based out -of-plane ferromagnetic resonance \nsetup . (b) Out -plane ferromagnetic resonance spectra recorded for the well -ordered LP673K \nsample at different temperatures 𝑓𝑓=10 GHz . \n \n \n17 \n Figure 2 \nFig. 2. (a) Magnetization vs. temperature plots measured on the CFA films with different \ndegree of atomic order. (b) Theoretically calculated magnetization vs. temperature curves for \nCFA phases with different degree of atomic order, where 50 % (100 %) Fe atoms on Heusler \nalloy 4a sites indicate B2 (L2 1) ordered phase, and the rest are intermediate B2 & L2 1 mixed \nordered phases. \n \n \n \n18 \n Figure 3 \nFig. 3. Resonance field vs. in -plane orientation of the applied magnetic field of (a) 𝑇𝑇𝑆𝑆=\n300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆=400℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇 𝑆𝑆=500℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇 𝑆𝑆=27℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100 𝑊𝑊 deposited films. Red lines \ncorrespond to fits to the data using Eq. (1). \n \n \n \n19 \n \nFigure 4 \nFig. 4. Line-width vs. frequency of (a) 𝑇𝑇𝑆𝑆=300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆=400℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇 𝑆𝑆=500℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇𝑆𝑆=27℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100 𝑊𝑊 deposited samples. Red lines correspond to fits to the data to extract the \nexperimental Gilbert damping constant and inhomogeneous line -width. Respective insets \nshow the experimentally determined temperature dependent Gilbert damping constants. \n \n \n \n \n20 \n Figure 5 \nFig. 5. Frequency vs. applied field of (a) 𝑇𝑇𝑆𝑆=300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆=\n400℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇𝑆𝑆=500℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇𝑆𝑆=27℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100 𝑊𝑊 deposited samples. Red lines correspond to Kittel’s fits to the data. Respective \ninsets show the temperature dependent effective magnetization a nd inhomogeneous line -width \nbroadening values. \n \n \n \n21 \n Figure 6 \nFig. 6. (a) Linewidth vs. frequency with and without a glass spacer between the waveguide \nand the sample. Red lines correspond to fits using Eq. (4). (b) Temperature dependent values \nof the radiative damping using Eq. (6). The lines are guide to the eye. (c) 𝛼𝛼−𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟≈𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 \nvs 𝛿𝛿2. The red line corresponds to a fit using Eq. (7) to extract the value of the correction \nfactor 𝐶𝐶. (d) Temperature dependent values of eddy current dampi ng using Eq. (7). The lines \nare guide to the eye. \n \n \n \n \n22 \n Figure 7 \nFig. 7. Experimental (a) and theoretical (b) results for the temperature dependent intrinsic \nGilbert damping constant for CFA samples with different degree of atomic order . The B2 & \nL21 mixed phase corresponds to the 75 % occupancy of Fe atoms on the Heusler alloy 4a \nsites. The lines are guide to the eye. \n \n \n \n23 \n Figure 8 \nFig. 8. Total and atom -resolved spin polarized density of states plots for various \ncompositional CFA phases; (a) A2, (b) B2 and (c) L2 1. \n \n \n \n" }, { "title": "2004.01237v2.Simulating_the_effect_of_weak_measurements_by_a_phase_damping_channel_and_determining_different_measures_of_bipartite_correlations_in_nuclear_magnetic_resonance.pdf", "content": "Simulating the e\u000bect of weak measurements by a phase damping channel and\ndetermining di\u000berent measures of bipartite correlations in nuclear magnetic\nresonance\nAkanksha Gautama, Varad R. Pandeb, Amandeep Singha,c, Kavita Doraia, Arvinda,1,\u0003\naDepartment of Physical Sciences, Indian Institute of Science Education & Research Mohali, Sector 81 SAS Nagar, Manauli PO 140306\nPunjab India.\nbDepartment of Physics, University of Maryland, Baltimore County (UMBC), Baltimore, Maryland 21250, USA.\ncShenzhen Institute for Quantum Science and Engineering, and Department of Physics, Southern University of Science and Technology,\nShenzhen 518055, China\nAbstract\nQuantum discord is a measure based on local projective measurements which captures quantum correlations that may\nnot be fully captured by entanglement. A change in the measurement process, achieved by replacing rank-one projectors\nwith a weak positive operator-valued measure (POVM), allows one to de\fne weak variants of quantum discord. In this\nwork, we experimentally simulate the e\u000bect of a weak POVM on a nuclear magnetic resonance quantum information\nprocessor. The two-qubit system under investigation is part of a three-qubit system, where one of the qubits is used as\nan ancillary to implement the phase damping channel. The strength of the weak POVM is controlled by varying the\nstrength of the phase damping channel. We experimentally observed two weak variants of quantum discord namely, super\nquantum discord and weak quantum discord, in two-qubit Werner and Bell-diagonal states. The resultant dynamics of\nthe states is investigated as a function of the measurement strength.\nKeywords: NMR quantum information processing, Quantum discord, Weak measurement, Weak positive\noperator-valued measure (POVM), Phase damping channel, Super quantum discord, Weak quantum discord\nPACS: 03.65.Ud, 03.67.Mn\n1. Introduction\nQuantum correlations play an important role in quan-\ntum communication and quantum information processing\n[1]. While quantum entanglement was discovered early on\nby Schr odinger [2], Ollivier and Zurek [3] and Henderson\nand Vedral [4], independently pointed out that a di\u000berent\ntype of quantum correlations can exist in bipartite sys-\ntems. The measure to quantify such correlations is termed\nas quantum discord (QD) [3, 4]. The evaluation of QD is\ncomputationally a hard task as it involves numerical opti-\nmization and hence, alternative measures have been pro-\nposed such as geometric quantum discord [5], Gaussian\ngeometric discord in continuous variable systems [6] and\nrelative entropy of discord [7]. The presence of nonclassical\ncorrelations in bipartite states was measured experimen-\ntally on a nuclear magnetic resonance (NMR) setup [8],\nwitnessed via a single-shot experiment [9] and its preser-\nvation has also been explored using NMR [10]. Several\n\u0003Corresponding Author\nEmail addresses: akankshagautam@iisermohali.ac.in\n(Akanksha Gautam), varadrpande@gmail.com (Varad R. Pande),\nsingh@sustech.edu.cn (Amandeep Singh),\nkavita@iisermohali.ac.in (Kavita Dorai),\narvind@iisermohali.ac.in (Arvind)\n1Tel & Fax:+91-172-2240266recent studies have explored the advantage of nonclassical\ncorrelations in quantum information processing even if the\nstate has almost null entanglement [11, 12, 13].\nIn classical information theory, the mutual information\nbetween two random variables has two equivalent expres-\nsions which can be computed from the respective Shan-\nnon entropies. In contrast, for quantum mutual informa-\ntion, these two de\fnitions give rise to di\u000berent values as\none of the expressions requires the von Neumann entropy\nconditioned by projective measurements. QD is de\fned\nas the di\u000berence between total mutual information and\nclassical mutual information, wherein classical mutual in-\nformation is found by determining the possible informa-\ntion gain about one subsystem while measuring the other\nsubsystem via projective measurements [4]. What if one\nreplaces these projective measurements by weak measure-\nments? The crux of a weak measurement lies in the weak\ninteraction between the system to be measured and the\nmeasuring apparatus. While originally the weak measure-\nments were used in the context of weak values [14], it was\nlater realized that any projective measurement can be ob-\ntained by a sequence of weak measurements [15]. These\nideas have been used in several applications including the\nobservation of spin Hall e\u000bect of light [16], direct mea-\nsurement of single photon wavefunction [17], protection\nPreprint submitted to Physics Letters A August 17, 2020arXiv:2004.01237v2 [quant-ph] 14 Aug 2020of quantum entanglement from decoherence [18], feedback\ncontrol of a quantum system [19], to amplify small trans-\nverse de\rections of an optical beam in a Sagnac interfer-\nometer [20] and for quantum state tomography [21, 22, 23].\nWeak measurements allow us to de\fne two di\u000berent\nvariants of quantum discord, which stem from the fact\nthat QD can be de\fned in two mathematically equivalent\nways [24]. Weak measurements have been utilized in a new\nde\fnition of bipartite correlations through a weak vari-\nant of QD termed super quantum discord (SQD) [25, 26],\nwhich is numerically greater than QD. Since then SQD has\nattracted interest in various contexts in quantum informa-\ntion processing [27, 28, 29, 30]. Recently, the question\nof whether weak measurements can be used to gain more\ninformation about correlations present in a quantum sys-\ntem, was explored by formulating another weak variant of\nQD termed weak quantum discord (WQD), which never\nexceeds QD [31, 32]. While projective measurements re-\nsult in a loss of coherence, the precise relationship between\ndecoherence and measurement needs to be clari\fed. Sim-\nulations of the phase damping (PD) channel have been\nrealized as projective measurements in NMR [33] and us-\ning this method quantum teleportation and the quantum\nZeno e\u000bect have been implemented in an NMR setup [34].\nIn this work we experimentally implement the e\u000bect of\na weak POVM via a phase damping channel, in two-qubit\nstates. The PD channel is a non-unitary operation used\nto model the decoherence process [1]. We have also inves-\ntigated the experimental behavior of two weak variants of\nQD namely, SQD and WQD. The determination of SQD\nand WQD from a bipartite state requires a weak mea-\nsurement on one subsystem instead of a projective mea-\nsurement. The result of a weak measurement on a Bell-\ndiagonal state and on a Werner state while \fnding SQD\nand WQD, a\u000bects the state in the same manner as if a PD\nchannel was acting on it. We mapped the weak measure-\nment to a PD channel and the weak measurement strength\nis controlled by tuning the strength of the PD channel. We\nuse three nuclear spins to encode three qubits, where the\ntwo qubits are used to prepare two-qubit states and the\nthird qubit acts as an ancillary qubit used to simulate a PD\nchannel acting on one of the qubit of two-qubit states [35].\nThis study reports the \frst experimental implementation\nof controlled weak measurements on an NMR hardware.\nWe successfully demonstrated that both SQD and WQD\napproach QD as the measurement strength is increased.\nThis article is organized as follows: Sec. 2.1 describes\nnonclassical correlations as quanti\fed by QD. Sec. 2.2 and\nSec. 2.2.2 describes weak measurement and its application\nin determining the SQD and WQD respectively, in two-\nqubit states. Sec. 3 details the mapping of weak measure-\nment to the PD channel and Sec. 4 contains experimental\nresults of implementing a PD channel to observe SQD and\nWQD. Sec. 5 contains some concluding remarks.2. Quantum Discord and Weak Measurements\n2.1. Quantum Discord\nIt was independently noted by Ollivier and Zurek [3]\nand by Henderson and Vedral [4] that certain types of\nmixed separable states have zero entanglement, yet contain\nnonclassical correlations. Mathematically, QD captures\nthese correlations and is the di\u000berence between the total\ncorrelation I(\u001aAB) and the classical correlation J(\u001aAjB):\nI(\u001aAB) =S(\u001aA) +S(\u001aB)\u0000S(\u001aAB)\nJ(\u001aAjB) =S(\u001aA)\u0000S(\u001aAjB) (1)\nwhereS(\u001aAB) =\u0000Tr(\u001aABlog2\u001aAB) is the von Neumann\nentropy of the quantum state, \u001aABis shared between two\npartiesAandB,\u001aA;B= TrB;A(\u001aAB) is the reduced den-\nsity operator of a subsystem, and S(\u001aAjB) is the condi-\ntional von Neumann entropy of subsystem Awhen sub-\nsystemBhas already been measured. Quantum discord\nis then de\fned as:\nQD(\u001aAB) =I(\u001aAB)\u0000maxJ(\u001aAjB) (2)\nwhere the maxima is taken over all possible projective mea-\nsurements on the subsystem B.\nIt turns out that we can also write quantum discord in\na mathematically equivalent form [24]:\nQD(\u001aAB) =I(\u001aAB)\u0000maxI(\u001a0\nAB) (3)\nwhere\u001a0\nABis the density operator of the combined sys-\ntem after a projective measurement over subsystem Bhas\nbeen carried out. As has been discussed in a lucid manner\nin [32] this alternative mathematical form that was origi-\nnally introduced for de\fning QD for multipartite systems,\nalso leads to an alternative interpretation of QD and is\nuseful in generalizing QD to the weak measurement sce-\nnario.\nFor a two-qubit system, projective measurements on\nthe single-qubit subsystem Bcan be characterized by the\nBloch sphere direction \u0012;\u001e. The corresponding projectors\n\u0005\u0012;\u001e\n1and \u0005\u0012;\u001e\n2can be constructed utilizing two orthogonal\nvectors as follows:\nj i\u0012;\u001e\n1= cos\u0012\n2j0i+ei\u001esin\u0012\n2j1i\nj i\u0012;\u001e\n2=\u0000sin\u0012\n2j0i+ei\u001ecos\u0012\n2j1i\n\u0005\u0012;\u001e\n1=j i\u0012;\u001e\n1h j\u0012;\u001e\n1 and \u0005\u0012;\u001e\n2=j i\u0012;\u001e\n2h j\u0012;\u001e\n2:(4)\nThe state of the subsystem Aafter a projective mea-\nsurement on subsystem Bgives a positive result for the\nprojector \u0005\u0012;\u001e\njand can be written as:\n\u001aAj\u0005\u0012;\u001e\nj=1\npjTrBh\u0010\nI\n\u0005\u0012;\u001e\nj\u0011\n\u001aAB\u0010\nI\n\u0005\u0012;\u001e\nj\u0011i\n(5)\nwherepjis the probability of the measurement outcome\ncorresponding to the projectors \u0005\u0012;\u001e\nj. The state of the\n2combined system after a measurement on the system B\ncan be written as:\n\u001a0\nAB=2X\nj=1h\u0010\nI\n\u0005\u0012;\u001e\nj\u0011\n\u001aAB\u0010\nI\n\u0005\u0012;\u001e\nj\u0011i\n: (6)\nThe conditional von Neumann entropy required in Eq. (1)\nis given by:\nS(\u001aAjB) =2X\nj=1pjS(\u001aAj\u0005\u0012;\u001e\nj): (7)\nQD can be computed by utilizing the above de\fned projec-\ntors in Eq.(7), followed by substituting the parameterized\nconditional entropy into Eq.(1):\nQD(\u001aAB) =I(\u001aAB)\u0000max\nf\u0012;\u001egJ(\u001aAjB) (8a)\n=I(\u001aAB)\u0000max\nf\u0012;\u001egI(\u001a0\nAB): (8b)\nThus one can compute QD in two di\u000berent ways: by max-\nimizing the J(\u001aAjB) or by maximizing I(\u001a0\nAB) (which for\nprojective measurements turn out to be the same) over\n\u00122[0;\u0019] and\u001e2[0;2\u0019], and substituting \u0012and\u001eback\ninto Eqs. (8a) & (8b).\n2.2. Weak Variants of Quantum Discord\nWhat happens if we replace the projective measure-\nment used to compute QD by a weak measurement? Weak\nmeasurements, where the system-apparatus interaction is\nweak, extract limited information from the system and\ncorrespondingly disturb the system in a limited way. Re-\npeated weak measurements lead to a strong or a projec-\ntive measurement. The positive operator valued measure\n(POVM) corresponding to the weak measurement is de-\n\fned through the operators:\nP(x) =r\n1\u0000tanhx\n2\u0005\u0012;\u001e\n 1+r\n1 + tanhx\n2\u0005\u0012;\u001e\n 2\nP(\u0000x) =r\n1 + tanhx\n2\u0005\u0012;\u001e\n 1+r\n1\u0000tanhx\n2\u0005\u0012;\u001e\n 2(9)\nwhere the strength of the weak measurement is parameter-\nized by the real parameter x\u00150. The POVM operators\nsatisfyP(x)yP(x) +P(\u0000x)yP(\u0000x) =I. Forx= 0,P(0)\nreduces to1p\n2Ii. e.no measurement at all and in the case\nofx!1 ,P(x) andP(\u0000x) reduce to the projectors \u0005\u0012;\u001e\n 2\nand \u0005\u0012;\u001e\n 1, respectively (as de\fned in Eq. (4)).\nAfter the weak measurement, the state of the combined\nsystem becomes:\n\u001ax\nAB= (I\nP(x))\u001aAB(I\nP(x)) +\n(I\nP(\u0000x))\u001aAB(I\nP(\u0000x))(10)\nThe post-measurement state of subsystem Afor each out-\ncome can be written as\n\u001aAjP(\u0006x)=1\np(\u0006x)TrB[(I\nP(\u0006x))\u001aAB(I\nP(\u0006x))]:\n(11)Wherep(\u0006x) are the probabilities for P(\u0006x). The ques-\ntion now is: how can we de\fne weak variants of quantum\ndiscord?\n2.2.1. Super quantum discord\nIf we take Eq. (8a) as the basic de\fnition of QD and re-\nplace the projective measurement with a weak POVM (as\nde\fned above), we obtain a straightforward generalization\nof QD. This can be written down by \frst computing the\nconditional entropy and classical information as:\nSx(\u001aAjB) =p(x)Sx(\u001aAjP(x)) +p(\u0000x)Sx(\u001aAjP(\u0000x))\nJx(\u001aAjB) =S(\u001aA)\u0000Sx(\u001aAjB): (12)\nIn terms of the above, a weak variant of quantum dis-\ncord can be de\fned which is called super quantum discord\n(SQD):\nSQD (\u001aAB) =I(\u001aAB)\u0000max\nf\u0012;\u001egJx(\u001aAjB) (13)\nwhich depends on the measurement strength x. The value\nof SQD is always greater than QD because Sx(\u001aAjB) is al-\nways larger than S(\u001aAjB), as the weak measurement is per-\nformed on subsystem Bwhile disturbing the state weakly,\nand reveals less information about the subsystem A. In\nthe limiting case of no measurement performed on system\nBat all,Sx(\u001aAjB)!S(\u001aA) and SQD discord will be equal\nto the total correlations.\n2.2.2. Weak Quantum Discord\nSQD has the feature that it is always larger than QD\nand has hence been found somewhat dissatisfying. A weak\ngeneralization of QD is possible if we take Eq. (8b) to be a\nfundamental de\fnition of QD and replace the second term\nwith its weak equivalent [24]. This process allows us to de-\n\fne another weak variant of quantum discord called weak\nquantum discord (WQD), which for a two-qubit system\ncan be written as:\nWQD (\u001aAB) =I(\u001aAB)\u0000max\nf\u0012;\u001egI(\u001ax\nAB) (14)\nwhere\u001ax\nABis the density operator of the composite system\nafter a measurement of strength xhas been performed on\nthe subsystem Bas given in Eq. (10). WQD as a quanti\fer\nhas the nice property that it is always less than QD and\nin the limit x!0 it approaches 0, while for x! 1\nit approaches QD. In the strong measurement limit, both\nSQD and WQD become the same and are equal to QD. The\nkey observation which plays a role in these two di\u000berent\ngeneralizations of QD in the weak regime, is that the two\ndi\u000berent equivalent expressions for QD do not remain the\nsame in the weak measurement regime. For interesting\nand more detailed interpretations of SQD and WQD, the\nreader is referred to Ref. [32].\n33. Simulating Weak POVM Via a Phase Damping\nChannel\nA projective measurement collapses the state, thereby\nkilling all the o\u000b-diagonal terms (coherences) in the density\nmatrix, in the measurement basis [34]. The weak measure-\nment formalism, as described by Aharonov, Albert and\nVaidmann (AAV) [14] utilizes the weak interaction [36].\nThe weak interaction couples the system weakly with the\nmeasuring device, and therefore the state of the system re-\ntains its coherence partially, even after the measurement.\nLater, Oreshkov and Brun [15] showed that any general-\nized measurement can be modeled by a sequence of weak\nPOVMs and for the two-qubit case are given in Eq.(9).\nWe consider two types of states to investigate the be-\nhavior of the quantities SQD and WQD (de\fned in the\nprevious section) with respect to measurement strength\nand to compare it with QD, namely, the Werner states\nand Bell-diagonal states. The two-qubit Werner states are\nde\fned as:\n\u001aws\nAB=zj \u0000ih \u0000j+1\n4(1\u0000z)I (15)\nwhere 1\u0000zquanti\fes the amount of mixedness, 0 \u0014z\u00141\nandj \u0000i=1p\n2(j01i\u0000j10i). The two-qubit Bell-diagonal\nstates [37] are de\fned as:\n\u001abs\nAB=1\n4\"\nI\nI+3X\ni=1ci(\u001bi\n\u001bi)#\n(16)\nwhere (\u001b1;\u001b2;\u001b3) are the Pauli matrices and \u00001\u0014c1;c2;c3\u0014\n1.\nThe evaluation of SQD and WQD in both the states in-\nvolves an optimization over all possible projectors by vary-\ning\u00122[0;\u0019] and\u001e2[0;2\u0019] as given in Eq.(13). The op-\ntimization gives the highest possible classical correlations\nat\u0012=\u0019and\u001e=\u0019for the Werner states as well for the\nBell-diagonal states. On substituting the optimal values\nof\u0012and\u001einto Eq.(9), the weak POVMs get simpli\fed to:\nP(x) =r\n1\u0000tanhx\n2j0ih0j+r\n1 + tanhx\n2j1ih1j\nP(\u0000x) =r\n1 + tanhx\n2j0ih0j+r\n1\u0000tanhx\n2j1ih1j:\n(17)\nA single-qubit mixed state \u001aon the Bloch sphere can be\nexpressed as:\n\u001a=1\n2(I+rx\u001b1+ry\u001b2+rz\u001b3) (18)\nwhererx,ryandrzare the coordinates of the Bloch vector\nandIis the 2\u00022 identity matrix. The e\u000bect of the sim-\npli\fed weak POVM given in Eq.(17) on the single-qubit\nstate\u001acan be readily computed and in the matrix form iswritten as:\n\u001a=1\n2\u0014\n1 +rzrx\u0000iry\nrx+iry1\u0000rz\u0015\n+\n\u001a0\nwm=1\n2\u00141 +rz (rx\u0000iry) sechx\n(rx+iry) sechx 1\u0000rz:\u0015\n(19)\nIt is clear from the post weak-measurement state \u001a0\nwmthat\nthe o\u000b-diagonal terms are a monotonically decreasing func-\ntion of the measurement strength x, leading to decoher-\nence. The extent to which the weak measurement deco-\nheres the state \u001adepends on the measurement strength\nx.\nWe now turn to the PD channel, which causes loss of\ncoherence and leads to the decay of the o\u000b-diagonal terms\nof the density matrix and can be described by a completely\npositive trace preserving map described through the Kraus\noperators [38]:\n\u001a0\nPD=E0\u001aEy\n0+E1\u001aEy\n1\nE0=1+p1\u0000\u0015\n2I+1\u0000p1\u0000\u0015\n2\u001b3\nE1=p\n\u0015\n2I\u0000p\n\u0015\n2\u001b3 (20)\nwhere the parameter \u00152[0;1] represents the strength of\nthe PD channel. The action of the PD channel on a\ngeneral one-qubit state in the matrix form is given by:\n\u001a=1\n2\u00141 +rzrx\u0000iry\nrx+iry1\u0000rz\u0015\n+\n\u001a0\nPD=1\n2\u00141 +rz (rx\u0000iry)p\n1\u0000\u0015\n(rx+iry)p\n1\u0000\u0015 1\u0000rz:\u0015\nThe e\u000bect of the PD channel is similar to the weak POVM\non a single-qubit state, wherein the o\u000b-diagonal terms are\ndiminished. Since both `sech x' and `p\n1\u0000\u0015' are monoton-\nically decreasing functions, they can be mapped onto each\nother with an appropriate scaling factor. Therefore, the\naction of the PD channel is in one-to-one correspondence\nwith weak POVM described in Eq. (9).\nIn order to implement the PD channel we follow an\nindirect approach [35], wherein non-unitary operators can\nbe thought of as unitary operations on an extended quan-\ntum system built upon the Duality Quantum Comput-\ning (DQC) framework [39]. This framework requires an\nancillary qubit. It has been demonstrated [35] that the\nKraus operator Ekdescribing the non-unitary transfor-\nmation corresponding to the PD channel can be e\u000eciently\nimplemented on a qubit if the unitary operators V,W,\nU0andU1can be found in the two-qubit space, where V\nandWact on the ancilla qubit, and U0andU1act on the\ntarget qubit controlled by the ancilla qubit. The operators\nneed to satisfy:\nEk(k= 0;1) =1X\ni=0WkiVi0Ui (21)\n4j0i\nj0i\nj0i(a)\n(b)R\nRHV\nU1W\n1H\n19F\n13Cxyx x\nz\nx y x x x zx x\n\u001c23 \u001c13Decoupleyxy x y xy x x\nyy yy\nyy\nyx\nzz\nFigure 1: (a) The quantum circuit in the left block creates a Bell-\ndiagonal state and a Werner state ( \u001a0\n\u001a ) on two qubits ( \u001a ),\nwith the third qubit ( \u001a0) acting as ancillary; H denotes a Hadamard\ngate, R is a NOT gate in the case of the Werner state and I2iden-\ntity operation (no operation) in the case of the Bell-diagonal state.\nThe quantum circuit in the right block implements a phase damping\nchannel on one of the qubits of hence prepared two-qubit state using\nan ancillary qubit. (b) NMR pulse sequences corresponding to the\nquantum circuits where the un\flled rectangles denote\u0019\n2radiofre-\nquency (rf) pulses, the \flled rectangles denote \u0019rf pulses and the\nshaded rectangles denote \u0012rf pulses where \u0012=\u00002 sin\u00001q\n1\u0000p1\u0000\u0015\n2,\nand\u0015is the strength of the PD channel, lying between 0 and 1.\nThe phase of the rf pulse is given above each pulse and a bar over\na phase represents negative phase. The free evolution time intervals\n\u001c12and\u001c23are given by 1 =(2J12) and 1=(2J23) respectively, where\nJijrepresents the scalar coupling strength between qubits iandj.\nwhereWkiiskith element of Woperator,Vi0is an element\nof the \frst column of Voperator and Uiis the controlled\noperator. A comparison with the decomposition of Kraus\noperators of the PD channel is given in Eq.(20). The uni-\ntary operators V,W,U0andU1can be evaluated as:\nU0=I; U 1=\u001b3;\nV=W=1\n22\n4q\n1+p1\u0000\u0015\n2q\n1\u0000p1\u0000\u0015\n2 q\n1\u0000p1\u0000\u0015\n2\u0000q\n1+p1\u0000\u0015\n23\n5:(22)\nThe quantum circuit to implement the PD channel is\nshown in Fig. 1(a), where Werner states and Bell-diagonal\nstates (Eq.(15-16)) states are created on qubits 2 and 3,\nqubit 1 acts as an ancilla, and the the PD channel acts on\nqubit 3. The strength of the PD channel is controlled by\ntheVgate. The e\u000bect of the PD channel can be evaluated\nby tracing out the ancillary qubit. Physically this was\nachieved by performing the measurements on the state of\nthe two-qubit subsystem consisting of qubits 2 and 3, while\nignoring the qubit 1, as is shown in Fig. 1.\nWe are now ready to experimentally investigate the\nbehavior of SQD and WQD by varying the measurement\nstrength. It is important to mention here that in an NMR\nset up the measurement is already weak (termed as an\n(a)\n\u0017H= 3331:50Hz\n\u0017F=\u00001110992:88Hz\n\u0017C= 12891:11Hz\nJHF= 47:5Hz\nJHC= 161:6Hz\nJFC=\u0000191:8Hz\nTH\n1= 4:10\u00060:06s\nTF\n1= 6:30\u00060:06s\nTC\n1= 6:74\u00060:17s\nTH\n2= 1:18\u00060:12s\nTF\n2= 2:99\u00060:10s\nTC\n2= 3:37\u00060:19s(b)\n-196.3 -196.4 -196.5 -196.6 -196.7 ppmCurrent Data Parameters\nNAME st1_s2\nEXPNO 3\nPROCNO 1\nF2 - Acquisition Parameters\nDate_ 20190801\nTime 9.21 h\nINSTRUM spect\nPROBHD Z108349_0002 (\nPULPROG zgflqn\nTD 11294\nSOLVENT Acetone\nNS 1\nDS 0\nSWH 5647.590 Hz\nFIDRES 1.000105 Hz\nAQ 0.9998955 sec\nRG 128\nDW 88.533 usec\nDE 6.50 usec\nTE 296.2 K\nD1 1.00000000 sec\nTD0 1\nSFO1 564.6591364 MHz\nNUC1 19F\nP1 23.27 usec\nPLW1 42.27000046 W\nF2 - Processing parameters\nSI 524288\nSF 564.7701319 MHz\nWDW EM\nSSB 0\nLB 0.30 Hz\nGB 0\nPC 1.00\n19F+acetone d6\n84.2 84.4 84.6 84.8 85.0 85.2 85.4 85.6 85.8 86.0 86.2 86.4 86.6 86.8 ppmCurrent Data Parameters\nNAME st1_s2\nEXPNO 5\nPROCNO 1\nF2 - Acquisition Parameters\nDate_ 20190801\nTime 9.27 h\nINSTRUM spect\nPROBHD Z108349_0002 (\nPULPROG zg\nTD 3018\nSOLVENT Acetone\nNS 1\nDS 0\nSWH 1509.662 Hz\nFIDRES 1.000439 Hz\nAQ 0.9995616 sec\nRG 203\nDW 331.200 usec\nDE 6.50 usec\nTE 296.2 K\nD1 2.00000000 sec\nTD0 1\nSFO1 150.9380780 MHz\nNUC1 13C\nP1 15.75 usec\nPLW1 179.47000122 W\nF2 - Processing parameters\nSI 524288\nSF 150.9251876 MHz\nWDW EM\nSSB 0\nLB 1.00 Hz\nGB 0\nPC 1.40\n13C+acetone d6\n5.40 5.45 5.50 5.55 5.60 5.65 5.70 5.75 ppmCurrent Data Parameters\nNAME st1_s2\nEXPNO 1\nPROCNO 1\nF2 - Acquisition Parameters\nDate_ 20190801\nTime 9.11 h\nINSTRUM spect\nPROBHD Z108349_0002 (\nPULPROG zg\nTD 24036\nSOLVENT Acetone\nNS 1\nDS 0\nSWH 6009.615 Hz\nFIDRES 0.500051 Hz\nAQ 1.9997952 sec\nRG 32\nDW 83.200 usec\nDE 6.50 usec\nTE 296.2 K\nD1 1.00000000 sec\nTD0 1\nSFO1 600.2223331 MHz\nNUC1 1H\nP1 9.46 usec\nPLW1 18.13999939 W\nF2 - Processing parameters\nSI 524288\nSF 600.2190000 MHz\nWDW EM\nSSB 0\nLB 0.30 Hz\nGB 0\nPC 1.00\n1H+acetone d6\n-196.3 -196.4 -196.5 -196.6 -196.7 ppm19F+acetone d6\n84.4 84.6 84.8 85.0 85.2 85.4 85.6 85.8 86.0 86.2 86.4 86.6 ppmCurrent Data Parameters\nNAME st1_s2\nEXPNO 9\nPROCNO 1\nF2 - Acquisition Parameters\nDate_ 20190801\nTime 9.36 h\nINSTRUM spect\nPROBHD Z108349_0002 (\nPULPROG aka_pps_hfc_3C\nTD 3018\nSOLVENT Acetone\nNS 1\nDS 0\nSWH 1509.662 Hz\nFIDRES 1.000439 Hz\nAQ 0.9995616 sec\nRG 203\nDW 331.200 usec\nDE 6.50 usec\nTE 296.2 K\nD1 2.00000000 sec\nD2 0.00001500 sec\nD12 0.00263158 sec\nD13 0.00077399 sec\nD16 0.00020000 sec\nD23 0.00195618 sec\nTD0 1\nSFO1 150.9380784 MHz\nNUC1 13C\nP1 15.75 usec\nPLW1 179.47000122 W\nSFO2 564.6591365 MHz\nNUC2 19F\nP2 23.27 usec\nPLW2 42.27000046 W\nSFO3 600.2223331 MHz\nNUC3 1H\nP3 9.46 usec\nPLW3 18.13999939 W\nGPNAM[1] SINE.100\nGPZ1 35.00 %\nGPNAM[2] SINE.100\nGPZ2 25.00 %\nGPNAM[3] SINE.100\nGPZ3 25.00 %\nGPNAM[4] SINE.100\nGPZ4 30.00 %\nP16 1000.00 usec\nF2 - Processing parameters\nSI 524288\nSF 150.9251876 MHz\nWDW EM\nSSB 0\nLB 1.00 Hz\nGB 0\nPC 1.40\n13C+acetone d6\n5.40 5.45 5.50 5.55 5.60 5.65 5.70 5.75 ppmCurrent Data Parameters\nNAME st1_s2\nEXPNO 7\nPROCNO 1\nF2 - Acquisition Parameters\nDate_ 20190801\nTime 9.31 h\nINSTRUM spect\nPROBHD Z108349_0002 (\nPULPROG aka_pps_hfc_1H\nTD 24036\nSOLVENT Acetone\nNS 1\nDS 0\nSWH 6009.615 Hz\nFIDRES 0.500051 Hz\nAQ 1.9997952 sec\nRG 32\nDW 83.200 usec\nDE 6.50 usec\nTE 296.2 K\nD1 1.00000000 sec\nD2 0.00001500 sec\nD12 0.00263158 sec\nD13 0.00077399 sec\nD16 0.00020000 sec\nD23 0.00195618 sec\nTD0 1\nSFO1 600.2223331 MHz\nNUC1 1H\nP1 9.46 usec\nPLW1 18.13999939 W\nSFO2 564.6591365 MHz\nNUC2 19F\nP2 23.27 usec\nPLW2 42.27000046 W\nSFO3 150.9380784 MHz\nNUC3 13C\nP3 15.75 usec\nPLW3 179.47000122 W\nGPNAM[1] SINE.100\nGPZ1 35.00 %\nGPNAM[2] SINE.100\nGPZ2 25.00 %\nGPNAM[3] SINE.100\nGPZ3 25.00 %\nGPNAM[4] SINE.100\nGPZ4 30.00 %\nP16 1000.00 usec\nF2 - Processing parameters\nSI 524288\nSF 600.2190000 MHz\nWDW EM\nSSB 0\nLB 0.30 Hz\nGB 0\nPC 1.00\n1H+acetone d6 (c)1H\n19F\n13Cj11ij01i j 10ij00i\nj01ij00i j 10ij11i\nj01i j 00ij10i j11i1H\n19F\n13Cj00i\nj00i\nj00i5.7 5.55 5.4\n-196.3 -196.5 -196.7\n86.8 85.4 84.2\u0017H(ppm)\n\u0017F(ppm)\n\u0017C(ppm)5.7 5.55 5.4\n-196.3 -196.5 -196.7\n86.8 85.4 84.2\u0017H(ppm)\n\u0017F(ppm)\n\u0017C(ppm)1H\n19F13CFigure 2: ( a) Molecular structure of13C-labeled diethyl\ruoroma-\nlonate with the three qubits labeled as1H,19F and13C. NMR pa-\nrameters i.e. the chemical shift \u0017i(in Hz) of each nuclear spin, spin-\nspin coupling between them J ij(in Hz), spin-lattice relaxation times\nTi\n1and spin-spin relaxation times Ti\n2(in seconds). NMR spectrum\nof (b) thermal equilibrium state obtained after a\u0019\n2readout pulse\nand (c) pseudopure state. The resonance lines of each qubit in the\nspectra are labeled by the corresponding logical states of the other\nqubits.\nensemble weak measurement) since the interaction of the\nmeasuring rf coil with the nuclear spins is weak [33]. How-\never, we are not using that weak measurement here. Our\nweak measurement is simulated in a controlled way by the\nPD channel, which is implemented with the help of the\nancilla qubit.\n4. Experimental Implementation of a Weak POVM\nAs discussed earlier in Sec. 3, weak measurements can\nbe mapped onto a PD channel and the strength of the weak\nmeasurement can be varied by tuning the strength of the\nPD channel. For the experimental realization on an NMR\nquantum processor, we realize the three qubits as the three\nspin-1/2 nuclei of13C-labeled diethyl \ruoromalonate dis-\nsolved in acetone-D6. The1H,19F and13C nuclear spins\nare labeled as the \frst, second and third qubit, respec-\ntively. It should be noted here that two-qubit system was\nsimulated by19F and13C nuclear spins while1H spin was\nutilized as the ancillary qubit. The molecular structure\nalong with relevant experimental parameters and corre-\nsponding NMR spectrum of the thermal equilibrium state\nare shown in Figs. 2(a) and (b) respectively. The Hamil-\ntonian for a three-qubit system in a rotating frame under\n5(a) Bell-Diagonal State\nReal ImagBell-Diagonal State\nx= 1:2Real ImagTheory Expr.00011011111001000001101100011011\n00011011111001000001101100011011\n00011011111001000001101100011011\n00011011111001000001101100011011\n(b) Werner State\nReal ImagWerner State\nx= 1:2Real ImagTheory Expr.\n00011011111001000001101100011011\n00011011111001000001101100011011\n00011011111001000001101100011011\n00011011111001000001101100011011Figure 3: Real and imaginary parts of theoretically expected and the\nexperimentally reconstructed tomographs of (a) Bell-diagonal state\nand (b) Werner state before (left) and after (right) PD channel im-\nplementation at a measurement strength x= 1:2.\nthe weak approximation [40] is given by:\nH=\u00003X\ni=1\u0017iIi\nz+3X\ni>j;i =1JijIi\nzIj\nz (23)\nwherei;j=1,2 and 3 labels the qubit, virepresents the\nchemical shift of the respective nuclei, J ijis the scalar cou-\npling constant between the ithandjthnuclear spins and Ii\nz\ndenotes the zcomponent of the spin angular momentum\noperators for the ithnucleus. We used the spatial aver-\naging technique [41, 42] to achieve the initial three-qubit\npseudopure state (PPS) j000ifrom the thermal equilib-\nrium state, with the density operator \u001a000given by:\n\u001a000=(1\u0000\u000f)\n8I+\u000fj000ih000j (24)\nwhere\u000f\u001810\u00005represents the spin polarization at room\ntemperature and Iis the 8\u00028 identity operator. The iden-\ntity operator does not evolve and the measurable NMR\nsignal can be attributed to the deviation density matrix.\nThe NMR spectrum of the three-qubit PPS is shown in\nFig. 2(c). The experimentally prepared PPS was tomo-\ngraphed using full quantum state tomography [43]. The\nstate \fdelity was found to be 0 :981\u00060:006 and was com-\nputed using the Uhlmann-Jozsa measure [44, 45]:\nF =\u0014\nTr\u0012qp\u001ath\u001aexp\u001ath\u0013\u00152\n(25)\nwhere\u001athand\u001aexdenote the theoretical and experimental\ndensity operators, respectively and F is normalized using\nF!1 as\u001aex!\u001ath. All the experimental density matri-\nces were reconstructed by performing full quantum statetomography [43] using a set of seven preparatory pulses\nfIII;XXX;IIY;XYX;YII;XXY;IYY g, whereIrepre-\nsents `no-operation' and X(Y) denotes local\u0019\n2unitary ro-\ntation with phase x(y) which is implemented by applying a\nspin-selective\u0019\n2pulse. We performed all the experiments\nat room temperature 293K on a Bruker Avance-III 600\nMHz FT-NMR spectrometer equipped with a QXI probe.\nThe quantum circuit to implement a weak measure-\nment, simulated by a PD channel, is shown in Fig. 1(a).\nThe left block in the circuit creates a Werner or a Bell-\ndiagonal state ( \u001a0\n\u001a ) on two qubits ( \u001a ) with the third\nqubit (\u001a0) acting as ancillary; R is a NOT gate in the case\nof the Werner state and a `no-operation' for a Bell-diagonal\nstate. The right block in the circuit depicts a PD channel\nacting on the one of the qubits of two-qubit state using\nan ancillary qubit, and the strength of the PD channel be-\ning controlled by a local rotation achieved by the Vgate\nacting on the ancillary qubit.\nTable 1: Experimental (Exp) results of weak measurement strength\nxvaried in a Werner (W) and a Bell-diagonal (BD) state while im-\nplementing the PD channel.\nTheoryx Expx(W) Exp x(BD)\n0.00 0.091\u00060.047 0.069\u00060.036\n0.34 0.503\u00060.018 0.373\u00060.002\n0.55 0.666\u00060.015 0.548\u00060.005\n0.75 0.831\u00060.016 0.721\u00060.005\n0.95 1.016\u00060.016 0.907\u00060.004\n1.20 1.215\u00060.020 1.103\u00060.007\n1.50 1.479\u00060.024 1.350\u00060.007\n1.75 1.819\u00060.045 1.667\u00060.018\n2.00 2.122\u00060.046 1.937\u00060.018\n2.50 2.454\u00060.062 2.213\u00060.026\n3.00 3.242\u00060.151 2.795\u00060.042\n3.50 3.758\u00060.095 3.448\u00060.069\n4.00 4.107\u00060.167 4.259\u00060.123\n4.50 4.402\u00060.062 4.575\u00060.069\n5.00 4.839\u00060.137 5.030\u00060.180\nExperimentally, we prepared a Werner state with mixed-\nness strength z= 0:8. The second term on the RHS\nof Eq.(15) is a singlet state and was created experimen-\ntally, followed by quantum state tomography (QST). To\nobtain the Werner state of a desired zvalue the QST gen-\nerated singlet state was numerically added to the iden-\ntity and thus, a Werner state with \fdelity 0 :990\u00060:001\nwas created. Next the Bell-diagonal state with c1= 1,\nc2=\u00001 andc3= 1 was prepared experimentally with\n\fdelity 0:980\u00060:002. The next step is to perform a weak\nmeasurement on the second qubit (treated as a subsys-\ntem of the two-qubit system). As described in Sec. 3, the\nweak measurement is simulated by the PD channel and the\nstrength of PD channel is controlled by the real parame-\nter\u0015which is related to weak measurement strength xas\n60 1 2 3 4 50.20.40.60.81.01.2\n0 1 2 3 4 500.511.52(a)\nWerner State\nBell-Diagonal State0.0 1.0 2.0 3.0 4.0 5.0\nMeasurement strength ( x)!Discord !\n0.00.20.40.60.81.01.2\n(b)\n0.0 1.0 2.0 3.0 4.0 5.00.00.51.01.52.0\nMeasurement strength ( x)!Discord !\nQDSQDSQD\nQDFigure 4: Plot of SQD and QD with measurement strength ( x) for\nthe two-qubit experimentally prepared (a) Werner state and (b) Bell-\ndiagonal state. The solid and dashed lines represent the theoretically\ncomputed values while the \flled circles and triangles represent the\nexperimental values of SQD and QD, respectively.\n\u0015= 1\u0000sech2x. We implemented the PD channel on the\none of the qubits of the prepared Werner state using the\nancillary qubit and the strength of the PD channel was in-\ncreased corresponding to weak measurement strength xas\nshown in Table-1. We performed a similar experiment for\nthe Bell-diagonal states, which was directly prepared from\nthe left block in the circuit as depicted in Fig. 1(a) and the\nPD channel was implemented with increasing weak mea-\nsurement strength xas shown in Table-1. All the exper-\nimentally prepared three-qubit states were tomographed\nbefore and after implementing PD channel. Both the two-\nqubit Werner and Bell-diagonal states were reconstructed\nutilizing QST and tracing out the ancillary qubit. The to-\nmograph of one such experimentally reconstructed density\nmatrix of both initially prepared states is shown on the\nLHS of Fig. 3(a) and Fig. 3(b), respectively. The tomo-\ngraphs on the RHS of Fig. 3(a) and (b) depicts the states\nafter the action of the PD channel corresponding to the\nweak measurement strength x= 1:2.\nWe investigated theoretically and experimentally, the\nbehavior of SQD, WQD, and QD present in both Werner\nand Bell-diagonal states while increasing the measurement\nstrengthx, and the results are plotted in Fig. 4 and Fig. 5,\nrespectively. Our results show that in both types of states,\n1 2 3 4 500.10.20.30.40.50.6\n1 2 3 4 500.20.40.60.81(a)\nWerner State\nBell-Diagonal State0.0 1.0 2.0 3.0 4.0 5.0\nMeasurement strength ( x)!Discord !\n0.00.10.20.30.40.50.6\n(b)\n0.0 1.0 2.0 3.0 4.0 5.00.00.20.40.60.81.0\nMeasurement strength ( x)!Discord !QD\nWQD\nQD\nWQDFigure 5: Plot of WQD and QD with measurement strength ( x) for\nthe two-qubit experimentally prepared (a) Werner state and (b) Bell-\ndiagonal state. The solid and dashed lines represent the theoretically\ncomputed values while the \flled circles and triangles represent the\nexperimental values of WQD and QD, respectively.\nthe value of SQD is always greater than QD and is max-\nimum at zero measurement strength, which implies that\nthe measured state is undisturbed. On the other hand, as\na quanti\fer WQD is never greater than QD and allows the\nresearcher to navigate the region between x\u0000!0 when\nno measurement is performed (and quantum correlations\nare intact) and x\u0000!1 when a projective measurement is\nperformed (and quantum correlations are destroyed). Fur-\nthermore, as evidenced from Fig. 4 and Fig. 5, both SQD\nand WQD approach QD as the measurement strength in-\ncreases. Our experimental results are hence in consonance\nwith the theoretically expected behavior of the quanti\fers\nof discord for the case of a weak measurement.\n5. Concluding Remarks\nWe have implemented a weak POVM, exploiting its\nrelationship with the PD channel, on an NMR quantum\ninformation processor. The noise induced by the PD chan-\nnel has been exploited to mimic the disturbance intro-\nduced by a weak measurement process. The weak POVM\nwas experimentally applied to \fnd the SQD and WQD in\ntwo classes of bipartite quantum states, namely, the Bell-\ndiagonal state and the Werner state. The SQD and WQD\nwere contrasted against QD and it was observed that both\n7these quantities converge to QD as the strength of the mea-\nsurement increases. The monotonicity of SQD and WQD\nwas also con\frmed. Our results could be useful for itera-\ntive experimental information processing protocols which\nseek to disturb the state only slightly. 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Rev.\nA 83 (2011) 032324.\n9" }, { "title": "2305.10111v1.Material_Parameters_for_Faster_Ballistic_Switching_of_an_In_plane_Magnetized_Nanomagnet.pdf", "content": "arXiv:2305.10111v1 [cond-mat.mes-hall] 17 May 2023Journal of the Physical Society of Japan FULL PAPERS\nMaterial Parameters for Faster Ballistic Switching of an In -plane Magnetized\nNanomagnet\nToshiki Yamaji1*and Hiroshi Imamura1 †\n1National Institute of Advanced Industrial Science and Tech nology (AIST), Tsukuba, Ibaraki 305-8568, Japan\nHigh-speed magnetization switching of a nanomagnet is nece ssary for faster information processing. The ballistic\nswitching by a pulsed magnetic filed is a promising candidate for the high-speed switching. It is known that the switch-\ning speed of the ballistic switching can be increased by incr easing the magnitude of the pulsed magnetic field. However\nit is difficult to generate a strong and short magnetic field pulse in a sm all device. Here we explore another direction\nto achieve the high-speed ballistic switching by designing material parameters such as anisotropy constant, saturati on\nmagnetization, and the Gilbert damping constant. We perfor m the macrospin simulations for the ballistic switching of\nin-plane magnetized nano magnets with varying material par ameters. The results are analyzed based on the switching\ndynamics on the energy density contour. We show that the puls e width required for the ballistic switching can be re-\nduced by increasing the magnetic anisotropy constant or by d ecreasing the saturation magnetization. We also show that\nthere exists an optimal value of the Gilbert damping constan t that minimizes the pulse width required for the ballistic\nswitching.\n1. Introduction\nIn modern information technologies huge amount of data\nare represented as the direction of the magnetization in a sm all\nmagnet such as magnetic grains in magnetic tapes or hard\ndisk drives. To write information on the conventional mag-\nnetic recording media an external magnetic field is applied i n\nthe opposite direction of the magnetization to switch the di -\nrection of the magnetization. During the switching the mag-\nnetization undergoes multiple precessions around the loca l ef-\nfective field consisting of the external field, anisotropy fie ld,\nand demagnetizing field. The typical switching time or write\ntime is of the order of nanoseconds.\nTo meet the growing demand for fast information process-\ning it is important to develop a faster switching scheme. The\nballistic switching is a promising candidate for high-spee d\nswitching, and much e ffort has been devoted to developing\nthe ballistic switching both theoretically1–8)and experimen-\ntally.9–16)In ballistic switching a pulsed magnetic field is ap-\nplied perpendicular to the easy axis to induce the large-ang le\nprecession around the external magnetic field axis. The dura -\ntion of the pulse is set to a half of the precession period. Aft er\nthe pulse the magnetization relaxes to the equilibrium dire c-\ntion opposite to the initial direction. The switching speed of\nthe ballistic switching can be increased by increasing the m ag-\nnitude of the pulsed field. However, it is di fficult to generate a\nstrong and short field pulse in a small device. It is desired to\nfind a way to speed up the ballistic switching without increas -\ning magnetic field.\nThe magnetization dynamics of the ballistic switching is\ndetermined by the torques due to the external magnetic field,\nthe uniaxial anisotropy field, the demagnetizing field, and t he\nGilbert damping. The torques other than the external mag-\nnetic field torque are determined by the material parameters\nsuch as the anisotropy constant, the saturation magnetizat ion,\nand the Gilbert damping constant. There is room to speed up\n*toshiki-yamaji@aist.go.jp\n†h-imamura@aist.go.jpthe ballistic switching by designing the appropriate mater ial\nparameters.\nIn the early 2000s the several groups each independently\nreported the optical microscope measurements of the ballis -\ntic switching by picosecond pulse magnetic field.9–13)Then\nthe mechanism of a ballistic switching was analyzed in terms\nof the nonlinear dynamics concepts such as a fixed point, at-\ntractors, and saddle point.2, 3, 6)Especially the minimal field\nrequired for a ballistic switching was investigated by comp ar-\ning the so-called Stoner-Wohlfarth (SW) type.2, 3)The damp-\ning constant dependence of the minimal switching field was\nalso studied.2)The characteristics of the parameters of a pulse\nmagnetic field, i.e., magnitude, direction, and rise /fall time on\nthe mechanism of a ballistic switching had been also studied\nby the simulations and experiments.6, 7, 14, 15)\nAs described above, in 2000s and 2010s a ballistic switch-\ning technique had received much attention for the fast magne -\ntization reversal with ringing suppression by fine-tuning t he\nmagnetic pulse parameters. Due to the recent advance of an\nultra-fast measurement17)the studies of a ballistic switching\nhave attracted much attention again. Last year the in-plane\nmagnetization switching dynamics as functions of the pulse\nmagnetic field duration and amplitude was calculated and\nanalyzed by using the conventional Landau-Lifshitz-Gilbe rt\n(LLG) equation and its inertial form, the so-called iLLG\nequation.16)The solutions of both equations were compared\nin terms of the switching characteristics, speed and energy\ndensity analysis. Both equations return qualitatively sim ilar\nswitching dynamics. However the extensive material param-\neter dependences of a ballistic switching region have not\nyet been sufficiently explored. Therefore it is worth clearing\nthe extensive material parameter dependences of the ballis tic\nswitching of an in-plane magnetized nanomagnet.\nIn this paper, we study the ballistic switching of an in-\nplane magnetized nanomagnet with systematically varying\nthe material parameters by using the macrospin simulations .\nThe results show that the pulse width required for the bal-\nlistic switching can be reduced by increasing the magnetic\n1J. Phys. Soc. Jpn. FULL PAPERS\nHp\nmz\nyx(a)\n(c) my at t = 10 ns (b) \n(d) 0 200 400-1 01\nt [ps]my\ntp [ps]0 1 2 3 4 5tSW [ps] \n110 10 210 3\ntl tutSW \n0 1 -1 \n0 1 2 3 4 502.55.010.0\n7.5\ntp [ps]Hp [T] \nFig. 1. (a) Schematic illustration of the in-plane magnetized nano magnet.\nThe pulse field, Hp, is applied along the x-direction. The initial direction of\nthe magnetization is in the positive y-direction. (b) Gray scale map of myat\nt=10 ns as a function of the pulse field width, tp, and Hp. The black and\nwhite regions represent the success and failure of switchin g. The parameters\nareµ0Ms=0.92 T,µ0HK=0.1 T, andα=0.023. (c) Typical example of\nthe time evolution of mywhen the magnetization switches ( Hp=5 T and tp\n=0.4 ps). The switching time, tSW, is defined as the time when mychanges\nthe sign. (d) tpdependence of tSWalong the dashed horizontal line at Hp=5\nT shown in Fig. 1(b). tlandtuare 3.15 ps and 3.93 ps, respectively. tSWat\ntl≤tp≤tuis 1.7 ps.\nanisotropy constant or by decreasing the saturation magnet i-\nzation. There exists an optimal value of the Gilbert damping\nconstant that minimizes the pulse width required for ballis -\ntic switching. The simulation results are intuitively expl ained\nby analyzing the switching trajectory on the energy density\ncontour.\n2. Model and Method\nIn this section we show the theoretical model, the numer-\nical simulation method, and the analysis using the trajecto ry\nin the limit ofα→0. The macrospin model of the in-plane\nmagnetized noanomagnet and the equations we solve to simu-\nlate the magnetization dynamics are given in Sec. 2.1. In Sec .\n2.2 we show that the switching conditions can be analyzed by\nusing the trajectory on the energy density contour in the lim it\nofα→0 if theα≪1.\n2.1 Macrospin Model Simulation\nFigure 1(a) shows the schematic illustration of the in-\nplane magnetized nanomagnet. The pulsed magnetic field,\nHp, is applied along the x-direction. The unit vector m=\n(mx,my,mz) indicates the direction of the magnetization. The\nsize of the nanomagnet is assumed to be so small that the dy-\nnamics of mcan be described by the macrospin LLG equation\ndm\ndt=−γm×/parenleftBigg\nHeff−α\nγdm\ndt/parenrightBigg\n, (1)\nwhere tis time,γis the gyromagnetic ratio, αis the Gilbert\ndamping constant. The e ffective field, Heff=Hp+HK+Hd,\ncomprises the pulse field, Hp, the anisotropy field, HK, andthe demagnetizing field, Hd. The anisotropy field and the de-\nmagnetizing field are defined as\nHK=/bracketleftbig2K/(µ0Ms)/bracketrightbigmyey, (2)\nand\nHd=µ0Msmzez, (3)\nrespectively, where Kis the uniaxial anisotropy constant, µ0\nis the magnetic permeability of vacuum, Msis the saturation\nmagnetization, and ejis the unit vector along the j-axis ( j=\nx,y,z).\nThe switching dynamics are calculated by numerically\nsolving the LLG equation. The initial ( t=0) direction is set\nasmy=1. The rectangular shaped pulse magnetic field with\nduration of tpis applied at t=0. The time evolution of magne-\ntization dynamics are calculated for 10 ns. Success or failu re\nof switching is determined by whether my<−0.5 att=10\nns.\nFigure 1(b) shows the gray scale plot of myatt=10 ns\non the tp-Hpplane. Following Ref. 16 the parameters are as-\nsumed to beµ0Ms=0.92 T, K=2.3 kJ/m3, i.e.µ0HK=\n0.1 T, andα=0.023. The black and white regions represent\nthe success and failure of switching, respectively. The wid e\nblack region at upper right of Fig. 1(b) represents the balli stic\nswitching region (BSR). A typical example of the time evolu-\ntion of mywhen the magnetization switches is shown in Fig.\n1(c). The switching time, tSW, is defined as the time when my\nchanges the sign. Figure 1(d) shows the tpdependence of tSW\nalong the horizontal line shown in Fig. 1(b), i.e. at Hp=5\nT. The BSR indicated by shade appears between tl=3.15\nps and tu=3.93 ps, where tSW=1.7 ps independent of tp.\nThe lower and upper boundary of the BSR are represented by\ntlandtu, respectively. We investigate the material parameter\ndependence of tlandtuwith keeping Hp=5 T.\n2.2 Analysis of the Switching Conditions for α≪1\nIf the Gilbert damping constant is much smaller than unity\nthe approximate value of tlandtucan be obtained without\nperforming macrospin simulations. In the limit of α→0, the\ntrajectory is represented by the energy contour because the en-\nergy is conserved during the motion of m. The energy density,\nE, of the nanomagnet is defined as18)\nE=1\n2µ0M2\nscos2θ+K(1−sin2θsin2φ), (4)\nwhereθandφare the polar and azimuthal angles of the mag-\nnetization, respectively. The color plot of the energy dens ity\ncontour is shown in Fig. 2. The separatrix representing the\nenergy contour with E=Kis indicated by the white curve,\nwhich is expresses as\n1\n2µ0M2\nscos2θ−Ksin2θsin2φ=0. (5)\nThe green dot indicates the initial direction of matt=0. The\nblack curve represents the trajectory of munder the pulse field\nofHpin the limit ofα→0. Under the pulse field the energy\ndensity is given by\nE=1\n2µ0M2\nscos2θ+K(1−sin2θsin2φ)\n−µ0MsHpsinθcosφ. (6)\n2J. Phys. Soc. Jpn. FULL PAPERS\n01 5 4 3 26E/K\ntltu\nθ\nφ\nFig. 2. (Color online) Color plot of the energy density contour give n by\nEq. (4).θandφare the polar and azimuthal angles of the magnetization, re-\nspectively. The material parameters, MsandKare same as in Fig. 1. The\nseparatrix given by Eq. (5) is indicated by the white curve. T he initial direc-\ntion of mis indicated by the green dot at ( θ,φ)=(π/2,π/2). The black curve\nrepresents the trajectory of the magnetization under the fie ld of Hp=5 T in\nthe limit ofα→0, which is given by Eq. (7). The yellow stars indicate the\nintersection points of the separatrix and the trajectory, w hich correspond to tp\n=tlandtu. If the pulse is turned o ffattl≤t≤tu, the magnetization switches\nballistically. The yellow triangle indicates the turning p oint of the trajectory\nof the magnetization near mz=1, at whichφ=0.\nSince the energy density of the initial direction, θ=φ=π/2,\nisE=0, the trajectory under the pulse field is expressed as\n1\n2µ0M2\nscos2θ+K(1−sin2θsin2φ)\n−µ0MsHpsinθcosφ=0. (7)\nThe yellow stars indicate the points where the trajectory\ncrosses the separatrix surrounding the equilibrium point a t\nφ=−π/2. The upper and lower points indicates the direc-\ntion of mat the end of the pulse with tp=tuandtl, re-\nspectively. The corresponding angles ( θl,φl) and (θu,φu) can\nbe obtained by solving Eqs. (5) and (7) simultaneously. If\ntl≤tp≤tu, the magnetization relaxes to the equilibrium di-\nrection at (θ,φ)=(π/2,−π/2) after the pulse to complete the\nswitching. We can obtain the approximate expressions of tl\nandtuas follows. Assuming that the pulse field is much larger\nthan the other fields, the angular velocity of the precession ,ω,\nis approximated as γHp/(1+α2), and tlandtuare analytically\nobtained as\ntl=π−2θturn\nω−1\n2∆θ\nω, (8)\nand\ntu=π−2θturn\nω+1\n2∆θ\nω, (9)\nwhere∆θ=θu−θl, andθturnis the polar angle at the turning\npoint (φ=0) indicated by the yellow triangle.3. Results and Discussion\nIn this section we discuss the dependence of the BSR on\nthe material parameters by analyzing the numerical simula-\ntion results and Eqs. (8) and (9). The results for the variati on\nof the magnetic anisotropy constant, K, saturation magnetiza-\ntion, Ms, and the Gilbert damping constant, α, will be given\nin Secs. 3.1, 3.2, and 3.3, respectively.\n3.1 Anisotropy Constant Dependence of the BSR\nFigure 3(a) shows the anisotropy constant, K, dependence\nof the BSR. The parameters are Hp=5 T,µ0Ms=0.92 T, and\nα=0.023. The simulation results of tlandtuare indicated\nby the orange and blue dots, respectively. The analytical ap -\nproximations of tlandtuobtained by solving Eqs. (5),(7),(8),\nand (9) are represented by the orange and blue curves, respec -\ntively. The simulation and analytical results agree well wi th\neach other because the Gilbert damping constant is as small a s\n0.023. As shown in Fig. 3(a), tlis a monotonically decreasing\nfunction of Kwhile tuis a monotonically increasing function\nofK. As a result the width of the BSR, tu-tl, is a monoton-\nically increasing function of Kas shown in the inset of Fig.\n3(a).\nIn the left panel of Fig. 3(b) the separatrix and the trajecto ry\nwithα=0 for K=2.3 kJ/m3are shown by the blue and\nblack curves, respectively. The same plot for K=9.3 kJ/m3\nis shown in the right panel. As shown in these panels, the\nincrease of Kdoes not change the trajectory much. However,\nthe increase of Kchanges the separatrix significantly through\nthe second term of Eq. (5). Assuming that the angular velocit y\nof the precession is almost constant, the spread of the area\nsurrounded by the separatrix results in the spread of the tim e\ndifference between tlandtu. As a result the BSR is spread by\nthe increase of Kas shown in Fig. 3(a)\n3.2 Saturation Magnetization Dependence of the BSR\nFigure 4(a) shows the saturation magnetization dependence\nof the BSR obtained by the numerical simulation and the ana-\nlytical approximations. The horizontal axis represents th e sat-\nuration magnetization in unit of T, i.e µ0Ms. The parameters\nareHp=5 T,K=2.3 kJ/m3, andα=0.023. The symbols are\nthe same as in Fig. 3(a). The lower boundary of the BSR, tl,\nincreases as theµ0Msincreases while the upper boundary of\nthe BSR, tu, decreases with increase of µ0Ms. Therefore, the\nfaster switching is available for smaller Ms. Theµ0Msdepen-\ndence of the BSR ( tu-tl) is also shown in the inset of Fig. 4(a).\nThe BSR decreases with increase of µ0Ms. In other words, the\nwider BSR is obtained for smaller Ms.\nIn the right panel of Fig. 4(b) the separatrix and the trajec-\ntory withα=0 forµ0Ms=0.35 T are shown by the blue and\nblack curves, respectively. The same plot for µ0Ms=0.92 T is\nshown in the left panel. As shown in these panels, the increas e\nofMsdoes not change the trajectory much but decrease the\nseparatrix significantly through the first term of Eq. (5). As -\nsuming that the angular velocity of the precession is almost\nconstant, the reduction of the area surrounded by the separa -\ntrix results in the reduction of the time di fference between tl\nandtu. As a result the BSR decreases with increase of Msas\nshown in Fig. 4(a)\n3J. Phys. Soc. Jpn. FULL PAPERStl, t u [ps] \n0 10 20 30 40 2.03.04.05.0\nK [kJ/m3]ballistic switching region (a)\ntltu\ntu - t l [ps] \nK [kJ/m 3]0 10 20 30 40 0.0 0.5 1.0 1.5 2.0 2.5 \n0π\nπ/2 \n0 0 π π -π -π π /2 π/2 -π /2 -π /2 θ\nφ φˑˑ\nˑˑ(b) K = 2.3 kJ/m3K = 9.3 kJ/m3\ntltltutu\nFig. 3. (Color online) (a) Anisotropy constant, K, dependence of the BSR\n(orange shade). Simulation results of tlandtuare plotted by the orange and\nblue dots, respectively. The analytical results are indica ted by the solid curves\nwith the same color. The parameters are Hp=5 T,µ0Ms=0.92 T, andα=\n0.023. In the inset the simulation and analytical results of the width of the\nBSR, tu-tl, are plotted by the dots and the solid curve, respectively. ( b)\nTypical examples of the trajectory of the magnetization (bl ack curve) and the\nseparatrix (blue curve). The left and right panels show the r esults for K=2.3\nkJ/m3andK=9.3 kJ/m3, respectively. The orange and blue stars indicate\nthe direction at t=tlandtu, respectively. The green dots indicate the initial\ndirection of m.\n3.3 Gilbert Damping Constant Dependence of the BSR\nFigure 5(a) shows the simulation results of the Gilbert\ndamping constant, α, dependence of the BSR. The width of\nthe BSR is shown in the inset. The symbols are the same as\nin Fig. 3(a). The approximate values obtained by Eqs. (8) and\n(9) are not shown because the αis not limited toα≪1. The\nparameters are Hp=5 T, K=2.3 kJ/m3, andµ0Ms=0.92\nT. There exists an optimal value of αthat minimizes tl. The\noptimum value in Fig. 5 (a) is αopt=0.35.\nTo understand the mechanism for minimization of tlat a\ncertain value ofαone need to consider two di fferent effects of\nαon the magnetization dynamics. One e ffect is the decrease\nof the precession angular velocity with increase of α. The pre-\ncession angular velocity around the e ffective field of Heffis\ngiven by (γHeff)/(1+α2), which decreases with increase of α.\nThis effect causes the increase of tlandtu.\nThe other effect is the increase of the energy dissipation rate\nwith increase ofα. Let us consider the trajectory in the cases\nof small damping ( α=0.023) and large damping ( α=αopt).\nIn Fig. 5 (b) the typical examples of the trajectory for the\nsmall damping are shown by the yellow and green curves\nand dots on the energy density contour. The pulse widths are\ntp=tl(=3.15 ps) and 3.14 ps. The trajectories during the\npulse are represented by the solid curves and the trajectori es\nafter the pulse are represented by the dots. The white curve\nshows the separatrix and the black dot indicates the initial di-\ntl, t u [ps] \n2.03.04.05.0\n0.0 0.3 0.6 0.9 1.2\nμ0Ms [T]ballistic switching region \ntltu(a)\n(b) μ0Ms = 0.92 T μ0Ms = 0.35 T\n0π\nπ/2 \n0 0 π π -π -π π /2 π/2 -π /2 -π /2 θ\nφ φˑˑ\nˑˑ\ntl tltu tu1.5tu - t l [ps] \n0123\n0.0 0.3 0.6 0.9 1.2 \nμ0Ms [T] 1.5 \nFig. 4. (Color online) (a) Saturation magnetization dependence of the\nBSR. The horizontal axis represents the saturation magneti zation in unit of T,\ni.eµ0Ms. The parameters are Hp=5 T, K=2.3 kJ/m3, andα=0.023. The\nsymbols are the same as in Fig. 3 (a). (b) Typical examples of t he trajectory\nof the magnetization (black curve) and the separatrix (blue curve). The right\nand left panels show the results for µ0Ms=0.35 T and 0.92 T, respectively.\nThe symbols are the same as in Fig. 3 (b).\nrection. The yellow and green stars indicate the points wher e\nthe trajectories cross the separatrix surrounding the targ et and\ninitial states, respectively. The arrows indicate the dire ction\nof the movement of the magnetization. For the small damp-\ning, even very close to the separatrix around the target stat e at\nthe end of the pulse, the magnetization flows to the sepatrari x\naround the initial state and relax to the initial state after many\nprecessions with the slow energy dissipation.\nFigure 5 (c) shows the tpdependence of tSWat the large\ndamping (α=αopt). All parameters except αare the same\nas in Fig. 1 (d). t′\nl,tl, and tuare 0.82 ps, 1.98 ps, and 4.54\nps, respectively. t′\nlis the time when for the large damping the\nmagnetization goes across the e ffective separatrix around the\ninitial state during the pulse duration. In Fig. 5 (d) the typ ical\nexamples of the trajectory for the large damping are shown\nby the yellow ( tp=0.9 ps), green ( tp=tl=1.98 ps), and\npurple ( tp=4.55 ps) curves and dots on the energy density\ncontour. The symbols are the same as in Fig. 5 (b). In the\nregion 1 ( tptu) after the pulse is removed the magnetization\nmoves toward the separatrix around the initial state under Heff\nand relaxes to the initial state. We find that the BSR for the\nlarge damping can be explained by the anisotropic spread of\nthe effective separatrix with increasing α, which is fundamen-\ntally due to the breaking of the spatial inversion symmetry o f\nthe spin dynamics. The broken symmetry of the spatial inver-\nsion of the spin dynamics for the large damping can be easily\nconfirmed by comparing Fig. 5 (c) with Fig. 1 (d).\n4. Summary\nIn summary, we study the material parameter dependence\nof the ballistic switching region of the in-plane magnetize d\nnanomagnets based on the macrospin model. The results show\nthat the pulse width required for the ballistic switching ca n be\nreduced by increasing the magnetic anisotropy constant or b y\ndecreasing the saturation magnetization. The results also re-\nvealed that there exists an optimal value of the Gilbert damp -\ning constant that minimizes the pulse width required for the\nballistic switching. The simulation results are explained by\nanalyzing the trajectories on the energy contour. The resul ts\nare useful for further development of the high-speed inform a-\ntion processing using the ballistic switching of magnetiza tion.\nThis work is partially supported by JSPS KAKENHI Grant\nNumber JP20K05313.\n1) L. He and W. D. Doyle: J. Appl. Phys. 79(1996) 6489.\n2) Z. Z. Sun and X. R. Wang: Phys. Rev. B 71(2005) 174430.\n3) D. Xiao, M. Tsoi, and Q. Niu: Journal of Applied Physics 99(2006)\n013903.\n4) Y . Nozaki and K. Matsuyama: Jpn. J. Appl. Phys. 45(2006) L758.\n5) Y . Nozaki and K. Matsuyama: Journal of Applied Physics 100(2006)\n053911.\n6) Q. F. Xiao, B. C. Choi, J. Rudge, Y . K. Hong, and G. Donohoe: J ournal\nof Applied Physics 101(2007) 024306.\n7) P. P. Horley, V . R. Vieira, P. M. Gorley, V . K. Dugaev, J. Ber akdar, and\nJ. Barna´ s: Journal of Magnetism and Magnetic Materials 322(2010)\n1373.\n8) Y . B. Bazaliy: Journal of Applied Physics 110(2011) 063920.\n9) T. Gerrits, H. A. M. van den Berg, J. Hohlfeld, L. B¨ ar, and T . Rasing:\nNature 418(2002) 509.\n10) I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Siegmann , J. St¨ ohr,\nG. Ju, B. Lu, and D. Weller: Nature 428(2004) 831.\n11) H. W. Schumacher, C. Chappert, R. C. Sousa, P. P. Freitas, and J. Miltat:\nPhys. Rev. Lett. 90(2003) 017204.\n12) W. K. Hiebert, L. Lagae, J. Das, J. Bekaert, R. Wirix-Spee tjens, and\nJ. De Boeck: Journal of Applied Physics 93(2003) 6906.\n13) W. K. Hiebert, L. Lagae, and J. De Boeck: Phys. Rev. B 68(2003)\n5J. Phys. Soc. Jpn. FULL PAPERS\n020402.\n14) H. W. Schumacher: Appl. Phys. Lett. 87(2005) 042504.\n15) N. Kikuchi, Y . Suyama, S. Okamoto, O. Kitakami, and T. Shi matsu:\nAppl. Phys. Lett. 104(2014) 112409.\n16) K. Neeraj, M. Pancaldi, V . Scalera, S. Perna, M. d’Aquino , C. Serpico,\nand S. Bonetti: Phys. Rev. B 105(2022) 054415.17) K. Neeraj, N. Awari, S. Kovalev, D. Polley, N. Zhou Hagstr ¨ om, S. S.\nP. K. Arekapudi, A. Semisalova, K. Lenz, B. Green, J.-C. Dein ert,\nI. Ilyakov, M. Chen, M. Bawatna, V . Scalera, M. d’Aquino, C. S erpico,\nO. Hellwig, J.-E. Wegrowe, M. Gensch, and S. Bonetti: Nat. Ph ys.17\n(2021) 245.\n18) W. F. Brown: Phys. Rev. 130(1963) 1677.\n6" }, { "title": "1112.2362v1.Spin_polarized_current_effect_on_antiferromagnet_magnetization_in_a_ferromagnet___antiferromagnet_nanojunction__Theory_and_simulation.pdf", "content": "arXiv:1112.2362v1 [cond-mat.mtrl-sci] 11 Dec 2011Spin-polarized current effect on antiferromagnet\nmagnetization in a ferromagnet–antiferromagnet\nnanojunction: Theory and simulation\nE. M. Epshtein∗, Yu. V. Gulyaev, P. E. Zilberman\nV. A. Kotelnikov Institute of Radio Engineering and Electronics\nof the Russian Academy of Sciences, Fryazino, 141190, Russia\nAbstract\nSpin-polarized current effect is studied on the static and dy namic mag-\nnetization of the antiferromagnet in a ferromagnet–antife rromagnet nano-\njunction. The macrospin approximation is generalized to an tiferromag-\nnets. Canted antiferromagnetic configuration and resultin g magnetic mo-\nment are induced by an external magnetic field. The resonance frequency\nand damping are calculated, as well as the threshold current density cor-\nresponding to instability appearance. A possibility is sho wn of generating\nlow-damping magnetization oscillations in terahertz rang e. The fluctu-\nation effect is discussed on the canted antiferromagnetic co nfiguration.\nNumerical simulation is carried out of the magnetization dy namics of the\nantiferromagnetic layer in the nanojunction with spin-pol arized current.\nOutside the instability range, the simulation results coin cide completely\nwith analytical calculations using linear approximation. In the instabil-\nity range, undamped oscillations occur of the longitudinal and transverse\nmagnetization components.\n1 Introduction\nThe discovery of the spin transfer torque effect in ferromagn etic junctions\nunder spin-polarized current [1, 2] has stimulated a number of works in\nwhich such effects were observed as switching the junction ma gnetic con-\nfiguration [3], spin wave generation [4], current-driven mo tion of magnetic\ndomain walls [5], modification of ferromagnetic resonance [ 6], etc. It is\nwell known that the spin torque transfer from spin-polarize d electrons\nto lattice leads to appearance of a negative damping. At some current\ndensity, this negative damping overcomes the positive (Gil bert) damping\n∗E-mail: epshtein36@mail.ru\n1with occurring instability of the original magnetic configu ration. The cor-\nresponding current density is high enough, of the order of 107A/cm2.\nThis, naturally, stimulates attempts to lower this thresho ld. Various ways\nwere proposed, such as using magnetic semiconductors [7], i n which the\nthreshold current density can be lower down to 105–106A/cm2because\nof their low saturation magnetization. However, using of su ch materials\nrequires, as a rule, low temperatures because of low Curie te mperature.\nBesides, the ferromagnetic resonance frequency is rather l ow in this case.\nIn connection with these difficulties, the other approaches w ere pro-\nposed, based on high spin injection [8] or joint action of ext ernal magnetic\nfield and spin-polarized current [9, 10]. It seems promising , also, using\nmagnetic junction of ferromagnet–antiferromagnet type, i n which the fer-\nromagnet (FM) acts as an injector of spin-polarized electro ns. The anti-\nferromagnetic (AFM) layer, in which the magnetic sublattic es are canted\nby external magnetic field, may have very low magnetization t hat pro-\nmotes low threshold [11]. The AFM resonance frequency may be both\nlow and high reaching 1012s−1, i.e. terahertz (THz) range. However,\ninvestigation and application of THz resonances is prevent ed because of\ntheir large damping. Such a damping in ferromagnetic juncti ons can be\nsuppressed, as mentioned above, by means of spin-polarized current. The\nquestion arises about possibility of such a suppression in F M/AFM junc-\ntions. Note, that this problem has been paid attention of a nu mber of\nauthors [12]–[21].\nAnotherinterestingfeatureoftheFM/AFMjunctionswithsp in-polarized\ncurrent is the possibility of canting the AFM structure by sp in-polarized\ncurrent without magnetic field.\n2 The equations of motion\nLet us consider a FM/AFM junction (Fig. 1) with current flowin g perpen-\ndicular to layers, along xaxis. An ultrathin spacer layer is placed between\nthe FM and AFM layers to prevent direct exchange coupling bet ween the\nlayers. An external magnetic field is parallel to the FM magne tization\nand lies in the layer plane yz. The simplest AFM model is used with two\nequivalent sublattices.\nThe AFM energy (per unit area), with uniform and nonuniform e x-\nchange, anisotropy, external magnetic field, and the sdexchange inter-\naction of the conduction electrons with the magnetic lattic e taking into\naccount, takes the form [22]\nW=/integraldisplayLAFM\n0dx/braceleftBigg\nΛ(M1·M2)+1\n2α/braceleftBigg/parenleftbigg∂M1\n∂x/parenrightbigg2\n+/parenleftbigg∂M2\n∂x/parenrightbigg2/bracerightBigg\n+α′/parenleftbigg∂M1\n∂x·∂M2\n∂x/parenrightbigg\n−1\n2β/braceleftbig\n(M1·n)2+(M2·n)2/bracerightbig\n−β′(M1·n)(M2·n)−((M1+M2)·H)\n−αsd((M1+M2)·m)/bracerightBigg\n, (1)\n2FM AFMNM\nMFH−+n\nxyz\n0 LAFMj/e\nFigure 1: Scheme of the ferromagnet (FM)–antiferromagnet (AF M) junction;\nNM being a nonmagnetic layer. The main vector directions are shown.\nwhereM1,M2are the sublattice magnetization vectors, Λ is the uniform\nexchange constant, α, α′are the intra- and inter-sublattice nonuniform\nexchange constants, respectively, β, β′are the corresponding anisotropy\nconstants, nis the unit vector along the anisotropy axis, His the ex-\nternal magnetic field, mis the conduction electron magnetization, αsdis\nthe dimensionless sdexchange interaction constant. We do not include\ndemagnetization term because its contribution is small com pared to the\nuniform exchange. The integral is taken over the AFM layer th ickness\nLAFM. We are interested in the spin-polarized current effect on th e AFM\nlayer, so we consider a case of perfect FM injector with pinne d lattice\nmagnetization and without disturbance of the electron spin equilibrium,\nthat allows to not include the FM layer energy in Eq. (1).\nTwo mechanisms are known of the spin-polarized current effec t on\nthe magnetic lattice, namely, spin transfer torque (STT) [1 , 2] and an\nalternative mechanism [23, 24] due to the spin injection and appearance\nof nonequilibrium population of the spin subbands in the col lector layer\n(this is the AFM layer, in our case). In the case of antiparall el relative\norientation oftheinjectorandcollector magnetization ve ctors, suchastate\nbecomes energetically unfavorable at high enough current d ensity, so that\nthe antiparallel configuration switches to parallel one (su ch a process in\nFM junction is considered in detail in review [25]). The latt er mechanism\nis described with the sdexchange term in Eq. (1). As to the former\nmechanism, it is of dissipative character (it leads to negat ive damping),\nso that it is taken into account by the boundary conditions (s ee below),\nnot the Hamiltonian.\nThe equations of the sublattice motion with damping taking i nto ac-\ncount take the form\n∂Mi\n∂t−κ\nM0/bracketleftbigg\nMi×∂Mi\n∂t/bracketrightbigg\n+γ/bracketleftBig\nMi×H(i)\neff/bracketrightBig\n= 0 (i= 1,2),(2)\n3whereM0is the sublattice magnetization, κis the damping constant,\nH(i)\neff=−δW\nδMi(i= 1,2) (3)\nare the effective fields acting on the corresponding sublatti ces.\nFrom Eqs. (1)–(3) the equations are obtained for the total ma gnetiza-\ntionM=M1+M2and antiferromagnetism vector L=M1−M2:\n∂M\n∂t−1\n2κ\nM0/braceleftbigg/bracketleftbigg\nM×∂M\n∂t/bracketrightbigg\n+/bracketleftbigg\nL×∂L\n∂t/bracketrightbigg/bracerightbigg\n+γ[M×H]+γ[M×Hsd]\n+1\n2γ(β+β′)(M·n)[M×n]+1\n2γ(β−β′)(L·n)[L×n]\n+1\n2γ(α+α′)/bracketleftbigg\nM×∂2M\n∂x2/bracketrightbigg\n+1\n2γ(α−α′)/bracketleftbigg\nL×∂2L\n∂x2/bracketrightbigg\n= 0,(4)\n∂L\n∂t−1\n2κ\nM0/braceleftbigg/bracketleftbigg\nL×∂M\n∂t/bracketrightbigg\n+/bracketleftbigg\nM×∂L\n∂t/bracketrightbigg/bracerightbigg\n+γ[L×H]+γ[L×Hsd]−γΛ[L×M]\n+1\n2γ(β+β′)(M·n)[L×n]+1\n2γ(β−β′)(L·n)[M×n]\n+1\n2γ(α+α′)/bracketleftbigg\nL×∂2M\n∂x2/bracketrightbigg\n+1\n2γ(α−α′)/bracketleftbigg\nM×∂2L\n∂x2/bracketrightbigg\n= 0,(5)\nwhere\nHsd(x) =δ\nδM(x)/integraldisplayLAFM\n0/parenleftbig\nM(x′)·m(x′)/parenrightbig\ndx′(6)\nis the effective field due to sdexchange interaction. This field determines\nthe spin injection contribution to the interaction of the co nduction elec-\ntrons with the antiferromagnet lattice.\nTo findHsd(x) field, the conduction electron magnetization m(x) is\nto be calculated. The details of such calculations are prese nted in our\npreceding papers [26, 9]. Here we adduce the result for the ca se, where\nthe antiferromagnet layer thickness LAFMis small compared to the spin\ndiffusion length lwith the current flow direction corresponding to the\nelectron flux from FM to AFM:\nm= (m+∆m)ˆM,∆m=µBτQj\neLAFM/parenleftBig\nˆM(0)·ˆMF/parenrightBig\n, (7)\nwheremis the equilibrium (in absence of current) electron magneti zation,\n∆mis the nonequilibrium increment due to current, ˆM=M/|M|is the\nunit vector along the AFM magnetization, ˆMFis the similar vector for\nFM,µBis the Bohr magneton, eis the electron charge, τis the electron\nspin relaxation time, jis the current density.\nIt should have in mind in varying the integral (6), that the el ectron\nmagnetization mdepends on the vector Morientation relative to the FM\nmagnetization vector MF. From Eqs. (6) and (7) we have [9]\nHsd=αsdmˆM+αsdµBτQj\neLAFM/parenleftBig\nˆM(0)·ˆMF/parenrightBig\nˆM\n+αsdµBτQj\neˆMFδ(x−0). (8)\n4By substitution (8) into (4) and (5), we obtain\n∂M\n∂t−1\n2κ\nM0/braceleftbigg/bracketleftbigg\nM×∂M\n∂t/bracketrightbigg\n+/bracketleftbigg\nL×∂L\n∂t/bracketrightbigg/bracerightbigg\n+γ[M×H]+γαsdµBτQj\ne/bracketleftBig\nM׈MF/bracketrightBig\nδ(x−0)\n+1\n2γ(β+β′)(M·n)[M×n]+1\n2γ(β−β′)(L·n)[L×n]\n+1\n2γ(α+α′)/bracketleftbigg\nM×∂2M\n∂x2/bracketrightbigg\n+1\n2γ(α−α′)/bracketleftbigg\nL×∂2L\n∂x2/bracketrightbigg\n= 0,(9)\n∂L\n∂t−1\n2κ\nM0/braceleftbigg/bracketleftbigg\nL×∂M\n∂t/bracketrightbigg\n+/bracketleftbigg\nM×∂L\n∂t/bracketrightbigg/bracerightbigg\n+γ[L×H]+γαsdµBτQj\ne/bracketleftBig\nL׈MF/bracketrightBig\nδ(x−0)\n−γ/parenleftbigg\nΛ−αsdm\nM−αsdµBτQj\neLAFMM/parenleftBig\nˆM(0)·ˆMF/parenrightBig/parenrightbigg\n[L×M]\n+1\n2γ(β+β′)(M·n)[L×n]+1\n2γ(β−β′)(L·n)[M×n]\n+1\n2γ(α+α′)/bracketleftbigg\nL×∂2M\n∂x2/bracketrightbigg\n+1\n2γ(α−α′)/bracketleftbigg\nM×∂2L\n∂x2/bracketrightbigg\n= 0.(10)\n3 The boundary conditions\nThe equations of motion (9) and (10) contain derivative over the space\ncoordinate x. Therefore, boundary conditions at the AFM layer surfaces\nx= 0 and x=LAFMare need to find solutions. The way of derivation\nwas described in Ref. [9] in detail. The conditions depend on the electron\nspin polarization and are determined by the continuity requ irement of the\nspin currents at the interfaces.\nThe terms with the space derivative in Eq. (9) may be written i n the\nform of a divergency:\n1\n2γ(α+α′)/bracketleftbigg\nM×∂2M\n∂x2/bracketrightbigg\n+1\n2γ(α−α′)/bracketleftbigg\nL×∂2L\n∂x2/bracketrightbigg\n=∂\n∂x/braceleftbigg1\n2γ(α+α′)/bracketleftbigg\nM×∂M\n∂x/bracketrightbigg\n+1\n2γ(α−α′)/bracketleftbigg\nL×∂L\n∂x/bracketrightbigg/bracerightbigg\n≡∂JM\n∂x. (11)\nTheJMvector is the lattice magnetization flux density.\nLet us integrate Eq. (9) over xwithin narrow interval 0 < x < ε with\nsubsequent passing to ε→+0 limit. Then only the mentioned terms with\nthe space derivative and the singular term with delta functi on will con-\ntribute to the integral. As a result, we obtain an effective ma gnetization\nflux density with sdexchange contribution at the AFM boundary x= +0\ntaking into account:\nJeff(+0) =JM(+0)+γαsdµBτQj\ne/bracketleftBig\nM(+0)׈MF/bracketrightBig\n.(12)\n5The magnetization flux density coming from the FM injector is\nJ(−0) =µBQ\nejˆMF. (13)\nThe component J/bardbl=/parenleftBig\nJ(−0)·ˆM(+0)/parenrightBig\nˆM(+0) remains with the elec-\ntrons, while the rest,\nJ⊥=J(−0)−J/bardbl=µBQ\nej/braceleftBig\nˆMF−ˆM(+0)/parenleftBig\nˆMF·ˆM(+0)/parenrightBig/bracerightBig\n=−µBQ\neM2j/bracketleftBig\nM(+0)×/bracketleftBig\nM(+0)׈MF/bracketrightBig/bracketrightBig\n, (14)\nis transferred to the AFM lattice owing to conservation of th e magnetiza-\ntion fluxes [1, 2].\nBy equating the magnetization fluxes (12) and (14), we obtain\nJM=−µBQ\neM2j/bracketleftBig\nM×/bracketleftBig\nM׈MF/bracketrightBig/bracketrightBig\n−γαsdµBτQ\nej/bracketleftBig\nM׈MF/bracketrightBig\n,(15)\nall theMvectors being taken at x= +0.\nSince the AFM layer thickness is small compared to the spin di ffusion\nlength and the exchange length, we may use the macrospin appr oximation\nwhich was described in detail in Ref. [9]. In this approximat ion, the\nmagnetization changes slowly within the layer thickness. T his allows to\nwrite\n∂JM\n∂x≈JM(LAFM)−JM(+0)\nLAFM=−JM(+0)\nLAFM, (16)\nbecause the magnetization flux is equal to zero at the interfa ce between\nAFM and the nonmagnetic layer closing the electric circuit, JM(LAFM) =\n0. This allows to exclude the terms with space derivative fro m Eq. (9).\nIn the rest terms, M(x, t) andL(x, t) quantities are replaced with their\nvalues at x= 0. Then Eq. (9) takes a more simple form:\ndM\ndt−1\n2κ\nM0/braceleftbigg/bracketleftbigg\nM×dM\ndt/bracketrightbigg\n+/bracketleftbigg\nL×dL\ndt/bracketrightbigg/bracerightbigg\n+γ[M×H]\n+1\n2γ(β+β′)(M·n)[M×n]+1\n2γ(β−β′)(L·n)[L×n]\n+K/bracketleftBig\nM×/bracketleftBig\nM׈MF/bracketrightBig/bracketrightBig\n+P/bracketleftBig\nM׈MF/bracketrightBig\n= 0, (17)\nwhere\nK=µBQ\neLAFMM2j, P =γαsdµBτQ\neLAFMj. (18)\nThe term with delta function does not present here, since it i s taken into\naccount in the boundary conditions.\nNow we are to use again the macrospin approximation to exclud e the\nspace derivatives from Eq. (10), too.\nOwing to known relationships [22] between MandLvectors, namely,\nM2+L2= 4M2\n0and (M·L) = 0, we have the following conditions:\n/parenleftbigg\nM·∂M\n∂t/parenrightbigg\n+/parenleftbigg\nL·∂L\n∂t/parenrightbigg\n= 0,/parenleftbigg\nL·∂M\n∂t/parenrightbigg\n+/parenleftbigg\nM·∂L\n∂t/parenrightbigg\n= 0.(19)\n6By substituting Eqs. (10) and (17) in (19) we find that conditi ons (19)\nare fulfilled if the terms in (10)\n1\n2γ(α+α′)/bracketleftbigg\nL×∂2M\n∂x2/bracketrightbigg\n+1\n2γ(α−α′)/bracketleftbigg\nM×∂2L\n∂x2/bracketrightbigg\n≡X(20)\nsatisfy the following equations:\n(X·M)+K/parenleftBig\nL·/bracketleftBig\nM×/bracketleftBig\nM׈MF/bracketrightBig/bracketrightBig/parenrightBig\n+P/parenleftBig\nL·/bracketleftBig\nM׈MF/bracketrightBig/parenrightBig\n= 0,\n(X·L) = 0. (21)\nLet us decompose the considered Xvector on three mutually orthog-\nonal vectors:\nX=aM+bL+cγ[L×M]. (22)\nThe substitution (22) in (21) gives a=K/parenleftBig\nL·ˆMF/parenrightBig\n−P/parenleftBig\n[L×M]·ˆMF/parenrightBig\n,\nb= 0. As to ccoefficient, it is a current-induced correction to the co-\nefficient of γ[L×M] term in Eq. (10), i.e., a correction to the uniform\nexchange constant Λ. Let us estimate the correction. Multip lying (22)\nscalarly by [ L×M] with (20) taking into account gives\nc=1\nM2L2/parenleftbigg\n[L×M]·/braceleftbigg1\n2γ(α+α′)/bracketleftbigg\nL×∂2M\n∂x2/bracketrightbigg\n+1\n2γ(α−α′)/bracketleftbigg\nM×∂2L\n∂x2/bracketrightbigg/bracerightbigg/parenrightbigg\n=1\n2/braceleftbigg\n(α+α′)1\nM2/parenleftbigg\nM·∂2M\n∂x2/parenrightbigg\n−(α−α′)1\nL2/parenleftbigg\nL·∂2L\n∂x2/parenrightbigg/bracerightbigg\n. (23)\nIt is seen that c∼α/L2\nAFM, while Λ ∼α/a2, where ais the lattice\nconstant [22]. Since LAFM≫a, the mentioned correction to Λ may be\nneglected.\nAs a result, Eq. (10) takes the form\ndL\ndt−1\n2κ\nM0/braceleftbigg/bracketleftbigg\nL×dM\ndt/bracketrightbigg\n+/bracketleftbigg\nM×dL\ndt/bracketrightbigg/bracerightbigg\n+γ[L×H]−γΛ[L×M]\n+1\n2γ(β+β′)(M·n)[L×n]+1\n2γ(β−β′)(L·n)[M×n]\n+K/bracketleftBig\nL×/bracketleftBig\nM׈MF/bracketrightBig/bracketrightBig\n+P/bracketleftBig\nL׈MF/bracketrightBig\n= 0. (24)\nHere, Λ constant contains also the equilibrium contributio n of the con-\nduction electrons −αsdm/M.\nEquations (17) and (24) are the result of applying the macros pin con-\ncept to AFM. It is shown that such an approximation may be just ified\nformally for AFM layer. Earlier, it was justified for FM layer s [1, 2] and\ngeneralized [9] with spin injection taking into account. Th e macrospin\napproach corresponds well to experimental conditions and s implifies cal-\nculations substantially. The terms with Kcoefficient in Eqs. (17), (24)\ndescribe effect of STT mechanism, while the terms with Pcoefficient take\nthe spin injection effect into account.\n74 The magnetization wave spectrum and\ndamping\nWe assume that the easy anisotropy axis lies in the plane of AF M layer\nand is directed along yaxis, the FM magnetization vector is parallel to\nthe positive direction of zaxis, the external magnetic field is parallel to z\naxis too (see Fig. 1).\nWe are interesting in behavior of small fluctuations around t he steady\nstateM={0,0,Mz},L={0,Ly,0}, i.e. thesmallquantities Mx, My,/tildewiderMz=\nMz−Mz, Lx,/tildewideLy=Ly−Ly, Lz.\nLet us project Eqs. (17), (24) to the coordinate axes and take the\nterms up to the first order. The zero order terms are present on ly in the\nprojection of Eq. (24) to xaxis. They give\nMz=Hz+P\nγ\nΛ+1\n2(β−β′)≈Hz+P\nγ\nΛ,\nLy=±/radicalBig\n4M2\n0−M2\nz≈ ±2M0. (25)\nNote that the spin-polarized current takes part in creating magnetic\nmoment together with the external magnetic field dueto thesp in injection\ninduced interaction of the electron spins with the lattice [ 23, 24], which P\nparameter in Eq. (25) corresponds to. Suchaninteraction le ads toappear-\nance of an effective magnetic field parallel to the injector ma gnetization.\nAs a result, a canted antiferromagnet configuration may be cr eated with-\nout magnetic field. However, such a configuration correspond s to parallel\norientation of FM and AFM layers, M/bardblMF. As is shown below, the insta-\nbility does not occur with this orientation, so that an exter nal magnetic\nfield is to be applied to reach instability.\nWith Eq. (25) taking into account, the equations for the first order\nquantities take the form\ndMx\ndt−1\n2κ\nM0/braceleftbigg\n−MzdMy\ndt+LydLz\ndt/bracerightbigg\n+(γHz+P)My\n−1\n2γ(β+β′)MzMy−1\n2γ(β−β′)LyLz+KMzMx= 0,(26)\ndMy\ndt−1\n2κ\nM0MzdMx\ndt−(γHz+P)Mx+KMzMy= 0,(27)\nd/tildewiderMz\ndt+1\n2κ\nM0LydLx\ndt+1\n2γ(β−β′)LyLx= 0, (28)\ndLx\ndt−1\n2κ\nM0/braceleftBigg\nLyd/tildewiderMz\ndt−Mzd/tildewideLy\ndt/bracerightBigg\n−γHzLy\nMz/tildewiderMz= 0,(29)\nd/tildewideLy\ndt−1\n2κ\nM0MzdLx\ndt−1\n2γ(β−β′)MzLx= 0, (30)\n8dLz\ndt+1\n2κ\nM0LydMx\ndt+(γHz+P)Ly\nMzMx+KMzLz= 0.(31)\nThe set of equations (26)–(31) splits up to two mutually inde pendent\nsets with respect to ( Mx, My, Lz) and (Lx,/tildewideLy,/tildewiderMz). They describe two\nindependent spectral modes, one of them corresponds to prec ession of the\nAFMmagnetization vectoraroundthemagnetic field, while an other tope-\nriodic changes ofthevectorlength alongthemagnetic field. Webeginwith\nthe spectrum and damping of the first mode. We consider monoch romatic\noscillation with ωangular frequency and put Mx, My, Lz∼exp(−iωt).\nThen we obtain from Eqs. (26), (27), (31)\n/parenleftbig\n−iω+KMz/parenrightbig\nMx+/braceleftbigg\nγHz+P−1\n2γ(β+β′)Mz−1\n2iκω\nM0Mz/bracerightbigg\nMy\n−/braceleftbigg1\n2γ(β−β′)−1\n2iκω\nM0/bracerightbigg\nLyLz= 0, (32)\n/parenleftbig\n−iω+KMz/parenrightbig\nMy−/braceleftbigg\nγHz+P−1\n2iκω\nM0Mz/bracerightbigg\nMx= 0,(33)\n/parenleftbig\n−iω+KMz/parenrightbig\nLz+/braceleftbigg\nγΛ+1\n2γ(β−β′)−1\n2iκω\nM0/bracerightbigg\nLyMx= 0.(34)\nNote that aforementioned additivity (in the algebraic sens e, the sign\ntakingintoaccount) oftheexternalmagnetic field andthein jection-driven\neffective field takes place not only in the steady magnetizati on (25), but\nalso in the oscillations of the magnetization and antiferro magnetism vec-\ntors, so that both fields appear in Eqs. (32), (33) “on an equal footing”.\nUsually, Λ ≫β, β′. With these inequalities and stationary solu-\ntion (25) taking into account we find the dispersion relation for the mag-\nnetization oscillation\n(1+κ2)ω2+2iνω−ω2\n0= 0, (35)\nwhere\nω0=/radicalBig\n2γ2HAHE+(KMz)2+(γHz+P)2, (36)\nν=κγHE+KMz, (37)\nHE= ΛM0is the exchange field, HA= (β−β′)M0is the anisotropy\nfield. Formulae (36) and (37) (without current terms KMzandP) coin-\ncide with known ones [22, 27]. At HE∼106–107Oe,HA∼103Oe we\nhave oscillations in THz range, ω0∼1012s−1. In absence of current the\ndamping is rather high: at κ∼10−2\nν\nω0=κ/radicalbigg\nHE\n2HA∼1. (38)\n9Let us consider the contribution of spin-polarized current to the fre-\nquency and damping of AFM resonance. At first we consider STT m ech-\nanism effect [1, 2]. According to (18) and (25),\nKMz=µBQΛ\neLAFMHzj. (39)\nAtHz<0, that corresponds to direction of the magnetic field (and,\ntherefore, the AFM magnetization) opposite to the FM magnet ization,\nthis quantity is negative. The total attenuation becomes ne gative also\n(an instability occurs), if\nj >eκγM 0|Hz|LAFM\nµBQ≡j0. (40)\nAtκ∼10−2,γM0∼1010s−1,|Hz| ∼102Oe,LAFM∼10−6cm,\nQ∼1 we have j0∼105A/cm2. Atjnear toj0weakly damping THz\noscillation can be obtained. At j > j0, instability occurs which may lead\nto either self-sustained oscillations, or a dynamic statio nary state. The\nlatter disappears with the current turning off. To answer the question\nabout future of the instability it is necessary to go out the s cope of the\nlinear approximation. We have simulated numerically the be havior of\nthe AFM magnetization behind the linear approximation (see section 8\nbelow).\nThespin-polarized currentcontributesalsototheoscilla tion frequency.\nAt the mentioned parameter values, we have |KMz| ∼1012s−1that is\ncomparable with the frequency in absence of the current. Thi s allows\ntuning the frequency by the current or excite parametric res onance by\nmeans of the current modulation.\n5 Current-induced spin injection effect\nNow let us discuss the injection mechanism effect [23, 24]. As mentioned\nbefore, the role of the mechanism is reduced to addition of an effective\nfieldP/γto the external magnetic field. At reasonable parameter valu es,\nthat field is much less than the exchange field HE, so that it does not\ninfluence directly the eigenfrequency (36). Nevertheless, that field can\nmodify substantially the contribution of the STT mechanism , because\nEq. (39) with (25) taking into account now takes the form\nKMz=µBQΛ\neLAFM(Hz+P/γ)j. (41)\nSuch a modification leads to substantial consequences. At Hz<0,P <\nγ|Hz|the instability threshold (40) is lowered, since |Hz|−P/γdifference\nappears now instead of |Hz|. If, however, P > γ|Hz|then the AFM\nmagnetization steady state\nMz=Hz+P/γ\nΛ(42)\nbecomes positive that corresponds to the parallel (stable) relative orienta-\ntion of the FM and AFM layers. In this case, the turning on curr ent leads\n10to switching the antiparallel configuration (stated before hand by means\nof an external magnetic field) to parallel one. With turning o ff current,\nthe antiparallel configuration restores.\nSincethementionedinjection-drivenfielddependsonthecu rrent(see(18)),\nthe instability condition (40) is modified and takes the form\nj0\n1+η< j j 0/ηthe antiparallel configuration switches to parallel one. Th e\nrelative contribution of the injection mechanism is determ ined with η\nparameter. At typical values, αsd∼104,κ∼10−2,γM0∼1010s−1,\nτ∼10−12s, this parameter is of the order of unity, so that the injecti on\neffect may lower noticeably the instability threshold.\nNow let us return to the set of equations (26)–(31) and consid er the\nsecond mode describing with Eqs. (28)–(30). The current infl uences this\nmode by changing steady magnetization Mzdue to the injection effective\nfield effect (see (26)), while the STT mechanism does not influe nce this\nmode. A calculation similar to previous one gives the former dispersion\nrelation (35), but now\nω2\n0= 2γ2HEHAγHz\nγHz+P, (44)\nν=κγHEγHz\nγHz+P. (45)\nAtHz<0,P >|Hz|, thatcorrespondstocurrentdensity j > j0/η, the\ntotal attenuation becomes negative, while the frequency be comes imagi-\nnary, that means switching the antiparallel configuration t o parallel one.\n6 Easy plane type antiferromagnet\nLetusconsiderbrieflythesituation whereAFMhaseasy-plan eanisotropy.\nWe take the AFM layer yzplane as the easy plane and xaxis as the (hard)\nanisotropy axis. The magnetic field, as before, is directed a longzaxis.\nWithout repeating calculations, similar to previous ones, we present\nthe results. A formal difference appears only in Eq. (36) for t he eigenfre-\nquencyω0of the first of the modes considered above. We have for that\nfrequency\nω0=/radicalBig\n(γHz+P)2+(KMz)2. (46)\nThe damping has the former form (37), so that the instability threshold\nis determined with former formula (43).\nIn absence of the current ( K= 0, P= 0) with not too small damping\ncoefficient κ, the frequency appears to be much less than damping, so\nthat the corresponding oscillations are not observed. The c urrent effect\nincreases the frequency, on the one hand, and decreases the d amping (at\nHz<0), on the other hand, that allows to observe oscillation reg ime.\n117 Fluctuation effect\nIt follows from Eq. (43) that the threshold current density i s proportional\nto the external magnetic field strength |Hz|and decreases with the field\nlowering. A question arises about permissible lowest limit of the total field\n|Hz|+P/γ. In accordance with Eq. (25), such a limit may be the field\nwhich create magnetization |Mz|comparable with its equilibrium value\ndue to thermal fluctuations. Let us estimate this magnetizat ion and the\ncorresponding field.\nThe AFM energy change in Vvolume under canting the sublattice\nmagnetization vectors with θ <180◦angle between them is\n∆E= ΛM2\n0(1−cosθ)V=1\n2ΛVM2\nz, (47)\nthe anisotropy energy being neglected compared to the excha nge energy.\nThe equilibrium value of the squared magnetization is calcu lated using\nthe Gibbs distribution:\n/an}bracketle{tM2\nz/an}bracketri}ht=∞/integraltext\n−∞M2\nzexp/parenleftbigg\n−ΛVM2\nz\n2kT/parenrightbigg\ndMz\n∞/integraltext\n−∞exp/parenleftbigg\n−ΛVM2\nz\n2kT/parenrightbigg\ndMz=kT\nΛV(48)\n(strictlyspeaking, themagnetization maybechangedwithi n(−2M0,2M0)\ninterval, however, Λ VM2\n0≫kT, so that the integration limits may be\ntaken infinity).\nTo observe the effects described above, the magnetization Mzwhich\nappears under joint action of the external field and the curre nt (see (25))\nshould exceed in magnitude the equilibrium magnetization /an}bracketle{tM2\nz/an}bracketri}ht1/2. At\nthe current density j=j0/(1+η) corresponding to the instability thresh-\nold, this condition is fulfilled at magnetic field\n|Hz|>/radicalbigg\nΛkT\nV(1+η)≡Hmin. (49)\nAt Λ∼104,η∼1,LAFM∼10−6cm and lateral sizes of the switched\nelement 10 ×10µm2we have V∼10−12cm3andHmin≈30 Oe at room\ntemperature. This limit can be decreased under larger eleme nt size.\nIt should be mentioned also about other mechanisms of AFM can ting.\nThe most known and studied one is the relativistic Dzyaloshi nskii–Moria\neffect(see, e.g.[22,28]). Besides, possible mechanismsha vebeendiscussed\ndue to competition between sdexchange and direct exchange interaction\nof the magnetic ions in the lattice [29]. At the same time, the re are no\nindications, to our knowledge, about measurements of canti ng in conduc-\ntive AFM. So, present theory is related to conductive AFM, in which the\nlattice canting is determined with external magnetic field.\n8 Simulation\nWiththepurposeofsimulating, itisconvenienttomodifysl ightlyEqs.(17)\nand (24), namely: i) to use sublattice magnetizations M1,M2instead of\n12M,L, ii) to describe damping by the Landau–Lifshitz representa tion with\ndouble vector product, and iii) to introduce the following d imensionless\nvariables:\nˆMi=Mi/M0(i= 1,2),h=H/M0, T=γM0t\n1+κ2,\nK0=µBQj\neLAFMγM2\n0, P0=η0K0, η0=αsdγM0τ.\nWith these variables, the set of equations (17), (24) take th e form\ndˆMi\ndT=−/bracketleftBig\nˆMi×h(i)\neff/bracketrightBig\n−κ/bracketleftBig\nˆMi×/bracketleftBig\nˆMi×h(i)\neff/bracketrightBig/bracketrightBig\n, i= 1,2,(50)\nh(1)\neff=h−ΛˆM2+β/parenleftBig\nˆM1·n/parenrightBig\nn+β′/parenleftBig\nˆM2·n/parenrightBig\nn+P0ˆMF\n+K0/bracketleftBig/parenleftBig\nˆM1+ˆM2/parenrightBig\n׈MF/bracketrightBig\n/parenleftBig\nˆM1+ˆM2/parenrightBig2≡F/parenleftBig\nˆM1,ˆM2/parenrightBig\n, (51)\nh(2)\neff=F/parenleftBig\nˆM2,ˆM1/parenrightBig\n. (52)\nIt is seen from Eqs. (50)– (52) that the motions of two sublatt ices are cou-\npled each other. The sources of such a coupling are the unifor m exchange,\nintersublattice anisotropy (coefficient β′), as well the effective field due to\nthe spin torque (the last term in Eq. (51)).\nWe assume the following orders of magnitude of the used param e-\nters:κ∼10−3–10−2, Λ∼103–104,β, β′∼10−1(β/ne}ationslash=β′),M0∼103\nG,γM0∼1010s−1. Under such conditions, h= 1 corresponds to mag-\nnetic field H∼103Oe, and T= 1 corresponds to time t∼10−10s.\nWithαsd∼104,τ∼10−12s,Q∼1,LAFM∼10−7cm we have η0∼102,\nandK0= 1 value corresponds to current density j∼107A/cm2.\nEquations (50)– (52) written in coordinates represent six o rdinary dif-\nferential equations of the first order in the Cauchy form for ˆM1x,ˆM1y,\nˆM1z,ˆM2x,ˆM2y,ˆM2z. The equations are not mutually independent be-\ncause of the normalization conditions\n/vextendsingle/vextendsingle/vextendsingleˆM1/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingleˆM2/vextendsingle/vextendsingle/vextendsingle= 1. (53)\nNevertheless, all the six equations are used in our simulati on, while the\nmentioned normalization conditions serve for checking cor rectness of the\ncalculations.\nThe simulation was carried out by means of Simulink program i n\nMATLAB system with using Differential Equation Editor (DEE) . Right-\nhand sides of the equations resolved with respect to derivat ives were\nentered into the DEE block. The parameters Λ , β, β′, κ, K 0, P0were\ngiven as input signals, while three projections of the magne tization vec-\ntorˆMx=ˆM1x+ˆM2x,ˆMy=ˆM1y+ˆM2y,ˆMz=ˆM1z+ˆM2zwere\noutput to oscilloscope blocks. Besides,/vextendsingle/vextendsingle/vextendsingleˆM1/vextendsingle/vextendsingle/vextendsingle=/parenleftBig\nˆM2\n1x+ˆM2\n1y+ˆM2\n1z/parenrightBig1/2\n13and/vextendsingle/vextendsingle/vextendsingleˆM2/vextendsingle/vextendsingle/vextendsingle=/parenleftBig\nˆM2\n2x+ˆM2\n2y+ˆM2\n2z/parenrightBig1/2\nvalues were output to digital dis-\nplays; these values must be equal to 1 (or, at least, be close t o 1) under\ncorrect calculation.\nIn the present work, we assume that the magnetic field and curr ent are\nturned on at the initial time instant T= 0 and hold constant. However,\nthe procedure used allows to consider arbitrary time depend ence of these\nquantities, specifically, to vary turning on and turning off i nstants.\nWe began simulation with “verifying” results of the linear t heory (ac-\ntually, this was a test for the model adequacy). As above, we a ssume\nthat the AFM layer lies in yzplane, the current flows along xaxis, the\neasy anisotropy axis coincides with yaxis (n={0,1,0}), the current is\npolarized along the positive direction of the zaxis (ˆMF={0,0,1}), the\nmagnetic field is collinear with zaxis (h={0,0, hz}). In such a configu-\nration, the collinear relative orientation of the FM and AFM layer magne-\ntizations is stationary (although, possibly, unstable), s othat asmall initial\ndeviation from such orientation was given to imitate therma l fluctuations.\nThe initial value of the ˆMzcomponent was chosen to be equal to the\nequilibrium value ˆMz=hz/Λ in the given magnetic field without current.\nThus, at Λ = 103, hz=−1 initial conditions ˆM1z=ˆM2z=−5×10−4\n(so that ˆMz=−1×10−3),ˆM1x=ˆM2x=−5×10−6,ˆM1y=−ˆM2y=\n(1−ˆM2\n1x−ˆM2\n1z)1/2are used taking the normalization conditions (53) into\naccount.\nWith the dimensionless variables, the main results of the li near theory\ntake the following form.\nThe stationary magnetization in zdirection is\nˆMz≡ˆM1z+ˆM2z=hz+η0K0\nΛ+1\n2(β−β′)≈hz+η0K0\nΛ.(54)\nUnder deviation from the stationary value, two modes appear with di-\nmensionless (referred to γM0) frequencies ω1,2and damping ν1,2defined\nwith the following formulae:\nω2\n1= 2(β−β′)Λ+(hz+η0K0)2+/parenleftbiggΛK0\nhz+η0K0/parenrightbigg2\n,(55)\nν1=κΛ+ΛK0\nhz+η0K0, (56)\nω2\n2=2(β−β′)Λhz\nhz+η0K0, (57)\nν2=κΛhz\nhz+η0K0. (58)\nThe instability of the antiparallel relative orientation o f the FM and\nAFM layers at hz<0 occurs in the current density value range defined\nwith inequalities\nκ|hz|\n1+κη0< K0<|hz|\nη0. (59)\nTo start, the case of absence of the current ( K0= 0) has been con-\nsidered. Perfect agreement has been observed with Eqs. (55) –(58). In\nparticular, the magnetization oscillation frequency drop s abruptly when\n140 100 200 300 400 500 600−3.8−3.6−3.4−3.2−3−2.8−2.6x 10−4\nt, psMz/M0\nFigure 2: Oscillations of the longitudinal magnetization in the instability range\natK0= 7×10−3.\nwe putβ=β′, and the oscillations disappear completely, if we take, mor e-\nover,h= 0. At β/ne}ationslash=β′the observed frequency consists with Eq. (55).\nIn absence of the magnetic field, the frequencies of the two mo des coin-\ncide, so that the oscillation takes the form of a simple sinus oid. Under\nrising magnetic field, the frequencies become different, and beats appear\nbecause of interaction between the modes. The simulation re sults consist,\nalso, with Eq. (54) for the stationary magnetization in pres ence of the\nmagnetic field. Turning on the magnetic field at T= 0 instant leads to an\naperiodic transient process which decays completely by T= 0.4, following\nwhich the magnetization remains constant value determined by Eq. (54).\nAt Λ = 103, κ= 10−2, η0= 102, hz=−1 the instability predicted by\nthe linear theory must occur at K0= 5×10−3and disappear at K0=\n1×10−2. In our numerical experiments, increasing K0parameter from\nzero to the indicated threshold leads, in accordance with Eq . (56), to\ndecrease of the damping of the magnetization vector precess ion about z\naxis because of the negative damping caused by the STT mechan ism.\nIncidentally, the absolute value of the (negative) ˆMzcomponent decreases\nfrom 10−3to 10−4because of influence of the injection mechanism, that\nconsists, also, with Eq.(54). Thus, thesimulation results agree completely\nwith the theory in the range below the instability threshold .\nOf course, theinstability range ( K0>5×10−3), which is not described\n150100 200 300 400 500 600 700−2−1.5−1−0.500.511.52x 10−4\nt, psMx/M0\nFigure 3: Beats of the transverse magnetization in the instability ra nge atK0=\n8×10−3.\nby the linear theory, is of more interest. At K0= 5.1×10−3undamped os-\ncillations are observed. At K0= 7×10−3the precession oscillations of the\ntransverse components ˆMx,ˆMybecome almost sinusoidal with ∆ T≈0.2\nperiod (this corresponds to angular frequency ∼3×1011s−1at the chosen\nparameter values). The longitudinal component ˆMzoscillates periodically\nwith a negative stationary background, however, the oscill ation form is far\nfrom sinusoidal one (Fig. 2). At K0= 8×10−3the oscillations of ˆMx,ˆMy\ncomponents take on a form of beats (Fig. 3), while they again b ecome\nsinusoidal at K0= 9×10−3. AtK0= 1×10−2(this is the right bound-\nary of the instability range in the linear theory) all the thr ee components\noscillate around zero values. Further, at K0= 1.1×10−2the oscillations\nhold yet, but at K0= 1.2×10−2they disappear almost completely, and\nthey are absent at K0= 1.5×10−2.\nThe subsequent increasing of the current leads only to risin g magne-\ntization in the positive direction due to the injection mech anism. At K0=\n2Λ/η0, thelongitudinalcomponent ˆMzreachesthemaximalpossiblevalue ˆMz=\n2 (the sublattices are flipped to a parallel position), then t he (dimension-\nless) angular frequency is equal to 2Λ (this corresponds to ∼1013s−1).\nSuch a situation corresponds to rather high current density , it is estimated\nas 2×108A/cm2at the above-mentioned parameter values.\n169 Conclusions\nThe obtained results show a principal possibility of contro lling frequency\nand damping of AFM resonance in FM/AFM junctions by means of s pin-\npolarized current. Under low AFM magnetization induced by a n external\nmagnetic field perpendicular totheantiferromagnetism vec tor, thethresh-\nold current density corresponding to occurring instabilit y is less substan-\ntially than in the FM–FM case. Near the threshold, the AFM res onance\nfrequency increases, while damping decreases, that opens a possibility of\ngenerating oscillations in THz range.\nNumerical simulation allows to trace behavior of the FM/AFM junc-\ntion in the whole current density range. The instability ran ge predicted\nby the linearized theory is broadened only slightly because of nonlinear\neffects. In the instability range undampedoscillation of si nusoidal or more\ncomplicated form including beats.\nUnder magnetic fields low compared to the exchange field, the i nduced\nmagnetization is small incomparison withthesublattice ma gnetization, so\nthat the stationary oscillation amplitude beyond the insta bility threshold\nis low, too.\nThus, the following features may be expected in comparison w ith the\nsimilar effects in FM/FM junctions. First, the instability t hreshold is to\nbe lower because of the lower magnetization. This is afavora ble fact facili-\ntating observations. Second, the oscillation intensity be yondthethreshold\nalso lowers as a square of the magnetization. This may make th e effect\ndifficult to observe. Nevertheless, studying the current-dr iven nonlinear\noscillations in FM/AFM structure is of principal interest, because the cur-\nrent induced instability can occur at relatively low curren t density, ∼105\nA/cm2.\nThe simulation results reveal an interesting possibility o f a spin-flip\ntransition without magnetic field under the action of a high- density spin-\npolarized current only. Such a current overcomes the exchan ge forces and\naligns thesublattice moments in parallel. Undersuch condi tions, applying\na low alternating magnetic field can excite precession of the magnetization\nvector at the AFM resonance frequencies, which may be as high as 3×1013\ns−1or more. Such a THz resonator might be useful to detect and mea sure\nsignals in THz range.\nAcknowledgments\nThe authors are grateful to Prof. G. M. 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Swaving, R.A. Duine. Phys. Rev. B 83, 054428 (2011).\n[22] A.I. Akhiezer, V.G. Baryakhtar, S.V. Peletminslii. Sp in Waves,\nNorth-Holland Pub. Co., Amsterdam, 1968.\n[23] C. Heide, P.E. Zilberman, R.J. Elliott, Phys. Rev. B 63, 064424\n(2001).\n[24] Yu.V. Gulyaev, P.E. Zilberman, E.M. Epshtein, R.J. Ell iott, JETP\nLett.76, 155 (2002).\n[25] Yu.V. Gulyaev, P.E. Zilberman, A.I. Panas, E.M. Epshte in.\nPhysics — Uspekhi 52, 335 (2009).\n18[26] Yu.V. Gulyaev, P.E. Zilberman, E.M. Epshtein, R.J. Ell iott. J. Exp.\nTheor. Phys. 100, 1005 (2005).\n[27] A.G.Gurevich, G.A.Melkov.Magnetizsation Oscillati ons andWaves,\nCRC Press, Boca Raton, FL, 1996.\n[28] V.E. Dmitrienko, E.N. Ovchinnikova, J. Kokubun, K. Ish ida. JETP\nLett.92, 383 (2010).\n[29] J.M. Robinson, P. Erd¨ os. Phys. Rev. B 6, 3337 (1972).\n19" }, { "title": "2210.08429v1.Magnetic_damping_anisotropy_in_the_two_dimensional_van_der_Waals_material_Fe__3_GeTe__2__from_first_principles.pdf", "content": "Magnetic damping anisotropy in the two-dimensional van der Waals material\nFe3GeTe 2from \frst principles\nPengtao Yang, Ruixi Liu, Zhe Yuan, and Yi Liu\u0003\nThe Center for Advanced Quantum Studies and Department of Physics,\nBeijing Normal University, 100875 Beijing, China\n(Dated: October 18, 2022)\nMagnetization relaxation in the two-dimensional itinerant ferromagnetic van der Waals ma-\nterial, Fe 3GeTe 2, below the Curie temperature is fundamentally important for applications to\nlow-dimensional spintronics devices. We use \frst-principles scattering theory to calculate the\ntemperature-dependent Gilbert damping for bulk and single-layer Fe 3GeTe 2. The calculated damp-\ning frequency of bulk Fe 3GeTe 2increases monotonically with temperature because of the dominance\nof resistivitylike behavior. By contrast, a very weak temperature dependence is found for the damp-\ning frequency of a single layer, which is attributed to strong surface scattering in this highly con\fned\ngeometry. A systematic study of the damping anisotropy reveals that orientational anisotropy is\npresent in both bulk and single-layer Fe 3GeTe 2. Rotational anisotropy is signi\fcant at low tem-\nperatures for both the bulk and a single layer and is gradually diminished by temperature-induced\ndisorder. The rotational anisotropy can be signi\fcantly enhanced by up to 430% in gated single-layer\nFe3GeTe 2.\nI. INTRODUCTION\nNewly emerged intrinsic two-dimensional (2D) ferro-\nmagnetic (FM) van der Waals (vdW) materials1{6have\nbecome the subject of intense research. Weak vdW\nbonding facilitates the extraction of thin layers down to\natomic thicknesses, whereas strong magnetocrystalline\nanisotropy protects long-range magnetic order. These\nmaterials provide an exciting arena to perform funda-\nmental investigations on 2D magnetism and promis-\ning applications of low-dimensional spintronics devices.\nAmong these materials, Fe 3GeTe 2(FGT) is especially\nattractive for its itinerant ferromagnetism and metal-\nlicity, such that both spin and charge degrees of free-\ndom can be exploited for designing functional devices.\nBulk FGT has a relatively high Curie temperature ( TC)\nof approximately 220-230 K.7{11Atomically thin lay-\ners of FGT have lower TCs, which, however, have been\nraised to room temperature (by ionic gating4) and be-\nyond (by patterning12). As a FM metal at reasonably\nhigh temperature, FGT opens up vast opportunities for\napplications.13{23\nThe dynamical properties of FGT critically a\u000bect the\napplicability and performance of these proposed low-\ndimensional spintronics devices. The most salient of\nthese properties is the dynamical dissipation of mag-\nnetization. It is usually described using a phenomeno-\nlogical parameter called Gilbert damping, which char-\nacterizes the e\u000eciency of the instantaneous magneti-\nzation to align eventually with the e\u000bective magnetic\n\feld during its precessional motion. Although this pa-\nrameter has been extensively studied in conventional\nFM materials, such as 3 dtransition metals and alloys,\ntwo key issues with the Gilbert parameter of FGT re-\nmain to be addressed: the temperature dependence and\nanisotropy (one naturally expects anisotropic damping\nin FGT because of its layered structure and the strong\nmagnetocrystalline anisotropy). Temperature-dependentGilbert damping was \frst observed in Fe24and later more\nsystematically in Fe, Co and Ni.25{27A nonmonotonic\ntemperature dependence has been found, for which a so-\ncalled \\conductivitylike\" component decreases with in-\ncreasing temperature, usually at low temperatures, and a\n\\resistivitylike\" component increases with temperature,\nusually at high temperatures. This nonmonotonic be-\nhavior has been successfully described by the torque-\ncorrelation model28and reproduced by \frst-principles\ncomputations.29{32Anisotropic damping was \frst theo-\nretically predicted in FM metals33and in noncollinear\nmagnetic textures.34With di\u000berent orientation of the\nequilibrium magnetization with respect to the crystal-\nlographic axes, the damping parameter can be quanti-\ntatively di\u000berent in general. This is referred to as the\norientational anisotropy. Even for the same equilibrium\nmagnetization orientation in a single crystalline lattice,\nthe magnetization may precess instantaneously along\ndi\u000berent directions resulting in the so-called rotational\nanisotropy.33The orientational anisotropy of damping\nhas been observed in recent experiments on single-crystal\nFM alloys,35{37but the underlying physical mechanism\nremains unclear.\nThe dimensionless Gilbert damping parameter \u000bcan\nbe expressed in terms of a frequency \u0015via\u0015=\u000b\rM ,38\nwhereM=jMjis the magnetization magnitude and \ris\nthe gyromagnetic ratio. Despite of the di\u000berent dimen-\nsions, these two parameters are equivalent39and both\npresent in literature for experimental24{27,35{37and the-\noretical studies.28,29,31,33,34,40{42\nIn this study, we systematically investigate\ntemperature-dependent Gilbert damping in single-\nlayer (SL) and bulk FGT using \frst-principles scattering\ntheory. Considering that the magnetization perpen-\ndicular to the 2D atomic planes is favored by the\nstrong magnetocrystalline anisotropy, we calculate the\ndamping as a function of temperature below TCand\n\fnd nearly temperature-independent damping in thearXiv:2210.08429v1 [cond-mat.mes-hall] 16 Oct 20222\n(a)\nFeⅠ\nFeⅡ\nGe\nTe(b)\nFIG. 1. (a) Side and (b) top view of the lattice structure\nfor bulk Fe 3GeTe 2. The black dashed frame delineates the\nin-plane unit cell.\nSL and damping dominated by resistivitylike behavior\nin the bulk. Varying the equilibrium direction of the\nFGT magnetization produces a twofold symmetry in\ndamping. When the magnetization is aligned inside\nthe 2D planes, a remarkable rotational anisotropy in\nthe Gilbert damping is present for in- and out-of-plane\nrotating magnetization.\nThis paper is organized as follows. The crystalline\nstructure of SL and bulk FGT is brie\ry introduced in Sec.\nII, followed by a description of our theoretical methods\nand computational details. The calculated temperature-\ndependent damping in SL and bulk FGT is presented in\nSec. III. The two types of damping anisotropy, i.e., orien-\ntational and rotational anisotropy, are analyzed in Sect.\nIV. Conclusions are drawn in Sec. V.\nII. GEOMETRIC STRUCTURE OF FGT AND\nCOMPUTATIONAL METHODS\nThe lattice structure of FGT is shown in Fig. 1. Two\ndi\u000berent types of Fe atoms occupy inequivalent Wycko\u000b\nsites and are denoted as FeI and FeII. Five atomic layers\nstack along the caxis to form an SL of FGT: Ge and\nFeII constitute the central atomic layer perpendicular to\nthecaxis, and two FeI layers and two Te layers are lo-\ncated symmetrically above and beneath the central layer,\nrespectively. Single layers with ABAB :::stacking form\nthe bulk FGT, where Layer A is translated in plane with\nrespect to Layer B, such that the Ge atoms in Layer A\nlie on top of the Te and FeII atoms in Layer B.\nThe electronic structure of bulk and SL FGT has\nbeen determined using the linear augmented plane wave\nmethod43within the local density approximation (LDA).\nDi\u000berent types of exchange-correlation functionals have\nbeen investigated in the literature, among which LDA\nwas found to yield satisfactory structural and magnetic\nproperties for FGT.44We employ experimentally ob-\ntained lattice constants7for bulk FGT calculations and\nobtain magnetic moments of 1.78 \u0016Band 1.13\u0016Bfor\nthe two types of Fe, respectively. The initial structure\nof a single layer is taken from the bulk lattice and fully\nrelaxed, resulting in an in-plane constant a= 3:92\u0017A.\nA vacuum spacing of 11.76 \u0017A is chosen to exclude theinterlayer interaction under periodic boundary condi-\ntions. The magnetic moments for the Fe atoms in SL\nFGT are obtained as 1.72 \u0016Band 1.01\u0016B. All the\ncalculated magnetic moments are in good agreement\nwith experimental7,8,45,46and calculated values44,47,48re-\nported in the literature.\nThe Gilbert damping calculation is performed using\nthe scattering theory of magnetization dissipation pro-\nposed by Brataas et al.49Within this theory, a single do-\nmain FM metal is sandwiched between two nonmagnetic\n(NM) metal leads. The Gilbert damping that charac-\nterizes the energy dissipation during magnetization dy-\nnamics can be expressed in terms of a scattering ma-\ntrix and its derivative with respect to the magnetiza-\ntion direction. We thus construct a two-terminal trans-\nport structure as Au jFGTjAu, where the Au lattice is\nslightly deformed to match that of FGT: we use 3 \u00021 and\n4\u00021 unit cells (UCs) of Au (001) to match the UCs of\nSL and bulk FGT, respectively. To investigate the ef-\nfect of temperature on Gilbert damping, we use a frozen\nthermal lattice and spin disorder31,40,50to mimic lattice\nvibration and spin \ructuation at \fnite temperatures in\nFGT. The measured Debye temperature \u0002 D= 232 K\nand temperature-dependent magnetization for the bulk8\nand SL3are employed to model the lattice and spin disor-\nder. In the scattering calculations, lateral supercells are\nemployed to satisfy periodic boundary conditions perpen-\ndicular to the transport direction. The electronic poten-\ntials required for the transport calculation are calculated\nself-consistently using a minimal basis of tight-binding\nlinear mu\u000en-tin orbitals (TB-LMTOs), and the result-\ning band structures for SL and bulk FGT e\u000bectively re-\nproduce those obtained using the linear augmented plane\nwave method. Then, the scattering matrices consisting\nof re\rection and transmission probability amplitudes for\nthe Bloch wave functions incident from the NM leads are\ndetermined by the so-called \\wave function matching\"\nmethod, which is also implemented using TB-LMTOs.40\nOther computational details can be found in our previous\npublications.31,40{42In this work, we focus on the damp-\ning with collective magnetization dynamics in the long-\nwave limit corresponding to the reported values in ex-\nperiment via ferromagnetic resonance and time-resolved\nmagneto-optical Kerr e\u000bect. The damping with a \f-\nnite wavelength can be determined in our framework of\nscattering calculation42or using the torque-correlation\nmodel,51but the wavelength dependence of damping is\nbeyond the scope of the current study.\nIII. TEMPERATURE-DEPENDENT DAMPING\nThe strong magnetocrystalline anisotropy of FGT re-\nsults in the equilibrium magnetization being naturally\nperpendicular to the atomic layers. Slightly excited mag-\nnetization deviates from the plane normal (denoted as ^ z)\nand relaxes back by dissipating energy and angular mo-\nmentum, as schematized in the inset of Fig. 2(a). The3\n𝛂∥𝛂∥𝑴(𝑡)𝑥𝑦𝑧\n5101520F|| (10-3)\n0 0.2 0.4 0.6 0.8 1T/TC468Q|| (108 Hz)Lattice disorder only(b)(a)\nBulkSingle layer\nBulkSingle layer\nFIG. 2. The calculated dimensionless Gilbert damping pa-\nrameter\u000bk(a) and corresponding damping frequency \u0015k(b)\nfor single-layer and bulk Fe 3GeTe 2as a function of tempera-\nture. The relaxation of the instantaneous magnetization M(t)\nresults in a change in the in-plane magnetization component,\nwhich is parallel to the atomic planes, as schematized in the\ninset of (a). The empty symbols in (b) denote the damping\nfrequencies that are calculated considering only thermal lat-\ntice disorder. The green line indicates the linear temperature\ndependence.\nGilbert damping parameter \u000bkdescribes the e\u000eciency\nof such a dissipative process. The calculated \u000bkof SL\nand bulk FGT is plotted in Fig. 2(a) as a function of\ntemperature. The damping for both increases monoton-\nically with the temperature. This behavior resembles\nthe so-called \\resistivitylike\" damping observed in many\nsingle-crystal FM metals.24{26However, the damping \u000bk\nfor the bulk tends to diverge as the temperature ap-\nproachesTC. This divergence originates from vanishing\nmagnetization, as has been found in three-dimensional\nFM alloys.42Therefore, as temperatures approaching TC,\nit is more appropriate to use the damping frequency pa-\nrameter\u0015=\u000b\rM .\nThe calculated damping frequencies are shown in\nFig. 2(b). The damping of a SL FGT, \u0015S\nk, is larger\nthan the damping of the bulk, \u0015B\nk, especially at low\ntemperatures. This di\u000berence can be attributed to the\nstrong surface e\u000bect of highly con\fned SL FGT. The\nlowered symmetry at the surface signi\fcantly enhances\nspin-orbit coupling (SOC),52which enables the dissipa-\ntion of angular momentum from electronic spins to the\norbital degree of freedom and then into the lattice reser-voir. In addition, as the thickness of a single layer is\nconsiderably smaller than the electronic mean free path,\nconduction electrons in FGT are strongly scattered by\nthe surface. Therefore, the two necessary ingredients for\nGilbert damping, namely, SOC and electronic scattering,\nare both enhanced in the SL compared with the bulk, re-\nsulting in a larger damping for the SL.\nThe calculated damping frequency \u0015S\nkremains nearly\nconstant with increasing temperature, except for a mi-\nnor increase at T > 0:6TC. To gain further insight into\nthe temperature e\u000bect, we perform the damping calcu-\nlation considering only lattice disorder, where the calcu-\nlated\u0015S\nlatare plotted as red empty circles in Fig. 2(b).\nLattice-disorder-induced damping in the SL FGT, \u0015S\nlat,\nexhibits a very weak temperature dependence, indicating\nthat increasing lattice vibration does not in\ruence the\ndamping frequency. The di\u000berence between \u0015S\nlatand\u0015S\nk\nincreases slightly only near TC, which can be attributed\nto the strong spin \ructuation. The overall weak tem-\nperature dependence in the damping for a single layer\nindicates that a non-thermal disorder scattering mecha-\nnism is dominant: the strong surface scattering in such\na thin layer (only a few \u0017A) combined with the enhanced\nSOC at the surfaces is the main channel for the magnetic\ndamping in the SL FGT instead of spin \ructuation and\nlattice vibration. Gilbert damping with a similarly weak\ntemperature dependence has also been found in a permal-\nloy,40,53where chemical disorder scattering overwhelms\nthermally induced disorder.\nThe temperature dependence of the bulk damping fre-\nquency is signi\fcantly di\u000berent from that of the SL. The\ncalculated bulk damping, \u0015B\nk, (shown by the black solid\ndiamonds in Fig. 2(b)) increases linearly with the temper-\nature. This typical resistivitylike behavior suggests that\nthe interband transition in bulk FGT is the dominant\ndamping mechanism.54We also calculate the damping\nfrequency\u0015B\nlatconsidering only lattice disorder, as shown\nas the black empty diamonds in Fig. 2(b). Comparing the\nresults corresponding to the solid and empty diamonds\nleads us to conclude that both lattice and spin disorder\nsubstantially contribute to damping in bulk FGT. As the\ntemperature approaches TC, the bulk damping is compa-\nrable with that in the single layer.\nIV. ANISOTROPIC DAMPING\nThe damping torque exerted on the magnetization\nin the Landau-Lifshitz-Gilbert equation has the general\nform of M(t)\u0002[~\u000b\u0001_M(t)], where the Gilbert damping\nparameter ~\u000bor the corresponding frequency is a tensor.\nThis tensor and its elements depend on both the instan-\ntaneous M(t) and its time derivative _M(t), where the\nanisotropy has been extensively analyzed using theoret-\nical models55and \frst-principles calculations.33,34Fol-\nlowing the de\fnition given by Gilmore et al. ,33we call\nthe anisotropic damping that depends on the equilibrium\norientation of Meqthe orientational anisotropy and that4\n𝑴𝐞𝐪𝑥𝑦𝑧𝜃\n-U/2 -U/40U/4U/2V81216F|| (10-3)Single layerBulk\nFIG. 3. The calculated Gilbert damping parameter \u000bkfor SL\n(red circles) and bulk FGT (black diamonds) as a function\nof the angle between the equilibrium magnetization Meqand\nthe atomic layer normal (^ z) of Fe 3GeTe 2. The lines are \ftted\nusingC0+C2cos 2\u0012.\ndepending on _M(t) the rotational anisotropy. Consider-\ning the layered structure of vdW materials, the lowered\nsymmetry should result in remarkable anisotropy for the\nmagnetization relaxation. Both the orientational and ro-\ntational anisotropy in bulk and SL FGT have been sys-\ntematically analyzed in this section. Notably, the damp-\ning tensor is reduced to a scalar for the con\fguration\nshown in Fig. 2.\nUnder a large in-plane magnetic \feld, the perpendicu-\nlar magnetization of FGT can be tilted toward the exter-\nnal \feld direction, which is de\fned as the y-axis without\nloss of generality. Thus, the angle between the equilib-\nrium magnetization Meqand the plane normal ^ zis re-\nferred to as \u0012, as shown in the inset of Fig. 3. At \u0012= 0,\nas studied in Sec. III, \u000bxx=\u000byy=\u000bk. For\u00126= 0,\n\u000bxx=\u000bkstill holds, whereas the other diagonal element\n\u000byydepends on speci\fc values of \u0012. Here, we focus on \u000bk\nto study the orientational anisotropy of damping. The\ncalculated in-plane damping \u000bkis plotted as a function\nof\u0012in Fig. 3 for a SL at 77 K and bulk FGT at 100\nK. The temperature is chosen in this way to obtain the\nsame relative magnetization for the two systems, namely,\nM=M s= 88%, according to the experimentally measured\nmagnetization as a function of temperature.3,8The same\ntwofold symmetry is found for the damping parameters\nof both SL and bulk FGT, which can be e\u000bectively \ftted\nusing a cos 2 \u0012term. As the magnetization rotates away\nfrom the easy axis, \u000bkincreases and reaches a maximum\nwhen the magnetization aligns inside the FGT layer. The\nchanges, [\u000b(\u0012=\u0006\u0019=2)\u0000\u000b(\u0012= 0)]=\u000b(\u0012= 0), are 62% for\nthe SL and 39% for the bulk. A similar dependence of\nthe damping on the magnetization orientation has been\nrecently observed in single-crystal CoFe alloys.35,36The\npredicted anisotropic damping of FGT shown in Fig. 3\nshould analogously be experimentally observable.\nThe rotational anisotropy of damping33in FGT is most\nsigni\fcant when the equilibrium magnetization lies in-\nside the atomic plane of FGT (along the hard axis), i.e.,\n0 0.20.40.6 0.81T/TC120150180QC/Q|| (%)\n0 0.2 0.4 0.6 0.8 1T/TC10152025Q (108 Hz)Bulk\nSingle layerBulkQCQ||Single layer\nBulkBulk\n𝛂∥𝑴(𝐭)\n𝑥𝑦𝑧(a)𝜶\"(b)FIG. 4. (a) Schematic of damping with the equilibrium mag-\nnetization Meqlying inside the atomic plane. Then, the in-\nstantaneous magnetization M(t) dissipates both the in- and\nout-of-plane spin angular momentum. The two types of dis-\nsipation are denoted as \u000bk(\u0015k) and\u000b?(\u0015?). (b) The calcu-\nlated Gilbert damping frequency \u0015k(?)as a function of tem-\nperature for single-layer and bulk Fe 3GeTe 2. The inset shows\nthe ratio of the two frequencies \u0015?=\u0015k.\n\u0012=\u0006\u0019=2. As schematized in Fig. 4(a), the magne-\ntization M(t) loses its in- or out-of-plane components\ndepending on the instantaneous precessional direction\n_M(t). In this case, one has \u000bxx=\u000bkand\u000bzz=\u000b?,\nwhereas the o\u000b-diagonal elements of the damping ten-\nsor are guaranteed to remain zero by symmetry.40The\ncalculated\u0015kand\u0015?for SL and bulk FGT are shown\nas a function of temperature in Fig. 4(b). For SL FGT,\n\u000bk(as shown by the circles with horizontal hatching) is\nnearly independent of temperature, which is the same as\nforMeqalong the easy axis. This result suggests that de-\nspite the sizable orientational anisotropy in the damping\nof SL FGT, the temperature has very little in\ruence on\nthe speci\fc values of the damping frequency. The calcu-\nlated\u0015?for the SL (shown by the red circles with vertical\nhatching) is considerably larger than \u0015kat low tempera-\ntures but decreases with increasing temperature. \u0015?be-\ncomes comparable with \u0015knear the Curie temperature,\nindicating that the rotational anisotropy is signi\fcantly\ndiminished by temperature.\nThe calculated \u0015kfor bulk FGT with Meqalong the\nhard axis (shown by the black diamonds with horizontal\nhatching) is temperature-independent, in sharp contrast\nto the linear temperature dependence of \u0015kwith Meq\nalong the easy axis shown in Fig. 2(b). This result sug-\ngests that the damping is already saturated in this case5\n-0.4 -0.2 0 0.2 0.4\nE-EF (eV)100200300400500λ⊥/λ|| (%)50 K\n77 K\n100 KSingle layer\nFIG. 5. The calculated rotational damping anisotropy for\nsingle-layer Fe 3GeTe 2as a function of the Fermi energy at\ndi\u000berent temperatures.\nat a su\u000eciently large scattering rate, where saturated\ndamping has also been found in FM Ni.25The calculated\n\u0015?of bulk FGT is also larger than \u0015kat low tempera-\ntures and slightly decreases with increasing temperature.\nWe summarize the results for the rotationally anisotropic\ndamping frequency by plotting the ratio between \u0015?and\n\u0015kin the inset of Fig. 4(b). The ratio for both SL and\nbulk FGT decreases with increasing temperature and\napproaches unity near TC. This behavior is consistent\nwith the results calculated using the torque-correlation\nmodel,33where rotationally anisotropic damping disap-\npears gradually as the scattering rate increases. In highly\ndisordered systems, the damping is more isotropic, as in-\ntuitively expected.\nWe emphasize that the calculated \u0015?values are dis-\ntinct from those reported in previous studies in the\nliterature,55that is,\u0015?was found to vanish in single-\ncrystal monoatomic FM layers based on the breathing\nFermi surface model.56{58Interband scattering is ne-\nglected in the breathing Fermi surface model. However,\nthe resistivitylike behavior of our calculated \u0015kfor bulk\nFGT shows that interband scattering plays an important\nrole in this vdW FM material.\nOne of the unique advantages of 2D vdW materials\nis the tunability of the electronic structure via electri-\ncal gating.4,59To simulate such a scenario, we slightly\nadjust the Fermi level EFof SL FGT without changing\nthe band structure for simplicity. The calculated rota-\ntional anisotropy in the damping \u0015?=\u0015kof SL FGT is\nshown as a function of the Fermi energy in Fig. 5. At all\nthe temperatures considered, the anisotropy ratio \u0015?=\u0015k\nincreases dramatically as EFis lowered by 0.3 eV, es-\npecially at low temperatures, and only exhibits minor\nchanges when EFis increased. At 50 K, the ratio \u0015?=\u0015k\nbecomes as high as 430%, which is almost three times\nlarger than that obtained without gating. This result\nsuggests that a small quantity of holes doped into SLFGT at low temperatures remarkably enhances the rota-\ntional damping anisotropy.\nV. CONCLUSIONS\nWe have systematically studied Gilbert damping in a\n2D vdW FM material Fe 3GeTe 2by using \frst-principles\nscattering calculations where the temperature-induced\nlattice vibration and spin \ructuation are modeled by\nfrozen thermal lattice and spin disorder. When the mag-\nnetization is perpendicular to the 2D atomic plane, the\ndamping frequency of bulk FGT increases linearly with\nthe temperature, whereas that of SL FGT exhibits a\nweak temperature dependence. The di\u000berence can be\nattributed to surface scattering (which is absent in the\nbulk) dominating scattering due to temperature-induced\ndisorder in SLs, which have a thickness smaller than the\nelectronic mean free path. The anisotropy of Gilbert\ndamping in this 2D vdW material has also been thor-\noughly investigated. The orientational anisotropy, which\ndepends on the direction of the equilibrium magnetiza-\ntion with respect to the atomic planes, exhibits twofold\nrotational symmetry in both the bulk and SL. When\nthe equilibrium magnetization is parallel to the atomic\nplane, the damping is signi\fcantly enhanced compared to\nthat with the magnetization perpendicular to the atomic\nplane. The rotational anisotropic damping depending on\nthe direction of motion of the instantaneous magnetiza-\ntion is remarkable with the equilibrium magnetization ly-\ning inside the atomic plane. With an out-of-plane compo-\nnent in the timederivative of the precessional magnetiza-\ntion, the damping frequency ( \u0015?) is much larger than the\none where only in-plane magnetization is varying ( \u0015k).\nThe ratio\u0015?=\u0015kis larger than unity for both the bulk\nand a single layer and decreases with increasing temper-\nature. In SL FGT, \u0015?=\u0015kcan be enhanced up to 430%\nby slight holedoping at 50 K.\nAntiferromagnetic order has recently been discovered\nin 2D vdW materials (as reviewed in Ref. 60 and the ref-\nerences therein) and some intriguing properties are found\nin their damping behaviors.61,62Owing to the more com-\nplex magnetic order, more than a single parameter is\nnecessary in describing the damping in antiferromagnetic\ndynamics.63,64It would be very interesting to study the\nmagnetization relaxation in these 2D materials with more\ncomplex magnetic order.\nACKNOWLEDGMENTS\nThe authors are grateful to Professor Xiangang Wan\nat Nanjing University for his support and helpful dis-\ncussions. Financial support for this study was provided\nby the National Natural Science Foundation of China\n(Grants No. 11734004 and No. 12174028).6\n\u0003yiliu@bnu.edu.cn\n1Cheng Gong, Lin Li, Zhenglu Li, Huiwen Ji, Alex Stern,\nYang Xia, Ting Cao, Wei Bao, Chenzhe Wang, Yuan\nWang, Z. Q. Qiu, R. J. 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In the present work, we demonstrate the geometrical inter-\npretation of neutrino in a vacuum in the presence of decay, which transforms this circular trajectory\nof neutrino into a helical track that effectively makes the neutrino system mimic a classical damped\ndriven oscillator. We show that in the absence of the phase factor ξin the decay Hamiltonian,\nthe neutrino exactly behaves like the system of nuclear magnetic resonance(NMR); however, the\ninclusion of the phase part introduces a CPviolation, which makes the system deviate from NMR.\nFinally, we make a qualitative discussion on under-damped, critically-damped, and over-damped\nscenarios geometrically by three different diagrams. In the end, we make a comparative study of\ngeometrical picturization in vacuum, matter, and decay, which extrapolates the understanding of\nthe geometrical representation of neutrino oscillation in a more straightforward way.\nI. INTRODUCTION\nThe experimental shreds of evidence of neutrino os-\ncillation have proven the existence of tiny but nonzero\nmasses of neutrino, which leads to a new era of explo-\nration of fundamental physics [1–5]. But for a better\nunderstanding of neutrino oscillation visually, the geo-\nmetrical interpretation of neutrino oscillation had been\nproposed where neutrino oscillation was described as the\nrotation of a unit vector depicting the flavor state of the\nneutrinos around a magnetic field-like vector denoted by\nthe Hamiltonian of the neutrino system [6–8]. Analo-\ngously, this picturization has been compared with the\nprecession of the magnetic moment vector in the pres-\nence of an external magnetic field. The projection of the\nflavor state |να(t)⟩at time t, over the mass basis |νi⟩is\nconnected to the amplitude of oscillation [6].\nAlthough the geometrical interpretation makes it eas-\nier to understand the neutrino oscillation and realize it\npictorially, such representation is not unique. Depend-\ning on the different choices of basis, some mathematical\napproaches have come up with a heuristic overview of\nthe flavor state oscillation. For example, in Giunti. et\nal.[6] an orthogonal basis ei\nF,ei\nMdescribing the flavor\nand the mass basis of the neutrino has been investigated,\nwhere istands for 1,2,3. The flavor state νerevolves\naround the Hamiltonian vector Bwith an angle 2 θ. A\nsimilar kind of picturization has been done in Smirnov.\net al [8] with the choices of basis with axes ν1,Reν 1,\nImν 2to describe the two generation neutrino oscillation.\nOn the other hand, Kim. et al. [7, 9] discussed the ge-\nometrical interpretation of flavor oscillation of neutrino\nin three-dimensional Euclidean space deduced from the\ntwo-valued representation of the flavor space. They im-\nplemented their interpretation to study the MSW effect\n∗Electronic address: rajrupab@iitbhilai.ac.in\n†Electronic address: kirans@iitbhilai.ac.in\n‡Electronic address: sudhanwa@iitbhilai.ac.in\n§Electronic address: panigrahi.iiser@gmail.comin adiabatic and non-adiabatic scenarios.\nOscillation experiments [10–12] and extensive litera-\nture [13–15] ambiguously prove that neutrinos possibly\ndecay to lighter invisible states. The literature proclaims\nan extensive study of the analytical approach to neutrino\noscillation in the presence of decay [16, 17]. However,\ndue to the non-Hermiticity of decay-Hamiltonian, it is\nchallenging to analyze the probability of oscillation an-\nalytically, making it harder to visualize the picture of\nthe damped neutrino oscillation [16, 18]. Due to this, a\ngeometrical approach to understanding the oscillation in\nthe presence of decay Hamiltonian has not been proposed\nyet.\nIn the present work, we address the above adversity of\nthe non-hermitian nature of decay Hamiltonian by a sim-\nple mathematical approach analogous to the NMR sys-\ntem and illustrate the neutrino oscillation geometrically\nin the presence of decay Hamiltonian. The motivation\nfor adopting such a procedure is to consider neutrino as\nan open quantum system with a similar viewpoint to an\nNMR system where the external magnetic field acts as a\nbeam splitter; in the case of neutrino, the unitary trans-\nformation matrix performs the same [15, 19–21]. The re-\nsult of the damped oscillation of neutrino also goes hand\nin hand with the NMR. However, introducing the phase\nfactor ξwith the non-Harmitian decay Hamiltonian the\nintroduces CPviolation in two flavor neutrino oscillation\neven in absence of the Majorana phase which modifies the\ndynamic of the flavor state that introduces a significant\ndifference from the NMR. For a simplistic overview, we\nrestrict ourselves to only two-flavor neutrino oscillation.\nThe construction of the paper is as follows. We start\nour work with the general theoretical framework(Section\nII) for the neutrino oscillation, where we define the two\nmass eigen states |ν1⟩and|ν2⟩as a two-level system.\nWe also introduce the decay Hamiltonian along with it.\nNext, we follow the density matrix formalism to define\nthe neutrino as an open quantum system in the presence\nof decay. To picture the oscillation vectorially, we vector-\nized the operator using the bra-flipper operator. We use\nthe Pauli Matrix basis to represent the vectorized oper-\nator and construct the dynamical evolution equation. InarXiv:2312.08178v1 [hep-ph] 13 Dec 20232\nthe result and discussion session(section III), we evalu-\nate the probabilities and equation of motion. We explain\nour result with two different schemes. First, we approxi-\nmated the decay parameters to be zero for the ideal case.\nIn the next, we derive the equation of motion, including\nthe decay parameters. In this section, we also explain the\nresult graphically and present the same using geomet-\nrical schematics. In the successive section(section IV),\nwe compare geometrical interpretation in different bases\nfollowing [7, 9] with the present work. We also briefly\ndiscuss the matter effect and illuminate the geometrical\ninterpretation of neutrino oscillation in matter. The pa-\nper concludes with an overall summary of the total work\nand sheds light on some of the other perspectives of the\nwork for future reference.\nII. THEORETICAL FRAMEWORK\nConsider an ultra-relativistic neutrino (left-handed)\nwith flavor αwith α=e, µ, τ and momentum p. The\npresent theoretical framework extends to considering the\nfollowing two eigenstates of neutrinos, i.e., mass eigen-\nstate, which is the propagating eigenstate, and flavor\neigenstate, which is the interacting eigenstate. These two\neigenstates are expressed by the two state vectors |να⟩\nand|νi⟩which are related by the unitary transformation\ntransformation.\n|να⟩=X\nkU∗\nαk|νk⟩ (1)\nWhere |να⟩is the flavor state of the neutrino, and |νk⟩is\nthe mass eigenstate of the neutrino. As we deal only with\nthe two flavors, no additional CPphase part is needed.\nHence, the unitary transformation U∗will be replaced by\nthe orthogonal transformation OT. where,\nO=\u0012\ncosθsinθ\n−sinθcosθ\u0013\n(2)\nIn this work, we consider only Dirac-type neutrino. It\nis to be mentioned here that for Majorana type of neu-\ntrino CPviolation phase needed to be considered even\nfor the two-flavor neutrino oscillation as well[17], i.e.,\n|να⟩=OUph|νi⟩, where Uph=\u00121 0\n0eiϕ\u0013\n. Associated\nwith any physical system, there is a complex Hilbert\nspace known as the system’s state space. In Hilbert\nspace, the state vector entirely describes the system’s\nstate. These state vectors of a finite Hilbert space cor-\nrespond to the possible physical condition of the system,\nand they are the pure state.\nConsidering the fact that a column matrix gives any\nstate vector in Hilbert space, we consider the mass eigen-\nstates of the neutrino as a two-level system with state\n|ν1⟩=\u0012\n1\n0\u0013\nand state |ν2⟩=\u0012\n0\n1\u0013\n. In the present work,\nwe restrict ourselves to the two-flavor oscillation only. In\nHilbert space any operator is given by a square matrix,\nˆA=\u0012\n⟨1|ˆA|1⟩ ⟨1|ˆA|2⟩\n⟨2|ˆA|1⟩ ⟨2|ˆA|2⟩\u0013\nWith diagonal elements as the\neigenvalues and the off-diagonal elements as the transi-\ntion elements between the two levels. In resemblancewith the neutrino system states |1⟩and|2⟩replace by\nthe mass eigenstates |ν1⟩and|ν2⟩. To start with, con-\nsider two flavor neutrino states as ψ=\u0000\nνeνµ\u0001Tand\nthe corresponding effective Hamiltonian in vacuum is ex-\npressed in the limit of ultra-relativistic energy approxi-\nmation Ei≃p+m2\ni/2p≈E+m2\ni/2Eas,\nH0=\u0014\nE\u0012\n1 0\n0 1\u0013\n+1\n2E\u0012\nm2\n10\n0m2\n2\u0013\u0015\n(3)\nIt is well known that the term proportional to identity\nhas no effect on neutrino oscillation and hence can be\ndropped from now on. The neutrino mass eigenstates\nare related to the corresponding flavor eigenstates by a\nunitary mixing matrix. Thus, the vacuum part of effec-\ntive Hamiltonian in flavor basis is read as,\nHF=1\n2E\u0014\nm2\n1cos2θ+m2\n2sin2θ−cosθsinθ∆m2\n−cosθsinθ∆m2m2\n1sin2θ+m2\n2cos2θ\u0015\n=∆m2\n4E\u0012\n−cos 2θsin 2θ\nsin 2θcos 2θ\u0013\u0003\n(4)\nwith ∆ m2=m2\n2−m2\n1is the mass square difference be-\ntween the two mass eigenvalues of the neutrinos. From\nthe eigenvalue equation for mass eigenstate have the\neigenvalues\nH0|νk⟩=Ek|νk⟩ (5)\nE1=m2\n1\n2EandE2=m2\n2\n2E. Here we consider the same\nenergy approach for the ultra-relativistic process pi∼\nE+m2\ni\n2Eso that the oscillation phase reduces to ∆ ϕ=\n−∆p·L∼∆m2\n2EL[22]. It is sufficient to notice no transi-\ntion between the two energy levels in vacuum oscillation.\nIn Schrodinger’s picture, any neutrino state follows the\nevolution equation of the initial flavor αas\nid\ndt|να⟩=HF|να⟩ (6)\nWhere HFis the Hamiltonian of the flavor state. For\nsimplified of the mathematical description we takem2\n1\n2E=\na1andm2\n2\n2E=a2.\nFrom the experimental evidence of neutrino oscilla-\ntions, it has already been proven that neutrino has a\nfinite mass. All the particles having finite rest mass have\nsome finite decay width. The decay probability increases\nwith the mass of the particles [23, 24]. The decay prob-\nability becomes very small for a very low-mass particle,\nand we get quasi-stationary states i.e., the particles are\nlocalized for a long time. Without decay, each flavor\neigenstate is the coherent superposition of mass eigen-\nstates. The Hermitian nature of the Hamiltonian gives\nrise to real eigenvalues. Nevertheless, the non-Hermitian\nattribute of the decay Hamiltonian generates discrete\ncomplex eigenvalues. The Decay Hamiltonian for neu-\ntrino is given by[16, 17, 25, 26]\nΓ/2 =\u0012\nb11\n2ηeiζ\n1\n2ηe−iζb2\u0013\n(7)\nIncorporating the decay Hamiltonian with the standard\nHamiltonian term, we get the effective Hamiltonian Hm3\nin mass basis,\nHm=H0−iΓ/2\n=\u0012\na10\n0a2\u0013\n−\u0012\nib1i\n2ηeiζ\ni\n2ηe−iζib2\u0013(8)\nH0is the vacuum Hamiltonian, associated with the mass\nsquare terms and energy. Considering a finite decay\nwidth of the single neutrino system, a decay Hamiltonian\nis also associated with the vacuum Hamiltonian given\nby the Γ /2 matrix. As mentioned in [17] a bound of\nτν≥5.7×105s(mν/eV) is derived from the neutrino data\nof Supernova 1987A. which leads to Γ ν≡b≈10−21eV\nfor a neutrino of mass 1 eV. Also, in [27] suggested an\napproximate value of b. Although the literature suggests\na rough estimation of the value of the decay parame-\nter, it converges towards the ultrahigh-energy neutrino\nfrom the astrophysical sources [17]. Here, b1, b2andηare\nreal parameters depicting the cause of the decay of mass\neigenstate. Since there is no established numerical value\nof decay parameters, the motivation of the work is to vi-\nsualize the decay geometrically and following the defined\nvalue of the parameters and probability in[16, 17, 27, 28]\nplot the appearance and disappearance probabilities in\nthe presence of decay. For the presence work, we restrict\nourselves only to the invisible neutrino decay for which\nwe approximate ξ= 0[28]. Also, for analytical simplicity,\nwe consider b1=b2=b[17].\nThe claim is that the Hamiltonian of the system doesn’t\nchange with time. The decaying system makes the stan-\ndard Hamiltonian of the system, H0, non-commutative\nwith the decay Hamiltonian, Γ /2, as a non-hermitian off-\ndiagonal element ηis associated with the decay Hamilto-\nnian. However, in spite of the anti-Hermitian nature of\nthe diagonal element bof Γ/2, it commutes with H0.\nThis is so because, in the absence of the off-diagonal\nelement, the decay basis matches with the mass basis\nHence, the information about the eigenstates of such a\ndecaying is not explicitly deterministic. Hence, the intu-\nition is to find a basis for this modified Hamiltonian for a\nsystem of decaying neutrino oscillation and to interpret\nthe system’s dynamics geometrically. We represent the\nHamiltonian for the decaying system of the neutrino in\nthe mass basis as Hm, which comprise both H0and Γ /2,\nFiguratively,\nHm=\u0012\na10\n0a2\u0013\n−i\n2\u0012\nb0\n0b\u0013\n−i\n2\u0012\n0η/2\nη/2 0\u0013\n(9)\nWe redefine eq.(10) as(a1+a2)\n2σ0+(a1−a2)\n2σ3−ibσ0−iη\n4σ1,\n[17]. Explitice mathematical form of this description is,\nHm=ωσ3+ (−ib)σ0+ (−iη)/2σ1 (10)\nThe term, ω=∆m2\n4E, incorporates the mass square differ-\nences of the neutrinos, which is the most essential com-\nponent for the phenomenon of oscillation. We neglect\nhere the term(a1+a2)\n2inσ0as it has no explicit physical\nsignificance in flavor-changing neutrino oscillation. This\ndecomposition makes it possible to represent the Hamil-\ntonian structurally as the inner product of a vector with\nthe Pauli matrices. To address this affirmation, we in-\ntroduce a vector B, which denotes the vector represent-\ningHm.This Hamiltonian is now analogous to the innerproduct of two vectors. Now define a vector B, depicting\nthe Hamiltonian of the system. The above expression is\nanalogous to the inner product of the vectorized Hamil-\ntonian, Band the Pauli matrices, which gives as follows,\nHm=B ·σ (11)\nHowever, the ambiguity arises in defining the basis. In-\nstead of relating Cartesian coordinates to define the Pauli\nmatrix, we use the mass eigenstates, |ν1⟩and|ν2⟩to ex-\npound the Pauli matrices. This characterization is fea-\nsible for the present framework as we consider mass and\nflavor basis. In this context, we refurbish the Pauli ma-\ntrices,\nσx=|ν1⟩⟨ν2|+|ν2⟩⟨ν1|\nσy=i|ν1⟩⟨ν2| −i|ν2⟩⟨ν1|\nσz=|ν2⟩⟨ν2| − |ν1⟩⟨ν1|\nI2=|ν1⟩⟨ν1|+|ν2⟩⟨ν2|(12)\nEq.(12) suggests the operator representation of the Pauli\nmatrices. Consequently, this definition motivates us to\nrepresent a new basis for the Pauli matrix. However, two\npoints need to be highlighted. Firstly, the dimension of\nthis space, i.e., |νi⟩⟨νj|will differ from the previously de-\nfined state space |νi⟩. Secondly, a vector is represented by\na column matrix, whereas a square matrix represents an\noperator. When we modify the Hamiltonian operator for-\nmalism to vector representation, the square matrix must\nalso be converted to the column matrix. This is possible\nusing a bra-flipper operator ℧, [29],\n℧[|νi⟩⟨νj|] =|νi⟩ ⊗ |νj⟩∗≡ |νi, νj⟩⟩\nThis allows us to define a linear space of matrices, con-\nverting the matrices effectively into vectors [29, 30]. But\nthe representation also has an extra complex factor i,\nmaking it difficult to interpret the Bvectorially. Here, we\ninitiate the concept of quaternion [31]. The quaternion\nis the three-dimensional projection of a four-dimensional\ncomplex hyperspace, similar to stereographic projection\nin complex analysis [32]. It is noteworthy that Pauli ma-\ntrices form the quaternion groups [33]. Hence, it is physi-\ncally more relevant to use the Pauli matrices as the basis\nfor this vectorization of the operator, which inherently\ncarries the complex factor within. Observing such fact in\nPauli basis , the complex ifactor is nothing but a phase\nfactor e−iπ/2which denotes the rotation of the axis [34].\nThis implies that the inner product with Pauli matrices\nleads to the vectorization of the operator. Following this,\nthe Hamiltonian modifies to,\nB=ωˆσ3+ (−ib)σ0+e−iπ/2ηˆσ1 (13)\nˆσis the correspond Bdenotes the vectorized Hamilto-\nnian in mass basis. Therefore, moving it in the flavor\nbasis changes the basis vectors, i.e., the Pauli operator.\nThe rotational transformation of the Pauli basis which\nisσM\n0=σF\n0,σM\n1= sin 2 θσF\n3+ cos 2 θσF\n1,σM\n2=σF\n2,\nσM\n3= cos 2 θσF\n3−sin 2θσF\n1. Defining this, we next move\nto analyze the flavor state and its evolution on this new\nbasis.\nNow consider the neutrino is created in the flavor eigen4\nstate as electron neutrino in Hilbert space H1. We have\nalready mentioned that each flavor eigenstate of neutrino\ndoes not have any definitive eigenstate. Instead, it is\nthe linear superposition of the two different mass eigen-\nstates. The dynamical propagation of the mass basis\nacquires an amount of flavor basis during propagation,\ndue to which a certain amount of mass also gets mixed\nassociated with each flavor state. More details can be\nfollowed from [35, 36]. As we have already mentioned,\neach flavor state of neutrino can be thought of as a two-\nlevel system consisting of two orthonormal mass eigen-\nstates, |ν1⟩=|0⟩and|ν2⟩=|1⟩, defined over the Pauli\nbases. For the present discussion on the two-flavor os-\ncillation, we restrict the superposition to the two mass\neigenstates only, i.e., |να⟩= cos θ|0⟩+ sin θ|1⟩. Hence,\neach flavor state of neutrino is an entangled state of the\nmass eigenstate [19, 21]. From the basic postulates of\nthe quantum mechanics corresponding to every system’s\nphysical state, a state vector is associated with it. The\nstate vector represents the pure states, which specify all\nthe known physical information about the system. This\nis an ideal case. In most cases, the system is not in a pure\nstate. Most generally, we may know that a quantum sys-\ntem can be in one state of a set {|να⟩}with probabilities\nwi. In this case, the mathematical tool that describes\nour knowledge of the system is the density operator (or\ndensity matrix). Due to the oscillatory property of the\nneutrinos, instead of using a state vector for the flavor\nstate, we use the density operator ˆ ραwhere ˆ ρα∈ H 1\n[28]. Consequently, instead of investigating the dynam-\nics of single state vector we study the evolution equation\nthe density operator ραα=|να⟩, we use |να⟩⟨να|for pure\nstates and ραβ=|να⟩⟨νβ|for the mixed state, that de-\npicts an ensemble of bi-flavored neutrino system. The\nstate|να⟩is from Hilbert space H1and the state |νβ⟩is\nfrom Hilbert space H2. The dynamics of the combined\nsystem ρ=|να⟩⟨νβ|is defined over the Hilbert space\nH1⊗ H 2[21] which has the dimension of H2⊗ H 2. The\nmanifestation of this idea makes our choice basis more\ntangible. The density operator |νe⟩⟨νe|for pure electron\nneutrino state is,\n|νe⟩⟨νe|=ρm\nee= cos2θ|ν1⟩⟨ν1|+ sin θcosθ|ν1⟩⟨ν2|\n+ sin θcosθ|ν2⟩⟨ν1|+ sin2θ|ν2⟩⟨ν2|\n= cos2θ|00⟩+ cos θsinθ|01⟩\n+ cos θsinθ|10⟩+ sin2θ|11⟩\n(14)\nWhile propagation, the flavor state is mixed with the\nmass eigenstate, and since each flavor state consists of\ntwo mass eigenstates, the purity of the system is not\nconserved, and the neutrino oscillates from one flavor\nstate to the other. Redefined mass eigen state consists of\n|00⟩=|0⟩s⊗ |0⟩e,|11⟩=|1⟩s⊗ |1⟩e,|11⟩=|0⟩s⊗ |1⟩e\nand|10⟩=|1⟩s⊗|0⟩e. From eq.(14), the density operator\ndenoting the flavor state, ρee=\u0012cos2θcosθsinθ\ncosθsinθsin2θ\u0013\n.\nWe exploit the fact that R2⊗ R2is isomorphic to R4\nto vectorise the density operator. The objective is to\nvectorize the Hamiltonian and density operator over a\ncommon basis to study the dynamics pictorially. This\nwill give us a comparative understanding of the neutrinoevolution and the precision of an NMR in an external\nmagnetic field. To express the vectorized form of the\ndensity operator, we introduce a momentum-like vector\nFsuch that Fi= Tr\u00021\n2σiρ\u0003\n, where istands for the 1, 2,\nand 3 components.\nF1= Tr\u00141\n2σ1ρ\u0015\n= Tr\u00141\n2\u0012\n0 1\n1 0\u0013\u0012\nρ11ρ12\nρ21ρ22\u0013\u0015\n=1\n2(ρ12+ρ21)(15)\nSimilarly F2=i\n2(ρ12−ρ21) and F3=1\n2(ρ11−ρ22). Hence,\nthe density matrix is constructed as,\nρij=\u0012\nF3 F1−iF2\nF1+iF2−F3\u0013\n(16)\nρijstands for the components of the density operator\nin Pauli bases. Hence, Fsignifies a vector that stands\nfor the flavor state of the system that aligns toward the\nflavor basis. We consider the neutrinos created a pure\nelectron flavor state. The two flavor states correspond-\ning to the bi-flavored neutrino oscillation are orthogonal\nto each other. Accordingly, we take the νestate along\nσF\n3andνµalong −σF\n3. In flavor basis at t= 0, FF\n1= 0,\nFF\n2= 0 and FF\n3= 1. We move Ffrom flavor to mass\nbasis to study the evolution equation to avoid mathe-\nmatical ambiguity. However, time evolved FF(t) will be\napplicable for understanding the scenario at the proba-\nbility level. Constructing so, we move to evaluate the\nevolution of the density operator.\nNow we have two vectors FandBrepresenting the two\noperators, density operator ρand the total Hamiltonian\nincluding decay Hmrespectively, defined over the Pauli\nbasis. The Hamiltonian vector Bhas a Hermitian com-\nponent, Bh, which is associated with mass square differ-\nences and a non-hermitian component, Bnh, which incor-\nporates the decay component of B. Although we initi-\nated the study of the Schrodinger equation, which states\nthe dynamics of the flavor state, as soon as we intro-\nduced the concept of density operator, we moved from\nthe Schrodinger picture to Heisenberg’s picture, where\nthe operator has the dynamical evolution instead of a\nstate vector.\nid\ndtρ\u0000\nt\u0001\n=\u0012\nBhρ(t)−ρ(t)Bh\u0013\n+\u0012\nBnhρ(t)−ρ(t)B†\nnh\u0013\n(17)\nFor the non-hermitian nature of the decay Hamiltonian\nBnh̸=B†\nnh. However, discrepancies arise as the Hamil-\ntonian is vectored but not the density operator. To rec-\ntify this inconsistency, we convert the density operator ρ\nto the vectorized form mentioned previously by the vec-\ntorF. Using the form of ρfrom eq.(16) the form of LHS\nof eq.(17) modified to,\nd\ndtρ=\u0012\n0 −2iω(F1−iF2)\n2iω(F1+iF2) 0\u0013\n−2b\u0012\nF3 F1−iF2\nF1+iF2−F3\u0013\n+\u0012\n−ηF10\n0−ηF1\u0013(18)5\nThe RHS of the above equation changes its form follow-\ning the formd\ndtFi=Tr1\n2σid\ndtρ. The component wise\nevolution equation of Fis thus,\n˙F1=−ωF2−bF1\n˙Fy=ωF1−bF2\n˙F2=−2bF3(19)\nIn matrix form,\n\n˙F1\n˙F2\n˙F3\n=\n−b−ω0\nω−b0\n0 0 −2b\n\nF1\nF2\nF3\n (20)\nit is noteworthy that although we have vectorized the\ndensity operator as F, we do not mention explicitly on\nwhich basis it is defined. In general, Fillustrates the\ndensity operator of the flavor state of neutrino. We can\ndenote flavor part of the FasFF\niwhile in mass part FM\ni.\nAs we consider neutrino is created as the pure electron\nflavor, hence at t= 0, FF\n1= 0,FF\n2= 0, FF\n3= 1. In\nmass basis at t= 0,FM\n1= sin 2 θ,FM\n2= 0,FM\n3= cos 2 θ.\nOn that account eq.(20) donates the evolution equation\nofFMin order to find time evolved FMatt= 0 [6].\nRewriting eq.(20),\n\n˙FM\n1˙FM\n2˙FM\n3\n=\n−b−ω0\nω−b0\n0 0 −2b\n\nFM\n1\nFM\n2\nFM\n3\n (21)\nRecall we have mentioned earlier that ω=∆m2\n4E. Accord-\ningly, 2 ω=∆m2\n2E, which defines the standard oscillation\nfrequency of the flavor oscillation of neutrinos. Thus, we\nredraft 2 ωasωoscfor mathematical simplification. It is to\nbe noted that the time evolution of the density operator\nis not independent of η, i.e., the off-diagonal part of the\ndecay Hamiltonian, which is also referred to in eq.(18).\nHowever, the time evolution of F, i.e., vector depicting\nthe flavor state, is independent of the off-diagonal ele-\nment η. Choosing the Pauli basis makes the evolution\nequation independent of the off-diagonal element.\nIII. RESULTS AND DISCUSSIONS\nIn this section, we discuss the evolution of FManalyt-\nically and illustrate the result geometrically. We inter-\npret the dynamical equation as a second-order differen-\ntial equation, an easier way to understand the oscillation\npictorially. We scrutinize two different scenarios. Firstly,\nwe consider the damping parameters zero and give the\ngeometrical representation figuratively. Next, we include\nthe decay parameters to reduce the differential equation\nof neutrino oscillation to the modified form and represent\nit geometrically.\n•Scenario 1: We first consider the standard neu-\ntrino oscillation without including decay parame-\nters. For the standard neutrino oscillation, we set\nη= 0, b= 0. Hence, from the eq.(21), we can\nobserve that the evolution equation of the vector\ndescribing the density matrix Fbecomes ˙FM\n1=−ωoscFM\n2,˙FM\n2=ωoscFM\n1and ˙FM\n3= 0. Individ-\nually FM\n1andFM\n2satisfy the equation of simple\nharmonic oscillator. Hence, in compact form\n¨FM\n1,2=−ω2\noscFM\n1,2 (22)\nHowever, to have a more rigorous mathematical\npicture, it will be convenient to express them in\nterms of BandF, although the in eq.(22) ωoscis\nnothing but the coefficient of Bitself. Eq.(21) can\nbe expressed as, ˙FM\n1=−ωoscFM\n2ˆσ1and ˙FM\n2=\nωoscFM\n1ˆσ2, which is contracted to the form,\n˙F=−ωoscFM\n2ˆσ1+ωoscFM\n1ˆσ2\n=\u0012\nωoscˆσ3×FM\n2ˆσ2\u0013\n+\u0012\nωoscˆσ3×F1ˆσ1\u0013\n=B×F(23)\nThis leads to the standard form of the Bloch equa-\ntion with oscillation frequency ωosc= ∆m2/2E.\nAlso, eq.(23) implies the gyromagnetic ratio is 1 for\nthis precessional motion of neutrinos. The diagram\nin the right panel of fig.1 figuratively refers to this\nprecessional motion. At time t= 0, the initial elec-\ntron flavor state representing vector F(green line)\nis aligned along the σF\n3and is denoted by F(0). Af-\nter a certain time t, theFwill rotate to a position\nand be aligned to the position F(t). Considering\nthe ultra-relativistic limit for the neutrinos tcan\nbe substituted with the oscillation length. After\nthe completion of one Oscillation length, Fwill ac-\nquire the state F(0). In the absence of decay Hamil-\ntonian, this circular motion of Faround B(yellow\nline) will remain unaffected. Flavor and the mass\nbasis are mentioned in the plot legend. Bloch equa-\ntion represents the evolution equation of the spin-\nangular moment vector or magnetic moment vec-\ntor along a direction of the externally applied mag-\nnetic field. Analogically, it can be inferred that the\nHamiltonian in the mass basis acts as a magnetic\nfield, and the flavor state revolves around it with\nthe precessional frequency ω.\nThe applied magnetic field acts as a perturbation\nto the system, which exerts a torque in the equi-\nlibrium condition of the system, driving the system\nto rotate around. A similar phenomenon occurs for\nthe system of neutrino as well, where the Hamilto-\nnian, which contains the squared difference term,\nillustrates the linear combination of the two differ-\nent masses; one is the system mass, and the other\nis the mass acquired from the vacuum while prop-\nagation.\nUsually, this phenomenon cannot be seen when a\nparticle has a definite mass. A single flavor state\nconsisting of a single mass eigenstate may not pos-\nsibly execute the oscillation. This adds the unique\ndynamical feature of eq.(23), making it distinct\nfrom any other fermions. |να⟩=c1|ν1⟩+c2|ν2⟩.\nThis torque appears due to the superposition of two\nmass eigenstates. In addition to this, the existence\nof different flavor states makes the phenomenon\nmore prominent. Illustratively, it is described by6\n 𝜎𝑧𝑓 \n𝜎𝑥𝑓 𝜎𝑧𝑚𝑑 \n−𝜎𝑧𝑚𝑑 2𝜃 \n2𝜃 𝜎𝑧𝑓 \n𝜎𝑥𝑓 \nb 𝑩⃗⃗ 𝜈1 \n𝜈2 𝜈𝑒 \n𝜈𝜇 𝜎𝑥𝑚 \nDisplaced mass basis in presence of decay \nFlavour basis −𝜎𝑧𝑓 \nHamiltonian denoting vector 𝑩⃗⃗ \nFlavour basis in presence of decay \nFlavour state denoting vector 𝑭⃗⃗ \n 𝜎3𝐹 \n𝜎1𝐹 𝜎3𝑀 \n−𝜎3𝑀 2𝜃 2𝜃 𝑩⃗⃗ 𝜈1 \n𝜈2 𝜈𝑒 \n𝜈𝜇 𝜎1𝑀 \n Mass basis \nFlavour basis −𝜎3𝐹 \nHamiltonian denoting vector 𝑩⃗⃗ \nFlavour state denoting vector 𝑭⃗⃗ \nF(t=L osc) F(0) \n−𝜎1𝐹 \nFIG. 1: Geometrical representation of basis in flavor basis and mass basis in the presence of decay( left-panel ) and\ngeometrical representation of standard neutrino oscillation without decay( right-panel ). Here we follow the Pauli\nbasis, i.e., ˆ σ1, ˆσ2, ˆσ3as the basis vector by which we define the coordinate axes. Plot legends in the box define the\ncolor representation of different components of the illustration. As the diagonal terms of the decay Hamiltonian\nmatch with the mass bases, it acts as a displacement element to the ˆ σ3component of the mass basis. The\nHamiltonian depicting vector Bis along ˆ σ3. The σ3direction denotes the ν1mass eigenstate and −σ3directs toward\ntheν2. On the other hand, in the flavor basis, the two orthogonal flavor states νeandνµalign along positive and\nnegative σ3directions, respectively, as illustrated in the left figure. The figure in the right panel shows that in the\nabsence of decay, the flavor state representing vector Frotates around the Hamiltonian vector Bfollowing in the\nequation ˙F=B×F. The initial flavor state is of electron neutrino; hence, Fis along +ˆ σ3. After a finite time t,Fis\nalong F(t), which implies the mixed state. After one complete revolution, it comes back to the initial pure state. In\nthe relativistic limit t≡L, the angular velocity of the rotation is equivalent to the oscillation length Losc, shown in\nthe adjacent figure on the right.\nFig.1. We have considered that neutrino is created\nas a pure electron flavor state. In flavor basis, the\npure electron flavor state is lined up along σf\n3, and\nthe muon flavor state is along −σf\n3. Considering the\ntransformation of the basis vectors, the mass and\nthe flavor basis are inclined by an angle 2 θ. This\nis known as the mixing angle of neutrino. eq.(13)\nsuggests that in mass basis Bis aligned along σM\n3\nwith the states ν1inσM\n3andν1in−σM\n3. Impos-\ning the initial condition at t= 0, i.e., in flavor\nbasis FF\n1= 0,FF\n2= 0,FF\n3= 1 and in mass basis,\nFM\n1= sin 2 θ,FM\n2= 0,FM\n3= cos 2 θ, the solution ofeq.(23) is reduced to,\nFM\n1(t) =FM\n1(0) cos ωosct+FM\n2(0) sin ωosct\nFM\n2(t) =FM\n1(0) cos ωosct−FM\n2(0) sin ωosct\nFM\n3(t) =FM\n3(0)(24)\nWhich follows, FM\n1=−sin 2θcosωosct,FM\n2=\n−sin 2θsinωosct,FM\n3= cos 2 θ. So far, we are in\nthe mass basis where the vector defining the den-\nsity matrix is also defined over the mass basis. This\nrefers to the solution of the evolution equation of F\naround Bin mass basis. To understand how the dy-\nnamics affect the probability level FM(t) has to be\ntransformed into the flavor basis FF(t) and FF\n3(t)\nwill give the probability of obtaining the initial fla-\nvorνeafter a time t. Following the transforma-\ntion, ˆ σF\n3= cos 2 θˆσM\n3−sin 2θˆσM\n1, of the basis vec-7\n0 1000 2000 3000 4000 50000.00.20.40.60.81.0\nL(km)/E(GeV)Probability\nAppearance ProbabilityP eeAppearance ProbabilityP eμ\nTotal Probability P\n0 50 100 150 200 250 3000.00.20.40.60.81.0\nLength(km)Probability\nFIG. 2: Plot of appearance probability Pee(red line), disappearance probability Peµ(black line) and the total\nprobability P(Blue-dashed line). The plot in the right panel shows the probabilities with the variation of\nL(km)/E(GeV) in vacuum. In the absence of decay, the two-flavor state acts as a two-level system with a constant\noscillation amplitude, which keeps oscillating without deviation. Meanwhile, the figure in the right panel shows the\nprobabilities in the presence of decay. Graphically it describes that the decay Hamiltonian which coincides with the\nmass basis is responsible for the amplitude damping of probabilities according to the expression Pd\nee=e−2bLPee,\nPd\neµ=e−2bLPeµ,Pd=e−2bL. The plot legends mention the figure indicates the appearance, disappearance, and\ntotal probability variation. The figure manifests the fact that with the increasing length L(km), the amplitude of a\nPd\nee,Pd\neµdecreases in an oscillatory manner while the total probability Pddecreases exponentially, with E= 1GeV.\nThis figure also conveys the graphical layout of under-damped oscillation when the decay parameter magnitude of\nthe decay parameter ( b) is much less than angular velocity ωL. The value of the parameter bis 10−21eVfor\nconsidering 1 eV= ∆m2[17]\ntors, the Fcan be transformed into the flavor basis,\ni.e.,FF\ni(t). We are only interested in the σ3compo-\nnent of the Fbecause the initial flavor state is along\nthe ˆσ3component. The time-evolved component of\nFzprovides the flavor state, which eventually gives\nthe appearance probability of the particular flavor\nstate, i.e., Pα=FM\n3(t) = 1 −sin22θsin2∆m2L\n4E.\nThe plot at the left of Fig.2 depicts the same behav-\nior of standard oscillation. The appearance (black\nline) and disappearance (red line) probabilities are\nplotted with respect to the L(km)/E(GeV) ratio.\nThis figure points to the un-deviated oscillatory na-\nture of the precessional motion of neutrinos, men-\ntioned in Fig.1 which also implies the same pictur-\nization of any standard two-level system [30]. It\nalso predicts after which length we should get back\nthe same flavor. Intuitively, this refers to the fact\nthat the flavor state of the neutrino acts as a mag-\nnetic moment vector rotating along the magnetic\nfield where the Hamiltonian depicts the magnetic\nfield.\n•Scenario 2: η̸= 0, b̸= 0. In the next scenario, we\nanalyze considering both the decay parameters to\nbe non-zero of the decay Hamiltonian, which effec-\ntively modifies the evolution equation eq.(19) to,\n¨FM\n1=ω2\noscFM\n1−2b˙F1M\n¨FM\n2=ω2\noscFM\n2−2b˙F2M\n˙F3\n3=−2bFM\n3(25)\nAnalogically, the above equation is the equation of\nmotion of a damped harmonic oscillator. Physi-\ncally, one such instance can be noticed in NMR. Inthe case of NMR, the 2 bfactor acts as the relax-\nation time 1 /T. It is noted from eq.(25) that the η\nfactor doesn’t play a role here. If the off-diagonal\nterm is hermitian, it contributes to the evolution\nequation, which can be noted following [37]. For\nthe invisible neutrino decay, the Bloch equation is\nindependent of the diagonal terms of the pertur-\nbation Hamiltonian. Moreover, the contribution\nfrom the σ3component makes the equation differ-\nent from the standard oscillation, where no contri-\nbution comes from the σ3component. To visualize\nthe dynamics geometrically, let us look through the\ndifferential Bloch equation form of eq.(24).\nd\ndtF=B×F−2bF (26)\nEq.(25) implies that due to the presence of the\ndecay factor, the circular motion is converted to\na spiral motion coiling into the mass basis of the\nHamiltonian. Moreover, due to the occurrence of\ndamping along the σM\n3-axis BdragF, which en-\nforced Fto execute a helical motion in the σ3di-\nrection with gradually decreasing screw pitch. We\nmust quantify parameter b with an approximated\nvalue mentioned in [17, 27] for a complete pictorial\noverview. For consistency, bhas to be the dimen-\nsion of energy(eV). Next, we derive the oscillation\nprobability using eq.(25), the solution of which,\nFM\n1(t) =e−2btFM\n1(0) cos ωosct\n+e−2btFM\n2(0) sin ωosct\nFM\n2(t) =e−2btFM\n1(0) cos ωosct\n−e−2btFM\n2(0) sin ωosct\nFM\n3(t) =e−2btFM\n3(0)(27)8\n 𝜎3𝐹 \n𝜎1𝐹 𝜎3𝑀𝑑 \n𝜎1𝑀𝑑 \n−𝜎3𝑀𝑑 2𝜃 \n2𝜃 𝜎3𝐹 \n𝜎1𝐹 \nb F(0) \nF(t=Losc) F(t=L) \n𝑩⃗⃗ \n−𝜎1𝐹 \n 𝜎3𝐹 \n𝜎1𝐹 𝜎3𝑀𝑑 \n𝜎1𝑀 \n−𝜎3𝑀𝑑 𝜃 \n𝜃 𝜎3𝐹 \n𝜎1𝐹 \nb F(0) \nF(t=L osc) \nF(t=L) 𝜈1 𝜈𝑒 \n𝜈𝜇 \n−𝜎3𝐹 \n Mass basis \nFlavour basis \nHamiltonian denoting vector 𝑩⃗⃗ \nFlavour state denoting vector 𝑭⃗⃗ 𝜈2 −𝜎1𝐹 \nFIG. 3: Figures illustrate the geometrical representation of neutrino oscillation in the presence of decay for critically\ndamped case( left panel ) and under damped case( right panel ). For the under-damped case, the oscillation\namplitude damped down in an oscillatory way. Consequently, there is oscillation with decreasing amplitude, which\ndirects towards the fact that after one full revolution, it is impossible to get back to the initial flavor state as the\npresence of decay parameter b, which causes the flavor state depicting vector Fexecute a helical motion. Moreover,\nthis helical motion doesn’t confine to a plane because this decay parameter affects the ˆ σ3components as well, which\nmakes Fdrag toward ˆ σmd\n3which the mass basis in the presence of decay. In addition, with the gradual decrement of\nthe pure state, the flavor state dies down, demonstrating that it ultimately dissipates to the mass state and Faligns\nitself towards B. In a critically damped case, the magnitude of the damping parameter is equal to the natural\nfrequency of the oscillation. Due to this equivalence, the Fvector is set off for rotational motion, although it can not\ncomplete one full rotation and decays to the mass eigenstate.\nHere, we impose the same boundary condition\nas before,i.e., FM\n1(0) = −sin 2θ,FM\n2(0) = 0,\nFM\n2(0) = cos 2 θ. Transforming the basis vector\nmass to flavor basis following ˆ σF\n3= cos 2 θˆσM\n3−\nsin 2θˆσM\n1, furnish the expression of FF\n3(t) which is\nthe probability of detecting flavour state νe. Ex-\nplicit mathematics follows,\nFF\n3(t) = cos 2 θFM\n3−sin 2θFM\n1 (28)\nThis produces the oscillation probability in the\npresence of decay Hamiltonian\nFF\n3(t) =e−2bt\u0014\n1−sin22θsin2\u0012ωosct\n2\u0013\u0015\n(29)\nWe have approximated ωosc>> b , so the oscillation\nfrequency remains unchanged. Nevertheless, the in-\nclusion of bmodifies the oscillation top\nω2osc+b2.\nThe expression refers to the phenomenon of am-\nplitude damping of neutrino oscillation apart fromimposing a perturbation to the mass basis. The\nplot at the right panel of Fig.2 implies the same\nfact, while the figure in the right panel of Fig.3 im-\nplements the same factuality diagrammatically.\nA parallel comparison can be drawn with the NMR\nsystem. The nuclear spin system needs T1to reach\nthe thermodynamic equilibrium in the presence\nof the applied transverse external magnetic field.\nThis results in exponential damping. Moreover,\nthe relaxation time depends on the material of the\nmedium and the direction of the applied field [38].\nThe Bloch matrix for the NMR system can be fol-\nlowed from [37], which shows.\n\n˙Sx\n˙Sy\n˙Sz\n=\n−1\nT1∆ 0\n−∆−1\nT1ϵ\n0−ϵ−1\nT2\n\nSx\nSy\nSz\n+\n0\n0\nS0\nz\n(30)\nDependence of the relaxation time over the direc-\ntion of the applied magnetic field introduces the9\n 𝜎3𝐹 \n𝜎1𝐹 𝜎3𝑀𝑑 \n−𝜎3𝑀𝑑 𝜎1𝑀 2𝜃 \n2𝜃 𝜎3𝐹 \n𝜎1𝐹 \nb F(0) \nF(t=L) 𝑩⃗⃗ \n−𝜎1𝐹 F(t=Losc) \nFIG. 4: Figures illustrate the geometrical representation\nof neutrino oscillation in the presence of decay for the\nover-damped scenario. The oscillation amplitude\ndamped down in an oscillatory way in the presence of\ndecay. Consequently, there is oscillation with decreasing\namplitude, which directs towards the fact that after one\nfull revolution, it is impossible to get back to the initial\nflavor state as the presence of decay parameter b, which\ncauses the flavor state depicting vector Fexecute a\nhelical motion. Moreover, this helical motion doesn’t\nconfine to a plane because this decay parameter affects\nthe ˆσ3components as well, which makes Fdrag toward\nˆσmd\n3which the mass basis in the presence of decay. In\naddition, with the gradual decrement of the pure state,\nthe flavor state dies down, demonstrating that it\nultimately dissipates to the mass state and Faligns\nitself towards B. In a critically damped case, the\nmagnitude of the damping parameter is equal to the\nnatural frequency of the oscillation. Due to this\nequivalence, the Fvector is set off for rotational motion,\nalthough it can not complete one full rotation and\ndecays to the mass eigenstate. The scenario of decay of\nflavor state to the mass eigenstate is much more\ndominant for the over-damped case where Fdirectly\ndissipates to Bwithout executing any rotational motion\nas shown by the figure in the left panel.\ntwo relaxation times T1andT2. Equivalently, the\nsystem of the decay parameter defines the lifetime\nof the flavor state [16, 17] that follows a straight-\nforward analytic definition of the decay parameter,\nwhich is subjected to a dependence on the mass of\nthe neutrino mνand lifetime of neutrino τν. Since\nno definite magnetic field applies to the neutrino\nsystem, only one damping factor bexists.\n•Scenario 3: η̸= 0, b̸= 0, ξ̸= 0. This is the most\ngeneral scenario where we do not approximate the\nξto be zero for the invisible neutrino decay. Inclu-\nsion of ξterm in the off-diagonal part of the decay\nHamiltonian further bifurcates Bnhin the followingway,\nBnh=−i\n2(σ0b)−−i\n4cosξˆσ1+i\n4sinξˆσ2 (31)\nIn our present choice of basis, again, the ˆ σ1com-\nponent will be decoupled from the dynamics of\n˙FM. However, a contribution remains from the\nˆσ2component. From the commutation relation\nd\ndtρ=−sinξ\n2i\u0012\n−F1F3\nF3F1\u0013\nwhich modifies the the\nevolution equation of FMto,\n˙F1=−ωF2−bF1+ηsinξ\n2iF3\n˙F2=ωF1−bF2\n˙F3=−2bF3−ηsinξ\n2iF1(32)\nIn more compact form, the matrix representation\nof eq.(32) is written as,\n\n˙FM\n1˙FM\n2˙FM\n3\n=\n−b−ω ηsinξ\n2i\nω−b0\n−ηsinξ\n2i0−2b\n\nFM\n1\nFM\n2\nFM\n3\n (33)\nThe above equation indicates that the inclusion of\ntheξterm can not decouple the ηterm from the\nderivation, which is a clear-cut indication of the\npresence of CPphase even in the absence of the\nMajorana phase ϕ. The effect of the factor ξin\na more explicit way can be shown in the following\nway.\n˙F=\u0012\nωoscˆσ3×FM\n2ˆσ2\u0013\n+\u0012\nωoscˆσ3×FM\n1ˆσ1\u0013\n−2bF+\u0012\ne−iπ/2η\n2sinξˆσ2×FM\n3ˆσ3\n+e−iπ/2η\n2sinξˆσ2×FM\n1ˆσ1\u0013\n=B×F−2bF+e−iπ/2(E×F)(34)\nWhere we have denoted E=η\n2sinξˆσ2+η\n2sinξˆσ2. This\nabove equation depicts that in the presence of ξ,Fnot\nonly tends to execute circular motion around B, but also\nit does evolve around E. This modifies the evolution\nequation\n¨FM\n1+ 2b˙FM\n1+ω2\n1FM\n1= 0\n¨FM\n2+ 2b˙FM\n2+ω2\noscFM\n2= 0\n¨FM\n3+ 2b˙F3+ηsinξ\n2i˙FM\n1= 0(35)\nWhere, ω1=s\nω2osc+b2−\u0012\nη2sinξ2\n4\u0013\n. This clearly\nshows that the presence of ξprofoundly modifies the os-\ncillation frequency, affecting the system as a perturbation\nintroducing quantum fluctuation. The relative depen-\ndence of the values of the parameters ωandbguide us\nto probe the result in the following three categories.10\n1. Under damped oscillation :The most general\ncase is when the oscillation parameter ωosctakes over the\ndamping parameter b, i.e., ωosc>> ω 1. The larger value\nof the oscillation parameter than the damping parameter\nsignifies the existence of both appearance and disappear-\nance probability with an exponential decrease in ampli-\ntude. The amplitude of the flavor state gradually dies\ndown. Consequently, Ftries to align itself along the B,\nwhich signifies that the flavor state exponentially damps\ndown to the mass state. Geometrically, the phenomenon\nis depicted in the right panel of Fig.3. In the presence of\nthe decay, the σM\n3axis is displaced by an amount b(red\ndashed line). The yellow dashed line denotes the system’s\ntotal Hamiltonian, B. The initial flavor state is aligned\nalong σF\n3, i.e., F(0). The existence of decay enforces Fto\nexecute a spiral motion. It is to be mentioned here that\nfor the critically damped case, the angular velocity mod-\nifies top\nω2osc+b2, although ωosc>> b . Moreover, this\nmotion is not confined to the σ2σ1plane. The appear-\nance of e−2btalong the σ3drags F(t) towards σ3compel\nthe spiral rotation to follow a helical trajectory as shown\nin the figure.\n2.Critically damped oscillation :In this scenario,\nthe oscillation parameter and damping parameter, i.e.,\nωandbmerge together,i.e., ωosc=b. The appear-\nance probability exponentially damped down in this case.\nThere is no oscillation of the flavor state associated with\nit. This implies that the flavor state does not execute\nthe rotational motion around B. With the decrement\nof the appearance probability, the disappearance proba-\nbility increases and reaches a maximum, after which it\nreduces to zero. This implies that neutrinos tend to os-\ncillate, but due to high perturbation, they damped down\nquickly to the mass basis. The diagram in the left panel\nof Fig.3 interprets this phenomenon geometrically. De-\nspite the fact thatp\nω2osc+b2, as the damping factor 2 b\nequals the standard oscillation frequency ω, theFdamps\ndown quickly to the mass basis. The figure suggests that\nF(0) is set to rotate around B. But due to the high\ndamping coefficient, which merges with the standard os-\ncillation frequency, the Fcan not complete one complete\nrevolution and dies down quickly to the mass basis.\n3. Overdamped oscillation :the appearance proba-\nbility decreases rapidly in this scheme. Initially, the neu-\ntrino tends to oscillate. Due to the negligible value of the\ndisappearance probability, it can be neglected. Without\nspiraling in, the flavor state decays directly to the mass\nbasis and loses its identity, as explained by Fig.4. It\nrefers that due to the overtaking of the value of bthan ω,\nFdirectly falls onto the mass basis. Without spiraling\nin.\nFrom the above analysis, it is clear that the neutrino\nsystem behaves like a mechanical oscillator. Although\nthe neutrino has been treated as an ultra-relativistic par-\nticle with quantum mechanical properties throughout the\nderivation, it starts to behave as a classical oscillator in\nthe presence of an external perturbing Hamiltonian. For\na mechanical system, the equation of motion in the pres-\nence of the viscous drag is ¨ x+ 2γ˙x+ω0x= 0, where γ\nfactor is associated with the damping of the system. In\nthe case of the amplitude of displacement getting damped\ndown, as in the case of the neutrino, the amplitude of theoscillation probability dies down. This implies that the\npresence of decay neutrino starts to mimic the mechani-\ncal oscillator.\nThe above discussion encompasses the fact that the fla-\nvor state neutrino decays to its mass state, which causes\nthe flavor of the neutrino to disappear after a certain\ndistance completely, and the neutrino propagates as a\nflavorless massive particle. In the critically damped oscil-\nlation, the flavor state of the neutrino set off its journey\ntowards accomplishing the circular trajectories. How-\never, the marginal value of the decay parameter, which\ncoincides with the decay parameter, restricts its motion,\nand they could not complete one complete revolution.\nConsequently, there is some finite probability of disap-\npearance occurring, although the oscillation length can\nnot be achieved, and the trajectory spirals towards the\nmass basis similar to the Fibonacci sequence [39]. Fig.3\ndepicts the same phenomenology as described. For the\nover-damped oscillation, due to the substantial value of\nthe decay parameter b, i.e., greater than the standard os-\ncillation, the flavor state directly dies down to the mass\nbasis without performing oscillation. It is also to be\nmentioned that the system of the decaying flavor state\nof neutrino is precisely similar to that of a nuclear spin\nstate in an external Magnetic [37]. This also indicates\nthe non-adiabaticity of the flavor state oscillation in the\nappearance of a perturbing Hamiltonian in the neutrino\nsystem. However, in the probability regime, we have con-\nsidered the adiabatic transition. But figuratively, there\nis a clear-cut indication that the mixing angle should also\nvary in the presence of decay. The flavor vacuum oscilla-\ntion of the neutrino indicates that the flavor state picked\nup from the vacuum during propagation, and the initial\nflavor state of the neutrino acts as a coupled harmonic\noscillator.\nWe find that the appearance probability and disappear-\nance probabilities in the presence of decay are dependent\non phase ζ. The Majorana phase( ϕ) also appears in the\nprobability analysis if the off-diagonal decay terms are\npresent. Thus, we have two phases ζandϕwhich can\ninduce CP violation in a vacuum. The anti-neutrino os-\ncillation probabilities can be easily obtained by replac-\ningζ→ − ζandϕ→ − ϕ. To quantify the amount\nof CP violation associated with the presence of decay,\nwe define the quantity ACP\nαβ=Pdecay\nαβ−Pdecay\nαβ. The\nasymmetry associated with muon appearance probabil-\nity,ACP\neµis proportional to factor e−2btsin(ζ−ϕ)B, where\nB=sin 2θsin2(∆m2L/4E)\n∆m2/2E[17].\nIV. COMPARISON WITH NEUTRINO\nOSCILLATION IN VACUUM AND MATTER\nWe provide an overview of the geometric representa-\ntion of neutrino oscillations in both vacuum and in the\npresence of matter, as previously explored in references\n[6, 7, 40]. Subsequently, we compare these findings with\nour current analysis, which incorporates the geometric\nrepresentation of neutrino oscillations along with decay\neffects.11\nA. Description of neutrino oscillations in vacuum\nBefore going to our results for geometrical interpre-\ntation in the presence of decay, we provide an overview\nof the geometrical interpretation in vacuum and matter\nexplored in the literature.\nLet’s briefly outline the geometric representation of\ntwo-flavor neutrino oscillations, as initially discussed by\nKim et al. (1988) [7, 9]. This illustration utilizes the evo-\nlution equation for two generations of neutrinos. Using\northonormal unit vectors, the authors explained the ge-\nometrical picture of neutrino oscillations in vacuum and\nmatter. The propagation and time evolution of neutrino\nflavor states are described as follows:\ni∂\n∂t\u0014\nνe\nνµ\u0015\n=HF\u0014\nνe\nνµ\u0015\n=−1\n2− →σ·− →B\u0014\nνe\nνµ\u0015\n(36)\nwhere\n− →B=1\n2E\u0014\n−∆m2sin 2θˆx+ (∆ m2cos 2θ−A) ˆz\u0015\nhere A= 2√\n2GFNeEwith GFFermi constant, Nethe\nelectron number density in matter, and Eneutrino en-\nergy. The vector B in the presence of matter is linked to\nits value B0in vacuum ( A= 0) as\nB=B0−(A/2E)ˆz\nDefining ψψ†=1\n2\u0002\n⊮+− →σ·− →P\u0003\n, where− →Prepresents\nthe equivalent magnetic moment vector (or polarization\nvector), the pure electron neutrino state νecan be visu-\nally depicted with− →P= ˆz, while the pure muon neutrino\nstate νµis oriented along the negative z-axis (− →P=−ˆz).\nThe concept of neutrino oscillation can be interpreted as\nthe precession of the magnetic moment vector (− →P) about\nan external magnetic field− →B.\nd− →P\ndt=− →P×− →B . (37)\nB is a constant vector ( B0) in vacuum, and P describes\nthe cone’s surface with axis B and an opening angle 2 θ\nas shown in Fig.5. Within the core of the Sun, we have\nmainly neutrino associated with electrons such that− →Pis\ndirected along the z-axis. Also, due to high electron den-\nsity for the propagating neutrinos,− →Bis almost parallel\nto the negative z-axis, so the precession cone is close to\n180◦. One can visualize it as if P is rotating about −B\nin an anticlockwise direction with an angle close to zero.\nWhen the neutrinos emerge from the Sun, the effective\nmatter potential decreases, and− →Bshifts to its vacuum\nvalue− →B0. P will precise around− →B0with opening angle\n180◦for adiabatic migration. Thus, in the final state, we\nhave a neutrino state dominated by νµexplaining the ge-\nometrical picture of the MSW effect inside the Sun. The\nwork carried out by Kim et al. was mainly focused on\none initial neutrino flavor. However, it is useful to under-\nstand the geometrical representation for an ensemble of\nneutrinos with different flavors. We can define such inco-\nherent neutrino beam by using density matrix formalismused by [6]. A Hermitian density operator can define the\nneutrino beam.\nρ(x) =X\nα|να(x)⟩Wα⟨να(x)| (38)\nThe parameter Wαis defined as the initial statistical\nweight factor, which is a measure of the probability\nof particular neutrino flavor αatx=t= 0. The\n Z \nX −𝑋 \n2𝜃 𝑯 𝜈𝑒 \n𝜈𝜇 \n-Z \nP L \nB \nFIG. 5: Illustration of the geometrical interpretation of\ntwo flavor neutrino oscillations in vacuum and matter\neffect. The choice of orthonormal basis is in the\nCartesian coordinate system where the vectors σ,B,\nP≡Fetc. are derived. This is the picturization of the\nprecession of the polarisation vector or neutrino flavor\nstate around the effective Hamiltonian in vacuum as\nwell as in matter. Initially, the pure electron-neutrino\nstate is defined as P(t= 0) along the direction of the\nz-axis. The evolution of this flavor state can be\nunderstood by the quantity H×P, where His defined\nfor effective Hamiltonian for vacuum or matter. The\nother muon-neutrino flavor state is depicted in the -z\naxis. Kim.et.al (1988) [7, 9]\nkey property of density matrix operator is Tr\u0002\nρ(x)\u0003\n=P\na⟨νa(x)|ρ(x)|νa(x)|= 1 where\b\n|νa⟩\t\nis a complete\nset of states. The evolution equation of the density ma-\ntrix in the flavor basis is derived in [6] and is given by\nidρF\ndx=\u0002\nHF, ρF\u0003\n(39)\nwith initial boundary condition [ ρF]rs(0) = Wrδrs. With12\n 𝑒𝑧𝐹 \n𝑒𝑧𝑀 \n2𝜃 \n2𝜃 𝑒𝑥𝑀 𝑩⃗⃗ 𝜈1 \n𝜈2 𝜈𝑒 \n𝜈𝜇 −𝑒𝑥𝐹 \nMass basis \nFlavour basis −𝑒𝑧𝐹 \nHamiltonian denoting vector 𝑩⃗⃗ \nFlavour state denoting vector 𝑭⃗⃗ \n𝑒𝑦𝑀=𝑒𝑦𝐹 \n \n𝑒𝑥𝐹 S(0) \nS(Losc) \n−𝑒𝑧𝑀 \n 𝑒𝑧𝐹 \n𝑒𝑧𝑀 2𝜃 2𝜃 𝑒𝑥𝑀 \n𝑩⃗⃗ \n𝜈1 𝜈2 𝜈𝑒 \n𝜈𝜇 −𝑒𝑥𝐹 \nMass basis \nFlavour basis −𝑒𝑧𝐹 \nHamiltonian denoting vector 𝑩⃗⃗ \nFlavour state denoting vector 𝑭⃗⃗ \n𝑒𝑦𝑀=𝑒𝑦𝐹 \n 𝑒𝑥𝐹 S(0) S(Losc) −𝑒𝑧𝑀 \n−𝑒𝑥𝑀 \nFIG. 6: Geometrical representation of neutrino oscillations in vacuum (in left-panel) and in presence of matter\n(right-panel). The choice of basis adopted here is the flavor basis of neutrinos with the three orthonormal vectors\ne1F,e2Fande3Frepresented by black solid lines.The mass eigenstates of neutrinos are derived from equivalent\northonormal vectors represented by dashed magenta color lines and are related to flavor ones by mixing angle 2 θ.\nThe effective Hamiltonian in two neutrino systems in a vacuum is described in flavor basis by HF=−1/2σFwhile\nthe vector B=−∆m2sin 2θ\n2Ee1F+∆m2cos 2θ\n2Ee3Fis depicted in the second quadrant. The initial neutrino flavor state,\nlet us say νe, is oriented along e3Fdirection and is represented by a flavor state vector S(t= 0). The evolution of\nthe neutrino state vector Sfrom distance x= 0 to x=Loscis described by the quantity S×Band thereby, the\nprecision of Sabout Bdescribes neutrino oscillation.\ntwo flavor scenario with νe−νµ, the effective Hamiltonian\nHFin flavor basis can be written down as\nHF=−1\n2− →σF·− →BρF=1\n2\u0002\nσ0+− →σF·− →P\u0003\n. (40)\nHere, the choice of basis is essential and as adopted in\n[6],− →e1F,− →e2F,− →e3Fare three orthonormal basis vectors\nwith property− →eaF·− →ebF=δabin which flavor states of\nneutrinos can be constructed.\nThe components of vectors− →B(can be interpreted as a\nmagnetic field) defined in the flavor basis are\nB1F=−∆m2sin 2θ\n2E\nB2F= 0\nB3F=∆m2cos 2θ\n2E. (41)\nFor the evolution of the system in the presence of mat-\nter, the mass square difference(∆ m2) and the mixing\nangle( θ) should be replaced by the effective mass square\ndifference( ∆ ˜ m2) and modified mixing angle ( θM) in the\npresence of matter potential. The components for equiv-\nalently interpreted Polarisation vector (− →P) arising fromthe density matrix operator as,\nP1F= 2Re[ ρF]eµ\nP2F=−2Im[ρF]eµ\nP3F= [ρF]ee−[ρF]µµ. (42)\nWith the orthonormal set of basis vectors− →ea, the\nvarious vectors defined as− →σF=P3\na=1σa− →ea\nF,− →B=P3\na=1Ba\nF− →ea\nF, and− →P=P3\na=1Pa\nF− →ea\nF. They derived from\nevolution equations of the density matrix operator, and\nthe corresponding translated initial boundary conditions\nare− →P1F(0) = 0,− →P2F(0) = 0 and− →P3F(0) = We−Wµ.\nLet us understand the physical meaning of neutrino\noscillations in vacuum in terms of− →σ,− →Band− →P.\nAssume that the initial neutrino state is a pure elec-\ntron neutrino produced in Sun, i.e., We= 1 and Wµ= 0.\nThe Polarisation vector− →P(0) is aligned along− →e3Fdi-\nrection. For pure muon type neutrino state ( We= 0 and\nWµ= 1) the Polarisation vector− →P(0) will be aligned\nalong negative− →e3Fdirection. For an incoherent mix-\nture of electron and muon neutrino states, the orienta-\ntion of the Polarisation vector− →P(0) will be along ±− →e3F\ndepending upon the relative strength of Wrwith r=e, µ.13\nThe norm of the Polarization vector will decide whether\nthe neutrino beam is a pure state or incoherent mixture.\nThe precision of− →Paround the− →Bcan have different geo-\nmetrical viewpoints depending upon the value of matter\ndensity compared to the resonance density. We present\none such scenario in Fig.6, where the transition is adia-\nbatic for solar neutrinos. The right panel of Fig.6 sug-\ngests the geometrical interpretation of neutrino oscilla-\ntion in matter in over eF\niandeM\nias described in [6]. The\npresence of matter Hamiltonian A= 2√\n2GFNeE, the\nstandard Hamiltonian of the neutrino is dragged down\nas the figure indicates. However, the Ptends to follow\nthe precessional motion around B. In order to retain\nits circular motion Pstarts to revolves round −Bas a\nresults of which instead of ν1it is now coincide with ν2\nandPdescribes a narrow cone ( ∼180◦) round B. The\nresonance crossing of Pin the nonadiabatic case repre-\nsents ν2→ν1[6] We find that both the works by Kim\nand Guinti describe neutrino oscillations in vacuum and\nmatter but with different basis choices. The choice of\northogonal basis is not unique. For example, Mikheyev\nand Smirnov [41] have described the two-flavor neutrino\noscillation using an orthogonal mass eigenstates as the\nchoice of basis as ν1,Re ν 2, and Im ν 2. Nevertheless,\nin the following subsection, we present the geometrical\ninterpretation of neutrino oscillations in the presence of\ndecay, which is not yet clearly explored in the literature.\nB. Description of neutrino oscillation with decay\nThe standard Hamiltonian of the neutrino oscillation\nis defined as H0= \nm2\n1\n2E0\n0m2\n2\n2E!\n. Following[17] this Hamil-\ntonian can be decomposed as H0=1\n2(ω0I+ω·σz) with\nω0=(m2\n1+m2\n2)\n2Eandω=(m2\n1−m2\n2)\n2E. Since the ω0term\ndoesn’t contribute to the flavor-changing neutrino oscil-\nlation, the first term of the HMis ruled out. As we have\nalready discussed, we can unify the effect of invisible neu-\ntrino decay by the Hamiltonian,\nHd=−i\n2\u0012\nb η/ 2\nη/2b\u0013\n=−i\n2(bI+η·σ) (43)\nWhich modifies the description of the Hamiltonian of the\nneutrino system to Hm=1\n2(−ibI+ω·σ−iη/2·σ). This\nresults in a modification of the neutrino system’s Hamil-\ntonian to Hm=1\n2(−ibI+ω·σ−iη/2·σ). In summary,\nwhen considering the presence of decay, the Hamiltonian\nfor the neutrino system can be vectorized by introduc-\ning a vector that represents decay in addition to the\nstandard Hamiltonian for neutrino oscillation, denoted\nasHm=H0+Hd. This is mathematically elucidated in\nthe following equation.\nH0=1\n2(ω0I+ω·σ3)\nHd=−i\n2(bI+η·σ1)(44)\nFor a geometric representation, we have opted for the\nPauli matrix basis. This choice is motivated by the abil-\nity to expand the Pauli matrices over the mass basis,which can be conceptualized as a two-level system fea-\nturing two mass eigenstates, namely |ν1⟩and|ν2⟩. The\nexplicit form of σi(where i= 1,2,3) over the mass basis\nis provided in eq.(12). The unique aspect of the super-\nposition of two non-degenerate mass eigenstates over the\nflavor eigenstate of the neutrino system motivates the\nadoption of this formalism. This contrasts with other\nparticles, where a unique mass eigenstate exclusively de-\nscribes each flavor eigenstate. The selection of this basis\nallows for the vectorization of the Hamiltonian in the\nmass basis, represented by the basis vector ˆ σi. In the\nmass basis, we denote the vector representation of the\nHamiltonian for standard oscillation as Bhand the de-\ncay Hamiltonian as Bnh.\nBh=1\n2ωˆσ3\nBnh=−i\n2(bI+η/2ˆσ1)\nB=1\n2(ωˆσ3+e−iπ\n2(bI+ηˆσ1))(45)\nThe imaginary −ifactor physically represents the rota-\ntion of the coordinate axes by the angle of−iπ\n2. After\nelucidating the vectorized Hamiltonian of the neutrino\nsystem, we have introduced the density matrix formal-\nism to express the flavor states. Similarly, as we have de-\nnoted the Hamiltonian in vector representation, we have\ndemonstrated the density operator of the neutrinos de-\npicting the flavor state as,\nρ=F·σ=F1ˆσ1+F2ˆσ2+F3ˆσ3 (46)\nThe Magnitude of the FisFi= Tr1\n2σiρ. For neu-\ntrino system, ρ(t= 0) =\u0012cos2θcosθsinθ\ncosθsinθsin2θ\u0013\n=\n\u0012\nF3 F1−iF2\nF1+iF2−F3\u0013\nThe interpretation directly shows\nFin terms of the component of ρas,\nF1= (ρ12+ρ21)\nF2=i(ρ12−ρ21)\nF3= (ρ11−ρ22)(47)\nThus, two operators, Hamiltonian, and density operator,\nare represented by two vectors BandFrespectively. The\ndynamical evolution equation of Fdescribes how these\ntwo vectors are interrelated. In this demonstration, in-\nstead of denoting the Hamiltonian as an inner product\nwith the Pauli matrices on some arbitrary basis, we have\nexpressed the vectorized Hamiltonian in Pauli bases. In\nthis context, it is to be mentioned that the main reason\nfor introducing the Pauli basis is that during the change\nof basis from mass to flavor or vice-versa only these ba-\nsis vector ˆ σiwill change which makes the visualization\nof the phenomenon of oscillation and mathematics much\nsimpler. This flavor-mass bases conversion is given as,\nσM\n0=σF\n0\nσM\n1= sin 2 θσF\n3+ cos 2 θσF\n1\nσM\n2=σF\n2\nσM\n3= cos 2 θσF\n3−sin 2θσF\n1(48)14\nThe scenario also addresses that the Hamiltonian is re-\nsponsible for the evolution of flavor state with either\ntime. Following the Von-Neumann equation, the evolu-\ntion equation of the density operator is explicitly of the\nform,\nid\ndtρ(t) =−i\n2[H0, ρ]−1\n2{Hd, ρ} (49)\nThis follows from the Schrodinger equation, which gives\nthe dynamical evolution of the operator. In the present\nrepresentation, this equation is written asd\ndtFi=\nTr1\n2σid\ndtρwhich gives the dynamical flavor evolution over\nPauli basis in terms of state vector as,\nd\ndtF=F×B−2bF (50)\nIn this framework, the dependency of the evolution of\nthe state for the two-flavor neutrino oscillation over the\noff-diagonal terms of the decay Hamiltonian is automat-\nically canceled out. It is also to be mentioned here that\nthe flavor state representing vector Fis defined over the\nmass basis. At an initial time t= 0, we have considered\nthe neutrinos are produced in the pure electron flavor\nstate. Therefore, it is clear from eq.(19) that solving the\nequation and transforming it to the flavor basis following\nthe change of bases from eq.(48) give the appearance and\nthe disappearance probabilities.\nPeµ=e−2btPvac\neµ\nPee=e−2btPvac\nee(51)\nWhere Pvac\neµ= sin22θsin2ωL. The geometrical descrip-\ntion of the eq.(50) is given in Fig.7, which implements the\ndamped oscillatory nature of the neutrino flavor state.\nFiguratively, it suggests that the flavor eigenstate of neu-\ntrino decays to the mass eigenstates following helical tra-\njectory due to the decay factor e−2bt. The work’s unique-\nness lies in the choice, simple mathematical description,\nand effective geometrical visualization.\nThis illustration also shows that the neutrino system\nmimics the NMR system in the context of the similar for-\nmat of the Bloch equation. From [37], the Bloch equation\nfor an NMR system in an external magnetic field is\n\n˙Sx\n˙Sy\n˙Sz\n=\n−1\nT2∆ 0\n−∆−1\nT2ϵ\n0−ϵ−1\nT2\n\nSx\nSy\nSz\n+\n0\n0\nS0\nz\n (52)\nWhere the Hamiltonian of the system, including the per-\nturbation, is given by ˆH= ∆ˆSz+ϵSx.Sx,Sy,Szde-\nnotes the nuclear spin under the influence of an external\nmagnetic field with T1andT2as the dissipation and de-\nphasing relaxation parameters. Similarly, for the Bloch\nequation for the neutrino system as derived in eq.(20),\n\n˙Fx\n˙Fy\n˙Fz\n=\n−b−ω0\nω−b0\n0 0 −2b\n\nFx\nFy\nFz\n (53)\nWhere the role of the external magnetic field is played by\nthe mass square differences ωand the dissipation param-\neter is b. Although the xcomponent of the Hamiltonian\n 𝜎3𝐹 \n𝜎1𝐹 𝜎3𝑀𝑑 \n𝜎1𝑀 \n−𝜎3𝑀𝑑 𝜃 \n𝜃 𝜎3𝐹 \n𝜎1𝐹 \nb F(0) \nF(t=L osc) \nF(t=L) 𝜈1 𝜈𝑒 \n𝜈𝜇 \n−𝜎3𝐹 \n Mass basis \nFlavour basis \nHamiltonian denoting vector 𝑩⃗⃗ \nFlavour state denoting vector 𝑭⃗⃗ 𝜈2 −𝜎1𝐹 FIG. 7: Figures illustrate the geometrical representation\nof neutrino oscillation in decay under damped case in\nPauli basis. For the under-damped case, the oscillation\namplitude damped down in an oscillatory way.\nConsequently, there is oscillation with decreasing\namplitude, which directs towards the fact that after one\nfull revolution, it is impossible to get back to the initial\nflavor state as the presence of decay parameter b, which\ncauses the flavor state depicting vector Fexecute a\nhelical motion. Moreover, this helical motion doesn’t\nconfine to a plane because this decay parameter affects\nthe ˆσ3components as well, which makes Fdrag toward\nˆσMd\n3which the mass basis in the presence of decay. In\naddition, with the gradual decrement of the pure state,\nthe flavor state dies down, demonstrating that it\nultimately dissipates to the mass state and Faligns\nitself towards B.\nis present in both cases, for NMR, it is Hermitian; hence,\nit is present, ϵ, and for neutrino it is non-Hermitian hence\nis doesn’t show up in the Bloch matrix. However, in the\nLiouville super-operator formalism it will be present in\nthe matrix defining the Liouville super-operator [28].\nV. CONCLUSION\nWe discussed the geometrical interpretation of two-\nflavor neutrino oscillation in the presence of neutrino de-\ncay with a detailed analysis of the theoretical framework.\nA comparative study of the geometrical description of\nneutrino oscillation performed in the present work along\nwith earlier works [6, 7, 40] has been presented for bet-\nter understanding of the reader. We followed the density15\nmatrix formalism to define the neutrino as an open quan-\ntum system in the presence of decay. We vectorized the\noperators, the Hamiltonian, and the density operator of\nthe system to analyze the neutrino oscillation, including\ndecay geometrically. Implementing such formalism, we\ndevised a simple mathematical analysis following [37] to\nproduce the expression of probabilities. Using the Pauli\nbasis, we interpret the results schematically, which val-\nidates the decay of the flavor state to the mass state.\nwe also showed the modified evolution of the flavor state\ndepicting vector in presence of phase factor ξin the non-\nhermitian Hamiltonian. This introduced a CPviolation\nphase in two flavor neutrino oscillation. We also quali-\ntatively discussed the three scenarios of under-damped,\nover-damped, and critically-damped motion. To give the\nwork an overall completeness, we did a comparative study\nof the geometrical interpretation of neutrino oscillation\nin vacuum, matter, and decay that includes the MSW\neffect. In a nutshell, we have found that in the presence\nof decay, the flavor state depicting vector Fexecutes a\nspiral motion around the Hamiltonian depicting vector\nB, through which the flavor state of neutrino decays to\nthe mass eigenstate. The choice of Pauli basis has made\nthe interpretation of the phenomenon of decay in a more\nconvenient way. However, such a choice of basis has made\nthe oscillation probability independent of the off-diagonal\nfactor, which stipulates the event of amplitude damping\nof the oscillation in addition to introducing a perturba-\ntion to the mass eigenstate.\nThe geometrical interpretation of two-flavor neutrino\noscillation provides an easier way to understand why such\noscillations get enhanced in matter- an effect known as\nMSW (Mikheyev-Smirnov-Wolfenstein) effect that could\nexplain the well-known solar neutrino puzzle. This de-\nscription can also apply to understanding the governing\ndynamics of Supernovae and the fast flavor conversion ofSupernovae neutrinos [42–44]. The three-flavor neutrino\noscillations can be approximated with two-flavor neutrino\noscillations ( νeandνxwith x=µ, τ) within Supernovae\nas the muon and tau neutrinos have identical interactions\nand thereby, behaves similarly [45, 46]. Additionally, one\nmay consider neutrino as an open quantum system with\na similar viewpoint to the nuclear magnetic resonance\n(NMR) system [37], where the external magnetic field\nacts as a beam splitter. In contrast, in the case of neu-\ntrino, the unitary transformation matrix performs the\nsame. In addition to this, the proper characterization of\nthe decay parameter with its value and its consequence\non neutrino oscillation is still a broad area to investigate.\nFurthermore, the investigation of the proper reason be-\nhind the neutrino decay will make the understanding of\nneutrino oscillation more effective.\nAcknowledgments\nRajrupa Banerjee would like to thank the Ministry of\nElectronics and IT for the financial support through the\nVisvesvaraya fellowship scheme for carrying out this re-\nsearch work. RB is very thankful to Prof. Prasant. K.\nPanigrahi for the fruitful discussion carried from time to\ntime for the betterment of this work. KS acknowledges\nthe financial support received via the ”Bi-nationally Su-\npervised Doctoral Degree Program” from the German\nAcademic Exchange Service (DAAD). 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Rev.\nLett. 126(2021) no. 6, 061302, arXiv:2009.03337 ." }, { "title": "1804.09242v1.Generalisation_of_Gilbert_damping_and_magnetic_inertia_parameter_as_a_series_of_higher_order_relativistic_terms.pdf", "content": "Generalisation of Gilbert damping and magnetic\ninertia parameter as a series of higher-order\nrelativistic terms\nRitwik Mondalz, Marco Berritta and Peter M. Oppeneer\nDepartment of Physics and Astronomy, Uppsala University, P. O. Box 516, SE-751 20\nUppsala, Sweden\nE-mail: ritwik.mondal@physics.uu.se\nAbstract. The phenomenological Landau-Lifshitz-Gilbert (LLG) equation of motion\nremains as the cornerstone of contemporary magnetisation dynamics studies, wherein\nthe Gilbert damping parameter has been attributed to \frst-order relativistic e\u000bects.\nTo include magnetic inertial e\u000bects the LLG equation has previously been extended\nwith a supplemental inertia term and the arising inertial dynamics has been related\nto second-order relativistic e\u000bects. Here we start from the relativistic Dirac equation\nand, performing a Foldy-Wouthuysen transformation, derive a generalised Pauli spin\nHamiltonian that contains relativistic correction terms to any higher order. Using the\nHeisenberg equation of spin motion we derive general relativistic expressions for the\ntensorial Gilbert damping and magnetic inertia parameters, and show that these ten-\nsors can be expressed as series of higher-order relativistic correction terms. We further\nshow that, in the case of a harmonic external driving \feld, these series can be summed\nand we provide closed analytical expressions for the Gilbert and inertial parameters\nthat are functions of the frequency of the driving \feld.\n1. Introduction\nSpin dynamics in magnetic systems has often been described by the phenomenological\nLandau-Lifshitz (LL) equation of motion of the following form [1]\n@M\n@t=\u0000\rM\u0002He\u000b\u0000\u0015M\u0002[M\u0002He\u000b]; (1)\nwhere\ris the gyromagnetic ratio, He\u000bis the e\u000bective magnetic \feld, and \u0015is an\nisotropic damping parameter. The \frst term describes the precession of the local,\nclassical magnetisation vector M(r;t) around the e\u000bective \feld He\u000b. The second term\ndescribes the magnetisation relaxation such that the magnetisation vector relaxes to the\ndirection of the e\u000bective \feld until \fnally it is aligned with the e\u000bective \feld. To include\nzPresent address: Department of Physics, University of Konstanz, D -78457 Konstanz, GermanyarXiv:1804.09242v1 [cond-mat.other] 3 Apr 20182\nlarge damping, the relaxation term in the LL equation was reformulated by Gilbert [2, 3]\nto give the Landau-Lifshitz-Gilbert (LLG) equation,\n@M\n@t=\u0000\rM\u0002He\u000b+\u000bM\u0002@M\n@t; (2)\nwhere\u000bis the Gilbert damping constant. Note that both damping parameters \u000band\u0015\nare here scalars, which corresponds to the assumption of an isotropic medium. Both the\nLL and LLG equations preserve the length of the magnetisation during the dynamics and\nare mathematically equivalent (see, e.g. [4]). Recently, there have also been attempts\nMHeff\nPrecession\nNutationDamping\nFigure 1. Sketch of extended LLG magnetisation dynamics. The green arrow denotes\nthe classical magnetisation vector which precesses around an e\u000bective \feld. The red\nsolid and dotted lines depict the precession and damping. The yellow path signi\fes\nthe nutation, or inertial damping, of the magnetisation vector.\nto investigate the magnetic inertial dynamics which is essentially an extension to the\nLLG equation with an additional term [5{7]. Phenomenologically this additional term of\nmagnetic inertial dynamics, M\u0002I@2M=@t2, can be seen as a torque due to second-order\ntime derivative of the magnetisation [8{11]. The essence of the terms in the extended\nLLG equation is described pictorially in Fig. 1. Note that in the LLG dynamics the\nmagnetisation is described as a classical vector \feld and not as a quantum spin vector.\nIn their original work, Landau and Lifshitz attributed the damping constant \u0015to\nrelativistic origins [1]; later on, it has been more speci\fcally attributed to spin-orbit\ncoupling [12{15]. In the last few decades, several explanations have been proposed\ntowards the origin of damping mechanisms, e.g., the breathing Fermi surface model\n[16, 17], torque-torque correlation model [18], scattering theory formulation [19], e\u000bective\n\feld theories [20] etc. On the other hand, the origin of magnetic inertia is less discussed\nin the literature, although it's application to ultrafast spin dynamics and switching\ncould potentially be rich [9]. To account for the magnetic inertia, the breathing Fermi\nsurface model has been extended [11, 21] and the inertia parameter has been associated\nwith the magnetic susceptibility [22]. However, the microscopic origins of both Gilbert3\ndamping and magnetic inertia are still under debate and pose a fundamental question\nthat requires to be further investigated.\nIn two recent works [23, 24], we have shown that both quantities are of relativistic\norigin. In particular, we derived the Gilbert damping dynamics from the relativistic\nspin-orbit coupling and showed that the damping parameter is not a scalar quantity\nbut rather a tensor that involves two main contributions: electronic and magnetic\nones [23]. The electronic contribution is calculated as an electronic states' expectation\nvalue of the product of di\u000berent components of position and momentum operators;\nhowever, the magnetic contribution is given by the imaginary part of the susceptibility\ntensor. In an another work, we have derived the magnetic inertial dynamics from a\nhigher-order (1 =c4) spin-orbit coupling and showed that the corresponding parameter\nis also a tensor which depends on the real part of the susceptibility [24]. Both these\ninvestigations used a semirelativistic expansion of the Dirac Hamiltonian employing the\nFoldy-Wouthuysen transformation to obtain an extended Pauli Hamiltonian including\nthe relativistic corrections [25, 26]. The thus-obtained semirelativistic Hamiltonian was\nthen used to calculate the magnetisation dynamics, especially for the derivation of the\nLLG equation and magnetic inertial dynamics.\nIn this article we use an extended approach towards a derivation of the\ngeneralisation of those two (Gilbert damping and magnetic inertia) parameters from\nthe relativistic Dirac Hamiltonian, developing a series to fully include the occurring\nhigher-order relativistic terms. To this end we start from the Dirac Hamiltonian in\nthe presence of an external electromagnetic \feld and derive a semirelativistic expansion\nof it. By doing so, we consider the direct \feld-spin coupling terms and show that\nthese terms can be written as a series of higher-order relativistic contributions. Using\nthe latter Hamiltonian, we derive the corresponding spin dynamics. Our results show\nthat the Gilbert damping parameter and inertia parameter can be expressed as a\nconvergent series of higher-order relativistic terms and we derive closed expressions\nfor both quantities. At the lowest order, we \fnd exactly the same tensorial quantities\nthat have been found in earlier works.\n2. Relativistic Hamiltonian Formulation\nTo describe a relativistic particle, we start with a Dirac particle [27] inside a material,\nand, in the presence of an external \feld, for which one can write the Dirac equation\nasi~@ (r;t)\n@t=H (r;t) for a Dirac bi-spinor . Adopting furthermore the relativistic\ndensity functional theory (DFT) framework we write the corresponding Hamiltonian as\n[23{25]\nH=c\u000b\u0001(p\u0000eA) + (\f\u0000 1)mc2+V 1\n=O+ (\f\u0000 1)mc2+E; (3)\nwhereVis the e\u000bective unpolarised Kohn-Sham potential created by the ion-ion, ion-\nelectron and electron-electron interactions. Generally, to describe magnetic systems, an4\nadditional spin-polarised energy (exchange energy) term is required. However, we have\ntreated e\u000bects of the exchange \feld previously, and since it doesn't contribute to the\ndamping terms we do not consider it explicitly here (for details of the calculations\ninvolving the exchange potential, see Ref. [23, 25]). The e\u000bect of the external\nelectromagnetic \feld has been accounted through the vector potential, A(r;t),cde\fnes\nthe speed of light, mis particle's mass and 1is the 4\u00024 unit matrix. \u000band\fare the\nDirac matrices which have the form\n\u000b= \n0\u001b\n\u001b0!\n; \f = \n10\n0\u00001!\n;\nwhere\u001b= (\u001bx;\u001by;\u001bz) are the Pauli spin matrix vectors and 1is 2\u00022 unit matrix.\nNote that the Dirac matrices form the diagonal and o\u000b-diagonal matrix elements of\nthe Hamiltonian in Eq. (3). For example, the o\u000b-diagonal elements can be denoted as\nO=c\u000b\u0001(p\u0000eA), and the diagonal matrix elements can be written as E=V 1.\nIn the nonrelativistic limit, the Dirac Hamiltonian equals the Pauli Hamiltonian,\nsee e.g. [28]. In this respect, one has to consider that the Dirac bi-spinor can be written\nas\n (r;t) = \n\u001e(r;t)\n\u0011(r;t)!\n;\nwhere the upper \u001eand lower\u0011components have to be considered as \\large\" and \\small\"\ncomponents, respectively. This nonrelativistic limit is only valid for the case when the\nparticle's momentum is much smaller than the rest mass energy, otherwise it gives\nan unsatisfactory result [26]. Therefore, the issue of separating the wave functions of\nparticles from those of antiparticles is not clear for any given momentum. This is mainly\nbecause the o\u000b-diagonal Hamiltonian elements link the particle and antiparticle. The\nFoldy-Wouthuysen (FW) transformation [29] has been a very successful attempt to \fnd\na representation where the o\u000b-diagonal elements have been reduced in every step of the\ntransformation. Thereafter, neglecting the higher-order o\u000b-diagonal elements, one \fnds\nthe correct Hamiltonian that describes the particles e\u000eciently. The FW transformation\nis an unitary transformation obtained by suitably choosing the FW operator [29],\nUFW=\u0000i\n2mc2\fO: (4)\nThe minus sign in front of the operator is because of the property that \fandO\nanticommute with each other. With the FW operator, the FW transformation of the\nwave function adopts the form 0(r;t) =eiUFW (r;t) such that the probability density\nremains the same, j j2=j 0j2. In this way, the time-dependent FW transformed\nHamiltonian can be expressed as [26, 28, 30]\nHFW=eiUFW\u0012\nH\u0000i~@\n@t\u0013\ne\u0000iUFW+i~@\n@t: (5)5\nAccording to the Baker-Campbell-Hausdor\u000b formula, the above transformed Hamilto-\nnian can be written as a series of commutators, and the \fnally transformed Hamiltonian\nreads\nHFW=H+i\u0014\nUFW;H\u0000i~@\n@t\u0015\n+i2\n2!\u0014\nUFW;\u0014\nUFW;H\u0000i~@\n@t\u0015\u0015\n+i3\n3!\u0014\nUFW;\u0014\nUFW;\u0014\nUFW;H\u0000i~@\n@t\u0015\u0015\u0015\n+:::: : (6)\nIn general, for a time-independent FW transformation, one has to work with@UFW\n@t= 0.\nHowever, this is only valid if the odd operator does not contain any time dependency. In\nour case, a time-dependent transformation is needed as the vector potential is notably\ntime-varying. In this regard, we notice that the even operators and the term i~@=@t\ntransform in a similar way. Therefore, we de\fne a term Fsuch thatF=E\u0000i~@=@t.\nThe main theme of the FW transformation is to make the odd terms smaller in every\nstep of the transformation. After a fourth transformation and neglecting the higher\norder terms, the Hamiltonian with only the even terms can be shown to have the form\nas [26, 30{33]\nH000\nFW= (\f\u0000 1)mc2+\f\u0012O2\n2mc2\u0000O4\n8m3c6+O6\n16m5c10\u0013\n+E\u00001\n8m2c4[O;[O;F]]\n\u0000\f\n8m3c6[O;F]2+3\n64m4c8\b\nO2;[O;[O;F]]\t\n+5\n128m4c8\u0002\nO2;\u0002\nO2;F\u0003\u0003\n:(7)\nHere, for any two operators AandBthe commutator is de\fned as [ A;B] and the\nanticommutator as fA;Bg. As already pointed out, the original FW transformation\ncan only produce correct and expected higher-order terms up to \frst order i.e., 1 =c4\n[26, 30, 33]. In fact, in their original work Foldy and Wouthuysen derived only the\nterms up to 1 =c4, i.e., only the terms in the \frst line of Eq. (7), however, notably\nwith the exception of the fourth term [29]. The higher-order terms in the original FW\ntransformation are of doubtful value [32, 34, 35]. Therefore, the Hamiltonian in Eq. (7)\nis not trustable and corrections are needed to achieve the expected higher-order terms.\nThe main problem with the original FW transformation is that the unitary operators in\ntwo preceding transformations do not commute with each other. For example, for the\nexponential operators eiUFWandeiU0\nFW, the commutator [ UFW;U0\nFW]6= 0. Moreover, as\nthe unitary operators are odd, this commutator produces even terms that have not been\nconsidered in the original FW transformation [26, 30, 33]. Taking into account those\nterms, the correction of the FW transformation generates the Hamiltonian as [33]\nHcorr:\nFW= (\f\u0000 1)mc2+\f\u0012O2\n2mc2\u0000O4\n8m3c6+O6\n16m5c10\u0013\n+E\u00001\n8m2c4[O;[O;F]]\n+\f\n16m3c6fO;[[O;F];F]g+3\n64m4c8\b\nO2;[O;[O;F]]\t\n+1\n128m4c8\u0002\nO2;\u0002\nO2;F\u0003\u0003\n\u00001\n32m4c8[O;[[[O;F];F];F]]: (8)6\nNote the di\u000berence between two Hamiltonians in Eq. (7) and Eq. (8) that are observed\nin the second and consequent lines in both the equations, however, the terms in the\n\frst line are the same. Eq. (8) provides the correct higher-order terms of the FW\ntransformation. In this regard, we mention that an another approach towards the correct\nFW transformation has been employed by Eriksen; this is a single step approach that\nproduces the expected FW transformed higher-order terms [34]. Once the transformed\nHamiltonian has been obtained as a function of odd and even terms, the \fnal form\nis achieved by substituting the correct form of odd terms Oand even termsEin the\nexpression of Eq. (8) and calculating term by term.\nSince we perform here the time-dependent FW transformation, we note that the\ncommutator [O;F] can be evaluated as [ O;F] =i~@O=@t. Therefore, following the\nde\fnition of the odd operator, the time-varying \felds are taken into account through\nthis term. We evaluate each of the terms in Eq. (8) separately and obtain that the\nparticles can be described by the following extended Pauli Hamiltonian [24, 26, 36]\nHcorr:\nFW=(p\u0000eA)2\n2m+V\u0000e~\n2m\u001b\u0001B\u0000(p\u0000eA)4\n8m3c2+(p\u0000eA)6\n16m5c4\n\u0000\u0012e~\n2m\u00132B2\n2mc2+e~\n4m2c2(\n(p\u0000eA)2\n2m;\u001b\u0001B)\n\u0000e~2\n8m2c2r\u0001Etot\u0000e~\n8m2c2\u001b\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot]\n\u0000e~2\n16m3c4\u001a\n(p\u0000eA);@Etot\n@t\u001b\n\u0000ie~2\n16m3c4\u001b\u0001\u0014@Etot\n@t\u0002(p\u0000eA) + (p\u0000eA)\u0002@Etot\n@t\u0015\n+3e~\n64m4c4n\n(p\u0000eA)2\u0000e~\u001b\u0001B;~r\u0001Etot+\u001b\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot]o\n+e~4\n32m4c6r\u0001@2Etot\n@t2+e~3\n32m4c6\u001b\u0001\u0014@2Etot\n@t2\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002@2Etot\n@t2\u0015\n:\n(9)\nThe \felds in the last Hamiltonian (9) are de\fned as B=r\u0002A, the external magnetic\n\feld,Etot=Eint+Eextare the electric \felds where Eint=\u00001\nerVis the internal \feld\nthat exists even without any perturbation and Eext=\u0000@A\n@tis the external \feld (only\nthe temporal part is retained here because of the Coulomb gauge). It is clear that as the\ninternal \feld is time-independent, it does not contribute to the fourth and sixth lines\nof Eq. (9). However, the external \feld does contribute to the above terms wherever it\nappears in the Hamiltonian.\nThe above-derived Hamiltonian can be split in two parts: (1) a spin-independent\nHamiltonian and (2) a spin-dependent Hamiltonian that involves the Pauli spin matrices.\nThe spin-dependent Hamiltonian, furthermore, has two types of coupling terms. The\ndirect \feld-spin coupling terms are those which directly couples the \felds with the\nmagnetic moments e.g., the third term in the \frst line, the second term in the third\nline of Eq. (9) etc. On the other hand, there are relativistic terms that do not directly\ncouple the spins to the electromagnetic \feld - indirect \feld-spin coupling terms. These7\nterms include e.g., the second term of the second line, the \ffth line of Eq. (9) etc. The\ndirect \feld-spin interaction terms are most important because these govern the directly\nmanipulation of the spins in a system with an electromagnetic \feld. For the external\nelectric \feld, these terms can be written together as a function of electric and magnetic\n\feld. These terms are taken into account and discussed in the next section. The indirect\ncoupling terms are often not taken into consideration and not included in the discussion\n(see Ref. [36, 37] for details). In this context, we reiterate that our current approach of\nderiving relativistic terms does not include the exchange and correlation e\u000bect. A similar\nFW transformed Hamiltonian has previously been derived, however, with a general\nKohn-Sham exchange \feld [23, 25, 26]. As mentioned before, in this article we do not\nintend to include the exchange-correlation e\u000bect, while mostly focussing on the magnetic\nrelaxation and magnetic inertial dynamics.\n2.1. The spin Hamiltonian\nThe aim of this work is to formulate the spin dynamics on the basis of the Hamiltonian\nin Eq. (9). The direct \feld-spin interaction terms can be written together as electric or\nmagnetic contributions. These two contributions can be expressed as a series up to an\norder of 1=m5[36]\nHS\nmagnetic =\u0000e\nmS\u0001\"\nB+1\n2X\nn=1;2;3;4\u00121\n2i!c\u0013n@nB\n@tn#\n+O\u00121\nm6\u0013\n; (10)\nHS\nelectric =\u0000e\nmS\u0001\"\n1\n2mc2X\nn=0;2\u0012i\n2!c\u0013n@nE\n@tn\u0002(p\u0000eA)#\n+O\u00121\nm6\u0013\n; (11)\nwhere the Compton wavelength and pulsation have been expressed by the usual\nde\fnitions \u0015c=h=mc and!c= 2\u0019c=\u0015cwith Plank's constant h. We also have used\nthe spin angular momentum operator as S= (~=2)\u001b. Note that we have dropped\nthe notion of total electric \feld because the the involved \felds ( B,E,A) are external\nonly, the internal \felds are considered as time-independent. The involved terms in the\nabove two spin-dependent Hamiltonians can readily be explained. The \frst term in the\nmagnetic contribution in Eq. (10) explains the Zeeman coupling of spins to the external\nmagnetic \feld. The rest of the terms in both the Hamiltonians in Eqs. (11) and (10)\nrepresent the spin-orbit coupling and its higher-order corrections. We note that these\ntwo spin Hamiltonians are individually not Hermitian, however, it can be shown that\ntogether they form a Hermitian Hamiltonian [38]. As these Hamiltonians describe a\nsemirelativistic Dirac particle, it is possible to derive from them the spin dynamics of\na single Dirac particle [24]. The e\u000bect of the indirect \feld-spin terms is not yet well\nunderstood, but they could become important too in magnetism [36, 37], however, those\nterms are not of our interest here.\nThe electric Hamiltonian can be written in terms of magnetic contributions with\nthe choice of a gauge A=B\u0002r=2. The justi\fcation of the gauge lies in the fact8\nthat the magnetic \feld inside the system being studied is uniform [26]. The transverse\nelectric \feld in the Hamiltonian (10) can be written as\nE=1\n2\u0012\nr\u0002@B\n@t\u0013\n: (12)\nReplacing this expression in the electric spin Hamiltonian in Eq. (11), one can obtain a\ngeneralised expression of the total spin-dependent Hamiltonian as\nHS(t) =\u0000e\nmS\u0001h\nB+1\n21X\nn=1;2;:::\u00121\n2i!c\u0013n@nB\n@tn\n+1\n4mc21X\nn=0;2;:::\u0012i\n2!c\u0013n\u0012\nr\u0002@n+1B\n@tn+1\u0013\n\u0002(p\u0000eA)i\n: (13)\nIt is important to stress that the above spin-Hamiltonian is a generalisation of the two\nHamiltonians in Eqs. (10) and (11). We have already evaluated the Hamiltonian forms\nforn= 1;2;3;4 and assume that the higher-order terms will have the same form [36].\nThis Hamiltonian consists of the direct \feld-spin interaction terms that are linear and/or\nquadratic in the \felds. In the following we consider only the linear interaction terms,\nthat is we neglect the eAterm in Eq. (13). Here, we mention that the quadratic terms\ncould provide an explanation towards the previously unknown origin of spin-photon\ncoupling or optical spin-orbit torque and angular magneto-electric coupling [38{40].\nThe linear direct \feld-spin Hamiltonian can then be recast as\nHS(t) =\u0000e\nmS\u0001h\nB+1\n21X\nn=1;2;:::\u00121\n2i!c\u0013n@nB\n@tn\n+1\n4mc21X\nn=0;2;:::\u0012i\n2!c\u0013n\u001a@n+1B\n@tn+1(r\u0001p)\u0000r\u0012@n+1B\n@tn+1\u0001p\u0013\u001bi\n: (14)\nThis is \fnal form of the Hamiltonian and we are interested to describe to evaluate its\ncontribution to the spin dynamics.\n3. Spin dynamics\nOnce we have the explicit form of the spin Hamiltonian in Eq. (14), we can proceed to\nderive the corresponding classical magnetisation dynamics. Following similar procedures\nof previous work [23, 24], and introducing a magnetisation element M(r;t), the\nmagnetisation dynamics can be calculated by the following equation of motion\n@M\n@t=X\njg\u0016B\n\n1\ni~D\u0002\nSj;HS(t)\u0003E\n; (15)\nwhere\u0016Bis the Bohr magneton, gis the Land\u0013 e g-factor that takes a value \u00192 for electron\nspins and \n is a suitably chosen volume element. Having the spin Hamiltonian in Eq.9\n(14), we evaluate the corresponding commutators. As the spin Hamiltonian involves the\nmagnetic \felds, one can classify the magnetisation dynamics into two situations: (a) the\nsystem is driven by a harmonic \feld, (b) the system is driven by a non-harmonic \feld.\nHowever, in the below we continue the derivation of magnetisation dynamics with the\nharmonic driven \felds. The magnetisation dynamics driven by the non-harmonic \felds\nhas been discussed in the context of Gilbert damping and inertial dynamics where it was\nshown that an additional torque contribution (the \feld-derivative torque) is expected\nto play a crucial role [23, 24, 26].\nThe magnetisation dynamics due to the very \frst term of the Hamiltonian in Eq.\n(14) is derived as [24]\n@M(1)\n@t=\u0000\rM\u0002B; (16)\nwith the gyromagnetic ratio \r=gjej=2m. Here the commutators between two spin\noperators have been evaluated using [ Sj;Sk] =i~Sl\u000fjkl, where\u000fjklis the Levi-Civita\ntensor. This dynamics actually produces the precession of magnetisation vector around\nan e\u000bective \feld. To get the usual form of Landau-Lifshitz precessional dynamics, one\nhas to use a linear relationship of magnetisation and magnetic \feld as B=\u00160(M+H).\nWith the latter relation, the precessional dynamics becomes \u0000\r0M\u0002H, where\r0=\r\u00160\nde\fnes the e\u000bective gyromagnetic ratio. We point out that the there are relativistic\ncontributions to the precession dynamics as well, e.g., from the spin-orbit coupling due\nto the time-independent \feld Eint[23]. Moreover, the contributions to the magnetisation\nprecession due to exchange \feld appear here, but are not explicitly considered in this\narticle as they are not in the focus of the current investigations (see Ref. [23] for details).\nThe rest of the terms in the spin Hamiltonian in Eq. (14) is of much importance\nbecause they involve the time-variation of the magnetic induction. As it has been shown\nin an earlier work [23] that for the external \felds and speci\fcally the terms with n= 1\nin the second terms and n= 0 in the third terms of Eq. (14), these terms together\nare Hermitian. These terms contribute to the magnetisation dynamics as the Gilbert\nrelaxation within the LLG equation of motion,\n@M(2)\n@t=M\u0002\u0012\nA\u0001@M\n@t\u0013\n; (17)\nwhere the Gilbert damping parameter Ahas been derived to be a tensor that has mainly\ntwo contributions: electronic and magnetic. The damping parameter Ahas the form\n[23, 24]\nAij=\u0000e\u00160\n8m2c2X\n`;k\u0002\nhripk+pkrii\u0000hr`p`+p`r`i\u000eik\u0003\n\u0002\u0000\n1+\u001f\u00001\u0001\nkj; (18)\nwhere 1is the 3\u00023 unit matrix and \u001fis the magnetic susceptibility tensor that can be\nintroduced only if the system is driven by a \feld which is single harmonic [26]. Note\nthat the electronic contributions to the Gilbert damping parameter are given by the10\nexpectation value hripkiand the magnetic contributions by the susceptibility. We also\nmention that the tensorial Gilbert damping tensor has been shown to contain a scalar,\nisotropic Heisenberg-like contribution, an anisotropic Ising-like tensorial contribution\nand a chiral Dzyaloshinskii-Moriya-like contribution [23].\nIn an another work, we took into account the terms with n= 2 in the second term\nof Eq. (14) and it has been shown that those containing the second-order time variation\nof the magnetic induction result in the magnetic inertial dynamics. Note that these\nterms provide a contribution to the higher-order relativistic e\u000bects. The corresponding\nmagnetisation dynamics can be written as [24]\n@M(3)\n@t=M\u0002\u0012\nC\u0001@M\n@t+D\u0001@2M\n@t2\u0013\n; (19)\nwith a higher-order Gilbert damping tensor Cijand inertia parameter Dijthat have the\nfollowing expressions Cij=\r0~2\n8m2c4@\n@t( 1+\u001f\u00001)ijandDij=\r0~2\n8m2c4( 1+\u001f\u00001)ij. We note\nthat Eq. (19) contains two fundamentally di\u000berent dynamics { the \frst term on the\nright-hand side has the exact form of Gilbert damping dynamics whereas the second\nterm has the form of magnetic inertial dynamics [24].\nThe main aim of this article is to formulate a general magnetisation dynamics\nequation and an extension of the traditional LLG equation to include higher-order\nrelativistic e\u000bects. The calculated magnetisation dynamics due to the second and third\nterms of Eq. (14) can be expressed as\n@M\n@t=e\nmM\u0002h1\n21X\nn=0;1;:::\u00121\n2i!c\u0013n+1@n+1B\n@tn+1\n+1\n4mc21X\nn=0;2;:::\u0012i\n2!c\u0013n\u001a@n+1B\n@tn+1hr\u0001pi\u0000D\nr\u0012@n+1B\n@tn+1\u0001p\u0013E\u001bi\n: (20)\nNote the di\u000berence in the summation of \frst terms from the Hamiltonian in Eq. (14).\nTo obtain explicit expressions for the Gilbert damping dynamics, we employ a general\nlinear relationship between magnetisation and magnetic induction, B=\u00160(H+M).\nThe time-derivative of the magnetic induction can then be replaced by magnetisation\nand magnetic susceptibility. For the n-th order time-derivative of the magnetic induction\nwe \fnd\n@nB\n@tn=\u00160\u0012@nH\n@tn+@nM\n@tn\u0013\n: (21)\nNote that this equation is valid for the case when the magnetisation is time-dependent.\nSubstituting this expression into the Eq. (20), one can derive the general LLG equation\nand its extensions. Moreover, as we work out the derivation in the case of harmonic\ndriving \felds, the di\u000berential susceptibility can be introduced as \u001f=@M=@H. The\n\frst term ( n-th derivative of the magnetic \feld) can consequently be written by the11\nfollowing Leibniz formula as\n@nH\n@tn=n\u00001X\nk=0(n\u00001)!\nk!(n\u0000k\u00001)!@n\u0000k\u00001(\u001f\u00001)\n@tn\u0000k\u00001\u0001@k\n@tk\u0012@M\n@t\u0013\n; (22)\nwhere the magnetic susceptibility \u001f\u00001is a time-dependent tensorial quantity and\nharmonic. Using this relation, the \frst term and second terms in Eq. (20) assume\nthe form\n@M\n@t\f\f\f\n\frst=e\u00160\n2mM\u00021X\nn=0;1;:::\u00121\n2i!c\u0013n+1nX\nk=0n!\nk!(n\u0000k)!@n\u0000k( 1+\u001f\u00001)\n@tn\u0000k\u0001@k\n@tk\u0012@M\n@t\u0013\n;\n(23)\n@M\n@t\f\f\f\nsecond=e\u00160\n4m2c2M\u0002\n1X\nn=0;2;:::\u00121\n2i!c\u0013nnX\nk=0n!\nk!(n\u0000k)!h@n\u0000k( 1+\u001f\u00001)\n@tn\u0000k\u0001@k\n@tk\u0012@M\n@t\u0013\nhr\u0001pi\n\u0000D\nr\u0012\u001a@n\u0000k( 1+\u001f\u00001)\n@tn\u0000k\u0001@k\n@tk\u0012@M\n@t\u0013\u001b\n\u0001p\u0013Ei\n:(24)\nThese two equations already provide a generalisation of the higher-order magnetisation\ndynamics including the Gilbert damping (i.e., the terms with k= 0) and the inertial\ndynamics (the terms with k= 1) and so on.\n4. Discussion\n4.1. Gilbert damping parameter\nIt is obvious that, as Gilbert damping dynamics involves the \frst-order time derivative of\nthe magnetisation and a torque due to it, kmust take the value k= 0 in the equations\n(23) and (24). Therefore, the Gilbert damping dynamics can be achieved from the\nfollowing equations:\n@M\n@t\f\f\f\n\frst=e\u00160\n2mM\u00021X\nn=0;1;:::\u00121\n2i!c\u0013n+1@n( 1+\u001f\u00001)\n@tn\u0001@M\n@t; (25)\n@M\n@t\f\f\f\nsecond=e\u00160\n4m2c2M\u00021X\nn=0;2;:::\u00121\n2i!c\u0013nh\u0012@n( 1+\u001f\u00001)\n@tn\u0001@M\n@t\u0013\nhr\u0001pi\n\u0000D\nr\u0012\u001a@n( 1+\u001f\u00001)\n@tn\u0001@M\n@t\u001b\n\u0001p\u0013Ei\n: (26)\nNote that these equations can be written in the usual form of Gilbert damping as\nM\u0002\u0000\nG\u0001@M\n@t\u0001\n, where the Gilbert damping parameter Gis notably a tensor [2, 23]. The12\ngeneral expression for the tensor can be given by a series of higher-order relativistic\nterms as follows\nGij=e\u00160\n2m1X\nn=0;1;:::\u00121\n2i!c\u0013n+1@n( 1+\u001f\u00001)ij\n@tn\n+e\u00160\n4m2c21X\nn=0;2;:::\u00121\n2i!c\u0013nh@n( 1+\u001f\u00001)ij\n@tn(hrlpli\u0000hrlpii)i\n: (27)\nHere we have used the Einstein summation convention on the index l. Note that there\nare two series: the \frst series runs over even and odd numbers ( n= 0;1;2;3;\u0001\u0001\u0001),\nhowever, the second series runs only over the even numbers ( n= 0;2;4;\u0001\u0001\u0001). Eq. (27)\nrepresents a general relativistic expression for the Gilbert damping tensor, given as a\nseries of higher-order terms. This equation is one of the central results of this article. It\nis important to observe that this expression provides the correct Gilbert tensor at the\nlowest relativistic order, i.e., putting n= 0 the expression for the tensor is found to be\nexactly the same as Eq. (18).\nThe analytic summation of the above series of higher-order relativistic contributions\ncan be carried out when the susceptibility depends on the frequency of the harmonic\ndriving \feld. This is in general true for ferromagnets where a di\u000berential susceptibility\nis introduced because there exists a spontaneous magnetisation in ferromagnets even\nwithout application of a harmonic external \feld. However, if the system is driven by a\nnonharmonic \feld, the introduction of the susceptibility is not valid anymore. In general\nthe magnetic susceptibility is a function of wave vector and frequency in reciprocal space,\ni.e.,\u001f=\u001f(q;!). Therefore, for the single harmonic applied \feld, we use \u001f\u00001/ei!tand\nthen-th order derivative will follow @n=@tn(\u001f\u00001)/(i!)n\u001f\u00001. With these arguments,\none can express the damping parameter of Eq. (27) as (see Appendix A for detailed\ncalculations)\nGij=e\u00160\n4m2c2\u0014~\ni+hrlpli\u0000hrlpii\u0015\n( 1+\u001f\u00001)ij\n+e\u00160\n4m2c2\"\n(2!!c+!2)~\ni+!2(hrlpli\u0000hrlpii)\n4!2\nc\u0000!2#\n\u001f\u00001\nij: (28)\nHere, the \frst term in the last expression is exactly the same as the one that has been\nderived in our earlier investigation [23]. As the expression of the expectation value\nhripjiis imaginary, the real Gilbert damping parameter will be given by the imaginary\npart of the susceptibility tensor. This holds consistently for the higher-order terms\nas well. The second term in Eq. (28) stems essentially from an in\fnite series which\ncontain higher-order relativistic contributions to the Gilbert damping parameter. As\n!cscales with c, these higher-order terms will scale with c\u00004or more and thus their\ncontributions will be smaller than the \frst term. Note that the higher-order terms will\ndiverge when != 2!c\u00191021sec\u00001, which means that the theory breaks down at the\nlimit!!2!c. In this limit, the original FW transformation is not de\fned any more\nbecause the particles and antiparticles cannot be separated at this energy limit.13\n4.2. Magnetic inertia parameter\nMagnetic inertial dynamics, in contrast, involves a torque due to the second-order time-\nderivative of the magnetisation. In this case, kmust adopt the value k= 1 in the\nafore-derived two equations (23) and (24). However, if k= 1, the constraint n\u0000k\u00150\ndictates that n\u00151. Therefore, the magnetic inertial dynamics can be described with\nthe following equations:\n@M\n@t\f\f\f\n\frst=e\u00160\n2mM\u00021X\nn=1;2;:::\u00121\n2i!c\u0013n+1n!\n(n\u00001)!@n\u00001( 1+\u001f\u00001)\n@tn\u00001\u0001@2M\n@t2; (29)\n@M\n@t\f\f\f\nsecond=e\u00160\n4m2c2M\u00021X\nn=2;4;:::\u00121\n2i!c\u0013nn!\n(n\u00001)!h\u0012@n\u00001( 1+\u001f\u00001)\n@tn\u00001\u0001@2M\n@t2\u0013\nhr\u0001pi\n\u0000D\nr\u0012\u001a@n\u0000k( 1+\u001f\u00001)\n@tn\u0000k\u0001@2M\n@t2\u001b\n\u0001p\u0013Ei\n: (30)\nSimilar to the Gilbert damping dynamics, these dynamical terms can be expressed\nasM\u0002\u0010\nI\u0001@2M\n@t2\u0011\nwhich is the magnetic inertial dynamics [8]. The corresponding\nparameter has the following expression\nIij=e\u00160\n2m1X\nn=1;2;:::\u00121\n2i!c\u0013n+1n!\n(n\u00001)!@n\u00001( 1+\u001f\u00001)ij\n@tn\u00001\n+e\u00160\n4m2c21X\nn=2;4;:::\u00121\n2i!c\u0013nn!\n(n\u00001)!h@n\u00001( 1+\u001f\u00001)ij\n@tn\u00001(hrlpli\u0000hripli)i\n: (31)\nNote that as ncannot adopt the value n= 0, the starting values of nare di\u000berent in\nthe two terms. Importantly, if n= 1 we recover the expression for the lowest order\nmagnetic inertia parameter Dij, as given in the equation (19) [24].\nUsing similar arguments as in the case of the generalised Gilbert damping\nparameter, when we consider a single harmonic \feld as driving \feld, the inertia\nparameter can be rewritten as follows (see Appendix A for detailed calculations)\nIij=\u0000e\u00160~2\n8m3c4( 1+\u001f\u00001)ij\u0000e\u00160~2\n8m3c4\u0012\u0000!2+ 4!!c\n(2!c\u0000!)2\u0013\n\u001f\u00001\nij\n+e\u00160\n8m3c4~\ni(hrlpli\u0000hripli)\u001216!!3\nc\n(4!2\nc\u0000!2)2\u0013\n\u001f\u00001\nij: (32)\nThe \frst term here is exactly the same as the one that was obtained in our earlier\ninvestigation [24]. However, there are now two extra terms which depend on the\nfrequency of the driving \feld and that vanish for !!0. Again, in the limit !!2!c,\nthese two terms diverge and hence this expression is not valid anymore. The inertia\nparameter will consistently be given by the real part of the susceptibility.14\n5. Summary\nWe have developed a generalised LLG equation of motion starting from fundamental\nquantum relativistic theory. Our approach leads to higher-order relativistic correction\nterms in the equation of spin dynamics of Landau and Lifshitz. To achieve this, we have\nstarted from the foundational Dirac equation under the presence of an electromagnetic\n\feld (e.g., external driving \felds or THz excitations) and have employed the FW\ntransformation to separate out the particles from the antiparticles in the Dirac equation.\nIn this way, we derive an extended Pauli Hamiltonian which e\u000eciently describes the\ninteractions between the quantum spin-half particles and the applied \feld. The thus-\nderived direct \feld-spin interaction Hamiltonian can be generalised for any higher-order\nrelativistic corrections and has been expressed as a series. To derive the dynamical\nequation, we have used this generalised spin Hamiltonian to calculate the corresponding\nspin dynamics using the Heisenberg equation of motion. The obtained spin dynamical\nequation provides a generalisation of the phenomenological LLG equation of motion\nand moreover, puts the LLG equation on a rigorous foundational footing. The equation\nincludes all the torque terms of higher-order time-derivatives of the magnetisation (apart\nfrom the Gilbert damping and magnetic inertial dynamics). Speci\fcally, however, we\nhave focussed on deriving an analytic expression for the generalised Gilbert damping\nand for the magnetic inertial parameter. Our results show that both these parameters\ncan be expressed as a series of higher-order relativistic contributions and that they\nare tensors. These series can be summed up for the case of a harmonic driving \feld,\nleading to closed analytic expressions. We have further shown that the imaginary part\nof the susceptibility contributes to the Gilbert damping parameter while the real part\ncontributes to the magnetic inertia parameter. Lastly, with respect to the applicability\nlimits of the derived expressions we have pointed out that when the frequency of the\ndriving \feld becomes comparable to the Compton pulsation, our theory will not be valid\nanymore because of the spontaneous particle-antiparticle pair-production.\n6. Acknowledgments\nWe thank P-A. Hervieux for valuable discussions. This work has been supported\nby the Swedish Research Council (VR), the Knut and Alice Wallenberg Foundation\n(Contract No. 2015.0060), the European Union's Horizon2020 Research and\nInnovation Programme under grant agreement No. 737709 (FEMTOTERABYTE,\nhttp://www.physics.gu.se/femtoterabyte).15\nAppendix A. Detailed calculations of the parameters for a harmonic \feld\nIn the following we provide the calculational details of the summation towards the results\ngiven in Eqs. (28) and (32).\nAppendix A.1. Gilbert damping parameter\nEq. (27) can be expanded as follows\nGij=e\u00160\n2m1\n2i!c( 1+\u001f\u00001)ij+e\u00160\n4m2c2(hrlpli\u0000hrlpii) ( 1+\u001f\u00001)ij\n+e\u00160\n2m1X\nn=1;2;:::\u00121\n2i!c\u0013n+1\n(i!)n\u001f\u00001\nij+e\u00160\n4m2c21X\nn=2;4;:::\u00121\n2i!c\u0013n\n(hrlpli\u0000hrlpii) (i!)n\u001f\u00001\nij\n=e\u00160\n2m1\n2i!c( 1+\u001f\u00001)ij+e\u00160\n4m2c2(hrlpli\u0000hrlpii) ( 1+\u001f\u00001)ij\n+e\u00160\n2m1\n2i!c1X\nn=1;2;:::\u0012!\n2!c\u0013n\n\u001f\u00001\nij+e\u00160\n4m2c21X\nn=2;4;:::\u0012!\n2!c\u0013n\n(hrlpli\u0000hrlpii)\u001f\u00001\nij\n=e\u00160\n4m2c2\u0014~\ni+hrlpli\u0000hrlpii\u0015\n( 1+\u001f\u00001)ij\n+e\u00160\n4m2c2\"\n~\ni1X\nn=1;2;:::\u0012!\n2!c\u0013n\n+ (hrlpli\u0000hrlpii)1X\nn=2;4;:::\u0012!\n2!c\u0013n#\n\u001f\u00001\nij\n=e\u00160\n4m2c2\u0014~\ni+hrlpli\u0000hrlpii\u0015\n( 1+\u001f\u00001)ij\n+e\u00160\n4m2c2\u0014~\ni!\n2!c\u0000!+ (hrlpli\u0000hrlpii)!2\n4!2\nc\u0000!2\u0015\n\u001f\u00001\nij\n=e\u00160\n4m2c2\u0014~\ni+hrlpli\u0000hrlpii\u0015\n( 1+\u001f\u00001)ij\n+e\u00160\n4m2c2\"\n(2!!c+!2)~\ni+!2(hrlpli\u0000hrlpii)\n4!2\nc\u0000!2#\n\u001f\u00001\nij: (A.1)\nWe have used the fact that!\n!c<1 and the summation formula\n1 +x+x2+x3+:::=1\n1\u0000x;\u00001eκγM 0|Hz|LAFM\nµBQ≡j0. (40)\nAtκ∼10−2,γM0∼1010s−1,|Hz| ∼102G,LAFM∼10−6cm,Q∼1\nwe have j0∼105A/cm2. Atjnear toj0weakly damping THz oscilla-\ntion can be obtained. At j > j 0, instability occurs which may lead to\neither self-sustained oscillations, or a dynamic stationa ry state. The lat-\nter disappears with the current turning off. To answer the que stion about\nfuture of the instability it is necessary to go out the scope o f the linear\napproximation.\nThespin-polarized currentcontributesalsototheoscilla tion frequency.\nAt the mentioned parameter values, we have |KMz| ∼1012s−1that is\ncomparable with the frequency in absence of the current. Thi s allows\ntuning the frequency by the current or excite parametric res onance by\nmeans of the current modulation.\n5 Current-induced spin injection effect\nNow let us discuss the injection mechanism effect [22, 23]. As mentioned\nbefore, the role of the mechanism is reduced to addition of an effective\nfieldP/γto the external magnetic field. At reasonable parameter valu es,\nthat field is much less than the exchange field HE, so that it does not\ninfluence directly the eigenfrequency (36). Nevertheless, that field can\nmodify substantially the contribution of the STT mechanism , because\nEq. (39) with (25) taking into account now takes the form\nKMz=µBQΛ\neLAFM(Hz+P/γ)j. (41)\nSuch a modification leads to substantial consequences. At Hz<0,P <\nγ|Hz|the instability threshold (40) is lowered, since |Hz|−P/γdifference\nappears now instead of |Hz|. If, however, P > γ|Hz|then the AFM\nmagnetization steady state\nMz=Hz+P/γ\nΛ(42)\nbecomes positive that corresponds to the parallel (stable) relative orienta-\ntion of the FM and AFM layers. In this case, the turning on curr ent leads\nto switching the antiparallel configuration (stated before hand by means\n10of an external magnetic field) to parallel one. With turning o ff current,\nthe antiparallel configuration restores.\nSincethementionedinjection-drivenfielddependsonthecu rrent(see(18)),\nthe instability condition (40) is modified and takes the form\nj0\n1+η< j j 0/ηthe antiparallel configuration switches to parallel one. Th e\nrelative contribution of the injection mechanism is determ ined with η\nparameter. At typical values, αsd∼104,κ∼10−2,γM0∼1010s−1,\nτ∼10−12s, this parameter is of the order of unity, so that the injecti on\neffect may lower noticeably the instability threshold.\nNow let us return to the set of equations (26)–(31) and consid er the\nsecond mode describing with Eqs. (28)–(30). The current infl uences this\nmode by changing steady magnetization Mzdue to the injection effective\nfield effect (see (26)), while the STT mechanism does not influe nce this\nmode. A calculation similar to previous one gives the former dispersion\nrelation (35), but now\nω2\n0= 2γ2HEHAγHz\nγHz+P, (44)\nν=κγHEγHz\nγHz+P. (45)\nAtHz<0,P >|Hz|, thatcorrespondstocurrentdensity j > j0/η, the\ntotal attenuation becomes negative, while the frequency be comes imagi-\nnary, that means switching the antiparallel configuration t o parallel one.\nThus, current does not cause instability of that mode.\n6 Easy plane type antiferromagnet\nLetusconsiderbrieflythesituation whereAFMhaseasy-plan eanisotropy.\nWe take the AFM layer yzplane as the easy plane and xaxis as the (hard)\nanisotropy axis. The magnetic field, as before, is directed a longzaxis.\nWithout repeating calculations, similar to previous ones, we present\nthe results. A formal difference appears only in Eq. (36) for t he eigenfre-\nquencyω0of the first of the modes considered above. We have for that\nfrequency\nω0=/radicalBig\n(γHz+P)2+(KMz)2. (46)\nThe damping has the former form (37), so that the instability threshold\nis determined with former formula (43).\nIn absence of the current ( K= 0, P= 0) with not too small damping\ncoefficient κ, the frequency appears to be much less than damping, so\nthat the corresponding oscillations are not observed. The c urrent effect\nincreases the frequency, on the one hand, and decreases the d amping (at\nHz<0), on the other hand, that allows to observe oscillation reg ime.\n117 Fluctuation effect\nIt follows from Eq. (43) that the threshold current density i s proportional\ntotheexternalmagnetic fieldstrength |Hz|anddecreases withthefield. A\nquestion arises about permissible lowest limit of the total field|Hz|+P/γ.\nIn accordance with Eq. (25), such a limit may be the field which create\nmagnetization |Mz|comparable with its equilibrium value due to thermal\nfluctuations. Let us estimate this magnetization and the cor responding\nfield.\nThe AFM energy change in Vvolume under canting the sublattice\nmagnetization vectors with θ <180◦angle between them is\n∆E= ΛM2\n0(1−cosθ)V=1\n2ΛVM2\nz, (47)\nthe anisotropy energy being neglected compared to the excha nge energy.\nThe equilibrium value of the squared magnetization is calcu lated using\nthe Gibbs distribution:\n/an}bracketle{tM2\nz/an}bracketri}ht=∞/integraltext\n−∞M2\nzexp/parenleftbigg\n−ΛVM2\nz\n2kT/parenrightbigg\ndMz\n∞/integraltext\n−∞exp/parenleftbigg\n−ΛVM2\nz\n2kT/parenrightbigg\ndMz=kT\nΛV(48)\n(strictlyspeaking, themagnetization maybechangedwithi n(−2M0,2M0)\ninterval, however, Λ VM2\n0≫kT, so that the integration limits may be\ntaken infinity).\nTo observe the effects described above, the magnetization Mzwhich\nappears under joint action of the external field and the curre nt (see (25))\nshould exceed in magnitude the equilibrium magnetization /an}bracketle{tM2\nz/an}bracketri}ht1/2. At\nthe current density j=j0/(1+η) corresponding to the instability thresh-\nold, this condition is fulfilled at magnetic field\n|Hz|>/radicalbigg\nΛkT\nV(1+η)≡Hmin. (49)\nAt Λ∼104,η∼1,LAFM∼10−6cm and lateral sizes of the switched\nelement 10 ×10µm2we have V∼10−12cm3andHmin≈30 G at room\ntemperature. This limit can be decreased under larger eleme nt size.\nIt should be mentioned also about other mechanisms of AFM can ting.\nThe most known and studied one is the relativistic Dzyaloshi nskii–Moria\neffect(see, e.g.[21,27]). Besides, possible mechanismsha vebeendiscussed\ndue to competition between sdexchange and direct exchange interaction\nof the magnetic ions in the lattice [28]. At the same time, the re are no\nindications, to our knowledge, about measurements of canti ng in conduc-\ntive AFM. So, present theory is related to conductive AFM, in which the\nlattice canting is determined with external magnetic field.\n8 Conclusions\nThe obtained results show a principal possibility of contro lling frequency\nand damping of AMF resonance in FM–AFM junctions by means of s pin-\npolarized current. Under low AFM magnetization induced by a n external\n12magnetic field perpendicular totheantiferromagnetism vec tor, thethresh-\nold current density corresponding to occurring instabilit y is less substan-\ntially than in the FM–FM case. Near the threshold, the AFM res onance\nfrequency increases, while damping decreases, that opens a possibility of\ngenerating oscillations in THz range.\nAcknowledgments\nThe authors are grateful to Prof. G. M. Mikhailov for useful d iscussions.\nThe work was supported by the Russian Foundation for Basic Re -\nsearch, Grant No. 10-02-00030-a.\nReferences\n[1] J.C. Slonczewski. J. Magn. Magn. Mater. 159, L1 (1996).\n[2] L. Berger. Phys. Rev. B 54, 9353 (1996).\n[3] J.A. Katine, F.J. Albert, R.A. Buhrman, E.B. Myers, D.C. Ralph.\nPhys. Rev. Lett. 84, 3149 (2000).\n[4] M. Tsoi, A.J.M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, P.\nWyder Phys. Rev. Lett. 80, 4281 (1998).\n[5] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, T. Shinj o.\nPhys. Rev. Lett. 92, 077205 (2004).\n[6] J.C. Sankey, P.M. Braganca, A.G.F. Garcia I.N. Krivorot ov, R.A.\nBuhrman, D.C. Ralph. Phys. Rev. Lett. 96, 227601 (2006).\n[7] M. Watanabe, J. Okabayashi, H. Toyao, T. Yamaguchi, J. Yo shino.\nAppl. Phys. Lett. 92, 082506 (2008).\n[8] Yu.V. Gulyaev, P.E. Zilberman, A.I. Krikunov, E.M. Epsh tein.\nTechn. Phys. 52, 1169 (2007).\n[9] Yu.V. Gulyaev, P.E. Zilberman, A.I. Panas, E.M. Epshtei n. J. Exp.\nTheor. Phys. 107, 1027 (2008).\n[10] Yu.V. Gulyaev, P.E. Zilberman, S.G. Chigarev, E.M. Eps htein.\nTechn. Phys. Lett. 37, 154 (2011).\n[11] Yu.V.Gulyaev, P.E.Zilberman, E.M. Epshtein.J.Commu n.Technol.\nElectron. 56, 863 (2011).\n[12] V.K.Sankaranarayanan, S.M.Yoon, D.Y.Kim, C.O. Kim, C .G. Kim.\nJ. Appl. Phys. 96, 7428 (2004).\n[13] A. S. N´ u˜ nez, R. A. Duine, P. Haney, A.H. MacDonald. Phy s. Rev. B\n73, 214426 (2006).\n[14] Z. Wei, A. Sharma, A.S. Nunez, P.M. Haney, R.A. Duine, J. Bass,\nA.H. MacDonald, M. Tsoi. Phys. Rev. Lett. 98, 116603 (2007).\n[15] Z. Wei, A. Sharma, J. Bass, M. Tsoi. J. Appl. Phys. 105, 07D113\n(2009).\n[16] J. Basset, Z. Wei, M. Tsoi. IEEE Trans. Magn. 46, 1770 (2010).\n13[17] S. Urazhdin, N. Anthony. Phys. Rev. Lett. 99, 046602 (2007).\n[18] H.V. Gomonay, V.M. Loktev. Low Temp. Phys. 34, 198 (2008).\n[19] H.V. Gomonay, V.M. Loktev. Phys. Rev. B 81, 144127 (2010).\n[20] K.M.D. Hals, Y. Tserkovnyak, A. Brataas. Phenomenolog y of\ncurrent-induced dynamics in antiferromagnets, arXiv:101 2.5655v1\n[cond-mat.mes-hall].\n[21] A.I. Akhiezer, V.G. Baryakhtar, S.V. Peletminslii. Sp in Waves,\nNorth-Holland Pub. Co., Amsterdam, 1968.\n[22] C. Heide, P.E. Zilberman, R.J. Elliott, Phys. Rev. B 63, 064424\n(2001).\n[23] Yu.V. Gulyaev, P.E. Zilberman, E.M. Epshtein, R.J. Ell iott, JETP\nLett.76, 155 (2002).\n[24] Yu.V. Gulyaev, P.E. Zilberman, A.I. Panas, E.M. Epshte in.\nPhysics — Uspekhi 52, 335 (2009).\n[25] Yu.V. Gulyaev, P.E. Zilberman, E.M. Epshtein, R.J. Ell iott. J. Exp.\nTheor. Phys. 100, 1005 (2005).\n[26] A.G.Gurevich, G.A.Melkov.Magnetizsation Oscillati ons andWaves,\nCRC Press, Boca Raton, FL, 1996.\n[27] V.E. Dmitrienko, E.N. Ovchinnikova, J. Kokubun, K. Ish ida. JETP\nLett.92, 383 (2010).\n[28] J.M. Robinson, P. Erd¨ os. Phys. Rev. B 6, 3337 (1972).\n14" }, { "title": "1502.00989v1.On_the_Stability_of_Cylindrical_Tangential_Discontinuity__Generation_and_Damping_of_Helical_Waves.pdf", "content": "arXiv:1502.00989v1 [physics.plasm-ph] 1 Feb 2015On the Stability of Cylindrical Tangential Discontinuity,\nGeneration and Damping of Helical Waves\nA. I. Ershkovich and P.L. Israelevich\nDepartment of Geophysics and Planetary Sciences, Faculty o f Exact Sciences, Tel Aviv\nUniversity, Tel Aviv, 69978, Israel\nABSTRACT\nStability of cylindrical interface between two ideal incompressible flu ids, in-\ncluding the magnetic field, surface tension and gravitational field is s tudied in\nlinear approximation. We found that helical waves arising both in plasm a comet\ntails and on the vertical cylindrical water jet in the air are described by the same\ndispersion equation where the comet tail magnetic field plays the sam e stabilizing\nrole as surface tension for water jet. Hence they represent the same phenomenon\nof Kelvin-Helmholtz instability. Thus, helical waves in planetary and co metary\nmagnetotails as well as in astrophysical jets may be simulated in the la boratory.\nThe resonance nature of the instability damping is demonstrated.\nSubject headings: instabilities — comets: general\n1. Introduction\nStability of plane interface between two ideal incompressible fluids ha s been first con-\nsidered by Kelvin (1871) (see also (Landau & Lifshitz 1959; Milne-Tho mson 1960)).\nSmall oscillations in such a fluid are always potential in the first approx imation. There-\nfore, the velocity potential satisfies the Laplace equation, and fo r perturbations proportional\nto exp[i(kx−ωt)] we arrive at the dispersion equation\nω\nk=ρ1V1+ρ2V2\nρ1+ρ2±/bracketleftBigg\n−ρ1ρ2(V1−V2)2\n(ρ1+ρ2)2+ρ1−ρ2\nρ1+ρ2/parenleftBiggn\nk/parenrightBig\n+σk\nρ1+ρ2/bracketrightBigg1/2\n, (1)\nwhereρis the density, Vis the velocity, gnis the normal (to the interface) component\nof the gravitational acceleration, and σis the surface tension. Rayleigh (1892) was the\nfirst to consider a cylindrical interface but his stability analysis was r estricted by ”varicose”– 2 –\nperturbations (in modern nomenclature, sausage-like ones) also p roportional to exp[ i(kz−\nωt)], and hence could not describe kink-modes proportional to exp[ i(kz+mϕ−ωt)],m∝negationslash= 0.\nMore general analysis was required following a discovery of the geom agnetic tail by Ness\n(1965). It has been performed in (McKenzie 1970) for a cylindrical interface in compressible\nplasma in MHD approximation (the dispersion equation in this case is tra nscendental and\nmay be solved only numerically), and in (Ershkovich & Nusinov 1971, 19 72) for incompress-\nible plasma with the magnetic field (like the plasma bulk velocity) parallel t o the cylinder\naxisz. The solution of Laplace equation proportional to exp[ i(kz+mϕ−ωt)] describes\nhelical waves, like kink modes m= 1,2, etc., along with sausage mode m= 0 (the solution\nis single-valued only if mis an integer, positive or negative). If such solutions are possible\nthen helical waves may exist in the nature, identified and observed.\nIndeed, in contrast to the Earth magnetic tail, such oscillations hav e long been observed\nvisuallyin rectilinear comet plasma tails (type I tails). Bessel (1836) was the first to describe\nthese wave motions in detail. Alfv´ en (1957) assumed that they are MHD waves. But helical\nwaves arising due to interface instability are surface waves propag ating in both fluids (as\na whole). Hence, they cannot be Alfv´ en waves as the Alfv´ en veloc ity in the comet tails\n(where the plasma is heavy) is usually much less than in the neighboring solar wind. The\nquantitative MHD theory of helical wave origin in type I comet tails due to instability\nin plasma cylinder was suggested in (Ershkovich et.al. 1972; Ershkov ich & Chernikov 1973;\nErshkovich 1980). The corresponding dispersion equation for the model of plasma cylinder\nwith radius Ris\nω\nk=ρiVi−LmρeVe\nρi−Lmρe±/bracketleftBigg\nB2\ni−LmB2\ne\n4π(ρi−Lmρe)+Lmρiρe(Ve−Vi)2\n(ρi−Lmρe)2/bracketrightBigg1/2\n, (2)\nwhere indices ianderefer to internal and external plasmas, respectively, and the fu nction\nLm= [I′\nm(kr)Km(kr)]/[Im(kr)K′\nm(kr)] is taken at the unperturbed interface r=R.Im\nandKmare modified Bessel functions, a stroke denotes the derivative by argument kr. The\nfunction Lm(kR) is always negative: 0 ≥Lm≥ −1 (see Figure 1).\nBy using the same standard procedure of linearization described in m ore details in\n(Ershkovich & Nusinov 1972; Ershkovich 1980) we may also include th e gravitational field g\nand surface tension σ, arriving at the following dispersion equations: for cylindrical interf ace\nbetween two liquids\nω\nk=ρiVi−LmρeVe\nρi−Lmρe±/bracketleftBigg\nB2\ni−LmB2\ne\n4π(ρi−Lmρe)+Lmρiρe(Ve−Vi)2\n(ρi−Lmρe)2+gn\nk(ρi+Lmρe)\n(ρi−Lmρe)+σk\nρi−Lmρe/bracketrightBigg1/2\n,\n(3)– 3 –\nand for liquid-gas interface\nω\nk=ρiVi−LmρeVe\nρi−Lmρe±/bracketleftBigg\nB2\ni−LmB2\ne\n4π(ρi−Lmρe)+Lmρiρe(Ve−Vi)2\n(ρi−Lmρe)2+gn\nk(ρi+Lmρe)\n(ρi−Lmρe)+σk\nρi−Lmρe/parenleftbiggI′\nm\nIm/parenrightbigg/bracketrightBigg1/2\n.\n(4)\nThe difference in the last term between equations above is due to the fact that σ= 0 for the\ngas which is located outside the cylinder ( r > R).\n2. Discussion\nWithVe=Vi=V,gn= 0,σ= 0 equations (3) and (4) describe stable ”surface\nAlfv´ en waves” convected with the fluid bulk velocity V:ω/k=V±[(B2\ni−LmB2\ne)/(4πρi−\n4πLmρe)]1/2. Standard expression VA=B/√4πρis obtained from the expression above with\nρ=ρi≫ρe,B=Bi≫Be.\nIn the short wavelength limit kR≫1,Lm→ −1, and with g= 0,σ= 0 equation (3)\nreduces to the dispersion equation for plane interface obtained by Syrovatskii (1953) within\nMHD for incompressible plasma. With kR≫1 andB= 0 equation (3), naturally, reduces\nto the dispersion equation (1). Finally, with B= 0,gn= 0,Ve= 0 equation (4) is the\nsolution of the dispersion equation (2.1) in (Yarin 2011) for a vertica l cylindrical water jet in\nair. Strictly speaking, the dispersion equation (2.1) for water jet ( Yarin 2011) was obtained\nfor perturbations proportional to exp[ i(kz−ωt)]. Such perturbations are plane waves, which,\nin principle, cannot correspond to helical waves. But it is clear that t he structure of the\ndispersion equation for cylindrical jet, being dependent on combina tionsIm,Kmand their\nfirst derivatives, has the same form for all integer values of m.\nAsLm<0,I′\nm/Im>0 both magnetic field and surface tension terms are always\npositive and hence tend to stabilize the interface. The velocity shea r term under the radical\nis negative being responsible for the Kelvin-Helmholtz instability. The t erm proportional to\ngnresults in the flute (Rayleigh-Taylor) instability for heavy fluid over t he light one, and\nwithρi≫ρe,V= 0 it describes stable gravitational waves on deep water: ω2=kg.\nExternal gravitational field always tends to violate the cylindrical s ymmetry, because\nthe external normal to the cylindrical surface changes its sign (r elative to the field) at the\nopposite sides of the cylinder.\nThere are, however, several cases when gnis small or even vanishes. This is, of course,\nalmost rectilinear (and cylindrical) plasma tail of comets, where helica l waves are visually\nobserved (see e.g. (Ershkovich 1980) and references therein) a s well as astrophysical jets,– 4 –\ne.g. (Birkinshaw 1996). Formation of plasma comet tails is governed b y solar wind, and,\nhence, they are almost in antisolar direction, with rather small aber ration angle due to\norbital motion around the Sun. Another example (although hypoth etical) is self-gravitating\nastrophysical jet where the gravitational field is radial and, henc e, axially symmetric.\nStability of vertical water jets has been studied in laboratories for a long time (see, e.g.,\n(Yarin 2011; Leibovich & Stewardson 1983; Gallaire & Chomaz 2003). Let us now estimate\nthe minimal velocity shear Vmin(R) required for instability of vertical water jet in air. With\nρi≫ρe,B= 0,Ve= 0 (air is at rest), gn= 0,V=Vi, equation (4) yields\nω\nk=V±1√ρi/radicalbigg\nLmρeV2+σkI′m\nIm, (5)\nwhence one obtains the instability criterion in the form\n|Lm|ρeV2> σkI′\nm\nIm. (6)\nAs functions LmandI′\nm/Imdepend on kR, the value kRis to be estimated.\nObservations in plasma comet tails show that kR∼1 (e.g. (Ershkovich 1980), Table\nII). This fact is not unexpected as the cylinder radius R(more precisely, the circumference\n2πR) isthe onlycharacteristic length scale ofthe problem under conside ration. Observations\nin water jet are also indicative of kR∼1, and the first attempt to explain this phenomenon\nwas performed by Rayleigh (1892). It is indeed tempting to explain th is fact by means of\nbehavior of the instability growth rate, γ= Imω. But this is not so: the function γ(kR) has\nno maximum with kR= 1. According to equation (5),\nγ=k√ρi/radicalbigg\n|Lm|ρeV2−σkI′m\nIm. (7)\nHence,γ∝k/radicalbig\n|Lm|ifσ→0 or the velocity of jet, V, is large enough. The factor\nx/radicalbig\n|Lm(x)|is shown in Figure 2. It is seen that the dependence of the instability g rowth\nrate Imω=γ(kR) is monotonous, and has no maximum with kR∼1.\nFluid parameters are varying in broad ranges while kR∼1 remains almost the same.\nIt means that kR∼1 is a geometric characteristic, a peculiarity of a cylinder. Observat ion\nmentioned above seems not to be explained within a framework of infin itesimal amplitudes.\nOn the other hand, a finite amplitude theory (Ershkovich & Cherniko v 1973) shows that the\ncritical wave amplitude δcof non-linear stabilization, indeed reaches maximum with kR∼1\nasδc/R= (Λ1)−1/2, and the function Λ 1(kR) has here the sharp minimum (Figure 4 in\n(Ershkovich 1980) shows the function Λ m(kR) form= 0 and m= 1).– 5 –\nAccording to Laplace formula, an additional surface tension under the cylinder interface\nisσ/R, so that with kR= 1 the coefficient σ= 74 dyn cm−1(for a plane water surface in\nair) is double valued, and using equation (7) we arrive at the conclusio n that the instability\narises ifV2>(2σI′\nm/Im)/(ρeR|Lm|).\nFinally, with σ= 74 dyn cm−1,ρi/ρe= 770 for the water-air interface, |L1| ≈0.75 and\nI′\nm/Im≈1.25 for kink mode m=±1 ((Kruskal & Tuck 1958), Figure 1) one obtains\nVmin[ms−1] =/radicalBigg\n20\nR[cm]. (8)\nThus, the minimal initial jet velocity of water vertical free fall requ ired for instability is\n4.5 m s−1, 2.0 m s−1, and 1.4 m s−1for the water jet radius R= 1,5, and 10 cm, respectively.\nThe value Vminhappens to be too high in order to observe helical waves in the cylindr ical\nvertical jet from a water tap, but, instead, it is possible to observ e there stable surface waves\n(i.e. normal modes of oscillations) traveling along the jet both upstr eam and downstream\n(in the frame of reference moving with the velocity V) when the expression under the radical\nin the equation (5) is positive. This phenomenon is just the same as rip ples created by wind\non the lake surface.\nHelical waves in plasma comet tails may become visible only when their amp litude\nbecomes large enough. As a result, this phenomenon is observed re latively seldom. The\nstability conditions seem to be marginal. As helical waves in comet tail a nd in vertical water\njet obey almost the same dispersion equation this astrophysical ph enomenon may be (and,\nin our opinion, is to be) simulated in laboratory.\nWe did not consider here the effect of finite width, d, of a transition layer between two\nfluids. It is known to be small with kd≪1 , and as for cylindrical jet kR∼1, this effect\nis negligible if d≪Rwhich is the case for water jet in air and seems to be observed (as\nsharp decrease of brightness) for plasma comet tails. Model of cy lindrical comet tail with\ntransition layer of finite thickness dis treated by Chen & Liu (1982).\n3. Resonance damping of helical waves\nLandau (1944) found a sharp decrease of the Kelvin-Helmnoltz inst ability growth rate\nwhen the phase velocity of surface wave, Re ω/kis approaching the acoustic velocity, c, with\nfull damping γ= Imω= 0 when the phase velocity reaches√\n2c(see also (Landau & Lifshitz\n1959), ch.9, §84). A similar effect was described in (Ray & Ershkovich 1983) for mag netoa-\ncoustic velocity. We believe that there is a simple explanation: when th e phase velocity of– 6 –\nsurface wave, Re ω/k, approaches the characteristic velocity of normal mode of oscillat ions\nin the fluid, a resonance arises, and stable hydrodynamical or MHD m odes are generated in\nthe whole fluid volume. But the Kelvin-Helmholtz instability of the tange ntial discontinu-\nity is a surface phenomenon, with the amplitude of perturbation dec reasing (in the plane\ncase - exponentially) away from the interface. Thus, these waves , in some meaning, are\ntwo-dimensional, with relatively restricted stock of kinetic energy, supplied by the velocity\nshear. When this energy is transferred from 2D to 3D space gener ating stable waves ev-\nerywhere, the energy stock is rapidly exhausted, and instability is d amping. But if so, the\nsame phenomenon should exist in incompressible plasma while reaching t he Alfv´ en velocity,\nω/k≈VA. Indeed, this effect was described in (Ray & Ershkovich 1983; Lau & Liu 1981)for\nplane interface.\nLet us consider a cylindrical plasma jet (with the velocity Vi) immersed into plasma at\nrest, with the same parameters, i.e. ρi=ρe=ρ,Bi=Be=B, andVe= 0 (alternatively,\nwe may choose the frame of reference where Ve= 0). Then equation (2) yields the phase\nvelocity Re ω/k=V/(1−Lm), where V=Viis the velocity jump. The radical in equation\n(2) vanishes, and the interface becomes stable with V= (1−Lm)VA/|Lm|, and the phase\nvelocityω/k=V/(1−Lm) =VA/|Lm|= 1.15VAfor kink mode m= 1,Lm=−0.75. For\nplane interface Lm=−1, hence V= 2VA, andω/k=VA.\nA possibility of resonance generation of Alfv´ en waves in the whole vo lume of fluid\nwithω→kVAseems to be obvious. The fact that the magnetized shear layer is st able\nif its Alfv´ en speed is greater than half the velocity jump across the interface was found in\n(Ray & Ershkovich 1983; Lau & Liu 1981) (unfortunately, the reso nance nature of the insta-\nbility damping has not been mentioned therein). According to Ray & Er shkovich (1983), the\ninterface remains stable (despite the growing flow velocity V) whenV≥2candVA≥c. As\nthe phase velocity ω/k=V/2 , these conditions may be rewritten in the form ω/k≥c≤VA\n, which agrees with the resonance scenario above: the instability ce ases because the energy\nsupplied bythe velocity shear transfers(due tothe resonance) f orexcitation ofnormal modes\nof the fluid oscillation, first, of sound waves (as c≤VA), and then, of MHD waves (Alfv´ en\nand magnetosonic).\nSimilar resonance damping occurs with unstable capillary waves. In or der to demon-\nstrate this effect, let us assume that the liquid in a cylindrical jet (mo ving with the speed\nV) has almost the same density, ρi, as the ambient liquid at rest, i.e. |ρi−ρe| ≪ρi,e=ρ.\nThen equation (3), with B= 0,gn= 0 yields\nω\nk=V\n1−Lm±/bracketleftbiggLmV2\n(1−Lm)2+σk\nρ(1−Lm)/bracketrightbigg1/2\n.– 7 –\nThe radical vanishes if\nV=/parenleftbigg1−Lm\n|Lm|/parenrightbigg1/2/radicalBigg\nσk\nρ,\nand the phase velocity\nω\nk=V\n1−Lm=/radicalBigg\nσk\nρ|Lm|(1−Lm).\nHence, for a kink mode m= 1,Lm=−0.75 one obtains ω/k= 0.87/radicalbig\nσk/ρ. For the plane\ninterface Lm=−1 we find ω/k=/radicalbig\nσk/(2ρ) (which, naturally, may be obtained directly\nfrom equation(1)). The classical value for stable capillary waves is ω/k=/radicalbig\nσk/ρ(Kelvin\n1871; Landau & Lifshitz 1959).\nA small region of the cylindrical interface may be considered as plane for perturbations\nwithkR≫1, and for plane case the dispersion equation for perturbations ∼exp[i(kr−ωt)]\ndepends on scalar products kVandkB. This means that there are always directions along\nwhich the stabilizing role of the magnetic field becomes negligible. As sho rt wavelength\nperturbations may propagate in all directions a tangential discont inuity always remains un-\nstable. But this is not the case for helical waves propagating along t he cylinder axis.\n4. Conclusion\nDispersion equations (3) and (4) describe rather broad class of hy drodynamical and\nMHD instabilities and normal modes of oscillations of the cylindrical inte rface between two\nfluids, started with Alfv´ en waves and gravitational waves on deep water to flute and Kelvin-\nHelmholtz instabilities in planetary and comet tails and water jets in air ( including also\ncapillar instability in liquids). Although they were obtained in linear appro ximation (and\nhence each of these effects may be studied independently) the sta bility criterion Im ω= 0 de-\npends on the balance of allthe relevant terms under the radical. This balance is particularly\nimportant under marginal stability conditions when only their sum is ind icative of stability\nor instability of the interface.\nWe also drew attention to the fact that the instability growth rate o btained in linear\napproximation cannot explain the preferential generation of mode s withkR∼1. In partic-\nular, this fact refers to helical waves observed visually in comet plas ma tails. At the same\ntime, finite amplitude treatment (Ershkovich & Chernikov 1973; Ers hkovich 1980) seemed\nto explain these observations.\nBoth Alfv´ en and capillary waves arising due to Kelvin-Helmholtz instab ility on the\ncylindrical interface have been considered. We arrived at the conc lusion that sharp damping– 8 –\nof these helical waves occurs when their phase velocity approache s the characteristic velocity\nof normal modes of oscillation, so that it has resonance nature.\nFinally, wefoundthathelical waves bothinplasmacomettailsandinver tical cylindrical\nwater jet in the air are governed by almost the same dispersion equa tion (which means that,\nin fact, we deal with the same phenomenon). This fact allows us to su ggest an idea of\nlaboratory simulation of helical wave generation in cometary and plan etary magnetotails as\nwell as in astrophysical jets by using vertical water (or any other suitable liquid) jet.\nREFERENCES\nAlfv´ en, H., 1957, Tellus, 9, 92\nBessel, F. W., 1836, Astron. Nachr., 13 (302), 185\nBirkinshaw, M., 1996, Astrophys. Space Sci., 242, 17\nChen, D.-H., & Liu, L.-Z., 1982, Scientia Sinica (Series A), 25, 971\nErshkovich, A. I., 1980, Space Sci. Rev., 25, 3\nErshkovich, A. I. & Nusinov, A. A., 1971, Cosmic. Res., 9, 430\nErshkovich, A. I. & Nusinov, A. A., 1972, Cosmic Electrodyn., 2, 471\nErshkovich & Chernikov, A. A., 1973, Planet. Space Sci., 21, 663\nErshkovich, A. I., Nusinov, A. A. & Chernikov, A. A., 1972, Planet. S pace Sci., 20, 1235\nGallaire, F., & Chomaz, J.-M., 2003, J. Fluid. Mech., 494, 223\nKruskal, M., & Tuck, J. L., 1958, Proc. Roy. Soc. A, 245, 222\nLandau, L. D., 1944, C.R. Acad. Sci. USSR, 44, 139\nLandau, L. D. & Lifshitz, E. M., Fluid Mechanics, Oxford: Pergamon, 1959, Chap. VII\nLau, Y.-Y., & Liu, C.-S., 1981, Phys. Fluids, 23, 939\nLeibovich S., & Stewardson, K., 1983, J. Fluid. Mech., 126, 335\nLord Kelvin, (Thomson), W. 1871, Philosophical Magazine, 42, 362\nLord Rayleigh, (Strutt), J. W. 1892, Philosophical Magazine, Serie 5, 34 (207), 177– 9 –\nMcKenzie, J. F., 1970, J. Geophys. Res., 75, 5331\nMilne-Thomson, L. M., 1960, Theoretical Hydrodynamics, Macmillan, 1960, Chap. 14\nNess, N. F., 1965, J. Geophys. Res., 70, 2989\nRay, T. P., & Ershkovich, A. I., 1983, MNRAS, 204, 821\nSyrovatskii, S. I., 1953, Zhur. Exper. Teor. Fiz., 24, 622\nYarin, A. L., Handbook of Atomization and Sprays, Springer, 2011, 55\nThis preprint was prepared with the AAS L ATEX macros v5.2.– 10 –\nFig. 1.— The function Lm(kR) withm= 0 and m= 1 (curves aandb, respectively,\naccording to Ershkovich & Chernikov (1973)).– 11 –\nFig. 2.— The function x/radicalbig\n|Lm(x)|withm= 0 (solid line) and m= 1 (dashed line), x=kR." }, { "title": "2311.16268v2.Gilbert_damping_in_two_dimensional_metallic_anti_ferromagnets.pdf", "content": "Gilbert damping in two-dimensional metallic anti-ferromagnets\nR. J. Sokolewicz,1, 2M. Baglai,3I. A. Ado,1M. I. Katsnelson,1and M. Titov1\n1Radboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, the Netherlands\n2Qblox, Delftechpark 22, 2628 XH Delft, the Netherlands\n3Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden\n(Dated: March 29, 2024)\nA finite spin life-time of conduction electrons may dominate Gilbert damping of two-dimensional\nmetallic anti-ferromagnets or anti-ferromagnet/metal heterostructures. We investigate the Gilbert\ndamping tensor for a typical low-energy model of a metallic anti-ferromagnet system with honeycomb\nmagnetic lattice and Rashba spin-orbit coupling for conduction electrons. We distinguish three\nregimes of spin relaxation: exchange-dominated relaxation for weak spin-orbit coupling strength,\nElliot-Yafet relaxation for moderate spin-orbit coupling, and Dyakonov-Perel relaxation for strong\nspin-orbit coupling. We show, however, that the latter regime takes place only for the in-plane\nGilbert damping component. We also show that anisotropy of Gilbert damping persists for any\nfinite spin-orbit interaction strength provided we consider no spatial variation of the N´ eel vector.\nIsotropic Gilbert damping is restored only if the electron spin-orbit length is larger than the magnon\nwavelength. Our theory applies to MnPS 3monolayer on Pt or to similar systems.\nI. INTRODUCTION\nMagnetization dynamics in anti-ferromagnets con-\ntinue to attract a lot of attention in the context\nof possible applications1–4. Various proposals utilize\nthe possibility of THz frequency switching of anti-\nferromagnetic domains for ultrafast information storage\nand computation5,6. The rise of van der Waals magnets\nhas had a further impact on the field due to the pos-\nsibility of creating tunable heterostructures that involve\nanti-ferromagnet and semiconducting layers7.\nUnderstanding relaxation of both the N´ eel vector and\nnon-equilibrium magnetization in anti-ferromagnets is\nrecognized to be of great importance for the function-\nality of spintronic devices8–13. On one hand, low Gilbert\ndamping must generally lead to better electric control of\nmagnetic order via domain wall motion or ultrafast do-\nmain switching14–16. On the other hand, an efficient con-\ntrol of magnetic domains must generally require a strong\ncoupling between charge and spin degrees of freedom due\nto a strong spin-orbit interaction, that is widely thought\nto be equivalent to strong Gilbert damping.\nIn this paper, we focus on a microscopic analysis of\nGilbert damping due to Dyakonov-Perel and Elliot-Yafet\nmechanisms. We apply the theory to a model of a two-\ndimensional N´ eel anti-ferromagnet with a honeycomb\nmagnetic lattice.\nTwo-dimensional magnets typically exhibit either\neasy-plane or easy-axis anisotropy, and play crucial\nroles in stabilizing magnetism at finite temperatures17,18.\nEasy-axis anisotropy selects a specific direction for mag-\nnetization, thereby defining an axis for the magnetic or-\nder. In contrast, easy-plane anisotropy does not select a\nparticular in-plane direction for the N´ eel vector, allowing\nit to freely rotate within the plane. This situation is anal-\nogous to the XY model, where the system’s continuous\nsymmetry leads to the suppression of out-of-plane fluc-\ntuations rather than fixing the magnetization in a spe-\ncific in-plane direction19,20. Without this anisotropy, themagnonic fluctuations in a two-dimensional crystal can\ngrow uncontrollably large to destroy any long-range mag-\nnetic order, according to the Mermin-Wagner theorem21.\nRecent density-functional-theory calculations for\nsingle-layer transition metal trichalgenides22, predict the\nexistence of a large number of metallic anti-ferromagnets\nwith honeycomb lattice and different types of magnetic\norder as shown in Fig. 1. Many of these crystals may\nhave the N´ eel magnetic order as shown in Fig. 1a and are\nmetallic: FeSiSe 3, FeSiTe 3, VGeTe 3, MnGeS 3, FeGeSe 3,\nFeGeTe 3, NiGeSe 3, MnSnS 3, MnSnS 3, MnSnSe 3,\nFeSnSe 3, NiSnS 3. Apart from that it has been predicted\nthat anti-ferromagnetism can be induced in graphene by\nbringing it in proximity to MnPSe 323or by bringing it\nin double proximity between a layer of Cr 2Ge2Te6and\nWS224.\nPartly inspired by these predictions and recent\ntechnological advances in producing single-layer anti-\nferromagnet crystals, we propose an effective model to\nstudy spin relaxation in 2D honeycomb anti-ferromagnet\nwith N´ eel magnetic order. The same system was studied\nby us in Ref. 25, where we found that spin-orbit cou-\npling introduces a weak anisotropy in spin-orbit torque\nand electric conductivity. Strong spin-orbit coupling was\nshown to lead to a giant anisotropy of Gilbert damping.\nOur analysis below is built upon the results of Ref. 25,\nand we investigate and identify three separate regimes\nof spin-orbit strength. Each regime is characterized by\nqualitatively different dependence of Gilbert damping on\nspin-orbit interaction and conduction electron transport\ntime. The regime of weak spin-orbit interaction is dom-\ninated by exchange field relaxation of electron spin, and\nthe regime of moderate spin-orbit strength is dominated\nby Elliot-Yafet spin relaxation. These two regimes are\ncharacterized also by a universal factor of 2 anisotropy\nof Gilbert damping. The regime of strong spin-orbit\nstrength, which leads to substantial splitting of electron\nFermi surfaces, is characterized by Dyakonov-Perel relax-\nation of the in-plane spin component and Elliot-Yafet re-arXiv:2311.16268v2 [cond-mat.dis-nn] 28 Mar 20242\nFIG. 1. Three anti-ferromagnetic phases commonly found\namong van-der-Waals magnets. Left-to-right: N´ eel, zig-zag,\nand stripy.\nlaxation of the perpendicular-to-the-plane Gilbert damp-\ning which leads to a giant damping anisotropy. Isotropic\nGilbert damping is restored only for finite magnon wave\nvectors such that the magnon wavelength is smaller than\nthe spin-orbit length.\nGilbert damping in a metallic anti-ferromagnet can be\nqualitatively understood in terms of the Fermi surface\nbreathing26. A change in the magnetization direction\ngives rise to a change in the Fermi surface to which the\nconduction electrons have to adjust. This electronic re-\nconfiguration is achieved through the scattering of elec-\ntrons off impurities, during which angular momentum is\ntransferred to the lattice. Gilbert damping, then, should\nbe proportional to both (i) the ratio of the spin life-time\nand momentum life-time of conduction electrons, and (ii)\nthe electric conductivity. Keeping in mind that the con-\nductivity itself is proportional to momentum life-time,\none may conclude that the Gilbert damping is linearly\nproportional to the spin life-time of conduction electrons.\nAt the same time, the spin life-time of localized spins is\ninversely proportional to the spin life-time of conduc-\ntion electrons. A similar relation between the spin life-\ntimes of conduction and localized electrons also holds\nfor relaxation mechanisms that involve electron-magnon\nscattering27.\nOur approach formally decomposes the magnetic sys-\ntem into a classical sub-system of localized magnetic mo-\nments and a quasi-classical subsystem of conduction elec-\ntrons. A local magnetic exchange couples these sub-\nsystems. Localized magnetic moments in transition-\nmetal chalcogenides and halides form a hexagonal lat-\ntice. Here we focus on the N´ eel type anti-ferromagnet\nthat is illustrated in Fig. 1a. In this case, one can de-\nfine two sub-lattices A and B that host local magnetic\nmoments SAandSB, respectively. For the discussion of\nGilbert damping, we ignore the weak dependence of both\nfields on atomic positions and assume that the modulus\nS=|SA(B)|is time-independent.\nUnder these assumptions, the magnetization dynamics\nof localized moments may be described in terms of two\nfields\nm=1\n2S\u0000\nSA+SB\u0001\n,n=1\n2S\u0000\nSA−SB\u0001\n, (1)\nwhich are referred to as the magnetization and staggeredmagnetization (or N´ eel vector), respectively. Within the\nmean-field approach, the vector fields yield the equations\nof motion\n˙n=−Jn×m+n×δs++m×δs−, (2a)\n˙m=m×δs++n×δs−, (2b)\nwhere dot stands for the time derivative, while δs+and\nδs−stand for the mean staggered and non-staggered non-\nequilibrium fields that are proportional to the variation of\nthe corresponding spin-densities of conduction electrons\ncaused by the time dynamics of nandmfields. The en-\nergy Jis proportional to the anti-ferromagnet exchange\nenergy for localized momenta.\nIn Eqs. (2) we have omitted terms that are propor-\ntional to easy axis anisotropy for the sake of compact-\nness. These terms are, however, important and will be\nintroduced later in the text.\nIn the framework of Eqs. (2) the Gilbert damping can\nbe computed as the linear response of the electron spin-\ndensity variation to a time change in both the magneti-\nzation and the N´ eel vector (see e. g. Refs.25,28,29).\nIn this definition, Gilbert damping describes the re-\nlaxation of localized spins by transferring both total and\nstaggered angular momenta to the lattice by means of\nconduction electron scattering off impurities. Such a\ntransfer is facilitated by spin-orbit interaction.\nThe structure of the full Gilbert damping tensor can be\nrather complicated as discussed in Ref. 25. However, by\ntaking into account easy axis or easy plane anisotropy we\nmay reduce the complexity of relevant spin configurations\nto parameterize\nδs+=α∥\nm˙m∥+α⊥\nm˙m⊥+αmn∥×(n∥×˙m∥),(3a)\nδs−=α∥\nn˙n∥+α⊥\nn˙n⊥+αnn∥×(n∥×˙n∥), (3b)\nwhere the superscripts ∥and⊥refer to the in-plane\nand perpendicular-to-the-plane projections of the corre-\nsponding vectors, respectively. The six coefficients α∥\nm,\nα⊥\nm,αm,α∥\nn,α⊥\nn, and αnparameterize the Gilbert damp-\ning.\nInserting Eqs. (3) into the equations of motion of\nEqs. (2) produces familiar Gilbert damping terms. The\ndamping proportional to time-derivatives of the N´ eel vec-\ntornis in general many orders of magnitude smaller than\nthat proportional to the time-derivatives of the magneti-\nzation vector m25,30. Due to the same reason, the higher\nharmonics term αmn∥×(n∥×∂tm∥) can often be ne-\nglected.\nThus, in the discussion below we may focus mostly on\nthe coefficients α∥\nmandα⊥\nmthat play the most important\nrole in the magnetization dynamics of our system. The\nterms proportional to the time-derivative of ncorrespond\nto the transfer of angular momentum between the sub-\nlattices and are usually less relevant. We refer to the\nresults of Ref. 25 when discussing these terms.\nAll Gilbert damping coefficients are intimately related\nto the electron spin relaxation time. The latter is rel-\natively well understood in non-magnetic semiconductors3\nwith spin-orbital coupling. When a conducting electron\nmoves in a steep potential it feels an effective magnetic\nfield caused by relativistic effects. Thus, in a disordered\nsystem, the electron spin is subject to a random magnetic\nfield each time it scatters off an impurity. At the same\ntime, an electron also experiences precession around an\neffective spin-orbit field when it moves in between the\ncollisions. Changes in spin direction between collisions\nare referred to as Dyakonov-Perel relaxation31,32, while\nchanges in spin-direction during collisions are referred to\nas Elliot-Yafet relaxation33,34.\nThe spin-orbit field in semiconductors induces a char-\nacteristic frequency of spin precession Ω s, while scalar\ndisorder leads to a finite transport time τof the con-\nducting electrons. One may, then, distinguish two limits:\n(i) Ω sτ≪1 in which case the electron does not have\nsufficient time to change its direction between consec-\nutive scattering events (Elliot-Yafet relaxation), and (ii)\nΩsτ≫1 in which case the electron spin has multiple pre-\ncession cycles in between the collisions (Dyakonov-Perel\nrelaxation).\nThe corresponding processes define the so-called spin\nrelaxation time, τs. In a 2D system the spin life-time\nτ∥\ns, for the in-plane spin components, appears to be dou-\nble the size of the life-time of the spin component that\nis perpendicular to the plane, τ⊥\ns32. This geometric ef-\nfect has largely been overlooked. For non-magnetic 2D\nsemiconductor one can estimate35,36\n1\nτ∥\ns∼(\nΩ2\nsτ,Ωsτ≪1\n1/τ, Ωsτ≫1, τ∥\ns= 2τ⊥\ns. (4)\nA pedagogical derivation and discussion of Eq. 4 can\nbe found in Refs. 35 and 36. Because electrons are con-\nfined in two dimensions the random spin-orbit field is\nalways directed in-plane, which leads to a decrease in the\nin-plane spin-relaxation rate by a factor of two compared\nto the out-of-plane spin-relaxation rate as demonstrated\nfirst in Ref. 32 (see Refs. 36–40 as well). The reason is\nthat the perpendicular-to-the-plane component of spin is\ninfluenced by two components of the randomly changing\nmagnetic field, i. e. xandy, whereas the parallel-to-the-\nplane spin components are only influenced by a single\ncomponent of the fluctuating fields, i. e. the xspin pro-\njection is influenced only by the ycomponent of the field\nand vice-versa. The argument has been further general-\nized in Ref. 25 to the case of strongly separated spin-orbit\nsplit Fermi surfaces. In this limit, the perpendicular-to-\nthe-plane spin-flip processes on scalar disorder potential\nbecome fully suppressed. As a result, the perpendicular-\nto-the-plane spin component becomes nearly conserved,\nwhich results in a giant anisotropy of Gilbert damping in\nthis regime.\nIn magnetic systems that are, at the same time, con-\nducting there appears to be at least one additional energy\nscale, ∆ sd, that characterizes exchange coupling of con-\nduction electron spin to the average magnetic moment of\nlocalized electrons. (In the case of s-d model descriptionit is the magnetic exchange between the spin of conduc-\ntionselectron and the localized magnetic moment of d\norfelectron on an atom.) This additional energy scale\ncomplicates the simple picture of Eq. (4) especially in the\ncase of an anti-ferromagnet. The electron spin precession\nis now defined not only by spin-orbit field but also by\n∆sd. As the result the conditions Ω sτ≪1 and ∆ sdτ≫1\nmay easily coexist. This dissolves the distinction between\nElliot-Yafet and Dyakonov-Perel mechanisms of spin re-\nlaxation. One may, therefore, say that both Elliot-Yafet\nand Dyakonov-Perel mechanisms may act simultaneously\nin a typical 2D metallic magnet with spin-orbit coupling.\nThe Gilbert damping computed from the microscopic\nmodel that we formulate below will always contain both\ncontributions to spin-relaxation.\nII. MICROSCOPIC MODEL AND RESULTS\nThe microscopic model that we employ to calculate\nGilbert damping is the so-called s–dmodel that couples\nlocalized magnetic momenta SAandSBand conducting\nelectron spins via the local magnetic exchange ∆ sd. Our\neffective low-energy Hamiltonian for conduction electrons\nreads\nH=vfp·Σ+λ\n2\u0002\nσ×Σ\u0003\nz−∆sdn·σΣzΛz+V(r),(5)\nwhere the vectors Σ,σandΛdenote the vectors of Pauli\nmatrices acting on sub-lattice, spin and valley space,\nrespectively. We also introduce the Fermi velocity vf,\nRashba-type spin-orbit interaction λ, and a random im-\npurity potential V(r).\nThe Hamiltonian of Eq. (5) can be viewed as the\ngraphene electronic model where conduction electrons\nhave 2D Rashba spin-orbit coupling and are also cou-\npled to anti-ferromagnetically ordered classical spins on\nthe honeycomb lattice.\nThe coefficients α∥\nmandα⊥\nmare obtained using linear\nresponse theory for the response of spin-density δs+to\nthe time-derivative of magnetization vector ∂tm. Impu-\nrity potential V(r) is important for describing momen-\ntum relaxation to the lattice. This is related to the an-\ngular momentum relaxation due to spin-orbit coupling.\nThe effect of random impurity potential is treated pertur-\nbatively in the (diffusive) ladder approximation that in-\nvolves a summation over diffusion ladder diagrams. The\ndetails of the microscopic calculation can be found in the\nAppendices.\nBefore presenting the disorder-averaged quantities\nα∥,⊥\nm, it is instructive to consider first the contribution\nto Gilbert damping originating from a small number of\nelectron-impurity collisions. This clarifies how the num-\nber of impurity scattering effects will affect the final re-\nsult.\nLet us annotate the Gilbert damping coefficients with\nan additional superscript ( l) that denotes the number\nof scattering events that are taken into account. This4\n01234\u0016\u000b(i)\n?[\"\u001c]\n\u0016\u000b(0)\n?\u0016\u000b(1)\n?\u0016\u000b(2)\n? \u0016\u000b(1)\n?\n10\u0000210\u00001100101\n\u0015\u001c01234\u0016\u000b(i)\nk[\"\u001c]\n\u0016\u000b(0)\nk\u0016\u000b(1)\nk\u0016\u000b(2)\nk\u0016\u000b(1)\nk\nFIG. 2. Gilbert damping in the limit ∆ sd= 0. Dotted (green)\nlines correspond to the results of the numerical evaluation of\n¯α(l)\nm,⊥,∥forl= 0,1,2 as a function of the parameter λτ. The\ndashed (orange) line corresponds to the diffusive (fully vertex\ncorrected) results for ¯ α⊥,∥.\nm.\nmeans, in the diagrammatic language, that the corre-\nsponding quantity is obtained by summing up the ladder\ndiagrams with ≤ldisorder lines. Each disorder line cor-\nresponds to a quasi-classical scattering event from a sin-\ngle impurity. The corresponding Gilbert damping coeffi-\ncient is, therefore, obtained in the approximation where\nconduction electrons have scattered at most lnumber\nof times before releasing their non-equilibrium magnetic\nmoment into a lattice.\nTo make final expressions compact we define the di-\nmensionless Gilbert damping coefficients ¯ α∥,⊥\nmby extract-\ning the scaling factor\nα∥,⊥\nm=A∆2\nsd\nπℏ2v2\nfS¯α∥,⊥\nm, (6)\nwhere Ais the area of the unit cell, vfis the Fermi ve-\nlocity of the conducting electrons and ℏ=h/2πis the\nPlanck’s constant. We also express the momentum scat-\ntering time τin inverse energy units, τ→ℏτ.\nLet us start by computing the coefficients ¯ α∥,⊥(l)\nm in the\nformal limit ∆ sd→0. We can start with the “bare bub-\nble” contribution which describes spin relaxation without\na single scattering event. The corresponding results read\n¯α(0)\nm,⊥=ετ1−λ2/4ε2\n1 +λ2τ2, (7a)\n¯α(0)\nm,∥=ετ\u00121 +λ2τ2/2\n1 +λ2τ2−λ2\n8ε2\u0013\n, (7b)\nwhere εdenotes the Fermi energy which we consider pos-\nitive (electron-doped system).In all realistic cases, we have to consider λ/ε≪1,\nwhile the parameter λτmay in principle be arbitrary. For\nλτ≪1 the disorder-induced broadening of the electron\nFermi surfaces exceeds the spin-orbit induced splitting.\nIn this case one basically finds no anisotropy of “bare”\ndamping: ¯ α(0)\nm,⊥= ¯α(0)\nm,∥. In the opposite limit of substan-\ntial spin-orbit splitting one gets an ultimately anisotropic\ndamping ¯ α(0)\nm,⊥≪¯α(0)\nm,∥. This asymptotic behavior can be\nsummarized as\n¯α(0)\nm,⊥=ετ(\n1 λτ≪1,\n(λτ)−2λτ≫1,(8a)\n¯α(0)\nm,∥=ετ(\n1 λτ≪1,\n1\n2\u0000\n1 + (λτ)−2\u0001\nλτ≫1,(8b)\nwhere we have used that ε≫λ.\nThe results of Eq. (8) modify by electron diffusion. By\ntaking into account up to lscattering events we obtain\n¯α(l)\nm,⊥=ετ(\nl+O(λ2τ2) λτ≪1,\n(1 +δl0)/(λτ)2λτ≫1,(9a)\n¯α(l)\nm,∥=ετ(\nl+O(λ2τ2) λτ≪1,\n1−(1/2)l+1+O((λτ)−2)λτ≫1,(9b)\nwhere we have used ε≫λagain.\nFrom Eqs. (9) we see that the Gilbert damping for\nλτ≪1 gets an additional contribution of ετfrom each\nscattering event as illustrated numerically in Fig. 2. This\nleads to a formal divergence of Gilbert damping in the\nlimit λτ≪1. While, at first glance, the divergence looks\nlike a strong sensitivity of damping to impurity scatter-\ning, in reality, it simply reflects a diverging spin life-time.\nOnce a non-equilibrium magnetization mis created it\nbecomes almost impossible to relax it to the lattice in\nthe limit of weak spin-orbit coupling. The formal diver-\ngence of α⊥\nm=α∥\nmsimply reflects the conservation law\nfor electron spin polarization in the absence of spin-orbit\ncoupling such that the corresponding spin life-time be-\ncomes arbitrarily large as compared to the momentum\nscattering time τ.\nBy taking the limit l→ ∞ (i. e. by summing up the\nentire diffusion ladder) we obtain compact expressions\n¯α⊥\nm≡¯α(∞)\nm,⊥=ετ1\n2λ2τ2, (10a)\n¯α∥\nm≡¯α(∞)\nm,∥=ετ1 +λ2τ2\nλ2τ2, (10b)\nwhich assume ¯ α⊥\nm≪¯α∥\nmforλτ≫1 and ¯ α⊥\nm= ¯α∥\nm/2\nforλτ≪1. The factor of 2 difference that we observe\nwhen λτ≪1, corresponds to a difference in the elec-\ntron spin life-times τ⊥\ns=τ∥\ns/2 that was discussed in the\nintroduction32.\nStrong spin-orbit coupling causes a strong out-of-plane\nanisotropy of damping, ¯ α⊥\nm≪¯α∥\nmwhich corresponds to5\na suppression of the perpendicular-to-the-plane damping\ncomponent. As a result, the spin-orbit interaction makes\nit much easier to relax the magnitude of the mzcompo-\nnent of magnetization than that of in-plane components.\nLet us now turn to the dependence of ¯ αmcoefficients on\n∆sdthat is illustrated numerically in Fig. 3. We consider\nfirst the case of absent spin-orbit coupling λ= 0. In\nthis case, the combination of spin-rotational and sub-\nlattice symmetry (the equivalence of A and B sub-lattice)\nmust make Gilbert damping isotropic (see e. g.25,41). The\ndirect calculation for λ= 0 does, indeed, give rise to the\nisotropic result ¯ α⊥\nm= ¯α∥\nm=ετ(ε2+∆2\nsd)/2∆2\nsd, which is,\nhowever, in contradiction to the limit λ→0 in Eq. (10).\nAt first glance, this contradiction suggests the exis-\ntence of a certain energy scale for λover which the\nanisotropy emerges. The numerical analysis illustrated\nin Fig. 4 reveals that this scale does not depend on the\nvalues of 1 /τ, ∆sd, orε. Instead, it is defined solely by\nnumerical precision. In other words, an isotropic Gilbert\ndamping is obtained only when the spin-orbit strength\nλis set below the numerical precision in our model.\nWe should, therefore, conclude that the transition from\nisotropic to anisotropic (factor of 2) damping occurs ex-\nactly at λ= 0. Interestingly, the factor of 2 anisotropy is\nabsent in Eqs. (8) and emerges only in the diffusive limit.\nWe will see below that this paradox can only be re-\nsolved by analyzing the Gilbert damping beyond the in-\nfinite wave-length limit.\nOne can see from Fig. 3 that the main effect of finite\n∆sdis the regularization of the Gilbert damping diver-\ngency ( λτ)−2in the limit λτ≪1. Indeed, the limit of\nweak spin-orbit coupling is non-perturbative for ∆ sd/ε≪\nλτ≪1, while, in the opposite limit, λτ≪∆sd/ε≪1,\nthe results of Eqs. (10) are no longer valid. Assuming\n∆sd/ε≪1 we obtain the asymptotic expressions for the\nresults presented in Fig. 3 as\n¯α⊥\nm=1\n2ετ(2\n3ε2+∆2\nsd\n∆2\nsdλτ≪∆sd/ε,\n1\nλ2τ2 λτ≫∆sd/ε,(11a)\n¯α∥\nm=ετ(2\n3ε2+∆2\nsd\n∆2\nsdλτ≪∆sd/ε,\n1 +1\nλ2τ2λτ≫∆sd/ε,(11b)\nwhich suggest that ¯ α⊥\nm/¯α∥\nm= 2 for λτ≪1. In the op-\nposite limit, λτ≫1, the anisotropy of Gilbert damping\ngrows as ¯ α∥\nm/¯α⊥\nm= 2λ2τ2.\nThe results of Eqs. (11) can also be discussed in terms\nof the electron spin life-time, τ⊥(∥)\ns = ¯α⊥(∥)\nm/ε. For the\ninverse in-plane spin life-time we find\n1\nτ∥\ns=\n\n3∆2\nsd/2ε2τ λτ ≪∆sd/ε,\nλ2τ ∆sd/ε≪λτ≪1,\n1/τ 1≪λτ,(12)\nthat, for ∆ sd= 0, is equivalent to the known result of\nEq. (4). Indeed, for ∆ sd= 0, the magnetic exchange\n10\u0000310\u0000210\u00001100101\n\u0015\u001c10\u00001101103105\u0016\u000bm;k;?[\"\u001c]\n\u0001sd=\"= 0:1\u0001sd=\"= 0\u0016\u000bm;k\n\u0016\u000bm;?FIG. 3. Numerical results for the Gilbert damping compo-\nnents in the diffusive limit (vertex corrected)as the function\nof the spin-orbit coupling strength λ. The results correspond\ntoετ= 50 and ∆ sdτ= 0.1 and agree with the asymptotic\nexpressions of Eq. (11). Three different regimes can be dis-\ntinguished for ¯ α∥\nm: i) spin-orbit independent damping ¯ α∥\nm∝\nε3τ/∆2\nsdfor the exchange dominated regime, λτ≪∆sd/ε, ii)\nthe damping ¯ α∥\nm∝ε/λ2τfor Elliot-Yafet relaxation regime,\n∆sd/ε≪λτ≪1, and iii) the damping ¯ α∥\nm∝ετfor the\nDyakonov-Perel relaxation regime, λτ≫1. The latter regime\nis manifestly absent for ¯ α⊥\nmin accordance with Eqs. (12,13).\nplays no role and one observes the cross-over from Elliot-\nYafet ( λτ≪1) to Dyakonov-Perel ( λτ≫1) spin relax-\nation.\nThis cross-over is, however, absent in the relaxation of\nthe perpendicular spin component\n1\nτ⊥s= 2(\n3∆2\nsd/2ε2τ λτ ≪∆sd/ε,\nλ2τ ∆sd/ε≪λτ,(13)\nwhere Elliot-Yafet-like relaxation extends to the regime\nλτ≫1.\nAs mentioned above, the factor of two anisotropy in\nspin-relaxation of 2 Dsystems, τ∥\ns= 2τ⊥\ns, is known in the\nliterature32(see Refs.36–38as well). Unlimited growth of\nspin life-time anisotropy, τ∥\ns/τ⊥\ns= 2λ2τ2, in the regime\nλτ≪1 has been described first in Ref. 25. It can be qual-\nitatively explained by a strong suppression of spin-flip\nprocesses for zspin component due to spin-orbit induced\nsplitting of Fermi surfaces. The mechanism is effective\nonly for scalar (non-magnetic) disorder. Even though\nsuch a mechanism is general for any magnetic or non-\nmagnetic 2D material with Rashba-type spin-orbit cou-\npling, the effect of the spin life-time anisotropy on Gilbert\ndamping is much more relevant for anti-ferromagnets. In-\ndeed, in an anti-ferromagnetic system the modulus of m\nis, by no means, conserved, hence the variations of per-\npendicular and parallel components of the magnetization\nvector are no longer related.\nIn the regime, λτ≪∆sd/εthe spin life-time is de-\nfined by exchange interaction and the distinction between\nDyakonov-Perel and Elliot-Yafet mechanisms of spin re-\nlaxation is no longer relevant. In this regime, the spin-\nrelaxation time is by a factor ( ε/∆sd)2larger than the\nmomentum relaxation time.\nLet us now return to the problem of emergency of the6\n10\u00006410\u00005410\u00004410\u00003410\u00002410\u000014\n\u0015\u001c12\u0016\u000bk=\u0016\u000b?n= 32\nn= 64n= 96\nn= 128\nFIG. 4. Numerical evaluation of Gilbert damping anisotropy\nin the limit λ→0. Isotropic damping tensor is restored only\nifλ= 0 with ultimate numerical precision. The factor of 2\nanisotropy emerges at any finite λ, no matter how small it\nis, and only depends on the numerical precision n, i.e. the\nnumber of digits contained in each variable during computa-\ntion. The crossover from isotropic to anisotropic damping can\nbe understood only by considering finite, though vanishingly\nsmall, magnon qvectors.\nfactor of 2 anisotropy of Gilbert damping at λ= 0. We\nhave seen above (see Fig. 4) that, surprisingly, there ex-\nists no energy scale for the anisotropy to emerge. The\ntransition from the isotropic limit ( λ= 0) to a finite\nanisotropy appeared to take place exactly at λ= 0. We\ncan, however, generalize the concept of Gilbert damping\nby considering the spin density response function at a\nfinite wave vector q.\nTo generalize the Gilbert damping, we are seeking a\nresponse of spin density at a point r,δs+(r) to a time\nderivative of magnetization vectors ˙m∥and ˙m⊥at the\npoint r′. The Fourier transform with respect to r−r′\ngives the Gilbert damping for a magnon with the wave-\nvector q.\nThe generalization to a finite q-vector shows that the\nlimits λ→0 and q→0 cannot be interchanged. When\nthe limit λ→0 is taken before the limit q→0 one\nfinds an isotropic Gilbert damping, while for the oppo-\nsite order of limits, it becomes a factor of 2 anisotropic.\nIn a realistic situation, the value of qis limited from\nbelow by an inverse size of a typical magnetic domain\n1/Lm, while the spin-orbit coupling is effective on the\nlength scale Lλ= 2πℏvf/λ. In this picture, the isotropic\nGilbert damping is characteristic for the case of suffi-\nciently small domain size Lm≪Lλ, while the anisotropic\nGilbert damping corresponds to the case Lλ≪Lm.\nIn the limit qℓ≪1, where ℓ=vfτis the electron mean\n\u00002 0 2\nk[a.u.]\u00002:50:02:5energy [a.u.]\u0015=\u0001sd= 4\n\u00002 0 2\nk[a.u.]\u0015=\u0001sd= 2\n\u00002 0 2\nk[a.u.]\u0015=\u0001sd= 1FIG. 5. Band-structure for the effective model of Eq. (5)\nin a vicinity of Kvalley assuming nz= 1. Electron bands\ntouch for λ= 2∆ sd. The regime λ≤2∆sdcorresponds to\nspin-orbit band inversion. The band structure in the valley\nK′is inverted. Our microscopic analysis is performed in the\nelectron-doped regime for the Fermi energy above the gap as\nillustrated by the top dashed line. The bottom dashed line\ndenotes zero energy (half-filling).\nfree path, we can summarize our results as\n¯α⊥\nm=ετ\n\nε2+∆2\nsd\n2∆2\nsdλτ≪qℓ≪∆sd/ε,\n1\n3ε2+∆2\nsd\n∆2\nsdqℓ≪λτ≪∆sd/ε,\n1\n2λ2τ2 λτ≫��sd/ε,, (14a)\n¯α∥\nm=ετ\n\nε2+∆2\nsd\n2∆2\nsdλτ≪qℓ≪∆sd/ε,\n2\n3ε2+∆2\nsd\n∆2\nsdqℓ≪λτ≪∆sd/ε,\n1 +1\nλ2τ2λτ≫∆sd/ε,(14b)\nwhich represent a simple generalization of Eqs. (11).\nThe results of Eqs. (14) correspond to a simple behav-\nior of Gilbert damping anisotropy,\n¯α∥\nm/¯α⊥\nm=(\n1 λτ≪qℓ,\n2\u0000\n1 +λ2τ2\u0001\nqℓ≪λτ,(15)\nwhere we still assume qℓ≪1.\nIII. ANTI-FERROMAGNETIC RESONANCE\nThe broadening of the anti-ferromagnet resonance\npeak is one obvious quantity that is sensitive to Gilbert\ndamping. The broadening is however not solely defined\nby a particular Gilbert damping component but depends\nalso on both magnetic anisotropy and anti-ferromagnetic\nexchange.\nTo be more consistent we can use the model of Eq. (5)\nto analyze the contribution of conduction electrons to an\neasy axis anisotropy. The latter is obtained by expanding\nthe free energy for electrons in the value of nz, which has\na form E=−Kn2\nz/2. With the conditions ε/λ≫1 and\nε/∆sd≫1 we obtain the anisotropy constant as\nK=A\n2πℏ2v2(\n∆2\nsdλ 2∆sd/λ≤1,\n∆sdλ2/2 2∆ sd/λ≥1,(16)7\nwhere Ais the area of the unit cell. Here we assume\nboth λand ∆ sdpositive, therefore, the model natu-\nrally gives rise to an easy axis anisotropy with K > 0.\nIn real materials, there exist other sources of easy axis\nor easy plane anisotropy. In-plane magneto-crystalline\nanisotropy also plays an important role. For example,\nN´ eel-type anti-ferromagnets with easy-axis anisotropy\nare FePS 3, FePSe 3or MnPS 3, whereas those with easy\nplane and in-plane magneto-crystalline anisotropy are\nNiPS 3and MnPSe 3. Many of those materials are, how-\never, Mott insulators. Our qualitative theory may still\napply to materials like MnPS 3monolayers at strong elec-\ntron doping.\nThe transition from 2∆ sd/λ≥1 to 2∆ sd/λ≤1 in\nEq. (16) corresponds to the touching of two bands in the\nmodel of Eq. (5) as illustrated in Fig. 5.\nAnti-ferromagnetic magnon frequency and life-time in\nthe limit q→0 are readily obtained by linearizing the\nequations of motion\n˙n=−Jn×m+Km×n⊥+n×(ˆαm˙m), (17a)\n˙m=Kn×n⊥+n×(ˆαn˙n), (17b)\nwhere we took into account easy axis anisotropy Kand\ndisregarded irrelevant terms m×(ˆαn˙n) and m×(ˆαm˙m).\nWe have also defined Gilbert damping tensors such as\nˆαm˙m=α∥\nm˙m∥+α⊥\nm˙m⊥, ˆαn˙n=α∥\nn˙n∥+α⊥\nn˙n⊥.\nIn the case of easy axis anisotropy we can use the lin-\nearized modes n=ˆz+δn∥eiωt,m=δm∥eiωt, hence we\nget the energy of q= 0 magnon as\nω=ω0−iΓ/2, (18)\nω0=√\nJK, Γ =Jα∥\nn+Kα∥\nm (19)\nwhere we took into account that K≪J. The expression\nforω0is well known due to Kittel and Keffer42,43.\nUsing Ref. 25 we find out that α∥\nn≃α⊥\nm(λ/ε)2and\nα⊥\nn≃α∥\nm(λ/ε)2, hence\nΓ≃α∥\nm\u0012\nK+J/2\nε2/λ2+ε2τ2\u0013\n, (20)\nwhere we have simply used Eqs. (10). Thus, one may\noften ignore the contribution Jα∥\nnas compared to Kα∥\nm\ndespite the fact that K≪J.\nIn the context of anti-ferromagnets, spin-pumping\nterms are usually associated with the coefficients α∥\nnin\nEq. (3b) that are not in the focus of the present study.\nThose coefficients have been analyzed for example in Ref.\n25. In this manuscript we simply use the known results\nforαnin Eqs. (17-19), where we illustrate the effect of\nboth spin-pumping coefficient αnand the direct Gilbert\ndamping αmon the magnon life time. One can see from\nEqs. (19,20) that the spin-pumping contributions do also\ncontribute, though indirectly, to the magnon decay. The\nspin pumping contributions become more important in\nmagnetic materials with small magnetic anisotropy. The\nprocesses characterized by the coefficients αnmay also be\n10\u0000310\u0000210\u00001100101\n\u0015\u001c0:000:010:021=\u0016\u000bk\nm\u0015=\"= 0:04\n\u0015=\"= 0:02\n\u0015=\"= 0:01FIG. 6. Numerical evaluation of the inverse Gilbert damping\n1/¯α∥\nmas a function of the momentum relaxation time τ. The\ninverse damping is peaked at τ∝1/λwhich also corresponds\nto the maximum of the anti-ferromagnetic resonance quality\nfactor in accordance with Eq. (21).\ninterpreted in terms of angular momentum transfer from\none AFM sub-lattice to another. In that respect, the spin\npumping is specific to AFM, and is qualitatively differ-\nent from the direct Gilbert damping processes ( αm) that\ndescribe the direct momentum relaxation to the lattice.\nAs illustrated in Fig. 6 the quality factor of the anti-\nferromagnetic resonance (for a metallic anti-ferromagnet\nwith easy-axis anisotropy) is given by\nQ=ω0\nΓ≃1\nα∥\nmr\nJ\nK. (21)\nInterestingly, the quality factor defined by Eq. (21) is\nmaximized for λτ≃1, i. e. for the electron spin-orbit\nlength being of the order of the scattering mean free path.\nThe quantities 1 /√\nKand 1 /¯α∥\nmare illustrated in\nFig. 6 from the numerical analysis. As one would ex-\npect, the quality factor vanishes in both limits λ→0\nandλ→ ∞ . The former limit corresponds to an over-\ndamped regime hence no resonance can be observed. The\nlatter limit corresponds to a constant α∥\nm, but the reso-\nnance width Γ grows faster with λthan ω0does, hence\nthe vanishing quality factor.\nIt is straightforward to check that the results of\nEqs. (20,21) remain consistent when considering systems\nwith either easy-plane or in-plane magneto-crystalline\nanisotropy. Thus, the coefficient α⊥\nmnormally does not\nenter the magnon damping, unless the system is brought\ninto a vicinity of spin-flop transition by a strong external\nfield.\nIV. CONCLUSION\nIn conclusion, we have analyzed the Gilbert damping\ntensor in a model of a two-dimensional anti-ferromagnet\non a honeycomb lattice. We consider the damping mech-\nanism that is dominated by a finite electron spin life-time8\ndue to a combination of spin-orbit coupling and impu-\nrity scattering of conduction electrons. In the case of a\n2D electron system with Rashba spin-orbit coupling λ,\nthe Gilbert damping tensor is characterized by two com-\nponents α∥\nmandα⊥\nm. We show that the anisotropy of\nGilbert damping depends crucially on the parameter λτ,\nwhere τis the transport scattering time for conduction\nelectrons. For λτ≪1 the anisotropy is set by a geo-\nmetric factor of 2, α∥\nm= 2α⊥\nm, while it becomes infinitely\nlarge in the opposite limit, α∥\nm= (λτ)2α⊥\nmforλτ≫1.\nGilbert damping becomes isotropic exactly for λ= 0, or,\nstrictly speaking, for the case λ≪ℏvfq, where qis the\nmagnon wave vector.\nThis factor of 2 is essentially universal, and is a geomet-\nric effect: the z-component relaxation results from fluctu-\nations in two in-plane spin components, whereas in-plane\nrelaxation stems from fluctuations of the z-component\nalone. This reflects the subtleties of our microscopic\nmodel, where the mechanism for damping is activated\nby the decay of conduction electron momenta, linked to\nspin-relaxation through spin-orbit interactions.\nWe find that Gilbert damping is insensitive to mag-\nnetic order for λ≫∆sd/ετ, where ∆ sdis an effective\nexchange coupling between spins of conduction and local-\nized electrons. In this case, the electron spin relaxation\ncan be either dominated by scattering (Dyakonov-Perel\nrelaxation) or by spin-orbit precession (Elliot-Yafet re-\nlaxation). We find that the Gilbert damping component\nα⊥\nm≃ε/λ2τis dominated by Elliot-Yafet relaxation irre-\nspective of the value of the parameter λτ, while the other\ncomponent crosses over from α∥\nm≃ε/λ2τ(Elliot-Yafet\nrelaxation) for λτ≪1, to α∥\nm≃ετ(Dyakonov-Perel re-\nlaxation) for λτ≫1. For the case λ≪∆sd/ετthe spin\nrelaxation is dominated by interaction with the exchange\nfield.\nCrucially, our results are not confined solely to the N´ eel\norder on the honeycomb lattice: we anticipate a broader\napplicability across various magnetic orders, including\nthe zigzag order. This universality stems from our focus\non the large magnon wavelength limit. The choice of the\nhoneycomb lattice arises from its unique ability to main-\ntain isotropic electronic spectra within the plane, coupled\nwith the ability to suppress anisotropy concerning in-\nplane spin rotations. Strong anisotropic electronic spec-\ntra would naturally induce strong anisotropic in-plane\nGilbert damping, which are absent in our results.\nFinally, we show that the anti-ferromagnetic resonance\nwidth is mostly defined by α∥\nmand demonstrate that the\nresonance quality factor is maximized for λτ≈1. Our\nmicroscopic theory predictions may be tested for systems\nsuch as MnPS 3monolayer on Pt and similar heterostruc-\ntures.ACKNOWLEDGMENTS\nWe are grateful to O. Gomonay, R. Duine, J. Sinova,\nand A. Mauri for helpful discussions. This project has\nreceived funding from the European Union’s Horizon\n2020 research and innovation program under the Marie\nSklodowska-Curie grant agreement No 873028.\nAppendix A: Microscopic framework\nThe microscopic model that we employ to calculate\nGilbert damping belongs to a class of so-called s–dmod-\nels that describe the physical system in the form of a\nHeisenberg model for localized spins and a tight-binding\nmodel for conduction electrons that are weakly coupled\nby a local magnetic exchange interaction of the strength\n∆sd.\nOur effective electron Hamiltonian for a metallic\nhexagonal anti-ferromagnet is given by25\nH0=vfp·Σ+λ\n2[σ×Σ]z−∆sdn·σΣzΛz,(A1)\nwhere the vectors Σ,σandΛdenote the vectors of Pauli-\nmatrices acting on sub-lattice, spin and valley space re-\nspectively. We also introduce the Fermi velocity vf,\nRashba-type spin-orbit interaction λ.\nTo describe Gilbert damping of the localized field n\nwe have to add the relaxation mechanism. This is pro-\nvided in our model by adding a weak impurity potential\nH=H0+V(r). The momentum relaxation due to scat-\ntering on impurities leads indirectly to the relaxation of\nHeisenberg spins due to the presence of spin-orbit cou-\npling and exchange couplings.\nFor modeling the impurity potential, we adopt a delta-\ncorrelated random potential that corresponds to the\npoint scatter approximation, where the range of the im-\npurity potential is much shorter than that of the mean\nfree path (see e.g. section 3.8 of Ref. 44), i.e.\n⟨V(r)V(r′)⟩= 2πα(ℏvf)2δ(r−r′), (A2)\nwhere the dimensionless coefficient α≪1 characterizes\nthe disorder strength. The corresponding scattering time\nfor electrons is obtained as τ=ℏ/παϵ , which is again\nsimilar to the case of graphene.\nThe response of symmetric spin-polarization δs+to the\ntime-derivative of non-staggered magnetization, ∂tm, is\ndefined by the linear relation\nδs+\nα=X\nβRαβ|ω=0˙mβ, (A3)\nwhere the response tensor is taken at zero frequency25,45.\nThe linear response is defined generally by the tensor\nRαβ=A∆2\nsd\n2πSZdp\n(2πℏ)2\nTr\u0002\nGR\nε,pσαGA\nε+ℏω,pσβ\u0003\u000b\n,(A4)9\nwhere GR(A)\nε,pare standing for retarded(advanced) Green\nfunctions and the angular brackets denote averaging over\ndisorder fluctuations.\nThe standard recipe for disorder averaging is the diffu-\nsive approximation46,47that is realized by replacing the\nbare Green functions in Eq. (A4) with disorder-averaged\nGreen functions and by replacing one of the vertex op-\nerators σxorσywith the corresponding vertex-corrected\noperator that is formally obtained by summing up ladder\nimpurity diagrams (diffusons).\nIn models with spin-orbit coupling, the controllable dif-\nfusive approximation for non-dissipative quantities may\nbecome, however, more involved as was noted first in\nRef. 48. For Gilbert damping it is, however, sufficient to\nconsider the ladder diagram contributions only.\nThe disorder-averaged Green function is obtained by\nincluding an imaginary part of the self-energy ΣR(not\nto be confused here with the Pauli matrix Σ 0,x,y,z) that\nis evaluated in the first Born approximation\nIm ΣR= 2παv2\nfZdp\n(2π)2Im1\nε−H0+i0. (A5)\nThe real part of the self-energy leads to the renormaliza-\ntion of the energy scales ε,λand ∆ sd.\nIn the first Born approximation, the disorder-averaged\nGreen function is given by\nGR\nε,p=1\nε−H0−iIm ΣR. (A6)\nThe vertex corrections are computed in the diffusive\napproximation. The latter involves replacing the vertex\nσαwith the vertex-corrected operator,\nσvc\nα=∞X\nl=0σ(l)\nα, (A7)\nwhere the index lcorresponds to the number of disorder\nlines in the ladder.\nThe operators σ(l)\nαcan be defined recursively as\nσ(l)\nα=2ℏv2\nf\nετZdp\n(2π)2GR\nε,pσ(l−1)\nαGA\nε+ℏω,p, (A8)\nwhere σ(0)\nα=σα.\nThe summation in Eq. (A7) can be computed in the\nfull operator basis, Bi={α,β,γ}=σαΣβΛγ, where each\nindex α,βandγtakes on 4 possible values (with zero\nstanding for the unity matrix). We may always normalize\nTrBiBj= 2δijin an analogy to the Pauli matrices. The\noperators Biare, then, forming a finite-dimensional space\nfor the recursion of Eq. (A8).\nThe vertex-corrected operators Bvc\niare obtained by\nsumming up the matrix geometric series\nBvc\ni=X\nj\u00121\n1− F\u0013\nijBj, (A9)where the entities of the matrix Fare given by\nFij=ℏv2\nf\nετZdp\n(2π)2Tr\u0002\nGR\nε,pBiGA\nε+ℏω,pBj\u0003\n.(A10)\nOur operators of interest σxandσycan always be de-\ncomposed in the operator basis as\nσα=1\n2X\niBiTr (σαBi), (A11)\nhence the vertex-corrected spin operator is given by\nσvc\nα=1\n2X\nijBvc\niTr(σαBi). (A12)\nMoreover, the computation of the entire response tensor\nof Eq. (A4) in the diffusive approximation can also be\nexpressed via the matrix Fas\nRαβ=α0ετ\n8ℏX\nij[TrσαBi]\u0014F\n1− F\u0015\nij[TrσβBj],(A13)\nwhere α0=A∆2\nsd/πℏ2v2\nfSis the coefficient used in\nEq. (6) to define the unit of the Gilbert damping.\nIt appears that one can always choose the basis of\nBioperators such that the computation of Eq. (A13)\nis closed in a subspace of just three Bioperators with\ni= 1,2,3. This enables us to make analytical computa-\ntions of Eq. (A13).\nAppendix B: Magnetization dynamics\nThe representation of the results can be made some-\nwhat simpler by choosing xaxis in the direction of the\nin-plane projection n∥of the N´ eel vector, hence ny= 0.\nIn this case, one can represent the result as\nδs+=c1n∥×(n∥×∂tm∥) +c2∂tm∥+c3∂tm⊥+c4n,\nwhere ndependence of the coefficients cimay be param-\neterized as\nc1=r11−r22−r31(1−n2\nz)/(nxnz)\n1−n2z, (B1a)\nc2=r11−r31(1−n2\nz)/(nxnz), (B1b)\nc3=r33, (B1c)\nc4= (r31/nz)∂tmz+ζ(∂tm)·n. (B1d)\nThe analytical results in the paper correspond to the\nevaluation of δs±up to the second order in ∆ sdusing\nperturbative analysis. Thus, zero approximation corre-\nsponds to setting ∆ sd= 0 in Eqs. (A1,A5).\nThe equations of motion on nandmare given by\nEqs. (2),\n∂tn=−Jn×m+n×δs++m×δs−, (B2a)\n∂tm=m×δs++n×δs−, (B2b)10\nIt is easy to see that the following transformation leaves\nthe above equations invariant,\nδs+→δs+−ξn, δ s−→δs−−ξm, (B3)\nfor an arbitrary value of ξ.\nSuch a gauge transformation can be used to prove that\nthe coefficient c4is irrelevant in Eqs. (B2).\nIn this paper, we compute δs±to the zeroth order in\n|m|– the approximation which is justified by the sub-\nlattice symmetry in the anti-ferromagnet. A somewhat\nmore general model has been analyzed also in Ref. 25 to\nwhich we refer the interested reader for more technical\ndetails.\nAppendix C: Anisotropy constant\nThe anisotropy constant is obtained from the grand po-\ntential energy Ω for conducting electrons. For the model\nof Eq. (A1) the latter can be expressed as\nΩ =−X\nς=±1\nβZ\ndε g(ε)νς(ε), (C1)\nwhere β= 1/kBTis the inverse temperature, ς=±is\nthe valley index (for the valleys KandK′),GR\nς,pis the\nbare retarded Green function with momentum pand in\nthe valley ς. We have also defined the function\ng(ε) = ln (1 + exp[ β(µ−ε)]), (C2)\nwhere µis the electron potential, and the electron density\nof states in each of the valleys is given by,\nνς(ε) =1\nπZdp\n(2πℏ)2Im Tr GR\nς,p, (C3)\nwhere the trace is taken only over spin and sub-lattice\nspace,\nIn the metal regime considered, the chemical potential\nis assumed to be placed in the upper electronic band.\nIn this case, the energy integration can be taken only for\npositive energies. The two valence bands are always filled\nand can only add a constant shift to the grand potential\nΩ that we disregard.\nThe evaluation of Eq. (C1) yields the following density\nof states\nντ(ε) =1\n2πℏ2v2\nf\n\n0 0 < ε < ε 2\nε/2 +λ/4ε2< ε < ε 1,\nε ε > ε 1,(C4)where the energies ε1,2correspond to the extremum\npoints (zero velocity) for the electronic bands. These\nenergies, for each of the valleys, are given by\nε1,ς=1\n2\u0000\n+λ+p\n4∆2+λ2−4ς∆λnz\u0001\n, (C5a)\nε2,ς=1\n2\u0000\n−λ+p\n4∆2+λ2+ 4ς∆λnz\u0001\n(C5b)\nwhere ς=±is the valley index.\nIn the limit of zero temperature we can approximate\nEq. (C1) as\nΩ =−X\nς=±1\nβZ∞\n0dε(µ−ε)νς(ε). (C6)\nThen, with the help of Eq. (C1) we find,\nΩ =−1\n24πℏ2v2\nfX\nς=±\u0002\n(ε1,ς−µ)2(4ε1,ς−3λ+ 2µ)\n+(ε2,ς−µ)2(4ε2,ς+ 3λ+ 2µ)\u0003\n. (C7)\nBy substituting the results of Eqs. (C5) into the above\nequation we obtain\nΩ =−1\n24πℏ2v2\nfh\n(4∆2−4nz∆λ+λ2)2/3\n+(4∆2+ 4nz∆λ+λ2)2/3−24∆µ+ 8µ3i\n.(C8)\nA careful analysis shows that the minimal energy cor-\nresponds to nz=±1 so that the conducting electrons\nprefer an easy-axis magnetic anisotropy. 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E. Troncoso, W. Belzig, and A. Brataas,\nPhys. Rev. B 98, 184402 (2018).\n42C. Kittel, Phys. Rev. 82, 565 (1951).\n43F. Keffer and C. Kittel, Phys. Rev. 85, 329 (1952).\n44J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986).\n45I. Ado, O. A. Tretiakov, and M. Titov, Phys. Rev. B 95,\n094401 (2017).\n46J. Rammer, Quantum Transport Theory (CRC Press, New\nYork, 2018).\n47G. D. Mahan, Many-particle physics (Springer Science &\nBusiness Media, 2013).\n48I. Ado, I. Dmitriev, P. Ostrovsky, and M. Titov, EPL 111,\n37004 (2015)." }, { "title": "2306.02770v1.Damping_of_coronal_oscillations_in_self_consistent_3D_radiative_MHD_simulations_of_the_solar_atmosphere.pdf", "content": "Astronomy &Astrophysics manuscript no. waves ©ESO 2023\nJune 6, 2023\nDamping of coronal oscillations in self-consistent 3D radiative\nMHD simulations of the solar atmosphere\nP. Kohutova1,2, P. Antolin3, M. Szydlarski1,2, and M. Carlsson1,2\n1Rosseland Centre for Solar Physics, University of Oslo, P.O. Box 1029, Blindern, NO-0315 Oslo, Norway\ne-mail: petra.kohutova@astro.uio.no\n2Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029, Blindern, NO-0315 Oslo, Norway\n3Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle Upon Tyne NE1 8ST, UK\nReceived; accepted\nABSTRACT\nContext. Oscillations are abundant in the solar corona. Coronal loop oscillations are typically studied using highly idealised models\nof magnetic flux tubes. In order to improve our understanding of coronal oscillations, it is necessary to consider the e ffect of realistic\nmagnetic field topology and density structuring.\nAims. We analyse the damping of coronal oscillations using a self-consistent 3D radiation-MHD simulation of the solar atmosphere\nspanning from the convection zone into the corona, the associated oscillation dissipation and heating, and finally the physical processes\nresponsible for the damping and dissipation. The simulated corona formed in such a model does not depend on any prior assumptions\nabout the shape of the coronal loops.\nMethods. We analyse the evolution of a bundle of magnetic loops by magnetic field tracing.\nResults. We find that the bundle of magnetic loops shows damped transverse oscillations in response to perturbations in two separate\ninstances with oscillation periods of 177 s and 191 s, velocity amplitudes of 10 km s−1and 16 km s−1and damping times of 176 s\nand 198 s, respectively. The coronal oscillations lead to the development of velocity shear in the simulated corona resulting in the\nformation of vortices seen in the velocity field caused by the Kelvin-Helmholtz instability, contributing to the damping and dissipation\nof the transverse oscillations.\nConclusions. The oscillation parameters and evolution observed are in line with the values typically seen in observations of coronal\nloop oscillations. The dynamic evolution of the coronal loop bundle suggests the models of monolithic and static coronal loops with\nconstant lengths might need to be re-evaluated by relaxing the assumption of highly idealised waveguides.\nKey words. Magnetohydrodynamics (MHD) – Sun: corona – Sun: magnetic fields – Sun: oscillations\n1. Introduction\nCoronal structures act as waveguides for a variety of MHD os-\ncillation modes (Nakariakov & Verwichte 2005; Nakariakov &\nKolotkov 2020). There is extensive observational evidence that\ntransverse oscillations are ubiquitous in the solar corona, in both\nclosed coronal loops and open coronal structures (Tomczyk et al.\n2007; McIntosh et al. 2011; Anfinogentov et al. 2015). The trans-\nverse oscillations commonly observed in coronal loops are iden-\ntified as standing kink modes (Aschwanden et al. 1999; Nakari-\nakov et al. 1999). Coronal oscillations carry magnetic energy,\nwhich is deposited through the oscillation damping and dissipa-\ntion. Commonly proposed physical mechanisms responsible for\nthe oscillation damping and dissipation include resonant absorp-\ntion (Hollweg & Yang 1988; Goossens et al. 2002), phase mix-\ning (Heyvaerts & Priest 1983) and Kelvin-Helmholtz instability\n(Terradas et al. 2008; Antolin et al. 2014). Such mechanisms\nhave been mostly analysed using simplified models of coronal\nloops as straight magnetic flux-tubes clearly distinct from the\nsurroundings. There are several advantages to this approach: It is\nstraightforward to isolate individual e ffects and processes linked\nto wave evolution as well as to have full control over the param-\neters of the modelled loop. They also allow for very high spatial\nresolution, necessary for modelling of certain physical processes\nsuch as resonant absorption (Van Doorsselaere et al. 2004). Such\nmodels, however, also require making several assumptions aboutthe shape, density structure and morphology of coronal loops\nwhich might be too idealised.\nThe solar corona is in fact a dynamic environment with\ncomplex density structuring and the coronal magnetic field is\ncontinuously changing and evolving. Models of static and ide-\nalised coronal loops therefore neglect this evolution. In order to\naccount for realistic magnetic field configurations and density\nstructuring in the solar corona, a more self-consistent approach\nto modelling the evolution of coronal structures is necessary.\nThis can be achieved by taking advantage of realistic convection-\nzone-to-corona models (e.g. Carlsson et al. 2016; Cheung et al.\n2019; Kohutova & Popovas 2021; Breu et al. 2022). The evolu-\ntion of the corona in such models is self-consistently driven by\nthe dynamics of the lower solar atmosphere. This type of simu-\nlations therefore reflects the dynamic and continuously evolving\nnature of the solar corona.\nOne common feature of such simulations is the lack of\nclearly-identifiable coronal loops with well-defined boundaries.\nIn the realistic solar simulations the actual 3D structure of coro-\nnal features with increased emissivity in the optically thin coro-\nnal emission lines is much more complex, despite appearing as\nthin, well-defined loops in forward-modelled emission due to\nline-of-sight e ffects. Such structure of the corona is referred to as\n’the coronal veil’, and has been described using a self-consistent\nMURaM simulation spanning from the convection zone into the\nArticle number, page 1 of 9arXiv:2306.02770v1 [astro-ph.SR] 5 Jun 2023A&A proofs: manuscript no. waves\ncorona (Malanushenko et al. 2022). Di fferent codes capable of\nproducing self-consistent convection-zone-to-corona models, in-\ncluding MURaM (Rempel 2016) and Bifrost (Gudiksen et al.\n2011), seem to reproduce the ’coronal veil’ structure of the so-\nlar corona. The question remains how well does the simulated\ncorona, which forms in these models, represent the real solar\ncorona. On one hand, the coronal veil model can explain cer-\ntain properties of the coronal loops seen in Extreme Ultraviolet\n(EUV) observations such as loop cross-sections that appear to be\nconstant with height (Aschwanden & Nightingale 2005), which\nare di fficult to explain otherwise; we note that alternative expla-\nnations include a specific density and temperature distribution\nacross the magnetic field (Peter & Bingert 2012) and the pres-\nence of magnetic twist (Li et al. 2020). It would however also\nbring the applicability of coronal loop oscillation models into\nquestion, as most rely on the assumption of coronal loops being\nwell represented by idealised magnetic cylinders. On the other\nhand, the existence and omnipresence of transverse MHD waves\nsuch as kink waves directly points to the existence of struc-\ntures acting as waveguides that lead to collective behaviour in\nthe coronal volume. Also, the occurrence of coronal rain show-\ners (¸ Sahin & Antolin 2022) and long period intensity-pulsations\n(Froment et al. 2015), ubiquitous over active regions, indicate\nthe existence of coronal entities (that we refer as loops or loop\nbundles) with similar thermodynamic behaviour.\nMost models for wave dissipation rely on modelling coronal\nloops as straight magnetic flux-tubes. The mechanism of reso-\nnant absorption depends on conversion of a global kink mode\ninto local azimuthal Alfvén modes in an inhomogeneous layer\nwith an Alfvén speed gradient at the boundary of a cylindrical\ncoronal loop (Goossens et al. 2002; Pascoe et al. 2010; How-\nson et al. 2019). Similarly, the development of Kelvin-Helmholtz\nInstability vortices at the boundary of a transversely oscillating\nloop has been mostly studied using models of cylindrical loops\n(e.g. Antolin et al. 2014; Karampelas et al. 2017; Howson et al.\n2017, 2019) with a distinct density or magnetic interface sepa-\nrating them from the background plasma.\nThe main shortcoming of these models is that the observa-\ntional evidence for the proposed damping and dissipation mech-\nanisms is largely inconclusive. Signatures reminiscent of reso-\nnant absorption have been observed in oscillating prominence\nthreads (Antolin et al. 2015; Okamoto et al. 2015) and in trans-\nversely oscillating spicules (Antolin et al. 2018). To the best of\nour knowledge, these are the only observational evidence of such\nmechanisms to date. Models for damping of coronal oscillations\nwould therefore benefit from extending into more realistic se-\ntups. Numerical studies using setups which do not correspond to\nmagnetic cylinders were done for both oscillations in the chro-\nmosphere (Leenaarts et al. 2015; Khomenko & Cally 2012) and\nin the corona (Matsumoto 2018), the latter two however lack the\nself-consistent treatment of the lower solar atmosphere driving\nthe corona.\nIn this work we therefore focus on analysing coronal oscilla-\ntions in a more advanced numerical setup. We exploit the poten-\ntial of convection-zone-to-corona simulations with the radiation-\nMHD code Bifrost for coronal studies. We have previously\nshown that coronal oscillations are abundant in self-consistent\nconvection-zone-to-corona simulations and that the detected os-\ncillation modes and regimes match those seen in solar observa-\ntions (Kohutova & Popovas 2021). Here we analyse the evolu-\ntion of an oscillating bundle of magnetic loops which forms in\nthe simulation and focus on the oscillation damping. Despite the\ncomplex evolution of the bundle, variable length etc. the indi-\nvidual magnetic field lines contained in the bundle are found toexhibit collective evolution during extended periods of time, in-\ncluding collective oscillations triggered by impulsive events and\nsubsequent oscillation damping.\nThe manuscript is structured as follows. Section 2 describes\nthe numerical model. Section 3 describes the methods used for\nthe analysis of the evolution of a bundle of magnetic loops and of\nthe corresponding oscillatory behaviour seen in the simulation.\nIn Section 4 we focus on the damping of the oscillating loops\nand on the physical mechanism responsible for the damping. In\nSection 5 we discuss the results and implications for the coronal\nloop models. We summarize our conclusions in Section 6.\n2. Numerical model\nWe analyse the evolution, oscillatory behaviour and damping of\na bundle of magnetic loops in the numerical simulation of a mag-\nnetically enhanced network spanning from the upper convection\nzone to the corona using the Bifrost code (Gudiksen et al. 2011).\nThis simulation corresponds to the extended run of the pub-\nlic Bifrost simulation of the enhanced network (Carlsson et al.\n2016). The simulation subset with 2000 s duration analysed in\nthis work covers the time range from t =100 s to t =2100 s of\nthe extended run of the enhanced network simulation run from\nthe last snapshot of the public simulation, while having the same\nphysical setup.\nBifrost is a 3D radiation-MHD code which solves the resis-\ntive MHD equations and includes radiative transfer with scatter-\ning in the photosphere and low chromosphere, and parametrised\nradiative losses and heating in the upper chromosphere, transi-\ntion region and corona. The e ffects of field-aligned thermal con-\nduction and the non-equilibrium ionisation of hydrogen in the\nequation of state are included in the simulation.\nThe physical size of the simulation grid is 24 ×24×16.8\nMm and the grid resolution is 504 ×504×496. The grid spans\nfrom 2.4 Mm below the photosphere to 14.4 Mm in the corona.\nThe photosphere is located at z=0 surface and corresponds to\nthe (approximate) height where the optical depth τ500is equal\nto unity. The grid spacing is 48 km and uniform in the xandy\ndirection, while in the zdirection it varies from 19 km to 98 km\nin order to resolve steep density and temperature gradients in the\nlower solar atmosphere.\nThe simulation domain uses periodic boundaries in the x\nandydirections and open boundaries in the zdirection. The top\nboundary uses characteristic boundaries which transmit distur-\nbances with minimal reflection (Gudiksen et al. 2011). At the\nbottom boundary the flows are let through and the magnetic field\nis passively advected without introducing any additional mag-\nnetic field into the domain. The average horizontal pressure is\ndriven towards a constant value with a characteristic timescale\nof 100 s, creating a pressure node at the bottom boundary. This\nleads to acoustic wave reflection resembling the refraction of\nwaves in the deeper solar atmosphere, resulting in global radial\nbox oscillations with a period of 450 s, which are a simulation\ncounterpart of solar p-modes (Stein & Nordlund 2001; Carlsson\net al. 2016).\nThe photospheric magnetic field is concentrated in two\npatches of opposite polarity and has an average unsigned value\nof about 50 G (Fig. 1). The dipolar structure of the magnetic field\ncreates several magnetic loops in the simulated corona. The con-\nvective motions in the lower solar atmosphere lead to magnetic\nfield braiding. The Ohmic and viscous heating associated with\nthe braiding together maintain high temperatures in the chromo-\nsphere and corona. An artificial heating term is employed for\nplasma with temperatures below 2500 K, in order to prevent the\nArticle number, page 2 of 9Kohutova et al.: Damping of coronal oscillations in 3D MHD simulations\nFig. 1. Left: Bundle of magnetic loops and the photospheric magnetic field at t=880 s.Top right: The cut across the density structure at x=13\nMm intersecting the apex of the magnetic bundle. Bottom right: The cut across the temperature structure at x=13 Mm. The white points mark\nthe positions at which the individual loops in the bundle intersect the cuts. An animation of this figure is available online.\n0 5 10 15 20\nx[Mm]0510z[Mm]\n0 5 10 15 20\ny[Mm]0510z[Mm]\n0 5 10 15 20\nx[Mm]5101520y[Mm]\nFig. 2. A projected view of the magnetic bundle in the xz-plane (left), yz-plane (centre) and xy-plane (right) at t=880 s. An animation of this\nfigure is available online.\ntemperature from dropping too low in regions that are rapidly\nexpanding. Only few isolated regions in the simulation domain\nare affected by this, and the heating in the vast majority of the\ndomain is self-consistent. Contributing to the heating are small-\nscale reconnection events which heat the plasma through a com-\nbination of direct Ohmic dissipation and by inducing shear flows\nwhich are then converted into heat by viscous dissipation as well\nas dissipating oscillations. In order to ensure numerical stability\nthe code employs a di ffusive operator; this consists of a small\nglobal di ffusion term as well as of a directionally-dependent hy-\nper di ffusion component which enhances the di ffusion locally.\nFurther details of the numerical setup can be found in e.g. Carls-\nson et al. (2016); Kohutova et al. (2020).\n3. Evolution of a magnetic loop bundle\nThe corona in the simulation is filled with closed magnetic loops\nwhich can extend up to heights of 10 −14 Mm. The density\nstructure of the simulated corona is complex and there are sev-\neral structures with enhanced densities compared to the sur-\nroundings. Most of the overdense structures are filled by chro-\nmospheric evaporation in response to heating (Kohutova et al.\n2020). A cut across the simulation domain shows a lack of loopswith clearly defined cross-sections in the density or temperature\nstructure (Fig. 1).\nThe magnetic field configuration in the simulation domain is\ndriven by the dynamics of the lower solar atmosphere and the\nfootpoints of coronal structures are shu ffled and dragged around\nby the convective motions. The magnetic loops in the corona are\ntherefore continuously evolving and undergoing complex mo-\ntions including sideways displacement, oscillatory and torsional\nmotion and vertical expansion /contraction.\nWe focus on the evolution of a bundle of magnetic loops lo-\ncated in the centre of the simulation domain shown in Fig. 1\nreaching a height of around 10 Mm. Due to the evolving nature\nof the magnetic field in the simulated corona, in order to obtain\nthe evolution of such a bundle in three dimensions it is necessary\nto trace the evolution of the corresponding magnetic field lines\nin the bundle through both time and space. To do this, we use\na field-tracing method described in Leenaarts et al. (2015) and\nKohutova & Popovas (2021).\nA magnetic field line is defined as a curve in a 3D space\nr(s) parametrised by the arc length along the curve s, for which\ndr/ds=B/|B|and is therefore a representation of magnetic\nconnectivity of the coronal plasma. We trace the evolution of\nmagnetic field lines by inserting seed points into the simulation\nArticle number, page 3 of 9A&A proofs: manuscript no. waves\n0 250 500 750 1000 1250 1500 1750 2000\nt[s]7891011121314z[Mm]\nFig. 3. The evolution of the height of the individual loops in the bundle\natx=13 Mm as a function of time. Blue regions indicate the two\ninstances of damped oscillatory motion in the vertical direction.\ndomain at the apex of the magnetic bundle. Using the velocity\nat the seed point position the seed points are then passively ad-\nvected forward and backward in time. The magnetic field is then\ntraced through the instantaneous seed point position in order to\ndetermine the spatial coordinates of the traced field line at every\ntime-step. The accuracy of this method is given by the size of\nthe time-step between two successive simulation snapshots (i.e.,\n1 second in this case). We find that the method works well for 10\ns step-size and there are no major di fferences in field-line evo-\nlution between 1 s and 10 s time-step size. The 10 s time-step\nsize is therefore used for the field line tracing in the following\nanalysis. We note that this approach requires that the evolution\nis smooth and there are no large-amplitude velocity variations\noccurring on timescales shorter than the size of the time-step.\nSimilarly, in the instances where magnetic reconnection occurs,\nthis approach fails and the tracing leads to a jump in the field-line\nevolution. This is, however, not the case in the coronal part of the\nmagnetic bundle during the analysed time-period. We note there\nare 2 instances of rapid transverse displacement occuring close\nto the foootpoints of the traced loops at t =830 s and t =1100 s\ncaused either by an external perturbation or change of magnetic\nconnectivity in the lower solar atmosphere. These however do\nnot lead to discontinuities in the evolution of physical quantities\nin the coronal part of the analysed loops.\nThe magnetic bundle is shown in di fferent projections in Fig.\n2 with the bundle evolution shown in the online animation. The\nindividual magnetic loops display a large degree of collective be-\nhaviour and the magnetic bundle behaves as a coherent structure\nduring most of the duration of the simulation. The footpoints\nof the magnetic bundle are not static, and the bundle is continu-\nously changing and evolving. The lengths of the individual loops\nin the magnetic bundle change significantly over the duration of\nthe simulation, sometimes on a timescale of minutes.\n4. Oscillations and damping\nFull spatial coordinates of the magnetic loops in the bundle at\nany point in time enable us to obtain the evolution of the physical\nquantities along the loops, including both scalar and vector quan-\ntities. To aid the oscillation analysis, we decompose the velocity\nvector into three velocity components relative to the direction\n0 250 500 750 1000 1250 1500 1750−50050v[km s−1]vT vR vN\n0 250 500 750 1000 1250 1500 1750−6−4−2log (Q[W m−3]) QJoule Qvisc\n0 250 500 750 1000 1250 1500 1750\nt[s]102030L[Mm]5.505.756.006.25\nlog (T[K])\n200300400\nvA[km s−1]Fig. 4. Top: The evolution of the longitudinal (green), normal (orange)\nand the binormal velocity component (blue) at the apex of the bundle.\nMiddle: The evolution of the Joule volumetric heating rate (red dashed),\nthe viscous volumetric heating rate (red dotted) and the temperature\n(black) at the apex of the bundle. Bottom: The evolution of the average\nlength (blue) and the Alfvén velocity at the apex of the loops in the\nbundle. Blue regions indicate the time-range corresponding to the two\noscillations.\nof the magnetic field, vT,vNandvR. The longitudinal velocity\nvT=v·Tcorresponds to the velocity component aligned with the\ntangent vector of the magnetic field line given by T=B/|B|. The\nnormal velocity vN=v·Ncorresponds to the velocity component\nalong the normal vector of the field line given by N=dT\nds/|dT\nds|.\nIn the case of a closed magnetic loop, vNrepresents the motion\nin the plane of the loop and perpendicular to the loop tangent.\nFinally the third velocity component aligned with the binormal\nvector is given by vR=v·Rwhere R=T×N. In the case of a\nclosed magnetic loop, vRcorresponds to transverse motion per-\npendicular to the plane of the loop and to the loop tangent. Unit\nvectors T,N, and Rtogether form an orthogonal coordinate sys-\ntem known as the Frenet frame of reference. Such coordinate\nsystem is well-suited to analysing oscillations in complex 3D\nmagnetic field geometries (e.g., Carlsson & Bogdan 2006; Fe-\nlipe 2012; Leenaarts et al. 2015; González-Morales et al. 2019;\nKohutova & Popovas 2021).\nThe collective behaviour of the loop bundle is apparent from\nFig. 3, which shows the evolution of the height of the individ-\nual magnetic loops in the bundle at x=13 Mm. The evolution\nshows two clear instances of oscillatory behaviour, starting at\nt=400 s and t=1100 s and lasting about 400 s in both cases,\nas indicated marked by blue regions in Figs. 3 and 4. The os-\ncillations are damped and occur in a plane perpendicular to the\nbundle axis. From the evolution shown in animations of Figs.\n1 and 2 it is clear that these oscillations correspond to a trans-\nverse standing mode of the bundle. The point of maximum os-\ncillation displacement and hence maximum oscillation velocity\namplitude lies close to the apex of the bundle with the bundle\nfootpoints acting as nodes of the standing oscillation. The oscil-\nlations are triggered by impulsive events in the corona associated\nArticle number, page 4 of 9Kohutova et al.: Damping of coronal oscillations in 3D MHD simulations\n400 450 500 550 600 650 700 750 800−10−50510vN[km s−1]A = -10 km/s\nτ= 176 s\nP = 177 s\nk = -0.04\n1100 1150 1200 1250 1300 1350 1400−10010vN[km s−1]A = -16 km/s\nτ= 198 s\nP = 191 s\nk = -0.28\n400 450 500 550 600 650 700 750 800−6−5−4−3log (Q[W m−3])QJoule Qvisc\n1100 1150 1200 1250 1300 1350 1400−5.0−4.5−4.0−3.5−3.0log (Q[W m−3])QJoule Qvisc\n400 450 500 550 600 650 700 750 800\nt[s]5.75.85.96.0log (T[K])\n1100 1150 1200 1250 1300 1350 1400\nt[s]6.056.106.15log (T[K])\n0.030.040.05\nω[m s−1]\n0.030.040.050.060.07\nω[m s−1]|ωx||ωz|\nFig. 5. Top: The evolution of the detrended normal velocity component at the bundle apex averaged over the oscillating fieldlines in the bundle.\nBest-fit is shown in red. Middle: The evolution of the Joule (dashed line) and viscous volumetric heating rate (dotted line) at the bundle apex.\nBottom: Temperature (black), the x-component of the vorticity averaged in the vicinity of the bundle apex (solid blue line) and the z-component\nof the vorticity averaged in the vicinity of the right bundle footpoint (dotted blue line). The time intervals shown correspond to the time ranges\nhighlighted in blue in Figs. 3 and 4.\nwith a peak in the Joule heating and subsequent large-scale dis-\nplacement of the analysed bundle.\nWe analyse the evolution of the vRandvNcomponents by\naveraging the velocity components over the oscillating loops in\nthe bundle at the loop apex, which we define as the point halfway\nbetween the two footpoints of the loop (Fig 4). The oscillations\nare most clearly detectable in the vNcomponent. In both cases\nthe oscillations follow large scale perturbations of the magnetic\nbundle.\nTo remove the large-scale trends in the vNevolution corre-\nsponding to a bulk motion of the bundle, the oscillation time-\nseries are detrended by subtracting a best-fit second-degree poly-\nnomial. The time-series are then fitted with the function\nv(t)=v0exp\u0012\n−t\nτ\u0013\nsin\u00122πt\nP+kt−Φ\u0013\n(1)\nwhich includes both the oscillation damping and linear change\nin the oscillation period (Fig. 5). Here v0corresponds to the ve-\nlocity amplitude, τis the decay time, Pis the oscillation period,\nkis a parameter controlling the linear change in period and Φis\nthe phase. We find that the initial velocity amplitudes are 10 km /s\nand 16 km /s, these are damped with decay times of 176 s and 198\ns respectively, corresponding to roughly 3 oscillation periods be-\ning detectable before the oscillation decays. The periods for the\nfirst and second oscillation are 177 s and 191 s respectively. The\noscillation periods in the both cases decrease over the duration of\nthe oscillation; by approximately 15 s for oscillation 1 and 100 s\nfor oscillation 2. The Joule and viscous volumetric heating ratesshown in Fig. 5 are also averaged over the oscillating loops at the\napex bundle. Over the duration of oscillations the heating rates at\nthe apex of the bundle are variable, but overall have an increasing\ntrend during the later stages of both oscillations. Similarly, the\naverage temperature at the bundle apex increases by around 0.5\nMK over the duration of the oscillation in both cases. Finally, we\nshow the evolution of the absolute value of the x-component of\nthe vorticity ωxin the y−zplane at x=13 Mm, averaged over the\narea surrounding the bundle apex. The ωx-component is chosen\nbecause the axis of the loop at the apex is roughly aligned with\nthex-axis and the oscillatory motion mostly occurs in the plane\nperpendicular to the loop axis. During the first oscillation, the\nvorticity in the surroundings of the oscillating bundle increases\nto reach the maximum value around 170 s after the oscillation\nonset, followed by a gradual decrease to the pre-oscillation val-\nues. In the later case, the vorticity in the vicinity of the bundle\napex decreases during the entire oscillation duration. For the sec-\nond oscillation we also calculate the evolution of the ωzvorticity\ncomponent at the right footpoint at the height of 3 Mm, as this\nundergoes strong transverse displacement. Similar vorticity peak\nas at the apex position of the first oscillation can be seen in the\nevolution here.\nTo further understand the evolution of the vorticity close\nto the oscillating bundle, we show the velocity field given by\nvyey+vzezin the y−zplane at x=13 Mm which is nearly per-\npendicular to the bundle axis and in the plane of the bundle os-\ncillations (Fig 6). We calculate the velocity field before the onset\nof the oscillation, during the oscillation and at the end of the os-\nArticle number, page 5 of 9A&A proofs: manuscript no. waves\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 240 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 500 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 700 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 900 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 1100 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 1390 s\n−40−2002040\nvz[km s−1]\n−40−2002040\nvz[km s−1]\nFig. 6. Top: The vzvelocity component and the corresponding velocity field streamlines in cuts parallel to the y−zplane at x=13 Mm shown before\nthe oscillation onset (left), during the oscillation (middle) and at the end of the oscillation (right). Bottom: Same as above but for oscillation 2. The\nred cross markers correspond to the positions at which the individual loops in the bundle intersect the cut. The locations of the enhanced shear\nin the vzvelocity component correspond to strong counter-directional flows. In both cases several vortices develop in the vicinity of oscillating\nstructures.\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 240 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 500 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 700 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 900 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 1100 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 1390 s\n0200400600800\nvA[km s−1]\n0200400600800\nvA[km s−1]\nFig. 7. Top: The Alfvén speed and the corresponding velocity field streamlines in cuts parallel to the y−zplane at x=13 Mm shown before the\noscillation onset (left), during the oscillation (middle) and at the end of the oscillation (right). Bottom: Same as above but for oscillation 2. The\nred cross markers correspond to the positions at which the individual loops in the bundle intersect the cut.\ncillation. The shear flows are abundant in the vicinity of the bun-\ndle, this shows as counter-directional large magnitude flows in\nthevzcomponent of the velocity. We find that the velocity shear\nis strongest before and during the oscillation. Associated with\nthis is the development of several vortices visible in the velocity\nstreamlines in the close proximity of the oscillating structures.\nThe velocity shear in this region weakens and the vortices mostly\ndisappear once the oscillations have decayed. The vortices visi-\nble in the velocity streamlines are always located in the regions\nwith strong velocity shear, suggesting they originate due to the\ndevelopment of the Kelvin-Helmholtz instability. The size of the\nvortices is of the order of few Mm, which matches the transverse\nlength scale in the Alfvén speed variation (Fig. 7), as predicted\nby the straight flux-tube models (Antolin & Van Doorsselaere2019). The Kelvin Helmholtz instability in magnetised plasmas\nis inhibited by the magnetic tension, in this case however, the di-\nrection of the magnetic field is perpendicular to the flow velocity\nvector due to the oscillatory motion. The vortices can therefore\ndevelop without distorting the magnetic field, which would lead\nto stabilizing magnetic tension (Hillier et al. 2019; Barbulescu\net al. 2019). We note that both presence of vortices and shear\nflows will contribute to the increased values of ωxin the anal-\nysed region.\nArticle number, page 6 of 9Kohutova et al.: Damping of coronal oscillations in 3D MHD simulations\n5. Discussion\n5.1. Oscillation parameters\nDisplacement and velocity amplitudes in both instances of bun-\ndle oscillations are in agreement with the values typically ob-\nserved in active region coronal loops (e.g. White & Verwichte\n2012), although this also depends on the type and magnitude of\nthe perturbation responsible for the excitation of the oscillation.\nThe detected oscillation periods are also within the range com-\nmonly seen in the observations (Nechaeva et al. 2019). We note\nthat the oscillation periods decrease over the duration of the os-\ncillation in both cases. This can be explained by changing phys-\nical properties in the magnetic loop bundle, such as changing\nloop length or distribution of the plasma along the loop. Such\nevolution is not surprising in a dynamic environment like this, as\nthe length of the oscillating loops as well as the values of Alfvén\nspeed can change on timescales comparable to the duration of\nthe oscillation. The expected values of the oscillation periods es-\ntimated from P∼2L/vAusing the values for the average length\nand the Alfvén speed at the apex of the oscillating bundle loops\nat the beginning of each oscillation are 205 s and 175 s for os-\ncillation 1 and 2 respectively. This roughly agrees with the de-\ntected initial periods, the change in the loop properties during\nthe oscillation duration will however a ffect the oscillation peri-\nods. Changes in oscillation period over the duration of the oscil-\nlation due to changes in properties of the oscillating loop have\nboth been seen in observations and been reproduced by numer-\nical modelling (Kohutova & Verwichte 2017; Verwichte & Ko-\nhutova 2017; Su et al. 2018). Such shift in oscillation properties\nhas a diagnostic potential using coronal seismology methods. A\nmore detailed test of coronal seismology methods in convection-\nzone-to-corona simulations involving synthetic observables will\nbe addressed in a follow-up study. Oscillation damping times are\ncomparable to oscillation periods in both cases and in agreement\nwith the observations of damped transverse oscillations, which\nare typically observed to decay within 3-4 oscillation periods\n(Goddard et al. 2017). We do not analyse the detailed profile of\nthe oscillation damping, that is whether the damping profile is\nbetter represented by an exponential profile, Gaussian profile or\na transition from Gaussian to an exponential profile as described\nin Hood et al. (2013); Pascoe et al. (2016). Such a model is based\non an assumption of a monolithic cylindrical coronal loop with\na non-homogeneous boundary layer at the loop interface, which\nis not representative of coronal structures in the self-consistent\nconvection-zone-to-corona simulations.\n5.2. Collective behaviour\nThe bundle of magnetic loops in the simulation does not oscil-\nlate in isolation. It is in fact di fficult to isolate the oscillating\nstructures from a static, background plasma, as there is a large\ndegree of collective behaviour among the structures in the sim-\nulated corona. The evolution of the individual magnetic loops in\nthe bundle is not identical, but averaging the physical quantities\nat the bundle apex gives us the overall evolution of the bundle.\nThis implies that care should be taken when modelling individ-\nual coronal loops as isolated structures. As the coronal loop evo-\nlution is coupled to the environment, the arcade model described\nby Hindman & Jain (2021) is more representative of the evolu-\ntion of the coronal structures in our simulation. The observations\nalso suggest that the oscillations of individual coronal loops are\noften coupled to the oscillation of nearby magnetic structuresand rarely occur in isolation (Verwichte et al. 2004, 2009; Jain\net al. 2015).\nRegardless of the geometry, oscillating large scale structures\ncreate shear flows that lead to Kelvin-Helmholtz instability and\nthe subsequent development of vortices if the instability is not\ninhibited by the magnetic tension. The shear created by large-\nscale translational motions also contributes to the development\nof Kelvin-Helmholtz instability, provided the direction of the\nmotion is perpendicular to the coronal magnetic field (Hillier\net al. 2019; Barbulescu et al. 2019). The development of vor-\ntices in our simulation may therefore be a result of a com-\nplex interplay of oscillatory and translational motions of coronal\nstructures. The traditional models of Kelvin-Helmholtz unstable\nloops (e.g. Terradas et al. 2008; Antolin et al. 2014; Karampelas\net al. 2017) assuming well-defined oscillating cylindrical loops\nembedded in a static plasma are however too idealised, as there\nare no loops with clearly defined cross-sections in our simula-\ntion. Because of a lack of clear loop boundaries the vortices are\ninstead developed at the locations of maximum velocity shear.\n5.3. Oscillation damping and dissipation\nWe stress the importance of the oscillation damping time τas\nthis corresponds to the rate at which the wave energy is either\nconverted into another wave mode or dissipated. Empirical scal-\ning of the damping time with the loop oscillation period is com-\nmonly cited as an indirect evidence for resonant absorption being\nthe primary mechanism responsible for loop oscillation damp-\ning (see e.g. review by Nakariakov & Kolotkov (2020)). It has\nhowever been argued that the use of such scaling laws for dis-\ncriminating between di fferent damping mechanisms is question-\nable due to the inherent dependence of the damping time on the\nloop parameters (Arregui et al. 2008). We note that traditionally\nused scaling laws for resonant absorption assume large density\ncontrast between the loop and the surrounding plasma (Ofman\n& Aschwanden 2002), which is not the case in our simulation,\nas shown in Fig. 1. Oscillation damping times are further ex-\npected to depend on the magnetic Reynolds number given by\nRm=UL/η, where U is a typical velocity scale, L is a typical\nlength scale and ηis magnetic di ffusivity. Rmvaries in the sim-\nulation as Bifrost uses η(r,t) that is spatially and temporally de-\npendent; Rmin the simulation is however several orders of mag-\nnitude smaller than the estimated values in the solar corona.\nWe note that the traditional models for oscillation damping\ndue to resonant absorption rely on a presence of an inhomoge-\nneous layer at the boundary of a thin cylindrical loop with an\nAlfvén speed gradient (Ruderman & Roberts 2002). This is not a\nvalid approximation for the loops in our simulation, event though\nAlfvén speed gradients are abundant in the simulated corona. We\nsee no clear evidence of mode conversion in the oscillating bun-\ndle (that is, conversion to clearly identifiable spatially localised\nazimuthal oscillations); instead, the evolution of the physical\nquantities at the apex of the loop bundle suggests that the wave\nenergy is dissipated into heat through the development of shear\nflows, leading to an increase in the viscous and resistive dissi-\npation over the duration of the oscillation (in MHD models, the\nresistive and viscous dissipation terms are a parametrisation of\nprocesses operating on kinetic scales). Velocity shear leads to de-\nvelopment of Kelvin-Helmholtz vortices which are detectable in\nthe velocity field in the plane perpendicular to the bundle cross-\nsection. The increase of the ωxcomponent in the vicinity of the\nbundle apex during the first oscillation decay suggests that the\nshear due to the oscillatory motion drives the development of\nvortices which then dissipate, leading to subsequent ωxdecrease\nArticle number, page 7 of 9A&A proofs: manuscript no. waves\neffectively explaining oscillation damping and dissipation. How-\never, the absence of such a clear vorticity peak at the loop apex\nduring the second oscillation decay suggests the picture in this\ncase is not as clear and multiple processes can contribute to the\noscillation damping. The evolution of loop oscillations might be\nfurther a ffected by the motion of the loop footpoints, as these are\nnot static. We note that the wave dissipation is only one of the\nseveral possible mechanisms contributing to the large tempera-\nture increase seen at the apex of the oscillating loops.\nThe onset of Kelvin-Helmholtz instability in models of trans-\nversely oscillating loops is linked to the development of turbu-\nlence leading to the formation of small scales which allows for\nfast dissipation of the wave energy (Hillier et al. 2020). For a\nloop to be considered truly ’turbulent’ it is however necessary to\ndemonstrate a non-linear cascade of energy to small scales. We\ncannot draw any conclusions about the presence of turbulent be-\nhaviour /lack thereof in our simulation due to limits posed by spa-\ntial resolution as well as by the magnetic di ffusivity and viscos-\nity. We also note that low Reynolds numbers in self-consistent\nMHD simulations artificially restrict the cascade to small scales\n(Howson et al. 2017).\nKelvin-Helmholtz vortices developed during oscillation\ndamping have also been proposed as being responsible for coro-\nnal oscillations appearing as decayless in certain emission lines\n(Antolin et al. 2016). In such model, however, the KHI vortices\nwere formed at much smaller spatial scales due to the sharp den-\nsity (and Alfvén speed) contrast at the loop boundary which is\nnot the case for the loop bundle analysed in this work.\n5.4. Implication for coronal loop models\nWe find that oscillation parameters and evolution observed in\ncoronal loops are reproduced by self-consistent simulations\nwhich include complex magnetic field geometry and density\nstructuring and which do not contain well-defined coronal loops.\nThis approach provides insight into how does the oscillating loop\nbundle actually evolve in three dimensions, including the de-\ntailed evolution of the magnetic field, into the degree of collec-\ntive oscillation of the surrounding plasma and into the physical\nmechanisms associated with the oscillation damping and dissi-\npation. This type of dynamics is impossible to capture by ide-\nalised straight flux tube models. This might have widespread im-\nplications for the accuracy of the coronal seismology methods,\nwhich are mostly based on cylinder approximations for coro-\nnal loops. Even recent numerical studies of the evolution of ini-\ntially homogeneous coronal loops in response to transverse mo-\ntions suggest that the highly idealised picture of coronal loops as\nmonolithic plasma cylinders is unlikely to be realistic in the first\nplace (Antolin & Van Doorsselaere 2019). The question remains\nhow realistic our current self-consistent simulations really are\nwhen it comes to reproducing detailed characteristics of the so-\nlar corona. However, despite the complex collective behaviour in\nsuch self-consistent simulations some results from simple coro-\nnal loop models are reproduced, namely the generation of the\nKelvin-Helmholtz vortices by the transverse motions. The sim-\nple coronal loop models provide a lot of value for understanding\nfundamental physical processes at play. Caution should however\nbe taken when drawing conclusions from observations whether\nthe assumptions made in the models are still applicable in the\nanalysed scenario. Combination of simple and self-consistent\nmodels is necessary for detailed understanding of the oscillatory\nbehaviour in the corona.\nFinally, we note that the length of loops we can simulate in\nthis type of setup is limited by the size of the simulation domain,with the maximum length of magnetic loops in the simulation\nused in this work being 20 - 30 Mm. This a ffects the parameter\nspace that is accessible by such models, as several oscillation\nparameters scale with loop length. Larger domains are therefore\nneeded for more accurate one-to-one comparison of oscillation\nparameters with observations.\n6. Conclusions\nFor the first time, we analysed the damping of coronal oscilla-\ntions using a self-consistent 3D radiation-MHD model of the\nsolar atmosphere spanning from the convection zone into the\ncorona, the associated oscillation dissipation and heating, and\nfinally the physical processes responsible for the damping and\ndissipation. The simulated corona formed in such a model does\nnot depend on any prior assumptions about the shape of the coro-\nnal loops. Using magnetic field tracing we analysed the evolution\nof a bundle of magnetic loops in the centre of the simulation do-\nmain. The magnetic bundle shows dynamic evolution and a large\ndegree of collective behaviour of the individual loops in the bun-\ndle. We find that the bundle of magnetic loops shows damped\ntransverse oscillations in response to perturbations in two sepa-\nrate instances with oscillation periods of 177 s and 191 s, veloc-\nity amplitudes of 10 km s−1and 16 km s−1and damping times\nof 176 s and 198 s, respectively. The oscillation parameters and\nevolution observed are in line with the values typically seen in\nobservations of coronal loop oscillations. The oscillation peri-\nods decrease in both instances during the oscillations. We find\nthat transverse coronal oscillations lead to the development of\nvelocity shear in the simulated corona resulting in the formation\nof vortices with sizes around one Mm seen in the velocity field\ncaused by the Kelvin-Helmholtz instability, contributing to the\ndamping and dissipation of the transverse oscillations into heat.\nAssuming the structure of the corona in the self-consistent mod-\nels is indeed realistic, our models of monolithic and static coro-\nnal loops with constant lengths might need to be reevaluated in\nfavour of more realistic models accounting for loop properties\nchanging with time and relaxing the assumption of highly ide-\nalised waveguides.\nAcknowledgements. PK acknowledges funding from the Research Council of\nNorway, project no. 324523. 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S. & Verwichte, E. 2012, A&A, 537, A49\n¸ Sahin, S. & Antolin, P. 2022, ApJL, 931, L27\nArticle number, page 9 of 9" }, { "title": "1210.2436v1.Comment_on__Thermal_fluctuations_of_magnetic_nanoparticles___arXiv_1209_0298_.pdf", "content": "arXiv:1210.2436v1 [cond-mat.mes-hall] 8 Oct 2012Commenton“Thermal fluctuations ofmagneticnanoparticles ”, cond-mat/arXiv:1209.0298\nJ.-L. Déjardin, H. Kachkachi, and J.-M. Martinez\nPROMES CNRS UPR8521, Université de Perpignan Via Domitia, 5 2 avenue Paul Alduy, 66860 Perpignan, France\nWe comment on some misleading and biased statements appeari ng in the manuscript “Thermal fluctuations\nof magnetic nanoparticles”, cond-mat/arXiv:1209.0298, a bout the use of the damped Landau-Lifshitz equation\ninconjunction withthekineticLangertheoryforthecalcul ationoftherelaxationrateofmagneticnanoclusters.\nWe provide simple scientific arguments, part of which is well known to the whole community, demonstrating\nthatthe authors overstate the issue and contradict a work th eyhave co-published earlier.\nHere we would like to draw the reader’sattention to the follo wingscientifically misleading and biased statement on page 46\nofthemanuscriptcond-mat/arXiv:1209.0298: “..... Unfortunately,some authors(see,e.g.,Ref. 137and138)ha veignoredthis\npropertyoftheLandau-Lifshitzequationand,inconsequen ce,haveusedthisintrinsicallyunder-dampedequationinc onjunction\nwiththeintrinsicallyIHDLangerformulaforthecalculati onoftheescaperateinalldampingranges. Thustheensuinge scape\nrate formulas [Refs. 137, 138] are misleading and not valid f or experimental comparison both at low damping, where they\ncoincide with the TST rate, and also in the IHD range, α/greaterorsimilar1, where they predict nonphysical behavior of the rate, namel y, a\nratein excessoftheTST one. ” Inthesequel,thisreviewpaperwill bereferredto asCK.\nLetusnowgiveabriefaccountofourscientificpointofviewc oncerningtheissueofdampingintheLandau-Lifshitzequat ion\nand its use in [1–3], Refs. (137, 138), together with Langer’ s approach. This is, of course, known to the whole community\nworkinginthisarea,andwe apologizeforhavingtoreiterat eit onceagain[17].\nItgoeswithoutsayingthatthedampingissueisasubtleonea ndtakesonspecialrelevanceinmagnetism. Theissuedatesb ack\nto 1935when Landauand Lifshitz publishedtheir seminal pap eron magnetics, andhas beensince a strongpoint of debatean d\ncontroversythroughhundredsofpublicationsandconferen ceproceedings,especiallyafterGilbertproposed,inthe1 955MMM\nconferenceproceedings,a new formfor the magnetizationda mping. Despite this longperiodof investigation,the issue has not\nbeen settled yet and the very origin of damping eludes any sim ple interpretation. The main reason is that damping is roote d\nin various kinds of correlation processes, both intrinsic a nd extrinsic, which cannot be captured by a few phenomenolog ical\nparameters added to the equation of motion. It would be too lo ng, if not impossible, to give a fair account of the divers\napproaches and interpretations of damping in magnetic syst ems. A brief account can be found in Ref. 4 [ Landau-Lifshitz or\nGilbert damping? Thatisthe question ]andRef. 5 [ Originof intrinsicGilbert damping ].\nThe equation of motion describing the magnetization dynami cs with a phenomenological damping parameter can be repre-\nsentedasoneofthe followingtwo well-knownforms:\n1. TheLandau-Lifshitzequation(LLE)\nd/vectorM\ndt=−γ/vectorM×/vectorH−λγ\nM/vectorM×/parenleftBig\n/vectorM×/vectorH/parenrightBig\n, (1)\nwithλbeinga(dimensionless)dissipationparameterand /vectorHtheeffectivefield.\n2. TheLandau-Lifshitz-Gilbertequation(LLGE)\nd/vectorM\ndt=−γ/vectorM×/vectorH+α\nM/vectorM×d/vectorM\ndt. (2)\nwhereαisanother(dimensionless)dissipationparameter.\nMathematically,thetwoequations(1)and(2)areequivalen t. Indeed,substitutingfor d/vectorM/dtontheright-handsideofEq. (2)\nthesameright-handsideandworkingouttheresultingdoubl ecrossproduct /vectorM×/parenleftBig\n/vectorM×d/vectorM\ndt/parenrightBig\n,andusing /vectorM·d/vectorM\ndt=0(becausethe\nmoduleof /vectorMisconstant),we obtain\nd/vectorM\ndt=−/parenleftbiggγ\n1+α2/parenrightbigg\n/vectorM×/vectorH−/parenleftbiggα\n1+α2/parenrightbiggγ\nM/vectorM×/parenleftBig\n/vectorM×/vectorH/parenrightBig\n(3)\nwhichisjust Eq. (1) uponmakingthe followingsubstitution s\nγ\n1+α2→γ,α\n1+α2→λ (4)\nin the first and second terms, respectively. Note that this tr ansformation depends on the normalization used in both equa tions\n[see e.g. http://en.wikipedia.org/wiki/Landau-Lifshit z-Gilbert_equation]. AnotherusefulformofEq. (3)consis tsin rewritingit\nintermsofthe re-scaledtime τ=t//parenleftbig\n1+α2/parenrightbig\n,andthusalso re-scalingtheNéelfreediffusiontime τN[9].2\nFurtherdiscussionof the two equationsandtheir compariso ncanbe foundin the textbooks[6, 7]. It is worthmentioningt he\nwork in Ref. [8] where it is rigorouslyshown that the LLGE dam pingterm is a mere re-scaling of time by a complexconstant.\nMoreover,itcanbeeasilyshown[9]thattheFokker-Plancke quationsassociatedwiththestochasticanalogsofthetwoe quations\n(1, 2) arealsoidentical.\nFrom the experimental point of view, there is no clear cut pro of as to which equation has to be used in general. In practice,\nbasedonmanyinvestigations,it hasbeenagreeduponthatfo rsmall damping,LLE andLLGE arealmostthe sameandthereby\nthe former is then assumed to be more suited to small damping r egimes. Indeed, for small damping, the transformation in Eq .\n(4) boils down to identity. However, many workers obtain the LLE damping for low frequency, long wavelength dynamics\n[4]. For high damping one would expect a damping-dependentg yromagnetic ratio, but this effect has still to be confirmed b y\nexperiments.\nNow, the work [3] (Ref. 137 in CK) uses the LLE for obtaining th e attempt frequency that enters the prefactor of Langer’s\nrelaxation rate. Had we used the Landau-Lifshitz-Gilbert d amping instead we would have obtained expressions that can b e\nrecovered by making the substitution (4). A concrete exampl e illustrating this procedure is provided by the work publis hed in\nRef. 10. In this reference,AppendixB summarizesthe analyt ical expressionsobtainedin Ref. 1 (1stpaper in Ref. 137 in CK),\nfor a system of two exchange-coupled magnetic moments using Langer’s approach. Then, these analytical expressions wer e\ncompared in Figs. 2 and 3 of Ref. 10 to the results of the fully i ndependent numerical method of matrix continued fractions ,\nwitha fairlygoodagreement.\nLast butnotleast,let usmentionafewpointsaboutthisrevi ewthatdeservespecialattentionfromthereader.\n•It is curious how the authors’ select their referenceswhen t hey write “ ...some authors (see, e.g., Ref. 137 and 138) have\nignoredthis property of the Landau-Lifshitzequation”; this is a rather biased and non objective mannerin reviewing the\nliterature, at variance with what a reader expects from a rev iew article. Indeed, one of the well-known specialist in thi s\narea,DmitryGaranin,andwhoisacknowledgedbytheauthors forhis“directorindirect ”contributiontothisreview,has\npublishedfundamentalandwell-knowncontributionswiths trongimpactonthedevelopmentsinthisareaofphysics. The\nauthors seem to ignore the fact that all of Garanin’s papers e xclusively use the Landau-Lifshitz equation. The reason is ,\nof course, scientifically motivated and is as explained abov e. In the work [11] a kind of phase diagram was obtained for\nuniaxialanisotropywith precisecrossoversbetweenvario usdampingregimes.\n•The authors claim that the analytical expressions publishe d in Refs.137, 138 are misleading and “ not valid for exper-\nimental comparison ”. It is indeed very important to show some care for compariso n between theory and experiments.\nHowever, this manuscript which is a big review that covers at least two decades fails to provide a single comparison of\nexperiments with any of the authors’ own theoretical work th at goes beyond the Néel-Brown model. The only two fig-\nures 8 and 22 that show a comparisonbetween experimentsand N éel-Brown model, are borrowed from the literature. A\nsuccessfulcomparisonbetweentheNéel-Brownmodelandexp erimentalmeasurementsonsinglemagneticnanoparticles\nwasachievedmanyyearsagobyW. Wernsdorferet al. in thesem inalwork[12–14].\n•The 2ndarticle in Ref. 137 in CK (which is Ref. 2) was published as a re view article in the special edition of the Journal\nofMolecularLiquidsthatwaseditedandprefacedbythefirst authorofthereviewCK. Thisarticle summarizedthemain\nstepsofLanger’scalculationofthe relaxationrate [15, 16 ] andclearlystarted the validityofthe approachwith respe ct to\ndamping.\nItis regretfulthatthisbigreviewdoesnotprovideawidera ndmoreobjectiveviewoftheworkavailableinthe literatur eonthe\ndynamics of magnetic nanoclusters, for the benefit of a new co mer to the field. It is also unfortunate that the authors have n ot\nprovidedadiscussionofthehugeamountofexperimentalwor kthatshowsthestate-of-the-artunderstandingofthereal situation\naboutthese systems.\n[1] H.Kachkachi, Eur.Phys.Lett. 62, 650 (2003).\n[2] H.Kachkachi, J.Mol. Liquids 114, 113(2004).\n[3] A.F.Franco, J.M.Martinez, J. L.Déjardin, H.Kachkachi , Phys.Rev. B 84, 134423 (2011).\n[4] W.M.Saslow, J.Appl. Phys. 105, 07D315 (2009).\n[5] M.C. Hickeyand J.S.Moodera, Phys. Rev. Lett. 102, 137601 (2009).\n[6] A.G.Gurevichand G.A.Melkov, Magnetization oscillations and waves (CSCPress,Florida,1996).\n[7] J.Stöer andH.C. Siegmann, Magnetism: from fundamentals tonanoscale dynamics (Springer, Berlin,2006).\n[8] M.Lakshmanan and K.Nakamura, Phys. Rev. Lett. 53, 2497 (1984).\n[9] J.L.Garcia-Palacios, Adv. Chem. Phys. 112, 1(2000).\n[10] S.Titov,H. Kachkachi, Yu. Kalmykov, W.T.Coffey, Phys .Rev. B72, 134425 (2005).\n[11] D.A.Garanin, E.Kennedy, D.S.F.Crothers, andW.T. Cof fey,Phys. Rev. E 60, 6499 (1999).3\n[12] W. Wernsdorfer, E. Boner Orozco, K. Hasselbach, A. Beno it, B. Barbara, N. Demoncy, A. Loiseau, H. Pascard, D. Mailly , Phys. Rev.\nLett.78, 1791 (1997).\n[13] W. Wernsdorfer, K. Hasselbach, A. Benoit, B. Barbara, B . Doudin, J. Meier, J.-Ph. Ansermet, and D. Mailly, J. Mag. Ma g. Mat.55,\n11552 (1997).\n[14] M.Jamet, W.Wernsdorfer, C.Thirion, D.Mailly, V. Dupu is, P.Mélinon, and A.Pérez,Phys. Rev. Lett. 86, 4676 (2001).\n[15] J.S.Langer, Phys. Rev. Lett. 21, 973 (1968).\n[16] J.S.Langer, Ann. Phys. (N.Y.) 54, 258 (1969).\n[17] We deem it our duty to inform the reader that the second au thor has already sent to Phys. Rev. B a comment on the article [ 3], Ref. 138\nin CK. After we sent our reply and after one more round in the re ferral process of Phys. Rev. B, we decided that the various co mments\nandreplies be sent toa thirdreferee. Tothe best of our knowl edge, Kalmykov’s comment has not appeared inPhys. Rev. B." }, { "title": "2101.02064v1.The_effect_of_flow_on_resonant_absorption_of_slow_MHD_waves_in_magnetic_flux_tubes.pdf", "content": "arXiv:2101.02064v1 [physics.plasm-ph] 3 Jan 2021The effect of flow on resonant absorption of slow\nMHD waves in magnetic flux tubes\nMohammad Sadeghi1, Karam Bahari2, Kayoomars Karami1\n1Department of Physics, University of Kurdistan, Pasdaran Stree t, P.O. Box 66177-15175, Sanandaj, Iran\n2Physics Department, Faculty of Science, Razi University, Kerman shah, Iran\nJanuary 7, 2021\nAbstract\nIn this paper, we study kink and sausage oscillations in the presence of longitudinal\nbackground flow. We study resonant absorption of the kink and sa usage modes in the\nslow continuum under magnetic pore conditions in the presence of flo w. we determine the\ndispersion relation then solve it numerically, and find the frequencies and damping rates of\nthe slowkink and sausagesurfacemodes. We also, obtain analytical solutionfor the damping\nrate of the slow surface mode in the long wavelength limit. We show tha t in the presence of\nplasma flow, resonanceabsorptioncan result in strongdamping for forwardwavesand can be\nconsidered as an efficient mechanism to justify the extremely rapid d amping of slow surface\nsausage waves observed in magnetic pores. Also, the plasma flow re duces the efficiency of\nresonance absorption to damp backward waves. Furthermore, f or the pore conditions, the\nresonance instability is avoided in our model.\n1 Introduction\nThe mechanism of the heating of the solar corona (and the coro na of the stars) is not yet fully\nunderstood. Several non-thermal mechanisms have been prop osed to explain this phenomenon,\nand the problem of justifying this phenomenon remains. Sure ly the heating must be tied to the\nmagnetic field, because it is obvious that the heated areas ha ve a non-potential magnetic field.\nPlasma is bounded by magnetic field lines and can form many typ es of visible structures. One of\nthese is the propagation of magnetohydrodynamic (MHD) wave s and their damping. Resonant\nabsorption proposed as the damping mechanism of MHD waves fo r the first time by Ionson [1].\nWith the launch of space satellites, the interest of theoret ical physicists in studying waves in\nthe solar atmosphere, and especially the use of resonance ab sorption, increased. Nakariakov re-\nported transverse oscillations in coronal loops with high d ampingrate [2] . Ruderman& Roberts\nexpressed the idea that the observed period of oscillation a nd their damping time can be used\nto determine the transverse density distribution in a coron al magnetic loop [3]. This method\nwas later used by many researchers (e.g., [4]- [17]).\nBecause the source of the high-temperature energy of the cor ona originates from the convec-\ntion zone below the surface of the sun, it is important to stud y the dynamics of MHD waves in\nthe photosphere and chromosphere (e.g., [18]; [19]). In the photosphere, in addition to Alfv´ en\nresonance, energy transfer by slow resonance absorption ca n be of particular importance. Yu et\nal. showed that slow resonance absorption can affect the dampi ng of waves in the photosphere\n[21] . They also found that the resonant damping of the fast su rface kink mode is much stronger\n1than that of the slow surface kink mode. Yu et al. [20] conside red linear profile for density\nand pressure in the transitional layers [20]. They showed in the cases where damping by Alfv´ en\ncontinuum is weak, the resonant absorption in slow continuu m can be an effective mechanism\nfor damping sausage and kink slow surface modes. Sadeghi & Ka rami investigated resonance\nabsorption in the presence of a weak magnetic twist in the pho tosphere condition [22]. They\nconcluded that a magnetic twist could be effective on more inte nse damping. In this paper, we\nstudy effect of flow on the slow sausage and kink MHD waves, which have been observed by\nDunn Solar Telescope [23].\nObservations by Brekke et al. and Tian et al. show that plasma flows in magnetic flux tubes\nare present everywhere in the solar atmosphere [24] and [31] . Soler et al. reported that the flow\nvelocities are usually less than 10% of the plasma Alfv´ en sp eed [32]. Grant et al. investigated\nwave damping observed in upwardly propagating sausage mode oscillations contained within a\nmagnetic pore [23]. They showed that the waves propagate onl y through 0.25 of it’s wavelength\nalong the before they damp whereas theory would expect the wa ve to survive for the distance\nof a few wavelengths. They also showed that the average upflow speed in photosphere is about\n1/3 Alfv´ en speed. Although higher speeds have been observed u p to about 1.15 Alfv´ en speeds.\nMHD oscillations of flowing plasma have been investigated by a number of researchers [33] and\n[35]. Joarder et al. (1977) [36] investigated resonant inst ability of MHD waves in the presence of\nplasma flow. They showed that if theplasma velocity is greate r than a certain value, it will cause\ninstability. Soler et al. studied analytically and numeric ally the damping length of resonantly\ndamped kink in static flux tubes including nonuniform transi tional layer [32]. They showed\nthat flow affects the wavelength and the damping length due to re sonant absorption. Bahari\nconsidered propagating kink MHD waves in the presence of mag netic twist and plasma flow [37].\nHe showed that the damping of the waves depend on the directio n of plasma flow and the wave\nnumber of the wave. Bahari et al. studied the propagation and instability of kink waves in a\ntwisted magnetic tube in the presence of flow [38]. They showe d that for particular values of\nflow speed in coronal flux tubes the kink MHD waves propagate wi thout damping. Ruderman\n& Petrukhin investigated the effect of flow on the damping of sta nding kink waves in the cold\nplasma approximation [34]. They concluded that the effect of fl ow on coronal seismography\nis weak but has a significant effect on prominences. Recently Ge eraerts et al. studied the\neffect of electrical resistivity on the damping of slow surfac e sausage modes. They showed that\nelectrical resistivity can play an important role in wave da mping and greatly reduce the number\nof oscillations [39].\nOur aim in the present work is to investigate the effect of flow on the oscillation and damping\nof slow surface sausage and kink modes in the magnetic pore co nditions. To study the effect\nof flow, we consider a model similar to the model of Yu et al. [20 ], in which the plasma flow\nhas been included too. In section 2, this model and the equati ons of motion governing the\nsurface modes are presented. We find the dispersion relation in the case of no inhomogeneous\nlayer in section 3. Then in section 4, we obtain the dispersio n relation in the presence of the\ninhomogeneous layer using the connection formula for slow c ontinuum. In Section 5, numerical\ncalculations for magnetic pore conditions are shown. Final ly, we conclude the paper in Section\n6.\n22 Equations of Motion and Model\nThe linear perturbations of homogeneous flowing magnetized plasma are governed by the fol-\nlowing equations [40]\n(1a) ρ/parenleftbigg∂\n∂t+v·∇/parenrightbigg2\nξ=−∇δp−1\nµ0/parenleftBig\nδB×(∇×B)+B×(∇×δB)/parenrightBig\n,\n(1b) δp=−ξ·∇p−γp∇·ξ,\n(1c) δB=−∇×(B×ξ),\nwhereρ,p,vandBare the background density, kinetic pressure, plasma veloc ity and magnetic\nfield, respectively. Also ξis the Lagrangian displacement vector, δpandδBare the Eulerian\nperturbations of the pressure and magnetic field, respectiv ely. Here, γis the ratio of specific\nheats (taken to be 5 /3 in this work), and µ0is the permeability of free space.\nWe consider a flux tube model with a unidirectional magnetic fi eld which is in the direction\nof the tube axis. The model consists of interior and exterior regions in which the equilibrium\nand stationary quantities are constant and transitional la yer in which the background quantities\nvary continuously. In the cylindrical coordinate the magne tic field is\n(2) B=/parenleftBig\n0,0,Bz(r)/parenrightBig\n.\nPlasma pressure and magnetic field must be satisfied in the hyd rostatic equilibrium equation\n(3)d\ndr/parenleftbigg\np+B2\nz\n2µ0/parenrightbigg\n= 0.\nHere the background plasma density and magnetic field are ass umed to be the same as those\nconsidered by Sadeghi & Karami (2019) [22]\nρ(r) =\n\nρi, r /lessorequalslantri,\nρi+(ρe−��i)/parenleftBig\nr−ri\nre−ri/parenrightBig\n, ri< r < r e,\nρe, r /greaterorequalslantre,(4)\nwhereri=R−l/2 andre=R+l/2. Here, Randlare the tube radius and the thickness of\nthe inhomogeneous layer, respectively,\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\ntube, respectively. Also BziandBzeare the interior and exterior constant longitudinal magnet ic\nfields, respectively. Putting Eqs. (5) into the magnetohydr ostatic equation (3), we obtain the\nbackground gas pressure as follows\np(r) =\n\npi, r /lessorequalslantri,\npi+(pe−pi)/parenleftBig\nr−ri\nre−ri/parenrightBig\n, ri< r < r e,\npe, r /greaterorequalslantre,(6)\n3where\npe=pi+/parenleftbig\nB2\nze−B2\nzi/parenrightbig\n2µ0, (7)\nandpiis an arbitrary constant. The plasma flow is considered to be i n the direction of the\nmagnetic field lines. as follows\nvz(r) =\n\nvzi, r /lessorequalslantri,\nvzi+(vze−vzi)/parenleftBig\nr−ri\nre−ri/parenrightBig\n, ri< r < r e,\nvze, r /greaterorequalslantre,(8)\nwherevziandvzeare the constant flow of the interior and exterior regions of t he flux tube,\nrespectively. In addition, we define the following quantiti es\nv2\nA(i,e)≡B2\nz(i,e)\nµ0ρ(i,e), (9)\nv2\ns(i,e)≡γp(i,e)\nρ(i,e), (10)\nv2\nc(i,e)≡v2\ns(i,e)v2\nA(i,e)\nv2\ns(i,e)+v2\nA(i,e), (11)\nwherevA(i,e),vs(i,e)andvc(i,e)are the interior/exterior Alfv´ en, sound and cusp velociti es, re-\nspectively.\nSincethehydrostaticequilibriumisonlyafunctionofr, al ltheperturbedquantities including\nξandδPTcan be Fourier analyzed\n(12) (ξ,δPT)∝ei(mφ+kzz−ωt),\nwhereωistheoscillation frequency, mistheazimuthal wavenumber forwhichonly integer values\nare allowed and, kz, is the longitudinal wavenumber in the zdirection. We study both forward\nand backward waves which propagate in the positive and negat ive z directions respectively,\nfor both the waves the longitudinal wavenumber is restricte d to positive values, the oscillation\nfrequency is positive for forward waves and is negative for b ackward wave. The perturbed\nquantity δPT=δp+B.δB/µ0is the Eulerian perturbation of total (gas and magnetic) pre ssure.\nPutting Eq. (12) into (1a)-(1c), we obtain the two coupled fir st order differential equations\n(13a) Dd(rξ)\ndr=−rC2δPT,\n(13b) DdδPT\ndr=C3ξ.\nThe above equations derived earlier by Appert et al. [41] and later by Hain & Lust [42],\nGoedbloed [43] and Sakurai et al. [44]. Here, the multiplica tive factors are defined as\n(14a) D ≡ρ/parenleftbig\nv2\ns+v2\nA/parenrightbig/parenleftbig\nΩ2−ω2\nA/parenrightbig/parenleftbig\nΩ2−ω2\nA/parenrightbig\n,\n(14b) C2≡Ω4−/parenleftbigg\nk2\nz+m2\nr2/parenrightbigg/parenleftbig\nv2\ns+v2\nA/parenrightbig/parenleftbig\nΩ2−ω2\nA/parenrightbig\n,\n(14c) C3≡ρD/parenleftbig\nΩ2−ω2\nA/parenrightbig\n,\nin which\n4fB≡kzBz,\nω2\nA≡f2\nB\nµ0ρ.\nand\nω2\nc≡/parenleftbiggv2\ns\nv2\nA+v2s/parenrightbigg\nω2\nA,\nHere Ω = ω−ωfis the Doppler shifted frequency which ωf(=kzvz.) is the flow frequency,\nωA(=kzvA) is the Alfv´ en oscillation frequency and ωc(=kzvc) is the cusp oscillation frequency.\nAlsovA=|Bz|/√µ0ρis the Alfv´ en speed, vs=/radicalbig\nγp/ρis the sound speed, and vc=vsvA\n(v2s+v2\nA)1/2\nis the cusp speed.\nCombiningEqs. (13a)and(13b), onecanobtainasecond-orde rordinarydifferentialequation\nfor radial component of the differential equation for δPTas [45]\n(15)d2δPT\ndr2+1\nrdδPT\ndr−/parenleftbigg\nk2\nr+m2\nr2/parenrightbigg\nδPT= 0,\nwhere\nk2\nr≡(ω2\ns−Ω2)(ω2\nA−Ω2)\n(v2\nA+v2s)(ω2c−Ω2), (16)\nsolutions of Eq. (15) in the interior ( r/lessorequalslantri) and exterior ( r/greaterorequalslantre) regions are given by\n(17a) δPTi(r) =AiIm(krir),\n(17b) δPTe(r) =AeKm(krer),\nwhereAiandAeare constant. Also I(.) andK(.) are the modified Bessel function of the second\nkind respectively. Replacing the solutions (17a) and (17b) into Eq. (13b) radial displacement\ncan be determined as\n(18a) ξri(r) =Ai\nρi(Ω2−ω2\nAi)I′\nm(krir),\n(18b) ξre(r) =Ae\nρi(Ω2−ω2\nAe)K′\nm(krir),\nin which prime denotes differentiation of the function with re spect to its argument. These\nsolutionsareusedinthenextsectionstodeterminethedisp ersionrelationofthetubeoscillations.\n3 Dispersion relation for the case of no inhomogeneous layer\nInthissection weconsiderafluxtubewithouttheinhomogene ouslayer andobtain thedispersion\nrelation of oscillations. For this purpose, the solutions o btained for ξrandδPTin the last section\ninside and outside the tube (i.e Eqs. (17a)-(18b)) must be sa tisfied in the following boundary\nconditions\n(19a) ξri/vextendsingle/vextendsingle/vextendsingle\nr=R=ξre/vextendsingle/vextendsingle/vextendsingle\nr=R,\n(19b) δPTi/vextendsingle/vextendsingle/vextendsingle\nr=R=δPTe/vextendsingle/vextendsingle/vextendsingle\nr=R,\nwhereRis the tube radius. Then the dispersion relation can be deter mined after some algebra\nas\nρi/parenleftbig\nΩ2\ni−ω2\nAi/parenrightbig\n−kri\nkreρe/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbig\nQm= 0, (20)\n50 2 4 6 8 10\nkz R0.790.80.810.820.830.840.85ci/si/si\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(a)0 2 4 6 8 10\nkz R0.790.80.810.820.830.840.85ci/si/si\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(b)\n0 2 4 6 8 10\nkz R-ci/si-0.862-0.86-0.858-0.856-0.854-0.852-0.85-0.848-0.846-0.844/sivzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(c)0 2 4 6 8 10\nkz R-ci/si-0.862-0.86-0.858-0.856-0.854-0.852-0.85-0.848-0.846-0.844/sivzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(d)\nFigure 1: The Dopller shifted phase speed Ω /ωsi, Eq. (20), of the slow surface sausage and kink\nmodesversus kzRforvariousflowparameters vzi/vsiforforwardandbackwardwaves. Panels(a)\nand (b) are for forward sausage and kink modes and panels (c) a nd (d) are for backward sausage\nand kink modes respectively. Under the magnetic pore condit ions, following [23] the auxiliary\nparameters are taken as vAi= 12 km s−1,vAe= 0 km s−1(i.e.Bze= 0),vsi= 7 km s−1,\nvse= 11.5 km s−1,vci= 6.0464 km s−1(≃0.8638vsi) andvce= 0 km s−1.\n6where\nQm=I′\nm(kriR)Km(kreR)\nIm(kriR)K′\nm(kreR).\nFor the case with no flow (Ω i= Ωe=ω), the dispersion relation reduces to the result\nobtained by Edwin & Roberts [45] and Yu et al. [20].\nHere we solve the dispersion relation (20) numerically and t he phase speed Ω /ωsiof the slow\nsurface sausage ( m= 0) and kink ( m= 1) modes versus kzRfor various values of the flow\nparameters vzi/vsiare displayed in Fig. 1. Panels (a) and (b) are for forward sau sage and kink\nmodes and panels (c) and (d) are for backward sausage and kink modes respectively. The figure\nshows that (i) for a given value of kzR, for forward waves when the flow speed increases the\nDoppler shifted phase speed decreases and for backward wave s the magnitude of the phase speed\nincreases. (ii) For a given flow speed vzi/vsiaskzRincreases the Doppler shifted phase speed\nfor forward decreases and magnitude of the Doppler shifted p hase speed for backward increases.\n(iii) ForkzR≪1, for both the forward and backward waves Ω /ωsitends to ωci/ωsi. (iv) These\nresults show that for specific values of the flow speed, the Dop pler shifted phase speed is between\nthe internal and external values of the cusp speed of the flux t ube. (vi) For the case of no flow,\nthe result of Yu et al. [20] is recovered.\n4 Dispersion relation in the presence of inhomogeneous laye r\nand resonant absorption\nIn this section we consider a flux tube with an inhomogeneous b oundary layer. According to\nEquations (4)-(6), the density, magnetic field and pressure change continuously from the inside\ntotheoutsideofthetube, sointhiscase, theDopller shifte d(Ω) ofthewaves may beequal tothe\ncusp (ωc) or Alfv´ en ( ωA) frequency. According to Yu et al. [20], under photosphere c onditions\nthe oscillation frequency will be equal to the cusp frequenc y at a point in the boundary layer\nwhich causes a singularity in the equations of motion. This p henomenon is called cusp resonant\nabsorption.\nSakurai et al. [44] showed that under the thin boundary appro ximation, the solutions inside\nand outside the tube can be connected using the connection fo rmula\n(21a)[ξr]≡ξre(re)−ξri(ri)\n=−iπSign Ω\n|∆c|µω4\nc\nrB2ω2\nA/vextendsingle/vextendsingle/vextendsingle\nr=rcδPTi,\n(21b)[δPT]≡δPTe(re)−δPTi(ri)\n= 0,\nwhere [ξr] and [δPT] represent the jumps for the Lagrangian radial displacemen t and total pres-\nsure perturbation across the inhomogeneous (resonant) bou ndary, which connects the solutions\ninside and outside of the flux tube. The subscript cin ∆cshows that the quantity must be\ncalculated in the surface where the cusp resonance occurs. W e will determine the location of the\ncuspresonance, rclater. We obtain thedispersionrelation inthe presenceof fl ow by substituting\nthe solutions (17a)-(18b) into the connection formula (21a ) and (21b), the result is\n(22)ρi/parenleftbig\nΩ2\ni−ω2\nAi/parenrightbig\n−ρe/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbigki\nkeQm\n+iπSign Ω\n|∆c|k2\nz\nρ/parenleftbiggv2\ns\nv2s+v2\nA/parenrightbigg2\nρiρe/parenleftbig\nΩ2\ni−ω2\nAi/parenrightbig/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbigGm\nke= 0,\n70 0.2 0.4 0.6 0.8 100.20.40.60.8611.21.41.641.71v/vsi\nvcevcivsivsevAi\nvs\nvA\nvc\n(a)0 0.2 0.4 0.6 0.8 1-1.71-1.64-1.4-1.2-1-0.86-0.6-0.4-0.2-0v/vsi\nvAivsevsivcivce\nvs\nvA\nvc\n(b)\nFigure 2: Variations the sound vs(−vs), Alfv´ en vA(−vA) and cusp speeds vc(−vc) versus\nδin the annulus layer under magnetic pore conditions ( vAi= 12 km s−1,vAe= 0 km s−1,\nvze= 0 km s−1,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). When vci≤v≤vcmresonance absorption occurs for the slow body modes\nand when v < vciresonance absorption occurs for slow surface modes in the sl ow continuum.\nwhereGm=Km(krere)\nK′\nm(krere). It is clear that in the absence of plasma flow this equation re duces to\nthe dispersion relation obtained by Yu et al. [20].\nTo display the background quantities in the boundary layer w e define the variable δ≡r−ri\nre−riwhich varies from 0 to 1 in the boundary layer. Using Eqs. (4) t o (6), one can write the\nquantities vs=/radicalbig\nγp/ρ,vA=|Bz|/√µ0ρin the inhomogeneous boundary layer as functions of\nδas\n(23) v2\ns=v2\nsi/bracketleftbigg1+δ(χv2\nsei−1)\n1+δ(χ−1)/bracketrightbigg\n,\n(24) v2\nA=v2\nAi/bracketleftbigg1+δ(χv2\nAei−1)\n1+δ(χ−1)/bracketrightbigg\n,\nand the cusp velocity vc≡vsvA\n(v2s+v2\nA)1/2in the inhomogeneous layer ( ri< r < r e) as\n(25) v2\nc=v2\nsiv2\nAi/bracketleftBig\n1+δ(χv2\nsei−1)/bracketrightBig/bracketleftBig\n1+δ(χv2\nAei−1)/bracketrightBig\n/bracketleftBig\n1+δ(χ−1)/bracketrightBig/bracketleftBig\nv2\nsi/parenleftBig\n1+δ(χv2\nsei−1)/parenrightBig\n+v2\nAi/parenleftBig\n1+δ(χv2\nAei−1)/parenrightBig/bracketrightBig,\nwhereχ≡ρe/ρi,vsei≡vse/vsi,vAei≡vAe/vAi. Using Eqs. (23)-(25) we plot the sound,\nAlfv´ en and cusp velocities under magnetic pore conditions in Fig. 2. The Figure shows that\nforvc< vciandvci< vc< vcmax, the surface and body sausage modes can resonantly damp\nin the slow continuum respectively. Here, vcmaxis the maximum value of the cusp speed in the\ntransition layer.\nNote that according to Yu et al. [20], the position of the cusp resonance point rcis obtained\nby setting Ω2=ω2\nc/vextendsingle/vextendsingle/vextendsingle\nr=rc≡k2\nzv2\nc/vextendsingle/vextendsingle/vextendsingle\nr=rc. Consequently, the resulting equation in terms of the\n8variable δc≡δ/vextendsingle/vextendsingle/vextendsingle\nr=rc=rc−ri\nre−riyields the following second order equation\nAδ2\nc+Bδc+C= 0, (26)\nwhereA,BandCare similar to the constants defined in Eqs. (55)-(57) in Yu et al. 2017 [20].\nThe solutions for δc(see the curve vcin Fig. 2)\nδc1=−B\n2A+√\nB2−4AC\n2A, (27)\nδc2=−B\n2A−√\nB2−4AC\n2A. (28)\nFor the slow surface sausage and kink mode due to having reson ance absorption, Ω /ωsishould\nbe below vci, which means that only δc2satisfies this condition [20].\nNext, we turn to calculate the parameter ∆ cappeared in the dispersion relation (22). To this\naim, using Eq. (25) 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\n(ω−ωf)dωf\ndr+ωcdωc\ndr/parenrightbigg\nr=rc\n=−2(ω−ωf(rc))ωfe−ωfi\nl−/parenleftBigg\nω2\nc(rc)\nl/parenrightBigg/braceleftBigg/parenleftbig\nχv2\nsei−1/parenrightbig\n1+δ/parenleftbig\nχv2\nsei−1/parenrightbig−(χ−1)\n1+δ(χ−1)\n+(χv2\nAei−1)\n1+δ/parenleftbig\nχv2\nAei−1/parenrightbig (29)\n−v2\nsi/parenleftBig\nχv2\nsei−1/parenrightBig\n+v2\nAi/parenleftBig\nχv2\nAei−1/parenrightBig\nv2\nsi/bracketleftBig\n1+δ/parenleftBig\nχv2\nsei−1/parenrightBig/bracketrightBig\n+v2\nAi/bracketleftBig\n1+δ/parenleftbig\nχv2\nAei−1/parenrightbig/bracketrightBig/bracerightBigg\nr=rc,\nwhereωf=kzvz.\n4.1 Weak Damping Limit—Slow Continuum\nHere, we study the dispersion relation (22) in the weak dampi ng limit. We first rewrite the\ndispersion relation as\nDAR+iDAI= 0, (30)\nwhereDARandDAIare the real and imaginary parts of Eq. (22) respectively, gi ven by\n(31) DAR=ρi(Ω2\ni−ω2\nAi)−ρe(Ω2\ne−ω2\nAe)kri\nkreQm,\n(32) DAI=πρiρek2\nz\nkreSign Ω\nρc|∆c|/vextendsingle/vextendsingle/vextendsingle\nr=rc/parenleftBigv2\nsc\nv2\nAc+v2sc/parenrightBig2\n(Ω2\ni−ω2\nAi)(Ω2\ne−ω2\nAe)Gm.\nNote that in Eqs. (31) and (32) we have the complex frequency ω=ωr+iγ, in which ωrandγ\nare oscillation frequency and the damping rate, respective ly. In the limit of weak damping, i.e.\nγ≪ωr, the damping rate γis given as [33]\nγmc=−DAI(ωr)/parenleftbigg∂DAR\n∂ω/vextendsingle/vextendsingle/vextendsingle\nωr/parenrightbigg−1\n. (33)\n9Here, we want to simplify Eq. (33), to obtain the damping rate of surface sausage modes in the\nweak damping limit, i.e. γ≪ωr. To this aim, we first calculate∂DAR\n∂ωfrom Eq. (31) as follows\n∂DAR\n∂ω= 2ρiΩi−2ρeΩekri\nkreQm−ρe/parenleftbig\nΩe−ω2\nAe/parenrightbig/parenleftbigg1\nkredkri\ndω−kri\nk2redkre\ndω/parenrightbigg\nQm−ρe/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbigkri\nkredQm\ndω.\n(34)\nNow from Eq. (16), one can obtain\ndkri\ndw=−Ω3\ni(Ω2\ni−2ω2\nci)\n(v2\nsi+v2\nAi)(Ω2\ni−ω2\nci)2kri, (35)\ndkre\ndw=−Ω3\ne(Ω2\ne−2ω2\nce)\n(v2se+v2\nAe)(Ω2e−ω2ce)2kre. (36)\nWith the help of Eqs. (35) and (36)\ndQm\ndω=xPmΩ3\ni(Ω2\ni−2ω2\nci)\n(ω2\nsi−Ω2\ni)(ω2\nAi−Ω2\ni)(Ω2\ni−ω2\nci)+ySmΩ3\ne(Ω2\ne−2ω2\nce)\n(ω2se−Ω2e)(ω2\nAe−Ω2e)(Ω2e−ω2ce).(37)\nReplacing this into Eq. (34) yields\n∂DAR\n∂ω= 2ρiΩi−2ρeΩekri\nkreQm−ρe/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbigkri\nkre/parenleftbigg(Qm+xPm)(Ω2\ni−2ω2\nci)Ω3\ni\n(ω2\nsi−Ω2\ni)(ω2\nAi−Ω2\ni)(Ω2\ni−ω2\nci)−(Qm−ySm)(Ω2\ne−2ω2\nce)Ω3\ne\n(ω2se−Ω2e)(ω2\nAe−Ω2e)(Ω2e−ω2ce)/parenrightbigg\n,(38)\nwhere\nPm≡/parenleftBigg\nI′′\nm(x)\nIm(x)−I′\nm(x)2\nIm(x)2/parenrightBigg\nKm(y)\nK′m(y),\nSm≡/parenleftbigg\n1−K′′\nm(y)Km(y)\nK′m(y)2/parenrightbiggI′\nm(x)\nIm(x), (39)\nandx=kririandy=krere. Finally, substituting Eqs. (32) and (38) into Eq. (33) one c an get\nthe damping rate γin the limit of weak damping for the surface modes in the slow c ontinuum\nas\nγmc/vextendsingle/vextendsingle/vextendsingle\nω=ωr=−πρek2\nz\nkreρcSignΩ\n|∆c|/vextendsingle/vextendsingle/vextendsingle\nr=rc/parenleftBig\nv2\ns\nv2\nA+v2s/parenrightBig2\n(Ω2\ni−ω2\nAi)(Ω2\ne−ω2\nAe)Gm\n2/parenleftBig\nΩi−χΩekri\nkreQm/parenrightBig\n−χTm, (40)\nwhere\nTm=/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbigkri\nkre/parenleftbigg(Qm+xPm)(Ω2\ni−2ω2\nci)Ω3\ni\n(ω2\nsi−Ω2\ni)(ω2\nAi−Ω2\ni)(Ω2\ni−ω2\nci)−(Qm−ySm)(Ω2\ne−2ω2\nce)Ω3\ne\n(ω2se−Ω2e)(ω2\nAe−Ω2e)(Ω2e−ω2ce)/parenrightbigg\n.\n(41)\nEquation (40) can be more simplified in the long wavelength li mit which we do in the next\nsubsection.\n104.2 Weak damping rate in long wavelength limit - slow continu um\nIn the limit kzR≪1 i.e.kriR(kreR)≪1 we can obtain a more simplified expansion for the\ndamping rate γ, by using the asymptotic expansion of Qm,Gm,PmandSm. For the sausage\n(m= 0) mode in the slow continuum we obtain (see Appendix A)\nγ0c=2πχ3SignΩ\n|∆c|R/bracketleftBigg\nω7\nciω2\nsi/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbig3\n3ω10\nAiω2\nci+8χω8\nAiω2\nsi/parenleftbig\nΩ2e−ω2\nAe/parenrightbig\nln(kzR)/bracketrightBigg\n(kzR)4ln3(kzR).(42)\nFor the kink ( m= 1) mode in the slow continuum we obtain (see Appendix B)\nγ1c=−πχ2Sign Ω\n8|∆c|Rω11\nci/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbig2\nω4\nAi/parenleftbig\nω2\nciω2\nAi−χω2\nsi/parenleftbig\nΩ2e−ω2\nAe/parenrightbig/parenrightbig2(kzR)4. (43)\nUnder magnetic pore condition ( vAe= 0)\nγ0c=2πχ3Sign Ω\n|∆c|R/bracketleftbiggω7\nciω2\nsiΩ6\ne\n3ω10\nAiω2\nci+8χω8\nAiω2\nsiΩ2eln(kzR)/bracketrightbigg\n(kzR)4ln3(kzR), (44)\nγ1c=−πχ2Sign Ω\n8|∆c|Rω11\nciΩ4\ne\nω4\nAi/parenleftbig\nω2\nciω2\nAi−χω2\nsiΩ2e/parenrightbig2(kzR)4. (45)\nIn the absence of flow ( vzi=vze= 0), Ω e=ωciso\nγ0c=2πχ3Sign Ω\n|∆c|R/bracketleftbiggω1\nci1ω2\nsi\n3ω10\nAi+8χω8\nAiω2\nsiln(kzR)/bracketrightbigg\n(kzR)4ln3(kzR), (46)\nγ1c=−πχ2Sign Ω\n8|∆c|Rω11\nci\nω4\nAi/parenleftbig\nω2\nAi−χω2\nsi/parenrightbig2(kzR)4, (47)\nwhere these relations are the same Eqs. (79) in [22] and (38) i n [20] respectively.\n5 Numerical results\nIn this section we solve the dispersion relation (Eq. (22)) n umerically to obtain the frequencies\nand damping rates of the slow surface sausage and kink modes a nd we compare the analytical\nresults (Eq. 40) with the numerical results. Under the magne tic pore conditions, following\n[23] we set again the model parameters as vAi= 12 km s−1,vAe= 0 km s−1(i.e.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.\nWe have assumed the flow outside the tube to be zero ( vze= 0 km s−1). Note that the disper-\nsion relations, Eqs. (20) and (22), are symmetric under the e xchange ( ω,vz) with (−ω,−vz).\nTherefore, it is sufficient to consider only the positive valu es of flow velocity with both positive\nand negative values of oscillation frequency, i.e. forward and backward waves in the presence of\nupward plasma flow. Our numerical results are shown in Figs. 3 to 10.\nFigures3and4represent variations of thephasespeed(or no rmalized frequency) v/vsi≡ωr/ωsi,\nDoppler shifted phase speed Ω /ωsiand the damping rate −γ0c/ωr(γ0c/ωr) of the slow surface\nsausage modes for forward and backward waves versus kzRfor various flow parameters and var-\nious thickness of the inhomogeneous layer l/R= (0.1,0.2). The left panels of these figures clear\n11that for forward wave and various flow parameters vzi/vsi= (10−5,0.2,0.4,0.6,0.8) (i) The value\nof the phase speed v/vsiincreases with increasing the flow parameter vzi/vsi. (ii) The minimum\nvalue of the Doppler shifted phase speed decreases with incr easing the flow. (iii) The maximum\nvalue of−γ0c/ωrincreases, and for low flow parameter correspond to smaller kzRwhenvzi/vsi\nincreases but for high flow parameter correspond to larger kzRwhenvzi/vsiincreases. (iv) The\ndashed-line curves in these figures represent the analytica l results of the damping rate −γ0c/ωr\nevaluated by Eq. (40). These curves show that for the weak dam ping (i.e. γ0c≪ωr) and in\nthe long wavelength limit (i.e. kzR≪1) the oscillation frequency is not affected by the pres-\nence of the transitional layer. This is also confirmed by our n umerical results. (vi) For a given\nl/R, the minimum value of the damping time to period ratio τD/T= 2π/|γ0c|decreases with\nincreasing vzi/vsi. For instance, for the case where l/R= 0.1 andkzR= 1, the value of τD/T\nforvzi/vsi= 0.8 changes by ∼95% less than the case where there is no flow. So, the relation\nbetween the damping rate (time) and the flow is of interest. Se veral researcher obtained similar\nresults for the sausage modes in photospheric conditions. Y u et al. showed that for l/R= 0.1\nthe minimum value of the damping time to period ratio is τD/T= 14.11 [20] and [22] showed\nthat for l/R= 0.1 the minimum value of the damping time to period ratio is τD/T= 10.2 for\ntwist parameter Bφi/Bzi= 0.3, while our results show that the minimum value of the dampin g\ntime to period ratio forhigh upflow is much lower. vii) For kzR→0, we see that the damping\nrate go to zero for finite values of the flow parameter, and it is an agreement with analytical\nrelation Eq. (44).\nThe right panels in Figs. 3 and 4 we plot the phase speed (or nor malized frequency) v/vsi≡\nωr/ωsi, normalized Doppler Shifted Ω /ωsiand the damping rate γ0c/ωrof the slow surface\nsausagemodesforbackwardwaveversus kzRforvariousflowparameters vzi/vsi= (10−5,0.1,0.2,0.3)\nand various thickness of the inhomogeneous layer l/R= (0.1,0.2). The figures show that (i)\nthe magnitude of the phase velocity decreases with increasi ng flow. (ii) The magnitude of the\nDoppler shifted phase speed increases with increasing flow. (iii) The maximum value of γ0c/ωr\ndecreases, and it corresponds to smaller kzRwhenvzi/vsiincreases. (vi) For a given l/R, the\nminimum value of τD/Tincreases with increasing vzi/vsi. For instance, for the case where\nl/R= 0.1 andkzR= 1, the value of τD/Tforvzi/vsi= 0.3 changes by ∼238% more than the\ncase where there is no flow. Due to the fact that at high flow para meters for backward waves,\nDoppler shifted frequencies out of the resonant region, so i t is plotted up to a flow parameters\nof 0.3.\nFigures 5 and 6 show the variations of thephase speed(or norm alized frequency) v/vsi≡ωr/ωsi,\nphase Doppler Shifted Ω /ωsiand the damping rate −γ0c/ωr(γ0c/ωr) of the slow surface sausage\nmodes for forward and backward wave versus the inhomogeneou s layer (l/R) for various flow\nparameters and kzR= (0.5,2). The left panels of figures 5 and 6 show that for forward wave s\nand various flow parameters vzi/vsi= (10−5,0.2,0.4,0.6,0.8) (i) the frequency increases with in-\ncreasing the flow ( vzi/vsi). (ii) With increasing l/RforkzR≪1, the Doppler shifted frequency\nincreases, but for kzR≫1 the Doppler shifted frequency reaches a peak value then ten ds to\nvci/vsi. (iii) For kzR≪1, the Doppler shifted frequency decrease when the flow incre ases, for\nkzR≫1, when the Doppler shifted frequency reaches above vci/vsi, it decreases with increasing\nflow and tends to the value of vci/vsi. (iv) For a given kzR, the damping rate values increases\nand the damping time to period ratio values decreases with in creasing flow. For example, for\nkzR= 2 the minimum value of τD/Tforvzi/vsi= 0.8 decreases ∼93% with respect to the case\nwhere there is no flow.\nThe right panels of figures 5 and 6 show the variations of the ph ase speed (or normalized fre-\nquency) v/vsi≡ωr/ωsi, Doppler shifted phase speed Ω /ωsiand the damping rate γ0c/ωrof\nthe slow surface sausage modes for backward wave versus the i nhomogeneous layer ( l/R) for\n12various flow parameters vzi/vsi= (10−5,0.1,0.2,0.3) andkzR= (0.5,4). The figures show that\n(i) the magnitude of the phase velocity increases with incre asing flow. (ii) With increasing l/R\nforkzR≪1, the Doppler shifted frequency decreases, but for kzR≫1 the Doppler shifted\nfrequency reaches a minimum value then tends to −vci/vsi. (iii) For kzR≪1, the magnitude\nof the Doppler shifted frequency increase when the flow incre ases. For kzR≫1, the Doppler\nshifted frequency reaches −vci/vsi, it decreases with increasing flow and tends to the value of\n−vci/vsi. (iv) For a given kzR, the values of the damping rate decrease and the values of the\ndamping time to period ratio increase with increasing flow. F or example, for kzR= 2 the mini-\nmum value of τD/Tforvzi/vsi= 0.3 increases ∼278% than the case where there is no flow.\nWe plot the results for kink waves in Figs. 7 and 8. Same as the c ase of sausage modes in Figs.\n3 and 4 for the forward wave (the left panels) the maximum valu e of−γ1c/ωrincreases, and for\nlow flow parameter correspondto smaller kzRwhenvzi/vsiincreases but for high flow parameter\ncorrespond to larger kzRwhenvzi/vsiincreases. For a given l/R, the minimum value of τD/T\nincreases with increasing vzi/vsi. For instance, for the case where l/R= 0.2 the minimum value\nofτD/Tforvzi/vsi= 0.8 changes by ∼57% less than the case where there is no flow. Yu et al.\n[20] showed that for l/R= 0.2, a minimum value of τD/Tis about 18.8 but our result gives value\nabout 2.8. It is now that Soler et al. [46] have obtained this n umber about 1000 for l/R= 0.2.\nAlso for the backward wave (the right panels) the maximum val ue ofγ1c/ωrincreases, and its\nposition moves to smaller kzRwhenvzi/vsiincreases.\nFigures 9-10 are similar to Figs. 5-6 but for kink modes. The r esults show that the effect of flow\non the slow resonance absorption of sausage and kink modes is almost the same. The effect of\nthe slow resonance in the presence of flow on the wave damping i s significant under photospheric\nconditions.\nIt should be noted that for the case there is no flow, the result s are similar to the results of [20].\nWhen the flow is very small (i.e vzi/vsi= 10−5) the results overlap with the no flow case.\nFigure 11 shows the minimum value of damping time to period ra tio (τD/T) for the forward\nwave of the slow surface sausage (solid line) and kink (dashe d line) modes versus upflow veloc-\nity (vzi/vsi). This figure shows that when the upflow velocity increases, t he minimum value of\ndamping time to period ratio can be considerably reduced. Fo r instance, for the upflow velocity\nvaluevzi/vsi= 0.87, the damping time to period ratio of the surface sausage mo de will reach\nabout 0.30. This confirms that the resonant absorption in the presence of flow can be considered\nas an effective mechanism to justify the rapid damping of slow s urface sausage mode observed\nby [23]. Note that for all the results indicated in Fig. 11, th e longitudinal wave number is\nin the observational range i.e. kzR/lessorequalslant5. In the observational range, the minimum number of\noscillations increases slightly for large values of vzi/vsi.\n6 Conclusions\nIn this paper we studied the effect of the flow parameter on the fr equencies, the damping rates in\nslow continuum of slow sausage and kink waves in magnetic flux tubes under solar photospheric\n(or magnetic pore) conditions. We considered a straight cyl indrical flux tube with tree region\ninside, annulus and outside in which the linear density, squ ared magnetic field (linear pressure)\nand linear flow profiles are considered in the annulus region o r transitional layer. In addition,\nwe numerically solved the dispersion relation and obtained the phase speed (or normalized fre-\nquency) v/vsi≡ωr/ωsi, the normalized Doppler shifted frequency, the damping rat eγmc/ωr,\nand the damping time to period ratio τD/Tof the slow surface sausage and kink modes for\nforward and backward waves under photospheric (magnetic po re) conditions. Our results show\nthat:\n13•For forward waves, the frequency and the damping rate increa se when the flow parameter\nincreases but for backward waves, the frequencies and the da mping rate decreases when\nthe flow parameter increases.\n•For forward waves, the damping time to period ratio decrease s when the flow parameter\nincrease but for backward waves, the damping time to period r atio increase when the flow\nparameter increases.\n•For a given l/R, the Doppler shifted frequency, approach Ω /ωsi→vci/vsifor forward\nwaves and approach Ω /ωsi→ −vci/vsifor backward waves and γmc/ωr→0 for both\nforward and backward waves, in the long and short-wavelengt h limit.\n•For a given kzR, the maximum value of γmc/ωr(or minimum value of τD/T) increases (or\ndecreases) for forward waves and decreases (increases) for backward waves.\n•For the case where l/R= 0.1, the minimum value of τD/Tforvzi/vsi= 0.6, for instance,\nchanges∼89% less for forward sausage waves and for backward sausage w aves the mini-\nmum value of τD/Tforvzi/vsi= 0.3, changes ∼204% more with respect to the case where\nthere is no flow. Also, for kink mode changes ∼83% less for forward waves and ∼272%\nmore for backward waves with respect to the case where there i s no flow. According to\nthese results, it can be said that the flow has a significant effec t on the resonant absorption\nof the slow surface sausage and kink modes in magnetic flux tub es under magnetic pore\nconditions.\n•For the case of l/R= 0.1 andvzi/vsi= 0.87, the damping time to period ratio of the\nsurface sausage mode can reach τD/T= 0.30. For comparison, for a static tube (no flow)\nwithl/R= 0.1, [20] obtained τD/T= 14.11. This confirms that the resonant absorption\nin the presence of plasma flow can justify the extremely rapid damping of the slow surface\nsausage mode observed by [23].\nAppendices\nA Weak damping rate in long wavelength limit for the sausage\nmode\nFor the sausage mode m= 0, we have\n(48)Q0=I′\n0(x)K0(y)\nI0(x)K′\n0(y)\n=−I1(x)K0(y)\nI0(x)K1(y)\n≈xy(ln(y/2)+γe)\n2,\n14(49)G0=K0(y)\nK′\n0(y)\n=K0(y)\n−K1(y)\n≈−ln(y/2)−γe\n−1/y\n=y(ln(y/2)+γe),\n(50)P0=/parenleftBigg\nI′′\n0(x)\nI0(x)−I′\n0(x)2\nI0(x)2/parenrightBigg\nK0(y)\nK′\n0(y)\n≈y/parenleftbigg1\n2−3y2\n16/parenrightbigg\n(ln(y/2)+γe),\n(51)S0=/parenleftBigg\n1−K′′\n0(y)K0(y)\nK′\n0(y)2/parenrightBigg\nI′\n0(x)\nI0(x)\n≈x\n2(1+ln(y/2)+γe).\nInserting Equations (48)-(51) into Equation (41) yields\nT0=/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbigkri\nkre/parenleftBiggxyln(y)/parenleftBig\n1−3y2\n16/parenrightBig\n(Ω2\ni−2ω2\nci)Ω3\ni\n(ω2\nsi−Ω2\ni)(ω2\nAi−Ω2\ni)(Ω2\ni−ω2\nci)\n+xy(Ω2\ne−2ω2\nce)Ω3\ne\n2(ω2se−Ω2e)(ω2\nAe−Ω2e)(Ω2e−ω2ce)/parenrightBigg\n, (52)\nwhere ln( y/2) +γe= ln(y). In the limit kzR <<1 (Ωi≈ωci) above relation becomes singular.\nTo avoid singularity, we need to evaluate the quantity α. To this aim, following [20] we first\nreplace Ω2\ni=ω2\nci−αinto Eq. (16) 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, (53)\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 (20) in long w avelength limit ( kzR≪1) reads\nρi/parenleftbig\nΩ2\ni−ω2\nAi/parenrightbig\n−kri\nkreρe/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbigxyln(y)\n2= 0. (54)\nNow, replacing k2\nrifrom Eq. (53) into (54), the quantity αcan be obtained as follows\nα=χ\n2ω4\nci\nω4\nAi/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbig\nk2\nzR2ln(kzR), (55)\nwhere Ω e=ωci+kz(vzi−vze) and\nk2\nri=−2ω2\nAiω2\nci\nχω2\nsi/parenleftbig\nΩ2e−ω2\nAe/parenrightbig\nR2ln(kzR), (56)\n15T0=/parenleftBigg\nx2ln(y)ω5\nci/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbig\n(ω2\nsi−ω2\nci)(ω2\nAi−ω2\nci)α\n−3x4ln(y)ω5\nci/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbig\n16(ω2\nsi−ω2\nci)(ω2\nAi−ω2\nci)α\n+x2(Ω2\ne−2ω2\nce)Ω3\ne\n2(ω2se−Ω2e)(ω2\nAe−Ω2e)(Ω2e−ω2ce)/parenrightBigg\n. (57)\nNow, replacing Eqs. (55) and (56) into Eq. (57) we obtain\nT0=/parenleftBigg\n−4ω6\nAi\nχ2ωciω2\nsi/parenleftbig\nΩ2e−ω2\nAe/parenrightbig\nk2zR2\n−3ω8\nAiωciln(y)\n2χ3ω4\nsi/parenleftbig\nΩ2e−ω2\nAe/parenrightbig2k2zR2ln2(kzR)\n−ω4\nciω2\nAi(Ω2\ne−2ω2\nce)Ω3\ne\nχω2\nsi(ω2se−Ω2e)/parenleftbig\nΩ2e−ω2\nAe/parenrightbig\nln(kzR)/parenrightBigg\n, (58)\nand Finally we reach\nT0=−3ω8\nAiω2\nci+8χω2\nsiω6\nAi/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbig\nln(kzR)\n2χ3ωciω4\nsi/parenleftbig\nΩ2e−ω2\nAe/parenrightbig2k2zR2ln2(kzR), (59)\nsubstituting Eq. (59) in (40) we have\nγ0c=−πρek2\nz\nkreSignΩ\nρc|∆c|/vextendsingle/vextendsingle/vextendsingle\nr=rc/parenleftBig\nv2\ns\nv2\nA+v2s/parenrightBig2\n(ω2\nci−ω2\nAi)(Ω2\ne−ω2\nAe)kzR\nχ3ω8\nAiω2\nci+8χω2\nsiω6\nAi(Ω2e−ω2\nAe)ln(kzR)\n2χ3ωciω4\nsi(Ω2e−ω2\nAe)2k2zR2ln2(kzR), (60)\nand after some algebra we get\nγ0c=2πχ3Sign Ω\n|∆c|R/bracketleftBigg\nω7\nciω2\nsi/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbig3\n3ω10\nAiω2\nci+8χω8\nAiω2\nsi/parenleftbig\nΩ2e−ω2\nAe/parenrightbig\nln(kzR)/bracketrightBigg\n(kzR)4ln3(kzR).(61)\nB Weak damping rate in long wavelength limit for the kink\nmode\nFor the kink mode m= 1, we have\n(62)Q1=I′\n1(x)K1(y)\nI1(x)K′\n1(y)\n=−K1(y)(I0(x)+I2(x))\nI1(x)(K0(y)+K2(y))\n≈ −/parenleftBigy\nx+xy\n4/parenrightBig\n,\n16(63)G1=K1(y)\nK′\n1(y)\n=−2K1(y)\nK0(y)+K2(y)\n≈1\ny−1\n4+1\n2(ln(y/2)+γe)\n−1\ny2+1\n4+1\n2(ln(y/2)+γe)\n=−y,\n(64)P1=/parenleftBigg\nI′′\n1(x)\nI1(x)−I′\n1(x)2\nI1(x)2/parenrightBigg\nK1(y)\nK′\n1(y)\n≈ −y/parenleftbigg1\n4−1\nx2/parenrightbigg\n,\n(65)S1=/parenleftBigg\n1−K′′\n1(y)K1(y)\nK′\n1(y)2/parenrightBigg\nI′\n1(x)\nI1(x)\n≈ −1+(1+3ln( y))y2\nx.\nInserting Eqs. (62)-(65) into (41) yields\nT1=/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbigkri\nkre/parenleftBigg/parenleftbig\n−/parenleftbigy\nx+xy\n4/parenrightbig\n−xy/parenleftbig1\n4−1\nx2/parenrightbig/parenrightbig\n(Ω2\ni−2ω2\nci)Ω3\ni\n(ω2\nsi−Ω2\ni)(ω2\nAi−Ω2\ni)(Ω2\ni−ω2\nci)\n−/parenleftBig\n−/parenleftbigy\nx+xy\n4/parenrightbig\n+y1+(1+3ln( y))y2\nx/parenrightBig\n(Ω2\ne−2ω2\nce)Ω3\ne\n(ω2se−Ω2e)(ω2\nAe−Ω2e)(Ω2e−ω2ce)/parenrightBigg\n.(66)\nFor Ω2\ni=ω2\nci−α, we obtain\nT1=/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbig/parenleftBigg\n−x2ω5\nci\n2(ω2\nsi−Ω2\ni)(ω2\nAi−Ω2\ni)α\n−/parenleftBig\n−x2\n4+(1+3ln( y))y2/parenrightBig\n(Ω2\ne−2ω2\nce)Ω3\ne\n(ω2se−Ω2e)(ω2\nAe−Ω2e)(Ω2e−ω2ce)/parenrightBigg\n. (67)\nIn the next, the dispersion relation (20) in long wavelength limit (kzR≪1) form= 1 reads\nρi/parenleftbig\nω2\nci−ω2\nAi/parenrightbig\n+ρe/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbig/bracketleftbigg\n1+x2\n4/bracketrightbigg\n= 0, (68)\nnow, replacing k2\nrifrom Eq. (53) into (54), the quantity αcan be obtained as follows\nα=χ\n4ω6\nci/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbig\nω2\nciω4\nAi−χω2\nsiω2\nAi/parenleftbig\nΩ2e−ω2\nAe/parenrightbigk2\nzR2, (69)\nk2\nri=4\nχω2\nciω2\nAi−χω2\nsi/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbig\nω2\nsi/parenleftbig\nΩ2e−ω2\nAe/parenrightbig\nR2, (70)\n17putting Eqs. (69) and (70) in (67) and keeping only the senten ce proportional to the sentence\n1\nk2zR2we obtain\nT1=8ω2\nsiω2\nAi/parenleftbig\nΩ2\ne−ω2\nAe/parenrightbig\nω5\ncik2zR2/parenleftBigg\nω2\nciω2\nAi\nχω2\nsi/parenleftbig\nΩ2e−ω2\nAe/parenrightbig−1/parenrightBigg2\n. (71)\nIn the following with the help of Eq. 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L., & Goossens, M. 2 009, ApJL, 695, L166\n200 2 4 6 8 10\nkz R0.80.911.11.21.31.41.51.61.71.8v/vsiForward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(a)0 2 4 6 8 10\nkz R-0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55v/vsiBackward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(b)\n0 2 4 6 8 10\nkzR0.450.50.550.60.650.70.750.80.850.90.95/siForward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(c)0 2 4 6 8 10\nkzR-0.868-vci-0.86-0.855-0.853/siBackward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(d)\n0 2 4 6 8 10\nkz R00.050.10.150.20.25-0c/rForward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(e)0 2 4 6 8 10\nkz R00.0020.0040.0060.0080.010.0120c/rBackward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(f)\nFigure 3: The left panels are for the forward sausage waves an d the diagrams in (a), (b) and\n(c) represent the phase speed v/vsi≡ωr/ωsi, the Doppler Shifted phase speed Ω /��siand the\ndamping rate −γ0c/ωras functions of kzRfor various values of plasma flow. The right panels\nare the same as the left panels for the backward sausage waves . For the damping rate the\ndashed curves represent the analytical solutions determin ed from Eq. (40). The dashed curves\nin the other diagrams show the results obtained in the case of no boundary layer i.e. Eq. (20).\nOther parameters of the tube are l/R= 0.1,vAi= 12 km s−1,vAe= 0 km s−1(i.e.Bze= 0),\nvze= 0 km s−1,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.\n210 2 4 6 8 10\nkz R0.80.911.11.21.31.41.51.61.71.8v/vsiForward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(a)0 2 4 6 8 10\nkz R-0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55v/vsiBackward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(b)\n0 2 4 6 8 10\nkzR0.650.70.750.8vci/vsi0.90.95/siForward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(c)0 2 4 6 8 10\nkzR-0.866-vci-0.86-0.855/siBackward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(d)\n0 2 4 6 8 10\nkz R00.020.040.060.080.10.120.140.160.180.2-0c/rForward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(e)0 2 4 6 8 10\nkz R00.0010.0020.0030.0040.0050.0060.0070.0080.0090.010c/rBackward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(f)\nFigure 4: Same as Fig. 3 , but for l/R= 0.2.\n220 0.5 1 1.5 2\nl/R0.80.911.11.21.31.41.51.61.7v/vsiForward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=1.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(a)0 0.5 1 1.5 2\nl/R-0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55v/vsiBackward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(b)\n0 0.5 1 1.5 2\nl/R0.810.820.830.840.85vci/vsi0.87/siForward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=1.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(c)0 0.5 1 1.5 2\nl/R-0.864-vci-0.863-0.8625-0.862-0.8615-0.861-0.8605-0.86/siBackward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(d)\n0 0.5 1 1.5 2\nl/R00.0020.0040.0060.0080.010.0120.0140.0160.018-0c/rForward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=1.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(e)0 0.5 1 1.5 2\nl/R00.511.522.530c/r10-3 Backward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(f)\nFigure 5: The left panels are for the forward sausage waves an d the diagrams in (a), (b) and\n(c) represent the phase speed v/vsi≡ωr/ωsi, the Doppler Shifted phase speed Ω /ωsiand the\ndamping rate −γ0c/ωras functions of l/Rfor various values of plasma flow. The right panels\nare the same as the left panels for the backward sausage waves . For all panels we have assumed\nkzR= 0.5, other parameters are the same as Fig. 3.\n230 0.5 1 1.5 2\nl/R0.80.911.11.21.31.41.51.61.71.8v/vsiForward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(a)0 0.5 1 1.5 2\nl/R-0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55v/vsiBackward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(b)\n0 0.5 1 1.5 2\nl/R0.70.750.80.8637vci/vsi0.90.95/siForward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=1.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(c)0 0.5 1 1.5 2\nl/R-0.866-vci-0.862-0.86-0.858-0.856-0.854-0.852/siBackward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(d)\n0 0.5 1 1.5 2\nl/R00.020.040.060.080.10.120.14-0c/rForward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(e)0 0.5 1 1.5 2\nl/R01234567890c/r10-3 Backward Sausage Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(f)\nFigure 6: Same as Fig. 5 , but for kzR= 2.\n240 2 4 6 8 10\nkz R0.80.911.11.21.31.41.51.61.71.8v/vsiForward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(a)0 2 4 6 8 10\nkz R-0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55v/vsiBackward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(b)\n0 2 4 6 8 10\nkzR0.70.750.8vci/vsi0.90.95/siForward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(c)0 2 4 6 8 10\nkzR-0.868-0.866-vci-0.862-0.86-0.858-0.856/siBackward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(d)\n0 2 4 6 8 10\nkz R00.020.040.060.080.10.120.140.16-1c/rForward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(e)0 2 4 6 8 10\nkz R00.0020.0040.0060.0080.010.0121c/rBackward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(f)\nFigure 7: The left panels are for the forward kink waves and th e diagrams in (a), (b) and\n(c) represent the phase speed v/vsi≡ωr/ωsi, the Doppler Shifted phase speed Ω /ωsiand the\ndamping rate −γ0c/ωras functions of kzRfor various values of plasma flow. The right panels\nare the same as the left panels for the backward kink waves. Fo r the damping rate the dashed\ncurves represent the analytical solutions determined from Eq. (40). The dashed curves in the\nother diagrams show the results obtained in the case of no bou ndary layer i.e. Eq. (20). For all\npanels we have assumed l/R= 0.1, other parameters are the same as Fig. 3.\n250 2 4 6 8 10\nkz R0.811.21.41.61.8v/vsiForward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(a)0 2 4 6 8 10\nkz R-0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55v/vsiBackward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(b)\n0 2 4 6 8 10\nkzR0.80.820.84vci/vsi0.880.90.92/siForward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(c)0 2 4 6 8 10\nkzR-0.87-0.865-0.86-0.855/siBackward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(d)\n0 2 4 6 8 10\nkz R00.010.020.030.040.050.06-1c/rForward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(e)0 2 4 6 8 10\nkz R00.0010.0020.0030.0040.0050.0060.0070.0080.0090.011c/rBackward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(f)\nFigure 8: Same as Fig. 7 , but for l/R= 0.2\n260 0.5 1 1.5 2\nl/R0.80.911.11.21.31.41.51.61.7v/vsiForward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(a)0 0.5 1 1.5 2\nl/R-0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55v/vsiBackward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(b)\n0 0.5 1 1.5 2\nl/R0.8550.8560.8570.8580.8590.860.8610.862vci/vsi/siForward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(c)0 0.5 1 1.5 2\nl/R-0.864-vci-0.8635-0.863-0.8625-0.862-0.8615/siBackward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(d)\n0 0.5 1 1.5 2\nl/R00.511.522.53-1c/r10-3 Forward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(e)0 0.5 1 1.5 2\nl/R00.511.51c/r10-3 Backward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(f)\nFigure 9: The left panels are for the forward kink waves and th e diagrams in (a), (b) and\n(c) represent the phase speed v/vsi≡ωr/ωsi, the Doppler Shifted phase speed Ω /ωsiand the\ndamping rate −γ0c/ωras functions of l/Rfor various values of plasma flow. The right panels\nare the same as the left panels for the backward kink waves. Fo r all panels we have assumed\nkzR= 0.5, other parameters are the same as Fig. 3.\n270 0.5 1 1.5 2\nl/R0.80.911.11.21.31.41.51.61.7v/vsiForward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(a)0 0.5 1 1.5 2\nl/R-0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55v/vsiBackward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(b)\n0 0.5 1 1.5 2\nl/R0.70.720.740.760.780.80.820.84vci/vsi0.88/siForward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(c)0 0.5 1 1.5 2\nl/R-0.866-vci-0.862-0.86-0.858-0.856-0.854-0.852/siBackward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(d)\n0 0.5 1 1.5 2\nl/R00.010.020.030.040.050.06-1c/rForward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.2\nvzi/vsi=0.4\nvzi/vsi=0.6\nvzi/vsi=0.8\n(e)0 0.5 1 1.5 2\nl/R0123456781c/r10-3 Backward Kink Waves\nvzi/vsi=0\nvzi/vsi=10-5\nvzi/vsi=0.1\nvzi/vsi=0.2\nvzi/vsi=0.3\n(f)\nFigure 10: Same as Fig. 9 , but for kzR= 2.\n280 0.2 0.4 0.6 0.8 1\nvzi/vsi0.1151015D/TForward Waves\nSausage\nKink\nFigure 11: The minimum value of the damping time to period rat io (τD/T) for the forward\nwaves including the slow surface sausage (solid line) and ki nk (dashed line) modes versus upflow\nvelocity ( vzi/vsi) forl/R= 0.1. Here, we have kzR≤5 and other parameters are the same as\nFig. 3.\n29" }, { "title": "2010.05650v1.Line_drag_damping_of_Alfvén_waves_in_radiatively_driven_winds_of_magnetic_massive_stars.pdf", "content": "MNRAS 000, 1–11 (2020) Preprint 13 October 2020 Compiled using MNRAS L ATEX style file v3.0\nLine-drag damping of Alfvén waves in radiatively driven winds of\nmagnetic massive stars\nF. A. Driessen1★, N. D. Kee1, J. O. Sundqvist1, and S. P. Owocki2\n1Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium\n2Bartol Research Institute, Department of Physics & Astronomy, University of Delaware, Newark, DE 19716, USA\nAccepted XXX. Received YYY; in original form ZZZ\nABSTRACT\nLine-driven stellar winds from massive (OB) stars are subject to a strong line-deshadowing\ninstability.Recently,spectropolarimetricsurveyshavecollectedampleevidencethatasubsetof\nGalacticmassivestarshostsstrongsurfacemagneticfields.Weinvestigateherethepropagation\nandstabilityofmagneto-radiativewavesinsuchamagnetised,line-drivenwind.Ouranalytic,\nlinear stability analysis includes line-scattering from the stellar radiation, and accounts for\nbothradialandnon-radialperturbations.Weestablishabridginglawforarbitraryperturbation\nwavelength after which we analyse separately the long- and short-wavelength limits. While\nlong-wavelength radiative and magnetic waves are found to be completely decoupled, a key\nresult is that short-wavelength, radially propagating Alfvén waves couple to the scattered\nradiation field and are strongly damped due to the line-drag effect. This damping of magnetic\nwavesinascattering-line-drivenflowcouldhaveimportanteffectsonregulatingthenon-linear\nwind dynamics, and so might also have strong influence on observational diagnostics of the\nwind structure and clumping of magnetic line-driven winds.\nKeywords: radiativetransfer–waves–instabilities–stars:early-type–stars:magneticfield\n– stars: winds, outflows\n1 INTRODUCTION\nThepowerfulradiationfrommassive(OB)starsisabletotransferits\nmomentum to the stellar wind plasma by absorption and scattering\nin spectral lines. A quantitative description of this line-driving has\nbeen provided by the seminal work of Castor, Abbott & Klein\n(1975, hereafter CAK). In formulating this nowadays widely used\ntheory, CAK relied on the Sobolev approximation for describing\ntheradiativeaccelerationofthewind.Thisapproximationassumes\nthat in a highly supersonic outflow the spectral line transport can\nbe described based on purely localinformation (Sobolev 1960)\nmeaning that the hydrodynamic flow quantities are constant over a\nSobolevlength 𝐿Sob\u0011𝑣th¹𝑑𝑣𝑛𝑑𝑛º(with𝑣ththeionthermalspeed\nand𝑑𝑣𝑛𝑑𝑛theprojectedvelocitygradientinradiationdirection 𝑛).\nIt turns out that the line-driving mechanism is subject to a ra-\ndiativeinstabilityasfirstsuggestedbyLucy&Solomon(1970).This\nvery strong, intrinsic instability is known as the line-deshadowing\ninstability (LDI). MacGregor, Hartmann & Raymond (1979) and\nCarlberg (1980) performed a linear stability analysis under the as-\nsumption of optically thin perturbations finding an unstable wind\nduetotheLDI.Ontheotherhand,however,Abbott(1980)assumed\nthatperturbationsfollowtheSobolevapproximationandfoundthat\n★E-mail: florian.driessen@kuleuven.bethe instability vanishes. Instead, a stable, inward propagating wave\n(Abbott wave) arises.\nThesecontradictoryresultswerereconciledbyOwocki&Ry-\nbicki(1984,hereafterOR84)showingthatbothcasesoccurdepend-\ning on perturbation wavelength, i.e. short-wavelength (instability)\nandlong-wavelength(waves).TheiranalysisillustratesthattheLDI\nis inherently acting on spatial scales below or near the Sobolev\nlength. This naturally explains why the LDI does not occur in dy-\nnamical CAK models that rely on the Sobolev approximation to\ncompute the radiative acceleration. Later analytic work performed\nby Owocki & Rybicki extended these results to clarify the effects\nofline-scatteringonthegrowthrateoftheLDI(Owocki&Rybicki\n1985, hereafter OR85), what the spatial and temporal evolution of\nthe LDI is (Owocki & Rybicki 1986), the effect of non-radial per-\nturbations on the growth rates (Rybicki, Owocki & Castor 1990,\nhereafter ROC90), and the growth rates in flows that have an opti-\ncally thick continuum (Owocki & Rybicki 1991).\nA calculation of OR84 shows that the LDI quickly attains a\nnon-linear regime, having a typical growth rate on the order of\n𝑒100(includingline-scatteringthisbecomes 𝑒50)inatypicalO-star\nwind.Subsequentnumericalsimulationsofthenon-linearevolution\nof the LDI (e.g. Owocki, Castor & Rybicki 1988; Owocki & Puls\n1999;Feldmeier&Thomas2017;Sundqvist,Owocki&Puls2018;\nDriessen, Sundqvist & Kee 2019) have shown that the wind frag-\nmentsandformslarge-scaleslow,overdense, clumpystructuresthat\n©2020 The AuthorsarXiv:2010.05650v1 [astro-ph.SR] 12 Oct 20202 F. A. Driessen, N. D. Kee, J. O. Sundqvist, and S. P. Owocki\nare separated by a fast, nearly void medium and is quite different\nfrom the homogeneous winds predicted by CAK.\nThe cause of large-scale structures seen in non-linear numeri-\ncal simulations of the LDI is complex. The initial instability is due\nto short-wavelength perturbations as predicted by linear analysis.\nRecently, Feldmeier & Thomas (2017) suggested that in the accel-\nerating wind additionally these wave perturbations get coherently\nstretched over many Sobolev lengths thereby allowing the non-\nlineargrowthoftheLDItoproceedtohighamplitudes.Thisoverall\nprocess would lead then to the typical wind structure of clumps\nseparated by a nearly void medium in numerical simulations.\nObservations of line-driven winds from massive stars have\nindeedprovidedevidenceofclumpystructures.Forexample,innon-\nmagnetic massive star winds it is well known that the LDI leads to\nsignificant wind clumping , with important effects on observational\ndiagnostics(e.g.,Puls,Sundqvist&Markova2015,foranoverview).\nThis has lead to a basic understanding of observed phenomena\nsuch as soft X-ray emission (Berghoefer et al. 1997, Feldmeier,\nPuls & Pauldrach 1997), extended regions of zero residual flux in\nresonanceUVspectrallines(Lucy1983;Sundqvist,Owocki&Puls\n2012), and migrating subpeaks in optical recombination spectral\nlines(Eversberg,Lépine&Moffat1998;Dessart&Owocki2002).\nItisimportanttorecognizethattheabovetheoreticalandobser-\nvationalanalyseshaveallbeencarriedoutfornon-magneticmassive\nstar winds. Yet, two decades ago spectropolarimetric observations\nhavemadeclearthatsomemassivestarsinourGalaxyaremagnetic\n(Donati et al. 2002). Over the years dedicated spectropolarimetric\nsurveys of massive stars in our Galaxy (Fossati et al. 2015 (BOB);\nWadeetal.2016(MiMeS))havegatheredampleobservationalevi-\ndencethatamodestfraction( \u001810%)ofmassivestarsharbourstrong\n(\u0018kG), organised (often dipolar) surface magnetic fields. The oc-\ncurrence of such surface magnetic fields is none the less puzzling\nbecausemassivestarslackanysubsurfacehydrogen/heliumrecom-\nbinationzonesthatarethoughttogeneratedynamomagneticfields\nas, for example, is the case for the Sun. The accepted view is that\nthesurfacemagneticfieldsareofprimordialorigin,butthephysical\nmechanism that can explain the generation and incidence of these\nfields is not yet settled (e.g., see Schneider et al. 2019, for recent\ninsights).\nPrevioustheoreticalworkonline-drivenwindsfrommagnetic\nmassive stars studied already the dynamics in various settings (ud-\nDoula&Owocki2002;ud-Doulaetal.2006,2008).Amajorresult\nfrom these studies is that the stellar wind is channeled along the\nmagnetic field lines forming a so-called circumstellar magneto-\nspherethat can significantly alter the mass-loss rate of the stars.\nHowever, all these theoretical works applied CAK theory to study\nthe global wind dynamics in these magnetic environments. As dis-\ncussed above, this means the dynamics of the LDI is neglected in\nthosemodels.Reasonsofomissionhavelargelybeenduetothesig-\nnificant computational complexity of non-linear LDI simulations\nalready for non-magnetic line-driven winds. With the increase in\ndetections of magnetic massive stars, however, it has become im-\nportant to undertake an investigation of the interplay between the\nLDI and stellar magnetic fields.\nFrom a theoretical point of view several questions arise on\nwhattheinfluenceisofthemagneticfieldonthedevelopmentofthe\ninstability. In environments with sufficiently strong magnetic fields\nthe fluid flow will be confined to the magnetic field lines. Such\nphysics might severely constrain, or even prevent, any horizontal\nfragmentation of matter, thereby reducing the clumpiness of the\nwind. If true, this would imply that the typical structure of dense,\nclumps separated by a nearly void medium might be replaced bylarge, dense shells of matter separated by a nearly void medium.\nAnother specific question to address is how the additional waves\nassociated with the magnetic field behave in the wind: What are\ntheir propagation properties? How do they couple to the radiation?\nAnd,mostimportantly,dotheyamplifyorattenuateanyhorizontal\nfragmentation of matter?\nAs a first step, in the present paper we perform an analysis\nof linear perturbations in a magnetic line-driven wind. An initial\nunderstanding of the magneto-radiative wave propagation and in-\nstabilitygrowthcanbegainedinthe linearregimeusinganalytical\ntools.Thisservestointerpretfuture non-linear magneticnumerical\nsimulations of the LDI.\n2 THEORETICAL FORMULATION\n2.1 Assumptions\nToreducethecomplexityoftheproblemweadoptasetofassump-\ntions to make calculations tractable while still retaining the core\nphysics.Thisisdonebothforthestellarwindplasmaandtheradia-\ntionfieldbytreatingalocalvolumeofwindplasmaabovemagnetic\npole of a stellar dipole field.\nWe take standard OB star wind assumptions for the back-\nground, mean wind and neglect gas pressure gradient and contin-\nuum opacities. The mean magnetic field above the pole is taken to\nbepurelyradialwhileradiationfromapointstarisradiallystream-\ning and interacts with one isolated pure scattering line for linear\nstability. The line-scattering is taken to be isotropic with complete\nfrequency redistribution. All perturbations are linear and act on\nWKB order, i.e. perturbation wavelengths are much smaller than\nthe typical spatial scales of the wind.\nRecall that in a magnetic flow the basic modes are due to\nmagnetic tension (Alfvén mode) and magnetic pressure (fast and\nslow mode) from the magnetic field line. In the absence of ther-\nmal pressure (zero-sound-speed limit), however, the slow mode\nvanisheswhilethefastmodepropagatesisotropicallyattheAlfvén\nspeed.Therefore,whatremainsinourpresentproblemareradiative,\nAlfvén, and fast modes.\nOur assumption of a point star and the omission of thermal\npressure have additional consequences for the nature of the flow\ncriticalpoints(Abbott1980).However,wedonotanalysethesolu-\ntion topology in this work, and as such here these assumptions are\nrelatively minor.\n2.2 Magnetohydrodynamics of line-driven winds\nWefirstpresentastandardlinearperturbationanalysisforthemag-\nneticflowinthepresenceofaradiativeforce.Readersfamiliarwith\nthis may wish to proceed to §2.3 where we discuss the essential\nbackground on the radiation field.\n2.2.1 Governing equations\nWe adopt the equations of ideal magnetohydrodynamics (MHD)\n(appropriate for a highly ionised medium like a line-driven wind)\nto describe the magnetic wind dynamics. The energy equation is\nomitted by assuming an isothermal wind. Since thermal pressure\nis neglected in the present paper no pressure dependencies arise.\nMNRAS 000, 1–11 (2020)Alfvén wave damping in magnetic line-driven winds 3\nFurthermore,wedescribeline-drivingbasedontheradiativeaccel-\neration due to a single line and include gravity\n𝜕𝜌\n𝜕𝑡¸𝜌𝜕𝑣𝑖\n𝜕𝑟𝑖¸𝑣𝑖𝜕𝜌\n𝜕𝑟𝑖=0 (1)\n𝜌\u0012𝜕𝑣𝑖\n𝜕𝑡¸𝑣𝑗𝜕𝑣𝑖\n𝜕𝑟𝑗\u0013\n=\u0000𝜌𝐺𝑀 eff𝑟𝑖\n𝑟3¸𝜌𝑔𝑖¸1\n4𝜋𝜖𝑖𝑗𝑘𝜖𝑗𝑙𝑚𝜕𝐵𝑚\n𝜕𝑟𝑙𝐵𝑘(2)\n𝜕𝐵𝑖\n𝜕𝑡=𝜖𝑖𝑗𝑘𝜖𝑘𝑙𝑚𝜕\n𝜕𝑟𝑗¹𝑣𝑙𝐵𝑚º (3)\n𝜕𝐵𝑖\n𝜕𝑟𝑖=0 (4)\nwith𝜌the density,𝑣𝑖the gas velocity, and 𝐵𝑖the magnetic field\n(spatial coordinate indices 𝑖run from 1!3). Furthermore, 𝐺is\nthe gravitational constant, 𝑀eff=¹1\u0000Γ𝑒º𝑀★the effective stel-\nlar mass reduced by the effect of (constant) electron scattering in\nthewinddescribedbyEddington’sgamma Γ𝑒=𝜅𝑒𝐿★¹4𝜋𝐺𝑀★𝑐º\nwith electron scattering opacity 𝜅𝑒=034cm2g\u00001, and𝜖is the\nthree-dimensional Levi–Civita tensor. All fluid quantities are as-\nsumed to be a function of position 𝑟𝑖and time𝑡. In the momentum\nequationthesourcetermsaregravity,theradiativeaccelerationdue\nto lines𝑔𝑖, and the Lorentz force from the magnetic field, respec-\ntively. The only unknown term here is 𝑔𝑖for which one could use,\ne.g.theSobolevapproximationforthemeanflow,butasalludedin\ntheIntroductionthisisnotvalidonthesmallspatialscaleswherethe\nLDIoperates.Determiningthissourcetermforlinearperturbations\nis therefore important, and is discussed in §2.3.\n2.2.2 Linear perturbations in three dimensions\nWe apply small (linear) perturbations on the above ideal MHD\nequations. We write the variables such that they consist of an un-\nperturbed, steady-state part (subscript 0) and a perturbed, evolving\npart(denotedbyaleading 𝛿).Bydefinitionaperturbedquantity 𝛿𝑞\nhas to satisfyj𝛿𝑞j𝑞0\u001c1. This leads to the following definitions\n𝜌\u0011𝜌0¹𝑟𝑗º¸𝛿𝜌¹𝑟𝑗𝑡º𝑣𝑖\u0011𝑣𝑖0¹𝑟𝑗º¸𝛿𝑣𝑖¹𝑟𝑗𝑡º\n𝐵𝑖\u0011𝐵𝑖0¹𝑟𝑗º¸𝛿𝐵𝑖¹𝑟𝑗𝑡º 𝑔𝑖\u0011𝑔𝑖0¹𝑟𝑗º¸𝛿𝑔𝑖¹𝑟𝑗𝑡º(5)\nFromourlocal,WKBorderanalysisdensitystratificationsare\nunderstood to be small such that perturbations on gravity can be\nneglected (this is valid if perturbations are much smaller than the\ndensity scale height). Substituting these expressions into (1) – (4)\nand keeping terms up to first order gives on WKB order the MHD\nperturbed state\n𝜕𝛿𝜌\n𝜕𝑡¸𝜌0𝜕𝛿𝑣𝑖\n𝜕𝑟𝑖=0 (6)\n𝜌0𝜕𝛿𝑣𝑖\n𝜕𝑡=𝜌0𝛿𝑔𝑖¸1\n4𝜋𝜖𝑖𝑗𝑘𝜖𝑗𝑙𝑚𝜕𝛿𝐵𝑚\n𝜕𝑟𝑙𝐵𝑘0 (7)\n𝜕𝛿𝐵𝑖\n𝜕𝑡=𝜖𝑖𝑗𝑘𝜖𝑘𝑙𝑚𝜕\n𝜕𝑟𝑗¹𝛿𝑣𝑙𝐵𝑚0º (8)\n𝜕𝛿𝐵𝑖\n𝜕𝑟𝑖=0 (9)\nin a frame locally comoving with the flow. On WKB order a lo-\ncal spherical coordinate system is indistinguishable from a localCartesian coordinate system. Therefore, the remaining analysis of\nthe perturbed state uses Cartesian 𝑥𝑗in place of spherical 𝑟𝑗.\nThe perturbed MHD equations (6) – (9) are linear and have\nconstant coefficients. This implies that on an unbounded domain\ntheir solution can be decomposed into plane waves\n©\n«𝛿𝜌¹𝑥𝑗𝑡º\n𝛿𝑣𝑖¹𝑥𝑗𝑡º\n𝛿𝐵𝑖¹𝑥𝑗𝑡º\n𝛿𝑔𝑖¹𝑥𝑗𝑡ºª®®®\n¬=©\n«𝛿˜𝜌\n𝛿˜𝑣𝑖\n𝛿˜𝐵𝑖\n𝛿˜𝑔𝑖ª®®®\n¬𝑒𝑖¹𝑘𝑗𝑥𝑗\u0000𝜔𝑡º (10)\nwith𝛿˜𝜌,𝛿˜𝑣,𝛿˜𝐵, and𝛿˜𝑔the real amplitude of the corresponding\nperturbed variable (from now on the tilde will be dropped). The\nwaveisdescribedbyawavevector 𝑘𝑗andawavefrequency 𝜔.For\nreal𝑘𝑗,𝜔canbereal,imaginary,orcomplex: 𝜔\u0011𝜔𝑅¸𝑖𝜔𝐼.Under\ntheplanewaveAnsatztherealpart 𝜔𝑅yieldsthewavepropagation\nspeed (phase speed) while the imaginary part 𝜔𝐼is the temporal\ngrowthrateandsignifieswavegrowth ¹instability𝜔𝐼¡0ºorwave\ndamping¹𝜔𝐼0º.\nWith plane waves the partial differential equations of the per-\nturbed MHD state (6) – (9) reduce to a set of algebraic equations\n\u0000𝑖𝜔𝛿𝜌¸𝑖𝜌0𝑘𝑖𝛿𝑣𝑖=0 (11)\n\u0000𝑖𝜔𝜌 0𝛿𝑣𝑖=𝜌0𝛿𝑔𝑖¸𝑖1\n4𝜋¹𝐵𝑗0𝑘𝑗𝛿𝐵𝑖\u0000𝐵𝑗0𝛿𝐵𝑗𝑘𝑖º (12)\n\u0000𝑖𝜔𝛿𝐵𝑖=𝑖\u0000𝑘𝑗𝐵𝑗0𝛿𝑣𝑖\u0000𝑘𝑗𝛿𝑣𝑗𝐵𝑖0\u0001 (13)\n𝑖𝑘𝑖𝛿𝐵𝑖=0 (14)\nwhere we have used the relation 𝜖𝑖𝑗𝑘𝜖𝑘𝑙𝑚=𝛿𝑖𝑙𝛿𝑗𝑚\u0000𝛿��𝑚𝛿𝑗𝑙to\nexpress the Levi–Civita tensor in terms of the Kronecker delta.\nThe 1st equation (perturbed continuity) is decoupled from the\nother equations. Likewise the 4th equation (perturbed divergence-\nfree condition) provides a constraint on 𝛿𝐵𝑖. The latter reads that\nwaves propagating along the mean magnetic field 𝐵𝑖will result in\na perturbed magnetic field perpendicular to the wave propagation.\nThe 2nd (momentum) and 3rd (induction) equation are coupled to\neachother.Toinvestigatethestabilitypropertiesoftheflowitisben-\neficial to work in a perturbed velocity representation. Additionally,\nthis naturally expresses what the linear response of the radiative\nacceleration is due to a velocity perturbation, i.e. it accounts for\nDoppler shifts,\n(\u0014\n𝜔2\u00001\n4𝜋𝜌0¹𝑘𝑙𝐵𝑙0º2\u0015\n𝛿𝑖𝑗\u0000𝑖𝜔𝛿𝑔𝑖\n𝛿𝑣𝑗\n¸1\n4𝜋𝜌0h\n¹𝑘𝑙𝐵𝑙0º\u0000𝑘𝑖𝐵𝑗0¸𝐵𝑖0𝑘𝑗\u0001\u0000𝐵2\n0𝑘𝑖𝑘𝑗i)\n𝛿𝑣𝑗=0(15)\nor the equivalent eigenproblem\nM𝑖𝑗𝛿𝑣𝑗=0 (16)\nOnce the velocity perturbation is set, the eigenproblem estab-\nlishes a dispersion relation between 𝜔and𝑘𝑗and allows to study\nwavepropagationandinstabilitygrowth.Infullgenerality M𝑖𝑗isa\nrather involved eigentensor, preventing a consideration of its com-\nplete representation. Instead, we here invoke our assumption of a\nradialmeanmagneticfieldchosentopointalongthe 𝑥3-axis(𝑧-axis)\nofthelocalCartesiancoordinatesystem.Withoutlossofgenerality,\nMNRAS 000, 1–11 (2020)4 F. A. Driessen, N. D. Kee, J. O. Sundqvist, and S. P. Owocki\nwe consider an arbitrary wave propagation in the two-dimensional\n𝑥1-𝑥3planearoundtheradialaxis,i.e. 𝑘1=𝑘sin𝜃and𝑘3=𝑘cos𝜃,\nwith𝜃the angle between the wave vector and the mean magnetic\nfield. Symmetry properties make the physical interpretation in the\n𝑥2-𝑥3plane the same. Under these simplifications the eigentensor\nbecomes\nM11=𝜔2\u0000𝑣2\n𝐴¹𝑘2\n1¸𝑘2\n3º\u0000𝑖𝜔𝛿𝑔1\n𝛿𝑣1\nM22=𝜔2\u0000𝑣2\n𝐴𝑘2\n3\u0000𝑖𝜔𝛿𝑔2\n𝛿𝑣2\nM33=𝜔2\u0000𝑖𝜔𝛿𝑔3\n𝛿𝑣3\nM𝑖𝑗=\u0000𝑖𝜔𝛿𝑔𝑖\n𝛿𝑣𝑗 𝑖≠𝑗(17)\nand the Alfvén speed is introduced as 𝑣𝐴=𝐵0√︁\n4𝜋𝜌0.\n2.3 Response of the radiative acceleration to a velocity\nperturbation\nIn the dispersion relation (15) we require the linear response of\nthe radiative acceleration to a velocity perturbation, 𝛿𝑔𝑖𝛿𝑣𝑗. This\ntensorexpressionwasfirstderivedinROC90showingthatthemulti-\ndimensional nature adds additional subtle radiation effects. As this\ntensor expressing 𝛿𝑔𝑖𝛿𝑣𝑗lies at the core of the present paper, we\ndiscuss here the important physical terms it consists of. The reader\ninterested in a more fundamental understanding of its nature and\nthe techniques required to derive is referred to Appendix A.\nFollowing ROC90 the perturbed radiative acceleration tensor\nfor a single line is\n𝛿𝑔𝑖\n𝛿𝑣𝑗=𝑔0\n𝑣thh𝜇D𝜇𝑝𝜇i\n\u0002\u001c\n𝑖𝑛𝑖\u0012\nD𝜇\u0000¹1\u0000𝑠ºhD𝜇𝑝𝜇i\nh𝑝𝜇i\u0013𝑛𝑙𝑘𝑙𝑛𝑗\n𝑄0𝑄0𝑘𝐿\n1¸𝑖𝑛𝑙𝑘𝑙𝑄0\u001d\n\n(18)\nwhere the meaning of the variables is introduced in the Appendix.\nRecallingourassumptions(§2.1)thatincludethepoint-starapprox-\nimation,D𝜇=𝛿¹𝜇\u00001º, and that linear stability of the flow is\nconsidered for a single optically thick line, 𝜏𝜇\u001d1, such that the\nescape probability becomes 𝑝𝜇\u00191𝜏𝜇=𝑄0𝑘𝐿, the prefactor\noutside angle brackets in Eq. (18) reduces to\n𝑔0\n𝑣thh𝜇D𝜇𝑄0i=2𝜔★\n𝜒★ (19)\nWe define the growth rate 𝜔★, characterizing how fast the\ninstability grows per unit time, as the ratio of the meanradiative\nacceleration of a single line 𝑔0to the thermal speed of ions in the\nwind𝑣th.Thequantity 𝜒\u00001\n★=1¹2h𝜇D𝜇𝑄0iºisthebridginglength\nfor a single optically thick line and is comparable to the Sobolev\nlength (OR84). It can be interpreted as the typical spatial scale\noverwhichaphotongetsabsorbedathighfrequencieswithinaline\nprofile.\nAnother important term is contained within the parentheses.\nThe quantity 𝑠is the line photon destruction probability for an\ninteractionbetweenaphotonandasingleline.Thisphotondestruc-\ntion probability allows pure absorption ¹𝑠=1ºand pure scattering\n(𝑠=0) to be treated separately. In the present paper we only take\nintoaccountpurescatteringasline-drivenwindsaremainlydriven\nby line-scattering processes. This then naturally splits Eq. (18) in\na contribution from direct radiation D𝜇and scattered radiationhD𝜇𝑝𝜇ih𝑝𝜇i. In particular, the scattered radiation has important\nphysicalconsequences for radiativeinstabilities byacting asa drag\nforce. In a purely radial flow the scattering reducesthe absolute\ngrowth of the LDI (Lucy 1984, OR85). On the other hand, when\ntakingintoaccountlateralradiation,scatteringactsto damptheLDI\nin the lateral directions (ROC90).\nWith this lateral damping from line-drag in mind one might\nexpectapriorisomeconsequencesinamagneticwind.Forexample,\nAlfvénwavespropagateastransversalwavesalongamagneticfield\nand they might be expected to experience effects related to this\ndamping. Whether line-drag affects the magnetic waves, and if so,\nin what way, is investigated next.\n3 STABILITY–DISPERSION ANALYSIS\n3.1 Dependence on perturbation wavelength: bridging law\nTogaininsightintothecouplingbetweentheradiationandthemag-\nneticfieldwehereestablishageneraldependenceofthedispersion\nrelation on perturbation wavelength. Under our adapted assump-\ntions, a magneto-radiative coupling only occurs in two eigentensor\nelementsinEq.(17)(withoutlossofgeneralitywefocuson M22).\nThe general dispersion relation (15) in the radial and lateral\ndirectionbecomesaquadraticpolynomialwitheigenmodesolutions\n𝜔33=0 𝜔 33=𝑖𝛿𝑔3\n𝛿𝑣3(20)\n𝜔22\u0006=1\n2266664𝑖\u0012𝛿𝑔2\n𝛿𝑣2\u0013\n\u0006√︄\n4𝑣2\n𝐴𝑘2\n3\u0000\u0012𝛿𝑔2\n𝛿𝑣2\u00132377775 (21)\nwhere we point out that most of the elements 𝛿𝑔𝑖𝛿𝑣𝑗=0if𝑖≠𝑗.\nThe latter follows from symmetry arguments of the radiation in\nthe perturbed radiative acceleration tensor (18). Below this will be\nexplicitlydemonstratedwhenwestudylong-andshort-wavelength\nperturbations. We now discuss the solutions in the radial and hori-\nzontal direction separately.\n3.1.1 Radial radiation\nIn the radial direction no coupling between the magnetic field and\nthe radiation occurs (cf. Eq. (17)). The perturbed radiative acceler-\nation tensor becomes\n𝛿𝑔3\n𝛿𝑣3=𝜔★𝑖𝑘3𝜒★\n1¸𝑖𝑘3𝜒★\u0000𝜔★\u00121¸𝜎\n1¸𝜎3\u0013 \u001c𝑖𝜇3𝑘3𝜒★\n1¸𝑖𝜇𝑘3𝑄0\u001d\n(22)\nwhere𝜎is related to the wind expansion (see Eq. (A13)). The first\ntermontheright-handsidecomesfrompureabsorptionasfoundby\nOR84 while the second term on the right-hand side now also takes\nintoaccountline-drag.Althoughthissolutionisforapointstar,we\nrefrain ourselves from further discussing it, because its properties\nare quite similar to that of a finite-coned star (OR85).\n3.1.2 Horizontal radiation\nThe perturbed radiative acceleration tensor is\n𝛿𝑔2\n𝛿𝑣2=\u0000𝜔★\n2\u00121¸𝜎\n1¸𝜎3\u0013 \u001c¹1\u0000𝜇2º𝑖𝜇𝑘3𝜒★\n1¸𝑖𝜇𝑘3𝑄0\u001d\n(23)\nMNRAS 000, 1–11 (2020)Alfvén wave damping in magnetic line-driven winds 5\n10−210−1100101102103104\nk/χ⋆10−510−410−310−210−1100101ω+/ω⋆vA/v0= 10\n1\n0.1\n0.01\n0.001vA/v0≥0.01\n10−210−1100101102103104\nk/χ⋆10−510−410−310−210−1100101ω−/ω⋆\nFigure 1. Stability-dispersion solutions for magneto-radiative modes for various values of 𝑣𝐴𝑣0at a point𝑟=5𝑅★in the wind. (Left panel) Outward\npropagatingmodes 𝜔¸.(Rightpanel)Inwardpropagatingmodes 𝜔\u0000.ThejRe»𝜔\u0006¼j(solidlines)givethewavespeed ¹𝜔\u0006𝑘ºwhilejIm»𝜔\u0006¼j(dashedlines)\nsets the damping rate. At small 𝑣𝐴𝑣0001(red, solid and red, dashed line) a splitting of inward and outward modes occurs while the phase speed of waves\ntends to zero. All curves are shown only for the case 𝑘3=𝑘, i.e. a perturbation along the mean magnetic field ¹𝜃=0\u000eº.\nand𝛿𝑔1𝛿𝑣1, i.e. radiation contributions in eigentensor element\nM11thatdescribesthefastmode,isexactlythesamewhenmaking\nthe substitution 𝑘3!𝑘, and𝑘=√︃\n𝑘2\n1¸𝑘2\n3.\nFig. 1 displays the behaviour of the real and imaginary wave\nfrequency versus wave number for a range of values of the ratio of\ntheAlfvénspeed 𝑣𝐴totheradiativewavespeed 𝜔★𝜒★.Thelatteris\napproximatelyequaltothemeanflow(Abbott)speed 𝑣0\u0011𝜔★𝜒★.\nContrarytoanon-magneticline-drivenwind,wheretheAbbott\nspeedismuchhigherthanthesoundspeed( 𝑣0\u001d𝑎,i.e.asupersonic\nflow), in the case of a magnetic wind 𝑣𝐴¡𝑣0,𝑣𝐴'𝑣0, or𝑣𝐴𝑣0\ndepending on the local conditions of magnetic field and density.\nTherefore, in a magnetic line-driven wind any of the curves shown\ninFig.1isaviablesolutiontodescribethemagneto-radiativewave\npropagation and instability. This is quite different from the non-\nmagnetic radiative-acoustic case studied in OR84.\nAt very high Alfvén speeds ¹𝑣𝐴\u001d𝑣0ºcurves of Re»𝜔\u0006¼\nare non-dispersive ordinary Alfvén waves over the full wave-\nlength range. When 𝑣𝐴𝑣0is lowered ordinary Alfvén waves re-\nmain ifjRe»𝜔\u0006¼j¡ 𝜔★but are modified by the radiation force for\njRe»𝜔\u0006¼j 𝜔★and become dispersive magneto-radiative waves\ninstead. In particular, when 𝑣𝐴\u0018001𝑣0waves with wavelengths\nnear the bridging length ¹𝑘\u0018𝜒★ºare modified by the radiative\nforcesuchthattheirpropagationcharacteristicsbecomemorecom-\nplicated. When 𝑣𝐴001𝑣0magneto-radiative waves are strongly\nmodified in their propagation.\nAn important result of Fig. 1 can be realized by further con-\nsidering Im»𝜔\u0006¼. All modes undergo net damping , i.e. they have a\nnegativegrowthrate(thisappearspositiveinthefigurebecausewe\nareplottingjIm»𝜔\u0006¼jonalogarithmicscale),whichfollowsdirectly\nfrom the fact that 𝛿𝑔2𝛿𝑣20for all perturbation wavelengths in\nEq. (23). For 𝑣𝐴&001𝑣0the damping is the same for all modes\nwhile it is slightly modified for 𝑣𝐴.001𝑣0. Specifically, in thelattercasetheinwardandoutwardmodesbecomedistinctandhave\ndifferent damping properties. The overall effect still remains and\nall modes undergo net damping. Finally, irrespective of 𝑣𝐴𝑣0, the\ndamping rate becomes vanishingly small at long-wavelengths such\nthat the magneto-radiative modes become marginally stable.\nItisinterestingtonotethattheratio 𝑣𝐴𝑣0hastoreducetoorder\n0.01forachangeinmodebehaviourtooccur.Examiningtheradical\nofEq.(21)showsthatsuchsituationmeansthat 𝑣𝐴𝑣0\u00181𝑘.Since\nthe mode switch manifests itself near ¹𝑘\u0018𝜒★ºthis indicates that\n𝑣𝐴𝑣0is effectively approaching the bridging length. The latter is\na characteristic length of the wind and is on the order of 𝐿Sob\u0019\n001𝑅★.Thisshowsthatforamajorityofexpectedconditionsinthe\nwinds of magnetic massive stars the damping of short-wavelength\nmagnetic waves should be very effective.\nA key result of this global analysis suggests thus that short-\nwavelength magnetic waves are always strongly damped by radia-\ntion.Thisdampingcomesfromahorizontalline-dragforceinduced\nbytheradiationontothemagneticwavesasillustratedinFig.2.Al-\nthough the line-drag in reality is much more subtle than this very\nsimple cartoon, the figure nevertheless sketches the basic effect in\na somewhat intuitive way. To gain more insight into the underlying\nphysics of the bridging law we now analyse separately the limiting\ncases of long- and short-wavelength perturbations.\n3.2 Long-wavelength perturbations\n3.2.1 Perturbed radiative acceleration tensor\nThe non-magnetic limit of long-wavelength perturbations ¹𝜆\u001d\n𝐿Sobºhas first been investigated by Abbott (1980). In the present\nnotation the long-wavelength limit is 𝑘𝑙\u001c𝑄0making the denom-\ninator¹1¸𝑖𝑛𝑙𝑘𝑙𝑄0ºin Eq. (18) approach unity. Therefore, the\nMNRAS 000, 1–11 (2020)6 F. A. Driessen, N. D. Kee, J. O. Sundqvist, and S. P. Owocki\nmagneticAAACAXicbVDLSsNAFJ3UV62vqBvBzWARuipJFXRZcOOygn1AE8pketMOnUnCzEQooW78FTcuFHHrX7jzb5y0WWjrgYHDOffeufcECWdKO863VVpb39jcKm9Xdnb39g/sw6OOilNJoU1jHsteQBRwFkFbM82hl0ggIuDQDSY3ud99AKlYHN3raQK+IKOIhYwSbaSBfeIJosdSZB6IZJzlLmhGZ7OBXXXqzhx4lbgFqaICrYH95Q1jmgqINOVEqb7rJNrPiDTjOMwqXqogIXRCRtA3NCIClJ/NL5jhc6MMcRhL8yKN5+rvjowIpaYiMJX5vmrZy8X/vH6qw2s/Y1GSaojo4qMw5VjHOI8DD5kEqvnUEEIlM7tiOiaSUG1Cq5gQ3OWTV0mnUXcv6o27y2qzVsRRRqfoDNWQi65QE92iFmojih7RM3pFb9aT9WK9Wx+L0pJV9ByjP7A+fwDGEpeo\u0000v2\nAAAB8XicbVBNS8NAEJ34WetX1aOXxSJ4kJJUQY8FLx4r2A9sQ9lsNu3SzSbsTgql9F948aCIV/+NN/+N2zYHbX0w8Hhvhpl5QSqFQdf9dtbWNza3tgs7xd29/YPD0tFx0ySZZrzBEpnodkANl0LxBgqUvJ1qTuNA8lYwvJv5rRHXRiTqEccp92PaVyISjKKVnrohl0jJqFftlcpuxZ2DrBIvJ2XIUe+VvrphwrKYK2SSGtPx3BT9CdUomOTTYjczPKVsSPu8Y6miMTf+ZH7xlJxbJSRRom0pJHP198SExsaM48B2xhQHZtmbif95nQyjW38iVJohV2yxKMokwYTM3ieh0JyhHFtCmRb2VsIGVFOGNqSiDcFbfnmVNKsV76pSfbgu1y7zOApwCmdwAR7cQA3uoQ4NYKDgGV7hzTHOi/PufCxa15x85gT+wPn8Aep9kFk=B0\nAAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4kJJUQY9FLx4r2g9oQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dlZW19Y3Ngtbxe2d3b390sFhU8epYthgsYhVO6AaBZfYMNwIbCcKaRQIbAWj26nfekKleSwfzThBP6IDyUPOqLHSw03P7ZXKbsWdgSwTLydlyFHvlb66/ZilEUrDBNW647mJ8TOqDGcCJ8VuqjGhbEQH2LFU0gi1n81OnZBTq/RJGCtb0pCZ+nsio5HW4yiwnRE1Q73oTcX/vE5qwms/4zJJDUo2XxSmgpiYTP8mfa6QGTG2hDLF7a2EDamizNh0ijYEb/HlZdKsVryLSvX+slw7z+MowDGcwBl4cAU1uIM6NIDBAJ7hFd4c4bw4787HvHXFyWeO4A+czx+yiY1X\nx3\nAAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgQcJuIugx4MVjRPOAZAmzk95kyOzsMjMrhpBP8OJBEa9+kTf/xkmyB00saCiquunuChLBtXHdbye3tr6xuZXfLuzs7u0fFA+PmjpOFcMGi0Ws2gHVKLjEhuFGYDtRSKNAYCsY3cz81iMqzWP5YMYJ+hEdSB5yRo2V7p961V6x5JbdOcgq8TJSggz1XvGr249ZGqE0TFCtO56bGH9CleFM4LTQTTUmlI3oADuWShqh9ifzU6fkzCp9EsbKljRkrv6emNBI63EU2M6ImqFe9mbif14nNeG1P+EySQ1KtlgUpoKYmMz+Jn2ukBkxtoQyxe2thA2poszYdAo2BG/55VXSrJS9arlyd1mqXWRx5OEETuEcPLiCGtxCHRrAYADP8ApvjnBenHfnY9Gac7KZY/gD5/MHCWiNkA==x2\nAAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4kJJUQY8FLx4r2g9oQ9lsN+3SzSbsTsQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IJHCoOt+Oyura+sbm4Wt4vbO7t5+6eCwaeJUM95gsYx1O6CGS6F4AwVK3k40p1EgeSsY3Uz91iPXRsTqAccJ9yM6UCIUjKKV7p961V6p7FbcGcgy8XJShhz1Xumr249ZGnGFTFJjOp6boJ9RjYJJPil2U8MTykZ0wDuWKhpx42ezUyfk1Cp9EsbalkIyU39PZDQyZhwFtjOiODSL3lT8z+ukGF77mVBJilyx+aIwlQRjMv2b9IXmDOXYEsq0sLcSNqSaMrTpFG0I3uLLy6RZrXgXlerdZbl2nsdRgGM4gTPw4ApqcAt1aACDATzDK7w50nlx3p2PeeuKk88cwR84nz8H5I2Pmagnetic + radiationAAACD3icbVDLSsNAFJ34rPUVdekmWJSCpSRV0GXBjcsK9gFNKJPJbTt0JgkzE6GE/oEbf8WNC0XcunXn3zhJs9DWAwOHc+69c+/xY0alsu1vY2V1bX1js7RV3t7Z3ds3Dw47MkoEgTaJWCR6PpbAaAhtRRWDXiwAc59B15/cZH73AYSkUXivpjF4HI9COqQEKy0NzDOXYzUWPHWBx+M0c0FR4tbO3ZrAAc3LZrOBWbHrdg5rmTgFqaACrYH55QYRSTiEijAsZd+xY+WlWOjhDGZlN5EQYzLBI+hrGmIO0kvze2bWqVYCaxgJ/UJl5ervjhRzKafc15XZ9nLRy8T/vH6ihtdeSsM4URCS+UfDhFkqsrJwrIAKIIpNNcFEUL2rRcZYYKJ0hGUdgrN48jLpNOrORb1xd1lpVos4SugYnaAqctAVaqJb1EJtRNAjekav6M14Ml6Md+NjXrpiFD1H6A+Mzx/7hJ0q\u0000g2/\u0000\u0000v2\nAAACCHicbVDLSgMxFM3UV62vUZcuDBbBhZaZUdBlwY3LCvYBnWHIZDJtaCYZkkyhlC7d+CtuXCji1k9w59+YtrPQ1gOBwzn3cnNOlDGqtON8W6WV1bX1jfJmZWt7Z3fP3j9oKZFLTJpYMCE7EVKEUU6ammpGOpkkKI0YaUeD26nfHhKpqOAPepSRIEU9ThOKkTZSaB/7MWEawV7oQT+TItMCXsBCHIZeaFedmjMDXCZuQaqgQCO0v/xY4DwlXGOGlOq6TqaDMZKaYkYmFT9XJEN4gHqkayhHKVHBeBZkAk+NEsNESPO4hjP198YYpUqN0shMpkj31aI3Ff/zurlOboIx5VmuCcfzQ0nOoEk7bQXGVBKs2cgQhCU1f4W4jyTC2nRXMSW4i5GXScuruZc17/6qWj8v6iiDI3ACzoALrkEd3IEGaAIMHsEzeAVv1pP1Yr1bH/PRklXsHII/sD5/AKzDmGc=\u0000g2/\u0000\u0000v2\nAAACCHicbVDLSgMxFM3UV62vUZcuDBbBhZaZUdBlwY3LCvYBnWHIZDJtaCYZkkyhlC7d+CtuXCji1k9w59+YtrPQ1gOBwzn3cnNOlDGqtON8W6WV1bX1jfJmZWt7Z3fP3j9oKZFLTJpYMCE7EVKEUU6ammpGOpkkKI0YaUeD26nfHhKpqOAPepSRIEU9ThOKkTZSaB/7MWEawV7oQT+TItMCXsBCHIZeaFedmjMDXCZuQaqgQCO0v/xY4DwlXGOGlOq6TqaDMZKaYkYmFT9XJEN4gHqkayhHKVHBeBZkAk+NEsNESPO4hjP198YYpUqN0shMpkj31aI3Ff/zurlOboIx5VmuCcfzQ0nOoEk7bQXGVBKs2cgQhCU1f4W4jyTC2nRXMSW4i5GXScuruZc17/6qWj8v6iiDI3ACzoALrkEd3IEGaAIMHsEzeAVv1pP1Yr1bH/PRklXsHII/sD5/AKzDmGc=\nFigure 2. Cartoon depicting the propagation of a short-wavelength Alfvén\nwave resulting from a horizontal velocity perturbation 𝛿𝑣2along the radial\nmeanmagneticfield.(Abovedashedline)Dampingoftheshort-wavelength\nAlfvén wave due to the line-drag 𝛿𝑔2𝛿𝑣2. (Below dashed line) Without\nradiation field a short-wavelength Alfvén wave propagates along the radial\nmagnetic field as a stable transversal wave.\nradiative acceleration tensor is\n𝛿𝑔𝑖\n𝛿𝑣𝑗=2𝜔★\n𝜒★\u001c\n𝑖𝑛𝑖𝑛𝑙𝑘𝑙𝑛𝑗\u0012\nD𝜇\u0000hD𝜇𝑝𝜇i\nh𝑝𝜇i\u0013\u001d\n (24)\nThe term associated with scattering in the above equation al-\nways vanishes in the long-wavelength limit. Specifically, the net\nangular integrand is odd in the unit normal 𝑛such that it vanishes\n\u001c\n𝑛𝑖𝑛𝑙𝑘𝑙𝑛𝑘hD𝜇𝑝𝜇i\nh𝑝𝜇i\u001d\n=0 (25)\nHence for a pure scattering-driven wind the radiative acceler-\nation tensor becomes\n𝛿𝑔𝑖\n𝛿𝑣𝑗=𝑖2𝜔★\n𝜒★\n𝑛𝑖𝑛𝑙𝑘𝑙𝑛𝑗D𝜇\u000b\n=𝑖2𝜔★\n𝜒★𝑘𝑙T𝑖𝑗𝑙 (26)\nWhat remains is to solve for the tensor T𝑖𝑗𝑙which simpli-\nfies significantly for a point star. Only radiation directly from the\nstar contributes and this is purely radial and spherically symmet-\nric. Necessarily the two non-radial radiation angles have to vanish,\ni.e.𝑛1𝑛2!0. The only remaining component of the radiation\nfieldunitnormalistheradialdirection 𝑛3.RecallthatD𝜇isaDirac\ndeltafunctionsuchthatmosttermsvanishintheangularintegration\nofT𝑖𝑗𝑙unless𝑛𝑗k𝑟3.Fromthisweconcludethat 𝑛𝑖=𝑛𝑗=𝑛𝑙=𝑛3\nand the only non-vanishing tensor component becomes\nT333=h𝑛3\n3𝛿¹𝜇\u00001ºi=1\n2 (27)\nConsequently, the only surviving component of the perturbed\nradiative acceleration is purely radial\n𝛿𝑔3\n𝛿𝑣3=𝑖𝜔★\n𝜒★𝑘3 (28)Thisresultcanalsoberecoveredwhenconsideringthegeneral\nbridging law in the radial direction derived in Eq. (22). Indeed, in\nthelong-wavelengthlimit, 𝑘3𝜒★\u001c1,thebridginglawreducesto\nthe expression derived here.\n3.2.2 Physical interpretation of long-wavelength perturbations\nThe derivation above shows that most radiation associated eigen-\ntensor elements in (17) vanish such that\nM11=𝜔2\u0000𝑣2\n𝐴¹𝑘2\n1¸𝑘2\n3º\nM22=𝜔2\u0000𝑣2\n𝐴𝑘2\n3\nM33=𝜔2\u0000𝑖𝜔𝛿𝑔3\n𝛿𝑣3\nM𝑖𝑗=0 𝑖≠𝑗(29)\nThe resulting eigenproblem has a unique, non-trivial solution\nwhenever the determinant of the eigentensor is zero\n\u0012\n𝜔2¸𝜔𝜔★\n𝜒★𝑘cos𝜃\u0013\u0010\n𝜔2\u0000𝑣2\n𝐴𝑘2\u0011 \u0010\n𝜔2\u0000𝑣2\n𝐴𝑘2cos2𝜃\u0011\n=0(30)\nwhere we have substituted the expressions 𝑘1=𝑘sin𝜃,𝑘3=\n𝑘cos𝜃, and that𝑘=√︃\n𝑘2\n1¸𝑘2\n3. This sextic polynomial with real\ncoefficients has six independent real solutions:\n𝜔¹1º=\u0000𝑣0𝑘cos𝜃 𝜔¹2º=0 𝜔¹3º\n¸=𝑣𝐴𝑘cos𝜃\n𝜔¹4º\n\u0000=\u0000𝑣𝐴𝑘cos𝜃 𝜔¹5º\n¸=𝑣𝐴𝑘 𝜔¹6º\n\u0000=\u0000𝑣𝐴𝑘(31)\nand the ratio of growth rate to bridging length 𝜔★𝜒★yields the\nmeanflowspeed 𝑣0.Allthederivedeigenfrequenciesarerealmean-\ning that the corresponding waves are stable. All eigenfrequencies\nalso linearly depend on wave number such that the waves are of\nnon-dispersive type.\nWe find one eigenfrequency \u0000𝑣0𝑘cos𝜃with eigenvector\n¹001ºrepresentingavelocityfluctuationpolarisedintheparallel\ndirection. This is the Abbott wave propagating inward at the mean\nflow speed (Abbott 1980). Additionally, four magnetic waves are\nfoundwitheigenfrequencies \u0006𝑣𝐴𝑘cos𝜃and\u0006𝑣𝐴𝑘thatdescribein-\nward/outwardpropagatingAlfvénandfastMHDwaveswitheigen-\nvectors¹010ºand¹100º,respectively.Allthesemagneticwaves\nhave velocity fluctuations that are horizontally polarised compared\ntothemeanmagneticfield.Finally,wefindagenuineneutralmode\nwitheigenvector¹000ºthatinvolvesadensityperturbationalone.\nInterestingly, these results also show that for long-wavelength per-\nturbationsthereisacompletedecouplingbetweentheradiativeand\nmagnetic waves such that they do not interact with each other di-\nrectly. Notice that when we take only radially propagating waves\n(𝑘1=0)thenthefastwavevanishes(solutions5&6)andbecomes\nan Alfvén wave. In this case, therefore, a degenerate pair of inward\nandoutwardAlfvénwavesremainstogetherwiththeinwardAbbott\nwave. On the other hand, in the limit of vanishing magnetic field\n(𝑣𝐴!0)onlyanAbbottwaveremainsasfoundbyAbbott(1980).\n3.3 Short-wavelength perturbations\n3.3.1 Perturbed radiative acceleration tensor\nInitialstabilityconsiderationsoftheradiation-hydrodynamicequa-\ntions of a line-driven wind have been performed by MacGregor\net al. (1979) and Carlberg (1980) in the short-wavelength limit\n(𝜆\u001c𝐿Sob). In terms of the radiative acceleration tensor in this\nMNRAS 000, 1–11 (2020)Alfvén wave damping in magnetic line-driven winds 7\nlimit the scattering term will not vanish in Eq. (18). Indeed, for\nshort-wavelength perturbations ¹𝑘𝑙\u001d𝑄0º\n𝛿𝑔𝑖\n𝛿𝑣𝑗=2𝜔★\n𝜒★\u001c\n𝑛𝑖𝑛𝑗\u0012\nD𝜇\u0000hD𝜇𝑝𝜇i\nh𝑝𝜇i\u0013\n𝑄0\u001d\n=2𝜔★\n𝜒★T𝑖𝑗 (32)\nThe non-vanishing of scattering contributions at short-\nwavelengths induces some properties onto the tensor T𝑖𝑗: (i) be-\ncause𝑛isaunitvector 𝑛𝑖𝑛𝑗=𝛿𝑖𝑗suchthatT𝑖𝑗mustbea diagonal\ntensor, (ii)T𝑖𝑗being a diagonal tensor implies that\nT𝑖𝑖=\u001c\u0012\nD𝜇\u0000hD𝜇𝑝𝜇i\nh𝑝𝜇i\u0013\n𝑄0\u001d\n=0 (33)\nmakingT𝑖𝑗atraceless,diagonaltensor,and(iii)radiationhassym-\nmetry around the radial axis such that the lateral and azimuthal\nradiation angles are indistinguishable , or\nT11=T22=1\n2¹T𝑖𝑖\u0000T33º=\u00001\n2T33 (34)\nwhere the latter equality comes from the traceless condition (prop-\nerty ii). Fromthese properties it follows thatthe radial tensor com-\nponent\nT33=1\u0000¹13¸𝜎5º\n1¸𝜎3 (35)\nand so\nT11=T22=\u00001\n2\u0014\n1\u0000¹13¸𝜎5º\n1¸𝜎3\u0015\n (36)\nThe surviving perturbed radiative acceleration to perturbed\nvelocity tensor elements in the pure scattering limit are then\n𝛿𝑔1\n𝛿𝑣1=\u00001\n2𝜔★T33𝛿𝑔2\n𝛿𝑣2=\u00001\n2𝜔★T33𝛿𝑔3\n𝛿𝑣3=𝜔★T33\n(37)\nThenon-vanishinghorizontalperturbedradiativeacceleration\nterms here leads to the coupling with magnetic waves in the eigen-\ntensorandthedampingphenomenonasshowninFig.1.Thephys-\nical interpretation of the damping can be attributed to a line-drag\n(Lucy 1984) as captured by the T𝑖𝑖term. This line-drag reduces\nthe instability growth in the radial direction, but dampsit in the\nhorizontal directions.\n3.3.2 Physical interpretation of short-wavelength perturbations\nThe eigenproblem is nearly the same as for the long-wavelength\nlimit, except that all diagonal elements of the eigentensor (17) will\nretain the radiative component\nM11=𝜔2\u0000𝑣2\n𝐴¹𝑘2\n1¸𝑘2\n3º\u0000𝑖𝜔𝛿𝑔1\n𝛿𝑣1\nM22=𝜔2\u0000𝑣2\n𝐴𝑘2\n3\u0000𝑖𝜔𝛿𝑔2\n𝛿𝑣2\nM33=𝜔2\u0000𝑖𝜔𝛿𝑔3\n𝛿𝑣3\nM𝑖𝑗=0 𝑖≠𝑗(38)\nThiscouplingbetweentheradiationandthemagneticfieldhas\nimportanteffectsaswewillshowbelow.Settingthedeterminantof\nthe eigentensor to zero to retrieve the non-trivial solutions yields\n\u0010\n𝜔2\u0000𝑖𝜔★T33𝜔\u0011\u0012\n𝜔2\u0000𝑣2\n𝐴𝑘2¸𝑖𝜔1\n2𝜔★T33\u0013\n\u0002\u0012\n𝜔2\u0000𝑣2\n𝐴𝑘2cos2𝜃¸𝑖𝜔1\n2𝜔★T33\u0013\n=0(39)where we have substituted the expressions 𝑘1=𝑘sin𝜃,𝑘3=\n𝑘cos𝜃, and that𝑘=√︃\n𝑘2\n1¸𝑘2\n3. This is a sextic polynomial with\nmixedrealandimaginarycoefficientssoithassixrootsthatcanbe\nreal, imaginary, or appearing in complex conjugate pairs.\nInstead of writing out the eigenmodes of Eq. (39) it is\nmore insightful to make the substitution 𝑤=\u0000𝑖𝜔¹𝜔★T33º=\nIm»𝜔¼¹𝜔★T33ºand𝐾=𝑣𝐴𝑘¹𝜔★T33ºgiving\n𝑤¹𝑤\u00001º\u0012\n𝑤2¸1\n2𝑤¸𝐾2cos2𝜃\u0013 \u0012\n𝑤2¸1\n2𝑤¸𝐾2\u0013\n=0(40)\nIn Fig. 3 the solution of the sextic equation in 𝑤is shown\nover a range of wave numbers 𝐾for each mode. There is always\none radial radiative mode (LDI) from a radial velocity polarisation\nthat is unstable¹�� ¡ 0ºfor any angle 𝜃alongside a neutral, non-\npropagating density mode. It can be seen that the magnetic modes\n(Alfvén and fast) resulting from a horizontal velocity polarisation\nexperience net damping for any angle 𝜃¹𝑤 0º.\nThesolutioncurvesofthemagneticmodesassumeapitchfork\nshape that is set by a critical perturbation wavelength 𝐾crit. This\ncritical perturbation wavelength essentially sets the behaviour of\ninward and outward magnetic modes. For 𝐾 ¡ 𝐾 critall magnetic\nmodes are damped at the damping rate of the line-drag while in-\nward and outward modes propagate identically. These propagation\ncharacteristicsareonlymodifiedonce 𝐾 𝐾 critandtheinwardand\noutwardmodesbecomedistinct(seealsodiscussionaroundFig.1).\nInlightoftherangeofvalidityofEq.(39),i.e.theshort-wavelength\nlimit¹𝑘\u001d𝑄0º,andthat𝑄0\u0018𝜒★and𝜔★T33\u0019𝜔★thewavenum-\nber variable 𝐾satisfies𝐾\u001d𝑣𝐴𝑣0. Thus depending on the local\nconditions in the wind, part of or the complete magnetic solution\ncurves in Fig. 3 apply to describe the magnetic mode damping.\nThe results here thus confirm the bridging law that short-\nwavelength magnetic waves with horizontal velocity polarisations\nundergo strong damping due to the line-drag. Overall the wind re-\nmains highly unstable, and the most unstable mode has the most\nradialvelocitypolarisation.Theexacteffectsofthelateraldamping\nin regulating the wind dynamics and wind fragmentation are spec-\nulative and have to be further investigated by means of numerical\nsimulation.\n4 CONCLUSIONS & FUTURE OUTLOOK\nInthispaperwehavestudiedthelinearstabilityofaline-scattering-\ndriven magnetic stellar wind. We have considered both radial\nand non-radial wave propagation in the limiting cases of long-\nwavelength and short-wavelength compared to the Sobolev length\nof the wind. In addition, we have also investigated the dependency\non perturbation wavelength by establishing a bridging law. Within\nourassumptionofaradialmagneticfieldwehavefoundthatstable,\nlong-wavelengthradiativeandmagneticwavesaredecoupled.Akey\nresult is that short-wavelength Alfvén waves are strongly damped\ndue to a drag force from line-scattering. Additionally, when the\nratio of Alfvén speed to radiative wave speed becomes small and\napproaches a scale comparable to the Sobolev length, the damping\nproperties of magnetic waves are slightly modified. This situation\nis more likely to play a role in weakly magnetic environments, or\nin a very dense wind where low Alfvén speeds can occur. Never-\ntheless, the overall result remains that short-wavelength magnetic\nwaves always undergo damping.\nPossibleanalyticalextensionstotheanalysishereincludeintro-\nducing thermal pressure or considering non-radial magnetic fields.\nSince the mean flow speed 𝑣0\u001d𝑎in most parts of line-driven\nMNRAS 000, 1–11 (2020)8 F. A. Driessen, N. D. Kee, J. O. Sundqvist, and S. P. Owocki\n10−1100101\nK−0.50.00.51.0w\nθ= 0◦\n10−1100101\nK−0.50.00.51.0\nθ= 45◦LDI neutral Alfv´ en fast\n10−1100101\nK−0.50.00.51.0\nθ= 90◦\nFigure3. Growthrate𝑤(inunitsof𝜔★T33)formagneto-radiativemodesasfunctionofwavenumber 𝐾(inunitsof𝜔★T33𝑣𝐴).Allpanelsshowadifferent\norientationoftheinitialwavepropagationvectorforeachmode.At 𝜃=0\u000etheAlfvénandfastmodeoverlapbeforethecollapseofthepitchforkbranches.For\nclarity only half of each mode branch is shown here. At 𝜃=90\u000ethe Alfvén mode vanishes leading to an additional neutral mode and a radiative mode.\nwinds, such sound speed terms are typically not important for the\ndynamics of the flow. However, in a magnetic line-driven wind the\nstabilitypropertiesmayalsodependonthecomparisonbetweenthe\nAlfvén speed and the sound speed. This could potentially alter the\ninstabilitygrowthanddampingforshort-wavelengthperturbations.\nThe effect of non-radial mean magnetic field components might\npotentially alter the coupling properties of radiative and magnetic\nwaves and their linear (in)stability. Such non-radial magnetic field\ncomponentscanoccurinrapidlyrotatingmagneticmassivestarsor\ninregionsofstrongfieldlinecurvatureinthecircumstellarmagne-\ntosphere near the magnetic equatorial plane.\nFurthermore, we plan to present non-linear magnetohydrody-\nnamic instability simulations that study the development and dy-\nnamicsofunstableLDIstructures.Thesesimulationsaimtoinves-\ntigate both the globalandlocaldynamics. In particular, the latter\nwill be able to study the damping of Alfvén waves as found in the\npresentpaperwiththeinclusionofthelateralperturbedradiativeac-\nceleration.Suchphysicsmightprevent,oralter,thelateralfragmen-\ntation seen in non-magnetic 2D instability simulations (Sundqvist\netal.2018)andaffectthewindclumpingproperties.Theinclusion\nof magnetic fields might also affect the growth rate of the insta-\nbility and the wave-stretching mechanism (Feldmeier & Thomas\n2017)thatcausestheseparationbetweenclumpsandtheinterclump\nmedium. Our first tentative global simulations already suggest that\nlateralbreak-upofwindmaterialisstronglymodifieddependingon\nthe background stellar magnetic field strength (Driessen et. al., in\nprep.). The instability dynamics in such a setting and its effect on\nobservational diagnostics will be reported in a future work.ACKNOWLEDGEMENTS\nFADthanksJonasBerxforsharingsomeofhisMathematicatricks\nuseful for this work. FAD and JOS acknowledge support from the\nOdysseus program of the Belgian Research Foundation Flanders\n(FWO) under grant G0H9218N. NDK acknowledges support from\nthe KU Leuven C1 grant MAESTRO C16/17/007. We thank the\nreferee,Prof.AchimFeldmeier,forconstructivecriticismandcom-\nments to the manuscript.\nDATA AVAILABILITY\nThe data underlying this article are available in the article.\nREFERENCES\nAbbott D. C., 1980, ApJ, 242, 1183\nBerghoefer T. W., Schmitt J. H. M. M., Danner R., Cassinelli J. P., 1997,\nA&A, 322, 167\nCarlberg R. G., 1980, ApJ, 241, 1131\nCastor J. I., 1970, MNRAS, 149, 111\nCastor J. I., Abbott D. C., Klein R. I., 1975, ApJ, 195, 157\nDessart L., Owocki S. P., 2002, A&A, 383, 1113\nDonatiJ.-F.,BabelJ.,HarriesT.J.,HowarthI.D.,PetitP.,SemelM.,2002,\nMNRAS, 333, 55\nDriessen F. A., Sundqvist J. O., Kee N. D., 2019, A&A, 631, A172\nEversberg T., Lépine S., Moffat A. F. J., 1998, ApJ, 494, 799\nFeldmeier A., Thomas T., 2017, MNRAS, 469, 3102\nFeldmeier A., Puls J., Pauldrach A. W. A., 1997, A&A, 322, 878\nFossati L., et al., 2015, A&A, 582, A45\nLucy L. B., 1983, ApJ, 274, 372\nLucy L. B., 1984, ApJ, 284, 351\nLucy L. B., Solomon P. M., 1970, ApJ, 159, 879\nMNRAS 000, 1–11 (2020)Alfvén wave damping in magnetic line-driven winds 9\nMacGregor K. B., Hartmann L., Raymond J. C., 1979, ApJ, 231, 514\nOwocki S. P., Puls J., 1996, ApJ, 462, 894\nOwocki S. P., Puls J., 1999, ApJ, 510, 355\nOwocki S. P., Rybicki G. B., 1984, ApJ, 284, 337\nOwocki S. P., Rybicki G. B., 1985, ApJ, 299, 265\nOwocki S. P., Rybicki G. B., 1986, ApJ, 309, 127\nOwocki S. P., Rybicki G. B., 1991, ApJ, 368, 261\nOwocki S. P., Castor J. I., Rybicki G. B., 1988, ApJ, 335, 914\nPulsJ.,SundqvistJ.O.,MarkovaN.,2015,inMeynetG.,GeorgyC.,GrohJ.,\nSteeP.,eds,IAUSymposiumVol.307,NewWindowsonMassiveStars.\npp 25–36 ( arXiv:1409.3582 ), doi:10.1017/S174392131400622X\nRybicki G. B., Owocki S. P., Castor J. I., 1990, ApJ, 349, 274\nSchneider F. R. N., Ohlmann S. T., Podsiadlowski P., Röpke F. K., Balbus\nS. A., Pakmor R., Springel V., 2019, Nature, 574, 211\nSobolev V. V., 1960, Moving envelopes of stars. Harvard University Press\nSundqvist J. O., Owocki S. P., Puls J., 2012, in Drissen L., Robert C.,\nSt-Louis N., Moffat A. F. J., eds, Astronomical Society of the Pacific\nConference Series Vol. 465, Proceedings of a Scientific Meeting in\nHonor of Anthony F. J. Moffat. p. 119 ( arXiv:1110.0485 )\nSundqvist J. O., Owocki S. P., Puls J., 2018, A&A, 611, A17\nWade G. A., et al., 2016, MNRAS, 456, 2\nud-Doula A., Owocki S. P., 2002, ApJ, 576, 413\nud-Doula A., Townsend R. H. D., Owocki S. P., 2006, ApJ, 640, L191\nud-Doula A., Owocki S. P., Townsend R. H. D., 2008, MNRAS, 385, 97\nAPPENDIX A: THE PERTURBED RADIATIVE\nACCELERATION TENSOR\nIn this Section we perform a detailed derivation on how to obtain\nthe perturbed radiative acceleration tensor in Eq. (18).\nA1 Equation of radiative transfer\nWe describe the radiation field by the spherically symmetric time-\nindependent equation of transfer for a single line in the comoving\nframe into direction 𝑛𝑗(OR85) :\n𝑛𝑗𝜕𝐼\n𝜕𝑟𝑗\u0000𝑄𝜕𝐼\n𝜕𝑥=\u0000𝑘¹𝑥º\n𝐿¹𝐼\u0000𝑆𝐿º (A1)\nwhere we neglect aberration and other terms on the order O¹𝑣𝑐º.\nThetransferequationdescribestheevolutionofintensity 𝐼¹𝑛𝑗𝑟𝑖𝑥º\nfor a given line-source function 𝑆𝐿¹𝑟𝑖º(we will drop these de-\npendencies again for notational clarity). Radiation propagation is\ndescribed by a unit vector 𝑛with components\n𝑛1=sinΘcosΦ 𝑛 2=sinΘsinΦ 𝑛 3=cosΘ(A2)\nwithΘthelateralangleand Φtheazimuthalangle.Westressthat Θ,\nΦare part of the momentum coordinates of the radiation field and\nunrelatedtotherealspacecoordinates 𝜃,𝜙ofanyglobalbackground\nspherical flow.\nThe normalised comoving frame frequency is\n𝑥\u0011𝑥CMF=𝜈CMF\u0000𝜈0\nΔ𝜈𝐷Δ𝜈𝐷=𝜈0𝑣th\n𝑐 (A3)\nwith𝜈0the line-centre frequency, 𝑣tha thermal speed, 𝑐the speed\nof light, andΔ𝜈𝐷the Doppler width of a line.\nLine-extinction in the wind is defined according to\n𝑘¹𝑥º\n𝐿=𝑘𝐿𝜙¹𝑥º (A4)\nfor a normalised line-profile function\n∫¸1\n\u00001𝑑𝑥𝜙¹𝑥º=1 (A5)Thefrequency-integratedline-extinction 𝑘𝐿(incm\u00001)depends\nontheupper 𝑢andlowerlevel 𝑙populationsandoscillatorstrength\n𝑓𝑙𝑢of the line\n𝑘𝐿=1\nΔ𝜈𝐷𝜎cl𝑛𝑙𝑓𝑙𝑢 (A6)\nforclassicalcross-section 𝜎cl,numberdensityofthelowerlevel 𝑛𝑙,\nandwherewehaveneglectedstimulatedemission.Furthermore, 𝑘𝐿\ncanberelatedtoamass-absorptioncoefficient 𝜅suchthat𝑘𝐿=𝜅𝜌.\nIn complete frequency redistribution it follows that for pure\nscattering\n𝑆𝐿=¯𝐽=h¯𝐼i (A7)\nwhere the mean intensity is defined as\n¯𝐽¹𝑟𝑖º\u00111\n2∫¸1\n\u00001𝑑𝜇¯𝐼¹𝑛𝑗𝑟𝑖𝑥º=h¯𝐼i (A8)\nwith𝜇=cosΘ, and the frequency-integrated intensity\n¯𝐼¹𝑛𝑗𝑟𝑖º\u0011∫¸1\n\u00001𝑑𝑥𝐼¹𝑛𝑗𝑟𝑖𝑥º (A9)\nThe factor𝑄in Eq. (A1) above contains the projection of the\nrate of strain tensor 𝜕𝑣𝑖𝜕𝑟𝑗into a direction 𝑛𝑗\n𝑄=1\n𝑣th𝜕𝑣𝑖\n𝜕𝑟𝑗𝑛𝑖𝑛𝑗 (A10)\nhere it is implicitly assumed that the mean flow is monotonically\nincreasing, i.e. 𝑄 ¡ 0. In a spherically symmetric flow the mean\nstate𝑄0is\n𝑄0=1\n𝑣th\u0014\n𝜇2𝑑𝑣0\n𝑑𝑟¸¹1\u0000𝜇2º𝑣0\n𝑟\u0015\n=𝑣0\n𝑣th𝑟¹1¸𝜎𝜇2º (A11)\nInthelatterequalitywehaveintroducedthehomologeouswind\nexpansion quantity 𝜎\u0011𝑑¹ln𝑣0º𝑑¹ln𝑟º\u00001(Castor 1970). If we\ndescribe the mean wind flow by a typical velocity law\n𝑣0¹𝑟º=𝑣1\u0012\n1\u0000𝑅★\n𝑟\u0013𝛽\n (A12)\nwith𝛽=05for a point star and 𝑣1the terminal wind speed, then\nit follows that\n𝜎¹𝑟º=¹𝛽¸1º𝑅★\u0000𝑟\n𝑟\u0000𝑅★ (A13)\nThe perturbed 𝛿𝑄, on the other hand, is a truly multidimen-\nsional quantity because perturbations are allowed to happen into\nany direction\n𝛿𝑄=1\n𝑣th𝜕𝛿𝑣𝑖\n𝜕𝑟𝑗𝑛𝑖𝑛𝑗 (A14)\nFinally,withthesedefinitionsoftheradiationfieldtheradiative\nline-force (per unit mass) becomes\n𝑔𝑖¹𝑟𝑗º\u00114𝜋𝑘𝐿\n𝜌𝑐¯𝐻𝑖¹𝑟𝑗º (A15)\nwhere ¯𝐻is the Eddington line-flux defined as the first moment of\nthe angle-averaged, frequency-integrated intensity\n¯𝐻𝑖¹𝑟𝑗º\u0011h𝑛𝑖¯𝐼i (A16)\nWe are now ready to determine how linear velocity perturba-\ntions affect the radiation field by finding the tensor 𝛿𝑔𝑖𝛿𝑣𝑗.\nMNRAS 000, 1–11 (2020)10 F. A. Driessen, N. D. Kee, J. O. Sundqvist, and S. P. Owocki\nA2 Linear perturbations in three dimensions\nFollowing the same procedure as for the magnetic flow, we apply\nperturbations on Eq. (A1) of the form\n𝐼\u0011𝐼0¹𝑛𝑗𝑟𝑖𝑥º¸𝛿𝐼¹𝑛𝑗𝑟𝑖𝑥º 𝑘¹𝑥º\n𝐿\u0011𝑘¹𝑥º\n𝐿0\n𝑆𝐿\u0011𝑆𝐿0¹𝑟𝑖º¸𝛿𝑆𝐿¹𝑟𝑖º 𝑄\u0011𝑄0¹𝜇𝑟º¸𝛿𝑄¹𝑛𝑗𝑟𝑖º(A17)\nUp to first order the mean radiation field is described by\n𝑄0𝜕𝐼0\n𝜕𝑥=𝑘¹𝑥º\n𝐿0¹𝐼0\u0000𝑆𝐿0º (A18)\nwhileintheWKBlimitanduptofirstordertheperturbedradiation\nfield is\n𝑛𝑗𝜕𝛿𝐼\n𝜕𝑟𝑗\u0000𝑄0𝜕𝛿𝐼\n𝜕𝑥\u0000𝑄0𝜕𝐼0\n𝜕𝑥=\u0000𝑘¹𝑥º\n𝐿0¹𝛿𝐼\u0000𝛿𝑆𝐿º (A19)\nA3 Solution of the mean radiation field\nTo solve for the mean radiation field in Eq. (A18) a suitable (blue-\nwing) boundary condition is\n𝐼0¹𝑛𝑗𝑟𝑖𝑥!¸1º=𝐼★D¹𝑛𝑗𝑟𝑖º (A20)\nwhere𝐼★is the stellar core intensity and D¹𝑛𝑗𝑟𝑖º\u0011D𝜇is an\nangular function capturing possible intensity variations across the\nstellar disk (a Dirac delta function for a point star). As the Sobolev\napproximation holds for the mean flow the solution of Eq. (A18) is\nsimplified when introducing the angle-dependent Sobolev optical\ndepth𝜏𝜇=𝑘𝐿𝑄0or\n𝜏𝜇=𝜏𝜇=0\n1¸𝜎𝜇2 (A21)\nwhere𝜏𝜇=0=𝑘𝐿𝑣th𝑟𝑣0is the Sobolev optical depth in the lateral\ndirection. Furthermore, we define the integration function\nΦ¹𝑥º\u0011∫¸1\n𝑥𝑑𝑥0𝜙¹𝑥0º (A22)\nso that\n𝜏𝜇Φ¹𝑥º=𝑘𝐿\n𝑄0∫¸1\n𝑥𝑑𝑥0𝜙¹𝑥0º=1\n𝑄0∫¸1\n𝑥𝑑𝑥0𝑘¹𝑥0º\n𝐿0(A23)\nThemeanradiationfield,Eq.(A18),canbesolvedwhenusing\nan integrating factor exp»\u0000𝜏𝜇Φ¹𝑥º¼\n𝐼0=𝐼★D𝜇𝑒\u0000𝜏𝜇Φ¹𝑥º¸𝑆𝐿0¹1\u0000𝑒\u0000𝜏𝜇Φ¹𝑥ºº (A24)\nWe proceed to determine the mean Eddington line-flux by\ncomputing the frequency-integrated mean intensity and insert this\ncondition into Eq. (A24). It follows that\n¯𝐼0=𝐼★D𝜇𝑝𝜇¸𝑆𝐿0¹1\u0000𝑝𝜇º (A25)\nwhere we define the escape probability of a single line as\n𝑝𝜇\u0011∫¸1\n\u00001𝑑𝑥𝜙¹𝑥º𝑒\u0000𝜏𝜇Φ¹𝑥º=1\u0000𝑒\u0000𝜏𝜇\n𝜏𝜇 (A26)\nTaking an angular average of the frequency-integrated mean\nintensity yields\n¯𝐽=h¯𝐼0i=𝐼★hD𝜇𝑝𝜇i¸𝑆𝐿0\u00001\u0000h𝑝𝜇i\u0001 (A27)\nand by virtue of Eq. (A16) it follows that\n¯𝐻𝑖0=𝐼★h𝑛𝑖D𝜇𝑝𝜇i (A28)\nNotice that in the latter expression any dependence on the\nangle-dependent line-source function has vanished. This followsfrom the fact that in the Sobolev approximation the escape prob-\nability (and so the Sobolev optical depth) is an even function in\nthe vector𝑛. Therefore, in the mean flow the diffuse flux vanishes\nandtheEddingtonline-fluxissolelyduetoabsorptionofthedirect\ncomponent of intensity.\nFinally, the mean radiative line-force follows from Eq. (A15)\n𝑔0=\u00124𝜋𝑘𝐿\n𝜌0𝑐\u0013\n¯𝐻0=\u00124𝜋𝜅𝐼★\n𝑐\u0013\nh𝜇D𝜇𝑝𝜇i (A29)\nwherewehaveusedtheassumptionofradiallystreamingradiation.\nA4 Solution of the perturbed radiation field\nFor the solution of the perturbed transfer equation, Eq. (A19), a\ncomplication occurs due to the appearance of 𝜕𝐼0𝜕𝑥which itself\ndependsonthemeanstateline-sourcefunction 𝑆0(seeEq.(A18)).\nIndeed,𝜕𝐼0𝜕𝑥/¹𝐼0\u0000𝑆𝐿0ºwhich makes the perturbed transfer\nequation (A19)\n1\n𝑄0𝑛𝑗𝜕𝛿𝐼\n𝜕𝑟𝑗\u0000𝜕𝛿𝐼\n𝜕𝑥¸𝜏𝜇𝛿𝐼=𝜏𝜇\u0012\n𝛿𝑆𝐿¸¹𝐼0\u0000𝑆𝐿0º𝛿𝑄\n𝑄0\u0013\n(A30)\nAsolutionfor 𝐼0hasbeenfoundinEq.(A24).However,rewrit-\ning Eq. (A24) shows that ¹𝐼0\u0000𝑆𝐿0º / \u0000𝑆𝐿0exp»\u0000𝜏𝜇Φ¹𝑥º¼.\nTherefore, we cannot solve the perturbed transfer equation until\nanexpressionforthemeanstateline-sourcefunction 𝑆𝐿0isfound.\nUsing again the Sobolev approximation and applying the isotropic\nscattering condition, 𝑆𝐿0=¯𝐽, on Eq. (A27) results in\n𝑆𝐿0=𝐼★hD𝜇𝑝𝜇i\nh𝑝𝜇i (A31)\nwhich allows us to rewrite Eq. (A24) without any additional line-\nsource function dependency\n𝐼0\u0000𝑆𝐿0=𝐼★𝑒\u0000𝜏𝜇Φ¹𝑥º\u0012\nD𝜇\u0000hD𝜇𝑝𝜇i\nh𝑝𝜇i\u0013\n (A32)\nHavingthisexpression,thesolutionoftheperturbedradiation\nfield can be found. In analogy with the treatment of the magnetic\nflow,weapplyaplanewavedecompositionontotheradiationfield,\ni.e.𝛿𝐼=𝛿˜𝐼exp»𝑖¹𝑘𝑗𝑟𝑗\u0000𝜔𝑡º¼. The latter will only affect the first\nterm on the left-hand side of Eq. (A19) making a term 𝑖𝑘𝑗appear.\nThe solution strategy is similar to that of the mean radiation field\nsolution with boundary condition,\n𝛿𝐼¹𝑛𝑗𝑟𝑖𝑥!¸1º=0 (A33)\nandusinganintegratingfactor exp»𝑖𝑛𝑗𝑘𝑗𝑥𝑄0\u0000𝜏𝜇Φ¹𝑥º¼suchthat\n𝛿𝐼=𝜏𝜇𝛿𝑆𝐿𝑒𝑖¹𝑛𝑗𝑘𝑗𝑄0º𝑥\u0000𝜏𝜇Φ¹𝑥º∫¸1\n𝑥𝑑𝑥0𝜙¹𝑥0º𝑒\u0000𝑖¹𝑛𝑗𝑘𝑗𝑄0º𝑥0¸𝜏𝜇Φ¹𝑥0º\n¸𝜏𝜇𝐼★\u0012\nD𝜇\u0000hD𝜇𝑝𝜇i\nh𝑝𝜇i\u0013𝛿𝑄\n𝑄0𝑒𝑖¹𝑛𝑗𝑘𝑗𝑄0º𝑥\u0000𝜏𝜇Φ¹𝑥º\n\u0002∫¸1\n𝑥𝑑𝑥0𝜙¹𝑥0º𝑒\u0000𝑖¹𝑛𝑗𝑘𝑗𝑄0º𝑥0\n(A34)\nIn a similar fashion as in the previous Section we take the\nfrequency-integration of the perturbed intensity\n𝛿¯𝐼=𝛿𝑆𝐿𝜉¸𝐼★\u0012\nD𝜇\u0000hD𝜇𝑝𝜇i\nh𝑝𝜇i\u0013𝛿𝑄\n𝑄0𝜁 (A35)\nMNRAS 000, 1–11 (2020)Alfvén wave damping in magnetic line-driven winds 11\nwith\n𝜉\u0011𝜏𝜇∫¸1\n\u00001𝑑𝑥𝜙¹𝑥º𝑒𝑖¹𝑛𝑗𝑘𝑗𝑄0º𝑥\u0000𝜏𝜇Φ¹𝑥º\n\u0002∫¸1\n𝑥𝑑𝑥0𝜙¹𝑥0º𝑒\u0000𝑖¹𝑛𝑗𝑘𝑗𝑄0º𝑥0¸𝜏𝜇Φ¹𝑥0º(A36)\n𝜁\u0011𝜏𝜇∫¸1\n\u00001𝑑𝑥𝜙¹𝑥º𝑒𝑖¹𝑛𝑗𝑘𝑗𝑄0º𝑥\u0000𝜏𝜇Φ¹𝑥º\n\u0002∫¸1\n𝑥𝑑𝑥0𝜙¹𝑥0º𝑒\u0000𝑖¹𝑛𝑗𝑘𝑗𝑄0º𝑥0(A37)\nNotice the appearance of the perturbed line-source function\nin Eq. (A35) that still remains from Eq. (A19). For the perturbed\nline-sourcefunctionwefollowtheprocedureinEq.(A31)andfrom\nEq. (A35) we obtain\n𝛿𝑆𝐿=𝐼★\n1\u0000h𝜉i\u001c\u0012\nD𝜇\u0000hD𝜇𝑝𝜇i\nh𝑝𝜇i\u0013𝛿𝑄\n𝑄0𝜁\u001d\n (A38)\nThe perturbed Eddington line-flux becomes\n𝛿¯𝐻𝑖=𝐼★\u001c\n¹𝑛𝑖¸𝑖𝜂𝑖º\u0012\nD𝜇\u0000hD𝜇𝑝𝜇i\nh𝑝𝜇i\u0013𝛿𝑄\n𝑄0𝜁\u001d\n (A39)\nFollowing OR85, in the latter expression we have used the\nfact that Re»𝜉¼=Re»\u0000𝜉¼such thath𝜉iis always real, and that\nIm»𝜉¼≠Im»\u0000𝜉¼makingthath𝑛𝑖𝜉iisalwaysimaginary.Thisallows\nus to rewrite the appearance of 𝜉into a new quantity\n𝜂𝑖\u0011h𝑛𝑖Im»𝜉¼i\n1\u0000hRe»𝜉¼i=\u0000𝑖h𝑛𝑖𝜉i\n1\u0000h𝜉i (A40)\nPhysically speaking 𝜂𝑖contains contributions from gradients\nintheline-sourcefunction,whichpotentiallyaltersthepropagation\nproperties of unstable structures (Owocki & Puls 1996, 1999). It\nvanishes both in the short-wavelength and long-wavelength limit\nof perturbations (OR85) such that it will not affect any growth or\ndampingpropertiesofradiativewaves.Thelatterbeingthefocusof\nthis work, we will neglect it from now on making that\n𝛿¯𝐻𝑖=𝐼★\n𝑣th\u001c\n𝑖𝑛𝑖\u0012\nD𝜇\u0000hD𝜇𝑝𝜇i\nh𝑝𝜇i\u0013𝑛𝑙𝑘𝑙𝑛𝑗\n𝑄0𝜁\u001d\n𝛿𝑣𝑗 (A41)\nwhere we have inserted Eq. (A14) under a plane wave perturbation\ntoreplacetheperturbed 𝑄-factor.Inordertoevaluatetheperturbed\nEddingtonline-fluxthequantity 𝜁hastobeknown.Ingeneral 𝜁can\nbesolvedusingFourierintegrals,butitismoreconvenient(OR85)\nto apply the exponential shadowing approximation\n𝜁\u00191𝜏𝜇\n1¸𝑖𝑛𝑗𝑘𝑗¹2𝑥𝜇𝑄0º=𝑄0𝑘𝐿\n1¸��𝑛𝑗𝑘𝑗𝑄0 (A42)\nwhere𝑥¹𝜇ºis the blue-edge absorption frequency (OR85). In the\nlast equality we have made use of the fact that 𝑥𝜇\u0019𝜙¹𝑥𝜇º𝜏𝜇2is\na slowly varying function of 𝜇and of order unity (ROC90). Since\n𝑔0/¯𝐻0it follows that\n𝛿𝑔𝑖\n𝑔0=𝛿¯𝐻𝑖\n¯𝐻0 (A43)\nAs calculated above, under a plane wave perturbation 𝛿¯𝐻𝑖/\n𝛿𝑣𝑗and hence𝛿𝑔𝑖/𝛿𝑣𝑗which is the sought relation in Eq. (15).\nThe perturbed line-force to perturbed velocity tensor satisfies\n𝛿𝑔𝑖\n𝛿𝑣𝑗=𝑔0\n𝑣thh𝜇D𝜇𝑝𝜇i\n\u0002\u001c\n𝑖𝑛𝑖\u0012\nD𝜇\u0000hD𝜇𝑝𝜇i\nh𝑝𝜇i\u0013𝑛𝑙𝑘𝑙𝑛𝑗\n𝑄0𝑄0𝑘𝐿\n1¸𝑖𝑛𝑙𝑘𝑙𝑄0\u001d\n(A44)Thisistheradiativeaccelerationtensorforascattering-driven\nflow and is a special case of the general tensor relation of ROC90\nwhereby we have only considered pure scattering.\nThis paper has been typeset from a T EX/LATEX file prepared by the author.\nMNRAS 000, 1–11 (2020)" }, { "title": "1702.08394v4.Magnetization_reversal_by_superconducting_current_in___varphi_0__Josephson_junctions.pdf", "content": "arXiv:1702.08394v4 [cond-mat.supr-con] 17 Apr 2017Magnetization reversal by superconducting current in ϕ0Josephson junctions\nYu. M. Shukrinov,1,2I. R. Rahmonov,1,3K. Sengupta,4and A. Buzdin5\n1)BLTP, JINR, Dubna, Moscow Region, 141980, Russia\n2)Dubna State University, Dubna, 141980, Russia\n3)Umarov Physical Technical Institute, TAS, Dushanbe, 73406 3, Tajikistan\n4)Theoretical Physics Department, Indian Association for th e Cultivation of Science, Jadavpur, Kolkata 700032,\nIndia\n5)University Bordeaux, LOMA UMR-CNRS 5798, F-33405 Talence C edex, France\n(Dated: 4 September 2018)\nWe study magnetization reversal in a ϕ0Josephson junction with direct coupling between magnetic moment\nand Josephson current. Our simulations of magnetic moment dynam ics show that by applying an electric\ncurrent pulse, we can realize the full magnetization reversal. We pr opose different protocols of full magneti-\nzation reversal based on the variation of the Josephson junction and pulse parameters, particularly, electric\ncurrent pulse amplitude, damping of magnetization and spin-orbit int eraction. We discuss experiments which\ncan probe the magnetization reversal in ϕ0-junctions.\nKeywords: Superconducting electronics, ϕ0-junction, magnetization reversal, spin-orbit interaction\nSpintronics, which deals with an active control of spin\ndynamicsinsolidstatesystems,isoneofthemostrapidly\ndeveloping field of condensed matter physics1. An im-\nportant place in this field is occupied by superconduct-\ning spintronics dealing with the Josephson junctions (JJ)\ncoupled to magnetic systems2. The possibility of achiev-\ning electric control over the magnetic properties of the\nmagnet via Josephson current and its counterpart, i.e.,\nachieving magnetic control over Josephson current, re-\ncently attracted a lot ofattention3–5. Spin-orbit coupling\nplaysamajorroleinachievingsuchcontrol. Forexample,\nin superconductor/ferromagnet/superconductor(S/F/S)\nJJs, its presence in a ferromagnet without inversion sym-\nmetryprovidesamechanismforadirect (linear)coupling\nbetween the magnetic moment and the superconducting\ncurrent. In such junctions, called hereafter ϕ0-junction,\ntime reversal symmetry is broken, and the current–phase\nrelation is given by I=Icsin(ϕ−ϕ0), where the phase\nshiftϕ0is proportional to the magnetic moment per-\npendicular to the gradient of the asymmetric spin-orbit\npotential, and also to the applied current.6–8. Thus such\nJJs allow one to manipulate the internal magnetic mo-\nment by Josephson current6,9. The static properties of\nS/F/Sstructures are well studied both theoretically and\nexperimentally; however, the magnetic dynamics of these\nsystems has not been studied in detail beyond a few the-\noretical works6,9–14.\nThe spin dynamics associated with such ϕ0-junctions\nwas studied theoretically in Ref. 9. The authors con-\nsidered a S/F/S ϕ 0-junction in a low frequency regime\nwhich allowed usage of quasi-static approach to study\nmagnetizationdynamics. It wasdemonstrated that a DC\nsuperconductingcurrentproducesastrongorientationef-\nfect on the magnetic moment of the ferromagnetic layer.\nThus application of a DC voltage to the ϕ0-junction is\nexpected to lead to current oscillations and consequently\nmagnetic precession. This precession can be monitored\nby the appearance of higher harmonics in current-phase\nrelation; in addition, it also leads to the appearance ofa DC component of the current which increases near a\nferromagnetic resonance9. It is then expected that the\npresence of external radiation in such a system would\nlead to several phenomena such as appearance of half-\ninteger steps in the current-voltage (I-V) characteristics\nof the junction and generation of an additional magnetic\nprecession with frequency of external radiation9.\nIn this paper we study the magnetization reversal in\nϕ0-junction with direct coupling between magnetic mo-\nment and Josephson current and explore the possibility\nofelectricallycontrollablemagnetizationreversalinthese\njunctions. We carry out investigations of the magneti-\nzation dynamics for two types of applied current pulse:\nrectangular and Gaussian forms. An exact numerical\nsimulation of the dynamics of magnetic moment of the\nferromagnetic layer in the presence of such pulses allows\nus to demonstrate complete magnetization reversal in\nthese systems. Such reversal occurs for specific param-\neters of the junction and the pulse. We chart out these\nparameters and suggest a possible way for determination\nof spin-orbit coupling parameter in these systems. We\ndiscuss the experiment which can test our theory.\nIn order to study the dynamics of the S/F/S system,\nwe use the method developed in Ref. 9. We assume that\nthe gradient of the spin-orbit potential is along the easy\naxis of magnetization taken to be along ˆ z. The total\nenergy of this system can be written as\nEtot=−Φ0\n2πϕI+Es(ϕ,ϕ0)+EM(ϕ0),(1)\nwhereϕis the phase difference between the supercon-\nductors across the junction, Iis the external current,\nEs(ϕ,ϕ0) =EJ[1−cos(ϕ−ϕ0)], andEJ= Φ0Ic/2πis\nthe Josephson energy. Here Φ 0is the flux quantum, Icis\nthe critical current, ϕ0=lυsoMy/(υFM0),υFis Fermi\nvelocity, l= 4hL//planckover2pi1υF,Lis the length of Flayer,his\ntheexchangefieldofthe Flayer,EM=−KVM2\nz/(2M2\n0),\nthe parameter υso/υFcharacterizesa relative strength of\nspin-orbit interaction, Kis the anisotropic constant, and\nVis the volume of the Flayer.2\nThe magnetization dynamics is described by the\nLandau-Lifshitz-Gilbert equation3(see also Supplemen-\ntary Material) which can be written in the dimensionless\nform as\ndmx\ndt=1\n1+α2/braceleftbig\n−mymz+Grmzsin(ϕ−rmy)\n−α/bracketleftbig\nmxm2\nz+Grmxmysin(ϕ−rmy)/bracketrightbig/bracerightbig\n,\ndmy\ndt=1\n1+α2/braceleftbig\nmxmz\n−α/bracketleftbig\nmym2\nz−Gr(m2\nz+m2\nx)sin(ϕ−rmy)/bracketrightbig/bracerightbig\n,\ndmz\ndt=1\n1+α2/braceleftbig\n−Grmxsin(ϕ−rmy)\n−α/bracketleftbig\nGrmymzsin(ϕ−rmy)−mz(m2\nx+m2\ny)/bracketrightbig/bracerightbig\n,(2)\nwhereαis a phenomenological Gilbert damping con-\nstant,r=lυso/υF, andG=EJ/(KV). Themx,y,z=\nMx,y,z/M0satisfy the constraint/summationtext\nα=x,y,zm2\nα(t) = 1. In\nthis system of equations time is normalized to the in-\nverse ferromagnetic resonance frequency ωF=γK/M 0:\n(t→tωF),γis the gyromagnetic ratio, and M0=/ba∇dblM/ba∇dbl.\nIn what follows, we obtain time dependence of magne-\ntizationmx,y,z(t), phase difference ϕ(t) and normalized\nsuperconducting current Is(t)≡Is(t)/Ic= sin(ϕ(t)−\nrmy(t)) via numerical solution of Eq.(2).\nLet us first investigate an effect of superconducting\ncurrent on the dynamics of magnetic momentum. Our\nmain goal is to search for cases related to the possibility\nof the full reversal of the magnetic moment by supercon-\nducting current. In Ref.9 the authors have observed a\nperiodic reversal, realized in short time interval. But, as\nwe see in Fig. 1, during a longtime interval the character\nofmzdynamics changes crucially. At long times, /vector mbe-\ncomes parallel to y-axis, as seen from Fig.1(b)) demon-\nstrating dynamics of my. The situation is reminiscent\nof Kapitza pendulum (a pendulum whose point of sus-\npension vibrates) where the external sinusoidal force can\ninvert the stability position of the pendulum.15Detailed\nfeatures of Kapitza pendulum manifestation will be pre-\nsented elsewhere.\ntmz\n050100150200-1-0.500.51(a)\nr=0.1,G=500 π,\nα=0.1, ω=5 mz\n0 0.5-101\ntmy\n010203000.20.40.60.81(b)\ntmy\n0 0.501\nFIG. 1. (a) Dynamics of mzin case of ωJ= 5,G= 500π,r=\n0.1,α= 0.1. The inset shows the character of time depen-\ndence in the beginning of the time interval; (b) The same as\nin (a) for my.\nThe question we put here is the following: is it pos-\nsible to revers the magnetization by the electric current\npulse and then preserve this reversed state. The answer\nmay be found by solving the system of equations (2) to-\ngether with Josephson relation dϕ/dt=V, written inthe dimensionless form. It was demonstrated in Ref.13\nthat using a specific time dependence of the bias voltage,\napplied to the weak link leads to the reversal of the mag-\nnetic moment of the nanomagnet. The authors showed\nthe reversal of nanomagnet by linearly decreasing bias\nvoltageV= 1.5−0.00075t(see Fig.3 in Ref.13). The\nmagnetization reversal, in this case, was accompanied by\ncomplex dynamical behavior of the phase and continued\nduring a sufficiently long time interval.\nIncontrast,inthepresentworkweinvestigatethemag-\nnetization reversal in the system described by the equa-\ntions (2) under the influence of the electric current pulse\nof rectangular and Gaussian forms. The effect of rectan-\ngular electric current pulse are modeled by Ipulse=As\nin the ∆ttime interval ( t0−∆t\n2,t0+∆t\n2) andIpulse= 0\nin other cases. The form of the current pulse is shown in\nthe inset to Fig.2(a).\nHere we consider the JJ with low capacitance C (\nR2C/LJ<<1, where LJis the inductance of the JJ and\nRis its resistance), i.e., we do not take into account the\ndisplacement current. So, the electric current through\nJJs is\nIpulse=wdϕ\ndt+sin(ϕ−rmy) (3)\nwherew=VF\nIcR=ωF\nωR,VF=/planckover2pi1ωF\n2e,Ic- critical current, R-\nresistance of JJ, ωR=2eIcR\n/planckover2pi1- characteristic frequency.\nWe solved the system of equations ( 2) together with\nequation ( 3) and describe the dynamics of the system.\nTime dependence of the electric current is determined\nthrough time dependence of phase difference ϕand mag-\nnetization components mx,my,mz.\nWe first study the effect ofthe rectangularpulse shown\nin the inset to Fig.2(a). It is found that the reversal of\nmagnetic moment can indeed be realized at optimal val-\nues of JJ ( G,r) and pulse ( As,∆,t0) parameters . An\nexample of the transition dynamics for such reversal of\nmzwith residual oscillation is demonstrated in Fig. 2(a);\nthe corresponding parameter values are shown in the fig-\nure.\nDynamics of the magnetic moment components, the\nphase difference and superconducting current is illus-\ntrated in Fig.2(b). We see that in the transition region\nthe phase difference changes from 0 to 2 πand, corre-\nspondingly, the superconducting current changes its di-\nrection twice. This is followed by damped oscillation of\nthe superconducting current. There are some character-\nistic time points in Fig.2(b), indicated by vertical dashed\nlines. Line 1 correspondsto a phase difference of π/2 and\nindicate maximum of superconducting current Is. The\nline 1′which corresponds to the maximum of my, and\nmz= 0 has a small shift from line 1. This fact demon-\nstrates that, in general, the characteristic features of mx\nandmytime dependence do not coincide with the fea-\ntures on the Is(t), i.e., there is a delayin reaction ofmag-\nnetic moment to the changes of superconducting current.\nAnother characteristic point corresponds to the ϕ=π.\nAt this time line 2 crosses points Is= 0,my= 0, and3\ntmi, Is\nϕ\n222426283032-1-0.500.51\n0123456mz\nIs\nmyϕ\nπ/2π3π/21’23\n1\nOn Off(b)\nFIG. 2. Transition dynamics of the magnetization compo-\nnentmzfor a system with rectangular current pulse shown\nin the inset; (b) Dynamics of magnetization components to-\ngether with the phase difference ϕand superconducting cur-\nrentIs. Arrows indicate the beginning and end of the electric\ncurrent pulse. Vertical dashed lines indicate the common fe a-\ntures while the horizontal ones mark the corresponding valu es\nof the phase difference.\nminimum of mz. At time moment when ϕ= 3π/2 line\n3 crosses minimum of Is. When pulse is switched off,\nthe superconducting current starts to flow through the\nresistance demonstrating damped oscillations and caus-\ningresidualoscillationsofmagneticmomentcomponents.\nNote also, that the time at which the current pulse ends\n(t= 28) is actually does not manifest itself immediately\nin themy(and not shown here mx) dynamics. They\ndemonstrate continuous transition to the damped oscil-\nlating behavior.\nFig. 2(b) provides us with a direct way of de-\ntermining the spin-orbit coupling strength in the\njunction via estimation of r. For this, we note\nthat the ϕ(t) =ϕ00+/integraltextt\n0V(t′)dt′can be deter-\nmined, up to a initial time-independent constant\nϕ00, in terms of the voltage V(t) across the junc-\ntion. Moreover, the maxima and minima of Isoccurs\nat times tmaxandtmin(see Fig. 2(b)) for which\nsin/bracketleftBig\nϕ00+/integraltexttmax[tmin]\n0V(t′)dt′−rmy(tmax[tmin])/bracketrightBig\n=+[−]1.Eliminating ϕ0from these equations, one gets\nsin1\n2/bracketleftbigg/integraldisplaytmin\ntmaxV(t′)dt′+r[my(tmax)−my(tmin)]/bracketrightbigg\n= 1\n(4)\nwhich allows us, in principal, to determine rin terms\nof the magnetization myat the position of maxima and\nminima of the supercurrent and the voltage Vacross the\njunction. We stress that for the experimental realization\nof proposed method one would need to resolve the value\nofthe magnetizationat the time difference ofthe orderof\n10−10-10−9c. Atthepresentstagethestudyofthemag-\nnetization dynamics with such a resolution is extremely\nchallenging. To determine the spin-orbit coupling con-\nstantrexperimentally it may be more convenient to vary\nthe parameters of the current pulse I(t) and study the\nthreshold of the magnetic moment switching.\nThe dynamics of the system in the form of magneti-\nzation trajectories in the planes my−mxandmz−mx\nduring a transition time interval at the same parameters\nof the pulse and JJ at α= 0 is presented in Fig.3. We\nmxmy\n0 0.5 1-1-0.500.51\nB\nA’\nCA\nQ(a)\nmxmz\n0 0.5 1-1-0.500.51\nB\nA’QA\nC(c)\nFIG. 3. Trajectories of magnetization components in the\nplanesmy−mxin the transition region: (a) during electric\npulse action (between points AandC), (b)after switchingthe\npulse off; In (c) and (d) the same is shown for the my−mz\nplane. Parameters of the pulse and the JJ are the same as in\nFig.2(a) at α= 0.\nsee that magnetic moment makes a spiral rotation ap-\nproaching the state with mz=−1 after switching off the\nelectric current pulse. The figures show clearly the spe-\ncific features of the dynamics around points B,A′and\nQand damped oscillations of the magnetization compo-\nnents (see Fig.3(b) and Fig.3(d)). The cusps at point B\nin Fig.3(a) corresponds just to the change from an in-\ncreasing of absolute value of mxto its decreasing and,\nopposite, at point A′in Fig.3(c). The behavior of mag-\nnetic system happens to be sensitive to the parameters4\nof the electric current pulse and JJ. In the Supplement\nwe show three additional protocols of the magnetization\nreversal by variation of As,Gandr.\nIt is interesting to compare the effect of rectangular\npulse with the Gaussian one of the form\nIpulse=As1\nσ√\n2πexp/parenleftbigg\n−(t−t0)2\n2σ2/parenrightbigg\n.(5)\nwhereσdenotes the full width at half-maximum of the\npulse and Ais its maximum amplitude at t=t0. In this\ncase we also solve numerically the system of equations\n(2) together with equation (3) using (5). An example\nof magnetic moment reversal in this case is presented in\nFigure 4, which shows the transition dynamics of mzfor\nthe parameters r= 0.1,G= 10,As= 5,σ= 2 at\nsmall dissipation α= 0.01. We see that the magneti-\ntmz\n0 50 100 150-1-0.500.51\nα=0.01\nG=10,r=0.1tIsignal\n152025303500.51t0=25\nσ=2As=5\nFIG. 4. Demonstration of transition dynamics of mzfor a\nGaussian electric current pulse (shown in the inset).\nzation reversal occurs more smoothly in compare with a\nrectangular case.\nWe also note that very important role in the reversal\nphenomena belongs to the effect of damping. It’s de-\nscribed by term with αin the system of equations (2),\nwhereαis a damping parameter. The examples of the\nmagnetization reversal at G= 50,r= 0.1 and different\nvalues of αare presented in Fig.5. We see that dissi-\ntmz\n0204060-1-0.500.51\nα=0.07(b)\nFIG. 5. Magnetization dynamics under rectangular pulse\nsignal in the system at different values of the dissipation pa -\nrameter α.\npation can bring the magnetic system to full reversal,even if at α= 0 the system does not demonstrate re-\nversal. Naturally, the magnetic moment, after reversal,\nshows some residual oscillations as well. We stress that\nthe full magnetization reversal is realized in some fixed\nintervals of dissipation parameter. As expected, the vari-\nation of phase difference by πreflects the maxima in the\ntime dependence of the superconducting current. Fig.6\ndemonstrates this fact. The presented data shows that\ntIs\nϕ\n0 20 40-1-0.500.51\n05101520ϕ\nIs5ππ\n6π2πG=50,\nr=0.1,\nAs=2,\ndt=10,\nt0=25\nα=0.07\nFIG. 6. Transition dynamics of the phase difference and the\nsuperconducting current for the case presented in Fig.5(b) .\nthe total change of phase difference consists of 6 π, which\ncorresponds to the six extrema in the dependence Is(t).\nAfterthefullmagnetizationreversalisrealized,thephase\ndifference shows the oscillations only.\nOne of the important aspect of the results that we ob-\ntain here is the achievement of a relatively short switch-\ning time interval for magnetization reversal. As we have\nseen in Figs. 2(a) and 4, the time taken for such reversal\nisωFt≃100 which translate to 10−8seconds for typical\nωF≃10GHz. We note that this amounts to a switching\ntime which is 1 /20thof that obtained in Ref. 13.\nExperimental verification of our work would involve\nmeasurement of mz(t) in aϕ0junction subjected to a\ncurrent pulse. For appropriate pulse and junction pa-\nrameters as outlined in Figs. 4 and 5, we predict obser-\nvation of reversal of mzat late times ωFt≥50. More-\nover, measurement of myat times tmaxandtminwhereIs\nreaches maximum and minimum values and the voltage\nV(t) acrossthe junction between these times would allow\nfor experimental determination of rvia Eq. 4.\nAs a ferromagnet we propose to use a very thin F\nlayer on dielectric substrate. Its presence produces the\nRashba-type spin-orbit interaction and the strength of\nthis interaction will be large in metal with large atomic\nnumber Z. The appropriate candidate is a permalloy\ndoped with Pt.16InPtthe spin-orbit interaction play\na very important role in electronic band formation and\nthe parameter υso/υF, which characterizes the relative\nstrength of the spin-orbit interaction is υso/υF∼1. On\nthe other hand, the Ptdoping of permalloy up to 10 %\ndid not influenced significantly its magnetic properties16\nand then we may expect to reach υso/υFin this case 0.1\nalso. If the length of Flayer is of the order of the mag-5\nnetic decaying length /planckover2pi1υF/h, i.e.l∼1, we have r∼0.1.\nAnother suitable candidate may be a Pt/Cobilayer, fer-\nromagnetwithoutinversionsymmetrylikeMnSiorFeGe.\nIf the magnetic moment is oriented in plane of the F\nlayer,thanthespin-orbitinteractionshouldgeneratea ϕ0\nJosephson junction6with finite ground phase difference.\nThe measurement of this phase difference (similar to the\nexperiments in Ref.17) may serve as an independent way\nfor the parameter revaluation. The parameter Ghas\nbeen evaluated in Ref.9 for weak magnetic anisotropy of\npermalloy K∼4×10−5K·˚A−3(see Ref.18) and S/F/S\njunction with l∼1 andTc∼10K asG∼100. For\nstronger anisotropy we may expect G∼1.\nIn summary, we have studied the magnetization rever-\nsal inϕ0-junction with direct coupling between magnetic\nmoment and Josephson current. By adding the electric\ncurrent pulse, we have simulated the dynamics of mag-\nnetic moment components and demonstrate the full mag-\nnetization reversal at some parameters of the system and\nexternal signal. Particularly, time interval for magneti-\nzation reversal can be decreased by changing the ampli-\ntude of the signal and spin-orbit coupling. The observed\nfeatures might find an application in different fields of\nsuperconducting spintronics. They can be considered as\na fundamental basis for memory elements, also.\nSee supplementary material for demonstration of dif-\nferent protocols of the magnetization reversal by vari-\nation of Josephson junction and electric current pulse\nparameters.\nAcknowledgment : The authors thank I. Bobkova and\nA. Bobkov for helpful discussion. The reported study\nwas funded by the RFBR research project 16–52–\n45011India, 15–29–01217, the the DST-RFBR grant\nINT/RUS/RFBR/P-249, and the French ANR projects\n”SUPERTRONICS”.\n1I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323\n(2004).\n2Jacob Linder and W. A. Jason Robinson, Nature Physics, 11,\n307 (2015).\n3A. I. Buzdin, Rev. Mod. Phys., 77, 935 (2005).\n4F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod. Phys. ,\n77, 1321 (2005).\n5A. A. Golubov, M. Y. Kupriyanov, and E. Ilichev, Rev. Mod.\nPhys. ,76, 411 (2004).\n6A. Buzdin, Phys. Rev. Lett., 101, 107005 (2008).\n7I. V. Krive, A. M. Kadigrobov, R. I. Shekhter, and M. Jonson,\nPhys. Rev. B 71, 214516 (2005).\n8A. A. Reynoso, G. Usaj, C. A. Balseiro, D. Feinberg, and M.\nAvignon, Phys. Rev. Lett. 101, 107001 (2008).\n9F. Konschelle, A. Buzdin, Phys. Rev. Lett ., 102, 017001 (2009).\n10X. Waintal and P. W. Brouwer, Phys. Rev., B 65, 054407 (2002).\n11V. Braude and Ya. M. Blanter, Phys. Rev. Lett., 100, 207001\n(2008).\n12J. Linder and T. Yokoyama, Phys. Rev., B 83, 012501 (2011).\n13Liufei Cai and E. M. Chudnovsky, Phys. Rev., B 82, 104429\n(2010).\n14Eugene M. Chudnovsky, Phys. Rev., B 93, 144422 (2016).\n15P. L. Kapitza, Soviet Phys. JETP, B21, 588-592 (1951); Usp.\nFiz. Nauk, B44, 7-15 (1951).\n16A. Hrabec, F. J. T. Goncalves, C. S. Spencer, E. Arenholz, A. T .\nNDiaye, R. L. Stamps and Christopher H. Marrows, Phys. Rev.\nB93, 014432 (2016).17D. B. Szombati, S. Nadj-Perge, D. Car, S. R. Plissard, E. P.\nA. M. Bakkers and L. P. Kouwenhoven, Nture Physics 12, 568\n(2016).\n18A. Yu. Rusanov, M. Hesselberth, J. Aarts, A. I. Buzdin, Phys.\nRev. Lett. 93, 057002 (2004).6\nSupplementary Material to “Magnetization reversal by\nsuperconducting current in ϕ0Josephson junctions”\nI. GEOMETRY AND EQUATIONS\nGeometry of the considered ϕ0-junction9is presented\nin Fig. 7. The ferromagnetic easy-axis is directed along\nthe z-axis, which is also the direction n of the gradient of\nthe spin-orbit potential. The magnetization component\nmyis coupled with Josephson current through the phase\nshift term ϕ0∼(− →n[− →m− →∇Ψ]), where Ψ is the supercon-\nducting order parameter (− →∇Ψ is along the x-axis in the\nsystem considered here).\nSyz,n\nx, IsM\nF S\nFIG. 7. Geometry of the considered ϕ0-junction.\nIn order to study the dynamics of the S/F/S system,\nwe use the method developed in Ref. 9. We assume that\nthe gradient of the spin-orbit potential is along the easy\naxis of magnetization taken to be along ˆ z. The total\nenergy of this system is determined by the expression\n(S1) in the main text.\nThe magnetization dynamics is described by the\nLandau-Lifshitz-Gilbert equation3\ndM\ndt=−γM×Heff+α\nM0/parenleftbigg\nM×dM\ndt/parenrightbigg\n(6)\nwhereγis the gyromagnetic ratio, αis a phenomeno-\nlogical Gilbert damping constant, and M0=/ba∇dblM/ba∇dbl. The\neffective field experienced by the magnetization Mis de-\ntermined by Heff=−1\nV∂Etot\n∂M, so\nHeff=K\nM0/bracketleftbigg\nGrsin/parenleftbigg\nϕ−rMy\nM0/parenrightbigg\n/hatwidey+Mz\nM0/hatwidez/bracketrightbigg\n(7)\nwherer=lυso/υF, andG=EJ/(KV).\nUsing (6) and (7), we obtain the system of equations\n(2) in main text, which describes the dynamics of the\nSFS structure.II. MAGNETIZATION REVERSAL UNDER ELECTRIC\nCURRENT PULSE\nMagnetic system is very sensitive to the parameters of\nthe electric current pulse and Josephson junction. Here\nwe show three additional protocols of the magnetization\nreversal by variation of As,Gandr.\nA. Effect of As-variation\nFigure 8 demonstrates the magnetization reversal by\nchanging pulse parameter As. We see that change of\nV1V2\n26.52727.5-1.01-1-0.99-0.98\n1.38\n1.41.44\n1.42\ntmz\n0 50 100 150-1-0.500.51\n1.3\n1.4G=10,r=0.1,\n∆t=6\n1.5DifferentAs\nFIG. 8. Magnetization reversal by changing pulse parameter\nAs. The number near curve shows value of As.\npulse amplitude As= 1.3 toAs= 1.4 reverses magnetic\nmoment. At As= 1.5 this feature is still conserved, but\ndisappears at larger values.\nB. Effect of G-variation\nFigure 9 demonstrates the magnetization reversal by\nchanging Josephson junction parameter G.\nC. Effect of r-variation\nFigure 10 demonstrates the magnetization reversal by\nchanging Josephson junction parameter of spin-orbital\ncoupling r. Figure 10 demonstrates the magnetization\nreversal by changing Josephson junction parameter of\nspin-orbital coupling r.We see that there is a possibil-\nity of magnetization reversal around G= 10. In this\ncase a decrease of spin-orbit parameter may lead to the\nmagnetization reversal also. The magnetization reversal7\ntmz\n0 50 100 150-1-0.500.51\n7\n10DifferentG\nr=0.1,\nAs=1.4,∆t=6\nFIG. 9. Magnetization reversal by changing Josephson junc-\ntion parameter G. The number near curve shows value of\nG.tmz\n0 50 100 150-1-0.500.51\n0.10.5\nDifferentr\nG=10,\nAs=1.4,∆t=60.3\nFIG. 10. Magnetization reversal by changing Josephson junc -\ntion parameter of spin-orbital coupling r. The number near\ncurve shows value of G.\ndepends on the other parameters of the system and, nat-\nurally, the minimal value ofparameter rdepends ontheir\nvalues. Inparticularcasepresentedhereitisaround0.05." }, { "title": "1611.05798v2.Inductive_detection_of_field_like_and_damping_like_AC_inverse_spin_orbit_torques_in_ferromagnet_normal_metal_bilayers.pdf", "content": "arXiv:1611.05798v2 [cond-mat.mtrl-sci] 26 Oct 2017Inductive detection of field-like and damping-like AC inver se spin-orbit torques in\nferromagnet/normal metal bilayers\nAndrew J. Berger,1Eric R. J. Edwards,1Hans T. Nembach,1\nAlexy D. Karenowska,2Mathias Weiler,3,4and Thomas J. Silva1,∗\n1Quantum Electromagnetics Division, National Institute of Standards and Technology, Boulder, CO 80305, U.S.A.†\n2Department of Physics, University of Oxford, Oxford, U.K.\n3Walther-Meißner-Institut, Bayerische Akademie der Wisse nschaften, Garching, Germany\n4Physik-Department, Technische Universit¨ at M¨ unchen, Ga rching, Germany\n(Dated: October 9, 2018)\nFunctional spintronic devices rely on spin-charge interco nversion effects, such as the reciprocal\nprocesses of electric field-driven spin torque and magnetiz ation dynamics-driven spin and charge\nflow. Both damping-like and field-like spin-orbit torques ha ve been observed in the forward process\nof current-driven spin torque and damping-like inverse spi n-orbit torque has been well-studied via\nspin pumping into heavy metal layers. Here we demonstrate th at established microwave transmis-\nsion spectroscopy of ferromagnet/normal metal bilayers un der ferromagnetic resonance can be used\nto inductively detect the AC charge currents driven by the in verse spin-charge conversion processes.\nThis technique relies on vector network analyzer ferromagn etic resonance (VNA-FMR) measure-\nments. We show that in addition to the commonly-extracted sp ectroscopic information, VNA-FMR\nmeasurements can be used to quantify the magnitude and phase of all AC charge currents in the\nsample, including those due to spin pumping and spin-charge conversion. Our findings reveal that\nNi80Fe20/Pt bilayers exhibit both damping-like and field-like inver se spin-orbit torques. While the\nmagnitudes of both the damping-like and field-like inverse s pin-orbit torque are of comparable scale\nto prior reported values for similar material systems, we ob served a significant dependence of the\ndamping-like magnitude on the order of deposition. This sug gests interface quality plays an impor-\ntant role in the overall strength of the damping-like spin-t o-charge conversion.\nI. INTRODUCTION\nSpin-charge transduction effects for ferromag-\nnet/nonmagnet (FM/NM) multilayers couple electric\nfields to magnetic torques in the forward process\n(so-called spin-orbit torque (SOT)), and they couple\nmagnetization dynamics to currents in the inverse\nprocess (iSOT). In general, these torques can be\nphenomenologically separated into two components:\ndamping-like and field-like. Both are perpendicular to\nthe FM magnetization, but the damping-like torque\nis odd under time-reversal and dissipative, whereas\nthe field-like torque is even under time-reversal and\nconservative1. A classic example of a field-like torque is\nthe action of an Oersted field on a FM magnetization\ndue to a charge current in an adjacent conducting layer.\nBy Onsager reciprocity, the inverse process is captured\nby Faraday’s law: magnetization dynamics in the FM\ngeneratechargecurrentsin the NM. Recently, it hasbeen\nfound that spin-orbit coupling (SOC) in multilayers can\ngive rise to both field- and damping-like SOTs2,3, but\nwith substantially different scaling than that achieved\nwith Oersted fields. Unlike the Oersted effect, these\nspin-orbitronic effects are short-range, making them\nhighly advantageous for microelectronic applications\nthat require device scaling to high densities such as\n∗thomas.silva@nist.gov\n†Contribution of the National Institute of Standards and Tec h-\nnology; not subject to copyright.nonvolatile memory and alternative state-variable\nlogic4,5.\nDamping-like torquesdue to the spin Hall effect (SHE)\nin heavy NM layers such as Pt and β-Ta are well-studied\nand understood, and have been investigated in both\nforward4and inverse configurations6–8. Substantial field-\nlike torques have also been measured for FM/NM inter-\nfaces in the forward configuration2,9–11. However, an in-\nversemeasurement ofthe field-like torque in Ni 80Fe20/Pt\nhas not yet been unambiguosly reported12. Here, we\npresent simultaneous measurements of inverse field-like\nand damping-like torques in Ni 80Fe20/Pt bilayers via\nwell-established coplanar waveguide (CPW) ferromag-\nnetic resonance (FMR). Time-varying magnetic fields\nproduced by a FM/NM sample under FMR excitation\nwill inductively couple into the CPW, altering the to-\ntal inductance of the microwave circuit. Such fields\nare produced by: (1) the Py precessing magnetization,\n(2) Faraday effect induced AC currents in the Pt layer,\nand (3) spin-orbit AC currents due to damping-like and\n(4) field-like processes. We show that through proper\nbackground normalization, combined with Onsager reci-\nprocity for the specific phenomenology of these measure-\nments, commonly-used vector network analyzer (VNA)\nFMR spectroscopy allows accurate identification of the\nprocesses that contribute to spin-charge conversion.\nThe paper is organized as follows. In Sec. II, by\nappealing to Onsager reciprocity we provide the phe-\nnomenological background relating the forward and in-\nverseprocessesthatproducemagnetictorquesandcharge\nflow in a ferromagnet/normal metal system under elec-2\ntrical bias or with excited magnetization dynamics. Sec.\nIII describes the quantitative VNA-FMR technique, and\nderives the expressions we use to calculate the sample’s\ncomplex inductance. This section also introduces the ef-\nfective conductivity ˜ σNMthat quantifies the magnitude\nand symmetry of magnetic torques due to applied charge\ncurrents,andreciprocally,oftheACchargecurrentsflow-\ning in a sample in response to the driven magnetization\ndynamics. In Sec. IV, we present data acquired from\nNi80Fe20/Pt bilayers and Ni 80Fe20/Cu control samples.\nThe magnitude of the phenomenological parameter ˜ σNM\nextracted from these data is well within the range of re-\nported values, and it obeys the usual symmetry proper-\ntiesassociatedwith thestackingorderoftheNi 80Fe20and\nPt layers. Finally, we discuss the results in Sec. V by\ncomparing our extracted iSOT parameters to the micro-\nscopic spin-chargeconversionparameters of spin Hall an-\ngle and Rashba parameter. In all cases, the magnitudes\nof the extracted spin Hall angle and Rashba parameter\nare in rough agreement with what has been reported in\nthe literature, though this agreementis contingenton the\nassumption of typical values for the interfacial and bulk\nspin transport parameters. However, we find that theextracted spin Hall angle changes by a factor of almost 4\ndepending on the growth order of the multilayer stacks,\nwith a larger spin Hall angle when the Pt is grown on\ntop of the Ni 80Fe20. This suggests that the spin trans-\nport parameters are in actuality highly dependent on the\nstack growth order.\nII. ONSAGER RELATIONS FOR SPIN-ORBIT\nTORQUE\nOnsager reciprocity relations13are well known for cer-\ntain pairs of forces and flows. For example, for thermo-\nelectriceffects, appliedelectricfieldsorthermalgradients\ncan drive both charge and heat flow. In this section,\nwe establish Onsager relations for charge current and\nmagnetic torque as the flows that are driven by applied\nelectric fields and magnetization dynamics in a FM/NM\nmultilayer1.\nBy analogy to Ohm’s Law, J=σE, we can write a\ngeneral matrix equation relating driving forces (magne-\ntization dynamics ∂ˆm/∂tand electric field E) to flows\n(magnetic torque density Tand charge current density\nJ)1:\n\n/parenleftbigg2e\n/planckover2pi1/parenrightbigg\n+dFM/integraldisplay\n0T(z)dz\n\n\n+dFM/integraldisplay\n−dNMJ(z)dz\n\n=\nG\nGmag sgn(ˆz·ˆn)/parenleftbig\n−σF\ne+σSOT\ne−σSOT\no[ˆm×]/parenrightbig\nsgn(ˆz·ˆn)/parenleftbig\n−σF\ne+σSOT\ne−σSOT\no[ˆm×]/parenrightbig\n−1\nZeff\n\n∗\n/parenleftbigg/planckover2pi1\n2e/parenrightbigg∂ˆm\n∂t\nˆz×E\n(1)\nwhere ˆmis the magnetization unit vector, /planckover2pi1is Planck’s\nconstant divided by 2 π,eis the electron charge, dFMand\ndNMare the FM and NM thicknesses. The terms in the\n2×2 conductivity matrix are described below. The sign\nof the off-diagonal terms are determined by sgn(ˆ z·ˆn),\nwhere ˆnis an interface normal pointing into the FM.\nThe coordinate unit vector ˆ zis defined by the sample\nplacement on the CPW, as shown in Fig. 1(a), and z= 0\nis defined by the FM/NM interface. Gis a 2×2 matrix\nimposing geometrical constraints: (1) magnetic torques\nareorthogonalto ˆ mand(2) chargecurrentscanflowonly\nin thex,yplane:\nG=/bracketleftbigg\n[ˆm×] 0\n0 [ˆz×]/bracketrightbigg\n(2)Thediagonalelementsoftheeffectiveconductivityma-\ntrix describe the Gilbert damping of the FM and charge\nflow in the metallic bilayer in response to an applied elec-\ntric field. That is,\n/parenleftbigg2e\n/planckover2pi1/parenrightbigg\n+dFM/integraldisplay\n0T(z)dz\n=/parenleftbigg/planckover2pi1\n2e/parenrightbigg\nGmag/parenleftbigg\nˆm×∂ˆm\n∂t/parenrightbigg\n(3)\n\n+dFM/integraldisplay\n−dNMJ(z)dz\n=−1\nZeffˆz×(ˆz×E) (4)\nwhereGmag≡ −dFM(2e//planckover2pi1)2(αMs/γ),αis the Gilbert\ndamping parameter, and γis the gyromagnetic ratio,3\nsuch that Eq. 3 is the usual Gilbert damping term from\nthe Landau-Lifshitz-Gilbert equation:\n∂ˆm\n∂t=−γµ0ˆm×H−/parenleftbiggγ\nMsdFM/parenrightbigg+dFM/integraldisplay\n0T(z)dz(5)In Eq. 4, Zeffis the effective frequency-dependent\nimpedance of the bilayer. Eq. 4 reduces to Ohm’s Law\nin the DC limit ( Zeff→R/squareasf→0).\nThe off-diagonal terms describe the electromagnetic\nreciprocity between Faraday’s and Ampere’s Law14,15, as\nwell as spin-orbit torques (SOT) and their inverse, using\nthe effective conductivities σF\ne,σSOT\ne, andσSOT\no.\n/parenleftbigg2e\n/planckover2pi1/parenrightbigg\n+dFM/integraldisplay\n0T(z)dz\n= sgn(ˆz·ˆn)ˆm×/parenleftbig\n−σF\ne+σSOT\ne−σSOT\no[ˆm×]/parenrightbig\n(ˆz×E) (6)\n\n+dFM/integraldisplay\n−dNMJ(z)dz\n= sgn(ˆz·ˆn)/parenleftbigg/planckover2pi1\n2e/parenrightbigg\nˆz×/parenleftbig\n−σF\ne+σSOT\ne−σSOT\no[ˆm×]/parenrightbig∂ˆm\n∂t(7)\nHere, the superscripts indicate the source of the torque\nor current as due to the Faraday effect or SOT. The sub-\nscripts indicate “even” or “odd” with respect to time-\nreversal, which determines the torque direction or phase\nof the corresponding current with respect to the driving\nelectric field or magnetization dynamics.\nFirst consider the Faraday conductivity, σF\ne. In the\nforward process an electric field Eproduces a charge cur-\nrent, which by Ampere’s Law produces a magnetic field.\nThis field exerts a torque Ton the magnetization of the\nFM layer. In the reverse process, magnetization dynam-\nics∂tˆmproduce an AC magnetic field, which by Fara-\nday’s Law induces a chargecurrent Jin the NM layer. In\nthis way, σF\nequantifies the reciprocity between Ampere’s\nand Faraday’s Law (see Eq. 31 for an estimate of the σF\ne\nmagnitude based on material properties). Inclusion of\nthe terms in Eq. 1 due to electrodynamic reciprocity is\ncritical for the proper interpretation of inverse spin orbit\ntorque experiments12.\nAlsopresentin the off-diagonaltermsareSOTconduc-\ntivities due to spin-charge conversion. In Eq. 6, these\nmanifest as electric-field driven damping-like torques,\nwhich are proportional to ˆ m×(ˆm×(ˆz×E)), and field-\nlike torques, which are proportional to ˆ m×(ˆz×E). The\nconstantsofproportionalitybetweenappliedelectricfield\nand SOTs are σSOT\noandσSOT\ne. In the reverse direction\n(Eq. 7), these effects are responsible for spin-to-charge\nconversion (e.g., inverse spin Hall effect (iSHE)16or in-\nverse Rashba-Edelstein effect (iREE)17).\nReporting effective conductivities, as opposed to spin-\ncharge conversion parameters like the spin Hall angle,\ndirectly relates the microwave inputs and charge current\noutputs of an iSOT device without the need for separate\ncharacterization of spin-mixing conductance or spin dif-\nfusion length. Reciprocally, in a spin torque experiment\nwith charge current inputs and magnetization dynam-\nics (or switching) as output, the effective conductivities\nprovide the ideal figure of merit for determining magne-\ntization oscillation and switching thresholds of the ap-plied current. To estimate the critical current density Jc\nneeded to switch the magnetization of a ferromagnetic\nlayer18,19, one simply needs to equate the Gilbert damp-\ningtorque(Eq. 3)andodd(anti-damping-like)SOT(Eq.\n6):\nJc=αMsdFMω\nγ2e\n/planckover2pi1/parenleftbiggσ\nσSOTo/parenrightbigg\n(8)\nwhereωis the FMR frequencywith noapplied fields (e.g.\nfor in-plane magnetization, ω=µ0γ/radicalbig\nHk(Ms+Hk),\nwith anisotropy field Hk). Using αas determined for\nthese Ni 80Fe20/Pt films (see SI), Ms= 700kA /m,Hk=\n160kA/m (for thermal stability considerations), bulk Pt\nresistivity20, and the measured σSOT\no(see Table I), we\nestimate a critical current density of 2 ×1012A/m2for a\n2nm Ni 80Fe20film.\nWhile the effective conductivity is the directly mea-\nsured quantity, in Sec. VA we nevertheless derive ex-\npressions relating the effective conductivities to micro-\nscopic spin-charge conversion parameters. Extraction of\nthe microscopic parameters is necessarily contingent on\nthe details of the model employed and parameters as-\nsumed.\nThe effective conductivities can also be related to the\noften-used quantity of effective flux density per unit cur-\nrent density21Beff/J, with units of Tm2A−1via the\nequation Beff/J=σSOT\ne,o/planckover2pi1/(2MsσdFMe) (where σis the\nordinary charge conductivity of the NM). However, our\ndefinitionfortheeffectiveconductivityismoregeneralin-\nsofaras it allowsone to calculate the actual SOT without\nthe need to independently determine the sample magne-\ntization, conductivity, or actual thickness.\nEq. 1 is consistent with the phenomenological formu-\nlation presented by Freimuth, Bluegel, and Mokrousov1,\nalthough it has been expanded to include the purely elec-\ntrodynamic contributions. Our use of the descriptors\n“even” and “odd” are different from that of Freimuth,4\net al., who use the symmetry of the spin orbit torques\nwith respect to magnetization-reversal as the symmetry\nidentifier. We instead use the symmetry of the torque\nwith respect to time-reversal because this is the relevant\nsymmetry with regard to the off-diagonal components in\nthe phenomenological Eq. 1. Any process for which the\ntorque is odd under time-reversal qualifies as microscop-\nically non-reversible in the sense of Onsager reciprocity,\nwhere microscopic reversibility pertains solely to forces\nthat are even functions of velocity, as well as position13.\n(We also note that all axial vectorssuch as magnetic field\nare odd under time reversal.)\nIII. EXPERIMENTAL TECHNIQUE\nThe broadband, phase-sensitive FMR measurements\nutilize a coplanar waveguide (CPW) as both the exci-\ntation and detection transducer (see Fig. 1(a)). Any\nsource of AC magnetic flux generated by the bilayer is\ninductively detected in the CPW. Therefore, the induc-\ntive load that the sample contributes to the CPW circuit\nconsists of four terms: (1) The real-valued L0due to the\noscillating magnetic dipolar fields produced by the res-\nonating FM magnetization, (2) the Faraday-effect cur-\nrents induced in the NM layer by the precessing FM\nmagnetization, (3) currents produced by damping-like\niSOT effects (e.g., spin pumping + iSHE), and (4) cur-\nrents produced by field-like iSOT effects (e.g., iREE).\nThe latter three inductances, which we collectively de-\nfine as complex-valued LNM, are produced by currents in\nthe NM which generate Oersted fields that inductively\ncouple to the CPW. We quantify these currents with the\neffective conductivities σF\ne,σSOT\no, andσSOT\ne, described\nabove. Importantly, as shown below, while L0is inde-\npendent of frequency, LNMis linear in frequency, as the\ncurrents in the NM are driven by ∂tˆm. Hence, frequency-\ndependent measurements allow us to disentangle L0and\nLNM.\nFigure1(b)and(c)showschematicsofthesefoursignal\nsources at two instants in time: when the dipolar and\neven SOT effects are maximal (Fig. 1(b)) and when the\nodd SOT effect is maximal (Fig. 1(c)). Fig. 1(d) shows\nthe time dependence of each of these signal sources, and\ntheir distinct phase relationships to the driving field hy,\nwhich we exploit below to determine their contributions\nseparately.\nFor our measurements, we place samples onto a copla-\nnar waveguide (CPW) with the metallic film side fac-\ning down (see Fig. 1). This setup is positioned be-\ntween the pole pieces of a room-temperature electromag-\nnet capable of producing fields up to ≈2.2T. Using aVNA, we measure the change in microwave transmis-\nsion through the CPW loaded with the bilayer sample\nas an out-of-plane DC magnetic field ( H0∝ba∇dblˆz) is swept\nthrough the FMR condition of the Ni 80Fe20(Permalloy,\nPy) layer. We acquire the microwave transmission S-\nparameter S21≡Vin,2/Vout,1whereVin(out),1(2)is the\nvoltage input (output) at port 1 (2) of the VNA. Field\nsweeps were repeated to average the transmission data\nuntil an appropriate signal-to-noise ratio was obtained.\nTypically, VNA-FMR measurements focus on the res-\nonance field and linewidth. Our method additionally\nmakes use of the signal magnitude and phase in order to\ndirectlyprobethe ACchargecurrentsproducedbyiSOT.\nPrevious studies of AC charge currents in spin pumping\nexperiments have relied on intricate experimental setups\nor techniques that suppress or are insensitive to spurious\nbackground signals12,22,23. Our technique remains sensi-\ntive to currents induced by the Faradayeffect, but is able\nto separate them from spin-charge conversion currents\nthrough the combination of phase-sensitive analysis and\ncomparison to control samples in which the heavy metal\nNM (here, Pt) is substituted with a Cu layer of nomi-\nnally negligible intrinsic spin-orbit effects. Furthermore,\nbecause the CPW is inductively coupled to the sample,\nno electrical connections need to be made directly to the\nFM/NM sample.\nThe sampleaddsa complexinductance Lin serieswith\nthe impedance of the bare CPW, Z0(here, 50Ω). The\nchange in microwave transmission ∆ S21is therefore that\nof a simple voltage divider24:\n∆S21=−1\n2/parenleftbiggiωL\nZ0+iωL/parenrightbigg\n≈−iωL\n2Z0(9)\nforZ0>> ωL, whereωis the microwave frequency. The\nfactor of 1 /2 is needed because the port 2 voltage mea-\nsurement is between the CPW signal and ground (and\nnot between port 2 and port 1).\nA. Inductance Derivations\nIn order to extract values for the SOT effects from the\nmeasured ∆ S21, we derive expressions for each contribu-\ntion toL.\n1. Inductance due to dipole field of dynamic magnetization\nTo derive the inductance due to AC dipolar fields pro-\nduced by the precessing FM magnetization, we follow\nRef. 24.5\n(a) (c)\n(b)(d)\nFigure 1. (a) Sample on CPW, showing out-of-plane field H0and sample length l. The microwave driving field points primarily\nalong ˆyat the sample. (b) Schematic of the bilayer, with precessing magnetization m(t) at time t0whenm=∝angbracketleftmx,0,mz∝angbracketright.\nBilayer is insulated from CPW using photoresist spacer laye r (not shown). At this instant in time, JF\ne(due to the Faraday\neffect in the NM) and JSOT\ne(e.g., due to inverse Rashba-Edelstein effect) are maximal a long±ˆx, andhyis also at its maximum\nstrength. The corresponding Oersted fields from JF\neandJSOT\neare superposed. The spin accumulation (with orientation ˆ s)\nandJSOT\neare produced at the FM/NM interface. Interface normal is giv en by ˆn. (c) Same as (b), except at time t1when\nm=∝angbracketleft0,my,mz∝angbracketright. Here, odd-symmetry SOT current JSOT\no(e.g., due to inverse spin Hall effect), and the dynamic fields HSOT\no\nandHdipoleare at maximum amplitude. Note that the dipolar signal is pro portional to ∂t(Hdipole·ˆy), and not simply to Hdipole.\nSpin flow direction ˆQˆsdue to spin pumping into the NM is also shown. (d) Amplitude of driving field hyand different signal\nsources as a function of time (left), and viewed in the comple x plane at time t0(right). Relative amplitudes not indicated. For\nfurther discussion of signal phases, see SI Sec. III.6\nL0=µ0ℓ\nWwgdFMI2\n+∞/integraldisplay\n−∞dydFM+dwg/integraldisplay\ndwgdz[q(y,z)·χ(ω,H0)·h1(y,z,I)]\n\n∗\n+∞/integraldisplay\n−∞dydFM+dwg/integraldisplay\ndwgdz[q(y,z)·h1(y,z,I)]\n\n∼=µ0ℓ\nWwgdFMI2χyy(ω,H0)h2\ny(I,z)d2\nFMW2\nwg\n∼=µ0ℓ\nWwgdFMI2χyy(ω,H0)/parenleftbiggI\n2Wwgη(z,Wwg)/parenrightbigg2\nd2\nFMW2\nwg\n=µ0ℓdFM\n4Wwgχyy(ω,H0)η2(z,Wwg) (10)\nwhereµ0isthe vacuumpermeability, lthe samplelength,\ndFMthe FM thickness, Wwgthe width of the CPW\nsignal line (here, 100 µm), and χyy(ω) the frequency-\ndependent magnetic susceptibility. η(z,Wwg)≡\n(2/π)arctan(Wwg/2z) is the spacing loss, ranging from 0\nto1, duetoafinitedistance zbetweensampleandwaveg-\nuide. We have assumed the coordinate system described\nin Fig. 1 (ˆ xalong the CPW signal propagation direction,\nˆzalong the CPW and sample normal). The function\nq(y,z) describes the normalized spatial amplitude of the\nFMR mode. For the uniform mode, q(y,z) = 1 over the\nentire sample. The first set of integrals in brackets cap-\ntures the integrated amplitude of the mode as excited by\nthe driving microwave field h1=hyˆy, while the second\ndescribes the inductive pickup sensitivity of the CPW.\nThe firstapproximationassumesuniform microwavefield\nover the sample dimensions. The second approximation\nutilizes the Karlqvist equation25to approximate the mi-\ncrowave field as hy(I,z)∼=I/(2Wwg)η(z,Wwg).\n2. Inductance due to AC current flow in NM\nFollowing Rosa26, we model the sample and CPW as\ntwo thin current-carrying sheets of width w=Wwg, sep-\narationz, and length l, so that the mutual inductance is\ngiven by\nL12=µ0\n4π2l/bracketleftbigg\nln/parenleftbigg2l\nR/parenrightbigg\n−1/bracketrightbigg\n(11)\nwhereRis defined as\nR≡/radicalbig\nw2+z2/parenleftbiggz√\nw2+z2/parenrightbigg(z\nw)2\n∗exp/parenleftbigg2z\nwarctan/parenleftigw\nz/parenrightig\n−3\n2/parenrightbigg\n(12)Viewing the sample-CPW system as a voltage trans-\nformer (two mutually-coupled inductors), the voltage in-\nduced in the CPWdue to current INMin the NM and the\nmutual inductance L12is given by V=−L12(∂INM/∂t).\nIf instead we consider the system to be a single lumped-\nelement inductor, the voltage due to the self-inductance\ncontributed by the sample LNMand applied current\nICPWisV=LNM(∂ICPW/∂t). Therefore, we can cal-\nculateLNMas\nLNM=−L12INM\nICPW(13)\nThe charge current we are interested in is that driven\nby the magnetization dynamics of the FM layer, and\ngiven by the off-diagonal term of Eq. 1:\nINM= ˆx·\n+dFM/integraldisplay\n−dNMJ(z)dz\nWwg\n= ˆx·/bracketleftbig\nˆz×(−σF\ne+σSOT\ne−σSOT\no[ˆm×])∂tˆm/bracketrightbig\n∗sgn(ˆz·ˆn)/parenleftbigg/planckover2pi1\n2e/parenrightbigg\nWwg (14)\nAssuming a linear solution to the Landau-Lifshitz-\nGilbert equation of motion for the magnetization, we\nwrite a simple relation between the dynamic component\nof the magnetization mand microwave field h1.\n∂tˆm=iωχ\nMsh1 (15)\nTo convert the vector cross products in Eq. 14 to the\ncomplex plane, we use χin the frequency domain27:7\nχ=γµ0Ms\nω2res−ω2+iω∆ω/bracketleftbigg/parenleftbig\n1+α2/parenrightbig\nωy−iαω iω\n−iω/parenleftbig\n1+α2/parenrightbig\nωx−iαω/bracketrightbigg\n(16)\nwhereωx,y≡γµ0Hx,y,Hx,yis the stiffness field in the x\norydirection(includingexternal,anisotropy,anddemag-\nnetizing fields), ωres≡√ωxωy, and ∆ω≡α(ωx+ωy).\nFor compactness in the following derivation, we utilize\nthe tensor components of the susceptibility as defined in\nEq. S1.\nEq. 14 has even terms along ˆ z×∂tˆmand odd terms\nalong ˆz×(ˆm×∂tˆm). Using Eq. 15 for ∂tˆm, we can\nwork out these cross products assuming ˆ m∝ba∇dblˆz(small-\nangle FMR excitation). The vector components of the\neven terms are given by:\nˆz×∂tˆm= ˆz×/parenleftbigg/bracketleftbigg\nχxxχxy\nχyxχyy/bracketrightbigg/bracketleftbigg\n0\nhy/bracketrightbigg/parenrightbigg/parenleftbiggiω\nMs/parenrightbigg\n= ˆz×(χxyhyˆx+χyyhyˆy)/parenleftbiggiω\nMs/parenrightbigg\n= (−χyyhyˆx+χxyhyˆy)/parenleftbiggiω\nMs/parenrightbigg\n(17)\nSimilarly, we find for the odd terms:\nˆz×(ˆm×∂tˆm) = ˆz×(−χyyhyˆx+χxyhyˆy)/parenleftbiggiω\nMs/parenrightbigg\n= (−χxyhyˆx−χyyhyˆy)/parenleftbiggiω\nMs/parenrightbigg\n(18)\nNoting from Eq. 16 that χxy=iχyy(ignoring terms\nof order αorα2, and working near resonance such that\nωx=ω), the vector relationships of Eq. 17 and 18 are\nsubstituted into Eq. 14. After evaluating the ˆ xprojec-\ntion as prescribed by Eq. 14 and grouping even and odd\nterms, we find:\nINM=/bracketleftbig\n(σF\ne−σSOT\ne)+iσSOT\no/bracketrightbig\nsgn(ˆz·ˆn)iχyyhy\nMs/parenleftbigg/planckover2pi1ω\n2e/parenrightbigg\nWwg\n(19)\nfrom which we define ˜ σNM= (σF\ne−σSOT\ne)+iσSOT\no. On\nresonance, χyy=−iγµ0Ms/(2αeffωres), such that Eq. 19\nproduces the current phases depicted in Fig. 1.\nFinally, using the Karlqvist equation25, we approxi-\nmate the field of the CPW. With these substitutions into\nEq. 13, we arrive at the final result for the inductance\ndue to all AC currents in the NM:\nLNM= sgn(ˆz·ˆn)L12(z,Wwg,l)η(z,Wwg)\n∗/planckover2pi1ω\n4Mseiχyy(ω,H0)˜σNM(20)\nThe different frequency dependencies of L0andLNMis\ncritical for our analysis. When normalized to χyy(ω,H0),L0is a frequency-independent inductance. By contrast,\nLNMhas an extra factor of ω, reflecting the fact that\nboth Faraday and SOT effects are driven by ∂tˆm, rather\nthan bym(t) itself.\nCareful attention needs to be paid to the signal phase\nin order to properly add the inductive effects of L0and\nLNM. As discussed in detail in the SI Sec. III, it is the\ncurrent phase in the CPW that determines the propa-\ngating signal phase. Using the excitation current in the\nCPW as the phase reference, we work out the phase of\nthe induced currents due to the perturbative inductance\nof the sample-on-CPW, and find that the inductances\nadd according to L=L0−iLNM.\nAfter normalizing by the fitted susceptibility ˜L≡\nL/χyy(ω,H0), the real and imaginary normalized induc-\ntance amplitudes are given by:\nRe(˜L) =µ0l\n4/bracketleftbiggdFM\nWwgη2(z,Wwg)−sgn(ˆz·ˆn)η(z,Wwg)\n∗L12(z,Wwg,l)\nµ0lMs/planckover2pi1ω\ne(σF\ne−σSOT\ne)/bracketrightbigg\n(21)\nIm(˜L) =−µ0l\n4/bracketleftbigg\nsgn(ˆz·ˆn)η(z,Wwg)\n∗L12(z,Wwg,l)\nµ0lMs/planckover2pi1ω\neσSOT\no/bracketrightbigg\n(22)\nNote that when the stacking order of FM and NM is\nreversed, so is the sign of the SOT and Faraday currents\n(and therefore their inductance contributions).\n3. Magnetization dynamics driven by forward SOT\nFrom the transformer analogy developed above and\ndiscussed in SI Sec. III, we see that “image currents”\nare produced in the CPW when currents flow in the con-\nducting sample. Reciprocity requires that the excitation\ncurrents in the CPW drive image currents in the sam-\nple. This current will produce Amperian torque and\nforward SOT effects according to Eq. 6, exciting ad-\nditional magnetization dynamics which are then picked\nup by the CPW. This series of transduction effects is\nfully reciprocal with the Faraday and iSOT sequence de-\nscribed above. In the first case, a drive current in the\nCPW excites magnetization dynamics (via the coupling\nfactor,η(z,Wwg)). Those magnetization dynamics drive\ncharge current in the NM via ˜ σNM. Finally, these charge\ncurrents couple into the CPW via the mutual inductance\nL12(z,Wwg,l). In the second case, the order is simply\nreversed: the CPW currents create image currents in the\nNM (via L12(z,Wwg,l)), which drive magnetization dy-\nnamics (via ˜ σNM), which are picked up by the CPW (via8\nη(z,Wwg)). It can be shown that the induced signal due\nto forward Amperian or SOT-driven magnetization dy-\nnamics add together in-phase with their inverse counter-\nparts, increasing the inductive response from each con-\ntribution by a factor of 2. The inductance in Eq. 20\n(and hence 21 and 22) is therefore too small by a factor\nof 2. Therefore, in the below calculation of ˜ σNMbased\non measured values of ˜LNM, we include this factor.\nB. Background Correction\nTo make use of the phase and amplitude information\nin the VNA-FMR spectra, we first fit the raw spectra to:\nS21(ω,H0) =Aeiφχyy(ω,H0)+C0+C1H0(23)\nwhereAis the signal amplitude, φis the raw phase (in-\nclusive of signal line delay), and C0andC1are complex\noffset and slope corrections to the background. Utiliz-\ning the information in this complex background is key to\nour data processing method. The background-corrected\nsignal can be plotted from the measured values of S21as:\n∆S21(ω,H0) =S21(ω,H0)−(C0+C1H0)\nC0+C1H0(24)\nThis corrects the signal phase for the finite length of the\nsignal line between the VNA source and receiver ports\nand the sample, effectively placing the ports at the sam-\nple position. Additionally, it normalizesthe signal ampli-\ntude by the frequency-dependent losses due to the com-\nplete microwave circuit (cables + CPW + sample). In\nFig. 2(a) and (b), we plot the raw and de-embedded\ndata, respectively. The large complex offset on top of\nwhich the resonance signal is superimposed in (a) repre-\nsentsC0andC1.\nComparison of Eqs. 23 and 24 shows that the change\nin microwave transmission can be written as:\n∆S21(ω,H0) =Aeiφ\nC0+C1H0χyy(ω,H0) (25)\nUsing this form for the background-corrected ∆ S21,\nthe inductance amplitude ˜L(f) is calculated as\n[∆S21/χyy(ω,H0)][i2Z0/(2πf)]. When ˜Lis plotted vs.\nfrequency as in Fig. 4, we note that there can be a small\nphase error that causes Im( ˜L)(f→0)∝negationslash= 0. The correc-\ntion for this phase error is discussed in SI Sec. IV.\nC. Calculation of ˜σNMfrom measured L\nUsing the results for Re( ˜L) and Im( ˜L) (Eqs. 21 and\n22), wecanisolatethe ˜ σNMcontributionasfollows. First,\nthe slope of ˜Lis used to isolate the contribution of ˜LNM:-0.7682-0.7680-0.7678-0.7676-0.7674-0.7672-0.7670Re(S21)\n-0.2032-0.2030-0.2028-0.2026-0.2024-0.2022-0.2020Im(S21)Py/Pt @ 20 GHz\nRaw data\nRe(S21)\nIm(S21)\n-0.0015-0.0010-0.00050.00000.00050.0010\u0001S21\n1.60 1.56 1.52 1.48\u00020 H0 (T)Re(\u0000S21)\nIm(\u0003S21)(a)\n(b)\nDe-embedded data\nFigure 2. Example S21spectrum, acquired at f = 20 .0GHz.\n(a) Raw data, with fits. Note the different background offsets\nof the Re and Im data (left and right axes). (b) De-embedded\n∆S21signal.\nd˜L\ndf=−1\n2sgn(ˆz·ˆn)η(z,Wwg)L12(z,Wwg,l)\nMs\n∗h\ne/bracketleftbig\n(σF\ne−σSOT\ne)+iσSOT\no/bracketrightbig\n(26)\nWe normalize d˜L/dfby˜L0in order to remove any resid-\nual differences in sample-CPW coupling from sample to\nsample. Variation in ˜L0(e.g., as seen in Fig. 4) can be\ncaused by sample-to-sample variations in magnetization,\nincluding dead layer effects at the various FM/NM inter-\nfaces, as well as measurement-to-measurement variations\nin the sample-waveguidespacing, which could be affected\nby small dust particles in the measurement environment.\nFinally, we solve for the effective conductivity.\n/bracketleftbig\n(σF\ne−σSOT\ne)+iσSOT\no/bracketrightbig\n=−sgn(ˆz·ˆn)\nd˜L\ndf\n2˜L0\n\n∗µ0l\nL12(z,Wwg,l)MsdFM\nWwgη(z,Wwg)e\nh(27)\nD. Analysis Protocol\nOur quantitative VNA-FMR analysis protocol is sum-\nmarized below28.9\n1. Complex VNA-FMR data is collected and fit with\nEq. 23.\n2. ∆S21is calculated with Eq. 25 to de-embed the\nsample contribution to the inductance.\n3. ∆S21is converted to sample inductance Lusing\nEq. 9.\n4.Lis normalized by magnetic susceptibility χyy,\nyielding the complex inductance amplitude given\nby Eqs. 21 and 22 (Re( ˜L) and Im( ˜L)).\n5. The phase error of ˜Lis corrected as described in SI\nSec. IV.\n6. Linear fits of ˜L(ω) (using Eqs. 21 and 22) are used\nto extract ˜L0and˜LNM(ω).\n7. The effective conductivities σSOT\noand (σF\ne−σSOT\ne)\nareobtainedfrom( ∂˜L/∂f)/˜L0accordingtoEq. 27.\nIV. DATA AND ANALYSIS\nTo demonstrate the quantitative VNA-FMR tech-\nnique, we measured FMR in metallic stacks consisting\nof substrate/Ta(1.5)/Py(3.5)/NM/Ta(3) and inverted\nstacks of substrate/Ta(1.5)/NM/Py(3.5)/Ta(3) (where\nthenumbersinparenthesesindicatethicknessinnanome-\nters). We focus on a Pt(6) NM layer due to its large in-\ntrinsic SOC, and use Cu(3.3) as a control material with\nnominally negligible SOC16,29,30. We collected room-\ntemperature FMR spectra as a function of out-of-plane\nexternal magnetic field H0with microwave frequencies\nfrom 5GHz to 35GHz and VNA output power of 0 dBm.\nExemplary Re(∆ S21) spectra are shown in Fig. 3. Each\nraw spectrum has been normalized by the complex sig-\nnal background (see Sec. IIIB). In the following discus-\nsion, we use a notation for the bilayers which indicates\nthe sample growth order as read from left-to-right. For\nexample, Py/Pt indicates Py is first deposited onto the\nsubstrate, followed by Pt.\nBoth Py/Cu and Cu/Py samples exhibit a mostly real\nnormalized inductance amplitude (symmetric Lorentzian\ndip for Re(∆ S21) in Fig. 3(a) and (b)) with a magni-\ntudelargelyindependentoffrequency,inaccordancewith\n˜LNM≈0. That is, the signal is dominated by the dipolar\ninductance. In contrast, the lineshape and magnitude of\nthe Py/Pt and Pt/Py data in Fig. 3(c) and (d) exhibit a\nclear frequency dependence as expected for ˜LNM∝negationslash= 0. In\nparticular, the data for Py/Pt indicate that ˜LNMadds\nconstructively with L0, such that Re( ˜L) increases with\nincreasing f. The Pt/Py inductance evolves in an oppo-\nsite sensedue to the stackinversion, leadingtoa decrease\nand eventual compensation of Re( ˜L) at high f. The in-\ncreasingly antisymmetric lineshape for both Py/Pt and\nPt/Pyrevealsthat the magnitude ofIm( ˜L) alsoincreases\nwith frequency, with a sign given by the stacking order.\nBy normalizing the spectra in Fig. 3 to the magnetic\nsusceptibility χ(ω,H0) defined in Eq. S2, we extract the\nRe(ΔS21)\n2.0 1.8 1.6 1.4 1.2\nμ0 H0 (T)-2.0-1.5-1.0-0.50.0x10-3\nPy/Cu(a)\n-2.0-1.5-1.0-0.50.0x10-3\n-2.0-1.5-1.0-0.50.0x10-3\n7.5 GHz 35 GHzPy/Pt\nPt/Py(c)\n(d)-2.0-1.5-1.0-0.50.0x10-3\nCu/Py(b)\nFigure 3. FMR spectra for FM/NM bilayers. Re(∆ S21)\nat several excitation frequencies for different samples: (a )\nPy/Cu, (b) Cu/Py, (c) Py/Pt, and (d) Pt/Py. The change\nin lineshape and amplitude for Py/Pt and Pt/Py clearly\nshows the presence of frequency-dependent inductive terms\nnot present in the Py/Cu and Cu/Py control samples. The\ncolored circles indicate the value of Re(∆ S21)∝Re(L) when\nH0satisfies the out-of-plane FMR condition.\ncomplex inductance amplitude ˜L. Re(˜L) and Im( ˜L) are\nshown in Fig. 4 for all investigated bilayers with a length\nlof 8mm. As shown in Eqs. 21 and 22, Re( ˜L) pro-\nvidesinformationaboutthedipolarinductance( ˜L0, zero-\nfrequency intercept), and −(σF\ne−σSOT\ne) (slope). Simi-\nlarly, the slope of Im( ˜L) reflects −σSOT\no. Immediately10\n504 \u0004\n30\n20\n10\n0Py/Pt\nPt/Py\nPy/Cu\nCu/Py- \u0005 \u0006-30-20-100\n35 30 25 20 15 10 5 0F \u0007 \b \t \n \u000b \f \r \u000ePy/Pt\nPt/Py\nPy/Cu\nCu/Py(a)\n(b)\nR\u000f\u0010\u0011\u0012\u0013\u0014\u0015\u0016\n~I\u0017\u0018\u0019\u001a\u001b\u001c\u001d\u001e\n\u001f\nFigure 4. Frequency dependence of real and imaginary induc-\ntances extracted from S21spectra (symbols) and fits to Eqs.\n21 and 22 (lines). (a) Re( ˜L) for all samples with l= 8mm.\nZero-frequency intercept indicates the dipolar inductive cou-\npling, while the linear slope reflects ( σF\ne−σSOT\ne). (b) Im( ˜L)\nfor all samples, as a function of frequency, where the linear\nslope is governed by σSOT\no.\nevident is the reversalofthe slopefor Py/Ptcomparedto\nPt/Py, which is captured by the sgn function (where ˆ nis\nthe FM/NM interface normal, pointing into the FM, and\nˆzis defined by the coordinate system in Fig. 1). This\nsign-reversal is consistent with the phenomenology ex-\npected for interface-symmetrysensitive effects, e.g., com-\nbined spin pumping and iSHE, as well as iREE. There\nis also a marked difference in the slope magnitude for\nPy/Pt and Pt/Py in panel (b), the implications of which\nare discussed below.\nEach of the inductance terms has some dependence on\nsample length: linear for the dipolar contribution, and\nslightly non-linear for the inductances due to charge flow\nin the NM (see Eqs. 10 and 11). We therefore repeated\nthe measurements shown in Fig. 4 for a variety of sample\nlengths from 4 to 10mm. Fig. 5 shows the measured\ninductance terms ˜L0,∂Re(˜L)/∂f(intercept and slope of\ncurves in Fig. 4(a)), and ∂Im(˜L)/∂f(slope of curves\nin Fig. 4(b)) as a function of sample length. Following\nnormalization by its corresonding ˜L0, each data point in\nFig. 5(b) provides a value of ( σF\ne−σSOT\ne) (see Eq. 27).\nSimilarly, datapoints in panel(c) providevaluesof σSOT\no.(a)\n(b)\n(c)\n !\"\nFigure 5. ˜L(f= 0) and ∂˜L/∂fextracted from data as in Fig.\n4 vs. sample length for all samples. (a) Dipolar inductive\ncoupling ˜L0. (b) From ∂[Re(˜L)]/∂f, we extract ( σF\ne−σSOT\ne).\n(c) From ∂[Im(˜L)]/∂f, we extract σSOT\no. Dashed lines are\nguides based on Eqs. M8 and M9 with values of σSOT\noand\n(σF\ne−σSOT\ne) calculated as described in the Methods. Several\nmeasurements were repeated to demonstrate reproducibilit y.\nThese valuesareaveragedto provideasingle ( σF\ne−σSOT\ne)\nandσSOT\nofor each sample type. Results are summarized\nin Table I. The dashed lines in Fig. 5(b) and (c) are\ncalculated curves based on these average values and the\nlength dependence of ˜L.\nBecause σSOT\neandσF\nehave the same phase and fre-\nquency dependence, we use control samples where we re-\nplace the Pt with Cu, wherein it is generally accepted\nthat both the SHE for Cu and the REE at the Py/Cu in-\nterface are negligible16,29,30. Furthermore, the Cu thick-11\nnessischosensothat itexhibits thesamesheetresistance\nas the Pt layer, so that the two samples have identical σF\ne\n(see Eq. 31). Subtraction of the time-reversal-even con-\nductivity for the Py/Cu control samples from the time-\nreversal-even conductivity for the Py/Pt samples there-\nfore isolates σSOT\nespecifically for the Py/Pt interface.\nLikewise, any damping-like contributions to σSOT\nodue to\nthe Ta seed layer should also be removed by subtraction\nof the Py/Cu inductance data.\nAdditional data collected for varied NM thickness (to\nbe presented in a future publication) indicates that the\ncharge currents produced by iSOT effects experience a\nshunting effect, whereby some fraction of the interfacial\ncharge current flows back through the sample thickness,\nreducing the inductive signal. This can be modeled as a\ncurrent divider with some of the iSOT-generated current\ncoupling to the 50Ω CPW via image currents, and the\nremainder shunted by the sheet conductance of the sam-\nple. Final values of the extracted conductivities reported\nin Table I have been corrected to account for current\nshunting (see SI Sec. V for more details). Comparison\nof the shunt-corrected SOT conductivities makes evident\nthatthefield-likechargecurrentsarecomparabletothose\ndue to damping-like spin-charge conversion processes.\nWe can compare our measured values of σSOT\neand\nσSOT\noto measurements made by other groups using dif-\nferent techniques. Garello, et al.9use the harmonic\nHall technique and Miron, et al.2investigate domain\nwall nucleation to quantify the spin-orbit torque ex-\nerted on Co sandwiched between Pt and AlO x. Con-\nverting their measured values of field-like SOT field\nper unit current density to our metric σSOT\ne, they find\n1.1×106Ω−1m−1and 1.9×107Ω−1m−1. Nguyen, et\nal.21find a similar value of ≈1.3×106Ω−1m−1for a\nPt/CobilayerusingharmonicHallmethods. TheGarello\nand Nguyen results are within an order of magnitude of\nour findings ( −1.48±0.07×105Ω−1m−1for Pt/Py and\n−1.8±0.2×105Ω−1m−1for Py/Pt).\nGarello and Nguyen also report damping-like values\nfor their effective SOT fields. Converted to σSOT\no,\nthey find 5 .8×105Ω−1m−1and≈2.9×105Ω−1m−1, re-\nspectively, which are again within an order of magni-\ntude of our values: 2 .4±0.3×105Ω−1m−1(Py/Pt) and\n0.6±0.2×105Ω−1m−1(Pt/Py).\nV. DISCUSSION\nFor comparison to previous measurements and to the-\nory, we can relate the effective conductivities σSOT\neand\nσSOT\noto microscopic spin-charge conversion parameters\nunderthe assumptionsthat the damping-likeiSOTisdue\nto iSHE only, and the field-like iSOT is from iREE only.\nWe alsorelate the Faradaycontribution to the AC charge\ncurrents in the NM—that is, σF\ne—to sample properties.A. Contributions to effective conductivity, ˜σNM\n1. Effective Faraday conductivity, σF\ne\nTo relate the effective Faraday conductivity, σF\ne, to\nsample parameters, we isolate the Faraday component\nof the induced charge current from Eq. 7:\n\n+dFM/integraldisplay\n−dNMJF(z)dz\n=−sgn(ˆz·ˆn)/parenleftbigg/planckover2pi1\n2e/parenrightbigg\nσF\ne(ˆz×∂tˆm) (28)\nThe charge current is driven by the induced e.m.f., Vx,\naccording to:\nˆx·\n+dFM/integraldisplay\n−dNMJF(z)dz\n=Ix\nw\n=Vx\nZeffl(29)\nThe induced e.m.f. is derived from inductive\nreciprocity31\nVx=−∂φ\n∂t=−µ0Ms/integraldisplay\nVFM[h(r)·∂tˆm]d3r(30)\nwhereh(r) is the magnetic sensitivity function for a cur-\nrent of unit amplitude in the NM layer. We assume this\nfield can be approximated with the Karlqvist equation,\nand use the results for ∂tˆmfrom Sec. IIIA. Subsituting\nEq. 30 into Eq. 29, and equating the result with Eq. 28\nyields the final expression for σF\ne:\nσF\ne=eµ0MsdFM\n/planckover2pi1Zeff(31)\n2. Rashba parameter and σSOT\ne\nWe can relate the even spin-orbit torque conductivity\nσSOT\neto the Rashba parameter αR. We start from the\nfield-like interfacial spin torque per spin tflintroduced by\nKim, et al. (Eq. 12 in Ref. 32):\ntfl= sgn(ˆz·ˆn)kRvs/bracketleftig\nˆm×(ˆj׈z)/bracketrightig/parenleftbigg/planckover2pi1\n2/parenrightbigg\n(32)\nwherekR= 2αRme//planckover2pi12is a wavevector corresponding to\nthe Rashba energy parameter αR,meis the mass of the\nelectron, and vs=PJintgµB/(2eMs) is the spin velocity,\nwith charge current density Jintat the FM/NM interface\nat which the Rashba effect is present, spin polarization\nof the charge current P, Land´ e g-factor g, and Bohr12\nSample ( σF\ne−σSOT\ne)meas (σSOT\no)meas (σSOT\ne)corr (σSOT\no)corr\nPy/Pt −0.45±0.03 1 .0±0.1 −1.48±0.07 2 .4±0.3\nPt/Py −0.69±0.05 0 .31±0.06 −1.8±0.2 0 .6±0.2\nPy/Cu 0 .143±0.006 0 .07±0.03\nCu/Py 0 .04±0.03 0 .06±0.01\nTable I.Effectiveconductivities(inunitsof105Ω−1m−1)andmicroscopic spin-charge conversionparameters (Rash baparameter\nαRand spin Hall angle θSH). Measured values are calculated from measured inductance s (Fig. 5). Corrected values are\ncalculated by subtraction of Cu control to remove the Farada y contribution (in the case of σe) and any contribution from the\nTa interfaces, followed by application of the shunting corr ection (see SI Sec. V).\nmagneton µB. Note that tfl/(/planckover2pi1/2) has units of Hz; that\nis, the same units as ∂tˆm. We can therefore relate Eq.\n32 to the volume-averaged magnetic torque density T\nfrom Eqs. 5 and 6 through the time rate of change of\nthe magnetization: tfldintδ(z)/(/planckover2pi1/2) =∂tˆm, where we\nhave added dintδ(z) to account for the interfacial nature\nof this torque (where dintis an effective thickness of the\ninterface).\n2\n/planckover2pi1dFM/integraldisplay\n0tfldintδ(z)dz=−γ\nMsdFM/integraldisplay\n0T(z)dz (33)\nkRvsˆm×(ˆj׈z)dint=−γ\nMs/planckover2pi1\n2eσSOT\neˆm×(ˆz×E)(34)\nThe final line results from subsituting Eq. 32 and the\neven SOT term from Eq. 6 into Eq. 33. Making the\nsubstitutions for kRandvs, and using E= (Jint/σint)ˆj\nyields:\nαR=/planckover2pi12\n2meσSOT\ne\nσint1\nPdint(35)\nHere,σintis the interfacial conductivity of the FM/NM\ninterface(extractedbymeasuringresistancevs. Pythick-\nness; see SI Sec. VI) and Pis the spin polarization at\nthe FM/NM interface. We use P= 0.6 as determined\nvia spin-wave Doppler measurements in Ref. 33, and as-\nsumedintis one Py lattice constant (0 .354nm)34. We\ntherefore find αR=−5.8±0.3meVnm for the Py/Pt\nsample, and −7.5±0.7meVnm for Pt/Py. These values\naresmallerthan thosemeasuredwith angle-resolvedpho-\ntoelectron spectroscopy (ARPES) for the surface state\nof Au(111) (33meVnm)35, Bi(111) (56meVnm)36, and\nGe(111) (24meVnm)37, and much smaller than the\nBi/Ag(111) interface (305meVnm)38.\nWe can also compare our results for the Rashba\nparameter to a recent theoretical calculation. Kim,\nLee, Lee, and Stiles (KLLS)32have shown that SOT\nand the Dzyaloshinskii-Moriya interaction (DMI) at a\nFM/NM interface are both manifestations of an under-\nlying Rashba Hamiltonian, and predict a straightfoward\nrelationship between the Rashba parameter αR, inter-\nfacial DMI strength Dint\nDMI, and the interfacial field-like\nSOT per spin tfl:αR=/planckover2pi12\n2me/parenleftbiggDint\nDMI\n2A/parenrightbigg\n=/planckover2pi1\nme/parenleftbiggtfl\nvs/parenrightbigg\n(36)\nwhereAis the exchange stiffness.\nFor the Pt/Py stack, the ratio of interfacial DMI,\nDint\nDMI, to bulk exchange Awas previously measured via\na combination of Brillouin light scattering (BLS) and\nsuperconducting quantum interference device (SQUID)\nmagnetometry for samples prepared under nearly identi-\ncal growth conditions, albeit with a stack geometry that\nwas optimized for optical BLS measurements39. The ra-\ntio is a constant value of −0.25±0.01nm−1over a Py\nthickness range of 1.3 to 15nm. As such, this material\nsystem is an ideal candidate to test the quantitative pre-\ndiction of the KLLS theory. Using the experimentally-\ndetermined value for Dint\nDMI/Awith Eq. 36 predicts a\nRashbastrengthof −4.8±0.2meVnm, whichagreeswell\nin sign and magnitude with the result of our iSOT mea-\nsurement for the Pt/Py sample of the same stacking or-\nder, as well as the Py/Pt sample with opposite stacking\norder. Together, the spin wave spectroscopy and iSOT\nmeasurementsclarifytheroleoftheRashbaspin-orbitin-\nteraction as the underlying physical mechanism for both\nDMI and field-like SOT in the Py/Pt system.\n3. Spin Hall angle and σSOT\no\nIn order to develop intuition for Eq. 7 we first derive\nan approximate relationship between σSOT\noand the spin\nHall angle, θSH, applicable when the NM thickness is\nmuch thicker than its spin diffusion length. We assume\nseries resistors 1 /G↑↓+ 1/Gext(interfacial spin-mixing\nconductance + spin conductance of the NM) in a voltage\ndivider model for the spin accumulation at the FM/NM\ninterface due to spin pumping\nµs(z= 0+)ˆs=/planckover2pi1\n2/parenleftbigg\nˆm×∂ˆm\n∂t/parenrightbigg/parenleftbiggG↑↓\nG↑↓+Gext/parenrightbigg\n(37)\nwhereµs(z= 0+) is the spin accumulation at the\nFM/NM interface. Using the result of Eq. 6 from Ref.\n40 for the effective one-dimensional spin conductance of\na NM (where we have set GNM\n2= 0 because we are inter-\nested in only a FM/NM bilayer, not a FM/NM1/NM2\nmultilayer):13\nGext=σ\n2λstanh/parenleftbiggdNM\nλs/parenrightbigg\n(38)\nwhereλsis the spin diffusion length in the NM. The\nintegrated charge current in the NM layer driven by the\nresulting spin chemical potential gradient −∇µs=Qs\nand the inverse spin Hall effect ( Jc∝Qs׈s) is given by\ndNM/integraldisplay\n0Jc(z)dz=dPt/integraldisplay\n0/bracketleftbigg\nσSH−∇µs(z)\ne׈s/bracketrightbigg\ndz(39)\n=σSHµs(z= 0+)\ne(−ˆz׈s) (40)\nassuming dNM>> λ s. The spin Hall conductivity is\nrelated to the spin Hall angle via the Pt charge conduc-\ntivity:σSH=θSHσPt. If we combine Eqs. 37, 38, and\n40 and equate the integrated charge current to that from\nσSOT\noin Eq. 7 we arrive at the final result:\nσSOT\no=σ\n\nθSHRe\nG↑↓\nσ\n2λstanh/parenleftbiggdNM\nλs/parenrightbigg\n+G↑↓\n\n\nǫ(41)\nThe model also accounts for less-than-unity efficiency ǫ\nfor spin transmission into the NM (such that (1 −ǫ) is the\nspin loss fraction, which has been attributed to processes\nsuch as spin memory loss41or promixity magnetism42).\nA more accurate version of Eq. 41 is obtained by re-\nplacing the unitless term in curly brackets with Eq. 11\nfrom Ref. 43:\nσSOT\no=σ/braceleftbigg\nθSH(1−e−dNM/λs)2\n(1+e−2dNM/λs)\n∗|˜G↑↓|2+Re(˜G↑↓)tanh2/parenleftbiggdNM\nλs/parenrightbigg\n|˜G↑↓|2+2Re(˜G↑↓)tanh2/parenleftbiggdNM\nλs/parenrightbigg\n+tanh4/parenleftbiggdNM\nλs/parenrightbigg\n\nǫ\n(42)\nwhere˜G↑↓=G↑↓2λstanh(dNM/λs)/σ. This properly ac-\ncounts for the boundary condition that the spin current\ngoes to zero at the distant surface of the NM.\nEq. 42canbe usedto calculate θSHifweassumevalues\nforλs,G↑↓, andǫ. Iftheseparametersarepresumediden-\ntical for the two stacking orders, we would find spin Hall\nangles that differ by a factor of 4 depending on whether\nPt is deposited on Py, or vice versa. Instead, the large\ndiscrepancy in σSOT\nofor the two stacking orders suggests\ndifferences in the FM/NM interface that affect G↑↓and\nǫ. Given the data presented here, it is possible for us to\nestimate the efficiency with which spins are pumped intothe Pt layer as follows. The total Gilbert damping αtot\nis the sum of intrinsic processes αint, spin pumping into\nthe Pt and Ta layers αPt(Ta), and possible spin memory\nlossαSML.\nαtot=αint+αPt+αTa+αSML (43)\nWe can apply Eq. 43 to each of the stacking or-\nders (Py/Pt and Pt/Py), and use the damping measure-\nments for Py/Cu and Cu/Py control samples as a mea-\nsure ofαint+αTafor Py/Pt and Pt/Py, respectively.\nWe note that that the total Gilbert damping for the\ntwo stacking orders differs by only 8% (see Table S2),\nwhile the odd SOT conductivity differs by a factor of\n4. This suggests that the damping-like procceses con-\ntributing to σSOT\no(i.e. iSHE) add only a small amount\nof enhanced damping, while the majority of spin current\npumped out of the FM is lost and not available for iSHE\nconversion41. If we therefore assume that αSMLis iden-\ntical for the two stacking orders, and that the difference\ninσSOT\nofor the two stacks is due entirely to a differ-\nence in spin-mixing conductance, such that αPt(Py/Pt)\n= 4αPt(Pt/Py), then the resulting system of equations\nis solvable for αPt(Py/Pt) and αPt(Pt/Py), as well as\nαSML. Using the results, we can estimate the spin pump-\ning efficiency factor ǫ≡αsp/(αsp+αSML). We find that\nonly 33% or 13% of the spin current pumped through the\nPt interface is available for iSHE conversion, for Py/Pt\nand Pt/Py samples respectively.\nA more rigorous calculation can be done to estimate\nG↑↓,ǫ, andθSHby simultaneously fitting Eq. 42 and Eq.\n43 for the two stacking orders (using the corrected values\n(σSOT\no)corrfrom Table I and total damping values from\nTableS2). Toperformthisoptimization, weusethefunc-\ntional form for the spin pumping damping contributions\nas presented in Ref. 40, such that αPt(Ta)depends on λs,\nG↑↓, andσin order to implement the spin current back-\nflow correction. We obtained a value for the Pt charge\nconductivity σ= 4.16×106Ω−1m−1from four-probe re-\nsistance measurement on a series of Py/Pt samples with\nvarying Pt thickness, to allow isolation of the Pt contri-\nbution to the total conductivity. Using a value of λs=\n3.4nm from Ref. 41, we obtain a spin Hall angle of θSH\n= 0.28. This falls within the range of published values\nfrom DC spin Hall measurements (0.01–0.33)7,12,44–50.\nIn good agreement with the estimate above, we find\nefficiencies of 34% and 18% for Py/Pt and Pt/Py re-\nspectively. Furthermore, this optimization yields G↑↓=\n8.9×1014Ω−1m−2(for Py/Pt) and 2 .3×1014Ω−1m−2\n(for Pt/Py). Both of these values are below the Sharvin\nconductance51(G↑↓=1×1015Ω−1m−2), which serves as\nthe theoretical upper bound for the spin-mixing conduc-\ntance. This result demonstrates clearly that when Py is\ndeposited on Pt, the FM/NM interface is detrimental to\nspin transport.14\nVI. CONCLUSION\nIn summary, we have quantified both field- and\ndamping-like inverse spin-orbit torques in Ni 80Fe20/Pt\nbilayers using phase-sensitive VNA-FMR measurements\nand an analysis of the sample’s complex inductance that\narises in part from the AC currents due to spin-charge\nconversion. The magnitude of these currents is deter-\nmined by their respective SOT conductivities, a key fig-\nure of merit for characterizating and optimizing oper-\national spintronic devices. Because our technique en-\ntails straightforward post-measurement data processing\nfor an experimental technique that is well-established in\nthe field, it provides a powerful way to unpick a highly\ncomplex experimental system and represents a broadlyapplicable tool for studying strong SOC material sys-\ntems. The technique could evenbe applied to previously-\nacquired VNA-FMR data sets in which only spectro-\nscopic analysis was performed. The measurements pre-\nsented here demonstrate that both Rashba-Edelsteinand\nspin Hall effects must be considered in FM/NM metal-\nlic bilayers. Together with the observation of significant\nvariation in σSOT\nowith respect to FM/NM stacking or-\nder, these results point to interfacial engineering as an\nopportunity forenhancing current-controlledmagnetism.\nACKNOWLEDGMENTS\nThe authors would like to thank Mark Stiles and Mark\nKeller for many helpful discussions and illuminating in-\nsights.\n1F. Freimuth, S. Bl¨ ugel, and Y. 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Buhrman,\nPhysical Review Letters 106, 036601 (2011).\n48M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pern-\npeintner, S. Meyer, H. Huebl, R. Gross, A. Kamra, J. Xiao,\nY.-T. Chen, H. Jiao, G. E. W. Bauer, and S. T. B. Goen-\nnenwein, Physical Review Letters 111, 176601 (2013).\n49M. Obstbaum, M. H¨ artinger, H. G. Bauer, T. Meier,\nF. Swientek, C. H. Back, and G. Woltersdorf,\nPhysical Review B 89, 060407 (2014).\n50C.-F. Pai, Y. Ou, L. H. Vilela-Le˜ ao, D. C. Ralph, and\nR. A. Buhrman, Physical Review B 92, 064426 (2015).\n51Y. Liu, Z. Yuan, R. Wesselink, A. A. Starikov, and P. J.\nKelly, Physical Review Letters 113, 207202 (2014).1\nSupplementary Information\nI. SAMPLE FABRICATION\nSampleDeposition Order\nPy/PtSubstrate/Ta(1.5)/Py(3.5)/Pt(6)/Ta(3)\nPt/PySubstrate/Ta(1.5)/Pt(6)/Py(3.5)/Ta(3)\nPy/CuSubstrate/Ta(1.5)/Py(3.5)/Cu(3.3)/Ta(3)\nCu/PySubstrate/Ta(1.5)/Cu(3.3)/Py(3.5)/Ta(3)\nTable S1. Sample deposition orders and metallization thick -\nnesses (in nanometers).\nAll samples were prepared by DC magnetron sputter-\ning in an Ar base pressure of ≈0.07Pa (≈0.5mTorr) and\na chamber base pressure of 3 ×10−6Pa (2×10−8Torr)\non 3-inch wafers of thermally oxidized (100) Si (nominal\nresistivity = 3Ωcm). The wafers were rotated at 1Hz\nto 2Hz during deposition to eliminate growth-induced\nanisotropy, and the sample holder was held at room tem-\nperature. All samples were grown on a 1 .5nm Ta seed\nlayer to promote (111) textured growth, which was then\nfollowed by the FM/NM (or NM/FM) bilayer. X-ray\ndiffraction shows that the Ta seed layer is unordered. A\n3nm Ta cap layer prevents oxidation of the FM and NM\nlayers. It is expected that 1nm to 2nm of the cap layer\nforms the insulator TaO when exposed to air. Depositionorder and film thicknesses are shown in Table S1. The Pt\nand Cu thicknesses were chosen so that the DC conduc-\ntivities (as characterized by a four-probe measurement)\nofthesampleandcontrolwereequal,toensureequalityof\nFaraday induced currents. The wafers were subsequently\ncoatedwith 8 µm ofphotoresistto provideelectricalinsu-\nlation from the CPW and reduce the capacitive coupling\nof the CPWto the metallic layers. The waferswere diced\nto precise sizes using an automatic dicing saw.\nII. MAGNETIC CHARACTERIZATION\nA. Magnetic Susceptibility\nForourgeometry,thedrivingmicrowavemagneticfield\nlies primarily along ˆ y, and we are concerned with the\nAC component of magnetization along ˆ y(see Fig. 1 in\nthe main text for coordinate system). Therefore, the S21\nspectra are fit to the χyycomponent of the complex mag-\nnetic Polder susceptibility tensor in order to extract res-\nonance field, linewidth, amplitude, and phase.\n/bracketleftbigg\nMx\nMy/bracketrightbigg\n=/bracketleftbigg\nχxxχxy\nχyxχyy/bracketrightbigg/bracketleftbigg\nhx\nhy/bracketrightbigg\n(S1)\nχ(ω,H0) =Ms/parenleftigg\n(H0−Meff)2−/parenleftbiggω\nγµ0/parenrightbigg2\n+i2αeffω(H0−Meff)\nγµ0/parenrightigg\n(H0−Meff)iω\nγµ0\n−iω\nγµ0(H0−Meff)\n (S2)\nwhereH0is the externally applied DC field, Meff=Ms−\nH⊥\nkis the effective magnetization, Msis the saturation\nmagnetization, H⊥\nkis the perpendicular anisotropy field,\nωis the drivingfrequency, γis the gyromagneticratio, µ0\nthe vacuum permeability, and αeff=α+γµ0∆H0/(2ω) is\nthe effective damping parameter, with Gilbert damping\nconstant αand inhomogeneous broadening ∆ H0.\nThe frequency dependence of the resonant field Hres\nand linewidth ∆ Hallow extraction of the effective mag-\nnetization Meff=Ms−H⊥\nk, spectroscopic g-factor g,\ninhomogeneous broadening ∆ H0, and Gilbert damping\nparameter α. We used SQUIDmagnetometrytomeasure\nthe magnetization per unit area for all samples. Magne-\ntization, g-factor, and damping values are summarized in\nTable S2.B. Resonance Field Dispersion\nFrom the susceptibility fits to the S21spectra, we ex-\ntract the resonance field as a function of microwave fre-\nquency. This is expected to followthe Kittel dispersionS1\nfor out-of-plane field H0.\nω=µ0γ(Hres−Meff) (S3)\nA plot of µ0Hresvs.f=ω/2πis shown in Fig. S1,\nwith slope set by the gyromagnetic ratio γ=gµB//planckover2pi1, and\ny-intercept set by µ0Meff.2\n2.0\n1.5\n1.0\n0.5\n0.0\n#0Hres(T)\n35302520151050\nFreq(GHz)Py/Pt\nPt/Py\nPy/Cu\nCu/Py\nFigure S1. Resonance field vs. frequency dispersion, to ex-\ntract spectroscopic g-factor, and Meff.\n/s32/s33\n/s34/s33\n/s35/s33\n/s36/s33\n/s37/s33\n/s38/s33\n/s33/s39/s33/s40/s41/s42/s40/s43/s44/s45/s46\n/s36/s34/s40/s47/s38/s33/s48 /s36/s33/s37/s34/s37/s33/s38/s34/s38/s33/s34/s33\n/s49/s50/s51/s52/s40/s43/s53/s42/s54/s46/s40/s55/s56/s57/s55/s58/s40\n/s40/s55/s58/s57/s55/s56/s40\n/s40\n/s40/s55/s56/s57/s59/s60/s40\n/s40/s59/s60/s57/s55/s56/s40\n/s40\nFigure S2. Resonance linewidth vs. frequency, to extract\nGilbert damping constant αand inhomogeneous broadening.\nC. Linewidth and Damping\nThe resonance linewidth is determined by the Gilbert\ndamping constant αand inhomogeneous broadening\n∆H0according to\nµ0∆H=µ0∆H0+2ωα\nγ(S4)\nData and fits of Eq. S4 for the 6mm long samples for\neach deposition order are shown in Fig. S2.\nD. SQUID Measurement\nWe measured in-plane hysteresis curves at room tem-\nperaturetodetermine the saturationmoment ofoursam-ples. This total moment was normalized by the sample\narea to obtain MsdFM(see Table S2).\nIII. DETERMINATION OF SIGNAL PHASE\nWe consider the sample and CPW in a lumped ele-\nment circuit model, in which the sample contributes an\nimpedance iωLto the circuit, in series with the char-\nacteristic impedance Z0of the CPW. Therefore, at the\nsample (ordevice-under-test), the currentis simply given\nby:\nIDUT=V1\nZ0+iωL\n≈V1\nZ0/parenleftbigg\n1−iωL\nZ0/parenrightbigg\n(S5)\n=ICPW+∆I\nforωL << Z 0, and where ICPWis the current in the\nunloaded CPW (with a positive Real current flowing in\nthe +ˆxdirection). Therefore:\n∆I=−/parenleftbiggiωL\nZ0/parenrightbigg\nICPW (S6)\nUsing the dipolar inductance of Eq. 10, and considering\nthe current response at the FMR condition, such that\nχyy=−iγµ0Ms/(2αeffωres) (for CCW precession), we\nfind:\n∆Idip=−γµ2\n0lMsdFMη(z,Wwg)\n8Z0αeffWwgICPW (S7)\nFrom Eq. S7 we see that the change in current is in-\nphase with, but opposite in sign to the current responsi-\nble forhy(asdepicted in Fig. S3(a)). This changein cur-\nrent could be viewed as a change in the CPW resistance.\nThat is, the sample inductance creates a purely dissipa-\ntive response at the FMR condition, which is clearly seen\nin Fig. 3(a) and (b), and is expected for a spin system\non resonance.\nLet us now consider the phase of the currents in the\nCPW due to currents in the NM (from the Faraday and\niSOT processes). These effects are captured by Fig. 1(b-\nd) and the derivation of Sec. IIIA2. For simplicity, we\nfirst focus on the Faraday-type currents in the NM. At\ntimet0, this current is maximum along the ˆ xdirection.\nViathe mutualinductance betweensampleandCPW,an\n“image current” flows in the CPW opposite to the Fara-\ndaycurrentin the NM. Extending this logicto all current\nsources in the NM layer, we produce the phasor diagram\nof Fig. S3(b). This demonstrates clearly that at the\nFMR condition, currents with even time-reversal sym-\nmetry create a dissipative response in the CPW, while\nodd-symmetry currents create a reactive response. The3\nSample Meff(kA/m) g µ0∆H0(mT) α M sdFM(µA)\nPy/Pt 663 .5±0.7 2 .079±0.001 1 .2±0.8 0 .0261±0.0003 2069 ±1\nPt/Py 647 ±1 2 .079±0.003 2 ±2 0 .0241±0.0008 2121 ±1\nPy/Cu 674 ±1 2 .075±0.001 1 .1±0.5 0 .0115±0.0001 2341 ±2\nCu/Py 642 ±1 2 .077±0.001 1 .7±0.9 0 .0129±0.0002 2077 .0±0.4\nTable S2. FMR and SQUID parameters for Py/Pt and Py/Cu bilaye rs.\n(a) (b)\nFigure S3. (a). Phasor diagram describing phase of current\ndue to dipolar coupling to precessing magnetization m, rela-\ntive tohyat the FMR condition. The current ∆ Idipcreates\na dissipative response. (b) Same as (a), but for currents in\nthe CPW due to currents INMcaused by Faraday and iSOT\neffects. Even currents appear dissipative or resistive, odd cur-\nrents appear reactive. Note that all currents are defined suc h\nthat a positive Real current in the CPW flows in the +ˆ xdi-\nrection, and relative magnitudes are not indicated.\nSample φcorr(deg)\nPy/Pt 12 ±1\nPt/Py 11 .6±0.4\nPy/Cu 1 .8±0.8\nCu/Py 7 .2±0.3\nTable S3.\ncontribution of even and odd currents to dissipative or\nreactive response changes as field is swept through the\nresonance condition, resulting in the evolving lineshapes\nobserved in Fig. 3(c) and (d).\nIn order to coherently add the perturbative currents\ndue toL0andLNMto satisfy the above discussion (i.e.\nto combine the effects of Fig. S3(a) and (b) with the\nproper phase assignment), we find:\n∆Itot= ∆IL0+∆ILNM\n=/parenleftbigg\n−iωL0\nZ0−ωLNM\nZ0/parenrightbigg\nICPW (S8)\n=−/parenleftbiggiωLtot\nZ0/parenrightbigg\nICPW (S9)\nwhereLtot≡L0−iLNM. Using this result, we recover\nthe complex inductance relationships given by Eqs. 21\nand 22.\nIV. PHASE ERROR OF ∆S21\nThe background correction procedure of Sec. IIIB re-\nquires one further phase correction in order to enforce\nRe(L) (fH)40\n30\n20\n10\n0Im(L) (fH)\n-30-20-100\n35 30 25 20 15 105 0\nFreq (GHz)Py/Pt\nPt/Py\nPy/Cu\nCu/Py\n35 30 25 20 15 105 0\nFreq (GHz)Raw Phase-Corrected~ ~\nFigure S4. Correction of phase to enforce Im( ˜L)(f= 0) = 0\nforl= 6mm sample. Raw data (left panels) show a small,\nnon-zero component of Im( ˜L) atf= 0, which is unphysical.\nWe therefore apply a small correction to eliminate this non-\nzeroy-intercept, resulting in the phase-corrected data (right\npanels).\nthat Im( ˜L)(f= 0) = 0, as any finite Im( ˜L) at zero fre-\nquency would be unphysical. However, as can be seen\nin the raw data of Fig. S4, the intercept of Im( ˜L) at\nf= 0 is indeed a small, finite number (left panels). In\naddition to the background phase correction described in\nEq. 25, we therefore force an additional phase correction\nφcorr= arctan[Im( ˜L)(f= 0)/Re(˜L)(f= 0)]. The φcorr\nnecessary for each sample is shown in Table S3.\nV. SHUNTING CORRECTION\nOursamplesexhibitashuntingeffectwhenthemetallic\nthicknesses are such that the sheet resistance of the sam-\nple drops below 50Ω ( Z0, the characteristic impedance\nof our CPW). This is similar to the shunting effect de-\nscribed in Ref. S2. However, in that case, the atten-\nuation of voltage signals as sample thickness increases\nfollows immediately from Ohm’s law and the decreasing\nresistance across which the iSHE voltage is measured.\nIn our inductive measurements, the AC currents driven\nbyiSOTgeneratesignalvoltagesacrossthecharacteristic\nimpedanceoftheCPW, Z0. However,whenthesampleis\nthick enough, there is also a current return path through\nthe thickness of the sample. For very thick samples, the4\nintegrated current through the sample thickness is zero\n(equal forward and return currents), and the inductive\nsignal drops to zero.\nWethereforemodeltheiSOTeffectsasacurrentsource\nwhich drives current through parallel resistances Z0and\nRs, whereRsis the measuredsheet resistanceofoursam-\nple. For all samples in this study Rswas found to be\n≈34Ω. In this model, only the fraction of the total cur-\nrent generatedby iSOT that flowsthroughthe Z0branch\ncan generatean inductive signal, correspondingto a frac-\ntionRs/(Z0+Rs)≈0.4ofthe totalcurrent. Wetherefore\nscaleσSOT\neandσSOT\noby≈2.5. Note that the Faraday\neffect acts as a source of emf, such that the currents due\nto the Faraday effect are observed to increase linearly\nwith sample thickness, in accordance with Ohm’s Law.\nTherefore, we do not correct σF\neby the same shunting\nfactor.\nVI. MEASUREMENT OF PERMALLOY\nRESISTIVITY\nIn order to determine the interface conductivity σint\nused for determination of αRin main text Eq. 35, wemeasuredthe resistivity ofTa(1.5)/Py( dPy)/Pt(6)/Ta(3)\nand Ta(1.5)/Pt(6)/Py( dPy)/Ta(3) films (thicknesses in\nnanometers) as a function of Py film thickness, dPy(Fig.\nS5). Ineachcase, wefindthatthedataarewell-described\nby a simple model in which the Py resistivity is indepen-\ndent of thickness, and adds as a parallel resistance with\nthe Pt and Ta conducting layers. That is, the total sheet\nresistance Rsis given by: 1 /Rs=dPy/ρ0+ 1/Rother,\nwhereρ0is the Py bulk resistivity, and Rotheris the com-\nbined sheet resistance of the Pt and Ta layers. We mul-\ntiply the measured sheet resistance by the Py thickness,\nsuch that\nRsdPy=dPy\ndPy\nρ0+1\nRother(S10)\nFrom the fits shown in Fig. S5, we find ρ0=\n21.9±0.2×10−8Ωm and Rother= 49.5±0.4Ω for the\nPy/Pt sample, and , ρ0= 22.78±0.04×10−8Ωm and\nRother= 60.7±0.1Ω for the Pt/Py sample. To calulate\nσintfor Eq. 35, we simply use the inverse of these bulk\nresistivity values.\n[S1] C. Kittel, Introduction to Solid State Physics , 8th ed.\n(Wiley, 2004).\n[S2] H. Jiao and G. E. W. Bauer,\nPhysical Review Letters 110, 217602 (2013)./s32/s33/s34/s35/s36/s32/s34/s37/s38\n/s32/s34/s34\n/s33/s34\n/s34/s39/s40/s35/s41/s42/s43/s35/s44/s45/s35/s46/s47\n/s48/s34/s32/s33/s32/s34/s33/s34\n/s41/s42/s43/s35/s44/s49/s46/s47/s35/s42/s50/s44/s51/s47/s52/s42/s43/s44/s41/s42/s43/s47\n/s35/s42/s43/s44/s41/s42/s43/s47/s52/s42/s50/s44/s51/s47\n/s35\n/s35\nFigure S5. Measured sheet resistance vs. Py\nthickness dPyfor both stacking orders of Py\nand Pt: Ta(1.5)/Py( dPy)/Pt(6)/Ta(1.5) and\nTa(1.5)/Pt(6)/Py( dPy)/Ta(1.5). Eq. S10 is used as the\nfit function." }, { "title": "1911.00744v2.Soft_contribution_to_the_damping_rate_of_a_hard_photon_in_a_weakly_magnetized_hot_medium.pdf", "content": "Soft contribution to the damping rate of a hard photon in a weakly magnetized\nhot medium\nRitesh Ghosh,1, 2,\u0003Bithika Karmakar,1, 2,yand Munshi G Mustafa1, 2,z\n1Theory Division, Saha Institute of Nuclear Physics,\n1/AF, Bidhannagar, Kolkata 700064, India\n2Homi Bhabha National Institute, Anushaktinagar,\nMumbai, Maharashtra 400094, India\nWe consider weakly magnetized hot QED plasma comprising electrons and positrons.\nThere are three distinct dispersive (longitudinal and two transverse) modes of a photon in a\nthermomagnetic medium. At lowest order in the coupling constant, a photon is damped in\nthis medium via Compton scattering and pair creation process. We evaluate the damping\nrate of hard photon by calculating the imaginary part of the each transverse dispersive modes\nin a thermomagnetic QED medium. We note that one of the fermions in the loop of one-loop\nphoton self-energy is considered as soft and the other one is hard. Considering the resummed\nfermion propagator in a weakly magnetized medium for the soft fermion and the Schwinger\npropagator for hard fermion, we calculate the soft contribution to the damping rate of hard\nphoton. In weak \feld approximation the thermal and thermomagnetic contributions to\ndamping rate get separated out for each transverse dispersive mode. The total damping rate\nfor each dispersive mode in presence of magnetic \feld is found to be reduced than that of\nthe thermal one. This formalism can easily be extended to QCD plasma.\n\u0003ritesh.ghosh@saha.ac.in\nybithika.karmakar@saha.ac.in\nzmunshigolam.mustafa@saha.ac.inarXiv:1911.00744v2 [hep-ph] 7 Mar 20202\nI. INTRODUCTION\nAstrophysical plasma is almost always immersed in magnetic \feld. Extreme, magnetized plasma\nis found in interiors of neutron star, magnetospheres of magnetars and central engines of super-\nnovae and gamma ray bursts [1]. The propagation of photon through the hot magnetized plasma,\nviz., electron-positron plasma (EPP), is of great interest. Because the magnetar phenomena are\nfound by analyzing the high-energy radiation detected at earth. Thus it is very important to have a\ngood understanding of the propagation of photon through the EPP. Furthermore, the phenomenon\nof Faraday rotation i.e., change of polarization of photon while propagating through a medium\nhas been studied in Ref. [2] in a \feld theoretical viewpoint. This has also been detected in several\nastrophysical objects [3]. Also high-intensity laser beams are used to create ultrarelativistic EPP\nof temperature around 10 MeV [4]. This EPP may play an important role in various astrophysical\nsituations. Some properties of such plasma, viz., the equation of state, dispersion relation of collec-\ntive plasma modes of photon and electron, damping rates, mean free paths, transport coe\u000ecients\nand particle production rates, are studied using QED at \fnite temperature [5, 6].\nOn the other hand in noncentral heavy ion collisions, the magnetic \feld as high as (15 \u000020)m2\n\u0019\ncan be generated [7] at LHC energies. After a few fm/ cof the collision, the magnetic \feld strength1\ndecreases to (1\u00002)m2\n\u0019. The e\u000bect of magnetic \feld on the properties of the QCD matter [ viz.\nquark-gluon plasma(QGP)] and on the phase diagram of QCD is of great interest. Recently, several\nstudies have found the e\u000bect of magnetic catalysis [13{16], i.e., the enhancement of phase transition\ntemperature of QCD matter in presence of external magnetic \feld, whereas, some results of inverse\nmagnetic catalysis [17{25] have been reported. Various properties of QCD matter at weak coupling\nin presence of magnetic \feld is being studied including the equation of state [26{28], transport\nproperties [29{31]. Modi\fcation of QCD Debye mass and the two point correlation functions of\nquarks [32] and gluons [33{35] i.e., partons have been analyzed recently. Dilepton production rate\nfrom a hot magnetized QCD plasma [12, 36{42] has been calculated. The photon is also considered\nas a good probe of the QGP medium as photon only interacts electromagnetically. Thus, it comes\nout of the hot QCD system without interacting much. The damping rate of the hard photon is\nassociated with the mean free path of photon [43] and hard photon production rate in QGP [44].\nDamping rate of photon is related to the imaginary part of photon dispersion in the medium [45]\nwhich is again related to the scattering crosssection of the process that we \fnd by cutting the pho-\n1The initial magnitude of this magnetic \feld can be very high at the initial time of the heavy-ion collisions and\nthen it decreases very fast, being inversely proportional to the square of time [8, 9]. However for a di\u000berent point\nof view, see Refs. [10{12], where the time dependence of magnetic \feld is shown to be adiabatic due to the high\nconductivity of the medium.3\nton self-energy diagram [46]. In lowest order coupling constant, photons are damped by Compton\nscattering and pair creation process. In case of low momentum transfer, the damping rate shows\ninfrared singularity. Thus one should consider the e\u000bective resummed propagator instead of bare\npropagator for soft momentum of fermion. We will call this as the soft contribution to the damping\nrate of photon. The hard contribution refers to the case where all the fermions in loop have momen-\ntum order of or much greater than the system temperature T. Both soft and hard contributions\nto the damping rate of hard photon in thermal medium have been calculated in Ref. [45]. The\ndispersion relations of photon are modi\fed for a hot magnetized medium [33]. So the damping rate\nof photon will also get modi\fed in a thermomagnetic medium. In this article we intend to compute\nthe soft contribution to the hard photon damping rate for a weakly magnetized hot medium in\none loop approximation of photon self-energy. In a thermomagnetic medium this would be a good\nindicator as higher loop calculation contributing to higher order would be extremely involved.\nWe consider hard photon of momentum P\u0016= (p0;p) wherep=jpj\u001dTin a relativistic\nhot magnetized QED medium. To \fnd the soft contribution of the damping rate we introduce\na separation scale \u0003 where eT\u001c\u0003\u001cT(gT\u001c\u0003\u001cTin case of QCD). In the soft part of\nthe damping rate, the contribution from soft loop momentum involving a fermion is taken into\naccount up to the separation scale \u0003 . Here we assume that the magnetic \feld strength is weak\ni.e.,p\neB < eT < T (pqfB < gT < T for QCD). We use the recently obtained e\u000bective fermion\npropagator [32] in presence of weak magnetic \feld for the soft fermion and Schwinger propagator\nfor the hard fermion in the loop. The Braaten-Pisarki-Yuan formalism [47] has been used here to\ncalculate the imaginary part of photon self-energy. Extension to the case of damping rate of hard\nphoton in weakly magnetized hot QCD medium is straightforward. We need to consider the loop\nfermions as quark and antiquark in that case.\nIn Sec. II we describe the set up to calculate the photon damping rate associated with imaginary\npart of photon self-energy. In Sec. III the self-energy is obtained in a weak \feld approximation.\nThe imaginary parts of various components of photon self-energy is obtained in Sec. IV. Results\nare given in Sec. V. We conclude in Sec. VI.\nII. SETUP\nWe consider plasma of electrons and positrons at temperature T. Thez-axis of the lab frame\nis oriented along the magnetic \feld. The general structure of the gauge boson self-energy and\ncorresponding e\u000bective propagator have been evaluated in Ref. [33]. The general covariant structure4\nof photon self-energy in a magnetized hot medium can be written as\n\u0005\u0016\u0017=\fB\u0016\u0017+\u001bR\u0016\u0017+\u000eQ\u0016\u0017+\u000bN\u0016\u0017; (1)\nwhere various form factors can be written as\n\f=B\u0016\u0017\u0005\u0016\u0017;\n\u001b=R\u0016\u0017\u0005\u0016\u0017;\n\u000e=Q\u0016\u0017\u0005\u0016\u0017;\n\u000b=1\n2N\u0016\u0017\u0005\u0016\u0017: (2)\nThe general covariant structure of photon propagator can be obtained [33] as\nD\u0016\u0017=\u0018P\u0016P\u0017\nP4+(P2\u0000\u000e)B\u0016\u0017\n(P2\u0000\f)(P2\u0000\u000e)\u0000\u000b2+R\u0016\u0017\nP2\u0000\u001b+(P2\u0000\f)Q\u0016\u0017\n(P2\u0000\f)(P2\u0000\u000e)\u0000\u000b2\n+\u000bN\u0016\u0017\n(P2\u0000\f)(P2\u0000\u000e)\u0000\u000b2: (3)\nWe note that the thermal medium (absence of magnetic \feld) has two dispersive modes of photon\ni.e., one degenerate transverse mode and one medium induced plasmon mode due to breaking of\nboost invariance. Now breaking of rotational invariance in the presence of a magnetic \feld leads\nto three dispersive modes of photon by lifting the degeneracy of the transverse modes. These three\ndispersive modes can be seen from the pole of Eq. (3). Now, the dispersion relations can be written\nas\nP2\u0000\u001b= 0; (4)\n(P2\u0000\u000e)(P2\u0000\f)\u0000\u000b2=\u0012\nP2\u0000\f+\u000e+p\n(\f\u0000\u000e)2+ 4\u000b2\n2\u0013\n\u0002\u0012\nP2\u0000\f+\u000e\u0000p\n(\f\u0000\u000e)2+ 4\u000b2\n2\u0013\n= 0: (5)\nIn weak magnetic \feld approximation \u000bdoes not contribute upto O[(eB)]2, one gets simple form\nof the above dispersive modes [26]\nP2\u0000\u001b= 0;\nP2\u0000\f= 0;\nP2\u0000\u000e= 0: (6)\nDamping rate is de\fned as the imaginary part of photon dispersion relation. The medium induced\nlongitudinal (plasmon) mode does not contribute to the damping rate2and the dispersion relations\n2The longitudinal dispersive mode merges with the light cone at high photon momentum.5\nfor two transverse modes of a photon are given, respectively, as\nP2\u0000\u001b= 0; P2\u0000\u000e= 0; (7)\nDamping rates \r\u000e(p) and\r\u001b(p) (for no overdamping i:e: \ri\u001cp0wherei=\u000e;\u001b) of hard photon\nare given by imaginary part of the form factors as [48]\n\r\u001b(p) =\u00001\n2pIm\u001b(p0=p); (8)\n\r\u000e(p) =\u00001\n2pIm\u000e(p0=p): (9)\nThe tensor structures of R\u0016\u0017andQ\u0016\u0017[33] are given as\nR\u0016\u0017=0\nBBBBBB@0 0 0 0\n0 0 0 0\n0 0\u00001 0\n0 0 0 01\nCCCCCCA; Q\u0016\u0017=0\nBBBBBB@0 0 0 0\n0\u0000cos2\u0012p0 sin\u0012pcos\u0012p\n0 0 0 0\n0 sin\u0012pcos\u0012p0\u0000sin2\u0012p1\nCCCCCCA: (10)\nUsing Eq.(10) in Eq.(2) we can write the form factors \u001band\u000ein weak \feld approximation as\n\u001b=\u0000\u0010\n\u000522\n0+ \u000522\n2\u0011\n; (11)\n\u000e=\u0000cos2\u0012p\u0010\n\u000511\n0+ \u000511\n2\u0011\n\u0000sin2\u0012p\u0010\n\u000533\n0+ \u000533\n2\u0011\n+ 2 sin\u0012pcos\u0012p\u0010\n\u000513\n0+ \u000513\n2\u0011\n: (12)\nCombining Eq.(8) with Eq.(11) and Eq.(9) with Eq.(12), the damping rates become\n\r\u001b(p) =1\n2p\u0010\nIm\u000522\n0+ Im\u000522\n2\u0011\n; (13)\n\r\u000e(p) =1\n2ph\ncos2\u0012p\u0010\nIm\u000511\n0+ Im\u000511\n2\u0011\n+ sin2\u0012p\u0010\nIm \u000533\n0+ Im \u000533\n2\u0011\n\u00002 sin\u0012pcos\u0012p\u0010\nIm \u000513\n0+ Im \u000513\n2\u0011i\n(14)\nThe damping rates in Eqs.(13) and (14) can now be written as\n\r\u001b(p) =\rth(p) +\rB\n\u001b(p); (15)\n\r\u000e(p) =\rth(p) +\rB\n\u000e(p): (16)\nwhere\rthis theO[(eB)0] contribution or thermal contribution is given as\n\rth(p) =1\n2pIm\u000522\n0=1\n2ph\ncos2\u0012pIm\u000511\n0+ sin2\u0012pIm \u000533\n0\u00002 sin\u0012pcos\u0012pIm \u000513\n0i\n: (17)\nThe thermomagnetic corrections of O[(eB)2] are given as\n\rB\n\u001b(p) =1\n2pIm\u000522\n2; (18)\n\rB\n\u000e(p) =1\n2ph\ncos2\u0012pIm\u000511\n2+ sin2\u0012pIm \u000533\n2\u00002 sin\u0012pcos\u0012pIm \u000513\n2i\n: (19)\nWe need to obtain the imaginary parts of 11, 22, 33 and 13 components of the photon self-energy\n\u0005\u0016\u0017which are computed in the following sections.6\nIII. PHOTON SELF-ENERGY IN HOT MAGNETIZED MEDIUM\nThe photon self-energy as shown in Fig. 1 can be written as\n\u0005\u0016\u0017(P) =ie2Zd4K\n(2\u0019)4\u001a\nTr[\r\u0016S\u0003(K)\r\u0017S(Q)] + Tr[\r\u0017S\u0003(K)\r\u0016S(Q)]\u001b\n: (20)\nwhereS\u0003(K) is e\u000bective electron propagator and S(K) is Schwinger propagator for bare electron.\nAs the external photon is hard, we consider one bare and one e\u000bective fermion propagator in the\nloop. In the following we would obtain the propagators for fermion.\nPK\nQ=K−P\nFIG. 1: Photon self-energy where the blob represents the e\u000bective electron propagator in\nmagnetic \feld and double line represents bare electron propagator in magnetic \feld\nA. Fermion propagator in weak \feld approximation\nIn the weak magnetic \feld limit, i.e.,p\neB < m th\u0018eT < T the Schwinger propagator for\nfermion can be written up to O[(eB)2] as [49]\nS(K) == K+mf\nK2\u0000m2\nf+i\r1\r2= Kq+mf\n(K2\u0000m2\nf)2(eB) + 2\"\nf(K\u0001u)= u\u0000(K\u0001n)= ng\u0000= K\n(K2\u0000m2\nf)3\u0000k2\n?(= K+mf)\n(K2\u0000m2\nf)4#\n(eB)2\n+O\u0002\n(eB)3\u0003\n=S0+S1+S2+O[(eB)3]: (21)\nThe general form of fermion self-energy in a weakly magnetized medium can be written as [32]\n\u0006(K) =\u0000a= K\u0000b= u\u0000b0\r5= u\u0000c0\r5= n : (22)\nIn one loop order, the form factors are given as\na(k0;k) =\u0000m2\nth\nk2Q1\u0012k0\nk\u0013\n; (23)\nb(k0;k) =m2\nth\nk\u0014k0\nkQ1\u0012k0\nk\u0013\n\u0000Q0\u0012k0\nk\u0013\u0015\n; (24)\nb0(k0;k) = 4e2M2(T;mf;eB)k3\nk2Q1\u0012k0\nk\u0013\n; (25)7\nc0(k0;k) = 4e2M2(T;mf;eB)1\nkQ0\u0012k0\nk\u0013\n; (26)\nwhere Legendre function of second kind are given as\nQ0(x) =1\n2ln\u0012x+ 1\nx\u00001\u0013\n; (27)\nQ1(x) =xQ0(x)\u00001; (28)\nand the thermomagnetic mass is given as\nM2(T;mf;eB) =eB\n16\u00192\u0014\nln 2\u0000\u0019T\n2mf\u0015\n; (29)\nwhereas thermal mass is given as\nm2\nth=1\n8e2T2: (30)\nThe e\u000bective fermion propagator can be written [32] as\nS\u0003(K) =P\u0000= L(K)\nL2P++P+= R(K)\nR2P\u0000\n=S\u0003\nL(K) +S\u0003\nR(K); (31)\nwhere chirality projection operators are given by\nP\u0006=1\n2(1\u0006\r5); (32)\nandL\u0016andR\u0016are given as\nL\u0016= (1 +a)K\u0016+ (b+b0)u\u0016+c0n\u0016; (33)\nR\u0016= (1 +a)K\u0016+ (b\u0000b0)u\u0016\u0000c0n\u0016: (34)\nFor simplicity of calculation we expand the e\u000bective fermion propagator in Eq. (31) in powers\nofeBand keep terms up to O[(eB)2] as\nS\u0003(K) =S\u0003\n0(K) +S\u0003\n1(K) +S\u0003\n2(K) +O[(eB)3]; (35)\nwhereS\u0003\n0(K) isO[(eB)0] and given as\nS\u0003\n0(K) =(1 +a)= K+b= u\nD2=(1 +a)= K+b= u\nD+D\u0000; (36)\nwhereD\u0006= (1 +a)(k0\u0007k) +b.\nEquation (36) is the e\u000bective HTL fermion propagator [50, 51] in thermal medium. The O[(eB)]\nis obtained as\nS\u0003\n1(K) =1\nD4\u0014\n2(1 +a)= K\r5\u001a\n\u0000(1 +a)k3c0\u0000(1 +a)k0b0\u0000bb0\u001b8\n+= u\r5\u001a\u0010\n(1 +a)2K2\u0000b2\u0011\nb0\u00002(a+ 1)bc0k3\u001b\n+c0= n\r5\u001a\u0010\n2(1 +a)k0+b\u0011\nb+ (a+ 1)2K2\u001b\u0015\n; (37)\nwhereasO[(eB)2] is obtained as\nS\u0003\n2(K) =\u0014\u0010\n2b0\b\n(1 +a)k0+b\t\n+ 2c0k3(1 +a)\u00112\nD6\u0000b02\u0000c02\nD4\u0015\u001a\n(1 +a)= K+b= u\u001b\n\u0000\u0010\n2b0\b\n(1 +a)k0+b\t\n+ 2c0k3(1 +a)\u0011\u0010\nb0= u+c0= n\u0011\nD4\n=\u0012h2(k0;k?;k3)\nD6\u0000h0\nD4\u0013n\n(1 +a)= K+b= uo\n\u0000h(k0;k?;k3)\nD4\u0010\nb0= u+c0= n\u0011\n; (38)\nwhereh= 2b0\b\n(1 +a)k0+b\t\n+ 2c0k3(1 +a) andh0=b02\u0000c02.\nB. Photon self-energy in weak magnetic \feld\nNow theO[(eB)0] contribution of \u0005\u0016\u0017given in Eq. (20) can be written as\n\u0005\u0016\u0017\n0=ie2Zd4K\n(2\u0019)4\u001a\nTr[\r\u0016S\u0003\n0(K)\r\u0017S0(Q)] + Tr[\r\u0017S\u0003\n0(K)\r\u0016S0(Q)]\u001b\n=ie2Zd4K\n(2\u0019)4(\nTr\u0014\n\r\u0016\u0012\r0\u0000~ \r\u0001^k\n2D++\r0+~ \r\u0001^k\n2D\u0000\u0013\n\r\u0017\u0012\nf(1)\n0\r0\u0000f(0)\n0~ \r\u0001~ q\u0013\u0015\n+ Tr\u0014\n\r\u0017\u0012\r0\u0000~ \r\u0001^k\n2D++\r0+~ \r\u0001^k\n2D\u0000\u0013\n\r\u0016\u0012\nf(1)\n0\r0\u0000f(0)\n0~ \r\u0001~ q\u0013\u0015)\n= 8ie2Zd4K\n(2\u0019)4(1 +a)\b\u0000\nK\u0016Q\u0017+K\u0017Q\u0016\u0001\n\u0000g\u0016\u0017K\u0001Q\t\n+b\b\u0000\nQ\u0016u\u0017+Q\u0017u\u0016\u0001\n\u0000g\u0016\u0017Q\u0001u\t\nD+D\u0000Q2;\n(39)\nwhere\nf(1)\n0=q0\nQ2; f(0)\n0=1\nQ2;\nf(1)\n1=q0\nQ4; f(0)\n1=1\nQ4: (40)\nTheO[(eB)] contribution of \u0005\u0016\u0017is given as\n\u0005\u0016\u0017\n1=ie2Zd4K\n(2\u0019)4\u001a\nTr[\r\u0016S\u0003\n0(K)\r\u0017S1(Q)] + Tr[\r\u0017S\u0003\n0(K)\r\u0016S1(Q)]\n+ Tr[\r\u0016S\u0003\n1(K)\r\u0017S0(Q)] + Tr[\r\u0017S\u0003\n1(K)\r\u0016S0(Q)]\u001b\n; (41)\nwhich becomes zero.9\nTheO[(eB)2] contribution of \u0005\u0016\u0017is given as\n\u0005\u0016\u0017\n2=ie2Zd4K\n(2\u0019)4\u001a\nTr[\r\u0016S\u0003\n1(K)\r\u0017S1(Q)] + Tr[\r\u0017S\u0003\n1(K)\r\u0016S1(Q)] + Tr[\r\u0016S\u0003\n0(K)\r\u0017S2(Q)]\n+ Tr[\r\u0017S\u0003\n0(K)\r\u0016S2(Q)] + Tr[\r\u0016S\u0003\n2(K)\r\u0017S0(Q)] + Tr[\r\u0017S\u0003\n2(K)\r\u0016S0(Q)]\u001b\n: (42)\nWe calculate the above mentioned trace as follows. The trace of the \frst and second terms of\nEq. (42) can be calculated as\nTr[\r\u0016S\u0003\n1(K)\r\u0017S1(Q)] + Tr[\r\u0017S\u0003\n1(K)\r\u0016S1(Q)]\n=8 (eB)\nD2\u0010\nQ2\u0000m2\nf\u00112\"\nb0n\n(u\u0016n\u0017+u\u0017n\u0016)(Q\u0001u)\u00002u\u0016u\u0017(Q\u0001n) +g\u0016\u0017(Q\u0001n)o\n+c0n\n2n\u0016n\u0017(Q\u0001u)\u0000(n\u0016u\u0017+n\u0017u\u0016)(Q\u0001n) +g\u0016\u0017(Q\u0001u)o#\n\u00008 (eB)\nD4\u0010\nQ2\u0000m2\nf\u00112\n\u0002\"\nh\u001a\u0000\n1 +a\u0001\u0012\ng\u0016\u0017\u0010\n(K\u0001u)(Q\u0001n)\u0000(K\u0001n)(Q\u0001u)\u0011\n\u0000(K\u0016u\u0017+K\u0017u\u0016)Q\u0001n\n+ (K\u0016n\u0017+K\u0017n\u0016)Q\u0001u\u0013\n+b\u0012\ng\u0016\u0017Q\u0001n+ (u\u0016n\u0017+u\u0017n\u0016)Q\u0001u\u00002u\u0016u\u0017Q\u0001n\u0013\u001b#\n: (43)\nThe trace of third and fourth terms in Eq. (42) can be obtained as\nTr[\r\u0016S\u0003\n0(K)\r\u0017S2(Q)] + Tr[\r\u0017S\u0003\n0(K)\r\u0016S2(Q)]\n=8(eB)2\nD+(Q2\u0000m2\nf)3\u0014\nq0\u0010\n^K\u0016g0\u0017+^K\u0017g0\u0016\u0000g\u0016\u0017\u0011\n\u0000q3\u0010\n^K\u0016g3\u0017+^K\u0017g3\u0016\u0000g\u0016\u0017^k3\u0011\n\u0000\u0010\n^K\u0016Q\u0017+^K\u0017Q\u0016\u0000g\u0016\u0017^K\u0001Q\u0011\u0015\n\u00008(eB)2q2\n?\nD+(Q2\u0000m2\nf)4\u0014\n^K\u0016Q\u0017+^K\u0017Q\u0016\u0000g\u0016\u0017^K\u0001Q\u0015\n+8(eB)2\nD\u0000(Q2\u0000m2\nf)3\u0014\nq0\u0010\n^K0\u0016g0\u0017+^K0\u0017g0\u0016\u0000g\u0016\u0017\u0011\n\u0000q3\u0010\n^K0\u0016g3\u0017+^K0\u0017g3\u0016+g\u0016\u0017^k3\u0011\n\u0000\u0010\n^K0\u0016Q\u0017+^K0\u0017Q\u0016\u0000g\u0016\u0017^K0\u0001Q\u0011\u0015\n\u00008(eB)2q2\n?\nD\u0000(Q2\u0000m2\nf)4\u0014\n^K0\u0016Q\u0017+^K0\u0017Q\u0016\u0000g\u0016\u0017^K0\u0001Q\u0015\n;(44)\nwhere ^K0\u0016= (1;\u0000^k).\nThe trace of \ffth and sixth terms in Eq. (42) are obtained as\nTr[\r\u0016S\u0003\n2(K)\r\u0017S0(Q)] + Tr[\r\u0017S\u0003\n2(K)\r\u0016S0(Q)]\n=8\n(Q2\u0000m2\nf)\"\u0010h2\nD6\u0000h0\nD4\u0011\u0010\n1 +a\u0011\u0010\nK\u0016Q\u0017+K\u0017Q\u0016\u0000g\u0016\u0017K\u0001Q\u0011\n+\u0010\nb\u0010h2\nD6\u0000h0\nD4\u0011\n\u0000b0h\nD4\u0011\n\u0002\n\u0010\ng\u00160Q\u0017+g\u00170Q\u0016\u0000g\u0016\u0017q0\u0011\n\u0000c0h\nD4\u0010\ng\u00163Q\u0017+g\u00173Q\u0016\u0000g\u0016\u0017q3\u0011#\n: (45)10\nThe photon self-energy in weak \feld approximation now can be decomposed using Eqs.(39),(41),(42)\nas\n\u0005\u0016\u0017(P) = \u0005\u0016\u0017\n0(P) + \u0005\u0016\u0017\n2(P); (46)\nwhere the \frst term is a pure thermal( O[(eB)0]) contribution and second term is thermomagnetic\ncorrection ofO[(eB)2].\nNow theO[(eB)0] expression of \u000511, \u000522, \u000533, and \u000513can be written from Eq. (39) as\n\u000511\n0(p0;p) = 8ie2Zd4K\n(2\u0019)4(1 +a)(k0q0+ 2k1q1\u0000~k\u0001~ q) +bq0n\u0000\n(1 +a)k0+b\u00012\u0000(1 +a)2k2o\nQ2\n=\u00004e2XZ\"\u0012f(1)\n0\nD++f(1)\n0\nD\u0000\u0013\n+\u0000\n2^k1q1\u0000^k\u0001q\u0001\u001af(0)\n0\nD+\u0000f(0)\n0\nD\u0000\u001b#\n;\n\u000522\n0(p0;p) = 8ie2Zd4K\n(2\u0019)4(1 +a)(k0q0+ 2k2q2\u0000~k\u0001~ q) +bq0n\u0000\n(1 +a)k0+b\u00012\u0000(1 +a)2k2o\nQ2\n=\u00004e2XZ\"\u0012f(1)\n0\nD++f(1)\n0\nD\u0000\u0013\n+\u0000\n2^k2q2\u0000^k\u0001q\u0001\u001af(0)\n0\nD+\u0000f(0)\n0\nD\u0000\u001b#\n;\n\u000533\n0(p0;p) = 8ie2Zd4K\n(2\u0019)4(1 +a)(k0q0+ 2k3q3\u0000~k\u0001~ q) +bq0n\u0000\n(1 +a)k0+b\u00012\u0000(1 +a)2k2o\nQ2\n=\u00004e2XZ\"\u0012f(1)\n0\nD++f(1)\n0\nD\u0000\u0013\n\u0000\u0000^k\u0001q\u00002^k3q3\u0001\u001af(0)\n0\nD+\u0000f(0)\n0\nD\u0000\u001b#\n;\n\u000513\n0(p0;p) = 8ie2Zd4K\n(2\u0019)4(1 +a)(k1q3+q1k3)n\u0000\n(1 +a)k0+b\u00012\u0000(1 +a)2k2o\nQ2\n=\u00004e2XZ\"\n(^k1q3+q1^k3)\u001af(0)\n0\nD+\u0000f(0)\n0\nD\u0000\u001b#\n: (47)\nUsing Eqs.(42),(43),(44) and (45), one can write the O[(eB)2] expression of \u000511, \u000522, \u000533, and\n\u000513as\n\u000511\n2=\u0000e2XZ\"\n\u00008eB\nD2(Q2\u0000m2\nf)2\u001a\nb0q3+c0q0\u001b\n+8eB\nD4(Q2\u0000m2\nf)2h\u001a\n(1 +a)(k0q3\u0000k3q0) +bq3\u001b\n+8(eB)2\nD+(Q2\u0000m2\nf)3\u0010\n^k2q2\u0000^k1q1\u0011\n\u00008(eB)2q2\n?\nD+(Q2\u0000m2\nf)4\u0010\nq0\u0000^k\u0001q+ 2^k1q1\u0011\n\u00008(eB)2\nD\u0000(Q2\u0000m2\nf)3\u0010\n^k2q2\u0000^k1q1\u0011\n\u00008(eB)2q2\n?\nD\u0000(Q2\u0000m2\nf)4\u0010\nq0+^k\u0001q\u00002^k1q1\u0011\n+8\n(Q2\u0000m2\nf)\u001a\u0010h2\nD6\u0000h0\nD4\u0011\n(1 +a)(2k1q1+K\u0001Q) +\u0012\nb\u0010h2\nD6\u0000h0\nD4\u0011\n\u0000b0h\nD4\u0013\nq0\n\u0000c0h\nD4q3\u001b#\n; (48)11\n\u000522\n2=\u0000e2XZ\"\n\u00008eB\nD2(Q2\u0000m2\nf)2\u001a\nb0q3+c0q0\u001b\n+8eB\nD4(Q2\u0000m2\nf)2h\u001a\n(1 +a)(k0q3\u0000k3q0) +bq3\u001b\n+8(eB)2\nD+(Q2\u0000m2\nf)3\u0010\n^k1q1\u0000^k2q2\u0011\n\u00008(eB)2q2\n?\nD+(Q2\u0000m2\nf)4\u0010\nq0\u0000^k\u0001q+ 2^k2q2\u0011\n\u00008(eB)2\nD\u0000(Q2\u0000m2\nf)3\u0010\n^k1q1\u0000^k2q2\u0011\n\u00008(eB)2q2\n?\nD\u0000(Q2\u0000m2\nf)4\u0010\nq0+^k\u0001q\u00002^k2q2\u0011\n+8\n(Q2\u0000m2\nf)\u001a\u0010h2\nD6\u0000h0\nD4\u0011\n(1 +a)(2k2q2+K\u0001Q) +\u0012\nb\u0010h2\nD6\u0000h0\nD4\u0011\n\u0000b0h\nD4\u0013\nq0\n\u0000c0h\nD4q3\u001b#\n; (49)\n\u000533\n2=\u0000e2XZ\"\n8eB\nD2(Q2\u0000m2\nf)2\u001a\n\u0000b0q3+c0q0\u001b\n+8eB\nD4(Q2\u0000m2\nf)2h\u001a\n(1 +a)(k0q3+k3q0) +bq3\u001b\n+8(eB)2\nD+(Q2\u0000m2\nf)3\u0010\n\u0000q3^k3+^k\u0001q\u0011\n\u00008(eB)2q2\n?\nD+(Q2\u0000m2\nf)4\u0010\nq0\u0000^k\u0001q+ 2q3^k3\u0011\n+8(eB)2\nD\u0000(Q2\u0000m2\nf)3\u0010\nq3^k3\u0000^k\u0001q\u0011\n\u00008(eB)2q2\n?\nD\u0000(Q2\u0000m2\nf)4\u0010\nq0+^k\u0001q\u00002q3^k3\u0011\n+8\n(Q2\u0000m2\nf)\u001a\u0010h2\nD6\u0000h0\nD4\u0011\n(1 +a)(2k3q3+K\u0001Q) +\u0012\nb\u0010h2\nD6\u0000h0\nD4\u0011\n\u0000b0h\nD4\u0013\nq0\n+c0h\nD4q3\u001b#\n; (50)\n\u000513\n2=\u0000e2XZ\"\n8eB\nD4(Q2\u0000m2\nf)2h(1 +a)(k1q0)\u00008(eB)2\nD+(Q2\u0000m2\nf)3^k3q1\u00008(eB)2q2\n?\nD+(Q2\u0000m2\nf)4\n\u0002\u0010\n^k1q3+^k3q1\u0011\n+8(eB)2\nD\u0000(Q2\u0000m2\nf)3^k3q1+8(eB)2q2\n?\nD\u0000(Q2\u0000m2\nf)4\u0010\n^k1q3+^k3q1\u0011\n+8\n(Q2\u0000m2\nf)\n\u0002\u001a\u0010h2\nD6\u0000h0\nD4\u0011\n(1 +a)\u0010\nk1q3+k3q1\u0011\n+c0h\nD4q1\u001b#\n: (51)\nIV. IMAGINARY PARTS OF THE COMPONENTS OF THE PHOTON SELF-ENERGY\nBefore obtaining the imaginary parts, we discuss below the various approximations used in this\ncalculation.\n1. We have considered the momentum of photon as hard ( p\u001dT). The momentum of soft\nfermionk\u001cT. Thus we can take the following approximations:\nnF(!)\u00181; nF(p\u0000!)\u0018e\u0000p=T; e\u0000p=T\u00180: (52)12\n2. An upper cuto\u000b \u0003( < T) of the soft fermion momentum khas been introduced in the inte-\ngrations.\n3. We consider mf=mthfor electron.\nθ\nxyz\nk\np\nφθp\nFIG. 2: Choice of reference frame for computing the various components of photon self-energy.\nThe magnetic \feld is along z-direction and \u0012pis the angle between momentum of photon and the\nexternal magnetic \feld.\n4. To perform the various integrations we choose a frame of reference as shown in Fig. 2 in\nwhich the external momentum of the photon in xzplane with 0 < \u0012p< \u0019= 2. So one can\nwrite\n~ p\u0011(psin\u0012p;0; pcos\u0012p); (53)\nand then the loop momentum as\n~k\u0011(ksin\u0012cos\u001e; ksin\u0012sin\u001e; kcos\u0012): (54)\nIn the following subsection we will obtain imaginary parts of various self-energy components.\nA. Imaginary parts of the magnetic \feld independent part, i:e:O[(eB)0]\nWe evaluate the imaginary parts of \u000511\n0, \u000522\n0, \u000533\n0, and \u000513\n0using the Braaten-Pisarski-Yuan\nmethod [45, 47].13\nIm \u000511\n0=\u00004e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\u001a\n\u001a(1)\n0(!0)\u0010\n\u001aD+(!)\n+\u001aD\u0000(!)\u0011\n\u0000\u0000^k\u0001q\u00002^k1q1\u0001\u0010\n\u001aD+(!)\u0000\u001aD\u0000(!)\u0011\n\u001a(0)\n0(!0)\u001b\n\u000e(!+!0\u0000p); (55)\nIm \u000522\n0=\u00004e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\u001a\n\u001a(1)\n0(!0)\u0010\n\u001aD+(!)\n+\u001aD\u0000(!)\u0011\n\u0000\u0000^k\u0001q\u00002^k2q2\u0001\u0010\n\u001aD+(!)\u0000\u001aD\u0000(!)\u0011\n\u001a(0)\n0(!0)\u001b\n\u000e(!+!0\u0000p); (56)\nIm \u000533\n0=\u00004e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\u001a\n\u001a(1)\n0(!0)\u0010\n\u001aD+(!)\n+\u001aD\u0000(!)\u0011\n\u0000\u0000^k\u0001q\u00002^k3q3\u0001\u0010\n\u001aD+(!)\u0000\u001aD\u0000(!)\u0011\n\u001a(0)\n0(!0)\u001b\n\u000e(!+!0\u0000p); (57)\nIm \u000513\n0=\u00004e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\u001a\n(^k1q3+q1^k3)\n\u0002\u0010\n\u001aD+(!)\u0000\u001aD\u0000(!)\u0011\n\u001a(0)\n0(!0)\u001b\n\u000e(!+!0\u0000p); (58)\nwhere\u001a(0)\n0;\u001a(1)\n0;\u001a(0)\n1;\u001a(1)\n1;\u001aD+and\u001aD\u0000are spectral representation of f(0)\n0;f(1)\n0;f(0)\n1;f(1)\n1;1=D+and\n1=D\u0000respectively. These spectral functions are obtained in Appendix A. We know that both \u001aD+\nand\u001aD\u0000have pole containing the mass shell \u000e-function + Landau cut part in space like region\nwhereas\u001a(0)\n0;\u001a(1)\n0;\u001a(0)\n1;\u001a(1)\n1have only pole containing the mass shell \u000efunction. Since imaginary\nparts of various components of the self-energy contain the product of two spectral functions, it\nwould then have the pole-pole and the pole-cut contributions.\nThe pole-pole parts of Im\u0005 11, Im\u0005 22, Im\u0005 33and Im\u0005 13contain\u000e(p\u0000!\u0006\u0000q) where!\u0006is the\nenergy of the fermion quasiparticle, ~kand~ q=~k\u0000~ pare the momenta of soft and hard fermion,\nrespectively. Hence !\u0006>k. The\u000e-function yields\np\u0000!\u0006\u0000q= 0\ncos\u001e\u0019!\u0006=k\u0000cos\u0012cos\u0012p\nsin\u0012sin\u0012p:\nThe value of!\u0006=k\u0000cos\u0012cos\u0012p\nsin\u0012sin\u0012pexcludes the range [ \u00001;1] for all values of the parameters \u0012and\u0012p\n. This restriction is valid for both thermal and the magnetic case. Thus pole-pole parts do not\ncontribute in this calculation [44, 45]. In O[(eB)0] the contribution comes only from the pole-cut\npart.14\n1. Pole-cut part of O[(eB)0]\nNow we would \fnd the pole-cut part of the above self-energy components in Eqs.(55), (56), (57)\nand (58) as\nIm \u000511\n0\f\f\f\f\npole\u0000cut\n= 2e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\u001a\n\u000e(!0\u0000q)\u0002(k2\u0000!2)\n\u0002\u0010\n\f+(!) +\f\u0000(!)\u0011\n\u00001\nq(^k\u0001q\u00002^k1q1)\u000e(!0\u0000q)\u0002(k2\u0000!2)\u0010\n\f+(!)\u0000\f\u0000(!)\u0011\u001b\n\u0002\u000e(p\u0000!\u0000!0);\n=\u0000e2\u0019Z\u0003\n0k2dk\n2\u00192Z\u0019\n01\n2sin\u0012d\u0012Z2\u0019\n0d\u001e\n2\u0019Zk\n\u0000kd!\u001a\u0010\n\f+(!) +\f\u0000(!)\u0011\n\u00001\nq(^k\u0001q\u00002^k1q1)\n\u0002\u0010\n\f+(!)\u0000\f\u0000(!)\u0011\u001b\n\u000e(p\u0000!\u0000q); (59)\nIm \u000522\n0\f\f\f\f\npole\u0000cut\n= 2e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\u001a\n\u000e(!0\u0000q)\u0002(k2\u0000!2)\n\u0002\u0010\n\f+(!) +\f\u0000(!)\u0011\n\u00001\nq(^k\u0001q\u00002^k2q2)\u000e(!0\u0000q)\u0002(k2\u0000!2)\u0010\n\f+(!)\u0000\f\u0000(!)\u0011\u001b\n\u0002\u000e(p\u0000!\u0000!0)\n=\u0000e2\u0019Z\u0003\n0k2dk\n2\u00192Z\u0019\n01\n2sin\u0012d\u0012Z2\u0019\n0d\u001e\n2\u0019Zk\n\u0000kd!\u001a\u0010\n\f+(!) +\f\u0000(!)\u0011\n\u00001\nq(^k\u0001q\u00002^k2q2)\n\u0002\u0010\n\f+(!)\u0000\f\u0000(!)\u0011\u001b\n\u000e(p\u0000!\u0000q); (60)\nIm \u000533\n0\f\f\f\f\npole\u0000cut\n= 2e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\u001a\n\u000e(!0\u0000q)\u0002(k2\u0000!2)\n\u0002\u0010\n\f+(!) +\f\u0000(!)\u0011\n\u00001\nq(^k?q?\u0000^k3q3)\u000e(!0\u0000q)\u0002(k2\u0000!2)\u0010\n\f+(!)\u0000\f\u0000(!)\u0011\u001b\n\u0002\u000e(p\u0000!\u0000!0)\n=\u0000e2\u0019Z\u0003\n0k2dk\n2\u00192Z\u0019\n01\n2sin\u0012d\u0012Z2\u0019\n0d\u001e\n2\u0019Zk\n\u0000kd!\u001a\u0010\n\f+(!) +\f\u0000(!)\u0011\n\u00001\nq(^k?q?\u0000^k3q3)\n\u0002\u0010\n\f+(!)\u0000\f\u0000(!)\u0011\u001b\n\u000e(p\u0000!\u0000q); (61)15\nIm \u000513\n0\f\f\f\f\npole\u0000cut\n= 2e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\u001a^k1q3+q1^k3\nq\u000e(!0\u0000q)\n\u0002\u0010\n\f+(!)\u0000\f\u0000(!)\u0011\n\u0002(k2\u0000!2)\u001b\n\u000e(p\u0000!\u0000!0)\n=\u0000e2\u0019Z\u0003\n0k2dk\n2\u00192Z\u0019\n01\n2sin\u0012d\u0012Z2\u0019\n0d\u001e\n2\u0019Zk\n\u0000kd!1\nq\u0010\n^k1q3+q1^k3\u0011\u0010\n\f+(!)\u0000\f\u0000(!)\u0011\n\u0002\u000e(p\u0000!\u0000q):\n(62)\nHere we note that the terms with \u000e(!0+q)\u000e(p\u0000!\u0000!0) \u0002(k2\u0000!2) will not contribute because\nk2\u0000(p+q)2can not be greater than zero. So we have excluded those terms.\nB. Imaginary part of magnetic \feld dependent part of O[(eB)2]\nSimilar toO[(eB)0] case, the imaginary part of \u000511\n2, \u000522\n2, \u000533\n2and \u000513\n2can be written as\nIm \u000511\n2=\u00008e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\n\u0002\u0014\neBn\nq3\u001a(0)\n1\u0010\n\u001a(1)\n9+\u001a10\u0000\u001a7\u0011\n\u0000\u001a(1)\n1\u0010\n\u001a8+k3\u001a(0)\n9\u0011o\n+ (eB)2\u0010\n^k1q1\u0000^k2q2\u0011\n\u001a(0)\n2\n\u0002\u0010\n\u001aD\u0000\u0000\u001aD+\u0011\n\u0000(eB)2q2\n?n\n\u001a(1)\n3\u0010\n\u001aD++\u001aD\u0000\u0011\n+ (2^k1q1\u0000^k\u0001q)\u001a(0)\n3\u0010\n\u001aD+\u0000\u001aD\u0000\u0011o\n+\u001a(1)\n0\u0010\n\u001a(1)\n15\u0000\u001a(1)\n14+\u001a16\u0000\u001a13\u0000\u001a11\u0011\n+ (2k1q1\u0000~k\u0001~ q)\u001a(0)\n0\u0010\n\u001a(0)\n15\u0000\u001a(0)\n14\u0011\n\u0000q3\u001a(0)\n0\u001a12\u0015\n\u0002\u000e(p\u0000!\u0000!0);\n(63)\nIm \u000522\n2=\u00008e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\n\u0002\u0014\neBn\nq3\u001a(0)\n1\u0010\n\u001a(1)\n9+\u001a10\u0000\u001a7\u0011\n\u0000\u001a(1)\n1\u0010\n\u001a8+k3\u001a(0)\n9\u0011o\n+ (eB)2\u0010\n^k2q2\u0000^k1q1\u0011\n\u001a(0)\n2\n\u0002\u0010\n\u001aD\u0000\u0000\u001aD+\u0011\n\u0000(eB)2q2\n?n\n\u001a(1)\n3\u0010\n\u001aD++\u001aD\u0000\u0011\n+ (2^k2q2\u0000^k\u0001q)\u001a(0)\n3\u0010\n\u001aD+\u0000\u001aD\u0000\u0011o\n+\u001a(1)\n0\u0010\n\u001a(1)\n15\u0000\u001a(1)\n14+\u001a16\u0000\u001a13\u0000\u001a11\u0011\n+ (2k2q2\u0000~k\u0001~ q)\u001a(0)\n0\u0010\n\u001a(0)\n15\u0000\u001a(0)\n14\u0011\n\u0000q3\u001a(0)\n0\u001a12\u0015\n\u0002\u000e(p\u0000!\u0000!0);\n(64)\nIm \u000533\n2=\u00008e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)16\n\u0002\u0014\neBn\nq3\u001a(0)\n1\u0010\n\u001a(1)\n9+\u001a10\u0000\u001a7\u0011\n+\u001a(1)\n1\u0010\n\u001a8+k3\u001a(0)\n9\u0011o\n+ (eB)2\u0010\nq3^k3\u0000^k\u0001q\u0011\n\u001a(0)\n2\n\u0002\u0010\n\u001aD\u0000\u0000\u001aD+\u0011\n\u0000(eB)2q2\n?n\n\u001a(1)\n3\u0010\n\u001aD++\u001aD\u0000\u0011\n+ (2q3^k3\u0000^k\u0001q)\u001a(0)\n3\u0010\n\u001aD+\u0000\u001aD\u0000\u0011o\n+\u001a(1)\n0\u0010\n\u001a(1)\n15\u0000\u001a(1)\n14+\u001a16\u0000\u001a13\u0000\u001a11\u0011\n+ (2k3q3\u0000~k\u0001~ q)\u001a(0)\n0\u0010\n\u001a(0)\n15\u0000\u001a(0)\n14\u0011\n+q3\u001a(0)\n0\u001a12\u0015\n\u0002\u000e(p\u0000!\u0000!0);\n(65)\nIm \u000513\n2=\u00008e2\u0019(1\u0000ep=T)Zd3k\n(2\u0019)3Z1\n\u00001Z1\n\u00001d!d!0nF(!)nF(!0)\n\u0002\u0014\neBn\n\u001a(1)\n1k3\u001a(0)\n9o\n+ (eB)2^k3q1\u001a(0)\n2n\n\u001aD\u0000\u0000\u001aD+o\n\u0000(eB)2q2\n?n\n(^k1q3+^k3q1)\u001a(0)\n3\n\u0002\u0010\n\u001aD+\u0000\u001aD\u0000\u0011o\n+ (k1q3+k3q1)\u001a(0)\n0\u0010\n\u001a(0)\n15\u0000\u001a(0)\n14\u0011\n+q1\u001a(0)\n0\u001a12\u0015\n\u000e(p\u0000!\u0000!0): (66)\nVarious spectral functions are obtained in Appendix A. As discussed before we also note that the\nimaginary part of various components of the self-energy contain the pole-pole and the pole-cut\ncontributions. As explained earlier the phase space does not allow the pole-pole part to contribute\nin this order. InO[(eB)2] the contribution comes only from the pole-cut part.\n1. Pole-cut part of O[(eB)2]\nNow the expressions of pole-cut parts of Eqs. (63), (64), (65) and (66) after using the approxi-\nmations, are given below:\nIm \u000511\n2\f\f\f\npole\u0000cut\n= 4e2\u0019Z\u0003\n0k2dk\n2\u00192Z1\n2sin\u0012d\u0012Zd\u001e\n2\u0019Zk\n\u0000kd!Z1\n\u00001d!0\u0014\n\u000e000(!0\u0000!q)\u001a\n\u0000(eB)2q2\n?\n96!3q\n\u0002\u0010\n\f++\f\u0000\u0011\n\u0000(eB)2q2\n?(2^k1q1\u0000^k\u0001q)\n96!4q\u0010\n\f+\u0000\f\u0000\u0011\u001b\n\u0000\u000e00(!0\u0000!q)\u001a3(eB)2\n64!3q\n\u0002\u0010q2\n?(^k1q1\u0000^k\u0001q)\n!2q\u0000(^k1q1\u0000^k2q2)\u0011\n\u0002\u0010\n\f+\u0000\f\u0000\u0011\n+3(eB)2q2\n?\n128!4q\u0010\n\f++\f\u0000\u0011\u001b\n+\u000e0(!0\u0000!q)\u001aeBq 3\n4!2q\u0010\n\f(1)\n9+\f10\u0000\f7\u0011\n\u0000eB\n4!q\u0010\n\f8+k3\f(0)\n9\u0011\n\u0000(eB)2\n16!4q\u0010\n^k2q2\u0000^k1q1\n+5q2\n?(^k1q1\u0000^k\u0001q)\n2!2q\u0011\u0010\n\f+\u0000\f\u0000\u0011\n\u0000(eB)2q2\n?\n32!5q\u0010\n\f++\f\u0000\u0011\u001b\n+\u000e(!0\u0000!q)\u001aeBq 3\n4!3q\n\u0002\u0010\n\f(1)\n9+\f10\u0000\f7\u0011\n\u0000(eB)2\n16!5q\u0010\n3^k2q2\u00003^k1q1+5q2\n?(^k1q1\u0000^k\u0001q)\n2!2q\u0011\u0010\n\f+\u0000\f\u0000\u0011\n\u00001\n2\u0012\n\f(1)\n15+\f16\u0000\u0010\n\f(1)\n14+\f11+\f13\u0011\u0013\n\u0000(2k1q1\u0000~k\u0001~ q)\n2!q\u0012\n\f(0)\n15\u0000\f(0)\n14\u0013\n+q3\n2!q\f12\u001b\u001517\n\u0002\u000e(p\u0000!\u0000!0); (67)\nIm \u000522\n2\f\f\f\npole\u0000cut\n= 4e2\u0019Z\u0003\n0k2dk\n2\u00192Z1\n2sin\u0012d\u0012Zd\u001e\n2\u0019Zk\n\u0000kd!Z1\n\u00001d!0\u0014\n\u000e000(!0\u0000!q)\u001a\n\u0000(eB)2q2\n?\n96!3q\n\u0002\u0010\n\f++\f\u0000\u0011\n\u0000(eB)2q2\n?(2^k2q2\u0000^k\u0001q)\n96!4q\u0010\n\f+\u0000\f\u0000\u0011\u001b\n\u0000\u000e00(!0\u0000!q)\u001a3(eB)2\n64!3q\n\u0002\u0010q2\n?(^k2q2\u0000^k\u0001q)\n!2q\u0000(^k2q2\u0000^k1q1)\u0011\n\u0002\u0010\n\f+\u0000\f\u0000\u0011\n+3(eB)2q2\n?\n128!4q\u0010\n\f++\f\u0000\u0011\u001b\n+\u000e0(!0\u0000!q)\u001aeBq 3\n4!2q\u0010\n\f(1)\n9+\f10\u0000\f7\u0011\n\u0000eB\n4!q\u0010\n\f8+k3\f(0)\n9\u0011\n\u0000(eB)2\n16!4q\u0010\n^k1q1\u0000^k2q2\n+5q2\n?(^k2q2\u0000^k\u0001q)\n2!2q\u0011\u0010\n\f+\u0000\f\u0000\u0011\n\u0000(eB)2q2\n?\n32!5q\u0010\n\f++\f\u0000\u0011\u001b\n+\u000e(!0\u0000!q)\u001aeBq 3\n4!3q\n\u0002\u0010\n\f(1)\n9+\f10\u0000\f7\u0011\n\u0000(eB)2\n16!5q\u0010\n3^k1q1\u00003^k2q2+5q2\n?(^k2q2\u0000^k\u0001q)\n2!2q\u0011\u0010\n\f+\u0000\f\u0000\u0011\n\u00001\n2\u0012\n\f(1)\n15+\f16\u0000\u0010\n\f(1)\n14+\f11+\f13\u0011\u0013\n\u0000(2k2q2\u0000~k\u0001~ q)\n2!q\u0012\n\f(0)\n15\u0000\f(0)\n14\u0013\n+q3\n2!q\f12\u001b\u0015\n\u0002\u000e(p\u0000!\u0000!0); (68)\nIm \u000533\n2\f\f\f\npole\u0000cut\n= 4e2\u0019Z\u0003\n0k2dk\n2\u00192Z1\n2sin\u0012d\u0012Zd\u001e\n2\u0019Zk\n\u0000kd!Z1\n\u00001d!0\u0014\n\u000e000(!0\u0000!q)\u001a\n\u0000(eB)2q2\n?\n96!3q\n\u0002\u0010\n\f++\f\u0000\u0011\n\u0000(eB)2q2\n?(^k3q3\u0000^k?q?)\n96!4q\u0010\n\f+\u0000\f\u0000\u0011\u001b\n\u0000\u000e00(!0\u0000!q)\u001a3(eB)2\n64!3q\n\u0002\u0010q2\n?(^k3q3\u0000^k?q?)\n!2q+^k?q?\u0011\n\u0002\u0010\n\f+\u0000\f\u0000\u0011\n+3(eB)2q2\n?\n128!4q\u0010\n\f++\f\u0000\u0011\u001b\n+\u000e0(!0\u0000!q)\u001aeBq 3\n4!2q\u0010\n(\f(1)\n9+\f10\u0000\f7\u0011\n+eB\n4!q\u0010\n\f8+k3\u001a(0)\n9\u0011\n\u0000(eB)2\n16!4q\u0010\n^k?q?\n+5q2\n?(^k3q3\u0000^k?q?)\n2!2q\u0011\u0010\n\f+\u0000\f\u0000\u0011\n\u0000(eB)2q2\n?\n32!5q\u0010\n\f++\f\u0000\u0011\u001b\n+\u000e(!0\u0000!q)\n\u0002\u001aeBq 3\n4!3q\u0010\n\f(1)\n9+\f10\u0000\f7\u0011\n\u0000(eB)2\n16!5q\u0010\n3^k?q?+5q2\n?(^k3q3\u0000^k?q?)\n2!2q\u0011\n\u0002\u0010\n\f+\u0000\f\u0000\u0011\n\u00001\n2\u0012\n\f(1)\n15+\f16\u0000\u0010\n\f(1)\n14+\f13+\f11\u0011\u0013\n\u0000(2k3q3\u0000~k\u0001~ q)\n2!q\n\u0002\u0012\n\f(0)\n15\u0000\f(0)\n14\u0013\n\u0000q3\n2!q\f12\u001b\u0015\n\u000e(p\u0000!\u0000!0); (69)\nIm \u000513\n2\f\f\f\npole\u0000cut18\n= 4e2\u0019Z\u0003\n0k2dk\n2\u00192Z1\n2sin\u0012d\u0012Zd\u001e\n2\u0019Zk\n\u0000kd!Z1\n\u00001d!0\u0014\n\u000e000(!0\u0000!q)\n\u0002\u001a\n\u0000(eB)2q2\n?(^k1q3+^k3q1)\n96!4q\u0010\n\f+\u0000\f\u0000\u0011\u001b\n\u0000\u000e00(!0\u0000!q)\u001a3(eB)2\n64!3q\u0010q2\n?(^k1q3+^k3q1)\n!2q\n\u0000^k3q1\u0011\u0010\n\f+\u0000\f\u0000\u0011\u001b\n+\u000e0(!0\u0000!q)\u001aeB\n4!qk1\f(0)\n9\u0000(eB)2\n16!4q\u0010\n\u0000^k3q1+5q2\n?(^k1q3+^k3q1)\n2!2q\u0011\n\u0002\u0010\n\f+\u0000\f\u0000\u0011\u001b\n+\u000e(!0\u0000!q)\u001a\n\u0000(eB)2\n16!5q\u0010\n\u00003^k3q1+5q2\n?(^k1q3+^k3q1)\n2!2q\u0011\u0010\n\f+\u0000\f\u0000\u0011\n\u0000(k1q3+k3q1)\n2!q\u0012\n\f(0)\n15\u0000\f(0)\n14\u0013\n\u0000q1\n2!q\f12\u001b\u0015\n\u000e(p\u0000!\u0000!0): (70)\nV. RESULTS\nWe perform the integrations in Eq. (59),(60),(61),(62),(67),(68),(69) and (70) numerically. In\nthis calculation we have taken m\u0019= 0:14 GeV. The results are shown for \u0003 = 0 :25 GeV which\nsatis\feseT\u001c\u0003\u001cT.\np=3GeV, T=0.5GeV, eB=m π2/4γδ\nγσ\nπ/8 π/4 3π/8 π/28.2×10-68.3×10-68.4×10-68.5×10-68.6×10-68.7×10-68.8×10-68.9×10-6\nθpγ[GeV]\nFIG. 3: Plot of damping rate of photon with the propagation angle \u0012pforp= 3 GeV,T= 0:5\nGeV andeB=m2\n\u0019=4.\nThe damping rate of photon in presence of magnetic \feld depends on the angle, \u0012p, between the\nmomentum of photon and the magnetic \feld. Figure 3 shows the variation of the damping rate of\na hard photon with the propagation angle. It increases with the increasing propagation angle. One\ncan see that the two transverse modes of a hard photon are damped in a similar fashion. Since the\nmagnetic \feld strength is very weak, this di\u000berence appears to be very small. We note that the19\nmagnetic correction is \u0018O[(eB)2] and switching the magnetic \feld from zto\u0000zdirection would\nnot a\u000bect the result. These two orientations of the magnetic \feld correspond to the propagation\nangle of photon \u0012pand\u0019\u0000\u0012p. These two situations are identical which correspond to the same\ndamping rates of photon at \u0012pand\u0019\u0000\u0012p.\n�=�� ������=� π�/�θ�=π/��θ�=π/�γ��\nγδ\nγσ\n���������������������×��-����×��-����×��-����×��-����×��-�����×��-�����×��-�\n�[���]γ[���]\nFIG. 4: Plot of damping rate of photon with the energy for T= 0:5 GeV and eB=m2\n\u0019=4 at\npropagation angles \u0012p=\u0019=10 and\u0019=2.\nIn Fig. 4 we display the damping rate as a function of photon momentum for two propagation\nangles\u0019=10 and\u0019=2. The soft contribution of the damping rate in a thermal medium agrees well\nwith that obtained in Ref. [45]. In presence of a thermomagnetic medium, the soft contribution\nto the damping rate is found to be reduced than that of the thermal one. For small propagation\nangle, the reduction of the damping rate is more compared to that of thermal medium. For\nhigher momentum the damping rate approaches the thermal value as the temperature becomes the\ndominant scale as compared to the strength of the magnetic \feld considered.\nFigure 5 displays the variation of damping rate with temperature for a speci\fc value of mo-\nmentum and magnetic \feld for two propagation angles \u0019=10 and\u0019=2. It is found that the soft\ncontribution to the damping rate increases with the increase in temperature both in thermal and\nthermomagnetic medium. For small propagation angle the damping rate is more reduced compared\nto that of large propagation angle. This observation is consistent with Fig. 4.\nFigure 6 shows the variation of the damping rate with the magnetic \feld strength for speci\fc\nvalues of photon momentum and temperature for two propagation angles. The thermal damping\nrate (O[(eB)0]) is represented by the black dashed horizontal line. The thermomagnetic damping20\n�=�������=� π�/�θ�=π/��θ�=π/�\nγ��\nγδ\nγσ\n���� ���� ��� ���� ������×��-����×��-����×��-����×��-����×��-�����×��-�\n�[���]γ[���]\nFIG. 5: Plot of damping rate of the hard photon with temperature at p= 3 GeV and eB=m2\n\u0019=4\nfor two propagation angles \u0019=10 and\u0019=2.\n�=������=������θ�=π/��θ�=π/�\nγ��\nγδ\nγσ\n����π�����π�����π�����π����×��-����×��-����×��-����×��-����×��-�\n��[����]γ[���]\nFIG. 6: Plot of damping rate of the hard photon with the magnetic \feld strength at T= 0:5 GeV\nandp= 3 GeV for two propagation angles \u0019=10 and\u0019=2.\nrate decreases with the increasing magnetic \feld. At smaller propagation angles the photons are\nless damped than that of higher propagation angles which are consistent with Fig. 4.\nFig. 7 shows the variation of the photon damping rate with the separation scale \u0003 keeping\nthe scale hierarchy eT\u001c\u0003\u001cT. As the allowed phase space increases with the increase of \u0003,\nthe damping rate is also found to increase with it3. The magnetic correction to the thermal\n3Nevertheless, the damping rate is expected to be \u0003 independent when hard contribution is added.21\np=3GeV, T=0.5GeV, eB=m π2/4θp=π/4 γth\nγδ\nγσ\n0.18 0.21 0.24 0.27 0.304×10-66×10-68×10-610×10-612×10-6\nΛ[GeV]γ[GeV]\nFIG. 7: Plot of damping rate of photon with \u0003 for \u0012p=\u0019=4,p= 3 GeV,T= 0:5 GeV and\neB=m2\n\u0019=4.\ndamping rate is negative. So, the di\u000berence between the thermal and thermomagnetic damping\nrate increases with \u0003.\nVI. CONCLUSION\nWe have calculated the soft contribution to the damping rate of a hard photon in a weakly\nmagnetized QED medium where momentum of one of the fermion in the loop is considered as soft.\nThe two degenerate transverse modes of photon in thermal medium are damped in a similar fashion\nin presence of weak magnetic \feld as shown in Fig. 3. The di\u000berence between two transverse modes\nis very marginal due to weak \feld approximation. The soft contribution to the damping rate in\nthermomagnetic medium is reduced compared to that of thermal medium. When the magnetic\n\feld is switched o\u000b thermomagnetic damping modes reduce to its thermal value. The e\u000bect of\nmagnetic \feld is found to be dominant at low temperature and low photon momentum.\nThe soft contribution to the hard photon damping rate is \u001810\u00006GeV. Thus, a photon of a few\nGeV energy traversing in the QED medium of temperature \u00180:5 GeV and background magnetic\n\feld\u00180:005 GeV2has a mean free path ( \u0015=\r\u00001=2) of a few \u0017A. When the present calculation is\nextended to the case of relativistic heavy ion collisions, the mean free path of photon is found to\nbe of a few hundred fm. This con\frms that the mean free path of photon is larger than the size of\nthe \freball and photon can be treated as a direct probe.22\nThe damping rate is found to be dependent on the separation scale \u0003. One needs to add the\nhard contribution with the soft contribution to cancel the \u0003 dependence of the result. The hard\ncontribution to the photon damping rate comes from two-loop order with hard particles in the loop\nhaving momentum of the order of or higher than the temperature. This itself is a huge calculation\nwhich is in progress.\nAcknowledgement: RG is funded by University Grants Commission (UGC). BK and MGM were\nfunded by Department of Atomic Energy (DAE), India via the project TPAES. RG and BK would\nlike to thank Aritra Das for useful discussions.\nAppendix A: Spectral representation of the propagators\nlim\n\u000f!0Z1\n\u00001dx\u000f\nx2+\u000f2f(x)\u0019f(0)Z1\n\u00001dx\u000f\nx2+\u000f28\n><\n>:signi\fcant contribution comes from\nintegration ;wherex'0\n=f(0)\u000fZ1\n\u00001dx1\nx2+\u000f2=\u0019f(0); (A1)\nwheref(x) is a test function.\nFrom the above equations we can write,\nlim\n\u000f!0\u000f\nx2+\u000f2=\u0019\u000e(x); (A2)\nlim\n\u000f!0Im1\nx+i\u000f=1\n2ilim\n\u000f!0\u00141\nx+i\u000f\u00001\nx\u0000i\u000f\u0015\n=1\n2ilim\n\u000f!0\u00002i\u000f\nx2+\u000f2=\u0000\u0019\u000e(x): (A3)\nlim\n\u000f!0Z1\n\u00001dx2\u000fx\n(x2+\u000f2)2f(x) = lim\n\u000f!0Z1\n\u00001dx\u000ff (x)d\ndx\u0014\n\u00001\n(x2+\u000f2)\u0015\n= lim\n\u000f!0Z1\n\u00001dxf0(x)\u000f\nx2+\u000f2\n=\u0019f0(0) =\u0019Z\ndxf0(x)\u000e(x)\n=\u0000\u0019Z\ndxf(x)\u000e0(x) (A4)\nFrom the above equation we \fnd,\nlim\n\u000f!02\u000fx\n(x2+\u000f2)2=\u0000\u0019\u000e0(x): (A5)23\nNow using Eq. (A5) one can calculate,\nlim\n\u000f!0Im1\n(x+i\u000f)2=1\n2ilim\n\u000f!0\u00141\n(x+i\u000f)2\u00001\n(x\u0000i\u000f)2\u0015\n=1\n2ilim\n\u000f!0\u0000i4\u000fx\n(x2+\u000f2)2=\u0019\u000e0(x): (A6)\nSimilarly,\nlim\n\u000f!0Z1\n\u00001dx\u000f3\u00003x2\u000f\n(x2+\u000f2)3f(x) = lim\n\u000f!0Z1\n\u00001dx\u000f3\n(x2+\u000f2)3f(x)\u0000lim\n\u000f!0Z1\n\u00001dx3x2\u000f\n(x2+\u000f2)3f(x)\n=I1+I2; (A7)\nwhere\nI1= lim\n\u000f!0Z1\n\u00001dx\u000f3f(x)\n(x2+\u000f2)3\u0019lim\n\u000f!0\u000f3f(0)Z1\n\u00001dx1\n(x2+\u000f2)3= lim\n\u000f!0\u000f3f(0)3\n81\n\u000f5= lim\n\u000f!03\n8\u000f2f(0);\nI2=\u00003 lim\n\u000f!0Z1\n\u00001dxx2\u000ff(x)\n(x2+\u000f2)3=\u00003 lim\n\u000f!0\u000fZ\ndxf(x)\u00141\n8d2\ndx2\u00121\nx2+\u000f2\u0013\n+1\n82\n(x2+\u000f2)2\u0015\n=\u0000lim\n\u000f!03\n8\u000f2f(0)\u00003\n8lim\n\u000f!0\u000fZ\ndxf(x)d2\ndx2\u00121\nx2+\u000f2\u0013\n: (A8)\nSo ,\nI1+I2=\u00003\n8lim\n\u000f!0\u000fZ\ndxf(x)d2\ndx2\u00121\nx2+\u000f2\u0013\n=3\n8lim\n\u000f!0Z\ndx\u000ff0(x)d\ndx\u00121\nx2+\u000f2\u0013\n=\u00003\n8lim\n\u000f!0Z\ndx\u000ff00(x)1\nx2+\u000f2=\u00003\n8\u0019Z\ndxf00(x)\u000e(x) =\u00003\n8\u0019Z1\n\u00001dxf(x)\u000e00(x):(A9)\nWe can conclude from the last few steps that,\nlim\n\u000f!0\u000f3\u00003x2\u000f\n(x2+\u000f2)3=\u00003\n8\u0019\u000e00(x): (A10)\nUsing Eq. (A10) we can \fnd\nlim\n\u000f!0Im1\n(x+i\u000f)3=1\n2ilim\n\u000f!0\u00141\n(x+i\u000f)2\u00001\n(x\u0000i\u000f)3\u0015\n= lim\n\u000f!0\u000f3\u00006\u000fx2\n(x2+\u000f2)3=\u00003\n8\u0019\u000e00(x) (A11)\nNow ,\nlim\n\u000f!0Im1\n(x+i\u000f)4=1\n2ilim\n\u000f!0\u00141\n(x+i\u000f)4\u00001\n(x\u0000i\u000f)4\u0015\n= lim\n\u000f!04x\u000f3\u00004x3\u000f\n(x2+\u000f2)4; (A12)\nlim\n\u000f!0Z1\n\u00001dxf(x)4x\u000f3\u00004x3\u000f\n(x2+\u000f2)4= lim\n\u000f!04\u000fZ1\n\u00001dxf(x)1\n24d3\ndx3\u00121\nx2+\u000f2\u0013\n=\u0000lim\n\u000f!0\u000f\n6Z1\n\u00001dxf000(x)1\nx2+\u000f2=\u0000\u0019f000(0)\n6=\u0019\n6Z1\n\u00001dxf(x)\u000e000(x): (A13)\nThus, lim\n\u000f!0Im1\n(x+i\u000f)4=\u0019\n6\u000e000(x): (A14)24\nNow we write the spectral representations for the free propagators as\n\u001a(1)\n0(!0;q) =1\n2\u0019lim\n\u000f!0Im\u00141\n!0+i\u000f+!q+1\n!0+i\u000f\u0000!q\u0015\n=\u00001\n2\u0014\n\u000e(!0+!q) +\u000e(!0\u0000!q)\u0015\n;(A15)\n\u001a(0)\n0(!0;q) =1\n2\u0019!qlim\n\u000f!0Im\u00141\n!0+i\u000f\u0000!q\u00001\n!0+i\u000f+!q\u0015\n=1\n2!q\u0014\n\u000e(!0+!q)\u0000\u000e(!0\u0000!q)\u0015\n;(A16)\n\u001a(1)\n1(!0;q) =1\n4\u0019!qlim\n\u000f!0Im\u00141\n(!0+i\u000f\u0000!q)2\u00001\n(!0+i\u000f+!q)2\u0015\n=\u00001\n4!q\u0014\n\u000e0(!0+!q)\u0000\u000e0(!0\u0000!q)\u0015\n; (A17)\n\u001a(0)\n1(!0;q) =1\n4\u0019!2qlim\n\u000f!0Im\u00141\n(!0+i\u000f+!q)2+1\n(!0+i\u000f\u0000!q)2\u00001\n!q\u00121\n!0+i\u000f\u0000!q\n\u00001\n!0+i\u000f+!q\u0013\u0015\n=1\n4!2q\u0014\n\u000e0(!0+!q) +\u000e0(!0\u0000!q)\u00001\n!q\u0012\n\u000e(!0+!q)\u0000\u000e(!0\u0000!q)\u0013\u0015\n; (A18)\n\u001a(1)\n2(!0;q) =1\n8\u0019!2qlim\n\u000f!0Im\u00141\n(!0+i\u000f+!q)3+1\n(!0+i\u000f\u0000!q)3+1\n2!q\u001a1\n(!0+i\u000f+!q)2\n\u00001\n(!0+i\u000f\u0000!q)2\u001b\u0015\n=1\n8!2q\u0014\n\u00003\n8\u0012\n\u000e00(!0+!q) +\u000e00(!0\u0000!q)\u0013\n+1\n2!q\u0012\n\u000e0(!0+!q)\u0000\u000e0(!0\u0000!q)\u0013\u0015\n;(A19)\n\u001a(0)\n2(!0;q) =\u00001\n8\u0019!3qlim\n\u000f!0Im\u00141\n(!0+i\u000f+!q)3\u00001\n(!0+i\u000f\u0000!q)3+1\n2!q\u001a1\n(!0+i\u000f+!q)2\n+1\n(!0+i\u000f\u0000!q)2+3\n!q\u00121\n!0+i\u000f+!q\u00001\n!0+i\u000f\u0000!q\u0013\u001b\u0015\n=\u00001\n8!3q\u0014\n\u00003\n8\u0012\n\u000e00(!0+!q)\u0000\u000e00(!0\u0000!q)\u0013\n+1\n2!q\u001a\n\u000e0(!0+!q) +\u000e0(!0\u0000!q)\n\u00003\n!q\u0012\n\u000e(!0+!q)\u0000\u000e(!0\u0000!q)\u0013\u001b\u0015\n; (A20)\n\u001a(1)\n3(!0;q) =\u00001\n16\u0019!3qlim\n\u000f!0Im\u00141\n(!0+i\u000f+!q)4\u00001\n(!0+i\u000f\u0000!q)4+1\n!q\u001a1\n(!0+i\u000f+!q)3\n+1\n(!0+i\u000f\u0000!q)3+1\n2!q\u00121\n(!0+i\u000f+!q)2\u00001\n(!0+i\u000f\u0000!q)2\u0013\u001b\u0015\n(A21)\n=\u00001\n16!3q\u00141\n6\u0012\n\u000e000(!0+!q)\u0000\u000e000(!0\u0000!q)\u0013\n+1\n!q\u001a\n\u00003\n8\u0012\n\u000e00(!0+!q)\n+\u000e00(!0\u0000!q)\u0013\n+1\n2!q\u0012\n\u000e0(!0+!q)\u0000\u000e0(!0\u0000!q)\u0013\u001b\u0015\n; (A22)\n\u001a(0)\n3(!0;q) =1\n16\u0019!4qlim\n\u000f!0Im\u00141\n(!0+i\u000f+!q)4+1\n(!0+i\u000f\u0000!q)4+1\n2!q\u001a4\n(!0+i\u000f+!q)3\n\u00004\n(!0+i\u000f\u0000!q)3+1\n2!q\u001210\n(!0+i\u000f+!q)2+10\n(!0+i\u000f\u0000!q)2+1\n2!q\n\u0002\u001220\n!0+i\u000f+!q\u000020\n!0+i\u000f\u0000!q\u0013\u0013\u001b\u0015\n(A23)25\n=1\n16!4q\"\n1\n6\u0012\n\u000e000(!0+!q) +\u000e000(!0\u0000!q)\u0013\n+1\n2!q\u001a\n\u00003\n2\u0012\n\u000e00(!0+!q)\n\u0000\u000e00(!0\u0000!q)\u0013\n+5\n!q\u0012\n\u000e0(!0+!q) +\u000e0(!0\u0000!q)\u00001\n!q\u0010\n\u000e(!0+!q)\n\u0000\u000e(!0\u0000!q)\u0011\u0013\u001b#\n: (A24)\nThe e\u000bective propagators are given as,\n1\nD2=1\nD+D\u0000=X\ni\u0012@D2\n@!\u0013\u00001\f\f\f\f\n!=!i1\n(!\u0000!i); (A25)\n1\nD4=1\n(D+D\u0000)2=X\ni\"\u0012@D2\n@!\u0013\u00002\f\f\f\f\n!=!i1\n(!\u0000!i)2+@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i1\n(!\u0000!i)#\n=X\ni\"\u0012@D2\n@!\u0013\u00002\f\f\f\f\n!=!i1\n(!\u0000!i)2\u0000@2D2\n@!2\u00121\n3!@3D6\n@!3\u0013\u00001\f\f\f\f\f\n!=!i1\n(!\u0000!i)#\n;(A26)\n1\nD6=1\n(D+D\u0000)3=X\ni\"\u0012@D2\n@!\u0013\u00003\f\f\f\f\n!=!i1\n(!\u0000!i)3+@\n@!\u001a(!\u0000!i)3\nD6\u001b\f\f\f\f\n!=!i1\n(!\u0000!i)2\n+1\n2@2\n@!2\u001a(!\u0000!i)3\nD6\u001b\f\f\f\f\n!=!i1\n(!\u0000!i)#\n=X\ni\"\u0012@D2\n@!\u0013\u00003\f\f\f\f\n!=!i1\n(!\u0000!i)3\u00003\n2@2D2\n@!2\u00121\n4!@4D8\n@!4\u0013\u00001\f\f\f\f\f\n!=!i1\n(!\u0000!i)2\n\u00003\n5 \n@3D2\n@!3\u001a\n6\u0012@4D8\n@!4\u0013\u00001\n+7\n12\u0012@D2\n@!\u0013\u00004\u001b\n+ 6@2D2\n@!2@\n@!\u0012@4D8\n@!4\u0013\u00001!\f\f\f\f\f\n!=!i\n\u00021\n(!\u0000!i)#\n; (A27)\nwhere!i=\u0006!\u0006are the poles of D+andD\u0000.\nThe spectral functions of the dressed propagators are given as\n\u001aD\u0006=\u0000(!2\u0000k2)\n2m2\nth\u0014\n\u000e(!\u0000!\u0006) +\u000e(!+!\u0007)\u0015\n+\f\u0006\u0002(k2\u0000!2); (A28)\nwhere\n\f\u0006=\u00001\n2(k\u0007!)m2\nth\u0014\nk(!\u0007k)\u0000m2\nth\u001a\nQ0(!\nk)\u0007Q1(!\nk)\u001b\u00152\n+\u0014\n1\n2(1\u0007!\nk)m2\nth\u0019\u00152; (A29)\nwhere we use the Legendre function of second kind\nQ0\u0012!\nk\u0013\n=1\n2ln\f\f\f\f!+k\n!\u0000k\f\f\f\f\u0000i\u0019\n2\u0002(k2\u0000!2): (A30)\n\u001a4(!;k) =1\n\u0019Im\u00101\nD2\u0011\n=\u0000X\ni\u0012@D2\n@!\u0013\u00001\f\f\f\f\n!=!i\u000e(!\u0000!i) +\f4\u0002(k2\u0000!2)26\n=!2\u0000k2\n4m2\nth(k2\u0000!2+m2\nth)\u0014\n(!\u0000k)\u0012\n\u000e(!\u0000!+) +\u000e(!+!\u0000)\u0013\n+ (!+k)\u0012\n\u000e(!\u0000!\u0000)\n+\u000e(!+!+)\u0013\u0015\n+\f4\u0002(k2\u0000!2); (A31)\n\u001a5(!;k) =1\n\u0019Im\u00101\nD4\u0011\n=X\ni\"\u0012@D2\n@!\u0013\u00002\f\f\f\f\n!=!i\u000e0(!\u0000!i) +@2D2\n@!2\u00121\n3!@3D6\n@!3\u0013\u00001\f\f\f\f\f\n!=!i\n\u0002\u000e(!\u0000!i)#\n+\f5\u0002(k2\u0000!2); (A32)\n\u001a6(!;k) =1\n\u0019Im\u00101\nD6\u0011\n=X\ni\"\n\u00003\n8\u0012@D2\n@!\u0013\u00003\f\f\f\f\n!=!i\u000e00(!\u0000!i)\u00003\n2@2D2\n@!2\u00121\n4!@4D8\n@!4\u0013\u00001\f\f\f\f\f\n!=!i\n\u0002\u000e0(!\u0000!i) +3\n5 \n@3D2\n@!3\u001a\n6\u0012@4D8\n@!4\u0013\u00001\n+7\n12\u0012@D2\n@!\u0013\u00004\u001b\n+ 6@2D2\n@!2@\n@!\u0012@4D8\n@!4\u0013\u00001!\f\f\f\f\f\n!=!i\n\u0002\u000e(!\u0000!i)#\n+\f6\u0002(k2\u0000!2); (A33)\n\u001a7(!;k) =1\n\u0019Im\u0012b0\nD2\u0013\n=\u0000b0X\ni\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00001\n\u000e(!\u0000!i) +\f7\u0002(k2\u0000!2); (A34)\n\u001a8(!;k) =1\n\u0019Im\u0012c0\nD2\u0013\n=\u0000c0X\ni\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00001\n\u000e(!\u0000!i) +\f8\u0002(k2\u0000!2); (A35)\n\u001a(0)\n9(!;k) =1\n\u0019Im\u0012h(1 +a)\nD4\u0013\n=h(1 +a)\u0002X\ni\"\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00002\n\u000e0(!\u0000!i)\n\u0000@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i\u000e(!\u0000!i)#\n+\f(0)\n9\u0002(k2\u0000!2); (A36)\n\u001a(1)\n9(!;k) =1\n\u0019Im\u0012h(1 +a)!\nD4\u0013\n=!h(1 +a)\u0002X\ni\"\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00002\n\u000e0(!\u0000!i)\n\u0000@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i\u000e(!\u0000!i)#\n+\f(1)\n9\u0002(k2\u0000!2); (A37)\n\u001a10(!;k) =1\n\u0019Im\u0012hb\nD4\u0013\n=hb\u0002X\ni\"\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00002\n\u000e0(!\u0000!i)\u0000@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i\n\u0002\u000e(!\u0000!i)#\n+\f10\u0002(k2\u0000!2); (A38)\n\u001a11(!;k) =1\n\u0019Im\u0014hb0\nD4\u0015\n=hb0\u0002X\ni\"\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00002\n\u000e0(!\u0000!i)\u0000@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i\n\u0002\u000e(!\u0000!i)#\n+\f11\u0002(k2\u0000!2); (A39)\n\u001a12(!;k) =1\n\u0019Im\u0014hc0\nD4\u0015\n=hc0\u0002X\ni\"\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00002\n\u000e0(!\u0000!i)\u0000@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i27\n\u0002\u000e(!\u0000!i)#\n+\f12\u0002(k2\u0000!2); (A40)\n\u001a13(!;k) =1\n\u0019Im\u0014h0b\nD4\u0015\n=h0b\u0002X\ni\"\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00002\n\u000e0(!\u0000!i)\u0000@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i\n\u0002\u000e(!\u0000!i)#\n+\f13\u0002(k2\u0000!2); (A41)\n\u001a(0)\n14(!;k) =1\n\u0019Im\u0014h0(1 +a)\nD4\u0015\n=h0(1 +a)X\ni\"\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00002\n\u000e0(!\u0000!i)\n\u0000@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i\u000e(!\u0000!i)#\n+\f(0)\n14\u0002(k2\u0000!2); (A42)\n\u001a(1)\n14(!;k) =1\n\u0019Im\u0014k0h0(1 +a)\nD4\u0015\n=!h0(1 +a)X\ni\"\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00002\n\u000e0(!\u0000!i)\n\u0000@\n@!\u001a(!\u0000!i)2\nD4\u001b\f\f\f\f\n!=!i\u000e(!\u0000!i)#\n+\f(1)\n14\u0002(k2\u0000!2); (A43)\n\u001a(0)\n15(!;k) =1\n\u0019Im\u0014h2(1 +a)\nD6\u0015\n=h2(1 +a)\u0002X\ni\"\n\u00003\n8\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00003\n\u000e00(!\u0000!i)\n+@\n@!\u001a(!\u0000!i)3\nD6\u001b\f\f\f\f\n!=!i\u000e0(!\u0000!i)\u0000@2\n@!2\u001a(!\u0000!i)3\nD6\u001b\f\f\f\f\n!=!i\u000e(!\u0000!i)#\n+\f(0)\n15\u0002(k2\u0000!2); (A44)\n\u001a(1)\n15(!;k) =1\n\u0019Im\u0014h2(1 +a)k0\nD6\u0015\n=!h2(1 +a)\u0002X\ni\"\n\u00003\n8\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00003\n\u000e00(!\u0000!i)\n+@\n@!\u001a(!\u0000!i)3\nD6\u001b\f\f\f\f\n!=!i\u000e0(!\u0000!i)\u0000@2\n@!2\u001a(!\u0000!i)3\nD6\u001b\f\f\f\f\n!=!i\u000e(!\u0000!i)#\n+\f(1)\n15\u0002(k2\u0000!2); (A45)\n\u001a16(!;k) =1\n\u0019Im\u0014h2b\nD6\u0015\n=h2b\u0002X\ni\"\n\u00003\n8\u0012@D2\n@!\f\f\f\f\n!=!i\u0013\u00003\n\u000e00(!\u0000!i) +@\n@!\u001a(!\u0000!i)3\nD6\u001b\f\f\f\f\n!=!i\n\u0002\u000e0(!\u0000!i)\u0000@2\n@!2\u001a(!\u0000!i)3\nD6\u001b\f\f\f\f\n!=!i\u000e(!\u0000!i)#\n+\f16\u0002(k2\u0000!2); (A46)\nwhere cut parts of the spectral functions are given as\n\f4=1\n\u0019Im\u00121\nD2\u0013\n=\u00001\n\u0019ImD2\n(ReD2)2+ (ImD2)2; (A47)\n\f5=1\n\u0019Im\u00121\nD4\u0013\n=\u00001\n\u00192 ReD2ImD2\nh\n(ReD2)2+ (ImD2)2i2; (A48)\n\f6=1\n\u0019Im\u00121\nD6\u0013\n=1\n\u0019\u0010\nImD2\u00113\n\u00003\u0010\nReD2\u00112\nImD2\nh\n(ReD2)2+ (ImD2)2i3; (A49)28\n\f7=1\n\u0019Im\u0012b0\nD2\u0013\n=1\n\u0019\u0000Reb0ImD2+ Imb0ReD2\n(ReD2)2+ (ImD2)2; (A50)\n\f8=1\n\u0019Im\u0012c0\nD2\u0013\n=1\n\u0019\u0000Rec0ImD2+ Imc0ReD2\n(ReD2)2+ (ImD2)2; (A51)\n\f(0)\n9=1\n2Im\u0012h(1 +a)\nD4\u0013\n=1\n\u0019Im\u0000\nh(1 +a)\u0001\u0014\n(ReD2)2\u0000(ImD2)2\u0015\n\u00002 ReD2ImD2Re\u0000\nh(1 +a)\u0001\nh\n(ReD2)2+ (ImD2)2i2; (A52)\n\f(1)\n9=1\n2Im\u0012h(1 +a)k0\nD4\u0013\n=1\n\u0019Im\u0000\nhk0(1 +a)\u0001\u0014\n(ReD2)2\u0000(ImD2)2\u0015\n\u00002 ReD2ImD2Re\u0000\nhk0(1 +a)\u0001\nh\n(ReD2)2+ (ImD2)2i2; (A53)\n\f10=1\n2Im\u0012hb\nD4\u0013\n=1\n\u0019Im\u0000\nhb\u0001\u0014\n(ReD2)2\u0000(ImD2)2\u0015\n\u00002 ReD2ImD2Re\u0000\nhb\u0001\nh\n(ReD2)2+ (ImD2)2i2; (A54)\n\f11=1\n2Im\u0012hb0\nD4\u0013\n=1\n\u0019Im\u0000\nhb0\u0001\u0014\n(ReD2)2\u0000(ImD2)2\u0015\n\u00002 ReD2ImD2Re\u0000\nhb0\u0001\nh\n(ReD2)2+ (ImD2)2i2; (A55)\n\f12=1\n2Im\u0012hc0\nD4\u0013\n=1\n\u0019Im\u0000\nhc0\u0001\u0014\n(ReD2)2\u0000(ImD2)2\u0015\n\u00002 ReD2ImD2Re\u0000\nhc0\u0001\nh\n(ReD2)2+ (ImD2)2i2; (A56)\n\f13=1\n2Im\u0012h0b\nD4\u0013\n=1\n\u0019Im\u0000\nh0b\u0001\u0014\n(ReD2)2\u0000(ImD2)2\u0015\n\u00002 ReD2ImD2Re\u0000\nh0b\u0001\nh\n(ReD2)2+ (ImD2)2i2; (A57)\n\f(0)\n14=1\n2Im\u0012h0(1 +a)\nD4\u0013\n=1\n\u0019Im\u0000\nh0(1 +a)\u0001\u0014\n(ReD2)2\u0000(ImD2)2\u0015\n\u00002 ReD2ImD2Re\u0000\nh0(1 +a)\u0001\nh\n(ReD2)2+ (ImD2)2i2; (A58)\n\f(1)\n14=1\n2Im\u0012h0(1 +a)k0\nD4\u0013\n=1\n\u0019Im\u0000\nh0(1 +a)k0\u0001\u0014\n(ReD2)2\u0000(ImD2)2\u0015\n\u00002 ReD2ImD2Re\u0000\nh0(1 +a)k0\u0001\nh\n(ReD2)2+ (ImD2)2i2;\n(A59)\n\f(0)\n15=1\n\u0019Im\u0012h2(1 +a)\nD6\u001329\n=1\n\u0019Im (h2(1 +a))\u0014\n(ReD2)3\u00003ReD2(ImD2)2\u0015\n+ Re (h2(1 +a))\u0014\n(ImD2)3\u00003ImD2(ReD2)2\u0015\nh\n(ReD2)2+ (ImD2)2i3\n(A60)\n\f(1)\n15=1\n\u0019Im\u0012h2(1 +a)k0\nD6\u0013\n=1\n\u0019Im (h2(1 +a)k0)\u0014\n(ReD2)3\u00003ReD2(ImD2)2\u0015\n+ Re (h2(1 +a)k0)\u0014\n(ImD2)3\u00003ImD2(ReD2)2\u0015\nh\n(ReD2)2+ (ImD2)2i3\n(A61)\n\f16=1\n\u0019Im\u0012h2b\nD6\u0013\n=1\n\u0019Im (h2b)\u0014\n(ReD2)3\u00003ReD2(ImD2)2\u0015\n+ Re (h2b)\u0014\n(ImD2)3\u00003ImD2(ReD2)2\u0015\nh\n(ReD2)2+ (ImD2)2i3\n(A62)\nwhere\nIm(D2) =\u0000\u0019m4\nth\nk2\u0014!\nk+\u0010\n1\u0000!2\nk2\u0011\nQ0\u0010!\nk\u0011\u0015\n; (A63)\nRe(D2) =!2\u0000k2\u00002m2\nth+m4\nth\nk2\u00122!\nkQ0\u0012!\nk\u0013\n\u00001\u0013\n+m4\nth\nk2\u0010\n1\u0000!2\nk2\u0011\u0012\nQ2\n0\u0012!\nk\u0013\n\u0000\u00192\n4\u0013\n; (A64)\nIm(b0) =\u00004e2M2\u0019k3!\n2k3; (A65)\nRe(b0) = 4e2M2k3\nk2\u0014!\nkQ0\u0012!\nk\u0013\n\u00001\u0015\n; (A66)\nIm(c0) =\u00004e2M2\u0019\n2k; (A67)\nRe(c0) = 4e2M21\nkQ0\u0012!\nk\u0013\n; (A68)\nIm\u0010\nh(1 +a)\u0011\n=4\u0019e2M2k3\n2k7\u0012\n\u00002k6\u00002k4\u0000\nk2\n0+ 3m2\nth\u0001\n+ 12k3k0m2\nthQ0\u00004k2m2\nth\u0000\nk2\n0+m2\nth\u0001\n+ 4kk0m2\nthQ0\u0000\nk2\n0+ 4m2\nth\u0001\n+k2\n0m4\nth\u0000\n\u00192\u000012Q2\n0\u0001\u0013\n;\nRe(h(1 +a)) =4e2M2k3\n2k7\u0012\n4k6Q0\u00004k5k0+ 4k4Q0\u0000\nk2\n0+ 3m2\nth\u0001\n+k3k0m2\nth\u0000\n\u000012Q2\n0+ 3\u00192\u00004\u0001\n+ 8k2m2\nthQ0\u0000\nk2\n0+m2\nth\u0001\n+kk0m2\nth\u0000\n\u00192\u00004Q2\n0\u0001\u0000\nk2\n0+ 4m2\nth\u0001\n+ 2k2\n0m4\nthQ0\n\u0002\u0000\n4Q2\n0\u00003\u00192\u0001\u0013\n; (A69)\nIm(hb) =4\u0019e2M2k3m2\nth\n2k7\u0012\n4k2k0\u0000\nk2\n0+m2\nth\u0001\n+ 4kQ0\u0000\nk4+ 2m2\nth\u0000\nk2\u00002k2\n0\u0001\n\u0000k4\n0\u0001\n+ 12k0m2\nthQ2\n0(k0\u0000k)(k+k0) +\u00192k0m2\nth(k\u0000k0)(k+k0)\u0013\n; (A70)30\nRe(hb) =4e2M2k3m2\nth\n2k7\u0012\nk5\u0000\n\u00192\u00004Q2\n0\u0001\n+ 2k3\u0000\n2k2\n0+m2\nth\u0000\n\u00192\u00004Q2\n0\u0001\u0001\n\u00002k2k0Q0\n\u0002\u0010\n4k2\n0+m2\nth\u0000\n\u00004Q2\n0+ 3\u00192+ 4\u0001\u0011\n\u0000kk2\n0\u0000\n\u00192\u00004Q2\n0\u0001\u0000\nk2\n0+ 4m2\nth\u0001\n+ 2k3\n0m2\nthQ0\n\u0002\u0000\n3\u00192\u00004Q2\n0\u0001\u0013\n; (A71)\nIm(hb0) =16\u0019e4M4k2\n3\n2k7\u0012\n2k4\u00004k3k0Q0+ 4k2\u0000\nk2\n0+m2\nth\u0001\n\u00004kk0Q0\u0000\nk2\n0+ 4m2\nth\u0001\n\u0000k2\n0m2\nth\u0000\n\u00192\u000012Q2\n0\u0001\u0013\n; (A72)\nRe(hb0) =\u000016e4M4k2\n3\n2k7\u0012\n4k4Q0+k3k0\u0000\n\u00192\u00004\u0000\nQ2\n0+ 1\u0001\u0001\n+ 8k2Q0\u0000\nk2\n0+m2\nth\u0001\n+kk0\u0000\n\u00192\u00004Q2\n0\u0001\u0000\nk2\n0+ 4m2\nth\u0001\n+ 2k2\n0m2\nthQ0\u0000\n4Q2\n0\u00003\u00192\u0001\u0013\n; (A73)\nIm(hc0) =\u000016\u0019e4M4k3\n2k5\u0000\n4k3Q0\u00002k2k0+ 4kQ0\u0000\nk2\n0+ 2m2\nth\u0001\n+k0m2\nth\u0000\n\u00192\u000012Q2\n0\u0001\u0001\n; (A74)\nRe(hc0) =\u000016e4M4k3\n2k5\u0012\nk3\u0000\n\u00192\u00004Q2\n0\u0001\n+ 4k2k0Q0+k\u0000\n\u00192\u00004Q2\n0\u0001\u0000\nk2\n0+ 2m2\nth\u0001\n+ 2k0m2\nthQ0\u0000\n4Q2\n0\u00003\u00192\u0001\u0013\n; (A75)\nIm(h0b) =8\u0019e4M4m2\nth\nk9\u0012\nk6\u0012\u00192\n4\u00003Q2\n0\u0013\n\u00002k5!Q0+k4\u0010\nk2\n3\u0000!2\u0012\u00192\n4\u00003Q2\n0\u0013\u0011\n\u00004k3!k2\n3Q0\u0000k2!2k2\n3\u0012\n\u00003Q2\n0+\u00192\n4+ 3\u0013\n+ 6k!3k2\n3Q0+!4k2\n3\u0012\u00192\n4\u00003Q2\n0\u0013\u0013\n; (A76)\nRe(h0b) =16e4M4m2\nth\nk9 \nk6\u0012\nQ3\n0\u00003\u00192\n4Q0\u0013\n+k5!\u0012\nQ2\n0\u0000\u00192\n4\u0013\n\u0000k4Q0\u0010\n!2\u0010\nQ2\n0\u00003\u00192\n4\u0011\n+k2\n3\u0011\n+k3!k2\n3\u0012\n2Q2\n0\u0000\u00192\n2\u00001\u0013\n+k2!2k2\n3Q0\u0012\n\u0000Q2\n0+3\u00192\n4+ 3\u0013\n+ 3k!3k2\n3\u0010\u00192\n4\u0000Q2\n0\u0011\n+!4k2\n3Q0\u0012\nQ2\n0\u00003\u00192\n4\u0013!\n; (A77)\nIm(h0(1 +a)) =8\u0019e4M4\nk9\u0012\n2k7Q0+ 2k5m2\nthQ0+k4!\u0010\n2k2\n3+m2\nth\u0010\u00192\n4\u00003Q2\n0\u0011\u0011\n\u00002k3!2k2\n3Q0\n+ 3k2!k2\n3m2\nth\u00006k!2k2\n3m2\nthQ0\u0000!3k2\n3m2\nth\u0010\u00192\n4\u00003Q2\n0\u0011\u0013\n; (A78)\nRe(h0(1 +a)) =16e4M4\nk9\u0012\nk7\u0010\u00192\n4\u0000Q2\n0\u0011\n+k5\u0010\nk2\n3+m2\nth\u0010\u00192\n4\u0000Q2\n0\u0011\u0011\n+k4!Q0\n\u0002\u0010\nm2\nth\u0010\nQ2\n0\u00003\u00192\n4\u0011\n\u00002k2\n3\u0011\n+k3k2\n3\u0012\n!2\u0010\nQ2\n0\u0000\u00192\n4\u0011\n+m2\nth\u0013\n\u00003k2!k2\n3m2\nthQ0\n+ 3k!2k2\n3m2\nth\u0012\nQ2\n0\u0000\u00192\n4\u0013\n\u0000!3k2\n3m2\nthQ0\u0010\nQ2\n0\u00003\u00192\n4\u0011\u0013\n; (A79)\nIm\u0010\nh2(1 +a)\u0011\n=16\u0019e4M4k2\n3\n2k11 \n\u00008k9Q0+ 8k8k0\u00008k7Q0\u0000\n2k2\n0+ 5m2\nth\u0001\n+k6k0\u0012\n8k2\n031\n+m2\nth\u0000\n60Q2\n0\u00005\u00192+ 24\u0001\u0013\n\u00008k5Q0\u0000\nk4\n0+ 12k2\n0m2\nth+ 8m4\nth\u0001\n+ 2k4k0m2\nth\n\u0002\u0012\nk2\n0\u0000\n36Q2\n0\u00003\u00192+ 6\u0001\n\u00008m2\nth\u0000\n\u000012Q2\n0+\u00192\u00001\u0001\u0013\n\u00008k3m2\nthQ0\u0012\n3k4\n0\n\u00004k2\n0m2\nth\u0000\n\u00004Q2\n0+\u00192\u00003\u0001\n+ 4m4\nth\u0013\n\u0000k2k0m2\nth\u0000\n\u00192\u000012Q2\n0\u0001\u0010\nk4\n0+ 12k2\n0m2\nth\n+ 12m4\nth\u0011\n+ 16kk2\n0m4\nthQ0\u0000\n\u00192\u00004Q2\n0\u0001\u0000\nk2\n0+ 3m2\nth\u0001\n+k3\n0m6\nth\u0000\n80Q4\n0\u000040\u00192Q2\n0+\u00194\u0001!\n; (A80)\nRe\u0010\nh2(1 +a)\u0011\n=\u000016e4M4k2\n3\nk11 \nk9\u0000\n\u00192\u00004Q2\n0\u0001\n+ 8k8k0Q0+k7\u0010\n2k2\n0\u0000\n\u00004Q2\n0+\u00192\u00002\u0001\n+ 5m2\nth\u0000\n\u00192\u00004Q2\n0\u0001\u0011\n+k6k0Q0\u0010\n8k2\n0+m2\nth\u0000\n20Q2\n0\u000015\u00192+ 24\u0001\u0011\n+k5\u0010\nk4\n0\u0000\n\u00192\u00004Q2\n0\u0001\n+ 4k2\n0m2\nth\u0000\n\u000012Q2\n0+ 3\u00192\u00001\u0001\n+ 8m4\nth\u0000\n\u00192\u00004Q2\n0\u0001\u0011\n+ 2k4k0m2\nthQ0\u0012\nk2\n0\u0010\n12Q2\n0\u00009\u00192+ 6\u0011\n+ 8m2\nth\u0000\n4Q2\n0\u00003\u00192+ 1\u0001\u0013\n+k3m2\nth\u0012\n3k4\n0\u0000\n\u00192\u00004Q2\n0\u0001\n\u00002k2\n0m2\nth\u0010\n8\u0000\n2Q2\n0+ 3\u0001\nQ2\n0\u00006\u00192\u0000\n4Q2\n0+ 1\u0001\n+\u00194\u0011\n+ 4m4\nth\u0000\n\u00192\u00004Q2\n0\u0001\u0013\n+k2k0m2\nthQ0\u0000\n4Q2\n0\u00003\u00192\u0001\u0000\nk4\n0+ 12k2\n0m2\nth+ 12m4\nth\u0001\n\u0000kk2\n0m4\nth\u0000\n16Q4\n0\u000024\u00192Q2\n0+\u00194\u0001\u0000\nk2\n0+ 3m2\nth\u0001\n+k3\n0m6\nthQ0\u0000\n16Q4\n0\u000040\u00192Q2\n0+ 5\u00194\u0001!\n; (A81)\nIm(h2b) =16\u0019e4M4k2\n3m2\nth\n2k11 \nk8\u0000\n\u0000\u0000\n\u00192\u000012Q2\n0\u0001\u0001\n\u00008k7k0Q0\u0000k6\u0010\nk2\n0\u0000\n\u000012Q2\n0+\u00192+ 4\u0001\n+ 4m2\nth\u0000\n\u00192\u000012Q2\n0\u0001\u0011\n+ 16k5k0Q0\u0000\nk2\n0+m2\nth\u0000\n\u00192\u00004Q2\n0\u0001\u0001\n+k4\u0012\nk4\n0\u0010\n\u00192\u000012\u0000\nQ2\n0+ 1\u0001\u0011\n\u000016k2\n0m2\nth\u00004m4\nth\u0000\n\u00192\u000012Q2\n0\u0001\u0013\n+ 8k3k0Q0\u0000\n3k4\n0+ 12k2\n0m2\nth+ 4m4\nth\u0000\n\u00004Q2\n0+\u00192+ 1\u0001\u0001\n+k2k2\n0\u0012\nk4\n0\u0000\n\u00192\u000012Q2\n0\u0001\n+ 12k2\n0m2\nth\u0000\n\u00192\u000012Q2\n0\u0001\n+m4\nth\u0010\n80Q4\n0\u00008\u0000\n18 + 5\u00192\u0001\nQ2\n0\n+\u00192\u0000\n12 +\u00192\u0001\u0011\u0013\n\u000016kk3\n0m2\nthQ0\u0000\n\u00192\u00004Q2\n0\u0001\u0000\nk2\n0+ 3m2\nth\u0001\n\u0000k4\n0m4\nth\u0000\n80Q4\n0\u000040\u00192Q2\n0+\u00194\u0001!\n; (A82)\nRe(h2b) =16e4M4k2\n3m2\nth\nk11 \nk8\u0000\n3\u00192Q0\u00004Q3\n0\u0001\n\u0000k7k0\u0000\n\u00192\u00004Q2\n0\u0001\n+k6Q0\u0010\nk2\n0\u0000\n\u00004Q2\n0+ 3\u00192+ 4\u0001\n+ 4m2\nth\u0000\n3\u00192\u00004Q2\n0\u0001\u0011\n+k5k0\u0000\n2k2\n0\u0000\n\u00004Q2\n0+\u00192\u00002\u0001\n+m2\nth\u0000\n16Q4\n0\u000024\u00192Q2\n0+\u00194\u0001\u000132\n+k4Q0\u0012\nk4\n0\u0000\n4Q2\n0\u00003\u00192+ 12\u0001\n+ 16k2\n0m2\nth+ 4m4\nth\u0000\n3\u00192\u00004Q2\n0\u0001\u0013\n+k3k0\u0012\n3k4\n0\u0000\n\u00192\u00004Q2\n0\u0001\n+ 12k2\n0m2\nth\u0000\n\u00192\u00004Q2\n0\u0001\n+ 2m4\nth\u0000\n16Q4\n0\u00008\u0000\n1 + 3\u00192\u0001\nQ2\n0+\u00192\u0000\n2 +\u00192\u0001\u0001\u0013\n+k2k2\n0Q0\n\u0002\u0012\n4Q2\n0\u0000\nk4\n0+ 12k2\n0m2\nth+ 2\u0000\n6 + 5\u00192\u0001\nm4\nth\u0001\n\u0000\u00192\u0000\n3k4\n0+ 36k2\n0m2\nth+\u0000\n36 + 5\u00192\u0001\nm4\nth\u0001\n\u000016m4\nthQ4\n0\u0013\n\u0000kk3\n0m2\nth\u0000\n16Q4\n0\u000024\u00192Q2\n0+\u00194\u0001\u0000\nk2\n0+ 3m2\nth\u0001\n+k4\n0m4\nthQ0\u0000\n16Q4\n0\n\u000040\u00192Q2\n0+ 5\u00194\u0001!\n; (A83)\nwhereM2is de\fned in Eq. 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Our theory shows th at non-equilibrium spin accumulation\ninduces a spin torque and the electron bath leads to a damping of the magnetization. For the\ntwo-dimensional magnetization thin film coupled to the elec tron gas with Rashba spin-orbit cou-\npling, the result for the spin-orbit torques is consistent w ith the previous semi-classical theory. Our\ntheory predicts a damping of the magnetization, which is abs ent in the semi-classical theory. The\nmagnitude of the damping due to the electron bath is comparab le to the intrinsic Gilbert damping\nand may be important in describing the magnetization dynami cs of the system.\nI. INTRODUCTION\nIn study of spin transfer torque (STT), it has been\nproposed [1, 2] to manipulate magnetic order parameter\ndynamics by using non-equilibrium electron bath instead\nof external magnetic fields. The proposal has already led\nto commercial products in spintronics engineering. Re-\ncently, there has been much attention on the ”spin-orbit\ntorque”(SOT), which was first proposed in theory[3, 4],\nand later confirmed in experiments[5–8] (see Ref. 9 and\n10 for a comprehensive review). After applying an ex-\nternal electric field to the electron gas with spin-orbit in-\nteraction(SOI), a component ofthe accumulated electron\nspin density mis-aligned with the ferromagnetic ordering\ncan be created[3, 4], which then will induce a field-like\ntorque. The SOT opens the possibility of manipulat-\ning the magnetic order parameter in collinear magnetic\nstructures and may efficiently reduce the critical current\ndensity for magnetization switching[3, 4]. In the theoret-\nical side, a full quantum theory has been proposed and\ndeveloped to describe the dynamics of a single domain\nmagnetunderthecontinuousscatteringbyspin-polarized\nelectrons. The quantum STT theory recovers the results\nof the semiclassical STT theory, and has revealed more\ndetailsaboutthemagnetizationdynamicsintheSTT[11–\n13]. Therefore, it will be natural to apply a full quantum\ntheory to study the magnetization dynamics influenced\nby the SOI electron gas. This may be an extension of\nthe quantum STT theory to SOT. In the full quantum\ntheory, the quantum dynamics of the magnetization can\nbe described by the evolution of its density matrix under\nthe influence of the electron gas, which can be tuned by\nthe external electric field. This treatment will not only\ngive the mean-field effect on the magnetization dynamics\nby the electron bath, but also include the damping of the\nmagnetization due to the fluctuation of the electron spin.\nThe similarstrategyhas been exploited to investigatethe\nphoto-excited dynamics of the order parameter in Peierls\nchain[14].\nThis paper is organized as follows. In section II, we\napply density matrix technique to derive general formal-E\nFIG. 1. (Color online). Schematic diagram for the lattice of\nlocalized spins (orange) coupled to the conductions electr ons\n(blue) through s-d exchange interaction. An external elect ric\nfieldEcan be applied to tune the electron bath.\nism for the magnetization dynamics driven by the elec-\ntron bath through s-d exchange interaction. In section\nIII, we apply the general formalism to the special case\nwhere the spatially uniform magnetization is coupled to\na two dimensional electron gas with Rashba SOI, and\ncalculate the spin-orbit torque and the damping effect of\nthe electron bath. The main results are summarized and\ndiscussed in section IV.\nII. GENERAL FORMALISM\nWe apply density matrix technique to study dynamics\nof the magnetization driven by the electron bath via an\ns-d exchange interaction. The system is schematically\nillustratedinFig.1, wheretheelectronbathcanbe tuned\nbyanexternalelectricfield. TheHamiltonianofthetotal\nsystem is formally written as\nH=HM+He+Hsd. (1)\nHere,HMis the Hamiltonian for the magnetization sub-\nsystem in terms of the local spin operators /hatwideSi,µat sitei\nwith spin directions µ(=x,y,z);Heis the Hamiltonian\noftheelectronsubsystem; Hsddescribesthes-dexchange\ninteraction between the magnetization and the electron,2\nwhere\nHsd=J/summationdisplay\ni,µ/hatwideSi,µ/hatwideσi,µ. (2)\nHere,/hatwideσi,µrepresents the electron spin operator at site i\nwithout /planckover2pi1/2, andJis the exchange coupling strength.\nNote that we have not specified the forms of HMandHe\nyet, thus the results below will be quite general.\nThe effect of the s-d exchange interaction Hsdis\ntwofold. On one hand, the magnetization dynamics is\ndriven by the electron bath via Hsd; on the other hand,\nthe electron states are also affected by the magnetization\nconfiguration in turn due to Hsd. Since the time scale of\nthe electron dynamics is usually much faster than that\nof the magnetization dynamics, we may assume that the\nelectrons under the bias voltage establish a stationary\nnon-equilibrium distribution in a very short time inter-\nval, during which the change of the magnetization con-\nfiguration is negligible and the non-equilirium electron\nbath is approximated to be constant. The validity of\nthis assumption only holds if the spin-lattice interaction\nis stronger than the s-d exchange interaction to relax\nthe electron spin. Consider a short time interval [ t0,t],\nwhere the initial density matrices of the magnetization\nand the electron bath are ρM(t0) andρe(t0) respectively.\nThen the initial magnetization configuration at each site\nisSi,µ(t0) = Tr[/hatwideSi,µρM(t0)], and the initial electron den-\nsity matrix ρe(t0) is determined by the bath Hamiltonian\nHB=He+J/summationtext\ni,µSi,µ(t0)/hatwideσi,µand the open boundary\nconditions.\nIn order to investigate the magnetization dynamics\nduring the time interval [ t0,t] defined above, we rede-\nfine the local spin operators /hatwideSi,µ=Si,µ(t0) +/hatwidesi,µ, then\nthe Hamiltonian Hin Eq. (1) can be rewritten as\nH=HM+HB+Vsd, (3)\nwith the interaction term\nVsd=J/summationdisplay\ni,µ/hatwidesi,µ/hatwideσi,µ. (4)\nDuring this time interval, the electron density matrix ρe\nmay be approximated to be constant because of the neg-\nligible change of the magnetization, and this can be jus-\ntified in the limit t→t0. Assuming the total density\nmatrix as ρ(t) =ρM(t)⊗ρe(t0) and to the second or-\nder of interaction strength, the equation for the density\nmatrix/tildewideρM(t) in the interaction picture is[15]\nd\ndt/tildewideρM(t) =J\ni/planckover2pi1/summationdisplay\ni,µσi,µ(t)[/tildewidesi,µ(t),/tildewideρM(t0)]\n+(J\ni/planckover2pi1)2/summationdisplay\ni,µ;j,ν/integraldisplayt\nt0dτ{Ci,µ;j,ν(t,τ)[/tildewidesi,µ(t),/tildewidesj,ν(τ)/tildewideρM(τ)]\n−Cj,ν;i,µ(τ,t)[/tildewidesi,µ(t),/tildewideρM(τ)/tildewidesj,ν(τ)]}. (5)\nHere,/tildewider···denotes the operators in the interaction\npicture; the electron spin polarization is σi,µ(t) =Tre[/tildewideσi,µ(t)/tildewideρe(t0)]; the electron spin-spin correlation func-\ntion isCi,µ;j,ν(t,τ) = Tr e[/tildewideσi,µ(t)/tildewideσj,ν(τ)/tildewideρe(t0)], which\nis a function of t−τonly and satisfies the relation\nCi,µ;j,ν(t,τ) =C∗\nj,ν;i,µ(τ,t). In priciple, the solution of\nEq. (5) gives the density matrix of the magnetization in\nthe time interval [ t0,t] under the influence of the electron\nbath, and can be applied to study the physical qualities\nthat we are particularly interested in.\nBased on Eq. (5), the dynamical equation for Sl,λ(t) =\nTrM[/tildewideSl,λ(t)/tildewideρM(t)] is obtained as\nd\ndtSl,λ(t) =1\ni/planckover2pi1/an}b∇acketle{t[/hatwideSl,λ,HM]/an}b∇acket∇i}htt+J\n/planckover2pi1/summationdisplay\nµ,νǫλµνσl,µ(t)Sl,ν(t)\n+i(J\ni/planckover2pi1)2/summationdisplay\nj,µ,ν,ξǫλµν/integraldisplayt\nt0dτ{Cl,µ;j,ξ(t,τ)/an}b∇acketle{t/hatwideSl,ν(t)/hatwidesj,ξ(τ)/an}b∇acket∇i}htτ\n−Cj,ξ;l,µ(τ,t)/an}b∇acketle{t/hatwidesj,ξ(τ)/hatwideSl,ν(t)/an}b∇acket∇i}htτ}. (6)\nHere,/an}b∇acketle{t···/an}b∇acket∇i}htt≡TrM[···ρM(t)], and the spin commutation\nrelation [/hatwideSl,λ,/hatwideSi,µ] =iδli/summationtext\nνǫλµν/hatwideSl,νhas been exploited.\nThe first term in the r.h.s.of Eq. (6) gives the intrin-\nsic magnetization dynamics due to HM; the second term\nis the spin torque term due to the accumulation of the\nelectron spin density; the third term gives the damping\neffect of the electron bath. If the operator /hatwideSl,ν(t) in the\ndamping term is approximately replaced by its expecta-\ntion value Sl,ν(t), Eq. (6) becomes\nd\ndtSl,λ(t) =1\ni/planckover2pi1/an}b∇acketle{t[/hatwideSl,λ,HM]/an}b∇acket∇i}htt+J\n/planckover2pi1/summationdisplay\nµ,νǫλµνσl,µ(t)Sl,ν(t)\n+2J2\n/planckover2pi12/summationdisplay\nj,µ,ν,ξǫλµνSl,ν(t)/integraldisplayt\nt0dτKl,µ;j,ξ(t−τ)sj,ξ(τ),(7)\nwhereKl,µ;j,ξ(t−τ) is the imaginary part of Cl,µ;j,ξ(t,τ),\nandsj,ξ(τ) =/an}b∇acketle{t/hatwidesj,ξ/an}b∇acket∇i}htτ. We introduce the kernel func-\ntionγ(t) which satisfies the relation dγl,µ;j,ξ(t)/dt=\nKl,µ;j,ξ(t). The integral in the last term in Eq. (7) is\nrewrittenas/integraltextt\nt0dτγl,µ;j,ξ(t−τ)˙Sj,ξ(τ) afterintegratingby\nparts and neglecting the boundary terms in the limiting\ncaset→t0. It can be further simplified as Γ l,µ;j,ξ˙Sj,ξun-\nder the Markovian approximation ˙Sj,ξ(τ)≈˙Sj,ξ(t), with\nthe coefficient Γ l,µ;j,ξ=/integraltextδt\n0dτγl,µ;j,ξ(τ) forδt=t−t0.\nBased on the discussions above, Eq. (7) can be written\nin a compact form\nd\ndtSl(t) =1\ni/planckover2pi1/an}b∇acketle{t[/hatwideSl,HM]/an}b∇acket∇i}htt+γeBl(t)×Sl(t),(8)\nwhereγeis the gyromagnetic ratio; Blis the effective\nmagnetic field on the the local spin Sloriginating from\nthe electron bath. The µ-component of Blis expressed\nas\nBl,µ(t) =J\nγe/planckover2pi1σl,µ(t)+2J2\nγe/planckover2pi12/summationdisplay\nj,ξΓl,µ;j,ξ(t)˙Sj,ξ(t).(9)\nThe first term in (9) will give the torque term due to the\nelectron spin accumulation, which has been discussed ex-\ntensively in previous studies; the second term will give3\nthe damping effect of the electron bath on the magne-\ntization dynamics, which only emerges in the quantum\ntreatment. The non-local feature of the damping term\ncan be found here, which depends on the spatial correla-\ntion of Γ l,µ;j,ξ.\nSo far we have established a general dynamical equa-\ntion for the magnetization when it is coupled to the elec-\ntron bath via s-d exchange interaction. Here, both the\nHamiltonian for the magnetization subsystem HMand\nthe Hamiltonian for the electron subsystem Hehave not\nbeen specified yet. The treatment is similar to the previ-\nous work on the order parameter dynamics in the photo-\nexcited Peierls chain[14]. In the next section, we apply\nthis general formula to study the magnetization dynam-\nics of a two-dimensional ferromagnetic thin film under\nthe influence of an electron gas with Rashba SOI, i.e. a\nmodel system for “spin-orbit torque”.\nIII. SPIN-ORBIT TORQUE\nA. Electron Bath with Rashba SOI\nWe consider a special system studied by Manchon and\nZhang[3] for the spin-orbit torque. The two-dimensional\nmagnetization thin film in x-y plane consists of N=\nM×Nlattice sites with the lattice constant a, and we\nwill use the discrete notations in both real and reciprocal\nspace. The magnetization is assumed to be uniform due\nto strong exchange interaction. The lack of inversion\nsymmetry in z-direction induces the Rashba spin-orbit\ninteraction in the two-dimensional electron gas. In this\ncase, the Hamiltonian for the electron bath is given as[3]\nHB=/hatwidep2\n2m∗e+αR\n/planckover2pi1(/hatwidep×/hatwidez)·/hatwideσ+JS·/hatwideσ,(10)\nwhere/hatwidepis the electron momentum operator; m∗\neis the\neffective mass of electrons; αRis the Rashba interaction\nstrength; S=Siis the localized spin at each site. For\nS=S(sinθcosφ,sinθsinφ,cosθ), the energy dispersion\nrelation of the electron is\nEk,±=/planckover2pi12k2\n2m∗e±∆k. (11)\nHere, we have denoted the electron wavevector k=\nk(cosϕ,sinϕ), and\n∆k=/radicalBig\nJ2S2+α2\nRk2−2JSαRksinθsin(φ−ϕ).\nThe corresponding electron eigenstates |k,±/an}b∇acket∇i}htare\n|k,±/an}b∇acket∇i}ht=1√\nNeik·r/parenleftBigg\ncosΘk,±\n2e−iΦk\nsinΘk,±\n2/parenrightBigg\n,(12)where the angles Θ k,±and Φ kare determined by\ncosΘk,±\n2=/radicalbig\n∆2\nk−J2S2cos2θ/radicalbig\n2∆2\nk∓2JS∆kcosθ,\nsinΘk,±\n2=±∆k−JScosθ/radicalbig\n2∆2\nk∓2JS∆kcosθ,\ncosΦk=JSsinθcosφ+αRksinϕ/radicalbig\n∆2\nk−J2S2cos2θ,\nsinΦk=JSsinθsinφ−αRkcosϕ/radicalbig\n∆2\nk−J2S2cos2θ.\nThespinpolarizationvectorforthestate |k,±/an}b∇acket∇i}htisPk,±=\n(sinΘk,±cosΦk,sinΘk,±sinΦk,cosΦk,±).\nThe statistical properties of the electron bath are de-\ntermined by the probability distribution function fk,sfor\nthe state |k,s=±/an}b∇acket∇i}ht, which can be tuned by the external\nfield. If an electric field Eis applied, the non-equilibrium\ndistribution of the electron states will be established due\nto the random scattering potential by impurities[3]. The\ndistribution function fk,sis determined by the Boltz-\nmann equation\n−eE\n/planckover2pi1·∇kfk,s=Sc[fk,s]. (13)\nThe collision integral Sc[fk,s] describes the relaxation of\nthe occupied state |k,s/an}b∇acket∇i}htand can be treated by the relax-\nation time approximation, namely,\nSc[fk,s] =−fk,s−f0\nk,s\nτ. (14)\nHere,f0\nk,sis the equilibrium distribution function, and\nan isotropic relaxation time τhas been assumed[3]. To\nthe first orderofthe electric field, the solution of Eq. (13)\nisfk,s=f0\nk,s+gk,s, where the out of equilibrium part\ninduced by the external electric field is\ngk,s=∂f0\nk,s\n∂EeE·vk,sτ, (15)\nwith the electron velocity vk,s=1\n/planckover2pi1∇kEk,s. Such a treat-\nment of the non-equilirium electron distribution was also\nexploited in the previous semiclassical theory[3].\nB. Electron Spin Polarization and Torque\nWith the non-equilibrium distribution function fk,s\ngiven above, the electron spin polarization σl,µat sitel\nand the correlationfunction Cl,µ;j,ξ(t,τ) in Eq. (9) can be\ncalculated, and the torque and damping effect due to the\nelectron bath can be obtained. In the second quantiza-\ntion representation of the basis set {|k,s/an}b∇acket∇i}ht}, the operator\n/hatwideσl,µis expressed as\n/hatwideσl,µ=1\nN/summationdisplay\nk,s;k′,s′χµ\nk,s;k′,s′ei(k′−k)·rl/hatwidec†\nk,s/hatwideck′,s′,4\nwhere the matrix element\nχµ\nk,s;k′,s′= (cosΘk,s\n2eiΦk,sinΘk,s\n2)σµ/parenleftBigg\ncosΘk′,s′\n2e−iΦk′\nsinΘk′,s′\n2/parenrightBigg\n.\nThen the electron spin polarization σl,µis\nσl,µ=1\nN/summationdisplay\nk,sχµ\nk,s;k,sfk,s=1\nN/summationdisplay\nk,sPµ\nk,sfk,s.(16)\nFor the physically relevant case αRk≪JS, the approx-\nimate value of Pk,±to the first order ofαRk\nJSis\nPk,±=±\nSx+αR\nJSSxSykx+αR\nJS(1−S2\nx)ky\nSy−αR\nJS(1−S2\ny)kx−αR\nJSSxSyky\nSz+αR\nJSSySzkx−αR\nJSSxSzky\n.\nHere, the unit vector for the magnetization is denoted as\nS= (sinθcosφ,sinθsinφ,cosφ).\nFor the electric current density je=je(cosϑ,sinϑ,0),\nthe non-equilibrium spin polarization δσlwhich is per-\npendicular to Sis calculated to be (Appendix A)\nδσl=−αRm∗\nejea3\ne/planckover2pi1Ef\ncosϑSxSy+sinϑ(1−S2\nx)\n−cosϑ(1−S2\ny)−sinϑSxSy\ncosϑSySz−sinϑSxSz\n,\nwhereEfdenotes the Fermi energy. The torque Tlis\nthen obtained as\nTl=JSαRm∗\nejea3\ne/planckover2pi12Ef\ncosϑSz\nsinϑSz\n−cosϑSx−sinϑSy\n\n=JαRm∗\nea3\ne/planckover2pi12Ef(/hatwidez×je)×Sl.\nThis result reproduces the form of SOT obtained\nbefore[3], but the magnetization vector is not restricted\nin two-dimensional x-y plane in our derivations. It is\neasily understood from the effective Hamiltonian (10),\nwhere the non-equilibrium distribution of electron states\nwill produce an extra electron spin polarizationalong the\ndirection/hatwidez×je.\nC. Correlation Function and Damping\nWe now calculate the correlation function Cl,µ;j,ξ(t,τ),\nwhich gives the damping term for the magnetization\ndynamics due to the electron bath. Since /hatwideck,s(t) =\n/hatwideck,se−iEk,st//planckover2pi1, the correlation function Cl,µ;j,ξ(t,τ) is for-\nmally written as\nCl,µ;j,ξ(t,τ)\n=1\nN2/summationdisplay\nk,s;k′,s′/summationdisplay\nk′′,s′′;k′′′,s′′′ei(k′��k)·rlei(k′′′−k′′)·rj\n×ei(Ek,s−Ek′,s′)t//planckover2pi1ei(Ek′′,s′′−Ek′′′,s′′′)τ//planckover2pi1\n×χµ\nk,s;k′,s′χξ\nk′′,s′′;k′′′,s′′′/an}b∇acketle{t/hatwidec†\nk,s/hatwideck′,s′/hatwidec†\nk′′,s′′/hatwideck′′′,s′′′/an}b∇acket∇i}ht.(17)We see that Cl,µ;j,ξ(t,τ) is the function of rl−rjand\nt−τ, due to the space and time translation invari-\nance for the investigated system. For simplicity, we es-\ntimateCl,µ;j,ξ(t,τ) with several approximations. Firstly,\nwe assumethat the phase factor ei(k′−k)·(rl−rj)will cause\nthe cancellation of the summations over kandk′if\nrl/ne}ationslash=rj, thusCl,µ;j,ξ=Cµξδlj. Secondly, χµ\nk,s;k′,s′are\ncalculated to the zeroth order ofαRk\nJSfor the relevant\ncaseαRk≪JS, where the electron spin states are k-\nindependent, i.e.\nχ±±=±(sinθcosφ,sinθsinφ,cosθ),\nχ+−= (−cosθcosφ−isinφ,−cosθsinφ+icosφ,sinθ).\nFurthermore, we calculate the correlation function\n/an}b∇acketle{t/hatwidec†\nk,s/hatwideck′,s′/hatwidec†\nk′′,s′′/hatwideck′′′,s′′′/an}b∇acket∇i}htwith the electron bath at equi-\nlibrium, where the effect of the non-equilibrium electric\ncurrent induced by the external field will be neglected.\nThis enable us to apply the Wick contraction[16] to sim-\nplify the calculations. The negligence of the dependence\nof the damping coefficient on the Rashba SOI and the\nnon-equilibrium electric current is valid if the dynamical\nequation (8) is kept to the first order of these two factors.\nWith the above approximations, we get\nCµξ(t)\n=1\nN2/summationdisplay\nk,sχµ\nssχξ\nssfk,s+1\nN2/summationdisplay\nk,s;k′,s′χµ\nssχξ\ns′s′fk,sfk′,s′\n+1\nN2/summationdisplay\nk,s;k′,s′ei(Ek,s−Ek′,s′)t//planckover2pi1χµ\nss′χξ\ns′sfk,s(1−fk′,s′),\nwhere|k,s/an}b∇acket∇i}htand|k′,s′/an}b∇acket∇i}htare different states.\nSince the kernel function γl,µ;j,ξ(t) is given by the\nrelation dγl,µ;j,ξ(t)/dt=Kl,µ;j,ξ(t), where Kl,µ;j,ξ(t) =\nℑ[Cl,µ;j,ξ(t)], their Fourier transformations are related by\nγl,µ;j,ξ(ω) =i\nωKl,µ;j,ξ(ω). The Fourier transformation of\nKl,µ;j,ξ(t) is obtained as (Appendix B)\nKl,µ;j,ξ(ω)\n=δlj(m∗\nea2\n2π/planckover2pi12)2/planckover2pi1\n2i/summationdisplay\ns,s′[χµ\nss′χξ\ns′sgs(ω)−(χµ\nss′χξ\ns′s)∗gs(−ω)],\nwhere the function gs(ω) is defined as\ngs(ω) =\n\n0,/planckover2pi1ω <0;\n/planckover2pi1ω ,0 E f−sJS.\nThen the damping kernel function γl,µ;j,ξ(t) can be\ncalculated by the inverse Fourier transformation from\nγl,µ;j,ξ(ω), which results in (Appendix B)\nγl,µ;j,ξ(t) =δlj(m∗\nea2\n2π/planckover2pi1)21\n2/summationdisplay\ns(δµξg−\ns(t)+is/summationdisplay\nνǫµξνSνg+\ns(t)).\n(18)\nHere,g±\ns(t) =/integraltext+∞\n−∞dωg±\ns(ω)e−iωtandg±\ns(ω) =\n1\n/planckover2pi1ω(gs(ω)±gs(−ω)), as schematically shown in Fig. 2.5\nThen the coefficient Γ l,µ;j,ξin Eq. (9) is obtained as\nΓl,µ;j,ξ=δlj(m∗\nea2\n2π/planckover2pi1)2(Γ(1)δµξ+Γ(2)/summationdisplay\nνǫµξνSν),(19)\nwith Γ(1)=1\n2/summationtext\ns/integraltextδt\n0dτg−\ns(τ) and Γ(2)=\ni\n2/summationtext\nss/integraltextδt\n0dτg+\ns(τ). Then the damping part in Eq. (8)\ncan be explicitly written as\nDl= 2(Jm∗\nea2\n2π/planckover2pi12)2(Γ(1)˙Sl×Sl+SΓ(2)˙Sl),(20)\nwhich is independent of the Rashba constant and the\nelectric current due to our approximations above.\n0−1−0.500.51\nhωgs+(ω)(a)\nEf+JSs = −s = +\nEf−JS\n000.51\nhωgs−(ω)(b)\ns = +\ns = −Ef−JSEf+JS\nFIG. 2. (Color online). Schematic diagram for g±\ns(ω). Blue\nline fors= +, and red line for s=−. Notice that g+\nsis an\nodd function of ωandg−\ns(ω) is an even function of ω, and\nthey approach to 0 when |ω| → ∞.\nThe first term in (20) will give the damping effect\nwhich drivesthe local spin towardsthe direction with the\nlower energy; while the second term in (20) will give a\nrenormalized factor in Eq. (8). Assuming that J∼1 eV,\nm∗\ne∼me,a∼1˚A, one gets the rough estimation of the\nmagnitudeorderforthe factor(Jm∗\nea2\n2π/planckover2pi12)2∼10−3, thusthe\ndamping effect due to the electron bath is comparable to\nthe intrinsic Gilbert damping of some ferromagnetic ma-\nterials. This damping effect can become important to\nunderstand the dissipative features of the magnetization\ndynamics driven by spin-orbit torque.\nIV. CONCLUSION\nIn conclusion, we have applied density matrix tech-\nnique to formulate the magnetization dynamics of a sys-\ntem consisting of local magnetic moments influenced by\nan electron gas through s-d exchange interaction. In\nthis approach, the magnetic subsystem is treated as an\nopen quantum system and the electron gas acts as a non-\nequilibrium bath tuned by the external electricfield. The\nspin torque due to the non-equilibrium electron spin ac-\ncumulation and the damping effect of the electron bath\nhave been taken into account simultaneously. We ap-\nply the developed formula to the model system for spin-\norbit torque, where the two-dimensional magnetization\nfilm is coupled to the Rashba electron gas through s-dexchange interaction. We have calculated the spin-orbit\ntorque and the results are consistent with the previous\nstudy. However, our method does not require the mag-\nnetization direction to be in the two-dimensional plane\nas in the previous study. Our approach enables us to ob-\ntain the damping effect due to the electron bath, which\nis a new feature absent in the semiclassical theory. The\ndamping caused by the electron bath is estimated to be\ncomparableto the intrinsic Gilbert damping, and may be\nimportanttodescribethemagnetizationdynamicsdriven\nby spin-orbit torque. In brief, this work has extended\nthe previous semiclassical theory for spin-orbit torque\nto a more complete description. Further applications of\nthis approach are expected to understand and to manip-\nulate the magnetization dynamics through electron gas\nin other complex cases.\nACKNOWLEDGMENTS\nThis work was supported in part by the Hong Kong’s\nUniversity Grant Council via grant AoE/P-04/08. This\nwork is also partially supported by National Basic Re-\nsearch Program of China (No. 2014CB921203), NSFC\ngrant (No.11274269), and NSFC grant (No.11204186).\nAppendix A: Electron Spin Polarization\nWe first assume that the electric field is applied along\nx-direction, then\nδσl=1\nN/summationdisplay\nk,sgk,sPk,s=1\nN/summationdisplay\nk(gk,+−gk,−)kxαR\nJSΣx,\nwhereΣx= (SxSy,−(1−S2\ny),SySz). The corresponding\nelectric current density is\nje=−e\nNa3/summationdisplay\nk,sgk,s(vk,s)x≈ −e/planckover2pi1\nm∗e1\nN/summationdisplay\nk,sgk,skx,\nand the spin current density is\njs=/planckover2pi1\n2Na3/summationdisplay\nk,sgk,s(vk,s)xPk,s\n≈/planckover2pi12\n2m∗e1\nNa3/summationdisplay\nk(gk,+−gk,−)kxS.\nThus a rough relation is obtained as\nδσl=−αRm∗\nejea3\ne/planckover2pi1EfΣx,\nwhere the relation js≈ −/planckover2pi1JS\n2eEfjeShas been used here.\nSimilarly, if the electric field is applied along the y-\ndirection, the non-equilibrium spin polarization will be\nδσl=−αRm∗\nejea3\ne/planckover2pi1EfΣy,6\nwithΣy= (1−S2\nx,−SxSy,−SxSz). Therefore, for the\nelectric current density je=je(cosϑ,sinϑ,0), we get\nδσl=−αRm∗\nejea3\ne/planckover2pi1Ef\ncosϑSxSy+sinϑ(1−S2\nx)\n−cosϑ(1−S2\ny)−sinϑSxSy\ncosϑSySz−sinϑSxSz\n.\nAppendix B: Correlation Function and Damping\nKernel\nThe imaginary part of Cµξ(t) is given as\nKµξ(t)\n=ℑ[Cµξ(t)]\n= (m∗\nea2\n2π/planckover2pi12)2/summationdisplay\ns,s′ℑ[χµ\nss′χξ\ns′s/integraldisplayEf\nsJSdǫ/integraldisplay∞\nEfdǫ′ei\n/planckover2pi1(ǫ−ǫ′)t]\n= (m∗\nea2\n2π/planckover2pi12)2/summationdisplay\ns,s′/integraldisplayEf\nsJSdǫ/integraldisplay∞\nEfdǫ′[−i\n2χµ\nss′χξ\ns′sei\n/planckover2pi1(ǫ−ǫ′)t+h.c.].\nHere,fk,sis approximated as the zero-temperature\nFermi distribution function, and the relation1\nN/summationtext\nk→\na2\n(2π)2/integraltext\nd2k=m∗\nea2\n2π/planckover2pi12/integraltext\ndǫhas been used. Its Fourier trans-formation Kµξ(ω) is then\nKµξ(ω)\n=1\n2π/integraldisplay+∞\n−∞dtKµξ(t)eiωt\n= (m∗\nea2\n2π/planckover2pi12)2/summationdisplay\ns,s′/integraldisplayEf\nsJSdǫ/integraldisplay∞\nEfdǫ′\n×[−i\n2χµ\nss′χξ\ns′sδ(ω+ǫ−ǫ′\n/planckover2pi1)+i\n2(χµ\nss′ξξ\ns′s)∗δ(ω+ǫ′−ǫ\n/planckover2pi1)]\n=−(m∗\nea2\n2π/planckover2pi12)2i/planckover2pi1\n2/summationdisplay\ns,s′[χµ\nss′χξ\ns′sgs(ω)−(χµ\nss′χξ\ns′s)∗gs(−ω)],\nwhere the function g(ω) is defined as\ngs(ω) =\n\n0,/planckover2pi1ω <0;\n/planckover2pi1ω ,0 E f−sJS.\nTherefore,\nγl,µ;j,ξ(ω)\n=δlj(m∗\nea2\n2π/planckover2pi1)21\n2/summationdisplay\ns,s′[ℜ(χµ\nss′χξ\ns′s)g−\ns(ω)+iℑ(χµ\nss′χξ\ns′s)g+\ns(ω)],\nwhereg±\ns(ω) =1\n/planckover2pi1ω(gs(ω)±gs(−ω)), andγl,µ;j,ξ(t) is cal-\nculated as\nγl,µ;j,ξ(t)\n=/integraldisplay+∞\n−∞dωγl,µ;j,ξ(ω)e−iωt\n=δlj(m∗\nea2\n2π/planckover2pi1)21\n2/summationdisplay\ns,s′[ℜ(χµ\nss′χξ\ns′s)g−\ns(t)+iℑ(χµ\nss′χξ\ns′s)g+\ns(t)]\n=δlj(m∗\nea2\n2π/planckover2pi1)21\n2/summationdisplay\ns(δµξg−\ns(t)+is/summationdisplay\nνǫµξνSνg+\ns(t))\n≈δljδµξ(m∗\nea2\n2π/planckover2pi1)2g−(t),\nwhereg±\ns(t) =/integraltext+∞\n−∞dωg±\ns(ω)e−iωtand we have used the\nexpressions\nχµ\n+,+χξ\n+,+=χµ\n−,−χξ\n−,−=SµSξ.\nχµ\n+,−χξ\n−,+= (χµ\n−,+χξ\n+,−)∗=δµξ−SµSξ+i/summationdisplay\nνǫµξνSν.\n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405\n(2008).\n[4] A. Manchon and S. Zhang, Phys. Rev. B 79, 094422\n(2009).\n[5] A. Chernyshov, M. Overby, X. Liu, J. K. Furdyna, Y.\nLyanda-Geller, and L. P. Rokhinson, Nat. Phys. 5, 656(2009).\n[6] I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A.\nSchuhl, S. Pizzini, J. Vogel, and P. Gambardella, Nat.\nMater.9, 230 (2010).\n[7] K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y.\nMokrousov, S. Bl¨ ugel, S. Auffret, O. Boulle, G. Gaudin,\nand P. Gambardella, Nat. Nanotechnol. 8, 587 (2013).\n[8] Y.B. Fan, P. Upadhyaya,X.F. Kou, M. R.Lang, S.Takei,\nZ.X. Wang, J. S. Tang, L. He, L.-T. Chang, M. Montaz-7\neri, G.Q. Yu, W. J. Jiang, T. X. Nie, R. N. Schwartz,\nY. Tserkovnyak, and K. L. Wang, Nat. Mater. 13, 699\n(2014).\n[9] P. Gambardella and L. M. Miron, Phil. Trans. R. Soc. A\n369, 3175 (2011).\n[10] A. Brataas and K. M. D. Hals, Nat. Nanotechnol. 9, 86\n(2014).\n[11] Y. Wang and L. J. Sham, Phys. Rev. B 85, 092403\n(2012).\n[12] Y. Wang and L. J. Sham, Phys. Rev. B 87, 174433\n(2013).[13] T. Tay and L. J. Sham, Phys. Rev. B 87, 174407 (2013).\n[14] Y. Wang, W.Q. Chen, and F.C. Zhang, Phys. Rev. B 90,\n205110 (2014).\n[15] K. Blum, Density Matrix Theory and Applications\n(Springer-Verlag, Berlin Heidelberg, 2012).\n[16] A. L. Fetter and J. D. Walecka, Quantum Theory of\nMany-Particle Systems , (McGraw-Hill, New York, 1971)." }, { "title": "1802.05548v1.Damping_s_effect_on_the_magnetodynamics_of_spin_Hall_nano_oscillators.pdf", "content": "Damping's e\u000bect on the magnetodynamics of spin Hall nano-oscillators\nYuli Yin,1, 2,\u0003Philipp D urrenfeld,2Mykola Dvornik,2Martina\nAhlberg,2Afshin Houshang,2Ya Zhai,1and Johan \u0017Akerman2, 3\n1Department of Physics, Southeast University, 211189 Nanjing, China\n2Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden\n3Department of Materials and Nano Physics, School of Information and Communication Technology,\nKTH Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden\n(Dated: July 27, 2021)\nWe study the impact of spin wave damping ( \u000b) on the auto-oscillation properties of nano-\nconstriction based spin Hall nano-oscillators (SHNOs). The SHNOs are based on a 5 nm Pt layer\ninterfaced to a 5 nm Py 100\u0000x\u0000yPtxAgymagnetic layer, where the Pt and Ag contents are co-varied\nto keep the saturation magnetization constant (within 10 %), while \u000bvaries close to a factor of\nthree. We systematically investigate the in\ruence of the Gilbert damping on the magnetodynamics\nof these SHNOs by means of electrical microwave measurements. Under the condition of a constant\n\feld, the threshold current scales with the damping in the magnetic layer. The threshold current as\na function of \feld shows a parabolic-like behavior, which we attribute to the evolution of the spatial\npro\fle of the auto-oscillation mode. The signal linewidth is smaller for the high-damping materials\nin low magnetic \felds, although the lowest observed linewidth was measured for the alloy with least\ndamping.\nPACS numbers: 75.70.-i, 76.50.+g, 75.78.-n\nINTRODUCTION\nSpin Hall nano-oscillators (SHNO) are spintronic de-\nvices in which magnetization oscillations are induced by\npure spin currents [1]. These pure spin currents can be\nexperimentally realized via the spin Hall e\u000bect (SHE)\nin an adjacent heavy metal layer [2{4] or by non-local\nspin injection [5, 6]. SHNOs, which use the SHE in\na heavy metal layer, have been fabricated in a vari-\nety of device layouts, which all utilize the focusing of\ncharge current into a region with a lateral size of tens\nto hundreds of nanometers. This focusing is commonly\ndone via a nano-gap between two highly conductive elec-\ntrodes [3, 7, 8], with a nanoconstriction [9{13], or with\na nanowire [14, 15]. Most recently, nanoconstriction-\nSHNOs have attracted large interest, due to their rel-\native ease of fabrication, their direct optical access to\nthe magnetization oscillation area, and their potential for\nlarge scale and large distance synchronization of multiple\nSHNOs [16, 17].\nNanoconstriction-SHNOs consist of a bilayer of a fer-\nromagnetic free layer and a SHE inducing heavy metal\nlayer. Since the SHE and the concomitant spin accumu-\nlation at the bilayer interface are only in\ruenced by the\ncurrent density in the heavy metal layer, magnetization\noscillations of the device under a constant current can\nbe directly linked to the magnetodynamic properties of\nthe magnetic free layer. Until now, the variety of materi-\nals from which SHNOs has been fabricated is limited to a\nfew standards like permalloy (Py, Ni 80Fe20), (Co,Fe)B, or\nyttrium iron garnet (YIG). However, these materials are\ndi\u000berent from each other in every one of the key magneto-\ndynamic parameters, such as magnetization ( M), Gilbertdamping (\u000b), or exchange constant ( A).\nIn a recent study, we have shown how the magneto-\ndynamic properties of Py can be engineered by alloying\nwith the noble metals Pt, Au, and Ag [18]. While alloy-\ning with Pt leads to a large increase in damping but only\na small decrease in magnetization, alloying with Ag has\nonly a weak e\u000bect on the damping but reduces the mag-\nnetization relatively strongly. Co-alloying with both ele-\nments Pt and Ag thus results in Py 100\u0000x\u0000yPtxAgy\flms,\nwhoseMand\u000bcan be tuned independently, e.g. the\nmagnetization can be kept constant, while the damping\nis strongly increased with increasing Pt concentration.\nHere, we employ a series of alloyed Py 100\u0000x\u0000yPtxAgy\nthin \flms in nanoconstriction-SHNOs, where we vary the\ne\u000bective damping of the free layer by a factor of three,\nwhile we keep the magnetization of the \flms constant.\nBased on these \flms, we fabricate geometrically identical\nnanoconstriction-SHNOs and compare their microwave\nauto-oscillation characteristics. This allows us to directly\nanalyze the in\ruence of one single magnetodynamic prop-\nerty, namely the Gilbert damping, on the spectral char-\nacteristics, i.e. the onset current ( Ith\nDC), the output power\n(P), and the linewidth (\u0001 f).\nSPIN HALL NANO-OSCILLATOR DEVICES\nBilayers of 5 nm Py 100\u0000x\u0000yPtxAgyand 5 nm Pt were\nsputter-deposited onto sapphire substrates in a high-\nvacuum chamber with a base pressure of less than\n3\u000210\u00008Torr. The deposition was carried out with\n3 mTorr argon gas at a \row rate of 30 sccm. The alloyed\nlayers were co-sputtered from up to 3 targets, and the\nPy target power was kept constant at 350 W, while thearXiv:1802.05548v1 [cond-mat.mes-hall] 15 Feb 20182\n150 nm\n200 nm\n(a) (c)\n(b)\nHIPCurrent\nFIG. 1. (a) Schematic representation of the sputtered bilayer\nstructure. (b) SEM micrograph of a nanoconstriction-SHNO\nshowing the relative orientations of current and \feld. (c) Op-\ntical micrograph showing the microwave wave guide used for\ncontacting the SHNOs.\nnoble metal sputtering powers and the sputtering time\nwas adjusted for composition and thickness, respectively.\nThe top Pt layer was magnetron sputtered with a dc\npower of 50 W. The alloy compositions are Py 84Ag16\n(S01), Py 77:5Pt10Ag12:5(S02), Py 75Pt15Ag10(S03), and\nPy73Pt19Ag8(S04), chosen to result in a constant satura-\ntion magnetization throughout the series of SHNOs [18].\nDevices for electrical measurements were fabricated\nfrom these bilayers by electron beam lithography and ar-\ngon ion beam etching, using the negative resist as an etch-\ning mask. Nanoconstrictions were formed by two sym-\nmetrical indentations with a 50 nm tip radius into 4 µm\nwide stripes, see Fig. 1(b). The width of the nanocon-\nstrictions is 150 nm. Finally, 1 µm thick copper waveg-\nuides with a 150 µm pitch were fabricated by optical\nlithography and lift-o\u000b, see Fig. 1(c).\nFILM CHARACTERIZATION\nCharacterization of the extended bilayer samples was\nperformed by ferromagnetic resonance (FMR), and two-\npoint anisotropic magneotresistance (AMR) measure-\nments. The FMR was carried out with in-plane applied\n\felds using a NanOsc Instruments PhaseFMR with a\n200µm wide coplanar waveguide (CPW). An asymmet-\nric Lorentzian was \ft to the absorption peaks. The fre-\nquency dependence of the determined resonance \felds\nand linewidths was subsequently used to extract the ef-\nfective magnetization ( \u00160Me\u000b) and the damping param-\n0.0\n0.2\n0.4\n0.6\n0.8\nμ0M eff\nα\n0.02\n0.03\n0.04\n0.05\n0.06\nα \n0\n5\n10\n15\n20\n0.3\n0.4\n0.5\nAMR (%)\nxPt(%)φ (°)\n0\n90\n180\n270\n360\n0.0\n0.1\n0.2\n0.3\n0.4\nMR (%)\nS01μ0Meff(T) μ0Meff = 0.617 TFIG. 2. (a) Magnetization and damping of the alloyed \flms\nin the bilayer as measured by CPW-based FMR. (b) AMR of\nthe extended layer structure. The inset shows the angular-\ndependent relative resistance of the Py 84Ag16/Pt (S01) bi-\nlayer, together with a \ft to a cos2-function.\neter (\u000b), respectively [18]. Figure 2(a) shows the two\nparameters, \u00160Me\u000band\u000b, as a function of Pt concen-\ntration. The magnetization is constant throughout the\nsample series ( \u00160Me\u000b= 0.617(34) T), while the damp-\ning increases linearly from 0 :023(1) to 0 :058(3) as the\nPt concentration increases from 0 (Py 84Ag16) to 19 %\n(Py73Pt19Ag8). The small layer thickness compared to\nthe \flms in Ref. 18 results in a slightly lower magnetiza-\ntion, whereas the damping is enhanced as a consequence\nof spin pumping into the adjacent Pt layer [19{21].\nThe AMR was determined by probing the resistance\nof 4µm wide stripes in a rotating 90 mT in-plane mag-\nnetic \feld. A representative AMR measurement is pre-\nsented in the inset of Fig. 2(b), together with a \ft of a\ncos2-function to the data. The angle '= 0\u000edenotes a\nperpendicular orientation between current and \feld, and\nthe AMR (Fig. 2(b)) is calculated by the di\u000berence in\nresistance at perpendicular and parallel alignments via\nAMR =Rk\u0000R?\nR?. The AMR is below 1 %, which is a re-\nsult of the majority of the current \rowing through the\nnonmagnetic platinum layer, which has a higher conduc-\ntivity than the Py alloys. The AMR reduces by \u001930 %\nacross the samples series, but the absolute resistance of\nthe bilayers decreases by less than 5 %. The AMR magni-\ntude is therefore most likely governed by the alloy compo-\nsition, since the amount of current in the magnetic layer\ndoes not change signi\fcantly.3\nf(GHz)\nCurrent (mA)\n2.2 2.4 2.6 2.8 3.0 3.2 3.45.96.06.1\n2468S01, H = 500 mT\n5.85 5.95 6.05 6.150246\nf(GHz)PSD (nV2/Hz)\nFIG. 3. Power spectral density (PSD) of the Py 84Ag16/Pt\n(S01) SHNO as a function of current in an external \feld of\n\u00160Hext= 0:5 T, tilted 80\u000eOOP. The inset shows the PSD at\nIDC= 3:26 mA and the solid line is a Lorentzian \ft resulting\nin \u0001f= 5:98 MHz and P= 1:02 pW.\nMICROWAVE EMISSION MEASUREMENTS\nAND DISCUSSION\nThe microwave measurements were conducted with the\ndevices placed in a magnetic \feld oriented at an out-of-\nplane (OOP) angle of 80\u000efrom the \flm plane, and an\nin-plane angle of '= 0\u000e. The in-plane component of the\nmagnetic \feld ( HIP\next) was thus perpendicular to the cur-\nrent \row direction, as sketched in Fig. 1(b). The relative\norientation of the current and HIP\nextyields a spin-torque\ncaused by the spin current from the Pt layer, which re-\nduces the damping in the Py layer and leads to auto-\noscillations for su\u000eciently large positive applied dc cur-\nrents (IDC) [22]. The current was applied to the samples\nvia the dc port of a bias-tee and the resulting microwave\nsignals from the devices were extracted from the rf port of\nthe bias-tee. The microwave signals were then ampli\fed\nby a broadband (0 :1 to 40 GHz) low-noise ampli\fer with\na gain of +32 dB before being recorded by a spectrum an-\nalyzer (Rohde&Schwarz FSV-40) with a resolution band-\nwidth of 500 kHz. All measurements were carried out at\nroom temperature.\nA typical microwave measurement of a Py 84Ag16/Pt\n(S01) device in a constant \feld of \u00160Hext= 0:5 T and\na varying current is displayed in Fig. 3. The peak fre-\nquency \frst decreases slightly after the oscillation onset\natIth\nDC= 2:26 mA, then reaches a minimum at \u00182:6 mA,\nand \fnally increases up to the maximum applied current\nof 3:4 mA. A Lorentzian peak function \fts well to the\nauto-oscillation signal, see inset of Fig. 3, allowing for de-\ntermination of the full-width at half-maximum linewidth\n(\u0001f) and the integrated output power ( P). Besides the\n10\n100\nΔf(MHz)\n5.9\n6.0\n6.1\nS01\nS02\nS03\nS04\nf(GHz)\n2.2\n2.4\n2.6\n2.8\n3.0\n3.2\n3.4\n0.1\n1\nPower (pW)\nI (mA)(a)\n(b)\n(c)FIG. 4. (a) Frequency, (b) linewidth, and (c) integrated power\nof the microwave auto-oscillations as a function of current for\nfour di\u000berent SHNOs with increasing damping. The applied\n\feld is\u00160Hext= 0:5 T, tilted 80\u000eOOP.\nhighly coherent auto-oscillation mode, no other modes\nare excited under these \feld conditions.\nFigure 4 shows the determined auto-oscillation char-\nacteristics of SHNOs with di\u000berent alloy composition\nand damping. The measurements were again made in\na constant \feld of 0 :5 T. The oscillation frequencies in\nFig. 4(a) lie around 6.0 \u00060.1 GHz for all samples, and the\ncurrent-frequency dependence is virtually identical above\nthe individual threshold currents. However, the current\nrange where fdecreases with IDCis missing for the S04\nsample, which suggests that the threshold current is un-\nderestimated for this device. The comparable frequencies\nof all samples con\frm that the saturation magnetization\nis constant throughout the alloy series. Furthermore, the\nquantitatively similar current tunability implies that the\nincreased damping does not change the fundamental na-\nture of the excited auto-oscillation mode.\nThe linewidth of the SHNOs decreases rapidly after the\nauto-oscillation onset and then levels o\u000b for higher IDC\nvalues, as shown in Fig. 4(b). This behavior is consistent\nwith previous studies on nanoconstriction-SHNOs made4\n300\n400\n500\n600\n700\n800\n2.0\n2.4\n2.8\n3.2\nS02\nS01\nS03\nS04\nIth\nDC(mA)\nField (mT)\nFIG. 5. Threshold current ( Ith\nDC) as a function of external\nmagnetic \feld for the four devices of this study.\nof permalloy \flms [11, 17]. The low-damping device S01\nreaches its minimum level at \u0001 f\u001811 MHz, while the\nSHNOs with higher damping materials all have a simi-\nlar minimum linewidth of \u0001 f\u00185 MHz. The linewidth\nis inversely proportional to the mode volume [23], and\nthe decrease in \u0001 fcan therefore be attributed to a spa-\ntial growth of the auto-oscillation region as the damping\nincreases. Nevertheless, the active area of the device is\ncon\fned to the nanoconstriction, which limits the reduc-\ntion in linewidth.\nThe output power of the four nanoconstriction-SHNOs\nis shown as a function of IDCin Fig. 4(c). The power\ngrows almost exponentially with increasing current for\nall samples. However, Pdrops dramatically as the Pt\nconcentration increases. The AMR also decreases in the\nhigher damping samples, but the reduction is too small\nto fully account for the drop in power. Together with the\ntrend in linewidth, the evolution of the power contradicts\nthe general assumption \u0001 f/\u000b=P [23{25]. This equa-\ntion is only valid in the vicinity of the threshold current\nand a direct comparison to the data is problematic, due\nto the experimental di\u000eculties of determining Ith\nDC. Still,\nthe direct relation between the intrinsic oscillator power\nand the electrically measured power is put into question\ndue to the remarkable decrease in the measured P. A\nnumber of factors could in\ruence the signal strength, e.g.\nrecti\fcation, spin-pumping, and the inverse spin-Hall ef-\nfect.\nThe onset current for auto-oscillations was determined\nby current scans for external \felds ranging between 0 :3 T\nand 0:8 T, and the results are shown in Fig. 5. The\n\feld dependence of Ith\nDCis parabola-like for all samples.\nThis kind of behavior has been predicted in a numerical\nstudy by Dvornik et al. [13]. The non-monotonic behav-\nior of threshold current as a function of applied \feld is\na result of a re-localization of the auto-oscillation mode\nand a corresponding change in the spin-transfer-torque\n(STT) e\u000eciency. In weak oblique magnetic \felds, the\nmode is of edge type and samples a signi\fcant portion of\nthe pure spin current, which is largest at the nanocon-striction edges due to the inhomogeneous current den-\nsity. When the \feld strength increases, the mode shows\nan even stronger localization towards the region of the\nhigher current density. Thereby, the STT e\u000eciency in-\ncreases and the threshold current drops. When the \feld\nstrength increases further, the mode detaches from the\nedges and eventually transforms to the bulk type. As\nthis transformation gradually reduces the spatial corre-\nlation between the spin current density and the location\nof the mode, the STT e\u000eciency drops and the threshold\ncurrent increases. The lower \feld tunability of Ith\nDCof\nthe high damping samples imply an initially larger mode\nvolume, which also was suggested by the evolution of the\nlinewidth.\nThe \feld and current range with detectable auto-\noscillations is strongly dependent on \u000b. The threshold\ncurrent should increase linearly with damping [13] and\nthe minimum Ith\nDCindeed scales with \u000b. The enhance-\nment is smaller than predicted (a factor of three), which\nindicates that the increase in damping is accompanied\nwith a higher STT e\u000eciency. A possible reason for the\nimproved e\u000eciency is a larger SHE through a more trans-\nparent interface for alloyed \flms. The origin of the ob-\nserved damping dependence of the threshold \feld is un-\nclear at this stage, calling for a closer inspection of the\nimpact of the applied \feld on the spectral characteristics.\nThus, a further investigation of our devices is targeted\ntowards the microwave emission as a function of \feld\nwith a constant IDC= 3.2 mA, i.e. above or at the pre-\nviously measured auto-oscillation threshold for all \felds.\nWhile the peak frequencies are virtually identical for all\nthe samples, see Fig. 6(a), the varied damping manifests\nin a clear pattern in Pand \u0001f. The microwave power,\nshown in Fig. 6(c), \frstly increases for all samples with\nincreasing \feld, peaks for an intermediate \feld, and \f-\nnally drops relatively sharp until a point where no more\noscillations are detectable. An opposite behavior can be\nseen for \u0001f, which shows a minimum for intermediate\n\felds. The \feld at which the SHNOs emit their maxi-\nmum output power decreases monotonically from 0.64 T\nto 0.4 T with increasing damping. The same trend is\nvisible for the point of minimum linewidth, which de-\ncreases with increased damping from 0.71 T to 0.49 T,\nand is therefore at a typically \u00180.1 T larger \feld than the\nrespective maximum power. The lowest overall linewidth\ncan be achieved for the lowest damping SHNO (S01) at\nhigh \felds, where only this device still shows a detectable\nsignal, i.e., \u0001 f= 1:2 MHz at\u00160Hext= 0:71 T. How-\never, at low applied \felds \u00160Hext\u00140:48 T a clear trend\nis noticeable towards smaller linewidths for the alloyed\npermalloy \flms with larger damping.\nIn light of this inverse trend, we can argue that auto-\noscillations in nanoconstriction-SHNO should also be de-\nscribed in the framework of non-linear auto-oscillators,\nalthough the study in Ref. 13 has shown that oscilla-\ntions in nanoconstriction-SHNOs emerge from a local-5\nΔf(MHz)\n(b)\n(c)\n4\n6\n8\n10\nS01\nS02S03\nS04\n1\n10\n100\n300\n400\n500\n600\n700\n800\n0.1\n1\nPower (pW)\nField (mT)(a)f(GHz)\nFIG. 6. (a) Frequency, (b) linewidth, and (c) integrated power\nof the auto-oscillations as a function of the applied external\nmagnetic \feld at a constant drive current IDC= 3:2 mA.\nized linear mode. The generation linewidth of nanocon-\ntact spin torque oscillators, which are a prime example\nof non-linear auto-oscillators, has been studied analyti-\ncally [23, 26] and experimentally [27]. The linewidth as\na function of current and magnetic \feld angle was shown\nto follow the expression:\n\u0001f=\u00000\n2\u0019\u0012kBT\nE0\u0013\"\n1 +\u0012N\n\u0000e\u000b\u00132#\n; (1)\nwherekB,T, andE0(IDC=Ith\nDC) are the Boltzmann con-\nstant, temperature and the average oscillator energy, re-\nspectively. Nis the nonlinear frequency shift, a material\nproperty that is determined by the internal magnetic \feld\nand the magnetization [28]. \u0000 e\u000bis the e\u000bective nonlinear\ndamping rate and \u0000 0is the positive damping rate, and\nboth have an explicit linear dependence on the Gilbert\ndamping\u000b[23]. Assuming everything else equal amongst\nour devices, a decrease of the linewidth with \u000bcan be\nthus expected, when the second term in the brackets in\nEq. 1 dominates. This is likely for low to intermediate\felds, since Ncan be calculated to take up the largest\nvalues under these conditions [28], which are thus in ac-\ncordance with the range of \felds, where we observe the\ndiscussed linewidth vs. damping behavior in our devices.\nCONCLUSIONS\nIn conclusion, we have fabricated a series of sam-\nples where the magnetization is constant, while the\nspin wave damping is varied by a factor of three. We\nhave shown that the damping of the magnetic layer in\nnanoconstriction-SHNOs has an important in\ruence on\nall the spectral characteristics of the devices. The re-\nsults of our study will encourage the application of tai-\nlored materials in SHNOs and can be used for a further\nunderstanding of the magnetodynamics in nanodevices,\ne.g. the coupling mechanisms in mutually synchronized\nSHNOs.\nACKNOWLEDGMENTS\nWe acknowledge \fnancial support from the China\nScholarship Council (CSC), the G oran Gustafsson\nFoundation, the Swedish Research Council (VR), the\nKnut and Alice Wallenberg Foundation (KAW), and\nthe Swedish Foundation for Strategic Research (SSF).\nThis work was also supported by the European Re-\nsearch Council (ERC) under the European Communitys\nSeventh Framework Programme (FP/2007-2013)/ERC\nGrant 307144 \\MUSTANG\".\n\u0003yuri@seu.edu.cn\n[1] T. Chen, R. K. Dumas, A. Eklund, P. K. Muduli,\nA. Houshang, A. A. Awad, P. D urrenfeld, B. G. Malm,\nA. Rusu, and J. \u0017Akerman, \\Spin-Torque and Spin-Hall\nNano-Oscillators,\" Proc. IEEE 104, 1919 (2016).\n[2] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. 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Moodera\nFrancis Bitter Magnet Laboratory, Massachusetts Institute of Technology,\n150 Albany Street, Cambridge, Massachusetts 02139 USA.\nThe damping of magnetization, represented by the rate at which it relaxes to equilibrium, is\nsuccessfully modeled as a phenomenological extension in the Landau-Lifschitz-Gilbert equation.\nThis is the damping torque term known as Gilbert damping and its direction is given by the vector\nproduct of the magnetization and its time derivative. Here we derive the Gilbert term from \frst\nprinciples by a non-relativistic expansion of the Dirac equation. We \fnd that this term arises when\none calculates the time evolution of the spin observable in the presence of the full spin-orbital\ncoupling terms, while recognizing the relationship between the curl of the electric \feld and the time\nvarying magnetic induction.\nPACS numbers: 76.20.-m, 75.30.-m and 75.45.+j\nThe Gilbert damping torque in magnetic systems de-\nscribes the relaxation of magnetization and it was intro-\nduced into the Laudau-Lifschitz equation [1, 2] for de-\nscribing spin dynamics. Gilbert damping is understood\nto be a non-linear spin relaxation phenomenon and it con-\ntrols the rate at which magnetization spins reach equilib-\nrium. The introduction of this term is phenomenological\nin nature [3] and the question of whether it has an in-\ntrinsic physical origin has not been fully addressed, in\nthe face of rather successful modeling of the relaxation\ndynamics of measured systems. Correlating ferromag-\nnetic resonance spectral line-widths [4, 5] in magnetic\nthin \flms with the change in damping has been success-\nful for con\frming the form of the damping term in the\nunderlying dynamical equations. The intrinsic origin of\nthe damping itself is still an open question. The damping\nconstant,\u000bis often reformulated in terms of a relaxation\ntime, and the dominant relaxation processes are invoked\nto calculate this, but this approach presupposes preces-\nsional damping torque.\nIt has been long thought that intrinsic Gilbert damp-\ning had its origin in spin-orbital coupling because this\nmechanism does not conserve spin, but it has never been\nderived from a coherent framework. Non-local spin re-\nlaxation processes [6] and disorder broadening couple to\nthe spin dynamics and can enhance the Gilbert damp-\ning extrinsically in thin \flms and heterostructures. This\ntype of spin relaxation, which is equivalent to ensemble\ndephasing [7], is modeled as the (S-S 0)/T\u0003\n2decay term\nin the dynamical Bloch equation, where T\u0003\n2is the decay\ntime of the ensemble of spins. Crudely speaking, during\nspin relaxation, some spins lag behind the mean mag-\nnetization vector and the exchange and magnetostatic\n\felds then exert a time dependent torque. Calculations\non relaxation driven damping of this kind presuppose the\nGilbert damping term itself which begs the question.\nThe inhomogeneous damping term can be written as\nM\u0002dr2M=dtwhich gives rise to non-local e\u000bects such\n\u0003Electronic mail : hickey@mit.eduas spin wave dissipation [6, 8]. These non-local theo-\nries are successful in quantifying the enhancement of the\nGilbert damping, but do not derive the intrinsic Gilbert\nterm itself. There are models [9, 10] which deal with\nthe scattering of electron spins from thermal equilibrium\nin the presence of phonon and spin-orbital interactions\nwhich is a dynamic interaction and this allows us to de-\ntermine the strength of the Gilbert damping for itiner-\nant ferromagnetic metals, generalizing the Gilbert damp-\ning response to a tensorial description. Both the s-d\nexchange relaxation models [11, 12] and the Fermi sur-\nface breathing models of Kambersky [9, 13] either pre-\nsuppose a Gilbert damping term in the dynamical equa-\ntion or specify a phenomenological Hamiltonian H = -\n1/(\rMs)^\u000b.dM/dt. While this method is ab initio from\nthe point of view of electronic structure, it already as-\nsumes the Gilbert term ansatz. Hankiewicz et al. [14]\nconstruct the inhomogeneous Gilbert damping by con-\nnecting the spin density-spin current conservation law\nwith the imaginary part of magnetic susceptibility ten-\nsor and show that both electron-electron and impurity\nscattering can enhance the damping through the trans-\nverse spin conductivity for \fnite wavelength excitations\n(q6= 0). In previous work [15], there are derivations\nof the Gilbert constant by comparing the macroscopic\ndamping term with the torque-torque correlations in ho-\nmogeneously magnetized electron gases possessing spin\norbital coupling. For the case of intrinsic, homogeneous\nGilbert damping, it is thought that in the absence of\nspin-orbital scattering, the damping vanishes. We aim to\nfocus on intrinsic, homogeneous damping and its physical\norigin in a \frst-principles framework and the question as\nto whether spin in a homogeneous time-varying magne-\ntization can undergo Gilbert damping is addressed.\nIn this work, we show that Gilbert damping does indeed\narise from spin-orbital coupling, in the sense that it is\ndue to relativistic corrections to the Hamiltonian which\ncouple the spin to the electric \feld and we arrive at the\nGilbert damping term by \frst writing down the Dirac\nequation for electrons in magnetic and electric potentials.\nWe transform the Hamiltonian in such a way as to write\nit in a basis in which the canonical momentum terms arearXiv:0812.3184v2 [cond-mat.other] 1 Apr 20092\neven powers. This is a standard approach in relativistic\nquantum mechanics and we do this in order to calculate\nthe terms which couple the linear momentum to the spin\nin a basis which is diagonal in spin space. This is often\nreferred to as a non-relativistic expansion of the Dirac\nequation. This allows us to formulate the contributions\nas a perturbation to an otherwise non-relativistic parti-\ncle. We then wish to calculate the rate equation for the\nspin observable with all of the spin-orbital corrections in\nmind.\nNow, we start with a purely relativistic particle, a Dirac\nparticle and we write the Dirac-Pauli Hamiltonian, as\nfollows :\nH=c\u000b:(p\u0000eA\nc) +\fm 0c2+e\u001e (1)\n=O+\fm 0c2+\" (2)\nwhere Aand\u001eare the magnetic vector potential and the\nelectrostatic potential, respectively and\n\u000b=\u0012\n0\u001bi\n\u001bi0\u0013\nwhile\n\f=\u0012\n1 0\n0\u00001\u0013\n:\nWe observe immediately that \fO=\u0000O\f.Ois the Dirac\ncanonical momentum , c and e are the speed of light in\na vacuum and the electronic charge, respectively.\nWe now need to rewrite the Hamiltonian in a basis where\nthe odd operators (whose generators are o\u000b diagonal in\nthe Pauli-Dirac basis : \u000bi,\ri,\r5..) and even operators\n(whose generators are diagonal in the Pauli-Dirac basis :\n(1,\f, \u0006,.. ) are decoupled from one another.\nIf we are to \fnd S so that H0does not contain odd powers\nof spin operators, we must chose the operator S, in such\na way as to satisfy the following constraint :\n[S;\f] =\u0000O\nim0c2(3)\nIn order to satisfy cancelation of the odd terms of O\nto \frst order, we require S=\u0000iO\f\n2m0c2and this is known\nas the Foldy-Wouthuysen transformation in relativistic\nquantum mechanics and it is treated in some detail in,\nfor example, reference [16]. We now would like to collect\nall of the terms into the transformed Hamiltonian, and\nthis is written as\nH0=\f\u0012\nm0c2+O2\n2m0c2\u0000O4\n8m3\n0c6\u0013\n+\"\u00001\n8m2\n0c4[O;[O;\"]] +\f\n2m0c2[O;\"]\u0000O3\n3m2\n0c4\nThe expression above contains odd powers of the canon-\nical momentumO, so we rede\fne the canonical momen-\ntum to encapsulate all of these odd power terms. So wenow apply the procedure of eliminating odd powers once\nagain :\nS0=\u0000i\f\n2m0c2O0=\u0000i\f\n2m0c2\u0012\f\n2m0c2[O;\"]\u0000O3\n3m2\n0c4\u0013\n(4)\nH00=eiS0\nH0e\u0000iS0\n=\fm 0c2+\"0+O00; (5)\nwhereO00is now O(1\nm2\n0c4), which can be further elimi-\nnated by applying a third transformation (S00=\u0000i\fO00\n2m0c4),\nwe arrive at the following Hamiltonian :\nH000=eiS00\u0010\nH00\u0011\ne\u0000iS00\n=\fm 0c2+\"0\n=\f\u0012\nm0c2+O2\n2m0c2\u0000O4\n8m3\n0c6\u0013\n+\n\"\u00001\n8m2\n0c4[O;[O;\"]]\nThus we have the fully Foldy-Wouthuysen transformed\nHamiltonian :\nH000=\f\u0012\nm0c2+(p\u0000eA=c)2\n2m0\u0000p4\n8m3\n0c6\u0013\n+e\b\n\u0000e~\n2m0c2\f\u0006:B\u0000ie~2\n8m2\n0c2\u0006:(r\u0002E)\n\u0000e~\n4m2\n0c2\u0006:E\u0002p\u0000e~2\n8m2\n0c2(r:E)\nThe terms which are present in the above Hamiltonian,\nshow us that we have a p4kinetic part which is the rela-\ntivistic expansion of the mass of the particle. The terms\nwhich couple to the spin \u0006 are of importance and we see\nthat these terms correspond to the Zeeman, spin-orbital\n(comprising momentum and electric \feld curl terms) and\nthe Darwin term, respectively. Strictly speaking, the\npresence of the scalar potential \u001ebreaks the gauge invari-\nance in the Pauli-Dirac Hamiltonian and a fully gauge in-\nvariant theory would require that this contain the gauge-\nfree electromagnetic \feld energy. We omit the term\ne2~\n4m2c3\u0006:(A\u0002E) (which establishes gauge invariance in\nthe momentum terms) in this rotated Hamiltonian, as it\nis O(1=m2c3) and we are only interested in calculating\nsemiclassical rate equations for \felds, which are mani-\nfestly gauge-invariant, and not wavefunctions or energy\neigenvalues. We can now de\fne the spin dependent cor-\nrections to a non-relativistic Hamiltonian :\nH\u0006=\u0000e~\n2m0c2\f\u0006:B\u0000e~\n4m2\n0c2\u0006:E\u0002p\u0000ie~2\n8m2\n0c2\u0006:(r\u0002E):\n(6)\nwhere\n\u0006=\u0012\n\u001bi0\n0\u001bi\u0013\n\u0011^Si:3\nand\u001biare the Pauli matrices. Note that the last\ntwo terms in Equation 6 encapsulate the entire spin\norbital coupling in the sense that these terms couple\nthe particle's linear momentum to the spin ^Si. The\n\frst spin-orbital term in the Hamiltonian is well known\nand give rise to momentum dependent magnetic \felds.\nWhen the ensuing dynamics are calculated for this\ncase, it gives rise to spin relaxation terms which are\nlinear in spin [17]. Note that, while neither spin-orbital\nterm is Hermitian, the two terms taken together are\nHermitian and so the particles angular momentum\nis a conserved quantity and the total energy lost in\ngoing from collective spin excitations (spin waves) to\nsingle particles states via spin-orbital coupling is gained\nby the electromagnetic \feld. Recognizing the curl of\nthe electric \feld in the last term, we now rewrite this\nthe time varying magnetic \feld as given by Maxwells\nequations asr\u0002E=\u0000@B\n@t. We now have an explicitly\ntime-dependent perturbation on the non-relativistic\nHamiltonian. We can write the time-varying magnetic\n\feld seen by the spin (in, for example a magnetic\nmaterial) as@B\n@t=@B\n@M\u0001@M\n@t=\u00160(1 +\u001f\u00001\nm)@M\n@t. We now\nhave the spin dependent Hamiltonian :\nHS=\u0000e~\n2m0c2\fS:B\u0000e~\n4m2\n0c2S:E\u0002p\n+ie~2\u00160\n8m2\n0c2S:\u0000\n1 +\u001f\u00001\nm\u0001\n:dM\ndt=\nHS=HS\n0+HS(t):\nWe focus our attention on the explicitly time-dependent\npart of the Hamiltonian HS(t) ;\nHS(t) =ie~2\u00160\n8m2\n0c2S:\u0000\n1 +\u001f\u00001\nm\u0001\n:dM\ndt: (7)\nIn this perturbation scheme, we allow the Hermitian\ncomponents of the Hamiltonian to de\fne the ground sate\nof the system and we treat the explicitly time-dependent\nHamiltonian (containing the spin orbital terms) as a time\ndependent perturbation. In this way, the rate equation is\nestablished from a time dependent perturbation expan-\nsion in the quantum Liouville description. We now de\fne\nthe magnetization observable as ^M=X\n\u000bg\u0016B\nVTr\u001a^S\u000b(t)\nwhere the summation is taken over the site of the magne-\ntization spin \u000b. We now examine the time dependence of\nthis observable by calculating the rate equation according\nto the quantum-Liouville rate equation ;\nd\u001a(t)\ndt+1\ni~[^\u001a;H] = 0 (8)\nThis rate equation governs the time-evolution of the\nmagnetization observable as de\fned above, in the non-\nequilibrium regime. We can write the time derivative ofthe magnetization [18], as follows ;\ndM\ndt=X\nn;\u000bg\u0016b\nVh\tn(t)j1\ni~[\u001aS\u000b;H] +@\u001a\n@tS\u000b+\u001a@S\u000b\n@tj\tn(t)i;\nand we can use the quantum Liouville rate equation as\nde\fned by Equation 8 to simplify this expression and we\narrive at the following rate equation :\ndM\ndt=X\n\u000bg\u0016b\nV1\ni~Trf\u001a[S\u000b;HS(t)]g (9)\nIn the case of the time dependent Hamiltonian derived\nin equation 7, we can assume a \frst order dynamical\nequation of motion given bydM\ndt=\rM\u0002Hand calculate\nthe time evolution for the magnetization observable :\ndM\ndt=X\n\u000b;\fg\u0016B\nV1\ni~Tr\u001a[Si\n\u000b;ie~2\u00160\n8m2c2Sj\n\f]:(1 +\u001f\u00001\nm) !@M\ndt\n=X\n\u000bg\u0016B\nVie~2\u00160\n8m2c21\ni~Tr\u001ai ~\u000fijkSk\n\u000b\u000e\u000b\f(1 +\u001f\u00001\nm)\u000ejl !@ Ml\ndt\n=\u0000ie~\u00160\n8m2c2(1 +\u001f\u00001\nm)M\u0002 !@M\ndt;\nwhere, in the last two steps, we have used the fol-\nlowing commutation relations for magnetization spins :\n[Si\n\u000b;Sj\n\f] =i~\u000fijkSk\n\u000b\u000e\u000b\fwhich implies that the theory pre-\nsented here is that which relates to local dynamics and\nthat the origin of the damping is intrinsic. We now rec-\nognize the last equation as the which describes Gilbert\ndamping, as follows :\ndM\ndt=\u0000\u000b\nMs:M\u0002 !@M\n@t(10)\nwhereby the constant \u000bis de\fned as follows :\n\u000b=ie~\u00160Ms\n8m2\n0c2\u0000\n1 +\u001f\u00001\nm\u0001\n(11)\nThe\u000bde\fned above corresponds with the Gilbert\ndamping found in the phenomenological term in the\nLandau-Lifschitz-Gilbert equation and \u001fmis the mag-\nnetic susceptibility. In general, the inverse of the suscep-\ntibility can be written in the form [19],\n\u001f\u00001\nij(q;!) = ~\u001f\u00001\n?(q;!)\u0000!ex\n\r\u00160M0\u000eij; (12)\nwhere the equilibrium magnetization points along the z-\naxis and!exis the excitation frequency associated with\nthe internal exchange \feld. The \u000eijterm in the in-\nverse susceptibility does not contribute to damping mech-\nanisms as it corresponds to the equilibrium response.4\nIn the basis (M x\u0006iMy,Mz), we have the dimensionless\ntransverse magnetic susceptibility, as follows :\n~\u001fm?(q;!) =\r\u00160M0\u0000i\r\u001b?q2\n!0\u0000!\u0000i\r\u001b?q2!0=M0\nThe \frst term in the dimensionless Gilbert coe\u000ecient\n(Equation 11) is small ( \u001810\u000011) and the higher damp-\ning rate is controlled by the the inverse of the suscep-\ntibility tensor. For uniformly saturated magnetization,\nthe damping is critical and so the system is already at\nequilibrium as far as the Gilbert mechanism is concerned\n(dM/dt = 0 in this scenario). The expression for the\ndimensionless damping constant \u000bin the dc limit ( !=0\n) is :\n\u000b=e~\u00160Ms\n8m2\n0c2Im0\n@!0\n\r\u00160M0\u0000i\u001b?q2!0\n\u00160M2\n0\n1\u0000i\r\u001b?q2=M01\nA; (13)\nand we have the transverse spin conductivity from the\nfollowing relation (in units whereby ~=1) :\n\u001b?=n\n4m\u0003!2\n0\u00121\n\u001cdis\n?+1\n\u001cee\n?\u0013\n;\nwhere\u001cdis\n?and\u001cee\n?are the impurity disorder and electron\nelectron-electron scattering times as de\fned and param-\neterized in Reference [14]. We calculate the extrinsically\nenhanced Gilbert damping using the following set of pa-\nrameters as de\fned in the same reference ; number den-\nsity of the electron gas, n=1.4 \u00021027m\u00003, polarization p,\nequilibrium magnetization M 0=\rpn/2, equilibrium ex-\ncitation frequency !0=EF[(1 +p)2=3\u0000(1\u0000p)2=3] and\nwave-number de\fned as q = 0.1 k F, where E Fand kF\nare the Fermi energy and Fermi wave number, respec-\ntively. m\u0003is taken to be the electronic mass. Using these\nquantities, we evaluate \u000bvalues and these are plotted as\na function of both polarization and disorder scattering\nrate in Figure 1.\nIn general, the inverse susceptibility \u001f\u00001\nmwill deter-\nmine the strength of the damping in real inhomogeneous\nmagnetic systems where spin relaxation takes place, sub-\nbands are populated by spin orbit scattering and spin\nwaves and spin currents are emitted. The susceptibil-\nity term gives the Gilbert damping a tensorial quality,\nagreeing with the analysis in Reference [10]. Further, the\nconnection between the magnetization dynamics and the\nelectric \feld curl provides the mechanism for the energy\nloss to the electromagnetic \feld. The generation of radi-\nation is caused by the rotational spin motion analog of\nelectric charge acceleration and the radiation spin inter-\naction term has the form :\nHS(t) =ie~2\u00160\n8m2\n0c2X\n\u000b\u0000\n1 +\u001f\u00001\nm\u0001\nS\u000b:dM\ndt: (14)In conclusion, we have shown that the Gilbert term,\nheretofore phenomenologically used to describe damping\nFIG. 1: (Color Online) Plot of the dimensionless Gilbert\ndamping constant \u000bin the dc limit ( !=0), as a function of\nelectron spin polarization and disorder scattering rate.\nin magnetization dynamics, is derivable from \frst prin-\nciples and its origin lies in spin-orbital coupling. By a\nnon-relativistic expansion of the Dirac equation, we show\nthat there is a term which contains the curl of the elec-\ntric \feld. By connecting this term with Maxwells equa-\ntion to give the total time-varying magnetic induction,\nwe have found that this damping term can be deduced\nfrom the rate equation for the spin observable, giving the\ncorrect vector product form and sign of Gilberts' origi-\nnal phenomenological model. Crucially, the connection\nof the time-varying magnetic induction and the curl of\nthe electric \feld via the Maxwell relation shows that\nthe damping of magnetization dynamics is commensu-\nrate with the emission of electromagnetic radiation and\nthe radiation-spin interaction is speci\fed from \frst prin-\nciples arguments.\nAcknowledgments\nM. C. Hickey is grateful to the Trinity and the uni-\nformity of nature. We thank the U.S.-U.K. Fulbright\nCommission for \fnancial support. The work was sup-\nported by the ONR (grant no. N00014-09-1-0177), the\nNSF (grant no. DMR 0504158) and the KIST-MIT pro-\ngram. The authors thank David Cory, Marius Costache\nand Carlos Egues for helpful discussions.5\n[1] L. Landau and E. Lifshitz, Phys. Z. Sowiet. Un. 8, 153\n(1935).\n[2] E. M. Lifschitz and L. P. Pitaevskii, Statistical Physics\nPart 2 (Pergamon Press, Oxford, United Kingdom,\n1980).\n[3] T. Gilbert, Magnetics, IEEE Transactions on 40, 3443\n(2004).\n[4] C. E. Patton, C. H. Wilts, and F. B. Humphrey, J. Appl.\nPhys. 38, 1358 (1967).\n[5] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando,\nA. Sakuma, and T. Miyazaki, Japanese Journal of Ap-\nplied Physics 45, 3889 (2006).\n[6] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n[7] Y. Tserkovnyak, G. A. Fiete, and B. I. Halperin, Appl.\nPhys. Lett. 84, 5234 (2004).\n[8] G. Eilers, M. L uttich, and M. M unzenberg, Phys. Rev.\nB74, 054411 (2006).\n[9] V. Kambersk\u0012 y, Canadian Journal of Physics 48, 2906\n(1970).[10] D. Steiauf and M. F ahnle, Phys. Rev. B 72064450\n(2005).\n[11] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[12] M. F ahnle, R. Singer, D. Steiauf, and V. P. Antropov,\nPhys. Rev. B 73, 172408 (2006).\n[13] J. Kune\u0014 s and V. Kambersk\u0013 y, Phys. Rev. B 65, 212411\n(2002).\n[14] E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak,\nPhys. Rev. B 78, 020404 (2008).\n[15] E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak,\nPhys. Rev. B 75, 174434 (2007).\n[16] W. Greiner, Relativistic Quantum Mechanics (Springer-\nVerlag, Berlin, Germany, 1987).\n[17] H.-A. Engel, E. I. Rashba, and B. I. Halperin, Phys. Rev.\nLett. 98, 036602 (2007).\n[18] J. Ho, F. C. Khanna, and B. C. Choi, Phys. Rev. Lett.\n92, 097601 (2004).\n[19] Z. Qian and G. Vignale, Phys. Rev. Lett. 88, 056404\n(2002)." }, { "title": "2208.12012v1.Polynomial_energy_decay_rate_of_a_2D_Piezoelectric_beam_with_magnetic_effect_on_a_rectangular_domain_without_geometric_conditions.pdf", "content": "arXiv:2208.12012v1 [math.AP] 25 Aug 2022POLYNOMIAL ENERGY DECAY RATE OF A 2D PIEZOELECTRIC BEAM WITH\nMAGNETIC EFFECT ON A RECTANGULAR DOMAIN WITHOUT GEOMETRIC\nCONDITIONS\nMOHAMMAD AKIL1AND VIRGINIE R ´EGNIER1\n1UNIV. POLYTECHNIQUE HAUTS-DE-FRANCE, INSA HAUTS-DE-FRAN CE, CERAMATHS-LABORATOIRE DE\nMAT´ERIAUX C ´ERAMIQUES ET DE MATH ´EMATIQUES, F-59313 VALENCIENNES, FRANCE\nEMAIL: MOHAMMAD.AKIL@UPHF.FR, VIRGINIE.REGNIER@UPHF.F R\nAbstract. In this paper, we investigate the stability of coupled equat ions modelling a 2D piezoelectric beam\nwith magnetic effect with only one local viscous damping on a r ectangular domain without geometric condi-\ntions. We prove that the energy of the system decays polynomi ally with the rate t−1.\nKeywords. coupled wave equations; viscous damping; C0-semigroup; polynomial stability; rectangular do-\nmains.\nContents\n1. Introduction 1\n2. Stability of a 2D piezoelectric beam on a rectangular domain 4\n2.1. Well-Posedness 4\n2.2. Polynomial Stability 5\n3. Conclusion and open problems 17\nReferences 18\n1.Introduction\nIt is known, since the 19th century, that materials such as quartz , Rochelle salt and barium titanate under\npressure produce electric charge/voltage: this phenomenon is ca lled the direct piezoelectric effect and was\ndiscovered by brothers Pierre and Jacques Curie in 1880. These sa me materials, when subjected to an electric\nfield, produce proportional geometric tension. Such a phenomeno n is known as the converse piezoelectric effect\nand was discovered by Gabriel Lippmann in 1881.\nMorris and Ozer proposed a piezoelectric beam model with a magnetic effect, based on the Euler-Bernoulli\nand Rayleigh beam theory for small displacement (the same equation s for the model are obtained if Midlin-\nTimoshenko small displacement assumptions are used). They consid ered an elastic beam covered by a piezo-\nelectric material on its upper and lower surfaces, isolated by the ed ges and connected to an external electrical\ncircuit to feed charge to the electrodes. As the voltage is prescrib ed at the electrodes, the following Lagrangian\nis considered\n(1.1) L=/integraldisplayT\n0[K−(P+E)+B+W]dt,\nwhereK,P+E,BandWrepresent the (mechanical) kinetic energy, total stored energy , magnetic energy\n(electrical kinetic) of the beam and the work done by external for ces, respectively, for a beam with length\nLand thickness hand considering v=v(x,t),w=w(x,t) andp=p(x,t) as functions that represent the\n1longitudinal displacement of the center line, transverse displaceme nt of the beam and the total load of the\nelectric displacement along the transverse direction at each point x, respectively. So, one can assume that\n(1.2)P+E=h\n2/integraldisplayL\n0/bracketleftbigg\nα/parenleftbigg\nv2\nx+h2\n12w2\nxx−2γβvxpx+βp2\nx/parenrightbigg/bracketrightbigg\ndx,B=µh\n2/integraldisplayL\n0p2\ntdx,\nK=ρh\n2/integraldisplayL\n0/bracketleftbigg\nv2\nt+/parenleftbiggh2\n12+1/parenrightbigg\nω2\nt/bracketrightbigg\ndx, W=−/integraldisplayL\n0pxV(t)dx,\nwhereV(t) is the voltage applied at the electrode. From Hamilton’s principle for a dmissible displacement\nvariations {v,w,p}ofLand observing that the only external force acting on the beam is th e voltage at the\nelectrodes (the bending equation is decoupled, see [ 11,12]), they got the system\n(1.3)ρvtt−αvxx+γβpxx= 0,\nµptt−βpxx+γβvxx= 0,\nwhereρ,α,γ,µ andβdenote the mass density, elastic stiffness, piezoelectric coefficient , magnetic permeability,\nwater resistance coefficient of the beam and the prescribed voltag e on electrodes of beam, respectively, and in\naddition, the relationship\n(1.4) α=α1+γ2β.\nThey assumed that the beam is fixed at x= 0 and free at x=L, and thus they got (from modelling) the\nfollowing boundary conditions\n(1.5)v(0,t) =αvx(L,t)−γβpx(L,t) = 0,\np(0,t) =βpx(L,t)−γβvx(L,t) =−V(t)\nh.\nThen, the authors considered V(t) =kpt(L,t) (electrical feedback controller) in ( 1.5) and established strong\nstabilization for almost all system parametersand exponential sta bility for system parametersin a null measure\nset. In [13] Ramos et al. inserted a dissipative term δvtin the first equation of ( 1.3) , where α >0 is a constant\nand considered the following boundary conditions\n(1.6)v(0,t) =αvx(L,t)−γβpx(L,t) = 0,\np(0,t) =βpx(L,t)−γβvx(L,t) = 0.\nThe authors showed, by using energy method, that the system’s e nergy decays exponentially. This means that\nthe friction term and the magnetic effect work together in order to uniformly stabilize the system. In [ 15], the\nauthors considered a one-dimensional piezoelectric beam with magn etic effect damped with a weakly nonlinear\nfeedback in the presence of a nonlinear delay term.They established an energy decay rate under appropriate\nassumptions on the weight of the delay. In [ 5], the authors studied the stability of piezoelectric beams with\nmagnetic effects of fractional derivative type and with/ without th ermal effects of Fourier’s law. They obtained\nan exponential stability by taking two boundary fractional damping s and additional thermal effects. In [ 18],\nthe authors studied the longtime behavior of a kind of fully magnetic e ffected nonlinear multi-dimensional\npiezoelectric beams with viscoelastic infinite memory. An exponential decay of the solution to the nonlinear\ncoupled PDE’s system is established by the energy estimation method under certain conditions. In [ 2], the\nauthor investigates the stabilization of a system of piezoelectric be ams under (Coleman or Pipkin)-Gurtin\nthermal law with magnetic effect. First, he study the Piezoelectric- Coleman-Gurtin system and he obtain\nan exponential stability result. Next, he consider the Piezoelectric -Gurtin-Pipkin system and he establish a\npolynomial energy decay rate of type t−1. In [1], the authors considered a one-dimensional dissipative system\nof piezoelectric beams with magnetic effect and localized viscous damp ing. They proved that the system is\nexponentially stable using a damping mechanism acting only on one comp onent and on a small part of the\nbeam.\nWhat happens if we go to dimension 2? For this aim, let Ω = /square= (0,1)2be the unit square and ∂Ω = Γ 0∪Γ1\n(See Figure 1), such that\nΓ0,1= ({0}×(0,1))∪((0,1)×{0}) and Γ 1,1= ({1}×(0,1))∪((0,1)×{1}).\nIn the present work, we consider the fully dynamic magnetic effects in a model for a piezoelectric beam with\nonly one viscous damping on a rectangular domain, whose dynamic beh aviour is described by an elasticity\n2equation and a charge equation coupled via the piezoelectric consta nts, which is given as follows\n(1.7)/braceleftiggρvtt(X,t)−α∆v(X,t)+γβ∆p(X,t)+d(·)vt(X,t) = 0, X= (x,y)∈Ω, t >0,\nµptt(X,t)−β∆p(X,t)+γβ∆v(X,t) = 0, X= (x,y)∈Ω, t >0,\nwherev(X,t) andp(X,t) respectively denote the transverse displacement of the beam an d the total load\nof the electric displacement along the transverse direction at each pointx∈Ω. The constant coefficients\nρ,α,γ,µ,β > 0 are the mass density per unit volume, elastic stiffness, piezoelectr ic coefficient, magnetic\npermeability, impermeabilitycoefficientofthebeam, respectivelyand satisfyα > γ2β. System( 1.7)issubjected\nto the following boundary conditions\n\n\nv(X,t) =p(X,t) = 0, X= (x,y)∈Γ0, t >0,\nα∂v\n∂ν(X,t)−γβ∂p\n∂ν(X,t) =β∂p\n∂ν(X,t)−γβ∂v\n∂ν(X,t) = 0, X= (x,y)∈Γ1, t >0,\nUsing the fact that α=α1+γ2β(see (1.4)), then the above boundary conditions can be replaced by\n(1.8)\n\nv(X,t) =p(X,t) = 0, X= (x,y)∈Γ0, t >0,\n∂v\n∂ν(X,t) =∂p\n∂ν(X,t) = 0, X= (x,y)∈Γ1, t >0.\nSystem ( 1.7) is considered with the following initial data\n(1.9) v(X,0) =v0(X), vt(X,0) =v1(X), p(X,0) =p0(X), pt(X,0) =p1(X), X= (x,y)∈Ω.\nLetd∈L∞(0,1), depending only on xsuch that\n(1.10) d(x)≥d0>0 in (a,b) and d(x) = 0 in (0 ,1)\\(a,b).\nWe setωd= (suppd)◦×(0,1) (See Figure 1).\nΓ0,1 Γ1,1\n•\na•\nb•\n1•\n0•1\nωd\nFigure 1. Model Describing Ω.\nThere exist a few results concerning wave equations or coupled wav e equations on a rectangular or cylindrical\ndomain with different kinds of damping [ 10,16,6,17,9,4,3]. But to the best of our knowledge, it seems that\nno result in the literature exists concerning the case of a 2D or multid imensional Piezoelectric beam with local\ndamping.\nThis paper is organized as follows: the second section is devoted to t he case of a rectangular domain. The\n3well-posedness of the system is proved by using semigroup approac h. Next, by combining an orthonormal basis\ndecomposition with frequency-multiplier techniques using appropria te cut-off functions, we prove a polynomial\nenergy decay rate of type t−1.\n2.Stability of a 2D piezoelectric beam on a rectangular domain\nIn this section, we study the stability result of a 2 Dpiezoelectric beam with magnetic effect on a rectangular\ndomain.\n2.1.Well-Posedness. The energy of System ( 1.7)−(1.9), is given by\nE1(t) =1\n2/integraldisplay\nΩ1(ρ|vt|2+α1|∇v|2)dX+1\n2/integraldisplay\nΩ1/parenleftbig\nµ|pt|2+β|γ∇v−∇p|2/parenrightbig\ndX.\nLemma 2.1. LetU= (v,vt,p,pt)be a regular solution of system (1.7)-(1.9). Then, the energy E1(t)satisfies\nthe following estimation\n(2.1)d\ndtE1(t) =−/integraldisplay\nΩ1d(x)|vt(X,t)|2dX.\nProof.Multiplying the first and the second equation of ( 1.7) byutetytrespectively, integrating by parts over\nΩ1, we get\n(2.2)1\n2d\ndt/parenleftbigg\nρ/integraldisplay\nΩ1|vt|2dX+α/integraldisplay\nΩ1|∇v|2dX/parenrightbigg\n−γβℜ/parenleftbigg/integraldisplay\nΩ1∇p·∇vtdX/parenrightbigg\n+/integraldisplay\nΩ1d(x)|vt(X,t)|2dX= 0\nand\n(2.3)1\n2d\ndt/parenleftbigg\nµ/integraldisplay\nΩ1|yt|2dX+β/integraldisplay\nΩ1|∇y|2dX/parenrightbigg\n−γβℜ/parenleftbigg/integraldisplay\nΩ1∇u·∇ytdX/parenrightbigg\n= 0.\nAdding ( 2.2) and (2.3), and using the fact that α=α1+γ2β, we get the desired equation ( 2.1). The proof has\nbeen completed. /square\nFrom (2.1), it follows that System ( 1.7)-(1.9) is dissipative. Now, let us define the energy space Hby\n(2.4) H/square=/parenleftbig\nW1×L2(Ω1)/parenrightbig2,\nwhereW1={f∈H1(Ω1);f= 0 on Γ 0,1}.H/squareis a Hilbert space, equipped with the inner product defined by\n(2.5) /a\\}b∇acketle{tU,U1/a\\}b∇acket∇i}htH/square=/integraldisplay\nΩ1(α1∇v·∇v1+ρzz1+β(γ∇v−∇p)·(γ∇v1−∇p1)+µqq1)dX,\nfor allU= (v,z,p,q)⊤andU1= (v1,z1,p1,q1)⊤inH/square. The expression /ba∇dbl·/ba∇dblH/squarewill denote the corresponding\nnorm. We define the unbounded linear operator A/square:D(A/square)⊂ H/square→ H/squareby\nD(A/square) :=/braceleftbigg\nU:= (v,z,p,q)∈ H/square;z,q∈W1,∆v,∆z∈L2(Ω1) and∂v\n∂ν=∂p\n∂ν= 0 on Γ 1,1/bracerightbigg\n.\nand\n(2.6) A/square(v,z,p,q) =/parenleftbigg\nz,1\nρ(α∆v−γβ∆p−dp),q,1\nµ(β∆p−γβ∆v)/parenrightbigg\n.\nIfU= (v,vt,p,pt)⊤is the state of System ( 1.7)−(1.9), then this system is transformed into the first order\nevolution equation on the Hilbert space H/squaregiven by\n(2.7) Ut=A/squareU, U(0) =U0,\nwhereU0= (v0,v1,p0,p1)⊤. It is easy to see that for all U= (v,z,p,q)∈D(A/square), we have\n(2.8) ℜ/parenleftig\n/a\\}b∇acketle{tA/squareU,U/a\\}b∇acket∇i}htH/square/parenrightig\n=−/integraldisplay\nΩ1d(x)|z(X)|2dX≤0,\nwhich implies that A/squareis dissipative. Now, let F= (f1,f2,f3,f4)∈ H, using the Lax-Milgram Theorem, one\nproves the existence of U∈D(A/square), solution of the equation\n−A/squareU=F.\n4Then, the unbounded linear operator A/squareism−dissipativein the energy space H/squareand consequently 0 ∈ρ(A/square).\nThus,A/squaregenerates a C0−semigroup of contractions/parenleftbig\netA/square/parenrightbig\nt≥0following the Lumer-Phillips theorem. The\nsolution of the Cauchy problem ( 2.7) admits the following representation\nU(t) =etA/squareU0, t≥0,\nwhich leads to the well-posedness of ( 2.7). Hence, we have the following result.\nTheorem 2.2. LetU0∈ H/square, then System (2.7)admits a unique weak solution Usatisfying\nU∈C0(R+,H/square).\nMoreover, if U0∈D(A/square), then Problem (2.7)admits a unique strong solution Usatisfying\nU∈C1(R+,H/square)∩C0(R+,D(A/square)).\n2.2.Polynomial Stability. This subsection is devoted to showing the polynomial stability of Syst em (1.7)-\n(1.9). Our main result in this subsection is the following theorem.\nTheorem 2.3. There exists a constant C >0independent of U0, such that the energy of System (1.7)-(1.9)\nsatisfies the following estimation\n(2.9) E(t)≤C\nt/ba∇dblU0/ba∇dbl2\nD(A/square),∀t >0,∀U0∈D(A/square).\nTo prove this theorem, let us first introduce the following sufficient a nd necessary condition on the polynomial\nstability of a semigroup proposed by Borichev-Tomilov in [ 8] (see also [ 7], [10], and the recent paper [ 14]).\nTheorem 2.4. Assume that Ais the generator of a strongly continuous semigroup of contractio ns/parenleftbig\netA/parenrightbig\nt≥0\non a Hilbert space H. If\n(2.10) iR⊂ρ(A),\nthen for a fixed ℓ >0 the following conditions are equivalent\n(2.11) limsup\nλ∈R,|λ|→∞1\n|λ|ℓ/ba∇dbl(iλI−A)−1/ba∇dblL(H)<∞.\n(2.12) /ba∇dbletAX0/ba∇dbl2\nH≤C\nt2\nℓ/ba∇dblX0/ba∇dbl2\nD(A), X0∈D(A),for some C >0.\nAccording to Theorem 2.4, to prove Theorem 2.3, we need to prove that ( 2.10) and (2.11) hold, where ℓ= 2.\nFor the technique, we use the orthonormal basis decomposition. T o this aim, let ej(y) =√\n2sin(ξjy), where\nξj=(2j+1)π\n2,j∈N∗andy∈(0,1). We may expand vinto a series of the form\n(2.13) v(X) =∞/summationdisplay\nj=1vj(x)ej(y),(x,y)∈Ω1.\nSimilarly, z,pandqcan be expanded into a series of the same form as that in ( 2.13) with, respectively, the\ncoefficients vj(x),zj(x),pj(x) andqj(x). The energy Hilbert space ( 2.4) is given by\n(2.14) H/square= (/hatwiderW1×L2(0,1))2\nwhere\n/hatwiderW1={f∈H(0,1);f(0) = 0}\nequipped with the following norm\n/ba∇dblU/ba∇dbl2\n/hatwidestH/square=∞/summationdisplay\nj=1/parenleftbig\nα1(/ba∇dblv′\nj/ba∇dbl2+ξ2\nj/ba∇dblvj/ba∇dbl2)+ρ/ba∇dblzj/ba∇dbl2+β/parenleftbig\n/ba∇dblγv′\nj−p′\nj/ba∇dbl2+ξ2\nj/ba∇dblγvj−pj/ba∇dbl2/parenrightbig\n+µ/ba∇dblqj/ba∇dbl2/parenrightbig\n,\nwhere/ba∇dbl·/ba∇dbl:=/ba∇dbl·/ba∇dblL2(0,1)and (for later) /ba∇dbl·/ba∇dbl∞:=/ba∇dbl·/ba∇dblL∞(0,1). This gives rise to the functions\n(2.15) ( vj,zj,pj,qj)∈/parenleftig/parenleftig\nH2(0,1)∩/hatwiderW1/parenrightig\n×/hatwiderW1/parenrightig2\nandv′\nj(1) =p′\nj(1) = 0,\n5where ”′” represents the derivative with respect to x. The operator A/squaredefined in ( 2.6) can be written as\n(2.16) A/square\nvj\nzj\npj\nqj\n=\nzj\n1\nρ/parenleftig\nα(v′′\nj−ξ2\njvj)−γβ(p′′\nj−ξ2\njpj)−dzj/parenrightig\nqj\n1\nµ/parenleftig\nβ(p′′\nj−ξ2\njpj)−γβ(v′′\nj−ξ2\njvj)/parenrightig\n.\n.\nProposition 2.5. iR⊂ρ(A/square).\nProof. To prove iR⊂ρ(A/square) it is sufficient to prove that σ(A/square)∩iR=∅. Since the resolvent of A/squareis\ncompact in H/squarethenσ(A/square) =σp(A/square). In the previous section, we already proved that 0 ∈ρ(A/square). It remains\nto show that σ(A/square)∩iR∗=∅. For this aim, suppose by contradiction that there exists a real nu mberλ/\\e}atio\\slash= 0\nandU= (v,z,p,q)⊤∈D(A/square)\\{0}such that\n(2.17) A/squareU=iλU.\nUsing the orthonormal basis decomposition, ( 2.16) and detailing ( 2.17), we get the following system\nzj=iλvj, (2.18)\niλρzj−α(v′′\nj−ξ2\njvj)+γβ(p′′\nj−ξ2\njpj)+dzj= 0, (2.19)\nqj=iλpj, (2.20)\niλµqj−β(p′′\nj−ξ2\njpj)+γβ(v′′\nj−ξ2\njvj) = 0. (2.21)\nFrom (2.8) and (2.17), we have\n(2.22) 0 = ℜ/parenleftbig\niλ/ba∇dblU/ba∇dblH/square/parenrightbig\n=ℜ/parenleftig\n/a\\}b∇acketle{tA/squareU,U/a\\}b∇acket∇i}htH/square/parenrightig\n=−/integraldisplay\nΩ1d(x)|z(X)|2dX=−/integraldisplay1\n0d(x)|zj(x)|2dx≤0.\nThus, from ( 2.18), (2.22) and the fact that λ/\\e}atio\\slash= 0, we have\n(2.23) d(x)zj(x) = 0 in (0 ,1) and consequently zj=vj= 0 in (a,b).\nUsing the fact that α=α1+γ2βand (2.23) in (2.19), we get\n(2.24) iλρzj−α1(v′′\nj−ξ2\njvj)−γ/parenleftig\nγβ(v′′\nj−ξ2\njvj)−β(p′′\nj−ξ2\njpj)/parenrightig\n= 0,in (0,1).\nCombining ( 2.24) and (2.21), we get\n(2.25) iλ(ρzj+γµqj)−α1/parenleftig\nv′′\nj−ξ2\njvj/parenrightig\n= 0,in (0,1).\nUsing (2.23) in (2.25) and the fact that λ/\\e}atio\\slash= 0, then using ( 2.20), we get\n(2.26) qj=pj= 0 in ( a,b).\nSincevj,pj∈H2(a,b)⊂C1([a,b]), we get\n(2.27) vj(ξ) =v′\nj(ξ) =pj(ξ) =p′\nj(ξ) = 0 where ξ∈ {a,b}.\nInserting ( 2.18) and (2.20) in (2.25), then combining with ( 2.21), we get the following system\nv′′\nj−ξ2\njvj=−λ2\nα1(ρvj+γµpj),in (0,1) (2.28)\np′′\nj−ξ2\njpj=−λ2\nα1/parenleftbigg\nγρvj+µα\nβpj/parenrightbigg\n,in (0,1). (2.29)\nLet/tildewideU= (vj,v′\nj,pj,p′\nj)⊤. From ( 2.27), we get/tildewideU(b) = 0. Now, system ( 2.28)-(2.29) can be written in ( b,L) as\nthe following\n(2.30) /tildewideUx=B/tildewideUin (b,L),\n6where\nB=\n0 1 0 0\nα1ξ2\nj−ρλ2\nα10−γµλ2\nα10\n0 0 0 1\n−λ2ργ\nα10α1βξ2\nj−λ2µα\nα1β0\n.\nThe solution of the differential equation ( 2.30) is given by\n(2.31) /tildewideU(x) =eB(x−b)/tildewideU(b) = 0 in ( b,L).\nIn the same way, we prove that /tildewideU= 0 in (0 ,a). Consequently, we get vj=pj= 0 in (0 ,L) therefore U= 0 in\n(0,L). The proof is thus complete. /square\nAs condition ( 2.10) is already proved in Proposition 2.5, we only need to prove condition ( 2.11). Here, we use\na contradiction argument. Namely, suppose that ( 2.11) is false, then there exists\n/braceleftig\n(λn,U(n):= (v(n),z(n),p(n),q(n)))/bracerightig\nn≥1⊂R∗\n+×D(A/square),\nwith\n(2.32) λn→ ∞asn→ ∞and/ba∇dblU(n)/ba∇dblH/square=/ba∇dbl(v(n),z(n),p(n),q(n))/ba∇dblH/square= 1,∀n∈N,\nsuch that\n(2.33) λℓ\nn(iλn−A)U(n)=F(n):= (f1,(n),f2,(n),f3,(n),f4,(n))→0 inH/square,asn→ ∞.\nFor simplicity, we drop the index n. Detailing equation ( 2.33), we get\n(2.34)\n\niλv−z=λ−2f1,\niλρz−α∆v+γβ∆p+dz=λ−2ρf2,\niλp−q=λ−2f3,\niλµq−β∆p+γβ∆v=λ−2µf4.\nUsing the orthonormal basis decomposition, system ( 2.34) turns into the system of one-dimensional equations\niλvj−zj=f1\nj\nλ2, (2.35)\niλρzj−α(v′′\nj−ξ2\njvj)+γβ(p′′\nj−ξ2\njpj)+dzj=ρf2\nj\nλ2, (2.36)\niλpj−qj=f3\nj\nλ2, (2.37)\niλµqj−β(p′′\nj−ξ2\njpj)+γβ(v′′\nj−ξ2\njvj) =µf4\nj\nλ2,. (2.38)\nInserting ( 2.35) and (2.37) in (2.36) and (2.38), we get the following system\n/parenleftbig\nλ2ρ−αξ2\nj/parenrightbig\nvj+αv′′\nj−γβ/parenleftig\np′′\nj−ξ2\njpj/parenrightig\n−iλdvj=F1\nj, (2.39)\n/parenleftbig\nλ2µ−βξ2\nj/parenrightbig\npj+βp′′\nj−γβ/parenleftig\nv′′\nj−ξ2\njvj/parenrightig\n=F2\nj, (2.40)\nwhere\n(2.41) F1\nj:=−/parenleftigg\nρf2\nj\nλ2+iρf1\nj\nλ+df1\nj\nλ2/parenrightigg\nandF2\nj:=−/parenleftigg\nµf4\nj\nλ2+iµf3\nj\nλ/parenrightigg\n.\nUsing the fact that α=α1+γ2βin (2.39), we get\n(2.42)/parenleftbig\nλ2ρ−α1ξ2\nj/parenrightbig\nvj+α1v′′\nj+γ/parenleftig\nγβ(v′′\nj−ξ2\njvj)−β(p′′\nj−ξ2\njpj)/parenrightig\n−iλdvj=F1\nj.\n7Now, combining ( 2.40) and (2.42), we get\n(2.43) α1/parenleftig\nv′′\nj−ξ2\njvj/parenrightig\n=−λ2ρvj−γλ2µpj+iλdvj+F1\nj+γF2\nj.\nInserting ( 2.43) in (2.40), we get the following system\n/parenleftbig\nλ2ρ−α1ξ2\nj/parenrightbig\nvj+α1v′′\nj+γµλ2pj−iλdvj=F3\nj, (2.44)\n/parenleftbig\nλ2µα−α1βξ2\nj/parenrightbig\npj+α1βp′′\nj+ργβλ2vj−iλγβdv j=F4\nj, (2.45)\nvj(0) =pj(0) =v′\nj(1) =p′\nj(1) = 0 , (2.46)\nwhere\n(2.47) F3\nj=F1\nj+γF2\njandF4\nj=αF2\nj+γβF1\nj.\nBefore going on, let us first give the consequence of the dissipative ness property of the solution ( vj,zj,pj,qj)\nof the system ( 2.35)-(2.38).\nLemma 2.6. The solution (v,z,p,q)of system (2.34)satisfies the following estimations\n(2.48)∞/summationdisplay\nj=1/ba∇dbl√\ndzj/ba∇dbl2=o(λ−2)and∞/summationdisplay\nj=1/ba∇dblλ√\ndvj/ba∇dbl2=o(λ−2).\nProof.First, taking the inner product of ( 2.33) withUinH, using the fact that /ba∇dblU/ba∇dblH= 1 and /ba∇dblF/ba∇dblH=o(1),\nwe get\n(2.49) /ba∇dbl√\ndz/ba∇dbl2\nL2(Ω1)=−ℜ/parenleftig\n/a\\}b∇acketle{tA/squareU,U/a\\}b∇acket∇i}htH/square/parenrightig\n=ℜ/parenleftig\n/a\\}b∇acketle{t(iλI−A/square)U,U/a\\}b∇acket∇i}htH/square/parenrightig\n=λ−2ℜ/parenleftig\n/a\\}b∇acketle{tF,U/a\\}b∇acket∇i}htH/square/parenrightig\n=o(λ−2).\nThus, by the orthonormal basis decomposition, we get the first es timation in ( 2.48). Now, multiplying ( 2.35)\nby√\ndand using the first estimation in ( 2.48) and that /ba∇dblF/ba∇dblH=o(1), we get the second estimation in ( 2.48).\nThe proof has been completed. /square\nFor all 0 < ε 0 inωandd(r) = 0 in Ω \\ω,\nwhereω:=/braceleftig\nX= (x,y)∈R2;r20 d(X) = 0\nd(X) = 0\nFigure 5. Model Describing Ω, Γ 0, Γ1andωin (OP3).\nReferences\n[1] M. Afilal, A. Soufyane, and M. de Lima Santos. Piezoelectr ic beams with magnetic effect and localized damping. Mathematical\nControl & Related Fields , 0:–, 2021.\n[2] M. Akil. Stability of pizoelectric beam with magnetic eff ect under (coleman or pipkin)-gurtin thermal law, 2022.\n[3] M. Akil, H. Badawi, S. Nicaise, and V. R´ egnier. Stabiliz ation of coupled wave equations with viscous damping on cyli ndrical\nand non-regular domains: Cases without the geometric contr ol condition, 2021.\n[4] M. Akil, I. Issa, and A. Wehbe. A n-dimensional elastic \\viscoelastic transmission problem with Kelvin-Voigt damp ing and\nnon smooth coefficient at the interface. SeMA Journal , May 2022.\n[5] Y. An, W. Liu, and A. Kong. Stability of piezoelectric bea ms with magnetic effects of fractional derivative type and\nwith/without thermal effects. 2021.\n[6] C. Batty, L. Paunonen, and D. Seifert. Optimal energy dec ay for the wave-heat system on a rectangular domain. SIAM\nJournal on Mathematical Analysis , 51(2):808–819, 2019.\n[7] C. J. K. Batty and T. Duyckaerts. Non-uniform stability f or bounded semi-groups on Banach spaces. J. Evol. Equ. , 8(4):765–\n780, 2008.\n[8] A. Borichev and Y. Tomilov. Optimal polynomial decay of f unctions and operator semigroups. Math. Ann. , 347(2):455–478,\n2010.\n[9] A. Hayek, S. Nicaise, Z. Salloum, and A. Wehbe. A transmis sion problem of a system of weakly coupled wave equations wit h\nKelvin-Voigt dampings and non-smooth coefficient at the inte rface.SeMA J. , 77(3):305–338, 2020.\n[10] Z. Liu and B. Rao. Characterization of polynomial decay rate for the solution of linear evolution equation. Z. Angew. Math.\nPhys., 56(4):630–644, 2005.\n[11] K. Morris and A. A. ¨Ozer. Strong stabilization of piezoelectric beams with mag netic effects. pages 3014–3019, 2013.\n[12] K. A. Morris and A. ¨O.¨Ozer. Modeling and stabilizability of voltage-actuated pi ezoelectric beams with magnetic effects.\nSIAM J. Control. Optim. , 52:2371–2398, 2014.\n[13] Ramos, Anderson J.A., Gon¸ calves, Cledson S.L., and Co rrˆ ea Neto, Silv´ erio S. Exponential stability and numeric al treatment\nfor piezoelectric beams with magnetic effect. ESAIM: M2AN , 52(1):255–274, 2018.\n[14] J. Rozendaal, D. Seifert, and R. Stahn. Optimal rates of decay for operator semigroups on Hilbert spaces. Advances in\nMathematics , 346:359 – 388, 2019.\n[15] A. Soufyane, M. Afilal, and M. L. Santos. Energy decay for a weakly nonlinear damped piezoelectric beams with magneti c\neffects and a nonlinear delay term. Zeitschrift f¨ ur angewandte Mathematik und Physik , 72(4), Aug. 2021.\n[16] R. Stahn. Optimal decay rate for the wave equation on a sq uare with constant damping on a strip. Zeitschrift f¨ ur angewandte\nMathematik und Physik , 68(2), Feb. 2017.\n18[17] K. Yu and Z.-J. Han. Stabilization of wave equation on cu boidal domain via Kelvin–Voigt damping: A case without geom etric\ncontrol condition. SIAM Journal on Control and Optimization , 59(3):1973–1988, 2021.\n[18] H.-E. Zhang, G.-Q. Xu, and Z.-J. Han. Stability of multi -dimensional nonlinear piezoelectric beam with viscoelas tic infinite\nmemory. Zeitschrift f ˜Ar angewandte Mathematik und Physik , 73(4):159, Jul 2022.\n19" }, { "title": "1603.00190v1.Ferromagnetic_resonance_and_magnetic_damping_in_C_doped_Mn5Ge3.pdf", "content": "arXiv:1603.00190v1 [cond-mat.mtrl-sci] 1 Mar 2016Ferromagnetic resonance and magnetic damping in\nC-doped Mn 5Ge3\nC-E Dutoit\nAix-Marseille Universit´ e, CNRS, IM2NP UMR7334, F-13397 M arseille Cedex\n20, France\nV.O. Dolocan\nE-mail:voicu.dolocan@im2np.fr\nAix-Marseille Universit´ e, CNRS, IM2NP UMR7334, F-13397 M arseille Cedex\n20, France\nM. D. Kuz’min\nAix-Marseille Universit´ e, CNRS, IM2NP UMR7334, F-13397 M arseille Cedex\n20, France\nL. Michez\nAix-Marseille Universit´ e, CNRS, CINaM UMR7325, 13288, Ma rseille, France.\nM. Petit\nAix-Marseille Universit´ e, CNRS, CINaM UMR7325, 13288, Ma rseille, France.\nV. Le Thanh\nAix-Marseille Universit´ e, CNRS, CINaM UMR7325, 13288, Ma rseille, France.\nB.Pigeau\nUniversit´ e Grenoble Alpes, CNRS, Inst. N´ eel, F-38042, Gr enoble, France\nS.Bertaina\nAix-Marseille Universit´ e, CNRS, IM2NP UMR7334, F-13397 M arseille Cedex\n20, France\nAbstract. Ferromagnetic resonance (FMR) was used to investigate the s tatic\nand dynamic magnetic properties of carbon-doped Mn 5Ge3(C0.1and C 0.2) thin\nfilms grown on Ge(111). The temperature dependence of magnet ic anisotropy\nshows an increased perpendicular magneto-crystalline con tribution at 80K with\nan in-plane easy axis due to the large shape contribution. We find that our\nsamples show a small FMR linewidth (corresponding to an intr insic magnetic\ndamping parameter α=0.005), which is a measure of the spin relaxation and\ndirectly related with the magnetic and structural quality o f the material. In the\nperpendicular-to-plane geometry, the FMR linewidth shows a minimum at around\n200K for all the samples, which seems to be not correlated to t he C-doping.Ferromagnetic resonance and magnetic damping in C-doped Mn 5Ge3 2\nThe magnetic relaxation parameters have been determined an d indicate the two-\nmagnon scattering as the main extrinsic contribution. We ob serve a change in the\nmain contribution from scattering centres in Mn 5Ge3C0.2at low temperatures,\nwhich could be related to the minimum in linewidth.\nPACS numbers: 76.50.+g,75.70.Ak,75.40.Gb,76.60.Es\nKeywords : ferromagnetic resonance, magnetic anisotropy, magnetic damp ing,\nthin filmsFerromagnetic resonance and magnetic damping in C-doped Mn 5Ge3 3\n1. INTRODUCTION\nThe field of semiconductor spintronics is rapidly de-\nveloping nowadays. The idea to combine the well es-\ntablished data processingcapabilities ofsemiconductor\nelectronics with ferromagnetism may lead to new func-\ntionalities and low power consumption of devices[1, 2].\nOne of the main obstacle for spin injection into a semi-\nconductoristheconductivitymismatchattheinterface\nof the ferromagnetic metal and the semiconductor[3].\nOne way to avoid it is to use a thin insulating layer\nacting as a tunnel barrier between the two materials.\nAnother approach is to design the spin injecting inter-\nface with a similar structure and properties by alloying\nor doping the semiconductor with a magnetic element.\nThe intermetallic magnetic Mn 5Ge3could pro-\nvide the desired solution as it grows directly onto Ge\nsubstrate[4], therefore being compatible with existing\nsemiconductor technology. Mn 5Ge3shows ferromag-\nnetism with a Curie temperature (T c) around room\ntemperature[5] and an important spin polarization (up\nto 42%)[6, 7]. The Mn 5Ge3hexagonal cell contains\n10 Mn atoms which are arranged in two different sub-\nlattices (Mn Iand Mn II) due to different coordina-\ntion. Inserting carbon atoms into interstitial voids\nof Mn IIoctahedra leads to an increase of T cup to\n450K, supplying a solution for the room temperature\nspin injection[8]. Ab-initio calculations indicate that\nthe structural distortions have a small influence on the\nincreased T cin Mn 5Ge3Cx(the lattice is compressed\ncompared to pure Mn 5Ge3), with the enhanced ferro-\nmagnetism attributed to a 90◦ferromagnetic superex-\nchange mediated by carbon[9].\nSeveral preparation methods were used to grow\nMn5Ge3thin films. The most common growth\nmethod is the solid phase epitaxy which consists\nin the deposition of Mn or Mn and C on a\nGe(111) layer followed by an annealing leading to\nthe formation of the Mn 5Ge3or Mn 5Ge3Cxfilms.\nDue to the low Mn solubility in Ge, secondary\nprecipitates or Mn-rich regions/clusters frequently\nappear inside the Mn 5Ge3films. Mn atoms also\ndiffuse in the underlying Ge(111) substrate which\ndeteriorates the interface quality. In this article,\nwe report on the structural and magnetic properties\nof thin films C-doped Mn 5Ge3epitaxially grown on\nGe(111) by reactive deposition epitaxy (RDE) at\nroom temperature. The low growth temperature\nreduces segregation and allows the formation of thinfilms of excellent crystalline quality suitable for the\ndetermination of various magnetic parameters by\nFMR: magnetic anisotropy, magnetisation and the g-\nfactor which were quantitatively determined and theirs\ndependence on carbon content and temperature was\nidentified. From the study of the FMR linewidth,\nthe magnetic relaxation process is investigated and the\nmagnetic relaxation parameters are found. The main\nrelaxationchannelswe identify arethe intrinsicGilbert\ndamping and the two-magnonscattering. The intrinsic\nmagnetic damping measured by FMR determines the\ntime scale of the dissipation of magnetic energy into\nthe lattice. We determine that C-doped Mn 5Ge3\nhas a very low damping of spin motions, with an\nintrinsic Gilbert damping constant αbetween 0.005\nand 0.01, which is one of the lowest known value\nfor ferromagnetic Mn-based thin-film compounds. In\nthe two-magnon scattering process, the uniform FMR\nmode (k=0) can scatter to degenerate non-uniform\nmodes (k/negationslash= 0) with an effective interaction matrix\nproportional to the Fourier transform of samples\ninhomogeneities[10]. We observe that the role of\nthe scattering centres changes at low temperatures\nin Mn 5Ge3C0.2where a superposition of twofold and\nfourfold symmetry dominates and not the usual six-\nfold as at room temperature. The ferromagnetic\nresonance measurements demonstrate the very good\nstructuralqualityandthelowmagneticdampinginthe\nC-doped Mn 5Ge3, paving the way for heterostructure\nintegration and spintronic applications.\n2. Experimental details\nThe sample preparation as well as the reflection high-\nenergy electron diffraction (RHEED) measurements\nwere performed in a UHV setup with a base pressureof\n2.7×10−8Pa. Mn 5Ge3Cxlayersweregrownepitaxially\non Ge(111) substrates[4, 11]. These substrates were\nchemically cleaned before introduction in the UHV\nchamber. Then we did a degassing of the Ge(111)\nsubstrates by direct heating up to 720 K for 12 h and\nflashedafterwardsat1020Ktoremovethenativeoxide\nlayer. After repeated flashes at 1020 K and a cooling\ndown at 770 K, a 15 nm thick Ge buffer layer was\ndeposited on the Ge(111) substrates to make sure that\nthe startingsurfaceoftheMn 5Ge3Cxthin films growth\nis of good quality. The quality of this starting surface\nwas checked in-situby RHEED. Eventually the sample\nwas cooled down to room temperature (RT).Ferromagnetic resonance and magnetic damping in C-doped Mn 5Ge3 4\nToformtheMn 5Ge3Cxlayersweusedthe reactive\ndeposition epitaxy method[12]. Using this method the\nMn5Ge3Cxlayers are created by phase nucleation at\nthe surface of the sample during the epitaxial growth.\nNo diffusion phenomenon is required for the growth\nunlike the solid phase epitaxy process which is usually\nemployed to form the Mn 5Ge3Cxfilms on Ge(111).\nHowever a good control of the different flows is needed\nto match the stoichiometry of the desired compound :\nGe and Mn were evaporated using Knudsen cells and\nC atomic flow was obtained thanks to a high purity\npyrolytic graphite filament source (SUKO) from MBE-\nKomponenten. The Ge and Mn flows were calibrated\nwith a water-cooled quartz crystal microbalance and\nthe C flowwascalibratedusingthe structuretransition\nbetween the Si(001) (2 ×1) and c(4 ×4) reconstructions\nwhich occurs for a C deposited thickness of 0.4 atomic\nmonolayer on Si(001) surfaces[13]. The growth of the\nMn5Ge3Cxfilms was monitored in-situby RHEED :\nthe Mn 5Ge3Cxfilms growing epitaxially on a Ge(111)\nsurface exhibit an easily identifiable RHEED (√\n3×√\n3)R30◦patternwhichischaracteristicoftheMn 5Ge3\nand Mn 5Ge3Cxcompounds[11, 14].\nThe saturation magnetisation and the estimated\nCurie temperatures of all samples were determined\nby SQUID measurements. A SQUID magnetometer\nQuantum Design MPMSXL working in a temperature\nrange 1.8K to 300K and in a magnetic field up\nto 5T was used. The FMR measurements were\nperformed with a conventional X-band (9.39GHz)\nBruker EMX spectrometer in the 80K to 300K\ntemperature range. The samples (2 ×2mm2) were\nglued on a quartz suprazil rode and mounted in the\ncentre of a rectangular cavity (TE 102). To improve\nthe signal-to-noise ratio, the FMR measurements were\ncarriedout usingamodulation field of100kHzand5Oe\namplitude with a lock-in detection. The FMR spectra\nwere measured with the applied magnetic field rotated\nin plane and out-of-plane. The FMR spectra were\nfitted to a Lorentzian profile and the resonance field\nand FWHM linewidth were subsequently extracted.\nTypicalspectraatRTareshowninfigure1(a)for12nm\nthick films. The signal-to-noiseratio(SNR) ofMn 5Ge3\nat 300K (figure 1) is 38 which is the lowest SNR of all\nmeasured spectra and is due to the proximity to T c.\nThe SNRs for the other spectra of figure 1 are 115\nand 250 for Mn 5Ge3C0.2and Mn 5Ge3C0.1respectively\nwhich have higher T c.\n3. Model and geometry\nThe FMR spectra were analyzed with the Smit-Beljers\nformalism for a thin film with uniaxial (hexagonal)\nsymmetry[15]. Foraferromagneticfilmwithhexagonal\nsymmetry, the free energy density including the\nFigure 1. (Color online) (a) Typical spectra at RT for\n12nm thick Mn 5Ge3, Mn5Ge3C0.1and Mn 5Ge3C0.2thin\nfilms. (b) Schema of the coordinate system used in the FMR\nmeasurements.\nZeeman energy, the demagnetizing energy and the\nanisotropy energy density is written as:\nF=−MH[sinθsinθHcos(ϕ−ϕH)+cosθcosθH]\n−(2πM2−K2)sin2θ+K4sin4θ+K6⊥sin6θ\n+K6/bardblsin6θcos6ϕ (1)\nwhereθH,ϕHare the polar and azimuthal angle of\nthe external field with respect to the surface normal\nof the thin film (the [001] direction) and, respectively,\nthe [100]direction, θandϕarethe polarandazimuthal\nangleofthemagnetisationwithrespectsamedirections\n(figure 1(b)) and K iare the anisotropy constants to\nsixth order. The resonance condition, neglecting the\ndamping effects and considering the magnetisation at\nequilibrium under steady field, is given by:\n/parenleftBigω\nγ/parenrightBig2\n=H1·H2 (2)\nwhereH1andH2represent the stiffness fields evalu-\nated at the equilibrium angles of the magnetisation:\nH1=1\nM∂2F\n∂θ2(3)\nH2=1\nMsin2θ∂2F\n∂ϕ2(4)Ferromagnetic resonance and magnetic damping in C-doped Mn 5Ge3 5\nEquation (2) is valid for a high-symmetry case,\nwhere the mixed second derivative of the free energy is\nnil. Our experiments were carried out in two distinct\ngeometries:\n(i) out-of-plane geometry ( ϕH= 0◦,θHvariable).\nThe stiffness fields are the following:\nHout\n1=Hrcos(θ−θH)−4πMeffcos2θ\n+2K4\nM(cos2θ−cos4θ)+30(K6⊥+K6/bardbl)\nMsin4θ\n−36(K6⊥+K6/bardbl)\nMsin6θ (5)\nHout\n2=Hrcos(θ−θH)−4πMeffcos2θ\n+4K4\nM(cos2θ−cos4θ)+6(K6⊥+K6/bardbl)\nMsin4θcos2θ\n−36K6/bardbl\nMsin6θ (6)\n(ii) in-plane geometry ( θH= 90◦,ϕHvariable). The\nstiffness fields are:\nHin\n1=Hrcos(ϕ−ϕH)+4πMeff−4K4\nM−6K6⊥\nM\n−6K6/bardbl\nMcos6ϕ (7)\nHin\n2=Hrcos(ϕ−ϕH)−36K6/bardbl\nMcos6ϕ (8)\nHere 4πMeff= 4πM−2K2/M,ωthe angular\nfrequency and γ= gµB//planckover2pi1the gyromagnetic ratio.\nThe FMR linewidth is analyzed by including the\nintrinsic and extrinsic damping mechanisms[16, 17, 18]\n:\n∆H= ∆Hintr+∆Hextr (9)\nIn this expression, the intrinsic contribution due\nto the magnon-electron interaction can be described\nby the dimensionless Gilbert damping parameter α[19,\n20]:\n∆Hintr=2αω\nγΨ(10)\nwhere Ψ =1\nH1+H2d(ω2/γ2)\ndHris the dragging function\nas the magnetisation Mis dragged behind Howing\nto anisotropy. When MandHare parallel, this\ncontribution vanishes. As generally the in-plane\nand out-of-plane linewidth are not equal, extrinsic\ncontribution have to be taken into account. The\nextrinsic contribution generally include the magnetic\nrelation due to magnon-magnon interaction, the two-\nmagnon interaction, which is given by[21, 22, 23, 24]:\n∆H2mag=Γ\nΨ(11)with Γ the two-magnon scattering rate. The two-\nmagnoncontributionusuallyvanishesforacriticalout-\nof-plane angle θ <45◦. Short-range fluctuations in\nmagnetic properties due to sample imperfections lead\nto two-magnon scattering. Inhomogeneous broadening\neffects (superposition of local resonances) due to\nlong-range fluctuations of magnetic properties also\nparticipate to the extrinsic linewidth, especially at\nintermediate angles as the resonance local field can\nvary. We consider here three types of inhomogeneous\nbroadening: ∆ Hmos,∆Hintand ∆Hinhom. The first\nterm is the mosaicity term due to the distribution of\neasy axes directions[16, 20]:\n∆Hmos=/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂Hr\n∂βH/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆βH (12)\nwithβH= (θH,ϕH). The second term represents the\ninhomogeneity of the internal fields in the sample[18]:\n∆Hint=/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂Hr\n∂(4πMeff)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆(4πMeff) (13)\nFinally, the last term which can contribute to\nthe linewidth is a residual frequency and angular\nindependent inhomogeneous linewidth that cannot be\nput in other form and is also due to long-range\nfluctuations of magnetic properties.\nThe procedure used to determine the magnetic\nparameters was as follows: the anisotropy fields\nwere determined using the system of equations (5)-\n(8) applied at high symmetry directions (along\neasy/hard axes) together with the corresponding\nmeasured resonance fields (fixed frequency) at a fixed\ng-factor. Afterwards, the polar and azimuthal angular\ndependenceoftheresonancefieldwasfittedtothesame\nequations and the equilibrium condition of the free\nenergy allowing for a variable g-factor as parameter.\nThe iteration was repeated several times until a good\nfit was obtained. This analysis yields the g-factor, the\nanisotropy constants and the magnetisation direction\nθ. These values serve in the fit of the angular variation\nof the linewidth which allows the numerical evaluation\nofα, Γ andthe inhomogeneouscontributionsusing (2)-\n(13).\n4. Results and discussion\nIn this section, experimental results of C-doped\nMn5Ge3thin films investigated by ferromagnetic\nresonance and SQUID magnetometry are presented.\nUsing samples with different carbon content, we\ndetermined the magnetic anisotropy energy, the\ng-factor, magnetisation and magnetic relaxation\nparameters.Ferromagnetic resonance and magnetic damping in C-doped Mn 5Ge3 6\n4.1. Magnetic anisotropy\nTo determine the magnetic energy anisotropy (in\nabsolute units), FMR measurements were carried out\nat a frequency of 9.4GHz. The FMR spectra were\nrecorded as a function of the polar and azimuthal\nangles of the external magnetic field at different\ntemperatures. The saturation magnetisation was\ndetermined fromSQUIDmeasurements. In figure2(d),\nthe temperature dependence of the magnetisation up\nto 300K is shown for Mn 5Ge3, Mn5Ge3C0.1and\nMn5Ge3C0.2. The Curie temperature was estimated\nfrom these curves using a mean-field approximation\nwhich gives a good approximation of the temperature\ndependence of the magnetisation. The full line\ncorrespond to a fit with the Brillouin function B 1.5\n(S=3/2)and the dotted line to afit with B 1(S=1) that\nfollows better the experimental points of Mn 5Ge3C0.2.\nThe estimated values of T care 315K, 345K and 450K.\nThe error bars correspond to ±10K for Mn 5Ge3and\nMn5Ge3C0.1as the experimental points cover a larger\ntemperature range and superpose closely with B 1.5.\nThe experimental points for Mn 5Ge3C0.2cover only\na small part of the temperature range and the error\nbars are estimated to be of ±30K.\nThe out-of-plane angular variation for the reso-\nnance field H ris shown in figure 2(a)-(c) for Mn 5Ge3,\nMn5Ge3C0.1and Mn 5Ge3C0.2at room temperature.\nThe H r(θH) curves indicate an in-plane easy axis with\na minimum resonance field of 1.6kOe, 2.3kOe and\n2.7kOe for Mn 5Ge3C0.2, Mn5Ge3C0.1and Mn 5Ge3re-\nspectively for ϕH= 0◦. The hard axis is perpen-\ndicular to plane ([001] direction) and has the highest\nHrof 8.6kOe, 6kOe and 5kOe. The azimuthal angu-\nlar dependence of the resonance field for Mn 5Ge3C0.2,\nrecorded also at 300K is shown in figure 2(e). The\nsixfold (hexagonal) symmetry in the azimuthal angu-\nlar dependence indicates that an in-plane hexagonal\nanisotropy exists with one easy axis along the [100] di-\nrection of the film. The experimental FMR data of\nout-of-plane and in-plane dependence of the resonance\nfield can be well simulated with (2) and the anisotropy\nfields can be extracted. The anisotropy constants can\nbe found in absolute units by using the sample mag-\nnetisation determined from SQUID measurements.\nThe resulting anisotropy constants are sum-\nmarised in Table 1 along with the g-factor at several\ntemperatures. The positive sign of K 2indicates that\nthis term favors an out-of-plane easy axis of magneti-\nsation while the shape anisotropy dominates[25]. In\nthe very thin film limit, K 2could overcome the shape\nanisotropy resulting in an out-of-plane anisotropy axis.\nThe different K ihave a different temperature depen-\ndence. For Mn 5Ge3and Mn 5Ge3C0.1, the sixfold in-\nplane symmetry is to low to be extracted, therefore\nonly the K 2and K 4constants were determined fromthe angular measurements. K 2is positive for Mn 5Ge3\nandC-dopedMn 5Ge3atalltemperaturesandincreases\nat low temperature. K 4decreases (increases in abso-\nlute values) for Mn 5Ge3, but for the C-doped com-\npounds has a minimum or a maximum at an inter-\nmediate temperature. The sixfold in-plane anisotropy\nin Mn 5Ge3C0.2increases at 250K from the room tem-\nperature value, while at lower temperature becomes to\nsmall or a transition to a fourfold in-plane anisotropy\narises as will be inferred from the linewidth temper-\nature dependence discussed in the next section. The\nerror bars for the anisotropy constants presented in\nTable 1 are around 5%, which come from the com-\nputation of M from SQUID measurements due to the\nuncertainty in the volume of our samples.\nTheg-factor can be estimated from the angular\ndependence of the resonance field. Its value indicates\nthe influence of the orbital contribution to the total\nmagnetic moment. The ratio of the orbital to the\nspin magnetic moment can be inferred from the Kittel\nformula and is equal to the deviation of the g-factor\nfrom the free electron value. The value of the g-\nfactor for Mn 5Ge3and Mn 5Ge3C0.1is 2.0005, while\nfor Mn 5Ge3C0.2this value increases to 2.0291 meaning\nan increased orbital contribution with carbon doping\n(1.5% of the spin magnetic moment).\n4.2. Magnetic relaxation\nThe linewidth of the resonant signal ∆ Hris directly\nrelated to the magnetic and structural quality of\nthe films and provide information about the different\nrelaxation channels in magnetic damping. In figure 3,\nthe temperature dependence of the FMR linewidth\nis shown for the perpendicular to plane direction\n(θH= 0◦) for Mn 5Ge3and C-doped Mn 5Ge3. A\nshallow minimum is observed for all three compounds\naround 200K and a sharp peak close to T c. At\nlower temperature, the FMR linewidth increases and\nsaturatesforMn 5Ge3(measuredto6K).The minimum\nin the linewidth seems not related with the C-\ndoping. It occurs around the same absolute value of\ntemperature and could be related with a small in-\nplane transition to a fourfold anisotropy from sixfold\nanisotropy (tetragonal distortion) or to a constriction\nby the substrate. The increase of linewidth at\nlow temperature was explained as an inhomogeneous\nbroadening due to the increase of the anisotropy\nconstants (K 2) with decreasing temperature[17]. The\nerror bars for the measured FMR linewidth are less\nthan 2% in all cases.\nFigure 4 and figure 5(a) show the out-of-plane\nvariation of the FMR linewidth for the C-doped\nMn5Ge3at room and low temperatures. The shape of\nthe curvesshowsthe characteristicdependence for thin\nfilms with a maximum of the linewidth at intermediateFerromagnetic resonance and magnetic damping in C-doped Mn 5Ge3 7\nFigure 2. (Color online) Out-of-plane angular variation of the reson ance field at 300K for (a) Mn 5Ge3, (b) Mn 5Ge3C0.1, (c)\nMn5Ge3C0.2. The temperature dependence of the magnetisation is shown i n (d) in normalised coordinates. The full and dotted\nlines correspond to fits with a Brillouin function. The estim ated T cs are 315K, 345K and 450K. (e) In-plane angular dependence\nof the resonance field for Mn 5Ge3C0.2at room temperature. The diagram centre and the dotted circl es correspond to H rvalues of\n1605, 1606 and 1607 Oe respectively. The error bars are ±0.07 Oe. The line represents a fit to (2).\nFigure 3. (Coloronline)Temperature variationofthe resonance\nlinewidth for Mn 5Ge3, Mn5Ge3C0.1and Mn 5Ge3C0.2.\nangles. Our films have an in-plane easy axis at all\ntemperatures, therefore the magnetisation lags behind\nthe applied field when the field is rotated out of the\nplane. Thepeakinthelinewidth occursfor θHbetween\n20◦at room temperature and 10◦at low temperature,corresponding to the largest interval between Mand\nH. From the theoretical fits of the data (solid lines)\nobtained using (2)-(13), the relaxation parameters are\nextracted and listed in Table 2.\nFor all compounds, the perpendicular to plane\nlinewidth is always smaller than the in-plane one\nindicating the presence of two-magnon scattering\nand other extrinsic contributions in the samples.\nThe intrinsic damping alone cannot explain the\nout-of-plane shape of the ∆ Hr(θH) dependence as\nobserved in figure 4, where the two-magnon and\nintrinsic Gilbert contribution are shown individually\nfor comparison. The respective values of the two\ncontributions are further compared in Table 2 by\ncomputing the Gilbert damping parameter G and the\ntwo-magnon scattering contribution γΓ2magin s−1.\nThe contribution of the intrinsic linewidth (Gilbert)\nto the total linewidth varies between 7% and 33%.\nThe mosaicity contribution is low for all cases and its\nangulardependence induces the V-shaped form around\nθH=0◦. A typical mosaicity contribution is shown in\nfigure 4 for Mn 5Ge3C0.1at 200K. The inhomogeneity\nof the internal field gives mainly a low contributionFerromagnetic resonance and magnetic damping in C-doped Mn 5Ge3 8\nFigure 4. (Color online) Out-of plane angular dependence\nof the resonance linewidth for Mn 5Ge3C0.1at different\ntemperatures. The full lines represent a fit with intrinsic a nd\nextrinsiccontributions, thedashed linesindicate the two -magnon\ncontribution while the dash-dotted lines indicate the intr insic\ncontribution. The symbols indicate experimental data. The\ntypical mosaicity and internal fields (∆(4 πMeff)) contributions\nare shown at 200K and 100K respectively.\naroundθH=0◦, with the exception of Mn 5Ge3C0.1at\nroom temperature where it gives a larger contribution\nforθHclose to the plane. A typical variation of this\ncontribution is shown in figure 4 at 100K.\nThe estimated intrinsic damping αis considered\nisotropic and independent of temperature in the\nconsidered temperature range (100K-300K). We prefer\nusing the dimensionless parameter αwhich varies\nbetween 0.005 and 0.01 over the Gilbert damping\nparameter G given by α=G/γM as the latter will\nimply a strong temperature dependence. The\nGilbert damping represents the decay of magnetisation\nby direct viscous dissipation to the lattice as\nit is introduced in the Landau-Lifschitz-Gilbert\nequation[19]. The spin-orbit coupling is assumed to be\nat the origin of spin-lattice relaxation in ferromagnets.\nAb-initio calculations that include the spin-orbit\ncoupling explicitly show a weak dependence of αwith temperature in a large range of temperatures[26,\n27]. Two different mechanisms contribute to the\ntemperature dependence[10], one conductivity-like and\none resistivity-like with a transition between the two\nat intermediate temperature. Sometimes these two\ncontributions have an equal influence on the damping.\nOther models predict an anisotropic tensorial damping\nconstant ˆ α[28] and its proportionality to 1/M[29],\nleading to a strong variation close to T c.\nWe estimated the value of αfor each compound\nby fitting the out-of-plane angular dependence of\n∆Hrat a temperature corresponding to the minimum\nof the curves in figure 3 (around 200K). For\nthis specific temperature, the estimation corresponds\nto the maximum possible value of αconsidering\nsmall inhomogeneous broadening (∆ Hintand ∆Hinh).\nAlthough we consider a constant α, as it is observed\nfrom Table 2, at room and low temperature the\nlinewidth (and correspondingly the inhomogeneous\nresidual field) increases for Mn 5Ge3C0.1which could\nbe explained by an increase of αat least at low\ntemperature. The room temperature increasing in the\nlinewidth is usually explained as a breakdown of the\nuniform precession due to thermal excitations (spin-\nwaves)[30]. The increasing of the linewidth at low\ntemperature is smaller for Mn 5Ge3and Mn 5Ge3C0.2\nin the 100-300K temperature range being compatible\nwith a constant αas considered for this temperature\nrange.\nThe second relaxation mode that influence the\nFMR linewidth is the two magnon scattering. The\nuniform mode can couple with degenerate spin-wave\nmodes due to fluctuations in the local effective field\nthat can arise from surface defects, scattering centres,\nfluctuation in the anisotropy from grain to grain\nor other inhomogeneities[21, 23]. The two magnon\nscattering rate Γ depends on the angle θH(out-of-\nplane geometry) and on the resonance field H res. A\ndetailed analysis based on the effect of the defects on\nthe response functions of thin films was performed in\n[22] and [31] for the case when the magnetisation is\ntipped out-of-plane. We consider here the same type\nof angular dependence of Γ as in [31] (see (8)). Γ\ndepends on the nature and shape of the defects that\nactivate the scattering mechanism. The values for\nthe Mn 5Ge3compounds, extracted from the fitting\nof the linewidth curves, are shown in Table 2 as a\nfunction of temperature. From the calculated value\nΓ2mag=8HKb2p/πD, the exchange spin-wave stiffness\nD can be inferred if details of the defects are known\nsuch as the covered fraction of the surface por the\neffective height b(HKthe anisotropy field). Atomic\nforce microscopy measurements were performed on\nthe samples, from which the rms surface roughness\nwas determined: for Mn 5Ge3the surface roughnessFerromagnetic resonance and magnetic damping in C-doped Mn 5Ge3 9\nwas about 1.5-2nm, while for Mn 5Ge3Cxit was close\nto 1nm. Therefore, at room temperature, the spin-\nwave stiffness was estimated as 0.12 ×10−8G cm2\nfor Mn 5Ge3, 0.16×10−8G cm2for Mn 5Ge3C0.1and\n0.39×10−8G cm2for Mn 5Ge3C0.2considering a defect\nratio of 50%. These values are only estimates as a\npreciseidentificationofthedefectsisdifficulttoobtain.\nAs observed from Table 2, the other extrinsic\ncontributionstothelinewidth haveonlyasmallimpact\non the fitted curves. The mosaicity is very small,\ninferior to 0.1◦, being a testimony of the good quality\nof our samples. Also the inhomogeneity of the internal\nfields is almost negligible in the majority of cases,\nonly for Mn 5Ge3C0.1at room temperature it seems to\nhave a larger influence. The higher values of H intare\nneeded to explain the small peak observed around θH\n= 0◦for both Mn 5Ge3and Mn 5Ge3C0.1and for the\nincrease of the linewidth at intermediate angles until\nθH= 90◦for Mn 5Ge3C0.1at room temperature. The\nvalues of the residual inhomogeneous contribution are\ngenerallysmall, the largervalues canalsobe attributed\nto a temperature dependent intrinsic contribution as\ndiscussed above.\nWe now discuss the case of Mn 5Ge3C0.2, for which\nboth out-of-plane and in-plane data were fitted, as\nshown in figure 5. The panel (a) show the out-of-\nplane dependence of the FMR linewidth. The 300K\nand 250K data are well fitted close to θH= 0◦and at\nlarger angles but not at the peaks that correspond to\nthe largest interval between MandH(critical angle).\nThe dashed line at T=300K corresponds to a fit with\nthe parameters indicated in Table 2 and ∆ θH= 0.05◦,\nwhile the full line to a fit with ∆ θH= 0.2◦. Although\nincreasing the mosaicity contribution fits better the\npeaks, the fitted curve becomes V-shaped between the\npeaks in total contradiction with the data. We believe\nthat the mosaicity is small (0.05◦) and the discrepancy\nat the critical angle at 300K is due to some other effect\n(the FMR line being strongly distorted at this angle).\nWe alsotriedto fitthe 300Kcurveintroducingin-plane\nsecond and fourth order anisotropy constants (K 2/bardbland\nK4/bardbl) without a better result (not shown). The low\ntemperature curves are nicely fitted with the presented\nmodel for all angles.\nFor the in-plane dependence of ∆ Hr, the only\ncontributions that were considered were from the\nisotropic intrinsic damping and the two-magnon\ncontribution which wasexpressed asfollows[21, 20, 31]:\n∆H2mag=/summationtext\niΓif(ϕi)\nΨ×\narcsin/parenleftBigg/radicalBigg/radicalbig\nω2r+(ω0/2)2−ω0/2/radicalbig\nω2r+(ω0/2)2+ω0/2/parenrightBigg\n(14)\nwithω0=γMeffand Γ if(ϕi) characterise theanisotropyofthetwo-magnonscatteringalongdifferent\ncrystallographic in-plane directions. At 300K and\n250K (figure 5(b)), the FMR linewith has the same\nsix-fold symmetry as the angular dependence of Hr\n(figure 2(e)). If the scattering centres are given\nby lattice defects (dislocation lines), the azimuthal\ndependence should reflect the lattice symmetry[32, 20].\nThe angular dependence of the scattering was fitted\nwith Γ if(ϕi) = Γ 0+ Γ2cos2(ϕ−ϕ2) + Γ6cos6(ϕ−\nϕ6) at 250K and 300K and with Γ if(ϕi) = Γ 0+\nΓ2cos2(ϕ−ϕ2)+Γ4cos4(ϕ−ϕ4) at 150K and 100K.\nThe parameters Γ 2and Γ 4are phenomenologically\nintroduced to account for the observed angular\nvariation. Γ 6is expected from the sixfold symmetry.\nThe in-plane anisotropies are very small as observed\nform their values in Table 1, therefore ϕM≈ϕH\nand the dragging function is very close to unity and\nneglected. A change of symmetry of the scattering\n(or of the dominant contribution to the scattering)\nseems to take place around 200K corresponding to\nthe minimum in figure 3. At lower temperature\na superposition of twofold and fourfold symmetry\ndominates the angular dependence of the in-plane\nlinewidth. This can be related to the shape and\norientation of the defects (rectangular) that cause the\ntwo-magnon scattering[21, 32, 33]. The two-magnon\nscattering intensity depends on the direction of the\nmagnetisation in respect to the symmetry axes of\nthe magnetic defects and on the angle between the\nmagnetisation and the crystallographic axes. One can\nthink that the twofold and fourthfold symmetry can be\nrelated to the magnetic symmetry of the defects and to\nthe lattice defect symmetry at low temperature with\nthe defects oriented mostly along the [110] direction.\nThis is also consistent with the increase in the two-\nmagnon scattering rate (see Table 2 and Table 3) at\nlower temperature and with the variation in the two-\nmagnon linewidth contribution to the total linewidth\n(from around 30% at 300K to 66% at 100K).\n5. Conclusion\n12nm thick Mn 5Ge3and Mn 5Ge3Cxfilms were grown\nby reactive deposition epitaxy on Ge(111) substrates.\nDetailed FMR measurements were performed on the\nsamples at different temperatures. Both Mn 5Ge3\nand C-doped Mn 5Ge3show perpendicular uniaxial\nmagneto-crystalline anisotropy and an in-plane easy\naxisofmagnetisationduetothelargeshapeanisotropy.\nDoping with carbon increases the anisotropy fields\nwithout diminishing the quality of the sample as\ndemonstrated by the small FMR linewidth of the films.\nFrom the angular dependence of the resonance field\nand of the linewidth, the anisotropy fields, g-factor\nand magnetic relaxation parameters are obtained. TheFerromagnetic resonance and magnetic damping in C-doped Mn 5Ge3 10\ncontributions to the broadening of the FMR linewidth\ncome primarily from the intrinsic Gilbert damping\nand two-magnon scattering. In the temperature\ndependenceoftheout-of-planelinewidthaveryshallow\nandwideminimumisobservedcloseto200 ±40K.One\ncansupposethatthisminimumandthechangingofthe\ndominating contribution in the two-magnon scattering\nfrom centres with different symmetry are connected\none to each other. Nevertheless this supposition\nrequires additional evidences.\nAcknowledgments\nThis work has been carried out thanks to the support\nof the A*MIDEX project (No. ANR-11-IDEX-\n0001-02) funded by the ”Investissements d’Avenir”\nFrench Government program, managed by the French\nNational Research Agency (ANR). This work is part\nof A*MIDEX through the HIT project APODISE. We\nalso want to thank the interdisciplinary French EPR\nnetwork RENARD (CNRS - FR3443).\nReferences\n[1] I. Zutic, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76,\n323 (2004).\n[2] D. D. Awschalom and M. E. Flatt´ e, Nature Phys. 3, 153\n(2007).\n[3] G. Schmidt, D. Ferrand, L. W. Molenkamp, A. T. Filip,\nand B. J. van Wees, Phys. Rev. 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Heinrich, Phys. Rev. B 69, 184417\n(2004).\n[33] K. Lenz, H. Wende, W. Kuch, K. Baberschke, K. Nagy and\nA. Janossy, Phys. Rev. B 73, 144424 (2006).Ferromagnetic resonance and magnetic damping in C-doped Mn 5Ge3 11\nTable 1. Magnetic parameters for Mn 5Ge3, Mn5Ge3C0.1and Mn 5Ge3C0.2at different temperatures obtained from the FMR.\nSample T(K) 4 πMeff(kOe) K 2(erg/cm3) K 4(erg/cm3) K 6/bardbl(erg/cm3)γ/2π(GHz/kOe)\nMn5Ge3300 1.5 0.37 ×1062832 2.8\n200 4.0 2.01 ×106-2.28×1052.8\n150 4.8 2.36 ×106-2.93×1052.8\nMn5Ge3C0.1300 2.6 1.65 ×1063.85×1042.8\n250 3.8 2.71 ×106-1901 2.8\n200 4.4 3.37 ×106-5131 2.8\n100 5.0 4.29 ×1062.58×1042.8\nMn5Ge3C0.2300 5.3 4.39 ×1064.41×10427 2.84\n250 5.8 4.78 ×1065.53×104134 2.84\n150 6.6 5.19 ×1065.35×1042.84\n100 7.0 5.28 ×1064.61×1042.84\nTable 2. Magnetic relaxation parameters for Mn 5Ge3, Mn5Ge3C0.1and Mn 5Ge3C0.2determined from the out-of-plane angular\nvariation of the FMR at different temperatures.\nSample T(K) αG(108s−1) Γ2mag(Oe)γΓ2mag(108s−1) ∆θH(deg) ∆(4 πMeff)(Oe) ∆ Hinh(Oe)\nMn5Ge3300 0.01 0.54 150 26.38 0.05 20 270\n200 0.01 1.29 340 59.81 0.1 10 70\n150 0.01 1.41 550 96.76 0.1 10 10\nMn5Ge3C0.1300 0.005 0.55 210 36.94 0.05 80 80\n250 0.005 0.73 280 49.26 0.1 5 15\n200 0.005 0.81 320 56.29 0.1 5 5\n150 0.005 0.88 400 70.37 0.1 10 5\n100 0.005 0.92 430 75.64 0.1 15 70\nMn5Ge3C0.2300 0.01 1.92 220 39.25 0.05-0.2 10 5\n250 0.01 2.02 300 53.53 0.05 10 5\n150 0.01 2.16 500 89.22 0.05 10 5\n100 0.01 2.21 450 80.30 0.05 10 5\nNote: The error bars are around 10%.\nTable 3. Magnetic relaxation parameters for Mn 5Ge3C0.2at different temperatures determined from the in-plane angu lar variation\nof FMR.\nT(K) Γ 0(Oe) Γ 2(Oe) Γ 4(Oe) Γ 6(Oe)ϕ2ϕ4ϕ6\n300 72.75 1.5 1.5 90 30\n250 97.5 1.7 1.5 90 30\n150 254.2 8.6 5.5 57 166\n100 291.4 12.4 8.6 57 167Ferromagnetic resonance and magnetic damping in C-doped Mn 5Ge3 12\nFigure 5. (Color online) Out-of plane (a) and in-plane (b) angular dep endence of the resonance linewidth for Mn 5Ge3C0.2at\ndifferent temperatures. The lines represent fits with intrin sic and extrinsic contributions. The error bars in panel (b) are±0.2 Oe\nat 300K and ±0.8 Oe at 100K." }, { "title": "2211.04496v1.On_the_injection_scale_of_the_turbulence_in_the_partially_ionized_very_local_interstellar_medium.pdf", "content": "DRAFT VERSION NOVEMBER 10, 2022\nTypeset using L ATEXpreprint2 style in AASTeX63\nOn the injection scale of the turbulence in the partially ionized very local interstellar medium\nSIYAO XU(\u001de)1AND HUILI(NV)2\n1Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA; sxu@ias.edua\n2Los Alamos National Laboratory, NM 87545, USA; hli@lanl.gov\nABSTRACT\nThe cascade of magnetohydrodynamic (MHD) turbulence is subject to ion-neutral collisional damping and\nneutral viscous damping in the partially ionized interstellar medium. By examining the damping effects in\nthe warm and partially ionized local interstellar medium, we find that the interstellar turbulence is damped by\nneutral viscosity at \u0018261au and cannot account for the turbulent magnetic fluctuations detected by Voyager\n1and2. The MHD turbulence measured by Voyager in the very local interstellar medium (VLISM) should be\nlocally injected in the regime where ions are decoupled from neutrals for its cascade to survive the damping\neffects. With the imposed ion-neutral decoupling condition and the strong turbulence condition for the observed\nKolmogorov magnetic energy spectrum, we find that the turbulence in the VLISM is sub-Alfv ´enic, and its\nlargest possible injection scale is \u0018194au.\n1.INTRODUCTION\nTurbulent magnetic fluctuations following a Kolmogorov\nspectrum were observed by Voyager 1 and2in the outer he-\nliosheath (Burlaga et al. 2018; Zhao et al. 2020; Lee & Lee\n2020; Fraternale & Pogorelov 2021; Burlaga et al. 2022).\nWhile the source(s) of the turbulence in the very local in-\nterstellar medium (VLISM) are under debate (Holzer 1989;\nZank 2015; Zank et al. 2019), it is believed to play a crucial\nrole in affecting the transport of energetic particles and cos-\nmic rays (Lazarian & Opher 2009; Stone et al. 2013; Krim-\nigis et al. 2013; L ´opez-Barquero et al. 2017; Fraternale et al.\n2022), and the structure of the Interstellar Boundary Explorer\n(IBEX) ribbon (Giacalone & Jokipii 2015; Zirnstein et al.\n2020). In particular, modeling by Zirnstein et al. (2020) sug-\ngests that the turbulent magnetic fields with scales <100\nau are important for producing the ribbon structure similar\ntoIBEX observations, and such turbulence is likely not of\npristine interstellar origin, whereas turbulent fluctuations at\nscales\u0015100au produce features inconsistent with IBEX ob-\nservations.\nTurbulence and turbulent magnetic fields are ubiquitous in\nthe ISM (e.g., Armstrong et al. 1995; Chepurnov & Lazar-\nian 2010; Gaensler et al. 2011; Xu & Zhang 2017; Lazarian\net al. 2018), with the turbulent energy injected by supernova\nexplosions on large length scales ( \u0018100pc; Breitschwerdt\net al. 2017) and cascading down toward smaller and smaller\nscales. In the warm and partially ionized local interstellar\nmedium (LISM) (Frisch et al. 2011; Slavin & Frisch 2008),\nthe magnetohydrodynamic (MHD) turbulence is subject to\nthe damping effects due to the frictional collisions between\nions and neutrals (i.e., ion-neutral collisional damping) and\namong neutrals (i.e. neutral viscosity) (Xu & Lazarian 2017).\nThe damping effects cause the cutoff of MHD turbulence cas-\naNASA Hubble Fellowcade when the damping rate exceeds the cascading rate of\nMHD turbulence.\nThe linear analysis of MHD waves in a partially ion-\nized medium was performed by Kulsrud & Pearce (1969)\nand more recently by e.g., Pudritz (1990); Balsara (1996);\nMouschovias et al. (2011); Zaqarashvili et al. (2011); Soler\net al. (2013). Different from linear MHD waves, MHD tur-\nbulence is characterized by the nonlinear cascade of turbu-\nlent energy, with scale-dependent turbulence anisotropy and\nlimited timescale of turbulent motions (Goldreich & Srid-\nhar 1995; Lazarian & Vishniac 1999). The ion-neutral col-\nlisional (IN) damping of MHD turbulence in a highly ion-\nized medium at a high plasma \fwas studied by Lithwick\n& Goldreich (2001). The neutral viscous (NV) damping of\nMHD turbulence was analyzed by Lazarian et al. (2004),\nwhich tends to dominate over the IN damping toward a higher\nionization fraction and a higher temperature (Xu & Lazarian\n2017). Xu et al. (2015, 2016) performed a general analysis\nincluding both damping effects of MHD turbulence in differ-\nent interstellar phases with varying ionization fractions.\nThe range of length scales for the existence of MHD tur-\nbulence depends on the coupling state between ions and neu-\ntrals and the corresponding damping effects. Irrespective\nof the origin, the MHD turbulence measured in the VLISM\nshould survive the damping in the partially ionized medium.\nIn this work, we will examine the damping of the interstellar\nMHD turbulence driven at large scales. More importantly, we\nwill explore the constraint imposed by the damping effects on\nthe locally driven MHD (LMHD) turbulence in the VLISM,\nas observed by Voyager . In Section 2, we analyze the damp-\ning of the interstellar turbulence driven in the strongly cou-\npled regime and the injection of the LMHD turbulence in\nthe decoupled regime with weak damping. In Section 3,\nwe determine the turbulence regime and the largest injection\nscale of the LMHD turbulence constrained by the ion-neutral\ndecoupling condition. Discussion and conclusions are pre-\nsented in Sections 4 and 5.arXiv:2211.04496v1 [physics.space-ph] 8 Nov 20222\n2.SOURCE FOR TURBULENCE IN THE VLISM\nThere are three possible sources for the turbulent magnetic\nfluctuations measured by Voyager , including the interstellar\nturbulence driven by supernova explosions at \u0018100pc, the\nturbulence driven in the Local Interstellar Cloud at \u00182pc\n(Zank et al. 2019), and the LMHD turbulence in the VLISM.\nWe will first focus on the former two possible sources, for\nwhich the turbulence is driven on large scales in the regime\nwhere neutrals and ions are strongly coupled together (Sec-\ntion 2.1). For the local source, the turbulence with a small en-\nergy injection scale is expected to arise in the regime where\nions are decoupled from neutrals (Section 2.2).\n2.1. Damping of interstellar turbulence driven in strongly\ncoupled regime\nThe interstellar turbulence is driven by supernova explo-\nsions atL\u0018100pc (Breitschwerdt et al. 2017) and cascades\ndown to smaller and smaller scales (Chepurnov et al. 2010;\nYuen et al. 2022). Its damping scale varies in different in-\nterstellar phases and regions, depending on the local physical\nconditions. Given the approximate energy equipartition be-\ntween turbulence and magnetic fields in the ISM expected\nfrom small-scale nonlinear turbulent dynamo (Xu & Lazar-\nian 2016) and indicated by observational measurements (Pat-\ntle et al. 2022), we consider that the interstellar turbulence\nis trans-Alfv ´enic, i.e.,MA=VL=VA\u00191. HereMAis the\nturbulent Alfv ´en Mach number, VLis the turbulent velocity\nat the injection scale Lof turbulence, VA=B=p4\u0019\u001ais the\nAlfv ´en speed,Bis the magnetic field strength, \u001a=\u001ai+\u001an\nis the total mass density, \u001ai=mHni=mHneis the mass\ndensity of ions, mHis the hydrogen atomic mass, niandne\nare the number densities of ions and electrons, \u001an=mHnH\nis the mass density of neutrals, and nHis the number density\nof neutrals.\nIn a partially ionized medium, MHD turbulence is subject\nto both IN damping and NV damping (Lazarian et al. 2004;\nXu & Lazarian 2017). They both depend on the coupling\nstate between ions and neutrals. When the driving rate of\nturbulence is lower than the neutral-ion collisional frequency\n\u0017ni=\rd\u001ai, where\rd= 5:5\u00021014cm3g\u00001s\u00001is the drag\ncoefficient (Draine et al. 1983; Shu 1992), the MHD turbu-\nlence is driven in strongly coupled ions and neutrals, i.e., the\nstrongly coupled regime. Neutrals decouple from ions when\nthe cascading rate of MHD turbulence \u001c\u00001\ncasbecomes larger\nthan\u0017ni. The neutral-ion decoupling entails significant colli-\nsional friction, causing the damping of MHD turbulence and\nthe cutoff of its energy cascade. The corresponding damping\nscale for Alfv ´enic turbulence is (Xu et al. 2015, 2016; see\nAppendix B for the derivation)\nldam,IN;?=\u00102\u0017ni\n\u0018n\u0011\u00003\n2L\u00001\n2V3\n2\nL; (1)\nwhere\u0018n=\u001an=\u001a. The subscript?means that the length\nscale is measured perpendicular to the local magnetic field.\nAs the energy cascade of MHD turbulence is mostly in the\nperpendicular direction, ldam,IN;?corresponds to the cutoff\nscale of MHD turbulence.In the case when the NV damping dominates over the\nIN damping, the damping of MHD turbulence occurs in\nthe strongly coupled regime, with the damping scale (Xu &\nLazarian 2017; see Appendix B)\nldam,NV;?=\u0010\u0018n\n2\u00113\n4\u00173\n4nL1\n4V\u00003\n4\nL; (2)\nwhere\u0017n=vth=(nH\u001bnn)is the kinematic viscosity in neu-\ntrals,vthis the neutral thermal speed, and \u001bnnis the colli-\nsional cross-section of neutrals. We note that ldam,NV;?is in\nfact the damping scale of the turbulent kinetic energy spec-\ntrum. The magnetic fluctuations in the sub-viscous range\nbelowldam,NV;?is termed new regime of MHD turbulence\n(Lazarian et al. 2004) (see Section 4.1).\nWe now discuss whether the magnetic turbulence in the\nVLISM can come from the interstellar turbulence that is in-\njected by supernova explosions. The LISM near the sun is\nwarm, low-density, and partially ionized, with the tempera-\ntureT\u00196300 K,nH\u00190:2cm\u00003, andne\u00190:07cm\u00003\n(Slavin & Frisch 2008; Swaczyna et al. 2020). In addition,\nwe adoptLISM\u0019100pc,VL;ISM\u0019VA,B\u00195\u0016G as the\ntypical driving conditions of interstellar turbulence and inter-\nstellar magnetic field strength (Crutcher et al. 2010). With\nthese high temperature and moderate ionization fraction, we\nfind that\nldam,IN;?\u00197:6\u00021013cm\u0010ne\n0:07cm\u00003\u0011\u00003\n2\u0010nH\n0:2cm\u00003\u0011\u00003\n4\n\u0010LISM\n100pc\u0011\u00001\n2\u0010B\n5\u0016G\u00113\n2\n(3)\nis much smaller than\nldam,NV;?\u00193:9\u00021015cm\u0010T\n6300 K\u00113\n8\u0010nH\n0:2cm\u00003\u0011\u00003\n8\n\u0010LISM\n100pc\u00111\n4\u0010B\n5\u0016G\u0011\u00003\n4;\n(4)\nwhere we assume nH+ne\u0018nH, and\u001bnn\u001910\u000014\ncm2(Krstic & Schultz 1998; Vranjes & Krstic 2013) is\nadopted. We note that the gradients of quantities in the\nouter heliosheath (Zank et al. 2013) cause uncertainties in\nthe above estimates. By assuming the uncertainties \u001b(ne)\u0018\n0:01cm\u00003,\u001b(nH)\u00180:01cm\u00003,\u001b(B)\u00181\u0016G,\u001b(T)\u0018\n1000 K, and\u001b(LISM)\u001810pc, we find \u001b(ldam,IN;?)\u0018\n2:8\u00021013cm and\u001b(ldam,NV;?)\u00186:4\u00021014cm.\nFor the interstellar turbulence driven in the Local Interstel-\nlar Cloud, the turbulent velocity of a few km s\u00001is similar to\nthat from the cascade of supernova-driven turbulence at \u00182\npc (Spangler et al. 2011). It might be a part of the global cas-\ncade of the interstellar turbulence. Therefore, it is also sub-\nject to the damping effects in the partially ionized LISM and\ndamped at the damping scale similar to that of the supernova-\ndriven interstellar turbulence.\nThe above calculations show that the damping of the MHD\nturbulence in the LISM from the interstellar origin is dom-\ninated by NV in the strongly coupled regime. The corre-\nsponding damping scale is \u00183:9\u00021015cm, i.e., 261au.3\nThis means that the turbulence observed by Vogayer 1 and2\nis unlikely due to the pristine interstellar origin.\n2.2. LMHD turbulence in decoupled regime\nAs discussed above, for the MHD turbulence injected in\nthe strongly coupled regime, its cascade is cut off either by\nthe IN or NV damping. When ions are decoupled from neu-\ntrals, however, the driven MHD turbulence is no longer sub-\nject to the NV damping, and the IN damping becomes con-\nstantly weak (Xu et al. 2016). The MHD turbulence injected\nin ions that are decoupled from neutrals, i.e., LMHD turbu-\nlence in the decoupled regime, is not cut off due to the damp-\ning effects arising in a partially ionized medium.\nBased on the in-situ measurements of turbulent magnetic\nenergy spectrum by Voyager 1 in the VLISM (Lee & Lee\n2020), we have the ratio of the turbulent component to the\naverage magnetic field strength \u000eBobs=B\u00190:06measured at\nlobs\u00193\u00021014cm (corresponding to the smallest wavenum-\nber of the measured spectrum and assumed to be in the in-\nertial range of turbulence). By assuming that the magnetic\nfluctuations are mainly induced by Alfv ´enic turbulence (Cho\n& Lazarian 2002; Hu et al. 2022; Lee & Lee 2020), the local\nturbulent velocity can be estimated as\nvobs=\u000eBobs\nBVAi\n= 2:5km s\u00001\u0010\u000eBobs=B\n0:06\u0011\u0010ne\n0:07cm\u00003\u0011\u00001\n2\u0010B\n5\u0016G\u0011\n:\n(5)\nBy assuming \u001b(\u000eBobs)\u00180:1\u0016G and the uncertainties\nof other parameters (see Section 2.1), we find \u001b(vobs)\u0018\n0:85km s\u00001. The large uncertainties in our calculations are\nmainly caused by the large uncertainties in magnetic field\nstrength measurements (Burlaga et al. 2018). Such a large\nturbulence level at the measured length scale cannot be ac-\ncounted for by the interstellar turbulence, which is cut off\nat a larger length scale ldam,NV;?(Eq. (4)). So the mea-\nsured turbulence is likely to be driven in the VLISM. Given\nvobs=lobs(\u00198:3\u000210\u000010s\u00001)>\u0017in(\u00191:8\u000210\u000010s\u00001), the\nLMHD turbulence measured by Voyager 1 should be injected\nin the decoupled regime.\n3.CONSTRAINT ON THE INJECTION SCALE OF THE\nLMHD TURBULENCE IN THE VLISM\nWe now discuss the constraint on the injection scale Lof\nthe LMHD turbulence. The three cases we consider are\n(a) super-Alfv ´enic turbulence with isotropic injection scale,\n(b) sub-Alfv ´enic turbulence with isotropic injection scale,\nand (c) sub-Alfv ´enic turbulence with anisotropic injection\nscale. In Fig. 1, we illustrate the scalings of super- and sub-\nAlfv ´enic turbulence (see also Appendix A) for isotropic and\nanisotropic injection.\nThe condition for the LMHD turbulence to arise in ions\nalone is\n\u0000L>\u0017in; (6)where \u0000Lis the driving rate of the turbulence, i.e., the cas-\ncading rate of turbulence at L, and\u0017in=\rd\u001anis the ion-\nneutral collisional frequency.\nCase (a). If the driven turbulence is super-Alfv ´enic, i.e.,\nMA=VL=VAi>1, whereVAi=B=p4\u0019\u001aiis the Alfv ´en\nspeed in ions, VLis related to vlby (Lazarian & Vishniac\n1999)\nvl=VL\u0010l\nL\u00111\n3; (7)\nwherevlis the local turbulent velocity measured at length\nscalel(< L). With the cascade of turbulent energy, the tur-\nbulent velocity becomes equal to VAiat the Alfv ´enic scale\nlA=LM\u00003\nA(Lazarian 2006). Below lA, the effect of mag-\nnetic fields on turbulence becomes important, resulting in tur-\nbulence anisotropy (see Fig. 1). Therefore, lin the above\nequation should be replaced by l?whenlis smaller than\nlA, wherel?is the length scale measured perpendicular to\nthe local magnetic field (Cho & Vishniac 2000). By using\n\u0000L=VL=Land the scaling relation in Eq. (7), the condition\nEq. (6) in this case becomes\nL<\u0010\n\u0017inv\u00001\nll1\n3\u0011\u00003\n2; lA1, isotropic injection\n (b)MA<1, isotropic injection\n (c)MA<1, anisotropic injection\nFigure 1. Illustration for turbulent eddies in super-Alfv ´enic (a) and sub-Alfv ´enic (b,c) turbulence. Strong MHD turbulence is indicated by\nshaded regions. With anisotropic injection scale of turbulence in (c), the entire turbulent cascade can be in the strong turbulence regime. The\nblue line indicates the magnetic field.\nwith the estimated uncertainty \u001b(L)\u00181:5\u00021015cm. The\nsuper-Alfv ´enic condition MA>1requiresL > lA, which\nis not satisfied by the above values. We conclude that the\nLMHD turbulence cannot be super-Alfv ´enic.\nCase (b). If the driven turbulence is sub-Alfv ´enic, i.e.,\nMA=VL=VAi<1, the MHD turbulence is weak with weak\ninteractions between counterpropagating Alfv ´en wave pack-\nets (Galtier et al. 2000) over scales [L;l tran], whereltran=\nLM2\nAis the perpendicular transition scale from weak to\nstrong MHD turbulence (Lazarian & Vishniac 1999) (see Fig.\n1). For weak turbulence, there is no parallel cascade. Its cas-\ncade to smaller perpendicular scales strengthens until the cas-\ncade becomes strong (Goldreich & Sridhar 1997). The scal-\ning of weak turbulence follows vl=VL(l?=L)1=2(Lazarian\n& Vishniac 1999), while in the strong turbulence regime, the\nlocal turbulent velocity follows the scaling (Lazarian & Vish-\nniac 1999)\nvl=VL\u0010l?\nL\u00111\n3M1\n3\nA: (12)\nFor the weak turbulence at L, there is\n\u0000L=VL\nLMA: (13)\nBy combining Eqs. (12) and (13), the condition in Eq. (6)\nbecomes\nL<\u0017\u00002\ninv3\nll\u00001\n?V\u00001\nAi: (14)\nThe spectral indices of magnetic fluctuations in weak and\nstrong MHD turbulence are \u00002and\u00005=3, respectively. With\nthe Kolmogorov slope ( \u00005=3) found for the measured mag-\nnetic energy spectrum (Burlaga et al. 2018; Lee & Lee\n2020), we consider that the locally measured turbulence is\nin the strong MHD turbulence regime. For the driven sub-\nAlfv ´enic turbulence, the condition in Eq. (14) gives\nL<3:6\u00021014cm\u0010ne\n0:07cm\u00003\u00111\n2\u0010nH\n0:2cm\u00003\u0011\u00002\n\u0010vl\n2:5km s\u00001\u00113\u0010l?\n3:0\u00021014cm\u0011\u00001\u0010B\n5\u0016G\u0011\u00001\n;(15)with\u001b(L)\u00183:8\u00021014cm. By using Eqs. (12) and (13),\nthe condition in Eq. (6) can also be written as\nVL<(\u0017\u00001\ninv3\nll\u00001\n?)1\n2\n<5:2km s\u00001\u0010nH\n0:2cm\u00003\u0011\u00001\n2\u0010vl\n2:5km s\u00001\u00113\n2\n\u0010l?\n3:0\u00021014cm\u0011\u00001\n2;(16)\nwith\u001b(VL)\u00182:7km s\u00001. The corresponding MAis\nMA=VL\nVAi<0:13\u0010ne\n0:07cm\u00003\u00111\n2\u0010nH\n0:2cm\u00003\u0011\u00001\n2\n\u0010vl\n2:5km s\u00001\u00113\n2\u0010l?\n3:0\u00021014cm\u0011\u00001\n2\u0010B\n5\u0016G\u0011\u00001\n;\n(17)\nwith\u001b(MA)\u00180:072. Given such a small MAvalue, the\nimpliedltran=LM2\nAis.6:1\u00021012cm. This means that\nthe observed turbulence would be in the weak MHD turbu-\nlence regime with lobs> l tran. This is inconsistent with the\nobserved Kolmogorov spectrum for strong MHD turbulence.\nCase (c). In the above calculations we assume that the\ninjection scale of turbulence is isotropic. In the presence of\nstrong background magnetic fields, the driven sub-Alfv ´enic\nturbulence is likely to have anisotropic L(see e.g., Pogorelov\net al. 2017). With a sufficiently small perpendicular compo-\nnent of injection scale L?, the shear in the direction perpen-\ndicular to the magnetic field can cause significant distortions\nof magnetic field lines within the Alfv ´en wave period. If\nthe anisotropy is sufficiently large so that the nonlinear in-\nteraction between counterpropagating Alfv ´en wave packets\nis strong and thus the critical balance relation (Goldreich &\nSridhar 1995) is satisfied at L, the entire turbulent cascade\nwould be in the strong MHD turbulence regime (see Fig. 1).\nIn this case, we have\nvl=VL\u0010l?\nL?\u00111\n3; (18)5\nand\u0000L=VL=L?. The condition in Eq. (6) leads to the\nconstraint on the perpendicular injection scale,\nL?<\u0010\n\u0017inv\u00001\nll1\n3\n?\u0011\u00003\n2\n\u00192:9\u00021015cm\u0010nH\n0:2cm\u00003\u0011\u00003\n2\u0010vl\n2:5km s\u00001\u00113\n2\n\u0010l?\n3:0\u00021014cm\u0011\u00001\n2;\n(19)\nwith the uncertainty \u001b(L?)\u00181:5\u00021015cm, and the same\nconstraint on VLas in Eq. (16).\nAmong the three cases, only Case (c), i.e., sub-Alfv ´enic\nturbulence with anisotropic injection scale, provides the self-\nconsistent result. In Fig. 2, we present 2\u0019=L?;maxtogether\nwith the observationally measured magnetic energy spectrum\ntaken from Lee & Lee (2020) for the period from 2012 Au-\ngust to 2019 December and Burlaga et al. (2018) for inter-\nvals 2013.3593-2014.6373 and 2015.3987-2016.6759. Here\nL?;max(Eq. (19)) is the largest possible perpendicular injec-\ntion scale of the LMHD turbulence in the VLISM. We find\nthatL?;maxis close toldam,NV;?of the interstellar turbulence\ngiven by Eq. (4).\n4.DISCUSSION\n4.1. New regime of MHD turbulence\nIn the case when the NV damping dominates over the IN\ndamping (see Section 2.1), the turbulent motions are damped\nat the neutral viscous scale, but the magnetic fluctuations can\nexist on smaller scales. There is a new regime of MHD tur-\nbulence in the sub-viscous range (Cho et al. 2002; Lazarian\net al. 2004; Xu & Lazarian 2016), where the kinetic energy\nspectrum is steep with the spectral index \u00004and the mag-\nnetic energy spectrum is flat with the spectral index \u00001. We\nnote that such a flat magnetic energy spectrum corresponds\nto scale-independent magnetic fluctuations. The sub-viscous\nmagnetic fluctuations are caused by the shear of viscous-\nscale turbulent eddies in the direction perpendicular to the\nmagnetic field. As the IN damping suppresses magnetic\nfluctuations, the damping scale of the sub-viscous magnetic\nfluctuations is determined by the balance between the eddy-\nturnover rate at ldam,NV;?and the IN damping rate,\nv\u0017\nldam,NV;?=!d,IN; (20)\nwherev\u0017=VL;ISM(ldam,NV;?=LISM)1=3is the turbulent ve-\nlocity atldam,NV;?. The IN damping rate in this case is\n!d,IN=\u0018nk2\n?(\u000eVA)2\n2\u0017ni; (21)\nwhere\u000eVA=\u000eB=p4\u0019\u001a, andk2\n?(\u000eVA)2is the magnetic\nforce per unit mass per unit displacement corresponding to\nthe sub-viscous magnetic fluctuation \u000eBperpendicular to the\nmagnetic field. Eq. (21) is different from the expression for\nIN damping rate of Alfv ´en waves in Kulsrud & Anderson(1992), as over sub-viscous scales, there are no Alfv ´en wave\nmotions of magnetic fields. By assuming that the Alfv ´enic\ncomponent dominates the turbulent motion, we approxi-\nmately have \u000eVA\u0019v\u0017. Then we obtain from Eq. (20)\nldam,IN;?;sub-v=\u0018n\n2\u0017\u00001\n2\nni\u00171\n2n\n\u00191:0\u00021015cm\u0010T\n6300 K\u00111\n4\u0010ne\n0:07cm\u00003\u0011\u00001\n2\n\u0010nH\n0:2cm\u00003\u0011\u00001\n2;\n(22)\nwith\u001b(ldam,IN;?;sub-v)\u00188:5\u00021013cm, as the IN damping\nscale of sub-viscous magnetic fluctuations (see Fig. 2). We\nsee thatldam,IN;?;sub-vis slightly smaller than ldam,NV;?in Eq.\n(4). So the cutoff scale of magnetic fluctuations is close to the\ncutoff scale of the kinetic energy spectrum of the interstellar\nturbulence.\nFor the LMHD turbulence driven in the VLISM, the new\nregime of MHD turbulence is also expected below the (ef-\nfective) ion viscous scale. As pointed out in Fraternale &\nPogorelov (2021), the small-scale spectral flattening with the\nspectral index\u00001(see also Fig. 2) may be accounted for by\nthe sub-viscous magnetic fluctuations in the new regime of\nLMHD turbulence (Cho et al. 2002; Xu & Lazarian 2016).\n4.2. Compressibility of the LMHD turbulence\nIn our calculations, we assume that the Alfv ´enic compo-\nnent carries most of the turbulent energy in compressible\nMHD turbulence. This is supported by compressible MHD\nturbulence simulations (Cho & Lazarian 2002; Hu et al.\n2022), as well as the higher power of perpendicular magnetic\nfluctuations compared to that of parallel magnetic fluctua-\ntions found in Lee & Lee (2020). We consider strong MHD\nturbulence with strong nonlinear interactions between oppo-\nsitely directed Alfv ´en wave packets and balanced cascade\n(Goldreich & Sridhar 1995) based on the Kolmogorov mag-\nnetic energy spectrum reported in e.g., Burlaga et al. (2018);\nLee & Lee (2020). But we also note the existence of large-\nscale compressive component of magnetic fluctuations with\nthe spectral index close to \u00002(Fraternale & Pogorelov 2021),\nwhich probably reflects the discontinuities in magnetic field\ndistribution associated with shock/compression waves. As\nthe conversion from Alfv ´en modes to compressive modes of\nMHD turbulence is inefficient (Cho & Lazarian 2002), the\ncompressive component is likely to originate from the turbu-\nlence injection. By combining the in-situ data of magnetic\nand electron density fluctuations, Lee & Lee (2020) found\nthat the observations cannot be explained by the linear mag-\nnetohydrodynamic modes alone. In addition, within the lim-\nited range of measured frequencies/scales, it is difficult to\ndistinguish between the spectral indices \u00005=3and\u00002in ob-\nservations, and thus the possibility of the existence of weak\nturbulence cannot be completely excluded.\nDue to the damping effects in the presence of neutrals, we\nfind that the measured turbulence in the VLISM is likely to\nbe entirely of heliospheric origin. Moreover, the turbulent6\n10-1610-1410-1210-1010-8\nk, cm-110-810-610-410-2100102Spectral density G2 cm\nldam,IN, ,sub-v\nk-5/3L,max\nldam,NV,\nFigure 2. The data for the magnetic energy spectrum are taken from Lee & Lee (2020) (black) and Burlaga et al. (2018) (red and blue).\nThe dashed line indicates the Kolmogorov slope. The vertical dashed and dash-dotted lines correspond to the damping scales of interstellar\nturbulence ( ldam,NV ;?, Eq. (4)) and interstellar magnetic fluctuations ( ldam,IN ;?;sub-v, Eq. (22)). The vertical solid line corresponds to the largest\npossible perpendicular injection scale ( L?;max, Eq. (19)) of the locally driven sub-Alfv ´enic turbulence in the VLISM.\nenergy transfer rate of the LMHD turbulence is \u000fLMHD\u0018\n\u001aiV3\nL=L?\u00195:7\u000210\u000024erg cm\u00003s\u00001, which is much larger\nthan\u000fISM\u0018\u001aV3\nL;ISM=LISM\u00191:4\u000210\u000026erg cm\u00003s\u00001of\nthe interstellar turbulence. This is consistent with the find-\ning in Lee & Lee (2020); Ocker et al. (2021) that the Kol-\nmogorov electron density spectrum measured by Voyager 1\nhas a significantly higher intensity than that measured in the\ninterstellar warm ionized medium (Armstrong et al. 1995;\nChepurnov & Lazarian 2010). The quasiperiodic structures\nseen in the magnetic field fluctuations also indicate the he-\nliospheric forcing of the measured turbulence in the VLISM\n(Fraternale & Pogorelov 2021). Both solar rotation and solar\ncycle may play an important role in driving the heliospheric\nturbulence (Zank et al. 2019).\nWith large observational uncertainties, the transition from\nthe magnetic fluctuations parallel to the mean magnetic field\nin an earlier interval to those transverse to the mean mag-\nnetic field in later intervals was found from Voyager 1 and2\nmeasurements (Burlaga et al. 2018; Zhao et al. 2020; Burlaga\net al. 2022). Magnetic fluctuations parallel to the local mag-\nnetic field arising from slow and fast modes in compress-\nible MHD turbulence and pseudo-Alfv ´enic modes in incom-\npressible MHD turbulence are important for the mirror diffu-\nsion of particles (Xu & Lazarian 2020; Lazarian & Xu 2021).\nZank et al. (2019) has shown that the fluctuations measured\nbyVoyager 1 and2can be explained by a transmission of\nfast modes from inner heliosheath with a conversion to in-\ncompressible Alfv ´en waves. They further assumed that thethe outer scale of this heliosphere-originated turbulence is\naround 120 au, and the interstellar turbulence will continue\ndown to scales smaller than those measured by Voyager . Our\nwork, by considering the ion-neutral interactions in the LISM\nregion, suggests that the interstellar turbulence should have a\ncutoff around 261au, and a new heliosphere-related turbu-\nlence (such as the scenario described by Zank et al. (2019))\nwith an injection scale smaller than \u00182:9\u00021015cm (\u0019194\nau) is needed to explain the Voyager measurements.\n5.CONCLUSIONS\nThe damping effects in the partially ionized LISM deter-\nmine the range of length scales for the existence of inter-\nstellar MHD turbulence and the LMHD turbulence. Due to\nthe high temperature and moderate ionization fraction in the\nLISM, we find that the dominant damping mechanism of the\ninterstellar MHD turbulence is the NV damping. The NV\ndamping scale of the interstellar turbulence is about 261au.\nBelow the NV damping scale of turbulent cascade, the new\nregime of MHD turbulence with constant magnetic fluctua-\ntions (Lazarian et al. 2004) is expected to rise. We find that\nthe sub-viscous magnetic fluctuations are cut off due to the\nIN damping at a scale slightly smaller than the NV damping\nscale.\nFor the LMHD turbulence, when the injection occurs in\nthe regime with ions decoupled from neutrals, the turbulent\ncascade can not be cut off by the damping related to partial\nionization. Given the turbulent velocity at the largest ob-7\nserved length scale inferred from in-situ measurements, after\napplying the ion-neutral decoupling condition, we find that\nthe LMHD turbulence in the VLISM is sub-Alfv ´enic with\nthe injected turbulent energy smaller than the magnetic en-\nergy. With the trajectory of Voyager 1 approximately per-\npendicular to the background magnetic field, by assuming\nanisotropic injection scale of the LMHD turbulence, we fur-\nther find the upper limit of the perpendicular injection scale\nL?;max\u00192:9\u00021015cm\u0019194au, which is close to the\nNV damping scale of the interstellar turbulence. Our es-\ntimated largest outer scale of LMHD turbulence is compa-\nrable to the extent of the heliosphere in the upstream di-\nrection (Pogorelov et al. 2017) and one order of magnitude\nsmaller than the estimate given in Burlaga et al. (2018); Lee\n& Lee (2020) by extrapolating the power-law slope to the\nequipartition between the turbulent fluctuation and the aver-\nage magnetic field strength. Note that other considerations\nsuch as mode conversion (Zank et al. 2019) and IBEX mod-eling (Zirnstein et al. 2020) could further limit the injection\nscale down to tens of au.\nACKNOWLEDGMENTS\nWe thank the referees for very detailed and helpful com-\nments, which improved the manuscript perceptibly. S.X. ac-\nknowledges inspiring discussions with Alex Lazarian, Ethan\nVishniac, and the support for this work provided by NASA\nthrough the NASA Hubble Fellowship grant # HST-HF2-\n51473.001-A awarded by the Space Telescope Science Insti-\ntute, which is operated by the Association of Universities for\nResearch in Astronomy, Incorporated, under NASA contract\nNAS5-26555. H.L. acknowledges useful discussions with\nFan Guo, Eric Zirnstein, and the support by LANL LDRD\nprogram.\nSoftware: MATLAB (MATLAB 2021)\nAPPENDIX\nA.ANISOTROPY OF STRONG MHD TURBULENCE\nThe theoretically established scale-dependent anisotropy of trans-Alfv ´enic turbulence (Goldreich & Sridhar 1995) and sub- and\nsuper-Alfv ´enic turbulence (Lazarian & Vishniac 1999) has been tested by MHD turbulence simulations (Cho & Lazarian 2002,\n2003) and supported by spacecraft measurements in the solar wind (Horbury et al. 2008; Luo & Wu 2010; Forman et al. 2011).\nHere we briefly review the anisotropy of strong Alfv ´enic turbulence.\nThe cascading rate of Alfv ´enic turbulence in strong MHD turbulence regime is\n\u001c\u00001\ncas=vll\u00001\n?=VstL\u00001\n3\nstl\u00002\n3\n?; (A1)\nwhereVstis the turbulent velocity at the outer scale Lstof strong MHD turbulence. More specifically, there is\nVst=VA;Lst=lA=LM\u00003\nA (A2)\nfor super-Alfv ´enic turbulence with MA>1, and\nVst=VLMA;Lst=ltran=LM2\nA (A3)\nfor sub-Alfv ´enic turbulence with MA<1,. We note that VAshould be replaced by VAiwhen the turbulence is injected in the\ndecoupled regime.\nBy combining the expression of \u001c\u00001\ncaswith the critical balance relation\n\u001c\u00001\ncas=VA\nlk; (A4)\nwe can obtain the anisotropic scaling relation of strong Alfv ´enic turbulence,\nlk=VA\nVstL1\n3\nstl2\n3\n?: (A5)\nIt shows that smaller-scale turbulent eddies are more elongated along the local magnetic field.\nB.DAMPING SCALES OF MHD TURBULENCE CASCADE IN A PARTIALLY IONIZED MEDIUM\nUnder the consideration of both IN and NV damping effects in a partially ionized medium, Xu et al. (2015, 2016); Xu &\nLazarian (2017) derived the general expression of the damping rate of Alfv ´enic turbulence. Here we briefly review its approximate\nform in different coupling regimes.8\nIn the weakly coupled regime, there is only the IN damping, with the damping rate\n!d=\u0017in\n2: (B6)\nFor the MHD turbulence injected in the decoupled regime, as the cascading rate is always larger than \u0017inand thus larger than !d,\nthe cascade is not cut off by the damping in a partially ionized medium.\nIn the strongly coupled regime, the damping rate can be approximately written as\n!d=\u0018n\n2\u0010\nk2\u0017n+k2\n?(\u000eVA)2\n\u0017ni\u0011\n=\u0018n\n2\u0010\nk2\u0017n+k2\nkV2\nA\n\u0017ni\u0011\n; (B7)\nwhere we assume that the magnetic fluctuations are mainly induced by Alfv ´enic turbulence and apply the critical balance relation\nfor strong MHD turbulence, k?\u000eVA=k?vk=kkVA. Herevkis the turbulent velocity at wavenumber k. The first and second\nterms of!dcorrespond to NV and IN damping, respectively.\nFor the MHD turbulence injected in the strongly coupled regime, when the damping rate exceeds the cascading rate, MHD\nturbulence cascade is damped. We consider that the damping scale is in the strong MHD turbulence regime. By comparing Eq.\n(B7) with Eq. (A1), we find the damping scale\nldam,NV;?=\u0010\u0018n\n2\u00113\n4\u00173\n4nL1\n4\nstV\u00003\n4\nst (B8)\nwhen the NV damping dominates over the IN damping, where we assume k\u0018k?, and\nldam,IN;?=\u00102\u0017ni\n\u0018n\u0011\u00003\n2L\u00001\n2\nstV3\n2\nst (B9)\nin the opposite case. If the MHD turbulence is damped due to neutral viscosity in the strongly coupled regime, there is the\nso-called new regime of MHD turbulence on scales below the NV damping scale (Lazarian et al. 2004). 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J., Giacalone, J., Kumar, R., et al. 2020, ApJ, 888, 29,\ndoi: 10.3847/1538-4357/ab594d" }, { "title": "1804.00554v1.Anisotropic_Gilbert_damping_in_perovskite_La___0_7__Sr___0_3__MnO___3___thin_film.pdf", "content": "Anisotropic Gilbert damping in perovskite La 0:7Sr0:3MnO 3thin film\nQing Qin,1Shikun He*,2Haijun Wu,1Ping Yang,1, 3Liang Liu,1\nWendong Song,2Stephen John Pennycook,1and Jingsheng Chen*1\n1Department of Materials Science and Engineering,\nNational University of Singapore, Singapore 117575\n2Data Storage Institute, Agency for Science, Technology and Research (A*STAR),\n2 Fusionopolis Way 08-01 Innovis, Singapore 138634\u0003\n3Singapore Synchrotron Light Source (SSLS),\nNational University of Singapore, 5 Research Link, Singapore 117603\n1arXiv:1804.00554v1 [cond-mat.mtrl-sci] 2 Apr 2018Abstract\nThe viscous Gilbert damping parameter governing magnetization dynamics is of primary importance for\nvarious spintronics applications. Although, the damping constant is believed to be anisotropic by theories.\nIt is commonly treated as a scalar due to lack of experimental evidence. Here, we present an elaborate\nangle dependent broadband ferromagnetic resonance study of high quality epitaxial La 0:7Sr0:3MnO 3films.\nExtrinsic effects are suppressed and we show convincing evidence of anisotropic damping with twofold\nsymmetry at room temperature. The observed anisotropic relaxation is attributed to the magnetization\norientation dependence of the band structure. In addition, we demonstrated that such anisotropy can\nbe tailored by manipulating the stain. This work provides new insights to understand the mechanism of\nmagnetization relaxation.\nA. INTRODUCTION\nThe magnetization relaxation process determines the speed of magnetization relaxation and the\nenergy required for current-induced magnetization reversal [1–6]. Understanding the mechanism\nand controlling of magnetization relaxation [7–12], including intrinsic Gilbert damping and extrinsic\neffects, pave the way for ultra-low power and high performance spintronic devices based on spin\ntransferandspinorbittorques[13–15]. IthasbeendemonstratedthatGilbertdampingconstant( \u000b)\ncanbetunedeffectivelybyengineeringthedensityofstatesandspinorbitcoupling(SOC)[9,16–18].\nIn addition, magnetization relaxations subjected to finite size and interfacial effects have also been\nextensively investigated [8, 19, 20]. However, it is still an open question that if magnetic damping\nis anisotropic. In principle, \u000bis magnetization orientation dependent and should be a 3 \u00023 tensor\nin the phenomenological Gilbert equation [21, 22], yet it is often treated as a scalar (isotropic). In\nthe case of polycrystalline thin films prepared by sputtering, such treatment is reasonable due to\nthe smearing of long range structural order. Whereas for single crystal thin films, it is still difficult\nto draw a conclusion due to the lack of convincing experimental evidence. From the view of theo-\nries, the Gilbert damping is determined by two scattering processes, the interband resistivity-like\nscattering and the intraband conductivity-like scattering [12]. Both terms vary with temperature\nthrough their dependence on electron relaxation time. The interband scattering which dominates\ndamping in most ferromagnets becomes isotropic at room temperature [23]. Therefore, anisotropic\nlinewidth in 3d magnetic metals was only observed at low temperature[24]. From the aspect of\nexperimental technique, Seib et al. have predicted that the precession trajectory of magnetization\nin a ferromagnetic resonance (FMR) measurement (standard technique for measuring damping)\nmay partially average out the anisotropy [25]. Hence, detecting the anisotropy in Gilbert damping\nis extremely difficult. Furthermore, the existence of several angle dependent extrinsic contributions\nto damping in most materials further hinders the determination of a possible weak anisotropic\ndamping [11, 26–28]. We note that in a ferromagnet with nearly half-metallic band structure, the\nisotropic interband term is suppressed [29] and the damping can be dominated by the anisotropic\nintraband contribution[23]. Recent reports have claimed the observation of anisotropic damping in\nhalf-metallic Heusler alloy[30, 31]. However, unavoidable chemical disorder [32, 33]of Heusler alloy\nintroduces extrinsic effects such as spin wave scattering hence complicates the verification procedure\nof such anisotropy.\n\u0003heshikun@gmail.com msecj@nus.edu.sg\n2La0:7Sr0:3MnO 3(LSMO) is an oxide perovskite material exhibited half-metallic band structure\nand ultra-low damping at room temperature [34, 35]. In this work, we studied the magnetiza-\ntion relaxation of LSMO films deposited on NdGaO 3(NGO) (110) substrates using angle-resolved\nbroadband ferromagnetic resonance. The purpose of choosing NGO (110) substrates is to utilize its\nnon-equalaandbaxis value. Such asymmetry will potentially lead to non-spherical Fermi surface.\nTwo types of high quality samples with different static magnetic anisotropies were investigated. The\nnormal LSMO film (hereafter denoted as S-LSMO) exhibited weak uniaxial magnetic anisotropy\nwhereas the other with modulated strain relaxation mode (hereafter denoted as W-LSMO) have\nboth uniaxial and cubic anisotropy fields. The angle dependence of the in-plane intrinsic Gilbert\ndampingshowedtwo-foldsymmetryinbothtypeofsamples. Strikingly, theorientationofminimum\ndamping differs 90 degree. This work provided strong evidence of anisotropic nature of magneti-\nzation relaxation and demonstrated the tuning of anisotropy in damping through stress relaxation\nengineering.\nB. RESULTS\nEpitaxial growth of LSMO\nPulsed laser deposition (PLD) was used to deposit LSMO thin films with a thickness of 25nm\non (110) NGO substrates. The energy and repetition frequency of KrF laser (248nm) were 225mJ\nand 2Hz, respectively. During deposition, the substrate temperature was fixed at 950\u000eC. The\noxygen pressure was 225mTorr for S-LSMO and 200mTorr for W-LSMOAfter deposition, S-LSMO\nwas cooled down to room temperature at 10K/min under the oxygen pressure of 1 Torr, whereas\nW-LSMO at 5K/min under the oxygen pressure of 100 Torr in order to promote the modification\nof strain hence micro-structurestructure.\nCrystalline quality analysis\nThe crystallographic structures of the films were characterized by synchrotron high resolution\nX-ray diffraction. Reciprocal space maps (RSMs) taken at room temperature around {013} pc(here\nthe subscript pc stands for pseudocubic) reflections confirm the epitaxial growth of LSMO layers\non the NGO substrate as shown in Fig. 1 (a). The vertical alignment of LSMO and NGO reciprocal\nlatticepointclearlyshowsthattheLSMOfilmiscompletelystrainedontheNGOsubstrate. Lattice\nmismatch along [100] pcand [010] pcare 1.03% and 0.8%, respectively. Considering the position of\nthe LSMO reciprocal lattice point in the {013} pcmappings, equal Lvalues of (103) pcand (-103) pc\nindicates the perpendicular relation between vector aandcin the lattice, whereas different L values\nfor (013) pcand (0-13) pcshows that the angle between bandcis not equal to 90Âř. Thus, the LSMO\nis monoclinic phase which is consistent with previous reports [36]. The good crystalline quality was\nfurther verified by aberration-corrected scanning transmission electron microscopy (AC-STEM).\nFig. 1 (b, c) are the simultaneously acquired high angle annular dark field (HAADF) and annular\nbright field (ABF) images of S-LSMO along [100]pc direction, while Fig. 1 (d, e) are for [010] pc\ndirection. The measurement directions can be differentiated from the diffraction of NGO substrate:\n31/2[010] superlattices for [100] pcdirection (inset of Fig. 1(c)) and 1/2[101] superlattices for [100] pc\ndirection (inset of Fig. 1(e)). High quality single crystalline films are essential for the present\npurposes because high density of defects will result in spin wave scattering [26].\nMagnetic anisotropy fields\nThe magnetic dynamic properties were investigated by a home-built angle-resolved broadband\nFMR with magnetic field up to 1.5T. All measurements were performed at room temperature.\nShown in Fig. 2(a) is the color-coded plot of the transmission coefficient S21 of the S-LSMO sample\nmeasured at 10GHz. 'His the in-plane azimuth angle of the external magnetic field counted\nfrom [010] pcdirection (Fig. 2(b)). This relative orientation was controlled by a sample mounting\nmanipulator with a precision of less than 0.1\u000e. The olive shape of the color region indicates the\nexistence of anisotropy field, whereas the very narrow field region of resonances is an evidence of low\ndamping. Three line cuts at 'H=0, 45 and 90 degrees are plotted in Fig. 2(c), showing the variation\nof both FMR resonance field ( Hres) and line shape with 'H. All curves are well fitted hence both\ntheHresand resonance linewidth \u0001H are determined. The 'Hdependence of H resat two selected\nfrequencies (20 and 40 GHz) are shown in Fig. 2(d) for S-LSMO. The angle dependencies of the\nresonance field Hres('H)is calculated starting from the total energy [37]:\nE=\u0000MH [cos\u0012Hcos\u0012M+ sin\u0012Hsin\u0012Mcos('M\u0000'H)] + 2\u0019M2cos2\u0012M\u00001\n2MH 2?cos2\u0012M\n\u00001\n4MH 4?cos4\u0012M\u00001\n2MH 2ksin2\u0012Mcos2('M\u0000\u001e2IP)\u00001\n4MH 4k3+cos 4('M\u0000\u001e4IP)\n4sin4\u0012M(1)\nwhere\u0012Mand'Mare the polar angle and the azimuth angle of the magnetization ( M),H2?,\nH4?,H2k,H4kare the uniaxial and cubic out-fo-plane and in-plane anisotropy fields. The easy axes\nof in-plane anisotropies are along \u001e2IPand\u001e4IP, respectively. According to Smit-Beljers equation\nthe resonance condition for \u0012M=\u0019/2 is [38]:\n2\u0019f=\r\nMsin\u0012p\nE\u0012\u0012E'' (2)\nHere,E\u0012\u0012=Hrescos('M\u0000'H) + 4\u0019Me\u000b\u0000H2kcos2('M\u0000\u001e2IP) +H4k(3 + cos 4('M\u0000\u001e4IP)=4)and\nE''=Hrescos('M\u0000'H)+H2kcos 2('M\u0000\u001e2IP)+H4kcos 4('M\u0000\u001e4IP)aresecondpartialderivatives\nof the total energy with respect to the polar and azimuth angles. \r=1.76\u0002107s\u00001G\u00001denotes the\ngyromagnetic ratio, 4\u0019Me\u000b= 4\u0019M\u0000H2?is the effective magnetization. The resonance field of\nS-LSMO shows pronounced minimum at 'H=n\u0001\u0019, indicating the existence of uniaxial magnetic\nanisotropy with easy axis along \u001e2IP= 0or [010] pcdirection. Cubic anisotropy is negligible hence\nH4k=0. Such uniaxial anisotropy observed in S-LSMO is consistent with previous reports [39],\nwhich is attributed to anisotropic strain produced by the NGO(110) substrate [40–42]. Compared\nto the resonance fields in our measurement, the magnetic anisotropy fields are orders of magnitude\nsmaller. Therefore, the calculated difference between 'Hand'Mare always smaller than 1\u000eand\n'='H='Mis assumed in the following discussion.\n4Magnetization orientation dependence of Gilbert damping\nIn order to study the symmetry of magnetization relaxation of the sample. The FMR linewidth\n\u0001Hfor a matrix of parameter list (72 field orientations and 36 frequency values) are extracted.\nThe results are shown by 3-D plots in Fig. 3(a) . Here, zaxis is \u0001Handx,yaxes aref\u0001cos'\nandf\u0001sin', respectively. The figure clearly shows that the linewidth depends on magnetization\norientation. At a given frequency, the linewidth is maximum (minimum) at '= 0('=\u0019=2) for\nS-LSMO. Fig. 3(c) shows the \u0001Hversus frequency for three field orientations. The FMR linewidth\ndue to intrinsic magnetic damping scales linearly with frequency \u0001HGL= 4\u0019\u000bf=\r cos ('M\u0000'H)\naccording to Laudau-Lifshitz-Gilbert phenomenological theory [43, 44]. However, a weak non-\nlinearity in the low frequency range can be identified. In general, extrinsic linewidth contributions\nsuch as inhomogeneity and magnon scattering will broaden the FMR spectrum hence result in\nadditionallinewidthcontributionsscalesnon-linearlywithfrequency[9,11]. Theinterfacialmagnon\nscattering is suppressed due to relative large film thickness (25 nm) and the bulk magnon scattering\ncontribution to the linewidth is negligible in our samples with very good atomic order. However, the\nstatic magnetic properties of the thin film may vary slightly in the millimeter scale. Since the FMR\nsignal is an averaged response detected by the coplanar waveguide (5mm long), a superposition of\nlocation resonance modes broadens the FMR spectrum. Such well-known contribution to linewidth,\ndefined as \u0001Hinhom, are generally treated as a constant [9, 44, 45]. However, it is frequency\ndependent for in-plane configuration and need to be treated carefully for samples with ultra-low\ndamping. Here, we fit the data with \u0001H= \u0001HGL+ \u0001Hinhom, taking into account the frequency\nand orientation dependence of \u0001Hinhom. As can be seen from Fig. 3(c), the data are well reproduced\nfor every field orientations. Hence, the magnetization orientation dependence of intrinsic damping\nconstant is determined and plotted in Fig. 3(e). Remarkably, the damping constant shows two-fold\nsymmetry. The lowest damping of S-LSMO with in-plane magnetization, observed at '= 0and\n'=\u0019, is(8:4\u00060:3)\u000210\u00004and comparable to the value measured under a perpendicular field\n(Tbl. I). The maximum damping at '=\u0019=2and'= 3\u0019=2is about 25% higher.\nSince the magnetization damping and resonance field of the S-LSMO sample exhibited identical\nsymmetry (Fig. 2 (d) and Fig. 2(e)), it seems that the observed anisotropic damping is directly\nrelatedtocrystallineanisotropy. Therefore, wepreparedtheW-LSMOsamplewithslightlydifferent\nstructureandhencemodifiedstaticmagneticanisotropyproperties. TheW-LSMOsampleexhibited\n1D long range atomic wave-like modulation [36] (twining domain motif) along [100] pcaxis near the\ninterface between substrate and film. Due to different strain relaxation mechanism as compared to\nS-LSMO, the 'Hdependence of Hresfor the W-LSMO have additional features and can only be\nreproduced by including both H2k(13:9\u00060:9Oe) andH4k(11:8\u00061:2Oe) terms. The easy axis of\nthe uniaxial anisotropy ( \u001e2IP=0 ) is the same as S-LSMO whereas the additional cubic anisotropy\nis minimum at \u001e4IP=45Âř. The magnetization orientation dependence of the FMR linewidth for\nW-LSMO is significantly different (Fig. 3(b)) as compared to S-LSMO. Such change in trend can be\nclearlyidentifiedfromthefrequencydependenceoflinewidthforselectedmagnetizationorientations\nshown in Fig. 3(d). Magnetization damping values are extracted using the same procedure as S-\nLSMObecausethespinwavecontributionisexcluded. Thedampingconstantagainshowedtwo-fold\nin-plane symmetry. However, in contrast to S-LSMO, the maximum damping value of W-LSMO is\nobserved at '= 0and'=\u0019.\n54\u0019Meff(T)H2k(Oe)H4k(Oe) \u000b? \u000b('= 0)\u000b('=\u0019=2)\nS-LSMO 0.3280 \u00060.0011 37\u000640 (8:6\u00060:5)\u000210\u00004(8:4\u00060:3)\u000210\u00004(11\u00060:6)\u000210\u00004\nW-LSMO 0.3620 \u00060.002513.9\u00060:911.8\u00061:2(4:7\u00060:7)\u000210\u00004(6:5\u00060:3)\u000210\u00004(5:3\u00060:3)\u000210\u00004\nTable I. Summary of the parameters for S-LSMO and W-LSMO samples.\nC. DISCUSSION\nAnisotropy in linewidth at low temperatures have been reported decades ago, however, data in\nmost early publications were taken at a fixed frequency in a cavity-based FMR [24, 46]. Due to\nlack of frequency dependence information, it is not clear if the anisotropy in linewidth is due to\nintrinsic damping or extrinsic effects [47–49]. In this study, besides wide range of frequencies, we\nalso adopted samples with effective anisotropy orders of magnitude smaller than the external field.\nTherefore, the field dragging effect and mosaicity broadening, both of which are anisotropic in natur\ne[50], are negligibly small and the Gilbert damping constant is determined reliably. Furthermore,\nthemechanisminthissimplesystemisdifferentfrompreviousreportsrelatedtointerfacialexchange\ncoupling and spin pumping[51, 52]. Since both S-LSMO and W-LSMO exhibited in-plane uniaxial\nmagnetic anisotropy, the opposite trends observed in these two samples exclude the existence of a\ndirect link between anisotropic damping and effective field. Both magnetic anisotropy and damping\nare related to the band structure but in quite different ways. According to perturbation theory,\nthe magnetic anisotropy energy is determined by the matrix elements of the spin-orbit interaction\nbetween occupied states. Hence, the contributions from all the filled bands must be considered to\ncalculate the absolute value of magnetic anisotropy. On the other hand, the magnetic damping is\nrelated to the density of states at the Fermi level.\nThe damping term in the Landau-Lifshitz-Gilbert equation of motion is\u000b\njMj\u0000\nM\u0002dM\ndt\u0001\n, there-\nfore, anisotropy in damping can have two origins, one related to the equilibrium orientation of\nmagnetization M(orientation anisotropy) and the other depends on the instantaneous change in\nmagnetization dM=dt(rotational anisotropy). In FMR experiments the magnetization vector ro-\ntates around its equilibrium position, therefore, the rotational anisotropy may be smeared out [25].\nThe orientation anisotropy is described by both interband and intraband scattering process. Ac-\ncording to Gilmore et al.[23], the latter is isotropic at sufficiently high scattering rates at room\ntemperature. We suspect that the anisotropic damping in LSMO is due to its half-metallic band\nstructure. As a result of high spin polarization, interband scattering is suppressed and the room\ntemperature damping is dominated by intraband scattering. The intraband contribution to damp-\ning exhibit anisotropy for all scattering rates [23] which agree well with our experiments. The\nsuppression of interband scattering is evidenced by the ultra-low damping in the order of 10\u00004.\nNotably, the absolute value of the observed anisotropy, 2.6 \u000210\u00004for S-LSMO and 1.2 \u000210\u00004for\nW-LSMO, is so small that could not be identified reliably for a material with typical damping\nvalues between 5 \u000210\u00003to 2\u000210\u00002.\nIn a microscopic picture, the Gilbert damping is proportional to the square of SOC constant ( \u0018)\nand density of states at the Fermi level, \u000b\u0018\u00182D(EF). The shape of the Fermi surface depends on\nthe orientation of the magnetization due to SOC. 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Back, Nature Physics (2018).\n9Figure 1. Structure characterization of S-LSMO sample. (a) and (b) XRD profiles around S-LSMO\n(00L) reflections (L=1,2,3,4) with the incident beam aligned along the [100] pcand [010] pc, respectively.\n(b) and (c) STEM-HAADF/ABF lattice images of S-LSMO along [100] pcdirection. (d) and (e) STEM-\nHAADF/ABF images of S-LSMO along [010] pcdirection. the insets are the intensity profile and FFT\nimage; The red dashed line indicates the interface.\n10(b)\n(c)\n10GHz(a)\n30\n21060\n24090\n270120\n300150\n330180 010GHz\n242022002420Hres(Oe)\n5.625.70\n12.6512.712.75\nHres(kOe)(d)\n40GHz\n20GHzS-LSMO\n0 90 180 270 360\n0\n0 fit\n45\n45 fit\n90\n90 fitS21 (a.u.)\n2200 2300 2400\nHres(Oe)Figure 2. Magnetic anisotropy characterization. (a) The 2D polar color plot of the FMR spectra of\nS-LSMO. The frequency is 10GHz. (b) Schematics of the FMR setup and the definition field orientation.\n(c) FMR spectra for 'H=0, 45 and 90 degrees for S-LSMO. (d) Field orientation ( 'H) dependence of the\nresonance fields ( Hres) of the S-LSMO sample at f=20 and 40GHz. The solid lines in (c) and (d) are\ncalculated values.\n11404040\n0 060\n-40\n40 4020\n0 030\n-40\nf (GHz) f (GHz)0 10 20 30 40204060ΔH (Oe)\n102030ΔH (Oe)\n0 10 20 30 40\n810120\nfit\n45\n90fit\nfit0\nfit\n45\n90fit\nfit(a) (b)\n(c) (d)\n(e) (f)ΔH (Oe)\nΔH (Oe)\nf (GHz) f (GHz)α (10-4)\n0 360 270 180 90\nφ567\n0 90 180 270 360α (10-4)\nφf (GHz) f (GHz)Figure 3. Anisotropic linewidth and damping: (a)-(b) 3-D plot of frequency and in-plane field ori-\nentation dependence of FMR linewidth. (c)-(d) frequency dependence of FMR linewidth for seleted field\norientations. Solid symbols are experimental data and the lines are calculated value. (e)-(f) Damping\nconstant as a function of '. (a),(c), (e) are for S-LSMO and (b),(d), (f) are for W-LSMO.\n12" }, { "title": "1707.06087v2.Engineering_elliptical_spin_excitations_by_complex_anisotropy_fields_in_Fe_adatoms_and_dimers_on_Cu_111_.pdf", "content": "Engineering elliptical spin-excitations by complex anisotropy\n\felds in Fe adatoms and dimers on Cu(111)\nFilipe S. M. Guimar~ aes,\u0003Manuel dos Santos\nDias, Benedikt Schwe\ringhaus, and Samir Lounisy\nPeter Gr unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J ulich & JARA, D-52425 J ulich, Germany\n(Dated: November 20, 2021)\nAbstract\nWe investigate the dynamics of Fe adatoms and dimers deposited on the Cu(111) metallic surface\nin the presence of spin-orbit coupling, within time-dependent density functional theory. The ab\ninitio results provide material-dependent parameters that can be used in semiclassical approaches,\nwhich are used for insightful interpretations of the excitation modes. By manipulating the sur-\nroundings of the magnetic elements, we show that elliptical precessional motion may be induced\nthrough the modi\fcation of the magnetic anisotropy energy. We also demonstrate how di\u000berent\nkinds of spin precession are realized, considering the symmetry of the magnetic anisotropy energy,\nthe ferro- or antiferromagnetic nature of the exchange coupling between the impurities, and the\nstrength of the magnetic damping. In particular, the normal modes of a dimer depend on the\ninitial magnetic con\fguration, changing drastically by going from a ferromagnetic metastable state\nto the antiferromagnetic ground state. By taking into account the e\u000bect of the damping into their\nresonant frequencies, we reveal that an important contribution arises for strongly biaxial systems\nand specially for the antiferromagnetic dimers with large exchange couplings. Counter intuitively,\nour results indicate that the magnetic damping in\ruences the quantum \ructuations by decreasing\nthe zero-point energy of the system.\n\u0003Electronic address: f.guimaraes@fz-juelich.de\nyElectronic address: s.lounis@fz-juelich.de\n1arXiv:1707.06087v2 [cond-mat.mes-hall] 2 Oct 2017I. INTRODUCTION\nFuture technological devices demand an understanding of quantum mechanisms in nanos-\ntructures such as single atoms and small clusters [1, 2]. Recent atomic manipulation and\nspectroscopy experiments utilizing the scanning tunneling microscope (STM) pushed for-\nward the frontiers of this area by the development of logic operations based on atomic spin\nmanipulation [3], subnanometer-sized sensors [4], magnetic stability of single adatoms [5],\nand many other atomic-scale realizations [6{11]. Additionally to miniaturizing components,\nadvanced spintronic devices also require ultrafast manipulation of the magnetic units. With\nthat aim, it is natural to investigate the dynamic processes of those magnetic building blocks\n[11{20].\nMany of those atomic-scale investigations are made on insulating surfaces, where the\nhost weakly interacts with the magnetic units (see, e.g., Refs. 5, 6, 11, 18, 21{24). On\nthe other hand, for metallic hosts, the surface and deposited structures hybridize strongly,\nmodifying the original electronic states of the isolated subsystems [15{17, 25, 26]. This\nstrong coupling leads to a noninteger magnetic moment and to broadened spin excitation\nspectra of the deposited nanostructures. It also a\u000bects the magnetic excitations of those\nsystems, changing relaxation times [27] and in\ruencing the polarization of spin currents\npumped out of the magnetic unit into the substrate [28]. These spin currents may be used\nto excite other magnetic units deposited on the surface [29], or, in the presence of spin-orbit\ncoupling, to generate charge currents [30, 31].\nWhen two or more magnetic atoms or clusters are brought together, their mutual interac-\ntion can lead to ferromagnetic, collinear antiferromagnetic, or even more complex magnetic\nstructures. Their excitation modes depend on the local environment and also on the ex-\nchange coupling between the components. In particular, antiferromagnetic systems have\nbeen studied for decades [32{34] and still generate interest in its ground state description\nand possible excitations [35, 36]. These structures play a central role in the rising and\npromising \feld of antiferromagnetic spintronics [37], in which excitations and switching can\nbe induced by spin-orbit torques, when charge currents are applied to adjacent heavy metal\nlayers [38, 39].\nIn this paper, we investigate magnetic excitations of the smallest possible nanostructures\n| adatoms and dimers, primer constituents of any ferromagnetic or antiferromagnetic sys-\n2tem | deposited on metallic surfaces. To correctly capture the mixing e\u000bects discussed\nabove, we employ \frst principles calculations based on time-dependent density functional\ntheory [25, 40, 41] taking the e\u000bects of the spin-orbit coupling into account, which can lead to\nnontrivial magnetic anisotropy \felds. We focus on the dynamic transverse magnetic suscep-\ntibility that describes the density of spin excitations and is directly related to the measured\nconductance in inelastic scanning tunneling spectroscopy (ISTS) experiments [42]. This\nquantity is de\fned as the magnetic response of the system to an oscillatory transverse mag-\nnetic \feld, as in ferromagnetic resonance experiments. We also make use of a semiclassical\nmodel to interpret the features of the precessional modes and to de\fne e\u000bective parameters.\nWe chose a prototypical system composed by Fe adatoms and dimers deposited on the\nCu(111) surface, but our results apply more generally to transition metals deposited on other\nmetallic surfaces, in a qualitative way. For a single magnetic adatom, we show how external\nmagnetic \felds can stabilize di\u000berent magnetization directions and drastically change the\nprecession shape of the motion. We also demonstrate how similar e\u000bects can be obtained by\natomic engineering [6, 43, 44], namely by bringing nonmagnetic Cu atoms to the vicinity of\nthe magnetic Fe atom, and thus modifying the magnetic anisotropy energy landscape of the\nsystem. Tilted anisotropies are also important to break the symmetries of spin-orbit torques,\nleading to deterministic switching of the magnetic unit [45]. When a second magnetic atom\nis placed close to the \frst one, the rotational symmetry is naturally broken and a biaxial\nanisotropy is induced. We also take advantage of the distance-dependent oscillatory exchange\ninteraction [7] to design structures with ferromagnetic (FM) and antiferromagnetic (AFM)\nalignments, showing how the spin excitations change by varying just the starting magnetic\ncon\fguration. For such nanosized magnetic structures, zero-point spin \ructuations can be\nof concern, and so we shall also discuss their role.\nThis paper is organized as follows. In Sec. II, we describe the \frst principles approach us-\ning time-dependent density functional theory (TDDFT) that we use to obtain the excitation\nspectra of the magnetic structures. We also derive a phenomenological formalism to aid in\nthe analysis of the ab initio results. By \ftting the results of the \frst-principles calculations\nto this model, we obtain material-dependent parameters that may be used in semiclassical\napproaches. Section III is devoted to the single adatom excitations. The magnetization\ndynamics of 3 dadatoms on the Cu(111) surface were explored in Ref. 14. Here we expand\nthis study and explore how the anisotropies a\u000bect the magnetization dynamics, focusing\n3on how they lead to elliptical precession. In Sec. IV, we investigate the excitations of Fe\ndimers, with large and relatively small interatomic magnetic couplings. In the latter case, we\nanalyze the excitations starting from two di\u000berent states: when the magnetic moments are\naligned ferromagnetically (metastable state) and antiferromagnetically (ground state). We\nshow how a simple change of the starting alignment drastically alters the excitation spectra\ndepending on the coupling, anisotropy, and damping. Finally, in Sec. V we summarize our\nresults.\nII. THEORETICAL FRAMEWORK\nA. Spin excitations from \frst-principles\nOur \frst-principles description of spin excitations is based on time-dependent density\nfunctional theory, using the linear response approach. The central object is the dynamical\nmagnetic susceptibility, which is closely related to the inelastic tunneling conductance mea-\nsured experimentally, as explained in Ref. 42. This approach requires two steps: First the\nself-consistent ground state electronic structure must be found, and then the linear response\nof the ground state to a magnetic perturbation is evaluated. These steps are brie\ry outlined\nbelow.\nThe ground state electronic structure is obtained from DFT calculations, based on the\nKorringa-Kohn-Rostoker Green function method [46] (KKR-GF), in the local spin density\napproximation (LSDA), as parametrized by Vosko, Wilk, and Nusair [47]. The scattering\nproblem is solved in the atomic sphere approximation (ASA) with `max= 3 cuto\u000b, with\nsubsequent use of the full charge density. Energy integrations are performed with a rectan-\ngular contour in the upper complex energy plane using 40 points, including \fve Matsubara\nfrequencies with temperature T= 50 K [48].\nThe electronic structure of the clusters on the Cu(111) surface is calculated in two steps.\nFirst the pristine surface is simulated using a 22-layer slab of Cu(111) planes, augmented\nwith two vacuum regions with the thickness of four bulk layers. The in-plane lattice con-\nstant is the experimental one, a= 3:615=p\n2\u0017A = 2:556\u0017A. No relaxation of the interlayer\ndistance has been considered for the slab calculation. A k mesh with 180 \u0002180 points in the\nwhole two-dimensional Brillouin zone is employed. Next, a real-space cluster is embedded\n4at the surface, including the adatoms and all surrounding nearest-neighbor positions (Cu\natoms and vacuum), and treated self-consistently. Structural optimization of the supported\nnanostructures is accounted for by a vertical relaxation of the structure towards the surface\nby 14% of the bulk interlayer distance [49, 50].\nThe key quantity for the description of spin excitations is the dynamical magnetic sus-\nceptibility. In linear response, when an external monochromatic magnetic \feld perturbs the\nsystem, this susceptibility describes the linear change to the spin density that it causes\n\u000eM\u0016(r;!) =X\n\u0017Z\ndr0\u001f\u0016\u0017(r;r0;!)\u000eB\u0017(r0;!): (1)\nHere\u0016;\u0017 =x;y;z are the cartesian components of the spin density. Within TDDFT,\nthe magnetic susceptibility is related to the one of the Kohn-Sham electrons through the\nHartree-exchange-correlation kernel via a Dyson-like equation:\n\u001f(!) =\u001fKS(!) +\u001fKS(!)KHxc\u001f(!); (2)\nwhere the spatial dependence has been omitted. The kernel KHxcincludes the Hartree and\nthe exchange-correlation contributions. The adiabatic LSDA was implied, which leads to a\nfrequency-independent kernel. Further details on the formalism and its implementation can\nbe found in Refs. 14, 25, 41, 51, 52.\nSpin-orbit coupling leads to new aspects in the calculation of the dynamical magnetic\nsusceptibility. Let M(r) be the ground state vector spin density. Then one may de\fne a\npointwise transformation of the global cartesian axes such that in this new, so-called local\nspin frame of reference, the ground state spin density has only one component,\nM(r)^ nz=R(r)M(r): (3)\nThroughout this paper, vector components in the global frame f^ n\u000b0gare primed, while in\nthe local framef^ n\u000bgthey are unprimed ( \u000b=x;y;z ). In the local frame, the Dyson-like\nequation for the dynamic susceptibility has the following matrix structure:\n0\nB@\u001fTT\u001fTL\n\u001fLT\u001fLL1\nCA\u00001\n=0\nB@\u001fKS\nTT\u001fKS\nTL\n\u001fKS\nLT\u001fKS\nLL1\nCA\u00001\n\u00000\nB@Kxc\nT 0\n0KHxc\nL1\nCA; (4)\nwhere the 2\u00022 blocks correspond to T = fx;ygand L =fz;ng, and the index npertains\nto the charge density. Frequency and spatial dependence were omitted for clarity. Note that\n5the Hartree contribution only appears in the longitudinal part of the kernel, as indicated\nby the superscript `Hxc'. The o\u000b-diagonal blocks of the KS susceptibility, \u001fKS\nLTand\u001fKS\nTL,\narise due to spin-orbit coupling and/or to noncollinear magnetic structures. If both of these\nare weak or absent, we can restrict the calculation only to the purely transverse part of\nthe dynamical susceptibility, \u001fTT, provided the transformation to the local frame has been\nperformed. This is the situation encountered for the systems described in this work.\nThe dynamical magnetic susceptibility is a very complex object, containing the infor-\nmation about all kinds of spin excitations as well as their detailed spatial and frequency\ndependence. A more intuitive physical picture is a\u000borded by de\fning atomic like quantities,\nby integrating out the spatial dependence over a region of space assigned to a given magnetic\natomi(j):\n\u001fi\u0016;j\u0017(!) =Z\nVidrZ\nVjdr0\u001f\u0016\u0017(r;r0;!): (5)\nIf the external perturbing magnetic \feld is taken to be uniform within each atomic volume\nVi, we arrive at an e\u000bective atomic description:\n\u000eMi\u0016(!) =X\nj\u0017\u001fi\u0016;j\u0017(!)\u000eBext\nj\u0017(!): (6)\nIn the low frequency regime, we can hope to make a connection to atomistic spin dynamics,\nas explained in the following.\nB. Phenomenological approach\nTo interpret the results obtained using the TDDFT formalism described above, we also\nemploy a phenomenological description of the magnetization dynamics given by the Landau-\nLifshitz-Gilbert (LLG) equation [27]. The 2 \u00022 transverse magnetic susceptibility \u001fTTcan\nbe obtained by assuming small deviations around the local equilibrium direction, which\nde\fnes the ^ nzaxis. Using this approach, we also extract all the relevant parameters directly\nfrom the \frst-principles dynamical susceptibility [14].\nThe equation of motion for the magnetic moment of atom iis given by\ndMi\ndt=\u0000\rMi\u0002Be\u000b+\u000b\nMiMi\u0002dMi\ndt: (7)\nThe \frst term on the right-hand side represents the torque due to an e\u000bective \feld Be\u000b\ni=\n\u0000@E=@ Miobtained from the energy functional E(fMig) of the system. The last term\n6describes relaxation e\u000bects that push the magnetization back to the equilibrium orientation.\n\ris the gyromagnetic ratio, and the damping is characterized by the Gilbert parameter \u000b.\nThe latter is, in principle, a 3 \u00023 matrix that captures its possible anisotropic behavior [53].\nNevertheless, for the purpose of the discussion set forth in this work, a scalar quantity is\nsu\u000ecient.\nConsidering an energy functional that includes magnetic anisotropies, coupling between\nthe di\u000berent components, and an external magnetic \feld Bextwe can write\nE(fMig) =X\niEi(Mi)\u0000J\nM2M1\u0001M2: (8)\nwherei= 1;2 labels the magnetic atoms, and\nEi(Mi) =MT\niKi\nM2\niMi\u0000Bext\u0001Mi (9)\nis the single atom energy containing the local anisotropy and Zeeman energies. For the\nsystems we investigate, the general 3 \u00023 matrix describing the on-site anisotropy can always\nbe brought to the diagonal form\nKi=0\nBBB@Kix0 0\n0Kiy0\n0 0 01\nCCCA(10)\nby a suitable de\fnition of the local frame of reference, as explained in connection with\nEq. (3).\nWhen Miis in equilibrium, i.e., pointing along the ^ nzdirection of the local frame of\nreference, the e\u000bective magnetic \feld is given by\nBe\u000b\ni=\u0012J\nM+Bext\u0013\n^ nz; (11)\nwhere we assume that the magnetization is aligned with the external \feld.\nThe spin excitations can be described by the transverse dynamical magnetic suscepti-\nbility. This quantity can be formally derived by calculating the small oscillations of the\nmagnetization Mi(t) =Mi^ nz+\u000eMi(t) induced by a transverse oscillatory external \feld\n\u000eBext(t) =\u000eBext\n0[cos(!t)^ nx+ sin(!t)^ ny]. In the frequency domain, the change in the mag-\nnetization is given by Eq. (6). We can then map the results obtained from the \frst-principles\ncalculations to the analytical phenomenological expressions (listed in Appendix A). The\n7anisotropy constants, exchange coupling, gyromagnetic ratio, and Gilbert damping for each\ncase is obtained by \ftting the appropriate functional form to the components of the trans-\nverse susceptibility \u001fTTcalculated using TDDFT close to != 0 [14].\nIt is convenient to work with the local circular basis ^ n\u0006=^ nx\u0006i^ ny. Using general\ncomplex components containing information on the amplitude and phase of the oscillation,\nthe transverse vectors are then transformed from cartesian ( vx;vy) to circular ( v\u0000;v+) as\nv\u0006=vx\u0006ivy, and\nv(t) = Re\u0002\n(vx^ nx+vy^ ny)e\u0000i!t\u0003\n=1\n2Re\u0002\n(v\u0000^ n++v+^ n\u0000)e\u0000i!t\u0003; (12)\nwhere v(t) represents any of the following: the external perturbing \feld \u000eBext(t), the trans-\nverse magnetization components \u000eMi(t), or the e\u000bective \feld \u000eBe\u000b\ni(t). Equation (7) can\nthen be written in the form\n0\n@\u000eMi\u0000\n\u000eMi+1\nA=X\nj0\n@\u001fi\u0000;j+\u001fi\u0000;j\u0000\n\u001fi+;j+\u001fi+;j\u00001\nA0\n@\u000eBext\nj\u0000\n\u000eBext\nj+1\nA: (13)\nA counterclockwise circularly polarized excitation \feld is described by the cartesian compo-\nnents\u000eBext\nx=\u000eBand\u000eBext\ny= i\u000eB, where\u000eBis a real value that describes the amplitude\nof oscillation. In the circular basis, this \feld is described by \u000eBext\n\u0000= 2\u000eBand\u000eBext\n+= 0.\nC. Precessional motion\nSpin excitations of small magnetic nanostructures can have a complex precessional nature,\nthe most general form being elliptical. In order to gain useful insights, here we explain\nhow the excitations can be described using only three parameters: the amplitude A, the\neccentricity e, and the tilt angle \u001e, which are all dynamical.\nWe \frst begin by writing the time-dependent magnetic moment in the local frame\nof reference as M(t) =Mcos\u0012(t)^ nz+Msin\u0012(t) [cos'(t)^ nx+ sin'(t)^ ny], where\u0012;'are\nthe usual spherical angles. Assuming small deviations from the equilibrium orientation,\nthis becomes M(t)'M^ nz+\u000eM(t), with the small transverse components \u000eM(t) =\nM\u000e\u0012(t) [cos'(t)^ nx+ sin'(t)^ ny]. We then re-express \u000eM(t) using the circular components\nof Eq. (12), parametrized as \u000eM\u0000=M\u000e\u0012ei(\u001e0\u0000\u001e)cos\u0018and\u000eM+=M\u000e\u0012ei(\u001e0+\u001e)sin\u0018(see Ap-\npendix B). Here, \u000e\u0012is the small opening angle of the precessional cone, and \u0018sets the aspect\n8ratio of the ellipse: Its semiaxes are A\u0006=M\u000e\u0012(cos\u0018\u0006sin\u0018)=2. Referring to Fig. 1, the tilt\nof the ellipse away from the x0axis is given by \u001eand\u001e0is the initial phase of the motion.\nThe main parameters of the ellipse can then be obtained from the circular components as\nA\u0006=j\u000eM\u0000j\u0006j\u000eM+j\n2\n\u001e=1\n2arg\u0012\u000eM+\n\u000eM\u0000\u0013: (14)\nA\u0000A+\u0000xx0y0y\n\u0000m(t)\u00000\n\u0000m(0)\nFigure 1: Schematic diagram of the elliptical precession. The time-dependent transverse magneti-\nzation vector \u000eM, illustrated in red, performs an elliptical motion starting at an azimuthal angle\n'=\u001e0fort= 0. The ellipse has major and minor semiaxes given by A\u0006=M\u000e\u0012(cos\u0018\u0006sin\u0018)=2\nand is rotated by an angle \u001ewith respect to the x0axis.\nIt is instructive to consider the simple scenario where \u001e= 0, i.e., when the axes of\nthe ellipse are aligned with the x0andy0directions. For ! > 0, the di\u000berent kinds of\nprecessional motion are: a counterclockwise circular polarization ( \t) for\u0018= 0, anxlinear\npolarization ($) for\u0018=\u0019=4, a clockwise circular polarization ( \b) for\u0018=\u0019=2, and ay\nlinear polarization ( l) for\u0018= 3\u0019=4. Values of \u0018between these angles represent ellipses with\ndi\u000berent eccentricities.\nImagine that the magnetic system is disturbed by a counterclockwise circularly polarized\nmagnetic \feld with angular frequency !. Then, the components of the dynamical magneti-\nzation are given by Eq. (13), and so the semiaxes of the ellipse and the tilt angle can then\n9be obtained from the appropriate susceptibilities as\nA\u0006=\u000eBext\n0\u0012j\u001f\u0000+j\u0006j\u001f++j\n2\u0013\n\u001e=1\n2arg\u0012\u001f++\n\u001f\u0000+\u0013 : (15)\nWe shall characterize the elliptical motion by focusing on its frequency-dependent amplitude\nAand eccentricity ede\fned as\nA(!) =r\nA2\n+(!) +A2\n\u0000(!)\n2\ne(!) =s\n1\u0000jA\u0000(!)j2\njA+(!)j2: (16)\nCircular precession is described by e= 0, while the motion describes a linear oscillation for\ne= 1.\nIn the following sections, we study the results obtained from \frst principles calculations,\nby making use of the analytical phenomenological expressions and the generic motion quanti-\nties detailed above, to describe the magnetization dynamics of single Fe adatoms and dimers\ndeposited on a Cu(111) surface.\nIII. SINGLE MAGNETIC IMPURITY\nWe start our investigation with a Fe adatom in three di\u000berent structures, as depicted in\nFig. 2, and how to describe their dynamics.\nmBext\nmBext=0\nmBext=0y0z0x0\nFigure 2: Diagrams of the three di\u000berent Fe adatoms con\fgurations deposited on Cu(111). Single\nFe adatom without external \feld (blue), single Fe adatom with large \feld along ^ n0\ny(red), and\nFe-Cu dimer (green). On the right, we display the global frame of reference.\n10A. Out-of-plane magnetization\nFirst, consider a single magnetic Fe adatom deposited on the Cu(111) surface. The\nenergy of the system can be mapped into a simple model given by Eq. (9), where the spin\nmagnetic moment is M= 3:2\u0016Bwhile the orbital magnetic moment is Morb= 0:55\u0016B. Due\nto theC3vsymmetry of this system, it presents uniaxial anisotropy with Kx=Ky=K. In\nFig. 3, we show the band energy variation with respect to a self-consistent calculation for\nMk^ nz0, i.e., \u0001E=Eband(M)\u0000Eband(M^ nz0), as a function of the magnetization direction.\nFollowing the magnetic force theorem [54], we \fnd a magnetic anisotropy energy constant\nofK\u0001E= 4:95 meV, which corresponds to an easy axis uniaxial anisotropy, with the easy\naxis being the normal to the surface. Throughout the text, we use the superscript \u0001 Eto\nindicate values obtained from band energy variations.\n03060901201501800246\nRotation angle (degrees)z0!x0x0!y01\u0000E(meV)\nFigure 3: Variation of the band energy for a single Fe atom on the Cu(111) surface. The energy is\nplotted as a function of the magnetization direction, taking as a reference the self-consistent calcu-\nlation where Mk^ nz0, i.e., \u0001E=Eband(M)\u0000Eband(M^ nz0). The system is schematically illustrated\nin the central diagram, with dark gray spheres representing Cu atoms, while transparent spheres\nare vacuum positions around the Fe adatom (red). For such C3vsymmetry, the uniaxial anisotropy\nconstantK\u0001E= 4:95 meV de\fned by the model Hamiltonian given in Eq. (9) can be obtained\ndirectly from the variation of the energy when the magnetization is rotated from out-of-plane to\nin-plane, in the z0x0plane (red circles). There is no energy variation when the magnetization\nis rotated in the ( x0y0) surface plane (blue triangles), con\frming the uniaxial character of the\nanisotropy.\n11We imagine that an STM tip is placed above the Fe adatom and that the tunneling\nelectrons induce spin-\rip excitations locally, which then lead to an inelastic contribution\nto the tunneling current [42]. We associate the dynamical spin excitations driven by this\nprocess with the local susceptibility \u001f\u0000+. For theC3vsymmetry, the circular components\nof the e\u000bective \feld simplify to \u000eBe\u000b\n\u0006=\u00002K\u000eM\u0006=M2+\u000eBext\n\u0006. The equations for \u000eM+and\n\u000eM\u0000decouple and the susceptibility matrix given in Eq. (13) has only diagonal components\n\u001f\u0000+(!) =\u000eM\u0000\n\u000eBext\n\u0000=\u0000\r+M\u0000\n!\u0000\r+Bk\u0001; (17)\nand\u001f+\u0000(!) = [\u001f\u0000+(\u0000!)]\u0003, where\r+=\r\n1+i\u000bandBk=Bext\nz+ 2K=M . Note that each\nsusceptibility has a single pole, located at !0=\r+Bkand!0=\u0000\r\u0003\n+Bk, respectively.\nFigure 4(a) (blue curve) shows the density of spin excitations from the TDDFT calculation\n(obtained from Im \u001f\u0000+) when no static magnetic \feld is applied to the sample. This quantity\nis related to the steplike features in the tunneling conductance measured in ISTS experiments\n[42].\nThe values of the anisotropy constant K, the e\u000bective gyromagnetic ratio \r, and the\ndamping parameter \u000bcan be extracted by \ftting the linear behavior of the real and imag-\ninary part of [ \u001f\u0000+]\u00001, shown in Eq. (A1), close to != 0 [14]. These values are listed in\nTable I. The \ftted values of Kdi\u000ber slightly from the ones calculated using the band energy\nvariation | while the latter captures the energy di\u000berence between two orthogonal direc-\ntions of the magnetic moment, the former represents the curvature of the energy around\nits equilibrium direction. Note, however, that the gyromagnetic factor \rshifts considerably\nfrom the expected value of 2. This induced delay occurs due to the high hybridization with\nthe surface [14], which leads to large spin currents pumped out of the adatom [28]. This\nstrong hybridization is also responsible for the high value obtained for the Gilbert damping\n\u000b. The shift of the resonance energy to imaginary values, Im \r+Bk=\u0000\u000b\rBk\n1+\u000b2, is proportional\nto the damping parameter and is responsible for the broadening of the response functions\nseen in Fig. 4(a). This e\u000bect can also be seen from the time dependence of the transverse\nmagnetization components obtained in Eq. (B7), where this contribution appears as an\nexponential decay of their amplitude.\nSince the susceptibility matrix is diagonal, the amplitude of oscillation A(!) =jA+(!)j=\njA\u0000(!)j, and the eccentricity e= 0 for every frequency. These quantities are shown in the\n12(c)(b)(a)0246\n-10-505100102-10-50-0.30Im\u0000\u0000+(meV\u00001)EccentricityeEnergy (meV)LCE}A(meV\u00001)Fe\"Fe!Fe-CuFigure 4: Dynamical excitations of single magnetic impurities. (a) Imaginary part of the transverse\ndynamical susceptibility \u001f\u0000+as a function of the energy. The inset displays a magni\fcation of\nthe responses for negative energies. (b) and (c) illustrate the amplitude Aand eccentricity of the\nprecessione, as de\fned in Eq. (16), as a function of the energy when a circularly polarized magnetic\n\feld is applied. L, E, and C indicate the regions with linear, elliptical, and circular oscillations. The\ncurves for the three di\u000berent magnetic adatom structures are color coded as follows: isolated Fe\nadatom (blue), isolated Fe adatom with in-plane magnetic \feld (red), Fe adatom with a neighboring\nCu adatom (green). See Fig. 2 for an illustration.\nblue curves of Figs. 4(b) and (c), and describe a circular precessional motion | re\recting an\ne\u000bective \feld with equal transverse components acting on the magnetization. The maximum\namplitude of oscillation presented in Fig. 4(b) is close to the maximum density of spin\nexcitations obtained in Fig. 4(c), although the former presents a much broader peak due to\ncontribution of the real part of \u001f\u0000+in this quantity, see Eq. (15).\nB. In-plane magnetization\nIn the presence of spin-orbit coupling, the symmetry of the system can be broken with\nexternal magnetic \felds. By applying a large Bextalong the ^ ny0direction of the global frame\n13Table I: Parameters from the ground state DFT calculations (upper half) and from \ftting the\nTDDFT susceptibility (lower half) for the three di\u000berent Fe adatom structures deposited on\nCu(111) investigated: single Fe adatom without external \feld, single Fe adatom with large \feld\nalong ^ ny0, Fe-Cu dimer. For the adatom in the presence of magnetic \feld, \u000bis highly anisotropic:\n\u000bxx= 0:34, while\u000byy= 0:39, in the local frame of reference.\nFe adatom\n(no Bext)\n\"Fe adatom\n(Bext)\n!Fe-Cu dimer\n(no Bext)\n-\nM(\u0016B) 3.24 3.24 3.19\nMorb(\u0016B) 0.55 0.24 0.41\nK\u0001E\nx(meV) 4.95 0.02 2.48\nK\u0001E\ny(meV) 4.95 -5.46 3.40\nKx(meV) 4.98 0.07 2.49\nKy(meV) 4.98 -5.43 3.37\n\r 1.72 1.71 1.71\n\u000b 0.33 0.39* 0.29\nof reference, we rotate the magnetization of the adatom to lie in the surface plane along this\ndirection (which de\fnes the local ^ nz). We chose a static magnetic \feld of 90 T such that the\nspin excitation resonance is similar to the one obtained for the out-of-plane magnetization.\nAlthough this represents an arti\fcially large magnetic \feld, it enables an easy comparison\nbetween both cases.\nThe strong magnetic \feld has only a minor impact on the ground state properties of\nthe adatom. The spin moment is unchanged, but the orbital moment is less than half\nof the value of the out-of-plane case. This reduction occurs as the spin moment is now\nperpendicular to the C3vsymmetry axis, which, due to the spin-orbit coupling, leads to a\nlowering of symmetry. In the local frame of reference, the magnetic anisotropy matrix can\nnow be described by Eq. (10) with Kx= 0,Ky=\u0000K.\nThe density of spin excitations is depicted by the red curve of Fig. 4(a). For positive\nfrequencies, the curves are very similar with a slightly larger broadening when the magne-\n14tization points in the plane. A more detailed picture is provided by the parameters listed\nin Table I, obtained from the \fts of the di\u000berent components of the inverse susceptibilities\nto Eq. (A2). The shift of the gyromagnetic factor away from 2 is very similar to the one\nobtained before, indicating that the spin pumping mechanism is essentially isotropic. In\ncontrast, the damping parameter is substantially anisotropic, with ( \u000bxx\u0000\u000byy)=\u000byy\u001813%,\nwhere\u000b\u0016\u0016was obtained from the \u001f\u00001\n\u0016\u0016component of the susceptibility.\nSurprisingly, there appears to be a new excitation peak manifested at negative energies.\nEven though the value of the response function is small, it represents a fundamental di\u000ber-\nence in the spin dynamics and in the precessional shape. It originates from the anisotropic\ne\u000bective \feld along the transversal directions. This can be understood from the expression\nfor the transverse susceptibility matrix, Eq. (A2). We \fnd\n\u001f\u0000+=\r+M\u0000\n!+\r\u0003\n+B\u0003\nk\u0001\n!+\u0000!\u0000\u00121\n!\u0000!\u0000\u00001\n!\u0000!+\u0013\n: (18)\nwhere the components of the e\u000bective \feld are Bk=Bext\u0000K=M andB?=K=M , de\fning\nthe poles\n!\u0006= i Im(\r+Bk)\u0006q\n[Re(\r+Bk)]2\u0000j\r+B?j2: (19)\nThis result explains why, besides the resonance at !+, there is also a signal at !\u0000, as seen\nin the red curve of the inset of Fig. 4(a). For the uniaxial case ( B?= 0), the second peak\nis absent because the numerator is proportional to !+\r\u0003\n+B\u0003\nk=!\u0000!\u0000and so the \frst\nterm in Eq. (18) becomes a constant. Furthermore, the precession becomes circular. We\ncan then distinguish circular and elliptical precession directly from \u001f\u0000+, by inspecting how\nmany resonances it displays. In addition, B?re\rects the ellipticity of the transverse e\u000bective\n\feld, and it lowers the resonance frequency in comparison to the uniaxial case. As before,\nthe imaginary part obtained in Eq. (19) is proportional to the damping parameter \u000band is\nresponsible for the decay of the magnetization back into the equilibrium direction.\nAs described in Appendix B, even though there are two resonant energies of the system,\nthey both represent a single elliptical precession. The amplitude of the motion followed by\nthe transverse component of the magnetization is illustrated in the red curve of Fig. 4(b).\nAlthough the amplitude is smaller than the out-of-plane case at the resonance, the peak\nis broader and the response is larger for negative frequencies close to !=!\u0000. In this\nregion, the oscillation is close to linear, as evidenced by the eccentricity close to e= 1\n15displayed in Fig. 4(c). For higher frequencies, the precession amplitude and eccentricity\ndecays, approaching a circular shape.\nFor this last scenario, we applied a large magnetic \feld to rotate the magnetization and\nmake use of di\u000berent anisotropy energies along the transverse directions to induce elliptical\nmotion of the magnetization precession. In the next section, we show how this can be\nachieved by manipulating the surroundings of the magnetic unit.\nC. Engineering magnetic anisotropy\nAs an alternative to an external magnetic \feld, we now explore the impact of a neighbor-\ning nonmagnetic Cu atom on the spin excitations of the Fe atom. The rotational symmetry\nis broken in real space and, due to the spin-orbit coupling, also in spin space, resulting in\na tilt of the equilibrium direction of the spin moment away from the surface normal. The\nspin and orbital magnetic moments decrease compared to the out-of-plane case, reaching\nM= 3:19\u0016BandMorb= 0:41\u0016B. The latter represent a 25% decrease in the orbital mo-\nment, which indicates a strong in\ruence from the additional atom. The symmetry lowering\nis con\frmed by the variation of band energy of the system when the magnetization angle is\nrotated along the three di\u000berent axes, illustrated in Fig. 5. The intricate energy landscape\nleads to a tilt of the spin moment of \u0012\u001818\u000eand\u001e\u0018177\u000e, in spherical coordinates, close\nto thex0z0plane. In the local frame of reference, K\u0001E\nx= 2:48 meV and K\u0001E\ny= 3:40 meV,\nwhich corresponds to a biaxial anisotropy, with the magnitude of the anisotropy constants\nreduced from the isolated Fe adatom case. Extra Cu atoms decrease their values even more.\nThis is also re\rected on the spin excitation spectra, green curve in Fig. 4(a), in which the\nresonance is shifted to lower frequencies, leading also to a higher amplitude of precession,\nFig. 4(b). The phenomenological parameters are again obtained by \ftting the \frst-principles\nresults to Eq. (A3), with the values listed in Table I. The gyromagnetic factor and the damp-\ning parameter are very similar to the out-of-plane adatom case, but the biaxial anisotropy\nis manifested in the elliptical precession indicated by the peak at negative energies [inset\nof Fig. 4(a)]. It is also seen from the eccentricity plotted in Fig. 4(c) that reaches close\ntoe= 1 | linear oscillation | at !=!\u0000and decreases away from this frequency. Our\nresults demonstrate that atomic manipulation not only of the magnetic unit but also of its\nenvironment can be used to control and tune possible magnetic excitations and may be used\n16030609012015018002\nRotation angle (degrees)12\u0000E(meV)z0!x0x0!y0z0!y0Figure 5: Variation of the band energy for a Fe-Cu dimer on the Cu(111) surface. The energy\nis plotted as a function of the magnetization direction, taking as a reference the self-consistent\ncalculation where Mk^ nz0, i.e., \u0001E=Eband(M)\u0000Eband(M^ nz0). The system is schematically\nillustrated in the central diagram, with dark gray spheres representing Cu atoms and transparent\nspheres are vacuum positions around the Fe (1) and Cu (2) dimer. The anisotropy constants\nde\fned for the model Hamiltonian given in Eq. (9) are obtained by \ftting the energy curves when\nthe magnetization is rotated along the z0x0plane (red circles), the z0y0plane (green squares), and\nalong thex0y0plane (blue triangles). The values of the magnetic anisotropy components in the\nlocal frame of reference of the ground state magnetization are listed in Table I.\nto operate static and dynamical aspects of nanostructures.\nIV. MAGNETIC DIMERS\nWe focus now on magnetic dimer structures and how their ground state properties a\u000bect\nthe possible excitation modes of the system. The energy of these structures may be mapped\ninto a model given by Eq. (8). By manipulating the separation between the Fe adatoms\nand taking advantage of the distance-dependent oscillatory exchange interaction between\nthe adatoms, we design two di\u000berent structures:\n\u000fDimer I | adatoms separated by the nearest-neighbor distance on the surface have a\nstrong ferromagnetic coupling.\n\u000fDimer II | adatoms separated by twice the nearest-neighbor distance on the surface\n17have a weak antiferromagnetic coupling.\nFor dimer II, we investigate excitations starting from the metastable FM state and from the\nAFM ground state. The three cases we consider are depicted schematically in Fig. 6.\nm1m2\nm1m2\nm1m2\nFigure 6: Diagrams of the three di\u000berent Fe dimers deposited on Cu(111). Nearest neighbor\ndistance with large ferromagnetic coupling (blue); twice the nearest neighbor distance with small\nantiferromagnetic coupling, starting from a ferromagnetic state (red) or from an antiferromagnetic\nstate (green).\nWhen bringing two adatoms together to form a dimer, the symmetry is lowered from C3v\ntoCs, leaving only one mirror plane. The system presents a biaxial anisotropy that can be\nmapped into the Kmatrix given by Eq. (10). The values of K\u0001E\nxandK\u0001E\ny(per Fe atom)\nare obtained from the change of the band energy when the magnetic moments are rotated\nsimultaneously along the di\u000berent directions.\nIn dynamical studies of nanostructured systems, a central role is played by the coupling\nbetween the magnetic moments. For the dimers considered in this paper, the spin-orbit inter-\naction does not lead to appreciable anisotropic pair interactions, such as the Dzyaloshinskii-\nMoriya interaction. Their values, obtained from the calculated susceptibilities, are two orders\nof magnitude smaller than the exchange interaction | as one would expect for systems with\nlow spin-orbit coupling such as Cu. Hence, we consider only the simpler isotropic Heisenberg\nexchange in our phenomenological model. As there are several ways of estimating J, we \frst\nprovide an overview of the methods before analyzing each dimer structure.\nA. Exchange coupling\nA \frst de\fnition of Jis given by assuming that the Heisenberg coupling appropriately\ndescribes the energetics of our dimers. Identifying the \frst-principles total energy di\u000berence\nbetween the antiferro- and the ferromagnetic states with the value expected from the model\n18we \fnd\n2J\u0001E=EAFM\u0000EFM: (20)\nIn this convention, a positive (negative) value of J\u0001Efavors a ferromagnetic (antiferromag-\nnetic) ground state.\nThe mapping between the Heisenberg coupling and the total energy di\u000berence of the\ntwo magnetic states assumes that the coupling constant (and the electronic structure) is not\nsubstantially a\u000bected by the angle between the two magnetic moments | which, in practice,\nmay not be a reasonable assumption. To avoid this problem, the exchange coupling may\nalso be derived via in\fnitesimal rotation of the magnetic moments taking into account\nthe magnetic force theorem [55, 56], which can be written for ferro- and antiferromagnetic\nalignments as\nJ0=\u0007(MKxc\nT)2\u001fKS\n1\u0000;2+: (21)\nSee Sec. II A for a description of these quantities.\nThis result, however, does not take into account many-body e\u000bects. Here we introduce\nyet another method to obtain the coupling parameter, based on the mapping of the suscep-\ntibilities calculated in TDDFT to those obtained from the LLG model, Eqs. (A4) and (A5).\nFor ferromagnetic or antiferromagnetic ground states, one \fnds\nJ=\u0007M2\u0002\n\u001f\u00001\u0003\n1\u0000;2+; (22)\nwhere 1 and 2 label the two magnetic moments for which the coupling constant is determined,\nand the sign\u0000(+) is used for the ferro- (antiferro-) magnetic alignment.\nThe expressions for the exchange couplings obtained in Eqs. (21) and (22) are related by\nJ=J0\u0014\n1\u00062J0\nM2Kxc\nT\u0015\u00001\n; (23)\nwhere + and\u0000signs account for a FM and an AFM alignment of the two involved magnetic\nmoments, respectively. The form of Eq. (23) has been discussed in the literature [57, 58]:\nSimilar to the connection between \u001fand\u001fKSviaKxc\nT, the exchange coupling constant Jcan\nbe seen as a renormalization of J0byKxc\nT, the exchange-correlation kernel. Since the spin\nsplitting is much larger than the spin excitation energies for the systems we investigate, we\nexpect the values of J0andJto be quite similar to each other [58]. In the following sections,\nwe analyze the ground state and dynamical properties of the di\u000berent dimer con\fgurations\n19and how the coupling constant and the initial magnetic con\fguration a\u000bect the excitation\nenergies.\nB. Dimer I\nThe spin and orbital magnetic moments per site, MandMorbare given in Table II.\nComparing with the isolated Fe adatom case (see Table I), the spin moment decreases\nslightly while the orbital moment is drastically reduced by 65%. Due to the broken rotation\nsymmetry in the plane, the system presents biaxial anisotropy | as demonstrated by the\nband energy variation when the moments are rotated simultaneously along all directions,\nshown in Fig. 7. The anisotropy constants per atom are listed in Table II. Their magnitudes\nare 3-4 times smaller than for the isolated adatom and about half of those for the Fe-Cu\ndimer, revealing the impact of the strong hybridization between the dorbitals of the two Fe\natoms.\nThe strong d-dhybridization is also evident in the large values of the ferromagnetic\ncouplingJ. From the total energy di\u000berence, Eq. (20), we obtain J\u0001E= 239 meV, while\nfrom the magnetic force theorem, Eq. (21), J0= 193 meV. This sizable di\u000berence re\rects\nthe change in the electronic structure going from the FM to the AFM state.\nWe also obtain the phenomenological parameters by \ftting the results to the inverse\nsusceptibility obtained in Eq. (A4). Their values are listed in Table II. The anisotropies\nare in very good agreement with the ones obtained by the band energy variation, and Jis\nvery close to J0, as expected. We obtain relatively small values for the damping parameter,\n\u000b= 0:12, which will be discussed below. The gyromagnetic ratio \ris closer to 2 than before,\nwhich indicates that the spin pumping mechanism is less e\u000ecient.\nFor a ferromagnetic dimer, we expect two precessional modes: a uniform mode where\nthe magnetic moments precess in-phase (acoustic mode), and one mode where they precess\nwith a phase di\u000berence of \u0019(optical mode). By linearizing the equation of motion given in\nEq. (A4), the poles corresponding to the acoustic and optical mode are\n!\u0006\nac\n\r0=\u0000i\u000b(Kx+Ky)\u0006q\n4KxKy\u0000\u000b2(Kx\u0000Ky)2\n!\u0006\nop\n\r0=\u0000i\u000b(Kx+Ky+ 2J)\n\u0006q\n4(Kx+J)(Ky+J)\u0000\u000b2(Kx\u0000Ky)2; (24)\n20Table II: Parameters from the ground state DFT calculations (upper half) and from \ftting the\nTDDFT susceptibility (bottom half) for the three Fe dimer structures deposited on Cu(111) inves-\ntigated: nearest neighbor distance and twice the nearest neighbor distance with both ferromagnetic\nand antiferromagnetic alignments. Values are given per atom.\nDimer I\n\"\"Dimer II\n\" \"Dimer II\n\" #\nM(\u0016B) 3.12 3.24 3.24\nMorb(\u0016B) 0.19 0.53 0.55\nK\u0001E\nx(meV) 1.79 4.36 4.89\nK\u0001E\ny(meV) 1.12 4.38 4.80\nJ\u0001E(meV) 239 -3.9 -3.9\nJ0(meV) 193 -2.4 -3.9\nKx(meV) 1.82 4.45 4.91\nKy(meV) 1.14 4.44 4.82\nJ(meV) 206 -2.8 -4.3\n\r 1.85 1.69 1.71\n\u000b 0.12 0.32 0.30\nwhere\r0=\r\nM(1+\u000b2). SinceJ\u001dKx;Ky, the optical mode is located at 2 \rJ=M\u0018250 meV.\nIf!\u001cJ, the two spin moments stay parallel to each other, and the system behaves as a\nmacrospin with biaxial magnetic anisotropy. In this case, \u001f\u0000+can be described by Eq. (18)\nwithBk= (Kx+Ky)=MandB?= (Kx\u0000Ky)=M, and the excitations follow similar dynamics\nas the adatom with magnetization in-plane and the Fe-Cu dimer described in Sec. III. The\nexpressions for the acoustic and optical frequencies, Eq. (24), show that in biaxial systems\nthe damping may play an important role, lowering the resonance frequency. These results\nare not captured by the so-called Kittel's formula [59], which neglects damping e\u000bects.\nWe imagine that an STM tip is placed above atom 1 and that the tunneling electrons\ninduce spin-\rip excitations locally, which then lead to an inelastic contribution to the tun-\nneling current [42]. We associate the dynamical spin excitations driven by this process in\natom 1 with the local susceptibility \u001f1\u0000;1+\u0011\u001f\u0000+. In Fig. 8(a) we show the imaginary part\n210306090120150180024z0!x0x0!y0z0!y0\nRotation angle (degrees)12\u0000E(meV)Figure 7: Variation of the band energy for the dimer I structure. The energy is plotted as a\nfunction of the magnetization direction of the parallel alignment, M= (M1+M2)^ n, taking as\na reference the self-consistent calculation where Mk^ nz0, i.e., \u0001E=Eband(M)\u0000Eband(M^ nz0).\nThe dimer is composed by two nearest-neighbor Fe atoms (1 and 2) on the Cu(111) surface, as\nillustrated in the central diagram. Dark gray spheres represent Cu atoms and transparent spheres\nare vacuum positions around the nearest neighbor Fe-Fe dimer. The anisotropy constants are\nobtained by \ftting the energy curves when Mis rotated along the z0x0plane (red circles), the\nz0y0plane (green squares), and along the x0y0plane (blue triangles). The values of the magnetic\nanisotropy components in the local frame of reference are listed in Table II.\nof the local \u001f\u0000+component as a function of the frequency for the nearest neighbor dimer\n(blue curve). The optical mode is out of the range of the \fgure. As before, the small peak\nat negative frequencies reveals an elliptical precession. The properties of the precessional\nmotion can be gleaned from Figs. 8(b) and (c). For the peak at positive frequencies, the\namplitude is large, while the movement is slightly elliptical ( e\u00180:5). On the other hand, for\nthe negative resonance, the amplitude is small but signi\fcant, while the precession becomes\nlinear for frequencies close to !\u0000. The low value of the damping obtained for this dimer\nleads to a relatively sharp peak in the blue curve of Fig. 8(a). This is due to the strong d-d\nhybridization, which creates bonding and antibonding states, thus lowering the density of\nstates near the Fermi energy. As the damping parameter is very sensitive to the electronic\nstructure around the Fermi energy [52], this explains its reduction to about a third of the\nvalues for the Fe adatoms.\n22051015\n-10-5051001048-10-50-0.30\nEnergy (meV)Im\u0000\u0000+(meV\u00001)EccentricityeLCE}A(meV\u00001)(c)(b)(a)\nDimer IDimer II FMDimer II AFMFigure 8: Dynamical excitations of dimers. (a) Imaginary part of the local transverse dynamical\nsusceptibility \u001f\u0000+as a function of the energy. The inset displays a magni\fcation of the responses\nfor negative energies. (b) and (c) illustrates the amplitude Aand eccentricity of the precession\ne, as de\fned in Eq. (16), as a function of the energy when a circularly polarized magnetic \feld\nis applied. L, E, and C indicate the regions with linear, elliptical, and circular oscillations. The\ncurves for the three di\u000berent dimer structures are color coded as follows: dimer I (blue), dimer II\nwith FM alignment (red), dimer II with AFM alignment (green). See Fig. 6 for an illustration.\nIn the following section, we uncover the dynamical behavior when the interatomic ex-\nchange coupling is now of the same order of magnitude as the magnetic anisotropy constants.\nThe two spin moments are no longer forced to be parallel to each other, and this has impor-\ntant consequences for the precessional motion.\nC. Dimer II\nWhen the two Fe atoms are pulled apart, their properties quickly recover those of isolated\nunits. For a separation of twice the nearest neighbor distance, the spin and orbital moments\nare very close to the ones obtained for the single Fe adatom, as seen in Table II. Their\ncoupling is weakened and even changes sign, as indicated by the value obtained from the\n23total energy di\u000berence J\u0001Eand from the magnetic force theorem J0. As these values are\nlower than the anisotropy constants of the isolated adatoms, the interplay between these\ntwo kinds of magnetic interactions may give rise to a metastable state. This is con\frmed by\nFig. 9, where we plot the band energy variations starting from two self-consistent states, FM\n(red curve) and AFM (blue curve), as a function of the angle between the magnetic moments.\nWe see that the AFM alignment is the ground state, but also that the FM alignment is a\nlocal minimum of the energy.\n0306090120150180-8-6-4-202\nRotation angle (degrees)\nm1m2\nm1m2\u0000E(meV)\nFigure 9: Stability of the two possible magnetic alignments for dimer II from the variation of the\nband energy. The energy is plotted as a function of the angle between the magnetic moments of the\nimpurities with M1\fxed to ^ nz0while M2is rotated in the x0z0plane. We take as a reference the\nferromagnetic state, i.e., \u0001 E=Eband(M1^ nz0;M2)\u0000Eband(M1^ nz0;M2^ nz0). Red and blue curves are\nobtained starting from a self-consistent ferromagnetic and antiferromagnetic states, respectively.\nAs both magnetic states are accessible, we characterize their magnetic anisotropic energy\nas done before for dimer I. The variation of the energy when the moments are rotated\ntogether in the FM or the AFM alignments, shown in Fig. 10, indicates an almost uniaxial\nmagnetic anisotropy energy, with Kx'Ky. The obtained values are given in Table II. The\nanisotropy constants are essentially independent from the magnetic alignment and are very\nclose to the values found for the isolated adatom, which shows that the Fe adatoms weakly\ndisturb each other.\nWe can access two qualitatively di\u000berent kinds of spin excitations, by starting either\nfrom the AFM ground state or from the FM metastable state. To compare with dimer I, we\nconsider again the local susceptibility \u001f1\u0000;1+\u0011\u001f\u0000+and we \frst analyze the FM case. The\n2403060901201501800510z0!x0x0!y0z0!y0\n03060901201501800510\nRotation angle (degrees)12\nm1m2\nm1m2(b)(a)\u0000E(meV)\u0000E(meV)z0!x0x0!y0z0!y0Figure 10: Variation of the band energy for magnetic anisotropy energy of the dimer II structure.\n(a) The energy is plotted as a function of the magnetization direction of the parallel alignment\n(as in the central diagram), M= (M1+M2)^ n, taking as a reference the self-consistent calcu-\nlation where Mk^ nz0, i.e., \u0001E=Eband(M)\u0000Eband(M^ nz0). (b) Change of the band energy as\na function of the magnetization direction of the antiparallel alignment (as in the central dia-\ngram), M= (M1\u0000M2)^ n, taking as a reference the self-consistent calculation where Mk^ nz0, i.e.,\n\u0001E=Eband(M)\u0000Eband(M^ nz0). The dimer is composed of two Fe atoms (1 and 2) situated at\ntwice the nearest neighbor distance on the Cu(111) surface, as illustrated in the central diagram.\nDark gray spheres represent Cu atoms and transparent spheres are vacuum positions around the\nFe sites. The anisotropy constants are obtained by \ftting the energy curves when the magneti-\nzation of both sites are rotated simultaneously along the z0x0plane (red circles), the z0y0plane\n(green squares), and along the x0y0plane (blue triangles). The values of the magnetic anisotropy\ncomponents in the local frame of reference are listed in Table II.\n25corresponding density of spin excitations is the red curve of Fig. 8(a). As before, we have\nthe two modes, acoustic and optical, described by Eq. (24). Since J <0, the optical mode\nhas lower energy than the acoustic mode: While the latter is located at \u00185 meV, the former\nresonates at\u00182 meV. Note that, unlike the biaxial case of dimer I, no signal can be seen\nfor negative energies in the inset of Fig. 8(a), indicating a nearly circular precession. The\namplitude of the motion for the optical mode is larger than for the acoustic, see Fig. 8(b).\nThe eccentricity of precession, represented by the red curve of Fig. 8(c), reaches a maximum\nvalue ofe\u00180:5 close to!\u0000\nop. This con\frms that the precession is slightly elliptical.\nA completely di\u000berent excitation spectrum is obtained when the initial state is, instead,\nthe AFM ground state, as seen in the green curve of Fig. 8(a). Note that even for the almost\nuniaxial anisotropy case considered here, the transverse susceptibility presents a peak at\nnegative energies. However, Fig. 8(b) shows that the amplitude of precession is signi\fcant\nonly for the peak at positive frequencies and is featureless for negative frequencies. In\naddition, the precessional motion is only slightly elliptical ( e<0:5), see Fig. 8(c).\nWe can interpret this behavior as re\recting the di\u000berent preferred precessional senses\nof the two Fe atoms composing the dimer. If the two atoms were uncoupled ( J= 0),\nthe intrinsic precessional sense of atom 1 would be counterclockwise, while atom 2 (being\nantiparallel to atom 1) would naturally precess clockwise, in a common frame of reference.\nWe assume that the tunneling current is exciting the precessional motion of atom 1. For\npositive frequencies, the excitation spectrum of atom 1 is similar to that of the isolated\nadatom, cf. Fig. 4(a), which indicates that the coupling Jto atom 2 is not playing a\nsigni\fcant role. For negative frequencies, if atom 1 was isolated, we would not expect any\nresonant behavior. However, the unfavorable precession driven in atom 1 is transferred to\natom 2 via J, triggering its natural precessional motion, which then feeds back to atom\n1 (again via J) leading to the observed enhanced response. In fact, the presence of the\nexcitation at negative energies indicates that the classical antiferromagnetic state j\"#i, for\nwhich we obtain the spectra, is not the true ground state of the system [33].\nAs done for dimer I, we can obtain analytical expressions for the resonances considering\n26biaxial anisotropy and damping:\n!\u0006\n1\n\r0=\u0000i\u000b(Kx+Ky\u0000J)\n\u0006q\n4Kx(Ky\u0000J)\u0000\u000b2(J+Kx\u0000Ky)2\n!\u0006\n2\n\r0=\u0000i\u000b(Kx+Ky\u0000J)\n\u0006q\n4Ky(Kx\u0000J)\u0000\u000b2(J\u0000Kx+Ky)2; (25)\nwhere\r0=\r\nM(1+\u000b2). Here,Kx;Ky>0 andJ < 0 for the AFM ground state. There are\nfour solutions that correspond to two distinct precessional modes (the + and \u0000solutions\ngenerate the same precessional motion, as explained in Appendix B). If the biaxial character\nis very weak, Kx\u0018Ky, the solutions become degenerate and we observe only two peaks in\nthe spectrum instead of the expected four. This is precisely the behavior that we obtain for\ndimer II.\nThe damping plays a much more important role for antiferromagnets than for ferromag-\nnets, especially when the coupling is relatively large. Increasing the damping strength, the\nresonance frequency lowers, as follows from Eq. (25). For simplicity, we explore this scenario\nfor uniaxial anisotropy, Kx=Ky=K. When\u000b > 2p\nK(K+jJj)=jJj, one of the modes\nmoves to zero frequency, recovering the Goldstone mode, with the other one moving to the\nimaginary frequency axis, corresponding to an overdamped precession. Although this con-\ndition is not ful\flled for our case, it may happen for dimers with the coupling jJj\u001djKj\n(for example, a Mn dimer deposited on metallic substrates).\nFinally, we consider the possible impact of the spin excitations on the stability of the\nclassical AFM ground state. In Ref. 35, it was argued that the AFM dimer was able to access\nthe two degenerate N\u0013 eel states j\"#iandj#\"i, since the zero-point \ructuations (involving the\ncouplingJ) were larger than the energy barrier between them (proportional to K). In that\nwork, the energy of the zero-point \ructuations was connected to the energy of the lowest spin\nexcitation mode. This description is also in accordance with Ref. 33, where the zero-point\nenergy was found to vanish for an AFM dimer with zero anisotropy. Nevertheless, neither of\nthese works have considered the e\u000bects of damping. Our results obtained in Eq. (25) show\nthat the damping reduces the frequency of the lowest excitation mode for antiferromagnetic\ndimers with large coupling. Counterintuitively, this implies that the zero-point energy is\nalso lowered when the damping increases, following the argument of Ref. 35. This may then\n27prevent the \ructuations over the energy barrier between the two N\u0013 eel states and stabilize\nthem. This can be contrasted with the behavior found in Ref. 60, in which the damping\nincreases the zero-point spin \ructuations of single magnetic adatoms deposited on metallic\nsurfaces.\nV. CONCLUSIONS\nIn this paper, we have presented a semiclassical interpretation of the dynamical spin ex-\ncitations in magnetic nanostructures computed using a \frst-principles approach. A crucial\nrole is played by the spin-orbit coupling, responsible for nontrivial magnetic anisotropies,\nwhich in turn lead to complex precessional motion. A description of the general elliptical\nprecession was provided and connected to the spectral features of the transverse magnetic\nsusceptibility. As the latter is intimately related to the inelastic contribution to the tunnel-\ning conductance, we believe this formalism can provide useful insights on the nature and\nproperties of spin excitations detected experimentally.\nConsidering a single Fe adatom deposited on the Cu(111) surface, we showed how the\nground state and also the excitation properties can be controlled by an external magnetic\n\feld or by atomic manipulation of its environment (the formation of a Fe-Cu dimer). We\nfound that the signature of noncircular precessional motion is the appearance of a secondary\npeak at negative frequencies in the density of spin excitations. In the vicinity of this peak, the\nprecession becomes highly elliptical. Our results indicate that the spin pumping mechanism\nis quite isotropic, while the precessional damping is large, anisotropic and tunable.\nWe next considered a di\u000berent kind of atomic manipulation, where the Cu atom is re-\nplaced by a second Fe atom. When the two Fe atoms are nearest neighbors, they are strongly\nferromagnetically coupled, behaving as a single magnetic unit when the frequency is much\nlower than their coupling. The magnetic anisotropy is now of biaxial nature, leading to\nelliptical spin excitations. The dynamical properties can be understood from the strong d-d\nhybridization, which is responsible for lowering both the spin pumping e\u000eciency and the\nmagnetic damping.\nWhen the Fe adatoms are pulled apart, their coupling weakens and becomes antiferro-\nmagnetic, being now comparable in magnitude to the magnetic anisotropy energy. This\nspecial con\fguration gives access to two di\u000berent magnetic states: the antiferromagnetic\n28ground state and the metastable ferromagnetic state. The two Fe atoms weakly in\ruence\neach other, and their local properties are very similar to those of isolated Fe adatoms. The\nmetastable FM state leads to two kinds of spin excitations: an acoustic mode where the spins\nprecess in-phase and an optical mode where they precess in anti-phase. The optical mode\nactually has lower energy than the acoustic one, as a consequence of the metastable nature\nof the ferromagnetic alignment. By inverting the spin alignment, we arrive at the antifer-\nromagnetic ground state, which has completely di\u000berent excitation characteristics. We \fnd\nonly one broad excitation peak at positive energies, instead of the two for the ferromagnetic\nalignment. The antiparallel Fe atoms have opposite intrinsic precessional motion, which\nleads to a secondary peak at negative energy. This does not represent elliptical precession,\nin contrast to the strongly ferromagnetic dimer. We obtain the excitation energies of the\nsystem, and show that the damping contribution lowers the resonance frequency, specially\nfor antiferromagnetic dimers with large coupling. The lowest energy mode is connected to\nthe zero-point \ructuation energy, which indicates that this lowering may inhibit \ructuations\nover the barrier between di\u000berent N\u0013 eel states.\nOur results shed light on how the spin excitations can be engineered by bringing atoms\ntogether or separating them apart. We demonstrate how the external \felds, magnetic\nanisotropy energies, exchange couplings and damping, as well as the initial alignment be-\ntween the magnetic units can be used to design a diverse range of precessional motions.\nTwo particular outcomes may be used as experimental guidance. First, we propose a pump-\nprobe-like experiment [43] to access the in\ruence of the magnetic alignment on the dynamical\nspin excitations, while keeping all the other quantities essentially unchanged: The system is\nput into the metastable state by an initial perturbation (pump) followed by a measurement\nof its excitations by a probe. They can be compared to the excitations from the ground\nstate, when the pump is switched o\u000b. Second, the e\u000bect of the damping on the zero-point\n\ructuation energy indicate that STM experiments made on dimers with similar magnetic\ninteractions but di\u000berent damping strengths (e.g., deposited on insulators or metals) may\npresent rather di\u000berent magnetic signals. The e\u000bect of the damping on the excitation modes\nalso impacts the \feld of antiferromagnetic spintronics. A naive expectation is that ultra-\nfast antiferromagnetic devices shall involve large coupling between the units, to make them\nswitch together, and high damping, to quickly relax the magnetization to a new switched\nstate. Our results demonstrate that the correct picture is more subtle, and a combination\n29of anisotropy, coupling and damping must be taken into account.\nThe interplay between the di\u000berent magnetic interactions o\u000bers multiple tools to control\nprocessing speeds and polarization of magnetic units and emitted spin currents, which may\nlay the foundations of the building blocks of future devices. Our \frst-principles description\nof the dynamical properties of magnetic nanostructures provides a predictive approach to\nthe design and engineering of those building blocks.\nAcknowledgments\nThis work is supported by the European Research Council (ERC) under the European\nUnion's Horizon 2020 research and innovation programme (ERC-consolidator Grant No.\n681405 { DYNASORE).\nAppendix A: Phenomenological expressions for the dynamical susceptibiltiies of\nnanostructures\nBy linearizing the LLG equation of motion [Eq. (7)], they can be written in the local frame\nof reference as [ \u001f\u00001(!)]\u000eM=\u000eBext. In this appendix, we list the inverse susceptibilities \u001f\u00001\nused to \ft the TDDFT results and obtain the parameters of the phenomenological model.\nIn the following, we use \r\u0006=\r\n1\u0006i\u000b.\n1. Single atom\nFor adatoms, \u000eMand\u000eBextare vectors containing the transverse circular components\n+;\u0000of the magnetization and the external \feld, respectively.\na. Uniaxial, no magnetic \feld\nWhen the magnetization points perpendicularly to the surface, the symmetry of the\nsystem isC3vand the anisotropy is uniaxial | in our convention, Kx=Ky=Kand\nKz= 0. The inverse susceptibility for this case is given by\n30\u001f\u00001(!) =0\n@2K\nM2+!\n\r\u0000M0\n02K\nM2\u0000!\n\r+M1\nA: (A1)\nb. Uniaxial, magnetic \feld along ^ ny0\nThe in-plane magnetic \feld saturates the magnetization along the ^ ny0direction in the\nglobal frame of reference. Transforming to the local frame of reference, where it points along\n^ nz, we \fndKx=Kz= 0 andKy=\u0000K. The inverse susceptibility is\n\u001f\u00001(!) =0\n@MBext\n0\u0000K\nM2+!\n\r\u0000MK\nM2\nK\nM2MBext\n0\u0000K\nM2\u0000!\n\r+M1\nA: (A2)\nc. Biaxial\nPlacing a Cu atom close to the Fe adatom, we break the symmetry and change the\nmagnetic anisotropy landscape of the system. By a suitable choice of the local frame of\nreference, the anisotropy matrix can be written in the diagonal form given in Eq. (10). \u001f\u00001\nis then\n\u001f\u00001(!) =0\n@Kx+Ky\nM2+!\n\r\u0000MKx\u0000Ky\nM2\nKx\u0000Ky\nM2Kx+Ky\nM2\u0000!\n\r+M1\nA: (A3)\n2. Dimer\nDimer structures naturally break the symmetry of the system. The linearized equation\nof motion includes the transverse components of both magnetic units in the global frame of\nreference,\u000eM= (M1+;M2+;M1\u0000;M2\u0000).\n31a. Parallel alignment\nWhen the moments are pointing parallel to each other, the inverse susceptibility is\n\u001f\u00001(!) =0\nBBBBB@(Kx+Ky)+J\nM2 +!\n\r\u0000M\u0000J\nM2(Kx\u0000Ky)\nM2 0\n\u0000J\nM2(Kx+Ky)+J\nM2 +!\n\r\u0000M0(Kx\u0000Ky)\nM2\n(Kx\u0000Ky)\nM2 0(Kx+Ky)+J\nM2\u0000!\n\r+M\u0000J\nM2\n0(Kx\u0000Ky)\nM2\u0000J\nM2(Kx+Ky)+J\nM2\u0000!\n\r+M1\nCCCCCA\n(A4)\nb. Antiparallel alignment\nIn the case of antiparallel alignment of the magnetic moments, we have\n\u001f\u00001(!) =0\nBBBBB@(Kx+Ky)\u0000J\nM2 +!\n\r\u0000MJ\nM2(Kx\u0000Ky)\nM2 0\nJ\nM2(Kx+Ky)\u0000J\nM2\u0000!\n\r+M0(Kx\u0000Ky)\nM2\n(Kx\u0000Ky)\nM2 0(Kx+Ky)\u0000J\nM2\u0000!\n\r+MJ\nM2\n0(Kx\u0000Ky)\nM2J\nM2(Kx+Ky)\u0000J\nM2 +!\n\r\u0000M1\nCCCCCA:\n(A5)\nAppendix B: Elliptical mode of single atoms in a circular basis\nTo obtain the natural precessional modes of a single adatom, we linearize the time-\ndependence of the magnetic moment (see Sec. II C). The small transverse components can be\nfound from the associated LLG equation \u001f\u00001\u000eM=\u000eBext. The normal modes of precession\nare obtained by solving the secular problem \u000eBext= 0, which leads to the eigenvalue problem\nDjui=!jui, or\n0\n@\r+Bk\r+B?\n\u0000\r\u0000B\u0003\n?\u0000\r\u0000B\u0003\nk1\nA0\n@u1\nu21\nA=!0\n@u1\nu21\nA: (B1)\nDis the dynamical matrix written in the circular basis with eigenmodes given by the poles\nof the susceptibility, obtained in Eq. (19). Since !\u0000=\u0000(!+)\u0003, the eigenvectors are obtained\nfrom\nju+i: (\r+Bk\u0000!+)u+\n1+\r+B?u+\n2= 0\nju\u0000i: (\r+Bk\u0000!+)u\u0000\u0003\n2+\r+B?u\u0000\u0003\n1= 0; (B2)\n32which shows that u\u0000\u0003\n2=u+\u0003\n1andu\u0000\u0003\n1=u+\n2. Therefore, we can write for the normalized\neigenvectors\nju+i=0\n@ei(\u001e0\u0000\u001e)cos\u0018\nei(\u001e0+\u001e)sin\u00181\nA;ju\u0000i=0\n@e\u0000i(\u001e0+\u001e)sin\u0018\ne\u0000i(\u001e0\u0000\u001e)cos\u00181\nA; (B3)\nwhere we have de\fned\nei(\u001e0\u0000\u001e)cos\u0018=\u0000\r+B?p\nj\r+B?j2+j\r+Bk\u0000!+j2\nei(\u001e0+\u001e)sin\u0018=\r+Bk\u0000!+\np\nj\r+B?j2+j\r+Bk\u0000!+j2: (B4)\nNotice that, since the dynamical matrix Dis not Hermitian, the eigenvectors ju+iand\nju\u0000iare not orthogonal to each other but rather to the left eigenvectors. These may be\nobtained byhvjD=!hvj.\nhv+j=\u0010\ne\u0000i(\u001e0\u0000\u001e)cos\u0018\u0000e\u0000i(\u001e0+\u001e)sin\u0018\u0011\nhv\u0000j=\u0010\n\u0000ei(\u001e0+\u001e)sin\u0018 ei(\u001e0\u0000\u001e)cos\u0018\u0011: (B5)\nThe left and right eigenvectors are orthogonal, i.e., hv+ju\u0000i=hv\u0000ju+i= 0, andhv+ju+i=\nhv\u0000ju\u0000i= cos2\u0018\u0000sin2\u0018. The dynamical matrix can then be written as\nD=!+ju+ihv+j\nhv+ju+i+!\u0000ju\u0000ihv\u0000j\nhv\u0000ju\u0000i: (B6)\nSinceDis written in the circular basis, the eigenvectors represent the transverse com-\nponents of the magnetization \u000eM\u0000and\u000eM+. Bothju+iandju\u0000ilead to the same time\ndependence of the magnetization. Making use of Eq. (12), we \fnally \fnd\n\u000eM(t) =M\u000e\u0012\n2Re\b\u0002\u0000\ne\u0000i\u001ecos\u0018+ei\u001esin\u0018\u0001\n^ nx\n+i\u0000\ne\u0000i\u001ecos\u0018\u0000ei\u001esin\u0018\u0001\n^ ny\u0003\nei(\u001e0\u0000!0t)o\ne!00t: (B7)\nwhere we have used !\u0006=\u0006!0+ i!00, with!0and!00the real and imaginary part of the\nfrequency ( !00<0), respectively.\nIn general, the modes are elliptical, as both \u000eM+and\u000eM\u0000are \fnite. 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Lounis, Nano Lett. 16, 4305 (2016).\n37" }, { "title": "2311.08628v1.Integrated_Local_Energy_Decay_for_Damped_Magnetic_Wave_Equations_on_Stationary_Space_Times.pdf", "content": "INTEGRATED LOCAL ENERGY DECAY FOR DAMPED MAGNETIC WAVE\nEQUATIONS ON STATIONARY SPACE-TIMES\nCOLLIN KOFROTH\nAbstract. We establish local energy decay for damped magnetic wave equations on stationary, asymptoti-\ncally flat space-times subject to the geometric control condition. More specifically, we allow for the addition\nof time-independent magnetic and scalar potentials, which negatively affect energy coercivity and may add\nin unwieldy spectral effects. By asserting the non-existence of eigenvalues in the lower half-plane and res-\nonances on the real line, we are able to apply spectral theory from the work of Metcalfe, Sterbenz, and\nTataru and combine with a generalization of prior work by the present author to extend the latter work and\nestablish local energy decay, under one additional symmetry hypothesis. Namely, we assume that either the\nimaginary part of the magnetic potentials are uniformly small or, more interestingly and novelly, that the\ndamping term is the dominant principal term in the skew-adjoint part of the damped wave operator within\nthe region where the metric perturbation from that of Minkowski space is permitted to be large. We also\nobtain an energy dichotomy if we do not prohibit non-zero real resonances. In order to make the structure\nof the argument more cohesive, we contextualize the present work within requisite existing theory.\nContents\n1. Introduction 1\n1.1. Background and Problem Set-Up 1\n1.2. Local Energy Spaces and Estimates 4\n1.3. Past Results 6\n1.4. Statement of Present Results 6\n2. Frequency Analyses and Two-Point Local Energy Decay 9\n2.1. High Frequency Analysis 9\n2.2. Remaining Frequency Analyses and Two-Point Local Energy Decay 16\n2.3. An Energy Dichotomy 18\n3. Resolvent Theory and Local Energy Decay 20\nReferences 24\n1.Introduction\n1.1.Background and Problem Set-Up. In this work, we will establish (integrated) local energy decay\nfor a broad class of damped wave equations on stationary, asymptotically flat space-times, including those\nwhich possess magnetic and scalar potential terms. This generalizes the result of [Kof23], which lacked\npotential terms and assumed that the damping was compactly-supported. The presence of the potentials\nallow for interaction with the damping and the existence of both complex eigenvalues and real resonances\nembedded in the continuous spectrum, all of which stand to inhibit local energy decay. We will discuss each\nof these obstructions carefully. Along the way, we also establish an energy dichotomy when non-zero real\nresonances are permissible.\nTo set up the problem which we will study, let ( R4, g) be a Lorentzian manifold with coordinates ( t, x)∈\nR×R3and metric signature ( −+ ++) .We will consider damped wave operators of the form\nP=2A,g+iaDt+V, 2A,g= (Dα+Aα)gαβ(Dβ+Aβ)\nDate : November 16, 2023.\n2020 Mathematics Subject Classification. 〈35L05, 58J45, 35P25, 35Q93 〉.\nKey words and phrases. Local energy estimates; Asymptotically flat; Damped wave equation; Geometric control; Trapping.\n1arXiv:2311.08628v1 [math.AP] 15 Nov 2023where a, V, and the components of Aare smooth and complex-valued; here, we are utilizing the notation\nDα=−i∂α, which should be interpreted as an operator. We will call Athe magnetic potential, Vthe scalar\npotential, and athe damping. If a≡0, then we will simply refer to Pas a wave operator . We will require\nthatPisasymptotically flat . More precisely, we first define the family of norms\n|||h|||k=X\n|α|≤k\r\r\r⟨x⟩|α|∂αh\r\r\r\nℓ1\njL∞([0,T]×Aj),\nwhere Aj={⟨x⟩ ≈2j}forj≥0 denote inhomogeneous dyadic annuli, ∂= (∂t,∇x) denotes the space-time\ngradient, and ⟨x⟩= (1 + |x|2)1/2denotes the Japanese bracket of x. The notation A≲Bindicates that\nA≤CBfor some C >0,and the notation A≈Bmeans that B≲A≲B. In the definition of the Aj’s, we\nrequire that these implicit constants are compatible to cover R3.We may now define the AFnorm topology\nvia\n∥(h, A, a, V )∥AF=|||h|||2+|||⟨x⟩A|||1+|||⟨x⟩a|||1+\f\f\f\f\f\f\f\f\f⟨x⟩2V\f\f\f\f\f\f\f\f\f\n0.\nDefinition 1.1. We say that Pis asymptotically flat if ∥(g−m, A, a, V )∥AF<∞, and\n\r\r\r⟨x⟩|α|∂αg\r\r\r\nℓ1\njL∞([0,T]×Aj)≲α1,|α| ≥3,\n\r\r\r⟨x⟩|α|+1∂αA\r\r\r\nℓ1\njL∞([0,T]×Aj)+\r\r\r⟨x⟩|α|+1∂αa\r\r\r\nℓ1\njL∞([0,T]×Aj)≲α1,|α| ≥2,\n\r\r\r⟨x⟩|α|+2∂αV\r\r\r\nℓ1\njL∞([0,T]×Aj)≲α1,|α| ≥1,\nwhere mdenotes the Minkowski metric.\nThe latter conditions are placed to work in standard symbol classes, as opposed to those with limited\nregularity. They extend the flavor of the estimate present in the definition of asymptotic flatness, but one\ndoes not require summability over the differentiation indices.\nWe will primarily be interested in when Pisstationary , which occurs when ( g, A, a, V ) are independent\noft. In particular, ∂tis a Killing field for gwhen Pis stationary. When Pis a damped wave operator,\nwe will always assume that ais independent of t,ais non-negative for |x| ≤2R0, and ais positive on an\nopen subset of {|x| ≤R0}. For |x| ≤2R0, the damping term in Pbehaves as typical viscous damping,\nand we consider it as a general time-independent first-order perturbation term for |x|>2R0. Further, we\nwill assume throughout ∂t, while dttthe constant time slices are uniformly space-like (i.e. dtis uniformly\ntime-like). The former condition implies an ellipticity condition on the terms in Pwhich are independent\nofDt(see Section 1.2 for more), while the latter guarantees that g00≳−1, allowing Pto be reduced to a\nnormal form of g00=−1.\nNext, we introduce parameters that quantify various aspects of asymptotic flatness. Namely, we instantiate\n•the parameters M0,R0andc, which are such that\n∥(g−m, A, a, V )∥AF≤M0, ∥(g−m, A, a, V )∥AF>R0≤c≪1,\nwhere the subscript denotes the restriction of the norm to {|x|> R 0}.The parameter cshould be\nviewed as being fixed first, after which we find a suitable R0for which the above holds.\n•the sequence ( cj)j≥log2R0of positive real numbers satisfying that\n∥(g−m, A, a, V )∥AF(Aj)≲cj,X\njcj≲c,\nwhere ∥·∥AF(Aj)denotes the restriction of the AFnorm to the dyadic region Aj.We may assume\nwithout any loss of generality that the sequence is slowly-varying, i.e.\ncj/ck≤2δ|k−j|, δ ≪1.\nThe integers 0 ≤j 0 independent of the previously-discussed parameters. ■\nRemark 1.5. The weakly ε-damping dominant condition, while seemingly esoteric, will arise naturally in\nour high frequency analysis and constitutes a sharpening of the more concrete and directly-verifiable strongly\nε-damping dominant condition. First, we remark that we will need the latter to hold for sufficiently small\nε >0 in order to obtain our results, whereas we only need the former to hold for some ε >0. The range\nofεwhich are sufficient for our results to hold for strongly ε-damping dominant Pwill guarantee that the\nweaker condition holds (for a different ε). We will now discuss their relationship more explicitly.\nIn the language of Section 2.1, the weakly ε-damping dominant condition may be written as\nb±\nb±−b∓(a+g0αImAα) +ξk\nb±−b∓gkαImAα≥εa, |x| ≤2R0, ξ ̸= 0,\nwhere\nb+(x, ξ)>0> b−(x, ξ), b±(x, ξ)≈ ±|ξ|, ξ ̸= 0.\nHence, the weakly ε-damping dominant condition may be recast (up to a fixed constant coefficient on the\nlower-bound side) in more simple terms as\na+g0αImAα±ξk\n|ξ|gkαImAα≥εa.\nIfPis strongly ε-damping dominant, then\na+g0αImAα±ξk\n|ξ|gkαImAα≥a−2(∥g−m∥L∞+ 1)|ImA|\n≥(1−2ε(∥g−m∥L∞+ 1)) a,|x| ≤2R0, ξ̸= 0.\nIfε <(2∥g−m∥L∞+ 2)−1,then the weakly ε′-damping dominant condition holds with\nε′= 1−2ε(∥g−m∥L∞+ 1)>0.\nIn particular, 0 < ε′<1.\nOn the other hand, if g=m, then the weakly ε-damping dominant condition stipulates that\n1\n2(a−ImA0)±1\n2|ξ|3X\nk=1ξkImAk≥εa, |x| ≤2R0, ξ ̸= 0.\n3By considering when ξis a standard basis vector ejand summing the results over j, we obtain that\n3(a−ImA0)±3X\nk=1ImAk≥6εa, |x| ≤2R0.\nFocusing the minus case, this yields that\na≥(3(1−2ε))−1(2 Im A0+|ImA|),|x| ≤2R0.\nIf Im A0≥0,then Pisε′-strongly damping dominant, with ε′= 3(1 −2ε).Hence, the conditions are\nequivalent in the flat case for non-negative Im A0and appropriate (and different) ε. ■\nWe note that Im Ais already guaranteed to be small for |x|> R 0by asymptotic flatness, so the ε-weakly\nmagnetic condition is more so a condition for |x| ≤R0. This is a weakening of the more standard ε-almost\nsymmetry condition, which will arise briefly in Section 2.3. The ε-damping dominant conditions are, to\nour knowledge, more original and seemingly more natural; we will only discuss the strong version for now\nsince it is simpler to parse and easier to verify (as well as stronger). It stipulates that the damping must\nbe the dominant skew-adjoint term (at the principal level) for |x| ≤2R0.Thus, the magnetic potential is\npermitted to have large imaginary components for |x| ≤2R0,so long as the damping is more significant,\nin the sense stated above. Moreover, the εin the strongly ε-damping dominant condition must satisfy that\nε <(2∥g−m∥L∞+ 2)−1for our results to hold in this work (in which case the weakly ε-damping dominant\ncondition holds with a different ε, as discussed in the previous remark). This indicates that if the metric\nperturbation is sufficiently large within {|x| ≤R0}, then |ImA|must be sufficiently smaller than a. If the\nperturbation is small, then |ImA|andaare permitted to be close for |x| ≤2R0.\nFinally, we remark that neither the ε-weakly magnetic nor the strongly ε-damping dominant condition\nis stronger than the other, although working with the conditions is somewhat similar in the high frequency\nanalysis; the same is true for the ε-weakly magnetic condition and the weakly ε-damping dominant condition,\nfor similar reasons. One should view the damping condition as allowing |ImA|to be potentially quite large\nfor|x| ≤2R0, so long as ais sufficiently larger (although, of course, both are inherently limited by the M0\nparameter, as well). However, let us say that ais zero outside of a ball of radius δ≪1 and is small within the\nball. Then, the strongly ε-damping dominant condition would require that Im Abe zero for δ≤ |x| ≤2R0\nand small everywhere . The ε-weakly magnetic condition allows for |ImAα|>0 for|x| ≤2R0,so long as we\ncontrol the AFsize uniformly.\n1.2.Local Energy Spaces and Estimates. The study of localized energy estimates dates back to the\nwork of [Mor66, Mor68, Mor75], [MRS77] on Minkowski space-time. Local energy estimates constitute\na powerful measure of dispersion, implying Strichartz estimates ([BT07, BT08], [JSS90, JSS91], [MMT08],\n[MMTT10], [MT09, MT12], [Tat08], [Toh12]) and pointwise decay estimates (e.g. [MTT12], [Tat13], [Mor20],\n[MW21], [Loo23], [Hin23]). The latter has been used to tackle various generalizations of Price’s law which,\nin its simplest form, conjectured a t−3pointwise decay rate of waves on non-rotating black hole space-times;\nsee the listed references on pointwise decay and [Hin22] (which settles the conjecture affirmatively in full\ngenerality) for more. The particular local energy estimate of interest in this work, integrated local energy\ndecay, is a powerful quantitative statement given in the form of an inequality which may be qualitatively\ninterpreted as expressing that the energy of a wave must disperse quickly enough to be time-integrable within\nany compact region of space.\nIn order to define the relevant energy inequalities explicitly, we will first define the local energy norms\n∥u∥LE= sup\nj≥0\r\r\r⟨x⟩−1/2u\r\r\r\nL2L2\u0000\nR+×Aj\u0001, ∥u∥LE1=∥∂u∥LE+\r\r⟨x⟩−1u\r\r\nLE.\nThe predual norm to the LEnorm is the LE∗norm, which is defined as\n∥f∥LE∗=∞X\nj=0\r\r\r⟨x⟩1/2f\r\r\r\nL2L2\u0000\nR+×Aj\u0001.\nIf we wish for the time interval to be e.g. [0 , T] in the above norms, then we will use the notation e.g.\n∥u∥LE[0,T]. A subscript of con any of these spaces denotes compact spatial support, whereas a subscript of\n0 denotes the closure of C∞\ncin the relevant norm.\n4Analogous to the local energy spaces, we define the spatial local energy spaces LE,LE1,LE∗when the\ntime variable is fixed (and there is no time derivative nor integral involved in the norms, either). We will\nalso require spaces which allow us to track dependence on a spectral parameter ω, namely\nLE1\nω=LE1∩ |ω|−1LE, ˙H1\nω=˙H1∩ |ω|−1L2.\nThese spaces are equipped with the norms\n∥u∥LE1\nω=∥u∥LE1+|ω|∥u∥LE, ∥u∥˙H1ω=∥u∥˙H1+|ω|∥u∥L2,\nrespectively. They will become relevant in this work when we introduce the resolvent formalism in Section\n3. Now, we will define local energy decay.\nDefinition 1.6. We say that (integrated) local energy decay holds for an asymptotically flat damped wave\noperator if\n(1.1) ∥u∥LE1[0,T]+∥∂u∥L∞L2[0,T]≲∥∂u(0)∥L2+∥Pu∥LE∗+L1L2[0,T],\nwith the implicit constant being independent of T.\nThe next estimate is a weaker variant of local energy decay which does not see complex eigenvalues nor\nnon-zero real resonances (to be defined in Section 3).\nDefinition 1.7. We say that two-point local energy decay holds for an asymptotically flat damped wave\noperator if\n(1.2) ∥u∥LE1[0,T]+∥∂u∥L∞L2[0,T]≲∥∂u(0)∥L2+∥∂u(T)∥L2+∥Pu∥LE∗+L1L2[0,T],\nwith the implicit constant being independent of T.\nThe last estimate is a standard uniform energy bound, which embodies the behavior that one observes\nwhen the energy of the system is non-increasing.\nDefinition 1.8. We say that uniform energy bounds hold for an asymptotically flat damped wave operator\nif\n(1.3) ∥∂u∥L∞L2[0,T]≲∥∂u(0)∥L2+∥Pu∥L1L2[0,T],\nwith the implicit constant being independent of T.\nNotice that the uniform energy bounds provide a link between the two-point local energy decay estimate\nand the local energy decay estimate. In order to discuss the uniform estimate further, let Pu=f, and\nassume that Pis stationary, asymptotically flat, and of symmetric wave type. Consider the sesquilinear map\nEon the energy space E:=˙H1×L2defined by\nE[u,v](t) =Z\nR3P0u¯v−g00∂tu∂tv dx, P 0=P\f\f\nDt=0,w= (w, ∂tw)∈ E.\nThis induces a quadratic energy functional, i.e. an energy form\nE[u](t) :=E[u,u](t).\nWhen ∂tis a uniformly time-like vector field (which we will assume throughout), P0is guaranteed to be\nelliptic (in the principal sense). Direct integration by parts gives that\nd\ndtE[u](t) = 2 ReZ\nR3∂tu¯f dx−2 ReZ\nR3a|∂tu|2dx.\nIfais real-valued and non-negative, then it follows that\nE[u](t)≲E[u](0) +TZ\n0Z\nR3|f∂tu|dxdt, 0≤t≤T.\nIfA, V≡0, then P0isuniformly elliptic (in the sense of the full symbol ), which implies that the energy is\ncoercive, i.e.\nE[u](t)≈ ∥∂u(t)∥2\nL2.\n5Energy coercivity applied to the above now provides the uniform energy bound (1.3).\nWhen AandVare non-zero, one is not guaranteed energy coercivity (and thus not guaranteed uniform\nenergy bounds), even if ais non-negative. However, one does get an almost -coercive energy statement from\nthe ellipticity of P0and the asymptotic flatness assumption:\n(1.4) ∥∂u(t)∥2\nL2≲E[u](t) +∥u(t)∥2\nL2c.\nWhen Pis not of symmetric wave type, then one redefines the energy by replacing P0with its symmetric\npart (see Section 2.3); the estimate (1.4) still holds for the energy associated to P0by ellipticity regardless of\nthe symmetry. The uniform bound (1.3) still need not hold when P0is replaced by its symmetric part. Thus,\nit is not straightforward to transition from (1.2) to (1.1) when A, V̸≡0, even for well-signed dampings; this\nis true even in the symmetric wave type case.\n1.3.Past Results. In [MST20], the authors proved (amongst other results) that local energy decay for\nstationary AFwave operators is equivalent to an absence of geodesic trapping ,negative eigenfunctions , and\nreal resonances . We will describe trapping in Section 2.1 and the spectral objects in Section 3. In short,\ntrapping occurs when there exist bicharacteristic rays which live within a compact set for all time. The\nspectral obstructions correspond to singular behavior of the resolvent - negative eigenfunctions live in L2\nand have corresponding eigenvalue in the lower half-plane, whereas real resonances lie on the real line and\nhave a corresponding resonant state which lives in a local energy space (one must distinguish between zero\nand non-zero resonances). In [MST20], the authors also did not necessarily possess a coercive energy; they\nproved local energy decay by establishing (1.2) using local energy estimates in different frequency regimes,\nthen they utilized resolvent estimates.\nThe work [BR14] utilized dissipative Mourre commutator methods to establish that if the space-time is\nstationary and asymptotically Euclidean (i.e. (R4, g) is a product manifold and hence possesses no non-trivial\nmetric cross terms dt⊗dxj), then one has local energy decay for AFstationary damped wave operators with\nA, V≡0 and abeing a non-negative short-range potential, provided that asatisfied a dynamical hypothesis\ncalled geometric control . This condition requires that all trapped null bicharacteristic rays intersect where\na > 0 (see Definition 2.1 for a precise definition), although the authors of [BR14] only required this for\ntrapped geodesics due to the product manifold structure. Geometric control dates back to [RT74], which\nutilized it to obtain exponential energy decay (i.e. uniform stabilization ) for dissipative problems on compact\nmanifolds. In [Kof23], we generalized the work of [BR14] to the asymptotically flat case (i.e. allowed the\nmetric to possess cross terms). To be precise, we will record this result, which is Theorem 1.9 in [Kof23]\n(adding in the missing assumption of uniformly space-like time slices).\nTheorem 1.9. LetPbe a stationary, asymptotically flat damped wave operator satisfying the geometric\ncontrol condition with A, V≡0andsupp a⊂ {|x| ≤R0}, and suppose that ∂tis uniformly time-like while\nthe constant time slices are uniformly space-like. Then, local energy decay holds, with the implicit constant\nin (1.1) independent of T.\nIn [Kof23], uniform energy bounds held as a result of the conditions on the damping and lack of potentials,\nwhich made it sufficient to prove the two-point bound in order to establish local energy decay. In order to\nestablish the former estimate, [Kof23] followed the strategy set forth in [MST20]:\n(1) Establish local energy estimates that imply local energy decay for Schwartz functions, whose cor-\nresponding function space we denote as S, which are cutoff to high, medium, and low frequency\nregimes.\n(2) Utilize a time frequency partition of unity to prove the estimate\n∥u∥LE1≲∥Pu∥LE∗, u ∈ S.\n(3) Apply an extension procedure to add back in the energy at times 0 and T.\n1.4.Statement of Present Results. We will generalize the work of [Kof23] to include the lower-order\nmagnetic and scalar potentials, along with more general damping functions. Our first result is an extension\nof the high frequency estimate present in [MST20] and [Kof23] to our setting.\nTheorem 1.10. LetPbe a stationary, asymptotically flat damped wave operator which satisfies the geo-\nmetric control condition and is either weakly ε-damping dominant for some ε >0orε-weakly magnetic with\n6ε≪R0,M01. Additionally, assume that ∂tuniformly time-like while the constant time slices are uniformly\nspace-like. Then, the high frequency estimate\n(1.5) ∥u∥LE1[0,T]+∥∂u∥L∞L2[0,T]≲∥∂u(0)∥L2+\r\r⟨x⟩−2u\r\r\nLE[0,T]+∥Pu∥L1L2+LE∗[0,T]\nholds with an implicit constant which is independent of T.\nThis theorem is the one of the primary results whose proof must be adapted from [Kof23] to account\nfor the lower-order terms. It is also where one requires the most substantial deviation from [MST20], since\ntrapping is high frequency. We utilize the symmetry-based assumptions in the theorem so that we may deal\nwith the additional lower-order terms. In particular, this is where the conditions on the interaction between\nthe damping and magnetic potentials come into play. In order to leverage the sign of the damping to mitigate\nthe harmful effects of the trapping, we must limit the magnetic potential appropriately.\nThe medium and low frequencies are not affected by the damping (which may simply be viewed as a\ngeneral sub-principal AFterm, as opposed to being leveraged like in the high frequency setting) and follow\ndirectly from the work in [MST20]. This allows us to establish the two-point local energy estimate under\nthe hypothesis that zero is not a resonance, which is needed in the low frequency regime.\nTheorem 1.11. LetPbe a stationary, asymptotically flat damped wave operator which satisfies the zero\nnon-resonance and geometric control conditions and is either weakly ε-damping dominant for some ε >0or\nε-weakly magnetic with ε≪R0,M01. Additionally, assume that ∂tis uniformly time-like while the constant\ntime slices are uniformly space-like. Then, two-point local energy decay holds, with the implicit constant in\n(1.2) independent of T.\nWe will define the zero non-resonance condition in Section 2.2 and its relation to zero resonant states in\nSection 3. If we additionally impose that Pis of ε-almost symmetric wave type or that Psatisfies the ε-\ndamping dominant as well as the analogous estimate with Areplaced by V, then we obtain a straightforward\nenergy dichotomy (just as in [MST20], where it is Theorem 2.16) as a consequence of the two-point local\nenergy estimate and a uniform energy relation. The given symmetry conditions fulfill a similar role to a\ncondition on the absence of non-zero embedded resonances, and it is needed to obtain the aforementioned\nuniform energy relation (i.e. an almost-conserved energy property if Pu= 0) which will appear in the proof.\nTheorem 1.12. LetPbe a stationary, asymptotically flat damped wave operator which is either\n(a)ε-almost symmetric wave type for sufficiently small ε≪R0,M01\nor\n(b) strongly ε-damping dominant with respect to the magnetic potential, the gradient of the magnetic\npotential, and the scalar potential for sufficiently small ε≪R0,M01, equivalently\na(x)≥ε−1(|A(x)|+|∇A(x)|+|V(x)|),|x| ≤2R0,\nand satisfies the zero non-resonance and geometric control conditions. Additionally, assume that ∂tis uni-\nformly time-like while the constant time slices are uniformly space-like. Then, there exists an α >0so that\nany solution to\nPu=f, u [0]∈ E, f ∈LE∗+L1L2\nsatisfies one of the following two properties:\n(1) Exponential growth asymptotics in terms of the data and forcing:\n∥∂u(t)∥L2≳eαt\u0010\n∥∂u(0)∥L2+∥f∥LE∗+L1L2[0,∞)\u0011\n, t ≫1.\n(2) Local energy decay:\n∥u∥LE1[0,∞)+∥∂u∥L∞L2[0,∞)≲∥∂u(0)∥L2+∥f∥LE∗+L1L2[0,∞).\nRemark 1.13. Condition (b) on the damping can be readily weakened to assuming that Pisε-weakly\ndamping dominant (in order for Theorem 1.11 to apply) and that, for εsmall enough and all t >0, the\nestimate\n−2 RetZ\n0Z\nR3∂suPau dxds −2iImtZ\n0Z\nR3∂suPa\n0u dx ds ≤ −C(ε,c)tZ\n0Z\nB2R0(0)a|∂su|2dxds +O(c)∥u∥2\nLE1\n7holds for some C(ε,c)>0, where PaandPa\n0are the skew-adjoint parts of PandP0, respectively. The\nassumption given in the dichotomy was stated as such merely for tractability, as it will both satisfy the\nrelevant hypothesis in Theorem 1.11 and allows the above bound to hold as a consequence of H¨ older’s\ninequality, Young’s inequality for products, and asymptotic flatness. ■\nAt least at a heuristic level, the solutions which exhibit case (1) behavior stem from eigenvalues in the lower\nhalf-plane of the corresponding stationary problem (see Section 3), which represent poles of the resolvent.\nThe resolvent has meromorphic continuation to the entire lower half-plane, and the poles must occur within\na relatively compact subset of frequencies. In particular, there are only finitely many such eigenvalues,\nand each generalized eigenspace has finite dimension by Fredholm theory. We also remark that we do not\nobtain improvements from the corresponding result in [MST20] here, nor do we obtain versions of their\nnon-stationary results since our high frequency work exploits the stationarity. While more refined energy\nspace decompositions are likely (and are also present in [MST20] within the non-damped setting), the proper\nstatements and results in the context of damped waves are not clear to us at this time.\nFinally, we have local energy decay. Here, we must further assume that Psatisfies various spectral\nhypotheses, which are defined in Section 3.\nTheorem 1.14. LetPbe a stationary, asymptotically flat damped wave operator which satisfies the zero\nnon-resonance and geometric control conditions and is either weakly ε-damping dominant for some ε >0\norε-weakly magnetic with ε≪R0,M01. Suppose further that Phas no negative eigenfunctions nor real\nresonances and that ∂tis uniformly time-like while the constant time slices are uniformly space-like. Then,\nlocal energy decay holds, with the implicit constant in (1.1) independent of T.\nIn order to prove this result, we cannot necessarily rely on uniform energy bounds to pass from the\ntwo-point local energy estimate to local energy decay like in [Kof23] and instead rely on spectral theory as\nin [MST20]. The structure of proving Theorem 1.14, and hence of the overall paper, is as follows and is\nmotivated by [MST20]:\n(1) Section 2.1: Extend the high frequency analysis from [Kof23] to the present context. One must take\ncare to consider how the damping interacts with the remaining principal skew-adjoint terms; this\nis the purpose of the ε-weakly magnetic and ε-damping dominant conditions. We will explain the\nresults that we borrow and why they apply here.\n(2) Section 2.2: Provide an overview of the low and medium frequency analysis; these do not require\nchange from [MST20], since the damping is simply treated as a first-order AFperturbation term.\nFrom here, Theorem 1.11 follows readily from the work in [MST20], [Kof23]. As an immediate\nconsequence, we will establish Theorem 1.12 in Section 2.3, although this has no bearing on the\nproof of Theorem 1.14.\n(3) Section 3: Summarize the necessary resolvent theory in [MST20] required to prove Theorem 1.14.\nThis will require Theorem 1.10 and Theorem 1.11. Once one is armed with the relevant frequency\nestimates, the work in [MST20] applies rather directly. We will summarize and/or provide many\n(but not all) of their arguments for the required results, especially where we believe that further\nelucidation would be beneficial for the sake of exposition. We require little deviation from their\ntheory in our present work.\nRemark 1.15. As a consequence of Remark 1.5, Theorems 1.10, 1.11, and 1.14 hold if the symmetry\nconditions are replaced by the strongly ε-damping dominant condition with 2 ε <(∥g−m∥L∞+ 1)−1.■\nRemark 1.16. As opposed to writing the d’Alembertian in divergence form (i.e. 2g=DαgαβDβ) and\nutilizing the volume form dV=dxdt for our analysis, one could work with the geometric d’Alembertian\n(that is, the Laplace-Beltrami form)\ne2g=|g|−1/2Dα|g|1/2gαβDβ,|g|=|detgαβ|\nwith the volume form dV=|g|1/2dxdt. Each d’Alembertian is symmetric with respect to the associated\nvolume form on L2(dV). One can transition from the latter framework to the former by conjugating the\noperator by |g|1/4(see e.g. [Tat13], [Mor20]); lower-order terms arise, but they are permissible in view of\nthe magnetic and scalar potential terms already allowable in P. For this reason, we are working with the\nformer case. ■\n8Acknowledgements. The author would like to thank Jason Metcalfe for helpful discussions. He would\nalso like to thank both anonymous reviewers for their feedback, including the previously-missing assumption\nof uniformly space-like time-slices, which is also needed in [Kof23].\nDeclarations. The author reports there are no competing interests to declare.\n2.Frequency Analyses and Two-Point Local Energy Decay\nIn this section, we will prove Theorem 1.11 using high, low, and medium frequency analyses. The low and\nmedium frequency work follows directly from that in [MST20], while the high frequency work is a variation\non [Kof23] and requires the symmetry assumptions as described in Section 1.1.\nTo start, we will define cutoff notation which we will use throughout the duration of the paper. Namely,\nwe will fix χ∈C∞\ncnon-increasing , χ≡1 for|x| ≤1, χ≡0 for|x|>2 and define χR= 1−χ0. (2.1)\nIn order to leverage the sign of the damping, [Kof23] utilized a scaling argument which we will also exploit.\nNamely, if usolves Pu=f, we consider\n˜u(t, x) :=u(γt, γx ), γ > 0.\nFor this discussion, a tilde over a function will denote dilation by γin each coordinate, as done to define ˜ u.\nIf we call\n˜P= (Dα+γ˜Aα)˜gαβ(Dβ+γ˜Aβ) +iγ˜aDt+γ2˜V ,\nthen\n˜P(γ−2˜u) =˜f if and only if Pu=f.\nThe benefit of this scaling is that we obtain an arbitrarily large constant in front of the damping function.\nHowever, we underscore that such a large constant is also inherited by the magnetic potential.\nAnalogous Hamiltonian systems and trapped/non-trapped sets exist for the principal symbol ˜ pof˜P,\nwhich amounts to simply dilating the coordinates of g. If we assume that geometric control holds for the\nflow generated by p, then it is proven in [Kof23] that it holds for the flow generated by ˜ p. Since the proof is\nstraightforward, we will omit it here and only record the result, which is Proposition 2.6 in [Kof23].\nProposition 2.2. Assume that the geometric control condition (2.1) holds. Then, for any γ > 0, (2.1)\nholds for the flow generated by ˜p, with areplaced by ˜a.\nSince the lower-order terms AandVare not at the principal level, they do not affect pnor ˜p. Henceforth,\nwe will fix a large γ > 0 and study the problem from the scaled perspective while reverting back to our\noriginal notation (e.g. no tildes). It is readily seen that it is equivalent to prove Theorem 1.10 for the scaled\nproblem, where we now have a large constant in front of the damping term.\nThe proof of the version of Theorem 1.10 present in [Kof23] (i.e. A, V≡0,supp a⊂ {|x| ≤R0}) is a\npositive commutator argument. At the symbolic level, this requires the construction of an escape function\nand a lower-order correction term. Let pandsskew represent the principal symbols of the self and skew-\nadjoint parts of P, respectively. Namely,\np(x, τ, ξ ) =−(τ2−2τg0j(x)ξj−gij(x)ξiξj)\nsskew(x, τ, ξ ) =iγ\u0000\nImAα(x)(gα0(x)τ+gαk(x)ξk) +τRea(x)\u0001\n.\nThe multiplication by γinsskew will prove advantageous for a bootstrapping argument, which is precisely\nwhy we implement the γ-scaling. However, the imaginary part of Ahas interaction with the damping and\nalso features multiplication by γ. The ε-damping dominant and ε-weakly magnetic conditions are applied\nin order to retain the positivity effects of the damping.\nThe precise escape function construction is as follows.\nLemma 2.3. For all λ >1, there exist symbols qj∈Sj(T∗R3)andm∈S0(T∗R3), all supported in |ξ| ≥λ,\nso that\nHpq−2isskewq+pm≳χ|ξ|>λ⟨x⟩−2\u0000\nτ2+|ξ|2\u0001\n,\nwhere q=τq0+q1.\nHere, Hpis the Hamiltonian vector field induced by p, and Sm(T∗R3) denotes the standard Kohn-\nNirenberg symbol class of order m. To each symbol b∈Sm(T∗R3), we have the associated Weyl quantization\nofb, denoted bw, which is a pseudodifferential operator of order mdefined by the action\nbw(x, D)u(x) = (2 π)−3Z\nR3Z\nR3ei(x−y)·ξb\u0012x+y\n2, ξ\u0013\nu(y)dydξ, u ∈ S(R3).\nWe will use Ψm(R3) to denote the space of pseudodifferential operators on R3of order m, and write\nΨ−∞(R3) :=\\\nm∈RΨm(R3)\nfor the space of smoothing operators on R3.\n10Lemma 2.3 was proven in [Kof23] in the special case of A, V≡0 and supp a⊆ {|x| ≤R0}, leading to a\nsimplified sskew. As in the aforementioned work, we will work with the half-wave decomposition; it is proven\nin [Kof23] that the null bicharacterstics are equivalent through a reparameterization argument, and this fact\ncontinues to hold here without any change (lower-order terms do not affect pand hence will not affect its\ninduced bicharacteristic flow). That is, we factor pas\np(τ, x, ξ ) =−(τ−b+(x, ξ))(τ−b−(x, ξ)),\nwhere\nb±(x, ξ) =g0j(x)ξj±q\n[g0j(x)ξj]2+gij(x)ξiξj.\nUsing that ∂tis uniformly time-like, it is readily seen that b±are both positively homogeneous of degree 1\ninξ, and\nb+(x, ξ)>0> b−(x, ξ)\nwhenever ξ̸= 0. The Hamiltonians p±:=τ−b±also generate bicharacteristic flows\nφ±\ns(w) =\u0000\nt±\ns(w), x±\ns(w), τ±\ns(w), ξ±\ns(w)\u0001\nonR×T∗R4which solve the Hamiltonian systems\n(\n˙t±\ns= 1, ˙τ±\ns= 0,\n˙x±\ns=−∇ξb±(φ±\ns(w)), ˙ξ±\ns=∇xb±(φ±\ns(w)),\nwith initial data w∈T∗R4.Observe that the ( t, τ) and ( x, ξ) systems are decoupled, allowing us to project\nonto the ( x, ξ) components of the flow without losing information. Notice that, after we project, we are\nno longer looking at null bicharacteristics but, rather, bicharacteristics with initial data having non-zero ξ\ncomponent.\nNow, we may define the forward and backward (denoted by the ±notation) trapped and non-trapped\nsets for the half-wave flows as\nΓ±\ntr=\u001a\nw∈T∗R3\\o: sup\ns≥0|x±\ns(w)|<∞\u001b\n,\nΛ±\ntr=\u001a\nw∈T∗R3\\o: sup\ns≥0|x±\n−s(w)|<∞\u001b\n,\nΓ±\n∞=\b\nw∈T∗R3\\o:|x±\ns(w)| → ∞ ass→ ∞\t\n,\nΛ±\n∞=\b\nw∈T∗R3\\o:|x±\n−s(w)| → ∞ ass→ ∞\t\n.\nThe trapped and non-trapped sets are\nΩ±\ntr= Γ±\ntr∩Λ±\ntrand Ω±\n∞= Γ±\n∞∩Λ±\n∞,\nΩtr= Ω+\ntr∪Ω−\ntrand Ω ∞= Ω+\n∞∪Ω−\n∞.\nAs a consequence of the factoring, we have the identities\nΩtr= Π x,ξ(Ωp\ntr) and Ω ∞= Π x,ξ(Ωp\n∞),\nwhere Π x,ξ(t, x, τ, ξ ) = (x, ξ).Additionally, we may re-state geometric control in terms of the factored flow.\nIfw∈Ωtr, then it is either trapped with respect the flow generated by p+orp−. If it is trapped with respect\ntop+,then there is a time so that wis flowed along a p+-bicharacteristic ray to a point where the damping\nis positive, and similarly if it is trapped with respect to p−.\nSince AandVdo not occur at the principal level for P, they will not affect the individual components of\nthe escape function construction. Hence, the construction in [Kof23] applies directly. The methodology in\n[Kof23], motivated by a combination of [BR14], [MST20], and [MMT08], is performed in the following steps:\n(1)On the characteristic set. We will refer to {|x| ≤R0}as the interior and{|x|> R 0}as the\nexterior .\n(a)Interior, semi-bounded null bicharacteristics. Here, one considers semi-bounded null\nbicharacteristics with initial data living in the interior region. Working with semi-bounded\ntrajectories is favorable since they include both trajectories that are trapped and those which\n11escape slowly. Additionally, geometric control extends to such trajectories. This is where\ngeometric control is used.\n(b)The remainder of the interior region. In this region, all of the trajectories escape both\nforward and backward in time. This region is more classical, but care must be taken both to\navoid the trapping and incorporate the half-wave structure.\n(c)The exterior region. As a consequence of asymptotic flatness, there are no trapped trajecto-\nries in this region. Here, one appeals to geometrically-adapted flat wave theory. The multiplier\nalso allows for the absorption of an error term which arises in the prior region.\n(2)On the elliptic set. Here, one requires a lower-order symbol which provides no contribution on\nthe characteristic set and provides positivity off of it. This essentially follows from the minimization\nof an appropriate quadratic in the dual time variable τ.\nNow, we cite the specific results from [Kof23] (namely, Lemmas 2.13 and 2.16, respectively). First, we\ndefine the interior, semi-trapped set\nΩ±\nR0:=\u0000\nΓ±\ntr∪Λ±\ntr\u0001\n∩ {|x| ≤R0}.\nThis is related to step (1a). Next, we define the function\nΦ±(x, ξ) =\u0012\nx,ξ\n|b±(x, ξ)|\u0013\n.\nIt can be shown directly that(\nx±\ns(x, ξ) =x±\ns(x, λξ),\nλξ±\ns(x, ξ) =ξ±\ns(x, λξ)\nfor any λ >0 due to the homogeneity of b±. Since |ξ/b±(x, ξ)| ≈1 for ξ̸= 0 and b±is a constant of motion\nfor the flow generated by p±(which explicitly utilizes that gis stationary), the function Φ±provides a lifting\nwhich is useful to pair with scaling arguments. The first portion of the construction (1a) is contained in\nthe following lemma, which was motivated by [BR14]. This is where geometric control is utilized. (More\nprecisely, geometric control also applies to semi-trapped trajectories; see [Kof23].)\nLemma 2.4 (Semi-bounded Escape Function Construction) .There exist q±\n1∈C∞(T∗R3\\o), an open set\nV±\nR0⊃Ω±\nR0, and C±∈R+so that\nHp±q±\n1+C±aχ R 0}and\nHp±q±\n2≳cj2−j1W±,|x| ≈2j.\nFurther, q±\n2=q±\nin+q±\nout,where q±\nin= ˜q±\nin◦Φ±, with ˜q±\nin∈C∞(T∗R3\\o)being supported in {|x| ≤4R}and\nq±\nout∈S0\nhom(T∗R3\\o).\nWe remark that the behavior of ( cj) does not matter so much for 0 ≤j < log2R0,as long as each\ncorresponding cjis positive.\nTo complete the remaining steps and prove Lemma 2.3, we proceed similarly to the work in [Kof23], with\nspecial attention paid to the new contributions of sskew.The presence of Im Ainsskew did not occur in\n[Kof23] and must be dealt with here. The symmetry conditions arise when one must balance the ability to\nleverage the sign on the damping for |x| ≤2R0with the necessity to absorb the unsigned magnetic terms.\nProof of Lemma 2.3. First, we truncate the symbols to stay away from ξ= 0:\nq±\nj,>λ=e−σjq±\njχ|b±|>λ, j= 1,2,\nwhere σ1, σ2≫1. Unlike [Kof23], we need two parameters σ1andσ2, as opposed to just one parameter; the\nadditional parameter is needed to deal with unsigned first-order errors. The exponentiation is implemented\nfor bootstrapping: Taking derivatives of the exponentials will provide multiplication by σ1andσ2. It\n12is readily seen that q±\nj,>λ∈S0(T∗R3) via the chain rule. We combine the symbols constructed on the\nindividual light cones together as\nq(x, τ, ξ ) = (τ−b+)(q−\n1,>λ+q−\n2,>λ) + (τ−b−)(q+\n1,>λ+q+\n2,>λ).\nCalling\nqj= (τ−b+)q−\nj,>λ+ (τ−b−)q+\nj,>λ,\nwe can see that\n\u0000\nHpq+ 2γ\u0000\nτRea+ Im Aα(gα0τ+gαkξk)\u0001\nq\u0001\f\f\nτ=b±\n=\u0000\nHpq1+ 2γ\u0000\nτRea+ Im Aα(gα0τ+gαkξk)\u0001\nq1\u0001\f\f\nτ=b±+\u0000\nHpq2+ 2γ\u0000\nτRea+ Im Aα(gα0τ+gαkξk)\u0001\nq2\u0001\f\f\nτ=b±\n=Hpq1\f\f\nτ=b±±2γ(b+−b−)\u0000\nb±(Rea+ Im Aαgα0) + Im Aαgαkξk|τ=b±\u0001\nq±\n1,>λ\n+Hpq2\f\f\nτ=b±±2γ(b+−b−)\u0000\nb±(Rea+ Im Aαgα0) + Im Aαgαkξk|τ=b±\u0001\nq±\n2,>λ.\nWe will work with each term in the last equality separately. First, we compute that\nHpqj\f\f\nτ=b±=−(b+−b−)2Hp±q±\nj,>λ−(b±−b∓)q±\nj,>λ(b±\nξjb∓\nxj−b±\nxjb∓\nξj)\n=σj(b+−b−)2q±\nj,>λHp±q±\nj−(b±−b∓)q±\nj,>λ(b±\nξjb∓\nxj−b±\nxjb∓\nξj).\nBy making σ1, σ2sufficiently large, we get that\nHpq1\f\f\nτ=b±+Hpq2\f\f\nτ=b±≥1\n2σ1(b+−b−)2q±\n1,>λHp±q±\n1+1\n2σ2(b+−b−)2q±\n2,>λHp±q±\n2,\nand so\n\u0000\nHpq+ 2γ\u0000\nτRea+ Im Aα(gα0τ+gαkξk)\u0001\nq\u0001\f\f\nτ=b±\n≳σ1(b+−b−)2q±\n1,>λHp±q±\n1±2γ(b+−b−)\u0000\nb±(Rea+ Im Aαgα0) + Im Aαgαkξk|τ=b±\u0001\nq±\n1,>λ\n+σ2(b+−b−)2q±\n2,>λHp±q±\n2±2γ(b+−b−)\u0000\nb±(Rea+ Im Aαgα0) + Im Aαgαkξk|τ=b±\u0001\nq±\n2,>λ.\nWe will consider each line in the above lower bound separately. First, observe that\nb±\nb±−b∓≈1.\nCase 1: Pis weakly ε-damping dominant. By using the weakly ε-damping dominant condition and\nchoosing γsufficiently larger than ε−1σ1, we may apply Lemma 2.4 to obtain that, for |x| ≈2ℓ,\nσ1(b+−b−)2q±\n1,>λHp±q1±2γ(b+−b−)\u0000\nb±(Rea+ Im Aαgα0) + Im Aαgαkξk|τ=b±\u0001\nq±\n1,>λ(2.2)\n=σ1(b+−b−)2q±\n1,>λ\u0012\nHp±q±\n1+\u00122γ\nσ1\u0013b±\nb±−b∓\u0000\nRea+g0αImAα\u0001\n+\u00122γ\nσ1\u0013ξk\nb±−b∓ImAαgαk\u0013\n≳σ1(b+−b−)2q±\n1,>λ\u0012\nHp±q±\n1+2γ\nσ1(εReaχR0)\u0013\n≳σ1χ|ξ|>λ|ξ|2\u0012\n1V±\nR0−2γ\nσ1cℓ2−ℓχ>R0\u0013\n.\nNext, we use the weakly ε-damping dominant, use that the damping is non-negative for |x| ≤2R0, use\nLemma 2.5, and choose σ2sufficiently larger than γto give that, for |x| ≈2ℓ,\nσ2(b+−b−)2q±\n2,>λHp±q2±2γ(b+−b−)\u0000\nb±(Rea+ Im Aαgα0) + Im Aαgαkξk|τ=b±\u0001\nq±\n2,>λ(2.3)\n=σ2χλ\u0012\nHp±q±\n2+\u00122γ\nσ2\u0013b±\nb±−b∓\u0000\nRea+g0αImAα\u0001\n+\u00122γ\nσ2\u0013ξk\nb±−b∓ImAαgαk\u0013\n+σ2χ>R0(b+−b−)2q±\n2,>λ\u0012\nHp±q±\n2+\u00122γ\nσ2\u0013b±\nb±−b∓\u0000\nRea+g0αImAα\u0001\n+\u00122γ\nσ2\u0013ξk\nb±−b∓ImAαgαk\u0013\n≳σ2χ|ξ|>λχ|x|λχ|x|>R0cℓ2−ℓ|ξ|2\u0012\n1−2γ\nσ2\u0013\n13≳σ2χ|ξ|>λχ|x|λχ|x|>R0cℓ2−ℓ|ξ|2.\nCase 2: Pisε-weakly magnetic. By using the ε-weakly magnetic condition and choosing γsufficiently\nlarger than σ1, we may apply Lemma 2.4 to obtain that, for |x| ≈2ℓ,\nσ1(b+−b−)2q±\n1,>λHp±q1±2γ(b+−b−)\u0000\nb±(Rea+ Im Aαgα0) + Im Aαgαkξk|τ=b±\u0001\nq±\n1,>λ(2.4)\n=σ1(b+−b−)2q±\n1,>λ\u0012\nHp±q±\n1+\u00122γ\nσ1\u0013b±\nb±−b∓\u0000\nRea+g0αImAα\u0001\n+\u00122γ\nσ1\u0013ξk\nb±−b∓ImAαgαk\u0013\n≳σ1(b+−b−)2q±\n1,>λ\u0012\nHp±q±\n1+2γ\nσ1(ReaχR0)\u0013\n≳σ1χ|ξ|≥λ|ξ|2\u0012\n1V±\nR0−2γ\nσ1\u0000\nε(1 +∥g−m∥L∞)χR0\u0001\u0013\n.\nThe poorly-signed interior term above is somewhat problematic and will dictate the choice of ε.\nNext, we use the ε-weakly magnetic condition, apply Lemma 2.5, and choose σ2sufficiently larger than γ\nto give that, for |x| ≈2ℓ,\nσ2(b+−b−)2q±\n2,>λHp±q1±2γ(b+−b−)\u0000\nb±(Rea+ Im Aαgα0) + Im Aαgαkξk|τ=b±\u0001\nq±\n2,>λ(2.5)\n=σ2χλ\u0012\nHp±q±\n2+\u00122γ\nσ2\u0013b±\nb±−b∓\u0000\nRea+g0αImAα\u0001\n+\u00122γ\nσ2\u0013ξk\nb±−b∓ImAαgαk\u0013\n+σ2χ>R0(b+−b−)2q±\n2,>λ\u0012\nHp±q±\n2+\u00122γ\nσ2\u0013b±\nb±−b∓\u0000\nRea+g0αImAα\u0001\n+\u00122γ\nσ2\u0013ξk\nb±−b∓ImAαgαk\u0013\n≳σ2χ|ξ|>λχ|x|λχ|x|>R0cℓ2−ℓ|ξ|2\u0012\n1−2γ\nσ2\u0013\n≳σ2χ|ξ|>λχ|x|λχ|x|>R0cℓ2−ℓ|ξ|2.\nWe claim that, in both cases, the estimate\n(2.6)\u0000\nHpq+ 2γ\u0000\nτRea+ Im Aα(gα0τ+gαkξk)\u0001\nq\u0001\f\f\nτ=b±≳ 1|ξ|≥λcℓ2−ℓ|ξ|2,|x| ≈2ℓ\nholds. Indeed, recall that V±\nR0∪W±=T∗R3\\o. In Case 1, we combine (2.2) and (2.3) and, if necessary,\nfurther increase σ2to directly get (2.6). In Case 2, we obtain (2.6) by combining (2.4) and (2.5) together, then\nchoosing ε≪R0,M01 and, if necessary, further increase σ2. From (2.6), we use that ( cℓ) is a slowly-varying,\nsummable sequence to conclude that\n\u0000\nHpq+ 2γ\u0000\nτRea+ Im Aα(gα0τ+gαkξk)\u0001\nq\u0001\f\f\nτ=b±≳χ|ξ|>λ⟨x⟩−2|ξ|2.\nThe work on the elliptic set in e.g. [MST20], [Kof23] applies without modification. Summarily, if we write\n(2.7) Hpq+2γ\u0000\nτRea+ Im Aα(gα0τ+gαkξk)\u0001\nq+pm= (a0−m)τ2+\u0000\na1+ (b++b−)m\u0001\nτ+(a2−b+b−m),\nwhere aj∈Sj, then we choose\nm=−(b+−b−)−2\u0000\na1(b++b−) + 2( a0b+b−+a2)\u0001\n.\nSuch an mensures that the quadratic polynomial (2.7) in τis concave up and has no real zeros. □\nThe proof of Theorem 1.10 is highly similar to the proof in [Kof23] for the analogous result. There are\na few additional terms to deal with, but the aforementioned work demonstrates how to deal with them, as\nthey are lower-order. We will briefly summarize the argument in [Kof23] and add in additional details for\nthe new terms. First, we remark that one can readily reduce the theorem to a simplified estimate, which we\nstate as a proposition.\nProposition 2.6. In order to prove Theorem 1.10, it suffices to prove that\n(2.8) ∥v>λ∥LE1\n<2R0≲C(λ, γ)\u0010\n∥Pv∥1/2\nLE∗\nc∥v∥1/2\nLE1+∥v∥L2L2\u0011\n+γλ−1/2∥v∥LE1\nfor all vsupported in {|x| ≤2R0},where vλ=χ|ξ|>λ(Dx)v.\n14This reduction was shown in [MST20] and [Kof23]. We will not reproduce the proof here, but the idea is\nas follows:\n(1) Use asymptotic flatness to reduce to the case of Puandu[0] compactly-supported in {|x| ≤2R0}.\n(2) Utilize a unit time interval and Duhamel’s theorem to reduce to u[0] = 0 and Pu∈LE∗\nc.\n(3) Remove the upper bound on the time integrals using a cutoff argument, making it sufficient to\nintegrate in tfrom−∞to∞.\n(4) Reduce to solutions supported in {|x| ≤2R0}via standard exterior wave estimates.\n(5) Add back in the low frequencies, then take λ≫γand apply Young’s inequality for products in order\nto bootstrap the LE1terms on the right into the left.\nNow, we prove Theorem 1.10. During the proof, we will use ⟨·,·⟩to denote the L2L2inner product. We also\nrecall the γ-scaling that was introduced after Definition 2.1.\nProof of Theorem 1.10. We compute that\n2Im\u001c\nPv,\u0012\nqw−i\n2mw\u0013\nv\u001d\n+iγ\n2\n[(ImAαgαβDβ+DαgαβImAα), mw]v, v\u000b\n+iγ\n2⟨[aDt, mw]v, v⟩(2.9)\n−iγ\n[(ReAαgαβDβ+DαgαβReAβ), qw], v, v\u000b\n−γ\n2\n\u0000\nmw(ReAαgαβDβ+DαgαβReAβ) + (Re AαgαβDβ+DαgαβReAβ)mw\u0001\nv, v\u000b\n+iγ2\n[ImAαgαβImAβ−ReAαgαβReAβ, qw]v, v\u000b\n−2γ2\n\u0000\nqw(ImAαgαβReAβ) + (Im AαgαβReAβ)qw\u0001\nv, v\u000b\n−γ2\n2\n\u0000\nmw(ReAαgαβReAβ) + (Re AαgαβReAβ)mw\u0001\nv, v\u000b\n+γ2\n2\n\u0000\nmw(ImAαgαβImAβ) + (Im AαgαβImAβ)mw\u0001\nv, v\u000b\n+γ2\n[ReAαgαβImAβ, mw]v, v\u000b\n+iγ2⟨[qw,ReV]v, v⟩+iγ\n2⟨[ImV, mw]v, v⟩\n−γ2\n2⟨(mwReV+ Re V mw)v, v⟩ −γ2⟨(qwImV+ Im V qw)v, v⟩\n+γ⟨ImaDtmwv, v⟩+γ⟨[Ima, qw]Dtv, v⟩\n=⟨i[2g, qw]v, v⟩+1\n2⟨(2gmw+mw2g)v, v⟩\n+γ\n\u0000\nqw(ReaDt+ Im AαgαβDβ+DαgαβImAβ) + (Re aDt+ Im AαgαβDβ+DαgαβImAβ)qw\u0001\nv, v\u000b\n.\nNotice that the right-hand side of (2.9) may be written as\n⟨(Hpq−2isskewq+mp)wv, v⟩+⟨R0v, v⟩, where R0∈Ψ0(R3).\nSplit vas\nv=v>>λ+v<<λ:=χ|ξ|+|τ|>λ(∂)v+χ|ξ|+|τ|<λ(∂)v,\nand recall that q, mare both supported at frequencies |ξ| ≥λ.By choosing γlarge enough, the sharp G˚ arding\ninequality yields that\n⟨(Hpq−2isskewq+mp)wv, v⟩≳D\u0000\nχ|ξ|>λ⟨x⟩−2(|ξ|2+τ2)\u0001wv>>λ, v>>λE\n− ∥v>>λ∥2\nH1/2\nt,x+⟨Sv, v⟩, where S∈Ψ−∞(R3).\nIntegrating by parts one time gives that\nD\u0000\nχ|ξ|>λ⟨x⟩−2(|ξ|2+τ2)\u0001wv>>λ, v>>λE\n≳∥∂v>λ∥2\nLE<2R0+⟨R1v>>λ, v>>λ⟩, where R1∈Ψ1(R3).\nAll together, the right-hand side of (2.9) is bounded below by a multiple of\n∥∂v>λ∥2\nLE<2R0− |⟨R1v>>λ, v>>λ⟩| − ∥ v>>λ∥2\nH1/2\nt,x− |⟨R0v, v⟩|.\n15Bounding the errors is performed using Plancherel’s theorem. With more specificity, one may utilize the\nstandard Sobolev mapping properties of pseudodifferential operators, the frequency localization, and the\ncompact spatial support of vto obtain that\n|⟨R1v>>λ, v>>λ⟩|+∥v>>λ∥2\nH1/2\nt,x≲λ−1∥v∥2\nLE1.\nOne shows that\n|⟨R0v, v⟩|≲C(λ)∥v∥2\nL2L2\nin a similar manner, except that one cannot leverage frequency localization and may incur an implicit\nconstant which depends on λ. Such a term will appear on the upper bound side of (2.8), so having such an\nimplicit constant is permissible.\nSummarizing, we have shown that the right-hand side of (2.9) is bounded below by a multiple of\n∥∂v>λ∥2\nLE<2R0−C(λ)∥v∥2\nL2L2−λ−1∥v∥2\nLE1. (2.10)\nNext, we consider the left-hand side of (2.9). By the Cauchy-Schwarz and Plancherel’s theorem,\n\u001c\nPv,\u0012\nqw−i\n2mw\u0013\nv\u001d\n=\u001c\nPv,\u0012\nqw−i\n2mw\u0013\nv>>λ\u001d\n+⟨Sv, v⟩, S ∈Ψ−∞(R3)\n≲C(λ)\u0010\n∥Pv∥LE∗c∥v∥LE1+∥v∥2\nL2L2\u0011\n.\nBy once again applying frequency splitting, the remaining inner products on the left-hand side of (2.9) are\nof the form\n(γ+γ2)\u0010D\neR0v, vE\n+D\neR1v>>λ, v>>λE\u0011\n,eRj∈Ψj(R3).\nWe have discussed how to bound both of these terms; namely,\f\f\fD\neR0v, vE\f\f\f+\f\f\fD\neR1v>>λ, v>>λE\f\f\f≲C(λ)∥v∥2\nL2L2+λ−1∥v∥2\nLE1.\nFactoring in the scalar coefficients of these inner products, we have demonstrated that the left-hand side of\n(2.9) is bounded above by a multiple of\nC(λ, γ)\u0010\n∥Pv∥LE∗c∥v∥LE1+∥v∥2\nL2L2\u0011\n+γ2λ−1∥v∥2\nLE1 (2.11)\nCombining (2.10)-(2.11) in application to (2.9) and completing the LE1norm on the lower-bound side\nprovides (2.8). □\n2.2.Remaining Frequency Analyses and Two-Point Local Energy Decay. In order to establish\nTheorem 1.11, we require similar estimates in the low and medium frequency regimes. The damping does\nnot play a meaningful role in either regime, as it may be treated as a lower-order perturbation term. Like\nin [Kof23], the relevant estimates from [MST20] carry through. We will briefly summarize why this is the\ncase, in lieu of full proofs.\nAt low frequencies, the obstruction to local energy decay arises when Phas a resonance at frequency\nzero (see Section 3 for a precise definition and further discussion on spectral obstructions to local energy\ndecay). A quantitative condition on the existence of corresponding zero resonant states is the following zero\nnon-resonance condition .\nDefinition 2.7. Pis said to satisfy a zero non-resonance condition if there exists some K0, independent of\nt, such that\n∥u∥˙H1≤K0∥P0u∥˙H−1∀u∈˙H1. (2.12)\nThe elliptic operator\nP0=P\f\f\nDt=0= (Dj+Aj)gjk(Dk+Ak) +A0g0j(Dj+Aj) + (Dj+Aj)gj0A0+ (A0)2g00+V\nrepresents Pat time frequency zero, and we underscore that the damping does not appear. Hence, the\ndamping has no bearing on whether or not the zero non-resonance condition holds. For example, if Pis\nstationary and asymptotically flat with Im A≡0,V >0,andA0= 0, then Psatisfies the zero non-resonance\ncondition. This follows from Lemma 6.2iii in [MST20], which also features a more general condition. The\nrelevant low frequency estimate is the following, and the corresponding theorem in [MST20] is Theorem 6.1.\n16Theorem 2.8. LetPbe an asymptotically flat damped wave operator which satisfies the zero non-resonance\ncondition, and suppose that ∂tis uniformly time-like. Then, the bound\n(2.13) ∥u∥LE1≲∥∂tu∥LE1c+∥Pu∥LE∗\nfor all u∈ S(R4).\nThe proof of Theorem 2.8 leverages weighted elliptic estimates for the flat Laplacian ∆ in order to get\nsimilar estimates for AFperturbations. Once again, the damping does not play a meaningful role. At\nfrequency zero, it provides no contribution, and near frequency zero, it is absorbed by the error term in\n(2.13); the damping arises when estimating P0ubyPuwithin a compact spatial set.\nAt medium frequencies, we require a weighted estimate which implies local energy decay for solutions\nsupported at any range of time frequencies bounded away from both zero and infinity. This is rooted in\nCarleman estimates. The Carleman weights which we need are constructed in e.g. [Boo18], [KT01]. The\nmain medium frequency estimate is the following, and the corresponding theorem in [MST20] is Theorem\n5.4. We remark that the theorem does not imply an absence of embedded eigenvalues/resonances on the real\nline.\nTheorem 2.9. LetPbe an asymptotically flat damped wave operator, and suppose that ∂tis uniformly\ntime-like. Then, for any δ >0, there exists a bounded, non-decreasing radial weight φ=φ(ln(1 + r))so that\nfor all u∈ S(R4), we have the bound\n(2.14)\r\r\r(1 +φ′′\n+)1/2eφ\u0010\n∇u,⟨r⟩−1(1 +φ′)u\u0011\r\r\r\nLE+\r\r\r(1 +φ′)1/2eφ∂tu\r\r\r\nLE\n≲∥eφPu∥LE∗+δ\u0010\r\r\r(1 +φ′)1/2eφu\r\r\r\nLE+\r\r\r⟨r⟩−1(1 +φ′′\n+)1/2(1 +φ′)eφ∂tu\r\r\r\nLE\u0011\n,\nwith the implicit constant independent of δ.\nThe proof of this theorem utilizes two intermediate Carleman estimates within two different regions of\nspace, which may be combined using a cutoff argument in order to prove Theorem 2.9.\n(1)Within a large compact set: The damping term is well-signed and readily absorbable as a\nperturbation due to the conditions on the weight φ,which will be convex.\n(2)Outside of a large compact set: Here, the damping is a small AFperturbation, so the proof\nin [MST20] follows through without any modification. Within this region, the authors of [MST20]\nbend the weight to be constant near infinity in order to apply exterior wave estimates. This leads\nto breaking this case into three sub-regions: one where the Carleman weight is convex, a transition\nregion where the conditions break in order to bend the weight to be constant near infinity, and a\nregion near infinity where the weight is constant.\nThe proofs of the Carleman estimates in the above regions are based on positive commutator arguments\nutilizing the self- and skew-adjoint parts of the conjugated operator Pφ=eφPe−φ.\nThe high, low, and medium frequency estimates are the key ingredients needed to establish Theorem\n1.11. As in [MST20], [Kof23], it sufficient to remove the Cauchy data at times 0 and Tin order to prove\nTheorem 1.11; we will elaborate on this momentarily. This makes it significantly easier to perform frequency\nlocalization. The pertinent result in [MST20] is Theorem 7.1.\nTheorem 2.10. LetPbe a stationary, asymptotically flat damped wave operator satisfying the geometric\ncontrol condition (2.1), and suppose that ∂tis uniformly time-like. Then, the estimate\n∥u∥LE1≲∥Pu∥LE∗ (2.15)\nholds for all u∈ S(R4).\nProof. We will utilize a time-frequency partition of unity. Let 0 < τ0≪1 and τ1≫1, which will be chosen\nwith more precision shortly. Then, we can write\nu=χ|τ|<τ0(Dt)u+χτ0<|τ|<τ1(Dt)u+χ|τ|>τ1(Dt)u=:Q1u+Q2u+Q3u.\nSince Pis stationary, it commutes with each Qj,and so it suffices to show that\n(2.16) ∥Qju∥LE1≲∥Pu∥LE∗, j = 1,2,3.\n17First, we apply Theorem 2.13 to Q1uand appeal to Plancherel’s theorem in order to obtain that\n∥Q1u∥LE1≲∥∂t(Q1u)∥LE1c+∥P(Q1u)∥LE∗≲τ0∥Q1u∥LE1c+∥Pu∥LE∗.\nIfτ0is sufficiently small, then we may absorb the error term on the upper-hand side into lower-bound side,\nwhich provides (2.16) for j= 1.\nWe proceed similarly with Q2uvia Theorem 2.9:\n\r\r\r(1 +φ′′\n+)1/2eφ\u0010\n∇Q2u,⟨r⟩−1(1 +φ′)Q2u\u0011\r\r\r\nLE+\r\r\r(1 +φ′)1/2eφ∂tQ2u\r\r\r\nLE\n≲∥eφP(Q2u)∥LE∗+δ\u0010\r\r\r(1 +φ′)1/2eφQ2u\r\r\r\nLE+\r\r\r⟨r⟩−1(1 +φ′′\n+)1/2(1 +φ′)eφ∂tQ2u\r\r\r\nLE\u0011\n≲∥eφPu∥LE∗+δ\nτ0\r\r\r(1 +φ′)1/2eφ∂tQ2u\r\r\r\nLE+δτ1\r\r\r⟨r⟩−1(1 +φ′′\n+)1/2(1 +φ′)eφQ2u\r\r\r\nLE.\nBy choosing δsufficiently small, last two terms absorb into the left-hand side. Since φis bounded and φ′≥0,\nwe obtain (2.16) for j= 2.\nFinally, we apply Theorem 1.10 to Q3u(t−T/2):\n∥Q3u∥LE1[−T/2,T/2]≲∥∂(Q3u)(−T/2)∥L2+\r\r\r⟨x⟩−2u\r\r\r\nLE[−T/2,T/2]+∥P(Q3u)∥LE∗[−T/2,T/2].\nTaking the limit as T→ ∞ and then applying Plancherel’s theorem give that\n∥Q3u∥LE1≲τ−1\n1∥Q3u∥LE1+∥Pu∥LE∗.\nIfτ1is large enough, then the error term on the right absorbs into the left, giving (2.16) for j= 3. □\nWe underscore how important it is that δmay be chosen arbitrarily in Theorem 2.9: It allowed for com-\npatibility with the high and low frequency estimates regardless of how high or low the frequency thresholds\nbecame, respectively (so long as they were away from zero and infinity).\nAs in Section 7 of [MST20], one proves that Theorem 2.10 implies Theorem 1.11 by fixing uand con-\nstructing a function vwhich matches the Cauchy data of uat times 0 and T(and satisfies an appropriate\nbound) which allows one to apply (2.15) to u−v.This construction is performing using a partition of unity\non the support of u[0],u[T], and Pu.In particular, one splits into an interior region {|x|<4R0}and an\nexterior region {|x|>2R0}. The damping is non-problematic in the interior and is small in the exterior (as\nwith the other lower-order terms). The latter fact is important since a time reversal argument is used, which\nmakes the damping a driving force.\nIn the interior, one utilizes a unit time interval partition of unity {χj}and analyzes the equations\nPvj=χjf,matching the data at times 0 and Twith the first and last of elements in the partition,\nrespectively. One then generates the desired approximate of uviaPχjvj.In the exterior region, one\nchooses an appropriate small AFperturbation of 2which matches Pin the exterior. If one considers the\nsame differential equation (same data and forcing) but replaces Pby the perturbation, one obtains good\nbounds via local energy decay. By truncating the solution appropriately to |x|> R 0andt < T, one obtains\nthe desired approximate in the exterior.\n2.3.An Energy Dichotomy. Here, we apply Theorem 1.11 in order to prove the energy dichotomy present\nin Theorem 1.12. Very little deviation is needed from the strategy given in [MST20], although we must take\nadvantage of the damping in the damping-dominant case. In particular, we must use the damping to absorb\ntime derivative error terms for |x| ≤2R0, outside of which we may use asymptotic flatness (for space\nderivative terms, one can use Young’s inequality for products, at the expense of shrinking ε). In the ε-almost\nsymmetric case, such interior errors are automatically small.\nProof. Split PandP0into self- and skew-adjoint parts\nP=Ps+Pa, P 0=Ps\n0+Pa\n0,\nrespectively, and define the energy of the symmetric part of Pas\nEs[u](t) =Z\nR3Ps\n0u¯u−g00|∂tu|2dx.\n18Then,\nEs[u](t) =Es[u](0) + 2 RetZ\n0Z\nR3∂suPsu dxds.\nApplying this to\nE[u](t) =Z\nR3P0u¯u−g00|∂tu|2dx\nand utilizing the symmetry assumptions yield that\nE[u](t) =Es[u](0) + 2 RetZ\n0Z\nR3∂suPsu dxds +Z\nR3Pa\n0u¯u dx(2.17)\n=E[u](0)−Z\nR3Pa\n0u(0)u(0)dx+ 2 RetZ\n0Z\nR3∂suf dxds −2 RetZ\n0Z\nR3∂suPau dxds +Z\nR3Pa\n0u¯u dx\n=E[u](0) + 2 RetZ\n0Z\nR3∂suf dxds −2 RetZ\n0Z\nR3∂suPau dxds +tZ\n0Z\nR3∂s(Pa\n0u¯u)dx ds\n=E[u](0) + 2 RetZ\n0Z\nR3∂suf dxds −2 RetZ\n0Z\nR3∂suPau dxds −2iImtZ\n0Z\nR3∂suPa\n0u dx ds\n≤E[u](0) + 2 RetZ\n0Z\nR3∂suf dxds −C(ε,c)tZ\n0Z\nB2R0(0)Rea|∂su|2dxds +D(ε,c)∥u∥2\nLE1\nwhere we have used the assumed symmetry assumptions in the last line, shrinking εif necessary. In the above,\nC(ε,c)>0, and D(ε,c) =O(ε) in the almost-symmetric case (and C(ε,c) = 2), whereas D(ε,c) =O(c) in\nthe damping-dominant case.\nDropping the damping term from the above and pairing the result with (1.4) gives that\n∥∂u∥2\nL∞L2[0,T]≲∥∂u(0)∥2\nL2+\r\r∂≤1u(T)\r\r2\nL2c+D(ε,c)∥u∥2\nLE1[0,T]+TZ\n0Z\nR3|∂tu||f|dxdt,\nwhere ∂≤1u:=P\n|α|≤1∂αu.The Schwarz inequality and H¨ older’s inequality imply that\n∥∂u∥L∞L2[0,T]≲∥∂u(0)∥L2+\r\r∂≤1u(T)\r\r\nL2c+ (δ+D(ε,c))∥u∥LE1[0,T]+δ−1∥f∥LE∗+L1L2[0,T], (2.18)\nwhere δ >0 is arbitrary. Now, we will make use of the two-point local energy estimate (1.2). By combining\n(2.18) with (1.2) and choosing δsufficiently small (and shrinking εif necessary in the almost-symmetric\ncase), we obtain that\n∥∂u(T)∥2\nL2≲∥∂u(0)∥2\nL2+\r\r∂≤1u(T)\r\r2\nL2c+∥f∥2\nLE∗+L1L2[0,∞).\nOn the other hand, (1.2) directly gives that\n\r\r∂≤1u(t)\r\r2\nL2L2c≲∥u∥2\nLE1c[0,T]≲∥∂u(0)∥2\nL2+∥∂u(T)∥2\nL2+∥f∥2\nLE∗+L1L2[0,∞),\nand so\n∥∂u∥2\nL2L2[0,T]≲∥∂u(T)∥2\nL2+ (T+ 1)\u0010\n∥∂u(0)∥2\nL2+∥f∥2\nLE∗+L1L2[0,∞)\u0011\n. (2.19)\nNotice that if we call E(T) =∥∂u∥2\nL2L2[0,T],then (2.19) gives that\nE′(T) =∥∂u(T)∥2\nL2≥αE(T)−(T+ 1)\u0010\n∥∂u(0)∥2\nL2+∥f∥2\nLE∗+L1L2[0,∞)\u0011\n, (2.20)\n19where α >0 is a constant.\nWe will consider two cases. First, we assume that\nE(T)<2α−1(T+ 1)\u0010\n∥∂u(0)∥2\nL2+∥f∥2\nLE∗+L1L2[0,∞)\u0011\nfor all T >0. In such a case, we have that, in particular,\nT−1∥∂u∥2\nL2L2[0,T]≲∥∂u(0)∥2\nL2+∥f∥2\nLE∗+L1L2[0,∞).\nBy the mean-value theorem for integrals, there exists a sequence ( Tj) so that Tj→ ∞ asj→ ∞ , and\n∥∂u(Tj)∥2\nL2≲∥∂u(0)∥2\nL2+∥f∥2\nLE∗+L1L2[0,∞).\nWe may use (1.2) and let Tj→ ∞ to conclude that local energy decay holds.\nNext, we consider if\nE(T′)≥2α−1(T′+ 1)\u0010\n∥∂u(0)∥2\nL2+∥f∥2\nLE∗+L1L2[0,∞)\u0011\n(2.21)\nfor some T′>0. Then, E(T) must bound (2.21) from above for all T≥T′, since (2.20) implies that E(T)\nis increasing for all T > T′.By applying an integrating factor argument to the differential inequality (2.20)\nand using (2.21), we get that\nE(T)≳T′eαT\u0010\n∥∂u(0)∥2\nL2+∥f∥2\nLE∗+L1L2[0,∞)\u0011\n, T ≥T′, T′≫1.\nCombining this with (2.19) gives the exponential growth for large enough T. □\n3.Resolvent Theory and Local Energy Decay\nIn this section, we will introduce the spectral theory required to prove Theorem 1.14. This follows from\nthe resolvent formalism introduced in [MST20] and the corresponding scattering theory. We will summarize\nthe relevant parts of their work here and provide details that were either omitted or are instructive to repeat.\nIn particular, the results necessary to prove Theorem 1.14 are based on the same frequency estimates present\nin both [MST20] and here, so our work requires little deviation. Throughout this section, we assume that P\nis stationary.\nConsider Pu= 0.One arrives at the stationary problem by studying “mode solutions” of the form\nu(t, x) =uω(x)eiωt, where ω∈C(equivalently, one replaces Dtbyω). Plugging such uinto the given\nhomogeneous equation generates the stationary equation\nPωuω= 0, where Pω= ∆ g,A+W(x, D x) +ωB(x, D x) +g00ω2,\nand\n∆g,A= (Dj+Aj)gjk(Dk+Ak),\nW(x, D x) =A0g0j(Dj+Aj) + (Dj+Aj)gj0A0+ (A0)2g00+V,\nB(x, D x) =g0j(Dj+Aj) + (Dj+Aj)gj0+ 2A0g00+ia.\nTheresolvent Rωis defined as the inverse of Pωwhen such an inverse exists. More explicitly, if we consider\nthe homogeneous Cauchy problem\nPu= 0, u (0) = 0 ,−g00∂tu(0) = f,\nthen we may formally define Rωvia the Fourier-Laplace transform of u, i.e.\nRωf=∞Z\n0e−iωtu(t)dt=:Ft→ω( 1[0,∞)(t)u), ω ∈C.\nOne may check via formal integration by parts that both definitions of Rωare consistent. In this section,\nwe will take fto be in either L2orLE∗, and it will be clear from context which is the case. It remains to\nmake rigorous sense of Rωas a well-defined bounded operator.\nFrom the global energy bounds and Gronwall’s inequality, it follows that usatisfies the crude estimate\n(3.1) ∥∂u(t)∥L2≲ect∥f∥L2, c ≥0.\n20Using (3.1) and the Minkowski integral inequality, we obtain that\n∥Rωf∥˙H1≤∞Z\n0eImωt∥∇u(t,·)∥L2dt≲∞Z\n0eImωtect∥f∥L2dt≲|Imω+c|−1∥f∥L2, Imω+c <0. (3.2)\nMeanwhile, integrating by parts once provides that ωRωf=−iFt→ω( 1[0,∞)∂tu).Taking the L2norm of the\nabove and performing the same work as in (3.2) yields an identical upper bound. Combining these estimates\ntogether gives the inequality\n∥Rωf∥˙H1ω≲|Imω+c|−1∥f∥L2, Imω+c <0. (3.3)\nHence, we may validly define the resolvent as a bounded operator from L2to˙H1\nω, provided that ωis in the\nrange given in (3.3). Notice that if the uniform energy bound (1.3) holds, then the resolvent is holomorphic\nin the lower half-plane H:={ω∈C: Imω <0}and satisfies the bound\n∥Rω∥L2→˙H1ω≲|Imω|−1, ω ∈ H. (3.4)\nIf the uniform energy bound does not hold, then one is only guaranteed meromorphic continuation to H.Due\nto this tie-in with uniform energy bounds, we will refer to (3.4) as the uniform energy resolvent bound . There\nis also an analogous resolvent bound to local energy decay, which we state as a theorem and will not prove\nhere; see the proof of Theorem 2.3 in [MST20] for more. It is largely a consequence of Plancherel’s theorem,\nalong with utilization of facts which we will discuss after the statement of the theorem. The damping plays\nno meaningful role here.\nTheorem 3.1. Local energy decay holds for a stationary damped wave operator Pif and only if Rωsatisfies\nthe local energy resolvent bound\n∥Rω∥LE∗→LE1ω≲1, ω ∈ H. (3.5)\nWe observe that if the uniform energy resolvent bound holds, then the local energy resolvent bound holds\nfor Im ω≲−1, since\n∥Rωf∥LE1\nω≲∥Rωf∥˙H1+|ω|∥Rωf∥L2=∥Rωf∥˙H1ω≲|Imω|−1∥f∥L2≲∥f∥LE∗. (3.6)\nOn the other hand (3.5) implies (3.4); Fredholm theory implies that Rωis bounded and holomorphic from\nL2to˙H1\nω, and the precise operator norm bound can by obtained by splitting f∈L2as\nf=χ<|Imω|−1f+χ>|Imω|−1f,\nthen using (3.5) and (3.3). Hence, the uniform energy and local energy resolvent bounds have close relation.\nWe previously alluded to two spectral obstructions to local energy decay (and hence local energy resolvent\nbounds), which we outline more precisely now.\nElements in the kernel of Pωliving in L2correspond to finite rank poles ωofRω. Given that Pωis an\nelliptic operator, such eigenfunctions live in Hsfor all s∈R, yet the corresponding mode solution to Pu= 0\nmust possess exponential growth in time since ω∈ H. In particular, the initial energy is finite, yet both\nterms on the lower-bound side of the local energy decay estimate (1.1) are unbounded as T→ ∞ . This\nbehavior violates both uniform energy bounds and local energy decay. Since the eigenvalues have negative\nimaginary part, we will call these negative eigenfunctions .\nDefinition 3.2. A negative eigenfunction for Pis a non-zero uω∈L2such that Pωuω= 0, with ω∈ H.\nThe corresponding eigenvalues are isolated and contained within a relatively compact set in H, which\nfollows due to (3.3) and the high frequency estimate (1.5).\nNext, we notice that the local energy resolvent bound must hold uniformly up to the real line in order to\nobtain local energy decay, per Theorem 3.1. To that end, we have another potential obstruction - a failure\nto obtain the continuous extension of RωtoR. First, we consider when one takes the limit as the spectral\nparameter approaches a non-zero real value.\nDefinition 3.3. An embedded resonant state for Pis a non-zero uω∈ LE1\nω, with ω∈R\\ {0}being the\ncorresponding embedded resonance, satisfying the outgoing/Sommerfeld radiation condition\n(3.7)\r\r\r⟨x⟩−1/2(∂r+iω)uω\r\r\r\nL2(Aj)→0 as j→ ∞\n21such that\nPωuω= 0.\nThe outgoing radiation condition as stated here is a variant of the standard one, adapted to the dyadic\nstructure of our spaces. It acts as a boundary condition at infinity to ensure unique solutions to the above\nproblem for a fixed ω. Such states are subtle obstructions, since the corresponding mode solution does not\ngenerally have finite energy (as uωis only guaranteed to live in LE1\nω). However, as explained in [MST20], one\nmay perform a truncation procedure and utilize normalized/Regge-Wheeler-type coordinates (see [Tat13])\nto produce functions χ>1(t−r)eiωtuωwhose energy exhibits a growth rate of t1/2yet whose image under\nPlives in L1L2.Hence, they represent an explicit obstruction to uniform and local energy bounds. The\nnon-existence of embedded resonant states is equivalent to (3.5) near the punctured real line and to a limiting\nabsorption principle. In our context, the corresponding result is as follows.\nTheorem 3.4. LetPbe a stationary, asymptotically flat damped wave operator which satisfies the geomet-\nric control condition and is either weakly ε-damping dominant for some ε >0orε-weakly magnetic with\nε≪R0,M01. Additionally, assume that ∂tis uniformly time-like. For any ω0∈R\\ {0}, the following are\nequivalent:\n(1)ω0is not a resonance.\n(2) The bound\n∥u∥LE1ω0≲∥Pω0u∥LE∗\nholds for all u∈ LE1\nω0satisfying the outgoing radiation condition (3.7).\n(3) The local energy resolvent bound (3.5) holds uniformly for ω∈ H near ω0, and the limit\nRω0f= lim\nH∋ω→ω0Rωf, f ∈ LE∗\nconverges strongly on compact sets and satisfies the outgoing radiation condition (3.7).\nWe sketch the proof, as it is similar to that given in [MST20] for their version of the result (which is\nProposition 2.5 in their work); only minor alterations are necessary to account for the damping and the\nlack of the non-trapping hypothesis. The result is perturbative of the work in [Kof23], just as the version in\n[MST20] was perturbative of the case where P=2g.\nProof. First, consider when A, V≡0 and a≡0 for|x|>2R0.In this scenario, we are in the setting of\n[Kof23] and obtain full local energy decay. 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Staudinger, P.A. Kolodin, and C.E. Patton, \nJ. Appl. Phys. 84, 3776 (1998). \n[34] M.P. Kostylev, A.A. Serga, T. Schneid er, B. Leven, B. Hillebrands, and R.L. \nStamps, Phys. Rev. B (to be published). \n[35] A.G. Gurevich and G.A. Melkov, “ Magnetization oscillations and waves ” \n(CRC Press, New York, 1996). \n[36] A.A. Serga, M. Kostylev, and B. Hillebrands, arXiv:0704.0024 \n(http://arxiv.org/abs /0704.0024 ); M. Kostylev, A.A. Serga, and B. Hillebrands, \n“Two dimensional microwave nonlinear sp in-wave pulses in in-plane confined \nmagnetic films”, Technical Digests of International Magnetic Conference \n(Intermag’2006), May 8-12, 2006, San Diego, California, FV03 (2006). \n[37] V.E. Demidov, U.-F. Hansen, O. Dz yapko, N. Koulev, S.O. Demokritov, and \nA.N. Slavin, Phys. Rev. B 74, 092407 (2006). 20FIGURE CAPTIONS \nFIG. 1. (Color online) Frequency depe ndence of the microwave absorption by the \nsquare magnetic element for H=800 Oe and P=1 mW. The resonant peaks \nappear at the frequencies of the eigenmodes and are labeled with two indexes \n(n,m) corresponding to the number of antinodes in the standing waves in the \ndirections parallel and perp endicular to the static magnetic field, respectively \nFIG. 2. (Color online) Meas ured distributions of the dynamic magnetization and their \ntwo-dimensional spatial Fourier spectra for the eigenmodes (2,1) (a) and (4,1) \n(b) in the linear and strongly nonlinear regimes corresponding to the excitation \npower equal to 1 and 200 mW, respectively. \nFIG. 3. (Color online) Experimental depende nces of the frequencies of the eigenmodes \n(2,1) and (4,1) and the average BLS in tensity obtained from the corresponding \ntwo-dimensional distributions on the excitation power. \nFIG. 4. (Color online) Distributions of the dynamic magnetization and their two-\ndimensional Fourier spectra calculat ed neglecting the nonlinear damping ( τ=0) \nfor the eigenmodes (2,1) (a) and (4,1) (b ). The distributions correspond to the \nstrongly nonlinear case ϕ=2.8°. \nFIG. 5. (Color online) Distributions of the dynamic magnetization and their two-\ndimensional Fourier spectra for the ei genmode (2,1) calculated taking into \naccount the nonlinear damping. The dist ributions are characterized by the \nmean precession angle ϕ, indicated close to the corresponding panels. \nFIG. 6. (Color online) Distributions of the dynamic magnetization and their two-\ndimensional Fourier spectra for the ei genmode (4,1) calculated taking into \naccount the nonlinear damping. 21\nFIG. 1 \nFIG. 2 22\nFIG. 3 \nFIG. 4 23 \nFIG. 5 \nFIG. 6 " }, { "title": "1704.01559v1.Relativistic_theory_of_magnetic_inertia_in_ultrafast_spin_dynamics.pdf", "content": "Relativistic theory of magnetic inertia in ultrafast spin dynamics\nRitwik Mondal,\u0003Marco Berritta, Ashis K. Nandy, and Peter M. Oppeneer\nDepartment of Physics and Astronomy, Uppsala University, P. O. Box 516, SE-75120 Uppsala, Sweden\n(Dated: November 12, 2018)\nThe influence of possible magnetic inertia effects has recently drawn attention in ultrafast mag-\nnetization dynamics and switching. Here we derive rigorously a description of inertia in the\nLandau-Lifshitz-Gilbert equation on the basis of the Dirac-Kohn-Sham framework. Using the Foldy-\nWouthuysen transformation up to the order of 1=c4gives the intrinsic inertia of a pure system\nthrough the 2ndorder time-derivative of magnetization in the dynamical equation of motion. Thus,\nthe inertial damping Iis a higher order spin-orbit coupling effect, \u00181=c4, as compared to the\nGilbert damping \u0000that is of order 1=c2. Inertia is therefore expected to play a role only on ultra-\nshort timescales (sub-picoseconds). We also show that the Gilbert damping and inertial damping\nare related to one another through the imaginary and real parts of the magnetic susceptibility tensor\nrespectively.\nPACS numbers: 71.15.Rf, 75.78.-n, 75.40.Gb\nI. INTRODUCTION\nThe foundation of contemporary magnetization dy-\nnamics is the Landau-Lifshitz-Gilbert (LLG) equation\nwhich describes the precession of spin moment and a\ntransverse damping of it, while keeping the modulus of\nmagnetization vector fixed [1–3]. The LLG equation of\nmotion was originally derived phenomenologically and\nthe damping of spin motion has been attributed to rela-\ntivistic effects such as the spin-orbit interaction [1, 4–6].\nIn recent years there has been a flood of proposals for the\nfundamental microscopic mechanism behind the Gilbert\ndamping: the breathing Fermi surface model of Kamber-\nský, where the damping is due to magnetization preces-\nsion and the effect of spin-orbit interaction at the Fermi\nsurface [4], the extension of the breathing Fermi surface\nmodel to the torque-torque correlation model [5, 7], scat-\ntering theory description [8], effective field theories [9],\nlinear response formalism within relativistic electronic\nstructure theory [10], and the Dirac Hamiltonian theory\nformulation [11].\nFor practical reasons it was needed to extend the orig-\ninal LLG equation to include several other mechanisms\n[12, 13]. To describe e.g. current induced spin-transfer\ntorques, the effects of spin currents have been taken\ninto account [14–16], as well as spin-orbit torques [17],\nand the effect of spin diffusion [18]. A different kind of\nspin relaxation due to the exchange field has been intro-\nduced by Bar’yakhtar et al.[19]. In the Landau-Lifshitz-\nBar’yakhtar equation spin dissipations originate from the\nspatial dispersion of exchange effects through the second\norder space derivative of the effective field [20, 21]. A\nfurther recent work predicts the existence of extension\nterms that contain spatial as well temporal derivatives of\nthe local magnetization [22].\nAnother term, not discussed in the above investiga-\ntions, is the magnetic inertial damping that has recently\n\u0003ritwik.mondal@physics.uu.sedrawnattention[23–25]. Originally, magneticinertiawas\ndiscussed following the discovery of earth’s magnetism\n[26]. Within the LLG framework, inertia is introduced\nas an additional term [24, 27–29] leading to a modified\nLLG equation,\n@M\n@t=\u0000\rM\u0002He\u000b+M\u0002\u0012\n\u0000@M\n@t+I@2M\n@2t\u0013\n;(1)\nwhere \u0000is the Gilbert damping constant [1–3], \rthe gy-\nromagnetic ratio, He\u000bthe effective magnetic field, and I\nis the inertia of the magnetization dynamics, similar to\nthe mass in Newton’s equation. This type of motion has\nthe same classical analogue as the nutation of a spinning\nsymmetric top. The potential importance of inertia is il-\nlustrated in Fig. 1. While Gilbert damping slowly aligns\nthe precessing magnetization to the effective magnetic\nfield, inertial dynamics causes a trembling or nutation of\nthe magnetization vector [24, 30, 31]. Nutation could\nconsequently pull the magnetization toward the equa-\ntor and cause its switching to the antiparallel direction\n[32, 33], whilst depending crucially on the strength of\nthe magnetic inertia. The parameter Ithat character-\nizes the nutation motion is in the most general case a\ntensor and has been associated with the magnetic suscep-\ntibility [29, 31, 33]. Along a different line of reasoning,\nFähnle et al.extended the breathing Fermi surface model\nto include the effect of magnetic inertia [27, 34]. The\ntechnological importance of nutation dynamics is thus\nits potential to steer magnetization switching in memory\ndevices [23–25, 32] and also in skyrmionic spin textures\n[35]. Magnetization dynamics involving inertial dynam-\nics has been investigated recently and it was suggested\nthat its dynamics belongs to smaller time-scales i.e., the\nfemtosecond regime [24]. However, the origin of inertial\ndamping from a fundamental framework is still missing,\nand, moreover, although it is possible to vary the size of\nthe inertia in spin-dynamics simulations, it is unknown\nwhat the typical size of the inertial damping is.\nNaturally the question arises whether it is possible\nto derive the extended LLG equation including iner-arXiv:1704.01559v1 [cond-mat.other] 20 Mar 20172\nMHeff\nPrecession\nNutation\nFigure 1. (Color online) Schematic illustration of magnetiza-\ntion dynamics. The precessional motion of Maround He\u000bis\ndepicted by the blue solid-dashed curve and the nutation is\nshown by the red curve.\ntia while starting from the fully relativistic Dirac equa-\ntion. Hickey and Moodera [36] started from a Dirac\nHamiltonian and obtained an intrinsic Gilbert damping\nterm which originated from spin-orbit coupling. How-\never they started from only a part of the spin-orbit cou-\npling Hamiltonian which was anti-hermitian [37, 38]. A\nrecent derivation based on Dirac Hamiltonian theory for-\nmulation [11] showed that the Gilbert damping depends\nstrongly on both interband and intraband transitions\n(consistent with Ref. [39]) as well as the magnetic sus-\nceptibility response function, \u001fm. This derivation used\nthe relativistic expansion to the lowest order 1=c2of the\nhermitian Dirac-Kohn-Sham (DKS) Hamiltonian includ-\ning the effect of exchange field [40].\nIn this article we follow an approach similar to that of\nRef. [11] but we consider higher order expansion terms\nof the DKS Hamiltonian up to the order of 1=c4. This is\nshown to lead to the intrinsic inertia term in the modi-\nfied LLG equation and demonstrates that it stems from\na higher-order spin-orbit coupling term. A relativistic\norigin of the spin nutation angle, caused by Rashba-like\nspin-orbit coupling, was previously concluded, too, in the\ncontext of semiconductor nanostructures [41, 42].\nIn the following, we derive in Sec. II the relativistic\ncorrection terms to the extended Pauli Hamiltonian up\nto the order of 1=c4, which includes the spin-orbit inter-\naction and an additional term. Then the corresponding\nmagnetization dynamics is computed from the obtained\nspin Hamiltonian in Sec. III, which is shown to contain\nthe Gilbert damping and the magnetic inertial damping.\nFinally, we discuss the size of the magnetic inertia in re-\nlation to other earlier studies.II. RELATIVISTIC HAMILTONIAN\nFORMULATION\nWe start our derivation with a fully relativistic par-\nticle, a Dirac particle [43] inside a material and in the\npresence of an external field, for which we write the DKS\nHamiltonian:\nH=c\u000b\u0001(p\u0000eA) + (\f\u0000 1)mc2+V 1\n=O+ (\f\u0000 1)mc2+E; (2)\nwhereVis the effective crystal potential created by the\nion-ion, ion-electron and electron-electron interactions,\nA(r;t)is the magnetic vector potential from the external\nfield,cis the speed of light, mis particle’s mass and 1\nis the 4\u00024unit matrix. \u000band\fare the Dirac matrices\nthat have the form\n\u000b=\u0012\n0\u001b\n\u001b0\u0013\n; \f =\u0012\n10\n0\u00001\u0013\n;\nwhere\u001bis the Pauli spin matrix vector and 1is2\u00022unit\nmatrix. TheDiracequationisthenwrittenas i~@ (r;t)\n@t=\nH (r;t)for a Dirac bi-spinor . The quantityO=c\u000b\u0001\n(p\u0000eA)defines the off-diagonal, or odd terms in the\nmatrix formalism and E=V 1are the diagonal, i.e., even\nterms. The latter have to be multiplied by a 2\u00022block\ndiagonal unit matrix in order to bring them in a matrix\nform. To obtain the nonrelativistic Hamiltonian and the\nrelativistic corrections one can write down the Dirac bi-\nspinor in double two component form as\n (r;t) =\u0012\n\u001e(r;t)\n\u0011(r;t)\u0013\n;\nand substitute those into the Dirac equation. The up-\nper two components represent the particle and the lower\ntwo components represent the anti-particle. However the\nquestion of separating the particle’s and anti-particle’s\nwave functions is not clear for any given momentum. As\nthe part\u000b\u0001pis off-diagonal in the matrix formalism, it\nretains the odd components and thus links the particle-\nantiparticle wave function. One way to eliminate the an-\ntiparticle’s wave function is by an exact transformation\n[44] which gives terms that require a further expansion in\npowers of 1=c2. Another way is to search for a represen-\ntation where the odd terms become smaller and smaller\nand one can ignore those with respect to the even terms\nand retain only the latter [45]. The Foldy-Wouthuysen\n(FW) transformation [46, 47] was the very successful at-\ntempt to find such a representation.\nIt is an unitary transformation obtained by suitably\nchoosing the FW operator,\nUFW=\u0000i\n2mc2\fO: (3)\nThe minus sign in front of the operator is because \fand\nOanti-commute with each other. The transformation of\nthewavefunctionadoptstheform 0(r;t) =eiUFW (r;t)3\nsuch that the probability density remains the same,\nj j2=j 0j2. The time-dependent FW transformation\ncan be expressed as [45, 48]\nHFW=eiUFW\u0012\nH\u0000i~@\n@t\u0013\ne\u0000iUFW+i~@\n@t:(4)\nThe first term can be expanded in a series as\neiUFWHe\u0000iUFW=H+i[UFW;H] +i2\n2![UFW;[UFW;H]]\n+::::+in\nn![UFW;[UFW;:::[UFW;H]:::]] +::: :(5)\nThe time dependency enters through the second term of\nEq. (4) and for a time-independent transformation one\nworks with@UFW\n@t= 0. It is instructive to note that the\naim of the whole procedure is to make the odd termssmaller and one can notice that as it goes higher and\nhigher in the expansion, the corresponding coefficients\ndecrease of the order 1=c2due to the choice of the unitary\noperator. After a first transformation, the new Hamilto-\nnian will contain new even terms, E0, as well as new odd\nterms,O0of1=c2or higher. The latter terms can be used\nto perform a next transformation having the new unitary\noperator as U0\nFW=\u0000i\n2mc2\fO0. After a second transfor-\nmation the new Hamiltonian, H0\nFWis achieved that has\nthe odd terms of the order 1=c4or higher. The trans-\nformation is a repetitive process and it continues until\nthe separation of positive and negative energy states are\nguaranteed.\nAfter a fourth transformation we derive the new trans-\nformed Hamiltonian with all the even terms that are cor-\nrect up to the order of1\nm3c6as [48–50]\nH000\nFW= (\f\u0000 1)mc2+\f\u0012O2\n2mc2\u0000O4\n8m3c6\u0013\n+E\u00001\n8m2c4h\nO;[O;E] +i~_Oi\n+\f\n16m3c6fO;[[O;E];E]g+\f\n8m3c6n\nO;h\ni~_O;Eio\n+\f\n16m3c6n\nO;(i~)2Oo\n: (6)\nNote that [A;B]defines the commutator, while fA;Bgrepresents the anti-commutator for any two operators Aand\nB. A similar Foldy-Wouthuysen transformation Hamiltonian up to an order of 1=m3c6was derived by Hinschberger\nand Hervieux in their recent work [51], however there are some differences, for example, the first and second terms in\nthe second line of Eq. (6) were not given. Once we have the transformed Hamiltonian as a function of odd and even\nterms, the final form is achieved by substituting the correct form of odd terms Oand calculating term by term.\nEvaluating all the terms separately, we derive the Hamiltonian for only the positive energy solutions i.e. the upper\ncomponents of the Dirac bi-spinor as a 2\u00022matrix formalism [40, 51, 52]:\nH000\nFW=(p\u0000eA)2\n2m+V\u0000e~\n2m\u001b\u0001B\u0000(p\u0000eA)4\n8m3c2\u0000e~2\n8m2c2r\u0001Etot+e~\n8m3c2n\n(p\u0000eA)2;\u001b\u0001Bo\n\u0000e~\n8m2c2\u001b\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot]\n\u0000e~2\n16m3c4f(p\u0000eA);@tEtotg\u0000ie~2\n16m3c4\u001b\u0001[@tEtot\u0002(p\u0000eA) + (p\u0000eA)\u0002@tEtot]; (7)\nwhere@t\u0011@=@tdefines the first-order time derivative.\nThehigherorderterms( 1=c6ormore)willinvolvesimilar\nformulations and more and more time derivatives of the\nmagnetic and electric fields will appear that stem from\nthe time derivative of the odd operator O[48, 51].\nThe fields in the last Hamiltonian (7) are defined as\nB=r\u0002A, the external magnetic field, Etot=Eint+\nEextare the electric fields where Eint=\u00001\nerVis the\ninternal field that exists even without any perturbation\nandEext=\u0000@A\n@tis the external field (only the temporal\npart is retained here because of the Coulomb gauge).\nThe spin Hamiltonian\nThe aim of this work is to formulate the magnetiza-\ntion dynamics on the basis of this Hamiltonian. Thus,we split the Hamiltonian into spin-independent and spin-\ndependentpartsandconsiderfromnowonelectrons. The\nspin Hamiltonian is straightforwardly given as\nHS(t) =\u0000e\nmS\u0001B+e\n4m3c2n\n(p\u0000eA)2;S\u0001Bo\n\u0000e\n4m2c2S\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot]\n\u0000ie~\n8m3c4S\u0001[@tEtot\u0002(p\u0000eA) + (p\u0000eA)\u0002@tEtot];\n(8)\nwhere the spin operator S= (~=2)\u001bhas been used. Let\nus briefly explain the physical meaning behind each term\nthat appears inHS(t). The first term defines the Zee-\nman coupling of the electron’s spin with the externally\napplied magnetic field. The second term defines an indi-\nrect coupling of light to the Zeeman interaction of spin\nand the optical B-field, which can be be shown to have4\nthe form of a relativistic Zeeman-like term. The third\nterm implies a general form of the spin-orbit coupling\nthat is gauge invariant [53], and it includes the effect of\nthe electric field from an internal as well as an external\nfield. Thelasttermisthenewtermofrelevanceherethat\nhas only been considered once in the literature by Hin-\nschberger et al.[51]. Note that, although the last term\nin Eq. (8) contains the total electric field, only the time-\nderivative of the external field plays a role here, because\nthe time derivative of internal field is zero as the ionic\npotential is time independent. In general if one assumes\na plane-wave solution of the electric field in Maxwell’s\nequation asE=E0ei!t, the last term can be written as\ne~!\n8m3c4S\u0001(E\u0002p)and thus adopts the form of a higher-\norder spin-orbit coupling for a general E-field.\nThe spin-dependent part can be easily rewritten in a\nshorter format using the identities:\nA\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002A= 2A\u0002(p\u0000eA)\n+i~r\u0002A (9)\nA\u0002(p\u0000eA) + (p\u0000eA)\u0002A=\u0000i~r\u0002A(10)\nfor any operator A. This allows us to write the spin\nHamiltonian as\nHS=\u0000e\nmS\u0001B+e\n2m3c2S\u0001B\u0014\np2\u00002eA\u0001p+3e2\n2A2\u0015\n\u0000e\n2m2c2S\u0001\u0002\nEtot\u0002(p\u0000eA)\u0003\n+ie~\n4m2c2S\u0001@tB\n+e~2\n8m3c4S\u0001@ttB: (11)\nHere, the Maxwell’s equations have been used to derive\nthe final form that the spatial derivative of the electric\nfield will generate a time derivative of a magnetic field\nsuch that r\u0002Eext=\u0000@B\n@t, whilst the curl of a internal\nfield results in zero as the curl of a gradient function is\nalways zero. The final spin Hamiltonian (11) bears much\nimportance for the strong laser field-matter interaction\nas it takes into account all the field-spin coupling terms.\nIt is thus the appropriate fundamental Hamiltonian to\nunderstand the effects of those interactions on the mag-\nnetization dynamics described in the next section.\nIII. MAGNETIZATION DYNAMICS\nIn general, magnetization is given by the magnetic mo-\nment per unit volume in a magnetic solid. The magnetic\nmomentisgivenby g\u0016BhSi,wheregistheLandég-factor\nand\u0016Bis the unit of Bohr magneton. The magnetization\nis then written\nM(r;t) =X\njg\u0016B\n\nhSji; (12)\nwhere \nis the suitably chosen volume element, the sum\njgoes over all electrons in the volume element, and h::i\nis the expectation value. To derive the dynamics, wetake the time derivative in both the sides of Eq. (12)\nand, withintheadiabaticapproximation, wearriveatthe\nequation of motion for the magnetization as [36, 54, 55]\n@M\n@t=X\njg\u0016B\n\n1\ni~h\u0002\nSj;HS(t)\u0003\ni:(13)\nNow the task looks simple, one needs to substitute the\nspin Hamiltonian (11) and calculate the commutators in\norder to find the equation of motion. Note that the dy-\nnamics only considers the local dynamics as we have not\ntaken into account the time derivative of particle density\noperator (for details, see [11]). Incorporating the latter\nwould give rise the local as well as non-local processes\n(i.e., spin currents) within the same footing.\nThe first term in the spin Hamiltonian produces the\ndynamics as\n@M(1)\n@t=\u0000\rM\u0002B; (14)\nwith\r=gjej=2mdefines the gyromagnetic ratio and\nthe Landé g-factor g\u00192for spins, the electronic charge\ne < 0. Using the linear relationship of magnetization\nwith the magnetic field B=\u00160(H+M), the latter is\nreplaced in Eq. (14) to get the usual form in the Landau-\nLifshitz equations, \u0000\r0M\u0002H, where\r0=\u00160\ris the\neffective gyromagnetic ratio. This gives the Larmor pre-\ncessionofmagnetizationaroundaneffectivefield H. The\neffective field will always have a contribution from a ex-\nchange field and the relativistic corrections to it, which\nhas not been explicitly taken into account in this article,\nas they are not in the focus here. For detailed calcula-\ntions yet including the exchange field see Ref. [11].\nThe second term in the spin Hamiltonian Eq. (11) will\nresult in a relativistic correction to the magnetization\nprecession. Within an uniform field approximation (A=\nB\u0002r=2), the corresponding dynamics will take the form\n@M(2)\n@t=\r\n2m2c2M\u0002BD\np2\u0000eB\u0001L+3e2\n8(B\u0002r)2E\n;\n(15)\nwithL=r\u0002pthe orbital angular momentum. The\npresence of \r=2m2c2implies that the contribution of this\ndynamics to the precession is relatively small, while the\nleading precession dynamics is given by Eq. (14). For\nsake of completeness we note that a relativistic correc-\ntion to the precession term of similar order 1=m2c2was\nobtained previously for the exchange field [11].\nThe next term in the Hamiltonian is a bit tricky to\nhandle as the third term in Eq. (11) is not hermitian, not\neven the fourth term which is anti-hermitian. However\ntogether they form a hermitian Hamiltonian [11, 37, 38].\nTherefore one has to work together with those terms and\ncannot only perform the dynamics with an individual\nterm. In an earlier work [11] we have shown that taking\nanuniformmagneticfieldalongwiththegauge A=B\u0002r\n2\nwill preserve the hermiticity. The essence of the uniform\nfield lies in the assumption that the skin depth of the5\nelectromagnetic field is longer than the thickness of the\nthin-film samples used in experiments. The dynamical\nequation of spin motion with the second and third terms\nthus thus be written in a compact form for harmonic ap-\nplied fields as [11]\n@M(3;4)\n@t=M\u0002\u0012\nA\u0001@M\n@t\u0013\n; (16)\nwith the intrinsic Gilbert damping parameter Athat is\na tensor defined by\nAij=\r\u00160\n4mc2X\nn;kh\nhripk+pkrii\u0000hrnpn+pnrni\u000eiki\n\u0002\u0010\n1+\u001f\u00001\nm\u0011\nkj:(17)\nHere\u001fmis the magnetic susceptibility tensor of rank 2\n(a3\u00023matrix) and 1is the 3\u00023unit matrix. Note that\nfor diagonal terms i.e., i=kthe contributions from the\nexpectation values of rkpicancel each other. The damp-\ning tensor can be decomposed to have contributions from\nan isotropic Heisenberg-like, anisotropic Ising-like and\nDzyaloshinskii-Moriya-like tensors. The anti-symmetric\nDzyaloshinskii-Moriya contribution has been shown to\nlead to a chiral damping of the form M\u0002(D\u0002@M=@t)\n[11]. Experimental observations of chiral damping have\nbeen reported recently [56]. The other cross term having\nthe formE\u0002Ain Eq. (11) is related to the angular mo-\nmentum of the electromagnetic field and thus provides\na torque on the spin that has been at the heart of an-\ngular magneto-electric coupling [53]. A possible effect in\nspin dynamics including the light’s angular momentum\nhas been investigated in the strong field regime and it\nhas been shown that one has to include this cross term\nin the dynamics in order to explain the qualitative and\nquantitative strong field dynamics [57].\nFor the last term in the spin Hamiltonian (11) it is\nrather easy to formulate the spin dynamics because it is\nevidently hermitian. Working out the commutator with\nthe spins gives a contribution to the dynamics as\n@M(5)\n@t=\u000eM\u0002@2B\n@t2; (18)\nwith the constant \u000e=\r~2\n8m2c4.\nLet us work explicitly with the second-order time\nderivative of the magnetic induction by the relation B=\n\u00160(H+M), using a chain rule for the derivative:\n@2B\n@t2=@\n@t\u0010@B\n@t\u0011\n=\u00160@\n@t\u0010@H\n@t+@M\n@t\u0011\n=\u00160\u0010@2H\n@t2+@2M\n@t2\u0011\n: (19)\nThis is a generalized equation for the time-derivative of\nthe magnetic induction which can be used even for non-\nharmonic fields. The magnetization dynamics is then\ngiven by\n@M(5)\n@t=\u00160\u000eM\u0002\u0010@2H\n@t2+@2M\n@t2\u0011\n:(20)Thus the extended LLG equation of motion will have\nthese two additional terms: (1) a field-derivative torque\nand (2) magnetization-derivative torque, and they ap-\npear with their 2ndorder time derivative. It deserves to\nbe noted that, in a previous theory we also obtained a\nsimilar term–a field-derivative torque in 1storder-time\nderivative appearing in the generalized Gilbert damping.\nSpecifically, the extended LLG equation for a general\ntime-dependent field H(t)becomes\n@M\n@t=\u0000\r0M\u0002H+M\u0002h\n\u0016A\u0001\u0010@H\n@t+@M\n@t\u0011i\n+\u00160\u000eM\u0002\u0010@2H\n@t2+@2M\n@t2\u0011\n; (21)\nwhere \u0016AisamodifiedGilbertdampingtensor(fordetails,\nsee [11]).\nHowever for harmonic fields, the response of the ferro-\nmagnetic materials is measured through the differential\nsusceptibility, \u001fm=@M=@H, because there exists a net\nmagnetization even in the absence of any applied field.\nWith this, the time derivative of the harmonic magnetic\nfield can be further written as:\n@2H\n@t2=@\n@t\u0010@H\n@M@M\n@t\u0011\n=@\n@t\u0010\n\u001f\u00001\nm\u0001@M\n@t\u0011\n=@\u001f\u00001\nm\n@t\u0001@M\n@t+\u001f\u00001\nm\u0001@2M\n@t2: (22)\nIn general the magnetic susceptibility is a spin-spin re-\nsponse function that is wave-vector and frequency depen-\ndent. Thus, Eq. (18) assumes the form with the first and\nsecond order time derivatives as\n@M(5)\n@t=M\u0002\u0012\nK\u0001@M\n@t+I\u0001@2M\n@t2\u0013\n;(23)\nwhere the parameters Iij=\u00160\u000e\u0000\n1+\u001f\u00001\nm\u0001\nijandKij=\n\u00160\u000e@t(\u001f\u00001\nm)ijare tensors. The dynamics of the second\nterm is that of the magnetic inertia that operates on\nshorter time scales [25].\nHaving all the required dynamical terms, finally the\nfull magnetization dynamics can be written by joining\ntogether all the individual parts. Thus the full magneti-\nzation dynamics becomes, for harmonic fields,\n@M\n@t=M\u0002\u0012\n\u0000\r0H+ \u0000\u0001@M\n@t+I\u0001@2M\n@t2\u0013\n:(24)\nNote that the Gilbert damping parameter \u0000has two con-\ntributions, one from the susceptibility itself, Aij, which\nis of order 1=c2and an other from the time derivative of\nit,Kijof order 1=c4. Thus, \u0000ij=Aij+Kij. However we\nwill focus on the first one only as it will obviously be the\ndominant contribution, i.e., \u0000ij\u0019Aij. Even though we\nconsider only the Gilbert damping term of order 1=c2in\nthe discussions, we shall explicitly analyze the other term\nof the order 1=c4. For an ac susceptibility i.e., \u001f\u00001\nm/ei!t\nwe find thatKij/\u00160\u000e@t(\u001f\u00001\nm)ij/i\u00160!\u000e\u001f\u00001\nm, which\nsuggests again that the Gilbert damping parameter of6\nthe order 1=c4will be given by the imaginary part of the\nsusceptibility,Kij/\u0000\u00160!\u000eIm\u0000\n\u001f\u00001\nm\u0001\n.\nThe last equation (24) is the central result of this\nwork, as it establishes a rigorous expression for the in-\ntrinsic magnetic inertia. Magnetization dynamics in-\ncluding inertia has been discussed in few earlier articles\n[24, 30, 31, 58]. The very last term in Eq. (24) has been\nassociated previously with the inertia magnetization dy-\nnamics [32, 59, 60]. As mentioned, it implies a magne-\ntization nutation i.e., a changing of the precession angle\nas time progresses. Without the inertia term we obtain\nthe well-known LLG equation of motion that has already\nbeen used extensively in magnetization dynamics simu-\nlations (see, e.g., [61–65]).\nIV. DISCUSSIONS\nMagnetic inertia was discussed first in relation to the\nearth’s magnetism [26]. From a dimensional analysis,\nthe magnetic inertia of a uniformly magnetized sphere\nundergoing uniform acceleration was estimated to be of\nthe order of 1=c2[26], which is consistent with the here-\nobtained relativistic nature of magnetic inertia.\nOur derivation based on the fundamental Dirac-Kohn-\nSham Hamiltonian provides explicit expressions for both\nthe Gilbert and inertial dampings. Thus, a comparison\ncan be made between the Gilbert damping parameter\nand the magnetic inertia parameter of a pure system.\nAs noticed above, both the parameters are given by the\nmagnetization susceptibility tensor, however it should be\nnoted that the quantiy hr\u000bp\fiis imaginary itself, because\n[11],\nhr\u000bp\fi=\u0000i~\n2mX\nn;n0;kf(Enk)\u0000f(En0k)\nEnk\u0000En0kp\u000b\nnn0p\f\nn0n:(25)\nThus the Gilbert damping parameter should be given by\nthe imaginary part of the susceptibility tensor [36, 66].\nOn the other hand the magnetic inertia tensor must be\ngiven by the real part of the susceptibility [31]. This is\nin agreement with a recent article where the authors also\nfound the same dependence of real and imaginary parts\nof susceptibility to the nutation and Gilbert damping re-\nspectively [33]. In our calculation, the Gilbert damping\nand inertia parameters adopt the following forms respec-tively,\n\u0000ij=i\r\u00160\n4mc2X\nn;k[hripk+pkrii\u0000hrnpn+pnrni\u000eik]\n\u0002Im\u0000\n\u001f\u00001\nm\u0001\nkj\n=\u0000\u00160\r~\n4mc2X\nn;k\u0014hripk+pkrii\u0000hrnpn+pnrni\u000eik\ni~\u0015\n\u0002Im\u0000\n\u001f\u00001\nm\u0001\nkj\n=\u0000\u0010X\nn;k\u0014hripk+pkrii\u0000hrnpn+pnrni\u000eik\ni~\u0015\n\u0002Im\u0000\n\u001f\u00001\nm\u0001\nkj;(26)\nIij=\u00160\r~2\n8m2c4h\n1+Re\u0000\n\u001f\u00001\nm\u0001\nkji\n=\u0010~\n2mc2h\n1+Re\u0000\n\u001f\u00001\nm\u0001\nkji\n; (27)\nwith\u0010\u0011\u00160\r~\n4mc2. Note that the change of sign from damp-\ningtensortotheinertiatensorthatisalsoconsistentwith\nRef. [33], and also a factor of 2 present in inertia. How-\never, most importantly, the inertia tensor is ~=mc2times\nsmallerthan the damping tensor as is revealed in our\ncalculations. Considering atomic units we can evaluate\n\u0010\u0018\u00160\n4c2\u00180:00066\n4\u00021372\u00188:8\u000210\u00009;\n\u0010~\n2mc2\u0018\u0010\n2c2\u00188:8\u000210\u00009\n2\u00021372\u00182:34\u000210\u000013:\nThis implies that the intrinsic inertial damping is typi-\ncally 4\u0002104times smaller than the Gilbert damping and\nit is not an independently variable parameter. Also, be-\ncause of its smallness magnetic inertial dynamics will be\nmore significant on shorter timescales [24].\nA further analysis of the two parameters can be made.\nOnecanusetheKramers-Kronigtransformationtorelate\nthe real and imaginary parts of a susceptibility tensor\nwith one another. This suggests a relation between the\ntwo parameters that has been found by Fähnle et al.[34],\nnamelyI=\u0000\u0000\u001c,where\u001cisarelaxationtime. Weobtain\nhere a similar relation, I/\u0000 \u0000\u0016\u001c, where \u0016\u001c=~=mc2has\ntime dimension.\nEven though the Gilbert damping is c2times larger\nthan the inertial damping, the relative strength of the\ntwo parameters also depends on the real and imaginary\nparts of the susceptibility tensor. In special cases, when\nthe real part of the susceptibility is much higher than\nthe imaginary part, their strength could be comparable\nto each other. We note in this context that there exist\nmaterials where the real part of the susceptibility is 102\u0000\n103times larger than the imaginary part.\nFinally, we emphasize that our derivation provides the\nintrinsic inertial damping of a pure, isolated system. For\nthe Gilbert damping it is already well known that en-\nvironmental effects, such as interfaces or grain bound-\naries, impurities, film thickness, and even interactions of7\nthe spins with quasi-particles, for example, phonons, can\nmodify the extrinsic damping (see, e.g., [67–69]). Simi-\nlarly, it can be expected that the inertial damping will\nbecome modified through environmental influences. An\nexample of environmental effects that can lead to mag-\nnetic inertia have been considered previously, for the case\nof a local spin moment surrounded by conduction elec-\ntrons, whose spins couple to the local spin moment and\naffect its dynamics [31, 32].\nV. CONCLUSIONS\nIn conclusion, we have rigorously derived the magne-\ntization dynamics from the fundamental Dirac Hamilto-\nnian and have provided a solid theoretical framework for,\nand established the origin of, magnetic inertia in pure\nsystems. We have derived expressions for the Gilbert\ndamping and the magnetic inertial damping on the same\nfooting and have shown that both of them have a rela-\ntivistic origin. The Gilbert damping stems from a gen-\neralized spin-orbit interaction involving external fields,\nwhiletheinertialdampingisduetohigher-order(in 1=c2)\nspin-orbit contributions in the external fields. Both have\nbeen shown to be tensorial quantities. For general time\ndependent external fields, a field-derivative torque with\na 1storder time derivative appears in the Gilbert-type\ndamping, and a 2ndorder time-derivative field torque ap-\npears in the inertial damping.\nIn the case of harmonic external fields, the expressions\nof the magnetic inertia and the Gilbert damping scalewith the real part and the imaginary part, respectively,\nof the magnetic susceptibility tensor, and they are op-\nposite in sign. Alike the Gilbert damping, the magnetic\ninertia tensor is also temperature dependent through the\nmagnetic response function and also magnetic moment\ndependent. Importantly, we find that the intrinsic iner-\ntial damping is much smaller than the Gilbert damping,\nwhich corroborates the fact that magnetic inertia was\nneglected in the early work on magnetization dynamics\n[1–3, 19]. This suggests, too, that the influence of mag-\nnetic inertia will be quite restricted, unless the real part\nof the susceptibility is much larger than the imaginary\npart. Another possibility to enhance the magnetic iner-\ntia would be to use environmental influences to increase\nits extrinsic contribution. 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Hals1, Yaroslav Tserkovnyak2, and Arne Brataas1\n1Department of Physics, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway\n2Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nWe derive a phenomenological theory of current-induced staggered magnetization dynamics in\nantiferromagnets. The theory captures the reactive and dissipative current-induced torques and the\nconventional e\u000bects of magnetic \felds and damping. A Walker ansatz describes the dc current-\ninduced domain-wall motion when there is no dissipation. If magnetic damping and dissipative\ntorques are included, the Walker ansatz remains robust when the domain-wall moves slowly. As in\nferromagnets, the domain-wall velocity is proportional to the ratio between the dissipative-torque\nand the magnetization damping. In addition, a current-driven antiferromagnetic domain-wall ac-\nquires a net magnetic moment.\nIn ferromagnets, a spin-polarized current can be used\nto manipulate magnetization via the exchange interac-\ntion. A misalignment between the polarization of the\ncurrent and the local magnetization direction causes a\nspin-transfer torque (STT) on the magnetization because\nof noncollinear spins that precess within the ferromag-\nnet. This e\u000bect was \frst theoretically predicted by Slon-\nczewski and Berger [1] and has since garnered abundant\nexperimental evidence (for a review, see Ref. [2]). The\nSTT e\u000bect is the reciprocal process of the charge currents\nthat are induced by a time-varying magnetic texture [3].\nA promising commercial application of the STT e\u000bect\nutilizes the spin-polarized currents to switch the ferro-\nmagnetic layers in spintronic devices, such as in magnetic\nrandom-access memory or to induce magnetic precession\nfor use in high-frequency oscillators in wireless communi-\ncation devices. A limitation of these applications of STT\nis the high levels of critical currents that are required to\nswitch the direction of the magnetization.\nAntiferromagnets are ordered spin systems in which\nthe magnetic moments of all electrons in each unit cell\ncompensate for each other in equilibrium. In spintronic\ndevices, antiferromagnets are commonly used to pin fer-\nromagnetic layers via the exchange bias e\u000bect. Recent\ntheoretical [4, 5] and experimental [6] works indicate,\nhowever, that current-induced torque e\u000bects also appear\nin antiferromagnets. Antiferromagnets also share an-\nother transport property with ferromagnets; namely, the\nanisotropic magnetoresistance (AMR) e\u000bect [7]. The an-\ntiferromagnetic AMR e\u000bect allows for a detailed experi-\nmental study of the current-induced switching of the an-\ntiferromagnetic layers and the motion of the spatially de-\npendent antiferromagnetic textures. Therefore, antifer-\nromagnets can replace ferromagnets for use in spintronics\ndevices. The applicability of antiferromagnets depends\non the critical currents that are needed to manipulate\nthe staggered magnetization as well as on the resistance\nvariation that such a reorientation of the magnetization\ninduces.\nMagnetization dynamics in ferromagnets is described\nby the Landau-Lifshitz-Gilbert (LLG) equation, which\nhas been extended to include STT [8]. In magnetictextures, a spin-polarized current contributes two di\u000ber-\nent terms to the LLG equation of motion: a reactive\ntorque term and a dissipative torque term [8]. The reac-\ntive torque term preserves the macroscopic time-reversal\ninvariance of the equation. This term arises from the\nout-of-equilibrium spin density that is induced by drift-\ning electrons, which have spins that are adiabatically\nguided by the magnetic texture. The dissipative torque\nbreaks the time-reversal symmetry and arises from the\nspin-dephasing processes. Except for those systems with\nstrong spin-orbit coupling [9], the magnitude of the dissi-\npative torque is typically much smaller than the reactive\ntorque, but it is not less important [8]. In some fer-\nromagnetic domain-wall systems, the dissipative torque\ndictates the current-induced domain-wall velocity [8].\nThe e\u000bect of the reactive STT on the staggered mag-\nnetization in antiferromagnets was recently discussed in\nRef. [5]. The torque has the same physical origin as that\nin ferromagnets. Based on the form of the dissipative\nSTT in ferromagnets, Ref. [5] also makes an educated\nguess about the form of the dissipative STT in antiferro-\nmagnets.\nIn the present paper, we develop a general phenomenol-\nogy that describes the coupled dynamics of currents and\nthe staggered order parameter in isotropic antiferromag-\nnets to the lowest order in spin-texture gradients and pre-\ncession frequency. The antiferromagnet is treated within\nthe exchange approximation. For the lowest order in rel-\nativistic interactions, the exchange forces only depend on\nthe relative orientation of the spins. This approximation\nis a good starting point for many conventional ferromag-\nnets and antiferromagnets [10], including disordered sys-\ntems. In these systems, impurities couple to the spin\ndegrees of freedom through random magnetic moments\nor spin-orbit coupling, but impurity averaging restores\nthe spin-rotational and sublattice symmetries. We in-\nclude the e\u000bects of damping, external magnetic \felds,\nand reactive and dissipative torque e\u000bects. Our results\ndi\u000ber from the postulated form of the dissipative torque\nin Ref. [5], and we explain why. We apply our theory\nto an antiferromagnetic domain-wall system, and \fnd an\nanalytic solution in the low current-density regime. Sim-arXiv:1012.5655v2 [cond-mat.mes-hall] 10 Mar 20112\nilar to ferromagnets, we \fnd that the domain-wall veloc-\nity is proportional to the ratio between the dissipative-\ntorque and a bulk damping coe\u000ecient. An interesting\nconsequence of the current-induced motion is that the\ndomain-wall develops a net magnetic moment. Current-\ninduced staggered magnetization dynamics can thus be\nobserved in two ways: via the AMR e\u000bect and via the\nout-of-equilibrium net magnetic moment.\nOur phenomenology is based on the theory of insulat-\ning antiferromagnets [11], which is extended to take into\naccount the current \row. An important aspect of our\nphenomenology is the exchange approximation that im-\nplies that the total energy is invariant during the simulta-\nneous rotation of all the magnetic moments [11]. Subse-\nquently, when considering the current-induced domain-\nwall motion, we include the magnetic anisotropy phe-\nnomenologically in the free energy, considering that these\nanisotropy energies are very small, e.g., on the scale of\nthe critical temperature.\nFor clarity, we restrict our treatment to systems in\nwhich each unit cell in the crystal lattice contains two\nequivalent magnetic sites. In this situation, the antifer-\nromagnet consists of two sublattices with magnetic mo-\nment densities m1(r;t) and m2(r;t), such that the total\nmagnetization is m(r;t) =m1(r;t) +m2(r;t) and the\nantiferromagnetic order parameter is l(r;t) =m1(r;t)\u0000\nm2(r;t). In equilibrium and in the absence of magnetic\n\felds and textures, mvanishes, and lis \fnite and ho-\nmogenous. In the following, we allow the antiferromag-\nnet to become distorted into metastable textured states,\nsuch as domain-walls or vortices, but we require that the\ntexture is smooth on the scale of relevant microscopic\nlength scales. The texture is parameterized by a slowly\nvarying unit vector n(r;t)\u0011l(r;t)=l(l\u0011jl(r;t)j). As-\nsuming sti\u000b antiferromagnetic ordering, the longitudinal\ndynamics of lcan be neglected so that the slow dissipa-\ntive dynamics of the system are fully described by the\ndirectional N\u0013 eel \feld n(r;t) along with the transverse\nmagnetization m(r;t), which physically corresponds to\nthe small relative canting of the magnetic sublattices.\nConstructing the phenomenological equations of motion\nwell below the N\u0013 eel temperature, we thus impose the\nconstraintsjnj= 1 and m\u0001n= 0. This starting point is\nanalogous to Haldane's mapping of the long-wavelength\nantiferromagnetic action into the nonlinear sigma model\n[12].\nIn addition to rotational invariance, the exchange ap-\nproximation requires that the free energy and the equa-\ntions of motion are invariant under the exchange of the\ntwo sublattices [11], i.e., that they are invariant under\nthe transformations n(r;t)7! \u0000n(r;t) and m(r;t)7!\nm(r;t). The leading-order free energy that satis\fes theappropriate symmetry requirements is thus [11]\nF=Z\ndr2\n41\n2am2+A\n2X\ni=x;y;z(@in)2\u0000H\u0001m3\n5:(1)\nHere, we expanded the free energy to the second order in\nthe gradients and the magnetization \feld m(r;t), which\nis coupled to an external magnetic \feld H. The equations\nof motion for m(r;t) and n(r;t) are found by expanding\ntheir slow dynamics to the lowest order in the e\u000bective\n\felds fn\u0011\u0000\u000enF=An\u0002(r2n\u0002n)\u0000m(H\u0001n) and\nfm\u0011 \u0000\u000emF=\u0000am+n\u0002(H\u0002n). To enforce the\nconstraintsjnj= 1 and m\u0001n= 0, we calculated the\nvariational derivatives \u000emFby varying mnormal to a\n\fxednand\u000enFby parallel transporting mon the sphere\nthat is parameterized by n. In the absence of electric\ncurrents, we obtain [13]\n_n= (\rfm\u0000G1_m)\u0002n; (2)\n_m= (\rfn\u0000G2_n)\u0002n+ (\rfm\u0000G1_m)\u0002m;(3)\nwhere\ris the e\u000bective gyromagnetic ratio. The dissipa-\ntion powerP\u0011_n\u0001fn+_m\u0001fm= (G1=\r)_m2+(G2=\r)_n2\u00150\nrequires that G1;2=\r\u00150. The nondissipative equa-\ntions withG1;2= 0 are derived in the linearized regime\nin Ref. [11]. In addition to the appending dissipa-\ntion, we have also added the second term in Eq. (3),\nwhich is quadratic in small deviations from the equi-\nlibrium, to enforce the constraint m\u0001n= 0. Note\nthat such a term naturally appears if one constructs\nthe antiferromagnetic equations of motion out of the\nferromagnetic LLG equations of the constituent mag-\nnetic sublattices. Eq. (2)-(3) can be reduced to a sin-\ngle equation for the N\u0013 eel \feld (without dissipation):\nn\u0002n=\r2an\u0002h\nAr2n\u0000H(H\u0001n)=a+_H\u0002n=\rai\n\u0000\n2\r(H\u0001n)_n. This equation agrees with the equation\nthat is derived in Ref. [10] from the Lagrangian density\nL= (_n=\r\u0000H\u0002n)2=2a\u0000A(rn)2=2.\nWe also make use of the linearized equations in the\nLandau-Lifshitz form:\n_n= ~\r(fm\u0002n+G1fn);_m= ~\r(fn\u0002n+G2fm);(4)\nwhere ~\r\u0011\r=(1 +G1G2). The Onsager reciprocity re-\nlations between the two \felds require the gyromagnetic\nratios to be the same in the two equations (see below).\nThe equations (2) and (3) describe the evolution of\nan electrically open antiferromagnet. Next, we include\nthe e\u000bect of itinerant electrons on the long-wavelength\ndynamics of n(r;t) and m(r;t) by adding torque terms\nthat arise from the currents that are induced by an ex-\nternal electric \feld E. To this end, we are guided by\nthe rotational symmetry requirements and the Onsager\nreciprocity relations. The Onsager reciprocity relations\napply to a system that is described by several param-\netersfqiji= 1;:::;Ngfor which the rate of change _ qi3\nis induced by the thermodynamic forces fi\u0011 \u0000@qiF,\nand state that the o\u000b-diagonal linear response coe\u000e-\ncients in the equations _ qi=PN\nj=1Lijfjare related by\nLij(H;M) =\u000fi\u000fjLji(\u0000H;\u0000M), where\u000fi= 1 (\u000fi=\u00001)\nifqiis even (odd) under time reversal. Here, Mrepre-\nsents any possible equilibrium magnetic order. The \felds\nthat describe the collective magnetic dynamics in the an-\ntiferromagnet are n(r;t) and m(r;t), and the associated\nconjugate forces are fnandfm, respectively. In the di\u000bu-\nsive regime, the charge transferred by the current-density\n|is conjugate to the electric \feld such that fq=E. The\nresponse coe\u000ecients that are needed are the response\nmatricesLn;qandLm;q, which describe the dynamics\nofnandmthat are induced by the electric \feld. Be-\ncause the magnetization is odd and the charge is even\nunder the time reversal, Onsager's theorem implies that\nLni(mi);qj(n;m) =\u0000Lqj;ni(mi)(\u0000n;\u0000m), whereLq;n(m)\ndescribes the charge currents that are pumped by fn(fm),\nand the (\u0006n;\u0006m) arguments denote an equilibrium tex-\nture.\nTo derive the STT terms, it is convenient to be-\ngin by phenomenologically constructing the magnetically\npumped charge current |pump, which yields Lq;n(m), and\nthen invoke Onsager's theorem to obtain Ln(m);q. For the\nlowest order of the space-time gradients and the magne-\ntization \feld m, we can write three pumping terms that\nsatisfy the appropriate exchange and spatial symmetries:\nn\u0001(_m\u0002@in),_n\u0001@in, andn\u0001(_n\u0002@im). However, because\nthe last term is quadratic in the small deviations from an\nequilibrium state (in the absence of magnetic \felds), we\ndisregard it in the following. Thus, the leading-order phe-\nnomenological expression for the pumped charge current\nis as follows:\n|pump\ni=\u001b=\u0011n\u0001(_m\u0002@in) +\f_n\u0001@in\n= ~\r[(\u0011+G1\f)@in\u0001fn+ (\f\u0000G2\u0011)n\u0002@in\u0001fm];(5)\nwhere we have utilized Eq. (4) and have scaled the cur-\nrent density with the conductivity \u001b. Here,\u0011(\f) is a\nphenomenological parameter. Later, it becomes clear\nthat\u0011(\f) parameterizes the adiabatic (non-adiabatic)\ntorque because the term is even (odd) under time re-\nversal. Eq. (5) yields the response coe\u000ecients Lqi;nj=\n\u001b~\r(\u0011+G1\f)@injandLqi;mj=\u001b~\r(\f\u0000G2\u0011)(n\u0002@in)j. Us-\ning the Onsager reciprocity relations and Ohm's law for\nthe drift current ( |=\u001bE), leads to the STT terms \u001cn=\n~\r(\u0011+G1\f)(|\u0001r)nand\u001cm=\u0000~\r(\f\u0000G2\u0011)n\u0002(|\u0001r)nfor\nthe N\u0013 eel and the magnetization \feld, respectively, which\nare added on the right side of the equations of motion\nin Eq. (4). Transforming these torques back to the LLG\nform of the equations, i.e., Eqs. (2) and (3), yields the\nfollowing:\n_n= (\rfm\u0000G1_m)\u0002n+\u0011\r(|\u0001r)n; (6)\n_m= [\rfn\u0000G2_n+\f\r(|\u0001r)n]\u0002n+\u001cnl:(7)\n\u001cnl= (\rfm\u0000G1_m)\u0002m\u0000\u0011\r[m\u0001(|\u0001r)n]nare the\nsimplest nonlinear terms that are added here to enforcethe constraint m\u0001n= 0. We disregard such higher-order\nterms in the following.\nFrom now on we use the simpli\fed notation for the\ne\u000bective forces [13], which allows to more readily solve\nformin terms of n. Combining Eqs. (6) and (7), we see\nthat the magnetization \feld is fully determined by the\norder parameter nand its dynamics:\nm=1\na\u0014\nn\u0002H+1\n~\r_n\u0000G1fn\u0000(\u0011+G1\f)(|\u0001r)n\u0015\n\u0002n:\n(8)\nSubstituting this into Eq. (7) allows us to derive a closed\nequation for the N\u0013 eel \feld to the linear order in the out-\nof-equilibrium deviations m,@tn,|, and H[14]:\nn=~\r=\u0000n\u0002_H+G1_fn+ (\u0011+G1\f)(_|\u0001r)n\n+a[\rfn\u0000G2_n+\r\f(|\u0001r)n]: (9)\nEq. (5)-(9) are our main results, which describe a gen-\neral phenomenological theory of weakly excited current-\ninduced dynamics in conducting antiferromagnets and\ncharge pumping that arises from moving textures. The\nreactive torque in Eq. (9), which is proportional to\n(\u0011+G1\f), was \frst found in Ref. [5]. The dissipative\ntorque term, which is proportional to a\f, and the ef-\nfects of magnetization damping are new terms that have\nnot been derived before. The consideration of the charge\npumping that occurs when moving antiferromagnetic tex-\ntures is also new. Ref. [5] suggests a dissipative torque of\nthe form\fn\u0002(_|\u0001r)nin Eq. (9). This term breaks the\nn7!\u0000ninvariance of the equation and therefore can-\nnot appear in the exchange approximation of equivalent\nmagnetic sublattices.\nAs an application of our theory, we consider an anti-\nferromagnetic domain-wall system and study the current-\ninduced domain-wall motion. For clarity, we set the ex-\nternal magnetic \feld to equal zero from this point of the\npaper on. Domain-walls can be created in systems with\nanisotropy, which is added phenomenologically to the free\nenergy as follows: F[m;n]!F[m;n] +W[n], where\nW[n] =R\ndr\u0000\nK?n2\ny=2\u0000Kzn2\nz=2\u0001\nis the anisotropy en-\nergy (K?;Kz>0). A local minima of the above en-\nergy functional is a N\u0013 eel wall that rotates in the xz\nplane, where the local magnetization direction is nx=\n1=cosh(z=\u0015w) andnz= tanh(z=\u0015w).\u0015w=p\nA=K zis\nthe domain-wall width. This equilibrium domain-wall\ntexture is denoted by n0(r;t) below.\nIn the following, we study how the domain-wall moves\nin response to a current along the zaxis. Let us \frst con-\nsider the case of no magnetization damping and \f= 0.\nIn this case, it follows from Eq. (6)-(7) that m(r;t) = 0\nandn(r;t) =n0(z\u0000rw(t)) is an exact solution of the\nequations, with the domain-wall velocity _ rw=\u0000\r\u0011|.\nBy including the magnetization damping and the dis-\nsipative torque, a local magnetic moment density devel-\nops. The torque \u001cm=\u0000~\r|(\f\u0000G2\u0011)n\u0002@zninduces\na magnetic moment density along the y-axis that should4\neventually approach a \fnite value due to the opposite\nacting damping term G2fmin Eq. (4). Thus, to \fnd\na stationary solution for Eq. (6)-(7), we use the ansatz\nn(r;t) =n0(z\u0000rw(t)) and m(r;t) =m0(t)n\u0002@zn. Here,\nm0(t) parameterizes the magnitude of the local magnetic\nmoment density. Substituting these two expressions into\nEq. (6)-(7), produces the following equations for the two\nparameters rw(t) andm0(t):\n_rw= ~\r[am0\u0000(\u0011+G1\f)|]; (10)\n_m0\n~\r= (G2\u0011\u0000\f)|\u0000G2a\u0012\n1 +_rwnz\n~\rG2a\u0015w\u0013\nm0:(11)\nThe second term inside the last parenthesis in Eq. (11)\nis position-dependent because of nz. When this term\nis negligible, the ansatz becomes a good approximation\nof Eq. (10)-(11). This low current-density regime cor-\nresponds to systems for which the characteristic intrin-\nsic relaxation time (~ \rG2a)\u00001of the antiferromagnetic\nsystem is much smaller than the timescale \u0015w=_rw; i.e.,\nthe domain-wall moves a small distance as compared\nto the domain-wall width during the relaxation time.\nIn this regime, m0approaches a \fnite stable value of\nm0=\u0000(\f\u0000G2\u0011)|=(G2a), and the domain-wall moves\nat a constant velocity _ rw=\u0000\r\f|=G 2. Similarly to that\nfor ferromagnets, the velocity is proportional to the ra-\ntio between the dissipative torque and a Gilbert damp-\ning coe\u000ecient. In contrast to ferromagnets, this result\nis independent of the uniaxial anisotropy K?such that\nthe Walker ansatz has a much wider range of applicabil-\nity. An interesting consequence of the current-induced\ndomain-wall motion is that the system develops a \fnite\nmagnetic moment in the domain-wall region. This e\u000bect\nmay be an alternative to the AMR e\u000bect for the mea-\nsurement of domain-wall motion.\nTo verify that the system approaches the above sta-\ntionary solution in the relevant regime, we conducted\na micromagnetic simulation of a one-dimensional sys-\ntem based on Eq. (6)-(7). For the numerical calcula-\ntion, we wrote the equations in a dimensionless form\nby scaling the zaxis with the lattice constant aland\nthe time axis with (~ \rA\u0003)\u00001. Here,A\u0003=A=\u0000\nla2\nl\u0001\n.\nWe considered a domain-wall system with a domain-\nwall width of \u0015w= 20al. The anisotropy and damp-\ning parameters are al=A\u0003= 10,Kz=(lA\u0003) = (20)\u00002,\nK?=(lA\u0003) = 0:1 andG1l=G2=l= 0:01. For the STT\ntorque parameters we used |(\u0011+G1\f)=(alA\u0003) = 0:1\nand|(\f\u0000G2\u0011)=(alA\u0003l) = 0:001. With these values,\nthe stationary solution implies that m\u0003\n0=m0=(all) will\napproach 0:01 and that the wall will move a distance of\n2alduring the relaxation time (~ \rG2a)\u00001. We therefore\nexpect the system to be in the relevant regime for which\nthe stationary solution is valid.\nFig. 1 shows the micromagnetic simulation of the above\nsystem. We see that the domain-wall velocity follows\nEq. (10) nearly perfectly and that velocity and m0ap-\nproach the expected stationary values. It should be noted\n0 20 40 60 80 100 120 140 160-0.25-0.2-0.15-0.1-0.050\nTimeVelocityStationary\na* m*0 - p*\nNumerical\n0 50 100 150-0.015-0.01-0.0050\nTimem*0Stationary\nm*0(t)FIG. 1: The blue line shows the domain-wall velocity, which\nwas determined using a micromagnetic simulation, as a func-\ntion of time when a current is applied at t= 0. The ve-\nlocity follows the analytic expression in Eq. (10) and ap-\nproaches the stationary value _ rw=\u0000\r\f|=G 2. Inset: The\ntime evolution of m 0. The parameter approaches the sta-\ntionary value m0=\u0000(\f\u0000G2\u0011)|=(G2a). All of the re-\nsults are given in dimensionless quantities. a\u0003=al=A\u0003, and\np\u0003=|(\u0011+G1\f)=(alA\u0003).\nthat our ansatz breaks down when (~ \rG2a)\u00001is not much\nsmaller than \u0015w=_rw; however, a more detailed study of\nthis regime is beyond the scope of this manuscript.\nThe typical values of the \f=G-ratio in antiferromagnets\nis an interesting issue for future experiments. This ratio\ncan be probed by measuring the domain-wall velocity as\na function of the current density or by measuring the\nreciprocal process, which is voltage that is induced by a\nmoving domain-wall. In the last case, one could initiate a\ndomain-wall motion using a short current pulse and then\nmeasure the voltage echo from the reciprocal process.\nIn conclusion, we have derived a general phenomeno-\nlogical theory of current-induced dynamics in antiferro-\nmagnets and have applied the theory to the study of\ncurrent-induced domain-wall motion. We found that the\ndomain-wall developed a net magnetic moment during\nthe current-induced motion and that the domain-wall ve-\nlocity was proportional to the ratio between the dissipa-\ntive torque parameter and a damping parameter.\nThis work was partially supported by the European\nUnion FP7 Grant No. 251759 \\MACALO\" (K. M. D. H.\nand A.B.), NSF Grant No. DMR-0840965, and DARPA\n(Y.T.). We would like to thank Erik Wahlstr om and\nAndr\u0013 e Kapelrud for helpful discussions.\n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996); L. Berger, Phys. Rev. B 54, 9353 (1996).5\n[2] D. C. Ralph and M. Stiles, J. Magn. Magn. Mater. 320,\n1190 (2008), and reference therein.\n[3] Y. Tserkovnyak and M. Mecklenburg, Phys. Rev. B 77,\n134407 (2008).\n[4] A. S. N\u0013 u~ nez, R. A. Duine, P. Haney, and A. H. MacDon-\nald, Phys. Rev. B 73, 214426 (2006); R. A. Duine, P. M.\nHaney, A. S. N\u0013 u~ nez, and A. H. MacDonald, Phys. Rev.\nB75, 014433 (2007); P. M. Haney et al. , Phys. Rev. B\n75, 174428 (2007); H. Gomonay and V. Loktev, J. Magn.\nSoc. Jpn. 32, 535 (2008); Y. Xu, S. Wang, and K. Xia,\nPhys. Rev. Lett. 100, 226602 (2008); P. M. Haney and\nA. H. MacDonald, Phys. Rev. Lett. 100, 196801 (2008);\nH. V. Gomonay and V. M. Loktev, Phys. Rev. B 81,\n144427 (2010).\n[5] A. C. Swaving and R. A. Duine, arXiv:0912.4519.\n[6] S. Urazhdin and N. Anthony, Phys. Rev. Lett. 99, 046602\n(2007).\n[7] A. B. Shick, S. Khmelevskyi, O. N. Mryasov, J. Wunder-\nlich, Phys. Rev. B 81, 212409 (2010); T. Jungwirth et al. ,\narXiv:1007.0177; B. G. Park et al. , arXiv:1011.3188v1.\n[8] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, J.\nMagn. Magn. Mater., 320, 1282 (2008), and referencetherein.\n[9] K. M. D. Hals, A. K Nguyen, and A. Brataas, Phys. Rev.\nLett. 102, 256601 (2009).\n[10] A. F. Andreev and V. I. Marchenko, Sov. Phys. Usp. 23,\n21 (1980).\n[11] E. M. Lifshitz and L. P. Pitaevskii Statistical Physics,\nPart 2 , Course of Theoretical Physics Vol. 9, (Pergamon,\nOxford, 1980).\n[12] A. Auerbach, Interacting Electrons and Quantum Mag-\nnetism (Springer-Verlag, New York, 1994).\n[13] It is useful to observe that calculating the forces fn\u0011\n\u0000\u000enF!Ar2nandfm\u0011\u0000\u000emF!\u0000am+Hnaively,\ni.e. not imposing the nonlinear constraints jnj= 1 and\nm\u0001n= 0, results in the correct equations (2) and (3) [as\nwell as our key equations (6) and (7)], upon substitution.\nIn the intermediate steps, such as Eqs. (4), however, we\nwill be using the fact that the proper forces are normal\nto the N\u0013 eel order, fn;fm?n.\n[14] Note that the second order di\u000berential equation for the\nN\u0013 eel order dynamics depends on m(t= 0) through the\ninitial condition for _ n(t= 0)." }, { "title": "1208.1462v1.Observation_of_Coherent_Helimagnons_and_Gilbert_damping_in_an_Itinerant_Magnet.pdf", "content": "arXiv:1208.1462v1 [cond-mat.str-el] 7 Aug 2012ObservationofCoherent HelimagnonsandGilbertdamping in anItinerant Magnet\nJ. D. Koralek1,∗,†, D. Meier2,∗,†, J. P. Hinton1,2, A. Bauer3, S. A. Parameswaran2,\nA. Vishwanath2, R. Ramesh1,2R. W. Schoenlein1, C. Pfleiderer3, J. Orenstein1,2\n1Materials Science Division, Lawrence Berkeley National La boratory, Berkeley, California 94720, USA\n2Department of Physics, University of California, Berkeley , California 94720, USA and\n3Physik Department E21, Technische Universit¨ at M¨ unchen, D-85748 Garching, Germany\n(Dated: DatedAugust 30, 2018)\nWe study the magnetic excitations of itinerant helimagnets by applying time-resolved optical spectroscopy\nto Fe0.8Co0.2Si. Optically excited oscillations of the magnetization in the helical state are found to disperse\nto lower frequency as the applied magnetic field is increased ; the fingerprint of collective modes unique to\nhelimagnets,knownashelimagnons. Theuseoftime-resolve dspectroscopyallowsustoaddressthefundamen-\ntal magnetic relaxation processes by directly measuring th e Gilbert damping, revealing the versatility of spin\ndynamics inchiralmagnets.\nTheconceptofchiralitypervadesallofscience,havingpro -\nfound implications in physics, chemistry and biology alike .\nIn solids, relativistic spin-orbit coupling can give rise t o the\nDzyaloshinskii-Moriya (DM) interaction,2,3imparting a ten-\ndency for the electron spins to form helical textures with a\nwell-definedhandednessincrystalslackinginversionsymm e-\ntry. Helical spinorderisespeciallyinterestingwhenthem ag-\nnetismarisesfromthesameelectronsresponsibleforcondu c-\ntion as is the case in doped FeSi which displays unconven-\ntional magnetoresitence,4,5helimagnetism,6and the recently\ndiscovered Skyrmion lattice.7,8The excitations of helimag-\nnets have been studied over the past 30 years, culminating\nrecently in a comprehensive theory of spin excitations call ed\nhelimagnons.9,10Signatures of helimagnons have been ob-\nserved in neutron scattering11and microwave absorption,12\nyet little is known about their magnetodynamics and relax-\nation phenomenaon the sub-picosecondtimescales on which\nmagnetic interactions occur. Understanding the dynamics,\nhowever,isofgreatimportanceregardingspintransfertor que\neffects in chiral magnets, and related proposed spintronic s\napplications.13–15\nIn this work we study the dynamics of collective spin ex-\ncitationsin the itineranthelimagnetFe 0.8Co0.2Si. Ouroptical\npump-probemeasurementsidentifyanomalousmodesatzero\nwavevector ( q=0) which we identify unmistakably as heli-\nmagnons. These helimagnons manifest as strongly damped\nmagnetization oscillations that follow a characteristic s caling\nrelation with respect to temperature and magnetic field. The\nsub-picosecond time resolution of our technique enables de -\nterminationof the intrinsic Gilbert dampingparameterwhi ch\nis foundto be oneorderof magnitudelargerthan in localized\nsystems, revealing the versatility of the spin-lattice int erac-\ntions available in the emergent class of DM-driven helimag-\nnets.\nDespite being a non-magnetic insulator, FeSi is trans-\nformed into an itinerant magnet upon doping with cobalt.4,16\nWe have chosen Fe 0.8Co0.2Si for our study because it can\neasily be prepared in high quality single crystals17with a\nreasonably high magnetic ordering temperature TN, and its\nexotic equilibrium properties are well characterized, ope n-\ning the door for non-equilibrium dynamical studies. Small-\nangle neutron scattering8was used to determine the phase\ndiagram and has revealed helimagnetic spin textures belowTN=30 K that emerge from the interplay between the fer-\nromagnetic exchange and DM interactions. In zero magnetic\nfield the spins form a proper helix with a spatial period of\n≈350˚A,18whereasfinitefieldscantthespinsalongthehelix\nwavevector, kh, (see Fig. 1(c))inducinga conicalstate witha\nnet magnetization. Sufficiently high fields, H≥Hc, suppress\nthe conical order in favor of field alignment of all spins. In\ntheexperimentsreportedhere,femtosecondpulsesoflinea rly\npolarized 1.5 eV photons from a Ti:Sapphire oscillator were\nused to excite a (100) oriented single crystal of Fe 0.8Co0.2Si\nat near normal incidence. The changes induced in the sam-\nple by the pump pulse were probed by monitoringthe reflec-\ntion and Kerr rotation of time-delayed probe pulses from the\nsame laser. In order to minimize laser heating of the sam-\nple the laser repetition rate was reduced to 20 MHz with an\nelectro-optic pulse picker, ensuring that thermal equilib rium\nwasreachedbetweensuccessivepumppulses. Signaltonoise\nwas improved by modulation of the pump beam at 100 KHz\nandsynchronouslock-indetectionofthereflectedprobe. Ke rr\nrotation was measured using a Wollaston prism and balanced\nphotodiode. All temperature and field scans presented in thi s\nwork were performed from low to high TandH||(100)after\nzero-fieldcooling.\nFig. 1 shows the transient reflectivity, ΔR/R, as a function\nof temperature and magnetic field. At high temperature we\nobserve a typical bolometric response from transient heati ng\nof the sample by the pump pulse (Fig 1 (a)).19This is char-\nacterized by a rapid increase in reflectivity, followed by tw o-\ncomponent decay on the fs and ps timescales, corresponding\nto the thermalizationtimes between differentdegreesof fr ee-\ndom (electron, spin, lattice, etc.).20As the sample is cooled\nbelowTN, the small thermal signal is beset by a much larger\nnegative reflectivity transient (Fig. 1 (b)) with a decay tim e\nof roughly τR≈175 ps at low temperature (Fig. 3 (b)). A\nnatural explanation for this is that the pump pulse weakens\nthe magnetic order below TN, which in turn causes a change\nin reflectivity via the resulting shift of spectral weight to low\nenergy.21Thetemperaturedependenceof the peak ΔR/Rval-\nues is plotted in Fig. 1 (c) for several applied fields, showin g\nonlyweakfielddependence.\nToaccessthemagnetizationdynamicsmoredirectlywean-\nalyze the polarizationstate of the probe pulses, which rota tes\nby an angle θKupon reflection from the sample surface, in2\nFIG.1: Timedependence ofthepump-inducedtransientreflec tivityΔR/Rinthe(a)paramagneticand(b)helimagneticstates. Thetem perature\ndependence of the maximum ΔR/Ris plottedin(c)for several applied magnetic fields.\nproportion to the component of the magnetization along the\nlight trajectory. The change in Kerr rotation induced by the\noptical pump, ΔθK, is shown in Fig. 2 as a function of tem-\nperature and field. The upper panels show temperature scans\nat fixedmagneticfield,while afieldscan atfixedtemperature\nisshowninpanel(d). Weobservethat ΔθKchangessignas H\nisreversed(notshown),andgoestozeroas Hgoesto zeroor\nas temperatureis raised above TN. Oscillationsof the magne-\ntization are clearly visible in the raw data below 25 K in the\nhelimagneticphase.\nIn order to analyze the magnetization dynamics, we use a\nsimple phenomenological function that separates the oscil la-\ntory and non-oscillatorycomponentsseen in the data. It con -\nsists ofadecayingsinusoidaloscillation,\nΔθK=e−t\nτK[A+Bsin(ωt)] (1)\nwitha timedependentfrequency,\nω(t)=2πf0/bracketleftBig\n1+0.8/parenleftBig\ne−t\nτK/parenrightBig/bracketrightBig\n(2)\nwhich decays to a final value ω0. We emphasize that there is\nonly a single decay time τKdescribing the magneto dynam-\nics, and it is directly related to the Gilbert damping parame -\nterα=(2πf0τK)−1. This function produces excellent fits to\nthe data as illustrated in Fig. 3 (a), allowing accurate extr ac-\ntion of the oscillation frequencies and decay times shown in\nFigs. 3 (b)-(d). The oscillation frequency is reduced as ei-\nther field or temperature is increased, while the decay time\nτKis roughly constant and equal to τRbelow 20 K. As the\ntemperature is raised towards the phase transition, the rel ax-\nation time τKdiverges, which can be understood in terms of\na diverging magnetic correlation length due to the presence\nof a critical point. The similarity between the decay times τR\nandτKwithin the ordered phase reflects strongly correlated\ncharge and spin degrees of freedom, and supports the notion\nthatΔR/Risdeterminedbythemagneticorder.\nThemagneticoscillationfrequencyreaches f0≈4.8GHzat\nlowtemperature,whichcorrespondstoaLarmorprecessiono f\nspinssubjectedtoafieldof170mT,whichisroughlythecrit-\nical field Hcrequiredto destroy the spin helix. This, togetherwith the fact that the oscillation frequencyis nonzero only in\nthehelicalstate,suggeststhattheoscillationsarecomin gfrom\nexcitations unique to the helical structure. It is well know n\nthat magnetization oscillations can be optically induced b y\nultrafast generation of coherent magnons,24–26however, or-\ndinary magnons cannot explain our data as their frequency\nwouldincreasewith H,oppositetowhatisseenin Fig. 3(c).\nBased on these observations, we propose the following in-\nterpretation of our results: In the helical magnetic phase, the\npump photons weaken the magnetic order through the ultra-\nfast demagnetization process.27As described above, this re-\nduction in magnetic order gives rise to a decrease in the re-\nflectivity at 1.5 eV which is nearly field independent. As a\nmagnetic field is applied the spins become canted along the\nhelix wavevector,giving rise to a macroscopic magnetizati on\nwhichweobserveinKerrrotationviaitscomponentalongthe\nprobelight trajectory. The demagnetizationfromthe pumpi s\nresponsible for the initial peak seen in the ΔθKtime traces,\nand is captured by the exponential component of our fitting\nfunction (green curve in Fig 3 (a)). The pump photons also\nlaunch a coherentspin wave, giving rise to the oscillations in\nΔθK(red curve in Fig. 3 (a)). The form of the oscillatory\ncomponent goes like sin (ωt)rather than [1−cos(ωt)], sug-\ngesting impulsive stimulated Raman scattering as the mecha -\nnism of excitation.25The anomalousfield dependenceshown\nin Fig. 3 (c) leads to the unambiguous conclusion that the\noptically excited spin waves are the fundamental modes of\nhelimagnetstermedhelimagnons.10Specifically,theoptically\naccessible helimagnon mode consists of the constituents of\nthe spin helix precessing in-phase about their local effect ive\nfield. Since this local effective field is reduced during the u l-\ntrafast demagnetizationprocess, the oscillation frequen cyde-\ncreases as a function of time delay as the field recovers, ne-\ncessitatingthetimedependentfrequencyinEq. 1. Theabili ty\nto resolve helimagnons with femtosecond time resolution at\nq=0isuniquetoouropticalprobe,andcomplimentsneutron\nscatteringwhichisrestrictedtomappinghelimagnonbands at\nhigherq. This regionof reciprocalspace is particularlyinter-\nestinginthecaseofhelimagnetsastheperiodicityintrodu ced\nbythehelicalspintexturegeneratesbandsthatarecentere dat\nq=±khand therefore have finite frequency modes at q=03\nFIG. 2: (a),(b),(c) Time dependence of the pump-induced cha nge in Kerr rotation, ΔθK, as a function of temperature for several applied\nmagneticfields. (d) ΔθKasafunctionofmagneticfieldforseveraltemperatures. Cur vesareoffsetforclarity. Alsoshownisaschematicphase\ndiagram, adapted from Reference 8,withredarrows illustra tingthe temperature and fieldscans usedin(a)-(d).\neven in the absence of a gap. This is in contrast to ordinary\nmagnonsinwhichthebandsaregenerallycenteredat q=0so\nthattheassociatedmodehaszerofrequency. Wenotethatour\nobservationsare in agreementwith previouswork on the col-\nlectivemodesofskyrmions28whichcoexistwithhelimagnons\nin theA-Phase(see Fig. 3).12Theappearanceofthese modes\nis not expected in our data as their corresponding oscillati on\nperiodsexceedtheobserveddampingtimein Fe 0.8Co0.2Si.\nInordertoquantitativelytestthehelimagnoninterpretat ion\nwetaketheexpressionforthe q=0helimagnonfrequencyin\nanexternalmagneticfield,\nf0=gµBHc/radicalbig\n1+cos2θ (3)\nwheregistheeffectiveelectron g-factor,µBistheBohrmag-\nneton,andπ\n2−θistheconicalanglei.e. theamountthespins\nare canted away from khby the applied field H. Ignoringde-\nmagnetization effects of the spin waves themselves, we can\nwrite sinθ=H\nHc, whereHcis the critical field at which the\nspinsall alignwiththe field andthe helimagnonceasesto ex-\nist asa well-definedmode. Thenweobtain,9\nf0=gµBHc/radicalBigg\n1−1\n2/parenleftbiggH\nHc/parenrightbigg2\n(4)\nwhich expresses the magnon frequency as a function of ap-\nplied field. This expression fits the data remarkably well as\nshown in Fig. 3 (c), capturingthe decrease in frequencywith\nincreasing Hwhich is unique to helimagnons. However, due\ntothefactthattheoscillationperiodexceedsthedampingt imeforfieldsabove75mT,itisnotpossibletoextractthevalueo f\nthecritical field Hcinthis system. The solidline in Fig. 3(d)\nisafittotheform f0∝/radicalBig\n1−T\nTNwhichgives TNasafunction\nofHin reasonableagreementwithpublisheddata.8\nThe Gilbert damping parameter can be directly obtained\nfrom the measured decay times through the relation α=\n(2πf0τK)−1, which gives a value of α≈0.4 for the heli-\nmagnetic phase of Fe 0.8Co0.2Si. This is an order of magni-\ntude larger than what was seen in insulating Cu 2OSeO3,12\nwhere helimagnetism arises from localized rather than itin -\nerant spins. The contrast in dynamics between these systems\nis critical in the context of potential spintronic applicat ions\nbasedonhelimagnetismwherethereisatradeoffbetweenfas t\nswitching which requires large damping, and stability whic h\nreliesonlowdamping.\nIn summary, this work demonstrates ultrafast coherent op-\ntical excitation of spin waves in an itinerant DM-driven spi n\nsystem and reveals the underlying spin dynamics. We iden-\ntifytheseexcitationsashelimagnonsthroughtheiranomal ous\nfield dependence and explain our observations with a com-\nprehensive model. Our experiments directly yield the intri n-\nsic Gilbertdampingparameter,revealingastrikingdiffer ence\nin spin relaxationphenomenabetweenitinerant andlocaliz ed\nhelimagnets. The results elucidate the dynamicsof collect ive\nmodes common to the actively studied B20 transition metal\ncompounds that codetermine their performance in potential\nspinbasedapplications.\nAcknowledgments: The work in Berkeley was supported\nby the Director, Office of Science, Office of Basic Energy4\nFIG.3: (a) Exemplary ΔθKoscillation data (blue circles) and fit (black line) using th e model described inthe text. The fitis decomposed into\nan exponential term (green curve) and an oscillatory term (r ed curve). The fitting function uses a single time constant τKfor all terms which\nis plotted in panel (b) as a function of temperature and field. For comparison we also plot the decay time of the reflectivity ,τR, averaged over\nall fields. The solid lines are guides to the eye. Panels (c) an d (d) show the reduced magnetization oscillation frequency for field scans and\ntemperature scans respectively, andsolidlines are fitstot he data as described inthe maintext.\nSciences,MaterialsSciencesandEngineeringDivision,of the\nU.S. Department of Energy under Contract No. DE-AC02-\n05CH11231. C.P. and A.B. acknowledge support through\nDFGTRR80(FromElectronicCorrelationstoFunctionality) ,\nDFG FOR960 (Quantum Phase Transitions), and ERC AdG\n(291079, TOPFIT). A.B. acknowledges financial support\nthrough the TUM graduate school. D.M. acknowledges sup-portfromtheAlexandervonHumboldtfoundationandS.A.P.\nacknowledgessupportfrom the SimonsFoundation. C.P. and\nA.B. also thank S. Mayr, W. M¨ unzer, and A. Neubauer for\nassistance.\n∗Theseauthorscontributedequallytothiswork.\n†Email address: jdkoralek@lbl.gov and meier@berkeley.edu\n2I. E.Dzyaloshinskii, Sov. Phys.JETP 5, 1259 (1957).\n3T. Moriya, Phys. Rev. 120, 91(1960).\n4N. Manyala etal., Nature404, 581 (2000).\n5N. Manyala etal., Nature Mater. 3, 255 (2004).\n6J. Beille, J. Voiron, and M. Roth, Solid State Commun. 47, 399\n(1983).\n7S.M¨ uhlbauer et al.,Science 323, 915 (2009).\n8W.M¨ unzer etal.,Phys. Rev. B 81, 041203(R) (2010).\n9M. Kataoka, J.Phys.Soc. Jap. 563635 (1987).\n10D.Belitz,T.R.Kirkpatrick,andA.Rosch,Phys.Rev.B 3,054431\n(2006).\n11M. Janoschek et al.,Phys.Rev. B 81, 214436 (2010).\n12Y. Onose, Y. Okamura, S.Seki,S.Ishiwata, and Y. Tokura, Phy s.\nRev. Lett. 109, 037603 (2012).\n13K. Everschor, M. Garst, B. Binz, F. Jonietz, S. M¨ uhlbauer,\nC. Pfleiderer,andA.Rosch, arXiv:1204.5051v1 (2012).\n14F.Jonietz et.al.,Science 330, 1648 (2010).\n15T. Schulz et.al., NaturePhys. 8, 301(2012).16G.AeppliandZ.Fisk,Comments Condens. MatterPhys. 16,155\n(1992).\n17A. Neubauer et.al.,Rev. Sci.Instrum. 82, 013902 (2011).\n18S.V. Grigoriev et al.,Phys.Rev. Lett. 102, 037204 (2009).\n19R.W.Schoenlein,W.Z.Lin,G.FujimotoandG.L.Besley,Phys .\nRev. Lett. 581680 (1987).\n20S. I. Anisimov, B. L. Kapeliovich and T. L. Perelman, Sov. Phy s.\nJETP39, 375 (1975).\n21F.P.Mena etal.,Phys. Rev. B 73, 085205 (2006).\n22S.A.Brazovskii,S.G.Dmitriev,Sov.Phys.JETP 42,497(1976).\n23M. Janoschek, M. Garst, A. Bauer, P. Krautscheid, R. Georgii ,\nP.B¨ oni, andC.Pfleiderer,arXiv:1205.4780v1(2012).\n24M. van Kampen etal.,Phys. Rev. Lett. 88, 227201 (2002).\n25A. M.Kalashnikova et al.,Phys.Rev. B 78, 104301 (2008).\n26D. Talbayev et al.,Phys. Rev. Lett. 101, 097603 (2008).\n27A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82,\n2731 (2010).\n28M. Mochizuk, Phys. Rev. Lett. 108, 017601 (2012)." }, { "title": "1908.11761v1.Magnetization_reversal__damping_properties_and_magnetic_anisotropy_of_L10_ordered_FeNi_thin_films.pdf", "content": "Magnetization reversal, damping properties and magnetic anisotropy of L 10\nordered FeNi thin \flms\nV. Thiruvengadam,1B. B. Singh,1T. Kojima,2K. Takanashi,2M. Mizuguchi,2,a)and S. Bedanta1,b)\n1)Laboratory for Nanomagnetism and Magnetic Materials (LNMM), School of Physical Sciences,\nNational Institute of Science Education and Research (NISER), HBNI, P.O.- Bhimpur Padanpur, Via Jatni, 752050,\nIndia\n2)Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577,\nJapan\n(Dated: August 2019)\nL10ordered magnetic alloys such as FePt, FePd, CoPt and FeNi are well known for their large magnetocrys-\ntalline anisotropy. Among these, L10-FeNi alloy is economically viable material for magnetic recording media\nbecause it does not contain rare earth and noble elements. In this work, L10-FeNi \flms with three di\u000berent\nstrengths of anisotropy were fabricated by varying the deposition process in molecular beam epitaxy system.\nWe have investigated the magnetization reversal along with domain imaging via magneto optic Kerr e\u000bect\nbased microscope. It is found that in all three samples, the magnetization reversal is happening via domain\nwall motion. Further ferromagnetic resonance (FMR) spectroscopy was performed to evaluate the damping\nconstant and magnetic anisotropy. It was observed that the FeNi sample with moderate strength of anisotropy\nexhibits low value of damping constant \u00184:9\u000210\u00003. In addition to this, it was found that the \flms possess\na mixture of cubic and uniaxial anisotropies.\nIn order to increase the storage density in magnetic\nrecording media it requires reduction in the bit size1. On\nthe other hand, in ferromagnetic materials, the super-\nparamagnetic (SPM) limit is inevitable at a critical ra-\ndius, below which the magnetic moment is thermally un-\nstable and become incapable of storing the information2.\nTherefore, magnetic material with high magnetocrys-\ntalline anisotropy is essential to overcome the SPM\nlimit3. In this context, the L10ordered magnetic alloys\nsuch as FePt, FePd, CoPt and FeNi have potential for\nultra-high density magnetic recording media because of\ntheir large uniaxial magnetocrystalline anisotropy energy\ndensity \u0018107erg cm\u00003.L10ordered alloy is a binary\nalloy system with face centered tetragonal (FCT) crys-\ntal structure where each constituent atomic layers are\nalternatively laminated along the direction of crystallo-\ngraphic c-axis4,5. In these materials the large anisotropy\nenergy is due to their tetragonal symmetry of L10crystal\nstructure6.\nL10-FeNi possess high values of saturation magnetiza-\ntion (1270 emu cm\u00003), coercivity (4000 Oe), and uniaxial\nanisotropy energy density (1.3 \u0002107erg cm\u00003)7{9. In ad-\ndition its Curie temperature is quite high \u0018550\u000eC and it\nexhibits excellent corrosion resistance7. All these above\nproperties make L10-FeNi a promising material for fab-\nricating information storage media and permanent mag-\nnets. Further, L10ordered FeNi alloy is free of noble\nas well as rare-earth elements. Also the constituent ele-\nments (i.e. Fe and Ni) are relatively inexpensive. There-\nfore L10ordered FeNi alloy is economically viable for\ncommercial applications4,10,11. It should be noted that\nalthough, L10- FeNi possess high uniaxial anisotropy,\na)Electronic mail: mizuguchi@imr.tohoku.ac.jp\nb)Electronic mail: sbedanta@niser.ac.inshape anisotropy becomes dominant in thin \flms11. Pre-\nviously the study of the order parameter (S) and Fe-Ni\ncomposition dependence on Kuhas been reported7. Pre-\nviously, magnetic damping constants for L10-FeNi and\ndisordered FeNi have been studied employing three kinds\nof measurement methods12. However, the e\u000bect of di\u000ber-\nent anisotropy values of L10- FeNi thin \flms on evo-\nlution of their magnetic domain structures and damping\nhas not been studied so far. Therefore, focus of this paper\nis to study the domain structures during magnetization\nreversal, anisotropy strength, damping properties of L10\n- FeNi thin \flms with di\u000berent anisotropy values.\nL10-FeNi thin \flms were prepared in a molecular beam\nepitaxy chamber with base pressure of 10\u00008Pa consist-\ning of e-beam evaporators (for Fe and Ni) and Knud-\nsen cells (for Cu and Au). First a seed layer of Fe (1\nnm) and Au (20 nm) were deposited at 80\u000eC on MgO\n(001) substrate followed by Cu (50 nm) layer deposited\nat 500\u000eC. It has been reported that Cu and Au layers\nare prone to form an alloy of Cu 3Au (001)9. In order\nto fabricate highly ordered L10-FeNi \flms, a bu\u000ber layer\nof Au 0:06Cu0:51Ni0:43(50 nm) was deposited on Cu 3Au\nlayer at 100\u000eC by co-deposition13. On top of this bu\u000ber\nlayer, FeNi layer was grown by alternative layer deposi-\ntion (Fe and Ni one after another) or co-deposition (de-\nposition of Fe and Ni simultaneously). Disordered-FeNi\n\flm (sample A) was obtained by co-deposition process\nwhereas L10-ordered FeNi \flms were fabricated by alter-\nnate deposition at 100\u000eC (sample B) and 190\u000eC (sam-\nple C), respectively. Finally, Au capping layer (3 nm)\nwas deposited on top of FeNi layer at 30-40\u000eC. To study\nmagnetization reversal and magnetic domain structures,\nwe have performed hysteresis measurements along with\nsimultaneous domain imaging using magneto optic Kerr\ne\u000bect (MOKE) based microscope manufactured by Evico\nMagnetics Ltd. Germany14. Kerr microscopy was per-arXiv:1908.11761v1 [physics.app-ph] 30 Aug 20192\nFIG. 1. Hysteresis loops measured at room temperature by longitudinal Kerr microscopy at 0\u000e, 30\u000e, 60\u000eand 90\u000efor samples\nA - C shown in (a) - (c), respectively..\nformed at room temperature in longitudinal geometry.\nAngle dependent hysteresis loops were measured by ap-\nplying the \feld for 0\u000e<\b<360\u000eat interval of 10\u000ewhere\n\b is the angle between the easy axis and the direction\nof applied magnetic \feld. Further, magnetization dy-\nnamics of the L10-FeNi thin \flms has been studied by\nferromagnetic resonance (FMR) spectroscopy technique\nin a \rip-chip manner using NanOsc Instrument Phase\nFMR15. Frequency range of the RF signal used in this\nexperiment was between 5 and 17 GHz. To understand\nthe symmetry of anisotropy and quantify the anisotropy\nenergy density, angle dependent FMR measurement has\nbeen performed by applying the magnetic \feld in the\nsample plane with a \fxed frequency of 7 GHz.\nMagnetic hysteresis loops measured using Kerr mi-\ncroscopy for samples A-C are shown in \fgure 1 for various\nvalues of = 0\u000e, 30\u000e, 60\u000eand 90\u000e. Square shaped hystere-\nsis loops have been observed for all three samples for all\nthe angles (0\u000e<\b<360\u000e). This indicates that the mag-\nnetization reversal occurs via nucleation and subsequent\ndomain wall motion16. It is observed that the coercivity\nvaries signi\fcantly among the samples e.g. HCof samplesA, B and C are 2.2, 1.7 and 25 mT, respectively. Kuof\nthese samples are -1.05 \u0002106erg cm\u00003(A), 3.47 \u0002106erg\ncm\u00003(B), and 4.86 \u0002106erg cm\u00003(C).\nDomain images captured near to coercive \feld HCfor\nall three samples at 0\u000e, 30\u000e, 60\u000eand 90\u000eare displayed in\n\fgure 2. The black and gray contrast in the domain im-\nages represent positive and negative magnetized states,\nrespectively. From the domain images it can be con-\ncluded that magnetization reversal occurs via domain\nwall motion. Sample A (\fgure 2 a to d) exhibits large\nwell-de\fned stripe domains which indicates the presence\nof weak magnetic anisotropy. These stripe domains are\nfound to be tilted for e.g. \b = 30\u000eand 60\u000e(\fgure 2\nb and c). For \b = 90\u000e(\fgure 2d) branched domains\nare observed. In addition to 180\u000edomain wall, sample\nB (\fgure 2 e to h) shows 90\u000edomain walls at certain\nangles of measurement (\fgure S1). Sample C (\fgure 2 i\nto l) shows narrow branched domains and are found to\nbe independent of \b , which indicates that the sample\nC is magnetically isotropic in nature. By comparing the\ndomain images at any particular \b among the samples,\nit is observed that the size (i.e. width) of the domain3\nFIG. 2. Domain images captured for samples A, B and C at angles, 0\u000e, 30\u000e, 60\u000eand 90\u000enear to coercivity are shown. Scale\nbars are shown on the domain images for each sample separately which are valid for the images recorded at other angles for\nthose respective samples.\ndecreases with increasing anisotropy strength.\nIn the following we have investigated the magnetiza-\ntion dynamics of the L10FeNi \flms by FMR. Measured\nFMR spectra (open symbol) for samples A and C at se-\nlective frequencies are shown in \fgure 3 (a) and (b), re-\nspectively. Resonance \feld (H res) and line width (\u0001 H)\nwere extracted by \ftting of Lorentzian shape function\n(equation 1) having anti-symmetric (\frst term) and sym-\nmetric components (second term) to the obtained FMR\nderivative signal17;\nS21=K14\u0001H(H\u0000Hres)\n[4(H\u0000Hres)2+ (\u0001H)2]2\n\u0000K2(\u0001H)2\u00004(H\u0000Hres)2\n[4(H\u0000Hres)2+ (\u0001H)2]2\n+ (slopeH ) +Offset(1)\nwhere S21is transmission signal, K1andK2are co-\ne\u000ecient of anti-symmetric and symmetric component,\nrespectively, slope H is drift value in amplitude of the\nsignal. Frequency ( f) dependence of Hresand \u0001 Hare\nplotted in \fgure 3 (a) and (b), respectively. From the\nfvs.Hresplot, parameters such as Lande g-factor, ef-\nfective demagnetization \feld (4 \u0019Meff), anisotropy \feld\nHKwere extracted by \ftting it to Kittel resonance\ncondition18;\nf=\r\n2\u0019q\n(HK+Hres)(HK+Hres+ 4\u0019Meff) (2)\nwhere\r=g\u0016B=~is gyromagnetic ratio, \u0016Bas Bohr\nmagneton, ~as reduced Plancks constant. The values ofdamping constant \u000bfor all three samples were extracted\nby using the equation17{19;\n\u0001H= \u0001H0+4\u0019\u000bf\n\r(3)\nwhere \u0001H0is called as inhomogeneous line width\nbroadening. Values of all the \ftting parameters obtained\nby using equations (1) and (2) are given in table 1. With\nincrease in anisotropy strength of the samples, 4 \u0019Meff\ndecreases whereas \u000bincreases. Sample A exhibits lowest\nvalue of\u000b, 4:9\u000210\u00003which is the same order with nor-\nmal FeNi alloy thin \flm ( \u000b= 1:2\u000210\u00003)20. The value\nof inhomogeneous line width broadening \u0001 H0, which de-\npends on the quality of the thin \flm17, is highest for\nsample C and lowest for sample A.\nApart from frequency dependent FMR, angle depen-\ndent FMR measurements at a \fxed frequency of 7 GHz\nwere performed to analyze the anisotropy energy density\nas well as symmetry of all three samples. Figure 4 shows\nthe angle dependent Hresplot for all three samples. Sam-\nples A and B clearly show the presence of mixed cubic\nand uniaxial anisotropies with minima in Hresapproxi-\nmately at angles 0\u000e, 90\u000e, 180\u000eand 270\u000e. From the an-\ngle dependent Hresplot, the value of uniaxial and cubic\nanisotropy energy constants have been estimated by \ft-\nting those data to solution of LLG equation that includes\nthe cubic and uniaxial anisotropies and it is written as21;4\nFIG. 3. FMR spectra of (a) sample A and (b) sample C measured at frequency 5 17 GHz. The solid lines in (a) and (b) are\n\ftted with equation 1. (c) fvs.Hresand (d) \u0001Hvs.fplots for samples A C extracted from the FMR frequency dependent\nspectra. The lines in (c) and (d) are the best \fts to the equations (2) and (3), respectively.\nTABLE I. The Parameters extracted from the \ftting of FMR experimental data (Figure 3) of all three samples using the\nequations (2) and (3)\nSample H K(Oe) 4\u0019Meff g-factor \u000b \u0001H0(Oe)\nA 49.41 \u00060.82 18279 \u0006365 1.97 \u00060.01 0.0049 \u00060.0001 42.85 \u00060.65\nB -52.39 \u00064.64 8900 \u0006193 1.93 \u00060.01 0.0277 \u00060.0007 66.89 \u00067.13\nC 680.29 \u0006106.88 3509 \u0006106 1.98 \u00060.01 0.0680 \u00060.0019 143.67 \u000616.20\nf=\r\n2\u0019\u0014\nH+2K2\nMscos2\u001e\u00004K4\nMscos4\u001e\u0015\n\u0002\u0014\nH+ 4\u0019Ms+2K2\nMscos2\u001e\u0000K4\nMs(3 +cos4\u001e)\u00151=2\n(4)\nwhere, K2andK4are cubic and uniaxial anisotropy\nenergy density constants, respectively, MSis saturation\nmagnetization, His the applied magnetic \feld, \b is the\nangle between applied \feld direction and easy axis of\nthe sample. The angle dependence of Hresis \ftted to\nequation 4. From the \ft, value of MS,K2andK4have\nbeen extracted and are shown in table 2. It has been\nfound that both cubic and uniaxial anisotropy energy\nconstants of sample A are greater than that of sample B\nby one order of magnitude. Further in sample A the ratioTABLE II. The parameters extracted from the \ftting of angle\ndependent FMR experimental data of two samples using the\nequation (4)\nSamplesMs(emu cm\u00003)K2(erg cm\u00003)K4(erg cm\u00003)\nA 1645 1.66 \u0002104-2.16\u0002104\nB 1026.8 4.8 \u0002103-1.2\u0002103\nC Magnetically isotropic behaviour\nbetween the cubic to uniaxial anisotropy is about 0.75\nwhereas for sample B the ratio is 4. Therefore for sample\nB the cubic anisotropy is four times stronger than the\nuniaxial anisotropy. This is the reason of occurrence of\n90\u000edomain walls for sample B as shown in Fig. S122. In\ncomparison to samples A and B, sample C shows isotropic\nbehavior which is in consistent with the results obtained5\nFIG. 4. Anisotropy symmetry plot and the \fts for (a) sample A, (b) sample B and (c) sample C measured using FMR by\nkeeping the frequency \fxed at 7 GHz. The red solid lines are the best \fts to equation (4).\nfrom Kerr microscopy.\nL10-FeNi thin \flms (samples A, B and C) show dif-\nferent strength of anisotropy, which were fabricated by\ndi\u000berent deposition processes in a MBE system. Do-\nmain images reveals that magnetization reversal for all\nthree samples occur via nucleation and subsequent do-\nmain wall motion. We have observed lowest values of \u000b\n\u00184:9\u000210\u00003for the sample A which shows moderate\nmagnetic anisotropy. Angle dependent FMR measure-\nment shows the mixed cubic and uniaxial anisotropy in\nsamples A and B, while sample C exhibits magnetically\nisotropic behavior. Our work demonstrates that vari-\nable anisotropic L10-FeNi thin \flms can be fabricated in\nwhich the damping constant and magnetization reversal\ncan be tuned. These results may be useful for future\nspintronics based applications.\nAcknowledgements: We acknowledge the \fnancial sup-\nport by department of atomic energy (DAE), Depart-\nment of Science and Technology (DST-SERB) of Govt.\nof India,DST-Nanomission (SR/NM/NS-1018/2016(G))\nand DST, government of India for INSPIRE fellowship.\nREFERENCES\n1Z. Z. Bandic and R. H. Victora, Proceedings of the IEEE 96,\n1749 (2008).\n2S. Bedanta and W. Kleemann, Journal of Physics D: Applied\nPhysics 42, 013001 (2008).\n3D. Weller and A. Moser, IEEE Transactions on magnetics 35,\n4423 (1999).\n4M. Kotsugi, M. Mizuguchi, S. Sekiya, M. Mizumaki, T. Kojima,\nT. Nakamura, H. Osawa, K. Kodama, T. Ohtsuki, T. Ohkochi,\net al. , Journal of Magnetism and Magnetic Materials 326, 235\n(2013).\n5S. Goto, H. Kura, E. Watanabe, Y. Hayashi, H. Yanagihara,Y. Shimada, M. Mizuguchi, K. Takanashi, and E. Kita, Scienti\fc\nreports 7, 13216 (2017).\n6T. Klemmer, C. Liu, N. Shukla, X. Wu, D. Weller, M. Tanase,\nD. Laughlin, and W. So\u000ba, Journal of magnetism and magnetic\nmaterials 266, 79 (2003).\n7T. Kojima, M. Ogiwara, M. Mizuguchi, M. Kotsugi, T. Ko-\nganezawa, T. Ohtsuki, T.-Y. Tashiro, and K. Takanashi, Journal\nof Physics: Condensed Matter 26, 064207 (2014).\n8K. Takanashi, M. Mizuguchi, T. Kojima, and T. Tashiro, Journal\nof Physics D: Applied Physics 50, 483002 (2017).\n9M. Mizuguchi, S. Sekiya, and K. Takanashi, Journal of Applied\nPhysics 107, 09A716 (2010).\n10M. Kotsugi, M. Mizuguchi, S. Sekiya, T. Ohkouchi, T. Kojima,\nK. Takanashi, and Y. Watanabe, in Journal of Physics: Con-\nference Series , Vol. 266 (IOP Publishing, 2011) p. 012095.\n11K. Mibu, T. Kojima, M. Mizuguchi, and K. Takanashi, Journal\nof Physics D: Applied Physics 48, 205002 (2015).\n12M. Ogiwara, S. Iihama, T. Seki, T. Kojima, S. Mizukami,\nM. Mizuguchi, and K. Takanashi, Applied Physics Letters 103,\n242409 (2013).\n13T. Kojima, M. Mizuguchi, T. Koganezawa, K. Osaka, M. Kot-\nsugi, and K. Takanashi, Japanese Journal of Applied Physics\n51, 010204 (2011).\n14\\EVICOmagnetics,\" http://www.evico-magnetics.de/\nmicroscope.html .\n15\\NanOsc FMR Spectrometers,\" https://www.qdusa.com/\nproducts/nanosc-fmr-spectrometers.html .\n16S. Mallick, S. Bedanta, T. Seki, and K. Takanashi, Journal of\nApplied Physics 116, 133904 (2014).\n17B. B. Singh, S. K. Jena, and S. Bedanta, Journal of Physics D:\nApplied Physics 50, 345001 (2017).\n18C. Kittel, Physical Review 73, 155 (1948).\n19B. Heinrich, J. Cochran, and R. Hasegawa, Journal of Applied\nPhysics 57, 3690 (1985).\n20Z. Zhu, H. Feng, X. Cheng, H. Xie, Q. Liu, and J. Wang, Journal\nof Physics D: Applied Physics 51, 045004 (2018).\n21S. Pan, T. Seki, K. Takanashi, and A. Barman, Physical Review\nApplied 7, 064012 (2017).\n22S. Mallik, N. Chowdhury, and S. Bedanta, AIP Advances 4,\n097118 (2014).\nSUPPLEMENTARY MATERIAL6\nFIG. 5. Magnetic domain images for sample B at various \felds\nclose to the reversal \feld. It shows that the magnetization\nreversal is happening through 90\u000edomain wall in sample B." }, { "title": "1901.05777v1.Influences_of_interfacial_oxidization_on_surface_magnetic_energy__magnetic_damping_and_spin_orbit_torques_in_Pt___ferromagnet___capping_structures.pdf", "content": "1 Influences of interfacial oxidiza tion on surface magnetic energ y, magnetic \ndamping and spin-orbit-torques in Pt / ferromagnet / capping st ructures \n \nD. J. Lee1,2, W. M. Jeong2,3, D. H. Yun2,4, S. Y . Park5, B.-K. Ju4, K.-J. Lee1,3, H. C. Koo1,2, \nB.-C. Min2,6, and O. J. Lee2* \n1KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, Korea \n2Center for Spintronics, Korea Institute of Science and Technology, Seoul 02792, Korea \n3Department of Materials Science and Engineering, Korea University, Seoul 02841, Korea \n4Department of Electrical Engineering, Korea University, Seoul 02841, Korea \n5Spin Engineering Physics Team, Korea Basic Science Institute, Daejeon 34133, Korea \n6Division of Nano and Information Technology, KIST School, Korea University of Science and Technology, \nSeoul 02792, Korea \n \nWe investigate the effect of capping layer (CAP) on the interfa cial magnetic anisotropy \nenergy density ( KS), magnetic damping (α), and spin-orbit torques (SOTs) in heavy -metal (Pt) \n/ ferromagnet (Co or Py) / CAP (MgO/Ta, HfO x, or TaN). At room temperature (RT) the CAP \nmaterials influence the effective magnitude of KS, which is associated with a formation of \ninterfacial magnetic oxides. The dynamical dissipation paramete r s o f C o a r e c o n s i d e r a b l y \ninfluenced by the CAP (especia lly MgO) while those of Py are no t. This is possibly due to an \nextra magnetic damping via spin-pumping process across the Co/C oO interface and \nincoherent magnon generation (spi n fluctuation) in the interfac ial CoO. It is also observed \nthat both anti-damping and field-like SOT efficiencies vary mar ginally with the CAP in the \nthickness ranges we examined. Our results reveal the crucial ro le of interfacial oxides on the \nperpendicular magnetic anisotropy, magnetic damping, and SOTs. \n \n* ojlee@kist.re.kr 2 Electrical manipulation of magnetization1,2,3,4,5 is of great interest because of the \nprospective application in low -power and high-speed spintronic devices, as well as the \nnotable scientific advancement. Recently, several works have de monstrated that SOTs2,3,4,5 \ninduced by in-plane charge currents can efficiently reverse the magnetization in multilayers \nof heavy-metal(HM)/ferromagnet(FM )/insulator-CAP with strong sp in-orbit coupling \n(SOC)6,7 (see Fig. 1a). The magnitudes of SOTs are generally characteriz ed by measurable \nfigure of merits that correspond to damping-like (DL, τDL) and field-like (FL, τFL) SOT-\nefficiencies, i.e., θDL and θFL, respectively. The dimensionless θDL is also referred to as a spin-\nHall angle2,4 that defines the conversion rati o from charge to spin currents in the HM. \nAlthough most of the research to date has focused on the HM/FM interface at which the \ngeneration of non-equilibrium spin-polarization2,4,7 or transmission8,9 of spin-current ( JS) and \nthe enhancement10 in α through the spin-pumping process occur, the results of se veral recent \nworks11,12,13,14 have suggested that the physical and chemical properties of th e CAP can also \nchange the characteristics of spi n torques via modification of reflection, absorption, and/or \nscattering of the JS at the FM/CAP interface. It is also expected that the differen t enthalpy of \nformation of the materials will dissimilarly promote an intermi xing (stoichiometry) and an \ninterfacial formation of magnetic oxides, which may change the magnetic and spintronic \nproperties. For instance, Ref. [11] investigated the process by which the oxide interface \nmodifies the τFL v i a t h e i n s e r t i o n o f a n u l t r a t h i n o x i d i z e d H f l a y e r a t t h e F e C oB/MgO \ninterface in perpendicularly magnetized HM/FeCoB/MgO heterostru ctures. Our previous \nwork15 demonstrated that the insertio n of an ultrathin magnetic dusti ng layer between FM \nand MgO layers can play an important role in the determination of α. Nevertheless, there is \nstill a dearth of experimental work studying how the FM/CAP int erface contributes to KS, α, \nθDL, and θFL in HM/FM/CAP heterostructures. 3 In this work, we study the effect s of interfacial magnetic oxid e formation on the magnetic \nand spintronic properties by examining six different series of layer stacks consisting of \nPt/FM/CAP (Fig. 1a). The FM is either Co or Py (=Ni 79Fe21) and the CAP is insulating \nMgO/Ta, HfO x or metallic TaN. As illustrated in Fig. 1(b), the presence of interfacial oxide \ncan give rise to a strong modification of the interfacial magne tic-anisotropy-energy density, \nmagnetic damping, and SOTs. We fi nd, by utilizing X-ray photoel ectron spectroscopy (XPS) \nand spin-torque ferromagnetic resonance (ST-FMR) measurements, that (i) the effective KS \nbecomes deteriorated with a subs tantial formation of interfacia l magnetic oxides; KS(MgO-\nCAP) < KS(HfO x-CAP) < KS(TaN-CAP) for both Co and Py samples. (ii) The α(Co) is \nsignificantly influenced by the CAP whereas the α(Py) is relati vely unaffected by the CAP \nmaterial; α(Pt/Co/HfO x) ≈ α (Pt/Co/TaN) << α(Pt/Co/MgO) whereas α(Pt/Py/MgO) ≈ \nα(Pt/Py/HfO x) ≈ α (Pt/Py/TaN). (iii) Both the θDL and θFL are weakly dependent on the CAP \nmaterials. Our results confirm that, though the SOTs originate primarily from the Pt/FM \ninterface, the perpendicular magnetic anisotropy (PMA) and the magnetic damping are \nsignificantly influenced by the interfacial formation of antiferromagnetic (AF) oxides. \nSix series of layer stacks were p repared on thermally oxidized Si substrates (Si/SiO x) \nusing dc/rf magnetron sputtering from two-inch planar targets a t room temperature (RT). The \nstructure consists of substrate/Pt(5)/FM( tFM)/CAP , where the FM is either Co or Py and the \nCAP is either MgO(2)/Ta(2), HfO x(3) or TaN(3) (nominal thic knesses in nm). The MgO layer \nis used as a standard tunnel ba rrier in MRAM technology because it facilitates very effective \nelectrical readout of magnetic cha nge. Therefore, it is necessa ry to investigate its influence \nfor a given HM/FM bilayer. The HfO x-CAP is an alternative oxide for comparison to MgO-\nCAP (MgO/Ta). The TaN-CAP is tested as a representative non-oxi de because it is stable, \nmetallic but also highly resistive. The thickness of the Co ( tCo) layer is varied from 3 nm to \n15 nm and that of the Py layer ( tPy) is varied from 2 nm to 10 nm. All of the sputtered 4 materials were deposited on the substrate with an oblique orien tation except for the MgO \ntarget that was faced towards it. The base pressure of the sput tering chamber was lower than \n5 × 10−8 Torr, and the deposition rates were lower than 0.5 Å /s. \nFor the ST-FMR measurement, optical lithography and ion-milling were used to pattern \nmultilayer films onto rectangular shaped bars with 15 μm width (ݓ )and 50 μm length ( ݈ .)In \na subsequent process step, a waveguide contact made of Ti (10 n m)/Au (100 nm) was defined \non top of the samples to facilitate the passage of a RF current through the devices. No post-\nannealing was carried out as the temperature ( T) of the samples was kept lower than 110 oC \nduring the fabrication process. It should be noted that the Ta( 2) protective CAP is expected to \nbe fully air-oxidized, so tha t both MgO(2)/Ta(2) and HfO x(3) are considered as insulators, i.e., \nno charge and spin currents flo w through the la yers (see Fig. 1 a). The TaN(3) layer is metallic \nbut since its resistivity is at least one or two orders of magn itude higher than that of Pt and \nF M ( ≈ 1 5 - 3 0 µ Ω-cm), the current shunting thro ugh the TaN layer is negligible in the ST-\nFMR analysis, which will be discussed in the following section. All the measurements were \ndone at RT except for the T-dependence of hysteresis loops. \nThe formation of the interfacial FM-oxide, which are dependent on the used CAP, was \nidentified using XPS on un-patter ned films. In-situ ion-etching was used to remove most of \nthe CAPs before investigating the composition of magnetic layer s. Figure 2a shows the Co \n2p3/2 XPS spectral region from the Pt/Co(5)/CAP samples. All of the spectra exhibit primary \npeaks at ~778 eV, confirming the existence of metallic Co. The secondary peak at ~780 eV \ncorresponding to CoO 2p 3/2 shows a variation in the amplitude depending on the CAP \nm a t e r i a l s . T h e f i l m w i t h t h e M g O - C A P h a s t h e l a r g e s t p e a k a t 7 8 0 eV, indicating the \npresence of a significant CoO at the Co/MgO interface; the film with the HfO x-CAP exhibits \na smaller peak, revealing less fo rmation of interfacial CoO; th e sample with the TaN-CAP \nshows no distinguishable peak. The Ni 2p 3/2 XPS spectral from the Pt/Py(5)/CAP films (Fig. 5 2b) show primary and satellite peaks at ~853 eV and ~860 eV, re spectively, both of which \noriginate from the metallic Ni. A secondary peak at ~855 eV cor responding to NiO is \nobserved only with the MgO-CAP whi ch demonstrates the existence of interfacial NiO. No \nnoticeable oxide peaks are observed for the HfO x-CAP and the TaN-CAP. The Fe 2p 3/2 XPS \nspectra also exhibited a very c onsistent dependence of XPS spec tra on the CAP (not shown). \nThe XPS results confirm the pres ence of the interfacial FM-oxid e state, in descending \norder of oxidation, in the MgO-, HfO x-, and TaN-CAP. A comparison of the relative \namplitudes of the secondary peak s between Pt/Co/MgO/Ta and Pt/P y/MgO/Ta films indicates \nthat the degree of oxi dation of Py is less than that of Co. Thi s is possibly due to the difference \nof the enthalpy of formation of oxides with each metallic eleme nt (Hf, Mg, Co, Ni, and Fe). \nFor instance, HfO 2 is a more stable oxide than MgO, since the enthalpy of formati on of HfO 2 \n(≈ −1120 kJ/mol) is much larger in magnitude than that of MgO ( ≈ −600 kJ/mol). The \nformation of the interfacial FM-oxide is relatively easier with the MgO-CAP than the HfO x-\nCAP. The TaN-CAP protects the FM surfaces from the oxidation as expected. \nW e e x a m i n e d t h e e f f e c t o f C A P o n t h e m a g n e t i c p r o p e r t i e s o f P t / FM(tFM)/CAP. The \nsaturation magnetization ( MS) and magnetic dead-layer thickness ( td) were characterized for \nthe six systems using a vibrating sample magnetometer (VSM). Th e effective thickness of \nFM is defined as tFMeff = t FM − tFMd. Figures 3a–b show the measured moment (per unit area) \nof the un-patterned films as a function of tCo or tPy. The linear fits provide the ( MS, td) of Co ≈ \n(1200 emu/cc, 0.4 nm), (1400 emu/cc, 0.3 nm) and (1430 emu/cc, 0.7 nm) for the MgO-, \nHfO x- and TaN-CAP respectively. In addition, the ( MS, td) of Py ≈ (870 emu/cc, 1.0 nm), \n(650 emu/cc, 0.0 nm) and (760 emu/cc, 0.5 nm) for the MgO-, HfO x- and TaN-CAP \nindividually (see Tabl e. 1). Note that the MS o f C o i n t h e P t / C o / M g O / T a l a y e r i s ~ 1 5 % \nsmaller than those in the Pt/Co/HfO x, and Pt/Co/TaN layers. This might be due to a \nsignificant Co-oxidation in Pt/Co/ MgO/Ta, as revealed by our XP S analysis, or to the 6 diffusion of Mg, Oxygen or Ta into the Co bulk16. The CAP material influences the MS of Py \nas well. \nNext, we systematically measured the frequency-dependent ST-FMR17,18,19 spectra from \nall the devices of the six serie s of Pt/FM/CAP systems. The spe ctra provide fitted parameters \nthat are used to quantify the magnitudes of the effective demag netization field ( 4πMeff), \nmagnetic damping constant (α), i nhomogeneous linewidth-broadeni ng (∆H0) and the DL- and \nFL-SOT efficiencies ( θDL and θFL) for each device. Figures 3c−d illustrate the 4πMeff of all the \ndevices as a function of 1/ tCoeff or 1/ tPyeff. The 4πMeff of both Co and Py decreases with \nincreasing 1/tFMeff regardless of the CAP. This is due to a presence of a strong KS t h a t \noriginates from the strong Pt 5 d – (Co,Fe) 3 d hybridization of the Pt/FM interface20 and in \npart, from the (Co,Fe) 3 d – O 2 p hybridization21 at the FM/oxide interface. The effective KS \n(Kseff) is estimated from the slope of the measured 4πMeff v s 1/tFMeff using the relation \n∂ሺ4ܯߨ ሻ/∂ሺ1/ݐிெሻൌെ ሺ 2 ܭ ௌ/ܯௌሻ. The obtained Kseff (Pt/Co/CAP) is as large as 1.0 \nerg/cm2, 1.2 erg/cm2 and 1.4 erg/cm2 for the MgO-, HfO x-, and TaN-CAP respectively. In \naddition, Kseff (Pt/Py/CAP) is as large as 0.16 erg/cm2, 0.22 erg/cm2, and 0.29 erg/cm2 for the \nsame CAPs. Notably, the experimental Kseff (Pt/Co/MgO/Ta) is very close to the previously \nreported20 KS ≈ 0.8–1.1 erg/cm2 in Pt/Co/MgO. \nWe notice an apparent dependence of Kseff on the CAP: Kseff (MgO-CAP) < Kseff (HfO x-\nCAP) < Kseff (TaN-CAP) for both Co and Py. The strength of PMA increases wi th a reduced \nformation of interfacial magnetic oxides. This is consistent wi th a recent work22 in which the \ninsertion of an ultrathin Hf lay er between FeCoB (Py) and MgO l ayers reduces the oxidation \nof FeCoB (Py) and thereby enhanc es the PMA. The reduction of th e PMA is also observed \nwhen there is an electric field induced O2- migration toward the Co layer in a Co/GdO x \nbilayer23,24. Based on these results, we cautiously assert that the interfa cial oxidation of the \nFM compensates the surface magnetic anisotropy energy provided a t t h e P t / F M . T h e 7 interfacial oxidation of FM deteri orates the PMA in the Pt/FM/C AP heterostructures. \nThe CAP material significantly in fluences the magnitudes of α a nd ∆H0 in Pt/Co/MgO/Ta \n(not in Pt/Py/MgO/Ta). Figures 4a−d present the obtained α and ∆H0 as a function of 1/ tCoeff \nor 1/ tPyeff where the thickness dependences of α and ∆H0 are quite different between Co and \nPy; both α and ∆H0 increase with decreasing tFMeff, but the increase is much more pronounced \nfor the Pt/Co/MgO than for the Pt/Py/MgO. For the thinnest Co s ample ( tCo = 3 nm), the \nPt/Co/MgO has α ≈ 0.06 which is at least three times as large a s the α of the other CAPs (≈ \n0.02). The α of different samples gets closer to each other as tCoeff increases. Within the \nexamined tFM ranges, α(Pt/Co/MgO) >> α(Pt/Co/HfO x) ≈ α(Pt/Co/TaN); α(Pt/Py/MgO) ≈ \nα(Pt/Py/HfO x) ≈ α(Pt/Py/TaN). Moreover, the α is proportional to 1/ teff2 for the Pt/Co/MgO \nsystem, whereas it is quasi-linearly proportional to 1/ teff for the rest of the system. A similar \nthickness dependence is observed with ∆H0. The results suggest that the Pt/Co/MgO system \nmust have different origins for the enhancement of α and ∆H0. \nThere are two interfaces in Pt/FM/oxide systems which influence α and ∆H0 . The Pt/FM \ninterface contributes the enhancements in α and ∆H0 not only for Pt/Py/MgO/Ta but also for \nPt/Co/CAP with the HfO x and TaN-CAPs, where both CAP mate rials negligibly contribute t o \nthe magnitudes of α and ∆H0. The major spin-dissipation channel is the spin-pumping (SP)10 \neffect across the Pt/FM interface, which is clearly observed wi th the samples having a thin \ntFMeff. In contrast, the FM/oxide interface considerably influences t he magnitudes of α and \n∆H0 for the Pt/Co/MgO, implying that the Co/MgO interface develops an additional \ndissipation of spin angular momentum. \nThe XPS and ST-FMR results show that the existence of a signifi cant CoO at the \nCo/MgO interface enhances α and ∆H0. The most probably mechanism for the additional \nrelaxation is a two magnon scattering (TMS) process: the inhomo geneous contribution ( ∆H0) \nbecomes increasingly dominant for a thinner tCoeff. Below the Néel temperature ( TN), the 8 interfacial magnetic oxides (CoO, NiO, and Fe 2O3) b e c o m e A F , b e c a u s e o f t h e s u p e r -\nexchange interaction in which the spins of the 3d electrons in FM ions are ordered, yet \noriented with respect to each other via the non-magnetic 2p ele ctrons in O2-. The AF-oxide \nopens additional relaxation pathwa ys and spatially non-uniform spin dynamics during the \nmagnetic precession via exchange coupling to the fluctuating sp ins of the AF oxide and the \nslow dragging of the AF-oxide domains. To support the above-mentioned speculation, we need an indepen dent evidence of AF \norder of the interfacial CoO. To investigate the AF order, the magnetic hysteresis loops of \nPt/Co( t\nCo=3 nm)/MgO/Ta film were measured as a function of T, from 400 K to 10 K, using a \nSQUID-magnetometer. As shown in Fig. 5a, the loop is symmetric at RT (300 K). As we \ndecrease T down to cryogenic temperatures (e.g., 30 K) the loop is shifte d (≈ 250 ±120 Oe) \nalong the axis of the applied in-plane magnetic field. This is an indicative of the exchange-\nbias effect between the Co and th e interfacial CoO layers. The exchange bias field ( Hex) \nincreases with a decrease in T (see Fig. 5b), thus confirming the AF nature of the interfacia l \nCoO layer. From the T-dependence of Hex, the blocking temperature ( TB) of exchanged-biased \nCo/CoO is estimated to be approximately 150 ±50 K. The hysteresis loops of the Pt/Co( tCo=3 \nnm)/HfO x film were also measured, but no clear loop shift was observed within the \nmeasurement temperature (from 10 K to RT) and field step (≈ 200 Oe). These observations \nare consistent with the XPS results in which the Pt/Co/HfO x system has less interfacial CoO \nthan the Pt/Co/MgO/Ta. \nA recent work25 has demonstrated that Py thin films with an air-oxidized surfa ce exhibit \na significant increase in α near the magnetic phase transition (TN) of the surface AF oxide \nlayer. This was explained based on a magnetic fluctuation proce ss. The surface AF layer \ndevelops magnon modes by enhancing the absorption of spin-angul ar momentum that is 9 pumped from the oscillating FM layer. Spin fluctuation26,27,28 theory suggests that the SP-\nefficiency is maximized at the TN of the AF-oxide, along with its long tail at higher T. This is \nbecause the magnetic susceptibilit y (χ) of an AF-material is qu ite large even at its \nparamagnetic state (i.e., χ-1 ∝ T+TN). Since TN is in general higher than TB, we expect 150 ±50 \nK < TN (CoO at Co/MgO) < 300 K (≈ TN of bulk29 CoO). Therefore, our CoO in the \nPt/Co/MgO/Ta will exhibit a significant SP-effect across the Co /CoO interface even at RT \nwhere our ST-FMR measurement was done (see Fig. 1b). This can e xplain the substantial \nincrease of damping in the Pt/Co/MgO/Ta. \nOur XPS studies also revealed that the Pt/Py/MgO/Ta has a non-n egligible amount of \ninterfacial NiO and FeO, but the ir degrees of oxidization are m uch less than that of the CoO \nin Pt/Co/MgO/Ta. The interfacial (Ni,Fe)-oxide does not substan tially enhance the α of the \nPt/Py/MgO/Ta at RT (Fig. 4), presumably because the TN of the interfacial NiO and FeO is \nmuch lower than that of CoO 25. The interfacial (Ni,Fe)-oxide would enhance the α as the T \napproaches the TN 25,30. For a more robust explanation of the linewidth broadening due to the \nFM/MgO interface, additional experiments are required such as a study of the FMR-\nlinewidths as functions of a wi de range of angles and frequency , as well as its T-dependence. \nFinally, the effective θDL and θFL were obtained from the six different series of layer stack. \nIn the ST-FMR measurement, the SOT-ratio efficiency ( θratio) is calculated from the \nsymmetric part ( VS) and the anti-symmetric part ( VA) of the FMR response using the \nexpression ߠ௧ൌೄ\nಲ\t\n4ܯߨ௦ݐிெݐ௧ቀ1ସగெ\nுೞቁଵ/ଶ\n, where is the reduced Planck \nconstant, e is the electron charge, Hres is the resonance field, and tPt is the thickness of Pt (= 5 \nnm). According to the ST-FMR theory14,18,31, the VS is proportional to ߬ and the VA is due \nto the sum of ߬ி and the Oersted field torque ( ߬ை). The relation between the effective θDL \nand θFL for a Pt/FM/CAP system is given as follows: 10 ଵ\nఏೌൌଵ\nఏವಽ൬1\nఏಷಽ\n௧ುଵ\nସగெೞ௧ಷಾ൰. ( 1 ) \nThen, the magnitude of θDL (θFL) is calculated from the intercept (slope) of linear fit to the \nplot of 1/θratio vs 1/ tFMeff . This analysis is valid provided that (i) the θDL and θFL do not have a \ndependence on tFMeff in the examined thickness range and (ii) no charge and spin cur rents \nflow through the CAP. Otherwise, the correct analysis should be done with the VS and VA \nmeasurement after a separate calibration of the microwave curre nt through the device using a \nnetwork analyzer14. \nFigures 5c–d present the results of 1/ θratio vs 1/ tCoeff or 1/ tPyeff for the six series of layer \nstacks. Interestingly, regardless of the CAP, the slope for Co samples is negative whereas it is \npositive for Py samples. This indicates that the τFL is opposite to the τOe (i.e., θFL < 0) for the \nPt/Co/CAP, whereas they are in the same direction for the Pt/Py /CAP (θFL > 0 ). F ro m th e \nlinear fits to Eq. (4), the θDL (θFL) for the Pt/Co/CAP systems are estimated to be 0.089 \n(−0.012), 0.074 (−0.016), and 0.082 (−0.012) for the MgO-, HfO x-, and TaN-CAP \nrespectively. In addition, the θDL (θFL) for the Pt/Py/CAP system are evaluated to be 0.084 \n(0.015), 0.052 (0.008), and 0.072 (0.013) for MgO-, HfO x-, and TaN-CAP respectively. We \nconclude that the CAP materials in our stack configurations hav e a minor role in the \ndetermination of θDL and θFL within the experimental accuracy of measurements. Thus the \nresults confirm that the SOTs originate mainly from the Pt/FM i nterface, and the contribution \nfrom the FM/CAP interface is very small. It is worth noting tha t the θDL of Pt/Co/CAP \nobtained by our work are slightly lower than the previously rep orted ones (≈ 0.12–0.14) by \nRef. [19], whereas the magnitudes of θDL of our Pt/Py/CAP are larger than 0.05 from the \nsame reference. Although both work s have utilized the same meas urement method (ST-FMR) \nfrom almost identical stacks, the estimated magnitudes could be different depending on \nwhether the contribution of τFL is taken into consideration or not in the ST-FMR analysis. In \naddition, the values of our θFL (Pt/Py/CAP) and θFL (Pt/Co/CAP) are in agreement with Ref. 11 [31] and Ref. [18], respectively. \nPresumably, the interfacial AF-oxide layer may lead a modificat ion of θDL and θFL with \nthe ultrathin FM ( tFM < 2 nm). The FM layers used in this work are thick enough to a bsorb \nmost of the spin-angular momentum transferred by the JS from the HM/FM interface, because \nthe examined tFM ranges are larger than the penetration depth ( λp) of the JS in the FM layers. \nThe λp is estimated to be 1–2 nm near the Fermi energy in 3d-FMs32,33; Ref. [33] suggests λp \n≈ 1.2 nm for Py, CoFeB, and Co . I n tFM > λp , t h e r o l e o f t h e i n t e r f a c i a l A F - o x i d e l a y e r \nwould be minimal in terms of determining the SOT-efficiencies. However, if tFM < λp, a \nportion of the JS will transmit to the spin-conducting AF-oxide layer11,13, and thereby the \nresultant SOTs might be different. The tFM range from 0.5 to 2 nm is of critical importance \nbecause most of the SOT-driven magnetic switching of perpendicu lar magnetization have \nbeen demonstrated with this tFM range6,7. \nSeveral works have reported field-free34,35,36 or beyond37 SHE-driven perpendicular \nmagnetization reversal in HM/FM/oxide multilayers, with some co rrelation to the FM-\noxidization states34,37. The presence of AF-oxide, if the measurement temperature is b elow \nthe TB of the AF-oxide, would lead to such field-free switching due t o the symmetry breaking \nin the multilayer system. Alternatively, the interfacial transm ission of JS into the AF-oxide \nand reflection back to FM might give rise to non-trivial SOT co ntribution to the FM layer \n(e.g., out-of-plane component of effective field or spin-polari zation). Our results encourage \nfurther theoretical work which takes into account the interfaci al AF-oxide in such \nHM/FM/oxide heterostructures (see Fig. 1b). \nIn summary, we illuminate an additional relaxation pathway of s pin current in \nPt/FM/CAP systems, namely the FM-oxide at the FM/CAP interface. The XPS showed the \npresence of interfacial FM-oxide, and ST-FMR demonstrated the d eterioration of PMA 12 associated with a formation of i nterfacial FM-oxides. The SOT-e fficiencies ( θDL and θFL) \nwere not substantially influenced by the CAP, confirming that b oth the DL- and FL-SOTs \noriginate mainly from the Pt/FM interface. The interfacial oxid e, especially CoO, \nsignificantly influences both α and ∆H0 indicating an extra magnetic damping, for instance, \nspin pumping process across the Co/CoO interface. This implies that the interfacial FM-oxide \nis a decent spin-current conductor, for instance, by incoherent magnon generation at the \ninterfacial CoO, even above its m agnetic ordering temperature ( TN). These results facilitate a \nbetter understanding of the interf acial-oxide contributions on the PMA, magnetic damping, \nand SOTs in Pt/FM/CAP systems. \n \nAcknowledgements \n \nThis work was supported by the National Research Council of Sci ence & Technology (NST) \ngrant (No. CAP-16-01-KIST) and the KIST Institutional Program ( 2E29410). K. -J. L. was \nsupported by the National Research Foundation of Korea (NRF) [N RF-2017R1A2B2006119] \nand KU-KIST School Project. K. -J. L. acknowledges the KIST Ins titutional Program \n(Project No. 2V05750). Experimental data in Fig. 5a is obtained from KBSI SQUID VSM. \n 13 Figure Captions \n \n \n \nFigure 1. (a) Conventional view of magnetization dynamics induced by the Js, and spin-\npumping process in a HM/FM/oxide hetero-structure. The influenc e of the FM/oxide \ninterface has been ignored. (b) A realistic view of the hetero- structure. There is a naturally \nformed AF-oxide layer at the interface between the FM and the o xide. A spin-fluctuation in \nthe AF-oxide layer can provide an extra magnetic damping via th e spin-pumping process \nacross the FM/AF-oxide interface. \n \n \n \n14 \n \n \n \nFigure 2. (a) XPS spectral of Co 2p3/2 from Pt/Co(5)/CAP samples. (b) XP S spectral of Ni \n2p3/2 from Pt/Py(5)/CAP samples (nominal thickness in nm, and P y=Ni 79Fe21). The CAP is \neither MgO/Ta (black), HfO x (blue) or TaN (red). \n \n \n \n \n \n \n15 \n \nFigure 3. (a) Measured moment (per unit ar ea) of un-patterned Pt/Py/CAP films as a function \nof tPy. (b) Measured moment (per unit area) of un-patterned Pt/Co/CAP films as a function of \ntCo. The linear fits provide s aturation magnetization ( MS) and magnetic dead-layer thickness \n(td) for the MgO-, HfO x- and TaN-CAP respectively. (c) Obtained 4πMeff as a function 1/ tPyeff \nfrom ST-FMR devices with Pt/Py/CAP multilayers. (d) Obtained 4πMeff as a function 1/ tCoeff \nfrom ST-FMR devices with Pt/Co/CAP multilayers. \n \n \n \n16 \n \nFigure 4 . (a)-(b) Obtained α and ∆H0 as functions of 1/ tPyeff f r o m S T - F M R d e v i c e s w i t h \nPt/Py/CAP multilayers. (c)-(d) Obtained α and ∆H0 as functions of 1/ tCoeff from ST-FMR \ndevices with Pt/Co/CAP multilayers. \n \n \n \n17 \n \nFigure 5 . (a) Hysteresis loops of the Pt/Co( tCo=3 nm)/MgO/Ta film at 30, 100 and 300 K, \nmeasured by a SQUID-magnetometer. (b) Obtained coercive field ( Hc) and loop center ( Hex) \nas a function of T. (c-d) Obtained 1/ θratio as a function of (c) 1/ tPyeff or (d) 1/ tCoeff for the six \nseries of layer stacks using ST-FMR measurement. The linear fit in the plot of 1/ θratio v s \n1/tFMeff provides the magnitude of θDL (θFL), which can be calculated from the intercept (slope) \nrespectively. The estimated θDL and θFL from all of the six layer stacks are summarized in \nTable. 1. \n \n18 Table \n \n td (nm) MS (emu/cc )KS (erg/cm2) θDL (×10−2) θFL (×10−2)\nPt / Py / MgO/Ta 1.0 ± 0.2 870 ± 30 0.16 ± 0.01 8.4 ± 0.4 1.5 ± 0.1 \nHfO x 0.0 ± 0.2 650 ± 20 0.22 ± 0.01 5.2 ± 0.5 0.8 ± 0.1 \nTaN 0.5 ± 0.1 760 ± 20 0.29 ± 0.01 7.2 ± 0.4 1.3 ± 0.1 \nPt / Co / MgO/Ta 0.4 ± 0.3 1200 ± 50 1.0 ± 0.1 8.9 ± 1.0 −1.2 ± 0.2\nHfO x 0.3 ± 0.3 1400 ± 40 1.2 ± 0.1 7.4 ± 2.4 −1.6 ± 0.5\nTaN 0.7 ± 0.4 1430 ± 60 1.4 ± 0.1 8.2 ± 3.4 −1.2 ± 0.5\n \nTable 1. 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Deorani, D.-H. Yang, W.-S . Noh, J.-H. Park, K.-J. Lee, \nH.-W. Lee, and H. Yang, Spin–orbi t-torque engineering via oxyge n manipulation, Nat. Nano. \n10, 333 (2015). " }, { "title": "2211.07744v2.Magnetization_Dynamics_in_Synthetic_Antiferromagnets_with_Perpendicular_Magnetic_Anisotropy.pdf", "content": "1 \n Magnetization Dynamics in Synthetic Antiferromagnets with Perpendicular \nMagnetic Anisotropy \n \nDingbin Huang1,*, Delin Zhang2, Yun Kim1, Jian-Ping Wang2, and Xiaojia Wang1,* \n1Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA \n2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN \n55455, USA \n \n \nABSTRACT: \nUnderstanding the rich physics of magnetization dynamics in perpendicular synthetic \nantiferromagnets (p-SAFs) is crucial for developing next-generation spintronic devices. In this \nwork, we systematically investigate the magnetization dynamics in p-SAFs combining time-\nresolved magneto -optical Kerr effect (TR -MOKE) measurements with theoretical modeling . \nThese model analyses, based on a Landau -Lifshitz -Gilbert approach incorporating exchange \ncoupling , provide detail s about the magnetization dynamic characteristics including the amplitude s, \ndirections, and phases of the precession of p-SAFs under varying magnetic fields . These model -\npredicted characteristics are in excellent quantitative agreement with TR-MOKE measurements \non an asymmetric p -SAF. We further reveal the damping mechanisms of two precession modes \nco-existing in the p -SAF and successfully identify individual contributions from different sources , \ninclud ing Gilbert damping of each ferromagnetic layer , spin pumping, and inhomogeneous \nbroadening . Such a comprehensive understanding of magnetization dynam ics in p -SAFs, obtained \n \n*Author s to whom correspondence should be addressed : huan1746@umn.edu and wang4940@umn.edu 2 \n by integrating high -fidelity TR -MOKE measurements and theoretical modeling, can guide the \ndesign of p-SAF-based architectures for spintronic applications . \n \nKEYWORDS: Synthetic antiferromagnets; Perpendicular magnetic anisotropy; Magnetization \nDynamics; Time -resolved magneto -optical Kerr effect; Spintronics 3 \n 1 INTRODUCTION \nSynthetic antiferromagnet ic (SAF) structures have attracted considerable interest for \napplications in spin mem ory and logic devices because of their unique magnetic configuration s [1-\n3]. The SAF structures are composed of two ferromagnetic (FM) layers anti -parallelly coupled \nthrough a non -magnetic (NM) spacer, offer ing great flexibilit ies for the manipulat ion of magnetic \nconfigurations through external stimuli (e.g., electric -field and spin-orbit torque , SOT) . This \npermit s the design of new architecture s for spintronic applications , such as magnetic tunnel \njunct ion (MTJ), SOT devices, domain wall devices, sky rmion devices, among others [4-7]. The \nSAF structures possess many advantages for such applications , including fast switching speeds \n(potentially in the THz regimes), low off set fields, small switching current s (and thus low energy \nconsumption) , high thermal stability, excellent resilience to perturbations from external magnetic \nfields, and large turnabilit y of magnetic properties [3,8-16]. \nA comprehensive study of the magnetization dynami cs of SAF structures can facilitate the \nunderstanding of the switching behavior of spintronic devices , and ultimately guide the design of \nnovel device architectures . Different from a single FM free layer, magnetization dynamics of the \nSAF structures involv es two modes of precession , namely high -frequency (HF) and low -frequency \n(LF) modes, that result from the hybridization of magnetizations precession in the two FM layers . \nThe relative phase and precession amplitude in two FM layers can significantly affect the spin-\npumping enhancement of magnetic damping [17], and thus play an important role in determining \nthe magnetization dynamic behaviors in SAFs. Heretofore, the exchange -coupling strength and \nmagnetic damping constant of SAFs have been studied by ferromagnetic resonance (F MR) [18-\n21] and optical metrolog y [22-25]. Most FMR -based experimental studies were limited to SAFs \nwith in -plane magnetic anisotropy (IM A). For device applications, perpendicular magnetic 4 \n anisotropy (PMA) gives better scalability [3,26] . Therefore , the characteristics of magnetization \ndynamics of perpendicular SAF (p-SAF) structures are of much valu e to investigat e. In addition, \nprior studies mainly focus ed on the mutual spin pumping between two FM layers [22,27,28] . A \nmore thorough understanding of the contribution s from various sources, including inhomogeneous \nbroadening [29], remains elusive . \nIn this paper, we report a comprehensive study of the magnetization dynamics of p -SAFs by \nintegrating high -fidelity experiments and theoretical modeling to detail the characteristic \nparameters. These parameters describe the amplitude, phase, and direction of magnetization \nprecess ion of both the HF and LF modes for the two exchange -coupled FM layers in a p -SAF. We \nconduct all -optical time -resolved magneto -optical Kerr effect (TR -MOKE) measurements [30-33] \non an asymmetric p -SAF structure with two different FM layers. The field-dependent amplitude \nand phase of TR -MOKE signal s can be well captured by our theoretical model, which in turn \nprovid es comprehensive physical insights into the magnetization dynamics of p -SAF structures. \nMost importantly, we show that inhomogeneous broadening plays a critical role in determining \nthe effective damping of both HF and LF modes, especially at low fields. We demonstrate the \nquantification of contributions from inhomogeneous broadening and mutual spin pumping (i.e., \nthe exchange of angular momentum between two FM layers via pumped spin currents ) [21] to the \neffect ive damping, enabl ing accurate determination of the Gilbert damping for individual FM \nlayers. Results of this work are beneficial for designing p-SAF-based architectures in spintronic \napplication s. Additionally, this work also serves as a successful example demonstrating that TR-\nMOKE, as an all -optical met rology, is a powerful tool to capture the magnetization dynamics and \nreveal the rich physics of complex structures that involve multilayer coupling . \n 5 \n 2 METHODOLOTY \n2.1 Sample preparation and characterization \nOne SAF structure was deposited onto thermally oxidize d silicon wafers with a 300 -nm SiO 2 \nlayer by magnetron sputtering at room temperature (RT) in a six -target ultra -high vacuum (UHV) \nShamrock sputtering system. The base pressure is below 5×10−8 Torr. The stacking structure of \nthe SAF is: [Si/SiO 2]sub/[Ta(5)/Pd(3)] seed/[Co(0.4)/Pd(0.7)/Co(0.4)] FM1/[Ru(0.6)/Ta(0.3)] NM/ \nCoFeB(1) FM2/[MgO(2)/Ta(3)] capping . The numbers in parentheses denote the layer thicknesses in \nnanometers. After deposition, the sample was annealed at 250 ℃ for 20 minutes by a rapid -\nthermal-annealing process. The two FM layers are CoFeB and Co/Pd/Co layers, separated by a \nRu/Ta spacer, forming an asymmetric p -SAF structure ( i.e., two FM layers having different \nmagnetic properties). The M-Hext loops were characterized by a physical propert y measurement \nsystem (PPMS) with a vibrating -sample magnetometer (VSM) module. The resulting M-Hext loops \nare displayed in Fig. 1(a). Under low out -of-plane fields ( Hext < 500 Oe), the total magnetic \nmoments in two FM layers of the SAF stack perfectly cancel out each other: M1d1 = M2d2 with Mi \nand di being the magnetization and thickness of each FM layer ( i = 1 for the top CoFeB layer and \ni = 2 for the bottom Co/Pd/Co laye r). The spin-flipping field ( Hf ≈ 500 Oe ) in the out -of-plane \nloop indicates the bilinear interlayer -exchange -coupling (IEC) J1 between the two FM layers : J1 = \n−HfMs,1d1 ≈ −0.062 erg cm-2 [34]. The values of Ms,1, Ms,2, d1, and d2 can be found in Table SI of \nthe Supplemental Material (SM) [35]. \n \n2.2 Theoretical foundation of magnetization dynamics for a p -SAF structure \nThe magnetic free energy per unit area for a p -SAF structure with uniaxial PMA can be \nexpressed as [36]: 6 \n 𝐹=−𝐽1(𝐦1⋅𝐦2)−𝐽2(𝐦1⋅𝐦2)2\n+∑2\n𝑖=1𝑑𝑖𝑀s,𝑖[−1\n2𝐻k,eff,𝑖(𝐧⋅𝐦𝑖)2−𝐦𝑖⋅𝐇ext] (1) \nwhere J1 and J2 are the strength of the bilinear and biquadratic IEC. mi = Mi / Ms,i are the normalized \nmagnetization vectors for individual FM layers ( i = 1, 2). di, Ms,i, and Hk,eff, i denote, respectively, \nthe thickness, saturation magnetization, and the effective anisotropy field of the i-th layer. n is a \nunit vector indicating the sur face normal direction of the film. For the convenience of derivation \nand discussion, the direction of mi is represented in the spherical coordinates by the polar angle θi \nand the azimuthal angle φi, as shown in Fig. 1 (b). \nThe equilibrium direction of magne tization in each layer (𝜃0,𝑖,𝜑0,𝑖) under a given Hext is \nobtained by minimizing F in the (𝜃1,𝜑1,𝜃2,𝜑2) space. The magnetization precession is governed \nby the Landau -Lifshitz -Gilbert (LLG) equation considering the mutual spin pumping between two \nFM layers [27,37 -40]: \n𝑑𝐌𝑖\n𝑑𝑡=−𝛾𝑖𝐌𝑖×𝐇eff,𝑖+(𝛼0,𝑖+𝛼sp,𝑖𝑖)\n𝑀s,𝑖𝐌𝒊×𝑑𝐌𝒊\n𝑑𝑡−𝛼sp,𝑖𝑗\n𝑀s,𝑖𝐌𝒊×(𝐦𝐣×𝑑𝐦𝒋\n𝑑𝑡)×𝐌𝒊 (2) \nOn the right -hand side of Eq. (2), the first term describes the precession with the effective field \nHeff,i in each layer, given by the partial derivative of the total free energy in the M space via 𝐇eff,𝑖=\n−∇𝐌𝑖𝐹. The second term represents the relaxation induced by Gilbert damping ( α) of the i-th layer, \nwhich includes the intrinsic ( 𝛼0,𝑖) and spin -pumping -enhanced ( 𝛼sp,𝑖𝑖) damping. For TR -MOKE \nmeasurements, 𝛼0,𝑖 and 𝛼sp,𝑖𝑖 are indistinguishable. Hence, we def ine 𝛼𝑖=𝛼0,𝑖+𝛼sp,𝑖𝑖 to include \nboth terms. The last term in Eq. (2) considers the influence of pumped spin currents from the layer \nj on the magn etization dynamics of the layer i. 7 \n The time evolution of Mi can be obtained by solving the linearized Eq. (2). Details are provided \nin Note 1 of the SM [35]. The solutions to Eq. (2) in spherical coordinates are: \n[𝜃1(𝑡)\n𝜑1(𝑡)\n𝜃2(𝑡)\n𝜑2(𝑡)]=[𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]+[Δ𝜃1(𝑡)\nΔ𝜑1(𝑡)\nΔ𝜃2(𝑡)\nΔ𝜑2(𝑡)]=[𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]+\n[ 𝐶𝜃,1HF\n𝐶𝜑,1HF\n𝐶𝜃,2HF\n𝐶𝜑,2HF] \nexp(𝑖𝜔HF𝑡)+\n[ 𝐶𝜃,1LF\n𝐶𝜑,1LF\n𝐶𝜃,2LF\n𝐶𝜑,2LF] \nexp(𝑖𝜔LF𝑡) (3) \nwith Δ𝜃𝑖 and Δ𝜑𝑖 representing the deviation angles of magnetization from its equilibrium direction \nalong the polar and azimuthal directions . The last two terms are the linear combination of two \neigen -solutions, denoted by superscripts HF (high -frequency mode) and LF (low -frequency mode). \nω is the complex angular frequencies of two modes, with the real and imaginary parts representing \nthe precession angular frequency ( 𝑓/2𝜋) and relaxation rate (1/ τ), respectively. For each mode, \nthe complex prefactor vector [𝐶𝜃,1,𝐶𝜑,1,𝐶𝜃,2,𝐶𝜑,2]𝑇 contains detailed information about the \nmagnetization dynamics. As illustrated in Fig. 1 (c), the moduli, |𝐶𝜃,𝑖| and |𝐶𝜑,𝑖| correspond to the \nhalf cone angles of t he precession in layer i along the polar and azimuthal directions for a given \nmode immediately after laser heating, as shown by Δ𝜃 and Δ𝜑 in Figs. 1 (b-c). The phase \ndifference between Δ𝜃𝑖 and Δ𝜑𝑖, defined as Arg(Δ𝜃𝑖/Δ𝜑𝑖)=Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖) with Arg \nrepresenting the argument of complex numbers, determines the direction of precession. If Δ𝜃𝑖 \nadvances Δ𝜑𝑖 by 90°, meaning Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=90°, the precession is counter -clockwise (CCW) \nin the θ-φ space (from a view against Mi).Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=−90°, on the contrary, suggests \nclockwise (CW) precession [ Fig. 1 (d)]. Further, the argument of 𝐶𝜃,2/𝐶𝜃,1 provides the relative \nphase in two FM layers. Arg(𝐶𝜃,2/𝐶𝜃,1)=0° corresponds to the precession motions in two FM \nlayers that are in -phase (IP) in terms of θ for a given mode. While the out -of-phase (OOP) \nprecession in terms of θ is represented by Arg(𝐶𝜃,2/𝐶𝜃,1)=180° [Fig. 1 (e)]. Given the precession 8 \n direction in each layer and the phase difference between the two FM layers in terms of θ, the phase \ndifference in terms of φ can be automatically determined. \n \n \nFIG. 1 (a) Magnetic hysteresis ( M-Hext) loops of the p -SAF stack. The magnetization is n ormalized \nto the saturation magnetization ( M/Ms). (b) Schematic illustration of the half cone angles (Δ θ and \nΔφ) and precession direction of magnetization. The precession direction is defined from a view \nagainst the equilibrium direction ( 0, φ0) of M. The representative precession direction in the \nschematic is counterclockwise (CCW). (c) The relation between precession half cone angles and \nthe prefactors. (d) The relation between precession direction and the prefactors. (e) The relative \nphase between two FM layers for different prefactor values. \n \nAs for the effective damping 𝛼eff=1/2𝜋𝑓𝜏, in addition to the intrinsic damping ( α0,i) and the \nspin-pumping contribution ( αsp,ii and αsp,ji) considered in Eq. (2), inhomogeneities can also bring \nsubstantial damping enhancement [32,33,41,42] . Here, we m odel the total relaxation rate as \nfollows: \n9 \n 1\n𝜏Φ=−Im(𝜔Φ)+1\n𝜏inhomoΦ (4) \nThe superscript Φ = HF or LF, representing either the high -frequency or low -frequency precession \nmodes. 𝜔Φ includes both the intrinsic and spin -pumping contributions. The inhomogeneous \nbroadening is calculated as: \n1\n𝜏inhomoΦ=∑1\n𝜋|𝜕𝑓Φ\n𝜕𝐻k,eff,𝑖|\n𝑖Δ𝐻k,eff,𝑖+∑1\n𝜋|𝜕𝑓Φ\n𝜕𝐽𝑖|\n𝑖Δ𝐽𝑖 (5) \nwhere the first summation represents the contrib ution from the spatial variation of the effective \nanisotropy field of individual FM layers (Δ Hk,eff, i). The second summation denotes the contribution \nfrom the spatial fluctuations of the bilinear and biquadratic IEC (Δ J1 and Δ J2). According to \nSlonczewski’s “thickness fluctuations” theory, Δ J1 generates J2 [43,44] . Therefore, the fact that J2 \n= 0 for our sample suggests that ΔJ1 is sufficiently small, allowing us to neglect the inhomogeneous \nbroadening from th e fluctuations of both the bilinear and biquadratic IEC in the following analyses . \n \n2.3 Detection of magnetization dynamics \nThe magnetization dynamics of the p -SAF sample is detected by TR -MOKE, which is \nultrafast -laser -based metrology utilizing a pump -probe configuration. In TR -MOKE, pump laser \npulses interact with the sample, initiating magnetization dynamics in magnetic layers via inducing \nultrafast thermal demagnetization. The laser -induced heating brings a rapid decrease to the \nmagnetic anisotropy fields and IEC [45,46] , which changes 𝜃0,𝑖, 𝜑0,𝑖 and initiates the precession. \nThe magnetizati on dynamics due to pump excitation is detected by a probe beam through the \nmagneto -optical Kerr effect. In our setup, the incident probe beam is normal to the sample surface \n(polar MOKE); therefore, the Kerr rotation angle ( 𝜃K) of the reflected probe beam is proportional \nto the z component of the magnetization [47]. More details about the experimental setup can be 10 \n found in Refs. [30,32] . For p -SAF, TR -MOKE signals contain two oscillating frequencies that \ncorrespond to the HF and LF modes (𝑓HF>𝑓LF). The signals are proportional to the change in \n𝜃K and can be analyzed as follows: \nΔ𝜃K(𝑡)=𝐴+𝐵𝑒−𝑡/𝜏T+𝐶HFcos(2𝜋𝑓HF𝑡+𝛽HF)𝑒−𝑡/𝜏HF+𝐶LFcos(2𝜋𝑓LF𝑡+𝛽LF)𝑒−𝑡/𝜏LF (6) \nwhere the exponential term 𝐵𝑒−𝑡/𝜏T is related to the thermal background with 𝜏T being the time \nscale of heat dissipation . The rest two terms on the right -hand side are the precession terms with \nC, f, β, and τ denoting , respectively, the amplitude, frequency, phase, and relaxation time of the \nHF and LF modes. \nAfter excluding the thermal background from TR -MOKE signals, the precession is modeled \nwith the initial conditions of step -function de creases in 𝐻k,eff,𝑖 and 𝐽𝑖, following the ultrafast laser \nexcitation [48]. This is a reasonable approximation since the precession period (~15 -100 ps for \nHext > 5 kOe) is much longer than the time scales of the laser excitation (~1.5 ps) and subsequent \nrelaxations among electrons, magnons, and lattice (~ 1 -2 ps) [49], but much shorter than the time \nscale of heat dissipation -governed recovery (~400 ps). With these initial conditions , the prefactors \nin Eq. ( 3) can be determined (see m ore details in Note 1 of the SM [35]). \nFor our SAF structure, 𝜃K detected by the probe beam contain s weighted contributions from \nboth the top and bottom FM layers: \n𝜃K(𝑡)\n𝜃K,s=𝑤cos𝜃1(𝑡)+(1−𝑤)cos𝜃2(𝑡) (7) \nwhere 𝜃K,s represents the Kerr rotation angle when the SAF s tack is saturated along the positive \nout-of-plane ( z) direction. w is the weighting factor, considering the different contributions to the \ntotal MOKE signals from two FM layers. w can be obtained from static MOKE measurements [50], \nwhich gives 𝑤= 0.457 (see more details in Note 2 of the SM [35]). 11 \n \n3 RESULTS AND DISCUSSION \n3.1 Field -dependent p recession frequencies and equilibrium magnetization directions \nTR-MOKE signals measured at varying Hext are depicted in Fig. 2 (a). The external field is \ntilted 15 ° away from in-plane [θH = 75°, as defined by Fig. 2 (c)] to achieve larger amplitdues of \nTR-MOKE signals [51]. The signals can be fitted to Eq. (6) to extract the LF and HF precession \nmodes. The field -dependen t precession frequenc ies of both modes are summarized in Fig. 2 (b). \nFor simplicity, when analyzing precession frequencies, magnetic damping and mutual spin \npumping are neglected due to its insignificant impacts on precession frequencies. By comparing \nthe experimental data and the prediction of ωHF/2π and ωLF/2π based on E q. (3), the effective \nanisotropy fields and the IEC strength are fitted as Hk,eff, 1 = 1.23 ± 0.28 kOe, Hk,eff, 2 = 6.18 ± 0.13 \nkOe, J1 = −0.050 ± 0.020 erg cm−2, and J2 = 0. All parameters and their determination methods are \nsummarized in Table SI of the SM [35]. The fitted J1 is close to that obtained from the M-Hext \nloops (~−0.062 erg cm-2). The inset of Fig 2 (b) shows the zoom ed-in view of field -dependent \nprecession frequencies around Hext = 8 kOe, where a n anti -crossing feature is observed: a narrow \ngap (~2 GHz) open s in the frequency dispersion curves of the HF and LF modes owing to the weak \nIEC between two FM layers. Without a ny IEC, the precession frequencies of two FM layers would \ncross at Hext = 8 kOe, as indicated by the green dashed line and blue dashed line in the figure. We \nrefer to t hese two sets of crossing frequencies as the single -layer natural frequencies of two FM \nlayers (FM 1 and FM 2) in the following discussions . 12 \n \nFIG. 2 (a) TR -MOKE signals under varying Hext when θH = 75° [as defined in panel (c)]. Circles \nare the experimental data and black lines are the fitting curves based on Eq. (6). (b) The precession \nfrequencies of the HF and LF modes as functions of Hext. Circles are experimental data and solid \nlines are fitting curves. The inset highlights the zoomed -in view of the field -dependent frequencies \naround 8 kOe, where the green dashed line and blue dashed line are the single -layer (SL) \nprecession frequencies of FM 1 and FM 2 without interlayer exchange coupling. (c) Schematic \nillustration of the definition of the equilibrium polar angles ( θ0,1 and θ0,2), and the direction of the \nexternal magnetic field ( θH). The illustration is equivalent to Fig. 1(b) due to symmetry. (d) θ0,1 \nand θ0,2 as functions of Hext. The dash -dotted line plots the difference between the two equilibrium \npolar angles. \n \n13 \n Based on the fitted stack properties ( Hk,eff,1, Hk,eff,2, J1, and J2), the equilibrium magnetization \ndirections in the two layers can be calculated. For SAFs with weak IEC compared with uniaxial \nPMA, the azimuthal angles of the magnetization in two FM layers are always the same as that of \nthe external field at equilibrium status. Therefore, two polar angles will be sufficient to describe \nthe equilibrium magnetization con figuration. Figure 2(c) illustrates the definition of the \nequilibrium polar angles of two FM layers ( θ0,1, θ0,2) and the external field ( θH). The values of θ0,1, \nθ0,2, and the difference between these two polar angles as functions of Hext are shown in Fig. 2(d). \nWhen Hext is low (< 1.6 kOe), magnetic anisotropy and antiferromagnetic coupling are dominant \nand |θ0,1 − θ0,2| is larger than 90 °. As Hext increases, both θ0,1 and θ0,2 approach θH. When Hext is \nhigh (> 15 kOe), the Zeeman energy becomes dominant and both M1 and M2 are almost aligned \nwith Hext. \n \n3.2 Cone angle, direction, and phase of magnetization precession revealed by modeling \nBesides the equilibrium configuration, using sample properties extracted from Fig. 2 (b) as \ninput parameters, the LLG -based modeling (described in section 2.2) also provide s information o n \nthe cone angle, direction, and phase of magnetization precession for each mode ( Fig. 1 ). The \ndiscussion in this section is limited to the case without damping an d mutual spin pumping . They \nwill be considered in Note 4 of the SM [35], sections 3.3, and 3.4. The calculation results are \nshown in Fig. 3 , which are categorized into three regions. At high external fields ( Hext > 1.6 kOe, \nregions 2 and 3), both FM layers precess CCW [ Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=90°], and the polar angles of \nmagnetization in two layers are in-phase [Arg(𝐶𝜃,2/𝐶𝜃,1)= 0°] for the HF mode and out-of-phase \n[Arg(𝐶𝜃,2/𝐶𝜃,1) = 180° ] for the LF mode. This is the reason for the HF mode (LF mode) also \nbeing called the acoustic mode ( optical mode) in the literature [23]. The criterion to differentiate 14 \n region 2 from region 3 is the FM layer that dominat es a given precessional mode (i.e., the layer \nwith larger precession cone angles) . In region 2 (1.6 kOe < Hext < 8 kOe) , the HF mode is \ndominated by FM 2 because FM 2 has larger cone angles than FM 1. This is reasonable since the \nhigher precession frequency is closer to the natural frequency of FM 2 [see Fig. 2(b)] in region 2. \nSimilarly, in region 3, the HF mode is dominated by FM 1 with larger precession cone angles. \nWhen Hext is low (region 1), the angle between two magnetizations is larger than 90° [ Fig. 2 (d)] \nowing to the more dominan t AF-exchange -coupling energy as compared with the Zeeman energy . \nIn this region, magnetization dynamics exhibits some unique features. Firstly, CW [ Arg(𝐶𝜃,𝑖/\n𝐶𝜑,𝑖)=−90°] precession emerges: for each mode, the dominant layer precesses CCW (FM 2 for \nthe HF mode and FM 1 for the LF mode) and the subservient layer precesses CW (FM 1 for the HF \nmode and FM 2 for the LF mode). This is because the effective field for the subservient layer [ e.g., \nHeff,1 for the HF mode, see Eq. (2)] precesses CW owing to the CCW precession of the dominant \nlayer when |𝜃0,1−𝜃0,2|>90° [Fig. 2(d)] . In other words, a low Hext that makes |𝜃0,1−𝜃0,2|>\n90° is a necessary condition for the CW precession. However, it is not a sufficient condition. In \ngeneral, certain degrees of symmetry breaking ( Hk,eff,1 ≠ Hk,eff,2 or the field is tilted away from the \ndirection normal to the easy axis ) are also needed to generate CW precession. For example, for \nsymmetric a ntiferromagnets ( Hk,eff,1 = Hk,eff,2) under fields perpendicular to the easy axis, CW \nprecession does not appear even at low fields (Fig. 2(a) in Ref. [52]). See Note 5 of the SM [35] \nfor more details. Secondly, as shown in Fig. 3 , the precession motions in two FM layers are always \nin-phase for both HF and LF modes; thus, there is no longer a clear differentiation between \n“acoustic mode” and “optical mo de”. Instead, the two modes can be differentiated as “right -handed” \nand “left -handed” based on the chirality [53]. Here, we define the chirality with respect to a \nreference direction taken as the projection of Hext or M2 (magnetization direction of the layer with 15 \n a higher Hk,eff) on the easy axis [ -z direction in Fig. 3 (c)]. Lastly, the shape of the precession cone \nalso varies in different regions. Δ θi and Δφi are almost the same for both modes in region 3, \nindic ating the precession trajectories are nearly circular. While in regions 1 and 2, Δ θi and Δφi are \nnot always equal, suggesting the precession trajectories may have high ellipticities. \n \n \nFIG. 3 The calculated half cone angle, direction, and phase of magnetization precession for (a) the \nHF mode and (b) the LF mode. In the top row, four curves represent the polar and azimuthal half \ncone angles of precession in two FM layers. All half cone angles are normalized with r espect to \nΔθ1. The middle row shows the value of Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖) under different Hext. A value of 90° (−90°) \nrepresents CCW (CW) precession. The bottom row is the phase difference of the polar angles in \ntwo layers. A value of 0° (180°) corresponds to the polar angles of the magnetization in two layers \nare IP (OOP ) during precession. Dashed lines correspon d to the reference case where damping is \nzero in both layers. (c) Schematic illustrations of the cone angle, direction, and phase of \n16 \n magnetization precession for the HF and LF modes in different regions, and their corresponding \ncharacteristics regarding ch irality and phase difference. \n \n3.3 Amplitude and phase of TR -MOKE signals \nActual magnetization dynamics is resolvable as a linear combination of the two eigenmodes \n(the HF and the LF modes ). By taking into account the initial conditions (i.e., laser excitation , see \nNote 1 of the SM [35]), we can determine the amplitude and phase of the two modes in TR -MOKE \nsignals . Figure 4 (a) summarizes the amplitudes of both HF and LF modes [CHF and CLF in Eq. (3)] \nunder different Hext. Noted that the y-axis represents Kerr angle ( θK) instead of the cone angle of \nprecession. The LF mode has a local minimum near 8 kOe, where the two FM layers have similar \nprecession cone angles but opposite phase s for the LF mode [ Fig. 3 (b)]. The amplitude s of both \nmodes decrease with Hext in the high -field region. This is similar to the single -layer case, where \nthe amplitudes of TR -MOKE signals decrease with Hext because the decrease in Hk,eff induced by \nlaser heating is not able to significantly alternate the equilibrium magnetization direction when the \nZeeman energy dominates [51]. The LF mode also has an amplitude peak at low fields ( Hext < 3 \nkOe), where the dominant layer of FM 1 changes its equilibrium direction dramatically with Hext \n(from ~75° to 170°) as shown in Fig. 2(d). \nTo directly compare the amplitudes of TR -MOKE signals and the LLG -based calculations , the \nweighting factor w and the initial conditions are needed. The initial conditions are determined by \n𝐻k,eff,1′,𝐻k,eff,2′, and 𝐽1′, representing the instantan eous effective anisotropy fields and IEC strength \nupon laser heating. These instantaneous properties are different from their corresponding room -\ntemperature values ( Hk,eff,1, Hk,eff,2, and J1). The accurate determination of𝐻k,eff,1′,𝐻k,eff,2′, and 𝐽1′ \ndemands the modeling of the laser heating process as well as the temperature dependence of stack \nproperties, which are challenging. Here, we treat these three variables as adjustable parameters and 17 \n determine their values by fitting the field -dependent amp litudes of TR -MOKE signals , which \nyields 𝐻k,eff,1′𝐻k,eff,1⁄=0.90±0.01, 𝐻k,eff,2′𝐻k,eff,2⁄=0.95±0.01, and 𝐽1′𝐽1⁄=0.83±0.01. \nIt is apparent that the field dependence of TR -MOKE signal amplitude is in excellent agreement \nwith the theoretical modeling , as s hown in Fig. 4 (a). \nFigure 4 (b) shows the calculated half polar cone angles for each mode in each FM layer. In \nTR-MOKE signals, the optical mode (the LF mode in regions 2 and 3) tends to be partially \ncanceled out because the two layers precess out -of-phase. Therefore, compared with Fig. 4 (a), the \ninformation in Fig. 4 (b) better reflects the actual intensity of both modes in FM 1 and FM 2. In Fig. \n4(b), the precession cone angles of both modes in FM 1 (Δ𝜃1HF,Δ𝜃1LF) have local maxima at the \nanti-crossing field (Hext ≈ 8 kOe). On the contrary, Δ𝜃2LF and Δ𝜃2HF of FM 2 have their maxima \neither above or below the anti-crossing field. This is because FM 2 has larger precession amplitudes \n(cone angles) than FM 1 at the anti-crossing field if there is no IEC [the dotted lines of FM 1 (SL) \nand FM 2 (SL) in Fig. 4 (b)]. With IEC, FM 2 with larger cone angles can drive the precession motion \nin FM 1 significantly near the anti-crossing field, where IEC is effective. Subsequently, the \nprecession amplitudes of FM 1 exhibit local maxima as its cone angle peaks at the anti-crossing \nfield [solid lines in Fig. 4(b)]. Also, compared with the uncoupled case [FM 1 (SL) in Fig. 4(b)], \nFM 1 in the SAF structure has a much larger cone angle at the boundary between regions 1 and 2 \n(Hext ≈ 1.6 kOe). This corresponds to the case where FM 1 fast switch ing is driven by Hext, as shown \nin Fig. 2( d). The energy valley of FM 1 created by IEC and uniaxial anisotropy is canceled out by \nHext. As a result, any perturbation in Hk,eff,1 or IEC can induce a large change in 𝜃1. \nBesides amplitude, the phase of TR -MOKE signals [ HF and LF in Eq. (6)] also provides \nimportant information about the magnetization dynamics in SAF [Fig. 4 (c)]. In Fig. 4 (c), the phase \nof the HF mode stays constant around π. However, the LF mode goes through a π-phase shift at 18 \n the transition from region 2 to region 3. Th is phase shift can be explained by the change of the \ndominant layer from region 2 to region 3 for the LF mode [ Fig. 3(c)]. As illustrated in Fig. 4 (d), \nthe LF mode (optical mode in regions 2 and 3) has opposite phases in FM 1 (~0°) and FM 2 (~180°). \nConsidering the two FM layers have comparable optical contributions to TR -MOKE signals ( w ≈ \n0.5), TR -MOKE signals will reflect the phase of the dominant layer for each mode. In region 3, \nFM 2 has larger p recession cone angles than FM 1 for the LF mode ; therefore, LF TR-MOKE signals \nhave the same phase as FM 2 (~180°). However, in region 2, the dominant layer shifts from FM 2 to \nFM 1 for the LF mode. Hence, the phase of LF TR-MOKE signals also change s by ~180° t o be \nconsistent with the phase of FM 1 (~0°). As for the HF mode, since the two layers always have \nalmost the same phase ( ~180°), the change of the dominant layer does not cause a shift in the phase \nof TR -MOKE signals. \nBy comparing Fig. 4 (d) and Fig. 3 (a-b), one can notice that the phase difference between two \nFM layers could deviate from 0° or 180° when damping and mutual spin pumping is considered \n[Fig. 4(d)]. The deviation of phase allows energy to be transferred from one FM layer to the other \nduring precession via exchange coupling [54]. In our sample system, FM 2 has a higher damping \nconstant ( 𝛼1= 0.020 and 𝛼2=0.060); therefore, the net transfer of energy is from FM 1 to FM 2. \nMore details can be found in Note 4 of the SM [35], which shows the phase of TR -MOKE signals \nis affected by Gilbert damping in both layers and the mutual spin pumping . By fitting the phase \n[Fig. 4(c)] and the damping [ Fig. 5(a) ] of TR -MOKE signals simultaneously, we obtained 𝛼sp,12 \n= 0.010 ± 0.004, 𝛼sp,21=0.007−0.007+0.009, 1= 0.020 ± 0.002, and 2 = 0.060 ± 0.008. Nonreciprocal \nspin pumping damping ( 𝛼sp,12≠𝛼sp,21) has been reported in asymmetric FM 1/NM/FM 2 trilayers \nand attributed to the different spin -mixing conductance ( 𝑔𝑖↑↓) at the two FM/NM interfaces [27], \nfollowing 𝛼sp,𝑖𝑗=𝑔𝑖𝜇B𝑔𝑗↑↓/(8𝜋𝑀s,𝑖𝑑𝑖), with 𝑔𝑖 the 𝑔-factor of the i-th layer and 𝜇B the Bohr 19 \n magneton [55]. The above equation neglects the spin -flip scattering in NM and assumes that the \nspin accumulation in the NM spacer equally flows back to FM 1 and FM 2 [37]. However, the \nuncertainties of our 𝛼sp,𝑖𝑗 are too high to justify the nonreciprocity of 𝛼sp,𝑖𝑗 (see Note 3 of the SM \n[35] for detailed uncertainty analyses). In fact, if the spin backflow to FM i is proportional to 𝑔𝑖↑↓, \nthen 𝛼sp,𝑖𝑗=𝑔𝑖𝜇B𝑔𝑖↑↓𝑔𝑗↑↓/[4𝜋𝑀s,𝑖𝑑𝑖(𝑔𝑖↑↓+𝑔𝑗↑↓)] (Eq. 1.14 in Ref. [56]). In this case, the different \nspin-mixing conductance at two FM/NM interfaces ( 𝑔1↑↓≠𝑔2↑↓) will not lead to nonreciprocal \n𝛼sp,𝑖𝑗. Although differences in 𝑔𝑖 and magnetic moment per area ( 𝑀s,i𝑑𝑖) can potentially lead to \nnonreciprocal 𝛼sp,𝑖𝑗, the values of 𝑔𝑖 and 𝑀s,i𝑑𝑖 for the two FM layers are expected to be similar \n(the net magnetization of SAF is zero without external fields). Therefore, nearly reciprocal 𝛼sp,𝑖𝑗 \nare plausible for our SAF stack. Assu ming 𝑔𝑖↑↓ values are similar at the two FM/NM interfaces \n(𝑔1↑↓≈𝑔2↑↓=𝑔↑↓), this yields 𝑔↑↓ =8𝜋𝑀s,𝑖𝑑𝑖𝛼sp,𝑖𝑗/(𝑔𝑖𝜇B) = 1.2 ~ 1.7 × 1015 cm−2. 𝑔↑↓ can also \nbe estimated from the free electron density per spin ( n) in the NM layer: 𝑔↑↓ ≈ 1.2𝑛2/3 [57]. With \nn = 5.2 × 1028 m−3 for Ru [58] (the value of n is similar for Ta [59]), 𝑔↑↓ is estimated to be 1.7 × 1015 \ncm−2, the same order as the 𝑔↑↓ value from TR -MOKE measurements, which justifies the 𝛼sp,𝑖𝑗 \nvalues derived from TR -MOKE are within a reasonable range. The values of 𝛼1 and 𝛼2 will be \ndiscussed in section 3.4. \n 20 \n \nFIG. 4 (a) Amplitudes of TR -MOKE signals a s functions of Hext. The circles and curves represent \nexperimental data and modeling fitting , respectively. (b) The calculated precession half cone \nangles at different Hext. Red curves and black curves represent the cone angles of the HF mode and \nthe LF mode in FM 1 (solid lines) and FM 2 (dash ed lines). Dotted lines are the precession cone \nangles of single -layer (SL) FM 1 and FM 2 without IEC. (c) Phases of TR -MOKE signals at varying \nHext. Circles and curves are experimental data and modeling fitting (𝛼sp,12=0.010, 𝛼sp,21=\n0.007, 𝛼1=0.020, 𝛼2=0.060). (d) Simulated precession phase of the HF mode (red curves) and \nthe LF mode (black curves) in FM 1 (solid lines) and FM 2 (dash ed lines). \n \n3.4 Magnetic damping of the HF and LF precession modes \nIn addition to the amplitude and phase of TR -MOKE signals for the p -SAF stack, the model \nanalyses also provide a better understanding of magnetic damping. Figure 5 (a) shows the effective \ndamping constant ( 𝛼eff=1/2𝜋𝑓𝜏) measured at different Hext (symbols), in comparison with \n21 \n model ing fitting (solid lines). The general Hext dependence of αeff can be well captured by the \nmodel. The fitted Gilbert damping, 1= 0.020 ± 0.002 and 2 = 0.060 ± 0.008 are close to the \nGilbert damping of Ta/CoFeB(1 nm)/MgO thin films (~0.017) [41,60] and Co/Pd multilayers with \na similar tCo/tPd ratio (~0.085) [61]. Other fitted parameters are Δ𝐻k,eff,1=0.26±0.02 kOe, \nΔ𝐻k,eff,2= 1.42±0.18 kOe, 𝛼12sp=0.010±0.004 𝛼21sp=0.007−0.007+0.009. Δ𝐽1 and Δ𝐽2 are set to be \nzero, as explained in Sec. 2.2. More details regarding the values and determination methods of all \nparameters involved in our data reduction are provided in Note 3 of the SM [35]. Dashed lines \nshow the calculated 𝛼eff without inhomogeneous broadening. At high Hext, the difference between \nthe solid lines and dashed lines approaches zero because the inhomogeneous broadening is \nsuppressed. At low Hext, the solid lines are significantly higher than the dashed lines , indicating \nsubstantial inhomogeneous broadening contributions . \nThe effective damping shows interesting features near the anti-crossing field. As shown in \nFig. 5(b), due to the effective coupling between two FM layers near the anti-crossing field, the \nhybridization of precession in two FM layers leads to a mix of damping with contributions from \nboth layers. The effective damping of the FM 1-dominant mode reaches a maximum within the \nanti-crossing region ( 7 Hext 10 kOe) and is higher than the single -layer (SL) FM 1 case. \nSimilarly, the hybridized HF and LF modes at 8.5 kOe exhibit a lower 𝛼eff (~0.073) compared to \nthe SL FM 2 case. eff consists of contributions from Gilbert damping ( 𝛼𝑖), mutual spin pumping \n(𝛼sp,𝑖𝑗, 𝑖≠𝑗), and inhomogeneous broadening ( Δ𝐻k,eff,𝑖 and Δ𝐽𝑖). To better understand the mixing \ndamping behavior, Fig. 5 (c) shows eff after excluding the inhomogeneous contribution ( 𝛼effinhomo). \nCompared to the SL layer c ase (green and blue dashed lines), the HF and LF modes (red and black \ndashed lines) clearly suggest that IEC effectively mixes the damping in two layers around the anti -\ncrossing field. Without the IEC, precession in FM 2 with a higher damping relaxes faster than that 22 \n in FM 1. However, the IEC provides a channel to transfer energy from FM 1 to FM 2, such that the \ntwo layers have the same precession relaxation rate for a given mode. Near the anti -crossing field, \ntwo layers have comparable precession cone angles; therefore, the damping values of the \nhybridized modes are roughly the average of two FM layers. In addition to the static IEC, dynamic \nspin pumping can also modify the damping of individual modes. The black and red solid lines \nrepresent the cases with mutu al spin pumping ( 𝛼sp,12 = 0.01 and 𝛼sp,21 = 0.007). Generally, in \nregions 2 & 3, mutual spin pumping reduces the damping of the HF mode and increases the \ndamping of the LF mode because the HF (LF) mode is near in -phase (out -of-phase). Overall, the \nstatic IEC still plays the essential role for the damping mix near the anti -crossing field. \n \n \nFIG. 5 (a) Effective damping constant under varying Hext. Circles are experimental data. Solid \nlines are fitting curves based on Eqs. (4 -5). Dashed lines denote eff after the removal of \ninhomogeneous -broadening contribution. (b) A zoomed -in figure of panel (a) between 5 kOe and \n15 kOe. Blue and green circles are measured effective damping of the mode dominated by FM 1 \nand FM 2, respectively. Blu e and green dashed lines are the 𝛼eff of FM 1 and FM 2 single layer \nwithout IEC. (c) Effective damping after excluding the inhomogeneous contribution as a function \nof Hext. The HF mode (red curves) and the LF mode (black curves) are represented by solid (o r \ndashed) curves when the mutual spin pumping terms ( 𝛼sp,12 and 𝛼sp,21) are considered (or \nexcluded). The d ashed green and blue lines are the SL cases for FM 1 and FM 2, respectively. \n \n \n \n23 \n 4 CONCLUSION \nWe systematically investigate d the magnetization dynamics excited by ultrafast laser pulses in \nan asymmetric p -SAF sample both theoretically and experimentally. We obtained d etailed \ninformation regarding magnetization dynamics, including the cone angles, directions, and phases \nof spin precession in each layer under different Hext. In particular, the dynamic features in the low -\nfield region (region 1) exhibiting CW precession, were revealed. The r esonance between the \nprecession of two FM layers occurs at the boundary between regions 2 an d 3, where an anti -\ncrossing feature is present in the frequency vs. Hext profile . The dominant FM layer for a given \nprecession mode also switches from region 2 to region 3. The amplitude and phase of TR -MOKE \nsignals are well captured by theoretical modeling . Importantly , we successfully quantified the \nindividual contributions from various sources to the effective damping , which enables the \ndetermination of Gilbert damping for both FM layers. At low Hext, the contribution of \ninhomogeneous broadening to the effective damping is significant. Near the anti-crossing field, \nthe effective damping of two coupled modes contains substantial contributions from both FM \nlayers owing to the strong hybridization via IEC . Although the analyses were made for an \nasymme tric SAF sample, this approach can be directly applied to study magnetization dynamics \nand magnetic properties of general complex material systems with coupled multilayers , and thus \nbenefits the design and optimization of spintronic materials via structural engineering. \n \nAcknowledgements \nThis work is primarily supported by the National Science Foundation ( NSF, CBET - 2226579). \nD.L.Z gratefully acknowledges the funding support from the ERI program (FRANC) “Advanced \nMTJs for computation in and near ra ndom access memory” by DARPA, and ASCENT, one of six 24 \n centers in JUMP (a Semiconductor Research Corporation program, sponsored by MARCO and \nDARPA). J.P.W and X.J.W also appreciate the partial support from the UMN MRSEC Seed \nprogram (NSF, DMR -2011401 ). D.B.H . would like to thank the support from the UMN 2022 -2023 \nDoctoral Dissertation Fellowship. The authors appreciated the valuable discussion with Prof. Paul \nCrowell. \n \nReferences \n[1] R. Chen, Q. Cui, L. Liao, Y. Zhu, R. Zhang, H. Bai, Y. Zhou, G. Xing, F. Pan, H. Yang et \nal., Reducing Dzyaloshinskii -Moriya interaction and field -free spin -orbit torque switching \nin synthetic antiferromagnets, Nat. Commun. 12, 3113 (2021). \n[2] W. Legrand, D. Maccariello, F. Ajejas, S. Collin, A. Vecchiola, K. Bouzehouane, N. \nReyren, V. Cros, and A. Fert, Room -temperature stabilization of antiferromagnetic \nskyrmions in synthetic antiferromagnets, Nat. Mater. 19, 34 (2020). \n[3] D.-L. Zhang, C. Sun, Y. Lv, K. B. Schliep, Z. Zhao, J. -Y. Chen, P. M. 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Magn. 48, 3288 (2012). \n 1 \n Supplement al Material for \nMagnetization Dynamics in Synthetic Antiferromagnets with Perpendicular \nMagnetic Anisotropy \n \nDingbin Huang1,*, Delin Zhang2, Yun Kim1, Jian -Ping Wang2, and Xiaojia Wang1,* \n1Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA \n2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN \n55455, USA \n \nSupplement al Note 1: Analyses of the magnetization precession in each Ferromagnetic (FM) \nlayer \nFor the convenience of derivation, mi is represented in the spherical coordinate s with the polar \nangle θi and the azimuthal angle φi, as shown in Fig. 1(b): \n𝐦𝑖=(sin𝜃𝑖cos𝜑𝑖,sin𝜃𝑖sin𝜑𝑖,cos𝜃𝑖) (S1) \nAccordingly, t he expressi on of Eq. ( 2) in the spherical coordinate s is: \n{ 𝜃̇1=−𝛾1\n𝑑1𝑀s,1sin𝜃1∂𝐹\n∂𝜑1−𝛼1sin𝜃1𝜑̇1+𝛼sp,12sin𝜃2cos(𝜃2−𝜃1)𝜑̇2\n𝜑̇1=𝛾1\n𝑑1𝑀s,1sin𝜃1∂𝐹\n∂𝜃1+𝛼1\nsin𝜃1𝜃̇1−𝛼sp,12\nsin𝜃1𝜃̇2\n𝜃̇2=−𝛾2\n𝑑2𝑀s,2sin𝜃2∂𝐹\n∂𝜑2−𝛼2sin𝜃2𝜑̇2+𝛼sp,21sin𝜃1cos(𝜃1−𝜃2)𝜑̇1\n𝜑̇2=𝛾2\n𝑑2𝑀s,2sin𝜃2∂𝐹\n∂𝜃2+𝛼2\nsin𝜃2𝜃̇2−𝛼sp,21\nsin𝜃2𝜃̇1 (S2) \n \n*Author s to whom correspondence should be addressed : huan1746@umn.edu and wang4940@umn.edu 2 \n where, a dot over variables represents a derivative with respect to time. When Mi precesses around \nits equilibrium direction: \n{𝜃𝑖=𝜃0,𝑖+Δ𝜃𝑖\n𝜑𝑖=𝜑0,𝑖+Δ𝜑𝑖 (S3) \nwith \ni and \ni representing the deviation angles of Mi from its equilibrium direction along the \npolar and azimuthal directions. Assuming the deviation is small, under the first -order \napproximation, the first -order partial derivative of F in Eq. (S2) can be expanded as: \n{ ∂𝐹\n∂𝜃𝑖≈∂2𝐹\n∂𝜃𝑖2Δ𝜃𝑖+∂2𝐹\n∂𝜑𝑖∂𝜃𝑖Δ𝜑𝑖+∂2𝐹\n∂𝜃𝑗∂𝜃𝑖Δ𝜃𝑗+∂2𝐹\n∂𝜑𝑗∂𝜃𝑖Δ𝜑𝑗\n∂𝐹\n∂𝜑𝑖≈∂2𝐹\n∂𝜃𝑖𝜕𝜑𝑖Δ𝜃𝑖+∂2𝐹\n∂𝜑𝑖2Δ𝜑𝑖+∂2𝐹\n∂𝜃𝑗∂𝜑𝑖Δ𝜃𝑗+∂2𝐹\n∂𝜑𝑗∂𝜑𝑖Δ𝜑𝑗 (S4) \nBy substituting Eq. ( S4), Equation ( S2) is linearized as [1]: \n[ Δ𝜃̇1\nΔ𝜑̇1\nΔ𝜃̇2\nΔ𝜑̇2] \n=𝐊[Δ𝜃1\nΔ𝜑1\nΔ𝜃2\nΔ𝜑2] (S5) \nwhere, K is a 4×4 matrix, con sisting of the properties of individual FM layers and the second -\norder derivatives of F in terms of 𝜃1,𝜑1,𝜃2,and𝜑2. Equation (S5) has four eigen -solutions, in the \nform of 𝐶exp(𝑖𝜔𝑡), corresponding to four precession frequencies: ±𝜔HF and ±𝜔LF. A pair of \neigen -solutions with the same absolute precession frequency are physically equivalent. Therefore, \nonly two eigen -solutions need to be considered: \n{Δ𝜃𝑖=𝐶𝜃,𝑖HFexp(𝑖𝜔HF𝑡)\nΔ𝜑𝑖=𝐶𝜑,𝑖HFexp(𝑖𝜔HF𝑡) and {Δ𝜃𝑖=𝐶𝜃,𝑖LFexp(𝑖𝜔LF𝑡)\nΔ𝜑𝑖=𝐶𝜑,𝑖LFexp(𝑖𝜔LF𝑡) (S6) \nAfter r earrange ment , the full solutions in the spherical coordinates are expressed as below (also \nEq. (3) in the main paper). 3 \n [𝜃1(𝑡)\n𝜑1(𝑡)\n𝜃2(𝑡)\n𝜑2(𝑡)]=\n[ 𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2] \n+[Δ𝜃1(𝑡)\nΔ𝜑1(𝑡)\nΔ𝜃2(𝑡)\nΔ𝜑2(𝑡)]=[𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]+\n[ 𝐶𝜃,1HF\n𝐶𝜑,1HF\n𝐶𝜃,2HF\n𝐶𝜑,2HF] \nexp(𝑖𝜔HF𝑡)+\n[ 𝐶𝜃,1LF\n𝐶𝜑,1LF\n𝐶𝜃,2LF\n𝐶𝜑,2LF] \nexp(𝑖𝜔LF𝑡) (S7) \nThe prefactors of these eigen -solutions provide information about magnetization dynamics of both \nthe HF and LF modes. Directly from solving Eq. (S2), one can obtain the relative ratios of these \nprefactors , which are [𝐶𝜑1HF/𝐶𝜃1HF,𝐶𝜃2HF/𝐶𝜃1HF,𝐶𝜑2HF/𝐶𝜃1HF] and [𝐶𝜑1LF /𝐶𝜃1LF ,𝐶𝜃2LF /𝐶𝜃1LF ,𝐶𝜑2LF /𝐶𝜃1LF ]. \nThese ratios provide precession information of each mode, as presented in Fig. 3. \nObtaining the absolute values of [𝐶𝜃,1,𝐶𝜑,1,𝐶𝜃,2,𝐶𝜑,2]𝑇 for each mode requires the initial \nconditions of precession , which i s necessary for fitting the actual precession amplitudes in TR -\nMOKE signals . In TR -MOKE measurements, magnetization precession is initiated by laser \nheating, which reduces the magnetic anisotropy of each FM layer and the interlayer exchange \ncoupling streng th between two FM layers [2]. Considering the laser heating process is ultrafast \ncompared with magnetization precession while the following cooling due to heat dissipation is \nmuch slower than magnetization dynamics, we approximately model the temporal profiles of \neffective anisotropy fields and exchange coupling as step functions. Owing to the sudden change \nin magnetic properties induced by laser heating , magnetization in each layer will establish a new \nequilibrium direction (𝜃0,𝑖′,𝜑0,𝑖′). In other words, M i deviates from its new eq uilibrium direction \nby Δ𝜃𝑖=𝜃0,𝑖−𝜃0,𝑖′, Δ𝜑𝑖=𝜑0,𝑖−𝜑0,𝑖′. Substituting 𝑡=0 to Eq. ( S7), one can get the initial \nconditions for magnetization dynamics: \nΔ𝜃𝑖(𝑡=0)=𝐶𝜃,𝑖HF+𝐶𝜃,𝑖LF=𝜃0,𝑖−𝜃0,𝑖′ \nΔ𝜑𝑖(𝑡=0)=𝐶𝜑,𝑖HF+𝐶𝜑,𝑖LF=𝜑0,𝑖−𝜑0,𝑖′=0 (S8) \nOnce the initial conditions are set, the absolute values of all prefactors can be obtained . \n 4 \n Supplementa l Note 2: Estimation of each layer’s contribution to total TR -MOKE signals \nThe contribution from each FM layer is estimated by static MOKE measurement. According \nto Ref. [3], the resu lt from this method matches well with that from the optical calculation. The \nsample is perpendicularly saturated before the static MOKE measurement. Then the out -of-plane \nM-Hext loop ( Fig. S1) is measured by static MOKE. As shown in the figure, two different \nantiferromagnetic (AF) configurations have different normalized MOKE signals, indicating the \ndifferent contribution s to the total signals by two layers. The weighting factor is calculated by: \n−𝑤+(1−𝑤)=0.085 (S9) \nwhich gives 𝑤=0.457. Considering the relatively small layer thicknesses [FM 1: CoFeB(1), spacer: \nRu(0.6)/Ta(0.3), and FM 2: Co(0.4)/Pd(0.7)/Co(0.4)], it is reasonable that FM 1 and FM 2 make \ncomparable contributions to the total TR -MOKE signals ( i.e., w ≈ 0.5). \n \nFIG. S1 Static MOKE hysteresis loop. Magnetic fields are applied along the out -of-plane direction. \n \n \n \n5 \n Supplemental Note 3: Summary of the parameters and uncertainties for data reduction \nGiven that a number of variables are involved in the analysis, TABLE SI summarizes the major \nvariables discussed in the manuscript, along with their values and determinatio n methods. \nTABLE SI. Summary of the values and determination methods of parameters used in the data \nreduction. The reported uncertainties are one -sigma uncertainties from the mathematical model \nfitting to the TR -MOKE measurement data. \nParameters Values Determination Methods \nHf ~500 Oe VSM \nMs,1 1240 emu cm−3 VSM \nMs,2 827 emu cm−3 VSM \nd1 1 nm Sample structure \nd2 1.5 nm Sample structure \nHk,eff,1 1.23 ± 0.28 kOe Fitted from f vs. Hext [Fig. 2(b)] \nHk,eff,2 6.18 ± 0.13 kOe Fitted from f vs. Hext [Fig. 2(b)] \nγ1 17.79 ± 0.04 \nrad ns−1 kOe−1 Fitted from f vs. Hext [Fig. 2(b)] \nγ2 17.85 ± 0.04 \nrad ns−1 kOe−1 Fitted from f vs. Hext [Fig. 2(b)] \nJ1 −0.050 ± 0.020 \nerg cm−2 Fitted from f vs. Hext [Fig. 2(b)] \nJ2 0 Fitted from f vs. Hext [Fig. 2(b)] \nw 0.457 Static MOKE \n𝐻k,eff,1′/𝐻k,eff,1 0.90 ± 0.01 Fitted from Amp vs. Hext [Fig. 4(a)] \n𝐻k,eff,2′/𝐻k,eff,2 0.95 ± 0.01 Fitted from Amp vs. Hext [Fig. 4(a)] \n𝐽1′/𝐽1 0.83 ± 0.01 Fitted from Amp vs. Hext [Fig. 4(a)] \n𝛼1 0.020 ± 0.002 Fitted from eff vs. Hext [Fig. 5(a)] and vs. Hext [Fig. 4(c)] \n𝛼2 0.060 ± 0.008 Fitted from eff vs. Hext [Fig. 5(a)] and vs. Hext [Fig. 4(c)] \nΔ𝐻k,eff,1 0.26 ± 0.02 kOe Fitted from eff vs. Hext [Fig. 5(a)] \nΔ𝐻k,eff,2 1.42 ± 0.18 kOe Fitted from eff vs. Hext [Fig. 5(a)] \n𝛼sp,12 0.010 ± 0.004 Fitted from eff vs. Hext [Fig. 5(a)] and vs. Hext [Fig. 4(c)] \n𝛼sp,21 0.007−0.007+0.009 Fitted from eff vs. Hext [Fig. 5(a)] and vs. Hext [Fig. 4(c)] \n 6 \n Supplemental Note 4: Impacts of 𝜶𝟏, 𝜶𝟐, and mutual spin pumping on the phase \nWithout damping, the phase difference in the precession polar angles of two FM layers \n[Arg(𝐶𝜃2/𝐶𝜃1)] is always 0° or 180°, as shown in Fig. 3 of the main article. However, this does not \nnecessarily hold if either the damping or mutual spin pumping is considered. The changes in the \nphase difference due to damping are depicted in Fig. S2. When 1 = 2, the phase difference \nbetween two layers stays at 0° or 180° [ Fig. S2(a)], identical to the lossless case ( 1 = 2 = 0) in \nFig. 3. As a result, the initial phase of TR -MOKE signals ( ) also stays at 0° or 180° [ Fig. S2(b)]. \nHowever, when 𝛼1≠𝛼2, Arg(𝐶𝜃2/𝐶𝜃1) deviates from 0° or 180° especially at high fields ( Hext > \n5 kOe) [ Fig. S2(c,e)]. The layer with a higher damping [FM 1 in (c) or FM 2 in (e)] tends to have a \nmore advanced phase at high fields (regions 2 and 3). For example, in Fig. S2(e), 0° < Arg( 𝐶𝜃2/𝐶𝜃1) \n< 180° for both HF and LF modes in regions 2 and 3. The deviation from the perfect in -phase (0°) \nor out -of-phase (180°) condition allows the IEC to transfer energy from the low -damping layer to \nthe high -damping layer, such that the precession in both layers can damp at the same rate [4]. As \na result, the initial phase of the TR -MOKE signals also changes, which opens a negative or positive \ngap at high fields (> 10 kOe) for both modes, as shown in Fig. S2(d,f). This enables us to determine \nthe difference between 1 and 2 by analyzing the in itial phase of TR -MOKE signals. 7 \n \nFIG. S 2 Impact of 𝛼1 and 𝛼2 on the phase without mutual spin pumping. (a,c,e) The phase \ndifference between the polar angles in two layers for HF and LF modes. (b,d,f) The calculated \ninitial phase of TR -MOKE signals for each mode with 1 = 2 = 0.02 (a,b), 1 = 0.06 and 2 = \n0.02 (c,d), and 1 = 0.02 and 2 = 0.06 (e,f). The mutual spin pumping is set as 𝛼sp,12=𝛼sp,21 = \n0 for all three cases. The rest of the parameters used in this calculation can be found in TABLE SI. \n \nThe impact of mutual spin pumping on the precession phase is illustrated in Fig. S3, where \nthree different cases of either the one -way (𝛼sp,12 or 𝛼sp,21) or two -way (both 𝛼sp,12 and 𝛼sp,21) \nspin pumping are considered. A reference case without the consideration of mutual spin pumping \n(1 = 0.02, 2 = 0.06, and 𝛼sp,12= 𝛼sp,21 = 0) is also plotted (dashed curves) for the ease of \ncomparison. In general, it can be seen that mutual spin pumping could also change the phase \ndifference in the precession polar angles of two layers, and thus the initial phase o f TR -MOKE \nsignals noticeably. This can be explained by the damping modification resulting from spin \npumping. In regions 2 and 3, Eq. (2) can be approximately rearranged as: \n \n8 \n 𝑑𝐦𝑖\n𝑑𝑡≈−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+𝛼𝑖𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡−𝐶𝑗\n𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2)𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡\n≈−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+[𝛼𝑖−𝐶𝑗\n𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2)]𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡\n=−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+𝛼̅𝑖𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡 (S10) \nwhere 𝐶𝑗/𝐶𝑖 represents the ratio of the cone angles in the j-th FM layer to the i-th FM layer. 𝐶𝑗/𝐶𝑖 \nis positive for the in -phase mode and negative for the out -of-phase mode. θ0,1 and θ0,2 are the \nequilibrium polar angle s of M1 and M2, as defined in Fig. 2(c). Therefore, the mutual spin -pumping \nterm either enhances or reduces the damping depending on the mode. 𝛼̅𝑖 = 𝛼𝑖−\n𝐶𝑗\n𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2) represents the effective Gilbert damping in the i-th FM layer after \nconsidering the mutual spin -pumping effect. This modification to damping is more significant \nwhen the i-th layer is subservient with a smaller cone angle ( e.g., FM 2 for the HF mode in region \n3), while the j-th layer is dominant with a mu ch larger precession cone angle ( e.g., FM 1 for the LF \nmode in region 3), leading to a large ratio of |𝐶𝑗/𝐶𝑖|. \nIn Fig. S 3(a), only the spin current injected from FM 1 to FM 2 is considered. According to the \nabove analysis, 𝛼sp,21 can only bring noticeable modifications to the damping of FM 2 when FM 1 \nis the dominant layer. Based on Fig. 3 in the main article, the LF mode in region 2 and HF mode \nin region 3 satisfy this condition (FM 1 dominant and FM 2 subservient). As shown in Fig. S 3(a), \nthe phase difference noticeably deviates from the reference case without mutual spin pumping \n(dashed curves) in region 2 for the LF mode (black curves) and in region 3 for the HF mode (red \ncurves). For the LF mode in region 2, the precession motions in two layers are nearly o ut-of-phase \n(negative C1/C2); therefore, the spin pumping from FM 1 enhances the damping in FM 2. Since 1 \n(0.02) is less than 2 (0.06), the spin pumping from FM 1 to FM 2 further increases | 𝛼̅1 − 𝛼̅2| \nbetween the two layers. Consequently, the phase difference shifts further away from 180°. While 9 \n for the HF mode in region 3, 𝛼sp,21 reduces the damping of FM 2 because C1/C2 is positive resulting \nfrom the near in -phase feature of this mode. Hence, | 𝛼̅1 − 𝛼̅2| becomes smaller and the phase \ndifference gets closer to 0°. In Fig. S 3(c), only 𝛼sp,12 is considered, which requires FM 2 as the \ndominant layer (the HF mode in region 2 and LF mode in region 3) for noticeable changes in | 𝛼̅1 \n− 𝛼̅2|. For the HF mode in region 2, spin pumping from FM 2 reduces 𝛼̅1 given that the precession \nmotions in two layers are nearly in phase (positive C2/C1). Therefore, | 𝛼̅1 − 𝛼̅2| increases and the \nphase difference in Fig. S 3(c) shifts further away from 0° in region 2. However, for the LF mode \nin regions 3, the nearly out -of-phase precession in two FM layers (negative C1/C2) increases 𝛼̅1 \nand reduces | 𝛼̅1 − 𝛼̅2|. As a result, the phase difference in Fig. S 3(c) shifts toward 180°. When \nboth 𝛼sp,12 and 𝛼sp,21 are considered [ Fig. S 3(e)], a combined effect is expected for the phase \ndifference with noticeable changes for both the HF and LF modes in regions 2 and 3. \nThe impacts of mutual spin pumping on the phase difference between the HF and LF modes \nare reflected by the initial phase of TR -MOKE signals [ in Fig. S 3(b,d,f)]. Compared with the \nreference case without mutual spin pumping (dashed curves), the introduction of mutual spin \npumping tends to change the gap in between the two modes. As shown in Fig. S3(e,f), the values \nof two mutual -spin-pumping induced damping terms are chosen as 𝛼sp,12 = 0.013 and 𝛼sp,21 = \n0.004, such that the gap of the initial phase of TR -MOKE signals is closed at high fields \n(region 3). Therefore, the initial phase of TR -MOKE signals provides certain measurement \nsensitivities to 𝛼sp,12 and 𝛼sp,21, which enables us to extract the values of 𝛼sp,𝑖𝑗 from \nmeasurement fitting. Here, we acknowledge that the measurement sensitivity to 𝛼sp,𝑖𝑗 from TR -\nMOKE is limited, which subsequently leads to relatively large error bars for 𝛼sp,𝑖𝑗 (see Table SI). \n 10 \n \nFIG. S3 Impact of mutual spin pumping on the phase with fixed damping values of 1 = 0.02 and \n2 = 0.06. (a,c,e) The phase difference between the polar angles in two layers for HF and LF modes. \n(b,d,f) The calculated initial phase of TR -MOKE signals ( ) for each mode with 𝛼sp,12= 0 and \n𝛼sp,21 = 0.01 (a,b), 𝛼sp,12 = 0.01 and 𝛼sp,21 = 0 (c,d), and 𝛼sp,12 = 0.013 and 𝛼sp,21 = 0.004 (e,f). \nFor the third case (e,f), the values of mutual spin pumping are chosen to close the gap in panel \n(f) for Hext > 15 kOe. The rest of the parameters used in this calculation can be found in TABLE \nSI. Dashed lines represent the reference case without mutual spin pumping ( 1 = 0.02, 2 = 0.06, \nand 𝛼sp,12= 𝛼sp,21 = 0). \n \nSupplemental Note 5: Region diagram s for p -SAFs with different degrees of asymmetries \nFigure S4 shows the region diagrams for p -SAFs with different degrees of asymmetries , \nrepresented by the difference of Hk,eff in two FM layers. Hk,eff,1 = Hk,eff,2 corresponds to the \nsymmetric case (lowest asymmetry), as shown by Fig. S4(c). While the SAF in Fig. S4(a) has the \nhighest asymmetry: Hk,eff,1 = 2 kOe, Hk,eff,2 = 6 kOe. Figure S4 clearly shows that |𝜃0,1−𝜃0,2|>\n90° is a necessary but not sufficient condition for region 1 (CW precession). Because regions 2 or \n3 also appear to the left of the red cu rve (where |𝜃0,1−𝜃0,2|>90°), especially when θH is close \nto 90° and Hk,eff,1 is close to Hk,eff,2. \n11 \n \nFIG. S4 Region diagrams of p -SAFs with different degrees of asymmetries: Hk,eff,1 = 2 kOe, Hk,eff,2 \n= 6 kOe (a), Hk,eff,1 = 4 kOe, Hk,eff,2 = 6 kOe (b), Hk,eff,1 = 6 kOe, Hk,eff,2 = 6 kOe (c). The blue \nbackground represents region 1. The green background covers regions 2 and 3. The red curve \nshows the conditions where |𝜃0,1−𝜃0,2|=90°. |𝜃0,1−𝜃0,2|>90° to the left of the red curve. 𝛼1, \n𝛼2, 𝛼sp,12, and 𝛼sp,21 are set as zero. 𝛾1=𝛾2=17.8 rad ns−1 kOe−1. Values of the rest parameters are \nthe same as those in Table SI. \n \nReferences \n[1] Z. Zhang, L. Zhou, P. E. Wigen, and K. Ounadjela, Angular dependence of ferromagnetic \nresonance in exchange -coupled Co/Ru/Co trilayer structures, Phys. Rev. B 50, 6094 (1994). \n[2] W. Wang, P. Li, C. Cao, F. Liu, R. Tang, G. Chai, and C. Jiang, Temperature dependence \nof interlayer exchange coupling and Gilbert damping in synthetic antiferromagnetic \ntrilayers investigated using broadband ferromagnetic resonance, Appl. Phys. Lett. 113, \n042401 (2018). \n[3] G. Malinowski, F. Dalla Longa, J. H. H. Rietjens, P. V. Paluskar, R. Huijink, H. J. M. \nSwagten, and B. Koopmans, Control of speed and efficiency of ultrafast demagnetization \nby direct transfer of spin angular momentum, Nat. Phys. 4, 855 (2008). \n[4] D. H. Zanette, Energy exchange between coupled mechanical oscil lators: linear regimes, J. \nPhys. Commun. 2, 095015 (2018). \n \n" }, { "title": "1511.04593v2.Parametric_resonance_induced_chaos_in_magnetic_damped_driven_pendulum.pdf", "content": "arXiv:1511.04593v2 [physics.class-ph] 5 Aug 2016Parametric resonance induced chaos in magnetic damped driv en pendulum\nGiorgi Khomeriki\nVekua school of Physics & Mathematics, 9 Tchaikovsky, 0105 T bilisi, Georgia\nA damped driven pendulum with a magnetic driving force, appe aring from a solenoid, where ac\ncurrent flows is considered. The solenoid acts on the magnet, which is located at a free end of\nthe pendulum. In this system the existence and interrelatio n of chaos and parametric resonance\nis theoretically examined. Derived analytical results are supported by numerical simulations and\nconducted experiments.\nPACS numbers: 05.45.Ac, 42.65.Yj, 05.45.-a\nIntroduction: Chaos in damped driven pendulum system\nhas a long standing history (see e.g. Refs. [1, 2] and ref-\nerences therein) and is applicable in vast variety of con-\ndensed matter problems [3–5] ranging from Josephson\njunctions to easy-plane ferromagnets. Governing equa-\ntion is written in the standard form:\n¨α=−Ω2sinα−q˙α+fDsinωt (1)\nwhere Ω and qcoefficients are usually fixed and fDis\na one we control. Increasing control parameter fDpe-\nriod doubling [6, 7] bifurcation scenario and transition\nto chaos takes place [8–10]. In all the mentioned papers\ncontrol parameter is constant [11] or a driving force has a\ntime periodic singular character (kicked excited systems\n[12]). In the present paper driving force is position an-\ngleαdependent, particularly, here, a realistic example\nof driven damped pendulum model is considered. In this\ncontext, driving force is of a magnetic origin, particularly\na solenoid with ac current is acting on the magnet, which\nplays a role of a bob in a pendulum with a rigid rod (see\nFig. 1). Therefore the amplitude of a harmonic force fD\ngreatly depends on the distance between solenoid and\nthe magnet, making it angle dependent in a non-trivial\nmanner.\nIn the frames of the model (1) a possibility of onset of\nchaos has been examined analytically, numerically and\nexperimentally. Thesimilarmodelofmagneticpendulum\nhas been studied long before [13], particularly, different\norientation of solenoid and magnet has been considered,\nwhere the orientation of the solenoid is perpendicular to\nthe pendulum’s rod when the deviation angle is zero. In\nthis case one gets quantization of amplitudes with no in-\ndication of onset of chaos, while in our case with parallel\norientations of solenoid and pendulum in unperturbed\nposition (see again Fig. 1) for some values of ac field\nand/or distance between solenoid and magnet chaos is\nobserved due to the parametric resonance [14]. Thus the\nmain peculiarity of our model is that the existence of\nparametric resonance is a necessary condition for the on-\nset of chaos in the system.\nTheoretical Model: In my experiments and numerical\nsimulations the magnet is rigidly fixed in the place of\na bob of the pendulum in such a way that the directionsof its magnetic moment and the rod of pendulum coin-\ncides. Approximating solenoid and magnet as point-like\nmagnetic moments (− →L1and− →L2, respectively), one can\nreadilywrite down their dipolar interactionenergy as fol-\nlows:\nU=µ0\n4π/parenleftBigg\n3·(− →L1·− →r)(− →L2·− →r)\nr5−− →L1·− →L2\nr3/parenrightBigg\nwhere− →r≡(x, y) is a radius vector of magnet with\nrespect to the solenoid, r=/radicalbig\nx2+y2. Taking into ac-\ncount nowthat accurrentis flowing into the solenoidand\nthe magnet is attached at the free end of the pendulum\none can write for the components of magnetic moments\nfollowing expressions (see also Fig. 1):\nL1x= 0, L2x=−L2x\nℓ,\nL1y=L1(t), L2y=L2r0+ℓ−y\nℓ(2)\nwhereℓis the length of the pendulum and r0is distance\nbetween magnet and solenoid when the deviation angle\nfrom vertical direction is zero (that is a minimal distance\nposition between solenoid and magnet).\nPlugging (2) into (1) we find FxandFycomponents\nof the forces acting on the magnet:\nFx=−∂U\n∂xFy=−∂U\n∂y\nwe write Newton’s second law for tangential axis of the\npendulum as follows:\nm¨αℓ=Fxcosα+Fysinα−mgsinα−q˙α(3)\nwhere a damping proportional to velocity has been in-\ncluded and mis a mass of the magnet. We do not write\nhere explicit expressions for components of the force be-\ncause of their cumbersomeness, although their complete\nexpressions will be used for numerical simulations, while\nfor analytics we just linearize (3) for small deviation an-\nglesαand approximate r→r0:\n¨α=−α/parenleftbiggg\nℓ+12L1(t)L2\nmr5\n0+2L1(t)L2\nmℓ2r3\n0/parenrightbigg\n−q˙α(4)2\nD\nmgxFyF\n\u0014L\u0015L\nFIG. 1: Experimental setup (left) and schematics (right) fo r\ndriven damped magnetic pendulum. L1is a dipolar moment\nof solenoid, L2is a dipolar moment of the magnet. αis a\ndeviation angle of the pendulum from vertical. FxandFyare\nxandycomponents of magnetic force acting on the magnet.\nwhereL1(t)≡L0\n1cos2ωtbecause of the ac current (with\n2ωfrequency) flowing through the solenoid. Then let us\ndenote\nω0=/radicalbiggg\nℓ, h=L0\n1/parenleftbigg12L2\nmr5\n0+2L2\nmℓ2r3\n0/parenrightbigg\n(5)\nand reduce (4) to the following equation:\n¨α=−α(ω2\n0+hcos2ωt)−q˙α (6)\nwhich is just a Mathieu equation if one sets damping to\nzero.\nThe presence of parametric resonance in (6) is exam-\nined in Ref. [15] for driving frequencies ωclose to pendu-\nlum oscillation frequency ω0. Actually, similar analysis\ncould be done for arbitrary ωand the existence of para-\nmetric resonance in the system will cause undamped os-\ncillations, chaos and some more interesting phenomena.\nIn order to find out what conditions should be fulfilled\nfor this to occur, we should seek the solution of equation\n(6) in the following form:\nα=a(t)cosωt+b(t)sinωt (7)\nconsidering a(t) andb(t) as slow functions of time and\nneglecting their second derivatives, (6) is simplified to\nthe following form:\nXcosωt+Ysinωt= 0 (8)\nWherecoefficients XandYbothdependon a(t)andb(t).\nFor the equation to be true, both coefficients should be\nequal to zero. Thus we get a set of two equations, where\nour goal is to find the regimes of parametric resonance.For this, we should seek for the solution in the exponen-\ntial form a(t) =Aestandb(t) =Bestand two equations\nare derived:\nA·(2sω+qω)−B·(ω02+h\n2−ω2) = 0\nA·(ω2−h\n2−ω02)−B·(2sω+qω) = 0.(9)\nFinally we get from the compatibility condition:\ns=ω02+h\n2−ω2−qω\n2ω(10)\nConsidering parametric instability growth rate sto be\npositive, the instability condition will be:\nh≥2|w2−w02+2qω|. (11)\nThis defines the limits of existence of parametric res-\nonance and its dependence on various parameters, but\nall of these is valid for small angles. In order to get\nthe full dynamics we should solve differential equation\n(3) in a full range of angles. FxandFycomponents of\nmagneticforceareknownfromderivativeofdipole-dipole\nenergy. If we do not consider the angle as small, we will\nnot be able to make the approximations that has been\ndone before. In general, FxandFycomponents are very\ncomplicated expressions and it is impossible to solve the\nequation (3) analytically. Therefore I performed numer-\nical simulations using Matlab.\nNumerical simulations: Our next goal is to prove theo-\nretically the existence of chaos in the system, considering\ndeviation angles as arbitrary. The given equation of mo-\ntion (3) has been solved using ode45 toolbox of Matlab\nprogram with an initial guess that chaos should occur\nwhen parametric resonance for small angles takes place.\nAnd this appears to be true, because as the numerical\nsimulations show, when there is parametric instability in\nthe system, it is always chaotic. To prove the existence\nof chaos, the common way is to check, whether changing\nany parameter insignificantly, the difference between the\nfirst and second measurement of some variable increases\nexponentially in time. In other words, Lyapunov expo-\nnentshouldbecalculatedinordertoanalyzethebehavior\nof chaotic motion. To calculate the exponent, one has to\ndeviate e.g. initial angle α(0) by small value making it\nα′(0) and as time evolves, divide the resulting difference\nbetweenangles α(t)andα′(t)oninitialdeviation. Taking\nout logarithm from this, dividing on time and averaging\nthe results upon the initial deviations Lyapunov expo-\nnent of the process could be defined. Positive exponent\nis an obvious indication of the presence of chaos and one\nshould look at the simultaneous presence of parametric\nresonance condition in the system.\nAnother test to check the relation between parametric\nresonance and chaos in our case of magnetic pendulum\nis to look whether the system performs large angle oscil-\nlations starting from initial insignificant deviations. In3\n11.11.21.31.41.51.61.71.81.920100200300400500\nr (cm)2ω (Hz)\nChaosStable\nFIG. 2: Comparison of theoretical and numerical results for\nfinding the boundaries of parametric resonance and chaos.\nSolid line is theoretical curve according to (11) and red dot s\nare plotted using numerical simulations. Error bars show th e\nboundary area of chaos in experiments.\nother words, if we give the pendulum very small initial\nangle, for example 0.0001 rad, and after some time it\nstarts to oscillate with normal angles, this means that\nthere is parametric resonance and chaos in the system.\nThe latter scenario is observed in experiments when the\nsystem is in parametrically unstable regime. In Fig. 2\nsolidbluelineindicatestheoreticalboundarylineofpara-\nmetric resonance, so it is also boundary of chaos and\nstability. Red dots are boundaries of chaos from numer-\nical simulations, so the discrepancy between theory and\nnumerical calculations is really small. We also indicate\nby error bar experimental range where transition from\nstability to chaos occurs.\nIn Figs. 3 and 4 the evolution of relative differ-\nences of the pendulum angle αand angular velocity\n˙α(ω) are presented. For instance, in case of relative\nangle difference we use for its calculation the formula\n[α′(t)−α(t)]/[α′(0)−α(0)]. We evaluate the dynamics\nfrom two small initial values, e.g. α(0) = 0.0001 and\nα′(0) = 0.0001001 and average upon different initial de-\nviations. Lyapunov exponent values are as follows: for\nangleswegettheexponentvalueequalto3 .47andforan-\ngularvelocitiesit is3 .56whichingoodapproximationco-\nincides with parametric instability growth rate s= 3.63,\ncalculated from formula (10). As seen from upper graph\nof Fig. 3, in the beginning we have rapid growth of rela-\ntive angle and velocity differences, which is characterized\nby a value of Lyapunov exponent equal to 23, and it is\nquite different than theoretical growth rate. This effect\nhappens because in numerical experiments at t= 0 cur-\nrent starts to flow in solenoid abruptly, therefore a force\nof finite value instantly appears on magnet, and this is\nthe cause of strange behaviour of pendulum. In order to\nexclude such a scenario we multiply the dipolar moment\nof solenoid L1(t)≡L0\n1cos2ωton the time dependent fac-\ntor 1−exp(−t), modelling smooth growth of the current\nin the solenoid. One can observe the result on the bot-0 1 2 3 4 51001010\nTime (sec)Reltive difference00.20.40.60.811.21.41.61.8210−51001051010\nTime (sec)Relative difference\n[ω′(t)−ω(t)]/[ω′(0)−ω(0)]\n[α′(t)−α(t)]/[α′(0)−α(0)][ω′(t)−ω(t)]/[ω′(0)−ω(0)]\n[α′(t)−α(t)]/[α′(0)−α(0)]\nFIG. 3: In both graphs blue and red solid lines show rela-\ntive angle and angular velocity differences, respectively, ver-\nsus time. Black dashed line follows from theoretical estima te\nof parametric resonance growth rate (10) equal to s= 3.63.\nDriving frequency in both graphs is 29 Hz and pendulum pa-\nrameters are indicated in the text. Initial angle is very sma ll\n0.0001 rad. In case of upper panel initial large growth rate\nwith the exponent ≈23 (dotted-dashed line) is caused by\nthe fact that at t= 0 current starts to flow in solenoid in-\nstantly. Bottom graph shows the same relative differences of\nangles and velocities, but when we multiply dipolar moment\nof solenoid on the factor 1 −exp(−t), it prevents current (and\ntherefore force) to gain large values almost instantly.\ntom panel of Fig. 3. As seen, no rapid growthtakes place\nin the beginning of the time, because, the current (and\ntherefore force) starts to increase slowly and the value of\ntheLyapunovexponentcoincideswiththeoreticalgrowth\nrate (black dashed line).\nNo calculations of Lyapunov exponent were made us-\ning scenario displayed on the bottom panel, it is only\nused to prove and explain the reason of rapid growth in\nthe upper panel of Fig. 3. In calculations of Lyapunov\nexponent of angles and velocities the beginning of time\nwhere rapid growth takes place has not been considered,\nand calculated Lyapunov exponent coincides with theo-\nretical growth rate s= 3.63. Theoretical and numerical\nresults are really close in Fig. 3, that is because of the\nfact that the initial deviation angle of the pendulum is\nsmall.\nWhile in Fig. 4 the initial deviation angle is around 1\nrad and Lyapunov exponent is equal to 1 .55 and it does\nnot coincide with theoretical growth rate s= 3.63. This\nfact was predictable, because pendulum actually spends4\n0123456781001051010\nTime (sec)Relative difference[ω′(t)−ω(t)]/[ω′(0)−ω(0)][α′(t)−α(t)]/[α′(0)−α(0)]\nFIG. 4: The same as in Fig. 3 except initial angle, which\nis 1 rad. This is not a small angle and that is why numer-\nical results do not fit with theoretical growth rate indicate d\nby dashed line. Lyapunov exponent in this case is 1 .55 and\ntheoretical growth rate is s= 3.63.\nlittle time at small deviation angles where parametrical\ninstability is in force. However, the range of stabilty-\nchaos diagram plotted in Fig. 2 is still valid for large\ninitial deviation angles.\nHere are the parameters for figures 3 and 4: minimal\ndistance between magnet and solenoidis r0= 14mm, the\nlength of the pendulum is taken as ℓ= 28 cm, dipolar\nmoment amplitude of solenoid is L0\n1= 1.8A·m2, dipolar\nmoment of the magnet is L2= 0.2A·m2, mass of the\nmagnet is m= 0.05 kg, damping coefficient is taken as\nq= 0.01 and ac current frequency is 2 ω= 58π.\nIn Fig. 5 two Poincare graphs are displayed, both of\nthem express the system dynamics with the same initial\nparameters,except minimaldistancefromsolenoidtothe\nmagnet: for left graph r0= 22 mm, which corresponds\nto the regular evolution, while at the right the chaotic\n−2 −1 0 1 2−10−50510\nαω\n−2 −1 0 1 2−10−50510\nαω\nFIG. 5: Left: Poincare graph when the system is not chaotic\ncorresponding to the minimal distance between solenoid and\nmagnetr0= 22 mm. Right: Poincare graph in case of chaos\nwhenr0= 14 mm. The initial angle in both cases is large\n(α(0) = 1 rad), and time step is the period of oscillations of\nfree pendulum with an initial 1 rad angle. Other parameters\nof the system are given in the text.\u0003\n 1\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003 2\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003 3\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003 4\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003 5\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003 6\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003 7\u0003\u0003 \u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003\u0003 8\u0003\u0003Ͳ1.5 \u0003\u0003\u0003Ͳ 1\u0003\u0003Ͳ 0.5 \u0003\u0003 0\u0003\u0003\u0003 0.5 \u0003\u0003\u0003\u0003 1\u0003\u0003\u0003 1.5 \u0003\u0003\nTime \u0003(sec )An gle \u0003(rad )\u0003\nFIG. 6: Comparison of theoretical model (blue solid line) an d\nexperimental one in case of regular dynamics. Red dots are\nexperimental data and red dashed line is for guide of eyes.\nbehavior is observed for r0= 14 mm. In both cases\nthe initial angle is large (1 rad) and the time period of\nplotting dots is the own oscillation period ofthe free pen-\ndulum with the same parameters. If the time period of\nplotting dots were the oscillation period in the presence\nofmagneticfieldand notthe ownoscillationperiod ofthe\nfree pendulum, Poincare graph would be a point for the\nparameters of left graph of Fig 5 (non-chaotic regime).\nPoincare graphs are only displayed for the cases where\nthe initial angle is large, because in case of small angles\nit is clear that when parametric resonance occurs, the\nsystem is unavoidably chaotic.\nExperiments: In Fig. 1 experimental magnetic pendu-\nlum is displayed. All parameters are easily measurable\nexcept dipolar moments of solenoid and magnet. For this\npurpose we have used magnetic field sensor and the mea-\nsurements were made in different locations (more than\n50 locations). Applying then regression formula we have\nestimated values of dipolar moments. The experimen-\ntal parameters which are used in numerical simulations\nare given in the previous section. While conducting ex-\nperiments slow motion camera has been used in order to\ntrack pendulum motion. After the data has been pro-\ncessed on the computer and points have been plotted on\ntheoretical graph (see Fig. 6). It has been taken into\naccount that experimental pendulum is not a mathemat-\nical one, and thus (3) has been rewritten for the physical\npendulum case. Besides that, experimentally, damping\nproportional to the velocity is to be taken into account.\nThe damping coefficient qwas measured as follows: the\ntime needed for damping from initial angle is recorded,\nand then it is compared to numerical calculations made\nin Matlab with different damping coefficients. For the\ncoefficient q= 0.01 the damping took the same time as\nin the experiment. In Fig. 6 a comparison of theoretical\nmodel and experiment has been made in case of non-\nchaotic regime and one can clearly see that theory and\nexperiment is well-fitted. The difference between them\nof course grows with time, because there are some ex-\nperimental errors, which constantly act on the motion\ncharacteristics. The main error is that dipoles in reality5\nhave size, especially solenoid while in theoretical model\nwe have made an assumption that they are point-like.\nThe video in supplemental material is recorded for the\ncase when the system is chaotic. We have zero initial de-\nviation. When we let the current flow into the solenoid\nthe pendulum start large amplitude oscillations. From\na very small initial angle system stars large amplitude\noscillations, so it proves the existence of parametric res-\nonance and consequently the chaos in the system.\nConclusions: We have proved and examined the exis-\ntenceandinterrelationofparametricresonanceandchaos\nin the system of magnetic pendulum. Lyapunov expo-\nnents were calculated using numerical simulations and\nwere compared with theoretical growth rate. Lyapunov\nexponent for small angles (angular velocities) matches\nwith theoretical growth rate and for large angles it is dif-\nferent, as it was expected. The overall conclusion is that\nour magnetic pendulum system is chaotic only when the\nconditions for parametric resonance are fulfilled. Besides\nthat, experiments have been carried out and give a very\ngood agreement with theoretical model and numerical\nsimulations.\nI would like to give special thanks to T. Gachechiladze\nand G. Mikaberidze for very useful discussions.\n[1] G.L. Baker and J.P. Gollub, Chaotic Dynamics: An In-\ntroduction, Cambridge University Press, (1990).[2] G.L. Baker and J.A. Blackburn, The Pendulum: A Case\nStudy in Physics, Oxford University Press (2005).\n[3] G. Filatrellaa, B.A. Malomed, and S. Pagano, Phys. Rev.\nE65, 051116 (2002).\n[4] F. Cagnetta, G. Gonnella, A. Mossa, S. Ruffo, EPL (Eu-\nrophysics Letters), 111, 10002, (2015).\n[5] K.N. Alekseev and F.V. Kusmartsev, Physics Letters A,\n305, 211, (2002).\n[6] E. Sander and J.A. Yorke, Ergod. Th. Dynam. Sys, 31,\n1249, (2011).\n[7] J. Bevivino, Dynamics at the Horsetooth, 1, (2009).\n[8] J.H. Hubbard, Amer. Math. Monthly, 106, 741, (1999).\n[9] J. Isohtl, K.N. Alekseev, L.T. Kurki, and P. Pietilainen ,\nPhys. Rev. E, 71, 066206, (2005).\n[10] R. Harish, S. Rajasekar, and K. P. N. Murthy, Phys. Rev.\nE65, 046214, (2002).\n[11] H. Hauptfleisch, T. Gasenzer, K. Meier, O. Nachtmann,\nand J. Schemmel, American Journal of Physics, 78, 555\n(2010).\n[12] V. Damgov and I. Popov, OPA (Overseas Publishers As-\nsociation), 4, 99 (2000).\n[13] D.B. Doubochinski, Ya.B. Duboshinsky et al., Zh. Tech.\nFiz. 49, 1160 (1979) [Sov. Phys. - Tech. Phys. 24, 642\n(1979)]\n[14] A. Belyakov, A. Seyranian, A. Luongo, Physica D, 238,\n1589, (2009).\n[15] L.D. Landau, E.M. Lifshitz, Mechanics. Pergamon Press ,\n(1969)." }, { "title": "2108.03263v1.Magnon_transport_in___mathrm__mathbf_Y_3Fe_5O__12_____Pt_nanostructures_with_reduced_effective_magnetization.pdf", "content": "Magnon transport in Y 3Fe5O12/Pt nanostructures with reduced e\u000bective\nmagnetization\nJ. G uckelhorn,1, 2,\u0003T. Wimmer,1, 2M. M uller,1, 2S. Gepr ags,1H. Huebl,1, 2, 3R. Gross,1, 2, 3and M. Althammer1, 2,y\n1Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany\n2Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany\n3Munich Center for Quantum Science and Technology (MCQST), D-80799 M unchen, Germany\n(Dated: August 10, 2021)\nFor applications making use of magnonic spin currents damping e\u000bects, which decrease the spin\nconductivity, have to be minimized. We here investigate the magnon transport in an yttrium iron\ngarnet thin \flm with strongly reduced e\u000bective magnetization. We show that in a three-terminal\ndevice the e\u000bective magnon conductivity can be increased by a factor of up to six by a current\napplied to a modulator electrode, which generates damping compensation above a threshold current.\nMoreover, we \fnd a linear dependence of this threshold current on the applied magnetic \feld. We\ncan explain this behavior by the reduced e\u000bective magnetization and the associated nearly circular\nmagnetization precession.\nPure spin currents carried by magnons, the elemen-\ntary excitations of the spin system in magnetically or-\ndered insulators (MOIs), have drawn much attention due\nto their potential applications in information processing\nat a low dissipation level [1{4]. The MOI yttrium iron\ngarnet (Y 3Fe5O12, YIG) is a promising candidate for\nhosting e\u000ecient magnon based spin transport due to its\nlow Gilbert damping parameter even in nanometer-thin\n\flms [5] and its correspondingly large magnon propaga-\ntion length [6{9]. Amongst the device concepts enabling\nlogic operation with magnonic spin currents, transistor-\ninspired devices and even logic gates have been demon-\nstrated [4, 10{13]. Such transistor-like device concepts\ngenerally rely on spin-transfer torque for spin current\ngeneration. The latter can be realized in bilayers con-\nsisting of MOIs and heavy metals with strong spin-orbit\ncoupling via the spin Hall e\u000bect (SHE) [7, 14{21]. The\nmagnon transport in the MOI can be controlled via an\nelectrical charge current, and the resulting e\u000bect is typ-\nically represented by a change in the e\u000bective magnon\nconductivity [4, 10]. At a certain threshold current the\ninjected magnons can even counteract the magnetization\ndamping, which results in an abrupt increase of the ef-\nfective magnon conductivity. The present understanding\nis that this threshold e\u000bect scales with the saturation\nmagnetization Msand the magnetic anisotropy of the\nmaterials [10, 22]. This warrants to explore the impact\nof these parameters on controlling the magnon conduc-\ntivity in MOIs, which has not been pursued so far to the\nbest of our knowledge.\nIn this Letter, we investigate the di\u000busive magnon\ntransport in MOIs with signi\fcant perpendicular mag-\nnetic anisotropy \felds Hkand reduced Ms. To this end,\nwe biaxially strain the YIG thin \flm hosting the magnons\nby growing YIG on yttrium scandium gallium garnet\n(Y3Sc2Ga3O12, YSGG). Our \flms exhibit low Gilbert\ndamping comparable to YIG thin \flms grown on lattice-\nmatched substrates. By investigating the magnon trans-\nport in three-terminal devices, we \fnd that the thresholdcurrent, which de\fnes the onset of the regime with com-\npensated damping, depends linearly on the applied mag-\nnetic \feld. Moreover, we can corroborate the expected\nscaling with the e\u000bective magnetization of the MOI.\nOur experimental approach utilized to enhance the\nmagnon-based spin conductivity is based on the mini-\nmization of the ellipticity of the magnetization preces-\nsion. As sketched in Fig. 1(a), YIG thin \flms grown\non the lattice-matched substrate gadolinium gallium gar-\nnet (Gd 3Ga5O12, GGG) exhibit a \fnite in-plane e\u000bective\nmagnetization Me\u000b=Ms\u0000Hk>0, and thus an ellip-\ntical magnetization precession trajectory with the long\naxis aligned in the \flm plane, giving rise to nonlinear\ndamping e\u000bects via parametric pumping of higher fre-\nquency magnon modes [23]. Recent experiments reported\nthe minimization of the ellipticity of the magnetization\nprecession (approaching Me\u000b= 0) and thereby achieved\nspin-orbit torque induced coherent magnetization auto-\noscillations even in extended magnetic \flms [24, 25]. For\nour experiments, we also reduce the ellipticity of the mag-\nnetization precession by reducing the e\u000bective magneti-\nzation of YIG. Approaching Me\u000b= 0, a circular mag-\nnetization precession is expected and, hence, nonlinear\ndamping e\u000bects should be suppressed (cf. Fig. 1(b)). To\nbe able to control Me\u000b, we biaxially strain the tYIG=\n12:3 nm thick YIG \flm by growing it pseudomorphi-\ncally onto an YSGG substrate by pulsed laser deposi-\ntion (see the Supplemental Material (SM) [26] for growth\ndetails). The lattice mismatch of 0 :4 % between YIG\nand YSGG induces a biaxial in-plane tensile strain in\nthe YIG thin \flm. This strain can result in a strong\nHk[27], originating from the strain-induced magnetoelas-\ntic coupling [28]. Fig. 1(c) shows the 2 \u0012\u0000!x-ray di\u000brac-\ntion scan of the thin \flm con\frming the in-plane lattice\nstrain. The substrate (444) di\u000braction peak is clearly\nvisible at 2 \u0012= 50:7°, while the corresponding broad\n\flm peak is shifted to larger 2 \u0012values due to the ten-\nsile strain and appears as a shoulder in the di\u000bration\npattern. Note, that the large width and low inten-arXiv:2108.03263v1 [cond-mat.mes-hall] 6 Aug 20212\n(c) (d)\nbulk YIG\n(444)YSGG\n(444)\nYIG\nM\nPt\nYIG\nM(a) (b)\nGGG YSGG\nPt\nYIG//YSGG\n YIG//YSGG\nFigure 1. (a) Sketch of the ellipticity of the magnetization\nprecession in YIG thin \flms grown on lattice-matched GGG.\n(b) In biaxially strained YIG thin \flms grown on YSGG, the\nellipticity is minimized due to the reduced e\u000bective magne-\ntization. (c) X-ray di\u000braction of a 12 :3 nm thick YIG \flm\ngrown on a (111)-oriented YSGG substrate. The blue verti-\ncal line marks the calculated 2 \u0012-position of the (444) re\rection\nof bulk YIG. (d) Resonance \feld Hresand linewidth \u0001 Hex-\ntracted from FMR measurements of the YIG \flm on YSGG\nas a function of frequency. Via a Kittel \ft (gray line) we ex-\ntract\u00160Me\u000b= (56:0\u00060:2) mT and from a linear \ft to the\nlinewidth (blue line) we obtain \u00160\u000eH= (3:6\u00060:4) mT and\n\u000bG= (1:5\u00060:2)\u000210\u00003.\nsity of the \flm peak is due to the small \flm thickness.\nWe magnetically characterize the strained YIG \flm us-\ning broadband ferromagnetic resonance (FMR) as shown\nin Fig. 1(d). We determine the e\u000bective magnetization\n\u00160Me\u000b= (56:0\u00060:2) mT of the thin \flm by extracting\nthe resonance \feld \u00160Hresapplied in the out-of-plane di-\nrection as a function of the stimulus frequency of the mi-\ncrowave radiation and linear \ftting with the Kittel equa-\ntion. This value is about three times smaller compared\nto unstrained YIG \flms of similar thickness [10]. More-\nover, FMR enables us to determine the Gilbert damping\nparameter \u000bG(see Fig. 1(d)) [29]. By \ftting the FMR\nlinewidth \u0001 Hto\u00160\u0001H=\u00160\u000eH+ 4\u0019f\u000b G=\r(blue line)\nwith\r=g\u0016B\n~the gyromagnetic ratio with the Land\u0013 e\nfactorgand Bohr's magneton \u0016B, we obtain the inho-\nmogenous FMR linewidth \u00160\u000eH= (3:6\u00060:4) mT and\n\u000bG= (1:5\u00060:2)\u000210\u00003. Similar values for \u000bGwere ob-\ntained for an epitaxial high-quality YIG \flm grown on\nlattice-matched GGG under the same conditions [10].\nAs a next step, we deposit ex-situ 5 nm thick Pt\nstrips on top of the strained YIG \flm using electron\nbeam lithography and magnetron sputtering, allowing\nfor an all-electrical generation and detection of pure spin\ncurrents [4]. With this sample, we investigate di\u000busive\nmagnon transport using twin-strip structures as depicted\nin Fig. 2(a). In our experiments a DC charge current\n(a)\n(c)(b)\n+\n-+\n-\nPt\nYIG\nMV detI inj\nYSGGz\nxyH\nφ\n(d)detectorinjector\nw\ndetw\ninj 0.2 T\n0.1 T\n0.05 T\n0.02 T\n0.2 T0.1 T0.05 T0.02 Td\nc\n00.511.520123\nλm (µm)\nµ0H (T)0.511.522.530.1110Adet\nSHE (µV)\ndc (µm)ASHEdetFigure 2. (a) Sketch of the sample con\fguration with the elec-\ntrical connection scheme, and the coordinate system with the\nin-plane rotation angle 'of the applied magnetic \feld \u00160H.\n(b) Detector signal Vdet\nSHE plotted versus the magnetic \feld\norientation 'for di\u000berent magnetic \feld magnitudes \u00160H.\nThe red line is a \ft to Adet\nSHEcos2('). (c) The voltage ampli-\ntudesAdet\nSHE, as indicated in (b), plotted versus the distance\ndcfor di\u000berent magnetic \felds on a logarithmic scale. The\nred lines correspond to exponential \fts to the data points for\ndc\u00151µm. (d) Extracted magnon spin di\u000busion lengths \u0015m\nfrom the exponential \fts in (c) for di\u000berent magnetic \feld\nmagnitudes \u00160H. The red line is a \ft to Eq. (2), resulting in\na magnon di\u000busion constant D= (1:75\u00060:05)\u000210\u00004m2=s.\nIinj= 100 µA is fed through one Pt strip (injector) to\ninject magnons into the YIG via the SHE. The magnons\ndi\u000buse away from the injector and can then be electri-\ncally detected via the inverse SHE at the second Pt strip\n(detector) as a voltage signal Vdet. In our sample a con-\nstant injector width of winj= 500 nm is used, while the\ndetector width wdetand the center-to-center distance dc\nbetween injector and detector is varied. Using a cur-\nrent reversal technique we unambiguously can assign the\nmeasured detector voltage Vdet\nSHEto the magnons gener-\nated at the injector via the SHE [13, 16]. To character-\nize the magnon transport, we measure the voltage sig-\nnalVdet\nSHEas a function of the magnetic \feld orientation\n'(cf. Fig. 2(a)) with \fxed magnitude \u00160Hat a tem-\nperature of 280 K. The data is shown in Fig. 2(b) for\ndc= 2:2µm andwdet= 500 nm. The results show the\ndistinctive cos2(') angular variation expected for di\u000bu-\nsive transport of SHE-generated magnons from injector\nto detector [4, 16]. The angle dependence can be \ft-\nted with a simple Adet\nSHEcos2(') function, as exemplary\nshown for\u00160H= 200 mT , where Adet\nSHEcorresponds to\nthe amplitude of the SHE-induced magnon transport sig-\nnal. The quantity Adet\nSHEis plotted in Fig. 2(c) as a func-\ntion ofdcfor di\u000berent \u00160H. We observe a decrease of3\nAdet\nSHEwith increasing dcas expected for di\u000busive magnon\ntransport: at distances shorter than the magnon di\u000bu-\nsion length \u0015m,Adet\nSHEfollows a 1=dcdependence, while\nfor larger distances the magnon relaxation dominates and\nan exponential decay is observed [7, 19]. An exponential\n\ft to the data measured for dc>1µm (red lines), al-\nlows us to extract \u0015m. The extracted values are shown\nin Fig. 2(d) as a function of the magnetic \feld magni-\ntude\u00160H. The\u0015mvalues are of the order of 1 \u0016m and\nthus in good agreement with the values found for YIG\n\flms grown on lattice-matched GGG [10]. To discuss the\nphysics leading to the magnetic \feld dependence of \u0015m\nin more detail, we consider the magnon relaxation rate\n\u0000ip\nmr, which is given by\n\u0000ip\nmr= \n\u000bG+\u000eH\n2p\nH(H+Me\u000b)!\n\r\u00160\u0012\nH+Me\u000b\n2\u0013\n(1)\nfor an in-plane magnetized \flm [30]. Taking damping\ncontributions from inhomogenous broadening \u000eHinto ac-\ncount [31], the damping rate \u0000ip\nmrdiverges for a \fnite pos-\nitiveMe\u000bat\u00160H= 0. However, in the limit of Me\u000b= 0,\nthe relaxation rate is constant for \u00160H= 0 and we ex-\npect a strictly linear dependence on the magnetic \feld.\nTogether with \u0015m=pD\u001cmand\u001cm= 1=\u0000ip\nmrwithDthe\nmagnon di\u000busion constant and \u001cmthe magnon lifetime,\nwe can describe the magnetic \feld dependence of \u0015mde-\ntermined from the twin-strip transport measurements as\n\u0015m=s\nD\n\r\u00160\u0000\n\u000bGH+\u000eH\n2\u0001: (2)\nAs shown in Fig. 2(d), the experimental data can be\nwell \ftted by Eq. (2). Utilizing the values obtained from\nthe FMR measurements and neglecting the \feld depen-\ndence ofD, we obtain a magnon di\u000busion constant of\nD= (1:75\u00060:05)\u000210\u00004m2=s. Similar values were ob-\ntained for YIG \flms on GGG, supporting the quantita-\ntive understanding of the phenomenon [32].\nNext, we turn to three-terminal devices, which allow\nus to manipulate the magnon transport between injec-\ntor and detector via the center Pt strip acting as mod-\nulator (see Fig. 3(a)). In this con\fguration, we apply\na low-frequency (7 Hz) charge current Iinj\nac= 200 µA to\nthe injector strip, while a constant DC charge current\nImod\ndc is applied to the modulator strip. The detector\nvoltageVdetis recorded via lock-in detection, where the\n\frst harmonic voltage signal Vdet\n1!can be assigned to the\ntransport of magnons generated via the SHE at the injec-\ntor. We measure Vdet\n1!as a function of the magnetic \feld\norientation 'for di\u000berent external magnetic \feld mag-\nnitudes and di\u000berent modulator currents Imod\ndc. Exem-\nplary results for a structure with an edge-to-edge distance\nde= 200 nm and a modulator width wmod= 400 nm,\nwhilewinj=wdet= 500 nm and \u00160H= 50 mT, are plot-\nted in Fig. 3(b). For Imod\ndc= 0 (black data points), Vdet\n1!\n+\n-+\n-\nmodulator\ndetectorinjectorPt\nYIG+\n-V detI inj\nIdc mod\nYSGG(a) (b)\n(c)\n-20 mT\n-50 mT\n-80 mT\n-100 mT20 mT\n50 mT\n80 mT\n100 mTz\nxyH\nφ-0.6 mA\n-0.4 mA0.6 mA0.4 mA\n0 mA\nw\ndetw\ninj\nw\nmod\nMd\ne\nd\ne\ncurrent density (1011 A/m²)A1ωdetFigure 3. (a) Sketch of the sample con\fguration for a three-\nterminal device with the electrical connection scheme, and\nthe coordinate system with the in-plane rotation angle 'of\nthe applied magnetic \feld \u00160H. (b) Detector signal Vdet\n1!of\na structure with de= 200 nm and wmod= 400 nm plotted\nversus the magnetic \feld orientation with constant magnitude\n\u00160H= 50 mT for various modulator currents Imod\ndc. (c) The\nvoltage amplitudes Adet\n1!, as indicated in (b), versus the DC\ncharge current Imod\ndc. The gray lines indicate \fts to Eq. (3) for\ncurrent values below the threshold current.\nexhibits the same cos2(') variation as in our twin-strip\nstructures [4, 16]. As reported previously [10, 11], we ob-\nserve a signi\fcant enhancement of Vdet\n1!at'=\u0006180°\nforImod\ndc>0. This observation can be attributed to a\nmagnon accumulation underneath the modulator caused\nby the SHE-induced magnon chemical potential and ther-\nmally generated magnons due to Joule heating. This\naccumulation increases the magnon conductivity, result-\ning in a larger voltage signal Vdet\n1!. At'= 0°, the\nmagnon transport signal is slightly suppressed, originat-\ning from the nearly compensation of the magnon deple-\ntion caused by the SHE by thermally generated magnons.\nForImod\ndc<0, we observe a 180 °shifted angle depen-\ndence of the detector voltage signal, i.e. an increase at\n'= 0°and a reduction at '=\u0006180°. This behavior\nis fully consistent with the assumption that there are\nboth SHE and Joule heating contributions [4, 10, 11].\nFor a more quantitative analysis, we extract the signal\namplitudes Adet\n1!(+\u00160H) at'= 180 °andAdet\n1!(\u0000\u00160H)\nat'= 0°and plot them as a function of the modu-\nlator current Imod\ndc for di\u000berent \u00160Hin Fig. 3(c). For\f\fImod\ndc\f\f<0:25 mA, we observe the expected superposition\nof a linear and quadratic Imod\ndcdependence correspond-\ning to SHE induced magnons and thermally generated\nmagnons due to Joule heating, respectively [4, 10, 11].\nHowever, for larger Imod\ndca clear deviation from this be-4\nhavior is observed. In particular, we observe a strongly\nincreased signal amplitude Adet\n1!. This observation can\nbe attributed to an enhanced e\u000bective magnon conduc-\ntivity underneath the modulator, which causes a strong\nincrease of the detector signal at the same magnon injec-\ntion rate at the injector. As reported previously [10], this\nenhanced magnon conductivity can be explained by the\npresence of a zero e\u000bective damping state generated be-\nlow the modulator electrode via the SHE-mediated spin-\norbit torque. We here observe a maximum enhancement\nofAdet\n1!by a factor of 6, a twofold increase as compared\nto our previous experiments. This strong enhancement\ncan be attributed to the reduction in Me\u000band the as-\nsociated circular magnetization precession. For the two\nmagnetic \feld polarities, we observe an asymmetry for\f\fImod\ndc\f\f>0:25 mA in the amplitude signal Adet\n1!. This is\nin stark contrast to the results obtained for YIG \flms on\nlattice-matched GGG [10, 11]. At present, we can only\nspeculate about the detailed origin of this asymmetry. It\nmay be related to a combination of the following aspects:\n(i) a misalignement of the magnetic \feld due to trapped\n\rux from our 3D-vector magnet, (ii) a Joule heating in-\nduced modi\fcation of the device properties, or (iii) e\u000bects\nrelated to the crystalline-orientation of YIG. The pre-\nviously investigated YIG \flms were (001)-oriented [10],\nwhile here we use a (111)-orientation allowing us to make\nuse of the crystalline magnetic anisotropy.\nThe zero e\u000bective damping state and the correspond-\ning peak-like structure in the magnon conductivity at\nthe threshold value Imod\ncrit was recently discussed by S.\nTakei [33]. According to this model considerations, one\ncan express the expected dependence of Adet\n1!originating\nfrom the thermal and SHE injection of magnons by\nAdet\n1!\u0000\nImod\ndc;\u0006\u00160H\u0001\n=A+Bp\n1\u0007Imod\ndc=Imod\ncrit\n1 +Cp\n1\u0007Imod\ndc=Imod\ncrit;(3)\nwhere the proportionality factors account for the induced\nmagnon conductivity and A,B,CandImod\ncritare used as\n\ft parameters. Note that the model is only valid up to\nImod\ndc=Imod\ncritand we thus restrict the \ft with Eq. (3) to\nthis region. The \ft, indicated by gray lines in Fig. 3(c),\nreproduces well the measured data points. Although this\nmodel does not account for the amplitude asymmetry, it\nis well suited to extract the threshold current Imod\ncrit.\nFor a quantitative comparison of the strained YIG\n\flms with reduced Me\u000band conventional YIG thin \flms\non GGG, we rely on the dependence of Imod\ncritwith\u00160H\nin Fig. 4. For the discussed structure (blue circles), we\nobserve a linear increase of the critical current Imod\ncritwith\napplied magnetic \feld for \u00160H > 20 mT. This is in\ncontrast to the observations in Ref. [10] (black circles),\nwhere an increase in Imod\ncrit with\u00160His only observed\nfor\u00160H > 50 mT, while for \u00160H\u001450 mTImod\ncrit re-\nmains constant. We note that Imod\ncrit vs\u00160Hwas asso-\nciated with damping compensation [10]. Here, the spin\nde = 400 nm, wmod = 500 nm\nde = 200 nm, wmod = 400 nmFigure 4. Extracted critical currents Imod\ncrit, as indicated by\nthe black triangles in Fig. 3(c), as a function of the magnetic\n\feld magnitude \u00160H(blue circles). For comparison, the black\ndata points are taken from our previous work, where we in-\nvestigate an YIG thin \flm grown on lattice-matched GGG\nsubstrates [10]. The dashed lines correspond to \fts to Eq. (4)\nwith a \fnite Me\u000b, while the solid line is a \ft to the data in\nthe limit of Me\u000b= 0.\ninjection rate due to SHE results in an interfacial spin\ntransfer torque \u0000 ST/Imod\ndc, which balances the intrinsic\ndamping of the material. Hence, zero e\u000bective damping is\nachieved, when the condition \u0000ip\nmr= \u0000STis satis\fed [30].\nIn this regime, we can de\fne the critical modulator cur-\nrent as [10]\nImod\ncrit=~\ne\u001bPt\n2lstPtwmod\n\u0012SHtanh(\u0011)\u0012\n1 + 4\u0019MstYIG\u000be\u000b\n~\rge\u000b\u0013\n\u0002\r\u00160\u0012\nH+Me\u000b\n2\u0013\n;(4)\nwhereeis the elementary charge, \u0012SHthe spin Hall angle\nof Pt,\u000be\u000bthe \feld dependent damping rate [34] taking\ninto account inhomogenous broadening. Furthermore,\nge\u000bdenotes the e\u000bective spin mixing conductance [35],\nwhich depends on the interface spin mixing conductance\ng\"#, the spin di\u000busion length ls, thickness tPt, and elec-\ntrical conductivity \u001bPtof Pt (see SM [26]). The varia-\ntion ofImod\ncritwith the applied magnetic \feld taken from\nRef. [10] is quantitatively well described by the theoret-\nical model (dashed line). Fitting our data with Eq. (4),\nwe usewmod= 400 nm, Ms= 80 kAm\u00001(from SQUID\nmagnetometry measurements see SM [26]), \u0012SH= 0:11,\nls= 1:5 nm [36] and \u001bPt= 2:15\u0002106(\nm)\u00001. Fur-\nthermore, we use the values of \u000bGandMe\u000bextracted\nfrom the FMR measurements, while g\"#is the only free\n\ft parameter. We observe good quantitative agreement\nfor large magnetic \feld magnitudes, but \fnd a clear\ndeviation for \u00160H < 40 mT. However, if we assume\nMe\u000b\u00190, the \ft (solid line) corroborates our observed\nlinear magnetic \feld dependence of Imod\ncritin Fig. 4. More-\nover, the linear dependence on \u00160His in accordance\nwith the results by Evelt et al. , who studied Bi:YIG thin\n\flms with PMA and nearly vanishing Me\u000b[24]. Fitting\nthe data, we obtain g\"#= (1:7\u00060:2)\u00021019m\u00002and\ng\"#= (9:9\u00060:4)\u00021018m\u00002in the limit of Me\u000b= 0,\ncomparable to YIG/Pt structures on GGG [10]. Devi-\nations between \ft and data are potentially caused by5\nuncertainties in the \fxed parameters, as for example \u000bG\nandMe\u000bare determined from out-of-plane FMR.\nIn summary, we investigate magnon transport in\nYIG with strongly reduced Me\u000binduced via biaxial\nstrain from growth on YSGG substrates [27]. Perform-\ning angle-dependent measurements in twin- and three-\nterminal devices, we \fnd a quantitatively similar behav-\nior as observed for YIG \flms on GGG for small mod-\nulator currents Imod\ndc, while di\u000berences occur above the\nthreshold value Imod\ncrit when damping compensation is\nreached. Most importantly, we observe an increase of the\nmagnon induced detector signal by a factor of about 6,\nwhich is much larger than reported previously [10]. An-\nother important di\u000berence is the strictly linear \feld de-\npendence of Imod\ncrit. 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Parkin,\nRole of transparency of platinum-ferromagnet interface\nin determining intrinsic magnitude of spin hall e\u000bect, Na-\nture Physics 11, 496 (2015)." }, { "title": "2103.14614v1.Linear_damping_and_depletion_in_flowing_plasma_with_strong_sheared_magnetic_fields.pdf", "content": "arXiv:2103.14614v1 [math.AP] 26 Mar 2021LINEAR DAMPING AND DEPLETION IN FLOWING PLASMA WITH\nSTRONG SHEARED MAGNETIC FIELDS\nHAN LIU, NADER MASMOUDI, CUILI ZHAI, AND WEIREN ZHAO\nAbstract. In this paper, we study the long-time behavior of the solutio n for the linearized\nideal MHD around sheared velocity and magnetic field under St ern stability condition. We\nprovethatthevelocityandmagnetic fieldwill convergetosh earedvelocity andmagnetic field\nas time approaches infinity. Moreover a new depletion phenom enon is proved: the horizontal\nvelocity and magnetic field at the critical points will decay to 0 as time approaches infinity.\n1.Introduction\nThe appearance of large coherent structures is an important phenomena in the magnetic\nfluid. The study of the long-time behavior of MHD waves is a ver y active field in physics and\nmathematics [5, 11, 24].\n1.1.The linearized MHD system. Inthis paper, we consider themagnetic fluiddescribed\nby the two-dimensional incompressible ideal MHD equations on the periodic domain T2=\n{(x,y)|x∈T,y∈T}\n(1.1)\n\n∂tU+U ·∇U −H·∇H +∇P= 0,\n∂tH+U ·∇H−H·∇U = 0,\n∇·U= 0,∇·H= 0,\nwith initial data U(0,x,y) andH(0,x,y). HereU= (U1,U2),H= (H1,H2) andPdenote\nthe velocity field, magnetic field, and the total pressure of t he magnetic fluid, respectively.\nSystem (1.1) has an equilibrium Us= (u(y),0),Hs= (b(y),0),Ps= const.. We shall focus\non the asymptotic behavior of the linearized 2D MHD equation s around this equilibrium,\nwhich take the form\n(1.2)\n\n∂tU1+u∂xU1+∂xp+u′U2−b∂xH1−b′H2= 0,\n∂tU2+u∂xU2+∂yp−b∂xH2= 0,\n∂tH1+u∂xH1+b′U2−b∂xU1−u′H2= 0,\n∂tH2+u∂xH2−b∂xU2= 0,\n∇·U= 0,∇·H= 0.\nWe introduce the vorticity ω=∂xU2−∂yU1and the current density j=∂xH2−∂yH1\nwhich satisfy the following equations:\n(1.3)/braceleftbigg\n∂tω+u∂xω−b∂xj=u′′U2−b′′H2,\n∂tj+u∂xj−b∂xω=b′′U2−u′′H2+u′∂xH1−u′∂yH2+b′∂yU2−b′∂xU1.\nWe further introduce the stream function ψand the magnetic potential function φ,satisfying\nU= (∂yψ,−∂xψ),ω=−∆ψandH= (∂yφ,−∂xφ),j=−∆φ,which allow us to derive the\nfollowing system, satisfied by ( ψ,φ):\n(1.4)/braceleftbigg\n∂t(∆ψ)+u∂x(∆ψ)−b∂x(∆φ) =u′′∂xψ−b′′∂xφ,\n∂t(∆φ)+u∂x(∆φ)−b∂x(∆ψ) =b′′∂xψ−u′′∂xφ−2u′∂x∂yφ+2b′∂x∂yψ.\n12 HAN LIU, NADER MASMOUDI, CUILI ZHAI, AND WEIREN ZHAO\nTaking the Fourier transform in xand inverting the operator ( ∂2\ny−α2),we rewrite the system\nas\n(1.5)\n\n∂t/parenleftBig/hatwideψ\n/hatwideφ/parenrightBig\n(t,α,y) =−iαMα/parenleftBig/hatwideψ\n/hatwideφ/parenrightBig\n(t,α,y),\n(/hatwideU1,/hatwideU2) = (∂y/hatwideψ,−iα/hatwideψ),(/hatwideH1,/hatwideH2) = (∂y/hatwideφ,−iα/hatwideφ),\nwhereα/\\e}atio\\slash= 0 and\n(1.6) Mα=−∆−1\nα/bracketleftbigg\nu′′−u∆α −b′′+b∆α\nb∆α+b′′+2b′∂y−u∆α−u′′−2u′∂y/bracketrightbigg\n.\nFor the homogenous equilibrium u= 0, b(y) = const.,/hatwideU2and/hatwideH2satisfy a 1-D wave\nequation, which is stable but exhibits no decay. There are fe w rigorous mathematical results\non the non-flowing plasma with inhomogeneous sheared magnet ic field. In [18], Tataronis\nand Grossmann predicted that the vertical components of vel ocity and magnetic field may\ndecay by phase mixing, to which a mathematically rigorous pr oof was given by Ren and\nZhao in [15], under the condition that the magnetic field is po sitive and strictly monotone.\nIf the positivity assumption on the magnetic field is removed , which allows the direction of\nthe sheared magnetic field to change, then it turns out that ma gnetic reconnection occurs\nin infinite time, generating the magnetic island. This pheno menon was predicted by Hirota,\nTatsuno and Yoshida [6] and later justified by Zhai, Zhang and Zhao [22]. For the flowing\nplasmau/\\e}atio\\slash= 0, fewer mathematical rigorous results are available. We r efer to [6, 14, 22] for\nthe long time behaviors of the solutions to the MHD equations linearized around a flowing\nplasma.\nNotations: Let us specify the notations to be used throughout the paper. We denote by\nA/lessorsimilarBan estimate of the form A≤CBand byA∼Ban estimate of the form C−1B≤\nA≤CB,whereCis a constant. Given a function f(x,y),we denote its Fourier transform in\nx-variable as/hatwidef(α,y) =1\n2π/integraltext\nTe−ixαf(x,y)dx,whereαis the wave number. We shall use the\nJapanese bracket notation /a\\}b∇acketle{tx/a\\}b∇acket∇i}ht:=/radicalbig\n|x|2+1.\n1.2.Vertical damping and horizontal depletion. In this paper, we focus on the long\ntime behavior of the solution to the linearized MHD equation (1.5).\nOur first main result states as follows:\nTheorem 1.1 (Vertical damping) .Letu,b∈C3(T)be such that b>|u| ≥0and the critical\npoints of (u±b)are non-degenerate. Let α/\\e}atio\\slash= 0be a fixed wave number and let/parenleftBig\n/hatwideψ,/hatwideφ/parenrightBig\nsolve\n(1.5)with initial data/parenleftBig\n/hatwideψ0,/hatwideφ0/parenrightBig\n∈(H3×H3). Then the following space-time estimate holds:\n(1.7)/vextenddouble/vextenddouble/vextenddouble/parenleftBig\n/hatwideψ,/hatwideφ/parenrightBig/vextenddouble/vextenddouble/vextenddouble\nH1\ntL2y≤Cα/vextenddouble/vextenddouble/vextenddouble/parenleftBig\n/hatwideψ0,/hatwideφ0/parenrightBig/vextenddouble/vextenddouble/vextenddouble\nH3y.\nIn particular, lim\nt→∞/vextenddouble/vextenddouble/vextenddouble/parenleftBig\n/hatwideU2,/hatwideH2/parenrightBig/vextenddouble/vextenddouble/vextenddouble\nL2y= 0.\nRemark 1.2. The condition |u|<|b|is called Stern stability condition (see [16]).\nRemark 1.3. Formally, the space-time estimate may indicate that/vextenddouble/vextenddouble/vextenddouble/parenleftBig\n/hatwideU2,/hatwideH2/parenrightBig/vextenddouble/vextenddouble/vextenddouble\nL2/lessorsimilar1\n/a\\gbracketleftt/a\\gbracketrightβwith\nβ >1\n2. The study of the precise decay rate shall be our forthcoming w ork.3\nRemark 1.4. For the case of flowing plasma with constant velocity or non-fl owing plasma\n(u= const.), it holds that\n(1.8) /ba∇dblU(t),H(t)/ba∇dblHkxL2y∼ /ba∇dblU0,H0/ba∇dblHkxL2y, k≥0,\nwhich implies linear growth of vorticity and current densit y, i.e.,\n/ba∇dblω(t),j(t)/ba∇dblL2x,y/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht/ba∇dblω0,j0/ba∇dblL2x,y.\nThe proof can be found in Appendix C. By (1.8), the total energy is almost conserved. Hence,\nthe vertical damping in Theorem 1.1 shows that energy is trans ferred from the vertical direc-\ntion to the horizontal direction. Whether similar energy co nservation results are true for the\nflowing plasma with u/\\e}atio\\slash= const.remains an open question.\nThe vertical damping is induced by a certain mixing mechanis m similar to the vorticity\nmixingthatleadstoinvisciddampingforlinearizedEulere quations(see[9,10,19,20,21,23]).\nReaders may consult [2, 4, 7, 8, 12, 13] for recent progress in nonlinear inviscid damping.\nTo better illustrate the mixing mechanism, let us recall the system in terms of ( U1,H1):\n(1.9)/braceleftbigg\n∂tU1+u∂xU1−b∂xH1=L1,\n∂tH1+u∂xH1−b∂xU1=L2,\nwhere (L1,L2) :=/parenleftbig\nb′H2−u′U2−2∂x∆−1(b′∂xH2−u′∂xU2),u′H2−b′U2/parenrightbig\ncan be seen as\nnonlocal forcing terms depending on U2andH2.\nBy the incompressibility condition, we can check that\n/ba∇dbl(∂xU1,∂xH1)/ba∇dblL2xH−1\ny∼ /ba∇dbl(U2,H2)/ba∇dblL2x,y.\nThen the mixing of ( U1±H1) would lead to the linear damping of ( U2,H2).\nLet us consider a toy model, obtained by neglecting the nonlo cal forcing terms ( L1,L2) in\nthe linearized system (1.9), i.e.,\n(1.10)/braceleftbigg\n∂tU1+u∂xU1−b∂xH1= 0,\n∂tH1+u∂xH1−b∂xU1= 0.\nIt is then easy to see that the Els¨ asser variables Z±\n1:=U1±H1satisfy certain transport\nequations and then\n/parenleftBig\n/hatwideU1+/hatwideH1/parenrightBig\n(t,α,y) =/hatwideZ+\n1,in(α,y)e−iα(u−b)t,\n/parenleftBig\n/hatwideU1−/hatwideH1/parenrightBig\n(t,α,y) =/hatwideZ−\n1,in(α,y)e−iα(u+b)t,(1.11)\nwith/hatwideZ±\n1,indenoting the initial data.\nRegarding the toy model (1.10), we have the following conclu sions.\nLemma 1.5. Letu,b∈C3(T)be such that u±bhave only non-degenerate critical points.\nThen the solution of (1.10)with initial data (U1,in,H1,in)satisfies\n(1.12) /ba∇dbl(∂xU1,∂xH1)/ba∇dblL2xH−1\ny/lessorsimilar1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht1\n2/ba∇dbl(U1,in,H1,in)/ba∇dbl\nH1\n2xH1y.\nMoreover, if the initial data (U1,in,H1,in)vanish at all the critical points of (u±b), then it\nholds that\n(1.13) /ba∇dbl(∂xU1,∂xH1)/ba∇dblL2xH−1\ny/lessorsimilar1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht/ba∇dbl(U1,in,H1,in)/ba∇dblH−1\nxH2y.4 HAN LIU, NADER MASMOUDI, CUILI ZHAI, AND WEIREN ZHAO\nTheproof of (1.12) can befoundin [20]; via the same dual meth od one can prove (1.13). In\nfact, the decay rate of t−1/2for the toy model (1.10) is optimal, as we know, via the classi cal\nstationary phase approximation (see Chapter VIII of [17]), that there exists a class of initial\ndata such that the corresponding solutions satisfy\n/ba∇dbl(∂xU1,∂xH1)/ba∇dblL2xH−1\ny∼1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht1\n2.\nThe space-time estimate in Theorem (1.1), however, fails to hold for the toy model (1.10).\nExploring the mechanisms behind the enhanced damping for th e complete system (1.9), we\nfound a new dynamical phenomenon apart from velocity mixing : the depletion of horizontal\nvelocity and magnetic field ( U1,H1) at the critical points of u±b. This leads to our second\nmain result, which states as follows:\nTheorem 1.6 (Horizontal depletion) .Letu,bsatisfy the same assumptions as in Theorem\n1.1. Lety0be a critical point of (u+b)or(u−b), i.e.,u′(y0) =b′(y0)oru′(y0) =−b′(y0).\nLet/parenleftBig\n/hatwideU1,/hatwideH1/parenrightBig\ncorrespond to the solution to (1.5)with initial data/parenleftBig\n/hatwideψ0,/hatwideφ0/parenrightBig\n∈(H3×H3).\nThen it holds that\nlim\nt→∞/vextendsingle/vextendsingle/vextendsingle/parenleftBig\n/hatwideU1,/hatwideH1/parenrightBig\n(t,α,y0)/vextendsingle/vextendsingle/vextendsingle= 0.\nRemark 1.7. From(1.11), we can see that the horizontal depletion in Theorem 1.6 is not\ntrue for the toy model (1.10).\nFor a more precise description of the long time behavior of th e horizontal components, we\nconjecture that there exist some final states Z±\n1,∞and someκ>0such that\n/parenleftBig\n/hatwideU1+/hatwideH1/parenrightBig\n(t,α,y)∼/parenleftBig\n/hatwideZ+\n1,∞/parenrightBig\n(α,y)e−iα(u−b)t+O(t−κ),\n/parenleftBig\n/hatwideU1−/hatwideH1/parenrightBig\n(t,α,y)∼/parenleftBig\n/hatwideZ−\n1,∞/parenrightBig\n(α,y)e−iα(u+b)t+O(t−κ).(1.14)\nTheorem 1.6 reveals that /hatwideZ±\n1,∞vanish at all the critical points of (u±b).Our conjecture (1.14)\nshall imply the space-time estimate (1.7)and even a precise decay rate, provided that suitable\nregularity of the final states could be proven.\nThe parallel phenomenon exists in the realm of hydrodynamic s. Vorticity depletion for\nthe linearized 2D Euler equations around shear flows, first pr edicted by Bouchet and Morita\nin [3], was later mathematically proven in [20] by Wei, Zhang and Zhao. A similar vorticity\ndepletion result for the 2D Euler equation linearized aroun d a radially symmetric and strictly\ndecreasing vorticity distribution is due to Bedrossian, Co ti-Zelati and Vicol [1]. As far as\nwe know, this is the first paper studying the depletion of the horizontal velocity\nand magnetic field for the linearized MHD equations.\nComparing the toy model (1.10) with the complete system (1.9 ), we observe at least three\nsignificant effects of the nonlocal terms L1andL2:\n(1) Altering the final state, as /hatwideZ±\n1,inon the right hand side of (1.11) are changed into\nsome other final state /hatwideZ±\n1,∞on the right hand side of (1.14);\n(2) Causing/hatwideZ±\n1,∞to vanish at the critical points of ( u±b);\n(3) Enhancing the damping.\nThis demonstrates that the nonlocal terms L1andL2in (1.9) cannot simply be neglected or\nbe regarded as mere perturbations on the toy model.5\nRemark 1.8. The significance of the nonlocal terms can also be seen from the resolvent\nestimate. Indeed, if we consider the toy model (1.10)instead, the Sturmian equation (2.2)in\nSection 3 would become/parenleftbig\n(u−c)2−b2/parenrightbig\n∂yΦ =F,which is much simpler and yields an obvious\nestimate\n|∂yΦ|/lessorsimilar|(u−c)2−b2|−1.\nYet, as we shall see in Section 3, at any critical point y0of(u+b)or(u−b),the solution Φ\nto the actual Sturmian equation (2.2)enjoys a non-trivial estimate\n|∂yΦ(y0)|/lessorsimilar|(u(y0)−c)2−b(y0)2|−3\n4.\nThis1\n4-improvement, resulting from the effects of L1andL2,is the key to the depletion result.\nFor more details, we refer to Section 3 and 4.\nWe have the following comments on the results in Theorem 1.1 a nd Theorem 1.6.\nRemark 1.9. The vertical damping and horizontal depletion results also h old for the case of\nfinite channel under slip boundary condition, provided that the critical points do not appear\nat the boundary.\nRemark 1.10. To highlight the differences among the long time behaviors of the solutions\nto the linearized MHD equations in various cases, we show the following table:\nConditionsResults References\nMonotonicityUniform direction\nb>0Stern stability\n|u| ≤ |b|Other conditions\nYes Yes Yes u≡0 Damping [15]\nYes No Yes u(0) =b(0) = 0 Magnetic Island [22]\nYes No Nou=k1y, b=k2y\nk1>k2≥0Damping [14]\nNo YesYes\n|u|0such that for c∈(Ωǫ0\\(RanZ+∪RanZ−)),the\nsolution to (3.1)satisfies the following bound, uniform with respect to c\n/ba∇dblΦ(α,·,c)/ba∇dblL2+/ba∇dbl(Z−−c)(Z+−c)∂yΦ(α,·,c)/ba∇dblH1≤C/ba∇dblF(α,·,c)/ba∇dblH1.\nThe estimate on Φ can be continued (in c) up to the boundary Ran Z+∪RanZ−.7\nProposition 3.2. Forc∈(RanZ+∪RanZ−),there exist Φ±(α,·,c)∈L2such that as\nǫ→0+,Φ(α,·,c±iǫ)→Φ±(α,·,c)inLrwithr∈(1,2)and/vextenddouble/vextenddoubleΦ±(α,·,c)/vextenddouble/vextenddouble\nL2≤C/ba∇dblF(α,·,c)/ba∇dblH1.\nWe shall prove Proposition 3.1 by contradiction. Suppose th at the proposition is false,\nthen there exists a sequence {cn,Φn,Fn}∞\nn=1satisfying the equation\n(3.2) ∂y((Z−−cn)(Z+−cn)∂yΦn)−α2(Z−−cn)(Z+−cn)Φn=Fn,\nsuch that\n•cn∈/parenleftbig\nΩǫ0\\/parenleftbig\nRanZ+∪RanZ−/parenrightbig/parenrightbig\n,\n• /ba∇dblΦn/ba∇dblL2+/ba∇dbl(Z−−cn)(Z+−cn)∂yΦn/ba∇dblH1= 1,\n•Fn→0 inH1asn→ ∞.\nFor convenience, we shall use the notation qn:= (Z−−cn)(Z+−cn)∂yΦnfrom time to time\nin this section.\nAs|cn| ≤Cand/ba∇dblΦn/ba∇dblL2+/ba∇dblqn/ba∇dblH1= 1,we know that\n•cn→cfor somec∈Ωǫ0up to a subsequence,\n•Φn⇀Φ inL2up to a subsequence,\n•qn⇀qinH1up to a subsequence, for some q∈H1.\n(For convenience, we shall simply use the original cnand Φnto denote the elements in the\nsubsequence.) Moreover, q= (Z−−c)(Z+−c)∂yΦ,as seen from the identity\n/a\\}b∇acketle{tqn,f/a\\}b∇acket∇i}htL2=/a\\}b∇acketle{t(Z−−cn)(Z+−cn)∂yΦn,f/a\\}b∇acket∇i}htL2\n=−/a\\}b∇acketle{tΦn,Z′\n−(Z+−cn)f+(Z−−cn)Z′\n+f+(Z−−cn)(Z+−cn)f′/a\\}b∇acket∇i}htL2,∀f∈C∞\n0.\nWe shall show in the following passages that\n•the weak limit Φ ≡0,\n•in fact, strong convergences hold true, i.e., Φ n→0 inL2andqn→0 inH1,\nwhich contradict the very assumption that /ba∇dblΦn/ba∇dblL2+/ba∇dbl(Z−−cn)(Z+−cn)∂yΦn/ba∇dblH1= 1.\nFormally, by performing integration by parts on the limitin g equation\n(3.3) ∂y((Z−−c)(Z+−c)∂yΦ)−α2(Z−−c)(Z+−c)Φ = 0,\nwe can see that the weak limit Φ ≡0.To prove this, we have to show that Φ belongs to a\nspace for which the operation is allowed.\nThe limitcbelongs to either/parenleftbig\nΩǫ0\\(RanZ+∪RanZ−)/parenrightbig\nor (RanZ+∪RanZ−).\nLet us first consider the case when cn→c= Rec+iImc /∈(RanZ+∪RanZ−).In this\ncase, it’s straightforward to show that Φ n→Φ inL2andqn→q:= (Z−−c)(Z+−c)∂yΦ in\nH1,where Φ is the classical solution to (3.3). Taking the inner p roduct with Φ,integrating\nby parts and separating the real and imaginary parts, we obta in\n(3.4)/integraldisplay\nT/parenleftBig\n(u(y)−Rec)2−(b(y))2−(Imc)2/parenrightBig/parenleftbig\n|∂yΦ(y,c)|2+α2|Φ(y,c)|2/parenrightbig\ndy= 0,\nand/integraldisplay\nT(u(y)−Rec)/parenleftbig\n|∂yΦ(y,c)|2+α2|Φ(y,c)|2/parenrightbig\ndy= 0. (3.5)\nMultiplying (3.5) by 2Re cand adding it to (3.4) lead to/integraldisplay\nT/parenleftbig\nu(y)2−b(y)2−(Rec)2−(Imc)2/parenrightbig/parenleftbig\n|∂yΦ(y,c)|2+α2|Φ(y,c)|2/parenrightbig\ndy= 0.\nThe condition |u|0},\n(2) critical points, where Z′\ns= 0 and |Z′′\ns|>0.\nIn our analysis, the two situations require separate treatm ents. To better distinguish between\nthe two, we denote a critical point as y0.\n3.1.Weak convergence of {Φn}∞\nn=1to0.In this subsection, we establish several lemmas\nimplying that Φ n⇀Φ≡0 ascn→c. As previously mentioned, to this end we need to prove\nthat Φ is regular enough such that the desired integration by parts is justified. This is clear\nwheny/\\e}atio\\slash∈(Z−)−1(c)∪(Z+)−1(c),as Φ∈H3\nloc/parenleftbig\nT\\(Z±)−1(c)/parenrightbig\nthanks to |(Z−−c)(Z+−c)|>\nC >0.Therefore, our consideration starts from the case when yc∈(Zs)−1(c) lies in a region\nwhereZsis strictly monotone, s= + or−.\nLemma 3.3. Let{cn}∞\nn=1⊂(Ωǫ0\\(RanZ+∪RanZ−))be such that cn→casn→ ∞for\na certainc∈(RanZ+∪RanZ−).Let the triple {cn,Φn,Fn}∞\nn=1satisfy the equation (3.2)\nalong with the following conditions\n•Φn⇀ΦinL2and(Z−−cn)(Z+−cn)∂yΦn⇀(Z−−c)(Z+−c)∂yΦinH1inI,\n•Fn→FinH1inI,\nfor some interval I:= [y1,y2]⊂Tsuch that there exists yc∈(Zs)−1(c)inIwith|Z′\ns(yc)|>0;\nZ′\ns(yc)Z′\ns(y)>0,∀y∈ I,wheres= +or−.\nThenΦn→ΦinL2(I)and(Z−−cn)(Z+−cn)∂yΦn→(Z−−c)(Z+−c)∂yΦinH1(I).\nIn particular, if F≡0,thenΦ∈H1(I)and\n−/integraldisplay\nI(Z−−c)(Z+−c)∂yΦf′dy+α2/integraldisplay\nI(Z−−c)(Z+−c)Φfdy= 0,∀f∈H1\nw,0(I), (3.6)\nwhereH1\nw,0=/braceleftbig\nf∈L2: ((y−yc)∂yf)∈L2andf|y=y1=f|y=y2= 0/bracerightbig\n.\nProof.To facilitate the proof, let us assume that c∈RanZ+. Then for ǫ0 such that for any n>N,|cn−c|<ǫ.Thus there exists C >0 such that\n|Z−−cn|>C−1>0 forn>N. Without loss of generality, let us also assume that n>N\nand Imcn>0. Thus, there exists ycn∈ Isuch thatZ+(ycn) = Recnandycn→ycascn→c.\nWe recallqn= (Z−−cn)(Z+−cn)∂yΦnwithcn∈Ωǫ0\\(RanZ+∪RanZ−).Dividing (3.2)\nby (Z−−cn)(Z+−cn) and differentiating, we see that qnsolves the following equation\n(3.7)∂y/parenleftbigg∂yqn\n(Z−−cn)(Z+−cn)/parenrightbigg\n−α2/parenleftbiggqn\n(Z−−cn)(Z+−cn)/parenrightbigg\n=∂y/parenleftbiggFn\n(Z−−cn)(Z+−cn)/parenrightbigg\n.\nBy Equation (3.2), qnalso satisfies\nq′\nn=Fn+α2(Z−−cn)(Z+−cn)Φn, (3.8)\nq′′\nn−α2qn=F′\nn+α2Z′\n−(Z+−cn)Φn+α2(Z−−cn)Z′\n+Φn. (3.9)9\nFrom Equation (3.9) and the assumptions Φ n⇀Φ inL2, qn⇀q:= (Z−−c)(Z+−c)∂yΦ in\nH1andFn→FinH1,we can infer that\n(3.10) /ba∇dblq′′\nn/ba∇dblL2≤ /ba∇dblF′\nn/ba∇dblL2+α2(/ba∇dblΦn/ba∇dblL2+/ba∇dblqn/ba∇dblH1),\nwhich, along with the assumption that qn⇀(Z−−c)(Z+−c)∂yΦ inH1,implies strong\nconvergence, i.e.,\n(Z−−cn)(Z+−cn)∂yΦn→(Z−−c)(Z+−c)∂yΦ inH1.\nBy our assumption and Sobolev embedding, /ba∇dblqn/ba∇dblL∞(I)≤ /ba∇dblqn/ba∇dblH1(I)0,∀y∈ I0.For\nnsufficiently large, |Recn−c| ≤ |Imcn|andZ′′\ns(y0)(Recn−c)≥0.\n(2)There exists y0∈(Zs)−1(c)inI0at whichZ′\ns(y0) = 0;Z′′\ns(y0)Z′′\ns(y)>0,∀y∈ I0.For\nnsufficiently large, Z′′\ns(y0)(Recn−c)<0.\nThen fornsufficiently large,\n(1) (y−y0)∂yΦn∈L2,(y−y0)∂yΦ∈L2and\n−/integraldisplay\nI0(Z+−c)(Z−−c)/parenleftbig\n∂yΦf′+α2Φf/parenrightbig\ndy=/integraldisplay\nI0Ffdy,∀f∈H1\nw,0, (3.13)\nwhereH1\nw,0=/braceleftbig\nf∈L2: ((y−y0)∂yf)∈L2andf|y=y1=f|y=y2= 0/bracerightbig\n;\n(2)there exists a constant Cindependent of cnsuch that\n(3.14) |Φn(y0)| ≤C|(Z+(y0)−cn)(Z−(y0)−cn)|−1\n4;\n(3)there exists a constant Cindependent of cnsuch that\n(3.15) |∂yΦn(y0)| ≤C|(Z+(y0)−cn)(Z−(y0)−cn)|−3\n4.\nProof.We recall that we shall only prove the part of the lemma for Z+(y0) =c,as the proof\nwhenZ−(y0) =cis along the same lines. As Ran Z+∩RanZ−=∅,we can find some constant\nC >1 such that C−1<(Recn−Z−)0,i.e.,Z+(y0) is a local minimum.\nWe can choose h>0 small enough independent of nso thatI2h:= [y0−2h,y0+2h]⊂ I0.\nWe also denote Ih:= [y0−h,y0+h].Let us introduce a cut-off function χsatisfying\n(1)χ≡1 onIhandχ≡0 outside I2h,\n(2)χ∈C1and 0≤χ≤1,\nIntegrating by parts, we have/integraldisplay\nI0ΦnΦnχdy=−/integraldisplay\nI0(y−y0)Φn∂y/parenleftbig\nΦnχ/parenrightbig\ndy−/integraldisplay\nI0(y−y0)∂yΦnΦnχdy,\nwhich leads to\n(3.16) /ba∇dbl√χΦn/ba∇dblL2(I2h)≤2/ba∇dbl(y−y0)∂yΦn√χ/ba∇dblL2(I2h)+/ba∇dblΦn/ba∇dblL2(I2h\\Ih).11\nMultiplying Equation (3.2) by Φnχ,integrating by parts and taking the real part and the\nimaginary part separately, we obtain the following identit ies\n−/integraldisplay\nI0/parenleftbig\n(Z+−Recn)(Z−−Recn)−(Imcn)2/parenrightbig/parenleftbig\n|∂yΦn|2+α2|Φn|2/parenrightbig\nχdy\n= Re/a\\}b∇acketle{tFn,χΦn/a\\}b∇acket∇i}htL2(I0)+Re/integraldisplay\nI2h\\Ih(Z+−cn)(Z−−cn)Φn∂yΦnχ′dy,(3.17)\n−/integraldisplay\nI0Imcn(Z++Z−−2Recn)/parenleftbig\n|∂yΦn|2+α2|Φn|2/parenrightbig\nχdy\n= Im/a\\}b∇acketle{tFn,χΦn/a\\}b∇acket∇i}htL2(I0)+Im/integraldisplay\nI2h\\Ih(Z+−cn)(Z−−cn)Φn∂yΦnχ′dy.(3.18)\nTo estimate (3.18), we note that for hsmall enough and for nlarge enough, it holds that\n|Z+(y)−Recn| ≤Ch+dist(cn,RanZ+)≤ǫ0+Ch≤Ch,∀y∈ I2h,\nwhich leads to the following bound:\nZ++Z−−2Recn≤Z−(y)−Recn+|Z+−Recn| ≤ −C−1+Ch≤ −1\n2C−1.\nHence, it follows from (3.18) that\n|Imcn|/parenleftBig\n/ba∇dbl√χ∂yΦn/ba∇dbl2\nL2(I2h)+α2/ba∇dbl√χΦn/ba∇dbl2\nL2(I2h)/parenrightBig\n/lessorsimilar/integraldisplay\nI2h/vextendsingle/vextendsingleFnχΦn/vextendsingle/vextendsingledy+/ba∇dbl(Z+−cn)(Z−−cn)∂yΦn/ba∇dbl2\nL2(I2h\\Ih).(3.19)\nIn particular, (3.19) shows that/vextenddouble/vextenddouble/vextenddouble/radicalbig\n|Imcn|∂yΦn/vextenddouble/vextenddouble/vextenddouble\nL2(Ih)0,∀y∈ I0;|Recn−c| ≥ |Imcn|for sufficiently large nand\nZ′′\ns(y0)(Recn−c)≥0,wheres= +or−.\nThen(y−y0)∂yΦ∈L2and\n−/integraldisplay\nI0(Z+−c)(Z−−c)/parenleftbig\n∂yΦf′+α2Φf/parenrightbig\ndy=/integraldisplay\nI0Ffdy,∀f∈H1\nw,0, (3.25)\nwhereH1\nw,0=/braceleftbig\nf∈L2: ((y−y0)∂yf)∈L2andf|y=y1=f|y=y2= 0/bracerightbig\n.\nProof.As in Appendix A, we can construct a solution, which we denote byϕℓ\nn(orϕr\nn), to\nthe homogeneous equation\n∂y((Z+−cn)(Z−−cn)∂yϕ)−α2(Z+−cn)(Z−−cn)ϕ= 0, (3.26)\nϕ/parenleftBig\nyℓ\ncn/parenrightBig\n= 1, ∂yϕ/parenleftBig\nyℓ\ncn/parenrightBig\n= 0/parenleftbig\norϕ/parenleftbig\nyr\ncn/parenrightbig\n= 1, ∂yϕ/parenleftbig\nyr\ncn/parenrightbig\n= 0/parenrightbig\n, (3.27)\non an interval [ y1,y0] (or (y0,y2],respectively).\nWe note that yi\ncn→y0ascn→c,i=ℓorr.In turn, we denote by ϕℓ(orϕr) the solution\nto\n∂y((Z+−c)(Z−−c)∂yϕ)−α2(Z+−c)(Z−−c)ϕ= 0,\nϕ(y0) = 1, ∂yϕ(y0) = 0,\non [y1,y0] (or [y0,y2],respectively).\nWe shall use the following properties of ϕi\nnandϕi,i=ℓorr,as proven in Appendix A:\n(1)/ba∇dbl∂yϕi\nn/ba∇dblL∞+/ba∇dblϕi\nn/ba∇dblL∞≤Cand/ba∇dbl∂yϕi/ba∇dblL∞+/ba∇dblϕi/ba∇dblL∞≤C,\n(2)/vextendsingle/vextendsingleϕi\nn/vextendsingle/vextendsingle≥1\n2andϕi≥1,\n(3)/vextendsingle/vextendsingleϕi\nn(y)−1/vextendsingle/vextendsingle≤C/vextendsingle/vextendsingley−yi\ncn/vextendsingle/vextendsingle2, y∈[y1,yℓ\ncn] (or [yr\ncn,y2],respectively),\nand/vextendsingle/vextendsingleϕi(y)−1/vextendsingle/vextendsingle≤C|y−y0|2, y∈[y1,y0] (or [y0,y2],respectively),\n(4) lim\nn→∞ϕi\nn=ϕi.\nFory∈[y1,yℓ\ncn] we can solve the inhomogeneous equation (3.2) explicitly b y\nΦn(y) =ϕℓ\nn(y)\nϕℓn(y1)Φn(y1)+µℓ\nnϕℓ\nn(y)/integraldisplayy\ny11\n((Z+−cn)(Z−−cn)(ϕℓn)2)(y′)dy′\n+ϕℓ\nn(y)/integraldisplayy\ny1/integraltexty′\nyℓcn/parenleftbig\nFnϕℓ\nn/parenrightbig\n(y′′)dy′′\n((Z+−cn)(Z−−cn)(ϕℓn)2)(y′)dy′,(3.28)\nas (3.2) is equivalent to\n(3.29) ∂y/parenleftbigg\n(Z+−cn)(Z−−cn)/parenleftbig\nϕi\nn/parenrightbig2∂y/parenleftbiggΦn\nϕin/parenrightbigg/parenrightbigg\n=Fnϕi\nn, i=ℓorr.\nHere the coefficient µℓ\nnis given by\nµℓ\nn:= (Z+(y1)−cn)(Z−(y1)−cn)/parenleftbigg\n(ϕℓ\nnΦ′\nn)(y1)−/parenleftbigg/parenleftBig\nϕℓ\nn/parenrightBig′\nΦn/parenrightbigg\n(y1)/parenrightbigg\n−/integraldisplayy1\nyℓcn/parenleftBig\nFnϕℓ\nn/parenrightBig\n(y)dy.14 HAN LIU, NADER MASMOUDI, CUILI ZHAI, AND WEIREN ZHAO\nWe have for the last term in (3.28) that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleϕℓ\nn(y)/integraldisplayy\ny1/integraltexty′\nyℓcn/parenleftbig\nFnϕℓ\nn/parenrightbig\n(y′′)dy′′\n((Z+−cn)(Z−−cn)(ϕℓn)2)(y′)dy′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayy\ny11/parenleftbig\ny′+yℓcn/parenrightbigdy′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤Cln/parenleftBigg\ny+yℓ\ncn\ny1+yℓcn/parenrightBigg\n∈L2(y1,yℓ\ncn).(3.30)\nFor the second term on the right hand side of (3.28), we have th e point-wise limit for y∈\n[y1,y0),\nϕℓ\nn(y)/integraldisplayy\ny11\n((Z+−cn)(Z−−cn)(ϕℓn)2)(y′)dy′→ϕℓ(y)/integraldisplayy\ny11\n((Z+−c)(Z−−c)(ϕℓ)2)(y′)dy′.\nThe fact that Re( Z+−c)∼(y−y0)2implies that\nϕℓ(y)/integraldisplayy\ny11\n((Z+−c)(Z−−c)(ϕℓ)2)(y′)dy′∼1\n(y−y0)/\\e}atio\\slash∈L2(y1,y0). (3.31)\nOn the other hand, from the facts that /ba∇dblΦn/ba∇dblL2≤C,/ba∇dbl(Z+−cn)(Z−−cn)∂yΦn/ba∇dblH1≤C\nand/ba∇dblFn/ba∇dblL∞0 is already covered in Lemma 3.3.\n3.2.Strong convergence of {Φn}∞\nn=1neary0.As for strong convergence of {Φn}∞\nn=1and\n{(Z+−cn)(Z−−cn)∂yΦn}∞\nn=1whencn→c=Zs(y0) withZ′\ns(y0) = 0, s= + or−,we note\nthat for the scenarios in Lemma 3.4, by (3.22) and (3.33), it i s not difficult to show that as\nFn→0 inH1, Φn→0 and (Z+−cn)(Z−−cn)∂yΦn→0 inL2(I0) andH1(I0),respectively\nvia integration by parts, while a more delicate analysis is n eeded for the scenario in Lemma\n3.5, i.e.,Z′′\ns(y0)(Recn−c)≥0, s= + or−,and|Recn−c| ≥ |Imcn|.We shall postpone\nthe proof for the scenarios in Lemma 3.4 to Section 3.3 and foc us primarily on that for the\nscenario in Lemma 3.5.\nWe shall prove the following lemma.\nLemma 3.6. Lety0∈/parenleftbig\n(Z+)−1(c)∪(Z−)−1(c)/parenrightbig\nbe a critical point, i.e., Z′\ns(y0) = 0, and\nthe interval I0:= [y1,y2]be such that y0∈ I0andZ′′\ns(y0)Z′′\ns(y)>0, s= +or−.Let\n{(cn,Φn,Fn)}∞\nn=1andΦsatisfy the conditions as in Lemma 3.5, then Φn→ΦinL2inI0.\nIn addition, the following estimates hold\n|Φn(y0)| ≤C|(Z−(y0)−cn)(Z+(y0)−cn)|−1\n4, (3.35)\n|∂yΦn(y0)| ≤C|(Z−(y0)−cn)(Z+(y0)−cn)|−3\n4. (3.36)\nTo prove Lemma 3.6, we shall construct solutions to Equation (3.2) inI0and obtain its\nexplicit formula using ODE techniques.\nWithout loss of generality, we may assume that the critical p ointy0= 0.LetI0:= [y1,y2]\nbe such that Z+(y1) =Z+(y2) (orZ−(y1) =Z−(y2)) withy1<0< y2.We shall rewrite\nEquation (3.2) by introducing\nΦ∗\nn(y) := Φn(y)−LΦn(y) := Φn(y)−Φn(y2)−Φn(y1)\n(y2−y1)(y−y1)−Φn(y1),16 HAN LIU, NADER MASMOUDI, CUILI ZHAI, AND WEIREN ZHAO\nwhich leads to\n(3.37)/braceleftBigg\n∂y((Z+−cn)(Z−−cn)∂yΦ∗\nn)−α2(Z+−cn)(Z−−cn)Φ∗\nn=F∗\nn,\nΦ∗\nn(y1) = Φ∗\nn(y2) = 0,\nwith\nF∗\nn(y) =Fn(y)−Φn(y2)−Φn(y1)\n(y2−y1)(Z′\n+)(y)(Z−(y)−cn)\n−Φn(y2)−Φn(y1)\n(y2−y1)(Z′\n−)(y)(Z+(y)−cn)+α2(Z+−cn)(Z−−cn)LΦn(y).\nWe can see that to study Equation (3.37) is to study an equatio n of the following type\n(3.38)/braceleftBigg\n∂y((Z+−c∗)(Z−−c∗)∂yΦ∗)−α2(Z+−c∗)(Z−−c∗)Φ∗=F∗,\nΦ∗(y1) = Φ∗(y2) = 0.\nHerec∗∈(Ωǫ0\\(RanZ+∪RanZ−)) satisfies the conditions |Rec∗−Z+(0)| ≥ |Imc∗|and\nZ′′\n+(0)(Rec∗−Z+(0))≥0 (or|Rec∗−Z−(0)| ≥ |Imc∗|andZ′′\n−(0)(Rec∗−Z−(0))≥0), while\nF∗∈H1.Forc∗closeenoughto Z+(0) (orZ−(0)), wecanfindexactly twopoints yi\n∗,i=ℓorr,\nin a sufficiently small neighborhood of 0 such that yℓ\n∗≤0≤yr\n∗andZ+/parenleftbig\nyℓ\n∗/parenrightbig\n=Z+(yr\n∗) = Rec∗\n(orZ−/parenleftbig\nyℓ\n∗/parenrightbig\n=Z+(yr\n∗) = Rec∗). It turns out that the solution to Equation (3.38) enjoys th e\nfollowing estimate.\nLemma 3.7. LetΦ∗be the solution to Equation (3.38). There exists some C >0independent\nofc∗such that\n/ba∇dblΦ∗/ba∇dblL2(y1,y2)≤C/ba∇dblF∗/ba∇dblL∞.\nBefore we prove Lemma 3.7, we make some preparations. Recall that here we assume that\n0 is a critical point of Z+, i.e.Z′\n+(0) = 0 (or Z′\n−(0) = 0) and that we restrict ourselves to the\ncasec∗→Z+(0) with Im c∗>0, as the proof for the other cases are along the same lines.\nWithout loss of generality, we may also assume that Z′′\n+(0)>0,i.e.,Z+(0) is a local\nminimum. We define σ(c∗)∈C,which satisfies ( σ(c∗))2=c∗−Z+(0) with Im σ(c∗)>0.\nTo this end, we proceed to introduce the notations. We define t he function V(y) such that\n(V(y))2=Z+(y)−Z+(0),i.e.,\n(3.39) V(y) =/braceleftBigg\n−/radicalbig\nZ+(y)−Z+(0) on [y1,0],/radicalbig\nZ+(y)−Z+(0) on [0,y2].\nItcanbeverified viadirect computations that V∈C2(I0) ismonotone, and V′(0) =√\n2Z′′\n+(0)\n2.\nDenoting the solution to Equation (3.38) on [ y1,0] by Φℓ\n∗and that on [0 ,y2] by Φr\n∗,we\nnotice that Equation (3.38) is equivalent to\n(3.40) ∂y/parenleftBigg\n(Z−−c∗)(Z+−c∗)/parenleftbig\nϕj\n∗/parenrightbig2∂y/parenleftBigg\nΦj\n∗\nϕj\n∗/parenrightBigg/parenrightBigg\n=F∗ϕj\n∗, j=ℓorr,\nwhereϕr\n∗andϕℓ\n∗are the solutions to the homogeneous equation\n∂y((Z−−c∗)(Z+−c∗)∂yϕ∗)−α2(Z−−c∗)(Z+−c∗)ϕ∗= 0, (3.41)\nϕ∗/parenleftbig\nyℓ\n∗/parenrightbig\n= 1, ∂yϕ∗/parenleftbig\nyℓ\n∗/parenrightbig\n= 0,fory∈[y1,0], (3.42)\norϕ∗(yr\n∗) = 1, ∂yϕ∗(yr\n∗) = 0,fory∈[0,y2], (3.43)17\nwithϕr\n∗andϕℓ\n∗corresponding to condition (3.42) and corresponding to (3. 43), respectively.\nThe following properties of ϕj\n∗, j=ℓorr,shall be useful –\n(1)|ϕj\n∗|>1\n2,\n(2)|ϕj\n∗(y)−1| ≤C/vextendsingle/vextendsingle/vextendsingley−yj\n∗/vextendsingle/vextendsingle/vextendsingle2\n, y∈[y1,0] (or [0,y2]),\n(3)|∂yϕj\n∗(y)| ≤C/vextendsingle/vextendsingle/vextendsingley−yj\n∗/vextendsingle/vextendsingle/vextendsingle, y∈[y1,0] (or [0,y2]).\nWe can integrate twice and obtain explicit solution formula e to Equation (3.38). On [0 ,y2],\nthe solution to Equation (3.38) is given by\nΦr\n∗(y) =νr[F∗](c∗)ϕr\n∗(y)+µr[F∗](c∗)ϕr\n∗(y)/integraldisplayy\n01\n(Z−(y′)−c∗)(Z+(y′)−c∗)(ϕr∗(y′))2dy′\n+ϕr\n∗(y)/integraldisplayy\n0/integraltexty′\nyr∗(F∗ϕr\n∗)(z)dz\n(Z−(y′)−c∗)(Z+(y′)−c∗)(ϕr∗(y′))2dy′\n=/tildewideµr[F∗](c∗)ϕr\n∗(y)/integraldisplayy\ny21\n(Z−(y′)−c∗)(Z+(y′)−c∗)(ϕr∗(y′))2dy′\n+ϕr\n∗(y)/integraldisplayy\ny2/integraltexty′\nyr∗(F∗ϕr\n∗)(z)dz\n(Z−(y′)−c∗)(Z+(y′)−c∗)(ϕr∗(y′))2dy′,(3.44)\nwhile the solution to Equation (3.38) on [ y1,0] is given by\nΦℓ\n∗(y) =ϕℓ\n∗(y)/integraldisplayy\ny1/integraltexty′\nyℓ∗(F∗ϕℓ\n∗)(z)dz\n(Z−(y′)−c∗)(Z+(y′)−c∗)(ϕℓ∗(y′))2dy′\n+/tildewideµℓ[F∗](c∗)ϕℓ\n∗(y)/integraldisplayy\ny11\n(Z−(y′)−c∗)(Z+(y′)−c∗)(ϕℓ∗(y′))2dy′\n=µℓ[F∗](c∗)ϕℓ\n∗(y)/integraldisplayy\n01\n(Z−(y′)−c∗)(Z+(y′)−c∗)(ϕℓ∗(y′))2dy′+νℓ[F∗](c∗)ϕℓ\n∗(y)\n+ϕℓ\n∗(y)/integraldisplayy\n0/integraltexty′\nyℓ∗(F∗ϕℓ\n∗)(z)dz\n(Z−(y′)−c∗)(Z+(y′)−c∗)(ϕℓ∗(y′))2dy′.\nHere the coefficients µj[F∗](c∗),/tildewideµj[F∗](c∗) andνj[F∗](c∗), j=ℓorr,are determined by F∗\nandc∗in a way such that Φr\n∗and Φℓ\n∗are well-defined and satisfy the conditions\n(3.45)/braceleftBigg\nΦℓ\n∗(y1) = 0,Φr\n∗(y2) = 0,\nΦℓ\n∗(0) = Φr\n∗(0), ∂yΦℓ\n∗(0) =∂yΦr\n∗(0),\nwhich gives us\n\n\nµr[F∗](c∗) =/tildewideµr[F∗](c∗), µℓ[F∗](c∗) =/tildewideµℓ[F∗](c∗),\nIr(c∗)µr[F∗](c∗)+νr[F∗](c∗) =−Tr[F∗](c∗),\nIℓ(c∗)µℓ[F∗](c∗)−νℓ[F∗](c∗) =Tℓ[F∗](c∗),\nϕr\n∗(0)νr[F∗](c∗)−ϕℓ\n∗(0)νℓ[F∗](c∗) = 0,\nϕℓ\n∗(0)µr[F∗](c∗)−ϕr\n∗(0)µℓ[F∗](c∗)+(Z−(0)−c∗)(Z+(0)−c∗)(ϕℓ\n∗ϕr\n∗∂yϕr\n∗)(0)νr[F∗](c∗)\n−(Z−(0)−c∗)(Z+(0)−c∗)(ϕr\n∗ϕℓ\n∗∂yϕℓ\n∗)(0)νℓ[F∗](c∗) =L[F∗](c∗).18 HAN LIU, NADER MASMOUDI, CUILI ZHAI, AND WEIREN ZHAO\nWe rewrite the above set of equations in the form of matrix equ ation as\n\nIr0 1 0\n0Iℓ0 −1\n0 0 ϕr\n∗(0) −ϕℓ\n∗(0)\nϕℓ\n∗(0)−ϕr\n∗(0)H0(c∗)(ϕℓ\n∗ϕr\n∗∂yϕr\n∗)(0)−H0(c∗)(ϕr\n∗ϕℓ\n∗∂yϕℓ\n∗)(0)\n\nµr[F∗]\nµℓ[F∗]\nνr[F∗]\nνℓ[F∗]\n=\n−Tr[F∗]\nTℓ[F∗]\n0\nL[F∗]\n,\nwhereH0(c∗) := (Z−(0)−c∗)(Z+(0)−c∗) and\nIℓ(c∗) :=/integraldisplay0\ny11\n(Z−(y)−c∗)(Z+(y)−c∗)(ϕℓ∗(y))2dy, (3.46)\nIr(c∗) :=/integraldisplayy2\n01\n(Z−(y)−c∗)(Z+(y)−c∗)(ϕr∗(y))2dy, (3.47)\nL[F∗](c∗) :=ϕℓ\n∗(0)/integraldisplayyr\n∗\n0(F∗ϕr\n∗)(y)dy−ϕr\n∗(0)/integraldisplayyℓ\n∗\n0(F∗ϕℓ\n∗)(y)dy, (3.48)\nTℓ[F∗](c∗) :=/integraldisplayy1\n0/integraltexty\nyℓ∗(F∗ϕℓ\n∗)(z)dz\n(Z−(y)−c∗)(Z+(y)−c∗)(ϕℓ∗(y))2dy, (3.49)\nTr[F∗](c∗) :=/integraldisplayy2\n0/integraltexty\nyr∗(F∗ϕr\n∗)(z)dz\n(Z−(y)−c∗)(Z+(y)−c∗)(ϕr∗(y))2dy. (3.50)\nAnd hereafter, we denote\nD(c∗) :=(Z−(0)−c∗)(Z+(0)−c∗)ϕr\n∗(0)ϕℓ\n∗(0)/parenleftBig\nϕℓ\n∗(0)∂yϕr\n∗(0)−ϕr\n∗(0)∂yϕℓ\n∗(0)/parenrightBig\nIr(c∗)Iℓ(c∗)\n−(ϕr\n∗(0))2Ir(c∗)−(ϕℓ\n∗(0))2Iℓ(c∗),\nwhich is in fact the determinant of the matrix in the matrix eq uation above.\nContinuing solving for the coefficients, we obtain\nµr[F∗](c∗) =/tildewideµr[F∗](c∗)\n:=1\nD(c∗)(Z−(0)−c∗)(Z+(0)−c∗)ϕr\n∗(0)ϕℓ\n∗(0)/parenleftBig\n∂yϕℓ\n∗(0)−∂yϕr\n∗(0)/parenrightBig\nTr[F∗](c∗)Iℓ(c∗)\n−1\nD(c∗)/parenleftBig\nϕℓ\n∗(0)L[F∗](c∗)Iℓ(c∗)+(ϕr\n∗(0))2Tr[F∗](c∗)/parenrightBig\n−1\nD(c∗)/parenleftBig\nϕr\n∗(0)ϕℓ\n∗(0)Tℓ[F∗](c∗)/parenrightBig\n,(3.51)\nµℓ[F∗](c∗) =/tildewideµℓ[F∗](c∗)\n:=1\nD(c∗)(Z−(0)−c∗)(Z+(0)−c∗)ϕr\n∗(0)ϕℓ\n∗(0)/parenleftBig\n∂yϕℓ\n∗(0)−∂yϕr\n∗(0)/parenrightBig\nTℓ[F∗](c∗)Ir(c∗)\n+1\nD(c∗)/parenleftbigg\nϕr\n∗(0)L[F∗](c∗)Ir(c∗)−/parenleftBig\nϕℓ\n∗(0)/parenrightBig2\nTℓ[F∗](c∗)/parenrightbigg\n+1\nD(c∗)/parenleftBig\nϕr\n∗(0)ϕℓ\n∗(0)Tr[F∗](c∗)/parenrightBig\n,(3.52)\nνr[F∗](c∗) =:1\nD(c∗)/parenleftBig\nϕℓ\n∗(0)L[F∗](c∗)Ir(c∗)Iℓ(c∗)−ϕr\n∗(0)ϕℓ\n∗(0)Tℓ[F∗](c∗)Ir(c∗)/parenrightBig\n+1\nD(c∗)/parenleftbigg/parenleftBig\nϕℓ\n∗(0)/parenrightBig2\nTr[F∗](c∗)Iℓ(c∗)/parenrightbigg\n,(3.53)19\nνℓ[F∗](c∗) :=1\nD(c∗)/parenleftBig\nϕr\n∗(0)L[F∗](c∗)Ir(c∗)Iℓ(c∗)+ϕr\n∗(0)ϕℓ\n∗(0)Tr[F∗](c∗)Iℓ(c∗)/parenrightBig\n+1\nD(c∗)/parenleftBig\n(ϕr\n∗(0))2Tℓ[F∗](c∗)Ir(c∗)/parenrightBig\n.(3.54)\nThus, we can verify that\nΦ∗(y,c∗) =/braceleftBigg\nΦℓ\n∗(y,c∗) on [y1,0],\nΦr\n∗(y,c∗) on [0,y2]\nis well-defined and is the unique C1- solution to (3.38) on I0.\nTo facilitate the estimation of Ik(c∗) fork=rorℓ, we introduce also the following\nquantities\nIr\n1(c∗) :=/integraldisplayy2\n01\n(Z−(y)−c∗)(Z+(y)−c∗)/parenleftbigg1\n(ϕr∗(y))2−1/parenrightbigg\ndy+/integraldisplayy2\n01\n2b(y)(Z−(y)−c∗)dy,\nIℓ\n1(c∗) :=/integraldisplay0\ny11\n(Z−(y)−c∗)(Z+(y)−c∗)/parenleftBigg\n1\n(ϕℓ∗(y))2−1/parenrightBigg\ndy+/integraldisplay0\ny11\n2b(y)(Z−(y)−c∗)dy,\nIr\n2(σ(c∗)) :=−1\n2b(y2)V′(y2)ln/parenleftbigg|V(y2)−σ(c∗)|\n|V(y2)+σ(c∗)|/parenrightbigg\n+i\n2b(y2)V′(y2)arctan/parenleftbiggImσ(c∗)\n|V(y2)−Reσ(c∗)|/parenrightbigg\n+/integraldisplayy2\n0∂y/parenleftbigg1\n2b(y)V′(y)/parenrightbigg\nLog/parenleftbiggV(y)−σ(c∗)\nV(y)+σ(c∗)/parenrightbigg\ndy\n+i\n2b(y2)V′(y2)arctan/parenleftbiggImσ(c∗)\n|V(y2)+Reσ(c∗)|/parenrightbigg\n,\nIℓ\n2(σ(c∗)) :=1\n2b(y1)V′(y1)ln/parenleftbigg|V(y1)−σ(c∗)|\n|V(y1)+σ(c∗)|/parenrightbigg\n+/integraldisplay0\ny1∂y/parenleftbigg1\n2b(y)V′(y)/parenrightbigg\nLog/parenleftbiggV(y)−σ(c∗)\nV(y)+σ(c∗)/parenrightbigg\ndy\n+i\n2b(y1)V′(y1)arctan/parenleftbiggImσ(c∗)\n|V(y1)+Reσ(c∗)|/parenrightbigg\n+i\n2b(y1)V′(y1)arctan/parenleftbiggImσ(c∗)\n|V(y1)−Reσ(c∗)|/parenrightbigg\n−2iπ\n2b(y1)V′(y1)+2iπ\n2b(0)V′(0),\nwhere Log is the complex logarithm with the principal value o f the argument in ( −π,π].\nWe shall prove the following auxiliary estimates on Ik(σ(c∗)), k=ℓorrandD(c∗) which\nwill help us characterize the behaviors of the coefficients µk[F∗](c∗),/tildewideµk[F∗](c∗) andνk[F∗](c∗),\nk=ℓorr,in the solution formulae to Equation (3.38).\nLemma 3.8. Assume that Imc∗>0andImσ(c∗)>0.It holds that\n(3.55) 2 σ(c∗)Ik(c∗) =−iπ\nb(0)/radicalbig\n2Z′′\n+(0)+2σ(c∗)Ik\n1(c∗)+Ik\n2(σ(c∗)), k=ℓorr.\nMoreover, there exist some δ0>0and a constant Cdepending only on α,such that if\n|σ(c∗)|<δ0,then the following estimates are true for k=ℓorr\n|Ik\n1(c∗)| ≤C, (3.56)\n|Ik\n2(σ(c∗))| ≤C|σ(c∗)|1\n4, (3.57)\nC−1≤ |2σ(c∗)Ik(c∗)| ≤C. (3.58)20 HAN LIU, NADER MASMOUDI, CUILI ZHAI, AND WEIREN ZHAO\nIn particular,\nlim\nc∗→Z+(0)2σ(c∗)Ik(c∗) =−iπ\nb(0)/radicalbig\n2Z′′\n+(0), k=ℓorr.\nProof.Splitting the integral Ir(c∗) and utilizing the function V,we have\n2σ(c∗)Ir(c∗) =2σ(c∗)/integraldisplayy2\n01\n(Z−(y)−c∗)(Z+(y)−c∗)(ϕr∗(y))2dy\n=2σ(c∗)/integraldisplayy2\n01\n(Z−(y)−c∗)(Z+(y)−c∗)/parenleftbigg1\n(ϕr∗(y))2−1/parenrightbigg\ndy\n+2σ(c∗)/integraldisplayy2\n01\n2b(y)(Z−(y)−c∗)dy−2σ(c∗)/integraldisplayy2\n01\n2b(y)(Z+(y)−c∗)dy\n=2σ(c∗)Ir\n1(c∗)−/integraldisplayy2\n01\n2b(y)(V(y)−σ(c∗))dy+/integraldisplayy2\n01\n2b(y)(V(y)+σ(c∗))dy\n=2σ(c∗)Ir\n1(c∗)−/integraldisplayy2\n01\n2b(y)V′(y)∂y(Log(V(y)−σ(c∗)))dy\n+/integraldisplayy2\n01\n2b(y)V′(y)∂y(Log(V(y)+σ(c∗)))dy.\nIntegration by parts yields\n2σ(c∗)Ir(c∗) =2σ(c∗)Ir\n1(c∗)−Log(V(y)−σ(c∗))\n2b(y)V′(y)/vextendsingle/vextendsingle/vextendsingley=y2\ny=0+Log(V(y)+σ(c∗))\n2b(y)V′(y)/vextendsingle/vextendsingle/vextendsingley=y2\ny=0\n+/integraldisplayy2\n0∂y/parenleftbigg1\n2b(y)V′(y)/parenrightbigg\n(Log(V(y)−σ(c∗))−Log(V(y)+σ(c∗)))dy\n=2σ(c∗)Ir\n1(c∗)−1\n2b(y2)V′(y2)ln/parenleftbigg|V(y2)−σ(c∗)|\n|V(y2)+σ(c∗)|/parenrightbigg\n+/integraldisplayy2\n0∂y/parenleftbigg1\n2b(y)V′(y)/parenrightbigg\nLog/parenleftbiggV(y)−σ(c∗)\nV(y)+σ(c∗)/parenrightbigg\ndy\n+i\n2b(y2)V′(y2)arctan/parenleftbiggImσ(c∗)\n|V(y2)+Reσ(c∗)|/parenrightbigg\n+i\n2b(y2)V′(y2)arctan/parenleftbiggImσ(c∗)\n|V(y2)+Reσ(c∗)|/parenrightbigg\n−iπ\n2b(0)V′(0)\n=2σ(c∗)Ir\n1(c∗)−iπ\n2b(0)V′(0)+Ir\n2(σ(c∗)).\nSimilarly, we have\n2σ(c∗)Iℓ(c∗) =2σ(c∗)Iℓ\n1(c∗)−/integraldisplay0\ny11\n2b(y)(V(y)−σ(c∗))dy+/integraldisplay0\ny11\n2b(y)(V(y)+σ(c∗))dy\n=2σ(c∗)Iℓ\n1(c∗)−Log(V(y)−σ(c∗))\n2b(y)V′(y)/vextendsingle/vextendsingle/vextendsingley=0\ny=y1\n+/integraldisplay0\ny1∂y/parenleftbigg1\n2b(y)V′(y)/parenrightbigg\nLog(V(y)−σ(c∗))dy+Log(V(y)+σ(c∗))\n2b(y)V′(y)/vextendsingle/vextendsingle/vextendsingley=0\ny=y1\n−/integraldisplay0\ny1∂y/parenleftbigg1\n2b(y)V′(y)/parenrightbigg\nLog(V(y)+σ(c∗))dy21\n=2σ(c∗)Iℓ\n1(c∗)+1\n2b(y1)V′(y1)ln/parenleftbigg|V(y1)−σ(c∗)|\n|V(y1)+σ(c∗)|/parenrightbigg\n+/integraldisplay0\ny1∂y/parenleftbigg1\n2b(y)V′(y)/parenrightbigg\nLog/parenleftbiggV(y)−σ(c∗)\nV(y)+σ(c∗)/parenrightbigg\ndy\n−i\n2b(y1)V′(y1)/parenleftbigg\n2π−arctanImσ(c∗)\n|V(y1)+Reσ(c∗)|−arctanImσ(c∗)\n|V(y1)−Reσ(c∗)|/parenrightbigg\n−i\n2b(0)V′(0)/parenleftbigg\n−π+arctanImσ(c∗)\n|Reσ(c∗)|−arctanImσ(c∗)\n|Reσ(c∗)|/parenrightbigg\n=2σ(c∗)Iℓ\n1(c∗)−iπ\n2b(0)V′(0)+Iℓ\n2(σ(c∗)).\nWe have thus shown (3.55).\nWe have the bound on Ik\n1(c∗), k=ℓorr,in (3.56) for |σ(c∗)|< δ0≪1 by the fact that/vextendsingle/vextendsingleϕk\n∗/vextendsingle/vextendsingle≥1\n2and/vextendsingle/vextendsingleϕk\n∗(y)−1/vextendsingle/vextendsingle≤C/vextendsingle/vextendsingley−yk\n∗/vextendsingle/vextendsingle2, k=ℓorr.\nWe proceed to prove (3.57). As |V′(y2)|>0,by taking |Im(σ(c∗))| ≪ |Re(σ(c∗))|< δ0,\nwe obtain the following\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglei\n2b(y2)V′(y2)/parenleftbigg\narctanImσ(c∗)\n|V(y2)−Reσ(c∗)|+arctanImσ(c∗)\n|V(y2)+Reσ(c∗)|/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C|σ(c∗)|. (3.59)\nFor sufficiently small δ0and|σ(c∗)|<δ0,we have\n(3.60)1\n2b(y2)V′(y2)ln/parenleftbigg|V(y2)−σ(c∗)|\n|V(y2)+σ(c∗)|/parenrightbigg\n≤C|σ(c∗)|,\nas\n1\n2b(y2)V′(y2)ln/parenleftbigg|V(y2)−σ(c∗)|\n|V(y2)+σ(c∗)|/parenrightbigg\n=1\n2b(y2)V′(y2)ln/parenleftbigg\n1−2V(y2)Reσ(c∗)\n|V(y2)+Reσ(c∗)|2+|Im(σ(c∗))|2/parenrightbigg\n.\nWe split the integral as follows –\n/integraldisplayy2\n0∂y/parenleftbigg1\n2b(y)V′(y)/parenrightbigg\nLog/parenleftbiggV(y)−σ(c∗)\nV(y)+σ(c∗)/parenrightbigg\ndy\n=/integraldisplay\nE∂y/parenleftBig1\n2b(y)V′(y)/parenrightBig\nLog/parenleftbiggV(y)−σ(c∗)\nV(y)+σ(c∗)/parenrightbigg\ndy\n+/integraldisplay\nEc∂y/parenleftBig1\n2b(y)V′(y)/parenrightBig\nLog/parenleftbiggV(y)−σ(c∗)\nV(y)+σ(c∗)/parenrightbigg\ndy\n:=K1(σ(c∗))+K2(σ(c∗)),\nwhere the set Eis defined as E:=/braceleftBig\ny∈[0,y2] :V(y)0small enough,\nC−1\n|σ(c∗)|≤ |D(c∗)| ≤C\n|σ(c∗)|. (3.63)\nIn addition, if Imc∗>0,then\nlim\nc∗→Z+(0)σ(c∗)D(c∗) =iπ\nb(0)/radicalbig\n2Z′′\n+(0).\nRecalling (3.44) and its counterpart on [ y1,0],we are ready to prove Lemma 3.7\nProof of Lemma 3.7. It is easy to check that\n|L[F∗](c∗)| ≤C/parenleftBig\n|yr\n∗|+|yl\n∗|/parenrightBig\n/ba∇dblF∗/ba∇dblL∞.\nWe also have the following estimate for Tℓ[F∗](c∗),\n/vextendsingle/vextendsingle/vextendsingleTℓ[F∗](c∗)/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayy1\n0/integraltexty\nyℓ∗F∗(z)dz\n(Z−(y)−c∗)(Z+(y)−c∗)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayy1\n0/integraldisplayy\nyℓ∗/parenleftbiggF∗(z)\n(Z−(y)−c∗)(Z+(y)−c∗)/parenrightbigg/parenleftbiggϕℓ\n∗(z)\nϕℓ∗(y)2−1/parenrightbigg\ndzdy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤C/ba∇dblF∗/ba∇dblL∞/parenleftbigg/integraldisplayy1\n01\n|y|+|yℓ∗|dy+1/parenrightbigg\n≤C/parenleftBig\n1+/vextendsingle/vextendsingle/vextendsingleln/parenleftBig\n|yℓ\n∗|/parenrightBig/vextendsingle/vextendsingle/vextendsingle/parenrightBig\n/ba∇dblF∗/ba∇dblL∞.\nSimilarly,\n|Tr[F∗](c∗)| ≤C(1+|ln(|yr\n∗|)|)/ba∇dblF∗/ba∇dblL∞.\nBy Lemma 3.8, Corollary 3.9 and the fact that |yr\n∗| ∼ |yℓ\n∗|,we have the following bounds\non the coefficients µj[F∗](c∗),/tildewideµj[F∗](c∗) andνj[F∗](c∗), j=ℓorr,/vextendsingle/vextendsingleµj[F∗](c∗)/vextendsingle/vextendsingle≤C|σ(c∗)|(1+|ln(|yr\n∗|)|)/ba∇dblF∗/ba∇dblL∞, (3.64)/vextendsingle/vextendsingle/tildewideµj[F∗](c∗)/vextendsingle/vextendsingle≤C|σ(c∗)|(1+|ln(|yr\n∗|)|)/ba∇dblF∗/ba∇dblL∞, (3.65)/vextendsingle/vextendsingleνj[F∗](c∗)/vextendsingle/vextendsingle≤C(1+|ln(|yr\n∗|)|)/ba∇dblF∗/ba∇dblL∞. (3.66)\nWe recall the explicit formula of Φr\n∗given by (3.44)\nΦr\n∗(y) =νr[F∗](c∗)ϕr\n∗(y)+µr[F∗](c∗)ϕr\n∗(y)/integraldisplayy\n01\n(Z−(y′)−c∗)(Z+(y′)−c∗)(ϕr∗(y′))2dy′\n+ϕr\n∗(y)/integraldisplayy\n0/integraltexty′\nyr∗(F∗ϕr\n∗)(z)dz\n(Z−(y′)−c∗)(Z+(y′)−c∗)(ϕr∗(y′))2dy′=:J1+J2+J3,fory∈(0,yr\n∗);\nand\nΦr\n∗(y) =/tildewideµr[F∗](c∗)ϕr\n∗(y)/integraldisplayy\ny21\n(Z−(y′)−c∗)(Z+(y′)−c∗)(ϕr∗(y′))2dy′\n+ϕr\n∗(y)/integraldisplayy\ny2/integraltexty′\nyr∗(F∗ϕr\n∗)(z)dz\n(Z−(y′)−c∗)(Z+(y′)−c∗)(ϕr∗(y′))2dy′=:/tildewideJ1+/tildewideJ2,fory∈(yr\n∗,y2).24 HAN LIU, NADER MASMOUDI, CUILI ZHAI, AND WEIREN ZHAO\nFrom (3.66), it is clear that for y∈(0,yr\n∗),\n|J1| ≤C(1+|ln(|yr\n∗|)|)/ba∇dblF∗/ba∇dblL∞≤C(1+|ln(|y|)|)/ba∇dblF∗/ba∇dblL∞.\nFrom (3.64) and the fact that (1+ |ln(|yr\n∗|)|)/lessorsimilar|σ(c∗)|−γ,0<γ <1\n4,forc∗close toZ+(0),\nwe infer that\n|J2| ≤C/ba∇dblF∗/ba∇dblL∞|σ(c∗)|1−γ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayy\n01\n(|y′|+|σ(c∗)|)(y′−yr∗)dy′/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤C/ba∇dblF∗/ba∇dblL∞|σ(c∗)|γ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayy\n01\n|y′−yr∗|1+2γdy′/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤C|σ(c∗)|γ/ba∇dblF∗/ba∇dblL∞/parenleftbigg1\n|y−yr∗|2γ+1/parenrightbigg\n,(3.67)\nwhich shows that /ba∇dblJ2/ba∇dblL2≤C/ba∇dblF∗/ba∇dblL∞.Ina similar way, we can show that /ba∇dbl/tildewideJ1/ba∇dblL2≤C/ba∇dblF∗/ba∇dblL∞.\nThe termJ3enjoys the following estimate/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayy\n0/integraltexty′\nyr∗(F∗ϕr\n∗)(z)dz\n(Z−(y′)−c∗)(Z+(y′)−c∗)(ϕr∗(y′))2dy′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ /ba∇dblF∗/ba∇dblL∞(1+|ln(|yr\n∗|+|y|)|),\nand the estimate for /tildewideJ2is similar.\nTherefore, it holds that\n/ba∇dblΦr\n∗/ba∇dblL2(0,y2)≤ /ba∇dblJ1/ba∇dblL2(0,yr∗)+/ba∇dblJ2/ba∇dblL2(0,yr∗)+/ba∇dblJ3/ba∇dblL2(0,yr∗)+/ba∇dbl/tildewideJ1/ba∇dblL2(yr∗,y2)+/ba∇dbl/tildewideJ2/ba∇dblL2(yr∗,y2)\n≤C/ba∇dblF∗/ba∇dblL∞.\nSimilarly, we have /ba∇dblΦℓ\n∗/ba∇dblL2(y1,0)≤C/ba∇dblF∗/ba∇dblL∞,which concludes our proof. /square\nRemark 3.10. In the case that c∗∈RanZ−withImc∗>0, without loss of generality, we\nmay assume that Z′\n−(0) = 0andZ′′\n−(0)>0.We can instead let V(y)2=Z−(y)−Z−(0)and\n(σ(c∗))2=c∗−Z−(0).Then by the same argument as in the proof of Lemma 3.8, we obtain\nlim\nc∗→Z−(0)2σ(c∗)Ik(c∗) =iπ\nb(0)/radicalbig\n2Z′′\n−(0), k=ℓorr\nand\nlim\nc∗→Z−(0)σ(c∗)D(c∗) =−iπ\nb(0)/radicalbig\n2Z′′\n−(0).\nProof of Lemma 3.6. Recalling Lemma 3.3, which asserts the strong convergence\nΦn→Φ inL2and (Z−−cn)(Z+−cn)∂yΦn→(Z−−c)(Z+−c)∂yΦ inH1,\naway from the critical points in Z−1\n+(c),along with Equation (3.37), we know that\nF∗\nn→F∗\n∞:=F(y)−Φ(y2)−Φ(y1)\n(y2−y1)Z′\n+(y)(Z−(y)−c)\n−Φ(y2)−Φ(y1)\n(y2−y1)Z′\n−(y)(Z+(y)−c)+α2(Z+−c)(Z−−c)LΦ(y)(3.68)\ninH1from our assumption that Fn→FinH1asn→ ∞.\nFrom Lemma 3.8 and Corollary 3.9, we know that as cn→candF∗\nn→F∗\n∞inH1,\n/tildewideµr[F∗\nn](cn)ϕr\n∗(y)/integraldisplayy\ny21\n(Z−−cn)(Z+−cn)(ϕr∗(y′))2dy′→0 inL2(0,y2),25\nwhoseanaloguesalsoholdforthetermscorrespondingto /tildewideµℓ[F∗\nn](cn),µℓ[F∗\nn](cn)andµr[F∗\nn](cn).\nTherefore, we know from (3.44) that as cn→c=Z+(0) andF∗\nn→F∗\n∞inH1,\nΦ∗\nn→Φ∗\n∞inL2(y1,y2),with Φ∗\n∞(y) :=\n\nϕℓ(y)/integraltexty\ny1/integraltexty′\n0(F∗\n∞ϕℓ)(z)dz\n(Z−−c)(Z+−c)(ϕℓ(y′))2dy′on [y1,0),\nϕr(y)/integraltexty\ny2/integraltexty′\n0(F∗\n∞ϕr)(z)dz\n(Z−−c)(Z+−c)(ϕr(y′))2dy′on (0,y2],\nwhereϕℓandϕrarethe solutions to thecorrespondinghomogeneous equatio n, as constructed\nin Appendix A. It is then evident that Φ n→Φ inL2(I0).\nIt follows from (3.44) and (3.66) that\n|Φ∗\nn(0)| ≤|νr[F∗\nn](cn)| ≤C/parenleftBig\n1+/vextendsingle/vextendsingle/vextendsingleln/parenleftBig/radicalbig\n|(cn−Z+(0))(cn−Z−(0))|/parenrightBig/vextendsingle/vextendsingle/vextendsingle/parenrightBig\n/ba∇dblF∗\nn/ba∇dblL∞\n≤C|(Z+(0)−cn)(Z−(0)−cn)|−1\n4.\nDifferentiating (3.44) with c∗=cnaty= 0 yields\n|∂yΦ∗\nn(0)|=|∂yΦr\n∗(0,cn)|\n≤ |νr[F∗\nn](cn)∂yϕr\n∗(0)|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n(Z−(0)−cn)(Z+(0)−cn)ϕr∗(0)/parenleftBigg\nµr[F∗\nn](cn)+/integraldisplay0\nyr∗(F∗\nnϕr\n∗)(z)dz/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤C(1+|ln(|yr\n∗|)|)/parenleftBigg\n1\n|(Z−(0)−cn)(Z+(0)−cn)|1\n2+1/parenrightBigg\n≤C|(Z−(0)−cn)(Z+(0)−cn)|−3\n4.\nWe note that |(Z−−cn)(Z+−cn)| ≥C−1>0 aty1andy2as the two points are far away\nenough from ( Z+)−1(c).By the facts\n(Z−−cn)(Z+−cn)∂yΦn∈L∞and Φn∈L2,\nwe know that |Φn(y1)|+|Φn(y2)|+|∂yΦn(y1)|+|∂yΦn(y2)| ≤C.The estimates on Φ∗\nn(y0)\nand∂yΦ∗\nn(y0) then imply (3.35) and (3.36). /square\n3.3.Proofs of Proposition 3.1 and Proposition 3.2. As in previous proofs, we restrict\nourselves to the case c∈RanZ+.By our assumptions on uandb,givenc∈RanZ+,we can\nassume that\n(Z+)−1(c) ={yc,1,yc,2,...,yc,k;y0,1,y0,2,...,y0,m},\nwhereyc,i,i= 1,2,...,kare the points at which Z+is monotone, i.e., |Z′\n+(yc,i)|>0,whereas\ny0,i, i= 1,2,...,mare the critical points, where Z′\n+(y0,i) = 0 (and |Z′′\n+(y0,i)| /\\e}atio\\slash= 0).\nFor eachyc,i,there exists an interval Iisuch thatyc,i∈ IiandZ′\n+(yc,i)Z′\n+(y)>0,∀y∈ Ii,\nwhereas for each critical point y0,jwe may find an interval I0,jcontaining y0,jsuch that\nZ′′\n+(y0,j)Z′′\n+(y)>0,∀y∈ I0,j.The rest of T,consisting of regions far away from the set\n(Z+)−1(c),can also be covered by finitely many intervals, which we denot e as{Ia,i}/tildewiden\ni=1.The\nintervals Ii, i= 1,2,...,k,I0,i, i= 1,2,...,m,andIa,i, i= 1,2,...,/tildewiden,can be chosen in the\nway such that\n[−π,π] =/parenleftBig\n∪k\ni=1Ii/parenrightBig\n∪(∪m\ni=1I0,i)∪/parenleftBig\n∪/tildewiden\ni=1Ia,i/parenrightBig\n,\nwhile each of the intervals overlaps only the ones next to it, with the size of the overlap not\nexceeding1\n10min1≤i≤k|Ii|and1\n10min1≤i≤m|I0,i|.26 HAN LIU, NADER MASMOUDI, CUILI ZHAI, AND WEIREN ZHAO\nyc\nc\n−πyc,1\nI1\nIa,1y0,1\nI0,1\nIa,2yc,2\nI2\nIa,3y0,2\nI0,2\nIa,4yc,3\nI3\nIa,5Graph ofu+b\nyc,4\nI4\nIa,6π\nIa,7\nFigure 1. Partition of [ −π,π]\nWe can then construct a family of cut-off functions {χj}m+k+/tildewiden\nj=1forming a smooth partition\nof unity of Tsuch that each χjis supported in /tildewideIj,with/tildewideIjbeing one of the intervals from\n{Ii}k\ni=1∪{I0,i}m\ni=1∪{Ia,i}/tildewiden\ni=1.The choice of the intervals ensures that χj≡1 near (Z+)−1(c).\nSee Figure 1 for the partition of [ −π,π].\nThe proofs of the propositions are as follows.\nProof of Proposition 3.1 We note that on the intervals {Ia,i}/tildewiden\ni=1we do not encounter any\nissue in integration by parts. As a consequence of Lemma 3.3, q(yc) =q′(yc) = 0 implies\nthat∂yΦ(yc) = 0,allowing us to set the test function to be f= sgn(Z+−c)Φχjon/tildewideIjif\n/tildewideIj∈ {Ii}k\ni=1,whileLemma3.4andLemma3.5haveenabledusalsotoset ψ= sgn(Z+−c)Φχj\non/tildewideIjwhen/tildewideIj∈ {I0,i}m\ni=1.\nSumming all of the integral identities and noticing that the terms containing χ′\njcancel\neach other, we have\n/integraldisplayπ\n−π|Z+−c|(Z−−c)m+k/summationdisplay\nj=1χj/parenleftbig\n|∂yΦ|2+α2|Φ|2/parenrightbig\ndy= 0,\nthat is,/integraldisplayπ\n−π|Z+−c|(Z−−c)/parenleftbig\n|∂yΦ|2+α2|Φ|2/parenrightbig\ndy= 0.\nHence, the fact that ( Z−−c) is sign-definite implies that Φ ≡0, and Φ n⇀0 inL2.\nIt remains to be shown that {Φn}∞\nn=1and{(Z+−cn)(Z−−cn)∂yΦn}∞\nn=1converge strongly\ninL2andH1,respectively, which is clear outside the set ( Z+)−1(c).Moreover, by Lemma 3.3\nwe already know that the desired strong convergence result h olds outside the critical points\ny0,1,y0,2,...,y0,k.27\nAs for the critical points in ( Z+)−1(c),i.e.,y0,1,y0,2,...,y0,k,if{Φn,cn,Fn}∞\nn=1satisfies the\nconditions in Lemma 3.4, then by (3.21) and (3.33),\nΦn→Φ≡0 inL2(I0).\nRecallingqn= (Z+−cn)(Z−−cn)∂yΦn,which satisfies Equation (3.9), i.e.,\nq′′\nn−α2qn=F′\nn+α2/parenleftbig\nZ′\n+(Z−−cn)+Z′\n−(Z+−cn)/parenrightbig\nΦn,\nwe have, from our assumptions /ba∇dblΦn/ba∇dblL2+/ba∇dblqn/ba∇dblH1= 1 andFn→0 inH1,that\n/ba∇dblq′′\nn/ba∇dblL2/lessorsimilar/ba∇dblΦn/ba∇dblL2+/ba∇dblqn/ba∇dblL2+/ba∇dblFn/ba∇dblH1,\nwhich together with the week convergence qn⇀qimpliesqn→qinH1(I0),that is,\n(Z+−cn)(Z−−cn)∂yΦn→(Z+−c)(Z−−c)∂yΦ≡0 inH1(I0).\nFinally, if {Φn,cn,Fn}∞\nn=1satisfies the conditions in Lemma 3.5 instead, we recall Equa tion\n(3.37), in which both y1,y2/∈(Z+)−1(c)∪(Z−)−1(c) are chosen uniform in n. Then by (3.33),\nwe obtain that as n→ ∞,|Φn(y1)|+|Φn(y2)| →0,and thus /ba∇dblF∗\nn/ba∇dblL∞→0. Lemma 3.7 then\nensures that Φ n→0 inL2(I0).By the same argument as above, we obtain that qn→qin\nH1(I0).\nThus, we have shown that Φ n→0 inL2and (Z+−cn)(Z−−cn)∂yΦn→0 inH1,which\ncontradicts the assumption that /ba∇dblΦn/ba∇dblL2+/ba∇dbl(Z+−cn)(Z−−cn)∂yΦn/ba∇dblH1= 1.As the same\nargument applies to the case c∈Ran(Z−),it must be that the uniform estimate\n/ba∇dblΦ(·,c)/ba∇dblL2+/ba∇dbl(Z+−c)(Z−−c)∂yΦ(·,c)/ba∇dblH1≤C/ba∇dblF(·,c)/ba∇dblH1\nholds true for c∈(Ωǫ0\\(RanZ+∪RanZ+)).\n/square\nProof of Proposition 3.2 Suppose that there exists some c∈RanZ+for which the limit\nΦ+(y,c) do not exist, then we can find two sequences {cn,1}∞\nn=1and{cn,2}∞\nn=1such that\nImcn,j>0 and Recn,j=c, j=,1,2,|cn,1−cn,2| →0 asn→ ∞,while there exists some\nδ>0 such that\n/ba∇dblΦ(·,cn,1)−Φ(·,cn,2)/ba∇dblLr≥δ,1|Recn,j−c|forj= 1,2. (Note here Re cn,j=c.)\nTherefore, by Lemma 3.4, we have\n(y−y0)∂yΦ(y,cn,j)∈L2, j= 1 or 2,\nwhich by the compactness result in Appendix B, yields the str ong convergence locally near\ny0in this case, i.e.\nΦ(·,cn,j)→Φc,j(·) inLr,10>−C−1≥Z−(y),Z+(y1) =Z+(y2), Z′\n+(y0) = 0 for the\ncritical point y0∈(y1,y2) and|Z′\n+(y)|>0 fory/\\e}atio\\slash=y0. Let\nD0:={c∈[min{Z+(y0),Z+(y1)},max{Z+(y0),Z+(y1)}]}\nand\nDǫ0:={c=cr+iǫ, cr∈[min{Z+(y0),Z+(y1)},max{Z+(y0),Z+(y1)}],0<|ǫ|<ǫ0}.\nThenSǫ0=D0∪Dǫ0.\nGivencr∈[min{Z+(y0),Z+(y1)},max{Z+(y0),Z+(y1)}],when restricted to [ y0,y2],we\ncan findyr∈[y0,y2] such that Z+(yr) =cr.And when restricted to [ y1,y0],we can find\nyℓ∈[y1,y0] such that Z+(yℓ) =cr.\nProposition A.1. 1. Forc∈ Sǫ0, there exists a solution ϕr(y,c)∈C([y0,y2]×Sǫ0)of the\nSturmian equation (A.5)and∂yϕr(y,c)∈C([y0,y2]× Sǫ0). Moreover, there exists ǫ1>0\nsuch that for any ǫ0∈[0,ǫ1)and(y,c)∈[y0,y2]×Sǫ0,\n|ϕr(y,c)| ≥1\n2,|ϕr(y,c)−1| ≤C|y−yr|2,\nwhere the constants ǫ1,Cmay depend on α.\n2. Forc∈D0, for anyy∈[y0,y2], there is a constant C(depends on α) such that,\nϕr(y,c)≥ϕr(y′,c)≥1,fory0≤yr≤y′≤y≤y2ory0≤y≤y′≤yr≤y2.31\nProposition A.2. 1. Forc∈ Sǫ0, there exists a solution ϕℓ(y,c)∈C([y1,y0]×Sǫ0)of the\nSturmian equation (A.5)and∂yϕℓ(y,c)∈C([y1,y0]× Sǫ0). Moreover, there exists ǫ1>0\nsuch that for any ǫ0∈[0,ǫ1)and(y,c)∈[y1,y0]×Sǫ0,\n|ϕℓ(y,c)| ≥1\n2,|ϕℓ(y,c)−1| ≤C|y−yℓ|2,\nwhere the constants ǫ1,Cmay depend on α.\n2. Forc∈D0, for anyy∈[y1,y0], there is a constant C(depends on α) such that,\nϕℓ(y,c)≥ϕℓ(y′,c)≥1,fory1≤yℓ≤y′≤y≤y0ory1≤y≤y′≤yℓ≤y0.\nIn the following, we only give the proof of Proposition A.1 an d Proposition A.2 can be\nsimilarly proved. To prove the existence result for Equatio n (A.1), we introduce the following\nadapted norms.\nDefinition A.3. For a function f(y,c)on[y0,y2]×Sǫ0, we define\n/ba∇dblf/ba∇dblX0:= sup\n(y,c)∈[y0,y2]×Sǫ0/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(y,c)\ncosh(A(y−yr))/vextendsingle/vextendsingle/vextendsingle/vextendsingle,/ba∇dblf/ba∇dblY0:=/ba∇dblf/ba∇dblX0+1\nA/ba∇dbl∂yf/ba∇dblX0.\nIn order to give the solution formula, we introduce the follo wing integral operators.\nDefinition A.4. Lety∈[y0,y2].The Sturmian integral operator Sis defined by\nSf(y,c) :=S0◦S1f(y,c) =/integraldisplayy\nyr/integraltexty′\nyr(Z−(y′′)−c)(Z+(y′′)−c)f(y′′,c)dy′′\n(Z−(y′)−c)(Z+(y′)−c)dy′,\nwhere\nS0f(y,c) :=/integraldisplayy\nyrf(y′,r)dy′, S1f(y,c) :=/integraltexty\nyr(Z−(y′′)−c)(Z+(y′′)−c)f(y′′,c)dy′′\n(Z−(y)−c)(Z+(y)−c).\nLemma A.5. There exists a constant Cindependent of Asuch that\n/ba∇dblS0f/ba∇dblX0≤C\nA/ba∇dblf/ba∇dblX0,/ba∇dblS1f/ba∇dblX0≤C\nA/ba∇dblf/ba∇dblX0,/ba∇dblSf/ba∇dblX0≤C\nA2/ba∇dblf/ba∇dblX0.\nFurthermore, there holds\n/ba∇dblSf/ba∇dblY0≤C\nA2/ba∇dblf/ba∇dblY0.\nProof.Forc∈D0, we shall only prove the part of the lemma for Z+(y0) =c,as the proof\nwhenZ−(y0) =cis along the same lines. As Ran Z+∩RanZ−=∅,we can find some positive\nCsuch thatC−1<|Z−(y)−c|1\n2.\nFrom the integral formula of S,we have\n|ϕr(y,c)−1| ≤α2/integraldisplayy\nyr/integraldisplayy′\nyr|ϕr(z,c)|/vextendsingle/vextendsingle/vextendsingle/vextendsingle(Z−(y′′)−c)(Z+(y′′)−c)\n(Z−(y′)−c)(Z+(y′)−c)/vextendsingle/vextendsingle/vextendsingle/vextendsingledy′′dy′\n≤C/parenleftBig\nα,/ba∇dblϕr/ba∇dblL∞([y0,y2]×Sǫ0)/parenrightBig\n|y−yr|2.33\nAppendix B.A compactness lemma\nWe prove a useful compactness result.\nLemma B.1. Let{yn}∞\nn=1⊂ I:= [a,b]be such that yn→a+b\n2.Let{fn}∞\nn=1be a family of\nfunctions defined on Isatisfying the uniform bound\n/ba∇dblfn/ba∇dblLr+/ba∇dbl(y−yn)∂yfn/ba∇dblLp≤C,11,leads to\n/ba∇dblfn/ba∇dblp\nLp(Ih)≤Cp/ba∇dbl(y−yn)∂yfn/ba∇dblp\nLp+Ch/ba∇dblfn/ba∇dblp\nLp(I\\Ih)≤C.\nHence, we can simply estimate I∗∗by H¨ older’s inequality –\nI∗∗≤/integraldisplay\nIh(|fn(x+h)|r+|fn(x)|r)dx≤2/ba∇dblfn/ba∇dblr\nLp(I2h)hp−r\np≤Ch1−r\np.\nTherefore, the desired compactness result is true. /square\nAppendix C.proof of Remark 1.4\nForu= 0, taking the Fourier transform in x, we get for α/\\e}atio\\slash= 0,\n(C.1)\n\n∂t/hatwideU1+iα/hatwidep−iαb/hatwideH1−b′/hatwideH2= 0,\n∂t/hatwideU2+∂y/hatwidep−iαb/hatwideH2= 0,\n∂t/hatwideH1+b′/hatwideU2−iαb/hatwideU1= 0,\n∂t/hatwideH2−iαb/hatwideU2= 0,\niα/hatwideU1+∂y/hatwideU2= 0, iα/hatwideH1+∂y/hatwideH2= 0.34 HAN LIU, NADER MASMOUDI, CUILI ZHAI, AND WEIREN ZHAO\nAnd we can diagonalize it to obtain\n(C.2)/braceleftbigg\n∂tt/hatwideU2+α2Aα/hatwideU2= 0,\n∂tt(/hatwideH2/b)+α2Aα(/hatwideH2/b) = 0,\nwhereAα= (∂2\ny−α2)−1/parenleftbig\nb2(∂2\ny−α2)+2bb′∂y/parenrightbig\n, and the details can be seen in [15]. In order\nto prove the energy conservation law, we only need to prove th e energy conservation on each\nfrequency.\nAt first, we can show that\n/ba∇dblαk/hatwideU1/ba∇dbl2\nL2y+/ba∇dblαk/hatwideU2/ba∇dbl2\nL2y+/ba∇dblαk/hatwideH2/ba∇dbl2\nL2y+/ba∇dblαk(/hatwideH1−i(αb)−1b′/hatwideH2)/ba∇dbl2\nL2y\n=/ba∇dblαk/hatwideU1,in/ba∇dbl2\nL2y+/ba∇dblαk/hatwideU2,in/ba∇dbl2\nL2y+/ba∇dblαk/hatwideH2,in/ba∇dbl2\nL2y+/ba∇dblαk(/hatwideH1,in−i(αb)−1b′/hatwideH2,in)/ba∇dbl2\nL2y. (C.3)\nIndeed, taking L2inner product of (C.2)2withα2k∂t(∂2y−α2)(/hatwideH2/b), integrating by part,\ntaking the real part, we obtain\nRe/integraldisplay\nTαk∂tt(/hatwideH2/b)αk∂t(∂2y−α2)(/hatwideH2/b)dy\n=−Re/parenleftBig/integraldisplay\nTαk∂tt∂y(/hatwideH2/b)αk∂t∂y(/hatwideH2/b)dy+α2+2k/integraldisplay\nT∂tt(/hatwideH2/b)∂t(/hatwideH2/b)dy/parenrightBig\n=−1\n2d\ndt/ba∇dblαk∂t∂y(/hatwideH2/b)/ba∇dbl2\nL2y−1\n2d\ndt/ba∇dblα1+k∂t(/hatwideH2/b)/ba∇dbl2\nL2y\nand\nRe/integraldisplay\nTAα(/hatwideH2/b)α2k∂t(∂2y−α2)(/hatwideH2/b)dy\n= Re/integraldisplay\nT(b2(∂2\ny−α2)+2bb′∂y)(/hatwideH2/b)α2k∂t(/hatwideH2/b)dy\n=−Re/parenleftBig/integraldisplay\nTαkb2∂y(/hatwideH2/b)αk∂t∂y(/hatwideH2/b)dy+α2k+2/integraldisplay\nT/hatwideH2∂t/hatwideH2dy/parenrightBig\n=−1\n2d\ndt/ba∇dblαk∂y(/hatwideH2/b)/ba∇dbl2\nL2y−1\n2d\ndt/ba∇dblα1+k/hatwideH2/ba∇dbl2\nL2y,\nand then\nd\ndt/parenleftBig\n/ba∇dblαk∂t∂y(/hatwideH2/b)/ba∇dbl2\nL2y+/ba∇dblα1+k∂t(/hatwideH2/b)/ba∇dbl2\nL2y+/ba∇dblα1+k∂y(/hatwideH2/b)/ba∇dbl2\nL2y+/ba∇dblα1+k/hatwideH2/ba∇dbl2\nL2y/parenrightBig\n= 0.\nBy (C.1), we get (C.3) for α/\\e}atio\\slash= 0. Finally, we prove that for some constant C >0 independent\noft,α,\nC−1/ba∇dbl/hatwideH1/ba∇dblL2y≤ /ba∇dbl/hatwideH1−i(αb)−1b′/hatwideH2/ba∇dblL2y≤C/ba∇dbl/hatwideH1/ba∇dblL2y.\nBy the condition that /hatwideH2(t,α,y1) =/hatwideH2(t,α,y2) = 0, we have /ba∇dbl/hatwideH2/ba∇dblL2y≤C/ba∇dbl∂y/hatwideH2/ba∇dblL2y≤\nC/ba∇dblα/hatwideH1/ba∇dblL2y. Thus,\n/ba∇dblα−1+k∂y(/hatwideH2/b)/ba∇dblL2y≤C/ba∇dblα−1+k∂y/hatwideH2/ba∇dblL2y+C/ba∇dblα−1+k/hatwideH2/ba∇dblL2y≤C/ba∇dblαk/hatwideH1/ba∇dblL2y.\nAnd on the other hand,\n/ba∇dblαk/hatwideH1/ba∇dblL2y≤ /ba∇dblαk(/hatwideH1−i(αb)−1b′/hatwideH2)/ba∇dbl2\nL2y+/ba∇dblαk(αb)−1b′/hatwideH2/ba∇dbl2\nL2y\n≤C/ba∇dblα−1+k∂y(/hatwideH2/b)/ba∇dblL2y≤ /ba∇dblαk(/hatwideH1−i(αb)−1b′/hatwideH2)/ba∇dbl2\nL2y,35\nwhere we used the fact that −iαb∂y(/hatwideH2/b) =/hatwideH1−i(αb)−1b′/hatwideH2in the last inequality. Since\nCis independent of tandα, we obtain the proof of Remark 1.4 by Plancherel identity.\nFor the vorticity and current density, we have that by taking Fourier transform in x,\n(C.4)/braceleftbigg\n∂t/hatwideω=iαb/hatwidej−b′′/hatwideH2,\n∂t/hatwidej=iαb/hatwideω−2iαb′/hatwideU1+b′′/hatwideU2.\nBy takingL2inner product of (C.4)1with/hatwidewand (C.4)2with/hatwidej, and taking the real part, we\nobtain that\n1\n2d\ndt/parenleftBig\n/ba∇dbl/hatwideω/ba∇dbl2\nL2y+/ba∇dbl/hatwidej/ba∇dbl2\nL2y/parenrightBig\n= Re/parenleftBig\n−/integraldisplay\nTb′′/hatwideH2/hatwideωdy−/integraldisplay\nT2iαb′/hatwideU1/hatwidejdy+/integraldisplay\nTb′′/hatwideU2/hatwidejdy/parenrightBig\n≤ /ba∇dblb′′/hatwideH2/ba∇dblL2y/ba∇dbl/hatwideω/ba∇dblL2y+/ba∇dbl2αb′/hatwideU1/ba∇dblL2y/ba∇dbl/hatwidej/ba∇dblL2y+/ba∇dblb′′/hatwideU2/ba∇dblL2y/ba∇dbl/hatwidej/ba∇dblL2y.\nAnd then, we get\nd\ndt/parenleftBig\n/ba∇dbl/hatwideω/ba∇dblL2y+/ba∇dbl/hatwidej/ba∇dblL2y/parenrightBig\n≤C/parenleftBig\n/ba∇dbl/hatwideH2/ba∇dblL2y+/ba∇dblα/hatwideU1/ba∇dblL2y+/ba∇dbl/hatwideU2/ba∇dblL2y/parenrightBig\n,\nby integrating in time, it holds\n/ba∇dbl/hatwideω/ba∇dblL2y+/ba∇dbl/hatwidej/ba∇dblL2y≤/parenleftBig\n/ba∇dbl/hatwideω0/ba∇dblL2y+/ba∇dbl/hatwidej0/ba∇dblL2y/parenrightBig\n+C/integraldisplayt\n0/parenleftBig\n/ba∇dbl/hatwideH2/ba∇dblL2y+/ba∇dblα/hatwideU1/ba∇dblL2y+/ba∇dbl/hatwideU2/ba∇dblL2y/parenrightBig\ndτ.\nThus by using (C.3), we obtain the linear growth result of vor ticity and current density.\nNote that when considering the non-flowing plasma case, the s ystem can be diagonalized\nto one with a self-adjoint operator Aα.\nFor general case with constant background velocity ( u= const.), we have\n\n\n∂t/hatwideU1+iαu/hatwideU1+iα/hatwidep−iαb/hatwideH1−b′/hatwideH2= 0,\n∂t/hatwideU2+iαu/hatwideU2+∂y/hatwidep−iαb/hatwideH2= 0,\n∂t/hatwideH1+iαu/hatwideH1+b′/hatwideU2−iαb/hatwideU1= 0,\n∂t/hatwideH2+iαu/hatwideH2−iαb/hatwideU2= 0,\niα/hatwideU1+∂y/hatwideU2= 0, iα/hatwideH1+∂y/hatwideH2= 0.\nLet us introduce/hatwide˜U=eiαut/hatwideU,/hatwide˜H=eiαut/hatwideH,/hatwide˜p=eiαut/hatwidepand/hatwide˜ω=eiαut/hatwideω,/hatwide˜j=eiαut/hatwidej, then\n(/hatwide˜U,/hatwide˜H,/hatwide˜p) solves (C.1) and ( /hatwide˜ω,/hatwide˜j) solves (C.2).\nThus Remark 1.4 follows from the fact that/vextenddouble/vextenddouble/vextenddouble(/hatwide˜U,/hatwide˜H,/hatwide˜ω,/hatwide˜j)/vextenddouble/vextenddouble/vextenddouble\nL2y=/vextenddouble/vextenddouble/vextenddouble(/hatwideU,/hatwideH,/hatwideω,/hatwidej)/vextenddouble/vextenddouble/vextenddouble\nL2y.\nReferences\n[1] J. 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Zhang and W. Zhao, Linear inviscid damping and enhanced dissipation for the Ko lmogorov\nflow, Advances in Mathematics, 362(2020),106963.\n[22] C. Zhai, Z. Zhang and W. Zhao, Long-time behavior of Alfv´ en waves in a flowing plasma: Gene ration of\nthe magnetic island. arXiv:1809.08544, 2018.\n[23] C. Zillinger, Linear inviscid damping for monotone shear flows , Trans. Amer. Math. Soc., 369 (2017),\n8799-8855.\n[24] H. Zohm. Magnetohydrodynamic stability of tokamaks. W iley-VCH. 2015.\nDepartment of Mathematics, New York University Abu Dhabi, S aadiyat Island, P.O. Box\n129188, Abu Dhabi, United Arab Emirates.\nEmail address :hl4294@nyu.edu\nDepartment of Mathematics, New York University Abu Dhabi, S aadiyat Island, P.O. Box\n129188, Abu Dhabi, United Arab Emirates; Courant Institute of Mathematical Sciences, New\nYork University, 251 Mercer Street, New York, NY 10012, USA.\nEmail address :masmoudi@cims.nyu.edu\nSchool of Mathematics and Physics, University of Science an d Technology Beijing, 100083,\nBeijing, P. R. China.\nEmail address :zhaicuili035@126.com, cuilizhai@ustb.edu.cn\nDepartment of Mathematics, New York University Abu Dhabi, S aadiyat Island, P.O. Box\n129188, Abu Dhabi, United Arab Emirates.\nEmail address :zjzjzwr@126.com, wz19@nyu.edu" }, { "title": "2108.01143v1.Large_amplitude_longitudinal_oscillations_in_solar_prominences_simulated_with_different_resolutions.pdf", "content": "arXiv:2108.01143v1 [astro-ph.SR] 2 Aug 2021Astronomy&Astrophysics manuscript no. ms ©ESO 2021\nAugust 4, 2021\nLarge-amplitude longitudinal oscillations in solar promi nences\nsimulated with different resolutions\nV . Liakh1,2, M. Luna3,4, and E. Khomenko1,2\n1Instituto de Astrofísica de Canarias, E-38205 La Laguna, Te nerife, Spain\ne-mail:vliakh@iac.es\n2Departamento de Astrofísica, Universidad de La Laguna, E-3 8206 La Laguna, Tenerife, Spain\n3Departament de Física, Universitat de les Illes Balears, E- 07122, Palma de Mallorca, Spain\n4Institute of Applied Computing & Community Code (IAC3), UIB, Spain\nABSTRACT\nContext. Large-amplitude longitudinal oscillations (LALOs) in sol ar prominences have been widely studied in the last decades.\nHowever, their damping and amplification mechanisms are not well understood.\nAims. In this study, we investigate the attenuation and amplificat ion of LALOs using high-resolution numerical simulations w ith\nprogressively increasing spatial resolutions.\nMethods. We performed time-dependent numerical simulations of LALO s using the 2D magnetic configuration that contains a dipped\nregion. After the prominence mass loading in the magnetic di ps, we triggered LALOs by perturbing the prominence mass alo ng the\nmagnetic field. We performed the experiments with four value s of spatial resolution.\nResults. In the simulations with the highest resolution, the period s hows a good agreement with the pendulum model. The convergen ce\nexperiment revealed that the damping time saturates at the b ottom prominence region with improving the resolution, ind icating the\nexistence of a physical reason for the damping of oscillatio ns. At the prominence top, the oscillations are amplified dur ing the first\nminutes and then are slowly attenuated. The characteristic time suggests more significant amplification in the experime nts with\nthe highest spatial resolution. The analysis revealed that the energy exchange between the bottom and top prominence re gions is\nresponsible for the attenuation and amplification of LALOs.\nConclusions. The high-resolution experiments are crucial for the study o f the periods and the damping mechanism of LALOs. The\nperiod agrees with the pendulum model only when using high en ough spatial resolution. The results suggest that numerica l diffusion\nin simulations with insu fficient spatial resolution can hide important physical mecha nisms, such as amplification of oscillations.\nKey words. Sun: corona – Sun: filaments, prominences – Sun: oscillation s – methods: numerical\n1. Introduction\nSolar prominences are clouds of cold and dense plasma sus-\npended in the solar corona, supported against gravity by the\nmagnetic field. They are subject to various types of oscilla-\ntions. These periodic motions are classified as small-ampli tude\nand large-amplitude oscillations according to their veloc ity (see\nreviews by Oliver & Ballester 2002; Arregui et al. 2018). In\nthe large-amplitude oscillations (LAOs), a large portion o f the\nprominence oscillates with amplitudes that exceed 10 km s−1. If\nthe plasma motions are directed along the magnetic field, the\noscillations are classified as large-amplitude longitudin al oscil-\nlations (LALOs). On the contrary, oscillations with veloci ty di-\nrected perpendicular to the magnetic field are called transv erse\nLAOs. More on the classification and properties of the di fferent\ntypes of oscillations can be found in the last update of the li ving\nreview by Arregui et al. (2018).\nJing et al. (2003, 2006) have been first reported the LA-\nLOs in solar prominences. Since then, many observations pro -\nvided important information on the properties of LALOs (see ,\ne.g., Vršnak et al. 2007; Zhang et al. 2012; Li & Zhang 2012;\nLuna et al. 2014; Bi et al. 2014; Zhang et al. 2017). Observa-\ntions have shown that the period of the LALOs is around 1 hour,\nand the damping time is usually less than two periods, indica tingstrong attenuation of the motions. Luna et al. (2018) catalo ged\nthe filaments oscillations using several months of the Globa l Os-\ncillation Network Group (GONG) H αdata near the maximum\nof the solar cycle 24. The catalog provided a large sample of t he\nLALOs and statistics of the periods and damping times. It was\nobtained that, on average, the ratio of damping time to perio d\nis equal to 1.25, in agreement with the earlier observations. An\nintriguing phenomenon in the prominence oscillations is th e am-\nplification of motions. Contrary to the damping caused by en-\nergy loss, oscillations somehow acquire additional energy and\nincrease their amplitudes in time. The first amplification ev ent\nwas reported by Molowny-Horas et al. (1999) from time series\nof Hβfiltergrams of a polar crown prominence. They measured\nperiodic Doppler signal and found clear oscillations with p eri-\nods between 60 to 95 minutes and damping times between 119\nto 377 minutes at six locations of the prominence. At one of th e\nlocations, they found an oscillation with a period of 28 minu tes\nand a growing amplitude with a characteristic time of 140 min -\nutes. More recently, Zhang et al. (2017) reported observati on of\nLALOs, in which the oscillatory amplitude remains constant or\nincreases with time in some regions of the prominence. This u n-\nusual behavior of the LALOs amplitude has also been found in\nthe GONG catalog by Luna et al. (2018; see, e.g., Fig. 25).\nArticle number, page 1 of 12A&A proofs: manuscript no. ms\nIn the last decades, the LALOs have been studied in many\nanalytical and numerical works. These were dedicated to an-\nswering questions related to the restoring and damping mech -\nanisms. Luna & Karpen (2012) proposed a so-called pendulum\nmodel where the main contribution to the restoring force is t he\ngravity projected along the magnetic field. This model sugge sts\nthat the period is defined by the radius of curvature of the mag -\nnetic field (see also Roberts 2019, for another description o f the\nmodel). The following studies, based on the analytical calc ula-\ntions or the 2D and 3D simulations, confirmed that the pendu-\nlum model describes quite well the periods of longitudinal o s-\ncillations under typical prominence conditions (e.g., Lun a et al.\n2012, 2016; Zhou et al. 2018; Zhang et al. 2019; Liakh et al.\n2020; Fan 2020). Zhou et al. (2018) found a systematic discre p-\nancy of some 10% with respect to the pendulum model. The\nauthors suggested that the longitudinal oscillations prod uce dy-\nnamical deformations of the field structure. Under these cir cum-\nstances, the period of LALOs can be slightly longer than pre-\ndicted by the pendulum model due to the flattening of the mag-\nnetic field by the plasma motions. Zhang et al. (2019) found th at\nin the regime of a relatively weak magnetic field, when the gra v-\nity to Lorentz force ratio is close to unity, the trajectory o f the\nprominence plasma is not along the rigid field lines. The auth ors\nfound that these dynamic deformations of the magnetic struc -\nture contribute to the damping of the oscillations. Similar ly, Fan\n(2020) found discrepancies with respect to the pendulum mod el\nalso associated with the dynamical deformation of the field l ines\nduring the oscillations.\nThere have been proposed many candidates to damping\nmechanisms in order to explain the observed LALOs attenua-\ntion. Radiative losses and thermal conduction have been con sid-\nered by Zhang et al. (2012) and Zhang et al. (2019) as damping\nmechanisms for LALOs. These authors pointed out the impor-\ntance of the inclusion of the nonadiabatic e ffects into considera-\ntion. However, using numerical simulations, these authors have\nshown that nonadiabatic e ffects alone are not enough to explain\nthe observed damping (see also Zhang et al. 2020). Zhang et al .\n(2013) and Ruderman & Luna (2016) proposed the alternative\nmechanism for the attenuation of LALOs. Zhang et al. (2013)\nconsidered a possibility of LALOs decaying due to the mass\ndrainage, while Ruderman & Luna (2016) studied the damping\ndue to the mass accretion. Both studies showed that either th e\nmass drainage or mass accretion results in decreasing veloc ity\nof the prominence. As a consequence, this leads to the dampin g\nof oscillations. Zhang et al. (2019) found that in a situatio n of\na relatively weak magnetic field, the wave leakage is an impor -\ntant mechanism for LALOs damping, in addition to the nona-\ndiabatic effects. The motion of the prominence mass produces\nperturbations in the field structure that emanate in the form of\nMHD waves.\nIn order to explain both the damping and the amplification\nof prominence oscillations, Ballester et al. (2016) derive d an ex-\npression for the temporal variation of the background tempe ra-\nture, taking into account the radiative losses and thermal c on-\nduction. The authors concluded that using some combination\nof the characteristic times of the di fferent mechanisms and in-\ncreasing or decreasing the background temperature could le ad\nto the damping or amplification of the oscillations. Zhang et al.\n(2017) proposed an alternative mechanism to explain the am-\nplification: the threads located at the di fferent dips of the same\nflux tube interchange their energy. Using 1D numerical simu-\nlations, Zhou et al. (2017) studied the di fferent combinations of\nthe active-passive threads and demonstrated that the energ y ex-\nchange significantly a ffects the damping or amplification time ofthe LALOs. However, this mechanism cannot explain the ampli -\nfication of the threads that belong to the di fferent field lines.\nAll the studies of LALOs in 2D and 3D have been done\nusing numerical simulations. In these numerical experimen ts,\ndissipation is unavoidable. Terradas et al. (2016) investi gated\nthe influence of the numerical dissipation on the prominence\noscillations in the 3D model, increasing spatial resolutio n up\nto 300 km. They concluded that the energy losses associated\nwith the numerical dissipation are significantly reduced in the\nhigh-resolution experiments. In more recent 3D simulation s,\nAdrover-González & Terradas (2020) and Fan (2020) pointed\nout that the reduction of the numerical dissipation is cruci al to\nstudy the damping of LALOs. Terradas et al. (2013), Luna et al .\n(2016) and Zhang et al. (2019) performed 2D numerical exper-\niments of the prominence oscillation with spatial resoluti on up\nto 125 and 156 km, respectively. These numerical simulation s\nshowed significant damping which might be partly contribute d\nby numerical diffusivity. Liakh et al. (2020) studied the conver-\ngence of a 2.5D experiment of LALOs excited in a magnetic\nflux rope. The authors performed an experiment with a spatial\nresolution of 60 km and compared it to the one with a spatial\nresolution of 240 km. They found that the periods of LALOs are\nconsistent in the two experiments, although the damping tim e\nbecomes longer in the higher resolution experiment. Thus, t hey\nconcluded that in their experiments, the attenuation of LAL Os is\nassociated mainly with numerical dissipation.\nIn this work, we aim to understand the physical origin of the\nLALOs damping and the other processes usually hidden by the\nartificial dissipation in numerical experiments. To this en d, we\nhave performed 2D convergence experiments with progressiv ely\nincreasing spatial resolutions up to the highest value of 30 km.\nThis paper is organized as follows, in Sect. 2 the numerical\nmodel is described. In Sects. 3 and 4 we explain the temporal\nevolution of the plasma under the disturbance directed alon g the\nfield and compare the oscillatory parameters in the experime nts\nwith the different spatial resolution and the di fferent prominence\nlayers. In Sect. 5 we summarize the main results.\n2. Initial configuration\nWe assume a 2D prominence model as shown in Fig. 1. The\nmodel is defined in the Cartesian coordinate system in the xz-\nplane, where z-axis corresponds to the vertical direction. The\ninitial magnetic field is a potential configuration which con tains\na dipped part, suitable to support a prominence. It is compos ed\nof periodic arcades defined as:\nBx\nB0=cos(k1(x−x0))e−k1(z−z0)−cos(k2(x−x0))e−k2(z−z0), (1)\nBz\nB0=−sin(k1(x−x0))e−k1(z−z0)+sin(k2(x−x0))e−k2(z−z0),(2)\nwhere B0=33 G, x0=0 Mm and z0=−2 Mm. We also set\nk1=π\nDandk2=3k1where D=191.4 Mm, that is the half-\nsize of the numerical domain along the x-direction. The super-\nposition of the major and minor arcades form the magnetic fiel d\nstructure shown in Fig. 1 with the dipped part centered at x=0\nMm and between z=0 and 25 Mm. As shown in the figure, the\ncurvature of the magnetic field lines decreases with height. For\nz>27 Mm the magnetic field lines change from concave-up to\nconcave-down becoming ordinary loops. A detailed descript ion\nof this magnetic configuration is given by Terradas et al. (20 13)\nand Luna et al. (2016). The magnetic field strength increases\nfrom 9.0 to 12.5 G for zfrom 8 to 25 Mm. The magnetic field\nArticle number, page 2 of 12Liakh, Luna & Khomenko: Numerical simulations of LAOs\nFig. 1: Density distribution and magnetic field lines at the central part of the computational domain after the mass loading and r elaxation processes.\nThe dashed lines denote the initial magnetic field prior to th e mass loading.\nstrength lies in the range of the values obtained from the mag -\nnetic field measurements in the solar prominences (Leroy et a l.\n1983, 1984).\nWe assume an initial atmosphere consisting of a stratified\nplasma in hydrostatic equilibrium, including the chromosp here,\ntransition region (TR), and corona. The temperature profile is\ngiven by\nT(z)=T0+1\n2(Tc−T0)/bracketleftBigg\n1+tanh/parenleftBiggz−zc\nWz/parenrightBigg/bracketrightBigg\n. (3)\nWe choose Tc=106K,T0=104K,Wz=0.4 Mm, and zc=\n3.6 Mm. In this profile the temperature ranges from Tch=104\nK at the base of the chromosphere to Tc=106K in the corona.\nAs the plasma is stratified in the vertical direction the dens ity\nchanges fromρ=9×10−9kg m−3in the chromosphere to ρ=\n1.69×10−12kg m−3at the base of the corona at the height zc=\n3.6 Mm.\nWe numerically solve ideal magnetohydrodynamic (MHD)\nequations using the MANCHA3D code (Khomenko et al. 2008;\nFelipe et al. 2010; Khomenko & Collados 2012). The govern-\ning equations of mass, momentum, internal energy, inductio n\nequation, and the corresponding source terms are described in\nFelipe et al. (2010). The computation domain consists of a bo x\nof 384×108 Mm size. In order to study the influence of the nu-\nmerical diffusivity on the prominence oscillations, we use four\nspatial resolutions with ∆ = 240,120,60 and 30 km from\nthe coarse to the fine grids. We assume a periodic condition at\nthe left and right boundaries. At the bottom boundary, we ap-\nply the current-free condition for the magnetic field (see, e .g.,\nLuna et al. 2016) and symmetric condition for the temperatur e\nand pressure. At the bottom boundary, the density is fixed. At\nthe top boundary, the zero-gradient condition is applied to all the\nvariables except for Bx. We impose that Bxis antisymmetric.\nIn order to add the prominence mass to the dipped region\nof the magnetic field, we use an approach of the artificial mass\nloading described by Liakh et al. (2020). We use the source te rm\nin the continuity equation to increase the density in the dip ped\nregion of the magnetic field. The density source term distrib u-\ntion is a Gaussian function (see Eq. (9) from Liakh et al. (202 0)\nwork) centered at ( x,z)=(0,11.8) Mm. The mass loading starts\natt=0 seconds and ends at t=100 seconds. The resulting\nprominence has a density of 110 times the density of the ini-\ntial corona with the dimensions of 5 and 9 Mm in the horizontal\nand vertical directions, respectively.\nDuring the first 8.3 minutes, we use intensive artificial damp-\ning in order to minimize the undesirable motions caused by th eresponse of the magnetic field to the mass loading process. Fi g-\nure 1 shows the prominence after this relaxation phase. The\ndashed lines denote the initial magnetic field. As we can see,\nthe magnetic field lines are slightly elongated downwards du e\nto the heavy prominence mass. The prominence itself also has\nsome deformation, becoming more compressed toward the cen-\nter of the dip, and it slightly drops down with respect to the i nitial\nheight. At the end of the relaxation process, the system is cl ose\nto static equilibrium.\nSimilarly to our previous work (Liakh et al. 2020), we trig-\nger oscillation after the relaxation phase by applying an ex ternal\nforce over the prominence. This force is incorporated as a so urce\nterm in the momentum equations as\nSmx=ρvpertBx\ntpertBexp/parenleftBigg\n−(x−xpert)4\nσ4x−(z−zpert)4\nσ4z/parenrightBigg\n, (4)\nSmz=ρvpertBz\ntpertBexp/parenleftBigg\n−(x−xpert)4\nσ4x−(z−zpert)4\nσ4z/parenrightBigg\n, (5)\nwhere tpert=10 seconds is the duration of the disturbance that\nstarts at t=8.3 minutes. The parameters σx=σz=12 Mm\nare the half-sizes of the perturbed region in the horizontal and\nvertical directions centered at ( xpert,zpert)=(0,12) Mm. The\nforce given by Eqs (4) and (5) is directed along the local mag-\nnetic field in contrast to our previous work (Liakh et al. 2020 )\nwhere the perturbation is purely horizontal or vertical. Th e max-\nimum velocity of the perturbation is vpert=22 km s−1, which is\nin the range of the observed amplitudes of LAOs (see review by\nArregui et al. 2018). In order to study the long-term evoluti on of\nprominence motions, we run the experiments during 275 minut es\nof physical time.\n3. Influence of spatial resolution on prominence\ndynamics\nIn this work, we study the influence of the numerical di ffusiv-\nity in the LAOs by progressively increasing the spatial reso -\nlution as described in Sect. 1. Figure 2 shows an example of\nthe temporal evolution after the perturbation for ∆ = 30 km.\nThis kind of evolution is representative of all our numerica l ex-\nperiments. After the initial disturbance at t=8.3 minutes, the\nprominence moves to the right along the magnetic field and\nreaches its maximum initial displacement at t=17 minutes,\napproximately (Fig. 2a). We see that the maximum horizontal\ndisplacement is large at the upper prominence part, whereas the\nArticle number, page 3 of 12A&A proofs: manuscript no. ms\nFig. 2: Temporal evolution of v/ba∇dbl, magnetic field, and the prominence mass after the perturbat ion. Dashed black lines denote unperturbed magnetic\nfield lines; solid black lines are the actual magnetic field li nes at a given moment; orange lines are density isocontours, and purple arrows are the\nvelocity field. Spatial resolution: ∆=30 km.\n∆=240 km\n0 50 100 150 200 250\nTime (min.)81012141618\nZ(Mm)V||=22 km s−1∆=120 km\n0 50 100 150 200 250\nTime (min.)81012141618\nZ(Mm)V||=22 km s−1\n∆=60 km\n0 50 100 150 200 250\nTime (min.)81012141618\nZ(Mm)V||=22 km s−1∆=30 km\n0 50 100 150 200 250\nTime (min.)81012141618\nZ(Mm)V||=22 km s−1(a)\n0.01 0.50 1.00 1.50 2.00\nρ (×10−10 kg m −3)(b)\n(c) (d)\nFig. 3: Temporal evolution of v/ba∇dblat the center of mass at the selected field lines. The color bar denotes the maximum initial density at each field line.\nThe left vertical axis indicates the velocity amplitude sca le. The right vertical axis denotes the vertical location of the dips of selected magnetic\nfield lines.\nbottom part is only slightly displaced from the equilibrium po-\nsition. The reason is that, as the initial perturbation laun ches\nthe prominence mass along the field lines with similar veloci -\nties, the plasma reaches di fferent horizontal locations since the\nfield lines have different curvatures. Figure 2b shows the promi-\nnence at t=45.3 minutes. At this time, the prominence has\nreached its maximum displacement on the left side and begins\nto move again toward the right. The top part of the prominence\nslightly delays from the rest of the prominence body, sugges tinga longer oscillation period. Figures 2c and 2d show the promi -\nnence evolution after several cycles of oscillations. From the\nvelocity field displayed by the purple arrows, we can see that\nthe prominence layers oscillate out of phase producing coun -\nterstreaming flows. The counterstreaming flows are commonly\nobserved in the filaments (Schmieder et al. 1991; Zirker et al .\n1998; Lin et al. 2005). Chen et al. (2014) claimed that these\ncounterstreaming motions are associated with oscillation s. The\noscillation period depends on the physical conditions of th e mag-\nArticle number, page 4 of 12Liakh, Luna & Khomenko: Numerical simulations of LAOs\nnetic field line where the prominence plasma resides. This me ans\nthat the period can vary slightly for the neighboring field li nes.\nThis explains a phase di fference between the motions of the\nprominence layers, shown in Fig. 2c. Besides, these alterna te\nmotions can also be associated with processes of evaporatio n and\ncondensation in prominences (Zhou et al. 2020) or shock down -\nstreams produced by jets (Luna & Moreno-Insertis 2021).\nIn order to study the prominence plasma oscillations, we an-\nalyze the bulk motions of the plasma in the two possible polar -\nizations: transverse and longitudinal to the magnetic field . Fol-\nlowing our previous works (Luna et al. 2016; Liakh et al. 2020 )\nwe compute the velocities of the center of mass of independen t\nfield lines using Eqs. (5) and (6) from Luna et al. (2016). The\nparametersv/ba∇dblandv⊥are the longitudinal and transverse veloc-\nities of the center of mass of each field line. First, we select 20\nequally spaced field lines that permeate the prominence body at\nthe height from 7.2 Mm up to 17.6 Mm. Then, we start integrat-\ning the field lines in the chromosphere, where the magnetic fie ld\nremains unchanged according to the line-tying condition. A s our\nfluid is perfectly conducting, the frozen-in condition is fu lfilled.\nThis allows us to advect the motions in the same field line at\neach time moment. We selected an identical set of the field lin es\nfor each numerical experiment in order to compare the result ing\nvelocities.\nFigure 3 shows the temporal evolution of v/ba∇dblfor the four\nnumerical experiments with di fferent spatial resolutions. The\nglobal behavior of the prominence is relatively similar in e ach of\nthe experiments. After the initial perturbation at t=8.3 minutes,\nthe longitudinal velocity increases in all the lines. The lo n-\ngitudinal velocity reaches the highest value, v/ba∇dbl=22 km s−1,\nin the densest prominence part at height z=11.8 Mm. At\nt=15−35 minutes, we see a certain shift in the signal. This\nphase shift is related to the di fferent periods associated with the\ndifferent field lines. On the one hand, we see that the oscilla-\ntions are less attenuated with increasing the spatial resol ution,\nalthough the difference between the cases ∆= 60 and 30 km is\nless pronounced. On the other hand, the behavior of the damp-\ning changed with height. This is in agreement with the visual\nimpression from Fig. 2, discussed above. Oscillations at th e bot-\ntom and central part are strongly damped in all the experimen ts.\nAt the height of 13 −17 Mm, the oscillations with weaker attenu-\nation last for a longer time. In the highest-resolution simu lation,\nthe upper prominence part keeps oscillating with a significa nt\namplitude even at the final stage of the numerical experiment .\n3.1. Influence of spatial resolution on the period\nFigure 4 shows the periods of the longitudinal oscillations for\neach of the 20 selected field lines and each numerical experi-\nment with different spatial resolutions. The period is more or\nless uniform under 10 .2 Mm and above 16 .1 Mm. In contrast,\nfor heights z=10.2−16.1 Mm, we see an increase of the period\nwith the height of the magnetic dip. These heights correspon d\nto the prominence region with the highest density contrast. One\ncan observe in this figure that the slope of the curves depends\non the spatial resolution. The steepest slope corresponds t o the\nexperiment with the highest spatial resolution ( ∆= 30 km). In\ncontrast, for∆ = 240 km, the period is more uniform with a\nsmoother variation with height.\nUnder prominence conditions, the main restoring force\nof these oscillations is the solar gravity projected along t he\nmagnetic field (see e.g., Luna & Karpen 2012; Luna et al.\n2012, 2016; Zhou et al. 2018; Liakh et al. 2020; Fan 2020).\nLuna & Karpen (2012) suggested that the period of the longi-8 10 12 14 16\nZ (Mm)2530354045P (min.)\n ∆x=240 km\n ∆x=120 km\n ∆x=60 km\n ∆x=30 km\nFig. 4: Periods as a function of height of the magnetic dips. The sym-\nbols denote the longitudinal period obtained from the numer ical exper-\niments, and the solid lines are the periods predicted by the p endulum\nmodel. Different colors and symbols correspond to the experiments with\ndifferent spatial resolutions. The color gradation from dark to light cor-\nresponds to decrease of the density contrast.\ntudinal oscillations depends only on the radius of curvatur e of\nthe magnetic dip, Rc, as\nP=2π/radicalBigg\nRc\ng, (6)\nwhereg=274 m s−2is the solar gravitational acceleration. In\nour magnetic configuration, the curvature of the magnetic fie ld\nlines in the prominence area decreases with height. This imp lies\nthat the period of the di fferent field lines increases with height in\nagreement with Fig. 3. The radius of curvature of the dips of t he\nfield lines changes with time in response to plasma motion. In\norder to compare the results of the simulations with the pend u-\nlum model, we compute the time-averaged radius of curvature .\nThe values obtained are Rc=17.3−51.1 Mm for the selected\nfield lines from the bottom to the top of the prominence. With\nEq. (6) we computed the theoretical period shown in Fig. 4 as\na solid line. From the figure, we see that for the experiments\nwith∆ = 30,60 and 120 km, we have a good agreement be-\ntween the numerical results and the pendulum model at height s\nz=10.2−16.1 Mm, where the body of the prominence is located.\nWe also observe that even though all three resolutions have a\ngood agreement, the agreement is slightly better for the fine st\nresolution. In contrast, in the coarse case with ∆ = 240 km,\nthere is only agreement at the central part of the prominence .\nThis indicates a strong influence of the numerical di ffusivity on\nthe period of oscillations and that it is necessary to reach a cer-\ntain spatial resolution to have an agreement with the pendul um\nmodel. Using 3D numerical simulation, Zhou et al. (2018) and\nFan (2020) concluded that in their experiments a disagreeme nt\nwith pendulum model is about of 10 %. We suggest that high-\nresolution experiments can reduce this discrepancy.\nFigure 4 shows a clear deviation from the pendulum model\nbelow the prominence, z<10.2 Mm and above it, z>16.1\nMm, even for high spatial resolution. This behavior suggest s\nan existence of a physical reason for the deviation. Previou s\nworks (such as Zhou et al. 2018; Zhang et al. 2019; Liakh et al.\n2020; Fan 2020) have also found discrepancies with the pendu -\nlum model. Zhou et al. (2018) and Zhang et al. (2019) found tha t\nif the gravity to Lorentz force ratio is close to unity, the he avy\nprominence plasma can significantly deform the magnetic fiel d\nArticle number, page 5 of 12A&A proofs: manuscript no. ms\nlines. They defined a dimensionless parameter as the ratio be -\ntween the weight of the thread to the magnetic pressure as\nδ=2ρgl\nB2/2µ0, (7)\nwhere lis the half-length of the prominence thread. The authors\nargued that if the parameter δis close to unity, the weight of the\nprominence changes the field geometry dynamically. In this s it-\nuation, the actual trajectory of the plasma does not coincid e with\nthe field lines, and the trajectory of the plasma has a larger r a-\ndius of curvature than the one corresponding to an unperturb ed\nmagnetic field line (see Fig. 9 from Zhang et al. 2019). Thus, t he\nresulting period of LALOs can be longer than predicted by the\npendulum model. In our model, the magnetic field at the dipped\npart increases with z, from 9 G at the base of the prominence up\nto 12.5 G at its top. Using the parameters of our model, we obtain\nthat the maximum value δ=0.7 reached at around z=11.8 Mm,\nthat is around the densest region of the prominence. In that r e-\ngion, we have obtained a good agreement with the pendulum\nmodel. For the regions below and above the prominence, δis\nmuch smaller. We conclude that the weakness of the magnetic\nfield cannot explain the discrepancies between the pendulum\nmodel and our simulations. However, we have found that the\ninteraction between the longitudinal motion of the promine nce\nand the magnetic structure is not negligible. When the promi -\nnence moves, the magnetic field structure changes considera bly.\nThese perturbations of the magnetic field could be transmitt ed to\nthe rest of the field structure. This contributes to the dampi ng of\nthe LALOs due to wave leakage, as we discuss below in Sect. 4.\nLuna et al. (2012) showed that the restoring force of LALOs\nis a combination of the gravity projected along the magnetic field\nand gas pressure gradient. The relative importance of both r estor-\ning forces depends on the radius of curvature of the field line s\nwith respect to a characteristic radius Rlim. In a situation where\nRc≪Rlim, the gravity dominates, and the pendulum model is\nvalid. In contrast, for Rc/greaterorsimilarRlimthe gas pressure term dominates.\nThe radius Rlimdepends on several parameters of the structure\nRlim=l(L−l)χg\nc2sc, (8)\nwhere cscis coronal sound speed, Lis half-length of the field\nline,χis the density contrast between the prominence and the\nambient corona. In our situation, below the prominence ( z<10.2\nMm), the ratio Rc/Rlim>0.2, and it increases for smaller values\nofz. The main reason is that the density contrast χis close to\n1 in that region. This indicates that the contribution of the gas\npressure to the restoring force is important, and the pendul um\nmodel is no longer valid. Similarly, for z≥16.1 Mm, both the\ncurvature of the field lines and the density contrast decreas e with\nheight. Thus Rc/Rlimincreases in these field lines from 0 .4 up to\n5.9 for higher values of z. This explains why the period of the\nlongitudinal oscillations deviates from the pendulum peri od for\nlocations below and above the prominence. In turn, in the cen tral\nregion, z=10.2−16.1 Mm, the ratio is around of 0 .1.\nThe prominence is also subject to transverse oscillations.\nSimilarly to the longitudinal velocity, we obtained tempor al evo-\nlution ofv⊥using Eq. (6) from Luna et al. (2016) for each se-\nlected field line. Figure 5 shows the temporal evolution for t he\n∆= 30 km case. We only show the case with the finest spatial\nresolution because all experiments have a very similar temp o-\nral evolution of the transverse velocity. From the figure, we see\nthat the transverse motions are synchronized in all the field lines\nwith the same oscillatory period, P=4 minutes. This uniformity∆=30 km\n0 50 100 150 200 250\nTime (min.)81012141618\nZ(Mm)V⊥=5 km s−1\n0.01 0.50 1.00 1.50 2.00\nρ (×10−10 kg m −3)\nFig. 5: Temporal evolution of v⊥at the center of mass of the selected\nfield lines. The color bar denotes the maximum initial densit y at each\nfield line. The left vertical axis indicates the velocity amp litude scale.\nThe right axis denotes the height of the dips of the field lines .\nof the oscillations shows that the motion is related to a glob al\nnormal mode of the field structure. The oscillation in v⊥is not\nharmonic, and the amplitude is modulated every 30 minutes ap -\nproximately. This modulation is also synchronized in all th e field\nlines shown in the figure. The amplitude of the transverse osc il-\nlations is equal to 5 km s−1that is much smaller than for the lon-\ngitudinal motions. This modulation of the amplitude is prob ably\nrelated to the motion of the prominence mass. The global moti on\nof the plasma changes the characteristics of magnetic struc ture\nperiodically, resulting in a modulation of the oscillation . The sig-\nnal modulation of the transverse velocity can also indicate the\nlongitudinal to transverse mode conversion. It can be a subj ect\nfor future work. During the simulation time, the transverse oscil-\nlations show nearly no significant damping in the four numeri cal\nexperiments with di fferent∆, indicating that numerical dissipa-\ntion does not affect this kind of motion.\n3.2. Influence of spatial resolution on the damping\nIn order to study in detail the damping in the experiments wit h\ndifferent∆, we consider longitudinal velocity in two field lines\nwith the dips at z=11.8 Mm (central part of the prominence)\nandz=16.1 Mm (top part). The v/ba∇dblin the central and top field\nlines are shown in Fig. 6. The upper panel of Fig. 6 shows that\nthe attenuation is strongest in the experiment with the coar sest\nresolution (∆= 240 km). The oscillation signal is almost com-\npletely damped after 150 minutes. In contrast, the experime nt\nwith∆ = 30 km is where observed the weakest attenuation.\nComparing all the cases, we see that the damping time increas es\nfor increasing spatial resolution (i.e., decreasing ∆). The curves\nfor∆= 60 and 30 km are relatively close to each other. This\nindicates that the damping time is close to the saturation an d that\nthe high-resolution experiment shows indications for the p hysi-\ncal damping not associated with numerical di ffusivity. In order\nto quantify the damping time we fit the signal of v/ba∇dblby decaying\nsinusoidal function such as v/ba∇dbl=V0e−t/τdsin(2πt/P+φ). The val-\nues ofτdobtained from the best fit to v/ba∇dblare shown in Fig. 7 (top\npanel) as a function of spatial resolution, ∆. We observe that the\ndamping time increases when ∆decreases for experiments with\n∆=240,120,and 60 km. In these three experiments, the damp-\ning time shows a quadratic dependence on ∆denoted by the solid\nline in the top panel of Fig. 7. The trend marked by the solid li ne\nArticle number, page 6 of 12Liakh, Luna & Khomenko: Numerical simulations of LAOs\nZ=11.8 Mm\n50 100 150 200 250\nTime (min.)−30−20−100102030V|| (km s−1)\n∆=240 km\n∆=120 km\n∆=60 km\n∆=30 km\nZ=16.1 Mm\n50 100 150 200 250\nTime (min.)−30−20−100102030V|| (km s−1)\nFig. 6: Temporal evolution of v/ba∇dblat the center of mass of the selected\nfield lines. Top: field line close to the prominence center; bo ttom: field\nline at the top part of the prominence. The colors stand for th e experi-\nments with different spatial resolutions.\nseems to indicate the damping time of about τd=110 minutes\nas∆approaches zero. However, the damping time in the experi-\nment with∆=30 km deviates from this quadratic trend and has\na similar value to the one in the ∆= 60 km experiment. We fit-\nted an arctangent function to all four values of τd, including the\none in∆=30 km experiment. This new fit is shown as a dashed\nline at the top panel of Fig. 7. The trend, including the highe st\nresolution case, deviates from the quadratic trend as the cu rve\nbecomes flatter. This suggests that τdmay saturate for decreas-\ning values of∆showing the non-numerical (i.e., physical) origin\nof this damping. According to the dashed line, τd→100 minutes\nas∆approaches zero.\nWe repeated the same analysis for the magnetic field line lo-\ncated at z=16.1 Mm, which is at the top of the prominence.\nThe results are shown at the bottom panel of Fig. 6. The tempo-\nral evolution of the longitudinal velocity di ffers much from the\none at z=11.8 Mm. We can clearly distinguish two stages in\ntemporal evolution. In the first stage, the velocity is ampli fied\nafter the initial perturbation. The amplification is more si gnifi-\ncant and extended in time in the simulations with higher spat ial\nresolution. For the case of ∆= 30 km, the amplitude increases\nfromv/ba∇dbl=19.5 km s−1up to 23 km s−1during 130 minutes. This\namplification can be well traced in the snapshots of temporal\nevolution shown in Fig. 2d above. The oscillations are atten u-\nated in the bottom and central parts of the prominence. How-\never, at heights z>15 Mm, the displacement is not only com-\nparable but even larger than the initial displacement shown in\nFig. 2a. During the second stage, the oscillations begin to d e-\ncay slowly. At the end of simulated time ( t=275 minutes), theZ=11.8 Mm\n0 50 100 150 200 250\n∆ (km)60708090100110τ (min.)\nZ=16.1 Mm\n0 50 100 150 200 250\n∆ (km)−50005001000τ (min.)Second phase\nFirst phase\nFig. 7: Top: damping time of oscillations at height 11 .8 Mm as a func-\ntion of the spatial resolution ∆. The solid line denotes the quadratic\ndependence, and the dashed line shows fitting with an arctang ent func-\ntion. Bottom: the same for the height 16 .1 Mm. The solid lines denote\nfitting with an arctangent function.\nmotions are almost completely damped for the experiments wi th\n∆= 240,120 km. In the higher-resolution simulations, ∆= 30\nand 60 km, the upper prominence part oscillates undamped for\na longer time. In order to quantify the damping time for these\ncomplex oscillations, we split the velocity signal into two parts\nassociated with the amplification and damping. We then fit v/ba∇dbl\nwith the decaying and amplifying sinusoidal functions. For the\namplification stage, we assume the characteristic time, τA, to be\nnegative, and for the decaying stage, the damping time, τD, pos-\nitive. From the best fit to the longitudinal velocity we obtai ned\nthe values ofτAin the range [−600,−225] minutes, and those of\nτDin the range of [152 ,628] minutes. The higher values of both\nparameters correspond to better resolutions. The results o f this\nanalysis are shown at the bottom panel of Fig. 7. The characte ris-\ntic time of the amplification, τA, differs significantly in the high-\nand low-resolution experiments. We can observe that, in the ex-\nperiment with∆= 30 km,τA=−225 minutes, that is the most\nsignificant amplification. In addition, the bottom panel of F ig. 6\nalso shows that this amplification stage is more extended in t ime\nfor the finer spatial resolutions. The damping time in the sec ond\nphase reaches the value of τD=628 minutes for the simulation\nwith∆=30 km, indicating the weakest damping. The amplifica-\ntion stage is not so relevant for the experiment with ∆=240 km.\nFor this experiment, the amplification time τA=−600 minutes\nis the longest, and the oscillations have almost constant am pli-\ntudes during the first 67 .8 minutes. We performed a fit to both\nτAandτDusing an arctangent function, allowing us to find the\ntrend for both parameters (Fig. 7, solid lines at the bottom p anel).\nArticle number, page 7 of 12A&A proofs: manuscript no. ms\nWith these trends, we can roughly estimate that the real phys i-\ncal amplification for our system would have a characteristic time\nof−200 minutes, whereas the damping time would be longer\nthan 900 minutes. These results may indicate that our highes t-\nresolution experiment is about to resolve a physical mechan ism\nfor these effects.\n4. Physical reasons for the amplification and\ndamping of oscillations\nAs we have shown in Sect. 3.2 above, oscillations at the botto m\npart of the prominence damp quickly, and those at the top part\nare initially amplified and damped later. We have also seen th at\nthese phenomena may have a physical origin and are not relate d\nto numerical dissipation. In this section, we perform a deta iled\nanalysis to shed light on the possible physical mechanisms t hat\nproduce these effects.\nFirst, we study the contributions of the di fferent forces to the\nenergy balance in two regions of the prominence defined by the\norange rectangles in Fig. 1. Equations (9–11) show the tempo -\nral derivatives of the works done by the di fferent forces in the\nregion, which we denote as Σ. In addition, Eq. (12) gives the ki-\nnetic energy flow along the boundaries of the region, ∂Σ. These\nequations are:\n•\nWg=/iintegdisplay\nΣρg·vdσ, (9)\n•\nWp=−/iintegdisplay\nΣv·∇p dσ, (10)\n•\nWB=/iintegdisplay\nΣ1\nµ0(∇×B)×Bdσ, (11)\n•\nΠK=−/contintegraldisplay\n∂Σ1\n2ρv2v·ndl, (12)\nwhere nis the vector normal to the region boundaries. The com-\nbination of the four terms is equal to the temporal derivativ e\nof the kinetic energy integrated in the region Σ,•\nEK. Figure 8\nshows the temporal evolution of the terms given by Eqs. (9–12 ).\nAll these magnitudes are shown with respect to their values a t\nt=8.5 minutes. The top and bottom panels of the figure show\nthe quantities computed in the top and bottom region shown in\nFig. 1, respectively.\nFigure 8 shows a clear di fference in the evolution of the\nplasma in both regions of the prominence. At the top of the\nstructure, the kinetic energy decreases rapidly right afte r the ini-\ntial perturbation. The curve oscillates around an average v alue\nthat seems to increase slightly from 30 to 100 minutes, appro x-\nimately. The amplification of the oscillations at the upper p art\nof the prominence does not result in a significant increase of the\nkinetic energy because the region of integration is larger t han\nthe prominence size and includes the nonamplified regions. A f-\ntert=100 minutes the kinetic energy decreases. Figure 8 (top\npanel) shows that the gas pressure work is negative and that i t\nmakes the largest contribution to the kinetic energy losses af-\ntert=100 minutes. In contrast, the work done by the Lorentz\nforce in the first 100 minutes is positive, indicating that it accel-\nerates the prominence plasma. In turn, the kinetic energy in flow\nthrough the boundaries is relatively small compared to the o ther\nterms.\nAt the bottom part of the prominence (Fig. 8, bottom\npanel), the black line shows a substantial decrease in the fir st\n100 minutes coinciding with the significant attenuation of t he os-\ncillations in this region. The bottom panel shows that the ma incontribution to the kinetic energy losses is associated wit h the\nwork of the gas pressure and the Lorentz force. This can indic ate\nthe generation of fast MHD waves, which are lately emitted, c on-\ntributing to the damping of the oscillations, as already sug gested\nby Zhang et al. (2019). These fast waves are associated with t he\ngas pressure and the Lorentz forces. Additional evidence fo r the\nwave leakage is shown in Fig. 9. This figure shows the time-\ndistance diagram of the transverse velocity, v⊥, along the axis,\nx=0 Mm, during 10 minutes. The fast MHD waves produce\nperturbations of the v⊥field. The figure shows the waves that\ntravel upwards with a pattern of alternating positive and ne gative\nvalues ofv⊥. The inclination of the ridges is indicative of the\nspeed of waves. Since we performed the experiments in the low -\nβregime, the speed of the fast waves is approximately the Alfv én\nspeed,vA. The dashed lines represent locations of a wave front\nthat would propagate at a local Alfvén speed. We can see that t he\nwaves emitted from the prominence propagate with speed simi -\nlar to the Alfvén speed. Figures 8 and 9 indicate that an impor tant\nportion of the energy of the prominence oscillation is emitt ed in\nthe form of fast MHD waves.\nFigure 8 demonstrates that the increase of WBat the top may\nbe related to its decrease at the bottom part of the structure .\nThus, not all the energy losses at the bottom of the prominenc e\nare emitted in the form of waves. Instead, a part of this energ y\nis transferred to the top of the structure. This transfer can ex-\nplain the amplification of the oscillations at the upper part of the\nprominence. We computed the incoming Poynting flux into two\nregions shown by the orange lines in Fig. 1 as\n•\nΠmag=−/contintegraldisplay\n∂Σ/parenleftBiggB2v\nµ0−(B·v)B\nµ0/parenrightBigg\n·ndl. (13)\nFigure 10 shows the time integral of the incoming Poynting\nflux in both regions. We can see that during the time interval\n65−150 minutes, the incoming Poynting flux at the prominence\ntop dominates over the outcoming flux. This indicates that th e\nmagnetic energy increases at the upper prominence region. T he\nsituation is the opposite at the bottom region. The magnetic en-\nergy leaves the region, and the time integral of the Poynting flux\nhas a negative sign. In addition, the shapes of both curves ar e\nsimilar but antiphase with respect to each other, indicatin g that\nthe top part gains a significant part of the energy lost by the b ot-\ntom part. Apart from this energy transfer, the flux at the bott om\npart also shows the leakage of energy emitted to the ambient a t-\nmosphere in the form of fast waves. Overall, Fig. 10 indicate s\nthat magnetic energy seems to be partially transferred from the\nbottom to the top of the prominence.\nIn order to understand the physical reason for the transfer\nof the energy from the bottom to the upper layers of the promi-\nnence, we computed the time-distance diagrams of the di fferent\ncontributions to the total energy along the vertical direct ion in-\ntegrated in the region from x=−12 Mm to x=12 Mm. The\nresult of this calculation is shown in Fig. 11. The three pane ls\nshow the kinetic energy density, ek=ρv2/2 (Fig. 11a), the mag-\nnetic energy density, eB=B2/2µ0(Fig. 11b), and the gravita-\ntional energy density, eg=ρgz(Fig. 11c). We are interested in\nthe variation of these energies after the mass loading. Ther efore,\nFig. 11 shows the di fference with respect to the values of each\nof the energies at t=8.3 minutes. The light and dark fringes\nreflect the plasma motions and velocity variations. The figur e\nclearly shows the phase shift between the oscillations at di ffer-\nent heights inside the prominence. In addition, it can be obs erved\nthat the kinetic energy increases at t=60−130 minutes and\nat heights z>13 Mm. A similar increase can be seen for the\nArticle number, page 8 of 12Liakh, Luna & Khomenko: Numerical simulations of LAOs\nTop\n50 100 150 200 250\nTime (min.)-8-6-4-2024W (´1011 J m-1)\nBottom\n50 100 150 200 250\nTime (min.)-5-4-3-2-101W (´1011 J m-1)PK\nWg\nWp\nWB\nEK-EK0\nFig. 8: Work done by the gravity force (brown line), by the gas pressu re force (light green line), by the Lorentz force (light blue line), and the\nkinetic energy flow (pink line) integrated in two rectangles shown by the orange lines in Fig. 1. The black lines denotes a t otal kinetic energy\nvariation with respect to its value at t=8.5 minutes.\nFig. 9: Time-distance diagram of the transverse velocity, v⊥, along the\nvertical direction at horizontal position x=0 Mm. Black horizontal\nlines denote the position of the prominence. Dashed lines de note the po-\nsition of a hypothetical wave front propagating at a local Al fvén speed.\nmagnetic and gravitational energies in Figs. 11b and 11c. Th e\nmagnetic energy increase is associated with the energy exch ange\nbetween the bottom and top parts of the prominence, as was al-\nready shown in Fig. 10. The gravitational energy increases d ur-\ning the same period of time which is related to the amplificati on\nof the oscillation velocities at the top of the prominence. S ince50 100 150 200\nTime (min.)−4−2024Πmag (×1011 W s m−1 )\nTop\nBottom\nFig. 10: Time-integrated magnetic energy inflow through the bound-\naries of the domains indicated by orange lines in Fig. 1. The s hort-period\ncomponent has been filtered out.\nplasma is accelerated, it can reach higher heights along the mag-\nnetic field. As a result, the total gravitational energy incr eases.\nAt the bottom, in the region of the strong attenuation, we can see\nthe energy losses in all the panels. As mentioned before, som e\nfraction of the energy is taken away by the fast MHD waves, and\nsome other fraction is transferred to upper prominence laye rs.\nThe results above seem to indicate that the amplification of\nthe oscillations is related to the energy transfer from the b ot-\nArticle number, page 9 of 12A&A proofs: manuscript no. ms\nFig. 11: The time-distance diagram of the energy variations with res pect to their values right before the perturbation, e−e0, integrated in the region\nfrom x=−12 Mm to x=12 Mm. The white lines denote the heights where the prominenc e is located. (a) kinetic energy density, ek=ρv2/2; (b)\nmagnetic energy density, eB=B2/2µ0; (c) gravitational energy density, eg=ρgz.\ntom to the top parts of the structure. The magnetic field struc ture\nchanges with time thanks to the oscillations, and the actual tra-\njectory of the plasma motions is not along the unperturbed ma g-\nnetic field lines anymore. In a situation with a rigid magneti c\nfield, the trajectory coincides with the field line, and the Lo rentz\nforce has no projection along the trajectory of the plasma. H ow-\never, in our situation, the trajectory does not coincide wit h the\nunperturbed field lines. In such a case, the forces along the t ra-\njectory can become di fferent from the forces projected along the\nmagnetic field lines. In order to understand the detailed mec ha-\nnism for the amplification and damping, we study the motion of\nthe individual fluid elements by integrating the velocity fie ld at\neach time moment. We selected two plasma elements with the\ninitial coordinates ( x,z)=(0,16.1) and (0,11.8) Mm that corre-\nspond to the particles at the top and the central part of the pr omi-\nnence. In order to highlight the modifications of the magneti c\nfield lines due to the plasma motions, we multiply the magneti c\nfield perturbation by a factor of 10. This magnetic field is de-\nfined as B′=10(B−B0)+B0, where Bis the actual magnetic\nfield at time tandB0is the magnetic field at t=0 seconds. Simi-\nlarly, the displacement of the fluid elements is also multipl ied by\nthe same factor in order to fulfill the frozen-in condition. F igure\n12 shows the position of the fluid elements (orange diamonds)\nand centers of the dips (blue circles) of the magnetic field li nes\nat the different time moments. Figure 12a shows the maximum\ndisplacement of the fluid elements right after the perturbat ion.\nWe see that the field lines hosting the particles changes, and the\nposition of the dips follow the motion of the particles. In Fi g.\n12b, the fluid elements move to the left of the structure. The d ip\nat the lower line continues following the particle motion. I n con-\ntrast, at the upper line, the dip is located ahead of the parti cle.\nIn this situation, the upper particle reaches the dip in a pos ition\nthat is displaced to the left from the original dip. Thus, the upper\nparticle has gained an increment in velocity during this firs t half\nperiod of oscillation. This process is repeated during the f ollow-\ning period. In Fig. 12c, we observe that the particle has gain ed\nenergy and that the amplitude of the oscillation is larger. I n Fig.\n12d, the dip has moved away from the particle again, leading t o\nan increase of the amplitude. The same process is also produc ed\nin the reverse direction, as can be observed in Figs. 12e, f. T he\nmotion of the upper dip follows the motion of the dip at the bot -\ntom line. The reason is that the magnetic structure reacts to the\nbulk motion of the prominence. In this sense, the changes in t hemagnetic configuration at the bottom part of the structure a ffect\nthe top part of the structure.\nThe situation is right the opposite at the bottom of the struc -\nture. The plasma motions also modify the field lines, but the d ip\napproaches the particle in this case. In this way, the oscill atory\namplitude is reduced in each oscillation period. In Figs. 12 c and\n12d, we can observe the following situation: the motion of th e\nfluid element at the bottom causes displacement of the corre-\nsponding magnetic dip. As we can see from Figs. 12e and 12f,\nthe dip at the bottom continues following the fluid element in\nthe next period of oscillations. Thus, the particle continu es los-\ning energy in each period, and the oscillations damp quickly .\nIn order to study the oscillation amplification phenomena in\nmore detail and check how the motions at the upper and lower\npart of the prominence a ffect each other, we performed an al-\nternative experiment described below. We used the same prom i-\nnence model, but the height of the maximum of the perturbatio n\nwas shifted down to z=9 Mm, and the characteristic vertical\nsize of the perturbed region was reduced to σz=4.8 Mm. Us-\ning this numerical setup, we excited the prominence oscilla tions\nonly at the bottom, while the upper part of the prominence re-\nmained unperturbed. Then, we repeated the calculations of v/ba∇dbl,\nas it was done in Sect. 3. Figure 13 shows v/ba∇dblat the selected field\nlines. We can observe that the motions at the field lines at hei ghts\n7.2−10.2 Mm are produced directly by the perturbation. These\noscillations are significantly attenuated during the time i nterval\nshown in Fig. 13. No perturbation has been applied for the fiel d\nlines with z>10.2 Mm, and therefore v/ba∇dblat those field lines\nis initially zero. However, after 20 minutes of the evolutio n, we\ncan observe a signature of oscillations with a small amplitu de\nat the upper part. After 20 minutes, the oscillations fronts reach\nthe top of the prominence, and their amplitude increases. At the\ntime 100−150 minutes, we can clearly distinguish oscillation at\nz>10.2 Mm. We can also observe the phase shift of the signal\nbetween different heights that resembles Fig. 3. We additionally\nperformed the opposite experiment perturbing the top regio n and\nleaving the bottom and central part without initial perturb ation.\nThe analysis of motions revealed a similar result, that the o s-\ncillations at the top drives the oscillations at the bottom l ayers\nof the prominence. This experiment seems to indicate that th e\ntransfer of energy could be symmetric. However, in the regul ar\nexperiments where all the layers are excited simultaneousl y, the\nbottom part of the prominence drives the motion producing th e\nArticle number, page 10 of 12Liakh, Luna & Khomenko: Numerical simulations of LAOs\nFig. 12: Temporal evolution of the fluid elements and the magnetic fiel d lines after the perturbation. The orange diamonds denote t he positions of\nthe fluid elements, and the blue circles correspond to the loc ations of the center of the dips. The purple arrows denote the velocity field. Dotted\nlines are unperturbed magnetic field lines; solid lines are a ctual magnetic field lines at a given moment. Note that the mag netic field perturbation\nis multiplied by a factor of 10 for better visibility.\n∆=120 km\n0 50 100 150\nTime (min.)681012141618\nZ(Mm)V||=22 km s−1\n0.01 0.50 1.00 1.50 2.00\nρ (×10−10 kg m −3)\nFig. 13: Temporal evolution of v/ba∇dblat the center of mass of the selected\nfield lines in the experiment with the perturbation at the bot tom. The\ncolor bar denotes the maximum initial density at each field li ne. The left\nvertical axis indicates the velocity amplitude scale. The r ight vertical\naxis denotes the height of the dips of the field lines.\namplification of the top part. The coupling of the oscillatio ns of\nthe different regions of prominence is a very interesting subject\nfor future research.\nIn order to check if oscillation amplification is a common\nphenomenon of LALOs, we performed yet another experiment.\nIn this experiment when we did not use any external perturbat ion\nbut instead loaded the prominence mass at some distance from\nthe center of the magnetic dips. Since the mass is not in equil ib-\nrium, it starts to move under the action of gravity in the dire ction\ntoward the center of the dips. Thus, the plasma starts to osci llate\naround the equilibrium position, and the LALOs are establis hed\nwithout any disturbance. We found that the resulting oscill ations\nin this experiment are similar to those obtained in our regul ar ex-\nperiment. Namely, we observe that the velocity at the top of t he\nprominence increases while at the bottom of the prominence, itrapidly decreases. After several cycles of oscillations, t he veloc-\nity of the plasma at the upper prominence region becomes high\nenough to allow plasma to leave the shallow dips. Consequent ly,\nthe acceleration of the plasma at the top leads to mass draina ge\nfrom the upper prominence region.\nSummarizing all above, we conclude that the e ffect of am-\nplification of the plasma at the prominence top takes place in\nseveral alternative experiments, such as experiments wher e we\nperturb only the bottom part of the prominence or do not apply\nperturbation but place prominence in a nonequilibrium posi tion.\nThis means that the e ffect of amplification is a frequent ingredi-\nent of LALOs and deserves more investigation in the future.\n5. Summary and Conclusions\nWe have studied the properties of LALOs, including their pe-\nriods and damping mechanism, based on 2D numerical simula-\ntions. We have used a simple magnetic field configuration that\nincludes a dipped region. After the mass loading in the dips, we\napplied a perturbation to the prominence. This perturbatio n was\ndirected along the magnetic field. Our main goal was to invest i-\ngate the physical mechanism of the LALOs attenuation covere d\nby the numerical di ffusion. Therefore, we studied the oscilla-\ntions in the same numerical model gradually increasing spat ial\nresolution,∆.\nWe have analyzed the prominence motions and computed the\nperiods in the different prominence regions. The period of the\noscillations shows an increase with height. We have also fou nd\nthat the period depends on ∆. In the higher-resolution simula-\ntions, the period shows a strong dependence on height, while\nin the lower-resolution experiments, this dependence is le ss pro-\nnounced. The period shows a good agreement with the pendulum\nmodel in our simulations with the best resolution of 30 km.\nWe have studied the damping of oscillations in di fferent ex-\nperiments and at di fferent prominence regions. In all the experi-\nments, the prominence bottom part is characterized by stron g at-\ntenuation of oscillations. This attenuation is present eve n in the\nArticle number, page 11 of 12A&A proofs: manuscript no. ms\nhighest-resolution experiments. The experiments with the finest\nresolution,∆ = 60 and 30 km, demonstrated that further im-\nprovement of the spatial resolution does not significantly a ffect\nthe damping time. This means that our experiments reached th e\nresolution where the damping of oscillations is not associa ted\nanymore with the numerical dissipation but is rather caused by\nsome physical mechanism. In a real situation, additional e ffects\nas thermal conduction and radiative losses also contribute to the\ndamping. It is necessary to include these additional e ffects in or-\nder to understand which mechanism dominates in the LALOs\ndamping. This will be the subject of future research.\nOur experiments revealed that oscillations at the prominen ce\ntop are amplified during the first 130 minutes of simulations a nd\nlater are slowly attenuated. The amplification appears to be more\nefficient and extended in time in the high-resolution experimen ts.\nIn order to explain the strong attenuation of oscillations a t\nthe prominence bottom and their simultaneous amplification at\nthe prominence top, we have analyzed the evolution of di fferent\ntypes of energies in the corresponding regions. This analys is re-\nvealed that the damping of the oscillations is partially due to the\ncollective work done by the gas pressure and Lorentz force. T he\nenergy is emitted in the form of fast magneto-acoustic waves .\nThis result is in agreement with the work by Zhang et al. (2019 ).\nFurthermore, we have seen that the damping of the oscillatio ns\nis related to the strength of the magnetic field. The motion of the\nprominence plasma produces periodic changes in the magneti c\nfield. We have found that this e ffect leads to the generation of\nfast MHD waves. The time-distance diagram of the transverse\nvelocity provides another piece of evidence for the wave lea k-\nage. The inclination of the wave front ridges found in the tim e-\ndistance diagram is in agreement with the inclination predi cted\nby the Alfvén speed. Yet another conclusion from our analysi s\nis that the Lorentz force plays an important role in the dampi ng\nand amplification of LALOs. While at the bottom, it contribut es\nto the kinetic energy losses and acts to decelerate plasma, a t the\nprominence top, the work done by the Lorentz force is positiv e\nand provides the gain of the energy needed for amplification o f\noscillations. The analysis of the Poynting flux revealed tha t a\nsignificant portion of the energy leaving the bottom part goe s to\nthe top. These results suggest that the energy losses at the l ower\nprominence region are caused by both the wave leakage and the\nenergy and momentum transfer to the upper prominence region .\nOur study of LALOs based on 2D numerical simulations has\nshown that high spatial resolution is crucial for investiga ting the\nperiods of LALOs. The period agrees with the pendulum model\nonly when the spatial resolution is high enough. The high spa tial\nresolution is also important for the understanding of the da mping\nof LALOs. On the other hand, the numerical dissipation can hi de\nthe important physical mechanism as amplification of LALOs.\nIn the future, it would be necessary to study the attenuation -\namplification mechanisms further by using high-resolution ex-\nperiments with more complex 3D setups, which would allow us\nto take the mechanism of resonant absorption into considera tion.\nFurthermore, it is also desirable to include the nonadiabat ic ef-\nfects. This can allow us to study the relative importance of t he\nmechanisms described in this paper with respect to the reson ant\nabsorption and nonadiabatic e ffects. On the other hand, more ob-\nservations are needed to study further the phenomena of the a m-\nplified prominence oscillations.\nAcknowledgements. V . Liakh acknowledges the support of the Instituto de As-\ntrofísica de Canarias via an Astrophysicist Resident fello wship. M. Luna ac-\nknowledges support through the Ramón y Cajal fellowship RYC 2018-026129-I\nfrom the Spanish Ministry of Science and Innovation, the Spa nish National Re-\nsearch Agency (Agencia Estatal de Investigación), the Euro pean Social Fundthrough Operational Program FSE 2014 of Employment, Educat ion and Train-\ning and the Universitat de les Illes Balears. V . L. and M. L. al so acknowl-\nedges support from the International Space Sciences Instit ute (ISSI) via team\n413 on “Large-Amplitude Oscillations as a Probe of Quiescen t and Erupting\nSolar Prominences.” E. K. thanks the support by the European Research Coun-\ncil through the Consolidator Grant ERC-2017-CoG-771310-P I2FA and by the\nSpanish Ministry of Economy, Industry and Competitiveness through the grant\nPGC2018-095832-B-I00 is acknowledged. V . Liakh, M. Luna, a nd E. Khomenko\nthankfully acknowledge the technical expertise and assist ance provided by the\nSpanish Supercomputing Network (Red Española de Supercomp utacíon), as well\nas the computer resources used: the LaPalma Supercomputer, located at the\nInstituto de Astrofísica de Canarias. The authors thankful ly acknowledge the\ncomputer resources at MareNostrum4 and the technical suppo rt provided by\nBarcelona Supercomputing Center (RES-AECT-2020-1-0012 a nd RES-AECT-\n2020-2-0010)\nReferences\nAdrover-González, A. & Terradas, J. 2020, A&A, 633, A113\nArregui, I. n., Oliver, R., & Ballester, J. L. 2018, Living Re views in Solar\nPhysics, 15, 3\nBallester, J. L., Carbonell, M., Soler, R., & Terradas, J. 20 16, A&A, 591, A109\nBi, Y ., Jiang, Y ., Yang, J., et al. 2014, ApJ, 790, 100\nChen, P. F., Harra, L. K., & Fang, C. 2014, ApJ, 784, 50\nFan, Y . 2020, ApJ, 898, 34\nFelipe, T., Khomenko, E., & Collados, M. 2010, ApJ, 719, 357\nJing, J., Lee, J., Spirock, T. J., & Wang, H. 2006, Sol. Phys., 236, 97\nJing, J., Lee, J., Spirock, T. J., et al. 2003, ApJ, 584, L103\nKhomenko, E. & Collados, M. 2012, ApJ, 747, 87\nKhomenko, E., Collados, M., & Felipe, T. 2008, Sol. Phys., 25 1, 589\nLeroy, J. L., Bommier, V ., & Sahal-Brechot, S. 1983, Sol. Phy s., 83, 135\nLeroy, J. 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F. 1998, Nature, 396, 440\nArticle number, page 12 of 12" }, { "title": "1011.5175v2.Magnetohydrodynamic_kink_waves_in_two_dimensional_non_uniform_prominence_threads.pdf", "content": "arXiv:1011.5175v2 [astro-ph.SR] 27 Jul 2011Astronomy& Astrophysics manuscriptno.ms c/ci∇cleco√y∇tESO 2018\nOctober12,2018\nMagnetohydrodynamic kink wavesintwo-dimensional non-un iform\nprominence threads\nI. Arregui1,R.Soler2,J. L.Ballester1, and A.N.Wright3\n1Departament de F´ ısica,Universitat de les IllesBalears,E -07122, Palma de Mallorca, Spain\ne-mail:[inigo.arregui,joseluis.ballester]@uib.es\n2CentreforPlasmaAstrophysics,DepartmentofMathematics ,KatholiekeUniversiteitLeuven,Celestijnenlaan200B,3 001Leuven,\nBelgium\ne-mail:roberto.soler@wis.kuleuven.be\n3School of Mathematics and Statistics,Universityof St.And rews, St.Andrews, KY169SS,UK\ne-mail:andy@mcs.st-and.ac.uk\nReceived ,;accepted\nABSTRACT\nAims.Weanalysetheoscillatorypropertiesofresonantlydamped transversekinkoscillationsintwo-dimensionalprominen cethreads.\nMethods. The fine structures are modelled as cylindrically symmetric magnetic flux tubes witha dense central part withprominence\nplasmapropertiesandanevacuatedpart,bothsurroundedby coronalplasma.Theequilibriumdensityisallowedtovaryn on-uniformly\ninboththetransverseandthelongitudinaldirections.Wee xaminetheinfluenceoflongitudinaldensitystructuringon periods,damping\ntimes, and damping rates for transverse kink modes computed by numerically solving the linear resistive magnetohydrod ynamic\n(MHD) equations.\nResults.The relevant parameters are the length of the thread and the d ensity in the evacuated part of the tube, two quantities that\nare difficult to directly estimate from observations. We find that bot h of them strongly influence the oscillatory periods and damp ing\ntimes,andtoalesserextent thedampingratios.Theanalysi softhespatialdistributionofperturbations andoftheene rgyfluxintothe\nresonances allows us toexplain the obtained damping times.\nConclusions. Implications for prominence seismology, the physics of res onantly damped kink modes in two-dimensional magnetic\nfluxtubes, and the heatingof prominence plasmas are discuss ed.\nKey words. Magnetohydrodynamics (MHD) –Waves–Sun: filaments,promin ences\n1. Introduction\nQuiescent filaments /prominences are cool and dense magnetic\nand plasma structures suspended against gravity by forces\nthoughttobe ofmagneticorigin.Inspite oftheirphysicalp rop-\nerties, with temperatures and densities that are akin to tho se\nin the chromosphere, some as yet not well determined mecha-\nnismsprovidethe requiredthermalisolation fromthe surro und-\ning coronal plasma and mechanical support during typical li fe-\ntimesfromfew daysto weeks. The magneticfield that pervades\nthese structures is believed to play a key role in the nature a nd\nthe thermodynamic and mechanical stability of prominences .\nEarly observations carried out with good seeing conditions\npointed out that prominences consist of fine threads (deJage r,\n1959; Kuperus&Tandberg-Hanssen, 1967). More recent high-\nresolution Hαobservations obtained with the Swedish Solar\nTelescope (SST) in La Palma (Linet al., 2005) and the Dutch\nOpen Telescope (DOT) in Tenerife (Heinzel&Anzer, 2006)\nhaveallowedtofirmlyestablishthefilamentsub-structurin gand\nthe basic geometrical and physical properties of threads (s ee\nalsoEngvold,1998;Linet al.,2005,2008; Lin,2010).Thesu b-\nstructureofquiescentprominencesisoftencomposedbya my r-\niad of horizontal,dark and fine threads, made of cool absorbi ng\nmaterial,believedtooutlinemagneticfluxtubes(Engvold, 1998,\n2008;Lin,2004;Linetal.,2005,2008;Martinet al.,2008). The\ntubes are only partially filled with cool and dense plasma and\ntheir total length is probably much larger ( ∼105km) than thethreads themselves. The measured average width of resolved\nthreads is about 0.3 arcsec ( ∼210 km) while their length is be-\ntween5and40arcsec( ∼3500-28000km).Theabsorbingcool\nmaterialisusuallyvisibleforupto20minutes(Linet al.,2 005).\nThe measured widths are close to the current resolution limi t,\n∼0.16 arcsec at the SST, hence thinner structures are likely t o\nexist.\nSmall amplitude oscillations in prominence threads are fre -\nquently observed (see reviews by Oliver& Ballester, 2002;\nEngvold, 2004; Wiehr, 2004; Ballester, 2006; Banerjeeet al .,\n2007; Engvold, 2008;Oliver,2009;Ballester, 2010).Early two-\ndimensional observations of filaments (Yi &Engvold, 1991;\nYi et al., 1991) revealed that individual threads or groups o f\nthem oscillate with periods that range between 3 and 20 min-\nutes. Recent relevant examples are traveling waves propaga t-\ning along a number of threads with average phase speed of 12\nkm s−1, wavelength of 4 arcsec, and oscillatory periods that\nvary from 3 to 9 minutes (Linet al., 2007), the both propa-\ngating and standing oscillations detected over large areas of\nprominences by Terradaset al. (2002) and Lin (2004), as well\nas observations from instruments onboard space-crafts, su ch as\nSoHO (Blancoetal., 1999; R´ egnieretal., 2001; Pougetet al .,\n2006)andHinode(Okamotoetal.,2007;Terradaset al.,2008 b;\nNingetal., 2009). The transverse oscillation nature of som e of\nthese events has been clearly established by Linetal. (2009 )\nby combining H αfiltergrams in the plane of the sky with H α\nDopplergrams which allow to detect oscillations in the line -2 I.Arregui et al.:Magnetohydrodynamic kink waves intwo-d imensional non-uniform prominence threads\nof-sight direction. A recurrently observed property of pro mi-\nnence oscillations is their rapid temporal damping, with pe r-\nturbations decaying in time-scales of only a few oscilla-\ntory periods (Landmanet al., 1977; Tsubaki& Takeuchi, 1986 ;\nTsubaki, 1988; Wiehret al., 1989; Molowny-Horasetal., 199 9;\nTerradaset al.,2002; Lin, 2004; Ningetal.,2009).\nTransverse thread oscillations are commonly interpreted i n\ntermsofstandingorpropagatingmagnetohydrodynamic(MHD )\nkink waves. The measured periods are of the order of a few\nminutes and the wavelengths are in between 3000 - 20 000\nkm, although Okamotoet al. (2007) report larger wavelength s\nmoreconsistentwiththestandingwaveinterpretation.The mea-\nsured wave quantities allow us to derive phase speeds that\nare consistent with the kink speed in magnetic and plasma\nconfigurations with typical properties of prominence plasm as.\nThe MHD wave interpretation of thread oscillations has al-\nlowed the development of theoretical models (see Ballester ,\n2005,2006,forrecentreviews).Joarderet al.(1997);D´ ıa z et al.\n(2001, 2003) considered the MHD eigenmodes supported by a\nfilament thread modelled in Cartesian geometry. More realis -\ntic studies using cylindrical configurations have extended the\ninitial investigations (D´ ıazet al., 2002; Dymova& Ruderm an,\n2005; D´ ıaz &Roberts, 2006). These studies have determined\nthe frequencies and confinement properties of the perturba-\ntions as functions of the length and the width of the threads.\nTheoreticaldampingmechanismshavealsobeendeveloped(s ee\nBallester, 2010; Arregui&Ballester, 2010, for recent revi ews).\nA systematic comparative study of di fferent mechanisms has\nbeen presented by Soler (2010), who assesses the ability of\neach mechanism to reproduce the observed attenuation time-\nscales. The considered mechanisms include non-ideal e ffects,\nsuch as radiation and thermal conduction, partial ionisati on\nthrough ion-neutral collisions, ion-electron collisions , and res-\nonant absorption due to coupling to Alfv´ en and slow waves.\nNon-idealeffectsdonotseemtoprovidetherequiredattenuation\ntime-scales for kink oscillations (Ballai, 2003; Carbonel let al.,\n2004;Terradaset al.,2001,2005;Soleret al.,2007,2008). Soler\n(2010) finds that resonant absorption in the Alfv´ en contin-\nuum is the only mechanism able to produce the observed at-\ntenuation time-scales (see Soleret al., 2008, 2009a,b,c, f or de-\ntails). Resonant damping was first considered in this contex t by\nArreguietal.(2008),andhasbeensubsequentlystudiedinc om-\nbination with damping in the slow continuum and partial ioni -\nsationbySoleret al.(2009a,b,c).Thesestudieshaveconsi dered\none-dimensional models with a density variation in the tran s-\nverse direction only. On the other hand, theoretical studie s that\ntake into account the longitudinal density structuring, he nce in-\ncorporating the fact that magnetic tubes supporting thread s are\nonly partially filled with cool plasma, consider piece-wise ho-\nmogeneous models in the transverse direction, and hence rul e\noutresonantdamping.\nArecentinvestigationbySoleret al.(2010)istheexceptio n.\nThese authors have obtained analytical and semi-analytica l ap-\nproximationsforperiods,dampingtimes,anddampingratio sof\nstanding kink modes in a two-dimensional prominence thread\nmodel. However, the analysis by Soleret al. (2010) is restri cted\nto the thin tube and thin boundary approximations,radial in ho-\nmogeneityis constrainedtothe densecoolpartofthetubeon ly,\nandthedensityintheevacuatedpartofthetubeistakenequa lto\nthe coronaldensity,furtherlimitingthe prominencethrea dsthat\ncan be theoretically modelled. These limitations are remov ed\nin the present investigation. We adopt a fully inhomogeneou s\ntwo-dimensional prominence thread model. The density dist ri-\nbution is allowed to vary non-uniformly in both the transver seandlongitudinaldirections.Wealsoaddanotherrelevantp aram-\neter with seismological implications, namely, the density in the\nevacuatedpartofthetube.Bycombiningthetwoparameterst hat\ncharacterise the longitudinal structuring in prominence t hreads,\nthe length of the thread and the density in the evacuated part of\nthetube,awiderangeofprominencethreadswithverydi fferent\nphysical conditions can be modelled and their oscillatory p rop-\nerties characterised. In addition, a general parametric st udy is\nnumericallyperformed,thatallowsustogobeyondthethint ube\nandthinboundaryapproximations.\nBesides obtaining the influence of longitudinal structur-\ning on periods and damping times for kink modes in two-\ndimensional threads, and extracting conclusionsabout the ir im-\nplicationsforprominenceseismology,ourstudyaimsatexp lain-\ning the obtainedresults. For this reason, we have performed the\nanalysis of the spatial distribution of perturbations and t he en-\nergy flux into the resonances. These two analyses, which are\nnovelin the contextof prominenceoscillations, provideus with\na comprehensiveexplanationfortheobtainedparametricre sults\nandaddfurtherinsightstothephysicsofresonantlydamped kink\nmodesintwo-dimensionalequilibriumstates.\nThe paper is organised as follows. Section 2 describes\nthe thread model, the relevant parameters introduced by its\ntwo-dimensional character, the linear MHD wave equations t o\nbe solved, and the numerical method used for that purpose.\nSection 3 presents our analysis and results. We first show com -\nputations of periods, damping times, and damping ratios as a\nfunctionofthelengthofthethreadandthedensityintheeva cu-\natedpartofthetube.Implicationsforthedeterminationof phys-\nical parameters in prominences are discussed. Next, a quali ta-\ntive explanation of the obtained results is given by analysi ng\nthespatialdistributionofperturbations.Finally,wedes cribeour\nenergy analysis, that in combination with the spatial struc ture\nof eigenfunctionsfully explains the obtained dampingtime s. In\nSection4, ourconclusionsarepresented.\n2. Threadmodel,linearMHD waveequations,and\nnumericalmethod\nWe consider an individual and isolated prominence thread in\na gravity-free static equilibrium in the zero plasma- βapproxi-\nmation. The fine structure is modelled by means of a cylindri-\ncally symmetric flux tube of radius aand length L. In a sys-\ntemofcylindricalcoordinates( r,ϕ,z)withthe z-axiscoinciding\nwith the axis of the tube, the magnetic field is pointing in the\nz-direction and has a uniform field strength. Because of the as -\nsumed zero-βapproximation the density profile can be chosen\narbitrarily. The non-uniform thread is then modelled as a de n-\nsity enhancement with a two-dimensional distribution of de n-\nsity,ρ(r,z) (see Fig. 1). The density distribution has two non-\nuniform layers, with length landlzin ther- andz- directions,\nrespectively. The first is introduced so as to study the reson ant\ndamping of oscillations. The second produces irrelevant ph ys-\nical results, but enables us to avoid contact discontinuiti es and\nprovidesuswithacontinuousbackgrounddensity.Surfacep lots\nofthe densitydistributionfordi fferentthreadmodelsare shown\nin Figure 2, where we have made use of the symmetry of the\nsystem in the r- andz- directions and only plot their positive\nvalues. The dense part of the tube with prominence condition s,\ni.e., the thread, has a density ρfand occupies only part of the\nlarger magnetic flux tube. It extends over a length Lthreadin the\nz- direction.The rest of theinternalpart ofthe tube,with le ngth\nL−Lthreadinthelongitudinaldirection,isfilledwithplasmawith\na densityρev, with the subscript “ev” indicating the evacuatedI.Arregui et al.:Magnetohydrodynamic kink waves intwo-di mensional non-uniform prominence threads 3\nFig.1.Schematic representation of the model configuration adopte d in this work. The two-dimensional cylindrically symmetri c\nline-tied structure consist of a dense part, the thread, wit h lengthLthreadand densityρf, and two evacuated parts, with density ρev.\nBothregionsareseparatedbyanon-uniformlayerofwidth lzinthelongitudinaldirection.Transversenon-uniformity isconsidered\nonalayerofwidth l. Thestructureissurroundedbyplasmawithcoronalpropert iesanddensityρc.\nFig.2.Density distribution in the ( r,z)-plane in the domain r∈[0,rmax],z∈[0,L/2] for prominence thread models with non-\nuniform radial and longitudinal structuring. The threads a re defined by a cool and dense part with density, ρfand length Lthread\nand an evacuated part of the tube with density ρevand length L−Lthread. These two regions connect non-uniformlyto the coronal\nsurrounding medium with density ρc. Depending on the value of ρev, three different situations are possible: (a) ρev=ρc, (b)\nρc≤ρev≤ρf, and(c)ρev≤ρc.\npart of the tube. All lengths are normalised by taking a=1. In\ncontrast to Soleret al. (2010), the value of ρevcan be different\nfromthecoronaldensity, ρc, andmayhavevalueslowerthan ρc\nup to a valueequal to the filament density, ρf, case in which the\nfulltubeis occupiedwith densecoolplasmaandwe recoverth e\none-dimensional case. Both regions along the axis of the tub e\nare connected by means of a non-uniform transitional layer o f\nlengthlzto produce a smooth longitudinal profile. The density\nvariationat thislayer,with Lthread−lz/2≤z≤Lthread+lz/2,can\nbeexpressedas\nρz(z)=ρf\n2/bracketleftBigg/parenleftBigg\n1+ρev\nρf/parenrightBigg\n−/parenleftBigg\n1−ρev\nρf/parenrightBigg\nsinπ(z−Lthread)\nlz/bracketrightBigg\n, (1)\nforr≤a−l/2. As for the radial direction, the internal filament\nplasma, with density ρf, is connected to the external coronal\nplasma,with density ρc, bymeansofa non-uniformtransitional\nlayerofthickness l,definedintheinterval[ a−l/2,a+l/2],that\ncan vary in between l/a=0 (homogeneous tube) and l/a=2\n(fully non-uniform tube). In contrast to the one-dimension al\nmodel used by Arreguiet al. (2008) the dense plasma of the\nthread does not occupy the full length of the tube, hence theradial variation of the plasma density for z≤Lthread−lz/2 and\na−l/2≤r≤a+l/2isgivenby\nρr(r)=ρf\n2/bracketleftBigg/parenleftBigg\n1+ρc\nρf/parenrightBigg\n−/parenleftBigg\n1−ρc\nρf/parenrightBigg\nsinπ(r−a)\nl/bracketrightBigg\n. (2)\nRadial non-uniformity is not restricted to the dense part, a s in\nSoleret al. (2010), but can be present also in the evacuatedp art\nofthetube.Asaconsequence,thereisanoverlapregionofra dial\nand longitudinal non-uniform layers, for a−l/2≤r≤a+l/2\nandLthread−lz/2≤z≤Lthread+lz/2,wherethe densityis given\nby\nρrz(r,z)=ρz(z)\n2/bracketleftBigg/parenleftBigg\n1+ρc\nρz(z)/parenrightBigg\n−/parenleftBigg\n1−ρc\nρz(z)/parenrightBigg\nsinπ(r−a)\nl/bracketrightBigg\n.(3)\nIntheevacuatedpartofthetube,for z≥Lthread+lz/2anda−l/2≤\nr≤a+l/2,theradialdensityprofileisgivenby\nρr(r)=ρev\n2/bracketleftBigg/parenleftBigg\n1+ρc\nρev/parenrightBigg\n−/parenleftBigg\n1−ρc\nρev/parenrightBigg\nsinπ(r−a)\nl/bracketrightBigg\n. (4)\nNote that since the case ρev≤ρcis not excluded, the slope of\ntheradialdensityprofileintheevacuatedpartofthetubeca nbe4 I.Arregui et al.:Magnetohydrodynamic kink waves intwo-d imensional non-uniform prominence threads\npositive (ifρev<ρev), negative (ifρev>ρev) or zero (ifρev=\nρev). Finally, for r≥a+l/2 we reach the coronal medium and\nρ(r,z)=ρc.Themodeladoptedinthispaperremovesunrealistic\ndiscontinuitiesinthedensitydistribution,consideredi nprevious\nworks,andallowsthetheoreticalmodellingofalargenumbe rof\nthreadsbysimplyconsideringdi fferentgeometricalandphysical\nparameter values for, e.g., the three di fferent situations outlined\ninFigure2.\nThis paper is concernedwith standing kink waves in promi-\nnence threads. To study small amplitude thread oscillation s, we\nconsider the linear resistive MHD wave equations for pertur ba-\ntionsoftheform f(r,z)∼exp(i(ωt+mϕ))withconstantresistiv-\nity,η. Heremis the azimuthal wave-numberand ω=ωR+iωI,\nthe complexoscillatory frequency.For resonantlydampeds olu-\ntions, the real part of the frequency gives the period of the o s-\ncillation, P=2π/ωR, while the imaginary part is related to the\ndamping time,τd=1/ωI. Magnetic diffusion is only included\nhere to avoid the singularity of the MHD equations at the reso -\nnantposition,butdi ffusionhasnoeffectontheresonantdamping\ntime-scales. Oscillations are then governedby the followi ng set\nof partial differential equations for the two components of the\nvelocityperturbation, vrandvϕ,andthethreecomponentsofthe\nperturbedmagneticfield, br,bϕ,andbz,\niωvr=B\nµρ/parenleftBigg∂br\n∂z−∂bz\n∂r/parenrightBigg\n, (5)\niωvϕ=−B\nµρ/parenleftBiggim\nrbz−∂bϕ\n∂z/parenrightBigg\n, (6)\niωbr=B∂vr\n∂z+η/bracketleftBigg∂2br\n∂r2−m2\nr2br+∂2br\n∂z2+1\nr∂br\n∂r\n−2im\nr2bϕ−br\nr2/bracketrightBigg\n, (7)\niωbϕ=B∂vϕ\n∂z+η/bracketleftBigg∂2bϕ\n∂r2−m2\nr2bϕ+∂2bϕ\n∂z2+1\nr∂bϕ\n∂r\n+2im\nr2br−bϕ\nr2/bracketrightBigg\n, (8)\niωbz=−B/parenleftBigg∂vr\n∂r+vr\nr+im\nrvϕ/parenrightBigg\n+η/bracketleftBigg∂2bz\n∂r2−m2\nr2bz\n+∂2bz\n∂z2+1\nr∂bz\n∂r/bracketrightBigg\n. (9)\nEquations (5)–(9), together with the appropriate boundary con-\nditions, define an eigenvalue problem for resonantly damped\nmodes.Astheplasma- β=0,theslowmodeisabsentandthereare\nnomotionsparalleltotheequilibriummagneticfield, vz=0.We\nfurther concentrate on perturbations with m=1, which repre-\nsent kink waves that producethe transverse displacementof the\naxis of the tube. The MHD kink wave represents a wave mode\nwith mixed fast and Alfv´ en character, its Alfv´ enic nature being\ndominant in and around the resonant position (Goossenset al .,\n2009). The problem can be further simplified by making use of\nthedivergence-freeconditionfortheperturbedmagneticfi eld\n1\nr∂(rbr)\n∂r+1\nr∂bϕ\n∂ϕ+∂bz\n∂z=0, (10)\nwhichreducesthesystemofequationstobesolvedtofour,up on\nexpressing bϕintermsof brandbz.Solutionstotheseequations\nare obtained by performing a normal mode analysis. Becauseof the complexityof the problem when a two-dimensional den-\nsityρ(r,z) is considered, numerical solutions to the frequency\nand spatial structure of eigenfunctions in the ( r,z)- plane are\ncomputed using PDE2D (Sewell, 2005), a general-purposepar -\ntial differential equation solver. The code uses finite elements\nand allows the use of non-uniformly distributed grids, whic h\nare needed to properly resolve the large gradients that aris e in\nthevicinityofresonantpositions.Di fferentgridresolutionshave\nbeen tested so as to assure the proper computation of the res-\nonant eigenfunctions. Magnetic dissipation has no influenc e on\nthe dampingof kinkmodesdue toresonantabsorption,a condi -\ntion that has to be checked in all numerical computations, bu t\nallows us to properly compute the spatial distribution of pe r-\nturbations in the resonance. We have made use of the symme-\ntry of the system and solutions are computed in the domain\nr∈[0,rmax],z∈[0,L/2].Non-uniformgridsareusedinbothdi-\nrections,toproperlyresolvetheregionswith r∈[a−l/2,a+l/2]\nandz∈[Lthread/2−lz/2,Lthread/2+lz/2]. The first is located so\nas to include the non-uniformtransitional layer in the radi al di-\nrection,while thesecondembracesthenon-uniformtransit ional\nlayer along the tube. As for the boundaryconditions, we equa te\nthem to the spatial distribution of perturbations for the fu nda-\nmental kink mode. The two perturbed velocity components, vr\nandvϕand the compressive component of the perturbed mag-\nnetic field, bzhave vanishing longitudinal derivatives at z=0,\nwhich correspondsto the apex of the flux tube, while they van-\nish atz=L/2, because of the line-tying boundary condition at\nthe photosphere. In the radial direction, they also have van ish-\ning radial derivative at the axis of the tube, r=0, while we\nimpose the vanishing of the perturbed velocity components f ar\nawayfromthe tubein theradialdirection,hence( vr,vϕ)→0as\nr→∞, a condition that is accomplished by setting the pertur-\nbations equal to zero at r=rmax, wherermax, the upper limit of\nthe domain in the radial direction, has to be chosen to be su ffi-\ncientlyfartoproperlycomputethedrop-o ffrateofperturbations\nin the radial direction. In all our computations,we have con sid-\neredrmax=20a.\n3. Analysisandresults\nArreguietal. (2008), in their analysis of the damping of kin k\noscillations in one-dimensional thread models, showed tha t the\nparameters that determine the temporal attenuation of osci lla-\ntions are the density contrast, ρf/ρcand the width of the non-\nuniform transitional layer, l/a. The damping ratio is rather de-\npendent on the first parameter for low values of the density\ncontrast, but stops being dependent in the high contrast rat io\nregime, typical of prominenceplasmas. The strongest influe nce\ncomes from the width of the transitional layer, with the damp -\ning time rapidly decreasing for increasing values of l/a. Before\nwe deal with the additional parameters introduced by the lon -\ngitudinal density structuring in two-dimensionalthread m odels,\nsolutions to Equations (5)–(9) have been first obtained by us -\ningatwo-dimensional(2D)densitydistributionwith Lthread=L.\nThe purpose of these numerical experiments has been to check\nthe correctbehaviourof the code by reproducingthe results ob-\ntained by Arreguiet al. (2008). In addition, we have also con -\nsidered the magnetic Reynolds number, Rm=vAfa/η, which\nshould not affect the computed damping times, in the limit of\nlarge Reynoldsnumbers. Figure 3 displays the obtained resu lts.\nThedampingtimeofresonantlydampedkinkwavesisindepen-\ndentofthemagneticReynoldsnumber,aslongasthisquantit yis\nlargeenoughforresonanceabsorptiontobetheoperatingda mp-\ningmechanism.Thisregime(seetheplateauregions)isobta inedI.Arregui et al.:Magnetohydrodynamic kink waves intwo-di mensional non-uniform prominence threads 5\nFig.3.Dampingtime,inunitsoftheinternalfilamentAlfv´ encross ingtime,τAf=a/vAf,asfunctionofthethreerelevantparameters\nfor kink oscillations in one-dimensionalthread models: (a ) the magnetic Reynolds number, Rm=vAfa/η, (b) the density contrast,\nρf/ρc, and (c) the transverse inhomogeneity length-scale, l/a. Solid lines indicate the 1D solution, while symbols repres ent the\nnumerical2D solutionsobtainedwith Lthread=L. In (a) and (c) the density contrast is ρf/ρc=200.In (b), l/a=0.2.In (b) and(c)\nRm=106.Allcomputationshavebeenperformedinatwo-dimensional gridwith Nr=401andNz=51points,with250grid-points\nintheradialtransitionallayer.Lengthsarenormalisedto a=1 andL=50a.\nFig.4.(a)-(c): period, damping time, and damping ratio as a functi on of the length of the thread for thread models with l/a=0.2\nandfortwovaluesofthedensitycontrast.(d)-(f):thesame quantitiesforthreadmodelswith ρf/ρc=200andforthreevaluesofthe\ntransverseinhomogeneitylength-scale.Inall figures a=1,Rm=106,ρev=ρc, andlz=a.Timesareshowninunitsofthe internal\nfilament Alfv´ encrossingtime, τAf=a/vAf. Symbolscorrespondto fullynumerical2D computationswhi le thedifferentline styles\nrepresenttheapproximate2DsolutionsobtainedbySoleret al.(2010).Allcomputationshavebeenperformedinatwo-di mensional\ngridwith Nr=401andNz=51points,with250grid-pointsin theresonantlayer.Lengt hsarenormalisedto a=1andL=50a.\nfor different values of Rmwhen different transitional layers are\nconsidered. Figure 3a shows a perfect agreement between the\n1D resultsandthecurrentcomputationsusingthe 2D code.Th e\nperfect correspondence between 1D and 2D computations for\nthe dampingtime as a functionof the density contrast in the r a-\ndial directionis shownin Figure3b.Finally,the most impor tant\nparameter that determines the damping of transverse thread os-\ncillationsisthewidthofthenon-uniformtransitionallay erinthe\nradialdirection.Thedampingtimestronglydecreaseswhen this\nparameterisincreased,ascanbeseenin Figure3c,whichaga in\nshows a very good agreement between the values computed in\n2D and the previous 1D computations. These results were inagreementwithpreviousworksinthecontextofthedampingo f\ncoronal loop transverse oscillations (e.g. Goossenset al. , 1992;\nRuderman&Roberts, 2002; Goossensetal., 2002)\nOnce we are confident about the goodness of the code, we\nconsiderthenewingredientsintroducedbythetwo-dimensi onal\nnatureoftheprominencethreadmodelsconsideredinthiswo rk.\nThese new ingredients are the length of the thread, Lthread, i.e.,\nthe lengthof the part of the magneticflux tube filled with dens e\nabsorbing plasma, and the density in the evacuated part of th e\ntube,ρev. Thenon-uniformtransitionallayerin densitybetween\nbothregionsalongthetube, lz,hasanirrelevante ffectontheos-\ncillatory period for low harmonicsin the longitudinaldire ction,6 I.Arregui et al.:Magnetohydrodynamic kink waves intwo-d imensional non-uniform prominence threads\nas shown by D´ ıazet al. (2008). Our computations (not shown\nhere) confirmthe findingby D´ ıaz etal. (2008) on the negligib le\nimportanceofthisparameterfortheoscillatoryperiodand show\na similar irrelevance concerning the damping time by resona nt\nabsorption. For this reason, we have concentrated our analy sis\nontheremainingtwo parameters, Lthreadandρev.\n3.1. Periodsanddampingtimes\nWe first consider the influence of the length of the thread on\nthe periodand dampingof resonantlydampedtransverse thre ad\noscillations. An initial analysisonthis subject was prese ntedby\nSoleret al.(2010),whoconsideredthethintubeandthinbou nd-\nary approximations(TTTB) in a two-dimensional thread mode l\nwithtransverseinhomogeneityonlyinthedensepartofthet ube.\nOur analysis goes beyondthe TTTB approximationsby consid-\nering a fully inhomogeneoustwo-dimensional density distr ibu-\ntion and combining its influence with that of the density in th e\nevacuatedpartofthetube, ρev.\nWehavefirstanalysedthevariationofperiod,dampingtime,\nand damping ratio, τd/P, by settingρev=ρc, so that we mimic\nthecasestudiedbySoleret al.(2010)analytically.Westar twith\nafullyfilledtubeandgraduallydecreasethelengthoftheth read.\nThe obtained results, for two values of the density contrast be-\ntweenthefilamentandcoronalplasma,areshowninFigures4a -\nc.Theperiodisstronglydependentonthelengthofthethrea d.It\ndecreasesbyalmosta factoroftwowhengoingfrom Lthread=L\ntoLthread=0.1L. Figure 4a also shows that the oscillatory pe-\nriod is almost independentof the density contrast, once thi s pa-\nrameter is large enough. As for standing kink waves in one-\ndimensional thread models (Arreguiet al., 2008), the kink f re-\nquency is a weighted mean of the internal and external Alfv´ e n\nfrequencies.Regardless of the density contrast, the perio d is al-\nlowed to vary in a a narrow range determined by a factor that\ngoes from√\n2 to 1, when going from ρf/ρc=1 toρf/ρc→∞.\nFor typical density contrasts in prominence plasmas, the pe -\nriod can be considered independentof the density contrast. The\ndampingtimeproducedbyresonantabsorption(Fig.4b)also de-\ncreases remarkably when the length of the cool and dense part\nof the tube is decreased. The decrease is also around a factor\nof two in the considered range of values for Lthread. Soleret al.\n(2010) find that in the TTTBlimit the dependenceof the period\nandthedampingtimewiththelengthofthethreadisexactlyt he\nsame, hence any influence on the damping ratio, τd/P, is can-\ncelled out. Outside the TTTB approximations, we find that thi s\nis not the case (see Fig. 4c), although the damping ratio is al -\nmostindependentofthelengthofthreadandonlyforverysho rt\nthreadsaslight increaseinthedampingratioisfoundwhenf ur-\nther decreasing this parameter. In Figure 4 we overplot resu lts\nobtained by Soleretal. (2010) by solving their dispersion r ela-\ntion. We see that there is a very good agreement and the di ffer-\nences,that are duetothe simplifyingassumptionsofthe ana lyt-\nical treatment, are rather small. One can observe an anomalo us\nbehaviouronthedampingtimecomputedbySoleret al.(2010) ,\nforsmallvaluesof Lthread.Thisisduetothesimplifyingassump-\ntionsconsideredto obtain the semi-analyticsolution,tha t might\nnot be entirely valid outside the long-wavelengthlimit. Ov erall,\nourresults confirmthe validity of the analytical approxima tions\nobtained by Soler etal. (2010) concerning the influence of th e\nlength of the thread on periods and damping times. In physi-\ncal terms, the shortening of the length of the thread produce s\nshorter period oscillations, since the physical system is e quiv-\nalent to a fully filled tube, with the wavelength of oscillati onsreplaced by a shorter e ffective wavelength. A physical explana-\ntionofthedampingtimedependenceonthelengthofthethrea d\nisprovided,byusingenergyarguments,inSect. 3.3.\nSimilar conclusions can be extracted from the computa-\ntions we have performed by fixing the density contrast and for\nthree different values of the width of the inhomogeneouslayer.\nFigures 4d-f show the obtained results. They clearly show ho w\nstronglythe dampingtime and the dampingratio are influence d\nby the width of the transitional layer, also in 2D models, whi le\nthe period of the oscillations is almost una ffected by the value\nofl/a.In view ofthe results displayedin Figure 4, we conclude\nthatthelengthofthethreadisaveryimportantparameter.W hen\nallowing to vary from the limit of fully filled tube to 10% fille d\ntube, periodsand dampingtimes are decreasedas much as 50%\npercent. This is very relevant in connectionto prominences eis-\nmology.We mustnotethat,in principle,thelengthofthethr ead\ncanbeestimateddirectlyfromobservations.However,thel ength\nofthesupportingmagneticfluxtubeismuchmoredi fficulttoes-\ntimate,sinceits endpointsareusuallyunobservable.\nOur generaldensity modelenable us to analyse the e ffect of\nthe density in the evacuated part of the tube on the oscillato ry\nproperties, as well. The study of the influence of this parame ter\nwas not undertaken by Soleret al. (2010), and requires a full y\nnumerical approach. We allow for ρevto be different from the\ncoronaldensity,ρc. Whenchangingthe densityinthe evacuated\npart of the tube, in addition to the radial non-uniformlayer that\nconnectstheprominencematerialtothecoronaweareintrod uc-\ning an additional radial non-uniformlayer in between the ev ac-\nuated part of the tube and the corona.The densityprofile in th is\nlayer and its slope depend on the relative values of ρevandρc,\nas givenin Equation(4) andshown in Figs. 2b and c. Insteadof\nanalysingtheeffectofρevontheoscillatorypropertiesinasepa-\nratemanner,wehaveselectedthreerepresentativevaluesf orthe\nlengthofthethreadandhavecomputedperiods,dampingtime s,\nand dampingratios as a functionof the density in the evacuat ed\npart of the tube, measured in units of the coronal density. We\nhavesplitouranalysisintotwoparts.\nFirst, we consider that ρc≤ρev≤ρf, hence the parameter\nis allowed to vary in between the coronal and prominence den-\nsities. Fig.5displaystheobtainedresults.For ρev=200ρc=ρf,\nthe tube is fully filled with cool and dense plasma. Then, as we\ndecreaseρev, periods and damping times have a marked linear\ndecrease.When 40% of the tube is filled with cool plasma, they\ndecrease by about a 15%. When a 10% filled tube is consid-\nered a decrease of up to a 50% is obtained. In physical terms,\nthe period decrease when the density in the evacuated part of\nthe tube is gradually decreased can be understood if we think\nabout the different inertia of the system, with or without dense\nplasma at those locationsalongthe tube. For explainingthe dif-\nferent damping time-scales, energy argumentsthat combine the\nenergyof the mode and the energyflux into the resonance have\ntobeconsidered,seeSect.3.3.Thedecreaseinperiodandda mp-\ning time is very similar, but not exactly the same. For instan ce,\nFig.5cshowsthatthedampingratioisslightlydependenton the\ndensity in the evacuated part of the tube, for all the conside red\nvaluesintherange ρc≤ρev≤ρf.\nNext, we have considered the possibility of the density in\nthe evacuated part of the tube being lower than the coronal\ndensity. For instance, D´ ıazet al. (2002) considered a valu e of\nρev=0.6ρc. The computations shown in Fig. 5 are extended to\nlowervaluesofρev.Theobtainedresultsaredisplayedinthein-\nsetplotsofFig.5andshowthatthedensityintheevacuatedp art\nof the tube is irrelevant in relation to the period of the osci lla-I.Arregui et al.:Magnetohydrodynamic kink waves intwo-di mensional non-uniform prominence threads 7\nFig.5.Period,dampingtime,anddampingratioasafunctionofthed ensityintheevacuatedpartofthetubeforprominencethrea ds\nwithρf/ρc=200,a=1,Rm=106,lz=a, andl/a=0.2 and three values of the length of the thread. The main plots a re for\nρc≤ρev≤ρf, while the inset plots correspond to the range 0 .1ρc≤ρev≤ρc. Times are shown in units of the internal filament\nAlfv´ en crossing time, τAf=a/vAf. All computationshave been performed in a two-dimensional grid with Nr=401 andNz=51\npoints,with 250grid-pointsin theresonantlayer.Lengths arenormalisedto a=1andL=50a.\nFig.6.Longitudinalandradialdependenceoftheeigenfunctionst ransversevelocitycomponent, vr,azimuthalvelocitycomponent,\nvϕ, and compressive magnetic field component, bz, in prominence threads with ρf/ρc=200,l/a=0.2,a=1,ρev=ρc, and\nRm=106, for different values of the length of the thread. All computations ha ve been performed in a two-dimensional grid with\nNr=401andNz=51points,with250grid-pointsintheresonantlayer.\ntions and the dampingby resonant absorption in the consider ed\nrangeofvalueswith0 .1ρc≤ρev≤ρc.\nThese results show that the density in the evacuated part of\nthetubeis alsoa relevantparameterforprominencethreads eis-\nmology,because of its e ffect on periodsand dampingtimes and\nthe difficulty in being measured by direct means. When consid-\nered in combinationwith e ffects due to the length of the thread,\nout computations enable us to perform a more accurate promi-\nnenceseismology,applicabletoa largenumberofthreads.\n3.2. Spatialdistributionof eigenfunctions\nChanges in the longitudinal density structuring have a dire ct\nimpact on the oscillatory period, which is easy to understan d,\nbut also on the damping time by resonant absorption. Besidesobtaining the parametric behaviour of kink mode periods and\ndampingtimesasafunctionofthelongitudinaldensitystru ctur-\ning, we aim to explain the obtained results. As a first step, we\nhave analysed the spatial structure of eigenfunctions. The rel-\nevant perturbed quantities are the radial and azimuthal vel oc-\nity components, vrandvϕ, and the compressive component of\nthe perturbedmagneticfield, bz, directlyrelatedto the magnetic\npressureperturbation, PT=Bbz/µ.\nWe first considertheinfluenceof thelengthof the threadon\nthe profiles of the eigenfunctions in the radial and longitud inal\ndirections.Resultsaregivenintermsofthemodulusofthec om-\nplex eigenfunctions. Fig. 6 shows one-dimensional cuts alo ng\nthe longitudinal and radial directions of the eigenfunctio ns for\ndifferentvaluesofthelengthofthethread.Thelongitudinalpr o-\nfiles are shown at the axis ( r=0) forvr, and at the mean radius8 I.Arregui et al.:Magnetohydrodynamic kink waves intwo-d imensional non-uniform prominence threads\nofthetube( r=a)forvϕandbz. Inthelongitudinaldirectionall\nthreeeigenfunctionsdisplaya trigonometricdependencew ithz,\nwhenLthread=L, i.e., the case that mimics the one-dimensional\nthread.When the lengthof the thread is decreasedseveralin ter-\nesting effects occur. First, both perturbed velocity components\ndisplay a slightly improved confinement. The maximum values\nof the velocity perturbations still occur in the dense part o f the\ntube, but the drop-o ffrate changes, becomingalmost linear out-\nside the threadso that theysatisfy the boundaryconditiona t the\nfoot-point of the tube. When eigenfunctions are normalised to\nthevalueof|vr|attheapexofthetube,itisseenthatthedecrease\nof the lengthof the thread produceslargeramplitudesof the az-\nimuthalvelocitycomponent(seeFig.6b),relatedtotheAlf v´ enic\ncharacter of the mode. This amplitude almost doubles its val ue\nwhengoingfrom Lthread=LtoLthread=0.04L.Thez-component\nofthemagneticfieldperturbationgivesanindicationofthe com-\npressibility of the normal mode. Our results indicate that w hen\nthelengthofthethreadisdecreased,thelongitudinalprofi leofbz\n(Fig.6c)becomesstrictly confinedto thedensepart ofthe tu be,\nwhere it reachesits maximumvalue. As the thread gets shorte r,\nthe maximum value of bzincreases, hence the compressibility\nbecomes larger at the apex of the tube for shorter threads osc il-\nlations, though the kink mode remains being an almost incom-\npressible wave mode. Kink modes in fully filled magnetic flux\ntubes are almost incompressible. Longitudinal density str uctur-\ningbytheinclusionofadensecentralpartincreases(decre ases)\nkink mode compressibility in the dense (evacuated) parts of\nthe tube, in comparison to the one-dimensional flux tube kink\nmodes.\nA close look at the spatial structure of eigenfunctionsin th e\nradialdirection(Figs.6d-f)providesuswithaqualitativ eexpla-\nnation on why the change of the length of the thread a ffects the\ndamping times computed in the previous subsection (and also\nthosepresentedbySoleret al.2010).First,thechangeinth epe-\nriod produced by the change in the length of the thread a ffects\ntheresonantpositionintheradialdirectionandhencethed amp-\ning time. However, the slope of the density profile in the tran -\nsitional layer is very large, because of the high density con trast\nratiotypicaloffilamentthreadsandthise ffectisnotveryimpor-\ntant.Second,theeigenfunctionsintheresonantlayerarea ffected\nby the value of the length of the thread. Both e ffects are clearly\nseen in Figs. 6d-f, where the radial dependenceof vr,vϕ, andbz\nis plotted for different thread lengths. We first note that the per-\nturbed velocity components at the resonance have a larger am -\nplitude the shorter the length of the thread. We interpret th is as\nanindicationofanimprovede fficiencyoftheresonantdamping\nmechanism.Thisisparticularlyclearifwelookattheazimu thal\nvelocity profile in the resonant layer (Fig. 6e). Now, not onl y\nthe increase of the amplitude for shorter lengths is evident , but\nalso the transverselengthscale is seen to decreasewhen sho rter\nthreads are considered, hence thinner resonance widths are ob-\ntained. This detail of the resonance also shows a slight shif t to-\nwards the left hand side which is due to the change of the reso-\nnantpositionfordi fferentlengthsofthethreads(hencedi fferent\noscillatoryperiods).Althoughtheresonancesarenotexac tlylo-\ncated atrA=a, our numericalresultsindicate that this approxi-\nmation,usedbySoleret al.(2010),isfullyjustified.Final ly,the\nradialprofileofthecompressivecomponentofthemagneticfi eld\nperturbation (Fig. 6f) shows the increase in its amplitude a t the\nresonant position mentioned above, when considering short er\nthreads. This is anotherindication of the improvede fficiencyof\nresonant damping. Notice that although our study is limited to\nlinear MHD waves, in a realistic situation perturbations in side\ntheresonantlayerbecomenonlinear,asshowninstudiesbye .g.,Terradaset al. (2008a); Clack et al. (2009); Ballai & Ruderm an\n(2011).\nWe have next examined the spatial distribution of the rel-\nevant perturbed quantities as a function of the density in th e\nevacuated part of the tube. Fig. 7 shows the obtained profiles\nin the longitudinal and radial directions for a fixed value fo r\nthe length of the thread. As before, the longitudinal profile s are\nshown at the axis ( r=0) forvr, and at the mean radius of the\ntube (r=a) forvϕandbz. Our results indicate that the den-\nsity in the evacuated part of the tube has also a direct impact\nontheradialandlongitudinalprofilesofeigenfunctions.F orthe\nlongitudinal profiles, as we decrease the value of ρevfromρf,\nwe have a slightly improvedconfinementof the velocitypertu r-\nbations and a strict confinement of the longitudinal compone nt\noftheperturbedmagneticfieldto thedensepartofthe tube(s ee\nFigs.7a-c).Theamplitudeof bzattheapexofthetubeincreases,\nwhile in the evacuated part of the tube, and for a fixed length\nof the thread, compressibility decreases as ρevis decreased (see\nFig. 7c). Taking a look at the eigenfunctions in the radial di -\nrection (Figs. 7d-f), the amplitude of both perturbed veloc ity\ncomponentsincreases when the density in the evacuatedpart of\nthe tube is decreased. The increase in the resonance peak and\nthe shorteningof the transverse spatial scale are mainly ev ident\nfortheazimuthalvelocitycomponent,whichgivesitsreson antly\ndampedandAlfv´ eniccharactertothemode.Wefindanincreas e\nin the compressibility of the normalmode (Fig. 7f) in the den se\nprominenceplasmaregionasthedensityintheevacuatedpar tof\nthetubeisdecreased.\nOverall, the decrease of the density in the evacuated part\nof the tube, starting from a fully filled tube, produces simil ar\nqualitative effects on the radial and longitudinal profiles for the\neigenfunctions as the ones produced by the shortening of the\nlengthof thethread.In the parameterrangestudiedin this w ork\nthoseeffectsarequantitativelymoreimportantinthecaseofthe\nchanges of the length of the thread. The properties of the spa -\ntial structure of eigenfunctionsgive us a qualitative expl anation\nonwhychangesinthelongitudinaldensitystructuringinpr omi-\nnence threads have a significant e ffect on the damping time of\nkinkoscillationsin two-dimensionalthreadmodels.\n3.3. Energyanalysis\nOur analysis of the spatial structure of eigenfunctionsind icates\nthat the decrease of both the length of the thread and the den-\nsity in the evacuatedpart of the tube producea strengthenin gof\ntheresonanceabsorptionprocessthatbecomesapparentthr ough\nthe appearance of more pronounced resonant profiles and thin -\nner resonance widths in the perturbed velocity components a t\nthe resonance. This would explain the marked decrease of the\ndampingtimesfoundin Section3.1whenvaryingthese twopa-\nrameters.Thisresult showsthat resonantdampingmaystron gly\ndepend on the details of the longitudinal density structuri ng in\nrathergeneraltwo-dimensionaldensitymodels,suchasthe ones\nusedheretomodelprominencethreads.\nIn order to obtain a quantitativeexplanationan energyanal -\nysis is carried out. Our analysis involves the energy of the k ink\nmode and the energyflux into the resonance. The energyof the\nmode can directly be computed from the numerical eigenfunc-\ntionsas\nE=1\n2/parenleftBig\nρv2+b2/parenrightBig\n. (11)\nThis expression has to be integrated over the entire ( r,z)-\nplane except for the resonance layer, easily recognisable f romI.Arregui et al.:Magnetohydrodynamic kink waves intwo-di mensional non-uniform prominence threads 9\nFig.7.Longitudinalandradialdependenceoftheeigenfunctionst ransversevelocitycomponent, vr,azimuthalvelocitycomponent,\nvϕ, and compressive magnetic field component, bzin prominence threads with ρf/ρc=200,l/a=0.2,a=1,Lthread=0.2L,\nandRm=106for different values of the density in the evacuated part of the tube. All computations have been performed in a\ntwo-dimensionalgridwith Nr=401andNz=51points,with250grid-pointsin theresonantlayer.\nthe eigenfunctions displayed in Figs. 6 and 7, as this region\nwould include a contribution from the Alfv´ en waves. In one-\ndimensional equilibrium models, the energy flux into the res -\nonance is proportional to the magnetic pressure perturbati on\nsquared(Andrieset al., 2000; Andries&Goossens, 2001), a r e-\nsult that was used by Arreguiet al. (2007b) to analyse the infl u-\nenceof the internalstructuringof coronalloopson the damp ing\nby comparing the e fficiency of the process at internal and ex-\nternal layers. In two-dimensional equilibrium states, the energy\nfluxabsorbedataparticularfieldlineisproportionaltothe over-\nlap integral between PT(rA,z), the profile of PTalong the tube\nat the resonantposition,andthe resonantAlfv´ eneigenfun ctions\n(Thompson&Wright, 1993; Tirry&Goossens, 1995). This is\nwhythelongitudinalprofilesof bzandvϕ,andtheirmodification\nduetochangesinthelongitudinaldensitydistributionare sorel-\nevantindeterminingkinkmodedampingtimes.Ageneralmath -\nematicalexpression,validinthethree-dimensionalcase, isgiven\nby Wright& Thompson (1994). When adapted to our cylindri-\ncal equilibrium,this expression gives the energy flux at the res-\nonanceperunitϕintheform\nF=rA/integraldisplay/bracketleftbigSr(r−\nA)−Sr(r+\nA)/bracketrightbigdz (12)\n=πm2\n4µ2\n0×B2\nrA(/integraltext\nφbzdz)2\n/integraltext\nφρφdz×/parenleftBiggdωA\ndr/parenrightBigg−1\nrA.\nIn this expression, Sr(r−\nA) andSr(r+\nA) represent the values of the\nradial component of the time-averaged Poynting vector to th e\nleft and to the right of the resonance position, that we appro x-\nimate by rA=a,φis the Alfv´ en eigenfunction along the tube\nat the resonant position, and ωA(r) the Alfv´ en continuum fre-\nquency.In order to evaluate the integrals as well as the slop e oftheAlfv´ encontinuumattheresonantpositioninexpressio n(12),\nwe have first solved the following equation for the Alfv´ en co n-\ntinuummodes\nd2φ(rA,z)\ndz2+ω2\nA(rA)\nv2\nA(rA,z)φ(rA,z)=0, (13)\nwithboundaryconditions\ndφ\ndz(z=0)=0,φ(z=L/2)=0.\nFor each value of rA, this equation is solved for di fferent val-\nuesfortherelevantparameters Lthreadandρev,thatinturndefine\ndifferent profiles for v2\nA(rA,z). The Alfv´ en continua arise from\nrepeating the procedure for di fferent values of rAin the range\n[a−l/2,a+l/2].Oncethisisdone,theslopeoftheAlfv´ encon-\ntinuaattheresonantpositionandtheAlfv´ eneigenfunctio nsthat\ncorrespondto eachthreadmodel can be evaluated.By using th e\nnumerically computedprofiles for bz(rA,z), in order to evaluate\ntheoverlapintegralinEquation(12),andmultiplyingby2 π,the\nrequired total energy flux is obtained. The ratio of the energ y\nof the kink mode to the time-averaged energy flux into the res-\nonance gives the required damping time, for every considere d\nvalueofLthreadandρev.\nFigure 8 shows the obtained results. They have been nor-\nmalised to the value of the damping time for the fully filled\ntube.Weseeanexcellentagreementbetweenthedampingtime s\ncomputedthroughthe energyanalysisexplainedin detail ab ove\nand the numerically computed ones for both cases in which we\nchange the length of the thread and the density in the evacu-\nated part of the tube. The results obtained in Section 3.1 can\ntherefore be explained in terms of the energetics of the mode s\nand the resonant energy transfer. These results also show th e10 I.Arregui et al.:Magnetohydrodynamic kink waves intwo- dimensional non-uniform prominence threads\nFig.8.(a) Damping time, normalised to the full tube damping\ntime, as a function of the length of the thread, for ρev=ρc.\n(b) Dampingtime, normalised to the full tube dampingtime, a s\na function of the density in the evacuated part of the tube, fo r\nLthread=0.2L. In both figures lines correspond to the numeri-\ncally computed damping times and the symbols are the values\nobtainedthroughtheenergyanalysisdescribedin Section3 .3.\naccuracy and utility of the analytical expression derived b y\nWright&Thompson (1994) for the time-averaged energy flux\nin terms of eigenfunctions. Furthermore, we have just demon -\nstrated that damping time estimates can be obtained, using e n-\nergy arguments, without the need to solve the full non-unifo rm\nproblem, but instead by solving the much more simpler piece-\nwise uniformproblemin orderto obtain the real part of the fr e-\nquency and the longitudinal profiles for the eigenfunctions for\nkink modes in combinationwith the non-uniformcomputation s\nfortheAlfv´ encontinuummodes,usingthisinformationinc om-\nbination with Equations (11) and (12). This is possible beca use\ntherealpartofthefrequencyandthelongitudinalprofileso fthe\neigenfunctions are only slightly a ffected by resonant coupling,\nandbecausetheenergyinEquation(11)hastobeintegratede x-\ncludingtheresonantlayer.Ofcourse,thefullnumericalso lution\nprovides us with more accurate results, but as Fig. 8 illustr ates\ntheagreementbetweenbothmethodsisexcellent.\nAconvenientwaytofurtherunderstandtheenergyfluxatthe\nresonanceandthedynamicsofresonantlydampedkinkmodesi s\ntocomputethespatialdistributionofthetime-averagedPo ynting\nflux in our two-dimensional domain by making use of the gen-\neralexpression\n=1\n2Re(E×b∗), (14)Fig.9.(a) Spatial distribution in the ( r,z)-plane of the time-\naveraged Poynting flux given by Equations (15) and (16). (b)\nOhmicheatingdistribution,in arbitraryunits. Theseresu ltscor-\nrespondtothetransverseoscillationofaprominencethrea dwith\nρf/ρc=200,ρev=ρc,l/a=0.2,a=1, andLthread=0.2L. The\nmagneticReynoldsnumberis Rm=106.\nwithE=−(v×B)+ηjtheperturbedelectricfield, b∗=(b∗\nr,b∗\nϕ,b∗\nz)\nthe complex conjugate of the perturbed magnetic field, and j=\n(∇×b)/µthe current. In contrast to the energy analysis above,\nwe now make use of the full numerical computations. In terms\nof the wave fields analysed in Section 3.2, the two components\nof the time-averaged Poynting vector in the ( r,z)- plane can be\ncast as\n=1\n2B\nµRe(vrb∗\nz)+η\n2µ2Re/bracketleftBigg\nb∗\nz/parenleftBigg∂br\n∂z−∂bz\n∂r/parenrightBigg\n−b∗\nϕ/parenleftBigg∂bϕ\n∂r+bϕ\nr−im\nrbr/parenrightBigg/bracketrightBigg\n, (15)\n=−1\n2B\nµRe(vrb∗\nr+vϕb∗\nϕ)+η\n2µ2Re/bracketleftBigg\nb∗\nϕ/parenleftBiggim\nrbz−∂bϕ\n∂z/parenrightBigg\n−b∗\nr/parenleftBigg∂br\n∂z−∂bz\n∂r/parenrightBigg/bracketrightBigg\n. (16)\nBy making use of the divergence-free condition given by\nEquation (10) to compute bϕ, we have produced an example of\nthe two-dimensional distribution of the time-averagedPoy nting\nfluxaroundtheresonantlayer,fora partiallyfilled thread.\nThe arrow plot in Fig. 9a shows that energy is fed into the\nresonant layer by concentrating it over the dense thread sec -\ntion. The amount of energy flux in the radial direction is de-\ntermined by the jump in (see Stenuitet al., 1999, for aI.Arregui et al.:Magnetohydrodynamic kink waves intwo-di mensional non-uniform prominence threads 11\none-dimensional example in ideal MHD). The resistive layer is\non a scale where both the ideal and resistive terms of the per-\nturbed electric field can be of similar magnitude, but each of\nthem could be dominant over di fferent regions of the domain.\nFor instance, the jump in is determined by the ideal\nterm in Equation (15) which is dominant in the thread region\nand zero in the evacuatedpart of the tube. Hence the energyin -\nflow towards the resonance in the radial direction comes from\nthe interior of the tube and is determined by the spatial profi les\nof the transverse velocitycomponentand the compressivema g-\nneticfieldperturbation.Notethattheamplitudeofthekink mode\nis smaller outside the tube compared to inside and also that t he\nevacuatedpartofthetubeismuchlessdensethanthe thread.\nOnce in the layer,the energyflow is divertedalong the field\nlinesinamannerdeterminedby ,withasmallradialcon-\ntributionthatisduetotheresistivetermsinEquation(15) .Such\nas correspondsto the Alfv´ enic character of the mode inside the\nresonant layer, the dominant term in Equation (16) involves the\nazimuthal velocity and magnetic field perturbations, vϕb∗\nϕ. This\nquantity decreases linearly along the field lines, producin g the\nlessening in the parallel Poynting flux towards the foot-poi nt.\nIn our example, the radial non-uniformlayer is restricted t o the\ndensepartofthefluxtube,since ρev=ρc.However,energycon-\ncentratedin the resonantlayer ofthe threadbecauseof reso nant\nwavedampingcanflowalongthefieldlinesand,eventually,su p-\nplyheatingintheevacuatedregion,wherefieldalignedcurr ents\nare dominant. Although the energy inflow into the resonance,\ngiven by, and its subsequent divergence along the field\nlines, given by, are mostly determined by ideal terms\nin Equations(15) and (16), this does not mean that resistivi ty is\nnot important. For instance, the amount of heating, in the fo rm\nof Ohmic dissipation, will be determined by those currents, re-\nsistivity, and their spatial distribution. For this partic ular case,\nheatingis distributedin a constantmannerin the evacuated part\nof the tube, even if there is no resonant layer in that region ( see\nFig.9b).\n4. Summaryand Conclusions\nQuiescent filament fine structures are only partially filled w ith\ncold and dense absorbing material. The length of the threads\ncan in principle be measured in events showing transverse os -\ncillations, provided the lifetime of threads is su fficiently large\ncomparedtotheoscillatoryperiod.Thelengthofthesuppor ting\nmagneticfluxtubesare howevermuchlarger,andcannotbe ob-\nserved.Densitymeasurements,bothinthethreadasintheev ac-\nuated part of the supportingmagnetictube, are also challen ging\nfrom the observational point of view. It is therefore import ant\nto quantify the variations on wave properties due to changes in\nthese equilibrium parameters if we aim to perform an accurat e\nprominence seismology. It is essential to have computation s of\nperiods and damping times for a wide range of thread models\nto include regimes in which the applicability of simple anal yti-\ncal models could be of limited extent. For this reason we have\ncomputedtheoscillatorypropertiesofresonantlydampedt rans-\nverse kink oscillations in rather general two-dimensional fully\nnon-uniformprominencethreadmodels.Thisallowsforabro ad\nrange of prominence threads with very di fferent physical con-\nditions to be modelled and their oscillatory properties cha rac-\nterised.\nThelengthofthethreadandthedensityintheevacuatedpart\nof the tube definetheir longitudinaldensity structuring.W e find\nthat the length of the thread strongly influences the period a nd\ndampingtimeoftransversekinkoscillations,whilethedam pingratioisratherinsensitivetothisparameter.Theseresult sconfirm\nthevalidityoftheanalyticalapproximationsmadebySoler et al.\n(2010). In addition, our modelling has allowed us to identif y\na new physical parameter with seismological implications, the\ndensityintheevacuatedpartofthethread.Thisquantityal soin-\nfluencesperiodsanddampingtimes,andtoalesserextentdam p-\ning ratios, and must be taken into account in the inversion of\nphysicalparametersinthe contextofprominenceseismolog y.\nCurrently available inversion schemes for one-dimensiona l\ncoronal loops and prominence threads (Arreguiet al., 2007a ;\nGoossenset al., 2008; Arregui& Ballester, 2010; Soleret al .,\n2010) make use of observed periods and damping ratios. The\nfirst, influence the inferred values for the Alfv´ en speed, wh ile\nthe second determine the transverse density structuring. B ased\non our results, we can conclude that ignorance on the length o f\nthe thread, the length of the supporting magnetic flux tube, a nd\nthe density in the evacuated part of the tube will have a signi f-\nicant impact on the inferred values for the Alfv´ en speed (he nce\nmagneticfield strength)in thethread,dependingonwhether we\nusethoseone-dimensionalinversionschemesortheresults from\ntwo-dimensionalmodelshereobtained.Onthecontrary,bec ause\nof the smaller sensitivity of the damping ratio to changes in the\nlongitudinal density structuring, seismological estimat es of the\ntransverse density structuring will be less a ffected by our igno-\nrance about the longitudinal density structuring of promin ence\nthreads.\nOur study provides additional insight to the physics of res-\nonantly damped kink modes in two-dimensional equilibrium\nstates, by extending previous applications (e.g, Andriese t al.,\n2005; Arreguiet al., 2005) to morecomplexnon-separablede n-\nsity distributions. It also provides an example of the metho ds\nandusesofcombiningtheinformationfromthespatialdistr ibu-\ntionofeigenfunctionswiththatobtainedfromenergyargum ents.\nIn particular, our energy analysis has allowed us to explain the\ndecrease in damping times for shorter thread lengths found b y\nSoleret al. (2010). The length of the thread influences the en -\nergyof the kink mode,andhence its oscillatory period,but a lso\naffects the damping by resonant absorption, through the energy\nflux into the resonance. In an analogous way, the value of the\ndensityintheevacuatedpartofthetubealsodeterminesper iods\nand dampingtimes, since both the energyof the kink mode and\ntheenergyfluxintotheresonancevary.Thismeansthatchang es\nintheequilibriumconfigurationinanon-resonantdirectio npro-\nducevariationsinthedampingpropertiesofkinkmodes,are sult\nthatwasqualitativelyexplainedbyadetailedexamination ofthe\nradial and longitudinal profiles of the eigenfunctions. Bot h the\nshortening of the length of the thread and the decrease of the\ndensity in the evacuated part of the tube produce more marked\nresonances, with the amplitude of the velocity perturbatio ns at\nthe resonance and the compressibility of the mode in the thre ad\nbeing larger. Inside the resonant layers shorter transvers e spa-\ntial scales for the Alfv´ enic velocity component are obtain ed. In\ncombination with the analysis of the energy of the kink modes\nand the energy flux into the resonances a quantitative explan a-\ntion was obtained for both the damping properties obtained i n\nourstudyandthoseinSoleret al.(2010).\nThe damping of kink oscillations in two-dimensional fully\nnon-uniformequilibriumconfigurationscanbecomputedbyu s-\ningenergyargumentstogetherwiththesolutionofsimplerp rob-\nlems for kink mode and Alfv´ en continuum modes. This aspect\nis worth to be considered in future studies of resonant absor p-\ntion in 2D/3D models of solar atmospheric magnetic structures\ninvolvingchangesofequilibriumparametersthata ffecttheden-\nsity structuring in a non-resonant direction. The use of ene rgy12 I.Arregui et al.:Magnetohydrodynamic kink waves intwo- dimensional non-uniform prominence threads\nargumentswould allow to have a first indication about how im-\nportantagivenparameterthatmodifiestheequilibriuminan on-\nresonant direction is, while avoiding to solve the full prob lem\nuntilwe areinterestedin thedetails.\nThe two-dimensional distribution of the time-averaged\nPoynting flux shows how energy is fed into the resonance and\nsubsequently flows along the field lines by properties of wave\nfieldsassociatedtoidealprocesses.Thisisthe reasonwhyr eso-\nnantdampingisamechanismforwaveenergytransferinwhich\ntime-scales are independent of resistivity, in the limit of high\nmagnetic Reynolds numbers. Magnetic di ffusion plays its role\nonce energy is concentrated at small spatial scales, by prov id-\ning heating at locations that, as in our example for a partial ly\nfilled thread, are distributed in regions where no resonant l ayer\nis present. This result o ffers additional insights to the dynamics\nofresonantlydampedkinkmodesandtheheatingofprominenc e\nplasmas by wave transformation processes. It must be consid -\nered in detail and extended to density models relevant to oth er\nsolaratmosphericstructures.\nAcknowledgements. IA, RS, and JLB acknowledge the funding provided under\ntheprojectAYA2006-07637bySpanishMICINNandFEDERFunds andthedis-\ncussionswithintheISSITeamonSolarProminenceFormation andEquilibrium:\nNew Data, New Models. RS acknowledges support from a Marie Cu rie Intra-\nEuropeanFellowship withintheEuropeanCommission7thFra mework Program\n(PIEF-GA-2010-274716). 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Lacoste, 185– +\nWiehr, E., Balthasar, H., & Stellmacher, G. 1989, Hvar Obser vatory Bulletin,\n13, 131\nWright, A. N.&Thompson,M. J.1994, Physics of Plasmas, 1,69 1\nYi, Z.&Engvold, O.1991, Sol. Phys.,134, 275\nYi, Z.,Engvold, O.,&Keil, S.L.1991, Sol. Phys.,132, 63" }, { "title": "1009.1700v1.A_number_conserving_linear_response_study_of_low_velocity_ion_stopping_in_a_collisional_magnetized_classical_plasma.pdf", "content": "arXiv:1009.1700v1 [physics.plasm-ph] 9 Sep 2010A number-conserving linearresponsestudy oflow-velocity ionstopping ina collisionalmagnetized\nclassicalplasma\nHrachya B. Nersisyan,1,2,∗Claude Deutsch,3and Amal K. Das4\n1Institute of Radiophysics and Electronics, 0203 Ashtarak, Armenia\n2Centre of Strong Fields Physics, Yerevan State University, Alex Manoogian str. 1, 0025 Yerevan, Armenia\n3LPGP (UMR-CNRS 8578), Universit´ e Paris XI, 91405 Orsay, Fr ance\n4Department of Physics, Dalhousie University, Halifax, Nov a Scotia B3H 3J5, Canada\n(Dated: May 6, 2019)\nThe results of a theoretical investigation on the low-veloc ity stopping power of the ions moving in a magne-\ntized collisional plasma are presented. The stopping power for an ion is calculated employing linear response\ntheoryusing the dielectricfunction approach. Thecollisi ons, whichleads toa damping of the excitations inthe\nplasma,istakenintoaccount throughanumber-conservingr elaxationtimeapproximationinthelinearresponse\nfunction. Inordertohighlighttheeffectsofcollisionsan dmagneticfieldwepresentacomparisonofouranalyt-\nicalandnumericalresultsobtainedforanonzerodamping or magneticfieldwiththose foravanishingdamping\nor magnetic field. It is shown that the collisions remove the a nomalous frictionobtained previously [Nersisyan\net al., Phys. Rev. E 61, 7022 (2000)] for the collisionless magnetized plasmas at l ow ion velocities. One of\nmajor objectives of this study is to compare and contrast our theoretical results with those obtained through a\nnoveldiffusionformulationbasedonDufty-Berkovskyrela tionevaluatedinmagnetizedone-component plasma\nmodels framedon targetions andelectrons.\nPACS numbers: 52.40.Mj, 52.25.Xz, 52.25.Fi\nI. INTRODUCTION\nThe energy loss of ion beams and the related processes in magn etized plasmas are important in many areas of physics such\nas transport, heating and magnetic confinement of thermonuc lear plasmas. The range of the related topics includes ultra cold\nplasmas [1], cold electron setups used for ion beam cooling [ 2–4], as well as manyvery dense systems involvedin magnetiz ed\ntarget fusions [5], or inertial confinement fusion. This lat ter thermonuclear scheme presently advocates a highly rega rded fast\nignitionscenario[6],basedonfemtolaserproducedproton orheavierionbeamsimpinginga precompressedcapsulecont aining\nathermonuclearfuel[7]init. Then,themagneticfield Bvaluesupto 1010Gmaybereachedinthelaboratory[8]. Suchatopic\nis also of intense astrophysical concern [9]. These interac tion geometries highlight low ion velocity slowing down (LI VSD)\nas playing a fundamental role in asserting the confining capa bilities and thermonuclear burn efficiency in dense and stro ngly\nmagnetizedmedia.\nFor a theoreticaldescriptionof the energyloss of ionsin a p lasma, there exist two standardapproaches. The dielectric linear\nresponse (LR) treatment considers the ion as a perturbation of the target plasma and the stopping is caused by the polariz ation\nof the surrounding medium [10–15]. It is generally valid if t he ion couples weakly to the target. Alternatively, the stop ping\nis calculated as the result of the energy transfers in succes sive binary collisions (BCs) between the ion and the electro ns [16–\n19]. Here it is essential to consider appropriate approxima tions for the screening of the Coulomb potential by the plasm a [3].\nHowever,significant gapsbetween these approachesinvolve the crucial ion stoppingalong magneticfield Band perpendicular\nto it. In particular, at high Bvalues, the BC predicts a vanishingly parallel energy loss, which remains at variance with the\nnonzero LR one. Also challenging BC-LR discrepancies persi st in the transverse direction, especially for vanishingly small\nion projectile velocity vwhen the friction coefficient contains an anomalous term div erging logarithmically at v→0[11,\n12]. In general when vis smaller than target electron thermal velocity ve, the ratioS(v)/v, whereS(v)≡ −dE/dxis the\nstopping power (SP), usually monitors a linear stopping pro file for highly ionized plasma with B= 0[20] orB/negationslash= 0[3]. An\nalternativeapproach,particularlyin the absenceof anyre levantexperimentaldata, isto test varioustheoreticalme thodsagainst\ncomprehensivenumericalsimulations[3, 16–18]. Thelatte rexhibithighalevelofnumericalnoiseat largemagneticfie lds,and\ninthev→0limit,whilekeepinga plasmacouplingbelowunity,whichis preciselythedomainofmanyimportantapplications\nofcurrentinterest.\nWith this backgroundwe reporton a theoretical study of ener gyloss of a slow-velocityion in a magnetizedclassical plas ma\nthrough a linear response approach which is constructed suc h that it conserves particle number. Broadly speaking, ther e are\n∗Electronic address: hrachya@irphe.am2\ntwo objectivesof this paper. The first objectiveis to use the Bhatnagar-Krook-Gross(BGK) approachbasedon the Boltzma nn-\nPoissonequationsforacollisionalandmagnetizedclassic alplasmawhichistreatedasaone-componentplasma(OCP).W euse\nthisapproachtoderivethedielectricfunctioninanumber- conservingmannerandusethisdielectricfunctiontocalcu latevarious\naspects of energyloss of an ion movingin the plasma. This is d one in Sec. II. We would like to see how number-conservation\nand damping (due to collisions) affect the stopping power of an ion in a low velocity range, i.e., for a slow ion. We should\nmention that for a colliisonal quantum plasma, e.g., a degen erateelectron gas (DEG) without magnetic field, Mermin [21] and\nDas [22] consideredthe equationof motionfor a single-part icledensitymatrix and derivedthe dielectric functionin a number-\nconservingapproachandinrandomphaseapproximation(RPA ). Themainpartofourfirst objectiveistosee towhatextentt he\nBGK approach can address the SP of a slow ion in a magnetized on e-componentclassical plasma. We will show that number\nconservation and collisions in such OCP have interesting an d experimentally observable effects on low-velocity SP. No w, one\nmay expect various collision mechanisms in a magnetized pla sma. Because of their importance we have separately dealt wi th\nthem in Sec. III. In Sec. IV we present detailed calculations of low-velocity SP in the BGK approach. Although the LR is\nnormallyusedto calculatetheSP ofa fast ion,we showthat fo rtheBGK modelonecanobtainusefulandinsightfulresultsf or\nlow-velocitywithintheLRtheory.\nA second objective of our paper is to compare and contrast our theoretical results with those obtained through a differen t\nmethod. The latter is specifically aimed at a low-velocity SP which is expressed in terms of velocity-velocity correlati on and\nhence to a diffusion coefficient. We refer to Dufty and Berkov sky (DB) [23] for an exposition of this method. The later part\nof Sec. IV contains a brief account of the DB relation. Marche ttiet al.[24] and Cohen and Suttorp [25] have calculated the\nrelevantdiffusioncoefficientsinamagnetizedplasmathro ughahydrodynamicalandkineticapproaches,respectively . Recently\nDeutschet al.in Ref. [26] have suggested an alternative approach for a cal culation of low-velocity SP via the DB relation and\nbyemployingthediffusioncoefficientsfora magnetizedOCP [24, 25],featuringeitherthetargetelectronsortargetio ns. Since\nourmaintheoreticalresultsareobtainedinakineticequat ionapproachwehavedecidedtodevoteSec.Vtoanappraisalo fthese\ntwo approaches. In that section we discuss results for an unm agnetized and magnetized plasma in the two approaches. This\nservestohighlightthemeritsofthe kineticapproachversu sthoseina hydrodynamicalapproach.\nSectionVIcontainssomediscussionandoutlook. AppendixA canbeconsultedforanintegralrepresentationofthediele ctric\nfunction. TheCoulomblogarithmwherethedynamicalpolari zationeffectsareneglectedisbrieflydiscussedin Appendi xB.\nII. LINEARRESPONSEFORMULATION\nIn this section we considerthe main aspectsof the linear res ponse(LR) theoryforthe ion-plasmainteractionin the pres ence\nof an external magnetic field. Within the LR, the electron pla sma is described as a continuous, polarizable medium, which is\nrepresented by the distribution function of the electrons f(r,v,t). The evolution of f(r,v,t)is determined by the kinetic and\nPoisson equations. Usually only a mean-field interactionbe tween the electronsis consideredand hard collisions are ne glected.\nThis is valid for weakly coupled plasmas where the number of e lectrons in the Debye sphere ND= 4πn0eλ3\ne= 1/ǫ≫1is\nverylarge. Here ǫistheplasmaparameter, n0eandλe= (kBTe/4πn0ee2)1/2aretheequilibriumdensityandtheDebyelength\nofelectrons,respectively.\nWe consider a nonrelativistic projectile ion with charge Zeand with a velocity v, which moves in a magnetized collisional\nand classical plasma at an angle ϑwith respect to the constant magnetic field B. We ignore any role of the electron spin or\nmagneticmomentduetothenonrelativisticmotionoftheion andtheplasmaelectrons. Weshallconsiderherethelimitof heavy\nions and neglect recoil effects. The strength of the couplin g between the moving ion and the electron plasma is given by th e\ncouplingparameter\nZ=Z/ND\n(1+v2/v2e)3/2. (1)\nHereveis the thermal velocity of the electrons. The derivation of E q. (1) is discussed in detail in Ref. [27]. The parameter Z\ncharacterizestheion-targetcoupling,where Z ≪1correspondstoweak,almostlinearcouplingand Z/greaterorsimilar1tostrong,nonlinear\ncoupling.\nLet us now specify the kinetic equation for the collision-in clusive classical magnetized plasma. The effect of collisi ons on\nthe dielectric propertiesof the plasma is included, in a num ber-conservingapproximation,through a relaxation time τ= 1/γ,\nwhereγis the collision frequency[28]. For τ→ ∞the collision-inclusivekinetic equationreducesto the co llisionless Vlasov\nequation. Thusweconsiderthe kineticequationofthecolli sionalplasmawithinrelaxation-timeapproximation(RTA) inwhich\nthecollisiontermisoftheBhatnagar-Gross-Krook(BGK)-t ype[28],\n∂f\n∂t+v·∂f\n∂r−e\nme/bracketleftbigg\nE+1\nc[v×B]/bracketrightbigg\n·∂f\n∂v=−γ/bracketleftbigg\nf−ne\nn0ef0(v)/bracketrightbigg\n, (2)3\nwherethecollisionfrequency γisa measureofdampingofexcitationsintheplasma,and\nne(r,t) =/integraldisplay\nf(r,v,t)dv, n0e=/integraldisplay\nf0(v)dv. (3)\nHerene(r,t)isthedensityoftheelectrons, f0(v)istheequilibriumdistributionfunctionoftheelectronsi nanunperturbedstate.\nFor instance, the distribution function f0(v)of the plasma electrons may be given by the Maxwell distribut ion. The right hand\nside of Eq. (2) is the collision term in a relaxation-timeapp roximationwhich was introducedby BGK in a number-conservi ng\nscheme. It is easy to see that this form of collision term cons erves the total number of particles. Eis a self-consistent electric\nfield(see below)and Bistreatedasanexternalmagneticfield. τ= 1/γistherelaxationtime.\nForasufficientlysmallperturbations(LRtreatment)weass umef=f0+f1,ne=n0e+n1e, with\nn1e(r,t) =/integraldisplay\nf1(r,v,t)dv, (4)\nandthelinearizedkineticequationbecomes\n∂f1\n∂t+v·∂f1\n∂r−Ωe[v×b]·∂f1\n∂v=e\nmeE·∂f0\n∂v−γ/parenleftbigg\nf1−n1e\nn0ef0/parenrightbigg\n. (5)\nHereb=ez=B/Bis the unit vector along the magnetic field, Ωe=eB/mecis the cyclotron frequency of the electrons,\nE=−∇ϕ,ϕis theself-consistentelectrostaticpotentialwhichisde terminedbythePoisson equation\n∇2ϕ=−4πρ0(r,t)+4πe/integraldisplay\nf1(r,v,t)dv, (6)\nwhereρ0isthedensityofthe externalcharge.\nWe solve the system of equations (5) and (6) by space-time Fou rier transforms. Because of the cylindrical symmetry of\nthe problem around the magnetic field direction b, we introduce cylindrical coordinates for the velocity v=exv⊥cosθ+\neyv⊥sinθ+ezv/bardblandthewavevector k=exk⊥cosψ+eyk⊥sinψ+ezk/bardbl,wherethesymbols /ba∇dbland⊥denotethecomponents\nof the vectorsparallelor perpendicularto the externalmag neticfield, respectively. We next introducethe Fouriertra nsformsof\nf1(r,v,t),n1e(r,t)andϕ(r,t)with respect to variables randt,f1k,ω(v),n1e(k,ω)andϕ(k,ω). Then the linearized kinetic\nequation(5)forthedistributionfunctionbecomes\n∂f1kω(v)\n∂θ+i\nΩe(k·v−ω−iγ)f1kω(v) (7)\n=−ie\nmeΩeϕ(k,ω)/parenleftbigg\nk·∂f0\n∂v/parenrightbigg\n+γ\nn0en1e(k,ω)f0(v).\nAssumingaxiallysymmetricunperturbeddistributionfunc tion,f0(v) =f0(|v/bardbl|,v⊥),Eq.(7)canbeformallyintegratedandthe\nsolutionis\nf1kω(v) =γ\nΩen0ef0n1e(k,ω)/integraldisplayθ\n−∞exp/bracketleftbiggi\nΩeU(θ′)/bracketrightbigg\ndθ′(8)\n−ie\nmeΩeϕ(k,ω)/integraldisplayθ\n−∞exp/bracketleftbiggi\nΩeU(θ′)/bracketrightbigg/bracketleftbigg\nk/bardbl∂f0\n∂v/bardbl+k⊥cos(θ′−ψ)∂f0\n∂v⊥/bracketrightbigg\ndθ′,\nwherethelowerlimitof θ′-integrationischosenso asto taketheintegrandvanish. He re\nU(θ′) =/parenleftbig\nk/bardblv/bardbl−ω−iγ/parenrightbig\n(θ′−θ)+k⊥v⊥[sin(θ′−ψ)−sin(θ−ψ)]. (9)\nTheθ′-integration in Eq. (8) can be performed using the Fourier se ries representation of the exponential function [29]. Afte r\nstraightforwardintegrationwe obtain\nf1kω(v) =−e−izesin(θ−ψ)∞/summationdisplay\nn=−∞ein(θ−ψ)Jn(ze)\nk/bardblv/bardbl−ω−iγ+nΩe(10)\n×/bracketleftbiggiγ\nn0ef0n1e(k,ω)+e\nmeϕ(k,ω)/parenleftbigg\nk/bardbl∂f0\n∂v/bardbl+nΩe\nv⊥∂f0\n∂v⊥/parenrightbigg/bracketrightbigg\n,4\nwhereze=k⊥v⊥/Ωe,Jnis the Bessel functionof the nth order. It shouldbe emphasizedthat Eq. (10) is a formal sol ution of\nthelinearizedkineticequationbecausetheFouriertransf ormedelectronicdensity n1e(k,ω)remainsunknown. Wenowperform\nv-integration in Eq. (10) and solve the obtained algebraic eq uation with respect to the quantity n1e(k,ω). Substituting this\nquantityintoFouriertransformedPoissonequationfinally yields\nϕ(k,ω) =4πρ0(k,ω)\nk2εM(k,ω,γ), (11)\nwhereεM(k,ω,γ)isthecollision-inclusivelongitudinaldielectricfunct ionofthemagnetizedplasmawhichis givenby\nεM(k,ω,γ) = 1+(ω+iγ)[ε(k,ω,γ)−1]\nω+iγQ(k,ω,γ)(12)\nwithε(k,ω,γ) =εe(k,ω+iγ),Q(k,ω,γ) =Qe(k,ω+iγ)and\nεe(k,ω) = 1−ω2\ne\nk22π\nn0e/integraldisplay∞\n−∞dv/bardbl/integraldisplay∞\n0v⊥dv⊥ (13)\n×∞/summationdisplay\nn=−∞J2\nn(ze)\nk/bardblv/bardbl+nΩe−ω−i0/parenleftbigg\nk/bardbl∂f0\n∂v/bardbl+nΩe\nv⊥∂f0\n∂v⊥/parenrightbigg\n,\nQe(k,ω) =2π\nn0e/integraldisplay∞\n−∞dv/bardbl/integraldisplay∞\n0f0v⊥dv⊥∞/summationdisplay\nn=−∞/parenleftbig\nk/bardblv/bardbl+nΩe/parenrightbig\nJ2\nn(ze)\nk/bardblv/bardbl+nΩe−ω−i0. (14)\nHereεe(k,ω)is the usual longitudinaldielectric functionof the magnet izedcollisionless and purelyelectron plasma (see, e.g.,\nRef.[28])and ωe= (4πn0ee2/me)1/2istheplasmafrequencyoftheelectrons. Similarly, Qe(k,ω)referstotheelectronplasma.\nThedielectricfunction εM(k,ω,γ)givenbyEqs.(12)-(14)hasbeenobtainedintheBGKapproach whichisnumber-conserving.\nNote the exact relation Qe(k,0) = 1which holds independently of the initial distribution f0if the latter is normalized to the\nunperturbedelectronicdensity n0e, seethe secondrelationin Eq.(3).\nItiswellknown[28]thattheusualrelaxation-timeapproxi mationcanbeobtainedfromEq.(2)ifthecollisiontermiswr itten\nas−γ(f−f0)and is equivalent to replacing ωbyω+iγin the collisionless dielectric function εe(k,ω). This procedure is\ninadequatebecause it does not conserve the local particle n umberand doesnot lead to the Drude behaviorat long waveleng ths\n(k→0). This is remedied in the BGK approach. The k→0case of the number-conserving dielectric function is also o f\ninterest. Notingthat Qe(0,ω) = 0,fromEq.(12)we find\nεD(ω) =k2\n⊥\nk2ε⊥(ω)+k2\n/bardbl\nk2ε/bardbl(ω) (15)\nwith\nε⊥(ω) = 1+ω2\ne(ω+iγ)\nω[Ω2e−(ω+iγ)2], ε/bardbl(ω) = 1���ω2\ne\nω(ω+iγ). (16)\nThe above results are a generalization of the Drude dielectr ic function for magnetized plasmas. Equations (15) and (16) are\nknownalsoasa“cold”plasmaapproximation(see,e.g.,Ref. [28])andcanbealternativelyobtainedfromEqs.(12)-(14) assum-\ninginitialdistributionfunction f0(v) =n0eδ(v). Forasimplicityweshallcalltheexpressions(13)and(14) asaBessel-function\nrepresentationofthedielectricfunction. Formanypracti calapplications,however,itisimportanttorepresentthe dielectricfunc-\ntioninanalternativebutequivalentintegralform,see App endixA fordetails.\nLetusnowspecifytheinitialdistributionfunction f0oftheelectrons. WeconsidertheMaxwellisotropicdistrib utionfunction\nf0/parenleftbig\nv/bardbl,v⊥/parenrightbig\n=n0e\n(2π)3/2v3eexp/parenleftBigg\n−v2\n/bardbl+v2\n⊥\n2v2e/parenrightBigg\n, (17)\nwhereve= (kBTe/me)1/2is thethermalvelocityoftheelectrons. Thecollision-inc lusivedielectricfunctionthenreads\nε(k,ω,γ) = 1+1\nk2λ2e[F1(k,ω)+iF2(k,ω)]. (18)\nHere\nF1(k,ω) = 1+∞/summationdisplay\nn=−∞1\nω+nΩeΛn(βe)[ωG(xn,y)−γF(xn,y)], (19)\nF2(k,ω) =∞/summationdisplay\nn=−∞1\nω+nΩeΛn(βe)[ωF(xn,y)+γG(xn,y)], (20)5\n0 1 2 30.0 0.5 1.0 1.5 2.0 2.5 3.0 ELF \nw/we Drude \n Collisionless \n BGK k|| le = k⊥le = 0.1 \n0 1 2 30.0 0.5 1.0 1.5 2.0 2.5 3.0 ELF \nw/we Drude \n Collisionless \n BGK k|| le = k⊥le = 0.4 \nFIG. 1: (Color online) Generalized Drude (solid line), coll isionless (dashed line) and number-conserving BGK (dotted line) energy loss\nfunctions vs ω/ωeforΩe=ωe,γ= 0.1ωeandk/bardblλe=k⊥λe= 0.1(leftpanel), k/bardblλe=k⊥λe= 0.4(right panel).\nare the real and imaginary parts of the generalized dispersi on function of the collisional magnetized plasma, respecti vely, and\nxn= (ω+nΩe)/|k/bardbl|ve,y=γ/|k/bardbl|ve,βe=k2\n⊥a2\ne,ae=ve/Ωeis the cyclotron radius of the electrons, Λn(z) =e−zIn(z),\nIn(z)isthemodifiedBesselfunctionofthe nthorder. HerewehaveintroducedthegeneralizedFried-Con tedispersionfunctions\nforthecollisionalplasma\nG(x,y) =x√\n2π/integraldisplay∞\n−∞(t−x)e−t2/2dt\n(t−x)2+y2, (21)\nF(x,y) =xy√\n2π/integraldisplay∞\n−∞e−t2/2dt\n(t−x)2+y2. (22)\nAt vanishing γ(aty→0)thesefunctionsbecometheusualFried-Contedispersionf unctions[30]ofthecollisionlessplasma\nG(x,0) =x√\n2π/integraldisplay∞\n−∞e−t2/2dt\nt−x, (23)\nF(x,0) =/radicalbiggπ\n2xe−x2/2. (24)\nThe function Q(k,ω,γ)which determines the dielectric function (12) is evaluated by inserting Eq. (17) into Eq. (14). It is\neasyto seethat fortheMaxwellisotropicdistributionfunc tion(17)thequantity Q(k,ω,γ)isgivenby\nQ(k,ω,γ) =εe(k,ω+iγ)−1\nεe(k,0)−1=F1(k,ω)+iF2(k,ω), (25)\nwhere\nεe(k,0) = 1+1\nk2λ2e(26)\nisthestatic dielectricfunctionwhichisnotaffectedbyth eexternalmagneticfield.\nFor ion stopping considerations, it is worth defining the ene rgy loss function (ELF) Im[−1/ε(k,ω,γ)]. Figure 1 shows\nDrude, collisionless, and BGK energy loss functions vs scal ed frequency ω/ωewhenk/bardblλe=k⊥λe= 0.1(left panel) and\nk/bardblλe=k⊥λe= 0.4(right panel) for Ωe=ωe,γ= 0.1ωe. As has been mentioned above at small momentum k→0the\nBGK energyloss functionreproducesthe Drude energyloss fu nction. And this is seen on the left panel of Fig. 1. Also at lon g\nwavelengths(i.e.,atsmall k)theBGKenergylossfunctionisbroadenedduetothedamping comparedtotheELFwithvanishing\ndamping.\nThe collision-inclusivedielectric functionallows bothp hysical insight and usefulnumericalestimates of the influe nceof the\ncollisions on energy loss. In an unmagnetizedand degenerat e electron gas the predicted effect is a shorter lifetime and smaller\nmean free path of the plasmons resulting in considerable mod ifications of the ELF [31–35]. For the stopping of a single ion ,6\nthe broadening of the plasmon peak with increasing γshifts the threshold for the energy loss by plasmon excitati on towards\nlower projectile velocities. This increases the SP at low pr ojectile velocities, comparedto the collisionless result [31–34]. The\nsituation with a present case of a classical and magnetized p lasma including the collisions may be quite different altho ugh\ncollisionalbroadeningoftheELFoccursalsoin thiscase. T hissituationwill befurtherdiscussedinSec.IV.\nIII. COLLISIONFREQUENCYIN A MAGNETIZEDPLASMA\nIn Sec. II the effect of the collisions in a magnetized classi cal plasma has been introduced in the dielectric function th rough\na phenomenologicalbut number-conservingcollision termw ithin the LR theory. Themodelcollision frequency γin solidsand\nplasmas can be determinedexperimentallyor, alternativel ycan be calculated theoretically. For instance, in some inv estigations\nofionstoppinginsolidtargetsintheabsenceofamagneticfi eld,γwasdeterminedbyfitting −Im[ε−1(0,ω,γ)]toexperimental\noptical energy loss functions (see, e.g., Refs. [33, 34] and references therein). In addition the model relaxation time τ= 1/γ\ncan be estimated from the experimental data of the dc conduct ivity or the mobility in a plasma either with or without exter nal\nmagneticfield. It shouldbe emphasizedthat ingeneralthere area numberofphysicalmechanismswhichmaycontributeto t he\ndamping parameter γ. And contribution of each mechanism depends strongly on the specific plasma conditions. We have not\nattempted here to evaluate the damping parameter from first p rinciples in the most general case but regard it rather as a mo del\nparameter. Inprinciple γcanbecalculatedtovaryingdegreesofapproximationswhic hmayallowustoseehowtheSPdepends\nonthetargetpropertiesandthemagneticfieldthroughtheir influenceon γ.\nIn this section we briefly consider a fully ionized and a weakl y coupled plasma where the contributions of the Coulomb\ncollisions to the frequency γmay play a dominant role. This frequency, in our case, is dete rmined by electron-electron (e-e,\nγee) and electron-ion(e-i, γei) Coulombcollisions(if we do not considerimpurities). Thu s, in contrast to Sec. II, we deal with\na two-componentelectron-ionplasma (TCP) accounting for t he dynamics of plasma ions. The total effective frequency,i n the\nlimit of a weakly coupled plasma, can be approximated as a sum of e-e and e-i collisions, γ=γee+γei. In the absence of a\nmagneticfield the theoryofCoulombcollisionsin a plasmaha sbeenformulatedbySpitzer[36] (see also[37]). Inthelast four\ndecadesorso thetheoryhasbeenfurtherdevelopedandexten ded. Therecentbook[38]summarizestheresultsobtaineddu ring\nlastfourdecades. However,toourknowledge,therelaxatio nprocessesinamagnetizedplasmahavenotbeenstudiedinas much\ndetail as in an unmagnetized plasma, and only several theore tical attempts exist for this case [39–46] (see also the refe rences\ntherein). For a classical plasma more complete expressions for the collision frequencies valid at arbitrary (but non-q uantizing)\nmagnetic fields have been derived by Ichimaru et al.and Matsuda [41–44], and by Montgomery et al.[45] and by Silin [46]\nwithandwithoutallowingfordynamicalpolarizationeffec tsin plasma,respectively.\nIn Refs. [41, 42] only e-i relaxation is considered. The gene ralization to the e-e case is straightforward. The final resu lt is\nsummarizedbya formula\nγeα=8√\n2πq2\nαe4nαηeα\n3memαv3eαlnΛeα, (27)\nwhereα=e,iindicatesthe plasmaspecies, qe=−1,qi=Zi,Zieisthechargeofplasmaion, ηei= 1,ηee= 21/2,\nlnΛeα=1\n2(2π)3/2v3\neα/integraldisplay\ndk/integraldisplay∞\n−∞Ge(k,ω)Gα(k,ω)ω2dω\nk2\n/bardblk4|εei(k,ω)|2, (28)\nGα(k,ω) =∞/summationdisplay\nn=−∞Λn/parenleftbig\nk2\n⊥a2\nα/parenrightbig\nexp/bracketleftBigg\n−(ω−nΩα)2\n2k2\n/bardblv2α/bracketrightBigg\n, (29)\nv2\nα=kBTα/mα,v2\neα=v2\ne+v2\nα,v−2\neα=v−2\ne+v−2\nα,aα=vα/Ωα,Λn(z) =e−zIn(z). AlsoTi,mi,viandΩi=\nZieB/micarethetemperature,themass,thethermalvelocityandthec yclotronfrequencyoftheplasmaions( aiisthecyclotron\nradius),respectively. The quantity lnΛeαis the generalizedCoulomblogarithmfor a magnetizedplasm a. Hereεei(k,ω)is the\nlongitudinal dielectric function of a magnetized and colli sionless electron-ion TCP (see, e.g., Ref. [41]). The limit of the\nvanishing magnetic field in Eq. (29) is not trivial. An altern ative but equivalent integral form for the function (29) all owing\neasilythelimitofthefield-freecaseisderivedinAppendix A,seeEqs.(A14)and(A15). InEq.(28)thedynamicalpolariz ation\neffects are included in a dielectric function εei(k,ω). These effects guarantee the convergenceof the k-integration in Eq. (28)\nat large distances or at small k. But an upper cutoff kmax= 1/rmin(whererminis the effective minimum impact parameter)\nmust be introduced in Eq. (28) to avoid the logarithmic diver genceat large k. This divergencecorresponds to the incapability\nof the linearizedkineticequationto treat close encounter sbetweenthe plasma particlesproperly. Also it shouldbe em phasized\nthat for the e-i collisions there are two specific frequencie sγie=Ziγeiandγeiwhich describe the relaxation of ionic and7\n10 -1 10 010 110 210 310 40.4 0.6 0.8 1.0 1.2 1.4 1.6 ln Lee /ln L0,ee \nB/B s xe = 10 \n xe = 10 2\n xe = 10 3\n10 -1 10 010 110 210 310 410 510 60.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ln Lei /ln L0,ei \nB/B s x = 10 \n x = 10 2\n x = 10 3\nFIG.2: (Coloronline)TheCoulomblogarithms,Eqs.(B2)and (B6),normalizedtothefield-freevaluesfore-e(leftpanel )andelectron-proton\n(right panel) collisions vs scaled magnetic field B/Bsand for different values of the cutoff parameter. See the tex tfor explanations.\nelectronic temperatures to their equilibrium values, resp ectively [41, 42]. Thus the total e-i collision frequency is given by\n/tildewideγei=γei+γie= (Zi+1)γei.\nTo estimate the range of variation of the collision frequenc y with increasing magnetic field consider now some particula r\ncases. At vanishingmagnetic field from Eqs. (A14) and (A15) w e obtainGα(k,ω) = (|k/bardbl|/k)exp(−ω2/2k2v2\nα). In this limit\nwe denoteγeα=γ0,eαwithΛeα= Λ0,eαandfromEq.(28) wefind\nlnΛ0,eα=/radicalbigg\n2\nπ/integraldisplaykmax\n0dk\nk/integraldisplay∞\n0e−u2/2u2du\n|ε0,ei(k,kveαu)|2(30)\nthe Coulomb logarithm in the absence of a magnetic field. Here ε0,ei(k,ω)is the usual longitudinal dielectric function of the\nelectron-ion unmagnetized TCP without collisions. Equati on (30) has been derived by Ramazashvili et al. [37]. It involves\nthe dynamical polarization effects through the dielectric functionε0,ei(k,ω)and requires only an upper cutoff kmax= 1/rmin\nin a Fourier space. The Spitzer formula is recovered assumin gε0,ei(k,ω) = 1in Eq. (30) and introducing a lower cutoff\nkmin= 1/λD. Then performing u-integration in Eq. (30) we obtain the usual Coulomb logarit hm withΛ0,eα=λD/reα,min\ngeneralized for electron-ion plasmas (see, e.g., Ref. [35] ). Hereλ−2\nD=λ−2\ne+λ−2\niwhilereα,min= max[λLeα;λDB], where\nλα=vα/ωαandωα= (4πn0αq2\nαe2/mα)1/2are the Debye screening length and the plasma frequency for p lasma species α,\nrespectively. λLeαdenotesthe usual Landaulength λLee=λLei/|Zi|=e2/3kBTeandλDB=ℏ/2√mekBTe, the electron de\nBrogliewavelength,takingcareoftheintrinsicallyquant umbehaviorofthehigh-temperatureplasmaintheshortrang elimit.\nIn the opposite case of an infinitely strong magnetic field ( γeα=γ∞,eαwithΛeα= Λ∞,eα) from Eqs. (A14) and (A15)\none obtains Gα(k,ω) = exp( −ω2/2k2\n/bardblv2\nα). Inserting this formula into Eq. (28) it is straightforward to show that the collision\nfrequencyin astrongmagneticfieldisthe halfofthefrequen cyγ0,eα. Thus\nlnΛ∞,eα=1\n2lnΛ0,eα. (31)\nThecollisionfrequencieshavebeenalsoinvestigatedbyso meauthorsusingtheFokker-Planckkineticequationwithth eLan-\ndauintegralofthecollisions[45,46]. Asstated abovethis approachneglectsthedynamicalpolarizationeffects. InA ppendixB\nwe show briefly that starting with Eqs. (27) and (28) where εei(k,ω)is set= 1, one arrivesat the expressionsderivedby Silin\n[46]. To understand the importance of the dynamical polariz ation effects which are neglected in Eqs. (B2) and (B6) (and a lso\nin the formula of Spitzer) we note that the kinetic equation i n the form of Landau accounts for only close (almost) Coulomb\ncollisions,thuscompletelyneglectinglong-rangewave-p articleinteractions. Ithasbeenshownpreviouslythatint heabsenceof\nmagnetic field corrections ∆eαto the standard Coulomb logarithm ln(λD/reα,min)arising due to these interactions may be of\nthesameorderastheleadingterm[37]. However,thiseffect iscrucialonlyfore-iinteractionsandwave-particleinte ractionsare\nnot expected to make any essential change in the rate of e-e Co ulomb collisions. For instance, in an anisothermic electro n-ion\nplasma low-frequencyion-acoustic waves may provide an eff ective mechanism for electron-ion interactions which lead s to an\nenhancementofthestandardCoulomblogarithm.\nSimilar long-range and low-frequency collective effects a re responsible for a strong enhancement of the collision rat es in a\nmagnetized plasma. In this case it has been shown [41, 42] tha t e-i collision rate contains, in addition to the contributi on from8\nTABLEI: The Coulomb logarithm (32) for electron-proton pla sma normalized to lnΛ0,eifor some values of the scaled magnetic field B/Bs\nand fordifferent values of the cutoff parameter ξ=λD/rei,min= 10,102,103.\nB/Bs0.1 1 .0 10 .0 102103104105106\nξ= 10 0.05 0.53 3.11 6.79 10.55 14.31 18.07 21.82\nξ= 1020.03 0.27 1.55 3.40 5.28 7.16 9.03 10.91\nξ= 1030.02 0.18 1.04 2.27 3.52 4.77 6.02 7.27\ncloseCoulombcollisions(B6), a termwhichin ournotations isgivenby\nlnΛ∗\nei=1\n4ln/parenleftbiggmi\nme/parenrightbigg\nez[(1+z)K0(z)−zK1(z)], (32)\nwherez= 1/ζ2,ζ=λe/ae= Ωe/ωeis the scaled magnetic field, K0(z)andK1(z)are the modified Bessel functionsof the\nsecondkind. TheCoulomblogarithm(32)dependsessentiall yonthemagneticfield. AtlargeandsmallmagneticfieldsEq.( 32)\nbehaves as lnΛ∗\nei∼ln(Ωe/ωe)andlnΛ∗\nei∼Ωe/ωe, respectively, and vanishes at Ωe→0. Also we note the large factor\nln(mi/me)in Eq. (32) which diverges at mi→ ∞and appears due to strong electron-ion interactions via col lective plasma\nwaves and is typically /greaterorsimilar10. However,Eq. (B6) which accounts for only close Coulomb col lisions does not contain such term\nandremainsfinite at mi→ ∞.\nTheresultsofthenumericalevaluationofEqs.(B2)and(B6) areshowninFig.2. Thisfigureshowstheratio lnΛeα/lnΛ0,eα\nas a functionofthe scaled magneticfield B/Bsfor e-e(left panel)andelectron-proton(rightpanel)coll isionsandfordifferent\nvalues of the cutoff parameters ( ξe=λD/ree,min(e-e) andξ=λD/rei,min(e-i)). The quantity Bsintroduced above is\nBs=mcve/eλD. Also for an electron-protonplasma the normalizedCoulomb logarithm lnΛ∗\nei/lnΛ0,ei, Eq. (32), is given in\nTableIforsomevaluesofthescaledmagneticfield. Itisseen thatthemagneticfieldmayessentiallyincreasethecollisi onrates\nin plasma comparedto the field-free onesand this is more impo rtantfor electron-ioncollisions. As discussed above,Eqs . (B2)\nand(B6)andhencetheresultsshowninFig.2accountforonly unscreenedCoulombcollisionsneglectingdynamicpolariz ation\neffects. In a vanishingmagnetic field these effectsare impo rtantfor e-i collisions andthe situation with B/negationslash= 0requiresfurther\ninvestigations, in particular for e-e collisions. In the e- i case a major contribution is expected from low-frequency c ollective\nmodes given by Eq. (32). Table I shows that this contribution exceeds the Coulomb logarithm lnΛeiatB/Bs∼10,102,103\ndepending on the cutoff parameter. Also it should be emphasi zed that the validity of the regime (31) of a classically stro ng\nmagnetic field (the domains B/Bs>104andB/Bs>106in the left and right panels of Fig. 2, respectively) require s the\ncondition ℏΩe< kBTe(orB < Bc= (mc/eℏ)kBTe). Thus the results shown in Fig. 2 are valid up to Bc/Bs=kBTe/ℏωe.\nClearly the realization of the regime (31) requires high tem peratures and low densities and the enhancement of the colli sion\nrates atB∼1−102Bsmay not be accessible under certain conditions. However the recent analysis shows [39] (see also the\nreferencestherein) that in a quantizingmagnetic field with B > Bcthe field-dependenceof the collisional rates becomes even\nstronger and the enhancement of γeαshown in Fig. 2 may turn even more significant although the cla ssical expressions (27),\n(28), (B2) and(B6)areinvalidin thisregime.\nIV. LOW-VELOCITYSTOPPINGPOWER\nIn this section, with the collision-inclusive dielectric f unction derived in Sec. II we consider the stopping power (SP ) of a\nlow-velocity ion moving in a magnetized plasma for an arbitr ary angle with respect to the magnetic field. The regime of low\nvelocities is of particular importance for some physical si tuations, e.g., for electron cooling processes [3] and for m agnetized\ntarget plasma fusion researches [5]. Previously the SP in a m agnetized plasma at small ion velocity has been investigate d by\nemploying linear response (LR) theory [11] and Dufty-Berko vsky relation [26]. The latter approach (see below) reduces the\nproblem to a determination of the diffusion coefficient of th e magnetized plasma. In Ref. [11] it has been shown that in the\npresenceofa magneticfieldandintheabsenceofcollisions, thefrictioncoefficientcontainsananomaloustermwhichdi verges\natv→0likeln(ve/v)inadditiontotheusualconstantonewhilethehydrodynamic approachofRef.[26]doesnotcontainsuch\nterm. We shall commentonthisfeatureinthissection.\nThe stoppingpower Sof an ion with charge Zeandvelocity vis defined as the energyloss of the ion in a unit length due to\ninteractionwithaplasma. FromEq.(11)itisstraightforwa rdtocalculatetheelectricfield E=−∇ϕ(orE(k,ω) =−ikϕ(k,ω)\nintermsofFouriertransforms),andthestoppingforceacti ngontheion. Then,thestoppingpoweroftheprojectilepoin tlikeion9\nbecomes(see,e.g.,Refs. [10–12])\nS=Z2e2\n2π2v/integraldisplay\ndkk·v\nk2Im−1\nεM(k,k·v,γ). (33)\nFor the friction coefficient we have to consider S, given by Eq. (33) in a low-velocity limit, and thus the diele ctric function\n(12) with (18) and the functions F1(k,ω)andF2(k,ω)given by Eqs. (A10) and (A11), when ω=k·v. Now we have to\nwrite the Taylor expansion of Eq. (12) for small ω=k·v. Using expressions (12), (18), (25), (26), (A10), and (A11) for the\ncollision-inclusivedielectricfunctionat ω→0weobtain\nIm−1\nεM(k,ω,γ)≃kλ2\ne\n(k2λ2e+1)2ω\nve/integraldisplay∞\n0e−X(t)−ςtdt, (34)\nwhereς=γ/kve=ν/kλe,ν=γ/ωe. Thefunction X(t)isdeterminedbyEq.(A9). ItshouldbeemphasizedthatEq.(3 4)does\nnot contain any logarithmic singularity at vanishing k/bardbl→0as for the case of collisionless magnetized plasma, see Ref. [11].\nThis singularity which leads to an anomalousfriction in a ma gnetized plasma has been removed here due to the collisions a nd\nthe factore−ςtin Eq. (34) guarantees the convergenceof the tintegration at k/bardbl→0. Thus from Eqs. (33) and (34) we obtain\nusual(linearwith respectto v) frictionlaw\nS(ϑ)≃2Z2e2\n√\n2πλ2ev\nveR(ϑ), (35)\nwhereR(ϑ)isthe dimensionlessfrictioncoefficient,\nR(ϑ) =/integraldisplayκ\n0k3dk\n(k2+1)2/bracketleftbigg\nψ1(k)cos2ϑ+1\n2ψ2(k)sin2ϑ/bracketrightbigg\n. (36)\nHereκ=kmaxλeandϑis the angle between vandb. In Eq. (36) we have introduced a cutoff parameter kmax= 1/rmin\n(whererministheeffectiveminimumimpactparameter)inordertoavoid thelogarithmicdivergenceatlarge k. Thisdivergence\ncorrespondstotheincapabilityofthelinearizedkinetict heorytotreatcloseencountersbetweentheprojectileiona ndtheplasma\nelectronsproperly. For rminwe thususe the effectiveminimumimpactparameterof classi cal binaryCoulombcollisionswhich\nat low-velocitiesof the ion reads rmin=|Z|e2/mv2\ne. It is seen that the parameter κ= 4πn0eλ3\ne/|Z|= 1/Z ≫1, whereZis\ndeterminedbyEq.(1)at v≪ve. Alsothe otherquantitiesinEq.(36) are\nψn(k) =1\n2/integraldisplay∞\n0exp/bracketleftbigg\n−2k2\nζ2sin2(ζt)−2νt/bracketrightbigg\nΦn(kQ(t))dt\ntΥ(ζt)(37)\nwithn= 1,2,ζ=λe/ae= Ωe/ωe,Q(t) =√\n2tΥ(ζt),Υ2(t) = 1−(sint/t)2,Φ1(x) =x−2Φ(x),Φ2(x) = 2erf(x)−\nx−2Φ(x). Thefunction Φ(x)isdeterminedbyEq.(B5).\nIn many experimentalsituations, the ions move in a plasma wi th random orientationsof ϑwith respect to the magnetic field\ndirection b. The friction coefficient appropriateto this situation may be obtainedby carrying out a spherical average over ϑof\nR(ϑ)inEq.(36). We find\n/angb∇acketleftR(ϑ)/angb∇acket∇ight=1\n3/integraldisplayκ\n0k3dk\n(k2+1)2[ψ1(k)+ψ2(k)]. (38)\nLet us analyzethe generalexpression(36) forsome particul arcases. Forinstance, at vanishingmagneticfield ( ζ→0) using\ntherelationQ(t)≃/radicalbig\n2/3ζt2atζ→0,onefinds\nψ1(k) =1\n2ψ2(k) =1\n3A/parenleftbiggν√\n2k/parenrightbigg\n, (39)\nwhereA(z) =ez2erfc(z),erfc(z)is the complementary error function. In this case the fricti on coefficient is isotropic and\nbecomes\nR0(ϑ) =1\n3/integraldisplay∞\np0A(νk)dk\nk(2k2+1)2. (40)\nHere1/p0=√\n2κ. Inadditionat vanishingdamping,i.e. at ν→0,A(νk)→1andwerecovertheusuallow-velocitystopping\npowerinan unmagnetizedcollisionlessplasmawith africti oncoefficient(see,e.g.,[11,47])\nR0(ϑ) =1\n6U(κ)≡1\n6/bracketleftbigg\nln/parenleftbig\n1+κ2/parenrightbig\n−κ2\nκ2+1/bracketrightbigg\n. (41)10\nAt strongmagneticfield ( ζ→ ∞), the plasma becomeshighlyanisotropicandthe frictionco efficientdependsessentially on\nthe angleϑ. For an evaluation of the functions ψ1(k)andψ2(k)we note that Q(t)→√\n2tandΥ(ζt)→1asζ→ ∞. Then\nsubstitutingthese relationsintoEq.(37)andafterintegr ationbypartsoneobtains\nψ1(k) =1\n2A(a)+a2B(a)−a√π, (42)\nψ2(k) =/parenleftbig\n1−a2/parenrightbig\nB(a)−1\n2A(a)+a√π(43)\nwitha=ν/√\n2k, and\nB(z) =/integraldisplay∞\nzdt\ntA(t) =2z√π/integraldisplay∞\n0ln/parenleftBig\nt+/radicalbig\nt2+1/parenrightBig\ne−z2t2dt. (44)\nThenthe frictioncoefficientat infinitelystrongmagneticfi eldreads\nR∞(ϑ) =1\n2/integraldisplay∞\np0dk\nk(2k2+1)2/braceleftbigg\nsin2ϑ/bracketleftbigg/parenleftbig\n1−ν2k2/parenrightbig\nB(νk)−1\n2/parenleftbigg\nA(νk)−2νk√π/parenrightbigg/bracketrightbigg\n(45)\n+cos2ϑ/bracketleftbigg\nA(νk)−2√πνk+2ν2k2B(νk)/bracketrightbigg/bracerightbigg\n.\nSimilarlyforthe angularaveragedfrictioncoefficientwe o btain\n/angb∇acketleftR∞(ϑ)/angb∇acket∇ight=1\n3/integraldisplay∞\np0B(νk)dk\nk(2k2+1)2. (46)\nThe function B(z)involved in Eq. (45) at small zbehaves asB(z)≃ln(1/z)−C/2, whereC= 0.5772is the Euler’s\nconstant,anddivergeslogarithmicallywhen z→0. Usingasymptoticbehaviorofthisfunctionitisstraightf orwardtocalculate\nfromEq.(45)thefrictioncoefficientat vanishing γ. Inthislimit andintheleadingorderwe obtain\nR∞(ϑ) =1\n4/braceleftBigg\nsin2ϑ/bracketleftBigg/parenleftBigg\nln√\n2ωe\nγ−C+1\n2/parenrightBigg\nU(κ)+U1(κ)/bracketrightBigg\n+U(κ)cos2ϑ/bracerightBigg\n, (47)\nwhereU(κ)is givenbyEq.(41), and\nU1(κ) =U(κ)lnκ−1\n4/bracketleftbig\nln2/parenleftbig\nκ2+1/parenrightbig\n−2ln/parenleftbig\nκ2+1/parenrightbig/bracketrightbig\n−1\n2Li2/parenleftbiggκ2\nκ2+1/parenrightbigg\n. (48)\nHereLi2(z)is the dilogarithm function. Note that at large κ≫1, which is a requirement of a weak ion-plasma coupling, the\nfunctionsU1(κ)andU(κ)can be approximated by U1(κ)≃ln2κ−π2/12andU(κ)≃2lnκ−1, respectively. It is seen\nthat the first term in Eq. (47) diverges logarithmically at va nishingγ. It can be shown that the general expression (36) with\n(37) for the friction coefficient derived for arbitrary but fi nite magnetic field behaves similarly. This is a consequence due to\nthe magnetic field since the field-free result (40) remains fin ite asγ→0(see, e.g., Eq. (41)). The divergent term in Eq. (47)\nvanishes,however,whentheionmovesalongthemagneticfiel d(ϑ= 0). Thenthefrictioncoefficientissolelygivenbythelast\nterm ofEq. (47). In addition,the frictioncoefficientEq.(4 7) forstrongmagneticfieldsshowsan enhancementforionsmo ving\ntransverse(ϑ=π/2)tothe magneticfieldcomparedto thecase of thelongitudina lmotion(ϑ= 0). Thiseffectis inagreement\nwithparticle-in-cellsimulationresults[3].\nAsstatedinIntroductionweshallnowmakecontactwithadif ferentmethod. IthasbeenshownbyDuftyandBerkovsky[23]\nthatthelow-velocitySP ofanionina plasmais relatedto the diffusioncoefficient Dthrough\nS(v)\nv/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nv→0=kBTe\nD. (49)\nAsinRef.[26]weconsider Dtotheself-diffusioncoefficientinamagnetizedclassical one-componentplasma. FromEqs.(35)-\n(37) we can relate the friction coefficient R(ϑ)to the diffusion coefficient Dthrough Eq. (49). Cohen and Suttorp [25] have\ncalculated parallel (to the magnetic field) diffusion coeffi cientD/bardbl. These authors, like us but unlike Marchetti et al.[24], have\nused a kinetic equation method. At vanishing damping ( γ→0), it can be shown that D/bardblobtained from Eqs. (35)-(37) and\nforϑ= 0coincides with the result of Cohen and Suttorp [25]. In parti cular, atγ→0, it is found from Eqs. (41) and (47)\nthatR0(0)/R∞(0) =D∞,/bardbl/D0,/bardbl= 2/3in agreement with Ref. [25]. Here D∞,/bardblandD0,/bardblare the parallel self-diffusion11\n10 -1 10 010 110 20.5 1.0 1.5 2.0 2.5 3.0 R(J)\nWe/we J = 0 \n J = p/4 \n J = p/2 \n 〈R(J)〉\n0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.5 0.6 0.7 0.8 0.9 1.0 R(J)\ng/we J = 0 \n J = p/4 \n J = p/2 \n 〈R(J)〉\nFIG.3: (Coloronline)Thefrictioncoefficient R(ϑ)vsthescaledmagneticfield Ωe/ωe(leftpanel)anddampingparameter γ/ωe(rightpanel)\nforϑ= 0(solid line), ϑ=π/4(dashed line) and ϑ=π/2(dotted line), κ= 10,γ/ωe= 0.1(left panel), Ωe=ωe(right panel). The line\nwithsymbols corresponds to /angbracketleftR(ϑ)/angbracketright.\ncoefficients at infinite and vanishing magnetic fields, respe ctively. However at finite γand forϑ= 0, comparingEq. (40) with\n(45)we concludethatthe simplerelationcitedaboveisnoto beyedingeneral,duetodamping.\nAs an example, we show in Fig. 3 plots of the dimensionless fri ction coefficient R(ϑ)given by Eq. (36), as a function of\nthe scaled magnetic field Ωe/ωefor model parameter γ(γ/ωe= 0.1) (left panel). The right panel of Fig. 3 shows R(ϑ)as a\nfunctionofthe scaleddampingparameter γ/ωeforΩe=ωe(i.e. fora givenmagneticfield). It isseen that thelow-velo citySP\nincreases with an increase in the angle ϑand also with the magnetic field. In the latter case the SP asym ptotically tends to the\nvalue given by expression (45). In the opposite limit of a wea k magnetic field the friction coefficient tends to the value gi ven\nby Eq. (40) which is independent of the angle ϑ. Also Fig. 3 shows that the friction coefficient decreases wi th damping. It is,\ninterestingly,oppositeto the behaviorfoundforan unmagn etizedDEG, see. e.g., Refs. [31–34]. A decrease of R(ϑ)withγin\nthe present case of a classical plasma is not attributable to the applied magnetic field because the field-free friction co efficient\ngiven by (40) shows a similar behavior (not shown in Fig. 3). I n a degenerate plasma an enhancement of the low-velocity SP\nwithγisaquantumeffectwhichisabsentinourpresentstudy. Fora DEGthedomainofplasmonexcitationsisshiftedtowards\nsmaller ion velocities [31, 32]; this increases the SP in thi s velocity regime. But in the present case the domain of colle ctive\nexcitationsisshiftedtowardshighervelocities[47]andt hefrictioncoefficientdecreaseswith γ.\nThe resulting friction coefficient (47) may be compared with Eq. (44) of Ref. [11] where the friction coefficient in the col li-\nsionless plasma contains an anomalous term ln(ve/v)vanishing at ϑ= 0. The physical origin of such an anomalous friction\ncoefficient may be traced to the spiral motion of the electron s along the magnetic field lines. These electrons naturally t end to\ncouple strongly with long-wavelengthfluctuations(i.e., s mallk/bardbl) along the magnetic field. In addition, when such fluctuation s\nare characterized by slow variation in time (i.e., small ω=k·v), the contact time or the rate of energy exchangebetween the\nelectrons and the fluctuations will be further enhanced. In a plasma, such low-frequency fluctuations are provided by a sl ow\nprojectile ion. The above coupling can therefore be an effici ent mechanism of energy exchange between the electrons and t he\nprojectileion. At vanishingdampingandin thelimit of v→0, thefrequency ω=k·v→0tendstozeroas well. Thecontact\ntime∼ω−1thus becomesinfinite and the friction coefficient diverges. The collisions of the plasma particles play a stabilizing\nrole since the fluctuationsprovidedby the slow ion are dampe d. Thusat small velocities v→0the contacttime is finite and is\ndeterminedby ∼γ−1. Asa resultEq.(47) doesnotcontainatermlike ln(ve/v)butbehavesas ln(1/γ)at vanishingdamping.\nV. KINETICVERSUSHYDRODYNAMICAPPROACH\nUsing the theoretical results obtained in Sec. IV, we presen t here some comparative analysis, looking for some contacts\nbetween our linear-response(kinetic) formulationand the previoushydrodynamicmode-couplingtreatments based on t he self-\ndiffusion coefficients. As mentioned above such connection is established via Dufty-Berkovsky relation (49). The plas ma is\nmodeled as a collisional dielectric medium whose linear res ponse function, within RTA, is given by Eqs. (12), (18)-(20) with\nγas a model damping parameter. In order to document the LIVSD p hysics highlighted by the relation (49), we first briefly\npay attention to the unmagnetized B= 0limit. We consider it through the small ǫ= 1/(4πn0eλ3\ne)≪1plasma parameter\napproximationfor the self-diffusioncoefficient given by S j¨ ogrenet al. [48]. Employing Eqs. (35), (41) and (49) an inspection\nshowsthat atvanishingdamping( γ= 0)theself-diffusioncoefficientobtainedfromthese formul ascoincideswiththe resultof12\n0 20 40 60 80 100 -10.0 -9.9 -9.8 -9.7 -9.6 -9.5 -9.4 -9.3 ln[S(v)/v (eV s cm -2 )] \nWe/we J = 0 \n k BTe/D (0) \n|| \n0 20 40 60 80 100 -10 -8 -6 -4 -2 02468ln[S(v)/v (eV s cm -2 )] \nWe/we J = p/2 \n k BTe/D (0) \n⊥\n k BTe/D ⊥\n k BTe/D '\n⊥\nFIG. 4: (Color online) Proton LIVSD in a plasma with ne= 1.064×1016cm−3,Te= 1eV (ǫ= 0.02) in terms of Ωe/ωe. The lines with\nsymbols inleftand right panels represent parallel and tran sverse LIVSD,respectively. The solidlines were obtained f rom Eqs. (35)-(37)with\nϑ= 0(left)and ϑ=π/2(right).\nRef.[48]if theionicchargenumbersquare Z2inEq.(35)isreplacedbythequantity Z2→P(Z), where\nP(Z) =/parenleftbigg\nZ+1√\n2/parenrightbigg32Z2+75√\n2Z+50\n104Z2+111√\n2Z+59. (50)\nForaprotonprojectilewith Z= 1thisfactoris P(1) = 1.003andtheagreementbetweenbothapproachesisalmostperfect . The\nfactor(50)whichisnonlinearwithrespectto Zaccountsforthenonlinearcouplingbetweenanincomingion andthesurrounding\nplasma [48]. Howeverforhighlychargedionswith Z≫1thisfactorincreaseslinearlywith Z,P(Z) = (4/13)Z,while more\nrigoroustreatment shows that at strong ion-plasmacouplin g the energyloss of an ion scales with its chargeapproximate lylike\nZ1.5[27].\nConsidernextthecaseofamagnetizedandcollisionalplasm a. Forsimplicityweconsiderelectron-protonplasmaandap roton\nas a projectileparticle. Exploringfirst the moderatelymag netizeddomain, Ωe/greaterorequalslantωe, one can explicit the field-freeparallel and\nB-dependenttransversediffusions[24],\nD(0)\n/bardbl=3√πv2\np\nγc, D(0)\n⊥=r2\nLγc\n3√π, (51)\nwherev2\np=kBTe/mp,mpistheprotonmass,and γc=ωeǫln(1/ǫ)isthecollisionfrequencyintermsoftheplasmaparameter\nǫ,rL=vp/Ωeis the Larmor radius. Note that the collision frequency γcis related to the e-e collisional relaxation rate γeeas\nγee=/radicalbig\n2/9πγc(see Sec. III). The transverse diffusion coefficient given b y Eq. (51) correspondsto a classical region, where\nD⊥∼B−2,andisvalidfor γc<Ωe<0.4ωeY(ǫ)withY(ǫ) = [ǫ2ln(1/ǫ)]−1/2asexplainedinRef. [24].\nWithhighermagneticfieldvalues, Ωe/ωe>4Y(ǫ),onereachesthetransversehydro-Bohmregimewith D⊥∼B−1featuring\n[24]\nD⊥=D(0)\n⊥+v2\np\n2Ωeǫ2[ln(1/ǫ)]3/2. (52)\nTo the intermediate plateau regime with D⊥∼B0between transverse diffusion coefficients given by Eqs. (51 ) and (52),\ncorrespondsthediffusioncoefficient[24] validat 0.4Y(ǫ)<Ωe/ωe qisρe< ρp. Below we assume that ρpandρe\nare constant. The unit tangent vector to the tube axis is\nl= (x′(s),0,z′(s)), where the prime indicates the deriva-\ntive.Theprojectionofthegravityaccelerationonthetube\naxis isl·g=−gz′(s). The projection on the tube axis of\nthe gravityforceacting on the element ofthe dense promi-\nnence thread from stos+∆sis−gρpz′(s)∆s. Then the\ntotal projection of the gravity force acting on the thread\nis\nfg=−/integraldisplayq\npgρpz′(s)ds=gρp[z(p)−z(q)]. (2)\nWe assume that there is continuous plasma evapora-\ntion at the tube footpoints that creates the plasma flows\nat the two sides of the thread. The flow speed is v= const\nand the flowsaredirected towardthe thread at both sides.\nTheplasmafluxatbothsidesisthesameandequalto ρev.\nThe thread velocity is u, and we assume that the plasma\nflow speed is larger than the thread speed, v>|u|.\nOur main assumption is that the accreting material is\ninstantaneously absorbed by the thread, and its density,\ntemperature and velocity become the same as those of the\nthread material. Due to accretion the velocity of the left\nend of the thread is smaller than u, while the velocity of\nthe right end of the thread is larger than u. The relative\nvelocity of the rarefied plasma flow and the thread is v−˙p\nattheleftthreadend,and −(v+˙q)attherightthreadend,\nwhere the dot indicates the time derivative. Hence, the\nrateofthreadlengthincreaseattheleftendis ρe(v−˙p)/ρp,\nwhile at the right end it is ρe(v+ ˙q)/ρp. Then it follows\nthat\n˙p=u−ρe\nρp(v−˙p),˙q=u+ρe\nρp(v+ ˙q). (3)\nAs a result, we obtain\n˙p=ρpu−ρev\nρp−ρe,˙q=ρpu+ρev\nρp−ρe. (4)M. S. Ruderman & M. Luna: Damping by mass accretion 3\nThemassofthe threadis M(t) =ρp(q−p). Differentiating\nthis relation and using Eq. (4) we obtain\n˙M=ρp(˙q−˙p) =2ρpρev\nρp−ρe. (5)\nIntegrating this equation yields\nM=ρp(q0−p0)+2ρpρevt\nρp−ρe, (6)\nwherep=p0andq=q0att= 0.\nConsider a system consisting of the thread and two\nvolumes of rarefied plasma attached to the thread at the\nleft and the right. The length of the left volume is ( v−\n˙p)∆t, while the length of the right volume is ( v+ ˙q)∆t.\nThe linear momentum of this system is\nM(t)u(t)+ρev(v−˙p)∆t−ρev(v+ ˙q)∆t\n=M(t)u(t)−2ρpρeuv∆t\nρp−ρe. (7)\nAfter time ∆ tthe plasma in both volumes is absorbed by\nthe thread. The thread mass and velocity become M(t+\n∆t) andu(t+ ∆t). The change in the linear momentum\nis equal to the impulse of force fg∆t,\nM(t+∆t)u(t+∆t)−M(t)u(t)\n+2ρpρeuv∆t\nρp−ρe=gρp[z(p)−z(q)]∆t. (8)\nDividing this relation by ∆ tand taking ∆ t→0 we obtain\nd(Mu)\ndt=gρp[z(p)−z(q)]−2ρpρeuv\nρp−ρe. (9)\nUsing Eq. (6) we transform this equation to\n/parenleftbigg\nq0−p0+2ρevt\nρp−ρe/parenrightbigg\n˙u=g[z(p)−z(q)]−4ρeuv\nρp−ρe.(10)\nEquations (4) and (10) constitute the system of equations\nforu,pandq.\nWe consider the model presented in this section as one\nof the first steps in understanding the damping mecha-\nnisms of LALOs, and we are well aware of its limitations.\nIn this model we neglect many physical effect that, prob-\nably, exist in the reality. We consider the plasma motion\nin LALOs as one-dimensional while, in the reality, it is\nthree-dimensional. The account of variation of the plasma\nparameters across the magnetic tube can result, for ex-\nample, in the distortion of the boundary between the hot\nand cold plasmas which can complicate the process of hot\nplasma accretion. Probably, the most vulnerable assump-\ntion of our model is that the hot accreting plasma is in-\nstantaneously accommodated by the cold dense thread.\nObviouslythe realprocessofhot plasmaaccretionis much\nmore complex. The collision of flows of the hot and cold\nplasmas can cause formation of a very complex interac-\ntion region involving shocks. The relaxation of this region\nwould, probably, involve heat conduction, ionisation and\nrecombination. How much all these complicated processes\nwill affect the damping of LALOs is an open question.\nIt should be addresses in the future by considering more\nsophisticated models.3. Prominence oscillations in a magnetic tube\nwith two straight parts\nTo obtain the solution to the system of Eqs. (4) and (10)\nwe need to specify the magnetic tube shape. To make the\nproblemassimpleaspossibleweassumethatthetubeaxis\nis composed of an arc of a circle with two straight lines\nattached to its ends such that the whole curve is smooth\n(see Fig. 1). Hence, we take\nxz\nθr s\nFig.1.Sketch of the equilibrium. The magnetic tube axis\nconsists of an arc of a circle of radius rwith the two\nstraight lines attached. The dense prominence thread oc-\ncupies the shaded area. Recall that we consider a one-\ndimensional problem and neglect the variationof all quan-\ntities across the magnetic tube.\nx(s) =\n\n(s+rθ−ℓ/2)cosθ−rsinθ,\n0≤s<ℓ/2−rθ,\nrsin2s−ℓ\n2r,|s−ℓ/2| ≤rθ,\n(s−rθ−ℓ/2)cosθ+rsinθ,\nℓ/2+rθ ℓ/2 +rθat\nany time, that is the ends of the dense thread are always\non the straight parts of the magnetic tube. As a result, we\nobtain\nZ(P) =/parenleftbiggL\n2−P−Rθ/parenrightbigg\nsinθ+R(1−cosθ),\nZ(Q) =/parenleftbigg\nQ−Rθ−L\n2/parenrightbigg\nsinθ+R(1−cosθ),(16)\nwhereL=gℓ/v2andR=gr/v2. It follows from this\nequation that\nZ(P)−Z(Q) = (L−P−Q)sinθ. (17)\nSubstituting this result in Eq. (14), differentiating the ob-\ntained equation, and using Eq. (15), we obtain the equa-\ntion forU,\nd\ndT/parenleftbigg\n[(Q0−P0)(ζ−1)+2T]dU\ndT+4U/parenrightbigg\n=−2ζUsinθ.\n(18)\nIntroducing the new variable σ= (Q0−P0)(ζ−1)+2T\nwe rewrite this equation as\nσd2U\ndσ2+3dU\ndσ+κU= 0, (19)\nwhereκ=1\n2ζsinθ.Belowweassumethat Z(P0) =Z(Q0),\nwhich implies that P0+Q0=L. Then it follows from\nEqs. (14) and (17) that\n(Q0−P0)dU\ndσ=−4U0\nζ−1atT= 0, (20)\nwhereU0=U(0). As a result, we havethe following initial\nconditions for U:\nU=U0,dU\ndσ=−2U0\nσ0atσ=σ0, (21)\nwhereσ0= (ζ−1)(Q0−P0). The variable substitution\nξ= 2√κσ, U=σ−1W, (22)\nreduces Eq. (21) to the Bessel equation\nd2W\ndξ2+1\nξdW\ndξ+/parenleftbigg\n1−4\nξ2/parenrightbigg\nW= 0. (23)\nThe general solution to this equation is\nW(ξ) =C1J2(ξ)+C2Y2(ξ), (24)\nwhereJ2andY2are the Bessel functions of the first\nand second kind, and C1andC2are arbitrary constants.\nReturning to the original variables we obtain\nU(σ) =1\nσ/bracketleftbig\nC1J2/parenleftbig\n2√κσ/parenrightbig\n+C2Y2/parenleftbig\n2√κσ/parenrightbig/bracketrightbig\n, (25)\nSubstituting Eq. (25) in Eq. (21) yields\nC1J2/parenleftbig\n2√κσ0/parenrightbig\n+C2Y2/parenleftbig\n2√κσ0/parenrightbig\n=σ0U0, (26)\nC1J′\n2/parenleftbig\n2√κσ0/parenrightbig\n+C2Y′\n2/parenleftbig\n2√κσ0/parenrightbig\n=−/radicalbiggσ0\nκU0, (27)where the prime indicates the derivative of Bessel func-\ntion with respect to its argument. Using the identity\n(Abramowitz & Stegun 1972)\nJ2(x)Y′\n2(x)−J′\n2(x)Y2(x) =2\nπx, (28)\nwe obtain from Eq. (27)\nC1=πσ0U0/bracketleftbig√κσ0Y′\n2/parenleftbig\n2√κσ0/parenrightbig\n+Y2/parenleftbig\n2√κσ0/parenrightbig/bracketrightbig\n,\nC2=−πσ0U0/bracketleftbig√κσ0J′\n2/parenleftbig\n2√κσ0/parenrightbig\n+J2/parenleftbig\n2√κσ0/parenrightbig/bracketrightbig\n.(29)\nTypicallyζ∼100, while the initial length of the dense\nthread is of the orderofa few Mm, lp=q0−p0/greaterorsimilar2 Mm. If\nwetakev/lessorsimilar50km/sand g= 274m/s2,thenQ0−P0/greaterorsimilar0.2\nandσ0/greaterorsimilar20. In addition, the radius of curvature could be\nestimated as of the order of r= 60 Mm (see, Luna et al.\n2014). We are considering threads that are larger than the\narched part of the tube, then lp≥2rθ. Forlp/greaterorsimilar2 Mm\nthis inequality can be satisfied if we take θ= 1◦. Hence,\nbelow we assume that θ/greaterorsimilar1◦. With these considerations\nκ/greaterorsimilar0.8. Taking into account that σ≥σ0we arrive at the\nestimate 2√κσ/greaterorsimilar8. For such values of the argument we\ncanusetheasymptoticexpressionsfortheBesselfunctions\n(Abramowitz & Stegun 1972) namely\nJm(x)≈/radicalbigg\n2\nπxcos/parenleftbigg\nx−π(2m+1)\n4/parenrightbigg\n, (30)\nYm(x)≈/radicalbigg\n2\nπxsin/parenleftbigg\nx−π(2m+1)\n4/parenrightbigg\n. (31)\nThen it follows from Eq. (29) that\nC1≈σ0U0\nκ4/radicalbig\nπ2κσ0cos/parenleftbigg\n2√κσ0−5π\n4/parenrightbigg\n, (32)\nC2≈σ0U0\nκ4/radicalbig\nπ2κσ0sin/parenleftbigg\n2√κσ0−5π\n4/parenrightbigg\n. (33)\nSubstituting Eqs. (32) and (33) in Eq. (25) and using\nEqs. (30) and (31) we transform it to the approximate\nform\nU(σ) =U0/parenleftBigσ0\nσ/parenrightBig5/4\ncos/parenleftbig\n2√κσ−2√κσ0/parenrightbig\n. (34)\nThis equation can be rewritten as\nU(σ) =U0/parenleftbigg\n1+T\nX/parenrightbigg−5/4\n×cos/parenleftBig\n2/radicalbig\n2κ(X+T)−2√\n2κX/parenrightBig\n, (35)\nwhereX=1\n2(ζ−1)(Q0−P0). In Fig. 2 we have plotted\nfour examples of the temporal evolution of the oscillating\nthreads given by the previous equation.\nThe maximumdisplacement ofthe thread occurswhen\nthe argument of cosine is equal to 2 πm,m= 0,1,...at\ntimes\nT=Tm≡πm\n2κ/parenleftBig\nπm+2√\n2κX/parenrightBig\n. (36)M. S. Ruderman & M. Luna: Damping by mass accretion 5\nFig.2.Plotofthetemporalevolutionofthevelocityofthe\nthread given by Eq. (35) normalized to the initial velocity\nU0as function oftime, t. We haveassumed atypical situa-\ntion ofζ= 100,r= 60 Mm, with g= 274 ms−2. For clar-\nity, we have split the different cases studied in two panels.\nIn (a) a thread with lp= 2.1 Mm andθ= 1◦is considered\nwithv= 10 kms−1(black curve) and v= 30 kms−1\n(orange curve). In (b) a thread with lp= 5 Mm and\nθ= 3◦is considered with v= 10 kms−1(red curve) and\nv= 30 kms−1(blue curve).\nThen then-th oscillation period is given by\nΠn=Tn−Tn−1=π\n2κ/bracketleftBig\nπ(2n−1)+2√\n2κX/bracketrightBig\n,(37)\nwheren= 1,2,.... This indicates that the period of the\noscillation depends of the cycle of the oscillation. In gen-\neral this period, Π nincreases with time. In Fig. 2 the\nincrease of the period for each oscillation is clear. The\ndimensional period Pnis\nPn=v\ngΠn=Pshift+Pg\n=vπ2(2n−1)\ngζsinθ+π/radicalBigg\n2lp(ζ−1)\ngζsinθ. (38)\nThe second term in the expression for Pnis associated\nwith the gravity as a restoring force,\nPg=π/radicalBigg\n2lp(ζ−1)\ngζsinθ. (39)\nThis term gives the oscillation period when v= 0. We can\nrecover the oscillation period found by Luna & Karpen(2012) and Luna et al. (2012a) if we assume that θis\nsmall. Then sin θ≈θand\nPg≈π/radicalBigg\n2lp(ζ−1)\nθζg. (40)\nIn addition, Luna & Karpen (2012) and Luna et al.\n(2012a) assumed that the thread filled the dipped part of\nthe flux tube meaning that rθ=lp/2. Then, taking into\naccountthat, fortypicalprominences,thedensitycontrast\nis very large meaning that 1 −1/ζ∼1, we finally arrive\nat\nPg≈2π/radicalbiggr\ng. (41)\nNote that Eq. (38) gives a more general expression. The\nfirst term in Eq. (38), Pshift, introduces the period shift.\nDuring each cycle of the oscillation the period increases\nby\n∆P=2vπ2\ngζsinθ. (42)\nThe period shift is related to the accretion rate onto the\nthread associated with the rarefied plasma. When the ac-\ncretion rate increases so does the period shift. Pshift= 0\nwhen there is no accretion.\nWe define the damping time Tdby the condition that\nthe oscillation amplitude decreases etimes atT=Tdwith\nrespect to the value at T= 0 . Then, using Eq. (35), we\nobtain\nTd= (e4/5−1)X= 2.12X, (43)\nand, in the dimensional variables,\ntd=v\ngTd= 1.06(ζ−1)lp\nv. (44)\nThis relation indicates that the oscillation damping time\nis inverselyproportionaltothe accretionspeed. The larger\nvthe smallervalue of tdis, and, consequently,the stronger\nthe damping is. Similarly, the larger the thread length lp,\nthe weaker the damping is. The reason is that the term\n(ζ−1)lpis essentially the mass of the thread at t= 0. The\ndamping occurs due to the increase of the thread mass\nand the decrease of its momentum. Hence, the larger the\ninitial thread mass the more time is needed to damp its\nmovement.\nIn this section we have assumed that the cold thread is\nlarger than the arched part of the tube, lp≥2rθ. Typical\nprominence threads are equal or smaller than 10 Mm but\nlarger than a blob of 1 Mm. In our previous studies we\nhave found that rshould be of the order of tens Mm. In\nparticular, in Luna et al. (2014) we determined the radius\nof curvature of the dipped field lines of an observed fila-\nment as approximately equal to 60 Mm. Using these num-\nbers we obtain that the angle θ≤5◦. The speed of the\naccretion flow, v, depends on the coronal heating at the\nflux-tube footpoints (see, e.g. Karpen et al. 2003). Based\non the simulations by Luna et al. (2012b) and the results6 M. S. Ruderman & M. Luna: Damping by mass accretion\nby Karpen et al. (2005) we can estimate that the speed\nof the hot flows is of the order of 30 kms−1. We also\ntake a typical value of ζ= 100 (see Labrosse et al. 2010).\nTo plot Fig. 2 we have considered four sets of parame-\nters. Figure 2(a) corresponds to a thread of initial length\nlp= 2.1 Mm and θ= 1◦. The black line corresponds to\nthe accretion velocity v= 10 kms−1and the orange line\ncorresponds to v= 30 kms−1. The difference in the ac-\ncretion flow produces important changes in the damping\ntime,td(Eq. (44)), and in the period shift, ∆ P(Eq. (42)),\nbut not in the gravity period, Pg(Eq. (39)). The damp-\ning and the period shift is stronger for the high accretion\nvelocity (orange curve). For the case with v= 10 kms−1\n(black line) Pgis 48.8 minutes, the shift is 3.4 minutes\nand the damping time is 367 min., and for the case with\nv= 30 kms−1(orange line) Pgis also 48.8 minutes, the\nshift is 10.3 minutes and the damping time is 122minutes.\nDue to the shift the period defined as the time interval\nbetween two consecutive maxima changes for each oscilla-\ntion. With Eq. (38) we can compute the period of the nth\ncycle, asPn= 3.4(2n−1)+48.8 min. = 52.2, 55.7, 59.1,\n... min. for the first set of parameters (black line). For the\nsecond set of parameters corresponding to the orange line\nthenth period is Pn= 10.3(2n−1) + 48.8 min. = 59.1,\n69.5, 79.8,...min. From this panel we clearly see the de-\npendence of the damping time and the period shift on the\naccretion velocity. Larger values of the accretion velocity\nproducesstrongerdamping,thatisshorterdampingtimes,\nand larger period shifts. A similar result can be seen in\nFig. 2(b) for a larger thread of initial length lp= 5 Mm.\nIn this case the damping and the period shift is weaker\nthan in the case with shorter threads (Fig. 2(a)). In fact,\nstronger damping, that is smaller damping time, involves\nlarger period shifts. It is possible to combine Equations\n(39), (42), and (44) to obtain\ntd∆P= 1.06P2\ng, (45)\nwhere we have assumed that the density contrast is large\nenough and taken ζ−1≈ζ. This relation reflects the\nfact that, for a given gravity period Pg, a strong damping\n(smalltd) corresponds to a large period shift (large ∆ P)\nand vice versa. This behaviour is clear in both panels in\nFig. 2. In Fig. 2(a) both cases have the same gravity pe-\nriodPg. The oscillation plotted in the orange curve has\nstronger damping and also larger period shift than in the\ncase showed by the black curve. A similar effect can be\nseen in Fig. 2(b).\nAs we have already pointed out, it follows from\nEq. (44) that the damping time is proportional to the ini-\ntial length of the thread and inversely proportional to the\nspeed of the accretion flow. This means that the damp-\ning is stronger for smaller initial threads and for stronger\naccretion flows. Using Eq. (6) it is possible to relate the\ndamping time with the mass of the thread at the initial\ntime and the rate of mass accretion as\ntd= 2.12M(t= 0)\n˙M, (46)Thisindicatesthat thedampingisstrongerinlongitudinal\noscillations produced in prominences with small thread\nmass.\n4. Prominence oscillations in a circular arched dip\nxr\nθ θz\nq p\nFig.3.Sketch of the equilibrium. The magnetic tube axis\nis an arc of a circle of radius r. The prominence occu-\npies the shaded area. We again recall that we consider a\none-dimensional problem and neglect the variation of all\nquantities across the magnetic tube.\nIn this section we consider prominence oscillations in\na magnetic tube that has the shape of an arc of a circle\nof radiusr. The equilibrium state is shown in Fig. 3. We\nintroduce the angles θpandθqbetween the lines connect-\ning the centre of the circular arc and the ends of the dense\nprominence thread. These angles are given by\nθp=2p−ℓ\n2r, θq=2q−ℓ\n2r. (47)\nFor this geometry we have\nz(p) =r(1−cosθp), z(q) =r(1−cosθq). (48)\nIntroducing the dimensionless variables\nτ=t/radicalbiggg\nr,˜u=u√rg,˜v=v√rg, δ=lp\nr, (49)\nwe can rewrite the system of Eqs. (4) and (10) as\ndθp\ndτ=ζ˜u−˜v\nζ−1,dθq\ndτ=ζ˜u+ ˜v\nζ−1, (50)\n/parenleftbigg\nδ+2τ˜v\nζ−1/parenrightbiggd˜u\ndτ= cosθq−cosθp−4˜u˜v\nζ−1. (51)\nProminence threads have typical lengths of a few Mm and\nthe radiusofcurvatureofseveraltens ofMm. Then wecan\nassume that the length of the thread is much smaller than\nthe radius of the dip curvature, l/r≪1. This condition\nis equivalent to θq−θp≪1 meaning that we can use the\napproximate relation\ncosθq−cosθp=−2sinθq+θp\n2sinθq−θp\n2\n≈ −(θq−θp)sinφ, (52)M. S. Ruderman & M. Luna: Damping by mass accretion 7\nwhereφ= (θq+θp)/2. Since the typical value of ζis 100,\nbelow we neglect 1 in comparison with ζ. Then, using\nEq. (52), we obtain from Eq. (50)\n˜u=dφ\ndτ, θq−θp=δ+2τ˜v\nζ. (53)\nWith the aid of these results we reduce the Eq. (51) to\nd2φ\ndτ2+sinφ+4˜v\nζδ+2τ˜vdφ\ndτ= 0. (54)\nBelow we assume that initially the dense thread is in equi-\nlibrium and then it is pushed and starts to oscillate. In\naccordance with this we impose the initial conditions\nφ= 0,dφ\ndτ= 2χ0atτ= 0, (55)\nwhereχ0is a constant related to the initial impulse given\nto the thread by some external trigger.\n4.1. Linear theory with strong damping\nWe first consider small-amplitude oscillations and assume\nthatφ≪1. Thus, we can use the approximate relation\nsinφ≈φand reduce Equation (54) to\nd2φ\ndτ2+4˜v\nζδ+2τ˜vdφ\ndτ+φ= 0. (56)\nThe variable substitution\nξ=τ+ζδ\n2˜v, φ=ξ−1y, (57)\nreduces Equation (56) to\nd2y\ndξ2+y= 0. (58)\nThe general solution to this equation is a linear combi-\nnation of sin ξand cosξ. Then, returning to the original\nvariablesandusingtheinitialconditionsEq.(55),wewrite\nthe solution to Eq. (56) as\nφ(τ) =2χ0\n1+2˜vτ/ζδsinτ. (59)\nThis solution describes oscillations with constant period\nΠ = 2πinthedimensionlessvariables.Thus,inthismodel,\nthe period is constant and, in the dimensional variables,\nit is given by\nP= 2π/radicalbiggr\ng, (60)\nwhich recovers the result by Luna & Karpen (2012),\nLuna et al. (2012a), and Luna et al. (2016). Luna et al.\n(2016) numerically simulated the motion of perturbed\nlocalised cold plasma supported by a two-dimensional\ndipped magnetic field. They found that the back-reaction\nof the field on the plasma oscillation is very weak, vali-\ndating the simpler assumption of rigid flux tubes in thepresent study. In particular, they obtained that the oscil-\nlation period was practically the same as that found by\nLuna & Karpen (2012).\nThe amplitude of the oscillations is given by the initial\ndimensionless velocity 2 χ0=u0/√rg, and the damping\ntime depends of the factor 2˜ v/(ζδ). This solution implies\nthat the dampingratechangeswith time similarlytowhat\nwasfoundintheprevioussection.Thedampingisstronger\nat the initial stage of the oscillation close to τ= 0, then\nlaterit decreasesforlarger τ. It isnowconvenienttointro-\nduce another dimensionless time Θ = vt/r=vτ/√gr. As\nin the previous section we define the dimensionless damp-\ningtimeΘ dbytheconditionthattheoscillationamplitude\ndecreasesetimes at Θ = Θ d. Thus,\ne−1=ζδ\nζδ+2˜vΘd, (61)\nand we obtain\nΘd= (e−1)ζδ\n2˜v≈0.86ζδ\n˜v. (62)\nIn terms of dimensional variables we have\ntd= 0.86ζlp\nv, (63)\nwhich implies that the damping is sufficiently strong for\nsmall threads and for the large accretion speed. Using\nEq. (6) it is possible to rewrite Eq. (63) as\ntd= 1.72M(t= 0)\n˙M, (64)\nwhich implies that the damping time not only depends on\nthe mass accretion rate, but also on the initial mass of\nthe thread. Equation (64) is almost identical to Eq. (46).\nThey only differ by just a small difference in the constant\nat the front of the ratio of the initial mass and the mass\naccretion rate. Equation (63) shows that strong damping\nis associated with large accretion rates and small initial\nthread masses. In Fig. 4 we plot the temporal evolution\nofφfor several values of parameters lpandv. In all the\ncases the period is P= 49 minutes. We clearly see from\nthis figure that the larger lpthe weaker the damping is,\nthat is the largerthe damping time is. Similarly, the larger\nthe accretion velocity the stronger the damping is, that is\nthe smaller the damping time is.\nLuna & Karpen (2012) found that the temporal evolu-\ntion of the oscillation velocity was given by a Bessel func-\ntion of order 1 and the damping is produced by the phase\nin the argument of this function. However, here we have\nfound that the temporal evolution of the velocity is given\nby the sine divided by a linear function of its argument.\nThe phase of the argument is one half of that found in the\ncase studied by Luna & Karpen (2012). The difference be-\ntween the two models is in how the hot evaporated mass\nis deposited in the cool thread. Luna & Karpen (2012)\nassumed that the hot flows adapt to the motion of the\nthread and accretion at both sides of the thread is sym-\nmetric. Then, the momentum transferred to the thread by8 M. S. Ruderman & M. Luna: Damping by mass accretion\nthe hot evaporated flows is cancelled and the net momen-\ntum transfer is zero. In this case the damping is exclu-\nsively produced by the change of mass of the thread. In\ncontrast,in the currentworkweassumethat the hot evap-\norated flows are not affected by the motion of the thread.\nIn this case there is a net transfer of the hot plasma flow\nmomentum to the thread. This is given by the last term\non the right hand side of Eq. (7). If we drop this term\nand solve the differential Eq. (10), then we recover the\ntemporal evolution found by Luna & Karpen (2012). In a\nmore realistic scenario that also includes the thermody-\nnamic processes, the process of the momentum deposition\nis, probably, something in between these two extreme sce-\nnarios.\nFig.4.Plot of the angular position φ(τ) normalized to\nits amplitude 2 χ0as a function of tfor various values of\nthe parameters vandlp. We have taken g= 274 ms−2\nand typical values of ζ= 100 and r= 60 Mm. In all\ncases the period is 49 min. The black curve corresponds\ntov= 30kms−1andlp= 2 Mm, the green curve to v=\n10kms−1andlp= 2 Mm, the red curve to v= 40kms−1\nandlp= 5 Mm, and the blue curve to v= 10kms−1and\nlp= 5 Mm. The damping times for these combinations\nof parameters are td= 55.6,166.7,138.9,416.7 min., re-\nspectively.\n4.2. Nonlinear weakly damped oscillations\nWhen there is no accretion (˜ v= 0) Eq. (54) reduces to\nthe equation of nonlinear pendulum. Its small-amplitude\noscillation is described by φ(τ) =φ0sinτ, whereφ0is the\nconstant oscillation amplitude. In that case the charac-\nteristic time of the variation of function φ(τ) is 1. The\ncharacteristic time remains the same for nonlinear oscil-\nlations when the oscillation amplitude is smaller than or\nof the order of π/2. The damping of oscillations due to\naccretion can be considered as slow if the dimensionless\ndamping time is much larger than 1. Since 1 is approxi-mately equal to one sixth of the oscillation period, which\nis 2π, this implies that the damping can be considered as\nslow if it is largerthan or of the orderof the oscillation pe-\nriod. This observation inspires us to search for a solution\nto the Eq. (54) describing slowly damped nonlinear oscil-\nlations. To do this we introduce the “slow” time τ1=ǫτ,\nwhereǫ≪1 is of the order of the ratio of the characteris-\ntic oscillationtime to the damping time. Then we consider\nφas a function of two variables, τandτ1. The damping is\nslow when the last term on the right-hand side of Eq. (54)\nis small. In accordance with this we put ˜ v=ǫ˜v1. After\nthat Eq. (54) is transformed to\n/parenleftbigg∂2φ\n∂τ2+ 2ǫ∂2φ\n∂τ∂τ1+ǫ2∂2φ\n∂τ2\n1/parenrightbigg\n+sinφ\n+4ǫ˜v1\nζδ+2τ1˜v1/parenleftbigg∂φ\n∂τ+ǫ∂φ\n∂τ/parenrightbigg\n= 0. (65)\nBelow we assume that φis a periodic function of τwith\nthe period Π that will be determined later. Note that, in\ngeneral, Π can depend on τ1. We search for a solution to\nEq. (65) in the form of expansion\nφ=φ1+ǫφ2+... (66)\nSubstituting this expansion in Eq. (65) and collecting\nterms of the order of unity we obtain the equation of non-\nlinear pendulum\n∂2φ1\n∂τ2+sinφ1= 0. (67)\nUsing Eq. (55) we obtain the initial conditions for φ1,\nφ1= 0,∂φ1\n∂τ= 2χ0atτ= 0. (68)\nIt is straightforwardto obtain the first integral of Eq. (67)\nsatisfying the initial conditions Eq. (68),\n/parenleftbigg∂φ1\n∂τ/parenrightbigg2\n−2cosφ1= 4χ2−2. (69)\nThe quantity χ2is proportional to the energy of the oscil-\nlation. When there is no damping the energy is conserved\nandχ=χ0. However the energy decreases due to the\ndamping, meaning that χis a function of τ1. This func-\ntion satisfies the initial condition χ=χ0atτ1= 0.\nThe angle φ1takes it maximum when ∂φ1/∂τ= 0.\nThen it follows from Eq. (69) that the oscillation ampli-\ntude is\nA= maxφ1= 2arcsinχ. (70)\nBelow we assume that the oscillation amplitude does not\nexceedπ/2. This condition implies that χ≤√\n2/2. We\nintroduce the new dependent variable ψrelated toφ1by\nsinψ=1\nχsinφ1\n2,−π\n2≤ψ≤π\n2. (71)\nIt follows from Eq. (69) that the absolute value of the\nright-hand side of this equation does not exceed 1, so itM. S. Ruderman & M. Luna: Damping by mass accretion 9\nalways can be solved with respect to ψ. Now Eq. (69)\nreduces to\n/parenleftbigg∂ψ\n∂τ/parenrightbigg2\n= 1−χ2sin2ψ. (72)\nIt follows from this equation that\nτ=/integraldisplayψ\n0dψ′\n/radicalbig\n1−χ2sin2ψ′, (73)\nwhere we have imposed the condition that ψis an in-\ncreasingfunction of τ, which correspondsto the first quar-\nter of the first oscillation period. Then using the relation\n(Korn & Korn 1961) sn( τ;χ) = sinψ, where sn( τ;χ) is\nthe elliptic sine, and Eq. (71), we eventually obtain\nφ1= 2arcsin(χsn(τ;χ)). (74)\nThis equationis valid forany τ≥0.The oscillationperiod\nis four times the time needed for φ1to vary from 0 to A.\nSinceψ= 0 whenφ1= 0 andψ=π/2 whenφ1=A,\nit follows that the oscillation period is Π = 4 K(χ), where\nK(χ)isthe completeelliptic integralofthefirstkindgiven\nby (Korn & Korn 1961)\nK(χ) =/integraldisplayπ/2\n0dψ/radicalbig\n1−χ2sin2ψ. (75)\nTo account for the effect of accretion we go to the next\norder approximation. Recall that now χis a function of\nτ1. Collecting the terms of the order of ǫin Eq. (65) yields\n∂2φ2\n∂τ2+φ2cosφ1=−2∂2φ1\n∂τ∂τ1−4˜v1\nζδ+2τ1˜v1∂φ1\n∂τ.(76)\nSinceφis a periodic function of τwith the period Π,\nthe same is true for φ2. We multiply Eq. (76) by ∂φ1/∂τ\nand integrate with respect to τfrom 0 to Π. Then, using\nEq. (67) and the integration by parts, we obtain on the\nleft-hand side\n/integraldisplayΠ\n0/parenleftbigg∂2φ2\n∂τ2+φ2cosφ1/parenrightbigg∂φ1\n∂τdτ\n=/integraldisplayΠ\n0φ2∂\n∂τ/parenleftbigg∂2φ1\n∂τ2+sinφ1/parenrightbigg\ndτ= 0. (77)\nThis implies that the right-hand side is also zero, which\ngives the equation\nd\ndτ1/integraldisplayΠ\n0/parenleftbigg∂ψ1\n∂τ/parenrightbigg2\ndτ+4ζ˜v1\nζδ+2τ1˜v1/integraldisplayΠ\n0/parenleftbigg∂ψ1\n∂τ/parenrightbigg2\ndτ= 0.\n(78)\nIn this equation we use the ordinary derivative because\nthe integral in this equation only depends on τ1. Using\nEq. (69) yields\n/integraldisplayΠ\n0/parenleftbigg∂ψ1\n∂τ/parenrightbigg2\ndτ= 4/integraldisplayΠ\n0/parenleftbigg\nχ2−sin2φ1\n2/parenrightbigg\ndτ. (79)\nThen, with the aid of Eqs. (71) and (73) we obtain\n/integraldisplayΠ\n0/parenleftbigg∂ψ1\n∂τ/parenrightbigg2\ndτ= 4χ2/integraldisplayΠ\n0cos2ψdτ= 16Υ(χ),(80)where\nΥ(χ) =χ2/integraldisplayπ/2\n0cos2ψdψ/radicalbig\n1−χ2sin2ψ\n=E(χ)−(1−χ2)K(χ), (81)\nand the complete elliptic integral of the second kind E(χ)\nis given by (Korn & Korn 1961)\nE(χ) =/integraldisplayπ/2\n0dψ/radicalbig\n1−χ2sin2ψ. (82)\nUsing Eq. (80) we transform Eq. (78) to\ndΥ(χ)\ndτ1+4˜v1Υ(χ)\nζδ+2τ1˜v1= 0. (83)\nIt follows from this equation that\nΥ(χ) =ζ2δ2Υ(χ0)\n(ζδ+2Θ)2. (84)\nRecall that Θ = vt/r=τ1˜v1=τ˜v, andχ0is the value\nofχat the initial time ( τ1= 0). It follows from Eq. (81)\nthat Υ(χ) is a monotonically increasing function. Then it\nfollows from Eq. (84) that χdecreases with time. Using\nthe expression for the oscillation amplitude Ain terms\nofχwe conclude that Aalso decreases with time, which\nis an expected result. Again we define the dimensionless\ndamping time Θ das the time when the oscillation ampli-\ntude becomes etimes smaller than the initial amplitude\nA0. Using Eqs. (70) and (84) we obtain\nΘd=ζδ\n2/parenleftBigg/radicalBigg\nΥ(sin(A0/2))\nΥ(sin(A0/2e))−1/parenrightBigg\n. (85)\nRewriting this expression in the dimensional variables\ngives the expression for the dimensional damping time td,\ntd=ζlp\n2v/parenleftBigg/radicalBigg\nΥ(sin(A0/2))\nΥ(sin(A0/2e))−1/parenrightBigg\n. (86)\nThe theory becomes especially simple in the linear ap-\nproximation that we obtain assuming that χ≪1. Then\nA= 2χ, sn(τ;χ) = sinτ, andK(χ) =E(χ) =π/2. Using\nthese relations and Eq. (81) we obtain that Π = 2 πand\nΥ =π��2/2. Now we obtain from Eqs. (84) and (86) that\nχ=ζδχ0\nζδ+2vt/r, td=ζlp(e−1)\n2v= 0.86ζlp\nv. (87)\nFinally, it follows from Eq. (84)\nφ1= 2χsinτ=ζlpA0\nζlp+2vtsin/parenleftbig\nt/radicalbig\ng/r/parenrightbig\n, (88)\nwhereA0= 2χ0. We see that the expression for tdcoin-\ncideswith that givenbyEq.(62). Itis alsostraightforward\nto verify that Eq. (88) coincides with Eq. (59). Hence, we\nrecovered the results obtained in Subsection 4.2.\nIn Fig. 5 the dependence of Aon Θ forζ= 100,\nδ= 1/12, and two values of the initial amplitude A0=10 M. S. Ruderman & M. Luna: Damping by mass accretion\n2arcsinχ0,A0=π/8 andA0=π/2, are shown. We did\nnot show the curve obtained using the linear theory be-\ncause it practically coincides with that corresponding to\nA0=π/8. We see that the nonlinearity only slightly re-\nduces the damping time. For ζ= 100 and δ= 1/12 the\nlinear theory gives Θ d= 7.16, while the nonlinear theory\ngives Θd= 7.11 whenA0=π/8 and Θd= 6.52 when\nA0=π/2. Hence, even when A0=π/2 the nonlinearity\nreduces the damping time by less than 10%.\nFig.5.Dependence of the oscillation amplitude Aon the\ndimensionless time Θ = vt/r. The solid and dashed curves\ncorrespond to the initial amplitude A0=π/2 andA0=\nπ/8, respectively. The vertical lines indicate the damping\ntime Θd.\nWhen there is no damping the oscillation period is\nequal to 4K(χ0). However, due to damping χdecreases\nwith time.\nConsider the sequence {τn},n= 0,1,..., whereφ= 0\natτ=τ2n,φtakes its local maximum when τ=τ2n+1,\nand it takes its local minimum when τ=τ2n+3. Thenth\noscillation cycle corresponds to the variation of τfrom\nτ2n−2toτ2n+2. The angle φincreases from 0 to its lo-\ncal maximum when τvaries from τ2ntoτ2n+1, than it\ndecreases back to 0 when τvaries from τ2n+1toτ2n+2,\ncontinues to decrease to reach its local minimum when τ\nvaries from τ2n+2toτ2n+3, and finally return to 0 when\nτvaries from τ2n+3toτ2n+4. Hence, we split each oscil-\nlation period in four quarters. Since χis a slowly varying\nfunction of τwe can neglect it variation in any quarter\nof period. Each quarter of period corresponds to the vari-\nation ofψbyπ/2. Then, using Eqs. (73) and (75), we\nobtain the recurrence relation\nτn+1−τn=/integraldisplayπ/2\n0dψ/radicalBig\n1−χ2(τn)sin2ψ=K(χ(τn)).(89)\nThenth oscillation period is given by\nΠn=τ4n+4−τ4n=3/summationdisplay\nj=0K(χ(τ4n+j)). (90)Fig.6.Plot of the temporal evolution of the angle φde-\nscribed by Eq. (54) normalized to the initial dimension-\nless velocity 2 χ0as function of time, t. We have taken\ng= 274 ms−2and typical values of ζ= 100,r= 60 Mm,\nandlp= 2 Mm. The black, green, red, blue, and orange\ncorrespond to the initial velocities of u0= 2χ0√rg=\n36,72,108,144, and 180 kms−1, respectively.\nThe function χ(τ) is defined by Eq. (84).\nSinceχisa monotonicallydecreasingfunction of τand\nK(χ) is a monotonically decreasing function of χ, it fol-\nlows that {Πn}is a monotonically decreasing sequence.\nWhenτ→ ∞,χ→0,K(χ)→π/2, and Π n→2π.\nThe stronger the damping the faster χ(τ) decreases and,\nconsequently, the faster the sequence {Πn}decreases. The\nlarger the initial amplitude A0the largerχ0is and, con-\nsequently, the larger the difference between the initial pe-\nriod, Π 1, and the limiting period value 2 πis. However\nthis difference is not very big even for quite larger ini-\ntial oscillation amplitude. When A0=π/2 we obtain\nΠ1<4K(χ0)≈7.42, meaning that, even for this large\nvalue of the oscillation amplitude, the difference between\nΠ1and 2πis less than 20%.\nAs an example, using Eqs. (84), (89), and (90) we cal-\nculated oscillation periods Pn= Πn/radicalbig\nr/gforA0=π/2,\nζ= 100,g= 274 ms−2,r= 60 Mm, lp= 2 Mm,\nandv= 30 kms−1. We obtained P1= 53.9 min.,\nP2= 50.4 min.,P3= 49.7 min., andP4= 49.4 min..\nPn→2π/radicalbig\nr/g= 49 min.asn→ ∞. Hence, in this par-\nticular example the period only decreases by 10%.\nIn Figure 6 we have plotted the full numerical solu-\ntions for typical values of parameters. We clearly see the\nnonlinear effects but also we see that these effects are not\nsignificant. The orange curve corresponds to A0=π/2\nthat in dimensional variables correspondsto the initial ve-\nlocity equal to 180 kms−1. We see that it is only slightly\ndifferent from the black curve corresponding to the initial\nvelocity equal to 36 kms−1.\n5. Summary and conclusion\nIn this articlewe havestudied the damping oflongitudinal\noscillations ofa prominence thread caused by the mass ac-M. S. Ruderman & M. Luna: Damping by mass accretion 11\ncretion of the evaporated chromospheric plasma. We have\nconsidered a thin curved magnetic tube of an arbitrary\nshape. The prominence thread is in the central part of\nthe tube and it consists of a dense cold plasma. The parts\nof the tube at the two sides of the thread are filled with\na hot rarefied plasma. The restoring force in the promi-\nnence oscillation is the gravity projected on the flux tube.\nWe have assumed that there are flows of coronal rarefied\nplasma toward the thread. These flows are caused by the\nplasma evaporation at the magnetic tube footpoints. The\ncoronal heating is localized at the chromosphere and at\nthe bottom of the corona produces the evaporation. The\nhot evaporated plasma condenses in the already formed\nprominence thread by the thermal non-equilibrium insta-\nbility. Our main assumption is that the hot evaporated\nplasma is instantaneously accommodated by the thread\nwhen it arrives at the thread, and its temperature and\ndensity become equal to those of the thread. Then we de-\nrived the system of three ordinary differential equations\ndescribing the thread dynamics.\nThe equations describing the thread oscillation are\nvalid for an arbitrary shape of the magnetic tube axis.\nThe only restriction is that it is a planar curve in a verti-\ncal plane. Of course the oscillation properties depend on\na particular shape of the magnetic tube. We considered\ntwo particular models. In the first one the magnetic tube\naxis is composed of an arc of a circle with two straight\nlines attached to its ends in such a way that the whole\ncurve is smooth. A very important property of this model\nis that the equations describing the thread oscillations are\nlinear for any oscillation amplitude under the restriction\nthat the thread ends remain of the straight parts of the\ntube. We obtained the solution to the governingequations\nin terms of Bessel functions. We showed that, for typical\nparameters of solar prominences, the Bessel functions can\nbe approximated by trigonometric functions. Then we ob-\ntained the analytical expressions for the oscillation damp-\ning time and periods. We found that the damping time\nis inversely proportional to the accretion rate and pro-\nportional to the initial mass of the thread. The oscillation\nperiod depends strongly on the angle between the straight\nparts of the tube axis and the horizontal direction. The\nlarger this angle, the smaller the period is. We also found\nthat the period increases with time and, in each cycle,\nthe time of the maximum thread displacement is shifted.\nWe have found that the larger the damping the larger the\nperiod shift for a given oscillation period.\nIn the second model studied in this article the shape of\nthe tube axis is an arc of a circle. We have considered the\nlinear as well as the nonlinear regime. In the linear regime\nwe assumedthat the displacementofthe thread is smallin\ncomparisonwith the radius ofcurvatureof the dipped flux\ntube. We have found that the period is equal to the period\nof the pendulum oscillation and it does not change with\ntime. The damping time is inversely proportional to the\nmass accretion rate and proportional to the initial mass\nof the thread. In the nonlinear regime, we assumed that\nthe damping is slow meaning that the damping time ismuch larger that the characteristic oscillation time. It is\nimportanttonotethatthecharacteristicoscillationtimeis\nthe oscillation period divided by 2 π. This implies that the\ndampingcanbeconsideredasslowevenwhenthedamping\ntime is of the order of the oscillation period. To study the\nthread oscillations we used the two-scale approach where\nthe oscillations are described by the solution of the non-\nlinear pendulum problem with slowly varying amplitude.\nWe showed that the nonlinearity only slightly reduces the\ndamping time. Again the damping time is inversely pro-\nportional to the accretion speed and proportional to the\ninitial mass. In this model the oscillation periods decrease\nwith time. This behaviour is in contrast with that found\nin the first model. The larger the initial oscillation ampli-\ntude the larger the reduction in the oscillation periods is.\nHowever, even for the largest initial oscillation amplitude\nconsidered in our article this reduction does not exceed\n20%.\nWe conclude that the mass accretion can damp the\nmotion of the threads rapidly. Thus, this mechanism can\nexplain the observed strong damping of large-amplitude\nlongitudinal oscillations. In addition, the damping time\ncan be used to determine the mass accretion rate and in-\ndirectly the coronal heating. More work needs to be done\nto increasethe complexityofthe model by including strat-\nification of the plasma, the physical processes in conden-\nsation of the thermal instability, and consider 2D and 3D\nmodels of the magnetic geometry in order to understand\nthe interaction of the plasma with the magnetic field. In\naddition, the damping by radiative losses should be con-\nsidered in a full model. Zhang et al. (2013) found that\neffect of the radiative losses can be significant in these\noscillation. Recently, Ballester (2016) have found that a\ntemporalvariationofthe backgroundtemperaturein com-\nbination with radiative losses can produce period shifts\nand damping of the slow modes in a prominence. These\nimprovements to the model will be a topic for future re-\nsearch.\nAcknowledgements. This paper was inspired by two ISSI\nworkshops, Bern, Switzerland, March and November 2015.\nThe authors acknowledge support from the International\nSpace Science Institute (ISSI) to the Team 314 on “Large-\nAmplitude Oscillation in prominences” led by M. Luna. MR\nacknowledges the financial support from the Science and\nTechnology Facilities Council (STFC). M. Luna acknowl-\nedges the support by the Spanish Ministry of Economy and\nCompetitiveness through projects AYA2011-24808, AYA2010 -\n18029 and AYA2014-55078-P. This work contributes to the de-\nliverables identified in FP7 European Research Council gran t\nagreement 277829, “Magnetic Connectivity through the Sola r\nPartially Ionized Atmosphere” (PI: E. Khomenko).\nReferences\nAbramowitz, M. & Stegun,I. 1972, Handbookof Mathematical\nFunctions (National Bureau of Standards)\nAntiochos, S. K. & Klimchuk, J. A. 1991, ApJ, 378, 37212 M. S. Ruderman & M. Luna: Damping by mass accretion\nAntiochos, S. K., MacNeice, P. J., & Spicer, D. S. 2000, ApJ,\n536, 494\nBallester, J. L. 2016, A&A\nBi, Y., Jiang, Y., Yang, J., et al. 2014, ApJ, 790, 100\nJing, J., Lee, J., Spirock, T. J., & Wang, H. 2006, Sol. Phys.,\n236, 97\nJing, J., Lee, J., Spirock, T. J., et al. 2003, ApJ, 584, L103\nKarpen, J. T. & Antiochos, S. K. 2008, ApJ, 676, 658\nKarpen,J.T.,Antiochos,S.K.,Hohensee,M.,Klimchuk,J.A .,\n& MacNeice, P. J. 2001, ApJ, 553, L85\nKarpen, J. T., Antiochos, S. K., Klimchuk, J. A., & MacNeice,\nP. J. 2003, ApJ, 593, 1187\nKarpen, J. T., Tanner, S. E. 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F., & Ichimoto, K. 2014, ApJ,\n795, 130\nTripathi, D., Isobe, H., & Jain, R. 2009, Space Sci. Rev., 149 ,\n283\nVrˇ snak, B., Veronig, A. M., Thalmann, J. K., & ˇZic, T. 2007,\nA&A, 471, 295\nXia, C., Chen, P. F., Keppens, R., & van Marle, A. J. 2011,\nApJ, 737, 27\nXia, C., Keppens, R., Antolin, P., & Porth, O. 2014, ApJ, 792,\nL38\nZhang, Q. M., Chen, P. F., Xia, C., & Keppens, R. 2012, A&A,\n542, A52\nZhang, Q. M., Chen, P. F., Xia, C., Keppens, R., & Ji, H. S.\n2013, A&A, 554, A124" }, { "title": "2106.03332v1.Voltage_control_of_damping_constant_in_magnetic_insulator_topological_insulator_bilayers.pdf", "content": "arXiv:2106.03332v1 [cond-mat.mes-hall] 7 Jun 2021Voltage-control of damping constant in\nmagnetic–insulator/topological–insulator bilayers\nTakahiro Chiba,1Alejandro O. Leon,2and Takashi Komine3\n1)National Institute of Technology, Fukushima College, 30 Na gao, Kamiarakawa, Taira, Iwaki, Fukushima 970-8034,\nJapan\n2)Departamento de F´ ısica, Facultad de Ciencias Universidad Tecnol´ ogica Metropolitana, Las Palmeras 3360, ˜Nu˜ noa 780-0003,\nSantiago, Chile\n3)Graduate School of Science and Engineering, Ibaraki Univer sity, 4-12-1 Nakanarusawa, Hitachi, Ibaraki 316-8511,\nJapan\n(Dated: 8 June 2021)\nThe magnetic damping constant is a critical parameter for ma gnetization dynamics and the e fficiency of memory devices\nand magnon transport. Therefore, its manipulation by elect ric fields is crucial in spintronics. Here, we theoretically\ndemonstrate the voltage-control of magnetic damping in fer ro- and ferrimagnetic–insulator (FI) /topological–insulator\n(TI) bilayers. Assuming a capacitor-like setup, we formula te an effective dissipation torque induced by spin-charge\npumping at the FI/TI interface as a function of an applied voltage. By using rea listic material parameters, we find that\nthe effective damping for a FI with 10 nm thickness can be tuned by one order of magnitude under the voltage with\n0.25 V . Also, we provide perspectives on the voltage-induce d modulation of the magnon spin transport on proximity-\ncoupled FIs.\nV oltage or electric-field control of magnetic properties\nis fundamentally and technologically crucial for energeti -\ncally efficient spintronic technologies,1,2such as magnetic\nrandom-access memories (MRAMs),3spin transistors,4,5and\nspin-wave-based logic gates.6In these technologies, voltage-\ncontrol of magnetic anisotropy (VCMA) in thin ferromag-\nnets7–9promises energy-efficient reversal of magnetization by\na pulsed voltage10–12and manipulation of propagating spin\nwaves with lower power consumption.13The control of mag-\nnetic damping is also highly desirable to increase the per-\nformance of spintronic devices. For instance, low magnetic\ndamping allows small critical current densities for magnet i-\nzation switching and spin-wave excitation by current-indu ced\nspin-transfer14and spin-orbit torques.15,16On the other hand,\na high magnetic damping can be beneficial in reducing the\ndata writing time in MRAM devices. For magnonic devices,\nmagnetic damping is a key factor because it governs the life-\ntime of spin waves or magnons as information carriers.17Even\nif the magnetic damping is a vital material parameter that go v-\nerns magnetization dynamics in several spintronic devices , its\nvoltage-control is not fully explored except for a few exper i-\nments with ferro- and ferrimagnets.18–22\nThe main origin of magnetic dissipation is the spin-orbit\ninteraction (SOI), which creates relaxation paths of the sp in-\nangular momentum into conduction electrons and the lattice .\nHence, potential candidates to achieve the voltage-contro l of\nmagnetic damping are magnetic materials and /or strong SOI\nsystems. Three-dimensional topological insulators (3D TI s),\nsuch as Bi 2Se3, are characterized by band inversion due to\na strong SOI23,24and possess an ideally insulating bulk and\nspin–momentum locked metallic surface states. Recently,\nBi2−xSbxTe3−ySey(BSTS)25and Sn-doped Bi 2−xSbxTe2S26\nhave been reported to be ideal 3D TIs with two-dimensional\n(2D) Dirac electrons on the surface and a highly insulating\nbulk. For spintronics, the interface between a ferromagnet\nand a TI can enhance the magnitude of both spin and charge\ncurrents.27,28Some experiments reported29–32the spin-chargeconversion at room temperature33,34in a bilayer of TI/ferro-\nand ferrimagnetic-insulator (FI) such as Y 3Fe5O12(YIG) with\nvery low Gilbert damping constant ( α). An essential feature\nof the FI/TI bilayer is that the TI bulk behaves as a semicon-\nductor, enabling the control of the surface carrier density by\na voltage.35Also, magnetically doped TI exhibit VCMA.12,36\nHence, TIs are a promising candidate to achieve the voltage-\ncontrol of magnetic damping.\nIn this work, we theoretically demonstrate the voltage-\ncontrol of magnetic damping in FI /TI bilayers. We formulate\nan effective dissipation torque induced by spin-charge pump-\ning at the FI/TI interface as a function of a gate voltage VG.\nOur main result is that the voltage changes the e ffective damp-\ning by one order of magnitude for a FI with a perpendicular\nmagnetization configuration and 10 nm thickness. Also, we\nprovide perspectives on the modification of magnon scattrin g\ntime in a FI-based magnonic device.\nTo study the effective damping torque, we consider 2D\nmassless Dirac electrons on the TI surface with the magnetic\nproximity effect,29–32i.e., coupled to the magnetization of an\nadjacent FI. The exchange interaction between the surface\nelectrons and the FI magnetization is modeled by a constant\nspin splitting along the magnetization direction with unit vec-\ntorm=M/Ms(in which Mis the magnetization vector with\nthe saturation magnetization Ms).37Then, the following 2D\nDirac Hamiltonian provides a simple model for the FI /TI in-\nterface state38:\nˆH=vFσ·(ˆp׈z)+∆σ·m, (1)\nwhere/planckover2pi1is the reduced Planck constant, vFis the Fermi veloc-\nity of the Dirac electrons at zero applied voltage, ˆp=−i/planckover2pi1∇\nis the momentum operator, {ˆx,ˆy,ˆz}are the unit vectors along\nthe respective Cartesian axes, σ=(σx,σy,σz) is the vector\nof Pauli matrices for the spin, and ∆is the exchange interac-\ntion constant. For simplicity, we ignore here the particle– hole\nasymmetry and the hexagonal warping e ffect in the surface\nbands. Also,∆andvFare assumed to be temperature indepen-2\nNFUBMVG\nd\n5* m\n'* \nxz\ny\nNFUBM VG\n5* '* ). ). µL\tB\n \n\tC\nµRBeffh(t)\nd\nFIG. 1. (a) Schematic geometry (side view) of a capacitor-li ke de-\nvice comprising a ferromagnetic insulator (FI) film (with th ickness d)\nsandwiched by a TI and a normal metal as a top electric gate wit hVG.\nThe yellow line corresponds to the TI surface state. The red a rrow\ndenotes the precessional magnetization directions in the F MR driven\nby static ( Beff) and oscillating ( h(t)) magnetic fields. (b) Schematic\ngeometry of a transistor-like device comprising a FI film san dwiched\nby a TI and a normal metal. The red wave arrow represents a magn on\ncurrent driven by the di fference in spin accumulation ( µL−µR) in the\nattached left and right heavy metal (HM) leads.\ndent.40Note that we operate in the weak magnetic coupling\nlimit, and therefore self-consistent treatment for the ind uced\ngap (∆)39is not necessary.\nLet us begin by calculating the dissipation torque induced\nby the spin-charge pumping28,41–43of a dynamic magnetiza-\ntion in FI/TI bilayers. A precessing magnetization, driven by\nferromagnetic resonance (FMR), as shown in Fig. 1, can be\nregarded as an effective vector potential Aeff(t)=∆/(evF)ˆz×\nm(t) with the electron charge −e(e>0), which drives a\ncharge current via an e ffective electric field Eeff=−∂tAeff\n(see the supplementary material), i.e.,\nJP=∆\nevF/parenleftBigg\nσAH∂m\n∂t−σLˆz×∂m\n∂t/parenrightBigg\n, (2)\nwhereσLandσAH are longitudinal and transverse\n(anomalous-Hall) conductivities, respectively, and depe nd on\nthez−component of the magnetization ( mz)38. From the\nHamiltonian ( 1), the velocity operator ˆv=∂ˆH/∂ˆp=vFˆz×σ\ndepends linearly on σ. Therefore, the nonequilibrium spin\npolarization µP(in units of m−2) is a linear function of the\ncharge current JPon the TI surface, i.e., µP=ˆz×JP/(evF).This nonequilibrium spin polarization µPexerts a dissipa-\ntion torque on the magnetization, TSP=−γ∆/(Msd)µP×m,\nnamely\nTSP=(−αAHmz+αLm×)/parenleftBigg∂m\n∂t−∂mz\n∂tˆz/parenrightBigg\n, (3)\nwith\nαL(AH)=γ∆2\ne2v2\nFMsdσL(AH), (4)\nwhereγis the gyromagnetic ratio and dis the thickness of the\nFI layer. Equation ( 3) is equivalent to the charge-pumping-\ninduced damping-like torque that Ndiaye et al. derived using\nthe Onsager reciprocity relation for a current-induced spi n-\norbit torque.43The first term in Eq. ( 3) originates from the\nmagnetoelectric coupling (the Chern–Simons term)37,44and\nrenormalizes the gyromagnetic ratio. By using parameters\nlisted in TABLE I,∆=40 meV , and d=10 nm,αAH≈10−4is\nestimated even by using σAHat 0 K as the upper value.38Thus,\nwe disregard the renormalization of the gyromagnetic ratio . In\ncontrast, the second term in Eq. ( 3) stems from the Rashba–\nEdelstein effect due to the spin-momentum locking on the TI\nsurface41and contributes to magnetic damping. Since we are\ninterested in voltage-control of magnetic damping, we here -\nafter focus onαLin this study.\nAccording to Eq. ( 4), the electric field e ffect on the con-\nductivityσLcan be used to control the magnetic dissipation.\nNamely, the voltage–induced change of the interfacial dens ity\nof states inσLrenders the TI a more or less e fficient spin sink.\nThe damping enhancement αLdepends on the chemical po-\ntentialµ, measured from the original band-touching (Dirac)\npoint. At room temperature, or below it, the thermal energy i s\nmuch smaller than the Fermi one, kBT≪EF, with Tthe tem-\nperature and kBthe Boltzmann constant. Then, we can use the\nfollowing Sommerfeld expansion of the chemical potential µ\nµ(T)≈EF1−π2\n6/parenleftBiggkBT\nEF/parenrightBigg2 (5)\nwith the voltage-dependent Fermi energy,12EF=µ(0), given\nby\nEF(VG)=/planckover2pi1vF/radicalBigg\n4π/parenleftBigg\nnint+∆2\n4π(/planckover2pi1vF)2+ǫ\nedVG/parenrightBigg\n, (6)\nwhereǫis the permittivity of a FI and nint=/parenleftBig\nE2\nF(0)−∆2/parenrightBig\n//parenleftBig\n4π/planckover2pi12v2\nF/parenrightBig\nis the intrinsic carrier density, i.e., at\nVG=0. Note that we can define a voltage-dependent sur-\nface electron density nV(VG)≡nint+ǫVG/(ed)that shows\nthe underlying mechanism behind the voltage-control of in-\nterfacial phenomena in insulating bilayers with surface ca r-\nriers, which goes beyond topological materials. Namely, a\nvoltage increases or decreases the e ffective electron density\nand therefore enhances or weakens all e ffects that depend on\nthis density, including isotropic45and anisotropic46exchange\ninteractions, emergence of magnetization in metals,47perpen-\ndicular magnetic anisotropy,3,9and spin-orbit torques.12The3\nЋ-\u0001\tʷ\u0012\u0011 \u000e\u0014 \n\tB\n \tC\n \nЋ-\u0001\tʷ\u0012\u0011 \u000e\u0014 \n\tD\n \nEF\n/g21/g39 \n#VML\u0001DPOEVDUJPO\n#VML\u0001WBMFODF EF2Ec\nFIG. 2. Effective damping enhancement αLof a TI/FI bilayer as functions of VGandTforEF(VG=0)=140 meV: (a) mz=1, (b) mz=0.\n(c) V oltage modulation of EFin a TI. Insets represent schematic of massless (dashed line ) and massive (solid line) surface state dispersions in\nthe bulk band gap. In these graphs, we use parameters listed i n Table I,∆= 40 meV , and d=10 nm for a FI thickness. The details of the\ncalculations are given in the text.\nvoltage-generated change in the surface density is equival ent\nto an interfacial Fermi energy shift. In this work, we predic t\nthat the spin-charge pumping e fficiency is also modulated, an\neffect that may also appear in usual FI |normal metal bilayers\nsince the spin-mixing conductance depends on the electroni c\ndensity.48\nWe investigate the e ffect of electric-gate on the e ffective\ndampingαLso that we assume hereafter that the low-energy\nDirac Hamiltonian ( 1) is an accurate description for a mo-\nmentum cut kc=/radicalbig\nE2c−∆2/(/planckover2pi1vF), in which 2 Ecis the bulk\nbandgap of TIs50(see Fig. 2(c)). Sufficiently far from the\nDirac point (/planckover2pi1τ/EF≪1,τis the transport relaxation time),\nthe electron scattering can be treated by the first Born appro x-\nimation.51With this, the longitudinal conductivity reads52\nσL=e2\n2h/integraldisplayEc\n−EcdEkEkτ(Ek,T)\n/planckover2pi1E2\nk−∆2m2\nz\nE2\nk+3∆2m2z/parenleftBigg\n−∂fFD\n∂Ek/parenrightBigg\n,(7)\nwhere fFD=/bracketleftbigexp{(Ek−µ)/(kBT)}+1/bracketrightbig−1is the Fermi-Dirac\ndistribution, the energy Ekis the eigenvalue of Eq. ( 1), and\nτ(Ek,T) is the transport relaxation time of massless Dirac\nelectrons within the Born approximation for impurity and\nphonon scatterings. By applying the Matthiessen rule,\n1\nτ(Ek,T)=Ek(a+bkBT), (8)\nwhere a=nV2\n0//parenleftBig\n4/planckover2pi13v2\nF/parenrightBig\n(in units of eV−1s−1) parameterize\ncontribution of the impurity scattering,53,54nis the impurity\nconcentration, and V0is the scattering potential. Also, contri-\nbution to the transport relaxation time from the phonon scat -\ntering53,54can be approximated by b=D2\n0//parenleftBig\n4/planckover2pi13v2\nFρtsv2\nL/parenrightBig\n(in\nunits of eV−2s−1), whereρis the mass density of the quintu-\nple layer (QL) in the TI crystal structure, tsis the thickness of\none atomic layer in 1 QL of TIs, vLis the longitudinal phonon\nvelocity, and D0is the deformation potential constant.\nFigures 2(a) and (b) show the VGandTdependence of the\neffective damping enhancement αLfor out-of-plane ( mz=1)\nand in-plane ( mz=0) magnetization configurations, respec-\ntively. Also, Fig. 2(c) illustrates the voltage modulation ofTABLE I. Material parameters for the TI /FI bilayer.\nSymbol Value Unit\naBSTS Fermi velocity vF 4.0×105ms−1\naBSTS bulk band gap 2 Ec 300 meV\nbYIG gyromagnetic ratio γ 1.76×1011T−1s−1\nbYIG Gilbert damping constant α 6.7×10−5\nbYIG saturation magnetization Ms 1.56×105Am−1\ncYIG relative permittivity ǫ/ǫ 0 15\naReference 25,bReference 33,cReference 49.\nEFin TI. The bulk damping constant can be influenced by\nmaterial and device parameters, such as SOI and magnetic\nanisotropies34. However, we predict the voltage-modulation\nof the damping enhancement by spin-charge pumping. There-\nfore, our results are independent of the intrinsic dissipat ion\nmechanisms. At the FI /TI interface, orbital hybridization be-\ntween TI and the 3 dtransition metal in FI, such as YIG, de-\nforms the TI surface states, which might shift the Dirac poin t\nto the lower energy and lift up EF,55so that we consider rel-\natively high value EF(VG=0)=140 meV with the corre-\nsponding carrier density of the order of 1012cm−2. Also,∆is\nused within the values reported experimentally in FI-attac hed\nTIs.56,57For impurity parameters, we use n=1011cm−2and\nV0=0.15 keVÅ2based on an analysis of the transport prop-\nerties of a TI surface.52We could not find estimates of the\nphonon scattering for BSTS in the literature so that we adopt\nthose of non-substituted Bi 2Te3being vL=2.9×105ms−1,\nD0=35 eV , ts=0.16 nm, andρ=7.86×103kgm−3\nin Ref. 58. These scattering parameters describe a relatively\nclean interface with the sheet resistance ∼1 kΩ, which is one\norder less than that of experiments. In Figs. 2(a) and (b),αL\nmonotonically decreases with increasing Tat a fixed VGwhile\nit has peaks for changing VGat a fixed T>0 (see also the\ninset of Fig. 3). This feature reflects thermal excitation of sur-\nface carriers into the bulk states ( Ek>Ec), reducing the spin-\ncharge-pumping contribution. With the out-of-plane config u-4\nration,αLcan be tuned by one order of magnitude under the\nvoltage, whileαLchanges by less than a factor two with the in-\nplane state, which suggests that the out-of-plane configura tion\nis superior in controllability. The calculated Tdependence of\ndamping enhancement at VG=0 for the in-plane configura-\ntion agrees with a few experiments with the FI /TI bilayer.31,59\nNote that at much lower than EF(VG=−250 mV)≈40 meV ,\nour calculation with the in-plane configuration breaks down\nbecause of the finite level broadening due to the higher-orde r\nimpurity scattering.60The VG–dependent FMR is character-\nized by the Landau-Lifshitz-Gilbert theory in the suppleme n-\ntary material.\nThe electric manipulation of magnon spin transport is a rel-\nevant topic in spintronics. For example, in YIG with an in-\njector and a detector Pt contact, changes of the magnon spin\nconductivity can be obtained by using a third electrode that\nchanges the magnon density,61–63potentially providing a func-\ntionality similar to the one a field-effect transistors . Damping\ncompensation by current-driven torques64,65in magnetic het-\nerostructures also influences magnon transport. Here, we pr o-\nvide a perspective on the electric-field-induced modulatio n of\nmagnon scattering time, τm. Magnons can be injected and\ndetected by their interconversion with charge currents in a dja-\ncent heavy metals (HMs) through the direct and inverse spin-\nHall effects.33Similar to charge transport induced by an elec-\ntrochemical potential gradient, a magnon spin current can b e\ndriven by the gradient of a magnon chemical potential inject ed\nby an external source.66,67Magnon transport through a FI can\nbe controlled by the gate voltage that modulates the e ffective\ndamping in Eq. ( 4).\nSo far, the magnon spin transport in the FI /TI bilayer lacks\nmicroscopic theory with few exceptions.68,69However, from\nthe bulk of magnon spin transport,66the control ofτmresults\nin the modification of all transport properties, including t he\nmagnon spin conductivity. In the presence of a TI contact, in -\nterfacial magnons are scattered by conducting Dirac electr ons\non the TI surface.70Considering a very thin ferromagnet that\ncan be modeled by a 2D magnet. The inset of Fig. 3shows\nthat the damping enhancement is at least one order of magni-\ntud larger than the bulk one of YIG.33,34Accordingly, let us\nassume that interfacial magnons are absorbed by transferri ng\ntheir energy and angular momentum to Dirac electrons at a\nrate 1/τm∝αL.66While there is no know microscopic expres-\nsion for the magnon spin conductivity in the present system,\nbulk magnon transport obeys the relationship σm∝τm,61,62\nwhereσmis the magnon spin conductivity. In our case, the\nscattering timeτmis dominated by the magnon-relaxation pro-\ncess into the FI/TI interface. To estimate an e ffect of electric-\ngate on the magnon spin transport, we define the modulation\nefficiency\nηm=τm(VG)−τm(Vmax)\nτm(Vmax)=αL(Vmax)\nαL(VG)−1, (9)\nwhere Vmax(≈−68 mV for Fig. 3) gives the maximum value\nofαL(and therefore the minimum value of τm). In principle,\nτmdepends on VGthrough not onlyαLbut also via magnon\ndispersion relation, /planckover2pi1ωq,66including a VG–dependent mag-\nnetic anisotropy. However, this VG–dependence is quite smallFIG. 3. Modulation e fficiency (ηm) as a function of the gate volt-\nage. Inset shows the corresponding behavior of the e ffective damp-\ning enhancementαL. In these graphs, we use parameters listed in\nTable I,∆= 40 meV , and d=2 nm for a FI thickness. We also set\nEF(VG=0)=140 meV and T=300 K.\neven for a FI with 2 nm thickness (see the supplementary ma-\nterial), so that we disregard the influence of the magnon gap\nin the following calculation. Figure 3shows VG-dependence\nof the modulation e fficiency at room temperature in which the\nstrongly nonlinear behavior is interpreted as follows. Dow n\ntoVG≈−130 mV,αLis affected by the thermal excitation of\nsurface carriers, which makes a peak around VG≈−70 mV.\nFrom−130 mV to−250 mV, the thermal excitation is sup-\npressed, so thatαLmonotonically decreases with |VG|due to\nthe reduction of the Fermi surface. Hence, in this regime,\none can effectively modulate the magnon spin transport by the\nvoltage.\nIn summary, we have theoretically demonstrated the\nvoltage-control of magnetic damping in ferro- ferrimagnet ic\ninsulator (FI)/topological insulator (TI) bilayers. Assuming\na capacitor-like setup, we formulate an e ffective damping\ntorque induced by spin-charge pumping at the FI /TI interface\nas a gate voltage function. The presence of a perpendicular\nelectric field results in a shift of the Fermi level or, equiv-\nalently, a modified interfacial electron density, increasi ng or\ndecreasing the efficiency of the pumping process. We stud-\nied the consequences of this damping enhancement using re-\nalistic material parameters for FI and TI. We found that the\neffective damping with the out-of-plane magnetization con-\nfiguration can be modulated by one order of magnitude under\nthe voltage with 0.25 V . The present results motivate an ap-\nplication: the magnon scattering time can be tuned by a gate\nvoltage, potentially allowing for a magnon transistor type of\napplication. A complete quantitative description of the la t-\nter requires a microscopic theory of magnon spin transport\nin FI/TI bilayers, which might remain an unexplored issue.\nThe voltage-control of magnetic damping paves the way for\nlow-power spintronic and magnonic technologies beyond the\ncurrent-based control.\nSee the supplementary material for the calculation of the5\nspin-charge pumping in FI /TI bilayers, the characterization of\nthe FMR under several values of the applied voltage, the in-\nfluence of VG–dependence of the anisotropy in the magnon\ndispersion.\nWe thank Camilo Ulloa and Nicolas Vidal-Silva for fruitful\ndiscussions. This work was supported by Grants-in-Aid for\nScientific Research (Grant No. 20K15163 and No. 20H02196)\nfrom the JSPS and Postdoctorado FONDECYT 2019 Folio\n3190030.\nDATA AVAILABILITY\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable request.\n1H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. 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Rev. Lett. 117, 127202 (2016)." }, { "title": "1605.05211v1.Direct_observation_of_dynamic_modes_excited_in_a_magnetic_insulator_by_pure_spin_current.pdf", "content": " 1Direct observation of dynamic mode s excited in a magnetic insulator \nby pure spin current \n \nV. E. Demidov1*, M. Evelt1, V. Bessonov2, S. O. Demokritov1,2, J. L. Prieto3, M. \nMuñoz4\n, J. Ben Youssef5, V. V. Naletov6,7, G. de Loubens6, O. Klein8, M. Collet9, P. \nBortolotti9, V. Cros9 and A. Anane9 \n1Institute for Applied Physics and Center for Nanotechnology, University of Muenster, \n48149 Muenster, Germany \n2M.N. Miheev Institute of Metal Physics of Ural Branch of Russian Academy of \nSciences, Yekaterinburg 620041, Russia. \n3Instituto de Sistemas Optoelectr ónicos y Microtecnologa (UPM), Ciudad \nUniversitaria, Madrid 28040, Spain \n4IMM-Instituto de Microelectrónica de Madrid (CNM-CSIC), PTM, E-28760 Tres \nCantos, Madrid, Spain \n5Laboratoire de Magnétisme de Bretagne CNRS, Université de Br etagne Occidentale, \n29285 Brest, France \n6Service de Physique de l’ État Condensé, CEA, CNRS, Université Paris-Saclay, CEA \nSaclay, 91191 Gif-sur-Yvette, France \n7Institute of Physics, Kazan Federal University, Kazan 420008, Russian Federation \n8INAC-SPINTEC, CEA/CNRS and Univ. Gre noble Alpes, 38000 Grenoble, France \n9Unité Mixte de Physique CNRS, Thales, Un iv. Paris Sud, Université Paris-Saclay, \n91767 Palaiseau, France \n \n 2Abstract: \nExcitation of magnetization dynamics by pur e spin currents has been recently \nrecognized as an enabling mechanism for spintronics and magnonics, which allows \nimplementation of spin-torque devices based on low-damping insulating magnetic \nmaterials. Here we report the first spat ially-resolved study of the dynamic modes \nexcited by pure spin current in nanometer-thick microscopic insulating Yttrium Iron \nGarnet disks. We show that these modes exhibit nonlinear self-b roadening preventing \nthe formation of the self-localized magnetic bullet, which plays a crucial role in the \nstabilization of the single-mode magnetization oscillations in all-metallic systems. This peculiarity associated with the efficien t nonlinear mode coupling in low-damping \nmaterials can be among the main factors govern ing the interaction of pure spin currents \nwith the dynamic magnetization in high-quality magnetic insulators. \n \n 3One of the decisive advantages of pure spin currents for the emerging \ntechnologies, such as spintronics1-3 and magnonics4-6, is the possibility to excite the \nmagnetization dynamics in insulating magnetic materials by the spin-transfer torque7,8. \nThe breakthrough idea that the spin-transfer to rque can be utilized to induce spin waves \nin magnetic insulators suggested in the pioneering work by Kajiwara et al. (Ref. 7) had \nan enormous impact on the developments in the physics of magnetism and revived \ninterest to magnetic insulators, such as Y ttrium Iron Garnet (YIG). The main advantage \nof this material is the unmatched small dynamic magnetic damping, which is expected \nto result in more efficient operation of spin-torque devices by enabling significant \nreduction of the density of the driving current necessary for the onset of current-induced \nauto-oscillations. Although th is advantage should greatly simplify the implementation \nof YIG-based spin-torque device s, it took many years of inte nse research to achieve the \nspin-current induced excitation of cohere nt magnetization dynamics in this material9-10. \nIn contrast, significantly mo re rapid progress has been achieved in the studies of \nall-metallic spin-current driven systems, which has led in the recent years to the \ndemonstration of a large variety of nano-de vices exhibiting highl y coherent magnetic \noscillations driven by the pure spin current11-18. This progress was gr eatly facilitated by \nthe flexible geometry of spin-current driven systems enabling the direct magneto-optical \nimaging of the current-induced dynamics. In particular, the spatially-resolved \nmeasurements have shown that the significant role in the stabilization of the single-mode current-induced oscillati ons is played by the nonlin ear spatial self-localization \neffects resulting in the forma tion of the so-called bullet mode\n11,19, which is a two-\ndimensional standing analogue of a spin-wave soliton20. \nSince the dynamic nonlinearity of the spin system largely determines the \nbehaviors of the dynamic magnetization under th e influence of the spin current, the \npeculiarities of the nonlinear response of low-damping YIG can be among the most 4important factors affecting the current -induced excitation of the magnetization \ndynamics in this material. In fact, it is long-known that the low damping in YIG \nfacilitates the nonlinear couplin g between dynamic magnetic modes21. This \nphenomenon was found to strongly affect the formation of spin-wave solitons22 and \nresult in a suppression of the nonlin ear spatial self-loc alization phenomena23, which \nplay a crucial role in spin -torque driven nano-systems17. \nHere we study experimentally the eff ects of the dynamic nonlinearity on the \nspatial characteristics of dynamic modes exci ted by the pure spin current in microscopic \nYIG disks. By using the direct spatially-reso lved imaging of the modes, we show that \nthe dynamic magnetization exhibits strong spatia l localization at the onset of the auto-\noscillations. However, just above the onset , the amplitude of the auto-oscillations \nrapidly saturates and the excited mode experiences a spatial broadening. This \nphenomenon is opposite to the nonlinear self -localization observed in all-metallic \nsystems and can be the main factor determ ining the response of the magnetization in \nlow-damping materials to the spin current. We attribute these behaviors to the efficient \nnonlinear mode coupling, which prevents the growth of the amplitudes of the dynamic \nmagnetization to the level necessary for th e onset of nonlinear se lf-localization and \nformation of the bullet mode. \n \nResults \nTest devices. The schematic of our experiment is shown in Fig. 1. The studied \ndevice consists of a YIG(20 nm)/Pt(8 nm) disk with a 2 m diameter (details about the \npreparation of the YIG films can be found in Ref. 24). The Pt la yer is electrically \ncontacted by using two Au(80 nm) electrodes with a 1 m wide gap between them. By 5applying an electric voltage betw een the electrodes, we inject the dc electric current into \nthe Pt layer. Because of the large difference in the electric conductance of the electrodes \nand the Pt film, the current is injected into the Pt layer only in the electrode gap area. \nDue to the spin-Hall effect25,26, the in-plane electri cal current in the Pt film is converted \ninto a transverse spin accumulation. The asso ciated pure spin current is flowing in the \nout-of-plane direction and is injected into the YIG film7 resulting in a spin-transfer \ntorque (STT)27,28 on its magnetization M. The device is magnetized by the in-plane \nstatic magnetic field H0 applied perpendicular to the di rection of the dc current flow. \nFor positive currents, as defined in Fig. 1, the polarization of the spin current corresponds to the STT compensating the dyna mic damping in the magnetic film. For \nsufficiently large dc currents, the damping becomes completely compensated, which results in the excitation of magnetic auto-oscillations\n10. \n \nMagnetooptical measurements. To observe the oscillations induced by the spin \ncurrent we use micro-focus Br illouin light scattering (BLS)29 spectroscopy, which \nenables the spectrally- and the spatially -resolved measurements of the dynamic \nmagnetization. We focus probing laser light into a diffraction-limited spot on the \nsurface of YIG (Fig. 1) and analyze the spectru m of light inelastically scattered from the \nmagnetization oscillations. The resulting BL S signal at the selected frequency is \nproportional to the intensity of the dynamic magnetization at this frequency, at the \nlocation of the laser spot. The wavelength of the laser is chosen to be 473 nm, which \nprovides high sensitivity of the method for m easurements with ultra-thin YIG. The \npower of the probing light is as low as 0.05 mW, which gua rantees negligible laser-\ninduced heating of the sample. 6In Fig. 2a we show BLS spectra recorded by placing the probing laser spot in the \nmiddle of the YIG disk for different values of the dc current I. The spectrum shown by \nsolid black symbols recorded for I = 0 characterizes the magnetic excitations existing at \nroom temperature due to the thermal fluctuat ions. Similar to the all-metallic systems11, \nthe application of the dc current results in the gradual enhancement of the magnetic \nfluctuations followed at I = IC 8 mA by an abrupt emergen ce of a narrow intense peak \n(blue squares in Fig. 2a) marking the onset of the current-induced auto-oscillations of \nthe YIG magnetization. The intensity of the au to-oscillation peak exceeds that of the \nthermal fluctuations at I = 0 by about two orders of ma gnitude. Further increase in I \nresults in the lowering of the central frequency of the peak and in its spectral broadening \n(red diamonds in Fig. 2a). \nTo obtain information about the dynamical modes contributing to the current-\ninduced auto-oscillations, we perform two-dimensional mapping of the dynamic \nmagnetization in the YIG disk by rastering the probing laser spot in the two lateral \ndirections. In Fig. 2b we show a typical spatial map obtained for I = IC+1 mA. As seen \nfrom these data, the auto-oscillations are strongly localized in the y-direction in the area \ncorresponding to the gap between the electrode s, whose edges are marked in Fig. 2b by \nthe horizontal lines. This localization can be attributed to the formation of an effective confining potential due to the joint effects of the spin- and elec trical currents, as \ndiscussed below. In contrast, there is no pr onounced localization of the auto-oscillations \nin the x-direction. They clearly extend over the entire active device area, where the \ninjection of the spin current takes place. \nFigure 3 illustrates the spatial localization of the auto-oscillations for different \nvalues of the dc current. x-profiles of the oscillation in tensity recorded at different I \n(Fig. 3a) show that the auto-oscillations do not always occupy the entire active device 7area. At currents close to IC, they are strongly localized in the middle of the gap. \nHowever, with the increase in I, they rapidly expand in the x direction. This process is \ncharacterized in detail in Fig. 3b, which show s the current dependence of the full width \nat half maximum (FWHM) of the x-profiles. This width lin early increases from 0.5 m \nto about 1.4 m for I varying from IC to IC+1 mA and then saturate s at the value of 1.5 \nm. \nWe emphasize that these behaviors are in contradiction with the typical \nmanifestations of the dynamic magnetic nonlinea rity, which is expected to result in the \nspatial self-localizati on of the intense oscillations and lead to a formation of the spin-\nwave bullet19 experimentally observed in all-metal lic systems driven by the pure spin \ncurrent11,17. The absence of the self-localization in YIG can be attributed to a nonlinear \nlimitation of the amplitude of the auto-oscillations. Indeed, if this amplitude is limited by any other nonlinear phenomenon, the self -localization mechanism requiring the \ndynamic magnetization to be comparable with the static magnetization can be \nsuppressed. \nThis scenario is supported by the data of Fig. 3c. The maximum intensity of the \nauto-oscillations (up triangles in Fig. 3c) saturates immediately after the onset and then \nstays nearly constant over the entire used range of I. In contrast, the integral intensity \n(down triangles in Fig. 3c) exhi bits a gradual increase up to I = 9.2 mA. This indicates \nthat the amplitude of the dynamic mode excite d at the onset cannot grow above a certain \nlevel and that the further increase in I results in the energy flow into other dynamical \nmodes leading to the formation of a str ongly nonlinear spatially-extended mode, which \ncannot be treated as a linear combin ation of the eigenmodes of the system\n23. These \nbehaviors can be attributed to the efficient nonlinear mode coupling stimulated by the 8small dynamic damping in YIG. We emph asize that the same mechanism was \npreviously found to result in the formation of traveling dark solitons in a medium \ncharacterized by the attractiv e nonlinearity, where the se lf-localization effects are \nexpected to lead to the fo rmation of bright solitons22. \nAccording to the described scenario, the current-induced auto-oscillations are also \nexpected to demonstrate a spatial broadening in the y-direction. The corresponding \nexperimental data are shown in Fig. 4. We note that the y-profiles of the auto-\noscillations (Fig. 4a) are mostly determined by the confining potential, which prevents \nthe extension of the dynami c excitations outside the re gion of the gap between the \nelectrodes. Because of the limite d spatial resolution of the used technique, which can be \nestimated as 240-250 nm, one cannot quantit atively analyze the broadening of the y-\nprofiles with the increase in the dc current. However, from Fig. 4b it is seen, that the y-\nwidth also tends to increase with increasing I. \n \nDiscussion \nFinally, we address the nature of the lo calized dynamic mode, wh ich is excited in \nthe YIG disk at I = IC before the nonlinear broadening takes place. To understand why \nthis mode shows a strong localization at the center of the disk, we perform \nferromagnetic-resonance (FMR) measurements ba sed on the excitation of the system by \na dynamic magnetic field created by an additional 5 m wide stripe antenna aligned \nparallel to the gap between the electrodes (s ee also Supplementary Figure 1). In Figs. 5a \nand 5b we show the spatial maps of the field-driven FMR mode recorded by BLS \nwithout any dc current in Pt and at I = 7 mA < IC, respectively. The maps clearly \ndemonstrate that the FMR mode initially spread over the entire disk becomes localized 9at its center when the dc curr ent is applied. We emphasize that the measurements were \nperformed at sufficiently small microwav e powers to exclude any nonlinear self-\nlocalization effects30. Therefore, the observed localizati on can only be attributed to the \njoint effect of the Oersted field of th e dc current, the local reduction of the \nmagnetization due to the Joule heating, and the partial compensation of the dynamic \ndamping by the pure spin current injected into the YIG film. \nTo prove this assumption we perform micromagnetic simulations using the \nsoftware package MuMax3 (Ref. 31) (see Me thods for details). The obtained spatial \nmaps of the intensity of the dynamic magne tization in the FMR mode are shown in \nFigs. 5c and 5d for the cases of I = 0 and 7 mA, respectively. As seen from the \ncomparison with the experimental data (Fi g. 5a and 5b), the simulations reproduce well \nthe localization behavior and clear ly show that the effects of the driving current result in \na shrinking of the FMR mode in both lateral directions resulting in the formation of a \nstrongly localized linear mode of the system. It is this mode that is most probably first \nexcited by the pure spin at the onset of th e auto-oscillations (s ee corresponding profiles \nin Figs. 3a and 4a). It is to be noticed that these current-induced localization effects \ncounteract the nonlinear broadening. Therefore, in a properly designed system, they can \nbe utilized to suppress this detrimental phenomenon. \nIn conclusion, we demonstrate that th e dynamic nonlinearity of the low-damping \nYIG films leads to a nonlinear self-broad ening of the current -induced oscillations \ninstead of the nonlinear self -localization typical for me tallic ferromagnetic films \ncharacterized by a relatively large dynami c damping. Since the self-localization \nmechanisms play a crucial role for the stab ilization of the singl e-mode current-induced \nauto-oscillations, in order to achieve highly-coherent oscillat ions in the YIG films, it is \nnecessary to utilize additiona l confining mechanisms, such as the patterning of the film 10into a nano-size element or creation of an additional confining potential. Our results \nshine light on the physical mechanisms responsible for the peculiarities of the interaction of magnetization in YIG with pure spin currents and create a base for implementation of efficient insulato r-based spin-torque nano-devices. \n \nMethods \nSample fabrication. 20 nm thick YIG films were grown by the pulsed laser deposition \non Gadolinium Gallium Garnet (GGG) (111) substrates\n24. The 8 nm thick layer of Pt \nwas deposited using dc magnetron sputteri ng. The dynamical properties of bare YIG \nfilms and YIG/Pt bilayers have been de termined from broadband FMR measurements. \nThe YIG/Pt microdiscs and the Au(80 nm)/ Ti(20 nm) electrodes were defined by e-\nbeam lithography. The system was insulated by a 300 nm thick SiO 2 layer, and a \nbroadband 5 m wide microwave antenna made of 250 nm thick Au was defined on top \nof the system using optical lithography. \nMagneto-optical measurements. Micro-focus BLS measurements were performed by \nfocusing light produced by a co ntinuous-wave single-frequenc y laser operating at the \nwavelength of 473 nm into a diffraction-limite d spot. The light scat tered from magnetic \noscillations was analyzed by a six-pass Fabry-Perot interferometer TFP-1 (JRS \nScientific Instruments, Switzerland) to ob tain information about the BLS intensity \nproportional to the square of the amplitude of the dynamic magnetization at the location \nof the probing spot. By raster ing the spot over the surface of the sample using a closed-\nloop piezo-scanner, two-dimensional maps of the dynamic magnetizat ion were recorded \nwith the spatial step size of 100 nm. The positioning system was stabilized by custom-\ndesigned software-controlled active feedback, providing long-term spatial stability of \nbetter than 50 nm. All measurements we re performed at room temperature. \nSimulations. The micromagnetic simulations were performed by using the \nsoftware package MuMax3 (Ref. 31). The co mputational domain with dimensions of \n220.02 m3 was discretized into 10 1010 nm3 cells. The FMR mode was excited by 11a 50 ps wide pulse of the out-of-plane magne tic field. The spatia l map of the dynamic \nmagnetization was reconstructed based on the Fourier analysis of the dynamic response \nof the magnetization. Standard valu e for the exchange stiffness of 3.66 10-12 J/m was \nused, while the value of th e saturation magnetization 4 M0=2.2 kG and the Gilbert \ndamping constant of 2 10-3 were determined from the FMR measurements. For the case \nof the disk free from the influence of the driving current, the saturation magnetization \nM, the damping constant , and the static field H0=1000 Oe were considered to be \nuniform across the disk area. To take into acco unt the effects of the driving current, the \nparameters M, , and H0 were reduced in the central rectangular region corresponding \nto the area of the gap between the electrodes. The reduction of M was taken from the \nresults of the FMR measurements (Supplem entary Figure 1), the reduction of was \ncalculated assuming the linear dependence of effective damping on the current strength, \nand the reduction of the field was cal culated by using the Ampere’s law. \n 12References \n \n1. Maekawa, S. Concepts in Spin Electronics (Oxford University Press, Oxford, 2006). \n2. Ralph, D. C. & Stiles, M. D. Spin transfer torques. J. Magn. Magn. Mater. 320, \n1190–1216 (2008). \n3. Locatelli, N., Cros, V., & Grollier , J., Spin-torque building blocks. Nature Mater. 13, \n11-20 (2014). \n4. Kruglyak, V. V., Demokritov, S. O., & Grundler, D. Magnonics. J. Phys. D: Appl. \nPhys. 43, 264001 (2010). \n5. Urazhdin, S. et al. , Nanomagnonic devices based on the spin-transfer torque. Nature \nNanotech. 9, 509-513 (2014). \n6. Chumak, A.V., Vasyuchka, V.I., Serga, A.A. & Hillebrands, B. Magnon spintronics. \nNature Phys. 11, 453-461 (2015). \n7. Kajiwara, Y. et al. Transmission of electrical signals by spin-wave interconversion in \na magnetic insulator. Nature 464, 262–266 (2010). \n8. Xiao J., Bauer, G.E.W, Spin-Wave Ex citation in Magnetic Insulators by Spin-\nTransfer Torque. Phys. Rev. Lett. 108, 217204 (2012) \n9. Hamadeh, A. et al. , Full Control of the Spin-Wave Damping in a Magnetic Insulator \nUsing Spin-Orbit Torque, Phys. Rev. Lett. 113, 197203 (2014). \n10. Collet, M., et al. , Generation of coherent spin-wave modes in Yttrium Iron Garnet \nmicrodiscs by spin-orbit torque. Nat. Commun. 7, 10377 (2016). \n11. Demidov, V. E. et al. Magnetic nano-oscillator driven by pure spin current. Nature \nMater. 11, 1028-1031 (2012). 1312. Liu, L., Pai, C.-F., Ralph, D. C., & Buhrma n, R. A. Magnetic Os cillations Driven by \nthe Spin Hall Effect in 3-Termin al Magnetic Tunnel Junction Devices. Phys. Rev. \nLett. 109, 186602 (2012). \n13. Liu, R. H., Lim, W. L., & Urazhdin, S. Spectral Characteristics of the Microwave \nEmission by the Spin Hall Nano-Oscillator. Phys. Rev. Lett. 110, 147601 (2013). \n14. Demidov, V. E., Urazhdin, S., Zholud, A ., Sadovnikov, A. V. & Demokritov, S. O. \nNanoconstriction-based Spin-Hall nano-oscillator. Appl. Phys. Lett. 105, 172410 \n(2014). \n15. Duan, Z. et al. Nanowire spin torque oscillato r driven by spin orbit torques. Nat. \nCommun. 5, 5616 (2014). \n16. Demidov, V. E. et al. Spin-current nano-oscillator ba sed on nonlocal spin injection. \nSci. Rep. 5, 8578 (2015). \n17. Yang, L. et al. , Reduction of phase noise in nanow ire spin orbit torque oscillators. \nSci. Rep. 5, 16942 (2015). \n18. Demidov, V. E. et al. Excitation of coherent propa gating spin waves by pure spin \ncurrents. Nat. Commun. 7, 10446 (2016). \n19. Slavin, A. & Tiberkevich, V. Spin wave mo de excited by spin-polarized current in a \nmagnetic nanocontact is a standing self-localized wave bullet. Phys. Rev. Lett. 95, \n237201 (2005). \n20. Kalinikos, B. A., Kovshikov, N. G., & Slavin, A. N., Envelope solitons and \nmodulation instability of dipole-exchange magnetization waves in yttrium iron garnet \nfilms. Sov. Phys. JETP 67, 303-312 (1988). \n21. Suhl, H. The theory of ferromagne tic resonance at high signal powers. Journal of \nPhysics and Chemistry of Solids 1, 209-227 (1957). 1422. Scott, M. M., Kostylev, M.P., Kalinikos, B.A., & Patton, C.E., Excitation of bright \nand dark envelope solitons for magnetost atic waves with attractive nonlinearity. \nPhys. Rev. B 71, 174440 (2005). \n23. Demidov, V. E., Hansen, U.-H., & Demokr itov, S. O., Spin-wave eigenmodes of a \nsaturated magnetic square at different precession angles, Phys. Rev. Lett. 98, 157203 \n(2007). \n24. d’Allivy Kelly, O. et al. Inverse spin hall effect in nanometer-thick yttrium iron \ngarnet/Pt system. Appl. Phys. Lett. 103, 082408 (2013). \n25. Dyakonov, M. I. & Perel, V. I. Possibility of orienting electron spins with current. \nSov. Phys. JETP Lett. 13, 467-469 (1971). \n26. Hirsch, J.E. Spin Hall Effect. Phys. Rev. Lett. 83, 1834-1837 (1999). \n27. Slonczewski, J. C. Current-drive n excitation of magnetic multilayers. J. Magn. \nMagn. Mater. 159, L1–L7 (1996). \n28. Berger, L. Emission of spin waves by a ma gnetic multilayer trav ersed by a current. \nPhys. Rev. B 54, 9353–9358 (1996). \n29. Demidov, V. E. & Demokritov S. O. Ma gnonic waveguides studied by microfocus \nBrillouin light scattering. IEEE Trans. Mag. 51, 0800215 (2015). \n30. Jungfleisch, M. B. et al. Large Spin-Wave Bullet in a Ferrimagnetic Insulator \nDriven by the Spin Hall Effect. Phys. Rev. Lett. 116, 057601 (2016). \n31. Vansteenkiste, A., Leliaert, J., Dvornik, M., Helsen, M., Garcia-Sanchez, F., & Van \nWaeyenberge, B., The design and verification of mumax3, AIP Advances 4, 107133 \n(2014). \n * Corresponding author: Correspondence and requests for materials should be \naddressed to V.E.D. ( demidov@uni-muenster.de ). 15 \nAcknowledgements: We acknowledge E. Jacquet, R. Lebourgeois, R. Bernard and A. \nH. Molpeceres for their contribution to samp le growth, and O. d’Allivy Kelly and A. \nFert for fruitful discussion. This resear ch was partially supported by the Deutsche \nForschungsgemeinschaft, the ANR Grant Tr inidad (ASTRID 2012 program), and the \nprogram Megagrant № 14.Z50.31.0025 of the Russian Ministry of Education and \nScience. M.C. acknowledges DGA for fina ncial support. V. V. N. acknowledges \nsupport from the Competitive Growth of KFU. \n \nAuthor Contributions: VED, ME and VB performed BLS measurements and data \nanalysis. ME additionally performed micromagnetic simulations. AA and VC \nsupervised the growth of the YIG films. JBY performed the magnetic characterizations. \nMC, VVN, JLP, MM, GdL and AA designed, nanofabricated, and characterized the \nsamples. VVN, PB, JBY, VED, SOD, VC , AA, GdL and OK in itiated and conducted \nthe project. All authors co-wrote and discussed the manuscript. \n \nAdditional Information: The authors have no compe ting financial interests. 16Figure legends \n \nFigure 1. Experimental layout. YIG(20 nm)/Pt(8 nm) disk with the diameter of 2 m \nis electrically contacted by using tw o Au(80 nm) electrodes with a 1 m wide gap \nbetween them. The in-plane dc electrical current in the Pt fi lm is converted into the out-\nof-plane spin current by the spin-Hall effect. The spin current is injected into the YIG film and exerts the spin-transfer torque on its magnetization M resulting in the \nexcitation of magnetic auto-oscillations. Th e excited oscillations are detected by the \nmagneto-optical technique utilizing probing laser light focused through the sample \nsubstrate. \nFigure 2. Spectral and spatial characte ristics of the auto-oscillations. a, BLS spectra \nrecorded by placing the probing laser spot in the middle of the YIG disk for different \nstrength of the dc current I: black solid symbols – I = 0, blue open squares – I = I\nC 8 \nmA, red open diamonds – I = IC+1 mA. The intensity of the peak for I = IC is by about \ntwo orders of magnitude larg er compared to that for I = 0. Lines are the guides for the \neye. The spectral linewidth of the peaks is determined by the limited spectral resolution of the measurement apparatus. The data were recorded at H\n0 = 1000 Oe. b, Typical \nspatial map of the intensity of current-induced magnetic auto-oscillations in the YIG disk. The contours of the disk and the edges of the electrodes are shown by white lines. \nThe map was recorded for I = I\nC+1 mA and H0 = 1000 Oe. \nFigure 3. Dependence of the spatial localization of the auto-oscillations on the dc \ncurrent. a, Spatial profiles of the oscillation in tensity in the direction along the gap \nbetween the electrodes recorded for different dc currents, as labeled. b, Current 17dependence of the full width at half ma ximum (FWHM) of the spatial profiles. c, \nCurrent dependences of the maximum intensity of the auto-oscillations detected in the center of the gap and that of the spatially-integ rated intensity. The data were recorded at \nH\n0 = 1000 Oe. Symbols are experimental data , curves are guides for the eye. \nFigure 4. Spatial localization of the au to-oscillations across the gap between the \nelectrodes. a, Spatial profiles of the oscillation intensity recorded for different dc \ncurrents, as labeled. b, Current dependence of the full width at half maximum (FWHM) \nof the spatial profiles. The data were recorded at H0 = 1000 Oe. Symbols are \nexperimental data, curves are guides for the eye. \nFigure 5. Spatial characteristics of the fi eld-driven ferromagnetic resonance mode. \na, and b, Spatial maps of the FMR mode measured by BLS at I = 0 and 7 mA, \nrespectively. The contours of the disk and the edges of the electrodes are shown by \nwhite lines. The maps were recorded at H0 = 1000 Oe. c, and d, Normalized spatial \nmaps of the intensity of the dynamic magne tization in the FMR mode obtained from \nmicromagnetic simulations. c corresponds to I = 0 and d corresponds to I = 7 mA. \n 18\n\n \n Fig. 1 \nFig. 2 \nFig. 3 Fig. 4 x, m-1 0 1BLS intensity, a.u.\n0.00.51.0\nICa\nbIC+2 mA IC+1 mA\n89 1 0FWHM, m\n0.51.01.5\nIC\nCurrent, mA89 1 0Maximum BLS \nintensity, a.u.\n0123\nIntegral BLS \nintensity, a.u.\n01c\nBLS Intensity, a.u.0.00.51.0\nx, m-1 0 1y,m\n-101c\nx, m-1 0 1-101d\n(mz/mzmax)2\n0.00.51.0-1 0 1y,m\n-101a\n-1 0 1-101bFrequency, GHz3456BLS Intensity, a.u.\n0.00.51.01.5\nIC\nx, m-1 0 1y,m\n-101a\nb\nBLS Intensity, a.u. 0.00.51.0Thermal (x50)IC+1 mA\nH0\ny, m-1 0 1BLS intensity, a.u.\n0.00.51.0\nICaIC+2 mA IC+1 mA\nCurrent, mA89 1 0FWHM, m\n0.40.50.6b\nFig. 5 19 \nSupplementary figures. \n \nSupplementary Figure 1. BLS measurements of the ferromagnetic resonance in the \nYIG disk. In order to measure the ferromagnetic resonance (FMR) in the YIG disk, we \ntransmit a microwave current through an additional 5 m wide stripe antenna, which \ncreates a nearly unif orm in-plane dynamic magnetic fiel d across the YI G disk. By \nvarying the frequency of the excitation si gnal in the antenna and measuring the BLS \nintensity, we record the FMR curves and obtain the FMR frequency. Additionally, the \nelectrical dc current I R . Rows correspond to α0,α1andα2, expanded in powers of D/J. Above,\nC≈449. Remaining parameters are given as η0= 16π/3,ηD=η0+ (4πR/3)(D/J).\nTRANSPORT COEFFICIENTS\nTexture-independent part of the transport coefficients can be obtained using the Boltzmann equation within the\nrelaxation-time approximation in terms of the integral [1, 2]\nJij\nn=1\n(2π)3/planckover2pi1/integraldisplay\nd/epsilon1τ(/epsilon1)(/epsilon1−µ)n(−∂/epsilon1f0)/integraldisplay\ndS/epsilon1vivj\n|v|(6)\nasσ=J0and Π =−J1/J0. Above,τ(/epsilon1) is the relaxation time, /epsilon1(k) =/planckover2pi1ωk,vi=∂ωk/∂ki,dS/epsilon1is the area d2k\ncorresponding to a constant energy surface with /epsilon1(k) =/epsilon1,f0is the Bose-Einstein equilibrium distribution. Under the\nassumption that the scattering processes are dominated by Gilbert damping, we set τ(/epsilon1)≈1/2αω. By evaluating the\nintegral after these substitutions, we obtain σ2D≈F−1/6π2λ/planckover2pi1αin three dimensions ( d= 3), where λ=/radicalbig\n/planckover2pi1J/skBT\nis the wavelength of the thermal magnons, F−1=/integraltext∞\n0d/epsilon1/epsilon1d/2e/epsilon1+x/(/epsilon1+x)(e/epsilon1+x−1)2∼1 evaluated at the magnon gap\nx=/planckover2pi1ω0/kBT. Similarly for d= 2, we obtain σ2D≈F−1/4π/planckover2pi1α.\nThe spin Seebeck coefficient Lis given by−/planckover2pi1σΠ = /planckover2pi1J1, for which we obtain L3D≈F0kBT/6π2λαin 3D and\nL2D≈F0kBT/4παin 2D, where F0=/integraltext∞\n0d/epsilon1/epsilon1d/2/(e/epsilon1+x−1)2∼1. Ford >2 and small x, the numerical factor F0\ncan be expressed in terms of Riemann zeta function and Euler gamma function as ζ(d/2)Γ(d/2 + 1) [3]. In the main\ntext, the numerical factors F−1andF0are omitted.\n[1] N. Ashcroft and N. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976).\n[2] A. A. Kovalev and Y. Tserkovnyak, EPL (Europhysics Lett. 97, 67002 (2012).\n[3] R. K. Pathria, Statistical Mechanics (Butterworth-Heinemann, 1996), 2nd ed." }, { "title": "1812.07244v2.Thermal_gradient_driven_domain_wall_dynamics.pdf", "content": "arXiv:1812.07244v2 [cond-mat.mes-hall] 26 May 2019Thermal gradient driven domain wall dynamics\nM. T. Islam,1,2X. S. Wang,3,4and X. R. Wang1,5,∗\n1Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong\n2Physics Discipline, Khulna University, Khulna, Banglades h\n3School of Electronic Science and Engineering and State Key L aboratory of Electronic Thin Film and Integrated Devices,\nUniversity of Electronic Science and Technology of China, C hengdu 610054, China\n4Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Tr ondheim, Norway\n5HKUST Shenzhen Research Institute, Shenzhen 518057, China\nThe issue of whether a thermal gradient acts like a magnetic fi eld or an electric current in the\ndomain wall (DW) dynamics is investigated. Broadly speakin g, magnetization control knobs can\nbe classified as energy-driving or angular-momentum drivin g forces. DW propagation driven by a\nstatic magnetic field is the best known example of the former i n which the DW speed is proportional\nto the energy dissipation rate, and the current-driven DW mo tion is an example of the latter. Here\nwe show that DW propagation speed driven by a thermal gradien t can be fully explained as the\nangular momentum transfer between thermally generated spi n current and DW. We found DW-\nplane rotation speed increases as DW width decreases. Both D W propagation speed along the wire\nand DW-plane rotation speed around the wire decrease with th e Gilbert damping. These facts\nare consistent with the angular momentum transfer mechanis m, but are distinct from the energy\ndissipation mechanism. We further show that magnonic spin- transfer torque (STT) generated by a\nthermal gradient has both damping-like and field-like compo nents. By analyzing DW propagation\nspeed and DW-plane rotational speed, the coefficient ( β) of the field-like STT arising from the non-\nadiabatic process, is obtained. It is found that βdoes not depend on the thermal gradient; increases\nwith uniaxial anisotropy K/bardbl(thinner DW); and decreases with the damping, in agreement w ith the\nphysical picture that a larger damping or a thicker DW leads t o a better alignment between the\nspin-current polarization and the local magnetization, or a better adiabaticity.\nI. INTRODUCTION\nManipulating domain walls (DW) in magnetic nanos-\ntructures has attracted much attention because of its po-\ntential applications in data storage technology [ 1] and\nlogic gates [ 2]. The traditional DW control knobs,\nnamely magnetic fields and spin-polarized currents, have\ncertain drawbacks in applications. In the magnetic-field-\ndriven DW motion, energy dissipation is the main cause\nofDWpropagationwhosespeedisproportionaltotheen-\nergy dissipation rate [ 3,4], and the magnetic field tends\ntodestroyunfavorabledomainsandDWs, insteadofdriv-\ning a series of DWs synchronously [ 5–7]. An electrical\ncurrent drives a DW to move mainly through the angu-\nlar momentum transfer so that it pushes multiple DWs\n[8–11] in the same direction. To achieve a useful DW\nspeed, it requires high electrical current densities that\nresult in a Joule heating problem [ 12–14]. To avoid these\nproblems, spin-wave spin current has been proposed as a\nmoreenergy-efficientcontrolparameter[ 15–18]. Thermal\ngradient, a way to generate spin-wave spin current, is an\nalternative control knob of the DW motion. The inves-\ntigation on thermal-gradient-driven domain wall motion\nis meaningful not only for conventional applications, but\nalso for the understanding of spin wave and domain wall\ndynamics [ 16,17,20–23], as well as for possible recycling\nof waste heat [ 19,24].\n∗[Corresponding author:]phxwan@ust.hkTo understand the mechanism behind thermal-\ngradient-drivenDWdynamics, therearemicroscopicthe-\nories [15–17,25,26] and macroscopic thermodynamic\ntheories [ 21,22]. Briefly speaking, the microscopic theo-\nries suggest that magnons populated in the hotter region\ndiffuses to the colder region to form a magnon spin cur-\nrent. The magnon spin currentpassesthrough a DWand\nexerts a torque on the DW by transferring spin angular\nmomentum to the DW. Thus, magnons drive the DW\npropagating toward the hotter region of the nanowire,\nopposite to the magnon current direction [ 15,16,18].\nThe thermodynamic theories anticipate that a thermal\ngradient generates an entropy force which always drives\nthe DW towards the hotter region in order to minimize\nthe system free energy. The macroscopic theories do not\nprovide any microscopic picture about DW dynamics al-\nthough a thermal gradient is often considered as an effec-\ntive magnetic field to estimate DW speed [ 21,22] from\nfield-driven DW theories. Thus, one interesting issue is\nwhether a thermal gradient in DW dynamics acts like a\nmagnetic field or an electric current. DW propagation\nspeed should be sensitive to both DW width and types\nof a DW (transverse DW) under an energy-driving force\nwhile the speed should be insensitive to the DW and DW\nstructure in the angular-momentum-driving force. This\nis the focus of the current work.\nIn this paper, we investigate DW motion along a uni-\naxial wire with the easy axis along the wire direction\nunder a thermal gradient. We found that the DW al-\nways propagates to the hotter region with an accom-\npanied DW-plane rotation. DW propagation speed and2\nz\nxy\nFIG. 1. Schematic diagram of a uniaxial magnetic nanowire\nwith a head-to-head DW at the center under a thermal gra-\ndient∇T. Black (white) color represents colder (hotter) end\nof the sample.\nDW-plane rotation speed increases as the magnetic easy-\naxis anisotropy and damping decreases. We show that\nDW motion can be attributed to the angular momen-\ntum transfer between magnonic spin current and the\nDW. Thus, we conclude that a thermal gradient in-\nteracts with DW through angular-momentum transfer\nrather than through energy dissipation. Similar to an\nelectric current [ 27], a thermal gradient can generate\nboth damping-like (or adiabatic) STT and field-like (or\nnon-adiabatic) STT. From the damping-dependence and\nanisotropy-dependence of the average DW velocity and\nDW-plane rotation angular velocity, we extract field-like\nSTT coefficient ( β). It is found that βis independent\nof thermal gradient; is bigger for a thinner DW; and de-\ncreases with the damping coefficient. We also show that\nin the presence of a weak hard-axis anisotropy perpen-\ndicular to the wire, the DW still undergoes a rotating\nmotion. The DW propagation speed increases slightly\nwhile the DW-plane rotation speed decreases with the\nstrength of the hard-axis anisotropy.\nII. MODEL AND METHOD\nWe consider a uniaxial nanowire of length Lxand\ncross-section Ly×Lzalong the x-axis (easy axis) with\na head-to-head DW at the center, as shown in Fig. 1.\nLy,Lzis much smaller than the DW width ∆, and ∆\nis much smaller than Lx. A thermal gradient is applied\nalong the wire. The highest temperature is far below\nthe Curie temperature Tc. The magnetization dynam-\nics is governed by the stochastic Landau-Lifshitz-Gilbert\n(LLG) equation [ 28,29],\ndm\ndt=−γm×(Heff+hth)+αm×∂m\n∂t,(1)\nwherem=M/MsandMsare respectively the magne-\ntization direction and the saturation magnetization. α\nis the Gilbert damping constant and γis the gyromag-netic ratio. Heff=2A\nµ0Ms/summationtext\nσ∂2m\n∂x2σ+2K/bardbl\nµ0Msmxˆx+hdipoleis\nthe effective field, where Ais the exchange constant, xσ\n(σ= 1,2,3) denote Cartesian coordinates x,y,z,K/bardblis\nthe easy-axis anisotropy, and hdipoleis the dipolar field.\nhthis the stochastic thermal field.\nThe stochastic LLG equation is solved numerically by\nMUMAX3 package [ 30] in which we use adaptive Heun\nsolver. To balance stability and efficiency, we choose the\ntime step 10−14s with the cell size (2 ×2×2) nm3. Mag-\nnetic charges at the two ends of the wire are removed to\navoid their attraction to the DW. The saturation mag-\nnetization Ms= 8×105A/m and exchange constant\nA= 13×10−12J/m are used to mimic permalloy in\nour simulations. The thermal field follows the Gaussian\nprocess characterized by following statistics [ 31]\n/angb∇acketlefthth,ip(t)/angb∇acket∇ight= 0,\n/angb∇acketlefthth,ip(t)hth,jq(t+∆t)/angb∇acket∇ight=2kBTiαi\nγµ0Msa3δijδpqδ(∆t),(2)\nwhereiandjdenote the micromagnetic cells, and p,q\nrepresent the Cartesian components of the thermal field.\nTiandαiare respectively temperature and the Gilbert\ndamping at cell i, andais the cell size. kBis the Boltz-\nmann constant [ 28]. The numerical results presented in\nthis study are averaged over 15 random configurations\n(for DW velocity) and 4000-5000 random configurations\n(for spin current).\nUnderthethermalgradient ∇xT,magnetizationatdif-\nferent positions deviate from their equilibrium directions\ndifferently and small transverse components myandmz\nare generated. The transverse components vary spatial-\ntemporally and depend on the local temperature. This\nvariation generates a magnonic spin current [ 16]. This\nmagnonic spin current can interact with spin textures\nsuch as DWs. In the absence of damping (the thermal\nfield also vanishes), the spin currentalong the xdirection\ncan be defined from the spin continuity equation derived\nfrom Eq. ( 1) as follows [ 15],\n∂m\n∂t=−1\n1+α2m׈xmxK/bardbl−∂J\n∂x,(3)\nwhere\nJ(x) =2γA\nµ0Msm×∂m\n∂x, (4)\nis the spin current density along x-direction due to the\nexchangeinteraction. J(x) can be numerically calculated\n[15,23]. In the presence of damping as well as the ther-\nmal field, the contribution of the damping term and the\nthermal term is proportional to α, which is relatively\nsmall. More importantly, according to the fluctuation-\ndissipation theorem [ 28], the damping term and the ther-\nmal term should cancel each other after average over a\nlong time. Since the time scale of DW dynamics is much\nlonger than the thermal fluctuation, the combined con-\ntribution of damping and thermal terms should be very\nsmall.3\n/s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s52/s56/s49/s50/s49/s54/s50/s48\n/s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s50/s52/s54/s56/s49/s48/s49/s50/s45/s56/s48/s48 /s45/s52/s48/s48 /s48 /s52/s48/s48 /s56/s48/s48/s45/s48/s46/s48/s50/s45/s48/s46/s48/s49/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50\n/s32/s32\n/s32/s118\n/s115/s105/s109/s117\n/s32/s118\n/s99/s117/s114/s114/s101/s110/s116/s118 /s32/s40/s109/s47/s115/s41\n/s120/s84 /s32/s40/s75/s47/s110/s109/s41/s32/s32/s32\n/s120/s84 /s32/s40/s75/s47/s110/s109/s41\n/s32/s32\n/s48/s46/s48/s55/s32 /s75/s47/s110/s109\n/s48/s46/s49 /s75/s47/s110/s109\n/s48/s46/s49/s53/s32 /s75/s47/s110/s109\n/s48/s46/s50/s32 /s75/s47/s110/s109\n/s48/s46/s50/s53/s32 /s75/s47/s110/s109\n/s48/s46/s51/s32 /s75/s47/s110/s109/s74\n/s116/s111/s116/s40/s120/s41/s40\n/s115/s41/s41\n/s120 /s32/s40/s110/s109/s41\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s49/s52/s49/s54/s49/s56/s50/s48/s50/s50/s40/s100/s41/s40/s98/s41/s40/s97/s41\n/s32\n/s75 /s32/s40/s49/s48/s52\n/s32/s74/s47/s109/s51\n/s41/s118 /s32/s40/s109/s47/s115/s41/s40/s99/s41/s100 /s47/s100/s116/s32/s40/s100/s101/s103/s47/s110/s115/s41\n/s100 /s47/s100/s116/s32/s40/s100/s101/s103/s47/s110/s115/s41\n/s51/s54/s51/s57/s52/s50\n/s32\nFIG. 2. (a) The spatial dependence of spin current densities\nJtot(x) for various ∇xT. The DW center is chosen as x=\n0. (b) Thermal gradient dependence of DW velocity vsimu\nfrom micromagnetic simulations (open squares) and vcurrent\ncomputedfrom total spin current(solid squares). (c)Therm al\ngradient dependence of DW-plane rotation angular velocity\n(squares). In (a)(b)(c) model parameters are Lx= 2048 nm,\nLy=Lz= 4 nm, α= 0.004 and K/bardbl= 5×105J/m3. (d)\nvsimu(solid squares) and dφ/dt(open squares) as a function\nofK⊥forLx= 1024 nm and ∇xT= 0.5 K/nm.\nIntegrating the x−component of Eq. ( 3) over a space\nenclosed the DW in the center and noticing the absence\nof the first term on the right, we have\nvcurrent=1\n2/integraldisplayLx/2\n−Lx/2∂mx\n∂tdx\n=−2γA\nµ0Ms/bracketleftbig1\n2(Jx|left−Jx|right)].(5)\nwhere we have assumed the fluctuations in the domains\nare small and the DW is not far from a symmetric one.\nJx|left,Jx|rightmean the x-components of the total spin\ncurrent on the left and right sides of the DW. The equa-\ntion clearly shows that the DW propagates opposite to\nthe spin current. This is the theoretical DW velocity un-\nder the assumption of angular momentum conservation,\nand it will be compared with the directly simulated DW\nvelocity below.\nIII. RESULTS\nA. Average spin current and DW velocity\nTosubstantiateourassertionthatDWpropagationun-\nder a thermal gradient is through angular-momentum ef-\nfect instead of energy effect, we would like to compare\nthe DW velocity obtained from micromagnetic simula-\ntions and that obtained from total spin current based on/s49 /s50 /s51 /s52 /s53 /s54 /s55/s56/s49/s50/s49/s54/s50/s48\n/s52/s56/s49/s50/s49/s54/s50/s48\n/s32/s118\n/s115/s105/s109/s117\n/s32/s118\n/s99/s117/s114/s114/s101/s110/s116\n/s100 /s47/s100/s116/s32/s40/s100/s101/s103/s47/s110/s115/s41/s32/s118 /s32/s40/s109/s47/s115/s41\n/s32/s76\n/s120/s61/s50/s48/s52/s56/s32/s110/s109\nFIG. 3. Damping αdependence of the DW dynamics: vsimu\n(Open squares); vcurrent(solid squares ); and dφ/dt(solid\ncircles). Model parameters are ∇xT= 0.2 K/nm, K/bardbl= 5×\n105J/m3,Lx= 2048 nm and Ly=Lz= 4 nm.\nEq. (5). Eq. (4) is used to calculate Jx(x). Fig.2(a) is\nspatial distribution of the ensemble averaged Jx(x) with\nDW atx= 0 for various thermal gradients. The sud-\nden sign change of Jx(x) at the DW center is a clear\nevidence of strong angular-momentum transfer from spin\ncurrent to the DW. Technically, magnetizationof the two\ndomains separated by the DW point to the opposite di-\nrections, thus the spin current polarization changes its\nsign. In calculating DW velocity vcurrentfrom Eq. ( 5),\nthe spin currents before entering DW and after passing\nDW are the averages of Jx(x) overx∈[−2∆,−∆] and\nx∈[∆,2∆], where ∆ is the DW width which is 16 nm\nin the current case. The thermal gradient dependence\nofvcurrentis shown in Fig. 2(b) (solid squares). vcurrent\ncompares well with the velocity vsimu(open squares) ob-\ntained directly from simulations by extracting the speed\nof the DW center along x-direction. The DW veloc-\nity is linearly proportional to the temperature gradient\nv=C∇xT, with the thermal mobility C= 6.66×10−8\nm2s−1K−1forvsimuorC= 6.59×10−8m2s−1K−1for\nvcurrent. It is noted that vcurrentalmost coincides with\nvsimuexcept a small discrepancy at very high thermal\ngradient when the nonlinear effects is strong. The small\ndiscrepancy may be attributed to the large fluctuations\nas well as the contribution from the damping, the dipo-\nlar and stochastic fields. These observations are consis-\ntent with magnonic STT [ 15,16,25,26]. It is observed\nthat the DW-plane rotates around the x-axis counter-\nclockwise for head-to-head DW and clockwise for tail-to-\ntail DW during DW propagation. DW rotation speed\ndφ/dt(squares) is shown in Fig. 2(c)) as a function of\n∇xT.4\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54\n/s52/s54/s56/s49/s48/s49/s50/s49/s52/s32/s32\n/s75\n/s124/s124/s32/s40/s49/s48/s53\n/s32/s74/s47/s109/s51\n/s41/s32/s118\n/s115/s105/s109/s117\n/s32/s118\n/s99/s117/s114/s114/s101/s110/s116/s118 /s32/s40/s109/s47/s115/s41\n/s100 /s47/s100/s116/s40/s100/s101/s103/s47/s110/s115/s41/s32\nFIG. 4. Anisotropy K/bardbldependence of the DW dynamics:\nvsimu(open squares); vcurrent(solid squares); and dφ/dt(solid\ncircles). Model parameters are Lx= 2048 nm, Ly=Lz=4 nm,\nα= 0.004 and ∇xT= 0.2 K/nm.\nB. Damping and anisotropy dependence of DW\ndynamics\nAn energy-effect and angular-momentum-effect\nhave different damping-dependence and anisotropy-\ndependence of DW dynamics. To distinguish the roles of\nenergy and the angular-momentum in thermal-gradient\ndriven DW dynamics, it would be useful to probe how\nthe DW dynamics depends on αandK/bardbl. Damping have\ntwo effects on the spin currents: one is the decay of\nspin current during its propagation so that the amount\nof spin angular momentum deposited on a DW should\ndecrease with the increase of the damping coefficient.\nAs a result, the DW propagation speed and DW-plane\nrotation speed should also be smaller for a larger α.\nIndeed, this is what we observed in our simulations\nas shown in Fig. 3(a) for DW speed and DW-plane\nrotation speed (open squares for vsimu, solid circles for\nvcurrent, and stars for dφ/dt). The model parameters are\nLx= 2048, Ly=Lz= 4 nm, ∇xT= 0.2 K/nm and\nK/bardbl= 5×105J/m3. The second damping effect is that\nthe larger αhelps the spin current polarization to align\nwith the local spin. This second effect enhances the\nadiabatic process that is important for non-adiabatic\nSTT or field-like torque discussed in the next subsection.\nTherefore, α−dependence of DW dynamics supports\nthe origin of thermal driven DW dynamics to be the\nangular-momentum effect, not the energy effect that\nwould lead to a larger vsimuanddφ/dtfor a larger α\n[3,4,33–35] instead of a decrease observed here.\nHere we would like to see how the DW dynamics de-\npendsonuniaxialanisotropy K/bardbl. Fig.4showsboth vsimu\n(open squares), vcurrent(filled squares) and dφ/dt(cir-\ncles) for Lx= 2048 nm, α= 0.004 and ∇xT= 0.2. The\nDW propagation speed, vsimudecreases with K/bardblwhileDW-plane rotational speed increases with K/bardbl. These re-\nsults seem follow partially the behavior of magnetic-field\ninduced DW motion, in which DW propagation speed\nis proportional to DW width (∆ ∼/radicalBigg\nA\nK/bardbl) or decrease\nwithK/bardbl, and partially electric current driven DW mo-\ntion, in which DW-plane rotational speed increases with\nK/bardbl. Thus, one may tend to conclude that a thermal gra-\ndient behaves more like a magnetic field rather than an\nelectric current from the DW width dependence of DW\npropagation speed, opposite to our claim of the angular-\nmomentum effects of the thermal gradient. It turns out,\nthis is not true. The reason is that magnon spectrum,\nωk=2γ\nµ0Ms/parenleftbig\nAk2+K/bardbl/parenrightbig\n, has a gap in a system with mag-\nnetic anisotropy. The larger K/bardblis, the bigger the energy\ngap will be. Thus, it becomes harder to thermally excite\nmagnon. As a result, the spin current decreasesas K/bardblin-\ncreases. To see whether the thermal-gradient driven DW\nmotion is due to the angular-momentum transfer or not,\none should compare whether vsimuandvcurrentmaintain\na good agreement with each other as K/bardblvaries. Indeed,\na good agreement between vsimuandvcurrentis shown in\nFig.4. This conclusion is also consistent with existing\nmagnonic STT theories [ 33–35].\nC. Separation of adiabatic and non-adiabatic\ntorques\nWe have already demonstrated that a thermal gradi-\nent interacts with DW through magnonic STT rather\nthan through energy dissipation. It is then interesting to\nknow what kind of STTs a thermal gradient can gen-\nerate. Specifically, whether a magnonic spin current\ngenerates damping-like (adiabatic), or field-like (Non-\nadiabatic) torques, or both just like an electric current\n/s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s73/s32/s40/s49/s48/s49/s48\n/s32/s65/s47/s109/s50\n/s41\n/s32\n/s120/s84 /s32/s40/s75/s47/s110/s109/s41\n/s32/s32\nFIG. 5. Model parameters are K/bardbl= 5×105J/m3,α=\n0.004,Lx= 1024 and Ly=Lz=4 nm. Effective electric current\ndensityI(open squares) and β(solid squares) are plotted as\nfunctions of ∇xT.5\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55/s48/s46/s49/s52/s48/s46/s50/s49/s48/s46/s50/s56\n/s40/s49/s48/s45/s51\n/s41\n/s32/s32\n/s75\n/s124/s124/s32/s40/s49/s48/s53\n/s32/s74/s47/s109/s51\n/s41/s40/s97/s41\n/s40/s98/s41/s32/s32\nFIG. 6. Model parameters are ∇xT=0.5 K/nm, Lx= 1024\nnm and Ly=Lz=4 nm. (a) α-dependence of βforK/bardbl= 106\nand J/m3. (b)K/bardbl-dependence of βforα= 0.004.\n[27] does. To extract the STT generated from a thermal\ngradient, we approximate DW dynamics by the motion\nof its collective modes of DW center Xand the titled\nangleφof DW-plane. Subject to both damping-like and\nfield-like torques, using the travelling-wave ansatz [ 33–\n35], tan(θ/2) = exp[( x−X)/∆] where ∆ ∼/radicalbig\nA/K/bardbl, one\ncan derive the equations for X and φ,\nα\n∆dX\ndt+dφ\ndt=β\nαu,1\n∆dX\ndt−αdφ\ndt=u\nα.(6)\nFrom the above two equations, one can straightfor-\nwardly find DW propagating speed and DW-plane ro-\ntation speed,\nv=(1+αβ)\n(1+α2)u,˙φ=(β−α)\n(1+α2)u. (7)\nOne can extract βand equivalent electric current den-\nsityI= (2eMsu)/gµBPfromvanddφ/dtobtained in\nsimulations. For α= 0.004,K/bardbl= 106J/m3, theIandβ\nare obtained and plotted in Fig. 5as a function of ∇xT.\nIt is evident that Ilinearly increases with ∇xTandβ\nis independent of ∇xTas it should be. We then fixed\n∇xT= 0.5 K/nm, and repeat simulations and analysis\nmentioned above for various αandK/bardbl. Fig.6(a) and\n(b) shows βas a function of αandK/bardbl. From the figure,\nit is evident that βdecreases with α. This is because\nthe larger damping favors the alignment of spin current\npolarization with the local spin so that the non-adiabatic\neffect,β, becomes smaller. βincreases with K/bardblfor the\nsimilar reason: Larger K/bardblmeans a thinner DW so that\nit is much harder for the spin current polarization to re-\nverse its direction after passing through the thinner DW,\ni.e. a stronger non-adiabatic effect.\nIn some experiments, the temperature gradient is gen-\nerated by a laser spot[ 36]. The laser spot will induce a\nGaussian distribution of the temperature over the space\n[36,37]. In Fig. 7, weshowtheDWmotionin aGaussian\ntemperature profile T(x) =T0exp/parenleftBig\n−(x−xL)2\n2σ2/parenrightBig\nby plot-\nting the DW position against the time. Here we use the/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48\n/s32/s32/s68/s87/s32/s112/s111/s115/s105/s116/s105/s111/s110/s32/s40/s110/s109/s41\n/s116/s105/s109/s101/s32/s40/s110/s115/s41\nFIG. 7. Domain wall position versus time in a Gaussian tem-\nperature profile. The gray lines are raw data for different\nrandom seeds and the red line is the averaged result. The\ngreen dashed line is theoretical result using the thermal mo -\nbilityC= 6.66×10−8m2s−1K−1obtained from Fig. 2(b).\nsame parameters as those in Fig. 2(b), except a longer\nwireLx= 2048 nm, and T0= 400 K, σ= 200 nm,\nxL= 200 nm. Theoretically, if the instantaneous DW\nspeed under a Gaussian temperature is the same as that\nin the constant thermal-gradient case, we should expect\ndx\ndt=CdT\ndx, where the thermal mobility Cis the same as\nthat in Fig. 2(b). Using C= 6.66×10−8m2s−1K−1,\nthe above differential equation for x(t) can be numeri-\ncally solved with initial condition x(0) = 0. The result\nis plotted in Fig. 7in green dashed line. The simu-\nlated speed is smaller than this theoretical result. This\nis probably because, for the constant thermal-gradient,\nwe focus on the steady-state DW motion speed. In a\nGaussian temperature, the DW cannot immediately fol-\nlow the local temperature gradient. Before the DW can\nreach the steady-state speed corresponding to the local\ntemperature, it already moves to a position of smaller\ntemperature gradient. More details about DW motion in\nGaussian temperature profile may be an issue of future\nstudies.\nIV. DISCUSSION AND SUMMARY\nWe have studied the thermal gradient-driven DW dy-\nnamicsinanuniaxialnanowire. Inreality, thereisalways\ncertain hard anisotropy in a wire whose cross-section is\nnot a perfect ellipse. Thus, it is interesting to see how\nthe above results will change in a weak biaxial nanowire\nwith a small hard anisotropy K⊥= 1/2µ0M2\ns(Nz−Ny),\nsay along y-direction. Our simulations show that a DW\nstill propagatestowardsthe higher temperature region in\na similar way as that in a uniaxial wire. Interestingly, as\nshown in Fig. 2(d) for the K⊥-dependence of vsimu(solid\nsquares) and dφ/dt(open squares), DW speed increases6\nslightly with K⊥. This may be due to the increase of\ntorque along θ-direction [ 33] since Γ θis proportional to\n(Nz−Ny). This is also consistent with the early results\nforthe uniaxialwire that vsimu(which includes stochastic\nthermal field and demagnetisation fields) is always larger\nthanvcurrent(where the transverse fields are neglected).\nAt the meanwhile, dφ/dtdecreases with K⊥.\nThe main purpose of this paper is to study the\nmagnonic effects in thermal-gradient-driven domain wall\ndynamics. We consider the spin waves explicitly and\nall the material parameters (exchange constant A, crys-\ntalline anisotropy K, saturation magnetization Ms, and\nGilbert damping α) are assumed to be constant. Indeed,\nthe atomistic magnetic moments are independent of tem-\nperature. At the atomistic level, the exchange constant\nAoriginating from the Pauli exclusion principle and the\ncrystalline anisotropy Koriginating from the spin-orbit\ncoupling onlyweaklydepend on the temperature because\nof the vibration of atoms [ 39]. In micromagnetic models,\nbecause finite volumes that contains many magnetic mo-\nments are considered as unit cells, the parameters A,K,\nandMsdepend on the temperature. This is because the\nthermally excited spin waves with wavelengths shorter\nthan the length scale of the unit cells are included in\nthe effective A,K, andMsby doing an average [ 16,38].\nSince we use small mesh size 2 ×2×2 nm3, only spin\nwaves of very short wavelength affect the parameters A,\nK, andMsin our model. Those short-wavelength spin\nwavespossess high energyaswell as low density ofstates,\nso their contributions to the effective A,K, andMsare\nnot significant. The Gilbert damping αdepends on the\ntemperature non-monotonically [ 40–43]. The underlying\nmechanism is still under debate, but for many cases the\ndependence is not significant in a wide range of temper-\nature.\nIn summary, our results show that the uniform ther-\nmal gradient always drives a DW propagating towards\nthe hotter region and the DW-plane rotates around the\neasy axis. The DW velocity and DW-plane rotational\nspeed decrease with the damping coefficient. The DW\nvelocity obtained from simulation agrees with the veloc-\nity obtained from angular momentum conservation when\nthe magnon current density ( J(x)) from the simulation is\nusedtoestimatetheamountofangularmomentumtrans-\nferred from magnon current to the DW. All the above\nfindings lead to the conclusion that the thermal gradient\ninteracts with DW through angular-momentum transfer\nrather than through energy dissipation. Furthermore,\nwe demonstrated that the magnonic STT generated by\na thermal gradient has both damping-like and field-like\ncomponents. The field-like STT coefficient βis deter-\nmined from DW speed and DW-plane rotation speed. β\ndoes not depend on the thermal gradient as expected,\nbut increases with a decrease of DW width. This behav-\nior can be understood from the expected strongmisalign-\nment of magnon spin polarization and the local spin so\nthat non-adiabatic torque (also called field-like torque)\nis larger. For the same reason, a larger Gilbert dampingresults in a better alignment between spin current polar-\nization and the local spin, thus βshould decrease with\nα. The thermal gradientcan be a veryinteresting control\nknob for nano spintronics devices, especially those made\nfrom magnetic insulators.\nThis work was supported by the National Natural Sci-\nence Foundation of China (Grant No. 11774296) as well\nas Hong Kong RGC Grants Nos. 16300117, 16301518\nand 16301816. X.S.W acknowledges support from NSFC\n(GrantNo. 11804045),ChinaPostdoctoralScienceFoun-\ndation (Grant No. 2017M612932and 2018T110957),and\nthe Research Council of Norway through its Centres of\nExcellence funding scheme, Project No. 262633, “QuS-\npin.” M. T. I acknowledges the Hong Kong PhD fellow-\nship.7\n[1] Parkin S S P, Hayashi M and Thomas L 2008 Science\n320 190\n[2] Allwood D A, Xiong G, Faulkner C C, Atkinson D, Petit\nD and Cowburn R P 2005 Science309 1688\n[3] Wang X R, P Yan, Lu J and He C 2009 Ann. Phys. (N.\nY.)324 1815\n[4] Wang X R, Yan P and Lu J 2009 Europhys. Lett. 86\n67001\n[5] Atkinson D, Allwood D A, Xiong G, Cooke M D,\nFaulkner C C, and Cowburn R P 2003 Nat. Mater. 2\n85\n[6] Beach G S D, Nistor C, Knutson C, Tsoi M, and Erskine\nJ L 2005 Nat. Mater. 4 741\n[7] Hayashi M, Thomas L, Bazaliy Ya B , Rettner C, Moriya\nR, Jiang X, and Parkin S S P 2006 Phys. Rev. Lett. 96\n197207\n[8] Berger L 1996 Phys. Rev. B 54 9353\n[9] Slonczewski J 1996 J. Magn. Magn. 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B84 014412\n[43] Maier-Flaig H, Klingler S , Dubs C, Surzhenko O, Gross\nR, Weiler M, Huebl H, and Goennenwein S T B 2017\nPhys. Rev. B 95 214423" }, { "title": "2009.08152v1.Resonant_absorption__transformation_of_compressive_motions_into_vortical_motions.pdf", "content": "Astronomy &Astrophysics manuscript no. 38394final c\rESO 2020\nSeptember 18, 2020\nResonant absorption:\nTransformation of compressive motions into vortical motions\nM. Goossens1, I. Arregui2;3, R. Soler4;5, and T. Van Doorsselaere1\n1Centre for mathematical Plasma Astrophysics, KU Leuven, Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgium\n2Instituto de Astrofísica de Canarias, Vía Láctea s /n, E-38205 La Laguna, Tenerife, Spain\n3Departamento de Astrofísica Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain\n4Departament de Física, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain\n5Institut d0Aplicacions Computacionals de Codi Comunitari (IAC3), Universitat de les Illes Balears, E-07122 Palma de Mallorca,\nSpain\ne-mail: marcel.goossens@kuleuven.be\nSeptember 18, 2020\nABSTRACT\nThis paper investigates the changes in spatial properties when magnetohydrodynamic (MHD) waves undergo resonant damping in the\nAlfvén continuum. The analysis is carried out for a 1D cylindrical pressure-less plasma with a straight magnetic field. The e \u000bect of\nthe damping on the spatial wave variables is determined by using complex frequencies that arise as a result of the resonant damping.\nCompression and vorticity are used to characterise the spatial evolution of the MHD wave. The most striking result is the huge spatial\nvariation in the vorticity component parallel to the magnetic field. Parallel vorticity vanishes in the uniform part of the equilibrium.\nHowever, when the MHD wave moves into the non-uniform part, parallel vorticity explodes to values that are orders of magnitude\nhigher than those attained by the transverse components in planes normal to the straight magnetic field. In the non-uniform part of\nthe equilibrium plasma, the MHD wave is controlled by parallel vorticity and resembles an Alfvén wave, with the unfamiliar property\nthat it has pressure variations even in the linear regime.\nKey words. Magnetohydrodynamics (MHD) – Waves – Sun: corona – Sun: magnetic fields\n1. Introduction\nIn a recent paper, Goossens et al. (2019) have studied the prop-\nerties of magnetohydrodynamic (MHD) waves in non-uniform\nplasmas. They pointed out that in non-uniform plasmas, MHD\nwaves behave di \u000berently from their counterparts in uniform plas-\nmas of infinite extent. In the latter case, the MHD waves can\nbe separated into Alfvén waves and magneto-acoustic waves.\nThe Alfvén waves propagate vorticity and are incompressible.\nIn addition, they have no parallel displacement component. The\nmagneto-acoustic waves are compressible and in general have a\nparallel component of displacement, but do not propagate par-\nallel vorticity. Compression, parallel vorticity, and parallel dis-\nplacement are the characteristic quantities. In a uniform plasma\nof infinite extent, compression and parallel displacement on one\nhand and parallel vorticity on the other hand are mutually exclu-\nsive. In a pressure-less plasma, the parallel displacement is zero\nbecause the Lorentz force has no component along the magnetic\nfield. Hence, in a pressure-less plasma, the waves have only two\ncharacteristic quantities: compression and parallel vorticity. The\ndistinction between the waves is then based on compression or\nparallel vorticity.\nThe situation is di \u000berent in a non-uniform plasma, as was\npointed out by Goossens et al. (2019). The clear division be-\ntween Alfvén waves and magneto-acoustic waves is no longer\npresent. The MHD waves have mixed properties. The general\nrule is that MHD waves propagate both parallel vorticity, as in\nclassic Alfvén waves, and compression, as in classic magneto-\nacoustic waves. The present paper focusses on the properties ofMHD waves that undergo resonant absorption. Here we concen-\ntrate on resonant absorption in the Alfvén continuum. In order to\nkeep the mathematical analysis as simple as possible while still\nretaining the essential physics, we consider a straight magnetic\nfield, and in addition, we assume that the plasma is pressure-less.\nThis assumption removes the slow magneto-acoustic part of the\nspectrum and resonant absorption in the slow continuum.\nGoossens et al. (2019) also studied what happens with com-\npression and vorticity for frequencies in the slow and Alfvén\ncontinuum. The analysis was restricted to the driven problem of\nstationary waves with real frequencies. The authors determined\nthe dominant singularities in the ideal MHD solutions and the\ndominant dynamics for stationary waves. For another study in\nwhich resonant absorption in both Alfvén and slow continua are\nconsidered, see Soler et al. (2009). Studies on resonant absorp-\ntion have mainly focussed on the components of the displace-\nment, the amount of absorbed energy, and the damping rate.\nAnalytical solutions for the components of the Lagrangian dis-\nplacement in the dissipative layer have for example been de-\nrived by Sakurai et al. (1991) and Goossens et al. (1995) for\nresonant MHD waves in ideal and dissipative stationary MHD.\nRuderman et al. (1995) studied non-stationary incompressible\nresonant MHD waves in non-ideal MHD for a planar equilib-\nrium. Tirry & Goossens (1996) studied non-stationary resonant\nMHD waves for cylindrical plasmas in visco-resistive MHD.\nSoler et al. (2013) studied non-stationary MHD waves for cylin-\ndrical plasmas both in resistive MHD and ideal MHD. Their\nmathematical scheme for non-stationary ideal MHD followed\nthe scheme devised by Hollweg (1990b) for planar plasmas. Lit-\nArticle number, page 1 of 9arXiv:2009.08152v1 [astro-ph.SR] 17 Sep 2020A&A proofs: manuscript no. 38394final\ntle or no attention has been given to the change in the spatial\nbehaviour of fundamental quantities such as compression and\nvorticity. Goossens et al. (2012) were the first to point out that\nthe fundamental radial mode of kink waves propagates parallel\nvorticity in the non-uniform part of the loop required for reso-\nnant absorption to operate. In the present investigation, we fo-\ncus on the eigenvalue problem and try to understand what hap-\npens when the wave is actually damped in non-stationary MHD.\nWe take the frequency to be complex and relate the spatial be-\nhaviour to the damping properties of the MHD wave. Soler et al.\n(2013) concentrated on the components of Lagrangian displace-\nment and the Eulerian perturbation of total pressure. Goossens\net al. (2012) took the existence of a parallel vorticity component\nas the base for the physical interpretation of the fundamental ra-\ndial mode of kink waves in terms of surface Alfvén waves. Our\nanalytic investigation first presents the components of vorticity\nin the case of non-stationary ideal MHD close to the resonant\nposition. Then, the semi-analytic approach of Soler et al. (2013)\nin non-stationary MHD is used to verify the predictions based\non approximate analytic theory. The aim is to obtain a simple\nunderstanding of how the fundamental quantities compression\nand vorticity are a \u000bected by the non-stationary behaviour of the\nresonantly damped wave.\n2. Resonant absorption\nThis section collects results on resonant absorption that are used\nin our discussion of the spatial solutions of the MHD waves\nthat undergo resonant damping. Resonant absorption has a long\nhistory in fusion plasma physics, space plasma physics, so-\nlar physics, and astrophysics. A characterisation was given by\nParker (1991), who noted that resonant absorption in the Alfvén\ncontinuum is to be expected when a wave with a phase velocity\n!=kspans a region in which the variation of the Alfvén velocity\nvAacross the region provides the resonance condition !=k=vA.\nNon-uniformity is key to the process. The Alfvén velocity vAand\nAlfvén frequency !Adepend on position. The resonance occurs\nat the position rA, where the frequency of the wave !is equal\nto the local Alfvén frequency !=!A(rA). The local Alfvén fre-\nquency!Ais assumed to be an analytic function.\nSince 2002 (Ruderman & Roberts 2002; Goossens et al.\n2002a), resonant absorption of kink waves is a popular and plau-\nsible mechanism for explaining the rapid damping of standing\nand propagating MHD waves in coronal loops (see e.g. Montes-\nSolís & Arregui 2017, for a discussion). The simple model in-\nvokes MHD waves superimposed on a cylindrical plasma col-\numn in static equilibrium. Cylindrical coordinates ( r;';z) are\nused. The inhomogeneity necessary for resonant absorption to\noperate is usually provided by the equilibrium density \u001a0(r) that\nvaries from \u001aito\u001aein the interval [ R\u0000l=2;R+l=2]. The den-\nsity\u001a0is constant in the internal and the external parts of the\nloop with values \u001aiand\u001ae;\n\u001a0=(\u001aifor 0\u0014r\u0014R\u0000l=2;\n\u001aeforR+l=2\u0014r<+1:\nIn the non-uniform transitional layer of thickness l;the equi-\nlibrium density varies continuously from its internal value \u001aito\nits external value \u001ae.\nThe equilibrium magnetic field is assumed to be axial and\nconstant B0=B01z, with 1zthe unit vector in the z- direc-\ntion. The temporal dependence and the spatial dependence on\nthe ignorable ( ';z)- coordinates are given by exp( \u0000i!t) andexp(i(m'+kzz)), with mandkzthe azimuthal and axial wave\nnumbers, respectively. The local Alfvén frequency !Aand the\nAlfvén velocity vAare defined as\n!2\nA=(k\u0001B)2\n\u0016\u001a0=k2\nzv2\nA=kkv2\nA and v2\nA=B2\n0\n\u0016\u001a0: (1)\nThe fundamental radial mode of kink ( m=1) waves has its\nfrequency in the Alfvén continuum and is always resonantly\ndamped. Here we consider standing waves, which means that\nthe axial wave number kzis real and the frequency !is complex.\nHence\n!=!R+i\r;\nexp(\u0000i!t)=exp(\u0000i!Rt) exp(\rt)=exp(\u0000i!Rt) exp(\u0000t=\u001cD)\n: (2)\nThe Period =2\u0019=! R,\ris the decrement, and \u001cD=1=j\rj\nthe damping time. The variation in density is confined to a layer\nof thickness land has a steepness \u000b. The steepness \u000bis related to\nthe factor Fintroduced by Arregui et al. (2007, 2008a) and has\nbeen used many times before, for example, by Goossens et al.\n(2008), Arregui et al. (2008b), and Soler et al. (2009) as\nF=\u000b4\n\u00192: (3)\nWhen we aim to arrive at analytical expressions for the period\nand damping time, we can use the thin tube and thin boundary\n(TTTB) approximation so that kzR\u001c1,l=R\u001c1. Analytical\nexpressions for the damping time \u001cDwere derived by Hollweg\n(1990a), Goossens et al. (1992), Ruderman & Roberts (2002),\nVan Doorsselaere et al. (2004), Soler et al. (2013), and Arregui\n& Goossens (2019). We use the expression given by Arregui &\nGoossens (2019),\n\u001cD\nPeriod=1\njmj4\n\u00192\u000b\nl=R\u001ai+\u001ae\n\u001ai\u0000\u001ae: (4)\nIn this equation, the quantities l=Rand\u000barise from the adop-\ntion of variation in the equilibrium density in a non-uniform\nlayer of thickness land with steepness \u000bso that the derivative\nof the density at the resonant position rAis given by\nd\u001a0\ndr]r=Re(rA)=\u0000\u000b\u001ai\u0000\u001ae\nl: (5)\nWe first recall that !is complex, with an imaginary part \r;\nand strictly speaking, the position rAis in the complex plane.\nThe thin boundary (TB) approximation assumes that the abso-\nlute value of \ris low in comparison with !Rand that Im( rA) is\nneglected. In what follows, Im( rA) is neglected unless otherwise\nstated.\nThe steepness \u000bis 1 for a linear variation of density and \u0019=2\nfor a sinusoidal variation. For kink waves, m=1, but any non-\naxisymmetric wave ( m,0) with its frequency in the Alfvén con-\ntinuum is resonantly damped. Soler (2017) studied the resonant\ndamping of fluting modes ( jmj\u00152) and showed that these modes\ncan be heavily damped ( \u001cD=Period\u001c1). We note also that the\nnon-uniform layers are often thick and the condition l=R\u001c1 is\nnot satisfied (see e.g. Goossens et al. 2002a; Arregui et al. 2007;\nGoossens et al. 2008; Pascoe et al. 2019). The errors associated\nwith the use of the TTTB approximation beyond its theoretical\nrange of applicability have been estimated by Van Doorsselaere\net al. (2004) and Soler et al. (2014).\nArticle number, page 2 of 9M. Goossens et al.: Transformation of compressive motions into vortical motions\n3. MHD waves with mixed properties for a straight\nfield\nWe recall from Goossens et al. (2019) that the mixed properties\narise because the MHD equations are coupled. The coupling of\nthe MHD equations is controlled by the coupling functions CA\nandCS, which were first introduced by Sakurai et al. (1991). For\na straight field B0=B0(r)1z, they take the simple form\nCA=gBP0=m\nrB0P0;CS=P0; (6)\nwith P0the Eulerian perturbation of total pressure. This means\nthat the equations are coupled because of P0. This was already\nknown by Hasegawa & Uberoi (1982). They noted that\nThe basic characteristic of the ideal Alfvén wave is that\nthe total pressure in the fluid remains constant during the\npassage of the wave as a consequence of the incompress-\nibility condition. For inhomogeneous medium, however,\nthe total pressure, in general, couples with the dynamics\nof the motion, and the assumption of neglect of pressure\nperturbations becomes invalid.\nWe now concentrate on compression and parallel vorticity.\nWe also take non-axisymmetric waves m,0 because for m=0,\nCA=0;and there is no coupling of equations, no mixed proper-\nties, and no resonant damping. Because we study non-stationary\nMHD waves, the frequency !that appears in the following equa-\ntions is a complex quantity, as defined in Eq. (2). For a pressure-\nless plasma, the velocity of sound vS=0 and there are no par-\nallel motions \u0018z=\u0018k=0. We take v2\nS=0 in equations (45) of\nGoossens et al. (2019) and find for the components of the dis-\nplacement \u0018and for compression r\u0001\u0018\n\u0018r=1\n\u001a0(!2\u0000!2\nA)dP0\ndr;\n\u0018?=\u0018'=im=r\n\u001a0(!2\u0000!2\nA)P0;\n\u0018k=\u0018z=0;\nr\u0001\u0018=\u0000P0\n\u001a0v2\nA=\u0000P0\nB2\n0=\u00160: (7)\nSimilarly, we take v2\nS=0 in equations (53) of Goossens et al.\n(2019) and find for the components of vorticity r\u0002\u0018\n(r\u0002\u0018)r=kzm\nr1\n\u001a0(!2\u0000!2\nA)P0:\n(r\u0002\u0018)?=(r\u0002\u0018)'=ikz1\n\u001a0(!2\u0000!2\nA)dP0\ndr;\n(r\u0002\u0018)k=(r\u0002\u0018)z\n=\u0000im\nr1\nn\n\u001a0(!2\u0000!2\nA)o2d\ndrn\n\u001a0(!2\u0000!2\nA)o\nP0: (8)\nEquations (7) and (8) clearly show that P0plays the role of\nthe coupling function. The transverse components of vorticity(r\u0002\u0018)rand (r\u0002\u0018)'are always non-zero. The parallel component\n(r\u0002\u0018)k=(r\u0002\u0018)zis non-zero when\nd\ndrn\n\u001a0(!2\u0000!2\nA)o\n,0: (9)\nAll wave variables are non-zero in a non-uniform plasma.\nExcept that here \u0018k=0 because of the assumption v2\nS=0.\nThere are no pure magneto-acoustic waves and no pure Alfvén\nwaves in a non-uniform plasma. The MHD waves have mixed\nproperties, and they also behave di \u000berently in di \u000berent parts of\nthe plasma because of the inhomogeneity of the plasma, as al-\nready emphasised, for example, by Goossens (2008), Goossens\net al. (2002a), Goossens et al. (2006), Goossens et al. (2011),\nGoossens et al. (2012), and Goossens et al. (2019).\nWe note that for a straight constant field B0=B01z, the\nspatial variation of !2\nAis solely due to the spatial variation of the\nequilibrium density \u001a0, hence\nd\u001a0\ndr,0 (10)\nis the important quantity for the resonant behaviour. The expres-\nsion for the parallel component of vorticity then simplifies to\n(r\u0002\u0018)k=(r\u0002\u0018)z=\u0000im\nr!2\nn\n\u001a0(!2\u0000!2\nA)o2d\u001a0\ndrP0: (11)\nIn order to make clear, as was done by Goossens et al. (2019)\nin their equations (62, 63, and 64), that parallel vorticity and\ncompression go together in a non-uniform plasma, the previous\nEq. (11) can be rewritten as\n(r\u0002\u0018)k=(r\u0002\u0018)z=im\nr!2\nn\n\u001a0(!2\u0000!2\nA)o2d\u001a0\ndrB2\n0\n\u00160r\u0001\u0018:(12)\nWe note that\n\u001a0v2\nA=B2\n0\n\u00160=constant; \u001a 0!2\nA=k2\nzB2\n0\n\u00160=constant:\n4. Analysis close to the ideal resonant point\nThis section investigates the behaviour of the wave variables for\na standing damped resonant wave close to the ideal resonant\npoint rA. For a resonantly damped standing wave, Eq. (2) reads\n!=!R+i\r; \r =!I; ! R\u0019!A(rA);\nj\rj\n!R=1\n2\u0019Period\n\u001cD: (13)\nAn immediate consequence of the fact that the frequency,\n!, is complex in non-stationary ideal MHD is that the resonant\npoint, rA, is also a complex quantity. The higher the damping\nrate,\r, the larger the imaginary part of rA. The following analytic\ninvestigation assumes weak damping, so that the imaginary part\nofrAcan be ignored. We therefore treat rAas a real quantity as\nin stationary MHD. However, we are aware that this is just an\napproximation. In Section 5 we show that we need to consider\nrAto be complex in order to understand the behaviour of the\nperturbations when wave damping is not weak.\nArticle number, page 3 of 9A&A proofs: manuscript no. 38394final\nIt is instructive to summarise the main results on the be-\nhaviour of resonant MHD waves in ideal MHD. The fundamen-\ntal conservation law for resonant Alfvén waves for a straight field\nwas obtained by Sakurai et al. (1991),\nP0=constant;[P0]=0: (14)\nThis conservation law was later confirmed in dissipative station-\nary MHD by Goossens et al. (1995) for cylindrical plasmas, in\ndissipative non-stationary MHD for incompressible plasmas by\nRuderman et al. (1995) for planar geometry, and by Tirry &\nGoossens (1996) for compressible visco-resistive MHD in cylin-\ndrical geometry. Soler et al. (2013) showed that in non-stationary\nideal MHD, P0displays a logarithmic jump at the resonant point.\nThe jump is proportional to the imaginary part of rA, so that the\nconservation law of P0=constant is approximately valid if the\ndamping is only weak (see Section 5).\nIn view of the observed periods and damping times, it is an\naccurate approximation to use\nj\rj\n!R\u001c1: (15)\nIn the first instance we concentrate on the parallel component\nof vorticity. The analysis in Goossens et al. (2019) has shown\nthat in ideal MHD, the dominant dynamics resides in the per-\npendicular component of the displacement and the parallel com-\nponent of vorticity in ideal MHD. We use the following approx-\nimate results by retaining the dominant terms, that is, the lowest\norder terms inj\rj=!R:\n!2d\u001a0\ndr=!2\nR(1+i\r\n!R)2d\u001a0\ndr\u0019!2\nRd\u001a0\ndr;\n!2=(!R+i\r)2\u0019!2\nR+2i\r!R;\n!2\u0000!2\nA\u00192i\r!R;\nn\n\u001a0(!2\u0000!2\nA)o2\u0019 \u0000 4!4\nR(\u001a0(rA))2 j\rj\n!R!2\n;\n\u0000im\nr!2\nn\n\u001a0(!2\u0000!2\nA)o2d\u001a0\ndr\u0019im\nrA1\n\u001a0(rA)!2\nR\n\u00021\n\u001a0(rA)d\u001a0\ndr\u00192\u001a\u001cD\nPeriod\u001b2\n:(16)\nWith the use of Eq. (16), we obtain for the parallel compo-\nnent of vorticity\n(r\u0002\u0018)z\u0019im\nrA\u00192\nk2z1\n\u001a0(rA)d\u001a0\ndr1\nB2\n0=\u00160\u001a\u001cD\nPeriod\u001b2\nP0: (17)\nThe aim is to relate vorticity to compression. In particular,\nwe are interested in parallel vorticity and compression. The rea-\nson is that for a uniform plasma of infinite extent, these two\nquantities are mutually exclusive and characterise Alfvén waves\nand fast magneto-acoustic waves, respectively. Goossens et al.\n(2019) have already pointed out that all wave variables are non-\nzero in a non-uniform plasma so that there are no pure magneto-\nacoustic waves and no pure Alfvén waves.For a pressure-less plasma with a straight magnetic field, the\nexpression forr\u0001\u0018is simplyr\u0001\u0018=\u0000P0=(B2\n0=\u00160). Hence at\nr=rA\nj(r\u0002\u0018)zj\njr\u0001 \u0018j]rA\u0019jmj\nrA\u00192\nk2z1\n\u001a0(rA)jd\u001a0\ndrj\u001a\u001cD\nPeriod\u001b2\n: (18)\nSo far, we have not made any assumption about the variation\nin the equilibrium density in the non-uniform layer or about the\nthickness of the non-uniform layer. We try to obtain an estimate\nfor\n1\n\u001a0(rA)jd\u001a0\ndrj:\nWe assume that the variation in density is confined to a non-\nuniform layer of thickness land has a steepness \u000b;\nrA\u0019R; \u001a 0(rA)\u0019\u001ai+\u001ae\n2:\njd\u001a0\ndrj \u0019\u000b\u001ai\u0000\u001ae\nl=R1\nR;1\n\u001a0(rA)jd\u001a0\ndrj\u00192\u001ai\u0000\u001ae\n\u001ai+\u001ae\u000b\nl=R1\nR:\nHence\nj(r\u0002\u0018)zj\njr\u0001 \u0018j]rA\u0019jmj\u00192\n(kzR)22\u001ai\u0000\u001ae\n\u001ai+\u001ae\u000b\nl=R\u001a\u001cD\nPeriod\u001b2\n: (19)\nThis expression gives the dependence of j(r\u0002\u0018)zjonkzR,\nl=R, and density contrast for a variation in density with steep-\nness\u000b. Other prescriptions for the variation of density can be\nused. We note that there is a hidden dependence on kzR,l=R,\nand density contrast because \u001cD=Period also depends on these\nquantities. Nevertheless, this simple formula is relatively good\nfor the interpretation of numerical results obtained, for example,\nby Goossens et al. (2012). We improve this if we cheat slightly.\nIn the TT and TB approximation, analytic expressions exist for\n\u001cD=Period. With the use of expression (4) for \u001cD=Period, we can\nrewrite Eq. (19) as\nj(r\u0002\u0018)zj\njr\u0001 \u0018j]rA\u00191\njmj32\n\u001921\n(kzR)2\u001ai+\u001ae\n\u001ai\u0000\u001ae\u000b3\n(l=R)3: (20)\nWe now concentrate on the transverse components ( r\u0002\u0018)r\nand (r\u0002\u0018)'. Expressions for these two components are given in\nthe first two equations of expression (8).\nThese two quantities are non-zero everywhere. They do not\nrequire non-uniformity to be non-zero. We now try to determine\ntheir values at and close to the position r=rAwhere!=!A(rA).\nWe recall that at this position,\n\u001a0(!2\u0000!2\nA)\u00192i\r\n!R\u001a0(rA)!2\nR;\nso that for the radial component\n(r\u0002\u0018)r\u0019 \u0000 im\nkzrA\u0019\u001cD\nPeriod1\nB2\n0=\u00160P0\n\u0019im\nkzrA\u0019\u001cD\nPeriodr\u0001\u0018: (21)\nArticle number, page 4 of 9M. Goossens et al.: Transformation of compressive motions into vortical motions\nHence\nj(r\u0002\u0018)rj\njr\u0001 \u0018j]rA\u0019jmj\u0019\nkzR\u001cD\nPeriod; (22)\nwhere we have taken rA\u0019R.\nWith the approximate expression (4) for \u001cD=Period for a vari-\nation in density of steepness \u000b;this can be rewritten as\nj(r\u0002\u0018)rj\njr\u0001 \u0018j]rA\u00191\nkzR4\n\u0019\u001ai+\u001ae\n\u001ai\u0000\u001ae\u000b\n(l=R): (23)\nThe azimuthal component of vorticity can be written as\n(r\u0002\u0018)'\u00191\n\u0019kz\u001cD\nPerioddP0\ndr1\nB2\n0=\u00160(24)\nand\nj(r\u0002\u0018)'j\njr\u0001 \u0018j]rA\u00191\n\u0019kz\u001cD\nPeriod1\nP0jdP0\ndrj: (25)\nFor a straight field the Eulerian perturbation of total pres-\nsure is a conserved quantity (see e.g. Sakurai et al. 1991; Tirry\n& Goossens 1996). In non-stationary ideal MHD, it is approxi-\nmately conserved when the damping is weak (Soler et al. 2013)\nand does not undergo strong spatial variations. This is also veri-\nfied by numerical computations. This leads us to usedP0\ndr\u0019P0=R\nas an estimate for the derivative. In Fig. 1 the compression,\nwhich is proportional to P0, is plotted as a function of the ra-\ndial position for the fundamental radial mode. This figure shows\nthat a linear function is a very good representation for P0. With\nthis approximation we obtain\nj(r\u0002\u0018)'j\njr\u0001 \u0018j]rA\u00191\n\u0019kzR\u001cD\nPeriod: (26)\nWith the use of expression (4) for \u001cD=Period, we can rewrite\nEq. (26) as\nj(r\u0002\u0018)'j\njr\u0001 \u0018j]rA\u00191\njmj1\nkzR4\n\u00193\u001ai+\u001ae\n\u001ai\u0000\u001ae\u000b\n(l=R): (27)\nWe recall that these expressions only apply in the non-\nuniform part of the loop, and strictly speaking, they apply only\nclose to the position rAwhere!=!A(rA). These expressions\ngive us a good description for understanding what happens with\nthe spatial solutions when a wave undergoes resonant damping.\n5. Spatial variation for resonant absorption for a\ncoronal loop\nHere we apply our results from the previous two sections to un-\nderstand and predict the spatial behaviour of MHD waves that\nundergo resonant absorption in the equilibrium coronal loop de-\nfined in Section 3. The aim here is to obtain a simple understand-\ning of how the fundamental quantities compression and vorticity\nare influenced by the non-stationary behaviour of the resonantly\ndamped wave.\nCompression and transverse vorticity do not require non-\nuniformity. They are non-zero everywhere in the plasma col-\numn. However, parallel vorticity is zero in a uniform plasma.\nFig. 1. Absolute value of the compression as a function of the radial\nposition in a flux tube with l=R=0:5,kzR=0:1, and\u001ai=\u001ae=3. The\nnormalisation max fjr\u0001\u0018jg=1 has been used. The shaded zone de-\nnotes the non-uniform region.\nIt requires a spatial variation of the equilibrium magnetic field\nand/or the equilibrium density. For a constant axial magnetic\nfield, the spatial variation in density causes the non-zero parallel\nvorticity. Hence we arrive at the following situation for the res-\nonant damping of transverse waves in the Alfvén continuum in\nthe equilibrium model of Section 3: In the uniform internal part\nof the loop, 0\u0014r\u0014R\u0000l=2;and in the uniform external part of\nthe loop, R+l=2\u0014r<1;we expect compression and transverse\nvorticity. We do not expect parallel vorticity. In the non-uniform\npart of the loop, we expect compression and both transverse and\nparallel vorticity. In the vicinity of the resonant point r=rA, the\nparallel vorticity is expected to dominate the transverse vorticity,\nj(r\u0002\u0018)zj\u001dj(r\u0002\u0018)rj\u0019j(r\u0002\u0018)'j; (28)\nso that the MHD wave is almost a pure Alfvén wave.\nIn order to verify these predictions based on approximate an-\nalytic theory, now we consider the semi-analytic approach of\nSoler et al. (2013) in non-stationary ideal MHD. The method\nwas inspired by an early work by Hollweg (1990b) in a sim-\nplified Cartesian configuration. We give a brief summary of the\ntechnique. We express the perturbations in the non-uniform part\nof the loop as a Frobenius series that includes a singular term\naccounting for the e \u000bect of the Alfvén resonance. This series\nexpansion allows us to connect through the non-uniform layer\nthe perturbations in the internal medium to those in the external\nmedium. Thus, we obtain a dispersion relation for all trapped\nmodes with any value of the azimuthal wavenumber, m. How-\never, we focus on m=1. The dispersion relation is valid for\narbitrary values of l=R. When the nonuniform layer is thin, that\nis,l=R\u001c1, the obtained dispersion relation consistently pro-\nvides the same results as those of the TTTB approximation. The\nnumerical part of the method consists of numerically solving\nthe transcendental dispersion relation to obtain the complex fre-\nquency,!, for a fixed kz. When the frequency is known, we can\ncompute the spatial behaviour of the perturbations. We refer to\nSoler et al. (2013) for a more detailed explanation of the method.\nSoler et al. (2013) studied the components of Lagrangian dis-\nplacement\u0018r,\u0018';and the Eulerian perturbation of total pressure,\nP0. Compression and the components of vorticity have now been\ncomputed and are presented in Figs. 1-6. A similar exercise was\nArticle number, page 5 of 9A&A proofs: manuscript no. 38394final\nFig. 2. Absolute value of the radial (left), azimuthal (centre), and parallel (right) vorticity components as functions of the radial position in the\nsame flux tube as in Fig. 1.\nmade for the parallel vorticity in non-stationary resistive MHD\nby Goossens et al. (2012) (see their figure 4). The normalisation\nmaxfjr\u0001\u0018jg=1 was used in all figures. A sinusoidal variation\nfor the density in the nonuniform part was assumed.\nIn Figs. 1 and 2 we plot compression and the components of\nvorticity on the interval r=R2[0;3] for a loop with a relatively\nthin non-uniform layer with l=R=0:5. We also used kzR=0:1\nand\u001ai=\u001ae=3. This case can be directly compared with the ap-\nproximate TTTB results obtained in the previous sections. Fig-\nure 1 shows that nothing particular happens for compression.\nThe behaviour does not substantially di \u000ber from that found in the\nstepwise constant case. Figure 2 shows that the transverse com-\nponents of vorticity are non-zero in the whole domain; the par-\nallel component of vorticity is only non-zero in the non-uniform\npart of the loop. The parallel component of vorticity is far larger\nthan the transverse components. In Fig. 2 the maximum value\nof the parallel component is at least four orders of magnitude\nhigher than the maximum values of the transverse components.\nIn turn, the amplitudes of the two transverse components are of\nthe same order of magnitude. All these results agree with the\nanalytic predictions in the TTTB approximation.\nNext we consider l=R=0:5 as before and compute the com-\nponents of vorticity for various values of kzRand\u001ai=\u001ae. In Fig. 3\nwe considered three values of kzR=0.1, 0.3, and 0.5 and \u001ai=\u001ae\n=2, while in Fig. 4 we considered three values of \u001ai=\u001ae=2, 5,\nand 10 and kzR=0.1. Because the parallel vorticity is only non-\nzero in the non-uniform transitional layer, the components are\nplotted on the interval r2[R\u0000l=2;R+l=2] alone. The results\nare plotted in both linear and logarithmic scales for better visu-\nalisation, and we recall that max fjr\u0001\u0018jg=1 in all cases. The\nspatial profile of vorticity is largely una \u000bected when either kzR\nor\u001ai=\u001aeare modified. This is so because the spatial behaviour of\nvorticity is largely determined by the spatial variation of density,\nwhich is the same in all cases. However, the relative amplitudes\nof the three components of vorticity depend upon the considered\nkzRand\u001ai=\u001ae. The larger kzR, the lower the vorticity. The ampli-\ntude of the field-aligned component of vorticity decreases with\nkzRas(kzR)\u00002, approximately, as Eq. (19) predicts, while both\nradial and azimuthal components behave as (kzR)\u00001according\nto Eq. (22) and (26). Equivalently, the higher the density con-\ntrast,\u001ai=\u001ae, the lower the vorticity. The dependence on density\ncontrast is also consistent with the analytic TTTB formulas, al-\nthough there is no such clear dependence as in the case of kzR.\nWe have confirmed so far that the analytic predictions in the\nTTTB approximation are consistent with the full solution pro-\nvided by the method of Soler et al. (2013) when the non-uniformtransition is thin. Now, we can be more ambitious and fully ex-\nploit the method by computing results beyond the range of ap-\nplicability of the approximations. Figures 5 and 6 display com-\npression and the components of vorticity, respectively, for a loop\nwith a thick non-uniform layer of l=R=1:5. The other parame-\nters are the same as in Figs. 1 and 2.\nWhen we compare the plots of compression for a thin (Fig. 1)\nand a thick (Fig. 5) non-uniform transition, a distinct feature is\nobvious. In the case of a thick transition, compression displays\na small jump at the resonance position. In order to understand\nthe behaviour of compression, we recall that compression is pro-\nportional to P0and we resort to the Frobenius series of P0. In\ngeneral, the full solution provided by the method of Frobenius is\ndi\u000ecult to handle (see Soler et al. 2013). To illustrate the present\ndiscussion, it su \u000eces to consider the first non-zero terms in the\nFrobenius series of P0in the nonuniform layer,\nP0\u0019S0+0BBBB@A0+S0m2\n2r2\nAln(r\u0000rA)1CCCCA(r\u0000rA)2; (29)\nwhere A0andS0are constants. The neglected terms in Eq. (29)\nare of order (r\u0000rA)3and higher. We note that in non-stationary\nideal MHD, rAas defined by the resonant condition !=!A(rA)\nis a complex quantity because !is a complex quantity. As a\nconsequence of this, there is not an actual resonance but a quasi-\nresonance at the radial position where r=Re(rA). At the quasi-\nresonance position, r\u0000rA=\u0000iIm(rA),0. The imaginary part\nofrAowes its existence to the non-zero damping rate, \r. Equa-\ntion (29) has a term proportional to (r\u0000rA)2ln(r\u0000rA). This\nterm does not vanish at r=Re(rA) as it would do at r=rA\nifrAwere real. Instead, the logarithm jumps when r\u0000rAcrosses\nthe imaginary axis. In the case of a thin non-uniform layer,\n\ris small, so that Im( rA) is small and can be ignored. Then,\nP0\u0019constant as in stationary ideal MHD and as the thin bound-\nary approximation assumes (see Hollweg & Yang 1988). Con-\nversely, when the nonuniform layer is thick the wave is strongly\ndamped, non-stationarity is important, \ris not small, and Im( rA)\nis not negligible. Hence, the jump in P0. This explains the pres-\nence of the jump in compression seen in Fig. 5. Of course, if a\ndissipative process is taken into account, the jumps obtained in\nnon-stationary ideal MHD are replaced by smooth variations in\nnon-stationary dissipative MHD (see e.g., Mok & Einaudi 1985;\nRuderman et al. 1995; Wright & Allan 1996; Vanlommel et al.\n2002; Soler et al. 2012, 2013).\nOn the other hand, by comparing Figs. 2 and 6, we see that\nthe components of vorticity have a smaller amplitude in the case\nArticle number, page 6 of 9M. Goossens et al.: Transformation of compressive motions into vortical motions\nFig. 3. Absolute value of the radial (left), azimuthal (centre), and parallel (right) vorticity components in the non-uniform part of a flux tube with\nl=R=0:5 and\u001ai=\u001ae=2. The top panels are in linear scale, and the bottom panels are in logarithmic scale. The di \u000berent line styles denote kzR=0:1\n(solid black line), kzR=0:3 (dotted red line), and kzR=0:5 (dashed blue line).\nFig. 4. Same as Fig. 3, but with kzR=0:1 and di \u000berent values of the density contrast: \u001ai=\u001ae=2 (solid black line), \u001ai=\u001ae=5 (dotted red line), and\n\u001ai=\u001ae=10 (dashed blue line).\nof a thick transition. Remarkably, the amplitude of the field-\naligned component is more than an order of magnitude smaller.\nThe analytic TTTB approximations, although not strictly valid\nnow, are helpful to understand this result. According to Eq. (20),the parallel component should behave as (l=R)\u00003, while both ra-\ndial and azimuthal components should behave as (l=R)\u00001accord-\ning to Eqs. (23) and (27). The predicted behaviour in the TTTB\napproximation explains the decrease in amplitude of the vorticity\nArticle number, page 7 of 9A&A proofs: manuscript no. 38394final\nFig. 5. Same as Fig. 1, but with l=R=1:5.\ncomponents when l=Rincreases and why the parallel component\ndecreases more than the perpendicular components. Finally, we\nalso note that in Fig. 6 all the components of vorticity jump at\nthe resonance position, whereas the jumps were not apparent in\nFig. 2.\n6. Conclusions\nWe used linear non-stationary ideal MHD to investigate the spa-\ntial behaviour of compression and vorticity for MHD waves that\nundergo resonant absorption in the Alfvén continuum. In linear\nMHD there is no interaction between waves, and the behaviour\nthat we discussed is associated with a single MHD wave that\nlives in the whole space. Pure Alfvén waves and pure magneto-\nacoustic waves exist in a uniform plasma of infinite extent. In\na non-uniform plasma, the MHD waves combine the properties\nof the classic Alfvén waves and of the magneto-acoustic waves\nin a uniform plasma of infinite extent. In a non-uniform plasma,\nMHD waves propagate both compression and parallel and trans-\nverse vorticity. The properties of the MHD wave depend on the\nproperties of the background plasma. As an MHD wave prop-\nagates through a non-uniform plasma, its properties therefore\nchange. When an MHD wave moves from a uniform into a non-\nuniform plasma, it is transformed from a fast magneto-acoustic\nwave into a mixed fast - Alfvén wave.\nResonant absorption is a clear example of the phenomenon\nthat the properties of an MHD wave change when it travels\nthrough an inhomogeneous plasma. In the case of resonant\nAlfvén waves, the MHD wave eventually arrives at a position\nwhere it behaves as an almost pure Alfvén wave, but with the\nunfamiliar property that it has pressure variations. The total\npressure perturbation and compression are non-zero everywhere.\nThe pressure variations are essential for resonant absorption be-\ncause the amount of absorbed energy and the damping rate are\ndirectly related to the pressure variations (see e.g. Thompson\n& Wright 1993; Tirry & Goossens 1995; Andries & Goossens\n2001; Goossens et al. 2002b; Goossens 2008; Goossens et al.\n2011; Arregui et al. 2011).\nClassic Alfvén waves are not the only waves to propagate\nvorticity from the photosphere to outer space. MHD waves that\nundergo resonant absorption in a non-uniform plasma can also\nplay this role.\nAcknowledgements. M.G. was supported by the C1 grant TRACEspace of In-\nternal Funds KU Leuven (number C14 /19/089). I.A. was supported by projectPGC2018-102108-B-I00 from Ministerio de Ciencia, Innovación y Universi-\ndades and FEDER funds. R.S. acknowledges the support from grant AYA2017-\n85465-P (MINECO /AEI/FEDER, UE) and from the Ministerio de Economía, In-\ndustria y Competitividad and the Conselleria d0Innovacio, Recerca i Turisme del\nGovern Balear (Pla de ciència, tecnologia, innovació i emprenedoria 2013-2017)\nfor the Ramón y Cajal grant RYC-2014-14970. T.V .D. was supported by the\nEuropean Research Council (ERC) under the European Union’s Horizon 2020\nresearch and innovation programme (grant agreement No 724326) and the C1\ngrant TRACEspace of Internal Funds KU Leuven (number C14 /19/089).\nReferences\nAndries, J. & Goossens, M. 2001, A&A, 375, 1100\nArregui, I., Andries, J., Van Doorsselaere, T., Goossens, M., & Poedts, S. 2007,\nAstron. Astrophys., 463, 333\nArregui, I., Ballester, J., & Goossens, M. 2008a, Astrophys. J. Lett., 676, L77\nArregui, I. & Goossens, M. 2019, A&A, 622, A44\nArregui, I., Soler, R., Ballester, J. L., & Wright, A. 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Same as Fig. 2, but with l=R=1:5.\nArticle number, page 9 of 9" }, { "title": "1109.2636v1.Reduction_of_compressibility_and_parallel_transfer_by_Landau_damping_in_turbulent_magnetized_plasmas.pdf", "content": "arXiv:1109.2636v1 [physics.plasm-ph] 12 Sep 2011Reduction of compressibility and parallel transfer by Land au\ndamping in turbulent magnetized plasmas\nP. Hunana,1,∗D. Laveder,1T. Passot,1P. L. Sulem,1and D. Borgogno2\n1Universit´ e de Nice Sophia Antipolis,\nCNRS, Observatoire de la Cˆ ote d’Azur,\nBP 4229 06304, Nice Cedex 4, France\n2Dipartimento di Energetica, Politecnico di Torino,\ncorso Duca degli Abruzzi 24, 10138 Torino, Italy\n(Dated: June 6, 2021)\nAbstract\nThree-dimensional numerical simulations of decaying turb ulencein a magnetized plasma are per-\nformed using a so-called FLR-Landau fluid model which incorp orates linear Landau damping and\nfinite Larmor radius (FLR) corrections. It is shown that comp ared to simulations of compressible\nHall-MHD, linear Landau damping is responsible for signific ant damping of magnetosonic waves,\nwhich is consistent with the linear kinetic theory. Compres sibility of the fluid and parallel energy\ncascade along the ambient magnetic field are also significant ly inhibited when the beta parameter\nis not too small. In contrast with Hall-MHD, the FLR-Landau fl uid model can therefore correctly\ndescribe turbulence in collisionless plasmas such as the so lar wind, providing an interpretation for\nits nearly incompressible behavior.\n∗Electronic address: hunana@oca.eu\n1I. INTRODUCTION\nHydrodynamics and Magnetohydrodynamics (MHD) are the centra l descriptions used to\nstudyturbulenceinthesolarwindandinawiderangeofnaturalsyst ems. Specificallyforthe\nsolar wind, MHD description yielded a great success in our understan ding of observational\ndata (e.g. see reviews by Goldstein et al. [1], Tu and Marsch [2], Bruno and Carbone [3],\nHorbury et al. [4], Marsch [5], Ofman [6]). Observational studies show that the solar wind is\ntypically found to be only weakly compressible (e.g. see Matthaeus et al. [7], Bavassano and\nBruno [8] andthereviews citedabove) andusual turbulence models which predict theenergy\nspectra are derived in the framework of an incompressible MHD desc ription (Iroshnikov [9],\nKraichnan [10], Goldreich and Sridhar [11, 12], Galtier et al. [13, 14], Bold yrev [15, 16],\nLithwick etal. [17], Chandran[18], Perez andBoldyrev[19], Podesta an dBhattacharjee[20],\nseealsoreviewsbyChoetal. [21], Zhouetal. [22], Galtier[23]andSridh ar[24]). Theoretical\nmodelswhichdescribetheradialevolutionofspatiallyaveragedsolar windquantitiesarealso\nusually developed in the framework of incompressible MHD formulated in Els¨ asser variables\n(e.g. Zhou and Matthaeus [25–27], Marsch and Tu [28], Zank et al. [29], Smith et al. [30],\nMatthaeus et al. [31], Breech et al. [32]). It is however well known th at the solar wind\nis not completely incompressible and many phenomena which require co mpressibility are\nobserved in the solar wind, such as the evolution of density fluctuat ions (e.g. Spangler and\nArmstrong [33], Armstrong et al. [34], Coles et al. [35], Grall et al. [36], Woo and Habbal\n[37], Bellamy et al. [38], Wicks et al. [39], Telloni et al. [40]), magnetic holes, solitons\nand mirror mode structures (e.g. Winterhalter et al. [41], Fr¨ anz e t al. [42], Stasiewicz et\nal. [43], Stevens and Kasper [44]) or strong temperature anisotrop ies which trigger, and are\nlimited by, micro-instabilities such as mirror and fire-hose (e.g. Gary e t al. [45], Kasper\net al. [46], Hellinger et al. [47], Matteini et al. [48], Bale et al. [49]). The impo rtance of\ncompressibility was stressed by Carbone et al. [50], who compared th e observational data\nwith the energy flux scaling laws of Politano and Pouquet [51], which are exact relations of\nincompressible MHD. Carbone et al. determined that the scaling relat ions can better fit the\ndata if the relations are phenomenologically modified to account for c ompressibility. The\nincorporation of weakly compressional density fluctuations was pa rtially addressed by so-\ncalled nearly incompressible models, which expand the compressible eq uations with respect\nto small sonic Mach number (Matthaeus and Brown [52], Zank and Ma tthaeus [53–55])\n2and which were recently formulated in the presence of a static large -scale inhomogeneous\nbackground (Hunana and Zank [56], Hunana et al. [57, 58], see also Bh attacharjee et al.\n[59]). These models however specifically assume, and cannot explain, why the solar wind is\nonly weakly compressible. The theoretical compressible MHD models d eveloped in the wave\nturbulence formalism (e.g. Kuznetsov [60], Chandran [61]) also canno t address this issue.\nDescribing the solar wind with fully compressible MHD or Hall-MHD formalis ms yields\nseveral problems. Most importantly, these compressible descript ions introduce sound waves\nand slow magnetosonic waves. As elaborated by Howes [62], slow magn etosonic waves are\nstrongly damped by Landau damping in the kinetic Maxwell-Vlasov desc ription. The pres-\nence of fast and slow magnetosonic waves naturally implies higher leve l of compressibility\nand overestimates the parallel energy transfer. Also, numerical simulations of compressible\nHall-MHD performed by Servidio et al. [63] in the context of the magn etopause boundary\nlayer showedthatcomparedtotheusual turbulenceincompressib le MHDsimulations, which\nconsists of Alfv´ en waves, the Hall term is responsible for decouplin g of magnetic and velocity\nfield fluctuations. In compressible Hall-MHD regime, Servidio et al. obs erved spontaneous\ngeneration of magnetosonic waves which transform to a regime of q uasi perpendicular “mag-\nnetosonic turbulence”. This is in contrast with observational stud ies which typically show\nthat turbulence in the solar wind predominantly consists of quasi pe rpendicular (kinetic)\nAlfv´ en waves (e.g., Bale et al. [64], Sahraoui et al. [65, 66]). Finally, a ll usual MHD or\nHall-MHD models cannot address how the energy is actually dissipated at small scales. It\nis well known that solar wind plasma is almost collisionless and therefore the classical von\nKarman picture of energy being dissipated via viscosity is not applicab le to the solar wind.\nIt is evident that new and more realistic models have to be introduced which can overcome\nsome of these drawbacks.\nThe most realistic approach is of course the fully kinetic Vlasov-Maxw ell description. It\nis however analytically intractable, and even the biggest kinetic simula tions cannot resolve\nthe large-scale turbulence dynamics. Two leading approaches appe ar to be promising to\nsubstitute MHD in describing solar wind turbulence : Gyrokinetics and Landau fluids.\nGyrokinetics (e.g. Schekochihin et al. [67], Howes et al. [68]) was origina lly developed\nfor simulations of fusion in tokamaks. It is a kinetic-like description, w hich averages out\nthe gyro-rotation of particles around a mean magnetic field and the refore makes the kinetic\ndescription more tractable, mostly by eliminating fast time scales. De rived directly from the\n3kinetic theory, gyrokinetics has a crucial advantage of being asym ptotically correct. Landau\nfluid description, on the other hand, is a fluid-like extension of compr essible Hall-MHD, in\nwhichwavedissipationisincorporatedkineticallybythemodelingoflinea rLandaudamping,\nthus retaining a realistic sink of energy. Other linear kinetic effects s uch as finite Larmor\nradius corrections are also incorporated.\nThe simplest Landau fluid closure was considered by Hammett and Per kins [69] and the\nassociated dispersion relations were numerically explored by Jayant i, Goldstein and Vi˜ nas\n[70]. The Landau fluid model was further advanced by Snyder, Hamm ett and Dorland [71],\nwho considered the largest MHD scales, starting from the guiding ce nter kinetic equation.\nThe approach was reconsidered and refined to its present form wit h incorporation of Hall\nterm and finite Larmor radius (FLR) corrections by Passot, Sulem, Goswami and Bugnon\n[72–75]. This Landau fluid approach starts with the Vlasov-Maxwell e quations and derives\nnonlinear evolution equations for density, velocity and gyrotropic p ressures. In the simplest\nformulation the model is closed at the level of heat fluxes by matchin g with the linear\nkinetic theory in the low frequency limit. Kinetic expressions usually co ntain the plasma\ndispersion function which is not suitable for fluid-like simulations. Land au fluid closure is\ntherefore performed in a way as to minimize occurrences of this plas ma dispersion function\nand, where not possible, this function is replaced by a Pad´ e approx imant. This eliminates\nthe time non-locality and also results in the presence of a Hilbert tran sform with respect\nto the longitudinal coordinate (in the direction of the ambient magne tic field) which in\nthe fluid formalism is associated with linear Landau damping. Further d etails about the\ndevelopment of Landau fluid models are thoroughly discussed in the p apers cited above.\nThe Landau fluid approach should however be contrasted with the m ore classical gyrofluid\nmodels (e.g. Dorland and Hammett [76], Brizard [77], Scott [78]), which a re derived by\ntaking fluid moments of the gyrokinetic equation and for which a similar closure scheme is\napplied afterwards.\nThe Landau fluid approach has the following advantages. In contra st with Hall-MHD,\nit contains separate equations for parallel and perpendicular pres sures and heat fluxes. It\ntherefore allows for the development of temperature anisotropy , which is observed in the\nsolar wind. Noticeably, in contrast with gyrokinetics or gyrofluid mod els, the Landau fluid\nmodel does not average out the fast waves. Compared with these approaches, it also has\nan advantage in that the final equations including the FLR correctio ns are written for the\n4usual quantities measured in the laboratory frame. Existing spect ral MHD and Hall-MHD\ncodes can therefore be modified to Landau fluid description relative ly easily. Also impor-\ntantly, even though gyrokinetics is a reduced kinetic description, it is still 5-dimensional and\ntherefore naturally quite difficult to compute. While current largest numerical simulations\nof gyrokinetics (Howes et al. [79]) require thousands CPU-cores fo r a fluid-like 128 ×642res-\nolution, the FLR-Landau fluid model requires computational power only slightly larger than\nthe usual Hall-MHD simulations. The results presented here, which e mploy a resolution of\nN= 128 grid points in all three directions, were calculated using 32 CPU- cores.\nLandau fluid models can be developed with several levels of complexity . For these first\nLandau fluid simulations of three dimensional turbulence, we use a sim plified version of the\nmost general Landau fluid model [72], where we constrain ourselves to isothermal electrons\nand leading order corrections in terms of the ratio of the ion Larmor radius to the considered\nscales. A similar model was used by Borgogno et al. [80], who studied th e dynamics\nof parallel propagating Alfv´ en waves in a medium with an inhomogeneo us density profile.\nThey numerically showed that the observed Alfv´ en wave filamentat ion and later transition\nto the regime of dispersive phase mixing is consistent with particle-in- cell simulations. In\nthis paper we concentrate on freely decaying turbulence and comp are these to simulations\nof compressible Hall-MHD. Numerical integration of the full Landau fl uid model in one\nspace dimension was presented by Borgogno et al. [81], who investiga ted the dynamics very\nclose to the mirror instability threshold and showed the presence of magnetic holes. Results\nconsidering quasi-transverse one-dimensional propagation in the full Landau fluid model are\npresented in [82, 83].\nII. THE MODEL AND ITS NUMERICAL IMPLEMENTATION\nConsidering a neutral bi-fluid consisting of protons (ions) and isoth ermal electrons, the\nLandau fluid model consists of evolution equations for proton dens ityρ=mpn(where\nmpis the proton mass and nthe number density), proton velocity up, proton parallel\nand perpendicular pressures p/bardblp,p⊥pand heat fluxes q/bardblp,q⊥p, together with the induction\nequation for magnetic field b. The equations are normalized and density, magnetic field\nand proton velocity are measured in units of equilibrium density ρ0, ambient magnetic field\nB0, and Alfv´ en speed VA=B0/√4πρ0, respectively. Pressures are measured in units of\n5initial proton parallel pressure p(0)\n/bardblpand heat fluxes in units of p(0)\n/bardblpVA. The total pressure\nin the momentum equation has the form of a tensor. Defining ˆb=b/|b|as a unit vector\nin the direction of local magnetic field, the proton pressure can be c ast in the form pp=\np⊥pn+p/bardblpτ+Π, whereτ=ˆb⊗ˆb, andn=I−ˆb⊗ˆb, withIbeing the unit tensor. Finite\nLarmor radius corrections to the gyrotropic pressures are repr esented by Π. Operator ⊗\nrepresents the usual tensor product and in the index notation, f or example, τij=ˆbiˆbj.\nElectrons are assumed to be isothermal with the scalar pressure pe=nT(0)\ne, whereT(0)\neis\nthe electron temperature. Parameter Ri, whose inverse multiplies the Hall-term and also\nthe FLR corrections, is defined as Ri=L/di, wherediis the ion inertial length and Lis\nthe unit length. The proton plasma beta is defined with respect to pa rallel pressure and\nβ/bardbl= 8πp(0)\n/bardblp/B2\n0. The density, momentum and induction equations of the FLR-Landa u fluid\nmodel can then be expressed as\n∂ρ\n∂t+∇·(ρup) = 0, (1)\n∂up\n∂t+up·∇up+β/bardbl\n2ρ∇·(p⊥pn+p/bardblpτ+Π+peI)\n−1\nρ(∇×b)×b= 0, (2)\n∂b\n∂t=∇×(up×b)−1\nRi∇×/bracketleftbigg1\nρ(∇×b)×b/bracketrightbigg\n. (3)\nDropping, for simplicity, indices pfor proton velocity upand proton pressures p⊥p,p/bardblp, the\nevolution equations for perpendicular and parallel pressures read s (neglecting the work done\nby the FLR stress forces)\n∂p⊥\n∂t+∇·(p⊥u)+p⊥∇·u−p⊥ˆb·(∇u)·ˆb\n+∇·(q⊥ˆb)+q⊥∇·ˆb= 0, (4)\n∂p/bardbl\n∂t+∇·(p/bardblu)+2p/bardblˆb·(∇u)·ˆb+∇·(q/bardblˆb)−2q⊥∇·ˆb= 0. (5)\nAssuming an ambient magnetic field of amplitude B0in the positive z-direction, a semi-\nlinear description of the finite Larmor radius corrections in the pres sure tensor neglecting\n6heat flux contributions can be expressed as\nΠxx=−Πyy=−/an}bracketle{tp⊥/an}bracketri}ht\n2Ri(∂yux+∂xuy), (6)\nΠxy= Πyx=−/an}bracketle{tp⊥/an}bracketri}ht\n2Ri(∂yuy−∂xux), (7)\nΠyz= Πzy=1\nRi/bracketleftbig\n2/an}bracketle{tp/bardbl/an}bracketri}ht∂zux+/an}bracketle{tp⊥/an}bracketri}ht(∂xuz−∂zux)/bracketrightbig\n, (8)\nΠxz= Πzx=−1\nRi/bracketleftbig\n2/an}bracketle{tp/bardbl/an}bracketri}ht∂zuy+/an}bracketle{tp⊥/an}bracketri}ht(∂yuz−∂zuy)/bracketrightbig\n, (9)\nΠzz= 0, (10)\nwhere/an}bracketle{tp⊥/an}bracketri}htand/an}bracketle{tp/bardbl/an}bracketri}htrepresents the instantaneous averaged ion pressures over the entire\ndomain, whose time variation is aimed to take into account the evolutio n of the global prop-\nerties of the plasma. Finally, parallel and perpendicular heat fluxes q/bardbl,q⊥evolve according\nto\n/parenleftBigg\nd\ndt+/radicalbigπβ/bardbl\n4(1−3π\n8)H∂z/parenrightBigg\nq/bardbl=1\n1−3π\n8β/bardbl\n2∂z(p/bardbl−ρ), (11)\n/parenleftBigg\nd\ndt−/radicalbigπβ/bardbl\n2H∂z/parenrightBigg\nq⊥=\nβ/bardbl\n2T(0)\n⊥p\nT(0)\n/bardblp∂z/bracketleftBigg/parenleftBigg\n1−T(0)\n⊥p\nT(0)\n/bardblp/parenrightBigg\n|b|−/parenleftBigg\nT(0)\n/bardblp\nT(0)\n⊥pp⊥−ρ/parenrightBigg/bracketrightBigg\n, (12)\nwhereT(0)\n⊥p,T(0)\n/bardblpare the initial perpendicular and parallel proton temperatures and d/dtis\nthe convective derivative. The operator H, which is defined as\nHf(z) =−1\nπVP/integraldisplay+∞\n−∞f(z′)\nz−z′dz′, (13)\nreduces in the Fourier space to a simple multiplication by ikz/|kz|and, is the signature of\nthe linear Landau damping.\nTo gain physical insight into a quite complicated model (1)-(12), it is u seful to momentar-\nily consider just the largest scales by putting 1 /Ri→0, which eliminates the nongyrotropic\ncontributions to the pressure tensor and which also eliminates the H all term. The result-\ning set of equations still contains the linear Landau damping and with t he exception of\nisothermal electrons, it is analogous to the model of Snyder, Hamm ett and Dorland [71]. If\nthis model is further simplified by elimination of eq. (11), (12) and by in stead prescribing\nq/bardbl=q⊥= 0, the resulting model collapses to the double adiabatic model (Che w et al. [84],\n7see also Kulsrud [85]) and the Landau damping disappears. The prese nce of ion Landau\ndamping in the system (1)-(12) is therefore a result of closure equ ations (11), (12) for the\nheat fluxes, which contain the operator H. At least in the static limit, this closure can be\nviewed as a modified Fick’s law where the gradient operator that usua lly relate the heat flux\nto the temperature fluctuations is here replaced by a Hilbert trans form that is a reminis-\ncence of the plasma dispersion function arising in the linear kinetic the ory, and is a signature\nof Landau (zero-frequency) wave-particle resonance. The effe ct of Landau damping in the\nsystem (1)-(12) might be better understood by solving the lineariz ed set of equations. In\nthe Appendix we consider waves which propagate parallel to the amb ient magnetic field.\nIt is shown that except the Alfv´ en waves, linear waves have a freq uency with a negative\nimaginary part, and are therefore damped. This corresponds to lin ear Landau damping.\nThe model used for the simulations presented here has several limit ations. First of all, it\ncontains electrons which are assumed isothermal, a regime in fact of ten assumed in hybrid\nsimulations which provide a kinetic description of the ions and a fluid des cription of the\nelectrons. In the future, more realistic simulations will be performe d with inclusion of\nindependent evolution equations for parallel and perpendicular elec tron pressures and heat\nfluxes. This will also result in the presence of electron Landau dampin g, which is absent in\nthemodelpresented hereandwhichseemstoplayanimportantrole insolarwindturbulence.\nAnother main limitation appears to be the form of finite Larmor radius corrections, which\nare derived as a large-scale limit of FLR corrections of the full Landa u fluid model and\nwhich are therefore significantly simplified. The FLR corrections (6) -(10) are sufficient for\nthe simulations of freely-decaying turbulence, which do not lead to s ignificant temperature\nanisotropies. However, our preliminary simulations which employ forc ing and lead to strong\ntemperature anisotropies show that the FLR corrections (6)-(1 0) are overly simplified and\nmight lead to artificial numerical instabilities. For simulations with stro ng temperature\nanisotropies, a more refined description ofFLR corrections is requ ired (see Passot andSulem\n[72], Borgogno et al. [81]).\nTo explore the behavior of the FLR-Landau fluid model, we performe d simulations of\nfreely decaying turbulence. The code we used is based on a pseudo- spectral discretization\nmethod, where spatial derivatives are evaluated in Fourier space. The time stepping is\nperformed in real space with a 3rd order Runge-Kutta scheme. Sp atial resolution is N3=\n1283and the size of the simulation domain is L= 16×(2π) in each direction. The Hall\n8parameter is Ri= 1, implying that the lengths are measured in the units of ion inertial\nlengthdi. Velocity and magnetic field fluctuations of mean square root amplitu de/an}bracketle{tu2/an}bracketri}ht1/2=\n/an}bracketle{tb2/an}bracketri}ht1/2= 1/8 are initialized in Fourier space on the first four modes kdi=m/16, where\nm∈[1,4], with flat spectra and random phases and are constructed to be divergence free.\nConstant pressures p⊥=p/bardbl= 1, density ρ= 1 and heat fluxes q⊥=q/bardbl= 0 are initialized\nin the entire domain. The temperature of the electrons is T(0)\ne= 1 and is therefore equal\nto proton temperatures, which can be defined as T⊥=/an}bracketle{tp⊥/ρ/an}bracketri}htandT/bardbl=/an}bracketle{tp/bardbl/ρ/an}bracketri}ht. In the\nsimulations presented here, both T⊥andT/bardblstay rather close to their initial value. The\ncompressible Hall-MHD model consists of eq. (1)-(3), where the div ergence of the pressure\ntensor in eq. (2) is substituted with the usual gradient of scalar pr essure and, assuming\nthe adiabatic law p=ργ, in the normalized units it is equal to β0/γ∇ργ, whereβ0is the\nusual plasma beta defined as β0=c2\ns/V2\nAand the sound speed c2\ns=γp/ρ. Adiabatic index\nγ= 1.66 is used.\nAs shown for example by Hirose et al. [86] and Howes [62], it is not str aightforward to\ncompare Hall-MHD and Vlasov-Maxwell kinetic theory because the as sociated dispersion\nrelations are quite different if in the kinetic description the proton (io n) temperature Tp\nis not negligible with respect to the electron temperature Te. We choose to follow Howes\n[62], who, in order to compare Hall-MHD with the kinetic theory (which is represented\nhere by the FLR-Landau fluid model), defined the necessary relatio n between β0andβ/bardbl\nasβ0=β/bardbl(1+Tp/Te)/(2Tp/Te). For equal proton and electron temperatures Tp=Tethis\nrelation yields β0=β/bardbl. Landau damping alone is not sufficient to run the code and some\nkind of artificial dissipation is needed to terminate the cascade. We h ere resorted to use a\nfiltering in the form of1\n2{1−tanh[(m−0.8N/4)/3]}, wheremis the mode index, applied\neach time step on all fields for both Hall-MHD and Landau fluid regimes. Because of the\nfiltering, it is crucial to have identical time steps dt= 0.128 in both models. In most of\nthe simulations we used β0=β/bardbl= 0.8. In normalized units, the Alfv´ en speed is equal to\nunity, the usual MHD sound speed cs=√β0= 0.894 and the turbulent sonic Mach number\nMs=/an}bracketle{tu2/an}bracketri}ht1/2/cs= 0.14. In section IV., we also consider simulations with β0=β/bardbl= 0.25\n(Ms= 0.25) andβ0=β/bardbl= 0.1 (Ms= 0.4).\n910-1610-1410-1210-1010-810-6\n10-210-1100power\nfrequency ( ω)HMHD\nbx, θ =0°\n10-1610-1410-1210-1010-810-6\n10-210-1100power\nfrequency ( ω)Landau\nbx, θ =0°\nFIG. 1: Left and right polarized Alfv´ en waves for the propag ation angle of 0◦for Hall-MHD\n(left) and Landau fluid (right), as revealed by frequency ana lysis ofbxmodes with wavenumbers\nkx= 0,ky= 0,kzdi=m/16, where m= 1 (red), m= 2 (green), m= 4 (blue), m= 8 (black).\nTheoretical predictions for peaks calculated from dispers ion relations for given kare shown on the\ntop axis.\nIII. IDENTIFICATION OF THE MHD MODES\nA wave analysis procedure was implemented in the code, which consist s in choosing few\nmodes in spatial Fourier space for each field and recording their valu e every 20 time steps.\nAfter the run, the time Fourier transform is performed and frequ ency-power spectra are\nobtained for each mode. This procedure makes it possible to identify at which frequency\nthere is maximum power for each mode, and by comparing these with f requencies obtained\nfrom theoretical dispersion relations it allows to uniquely identify whic h waves are present\nin the system. This method was previously used for incompressible MH D by Dmitruk and\nMatthaeus [87], therefore detecting Alfv´ en waves. Considering t he dynamics associated\nwith waves propagating in the direction parallel to the ambient magne tic field (propagation\nangleθ= 0◦), Fig. 1 shows frequency-power spectra of four modes with wave numbers\nkx= 0,ky= 0,kzdi=m/16, with m= 1,2,4,8, recorded from the component bxfor Hall-\nMHD (left) and for FLR-Landau fluid (right). For Hall-MHD, these co rrespond to polarized\nAlfv´ en waves which obey the dispersion relation ω=±k2/(2Ri) +k/radicalbig\n1+(k/2Ri)2. The\ntheoretical frequency values which are expected from this disper sion relation for given kare\n10plottedonthe topaxisof thefigure asblack vertical lines. Figure1 s hows that the resolution\nof 1283is sufficient to clearly distinguish between left and right polarized Alfv´ en waves. In\ncontrast with Hall-MHD, for which it is possible to write the general dis persion relation\nfor frequency ωas a relatively simple polynomial of 6th order, for FLR-Landau fluids t he\ngeneral dispersion relation would be uneconomically large to write dow n. In general, it is\nnecessary to numerically solve the determinant obtained from linear ized equations (1)-(12)\nfor a given wavenumber kafter assuming linear waves. FLR-Landau fluid (1)-(12) consists\nof 11 evolution equations in 11 variables and, together with the diver gence free constraint\nfor the magnetic field, therefore yields general dispersion relation for frequency ωin the\nform of a complex polynomial of 10th order. This represents 5 forw ard and 5 backward\npropagating waves, with some solutions having highly negative imagina ry part and which\nare therefore strongly damped. A similar situation is encountered in the Vlasov-Maxwell\nkinetic theory which essentially yields an infinite number of strongly da mped solutions. For\nthe propagation angle θ= 0◦and additional constraint T⊥=T/bardbl= 1, it is however possible\nto obtain an analytic solution for the circularly polarized Alfv´ en wave s as\nω=±k2\n2Ri/parenleftbigg\n1+β/bardbl\n2/parenrightbigg\n+k/radicalBigg\n1+/parenleftbiggk\n2Ri/parenrightbigg2/parenleftbigg\n1−β/bardbl\n2/parenrightbigg2\n, (14)\nwith two other solutions obtained by substituting ωwith−ω. The dispersion relation is\nquite similar to that of Hall-MHD with additional terms proportional to β/bardbland resulting\nfrom the finite Larmor radius corrections. Expected theoretical frequencies obtained from\nthis analytic solution are plotted on the top axis of Fig. 1 for the Land au fluid regime\n(right). They again match quite precisely. Note also the moderately strong damping of\nparallel Alfv´ en waves in Landau fluid regime, which can be seen for th e last mode m= 8.\nLandau damping does not act directly on linear constant amplitude Alf v´ en waves obeying\nrelation (14), which are exact solutions of linearized FLR-Landau flu id model. However,\nnonlinear parallel Alfv´ en waves, which are of course present in the full model, cause produc-\ntion of density (sound) fluctuations. Sound waves in the FLR-Land au fluid model, as well\nas in the kinetic theory, are heavily damped by Landau damping as sho wn below and this\nprocess therefore also results in damping of Alfv´ en waves. Mjølhu s and Wyller [88] stud-\nied the kinetic derivative nonlinear Schr¨ odinger equation (KDLNS) f or parallel propagating\nlong-wavelength Alfv´ en waves where they refer to this effect as n onlinear Landau damping,\nbecause it is acting on Alfv´ en waves in a nonlinear way.\n1110-1610-1410-1210-1010-810-6\n10-210-1100power\nfrequency ( ω)HMHD\nuz, θ =0°S\nA\n10-1610-1410-1210-1010-810-6\n10-210-1100power\nfrequency ( ω)SALandau\nuz, θ =0°\nFIG. 2: Sound waves for the propagation angle of 0◦for Hall-MHD (left) and Landau fluid (right),\nfrom frequency analysis of uzmodes with the same wavenumbers as in Fig. 1. Sound waves (S)\nare heavily damped for the Landau fluid, which is consistent w ith the kinetic theory. The spectra\nalso show weak presence of Alfv´ en waves (A), which are visib le for the first two modes.\nFurther exploring propagation angle of θ= 0◦, Fig. 2 shows frequency power spectra\nobtained from component uzand which should therefore predominantly display sound waves\n(almost identical spectra can be obtained from component ρ). The same modes with kzdi=\nm/16, where m= 1,2,4,8 are shown as in Fig. 1. For Hall-MHD, parallel sound waves\nobey the dispersion relation ω=kcs, where the sound speed cs=√β0. Sound waves (S)\nare clearly presented in Fig. 2 for Hall-MHD (left) with quite sharp pea ks which match\nthe theoretical dispersion values shown on the top axis. Weak pres ence of polarized Alfv´ en\nwaves (A) for modes m= 1,2 is also visible in this component generated by nonlinear\ncoupling. For the Landau fluid regime (Fig. 2 right), it is not possible to obtain simple\nanalytic dispersion relation which corresponds to sound waves and c orrect values must be\nobtained numerically as explained above (see also the Appendix). This yields 5 frequencies\nfor forward propagating waves, with 2 solutions corresponding to the polarized Alfv´ en waves\n(14) and 3 solutions which are highly damped. The sound wave was cho sen as the least\ndamped solution. Dispersion relation ω=k/radicalBig\n(3+T(0)\ne)β/bardbl/2 obtained from the double\nadiabatic model, to which Landau fluid description collapses after ass umption of zero heat\nfluxes and large scales, can also be used as a heuristic guide to deter mine which out of\nthe 3 damped solutions represents the sound wave frequency. Ob tained frequency values\n12are again plotted on the top axis. Figure 2 shows that the sound wav es (S) are heavily\ndamped for the Landau fluid regime (Fig. 2 right) with the nonlinearly g enerated Alfv´ en\nwaves (A) completely overpowering the spectra. Inhibition of soun d waves is consistent with\nthe kinetic theory. As elaborated by Howes [62], sound waves are ov erestimated by MHD\nand Hall-MHD descriptions as they represent solutions which are str ongly damped by the\nLandau resonance. Note that the calculated sound wave frequen cies in the FLR-Landau\nfluid regime are actually higher than the corresponding Alfv´ en wave frequencies. A similar\nresult was obtained by Howes [62] who numerically compared dispers ion relations for Hall-\nMHD and kinetic theory and who, for the nearly parallel propagation withTi=Teand\nβ0=β/bardbl= 1, noted that the kinetic solution corresponding to the slow wave h as a higher\nphase speed than the kinetic solution corresponding to the Alfv´ en wave.\nConsidering the propagation angle of θ= 45◦, Fig. 3 shows frequency-power spectra in\ncomponent bxfor wavenumbers ky= 0,kxdi=kzdi=m/16, where m= 1,2,4,6 and which\ntherefore predominantly shows slow (S) and fast (F) magnetoson ic waves. Spectra also show\nweaker presence of Alfv´ en waves (A). Again for this angle of prop agation, the slow waves\nare strongly damped. In both Hall-MHD and FLR-Landau fluid simulatio ns, the associated\ndispersion relations had to be solved numerically and the predicted fr equencies for spectral\npeaks are shown on the top axis. Slow waves (S) in the FLR-Landau fl uid regime were again\nidentified as the least damped frequency out of the 3 highly damped s olutions. For identical\nwavenumbers, the presence of Alfv´ en waves can be better explo red in the component byand\ncorresponding spectra are shown in Fig. 4.\nFinally, considering purely perpendicular propagation with θ= 90◦, Fig. 5 shows\nfrequency-power spectra obtained from the density field ρand wavenumbers ky= 0,kz=\n0,kxdi=m/16, where m= 1,2,4,8. For Hall-MHD regime (Fig. 5 left) the spectral peaks\ncorrespond to magnetosonic waves with the usual dispersion relat ionω=k√1+β0. In\nthe Landau fluid regime, linearized set of equations (1)-(12) with th e additional constraint\nT⊥=T/bardbl= 1 can be shown to yield the dispersion relation for perpendicular mag netosonic\nwaves in the form\nω=k/radicaltp/radicalvertex/radicalvertex/radicalbt1+β/bardbl/parenleftBigg\n1+T(0)\ne\n2/parenrightBigg\n+/parenleftbiggβ/bardblk\n4Ri/parenrightbigg2\n. (15)\nThe dispersion relation clearly shows the effect of inclusion of isother mal electrons (for\nsimulations presented here T(0)\ne= 1) and also the effect of finite Larmor radius corrections\n1310-1610-1410-1210-1010-810-6\n10-210-1100power\nfrequency ( ω)HMHD\nbx, θ =45°S\nAF\n10-1610-1410-1210-1010-810-6\n10-210-1100power\nfrequency ( ω)Landau\nbx, θ =45°\nSAF\nFIG. 3: Slow (S) and fast (F) magnetosonic waves for the propa gation angle of 45◦for Hall-\nMHD (left) and Landau fluid (right), from frequency analysis ofbxmodes with wavenumbers\nky= 0,kxdi=kzdi=m/16, where m= 1 (red), m= 2 (green), m= 4 (blue), m= 6 (black). The\npresence of Alfv´ en waves (A) is also visible. Theoretical p redictions for slow and fast waves are\nshown on the top axis.\nwhich are represented by the last quadratic term. Theoretical pr edictions from Hall-MHD\nand FLR-Landau fluid dispersion relations are again shown on the top axis. To clearly show\nthe shift of the peaks between the two regimes, we also added the t heoretical Hall-MHD\nfrequencies to the top axis of FLR-Landau fluid regime (Fig. 5 right) and represent them\nwith the small magenta lines.\nIV. FLOW COMPRESSIBILITY\nCompressibility of the flow can be evaluated by decomposing the veloc ity field into its\nsolenoidal and non-solenoidal components and by calculating the as sociated energies accord-\ning to\n/summationdisplay\nk|uk|2=/summationdisplay\nk|k×uk|2\n|k|2+/summationdisplay\nk|k·uk|2\n|k|2, (16)\nwhere the left-hand side corresponds to the total energy EUin velocity field, the first term in\ntheright-handsidecorrespondstotheenergy Eininthesolenoidalcomponentandthesecond\ntermEcoriginates from the compressible one. Relation (16) can be therefo re expressed as\nEU=Ein+Ecand the compressibility of the flow can be evaluated as a ratio of comp ressible\n1410-1610-1410-1210-1010-810-6\n10-210-1100power\nfrequency ( ω)HMHD\nby, θ =45°\n10-1610-1410-1210-1010-810-6\n10-210-1100power\nfrequency ( ω)Landau\nby, θ =45°\nFIG. 4: Alfv´ en waves for the propagation angle of 45◦for Hall-MHD (left) and Landau fluid\n(right), from frequency analysis of bymodes with the same wavenumbers as in Fig. 3. Theoretical\npredictions for the Alfv´ en waves frequencies are shown on t he top axis.\nand total energy Ec/EU. Time evolution of Ec/EUfor Hall-MHD and FLR-Landau fluid\nregime with β0=β/bardbl= 0.8 is presented in Fig. 6. The figure shows that the ratio Ec/EU\nwhich represents compressibility is significantly lower in Landaufluid re gime and is therefore\na result of presence of Landau damping. The question then arises o f the influence of the\nsonic Mach number in the compressibility evolution. The time evolution o fEc/EUfor\nβ0=β/bardbl= 0.25 (which corresponds to Ms= 0.25) is displayed in Fig. 7 (left), whereas\nthe time evolution of Ec/EUforβ0=β/bardbl= 0.1 (which corresponds to Ms= 0.40) is shown\nin Fig. 7 (right). The simulations were performed with the same time st epdt= 0.128\nand filtering, for approximately half the total integration time of th e previous regime with\nβ0=β/bardbl= 0.8 (Ms= 0.14). The figure shows that while the compressibility is still clearly\nreducedinLandaufluidsimulationwith β0=β/bardbl= 0.25,thisreductionisalmostinsignificant\nfor simulations with β0=β/bardbl= 0.1. This is an expected effect, as the strength of the Landau\ndamping is proportional to β/bardbl. We note that the turbulent sonic Mach number in the\nsolar wind is typically small and, for example, analysis of observationa l data performed by\nBavassano and Bruno [8] (from 0.3-1.0 AU) showed that the most pr obable value is between\nMs= 0.1−0.2 with the distribution having an extended tail to lower values.\nThe question also arises of the compared evolution of total energy for Hall-MHD and\nFLR-Landau fluid simulations. The total energy Etotcan be evaluated as the sum of the\n1510-1610-1410-1210-1010-810-6\n10-210-1100power\nfrequency ( ω)HMHD\nρ, θ =90°\n10-1610-1410-1210-1010-810-6\n10-210-1100power\nfrequency ( ω)Landau\nρ, θ =90°\nFIG. 5: Magnetosonic waves for the propagation angle of 90◦for Hall-MHD (left) and Landau fluid\n(right), from frequency analysis of density modes with wave numbers ky= 0,kz= 0,kxdi=m/16,\nwherem= 1 (red), m= 2 (green), m= 4 (blue), m= 8 (black). Theoretical predictions from\ndispersion relations are shown on the top axis (long black li nes). For comparison, on the right\npanel we also included the frequency predictions from Hall- MHD (small magenta lines).\n00.050.10.150.20.25\n0200040006000800010000compressibility\ntimeHMHD\nLandauβ0=β||=0.8\nFIG. 6: Compressibility for Hall-MHD (red line) and FLR-Lan dau fluid (blue line) evaluated as\n(/summationtext\nk|k·uk|2/|k|2)//summationtext\nk|uk|2forβ0=β/bardbl= 0.8. Both regimes start with the identical initial\ncondition where the velocity field is divergence free. The fig ure shows that the compressibility is\nclearly inhibited in the Landau fluid simulation.\n1600.050.10.150.20.25\n010002000300040005000compressibility\ntimeHMHD\nLandauβ0=β||=0.25\n00.050.10.150.20.25\n0100020003000400050006000compressibility\ntimeHMHD\nLandauβ0=β||=0.1\nFIG. 7: Compressibility for Hall-MHD (red line) and FLR-Lan dau fluid (blue line) evaluated as\n(/summationtext\nk|k·uk|2/|k|2)//summationtext\nk|uk|2withβ0=β/bardbl= 0.25 (left) and with β0=β/bardbl= 0.1 (right). The\nfigure shows that the compressibility is clearly inhibited i n Landau fluid for simulations with\nβ0=β/bardbl= 0.25, whereas for simulations with β0=β/bardbl= 0.1 the level of compressibility is almost\nidentical.\nkinetic energy Ekin, the magnetic energy Emagand the internal energy Eint. In Hall-MHD\nand FLR-Landau fluid model, the definitions of kinetic and magnetic en ergy are identical\nand equal to\nEkin=1\n2/integraldisplay\nρ|u|2dx3, E mag=1\n2/integraldisplay\n|b|2dx3. (17)\nHowever, the definition of internal energy is naturally different in ea ch model. In the Hall-\nMHD model, the internal energy is defined as\nHMHD: Eint=βo\nγ(γ−1)/integraldisplay\nργdx3, (18)\nwhereas in the FLR-Landau fluid model with isothermal electrons, t he internal energy is\ngiven by\nLandau: Eint=β/bardbl\n2/integraldisplay/parenleftBig\np⊥+p/bardbl\n2+T(0)\neρlnρ/parenrightBig\ndx3. (19)\nIt is emphasized that because of the filtering, the total energy Etotis not exactly preserved\nin the Hall-MHD and Landau fluid simulations. During the simulations with β0=β/bardbl= 0.8,\nthe total energy in Hall-MHD decreased by approximately 1.2%, wher eas in FLR-Landau\nfluid the decrease was approximately 0.5%. Filtering indeed dissipates kinetic and magnetic\nenergies without turning them into heat. Therefore, in contrast w ith Landau fluid models,\n1700.20.40.60.81\n0200040006000800010000Ekin + Emag\ntimeHMHD\nLandaunormalized energy Ekin + Emag\n11.0051.011.0151.02\n0200040006000800010000Eint\ntimeHMHDLandaunormalized internal energy Eint\nFIG. 8: Normalized mechanical fluctuations energy Ekin+Emag(left) and normalized internal\nenergyEint(right) for Hall-MHD (red line) and FLR-Landau fluid (blue li ne), in the case β0=\nβ/bardbl= 0.8. Landau damping transfers energy from Ekin+EmagintoEint.\nHall-MHD simulations do not have heating at all and the internal energ y could increase\nonly through development of density fluctuations. Time evolution of the sum of kinetic and\nmagnetic energies (normalized to their initial values) is displayed in Fig. 8 (left), where the\nenergy contained in the ambient magnetic field was subtracted. This figure shows that this\nsum decays faster for FLR-Landau fluid simulation. Time evolution of internal energy Eint\nis shown in Fig. 8 (right), where energies were again normalized to the ir initial value. The\ninitial jump observed in both simulations reflects a rapid adjustment from initial conditions\nthat are not close to an equilibrium state. Later on, the internal en ergy of Hall-MHD\nsimulation decreases, whereas the internal energy of FLR-Landa u fluid simulation more or\nless smoothly increases until around time t= 1500. During this time (which corresponds\nto over 104time steps) the Landau damping acts strongly and, by mainly damping slow\nwaves, converts the mechanical energy Ekin+Emaginto the internal energy Eint, which\nrepresents heatingoftheplasma. Thequestionalsoariseswhatfr actionofmechanical energy\nis dissipated directly by Landau damping and what fraction is dissipate d by the filtering\nprocess. Unfortunately, we are unaware of any technique how to address this question.\nWe note that for simulations of freely decaying turbulence the heat ing is quite weak, im-\nplying that driving the system is necessary to produce significant te mperature anisotropies.\nHowever, the absence of forcing, which yields only a small amount of heating, makes it easier\n18to precisely identify various waves in the system as was presented in Sec. III.\nV. ANISOTROPY OF THE ENERGY TRANSFER\nThe presence of Landau damping can also be seen in the usual waven umber velocity and\nmagnetic field spectra. Considering first simulations with β0=β/bardbl= 0.8, Fig. 9 shows the\nvelocity spectra with respect to perpendicular and parallel wavenu mbersk⊥,k/bardblwhich are\ndefined as EU=/integraltext\nEu(k⊥)dk⊥=/integraltext\nEu(k/bardbl)dk/bardbl. With respect to k⊥, the spectra for Hall-\nMHD and FLR-Landau fluid are almost identical (Fig. 9 left) whereas w ith respect to k/bardbl\n(Fig. 9 right), the spectra of Landau fluid are much steeper. Land au damping therefore\nsignificantly inhibits the parallel transfer. Even though low resolutio n does not allow to\nprecisely identify the slopes of the spectra, three straight lines we re added to figures and\ncorrespond to power law solutions ks, wheres=−3/2,−5/3 and−7/3. ForEu(k⊥), the\nclosest spectral index value appears to be −5/3, the spectral range being however quite\nlimited. The same conclusion with the inhibition of parallel transfer is als o obtained for\nthe magnetic field spectra, which are almost identical to the velocity spectra and are shown\nin Fig. 10. In contrast, for simulations with β0=β/bardbl= 0.1 the parallel spectrum E(k/bardbl)\ndisplays a similar behavior for Hall-MHD and Landau fluids. The velocity s pectra are shown\ninFig. 11andthemagnetic fieldspectra areshown inFig. 12. This iscon sistent withresults\npresented intheprevious sectionwhere it wasshown that theLand audamping is responsible\nfor significant reduction of compressibility for simulations with β0=β/bardbl= 0.8, whereas for\nsimulations with β0=β/bardbl= 0.1, the Landau damping was much weaker and the reduction\nof compressibility almost negligible.\nIt is useful to compare our compressible simulations to incompressib le simulations. Nat-\nurally, our compressible Hall-MHD code cannot be run in an incompress ible regime, for\nwhich the turbulent sonic Mach number Ms→0 and the sound speed cs→ ∞. Neverthe-\nless, incompressible MHD simulations of decaying turbulence were per formed, for example,\nby Bigot et al. [89]. These simulations showed that the combined veloc ity and magnetic\nfield spectra (they used Els¨ asser variable z+) are much steeper with respect to k/bardblthan with\nrespect to k⊥, if the ambient magnetic field is sufficiently strong. Our compressible s im-\nulations presented here have initially /an}bracketle{tb2/an}bracketri}ht1/2/B0= 1/8 and can therefore be compared to\nincompressible MHD simulations of Bigot et al. [89] (their Fig. 17 c,d), wh ere this ratio\n1910-810-710-610-510-410-3\n10-1100Eu (k⊥)\nk⊥ diEu (k⊥)\nβ0=β||=0.8\n10-810-710-610-510-410-3\n10-1100Eu (k||)\nk|| diEu (k||)\nβ0=β||=0.8\nFIG. 9: Velocity spectra for Hall-MHD (red) and Landau fluid ( blue) with respect to perpendicular\nwavenumber Eu(k⊥) (left) and with respect to parallel wavenumber Eu(k/bardbl) (right), for β0=β/bardbl=\n0.8. Spectra were taken at time t= 5248. Straight lines correspond to k−3/2,k−5/3andk−7/3. The\nfigure shows that spectra with respect to k/bardblare much steeper in Landau fluid simulation, which is\na result of Landau damping.\nis 1/5 and 1/15, respectively. Considering compressible Hall-MHD, our simulations showed\nthat spectra with respect to k/bardbl(e.g. Fig. 9 right, red line) are steeper than spectra with\nrespect to k⊥(Fig. 9 left, red line). However, these parallel spectra are nowher e near as\nsteep as the parallel spectra of Bigot et al. [89]. Interestingly, the ir parallel spectra more\nresemble our parallel spectra for Landau fluid model (Fig. 9 right, b lue line), where the Lan-\ndau damping was strong ( β/bardbl= 0.8). These results show that magnetosonic waves (which\nare not present in an incompressible MHD description) play an importa nt role in regulating\nthe parallel energy cascade. Steeper parallel spectra in Landau fl uid model are therefore a\nconsequence of damping of slow magnetosonic waves by Landau dam ping.\nVI. CONCLUSION\nWe have presented the first three-dimensional fluid simulations of d ecaying turbulence\nin a collisionless plasma in conditions close to the solar wind. For this purp ose, we used\nthe FLR-Landau fluid model that extends compressible Hall-MHD by in corporating low-\nfrequency kinetic effects such as Landau damping and finite Larmor radius corrections. It\n2010-810-710-610-510-410-3\n10-1100Eb (k⊥)\nk⊥ diEb (k⊥)\nβ0=β||=0.8\n10-810-710-610-510-410-3\n10-1100Eb (k||)\nk|| diEb (k||)\nβ0=β||=0.8\nFIG. 10: Magnetic field spectra for Hall-MHD (red) and Landau fluid (blue), for Eb(k⊥) (left) and\nEb(k/bardbl) (right) when β0=β/bardbl= 0.8. Spectra were taken at time t= 5248.\n10-810-710-610-510-410-3\n10-1100Eu (k⊥)\nk⊥ diEu (k⊥)\nβ0=β||=0.1\n10-810-710-610-510-410-3\n10-1100Eu (k||)\nk|| diEu (k||)\nβ0=β||=0.1\nFIG.11: Velocity spectraforHall-MHD(red)andLandaufluid (blue)withrespecttoperpendicular\nwavenumber Eu(k⊥) (left) and with respect to parallel wavenumber Eu(k/bardbl) (right) for β0=β/bardbl=\n0.1. Spectra were taken at time t= 5248. Straight lines correspond to k−3/2,k−5/3andk−7/3.\nwasshownthatinspiteoftheturbulentregime, itispossibletoprecis elyidentifylinearwaves\npresent in the system. Comparisons between compressible Hall-MHD and FLR-Landau\nfluid model showed that when beta is not too small, linear Landau damp ing yields strong\ndamping of slow magnetosonic waves in Landau fluid simulations. These waves are indeed\ndampedinkinetictheorydescribed bytheVlasov-Maxwell equations butnotincompressible\n2110-810-710-610-510-410-3\n10-1100Eb (k⊥)\nk⊥ diEb (k⊥)\nβ0=β||=0.1\n10-810-710-610-510-410-3\n10-1100Eb (k||)\nk|| diEb (k||)\nβ0=β||=0.1\nFIG. 12: Magnetic field spectra for Hall-MHD (red) and Landau fluid (blue), for Eb(k⊥) (left) and\nEb(k/bardbl) (right) when β0=β/bardbl= 0.1. Spectra were taken at time t= 5248. Spectra for Hall-MHD\nand Landau fluid model are again almost identical.\nMHD and Hall-MHD descriptions, which overestimate compressibility an d parallel transfer\ninmodeling weakly collisional plasmas. TheFLR-Landaufluidmodel cant hereforebeuseful\nfor simulating the solar wind, which is typically found to be only weakly co mpressible.\nAcknowledgements\nThe support of INSU-CNRS “Programme Soleil-Terre” is acknowledg ed. Computations\nwere performed on the Mesocentre SIGAMM machine hosted by the Observatoire de la\nCˆ ote d’Azur (OCA) and on the JADE cluster of the CINES computat ional facilities. PH\nwas supported by an OCA Poincar´ e fellowship. The work of DB was su pported by the\nEuropean Community under the contract of Association between E URATOM and ENEA.\nThe views and opinions expressed herein do not necessarily reflect t hose of the European\nCommission.\nAPPENDIX\nTo clearly understand how the Landau damping acts in the present s ystem (1)-(12), it is\nuseful to solve dispersion relations for linear waves propagating in p arallel direction to the\n22ambient magnetic field. A detailed analysis of linear waves for various p ropagation angles\nwas elaborated by Passot and Sulem [90]. To simplify the analytic expre ssions, we define the\nproton temperature anisotropy as T(0)\n⊥p/T(0)\n/bardblp≡ap, and the normalized electron temperature\nasT(0)\ne/T(0)\n/bardblp≡τ. It can be shown that for parallel propagation angle, the Landau fl uid\nmodel contains four dispersive Alfv´ en waves. Two waves obey the dispersion relation which\ncan be expressed as\nω=±k2\n2Ri/bracketleftBig\n1+β/bardbl/parenleftBig\n1−ap\n2/parenrightBig/bracketrightBig\n+k/radicalBigg\n1+β/bardbl\n2(ap−1)+/parenleftbiggk\n2Ri/parenrightbigg2/bracketleftBig\n1−β/bardbl/parenleftBig\n1−ap\n2/parenrightBig/bracketrightBig2\n,(20)\nwith another two solutions obtained by substituting ωwith−ω. Obviously, these Alfv´ en\nwaves are independent of the electron temperature τ, which is a consequence of electrons\nbeing modeled as isothermal. For a more general Landau fluid model w hich contains evo-\nlution equation for electron pressures and heat fluxes, the electr on temperature τenters\nthe dispersion relation for Alfv´ en waves. The solutions (20) can be come imaginary, if the\nexpression under the square root becomes negative. At large sca les (when 1 /Ri→0) the\ncondition 1 + β/bardbl(ap−1)/2<0 represents the well known criterion for fire-hose instability\n(see, for example, Ferri` ere and Andr´ e [91]). The Hall term and F LR corrections modify the\ninstability criterion. For isotropic temperatures ( ap= 1), the solution (20) naturally col-\nlapses to the solution (14). The four Alfv´ en waves (20) can be elimin ated from the general\ndispersion relation and this yields a complex polynomial of 6th order in f requency ω. Solu-\ntions of this polynomial represent 3 forward and 3 backward propa gating waves which have\na negative imaginary part and are therefore damped. Importantly , it is possible to eliminate\nthe dependence on β/bardbland wavenumber kand, after applying a substitution Ω = ω/(k/radicalbigβ/bardbl),\nthe polynomial of 6th order can be simplified to\nΩ6+Ω5i√π−4+3π\n−16+6π+Ω4−π(14+3τ)+24+8 τ\n−16+6π+Ω3i√πτ−π(9+3τ)/4\n−8+3π\n+Ω2π(15+5τ)/4−2\n−8+3π+Ω1i√π(5+3τ)/2\n−8+3π−1+τ\n−8+3π= 0. (21)\nBecause this polynomial in Ω does not depend on β/bardblork, the substitution implies that\nall 6 waves are linear with kand/radicalbigβ/bardbl. The polynomial (21) has to be solved numeri-\ncally for a given value of τ. The simulations presented here use τ= 1 and numerically\nsolving polynomial (21) yields Ω = ±1.48106−i0.36117, Ω = ±0.65467−i0.88285 and\n23Ω =±0.55104−i0.44311. These six waves therefore satisfy\nω=k/radicalBig\nβ/bardbl(±1.48106−i0.36117), (22)\nω=k/radicalBig\nβ/bardbl(±0.65467−i0.88285), (23)\nω=k/radicalBig\nβ/bardbl(±0.55104−i0.44311). (24)\nThe least damped solution (22) represents the sound wave. The so lutions of Landau fluid\nmodel (1)-(12) for parallel propagation angle are therefore 4 Alf v´ en waves (20), 2 sound\nwaves (22) and 4 waves (23), (24), which are highly damped. These 4 waves (23), (24)\ndo not have an analogy in Hall-MHD description and correspond to solu tions of kinetic\nMaxwell-Vlasov description, which contains an infinite number of highly damped solutions.\nInterestingly, thelast solution (24) is not dependent onthe valueo fτandit canbe expressed\nanalytically as Ω = ±√8−π/4−i√π/4. After eliminating these waves from eq. (21), the\npolynomial which contains the sound waves (22) and solutions (23) is now of 4th order in Ω\nand expressed as\nΩ4+i2√π\n−8+3πΩ3−1\n29π−16+(−8+3π)τ\n−8+3πΩ2−i√π(3+τ)\n−8+3πΩ+2(1+τ)\n−8+3π= 0.(25)\nItisofcoursepossibletouseFerrari-Cardano’srelationstosolvet hispolynomialanalytically,\nthe final result is however too complicated and it is still more convenie nt to solve (25)\nnumerically for a given value of τ.\nIf this wave analysis is repeated with the model (1)-(10) with heat fl ux equations q/bardbl= 0,\nq⊥= 0, the same dispersion relation (20) for Alfv´ en waves is obtained. However, the only\nother solutions present in the parallel direction are\nω=±k/radicalbigg\nβ/bardbl\n2(3+τ). 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In par-\ntially ionized astrophysical plasmas, the dynamo growth of magnetic energy strongly depends on the coupling\nstate between ions and neutrals and the ion-neutral collisi onal damping effect. A new damping stage of turbu-\nlent dynamo in a weakly ionized medium was theoretically pre dicted by Xu & Lazarian (2016). By carrying out\na 3D two-fluid dynamo simulation, here we for the first time num erically confirmed the physical conditions and\nthe linear-in-time growth of magnetic field strength of the d amping stage of dynamo. The dynamo-amplified\nmagnetic field has a characteristic length as the damping sca le, which increases with time and can reach the\ninjection scale of turbulence after around eight largest ed dy-turnover times given sufficiently low ionization\nfraction and weak initial magnetic field. Due to the weak coup ling between ions and neutrals, most turbulent\nenergy carried by neutrals cannot be converted to the magnet ic energy, resulting in a relatively weak magnetic\nfield at the end of dynamo. This result has important implicat ions for the growth of magnetic fields in the\npartially ionized interstellar medium and shock accelerat ion of Galactic cosmic rays.\nSubject headings: Physical data and processes: dynamo - turbulence - ISM: magn etic fields\n1.INTRODUCTION\nMagnetic fields pervade the Universe and are manifest in\ndiverse astrophysical systems (Han 2017). The turbulent dy -\nnamo, which both amplifies the strength of the magnetic field\nand increases its coherence length, is the most promising\nmechanism to account for the growth and maintenance of the\ncosmic magnetism (Brandenburg & Subramanian 2005). In\nparticular, the turbulent dynamo acting on scales comparab le\nor smaller than the driving scale of turbulence, i.e., the sm all-\nscale dynamo, is much more efficient than the large-scale dy-\nnamo, and also more generally operates in astrophysical env i-\nronments wherever the turbulent energy exceeds the magneti c\nenergy.\nDepending on the physical conditions, there are a variety\nof dynamo regimes (Xu & Lazarian 2016, hereafter XL16).\nIn the case of a large Prandtl number, which is the ratio of\nviscosity to resistivity, the kinematic regime of the small -\nscale dynamo at sub-viscous scales has been extensively stu d-\nied (e.g., Maron & Blackman 2002; Schekochihin et al. 2002;\nMaron et al. 2004). The concentration of the magnetic en-\nergy at the small resistive scale claimed in these theoreti-\ncal and low-resolution numerical studies was disproved by\nhigh-resolution dynamo simulations (Haugen et al. 2004).4\nMeanwhile, the nonlinear regime of the small-scale dynamo\nin the inertial range of turbulence has also been studied nu-\nmerically (Cho & Vishniac 2000; Cho et al. 2009; Beresnyak\n2012), which is found to be characterized by a very inef-\nficient linear-in-time growth of magnetic energy. Recent\ntheoretical and numerical advances in the study of mag-\nnetohydrodynamic (MHD) turbulence (Goldreich & Sridhar\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 Fellow\n3Department of Physics, University of Notre Dame, Notre Dame , IN\n46556, USA; sgarain@nd.edu, dbalsara@nd.edu\n4In fact, after a close inspection of, e.g., figure 1 of Maron & B lackman\n(2002) (as pointed out by Haugen et al. (2003)), figure 12 in Ma ron et al.\n(2004), one can easily see that their results also show the pe ak of the magnetic\nenergy spectrum significantly away from the resistive scale .1995; Lazarian & Vishniac 1999; Maron & Goldreich 2001;\nCho et al. 2002b; Kowal et al. 2009, 2012) enable us to con-\nstruct an analytical theory of the nonlinear turbulent dyna mo\n(XL16), which has been shown in quantitative agreement with\nnumerical measurements. In XL16, the turbulent diffusion o f\nmagnetic fields enabled by the turbulent magnetic reconnec-\ntion (Lazarian & Vishniac 1999) was identified as the physi-\ncal origin of the low efficiency of the nonlinear dynamo. Be-\nsides, XL16 also analytically discovered a transitional st age\nconnecting the kinematic and nonlinear regimes, where the\npeak of the magnetic energy spectrum shifts from the resisti ve\nscale to the viscous scale. Their theoretical prediction on the\nsub-viscous spectral tail k−1formed during the transitional\nstage is consistent with the numerical result in Haugen et al .\n(2004).\nIn astrophysical plasmas with a significant neutral compo-\nnent in e.g., the early Universe, cold phases of the interste llar\nmedium (ISM), protoplanetary disks, the solar chromospher e,\nboth MHD turbulence and turbulent dynamo are influenced by\nthe partial ionization (Xu & Lazarian 2017b). Ion-neutral c ol-\nlisional damping of linear MHD waves has been earlier stud-\nied by, e.g., Langer (1978); Balsara (1996); Zaqarashvili e t al.\n(2011). On the basis of the updated understanding of MHD\nturbulence mentioned above, the damping of MHD turbulence\ndue to ion-neutral collisions and the viscosity in neutrals has\nbeen studied both analytically (Lithwick & Goldreich 2001;\nLazarian et al. 2004; Xu et al. 2015, 2016; Xu & Lazarian\n2017b) and numerically (Tilley & Balsara 2008, 2011, 2010;\nMeyer et al. 2014; Burkhart et al. 2015).\nRegarding the small-scale dynamo in a partially ionized\nmedium, the damping effect due to ion-neutral collisions\non the efficiency of dynamo has been discussed in, e.g.,\nKulsrud & Anderson (1992); Subramanian (1998). The new\nfindings in XL16 include (i) a sub-viscous spectral tail k−1\nformed during the transitional stage at a relatively high io n-\nization fraction; (ii) a damping stage of dynamo character-\nized by a linear-in-time growth of magnetic field strength at\na relatively low ionization fraction; (iii) the nonlinear s tage\nof dynamo with a universal dynamo efficiency irrespective of2\nthe ionization fraction; (iv) a direct relation of the dampi ng\nof MHD turbulence to that of turbulent dynamo. These theo-\nretical findings have also been applied to studying the role o f\nmagnetic fields in, e.g. the star formation in the early Unive rse\n(XL16), cosmic ray acceleration at shocks (Xu & Lazarian\n2017a, hereafter XL17),\nIn this work, our purpose is to numerically test the damping\nstage of dynamo in a weakly ionized medium. Different from\nthe exponential growth of magnetic energy in the sub-viscou s\nrange (Kulsrud & Anderson 1992), XL16 demonstrated that\nthe damping stage of dynamo takes place within the inertial\nrange of turbulence. It arises at a sufficiently low ionizati on\nfraction so that (a) ions and neutrals are only weakly couple d,\nand thus most turbulent energy in neutrals is not involved in\nthe dynamo; (b) the ion-neutral collisional damping scale c o-\nincides with the dynamo driving scale; (c) the magnetic field\nstrength grows linearly with time; and (d) there is no equipa r-\ntition between the turbulent and magnetic energies. We will\npresent the first numerical test of the theoretical predicti on on\nthe damping stage of dynamo in XL16 by carrying out a 3D\ntwo-fluid numerical simulation. We use the two-fluid version\nof the RIEMANN code (Balsara 1998a,b, 2004, 2010, 2012;\nBalsara & Spicer 1999a,b) to simulate the weakly ionized tur -\nbulent plasma. The RIEMANN code has been widely used\nfor studying astrophysical problems in partially ionized p las-\nmas (e.g., Tilley & Balsara 2010, 2011; Meyer et al. 2014).\nIn general, a two-fluid MHD simulation requires extensive\ncomputational effort. Our two-fluid dynamo simulation is\neven more challenging in order to achieve (i) a low ionizatio n\nfraction to ensure the emergence of the damping stage; (ii) a\nlarge inertial range, as the damping scale increases with ti me;\nand (iii) a long simulation time to observe the entire dynamo\nevolution of magnetic fields. Despite its high computationa l\ncost, this numerical testing will provide direct evidence f or\nthe XL16 theory of the damping stage of dynamo and quanti-\ntatively reinforce our understanding of the dynamo physics in\na weakly ionized medium. It is also important for further ap-\nplications of the theory to studying the evolution and struc ture\nof magnetic fields in neutral dominated astrophysical envir on-\nments.\nThe paper is organized as follows. In Section 2, we describe\nthe physical conditions and the analytically derived evolu tion\nlaw of the magnetic field for the damping stage of dynamo.\nIn Section 3, we present the numerical results of the two-flui d\nsimulation and their comparisons with our theoretical pred ic-\ntions. In Section 4, we further examine the importance of the\ndamping stage of dynamo in the partially ionized ISM. The\ndiscussion about the effect of ion-neutral coupling on MHD\nturbulence and turbulent dynamo is in Section 5. The sum-\nmary follows in Section 6.\n2.DAMPING STAGE OF DYNAMO IN A WEAKLY\nIONIZED MEDIUM\nBy stretching magnetic field lines, turbulent motions can\namplify magnetic fields. Meanwhile, magnetic fields also\nundergo diffusion due to plasma or/and turbulence effects.\nThese two opposing processes, turbulence stretching and\nmagnetic field diffusion, together determine the dynamo ef-\nficiency.\nIn the kinematic dynamo regime, the magnetic energy is\nlower than the turbulent energy, and the magnetic field is dy-\nnamically unimportant. The diffusion only arises from plas ma\neffects. In the case of a weakly ionized plasma, i.e., molecu -\nlar clouds in the ISM, the diffusion in the kinematic dynamoregime mainly comes from the slippage between ions and\nneutrals. So the ion-neutral collisional damping is the dom i-\nnant damping process of magnetic fluctuations, whereas othe r\ndamping effects including the viscous damping and resistiv e\ndamping are negligible (Kulsrud & Anderson 1992; Xu et al.\n2016).\nHere we consider the damping stage of dynamo in a weakly\nionized medium, which was first identified by XL16. It is\nin the kinematic regime and subjected to severe ion-neutral\ncollisional damping.\n2.1. Physical conditions for the damping stage of dynamo\nDepending on the ionization fraction, the turbulent dynamo\nin a partially ionized medium undergoes different evolutio n-\nary stages. To observe a significant damping effect on the dy-\nnamo growth of magnetic energy in a damping stage, the ion-\nization fraction should be sufficiently small, so that the io n-\nneutral coupling is weak and the ion-neutral collisional da mp-\ning is strong. We note that unlike the strongly coupled regim e\nwhere ions and neutrals are strongly coupled together and th e\ndecoupled regime where the two species are decoupled from\neach other, in the weakly coupled regime considered here,\nneutrals are decoupled from ions, but ions can still collide\nwith surrounding neutrals in a weakly ionized medium, and\nthus the motions of ions and magnetic fields are most severely\ndamped (Xu et al. 2016). Next we detail the physical condi-\ntions for the damping stage of dynamo to arise.\nTABLE 1 List of main notations\nDescription Symbol\nmagnetic energy EM\nmagnetic energy spectrum M(k,t)\ndrag coefficient γd\nneutral-ion collision frequency νni\nion-neutral collision frequency νin\nion-neutral collisional damping scale ld\nneutral viscosity νn\nviscous damping scale lν\npeak scale of M(k,t) lp\ninjection scale of turbulence L\nturbulent velocity at L V L\neddy-turnover time at L τ eddy\nturbulent velocity at l v l\nstretching rate / turnover rate at length scale l Γl\nstretching rate / turnover rate at ld Γd\nstretching rate / turnover rate at lν Γν\nstretching rate / turnover rate at L ΓL\nstretching rate / turnover rate at lp Γp\nion mass density ρi\nneutral mass density ρn\ntotal mass density ρ\nneutral fraction ξn\nion-neutral coupling coefficient ηc\neffective density (Eq. (30)) ρeff\nAlfv´en speed of ionized fluid VAi\nAlfv´en Mach number of ionized fluid MAi\nAlfv´en speed of strongly coupled ions and neutrals VA,tot\nAlfv´en speed in terms of ρeff VA,eff\nCondition (1) : a sufficiently small ionization fraction\nThe damping stage of dynamo is characterized by the weak\ncoupling state between ions and neutrals and the consequent\nsevere ion-neutral collisional damping. Quantitatively, the\nneutral-ion collisional frequency νnishould be smaller than\nthe dynamo stretching rate Γlof magnetic fields to ensure the\nweak coupling between ions and neutrals (see Table 1 for the\nmain notations used in this paper). The former is given by\nνni=γdρi, with the drag coefficient γd(see e.g. Shu 1992)3\nand the ion density ρi. The latter is determined by the turbu-\nlence eddy-turnover rate vl/l, wherevlis the turbulent veloc-\nity at the length scale l. According to the Kolmogorov scaling\nof hydrodynamic turbulence, vldecreases with las\nvl=VL/parenleftBigl\nL/parenrightBig1\n3(1)\nalong the turbulent energy cascade, where VLis the turbulent\nvelocity at the injection scale Lof turbulence. It can be eas-\nily seen that smaller eddies have larger eddy-turnover rate s.\nSince the eddies at the ion-neutral collisional damping sca leld\nof magnetic fluctuations are the smallest ones that can effec -\ntively stretch magnetic field lines, they are mainly respons ible\nfor the dynamo action. The corresponding dynamo stretching\nrate isΓd=vd/ld, wherevdis the turbulent velocity at ld.\nThe above condition is formulated as (XL17)\n2\nCΓd<1, (2)\nwhere\nC=ξn\n3νni≈1\n3νni, (3)\nwhich imposes a constraint on the maximum value of the ion-\nization fraction. We note that the expression on the LHS\nof Eq. (2) is related to the Reynolds number at lddefined\nin Balsara (1996). Here the ratio between the neutral den-\nsity and the total density ξn=ρn/ρis approximately equal\nto unity in a weakly ionized medium. It implies that when\nthe ionization fraction is sufficiently small, neutrals col lide\nwith ions so infrequently that neutrals are basically decou pled\nfrom the dynamo-stretched field lines. On the other hand, in\na neutral dominated medium, ions can still collide with sur-\nrounding neutrals. Quantitatively, there is νin≫Γd, where\nνin=γdρnis the ion-neutral collisional frequency. It is re-\nlated toνnibyνin= (ρn/ρi)νni. Evidently, νinis much\nlarger than νniin a weakly ionized medium. Because of the\nweak coupling between ions and neutrals, the dynamo action\ncannot effectively convert the turbulent kinetic energy ca rried\nby neutrals to the magnetic energy.\nCondition (2) : sufficiently small magnetic energy\nAs mentioned earlier, the magnetic energy in the kinematic\ndynamo regime is smaller than the turbulent energy. At ld,\nwhere the local turbulent motions dominate the dynamo ac-\ntion, there should be\nEM<1\n2v2\nd (4)\nwhere\nEM=1\n2V2\nA (5)\nis the magnetic energy, and VAis the Alfv´ en speed. So the\nrelation in Eq. (4) is equivalent to VA< vd.\nMeanwhile, there exists the equalization between Γdand\nthe ion-neutral collisional damping rate ωINatld, whereωIN\nis given by (Kulsrud & Anderson 1992)\nωIN=Cl−2EM=C\n2l−2V2\nA. (6)\nFromΓd=ωINatld, we find\nld=C\n2V2\nAv−1\nd. (7)Combining the above expression with the condition in Eq. (2)\nyieldsVA< vd. It shows that under Condition (1) ,Con-\ndition (2) is naturally satisfied. In fact, due to the severe\ndamping effect at a small ionization fraction, the equipart ition\nbetween the magnetic and turbulent energies at ldcannot be\nreached. Any further growth of magnetic energy would break\nthe balanced condition Γd=ωINatlduntil the new balance\nis achieved at a larger ld. Hence the dynamo in the damping\nstage remains in the kinematic regime.\nCondition (3) : dominant ion-neutral collisional damping\nover the neutral viscous damping\nAs mentioned above, the ion-neutral collisional damping\nis the dominant damping effect for the damping stage of dy-\nnamo. But we note that as the ion-neutral collisional damp-\ning depends on the magnetic energy (Eq. (6)), to ensure\nωIN> ω NV, where\nωNV=l−2νn, (8)\nis the damping rate related to the kinematic viscosity νnin\nneutrals, we should have the magnetic energy (Eqs. (6) and\n(8))\nEM>C−1νn. (9)\nWhen we consider a small ionization fraction and the dynamo\ngrowth of magnetic energy, the above condition can be easily\nsatisfied.\nAlternatively, when the ion-neutral collisional damping\ndominates over the neutral viscous damping, ldshould be\nlarger than the viscous damping scale lν. The condition\nld> lνyields (Eq. (7))\nCEMv−1\nd> νnv−1\nν, (10)\nwhere the relation l−2\nννn=vν/lνis used, and vνis the tur-\nbulent velocity at lν. Sincevd> vν, there must be\nEM>C−1νn, (11)\nwhich recovers the condition in Eq. (9).\nUnder the above conditions (Eq. (2) and Eq. (9)), we ex-\npect that the turbulent dynamo in a weakly ionized medium\nundergoes a damping stage.\n2.2. Magnetic field evolution during the damping stage of\ndynamo\nIn the damping stage, the time evolution of magnetic fields\nstrongly depends on the ion-neutral collisional damping. A s\nmentioned earlier, the dynamo stretching rate is given by th e\neddy-turnover rate at ld,\nΓd=vd\nld=L−1\n3VLl−2\n3\nd, (12)\nwhere the Kolmogorov scaling in Eq. (1) is used. With the\nsame scaling, the expression of ldin Eq. (7) can be rewritten\nas\nld=C3\n4L1\n4V−3\n4\nLE3\n4\nM. (13)\nThe growth of EMresults in a stronger damping effect and a\nlargerld.\nThe magnetic fluctuations on length scales larger than ld\nfollow the Kazantsev spectrum (Kazantsev 1968) as a result\nof the dynamo stretching,\nM(k,t) =M1exp/parenleftbigg3\n4/integraldisplay\nΓddt/parenrightbigg/parenleftbiggk\nk1/parenrightbigg3\n2\n, (14)4\nwhereM1is the initial magnetic energy spectrum at some\nreference wavenumber k1. The Kazantsev spectrum has de-\npendence on both wavenumber kand time t. By integrating\nM(k,t)overk, we can derive EMas a function of t,\nEM(t) =1\n2/integraldisplaykd\n0M(k,t)dk. (15)\nCombining Eqs. (12)-(15) and after some straightforward al -\ngebra, we arrive at (XL16),\n/radicalbig\nEM=/radicalbig\nEM1+3\n23C−1\n2L−1\n2V3\n2\nL(t−t1), (16)\nwith the magnetic energy EM1at the beginning of the damp-\ning stage t=t1. As√EM∝B, whereBis the magnetic\nfield strength, the damping stage of dynamo is characterized\nby a linear-in-time growth of B.\nFrom Eqs. (14) and (15), we find\ndlnEM\ndt∝Γd. (17)\nHereΓd∝ E−1\n2\nMaccording to Eqs. (12) and (13), which re-\nsults from both the equalization Γd=ωINatldand the Kol-\nmogorov scaling of turbulence. Therefore, we have√EM∝\nt.\nFurthermore, after inserting Eq. (16) into Eq. (13), we can\nalso derive the time evolution of ld,\nld=/parenleftBig\nl2\n3\nd1+3\n23L−1\n3VL(t−t1)/parenrightBig3\n2, (18)\nwithld1att=t1. If the damping stage can proceed until ld\nincreases up to L,Condition (1) (Eq. (2)) should be satisfied\natL, that is,\n2L\nCVL<1. (19)\nCompared with the general form in Eq. (2), the above condi-\ntion requires a further smaller ionization fraction so that even\nthe largest eddy-turnover time is still smaller than the neu tral-\nion collisional time. With neutrals decoupled from the dy-\nnamo action on all length scales from the initial ldup toL,\nit ensures that the dynamo remains in the kinematic damp-\ning stage, and the unsaturated magnetic energy at the end of\ndynamo mainly comes from the turbulent energy carried by\nions.\nWhenld=L, the corresponding time is (Eq. (18)),\nt(ld=L) =t1+23\n3L1\n3V−1\nL(L2\n3−l2\n3\nd1). (20)\nGivenld1≪L, the entire damping stage of dynamo lasts for\naround7.7times the largest eddy-turnover time. The mag-\nnetic energy reached at ld=Lis (Eq. (13)),\nEM(ld=L) =C−1LVL. (21)\nIn the kinematic damping stage, there is\nEM(ld=L)<1\n2V2\nL, (22)\nwhich naturally recovers the condition in Eq. (19).3.NUMERICAL TEST OF THE DAMPING STAGE OF\nDYNAMO WITH A TWO-FLUID SIMULATION\nTo numerically test the above theory for the damping stage\nof dynamo, we perform a 3D two-fluid dynamo simulation\nby using the RIEMANN code (Balsara 1998a,b, 2004, 2010,\n2012; Balsara & Spicer 1999a,b). The neutral and ionized flu-\nids are separately treated with the isothermal Euler equati ons\nand isothermal MHD equations, respectively. Their couplin g\nis described by the ion-neutral friction term, which is intr o-\nduced using an operator-split method (Tilley & Balsara 2008 ;\nTilley et al. 2012). We solve the following equations (Drain e\n1986) using the above mentioned code:\n∂ρi\n∂t+∇·(ρivi) = 0,\n∂vi\n∂t+(vi·∇)vi=−c2\ns∇lnρi−1\n4πB×(∇×B)\n−γdρn(vi−vn),\n∂B\n∂t=∇×(vi×B),\n∂ρn\n∂t+∇·(ρnvn) = 0,\n∂vn\n∂t+(vn·∇)vv=−c2\ns∇lnρn−γdρi(vn−vi),(23)\nwhereviandvnare the velocities of the ionized and neu-\ntral fluids, and Bis the magnetic field. As the time step\nis restricted by the Alfv´ en time step for ions, a two-fluid\nsimulation at a low ionization fraction is computationally\nvery expensive. To reduce the computational cost, a “heavy\nion approximation” (HIA, Oishi & Mac Low 2006; Li et al.\n2008) with artificially decreased ion Alfv´ en speed and in-\ncreased ionization fraction is frequently adopted. Howeve r,\nTilley & Balsara (2010) showed that the HIA can unphysi-\ncally affect the dissipation characteristics of magnetic fl uc-\ntuations. It is also possible to numerically investigate th e par-\ntially ionized magnetized fluids using the single-fluid trea t-\nment by adding an additional diffusive term in the induc-\ntion equation (O’Sullivan & Downes 2006, 2007). However,\nthis approach is unable to capture the two-fluid effect in the\nweakly coupled regime (Balsara 1996; Xu et al. 2016), which\nis of key importance to study the damping stage of dynamo\nconsidered here. Therefore, we perform a full two-fluid sim-\nulation with realistic ion masses to obtain reliable numeri cal\nmeasurements.\n3.1. Simulation setup\nWe set initially uniform densities of both ions and neutrals ,\nwith the neutral density equal to unity. The ions and neutral s\nhave molecular weights as µi= 29 amu (corresponding to\nHCO+) and µn= 2.3amu (corresponding to H 2and He), re-\nspectively, as the mean molecular mass of ions and neutrals i n\nmolecular clouds (Shu 1992; Balsara 1996; Tilley & Balsara\n2010; Meyer et al. 2014; Burkhart et al. 2015). The RIE-\nMANN code has been used to simulate two-fluid magne-\ntized turbulence with an ionization fraction as low as 10−6\n(Tilley & Balsara 2008, 2010). Here we choose a value of\n10−4. The initial seed magnetic field for dynamo amplifi-\ncation is uniform (Cho et al. 2009) and aligned along the x-\ndirection. We drive hydrodynamical turbulence in this init ial\nsetup. The hydrodynamic turbulence is forced via driving ra n-\ndom Gaussian fluctuations in Fourier space, with the driving5\nT(k)\nM(k)\nkνk3/2\nkdkinjk-5/3\n(a) Dissipation-free stageT(k)\nM(k)\nk3/2k-5/3\nkdkνkinj\n(b) Damping stage\nFIG. 1.— Sketches of the magnetic energy spectrum M(k)and the turbulent kinetic energy spectrum T(k)for the dissipation-free and damping stages of\ndynamo, respectively. kinj,kd,kνare the wavenumbers corresponding to L,ld, andlν.\n 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10\n 1 10 100E(k)\nk / 2πk3/2k-5/3M(k,t) : t/ τeddy = 0.3 \n = 0.6 \n = 1.25\n = 3.29\n = 4.98\n = 7.49\nT(k)\n(a) 0.01 0.1 1 10 100 1000\n 1 10 100T(k) k5/3\nk / 2π\n(b)\nFIG. 2.— (a) Numerically measured M(k,t)at different times. T(k)is the steady turbulent energy spectrum. The short dashed li nes indicate the spectral\nscalings of the Kazantsev spectrum and Kolmogorov spectrum . (b) Compensated turbulent energy spectrum.\nt/τeddy0 1 2 3 4 5 6 7 8VA,eff / Vrms\n00.10.20.30.40.50.60.70.80.91\n(a)t/τeddy0 1 2 3 4 5 6 7 8ln (VA,eff / Vrms)\n-3.5-3-2.5-2-1.5-1-0.50\n(b)\nFIG. 3.— (a) Time evolution of the numerically measured VA,eff(normalized by Vrms, filled circles) in comparison with our theoretical predict ion, where the\ndashed and solid lines represent the dissipation-free (Eq. (29)) and damping (Eq. (32)) stages of dynamo, respectively . (b) Same as (a) but for the logarithm of\nVA,eff/Vrms.6\nTABLE 2\nSIMULATION PARAMETERS\nR L ρ i/ρnVrmscsMAi0νni\nVrms/L\n10243512 1.26×10−30.2 1 17 .7 0.08\nscale peaked at k/2π= 2 and spanning 1≤k/2π≤4, and\nan rms velocity of 0.2times the sound speed. The turbulence\nbecomes statistically steady after around three turnover t imes\nof the largest eddy. We continuously drive the turbulence in\nboth ions and neutrals to maintain a constant turbulent en-\nergy and a constant rms velocity throughout the simulation.\nThe turbulent energy cascades toward smaller scales and dis -\nsipates at the numerical dissipation scale. To ensure a clea r\nseparation between the driving scale of turbulence, the ion -\nneutral collisional damping scale that increases with time , and\nthe numerical dissipation scale of turbulence, our simulat ion\nhas a high resolution of 10243mesh points. It is performed in\na computational domain given by [0,1]×[0,1]×[0,1].\nTable 2 lists the numerical resolution (mesh points), the in -\njection scale (mesh points) where most turbulent energy is i n-\njected, the ratio between ρiandρn, the rms velocity, the sound\nspeed, the initial Alfv´ en Mach number MAi0=Vrms/VAi0\nof the ionized fluid, where VAi0=B0/√4πρiis the initial\nAlfv´ en speed in terms of the initial magnetic field strength\nB0and ion density ρi, the ratio between νniand the eddy-\nturnover rate Vrms/LatL. The large value of MAi0shows\nthat the initial magnetic energy contained in the ionized flu id\nis much smaller than the turbulent energy.\nWith our focus on the damping stage of dynamo, we desig-\nnate the values of above parameters in the simulation to sati sfy\nthe physical conditions presented in Section 2.\nCondition (1) :\nTo ensure that the dynamo stage can proceed until ld=L,\nwe have (Eq. (3), Eq. (19))\n2L\nCVL≈0.48<1, (24)\nwhere the values in Table 2 are adopted and we take VL=\nVrms. It shows that due to the low ionization fraction, neutrals\nare decoupled from the dynamo action on all length scales.\nCondition (2) :\nWe note that Condition (2) is naturally fulfilled given Con-\ndition (1) (see Section 2.1). Due to the low ionization frac-\ntion and strong ion-neutral collisional damping, there is n o\nequipartition between magnetic and turbulent energies on a ll\nlength scales. At the end of damping stage at ld=L, the un-\nsaturated magnetic energy mainly contained in ions is small er\nthan the turbulent energy at L.\nCondition (3) :\nThe initial magnetic energy contained in ions is\nEM0=1\n2V2\nAi0. (25)\nWe rewrite Eq. (9) in a dimensionless form and find\nEM0\nC−1νn=1\n61\nM2\nAi0VL/L\nνni/parenleftBigL\nlν/parenrightBig4\n3≈1.3>1, (26)\nwhere the viscous scale is\nlν=L1\n4V−3\n4\nLν3\n4n. (27)In our simulation, lνis determined by the numerical dissipa-\ntion scale, which is on the order of 10mesh points. With the\ngrowth of EM, we have the ion-neutral collisional damping as\nthe dominant damping effect and Condition (3) is satisfied.\n3.2. Comparison between theoretical predictions and\nnumerical measurements\nAt an early time of the simulation, before the turbulent en-\nergy spectrum is fully developed, due to the turbulent energ y\ncascade from large to small scales, the dynamo stretching\nscale, which determines the peak scale of magnetic energy\nspectrum, shifts toward smaller scales. The initial weak ma g-\nnetic field leads to the initially weak ion-neutral collisio nal\ndamping effect. Thus the dynamo is in the dissipation-free\nregime, which is characterized by an exponential growth of\nmagnetic energy. A Kazantsev magnetic energy spectrum on\nscales larger than the peak scale is expected, as seen earlie r\nin one-fluid dynamo simulations (e.g., Haugen et al. 2004;\nBrandenburg & Subramanian 2005).\nWith the growth of magnetic energy, the ion-neutral colli-\nsional damping becomes important, so that the magnetic en-\nergy spectrum peaks and is also damped at ld. The dynamo\nenters the damping stage. As analyzed in Section 2.2, we ex-\npect that the magnetic field strength grows linearly with tim e,\nand the spectral peak at ldmoves toward larger scales.\nIn Fig. 1, we illustrate M(k)in both the dissipation-free\nstage and the damping stage. As a comparison, the numer-\nically measured M(k,t)at different times are presented in\nFig. 2(a). As expected, the spectral peak of M(k,t)indeed\nfirst shifts to smaller scales and then back to larger scales. The\nascending spectral form on large scales is also consistent w ith\nthe Kazantsev spectrum ∼k3/2. Besides, we also present\nthe stationary and fully developed turbulent energy spectr um\nT(K), which is expected to follow the Kolmogorov spectrum\nk−5/3. The appearance of a bottleneck effect with a pileup\nof energy (Falkovich 1994) is observed on small scales of the\ninertial range, which can be more clearly seen in the compen-\nsated turbulent energy spectrum in Fig. 2(b).\n1. Dissipation-free stage of dynamo\nWhen the ion-neutral collisional damping effect is weak,\nthe dynamo stretching leads to an exponential growth of mag-\nnetic energy,\nEM=EM0exp(2Γ pt). (28)\nThe dynamo growth rate Γpcorresponds to the eddy-turnover\nrate at the peak scale lpofM(k,t). As the spectral peak shifts\ntoward smaller scales, Γpincreases with time.\nTo compare with the numerical result, we rewrite Eq. (28)\nin the form,\nVA1,eff\nVL=VA0,eff\nVLexp/parenleftBig/parenleftBigL\nlp/parenrightBig2\n3t\nτeddy/parenrightBig\n, (29)\nwhereτeddy=L/VLis the turnover time of the largest eddy\natL, andVA0,effandVA1,effare the effective Alfv´ en speeds in\nterms of the effective density ρeffat the beginning and the end\nof the dissipation-free stage. We define ρeffas\nρeff=ηcρi, (30)\nwhereηcis the coupling coefficient, as an indicator of the\ncoupling degree between ions and neutrals. When ions and\nneutrals are strongly coupled together, there is ηc=ρ/ρiand\nVA,eff=VA,tot, whereVA,totis the Alfv´ en speed in terms of\nthe total density. When ions and neutrals are decoupled from7\neach other, we have ηc= 1 andVA,eff=VAi. Here we\nare concerned with the weak coupling regime with ηc/greaterorsimilar1,\nwhere neutrals are decoupled from ions but ions are still cou -\npled with neutrals. The exact value of ηcwill be determined\nnumerically.\nBy adopting the values in Table 2, we present the above\ntheoretical calculation (Eq. (29)) in comparison with the n u-\nmerical result in Fig. 3. Approximately, we use a constant\nvalue oflp∼L/3as an estimate of the evolving lpand find\nVA1,eff\nVL≈0.36 (31)\nat the end of the dissipation-free stage at t=t1= 1.1τeddy.\nWe would like to stress here that the growth of magnetic\nenergy during the dissipation-free stage indeed enhances t he\ndamping effect, but the key and necessary condition for the\ndamping stage to arise is a sufficiently small ionization fra c-\ntion, i.e., Condition (1) (Eq. (24)).\n2. Damping stage of dynamo\nWe rewrite the evolution law of magnetic energy in the\ndamping stage of dynamo given by Eq. (16) in a dimension-\nless form\nVA2,eff\nVL=VA1,eff\nVL+3√\n2\n23/parenleftBig3νniL\nVL/parenrightBig1\n2(t−t1)\nτeddy. (32)\nWith the values of parameters in Table 2 used, the theoretica l\ncalculation is displayed in Fig. 3. By comparing with the\nnumerical measurement, we also found ηc≈2.45. Withηc\nbeing of the order of a few, ρeffis close to ρi(Eq. (30)). It\nshows that the growing magnetic energy mainly comes from\nthe turbulent energy contained in ions in the weak coupling\nregime, as discussed in Section 2.1.\nAt the end of the damping stage, the theoretical expectation\nin Eq. (21) yields\nVA2,eff\nVL=/parenleftBig\n6νniL\nVL/parenrightBig1\n2= 0.69<1. (33)\nThe corresponding time is (Eqs. (31), (32), and (33)),\nt2=t1+3.7τeddy= 4.8τeddy. (34)\nWe see in Fig. 3 that the damping stage observed in the nu-\nmerical simulation is slightly more extended than the above\nprediction, but the dynamo growth ceases soon after t=t2.\nMoreover, the time evolution of ldin the damping stage is\n(Eq. (18))\nld\nL=/bracketleftbigg/parenleftbiggld1\nL/parenrightbigg2\n3\n+3\n23(t−t1)\nτeddy/bracketrightbigg3\n2\n. (35)\nStarting from (Eqs. (13) and (31))\nld1\nL=/bracketleftbiggVL\n6νniLV2\nA1,eff\nV2\nL/bracketrightbigg3\n4\n= 0.38, (36)\nwe see that ldreachesLatt=t2(Eqs. (34) and (35)). Fig. 4\ndisplays the 2D magnetic field structure measured at the end\nof the simulation, which is dominated by large-scale magnet ic\nfield fluctuations. It confirms that the magnetic field resulti ng\nfrom the damping stage of dynamo has a characteristic length\nscale comparable to Lin our simulation.4.PHYSICAL CONDITIONS IN THE ISM FOR THE\nDAMPING STAGE OF DYNAMO\nAs illustrative examples for the applications of the above\ndynamo theory, here we examine the physical conditions\nin the partially ionized ISM for the damping stage of dy-\nnamo. Table 3 lists the typical parameters of the warm neutra l\nmedium (WNM), the cold neutral medium (CNM), molecu-\nlar clouds (MC) and dense cores in molecular clouds (DC),\nwherenHandneare number densities of the atomic hydro-\ngen and electrons, and Tis the temperature. Their values\nare taken from Draine & Lazarian (1998). Besides, we as-\nsumemi=mn=mHas the masses of ions and neutrals in\nWNM and CNM, and mi= 29mH,mn= 2.3mHin MC and\nDC (Shu 1992), where mHis the hydrogen atomic mass. We\nalso have νn=vth/(nnσnn), with the neutral thermal speed\nvth, the neutral number density nn, and the cross-section of a\nneutral-neutral collision σnn∼10−14cm2(Vranjes & Krstic\n2013). The drag coefficient is γd= 5.5×1014cm3g−1s−1in\nWNM and CNM, and γd= 3.5×1013cm3g−1s−1in MC and\nDC (Draine et al. 1983; Shu 1992). We next analyze the tur-\nbulent dynamo induced by (a) the globally driven interstell ar\nturbulence and (b) the locally excited turbulence in supern ova\nremnants (SNRs).\nTABLE 3\nTURBULENT DYNAMO IN THE PARTIALLY IONIZED ISM\nWNM CNM MC DC\nnH[cm−3]0.4 30 300 104\nne/nH 0.1 10−310−410−6\nT[K] 6000 100 20 10\nInterstellar turbulence\nld,cr[pc] - - 6.3×10−63.3×10−5\nτdam[kyr] - - 0.4 2 .3\nBdam[µG] - - 0.5 4 .7\nτnon[kyr] 1.9×1041.9×1041.9×1041.9×104\nBnon[µG] 3.0 25 .1 79 .5 458 .1\nPreshock turbulence\nld,cr[pc] 0.1\nτdam[kyr] 0.75\nBdam[µG]79.1 56 .6 415 .2 138 .2\n(a) Interstellar turbulence\nWe consider that the interstellar turbulence driven by su-\npernova explosions has a typical driving condition (Spitze r\n1978),\nL= 30 pc, VL= 10 km s−1. (37)\nAs a result of turbulent energy cascade, the interstellar tu r-\nbulence extends from Ltolν. Here we assume that the ini-\ntial seed magnetic field is sufficiently weak, so that the turb u-\nlent motions on all length scales can contribute to the dynam o\ngrowth. In partially ionized phases, to examine Condition (1) ,\nwe calculate the dynamo stretching rate Γνoflν-scale ed-\ndies and ΓLofL-scale eddies in comparison with C−1, as\npresented in Fig. 5(a). We find that in WNM and CNM, as\nCondition (1) is not satisfied in the entire inertial range [L,lν]\nof turbulence, the dynamo does not go through the damping\nstage, but instead has a nonlinear stage (see below). In MC\nand DC, the damping stage of dynamo can arise at lν, but can-\nnot proceed to LasCondition (1) atLis not met. Therefore,\nthe dynamo has both damping and nonlinear stages.8\nxz|B|\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.020.040.060.080.10.12\n(a)xy|B|\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.020.040.060.080.1\n(b)\nyz|B|\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.020.040.060.080.10.12\n(c)\nFIG. 4.— A 2D cross section through the middle of the computation al domain of the numerically measured magnetic field strengt h in the (a) xz plane, (b) xy\nplane, and (c) yz plane, corresponding to M(k,t)att= 7.49τeddyin Fig. 2(a).\nThe critical damping scale where the damping stage termi-\nnates can be determined by\n2\nCΓd,cr= 1, (38)\nwhere\nΓd,cr=L−1\n3VLl−2\n3\nd,cr. (39)\nIt yields (XL16)\nld,cr=/parenleftBigC\n2/parenrightBig3\n2L−1\n2V3\n2\nL. (40)\nBy inserting the above expression in Eq. (18), we obtain the\ntimescale of damping stage,\nτdam=t(ld=ld,cr)−t1(ld1=lν) =23\n3/parenleftBigC\n2−Γ−1\nν/parenrightBig\n.(41)\nHere we assume that the initial magnetic field is sufficiently\nweak and thus the damping stage starts from lν. The values\nofld,crandτdamfor MC and DC are listed in Table 3.At the end of damping stage, EMbecomes\nEM,dam=1\n2v2\nd,cr=1\n2V2\nLL−2\n3l2\n3\nd,cr. (42)\nBy inserting Eq. (40) into the above equation, we obtain\n(XL16)\nEM,dam=C\n4L−1V3\nL, (43)\nwhich can also be derived by combining Eq. (13) with Eq.\n(40). The corresponding field strength is\nBdam=/radicalbig\n8πρeffEM,dam. (44)\nAccording to Eq. (38), the ion-neutral coupling becomes\nstrong at the end of damping stage. By using ρeff=ρin the\nabove expression, we determine the values of Bdam, as pre-\nsented in Table 3. We see that due to the small length scale,\nthe short timescale, and the resulting weak magnetic field, t he\ndamping stage is not important for the dynamo process in-\nduced by the interstellar turbulence in the partially ioniz ed\nISM.\nAfter the short damping stage, the turbulent dynamo enters\nthe nonlinear regime. Both the dynamo stretching and tur-9\nbulent diffusion of magnetic fields mainly take place at lpof\nM(k,t), where (XL16)\nΓpEM=1\n2L−1V3\nL. (45)\nBy comparing ωINatlpwithΓp(Eqs. (6) and (45)),\nωIN(l=lp)\nΓp=Cl−2\npEM\nΓp=CΓp\n2, (46)\nwe see that since Condition (1) breaks down in the nonlinear\nstage, the above ratio is less than unity. As the nonlinear tu r-\nbulent dynamo is in a strongly coupled regime, the magnetic\nfield diffusion due to the slippage between ions and neutrals\nand the ion-neutral collisional damping are unimportant fo r\nthe nonlinear stage of dynamo (XL16).\nThe nonlinear turbulent dynamo leads to a scale-by-scale\nequipartition between the turbulent energy and the magneti c\nenergy. At the full saturation at L, all the turbulent energy\ncarried by strongly coupled ions and neutrals can be con-\nverted to the magnetic energy. The saturated field strength\nBnon=√4πρVLat the end of nonlinear stage is presented\nin Table 3, which provides the maximum magnitude of turbu-\nlent magnetic fields in the partially ionized ISM. These esti -\nmates are also consistent with the Zeeman measurements by\nCrutcher et al. (2010). It implies that the nonlinear turbul ent\ndynamo accounts for the turbulent magnetic fields observed\nin the ISM.\nThe timescale of nonlinear stage is (XL16)\nτnon=19\n3/parenleftbiggL\nVL−Γ−1\nν/parenrightbigg\n(47)\nin WNM and CNM, and\nτnon=19\n3/parenleftbiggL\nVL−C\n2/parenrightbigg\n. (48)\nin MC and DC (see Table 3). It is approximately 6τeddy, which\nis longer than τdamby several orders of magnitude.\n(b) Preshock turbulence in SNRs\nWhen an SNR shock sweeps through the ISM, the preshock\nturbulence can be driven by the interaction between the\ncosmic-ray pressure gradient and interstellar density inh omo-\ngeneities (Beresnyak et al. 2009). We consider the driving\ncondition as (XL17)\nL= 0.1pc, VL= 103km s−1. (49)\nHere we use the characteristic scale of the density structur e in\nthe cold ISM (Heiles & Troland 2003; Goodman et al. 1998)\nasL, andVLis of the order of the shock velocity. With a\nhigh dynamo stretching rate and Condition (1) satisfied in the\nentire inertial range [L,lν]of preshock turbulence (see Fig.\n5(b)), the preshock turbulent dynamo in all partially ioniz ed\nphases remains in the damping stage. ld,crin this case is equal\ntoL. Accordingly, the damping stage has a timescale\nτdam=t(ld=L)−t1(ld1=lν) =23\n3/parenleftBigL\nVL−Γ−1\nν/parenrightBig\n.(50)\nHere we again assume that the dynamo starts at lνwith suf-\nficiently weak seed field. As 1/Γνis negligibly small com-\npared with L/VL, the values of τdamin different phases are\napproximately the same (see Table 3). We note that τdam\nis sufficiently small compared to the precursor crossing tim eτc∼(c/v sh)L/VL, wherecandvshrepresent light speed and\nshock velocity, respectively (XL17). So the L-scale magnetic\nfield can be amplified within τc.\nEM,damat the end of damping stage is given by Eq. (21). As\nthe damping stage of dynamo is in a weakly coupled regime,\nwe adopt ρeff∼ρiand present Bdam∼/radicalbig\n8πρiEM,damas the\nlower limit of Bdamin Table 3. The dynamo-amplified mag-\nnetic field can confine energetic particles near the shock to\nfacilitate the shock acceleration. For example, the maximu m\nenergy of cosmic rays that can be confined by the resulting\npreshock magnetic field in the case of MC is\nECR,max=eBdamL= 38.4PeV. (51)\nThis already reaches the PeV knee of the cosmic ray spectrum\nand supports the Galactic origin of the cosmic rays below the\nknee. Besides, magnetic fields of the order of 100µG near the\nshock front of SNRs are also inferred from observations (e.g .,\nBamba et al. 2003, 2005b,a; Vink 2012).\n5.ION-NEUTRAL COUPLING IN MHD TURBULENCE\nAND IN TURBULENT DYNAMO\nIn a partially ionized medium, the coupling state between\nions and neutrals is crucial for determining the damping of\nMHD turbulence and the efficiency of turbulent dynamo.\nMHD turbulence. In the strong Alfv´ enic turbulence with\nthe magnetic energy in equipartition with the turbulent en-\nergy atL, there is a critical balance between the turbulent\ncascade rate, i.e., eddy-turnover rate, vl/l⊥and the Alfv´ en\nwave frequency ωA=VA,eff/l/bardbl(Goldreich & Sridhar 1995),\nwherel⊥andl/bardblare the perpendicular and parallel compo-\nnents of the length scale with respect to the local magnetic\nfield (Lazarian & Vishniac 1999). The anisotropic scaling re -\nsulting from the critical balance in the local reference system\nhas been confirmed in both one-fluid (e.g., Cho & Lazarian\n2002, 2003) and two-fluid (Burkhart et al. 2015) MHD simu-\nlations down to the dissipation scale of Alfv´ enic turbulen ce.\nThe ion-neutral collisional damping of the turbulent cas-\ncade depends on the coupling state between ions and neutrals .\nAs summarized in Table 4, in the low wave-frequency range\nwithωA< νni, Alfv´ en waves with VA,eff=VA,totpropa-\ngate in the strongly coupled ions and neutrals.5By contrast,\nat high wave frequencies with ωA> νin, ions and neutrals\nare essentially decoupled from each other, and Alfv´ en wave s\nwithVA,eff=VAican only propagate in ions. Within inter-\nmediate wave frequencies, neutrals are decoupled from ions ,\nbut ions are still collisionally coupled to neutrals. Accor d-\ningly, Alfv´ en waves propagating in the weakly coupled ions\nand neutrals have (Xu et al. 2015, 2016)\nω2\nA=k2\n/bardblV2\nAi[(1+χ)ν2\nni+k2\n/bardblV2\nAi]\n(1+χ)2ν2\nni+k2\n/bardblV2\nAi=k2\n/bardblV2\nA,eff, (52)\nwherek= 1/l,χ=ρn/ρi, and\nVA,eff=/radicaltp/radicalvertex/radicalvertex/radicalbt(1+χ)ν2\nni+k2\n/bardblV2\nAi\n(1+χ)2ν2\nni+k2\n/bardblV2\nAiVAi, (53)\nwhich depends on the length scale. MHD turbulence in the\nweak coupling regime is subjected to the severest ion-neutr al\ncollisional damping. As a result, both Alfv´ en waves and\n5In the strong Alfv´ enic turbulence, Alfv´ en waves can only p ropagate over\nthe distance of one wavelength due to their nonlinear intera ctions.10\nC−1[s−1]3×10-1110-103×10-10Γl[s−1]\n10-1510-1410-1310-1210-1110-1010-910-810-7\n(a) Interstellar turbulenceC−1[s−1]3×10-1110-103×10-10Γl[s−1]\n10-1110-1010-910-810-710-610-510-410-3\n(b) Preshock turbulence\nFIG. 5.— The shaded region shows the parameter space for the appe arance of damping stage of dynamo. The symbols represent the values for WNM (circle),\nCNM (square), MC (triangle), and DC (diamond). Filled and op en symbols correspond to ΓνandΓL, respectively.\nAlfv´ enic turbulent motions are damped in the weak coupling\nregime.\nIt is worth noting that the ambipolar diffusion scale fre-\nquently used in the literature (e.g., Mouschovias 1991)\nlAD=VA,tot\nνni(54)\nis only equivalent to the parallel neutral-ion decoupling s cale\nfor the anisotropic Alfv´ enic turbulence. Since the energy of\nAlfv´ enic turbulence cascades mainly along the direction p er-\npendicular to the local magnetic field, we are concerned with\nthe perpendicular neutral-ion decoupling scale, which is r e-\nlated tolADvia the critical balance mentioned above.\nTurbulent dynamo. Similarly, there also exist different ion-\nneutral coupling regimes for the turbulent dynamo, dependi ng\non the range of Γl(see Table 4). When Γl< νni, turbu-\nlence in the strongly coupled ions and neutrals induces the\ngrowth of magnetic energy, which can be expressed in terms\nofVA,eff=VA,tot. When Γl> νin, neutrals are not in-\nvolved in the dynamo process. The dynamo only operates\nin ions and results in the growth of magnetic energy in terms\nofVA,eff=VAi. For an intermediate Γlconsidered in this\nwork, the dynamo takes place in the weakly coupled ions and\nneutrals and is affected by the strongest ion-neutral colli sional\ndamping. As a result, the dynamo has a damping stage.\nTABLE 4\nION-NEUTRAL COUPLING IN MHD TURBULENCE AND TURBULENT\nDYNAMO\nCoupling state Strong coupling Weak coupling Decoupling\nMHD turbulence ωA< νniνni< ωA< νinωA> νin\nTurbulent dynamo Γl< νniνni<Γl< νinΓl> νin\nBesides ion-neutral collisional damping, the viscosity in\nneutrals also leads to the damping of MHD turbulence in\na partially ionized medium (Lazarian et al. 2004). The pa-\nrameter space for the dominance of neutral viscous damping\nand the appearance of the new regime of MHD turbulence in\nthe sub-viscous range (Cho et al. 2002a, 2003) is provided inXu & Lazarian (2017b). In the context of turbulent dynamo,\nthe damping stage of dynamo can only arise when the ion-\nneutral collisional damping is stronger than the neutral vi s-\ncous damping.\n6.SUMMARY\nWe have studied the turbulent dynamo in a weakly ionized\nmedium and numerically tested the damping stage of dynamo\nas theoretically predicted by XL16. Here we summarize our\nmain results.\n•We have explicitly provided the physical conditions un-\nder which the damping stage of dynamo can arise in a\npartially ionized medium. They are Eq. (2) and Eq.\n(9), and Eq. (19) for the damping stage to persist until\nthe damping scale reaches the injection scale Lof tur-\nbulence. With sufficiently small ionization fraction and\nseed magnetic field, the timescale of damping stage is\naround eight times the largest eddy-turnover time (Sec-\ntion 2).\n•By performing the two-fluid dynamo simulation un-\nder the above conditions (Eqs. (2), (9), and (19)) and\nquantitative comparisons between the theoretical pre-\ndictions and numerical measurements, we have numer-\nically confirmed the linear-in-time growth of magnetic\nfield strength due to the severe ion-neutral collisional\ndamping in the damping stage of dynamo. As a result\nof the weak coupling between ions and neutrals, most\nturbulent kinetic energy contained in neutrals cannot be\nconverted to the magnetic energy (Section 3).\n•We have examined the physical conditions for the\ndamping stage of dynamo in the partially ionized ISM\nand provided the parameter space for its appearance\n(Section 4). For the dynamo induced by the interstel-\nlar turbulence, the damping stage contributes insignif-\nicantly to the dynamo growth of magnetic energy. In-\nstead, the nonlinear stage is mainly responsible for the\ngrowth of the interstellar turbulent magnetic fields. By\ncontrast, the dynamo induced by the preshock turbu-\nlence in SNRs remains in the damping stage, which is11\nimportant for studying the magnetic field amplification\nand cosmic ray acceleration at shocks.\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\nby the Association of Universities for Research in Astron-omy, Incorporated, under NASA contract NAS5-26555. S.X.\nalso thanks Chris McKee for useful conversations. DSB ac-\nknowledges support via NSF grants NSF-ACI-1533850, NSF-\nDMS-1622457, NSF-ACI-1713765 and NSF-DMS-1821242.\nSupport from a grant by Notre Dame International is also ac-\nknowledged. A.L. acknowledges the support from grant NSF-\nDMS-1622353.\nREFERENCES\nBalsara, D. 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O. 2011 , A&A, 529,\nA82" }, { "title": "2303.03852v1.Electrically_tunable_Gilbert_damping_in_van_der_Waals_heterostructures_of_two_dimensional_ferromagnetic_metals_and_ferroelectrics.pdf", "content": "Page 1 of 15 \n Electrically tunable Gilbert damping in van der Waals heterostructures of two-\ndimensional ferromagnetic meta ls and ferroelectrics \nLiang Qiu,1 Zequan Wang,1 Xiao-Sheng Ni,1 Dao-Xin Yao1,2 and Yusheng Hou 1,* \nAFFILIATIONS \n1 Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, State \nKey Laboratory of Optoelectronic Materials and Technologies, Center for Neutron \nScience and Technology, School of Physics, Sun Yat-Sen University, Guangzhou, \n510275, China \n2 International Quantum Academy, Shenzhen 518048, China \n \nABSTRACT \nTuning the Gilbert damping of ferromagnetic (FM) metals via a nonvolatile way is \nof importance to exploit and design next-generation novel spintronic devices. Through \nsystematical first-principles calculations, we study the magnetic properties of the van \nder Waals heterostructure of two-dimensional FM metal CrTe 2 and ferroelectric (FE) \nIn2Te3 monolayers. The ferromagnetism of CrTe 2 is maintained in CrTe 2/In2Te3 and its \nmagnetic easy axis can be switched from in-plane to out- of-plane by reversing the FE \npolarization of In 2Te3. Excitingly, we find that the Gilbert damping of CrTe 2 is tunable \nwhen the FE polarization of In 2Te3 is reversed from upward to downward. By analyzing \nthe k-dependent contributions to the Gilbert damping, we unravel that such tunability \nresults from the changed intersections between the bands of CrTe 2 and Fermi level on \nthe reversal of the FE polarizations of In 2Te3 in CrTe 2/In2Te3. Our work provides a n \nappealing way to electrically tailor Gilbert dampings of two-dimensional FM metals by \ncontacting them with ferroelectrics. \n \n*Authors to whom correspondence should be addressed: \n[Yusheng Hou, houysh@mail.sysu.edu.cn] \n \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 2 of 15 \n Since the atomically thin long-range ferromagnetic ( FM) orders at finite \ntemperatures are discovered in CrI 31 monolayer (ML) and Cr 2Ge2Te62 bilayer, two-\ndimensional (2D) van der Waals (vdW) FM materials have attracted intensive \nattention.3-5 Up to now, many novel vdW ferromagnets such as Fe 3GeTe 2,6 Fe5GeTe 2,7 \nVSe 28,9 and MnSe 210 have been synthesized in experiments. Due to the intrinsic \nferromagnetism in these vdW FM materials, it is highly fertile to engineer emergent \nphenomena through magnetic proximity effect in their heterostructures.11 For instance , \nan unprecedented control of the spin and valley pseudospins in WSe 2 ML is reported in \nCrI 3/WSe 2.12 By contacting the thin films of three-dimensional topological insulators \nand graphene with CrI 3, high-temperature quantum anomalous Hall effect and vdW spin \nvalves are proposed in CrI 3/Bi2Se3/CrI 313 and CrI 3/graphene/CrI 3,14 respectively. On the \nother hand, the magnetic properties of these vdW FM materials can also be controlled \nby means of external perturbations such as gating and moiré patterns.3 In CrI 3 bilayer, \nHuang et al. observed a voltage-controlled switching between antiferromagnetic (AFM) \nand FM states.15 Via an ionic gate, Deng et al. even increased the Curie temperature \n(TC) of the thin flake of vdW FM metal Fe 3GeTe 2 to room temperature, which is much \nhigher than its bulk TC.6 Very recently, Xu et al. demonstrated a coexisting FM and \nAFM state in a twisted bilayer CrI 3.16 These indicate that vdW FM materials are \npromising platforms to design and implement spintronic devices in the 2D limit.4,11 \nRecently, of great interest is the emergent vdW magnetic material CrTe 2 which is \na new platform for realizing room-temperature intrinsic ferromagnetism.17,18 Especially, \nCrTe 2 exhibits greatly tunable magneti sm. In the beginning, its ground state is believed \nto be the nonmagnetic 2 H phase,19 while several later researches suggest that either the \nFM or AFM 1 T phases should be the ground state of CrTe 2.17,18,20- 23 Currently, the \nconsensus is that the structural ground state of CrTe 2 is the 1 T phase. With respect to its \nmagnetic ground state, a first-principles study shows that the FM and AFM ground \nstates in CrTe 2 ML depend on its in-plane lattice constants.24 It is worth noting that the \nTC of FM CrTe 2 down to the few-layer limit can be higher than 300 K,18 making it have \nwide practical application prospects in spintronics. \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 3 of 15 \n Building heterostructures of FM and ferroelectric (FE) materials offers an effective \nway to control nonvolatile magnetism via an electric field. Experimentally, Eerenstein \net al. presented an electric-field-controlled nonvolatile converse magnetoelectric effect \nin a multiferroic heterostructure La0.67Sr0.33MnO 3/BaTiO 3.25 Later, Zhang et al. reported \nan electric-field-driven control of nonvolatile magnetization in a heterostructure of FM \namorphous alloy Co40Fe40B20 and FE Pb(Mg 1/3Nb2/3)0.7Ti0.3O3.26 Theoretically, Chen et \nal. demonstrated based on first-principles calculations that the interlayer magnetism of \nCrI 3 bilayer in CrI 3/In2Se3 is switchable between FM and AFM couplings by the \nnonvolatile control of the FE polarization direction of In 2Se3.27 In spite of these \ninteresting findings, using FE substrates to electrically tune the Gilbert damping of \nferromagnets, an important factor determining the operation speed of spintronic devices, \nis rarely investigated in 2D FM/FE vdW heterostructures. Therefore, it is of great \nimportance to explore the possibility of tuning the Gilbert damping in such kind of \nheterostructures. \nIn this work, we first demonstrate that the magnetic ground state of 1 T-phase CrTe 2 \nML will change from the zigzag AFM (denoted as z-AFM) to FM orders with increasing \nits in-plane lattice constants. By building a vdW heterostructure of CrTe 2 and FE In2Te3 \nMLs, we show that the magnetic easy axis of CrTe 2 can be tuned from in-plane to out-\nof-plane by reversing the FE polarization of In 2Te3, although its ferromagnetism is kept . \nImportantly, we find that the Gilbert damping of CrTe 2 is tunable with a wide range on \nreversing the FE polarization of In 2Te3 from upward to downward. Through looking \ninto the k-dependent contributions to the Gilbert damping, we reveal that such tunability \noriginates from the changed intersections between the bands of CrTe 2 and Fermi level \nwhen the FE polarizations of In 2Te3 is reversed in CrTe 2/In2Te3. Our work demonstrates \nthat putting 2D vdW FM metals on FE substrates is an attractive method to electrically \ntune their Gilbert dampings. \nCrTe 2, a member of the 2D transition metal dichalcogenide family, can potentially \ncrystalize into several different layered structures such as 1 T, 1Td, 1H and 2 H phases.28 \nIt is believed that the 1 T phase is the most stable among all of the se possible phases in \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 4 of 15 \n both bulk and ML. This phase has a hexagonal lattice and belongs to the P3_m1 space \ngroup, with each Cr atom surrounded by the octahedrons of Te atoms (Fig. 1a). In view \nof the hot debates on the magnetic ground state in CrTe 2 ML, we establish a 2× 2√3 \nsupercell and calculate the total energies of several different magnetic structures (Fig. \nS1 in Supplementary Materials) when its lattice constant varies from 3.65 to 4.00 Å. As \nshown in Fig. 1b, our calculations show that z-AFM order is the magnetic ground state \nwhen the lattice constant is from 3.65 to 3.80 Å. By contrast, the FM order is the \nmagnetic ground state when the lattice constant is in the range from 3.80 to 4.00 Å. \nNote that our results are consistent with the experimentally observed z- AFM23 and \nFM29 orders in CrTe 2 with a lattice constant of 3.70 and 3.95 Å, respectively. Since we \nare interested in the Gilbert damping of ferromagnets and the experimentally grow n \nCrTe 2 on ZrTe 2 has a lattice constant of 3.95 Å,29 we will focus on CrTe 2 ML with this \nlattice constant hereinafter. \n \n \nFIG. 1. (a) Side (the top panel) and top (the bottom panel) views of CrTe 2 ML. The NN \nand second- NN exchange paths are shown by red arrows in the top view. (b) The phase \ndiagram of the magnetic ground state of CrTe 2 ML with different lattice constants. Insets \nshow the schematic illustrations of the z-AFM and FM orders. The up and down spins \nare indicated by the blue and red balls, respectively. The stars highlight the experimental \nlattice constants of CrTe 2 in Ref.23 and Ref.29. \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 5 of 15 \n To obtain an deeper understanding on the ferromagnetism of CrTe 2 ML, we adopt \na spin Hamiltonian consisting of Heisenberg exchange couplings and single-ion \nmagnetic anisotropy (SIA) as follows:30 \n𝐻 = 𝐽 1∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨𝑖𝑗⟩ + 𝐽 2∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨⟨𝑖𝑗⟩⟩ − 𝐴 ∑ ( 𝑆𝑖𝑧)2\n𝑖 (1) \nIn Eq. (1), J1 and J2 are the nearest neighbor (NN) and second- NN Heisenberg exchange \ncouplings. Note that a negative (positive) J means a FM (AFM) Heisenberg exchange \ncouples. Besides, A parameterizes the SIA term. First of all, our DFT calculations show \nthat the magnetic moment of CrTe 2 ML is 3.35 μB/Cr, consistent with previous DFT \ncalculations.31 As shown in Table I, the calculated J1 and J2 are both FM and J1 is much \nstronger than J2. Both FM J1 and J2 undoubtedly indicate that CrTe 2 ML has a FM \nmagnetic ground state. Finally, the SIA parameter A is obtained by calculating the \nenergy difference between two FM states with out-of-plane and in-plane magnetizations. \nOur calculations obtain A=1.81 meV/Cr, indicating that CrTe 2 ML has an out-of-plane \nmagnetic easy axis. Hence, our calculations show that CrTe 2 ML exhibits an out-of-\nplane FM order, consistent with experimental observations.29 \n \nTABLE I. Listed are the in-plane lattice constant s a, Heisenberg exchange couplings J \n(in unit of meV) and SIA (in unit of meV/Cr) of CrTe 2 ML and CrTe 2/In2Te3. \nSystem a (Å) J1 J2 A \nCrTe 2 3.95 -24.56 -0.88 1.81 \nCrTe 2/In2Te3(↑) 7.90 -20.90 -1.80 -1.44 \nCrTe 2/In2Te3(↓) 7.90 -19.33 -0.88 0.16 \n \nTo achieve an electrically tunable Gilbert damping in CrTe 2 ML, we establish its \nvdW heterostructure with F E In2Te3 ML. In building this heterostructure, w e stack a \n2×2 supercell of CrTe 2 and a √3 ×√3 supercell of In2Te3 along the (001) direction. \nBecause the magnetic properties of CrTe 2 ML are the primary topic and the electronic \nproperties of In2Te3 ML are basically not affected by a strain (Fig. S2), we stretch the \nlattice constant of the latter to match that of the former. Fig. 2a shows the most stable \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 6 of 15 \n stacking configuration in CrTe 2/In2Te3 with an upward FE polarization [denoted as \nCrTe 2/In2Te3(↑)]. At the interface in this configuration, one of four Cr atoms and one of \nfour Te atoms at the bottom of CrTe 2 sits on the top of the top-layer Te atom s of In2Te3. \nIn CrTe 2/In2Te3 with a downward FE polarization [denoted as CrTe 2/In2Te3(↓ )], the \nstacking configuration at its interface is same as that in CrTe 2/In2Te3(↑ ). The only \ndifference between CrTe 2/In2Te3(↑ ) and CrTe 2/In2Te3(↓ ) is that the middle-layer Te \natoms of In 2Te3 in the former is farther to CrTe 2 than that in the latter (Fig. 2a and 2c). \nIt is noteworthy that the bottom-layer Te atoms of CrTe 2 do not stay at a plane anymore \nin the relaxed CrTe 2/In2Te3 (see more details in Fig. S3), suggesting non-negligible \ninteractions between CrTe 2 and In 2Te3. \n \n \nFIG. 2. (a) The schematic stacking configuration and (b) charge density difference 𝛥ρ \nof CrTe 2/In2Te3(↑). (c) and (d) same as (a) and (b) but for CrTe 2/In2Te3(↓). In (b) and \n(d), color bar indicates the weight of negative (blue) and positive (red) charge density \ndifferences. (e) The total DOS of CrTe 2/In2Te3. (f) and (g) show the PDOS of CrTe 2 and \nIn2Te3 in CrTe 2/In2Te3, respectively. In (e)-(g), upward and downward polarizations are \nindicated by black and red lines, respectively. \n \nTo shed light on the effect of the FE polarization of In 2Te3 on the electronic \nproperty of CrTe 2/In2Te3, we first investigate the spatial distribution of charge density \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 7 of 15 \n difference \n2 2 3 2 2 3 CrTe In Te CrTe In Te = − − with different FE polarization directions. \nAs shown in Fig. 2b and 2d, we see that there is an obvious charge transfer at the \ninterfaces of both CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓), which is further confirmed by \nthe planar averaged 𝛥ρ (Fig. S4). Additionally, the charge transfer in CrTe 2/In2Te3(↑) \nis distinctly less than th at in CrTe 2/In2Te3(↓). Fig. 2e shows that the total density of \nstates (DOS) near Fermi level are highly different in CrTe 2/In2Te3(↑ ) and \nCrTe 2/In2Te3(↓). By projecting the DOS onto CrTe 2 and In 2Te3, Fig. 2f shows that the \nprojected DOS (PDOS) of CrTe 2 in CrTe 2/In2Te3(↑) is larger than that in CrTe 2/In2Te3(↓) \nat Fermi level. Interestingly, the PDOS of In 2Te3 in CrTe 2/In2Te3(↑) is larger than that \nin CrTe 2/In2Te3(↓) below Fermi level while the situation is inversed above Fermi level \n(Fig. 2g). By looking into the five Cr- d orbital projected DOS in CrTe 2/In2Te3(↑) and \nCrTe 2/In2Te3(↓) (Fig. S5), we see that there are obviously different occupations for xyd, \n22xyd− and 223zrd− orbitals near Fermi level. All of these imply that the reversal of the \nFE polarization of In 2Te3 may have an unignorable influence on the magnetic properites \nof CrTe 2/In2Te3. \nDue to the presence of the FE In 2Te3, the inversion symmetry is inevitably broken \nand nonzero Dzyaloshinskii-Moriya interactions (DMIs) may exist in CrTe 2/In2Te3. In \nthis case, we add a DMI term into Eq. (1) to investigate the magnetism of CrTe 2/In2Te3 \nand the corresponding spin Hamiltonian is in the form of32 \n𝐻 = 𝐽 1∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨𝑖𝑗⟩ + 𝐽 2∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨⟨𝑖𝑗⟩⟩ +∑ 𝑫𝑖𝑗⋅ (𝑆 𝑖× 𝑆 𝑗) ⟨𝑖𝑗⟩ − 𝐴 ∑ ( 𝑆𝑖𝑧)2\n𝑖 (2). \nIn Eq. (2), Dij is the DMI vector of the NN Cr-Cr pairs. As the C6-rotational symmetry \nwith respect to Cr atoms in CrTe 2 is reduced to the C3-rotational symmetry, the NN \nDMIs are split into four different DMIs (Fig. S6). For simplicity, the J1 and J2 are still \nregard ed to be six-fold. From Table I, we see that the NN J1 of both CrTe 2/In2Te3(↑) and \nCrTe 2/In2Te3(↓) are still FM but slightly smaller than that of free-standing CrTe 2 ML. \nMoreover, the second- NN FM J2 is obviously enhanced in CrTe 2/In2Te3(↑) compared \nwith CrTe 2/In2Te3(↓) and free-standing CrTe 2 ML. To calculated the NN DMIs, we build \na √3×√3 supercell of CrTe 2/In2Te3 and the four-state method33 is employed here. As \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 8 of 15 \n listed in Table S1, the FE polarization direction of In 2Te3 basically has no qualitative \neffect on the DMIs in CrTe 2/In2Te3 although it affects their magnitudes. More explicitly, \nthe magnitudes of the calculated DMIs range from 1.22 to 2.81 meV, which are about \none order smaller than the NN J1. Finally, we find that the SIA of CrTe 2/In2Te3 is \nstrongly dependent on the FE polarization of In 2Te3. When In 2Te3 has an upward FE \npolarization, the SIA of CrTe 2/In2Te3(↑) is negative, indicating an in-plane magnetic \neasy axis. However, when the FE polarization of In 2Te3 is downward, CrTe 2/In2Te3(↓) \nhas a positive SIA, indicating an out-of-plane magnetic easy axis. It is worth noting that \nCrTe 2/In2Te3(↓) has a much weak SIA than the free-standing CrTe 2 ML, although they \nboth have positive SIAs. The different Heisenberg exchange couplings, DMIs and SIAs \nin CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓) clearly unveil that the magnetic properties of \nCrTe 2 are tuned by the FE polarization of In2Te3. \nTo obtain the magnetic ground state of CrTe 2/In2Te3, MC simulations are carried \nout. As shown in Fig. S7, CrTe 2/In2Te3(↑) has an in-plane FM magnetic ground state \nwhereas CrTe 2/In2Te3(↓ ) has an out-of-plane one. Such magnetic ground states are \nunderstandable. Firstly, the ratios between DMIs and the NN Heisenberg exchange \ncouplings are small and most of them are out of the typical range of 0.1–0.2 for the \nappearance of magnetic skyrmions.34 Secondly, the SIAs of the CrTe 2/In2Te3(↑) and the \nCrTe 2/In2Te3(↓ ) prefer in-plane and out-of-plane magnetic easy axes, respectively . \nTaking them together, we obtain that the FM Heisenberg exchange couplings dominate \nover the DMIs and thus give rise to a FM magnetic ground state with its magnetization \ndetermined by the SIA,35 consistent with our MC simulated results. \nFigure 3a shows the Γ-dependent Gilbert dampings of CrTe 2/In2Te3 with upward \nand downward FE polarizations of In2Te3. Similar to previous studies,36,37 the Gilbert \ndampings of CrTe 2/In2Te3 decrease first and then increase as the scattering rate Γ \nincreases. Astonishingly, the Gilbert dampings of CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓) \nare distinctly different at the same scattering rate Γ ranging from 0.001 to 1.0 eV . To \nhave a more intuitive sense on the effect of the FE polarizations of In 2Te3 on the Gilbert \ndampings in CrTe 2/In2Te3, we calculate the ratio = at any given Γ, where \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 9 of 15 \n ( ) is the Gilbert damping of CrTe 2/In2Te3(↑) [CrTe 2/In2Te3(↓)]. As shown in Fig. \n3b, the ratio 𝜂 ranges from 6 to around 1.3 with increasing Γ. As the FE polarization \nof In 2Te3 can be switched from upward to downward by an external electric field, the \nGilbert damping of CrTe 2/In2Te3 is electrically tunable in practice. \n \n \nFIG. 3. (a) The Γ-dependent Gilbert dampings of CrTe 2/In2Te3 with upward (black line) \nand downward (red line) FE polarizations of In 2Te3. (b) The Gilbert damping ratio 𝜂 \nas a function of the scattering rate Γ. \n \nTo gain a deep insight into how the FE polarization of In 2Te3 tunes the Gilbert \ndamping in CrTe 2/In2Te3, we investigate the k-dependent contributions to the Gilbert \ndampings of CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓). As shown in Fig. 4a and 4b, the bands \naround Fermi level have qiute different intermixing between CrTe 2 and In 2Te3 states \nwhen the FE polarizaiton of In 2Te3 is reversed. Explicitly, there are obvious intermixing \nbelow Fermi leve in CrTe 2/In2Te3(↑) while the intermixing mainly takes place above \nFermi level in CrTe 2/In2Te3(↓). Especially, the bands intersected by Fermi level are at \ndifferent k points in CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓). Through looking into the k-\ndependent contributions to the ir Gilbert dampings (Fig. 4c and 4d), we see that large \ncontributions are from the k points (highlighted by arrows in Fig. 4) at which the bands \nof CrTe 2 cross Fermi level. In addition, these large contributions are different. Such k-\ndipendent contribution to Gilbert dampings is understandable. Based on the scattering \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 10 of 15 \n theory of Gilbert damping 38, Gilbert damping parameter is calculated using the \nfollowing Eq. (3) 36 \n( ) ( ) , , , , , , (3)kk\nk i k j k j k i F k i F k j\nk ij SE E E EM u u\n = − − −\nHH, \nwhere EF is Fermi level and Ek,i is the enery of band i at a given k point. Due to the delta \n( ) ( ) ,, F k i F k jE E E E −− , only the valence and conduction bands near Fermi level \nmake dominant contribution to the Gilbert damping. Additionally, their contributions \nalso depend on factor , , , ,kk\nk i k j k j k iuu \nHH. Overall , through changing the \nintersections between the bands of CrTe 2 and Fermi level, the reversal of the FE \npolarization of In 2Te3 can modulate the contributions to Gilbert damping. Consequently, \nthe total Gilbert dampings are different in CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓). \n \n \nFIG. 4. (a) Band structure calculated with spin-orbit coupling and (c) the k-dependent \ncontributions to the Gilbert damping in CrTe 2/In2Te3(↑). (b) and (d) same as (a) and (c) \nbut for CrTe 2/In2Te3(↓). In (a) and (b), Fermi levels are indicated by horizontal dash \nlines and the states from CrTe 2 and In 2Te3 are shown by red and blue, respectively. \n \n From experimental perspectives, the fabrication of CrTe 2/In2Te3 vdW \nheterostructure should be feasible. On the one hand, CrTe 2 with the lattice constant of \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 11 of 15 \n 3.95 Å has been successfully grown on ZrTe 2 substrate by the molecular beam epitaxy.29 \nOn the other hand, In 2Te3 is also synthesized.39 Taking these and the vdW nature of \nCrTe 2 and In 2Te3 together, a practical scheme of growing CrTe 2/In2Te3 is sketched in \nFig. S8 : first grow CrTe 2 ML on ZrTe 2 substrate29 and then put In2Te3 ML on CrTe 2 to \nform the desired CrTe 2/In2Te3 vdW heterostructure. \nIn summary, by constructing a vdW heterostructure of 2D FM metal CrTe 2 and FE \nIn2Te3 MLs, we find that the magnetic properties of CrTe 2 are engineered by the reversal \nof the FE polariton of In 2Te3. Although the ferromagnetism of CrTe 2 is maintained in \nthe presence of the FE In2Te3, its magnetic easy axis can be tuned from in-plane to out-\nof-plane by reversing the FE polarization of In 2Te3. More importantly, the Gilbert \ndamping of CrTe 2 is tunable with a wide range when reversing the FE polarization of \nIn2Te3 from upward to downward. Such tunability of the Gilbert damping in \nCrTe 2/In2Te3 results from the changed intersections between the bands of CrTe 2 and \nFermi level on reversing the FE polarizations of In 2Te3. Our work introduces a \nremarkably useful method to electrically tune the Gilbert dampings of 2D vdW FM \nmetals by contacting them with ferroelectrics, and should stimulate more experimental \ninvestigations in this realm. \n \nSee the supplementary material for the details of computational methods31,36,40- 50 \nand other results mentioned in the main text. \n \nThis project is supported by National Nature Science Foundation of China (No. \n12104518, 92165204, 11974432), NKRDPC-2018YFA0306001, NKRDPC-\n2022YFA1402802, GBABRF-2022A1515012643 and GZABRF-202201011118 . \nDensity functional theory calculations are performed at Tianhe- II. \n \nAUTHOR DECLARATIONS \nConflict of Interest \nThe authors have no conflicts to disclose. \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 12 of 15 \n Author Contributions \nLiang Qiu : Investigation (equal); Methodology (equal); Writing –original draft (equal). \nZequan Wang : Methodology (equal). Xiao -sheng Ni : Investigation (equal); \nMethodology (equal). Dao-Xin Yao : Supervision (equal); Funding acquisition (equal); \nInvestigation (equal); Writing – review &editing (equal). Yusheng Hou : \nConceptualization (equal); Funding acquisition (equal); Investigation (equal); Project \nadministration(equal); Resources (equal); Supervision (equal); Writing – review \n&editing (equal). \n \nDATA A V AILABILITY \nThe data that support the findings of this study are available from the \ncorresponding authors upon reasonable request. \n \n \nREFERENCES \n1 B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, \nD. Zhong, E. 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However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401" }, { "title": "2109.03123v2.Fluid_energy_cascade_rate_and_kinetic_damping__new_insight_from_3D_Landau_fluid_simulations.pdf", "content": "Draft version October 14, 2021\nPreprint typeset using L ATEX style AASTeX6 v. 1.0\nFLUID ENERGY CASCADE RATE AND KINETIC DAMPING: NEW INSIGHT FROM 3D LANDAU-FLUID\nSIMULATIONS\nR. Ferrand1, F. Sahraoui1, D. Laveder2, T. Passot2, P.L. Sulem2and S. Galtier1,3\n1Laboratoire de Physique des Plasmas, CNRS, \u0013Ecole polytechnique, Universit\u0013 e Paris-Saclay, Sorbonne Universit\u0013 e, Observatoire de\nParis-Meudon, F-91128 Palaiseau Cedex, France\n2Universit\u0013 e C^ ote d'Azur, Observatoire de la C^ ote d'Azur, CNRS, Laboratoire J.L. Lagrange, Boulevard de l'Observatoire, CS 34229,\n06304 Nice Cedex 4, France\nand\n3Institut Universitaire de France (IUF)\n(Dated: October 14, 2021)\nABSTRACT\nUsing an exact law for incompressible Hall magnetohydrodynamics (HMHD) turbulence, the energy\ncascade rate is computed from three-dimensional HMHD-CGL (bi-adiabatic ions and isothermal elec-\ntrons) and Landau \ruid (LF) numerical simulations that feature di\u000berent intensities of Landau damp-\ning over a broad range of wavenumbers, typically 0 :05.k?di.100. Using three sets of cross-scale\nsimulations where turbulence is initiated at large, medium and small scales, the ability of the \ruid\nenergy cascade to \\sense\" the kinetic Landau damping at di\u000berent scales is tested. The cascade rate\nestimated from the exact law and the dissipation calculated directly from the simulation are shown\nto re\rect the role of Landau damping in dissipating energy at all scales, with an emphasis on the\nkinetic ones. This result provides new prospects on using exact laws for simpli\fed \ruid models to an-\nalyze dissipation in kinetic simulations and spacecraft observations, and new insights into theoretical\ndescription of collisionless magnetized plasmas.\n1.INTRODUCTION\nThe understanding of turbulent astrophysical plasmas\nremains to date a challenging problem: their chaotic na-\nture and the complexity of the mechanisms at work in\nsuch media impose limitations to the methods one can\nuse to study them e\u000eciently. Yet, enhancing our under-\nstanding of turbulent plasmas would provide the keys\nto solve a variety of problems related to energy dissi-\npation, particle heating and acceleration. Examples of\nsystems where these processes are crucial include the\nsolar wind (SW) and planetary magnetospheres (Bruno\n& Carbone 2005; Matthaeus & Velli 2011; Goldstein\net al. 1995; Sahraoui et al. 2020), accretion \rows around\ncompact objects (Balbus & Hawley 1998; Quataert &\nGruzinov 1999) and fusion devices (Diamond et al. 2005;\nGarbet 2006; Fujisawa 2021). In the SW, the heating\nproblem is re\rected by the slow decline of ion tem-\nrenaud.ferrand@lpp.polytechnique.frperature (as function of the radial distance from the\nsun) in comparison with the prediction from the adia-\nbatic expansion model of the wind (Richardson et al.\n1995). Turbulence has long been proposed as a way to\nexplain this behavior (Matthaeus et al. 1999), through\nscale-by-scale transfer of energy (i.e., cascade) toward\nsmall (kinetic) scales where dissipation is more e\u000bec-\ntive (Schekochihin et al. 2009). A common tool used\nto estimate this energy dissipation is the formalism of\nexact law for fully developed turbulence \frst introduced\nby Kolmogorov (1941) to study incompressible neutral\n\ruids. In this formalism, energy is assumed to be in-\njected at large scales at a constant rate per unit volume\n\", which is assumed to be equal to the rate of cascade\nto smaller scales and to the rate of dissipation at those\nscales. Assuming statistical homogeneity and stationar-\nity of the turbulent \felds, and the existence of an iner-\ntial range in which both forcing and dissipation mech-\nanisms are negligible, the cascade rate \"must remain\nconstant in the inertial range (Kolmogorov 1941; MoninarXiv:2109.03123v2 [physics.plasm-ph] 13 Oct 20212\n1959; Antonia et al. 1997). The formalism of exact law\nhas been extended to (in)compressible magnetized plas-\nmas within various approximations (Politano & Pouquet\n1998; Galtier 2008; Banerjee & Galtier 2013; Andr\u0013 es &\nSahraoui 2017; Hellinger et al. 2018; Andr\u0013 es et al. 2018;\nFerrand et al. 2019).\nExact laws have been used successfully to measure\nthe energy cascade rate in the SW (Smith et al. 2006;\nPodesta et al. 2007; Sorriso-Valvo et al. 2007; MacBride\net al. 2008; Marino et al. 2008; Carbone et al. 2009;\nSmith et al. 2009; Stawarz et al. 2009; Osman et al. 2011;\nCoburn et al. 2015; Banerjee et al. 2016; Hadid et al.\n2017) and terrestrial magnetosheath (Hadid et al. 2018;\nAndr\u0013 es et al. 2019). In those studies, the estimated\ncascade rate was interpreted as the turbulence energy\ndissipation rate, and hence used to quantify the amount\nof plasma heating due to turbulence (Sorriso-Valvo et al.\n2007; Carbone et al. 2009; Banerjee et al. 2016). How-\never, as explained above, such an equivalence between\ninjection, cascade and dissipation rates stems only from\nthe hypothesis underlying exact laws derivation and can-\nnot be demonstrated in spacecraft observations. Indeed,\nwhile in numerical simulations the injection, cascade and\ndissipation rates can generally be estimated separately\nand compared to each other as done in this paper, es-\ntimating ( irreversible ) dissipation from spacescraft ob-\nservation is a challenge and, generally, only the cascade\nrate, which is directly linked to measurable quantities\nthrough the exact law, is accessible (Sorriso-Valvo et al.\n2007; Hadid et al. 2017). Thus, in spacecraft data, inter-\npreting the energy cascade rate as the actual dissipation\nrate is not straightforward. This is particularly true\nbecause of the weakly collisional nature of the SW: in\nsuch plasmas classical viscous and/or resistive e\u000bects are\nabsent, and dissipation is expected to occur via kinetic\ne\u000bects (e.g., Landau and cyclotron resonances) (Leamon\net al. 1998; Sahraoui et al. 2009; Sahraoui et al. 2010;\nHe et al. 2015; Chen et al. 2019) that are not captured\nby usual \ruid descriptions of plasmas. A fundamental\nquestion arises here: is the \ruid turbulent cascade rate\nestimated in simulations and spacecraft observations of\nspace plasmas representative of the actual kinetic dis-\nsipation in those media? It is the main goal of this\npaper to address this question, which impacts the use\nof \ruid models to interpret part of in-situ spacecraft ob-\nservations in the near-Earth space and the theoretical\n(\ruid vs. kinetic) modeling of weakly collisional plas-\nmas. In contrast with previous studies based on 2D\nhybrid particle-in-cell simulations (Hellinger et al. 2018;Bandyopadhyay et al. 2020), the use of 3D LF models\ngive the possibility to isolate the in\ruence of electron\nand ion Landau damping, neglecting all the other ki-\nnetic e\u000bects, and is therefore very suited to address the\nquestion of interest here.\n2.THEORETICAL MODEL\nAlthough we are dealing with weakly compressible\nregimes we chose, for the sake of simplicity, to use here\nthe exact law derived by Ferrand et al. (2019) for in-\ncompressible HMHD (see below about the use of more\ngeneral compressible models). Starting from the in-\ncompressible HMHD equations, and under the usual as-\nsumptions of time stationarity, space homogeneity and\nin\fnite (kinetic and magnetic) Reynolds numbers, one\ncan derive for the energy cascade rate in the inertial\nrange the expression \"=\"MHD+\"Hall, with\n\"MHD=\u00001\n4r`\u0001\n(j\u000evj2+j\u000ebj2)\u000ev\u00002(\u000ev\u0001\u000eb)\u000eb\u000b\n;\n(1)\n\"Hall=\u00001\n8dir`\u0001\n2(\u000eb\u0001\u000ej)\u000eb\u0000j\u000ebj2\u000ej\u000b\n; (2)\nwhere v,b=B=p\u00160\u001a0andj=r\u0002bare the velocity,\nmagnetic \feld and electric current in Alfv\u0013 en units ( \u001a0\nis the constant mass density) and diis the ion inertial\nlength. Fields are taken at points xandx0separated\nby a spatial increment `=x0\u0000x, and the notations\nv\u0011v(x) and v0\u0011v(x0) are adopted. We then de\fne\nthe increment operator \u000eas\u000ev=v0\u0000v, andr`as the\nderivative operator with respect to the increment `.\n3.SIMULATION DATA\n3.1. Presentation of the data\nIn this study, HMHD-CGL refers to a \ruid model\nwith anisotropic ion pressure whose gyrotropic compo-\nnents parallel and perpendicular to the local magnetic\n\feld obey nonlinear dynamical equations where the heat\n\ruxes are neglected (bi-adiabatic approximation intro-\nduced by Chew, Goldenberg and Low (Chew et al. 1956),\nthus the acronym). The electrons are assumed isother-\nmal. Di\u000berently, the LF model retains the nonlinear dy-\nnamics of the parallel and perpendicular pressures and\nheat \ruxes for both the ions and electrons, and involves a\nclosure at the level of the fourth-order moments, consis-\ntent with the low-frequency linear kinetic theory (Snyder\net al. 1997; Passot & Sulem 2007). The main assump-\ntion for modeling Landau damping consists in retain-\ning the imaginary contribution of the plasma response\nfunction in the closure relation which expresses the last3\nRunk?;fdiResolution \u0012 \u0017 =\u0011 \u000b\nCGL1 0:045 512383\u000e7:35\u000210\u0000880\nCGL2 0:045 512375\u000e7:35\u000210\u0000810\nCGL3 0:5 5122\u00021024 75\u000e10\u0000142.5\nCGL4 0:011 1024375\u000e3\u000210\u000035\nLF1 0:045 512383\u000e7:35\u000210\u000081\nLF2 0:045 512375\u000e7:35\u000210\u000081\nLF3 0:5 432375\u000e7\u000210\u0000141.5\nLF4 0:011 512375\u000e3\u000210\u000032\nTable 1 . List of runs and their relevant parameters, where\nCGLx and LFx refer to HMHD-CGL and LF simulations,\nrespectively. The ratio of the longitudinal to transverse box\nsizes is given by tan( \u0012).\nretained \ruid moment of the hierarchy in terms of the\nlower ones. In Fourier space, this procedure generates\nfactors of the form \\ isgn(kz)\" which, in physical space,\nidenti\fes with the Hilbert transform along the ambient\nmagnetic \feld (Hammett & Perkins 1990; Hunana et al.\n2019). It is then possible to generalize this formulation\nto take into account magnetic \feld line distortion, us-\ning the convolution form of the Hilbert transform (Sny-\nder et al. 1997). Its approximation in the numerical\ncode is discussed in Passot et al. (2014). In both mod-\nels, \fnite ion and electron Larmor radius corrections are\nneglected, thus reducing the kinetic e\u000bects to Landau\ndamping. The Ohm's law includes the Hall term and\nthe electron pressure contribution. Turbulence is forced\nwith counter-propagating kinetic Alfv\u0013 en waves (KAWs)\nmaking an angle \u0012with the ambient magnetic \feld, at\nthe largest scales of the simulation domain. This cor-\nresponds to transverse wavenumbers k?;f, whose val-\nues are summarized in Table 1. The amplitudes obey a\nLangevin equation, with an oscillation frequency given\nby the KAW linear dispersion relation (TenBarge et al.\n2014). We also introduce two thresholds in order to con-\nstrain the sum of perpendicular kinetic and magnetic\nenergies to stay within a certain range. Small-scale dis-\nsipation is ensured by the hyperviscosity and hyperdi\u000bu-\nsivity terms in the velocity and induction equations, of\nthe form d\u0017=\u0017(\u0001?+\u000b@2\nz)4vandd\u0011=\u0011(\u0001?+\u000b@2\nz)4b,\nwith\u000bbeing an anisotropy coe\u000ecient.\nIn all the simulations, \fi= 1 and the ion and electron\npressures are taken isotropic and equal initially. The\nother parameters are reported in Table 1. The simula-\ntions are performed using a desaliased spectral code (at\n2/3 of the maximum wavenumber) with a third-order\nRunge-Kutta scheme for time stepping.\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0\nk⊥di−0.20.00.20.40.60.81.0γ,ω/ωci\nω75\nγ75ω83\nγ83Figure 1 . Frequency !and damping rate \r(normalized by\nthe ion gyrofrequency !ci) of KAWs versus the normalized\ntransverse wavenumber k?di(wheredirefers to the ion iner-\ntial length) for the LF model at the two propagation angles\n\u0012= 83\u000eand 75\u000eused in driving the simulations ( \fi= 1,\nTi=Te).\nTwo di\u000berent propagation angles for the KAWs driven\nat the largest scale of each LF simulation were chosen,\nhence tuning Landau damping to two di\u000berent levels\n(Kobayashi et al. 2017). This can be seen in Fig. 1\nwhich compares the linear dispersion relation and damp-\ning rate of the KAWs: the higher the propagation an-\ngle, the lower the damping rate at a given scale. Note\nthat, while changing the angle, we do not change the\namplitude of the \ructuations at the driving scale (i.e.,\n!NLremains constant), and so the nonlinear parame-\nter\u001f=!NL=!Lalso varies: when the angle decreases,\nkkincreases and so does !L, thus\u001fis reduced. As the\nratio\r=!Lis approximately constant for high oblique\nangles\u0012(e.g., Fig. 7 in Sahraoui et al. (2012)), the\nstrength of the Landau damping relative to the cascade\nrate\r=!NL= (1=\u001f)(\r=!L) thus increases as the angle\ndecreases.\n3.2. Energy balance and time stationarity\nLetEtot(t) be the total energy of the system at time\nt,It(t) the injection rate due to the external forcing\non the perpendicular velocity components, and Dh(t)\nthe total dissipation rate due to the hyperviscous and\nhyperdi\u000busive terms. Since Landau damping does not\na\u000bect the total energy balance, total energy conservation\nimplies\nd\ndtEtot(t) =It(t)\u0000D h(t): (3)\nDenoting byEintandEkthe parts of the total energy as-\nsociated with the pressure components (internal energy)\nand the parallel velocity and magnetic \feld components\nentering the kinetic and magnetic parts, we can write\nd\ndtEtot(t)\u0000d\ndtEint(t)\u0000d\ndtEk(t)\u0011d\ndtE?(t)\u00190;(4)4\nwhereE?is the sum of the perpendicular kinetic and\nmagnetic energies, a quantity bound to remain nearly\nconstant by the forcing procedure.\nBecause of computational constraints, the time evolu-\ntion of the di\u000berent energy components is computed for\nlow resolution (LR) simulations analogs of runs CGL3\nand LF3 and shown in Fig. 2 (injection and dissipation\nrates needed to perform this extra study were not output\nat a high-enough frequency in the large-resolution simu-\nlations). From this \fgure it is conspicuous that the time\nevolution of total energy, injection and hyperdissipation\nis consistent with the energy conservation (3). More-\nover, one can see the driving procedure at play in keep-\ning the perpendicular energy E?roughly constant. Its\ntime stationarity is in practice established when the hy-\nperdissipation rate has reached a constant value. When\ncomparing CGL3-LR with LF3-LR, one notices that run\nLF3-LR requires a larger injection rate to maintain the\nsame level of turbulence on the magnetic and perpendic-\nular velocity than in run CGL3-LR, since Landau damp-\ning e\u000eciently converts a part of the injected energy into\ninternal energy. This is evidenced by the dashed green\ncurve in Fig. 2 (bottom), which shows that the increase\nof the internal energy is consistent with the heating by\nheat \ruxes. Moreover, the hyperdissipation rate is lower\non run LF3-LR, suggesting that part of the cascading en-\nergy is taken by Landau damping, as will be evidenced\nin next section.\n4.CALCULATION OF THE ENERGY CASCADE\nRATE\nUsing Eqs. (1)-(2), we compute the transverse energy\ncascade rate for each simulation as a function of the\nperpendicular increment by averaging over all spatial\npositions in the simulation box and di\u000berent increment\nvectors, at a time for which the simulations reached a\nstationary state. Increment vectors `= (`?;`k) are se-\nlected following the angle averaging method of Taylor\net al. (2003), and only the increments forming an angle\nof at least 45\u000ewith the parallel direction are retained.\nAs already mentioned, the exact law used is the one\nfor incompressible HMHD, whereas the simulations are\nweakly compressible. A comparison (not shown) with a\nfull compressible HMHD exact law (Andr\u0013 es et al. 2018)\nonly showed slight change ( .10% in the inertial range)\nof the cascade rate with respect to the current estimate\nfrom the incompressible model. The transverse cascade\nrate is then averaged over all increments of equal value\nof`?. The transverse hyperdissipation is computed in\nFigure 2 . Low resolution runs of CGL3 (CGL3-LR, top)\nand LF3 (LF3-LR, bottom): Time evolution of the total\nenergy injected in the systemRt\n0Itdt(solid red line), total\nenergyEtot(solid blue), internal energy Eint(solid green),\nperpendicular energy E?(solid magenta, roughly constant)\nand the time integrated hyperdissipationRt\n0Dhdt(dashed\nred). The piece of dashed green curve (right) starting at t=\n250, whose vertical position is arbitrary, displays the heating\ndue to heat \ruxes, which is consistent with the increase of\ninternal energy.\nFourier space as\n\"diss(`?) =Zk?\n0dk0\n?Z\nk08\n?(\u0011jb(k0)j2+\u0017jv(k0)j2)k0\n?d\u00120dk0\nz\n(5)\nwhere we use `?=\u0019=k?.\nFor simulations forced at intermediate scales, whose\nresults are reported in Fig. 3, the behavior of the MHD\nand Hall contributions to the energy cascade rate are\nsimilar, with the latter rising up at sub-ion scales, then\ndominating the former at about the ion inertial length.\nThe total energy cascade rate is roughly constant on\nmore than one decade of scales in the simulations with\n\u0012= 83\u000e, in particular in CGL1, which demonstrates the\nexistence of an inertial range. To highlight the e\u000bect\nof Landau damping on the cascade rate, we compare in\nFig. 4 the cascade rates from the LF simulations with5\n100101\n/lscript⊥/di10−810−710−610−510−4\nCGL1 (θ= 83◦) εMHD\nεHallε\nεdiss\n100101\n/lscript⊥/di10−810−710−610−510−4\nLF1 (θ= 83◦) εMHD\nεHallε\nεdiss\nFigure 3 . Energy cascade rate \", its ideal MHD and Hall\ncomponents together with the transverse hyperdissipation\ncomputed for runs CGL1 (top) and LF1 (bottom). Plain\nlines represent positive values and dashed lines negative val-\nues.\n\u0012= 75\u000eand\u0012= 83\u000e, normalized to the corresponding\nones from the CGL simulations. We observe a stronger\ndecrease (by up to a factor 5) in the normalized cascade\nrate at small scales for LF2 ( \u0012= 75\u000e), i.e. for the simu-\nlation with the strongest Landau damping, than for LF1\n(\u0012= 83\u000e) for which the normalized cascade rate remains\nnearly constant at all scales. This result clearly relates\nthe enhancement of Landau damping at kinetic scales\nto the decline of the energy cascade rate at these scales.\nWe note also the consistency between the (transverse)\nhyperdissipation and the cascade rate at the smallest\nscale of the simulation box (Fig. 3).\nWe complement our study with the cascade rates es-\ntimated from simulations forced at even smaller scales\n(LF3 and CGL3) with \u0012= 75\u000eand reported in Fig. 5.\nSimulation LF3 exhibits a strong decrease in \"MHDpar-\ntially compensated by a quick rising of the Hall compo-\nnent, giving no clear inertial range, in contrast to CGL3\nwhich still behaves similarly to the simulations forced at\nintermediate scales. As shown below, this e\u000bect may be\nattributed to the fact that Landau dissipation reaches\nhigh levels at the sub-ion scales of LF3, whereas CGL3\ncontains no dissipation mechanisms other than hyper\nviscosity and di\u000busivity, which are bound to act only\nat the smallest scales. Note that the sudden changes of\n100101\n/lscript⊥/di10−1100101\nεLF1/εCGL 1(θ= 83◦)\nεLF2/εCGL 2(θ= 75◦)Figure 4 . Ratios of the energy cascade rate computed for LF\nsimulations over the one for CGL simulations for a driving\nwave angle \u0012= 83\u000e(black) and \u0012= 75\u000e(red).\n10−1100\n/lscript⊥/di10−610−510−410−310−2\nCGL3 (θ= 75◦) εMHD\nεHallε\nεdiss\n10−1100\n/lscript⊥/di10−610−510−410−310−2\nLF3 (θ= 75◦) εMHD\nεHallε\nεdiss\nFigure 5 . Same as in Fig. 3 for runs CGL3 (top) and LF3\n(bottom).\nsign observed at large scales in some components of the\ncascade rates in Figs. 3 and 5 are likely to be due to\nthe proximity of the forcing. Those observed at small\nscales for the MHD component of run CGL3 would re-\nsult from numerical errors in the calculation of \"MHD\ngiven its very small magnitude at those scales.\nTo obtain a full picture as to how Landau damping af-\nfects the energy cascade rate, we performed simulations\nforced at large scales (LF4 and CGL4). Combining the\nruns CGL2-3-4 and LF2-3-4 we construct a multi-scale\nenergy cascade rate over nearly three decades of scales\nthat highlights the e\u000bect of Landau damping on it. As\nthe simulations were run at di\u000berent scales, the ampli-\ntude of the forcing was changed to ensure that each6\n10−1100101\n/lscript⊥/di10−710−610−5\nCGL3\nLF3CGL2\nLF2CGL4\nLF4\nεCGL(θ= 75◦)\nεLF(θ= 75◦)\nFigure 6 . Energy cascade rates reconstructed with CGL2-\n3-4 runs and LF2-3-4 runs. The ranges spanned by each\nsimulation are delimited by the black dotted lines. A slight\nirregularity is observed on the green curve at the transition\nbetween CGL2 and CGL3, which is caused by an insu\u000ecient\noverlap of the cascade rates at these scales.\nsimulation reaches a fully turbulent state. Therefore,\nwe renormalized the cascade rate \"obtained from the\ndi\u000berent simulations to match the one of intermediate\nruns CGL2 and LF2, while taking care to discard the\nsmallest scales of intermediate and large-scale forcing\ncascade rates to ensure that hyperviscosity is not acting\nat intermediate scales of the reconstructed energy cas-\ncade. Fig. 6 shows the full energy cascade rate for CGL\nand LF runs for the driving wave angle \u0012= 75\u000e. CGL\nruns exhibit an almost constant energy cascade rate over\ntwo and a half decades of scales, whereas \"LFdecreases\nsteadily over scales and reaches its minimum value at\nthe smallest ones, con\frming that the behavior already\nobserved in Fig. 4 remains valid over a broader range of\nscales.\n5.INFLUENCE OF LANDAU DISSIPATION\n5.1. Heating due to heat \ruxesIn the wake of the previous results an important ques-\ntion arises : can the drop in the energy cascade rate for\nLF runs be directly connected to Landau damping? For\nthis purpose, we calculate the heating due to heat \ruxes\nin presence of Landau damping. For each species, the\npressure equations with the Hall term and the gyrotropic\nheat \ruxes read\nd\ndtln\u0012pkjBj2\n\u001a3\u0013\n=\u00002c\njBjbb\u0001r\u0002EH\n\u00001\npk\u0010\n\u00002q?r\u0001bb+r\u0001(qkbb)\u0011\n;(6)\nd\ndtln\u0012p?\n\u001ajBj\u0013\n=c\njBjbb\u0001r\u0002EH\n\u00001\np?\u0010\nq?r\u0001bb+r\u0001(q?bb)\u0011\n: (7)\nWe de\fne the parallel, perpendicular and total en-\ntropies per unit mass\nsk=cV\n3ln(pkjBj2\n\u001a3); s?=2cV\n3ln(p?\n\u001ajBj); (8)\ns=sk+s?=cV\n3ln(pkp2\n?\n\u001a5); (9)\nwherecVis the speci\fc heat at constant volume. Denot-\ning byethe internal energy per unit mass, the internal\nenergy per unit volume reads E\u0011\u001ae=p?+1\n2pk=3\n2nT\nwhereT=1\n3(2T?+Tk). Frome=cVT, one gets\ncV=3\n2m(the Boltzmann constant is included in the\nde\fnition of temperature). The total entropy then obeys\n@t(\u001as) +r\u0001\u0012\n\u001asu+ (q?\nT?+qk\n2Tk)bb\u0013\n= (1\nTk\u00001\nT?)q?r\u0001bb\u0000\u0012q?\nT?(bb\u0001r) lnT?+qk\n2Tk(bb\u0001r) lnTk\u0013\n: (10)\nFrom the form of the right hand side of Eqs. (6)-(7),\nwe can conclude that the rates of change of the parallel\nand perpendicular entropies per unit mass ( sp\nkandsp\n?\nrespectively) associated with a production (or destruc-\ntion) and excluding transport or exchanges between the\nparallel and perpendicular directions (see e.g. Hazeltineet al. (2013)), are given by\nd\ndtsp\nk=1\n\u001aTkq?r\u0001bb\u0000qk\n2\u001aTk(bb\u0001r) lnTk (11)\nd\ndtsp\n?=\u00001\n\u001aT?q?r\u0001bb\u0000q?\n\u001aT?(bb\u0001r) lnT?:(12)\nThe associated rates of heat production per unit mass\nare related by dQk=dt =Tkdsp\nk=dtanddQ?=dt =\nT?dsp\n?=dt. We thus get, for the total heat production7\nQ=Qk+Q?\n@t(\u001aQ)+r\u0001(\u001aQu) =\u0000qk\n2(bb\u0001r) lnTk\u0000q?(bb\u0001r) lnT?:\n(13)\nThe global heating is thus given by\nH=\u0000Z\u0010qk\n2(bb\u0001r) lnTk+q?(bb\u0001r) lnT?\u0011\nd3x;\n(14)whereqkandq?are the heat \ruxes obtained from the\nintegration of the model closed at the level of the fourth-\nrank moments.\nWe can de\fne a spectral density for the heating rate\nH(also referred to as co-spectrum) in the form\nH(k) =\u00001\n2\u00121\n2Ffqkg(\u0000k)Fn\n(bb\u0001r) lnTko\n(k) +Ffq?g(\u0000k)Fn\n(bb\u0001r) lnT?o\n(k) +c:c:\u0013\n(15)\nwhereFdenotes the Fourier transform.\nA few remarks can be made here:\n1. In all the simulations we have performed, the volume\nintegrated heat production is observed to be positive but\nits pointwise value can be negative in relatively small re-\ngions of space. This contrasts with the (semi-)collisional\nregime where the heat \ruxes obey Fourier laws of the\nformq=\u0000\u0014(bb\u0001r)T, making the heat production pos-\nitive everywhere in space.\n2. Inserting in Eq. (14) the quantities q?andqkob-\ntained by the integration of the dynamical equation for\nthe heat \ruxes results in taking into account in the\nheating rate contributions originating from the heat \rux\npresent when a quasi-normal closure is implemented (i.e.\nwhere the fourth-rank cumulants are taken equal to zero,\nthus making the Landau damping disappear). In the\npresent simulations, this contribution does not exceed\n15% of the total heating rate. In order to only deal with\nthe heat \rux originating from the Landau damping, it\nwould be necessary to de\fne a conserved entropy for the\nquasi-normal closure and evaluate its rate of change due\nto the introduction of Landau damping. This is left for\nfuture work as it is not straightforward.\n3. More importantly, this heating rate takes into ac-\ncount the Landau damping on all the waves present\nin the simulations, including the magnetosonic waves.\nAt this level, it appears di\u000ecult to separate the contri-\nbutions of the KAW and to evaluate their dissipation\nby Landau damping. Nevertheless, these magnetosonic\nwaves get dissipated at large scales, thus at small enough\nscales the estimated heating rate mostly results from\nLandau damping of KAWs and it becomes possible to\ncompare it to the cascading energy. This particularity\nis also the reason why Landau damping appears to be\nacting at all scales in all the results presented above,\neven in simulations forced at large scales.\nFigure 7 . Spectral densities of the heating rate DL(k?)\n(red) and of the magnetic (blue) and kinetic (green) hyper-\ndissipation as functions of the transverse wavenumber k?for\nrun LF3. A straight line of slope 5.2 is supplemented, for\ncomparison with the scale-variation of the magnetic hyper-\ndissipation.\nThe fact that Landau damping is present at all scales\nin the simulation can be seen by estimating the spectral\ndensity of total heating rate at a given wavenumber k?,\nDL(k?) =R\nk?H(k)dkzd\u0012, whereH(k) is the sum of\nthe spectral densities given by equation (15) for both\nthe ions and the electrons. This spectral density is rep-\nresented in Fig. 7 along with the densities of hyperdis-\nsipation and hyperdi\u000busivity. One clearly sees that the\nheating rate due to the presence of heat \ruxes dominates\nhyper-dissipation over a broad range of scales due to the\ndissipation of KAWs and magnetosonic modes, the two\nbecoming comparable only at the smallest scales (note\nthat the magnetic hyper-dissipation dominates at small\nscales over the kinetic one).\n5.2. Dissipation due to Landau damping\nThe energyE?(t) of the (quasi-incompressible) KAWs\nthat cascade towards small scales, and which is the sub-8\nject of our study, obeys\nd\ndtE?(t) =IC(t)\u0000DC\nL(t)\u0000DC\nh(t); (16)\nwhereICis the part of the injection rate that con-\ntributes to the KAW cascade (the other part is trans-\nferred to magnetosonic modes which are dominantly dis-\nsipated at large scales), while DC\nLandDC\nhare the parts\nof the Landau and hyperviscous (and hyperdi\u000busive)\ndissipation that a\u000bect the cascading modes. Using cylin-\ndrical coordinates and assuming time stationarity, one\ncan write the integrated energy balance at each Fourier\nmode as (adopting roman scripts for spectral densities):\n\u000f(k?) =Zk?\n0\b\nIC(k0\n?)\u0000DC\nL(k0\n?)\u0000DC\nh(k0\n?)\t\ndk0\n?:\n(17)\nConsidering two wavenumbers k?1andk?2large enough\nso that the forcing (which is concentrated at large scales)\nleads toRk?1\n0IC(k0\n?)dk0\n?=Rk?2\n0IC(k0\n?)dk0\n?=IC, yet\nsmall enough for hyperviscous dissipation to be negligi-\nble, one obtains:\n\u000f(k?1)\u0000\u000f(k?2) =Zk?2\nk?1DC\nL(k0\n?)dk0\n?.Zk?2\nk?1DL(k0\n?)dk0\n?:\n(18)The inequality draws closer to an equality for values of\nk?large enough so that all magnetosonic modes have\nbeen dissipated.\nEquation (18) can be used to estimate a correction to\nthe energy cascade rate which would take into account\nthe energy lost due to Landau damping. We do so for\nrun LF3: using this equation we add to the transfer rate\nthe cumulative Landau dissipation between an arbitrary\nscale (chosen however to be not too large nor too small)\nand the running (smaller) scale l?. Two of these re-\nsulting corrected rates \"corrare shown in Fig. 8. They\nappear to be almost constant, and as such they behave\nvery similarly to the transfer rate of run CGL3 (Fig. 5).\nThe slight increase of \"corrtowards small scales proba-\nbly re\rects the (weak) contribution of some remaining\nmagnetosonic waves to the calculated Landau damping.\nThis clearly demonstrates that the energy lost along the\ncascade due to Landau damping is well captured by the\ndecline of the (\ruid) cascade rate at the corresponding\nscales.\nA complementary estimate of energy dissipation can\nbe done in Fourier space by also taking into account hy-\nperdissipation. Indeed, assuming stationarity, one can\nalso derive that\n\u000f(k?) =IC\u0000DC\nL+Z1\nk?DC\nL(k0\n?)dk0\n?\u0000Zk?\n0DC\nh(k0\n?)dk0\n?=Z1\nk?\b\nDC\nh(k0\n?) +DC\nL(k0\n?)\t\ndk0\n?: (19)\nEquation (19) indicates that, as expected, the rate of\nenergy transfer at the wavenumber k?identi\fes with\nthe sum of the rates of Landau and hyperdissipation be-\nyond this wavenumber. One can compare the second\nright-hand-side term of this equation to the energy cas-\ncade rate\u000f(k?) obtained from the IHMHD exact law,\nas displayed in Fig. 9. The di\u000berence between the two\ncurves, which is especially signi\fcant at large scales, is\ndue to the fact that the estimation of the dissipation\nincludes the Landau damping of magnetosonic modes,\nwhereas the cascade rate considers only incompressible\nmodes. At smaller scales however, where magnetosonic\nmodes have already been dissipated, the dissipation and\ncascade rates decreases parallel to each other: this indi-\ncates that, at scales not yet a\u000bected by hyperdissipation,\nthe decay of \u000f(k?) in a spectral interval identi\fes with\nLandau dissipation within this interval.\nFigs. 8 and 9 clearly demonstrates that, through the\ncascade, the energy lost due to Landau damping is well\n10−1100\n/lscript⊥/di10−610−510−410−310−2\nLF3 (θ= 75◦)εcorr\n0.2π\nεcorr\n0.1πε\nεdissFigure 8 . Energy cascade rate \"(red) and transverse hy-\nperdissipation (violet) for run LF3. The orange and brown\ncurves show the same \"corrected by Landau damping in-\ntegrated between `?and a reference scale `?= 0:1\u0019diand\n`?= 0:2\u0019direspectively.\ncaptured by the decline of the (\ruid) cascade rate at\nthe corresponding scales. Note that a similar decline of\nthe \ruid cascade rate at kinetic scales was reported in\n2D hybrid PIC simulations and spacecraft observations9\nFigure 9 . Energy cascade rate \u000f(k?) (black line) together\nwith Landau and hyper-dissipation (red line) computed with\nequation (19) for run LF3.\nin the SW and magnetosheath (Hellinger et al. 2018;\nBandyopadhyay et al. 2020). Also, Sorriso-Valvo et al.\n(2019) found a correlation between enhancement of a\nproxy of the local cascade rate and the signatures of\nwave-particle interactions in MMS data.\n6.CONCLUSION\nIn this study, we tackle a fundamental question about\nthe ability of \ruid exact laws to re\rect the presence of\nkinetic (Landau) damping. By constructing multi-scale\nenergy cascade and dissipation rates using the HMHD\nmodel on a variety of turbulence simulations bearing\ndi\u000berent intensities of Landau damping, we showed that\nthe presence of Landau damping at small (kinetic) scales\nis re\rected by the steady decline of the energy cascaderate at the same scales, which was found to be compa-\nrable to the e\u000bective Landau dissipation at those scales.\nBy demonstrating the ability of a \ruid exact law to pro-\nvide a correct estimate of kinetic dissipation in the sub-\nion range of numerical simulations, this work provides\na means to evaluate the amount of energy that is dissi-\npated into particle heating in spacecraft data: the de-\ncline of the cascade rate allows one to evaluate the ki-\nnetic dissipation as a function of scale. This should help\ninvestigating (at least partially) a longstanding problem\nin astrophysical plasmas about energy partition between\nions and electrons (Kawazura et al. 2019), which are gen-\nerally heated at di\u000berent scales.\nThe study presented in this paper only makes use\nof Landau damping. It would be interesting in future\nworks to extend these conclusions to a broad variety of\nkinetic e\u000bects and to test them on more general simu-\nlations of the SW, featuring a plasma turbulence driven\nby other types of waves than slightly perturbed KAWs.\nIt is also important to stress that, even if the oversimpli-\n\fed (yet fully nonlinear) \ruid models of turbulence can\nprovide good estimates of the amount of energy that is\ndissipated into particle heating, they do not specify how\nthis dissipation occurs. 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J., 697, 1119\nTaylor, M. A., Kurien, S., & Eyink, G. L. 2003, Phys. Rev. E,\n68, 026310\nTenBarge, J., Howes, G., Dorland, W., & Hammett, G. 2014,\nComp. Phys. Comm., 185, 578" }, { "title": "1607.07274v1.Damping_of_parametrically_excited_magnons_in_the_presence_of_the_longitudinal_spin_Seebeck_effect.pdf", "content": "arXiv:1607.07274v1 [cond-mat.mes-hall] 25 Jul 2016Damping of parametrically excited magnons in the presence o f the longitudinal spin\nSeebeck effect\nThomas Langner,1,∗Akihiro Kirihara,2Alexander A. Serga,1Burkard Hillebrands,1and Vitaliy I. Vasyuchka1\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\n2IoT Devices Research Laboratories, NEC Corporation, Tsuku ba 305-8501, Japan\n(Dated: September 26, 2018)\nThe impact of the longitudinal spin Seebeck effect (LSSE) on t he magnon damping in magnetic-\ninsulator/nonmagnetic-metal bilayers was recently discu ssed in several reports. However, results\nof those experiments can be blurred by multimode excitation within the measured linewidth. In\norder to avoid possible intermodal interference, we invest igated the damping of a single magnon\ngroup in a platinum covered Yttrium Iron Garnet (YIG) film by m easurement of the threshold of\nits parametric excitation. Both dipolar and exchange spin- wave branches were probed. It turned\nout that the LSSE-related modification of spin-wave damping in a micrometer-thick YIG film is too\nweak to be observed in the entire range of experimentally acc essible wavevectors. At the same time,\nthe change in the mean temperature of the YIG layer, which can appear by applying a temperature\ngradient, strongly modifies the damping value.\nSpin caloritronics, the research field focused on\nthe interaction between magnetic and thermal effects,\nhas attracted a lot of interest in recent investigation\nactivities1–3. The possibility to control and manipulate\nmagnetic processes by thermal means offers a high po-\ntential for application. Furthermore the finding of ways\nto reinvest waste heat is one of the major challenges to-\nwards green energy techniques4,5. The spin Seebeck ef-\nfect is one of the most fascinating phenomena in this\nresearch area. Here a spin current is generated by a tem-\nperature gradient across the interface between a mag-\nnetic material and a nonmagnetic metallic layer6. Usu-\nally the inverse spin Hall effect is used to detect this\nspin current7. This means that it is converted into a\ncharge current due to spin orbit interaction inside the\nnonmagnetic metal and, thus, an electric voltage can be\ndetected. Since its observation the spin Seebeck effect\nhas become the major tool in spincaloritronic research.\nAlthough there has been much effort to reveal the na-\nture of the spin Seebeck effect8,9and to develop possible\napplication schemes, there are still open questions. For\nexample, the influence of the spin Seebeck effect on the\nmagnetization dynamics is still under discussion.\nRecentresearchactivitieswerefocusedtofind evidence\nthat the spin Seebeck effect can partially compensate\nmagnetic damping10,11and can even enhance the magne-\ntization precession12in case of dipolar spin waves. It has\nbeen reported that a generated spin current establishes\nan additional spin torque to the ferromagnet13. If the\ntemperature of the non-magnetic metal layer is higher\nthan the temperature of the magnetic material a spin\nangular momentum is transferred into the ferromagnet\nand reduces the effective damping. In the opposite case\nthis angular momentum is absorbed by the nonmagnetic\nmetal layer and thus the effective magnetic damping is\nenhanced. In order to gain more insight into this pos-\nsible influence on the damping of excited magnons we\ninvestigate the threshold power of the parametric gener-\nation of magnons. Parametric pumping is a well estab-lished,powerfultooltoexciteandamplifymagnonsinthe\ndipolar and in the dipolar-exchange areas of a spin-wave\nspectrum14–16. Hereby an alternating magnetic field os-\ncillating with twice the magnon frequency is applied par-\nallel to the magnetization ofa ferromagnet. A microwave\nfrequency photon converts into two magnons with half\nthe frequency and opposite wavevectors. If the energy\ntransferred to the spin system overcomes the spin-wave\nlosses,parametricinstabilityoccursandthemagnonden-\nsity increases exponentially with time. Thus the para-\nmetric magnon generation is a threshold process. An\ninfluence of the spin Seebeck effect on the damping will\nmodify this pumping threshold. Therefore we investigate\nthe impact of a temperature gradient on a magnonic sys-\ntem in a bilayer of a ferrimagnetic film and an attached\nnonmagnetic metallic film with high spin orbit coupling.\nWe show that this influence of a temperature gradient\nis weak and cannot be detected within the experimen-\ntal errors. We compare these results with homogeneous\nvariation of the temperature. In that case the influence\non the damping is pronounced.\nIn order to perform the mesaurements, the experi-\nmental setup shown in Fig. 1 is used. The investigated\nsample is a multilayer of a 5nm thick Platinum (Pt)\nfilm, a 6.7 µm thick Yttrium Iron Garnet (YIG) film\nand a 500 µm thick Gallium Gadolinium Garnet (GGG)\nsubstrate. The YIG layer is grown on the GGG sub-\nstrate in (111)-orientation by liquid phase epitaxy. The\nplatinum film was sputtered on the YIG surface. With\nthe platinum layer at the bottom, the sample is placed\non top of a 50 µm wide copper microstrip. A thin coating\nlayer of Polymethyl Methacrylate (PMMA) electrically\nisolates the Pt layer from this microstrip. The microstrip\nis structured on a metallized aluminum nitride (AlN)\nsubstrate that is used due to its high thermal conduc-\ntivity at room temperature17of 285Wm−1K−1. Two\nseparately controlled Peltier elements, one below the\nAlN substrate and one directly on top of the sample, are\nused to create a temperature gradient perpendicular to2\nFIG. 1. Scheme of the experimental setup. The\nPt/YIG/GGG sample and the AlN substrate with microstrip\nline are clamped between two separately controlled Peltier\nelements that create a thermal gradient ∇Tacross the sam-\nple. The Pt layer is electrically isolated from the copper mi -\ncrostrip by a thin PMMA coating. The microstrip is part of\na microwave resonator. A microwave current applied to the\nresonator creates a dynamic Oersted field hdynwith a compo-\nnent parallel to the static magnetization of the sample. The\nwhole setup is mounted in a heat sink. On the right side the\nmeasurement scheme is shown. The reflected signal delivers\ninformation about the creation of magnons.\nthe sample plane, see Fig. 1. Using Peltier elements the\ntemperature configurations can be precisely controlled\nfor uniform temperatures and also for a temperature\ngradient in both directions. Nevertheless the maximal\npossible temperature difference in both directions is\nlimited to about 20◦C in the experimental setup at\naround room temperature due to heat dissipation. The\nback sides of the Peltier elements are connected to heat\nsinks that are clamped between the water-cooled poles\nof an electromagnet ensuring effective heat transfer in\nthe system. The externally applied magnetic field His\noriented in plane of the sample and perpendicular to the\nlong axis of the copper stripline.\nA stub tuner connected to the microstrip line creates\na tunable microwave resonator. 10 µs long microwave\npulses with a carrier frequency of 14GHz and a repeti-\ntion time of 10ms applied to this resonator create an al-\nternating Oersted field hdynaround the microstrip. The\nsmall width of the microstrip leads to a high microwave\ncurrent density and, thus, a strongly localized high mag-\nnetic field density. The reflected microwave signal from\nthis resonator is rectified by a semiconductor diode and\nthe envelope is shown on an oscilloscope. The microwave\npower applied to the resonator is controlled to exactly\nthat level where the reflected pulse starts showing a kink\nat the end of the pulse profile. This kink appears as\na consequence of a change in the quality factor of the\ntuned resonator due to the excitation of magnons, and\nthus gives evidence for the appearance of the parametric\ninstability18,19. In this case the applied microwavepower\nlevel is considered as the threshold power. Since the\namount ofgeneratedmagnonsis still low at the threshold\npower level and the pumping pulse is switched off closeto this point, we can neglect the additional heating of the\nsample by magnon-phonon transfer.\nIn our experiments the threshold power is measured in\na wide range of bias magnetic field values and thus is de-\ntermined for a wide range of magnon wavenumbers (see\nFig. 2a). Firstly the temperature of the YIG/Pt bilayer\nwas changed homogeneously in 20◦C steps from -5◦C to\n75◦C. The results are shown in Fig. 2b. In each case\nthere is the typical dependence of the threshold power\non the magnetic field19. Close to the ferromagnetic res-\nonance (FMR) with wavenumber k→0, at the critical\nfieldHcrit, the parametric excitation is most efficient due\nto the highest ellipticity in the precession. This ellip-\nticity is caused by the dynamic stray field. Thus, the\nthreshold pumping field strength is minimal. With de-\ncreasing external magnetic field in the range below the\ncritical field ( H < H crit), the threshold power increases\nslowly due to an increase in wavenumbers and a related\ndecrease in the ellipticity of the excited spin waves20.\nFor magnetic fields little below Hcrita sharp peak in the\nthreshold power can be found. It appears because of\nan increase of the threshold power due to interactions\nof the magnons with a transversal phonon mode. With\nincreasing external magnetic field above the critical field\nthe threshold power increases sharply19. The coupling of\nmicrowave photons to the corresponding magnons is re-\nduced since the angle of the amplified spin waves to the\nparallel pump field is smaller than 90◦. Moreover, since\nthese spin waves possess a nonzero group velocity com-\nponent parallel to the external field, they flow out of the\nstrongly localized amplification area of 50 µm width on\nthe pumping microstrip and therefore increase the effec-\ntive damping. It is important to notice that there is only\none distinct group of magnons parametrically excited for\neach magnetic field value. The spectral density of para-\nmetrically excited spin waves is typically of the order of\nseveralkilohertz21, what is much smallerthan the typical\nFMR linewidth of a YIG film.\nComparing the curves for different temperatures, we\nobserve a change in the magnetic field position of the\nminimum. This occurs due to a change in the satura-\ntion magnetization by changing the temperature. The\nsaturation magnetization is recalculated using Kittel’s\nformula22with the values of the critical fields. The re-\nsults are shown in Fig. 3. The linear fit has a slope of\n∆(4πMS)/∆T=-3.2G·K−1, what is in good agreement\nto previous results23–25. Beside the shift of the critical\nfield values in the threshold curve towards higher mag-\nnetic fields, a monotonous increase in the full range of\nthe magnetic field in the threshold power is obvious. For\neach 20◦C temperature step we observe a difference of\napproximately 1dB in applied microwave power to reach\nthethresholdvalue. AccordingtoCherepanov et al.26the\nintrinsic damping in YIG strongly depends on the abso-\nlute temperature due to Kasuya-LeCraw processes27,28.\nAhigher dampingleads to a higherthreshold powersince\nit has to be overcome by a higher pumping field.\nFrom the values of Fig. 3 we can estimate the change3\nFIG. 2. (a) Excited wavenumbers for different temperatures\nwith respect to the externally applied magnetic field. The\nspectrum was determined using the theory from Gurevich et\nal.15. (b) Measured dependencies of the threshold power on\nthe externally applied magnetic field for constant tempera-\ntures of the sample. Below the temperature dependent criti-\ncal fields Hcritthe generated magnon wavevector is oriented\nperpendicular to the applied magnetic field.\nin the spin-wave relaxation parameter Γ for the applied\nconstanttemperaturescomparedtothereferencevalueat\nTref=35◦C for the wavenumber k→0. We can deter-\nmine the relative change in the relaxation parameters15\nusing\n∆Γ =Γ(T)−Γ(Tref)\nΓ(Tref)·100%. (1)\nTaking into account the connection between the relax-\nation parameter, the magnetization dependent coupling\ncoefficient and the threshold field we can rewrite:\n∆Γ = (MS(T)\nMS(Tref)·/radicalBigg\nPthr(T)\nPthr(Tref)−1)·100% (2)\nwith the corresponding threshold microwave power Pthr.\nTheresultingrelativechangesareshownin Fig.3asopen\ncircles. In the case of a temperature of 75◦C the relax-\nationfrequencyis13.0%higher, for-5◦Cit is9.4%lower\nthan the relaxation frequency at the reference tempera-\nture of 35◦C. This means that the change in damping byFIG.3. Fullsquares: Calculated dependenceofthesaturati on\nmagnetization on the temperature of the YIG layer. The\nline is the result of a linear fit. Open circles: Temperature\ndependent change in relaxation parameter ∆Γ for the critica l\nfield relative to the corresponding relaxation parameter fo r\nthe reference temperature of Tref=35◦C.\nchanging the absolute temperature is very pronounced.\nAt the same time we can neglect the influence of the\nGGG substrate on the threshold power around tempera-\ntures of around 300K29. Therefore, all investigations on\nthe pure influence of a temperature gradient must neces-\nsarily avoid any changes in the mean temperature of the\nsystem.\nIn the next step, temperature gradients were created\nacross the sample thickness. In order to investigate the\npure influence of a temperature gradient on the paramet-\nric pumping process it is important to keep the average\ntemperature of the YIG layer, where magnons are ex-\ncited, constant. The bottom Peltier element has been\nadapted so that we obtain the same mean temperature\nof the YIG film of 35◦C in every case. In first approach\nthis mean temperature has been tuned by the electri-\ncal resistance of the Pt layer used as a thermometer and\nby an additional precise alignment of the FMR thresh-\nold minimum to the same magnetic field value in every\ncase. The temperature on the top of the sample is either\n17◦C higher or 15◦C lower than this base temperature,\ndepending on the direction of the gradient. The depen-\ndence of the threshold power on the magnetic field for a\nhomogeneous temperature of 35◦C is also used here for\ncomparison.\nA temperature gradient across the YIG/Pt bilayer inter-\nface leads to the longitudinal spin Seebeck effect (LSSE).\nIn our system its existence has been proven by direct\nmeasurements of the LSSE voltage between the lateral\nedges of the Pt layer, see the inset in Fig 4. For opposite\ndirectionsofthetemperaturegradientthe measuredvolt-\nageshaveoppositesignswithasignchangeatzerofield30.\nFor a homogeneous temperature of the sample of 35◦C\nno voltage is detected. At the same time the measure-\nment of the threshold powers in the parametric pumping\nprocessrevealthatthereisnomeasurableinfluenceofthe4\nFIG. 4. Measured dependencies of the threshold power on\nthe externally applied magnetic field for different tempera-\nture gradients perpendicular to the sample plane. The mean\ntemperature ofT=35◦Cofthesample is thesame for all three\ncurves. The inset shows the spin Seebeck voltage depending\non the external magnetic field for the same temperature gra-\ndients.\nlongitudinal spin Seebeck effect on the spin-wave damp-\ning parameter (see Fig. 4). All three curves shown are\non the same level within an experimental uncertainty.\nIn contrast to previous reports, where a broader range\nof magnons are excited, the parametric pumping process\ngenerates magnons of only one distinct magnon group\nwithin a very narrow bandwidth. Measurements of the\nferromagnetic resonance in thick macroscopic YIG sam-\nples might show a temperature dependent heterogeneous\nbroadening of the linewidth due to a relative shift of the\nmanymodeexcitationinthisprocess. Aninfluenceofthe\nspin Seebeck effect on the well-defined mode excited by\nparametric pumping in our experiments was not found.\nRecent theoretical investigations by Bender et al.31show\nthat the temperature difference at the interface between\nthe electrons in the platinum layer and the magnons in\nthe YIG layer needed to compensate the magnetic damp-\ning is at least in the order of the magnon energy ¯ hΩ.Hereby Ω is the magnon frequency, that is strictly lim-\nited to 2π×7GHz in our experiment. Therefore a visible\nchange in the damping in the order of 5% can be reached\nby a temperature difference ∆ Tmpof 16.8mK between\nthe magnon temperature in YIG and the electron tem-\nperature in Pt. Our calculations using the theory of Xiao\net al.32,33applied for a mean temperature of 35◦C and\nthe parameters of our experiment reveal a value of ∆ Tmp\n= 1.9mK. Thus, the influence of the spin Seebeck ef-\nfect on the damping of parametric spin waves is even in\ntheory too weak to be determined. Lauer et al.34have\nfound a spin-transfer torque by a spin polarized current\ncreated by the spin Hall effect that affects the parametric\npumpingthresholdinthinYIG/Ptbilayers. Nevertheless\nan influence of the spin Seebeck effect could also not be\nobserved in that report, what supports our findings.\nIn summary, the threshold power levels for the paral-\nlel parametric pumping process in YIG/Pt bilayers in a\nwide wavevector range have been investigated for differ-\nent thermal configurations. It has been shown, that the\nthreshold power strongly depends on the mean temper-\nature of the YIG layer, whereas a temperature gradient\ndoes not change the threshold power as long as there\nis no change in the mean temperature of the YIG film\nwith thickness in the order of micrometers. These ten-\ndencies are visible throughout the entire measured range\nof wavevectors. It has been confirmed that the longitudi-\nnal spin Seebeck effect is present, but finally its possible\ninfluence on the spin-wave damping is too weak to be\ndetermined in our experiments.\nACKNOWLEDGMENTS\nThe authors thank Tobias Fischer as well as Bert\nL¨ agel from the Nanostructuring Center (NSC) of TU\nKaiserslauternforthe preparationofthe samples. Finan-\ncial support by Deutsche Forschungsgemeinschaft(DFG)\nwithin priority program SPP1538 ”Spin Caloric Trans-\nport” (projects VA 735/1-2 and SE 1771/4-2) is grate-\nfully acknowledged.\n∗tlangner@rhrk.uni-kl.de\n1G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Spin\ncaloritronics , Nature Mater. 11, 391 (2012).\n2S. R. Boona, R. C. Myers, and J. P. Heremans, Spin\ncaloritronics , Energy Environ. Sci. 7, 885 (2014).\n3Y. Tserkovnyak, S. A. Bender, R. A. Duine, and B. 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Goennen-\nwein,Magnon, phonon, and electron temperature profiles\nand the spin Seebeck effect in magnetic insulator/normal\nmetal hybrid structures , Phys. Rev. B 88, 094410 (2013).\n34V. Lauer, D. A. Bozhko, T. Br¨ acher , P. Pirro,\nV. I. Vasyuchka, A. A. Serga, M. B. Jungfleisch,\nM. Agrawal, Yu. V. Kobljanskyj, G. A. Melkov, C. Dubs,\nB. Hillebrands, and A. V. Chumak, Spin-transfer torque\nbased damping control of parametrically excited spin waves\nin a magnetic insulator , Appl. Phys. Lett. 108, 012402\n(2016)." }, { "title": "1111.1219v1.Tunable_magnetization_relaxation_in_spin_valves.pdf", "content": "arXiv:1111.1219v1 [cond-mat.mes-hall] 4 Nov 2011Tunable magnetization relaxation in spin valves\nXuhui Wang∗and Aurelien Manchon\nPhysical Science & Engineering Division, KAUST, Thuwal 239 55-6900, Kingdom of Saudi Arabia\n(Dated: June 24, 2018)\nIn spin values the damping parameters of the free layer are de termined non-locally by the entire\nmagnetic configuration. In a dual spin valve structure that c omprises a free layer embedded be-\ntween two pinned layers, the spin pumping mechanism, in comb ination with the angular momentum\nconservation, renders the tensor-like damping parameters tunable by varying the interfacial and dif-\nfusive properties. Simulations based on the Landau-Lifshi tz-Gilbert phenomenology for a macrospin\nmodel are performed with the tensor-like damping and the rel axation time of the free layer mag-\nnetization is found to be largely dependent on while tunable through the magnetic configuration of\nthe source-drain magnetization.\nPACS numbers: 75.70.Ak, 72.25.Ba, 75.60.Jk, 72.25.Rb\nA thorough knowledge of magnetization relaxation\nholds the key to understand magnetization dynamics in\nresponse to applied fields1and spin-transfer torques.2,3\nIn the framework of Landau-Lifshitz-Gilbert (LLG) phe-\nnomenology, relaxation is well captured by the Gilbert\ndamping parameterthat is usuallycited asa scalarquan-\ntity. As pointed out by Brown half a century ago,4the\nGilbert damping for a single domain magnetic particle is\nin general a tensor.\nWhen a ferromagnetic thin film is deposited on a nor-\nmal metal substrate, an enhanced damping has been ob-\nserved ferromagnetic resonance experiments.5This ob-\nservation is successfully explained by spin pumping:6,7\nThe slow precession of the magnetization pumps spin\ncurrent into the adjacent normal metal where the dis-\nsipation of spin current provides a non-local mechanism\nto the damping. The damping enhancement is found\nto be proportional to spin mixing conductance, a quan-\ntity playing key roles in the magneto-electronic circuit\ntheory.7,8\nThepumped spincurrent Ip∝M×˙Misalwaysin the\nplaneformedbythefreelayermagnetizationdirection M\nand the instantaneous axis about which the magnetiza-\ntion precesses. Therefore, in a single spin valve, when M\nis precessingaround the source(drain) magnetization m,\nthe pumpingcurrentisalwaysinthe planeof mandM.9\nLet us assume an azimuth angle θbetween mandM. In\nsuch anin-plane configuration, the pumping current Ip\nhas a component Ipsinθthat is parallel to m. The spin\ntransfer torque acting on the source (drain) ferromagnet\nmis the component of spin current that is in the plane\nand perpendicular to m. To simplify the discussion, we\nconsider it to be completely absorbed by m. The lon-\ngitudinal (to m) component experiences multiple reflec-\ntion at the source (drain) contact, and cancels the damp-\ning torque by an amount proportional to Ipsin2θbut is\nstill aligned along the direction of M×˙M. Therefore\nthe total damping parameter has an angle θdependence\nbut still picks up a scalar (isotropic) form. This is the\nwell-known dynamic stiffness explained by Tserkovnyak\net al.9In the most general case, when the precessing axis\nof the free layer is mis-aligned with m, there is always anout-of-plane pumping torque perpendicular to the plane.\nIn the paradigmof Slonczewski, this out-of-plane compo-\nnent is not absorbed at the interface of the source (drain)\nferromagnetic nodes, while the conservation of angular\nmomentum manifests it as a damping enhancement that\nshows the tensor form when installed in the LLG equa-\ntion.\nStudies in lateral spin-flip transistors have suggested\na tensor form for the enhanced damping parameters.10\nIn spin valves, works based on general scattering the-\nory have discussed the damping in the framework of\nfluctuation-dissipation theorem11and shown that the\nGilbert dampingtensorcanbe expressedusingscattering\nmatrices,12thus enabling first-principle investigation.13\nBut explicit analytical expressions of the damping ten-\nsor, its dependence on the magnetic configuration as well\nas the material properties and particularly its impact on\nthe magnetization relaxation are largely missing.\nIn this paper, we investigate the Gilbert damping pa-\nrameters of the free layer in the so-called dual spin valve\n(DSV).14–16We analyze the origin of the damping tensor\nand derive explicit analytical expressions of its non-local\ndependence on the magnetic configuration and materials\nproperties. A generalization of our damping tensor to a\ncontinuous magnetic texture agrees well with the results\nin earlier works. Particularly, we show, in numeric sim-\nulations, that by tuning the magnetic configurations of\nthe entire DSV, the relaxation time of the free layer can\nbe increased or decreased.\nmLM mR\nReservoir Reservoir\nFIG. 1: A dual spin valve consists of a free layer (with magne-\ntization direction M) sandwiched by two fixed ferromagnetic\nlayers (with magnetization directions mLandmR) through\ntwo normal metal spacers. The fixed layer are attached to\nreservoirs.2\nTo analyze the spin and charge currents in a DSV, we\nemploy the magneto-electronic circuit theory and spin\npumping,7,8in combination with diffusion equations.17\nPillar-shaped metallic spin valves usually consist of\nnormal-metal ( N) spacers much shorter than its spin-flip\nrelaxation length, see for example Ref.[3,15]. To a good\napproximation, in the Nnodes, a spatially homogeneous\nspin accumulation is justified and the spin current ( Ii)\nconservation dictates/summationtext\niIi= 0 (where subscript iindi-\ncates the source of spin current).\nA charge chemical potential ( µ) and a spin accumula-\ntion (s) are assigned to every ForNnode. In a transi-\ntion metalferromagnet,astrongexchangefieldalignsthe\nspin accumulation to the magnetization direction. At ev-\neryF|Ninterface, the charge and spin currents on the N\nside are determined by the contact conductance and the\ncharge and spin distributions on both sides of the con-\ntact. For example, at the contact between the left lead\nferromagnet to the left normal metal N1, called L|N1\nthereafter, the currents are8\nIL=e\n2hGL[(µ1−µL)+PL(s1−sL)·mL],\nIL=−GL\n8π[2PL(µ1−µL)mL+(s1−sL)·mLmL\n+ηL(s1−s1·mLmL)]. (1)\nWe have used the notation G=g↑+g↓is the sum of\nthe spin- σinterface conductance gσ. The contact polar-\nisationP= (g↑−g↓)/(g↑+g↓). The ratio η= 2g↑↓/G\nis between the real part of the spin-mixing conductance\ng↑↓and the total conductance G. The imaginary part\nofg↑↓is usually much smaller than its real part, thus\ndiscarded.18The spin-coherence length in a transition\nmetalferromagnetisusuallymuchshorterthanthethick-\nness of the thin film,19which renders the mixing trans-\nmission negligible.7The precession of the free layer mag-\nnetization Mpumps a spin current Ip= (/planckover2pi1/4π)g↑↓\nFM×\n˙Minto the adjacent normal nodes N1andN2, which\nis given by the mixing conductance g↑↓\nFat theF|N1(2)\ninterface (normal metals spacers are considered identical\non both sides of the free layer).\nA back flow spin current at the F|N1interface reads\nI1=−GF\n8π[2PF(µ1−µF)M+(s1−sF)·MM\n+ηF(s1−s1·MM)] (2)\non theN1side. Therefore, a weak spin-flip scattering\ninN1demands IL+I1+Ip= 0, which is dictated by\nangularmomentum conservation. The sameconservation\nlaw rules in N2, whereIR+I2+Ip= 0.\nFor the ferromagnetic ( F) nodes made of transition\nmetals, the spin diffusion is taken into account properly.9\nIn a strong ferromagnet, any transverse components de-\ncay quickly due to the large exchange field, thus the\nlongitudinal spin accumulation sν=sνmν(withν=\nL,R,F) diffuses and decays exponentially at a length\nscale given by spin diffusion length ( λsd) as∇2\nxsν=sν/λsd. The difference in spin-dependent conductivty\nof majority and minority carriers is taken into account\nby enforcing the continuity of longitudinal spin current\nmν·Iν=−(D↑\nν∇xs↑\nν−D↓\nν∇xs↓\nν) at the every F|Nin-\nterface. We assume vanishing spin currents at the outer\ninterfaces to reservoirs.\nThe diffusion equations and current conservation de-\ntermine, self-consistently, the spin accumulations and\nspin currents in both NandFnodes . We are mainly\nconcerned with the exchange torque9T=−M×(IL+\nIR)×Macting on M. A general analytical formula is\nattainable but lengthy. In the following, we focus on two\nscenarios that are mostly relevant to the state-of-the-art\nexperiments in spin valves and spin pumping: (1) The\nfree layer has a strong spin flip (short λsd) and the thick-\nnessdF≥λsd, for which the permalloy (Py) film is an\nideal candidate;15(2) The free layer is a half metal, such\nas Co2MnSi studied in a recent experiment.20\nStrong spin flip in free layer. We assume a strong spin\nflip scattering in the free layer i.e., dF≥λsd. We leave\nthe diffusivity properties in the lead Fnodes arbitrary.\nThe total exchange torque is partitioned into two parts:\nAnisotropic part that is parallel to the direction of the\nGilbert damping M×˙Mand ananisotropic part that is\nperpendicular to the plane spanned by mL(R)andM(or\nthe projection of M×˙Mto the direction mL(R)×M),\ni.e.,\nT=/planckover2pi1g↑↓\nF\n4π(DL\nis+DR\nis)/parenleftBig\nM×˙M/parenrightBig\n+/planckover2pi1g↑↓\nF\n4πM×/bracketleftBig\n(DL\nanˆAL,an+DR\nanˆAR,an)˙M/bracketrightBig\n,(3)\nwhere the material-dependent parameters DL(R)\nisand\nDL(R)\nanare detailed in the Appendix A.\nMost interest is in the anisotropic damping described\nby a symmetric tensor with elements\nˆAij\nan=−mimj (4)\nwherei,j=x,y,z(we have omitted the lead index Lor\nR). The elements of ˆAanare given in Cartesian coordi-\nnates of the source-drain magnetization direction. The\nanisotropic damping appears as M׈Aan˙Mthat is al-\nways perpendicular to the free layer magnetization direc-\ntion, thus keeping the length of Mconstant.11It is not\ndifficult to show that when Mis precessing around m,\nthe anisotropic part vanishes due to ˆAan˙M= 0.\nWe generalizeEq.(4) toa continuousmagnetictexture.\nConsider here only one-dimensional spatial dependence\nand the extension to higher dimensions is straightfor-\nward. The Cartesian component of vector U≡M×\nˆAan˙MisUi=−εijkMjmkml˙Ml(whereεijkis the Levi-\nCivita tensor and repeated indices are summed). We as-\nsume the fixed layer and the free layer differ in space by\na lattice constant a0, which allows mk≈Mk(x+a0). A\nTaylor expansion in space leads to U=−a2\n0M×(ˆD˙M),\nwherethematrixelements ˆDkl= (∂xM)k(∂xM)landwe3\nhave assumed that the magnetization direction is always\nperpendicular to ∂xM. In this case, three vectors ∂xM,\nM×∂xMandMare perpendicular to each other. A\nrotation around Mbyπ/2 leavesMand˙Munchanged\nwhile interchanging ∂xMwithM×∂xM, we have\nˆDkl= (M×∂xM)k(M×∂xM)l, (5)\nwhich agrees with the so-called differential damping ten-\nsor Eq.(11) in Ref.[21].\nEq.(3) suggests that the total exchange torque on the\nfree layer is a linear combination of two independent ex-\nchange torques arsing from coupling to the left and the\nrightFnodes. This form arises due to a strong spin-\nflip scattering in the free layer that suppresses the ex-\nchange between two spin accumulations s1ands2in the\nNnodes. In the pursuit of a concise notation for the\nGilbert form, the exchange torque can be expressed as\nT=M×← →α˙Mwith a total damping tensor given by\n← →α=/planckover2pi1g↑↓\nF\n4π/parenleftBig\nDL\nis+DR\nis+DL\nanˆAL,an+DR\nanˆAR,an/parenrightBig\n.(6)\nThe damping tensor← →αis determined by the entire mag-\nnetic configuration of the DSV and particularly by the\nconductance of F|Ncontacts and the diffusive proper-\nties theFnodes.\nHalf metallic free layer . This special while experimen-\ntally relevant20case means PF= 1. Half-metallicity\nin combination with the charge conservation enforces a\nlongitudinal back flow that is determined solely by the\nbias current: The spin accumulations in Nnodes do\nnot contribute to the spin accumulation inside the free\nlayer, thus an independent contribution due to left and\nright leads is foreseen. We summarize the material spe-\ncific parameters in the Appendix A. When spin flip is\nweak in the source-drain ferromagnets, ξL≈0 leads to\nDis≈0. In this configuration, by taking a (parallel or\nanti-parallel)source-drainmagnetization direction as the\nprecessingaxis,thetotaldampingenhancementvanishes,\nwhich reduces to the scenario of ν= 1 in Ref.[9].\nMagnetization relaxation . To appreciate the impact of\nan anisotropic damping tensor on the magnetization re-\nlaxation, we perform a simulation, for the free layermag-\nnetization, using Landau-Lifshitz-Gilbert (LLG) equa-\ntion augmented by the tensor damping, i.e.,\ndM\ndt=−γM×Heff+α0M×dM\ndt\n+γ\nµ0MsVM×← →αdM\ndt.(7)\nα0is the (dimensionless) intrinsic Gilbert damping pa-\nrameter. Symbol γis the gyromagnetic ratio, Msis the\nsaturation magnetization, and Vis the volume of the\nfree layer. µ0stands for the vacuum permeability. The\ndynamics under the bias-driven spin transfer torque is\nnot the topic in this paper, but can be included in a\nstraightforward way.22We give in the Appendix B the\nexpressions of the bias-driven spin torques.We are mostly interested in the relaxation of the mag-\nnetization, instead of particular magnetization trajecto-\nries, in the presence of a tensor damping. The follow-\ning simulation is performed for the scenario where the\nfree layer has a strong spin flip, i.e., Case (1). We em-\nploy the pillar structure from Ref.[15] while consider-\ning the free layer (Py) to be 8nm thick (a thicker free\nlayer favors a better thermal stability.15) The source-\ndrain ferromagnets are cobalt (Co) and we expect the\nresults are valid for a larger range of materials selec-\ntions. The Py film is elliptic with three axes given by\n2a= 90 nm, 2 b= 35 nm,15andc= 8 nm. The de-\nmagnetizing factors Dx,y,zin the shape anisotropy en-\nergyEdem= (1/2)µ0M2\nsV/summationtext\ni=x,y,zDiM2\niareDx= 0.50,\nDy= 0.37 and Dz= 0.13. An external field Haleads\nto a Zeeman splitting EZee=−Vµ0MsHa·M. For Py\nfilms, we neglect the uniaxial anisotropy. The total free\nenergyET=EZee+Edemgives rise to an effective field\nHeff=−(1/VMsµ0)∂ET/∂M.\nThe spin-dependent conductivities in the bulk of Co\nandthe spin diffusion length λCo≈60nm aretaken from\nthe experimental data.24For Py, we take λPy≈4 nm.25\nTo have direct connection with experiments, the above\nmentioned bare conductance has to be renormalized by\nthe Sharvin conductance.26For Py/Cu the mixing con-\nductance, we take the value g↑↓\nFS−1≈15 nm−2,26which\ngivesRL(R)F≈1.0.\n56789100.950.960.970.980.991(a) Bz = 50 G; I.P. y−axis.\n \n(y,y)(y,x)(x,x)(x,z)(y,z)(z,z)369(b) Relaxation time\n(x,x)\n(y,y)\n(z,z)\nFIG. 2: (Color online) Mzas a function of time (in ns) in\npresence of differentsource-drain magnetic configurations and\napplied fields. (a) The external magnetic field Bz= 50 Gauss\nis applied along z-axis. The blue (dashed), red (solid) and\nblack (dotted dash) curves correspond to source-drain magn e-\ntization in configurations ( y,y), (x,x), and (z,z) respectively.\n(b) Magnetization relaxation times (in the unit of ns)versu s\nsource-drain magnetic configurations at different applied fi eld\nalongz-axis.:Bz= 10 G (red /square),Bz= 50 G (blue /circlecopyrt),\nBz= 200 G (green ▽),Bz= 800 G (black ���). Lines are a\nguide for the eyes. The initial position (I.P.) of the free la yer\nis taken along y-axis.\nThe relaxation time τris extracted from the sim-\nulations by demanding at a specific moment τrthe\n|Mz−1.0|<10−3, i.e., reaches the easy axis. In the\nabsence of bias, panel (a) of Fig.2 shows the late stage\nof magnetization relaxation from an initial position ( y-4\naxis) in the presence of an tensor damping, under various\nsource-drain (SD) magnetic configurations. The results\nare striking: Under the same field, switching the SD con-\nfigurations increases or decreases τr. In panel (b), the\nextracted relaxation times τrversus SD configurations\nunder various fields are shown. At low field Bz= 10 G\n(red/square), when switching from ( z,z) to (y,y),τris im-\nproved from 8.0 ns to 6.3 ns, about 21%. At a higher\nfieldBz= 800 G (black ♦), the improvement is larger\nfrom 5.2 at ( z,z) to 3.6 at ( y,y), nearly 31%. To a large\ntrend, the relaxation time improvement is more signifi-\ncant at higher applied fields.\nIn conclusion, combining conservation laws and\nmagneto-electronic circuit theory, we have analyzed the\nGilbert damping tensor of the free layer in a dual spin\nvalve. Analytical results of the damping tensor as func-\ntions of the entire magnetic configuration and material\nproperties are obtained. Numerical simulations on LLG\nequation augmented by the tensor damping reveal a tun-\nable magnetization relaxation time by a strategic selec-\ntion of source-drain magnetization configurations. Re-\nsults presented in this paper open a new venue to the\ndesign and control of magnetization dynamics in spin-\ntronic applications.\nX.Wang is indebted to G. E. W. Bauer, who has\nbrought the problem to his attention and offered invalu-\nable comments.\nAppendix A: Material dependent parameters\nIn this paper, RL(R)F≡g↑↓\nL(R)/g↑↓\nFis the mixing con-\nductance ratio and χL(R)≡mL(R)·M. The diffusivity\nparameter ξL(R)=φL(R)(1−P2\nL(R))/ηL(R), where for the\nleftFnode\nφL=1\n1+(σ↑\nL+σ↓\nL)λLe2\n4hSσ↑\nLσ↓\nLtanh(dL/λL)GL(1−P2\nL)(A1)\nwherehthe Planck constant, Sthe area of the thin film,\nethe elementary charge, λLthe spin diffusion length, dL\nthe thickness of the film, and σ↑(↓)the spin-dependent\nconductivity. φRisobtainedbysubstituting all LbyRin\nEq.(A1). Parameter ξFis given by ξF= (1−P2\nF)φF/ηF\nwith\nφF=1\n1+(σ↑\nF+σ↓\nF)λFe2\n4hSσ↑\nFσ↓\nFGF(1−P2\nF).(A2)\nThe material dependent parameters as appearing in the\ndamping tensor Eq.(6) are: (1) In the case of a strongspin flip in free layer,\nDL(R)\nis=RL(R)F\nLL(R)F/bracketleftBig\nξL(R)RL(R)F+ξL(R)ξF(1−χ2\nL(R))\n+ξF(1−χ2\nL(R))χ2\nL(R)/bracketrightBig\n,\nDL(R)\nan=RL(R)F\nLL(R)F(ξL(R)−1)[ξF(1−χ2\nL(R))+RL(R)F]\n1+RL(R)F,\nLL(R)F=(1+RL(R)Fχ2\nL)ξF(1−χ2\nL(R))\n+RL(R)F/bracketleftBig\n(1−χ2\nL(R))(1+ξL(R)ξF)\n+ξL(R)RL(R)F+ξFχ2\nL(R)/bracketrightBig\n; (A3)\n(2)In the case of a half metallic free layer\nDL(R)\nis=RL(R)FξL(R)\n(1−χ2\nL(R))+ξL(R)(χ2\nL+RL(R)F),\nDL(R)\nan=RL(R)F\n1+RL(R)F\n×ξL(R)−1\n(1−χ2\nL(R))+ξL(R)(χ2\nL+RL(R)F).(A4)\nAppendix B: Bias dependent spin torques\nThe full analytical expression of bias dependent spin\ntorques are rather lengthy. We give here the expres-\nsions, under a bias current I, for symmetric SD fer-\nromagnets (i.e., φL=φR=φthusξL=ξR=ξ)\nwith parallelor anti-parallelmagnetization direction. (1)\nWith a strong spin flip in the free layer, the parallel\nSD magnetization leads to vanishing bias-driven torque\nT(b)\n⇑⇑= 0; WhentheSDmagnetizationsareanti-parallelly\n(i.e.,mL=−mR≡m),\nT(b)\n⇑⇓=I/planckover2pi1Pφ\ne(1+R)L/bracketleftbig\n(ξF+RξFχ2+R)(1−χ2)\n+R(R+ξF(1−χ2)+χ2)/bracketrightbig\nmF×(m×mF).\n(B1)\n(2) When the free layer is half metallic, for symmetric\nSD ferromagnets , T(b)\n⇑⇑= 0 and\nT(b)\n⇑⇓=I/planckover2pi1\neφP\n(1−ξ)(1−χ2)+ξ(χ2+R)mF×(m×mF).\n(B2)\n∗Electronic address: xuhui.wang@kaust.edu.sa\n1L. D. Landau and E. M. Lifshitz, Statistical Physics ,Part\n2(Pergamon, Oxford, 1980); T. L. Gilbert, IEEE. Trans.\nMag.40, 2443 (2004).2J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996);\nL. Berger, Phys. Rev. B 54, 9353 (1996).\n3E. B. Myers, et al., Science 285, 867 (1999); J. A. 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Phys. Lett. 86, 152509 (2005).16P. Bal´ aˇ z, M. Gmitra, and J. Barna´ s, Phys. Rev. B 80,\n174404 (2009); P. Yan, Z. Z. Sun, and X. R. Wang, Phys.\nRev. B83, 174430 (2011).\n17T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993); A. A.\nKovalev, A. Brataas, and G. E. W. Bauer Phys. Rev. B\n66, 224424 (2002).\n18K. Xia,et al., Phys. Rev. B 65, 220401(R) (2002).\n19M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n20H. Chudo, et al., J. Appl. Phys. 109, 073915 (2011).\n21S. Zhang and S. -L. Zhang, Phys. Rev. Lett. 102, 086601\n(2010).\n22J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 72,\n014446 (2005).\n23J. Osborn, Phys. Rev. 67, 351 (1945).\n24J. Bass and W. P. Pratt, J. Magn. Magn. Mater. 200, 274\n(1999).\n25A. Fert and L. Piraux, J. Magn. Magn. Mater. 200, 338\n(1999).\n26G. E. W. Bauer, et al., Phys. Rev. B 67, 094421 (2003)." }, { "title": "1510.01894v1.Tunable_damping__saturation_magnetization__and_exchange_stiffness_of_half_Heusler_NiMnSb_thin_films.pdf", "content": "Tunable damping, saturation magnetization, and exchange sti\u000bness of half-Heusler\nNiMnSb thin \flms\nP. D urrenfeld,1F. Gerhard,2J. Chico,3R. K. Dumas,1, 4M. Ranjbar,1A. Bergman,3\nL. Bergqvist,5, 6A. Delin,3, 5, 6C. Gould,2L. W. Molenkamp,2and J. \u0017Akerman1, 4, 5\n1Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden\n2Physikalisches Institut (EP3), Universit at W urzburg, 97074 W urzburg, Germany\n3Department of Physics and Astronomy, Uppsala University, Box 520, 752 20 Uppsala, Sweden\n4NanOsc AB, 164 40 Kista, Sweden\n5Materials and Nano Physics, School of ICT, KTH Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden\n6Swedish e-Science Research Centre (SeRC), 100 44 Stockholm, Sweden\nThe half-metallic half-Heusler alloy NiMnSb is a promising candidate for applications in spin-\ntronic devices due to its low magnetic damping and its rich anisotropies. Here we use ferromagnetic\nresonance (FMR) measurements and calculations from \frst principles to investigate how the com-\nposition of the epitaxially grown NiMnSb in\ruences the magnetodynamic properties of saturation\nmagnetization MS, Gilbert damping \u000b, and exchange sti\u000bness A.MSandAare shown to have a\nmaximum for stoichiometric composition, while the Gilbert damping is minimum. We \fnd excellent\nquantitative agreement between theory and experiment for MSand\u000b. The calculated Ashows the\nsame trend as the experimental data, but has a larger magnitude. Additionally to the unique in-\nplane anisotropy of the material, these tunabilities of the magnetodynamic properties can be taken\nadvantage of when employing NiMnSb \flms in magnonic devices.\nI. INTRODUCTION\nInterest in the use of half-metallic Heusler and half-\nHeusler alloys in spintronic and magnonic devices is\nsteadily increasing,1{3as these materials typically exhibit\nboth a very high spin polarization4{8and very low spin-\nwave damping.9{12One such material is the epitaxially\ngrown half-Heusler alloy NiMnSb,13,14which not only has\none of the lowest known spin-wave damping values of\nany magnetic metal, but also exhibits an interesting and\ntunable combination of two-fold in-plane anisotropy15\nand moderate out-of-plane anisotropy,10all potentially\ninteresting properties for use in both nanocontact-\nbased spin-torque oscillators16{22and spin Hall nano-\noscillators23{27. To successfully employ NiMnSb in such\ndevices, it is crucial to understand, control, and tailor\nboth its magnetostatic and magnetodynamic properties,\nsuch as its Gilbert damping ( \u000b), saturation magnetiza-\ntion (MS), and exchange sti\u000bness ( A).\nHere we investigate these properties in Ni 1-xMn1+xSb\n\flms using ferromagnetic resonance (FMR) measure-\nments and calculations from \frst principles for compo-\nsitions of -0.1\u0014x\u00140.4.MSandAare shown experi-\nmentally to have a maximum for stoichiometric compo-\nsition, while the Gilbert damping is minimum; this is\nin excellent quantitative agreement with calculations of\nand experiment on MSand\u000b. The calculated Ashows\nthe same trend as the experimental data, but with an\noverall larger magnitude. We also demonstrate that the\nexchange sti\u000bness can be easily tuned over a wide range\nin NiMnSb through Mn doping, and that the ultra-low\ndamping persists over a wide range of exchange sti\u000b-\nnesses. This unique behavior makes NiMnSb ideal for\ntailored spintronic and magnonic devices. Finally, by\ncomparing the experimental results with \frst-principlescalculations, we also conclude that the excess Mn mainly\noccupies Ni sites and that interstitial doping plays only\na minor role.\nII. METHODS\nA. Thin Film Growth\nThe NiMnSb \flms were grown by molecular beam\nepitaxy onto InP(001) substrates after deposition of a\n200 nm thick (In,Ga)As bu\u000ber layer.15The \flms were\nsubsequently covered in situ by a 10 nm thick mag-\nnetron sputtered metal cap to avoid oxidation and sur-\nface relaxation.28The Mn content was controlled dur-\ning growth via the temperature, and hence the \rux, of\nthe Mn e\u000busion cell. Six di\u000berent samples (see table I)\nwere grown with increasing Mn concentration, sample 1\nhaving the lowest and sample 6 the highest concentra-\ntion of Mn. High-resolution x-ray di\u000braction (HRXRD)\nmeasurements give information on the structural proper-\nties of these samples, con\frming the extremely high crys-\ntalline quality of all samples with di\u000berent Mn concentra-\nSamplevertical\nlattice\nconstant ( \u0017A)thickness\n(nm)uniaxial\neasy axis2K1\nMS(Oe)\n1 5.94 38 [110] 170\n2 5.97 38 [110] 8.4\n3 5.99 40 [110] 0\n4 6.02 45 [1 \u001610] 9.0\n5 6.06 45 [1 \u001610] 14.2\n6 6.09 38 [1 \u001610] 25.5\nTable I. Overview of NiMnSb \flms investigated in this study.arXiv:1510.01894v1 [cond-mat.mtrl-sci] 7 Oct 20152\ntion, even in the far from stoichiometric cases (samples 1\nand 6).15The vertical lattice constant is found to increase\nwith increasing Mn concentration and, assuming a linear\nincrease,29we estimate the di\u000berence in Mn concentra-\ntion across the whole set of samples to be about 40 at. %.\nWe will thus represent the Mn concentration in the fol-\nlowing experimental results by the measured vertical lat-\ntice constant. Stoichiometric NiMnSb exhibits vertical\nlattice constants in the range of 5.96{6.00 \u0017A, leading to\nthe expectation of stoichiometric NiMnSb in samples 2\nand 3.15Finally, the layer thicknesses are also determined\nfrom the HRXRD measurements, giving an accuracy of\n\u00061 nm.\nB. Ferromagnetic Resonance\nBroadband \feld-swept FMR spectroscopy was per-\nformed using a NanOsc Instruments PhaseFMR system\nwith a coplanar waveguide for microwave \feld excitation.\nMicrowave \felds hrfwith frequencies of up to 16 GHz\nwere applied in the \flm plane, perpendicularly oriented\nto an in-plane dc magnetic \feld H. The derivative of\nthe FMR absorption signal was measured using a lock-\nin technique, in which an additional low-frequency mod-\nulation \feld Hmod<1 Oe was applied using a pair of\nHelmholtz coils parallel to the dc magnetic \feld. The\n\feld directions are shown schematically in Fig. 1(a) and\na typical spectrum measured at 13.6 GHz is given in the\ninset of Fig. 1(b). In addition to the zero wave vector\nuniform FMR mode seen at about H=2.1 kOe, an addi-\ntional weaker resonance is observed at a much lower \feld\nof about 500 Oe, and is identi\fed as the \frst exchange-\ndominated perpendicular standing spin wave (PSSW)\nmode. The PSSW mode has a nonzero wave vector point-\ning perpendicular to the thin \flm plane and a thickness-\ndependent spin-wave amplitude and phase.30,31This can\nbe e\u000eciently excited in the coplanar waveguide geome-\ntry due to the nonuniform strength of the microwave \feld\nacross the \flm thickness.32\nThe \feld dependence of the absorption spectra (inset\nof Fig. 1(b)) can be \ft well (red line) by the sum of a sym-\nmetric and an antisymmetric Lorentzian derivative:33,34\ndP\ndH(H) =\u00008C1\u0001H(H\u0000H0)\nh\n\u0001H2+ 4 (H\u0000H0)2i2\n+2C2\u0000\n\u0001H2\u00004(H\u0000H0)2\u0001\nh\n\u0001H2+ 4 (H\u0000H0)2i2; (1)\nwhereH0is the resonance \feld, \u0001 Hthe full width at\nhalf maximum (FWHM), and C1andC2\ftted param-\neters representing the amplitude of the symmetric and\nantisymmetric Lorentzian derivatives, respectively. Both\nthe FMR and the PSSW peaks can be \ftted indepen-\ndently, as they are well separated by the exchange \feld\n\u00160Hex/(\u0019=d)2, wheredis the thickness of the layer.\neasy axis\nH, Hmod\nhard axis\nhrf p = 0\n(FMR)p = 1\n(PSSW)z\nHexx\ny\nFMRPSSW(a)\n(b)\n0\n500\n1000\n1500\n2000\n2500\n0\n5\n10\n15\nf(GHz)\nField (Oe)\n400\n600\n2000\n2200\n-0.5\n0.0\n0.5\n1.0\n1.5\ndP/dH (a. u.)\nField (Oe)\nf= 13.6 GHzFigure 1. (a) Schematic diagram of the FMR measurement\nshowing \feld directions. In our setup, the FMR mode and the\n\frst PSSW mode are excited. (b) Frequency vs. resonance\n\felds of the PSSW (red) and uniform FMR (black) mode for\nsample 2. The solid lines are \fts to the Kittel equation, and\nboth modes are o\u000bset horizontally by Hex. Inset: Resonance\ncurves forf=13.6 GHz. The \frst PSSW mode on the left and\nthe FMR mode on the right were \ft with Eq. 1\nFor our chosen sample thicknesses, the di\u000berences in res-\nonance \felds are always much larger than the resonance\nlinewidths.\nThe \feld dependence of both resonances is shown in\nFig. 1(b) and can now be used to extract information\nabout the magnetodynamic properties and anisotropies\nof the \flms. The curves are \fts to the Kittel equation,\nincluding internal \felds from the anisotropy and the ex-\nchange \feld for the PSSW excitation:15,35\nf=\r\u00160\n2\u0019\u0014\u0012\nH0+2KU\nMS\u00002K1\nMS+Hex\u0013\n\u0002\u0012\nH0+2KU\nMS+K1\nMS+Hex+Me\u000b\u0013\u00151=2\n;(2)\nwhereH0is the resonance \feld, \r=2\u0019the gyromagnetic\nratio, and\u00160the permeability of free space. Me\u000bis the ef-\nfective magnetization, which has a value close to the satu-\nration magnetization MS. 2KU=MSand 2K1=MSstands\nfor the internal anisotropy \felds coming from the uniaxial\n(KU) and biaxial ( K1) anisotropy energy densities in the\nhalf-Heusler material. The e\u000bective magnetic \feld also\nincludes an exchange \feld \u00160Hex= (2A=M S)(p\u0019=d )2,3\nwhich is related to the exchange sti\u000bness A, the \flm\nthicknessd, and the integer order of the PSSW mode\np, wherep= 0 denotes the uniform FMR excitation and\np= 1 the \frst PSSW mode. This mode numbering re-\n\rects the boundary conditions with no surface pinning of\nthe spins, which is expected for the in-plane measurement\ngeometry.36\nWe stress that the expression for the anisotropy con-\ntribution in Eq. 2 is only valid for the case in which the\nmagnetization direction is parallel to the uniaxial easy\naxis and also parallel to the applied \feld. A full angular-\ndependent formulation of the FMR condition is described\nin Ref. 15. To ful\fll the condition of parallel alignment\nfor all resonances, we perform the FMR measurements\nwith the dc magnetic \feld being applied along the domi-\nnant uniaxial easy axis of each \flm, which changes from\nthe [110] crystallographic direction to the [1 \u001610]-direction\nwith increasing Mn concentration (see Table I).\nThe values of the biaxial anisotropy2K1\nMShave been de-\ntermined in a previous study by \fxed-frequency in-plane\nangular dependent FMR measurements,15and were thus\ntaken as constant values in the \ftting process for Eq. 2;\na simultaneous \ft of both contributions can yield arbi-\ntrary combinations of anisotropy \felds due to their great\ninterdependence. The values for the uniaxial anisotropy\n2KU\nMSobtained from the frequency-dependent \ftting are\nin very good agreement with the previously obtained val-\nues in Ref. 15. The gyromagnetic ratio was measured\nto be\r=2\u0019= (28.59\u00060.20) GHz/T for all investigated\nsamples, and was therefore \fxed for all samples to allow\nbetter comparison of the e\u000bective magnetization values.\nThe Gilbert damping \u000bof the \flms is obtained by\n\ftting the FMR linewidths \u0001 Hwith the linear depen-\ndence:37\n\u00160\u0001H=\u00160\u0001H0+4\u0019\u000b\n\rf; (3)\nwhere \u0001H0is the inhomogeneous linewidth broaden-\ning of the \flm. The parallel alignment between mag-\nnetization and external magnetic \feld ensures that the\nlinewidth is determined by the Gilbert damping process\nonly.38\nC. Calculations from First Principles\nThe electronic and magnetic properties of the NiMnSb\nhalf-Heusler system were studied via \frst-principles cal-\nculations. The material was assumed to be ordered in a\nface-centered tetragonal structure with an in-plane lat-\ntice parameter ak\nlat= 5.88 \u0017A, close to the lattice con-\nstant of the InP substrate, and an out-of-plane lattice\nconstant of a?\nlat= 5.99 \u0017A, matching the value for the\nstoichiometric composition. Fixed values for the lattice\nparameters were chosen since an exact relation between\nthe o\u000b-stoichiometric composition and the experimen-\ntally measured vertical lattice constants cannot be es-\ntablished. Moreover, calculations with a varying verticallattice parameter for a constant composition showed only\na negligible e\u000bect on M S,A, and\u000b. The calculations\nwere performed using the multiple scattering Korringa-\nKohn-Rostocker (KKR) Green's function formalism as\nimplemented in the SPRKKR package.39Relativistic ef-\nfects were fully taken into account by solving the Dirac\nequation for the electronic states, the shape of the poten-\ntial was considered via the Atomic Sphere Approximation\n(ASA), and the local spin density approximation (LSDA)\nwas used for the exchange correlation potential. The co-\nherent potential approximation (CPA) was used for the\nchemical disorder of the system.\nThe Gilbert damping \u000bof the material was calculated\nusing linear response theory40, including the temperature\ne\u000bects from interatomic displacements and spin \ructua-\ntions.41,42\nThe exchange interactions Jijbetween the atomic\nmagnetic moments were calculated using the magnetic\nforce theorem, as considered in the LKAG formalism.43,44\nThe interactions were calculated for up to 4.5 times the\nlattice constant in order to take into account any long-\nrange interactions. Given the interatomic exchange in-\nteractions, the spin-wave sti\u000bness Dcan be calculated.\nDue to possible oscillations in the exchange interactions\nas a function of the distance, it becomes necessary to in-\ntroduce a damping parameter, \u0011, to assure convergence\nof the summation. Dcan then be obtained by evaluating\nthe limit\u0011!0 of\nD=2\n3X\nijJijp\nMiMjr2\nijexp\u0012\n\u0000\u0011rij\nalat\u0013\n; (4)\nas described in [45]. Here, MiandMjare the local mag-\nnetic moments at sites iandj,Jijis the exchange cou-\npling between the magnetic moments at sites iandj,\nandrijis the distance between the atoms iandj. This\nformalism can be extended to a multisublattice system46.\nTo calculate the e\u000bect of chemical disorder on the ex-\nchange sti\u000bness of the system, the obtained exchange in-\nteractions were summed over a supercell with a random\ndistribution of atoms in the chemically disordered sub-\nlattice. The e\u000bect that distinct chemical con\fgurations\ncan have over the calculation of the exchange sti\u000bness\nwas treated by taking 200 di\u000berent supercells. The re-\nsults were then averaged and the standard deviation was\ncalculated. The cells were obtained using the atomistic\nspin dynamics package UppASD.47\nFinally, with the spin-wave sti\u000bness determined as de-\nscribed above, the exchange sti\u000bness Acan be calculated\nfrom:48\nA=DM S(T)\n2g\u0016B: (5)\nHere,gis the Land\u0013 e g-factor of the electron, \u0016Bthe Bohr\nmagneton, and MS(T) the magnetization density of the\nsystem for a given temperature T, which for T= 0 K\ncorresponds to the saturation magnetization.\nFrom the \frst-principles calculations, the magnetic\nproperties for ordered NiMnSb and chemically disordered4\n0.60.70.80.95\n.956 .006 .056 .100123(b) m0MS \nm0Meffm0M (T)(a)4 µB/u.f.KS (mJ/m2)v\nertical lattice constant (Å)\nFigure 2. (a) MSandMe\u000bas functions of vertical lattice\nconstant. The theoretical value of 4.0 \u0016B=u.f. is shown by\nthe blue dashed line. (b) The calculated surface anisotropy\ndensity follows from the di\u000berence between MSandMe\u000b.\nNi1-xMn1+xSb were studied. To obtain the values of the\nexchange sti\u000bness AforT= 300 K, the exchange interac-\ntions from the ab initio calculations were used in conjunc-\ntion with the value of the magnetization at T= 300 K\nobtained from Monte Carlo simulations.\nIII. RESULTS\nA. Magnetization\nThe values of \u00160Me\u000bare plotted in Fig. 2(a) as red\ndots. The e\u000bective magnetization is considerably lower\nthan the saturation magnetization \u00160MS, which was in-\ndependently assessed using SQUID measurements and al-\nternating gradient magnetometry (AGM). The values for\n\u00160MScorrespond to a saturation magnetization between\n3.5\u0016B=unit formula and 3.9 \u0016B=u.f., with the latter\nvalue being within the error bars of the theoretically ex-\npected value of 4.0 \u0016B=u.f. for stoichiometric NiMnSb.49\nA reduction of MSis expected in Mn-rich NiMnSb alloys,\ndue to the antiferromagnetic coupling of the Mn Nidefects\nto the Mn lattice in the C1 bstructure of the half-Heusler\nmaterial.29An even stronger reduction is observed for\nthe Ni-rich sample 1, which is in accordance with the\nformation of Ni Mnantisites.50\nWhile the measurement error for MSis comparatively\nlarge due to uncertainties in the volume determination,\nthe error bars for Me\u000b, as obtained from ferromagnetic\nresonance, are negligible. NiMnSb \flms have been shown\nto possess a small but substantial perpendicular mag-\nnetic anisotropy, which can arise from either interfacial\nanisotropy or lattice strain.10,12To quantify the di\u000ber-\nence observed between MSandMe\u000b, we assume a uniax-\nial perpendicular anisotropy due to a surface anisotropy\n2\n4\n6\n8\n10\n5.95\n6.00\n6.05\n6.10\n0\n2\n4\n(b)\nA (pJ/m)\n(a)\nα(10-3)\nvertical lattice constant (Å)\n2\n4\n6\n8\n10\n-0.1\n0.0\n0.1\n0.2\n0.3\n0.4\n0\n2\n4\n(d)\n(c)\nxantisitesFigure 3. (a) and (b) show respectively the exchange sti\u000bness\nand Gilbert damping constant obtained from FMR measure-\nments, plotted as a function of the vertical lattice constant.\n(c) and (d) show the corresponding values obtained from \frst-\nprinciple calculations for T= 300 K. Negative values for x\nimply the introduction of Ni Mnantisites and positive values\nare related to Mn Niantisite defects. The error bars in (c) are\nthe standard deviations from repeated \frst-principles calcu-\nlations with 200 randomized supercells.\nenergy density KS, which is known to follow the rela-\ntion:51\n\u00160Me\u000b=\u00160MS\u00002KS\nMSd: (6)\nTheKScalculated in this way has values between\n0.5mJ=m2and 1.5mJ=m2, as shown in Fig. 2(b); these\nare comparable to the surface anisotropies obtained in\nother crystalline thin \flm systems.52. Although the \flm\nthicknesses in our set vary unsystematically, we can ob-\nserve systematic behavior of KSwith the vertical lat-\ntice constant, with an apparent minimum under the con-\nditions where stoichiometric NiMnSb is expected|that\nis, for samples 2 and 3. The increasing values for o\u000b-\nstoichiometric NiMnSb can be thus attributed to the con-\ncomitant increase in lattice defects, and thus of surface\ndefects, in these \flms.\nB. Exchange Sti\u000bness and Gilbert Damping\nThe experimentally determined exchange sti\u000bness, as\na function of the vertical lattice constant, and the Gilbert\ndamping parameter are shown in Fig. 3(a) and (b), re-\nspectively. The minimum damping observed in our mea-\nsurements is 1 :0\u000210\u00003for sample 3, and so within sto-\nichiometric composition. Sample 1, with a de\fciency\nof Mn atoms, showed nonlinear linewidth behavior at\nlow frequencies, which vanished for out-of-plane measure-\nments (not shown). This is typical with the presence of\ntwo-magnon scattering processes.52However, the damp-\ning is considerably lower in all samples than in a permal-\nloy \flm of comparable thickness.\nThe exchange sti\u000bness and Gilbert damping ob-\ntained from the \frst-principles calculations are shown in5\nFig. 3(c) and (d), respectively. For both parameters, the\nexperimental trends are reproduced quantitatively, with\nAhaving a maximum and \u000ba minimum value at stoi-\nchiometry.\nAs the concentration of both Mn or Ni antisites in-\ncreases, the exchange sti\u000bness decreases. This behavior\ncan be explained by analyzing the terms in the expres-\nsion for the spin-wave sti\u000bness, Eq. 4. It turns out that\nthe new exchange couplings Jij, which appear when an-\ntisites are present, play a major role, whereas changes in\nthe atomic magnetic moments or the saturation magne-\ntization appear to be relatively unimportant. Mn anti-\nsites in the Ni sublattice (i.e., excess Mn) have a strong\n(2 mRy) antiferromagnetic coupling to the Mn atoms in\nthe adjacent Mn layers. This results in a negative contri-\nbution toDcompared to the stoichiometric case, where\nthis interaction is not present. On the other hand, Ni\nantisites in the Mn sublattice have a negative in-plane\nexchange coupling of 0.3 mRy to their nearest-neighbor\nMn atoms, with a frustrated antiferromagnetic coupling\nto the Ni atoms in the adjacent Ni plane. The net e\u000bect is\na decreasing spin-wave sti\u000bness as the composition moves\naway from stoichiometry. The calculated values of Aare\naround 30 % larger than the experimental results, which\nis the same degree of overestimation we recently observed\nin a study of doped permalloy \flms53. It thus seems to\nbe inherent in our calculations from \frst principles.\nThe calculated Gilbert damping also agrees well with\nthe experimental values. The damping has its minimum\nvalue of 1.0\u000210\u00003at stoichiometry and increases with\na surplus of Ni faster than with the same surplus of\nMn. Both Mn and Ni antisites will act as impurities and\nit is thus reasonable to attribute the observed increase\nin damping at o\u000b-stoichiometry to impurity scattering.\nWhile the damping at stoichiometry also agrees quanti-\ntatively, the increase in damping is underestimated in the\ncalculations compared to the experimental values.\nDespite the fact that the calculations here focus purely\non the formation of Mn Nior Ni Mnantisites, they are\nnonetheless capable of reproducing the experimental\ntrends well. However, interstitials|that is, Mn or Ni sur-\nplus atoms in the vacant sublattice|may also be a possi-\nble o\u000b-stoichiometric defect in our system.50We have cal-\nculated their e\u000bects and can therefore discuss about the\nexistence of interstitials in our samples. A large fraction\nof Mn interstitials seems unlikely, as an increase in the\nsaturation magnetization can be predicted through calcu-\nlations, contrary to the experimental trend; see Fig. 2(a).On the other hand, the existence of Ni interstitials may\nbe compatible with the observed experimental trend, as\nthey decrease the saturation magnetization|albeit at a\nslower rate than Ni antisites and slower than experimen-\ntally observed. Judging from the measured data, it is\ntherefore likely that excess Ni exists in the samples as\nboth antisites and interstitials.\nIV. CONCLUSIONS\nIn summary, we have found that o\u000b-stoichiometry in\nthe epitaxially grown half-Heusler alloy NiMnSb has a\nsigni\fcant impact on the material's magnetodynamic\nproperties. In particular, the exchange sti\u000bness can be\naltered by a factor of about 2 while keeping the Gilbert\ndamping very low ( \u00195 times lower than in permalloy\n\flms). This is a unique combination of properties and\nopens up for the use of NiMnSb in, e.g., magnonic cir-\ncuits, where a small spin wave damping is desired. At the\nstoichiometric composition, the saturation magnetization\nand exchange sti\u000bness take on their maximum values,\nwhereas the Gilbert damping parameter is at its mini-\nmum. These experimentally observed results are repro-\nduced by calculations from \frst principles. Using these\ncalculations, we can also explain the microscopic mecha-\nnisms behind the observed trends. We also conclude that\ninterstitial Mn is unlikely to be present in the samples.\nThe observed e\u000bects can be used to \fne-tune the mag-\nnetic properties of NiMnSb \flms towards their speci\fc\nrequirements in spintronic devices.\nACKNOWLEDGMENTS\nWe acknowledge \fnancial support from the G oran\nGustafsson Foundation, the Swedish Research Council\n(VR), Energimyndigheten (STEM), the Knut and Alice\nWallenberg Foundation (KAW), the Carl Tryggers Foun-\ndation (CTS), and the Swedish Foundation for Strate-\ngic Research (SSF). F.G. acknowledges \fnancial support\nfrom the University of W urzburg's \\Equal opportunities\nfor women in research and teaching\" program. This work\nwas also supported initially by the European Commission\nFP7 Contract ICT-257159 \\MACALO\". A.B acknowl-\nedges eSSENCE. 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B 92, 024427\n(2015)." }, { "title": "1106.4491v1.Tunable_Magnonic_Frequency_and_Damping_in__Co_Pd_8_Multilayers_with_Variable_Co_Layer_Thickness.pdf", "content": " 1 Tunable Magnonic Frequency and Damping in [Co/Pd] 8 \nMultilayers with Variable Co Layer Thickness \n \nS. Pal 1, B. Rana 1, O. Hellwig 2, T. Thomson, 2 and A.Barman 1,a) \n \n1Department of Material Sciences, S. N. Bose National Centre for Bas ic Sciences, Block \nJD, Sector III, Salt Lake, Kolkata 700 098, India \n2San Jose Research Center, Hitachi Global Storage Technologies, 3403 Yerba Bue na Rd., \nSan Jose, California 95135, USA \n \nAbstract \nWe report the experimental observation of collective p icosecond magnetization dynamics \nin [Co/Pd] 8 multilayers with perpendicular magnetic anisotropy. Th e precession \nfrequency shows large and systematic variation from ab out 5 GHz to about 90 GHz with \nthe decrease in the Co layer thickness from 1.0 nm to 0. 22 nm due to the linear increase \nin the perpendicular magnetic anisotropy. The damping coeff icient α is found to be \ninversely proportional to the Co layer thickness and a linea r relation between the \nperpendicular magnetic anisotropy and α is established. We discuss the possible reasons \nbehind the enhanced damping as the d-d hybridization at the interface and spin pumping. \nThese observations are significant for the application s of these materials in spintronics \nand magnonic crystals. \n \na) Electronic mail: abarman@bose.res.in 2 Magnetic multilayers (ML) with perpendicular magnetic anisotropy (PMA) have \nattracted attention due to their potential applications i n patterned magnetic media 1, spin \ntransfer torque magnetic random access memory (STT-MRAM ),2-3 and magnonic \ncrystals. 4 For applications in magnetic media and STT-MRAM devices, large precession \nfrequency associated with the large PMA and a reliable a nd low damping constant α are \ndesirable. On the other hand, for applications in magnoni c crystals, broadly tunable \nmagnonic frequencies and α with physical and material parameters are essential. All \npotential applications demand large and broadly tunable prec ession frequencies, small α \nvalues and a correlation between the PMA and α. PMA is believed to originate from the \ninterface anisotropy due to the broken symmetry and d-d hybridization 5 at the Co/Pd and \nCo/Pt interfaces. The competition between interface and volume anisotropies results in a \nvariation of PMA with the thickness of the Co layer ( t Co ) as has been reported in \ncontinuous 6 and patterned 7 magnetic multilayers. Consequently, a large variation in the \nprecession frequency in the picosecond magnetization dynam ics of these multilayers is \nexpected. On the other hand, it has been predicted recent ly 8 that there may be a linear \ncorrelation between PMA and damping based on existing the oretical works.9-10 The \nintrinsic Gilbert damping α and PMA both have their origins in the spin-orbit inte raction \nand are approximately proportional to ξ2/W, where ξ is the spin-orbit interaction energy \nand W is the d-band width. However, no clear correlation between the PMA and α has \nbeen observed so far. 8,11-12 Mizukami et al. 8 observed an increases in α with decrease in \ntCo but it was not inversely proportional to t Co . In this work we studied the picosecond \nmagnetization dynamics in [Co(t Co )/Pd(0.9 nm)] 8 multilayers with t Co varying between \n1.0 nm and 0.22 nm. We observed a systematic increase in the precession frequency and 3 α with the decrease in t Co . The extracted PMA, from the macrospin modeling of th e \nprecession frequency, shows a linear correlation with α. \n \nThe ML structures are deposited by dc magnetron sputtering.7 The base pressure \nof the deposition chamber was 2 × 10 -8 mbar and the deposition was performed at 3 \nmTorr of Ar pressure. A series of [Co(t Co )/Pd(0.9 nm)] 8 stacks were prepared using a \nTa(1.5 nm)/Pd(3.0 nm) seed layer, which ensured [111] textured ML with a mosaic \nspread of 7° full width at half maximum (FWHM). The Co la yer thickness is varied from \n0.13 nm to 1.0 nm in this experiment. The thickness of the C o and Pd layers were \nconfirmed by x-ray reflectivity, and the average roughness at the interface was found to \nbe about 0.05 nm. The magnetic hysteresis loops were meas ured by both polar magneto-\noptical Kerr effect (P-MOKE) and vibrating sample magne tometer (VSM) at room \ntemperature. Figure 1(a) shows the magnetic anisotropy fiel d (HK) increases \nsystematically with decrease in t Co and exhibits a maximum at 0.22 nm, beyond which it \ndecreases sharply. The saturation magnetization ( MS), on the other hand, decreases \nmonotonically with the decrease in t Co over the entire range. Figure 1(b) shows a typical \nsquare magnetic hysteresis loop for t Co = 0.36 nm. The ultrafast magnetization dynamics \nwas probed by time-resolved magneto-optical Kerr effect (TR-MOKE) measurements in \na two-color optical pump-probe setup. 11 The second harmonic ( λ = 400 nm) of a Ti-\nsapphire laser (pulse-width < 80 fs) was used to pump the sam ples, while the time-\ndelayed fundamental ( λ = 800 nm) laser beam was used to probe the dynamics by \nmeasuring the Kerr rotation by a balanced photo-diode detector, which completely \nisolates the Kerr rotation and the reflectivity signal s. The pump and the probe beams 4 were focused and spatially overlapped onto the sample surfac e by a microscope objective \nwith numerical aperture N. A. = 0.85 in a collinear geome try. A large magnetic field is \nfirst applied at a small angle (~ 10°) to the surface norm al of the sample to saturate its \nmagnetization. The magnetic field strength is then re duced to the bias field value ( H), \nwhich ensures that the magnetization remains saturated a long the bias field direction. The \npump beam was chopped at 2 kHz frequency and a phase sensitive detection of the Kerr \nrotation was used. Figure 1(c) shows typical time-resolve d reflectivity and the Kerr \nrotation data and the corresponding fast Fourier transfor m (FFT) spectra from the ML \nwith t Co = 0.5 nm at H = 1.72 kOe. \n \nThe precessional dynamics appears as an oscillatory sign al above the slowly \ndecaying part of the time-resolved Kerr rotation after a fast demagnetization within 500 \nfs, and a fast remagnetization within 10 ps. A bi-exponent ial background is subtracted \nbefore performing the FFT to find out the corresponding pow er spectra. The time-\nresolved reflectivity also shows an oscillation at abo ut 77 GHz originating from thermally \nexcited strain waves. The precessional frequency shows clear variation with the bias \nfields as opposed to the reflectivity signal. Figures 2(a) -(b) show the time-resolved Kerr \nrotations and the corresponding FFT spectra for six ML s amples with t Co = 1.0 nm, 0.75 \nnm, 0.5 nm, 0.36 nm, 0.28 nm and 0.22 nm at bias fields as shown i n the figure. For \nsamples with t Co < 0.22 nm, no clear precession is observed. All samples show a single \nprecession frequency due to the collective precession of th e whole stack as if it is a single \nmacrospin, which allows us to use the macrospin modeling of the Landau-Lifshitz-\nGilbert equation 13 to analyze the frequency and damping. The variation of prec ession 5 frequency with the bias field is plotted in Fig. 3(a) for various values of t Co . The \nprecession frequency increases sharply for t Co ≤ 0.75 nm indicating the sharp increase in \nthe PMA in this range. For the experimental geometry, a s shown in Fig. 3(b), the \nexpression for precession frequency is \n ( )\n\n\n− +\n+=S\nSeff MMKH f\n0202sin sin \n1 µ θβ\nαγµ \n [1], \nwhere γ is the gyromagnetic ratio, α is the damping coefficient, θ and β are the angles \nmade by the equilibrium magnetization ( M) and the bias field ( H) with x-axis and Keff is \nthe effective magnetic anisotropy. θ is obtained by minimizing the total energy of the \nsystem, while β is known from the experimental geometry. α is determined by fitting the \ntime-resolved magnetization with a damped sine function \n ( ) φ ωτ− =−\nt e M tMt\nsin ) 0 ( )( [2], \nwhere απτf21= , f is the experimentally obtained precession frequenc y and φ is the \ninitial phase of oscillation.14 The calculated frequencies are plotted as solid li nes in Fig. \n3(a) and are in good agreement with the experimenta l data. α is found to be inversely \nproportional to t Co over the entire range, as shown in Fig. 3(c). The extrapolation of the \nlinear fit to α vs. 1/t Co data upto 1/t Co = 0 gives α = 0.011, which is comparable with the \nvalue for bulk Cobalt (0.01). In Fig. 3(d) we plot Keff and MS as a function of t Co , as \nextracted from the macrospin modeling. Keff is also found to be inversely proportional to \ntCo similar to α, indicating a clear linear correlation between α and Keff . The values of MS \nobtained from the TR-MOKE measurements almost coinc ide with those obtained from \nthe VSM loops. We have also calculated the variatio n of MS with t Co and found that 6 consideration of slight magnetization of the Pd lay ers 15 (~ 210 emu/cc) gives good \nagreement between the experimental and the theoreti cal data. \n \nIn Fig. 4, we plot α as a function of Keff , which clearly shows that α is directly \nproportional to Keff with a slope of 4.33×10 -8 cc/erg. The values of α reported here are \nlower than the previously published works. 8,11,16 Below, we discuss the possible \nmechanisms responsible for the enhancement of α from its intrinsic value. One common \nchannel of dissipation of energy is by scattering o f the uniform precession with short \nwavelength magnons due to the presence of inhomogen eities including defects, which \nshould increase as the thickness reduces. However, it has been reported that for \nperpendicularly magnetized samples magnon scatterin g is less effective 17 and hence is \nruled out in materials with high PMA. The second po ssibility is spin pumping,18 caused \nby the spin current generated by the precession of magnetization of the Co layers entering \ninto the Pd layers and getting absorbed due to its small spin diffusion length, thereby \nenhancing α. This is usually accounted for by considering the variation of the relaxation \nfrequency G = αγ MS with 1/t Co.8, 18 The slope of G vs. 1/t Co obtained in our case is only \n3.2×10 8 rad/s as compared to the previously reported value s of about 13×10 8 rad/s for \nPt/Ni 80 Fe 20 /Pt and 34×10 8 rad/s in Pt/Co/Pt films. The third possibility is the decrease in \nbandwidth W of the Co atomic layer in contact with the Pd laye r due to the Co 3 d-Pd 5 d \nhybridization, 5 This is primarily an interface effect and effectiv ely increases both α and \nKeff , as discussed earlier. The observation of direct p roportionality between α and Keff \nstrongly indicates that this may be the primary mec hanism of enhancement of α in our \nexperiment. The fourth possibility is the roughness and alloying effects at the interface. 8 7 However, while interface roughness and alloying wou ld increase α, it would also \ndecrease Keff , which is opposite to our observation and hence th is possibility is also ruled \nout. Other possibilities such as dephasing of multi ple spin wave modes due to incoherent \nprecession of the constituent layers and formation of perpendicular standing waves are \nnegligible because of the observation of a collecti ve precession of all the layers in the \nstack and uniform excitation of the whole stack, re spectively. \n \nIn summary, we have studied the time-resolved magne tization dynamics in a \nseries of [Co(t Co )/Pd(0.9 nm)] 8 multilayers with variable Co layer thickness t Co . The \ndecrease in t Co increases the perpendicular magnetic anisotropy Keff , which effectively \nincreases the precession frequency and a broadly tu nable precession frequency between \nabout 5 GHz and 90 GHz is observed. The precession frequency was analyzed by \nmacrospin modeling of LLG equation and the saturati on magnetization MS, α, and Keff \nwere independently obtained from the dynamics. Both α and Keff are inversely \nproportional to t Co and hence are directly proportional. The enhanceme nt of α is possibly \ndue to spin pumping and the d-d hybridization at the Co/Pd interfaces as both effe cts are \ninversely proportional to t Co . However, only the later is directly correlated to the \nenhancement of Keff due to the decrease in bandwidth W of the Co atomic layer at the \ninterface, while the former has no contribution to Keff . Hence, we believe that in our case \nthe enhancement of α is caused primarily due to the d-d hybridization effect. The \nobservations of relatively low values of α associated with large Keff and their linear \ncorrelation are significant for their applications in the STT-MRAM devices and \nmagnonic crystals. 8 \nWe gratefully acknowledge the financial assistance from Department of Science \nand Technology, Govt. of India under the India-EU c ollaborative project \"DYNAMAG\" \n(grant number INT/EC/CMS (24/233552)) and the Nano Mission (grant number \nSR/NM/NS-09/2007). \n 9 References: \n1T. Thomson, G. Hu, and B. D. Terris, Phys. Rev. Let t. 96 , 257204 (2006); O. Hellwig, A. \nBerger, T. Thomson, E. Dobisz, H. Yang, Z. Bandic, D. Kercher, and E. E. Fullerton, \nAppl. Phys. Lett. 90 , 162516 (2007). \n2Y. Huai, F. Albert, P. Nguyen, M. Pakala, and T. Va let, Appl. Phys. Lett. 84 , 3118 \n(2004). \n3S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey , B. D. Terris, and E. E. Fullerton, \nNature Mater. 5, 210 (2006). \n4V. V. Kruglyak, S. O. Demokritov and D. Grundler, J . Phys. 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(a) Dependence of magnetic anisotropy field (HK) and the saturation \nmagnetization ( MS) on the Co layer thickness t Co in [Co/Pd] 8 multilayer films, as \nmeasured by static magnetometry. (b) A typical hyst eresis loop from the multilayer \nsample with t Co = 0.36 nm, obtained from polar MOKE measurement. (c) The time-\nresolved reflectivity and Kerr rotation signals and the corresponding FFT spectra from \nthe multilayer sample with t Co = 0.5 nm, showing the frequencies of phonon and th e \nprecession of magnetization, respectively. \nFIG. 2. (a) The time-resolved Kerr rotation data af ter a bi-exponential background \nsubtraction and (b) the corresponding FFT spectra a re shown for [Co/Pd] 8 films with \ndifferent Co layer thickness t Co . The solid lines in Fig. 2(a) correspond to the fi t with Eq. \n[2]. The applied bias fields are also shown in the figure. \nFIG. 3. (a) The bias field dependence of the experi mental precession frequencies \n(symbols) and the calculated frequencies (solid lin e) with Eq. [1] are shown for \nmultilayers with different Co layer thickness. (b) The geometry for the macrospin model \nis shown. (c) The damping coefficient α (symbols: experimental data, solid line: linear \nfit) is plotted as a function of 1/t Co . (d) The extracted perpendicular magnetic anisotro py \nKeff and the saturation magnetization MS (filled squares: from TR-MOKE, open circles: \nfrom static magnetometry) are plotted as a function of t Co . The dashed line shows the \ncalculated MS values, while the dotted line corresponds to the l inear fit to Keff vs t Co . \nFIG. 4. The damping coefficient α is plotted as a function of Keff (symbols) and the dotted \nline corresponds to the linear fit. 12 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Fig. 1 \n \nS. Pal et al. -4 -2 0 2 4-1.0 -0.5 0.0 0.5 1.0 \n M/M S\nH (kOe) tCo = 0.36 nm \n(b) -10 1 200 400 -4 -3 -2 -1 0 25 50 75 100 125 510 15 20 25 \n0 50 100 150 200 04812 \n0 25 50 75 100 02468 Kerr rotation (arb. unit) Time (ps) H = 1.72 kOe \n(c) Reflectivity (arb. unit) \n \nPower (arb. unit) \ntCo = 0.5 nm \n \n \n \n \nFrequency (GHz) 0.0 0.2 0.4 0.6 0.8 0510 15 20 25 30 \n200 300 400 500 600 700 800 HK (kOe) \ntCo (nm) (a) \n \nMS (emu/cc) \n-4 -2 0 2 4-1.0 -0.5 0.0 0.5 1.0 \n M/M S\nH (kOe) tCo = 0.36 nm \n(b) -10 1 200 400 -4 -3 -2 -1 0 25 50 75 100 125 510 15 20 25 \n0 50 100 150 200 04812 \n0 25 50 75 100 02468 Kerr rotation (arb. unit) Time (ps) H = 1.72 kOe \n(c) Reflectivity (arb. unit) \n \nPower (arb. unit) \ntCo = 0.5 nm \n \n \n \n \nFrequency (GHz) 0.0 0.2 0.4 0.6 0.8 0510 15 20 25 30 \n200 300 400 500 600 700 800 HK (kOe) \ntCo (nm) (a) \n \nMS (emu/cc) \n 13 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Fig. 2 \n \n \nS. Pal et al. 0 500 1000 1500 -1 01\n024\n0 500 1000 1500 -1 01\n01\n0 200 400 -1 01\n01\n0 50 100 150 -1 01\n02\n0 50 100 150 -1 01\n02\n0 50 100 150 -1 0\n0 50 100 150 036 \nH = 1.71 kOe \n 1.0nm \nH = 1.71 kOe 0.75nm \nH = 1.58 kOe 0.5nm \nH = 1.2 kOe \n 0.36nm H = 2 kOe \n 0.28nm Kerr rotation (arb. unit) \nPower (arb. unit) \nTime (ps) H = 2kOe \n \nFrequency (GHz) 0.22nm \n(a) (b) 14 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Fig. 3 \n \n \nS. Pal et al. 0.4 0.8 1.2 345678\n200 300 400 500 600 700 800 900 Keff (x 10 6 erg/cc)\ntCo (nm) \nMS (emu/cc) \n(d) \n 0 1 2 3 4 50.00 0.04 0.08 0.12 0.16 0.20 \n \nαααα\n1/t Co (nm -1 ) Experimental \n Linear Fit \n(c) \nθβMHKeff \n(b) xz\nyθβMHKeff \n(b) θβMHKeff θβMHKeff \n(b) xz\ny0.5 1.0 1.5 2.0 2.5 4812 1.0 1.5 2.0 2.5 4812 \n1.0 1.1 1.2 35 36 \n1.0 1.5 2.0 44 48 52 1.2 1.4 1.6 1.8 27 28 \n \n0.75nm \n \n \n1.0nm \n(a) \n \n 0.36nm \n \n 0.28nm \nH (kOe) Frequency (GHz) \n \n 0.5nm \n0.4 0.8 1.2 345678\n200 300 400 500 600 700 800 900 Keff (x 10 6 erg/cc)\ntCo (nm) \nMS (emu/cc) \n(d) \n 0 1 2 3 4 50.00 0.04 0.08 0.12 0.16 0.20 \n \nαααα\n1/t Co (nm -1 ) Experimental \n Linear Fit \n(c) \nθβMHKeff \n(b) xz\nyθβMHKeff \n(b) θβMHKeff θβMHKeff \n(b) xz\ny0.5 1.0 1.5 2.0 2.5 4812 1.0 1.5 2.0 2.5 4812 \n1.0 1.1 1.2 35 36 \n1.0 1.5 2.0 44 48 52 1.2 1.4 1.6 1.8 27 28 \n \n0.75nm \n \n \n1.0nm \n(a) \n \n 0.36nm \n \n 0.28nm \nH (kOe) Frequency (GHz) \n \n 0.5nm \n \n 15 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Fig. 4 \n \n \nS. Pal et al. \n3 4 5 6 7 80.04 0.08 0.12 0.16 0.20 αααα Expt. values \n Linear fit \n \nKeff ( x 10 6 erg/cc) " }, { "title": "1806.00376v2.Fluctuation_damping_of_isolated__oscillating_Bose_Einstein_condensates.pdf", "content": "Fluctuation-damping of isolated, oscillating Bose-Einstein condensates\nTim Lappe,1,\u0003Anna Posazhennikova,2,yand Johann Kroha1, 3,z\n1Physikalisches Institut and Bethe Center for Theoretical Physics,\nUniversit at Bonn, Nussallee 12, 53115 Bonn, Germany\n2Department of Physics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK\n3Center for Correlated Matter, Zhejiang University, Hangzhou, Zhejiang 310058, China\n(Dated: August 10, 2018)\nExperiments on the nonequilibrium dynamics of an isolated Bose-Einstein condensate (BEC)\nin a magnetic double-well trap exhibit a puzzling divergence: While some show dissipation-free\nJosephson oscillations, others \fnd strong damping. Such damping in isolated BECs cannot be\nunderstood on the level of the coherent Gross-Pitaevskii dynamics. Using the Keldysh functional-\nintegral formalism, we describe the time-dependent system dynamics by means of a multi-mode\nBEC coupled to \ructuations (single-particle excitations) beyond the Gross-Pitaevskii saddle point.\nWe \fnd that the Josephson oscillations excite an excess of \ructuations when the e\u000bective Josephson\nfrequency, ~!J, is in resonance with the e\u000bective \ructuation energy, ~ \"m, where both, ~ !Jand ~\"m,\nare strongly renormalized with respect to their noninteracting values. Evaluating and using the\nmodel parameters for the respective experiments describes quantitatively the presence or absence\nof damping.\nI. INTRODUCTION\nWhen a system of ultracold, condensed bosons is\ntrapped in a double-well potential with an initial pop-\nulation imbalance, it undergoes Josephson oscillations1\nbetween the wells and can, therefore, be referred to as\na Bose-Josephson junction (BJJ). Josephson oscillations\nwere observed in a number of experiments.2{5Since the\nexperimental systems are almost ideally separated from\nthe environment, a BJJ can serve as a prototype of a\nnonequilibrium closed quantum system. Because of the\nunitary time-evolution which prohibits the maximization\nof entropy, a closed quantum system cannot thermalize\nas a whole, once driven out of equilibrium. However,\nstrong damping of Josephson oscillations was observed\nin the experiments by LeBlanc et al. ,4whereas the ex-\nperiments by Albiez et al.2and by Spagnolli et al. clearly\ndisplayed undamped oscillations for extended periods of\ntime. Explaining this discrepancy and, thereby, giving\nguidelines for designing an experimental setup with or\nwithout damping and thermalization, is the aim of this\nwork.\nPreviously some of the present authors proposed the\ndynamical heat-bath generation (DBG) as a damping\nand thermalization mechanism:6,7For a su\u000eciently com-\nplex, isolated quantum system the Hilbert space dimen-\nsion is so large that only a small subset of the huge\namount of quantum numbers characterizing the system's\nstate vector can be determined in any given experiment.\nThis subset de\fnes a subspace of the total Hilbert space,\nreferred to as the \\subsystem\" S. Any measurement per-\nformed onSalone is partially destructive, in that the\nquantum numbers de\fning the Hilbert space of Sare\n\fxed (partial state collapse), but the remaining subspace\nof undetermined quantum numbers is traced out. This\nremaining subspace, R, becomes massively entangled8\nwith the states of the subsystem Svia the many-body\ndynamics and, hence, acts as a grand-canonical bath orreservoir. By the resulting, e\u000bectively grand canonical\ntime evolution of the subsystem S, it will naturally reach\na thermal state in the long-time limit,7if the system is\nergodic. Thus, the measurement process itself de\fnes\na division into subsystem and reservoir. For instance,\nwhen the population imbalance in a BJJ is measured,\nthe Bose-Einstein condensate (BEC) states comprise S,\nand all the many-body states involving incoherent ex-\ncitations outside the BEC comprise R. Note that this\nthermalization process is dynamical and is possible even\nwhen the bath states (for a BJJ, the incoherent excita-\ntions) are initially not occupied, hence the term dynami-\ncal bath generation. By contrast, the so-called eigenstate\nthermalization hypothesis9,10(ETH) requires the system\nto be near a many-body eigenstate of the total Hamilto-\nnian (microcanonical ensemble), i.e., it is stationary by\nconstruction. See Ref. [6] for a detailed discussion.\nThe DBG mechanism was corroborated for a BJJ with\narbitrary system parameters, where it was shown that\nincoherent excitations are e\u000eciently generated out of the\noscillating BEC due to a parametric resonance.7The\ncomplex thermalization dynamics of a BJJ involving sev-\neral time scales has been analyzed in detail in Refs. [6,7].\nIn particular, the thermalization time \u001cthis necessarily\nmuch larger than the BJJ oscillation period, because (1)\nthe incoherent \ructuations are created by the Joseph-\nson oscillations themselves and (2) because of the quasi-\nhydrodynamic long-time dynamics.7\nIn the present work we examine this damping\nmechanism for realistic experimental parameters and\nspeci\fc traps. Previous studies within the two-\nmode approximation1,4,11showed signi\fcant, interaction-\ninduced renormalizations of the Josephson frequency, ~ !J,\nbut did not explain the observed oscillation damping.4\nA multimode expansion of the Gross-Pitaevskii equation\n(GPE) in terms of the complete basis of single-particle\ntrap eigenmodes can describe the coherent part of the\ndynamics in principle exactly. However, the dynamicalarXiv:1806.00376v2 [cond-mat.quant-gas] 10 Aug 20182\nexcitation of higher trap levels also involves the creation\nofincoherent \ructuations which are not captured by the\nGPE saddlepoint dynamics. The excitation of higher\ntrap modes and the concatenated creation of incoher-\nent \ructuations is crucial for damping in realistic sys-\ntems. These \ructuations are captured by the systematic\nexpansion about the GPE saddlepoint (see Sec. II B),\ninvolving BEC as well as \ructuation Green's functions.\nWe \fnd that e\u000ecient coupling to higher trap modes oc-\ncurs if ~!Jis in resonance with the excitation energy of\none of the trap levels, ~ !J\u0019~\"m, where ~!J, as well as ~ \"m,\nare strongly renormalized and broadened by their mutual\ncoupling and by the interactions. Conversely, in the o\u000b-\nresonant regime, the Josephson oscillations remain un-\ndamped over an extended period of time. Our quantita-\ntive calculations reveal that the experimental parameters\nof LeBlanc et al.4are in the strongly damped regime and\nthose of Albiez et al.2in the undamped regime, in agree-\nment with the experimental \fndings. This reconciles the\napparent discrepancy between these two classes of exper-\niments and supports the validity of the DBG mechanism\nin Bose-Josephson junctions.\nThe article is organized as follows. In Sec. II we de-\nscribe the many-body action used to model the system\nand its representation in the trap eigenbasis. We develop\nthe nonequilibrium temporal dynamics by means of the\nKeldysh path integral. Sec. III contains the numerical\nanalysis: the resonance e\u000bect responsible for the damp-\ning, and a detailed application to the two exemplary ex-\nperiments, Refs. [2] and [4], respectively. This is followed\nby a discussion and concluding remarks in Sec. IV.\nII. FORMALISM\nThe model for an ultracold gas in a double-well trap\npotentialVext(r) with multiple single-particle levels is\nde\fned using the functional-integral formalism. It al-\nlows for a convenient distinction between the condensate\namplitudes in each level, de\fned by the time-dependent\nGross-Pitaevskii saddle point, and the non-condensate\nexcitations. The nonequilibrium dynamics will be de-\nscribed by the functional integral on the Keldysh time\ncontour.\nA. Multi-mode model\nThe action Sfor a trapped, atomic Bose gas with a\ncontact interaction reads in terms of the bosonic \felds\n (r;t); \u0003(r;t),\nS[ ; \u0003] =Z\nd3rdt\u0002\n \u0003(r;t)G\u00001\n0(r;t) (r;t)\n\u0000~g\n2 \u0003(r;t) \u0003(r;t) (r;t) (r;t)\u0003\n; (1)\nwhere the coupling parameter ~ g= 4\u0019~2as=mis propor-\ntional to the s-wave scattering length as,12,13and theinverse free Green function is\nG\u00001\n0(r;t) = i@t\u0000\u0012\n\u0000~2r2\n2m+Vext(r)\u0013\n: (2)\nThe spatial dependence of the \feld (r;t) may be re-\nsolved into the complete, orthonormal basis of single-\nparticle eigenfunctions f'\u0000(r);'+(r);'3(r);'4(r); :::g\nof the trap,7\n (r;t) = +(r;t) + \u0000(r;t) +MX\nm=3 m(r;t)\n='+(r)\u001e+(t) +'\u0000(r)\u001e\u0000(t) +MX\nm=3'm(r)\u001em(t);\n(3)\nwith time-dependent amplitudes \u001em(t) andMthe\nnumber of modes taken into account. The 'i(r)\nare the solutions of the stationary Schr odinger equa-\ntion with the potential Vext(r), with eigenfrequencies\nf\"\u0000; \"+; \"3; \"4;:::g. The wavefunctions '\u0000(r) and\n'+(r) are the two lowest-lying eigenfunctions of Vext(r)\nextending over both wells, with odd (-) and even (+)\nparity, respectively. In view of the anticipated dynamics\nwith di\u000berent occupation numbers in the two wells, it is\nuseful to de\fne the symmetric and antisymmetric super-\npositions'1;2(r) = ['\u0000(r)\u0006'+(r)]=p\n2, since they are\nlocalized in the left or right well, respectively. With the\nexpansion (3) the action takes the form S=S0+Sint,\nwith the noninteracting part,\nS0=Z\ndt(MX\ni=1[\u001e\u0003\ni(i@t\u0000\"i)\u001ei]\u0000J(\u001e\u0003\n1\u001e2+\u001e\u0003\n2\u001e1))\n;\n(4)\nand the interacting part\nSint=\u00001\n2Z\ndtMX\nijkl=1Uijkl\u001e\u0003\ni(t)\u001e\u0003\nj(t)\u001ek(t)\u001el(t);(5)\nwhere the\u001e1;2are the symmetric and antisymmetric su-\nperpositions of the time-dependent \felds \u001e\u0006(t). In this\nmode representation, the spatial dependence of the Bose\n\feld (r;t) is absorbed into the overlap integrals \"i,J,\nandUijkl, which are given by\n\"i=Z\nd3r'\u0003\ni(r)\u0012\n\u0000~2r2\n2m+Vext(r)\u0013\n'i(r); (6)\n\"1=\"2=1\n2(\"\u0000+\"+); (7)\nJ=Z\nd3r'\u0003\n1(r)\u0012\n\u0000~2r2\n2m+Vext(r)\u0013\n'2(r)\n=1\n2(\"\u0000\u0000\"+); (8)\nUijkl=~gZ\nd3r'\u0003\ni(r)'\u0003\nj(r)'k(r)'l(r); (9)3\nwhere the bound-state functions 'i(r) may be chosen\nreal. Note that a bare Josephson coupling Jexists only\nbetween the two lowest modes '1(r),'2(r), localized in\nthe left or right well, while the modes with i\u00153 are trap\neigenmodes and extended over the entire trap. Without\nloss of generality we may choose the zero of energy as\n\"1=\"2= 0.\nForM!1 the representation Eqs. (4){(9) in terms of\nthe single-particle trap eigenmodes is exact. Numerically,\nthe decomposition in Eq. (3) is analogous to a Galerkin\nmethod. Replacing the space-dependence by summations\nover eigenfunctions leads to a signi\fcant simpli\fcation of\nthe numerical initial-value problem when truncating the\ndecomposition at a \fnite value of M. In this work we will\ntakeM= 4;6, depending on the form of the external\npotentialVext(r), see section III.\nB. Nonequilibrium e\u000bective action\nIn this subsection, we are going to present the formal\nderivation of the equations of motion in the Bogoliubov-\nHartree-Fock (BHF) approximation that describe the\ncondensate and its exchange with a cloud of noncon-\ndensed particles.\nThe Keldysh technique14in path-integral\nformulation15is a particularly elegant tool for the\nconstruction of self-consistent approximations via the\ne\u000bective action, where both the condensate amplitudes\n\bi=h\u001eiiand the \ructuations above the condensate,\n\u000e\u001ei, are treated on an equal footing.\nFor the general derivation of the one-particle irre-\nducible (1PI) e\u000bective action, we will suppress the \feld\nindices and instead work with a time-dependent \feld \u001e\nwhich can in principle carry arbitrary quantum numbers.\nThe bosonic \felds should now be separated into \felds on\nthe forward branch C1of the Keldysh contour and \felds\non the backward branch C2, such that we can express the\naction as\nSK[\u001eC1;\u001e\u0003\nC1;\u001eC2;\u001e\u0003\nC2] =S[\u001eC1;\u001e\u0003\nC1]\u0000S[\u001eC2;\u001e\u0003\nC2]:\n(10)\nFrom this action we obtain SK[\u001ec;\u001e\u0003\nc;\u001eq;\u001e\u0003\nq] by perform-\ning the Keldysh rotation according to\n\u001eC1=1p\n2(\u001ec+\u001eq); \u001eC2=1p\n2(\u001ec\u0000\u001eq); (11)\nwherecstands for \"classical\" and qfor \"quantum\". This\nnomenclature stems from the fact that neglecting \ruc-\ntuations, the \feld \u001ecwill obey the classical equations of\nmotion which follow from the corresponding classical ac-\ntion. The \\quantum\" \feld \u001eqis the so-called \\response\"\n\feld describing all \ructuations (both classical and quan-\ntum). In the simplest classical limit, it essentially corre-\nsponds to a description of Gaussian white noise with zero\nmean through the characteristic functional known from\nprobability theory.De\fning complex \feld spinors \u001e= (\u001e;\u001e\u0003)Tand ex-\nternal sources j= (j;j\u0003)T, the partition function will\nbe\nZ[jc;jq] =Z\nD[\u001ec;\u001eq]eiSK[\u001ec;\u001eq]eiR\ndt(jy\nq\u001ec+jy\nc\u001eq);\n(12)\nwhere we have also introduced Keldysh classical and\nquantum components for the external sources. Taking\nthe logarithm of Z, we \fnd the cumulant-generating func-\ntional\nW[jc;jq] =\u0000i lnZ[jc;jq]: (13)\nDi\u000berentiation with respect to jgives the expectation\nvalue of the \feld in the presence of external sources,\n\bc;q=h\u001ec;qi=\u000eW\n\u000ej\u0003q;c: (14)\nand we de\fne \b= (\b;\b\u0003)T. By a Legendre transform16\nto these new variables, we arrive at the 1PI e\u000bective ac-\ntion\n\u0000[\bc;\bq] =W[jc;jq]\u0000Z\ndt(jy\nq\bc+jy\nc\bq);(15)\nwhich will be the main tool of our analysis, since it allows\nfor a rigorous derivation of self-consistent perturbation\ntheory. To this end, we \fnally decompose the \feld into\na \fnite average plus \ructuations according to\n\u001ec;q= \bc;q+\u000e\u001ec;q: (16)\nPlugging this into Eq. (15), and using (12), the source\nterms coupled to the averages \bvanish, and we are left\nwith\nei\u0000[\bc;\bq]=Z\nD[\u000e\u001ec;\u000e\u001eq] exp\b\niSK[\u001ec;\u001eq]\t\n\u0002exp(\n\u0000iZ\ndt \u0012\u000e\u0000\n\u000e\bc\u0013T\n\u000e\u001ec+\u0012\u000e\u0000\n\u000e\bq\u0013T\n\u000e\u001eq!)\n:\n(17)\nThis path integral supplements the \feld averages by \ruc-\ntuation terms in a similar way to a Ginzburg-Landau ap-\nproach. From it, one can easily generate the established\nBogoliubov-Hartree-Fock (BHF) approximation by keep-\ning only the quadratic \ructuations. In Appendix A, its\nconserving properties, that is, conservation of energy and\nparticle number, are proved explicitly for the two-mode\nmodel. As is well-known, the BHF approximation is not\ngapless and violates the Hugenholtz-Pines theorem.\nVariation of the e\u000bective action with respect to \b\u0003\niq\nresults in a modi\fed Gross-Pitaevskii equation (GPE),\nwhich gives the evolution of the classical average \b ic(t),\ndescribing the condensate,\n0 =\u000e\u0000\n\u000e\b\u0003\niq; (18)4\nwhile the average of the quantum component has to van-\nish identically,\n\biq(t) = 0: (19)\nSince we would like to investigate the occupation dy-\nnamics including the \ructuations, we have to consider\nthe Keldysh Green functions as well, which we write as\nGK\nij(t;t0) =\u0012\nGij(t;t0)gij(t;t0)\n\u0000g\u0003\nij(t;t0)\u0000G\u0003\nij(t;t0)\u0013\n=\u0000i\u0012\nh\u000e\u001eic(t)\u000e\u001e\u0003\njc(t0)i h\u000e\u001eic(t)\u000e\u001ejc(t0)i\nh\u000e\u001e\u0003\nic(t)\u000e\u001e\u0003\njc(t0)i h\u000e\u001e\u0003\nic(t)\u000e\u001ejc(t0)i\u0013\n;(20)\nwhere for the matrix elements we drop the Keldysh super-\nscript and explicitly keep the anomalous contributions,\ndesignated by a lowercase g. Since \b iq(t) = 0, in the\nfollowing we will simply write \b ic(t) = \bi. With these\nde\fnitions, in its most general form Eq. (18) will be given\nby\n0 = (i\u000eij@t\u0000hij) \bj\u0000Uijkl\n2[ \b\u0003\nj\bk\bl\n+ i\b\u0003\njgkl(t;t) + i\bkGjl(t;t) + i\blGjk(t;t) ];(21)\nwherehijrepresents the coe\u000ecients from the quadratic\npart of the action, and repeated indices are summed\nover. Without the contributions from the \ructuations,\nthis would be the standard GPE.\nIn order to determine the \ructuation Green functions,\nwe have to solve the respective Dyson equations,\nZ\nd\u0016t\u000e(t\u0000\u0016t)\u0010\n[GR\n0]\u00001\nij(t)\u0000\u0006R\nij(t)\u0011\nGK\njk(\u0016t;t0) = 0;\nZ\nd\u0016t\u000e(\u0016t\u0000t0)GK\nij(t;\u0016t)\u0010\n[GA\n0]\u00001\njk(t0)\u0000\u0006A\njk(t0)\u0011\n= 0;(22)\nself-consistently alongside Eq. (21). The inverse Green\nfunctions and self-energies can be obtained from the sec-\nond derivatives of the e\u000bective action,\n[GR\n0]\u00001\nij(t;t0)\u0000\u0006R\nij(t;t0) =0\n@\u000e2\u0000\n\u000e\b\u0003\niq(t)\u000e\bjc(t0)\u000e2\u0000\n\u000e\b\u0003\niq(t)\u000e\b\u0003\njc(t0)\n\u000e2\u0000\n\u000e\biq(t)\u000e\bjc(t0)\u000e2\u0000\n\u000e\biq(t)\u000e\b\u0003\njc(t0)1\nA:\nAt Hartree-Fock level, the self-energies are local in time,\nwhich leads to the temporal delta functions in (22). The\ninverse Green functions are\n[GR\n0]\u00001\nij(t) =\u0012\ni\u000eij@t\u0000hij 0\n0\u0000i\u000eij@t\u0000hij\u0013\n; (23)\n[GA\n0]\u00001\nij(t) = \n\u0000i\u000eij \u0000@t\u0000hij 0\n0 i\u000eij \u0000@t\u0000hij!\n; (24)\nand the retarded and advanced self-energies read\n\u0006R\nij(t) =\u0006A\nij(t) =\u0012\n\u0006ij(t)\u001bij(t)\n\u001b\u0003\nij(t) \u0006ij(t)\u0013\n; (25)\nwhere\n\u0006ij(t) =Uijkl[\b\u0003\nk(t)\bl(t) + iGkl(t;t)]; (26)\n\u001bij(t) =Uijkl\n2[\bk(t)\bl(t) + igkl(t;t)]: (27)This set of self-consistent BHF equations for the \feld\naverages and the Keldysh components of the Green func-\ntions, Eqs. (21) and (22), can be solved in the equal-\ntime limit by combining the retarded and advanced\nequations.17Speci\fcally, the upper left and right com-\nponents of the retarded Bogoliubov-matrix equation in\n(22) are\n0 = (i\u000eij@t\u0000hij\u0000\u0006ij(t))Gjk(t;t0) +\u001bij(t)g\u0003\njk(t;t0);\n0 = (i\u000eij@t\u0000hij\u0000\u0006ij(t))gjk(t;t0) +\u001bij(t)G\u0003\njk(t;t0);\n(28)\nrespectively. Accordingly, the upper left and right com-\nponents of the advanced equation in (22) are\n0 = (\u0000i\u000ejk@t0\u0000hjk\u0000\u0006jk(t0))Gij(t;t0)\u0000\u001b\u0003\njk(t0)gij(t;t0);\n0 = (i\u000ejk@t0\u0000hjk\u0000\u0006jk(t0))gij(t;t0)\u0000\u001bjk(t)Gij(t;t0):\n(29)\nNote the di\u000bering time derivatives and arguments of the\nself-energies. By subtracting the \frst of Eqs. (29) from\nthe \frst of Eqs. (28) and taking the equal-time limit, one\n\fnds equations for the Gij(t;t). Similarly, by adding the\nsecond of Eqs. (28) to the second of Eqs. (29), in the\nequal-time limit one obtains equations for the anomalous\nGreen functions gij(t;t).\nFurther details of the derivation are exempli\fed in Ap-\npendix A for the two-mode case.\nIII. APPLICATION TO EXPERIMENTS\nThis section is divided into three parts. The \frst part\nis dedicated to the quantitative calculation of the trap\nand interaction parameters for the experiments of Albiez\net al.2and LeBlanc et al.4, respectively. In the second\npart, by scanning through realistic trap-parameter val-\nues, we demonstrate numerically that e\u000ecient damping\ncan occur only if the resonance condition for the Joseph-\nson frequency ~ !Jand the broadened energy levels of the\nincoherent excitations ~ \"mis ful\flled. The third and \f-\nnal part contains our numerical results for experiments\nwith undamped2and strongly damped4Josephson oscil-\nlations, respectively.\nA. Realistic trap parameters and level\nrenormalization\nWe quantitatively analyze two classes experiments:\nthose of Albiez et al.2as an exemplary observation of\nundamped Josephson oscillations, hereafter referred to\nas experiment (A), and those by LeBlanc et al.4where\nstrong damping occurred, and which we will refer to\nas experiment (B). Both experiments were performed in\ndouble-well potentials, and the population imbalance z(t)\nbetween the two wells was traced as a function of time.5\nas= 000.1˜ε(in units of 103J)\nas= 98a0˜ωJ0.20.40.6\nas= 000.10.20.30.4˜ε(in units of 103J)\nas= 98a0˜ωJ46810\nFigure 1: Bare (left) vs. mean-\feld-shifted (right) single-\nparticle energies of the trapping potentials VA(r) from Ref. [2]\n(upper panels) and VB(r) from Ref. [4] (lower panels). The\n\frst ten levels are shown. Thick lines indicate nearly degener-\nate state pairs. The right panels show the initial renormaliza-\ntion of the levels due to the interaction ( as= 98a0, witha0\nthe Bohr radius). The solid (colored) lines in the right panels\nare the ones used for the time-dependent numerical calcula-\ntions (see text). The renormalized Josephson frequency ~ !J,\nas extracted from the time evolution of z(t), is also shown for\neach case.\nWhile the experiments (A) are well described by an ef-\nfective nonpolynomial Schr odinger equation,18in the ex-\nperiments (B) the Fourier transform of z(t) exhibits two\nor three frequencies in addition to damping,4indicating\ncontributions from more than two modes.\nIn order to reduce the numerical e\u000bort for the subse-\nquent, time-dependent computations, one should select\nthose levels which participate signi\fcantly in the dynam-\nics. To this end, it is important to realize that both,JUJ0U0N\nAlbiez et al.2\u00001:00:40\u00000:002 0:0001 1150\nLeBlanc et al.4\u00001:01:73\u00000:006 0:0001 4500\nTable I: Hamiltonian matrix elements involving the only left-\nand right-localized modes, and total particle number Nfor\nthe experiments (A) of Albiez et al. and (B) of LeBlanc et\nal., respectively.\nthe single-particle level energies and the Josephson fre-\nquency, are strongly renormalized by the interactions.\nWe calculate the level renormalizations within the BHF\napproximation at the initial time t= 0. The bare ( \")\nand the renormalized (~ \") single-particle levels are shown\nin Fig. 1 for the experiments (A) and (B), respectively, for\nthe example that all particles are initially condensed in\nthe left potential well. It is seen that the interactions even\nchange the sequence of the trap levels. In particular, the\ntwo low-lying left- or right-localized levels ( \u000b= 1;2) are\nshifted upward above the other renormalized levels. The\nreason is that the two lowest-lying single-particle orbitals\nare macroscopically occupied by the BEC atoms with\ncondensate population number N\u000b, so that the energy\nfor adding one additional particle in these levels is renor-\nmalized on the order of ~ \"\u000b\u0019\"\u000b+N\u000bU, with additional\ncontributions from the inter-level condensate interactions\nU0;J0. Similarly, the excited single-particle levels are\nrenormalized predominantly by their interaction with the\ncondensates as ~ \"n\u0019\"n+NKn,n= 3;4;5;:::, where\nN=N1+N2is the total condensate occupation number\nandKnsubstantially smaller than U. The interaction-\ninduced re-ordering of levels shown in Fig. 1 remains valid\nas long as the ground-state occupations N\u000b,\u000b= 1;2, are\nsubstantial. As a side result, this level re-ordering justi-\n\fes the frequently used Bogoliubov approximation,6,7,19\nwhere non-condensate amplitudes in the left- or right-\nlocalized ground modes \u000b= 1;2 are neglected, because\nsuch \ructuations are energetically suppressed. Our cal-\nculations show that di\u000berent initial BEC population im-\nbalancesz(0) do not signi\fcantly alter the renormalized\nlevel schemes. In particular, we \fnd that this remains\ntrue for the time evolution in both experiments, (A) and\n(B). Therefore, the initial level renormalization shown in\nFig. 1 may be used for selecting the relevant levels at all\ntimes during the evolution, see subsection III C.\n\"3\"4\"5\"6U3; U4U5; U6K3; K4K5; K6R\u000b3; R\u000b4R\u000b5; R\u000b6U0\n34U0\n56U0\n35;36;45;46\nAlbiez et al.2131:0133.0 { { 0.075 { 0.055 {\u00060:063 { 0.075 { {\nLeBlanc et al.4189:0191.0 381.0 383.0 0.56 0.48 0.33 0.25\u00060:43\u00060:19 0.56 0.48 0.29\nTable II: Model parameters involving at least one excited trap mode, n\u00153. Note that \"4;6=\"3;5+ 2jJj.\nThe trap potentials of the two experiments (A) and (B)\nhave di\u000berent shapes, VA(r) andVB(r), respectively, asgiven in Appendix B. In order to develop a quantitative\ndescription of the dynamics, we solve the (noninteract-6\nE3 = 200\nE3 = 300\nE3 = 500\nE3 = 1000\nE3 = 200\nE3 = 300\nE3 = 500\nE3 = 1000\nFigure 2: Population imbalance z(t) (left panels) and total fraction of \ructuations \u000eN(t)=N(right panels) for M= 3 modes\nwithN= 5000 particles, z(0) = 0:2, initial BEC phase di\u000berence \u0001 \u0012(0) = 0 and N3(0) = 0. In all plots, the parameters are\nU= 1:0,U0= 0,J0=\u00000:05,U3= 0:5K3= 0:1,R\u000b3= 0:01, in units ofjJj.\"3is varied and has values \"3= 200;300;500;1000\n(top to bottom), as indicated.\ning) Schr odinger equation with the potentials VA(r) and\nVB(r) for the \frst ten single-particle trap wave functions\n'i(r) and compute the matrix elements of the full Hamil-\ntonian in the trap eigenbasis according to Eqs. (6)-(9).\nThe general interaction matrix elements Uijklcan be clas-\nsi\fed into intra-level interactions ( U\u000b\u000b\u000b\u000b\u0011U,Unnnn\u0011\nUn), density-density interactions between di\u000berent lev-\nels (U\u000b\f\f\u000b\u0011U0,Unmmn\u0011U0\nnm,U\u000bmm\u000b\u0011Km) and\ninteraction-induced transitions between di\u000berent levels\n(U\u000b\u000b\u000b\f\u0011J0,U\u000b\u000b\u000bm\u0011R\u000bm). Here\u000b;\f= 1;2,\u000b6=\f,\ndenote the ground modes '1;2(r) localized in the left or\nright potential well and n; m = 3;4;5; ::: the higher\ntrap levels. See Appendix B for details of the de\fni-\ntions and calculations. The parameter values computed\nfor a87Rb gas (scattering length as\u001998a0)12in the\nexperimental setups (A) and (B) are listed in Tabs. I\nand II. The bare Josephson coupling Jturns out to be\napproximately equal for both experiments, (A) and (B),\nJ\u0019\u00002\u0019\u00020:16 Hz (see Eq. (8) and Appendix B). All\nenergies in this paper are given in units of jJj.\nB. Resonant single-particle excitations\nIn this subsection, we establish that incoherent excita-\ntions (\ructuations) out of the condensate are e\u000eciently\ncreated, and therefore that damping occurs, if the fre-\nquency of the Josephson oscillations is in resonance withone of the renormalized single-particle levels. The in-\nteractions not only renormalize the single-particle levels,\nbut also the Josephson frequency, !J!~!J. Within the\ntwo-mode model in the linear regime of Josephson oscil-\nlations, it is given by1\n~!J= 2Jp\n1 +NU= 2J : (30)\nIn the general case of multiple modes and inter-mode\ninteractions it is, however, not possible to give an ana-\nlytical expression. Therefore, we numerically evolve the\ninteracting system in time for a large number of oscilla-\ntions, using the Keldysh equation-of-motion method pre-\nsented in section II, and extract the renormalized Joseph-\nson frequency ~ !Jfrom the Fourier spectrum of the time-\ndependent BEC population imbalance z(t). To establish\nthe resonance condition for realistic experimental setups,\nwe consider an exemplary system of three modes with\ntypical parameter values for the experiments (A), (B),\ngiven in the caption of Fig. 2, and vary the bare en-\nergy\"3of the third mode above the two lowest modes,\nwhose bare energy we set to \"1=\"2= 0. The cor-\nresponding time traces of the BEC population imbal-\nancez(t) = [N1(t)\u0000N2(t)]=Nand of the total frac-\ntion of noncondensed particles \u000eN(t)=N(\ructuations)\nare shown in Fig. 2. For small and for large level spacings,\n\"3= 200;1000, essentially no \ructuations are generated\n(right panels), and the Josephson oscillations remain un-\ndamped (left panels). However, for intermediate level7\nspacings,\"3= 300, and more so for \"3= 500, we observe\ne\u000ecient excitation of \ructuations at a characteristic time\n\u001cc, and at the same time scale the oscillations become de-\npleted and irregular, but remain reproducible.19Inelas-\ntic interactions between these incoherent excitations (not\ntaken into account at the BHF level of approximation)\nwill lead to rapid damping and eventual thermalization of\nthe Josephson oscillations, as shown in Ref. [7]. To deter-\nmine the Josephson frequency of the interacting system,\n~!, we compute the magnitude spectrum of z(t) by fast\nFourier transform (FFT) of the time traces up to the\ntime\u001cc, i.e., before \ructuations are e\u000eciently generated,\nas shown in Fig. 3. ~ !Jis given by the position of the\npronounced peak in these spectra.\nTo analyze now the \ructuation excitation mechanism\nquantitatively, the renormalized level ~ \"3as well as the\ntime traces z(t),\u000eN(t)=Nare computed for a large num-\nber of bare \"3values, and the time-averaged fraction of\n\ructuations,h\u000eN=Ni, is plotted as a function of the ratio\n~\"3=~!J, see Fig. 4. The \fgure clearly exhibits resonant be-\nhavior: The \ructuation fraction reaches a broad but pro-\nnounced maximum when the renormalized level ~ \"3and\nthe renormalized Josephson frequency ~ !Jcoincide.\nWe note that this resonant \ructuation-creation mech-\nanism is closely related to, but more general than the\ndynamical mean-\feld instabilities reported in Refs. [20{\n22]. It leads to the highly nonlinear, abrupt creation cre-\nation of \ructuations19at the characteristic time \u001ccseen,\ne.g., in Fig. 2, panels of the second row ( \"3= 300). The\nfrequency ~!Jacts like the frequency of an external driv-\ning \feld for the subsystem of non-condensate excitations\n(\ructuations). However, in the present Josephson system\nthe driving is an intrinsic e\u000bect, not an external one as\nin Ref. [20]. Also, our approach is not restricted to the\ntwo-mode scenario,21but can be extended to any number\nof modes involved. We have tested this for various sets\nFigure 3: Magnitude spectrum (absolute value of the Fourier\ntransform) of the population imbalance z(t) for the parame-\nters given in Fig. 4 for \"3= 200 (solid line) and \"3= 300\n(dashed line), N= 5000. The time interval of the FFT was\ntruncated at the onset of the \ructuation regime: for \"3= 300,\nthe time trace was cut at Jt= 0:04, whereas for \"3= 200,\nthe entire displayed interval was used.\nE3 = 200E3 = 300E3 = 300E3 = 500\nE3 = 1000Figure 4: Time-averaged fraction of \ructuations as a func-\ntion of the ratio of the e\u000bective single-particle energy ~ \"3and\nthe e\u000bective Josephson frequency ~ !Jfor two di\u000berent parti-\ncle numbers (the small, gray squares are for N= 1000, the\nsmall, black circles are for N= 5000, corresponding to exper-\niments (A) and (B), respectively). The interaction parame-\nters areU= 1:0,U0= 0,J0=\u00000:05,U3= 0:5,K3= 0:1,\nR\u000b3= 0:01, and the initial conditions: initial population im-\nbalancez(0) = 0:2, initial phase di\u000berence \u0001 \u0012(0) = 0,M= 3,\nandN3(0) = 0. The renormalized Josephson frequency is ex-\ntracted from the Fourier transforms of z(t): ~!J\u00191571 for\nN= 5000, and ~ !J\u0019315 forN= 1000. The time average\nof the \ructuations was taken over the displayed time interval\nforN= 5000, whereas for N= 1000 an interval of 5 times\nthat size was used. The thick dots represent the results for\nthe time traces of Fig. 2 for the corresponding values of \"3,\nas indicated in the \fgures.\nof parameter values and system sizes.\nIncoherent excitations will lead to rapid damping and\neventual thermalization of the system.7In order to avoid\ndamping and to stabilize coherent motion, one needs to\ntune the away from the resonance. One way of achieving\nthis is to change the particle number N: While the exci-\ntation energies of the not macroscopically occupied levels,\n~\"n,n\u00153, are not strongly a\u000bected by N, ~!Jdepends\nsensitively on N[c.f. Eq.(30)], so that the resonance con-\ndition ~!J\u0019~\"n(c.f. Fig. 4) may easily be avoided.\nC. Comparison with experiments\nWe now examine how the experiments (A) and (B) \ft\ninto the resonant-\ructuation-creation scenario described\nabove.\nIn Fig. 5 we give the results of our calculations for the\nexperimental setup (A) of Albiez et al.2with the four rel-\nNz(0)N1N2N3N4N5N6\nAlbiez et al.21150 0.290 742 408 00{{\nLeBlanc et al.44500 0.116 2436 1914 757500\nTable III: Occupation numbers at time t= 0 used for the\nnumerical calculations.8\n−0.4−0.20.00.20.4z\n0.00 0.02 0.04 0.06 0.08\nJt0.00.10.20.30.40.50.6δN/N\nFigure 5: Population imbalance z(t) and relative fraction of\n\ructuations \u000eN(t)=Nfor the experiments (A).2The experi-\nmental data points (black dots) are taken from the reference.\nThe parameters and initial conditions for the calculations are\nlisted in Tabs. I { III.\nevant modes shown in Fig. 1 in direct comparison with\nthe experimental data points. Note that there is no \ft-\nting of parameters involved. We see that the agreement\nwith the experiment is very good regarding both the fre-\nquency and the amplitude of the Josephson oscillations.\nIn particular, no damping is observed in the experiment\nas well as in the calculation. The fraction of \ructuations\nremains below 10 %, indicating that this experimental\nsetup is away from the resonance discussed in Fig. 4.\nIn Fig. 6 we display the corresponding calculations for\nthe experiment (B).4We took six relevant modes into\naccount in our calculations, as explained in the discus-\nsion of Fig. 1. For this experiment we assume a small\ninitial condensate occupation of the modes m= 3;4, as\nlisted in Tab. III, because of the small excitation energy\nof these modes (see Fig. 1) with regard to the larger inter-\naction parameters of experiment (B). Here the agreement\nwith experiment is quantitatively not as good as for the\nexperiment (A).2However, the theoretical calculation re-\nproduces the strong amplitude reduction of z(t) after a\nshort time of only t\u00190:004Jin agreement with exper-\niment. At the same time, the calculation shows a fast\nand e\u000ecient excitation of \ructuations, which set in at a\ncharacteristic time scale7of\u001cc\u00190:0013Jand reach a\nmaximum amplitude of about \u000eNmax=N\u00190:5 near the\ntimet\u00190:0035J. This indicates that this experimental\nsetup is in the resonant regime. Importantly, we \fnd that\nthe e\u000ecient creation of \ructuations for the parameters of\nexperiment (B) is robust, independent of the small con-\ndensate occupation of the modes with m= 3;4 as well\nas the precise value of N.\nThe reason for the reduced quantitative agreement\nwith experiment can be understood from the behavior of\n−0.20.00.2z\n0.0000 0.0025 0.0050\nJt0.00.20.40.6δN/NFigure 6: Population imbalance z(t) and relative fraction of\n\ructuations \u000eN(t)=Nfor the experiments (B).4The experi-\nmental data points (black dots) are taken from the reference.\nThe parameters and initial conditions for the calculations are\nlisted in Tabs. I { III.\nthe \ructuation fraction. As seen in Fig. 6, lower panel,\nthe departure of the theoretical results from the exper-\nimental data points is signi\fcant for those times when\nthe non-condensate fraction \u000eN(t)=Nis large. A large\nfraction of \ructuations means that the BHF approxima-\ntion employed in the present work is not su\u000ecient, and\nhigher-order corrections should be taken into account.\nThey account for inelastic collisions of excitations and\nwill, therefore, lead to rapid damping,7as observed in\nexperiment (B).4\nIV. DISCUSSION AND CONCLUSION\nWe have considered Josephson oscillations of isolated,\natomic BECs trapped in double-well potentials and an-\nalyzed the impact of \ructuations, i.e. out-of-condensate\nparticle excitations, on the dynamics of the oscillations\nfor the two speci\fc experiments of Albiez et al. (A),2\nand of LeBlanc et al. (B).4While the \frst experiment\nis well described by Gross-Pitaevskii dynamics,1sug-\ngesting a negligible role played by the \ructuations, the\nlatter experiment exhibits fast relaxation of the oscilla-\ntions, which is not contained in the semiclassical Gross-\nPitaevskii description, even if multiple trap modes are\nconsidered. One therefore expects a sizable number of\nnon-condensate excitations created in this experiment.\nWe identi\fed a scenario for the resonant excitation of\n\ructuations. It indicates that, whenever any of the renor-\nmalized trap levels is close to the e\u000bective Josephson fre-\nquency, this leads to resonant creation of \ructuations and\na departure from the Gross-Pitaevskii dynamics. The9\ninteraction-induced renormalization of both the trap lev-\nels as well as the Josephson frequency is important for\nthis resonant e\u000bect to occur. By numerical calculations\nfor the realistic model parameters, we showed that indeed\nexperiment (A) is o\u000b resonance with only a small amount\nof \ructuations created, while experiment (B) is operated\nin the resonant regime and dominated by \ructuations.\nThis reconciles the qualitatively di\u000berent behavior of the\ntwo experiments. In another, more recent experiment5\nthe bare Josephson frequency !Jwas chosen smaller than\nthe trap level spacings (see Supplemental Information to\nRef. [5]), and the BJJ oscillation frequency was further\nreduced by tuning the interaction Uto become attrac-\ntive. Thus, this experiment is in the o\u000b-resonant regime.\nIndeed, it shows extended undamped oscillations. It is\nwell described by GPE dynamics alone5, as expected.\nAs a more general conclusion, for the design of\nlong-lived, coherent Josephson junctions it is essential\nto ensure that none of the renormalized and possiblyinteraction-broadened trap levels is on resonance with\nthe e\u000bective Josephson frequency. This can be achieved\nby either tuning the parameters of the trap or by ad-\njusting the total number of particles. In this way, Bose-\nJosephson junctions may serve as a device for studying\nthe departure from classicality due to quantum \ructua-\ntions in a controlled way.\nACKNOWLEDGMENTS\nWe would like to thank A. Nejati, B. Havers and\nM. Lenk for useful discussions and especially J. H.\nThywissen and L. J. LeBlanc for providing us with\nthe details of their trapping potential. This work was\nsupported by the Deutsche Forschungsgemeinschaft\n(DFG) through SFB/TR 185.\n\u0003Email: lappet@th.physik.uni-bonn.de\nyEmail: anna.posazhennikova@rhul.ac.uk\nzEmail: kroha@th.physik.uni-bonn.de\n1A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy,\nPhys. Rev. Lett. 79, 4950 (1997).\n2M. Albiez, R. Gati, J. F olling, S. Hunsmann, M. Cristiani,\nand M. K. Oberthaler, Phys. Rev. Lett. 95, 010402 (2005).\n3S. Levy, E. Lahoud, I. Shomroni, and J. Steinhauer, Na-\nture449, 579 (2007).\n4L. J. LeBlanc, A. B. Bardon, J. McKeever, M. H. T.\nExtavour, D. Jervis, J. H. Thywissen, F. Piazza, and\nA. Smerzi, Phys. Rev. Lett. 106, 025302 (2011).\n5G. Spagnolli, G. Semeghini, L. Masi, G. Ferioli,\nA. Trenkwalder, S. Coop, M. Landini, L. Pezz\u0012 e, G. Mod-\nugno, M. Inguscio, A. Smerzi, and M. Fattori, Phys. Rev.\nLett. 118, 230403 (2017).\n6A. Posazhennikova, M. Trujillo-Martinez, and J. Kroha,\nAnn. Phys. (Berlin) 530, 1700124 (2018).\n7A. Posazhennikova, M. Trujillo-Martinez, and J. Kroha,\nPhys. Rev. Lett. 116, 225304 (2016).\n8M. Lenk, T. Lappe, A. Posazhennikova, and J. Kroha, to\nbe published .\n9J. M. Deutsch, Phys. Rev. A 43, 2046 (1991).\n10M. Srednicki, Phys. Rev. E 50, 888 (1994).\n11D. Ananikian and T. Bergeman, Phys. Rev. A 73, 013604\n(2006).\n12J. M. Vogels, C. C. Tsai, R. S. Freeland, S. J. J. M. F.\nKokkelmans, B. J. Verhaar, and D. J. Heinzen, Phys. Rev.\nA56, R1067 (1997).\n13E. G. M. van Kempen, S. J. J. M. F. Kokkelmans, D. J.\nHeinzen, and B. J. Verhaar, Phys. Rev. Lett. 88, 093201\n(2002).\n14L. V. Keldysh, JETP 20, 1080 (1964).\n15L. M. Sieberer, M. 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Phys. 17, 87 (1975).10\nAPPENDIX A: TWO-MODE APPROXIMATION\nTo illustrate the details of the formalism, we present\nhere the derivation of the equations of motion for a two-\nmode system where, however, the out-of-condensate \ruc-\ntuations are taken into account in each mode. In this re-\nspect, the calculation goes beyond the two-mode model\nstudied at the semiclassical (Gross-Pitaevskii) level of ap-\nproximation in Refs. [1,11]. For clarity of presentation,\nwe here discard the nonlocal (inter-mode) interaction\nparameters. The important steps to be demonstrated\nin this appendix carry over to the general case used to\ndescribe the experiments (multi-mode, nonlocal interac-\ntions) in a straightforward manner. For the scope of this\nappendix, the action hence reads,\nS=S0+2X\n\u000b=1Sint[\u001e\u0003\n\u000b;\u001e\u000b];\nwhere\nS0=Z\ndt\"2X\n\u000b=1\u0000\n\u001e\u0003\n\u000bG\u00001\n0\u001e\u000b\u0001\n\u0000J(\u001e\u0003\n1\u001e2+\u001e\u0003\n2\u001e1)#\n;(31)\nand\nSint[\u001e\u0003;\u001e] =\u0000U\n2Z\ndtj\u001ej4: (32)\nWriting the corresponding Keldysh action explicitly, one\n\fnds\nSK[\bc;\bq] =Z\ndt\u001a2X\n\u000b=1[\u001e\u0003\n\u000bq(i@t\u0000\")\u001e\u000bc\n+\u001e\u0003\n\u000bc(i@t\u0000\")\u001e\u000bq]\u0000J\u0002\n\u001e\u0003\n1q\u001e2c+\u001e\u0003\n1c\u001e2q+ c.c.\u0003\n\u0000U\n22X\n\u000b=1\u0002\n\u001e\u0003\n\u000bc\u001e\u0003\n\u000bc\u001e\u000bc\u001e\u000bq+\u001e\u0003\n\u000bq\u001e\u0003\n\u000bq\u001e\u000bq\u001e\u000bc+ c.c.\u0003\u001b\n:(33)\nPerforming the variation according to Eq. (18) yields the\nmodi\fed Gross-Pitaevskii equation (GPE) as the saddle-\npoint equation of our action:\ni@t\b1c=\"\b1c+J\b2c+U\n2\b\u0003\n1c\b1c\b1c\n+U\n2(\b1q\b1q\b\u0003\n1c+ 2\b\u0003\n1q\b1q\b1c\n+ 2h\u000e\u001e1c\u000e\u001e\u0003\n1ci\b1c+h\u000e\u001e1c\u000e\u001e1ci\b\u0003\n1c\n+ 2h\u000e\u001e1c\u000e\u001e1qi\b\u0003\n1q+ 2(h\u000e\u001e1c\u000e\u001e\u0003\n1qi+ c.c.)\b1q\n+ 2h\u000e\u001e1q\u000e\u001e\u0003\n1qi\b1c+h\u000e\u001e1q\u000e\u001e1qi\b\u0003\n1c):\n(34)\nTaking into account that \b 1q= \b\u0003\n1q= 0, as well as the\nfact that all Green functions of two quantum \felds van-\nish because of the relation between (anti-) time-ordered,\ngreater and lesser Green functions, by letting \b \u000bc= \b\u000b\nwe obtain the \fnal form of our modi\fed GPE as\ni@t\b1=\"\b1+J\b2+U\n2\b\u0003\n1\b1\b1;\n+U\n2(h\u000e\u001e1c\u000e\u001e1ci\b\u0003\n1+ 2h\u000e\u001e1c\u000e\u001e\u0003\n1ci\b1); (35)which upon introduction of the \ructuation Green func-\ntions reads\ni@t\b1=(\"+U\n2\b\u0003\n1\b1)\b1+J\b2+ iU(G11\b1+1\n2g11\b\u0003\n1):\n(36)\nThe equation for the second \feld can be obtained by\nsubstituting 2 1 and vice versa. Next we calculate the\nsecond derivatives of the e\u000bective action and \fnd\n\u000e2\u0000\n\u000e\b\u0003\u000bq(t)\u000e\b\u000bc(t)= i@t\u0000\"\u0000U(\b\u0003\n\u000b\b\u000b+ iG\u000b\u000b);(37)\n\u000e2\u0000\n\u000e\b\u0003\u000bq(t)\u000e\b\u0003\u000bc(t)=\u0000U\n2\u0000\n\b2\n\u000b+ ig\u000b\u000b\u0001\n; (38)\nwhereas the o\u000b-diagonals in level space are simply\n\u000e2\u0000\n\u000e\b\u0003\n1q(t)\u000e\b2c(t)=\u0000J; (39)\n\u000e2\u0000\n\u000e\b\u0003\n1q(t)\u000e\b\u0003\n2c(t)= 0: (40)\nNow make the ansatz\n\b\u000b=p\n2N\u000bei'\u000b(41)\nfor the condensate \felds. Subtracting Eq. (22) and the\ncorresponding advanced equation, and taking the upper\nleft component of the matrices in Bogoliubov space, one\n\fnds, after performing the equal-time limit on the Green\nfunctionsG\u000b\f(t;t0), that\ni@tG11+N1U(e\u00002i'1g11+ c.c.) +J(G12\u0000G21) = 0;\ni@tG22+N2U(e\u00002i'2g22+ c.c.)\u0000J(G12\u0000G21) = 0;(42)\nwhereG\u000b\f=G\u000b\f(t) =G\u000b\f(T=t;\u001c= 0) depends only\non the average time T= (t+t0)=2 =tafter taking t0!t.\nThe same holds for the anomalous Green functions.\nAccordingly, adding Eq. (22) and the corresponding ad-\nvanced equation, and taking the upper right component\nin Bogoliubov space, one \fnds for the anomalous Green\nfunctions, e.g.\ni@tg\u000b\u000b\u00002(\"+ 2N\u000bU+ iUG\u000b\u000b)g\u000b\u000b\n\u0000U\u0000\n2N\u000be2i'\u000b+ ig\u000b\u000b\u0001\nG\u000b\u000b\u00002Jg12= 0:(43)\nThe remaining equations are\ni@tG12\u0000U(2(N1\u0000N2) + iG11\u0000iG22)G12+J(G11\u0000G22)\n+U\n2[g\u0003\n12(\b1\b1+ ig11) +g12(\b\u0003\n2\b\u0003\n2\u0000ig\u0003\n22) ] = 0;\n(44)\nand\ni@tg12\u00002\"g12+UX\n\u000b(2N\u000b+ iG\u000b\u000b)g12\u0000JX\n\u000bg\u000b\u000b\n+U\n2[G\u0003\n12(\b1\b1+ ig11)\u0000G12(\b\u0003\n2\b\u0003\n2+ ig22)] = 0;\n(45)11\ntogether with the identities G21(t) =\u0000G\u0003\n12(t) and\ng21(t) =g12(t).\nWithG\u000b\u000b=\u0000iF\u000b, where\nF\u000b= 2\u000eN\u000b+ 1; (46)\none obtains for the total number of \ructuations\n\u000e_N=\u000e_N1+\u000e_N2=\u00002X\n\u000b=1N\u000bU\n2(e\u00002i'\u000bg\u000b\u000b+ c.c.):(47)\nDe\fning the phase di\u000berence of the two condensates as\n\u0001'='2\u0000'1, from Eq. (36) one calculates\n_N1= +2Jp\nN1N2sin \u0001'+N1U\n2(e\u00002i'1g11+ c.c.);\n_N2=\u00002Jp\nN1N2sin \u0001'+N2U\n2(e\u00002i'2g22+ c.c.);\n(48)\nwhich resonates with the results from Ref. [1], with the\nadditional contributions from the \ructuations. It should\nbe noted here that J <0 in our convention.\nOne clearly sees from (47) and (48) that the total par-\nticle number Nis conserved,\n@tX\n\u000b(N\u000b+\u000eN\u000b) = _N= 0: (49)\nSimilarly, by employing the dynamical equations (42),\n(43) and (44), the total energy\nE=1\n2X\n\u000b(Ec\n\u000b+Eq\n\u000b); (50)\nwith the condensate energy\nEc\n\u000b= 2UN2\n\u000b+ 2UF\u000bN\u000b+U\n4(ig\u000b\u000b\b\u0003\n\u000b\b\u0003\n\u000b+ c.c.)\n+J(\b\u0003\n1\b2+ c.c.);(51)\nand the \ructuation energy\nEq\n\u000b=UF\u000b(2N\u000b+F\u000b) +U\n2g\u0003\n11g11\n+U\n4(ig\u000b\u000b\b\u0003\n\u000b\b\u0003\n\u000b+ c.c.) + iJ(G12\u0000G\u0003\n12);(52)\nmay be shown to be conserved,\ni@tE= 0: (53)\nAPPENDIX B: COMPUTATION OF TRAP\nPARAMETERS\nA. Diagonalization of trap potentials\nThe trap potential employed in experiment Ref. [2]\nreads\nVA(r) =m\n2\u0002\n!2\nxx2+!2\nyy2+!2\nzz2\u0003\n+V0\n2\u0002\n1 + cos\u00002\u0019x\nd\u0001\u0003\n;\n(54)\nwith frequencies given in Ref. [2]. Since a Hamiltonian\nwith this potential is separable, the eigenfunctions are theproducts of the eigenfunctions in each spatial dimension.\nHence, the diagonalization of the noninteracting trap sys-\ntem reduces to three separate diagonalizations, which can\nbe performed by applying standard library methods (e.g.\nRef. [23]), yielding all eigenvalues and eigenfunctions of\nthe trap.\nThe con\fning potential of the experiment Ref. [4] is\nmore involved,\nVB(r) =m0\nF~s\n\u000e(r)2+\u0012\u0016BgFBRF;?(r)\n2~\u00132\n+m\n2!2\nyy2;(55)\nwhere\u000e(r) =!RF\u0000j\u0016BgFBS(r)=~j, see Ref. [4]) for de-\ntails and the de\fnition of the parameters. We use the\nparameter values quoted there with \u000e= 2\u0019\u0002(\u00000:4).\nSince Eq. (55) is not separable along the spatial axes,\nthe Hamiltonian dimension is too large for direct numer-\nical diagonalization. In order to be as close to the actual\nexperiment as possible, we expressly do not approximate\nEq. (55) by an expression that would be easily accessible\nnumerically. Therefore, one has to resort to an algo-\nrithm that can handle very large matrices. We employ\nthe Jacobi-Davidson algorithm.24,25It is an iterative sub-\nspace method that iteratively returns the \frst few eigen-\nvalues and eigenvectors of a high-dimensional problem.\nNote that for the present analysis it is essential to in-\nclude higher trap states. As examples of the results, the\nwave functions of three di\u000berent trap eigenstates for the\nnonseparable potential, Eq. (55), are shown in Fig. 7.\nB. Computation of the interaction parameters\nFor the experiments (A) and (B), many of the pa-\nrameters of Eq. (9) turn out to be negligible, such\nthat, retaining only the signi\fcant parameters, the in-\nteracting part of the action can be simpli\fed to Sint=\n\u00001\n2(Sloc+S12+Smn+S\u000bm);with\nSloc=Z\ndt \nU2X\n\u000b=1j\u001e\u000bj4+MX\nm=3Umj\u001emj4!\n; (56)\nS12=Z\ndt[U0(\u001e\u0003\n1\u001e\u0003\n1\u001e2\u001e2+ 2\u001e\u0003\n1\u001e1\u001e\u0003\n2\u001e2)\n+ 2J0(\u001e\u0003\n1\u001e\u0003\n1\u001e1\u001e2+\u001e\u0003\n2\u001e\u0003\n2\u001e2\u001e1) + c.c. ];(57)\nSmn=Z\ndtMX\nm;n=3\nm6=n1\n2U0\nmn(\u001e\u0003\nm\u001e\u0003\nm\u001en\u001en+ 2\u001e\u0003\nm\u001em\u001e\u0003\nn\u001en+ c.c.);\n(58)\nS\u000bm=Z\ndt2X\n\u000b=1MX\nm=3[Km(\u001e\u0003\n\u000b\u001e\u0003\n\u000b\u001em\u001em+ 2\u001e\u0003\n\u000b\u001e\u000b\u001e\u0003\nm\u001em)\n+ 2R\u000bm\u001e\u0003\n\u000b\u001e\u0003\n\u000b\u001e\u000b\u001em+ c.c. ]: (59)12\nThe parameters introduced in Eqs. (56) { (59) are de-\n\fned by\nU= ~gZ\nd3r'4\n\u000b(r); \u000b = 1;2 (60)\nUm= ~gZ\nd3r'4\nm(r); m\u00153 (61)\nU0= ~gZ\nd3r'2\n1(r)'2\n2(r) (62)J0= ~gZ\nd3r'3\n1(r)'2(r) (63)\nU0\nmn= ~gZ\nd3r'2\nm(r)'2\nn(r); m; n\u00153 (64)\nKm= ~gZ\nd3r'2\n\u000b(r)'2\nm(r); \u000b = 1;2;m\u00153 (65)\nR\u000bm= ~gZ\nd3r'3\n\u000b(r)'m(r); \u000b = 1;2;m\u00153 (66)\nµz [ m]µy [ m]+\n++\n+µx [ m]\nµz [ m]\nµx [ m]\nµy [ m]y = 0 z = 0 x = 1.56 m µ\n+\nµz [ m] µy [ m]µy [ m]_+\n_+µx [ m]\nµz [ m]µx [ m]y = 0 z = 0 x = 1.56 m µ\n+\nµz [ m]µy [ m]+\n++\n++\n+ ++_\n__\n_\nµy [ m]µx [ m]µx [ m]\nµz [ m]y = 0 z = 0 x = 1.56 m µ\n+ + +_ _\nFigure 7: Spatial pro\fles of eigenfunctions of the trap potential Eq. (55), where the double-well is orianted along the zaxis. Top\nrow: the symmetric state '+(r); middle row: the antisymmetric state '\u0000(r); bottom row: the symmetric state corresponding\nto the uppermost thick level in Fig. 1 for as= 0. The cuts shown are along the y\u0000zplane forx=xmin= 1:56\u0016m (position\nof the trap minimum), along the z\u0000xplane fory= 0, and along the x\u0000yplane forz= 0, respectively, as indicated. The color\nor gray scale describes the wave function amplitude (arbitrary units). + ( \u0000) signs indicate the regions of positive (negative)\nextrema of the wave function. In the panels without sign change, the zero level of the wave function is represented by dark\nblue (dark gray), while in the panels with sign change, the zero level is represented by light green (light gray). Some of the\ntrap equipotential lines are shown as white lines, providing a guide to the eye where the minima of the trap potential are\nlocated. This solution was obtained with the Jacobi-Davidson algorithm for a spatial resolution of 64 grid points in each spatial\ndimension." }, { "title": "1907.04577v2.The_superior_role_of_the_Gilbert_damping_on_the_signal_to_noise_ratio_in_heat_assisted_magnetic_recording.pdf", "content": "The superior role of the Gilbert damping on the signal-to-noise ratio in\nheat-assisted magnetic recording\nO. Muthsam,1,a)F. Slanovc,1C. Vogler,1and D. Suess1\nUniversity of Vienna, Physics of Functional Materials, Boltzmanngasse 5, 1090 Vienna,\nAustria\n(Dated: 25 September 2019)\nIn magnetic recording the signal-to-noise ratio (SNR) is a good indicator for the quality of\nwritten bits. However, a priori it is not clear which parameters have the strongest in\ruence\non the SNR. In this work, we investigate the role of the Gilbert damping on the SNR. Grains\nconsisting of FePt like hard magnetic material with two di\u000berent grain sizes d= 5 nm and\nd= 7 nm are considered and simulations of heat-assisted magnetic recording (HAMR) are\nperformed with the atomistic simulation program VAMPIRE. The simulations display that\nthe SNR saturates for damping constants larger or equal than 0.1. Additionally, we can\nshow that the Gilbert damping together with the bit length have a major e\u000bect on the SNR\nwhereas other write head and material parameters only have a minor relevance on the SNR.\nI. INTRODUCTION\nThe next generation recording technology to increase\nthe areal storage density of hard drives beyond 1.5 Tb/in2\nis heat-assisted magnetic recording (HAMR)1{6. Higher\nareal storage densities (ADs) require smaller recording\ngrains. These grains need to have high anisotropy to be\nthermally stable. HAMR uses a heat pulse to locally\nenhance the temperature of the high anisotropy record-\ning medium beyond the Curie temperature. Due to the\nheating, the coercivity of the grain drops and it can be\nwritten with the available head \felds. After the grain\nis written, the medium is cooled and the information is\nsafely stored. A good indicator for the quality of the\nwritten bits is the so-called signal-to-noise ratio (SNR)\nwhich gives the power of the signal over the power of the\nnoise7. To achieve high areal storage densities, record-\ning materials that show good magnetic properties even\nat small grain sizes and thus yield high SNR values are\nneeded. However, a priori it is not clear which parame-\nters have the strongest in\ruence on the SNR.\nIn this work, we investigate the e\u000bect of a varying damp-\ning constant on the SNR. HAMR simulations with the\natomistic simulation program VAMPIRE8are performed\nfor cylindrical recording grains with two di\u000berent diame-\ntersd= 5 nm and d= 7 nm and a height h= 8 nm. The\nmaterial parameters of FePt like hard magnetic record-\ning media according to the Advanced Storage Technol-\nogy Consortium (ASTC)9are used. Damping constants\nbetween\u000b= 0:01 and\u000b= 0:5 are considered. Addition-\nally, we present an equation to include the in\ruence of\nthe bit length to the SNR. With this we can explain a\nSNR decrease of about 8.25 dB for 5- nm grains, which\nresults when changing the material and writing parame-\nters in the HAMR simulations from those used in former\nsimulations10{12to those according to the Advanced Stor-\nage Technology Consortium9, with the damping constant\nand the bit length only.\nThe structure of this paper is as follows: In Section II,\nthe HAMR model is introduced and it is explained how\na)Electronic mail: olivia.muthsam@univie.ac.atthe SNR is determined. In Section III, the results are\npresented and in Section IV they are discussed.\nII. HAMR MODEL\nCylindrical recording grains with height h= 8 nm and\ndiametersd= 5 nm and d= 7 nm are considered. One\ngrain can be interpreted as one grain of a state-of-the-\nart granular recording medium. A simple cubic crystal\nstructure is used. The exchange interaction Jijand the\ne\u000bective lattice parameter aare adjusted so that the sim-\nulations lead to the experimentally obtained saturation\nmagnetization and Curie temperature13,14. In the simu-\nlations, only nearest neighbor exchange interactions be-\ntween the atoms are included. A continuous laser pulse\nwith Gaussian shape and the full width at half maximum\n(FWHM) of 60 nm is assumed in the simulations. The\ntemperature pro\fle of the heat pulse is given by\nT(x;y;t ) = (Twrite\u0000Tmin)e\u0000x2+y2\n2\u001b2+Tmin (1)\n=Tpeak(y)\u0001e\u0000x2\n2\u001b2+Tmin (2)\nwith\n\u001b=FWHMp\n8 ln(2)(3)\nand\nTpeak(y) = (Twrite\u0000Tmin)e\u0000y2\n2\u001b2: (4)\nv= 15 m/s is the speed of the write head. xandylabel\nthe down-track and the o\u000b-track position of the grain,\nrespectively. In our simulations both the down-track po-\nsitionxand the o\u000b-track position yare variable. The\nambient and thus minimum temperature of all simula-\ntions isTmin= 300 K. The applied \feld is modeled as\na trapezoidal \feld with a \feld duration of 0.57 ns and a\n\feld rise and decay time of 0.1 ns, resulting in a bit length\nof 10.2 nm. The \feld strength is assumed to be +0 :8 T\nand\u00000:8 T inz-direction. Initially, the magnetization of\neach grain points in + z-direction. The trapezoidal \feldarXiv:1907.04577v2 [physics.app-ph] 24 Sep 20192\ntries to switch the magnetization of the grain from + z-\ndirection to\u0000z-direction. At the end of every simulation,\nit is evaluated if the bit has switched or not.\nThe material and write head parameters according to the\nAdvanced Storage Technology Consortium9are shown\nTable I.\nA. Determination of SNR\nTo calculate the signal-to-noise ratio, the read-back\nsignal of a written bit pattern has to be determined. To\nwrite the bit pattern and get the read-back signal from it,\nthe following procedure is used. First, a switching prob-\nability phase diagram is needed for the writing process of\nthe bit pattern. Since it is very time consuming to com-\npute a switching probability phase diagram with atom-\nistic or micromagnetic simulations, an analytical model\ndeveloped by Slanovc et al15is used in this work. The\nmodel uses eight input parameters (the maximum switch-\ning probability Pmax, the down-track jitter \u001bdown;the o\u000b-\ntrack jitter \u001bo\u000b;the transition curvature c, the bit length\nb, the half maximum temperature F50, the position p2of\nthe phase diagram in Tpeakdirection and the position p3\nof the phase diagram in down-track direction) to deter-\nmine a switching probability phase diagram. Slanovc et\nalshowed that the maximum switching probability Pmax\nand the down-track jitter \u001bdown are the input parameters\nwith the strongest in\ruence on the SNR. Note, that the\nbit lengthbalso has a strong in\ruence on the SNR. In\nthe further course of this work, an equation to include\nthe bit length to the SNR calculations is shown. Thus,\nthe bit length can be assumed constant during the SNR\ndetermination. The transition curvature cdid not show\nstrong in\ruence on the SNR for the used reader model\nand the o\u000b-track jitter \u001bo\u000bis neglectable since the reader\nwidth is with 30 :13 nm smaller than the track width with\n44:34 nm and thus does not sense the o\u000b-track jitter. p2\nandp3only shift the bit pattern and can thus be \fxed\nfor comparability. For this reason, it is reasonable to \fx\nthe input parameters, except for the maximum switch-\ning probability Pmaxand the down-track jitter \u001bdown.\nThe \fxed input parameters are determined by a least\nsquare \ft from a switching probability phase diagram\ncomputed with a coarse-grained Landau-Lifshitz-Bloch\n(LLB) model16for pure hard magnetic grains with mate-\nrial parameters given in Table I. The \ftting parameters\nare summarized in Table II for grain diameters d= 5 nm\nandd= 7 nm.\nFurther, it is necessary to compute the down-track jitter\n\u001bdown and the maximal switching probability Pmaxfor\nthe considered set of material and write head parameters,\nsee Table I. In the simulations, the switching probability\nof a recording grain at various down-track positions xat\na peak temperature Tpeak=Tc+ 60 K is calculated with\nthe atomistic simulations program VAMPIRE8, yielding\na down-track probability function P(x). To get the down-\ntrack jitter and the maximum switching probability, the\nswitching probability curve is \ftted with a Gaussian cu-mulative distribution function\n\b\u0016;\u001b2=1\n2(1 + erf(x\u0000\u0016p\n2\u001b2))\u0001Pmax (5)\nwith\nerf(x) =2p\u0019Zx\n0e\u0000\u001c2d\u001c; (6)\nwhere the mean value \u0016, the standard deviation \u001band\nthe mean maximum switching probability Pmax2[0;1]\nare the \ftting parameters. The standard deviation \u001b,\nwhich determines the steepness of the transition function,\nis a measure for the transition jitter and thus for the\nachievable maximum areal grain density of a recording\nmedium. The \ftting parameter Pmaxis a measure for\nthe average switching probability at the bit center. Note,\nthat the calculated jitter values \u001bdown only consider the\ndown-track contribution of the write jitter. The so-called\na\u0000parameter is given by\na=q\n\u001b2\ndown+\u001b2g (7)\nwhere\u001bgis a grain-size-dependent jitter contribution17.\nThe write jitter can then be calculated by\n\u001bwrite\u0019ar\nS\nW(8)\nwhereWis the reader width and S=D+Bis the grain\nsize, i.e. the sum of the grain diameter Dand the non-\nmagnetic boundary B15,18.\nFor each\u001bdown andPmaxcombination a switching prob-\nability phase diagram is computed with the analytical\nmodel. With the resulting phase diagram, the writing\nprocess of a certain bit pattern is simulated on granu-\nlar recording medium15. Here, the switching probabil-\nity of the grain is set according to its position in the\nphase diagram. The writing process is repeated for 50\ndi\u000berent randomly initialized granular media. Finally,\nthe read-back signal is determined with a reader model\nwhere the reader width is 30.13 nm and the reader res-\nolution in down-track direction is 13 :26 nm. The SNR\ncan then be computed from the read-back signal with\nthe help of a SNR calculator provided by SEAGATE19.\nThe resulting SNR value is given in dB (SNR dB). In the\nfollowing, the SNR dBis simply called SNR unless it is\nexplicitly noted di\u000berent.\nIII. RESULTS\nA. SNR Dependency on Damping\nFirst, the in\ruence of the damping constant on the\nSNR is investigated in more detail. The damping con-\nstant is varied from \u000b= 0:01 to\u000b= 0:5 for two di\u000berent\ngrain sizesd= 5 nm and d= 7 nm. All other parameters\nare taken from Table I. The bit length in the simula-\ntions is 10:2 nm and the track width is 44 :34 nm. The\ndown-track jitter curves are computed at Tpeak= 760 K\nand \ftted with eq. (5). In Figure 1, the SNR over the3\nCurie temp.\nTC[K]Damping\u000bUniaxial anisotropy\nku[J/link]Jij[J/link] \u0016s[\u0016B]v[m/s]\feld duration\n(fd) [ns]FWHM [nm]\n693.5 0.02 9:124\u000210\u0000236:72\u000210\u0000211.6 15 0.57 60\nTABLE I. Material and write head parameters of a FePt like hard magnetic granular recording medium accoring to the Advanced\nStorage Technology Consortium.\ngrain size\n5 nm 7 nm\n\u001bo\u000b[K] 22.5 14.4\nPmax 0.995 0.997\nF50[K] 602 628\nb[nm] 10.2 10.2\nc[10\u00004nm/K2]3.88 4.89\np2[K] 839 830\np3[nm] 27.5 25.8\nTABLE II. Reference parameters that are evaluated via least\nsquare \ft of the simulated phase diagrams for grain sizes 5 nm\nand 7 nm. Details of the parameters can be found in15.\ndamping constants for both grain sizes is visible. Note\nthat the SNR is proportional to the number of grains,\nmeaning that the number of grains per bit has to be kept\nconstant to determine a nearly constant SNR20. How-\never, since the dimensions of the granular media used for\nthe writing and reading process are \fxed, less grains form\none bit ford= 7 nm. Thus, the SNR values for the larger\ngrain size are smaller than for the small grains.\nThe results show that changing the damping constant\nfrom\u000b= 0:01 to\u000b= 0:02 already increases the SNR by\n3.66 dB for 5 nm-grains. For d= 7 nm, the SNR gain is\n1.65 dB. For 5 nm-grains, damping constants \u000b\u00150:1 lead\nto the best results with a total improvement of 6 dB com-\npared to\u000b= 0:01. Surprisingly, enhancing the damping\nconstant beyond 0 :1 does not show any further improve-\nment, the SNR saturates. This behavior is the same for\nthe 7 nm-grains. However, the total betterment of the\nSNR is only 2.24 dB for the larger grains. The SNR sat-\nuration results from the fact that Pmax= 1 for\u000b\u00150:1.\nSimultaneously, the down-track jitter \u001bdown varies only\nmarginally for \u000b\u00150:1 (see Table III) such that it does\nnot alter the SNR. The correlation between the SNR and\nthe maximum switching probability Pmaxis shown in Fig-\nure 2. It shows that the \ftted SNR curve reproduces the\ndata very well.\nBy further studying the switching dynamics of a 5 nm-\ngrain, one can show that the assumed pulse duration of\nthe heat pulse and the applied \feld strength are crucial\nfor the saturation of the SNR. In Figure 3, it is displayed\nhow the duration of the heat pulse in\ruences the maxi-\nmum switching probability and with it the SNR. During\nthe duration of the heat pulse the \feld is considered to\nconstantly point in \u0000z\u0000direction. The results demon-\nstrate that Pmaxdoes not saturate for small pulse du-\nrations. If longer pulse durations \u00150:5 ns are assumed,\naPmaxsaturation can be seen. A similar e\u000bect can be\nseen for a change of the \feld strength when the pulse\nduration is assumed to be 0 :5 ns (see Figure 4). For a\nsmall head \feld with a strength of 0 :5 T,Pmaxshows no\nsaturation whereas it does for larger head \felds. From\nFIG. 1. Resulting SNR for various damping constants \u000bfor\ngrains with two di\u000berent diameters d= 5 nm and d= 7 nm.\nFIG. 2. SNR and Pmaxdepending on the damping constant\n\u000bfor grain size with a diameter of 5 nm.\nthe simulations with varying duration of the heat pulse\nand \feld strength, it can also be seen that the SNR can\nbe improved for smaller damping constants if the dura-\ntion of the heat pulse is increased due to a smaller head\nvelocity or the \feld strength are enhanced.\nB. SNR Dependency on bit length\nThe in\ruence of the bit length on the SNR was al-\nready studied by Slanovc et al15. In this work, the fol-4\n5 nm 7 nm\n\u000b\u001bdown[nm]Pmax\u001bdown[nm]Pmax\n0.01 2.0 0.917 1.13 0.955\n0.02 0.9475 0.974 0.83 0.99\n0.05 0.7 0.989 0.549 1.0\n0.1 0.688 1.0 0.442 1.0\n0.3 0.495 1.0 0.48 1.0\n0.5 0.64 1.0 0.636 1.0\nTABLE III. Resulting down-track jitter parameters and mean maximum switching probability values for pure hard magnetic\nmaterial with di\u000berent damping constants \u000b.\nFIG. 3. Maximum switching probability Pmaxover damping\n\u000bfor di\u000berent pulse lengths of the heat pulse. A \feld strength\nof\u00000:8 T for grains with diameter d= 5 nm is assumed.\nlowing calculation is important. For the SNR calcula-\ntions a bit length b1= 10:2 nm is assumed since this is\nthe bit length resulting from the ASTC parameters. The\ntrack width in the simulations is again 44 :34 nm. How-\never, the bit length can change due to a variation of the\nwrite head parameters (\feld duration and head velocity).\nTherefore, the bit length for the former parameters10{12\nis 22 nm. To write a bit pattern with larger bit lengths\n(b >12 nm) the simulations of new granular media are\nrequired. This is computationally very expensive. Thus,\na di\u000berent approach is needed to qualitatively investi-\ngate the in\ruence of the bit length. For the SNR with\nSNR dB= 10 log10(SNR), there holds18\nSNR/\u0012b\na\u00132\u0012T50\nb\u0013\u0012W\nS\u0013\n(9)\nwith the bit length band the read-back pulse width T50\nwhich is proportional to the reader resolution in down-\ntrack direction. The ratio T50=bis called user bit density\nand is usually kept constant18. Further, the reader width\nWand the grain size Sare constant. Since the aim is to\nqualitatively describe the SNR for a bit length b2from\nSNR calculations with a bit length b1;the a-parameter a\nis also assumed to be constant. The SNR dBfor a di\u000berent\nbit lengthb2can then be calculated by\nFIG. 4. Maximum switching probability Pmaxover damping\n\u000bfor di\u000berent \feld strengths. The durations of the heat pulse\nof 0:5 ns for grains with diameter d= 5 nm is assumed.\nSNR dB(b2)\u0000SNR dB(b1)\n= 10 log10(SNR(b2))\u000010 log10(SNR(b1))\n= 10 log10(b2\n2)\u000010 log10(b2\n1) = 20 log10(b2\nb1) (10)\nsince all other parameters are the same for both bit\nlengths. Thus, one can compute the SNR dBvalue for\na varied bit length b2via the SNR dBof the bit length b1\nby\nSNR dB(b2) = SNR dB(b1) + 20 log10(b2\nb1): (11)\nThe curve achieved by eq. (11) with b1= 10:2 nm agrees\nqualitatively very well with the SNR(bit length) data\nfrom Slanovc et al15. It is thus reasonable to use this\nequation to include the bit length to the SNR.\nC. Combination of damping and bit length5\nCurie temp.\nTC[K]Damping\u000bUniaxial anisotropy\nku[J/link]Jij[J/link] \u0016s[\u0016B]v[m/s]\feld duration\n(fd) [ns]FWHM [nm]\n536.6 0.1 9:12\u000210\u0000235:17\u000210\u0000211.7 20 1.0 20\nTABLE IV. Material and write head parameters of a FePt like hard magnetic granular recording medium that were used in\nformer works10{12.\nParameter set diameter [nm] Tpeak[K]bit length [nm] Pmax\u001bdown [nm] SNR [dB]\nASTC 5 760 10.2 0.974 0.95 17.51\nParameters of former\nworks10{12 5 600 22 0.984 0.384 25.76\nASTC 7 760 10.2 0.99 0.83 15.35\nParameters of former\nworks10{12 7 600 22 1.0 0.44 22.75\nTABLE V. Resulting Pmax; \u001bdown and SNR values for the simulations with ASTC parameters and those used in former\nsimulations.\nThe simulations with write head and material parameters\naccording to the ASTC are compared to simulations with\nparameters used in former works10{12. Main di\u000berences\nto the currently used parameters are the bit length, the\ndamping constant, the height of the grain, the exchange\ninteraction, the atomistic spin moment, the full width at\nhalf maximum, the head velocity and the \feld duration.\nThese former parameters are summarized in Table IV.\nComparing the SNR values of both parameter sets shows\nthat ford= 5 nm the SNR is about 8.25 dB larger for the\nformer used parameters than for the ASTC parameters\nand ford= 7 nm it is\u00187:4 dB larger. The question is\nif the damping and bit length variation can fully explain\nthis deviation.\nIncreasing the damping constant from \u000b= 0:02 to\u000b=\n0:1, yields about +2 :25 dB ford= 5 nm and +0 :72 dB for\nd= 7 nm. Additionally, with the calculations from Sec-\ntion III B, one can show that by changing the bit length\nfromb1= 10:2 nm tob2= 22 nm gives\nSNR dB(b2) = SNR dB(b1) + 6:85 dB: (12)\nCombined, this shows that the di\u000berence in the SNR\ncan be attributed entirely to the damping and the bit\nlength enhancement. Moreover, simulations where the\nother material and write head parameters are changed\none by one con\frm this \fndings. The other write head\nand material parameters that are changed in the simu-\nlations have only minor relevance on the SNR compared\nto the damping constant and the bit length.\nIV. CONCLUSION\nTo conclude, we investigated how the damping con-\nstant a\u000bects the SNR. The damping constant was varied\nbetween\u000b= 0:01 and\u000b= 0:5 for two di\u000berent grain sizes\nd= 5 nm and d= 7 nm and the SNR was determined.\nIn practice, the damping constant of FePt might be in-\ncreased by enhancing the Pt concentration21,22. Another\noption would be to use a high/low Tcbilayer structure23\nand increase the damping of the soft magnetic layer by\ndoping with transition metals24{28. An interesting \fnd-\ning of the study is the enormous SNR improvement of6 dB that can be achieved for 5 nm-grains when enhanc-\ning the damping constant from \u000b= 0:01 to\u000b= 0:1\nand beyond. It is reasonable that the SNR improves\nwith larger damping. This results from the oscillatory\nbehavior of the magnetization for small damping dur-\ning switching. In fact, smaller damping facilitates the\n\frst switching but with larger damping it is more likely\nthat the grain will switch stably during the cooling of the\nthermal pulse29. This leads to a smaller switching time\ndistribution for larger damping constants and in the fur-\nther course to higher SNR values. However, an increase\nof the duration of the heat pulse due to a smaller head\nvelocity or an increase of the \feld strength can improve\nthe SNR even for smaller damping constant.\nFurthermore, the results display a SNR saturation for\ndamping constants \u000b\u00150:1. This SNR saturation can be\nexplained with the saturation of the maximum switching\nprobability and the only marginal change of the down-\ntrack jitter for \u000b\u00150:1. Indeed, one can check that for\nshorter pulse widths and smaller \feld strength, the be-\nhavior is di\u000berent and the SNR does not saturate. In\nthis case, the SNR rises for increasing damping constants.\nSummarizing, the SNR saturation for a varying damping\nconstant depends strongly on the used \feld strength and\nthe duration of the heat pulse.\nThe qualitative behavior for 7 nm-grains is the same. In-\nterestingly, the SNR change for a varying damping con-\nstant is not as signi\fcant as for grains with d= 5 nm.\nThis results from the higher maximum switching proba-\nbility and the smaller down-track jitter \u001bdown for 7 nm-\ngrains even for small damping constants. This is as\nexpected since larger grain sizes lead to an elevated\nmaximum switching probability11and smaller transition\njitter7compared to smaller grain sizes. This limits the\npossible increase of the recording performance in terms\nofPmaxand\u001bdown and thus the possible SNR gain. Ad-\nditionally, the SNR saturation value is smaller for 7 nm-\ngrains since one bit consists of fewer grains.\nThe overall goal was to explain the decrease of the SNR\nby about 8:25 dB and 7 :4 dB ford= 5 nm and d= 7 nm,\nrespectively, when changing from recording parameters\nused in former simulations10{12to the new ASTC pa-\nrameter. Indeed, together with the bit length variation,\nthe SNR variation could be fully attributed to the damp-6\ning enhancement. The other changed parameters like the\natomistic spin moment, the system height, the exchange\ninteraction and the full width at half maximum have only\na minor relevance compared to the in\ruence of the damp-\ning\u000band the bit length.\nIn fact, the variation of the bit length gave the largest\nSNR change. However, since an increase of the bit length\nis not realistic in recording devices, the variation of the\nmaterial parameters, especially the increase of the damp-\ning constant, is a more promising way to improve the\nSNR.\nV. ACKNOWLEDGEMENTS\nThe authors would like to thank the Vienna Sci-\nence and Technology Fund (WWTF) under grant No.\nMA14-044, the Advanced Storage Technology Consor-\ntium (ASTC), and the Austrian Science Fund (FWF)\nunder grant No. I2214-N20 for \fnancial support. The\ncomputational results presented have been achieved us-\ning the Vienna Scienti\fc Cluster (VSC).\n1L. Burns Jr Leslie and others. Magnetic recording system . Google\nPatents, December 1959.\n2G. W. Lewicki and others. Thermomagnetic recording and\nmagneto-optic playback system . Google Patents, December 1971.\n3Mark H. Kryder, Edward C. Gage, Terry W. McDaniel,\nWilliam A. Challener, Robert E. Rottmayer, Ganping Ju, Yiao-\nTee Hsia, and M. Fatih Erden. Heat assisted magnetic recording.\nProceedings of the IEEE , 96(11):1810{1835, 2008.\n4Robert E. Rottmayer, Sharat Batra, Dorothea Buechel,\nWilliam A. Challener, Julius Hohlfeld, Yukiko Kubota, Lei Li,\nBin Lu, Christophe Mihalcea, Keith Mount\feld, and others.\nHeat-assisted magnetic recording. IEEE Transactions on Mag-\nnetics , 42(10):2417{2421, 2006.\n5Hiroshi Kobayashi, Motoharu Tanaka, Hajime Machida, Takashi\nYano, and Uee Myong Hwang. Thermomagnetic recording .\nGoogle Patents, August 1984.\n6C. Mee and G. Fan. 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Journal of Applied Physics ,\n117(16):163913, 2015.\n12O. Muthsam, C. Vogler, and D. Suess. Noise reduction in heat-\nassisted magnetic recording of bit-patterned media by optimizing\na high/low Tc bilayer structure. Journal of Applied Physics ,\n122(21):213903, 2017.\n13Oleg N Mryasov, Ulrich Nowak, K Yu Guslienko, and Roy W\nChantrell. Temperature-dependent magnetic properties of fept:\nE\u000bective spin hamiltonian model. EPL (Europhysics Letters) ,\n69(5):805, 2005.\n14O Hovorka, S Devos, Q Coopman, WJ Fan, CJ Aas, RFL Evans,\nXi Chen, G Ju, and RW Chantrell. The curie temperature dis-\ntribution of fept granular magnetic recording media. Applied\nPhysics Letters , 101(5):052406, 2012.15Florian Slanovc, Christoph Vogler, Olivia Muthsam, and Dieter\nSuess. Systematic parameterization of heat-assisted magnetic\nrecording switching probabilities and the consequences for the\nresulting snr. arXiv preprint arXiv:1907.03884 , 2019.\n16Christoph Vogler, Claas Abert, Florian Bruckner, and Dieter\nSuess. Landau-Lifshitz-Bloch equation for exchange-coupled\ngrains. Physical Review B , 90(21):214431, December 2014.\n17Xiaobin Wang, Bogdan Valcu, and Nan-Hsiung Yeh. Transi-\ntion width limit in magnetic recording. Applied Physics Letters ,\n94(20):202508, 2009.\n18Gaspare Varvaro and Francesca Casoli. Ultra-High-Density Mag-\nnetic Recording: Storage Materials and Media Designs . CRC\nPress, March 2016.\n19S. Hernndez, P. Lu, S. Granz, P. Krivosik, P. Huang, W. Eppler,\nT. Rausch, and E. Gage. Using Ensemble Waveform Analysis to\nCompare Heat Assisted Magnetic Recording Characteristics of\nModeled and Measured Signals. IEEE Transactions on Magnet-\nics, 53(2):1{6, February 2017.\n20Roger Wood. The feasibility of magnetic recording at 1 terabit\nper square inch. IEEE Transactions on magnetics , 36(1):36{42,\n2000.\n21Satoshi Iihama, Shigemi Mizukami, Nobuhito Inami, Takashi Hi-\nratsuka, Gukcheon Kim, Hiroshi Naganuma, Mikihiko Oogane,\nTerunobu Miyazaki, and Yasuo Ando. Observation of Preces-\nsional Magnetization Dynamics in L10-FePt Thin Films with Dif-\nferent L10 Order Parameter Values. Japanese Journal of Applied\nPhysics , 52(7R):073002, June 2013.\n22Ji-Wan Kim, Hyon-Seok Song, Jae-Woo Jeong, Kyeong-Dong\nLee, Jeong-Woo Sohn, Toshiyuki Shima, and Sung-Chul Shin.\nUltrafast magnetization relaxation of L10-ordered Fe50pt50 al-\nloy thin \flm. Applied Physics Letters , 98(9):092509, February\n2011.\n23D. Suess and T. Schre\r. Breaking the thermally induced write er-\nror in heat assisted recording by using low and high Tc materials.\nApplied Physics Letters , 102(16):162405, April 2013.\n24W. Zhang, S. Jiang, P. K. J. Wong, L. Sun, Y. K. Wang, K. Wang,\nM. 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Applied Physics Letters ,\n100(10):102402, 2012." }, { "title": "2302.06099v3.Thickness_and_temperature_dependent_damping_in_La___0_67__Sr___0_33__MnO___3___epitaxial_films.pdf", "content": " \nThickness and temperature -dependent damping in La0.67Sr0.33MnO 3 \nepitaxial films \n \nYifei Wang,1 Xinxin Fan,1 Xiaoyu Feng,1 Xiaohu Gao,1 Yunfei Ke,1 Jiguang Yao,1,2 Muhan \nGuo,1 Tao Wang,1 Lvkang Shen,3,4 Ming Liu,3,4 Desheng Xue,1 and Xiaolong Fan1 \n \nAFFILIATIONS \n1Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou \nUniversity, Lanzhou 730000, People ’s Republic of China \n2Department of Physics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2 \n3School of Microelectronics, Xi ’an Jiaotong University, Xi ’an 710049, People ’s Republic of China \n4State Key Laboratory for Mechanical Behavior of Materials, Xi’an Jiaotong University, Xi ’an \n710049 , People ’s Republic of China \n \nABSTRACT \nThe damping of La 0.67Sr0.33MnO 3 (LSMO) epitaxial films as a function of thickness \nat different temperature s was studied . The competition between two scattering types \n(𝜌 like and 𝜎 like) with entirely distinct thickness and temperature dependencies \nresulted in complicated damping behavior. The behavior of 𝜎 like damping in LSMO \nfilms is consistent with the behavior in magnetic metal films. However, because 𝜌 like \ndamping is sensitive to the fine electron struc ture near the Fermi surface, the distortion \nof the oxygen octahedra controlled by the film thickness is an important factor in \ncontrolling the damping. Our study demonstrates that the complexity of damping in \nLSMO epitaxial films is a consequence of strong correlation effects, which are \ncharacteristic of complex transition metal oxides. \n \nSince the rapid development of spintronics for next generation memory and \nprocessor architectures ,1 the family of spintronic materials has extended from transition \nmetals to complex oxides with strongly correlated elec trical and magnetic properties .2,3 \nIn conventional metals, electron density and spin orbit coupling are limite d owing to \ntheir robust electron structure. In contrast, transition metal oxides with strongly coupled \ncharge, spin, and crystalline structure s are promising candidates for improved \nperformance and surprising spintronic effects .4–6 For example, a two dimensional \nelectron system with high mobility at oxide interfaces7 can achieve higher spin charge \ninterconversion efficiency than conventional heavy me tals;8 current induced \ndeterministic magnetic field free switching has been found in all oxide heterojunctions \nwith higher efficiency .9 \nAs a comple x transition metal oxide, LSMO film has attracted considerable attention \nin the field of spintronics because of its high Curie temperature (~360 K) among \nmanganese oxides10 and a spin polarization rate close to 100 %.11 The latter gives rise \nnot only to an ultra high tunnel magnetoresistance (1800% at 4 K)11 but also to an ultra \nlow Gilbert damping (5.2 × 10 4, grown on NdGaO 3 substrate)12 because of the \nrestricted spin dependent scattering around the Fermi surface with only one spin \noriented state .13 In spintronics, Gilbert damping is not only an essential parameter in \nmagnetization dynamics14 but also limits the threshold current for spin torque switching \nand auto oscillators15,16 and the decay length of diffusive spin waves .17 However, there \nis limited research on the damping mechanism of epitaxial LSMO films, and the \nspecifics of their damping tuning mechanism remain controversial. For instance, it is \ndisputed whether the temperature dependent damping of LSMO films i s monotonic .18,19 \nTo address this discrepancy, our study focus ed on the influence of strong correlation \neffects on the damping mechanism. Specifically, the correlation between the crystalline \nstructure and electronic momentum scattering was systematically studied in epitaxial \nLSMO films. \nLSMO films of different thicknesses ( 𝑡 77.1 38.4 nm) were epitaxially grown by \npulsed laser deposition using a KrF laser with a wavelength of 248 nm on (001) oriented \nSrTiO 3 (STO) substrates . The base pressure was greater than 3.6 × 10 8 Torr, and the \nfilms were grown at 620 °C with an oxygen pressure of 90 mTorr. The pulse energy \nwas 500 mJ with a frequency of 3 Hz. After deposition, the films were annealed in situ \nfor 1 h at 750 °C under an oxygen atmosphere of 300 To rr, and then cooled to 300 K at \na rate of 10 °C/min. The thickness es of the LSMO films were determined using X ray \nreflectometry, and their crystalline structure was characterized using high resolution X \nray diffraction (XRD) and reflection high energy electron diffraction (RHEED). \nSurface topography was characterized using atomic force microscopy (AFM). The \nmagnetic hysteresis loop s were measured using a vibrating sample magnetometer. The \ntemperature dependent ferromagnetic reson ance (FMR) and transport properties were \nmeasured using a self built testing system with a Cryogenic J4440 cryofree vector \nmagnet. \nFigure 1(a) shows the θ 2θ scan of XRD for LSMO (002) peaks, indicating the \nLSMO films are epitaxially grown on STO substrates. Local magnification s of the \nLSMO (002) peaks with thicknesses of 24.0 nm (green) and 38.4 nm (black) are shown \nin the insets. The clear Laue fringes demonstrate the flatness and uniformity of the \nepitaxial LSMO film. Moreover, a shift in the L SMO (002) peak position to a high er \ndiffraction angle can be observed, indicating that the values of the out of plane lattice \nconstant c decrease with increasing thickness. \nTo quantify the variation between the lattice constant and thickness, the diffracti on \nangles of LSMO (002) were determined using Jade software, and the values of c were \nobtained as follows: \n2𝑎𝑏𝑐 sin𝜃\n√ℎ2𝑏2𝑐2+𝑘2𝑎2𝑐2+𝑙2𝑎2𝑏2=𝜆 , (1) \nwhere 𝑎 and 𝑏 are the in plane lattice constants, h, k, l are Miller indices, and 𝜆71.54 Å \nis the X ray wavelength. As the diffraction peaks are spread out, the calculated lattice \nconstants are only meaningful as a statistical average. It can be seen from Fig. 1( b) that \nc decreases with increasing thickness and tends to be saturated. According to A. \nVailionis et al. ,20 the variation of the LSMO lattice constant with thickness is due to the \nspontaneous presence of regions with different l attice constants in the thickness \ndirection of the films. Here, we simplified this complicated situation into a bilayer \nmodel: an LSMO interfacial layer with N unit cells and a relatively large 𝑐i73.93 Å, \nand a bulk like layer for the remaining part with 𝑐b73.84 Å ,20 where i and b denote \ninterface and bulk like. Based on this model, the average lattice constant x (7a, b, c) as \na function of thickness 𝑡 is given by \n𝑥̅=N𝑐i𝑥i+(𝑡−N𝑐i)𝑥b\n𝑡 . (2) \nAs shown in Fig. 1( b), the fitting curve is consistent with the data and gives N≈9u.c. \n \nFIG. 1. (a) XRD patterns around the (002) peaks of the films with different thicknesses; the \ninset is the local magnification of the data for thicknesses of 24.0 nm (green) and 38.4 nm \n(black) . Average (b) out of plane, (c) in plane lattice constants and (d) effective strain as a \nfunction of thickness, followed by a fitted curve of Eq. (2). (e) Schematic i llustration of the \nbilayer model, where the oxygen octahedra in the interface layer (bulk like layer) are \ncompressive (tensile) strained. \n \n \nTo comprehensively evaluate the structural variation of the films with different \nthicknesses, we also performed XRD analysis of the LSMO (201) peak. Similar to the \ncalculation of 𝑐, the average in plane lattice constants 𝑎 are shown in Fig. 1(c) as a \nfunction of thickness. The parameter s 𝑎i=3.85 Å , 𝑎b=3.88 Å are obtained from the \nfitting of Eq. (2) with fixing N=9u.c.. Compared with the data shown in Fig. 1( b), \nthe values of 𝑐 and 𝑎 are different, exhibiting opposite trends with the thickness. The \nvariation in the LSMO film structure is due to the distortion of the oxygen octahedron , \nand the degree of distorti on varies with thickness. These deformations normally arise \nfrom two effects: the Jahn Teller effect and the strain caused by lattice mismatch .21,22 \nAs both effects break the symmetry of the oxygen octahedron, we use a total average \neffective strain 𝜀eff as the quantitative parameter for describing the structural detail in \nLSMO films of different thicknesses, given by 𝜀eff=√1\n6(2𝜀𝑧𝑧−𝜀𝑥𝑥−𝜀𝑦𝑦) ,23,24 \nwhere 𝜀𝑧𝑧 (𝜀𝑥𝑥 , 𝜀𝑦𝑦 ) is the out of plane (in plane) strain . The calculations are \npresented in Fig. 1(d). Based on the bilayer model, the effective strain can be fitted \nsimilarly by replacing x with the effective strain in Eq. (2) and the fitting gives 𝜀effi=\n1.79% (c>a, compressive) and 𝜀effb=−0.98% (c0. For\nsmalleV, all QD energy levels lie outside the bias-voltage\nwindow and there is no spin transport [see Fig. 6(a)].\nHence\u000b;\f!0. AteV=\u000f#the tunnel processes in\nFig. 6(b) come into play, leading to a finite STT and the\ncoefficient\u000bincreases while the coefficient \fhas a local\nminimum. In the voltage region specified by Eq. (25) for\n\u0016L, the coefficient \u000bapproaches a constant value while\nthe coefficient \fincreases. By increasing the bias voltage\ntoeV=\u000f#+!Lthe tunnel processes in Fig. 6(c) occur,\nleading to a decrease of \u000band a local maximum of \f. For\n\u000f#+!L0, when the magnitude of the external\nmagnetic field is large enough, the tunnel processes in\nFig. 6(f) take place due to the above-mentioned sym-\nmetries. These tunnel processes lead to a finite STT,8\n0.00.51.01.52.01234567\neV@e0Da@10-4D\ne¯+wLHaL\ne¯\nee-wLG=5e0G=3e0G=0.2e0G=0.02e0\n0.00.51.01.52.02.5-8-6-4-2024\neV@e0Db@10-4D\ne¯+wLHbL\ne¯\nee-wLG=5e0G=3e0G=0.2e0G=0.02e0\nFIG. 7: (Color online) (a) Gilbert damping coefficient \u000band (b) coefficient \fas functions of the applied bias voltage eV=\n\u0016L\u0000\u0016R, with\u0016R= 0, for different tunneling rates \u0000at zero temperature. Other parameters are \u0000L= \u0000R= \u0000=2,\u000f\"= 1:48\u000f0,\n\u000f#= 0:52\u000f0,S= 100,J= 0:01\u000f0,JSz= 0:8\u000f0, and!L= 0:16\u000f0. In the case of the smallest value of \u0000(red lines), \u000bapproaches\na constant value when \u0016Llies within the energy range specified by Eqs. (25) and (26). The coefficient \fhas one local minimum\nand one local maximum for the same energy range.\n-0.50.00.51.01.52.01234567\ne0@eVDa@10-4D\nmR=eHaL\nmR=e-wL\nmR=e¯mR=e¯+wL\nmL=e-wLmL=e\nmL=e¯+wL\nmL=e¯G=2.5eVG=1.5eVG=0.1eVG=0.01eV\n-0.50.00.51.01.52.0-6-4-2024\ne0@eVDb@10-4D\nmR=eHbL\nmR=e-wLmR=e¯+wL\nmR=e¯\nmL=e\nmL=e-wL\nmL=e¯+wL\nmL=e¯G=2.5eVG=1.5eVG=0.1eVG=0.01eV\nFIG. 8: (Color online) (a) Gilbert damping coefficient \u000band (b) coefficient \fas functions of the position of the electronic\nlevel\u000f0for different tunneling rates \u0000at zero temperature. The applied bias voltage is eV=\u0016L\u0000\u0016R, with\u0016R= 0. Other\nparameters are \u0000L= \u0000R= \u0000=2,\u000f\"\u0000\u000f0= 0:24eV,S= 100,J= 0:005eV,JSz= 0:4eV, and!L= 0:08eV. In the case of the\nsmallest value of \u0000(red lines), there are four regions in which the Gilbert damping and the change of the precession frequency\noccur. In each of these regions \u000f0satisfies the inequalities (25) and (26), and \u000bapproaches a constant value, while \fhas one\nlocal maximum and one local minimum.\na maximum for the Gilbert damping coefficient \u000b, and\na negative minimum value for the \fcoefficient. As !L\nincreases, the inequalities of Eqs. (25) and (26) are sat-\nisfied and the tunnel processes shown in Fig. 6(e) may\noccur. Hence, there is a contribution to the STT, but\nas is shown in Eq. (23), the Gilbert damping decreases\nwith increasing precession frequency. At larger values of\n!L, resulting in \u000f#+!L=\u0016L, the Gilbert damping co-\nefficient has a step increase towards a local maximum,\nwhile the coefficient \fhas a local maximum, as a conse-\nquence of the enhancement of the STT due to additional\nspin-flip processes occurring in this case. For even larger\nvalue of!L, the conditions (25) and (26) are no longerfulfilled and both coefficients vanish. It is energetically\nunfavorable to flip the spin of an electron against the\nantiparallel direction of the effective constant magnetic\nfieldBc\ne\u000b. Hence, as !Lincreases, more energy is needed\nto flip the electronic spin to the direction of the field.\nThis causes \u000bto decrease with increasing !L. Addition-\nally, the larger the ratio !L=\u0000, the less probable it is that\nspin-angular momentum will be exchanged between the\nmolecular spin and the itinerant electrons. For !L= 0,\nthe molecular spin is static, i.e.,_~S= 0. In this case\n~T(t) =~0. The coefficient \u000bthen drops to zero while the\ncoefficient\freaches a negative local maximum which is\nclose to 0. Both \u000band\freach an extremum value for9\n-4-20240123\nwL@e0Da@10-4DHaL\nmL=e-wL\nmL=emL=e¯\nmL=e¯+wL\nG=5e0G=3e0G=0.2e0G=0.02e0\n-4-2024-6-4-20\nwL@e0Db@10-4DHbL\nmL=e¯+wLmL=emL=e¯\nmL=e-wL\nG=5e0G=3e0G=0.2e0G=0.02e0\nFIG. 9: (Color online) (a) Gilbert damping coefficient \u000band (b) coefficient \fas functions of the precession frequency !L=\ng\u0016BBcof the spin ~Sof the SMM, with ~Bc=Bc~ ez, for different tunneling rates \u0000at zero temperature. The applied bias voltage\niseV=\u0016L\u0000\u0016R= 2\u000f0, with\u0016R= 0. The other parameters are the same as in Fig. 7. In the case of the smallest \u0000(red lines),\nthe coefficient \u000bhas a step increase towards a local maximum while the coefficient \fhas a local maximum or minimum at a\nvalue of!Lcorresponding to a resonance of \u0016Lwith one of the levels in the QD.\nlarge values of \u0000at this point. For !L<0and\u0000\u001cj!Lj\n(red lines), at the value of !Lfor which\u0016L=\u000f\"\u0000!L, the\ncoefficient\u000bhasastepincreasetowardsalocalmaximum\nwhilethecoefficient \fhasanegativelocalminimum. The\ncoefficient\u000bthen decreases with a further decrease of !L\nas long as\u000f#\u0014\u0016L\u0014\u000f\"\u0000!L. At the value of !Lfor\nwhich\u0016L=\u000f#,\u000bhas another step increase towards a\nlocal maximum while \fhas a maximum value. Accord-\ning to Eq. (23), the Gilbert damping also does not occur\nif~Sis perpendicular to ~Bc. In this case \f.0and the\nonly nonzero torque component \f_~S(t)acts in the oposite\ndirection than the molecular spin’s rotational motion.\nIV. CONCLUSIONS\nIn this paper we have first theoretically studied time-\ndependent charge and spin transport through a small\njunction consisting of a single-level quantum dot cou-\npled to two noninteracting metallic leads in the pres-\nence of a time-dependent magnetic field. We used the\nKeldysh nonequilibrium Green’s functions method to de-\nrive the charge and spin currents in linear order with\nrespect to the time-dependent component of the mag-\nnetic field with a characteristic frequency !. We then\nfocused on the case of a single electronic level coupled\nvia exchange interaction to an effective magnetic field\ncreated by the precessional motion of an SMM’s spin in\na constant magnetic field. The inelastic tunneling pro-\ncesses that contribute to the spin currents produce an\nSTT that acts on the molecular spin. The STT con-\nsists of a Gilbert damping component, characterized bythe coefficient \u000b, as well as a component, characterized\nby the coefficient \f, that acts as an additional effective\nconstant magnetic field and changes the precession fre-\nquency!Lof the molecular spin. Both \u000band\fdepend\non!Land show a nonmonotonic dependence on the tun-\nneling rates \u0000. In the weak coupling limit \u0000\u001c!L,\u000b\ncan be switched on and off as a function of bias and gate\nvoltages. The coefficient \fcorrespondingly has a local\nextremum. For \u0000!0, both\u000band\fvanish. Taking\ninto account that spin transport can be controlled by the\nbias and gate voltages, as well as by external magnetic\nfields, our results might be useful in spintronic applica-\ntions using SMMs. Besides a spin-polarized STM, it may\nbe possible to detect and manipulate the spin state of an\nSMM in a ferromagnetic resonance experiment56–59and\nthus extract information about the effects of the current-\ninduced STT on the SMM. Our study could be com-\nplemented with a quantum description of an SMM in a\nsingle-molecule magnet junction and its coherent prop-\nerties, as these render the SMM suitable for quantum\ninformation storage.\nAcknowledgments\nWe gratefully acknowledge discussions with Mihajlo\nVanevićandChristianWickles. 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Yuan,1Jikun Xie,1, 2and Rembert A. Duine1, 3\n1Institute for Theoretical Physics, Utrecht University, 3584CC Utrecht, The Netherlands\n2Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices,\nSchool of Physics, Xi'an Jiaotong University, Xi'an 710049, China\n3Department of Applied Physics, Eindhoven University of Technology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: January 24, 2023)\nHybrid quantum systems based on magnetic platforms have witnessed the birth and fast devel-\nopment of quantum spintronics. Until now, most of the studies rely on magnetic excitations in\nlow-damping magnetic insulators, particularly yttrium iron garnet, while a large class of magnetic\nsystems is ruled out in this interdisciplinary \feld. Here we propose the generation of a magnon\nbundle in a hybrid magnet-qubit system, where two or more magnons are emitted simultaneously.\nBy tuning the driving frequency of qubit to match the detuning between magnon and qubit mode,\none can e\u000bectively generate a magnon bundle via super-Rabi oscillations. In contrast with general\nwisdom, magnetic dissipation plays an enabling role in generating the magnon bundle, where the\nrelaxation time of magnons determines the typical time delay between two successive magnons. The\nmaximal damping that allows an antibunched magnon bundle can reach the order of 0.1, which may\nbreak the monopoly of low-dissipation magnetic insulators in quantum spintronics and enables a\nlarge class of magnetic materials for quantum manipulation. Further, our \fnding may provide a\nscalable and generic platform to study multi-magnon physics and bene\ft the design of magnonic\nnetworks for quantum information processing.\nI. INTRODUCTION\nQuantum information science utilizes the basic prin-\nciples of quantum mechanics for information process-\ning, and it has shown great potential in innovating our\ncomputing and communication technologies. Qubits, as\nthe fundamental element to store quantum information,\nlie at the heart of quantum information and has been\nrealized in superconducting systems, photonic systems,\nsolid-state vacancies etc [1]. In particular, photons and\nphonons can also be engineered in powerful quantum cir-\ncuits, even though they are continuous variable systems\nwith an in\fnite number of freedoms [2]. This rapidly de-\nveloping \feld is called continuous variable quantum in-\nformation [3, 4] and has become an intriguing approach\nto quantum communication and quantum computation\n[5].\nMagnons, quasi-particle excitation in ordered mag-\nnets, have recently entered the territory of continuous\nvariable quantum information. The rising \feld of so-\ncalled quantum magnonics manipulates the quantum\nstates of magnons and the integration of magnon plat-\nforms with other quantum systems, including supercon-\nducting qubits, photonic cavities, nitrogen-vacancy cen-\nters, and mechanical oscillations [6, 7]. A hybrid mag-\nnetic system bene\fts from the tunability of the magnon\nfrequency from gigahertz to terahertz regime, low re-\nlaxation rate of magnon modes, and abundant nonlin-\nearities that even exist at room temperature. Up till\nnow, there have been signi\fcant studies on the entangle-\nment among magnons, photons, phonons and qubits, and\nvarious quantum states of magnons, including squeezed\nstates [8{12], single magnon states [13, 14], Schr odinger\ncat states [15{17] have been proposed. The one that hasbeen demonstrated in the experiments is the indirect cou-\npling of magnon and superconducting qubit mediated by\nthe excitation of virtual photons in a three-dimensional\ncavity. Such coupling allows the detection of magnon ex-\ncitations down to the single magnon level by reading out\nthe state of the qubit in a delicate way [14]. However,\nthe scalability of this hybrid platform remains a chal-\nlenge. Moreover, almost all of the existing proposals use\nyttrium iron garnet as the magnetic medium to excite\nmagnons because of its ultralow damping [6, 18]. This\nrules out a large class of magnetic materials for quantum\nmagnonics. Whether we can generate and manipulate\nrobust magnon quantum states comparable with more\ndiverse magnon platforms is an open question.\nIn this article, we study a hybrid quantum system con-\nsisting of a magnetic sphere and a superconducting \rux\nqubit. Here the magnetic \rux generated by the mag-\nnetic sphere penetrates the superconducting circuit and\nproduces an e\u000bective coupling with both coherent and\ndispersive components. By driving the qubit with a fre-\nquency matching the gap of the multi-magnon energy\nlevel and qubit energy, we observe a strong multi-magnon\nemission, i.e., a magnon bundle, via the super-Rabi os-\ncillations. Interestingly, magnetic dissipation plays an\nenabling role in generating a sequence of magnon pairs\nwith strong quantum correlations. The time interval be-\ntween two magnons in a pair is characterized by the co-\nherence time of the magnons, while the temporal spacing\nof two magnon pairs is determined by the qubit's deco-\nherence time. The maximal damping to realize the an-\ntibunched magnon bundle can be as large as 0 :1, which\nis three orders of magnitude larger than that of yttrium\nitron garnet and thus readily enables magnetic materi-\nals ranging from insulators to metals with moderate andarXiv:2301.09095v1 [cond-mat.mes-hall] 22 Jan 20232\nlarge damping to be useful in quantum magnonics. More-\nover, our \fnding may provide a novel platform to study\nmulti-magnon physics and can be extended to engineer\nmagnonic networks for quantum information processing.\nII. MODEL AND METHODOLOGY\nLet us consider a hybrid quantum system composed of\na superconducting \rux qubit and Nmmagnetic spheres\ncirculating the circuit, as shown in Fig. 1. The total\nHamiltonian of the hybrid system is ^H=^Hm+^Hq+\n^Hint, where ^Hm;^Hq;^Hintrepresents the Hamiltonian for\nthe magnet, qubit and interaction between the magnet\nand qubit, respectively. Here, the Hamiltonian of the\nmagnetic spheres is\n^Hm=NmX\nl=10\n@JX\nS(l)\ni\u0001S(l)\nj\u0000X\njS(l)\nj\u0001H(l)1\nA;(1)\nwhere S(l)\njis thejth spin in the lth magnetic sphere, and\nthe \frst and second terms refer to the exchange energy\nbetween neighboring spins inside one magnet and Zee-\nman energy of spins subject to external \feld H(l). The\n\rux qubit is described by the Hamiltonian [19]\n^Hq=EC^N2+EJ(1\u0000cos ^') +EL\n2^'2; (2)\nwhereECis the cooper-pair charging energy, EJis the\nJosephson energy, and ELis the inductance of the cir-\ncuit. ^'is the phase di\u000berence of superconducting wave-\nfunction on the two sides of the Josephson junction while\n^N=\u0000i@=@' is the cooper pair number operator. Here\nthe phase and charge degrees of freedom are two con-\njugate variables that satisfy the commutation relation\n[ ^';^N] =i, resembling the position and momentum oper-\nator of a particle. The interaction of the qubit with the\nmagnetic sphere is through the magnetic \rux generated\nby the magnets, which penetrates the magnetic circuit\nand modi\fes the potential energy of the qubit in Eq. (2)\nas [20]\n^Hint=EL\n2 \n^'\u0000NmX\nl=1^'l(S(l))!2\n; (3)\nwhere ^'lis the magnetic \rux in the area of the cir-\ncuit generated by the lth magnet and is de\fned as\n^'l=R\nSCB(l)\u0001dSwithB(l)being the magnetic induc-\ntance generated by the lth magnet.\nNow, we are ready to quantize the total Hamiltonian.\nTo be clear in mathematical notation, we shall \frst focus\non a single magnetic sphere ( Nm= 1) and then gen-\neralize it to the case of several spheres. In general, we\napply a su\u000eciently strong magnetic \feld to magnetize\nthe sphere in a uniform state. This allows us to intro-\nduce a local frame in which the axis of the magnetization\nxyz\nz’\nx’y’\nO\nO2R hdzS(1)\nx yFIG. 1. Schematic of magnetic spheres coupled to a super-\nconducting \rux qubit composing of a Josephson junction (red\ncuboid). Here the magnetic sphere generates a magnetic \rux\nthat penetrates through the superconducting circuit and thus\ncouples magnon mode excited inside the magnets to the qubit\nstate.\nis along the z0axis while the x0andy0axes are de\fned in\na right-hand clockwise manner, as shown in Fig. 1. The\nmagnon excitation above the ground state are quantized\nby the standard Holstein-Primako\u000b (HP) transformation\n[21], i.e.,\nS+\ni=q\n2S\u0000ay\niaiai; (4a)\nS\u0000\ni=ay\niq\n2S\u0000ay\niai; (4b)\nSz0=S\u0000ay\niai; (4c)\nwhereS\u0006\ni=Sx0\u0006iSy0are the spin raising and lowering\noperators and we have removed the upper label ( l= 1)\nof spin operator for simplicity. By substituting the HP\ntransformation into the Hamiltonian (1) and transferring\nto momentum space via the Fourier transformation ^ ai=\n1=p\nNP^akeik\u0001rwithNbeing the total number of spins\nin the magnet, we obtain the e\u000bective Hamiltonian as\n^Hm=!k^ay\nk^ak. Here the magnon dispersion reads !k=\nH+ 2ZJSk2withZbeing hte coordinate number of the\nlattice.\nThe \rux qubit is quantized by treating the double-well\npotential in Eq. (2) as two parabolas [19], where an exter-\nnal \rux lifts the degeneracy of the potential and generates\nan e\u000bective two-level system. The e\u000bective Hamiltonian3\nreads\n^Hq=1\n2(\u000f^\u001bz+ \u0001^\u001bx); (5)\nwhere the energy level splitting and transverse \feld are,\nrespectively,\n\u000f= 2fELs\n6(EJ\u0000EL)\nEJ; (6a)\n\u0001 =p\nEC(EJ\u0000EL) exp\"\n\u000012(EJ\u0000EL)3=2\nE1=2\nCEL#\n:(6b)\nHeref='ex\u0000\u0019is the small phase di\u000berence between\nexternal \rux and \u0019, which allows the expansion of the\nanharmonic potential cos ^ 'up to the quartic orders of f.\nDue to the exponential factor in \u0001, it is usually much\nsmaller than the qubit resonance frequency ( \u000f) and will\nbe dropped below.\nWhen the magnetic \rux produced by the magnetic\nspheres is taken into account, it will further add an ad-\nditional term to the phase di\u000berence fas\n'm=1\n\b0Z\nSCB\u0001dS=2\u0019\n\b0~\r\u00160\n4\u0019dX\niSi\u0001I; (7)\nwhere \b 0is the quantum of magnetic \rux and I=\n(Ix;Iy;Iz) is a geometric vector that depends on the rel-\native distance between the magnet and superconducting\ncircuit, as shown in Fig. 2. When the magnetic sphere\nis deposited as the same surface of the superconducting\ncircuit (h= 0),Ix=Iy= 0;Iz\u00192 forR\u001dd. This\nimplies that only the ^Szcomponent of spin is coupled to\nthe qubit.\nWe note that the point dipole approximation is used in\nderiving Eq. (7). Thus, it does not quantitatively hold\nwhen the size of the magnetic sphere Rmand the dis-\ntance between the magnet and superconducting circuit d\nis comparable to the circuit size R. Furthermore, for non-\nuniform spin-wave modes ( k6= 0) excited in the magnet,\nthe sum of spin \ructuations at di\u000berent positions of the\nmagnet will cancel each other. Hence, they do not con-\ntribute to the net magnetic \rux. What remains is a static\n\rux generated by the stable magnetization and the uni-\nform \ructuation around the steady magnetization. Since\nthe static \rux can be absorbed into the term 'ex, we\nonly have to consider the ferromagnetic resonance mode\nk= 0. This allows us to remove the subindex iof spin\nposition in the HP transformation and treat each mag-\nnetic sphere as a macrospin.\nNow we write down the quantized form of the inter-\nacting Hamiltonian\n^Hint= ^\u001bz(g^a+g\u0003^ay) +G^\u001bz^ay^a; (8)\nwhere the \frst term and second terms denote contribu-\ntions from the transverse component ( x0y0) and longitudi-\nnal (z0) component of the spin \ructuations, respectively.TABLE I. E\u000bective coupling strength between\nthe \rux qubit and magnetic sphere. g0 =\n2ELp\n6(EJ\u0000EL)=EJ~\r\u00160=(4\b 0d).\nEquilibrium S0Linearg=g 0NonlinearG=g 0\nxp\n2SIx\u00002Iz\nyp\n2S(Iz\u0000iIx) 0\nz ip\n2SIz\u00002Ix\n(a)\n(b)\nFIG. 2. Geometric parameters of the hybrid system. Iy= 0\nfor magnetic sphere locating at ( R+d;0;0). The coupling\nstrength between a magnetic sphere and \rux qubit is propor-\ntional to these geometric parameters as listed in Table I.\nAgain, the coupling strengths gandG, as follows from\nthe magnetic \rux (7), depend on the magnetization di-\nrection and geometric factors of the hybrid system. Table\nI shows the coupling strength as a function of geometry\nfactors for stable magnetization along the x,y, andz\naxes, respectively.\nFinally, we arrive at the e\u000bective Hamiltonian of the\nhybrid magnet-qubit system\n^H=1\n2\u000f^\u001bz+!m^ay^a+ ^\u001bz(g^a+g\u0003^ay) +G^\u001bz^ay^a: (9)\nWhen the qubit is driven, an additional term ^Hd=\n\u0010(^\u001b+e\u0000i!dt+ ^\u001b\u0000ei!dt) has to be added, where ^ \u001b\u0006are\nPauli raising and lowering operators, \u0010is the driving am-4\nplitude and !dis the driving frequency.\nIII. SUPER-RABI OSCILLATION\nIn this section, we shall show how the hybrid magnet-\nqubit system can be used to generate a multi-magnon\nstate, or magnon bundle, by properly engineering the\ndriving amplitude and frequency of the qubit. We \frst\ntransfer to a rotation frame by performing the trans-\nformation ^V= exp(\u0000i!dt^\u001bz=2) and derive a time-\nindependent Hamiltonian reading\n^H=1\n2\u0001^\u001bz+!m^ay^a+^\u001bz(g^a+g\u0003^ay)+G^\u001bz^ay^a+\u0010^\u001bx;(10)\nwhere \u0001 = \u000f\u0000!dis the frequency detuning between\nthe qubit and driving microwave. To identify the role of\nthe driving term, we perform the displacement operation\n^U= exp(\u0000i(\u0011^a\u0000\u0011\u0003^ay)^\u001bz=2). By choosing \u0011= 2g=!m,\nthe Hamiltonian (10) is rewritten as\n^H=\u0001\n2^\u001bz+!m^ay^a+G^\u001bz^ay^a+\u0010(e\u0011\u0003^ay\u0000\u0011^a^\u001b++h:c:):(11)\nHere, the last term enables the magnon bundle. In\ngeneral, it can be expanded to be a sum of a series\nof termsgn^\u001b+(^ay)n+h:c: with the coupling strength\ngn= exp(\u0000\u0011\u0011\u0003=2)\u0010=n!. Such coupling implies that the\nexcitation of a multi-magnon state (^ ay)nis accompanied\nby the qubit excitation ^ \u001b+. To maximize this parametric\nexcitation process, one has to match the frequency of the\nmagnons, qubits, and drivings, i.e., !d=\u000f\u0000n!mwith-\nout considering the nonlinear Gterm in the Hamiltonian.\nThis condition allows us to tune the number of magnons.\nTo be speci\fc, let us study the process of n\u0000magnon\nexcitation governed by the Hamiltonian (11).\nDepending on the driving strength, two regimes can\nbe distinguished as below. In the weak driving regime,\nthe dominant transition of the system is between jg;0i\nandje;nias shown in Fig. 3(a), where jgiandjeirep-\nresent the ground and excited states of the qubit and\njnirefers to the magnon number state in Fock space.\nThis implies that the hybrid system is in a superpo-\nsition statej'i=cgjg;0i+ceje;ni. By solving the\nSchr odinger equation i@tj'i=^Hj'iunder the initial con-\nditionscg(0) = 1;ce(t= 0) = 0, one \fnds\njce(t)j2=4j\nwdj2\np\n4j\nwdj2+ (\u0001 +n!m+nG)2\nsin2\u0012t\n2p\n4j\nwdj2+ (\u0001 +n!m+nG)2\u0013\n:\n(12)\nAt the resonance condition \u0001 + n!m+nG= 0, one im-\nmediately hasjce(t)j2= sin2(j\nwdjt), where the Rabi fre-\nquency reads\n\nwd=1p\nn!\u0010exp\"\n\u00002\u0012g\n!m\u00132#\u00122g\n!m\u0013n\n; (13)which characterizes the emission rate of the n\u0000magnon\nstate. A typical super-Rabi oscillation in the hybrid sys-\ntem withn= 2 is shown in Fig. 3(c).\nIn the strong driving regime, the qubit has a larger\nprobability to jump from the ground state to the ex-\ncited state even in the absence of magnon excitations\nas shown in Fig. 3(b). Thus one has to include the\nenergy levelsje;0iandjg;niin the wavefunction, i.e.\nj'i=cgjg;0;i+c1eje;ni+c2eje;0i+c3ejg;ni. Following\na similar procedure as in the weak driving case, we derive\nthe occupation probability of je;nias\njc1e(t)j2=\u0003\u00004\u00102\u0000n!m\u0001\n2\nsc\u0003sin2(\nsdt); (14)\nwhere we assume G = 0 to get this ana-\nlytical result and the Rabi frequency \n sd = p\n(n!m)2+ 4\u00102+ \u00012+ 4\n2\nwd\u00002\u0003=2 with \u0003 =p\n(n!m)2(4\u00102+ \u00012) + 16\u00102\n2\nwd. To maximize the\nmaximal occupation probability, the resonance condition\nis now (n!m)2= 4\u00102+ \u00012, under which the Rabi\nfrequency is rewritten as\n\nsd= \n wd\u0001p\n4\u00102+ \u00012: (15)\nThe occupation probability of je;ninow becomes\njc1ej2=1\n2p\n4\u00102+ \u00012\u0000\u0001p\n4\u00102+ \u00012sin2(\nsdt): (16)\nFigure 3(d) shows the super-Rabi oscillation of the sys-\ntem and that the multi-magnon state are periodically\nemitted with a much higher rate as in the weak driv-\ning case. An alternative angle to understand this os-\ncillation is from the dressed basis. Here a strong\ndriving on the qubit will generate a hybrid of bare\nground and excited states of qubit as j+i=\u0000sin\u0012jgi+\ncos\u0012jei;j\u0000i = cos\u0012jgi+ sin\u0012jeiwith energy levels\n!\u0006=\u0006p\n\u00012+ 4\u00102=2, respectively. Taking into account\nof the magnon states, the hybrid system will oscillate be-\ntween the dressed states j\u0000;0iandj+;niunder resonant\ncondition!+\u0000!\u0000=n!m, which is exactly the same as\nthe resonance condition that we have obtained by solving\nthe Schr odinger equation.\nIV. MAGNON PAIR GENERATION\nIn a real system, both the superconducting qubit and\nmagnonic system will interact with the environment and\nare subject to relaxation and dephasing. Hence, it is\nmeaningful to study the in\ruence of these decoherence\nchannels on the generation and stability of the magnon\nbundle. Before discussing the in\ruence of decoherence,\nlet us \frst estimate the time scale of the hybrid sys-\ntem. The size of the superconducting loop is around\n2R= 10\u0016m, this gives a constraint on the size of5\n(a)\n(c)|g,0>|g,1>|g,2>|e,0>|e,1>|e,2>\n ωd ε ωm\n|g,0>|g,1>|g,2>|e,0>|e,1>|e,2>weak driving strong driving(b)\n(d)\nFIG. 3. (a-b) Schematic of energy level distribution and the\ndominate transitions in the weak and strong driving regimes\nof the hybrid magnon-qubit system. We take two magnon\nexcitation as an example here to indicate the main transition\nevents. (c-d) Super-Rabi oscillation in the weak and strong\ndriving regimes. Parameters are \u0010=2\u0019= 60 MHz for weak\ndriving and 500 MHz for strong driving. g= 50 MHz. \u0001 =\n\u0000p\n4!2m\u0000\u00102is taken at the two magnon resonance.\nthe magnetic particle to validate the approximation as a\npoint dipole. We assume d=Rm= 1\u0016m, which gives an\ne\u000bective coupling g=2\u0019= 50 MHz, for ferromagnetic res-\nonance frequency !m=2\u0019= 1 GHz and driving strength\n\u0010=2\u0019= 500 MHz. The Rabi frequency in the weak cou-\npling regime can be readily evaluated as \n =2\u0019= 3 MHz\nand corresponds to a period of 0 :33\u0016s. This time scale\nis smaller than the decoherence time of a well-designed\nqubit, which can be at the order of several \u0016s or even\nlonger [1, 22], and a detailed discussion of the decoher-\nence shall be done below.\nTo quantitatively describe the relaxation and dephas-\ning of magnons and qubit, we consider the master equa-\ntion that governs the dynamics of the system\n@^\u001a\n@t=\u0000i[^H;^\u001a] +\u0014q\n2L\u001b\u0000[^\u001a] +\rq\n2L\u001bz[^\u001a]\n+\u0014m\n2La[^\u001a] +\rm\n2Laya[^\u001a];(17)\nwhere ^\u001ais the density matrix of the hybrid system, and\nthe Lindblad super-operator L[23] is de\fned asL^A[^\u001a]\u0011\n2^A^\u001a^Ay\u0000^Ay^A^\u001a\u0000^\u001a^Ay^A. The coe\u000ecients \u0014q;\rq;\u0014m;\rm\ncharacterize the relaxation and pure dephasing rates of\nqubit and magnon mode, respectively. Since the \rux\nqubit works at a temperature around 10 mK, the de-\nphasing rate of magnons is estimated to be much smaller\nthan the relaxation rate [24]. Hence, its role in the steady\nstate may be safely neglected. To minimize the number\n1m 2m 3m 4m(a)\n(b)\nbunchingantibunchingmagnon coherence time\nqubit coherence timeFIG. 4. (a) Second and third order correlation functions of\nthe steady magnon states as a function of frequency detun-\ning between the driving and qubit modes. The dashed line\nindicates the position of multi-magnon process. Parameters\nareg= 50 MHz; \u0014m= 3\u000210\u00003!m; \u0014q= 2\u000210\u00005\u000f; \rq=\n0; \u0010= 60 MHz. (b) Illustration of magnon bundle antibunch-\ning (g(2)\n2<1) and bunching ( g(2)\n2>1).\nof variables, we present results for the zero dephasing rate\nof the qubits ( \rq= 0) below, while the general results are\nstill valid when the qubit dephasing is included.\nLet us \frst look at the correlation of magnons in the\nsteady state, quanti\fed by the nth-order correlation func-\ntion [25]\ng(n)(\u001c) =h^ay(0)^ay(\u001c)\u0001\u0001\u0001^ay(n\u001c)^a(n\u001c)\u0001\u0001\u0001^a(\u001c)^a(0)i\nh^ay(0)^a(0)i\u0001\u0001\u0001h ^ay(n\u001c)^a(n\u001c)i;\n(18)\nwhereh^Ai=tr(^\u001ass^A) with ^\u001assbeing the steady den-\nsity matrix of the hybrid system. Figure 4(a) shows\nthe behavior of zero-delay ( \u001c= 0) correlation func-\ntionsg(2)andg(3)as a function of the frequency detun-\ning between the driving and qubit mode. First, there\nis a strong dip in both g(2)andg(3)when \u0001 = !m,\nwhich suggests magnon antibunching. As the detuning\nincreases around \u0001 = 2 !m, a resonant dip resides inside\nthe huge superbunching peak, indicating the existence of\nstrong magnon correlations at the resonance. Neverthe-\nless,g(n)(\u001c) that characterizes the correlation of single\nmagnons cannot give further information about the cor-\nrelations of magnon pairs studied here. One may consider6\ngeneralizing the second-order correlation function as [26]\ng(2)\nN(\u001c) =h^ayN(0)^ayN(\u001c)^aN(\u001c)^aN(0)i\nh(^ayN^aN)(0)ih(^ayN^aN)(\u001c)i; (19)\nwhich can be reduced to the familiar form of g(2)for\nN= 1. In general, the value of g(2)\n2(0) represents the sta-\ntistical behavior of the magnon bundle. For g(2)\n2(0)<1,\nthe magnon bundle is antibunched, as shown in Fig. 4\n(b). Here a sequence of magnon pairs is evenly spaced in\ntime. This resembles the single magnon behavior charac-\nterized byg(2), while the basic emission unit now becomes\na two-magnon state. Indeed, the bundle correlation func-\ntiong(2)\n2shows a deep dip below one at the two magnon\nresonance (green line in Fig. 4(a)).\nTo gain more insight in how the two-magnon states\nare emitted in the dissipative hybrid system, we em-\nploy a quantum Monte Carlo simulation, which allows\nus to trace the evolution of the wavefunction [27, 28].\nIn this approach, the environment is continuously mon-\nitored and generates a series of quantum jumps of the\nwavefunctions. Figure 5 shows the evolution of the wave-\nfunction at the two-magnon resonance peak, where the\nvertical axis can be interpreted as the probability of the\nsystem lying in the corresponding state jg=e;ni. Initially,\nboth the qubit and magnon are at the ground states. Be-\nfore 270 ns, the states of magnon and qubit jg=e;0ikeep\noscillating. At the same time, there is a large excita-\ntion probability of the two magnon state je;2i, while the\nprobability of single magnon excitation jg=e;1iis much\nsmaller. At 270 ns, there appears a sudden jump of the\nstateje;1ito be close to one, which signals the emission\nof the \frst magnon. Within a time window of 250 ns, a\nsecond magnon is emitted characterized by the jump of\ntwo magnon state je;2ito a signi\fcantly lower probabil-\nity. Now the qubit is in the excited state ( je;0i) and later\nrelaxes toward the ground state at 880 ns, which also re-\nsets the high-probability of two magnon emissions (see\nje;2icurve). Then this process repeats, and two-magnon\npairs are emitted periodically.\nBased on the analysis of the quantum trajectory, the\nmagnon relaxation time characterizes the delay of the\nsecond magnon emission after the \frst magnon emission,\nwhile the qubit relaxation time represents the typical\ntime to restore the hybrid system to its initial state. Fig-\nure 6(a) shows the phase diagram of the two-magnon\nemission behaviors as we tune the relaxation rate of\nmagnons and qubits with the following features: (1) At\na moderate value of magnon relaxation rate ( \rm=!m=\n0:008, white line), the magnon bundle is antibunched\nwhen the qubit relaxation rate is below \r\u001b=\u000f= 0:0003,\nwhich guarantees that the magnon pair is well separated\nin time. As the qubit relaxes faster, the system falls\ninto the bunching regime of the magnon bundle. (2) At\na considerable value of the magnon relaxation rate, the\nsingle magnon emission will play a signi\fcant role even\nthough it is o\u000b-resonant and destroys the two magnon\ncorrelations in the steady state (green line). The maxi-\n1st 2ndFIG. 5. Quantum trajectory of the hybrid magnon-qubit\nstates at the two magnon resonance. Parameters are \u0014m=\n3\u000210\u00003!m; \u0014q= 2\u000210\u00004\u000f; \u0010 = 500 MHz to increase the\nemission rate of two magnon bundle.\nmal damping that allows for magnon bundle antibunch-\ning is located around \rm=!m= 0:04, which corresponds\nto magnon linewidth at 40 MHz or Gilbert damping at\n0.04. This maximal damping can be further increased to\nbe of the order of 0.1 by increasing the driving of the\nsystem, as shown in Fig. 6(b). This enables a large class\nof magnetic material with moderate and large dampings\nto be useful for generating a magnon bundle.\nV. MAGNONIC NOON STATE\nUp till now, we have focused on the hybrid system with\nonly one magnetic sphere. As we place two or more mag-\nnetic spheres around the superconducting circuit, more\nexotic quantum states can be generated. Here, we take\ntwo magnets as an example and show the generation of\na magnonic NOON state. Following a similar procedure\nas in the one magnet case, we derive the Hamiltonian of\nthe hybrid system as\n^H=1\n2\u000f^\u001bz+2X\nj=1!m;j^ay\nj^aj+ ^\u001bz2X\nj=1(gj^aj+g\u0003\nj^ay\nj)\n+ ^\u001bz2X\nj=1Gj^ay\nj^aj+\u0010(^\u001b+e\u0000i!dt+ ^\u001b\u0000ei!dt):(20)7\n(a)\ng(2)\n(b)2κκ\nFIG. 6. (a) Second-order bundle correlation function g(2)\n2\nas a function of magnon relaxation rate and qubit relaxation\nrate at the two-magnon resonance. \u0001 = \u00002p\n!2m\u0000\u00102; \rq=\n0; \u0010= 60 MHz. (b) The optimal and maximal magnetic\ndamping to have magnon bundle antibunching as a function\nof driving strength. The optimal damping is de\fned as the\ndamping that the g(2)\n2has the smallest value while the max-\nimal damping is the position that g(2)\n2begins to exceed one.\nThe qubit relaxation is \fxed at \u0014q= 2\u000210\u00005\u000fcorresponding\nto the decoherence time of 10 \u0016s.\nEmploying a rotation transformation ^V =\nexp(\u0000i!dt^\u001bz=2) and a displacement operation\n^U= exp(^\u001bz=2P2\nj=1(\u0011\u0003\nj^ay\nj\u0000\u0011j^aj)) with\u0011j= 2gj=!m;j,\nthe Hamiltonian (20) is recast as\n^H=\u0001\n2^\u001bz+2X\nj=1!m;j^ay\nj^aj+2X\nj=1Gj^\u001bz^ay\nj^aj\n+\u0010(eP2\nj=1\u0011\u0003\nj^ay\nj\u0000\u0011j^aj^\u001b++h:c:);(21)\nwhich resembles the single magnet case Eq. (9), but\nnow allows the excitation of a magnon bundle in each\nmagnet. Suppose a p\u0000magnon bundle and q\u0000magnon\nbundle are respectively generated at the two magnets\nby properly tuning the resonance frequency, then thehybrid system may oscillate among the states jg;0;0i,\nje;p;0i,je;0;qiandje;p;qiunder a weak driving. The\nstrength of these transitions is related to the driving as\n\u0010;\u0010\u0011p\n1=pp!;\u0010\u0011q\n2=pq!;\u0010\u0011p\n1\u0011q\n2=pp!q!. Since\u0011j= 2gj=!m;j\nis usually smaller than one, the joint excitation proba-\nbility of the state je;p;qiwill be a small quantity of or-\nder\u001b(\u00111\u00112) compared with the excitations of je;p;0iand\nje;0;qi. Therefore the wavefunction of the hybrid system\nmay be approximated as j'i=cgjg;0;0i+c1eje;p;0i+\nc2eje;0;qi. When the qubit is measured to be in the ex-\ncited state, the wavefunction of two magnets will collapse\ninto a NOON state j'iNOON =c1ejp;0i+c2ej0;qi. For\ntwo identical magnets with the same number of magnon\nexcitations ( p=q), the coe\u000ecients satisfy the relations\nc1e=c2e= 1=p\n2.\nVI. DISCUSSIONS AND CONCLUSIONS\nOur proposal launches magnonic systems as another\npromising platform to study multiparticle physics, in\nanalog with their photonic and phononic counterparts\n[26, 29, 30]. To detect the magnon excitation, one may\nperform state tomography on the magnon states to re-\ncover the system's density matrix and compare it with\nthe theoretical predictions [31, 32]. On the other hand,\none may couple the magnetic sphere to a single spin\nqubit, for example, a nitrogen-vacancy center, to read\nout the magnon states [33].\nIn conclusion, we have shown that the nonlinear inter-\naction between a magnetic sphere and a superconducting\nqubit can generate a magnon bundle. When dissipative\ne\u000bects are considered, a sequence of magnon pairs with\nstrong quantum correlations is generated. The time de-\nlay of two magnons inside a pair is determined by the\nmagnon lifetime, while the two neighboring magnon pairs\nare well separated by the decoherence time of the qubits.\nThis mechanism is robust over a large window of mag-\nnetic dissipation and allows for the generation of magnon\nquantum states in a broad class of magnetic systems with\nmoderate and large dampings. Moreover, our methods\ncan be generalized to manipulate magnon quantum states\nin a scalable magnonic network, including a periodic ar-\nray of superconducting qubits and magnetic spheres.\nACKNOWLEDGMENTS\nWe acknowledge helpful discussions with Marios\nKounalakis. H.Y.Y. acknowledges the European Union's\nHorizon 2020 research and innovation programme un-\nder Marie Sk lodowska-Curie Grant Agreement SPINCAT\nNo. 101018193. J.K.X. acknowledges the support of\nChina Scholarship Council. 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Prominences are partially ionized, magnetized plasmas emb edded in the solar corona.\nDamped oscillations and propagating waves are commonly obs erved. These oscillations have\nbeen interpreted in terms of magnetohydrodynamic (MHD) wav es. Ion-neutral collisions and\nnon-adiabatic effects (radiation losses and thermal conduction) have been pr oposed as damping\nmechanisms.\nAims.We studythe effectof the presence of helium on the time damping of non-adiab atic MHD\nwaves in a plasma composed by electrons, protons, neutral hy drogen, neutral helium (He i), and\nsinglyionized helium (He ii) inthe single-fluidapproximation.\nMethods. The dispersion relation of linear non-adiabatic MHD waves i n a homogeneous, un-\nbounded, and partially ionzed prominence medium is derived . The period and the damping time\nof Alfv´ en, slow, fast, and thermal waves are computed. A par ametric study of the ratio of the\ndamping time tothe period withrespect tothe helium abundan ce isperformed.\nResults.The efficiency of ion-neutral collisions as well as thermal conduct ion is increased by\nthe presence of helium. However, if realistic abundances of helium in prominences ( ∼10%) are\nconsidered, this effecthas a minor influence on the wave damping.\nConclusions. The presence of helium can be safely neglected in studies of M HD waves in par-\ntiallyionized prominence plasmas.\nKey words. Sun: oscillations –Sun: magnetic fields –Sun: corona –Sun: p rominences\n1. Introduction\nSmall-amplitude oscillations and propagating waves are co mmonly observed in both quiescent\nandactiveregionprominences /filaments.Theyhavebeeninterpretedintermsofmagnetohyd rody-\nnamic(MHD)eigenmodesofthemagneticstructureand /orpropagatingMHDwaves.Thereaderis\nSend offprint requests to : R.Soler2 R.Soler etal.: MHD waves ina partiallyionized prominence : Effectof helium (RN)\nreferredtosomerecentreviewsformoreinformationaboutt heobservationalandtheoreticalback-\ngrounds (Oliver& Ballester 2002; Engvold 2004; Ballester 2 006; Banerjeeet al. 2007; Engvold\n2008)\nProminence oscillations are known to be quickly damped, wit h damping times corresponding\nto a few oscillatory periods (this topic has been reviewed by Oliver 2009; Mackayet al. 2009).\nSeveral damping mechanisms of MHD waves have been proposed, non-adiabatic effects and ion-\nneutralcollisionsbeingthemoreextensivelyinvestigate d.Inordertounderstandindetailtheseef-\nfects,theyhavebeenstudiedinsimpleconfigurationssucha sunboundedandhomogeneousmedia.\nCarbonellet al. (2004) investigatedthe time dampingin a ho mogeneousprominencemediumtak-\ningnon-adiabatice ffects(opticallythinradiationlossesandthermalconducti on)intoaccount.Later\non,thespatialdampingwasstudiedbyCarbonellet al.(2006 )andtheeffectofabackgroundmass\nflow was analyzedbyCarbonelletal. (2009). Subsequently,s ome workshave extendedthese pre-\nvious results by consideringthe presence of the coronal med ium (Soleret al. 2007, 2008, 2009a).\nThecommonconclusionoftheseinvestigationsisthatonlys lowandthermalwavesaree fficiently\ndampedby non-adiabatice ffects, while fast wavesare veryslightly dampedand Alfv´ enw avesare\ncompletelyunaffected.\nOn the other hand, the influence of partial ionization on the p ropagationand time damping of\nMHDwaveshasbeenalsoinvestigatedinanunboundedmedium. Fortezaet al.(2007)followedthe\ntreatmentbyBraginskii(1965)andderivedthefullsetofMH Dequationsalongwiththedispersion\nrelation of linear waves in a partially ionized, single-flui d plasma (see also Pintoet al. 2008). The\npresenceofelectrons,protons,andneutralhydrogenatoms wastakenintoaccount,whereashelium\nand other species were not considered. In a subsequent work ( Fortezaetal. 2008), they extended\ntheir previous analysis by considering radiative losses an d thermal conduction by electrons and\nneutrals. Their main results with respect to the fully ioniz ed case (Carbonellet al. 2004) were,\nfirst, that ion-neutral collisions (by means of the so-calle d Cowling’s diffusion) can damp both\nAlfv´ en and fast waves but non-adiabatic e ffects remain only important for the damping of slow\nand thermal waves, and second, that there exist critical val ues of the wavenumber in which the\nreal part of the frequency vanishes, so wave propagation is n ot possible for larger wavenumbers.\nAgain,applicationstoamorecomplexcylindricalgeometry havebeenalsoperformed(Soleret al.\n2009b,c)\nOnthebasisofthesepreviousresults,it seemsclearthatpa rtialionizationplaysarelevantrole\non wave propagation in prominences. Prominences are roughl y composed by 90% hydrogen and\n10% helium but, to date, all the investigationsconsidered a pure hydrogenplasma. Therefore,the\neffect of the presence of helium on the propagationand dampingo f MHD waves is still unknown\nand is the motivation for the present work. Here, we consider an unbounded and homogeneous\nprominence medium permeated by a homogeneous magnetic field . The plasma is assumed to be\npartially ionized, electrons, protons, neutral hydrogen, neutral helium (He i), and singly ionized\nhelium (Heii) being the species taken into account. Recent studies by Gou ttebroze&Labrosse\n(2009)indicatethat forcentralprominencetemperatures, theratioof thenumberdensitiesof He ii\nto Heiisaround10%,whereasthe presenceof He iiiis negligible.This result allowsusto neglect\nHeiiiin this work. Extending the works by Fortezaet al. (2007, 200 8), the derivation of the basic\nMHD equationsfora non-adiabatic,partiallyionized,sing le-fluidplasmahasbeengeneralizedbyR.Soler etal.: MHD waves ina partiallyionized prominence: Effectof helium (RN) 3\nconsideringnowfivedi fferentspecies,allowingustostudyhowthepresenceofneutr alandsingly\nionizedheliumaffectstheirpreviousresults.\nThis paper is organizedas follows. The description of the eq uilibriumand the basic equations\naregiveninSect.2.TheresultsarediscussedinSect. 3.Fin ally,Sect.4containstheconclusionof\nthiswork.\n2. Equilibriumandbasicequations\nOur equilibrium configuration is a homogeneous and unbounde d partially ionized plasma com-\nposedbyelectrons,protons,neutralhydrogen,neutralhel ium,andsinglyionizedhelium.Hereafter,\nsubscripts e, p, H, He i, and Heiiexplicitly denote these species, respectively. The magnet ic field\nis also homogeneous and orientated along the x-direction, B0=B0ˆex, withB0=5 G. We adopt\nthe single-fluid approximation. Following Fortezaet al. (2 007, 2008) and neglecting the electron\ncontribution,wedefinethecenterofmassvelocity, v,asfollows,\nv≈ξpvp+ξHvH+ξHeivHei+ξHeiivHeii, (1)\nwithξαthe relative density of species α, andvαthe corresponding species velocity. Equivalently,\ntheequilibriumtotal density, ρ0, andgaspressure, p0, are,\nρ0≈ρp+ρH+ρHei+ρHeii, (2)\np0=2/parenleftBig\npp+pHeii/parenrightBig\n+pH+pHei. (3)\nSinceρα=ξαρ0, we get the relation ξp+ξH+ξHei+ξHeii≈1. We assume a strong thermal\ncoupling between species, so all the species have the same eq uilibrium temperature T0. Then, the\nthreeequilibriumquantitiesare relatedasfollows,\np0=ρ0R\n˜µT0, (4)\nwhereRistheidealgasconstantand ˜ µisthemeanatomicweight,\n˜µ=1\n2ξp+ξH+1\n4ξHei+1\n2ξHeii. (5)\nWiththe helpofsomedefinitions,we canexpress ˜ µina moreconvenientform,\n˜µ=˜µH\n1−/bracketleftBig\n(1+δHe)+(1+2δHe)1\n4˜µH/bracketrightBig\nξHei, (6)\nwith\n˜µH=ξp+ξH\n2ξp+ξH, δ He=ξHeii\nξHei. (7)\nThe quantity ˜µHis equivalent to the mean atomic weight of a pure hydrogen pla sma defined\nin Eq. (3) of Fortezaet al. (2007), and ranges between ˜ µH=0.5 for a fully ionized hydrogen\nplasma and ˜µH=1 for a fully neutral hydrogen gas. On the other hand, δHeindicates the he-\nlium ionization degree. A realistic value of this parameter isδHe=0.1 according to the results\nof Gouttebroze&Labrosse (2009). From Eq. (6) one can see tha t ˜µ>˜µHdue to the presence of\nhelium.Inthe absenceofhelium, ξHei=ξHeii=0so ˜µ=˜µH. Figure1 displaysthedependenceof\n˜µonξHeiforseveralvaluesof ˜ µH.4 R.Soler etal.: MHD waves ina partiallyionized prominence : Effectof helium (RN)\nFig.1.Meanatomicweight, ˜ µ,asafunctionoftherelativeneutralheliumdensity, ξHei,for ˜µH=0.5\n(dottedline), ˜µH=0.6(dashedline), ˜µH=0.8(solid line),and ˜µH=0.95(dash-dottedline).Inall\ncases,δHe=0.1.\nThe details of the derivation of the basic governing equatio ns for a non-adiabatic, partially\nionized, one-fluid plasma can be followed in, e.g., Braginsk ii (1965); Fortezaet al. (2007, 2008);\nPintoet al.(2008).Here,wefollowthesameprocedurebutge neralizetheanalysisofFortezaet al.\n(2007) by including additional species. In brief, the separ ate governing equations for the five\nspecies are added and a generalized Ohm’s law is obtained. Th ese basic equations correspond\nto Eqs. (1)–(6)of Fortezaet al. (2008), which are formallyi dentical in our case. A key step in the\npresentderivationistocomputethedensitycurrent, j,as\nj=e/parenleftBig\nnpvp+nHeiivHeii−neve/parenrightBig\n, (8)\nalong with the condition ne=np+nHeii, wherene,np, andnHeiiare the electron, proton, and\nHeiinumberdensities, respectively,and eis the electron charge.The resulting inductionequation\n(see Eq. [14] of Fortezaet al. 2007) contains several di ffusion terms whose coe fficients depend\non the collisional frequencies between species. The physic al meaning of these nonideal terms is\nexplainedindetailinPintoet al.(2008).Inparticular,io n-neutralcollisionsareresponsibleforthe\nso-called Cowling’s di ffusion, which is much more e fficient than Ohm’s di ffusion in a partially\nionized plasma. However, some terms are not relevant for our present application. Hall’s e ffect is\nnegligible in prominence conditions (Soleretal. 2009c), a nd the so-called “Biermann’s battery”\ntermisidenticallyzeroinahomogeneousmedium.Forthisre ason,ourfinalformoftheinduction\nequation (Eq. [21] of Fortezaet al. 2007) only contains the t erms corresponding to Ohm’s and\nambipolar(Cowling’s)di ffusion,alongwiththediamagneticcurrentterm.\nOhm’s,η, and Cowling’s,ηC, coefficients of magnetic di ffusion can be expressed in terms of\ntheircorrespondingconductivities,\nη=1\nµσ, η C=1\nµσC, (9)R.Soler etal.: MHD waves ina partiallyionized prominence: Effectof helium (RN) 5\nwithµ=4π×10−7N A−2. So, Ohm’s and Cowling’s conductivities, as well as the diam agnetic\ncurrentcoefficient,Ξ, whichappliesinourcase whenheliumis includedare:\nσ=e2n2\ne/parenleftBig\nαe−α2en/αn/parenrightBig, σ C=σ\n1+B2\n0(ξH+ξHei)2\nαnσ(10)\nΞ=(ξH+ξHei)\n˜µαn/parenleftBigg\nξpξH−1\n2ξHξHeii+7\n4ξpξHei+1\n4ξHeiξHeii/parenrightBigg\n. (11)\nIn addition,αe,αen, andαnare the electron, electron-neutral, and neutral friction c oefficients, re-\nspectively, whose expressions depend on the sum of the frict ion coefficients between particular\nspecies,\nαen=αeH+αeHei, (12)\nαe=αep+αeH+αeHei+αeHeii, (13)\nαn=αeH+αeHei+αpH+αpHei+αHeiiH+αHeiiHei. (14)\nEachparticularfrictioncoe fficient,αββ′,iscomputedas\nαββ′=nβmββ′��ββ′, (15)\nwithnβthe number density of the species β,νββ′the collisional frequency between species βand\nβ′, and\nmββ′=mβmβ′\nmβ+mβ′, (16)\nwithmβthemassparticleofthespecies β.AsgivenbyDe Pontieuet al.(2001),seealsoSoleret al.\n(2009b),the collisionalfrequenciesbetweenelectronsan dprotonsorHeiiare\nνei=3.7×10−6nilnΛ\nT3/2\n0, (17)\nwith i=p or Heiiand lnΛthe Coulomb logarithm, while the collisional frequency bet ween a\nchargedspecies,q =e,p,orHe ii,andaneutralspecies,n =H orHei,is\nνqn=nn/radicalBigg\n8kBT0\nπmqnΣqn, (18)\nwithkBthe Boltzmann’s constant, and Σqnthe collisional cross-section. Here, we consider the\nvaluesΣen=10−19m2,andΣpn=ΣHeiin=5×10−19m2.\nOntheotherhand,thethermalconductivityduetoneutrals( Eq.[16]ofFortezaet al.2008)has\ntoincludenowtheheliumcontribution.AccordingtoParker (1953),acorrectedexpressionforthe\nconductivityofneutralsin MKSunitsis\nκn=κH+κHei=/parenleftBig\n2.44×10−2ξH+3.18×10−2ξHei/parenrightBig\nT1/2\n0. (19)\nFinally, we assume an optically thin radiation (Hildner 197 4) to represent the hydrogen radiative\nlosses. According to Cox &Tucker (1969, see their Fig. 3), th e radiative losses by helium are\nseveral orders of magnitude smaller than those of hydrogen f or typical prominence temperatures\n(∼104K) and,therefore,irrelevantforthepresentinvestigatio n.\nHereafter, our analysis follows that of Fortezaet al. (2008 ). We linearize the basic equations\nandassumesmallperturbationsproportionaltoexp (iωt+ikxx+ikzz).Thentheresultingequations6 R.Soler etal.: MHD waves ina partiallyionized prominence : Effectof helium (RN)\nFig.2.Ratio of the damping time to the period, τD/P, versus the wavenumber, k, corresponding\nto the (a) Alfv´ en wave, ( b) fast wave, and ( c) slow wave forθ=π/4, ˜µH=0.8, andδHe=0.1.\nThedifferentlinestylesrepresent ξHei=0%(solidline),ξHei=10%(dottedline),and ξHei=20%\n(dashed line). The results for ξHei=10% andδHe=0.5 are plotted by means of symbols for\ncomparison. The shaded regions correspond to the range of ty pically observed wavelengths of\nprominenceoscillations.\n(Eq.[18]–[27]ofFortezaet al.2008)arecombinedandfinall ytwodifferent,uncoupleddispersion\nrelations,oneforAlfv´ enwaves(theirEq.[28])andanothe rformagnetoacousticandthermalwaves\n(their Eq. [30]) are obtained. Note that although our definit ions ofη,ηC,Ξ, andκncontain the\neffect of helium, the resulting dispersion relations are forma lly identical to those of Fortezaet al.R.Soler etal.: MHD waves ina partiallyionized prominence: Effectof helium (RN) 7\n(2008). For the sake of simplicity, we do not write again thes e expressions here and refer the\nreader to Fortezaetal. (2008). The dispersion relations ar e numerically solved for real values of\nthewavenumbermodulus, k=/radicalBig\nk2x+k2z,andtheangleθbetweenB0andk.Acomplexfrequency,\nω=ωR+iωI,isobtained.Theperiod, P,anddampingtime, τD,arerelatedtotherealandimaginary\npartsofthefrequencyasfollows,\nP=2π\nωR, τ D=1\nωI. (20)\n3. Results\nInthe followingcomputations,we considertypicalpromine nceconditions,ρ0=5×10−11kgm−3\nandT0=8000 K. Quantities ˜ µH,ξHei, andδHeare considered free parameters. We focus our\nattentionontheeffectofthe relativeneutralheliumdensity, ξHei, onthe ratioτD/P.\n3.1. Free propagationinan unboundedmedium\nFirst, we assumeθ=π/4. Figure 2 displays τD/Pas a functionof kfor the Alfv´ en, fast, and slow\nwaves. The results corresponding to several helium abundan ces are compared for hydrogen and\nhelium ionizationdegreesof ˜ µH=0.8 andδHe=0.1, respectively.We see that even in the case of\nthe largest quantity of helium considered( ξHei=20%), the presence of helium has a minor e ffect\non the results. In the case of Alfv´ enand fast waves (Fig. 2a, b), their critical wavenumber(i.e., the\nvalue ofkwhich causes the real part of the frequencyto vanish) is shif ted toward slightly smaller\nvalues. So, the larger ξHei, the smaller ka\nc. This result can be understood by considering that the\nAlfv´ enwavecritical wavenumber, ka\nc, givenbyEq.(38)ofFortezaetal. (2008) is,\nka\nc=2vA/parenleftbigηC+ηtan2θ/parenrightbigcosθ, (21)\nwithvA=B0/√µρ0the Alfv´ enspeed. Equation(21) is also approximatelyvali d for the fast wave\ncritical wavenumber. Then, we see that ka\ncis inversely proportional to Cowling’s di ffusivity,ηC.\nSinceηCis larger in the presence of helium than in the pure hydrogen c ase due to additional\ncollisions of neutral and singly ionized helium species, ka\ncis therefore smaller. Turning our atten-\ntion to the slow wave (Fig. 2c), we see that the maximum and the right-hand side minimum of\nτD/Pare also slightly shifted toward smaller valuesof k. Results fromCarbonellet al. (2004) and\nFortezaet al. (2008) indicate that thermal conductionis re sponsible for these maximum and min-\nimum ofτD/P. Thus, the additional contribution of neutral helium atoms to thermal conduction\n(Eq.19) causesthisdisplacementofthecurveof τD/P.AsforAlfv´ enandfast waves,thise ffectis\nofminorimportance.Forcomparison,equivalentresultswi thξHei=10%andδHe=0.5areplotted\nby means of symbols in Fig. 2. We see that for realistic values ofδHe, its role is almost irrelevant,\nmeaning that the presence of He iican be neglected.It is worth mentioningthat we have repeate d\nthesecalculationsforothervaluesof ˜ µHandsimilar resultshavebeenobtained.\nNext, we study the thermal mode. Since it is a purely damped, n on-propagating disturbance\n(ωR=0), we only plot the damping time, τD, as a function of kfor ˜µH=0.8 andδHe=0.1\n(Fig. 3). We can see that the e ffect of helium is di fferent in two ranges of k. Fork/greaterorsimilar10−4m−1,\nthermal conduction is the dominant damping mechanism. So, t he larger the amount of helium,\nthe smallerτDbecauseof the enhancedthermalconductionby neutralheliu matoms.On the other8 R.Soler etal.: MHD waves ina partiallyionized prominence : Effectof helium (RN)\nhand,radiativelossesaremorerelevantfor k/lessorsimilar10−4m−1.Inthisregion,thethermalmodedamping\ntimegrowsastheheliumabundanceincreases.Sincetheseva riationsofthedampingtimearevery\nsmall,wehavetoconcludeagainthatthedampingtimeobtain edintheabsenceofheliumdoesnot\nsignificantlychangewhenheliumistakenintoaccount.Comp utationswithothervaluesof ˜ µHand\nδHedonotmodifythisstatement.\nFig.3.Damping time,τD, of the thermal wave versus the wavenumber, k, withθ=π/4. The\ndifferent linestyles represent: ξHei=0% (solid line),ξHei=10% (dotted line), and ξHei=20%\n(dashedline).Inallcomputations, ˜ µH=0.8andδHe=0.1.TheresultforξHei=10%andδHe=0.5\nisplottedbymeansofsymbolsforcomparison.\n3.2. Constrainedpropagationby awaveguide\nWe can estimate the e ffect of a magnetic structure, say a slab or a cylinder, which wo uld act as a\nwaveguide. To do so, we set the wavenumber component in the pe rpendicular direction to mag-\nnetic field lines to a fixed value, kzL=π/2, withLa typical length-scale in the perpendicular\ndirection. Since high-resolution observations of filament s (see, e.g., Linet al. 2007, 2009) show\nfine-structures(threads)withatypicalwidthof ∼100km,weselect L=105masourperpendicu-\nlarlength-scale.Therefore,thepropagationangle θdependsnowon kx,\nθ=arctan/parenleftBiggπ/2\nkxL/parenrightBigg\n. (22)\nFigure 4 displays the results for the Alfv´ en, fast, and slow waves. We see that the behavior of\nthe three solutions is substantially di fferent from that of the free propagation case. The Alfv´ en\nmode now possesses an additional critical wavenumber for sm all values of kx, namely kc−\nx, which\nis independentof the ionization degree and the helium abund ance.It can be approximatedas (see\ndetailsin Soleret al.2009b,Eq.[38]),\nkc−\nx≈η\n2vAk2\nz=ηπ2\n8vAL2. (23)\nOntheotherhand,thefastwaveisnowmoreattenuatedinther elevantrangeofwavenumbersthan\nin the free propagation case, whereas the slow wave also has a new critical wavenumber, namelyR.Soler etal.: MHD waves ina partiallyionized prominence: Effectof helium (RN) 9\nFig.4.Ratio of the dampingtime to the period, τD/P, versusthe wavenumbercomponentparallel\nto magneticfield lines, kx, correspondingto the ( a)Alfv´ en wave, ( b)fast wave, and( c) slow wave\nforkzL=π/2,withL=105m, ˜µH=0.8,andδHe=0.1.Thedifferentlinestylesrepresent ξHei=0\n(solid line),ξHei=10% (dottedline), and ξHei=20% (dashedline). Thevertical dot-dashedlines\nin (a) and (c) correspond to the approximated critical wavenumbers give n by Eqs. (23) and (24),\nrespectively,forξHei=10%.\nkcs\nxwhich falls within the relevant range. An expressionfor the slow mode critical wavenumberis\nalsoprovidedbyEq.(48)ofSoleretal. (2009b),whichinour presentnotationis,\nkcs\nx≈csηC\n2v2\nAk2\nz=csηCπ2\n8v2\nAL2, (24)10 R.Soler etal.: MHD waves ina partiallyionized prominenc e: Effectof helium (RN)\nwherecs=/radicalbig\nγp0/ρ0is thesoundspeed,with γ=5/3theadiabaticindex.Theslow modecritical\nwavenumber is shifted toward larger values as the helium abu ndance increases. Note that there is\nnoadditionalcriticalwavenumberforthefastwave.Thesea pproximatedcriticalwavenumbersare\nindicated by means of vertical lines in Fig. 4. We see an excel lent agreement in the case of the\nAlfv´ enmodecriticalwavenumber(Eq.[23]).Fortheslowwa ve,theapproximatedvalue(Eq.[24])\nisslightlylargerthanthatnumericallyobtained.\nFinally, we have also computed the results in the case of the g uided thermal disturbance. We\nfind that the thermal mode behavioris the same in the waveguid ecase and in the free propagation\ncase. Hence, this mode is not a ffected by the variation of the propagation angle and no furthe r\ncommentsareneeded.\n4. Conclusion\nIn this work, we have studied the e ffect of helium (Heiand Heii) on the time dampingof thermal\nandMHD waves in a partiallyionizedprominenceplasma.This is an extensionof previousinves-\ntigations by Fortezaetal. (2007, 2008) in which helium was n ot taken into account. We conclude\nthat,althoughthe presenceof neutralheliumincreasesthe efficiencyofbothion-neutralcollisions\nandthermalconduction,itse ffectisnotimportantforrealisticheliumabundancesinprom inences.\nInaddition,duetotheverysmallHe iiabundanceforcentralprominencetemperatures,itspresen ce\nis irrelevant to the wave behavior. This conclusion applies both to the free propagation case and\nthe constrainedpropagationby a waveguidecase. Although t he role of Heii(or even Heiii) could\nbe larger for typical prominence-corona transition region temperatures, the present result allows\nfuturestudiesofMHD wavesandoscillationsinprominences to neglectthepresenceofhelium.\nAcknowledgements. We thank N. Labrosse for giving useful information about the helium ionization degree in promi-\nnences andI.Arreguifor someusefulcomments.Theauthors a cknowledge thefinancial supportreceived fromtheSpanish\nMICINN, FEDER funds, and the Conselleria d’Economia, Hisen da i Innovaci´ o of the CAIB under Grants No. AYA2006-\n07637 and PCTIB-2005GC3-03. RS thanks the CAIB for a fellows hip.\nReferences\nBallester, J.L.2006, Phil. Trans.R.Soc. A,364, 405\nBanerjee, D.,Erd´ elyi, R.,Oliver R., &O’Shea, E.2007, Sol . Phys.,246,3\nBraginskii, S.I.1965, Rev. 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Phys.,246, 65R.Soler etal.: MHD waves ina partiallyionized prominence: Effectof helium (RN) 11\nLin, Y., Soler, R., Engvold, O., Ballester, J. L., Langangen , Ø., Oliver, R., & Rouppe van der Voort, L. H. M. 2009, ApJ,\n704, 870\nMackay, D.H.,Karpen, J.T.,Ballester, J.L.,Schmieder, B. ,&Aulanier, G.2009, Space Sci Rev, submitted\nOliver, R.&Ballester, J.L.2002, Sol. Phys.,206, 45\nOliver, R.2009, Space Sci Rev, in press\nParker, E.N.1953, ApJ,117, 431\nPinto, C., Galli, D.,& Bacciotti, F.2008, A&A,484, 1\nSoler, R.,Oliver, R., &Ballester, J. L.2007, A&A,471, 1023\nSoler, R.,Oliver, R., &Ballester, J. L.2008, ApJ, 684,725\nSoler, R.,Oliver, R., &Ballester, J. L.2009a, NewA, 14,238\nSoler, R.,Oliver, R., &Ballester, J. L.2009b, ApJ,699, 155 3\nSoler, R.,Oliver, R., &Ballester, J. L.2009c, ApJ,submitt ed" }, { "title": "1708.02008v2.Chiral_damping__chiral_gyromagnetism_and_current_induced_torques_in_textured_one_dimensional_Rashba_ferromagnets.pdf", "content": "arXiv:1708.02008v2 [cond-mat.mes-hall] 31 Aug 2017Chiral damping, chiral gyromagnetism and current-induced torques in textured\none-dimensional Rashba ferromagnets\nFrank Freimuth,∗Stefan Bl¨ ugel, and Yuriy Mokrousov\nPeter Gr¨ unberg Institut and Institute for Advanced Simula tion,\nForschungszentrum J¨ ulich and JARA, 52425 J¨ ulich, German y\n(Dated: May 17, 2018)\nWe investigate Gilbert damping, spectroscopic gyromagnet ic ratio and current-induced torques\nin the one-dimensional Rashba model with an additional nonc ollinear magnetic exchange field. We\nfind that the Gilbert damping differs between left-handed and right-handed N´ eel-type magnetic\ndomain walls due to the combination of spatial inversion asy mmetry and spin-orbit interaction\n(SOI), consistent with recent experimental observations o f chiral damping. Additionally, we find\nthat also the spectroscopic gfactor differs between left-handed and right-handed N´ eel- type domain\nwalls, which we call chiral gyromagnetism. We also investig ate the gyromagnetic ratio in the Rashba\nmodel with collinear magnetization, where we find that scatt ering corrections to the gfactor vanish\nfor zero SOI, become important for finite spin-orbit couplin g, and tend to stabilize the gyromagnetic\nratio close to its nonrelativistic value.\nI. INTRODUCTION\nIn magnetic bilayer systems with structural inversion\nasymmetry the energies of left-handed and right-handed\nN´ eel-type domain walls differ due to the Dzyaloshinskii-\nMoriya interaction (DMI) [1–4]. DMI is a chiral interac-\ntion, i.e., it distinguishes between left-handed and right-\nhanded spin-spirals. Not only the energy is sensitive to\nthe chirality of spin-spirals. Recently, it has been re-\nported that the orbital magnetic moments differ as well\nbetween left-handed and right-handed cycloidal spin spi-\nrals in magnetic bilayers [5, 6]. Moreover, the experi-\nmental observation of asymmetry in the velocity of do-\nmain walls driven by magnetic fields suggests that also\nthe Gilbert damping is sensitive to chirality [7, 8].\nIn this work we show that additionally the spectro-\nscopic gyromagnetic ratio γis sensitive to the chirality\nof spin-spirals. The spectroscopic gyromagnetic ratio γ\ncan be defined by the equation\ndm\ndt=γT, (1)\nwhereTis the torque that acts on the magnetic moment\nmand dm/dtis the resulting rate of change. γenters\nthe Landau-Lifshitz-Gilbert equation (LLG):\ndˆM\ndt=γˆM×Heff+αGˆM×dˆM\ndt,(2)\nwhereˆMis a normalized vector that points in the direc-\ntionofthemagnetizationandthetensor αGdescribesthe\nGilbert damping. The chiralityofthe gyromagneticratio\nprovides another mechanism for asymmetries in domain-\nwall motion between left-handed and right-handed do-\nmain walls.\nNot only the damping and the gyromagnetic ratio\nexhibit chiral corrections in inversion asymmetric sys-\ntems but also the current-induced torques. Amongthese torques that act on domain-walls are the adia-\nbatic and nonadiabatic spin-transfer torques [9–12] and\nthe spin-orbit torque [13–16]. Based on phenomenologi-\ncal grounds additional types of torques have been sug-\ngested [17]. Since this large number of contributions\nare difficult to disentangle experimentally, current-driven\ndomain-wall motion in inversion asymmetric systems is\nnot yet fully understood.\nThe two-dimensionalRashbamodel with an additional\nexchange splitting has been used to study spintronics\neffects associated with the interfaces in magnetic bi-\nlayer systems [18–22]. Recently, interest in the role of\nDMI in one-dimensional magnetic chains has been trig-\ngered [23, 24]. For example, the magnetic moments in\nbi-atomic Fe chains on the Ir surface order in a 120◦\nspin-spiral state due to DMI [25]. Apart from DMI, also\nother chiral effects, such as chiral damping and chiral\ngyromagnetism, are expected to be important in one-\ndimensional magnetic chains on heavy metal substrates.\nThe one-dimensional Rashba model [26, 27] with an ad-\nditional exchange splitting can be used to simulate spin-\norbit driven effects in one-dimensional magnetic wires on\nsubstrates [28–30]. While the generalized Bloch theo-\nrem[31]usuallycannotbeusedtotreatspin-spiralswhen\nSOI is included in the calculation, the one-dimensional\nRashba model has the advantage that it can be solved\nwith the help of the generalized Bloch theorem, or with a\ngauge-field approach [32], when the spin-spiral is of N´ eel-\ntype. WhenthegeneralizedBlochtheoremcannotbeem-\nployed one needs to resort to a supercell approach [33],\nuse open boundary conditions [34, 35], or apply pertur-\nbation theory [6, 9, 36–39] in order to study spintronics\neffects in noncollinear magnets with SOI. In the case of\nthe one-dimensional Rashba model the DMI and the ex-\nchangeparameterswerecalculatedbothdirectlybasedon\nagauge-fieldapproachandfromperturbationtheory[38].\nThe results from the two approaches were found to be in\nperfect agreement. Thus, the one-dimensional Rashba2\nmodel provides also an excellent opportunity to verify\nexpressions obtained from perturbation theory by com-\nparisonto the resultsfromthe generalizedBlochtheorem\nor from the gauge-field approach.\nIn this work we study chiral gyromagnetism and chi-\nral damping in the one-dimensional Rashba model with\nan additional noncollinear magnetic exchange field. The\none-dimensional Rashba model is very well suited to\nstudy these SOI-driven chiral spintronics effects, because\nit can be solved in a very transparent way without the\nneed for a supercell approach, open boundary conditions\nor perturbation theory. We describe scattering effects by\nthe Gaussian scalar disorder model. To investigate the\nrole of disorder for the gyromagnetic ratio in general, we\nstudyγalso in the two-dimensional Rashba model with\ncollinear magnetization. Additionally, we compute the\ncurrent-induced torques in the one-dimensional Rashba\nmodel.\nThis paper is structured as follows: In section IIA we\nintroduce the one-dimensional Rashba model. In sec-\ntion IIB we discuss the formalism for the calculation\nof the Gilbert damping and of the gyromagnetic ratio.\nIn section IIC we present the formalism used to calcu-\nlate the current-induced torques. In sections IIIA, IIIB,\nand IIIC we discuss the gyromagnetic ratio, the Gilbert\ndamping, and the current-induced torques in the one-\ndimensionalRashbamodel, respectively. Thispaperends\nwith a summary in section IV.\nII. FORMALISM\nA. One-dimensional Rashba model\nThe two-dimensional Rashba model is given by the\nHamiltonian [19]\nH=−/planckover2pi12\n2me∂2\n∂x2−/planckover2pi12\n2me∂2\n∂y2+\n+iαRσy∂\n∂x−iαRσx∂\n∂y+∆V\n2σ·ˆM(r),(3)\nwhere the first line describes the kinetic energy, the first\ntwotermsin thesecondline describethe RashbaSOI and\nthe last term in the second line describes the exchange\nsplitting. ˆM(r) is the magnetization direction, which\nmay depend on the position r= (x,y), andσis the\nvector of Pauli spin matrices. By removing the terms\nwith the y-derivatives from Eq. (3), i.e., −/planckover2pi12\n2me∂2\n∂y2and\n−iαRσx∂\n∂y, one obtains a one-dimensional variant of the\nRashba model with the Hamiltonian [38]\nH=−/planckover2pi12\n2me∂2\n∂x2+iαRσy∂\n∂x+∆V\n2σ·ˆM(x).(4)\nEq. (4) is invariant under the simultaneous rotation\nofσand of the magnetization ˆMaround the yaxis.Therefore, if ˆM(x) describes a flat cycloidal spin-spiral\npropagating into the xdirection, as given by\nˆM(x) =\nsin(qx)\n0\ncos(qx)\n, (5)\nwe can use the unitary transformation\nU(x) =/parenleftBigg\ncos(qx\n2)−sin(qx\n2)\nsin(qx\n2) cos(qx\n2)/parenrightBigg\n(6)\nin order to transform Eq. (4) into a position-independent\neffective Hamiltonian [38]:\nH=1\n2m/parenleftbig\npx+eAeff\nx/parenrightbig2−m(αR)2\n2/planckover2pi12+∆V\n2σz,(7)\nwherepx=−i/planckover2pi1∂/∂xis thexcomponent of the momen-\ntum operator and\nAeff\nx=−m\ne/planckover2pi1/parenleftbigg\nαR+/planckover2pi12\n2mq/parenrightbigg\nσy (8)\nis thex-component of the effective magnetic vector po-\ntential. Eq. (8) shows that the noncollinearity described\nbyqacts like an effective SOI in the special case of the\none-dimensional Rashba model. This suggests to intro-\nduce the concept of effective SOI strength\nαR\neff=αR+/planckover2pi12\n2mq. (9)\nBased on this concept of the effective SOI strength\none can obtain the q-dependence of the one-dimensional\nRashba model from its αR-dependence at q= 0. That a\nnoncollinear magnetic texture provides a nonrelativistic\neffective SOI has been found also in the context of the\nintrinsic contribution to the nonadiabatic torque in the\nabsence of relativistic SOI, which can be interpreted as\na spin-orbit torque arising from this effective SOI [40].\nWhile the Hamiltonian in Eq. (4) depends on position\nxthrough the position-dependence of the magnetization\nˆM(x) in Eq. (5), the effective Hamiltonian in Eq. (7) is\nnot dependent on xand therefore easy to diagonalize.\nB. Gilbert damping and gyromagnetic ratio\nIn collinear magnets damping and gyromagnetic ratio\ncan be extracted from the tensor [16]\nΛij=−1\nVlim\nω→0ImGR\nTi,Tj(/planckover2pi1ω)\n/planckover2pi1ω, (10)\nwhereVis the volume of the unit cell and\nGR\nTi,Tj(/planckover2pi1ω) =−i∞/integraldisplay\n0dteiωt/angbracketleft[Ti(t),Tj(0)]−/angbracketright(11)3\nis the retarded torque-torque correlation function. Tiis\nthei-th component of the torque operator [16]. The dc-\nlimitω→0 in Eq. (10) is only justified when the fre-\nquency of the magnetization dynamics, e.g., the ferro-\nmagnetic resonance frequency, is smaller than the relax-\nationrateoftheelectronicstates. In thin magneticlayers\nand monoatomicchains on substratesthis is typically the\ncase due to the strong interfacial disorder. However, in\nvery pure crystalline samples at low temperatures the\nrelaxation rate may be smaller than the ferromagnetic\nresonance frequency and one needs to assume ω >0 in\nEq. (10) [41, 42]. The tensor Λdepends on the mag-\nnetization direction ˆMand we decompose it into the\ntensorS, which is even under magnetization reversal\n(S(ˆM) =S(−ˆM)), and the tensor A, which is odd un-\nder magnetization reversal ( A(ˆM) =−A(−ˆM)), such\nthatΛ=S+A, where\nSij(ˆM) =1\n2/bracketleftBig\nΛij(ˆM)+Λij(−ˆM)/bracketrightBig\n(12)\nand\nAij(ˆM) =1\n2/bracketleftBig\nΛij(ˆM)−Λij(−ˆM)/bracketrightBig\n.(13)\nOne can show that Sis symmetric, i.e., Sij(ˆM) =\nSji(ˆM), while Ais antisymmetric, i.e., Aij(ˆM) =\n−Aji(ˆM).\nThe Gilbert damping may be extracted from the sym-\nmetric component Sas follows [16]:\nαG\nij=|γ|Sij\nMµ0, (14)\nwhereMis the magnetization. The gyromagnetic ratio\nγis obtained from Λ according to the equation [16]\n1\nγ=1\n2µ0M/summationdisplay\nijkǫijkΛijˆMk=1\n2µ0M/summationdisplay\nijkǫijkAijˆMk.\n(15)\nIt is convenient to discuss the gyromagnetic ratio in\nterms of the dimensionless g-factor, which is related to\nγthrough γ=gµ0µB//planckover2pi1. Consequently, the g-factor is\ngiven by\n1\ng=µB\n2/planckover2pi1M/summationdisplay\nijkǫijkΛijˆMk=µB\n2/planckover2pi1M/summationdisplay\nijkǫijkAijˆMk.(16)\nDue to the presence of the Levi-Civita tensor ǫijkin\nEq. (15) and in Eq. (16) the gyromagnetic ratio and the\ng-factoraredetermined solelyby the antisymmetriccom-\nponentAofΛ.\nVarious different conventions are used in the literature\nconcerning the sign of the g-factor [43]. Here, we define\nthe sign of the g-factor such that γ >0 forg >0 and\nγ <0 forg <0. According to Eq. (1) the rate of change\nofthemagneticmomentisthereforeparalleltothetorqueforpositive gandantiparalleltothetorquefornegative g.\nWhile we are interested in this work in the spectroscopic\ng-factor, and hence in the relation between the rate of\nchange of the magnetic moment and the torque, Ref. [43]\ndiscusses the relation between the magnetic moment m\nandtheangularmomentum Lthatgeneratesit, i.e., m=\nγstaticL. Since differentiation with respect to time and\nuse ofT= dL/dtleads to Eq. (1) our definition of the\nsigns ofgandγagrees essentially with the one suggested\nin Ref. [43], which proposes to use a positive gwhen the\nmagnetic moment is parallel to the angular momentum\ngeneratingitandanegative gwhenthemagneticmoment\nis antiparallel to the angular momentum generating it.\nCombining Eq. (14) and Eq. (15) we can express the\nGilbert damping in terms of AandSas follows:\nαG\nxx=Sxx\n|Axy|. (17)\nIntheindependentparticleapproximationEq.(10)can\nbe written as Λij= ΛI(a)\nij+ΛI(b)\nij+ΛII\nij, where\nΛI(a)\nij=1\nh/integraldisplayddk\n(2π)dTr/angbracketleftbig\nTiGR\nk(EF)TjGA\nk(EF)/angbracketrightbig\nΛI(b)\nij=−1\nh/integraldisplayddk\n(2π)dReTr/angbracketleftbig\nTiGR\nk(EF)TjGR\nk(EF)/angbracketrightbig\nΛII\nij=1\nh/integraldisplayddk\n(2π)d/integraldisplayEF\n−∞dEReTr/angbracketleftbigg\nTiGR\nk(E)TjdGR\nk(E)\ndE\n− TidGR\nk(E)\ndETjGR\nk(E)/angbracketrightbigg\n.(18)\nHere,dis the dimension ( d= 1 ord= 2 ord= 3),GR\nk(E)\nis the retarded Green’s function and GA\nk(E) = [GR\nk(E)]†.\nEFis the Fermi energy. ΛI(b)\nijis symmetric under the\ninterchange of the indices iandjwhile ΛII\nijis antisym-\nmetric. The term ΛI(a)\nijcontains both symmetric and\nantisymmetric components. Since the Gilbert damping\ntensor is symmetric, both ΛI(b)\nijand ΛI(a)\nijcontribute to\nit. Since the gyromagnetic tensor is antisymmetric, both\nΛII\nijand ΛI(a)\nijcontribute to it.\nIn order to account for disorder we use the Gaus-\nsian scalardisordermodel, wherethe scatteringpotential\nV(r) satisfies /angbracketleftV(r)/angbracketright= 0 and /angbracketleftV(r)V(r′)/angbracketright=Uδ(r−r′).\nThis model is frequently used to calculate transport\nproperties in disordered multiband model systems [44],\nbut it has also been combined with ab-initio electronic\nstructure calculations to study the anomalous Hall ef-\nfect [45, 46] and the anomalous Nernst effect [47] in tran-\nsition metals and their alloys.\nIn the clean limit, i.e., in the limit U→0, the an-\ntisymmetric contribution to Eq. (18) can be written as4\nAij=Aint\nij+Ascatt\nij, where the intrinsic part is given by\nAint\nij=/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nn,m[fkn−fkm]ImTi\nknmTj\nkmn\n(Ekn−Ekm)2\n= 2/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nn/summationdisplay\nll′fknIm/bracketleftbigg∂/angbracketleftukn|\n∂ˆMl∂|ukn/angbracketright\n∂ˆMl′/bracketrightbigg\n×\n×/summationdisplay\nmm′ǫilmǫjl′m′ˆMmˆMm′.\n(19)\nThe second line in Eq. (19) expresses Aint\nijin terms of\nthe Berry curvature in magnetization space [48]. The\nscattering contribution is given by\nAscatt\nij=/planckover2pi1/summationdisplay\nnm/integraldisplayddk\n(2π)dδ(EF−Ekn)Im/braceleftBigg\n−/bracketleftbigg\nMi\nknmγkmn\nγknnTj\nknn−Mj\nknmγkmn\nγknnTi\nknn/bracketrightbigg\n+/bracketleftBig\nMi\nkmn˜Tj\nknm−Mj\nkmn˜Ti\nknm/bracketrightBig\n−/bracketleftbigg\nMi\nknmγkmn\nγknn˜Tj\nknn−Mj\nknmγkmn\nγknn˜Ti\nknn/bracketrightbigg\n+/bracketleftBigg\n˜Ti\nknnγknm\nγknn˜Tj\nkmn\nEkn−Ekm−˜Tj\nknnγknm\nγknn˜Ti\nkmn\nEkn−Ekm/bracketrightBigg\n+1\n2/bracketleftbigg\n˜Ti\nknm1\nEkn−Ekm˜Tj\nkmn−˜Tj\nknm1\nEkn−Ekm˜Ti\nkmn/bracketrightbigg\n+/bracketleftBig\nTj\nknnγknm\nγknn1\nEkn−Ekm˜Ti\nkmn\n−Ti\nknnγknm\nγknn1\nEkn−Ekm˜Tj\nkmn/bracketrightBig/bracerightBigg\n.\n(20)\nHere,Ti\nknm=/angbracketleftukn|Ti|ukm/angbracketrightare the matrix elements of\nthe torque operator. ˜Ti\nknmdenotes the vertex corrections\nof the torque, which solve the equation\n˜Ti\nknm=/summationdisplay\np/integraldisplaydnk′\n(2π)n−1δ(EF−Ek′p)\n2γk′pp×\n×/angbracketleftukn|uk′p/angbracketright/bracketleftBig\n˜Ti\nk′pp+Ti\nk′pp/bracketrightBig\n/angbracketleftuk′p|ukm/angbracketright.(21)\nThe matrix γknmis given by\nγknm=−π/summationdisplay\np/integraldisplayddk′\n(2π)dδ(EF−Ek′p)/angbracketleftukn|uk′p/angbracketright/angbracketleftuk′p|ukm/angbracketright\n(22)\nand the Berry connection in magnetization space is de-\nfined as\niMj\nknm=iTj\nknm\nEkm−Ekn. (23)\nThe scattering contribution Eq. (20) formally resembles\nthe side-jump contribution to the AHE [44] as obtainedfrom the scalar disorder model: It can be obtained by\nreplacing the velocity operators in Ref. [44] by torque\noperators. We find thatin collinearmagnetswithoutSOI\nthis scattering contribution vanishes. The gyromagnetic\nratio is then given purely by the intrinsic contribution\nEq. (19). This is an interesting difference to the AHE:\nWithout SOI all contributions to the AHE are zero in\ncollinear magnets, while both the intrinsic and the side-\njump contributions are generally nonzero in the presence\nof SOI.\nIn the absence of SOI Eq. (19) can be expressed in\nterms of the magnetization [48]:\nAint\nij=−/planckover2pi1\n2µB/summationdisplay\nkǫijkMk. (24)\nInserting Eq. (24) into Eq. (16) yields g=−2, i.e., the\nexpected nonrelativistic value of the g-factor.\nTheg-factor in the presence of SOI is usually assumed\nto be given by [49]\ng=−2Mspin+Morb\nMspin=−2M\nMspin,(25)\nwhereMorbis the orbital magnetization, Mspinis the\nspin magnetization and M=Morb+Mspinis the total\nmagnetization. The g-factor obtained from Eq. (25) is\nusually in good agreementwith experimental results [50].\nWhen SOI is absent, the orbital magnetization is zero,\nMorb= 0, and consequently Eq. (25) yields g=−2 in\nthat case. Eq. (16) can be rewritten as\n1\ng=Mspin\nMµB\n2/planckover2pi1Mspin/summationdisplay\nijkǫijkAijˆMk=Mspin\nM1\ng1,(26)\nwith\n1\ng1=µB\n2/planckover2pi1Mspin/summationdisplay\nijkǫijkAijˆMk. (27)\nFrom the comparison of Eq. (26) with Eq. (25) it follows\nthat Eq. (25) holds exactly if g1=−2 is satisfied. How-\never, Eq. (27) usually yields g1=−2 only in collinear\nmagnets when SOI is absent, otherwise g1/negationslash=−2. In the\none-dimensionalRashbamodel the orbitalmagnetization\nis zero,Morb= 0, and consequently\n1\ng=µB\n2/planckover2pi1Mspin/summationdisplay\nijkǫijkAijˆMk. (28)\nThe symmetric contribution can be written as Sij=\nSint\nij+SRR−vert\nij+SRA−vert\nij, where\nSint\nij=1\nh/integraldisplayddk\n(2π)dTr/braceleftbig\nTiGR\nk(EF)Tj/bracketleftbig\nGA\nk(EF)−GR\nk(EF)/bracketrightbig/bracerightbig\n(29)5\nand\nSRR−vert\nij=−1\nh/integraldisplayddk\n(2π)dTr/braceleftBig\n˜TRR\niGR\nk(EF)TjGR\nk(EF)/bracerightBig\n(30)\nand\nSAR−vert\nij=1\nh/integraldisplayddk\n(2π)dTr/braceleftBig\n˜TAR\niGR\nk(EF)TjGA\nk(EF)/bracerightBig\n,\n(31)\nwhereGR\nk(EF) =/planckover2pi1[EF−Hk−ΣR\nk(EF)]−1is the retarded\nGreen’s function, GA\nk(EF) =/bracketleftbig\nGR\nk(EF)/bracketrightbig†is the advanced\nGreen’s function and\nΣR(EF) =U\n/planckover2pi1/integraldisplayddk\n(2π)dGR\nk(EF) (32)\nis the retarded self-energy. The vertex corrections are\ndetermined by the equations\n˜TAR=T+U\n/planckover2pi12/integraldisplayddk\n(2π)dGA\nk(EF)˜TAR\nkGR\nk(EF) (33)\nand\n˜TRR=T+U\n/planckover2pi12/integraldisplayddk\n(2π)dGR\nk(EF)˜TRR\nkGR\nk(EF).(34)\nIn contrast to the antisymmetric tensor A, which be-\ncomes independent of the scattering strength Ufor suf-\nficiently small U, i.e., in the clean limit, the symmetric\ntensorSdepends strongly on Uin metallic systems in\nthe clean limit. Sint\nijandSscatt\nijdepend therefore on U\nthrough the self-energy and through the vertex correc-\ntions.\nIn the case of the one-dimensional Rashba model, the\nequations Eq. (19) and Eq. (20) for the antisymmet-\nric tensor Aand the equations Eq. (29), Eq. (30) and\nEq. (31) for the symmetric tensor Scan be used both\nfor the collinear magnetic state as well as for the spin-\nspiral of Eq. (5). To obtain the g-factor for the collinear\nmagnetic state, we plug the eigenstates and eigenvalues\nof Eq. (4) (with ˆM=ˆez) into Eq. (19) and into Eq. (20).\nIn the case of the spin-spiral of Eq. (5) we use instead the\neigenstates and eigenvalues of Eq. (7). Similarly, to ob-\ntain the Gilbert damping in the collinear magnetic state,\nwe evaluate Eq. (29), Eq. (30) and Eq. (31) based on\nthe Hamiltonian in Eq. (4) and for the spin-spiral we use\ninstead the effective Hamiltonian in Eq. (7).\nC. Current-induced torques\nThe current-induced torque on the magnetization can\nbe expressed in terms of the torkance tensor tijas [15]\nTi=/summationdisplay\njtijEj, (35)whereEjis thej-th component of the applied elec-\ntric field and Tiis thei-th component of the torque\nper volume [51]. tijis the sum of three terms, tij=\ntI(a)\nij+tI(b)\nij+tII\nij, where [15]\ntI(a)\nij=e\nh/integraldisplayddk\n(2π)dTr/angbracketleftbig\nTiGR\nk(EF)vjGA\nk(EF)/angbracketrightbig\ntI(b)\nij=−e\nh/integraldisplayddk\n(2π)dReTr/angbracketleftbig\nTiGR\nk(EF)vjGR\nk(EF)/angbracketrightbig\ntII\nij=e\nh/integraldisplayddk\n(2π)d/integraldisplayEF\n−∞dEReTr/angbracketleftbigg\nTiGR\nk(E)vjdGR\nk(E)\ndE\n− TidGR\nk(E)\ndEvjGR\nk(E)/angbracketrightbigg\n.(36)\nWe decompose the torkance into two parts that are,\nrespectively, even and odd with respect to magnetiza-\ntion reversal, i.e., te\nij(ˆM) = [tij(ˆM) +tij(−ˆM)]/2 and\nto\nij(ˆM) = [tij(ˆM)−tij(−ˆM)]/2.\nIn the clean limit, i.e., for U→0, the even torkance\ncan be written as te\nij=te,int\nij+te,scatt\nij, where [15]\nte,int\nij= 2e/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nn/negationslash=mfknImTi\nknmvj\nkmn\n(Ekn−Ekm)2(37)\nis the intrinsic contribution and\nte,scatt\nij=e/planckover2pi1/summationdisplay\nnm/integraldisplayddk\n(2π)dδ(EF−Ekn)Im/braceleftBigg\n/bracketleftBig\n−Mi\nknmγkmn\nγknnvj\nknn+Aj\nknmγkmn\nγknnTi\nknn/bracketrightBig\n+/bracketleftBig\nMi\nkmn˜vj\nknm−Aj\nkmn˜Ti\nknm/bracketrightBig\n−/bracketleftBig\nMi\nknmγkmn\nγknn˜vj\nknn−Aj\nknmγkmn\nγknn˜Ti\nknn/bracketrightBig\n+/bracketleftBig\n˜vj\nkmnγknm\nγknn˜Ti\nnn\nEkn−Ekm−˜Ti\nkmnγknm\nγknn˜vj\nknn\nEkn−Ekm/bracketrightBig\n+1\n2/bracketleftBig\n˜vj\nknm1\nEkn−Ekm˜Ti\nkmn−˜Ti\nknm1\nEkn−Ekm˜vj\nkmn/bracketrightBig\n+/bracketleftBig\nvj\nknnγknm\nγknn1\nEkn−Ekm˜Ti\nkmn\n−Ti\nknnγknm\nγknn1\nEkn−Ekm˜vj\nkmn/bracketrightBig/bracerightBigg\n.\n(38)\nis the scattering contribution. Here,\niAj\nknm=ivj\nknm\nEkm−Ekn=i\n/planckover2pi1/angbracketleftukn|∂\n∂kj|ukm/angbracketright(39)\nis the Berry connection in kspace and the vertex correc-\ntions of the velocity operator solve the equation\n˜vi\nknm=/summationdisplay\np/integraldisplaydnk′\n(2π)n−1δ(EF−Ek′p)\n2γk′pp×\n×/angbracketleftukn|uk′p/angbracketright/bracketleftbig\n˜vi\nk′pp+vi\nk′pp/bracketrightbig\n/angbracketleftuk′p|ukm/angbracketright.(40)6\nThe odd contribution can be written as to\nij=to,int\nij+\ntRR−vert\nij+tAR−vert\nij, where\nto,int\nij=e\nh/integraldisplayddk\n(2π)dTr/braceleftbig\nTiGR\nk(EF)vj/bracketleftbig\nGA\nk(EF)−GR\nk(EF)/bracketrightbig/bracerightbig\n(41)\nand\ntRR−vert\nij=−e\nh/integraldisplayddk\n(2π)dTr/braceleftBig\n˜TRR\niGR\nk(EF)vjGR\nk(EF)/bracerightBig\n(42)\nand\ntAR−vert\nij=e\nh/integraldisplayddk\n(2π)dTr/braceleftBig\n˜TAR\niGR\nk(EF)vjGA\nk(EF)/bracerightBig\n.(43)\nThe vertex corrections ˜TAR\niand˜TRR\niof the torque op-\nerator are given in Eq. (33) and in Eq. (34), respectively.\nWhile the even torkance, Eq. (37) and Eq. (38), be-\ncomes independent of the scattering strength Uin the\nclean limit, i.e., for U→0, the odd torkance to\nijdepends\nstrongly on Uin metallic systems in the clean limit [15].\nIn the case of the one-dimensional Rashba model, the\nequations Eq. (37) and Eq. (38) for the even torkance\nte\nijand the equations Eq. (41), Eq. (42) and Eq. (43) for\nthe odd torkance to\nijcan be used both for the collinear\nmagnetic state as well as for the spin-spiral of Eq. (5).\nTo obtain the even torkance for the collinear magnetic\nstate, we plug the eigenstates and eigenvalues of Eq. (4)\n(withˆM=ˆez) into Eq. (37) and into Eq. (38). In the\ncase of the spin-spiral of Eq. (5) we use instead the eigen-\nstates and eigenvalues of Eq. (7). Similarly, to obtain the\nodd torkance in the collinear magnetic state, we evaluate\nEq. (41), Eq. (42) and Eq. (43) based on the Hamilto-\nnian in Eq. (4) and for the spin-spiral we use instead the\neffective Hamiltonian in Eq. (7).\nIII. RESULTS\nA. Gyromagnetic ratio\nWe first discuss the g-factor in the collinear case, i.e.,\nwhenˆM(r) =ˆez. Inthis casetheenergybandsaregiven\nby\nE=/planckover2pi12k2\nx\n2m±/radicalbigg\n1\n4(∆V)2+(αRkx)2.(44)\nWhen ∆ V/negationslash= 0 orαR/negationslash= 0 the energy Ecan become\nnegative. The band structure of the one-dimensional\nRashba model is shown in Fig. 1 for the model param-\netersαR=2eV˚A and ∆ V= 0.5eV. For this choice of\nparameters the energy minima are not located at kx= 0\nbut instead at\nkmin\nx=±/radicalBig\n(αR)4m2−1\n4/planckover2pi14(∆V)2\n/planckover2pi12αR,(45)-0.4 -0.2 0 0.2 0.4\nk-Point kx [Å-1]00.511.5Band energy [eV]\nFIG. 1: Band structure of theone-dimensional Rashbamodel.\nand the corresponding minimum of the energy is given\nby\nEmin=−m(αR)4+1\n4/planckover2pi14\nm(∆V)2\n2/planckover2pi12(αR)2. (46)\nThe inverse g-factor is shown as a function of the SOI\nstrength αRin Fig. 2 for the exchange splitting ∆ V=\n1eV and Fermi energy EF= 1.36eV. At αR= 0 the\nscattering contribution is zero, i.e., the g-factor is de-\ntermined completely by the intrinsic Berry curvature ex-\npression, Eq. (24). Thus, at αR= 0 it assumes the value\n1/g=−0.5, which is the expected nonrelativistic value\n(see the discussion below Eq. (24)). With increasing SOI\nstrength αRthe intrinsic contribution to 1 /gis more and\nmore suppressed. However, the scattering contribution\ncompensates this decrease such that the total 1 /gis close\nto its nonrelativistic value of −0.5. The neglect of the\nscattering corrections at large values of αRwould lead in\nthis case to a strong underestimation of the magnitude\nof 1/g, i.e., a strong overestimation of the magnitude of\ng.\nHowever, at smaller values of the Fermi energy, the\ngfactor can deviate substantially from its nonrelativis-\ntic value of −2. To show this we plot in Fig. 3 the in-\nverseg-factor as a function of the Fermi energy when\nthe exchange splitting and the SOI strength are set to\n∆V= 1eV and αR=2eV˚A, respectively. As discussed in\nEq. (44) the minimal Fermi energyis negativ in this case.\nThe intrinsic contribution to 1 /gdeclines with increas-\ning Fermi energy. At large values of the Fermi energy\nthis decline is compensated by the increase of the vertex\ncorrections and the total value of 1 /gis close to −0.5.\nPrevious theoretical works on the g-factor have not\nconsidered the scattering contribution [52]. It is there-\nfore important to find out whether the compensation\nof the decrease of the intrinsic contribution by the in-7\n00.511.52\nSOI strength αR [eVÅ]-0.5-0.4-0.3-0.2-0.101/gscattering\nintrinsic\ntotal\nFIG. 2: Inverse g-factor vs. SOI strength αRin the one-\ndimensional Rashba model.\n0 1 2 3 4 5 6\nFermi energy [eV]-0.6-0.4-0.201/gscattering\nintrinsic\ntotal\nFIG. 3: Inverse g-factor vs. Fermi energy in the one-\ndimensional Rashba model.\ncrease of the extrinsic contribution as discussed in Fig. 2\nand Fig. 3 is peculiar to the one-dimensional Rashba\nmodel or whether it can be found in more general cases.\nFor this reason we evaluate g1for the two-dimensional\nRashba model. In Fig. 4 we show the inverse g1-factor\nin the two-dimensional Rashba model as a function of\nSOI strength αRfor the exchange splitting ∆ V= 1eV\nand the Fermi energy EF= 1.36eV. Indeed for αR<\n0.5eV˚A the scattering corrections tend to stabilize g1at\nits non-relativistic value. However, in contrast to the\none-dimensional case (Fig. 2), where gdoes not deviate\nmuch from its nonrelativistic value up to αR= 2eV˚A,\ng1starts to be affected by SOI at smaller values of αR\nin the two-dimensional case. According to Eq. (26) the\nfullgfactor is given by g=g1(1+Morb/Mspin). There-\nfore, when the scattering corrections stabilize g1at its00.511.52\nSOI strength αR [eVÅ]-0.5-0.4-0.3-0.2-0.101/g1\nscattering\nintrinsic\ntotal\nFIG. 4: Inverse g1-factor vs. SOI strength αRin the two-\ndimensional Rashba model.\nnonrelativistic value the Eq. (25) is satisfied. In the two-\ndimensional Rashba model Morb= 0 when both bands\nare occupied. For the Fermi energy EF= 1.36eV both\nbands are occupied and therefore g=g1for the range of\nparameters used in Fig. 4.\nThe inverse g1of the two-dimensional Rashba model\nis shown in Fig. 5 as a function of Fermi energy for the\nparameters ∆ V= 1eV and αR= 2eV˚A. The scattering\ncorrection is as large as the intrinsic contribution when\nEF>1eV. While the scattering correction is therefore\nimportant, it is not sufficiently large to bring g1close to\nits nonrelativistic value in the energy range shown in the\nfigure, which is a major difference to the one-dimensional\ncase illustrated in Fig. 3. According to Eq. (26) the g\nfactor is related to g1byg=g1M/Mspin. Therefore, we\nshow in Fig. 6 the ratio M/Mspinas a function of Fermi\nenergy. AthighFermienergy(whenbothbandsareoccu-\npied) the orbital magnetization is zeroand M/Mspin= 1.\nAt low Fermi energy the sign of the orbital magnetiza-\ntionis oppositeto the signofthe spin magnetizationsuch\nthat the magnitude of Mis smaller than the magnitude\nofMspinresulting in the ratio M/Mspin<1.\nNext, we discuss the g-factor of the one-dimensional\nRashba model in the noncollinear case. In Fig. 7 we\nplot the inverse g-factor and its decomposition into the\nintrinsic and scattering contributions as a function of\nthe spin-spiral wave vector q, where exchange splitting,\nSOI strength and Fermi energy are set to ∆ V= 1eV,\nαR= 2eV˚A andEF= 1.36eV, respectively. Since\nthe curves are not symmetric around q= 0, the g-\nfactor at wave number qdiffers from the one at −q, i.e.,\nthegyromagnetism in the Rashba model is chiral . At\nq=−2meαR//planckover2pi12theg-factorassumesthevalueof g=−2\nand the scattering corrections are zero. Moreover, the\ncurves are symmetric around q=−2meαR//planckover2pi12. These8\n0 2 4 6\nFermi energy [eV]-0.5-0.4-0.3-0.2-0.101/g1\nscattering\nintrinsic\ntotal\nFIG. 5: Inverse g1-factor 1 /g1vs. Fermi energy in the two-\ndimensional Rashba model.\n-2 0 2 4 6\nFermi energy [eV]00.511.52M/Mspin\nFIG. 6: Ratio of total magnetization and spin magnetization ,\nM/Mspin, vs. Fermi energy in the two-dimensional Rashba\nmodel.\nobservationscan be explained by the concept of the effec-\ntive SOI introduced in Eq. (9): At q=−2meαR//planckover2pi12the\neffective SOI is zero and consequently the noncollinear\nmagnet behaves like a collinear magnet without SOI at\nthis value of q. As we have discussed above in Fig. 2, the\ng-factor of collinear magnets is g=−2 when SOI is ab-\nsent, which explains why it is also g=−2 in noncollinear\nmagnets with q=−2meαR//planckover2pi12. If only the intrinsic con-\ntribution is considered and the scattering corrections are\nneglected, 1 /gvaries much stronger around the point of\nzero effective SOI q=−2meαR//planckover2pi12, i.e., the scattering\ncorrections stabilize gat its nonrelativistic value close to\nthe point of zero effective SOI.-2 -1 0 1\nWave vector q [Å-1]-0.8-0.6-0.4-0.201/g\nscattering\nintrinsic\ntotal\nFIG. 7: Inverse g-factor 1 /gvs. wave number qin the one-\ndimensional Rashba model.\n0 1 2 3 4\nScattering strength U [(eV)2Å]-0.4-0.200.20.4Gilbert Damping αG\nxx\nRR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 8: Gilbert damping αG\nxxvs. scattering strength Uin the\none-dimensional Rashba model without SOI. In this case the\nvertex corrections and the intrinsic contribution sum up to\nzero.\nB. Damping\nWe first discuss the Gilbert damping in the collinear\ncase, i.e., we set ˆM(r) =ˆezin Eq. (4). The xxcom-\nponent of the Gilbert damping is shown in Fig. 8 as\na function of scattering strength Ufor the following\nmodel parameters: exchange splitting ∆ V=1eV, Fermi\nenergyEF= 2.72eV and SOI strength αR= 0. All\nthree contributions are individually non-zero, but the\ncontribution from the RR-vertex correction (Eq. (30)) is\nmuchsmallerthanthe onefromthe AR-vertexcorrection\n(Eq. (31)) and much smaller than the intrinsic contribu-\ntion (Eq. (29)). However, in this case the total damping\nis zero, because a non-zero damping in periodic crystals\nwith collinear magnetization is only possible when SOI\nis present [53].9\n1 2 3 4\nScattering strength U [(eV)2Å]050100150200250300Gilbert Damping αG\nxx\nRR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 9: Gilbert damping αG\nxxvs. scattering strength Uin the\none-dimensional Rashba model with SOI.\nIn Fig. 9 we show the xxcomponent of the Gilbert\ndamping αG\nxxas a function of scattering strength Ufor\nthe model parameters ∆ V= 1eV, EF= 2.72eV and\nαR= 2eV˚A. ThedominantcontributionistheAR-vertex\ncorrection. The damping as obtained based on Eq. (10)\ndiverges like 1 /Uin the limit U→0, i.e., proportional\nto the relaxation time τ[53]. However, once the relax-\nation time τis larger than the inverse frequency of the\nmagnetization dynamics the dc-limit ω→0 in Eq. (10)\nis not appropriate and ω >0 needs to be used. It has\nbeenshownthattheGilbertdampingisnotinfinite inthe\nballistic limit τ→ ∞whenω >0 [41, 42]. In the one-\ndimensional Rashba model the effective magnetic field\nexerted by SOI on the electron spins points in ydirec-\ntion. Since a magnetic field along ydirection cannot lead\ntoatorquein ydirectionthe yycomponentoftheGilbert\ndamping αG\nyyis zero (not shown in the Figure).\nNext, we discuss the Gilbert damping in the non-\ncollinear case. In Fig. 10 we plot the xxcomponent\nof the Gilbert damping as a function of spin spiral\nwave number qfor the model parameters ∆ V= 1eV,\nEF= 1.36eV,αR= 2eV˚A, and the scattering strength\nU= 0.98(eV)2˚A. The curves are symmetric around\nq=−2meαR//planckover2pi12, because the damping is determined by\nthe effective SOI defined in Eq. (9). At q=−2meαR//planckover2pi12\nthe effective SOI is zero and therefore the total damp-\ning is zero as well. The damping at wave number qdif-\nfers from the one at wave number −q, i.e.,the damp-\ning is chiral in the Rashba model . Around the point\nq=−2meαR//planckover2pi12the damping is described by aquadratic\nparabola at first. In the regions -2 ˚A−1< q <-1.2˚A−1\nand 0.2˚A−1< q <1˚A−1this trend is interrupted by a W-\nshape behaviour. In the quadratic parabola region the\nlowest energy band crosses the Fermi energy twice. As\nshown in Fig. 1 the lowest band has a local maximum at-2-1.5-1-0.500.51\nWave vector q [Å-1]05101520Gilbert damping αxxG RR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 10: Gilbert damping αG\nxxvs. spin spiral wave number q\nin the one-dimensional Rashba model.\nq= 0. In the W-shape region this local maximum shifts\nupwards, approaches the Fermi level and finally passes it\nsuch that the lowest energy band crosses the Fermi level\nfour times. This transition in the band structure leads to\noscillations in the density of states, which results in the\nW-shape behaviour of the Gilbert damping.\nSince the damping is determined by the effective SOI,\nwe can use Fig. 10 to draw conclusions about the damp-\ning in the noncollinear case with αR= 0: We only need\nto shift all curves in Fig. 10 to the right such that they\nare symmetric around q= 0 and shift the Fermi energy.\nThus, for αR= 0 the Gilbert damping does not vanish\nifq/negationslash= 0. Since for αR= 0 angular momentum transfer\nfrom the electronic system to the lattice is not possible,\nthe damping is purely nonlocal in this case, i.e., angular\nmomentum is interchanged between electrons at differ-\nent positions. This means that for a volume in which\nthe magnetization of the spin-spiral in Eq. (5) performs\nexactly one revolution between one end of the volume\nand the other end the total angular momentum change\nassociated with the damping is zero, because the angu-\nlar momentum is simply redistributed within this volume\nand there is no net change of the angular momentum.\nA substantial contribution of nonlocal damping has also\nbeen predicted for domain walls in permalloy [35].\nIn Fig. 11 we plot the yycomponent of the Gilbert\ndamping as a function of spin spiral wave number qfor\nthe model parameters ∆ V= 1eV,EF= 1.36eV,αR=\n2eV˚A, and the scattering strength U= 0.98(eV)2˚A. The\ntotaldampingiszerointhiscase. Thiscanbeunderstood\nfrom the symmetry properties of the one-dimensional\nRashba Hamiltonian, Eq. (4): Since this Hamiltonian is\ninvariant when both σandˆMare rotated around the\nyaxis, the damping coefficient αG\nyydoes not depend on\nthe position within the cycloidal spin spiral of Eq. (5).10\n-3 -2 -1 0 1 2\nWave vector q [Å-1]-0.4-0.200.20.4Gilbert Damping αG\nyyRR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 11: Gilbert damping αG\nyyvs. spin spiral wave number q\nin the one-dimensional Rashba model.\nTherefore, nonlocal damping is not possible in this case\nandαG\nyyhas to be zero when αR= 0. It remains to be\nshown that αG\nyy= 0 also for αR/negationslash= 0. However, this fol-\nlows directly from the observation that the damping is\ndetermined by the effective SOI, Eq. (9), meaning that\nany case with q/negationslash= 0 and αR/negationslash= 0 can always be mapped\nonto a case with q/negationslash= 0 and αR= 0. As an alternative\nargumentation we can also invoke the finding discussed\nabovethat αG\nyy= 0 in the collinearcase. Since the damp-\ning is determined by the effective SOI, it follows that\nαG\nyy= 0 also in the noncollinear case.\nC. Current-induced torques\nWe first discuss the yxcomponent of the torkance. In\nFig. 12 we show the torkance tyxas a function of the\nFermi energy EFfor the model parameters ∆ V= 1eV\nandαR= 2eV˚A when the magnetization is collinear and\npoints in zdirection. We specify the torkance in units of\nthe positive elementary charge e, which is a convenient\nchoice for the one-dimensional Rashba model. When\nthe torkance is multiplied with the electric field, we ob-\ntain the torque per length (see Eq. (35) and Ref. [51]).\nSince the effective magnetic field from SOI points in\nydirection, it cannot give rise to a torque in ydirec-\ntion and consequently the total tyxis zero. Interest-\ningly, the intrinsic and scattering contributions are indi-\nvidually nonzero. The intrinsic contribution is nonzero,\nbecause the electric field accelerates the electrons such\nthat/planckover2pi1˙kx=−eEx. Therefore, the effective magnetic\nfieldBSOI\ny=αRkx/µBchanges as well, i.e., ˙BSOI\ny=\nαR˙kx/µB=−αRExe/(/planckover2pi1µB). Consequently, the electron\nspin is no longer aligned with the total effective magnetic\nfield (the effective magnetic field resulting from both SOI-1 0 1 2 3 4 5 6\nFermi energy [eV]-0.2-0.100.10.2Torkance tyx [e]scattering\nintrinsic\ntotal\nFIG. 12: Torkance tyxvs. Fermi energy EFin the one-\ndimensional Rashba model.\nand from the exchange splitting ∆ V), when an electric\nfield is applied. While the total effective magnetic field\nlies in the yzplane, the electron spin acquires an xcom-\nponent, because it precesses around the total effective\nmagnetic field, with which it is not aligned due to the\napplied electric field [54]. The xcomponent of the spin\ndensity results in a torque in ydirection, which is the\nreason why the intrinsic contribution to tyxis nonzero.\nThe scattering contribution to tyxcancels the intrinsic\ncontribution such that the total tyxis zero and angular\nmomentum conservation is satisfied.\nUsing the concept of effective SOI, Eq. (9), we con-\nclude that tyxis also zero for the noncollinear spin-spiral\ndescribed by Eq. (5). Thus, both the ycomponent of the\nspin-orbit torque and the nonadiabatic torque are zero\nfor the one-dimensional Rashba model.\nTo show that tyx= 0 is a peculiarity of the one-\ndimensional Rashba model, we plot in Fig. 13 the\ntorkance tyxin the two-dimensional Rashba model. The\nintrinsic and scattering contributions depend linearly on\nαRfor small values of αR, but the slopes are opposite\nsuch that the total tyxis zero for sufficiently small αR.\nHowever, for largervalues of αRthe intrinsic and scatter-\ning contributions do not cancel each other and therefore\nthe total tyxbecomes nonzero, in contrast to the one-\ndimensional Rashba model, where tyx= 0 even for large\nαR. Several previous works determined the part of tyx\nthat is proportionalto αRin the two-dimensionalRashba\nmodel and found it to be zero [21, 22] for scalar disor-\nder, which is consistent with our finding that the linear\nslopes of the intrinsic and scattering contributions to tyx\nare opposite for small αR.\nNext, we discuss the xxcomponent of the torkance\nin the collinear case ( ˆM=ˆez). In Fig. 14 we plot\nthe torkance txxvs. scattering strength Uin the one-11\n00.511.52\nSOI strength αR [eVÅ]-0.00500.0050.01Torkance tyx [e/Å]\nscattering\nintrinsic\ntotal\nFIG. 13: Nonadiabatic torkance tyxvs. SOI parameter αRin\nthe two-dimensional Rashba model.\ndimensional Rashba model for the parameters ∆ V=\n1eV,EF= 2.72eV and αR= 2eV˚A. The dominant con-\ntribution is the AR-type vertex correction (see Eq. (43)).\ntxxdiverges like 1 /Uin the limit U→0 as expected for\nthe odd torque in metallic systems [15].\nIn Fig. 15 and Fig. 16 we plot txxas a function of\nspin-spiral wave number qfor the model parameters\n∆V= 1eV,EF= 2.72eV and U= 0.18(eV)2˚A. In Fig. 15\nthe case with αR= 2eV˚A is shown, while Fig. 16 illus-\ntrates the case with αR= 0. In the case αR= 0 the\ntorkance txxdescribes the spin-transfer torque (STT). In\nthe case αR/negationslash= 0 the torkance txxis the sum of contribu-\ntions from STT and spin-orbit torque (SOT). The curves\nwithαR= 0 andαR/negationslash= 0 are essentially related by a shift\nof ∆q=−2meαR//planckover2pi12, which can be understood based on\nthe concept of the effective SOI, Eq. (9). Thus, in the\nspecial case of the one-dimensional Rashba model STT\nand SOT are strongly related.\nIV. SUMMARY\nWe study chiral damping, chiral gyromagnetism and\ncurrent-induced torques in the one-dimensional Rashba\nmodel with an additional N´ eel-type noncollinear mag-\nnetic exchange field. In order to describe scattering ef-\nfects we use a Gaussian scalar disorder model. Scat-\ntering contributions are generally important in the one-\ndimensional Rashba model with the exception of the gy-\nromagnetic ratio in the collinear case with zero SOI,\nwhere the scattering correctionsvanish in the clean limit.\nIn the one-dimensional Rashba model SOI and non-\ncollinearity can be combined into an effective SOI. Us-\ning the concept of effective SOI, results for the mag-\nnetically collinear one-dimensional Rashba model can be\nused to predict the behaviour in the noncollinear case.1 2 3 4\nScattering strength U [(eV)2Å]-6-4-20Torkance txx [e]\nRR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 14: Torkance txxvs. scattering strength Uin the one-\ndimensional Rashba model.\n-2 -1 0 1\nWave vector q [Å-1]-4-2024Torkance txx [e]RR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 15: Torkance txxvs. wave vector qin the one-\ndimensional Rashba model with SOI.\nIn the noncollinear Rashba model the Gilbert damp-\ning is nonlocal and does not vanish for zero SOI. The\nscattering corrections tend to stabilize the gyromagnetic\nratio in the one-dimensional Rashba model at its non-\nrelativistic value. 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Vehstedt, et al., Nature nanotechnology\n9, 211 (2014)." }, { "title": "0801.4583v1.Long_Term_Evolution_of_Magnetic_Turbulence_in_Relativistic_Collisionless_Shocks.pdf", "content": "arXiv:0801.4583v1 [astro-ph] 29 Jan 2008October 26, 2018 7:36 WSPC - Proceedings Trim Size: 11in x 8.5in proceedings\n1\nLong Term Evolution of Magnetic Turbulence in Relativistic Collisionless Shocks\nPhilip Chang, Anatoly Spitkovsky, and Jonathan Arons\nDepartment of Astronomy, Campbell Hall, University of Cali fornia, Berkeley, CA 94720\nDepartment of Astrophysical Sciences, Peyton Hall, Prince ton University, Princeton, NJ 08544\nEmail: pchang@astro.berkeley.edu, anatoly@astro.princ eton.edu, arons@astro.berkeley.edu\nWe study the long term evolution of magnetic fields generated by an initially unmagnetized collisionless relativistic\ne+e−shock. Our 2D particle-in-cell numerical simulations show that downstream of such a Weibel-mediated shock,\nparticle distributions are approximately isotropic, rela tivistic Maxwellians, and the magnetic turbulence is highl y\nintermittent spatially, nonpropagating, and decaying. Us ing linear kinetic theory, we find a simple analytic form for\nthese damping rates. Our theory predicts that overall magne tic energy decays like ( ωpt)−qwithq∼1, which compares\nfavorably with simulations, but predicts overly rapid damp ing of short wavelength modes. Magnetic trapping of\nparticles within the magnetic structures may be the origin o f this discrepancy. We conclude that initially unmagnetize d\nrelativistic shocks in electron-positron plasmas are unab le to form persistent downstream magnetic fields. These resu lts\nput interesting constraints on synchrotron models for the p rompt and afterglow emission from GRBs.\nKeywords : shock waves – turbulence – gamma ray: bursts – plasmas\n1. Introduction\nThe prompt emission and afterglows of gamma-ray\nbursts (GRBs) may be manifestations of ultrarela-\ntivistic shock waves. These shock waves may be me-\ndiated via the relativistic form of Weibel instability\n(Weibel 1959; Yoon and Davidson 1987; Medvedev\nand Loeb 1999; Gruzinov and Waxman 1999). The\nfree energy from strong plasma anisotropy in the\nshock transition layer generates strong magnetic\nfields (with strengths comparable to the available\nfree energy). However, these fields have very small\nspatialscales,i.e.,theorderoftheplasmaskindepth,\nc/ωp, whereωpis the plasma frequency. These ini-\ntially small-scale B-fields must survive for tens of\nthousands to millions of inverse plasma periods to\nserve as this source of the magnetization for syn-\nchrotron models of burst emission and afterglows\n(Gruzinov & Waxman 1999; Piran 2005ab; Katz,\nKeshet,&Waxman2007).Whether ornot thesefield\ncan is an open question.\nNumerical and analytic studies (Kazimura et al.\n1998; Silva et al.(2003); Frederiksen et al.2004;\nMedvedev et al.2005;Hededal et al.2005;Nishikawa\net al.2003, 2005; Spitkovsky this proceedings) have\nelucidated the basic physics. The instability initially\nforms filaments of electric current and Bfields,\nwhich then merge to inverse cascade magnetic en-\nergy to larger scales, but only in the foreshock re-\ngion. When the B-fields reach the magnetic trap-\nping limit (Davidson et al.1972; Kato 2005; also see\nMilosavljevic, Nakar, & Spitkovsky 2006; Milosavlje-vic & Nakar 2006a), particle orbits become chaotic,\ndisorganizing the filaments. The disorganized mag-\nnetic fluctuations scatter their supporting particles,\nwhich isotropizes and thermalizes the flow, within\ntens to hundreds of skin depths (Spitkovsky 2005).\nThe magnetic energy peaks in this layer at ∼10-20%\nof the bulk plasma flow energy.\nHowever, present simulations have not deeply\nfollowed the flow into the downstream region to ex-\nplore the long term behavior of these B-fields. Thus,\nthe question of the structure and long-term survival\nof the B-fields remains open (see for instance, Gruzi-\nnov & Waxman 1999; Gruzinov 2001ab; Medvedev\net al.2005).\nIn this proceeding, we discuss recent work\n(Chang, Spitkovsky, and Arons 2008; hereafter\nCSA08) which shows that this magnetic energy must\nrapidly decay in the downstream medium. We first\ndescribethe basicfeaturesofthe downstreamplasma\nfrom our numerical simulations. We then calculate\nthe evolution (decay) of the downstream plasma us-\ning Vlasov linear response theory and then compare\nthis evolution with simulations. While linear theory\ndoes reasonably well in estimating the decay rate of\nthetotalmagneticenergy,itoverestimatesthedamp-\ningrateofshorterwavelengthmodes.Wediscussthis\ndiscrepancyas a result of magnetic trapping.Finally,\nwe summarize our results.October 26, 2018 7:36 WSPC - Proceedings Trim Size: 11in x 8.5in proceedings\n2\n2. Simulation Results\nSpitkovsky (2005, this proceedings) and Spitkovsky\nand Arons (in prep) describe a series of 2D and 3D\nsimulations of relativistic shock waves in e+e−plas-\nmas. These are Particle-in-Cell (PIC) simulations,\nusing the code TRISTAN-MP . We simulate shocks\nby injecting cold relativistic plasma particles at one\nend of a largedomain that reflect off a fixed conduct-\ning wall at the other end. We use 2D boxes as large\nas 50,000 x 2048 cells with up to 1 .35×1010particles\nto study these shocks. We refer the reader to CSA08\nfor additional details.\nFig. 1. Snapshot of a region froma large 2D relativistic shoc k\nsimulationinthe downstreamframe.a)Densitystructure, n or-\nmalized in upstream units, in the simulation plane. b) Mag-\nnetic energy, normalized in terms of upstream energy of the\nincoming flow: ǫB=B2/4πγ1n1mc2. A power law scaling,\nǫ1/4\nB, was applied to stretch the color table to show weak field\nregions and is reflected in the colorbar. c) Plasma density av er-\naged in the transverse direction as a function of the distanc e\nalong the flow. d) Magnetic energy density averaged in the\ntransverse direction, as a function of distance along the flo w.\nFigure 1 shows the snapshots of density and\nmagnetic energy from a typical 2D simulation. Co-\nordinates are in units of the upstream skin depth,\nc/ωp1. In the simulation shown, the upstream flow\nmoves to the left with γ1= 15.\nOur simulations are large enough to permit the\ncomplete development of the shock and show the\nmain features of contemporary collisionless shock\nsimulations. For instance, we see the factor of ≈3.13\nincrease in density between the upstream and thedownstream (Fig. 1c), which is the expected com-\npression factor (Gallant et al.1992; Spitkovsky and\nArons, in prep). Current filaments show up as an en-\nhancement in the plasma density and magnetic en-\nergy density in the foreshock (Fig. 1ab). The scale of\nthe filaments growstowardsthe shock throughmerg-\ning.\nAtthe shocktransitionlayer,thefilamentsdisor-\nganizeand becomeclumps ofmagneticenergy.These\nmagnetic clumps lose intensity the further down-\nstream they are from the shock (Figure 1b). We also\nfindthattheparticledistributionfunctionchangesto\nan isotropic (in the downstream rest frame) thermal\npopulation, i.e., the difference between the perpen-\ndicular and parallel momentum is extremely small,\n<1%. As Figures 1 b and d suggest, the magnetic\nfields decay in the downstream region of the shock.\n3. Downstream Evolution of Magnetic\nTurbulence\nWe now attempt to analytically understand this de-\ncay of magnetic turbulence. The simulations show\nthat the downstream plasma is isotropic and the\ndownstream particle distribution function is well de-\nscribed by a relativistic Maxwellian. We will also as-\nsumethe downstream field amplitudes are so small\nthat particle orbits are almost straight lines.\nWe begin by deriving the linear plasma response\nis determined by the plasma susceptibility, χ(Stix\n1992). We evaluate the susceptibilty for distribution\nfunctions that are isotropic in two and three dimen-\nsions. Details can be found in CSA08. We set ωr= 0,\nbecause of the non-propagating nature of the mag-\nnetic clumps, which we infer from the simulations. In\nthe long wavelength limit, i.e., k≪ωp/c, wherekis\nthe wavenumber, we find:\n4πχ≈\n\niω2\np\n|k|cω2D\niπ\n4ω2\np\n|k|cω3D. (1)\nNote that the 2D and 3D results only vary by a\nnumerical factor. Long wavelength modes have the\nsame qualitative behavior in two and three dimen-\nsions.\nThe plasma susceptibility (eq.[1]) can be utilized\nto calculate the evolution of an initial field of fluctu-\nations. We refer the interested reader to CSA08 forOctober 26, 2018 7:36 WSPC - Proceedings Trim Size: 11in x 8.5in proceedings\n3\ndetails, but the result is\nd|δBk|2\ndt=−2γk|δBk|2, (2)\nwhereγk= (kc)2ω−1ℑ(4πχ)−1. The asymptotic\nforms of χfrom equation (1) for 2D and 3D gives\nγk=\n\n|kc|3\nω2\np2D\n4\nπ|kc|3\nω2p3D. (3)\nNote the strong cubic kdependence on the decay\nrate. Short wavelength modes rapidly damp, but\nlonger wavelength modes can persist.\nWe now compare these expectations to the nu-\nmericalsimulations.WetaketheFouriertransformof\nδBfrom our 2D numerical simulations from a down-\nstream region behind the shock front at x=x0. We\nevolve these spectra for 450 (red), 900 (green), and\n1350ω−1\np(blue) using equation (3). We compare this\nanalytically evolved spectra to Fourier transformed\nsnapshots taken from our numerical simulations at\nthese times in Figure 2, where x0= 840c/ωp. While\ntheory and simulation agree at very low wavenum-\nber (kyc/ωp/lessorsimilar0.2), theory overpredicts the cutoff in\npower at larger k. The discrepancy may be due to\nmagnetic trapping (see §4).\nFig. 2. Spectral evolution of magnetic field from the slice at\n840c/ωp.Initialfieldspectrum (black solidline)isplotted after\n450ω−1\np(red), 900 ω−1\np(green), and 1350 ω−1\np(blue) based on\nsimulation data. Dashed curves represent analytic evoluti on\nof the initial field.\nSince total B-field energy is dominated by long\nwavelength modes, we use equation (2) to find a sim-ple decay law for the total B-field. Again we refer the\ninterested reader to CSA08 for details, but if the ini-\ntial spatial spectrum is a power law in wave number\n|δBk|2∝k2p, then the B-field should decay like\nδB2∝t−2(p+1)/3. (4)\nFor a shock moving at constant velocity, we have\nxpeak−x∝t. HenceδB2∝(xpeak−x)−2(p+1)/3. Our\nnumerical simulations are extremely suggestive that\nthe magnetic energy density follows the a δB2/8π∝\nt−2/3decay expected for an initially flat magnetic\nspectrum at early times ( p= 0), then steepening to\nat−1decay at later times ( p= 1/2) as shown in\nFigure 3. We have analyzed additional simulations\nwith a large transverse spatial scale and they sug-\ngestp= 0. This difference in the index of the decay\nlaw expected from the theory and measured from the\nsimulations may also be due to magnetic trapping\n(see§4; see Gruzinov 2001b for an alternate expla-\nnation and CSA08 for a rebuttal).\nFig. 3. Magnetic energy density (in units of upstream kineti c\nenergy) as a function of position downstream of the shock. A\nbroken power law proportional ( x−xpeak)−2/3fits well at\nearly times, but a ( x−xpeak)−1power law fits better at later\ntimes.\n4. Magnetic Trapping\nWhile oursimulations and theoryareconsistent with\noneanotherinsuggestingoverallpowerlawdecay t−q\nwithq∼1, they disagree for short wavelengths.This\ndiscrepancy may arise from the nonlinear effects of\nmagnetic trapping. Our simulations also show that\nnot all particles follow straight line trajectories that\nareweaklyperturbed,but somearepartiallytrappedOctober 26, 2018 7:36 WSPC - Proceedings Trim Size: 11in x 8.5in proceedings\n4\nand strongly deflected. By following test particle or-\nbits in our simulations, we find that the Larmorradii\nof many of the test particles are of the same order\nof the sizes of these clumps or smaller. These strong\ndepartures from weakly perturbed particle dynamics\nmay be the cause of the decreased damping at large\nwavenumber found in the simulations. It may also\nmodify the overall decay away from t−2/3decay law\nexpected from simple linear theory.\n5. Discussion\nWe have studied the downstream evolution of mag-\nnetic turbulence in the context ofa collisionless e+e−\nshock both analytically and numerically. Our simu-\nlations show that the downstream region consist of\nnonpropagatingmagneticclumpsembeddedinquasi-\nhomogenous medium where the background parti-\ncle distribution function is an isotropic Maxwellian.\nIn such a background, we showed that magnetic en-\nergy will decay like t−qwithq∼1. However, linear\ntheory overpredicts the decay rates at short wave-\nlengths compared to simulations. Magnetic trap-\nping may play an role in resolving this discrepancy.\nRapid field decay puts severe constraints on GRB\nemission mechanism, but they may not be incon-\nsistent with GRB observations (see Pe’er & Zhang\n2006). Finally, if ion-electron collisionless shocks\nreach roughly equipartition with each other as sug-\ngest by recent large scale simulations (Spitkovsky\n2008), they would reproduce the physics of the e±\nshock and their B-fields would decay as well.\nAcknowledgements\nWe thank S. Cowley, D. Kocelski, M. Milosavljevic,\nA. Pe’er and E. Quataert for useful discussions. P.C.\nis supported by the Miller Institute for Basic Re-\nsearch. J.A. has benefited from the support of NSF\ngrant AST-0507813, NASA grant NNG06GI08G,\nand DOE grant DE-FC02-06ER41453, all at UC\nBerkeley; by the Department of Energy contract to\nthe Stanford LinearAcceleratorCenter no. DE-AC3-\n76SF00515; and by the taxpayers of California. A.S.\nis pleased to acknowledge that the simulations re-\nported on in this paper were substantially performed\nat the TIGRESS high performance computer center\nat Princeton University which is jointly supported\nby the Princeton Institute for Computational Sci-\nence and Engineering and the Princeton UniversityOffice of Information Technology.\nReferences\n1. Chang, P., Spitkovsky, A., & Arons, J. 2008, ApJ,\n674, 378 (CSA08)\n2. Davidson, R. C., Hammer, D. A., Haber, I., & Wag-\nner, C. E. 1972, Phys. Fluids, 15, 317\n3. Frederiksen, J. T., Hededal, C. B., Haugb` olle, T., &\nNordlund, . 2004, ApJ, 608, L13\n4. Gallant, Y. A., Hoshino, M., Langdon, A. B., Arons,\nJ., & Max, C. E. 1992, ApJ, 391, 73\n5. Gruzinov, A. 2001a, submitted to ApJ, astro-\nph/0111321\n6. Gruzinov, A. 2001b, ApJ, 563, L15\n7. Gruzinov, A. & Waxman E. 1999, ApJ, 511, 852\n8. Hededal, C. B., Trier Frederiksen, J., Haugboelle,\nT., & Nordlund, A. 2005, Neutrinos and Explosive\nEvents in the Universe, Proceedings of the 14th\nCourse of the International School of Cosmic Rays\nAstrophysics, a NATO Advanced Study Institute,\nheld in Erice, Italy, 2-13 July 2004. Edited by Mau-\nrice M. Shapiro, Stanev Todor, and John P. Wefel.\n9. Kato, T. N. 2005, Phys. Plasmas, 12, 80705\n10. Katz, B., Keshet, U., & Waxman, E. 2007, ApJ, 655,\n375\n11. Kazimura, Y., Sakai, J. I., Neubert, T., & Bulanov,\nS. V. 1998, ApJ, 498, L183\n12. Medvedev, M. V. & Loeb, A. 1999, ApJ, 526, 697\n13. Medvedev, M. V., Fiore, M., Fonseca, R. A., Silva,\nL. O.; Mori, W. B. 2005, ApJ, 618, L75\n14. Milosavljevic, M., Nakar, E., & Spitkovsky, A. 2006,\nApJ, 637, 765\n15. Milosavljevic, M. & Nakar, E. 2006a, ApJ, 641, 978\n16. Mikhailovskii, A. B. 1979, Plasma Phys., 22, 133\n17. Nishikawa, K.-I., Hardee, P., Richardson, G., Preece,\nR., Sol, H., & Fishman, G. J. 2003, ApJ, 595, 555\n18. Nishikawa, K.-I., Hardee, P., Richardson, G., Preece,\nR., Sol, H., & Fishman, G. J. 2005, ApJ, 622, 927\n19. Pe’er, A. & Zhang, B. 2006, ApJ, 653, 454\n20. Piran, T. 2005a, Rev. of Modern Phys., 76, 1143\n21. Piran, T. 2005b, Magnetic Fields in the Universe,\nAngra dos Reis, Brazil, Nov. 29-Dec 3, 2004, Ed. E.\nde Gouveia del Pino, AIP Conference Proceedings,\nv784 (New York:AIP), p164\n22. Silva, L. O., Fonseca, R. A., Tonge, J. W., Dawson,\nJ. M., Mori, W. B., & Medvedev, M. V. 2003, ApJ,\n596, 121\n23. Spitkovsky,A.2005, AIPConf. Proc, 801, 345; astro-\nph/0603211\n24. Spitkovsky, A. 2008, ApJ, 673, L38\n25. Stix, T.H. 1992, “Waves in Plasmas” (American In-\nstitute of Physics: New York)\n26. Weibel, E. S. 1959, Phys. Rev. Letts., 2, 83\n27. Yoon, P. H., & Davidson, R. C. 1987, Phys. Rev. A.,\n35, 2718" }, { "title": "1706.03325v1.Absorbing_boundary_layers_for_spin_wave_micromagnetics.pdf", "content": "Absorbing boundary layers for spin wave micromagnetics\nG. Venkata,\u0003, H. Fangohrb,c, A. Prabhakara\naDept. of Electrical Engineering, Indian Institute of Technology Madras, India 600036\nbFaculty of Engineering and the Environment, University of Southampton, UK.\ncEuropean XFEL GmbH, Holzkoppel 4, 22869 Schenefeld, Germany\nAbstract\nMicromagnetic simulations are used to investigate the e \u000bects of di \u000berent absorbing boundary layers (ABLs) on spin waves (SWs)\nreflected from the edges of a magnetic nano-structure. We define the conditions that a suitable ABL must fulfill and compare the\nperformance of abrupt, linear, polynomial and tan hyperbolic damping profiles in the ABL. We first consider normal incidence in a\npermalloy stripe and propose a transmission line model to quantify reflections and calculate the loss introduced into the stripe due\nto the ABL. We find that a parabolic damping profile absorbs the SW energy e \u000eciently and has a low reflection coe \u000ecient, thus\nperforming much better than the commonly used abrupt damping profile. We then investigated SWs that are obliquely incident at\n26:6\u000e, 45\u000eand 63:4\u000eon the edge of a yttrium-iron-garnet film. The parabolic damping profile again performs e \u000eciently by showing\na high SW energy transfer to the ABL and a low reflected SW amplitude.\nKeywords: Magnetization dynamics, micromagnetic simulations, magnonics, spin waves\n1. Introduction\nEasier access to computational resources over the last decade\nhas led to the development of many micromagnetic pack-\nages that solve the Landau-Lifshitz (LL) equation for magnetic\nnano-structures. These packages are being used to study spin\nwave mode profiles and spectra in a quest to build devices with\nnovel functionalities [1–3]. One approach to these studies is\nto perturb the ground state with a broadband excitation, and\nthen extract the spin wave (SW) dispersion characteristics [4–\n7]. However, simulation boundaries are known to a \u000bect the\ndissipative dynamics of the magnonic spectra in such studies\n[8, 9], and we artificially increase the damping \u000bat the bound-\naries, to absorb the SW reflections. The increase in \u000bcan be\nsmooth, e.g. using a hyperbolic tangent function [10], or abrupt\n[11]. The latter approach was used to attenuate SW reflec-\ntions, and to calculate the dispersion and scattering parameters\nin magnonic devices [12, 13]. More recently, an exponential\nincrease in damping was used to curb reflections in the study of\nskyrmions and the Dzyaloshinskii-Moriya interaction in mag-\nnetic nanostripes [14, 15].\nIn this article, we define the return loss using transmission\nline models, to study the impact of using artificial regions of\nhigh\u000b, or absorbing boundary layers (ABLs), at the edges of\nthe device. We propose a parabolic increase in \u000band show that\nit causes less spurious SW reflections than an abrupt increase in\n\u0003Corresponding author\nEmail address: guruvenkat7@gmail.com (G. Venkat)\nABL - Absorbing Boundary Layer; SW - Spin Wave; LL - Landau-Lifshitz;\nPML - Perfectly Matched Layer; FDTD - Finite Di \u000berence Time Domain; GPU\n- Graphics Processing Unit; FD - Finite Di \u000berence; YIG - Yttrium Iron Garnet;\nTransmission line - Tx line\u000b. We compare the parabolic profile against the abrupt, linear\nand the tan hyperbolic profile, for di \u000berent angles of incidence.\nThe parabolic profile also aligns the micromagnetic commu-\nnity more closely with the accepted polynomial form of per-\nfectly matched layers (PMLs) in finite di \u000berence time domain\n(FDTD) simulations of Maxwell’s equations [16].\nTo our knowledge, this is the first exhaustive study of ABLs\nusing the graphics processing unit (GPU) accelerated finite dif-\nference (FD) micromagnetic package MuMax3 [17]. We also\nprovide the codes for post processing the simulation data and\nraw data for the figures in a code repository for easy reproduc-\ntion [18].\n2. Normal incidence of spin waves\nThe time evolution of the magnetization is described by the\nLL equation [19, 20]\n@m\n@t=\r0[(m\u0002H)+\u000b(m\u0002(m\u0002H))]; (1)\nwhere m=M=MSis the normalized magnetization, and Mand\nHare the total magnetization and e \u000bective field at time t, re-\nspectively.\r0=\r\u00160=(1+\u000b2), with\r<0 being the electron gy-\nromagnetic ratio, \u000bthe phenomenological damping coe \u000ecient\nand\u00160the permeability of free space. We consider a stripe of\npermalloy (Ni 80Fe20) having dimensions 4000 \u00021000\u00025 nm3,\nas shown in Figure 1 (a). The structure was proposed as a mi-\ncromagnetic sample problem for studying SW dynamics and\ndispersion [7]. We choose a simple geometry with known solu-\ntions for the mode profiles.\nThe material parameters used for permalloy were the satu-\nration magnetization Ms=800 kA /m and exchange constantarXiv:1706.03325v1 [cond-mat.mes-hall] 11 Jun 2017A=13\u000210\u000012J/m [7]. No crystalline anisotropy was consid-\nered. The cell size was taken as 4 \u00024\u00025 nm3, such that the\ncell dimensions are less than the exchange length for permal-\nloy,lex'5:7 nm.\nFig. 1. (a) A magnonic waveguide with absorbing boundary layers along all\nedges. A SW excitation pulse hexc(t), is applied along ˆy, at the left edge. The\norigin is at the bottom left corner. (b) A snapshot of myatt=500 ns. The\ncolorbar is in linear scale.\nA harmonic field excites SWs at the left edge of the stripe\nso that they propagate along ˆx. ABLs along the top and bot-\ntom stripe edges confine the SWs to the centre of the stripe, as\nshown in Figure 1 (b). Now, consider di \u000berent spatial profiles\nfor damping, defined at the right end as:\nconstant and abrupt\n\u000ba(x)=8>><>>:0 x<3:8\u0016m\n0:1x\u00153:8\u0016m\ntan hyperbolic, with \u0001\u000b=0:5,x0=3:9\u0016m and\n\u001bx=40 nm, modified from [10]\n\u000bb(x)=8>><>>:0 x<3:8\u0016m\n\u0001\u000b(1+tanhx\u0000x0\n\u001bx)x\u00153:8\u0016m\npolynomial, with x0=3:8\u0016m\n\u000bc;n(x)=8>><>>:0 x<3:8\u0016m\na(x\u0000x0)nx\u00153:8\u0016mn=1;2\nIn each case the constants were chosen to obtain \u000b=1:0 at\nx=4\u0016m, as shown in Figure 2. \u000bc;1and\u000bc;2are linear and\nparabolic profiles respectively. We compare di \u000berent profiles\nover a constant ABL length of 200 nm. In the following sec-\ntions, we also show that 200 nm is su \u000ecient for the energy den-\nsity to decay by over 15 dB, for all the damping profiles.\n2.1. Simulation procedure\nWe apply a high bias magnetic field H0=804 kA /mˆx, with\nan artificially high damping (\u000b=0:5), and allow mto relax to\nits ground state. Since the magnetization in the stripe is satu-\nrated, we do not have any domain walls or vortices in the stripe.\nFig. 2. Spatial variation of di \u000berent damping profiles that were studied. \u000bc;2is\nthe parabolic damping profile.\nFig. 3. SW dispersion in the magnonic stripe, showing how fexc=39:96 GHz\nexcites only the fundamental mode.\nStarting with the ground state, an excitation magnetic field\nhexc(x;y;t)=h0sin(2\u0019fexct)cos\u0012\u0019\n2wy\u0000\u0019\n4\u0013\nˆy; (2)\nis applied at x<20 nm (in the region marked in red in Figure 1\n(a)) with h0=0:01H0,fexc=39:96 GHz and the width of the\nstripe w=1\u0016m. A low value of h0ensures that we excite small\namplitude SWs. The spatial form of cos\u0010\u0019\n2wy\u0000\u0019\n4\u0011\nwas chosen\nso that we preferentially excite the lowest order width mode.\nThe dispersion relation for the lowest SW mode in a back-\nward volume geometry ( kkH0) was derived by Kalinikos [21].\nIf we include exchange interactions, we get\n!=s\n!ex \n!ex+!M1\u0000e\u0000kh\nkh!\n;\n!ex=!0+!M\u0015exk2;(3)\nwhere his the stripe thickness, !0=\r\u00160H0is the uniform\nmode precession frequency and !M=\r\u00160MS.\u0015ex=2A\n\u00160M2s\nwhere Ais the exchange constant. k2=k2\nx+k2\nywhere kxis the\npropagation constant and ky=\u0010\nny+1\u0011\u0019\nwis the quantized wave\n2vector component along the width. We choose ny=0, and\npick fexc=39:96 GHz, to excite only the fundamental mode, as\nshown in Figure 3. m(x;y;z;t)is saved at all the nodes of the\nFD grid. The SWs take approximately 25 ns to reach the right\nend of the stripe. We allow the simulation to run till 500 ns so\nthat the SWs travel ten round trips in the stripe.\n2.2. Transmission line model for ABLs\nThe purpose of an ABL is three fold:\n1. The SWs should decay su \u000eciently by the end of the ABL\nto have no reflections from the stripe edge.\n2. The ABL causes minimum reflections back into the de-\nvice.\n3. Minimum energy is reflected into higher order modes.\nConsequently, we evaluate the di \u000berent ABLs, using as a metric\nthe energy density in the ABL and reflections from the ABL.\nThe energy density of the SWs propagating along the stripe\nis [17]\nE(x;y;t)=\u00001\n2M(x;y;t):B(x;y;t); (4)\nwhere Bis the instantaneous magnetic flux density. Figure 4\nshows the variation of the normalized energy density in the\nABL at t=500 ns.Edecays by over 15 dB within 200 nm\nfor all the profiles. We observe no significant reflections from\nthe structure edge, and hence we fix our ABL length at 200 nm\nfor all the profiles.\nFig. 4. The decay of the normalized SW energy density Ein the ABL. The\nenergy decays by more than 15 dB within 200 nm for all profiles.\nWe now investigate reflections that originate from the start of\nthe ABL at x=3:8\u0016m. When we make a transition from \u000b=0\nto\u000b,0, we observe SW reflections in a manner analogous to\nhaving an impedance mismatch along a transmission (Tx) line.\nWe model the wave propagation in the stripe as standing waves\nformed on a lossy Tx line. In its simplest form, the magnetiza-\ntion on this line takes the form (c.f. Appendix A)\nm(x)=m+h\ne\u0000\u0010xcos\fx+j\u0000je+\u0010xcos(\fx+\u001e)i\n; (5)\nwhere m+is the peak amplitude of the incident wave, \u0010is the\nloss per unit length, \fis the propagation constant of the standingwave and \u0000 =j\u0000jej\u001eis the complex reflection coe \u000ecient at the\nload end. We fit the standing wave my(x;hyi;t), in the stripe,\nto Eq. (Equation 5) for each of the di \u000berent damping profiles.\nThese fits are done for t=475 to t=500 ns, to obtain the mean\nand standard deviation for \u0010,j\u0000jand\u001e. One such fit is shown in\nFigure 5 for \u000bc;2att0=500 ns.\nFig. 5. A fit of Eq. (Equation 5) with the magnetization in the stripe at t0=\n500 ns, when the \u000bc;2profile is used in the ABL. The fit is used to estimate the\nreturn loss appearing in the line due to the introduction of the ABL.\nThe return loss in a Tx line is a measure of the power re-\nflected by a mismatched load and is given as [22]\nRL=\u000020 log10j\u0000jdB: (6)\nThe time averaged \u0010and RL values (along with the precision)\nare given for the di \u000berent profiles in Table. 1. A higher value\nof RL indicates a lower reflection coe \u000ecient and thus a more\nmatched load, and the parabolic profile shows a 1.5 dB higher\nRL than the commonly used abrupt profile. The value of RL\nfor the parabolic profile ( \u000bc;2) is comparable to that of the tan\nhyperbolic profile ( \u000bb) and therefore both appear to be e \u000ecient\nfor use in an ABL.\nTable 1\n\u0010and Return loss for the di \u000berent ABL profiles.\nS. No. Profile \u0010\u0010\n\u0016m\u00001\u0011\nRL(dB)\n1\u000ba 0:08\u00060:01 5:21\u00060:01\n2\u000bb 0:1\u00060:02 6:99\u00060:02\n3\u000bc;1 0:08\u00060:01 5:41\u00060:02\n4\u000bc;2 0:1\u00060:02 6:72\u00060:01\n3. Oblique incidence of spin waves\nIn FDTD simulations, the performances of PMLs are typ-\nically functions of the angles at which the electromagnetic\nwaves are incident on them. Having shown the performance of\nABLs for perpendicular incidence in section 2, we now investi-\ngate their e \u000bect when we have oblique incidence. Consider the\n3geometry recently used to simulate the Goos-Hanchen e \u000bect for\nSWs [23], in a yttrium iron garnet (YIG) film, which is shown in\nFigure 6. The dimensions of the film are 6000 \u00023000\u00025 nm3.\nThe material parameters used for YIG were Ms=194 kA /m\nandA=4\u000210\u000012J/m [23]. Again no crystalline anisotropy was\nconsidered.\nFig. 6. A thin film of YIG with ABLs all around the periphery and with an\nexcitation region at an angle. The origin is at the bottom left corner.\nThe\u000bc;2profile was applied along the left, top and right edges\nallowing us to focus on reflections o \u000bthe bottom edge. We\napply a magnetic field H0=558 kA /mˆxand allow mto relax\nto its ground state. We then choose an area at an angle \u0012, as\nshown in Figure 6, and apply [23]\nhexc\u0000x0;y0;t\u0001=h0e\u00002\u0012x0\u0000x0\n0\nlexc\u001bexc\u00132\nsin(2\u0019f t)ˆy;(7)\nwith h0=0:01H0.ˆyis the desired direction of SW propagation,\nat an angle\u0012, and ˆx0is the direction of spin wavefronts.\u0010\nx0\n0;y0\n0\u0011\nmarks the centre of the excitation region, and was chosen appro-\npriately for the di \u000berent angles of incidence considered below,\nand shown in Fig. 7. The choice of\u0010\nx0\n0;y0\n0\u0011\nensured that point\nof incidence was the same for each simulation.\nlexc=1\u0016m and wexc=5 nm are the length and width of\nthe excitation area, and we apply hexcto all mesh nodes that\nfall within this region. \u001bexc=0:4 decides the spread of the\nGaussian envelope. We tested the ABL for sinusoidally pumped\nspin waves with f=35 GHz [23].\nWe observed that proper SW collimation was obtained when\nthe SW propagation angle (\u0012)was related to the cell edge\nlengths, \u0001xand\u0001y, by tan\u0012=\u0001y\n\u0001x. Consequently, we con-\nsidered three cases where we took \u0001x=5 nm and \u0001y=\n2:5;5 and 10 nm. Each of these edge lengths is smaller than the\nexchange length of YIG ( lex\u001913 nm). For these three cases,\ntan\u0012=0:5;1 and 2 which lead to \u0012=26:6\u000e;45\u000eand 63:4\u000ere-\nspectively.\nThe snapshots for the \u000baand\u000bc;2profiles, for the di \u000berent\n\u0012, are shown in Figure 7. We see significant reflections when\n\u000bais used whereas \u000bc;2hardly shows any reflections for the\nthree angles of incidence. The larger reflections from \u000baleads\nto regions of constructive and destructive interference close to\nthe point of incidence. Such artifacts are avoided with \u000bc;2.\nFigure 8 shows the cumulative energy density, from Eq.\n(Equation 4), in the ABL for the di \u000berent profiles at \u0012=63:4\u000e.\n\u000bc;2leads to maximum absorption of SWs in the ABL and thus\nis the most e \u000ecient of all the profiles we have considered. Fig-\nure 9 shows the magnetization scanned along a wavefront of the\nreflected wave, which is shown by the red line in Figure 7. Here\ntoo the amplitude of the reflected SW beam is low for \u000bc,2.\nFig. 7. SWs in the YIG film for di \u000berent incident angles. Columns (a) and (b)\nhave the\u000baand\u000bc;2profiles in the ABL respectively. The colorbar is in lin-\near scale.\u0010\nx0\n0;y0\n0\u0011\nis the centre of the excitation region and was appropriately\nchosen for each excitation angle. The magnetization is scanned along the wave-\nfront (red line) to obtain the plot in Figure 9. \u000bc;2causes minimal reflections\nfor all three angles of incidence.\n4. Summary and conclusions\nReducing unwanted reflections from boundaries is important\nfor accurate simulations of magnonic devices. Shorter ABLs\nwith abrupt changes in \u000bcan cause spurious artifacts. We cal-\nculated the return loss introduced in a permalloy stripe due to\nthe SWs normally incident on a ABL, using a transmission line\nmodel. The parabolic damping profile yields a higher return\nloss, 1:5 dB higher than an abrupt ABL.\nWe then considered SWs obliquely incident on the ABL at\ndi\u000berent angles of incidence. Even at a large incidence angle of\n63:4\u000e, the parabolic profile \u000bc;2causes minimal reflections and\nleads to the largest SW energy transfer to the ABL. The per-\nFig. 8. The cumulative energy density as a function of time in the ABL region\nfor the di \u000berent profiles at \u0012=63:4\u000e.\u000bc;2shows the largest energy transfer to\nthe ABL.\n4Fig. 9. The variation of the magnetization along the wavefront (red line in\nFigure 7; simulations were run for \u000bband\u000bc;1also). The\u000bband\u000bc,2profiles\nshow least reflections from the ABL.\nformance of the tan hyperbolic damping profile is comparable\nto that of the parabolic profile. Yet we urge the micromagnetic\ncommunity to adopt the latter so as to align ourselves with the\nestablished use of PMLs in FDTD simulations.\nExample scripts to analyze the data, as well as raw data for\nthe figures, are available in the associated electronic supple-\nmentary material [18].\nAcknowledgments\nThe authors would like to thank Malathi M., Manas Srivas-\ntava and Rajavardhan T for fruitful discussions and the High\nPerformance Computing Centre at IIT Madras for the use of\ntheir GPU cluster. This work was supported in part by the\nDepartment of Science and Technology, Government of India\nsanction number SB /S3/EECE /011/2014 (IITM).\nAppendix A. Standing waves on a transmission line\nConsider a lossy transmission line extending from x=0 to\nx=x0. If a wave is launched on a lossy transmission line\ntowards the right at x=0 and the line is terminated by an\nunmatched load, standing waves will be formed on the line at\nsteady state [Pozar 1997]. For small signal magnetization, as-\nsuming linear systems, the standing waves are written as a sum\nof incident and reflected waves as\nmtot(x)=m+e\u0000\u0010xe\u0000j\fx+m\u0000e+\u0010xe+j\fx;\nwhere m+,m\u0000,\u0010and\fare the maximum amplitude of incident\nand reflected waves, the loss per unit length and the propagation\nconstant of the wave respectively. We then have\nmtot(x)=m+h\ne\u0000\u0010xe\u0000j\fx+ \u0000e+\u0010xe+j\fxi\n;\nwhere \u0000 =j\u0000jej\u001eis the reflection coe \u000ecient at the load end\nx=x0. The real part of mtot(x)is\nm(x)=m+h\ne\u0000\u0010xcos\fx+j\u0000je+\u0010xcos(\fx+\u001e)i\n:Appendix B. Implementation of the ABL in Mumax3\nTo assist the interested reader, we reproduce the MuMax3\ncode for setting the parabolic damping profile at the edge of\nthe stripe in Figure 1 (a). We define each cell in the ABL as a\nregion and set the parabolic damping in it. We define the start\nand stop damping values, and the range of xvalues.\nalstart := 0.0 //alpha at start of ABL\nalstop := 1.0 //alpha at stop of ABL\nxstart := 3800 //x at start of ABL in nm\nxstop := 4000 //x at stop of ABL in nm\nn := 2 //Polynomial order\na := (alstop-alstart)/ //Polynomial coefficient\n(Pow((xstop-xstart), nxp))\ncX := 5e-9 //Cellsize along x\nNB := ((xstop-xstart)*1e-9)/cX\n//No. of cells in ABL\n//Set the damping cellwise\nfor i :=0; i1\u0002104\n8 200\u0006100 150\u000620 700\u00061000 25\u00062 900\u0006100\n10 140\u000640 200\u000620 160\u000650 33\u00061 800\u0006200\n15 9\u00062 800\u0006100 10\u000620 100\u000680 >1\u0002104\n25 0\u00065 >1\u000210460\u000630 >1\u000210410.41\u00060.01\nAg/Fe6 0\u000640 >1\u0002104150\u000640 >1\u000210411.7\u00060.7\n8 0\u000630 >1\u0002104170\u000650 >1\u000210412\u00064\n10 6\u00061 1500\u0006300 8\u000640 200\u0006500 >1\u0002104\n15 2\u00062 4000\u00063000 3\u00069 500\u0006900 >6\u0002103\n25 0\u00066 >1\u0002104140\u000650 >1\u000210415\u00066\nthe mechanisms of extrinsic damping in structurally disordered thin \flms.\nACKNOWLEDGMENTS\nS.W. acknowledges support by the ICTAS Junior Faculty Program. D.A.S. and S.E.\nacknowledge support by the National Science Foundation, Grant No. DMR-2003914. P.\nN. would like to acknowledge support through NASA Grant NASA CAN80NSSC18M0023.\nA. R. would like to acknowledge support through the Defense Advanced Research Project\nAgency (DARPA) program on Topological Excitations in Electronics (TEE) under Grant\nNo. D18AP00011. This work was supported by NanoEarth, a member of National Nan-\notechnology Coordinated Infrastructure (NNCI), supported by NSF (ECCS 1542100).\n18Appendix A: Details of the Two-Magnon Scattering Model\nIn the model developed by McMichael and Krivosik, the two-magnon scattering contri-\nbutionΔ𝐻TMS to the FMR linewidth is given by25,52,53\nΔ𝐻TMS=𝛾2𝐻2\n𝑎\n2𝜋𝑃𝐴¹𝜔º∫\nΛ0𝑘𝐶𝑘¹𝜉º𝛿𝛼¹𝜔\u0000𝜔𝑘ºd2𝑘 (A1)\nwhere𝜉is correlation length, 𝐻𝑎is the e\u000bective anisotropy \feld of the randomly oriented\ngrain.𝑃𝐴¹𝜔º=𝜕𝜔\n𝜕𝐻\f\f\n𝐻=𝐻FMR=√︃\n1¸¹4𝜋𝑀𝑠\n2𝜔𝛾º2accounts for the conversion between the fre-\nquency and \feld swept linewidth. Λ0𝑘represents the averaging of the anisotropy axis \ruc-\ntuations over the sample. It also takes into account the ellipticity of the precession for both\nthe uniform FMR mode and the spin wave mode52. The detailed expression of Λ0𝑘can\nbe found in the Appendix of Ref.52. The coe\u000ecients in the expression of Λ0𝑘depend on\nthe type of anisotropy of the system. Here, we used \frst-order cubic anisotropy for bcc Fe.\n𝛿𝛼¹𝜔\u0000𝜔𝑘ºselects all the degenerate modes, where 𝜔represents the FMR mode frequency\nand𝜔𝑘represents the spin wave mode frequency. The detailed expression of 𝜔𝑘can be found\nin Ref.25. In the ideal case where Gilbert damping is 0, 𝛿𝛼is the Dirac delta function. For a\n\fnite damping, 𝛿𝛼¹𝜔0\u0000𝜔𝑘ºis replaced by a Lorentzian function1\n𝜋¹𝛼IP𝜔𝑘𝛾º𝜕𝜔𝜕𝐻\n¹𝜔𝑘\u0000𝜔º2¸»¹𝛼IP𝜔𝑘𝛾º𝜕𝜔𝜕𝐻¼2,\nwhich is centered at 𝜔and has the width of ¹2𝛼IP𝜔𝑘𝛾º𝜕𝜔𝜕𝐻.\nFinally,𝐶𝑘¹𝜉º(or𝐶𝑘¹𝜉𝜆º) is the Fourier transform of the grain-to-grain internal \feld\ncorrelation function, Eq. (3) (or Eq. (4)). For the description of magnetic inhomogeneity\nanalogous to the simple self-a\u000ene topographical surface58, the Fourier transform of the\ncorrelation function, Eq. (3), is\n𝐶𝑘¹𝜉º=2𝜋𝜉2\n»1¸¹𝑘𝜉º2¼3\n2 (A2)\nas also used in Refs.25,52,53. For the description analogous to the mounded surface, the\nFourier transform of the correlation function, Eq. (4), is58\n𝐶𝑘¹𝜉𝜆º=8𝜋3𝜉2\u0010\n1¸2𝜋2𝜉2\n𝜆2¸𝜉2\n2𝑘2\u0011\n\u0014\u0010\n1¸2𝜋2𝜉2\n𝜆2¸𝜉2\n2𝑘2\u00112\n\u0000\u0010\n2𝜋𝜉2\n𝜆𝑘\u00112\u001532 (A3)\nWhen𝜆\u001d𝜉, Eq. (A3) becomes\n𝐶𝑘¹𝜉º\u00198𝜋3𝜉2\n\u0010\n1¸𝜉2\n2𝑘2\u00112 (A4)\n19100102104106108101010-2410-2210-2010-1810-1610-1410-1210-10 \nSelf-affine \nMounded λ = 10 nm \nMounded λ = 100 nm \nMounded λ = 1000 nmCk [m2]k\n [m-1]ξ = 100 nmFIG. 7. Fourier transform of correlation function for mounded surfaces as a function of wavenumber\n𝑘for three di\u000berent 𝜆values. Fourier transform of correlation function for self-a\u000ene surfaces as a\nfunction of 𝑘is also included for comparison purpose. 𝜉is set as 100 nm for all curves.\nwhich has a similar form as Eq. (A2). This similarity can also be demonstrated graphically.\nFigure 7 plots a self-a\u000ene 𝐶𝑘curve (Eq. (A2)) at 𝜉=100 nm and three mounded 𝐶𝑘curves\n(Eq. (A3)) at 𝜆=10, 100, 1000 nm. 𝜉in mounded 𝐶𝑘curves is set as 100 nm as well. It\nis clearly shown in Fig. 7 that when 𝜆=1000 nm, the peak appearing in 𝜆=10 and 100\nnm mounded 𝐶𝑘curves disappears and the curve shape of mounded 𝐶𝑘resembles that of\nself-a\u000ene𝐶𝑘.\nThe hump feature in Fig. 4 is governed by both 𝛿𝛼and𝐶𝑘(see Eq. A1). 𝛿𝛼has the shape\nof1in reciprocal space ( 𝑘space), as shown in our videos in the Supplemental Material as\nwell as Fig. 5(b) of Ref.53and Fig 2 (b) of Ref.25. The size of the contour of the degenerated\nspin wave modes in 𝑘space increases as the microwave frequency 𝑓increases, which means\nthe number of available degenerate spin wave modes increases as 𝑓increases. As shown\nin Fig. 7, self-a\u000ene 𝐶𝑘is nearly constant with the wavenumber 𝑘until𝑘reaches\u00181𝜉.\nThis suggests that the system becomes e\u000bectively more uniform (i.e. weaker inhomogeneous\nperturbation) when the length scale falls below the characteristic correlation length 𝜉(i.e.,\n𝑘 ¡1𝜉). Because inhomogeneities serve as the scattering centers of two-magnon scattering\n20process, degenerate spin wave modes with 𝑘 ¡1𝜉are less likely to be scattered into.\nNow we consider the 𝑓dependence of the two-magnon scattering rate. When 𝑓is small,\nthe two-magnon scattering rate increases as 𝑓increases because more degenerate spin wave\nmodes become available as 𝑓increases. When 𝑓further increases, the wavenumber 𝑘of\nsome degenerate spin wave modes exceeds 1 𝜉. This will decrease the overall two-magnon\nscattering rate because the degenerate spin wave modes with 𝑘 ¡1𝜉are less likely to be\nscattered into, as discussed above. Furthermore, the portion of degenerate spin wave modes\nwith𝑘 ¡ 1𝜉increases as 𝑓continues to increase. When the impact of decreasing two-\nmagnon scattering rate for degenerate spin wave modes with high 𝑘surpasses the impact\nof increasing available degenerate spin wave modes, the overall two-magnon scattering rate\nwill start to decrease as 𝑓increases. Consequently, the nonlinear trend { i.e., a \\hump\" {\nin FMR linewidth Δ𝐻TMS vs𝑓appears in Fig. 4.\nHowever, the scenario discussed above can only happen when 𝜉is large enough, because\nthe wavenumber 𝑘of degenerate spin wave modes saturates (i.e., reaches a limit) as 𝑓\napproaches in\fnity. If the limit value of 𝑘is smaller than 1𝜉, the two-magnon scattering\nrate will increase monotonically as 𝑓increases. In that case the hump feature will not\nappear. See our videos in the Supplemental Material that display the 𝑓dependence of Λ0𝑘,\n𝛿𝛼¹𝜔\u0000𝜔𝑘º,𝐶𝑘¹𝜉º\n2𝜋𝜉2,Λ0𝑘𝐶𝑘¹𝜉º𝛿𝛼¹𝜔\u0000𝜔𝑘º\n2𝜋𝜉2 , andΔ𝐻TMS for various𝜉values.\nPrevious discussions of the hump feature are all based on the self-a\u000ene correlation func-\ntion (Eq. 3). The main di\u000berence between the mounded correlation function (Eq. 4) and the\nself-a\u000ene correlation function (Eq. 3) is that the mounded correlation function has a peak\nwhen𝜆is not much larger than 𝜉as shown in Fig. 7. This means when the wavenumber\n𝑘of degenerate spin wave modes enters (leaves) the peak region, two-magnon scattering\nrate will increase (decrease) much faster compared to the self-a\u000ene correlation function. In\nother words, the mounded correlation function can generate a narrower hump compared to\nthe self-a\u000ene correlation function in the two-magnon linewidth Δ𝐻TMS vs𝑓plot, which is\nshown in Fig. 6 (b, c).\n1Z. Diao, Z. Li, S. Wang, Y. Ding, A. Panchula, E. Chen, L.-C. 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Rezende, Phys. Rev. B 62, 5331 (2000).\n55W. K. Peria, X. Wang, H. Yu, S. Lee, I. Takeuchi, and P. A. Crowell, Phys. Rev. B 103,\nL220403 (2021).\n56A. E. Clark, K. B. Hathaway, M. Wun-Fogle, J. B. Restor\u000b, T. A. Lograsso, V. M. Keppens,\nG. Petculescu, and R. A. Taylor, J. Appl. Phys. 93, 8621 (2003).\n57E. M. Summers, T. A. Lograsso, and M. Wun-Fogle, J Mater Sci 42, 9582 (2007).\n58M. Pelliccione and T.-M. Lu, in Springer Series in Materials Science , Vol. 108 (Springer, 2008).\n59C. Scheck, L. Cheng, and W. E. Bailey, Appl. Phys. Lett. 88, 252510 (2006).\n24" }, { "title": "1605.06179v1.High_frequency_behavior_of_FeN_thin_films_fabricated_by_reactive_sputtering.pdf", "content": "1 \n High-frequency behavior of FeN thin films fabricated by \nreactive sputtering \n \nTae-Jong Hwang, Joonsik Lee, Ki Hyeon Kim, and Dong Ho Kim \nDepartment of Physics, Yeungnam University, Gyeongsan, 38541, Korea \n \nAbstract \nWe investigated high-frequency behavior of Fe N thin films prepared by reactive sputtering \nthrough ferromagnetic resonance (FMR) and its relationship with the static magnetic \nproperties. The FMR was observed in the freque ncy range from 2 to 18 GHz in the FeN films \nfabricated at proper nitrogen flow rate (NFR). In those FeN thin films, a decrease of the \nsaturation magnetization and the correspondi ng decrease of the FMR frequency were \nobserved as NFR was increased during the depos ition. The external field dependences of the \nFMR frequencies were well fit to the Kittel form ula and the Landé g-factors determined from \nthe fit were found to be very close to the free electron value. The high-field damping \nparameters were almost insensitive to the gr owth condition of NFR. However, the low-field \ndamping parameters exhibited high sensitivity to NFR very similar to the dependence of the \nhard-axis coercivity on NFR, suggesting that extrinsic mate rial properties su ch as impurities \nand defect structures could be important in deciding the low-field damping behavior. \n Corresponding author: Dong Ho Kim Email: dhkim@ynu.ac.kr2 \n Introduction \nWhen a magnetic material with magnetization M is placed in an ex ternal magnetic field H \nand M and H are initially not parallel, a torque is exerted on the magnetization causing a \nprecession around the external field direction. Th e precession frequency is given by so called \nthe Lamor frequency, H/2, where is the gyromagnetic ratio defined as \tγ ൌఓಳ\n. Here g is \nthe Landé g-factor and ߤ\tis the Bohr magneton. The precession motion will continue forever \nwhen there is no damping force. However, in the real materials, the magnetization will \neventually align to the minimum energy dire ction as the energy is transferred from the \nprecession motion to the environment through spin -spin or spin-lattice interaction. In the \nsmall damping limit, the motion of magnetizati on subject to an effective magnetic field Heff is \ndescribed by the Landau-Lifsch itz-Gilbert (LLG) equation [1] \n \n݀M\nݐ݀ൌെ ߛሺMൈHeffሻߙ\nܯ௦Mൈ݀M\nݐ݀\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t, ሺ1ሻ \n \nwhere 4MS is the saturation magnetization and is the Gilbert damping parameter. Due to \nthe anisotropy contributions, Heff is sum of the anisotropy field HK and applied external \nmagnetic field Hext. Solving equation (1) with Hext applied along the magnetization easy axis \nand a small uniform excitation field perpendicular to Hext gives complex magnetic \nsusceptibility of the material. Imaginary pa rt of susceptibility , corresponding to energy \nabsorption by the magnetic material, becomes maximal at a frequency defined as the \nresonance frequency fres. This phenomenon is named fe rromagnetic resonance (FMR) and fres \nis written by the Kittel formula [2] 3 \n \n݂௦ൌఊ\nଶగඥሺܪ௫௧ܪሻሺܪ௫௧ܪ4ܯߨ ௦ሻ\t.\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tሺ2ሻ . \n \nWith known 4 MS, the field dependence of fres can be fit to equa tion (2) to obtain HK and . \nUsing the linewidth f of the resonance peak, the Gilbert damping parameter can be \nestimated by the following relation \n \nαൌ2ߨ∆݂\nߛሾ2ሺܪ௫௧ܪሻ4ܯߨ ௦ሿ\t.\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t ሺ3ሻ \n \nThe iron-nitrogen system has been studied for many decades because of the excellent \nmechanical and magnetic properties. Especially, effect of nitrogen incl usion on the structural \nand magnetic properties of FeN film s have been one of the mostly investigated subjects. [3-10] \nOn the other hand, studies on high-frequency be haviors of FeN film have been seldom \nperformed. [9,10] In this study, we report the result s of intensive study of high-frequency \ncharacteristics of FeN thin films in which ni trogen contents were sy stematically varied \nthrough the deposition process. We obtained high-frequency parameters such as FMR \nfrequencies, damping parameters, and g-factors, and discuss their rela tionship with static \nmagnetic properties, electrical resistivity and crystal structures. \n \nExperimental \nThe FeN thin films were fabricated on single-c rystal Si (100) substr ates by reactive RF \nmagnetron sputtering. An external field of 400 Oe was applied to the in-plane direction in \norder to induce in-plane uniaxial anisotropy. The base pressure of the sputtering chamber was 4 \n below 310-7 Torr. The sputtering gas was mixture of Ar and N 2 and their mixing ratio was \ncontrolled by the flow rate of each gas. The Ar flow rate was fixed at 10.0 sccm while N 2 \nflow rate (NFR) was varied from 1.1 to 2.1 sccm with 0.1 sccm step to adjust the nitrogen \ncontent in FeN thin films. For any N 2/Ar flow-rate ratio, the total pressure of the gas mixture \nwas maintained at 3 mTorr. The sputtering pow er was 100 W and the substrate holders were \ncooled with coolant at 5 C. The deposition time was adjusted to fabricate 100 nm thick films \nand the actual thicknesses of the films measured by using a surface profiler were 100~105 nm. \nThe resistivity was measured by using the four-probe method. \nThe crystal structure was analyzed by X-ray diffraction (XRD) patterns. Static magnetic \nproperties such as the M-H loops were investigated by vibr ating sample magnetometer (VSM). \nHigh frequency behavior, FMR fr equency and frequency linewi dth, were obtained by using \nvector network analyzer (VNA) and grounded c oplanar waveguide (CPW) in the frequency \nrange from 100 MHz to 26.5 GHz. An external ma gnetic field up to 2 kOe has been applied \nparallel to the magnetization easy axis a nd the RF field along the hard axis. [11] \n \nResults and discussion \nFig. 1 shows the XRD patterns of -2 scan of representative FeN thin films deposited at \ndifferent nitrogen gas flow rate. The pure Fe thin film prepared by the same deposition \ncondition showed polycrystalline nature with a rather broad -Fe (110) peak at 2 = 44.36 in \nthe XRD pattern. With increase of NFR to 1.2 sccm, the (110) peak became much broader \nand the peak position was slightly shifted to a lower angular position of 2 = 44.12. For films \nprepared at NFR of 1.4 sccm, th e (110) peak is replaced by just a trace of the peak. With \nfurther addition of nitrogen of NFR higher than 2.0 sccm, a new broad peak appeared at 2 = 5 \n 43.15, which is assumed to be a (101) peak of -Fe 3N phase. [5] Atomic percentages of Fe \nand N elements were obtained by using X-ra y energy dispersion spectroscopy. The nitrogen \natom percentage was ~29% for FeN thin films prepared at NFR of 1.1 sccm, and it gradually \nincreased with increasing NFR, becoming ~35% for NFR of 1.4 sccm, and finally reached ~46% at NFR of 2.0 sccm. It means that addi tion of proper amount of nitrogen induces an \namorphous phase, in such, nitrogen atoms are random ly distributed on inte rstitial sites. But \nfurther incorporation of nitrog en yields a partial formati on of unwanted phase such as -Fe\n3N. \nThe electrical resistivity of FeN thin has al so been affected by th e nitrogen addition. The \nresistivity of pure Fe film was 10 cm, similar to that of the single crystalline Fe film. [4] It \nincreased monotonically reaching 70 cm at NFR of 1.3 sccm then saturated around 90 \ncm at higher NFR. The resistivities of the current FeN films are smaller than the reported \nvalues of ~200 cm for films prepared with Ar and N 2 gas flow rates of 12 and 3 sccm, \nrespectively. [10] \nThe role of nitrogen addition on the magnetic properties of FeN thin film can be clearly \nobserved in Fig. 2 which shows how M–H loops develop with increas ing NFR. The solid lines \nare the loops measured with fields along the easy axis of magnetizatio n and the dotted lines \nalong the hard axis. The pure Fe film exhibite d almost isotropic in-p lane magnetization and \nhigh saturation magnetization 4 MS of 23 kG. This value is compar able to or slightly greater \nthan the reported values in th e literatures, [5-7,10] implying that our growth condition for Fe \nthin film is well optimized. Increase of NFR generated monotonic reduction of the saturation \nmagnetization throughout the entire NFR range, cons istent with earlier st udies. [5,7,8] It has \nbeen reported that further increase of nitr ogen gas fraction during the growth process \neventually incurs a complete disappearance of ferromagnetism in FeN films. [7,8] 6 \n In the case of coercivities, non-monot onic dependence on NFR was observed. The \ncoercivities first decreased from 18 Oe of the pu re Fe film to reach the minimum value at the \nfilm prepared at NFR of 1.4 sccm. The corres ponding coercivities for easy and hard axis, Hce \nand Hch, were 1.5 Oe and 1.9 Oe, respectively, as can be seen in Fig. 2(c). With further \nincrease of NFR, Hce started to increase, reaching ~10 Oe for FeN thin films grown at NFR of \n1.7 sccm. Beyond NFR of 1.8 sccm, totally different magnetic behavior has been observed. A \ntypical M–H loop in this range is shown in Fig. 2(d) , where anisotropy exists, but of different \nshape, and the coercivities are much larger than those of pure Fe thin films. This happens \nprobably due to the appearance of -Fe 3N phase. \nIn the films with very small grains of ra ndomly oriented, for example in amorphous phase, \nthe effective anisotropy energy is reduced due to the fact that more grai ns are involved in the \nmagnetic coupling, which in turn results in the partial cancellation of the crystalline \nanisotropy. As a result, coercivities would be significantly reduced [12] and samples will \nexhibit soft magnetic propert ies. The dependences of 4 MS, Hce and Hch on NFR of soft and \nuniaxial in-plane anisotropic FeN thin films are summarized in Fig. 3. \nAs described earlier, the high- frequency dynamics of magnetiz ation can be well described \nby the phenomenological LLG formula in equatio n (1) with a character istic precession and \nthe Gilbert damping associated with the precessi on. When the frequency of RF field is equal \nto fres, resonance absorption takes place. In this work, the FMR parameters were obtained \nfrom standard microwave S-parameter measurements by using VNA and CPW as a function \nof frequency at various static magnetic fields. \nOnly the FeN films produced at NFR range fr om 1.3 to 1.7 sccm, also having uniaxial in-\nplane anisotropy, showed meaningful FMR signals . The data were analyzed assuming that the 7 \n dominant CPW signals were in the quasi-tra nsverse electromagnetic mode. Fig. 4(a) shows \nthe amplitude spectra of S21 of the FeN thin film prepared at NFR of 1.5 sccm. As the external \nfield was increased from 0 to 2,000 Oe, the re sonance curve moved to higher frequencies \nconsistent with the Kittel formula in equation (2); as Hext increases, so does fres. The other \nspecimens fabricated at different NER showed a very similar trend in magnetic fields. \nRepresentative spectra at 500 and 750 Oe for th ree films prepared at NFR of 1.3, 1.5, and 1.7 \nsccm, respectively, are shown in Fig. 4(b). It s hows that the spectra shift to lower frequency \nwith increasing NFR. The FMR frequency fres and the frequency linewidth f at the half \nmaximum of the resonance peak were determined from the fit of S21 spectrum to the \nLorentzian curve. Before the fit, the backgr ound signal from the Si substrate was subtracted. \nThe solid lines in Figs. 4(a) and (b) are the Lorentzian fit. \nThe fres as a function of Hext for three FeN thin films, for cl arity, are plotted in Fig. 5(a). The \ndecrease of fres with increasing NFR is due to the decrease of MS with increasing NFR (see Fig. \n3 and equation (2)). The solid lin es in Fig. 5(a) are fit to the Kittel formula. All the fres data \nshowed good fit to e quation (2). Since 4 MS had been determined from the VSM \nmeasurement, or g-factor and HK were used for the two fitting parameters. The NFR \ndependences of the fitting para meters are shown in Fig. 5(b). The g-factors were within 1.98 \nand 2.06, very close to the free electron value, and HK were found to be in the range from 30 \nto 38 Oe. The obtained HK values from the fit were in a goo d agreement with those measured \nfrom the M–H loops, for instance, HK of 35 Oe from the fit for NFR of 1.4 sccm is \ncomparable to the experimental result in Fig. 2(c). No systematic dependence of either g-factor or H\nK on NFR was observed. \nThe Gilbert damping parameters estimated from f and equation (3) for five FeN thin films 8 \n are shown in Fig. 6(a) as a function of external magnetic field. Logarithmic scale was used for \nfield axis to emphasize the low-field behavior. Note that the data points plotted at 0.5 Oe \ncorrespond to those measured at a zero external field. All the decreased with increasing \nfield. The high-field values for all the samples are almost the same in the range between \n0.0052 and 0.0058. These values of ~0.005 are compar able to those observed in as-grown \nFeCoB films. [13] However, the low-field values were vastly different from each other, \nshown as a function of NFR in Fig. 6(b). Before fully saturated, thin films used to have non-\nuniform magnetization or anisotropy which vari es not only from position to position because \nthe sample is not in a uniform magnetic state but from sample to sample since films are \nfabricated at different deposition conditions. The former is the origin for the enhancement of \ndamping at low fields and the latter is responsib le for the sample to sample variation of low-\nfield damping. Note that NF R dependence of low-field in Fig. 6(b) resembles to those of \ncoercivities in Fig. 3, that is, the higher coercivities the higher low-field values. More \nresemblance to Hch is observed presumably because the RF electric field is applied along the \nhard axis such that energy loss at RF frequency would be affected more by Hch. The intrinsic \norigin of the Gilbert damping has been intensively discussed in terms of spin-orbit coupling. [14,15] However, our finding of g-factor very cl ose to 2.00 indicates that damping due to the \nspin-orbit coupling is rather w eak. Furthermore, because of the fact that the spin-orbit \ncoupling effect would be almost field independ ent for applied fields < 2 kOe, the spin-orbit \neffect alone may not explain the enhanced lo w-field damping. Instea d, extrinsic material \nproperties such as impurities and defect st ructures that strong ly depend on deposition \nconditions could play more important roles in deciding the low-fiel d damping behavior. \n 9 \n Summary \nWe investigated the relationship between high frequency behavior and the static magnetic \nproperties of FeN thin films prepared at va rious nitrogen flow rates. The saturation \nmagnetization showed a monotonic decrease with increasing NFR, so does the resonance \nfrequency. On the other hand, the damping pa rameters at low field exhibited strong \ndependence on the nitrogen flow rate while high-field damping parameters were unaffected. \nWe observed a correlation between the hard -axis coercivities and low-field damping \nparameters. \n \nAcknowledgments \nThis work was supported by the National Rese arch Foundation of Korea (NRF) grant funded \nby the Korea government (MSIP) No. 2014R1A2A1A11051500. \n 10 \n References \n[1] T. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). \n[2] C. Kittel, Phys. Rev. 73, 155 (1948). \n[3] T. K. Kim and M. Takahashi, Appl. Phys. Lett. 20, 492 (1972). \n[4] Y. Sugita, H. Takahashi, M. Komuro, M. Ig arashi, R. Imura and T. Kambe, J. Appl. Phys. \n79, 5576 (1996). \n[5] S. Iwatsubo and M. Naoe, J. Appl. Phys. 87, 5245 (2000); S. Iwatsubo and M. Naoe, \nVacuum 66, 251 (2002). \n[6] X. Wang et al. Appl. Surface Sci. 220, 30 (2003). \n[7] Y. Utsushikawa and K. Niizuma, J. Alloys Compd. 222, 188 (1995). \n[8] G. Li, Y. Liu, R. Zhao, J. Shen, S. Wang, P. Shan, C. Zhen and D. Hou, Thin Solid Films 589, 22 (2015). \n[9] S. Okamoto, O. Kitakami and Y. Shimada, J. Magn. Magn. Mater. 208, 102 (2000). \n[10] H. Naganuma, R. Nakatami, Y. Endo, Y. Kawamura and M. Yamamoto, Surf. Technol. \nAdv. Mater. 5, 101 (2004). \n[11] D.-H. Kim, H.-H. Kim C. You and H. Kim, J. Magn. 16, 206 (2011). \n[12] G. Herzer, IEEE Trans. Magn. 26, 1397 (1990). \n[13] C. Bilzer, T. Devolder, J. Kim, G. Counil, C. Chappert, S. Cardoso and P. P. Freitas, J. \nAppl. Phys. 100, 053903 (2006). \n[14] V. Kambersky, Can. J. Phys. B 26, 2906 (1970). \n[15] M. C. Hickey and J. S. Moodera, Phys. Rev. Lett. 102, 137601 (2009). \n \n \n 11 \n Figure Captions \nFig. 1. XRD patterns of -2 scan of representative FeN th in films deposited at various \nnitrogen gas flow rates. \n Fig. 2. Magnetization hysteresis loops of FeN thin films fabricated at the nitrogen flow rates \nof (a) 0 sccm, (b) 1.2 sccm, (c) 1.4 sccm, and (d ) 2.0 sccm. The solid lines are easy-axis loops \nand the dotted lines are hard-a xis loops. Two films of (b) a nd (c) showed soft magnetic \nproperties as well as uniaxial in-p lane anisotropy. \n \nFig. 3. The dependences of 4 M\nS, Hce and Hch on the nitrogen flow ra te of soft and uniaxial \nin-plane anisotropic FeN thin films. \n Fig. 4. (a) The amplitude spectra of S\n21 measured at various external fields from 0 to 2,000 Oe \nfor the FeN thin film fabricated at the nitrogen flow rate of 1.5 sccm. (b) Representative \nspectra at 500 and 750 Oe for three film s prepared at NFR of 1.3, 1.5, and 1.7 sccm, \nrespectively. It shows that the spectra shift to lower frequency with in creasing NFR. The solid \nlines are fit to the Lorentzian curve. Fig. 5. (a) External field dependence of the fe rromagnetic resonance fre quencies of FeN thin \nfilms fabricated at 1.3, 1.5, and 1.7 sccm. The so lid lines are fit to the Kittel formula. (b) \nObtained g-factors and anisotropy fields from the fit. \nFig. 6. (a) Damping parameters of five FeN thin films as a function of external field. (b) 12 \n Dependence of the damping parameters on the nitr ogen flow rate at external fields of 0, 1,000, \nand 1,500 Oe. 13 \n \n \n \nFig. 1 \n14 \n \n \n Fig. 2 \n15 \n \n \nFig. 3 \n16 \n \n \nFig. 4 \n17 \n \n \nFig. 5\n18 \n \nFig. 6 \n" }, { "title": "1311.7344v1.Conservative_effects_in_spin_transfer_driven_magnetization_dynamics.pdf", "content": "arXiv:1311.7344v1 [cond-mat.mes-hall] 28 Nov 2013Conservative effects in spin-transfer-driven magnetizati on dynamics\nG. Bertotti,1C. Serpico,2I.D. Mayergoyz,3\n1INRIM, Istituto Nazionale di Ricerca Metrologica, Strada d elle Cacce 91, 10135 Torino, Italy\n2Department of Electrical Engineering, University of Naple s ”Federico II”, via Claudio 21, 80125 Napoli, Italy\n3Department of Electrical and Computer Engineering,\nUniversity of Maryland, College Park MD 20742\n(Received June 15, 2018)\nIt is shown that under appropriate conditions spin-transfe r-driven magnetization dynamics in a\nsingle-domain nanomagnet is conservative in nature and adm its a specific integral of motion, which\nis reduced to the usual magnetic energy when the spin current goes to zero. The existence of\nthis conservation law is connected to the symmetry properti es of the dynamics under simultaneous\ninversion of magnetisation and time. When one applies an ext ernal magnetic field parallel to the\nspin polarization, the dynamics is transformed from conser vative into dissipative. More precisely, it\nis demonstrated that there exists a state function such that the field induces a monotone relaxation\nof this function toward its minima or maxima, depending on th e field orientation. These results hold\nin the absence of intrinsic damping effects. When intrinsic d amping is included in the description,\na competition arises between field-induced and damping-ind uced relaxations, which leads to the\nappearance of limit cycles, that is, of magnetization self- oscillations.\nThe spin-transfer phenomenon and related spintronic\napplications have been the focus of considerable re-\nsearch in the past two decades [1–5]. This research has\nbeen dominated by experimental and theoretical stud-\nies of spin-transfer-induced magnetization switching [6–\n9], as well as spin-transfer-driven magnetization self-\noscillations [10–14]. These studies have all been based\non the seed idea that spin transfer manifests itself as\na non-conservative torque that competes with intrinsic\n(thermal) damping. In particular, it has been realized\nthat the mutual compensation ofnon-conservativeeffects\ncaused by spin transferand thermal damping is the phys-\nical mechanism for the formation of magnetization self-\noscillations [1, 13, 15].\nIt is demonstrated in this Letter that in single-domain\nnanomagnets spin transfer may act as a purely conser-\nvative torque when electron spin polarization is directed\nalong the intermediate (i. e., hard in-plane) anisotropy\naxis. Under these conditions, the following new phys-\nical features emerge: the appearance of purely conser-\nvative magnetization dynamics with closed precession-\ntype trajectories; the existence of a special integral of\nmotion for this conservative dynamics, which is reduced\nto the conventional magnetic energy at zero spin cur-\nrent; a very unique global bifurcation in magnetization\ndynamics occurring at a specific critical value of the in-\njected spin-polarized current; the conversion of the con-\nservative dynamics into monotone relaxation when an in-\nplane dc magnetic field is applied along the intermediate\nanisotropy axis; the existence of a Lyapunov function\ngoverning these field-induced relaxations as well as the\nappearance of field-induced interlacing of the basins of\nattractions of the critical points of the dynamics. The\norigin of all these new physical features can be traced\nback to the special symmetry of magnetization dynamics\nappearing in the case when both electron spin polariza-\ntion and applied dc magnetic field are directed along theintermediate axis of magnetic anisotropy.\nThe described new physical features of magnetization\ndynamics appear when intrinsic damping effects are ne-\nglected. When these damping effects are accounted for,\nthe mutual compensation of the non-conservative effects\ncaused by damping and the applied dc magnetic field\n(rather than spin-transfer) may lead to the formation of\nmagnetization self-oscillations. This suggests the intrin-\nsic controllability of these oscillations by the applied dc\nmagnetic field, a feature that may be potentially useful\nin the development of novel nano-magnetometers.\nTo discuss the essence of these phenomena, consider\na single-domain nanomagnet with total (i.e., crystal +\nshape) ellipsoidal anisotropy and principal axes along\nx,y,z. Theenergyofthesystemcanbewrittenindimen-\nsionless form as: gM(m) =/parenleftbig\nDxm2\nx+Dym2\ny+Dzm2\nz/parenrightbig\n/2.\nIn this expression, energy is measured in units of µ0M2\nsV\n(Vis the volume of the nanomagnet and Msis the spon-\ntaneous magnetization), while m= (mx,my,mz) rep-\nresents the normalized magnetization ( |m|2= 1 ) of\nthe nanomagnet. Assume that the x,y, andzaxes are\nthe easy, intermediate, and hard anisotropy axes, respec-\ntively. The magnetic anisotropy coefficients are then or-\ndered in the following manner: Dx< Dy< Dz. A typ-\nical case of interest is the disk-like free layer of a spin-\ntransfer nanopillar device with in-plane anisotropy, for\nwhichDx<0,Dy≃0,Dz≃1. Under these conditions,\nit is convenient to shift the zero of energy by the amount\nDy/2 and rewrite the energy as:\ngM(m) =1\n2/parenleftbig\nDzym2\nz−Dyxm2\nx/parenrightbig\n, (1)\nwhereDyx≡Dy−Dx>0,Dzy≡Dz−Dy>0, and use\nhas been made of the identity m2\nx+m2\ny+m2\nz= 1.\nAssume now that the nanomagnet is subjected to a\nspin-transfer torque of the form βm×(m×ey), due to2\na spin current with polarisationparallel to the intermedi-\nate axisey. The dimensionless parameter βmeasuresthe\nintensity of the spin current. The equation for the mag-\nnetization dynamics in the absence of intrinsic (thermal)\ndamping is:\ndm\ndt=−m×hM+βm×(m×ey),(2)\nwherehM≡ −∂gM/∂m=Dyxmxex−Dzymzez. Equa-\ntion (2) is also dimensionless, with time measured in\nunits of ( γMs)−1(γis the absolute value of the gyro-\nmagnetic ratio). The dynamics preserves the magnitude\nof magnetization and thus takes place on the surface of\nthe unit sphere |m|2= 1. Of special importance are the\ntwo points: my=±1,mx=mz= 0, which are critical\npoints of the dynamics for any arbitrary value of the spin\ncurrent.\nThe dynamics (2) is characterised by a conservation\nlaw. This follows from the fact that Eq. (2) is invariant\nunder the transformation: my→ −my,t→ −t. To ex-\nplain this, considerfirst the purely precessionaldynamics\nunder zero current: dm/dt=−m×hM. The trajecto-\nries of this dynamics on the unit sphere |m|2= 1 are\nconstant-level lines of the anisotropy energy (1). When\nconsidered as a function of mon the unit sphere, this\nenergy is an even function of my. Consequently, its max-\nima and minima lie on the plane my= 0 and all its\nconstant-level curves intersect the plane my= 0 at least\nonce.\nAs the spin current βis gradually increased from β=\n0, the property of these trajectories on the unit sphere\nof intersecting the plane my= 0 cannot be immediately\ndestroyedby the current, because ofcontinuity. Consider\none of these trajectories, and choose the time origin at\nthe moment when the plane my= 0 is crossed. Then,\nbecause of the mentioned ( my,t) reversal symmetry, the\ntrajectorywillconsistoftwoparts, aforward-in-timeand\na backward-in-timeparts, mirror-symmetricwith respect\nto the plane my= 0. Consequently, if that trajectory\ncrosses the plane my= 0 a second time, it is a closed\ntrajectory. Trajectorieswith more than two intersections\nwithmy= 0arenotpossible. Ifatrajectoryisnotclosed,\nthen, again because of the ( my,t) reversal symmetry, it\nnecessarily connects two critical points characterized by\noppositevaluesof my. Thesecriticalpointsarethepoints\n(mx= 0,my=±1,mz= 0), which are saddle points of\nthe dynamics if the current is not too large.\nTherefore,onearrivesattheimportantconclusionthat\nthe phase portrait consists only of closed trajectories\nor open trajectories connecting saddle points (so-called\nseparatrix trajectories) even under nonzero spin current.\nThese closed trajectories and separatrices can be inter-\npreted as constant-level curves of some conserved quan-\ntity [16], say ˜ gM(m;β). Consequently, there must exist\nan integrating factor f(m;β) reducing Eq. (2) to theconservative form:\ndm\ndt=1\nfm×∂˜gM\n∂m, (3)\nwhere the conserved quantity is ˜ gM(m;β) =\nf(m;β)gM(m) andf(m;0)≡1, in order to guarantee\nthat the dynamics is reduced to dm/dt=−m×hM\nwhen no spin current is injected.\nFIG. 1: Phase portrait of undamped spin-transfer-driven dy -\nnamics under zero external field and increasing spin current .\nContinuous lines: constant-level curves of integral of mot ion\n˜gM. Arrows: direction of magnetization change. Dashed\nlines: invariant trajectories w±wQ= 0. Black dots: crit-\nical points. (a) β= 0.75Q. (b)β=Q(bifurcation point),\nthebold continuous line representing thecritical line on w hich\ndm/dt= 0. (c) β= 1.15Q. Parameters: Dyx= 0.3,Dzy= 1,\nQ=/radicalbig\nDzyDyx.3\nTo derive this integrating factor, consider that ˜ gM, as a\nconserved quantity, must satisfy the condition: d˜gM/dt=\n(∂˜gM/∂m)·(dm/dt) = 0alongmagnetizationtrajectories,\nthat is (see Eq. (2)):\n/parenleftbigg\nhM−gM\nf∂f\n∂m/parenrightbigg\n·/bracketleftBig\nm×/parenleftbig\nhM−βm×ey/parenrightbig/bracketrightBig\n= 0.(4)\nThis condition is identically satisfied for any mif the\nvectorshM−(gM/f)∂f/∂mandhM−βm×eyareparallel,\nthat is, if:\ngM\nf∂f\n∂m=βm×ey. (5)\nThis equation yields the following differential equation:\n1\nfdf\ndw=β\nQ2wQ\nw2−w2\nQ, w=mz\nmx,(6)\nwhere:Q=/radicalbig\nDzyDyxandwQ=/radicalbig\nDyx/Dzy. Indeed,\nm×ey=m2\nx∂w/∂mandgM(m) =Dzym2\nx/parenleftbig\nw2−w2\nQ/parenrightbig\n/2\n(see Eq. (1)). ByintegratingEq. (6) under the condition\nthatf(w;0)≡1, one obtains:\nf(w;β) =/vextendsingle/vextendsingle/vextendsingle/vextendsinglew−wQ\nw+wQ/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ/Q\n. (7)\nBy using Eq. (5), Eq. (2) is transformed into Eq. (3),\nas anticipated. The phase portrait of the dynamics is\nstraightforwardlyobtained by drawing the constant-level\nlines of ˜gM(m;β)≡f(w;β)gM(m) (Fig. 1). A convenient\nrepresentation is obtained in terms of cylindrical coordi-\nnates (mz,φ), whose relation to the cartesian magnetiza-\ntion components ( mx,my,mz) is:mx=/radicalbig\n1−m2zcosφ,\nmy=/radicalbig\n1−m2zsinφ.\nFigure 1 illustrates the progressiverestructuring of the\ndynamics as the spin current is increased. A remarkable\nproperty of this restructuring is the invariance of zero-\nenergy trajectories. According to Eq. (1), constant-level\nlines on which gM= 0 are described by the equations:\nmz/radicalbigDzy±mx/radicalbigDyx= 0. It is easily verified that on\nthese curves hM=±Qm×ey. By substituting this ex-\npression into Eq. (2) and by taking into account that\nhM=−∂gM/∂m, one obtains:\ndm\ndt=/parenleftbigg\n1∓β\nQ/parenrightbigg\nm×∂gM\n∂m. (8)\nThis expression reveals that if gM= 0, then dgM/dt=\n(∂gM/∂m)·(dm/dt) = 0. In other words, constant-level\ncurves on which gM= 0 are always solutions of the dy-\nnamics, whatever the value of the spin current β. The\ncurrent only affects the rate at which the constant-levelcurve is traversed. This rate goes to zero when β=±Q.\nWhen this condition is met, the entire constant-level\ncurve becomes a critical line along which dm/dt= 0.\nThis occurs as a result of a complex bifurcation (see Fig.\n1(b)), at which a global transition from closed to open\nmagnetization trajectories occurs.\nThe dramatic restructuring of the phase portrait at\nβ=±Qaffects the integral of motion ˜ gM, which is single-\nvalued and continuous everywhere for β2< Q2, while it\ndiverges on the curve w+wQ= 0 when β > Qor on\nthe curve w−wQ= 0 when β <−Q. However, this\ndivergence can be eliminated by taking advantage of the\nfact that any arbitrary monotone function F(˜gM) can be\ntaken as integral of motion instead of ˜ gM. If one chooses\nF(˜gM) = arctan˜ gM, then Eq. (3) is transformed into:\ndm\ndt=1+ ˜g2\nM\nfm×∂\n∂marctan˜gM.(9)\nThe function arctan˜ gMappears similar to a stream func-\ntion: it is conservedalongmagnetization trajectoriesand\nexhibits a discontinuous jump of amplitude equal to π\nacross the curve on which ˜ gMdiverges.\nWhen a magnetic field haeyis applied along the in-\ntermediate axis ey, the energy of the system becomes\ng(m;ha) =gM(m)−haey·m, and the undamped spin-\ntransfer-driven dynamics is governed, instead of Eq. (3),\nby the equation:\ndm\ndt=1\nfm×∂˜gM\n∂m−ham×ey.(10)\nThe introduction of the field breaks the ( my,t) reversal\nsymmetry and thus destroys the property of ˜ gMof being\nan integral of motion. However, quite remarkably, it is\npossible to modify ˜ gMin order to obtain a state function\nthat actsasa globalLyapunovfunction [16, 17], that is, a\nfunction that monotonically increases or decreases under\nall circumstances during the magnetization process. We\nshall limit the discussion to the current interval β2<\nQ2, in which ˜ gMis a single-valued, well-behaved state\nfunction. Adifferent approach, notdiscussed here, would\nbe necessary to deal with the case when β2> Q2.\nThe time derivative of ˜ gMderived from Eq. (10) is:\nd˜gM/dt=haf dmy/dt. The term dmy/dtcan be com-\nputed from Eq. (10). One finds:\n1\n1−m2ydmy\ndt=−/parenleftbigg2Rw\n1+w2+β/parenrightbigg\n,(11)\nwherew=mz/mxandR= (Dzy+Dyx)/2. Equation\n(11) shows that the sign of dmy/dtis fully controlled by\nthe roots of the equation w2+2Rw/β+1 = 0, namely,\nw1,2=−/parenleftBig\nR∓/radicalbig\nR2−β2/parenrightBig\n/β. When β2< Q2< R2,4\nFIG. 2: Constant-level curves of function ˜ g(m;β,ha) forβ2<\nQ2. Dashed lines: curves w±wQ= 0 on which gM(m) = 0.\nBlackdots: critical points. Shadowedregion: regiondelim ited\nby curves w=w1andw=w2, in which ( f−f0)>0 and\n(w−w1)(w−w2)<0 (the opposite occurs in remaining non\nshadowed region). Parameters: Dyx= 0.3,Dzy= 1,β= 0.2,\nha= 0.1,wQ=/radicalbig\nDzy/Dyx.\nw2\n1< w2\nQandw2\n2> w2\nQ. Therefore, the curve w=w1\nlies in the region gM<0 and the curve w=w2in the\nregiongM>0, sincegM(m) =Dzym2\nx/parenleftbig\nw2−w2\nQ/parenrightbig\n/2 (see\nEq. (1) and Fig. 2).\nConsider now the function:\n˜g(m;β,ha) = (12)\n˜gM(m;β)−f0haey·m, f0=/braceleftbiggf(w1;β) ifgM≤0\nf(w2;β) ifgM>0.\nIts time derivative, computed from Eq. (10), is: d˜g/dt=\nha(f−f0)dmy/dt, ateverypointin statespaceatwhich\ngM/negationslash= 0. By combining this result with Eq. (11), one\narrives at:\nd˜g\ndt=−βha1−m2\ny\n1+w2/parenleftbig\nf−f0/parenrightbig/parenleftbig\nw−w1/parenrightbig/parenleftbig\nw−w2/parenrightbig\n.(13)\nBy definition of f(w;β) (Eq. (7)) and f0(Eq. (12)),\n(f−f0)>0 when ( w−w1)(w−w2)<0 and vice versa\n(see Fig. 2). Therefore, the function ˜ g(m;β,ha) will be\nan increasing or decreasing function of time, depending\non whether the product βhais positive or negative, re-\nspectively. In particular, the maxima and minima of ˜ g\nwill represent critical points of the dynamics (Fig. 2).\nWhenβ= 0 orha= 0, ˜gis reduced to the corresponding\nconserved quantity, g(m;ha) or ˜gM(m;β), respectively.\nEquation (13) is not valid when gM= 0, because ˜ gis\ndiscontinuous there as a consequence of the jump in f0.\nHowever, this discontinuity does not modify the conclu-\nsions of our analysis. It is sufficient to complement Eq.\n(13) with the information about the direction of cross-\ning of the boundary gM= 0. This information is readilyobtained from the dynamics of the ratio w=mz/mx.\nFrom Eq. (10) one finds that if w±wQ= 0, then\ndw/dt=−/parenleftbig\n1+w2\nQ/parenrightbig\nha. Thus, the boundary gM= 0\nis crossed in the sense of decreasing wwhenha>0 and\nincreasing wwhenha<0.\nThe existence of the function ˜ gimplies that the un-\ndamped spin-transfer-driven dynamics under nonzero\nfield is nothing but a field-induced relaxation process to-\nward ˜gminima or maxima, depending on the sign of the\nproduct βha. The function ˜ gis characterised by a pair\nof minima in the region gM<0 and a pair of maxima\nin the region gM>0 [18]. Therefore, there will exist\ntwo basins of attraction for the field-induced relaxation.\nThese basins exhibit some degree of interlacing, similarly\nto what one observes in conventional magnetization re-\nlaxation due to intrinsic damping [19], the smaller the\nfieldha, the finer the interlacing. An example, obtained\nby numerical integration of Eq. (10), is shown in Fig. 3.\nThe fine basin interlacing makes the field-induced relax-\nation probabilistic in nature whenever control of initial\nconditions is imperfect [20, 21]. We stress that intrinsic\ndamping has been neglected in the derivation of these\nresults.\nFIG. 3: Basins ofattraction ofundampedspin-transfer-dri ven\ndynamics under nonzero external magnetic field. Parameters :\nDyx= 0.3,Dzy= 1,β= 0.05,ha=−0.025.\nIntrinsic damping effects can be conveniently intro-\nducedinso-calledGilbertform[3,22,23], whichamounts\ntochanging dm/dtintodm/dt−αm×dm/dtinEq. (10).\nAs a consequence, Eq. (13) is modified as:\nd˜g\ndt=−αf/vextendsingle/vextendsingle/vextendsingle/vextendsingledm\ndt/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+ha(f−f0)dmy\ndt,(14)\nwhereαrepresents the damping constant. The last term\non the right-hand side of Eq. (14) is not exactly co-\nincident with the right-hand side of Eq. (13), because\ndmy/dtis slightly modified by the introduction of damp-\ning. However, this modification plays a secondary role if\nα≪1. In essence, Eq. (14) is controlledby twoterms, of5\nwhich the one due to damping is alwaysnegative whereas\nthe onedue to currentand field is approximatelyequal to\nthe right-hand side of Eq. (13) and has thus the sign of\nthe product βha. Hence, when current and field have op-\nposite sign, their action contributes jointly with intrinsic\ndamping to the stabilization of ˜ gminima, whereas when\nthey have identical sign their action competes with that\nof intrinsic damping.\nThere exist conditions under which damping-induced\nand field-induced relaxations balance each other, leading\nto the appearance of limit cycles, that is, of magneti-\nzation self-oscillations. A typical scenario, confirmed by\ncomputersimulations,istheformationofapairofattrac-\ntive/repulsive limit cycles through a semi-stable limit-\ncycle bifurcation [17, 24]. 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