Microscopic mechanisms of magnetization reversal Vladimir L. Safonov Center for Magnetic Recording Research, University of California -San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0401 (February 2, 2008) 4 0 0 Our aim in this paper is to develop a self-consistent 2 Twoprincipalscenariosofmagnetization reversalarecon- picture that describes the entire reversal process. We n sidered. Inthefirstscenarioallspinsperformcoherentmotion considertwopossible scenarios: 1)the magnetizationre- a and an excess of magnetic energy directly goes toa nonmag- verses uniformly, involving nonlinear dynamic damping; J netic thermal bath. A general dynamic equation is derived 2)themagnetizationreversesnonuniformly,involvingthe 8 which includes a tensor damping term similar to the Bloch- excitation of nonlinear spin waves. We give an explicit 2 Bloembergen formbutthemagnetization magnituderemains criterion for this nonuniform process for an untra-thin constant for any deviation from equilibrium. In the second ] reversal scenario, the absolute value of the averaged sample magnetic film. r Scenario #1: the total film magnetization M is con- e magnetization is decreased byarapid excitation of nonlinear | | h spin-wave resonances by uniform magnetization precession. stant during reversal. All spins perform a coherent mo- t tion (the role of non-uniform spin motions is neglected). o We have developed an analytic k-space micromagnetic ap- . proach that describes this entire reversal process in an ultra- Anexcessofmagnetic energygoesdirectly toanonmag- at thinsoftferromagneticfilmforupto90o deviationfromequi- netic thermal bath. We derive a general magnetization m librium. Conditions for the occurrence of the two scenarios dynamicequationfromanonlinearoscillatormodel. The are discussed. nonlineardampingfollowsfromthevarietyofwell-known - d physical damping mechanisms [7], [8]. Here we extend n our previousresults for uniaxialsymmetry to the caseof o non-uniaxial symmetry. c Scenario #2: the total film magnetization M de- [ | | creases. Experimentally it was observed in Ref. [10]. I. INTRODUCTION 1 Large angle magnetization motion can excite spin-wave v instabilities, which increase substantially the magnetiza- 0 Studies of magnetization reversal in ultra-thin ferro- tion reversalrate [11]. We explicitly evaluate the second 9 magnetic films under an applied external magnetic field orderSuhlinstabilityandconstructaself-consistentthe- 5 1 areofgreatimportanceinmagneticrecordingphysics. A ory of magnetization switching for up to 90o from equi- conventional theoretical tool to study magnetization re- 0 librium. versalis based on the phenomenologicalLandau-Lifshitz 4 The problem of nonlinear spin-wave excitation during 0 equation [1] or, its equivalent modification with the reversal has been explored by numerical simulations in / Gilbert form of relaxation [2]. These equations conserve t nanograins [12], [13], and thin films [14], [15]. All these a the absolute value of magnetization (M = const) in a simulationshavebeenperformedusingconventionallocal m single domain region. Both equation|s w|ere introduced micromagnetic modeling, which includes: a) the anal- - (a) for small magnetizationmotions and (b) for the case ysis of intra- and inter-cell interactions, b) analysis of d ofuniaxialmagneticsymmetry. Theenergylossesarede- phenomenological dynamic equations, and c) computer n fined by an isotropic phenomenological damping fitting o simulations. There are two principal problems in this parameter α (“damping constant”). c technique: 1) the physical problem of the introduction : Recently a theoretical approach[3], [4], [5], [6], [7] has of local phenomenological damping (and corresponding v been developed to correctthe limitations of the Landau- i magnetic noise) and 2) the computing problem in the X Lifshitz-Gilbert (LLG) theory. The main idea was to case of a large number of cells. r represent the magnetization dynamics as the motion of Bothproblemsoflocalmicromagneticmodelingcanbe a a damped nonlinear oscillator with the random force of avoided by developing the k-space micromagnetic mod- thermal fluctuations. The oscillator model is a conve- eling as we do in this work. Our theory includes: a) an nienttooltoestablisha“bridge”betweenthemicroscopic analysisofspin-wavespectraandinteractionsinanultra- physics,wheretherotationaloscillatorcomplexvariables thin film, b) the calculation of the effective scattering naturally describe spin excitations and the macroscopic processes(mostoftheaccumulatedenergyistobetrans- magnetization dynamics. It has been rigorously shown formed to nonlinear spin waves), c) the analysis of self- by including specific coupling of a magnetic system to consistent dynamic equations with microscopic damping a variety of loss mechanisms [8], [9] that for small oscil- (andnoise,ifnecessary). Notethatasimilartechniqueto lations near equilibrium the macroscopic damping term study nonlinear spin-wave dynamics has been developed reflects the anisotropy of the system. 1 ∗ ∗ ∗ inthetheoryofparametricmagnonexcitation(mainlyin a=uc+vc , a =uc +vc, (4) the bulk, see, e.g., [16], [17], [18], [19]). We have already +ω ω 0 0 considered k-space modeling in application to magnetic u= A , v = B A− . 2ω − 2ω noiseinathinfilm[20]. RecentlyDobinandVictora[21] r 0 |B|r 0 estimated the increment of the second order Suhl insta- The energy in terms of the normal mode coordinates c bility and correspondingeffective magnetizationreversal ∗ and c is simply: time for up to 25o deviationfromequilibrium in the film plane. Here we give an explicit analytic formulation to M V s ∗ describemagnetizationreversal(switching)forupto90o / =ω0c c, (5) E γ deviationfromequilibriuminultra-thinfilms intermsof (cid:18) (cid:19) spin-wave pair excitations. where ω = √ 2 2 = γ H H is the ferromag- 0 A −B x0 y0 netic resonance frequency. The dynamic equations fopr c and c∗ are independent II. SCENARIO #1: |M|=CONST and can be written as: dc dc∗ In this section we consider the magnetization rever- +ηc= iω c, +ηc∗ =iω c∗. (6) 0 0 sal without spin-wave excitations. The approach is to dt − dt transformthe magnetizationdynamics without damping Here η is the linear relaxation rate, which can be found to normal mode coordinates. Then we introduce nonlin- microscopically [8]. In the case of large magnetization ear damping, which has a connection with microscopic motionwecanwrite acorrespondingnonlinearoscillator physics and return back to magnetization coordinates. equation in the form: The analysis parallels the approach for low-level excita- tions. dc ∗ ∗ Let us consider small-amplitude magnetization mo- +η(N)c=G(c,c ), N c c. (7) dt ≡ tions of a single-domain ferromagnetic particle in the vicinity of equilibrium state M z , where z is the unit Here G(c,c∗) corresponds to the gyromagnetic term 0 0 || vector in the equilibrium direction. The magnetization γm H . The nonlinear relaxation rate η(N) can eff − × rotation around effective field in this case, in general, is be estimated from the known relaxation process for the b b elliptical and the magnetic energy can be represented uniform precession [7]. We assume that m = 1 and E | | as a quadratic form: therefore no spin waves are excited. Using back transformations (4) and (2), we can de- / MsV = γHx0m2 + γHy0m2 . (1) rive an equation (corresponding to (7)) in terms of m- E γ 2 x0 2 y0 components: (cid:18) (cid:19) Here m M/M , x and y are the unit orthogonal dm ↔ vectors in≡the planse pe0rpendic0ular to the equilibrium di- dt =−γm×Heff− Γ ·(m−z0), (8) rection,Ms is the satburationbmagnetizationandV is the mz0 0 0 npeasrst”iclfieevldosl,umweh.icHhxin0calunddeHbyo0tharme ipcorsoistcivoepiKciatntedl “shstaipffe- ↔Γ =2η(N) 1+0mz0 1+mmz0z0b 0 , 0 0 1 anisotropies and the external magnetic field. The pa-   rameter M V/γ ~S, where ~ is Planck’s constant and   s ≡ where S is the total spin of the film. From the Holstein-Primakoff transformation [22] we m2 m2 have: N = ωA(1−mz0)+ ωB 1x0+−m y0. (9) 0 0 z0 m+ =a 1+m , m =1 a∗a, (2) z0 z0 − Note that the Eq.(8) conserves the magnitude of m. − ∗ ± m =ap 1+mz0, m =mx0 ±imy0, For small deviations from equilibrium, when mz0 ≃ 1, this equations exactly correspondto Bloch-Bloembergen where a∗ and a dpescribe spin excitations. equations [24] with η(0) = 1/T and 1/T = 2/T . From 2 1 2 The magnetic energy (1) can be written in the Eq.(8)we see that one can expect the anomalously large quadratic form: damping in the case of 180o reversal when 1+m 0 z0 → (see,also[7],[25]). Simplenumericalanalysisshowsthat M V / s = a∗a+ B(aa+a∗a∗), (3) magnetization dynamics in the framework of Eq.(8) for E γ A 2 angles about 70o (and smaller) can be approximated by (cid:18) (cid:19) LLG dynamics. This can explain why the switching ex- where =γ(H +H )/2and =γ(H H )/2. The A x0 y0 B x0− y0 periment [10] with M = const was well fitted by LLG non-diagonal terms in (3) are eliminated by the linear | | equation. canonical transformation (e.g., [23]): 2 III. SCENARIO #2: |M|6=CONST where ∆() is the Kroneckerdelta function: ∆(q)=1, if · q=0 and ∆(q)=0 otherwise. Weshallconsideranultra-thinferromagneticfilm(τ Thequadraticuniaxialanisotropyenergy(z isaneasy L L ) with the magnetization: M(r)=M m(r), r=× axis, see, Fig. 1) in the k-space is: y z s × (y,z). Thevariationofthe unitvectormwithinthefilm thickness (−τ/2 ≤ x ≤ τ/2) will be neglected. Locally Eanis =−VK1 (−my0,ksinθ0+mz0,kcosθ0) k one has: m(r) = m2 +m2+m2 =1. X Inorde|rtoin|troqducexcollecytivemzagnetizationmotions, ×(−my0,−ksinθ0+mz0,−kcosθ0). (15) we assume that the film is periodic along both y and The Zeeman energy in the external magnetic field z directions with periods Ly and Lz, respectively. The H0 =(0, H0sinθH, H0cosθH) is: Fourier series representation can be written as = VM H [m sin(θ θ )+m cos(θ θ )]. m(r)= mkexp(ik r), (10) EZ − s 0 y0,0 H − 0 z0,0 H − 0 · (16) k X Ly Lz The demagnetization energy (see, [26]) is defined by : 1 mk = dy dz m(r)exp( ik r), L L − · y z Z0 Z0 Edmag =2πMs2V {G(kτ)mx0,kmx0,−k (17) k k = 2πn /L , k = 2πn /L ( < n ,n < ) are X thyeThweaveequvyeilcibtyorriucmzomisposnuepnpztosseizndtt−hoe∞bpelaaneu.ynifozrmly∞mag- +[1−G(kτ)][ kky0 2my0,kmy0,−k (cid:18) (cid:19) netized state, in which the magnetization is oriented in k 2 k k the (y,z) plane along an equilibrium axis z0. The trans- + kz0 mz0,kmz0,−k+2 yk02z0my0,kmz0,−k]}, formation from the (x,y,z) coordinates to equilibrium (cid:18) (cid:19) coordinates (Fig.1) (x0,y0,z0) is defined by where G(x)=[1 exp( x)]/x. − − y cosθ sinθ y = 0 0 0 (11) z sinθ cosθ z 0 0 0 B. Spin waves (cid:18) (cid:19) (cid:18)− (cid:19)(cid:18) (cid:19) Here θ determines a rotation in the film plane, x = x . 0 0 We shallutilize aclassicalformofthe spinrepresenta- Analogoustransformationshouldbeusedfor(m ,m ) y z → tion in terms of Bose operators introduced in Refs. [27], (m ,m ) and wave vector components (k ,k ) y0 z0 y z → [28]andconvenientfor 2Dsystems. For the unit magne- (k ,k ). Note that boththe absolute value ofthe wave y0 z0 tizationvectormthisrepresentationintermsofcomplex vector k = k and the scalar product k r are invariant | | · variables a and a∗ can be written as: in respect to choice of system of coordinates. a a∗ m =i − , (18a) x0 √2 A. Magnetic energy a+a∗ m = 1 m2 sin , (18b) eneTrhgey,menagerngeyticofeannerisgoytroofptyh,eZfielemmacnonetnaeinrgsythaendexdcehmanagge- y0 q − x0 (cid:18)a√+2a∗(cid:19) m = 1 m2 cos . (18c) netization energy: z0 − x0 √2 q (cid:18) (cid:19) = exch+ anis+ Z + dmag. (12) An expansionof (18a)-(18c)up to the fourth order gives E E E E E accuracy 6%forabout90o deviationfromequilibrium: The exchange energy A( m)2 can be represented ∼ as − ▽· a a∗ m =i − , (19a) x0 √2 Eexch =VA k2(my0,kmy0,−k+mz0,kmz0,−k), (13) a+a∗ a3+(a∗)3 3a∗a2 3(a∗)2a Xk my0 ≃ √2 + −6√2 − , (19b) where A is the exchange constant and V =τLyLz is the a4+(a∗)4 2a∗a3 2(a∗)3a film volume. To obtain (13) we have used the following m 1 a∗a − − . (19c) z0 ≃ − − 12 formula: The following Fourier representation for a(r) (and its Ly Lz 1 complex conjugate) will be used: dy dz exp[i(k+k )r]=∆(k+k ), (14) 1 1 L L · y z Z Z 0 0 3 a(r)= akexp(ikrj), (20) The spin-wave frequency, ωk, in an explicit form is k X 1 Ly Lz ωk =γ H0cos(θH −θ0)−HKsin2θ0 ak = dy dz a(r)exp( ik r). h k 2 1/2 LyLz Z0 Z0 − · +4πMs[1−G(kτ)] ky0 +2αEk2 (cid:18) (cid:19) i In general, the magnetic energy can be expressed as [H cos(θ θ )+4πM G(kτ)]1/2. (30) 0 H 0 s × − = (0)+ (1)+ (2)+ (3)+ (4)+..., (21) E E E E E E wherethe superscriptdenotes anorderintermsofa and C. Spin-wave interactions a∗. The zeroth order energy term is equal to Theinteractionenergycanberepresentedintheform: E(0) =−VK1cos2θ0−VMsH0cos(θH −θ0). (22) Eint = Ψ1(1,2,3)c1c2c3 (31) (M V/γ) 3 Theequilibriumuniformlymagnetizedstateisdefinedby s 1X,2,3h the condition: ∂E(0)/∂θ0 =0, which corresponds to +Ψ2(1,2; 3)c1c2c∗−3+c.c. ∆(1+2+3) − H sin2θ =2H sin(θ θ ). (23) 1 i K 0 0 H − 0 + Φ(1,2,3,4)c∗1c∗2c3c4∆(1+2 3 4). 2 − − Here HK =2K1/Ms is the anisotropy field. In order to 1,X2,3,4 have a stable stationary state, we need ∂2 (0)/∂θ2 > 0. The first order energy term (1) = 0 at thEe equili0brium Here for simplicity we use the following notations: k1 ≡ E 1, k2 2, etc., for example, k1+k2+k3 1+2+3. and this condition coincides with Eq.(23). ≡ ≡ The three-wave interaction amplitudes are The quadratic term has the form: 1 (2)/ MsV = ka∗kak+ Bk(aka−k+a∗ka∗−k) , Ψ1(1,2,3)= 2{ψ(1)(u1+v1)(u2v3+u3v2) E (cid:18) γ (cid:19) k (cid:20)A 2 (cid:21) +ψ(2)(v1u3+v3u1)(u2+v2) X (24) +ψ(3)(v1u2+v2u1)(u3+v3) . (32) } and where 1 k =γαEk2 γHK sin2θ0+γH0cos(θH θ0) Ψ2(1,2;3)= 2{ψ(1)(u1+v1)(u2u3+v2v3) A − 2 − +ψ(2)(u2+v2)(u1u3+v1v3) 2 k +2πγM [1 G(kτ)] y0 +G(kτ) , (25) +ψ(3)(u3+v3)(v1u2+v2u1) . (33) s" − (cid:18) k (cid:19) # } Here γ H k k Bk =γαEk2− γH2K sin2θ0 ψ(k)=−√2(cid:18) 2K sin2θ0+4πMs[1−G(kτ)] yk02z0(cid:19). k 2 (34) +2πγM [1 G(kτ)] y0 G(kτ) , (26) s " − (cid:18) k (cid:19) − # The four-waveinteraction amplitude can be expressed as: and α A/M . Using the following linear canonical E s ≡ transformation (e.g., [23]): Φ(1,2,3,4)=Φ (1,2,3,4)+Φ (1,2,3,4)+Φ (1,2,3,4), 0 s Q ∗ ∗ ∗ (35) ak =ukck+vkc−k, ak =ukck+vkc−k, (27) where k+ωk k k ωk Φ =[γH cos(θ θ )+γH cos2θ ] uk = A , vk = B A − , (28) 0 0 H − 0 K 0 2ωk − k 2ωk r |B |r u1u2v3v4+v1v2u3u4 × we obtain n1 [(u1u2+v1v2)(u3v4+v3u4) M V −2 (2) = s ωkc∗kck, ωk = 2k k2. (29) E γ Xk qA −B +(u1v2+v1u2)(u3u4+v3v4)] , (36) o 4 1 1 1 Φs = [ (1)+ (2)+ (3)+ (4)] + 4 P P P P × ω1+ω3−1 ω3 ω4+ω2−4 ω2 (cid:18) − − (cid:19) [(u1u2 v1v2)(v3v4 u3u4) Ψ (1,4 1,4)Ψ (3,2 3,2) × − − 2 2 − − − (u1v2+v1u2)(v3v4+u3u4) 1 1 − + −(u1u2+v1v2)(v3u4+u3v4) ×(cid:18)ω1+ω4−1−ω4 ω3+ω2−3−ω2(cid:19) 1 −2(v1u2+u1v2)(v3u4−u3v4)], (37) −Ψ2(2,3−2,3)Ψ2(4,1−4,1) 1 1 + × ω2+ω3−2 ω3 ω4+ω1−4 ω1 ΦQ =[ (1+2)+ (3+4)](u1u2u3u4+v1v2v3v4) Ψ (cid:18)(2,4 2,4)Ψ−(3,1 3,1) − (cid:19) Q Q 2 2 +[ (1+4)+ (2+3)](u1v2u3v4+v1u2v3u4) − − 1 − 1 Q Q + . (44) +[Q(1+3)+Q(2+4)](u1v2v3u4+v1u2u3v4), (38) ×(cid:18)ω2+ω4−2−ω4 ω3+ω1−3−ω1(cid:19) The energy (41) together with the Hamilton’s equa- P(k) α k2 HK sin2θ +2πM [1 G(kτ)] ky0 2, tions of motion for complex spin-wave variables E 0 s γ ≡ − 2 − k (cid:18) (cid:19) d γ ∂ (39) dt +ηk ck =−i M V ∂cE∗, (45) (cid:18) (cid:19) (cid:18) s (cid:19) k and supplemented by the (microscopic) relaxation rate ηk, representthebasisformagnetizationdynamicsmodeling Q(γk) ≡αEk2− H2K cos2θ0+2πMs[1−G(kτ)](cid:18)kkz0(cid:19)2. innettihzeatkio-snpadceev.iaCtiaolncuslaatki(ntg),cka(∗kt()t,)cw∗k(itt)h,wtheechanelpfinodfmbaacgk- transformation (27) and finally with Eqs. (19a)-(19c), (40) the averaged Ly Lz 1 D. Effective four-wave interactions m(r,t) = dy dz m(r,t)=m0(t), (46) h i L L y z Z Z 0 0 Unitary transformation(see, [29])makes it possible to whichgivesameasureofnon-uniformmagnetizationmo- eliminate forbidden three-magnon interaction terms in tions in the system. In general, m <1 andonly in the Eq.(31) and obtain effective interaction terms. As a re- | 0| case of coherent spin motion m =1. sult we have the following spin-wave energy | 0| M V s ∗ / = ωkckck (41) E. Example E γ (cid:18) (cid:19) k X +1 Φ(1,2,3,4)c∗1c∗2c3c4∆(1+2 3 4), Forsimplicityweshallconsidertheuniformmagnetiza- 2 − − tion precession interacting with one spin-wave pair with 1,2,3,4 X k = (0,0,k) along z . In this case the energy (Hamilto- e 0 where nian) of the system can be reduced to the form: b Φ(1,2,3,4)=Φ(1,2,3,4)+Φ (1,2,3,4)+Φ (1,2,3,4), = 0+ int+ p, (47) 1 2 H H H H (42) where e M V s ∗ ∗ ∗ Φ (1,2,3,4)= Ψ (1,2, 1 2)Ψ (3,4, 3 4) H0/ γ =ω0c0c0+ωk(ckck+c−kc−k) (48) 1 − 1 − − 1 − − (cid:18) (cid:19) 1 1 + , (43) describes the uniform precession and spin-wave pair, × ω1+ω2+ω−1−2 ω3+ω4+ω−3−4 (cid:18) (cid:19) Φ int 00 ∗ ∗ ∗ ∗ ∗ and (MHV/γ) = 2 c0c0c0c0+2Φ0kc0c0(ckck+c−kc−k) (49) s Φ2(1,2,3,4)=Ψ2(1,2,1+2)Ψ2(3,4,3+4) +Φkk(c∗kckc∗kck+c∗−kc−kc∗−kc−k)+2Φk,−kc∗kckc∗−kc−k 2 1 1 + describes nonlinear interactions in the system, Φ Ψ×2(cid:18)(1ω,31+ω12,3−)Ψω21(+42,2 ω43,+2)ω4−ω3+4(cid:19) Φ(0,0,0,0), Φ0k ≡ Φ(k,0,k,0), Φkk ≡ Φ(k,k,k00,k≡), − − − Φk,−k Φ(k, k,k, k), and ≡ − − e e e e 5 M V s ∗ ∗ ∗ ∗ IV. DISCUSSION Hp/ γ =Φp c0c0ckc−k+c0c0ckc−k (50) (cid:18) (cid:19) describes the spin-wave(cid:0)pair excitation by the(cid:1)uniform We have considered two principally different scenarios of magnetization reversal. In the first one the coherent precession, Φ Φ(k, k,0,0). p ≡ − rotationofallspinsistheprincipalmotioninthesystem. The uniform precession and spin-wave pair dynamics ThemagnetizationdynamicsisdefinedbyEq. (8),which are defined by e isreducedtoBloch-Bloembergenforminthecaseofsmall d ∗ magnetization motions. dt+η0 c0 =−iω0c0−2iΦpckc−kc0, (51a) In the second scenario we have considered an ultra- (cid:18) (cid:19) d thing ferromagnetic film with large dimensions in the +ηk ck = iωekck iΦp(c0)2c∗−k, (51b) plane. In this case the role of plane boundaries is negli- dt − − (cid:18) (cid:19) gible and the most convenient technique to describe the where e non-uniform spin motions is their spin-wave representa- ω0 =ω0+Φ00 c0 2+2Φ0k(ck 2+ c−k 2), (52) tioninthek-space. Takingintoaccountlinearspin-wave | | | | | | modes and their scattering, we have constructed a non- ωk =ωk+2Φ0k c0 2+Φkk ck 2+2Φk,−k c−k 2 (53) linear self-consistent theory of magnetization reversal as e e | | | | | | a decay of uniform magnetization precession and non- are nonlinear frequencies and η0, ηk are the relaxation raetes. e linearexcitationofspin-wavepairs. Thistheoryincludes aneffectiveenergy(41)anddynamicequations(45). The Fromtheenergysymmetrywehaveck =c−k (see,also mostimportantspin-wavemodesaredefinedbythereso- [19]). Thus, Eqs.(51a)-(53) represent a self-consistent nonlinear theory of magnetization reversalwith just two nancecondition2ω0 =ωk+ω−k(similartoSuhl’ssecond orderinstability). The excitationofallspinwavesoutof independent complex variables. Simple analysis for both ck(t) and c∗−k(t) exp(κt), where κ is an increment of this resonanceis semalleand teherefore they can be consid- ∝ ered as a part of a thermal bath. In the simple example instability, gives: we have demonstrated that strongly excited spin-wave κ=−ηk+ |Φpc20(0)|2+(ωk−ω0)2. (54) isnwsittacbhiilnitgyracaten. increase substantially the magnetization q This formula is similar to that obtained in Ref. [30] for Note that both scenarios are consistent which each parametric instabilities. The diffeerenceeis that the res- other: inthecaseof m 1Eqs.(45)canbereducedto 0 | |→ onance condition includes nonlinear frequencies: 2ω = the case of a nonlinear oscillator equation considered in 0 ωk+ω−k (all spin waves out of this equality can be in- scenario 1. We also emphasize that the relaxation rates cluded into a thermal bath) and ω is the uniform pre- of uniform precession and spin waves in both scenarios 0 e cession frequency (not a driving field frequency). The can be estimated from microscopic physics. e e onset of instability is defined by Φepc20(0) ≥ηk. Taking m (0) = 0, from Eqs.(18b), (18c) and (27) we can find x0 (cid:12) (cid:12) c0(0) and rewrite this criterion as(cid:12) (cid:12) ACKNOWLEDGMENTS θ (u0+v0) 2ηk/Φp , (55) The author wish to thank H. N. Bertram for many ≥ | | where θ tan−1(m /m )qis the initial deviation angle stimulatingandcriticaldiscussions. Ialsogreatlyappre- ≡ y0 z0 ciate helpful discussions with H. Suhl, C. E. Patton, T. from the equilibrium direction z . Here the parameters 0 J. Silva, P. Kabos, and J. P. Nibarger. This work was u ,v andΦ canbe directlycalculatedusing Eqs. (27), 0 0 p supportedinpartby the NationalInstitute ofStandards (50). The damping ηk can be estimated microscopically and Technology’s Nanotechnology Initiative and partly [7], [8]. supported by matching funds from the Center for Mag- Fig.2 shows two different evolutions in magnetic sys- netic Recording Research at the University of California tem calculated by Eqs.(51a), (51b). Spin-wave pair ex- - San Diego and CMRR incorporated sponsor accounts. citation in Fig.2a is not sufficiently strong during the switchingprocesstoaffecttheuniformmagnetizationdy- namics. The averaged m(r) =m , which gives a mea- 0 h i sure of non-uniform magnetization motions in the sys- tem is relatively small (1 m .0.05). Fig.2b demon- 0 −| | stratesastrongexcitationofspin-waveinstabilitybyuni- form precession and substantial increase of the magneti- [1] L.LandauandE.Lifshitz,Phys.Z.Sowjet.8,153(1935); zation reversalrate. In this case m0 reaches 0.6. For in LandauL. D.Collected Papers. editedby D.terHaar | | ≃ stronger coupling (Φp /ηk > 21) we observed beating (Gordon and Breach, New York,1967) p.101. | | between the uniform precession and the spin-wave pair. [2] T.L.Gilbert,Phys.Rev.100,1243(1955);T.L.Gilbert In this case it is necessary to include into consideration Ph.D.thesis,IllinoisInstituteofTechnology,Chicago,IL, the excitation of another resonance spin-wave pairs. 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