PHYSICAL REVIEW B VOLUME 54, NUMBER 5 1 AUGUST 1996-I Brillouin light scattering study of Fe/Cr/Fe 211... and 100... trilayers M. Grimsditch, S. Kumar, and Eric E. Fullerton Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439 Received 15 January 1996 The magnitude of the bilinear and biquadratic interlayer coupling strengths between Fe layers separated by Cr spacer layers is investigated by means of Brillouin light scattering, magneto-optic Kerr rotation, and magnetoresistance techniques. A data analysis scheme, which treats all three data sets on an equal footing, yields self-consistent anisotropy and interlayer coupling parameters extracted independently from the three techniques. The values of the bilinear and biquadratic coupling strengths are compared for simultaneously grown 211 and 100 Fe/Cr samples. The approach not only provides reliable values for the coupling strengths but also highlights the complementarity of these techniques in uniquely determining the magnetic parameters. S0163-1829 96 03630-2 I. INTRODUCTION pling strengths. Since Fe 211 layers possess a strong in- plane uniaxial anisotropy, the presence of biquadratic cou- Many systems consisting of magnetic layers separated by pling cannot be directly determined merely from an observed nonmagnetic layers exhibit oscillatory interlayer coupling 90° alignment of adjacent layers, but, requires quantitative across the nonmagnetic spacer layers.1 The coupling oscil- analysis of the magnetization and spin-wave spectra. We lates periodically between ferromagnetic and antiferromag- have combined magnetization, magnetoresistance MR , and netic AF with increasing spacer-layer thickness in the na- Brillouin light scattering BLS measurements to quantita- nometer scale region. The oscillatory nature of the coupling tively determine both the interlayer coupling constants and is explained by RKKY treatments of the spacer layer, the the anisotropies in a manner similar to that used by Krebs resulting period being inversely proportional to the length of et al.8 for 100 -Fe/Cr/Fe trilayers. We find quantitative spanning vectors which join extremal points of the Fermi agreement between anisotropy and interlayer coupling con- surface normal to the layering direction. In most treatments stants extracted using the three techniques independently. of the RKKY theory, it is assumed that the coupling is The only other determination of biquadratic coupling in a Heisenberg-like of the form J1m1*m2 so that only ferro- uniaxial system we are aware of is by Elmers et al.,9 who magnetic or AF coupling is possible. However, it has been used a novel geometry to study 110 -Fe/Cr/Fe trilayers. observed that when J We focus mainly on the results and analysis of our 1 is small, the magnetic moments of the layers sometimes align at 90° with respect to each other 211 -Fe films and 211 -Fe/Cr/Fe trilayers. Results on 001 e.g., Fe/Cr 001 Ref. 2 or Fe/Al 001 Ref. 3 . This type films and trilayers are presented with little emphasis on the of coupling can be described by introducing a phenomeno- analysis, since they follow closely the methods used for the logical interlayer coupling term J 211 samples. Results for the two orientations are contrasted 2(m1*m2)2, where J2 is the biquadratic coupling constant. This type of coupling has and compared with magnetization results for superlattices. In been attributed to either intrinsic properties of the spacer superlattices, however, there are additional effects originat- layer1,4 or to a variety of extrinsic factors, such as paramag- ing from the surface magnetic layers being exchange coupled netic impurities within the spacer layers4 or dipolar fields to only one neighboring magnetic layer while the interior resulting from rough interfaces.5 For Fe/Cr 100 superlat- magnetic layers are coupled to two neighboring layers. This tices, the presence of biquadratic coupling has been attrib- makes a quantitative comparison difficult. In Fe/Cr 211 su- uted to fluctuations in the Cr layer thickness which average perlattices, this can give rise to two spin-flop transitions out the short-period oscillations in Cr 100 .4 bulk and sometimes also surface when the applied field is Here we present an investigation aimed at studying the parallel to the uniaxial anisotropy axis.10 nature of the coupling in 211 -oriented Fe/Cr samples. Pre- The paper is structured as follows. Section II describes the vious studies of Fe/Cr 211 superlattices have found that the experimental procedures. Section III contains experimental bilinear interlayer coupling oscillates in sign with the same results. Section IV describes the data analysis of the magne- period 18 Å , phase, and strength as similarly prepared Fe/ tization, MR and BLS results, and includes details of the Cr 100 superlattices.6 This rather isotropic behavior has calculation of the spin-wave mode frequencies. Section V been attributed to spanning vectors across a ``lens'' feature compares the results obtained with the different techniques. of the bulk Cr Fermi surface.7 In the superlattice studies, the Section VI presents the results for 001 Fe/Cr/Fe trilayers. interlayer coupling strength was determined from the satura- Finally Sec. VII contains discussions and conclusions. tion field of the magnetic hysteresis loops, and no attempt was made to separate bilinear and biquadratic coupling con- II. EXPERIMENTAL PROCEDURES tributions. Although it is straightforward to determine the period of the oscillatory coupling, a more difficult problem is The 211 -Fe/Cr samples were made by dc magnetron obtaining reliable quantitative values of the interlayer cou- sputtering onto epitaxially polished single-crystal MgO 110 0163-1829/96/54 5 /3385 9 /$10.00 54 3385 © 1996 The American Physical Society 3386 M. GRIMSDITCH, S. KUMAR, AND ERIC E. FULLERTON 54 FIG. 2. SMOKE magnetization loop for H along the hard axis FIG. 1. Schematic diagram showing the orientation of M and of the single 211 -Fe layer. Symbols are experimental points, the H with respect to the Fe crystallographic orientation. The Fe 211 line is the fit described in the text. Parameters determined from the axis is normal to the layer, the hard 1¯11 and easy 01¯1 axes lie fit are given in Table I. in the layer plane. Although not shown in the figure, the magneti- zation is not constrained to lie in plane and forms an angle with trilayer: the squares and circles denote experimental data for the 211 axis. the hard and easy axes, while the solid lines are fits to be described below. The single Fe film exhibits the expected substrates using the same growth procedure outlined for uniaxial anisotropy with the easy axis parallel to the superlattices.6 A 200-Å Cr 211 base layer was grown at Fe 01¯1 axis. When H is applied along the easy axis, the 600 °C. The substrate was then cooled to 180 °C prior to Fe/Cr/Fe trilayer exhibits a spin-flop transition characteristic the growth of a 20-Å Cr layer, and either a 20-Å Fe layer or of a film in which there is a combination of AF coupling and an AF-coupled Fe 20 Å /Cr 11 Å /Fe 20 Å trilayer which uniaxial anisotropy. This can be seen as discrete jumps at were then capped with a 20-Å Cr layer. The samples grow H 1.5 kOe in both the magnetization and MR results. At with the Fe 211 along the surface normal and the in-plane low fields, the Fe layers are AF aligned along the easy axis. 1¯11 and 01¯1 directions parallel to MgO 1¯10 and 001 , Higher fields induce a first-order phase transition in which respectively. Magnetization studies have shown that the 1¯11 and 01¯1 directions are the hard and easy axes, re- spectively. The coordinate system we use is shown in Fig. 1. We define and as the angles that the magnetization (M) and the applied field (H), respectively, make with the hard axis in the plane of the film. is the angle the magne- tization makes with the 211 axis not shown in the figure . 100 -Fe/Cr samples were grown onto MgO 100 substrates simultaneously with the growth of the 211 samples; in the 100 samples the epitaxial orientation is Fe/ Cr 001 MgO 011 . All measurements were done at room temperature. For each technique, both the easy- and hard-axis behavior of each sample was investigated. Magnetic hysteresis loops were measured by both longitudinal surface magneto-optical Kerr effect SMOKE and by SQUID magnetometry. The MR was measured using a standard, four-terminal dc tech- nique. The spin-wave excitations were measured by BLS ex- periments using 250 mW of 5145-Å radiation from an Ar laser. The scattered radiation was analyzed with a tandem Fabry-Perot interferometer11 in 3 2 pass operation. The magnitude of the wave vector q in our experiments deter- mined from the scattering geometry is 0.65 105 cm 1. All techniques were used to study the same films; thereby elimi- nating any sample-to-sample variations. III. EXPERIMENTAL RESULTS FIG. 3. Hard squares and easy circles axis magnetization upper and magnetoresistance lower loops for the 211 -Fe/Cr/Fe Figure 2 shows hard-axis SMOKE results for a 211 - sample. Symbols are experimental points, the line is the fit de- oriented 20-Å Fe film. Figures 3 a and 3 b show the Kerr scribed in the text. Parameters determined from the fits are given in effect and MR results, respectively, for the 211 Fe/Cr/Fe Table I. 54 BRILLOUIN LIGHT SCATTERING STUDY OF . . . 3387 FIG. 4. Magnon frequency of a single 211 -Fe film for the field applied along the hard and easy directions. Symbols are experimen- tal points, the line is the fit described in the text. For clarity the FIG. 5. Spectrum obtained from the 211 -Fe/Cr/Fe sample with hard-axis results have been plotted along the negative field axis. H 0.5 kG along the hard axis. The arrows indicate the magnon Parameters determined from the fit are given in Table I. peaks. The central peak is the unshifted radiation attenuated by 105. the spins switch from being antiparallel along the easy axis to the spin-flop phase in which the spins reorient almost IV. DATA ANALYSIS 90° from the field direction but cant toward it. When H is We have fitted the field dependence of the magnetization, along the hard axis, the Fe layers continuously rotate to satu- MR and BLS results to extract magnetic parameters. The ration. In the quantitative analysis of the magnetization loops values for the magnetization and MR depend on the equilib- we will concentrate on SMOKE as opposed to SQUID re- rium magnetic configuration which can be calculated by sults. Analysis of the SQUID results were complicated by minimizing the total energy of the system. The magnon fre- the difficulty in uniquely separating the contributions from quencies are obtained by calculating the perturbations of the the substrate and/or sample holder from that of the Fe film. layers from their equilibrium state. As a result of the large We found that the magnetic signal from the MgO substrate demagnetizing fields, the equilibrium condition corresponds consisted of both a large diamagnetic response and a super- to the Fe moments in the plane of the film ( 90°). How- paramagnetic signal which arises from impurities which ever, since magnons involve the precession of the moments saturates at 2000 Oe with a moment equivalent to 3 ­4 Å out of the plane, to properly treat the frequencies the three- of Fe. dimensional total energy must be considered. Therefore, we BLS spectra from the single film show a single mode with first derive the most general energy expressions appropriate asymmetric Stokes and anti-Stokes intensities, as expected. to BLS, and then simplify them when fitting the magnetiza- The magnon frequency, for H both along the easy and hard tion and MR data. The energy per unit volume for a single 211 -Fe film, directions, is shown in Fig. 4; symbols are experimental points, lines are the fits to be described in the next section. The difference in the frequencies along the two field direc- tions depends only on the anisotropy. Representative spectra of the trilayer film are shown in Fig. 5; they show two modes resulting from in-phase and out-of-phase oscillations of the two Fe layers. The intensity of the weak mode in Fig. 5 is only about 2% of that of the intense mode. In principle, since the weak mode is antisymmetric, the contribution to the scat- tering cross section of the two layers should exactly cancel. However, we believe that the small attenuation as light traverses the outer layers or small differences in the anisotro- pies of the two layers explains why the mode is observable. The frequencies of the symmetric and antisymmetric modes plotted vs H are shown in Fig. 6. The difference at ``zero'' field between the magnon frequency of the upper mode for H along the hard and easy axes is real; it is due to the finite wave-vector correction, as discussed in the Appen- FIG. 6. Magnon frequencies of the 211 -Fe/Cr/Fe sample for dix. The discontinuity in the frequencies at 1.5 kG with the field applied along the hard and easy directions. Symbols are H along the easy axis reflects the spin-flop transition. The experimental points, the line is the fit described in the text. For frequency minimum at 5 kG reflects the saturation field clarity the hard-axis results have been plotted along negative field Hs . axis. Parameters determined from the fit are given in Table I. 3388 M. GRIMSDITCH, S. KUMAR, AND ERIC E. FULLERTON 54 which includes crystalline and uniaxial anisotropies, shape HMsd 2J1 4J2 2dKu dK1 cos anisotropy and Zeeman terms, is given by 7 8J cos3 . 5 2 1 2 3 dK1 E 1 1 K1 4cos4 3 cos3 cos sin 3sin4 cos4 Since the magnetization of the magnetic trilayer is given by 1 2Msd cos , and the MR is proportional to 1 cos2 , the 2cos2 sin2 sin2 2cos sin3 cos anisotropies and interlayer coupling constants can be readily least-squares fitted to the experimental data. Trivial modifi- 1 cations to the above expressions (J1 J2 0) allow them to 2 4sin4 sin4 Kucos2 2 Mscos2 be used to describe the hard-axis magnetization of the single film Fig. 2 . HMscos sin , 1 Similar arguments can be made for H applied along the easy axis. When H HSF , 1 90° and 2 90°; for where K1 is the cubic anisotropy constant, Ku is a uniaxial H Hs ( 2J1 4J2 2dKu dK1)/Msd we have anisotropy constant characteristic of Fe 211 films, and Ms is 1 2 90°, and for HSF H Hs and provided that J2 is the saturation moment of the layer. The angles are defined in not too large, a condition satisfied by our sample we have Fig. 1. The unusual cubic-anisotropy term in Eq. 1 results 1 180 2 which leads to from a rotation from the cubic axes in our reference frame. The energy per unit area of the bilayer system is given 4 by HM sd 2J1 4J2 2dKu 3 dK1 sin E d1E1 d2E2 J1m1*m2 J2 m1*m2 2, 2 7 8J 2 3 dK1 sin3 . 6 where Ei and di are the energy densities and thicknesses, respectively, of layers i 1,2. The product m1*m2 is given by It is clear from Eqs. 5 and 6 that if one of the param- eters is smaller than all others (K1 in our case it is less likely to be extracted reliably from magnetization data since m1*m2 sin 1sin 2cos 1 2 cos 1cos 2 . 3 the parameters always appear in combination with J2. Note also that Eqs. 5 and 6 contain only three distinct combi- A. Magnetization and magnetoresistance nations of parameters; it is therefore unreasonable to attempt to extract more than three parameters from fits to these ex- The condition for equilibrium requires that the derivative pressions. of E with respect to all the angles must be zero. Because of the magnitude of the demagnetizing field in Eq. 1 , it turns out that to a very good approximation, the equilibrium con- B. Brillouin light scattering dition is 90°. This condition greatly simplifies the energy There have been many derivations of magnon frequencies expression which can be used to evaluate the magnetization in coupled layer systems.12­14 However, in order to guaran- and the MR, viz., tee that the approximations made in describing the magneti- zation are identical to those used to describe the BLS, we 2 1 derived the BLS frequencies starting from the same energy E K 1 1di Kudicos2 i expression we used to derive the magnetization and MR. i 1 3cos4 i 4sin4 i This approach guarantees that any discrepancies between the HM BLS and magnetization results cannot be attributed to incon- sdicos i J1cos 1 2 sistent forms of the energy expression. J2 cos2 1 2 . 4 The formalism which we use to calculate the Brillouin frequencies basically follows that described Cochran et al.12 In general, we numerically minimize this equation to deter- By using this approach we implicitly assume that i there is mine 1 and 2, and adjust the parameters to fit the experi- no domain formation, and ii effects of intrafilm exchange mental SMOKE and MR results. The magnetization and MR are negligible. These approximations are expected to be valid are proportional to Ms(d1cos 1 d2cos 2) and because the Fe layers are so much thinner than the wave- 1 cos( 1 2), respectively. However, for our particular lengths of the excitations probed.15 The magnon modes sample with two equivalent Fe layers (d1 d2 d), the de- probed with BLS typically have a wave-vector component rivatives can be manipulated further to yield analytical ex- parallel to the surface of order of 2 / , where is the pressions which are more convenient in understanding the wavelength of light. An exact calculation of such modes can reliability of the constants extracted. For H applied along the be performed,16 but requires extensive use of numerical tech- hard axis, the condition 1 2 holds for all fields. Substi- niques in all but the simplest cases. Since solutions obtained tuting the value 1 2 into the equilibrium condition by numerical methods make fitting procedures unwieldy, dE/d 0, we get the following expressions relating to such solutions are not particularly useful when one desires to H. For H less than Hs ( 2J1 4J2 2dKu extract the physical constants from the experimental data. 4dK1 /3)/Msd, Approximate analytical expressions for the magnon frequen- 54 BRILLOUIN LIGHT SCATTERING STUDY OF . . . 3389 TABLE I. Parameters extracted for 211 single film and coupled trilayers. Asterisks indicate parameters which are fixed during fitting. K1 Ku 4 Ms J1 J2 ( 105 ergs/cm3) ( 105 ergs/cm3) kG ergs/cm2) ergs/cm2) Single 211 -Fe layer SMOKE 1.1 0.5 4.6 0.2 BLS 1.1* 4.6 0.2 16.5 0.5 211 -Fe/Cr/Fe layers SMOKE 1.1* 5.6 0.3 0.64 0.02 0.045 0.010 MR 1.1* 5.6 0.3 0.69 0.02 0.038 0.013 BLS 1.1* 5.0 0.3 18.0 0.1 BLS 1.1* 5.0* 18* 0.61 0.02 0.038 0.013 cies which we use to least-squares fit to the experimental some degree on how the data are fitted. Fitting separately the results are outlined in the Appendix. single-film BLS data along the hard and easy axes yields two In all our fits to the BLS data we use the bulk Fe value of values of Ku , both with large uncertainties. However, fitting 2.93 GHz/kG. We emphasize that in fitting the data to both data sets simultaneously produces a single value of Ku Eqs. A7 it is necessary to evaluate Eqs. A8 at the equi- with its uncertainty considerably reduced. Therefore, we fit librium angles. In our case this can be done analytically the hard- and easy-axis data simultaneously for all data sets. through Eqs. 5 and 6 . We separately fit the and modes in order to isolate the contribution of interlayer coupling from that of the an- C. Errors isotropy; senses only the anisotropy Eq. A7a while depends on both the interlayer coupling and anisotropy The equations obtained above allow least-squares fits to Eq. A7b . be performed on each of the data sets. Determining the con- The values for J fidence level of parameters thus extracted, especially when 1 and J2 determined from SMOKE, MR, and BLS are in reasonable agreement; the largest discrep- strong correlations exist between parameters, is often not ad- ancy is between the BLS and MR values of J dressed in the literature. It therefore deserves some attention. 1 which are slightly outside the estimated error bars. The values of J First the data were fitted with all parameters as variables, and 2 are self-consistent and considerably smaller than J the mean-square deviation calculated. Second, each param- 1. Although J eter in turn was varied by a small amount and the data refit 2 is small, it proved impossible to quantitatively fit any of the data without including it in the energy expression. Our with that parameter held fixed. The change which led to an values for K increase of 50% in the mean-square deviation was chosen u determined using the different techniques agree with each other for a given sample, but are slightly as the confidence level of the parameter in question. We outside estimated uncertainties when comparing the trilayer chose the somewhat arbitrary 50% increase because this typi- cally leads to a visibly discernible deterioration of the fit. to the single film. Both the anisotropy and J1 are consistent with previous superlattice results.6 Our values of 4 Ms are in the range reported for 20-Å Fe films. There is, however, a V. COMPARISON OF RESULTS small difference between the 4 Ms values for the single and All the curves in the figures presented in Sec. III corre- trilayers which is puzzling. This difference could be real and spond to least-squares fits to the data. Excellent fits are ob- due to subtle growth effects which give rise to different per- tained for all the data sets. The question we wish to address pendicular surface anisotropies, or could be an artifact of the here is that of self-consistency of the various determinations, approximations made in deriving the magnon frequencies. both between the single film and the coupled trilayers and between the different experimental techniques. VI. Fe/Cr/Fe 100... TRILAYERS Table I contains the parameters extracted from the single 211 -Fe and coupled 211 -trilayer films. To evaluate K and Having established that accurate numerical values can be J in Table I we have used M 1.6 kG and d 20 Å . We extracted using the techniques mentioned above, we also begin the comparison by stressing that we expect the relevant measured a single Fe film and a coupled trilayer which were properties of the single film to be close to those of the indi- deposited onto MgO 100 substrates simultaneously with vidual films in the coupled layers. From the magnetization the samples discussed above. The energy equation which in- loops for the single film, we can determine the cubic anisot- cludes the crystalline anisotropy, Zeeman, and interlayer ropy K coupling terms is 1 Eq. 5 with J1 J2 0 . It turns out however that the effects of cubic anisotropy are small and it is not possible to extract a reliable value for K 2 1 from any of our other mea- 1 surements. Therefore, we have simply fixed the value of E i 1 4 K1disin2 2 i HM sdicos i K1 in fitting all other data sets. The reliability of the parameters extracted depends to J1cos 1 2 J2cos2 1 2 . 7 3390 M. GRIMSDITCH, S. KUMAR, AND ERIC E. FULLERTON 54 FIG. 8. Magnon frequencies of the 100 -Fe/Cr/Fe sample for FIG. 7. Hard- squares and easy- circles axis magnetoresis- the field applied along the hard and easy directions. Symbols are tance for the 100 -Fe/Cr/Fe sample. Symbols are experimental experimental points, the line is the fit described in the text. For points, the line is the fit described in the text. Parameters deter- clarity the hard-axis results have been plotted along negative field mined from the fit are given in Table II. axis. Parameters determined from the fit are given in Table II. Even though the energy expression for this orientation is measurements on Fe/Cr/Fe 100 trilayers by Heinrich et al.15 simpler a fourfold anisotropy term and KU 0) the mag- and Gru¨nberg et al.17 and with the results of Elmers et al.9 netic phase diagram is more complex since there are now and Parkin et al.18 on Cr 110 spacer layers. Most studies of four easy directions for the magnetization of each layer. the bilinear interlayer coupling in Fe/Cr systems report a Nonetheless, following basically the same procedure as out- maximum value of J1 1 erg/cm2 for Cr thickness of lined above for the 211 samples, we have extracted the 8 Å , which appears to be independent of the crystallo- magnetic parameters for the 100 films from the BLS and graphic orientation. There are, however, some reports of MR data. lower J1 values in Fe/Cr/Fe 100 trilayers.8,14,19,20 The measurements were performed with the field along In contrast to the bilinear coupling for which J1 is similar the easy-axis 001 and hard-axis 011 in-plane directions. for the two orientations, it is surprising that the biquadratic The MR results are shown in Fig. 7, and the parameters of coupling is considerably higher in the 211 sample than in the least-squares fit are given in Table II. The difference in the 100 sample. Although we find the biquadratic coupling the saturation fields in Fig. 7 reflects the contribution of the to be quite small for the 100 sample investigated here, in crystalline anisotropy. Figure 8 shows the magnon frequen- similarly grown 100 superlattices with thicker Cr layers, cies and fits for the coupled 100 trilayer. The MR results the biquadratic coupling can dominate the bilinear are compared with the BLS results in Table II. The two coupling.21 techniques give the same values for both the anisotropy and In comparing to the literature values for J2, there is con- interlayer coupling. The largest discrepancy is in the deter- siderable spread in the reported values for the Fe/Cr 100 mination of J2, but the values are still just within the esti- system,15,17,19,20,22 and the values of J2 are sensitive to details mated uncertainties. of the growth conditions.23,24 Elmers et al.9 observe large values of J2 in Fe/Cr/Fe 110 trilayers for thin Cr spacers. VII. DISCUSSION AND CONCLUSIONS These results suggest that an extrinsic mechanism which de- pends sensitively on the interfacial structure is controlling In comparing the results for 211 - and 100 -coupling the magnitude of J2. In Slonczewski's4 fluctuation model, strengths we see that the values for J1 are very similar for the the relevant structural parameter is the lateral length of two orientations. This agrees with previous superlattice atomically smooth Cr layer terraces. This could explain the studies.6 The results for J1 are also in agreement with recent larger J2 values observed in the samples in which the short- TABLE II. Parameters extracted for 100 single-film and coupled trilayers. Asterisks indicate parameters which are fixed during fitting. K1 4 Ms J1 J2 ( 105 ergs/cm3) kG ergs/cm2) ergs/cm2) Single Fe layer BLS 2.2 0.3 20.6 0.3 Double Fe layer MR 2.4 0.3 0.57 0.02 0.016 0.013 BLS 2.2 0.2 19.8 0.1 BLS 2.2* 19.8* 0.57 0.02 0.003 0.003 54 BRILLOUIN LIGHT SCATTERING STUDY OF . . . 3391 period oscillations are also observed.15,17 The model may not where E , E , and E are the second derivatives of the be applicable to the 211 -oriented sample in which short energy per unit volume given in Eq. 1 with respect to the period oscillations, although theoretically predicted,7 have angles indicated in the subscript, and is the gyromagnetic not been observed. Careful measurement of the thickness and ratio. Since our Fe layers are only 20-Å thick, the correction temperature dependence of the biquadratic coupling in 211 which must be included to account for the finite wavelength coupled samples are needed to further understand its origin. of the magnons probed by Brillouin scattering can be treated In conclusion, we have used BLS in conjunction with as a perturbation. Following the approach of Cochran et al. it magnetization and magnetoresistance techniques to study can be shown that for magnon wave vectors (q) perpendicu- 211 - and 100 -Fe/Cr/Fe trilayers. We find that the cou- lar to the applied field, and to lowest-order terms in qd, the pling strengths extracted using each technique are self- following terms consistent. This not only provides reliable values for the cou- pling strengths but also highlights the complementarity of 2qdcos2 0 these techniques in uniquely determining the magnetic pa- 2 Ms 2 A2 rameters. In particular, combining magnetization measure- 0 2 Msqd ments which are proportional to the net magnetization in the field direction and the MR which is proportional to the must be added to Eq. A1 . cosine of the angle between the magnetization of the layers The general solution for the field along the hard axis, allows one to uniquely extract the magnetic configuration as which includes the region below saturation, where M and a function of applied field. This proves particularly useful in H are not parallel, is analyzing films where the two magnetic layers have different thicknesses and/or anisotropies. 2K / 2 Hcos U M 1 2sin2 S ACKNOWLEDGMENTS K 1 3M 29 sin2 28 sin4 4 We wish to thank Dr. S. Bader for helpful discussions and S a critical reading of the manuscript. This work was supported K1 by U.S. Department of Energy, Basic Energy Sciences- Hcos 4 MS 3M 11sin2 7sin4 4 S Materials Sciences under Contract No. W-31-109-ENG-38. 2 K1 2 M 9sin6 12 sin4 4 sin2 S APPENDIX 2 MS 22qdcos2 . A3 In this appendix we outline the approach to calculate ana- lytical expressions for the spin-wave frequencies for single In order to obtain the field dependence of one must also layers and coupled trilayers. use Eqs. 5 and 6 which relate to H. For the field along the easy axis, 90° for all fields and the expression for frequency is given by 1. Single layer It is known25 that the FMR frequency, which corresponds 2K K / 2 H U 1 H 4 M to the infinite wavelength magnon, is obtained by solving M S S MS E E i Ms/ sin 2 K1 2 2 MS 22qd. A4 E 0, A1 MS i M s / sin E 2. Double layer The general expression for the frequencies of the FMR modes of a double layer can be obtained by solving the following 4 4 determinant derived from the equations of motion assuming 1 2 90° and d1 d2 d): E E i M E 1 1 1 1 s / E 1 2 1 2 E i M E E 1 1 s / E 1 1 1 2 1 2 E 0, A5 E E E i M 2 1 2 1 2 2 2 2 s / E E E i M 2 1 2 1 2 2 s / E 2 2 where again E , E , and E with k 1,2 corresponding to layer 1 or 2 are the second-order derivatives of energy per k k k k k k unit volume with respect to the angles indicated in the subscript. The finite-wave-vector effects can be included for this case 3392 M. GRIMSDITCH, S. KUMAR, AND ERIC E. FULLERTON 54 by following an approach similar to that of the single layer. Keeping only the lowest-order terms in qd, the following matrix must be added to Eq. A5 : cos2 1 0 cos 1 cos 2 icos 1 0 1 icos 2 1 2 M2sqd cos . A6 1 cos 2 icos 2 cos2 2 0 icos 1 1 0 1 In our symmetric bilayer there are some additional sim- A7 is not valid below H SF when H is applied along the plifications which allow us to extract analytical expressions easy axis. The second derivatives of energy can be written for the frequencies. For all fields, when H 111 hard as axis we have 1 2; when H 0 11 easy axis and for fields above H 2K SF we have 1 180 2; both of these U conditions lead to the simplification E E Hcos 2cos2 1 E E M 1 1 2 2 , S E E E E . 1 1 2 2 , E 1 1 2 2 K The above conditions and the inversion symmetry of our 1 28cos4 27cos2 3 E , sample, that requires that the solutions of Eq. A5 be either 3MS 1 2 symmetric or antisymmetric, enable us to reduce the deter- A8a minant from a 4 4 to a 2 2, and consequently to obtain analytical solutions. Note that Eq. A5 , plus Eq. A6 does E J 1 2 1cos2 2J2 cos2 sin2 2 not have exact symmetric and antisymmetric solutions. How- ever, because Eq. A6 is small, the corrections are quadratic. 4 cos2 sin2 /d A8b Also the inversion symmetry condition does not hold for H H K1 SF when H is applied along the easy axis. The expan- E 3cos2 7 cos4 , sion of the determinant keeping only the first-order terms in Hcos 4 M S 3MS qd) leads to a quadratic equation in 2 whose solutions are A8c and 2 / 2 E E E cos2 1 2 E 4 M 2 2 s 2qd, A7a E 2 K1 M sin2 3cos2 1 2. A8d S 2 / 2 E E E , A7b The frequencies of the two modes can be calculated from 1 2 E Eqs. A7a and A7b by setting 0 and 90° for the hard where 1 2 for H 1¯11 and 1 180 2 for and easy axes, respectively, and by using Eqs. 5 and 6 to H 01¯1 and is the angle between H and 1¯11 . Equation calculate as a function of H. 1 See articles in Ultrathin Magnetic Structures I and II, edited by J. 10 R. W. Wang, D. L. Mills, E. E. Fullerton, J. E. Mattson, and S. D. A. C. Bland and B. Heinrich Springer, New York, 1994 . Bader, Phys. Rev. Lett. 72, 920 1994 . 2 M. Ru¨hrig, R. Scha¨fer, A. Hubert, R. 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