The effect of sample inhomogeneity on the ferromagnetic resonance linewidth is determined by diagonalization of a spin wave Hamiltonian for ferromagnetic thin films with inhomogeneities spanning a wide range of characteristic length scales. A model inhomogeneity is used that consist of size D grains and an anisotropy field Hp that varies randomly from grain to grain in a film with thickness d and magnetization Ms. The resulting linewidth agrees well with the two-magnon model for small inhomogeneity, HpD « πMsd. For large inhomogeneity, HpD » πMsd the precession becomes localized and the spectrum approaches that of local precession on independent grains

Kunz, Andrew

McMichael, R. D.

Twisselmann, D. J.

English

epublications@Marquette

Marquette Universitye-Publications@MarquettePhysics Faculty Research and Publications Physics, Department of6-1-2003Localized Ferromagnetic Resonance inInhomogeneous Thin FilmsR. D. McMichaelNational Institute of Standards & TechnologyD. J. TwisselmannNational Institute of Standards & TechnologyAndrew KunzMarquette University, andrew.kunz@marquette.eduPublished version. Physical Review Letters, Vol. 90, No. 22 ( June 2003). DOI. © 2003 AmericanPhysical Society. Used with permission.Localized Ferromagnetic Resonance in Inhomogeneous Thin FilmsR. D. McMichael and D. J. TwisselmannNational Institute of Standards & Technology, Gaithersburg, Maryland 20899, USAAndrew KunzPhysics Department, Lawrence University, Appleton Wisconsin 54912, USA(Received 23 December 2002; published 3 June 2003)The effect of sample inhomogeneity on the ferromagnetic resonance linewidth is determined bydiagonalization of a spin wave Hamiltonian for ferromagnetic thin films with inhomogeneitiesspanning a wide range of characteristic length scales. A model inhomogeneity is used that consistsof size D grains and an anisotropy field Hp that varies randomly from grain to grain in a film withthickness d and magnetization Ms. The resulting linewidth agrees well with the two-magnon model forsmall inhomogeneity, HpD Msd. For large inhomogeneity, HpD Msd, the precession becomeslocalized and the spectrum approaches that of local precession on independent grains.DOI: 10.1103/PhysRevLett.90.227601 PACS numbers: 76.50.+g, 75.40.Gb, 75.40.MgIn many ferromagnetic materials, a few nanosecondsare typically required for magnetization precession todamp out after the magnetization is rotated away fromthe equilibrium direction. As data rates approach 1 GHz,this nanosecond time scale for damping becomes a criti-cal issue for data storage and other magnetic applications.Accordingly, there has recently been a surge of interest inmagnetization damping, i.e., the coupling of the magne-tization to the thermal bath [1–8], and the related mag-netic fluctuations in submicron magnetic sensors [9,10].Ferromagnetic resonance (FMR) is the most promi-nently used technique to measure magnetization damp-ing. FMR spectra are typically measured either bymeasuring a pulse response in the time domain [11] or,more commonly, by driving the magnetization with aconstant frequency microwave field while slowly chang-ing the resonance frequency via an applied magnetic fieldH. Damping is measured through the linewidth H of theabsorption peak in the transverse susceptibility spectrum.A central difficulty in the measurement of magnetiza-tion damping by ferromagnetic resonance is that experi-mental values of damping time or FMR linewidth reflectthe effects of both inhomogeneity and damping. ThisLetter describes modeling of FMR line broadening ininhomogeneous thin films through calculations of theeigenmodes of the magnetization motion. The modeldescribed below unifies the local resonance model, whichis valid for large, strong inhomogeneities, and the two-magnon model, which is valid for small, weak inhomo-geneities.The resonant precession frequency of the magnetiza-tion depends on a number of experimental and materialsparameters, xi, including applied field, magnetization,surface anisotropy, film thickness, and magnetocrystal-line anisotropy. The simplest model of FMR linewidthand inhomogeneity attributes the inhomogeneous broad-ening to parameter variations xi that produce differentresonance conditions in independent parts of the film[12]. The resulting spread of resonance frequencies yieldsa field linewidth that is given byHlocal @!@H1Xi~~~~~~@!@xi~~~~~~xi 23p !; (1)where the last term represents the linewidth due to damp-ing and  is the phenomenological Gilbert dampingparameter. In the following, this model will be referredto as the ‘‘local resonance’’ model of linewidth. Thismodel has provided a good description of FMR linewidthdata as a function of magnetization angle in several cases[12–14]. Additionally, if the important inhomogeneouseffective fields Hp corresponding to parameters xi simplyadd to the applied field, the first term in (1) is a constant,and the linewidth is linear in frequency:Hlocal  Hp  23p !: (2)The frequency dependence of H in ultra-thin magneticmetal films is often found to be approximately linear inthis way [13,15–17].The essential assumption of the local resonance modelof linewidth is that neighboring regions of the film do notinteract significantly. However, exchange and dipolar in-teractions are essential characteristics of ferromagneticmaterials. The two-magnon model of linewidth [18–24],reviewed below, accounts for interactions through thespin wave dispersion relation and treats the inhomogene-ity as a perturbation. Two-magnon and local resonancelinewidths behave very differently, but the two-magnonmodel has also been verified by a different set of experi-mental results [18,25,26].In the following, we briefly review some of the proper-ties of spin waves, and use them as a basis for modelinglinewidth in inhomogeneous films.Our review of spin waves begins with a uniform mag-netic film lying in the x-z plane with an applied field HP H Y S I C A L R E V I E W L E T T E R S week ending6 JUNE 2003VOLUME 90, NUMBER 22227601-1 0031-9007=03=90(22)=227601(4)$20.00  2003 The American Physical Society 227601-1along the z direction. For small deviations of the magne-tization from its equilibrium, Msz^, we writeM r ’ mxrx^myry^ Msz^: (3)It is convenient to expand the magnetization in terms ofthe Fourier coefficients of the magnetization; mxr Pkmx;keikr, and similarly for myr. Neglecting a con-stant term, the energy of the uniform film can be ex-panded asE0  V02Xkhx;km2x;k  hy;km2y;k; (4)where V is the volume of the film. We assume that themagnetization is uniform across the thickness of the film;the wave vector k is therefore restricted to the x-z plane.The normalized stiffness fields, hx;k and hy;k, include theapplied field, dipolar fields, and exchange interactions.For a film of thickness d with exchange stiffness A[18,27],Mshx;k  H  2A=Msk2  1 NkMsk2x=k2; (5)Mshy;k  H  2A=Msk2  NkMs; (6)where Nk  1 ekd=kd is a k-dependent demagneti-zation factor [28] under the assumption that the magne-tization is uniform through the film thickness. The spinwave dispersion relation is given by!k  Ms0hx;khy;k1=2; (7)where  is the gyromagnetic ratio, gB= h; see Fig. 1.In an FMR measurement on an ideal uniform film, thek  0 uniform mode is resonantly driven by a uniformmicrowave field and the width of the resonance dependsonly on damping.The quantum mechanical raising and lowering opera-tors ayk and ak corresponding to the classical spin wavesare given bymxk  ih!k=20Vhx;kqak  ayk; (8)myk h!k=20Vhy;kqak  ayk: (9)For a simple model of inhomogeneity, we add a spa-tially varying anisotropy field [18,19]. The energy expres-sion (4) becomesEp  E0 0Vc2MsXrHprmxr2 myr2; (10)where the sum is over cells of a superimposed 2D grid andVc is the volume of a grid cell.The r-dependent terms in (10) are expanded in terms ofFourier coefficients, Hp;k, mx;k, and my;k, which are ex-panded using Eqs. (8) and (9) to obtain a spin waveHamiltonian for the inhomogeneous film [29],H  hXk!kaykak Xk;k0Ak;k0ayk0ak  H:c:; (11)where H.c. indicates the Hermitian conjugate. Inhomoge-neities such as surface pits or voids [20–22] or irregulardipolar and exchange interactions [23] yield Hamil-tonians of the same form. For the case of a varyingapplied field,Ak;k0  0 hMshx;khx;k0p  hy;khy;k0p2!k!k0p 0Hp;kk0 :(12)Terms of the form A0;ka0ayk are particularly important forFMR linewidth because they describe the coupling of theuniform mode to spin wave modes.Treating the A terms in (11) as a perturbation yieldsthe well-known ‘‘two-magnon’’ model for the apparentdecay rate 2mag of the k  0 magnon [18–24]. To lowestorder in Hp,2mag  2hXkjA0;kj2  h!k  h!0: (13)Peak-to-peak linewidth is taken from the extrema of thederivative spectrum. The corresponding two-magnoncontribution to the peak-to-peak, field-swept linewidth isH2mag  13p dHd!02mag: (14)Exchange and dipolar interactions are included in2mag through the spin wave dispersion relation, !k, butthe validity of (13) is limited to weak perturbations.Higher order extensions of the two-magnon model areneeded [30–32].Here, we move toward finding the linewidth to allorders in Hp by diagonalizing the Hamiltonian (11) nu-merically [33]. A similar approach was taken by Huberand Ching in a study of density of states and localizationin spin clusters with random axis anisotropy [34]. Out of106 eigenmodes of the discretized uniform film, wechoose as basis states the 3001 spin waves that are closestin frequency to!0, plotted in Fig. 1. For large enough sets 9.5 10 10.5 11 11.5 12 12.5 130x100 1x105 2x105 3x105 4x105 5x105Frequency (GHz)|k| (cm-1)FIG. 1. Precession frequencies of 3001 spin waves for a 3 nmthick rectangle of uniform Permalloy, 10 m 2:5 m. A114 mT field is applied in plane to give a uniform precessionfrequency of !k  0=2  10 GHz.P H Y S I C A L R E V I E W L E T T E R S week ending6 JUNE 2003VOLUME 90, NUMBER 22227601-2 227601-2of basis states, we find that the calculated linewidth is notsensitive to the number of selected modes. We modelHpr by assigning a random field to each grain in aperiodic ‘‘grain structure’’ made up of Voronoi polygons.The random field is taken from a Gaussian distributionwith root mean square value Hrmsp .The diagonalization routine yields the eigenfrequen-cies of the inhomogeneous film and its eigenmodes bi.These new eigenmodes are mixtures of the basis spinwaves ak; bi PkUi;kak where Ui;k is the unitary ma-trix that diagonalizes (11). Because the k  0 spin waveis generally a component of each eigenmode, each of theeigenmodes can couple directly to the applied uniformmicrowave field. The FMR intensity Ii of each eigenmodeis given by Ii  jUi;0j2.We illustrate this procedure by modeling the FMRresponse of 3 nm-thick films with Ms  800 kA=m(4Ms  10 kG) in an applied field of 0H  114 mTto give a resonant frequency of 10 GHz. Results are shownfor average grain sizes of 20 and 1400 nm in Figs. 2 and 3,respectively.The FMR intensity distribution is plotted as impulsesin Figs. 2(a) and 3(a). For uniform films, these distribu-tions would appear as single spikes at 10 GHz. Wesimulate FMR signals by replacing each eigenmodespike with a Lorentzian peak, each peak having inte-grated intensity Ii and having full width at half maxi-mum given by the Landau-Lifshitz-Gilbert linewidth!50  0Mshx;0  hy;0 and   0:01. The indi-vidual Lorentzian peaks are summed and the result is dif-ferentiated with respect to frequency to obtain the simu-lated FMR signal. The peak-to-peak frequency linewidth! is determined from the maximum and minimum ofthis signal, similar to the common experimental prac-tice. Finally, the field linewidth is obtained usingH  !dH=d!0.The grain structures and perturbation fields used inthese examples are shown in Figs. 2(b) and 3(b) whereHpr is lowest in dark grains and highest in light grains.The magnetization patterns associated with single ei-genmodes are shown in Figs. 2(c) and 3(c) with greyscale indicating mxr. These patterns are calculatedusing mx;ir /PkUi;k  Ui;keikr. Using this colorscheme, uniform precession would appear as a solid lightor dark color.There are a number of qualitative differences betweenthe results for large and small grains. For small grains(Fig. 2) the magnetization precession is on length scalesmuch larger than the grain size and the FMR signal looksLorentzian. For large grains, (Fig. 3), the magnetizationpattern is closely correlated with the grain structure andthe spectra are more complicated. These spectra are simi-lar to the results of local resonance calculations where theprecession frequency for each grain is calculated from thelocal applied field H Hpr and the intensity is propor-tional to the grain area. The FMR signals calculated fromthe 3001 eigenmodes and from the 51 grains are verysimilar. [Fig. 3(a)]. For the complicated, large-grain spec-tra, the peak-to-peak characterization of linewidth doesnot give consistent results for different grain structureswith the same average grain size. As an alternative line-width characterization for large grains, we calculate thestandard deviation of the eigenmode intensity distribu-tion, and report a peak-to-peak linewidth correspondingto a Gaussian peak with the same standard deviation. Thismethod fails for Lorentzian intensity distributions be-cause the standard deviation is not well defined.In Fig. 4, linewidth values from eigenmode calcula-tions, the two-magnon model, and the local resonancemodel are plotted as a function of grain size for threevalues of rms perturbation field. There is a transition fromFIG. 2. Calculated FMR response for a 3 nm thick film ofPermalloy with 20 nm grains and a 10 mT rms perturbationfield. (a) FMR intensity and simulated FMR signal, (b) half ofthe grain structure, and (c) the corresponding half of theeigenmode indicated by the arrow in (a) showing the collectiveresponse of this mode to a uniform driving field.FIG. 3. Calculated FMR response for a 3 nm thick film ofPermalloy with 1:4 m grains and a 10 mT rms perturbationfield. (a) FMR intensities and signals calculated from eigen-mode intensity and from the local resonance approximation,(b) half of the grain structure, and (c) the corresponding half ofthe eigenmode indicated by the arrow in (a) showing localprecession appearing predominantly in two of the grains.P H Y S I C A L R E V I E W L E T T E R S week ending6 JUNE 2003VOLUME 90, NUMBER 22227601-3 227601-3two-magnon behavior to local resonance behavior thatdepends on grain size and perturbation field.The characteristic combination of perturbation fieldand grain size for the transition from two-magnon be-havior to local resonance behavior can be estimated byconsidering a low-k spin wave in a grain of width Dwhere the applied field differs by Hp;c. If the spin wavewith frequency! is launched in the center of the grain weargue that it will behave locally if it is more than ap-proximately  out of phase at the grain boundaries;@ky@H~~~~!Hp;cD 22H MM2sdHp;cD  2: (15)This estimate of Hp;c is plotted in Fig. 4. In contrast tothis estimate, the Ioffe-Regel criterion for localization[35] suggests that there should never be localization inthis system because the mean free path for a magnon isalways less than the essentially infinite wavelength ofinterest.The eigenmode analysis presented here unifies the two-magnon model and the local resonance model and sug-gests that the two-magnon model describes not truedamping, but only the width of the FMR intensity dis-tribution that remains after inhomogeneities are effec-tively smoothed by dipolar and exchange interactions.[1] V. L. Safonov, J. Appl. Phys. 91, 8653 (2002).[2] V. L. Safonov and H. N. Bertram, cond-mat/0207721[J. Appl. Phys. (to be published)].[3] N. Smith, J. Appl. Phys. 92, 3877 (2002).[4] V. Kambersky´, Can. J. Phys. 48, 2906 (1970).[5] V. Korenman and R. E. Prange, Phys. Rev. B 6, 2769(1972).[6] L. Berger, J. Phys. Chem. Solids 38, 1321 (1977).[7] V. L. Safonov and H. N. Bertram, Phys. Rev. B 61, 14 893(2000).[8] S. E. Russek, P. Kabos, R. D. McMichael, C. G. Lee, W. E.Bailey, R. Ewasko, and S. C. Sanders, J. Appl. Phys. 91,8659 (2002).[9] N. Smith and P. Arnett, Appl. Phys. Lett. 78, 1448 (2001).[10] H. N. Bertram, V. L. Safonov, and Z. Jin, IEEE Trans.Magn. 38, 2514 (2002).[11] T. J. Silva, C. S. Lee, T. M. Crawford, and C.T. Rogers,J. Appl. Phys. 85, 7849 (1999).[12] C. Chappert, K. L. Dang, P. Beauvillain, H. Hurdequint,and D. Renard, Phys. Rev. B 34, 3192 (1986).[13] W. Platow, A. N. Anisimov, G. L. Dunifer, M. Farle, andK. Baberschke, Phys. Rev. B 58, 5611 (1998).[14] S. Mizukami, Y. Ando, and T. Miyazaki, Jpn. J. Appl.Phys. 40, 580 (2001).[15] Z. Celinski and B. Heinrich, J. Appl. Phys. 70, 5935(1991).[16] R. Karim, S. D. Ball, J. R. Truesdon, and C. E. Patton,J. Appl. Phys. 73, 4512 (1993).[17] D. J. Twisselmann and R. D. McMichael, J. Appl. Phys.93, 6903 (2003).[18] R. D. McMichael, M. D. Stiles, P. J. Chen, and W. F.Egelhoff, Jr., J. Appl. Phys. 83, 7037 (1998).[19] C.W. Haas and H. B. Callen, in Magnetism (AcademicPress, New York, 1963), Vol. 1, Chap. 10, pp. 480–497.[20] M. Sparks, R. Loudon, and C. Kittel, Phys. Rev. 122, 791(1961).[21] M. Sparks, Ferromagnetic Relaxation Theory, AdvancedPhysics Monograph Series (McGraw-Hill, New York,1964).[22] M. J. Hurben and C. E. Patton, J. Appl. Phys. 83, 4344(1998).[23] A. M. Clogston, H. Suhl, L. R. Walker, and P.W.Anderson, J. Phys. Chem. Solids 1, 129 (1956).[24] R. Arias and D. L. Mills, Phys. Rev. B 60, 7395 (1999).[25] R. D. McMichael, D. J. Twisselmann, J. E. Bonevich, A. P.Chen, W. F. Egelhoff, and S. E. Russek, J. Appl. Phys. 91,8647 (2002).[26] C. E. Patton, Phys. Rev. 179, 352 (1969).[27] B. A. Kalinikos and A. N. Slavin, J. Phys. C 19, 7013(1986).[28] K. J. Harte, J. Appl. Phys. 39, 1503 (1968).[29] Terms including aykayk0 and akak0 , which mostly affectthe ellipticity of the precession, have been dropped.[30] E. Schlo¨mann, Phys. Rev. 182, 632 (1969).[31] Q. H. F. Vrehen, J. Appl. Phys. 40, 1849 (1969).[32] Q. H. F. Vrehen, A. B. van Groenou, and J. G. M. de Lau,Solid State Commun.. 7, 117 (1969).[33] A. Kunz and R. D. McMichael, IEEE Trans. Magn. 38,2400 (2002).[34] D. L. Huber and W.Y. Ching, Phys. Rev. B 39, 4453(1989).[35] A. F. Ioffe and A. R. Regel, in Progress in Semicon-ductors, edited by A. F. Gibson (Wiley, New York,1960), Vol. 4, p. 237. 0 10 20 30 40 50 60 70 80 10  100  1000 0 5 10 15 20 25 30 35Linewidth (mT)µ 0Hprms , µ 0Hp,c (mT)Average grain size (nm)µ0Hprms = 35 mTµ0Hprms = 10 mTµ0Hprms = 3.5 mT2-magnonlocal res.Hp,cFIG. 4. Peak-to-peak (filled symbols) and standard devia-tion (open symbols) linewidths as a function of grain size.Results are shown for three values of rms perturbation field,Hrmsp . The characteristic value of Hp;c for the transition fromcollective to local behavior is shown. The left and right axesare related by Eq. (2).P H Y S I C A L R E V I E W L E T T E R S week ending6 JUNE 2003VOLUME 90, NUMBER 22227601-4 227601-4

Localized Ferromagnetic Resonance in Inhomogeneous Thin Films

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Localized Ferromagnetic Resonance in Inhomogeneous Thin Films

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