PHYSICAL REVIEW B VOLUME 56, NUMBER 5 1 AUGUST 1997-I Exchange and anisotropy effects on spin waves in epitaxial Co films M. Grimsditch and Eric E. Fullerton Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439 R. L. Stamps Department of Physics and Lima Campus, Ohio State University, Columbus, Ohio 43210 Received 20 February 1997 Using Brillouin scattering we have investigated the effect of large in-plane anisotropy on magnetic excita- tions in b axis-oriented epitaxial 50-nm Co films. For in-plane fields along the easy axis the magnon frequen- cies increase monotonically with increasing field. For in-plane fields along the hard axis, nonmonotonic behavior is observed in which the surface to bulk spin-wave character of one magnon mode changes as a function of applied field. Although fits to the data based on a full theoretical treatment are consistent with parameters in the literature and with parameters extracted from the magnetization data, we show that uncer- tainties in crystal orientation as small as 1° can produce dramatic effects in the quantitative data interpreta- tion. S0163-1829 97 06529-6 I. INTRODUCTION isotropy and magnetization. The exchange constant was again found to be inconsistent with the neutron data and the It is now possible to fabricate high-quality epitaxial g factor was considerably lower than expected for hcp Co. hcp-Co films with the easy magnetization axis the c axis of The samples used in Refs. 8 and 9 did not allow the the hcp structure lying in-plane. These films provide a effects of orientation to be investigated, in Ref. 9 because it unique opportunity to study spin-wave excitations in a sys- was polycrystalline and in Ref. 8 because the demagnetizing tem with a large in-plane anisotropy which is comparable in fields kept the magnetization in the basal plane. With our magnitude to the demagnetizing and applied fields. Infinite- epitaxial b-axis-oriented films we are able to observe the wavelength excitations observed by ferromagnetic resonance effects of the large anisotropy by probing different propaga- techniques,1 and the very short-wavelength excitations domi- tion and field directions. In addition, the film thickness is nated by exchange interaction, investigated by neutron such that some of the bulk ``exchange'' modes are close in scattering,2 are well understood in Co. The intermediate- frequency to the surface mode. We are thereby also able to wavelength regime, where exchange, anisotropy, applied, observe interactions between bulk and surface modes as the and demagnetizing fields all contribute significantly to the propagation direction with respect to the magnetization frequency of the excitations, has not yet been fully investi- changes. gated. In this wavelength regime, which can be investigated In the next section we present experimental data and com- using Brillouin scattering,3­6 effects of exchange and propa- pare our results with theoretical calculations. In particular, gation direction, not important in ferromagnetic resonance, the pronounced effects of the large in-plane anisotropy on must also be included. A detailed investigation, using Bril- the spin-wave frequencies and their field dependence in dif- louin scattering, of the effects of anisotropy on spin excita- ferent geometries is examined. Next, a sensitive dependence tions in single-crystal Fe has been presented in Ref. 6; the of the spin-wave frequencies on crystal and field orientation anisotropy, however, played only a small role. is described in Sec. III. Finally, the data taken from magne- Recently there has been considerable interest in cobalt tization and Brillouin-scattering experiments allow a deter- with special attention being paid to its metastable fcc and bcc mination of values for the magnetization, exchange stiffness, polymorphs. In the fcc and bcc forms the anisotropies are first- and second-order anisotropies, and g factor. These val- relatively small; Brillouin-scattering investigations7,8 of ues are discussed in Sec. IV and compared with results from these forms show the expected behavior similar to that found previous experiments. An apparent inconsistency in mea- for Fe.6 We are aware of only two investigations of hcp Co surements of the Co exchange constant is discussed. in the intermediate-wavelength regime probed by Brillouin scattering.8,9 The investigation of Ref. 9 utilized a Co film II. EXPERIMENTAL RESULTS AND COMPARISON which was a-axis oriented but polycrystalline in the plane. TO THEORY The polycrystalline nature of the material, which allowed the data to be analyzed assuming no anisotropy, produced results The Co films were grown by magnetron sputtering onto consistent with known values of the magnetization and gy- Cr 211 -buffered MgO 110 single-crystal substrates. The Cr romagnetic ratio. They did find however that the exchange buffer layers were deposited at 600 °C and provide a lattice- stiffness constant was not in agreement with previously ac- matched template for b-axis Co 101¯0 films which were cepted bulk values determined by neutron scattering.2 In Ref. grown at 300 °C. The Co films were then capped with a 8 a c-axis-oriented hcp Co film was investigated; it was 1.5-nm Cr layer to prevent oxidation. In addition to the Bril- found that the Brillouin data was consistent with known an- louin light-scattering studies, magnetization measurements 0163-1829/97/56 5 /2617 6 /$10.00 56 2617 © 1997 The American Physical Society 2618 M. GRIMSDITCH, ERIC E. FULLERTON, AND R. L. STAMPS 56 Brillouin spectra were recorded in the quasibackscattering geometry with the incident beam at 60° from the surface normal and the scattered light collected along the surface normal. A 3 4 pass Fabry-Perot interferometer was used to analyze the scattered 300 mW of 514.5 nm incident radia- tion. The scattered light was analyzed in crossed polarization to eliminate contributions to the spectra from surface phonons. All spectra were recorded at ambient temperature. The component of the wave vector parallel to the sample surface, determined by the scattering geometry, was 1.06 105 cm 1 and it was always perpendicular to the applied magnetic field. The sample could be rotated about its normal thereby allowing different propagation directions in the plane of the sample to be studied. The highest fields obtainable in our Brillouin experiments were just below 7 kG. Figure 2 a shows a spectrum obtained for a field of 1.5 FIG. 1. Magnetization for applied fields in plane along the easy- kG along the easy axis. Following Ref. 10, the features in the axis solid line and hard-axis open circles directions. The line spectrum are identified as a surface wave S or a bulk stand- through the hard-axis data is a fit to Eq. 1 and the fitted values are ing spin wave Bn , where the subscript n denotes the number given in Table I. of nodes across the film thickness. These latter modes corre- were made for in-plane fields H in the easy and hard direc- spond to guided ``bulklike'' magnons where the wave vector tions. Figure 1 shows the measured magnetization loops for a is quantized by the finite thickness of the sample. The sur- 50-nm Co film with H applied along the hard circles and face mode S is identified by a large Stokes/anti-Stokes asym- easy full line axes. The full line superimposed on the metry in the scattering intensities. circles is a fit to the hard-axis data based on the energy The surface mode corresponds to the Damon-Eshbach expression: mode on a semi-infinite ferromagnet. The Damon-Eshbach mode only propagates in directions near 90° as measured E K from the direction of the saturation magnetization. There is a 1sin2 K2 sin4 MHcos H , 1 critical propagation angle below which the Damon-Eschbach where K1 and K2 are the first- and second-order anisotropies, mode no longer exists. For the thin-film geometry considered is the angle between the magnetization M and the c axis, here, this critical angle corresponds to a change from local- and H is the angle between the applied field H and the c ized to bulk spin-wave behavior. This can be seen in the axis. The films are magnetized in-plane, so shape anisotro- spectrum shown in Fig. 2 b where a small field of 270 G is pies serve to keep the magnetization in-plane but do not oriented along a hard direction. With this small field strength otherwise enter the energy in Eq. 1 . The magnetization was the magnetization is nearly aligned along the same direction obtained from the saturation value, and the anisotropies were that the measured spin waves propagate. This means that no obtained by fitting the hard-axis magnetization curve. The surface mode exists and is evidenced by the nearly sym- resulting values were in good agreement with those of bulk metrical scattering intensities, indicating no localization to Co and are listed in Table I. Note that in order to reproduce the surface. the curvature of the magnetization in the hard direction it It is interesting to compare the behavior of the surface was necessary to include the higher-order anisotropy K2 . mode to the magnetostatic case considered in Ref. 15. The However, the value obtained for K2 determined from fitting neglect of exchange in the magnetostatic limit is a standard the hard-axis magnetization depends sensitively on the ori- approximation for thick films with a consequence that the entation of the sample and slight misalignments or mosaic surface mode merges into a bulk manifold at a critical value spread of the c axis with respect to the applied field produces of the angle describing the propagation direction. Anisotro- substantial changes in the fitted values. We shall return to pies can strongly influence the frequencies of the surface this point in the last section. mode and bulk band limits, leading to several interesting TABLE I. Magnetic parameters of Co determined by Brillouin light scattering BLS , neutron scattering NS , and magnetization Mag . Ms K1 K2 D Expt. G g (106 erg/cm3) (106 erg/cm3) (meV Å2) Mag 1430 100 3.4 0.4 1.1 0.4 this paper BLS 1350 50 2.13 0.04 3.4 0.1 0.80 0.05 460 70 this paper NS Ref. 2 1420 540 40 Mag Ref. 17 1450 3.0 1.3 BLS Ref. 8 1.86 0.02 3.4 0.4 435 35 BLS Ref. 9 1330 2.16 0.02 340 75 56 EXCHANGE AND ANISOTROPY EFFECTS ON SPIN . . . 2619 magnetization determined by competition of the anisotropies and the applied field Eq. 1 ; 2 writing the equations of motion describing torques acting on spins deviating slightly from the equilibrium orientation due to effective fields Heff ; 3 solving the linearized equations of motion, together with the appropriate electromagnetic and exchange boundary conditions, to obtain the frequencies of long-wavelength spin excitations. Effective fields are found from Heff "mF where F is an energy including exchange contributions due to spatial variations in the magnetization, the first- and second-order anisotropies, and the Zeeman energy. The time and spatially varying part of the magnetization is denoted by the vector m. Effects due to shape demagnetizing fields enter via Max- well's equations. The fields are evaluated with the static magnetization along the equilibrium direction specified by which is in general different from H except along special directions. Writing D as the usual exchange stiffness and q as the in- plane spin-wave wave vector, the torque equations are dmx /dt H cos H 2K1 /M 1 2 sin2 4K2 /M sin22 sin2 D/ q2 d2/dy2 my Mhy , 2 dmy /dt H cos H 2K1 /M cos2 K FIG. 2. Example Brillouin light-scattering spectra with the ap- 2 / M sin22 D/ q2 d2/dy 2 mx plied magnetic field along easy- and hard-axis directions. A 1.5 kG Mhx . 3 field is in the easy-axis direction in a and a small 270 G in-plane field is applied in a hard-axis direction in b . The large Stokes/anti- Here mx and my are the time and spatially varying compo- Stokes asymmetry visible in a clearly identifies the surface mode. nents of the magnetization along directions perpendicular to Note the absence of a surface mode in b due to the orientation of the equilibrium (z) direction and y is along the surface nor- the saturation magnetization along the direction of propagation. mal. The gyromagnetic ratio is 2 g B /h. The anisot- ropy contributions to the effective fields are consistent with phenomena as discussed in Ref. 15. In the present thin-film the energy as written in Eq. 1 . Dynamic demagnetizing case, there is no manifold of nearly degenerate bulk modes. fields, which originate from Maxwell's equations, are repre- Instead the surface mode can be uniquely identified as the sented by hx and hy . This means that in order to completely lowest-order bulk spin wave. The effect of dipolar interac- define the problem, in the magnetostatic limit the complete tions is to shift the energy of this mode and localize it to one equations of motion must also include "*(h 4 m) 0. surface of the film. These effects are clearly evident in the Figure 3 shows the frequency of the three modes observed frequency shifts and intensity ratios of the Brillouin spectra. in Fig. 2 as a function of field along the easy 3 a and hard We now describe the data analysis. At 50 nm thick, the 3 b axes. For H applied along the easy axis the frequency of Co film supports bulk standing spin waves and a surface all three modes increase monotonically with field; qualita- magnon with comparable frequencies. The frequencies of the tively this reflects the fact that at zero field the magnetization modes therefore include contributions from exchange, Zee- is already aligned along the easy axis so that the applied field man terms, anisotropies and ``demagnetizing'' fields, all of does not compete with the anisotropy but simply contributes which must be properly described in the theory used for the an additional torque. For H along the hard axis the frequen- data analysis. An appropriate theory is described in Refs. cies of all three modes exhibit nonmonotonic behavior re- 11­13 and was modified for this problem to include uniaxial flecting the ``reorientation'' of the sample magnetization as anisotropies and arbitrary in-plane orientations of the mag- the field is applied. It is interesting to note that, in our scat- netization with respect to the applied field. We note that tering geometry where q is perpendicular to H, the B0 mode previous related calculations for spin waves with an empha- which as discussed above has no nodes across the film sis on anisotropies presented in Refs. 14 and 15 did not in- thickness propagates parallel to M at zero field. At high clude exchange contributions to the spin-wave frequencies. fields B0 no longer exists but is replaced by a surface mode The theory discussed here is a generalization of these previ- that propagates perpendicular to M. The bulklike peaks B2 ous magnetostatic studies for thin-film geometries. and B3 whose wave vectors lie close to the surface normal The mode frequency calculations are based on a long- are always propagating perpendicular to M. The details of wavelength semiclassical approximation that involves the how the B0 mode transforms to a surface mode are discussed following steps: 1 finding the equilibrium orientation of the below. 2620 M. GRIMSDITCH, ERIC E. FULLERTON, AND R. L. STAMPS 56 analysis are listed in Table I together with estimated errors based on the range of allowable values that produce reason- able fits to the data. The range of applied fields between zero and 5 kG from the hard-axis data is interesting because of the behavior of the surface mode. As the field is increased the magnetization rotates towards the direction of H and away from q. During this process the lowest frequency mode transforms into the surface mode. Localization of the long-wavelength spin wave to the surface is due to magnetostatic energies, and these also increase the frequency of the mode. In this case, the increase in frequency leads to a hybridization between the surfacelike mode and the lowest-frequency bulk mode (B1). Expected details of the mode ``crossing'' with H applied along the hard axis are displayed in Fig. 4. Calculated spec- tra are shown in a and were calculated along the lines of the theory given in Ref. 16 using the parameters given above and a small damping term. The calculated behavior reproduces the main features observed in the actual data: at low fields the Stokes/anti-Stokes asymmetries indicate that all modes are bulklike whereas at fields above 3 kG a clearly identifi- able surface mode appears. The scattering intensities can be understood in greater de- tail by studying the amplitudes of the fluctuating magnetiza- tion as a function of position in the film associated with the different spin-wave modes. The amplitudes in the plane of the film the mx of Eqs. 2 and 3 for the two lowest frequency spin waves are shown as a function of depth in Fig. 4 b . The mode profile, determined by the thin-film ge- ometry, allows the modes to be classified by the number of FIG. 3. Spin-wave frequencies as functions of applied field. Ex- nodes n. At low fields it is the lowest-frequency mode which perimental results and theory are shown for the field aligned in the corresponds to n 0. This mode is localized to one surface in-plane easy-axis a and hard-axis b directions. The large circles when propagating near perpendicular with respect to the are data points taken from the spectra and the dotted lines are cal- culated using the theory described in the text. A mode crossing is magnetization, as discussed before. Between 5 and 5.5 kG evident in the hard-axis data b . This is due to the increase in the node structure changes as the magnetization rotates and it frequency of the lowest order bulk mode as the propagation direc- is the second mode which has n 0 indicating that it has tion changes. The mode becomes increasingly more surfacelike and become the surfacelike magnetic excitation. crosses the B1 bulk mode between 3 and 5 kG. It is interesting to note that in order for the node structure The small dots in Fig. 3 are the results of calculations to switch from one mode to another with increasing applied where the individual parameters were determined as follows. field, it is necessary for the amplitude at one surface to van- The slope of the frequencies as a function of field along the ish for some applied field strength. A maximum in the easy direction yields g. The position of the frequency mini- Stokes/anti-Stokes ratio can then be expected. The spectra mum 6.5 kG , which corresponds to the field at which the shown in Fig. 4 a exemplify this behavior; at 5 kG the first magnetization aligns with the field along the hard axis, is second mode is notably absent on the Stokes anti-Stokes determined by K side of the spectrum. The intensities below 4 kG do not 1 and K2 . This region is very sensitive to the orientation of the crystal, and the consequences will be indicate any strongly localized surface mode, as expected. discussed in Sec. III. The difference in the ``hard'' and Between 5 and 6 kG, where the two lowest modes interact, ``easy'' frequencies above saturation is also determined by large Stokes/anti-Stokes ratios appear. Above 6 kG the inten- K sity ratios are consistent with the middle frequency mode 1 and K2 . The overall frequency scale is determined mainly by 4 M. being a strongly localized surface mode. In addition to the anisotropies, frequency differences pro- Experimental verification of the above conclusions is vide information about exchange. The 50-nm Co film thick- hampered by the closeness of the two modes which makes ness allows the measurement of two bulk spin-wave frequen- them experimentally unresolved. Furthermore it is not clear cies in addition to the surface-mode frequency. This provides what the effects quadratic coupling between the magnetiza- a sensitive measure of the exchange since the difference in tion and the light might play in the calculated spectra. The the frequencies of the two bulk modes is determined by D results presented above compare well with those presented in whereas the surface mode is insensitive to D. Values of the Ref. 4 where mode repulsion between the two lowest modes various parameters determined from the light-scattering was also observed in the angular dependence for an iron film. 56 EXCHANGE AND ANISOTROPY EFFECTS ON SPIN . . . 2621 FIG. 5. Effects of a small misalignment on the hard-axis mag- netization measurements. The calculated ac susceptibility and mag- netization inset are shown as a function of applied field for H 89 and 90°. The open circles are the ac-susceptibility data for a Co film. Even the small misalignment of 1° has a dramatic effect on the susceptibility near saturation. Eq. 1 when H makes an angle H with the easy axis is dE/d 0 2K1sin cos 4K2sin3 cos MH sin H . 4 The measured magnetization is given by M cos( H). It is clear that for H 90° sin( H) cos( ) Eq. 4 can be simplified by dividing by cos and the solution requires solving a cubic equation in sin . However, since this solu- tion requires dividing by cos it is no longer strictly valid above saturation when 90°. The solution to Eq. 4 for a general H requires numerical techniques. The inset of Fig. 5 shows the calculated magnetization for H 90° and 89° near saturation. For H 90°, there is a well defined satura- tion field HS given by (2K1 4K2)/M. However, if the field is slightly misaligned from the hard axis, the magnetization approaches saturation asymptotically. This difference near saturation is clearly seen in the ac susceptibility shown in Fig. 5. For H 90° there is a discontinuity in the suscepti- bility at HS . For H 89° the discontinuity is rounded and in close agreement with the experimental results for a Co film. Although this rounding at saturation is consistent with mis- alignment of the film, the present results cannot rule out that other effects, e.g., an in-plane mosaic spread of Co crystal- lites, may also contribute to the rounding. It is, however, clear from Fig. 5 that anisotropy parameters especially K2 extracted from magnetization loops will be sensitive to align- FIG. 4. Calculated scattering intensities and mode profiles. Cal- ment even in the absence of mosaic spread. culated Stokes/anti-Stokes intensities are examined in a for the A comparison of the magnon's calculated field depen- crossing region seen in Fig. 3 b . A maximum in the surface mode dence when H is applied at 90° and 89° is shown in Fig. 6. intensity ratio is predicted for applied fields where the two lowest Here also it is clear that the frequency minima are dramati- frequency modes hybridize. This is due to the change between cally affected by slight sample misorientation. The position surface- and bulklike character of the two modes as illustrated in of the minimum is also slightly displaced to higher fields. It mx profiles shown in b . The mx amplitudes shown in b are is interesting to note that the sharpness of the minimum in- plotted as functions of position across the Co film. The horizontal dicates the accuracy of the alignment: the dip is much more lines are zero amplitude references. pronounced with good alignment. It is important to remem- ber however that the magnon frequencies at low fields are III. EFFECTS OF MISALIGNMENT also sensitive to the anisotropies. K2 is especially important in determining the low-field frequencies of the hard-axis As mentioned in the previous section the numerical values data, so that the light-scattering measurements provide valu- obtained for the magnetic parameters depend strongly on able additional data not contained in pure magnetization sample orientation. The equilibrium condition obtained from measurements. 2622 M. GRIMSDITCH, ERIC E. FULLERTON, AND R. L. STAMPS 56 values obtained by magnetization measurements and Bril- louin light scattering are consistent. The small differences between the anisotropies we measured and those typical of bulk Co are most likely due to subtle changes induced by growth strains or other thin film effects. More interesting are the lower values for D obtained in our and at least two other Brillouin works8,9 compared to a value of 500­ 580 meV Å2 obtained from neutron-scattering and magnetization measurements on bulk Co.2,17 This dis- crepancy was in fact first noticed by Vernon et al., in an earlier Brillouin light-scattering study of Co films with the suggestion that the average D measured in light scattering might be reduced in polycrystalline samples.9 A more likely explanation for the discrepancy, recently brought to our at- tention by Cochran,18 is based on pinning effects. Pinning FIG. 6. Effects of a small misalignment on the spin-wave fre- modifies the location of the surface antinodes of the standing quencies. Calculated spin-wave frequencies are shown as a function spin waves thereby changing their effective wavelength. of applied field for H 89 and 90°. The dark squares are for per- Since the Brillouin frequencies depend on Dq2 this translates fect alignment along the hard axis ( H 90°) and the open circles into a decrease in the value extracted for D. The very limited are for a 1° misalignment ( H 89°). The effect of misalignment is available information on pinning precludes, at present, re- to shift the position of the frequency dip and reduce its sharpness. finements of the calculations to include pinning effects in a reliable manner. The main conclusion is that the accurate determination of In conclusion, we have measured anisotropies, magnetiza- the anisotropies is largely limited by alignment. The limit tion, and exchange stiffness constants for a uniaxial Co film currently achievable in our laboratory corresponds to an un- grown with an in-plane c axis. Results for the anisotropies certainty of approximately 50 G for the values of the ef- found from Brillouin light-scattering data were consistent fective anisotropy field, as estimated by the uncertainty in with those determined by magnetization measurements. The the position of the minimum described above. This uncer- exchange stiffness was found to agree with previous light- tainty is less consequential for the determination of other scattering results on Co films, but most Brillouin results dis- parameters M, g, and D, since these are primarily measured agree with earlier neutron-scattering measurements on bulk from aspects of the data that are not strongly dependent on Co. Finally, we have also studied a hybridization between alignment, as discussed in Sec. III. The error in D originates spin-wave modes driven by the long-range dipolar interac- mainly from the uncertainty ( 5%) in the film thickness. tion. As a result, a change from bulk- to surfacelike character was observed as a function of applied field that strongly af- fected Stokes/anti-Stokes ratios. The experimental observa- IV. DISCUSSION AND CONCLUSIONS tions were consistent with theoretical calculations for the light-scattering intensities. The magnetic parameters determined with Brillouin light scattering in this study are summarized in Table I. Values obtained in the present study using magnetization measure- ACKNOWLEDGMENT ments and literature values for bulk Co are also given. Ex- The work at ANL was supported by the U.S. Department cept for one determination8 all values of g are within the of Energy, Basic Energy Sciences-Materials Sciences, under quoted uncertainties. It can be seen that our experimental Contract No. W-31-109-ENG-38. 1 M. B. Stearns, J. Appl. Phys. 53, 2436 1982 . 9 S. P. Vernon, S. M. Lindsay, and M. B. Stearns, Phys. Rev. B 8, 2 G. Shirane, V. J. Minkiewicz, and R. Nathans, J. Appl. Phys. 39, 4439 1984 . 383 1968 . 10 M. Grimsditch, A. Malozemoff, and A. Brunsch, Phys. Rev. Lett. 3 J. Sandercock, in Light Scattering in Solids III, edited by M. 43, 711 1979 . 11 Cardona and G. Guntherodt Springer, Berlin, 1982 , p. 173. R. E. Camley and D. L. Mills, Phys. Rev. B 18, 4821 1978 . 12 4 R. E. Camley and M. Grimsditch, Phys. Rev. B 22, 5420 1980 . R. L. Stamps and B. Hillebrands, Phys. Rev. B 43, 3532 1991 . 13 5 S. Subramanian, X. Liu, R. L. Stamps, R. Sooryakumar, and G. P. Grunberg, C. M. Mayr, W. Vach, and M. Grimsditch, J. Magn. A. Prinz, Phys. Rev. B 52, 10 194 1995 . Magn. Mater. 28, 319 1982 . 14 B. Schneider, Phys. Status Solidi B 51, 325 1972 . 6 G. Rupp, W. Wettling, R. S. Smith, and W. Jantz, J. Magn. Magn. 15 M. J. Hurben and C. E. Patton, J. Magn. Magn. Mater. 162, 39 Mater. 45, 404 1984 . 1996 . 7 P. Kramas, F. Lauks, R. L. Stamps, B. Hillebrands, and G. 16 J. R. Dutcher and J. F. Cochran, J. Magn. Magn. Mater. 72, 307 Guntherodt, Phys. Rev. Lett. 69, 3674 1992 . 1988 . 8 X. Liu, M. M. Steiner, R. Sooryakumar, G. A. Prinz, R. F. C. 17 R. Pauthenet, J. Appl. Phys. 53, 8187 1982 . Farrow, and G. Harp, Phys. Rev. B 53, 12 166 1996 . 18 J. Cochran private communication .