PHYSICAL REVIEW B 1 DECEMBER 2000-IIVOLUME 62, NUMBER 22

Acoustical and optical spin modes of multilayers with ferromagneticand antiferromagnetic coupling

R. Zivieri, L. Giovannini, and F. NizzoliDipartimento di Fisica and INFM Unita` di Ricerca di Ferrara, Universita` degli Studi di Ferrara,

Via Paradiso 12, I-44100 Ferrara, Italy~Received 25 July 2000!

A study of the in-phase~‘‘acoustical’’! and of the out-of-phase~‘‘optical’’ ! spin modes in layered ferro-magnetic structures with both ferromagnetic and antiferromagnetic coupling is presented, in particular for thesystem Fe~20 Å!/Cr~20 Å!/Fe~20 Å!/Cr~9 Å!/Fe~100 Å!. A calculation of the dynamic magnetization, assumedconstant in each film, gives an explanation of the behavior of the two kinds of modes at different magneticfields. The ‘‘acoustical’’ mode is the lowest frequency one for low applied fields, but its frequency switcheswith that of the ‘‘optical’’ modes at higher fields. It is found that the system exhibits soft modes as a functionof the applied field. One of these is a Goldstone mode.

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I. INTRODUCTION

Great attention has been given in recent years to the eof antiferromagnetic~AF! coupling on spin waves in layeresystems. Since its first observation by light scattering frspin waves in Fe/Cr and Fe/Cu structures1 and by neutrondiffraction in single-crystal Gd–Y superlattice,2 a series ofexperimental measurements and theoretical calculationsbeen performed.

Grunberg3 introduced, in a pair of ferromagnetic layewithout exchange coupling, the terms ‘‘uniform’’ and ‘‘optical’’ applied to magnetic modes: the first indicates anphase spins precession and the latter is related to an ouphase oscillation. Successively, Gru¨nberg4 has investigatedthe effect of the interlayer spacer on spin waves spectrmagnetic multilayers and Gru¨nberget al.1 have observed theexchange coupling in Fe double layers as a function of CAu interlayer spacers. They have shown that an AF intertion is present in a certain range of Cr spacer thicknesCochran and Dutcher5 have studied the effect of exchanginteraction on calculated Brillouin light scattering~BLS!spectra confirming Gru¨nberg results on the lack of depedence of the ‘‘uniform’’ mode on the exchange interactiand on the influence of interlayer coupling on the ‘‘opticamode. Wang and Mills6 have extended the analysis to colective spin wave modes in AF superlattices studyingmagnetic-field dependence of the spin wave spectrumsystem with an even number of ferromagnetic layers. Thclassification of spin waves distinguishes in low exterfield a ‘‘uniform’’ bulk mode from a couple of surface exchange bands and introduces the ‘‘acoustical’’ standing wresonances as modes at a lower energy than the ‘‘opticones. In analogy with phonons,7 Wang and Mills6 noticedthat, at least for low applied fields, the energy requiredexcite the ‘‘acoustical’’ wave is less than that needed for‘‘optical’’ one. More recently, studies of antiferromagnecally coupled Fe/Cr bilayers performed by Rezendeet al.8

have shown that an exchange between the ‘‘acoustical’’the ‘‘optical’’ modes occurs at a critical external field. Thmagnetization precession in the same direction correspoto a uniform atomic displacement within each unit cell in tcase of lattice dynamics, while opposite spin directions

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analogous of out-of-phase atoms displacement.In antiferromagnetically coupled multilayers, the ma

netic ground state develops as a function of the applied fiand the classification of ‘‘acoustical’’ and ‘‘optical’’ spinmodes as lower and higher frequency modes, respectivelnot generally valid. So it becomes interesting to discussmeaning of ‘‘acoustical’’ and ‘‘optical’’ modes in magnetisystems with AF interaction in different magnetic groustates.

This paper is organized as follows. Section II introducthe model and discusses the meaning of ‘‘acoustical’’ a‘‘optical’’ spin modes in AF multilayers. Section III presentresults and discussion on symmetry properties of spin moSection IV is devoted to soft and Goldstone modes. In Stion V conclusions are drawn.

II. THEORETICAL MODEL

In our model, applied to layered structures, we have csidered a mean-field Hamiltonian, expressed as an energunit area in the following form:

E5(j 51

N

~EAj 1EZ

j !1 (j 51

N21

Eexchj , ~1!

with N indicating the number of ferromagnetic layers. Tanisotropy energyEA

j of the j th ferromagnetic layer whichincludes magnetocrystalline, in-plane uniaxial and shaanisotropies for the case of the~211! multilayers is expressedas

EAj 5djFK1

j S 1

4cos4f j1

A2

3cos3f j cosu j sinf j

11

3sin4f j cos4u j1

1

2cos2f j sin2f j sin2u j

2A2 cosf j sin3f j cosu j sin2u j11

4sin4u j sin4f j D

1Kuj cos2u j12pM j

2 cos2f j G , ~2!

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whereK1j andKu

j are the magnetocrystalline and the uniaxanisotropy constants anddj is the thickness of thej th layer,while f j andu j are the angles formed by thej th static mag-netizationM j with respect to the normal to the surface andthe hard axis (x axis!, respectively. The ZeemanEZ

j contri-bution for thej th layer is

EZj 52HM jdj cos~u j2j!sinf j , ~3!

whereH is the applied field andj is the angle betweenH andthe hard axis, in the plane of the film. The last term of Eq.~1!is the exchange energy which for a generic couple of layassumes the following form:

Eexchj 52J1

j , j 11@sinf j sinf j 11 cos~u j2u j 11!

1cosf j cosf j 11#1J2j , j 11@sinf j sinf j 11

3cos~u j2u j 11!1cosf j cosf j 11#2. ~4!

J1j , j 11 and J2

j , j 11 are the bilinear and the biquadratic echange coupling constants between thej th and the (j11)th layers, respectively.

In the following the external fieldH is applied along theeasy axis (j5p/2). The films are assumed to be parallelthe x-z plane withH along thez axis and the surface wavvectorq along thex axis. Obviously from the experimentapoint of view the direction ofq is determined by the scatteing geometry. In the calculation of spin wave frequenc

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and of dynamic magnetizations a term has been added wtakes into account the effect of dipolar interaction betwethe films. In our model applied to thin films, it has beeassumed that the dynamic magnetization is constant aceach layer.

To obtain the static equilibrium magnetizations we haminimized the energy functional of Eq.~1! with respect tothe polarf and azimuthalu angles. For numerical purposthe steepest descent technique has been used. In ourdue to the strong demagnetizing field the static magnettion results in-plane (f j5p/2). The dynamic problem habeen faced solving the semiclassical Landau–Lifshitz eqtions of motion and keeping only the first-order terms in tcomplex dynamic magnetization. In our treatment, we hageneralized to a ferromagnetic trilayer the Cochrapproach,9 which was limited to a bilayer and, in additionwe have included the biquadratic exchange term.

We do not present explicit expressions for a ferromanetic bilayer,9 but we extend the study to the case of ttrilayer @N53 in Eq.~1!# expressing the equations of motioin the following matrix form:

(j 51

6

~a i j 1b i j !mj50, ~5!

where m1 , m3 , m5 are thex components of the dynamimagnetization for the three ferromagnetic layers, resptively, and similarlym2 , m4 , m6 are they components.

The matrix a, which is referred to the case of infinitmagnon wavelength, is given by

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wherev j85 i (v/g)M j , with j 5A,B,C labelling the three Fe films andg is the gyromagnetic ratio. The subscripts indicathe second derivates, with respect to the azimuthal angleu and to the polar anglef, of the energy per unit area given iEq. ~1! for the caseN53. The matrixb takes into account the dipolar correction to first order in the wave vectorq and isgiven by

14 952 PRB 62R. ZIVIERI, L. GIOVANNINI, AND F. NIZZOLI

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where K52pMAMBMC and cj5cos(uj2j) with j5A,B,C. The spin wave frequencies are obtained imposthat the determinant ofa1b, evaluated at the equilibriumanglesu j andf j , vanishes for different applied fieldsH. Thefollowing step is the calculation of in-plane and out-of-placomponents of the dynamic magnetizationm, mx , andmy ,respectively, for each ferromagnetic layer. It is importantnote that in the following discussion the components ofmare referred to the local frame, which is rigidly connectwith the static magnetization in each layer. Therefore,each layer,mx is referred to a newx-axis which is perpen-dicular to the static magnetization and belongs to the fiplane.

III. RESULTS AND DISCUSSION

Before introducing the rather complicated case ofasymmetric ferromagnetic trilayer we begin the discussconsidering the most simple case, corresponding toN52 inEq. ~1!. As a model system, we refer to a symmetric~211!–Fe~20 Å!/Cr~11 Å!/Fe~20 Å! bilayer with AF coupling,which has been already studied from the point of viewBrillouin scattering and of the behavior of magnon frequenas a function of the applied field.10 In order to extend ourinvestigation to the trilayer, it is interesting to analyze tsymmetry and the nature of the spin modes in such a rtively simple system. In the ground state without any extnal applied field, the two static magnetizations are antipalel to each other and directed along the easy axis.calculated complex dynamic magnetizations in the twolayers are of the same sign for the low-frequency modeof opposite sign for the high-frequency mode, for bothmxandmy . Therefore, the low-frequency mode is characterizby an in-phase spin precession which gives to it an ‘‘acotical’’ nature, while the high-frequency mode consists of oof-phase oscillations and it is classified as ‘‘optical.’’ In thcase there is a complete analogy to the case of phononhigh fields an exchange between the ‘‘optical’’ and t‘‘acoustical’’ mode occurs. This behavior can be quantitively understood within our model. It can be shown that,

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the long-wavelength limit, the frequencies of these twmodes are given by11

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51

M2@~EfAfB

7EfAfA!EuAuB

1~EfAfA2EfAfB

!EuAuA#, ~8!

where the symbols have the same meaning as in Sec. IIwith only two ferromagnetic films A and B. Here the minusign is referred to the ‘‘optical’’ mode, while the plus signreferred to the ‘‘acoustical’’ mode.M is the saturation magnetization of both iron layers. Therefore, the difference btween the squares of the frequency of the ‘‘optical’’ a‘‘acoustical’’ modes is

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whereEfAfAis always positive.EuAuB

52J1 cosDu, where

J1 is the negative bilinear exchange constant andDu is theangle between the static magnetizations. At low fieldsDu5p so thatEuAuB

is negative and the ‘‘optical’’ frequency isgreater than the ‘‘acoustical’’ one. Above some critical aplied field a spin–flop transition occurs so thatDu is lessthan p/2, cosDu becomes positive and the situation is rversed, e.g., the frequency of the ‘‘acoustical’’ moswitches with that of the ‘‘optical’’ one. Therefore, a simprule applies. The lowest frequency mode corresponds tonamic magnetizations that would tend to take the systemthe magnetic ground state occurring atH50.

The central scope of this paper is to study in detairather general case, i.e., an asymmetric~211!–Fe~20 Å!/Cr~20 Å!/Fe~20 Å!/Cr~9 Å!/Fe~100 Å! trilayer with a ferro-magnetic coupling between the first two Fe layers and ancoupling between the last two Fe layers. This choice is amotivated by the availability of experimental fitting parameters extracted from SQUID magnetometry, magnetoretance, and BLS measurements.12

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PRB 62 14 953ACOUSTICAL AND OPTICAL SPIN MODES OF . . .

In Fig. 1 the sample is depicted together with the refence frame; the direction of the external fieldH and of thestatic magnetizationsMA , MB , andMC in the ground statewithout any external field are also shown. Figure 2 showsground state configurations of the equilibrium static magtizations at different magnetic fields. In the ground state wH50 the static magnetizations reflect the overwelming iportance of the bilinear ferromagnetic and AF couplingstween the films A–B and B–C, respectively. In the preseof different applied fields Fig. 2 also shows the reorientatof the static magnetizations as a function of the increasfield. In Fig. 3 the frequencies of the three couples~Stokesand anti-Stokes! of spin waves are plotted as a function ofHand compared to the BLS data.12 The calculated frequenciehave been obtained applying the matrix approach introduin the preceding section to the asymmetric ferromagntrilayer. The splitting between Stokes and anti-Stokes mnons is, in general, small so that in the following we wrefer to three branches. The calculations presented in Rehave been refined so that the parameters extracted frombest fit are slightly different from those previously reporteThe values remain inside the error bars with the only exction of the uniaxial anisotropy constant of the layer C. Sin

FIG. 1. The sample geometry pictorially represented togewith the reference system. Film A, Fe~211! 5 20 Å; Cr 5 20 Å;film B, Fe~211! 5 20 Å; Cr 5 9 Å; film C, Fe~211! 5 100 Å. Thesubstrate is MgO. The arrows indicate the directions of the stmagnetizationsMA , MB , andMC , respectively, from the top to thebottom. The direction of the applied fieldH and of the magnonwave vectorq are also shown.

FIG. 2. Static equilibrium magnetization positions for four dferent ranges of the applied field of the films A, B, and C. Tmagnetization vectors lie in thex-z plane. The laboratory coordinate system and the direction of the external magnetic field areshown.

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the magnitude of the magnetocrystalline anisotropy is smit is impossible to extract a reliable value of the magnecrystalline anisotropy constantsK1

j . We have assumed thathe magnetocrystalline anisotropy is the same in each laand has the same numerical value as for the symmetriclayer, i.e.,K1

j 51.13105 erg/cm3.10 Indeed, we have foundthat this anisotropy does not change appreciably the behaof the ground state and of the dynamic magnetization. Tfitting procedure has been based on the adjustment of paeters to minimize the difference between the calculatedquencies and the BLS data (x2). The error for each parameter has been determined finding the change, after adjusall other parameters, such thatx2 increases by 50%. Fromthe best fit we have obtained the following values for tfitting parameters: 4pMA54pMB518.060.5 kG, 4pMC

517.360.8 kG, KuA5Ku

B5(5.560.6)3105 erg/cm3, KuC

5(3.060.6)3105 erg/cm3, J1A,B50.11460.025 erg/cm2,

J1B,C520.73560.060 erg/cm2, J2

A,B50.01060.010 erg/cm2, J2

B,C50.11060.040 erg/cm2. The low numerical valueof Ku

C, compared toKuA and Ku

B , is due to the geometricaasymmetry of the system which lowers the in-plane uniaxanisotropy of the thickest ferromagnetic layer. As shownFig. 3, the fit is generally good apart from a couple of BLpoints for the low-frequency mode close to the dip minimuat H51.6 kG. The reason for this disagreement coulddue to the critical dependence of the soft modes on theviation of these multilayers from the ideal situation implieby our theoretical model~presence of roughness, intermixing, etc.!.

We have found that an ‘‘acoustical’’ mode is not alwapresent. Four different regions where there is such a mmay be recognized. The first is located between zero mnetic field and H50.4 kG, the second in the rangeH50.4–1.1 kG, the third betweenH51.6 kG and H52.5 kG, and the last beyondH52.75 kG. It is interestingto note that in the regions where an ‘‘acoustical’’ modenot present there is a spin wave for which the dynamic mnetizations of two layers are in-phase, while that of the th

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FIG. 3. Magnon frequencies as a function of the applied mnetic fieldH. Full squares: BLS data points from Ref. 12. Full linecalculated dispersion for surface wave vectorq50.983105 cm21

as for the data points. Dashed line: calculated dispersion forlow-frequency branch atq50. In each region discussed in the tethe symmetry character of the dynamic magnetization is represepictorially by three arrows which refer to films A, B, and C, respetively. When the arrows are parallel the spin precession is in-photherwise it is out-of-phase.

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14 954 PRB 62R. ZIVIERI, L. GIOVANNINI, AND F. NIZZOLI

layer is out-of-phase, but much smaller than the other tWe analyze the first region where the system is in a grostate with ferromagnetic~films A–B! and AF ~films B–C!alignments. Within this region, we classify the three modas follows:~1! a low-frequency mode, for which the precesion of the dynamic magnetization is in-phase in the loreference system, therefore ‘‘acoustical,’’~2! anintermediate-frequency mode called low-frequency ‘‘opcal,’’ and ~3! a high-frequency mode called high-frequen‘‘optical.’’ The ‘‘optical’’ modes are characterized by thpresence of out-of-phase oscillations. The high-freque‘‘optical’’ mode is the result of out-of-phase precessionsthe dynamic magnetizations of both the couples A–B aB–C, so that it requires the highest energy to be excited.other ‘‘optical’’ mode exhibits a lower frequency, becauonly the spins A–B are out-of-phase.

A first-order phase transition takes place atH50.4 kGwhere, as depicted in Fig. 2,MA , from being antiparallel,becomes parallel toH. This ground state configuration pesists until aboutH51.6 kG. In the regionH50.4–1.1 kGthe intermediate-frequency mode acquires an ‘‘acoustic’’ture. In analogy to the bilayer, the ‘‘acoustical’’ mode nlonger has the lowest energy. The other two modes are ‘tical.’’ We are able to explain the symmetry of the lowefrequency mode according to the simple interpretatscheme introduced for the case of the bilayer. For this mthe precession of the spins B–C is in-phase, while the sA–B are out-of-phase favoring their ferromagnetic aligment typical of the ground state atH50.

In the rangeH51.6–2.5 kG the symmetry characterthe modes is again reversed and is similar to that oflow-field region. In this state the equilibrium configurationthe three static magnetizations is predominantly determiby the biquadratic exchange term that has the effect to mtain the static magnetizationMB nearly normal to the othetwo, as depicted in Fig. 2. Consequently, the bilinearchange term, responsible for the AF ground state atH50, isalmost ineffective in this configuration. Therefore, anphase precession of the spins maintains the ground stateimplies a low energy excitation. For the opposite reasonhigh-frequency modes are ‘‘optical.’’

For H.2.75 kG the three magnetizations are almost pallel to each other and withH. In this region the low-frequency mode is ‘‘optical.’’ It minimizes the excitatioenergy, because it is a combination of an in-phase precesof the dynamic magnetizationsMA and MB and an out-of-phase oscillation ofMB andMC which are AF coupled, buare forced in the ground state to stay parallel by the strapplied field. We have found that the frequency orderingthe two high-frequency modes~one ‘‘optical’’ and one‘‘acoustical’’! depends critically on the balance betweenZeeman and the dipole–dipole interaction. In particular,‘‘acoustical’’ mode exhibits in Fig. 3 the highest frequencbut the situation would be reversed in case the dipole inaction were neglected.

IV. SOFT AND GOLDSTONE MODES

An analysis of the low-frequency branch at various manetic fields and forq→0 opposed toqÞ0 ~Fig. 3! allows usto gain a deeper understanding of the spin excitations in

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long wavelength limit. It is known that an acoustical phonassumes the character of a Goldstone mode13 as its wavevector q→0, i.e., the energy required for its excitationzero. On the other hand, the Goldstone theorem predictsexistence of Goldstone modes in ferromagnetic materialsscribed by a Heisenberg Hamiltonian14 in presence of spontaneous broken symmetry. In the ground state of our sysat H50 the low ‘‘acoustical’’ branch excitation energy inot zero forq→0 due to the presence in the Hamiltonianin-plane anisotropy terms. In this case the ground statunique and possesses the same symmetry of the HamiltoIt can be seen in Fig. 3 that a Goldstone excitation is presat the beginning of the third region where the low-frequenmode becomes ‘‘acoustical’’ and its frequency vanishesq→0. For qÞ0 the Stokes and anti-Stokes modes are sand their calculated frequency does show a deep minimbut does not vanish. The experimental BLS data reproduin Fig. 3 are not available for frequency less than 5 GHbecause in this frequency range the inelastic peaks areden by the quasielastic scattering. Therefore, a comparwith the experiment is not possible for soft modes. In thecases the ferromagnetic resonance technique could be uFor H.1.6 kG we have found that there is a second-orphase transition. In our model, transitions involving thequadratic exchange term may be either first- or second-odepending on the values of the parameters. We have foin agreement with Bezerraet al.,15 that the transition is ofsecond order when the ratio betweenJ2

B,C and uJ1uB,C issmall. In our case this ratio is actually small being abo0.15. It is known that, in presence of a second-order phtransition, a spontaneous breaking of the symmetry maycur and Goldstone modes may be present.16 The competitionbetween the Zeeman and the biquadratic exchange intetion, which are dominant over the other terms, lowerssymmetry and causes the appearance of Goldstone moWang and Mills6 did not find in AF coupled multilayers aGoldstone mode in the same region, but a soft mode foq→0. However, the chaotic behavior typical of multilayewith a large number of repetitions17 could have masked aGoldstone excitation. The other minimum in the lowfrequency branch occurring atH53.16 kG does not correspond to a Goldstone mode, because it has an ‘‘opticnature. In fact, as shown in Fig. 3, it does not depend oqappreciably.

To complete the analysis we study what happens wheHis oriented along the hard axis. By increasing the exterfield the static magnetizations tend to progressively rottowards the hard axis until saturation without suddchanges of orientation. Although in this case there isabrupt change of the static magnetization neither phase tsition, we have found that the exchange between ‘‘acoucal’’ and ‘‘optical’’ modes whenH increases is still presenwith more or less the same sequence. The classificatiothe modes in four regions remains valid also for this geoetry. We have found that, in this configuration, a Goldstomode is not present due to the lack of a second-order phtransition.

It becomes interesting to study the behavior of the lofrequency mode forq→0 vs applied fieldH in the caseN52 of Eq. ~1! and of the symmetric~211!–Fe~20 Å!/Cr~11Å!/Fe~20 Å! bilayer with AF coupling discussed in the pre

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PRB 62 14 955ACOUSTICAL AND OPTICAL SPIN MODES OF . . .

ceding section.10 At H50 the two magnetizationsMA andMB are antiparallel, lie in the film plane and are alignalong the easy axis. AroundH.1.5 kG applied along theeasy axis, a first-order phase transition occurs leading90o jump of the magnetizationMB . MA and MB becomenearly perpendicular to each other in agreement withstudy of Bezerraet al.15 on Fibonacci sequences. The frquency of the lowest mode decreases with increasing exnal field and reaches a minimum, but does not vanish efor q→0. Therefore, there is no evidence for Goldstonecitations in this system. ForH.1.5 kG andq→0 the dis-persion is similar to the case of the trilayer with the preseof a second minimum near saturation. The absence of aishing frequency and therefore of a Goldstone mode is dua first-order phase transition which occurs atH.1.5 kG. Inorder to complete the analysis of Goldstone excitationsbilayer systems with AF coupling we have studied, in tlimit q→0, the behavior of the low-frequency modean asymmetric~100!–Fe~45 Å!/Cr~30 Å!/Fe~15 Å! bilayerwhich has been recently studied experimentally.18 Again, atH50.1 kG a first-order phase transition occurs andGoldstone mode is present for this system. ForH.0.1 kG

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the frequency of the lowest mode presents a behavior simto that of the symmetric bilayer.

V. CONCLUSIONS

In summary, we have found that in a layered system wboth ferromagnetic and antiferromagnetic couplings a clafication in terms of ‘‘optical’’ and ‘‘acoustical’’ modes maybe adopted over a wide range of values ofH. The frequencyexchange between ‘‘acoustical’’ and ‘‘optical’’ magnetmodes is a general property of layered magnetic structuwith at least an AF coupling, independently on the thickneof each film and on the orientation ofH. Our analysis hasalso shown that, for a critical applied field associated tosecond-order phase transition, a Goldstone mode maypresent, while in presence of a first-order transition, as inAF coupled ferromagnetic bilayer systems, there is no edence for Goldstone excitations.

ACKNOWLEDGMENTS

This work has been developed in the framework of tINFM Project SIMBRIS. One of us, R. Z., was supportedINFM.

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-

.

,

11This formula, similar to Eqs.~A7a! and~A7b! of Ref. 10, is validfor a symmetric bilayer without magnetocrystalline anisotrop

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