PHYSICAL REVIEW B VOLUME 53, NUMBER 5 1 FEBRUARY 1996-I Magnons in antiferromagnetically coupled superlattices R. W. Wang Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824 D. L. Mills Department of Physics and Astronomy, University of California, Irvine, California 92717-4575 Eric E. Fullerton, Sudha Kumar, and M. Grimsditch Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439 Received 21 July 1995 We present an experimental Brillouin-scattering study of the collective spin-wave modes in antiferromag- netically coupled Fe/Cr 211 superlattices. We show spectra as a function of external magnetic field, where the field is swept from very small values up to those required to saturate the sample in the ferromagnetic state. We thus explore excitations in the low-field antiferromagnetic state, the surface spin-flop regime, the symmetric ``bulk'' spin-flop regime, and the high-field state. The data are compared with theoretical studies based on a description of the collective modes developed previously. I. INTRODUCTION aligned 100 sheets of spins with antiferromagnetic coupling between such sheets, along with twofold in-plane anisotropy. Advances in deposition techniques have now made it pos- Application of an external magnetic field parallel to the easy sible to synthesize magnetic superlattices from diverse con- axis thus induces a spin-flop transition in the Fe/Cr 211 stituents, and thereby to realize new materials with macro- superlattices, very much as it does in MnFe2 or FeF2. Recent scopic properties and magnetic response characteristics studies2 of finite Fe/Cr 211 structures show that these ma- subject to design. Structures composed of thin films of a terials display not only the spin-flop transition, but in addi- ferromagnetic metal, such as Fe, separated by nonmagnetic tion to the surface spin-flop transition discussed many years spacer layers have been found to possess many unexpected ago in the literature on antiferromagnetism at surfaces.3,4 and intriguing properties. In such systems the spins within Here we present an investigation of dynamic effects around each ferromagnetic layer are coupled tightly by very strong both the bulk and surface spin-flop transitions. effective exchange interactions, so for many purposes one Magnetic superlattices possess collective excitations, can view the film as endowed with a fixed total magnetiza- whose dispersion relation and frequency spectrum are con- tion Ms , whose direction may vary in response to an exter- trolled by the combination of interfilm exchange, intrafilm nally applied magnetic field. anisotropy, and Zeeman interaction with an external mag- Both the weaker interfilm exchange couplings, mediated netic field. As noted many years ago,5 interfilm coupling via through spacer layers, along with the intrafilm crystalline macroscopic dipolar fields generated by spin motions within anisotropies can be controlled by the superlattice geometry, the film must also be included. It has been well known for the symmetry of the substrate, and the growth conditions. decades that in ferromagnetic resonance, the frequency of the This means one may produce materials with rich magnetic uniform mode of the film is affected by internal demagnetiz- phase diagrams. It has been found that many of these mate- ing fields which also have their microscopic origin in dipolar rials exhibit interfilm exchange coupling whose origin is still interactions.6 We will deal here with collective modes whose somewhat controversial but is clearly mediated by the spacer wave vector k parallel to the film surface is nonzero. In this layers. The strength and sign of the exchange coupling is circumstance, the dipolar fields leak out of a film within affected by the superlattice geometry, substrate, spacer layer, which they are generated, to provide interfilm coupling.5 For growth conditions, etc., and the above factors also affect the the particular case of Fe, for which 4 Ms 21 kG, the in- crystalline anisotropy fields within the layers. This great va- terfilm dipolar fields can be very substantial in strength. Re- riety and variability of magnetic parameters leads to a very cently, two of the present authors have developed the theory rich and fascinating behavior of the magnetic properties, of the collective modes of the Fe/Cr 211 structures, with with only modest fields required to alter the nature of the attention to the issues just mentioned.7 magnetic ground state. The purpose of this paper is to present an experimental An interesting example studied in detail recently is Fe/Cr Brillouin-scattering study of the collective spin-wave modes 211 superlattices.1 Here the Fe magnetizations lie in plane, in an Fe/Cr 211 structure. In these experiments, the modes and have a dominant in-plane uniaxial anisotropy by virtue excited have k 105 cm 1. In this regime, the modes are of the growth of these structures on MgO 110 surfaces. influenced very importantly by the interfilm dipolar cou- When the interfilm coupling is antiferromagnetic, the energy pling, in addition to interfilm anisotropy and exchange.7 In functional is very similar to that of the classical antiferro- order to interpret the experimental spectra we have gener- magnets MnF2 and FeF2, wherein one has ferromagnetically ated, within an approximation scheme described below, theo- 0163-1829/96/53 5 /2627 6 /$06.00 53 2627 © 1996 The American Physical Society 2628 WANG, MILLS, FULLERTON, KUMAR, AND GRIMSDITCH 53 retical light-scattering spectra which compare very favorably wave frequency. For bulk Fe, D 2.5 10 9 Oe cm2. If we with the data. Our investigation establishes that the rich assume this value applies to the Fe films in the superlattice, spectrum of collective modes in these structures, which can then l be easily altered by application of a modest magnetic field, is ex 100 A. When d lex , we may ignore the variation of m(z). This remains true so long as k well described by the theory developed in Ref. 7. is sufficiently small, compared to 1/d, a condition amply satisfied for modes excited in either Brillouin-scattering experiments II. EXPERIMENTAL DETAILS such as those discussed here, or ferromagnetic resonance. The sample employed in this study is an Fe/Cr 211 su- Ignoring interfilm spin-wave dipolar coupling, the theory of perlattice. The sample, prepared and characterized as de- the collective excitations of the superlattices is very simple: scribed in Ref. 1, consists of 22 double layers of Fe 40 Ć / it is isomorphic to that of a suitable one-dimensional line of Cr 11 Ć . The static magnetization properties of the sample spins with exchange coupling and single site anisotropy. In used in this investigation are described in Ref. 2. Ref. 7, we introduced a simple procedure for appending the The Brillouin spectra were recorded with a 5 4 pass tan- spin-wave generated interfilm dipolar coupling to this pic- dem interferometer,8 using 300 mW of 515 nm radiation. The ture. The calculation of both the frequencies and eigenvec- polarization of the scattered light was analyzed at 90° rela- tors of the superlattice collective excitations is calculated tive to that of the incident light. This isolates the purely rather straightforwardly in this picture. magnetic scattering, and greatly reduces any scattering from We wish to elaborate further on the remarks of the para- surface phonons Rayleigh waves , and bulk phonons with graph above. Quite a number of years ago, a continuum appreciable amplitude within the skin depth. The magnetic theory of spin waves in thin ferromagnetic films was devel- field was applied perpendicular to the scattering plane, oped, with focus on the regime where exchange, dipolar, and which contains both the incident and scattered photon wave Zeeman energies are all comparable in magnitude.9 The vectors and parallel to the easy axis of the sample: the spin theory was applied to early light-scattering studies of thin waves observed in the Brillouin spectra therefore have wave films, and a remarkable quantitative account of the relative vectors perpendicular to the easy axis. The scattering geom- intensities observed for various modes emerged from the etry is a backscattering configuration, where the incident and theory. Those experiments explored films with thickness scattered wave vectors of the light make angles of 45° and 0° d lex , so a full theory which accounts for the full spatial with respect to the surface normal, respectively. Typical data variation of m(z) was essential for a successful account of acquisition times were one-half to 2 h per spectrum. the data. The theory is rather complex to implement, since All experimental spectra show a discontinuity at around for each frequency, the response of the film is accounted for 0.2 cm 1. This artifact is caused by filter which must be by superimposing six waves, each with complex wave vector inserted into the beam as the spectrometer scans through the in the direction normal to the surface. Boundary conditions laser frequency. All spectral features which appear close to which account for the possible presence or absence of spin these frequencies must be treated with caution. pinning at each interface are required by the mathematical structure of the theory. Fortunately, the data that motivated the original work10 was accounted for nicely without invok- III. THEORETICAL DETAILS ing spin pinning; only bulk parameters were required. If a sample with an even number of Fe layers is placed in The theoretical structure described in the preceding para- a weak magnetic field H i.e., H lies below the field required graph, supplemented by anisotropy terms appropriate to ul- to initiate the surface spin-flop transition parallel to the easy trathin films, has been extended to superlattice structures by axis, then the structure resides in the simple antiferromag- Stamps and Hillebrands.11 To apply this theory, one requires netic ground state.2 Necessarily the magnetization in one out- the value of the intrafilm exchange stiffness D, along with ermost Fe film will be parallel to the applied field, while that information on the degree of spin pinning at each interface. on the opposite side of the finite structure will be antiparallel. The magnetic response of each film is described again by As H is increased the surface spin-flop transition is initiated superimposing six waves. It would be extremely involved to through interaction of H with the outer film whose magneti- use this full theory to account for the excitations of a super- zation is antiparallel to the magnetic field. lattice with complex ground states such as those we encoun- The theoretical model we have employed is based on the ter in the present analysis. theory of collective modes developed in Ref. 7, extended to In the end, at least so long as d lex , on physical grounds make contact with the Brillouin data. Quite generally, when a one knows that m(z) must be quite uniform across a given spin wave is excited, the magnetization within a given film film. Therefore, both the excitation spectrum and light- has the form scattering spectrum will be insensitive to many details in- cluded in the full theory of Ref. 11 and we may use the M x,y,z;t M simplified theory of Ref. 7 with quantitative accuracy. Once sn 0 m z exp ik *x i t , 1 the orientation of the ground-state moments in the superlat- where n 0 describes the equilibrium orientation, and m(z) tice is determined this is a challenging numerical task for perpendicular to n 0) is the fluctuation in magnetization as- the examples discussed here and in Ref. 2 , the spin-wave sociated with the spin wave. Here the z axis is normal to the eigenvectors and frequencies are generated very straightfor- film surface. The picture utilized here applies to the case wardly within the simple theory. where the ferromagnetic films have thickness d small com- Once the spin-wave frequencies and eigenvectors are pared to an effective exchange length lex (D/ )1/2, where known, we have calculated the light-scattering spectra as fol- D is the intrafilm exchange stiffness, and a typical spin- lows. The superlattice is approximated as an optically homo- 53 MAGNONS IN ANTIFERROMAGNETICALLY COUPLED SUPERLATTICES 2629 geneous material, i.e., we did not consider its microstructure parameters, in the notation of Ref. 2, used in the present and its influence on both the incident and scattered radiation. calculations are HE 1.9 kG, and HA 0.1 kG, and 4 M This is, in effect, an effective-medium approach appropriate 21 kG from bulk Fe. The calculated Brillouin spectra to the case where all films have a thickness small compared should therefore be considered to contain no adjustable pa- to the length scale of the optical field. The light scattering rameter except for a phenomenological ``width'' of the spin- process is described by the formalism developed earlier,5,9 wave modes. This ``width'' is introduced because, since no with the incident and scattered phonon treated as plane lifetime effects are considered, the calculated spin-wave waves. spectra are functions in frequency. In order to mimic the There are two sources of coupling between the photon experimental spectra we have broadened all functions by and fluctuations in the spin system. The first has origin in an experimental instrumental function modeled as a Loren- terms in the Hamiltonian quadratic in the electric field, and zian. linear in the magnetization. These interactions also are re- As a final point we mention that the sample on which the sponsible for the phenomenon of the Faraday rotation. There measurements were made had 22 Fe layers. However be- are, in addition, terms quadratic in both the electric-field and cause of the difficulty of calculating the spin arrangements in magnetic-field components. In magneto-optics, these lead to the finite superlattice reliably and accurately becomes in- the Cotton-Mouton effect. Noting that earlier analyses based creasingly difficult as the number of ferromagnetic layers is on only the linear terms provide a good account of the rela- made larger, particularly in the asymmetric surface spin-flop tive intensities of features in the Brillouin spectra of Fe- state, we limit the total number of layers to 16. based ferromagnets,9 we have retained only the Faraday terms. If we assume also that the Faraday tensor for the IV. RESULTS AND DISCUSSION superlattice can be approximated by the form appropriate to We begin by reminding the reader of the principal features bulk Fe, there is then only one nonzero coupling parameter. of the magnetic phase diagram of an Fe/Cr 211 superlattice This parameter controls only the overall intensity of the with antiferromagnetic coupling between adjacent films. In spectra. these materials, the Fe magnetization lies in plane, with an The above picture allows us to generate theoretical Bril- easy uniaxial axis in plane. For very small applied fields H, louin spectra by introducing no parameters beyond those em- the system remains in the antiferromagnetic state. If there are ployed to describe the magnetic ground state of the superlat- an even number of layers, upon increasing the external field, tice, and its spectrum of collective excitations. In our view, a field-induced phase transition into the surface spin-flop there are two approximations invoked above which are po- tentially troublesome. The first is our use of only the Faraday term to couple light to the fluctuations in the spin system. If we were to extend the treatment to a more realistic picture which recognizes the influence of the film geometry on the Faraday tensor, and also which includes the Cotton-Mouton terms, we would encounter several additional parameters in the Brillouin matrix element whose values are unknown for these structures. We shall see however that the calculations with the simple model account nicely for most features in the measured spectra, and it is doubtful if we could obtain deeper insight into the data with a very complex multiparam- eter matrix element. Modeling of the magnetic ground state and the spin-wave spectrum follows the approach employed in Refs. 2 and 7. Based on previous experiments on this sample2 the principal parameters which describe its magnetic behavior are a uniaxial in-plane anisotropy and a bilinear interfilm ex- change coupling. In calculating the spin-wave frequencies, we have, of course, included dipolar coupling whose strength depends on 4 M. Some studies of Fe/Cr have indicated that biquadratic exchange and cubic anisotropy may be appre- ciable in some of these structures.12 However, since for the sample investigated here the strength of the biquadric cou- pling and cubic anisotropy are not accurately known, their influence would have to be evaluated by introducing an ad- ditional parameter. We have not attempted this approach here. Although it may be concluded from the arguments in the preceding paragraph that there are two adjustable parameters in the calculation of Brillouin spectra, the numerical value of the parameters is constrained by the need to reproduce the FIG. 1. Brillouin spectra for an Fe/Cr 211 superlattice with 22 fields at which the surface and bulk spin flops occur. The Fe films for fields in the range 1.2 to 3.75 kG. 2630 WANG, MILLS, FULLERTON, KUMAR, AND GRIMSDITCH 53 FIG. 3. Theoretical light-scattering spectra calculated as de- scribed in the text, for the six applied fields indicated. The small vertical lines are placed at the frequencies of the collective modes of our model 16-layer slab. cal'' in two respects: there is a dramatic Stokes­anti-Stokes intensity asymmetry, and there are two salient peaks. The higher-frequency peak is expected to be due to a surface FIG. 2. The same as Fig. 1, for external magnetic fields in the Damon-Eshbach type magnon and it only appears on one range 0.25 to 1.0 kG. side of the spectrum. The lower frequency mode is due to a ``bulk'' or standing spin-wave resonance of the structure; it phase is initiated. By the mechanism described in Ref. 2, this appears on both sides of the spectra but with unequal inten- state evolves into a symmetric configuration which may be sity. As expected, on reversal of the magnetic field the spec- identified with the bulk spin-flop phase, modified near the tral features are found to change from the Stokes to anti- surface by virtue of the missing ``exchange bonds.'' At high Stokes side of the spectrum. Experimentally the two-mode fields the symmetric spin-flop configuration reaches satura- behavior persists down to fields close to 1.2 kG at which tion, where all Fe film moments are aligned parallel to the point additional features appear in the spectrum. external field. Experimentally the surface and bulk spin flops Figure 3 contains the high-field calculated spectra which are found at 0.5 and 1.1 kG, the calculations yield 0.45 and are to be compared with the experimental spectra in Fig. 1. 0.61 kG, respectively. The small vertical lines in the figure indicate the position of Because it is not possible from the light-scattering spectra the individual collective modes. In agreement with experi- alone to decide whether the magnetization at the illuminated ment, the calculations show spectra with two strong features, end is parallel or antiparallel to the field, one must rely on the frequencies agree to within around 10% recall that there the past history of the sample to do this. Based on the Kerr are no adjustable parameters to fit the frequencies , and the loops in Ref. 2 we do not know that the two surfaces switch features change side on reversal of the field. at plus and minus fields, respectively. To guarantee that the The split-off mode is the Damon-Eshbach surface spin- regime around the surface spin flop would be explored in the wave mode mentioned earlier, which for one sense of k is experiments, we swept the field through zero, from a value localized on the upper surface of the structure, and for the well above the surface spin-flop field, to a value well below. other is localized on the bottom surface. This mode is well Even then, it is not possible to tell from the light-scattering known to provide a dramatic asymmetry in the Stokes­anti- data alone if the transition at the illuminated surface will Stokes ratio. If D is the thickness of the structure, and if also occur for positive or negative field. k D 1 then the origin of the Stokes­anti-Stokes asymmetry Spectra were recorded as a function of monotonically resides in the fact that for one sign of k , the mode is local- changing field from 3.75 down to 3.75 kG. Figures 1 and ized on the surface exposed to the laser beam, and for the 2 show spectra in the high- and low-field regions, respec- other, it is localized on the ``dark side'' recall that in a Bril- tively. At high fields the magnetization of all layers are louin experiment, the Stokes and anti-Stokes sides of the roughly aligned with the field and hence it is not surprising spectrum sample opposite values of k ). In the present ex- that the observed spectra are very similar to those in conven- periments, however, we are in the regime k D 1, and the tional uncoupled superlattices.5,13­15 The spectra are ``typi- surface mode is thus not completely localized at a single 53 MAGNONS IN ANTIFERROMAGNETICALLY COUPLED SUPERLATTICES 2631 surface. In the system under study here, the large Stokes­ spectrum most notably the Stokes­anti-Stokes switch in the anti-Stokes asymmetry thus has its origin in the mechanism high-frequency peak. discussed some years ago by Camely and Grimsditch.16 In Figs. 4 a and 4 b , we show the theoretical spectra for The second, lower frequency feature is not produced by a the two cases H 0.25 kG, and H 0.25 kG, respec- single mode, but rather by modes that cluster together near tively. While the theoretical spectra contain more detailed the maximum frequency of the standing spin-wave band. If structure than found in the experimental data, the theory is in rather good accord with experiment. Note that the dominant M is this frequency, then for a semi-infinite ferromagnet, one may show that the density of modes diverges as ( contributions to the Brillouin spectrum are on the anti-Stokes M ) 1/2 as approaches side of the laser line, for both field directions. It is also the M from below. In the finite structure, the modes cluster near the top of the band, and the case in both theory and experiment that the spectrum is matrix element which couples light to these modes is large. broader at H 0.25 kG, than for H 0.25 kG. In the We remark that the spectra here are similar to earlier spectra theoretical plots, the small vertical marks indicate the fre- taken on Fe/Pd superlattices, by Hillebrands et al.17 Of quency of a collective spin-wave mode of the 16-layer su- course, in the present case, the frequencies and eigenvectors perlattice. We see that the feature in the experimental spec- of the modes are influenced strongly by the antiferromag- trum comes not from one particular mode, but is a structure netic interfilm coupling, ignored in earlier theories.9 formed from contributions of a spectrum of modes near the The ``two-mode'' structure present at 3.75 kG extends bottom of the spin-wave band. These have the character of down to lower fields, into the spin-flop regime. We see this standing spin-wave resonances of the whole structure. We from the data and theory at H 1.5 kG, where the system see a high-frequency mode, split off from the band of stand- is in the symmetric spin-flop configuration. We no longer ing spin-wave resonances, near 1.5 cm 1. This is a surface have the Damon-Eshbach mode split off from the bulk con- mode, which contributes only a weak line to the theoretical tinuum at this field, as one sees from the array of vertical spectra. There is no evidence of this mode in the experimen- dashes. There is a structure at both the maximum frequency tal data. We show theoretical spectra for H 0.5 kG in Figs. 4 c M , and the minimum frequency m of the collective spin- wave band. Once again, in the semi-infinite limit, we have and 4 d . In contrast to the theory and experiments at 0.25 van Hove singularities at both kG, we see that now reversal of the direction of the magnetic m and M , and the two mode structure here is a reflection of these two singularities. field shifts the dominant part of the scattering from the In the theory, this apparent ``two-mode'' structure contin- Stokes to the anti-Stokes side. The prominent asymmetric ues down to lower fields, as we see from the calculations for peak with maximum near 0.2 cm 1 is produced by scattering H 1.2 kG. However, by the time we reach 1.2 kG, struc- from low-lying acoustic spin-wave modes, and three rather ture not present in theory is evident in the data. We see no weak higher frequency peaks have their origin in the optical evidence, in the calculations, of the three mode spectrum modes. The sharp peak near 0.6 cm 1 is a surface mode, displayed in 1 g and 1 h . bound relatively weakly to the surface. Here again there is Below 1.2 but above 0.5 kG i.e., the region between the more structure in the theory than found in the data, but the bulk and surface spin flops the experimental spectra in Figs. principal features displayed by theory are evident in the ex- 1 and 2 are less well defined and noisier. The reason for this perimental spectra. For example, in the experimental spec- can be found in Figs. 4 c and 4 d which show the calcu- trum labeled H 0.50 kG, we see the surface mode peak lated spectra at 0.5 kG. For fields in the range between the near 0.6 cm 1 on the high-frequency side, and the lower- surface and the bulk spin flop, the orientation of the magne- frequency feature from the acoustic spin-wave band on both tization of the films forms very complicated arrangements sides. We do not perceive the structures from the optical similar to the one labeled H 1.492 kG shown in Fig. 1 b of mode region in the data; the intensities of these are also Ref. 2. The resulting spin-wave spectrum is also quite com- plicated and similar to the one at H 1.5 kG in Fig. 6 of Ref. 7; where the surfacelike modes are localized in the gap be- tween low-lying spin waves of acoustic character and higher frequency resonances of optical character. Just below 0.5 kG we know Kerr loops Ref. 2 that the sample undergoes a surface spin flop in which one of the surface layers aligns antiparallel to the field. However, as mentioned earlier, we do not know at which surface it oc- curs. Since the experimental spectra at 0.5 and 0.25 kG in Fig. 2 show no major changes and, what is more, the peaks do not change from the Stokes to the anti-Stokes side, it is a clear indication that it is the back layer which has switched. As the field is decreased, changed in sign, and brought to 0.25 kG we still observe no major change: this is consistent with the spins remaining in the antiferromag- netic alignment with the outermost layer being now antipar- FIG. 4. Theoretical light-scattering spectra calculated as de- allel to the field as inferred from the Kerr data. A further scribed in the text, for the four applied fields indicated. The small decrease in field to 0.5 kG produces the surface spin flop vertical lines are placed at the frequencies of the collective modes on the outermost surface and leads to the clear changes in the of our model 16-layer slab. 2632 WANG, MILLS, FULLERTON, KUMAR, AND GRIMSDITCH 53 rather weak in the theory. One sees a peak near 0.6 cm 1 in We are intrigued by this fascinating class of materials. the spectrum for H 0.50 kG in the data, but the peak is Previous studies have shown that the Fe/Cr 211 structures not fully resolved in this case. A more complete and quanti- have a rich magnetic phase diagram which includes a surface tative description of the matrix element which couples the spin-flop phase, which with increasing external field evolves evanescent light field with the collective spin wave may im- into a symmetric ``bulk'' spin-flop phase. There is a dramatic prove the correspondence between theory and experiment. even-odd effect, in that the surface spin-flop phase is absent Except for the difficulty mentioned in the last paragraph, from samples with an odd number of Fe layers. The present in our view the overall accord between theory and experi- paper shows that there is a rich range of dynamical responses ment is quite satisfactory. The principal trends are repro- displayed by these structures, as one traverses the various duced nicely by theory, and in fact we have quantitative field regimes. agreement for many features of the data. Both the phase diagram, and the dynamic response char- acteristics can be controlled and varied by the superlattice V. CONCLUDING REMARKS geometry. We thus have a new class of magnetic material, whose rich phase diagram can be scanned by very modest This paper compares experimental and theoretical Bril- excursions in the magnetic field. louin spectra for Fe/Cr 211 superlattices and shows that the principal features in the data agree with theory. A few minor differences as discussed above can be understood on the ba- ACKNOWLEDGMENTS sis of the simplifications in the theoretical model we have used. Improvements in the model would require introduction The research of R.W. and D.L.M. was supported by the of new fitting parameters higher-order anisotropies, biqua- Army Research Office, under Contract No. CS001028. Work dratic interlayer coupling, quadratic magneto-optic coupling at Argonne National Laboratory was supported by the U.S. whose values may not be determined uniquely by the data: Department of Energy, BES-Materials Sciences, under Con- for this reason, we have limited the complexity of the model. tract No. W-31-109-ENG-38. 1 E. E. Fullerton, M. J. Conover, J. E. Mattson, C. H. Sowers, and 10 M. Grimsditch, A. Malozemoff, and A. Brunsch, Phys. Rev. Lett. S. D. Bader, Phys. Rev. B 48, 15 755 1993 . 43, 711 1979 . 2 R. W. Wang, D. L. Mills, E. E. Fullerton, J. E. Mattson, and S. D. 11 R. L. Stamps and B. Hillebrands, Phys. Rev. B 44, 5095 1991 . Bader, Phys. Rev. Lett. 72, 920 1994 . 12 M. Grimsditch, S. Kumar, and E. E. Fullerton unpublished . 3 D. L. Mills, Phys. Rev. Lett. 20, 18 1968 ; F. Keffer and H. 13 For a review, and comments on the origin of the Stokes­anti- Chow, ibid. 31, 1061 1973 . Stokes asymmetries, see the article by F. Nizzoli and D. L. 4 W. Saslow and D. L. Mills, Phys. Rev. 71, 488 1968 . Mills, in Nonlinear Surface Electromagnetic Phenomena, edited 5 R. E. Camley, T. S. Rahman, and D. L. Mills, Phys. Rev. B 27, by G. I. Stegeman Elsevier, Amsterdam, 1991 , Chap. 7. 261 1983 . 14 M. Grimsditch, M. R. Khan, A. Kueny, and I. K. Schuller, Phys. 6 C. Kittel, Phys. Rev. 71, 270 1947 ; 73, 155 1948 . Rev. Lett. 51, 498 1983 . 7 R. W. Wang and D. L. Mills, Phys. Rev. B 50, 3931 1994 . 15 A. Kueny, M. R. Khan, I. K. Schuller, and M. Grimsditch, Phys. 8 J. R. Sandercock, in Light Scattering from Solids III, edited by M. Rev. B 29, 2879 1984 . Cardona and G. Gušntherodt, Topics in Applied Physics, Vol. 51 16 R. E. Camley and M. Grimsditch, Phys. Rev. 22, 5420 1979 . Springer, New York, 1978 . 17 B. Hillebrands, A. Boufelfel, C. M. Falco, P. Baumgart, G. 9 R. E. Camley, T. S. Rahman, and D. L. Mills, Phys. Rev. B 23, Gušntherodt, E. Zirngieble, and J. D. Thompson, J. Appl. Phys. 1226 1981 . 63, 3880 1988 .