This article describes calculations of ferromagnetic resonance damping rates due to coupling between the magnetization and lattice vibrations through inhomogeneities. The mechanisms we have explored include generation of shear phonons through inhomogeneous anisotropy and generation of both longitudinal and shear phonons through inhomogeneous magnetostriction. In both cases, inhomogeneities couple the uniform precession to finite wave vector phonons. For both coupling mechanisms, the predicted damping rate is on the order of 106 s-21 in transition metals. The damping rate by these mechanisms is inversely proportional to the fifth power of the shear phonon velocity, and may play a significant role in mechanically softer materials such as magnet/polymer nanocomposites

Kunz, Andrew

McMichael, R. D.

English

epublications@Marquette

Marquette Universitye-Publications@MarquettePhysics Faculty Research and Publications Physics, Department of5-1-2002Calculation of Damping Rates in ThinInhomogeneous Ferromagnetic Films Due toCoupling to Lattice VibrationsR. D. McMichaelNational Institute of Standards & TechnologyAndrew KunzMarquette University, andrew.kunz@marquette.eduPublished version. Journal of Applied Physics, Vol. 91, No. 10 (May 2002). DOI. © 2002 AmericanInstitute of Physics. Used with permission.Calculation of damping rates in thin inhomogeneous ferromagnetic filmsdue to coupling to lattice vibrationsR. D. McMichaela) and Andrew KunzNational Institute of Standards and Technology, Gaithersburg, Maryland 20899This article describes calculations of ferromagnetic resonance damping rates due to couplingbetween the magnetization and lattice vibrations through inhomogeneities. The mechanisms wehave explored include generation of shear phonons through inhomogeneous anisotropy andgeneration of both longitudinal and shear phonons through inhomogeneous magnetostriction. Inboth cases, inhomogeneities couple the uniform precession to finite wave vector phonons. For bothcoupling mechanisms, the predicted damping rate is on the order of 106 s21 in transition metals. Thedamping rate by these mechanisms is inversely proportional to the fifth power of the shear phononvelocity, and may play a significant role in mechanically softer materials such as magnet/polymernanocomposites. @DOI: 10.1063/1.1450831#I. INTRODUCTIONIt is well known that inhomogeneities can cause uniformprecession modes to scatter into finite-wave vector spinwaveexcitations in a process described by the two-magnonmodel.1–7 The two-magnon process produces an effectiveline width in ferromagnetic resonance experiments, but itmay be regarded as something other than true damping be-cause the energy stays within the magnetization without be-ing transferred to the lattice. The analogous process was il-lustrated in micromagnetic models of switching withoutdamping.8The inhomogeneities that allow the uniform, k50, pre-cession mode to drive nonuniform precession modes withwave vector kÞ0 will also allow uniform precession tocouple to kÞ0 sound waves in the film and substrate. Thetransfer of energy to sound waves would be a true dampingeffect, but its measurement by ferromagnetic resonancewould be masked by the presence of two-magnon broaden-ing. The purpose of this article is to estimate theoreticallywhether the inhomogeneities that enable two-magnon pro-cesses within the magnetic subsystem also enable a signifi-cant amount of true damping to sound waves in the lattice.We consider a Hamiltonian that includes the uniformmode of precession, a manifold of phonons and interactionterms of the formH\5v0a†a1(k,svs~k!ck,s† ck,s1(k,s~Fk,sack,s†1Fk,s* a†ck,s!, ~1!where a ,a† and ck,s ,ck,s† are the lowering and raising opera-tors for the uniform precession mode and for phonons, re-spectively. Phonon modes are identified by wave vector k,and polarization index s .The decay rate for the uniform precession, G, is given bythe golden rule of time dependent perturbation theory,G52p\ (k,s uFk,su2 d@\v02\vs~k!# . ~2!To proceed with this formulation, we must determine theinteraction coefficients Fk,s in terms of materials propertiesand the properties of the excitations.For small deviations of the magnetization from the equi-librium direction, we write the magnetization as M5M srˆ1muuˆ 1mffˆ . The equilibrium magnetization direction isalong the unit vector rˆ; uˆ and fˆ are chosen to lie along theprincipal axes of the uniform precession ~see Fig. 1!.We write the energy of the magnetization in a uniformfilm asE5m0V f~ 12 huumu21 12 hffmf2 ! , ~3!where huu and hff are normalized fields and V f is the vol-ume of the magnetic film. For magnetization aligned with anapplied field Ha in the plane of a film, huu5Ha /M s andhff5(Ha1M s)/M s .In terms of a and a†, mu and mf are given bymu ,051iA\v02m0V fhuu~a2a†!, ~4!a!Electronic mail: rmcmichael@nist.govFIG. 1. Coordinate system used for the calculations. The film and substratelie in the x – z plane. Unit vectors uˆ k and fˆ k are chosen as polarizationvectors for shear phonons.JOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 10 15 MAY 200286500021-8979/2002/91(10)/8650/3/$19.00mf ,05A \v02m0V fhff~a1a†!. ~5!The displacement of atoms in the film and substrate canbe written as9u~r!5(k,sA \2rVsvs~k!~cks1c2ks† !eˆs~k!eik"r, ~6!where r is the substrate density, Vs is the substrate volume,and vs(k) is the angular frequency of a phonon with wavevector k and polarization vector eˆs .In Sec. II, we calculate the damping rate with the anisot-ropy as the spin lattice coupling, and in Sec. III, we calculatethe direct-to-lattice damping rate due to magnetostriction.II. COUPLING VIA ANISOTROPYThe phonon-generating process can be described classi-cally. Imagine a small grain of anisotropic material in a uni-form host. The precessing magnetization and the lattice inthis grain will experience equal but opposite, time-dependenttorques due to the local anisotropy. The relaxation of thelattice in response to these torques leads to the dissipation ofmagnetic energy by this mechanism.For a material with an inhomogeneous anisotropy, thelocal energy density can be expanded in a Taylor series in muand mf . The first order terms give rise to magnetizationripple10 and the second order terms areEp5m0V fN (r @12 huu~r!mu~r!21huf~r!mu~r!mf~r!1 12 hff~r!mf~r!2# . ~7!Written in terms of Fourier coefficientsEp5m0V f(k@ f uu ,2kmu ,kmu ,01 f ff ,2kmf ,kmf ,01 f uf ,2k~mu ,kmf ,01mf ,kmu ,0!# , ~8!where only those terms involving the uniform precession arekept and the uniform and inhomogeneous parts of hi j(r) areseparatedhi j~r!5hi j1 (kÞ0f i j ,keik"r. ~9!In Eq. ~7!, and in what follows, mu and mf are smallmagnetization changes due to rotations of M relative to thelattice. The magnetization with respect to the direction of thelocal crystal axes becomes m5m01mp , where m0 is themagnetization due to uniform precession magnons only andmp is the effective magnetization due to local rotation of thelattice which is described by the rotation vector „3u:mp ,k51N (r M srˆ~„3u !e2ik"r5iA \2rVsvs~k!~cks1c2ks† !$M@k3 eˆs~k!#%. ~10!Assigning the polarization vectors eˆ1 and eˆ2 to uˆ k andfˆ k , respectively, the double cross product in Eq. ~10! can bewrittenM3k3uˆ k52M skcos~fk2f!uˆ , ~11!M3k3fˆ k52M sk@cos~uk!sin~f2fk!uˆ1sin~uk!fˆ # . ~12!The perturbation energy @Eq. ~8!# can now be expandedin terms of raising and lowering operators for magnons andphonons. Using Eqs. ~11! and ~12!, the u and f componentsof mp @Eq. ~10!# can be identified and substituted for mu ,kand mf ,k and Eqs. ~4! and ~5! can be substituted for mu ,0 andmf ,0 . For each polarization index s , the coefficients Fk,s ofterms containing ck†a can be identified by comparison withEq. ~1!. The result is(suFk,su25m0Vs~\kgm0M s2!24rV fvkv03$@ u f uu ,ku2hff1u f uf ,ku2huu#3@cos2~f2fkrm!1cos2 ukrmsin2~f2fkrm!#1@ u f ff ,ku2huu1u f uf ,ku2hff#sin2 ukrm%. ~13!In this expression we have assumed that the material hassymmetry such that ^ f i j(r) f kl(r1r8)&5^ f i j(r) f kl(r2r8)&where ^ f & is the spatial average of f and i , j , k , and l standfor u and f. With this assumption, f i j ,kf kl ,k* is real. Termsodd in f2fkrm have also been dropped because they willcancel in the sum over k.To proceed further, the properties of u f i j ,ku2 must be de-termined. As an example, we assumef i j~r!5H f i j~ri! u 0,y,t f0 u y>t f , ~14!where ri is a vector in the plane of the film, and we assumethat f i j(ri) is correlated over a distance D such that^ f i j~ri! f i j~ri1ri8!&5^ f i j2 ~ri!&e2uri8u/D. ~15!Under these assumptionsu f i j ,ku25V fVs2 S sin~kyt f /2!kyt f /2 D2 2pD2t f^ f i j2 ~ri!&@11~k iD !2#3/2, ~16!where k i is the in-plane component of k.To calculate G, the sum over k in Eq. ~2! is converted toan integral, (k→Vs /(2p)3*dk for proper evaluation of thedelta function. We assume a linear, isotropic dispersion rela-tion for the shear phonons, vk5cuku:G512p E dcos ukdfkdk m0M s2~gm0M s!2v024rc5 F sin~kyt f /2!kyt f /2 G23D2t f@11~k iD !2#3/2d~ uku2v0/c !$@^ f uu2 &hff1^ f uf2 &huu#3@cos2~f2fkrm!1cos2ukrmsin2~f2fkrm!#1@^ f ff2 &huu1^ f uf2 &hff#sin2ukrm%. ~17!8651J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 R. D. McMichael and A. KunzFor high frequencies, when the delta function in Eq. ~1! se-lects large values of k such that ukut f@1 and ukuD@1. Theterms on the second line of Eq. ~17! decrease with k , so thedamping rate will decrease like 1/v3 for large v.For small k such that ukut f!1 and ukuD!1, the terms onthe second line of Eq. ~17! are approximately D2t f , and thedamping rate can be approximated byG’m0M s2D2t f~gm0M s!2v28prc5 @huu~^ f ff2 &1^ f uf2 &!1hff~^ f uu2 &1^ f uf2 &!# . ~18!For low v, the damping rate is proportional to the square ofthe frequency. We expect maximum damping when the cor-relation length and/or film thickness are on the order of aphonon wavelength.Note that the damping rate depends sensitively on anumber extrinsic materials parameters including the correla-tion length D , and the thickness of the film. If we use order-of-magnitude magnetic parameters typical of Permalloyfilms, (M s583105 A/m, and D5t f550 nm! mechanicalproperties typical of silicon, (r52.33103 kg/m3, c533103 m/s! and v52p310 GHz, and if we allow for a verylarge perturbation field on the same order as M s , such that^ f i j2 &51, the damping rate due to generation of phonons isstill only on the order of 106 s21, much smaller than rates of109 s21 that are typically observed in ferromagnetic reso-nance experiments. Note that Eq. ~17! depends on the speedof sound for transverse waves as c25. For a magnet/polymercomposite with a lower speed of sound, the damping by thismechanism may be large enough to be dominant.III. COUPLING VIA MAGNETOSTRICTIONMagnetostrictive coupling between magnetization pre-cession modes and phonons have been observed as finestructure in the ferromagnetic resonance of the garnet due tostanding sound waves in the substrates.11–13 Abrahams andKittel14 calculated the damping rate in single crystals due toa process where magnon scattering is accompanied by emis-sion or absorption of a phonon. Suhl15 has calculated damp-ing by generation of shear sound waves in particles muchsmaller than the wavelength of sound and viscous behaviorof the lattice. In both of these works, the damping rate issmall compared to experimental values.We outline a calculation of the damping rate due to mag-netostrictive generation of phonons in polycrystalline filmsusing the the cubic magnetoelastic coupling energy given byEmagel5V fN (r @B1~exxa121eyya221ezza32!1B2~exya1a21eyza2a31ezxa3a1!# , ~19!where ei j is the strain tensor and a i is a direction cosine ofthe magnetization relative to the local crystallographic axes;a i5mi /M s . The strain is given byei j512 S ]ui]x j 1 ]u j]xi D . ~20!Because the directions of the crystallographic axes varyfrom grain to grain in the film, we define a unitary coordinatetransformation matrix Ur such that a i5( jUi j(r)M j /M s .The next step in the calculation would be to use Eqs. ~4!, ~5!,and ~6! to expand Eq. ~19! in terms of magnon and phononraising and lowering operators. Because a i is a linear com-bination of M s , mu , and mf , products a ia j will containcross terms proportional to mu /M s and mf /M s with coeffi-cients on the order of unity that depend on r. With thesesubstitutions, the coupling energy takes a form similar to Eq.~8! with terms Bg2k /(m0M s2) in place of f i j ,2k where g2k isa Fourier component of coefficients of the r-dependent coor-dinate transformation from local crystallographic axes to thelab coordinates.For transition metals with magnetostriction on the orderof 1025, B is typically of 106 J/m, the same order of mag-nitude as m0M s2. Based on this estimate, damping tophonons via magnetostriction and damping to phonons viamagnetocrystalline anisotropy are similar in magnitude. Per-haps the similar damping rates predicted for these twomechanisms should not be surprising since both magneto-striction and magnetocrystalline anisotropy have their originsin spin-orbit coupling.In conclusion, the results of these calculations indicatethat inhomogeneous coupling between the magnetization andlattice vibrations may be strong enough to give measurabledamping in transition metal thin films. For magnet/polymernanocomposites or other materials with low sound speeds,damping by phonon generation may play a much more sig-nificant role.ACKNOWLEDGMENTThe authors acknowledge the support of the NIST nano-technology program.1 A. M. Clogston, H. Suhl, L. R. Walker, and P. W. Anderson, J. Phys.Chem. Solids 1, 129 ~1956!.2 M. Sparks, R. Loudon, and C. Kittel, Phys. Rev. 122, 791 ~1961!.3 H. B. Callen, Fluctuation, Relaxation and Resonance in Magnetic Systems~Oliver and Boyd Ltd., Edinburgh, 1962!, Chap. 4, pp. 69–85.4 C. W. Hass and H. B. Callen, in Magnetism, edited by G. T. Rado and H.Suhl ~Academic, New York, 1963!, Chap. 10, pp. 480–497.5 M. J. Hurben and C. E. Patton, J. Appl. Phys. 83, 4344 ~1998!.6 R. D. McMichael, M. D. Stiles, P. J. Chen, and W. F. Egelhoff, Jr., J. Appl.Phys. 83, 7037 ~1998!.7 R. Arias and D. L. Mills, Phys. Rev. B 60, 7395 ~1999!.8 V. L. Safonov and H. N. Bertram, Phys. Rev. B 63, 094419 ~2001!.9 N. W. Ashcroft and N. D. Mermin, Solid State Physics ~Saunders College,West Washington Square, Philadelphia, PA 19105, 1976!, Apppendix L.10 K. J. Harte, J. Appl. Phys. 39, 1503 ~1968!.11 H. Do¨tch, Appl. Phys. 15, 167 ~1978!.12 R. J. Yeh, P. E. Wigen, and H. Do¨tch, Solid State Commun. 44, 1183~1982!.13 D. A. Romanov and E. G. Rudashevsky, IEEE Trans. Magn. 29, 3405~1993!.14 E. Abrahams and C. Kittel, Phys. Rev. 88, 1200 ~1952!.15 H. Suhl, IEEE Trans. Magn. 34, 1834 ~1998!.8652 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 R. D. McMichael and A. Kunz

Calculation of Damping Rates in Thin Inhomogeneous Ferromagnetic Films Due to Coupling to Lattice Vibrations

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Calculation of Damping Rates in Thin Inhomogeneous Ferromagnetic Films Due to Coupling to Lattice Vibrations

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