JOURNAL OF APPLIED PHYSICS VOLUME 84, NUMBER 2 15 JULY 1998 Studies of coupled metallic magnetic thin-film trilayers S. M. Rezende, C. Chesman, M. A. Lucena, A. Azevedo, and F. M. de Aguiara) Departamento de Fi´sica, Universidade Federal de Pernambuco, 50670-901 Recife, PE, Brazil S. S. P. Parkin IBM Research Division, Almanden Research Center, 650 Harry Road, San Jose, California 95120-6099 Received 5 January 1998; accepted for publication 24 March 1998 Results are reported of a detailed study of static and dynamic responses in symmetric systems consisting of two ferromagnetic films separated by a nonferromagnetic spacer layer. A comparison is made with experimental results for two systems grown by sputter deposition in an UHV chamber, namely, NiFe/Cu/NiFe and Fe/Cr/Fe. First, we present model calculations where the coupling between the magnetic films through magnetic dipolar, bilinear, and biquadratic exchange interactions are fully taken into account, together with surface, in-plane uniaxial, and cubic anisotropies. An analytical expression is given that can readily be used to consistently interpret magnetoresistance, magneto-optical Kerr effect, ferromagnetic resonance, and Brillouin light scattering BLS data in such trilayers. Application of the results to BLS data in Ni81Fe19(d)/Cu 25 Å Ni81Fe19(d), with d 200 and 300 Å, shows that it is essential to treat the dipolar interaction adequately in moderately thick systems. The results are also applied to interpret very interesting data in Fe 40 Å /Cr(s)/Fe(40 Å), with 5 Å s 35 Å, investigated by the four techniques mentioned above, at room temperature. It is shown that consistent values for all magnetic parameters can be extracted from the data with a theory that treats both static and dynamic responses on equal footing. © 1998 American Institute of Physics. S0021-8979 98 00913-X I. INTRODUCTION with a few atomic monolayers give rise to an effective ex- change coupling,5 described by an interaction energy bilinear Brillouin light scattering BLS and ferromagnetic reso- in the magnetizations of the films.6 This bilinear interaction nance FMR are among the best experimental techniques to is characterized by an exchange constant which can be posi- determine the interlayer exchange coupling between ferro- tive or negative, corresponding to ferromagnetic or antifer- magnetic films separated by a nonmagnetic metallic spacer. romagnetic alignment of the neighboring magnetizations. Measurement of this coupling as a function of spacer and More recently it was discovered that certain systems exhibit magnetic layer thicknesses, temperature, interface properties, and various material parameters, are essential to test theories an additional exchange coupling, modeled by a biquadratic underlying the coupling mechanisms and to study basic interaction energy,7,8 that can make the two magnetizations properties of artificial structures. In addition to the interlayer align at 90° to each other. coupling, the BLS and FMR data also provide unique infor- The spin-wave dispersion relations for trilayer structures mation on the magnetic anisotropy and magnetization of the have been calculated by several authors. The earlier ferromagnetic layers. calculations,3,4 made before the exchange coupling was dis- In order to interpret the BLS and FMR data it is neces- covered, considered only the effect of the dipolar interaction sary to have knowledge of the spin-wave dispersion rela- on the surface and volume magnetostatic modes in the tions. The BLS spectra provide information on the volume coupled films. The simultaneous presence of dipolar and ex- and surface modes with nonzero wave vector, q 0, whereas change interactions complicates the problem considerably. the FMR technique probes the surface modes in the uniform, The equations of motion involve three components of the q 0 limit.1 The spin-wave mode configurations and disper- magnetization and the dipolar magnetic field in each layer, sion relations are strongly dependent on the coupling be- matched by the appropriate boundary conditions at the inter- tween the magnetizations in the magnetic layers. This cou- faces. The full solutions including the bilinear exchange cou- pling arises from the dipolar stray fields and the exchange pling, but restricted to ferromagnetic alignment, were interaction. The dipolar coupling has been a familiar mecha- worked out by Hillebrands9 for an arbitrary number of mag- nism for several decades.2­4 It dominates the interaction be- netic layers. For a trilayer structure, formed by two magnetic tween the magnetic layers in relatively thick structures. With films separated by a nonmagnetic spacer, the dispersion re- the advance in fabrication techniques, it has become possible lation is obtained from a system of 16 linear equations. This to control the epitaxial growth layer by layer in atomic scale. requires the use of appropriate numerical tools which com- Nearly ten years ago, it was then found that metallic spacers plicate the interpretation of the observed spectra. This fact, added to the need to interpret data in systems with antiferro- a Electronic mail: fma@df.ufpe.br magnetic coupling, led several authors to develop alternative 0021-8979/98/84(2)/958/15/$15.00 958 © 1998 American Institute of Physics J. Appl. Phys., Vol. 84, No. 2, 15 July 1998 Rezende et al. 959 calculations, each with its own limitations or simplifying assumptions.10­19 In this paper we present a calculation of the spin-wave dispersions for a trilayer structure with bilinear and biqua- dratic coupling, which can readily be used to interpret BLS and FMR experiments. The calculation takes into account an applied static magnetic field in the film plane, surface and in-plane uniaxial and cubic crystalline anisotropies, and bi- linear and biquadratic exchange interactions with arbitrary signs and magnitudes. The approach used is based on the equations of motion for the small-signal magnetization de- viations from the equilibrium directions, similar to that of Cochran and co-workers.12,16 However, besides introducing the biquadratic coupling, our calculation differs from Co- chran's by an improved treatment of the dipolar interaction, which makes the results suitable for reasonably thick mag- netic layers. The following section is devoted to energy con- siderations while the spin-wave dispersion relations are de- rived in Sec. III. Section IV describes details of the experimental techniques and sample preparation. In Sec. V the theoretical results are applied to interpret BLS and FMR experiments in NiFe/Cu/NiFe trilayers with moderately thick magnetic layers, for which the coupling is mainly due to the dipolar interaction. Section VI presents a unified picture of quite interesting magneto-optical Kerr effect MOKE , MR, BLS, and FMR data in a series of sputtered single-crystal 100 Fe/Cr/Fe trilayers with varying Cr thickness. Finally, Sec. VII summarizes the main results. FIG. 1. Trilayer structure upper panel and coordinate systems lower panel used to represent the fields and magnetizations in the two magnetic layers. The axes z1 and z2 are chosen to coincide with the equilibrium II. ENERGY CONSIDERATIONS: STATIC PROPERTIES directions of M1 and M2 . The calculation presented here is based on the con- tinuum approach used by Hillebrands, Cochran et al., and other authors.9­12 The geometry and the coordinate system where the subscripts denote, in order, Zeeman, anisotropy, employed are shown in Fig. 1. We consider two magnetic exchange, and dipolar terms. The Zeeman energy per unit thin single-crystal films, 1 and 2, having cubic lattice struc- area is ture. They have thicknesses d 2 1 and d2 and are separated by a nonmagnetic spacer layer with thickness s. The coordinate Ez diMi­H0. 2 system is chosen so that the xz plane is parallel to the film i 1 surface, with the x and z axes along 100 and 001 crystal The anisotropy energy has three contributions: Eac is the directions, respectively. We study only the situation where cubic magnetocrystalline energy; Eau is an in-plane uniaxial the external static magnetic field H0 is applied in the plane of term due to distortions introduced by mismatches between the films, at an arbitrary angle H with respect to the 001 the lattices of the films and the substrate; Eas is a surface direction. In this case, the equilibrium directions of the mag- energy due to the broken cubic symmetry at the film sur- netizations of the two films, M1 and M2 , are also in the xz faces. These three contributions to the energy per unit area plane, characterized by the polar angles 1 and 2 . As is are well known, the static and dynamic responses result from the K i d competition of several interactions. In general, each interac- E 1 i 2 2 2 2 2 2 ac 4 M M M M M M , 3 tion tries to align the magnetizations of the two magnetic ix iy ix iz iy iz i Mi films along different directions. Since there is a close rela- i tionship between the static configuration and the dynamic K d E u i au response of the system, the frequencies of the magnetic ex- 2 Mi* ui 2, 4 i Mi citations depend strongly on the equilibrium configuration of i the magnetizations. Therefore our initial goal is to determine K E s 2 as , 5 the equilibrium values of M2 Miy 1 and 2 . i i The equilibrium directions of M1 and M2 are determined where K(i) is the first-order cubic anisotropy constant of film by the minima of the total free energy. We consider a free 1 i, K(i) is the uniaxial in-plane anisotropy constant in a direc- energy per unit area with four basic contributions: u tion defined by a polar angle (i) (i) u , u is the unit vector in E E (i) z Ea Eex Edip , 1 that direction, and Ks is the uniaxial surface anisotropy en- 960 J. Appl. Phys., Vol. 84, No. 2, 15 July 1998 Rezende et al. ergy constant. Note that the anisotropy constants K(i) 1 and this paper, the equilibrium configuration was determined nu- K(i) u have units of erg/cm3 and each one is associated with an merically by varying 1 and 2 , first in steps of 5°, to locate effective anisotropy field H(i) the range of minimum energy, and then in steps of 0.5° to a 2K(i)/ M i with units of Oe. Usually the anisotropy parameters are the same for both obtain accurate minima values. magnetic films, so that the superscript (i) can be dropped for Once 1 and 2 are found, we obtain normalized values simplicity. for the magnetoresistance from The exchange energy has the usual volume intralayer R H 1 cos exchange contribution plus the interlayer exchange coupling, 0 1 2 composed of bilinear E R 0 2 , 10 ex1 and biquadratic Eex2 terms, given by and for the magnetization component in the field direction M from E 1­M2 ex1 J1 M , 6 M H M 1M 2 0 1 cos 1 H M 2 cos 2 H M . 11 S M1 M2 E M1­M2 2 ex2 J2 M , 7 In 10 R(0) is the resistance in the absence of the external 1M 2 magnetic field, and in 11 Ms is the total saturation magne- where J1 and J2 are the so-called bilinear and biquadratic tization. These two equations will be used to fit MR and coupling constants, which have units of erg/cm2. They are MOKE data, respectively, as presented in Sec. VI. associated to effective exchange fields H(i) ex1 J1 / M idi and H(i) ex2 J2 / M idi with units of Oe. Note that J1 0 and J1 0 correspond, respectively, to ferromagnetic and antiferro- magnetic couplings. In the case of the biquadratic coupling, III. DERIVATION OF THE SPIN-WAVE DISPERSION J2 0 tends to make the magnetizations in the two films to In this section we derive the spin-wave dispersion rela- lie at 90° to one another. As will be shown later, in some tions for the trilayer structure which are used to interpret the systems the biquadratic coupling is sufficiently large and BLS and FMR data. The calculation is based on the torque dominates in the determination of the equilibrium configura- equations of motion for the continuous magnetizations of the tion of the magnetizations, having a marked influence on the two magnetic films. Similar calculations have been made by spin-wave frequencies. several authors.9­20 The novelty here is the introduction of Finally, the dipolar energy has surface and volume con- the biquadratic exchange coupling and the use of an im- tributions. The surface contribution, also called demagnetiz- proved approximation for the dipolar coupling, which makes ing energy, is given by the results valid for relatively thick magnetic layers. The equation of motion for the magnetization of film i is E 2 demag 2 diMiy, 8 written as i d which has the same form as the surface anisotropy energy i , 12 5 . The volume contribution is associated with the magnetic dt Mi iMi Heff field created by the spatial variations of the small-signal where i gi B / is the gyromagnetic ratio i/2 magnetization. Since it contributes to the dynamics of the 2.8 GHz/kOe for g (i) i 2 and Heff is the effective field act- system but not to the equilibrium configuration, we leave its ing on Mi . All fields and magnetizations are decomposed discussion for Sec. III. into a static part and a small-signal dynamic component. If the external magnetic field is applied in the plane of For each film we use a Cartesian coordinate system the films and the combination of the surface anisotropy and xiyizi , obtained from the one with the axes in the 100 demagnetizing effects has an easy-plane character, the two directions, by rotation about the y axis so that the zi axis magnetizations are confined to the xz plane. In this case coincides with the equilibrium direction of the magnetization Miy 0 so that the relevant energy per unit area to determine Mi , as shown in the lower panel of Fig. 1. Hence, the mag- the equilibrium configuration is, from Eqs. 1 ­ 8 : netization in film i can be written as 2 1 M y , 13 E d i i x imixi imiy z iM izi i M iH0 cos i H sin2 2 i i 1 4 K1 where it is assumed that mix , m . The transformation i iy M izi from the original variables is given by K i i u cos2 i u J1 cos 1 2 Mix Miz sin cos i i mixi i , 14 J2 cos2 1 2 . 9 Miy miy , 15 In simple situations the equilibrium configuration can be M cos sin obtained analytically by equating to zero the derivatives of iz M izi i mixi i . 16 the energy in Eq. 9 with respect to 1 and 2 . However, in Likewise, the effective field is written as more general situations this leads to transcendental equations i which cannot be solved analytically. In the cases studied in Heff x ihix y h . 17 i iy z iHizi J. Appl. Phys., Vol. 84, No. 2, 15 July 1998 Rezende et al. 961 The effective fields corresponding to the Zeeman, anisot- J1 ropy, bilinear, and biquadratic exchange energy contribu- hex ix m cos i d jx j 1 2 tions E iM 1M 2 are given by 2J2 H i mjx cos 2 1 2 M E d j i /di . 18 iM 1M 2 2J The calculation of the volume dipolar magnetic field is 22 mix sin2 1 2 , 26 more involved and requires some approximations in order to diMi i be carried out analytically, as will be shown later. Since the J 2J spin-wave frequencies are determined by the linearized equa- hex 1 2 iy mjy mjy cos 1 2 , 27 tions, in the transformation of the energy expressions 2 ­ 9 diM1M2 diM1M2 to the new variables 13 and 17 , only terms quadratic in J1 2J2 small-signal components have to be kept. Furthermore, in the Hex iz cos cos2 i d 1 2 d 1 2 . 28 calculation of the effective fields, only terms linear in small iM i iM i quantities in the x The treatment of the dipolar magnetic field is far more i and y i components and constant in the zi components need to be retained. Note also that it is not nec- involved and requires several approximations to be carried essary to expand the M out analytically. This field does not exist when the magneti- iz components into m , and m , as i ixi iyi prescribed in Ref. 12. It is simpler to evaluate the derivatives zation is uniform and lies in the film plane therefore it was of E not considered in the calculation of the equilibrium configu- with respect to M iz and enter the equation of motion i ration. However, when the magnetization deviates from equi- with the corresponding zi components of the effective fields. librium, its spatial variation creates uncompensated magnetic From the Zeeman energy we obtain only one relevant dipoles which in turn generate a magnetic field h that obeys field component, Maxwell's equations. This field can be calculated using the magnetostatic approximation, h 0, and defining a mag- 1 E Hz z netic potential through h . This potential satisfies a iz H i d 0 cos i H . 19 i M izi Poisson equation: 2 4 *M, 29 The contributions from the cubic anisotropy energy to the effective field are and can be obtained by standard boundary value problems methods, thus providing both surface and volume contribu- K i tions to the dipolar field. Gru¨nberg3 has worked out the prob- hac 1 ix 3 sin2 2 lem for two parallel magnetic films with the full boundary i M2 mix i 2 , 20 i i conditions from Maxwell's equations but without other inter- actions. The dipolar field couples the excitations in the two 2K i films, giving rise to in-phase and out-of-phase surface mag- hac 1 iy M2 miy , 21 non modes, also called, respectively, acoustic and optic i modes, propagating perpendicularly to the applied field. K i However, the presence of the exchange interaction between Hac 1 the films complicates the problem considerably. In order to iz sin2 2 i M i . 22 i treat the dipolar interaction in the same context as the other fields we follow the approach of Cochran et al.12 in consid- The relevant components of the uniaxial anisotropy field ering the field produced by one film and neglecting the effect are of the boundary conditions on the second film. Assume a single semi-infinite film of thickness d with surfaces in the 2K i xz plane located at y d/2 with the static field applied hau u i ix m , 23 i M2 sin2 i u ix along the z direction. Consider also that a magnetostatic sur- i i face wave propagates in the x direction, so that the deviation 2K i of the magnetization from equilibrium is given by Hau u i iz cos2 , 24 i M i u M i x x M xeiqx, My x Myeiqx. 30 Solution of Eq. 29 with Eq. 30 yields the following and from the surface anisotropy energy, the only component dipolar field components in the three regions of interest.12 is i Inside the film 2K i hx 2 Mxe qd/2 eqy e qy 4 Mx has s iy d 2 miy . 25 iM 2 iM i ye qd/2 eqy e qy eiqx, 31 h Finally, from the bilinear and biquadratic exchange en- y 2 iM xe qd/2 eqy e qy ergies we obtain the following components: 2 Mye qd/2 eqy e qy eiqx. 32 962 J. Appl. Phys., Vol. 84, No. 2, 15 July 1998 Rezende et al. ii To the right of the film (y d/2) H 1 H ac 1 1 H0 cos 1 H cos2 1 h 4 3 cos 4 1 Hau x 2 M x iM y e qd/2 eqd/2 e qy iqx , 33 1 1 u Has 4 M 1 1 qd1/2 D 1 q2 hy 2 iMx My e qd/2 eqd/2 e qy iqx . 34 H 1 ex1 cos 1 2 iii To the left of the film (y d/2) 2H 1 ex2 cos2 1 2 , 42 hx 2 Mx iMy e qd/2 eqd/2 e qy iqx , 35 H 2 2 2 Hex1 2 M 1qd1 1 qd2/2 e qs 2Hex2 cos 1 hy iMx My e qd/2 eqd/2 e qy iqx . 36 2 , 43 Clearly the nature of the dipolar field introduces a spatial H 1 3 H0 cos 1 H Hac cos 4 1 variation with y that complicates the problem and requires 1 1 further approximations. Cochran et al.12 expand the expo- Hau cos 2 1 u nential functions in the small parameters qy and qd to first 2 M order, simplifying the equations considerably. It turns out 1qd1 cos2 1 H D 1 q2 that in situations of interest this approximation is not satis- H 1 1 ex1 cos 1 2 2Hex2 cos 2 1 2 , 44 factory. Here we assume that the small-signal magnetizations 2 do not vary along y and replace the dipolar fields by suitable H4 Hex1 cos 1 2 2 M1qd1 1 qd2/2 e qs averages17 along y: cos 2 1 H cos 2 H 2Hex2 cos 2 1 2 , 1 d 45 1/2 hdip 1 d hdip y dy, 37 1 d1/2 H5 2 M1qd1 1 qd2/2 e qs cos 2 H , 46 1 d H6 2 M1qd1 1 qd2/2 e qs cos 1 H , 47 1/2 s d2 hdip 2 d hdip y dy. 38 2 d1/2 s and G1­ G6 are given by the same expressions as H1­ H6 with 1 2. Note that D(i) is the intralayer exchange stiffness Integration of Eqs. 31 ­ 36 yields the average dipolar constant for film i, which was introduced in the usual field in film 1, for arbitrary direction of the applied in-plane manner.9,17 The upper and lower signs in Eqs. 46 and 47 field H: correspond to the Stokes and anti-Stokes frequency shifts, respectively. As is well known1,17,19 these are equal in the hdip 1x 4 M 1x 1 1 e qd1 /qd1 ferromagnetic phase but are somewhat different in the anti- 2 iM ferromagnetic and spin-canted phases. Note that, besides the 2y M 2x 1 e qd1 generalization for arbitrary field direction and inclusion of 1 e qd2 e qs/qd1 cos 1 H 39 biquadratic exchange, expressions 41 ­ 47 differ from those in Ref. 12 in the dependence on the parameters qdi and hdip 1y 4 M 1y 1 e qd1 /qd1 2 iM 2x M 2y qs, and are identical only in the limit of vanishing qdi and qs. As a result, our treatment provides a better approxima- 1 e qd1 1 e qd2 e qs/qd1 . 40 tion for the dipolar field, and, as will be shown in Sec. V, is applicable to film thicknesses of several hundred angstroms. The final expressions are obtained by introducing the trans- The solutions of Eq. 41 are found by requiring that the formation 14 ­ 16 in 39 and 40 and expanding the ex- secular determinant vanishes. This leads to magnetic excita- ponential functions in the parameters qdi 1. Similar expres- tion frequencies which are given by the zeroes of the follow- sions follow for the dipolar field in film 2. Using the ing: resulting dipolar fields and the other field components 19 ­ 28 in Eq. 12 , we obtain the appropriate equations of mo- 4 tion for the small-signal magnetizations in the two films. 2 2 a 2 b c 0, 48 Assuming the time variation exp(i t), where is the angular 1 2 frequency, and retaining only terms to first order in small where quantities we obtain 2 a G2H4 G4H2 G5H5 G6H6 / 1 2 H1H3 / 2 i / 2 1 H1 iH5 H2 G1G3 / 1, 49 H3 i / 1 H4 iH6 m1x1m1y iG 0, b G3G5H2 G1G4H5 G1G6H4 G2G3H6 / 1 5 G2 i / 2 G1 m2x2 G4 iG6 G3 i / 2 m G 2y 6H2H3 G4H1H6 G5H1H4 G2H3H5 / 2 , 41 50 where and J. Appl. Phys., Vol. 84, No. 2, 15 July 1998 Rezende et al. 963 c G 2 1G6H3H5 G1G3H1H3 G5G6H5H6 2H2ex Hu Hex 4 Meff sin2 Hex G 5G6H2H4 G3G5H1H6 G2G4H2H4 G 4 M 2 H 2G3H2H3 G1G4H1H4 G2G4H5H6 . 51 eff Hex Hu Hex ex 2H For any given applied field, Eq. 48 has two real solu- ex 8 M eff Hu sin2 Hu 4 M eff . tions, corresponding to the acoustic and optic modes. Evi- 54 dently, in order to find the frequencies one must first Finally, when the field is increased beyond the critical determine the equilibrium angle, as indicated previously. value Hc 2Hex Hu , the magnetizations become aligned in Note that the FMR technique probes the q 0 modes and the z direction. In this saturated, or ferromagnetic, phase, the since they are not influenced by the dipolar coupling the FMR frequencies are calculation simplifies considerably. In this case H5 H6 2 G 5 G6 0 so that the term linear in in Eq. 48 van- H0 Hu H0 Hu 4 Meff , 55 ishes. This allows the FMR frequencies to be determined analytically. For 1 2 , we have 2 H0 Hu 2Hex H0 Hu 2Hex 4 Meff , 2 56 a0/2 a0/2 2 c0, 52 which are, again, in agreement with Ref. 15. This result shows that the difference between the fields for resonance of where the acoustic and optic modes in the ferromagnetic phase, observed at fixed frequency, yields a direct measurement of a0 G2H4 G4H2 H1H3 G1G3 q 0 , the exchange coupling field. In the case of the BLS technique the field is fixed and one observes the frequency shifts of the c two modes. One difficulty encountered in both FMR and 0 G1G3H1H3 G2G4H2H4 G2G3H2H3 BLS is that the optic mode intensity is much smaller than G1G4H1H4 q 0 . that of the acoustic mode. In fact, as is well known,1,14 for films of identical materials the intensity of the optic mode in Let us apply the result 52 to the simple case of a the ferromagnetic phase is theoretically zero. Fortunately, trilayer having two identical magnetic films, coupled through experimental films are always somewhat different from each a bilinear antiferromagnetic exchange. Consider further, that other and present various magnetic phases, so that although the in-plane anisotropy is uniaxial with easy axis in the z the optic mode is weak it can be observed in many situations. direction and that the external field H0 is applied in this direction. In this case there are three equilibrium phases de- IV. EXPERIMENTAL DETAILS AND SAMPLES pending on the field value, as determined by minimizing 9 with vanishing A. Techniques H , u , K1 , and J2 . For increasing field, the first phase is in the range 0 H0 Hu(2Hex Hu) HSF , The experiments were carried out with three standard where Hex is the absolute value of the antiferromagnetic bi- techniques, namely magneto-optic Kerr effect magnetometry linear exchange field and Hu is the uniaxial anisotropy field MOKE , Brillouin light scattering spectroscopy BLS , and for both films. In this phase the magnetizations are aligned ferromagnetic resonance FMR . Some samples were also antiferromagnetically, i.e., 1 0 and 2 180°, and the investigated by magnetoresistance MR measurements. All FMR frequencies for the optic and acoustic modes obtained samples studied are made of single crystal films as deter- from 52 are mined by X-ray diffractometry. The MOKE setup employs a He­Ne laser modulated in 2 amplitude at 50 kHz by an elasto-optic modulator placed 2 2 H0 Hu Hu Hex 2Hu 4 Meff between crossed polarizers. Magnetic hysteresis loops were measured in the longitudinal Kerr effect configuration, with H20 2Hu 4 Meff 2Hu 4 Meff 4Hex the light polarized along the dc magnetic field in the film plane. H2ex 4 Meff 2 1/2, 53 The BLS measurements were carried out in the back- scattering geometry,1,19,20 using a Sandercock model tandem where 4 Meff 4 Ms Has . This result is identical to that Fabry­Perot interferometer with active stabilization in a (2 obtained previously by Wigen et al.15 As the field is in- 3)-pass configuration. The light source was a single-mode creased beyond the spin­flop critical value HSF , the magne- stabilized argon ion laser operating at 5145 Å, with incident tizations become canted with angles 1 and 2 , power in the range 60­100 mW. Light detection was made where sin H0 /(2Hex Hu) . As the field increases in the by photon counting using a cooled EG&G photodetector spin­flop phase the magnetizations rotate towards the z axis with 40% quantum efficiency and an average dark noise of and the FMR frequencies vary with field as 1.5 cps. The spectra were stored in 256 channels, each with a 964 J. Appl. Phys., Vol. 84, No. 2, 15 July 1998 Rezende et al. gate length of 1 ms. Several hundred to a few thousand in- terferometer scans were used to record the spectra, depend- ing on the signal intensity. All measurements were done at room temperature with the sample between the poles of an electromagnet, with the field in the film plane and perpen- dicular to the incidence plane of the light beam. The sample was mounted on a goniometer to allow for two independent rotations: rotation around the normal as to vary the field direction in the film plane; rotation around the field direction to vary the incident angle and thus the spin-wave wave number q 2kL sin . Magnons with wave numbers in the range 0.4 105 cm 1 q 2.1 105 cm 1 could then be probed in a magnetic field varying from 0 to 6 kOe. The FMR measurements were carried out with a home- made microwave spectrometer employing a sweep oscillator with frequency stabilized at the cavity resonance. The spec- tra were obtained by sweeping the field at fixed frequency and monitoring the derivative of the absorption lines pro- vided by field modulation at 1 kHz with Helmholtz coils mounted on the cavity walls. In order to change the fre- quency several retangular cavities were constructed to oper- ate in the TE102 mode in the X and K microwave bands. All measurements were made at room temperature with the field applied in the film plane. In order to investigate the in-plane magnetic anisotropy the sample is rotated about the normal to its plane, maintaining the field in the plane. B. Samples FIG. 2. BLS spectra for several values of the external field H0 applied All studies reported in this paper were carried out along a hard magnetization axis, observed in Ni81Fe19 200 Å /Cu 25 Å / with single-crystal samples prepared at the IBM Alman- Ni81Fe19 200 Å grown onto 110 MgO. The grey scale indicates the mag- den Research Center. Two sets of trilayer samples were stud- netic field increase. ied: Ni81Fe19(d)/Cu 25 Å /Ni81Fe19 d and Fe 40 Å /Cr(s)/ Fe(40 Å). The films were grown by magnetron sputter depo- samples grow on the 100 plane with the Fe 100 along the sition in an ultrahigh vacuum chamber equipped with six MgO 110 direction. All samples have the same Fe layer 2-in. dc magnetron sputtering sources. The base pressure thickness, d 40 Å. Initial characterization of the coupling prior to deposition was typically 2 10 9 Torr and the sput- was made with a Cr wedged sample, with 0 s 70 Å. Then ter pressure was usually 3 10 3 Torr Ar. several samples were prepared with uniform Cr thickness The NiFe, or permalloy Py , films have relatively large varying from 5 to 35 Å, a range that corresponds to the first thicknesses, d 200 and 300 Å, and provide an excellent two antiferromagnetic peaks. The 100 Fe/Cr/Fe samples room for testing the spin-wave dispersion calculated with are used here to study the effect of the biquadratic coupling various approximations for the dipolar field. The Py/Cu/Py on the equilibrium configuration and on the spin-wave dis- trilayers were deposited onto polished, chemically cleaned persion. single-crystal 100 MgO, 110 MgO, and 0001 sapphire substrates, coated with Fe 5 Å /Pt 5 Å /Cu 50 Å buffer lay- ers. Symmetry and protection of the top Py layer were pro- V. APPLICATION TO MODERATELY THICK FILMS: vided by a 50 Å thick Cu overcoat layer. The thicknesses SPUTTERED Py/Cu/Py TRILAYERS were primarily determined from the deposition time for each In this section we present BLS data only on 100 and layer. The deposition rates ( 2 Å/s) were determined from 110 trilayers of Py(d)/Cu 25 Å /Py(d) with d 200 and the measured thicknesses of thick ( 1000 Å) calibration 300 Å, compare with the spin-wave theory developed in Sec. films grown along with multilayers. The values of the layer III, and demonstrate the reliability of the approximations thicknesses determined by this method were cross checked made in the calculation of the dipolar field. This is possible by x-ray reflectivity on selected samples. because with a Cu spacer layer 25 Å thick the two Py films The Fe/Cr/Fe trilayers were grown on MgO 100 sub- have negligible exchange interaction and then the nature of strates. Initially a Cr seed layer, at least 100 Å thick, was the coupled modes relies on the dipolar interaction.10 deposited on the substrate at temperatures ranging up to The BLS data, as well as the FMR, reveal that while the 525 °C in order to establish epitaxy. The subsequent Fe/ 100 samples are isotropic in the plane, as expected from the Cr/Fe layers were deposited at 150­180 °C and were capped small cubic magnetocrystalline anisotropy of Py (Hac with a thin Cr layer. As previously reported,21,22 the Fe/Cr/Fe 5 Oe), the 110 samples develop a strong uniaxial in- J. Appl. Phys., Vol. 84, No. 2, 15 July 1998 Rezende et al. 965 FIG. 4. The same as Fig. 3 a in isotropic 100 Ni81Fe19 200 Å /Cu 25 Å / Ni81Fe19 200 Å . Lines are results of calculations with Eq. 48 solid , theory in Ref. 12 dashed , and theory in Ref. 3 dotted . magnetic field of 1 kOe. For simplicity, the in-plane field angle is defined here with respect to the hard magnetiza- tion axis. In contrast to the flat isotropic behavior observed in FIG. 3. Magnon frequencies for q 1.73 105 cm 1 vs external field H MgO 100 films, the samples grown on MgO 110 exhibit a 0 , applied along the a hard magnetization axis ( 0) in Ni81Fe19 200 Å /Cu 25 Å /Ni81Fe19 200 Å grown onto 110 MgO. Symbols twofold symmetric response, with a maximum at 90°, are the BLS data: solid triangles for the lowest-order volume mode, open where the field is applied parallel to an easy magnetization circles for the surface optic mode, and open triangles for the surface acoustic direction. The results above were confirmed by room- mode. Lines are results of calculations with Eq. 48 solid , and theory in Ref. 12 dashed . b Frequency shift for the same magnon wave number temperature angle-dependent FMR experiments. The source and H0 1 kOe as a function of the in-plane field angle with respect to the of the uniaxial in-plane anisotropy is probably the stress in- hard axis. duced by lattice mismatch.21 Similar spectra were observed in the samples grown on MgO 110 with the field applied along an easy magnetiza- plane anisotropy.23 Figure 2 shows representative spectra in tion axis and in the samples grown on MgO 100 . However, the low-field low-frequency region for the sample with d in these cases all frequencies increase monotonically with 200 Å grown on MgO 110 , with the field applied parallel increasing field. This is seen in the dispersion relations mea- to a hard magnetization axis. All spectra were obtained with 500 scans, with laser power 80 mW at an angle of incidence of 45° corresponding to a scattering in-plane magnon wave number of 1.73 105 cm 1. The spectra display intense acoustic and optic surface modes, with equal Stokes and anti-Stokes shifts. These modes have an apparently intrigu- ing behavior with increasing field. The two inelastic scatter- ing peaks first approach each other for 0 H0 200 Oe, then they split with increasing field. This separation reaches a maximum at about 750 Oe and decreases monotonically as the frequencies of both modes increase at higher field values. Not shown in Fig. 2 is the volume mode line about 26 GHz apart from the laser line, well outside the free spectral range. The behavior above is not observed when the field is applied parallel to an easy magnetization axis and, as discussed later, is due to a large uniaxial anisotropy characteristic of the samples grown on MgO 110 . The measurements of the frequency shift versus magnetic field are shown by the sym- bols in the upper panel of Fig. 3. The calculated dispersion relations represented by the lines will be discussed later. The FIG. 5. The same as Fig. 4 in 110 Ni lower panel in Fig. 3 shows the frequency shifts as the 81Fe19 300 Å /Cu(25Å)/ Ni81Fe19 300 Å , with the field applied along an easy magnetization axis sample is rotated around the normal to the film plane, in a ( 90°). 966 J. Appl. Phys., Vol. 84, No. 2, 15 July 1998 Rezende et al. sured in the isotropic 100 sample with d 200 Å, shown in Fig. 4, and in the 300 Å 110 sample with H0 along the easy axis shown in Fig. 5. In order to demonstrate the importance of the dipolar interaction in the dynamics of the system we show in Fig. 4 the comparison of the data for the 100 Py 200 Å /Cu 25 Å /Py 200 Å sample with three theoretical results. The dot- ted line represents a fit with the calculation of Gru¨nberg3 which treats the dipolar coupling exactly in the limit of van- ishing exchange and anisotropy interactions. The parameters used for the fit are the nominal magnetic and spacer layers thicknesses, saturation magnetization 4 M 9.0 kG, spec- troscopic factor g 2.15 and spin-wave wave number q 1.73 105 cm 1. The solid lines represent the results of our calculation, Eq. 48 , using the same previous parameters and vanishing exchange and anisotropy fields. The volume mode curve is obtained by properly including anisotropy to the usual spin-wave expression, using an intralayer exchange stiffness D 1.9 10 9 Oe cm2. We conclude that for this FIG. 6. BLS frequency shift vs magnon wave number q in 100 magnetic layer thickness and wave number, corresponding to Ni81Fe19 300 Å /Cu 25 Å /Ni81Fe19 300 Å . Lines and symbols are as in a product qd 0.34 1, our calculation treats the dipolar in- Fig. 4. teraction quite well based on its agreement with the data and Gru¨nberg's results. The same is not true for the calculation sent, respectively, the calculations of Ref. 3, ours and the one of Cochran et al.,12 represented in Fig. 4 by the dashed lines. in Ref. 12. As clearly seen, all calculations give the same While the two calculations give the same result for the result for the vanishing wave number because the dipolar and acoustic mode, which does not depend on the coupling be- the intralayer exchange interactions vanish in this limit. tween the two magnetic layers, they depart from each other However, as q increases, the three calculations depart from considerably for the optic mode, which does depend on the each other. While our calculation agrees quite well with ex- coupling. periments, the results of Ref. 3 depart slightly from the data Note that the result obtained with the approximations in because it does not take into account the intralayer exchange. Ref. 12 becomes worse as the optic mode frequency de- On the other hand, the results of Ref. 12 depart considerably creases. Again this results from the crude approximations from the data for high q values because the approximations made in the dipolar field expansions, since the lower the made in the dipolar field calculation are not satisfactory in frequency the more important the dipolar energy is as com- this range. Note, however, that since the important parameter pared to the Zeeman contribution. This is also seen in the in the dipolar field expansion is the product qd, the calcula- comparison between the data for the anisotropic 200 Å tion in Ref. 12 would be quite satisfactory in the high q sample on MgO 110 in Fig. 3 and the calculated dispersion range for film thicknesses d 50 Å. relations. Our result solid line provides a good fit to the data for all modes using the parameters 4 M 9.0 kG, Hau 0.55 kOe, g 2.1, and D 2 10 9 Oe cm2. However, VI. BILINEAR AND BIQUADRATIC EXCHANGE IN the approach in Ref. 12 dashed line with the same param- 100... Fe/Cr/Fe TRILAYERS eters departs from the data for the optic mode, especially in Having established that the spin wave calculation pre- the low frequency region. Note that we do not show in Fig. 3 sented in Sec. III treats accurately the dipolar interaction the dispersion calculated with Gru¨nberg's theory because it between two magnetic layers, we now apply that calculation does not include the anisotropy energy. The departure be- to investigate the exchange coupling in several sputtered tween the theoretical approximations in Ref. 12 and data 100 Fe/Cr/Fe trilayers. Actually, for thicknesses of the fer- becomes larger for thicker magnetic layers. This is seen in romagnetic layer smaller than about 100 Å, as is the case of Fig. 5 which shows the data for the 300 Å sample on MgO the Fe/Cr/Fe samples studied here, a higher-order approxi- 110 and the dispersion relations calculated with our result mation to the dipolar field is shown to play an important role. solid line and with the approximations in Ref. 12 dashed As will be shown, it is also essential to include the effect of lines . the biquadratic exchange interaction. Since the dipolar interaction becomes more important Fe/Cr films can be prepared in single-crystal form by with increasing qd, it is interesting to compare the three several techniques, such as molecular beam epitaxy, electron theoretical results with the spin-wave dispersion as a func- beam deposition, and sputtering.1,16 Since the nature of the tion of the wave number. This is shown in Fig. 6 for the coupling between the magnetic layers depends on the chemi- 100 Py 300 Å /Cu 25 Å /Py 300 Å trilayer in a constant cal composition and on the details of the microstructure, in-plane field H0 1.0 kOe. The wave number is varied by Fe/Cr systems grown by different methods have become a changing the scattering incidence angle in the range 15° prototype for studies of coupling in magnetic multilayers. In 60°. Again, the dotted, solid, and dashed lines repre- particular, the peculiar biquadratic coupling has been ob- J. Appl. Phys., Vol. 84, No. 2, 15 July 1998 Rezende et al. 967 FIG. 7. Saturation field measured with MOKE magnetometry in a Fe/Cr/Fe wedge as a function of the Cr thickness Ref. 26 . The solid line is a guide to the eyes. served in Fe/Cr/Fe trilayers by several authors.24­28 Here we show that the static and dynamic properties of this system FIG. 8. Normalized magnetization vs external field H0 applied along 001 are very sensitive to the biquadratic interaction. This fact is ( H 0), in 100 Fe 40 Å /Cr 11 Å /Fe 40 Å . Open circles are MOKE used to measure both the bilinear and biquadratic couplings data. Lines are calculated. b Calculated equilibrium magnetization angles as a function of the spacer layer thickness in sputtered 100 1 and 2 vs H0 . Fe 40 Å /Cr(s)/Fe(40 Å). In order to determine the overall behavior of the cou- pling as a function of spacer thickness we have done easy fields M1 and M2 are aligned nearly opposite to each other axis MOKE magnetometry measurements in a wedged and perpendicular to H0 , in a canted-spin configuration. As trilayer structure. Figure 7 shows a plot of the saturation field the field increases and the Zeeman energy becomes more as a function of the Cr layer thickness. The saturation field is important, the magnetizations gradually rotate towards the defined arbitrarily as the field at which the moment reaches field. Saturation occurs at fields larger than a critical value 75% of the saturation value. Note that the coupling reaches HSAT . For identical magnetic films, this value can be ob- the first antiferromagnetic AF maximum at about s tained analytically from the condition of vanishing derivative 10 Å, crosses to ferromagnetic at s 15 Å, crosses back of Eq. 9 with respect to 1 2 , and is given by to AF at s 25 Å, and reaches the second AF peak at s 29 Å. Since this behavior is primarily determined by the HSAT 2 Hex1 4Hex2 Hac , 57 bilinear exchange parameter J1 , it is of interest to find out where the sign occurs when the field is applied along how the biquadratic exchange constant J2 varies as J1 an easy hard magnetization axis. The parameters above changes with s. In order to extract accurate values for J1 and lead to a critical field HSAT 1.49 kOe. Note that the pres- J2 we have studied three trilayer samples having uniform Cr ence of a small biquadratic exchange does not change the spacers with thickness s 11, 15, and 25 Å. nature of the equilibrium phase dashed line in Fig. 8 a . A positive J A. MOKE and FMR in 100... Fe 40 Å.../Cr 11 Å.../Fe 40 Å... 2 changes the curvature of the magnetization curve near the critical field and slightly increases the value of Consider first, the 100 Fe/Cr/Fe sample with Cr thick- HSAT . Furthermore, notice that in the case of Fig. 8 HSAT is ness s 11 Å, a value close to the first AF maximum. The no longer given by Eq. 57 when J2 0, since saturation is initial characterization is made with MOKE magnetometry. reached through a first-order transition. Figure 8 a shows data open circles measured with the field A remarkably distinct phase diagram occurs when the along the easy 001 axis. The solid line represents a theo- field H0 is applied along the hard 101 axis, i.e., H 45°. retical fit with the calculation described in Sec. II, using the Figure 9 shows the equilibrium angles and the magnetization following parameters: 4 M 19.5 kG, Hac 2K1 /M versus field for this case, where three distinct phases are 0.57 kOe, Hu Hs 0, Hex1 J1 /d1M 0.89 kOe (J1 observed. At fields below a certain critical value of 0.5 kOe, 0.55 erg/cm 2), Hex2 J2 /d1M 0.07 kOe J2 the magnetizations remain close to an antiferromagnetic 0.044 erg/cm2, J2 /J1 0.08 . The corresponding equi- AF alignment along the 001 axis. At this field there is a librium angles 1 and 2 are shown in Fig. 8 b . The arrows sudden transition to a spin­flop SF state. As the field in- represent the evolution of the spacial configuration of the creases above this value, the spins rotate towards the field magnetizations in the Fe films towards saturation. At low and only at 2.63 kOe the system attains saturation. Again, 968 J. Appl. Phys., Vol. 84, No. 2, 15 July 1998 Rezende et al. FIG. 10. Symbols: Variable-frequency FMR measurements in 100 Fe 40 Å /Cr 11 Å /Fe 40 Å with the external field H0 applied along 101 ( H 45°). Solid, dashed, and dotted lines are results of calculations. Inset: FMR spectrum at 9.5 GHz and H 30°. FIG. 9. Same as Fig. 8, with H0 applied along 101 ( H 45°). spin­flop SF for 0.5 kOe H0 2.63 kOe; and saturated or FM for H0 2.63 kOe. The inset shows a typical spectrum at 9.5 GHz and H 30°, where one acoustic mode and one note that the existence of a small positive J2 does not change optic mode resonance are seen. Note that the optic mode the nature of the phases, it only varies the curvature of M vs resonance expected for H0 HSAT is not observed because H0 and changes the values of the critical fields. its intensity in the FM phase is theoretically zero. As seen in There are three independent components in the critical Fig. 10, the behavior of the frequency versus field for both field given in Eq. 57 . It should be possible to determine the acoustic and optic modes is different in each phase. This three field parameters from the simultaneous fittings of the results from the fact that the frequencies of the magnetic two magnetization curves in Figs. 8 a and 9 a . However, excitations depend directly on the configuration of the mag- we find that the parameters that give the best least-square fit netizations in both films, as shown by Eqs. 53 ­ 56 . As in to the 101 data solid line in Fig. 9 a produce a slight Fig. 9 a the dashed line is obtained with Hex2 0, keeping departure between theory and data for the 001 direction the same values for the other parameters. As expected, this solid line in Fig. 8 a . The uncertainty in the value of Hex2 produces a large departure of the optic mode line from the is considerable when its magnitude is much smaller than data because the critical field for the SF­FM transition is Hex1 . very sensitive to the value of Hex2 . One may ask what hap- Better accuracy in the determination of all magnetic pa- pens if we keep Hex2 0 and vary the other parameters to rameters is achieved with BLS and FMR techniques. In order obtain a least-square fit to the data. The result is shown by to obtain reliable fits between theory and experiment it is the dotted line in Fig. 10, obtained with 4 M 19.5 kOe, necessary to measure the variation of some spin-wave quan- Hac 0.53 kOe, Hex1 0.82 kOe, and g 2.1. This shows tity as a function of a convenient parameter. In the case of that although Hex2 is small, it is not possible to fit the data to FMR one can measure the field for resonance at constant the theory without inclusion of the biquadratic exchange en- microwave frequency as a function of the azimuthal in-plane ergy in the calculation. The estimated maximum error in the angle H as the sample is rotated in its plane.1,16,29,30 Alter- values of the parameters extracted from the fit is 3% based natively, it is possible to employ several microwave cavities on the visual departure of the curves from the least-square fit. to vary the frequency discretely and obtain the q 0 spin- Another way to obtain data with the FMR technique is wave dispersion relations.13 However, in both cases it is nec- by keeping the frequency fixed and measuring the resonance essary to observe both acoustic and optic modes in order to field for the acoustic and optic modes as the sample is rotated extract accurate values for the exchange coupling constants. about its normal, maintaining the field in the plane. The mea- Figure 10 shows the dispersion relations for the q 0 sured field variations with angle exhibits fourfold symmetry modes with the magnetic field H0 applied along the hard due to the cubic anisotropy of Fe 100 . Figure 11 shows the 101 axis ( H 45°). The symbols represent the data and data obtained with a frequency of 9.5 GHz in one quadrant the lines are theoretical results obtained with Eq. 52 with together with theoretical lines calculated with Eq. 52 . The the same parameters used to fit the MOKE data in Fig. 9 a . solid line is the least-square fit obtained with parameters As the equilibrium state, the dispersion relations are charac- almost identical to the previous values, namely: terized by three distinct phases: AF for 0 H0 0.5 kOe; 4 M 19.5 kOe, Hac 0.57 kOe, Hex1 0.89 kOe, Hex2 J. Appl. Phys., Vol. 84, No. 2, 15 July 1998 Rezende et al. 969 FIG. 12. Comparison between FMR data at several frequencies and theory with the same parameters used to obtain the solid lines in Fig. 11. 11 with the parameters: 4 M 19.0 kG, Hac 0.55 kOe, FIG. 11. Resonance field vs in-plane field angle H in 100 Fe 40 Å /Cr 11 Hex1 0.15 kOe, Hex2 0.05 kOe. The exchange fields Å /Fe 40 Å . Open circles are for the measured FMR optic mode and the correspond to the parameters J1 0.09 erg/cm2 and J2 open triangles for the acoustic mode. The solid lines are the best theoretical fits obtained. The solid lines in a and b were obtained with the exchange 0.03 erg/cm2. This relatively large biquadratic coupling is field parameters (Hex1 ,Hex2) ( 0.89 kOe, 0.07 kOe). a the dashed dot- responsible for the two first-order phase transitions that pro- ted line was obtained with Hex2 0.07 kOe and Hex1 1.0 kOe duce the jumps in the data. As shown in Fig. 13 b , at H0 ( 0.80 kOe). b the dashed dotted line was obtained with Hex1 0.1 kOe the alignment changes from AF to nearly 90°; at 0.89 kOe and Hex2 0.14 kOe 0 kOe . H0 0.22 kOe it changes from 90° to FM alignment. 0.07 kOe, and g 2.1. As can be seen by the dashed and dotted lines in Figs. 11 a and 11 b , the resonance field increases with increasing Hex1 and Hex2 . In order to show the consistency of the set of parameters extracted from this fit, we show in Fig. 12 the comparison between data taken at four other frequencies with the theoretical prediction ob- tained with the same parameter values. Even the curious triple peaked shape of the data at 19 GHz is very well repro- duced. The dispersion relations obtained with these param- eters for q 0 are also in excellent agreement with the BLS data for this sample. B. 100... Fe 40 Å.../Cr 15 Å.../Fe 40 Å... When the sample with Cr thickness s 15 Å was first investigated it was expected to have a small antiferromag- netic coupling and correspondingly small critical fields for all magnetic phase transitions. However, besides the ex- pected low critical fields, the data furnished by the various techniques displayed surprisingly sudden jump as the field varied. This was soon found to be a result of abrupt phase transitions in the magnetic state produced by the relatively large biquadratic exchange.26,27 Figure 13 a shows MOKE measurements in the field range 0 H0 0.4 kOe, applied along the easy axis. The circles in the inset are MR data, obtained with a standard four-probe technique. Both data FIG. 13. a Open circles: easy-axis MOKE data in 100 Fe 40 Å /Cr 15 Å /Fe 40 Å . Inset: open circles: corresponding magnetoresistance data. without the coercive field are very well fitted by the theoret- Solid lines are fits with Eqs. 11 and 10 , respectively. b Calculated ical lines shown in Fig. 13 a , obtained with Eqs. 10 and equilibrium magnetization angles vs external field H0 . 970 J. Appl. Phys., Vol. 84, No. 2, 15 July 1998 Rezende et al. FIG. 15. Magnon frequencies for q 1.22 105 cm 1 vs external field H0 , applied along an easy magnetization axis in 100 Fe 40 Å /Cr 15 Å /Fe 40 Å . Symbols are BLS data: solid triangles for the acoustic mode and open circles for the optic mode. Solid lines are results of calculations with Eq. FIG. 14. Measured BLS spectra in 100 Fe 40 Å /Cr 15 Å /Fe 40 Å for 52 with the same parameters as in Fig. 13. The dotted lines in a were four different values of the applied field, corresponding to different regions obtained with Hex2 0. The dotted lines in b were obtained without the of the magnetization curve shown in Fig. 13. a AF phase at H volume contribution to the rf dipolar field. 0 0; b near 90° phase at 0.220 kOe; c mixed-domain state at 0.225 kOe; d saturated phase at 0.230 kOe. istic parameters used in Fig. 13. Again, there is excellent agreement between theory and data. Note that the inversion Steps due to first-order phase transitions are also ob- of the relative positions of the acoustic and optic modes at served in the spin-wave frequency versus field data obtained the second transition (H0 0.22 kOe) is predicted by theory. with BLS. Figure 14 shows spectra obtained at several fields The dotted lines in Fig. 15 a represent the frequencies cal- applied along the 001 axis in the backscattering geometry culated without the contribution of the biquadratic exchange with an incidence angle of 30°, corresponding to a magnon field in Eq. 48 . Actually, Hex2 has been kept in the energy wave number q 1.22 105 cm 1. The spectra were mea- expression 9 , to preserve the correct ground states. The sured with laser power of 100 mW and 2000 interferometer most pronounced effect of the biquadratic coupling in this scans. Only the anti-Stokes peaks are shown in Fig. 14. In case is to shift the frequency of the optic mode, downwards the AF phase, shown in Fig. 14 a at zero field, the two in the AF and FM phases and upwards in the 90° central magnon modes are separated in frequency by about 2 GHz, region. In order to further investigate the effect of the dipolar the optic mode having higher frequency than the acoustic coupling in the dynamics, we show in Fig. 15 b by dotted one. When the field is increased above 0.1 kOe the system lines the frequencies calculated with the same parameters, acquires the near 90° alignment and the separation between but neglect the volume contribution to the dipolar field. The the acoustic and optic modes suddenly increases. The two poor fit thus obtained is an indication of the important role peaks at H0 0.220 kOe Fig. 14 b are 5 GHz apart. At played by the dipolar interaction, even in these thinner films. H0 0.230 kOe the system reaches the FM state, and the We finally mention that the calculated resonance field versus acoustic mode frequency becomes higher than the optic in-plane azimuth angle, with the same parameters used to mode frequency. It is interesting to observe that at H0 obtain the solid lines in Fig. 15, is also in excellent agree- 0.225 kOe Fig. 14 c , the peaks corresponding to the 90° ment with FMR data.26 and FM phases are simultaneously present, probably due to the formation of a domains state. The overall behavior of the frequency shift as a function C. 100... Fe 40 Å.../Cr 25 Å.../Fe 40 Å... of the applied field is shown in Fig. 15 a . Open circles solid The last sample studied has a Cr thickness s 25 Å near triangles represent the measured optic-mode acoustic- the second AF­FM coupling transition. Figure 16 shows the mode frequencies. The solid lines are the spin-wave fre- calculated equilibrium angles 1 and 2 upper panel , and quencies calculated from Eq. 48 , with the same set of real- BLS data lower panel , obtained with the field applied along J. Appl. Phys., Vol. 84, No. 2, 15 July 1998 Rezende et al. 971 25 Å, respectively, corresponding to a field Hex1 0.89, 0.15, and 0.036 kOe. On the other hand, the biquadratic exchange parameter does not vary much; J2 0.044, 0.030, 0.024 erg/cm2, or Hex2 0.070, 0.050, and 0.036 kOe for s 11, 15, and 25 Å, respectively. The consistent experimen- tal support to our model calculations, provided by the samples and techniques discussed in this paper, has led us to exploit its richness in further studies. In particular, interest- ing phase diagrams for the equilibrium configuration of the magnetizations can be predicted. Results in this regard will be published elsewhere.31 ACKNOWLEDGMENTS The authors thank K. P. Roche for technical support, X. Bian for helping in some experiments, and P. Kabos for stimulating discussions. The work at UFPE has been sup- ported by CNPq, CAPES, PADCT, FINEP, and FACEPE Brazilian agencies . The work at IBM was partially sup- ported by the Office of Naval Research. 1 See, for example, P. Gru¨nberg, in Light Scattering in Solids V, edited by M. Cardona and G. Gu¨ntherodt, Topics in Applied Physics Vol. 66 Springer, Berlin, 1989 , Chap. 8; M. H. Grimsditch, ibid., Chap. 7; B. FIG. 16. a Calculated easy-axis equilibrium magnetization angles and b Heinrich and J. F. Cochran, in Ultrathin Magnetic Structures, edited by B. corresponding BLS data symbols and fit solid lines in 100 Fe 40 Å / Heinrich and J. A. C. Bland Springer, Berlin, 1994 , Vol. II, Chap. 3; J. Cr 25 Å /Fe 40 Å . R. Dutcher, in Linear and Nonlinear Spin Waves in Magnetic Films and Superlattices, edited by M. G. Cottam World Scientific, Singapore, 1994 , Chap. 6. 2 R. W. Damon and J. R. Eshbach, J. Phys. Chem. 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