PHYSICAL REVIEW B VOLUME 44, NUMBER 17 1 NOVEMBER 1991-I

Magnetoelastic instabilities in the ferrimagnetic resonance of magnetic garnet films

M. Ye and H. DotschUniversity of Osnabriick, P.O. Box 4469, D 450-0 Osnabriick, Federal Republic of Germany

(Received 21 March 1991)

The nonlinear magnetoelastic interaction in [111]-oriented films of yttrium iron garnet is investigatedtheoretically and experimentally. A system of nonlinear equations of motion, including boundary condi-tions, is introduced. An appropriate numerical method to solve such a boundary-value problem isdeveloped. It is shown that the ferrimagnetic resonance (FMR) couples to elastic resonances. This non-linear coupling leads to instability oscillations of the FMR or even to chaos, if the excitation levelexceeds a critical threshold. The numerical calculations agree very well with experimental results. Theinfluences of the material parameters on the nonlinear dynamics of the FMR have been investigated.

I. INTRODUCTION

The ferrimagnetic resonance (FMR) is a well-established tool to study the properties of magnetic gar-net films grown by liquid-phase epitaxy on paramagneticsubstrates. At high excitation levels the resonance be-comes unstable, resulting in a variety of phenomena:auto-oscillations in the frequency range between 10 kHzand several MHz as well as difFerent routes to chaos areobserved. ' The basic nonlinear interaction leading tothis behavior is the coupling between the FMR and spinwaves according to the original theory of Suhl. Inspheres Benner considers the interaction between theFMR and magnetostatic modes, while for thin filmsWigen, McMichael, and Jayaprakash' take into accountthe coupling of the FMR to exchange modes. The FMRof garnets can be used as an excellent example to studybasic principles of nonlinear dynamics.

In thin magnetic films biased perpendicular to the filmplane, FMR is located at the bottom of the spin-wavemanifold. Thus there are no spin waves to interact withthe FMR. However, because of the magnetoelastic in-teraction, a hybridization between spin and elastic wavestakes place, resulting in magnetoelastic waves. Suchwaves exist also below the spin-wave band and are excitedby the precessing magnetization. If these waves propa-gate perpendicular to the film plane, they penetrate intothe paramagnetic substrate region as pure elastic waves.ReAections at the free surfaces of the film and substratecause standing waves, and thus an elastic resonance, ifthe total thickness of the crystal equals an integer multi-ple of half the wavelength. As elastic losses in garnets arevery low, these resonances have high quality factors ofmore than 8000 at 1 GHz.

A strong interaction between the FMR and elastic res-onances is observed experimentally in the frequencyrange between 1 and 4 GHz. If the frequencies of theseresonances coincide, all the energy of the FMR is coupledinto the elastic resonator. This phenomenon is treated ina linear model, and excellent agreement between theoryand experiment is obtained. "

At high excitation the magnetoelastic interaction leadsto a strong nonlinear coupling between the FMR and

where f0 is the excitation frequency of the FMR, f„ thefrequency of the nearest elastic resonance, with f„&f0,and n is an integer typically below 10. If the frequencydifference fo —f,&

is below 100 KHz, a transition tochaos takes place. "

In the present paper [ill]-oriented films of yttriumiron garget (YIG) are used to study theoretically and ex-perimentally the nonlinear magnetoelastic coupling be-tween the FMR and elastic resonances. For this purposethe equations of motion are solved numerically and theresults are compared with measurements.

II. THEORY

A. Equations of motion and boundary conditions

The geometry used to describe the ferrimagnetic reso-nance of [ill]-oriented Y1G films is illustrated in Fig.1(a). A static induction Bo and a circularly polarized rfdriving induction b(t)=bee' ' are applied perpendicularand parallel to the film plane, respectively. The precess-ing magnetization excites elastic waves because of themagnetoelastic interaction. At an elastic resonance

elasticstanding wave Bo

b

[]11j~~ Z

(a)

[112] SUbd

Y 0 film

strate

(b)

FICx. 1. (a) Coordinate system for describing ferrimagneticresonance and (b) elastic resonance over crystal thickness.

e)astic resonances, and so it causes also instabilities. Inthis case the observed frequency spectrum consists ofdiscrete lines at frequencies

9458 1991 The American Physical Society

MAGNETOELASTIC INSTABILITIES IN THE. . . 9459

standing elastic waves build up as shown in Fig. 1(b). Itis assumed that a garnet 61m is grown only on one side ofthe substrate to simplify the theory.

The Gilbert form of the equation of motion is used todescribe the dynamical behavior of the magnetization:

M= —yMXB, /r+ (MXM),S

where y is the gyromagnetic ratio, a the magneticdamping parameter, and M, the saturation magnetiza-tion. The e6'ective induction B,ff can be calculated fromthe density F of the free energy by the variational deriva-tive with respect to the magnetization, '

eff

where F contains the following contributions, neglectingthe cubic anisotropy energy,

F=FZ + Fthm +Fex +Fe1+FME

c&j k& and 6pjk& denote the elements of elastic and magne-toelastic tensors. Because of the cubic symmetry of gar-nets, they have only three nonzero elements; using Voigtnotation, these are C», C,2, C~ and B»,8,2,B44. Equa-tions (3) have to be transformed into the coordinate sys-tem illustrated in Fig. 1.

The equation of motion of the elastic displacementu(t, r) in film and substrate is'

u.p =—g(1+5;)

21 aej

(4)

where p is the mass density of the elastic medium.It will be assumed that the elastic and magnetic excita-

tions are transverse plane waves propagating along the zdirection. Thus there is no dipole-dipole interaction,which would cause coupling between the FMR and de-generated spin waves. ' For the following discussion itis convenient to introduce complex amplitudes for themagnetic and elastic waves:

Fz is the Zeeman energy caused by the static and rf driv-ing induction,

Fz = —M, I bo[p„cos(cot )+plain(cot )]+p,80 J,

M (t,z)e'"'=M„(t, z)+ iM„(t,z),u '(t, z)e'"'=uf'(t, z)+iu '(t, z) .

(5)

FME

3

ijkl ij kli,j,k, I =1

3

b jk/P Pjekl'i j,k, l =1

(3)

F„ is the uniaxial anisotropy energy including the demag-netizing energy,

F„=( ,' poM, K„-)P, —

and F,„ is the exchange energy,

F,„=,'p, oM, Do[(VP—)+(V ) +(VP, ) ],where p is the unit vector in the direction of M and Dothe exchange constant. F,1 and FME denote the densityof elastic and magnetoelastic energy. The latter describesthe coupling between magnetization and elastic deforma-tion. These two contributions to F depend on the elasticstrain e;. of the medium. "

The elastic waves in the film and substrate are denoted byu and u', respectively. The complex amplitudes M(t, z)and uf'(t, z) in Eqs. (5) describe the behavior of the mag-netization and elastic displacement in a coordinate sys-tern, which rotates with the microwave frequency co

about the direction of the static induction Bo. In this ro-tating frame the instability oscillations of M and uf' willbe observed.

Using the approximation for the z component of themagnetization,

M +MM =[M —(M +M )]' =M-

Z S X P SS

MM2M,

one obtains the following coupled equations of motion forthe elastic displacements u '(t, z) and for the normalizedmagnetization m (t,z) =M(t, z)/M, up to the third orderin m and u:

Bm a2m . Xb2 auf(1 i a ) =—i [coo+co„,/t —(1 i a )co]m i yp—DDO

—z +i

aimm*

lp 1

ff+~+ 2 .+ 2 am* . ~~2 2 auf* .auf2 Bt 2M, Bz az

O~z~d,

a'uf . auf, a'uf, . f b2 am+2'(a, t+i) = Vf 2+co (1 2ia, t)uf+-

at2 at az2 'pf az

a(m m ), O~z~d,

2pf az

a2 s s a2 s

2 +2'(a„+i ) =V,2 +co (1—2ia„)u', d ~z~D,

at2 ' az'

9460 M. YE AND H. DOTSCH

where

~0 7+0 ~u eff= 2y

S

Po7

These instabilities of the FMR have already been ob-served experimentally. "

B. Numerical analysis neglecting boundary conditions

b2 =2b44 =2B44+ —,'(B)) B)2——2B44) .

u, f and cx„denote the damping parameters of elasticwaves in the film and substrate, respectively, while

Vf Q C44(f ) /p( f ) is the propagation velocity oftransverse elastic waves.

The magnetization m and elastic displacement u inthe film are coupled by the terms containing thecoefficient b2. Some of these are nonlinear with respectto m and uf; these nonlinear coupling terms lead to theinstability of the FMR at high excitation levels as will beshown below. As the frequencies of the instability oscil-lations are expected in the range of kHz to MHz, ' non-linear coupling terms containing the coefficient b, 4 havealready been omitted in Eqs. (6) because they are propor-tional to e ' '. The other nonlinear terms in Eqs. (6)cause the foldover effect. This can be proved by neglect-ing the magnetoelastic coupling and solving thedifferential equation analytically.

The equation of motion for the elastic waves u' in thesubstrate is not coupled directly to that of the film. How-ever, the coupling is caused by the boundary conditionsat the interface z =d between film and substrate, wherethe elastic stresses and displacements must be continuous.Furthermore, the elastic stresses vanish at the crystal sur-faces, z =0 and D. The magnetization is assumed un-pinned at both sides of the film. Thus the boundary con-ditions for the equations of motion (6) are as follows:

Omitting the boundary conditions, it will be demon-strated at first that the nonlinear magnetoelastic interac-tion in the equations of motion (6) already yields instabili-ties of the FMR. This situation corresponds to an infinitelarge YIG crystal. The inhuence of the boundary condi-tions on the propagation of magnetic and elastic waveswill be considered later.

The magnetic excitation m (t,z) is decomposed intotwo parts: the uniform precession mo(t) and spin wavesmk(t, z) Fur. thermore, the spin waves mk and elasticwaves u can be written as linear superpositions of lanewaves with time-dependent amplitudes mk(t) and u (t):

m (t,z)=mo(t)+ g mI, (t)e' ',k

For simplicity it is assumed that only two plane waveswith wave vectors k and —k and equal amplitudes in mand u are essentially excited. ' The amplitudes of mkare small compared to mo, so that the nonlinear terms ofthird order of mk can be neglected. Under these assump-tions one obtains from Eqs. (6) a system of coupled non-linear ordinary differential equations for the amplitudesmo, mk, and u)t, which can be integrated numerically.

Some results of the integration are presented in Fig. 2using the material parameters given in Table I. On theleft-hand side, m is plotted versus m, which are definedby

Ouf )fc

Qu t7lIBz 2

BufBz 2

—0 o

and

1m„= —Re m (t,z)dz6 . 0

1m =—Im J m(t z)dz

0

z =D: - =0;BQ

az

Bmz=O, d- =0 .BZ

This nonlinear boundary-value problem, differential equa-tions (6) and boundary conditions (7), is investigated nu-

merically. It will be seen in the next section that at highexcitation levels the solutions for magnetic and elasticwaves show periodic oscillations or chaotic behavior.

while the right-hand side shows the phase difference be-tween m(t, z) and driving induction. The component m„can be measured experimentally. When the driving in-duction exceeds a critical threshold, the motion of themagnetization exhibits periodic oscillations [limit cycles(a) and (b)], period doubling (c), and finally chaoticmotion (d). As can be seen from Fig. 2, the oscillationamplitude of the phase is about 20' —30', while that of theprecession angle is only about several tenths of a degree.These numerical results agree qualitatively with opticalmeasurements.

TABLE I. Parameters used for numerical integration of equations of motion without boundary conditions. The material parame-ters correspond to yttrium iron garnet.

M,(A/ )

b2(J/m')

y(GHz/T)

Dp(Am)

COp

(GHz)~u ea

(GHz) (GHz)k Vf Pf

(pm ') (km/s) (kg/m')+ef

1.43 X 10 4.66X 10 2m X28 4. 1 X 10 " 2m X 5.827 —2m X4.827 2m X 1.0036 1.702 3.849 5.17X 10 10 10

9461I~ITIES IN THE .MAGNETOEL&STIC

E—3.6O

~ —621 4 2 4 3.4 4time (pe)

I

9 3.502 mx

bb

:--- JJJJJJJJJJJJI62 —,

4.40.42.2time (pe)

"uf(&, z)e 'Uf(Q, z) =

&2

m(r z)eM(»z)= ~2

—1.2

E—2.o

o —2.80pT

I I

8 2.002 mx

bp=—3.6-I

1.6

ns of motion arequations

gUf—(1—~ )Q]M yap o ~zz M,[iso+~~ eIr

2 7 -b =2.093 PT

ie remaining(9) and (1O)roblem Is

Substituting Eqs., h boundary-value p ocondition at z =

.d contains only p

o~

am li-ary c

e film regioncte„f(t,z).tu et des m(t, z) and '

f tnotion (6) an d bouIldthe next stepf d into frequ

tions oaced, t, ns (7) are trans o

d the frequencYai-y con»o

h Fourler trans ormation, anetic and elastic

using eUf(Q, z ) of magne IM(Q, z) '"dpectra

'll be determined:in the film wi

o —3 ~ 5

I I3 9 —I I

4.64, 010 mx

eff ~ 2G=&a +

2 i 2m,G)+ 2

g2Uff 2(1 2, ~„)Q U + Vfgz&

b moG +) G4~2 ' 2

b,

Pf z 2P

10 x

a netiza-f normalized magthe motion o n'ations

FIG.xp

a netization an0 pf hase between mag

in boundary conddaonsical analysis including ounC Numer

ne an e as ich h u'. 1" f""'d

nditions (7) defi'g gof the elastic waves.

terac iont' n between the e as'

1 e uations (6)Considering the7y

large thickness is difference betweend 1 m, respective y, i1 ) it wouldotth n 1

.5 mm an pm,alculationsbe a van

h fllint eh fil o 1. Th

' in the substrate region is e eof u inthe boundary condition

z = U'(Q)cos[k, (D —z)]e' 'dc'',u'(t, z = (9)

where

I) and Q =co+cok = (1—ia„anS y

d' '

n uf(t, d)=u r,the excittng mic enc .

The condition udetermine the amp im litude U

f uf(t, d)e ' r .&2m cos[k, (D —d)]

(10)

z=: f +b, [M(Q, O) ,'G, —(Q—,O)]=O;z=O: C44 +b2 M

+b [M(Q, d) ——,'G, (Q, d)]z=d: Cf44 +bz M

amz =O, d:

—. ' Uf(Q d)k, tan(ksDs )' (13)=C44

s G. are given byThe nonlinear terms

M(Q, z)eM(Q, z)eeM (Q,z),Gi=

M(Q, z)eM(Q, z)e62= BUf*(Q,z)BZ

M(Q, z ) e M'(Q, z ) e+2. 8Uf(Q, z)Bz

(14)M(Q, z ) + M*(Q,z ),&2n.

M'(Q, z)] .6 = M(Q, )AM(Q, )ez[QM4 Nz

ution with respect tosterisk e denotes convolution wi

M(Q„zM(Q, z)eM(Q, z)=

ericaHy at discrete'll be solved numenca yThese equations wi e

of the exciting mi-b d d

c s ectrum o1-Th o odcrowave induc

'tion. e

tions are as foHows:

9462 M. YE AND H. DOTSCH

points in the 0-z plane, given by

o+j ~ I=z&=l hz, I =0, 1, . . . , nd,

nf+1co=Do+ j„hQ, j„=

(15)

b old2

~J' l f r I k(Mlyk 2 ij, k )2C44 k=o

(16)

The coef5cient matrixes I. and the nonlinear terms G& - I,which are third-order convolutions of M &, are given inthe Appendix.

The remaining equations can then be written in the fol-lowing form, which contains only the spectral com-ponents M. I

..1ld

M~ i= g Ai kD$, j=0, 1, . . . , nI, l =0, 1, . . . , nd .

k=0(17)

Here nf and nd are two integers. The frequency band-width taken into account is nf AQ with the lowest fre-quency Ao. The rf driving frequency co lies at the centerof the frequency region. According to the experimentalobservations, the bandwidth should be in the range of 10MHz.

The spatial derivatives in Eqs. (12) can be replaced byfinite differences. One obtains now a system of nonlinearequations for the spectral components M

&and U& &

atthe above-defined points (Q, z& ) in the 0-z plane.

It is of special importance to determine the com-ponents M

&because they are directly related to experi-

mental results. Therefore, the spectral components U&

are expressed in terms of the components M I, which canbe regarded as driving terms for the U. I ~ Using the di-mensionless notation 8 Jfl = Ujf1 I~z one has

1.0155 1.0223frequency (GHz)

1.0257

FIG. 3. Calculated amplitude of the FMR spectrum com-ponent as function of rf driving frequency below instabilitythreshold. rf induction: (a) 0.02 pT and (b) 1.05 pT.

function of microwave frequency. The two curves arecalculated with different rf driving inductions, and theparameters used are given in Table II. The elastic reso-nances are clearly visible. They appear periodically atfrequencies

V,el=j2D

where j is an integer. For a typical thickness of D =0.5mm, the period is b,f=3.6 MHz.

The FMR curves and inhuence of the elastic reso-nances are identical to that of an analytical solution ofthe linear magnetoelastic equations of motion. " As therf driving induction increases, the maximum of the FMRcurve (b) is shifted toward higher frequencies. This is thewell-known foldover effect. ' The results in Fig. 3 showthat Eqs. (17) yield indeed the expected results for low ex-citation levels. Thus they can be used to calculate thenonlinear behavior of magnetization at higher excita-tions.

The coefticient matrices 2 J and the driving vectors D~ inEq. (17) are given in the Appendix. The vectors DJ con-tain the spectrum of the rf driving induction and all thenonlinear terms as G» &

in Eq. (16). Equations (17) havethe form x'=F(x') and can be solved by using the fixed-point method:

x„+,=F(x„), lim x„=x' .

At a low excitation level, the nonlinear terms in the driv-ing vectors D~ are small and can be neglected. Thus thesolutions of Eq. (17) should agree with the solution of thelinear boundary-value problem, which can be solvedanalytically. " Figure 3 presents the amplitude ~M ~

ofJn

the FMR spectrum, averaged over the film thickness, as a

III. EXPERIMENTS

Thin films of yttrium iron garnet are grown on [111]-oriented gadolinium gallium garnet substrates by liquid-phase epitaxy. The film thicknesses are below 1 pm toensure that the FMR lies at the very bottom of the spin-wave manifold. The film thickness used for the measure-ments presented in this paper is O.S pm. The FMR is ex-cited by a short-circuited slot line and detected by a co-planar waveguide as described in Ref. 7. To observemagnetoelastic resonance, it is important to use suchbroadband microwave structures. The microwave signalof the FMR can be fed directly to a spectrum analyzer orit can be rectified by a diode and displayed on an oscillo-scope.

TABLE II. Parameters used for numerical calculation of the spectrum of FMR with boundary conditions.

Q)p

(6Hz)

2m' X 5.8485

u ea

(GHz)

—2~X 4.827

(G-Hz)

2m X 1.0213

d(pm)

0.5

D(mm)

0.45

O'es

10

V,(km/s)

3.57

ps(kg/m )

7.09 X 10

EQ(MHz)

2~X0.05 255 4

MAGNETQELASTIC INSTABILITIES IN THE. . . 9463

fp

fei tfp

1.0288I I

1.0313 1.0338frequency ( GHz )

1.0363A & I

feh I to

FIG. 4. M easured FMR curve with pumping power abovethe instability threshold.

1.029

iL

fFMR/

1.032 1.035frequency ( GHz)

1.038

IV. RESULTS AND DISCUSSION

Figure 4 shows a FMR signal measured in aure in a swept-q cy experiment, where the excitation level is above

instability threshold. The shaded areas indicate the re-gions of instability oscillations. It is noteworthy that themain region lies just above an elastic resonance.

Fi ure5 higure 5 shows two measured spectra of FMR signals.The excitation frequency lies within the instability re ion

e y, 0. t low power only one componentat the exciting frequency is observed (upper trace), whichcorres onds to thAbove a

p e uniform precession of magnetizat'a threshold of the input power of about —12.4

ion.

dBm, new spectral components appear (lower trace).The frequency difference between two adjacent com-ponents is constant and determines th b fe ase requency ofthe instability oscillation.

Because of the nonlinear magnetoelastic interaction,t e FMR is coupled to a neighboring elastic resonance.

the system will show instability if the pumping ower

This nonlinear coupling is demonstrated by the experi-mental results presented in Fig. 6. Th

' ' fe exciting frequen-cy is marked b

~ ~ ~

s ek y o. When it is continuously lowered th

pectral component corresponding to the coupled elasticresonance (marked by f,&

in Fig. 6) keeps its frequency

FIG. 6. Measureasured frequency spectra of the instability of theFMR with different microwave frequencies.

~ ~

position, because it is solely determined b th h' k'

e y e t ic nesso e crystal and propagation velocity of the elasticwaves. Thus the base frequency of the instability oscilla-tion is locked to the frequency difference between thevariable excitation frequency of the FMR and the fixedfrequency of the neighboring elastic resonance. Only ifthis frequency difference lies below about 100 kHz, doesthe behavior of the FMR become chaotic (upper trace in

ig.The strong inAuence of the nonlinear magnetoelastic

p ing e ween FMR and an elastic resonance on theinstability is also demonstrated b th f 11y e o owing experi-ment. If the back side of the crystal [z =D in Fig. 1(b)] is

1.37

f

0

fp

-12.5 dBm 1.13

1.0305

fo

-12.4 dBm

1.0320 1.0335 1.0350frequency ( G Hz )

1.0158

&ei

hM1.0182 1.0206

frequency

MR

1.0230 1.0254

(GHz)

FIG. 5. MMeasured frequency spectra of the FMR withdifferent pumping power.

FIG. 7. C. Calculated frequency spectra of the instability of theFMR with different microwave frequencies.

M. YE AND H. DQTSCH

(fo

ECDU

o —16-CL

——19-CL

EcL 22

I0

4

instability region

~ e+~ 0

0

e0

1.0335 1.0360 1.0385frequency f 6Hz I

FIG-. 8. Measured frequency spectra of the FMR showingperiod doubling. Driving frequency: (a) 1.0346 GHz and (b)1.0347 GHz.

1I I I

1.430 1.431 1.432 1.433

frequency (GHZ)

FIG. 10. Measured dependence of the threshold of the insta-bility of the FMR on microwave frequency.

roughened by mechanical grinding, the elastic resonatoris destroyed. In this case a large increase of the instabili-ty thresho1d is observed.

Calculated spectra of the magnetoelastic instability ofthe FMR are shown in Fig. 7 for different excitation fre-quencies, which are marked by fo. The frequency of theneighboring elastic resonance is marked by f,&, and therespective thresholds of the driving induction bo aregiven at each spectrum. These results are in excellentagreement with the measurements shown in Fig. 6. Also,the tendency of the FMR to become chaotic is observed ifthe excitation frequency is too close to the coupled elasticresonance.

The period doubling of the instability oscillation is atypical route to chaos for a nonlinear dynamical system.This can also be observed in the present experiments ifone of the experimental parameters is changed, for exam-ple, the static or rf driving induction or exciting frequen-cy. Figure 8 presents two measured spectra of the insta-bility oscillations of the FMR. Spectrum (a) shows thecoupling between the FMR and an elastic resonance simi-lar to the spectra of Fig. 6. If the exciting frequency isshifted by 0.1 MHz [spectrum (b)], new frequency com-ponents appear at the center between two adjacent com-ponents of spectrum (a). The base frequency of the insta-

bility oscillation is thus reduced to one-half, correspond-ing to period doubling. Two calculated spectra areshown in Fig. 9, where the driving frequencies differ by0.17 MHz. These calculations agree qualitatively withthe measurements shown in Fig. 8.

In Fig. 10 the measured instability threshold at con-stant static induction is plotted versus the rf driving fre-quency. The frequencies of elastic resonances are indicat-ed by arrows. For comparison the calculated critical rfinduction for instability is illustrated in Fig. 11. In bothcases the threshold decreases if the rf frequency is movedaway from elastic resonances and shows a minimum be-tween two of them.

The material parameters infIuence also the instabilitythreshold. As an example, the calculated dependence ofthe critical driving induction on the magnetic dampingparameter cx is given in Fig. 12. As n increases, theprecession angle of the magnetization and thus the exci-tation of elastic waves by the magnetoelastic interactionsdecreases. Correspondingly, a higher excitation level isnecessary to reach the instability threshold.

ACKNOW%'LEG GMENTS

Financial support by the Deutsche Forschungsgemein-schaft, Sonderforschungsberecih 225, is gratefully ac-knowledged. We thank A. Brockmeyer for the prepara-tion of the garnet films, and we are indebted to Wacker

I—1.7

1.0160

feI of

I

1.0195 1.0230 1.0265

0 1.5O

~ 1.3

-o1 1fel

frequency (GHz)

FIG. 9. Calculated frequency spectra of showing period dou-bling. Driving frequency: (a) 1.02128 GHz and (b) 1.02145GHz.

1.0197I I

1.021 1 1.0225 1.0390

frequency (GHz)

FIG. 11. Calculated dependence of the critical rf driving in-duction of the instability of the FMR on microwave frequency.

MAGNETOELASTIC INSTABILITIES IN THE. . . 9465

1 3

1000aC:

0.6 1.6 2.6 3.6damping parameter 10~ n

4.6

FIG. 12. Calculated dependence of the critical rf driving in-

duction of instability of the FMR on the magnetic damping pa-rameter e

LJ=

0 0

0 0

0 0 . . 1 2a- 1J

0 0 . 0 1 a +cj—2 0 0 0

1 0 —1 00 1 0 —1

0 0 00 0 00 0 0

a 1 0 0 0 0 0J

1 2a 1 0 . . . 0 0 0J

0 1 2a 1 0 0 0J

Chemitronic, Burghausen, for providing substrate crys-tals.

APPENDIX

0 0 00 0 0

Q Azaj = (1—2ia, f)

2Vf

00

1 0 —1

0 0 2

In Eq. (16) the spectral components Wf l of elasticwaves in a film are expressed in terms of the componentsMj&. The coe%cients L are the frequency-dependentcomplex matrices of dimension (n d+1) X(n d+1), inwhich the corresponding boundary conditions (13) havebeen taken into account:

C44c.= bz k, .tan(k, .D, ), j=0, 1, . . . , nf .J ( f

Using Eq. (16), the equations for spectral componentsM~. l can be written in the form of Eq. (17) with coefficientmatrices A J and driving vectors Dj. The (nd + 1)X(nd+1) complex matrices A are given by

d. —4aJ—2b

abL& o—b QAL»+dj

0

QAL( z—b

0

abL~) 3

abL) o aAL(, bahL(2+—d, ahL) 3b—

0

QEL~) „

ahL) „

0

QELjl „

abLl2„

0

QEL) „

abLl2„

QALJ I o

2acJL„'

QAL„'

2ac L„

QAL J

2QcJ L„d

QAL J

2acJ L„ 2acj L„' „&—2b 2acjL„„+d, —4a

yb2 C~a=, cj = hzk, tan(k, .D, ),

4M, C44 C f44

d =coo+co„,lr—Qj(1 ia )+—2b, b = 'YPoDo

hzJ J J

~LI, k—LI+~, k

(A2)

j Oy 1) ~ o ~ p nf p 1=1,2, . . . , nd —1 .

The driving vectors D consist of the spectrum B of the rf driving induction and all of the nonlinear terms G;:

bo 1 5yb2DJo =y B — G3'o + co ff f +in co GiJo+' G4Jo

DJ =y B—nd J

bo 1 5yb2 ~m

2 ~ d3j n + ~ueff f +~( mQ) G~j n +l G4J nM C 'd 2 'ds 44

y b,C44+ f b,zk,jtan(k, D, ) G2, „+g Ljd kG, J „ (A3)

bo CO ff+ iO! CO CXm yb 2

J~G3j, l + G1j,l+1 4j, l+ f 2j, l+ rf ~L1,kGlj, k8M, C

j=0, 1, . . . , nf, l =1,2, . . . , nd —1 .

9466 M. YE AND H. DOTSCH

The linewidth of the rf driving induction and thermal excitation of the magnetization are simulated by introducing twoconstants h, and h 2 in the spectrum B.:

B.=bo(5 J +h]5ii +, +h2)(nf+1), h2=10, h, =3X10

The nonlinear terms 67J I are discrete convolutions of spectrum Mj &. For example, 6;j &is given by

fG, . i= g M ]M ]M'+,. i, E=nf+1 .

O

(A4)

The numerical calculation of 6; &can be accelerated if one uses discrete Fourier transformation:

nf

G ~ ~ 2 ~ + e—7277jn/N

].j, l ~ ™n,l~n, len=0

nf

62 i= y m„[[m„t(w„i+]—w„t* ])+3m„"i(w„i+]—w„i ])]en=0

nf6 ~ e —72m jn/N

3j, l ~ mn, l~n, len=0

nf6 ~ 2 e —7'2' n /N

4j, 1 ~ ™n,I ~n, ln=0

j=0, 1, . . . , nf, I=0, 1, . . . , nd .

(A5)

The components m„ i, wfi, and t7, t in Eqs. (5) are defined in time space and can be calculated also by the discreteFourier transformation:

7'2~j n /Nf

n 1 ~ jlj=0

wf =—f f i2+jn /Nw„& —— Ie

j=0(A6)

ft7„]=—g M, ]sin

j=0

2]r(j—j„) 72m jn /N

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