Model of damping and anisotropy at elevated temperatures: application to granular FePt films
Mara Strungaru, Sergiu Ruta, Richard F L Evans, Roy W Chantrell
MModel of damping and anisotropy at elevated temperatures: application to granular FePt films
Mara Strungaru, Sergiu Ruta, Richard F. L. Evans, and Roy W. Chantrell Department of Physics, University of York, York, YO10 5DD, UK
Understanding the damping mechanism in finite size systems and its dependence on temperature is a criticalstep in the development of magnetic nanotechnologies. In this work, nano-sized materials are modeled via atom-istic spin dynamics, the damping parameter being extracted from Ferromagnetic Resonance (FMR) simulationsapplied for FePt systems, generally used for heat-assisted magnetic recording media (HAMR). We find that thedamping increases rapidly close to T C and the effect is enhanced with decreasing system size, which is ascribedto scattering at the grain boundaries. Additionally, FMR methods provide the temperature dependence of bothdamping and the anisotropy, important for the development of HAMR. Semi-analytical calculations show that,in the presence of a grain size distribution, the FMR linewidth can decrease close to the Curie temperature dueto a loss of inhomogeneous line broadening. Although FePt has been used in this study, the results presented inthe current work are general and valid for any ferromagnetic material. I. INTRODUCTION
The magnetic damping parameter is important from both afundamental and applications point of view as it controls thedynamic properties of the system such as magnetic relaxation,spin waves, domain wall propagation and magnetic reversalprocesses. Magnetic materials have a broad range of inter-est for nano-devices/nano-elements and exhibit a fast responseto external excitations. In information technologies, dampingplays a crucial role, especially for spin-transfer torque mag-netic random access memories (STT-MRAM) where it con-trols the switching current [1]. With the emerging field ofmagnetisation switching via ultrafast laser pulses, the damp-ing parameter can influence the fluence of the laser pulse nec-essary for demagnetising and switching of the sample [2].Spintronic devices such as race-track memories which arebased on domain wall propagation in magnetic nanowires arealso influenced by damping [3]. As current magnetic tech-nologies are based on nanostructures of smaller and smallersizes, the finite size effects become more important and cansignificantly influence the magnetic properties including thedamping. Therefore understanding the dependence of damp-ing on temperature in finite size systems is a critical step inthe development of magnetic nano-technologies.One of the technologies that is strongly influenced bydamping is magnetic recording, where the damping constantof the storage medium controls the writing speeds and bit er-ror rates [4, 5]. The next generation of ultra-high density stor-age technology is likely to be based on heat-assisted magneticrecording (HAMR) [6–9]. The main candidate for HAMRmedia is L ordered FePt [8, 10] due to its large perpendicu-lar anisotropy and low Curie temperature ( T C ). For HAMR ap-plications, information is stored at room temperature (300K)but the writing is done at elevated temperatures close to T C ,therefore providing a large range over which the temperaturedependence of the damping needs to be understood. As theareal density increases, the grain size decreases and finite-sizeeffects are becoming crucial. For this reason, FePt is an idealcandidate for studying temperature and finite size effects onthe damping. Although FePt has been used in this study, theresults presented in the current work are general and valid forany ferromagnetic material. First investigations of the Gilbert damping for FePt in-volved experimental measurements via optical pump-probetechniques. The damping measured at room temperaturevaries widely from one study to another. Becker et al. [11]reported an effective damping of 0.1, and an even larger value(0.21) was found by Lee et al. [12], while the measurmentsof Mizukami et al [13] gave a value of 0.055. It is importantto note that these values include both intrinsic and extrinsiccontributions, the purely intrinsic damping being even smallerthan the reported values [13, 14]. Recently, Richardson et al[15] reported experimental measurements of damping at ele-vated temperatures showing an unexpected decrease of damp-ing with temperature. A decrease in the effective damping canbe crucial in HAMR, as this can increase the switching time,affect the signal to noise ratio and negatively impact the per-formance of HAMR. Theoretical studies on how the dampingvaries at elevated temperatures and in finite sized systems istherefore a critically important problem.Ostler et al [16] have successfully calculated the tem-perature dependence of damping in FePt bulk and thin-filmsystems based on the Landau-Lifshitz-Bloch (LLB) equa-tion [17], showing an increased damping for thin-film sys-tems, in comparison with the bulk case. The LLB equationis derived for a bulk material. It is important to note that amajor contribution to damping, especially at elevated temper-atures, arises from magnon scattering. On the bulk scale theseprocesses are reproduced by the LLB equation, but with de-creasing linear dimension, finite size and surface effects be-come important. Since these are not accounted for by theLLB equation it is necessary to use atomistic spin dynamics(ASD) simulations [18] for nanoscale grains, as ASD calcula-tions include magnon processes. Using atomistic spin dynam-ics, ASD, we are able to calculate the FMR spectra for smallsystem sizes at elevated temperatures. Ferromagnetic reso-nance simulations are computationally very expensive, hencewe have developed a more efficient method of calculating thedamping of these systems via a grid search method. We showthat both of the methods agree, and furthermore, that theyare able to calculate both the dependence of damping andanisotropy as function of temperature. The dependence ofanisotropy as function of temperature is crucial for HAMR asit defines the temperature at which the writing process occurs. a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b II. FERROMAGNETIC RESONANCE USING ATOMISTICSPIN DYNAMICS
To calculate damping as function of temperature we per-form atomistic spin dynamics (ASD) simulations using thesoftware package
VAMPIRE [18]. ASD simulations assumea fixed lattice of atoms to which is associated a magneticmoment or spin S i = µ i / µ s that can precess in an effectivefield H i according to the Landau-Lifshitz Gilbert (LLG) equa-tion. In our model, the Hamiltonian of the system contains aHeisenberg exchange term of strength J i j , uniaxial energy ofstrength k u and a Zeeman term as shown in Eq. 1: H = − ∑ i , j J i j ( S i · S j ) − k u ∑ i ( S i · e ) − ∑ i µ i ( S i · B ) (1)The effective field can be calculated from the Hamiltonian ofthe model to which we add a thermal noise ξ i , that acts as aLangevin thermostat: H i = − µ i µ ∂ H ∂ S i + ξ i (2)The thermal field is assumed to be a white noise, with thefollowing mean, variance and strength, as calculated from theFokker-Planck equation: (cid:104) ξ i α ( t ) (cid:105) = , (cid:104) ξ i α ( t ) ξ j β ( s ) (cid:105) = D δ α , β δ i j δ ( t − s ) (3) D = λ k B T γ µ i µ (4)where λ represents the coupling to the heat bath, T the ther-mostat temperature, γ the gyromagnetic ratio. We note thatthe heat bath coupling constant λ is different from the effec-tive Gilbert damping α , as the latter includes contributionsfrom magnon scattering and other extrinsic processes such asinhomogeneous line broadening.After calculating the effective field that acts on each atom,the magnetisation dynamics is given by solving the LLG equa-tion (Eq. 5) applied at the atomistic level [19] using a numer-ical integration based on the Heun scheme. ∂ S i ∂ t = − γ ( + λ ) S i × ( H i + λ S i × H i ) (5)To calculate the damping using atomistic spin dynamics weapply an out-of-plane magnetic field ( B ) to the sample withan additional in-plane oscillating field ( B rf = B sin ( πν t ) ),which is the default setup for ferromagnetic resonance experi-ments. The oscillating field will induce a coherent precessionof the spins of the system which will result in an oscillatorybehaviour of the in-plane magnetisation. By sweeping the fre-quency of the in-plane field, the amplitude of the oscillationsof magnetisation will change, with a maximum correspondingto the resonance frequency (as shown in Fig. 1). By Fourier FIG. 1. Illustration of the setup used for ferromagnetic resonance ex-periments. An out of plane magnetic field ( B ) and an in-plane oscil-lating field ( B r f = B sin ( πν t ) ) is applied to the sample, as shownin the right inset. By Fourier Transformation of the in-plane mag-netisation the power spectrum as function of frequency is obtained.The simulation is performed for a single FePt spin at T=0K, havinga damping of 0.01. This is equivalent with simulating a macrcospinat T=0K with equivalent properties. By fitting the power spectrum,the input resonance frequency and damping can be reproduced. transformation of the in-plane magnetisation, the power spec-trum as a function of frequency is obtained. Fig. 1 shows theFMR spectra for a single spin of FePt at 0K. The spectra canbe fitted by a Lorentzian curve (Eq. 6) where w represents thewidth of the curve and A its amplitude. By fitting with Eq. 6,the effective Gilbert damping α and resonance frequency f can be extracted. L ( x ) = A π . w ( x − f ) + ( . w ) , α = . wf (6)The model parameters for FePt are listed in Tab. 1. L10FePt has a face-centred tetragonal structure formed of alter-nating layers of Fe and Pt, which can be approximated to abody-centered tetragonal structure with the central site occu-pied by Pt. The ab-initio calculations by Mryasov et al [20]showed that the Pt spin moment is found to be linearly de-pendent on the exchange field from the neighbouring Fe mo-ments. This dependence allows the Hamiltonian to be writtenonly considering the Fe degrees of freedom. Under these as-sumptions, by neglecting the explicit Pt atoms, the system canbe modelled as a simple cubic tetragonal structure with eachatomic site corresponding to an effective Fe+Pt moment. Themodel used for the FePt system is restricted only to nearestneighbour interaction to minimise the computational cost ofFMR calculations, in contrast with the full Hamiltonian givenby Mryasov et al [20]. The nearest-neighbour exchange valueis chosen to give a Curie temperature of FePt of 720K, to be inagreement with reported values for nearest and long-range ex-change magnetic Hamiltonian [21]. The damping parameterhas been chosen to approximate the experimentally measuredvalue in recording media provided by Advanced Storage Re-search Consortium (ASRC). The L1 phase of FePt has a verylarge uniaxial anisotropy, hence the increased thermal stabilityof the grains. The uniaxial anisotropy used in the simulationgives a anisotropy field of H k = k u / µ s = .
55 T, slightlylarger than the value used by Ostler et al (15.69 T). The FMRfields (0 .
05 T) used in our simulations are generally largerthan experimental FMR fields to allow more accurate simu-lations with enhanced temperature. Our tests confirm that nonon-linear modes are excited during the FMR simulations.
Quantity Symbol Value UnitsNearest-neighbours exchange J ij . × − JAnisotropy energy k u . × − JMagnetic moment µ S . µ B Thermal bath coupling λ . DC perpendicular field B TRF in-plane field B r f . TTABLE I. Parameters used for the initial calculations of the dampingconstant of FePt. At T = λ as no ther-mal scattering effects are present, hence the effective damp-ing of the system is given by the Gilbert damping which isthe coupling to the heat bath. For this simulation, we haveused an input heat bath constant of λ = .
01, which we thenrecover by performing FMR calculations at T = f Kittel = γ π · ( B + k u µ S ) , depending on the applied fieldstrength ( B ) and on the perpendicular anisotropy of our sys-tem ( H k = k u µ s ) . For an FePt system the resonance frequencywe obtain is 520 GHz within 1% fitting error, due to the ex-ceptionally large magnetic anisotropy of the system. III. GRID-SEARCH METHOD
The Gilbert damping can be also calculated by fitting thetime-traces of the magnetisation relaxation. The time-tracescan be obtained via pump-probe experiments [11], howeverthe dynamics of the magnetisation will include the effect ofthe laser pulse, such as heating and induced local magnetisa-tions due to the inverse Faraday effect. To avoid the contribu-tions to the damping from the laser pulse, damping can be cal-culated by taking the system out of equilibrium, letting it relaxand subsequently recording the time-trace of the magnetisa-tion. Ellis et al [22] have numerically studied the dampingof rare-earth doped permalloy using the transverse relaxationcurves, by fitting them with the analytical solutions of theLLG equations. In the case of large anisotropy, exchange in-teraction and applied field, there is no simple general solutionto the LLG equation. Pai et al [23] used an applied field muchlarger than the anisotropy field so that the dynamics closelyapproximate that of the LLG equation with no anisotropy.However, this approach is unsuitable for FePt due to the verylarge fields required and also the influence of strong magneticfields near the Curie temperature. Hence, we adopt a com-putational grid search method where we pre-calculate single
FIG. 2. χ map calculated using the grid-search method based onsingle-spin simulations at T=0.1K. (inset) The input and fitted mag-netisation relaxation curves showing the validation of the method. spin solutions for the LLG equation using ASD, build a database using these solutions and then build an algorithm thatcan identify the damping and anisotropy parameters from anytransverse relaxation curve. The method we chose simply in-volves sweeping through the parameter space, the solution be-ing given by minimising the sum of the squared residuals, amethod known as grid-search.The grid search method can be used to fit time-dependent m ( t ) curves in the case where analytical solutions do not exist.The numerical curves that need to be fitted are compared witheach of the pre-calculated numerical curves with the singlespin system. The best match will be given by the curve withlowest sum of squared residuals, the χ parameter, where χ is defined as: χ = N ∑ i = (cid:104) m ( t i ) − f ( t i , p ) (cid:105) (7)where m i ( t i ) is the value of the magnetisation at each momentin time t i , f ( t i , p ) are the pre-calculated single-spin depen-dences of the magnetisation at each moment t i , p is the listof parameters that have been varied (in our case p = ( K , α )).The minimum value of χ from all p parameters is the best-agreement numerical solution.Figure 2 shows the calculated χ as function of the main pa-rameters, specifically the anisotropy and damping, at T=0.1K.In order to construct the single spin simulation data-base, wechose a resolution of ∆ k u = . × − J for the anisotropyand ∆ α step = .
001 for the damping. It can be seen that theanisotropy is very well resolved: there is a sharp minimumat k u = . × − , which is the closest value to the inputanisotropy, k u = . × − taking into account the resolu-tion we use for the data base. In the case of damping, theminimum is wider, leading to an error of approximately 0 . FIG. 3. Comparison of FMR and grid-search fitting; a) Damping; b)Anisotropy;
IV. HIGH-TEMPERATURE FMR: DAMPING ANDANISOTROPY CALCULATIONS
In this section the damping and anisotropy are computed,from frequency dependent FMR spectra and via the gridsearch method. The aim is to investigate the damping closeto T c and in particular the effect of finite grain size. First,we test the effectiveness of the grid search method which waspresented in Section III. Fig. 3 shows the comparison betweenthe two methods of calculation of the damping as a functionof temperature for a granular system of 15 non-interactinggrains of 5nm diameter and 10nm height. The variations ofanisotropy with temperature agree very well between the twomethods, however the grid search method is far more com-putationally efficient. The enhanced computational efficiencycomes from the fact that instead of simulating multiple fre-quency points to obtain the FMR, a single transverse relax-ation simulation is needed to calculate the same parameters.The time-scales for the two simulations are also different: thefrequency dependent FMR requires around 3 ns for each datapoint in the FMR spectra to perform the FFT analysis, whilethe transverse relaxation method requires, depending on thematerial, less than 1 ns. Extracted damping values agree rea-sonably well between the two methods, within the error bars.For the grid search method, there will be a damping intervalthat gives the same value of χ . For the FMR experiment,the error bar is computed as the standard error of groups of 5non-interacting grains.Because of the large error bars, especially close to T C forthe grid search method, we use the direct FMR simulationsfor the remainder of the paper and later consider possiblemeans of improvement of the reliability of the grid searchmethod. For initial calculations we model a granular FePtsystem as a cylinder of 10nm height and 5nm diameter. Forcomparison, the bulk FePt system is modelled via a system of32 × ×
32 atoms with periodic boundary conditions. Closeto T C , the thermal fluctuations become increasingly large fornon-periodic systems and can lead to large errors in the de-termination of damping and anisotropy. For this reason, to re-duce the statistical fluctuations, a system of 15 non-interactinggrains is modelled. This significantly reduces the fluctua-tions in the magnetisation components and leads to statisti-cally improved results. The in-plane magnetisation time se-ries is Fourier transformed, and the damping is extracted as FIG. 4. Damping as function of normalised temperature for bulkand granular FePt system. The granular system shows overall largerdamping than the bulk system, due to additional magnon scatteringprocesses at the interface. (inset) Magnetisation curves for granularand bulk FePt. The Curie temperatures for the two systems are :grain- T C =690K, bulk - T C =720K. presented in Section II.Fig. 4 shows the damping as a function of temperature forbulk and granular systems. For comparison the temperatureis normalised to the Curie Temperature of the systems, whichdiffer due to finite size effects [21, 24]. The granular systemwill have a reduced Curie Temperature due to the cutoff in theexchange interactions at the surface. This is shown as an insetin Fig. 4, where the magnetisation as a function of temper-ature is computed for the two systems. The Curie tempera-tures for the two systems, determined from the susceptibilitypeak, are: T C for the grain =690K, and for the bulk T C =720K.The input Gilbert damping parameter is 0.05, this value be-ing reproduced at T=0K as expected due to the quenching ofmagnon excitations. FIG. 5. Damping as function of temperature for granular (5nm × × m ( T , D ) and T c ( D ) are computed numerically from the atomistic model. With increasing temperature, for both bulk and granularsystems, the effective damping increases. This can be under-stood as, with enhanced temperature, there is increasing exci-tation of magnons which can suffer more complex non-linear
FIG. 6. Temperature dependence of the damping constant for diameters of 4nm, 5nm and 6nm. Solid lines are calculations using the LLBdamping expression. Divergence from the LLB expression for small particle diameter is indicative of surface effects. scattering processes.Moving on to granular systems, the inclusion of the surfacewill add extra magnon modes into the system, leading to morescattering effects that will increase the effective damping. Inorder to effect a qualitative illustration of surface effects weuse the damping calculated from the Landau-Lifshitz-Blochequation [17]. An analytical solution to the variation ofdamping with temperature exists in the LLB description, asgiven by Garanin [17] and Ostler et al [16]. The effectivedamping as derived within the LLB description is given by: α ( T ) = λ m ( T ) (cid:18) − T T c (cid:19) , (8)where λ is the input coupling to the thermal bath used in atom-istic spin dynamics simulations, T C the Curie Temperature ofthe system, m ( T ) = M ( T ) / M s V the normalised magnetisa-tion. In principle Eq. 8 is strictly valid only for an infinite sys-tem. However, as a first approximation, finite size effects canbe introduced empirically using diameter dependent functions m ( T , D ) and T c ( D ) calculated using an atomistic model. In thedamping calculations considered here, the grain surfaces havetwo effects. Firstly, the loss of coordination at the surfacesdrive a reduction in T c and loss of criticality of the phase tran-sition. This effect can be accounted for by using numericallycalculated m ( T , D ) and T c ( D ) for a given diameter D. The sec-ond effect is the increased magnon scattering at the surfaceswhich is a dynamic effect and not included in the parameter-ization of the static properties. Thus it seems reasonable toassociate deviations from the parameterized version of Eq. 8with scattering at the grain surfaces.Consequently, we compare our numerical results for α ( T , D ) with the parameterised version of Eq. 8 - Fig. 5 ,where m ( T , D ) is calculated numerically with the ASD model(shown in Fig. 4, inset) and the Curie Temperature ( T c ( D ) )is calculated from the peak of the susceptibility. For thebulk system, the numerical damping calculated from the FMRcurves with the atomistic model agree well with the damp-ing calculated with the analytical formula given by Eq. 8.This is consistent with the first comparison of atomistic andLLB models [25] which showed that the mean-field treatmentof [17] agreed quantitatively well with atomistic model calcu-lations for the transverse and longitudinal damping. However, the granular system gives a consistently increased dampingcompared to the analytical formula. Following the earlier rea-soning, this enhancement can be attributed to the scatteringeffects at the grain surface.To systematically study the effect of the scattering at thesurface, we have calculated the damping as a function of thesystem size. For simplicity, we consider cubic grains witha volume varying from 4nm × × × × × × V. MODEL INCLUDING INHOMOGENEOUS LINEBROADENING
Realistic granular systems will present a distribution ofproperties. In the simplest case, the distribution of mag-netic properties can arise from a distribution of the size of thegrains, which can induce a distribution of T C , m and H k . Sinceit is computationally expensive to study a system of grains nu-merically within the ASD model we can, in the first instance,model the effect of the distributions analytically. In the caseof a distribution of grains of diameter D , the power spectrumof the system is expressed by: P sys ( f , T ) = (cid:90) + ∞ P ( f , D , T ) F ( D ) dD (9) FIG. 7. Field swept FMR for a lognormal distribution of grains of D = σ D = .
17. Input damping λ = .
01, input H k = . f = . ∆ H ) and (c) system magnetisation and anisotropy field as function oftemperature. Close to T C , the linewidth shows a decrease which translates to a decrease in the damping of the system. No magnetostatic orexchange interaction between grains is considered. The distribution of size, F ( D ) , is considered lognormal.The power spectrum of a grain of diameter D can be expressedby [16]: P ( f , B , D , T ) = C ˜ mD f γ B ˜ α ( ˜ α ˜ f ) + ( f − ˜ f ) (10)where ˜ m = m ( T , D ) , ˜ α = α ( T , D ) , ˜ f = f ( T , D ) = γ ( B + Hk ( T , D )) , C = π h . This allows to model both fre-quency swept FMR ( B = constant) and field swept FMR( f =constant).We note that a Distribution of grain size leads to distri-butions of further properties, starting, due to finite size ef-fects, with the Curie temperature. Each of these is introducedinto the analytical model as follows. Hovorka et al [21] haveshown via finite size scaling analysis that the relation betweenthe size of a grain and its Curie temperature is given by: T C ( D ) = T ∞ C ( − d / D ) / ν , (11)where d = .
71 and ν = .
79 [21] parametrised for nearestneighbours exchange systems and T ∞ C = K . The variationin T C will introduce a variation in the magnetisation curvesgiven by: m ( T , D ) = (cid:18) − TT C ( D ) (cid:19) β , β = . . (12)As a further consequence, the anisotropy will be dependenton the diameter. The uniaxial anisotropy energy K has a tem-perature dependence in the form of K ( T ) ∼ m ( T ) γ . For FePtit was found that the exponent is equal to 2.1 by experimen-tal measurements [26][27] in agreement with later ab-initio calculations [20] . Hence the anisotropy field will depend on m ( T , D ) with an exponent of 1.1; H K ( T , D ) = H K m ( T , D ) . , (13)leading to a dispersion of H K ( D , T ) .Finally the size distribution will produce different varia-tions of damping as a function of grain size, since α ( T , D ) = λ m ( T , D ) (cid:18) − T T c ( D ) (cid:19) . (14) In the presence of distributions of properties, the variationof damping with temperature can have a complex behaviour,especially close to T c where there is a strong variation ofmagnetic properties with temperature and size. On revertingto a monodispersed system by setting the distributions to δ -functions the damping is given by Eq. 14 resulting in an in-crease of linewidth consistent with temperature Fig. 6.Richardson et al [15] have shown that, in the case of agranular system of FePt, close to T C a decrease in damping/-linewidth is observed. This effect was attributed to the com-petition between two-magnon scattering and spin-flip magnonelectron scattering. We have shown via atomistic spin dynam-ics simulations that surface effects alone cannot be responsiblefor a decrease in damping, scattering at the surface leading toincreased damping at high temperatures.It is well known that, in the presence of a distribution ofproperties in the system, the linewidth broadens. In our casethe distribution of size will lead to a distribution of anisotropywhich increases the linewidth. Close to the Curie Temperatureof the system, some grains will become superparamagneticand will not contribute further to the FMR spectrum, hence itis possible that close to T c , the linewidth can decrease. Fig. 7presents a case where a decrease in linewidth appears within30-40K of the Curie Temperature of the system, a similartemperature interval as spanned by the experimental measure-ments [15]. Fig. 7 a) and b) show the variation of the FMRfield and linewidth as functions of temperature. The FMRspectra are calculated at constant frequency of f = . GHz ,consistent with the experimental value used in [15]. The aver-age magnetisation of distributed grains is calculated as M ( T ) = (cid:82) ∞ m ( T , D ) F ( D ) D dD (cid:82) ∞ F ( D ) D dD (15)and the anisotropy field is calculated as H K ( T ) = (cid:90) ∞ H K ( T , D ) F ( D ) dD . (16)The decrease in linewidth is associated with the fact that, closeto the Curie Temperature of the system, the small grains be-come superparamagnetic and do not contribute to the powerspectrum. The loss of the signal from the small grains is es-pecially pronounced due to the enhanced damping of smallergrains. VI. CONCLUSIONS AND OUTLOOK
We have calculated the temperature dependence of dampingand anisotropy for small FePt grain sizes. These parameterswere calculated within the ASD framework, via simulation ofswept frequency FMR processes and a fitting procedure basedon the grid search method. The grid search method offers amuch faster determination of damping and anisotropy, param-eters crucial for the development of future generation HAMRdrives. The method can be applied both for numerical data,as well for experimental relaxation curves obtained via pump-probe experiments. The damping calculations at large tem-peratures showed an increased damping for uncoupled gran-ular systems as expected due to increased magnon excitationat high temperature. Deviations from the parameterised ex-pression for the temperature dependence of damping from theLLB equation with decreasing grain size suggest that scatter-ing events at grain boundaries enhance the damping mecha-nism.This increase in damping, however, is not consistent withthe experimental data of Richardson et al [15] which showa decrease in linewidth at elevated temperatures. We havedeveloped a model taking into account inhomogeneous line broadening arising from the size distribution of the grainswhich gives rise to concomitant dispersions of T C , m and K .The model has been used to simulate swept field FMR as usedin the experiments. Calculations have shown that, under theeffect of distribution of properties, the linewidth can exhibit adecrease towards large temperatures, in accordance with theexperiments of Ref. [15]. The decrease is predominantly dueto a transition to superparamagnetic behaviour of small grainswith increasing temperature. This suggests inhomogeneousline broadening (likely a significant factor in granular films)as an explanation for the unusual decrease in linewidth mea-sured by Richardson et al [15]. As large damping is necessaryfor good performance of HAMR and MRAM devices withthis work we further stress the importance of experimentallycontrolling the size distributions of the media. VII. ACKNOWLEDGEMENTS
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