PHYSICAL REVIEW B VOLUME 55, NUMBER 18 1 MAY 1997-II Oscillatory biquadratic coupling in Fe/Cr/Fe 001... A. J. R. Ives, J. A. C. Bland, R. J. Hicken, and C. Daboo Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, United Kingdom Received 13 August 1996; revised manuscript received 3 December 1996 Polar Kerr measurements have been used to measure the dependence of the biquadratic coupling strength B12 on Cr thickness in an Fe/Cr/Fe trilayer. The overall behavior, which consists of a maximum coupling strength at dCr 5 Å 3.5 ML with a falloff at greater Cr thicknesses, is found to be consistent with in-plane Kerr and Brillouin light-scattering measurements performed on the same sample. The polar Kerr measurements suggest additionally that B12 increases from zero near zero Cr thickness, and that it oscillates in magnitude after the first peak, with a second peak in B12 occurring at about dCr 12 Å 8.3 ML . The positions and heights of the first and second biquadratic coupling maxima, in relation to the first bilinear coupling maximum, show excellent agreement with previous measurements by Ko¨bler et al. of the biquadratic coupling behavior in Fe/Cr/Fe, and also show good agreement with the predictions of an intrinsic biquadratic coupling mechanism due to Edwards et al. S0163-1829 97 09117-0 I. INTRODUCTION The usual bilinear coupling is proportional to the cosine of the angle between the magnetizations M1 and M2 of Antiferromagnetic AFM interlayer exchange coupling adjacent ferromagnetic layers, and has the form was discovered in Fe/Cr/Fe structures by Gru¨nberg et al.,1 2A12M 1*M 2 , where M 1 and M 2 are unit vectors along the and the Fe/Cr system has since become particularly impor- directions of M1 and M2 . It is known that there may also tant to the development of our understanding of the mecha- exist a so-called biquadratic interlayer coupling which is pro- nism of interlayer exchange coupling.2 Chromium is an in- portional to the cosine of the angle squared and can favor a teresting choice of spacer material because bulk Cr is known 90° alignment of the magnetizations of the ferromagnetic to exhibit incommensurate spin-density-wave antiferro- layers. The biquadratic coupling energy has the form magnetism.3 Fe/Cr structures grown on Fe 001 whiskers are 2B12(M 1*M 2)2. Biquadratic coupling was discovered in believed to have the flattest interfaces currently obtainable, Fe/Cr/Fe 001 magnetic trilayers by Ru¨hrig et al.12 using and it was in such structures that the interlayer coupling was magneto-optic Kerr effect microscopy, and has since observed to oscillate with a period of approximately two Cr been found to occur in a number of other systems, monolayers ML in addition to the previously discovered such as Fe/ Al,Au /Fe 001 ,13 Fe/ Cu,Ag /Fe 001 ,14 and long-period oscillations of about 18 Å 12.5 ML .4,5 It was NiFe/Ag/NiFe.15 The biquadratic coupling strength in subsequently demonstrated that these short-period coupling Fe/Cr/Fe structures has been observed to be of comparable oscillations are correlated with the AFM ordering of the Cr.6 magnitude to the bilinear coupling strength.8,16 There is now general agreement that the period of the cou- Several theories have been proposed to account for the pling oscillations is determined by the geometry of the Fermi biquadratic coupling, some of which are intrinsic to the elec- surface of the spacer material in the direction perpendicular tronic structure of the multilayer system, and are referred to to the layers, so that the discovery by Fullerton et al.7 that as intrinsic theories, and some of which rely on effects other spacer layers of two different orientations yielded identical than the electronic structure, such as structural imperfections, values for the strength, oscillation period and phase of the and are referred to as extrinsic theories. Three intrinsic theo- long-period oscillations was somewhat surprising. Studies ries due to Edwards, Ward, and Mathon,17 Barnas´,18 and using Fe whisker substrates have provided information con- Erickson, Hathaway, and Cullen19 predict that the biqua- cerning the phase of the short-period oscillations.8 Since an dratic coupling oscillates as a function of interlayer thickness odd number of Cr monolayers are expected to cause ferro- and decays in amplitude with increasing interlayer thickness magnetic alignment of adjacent Fe layers, the observation by in a similar way to the bilinear coupling. In all of Refs. Heinrich et al.8 and by Unguris, Celotta, Pierce6 of AFM 17­19, the phase and period of the biquadratic oscillations coupling after the growth of 5 ML ( 7 Å) of Cr grown on are found to be different from those of the bilinear coupling Fe whisker samples was again surprising, and is associated oscillations, and the amplitude of the biquadratic coupling is with a phase slip of the AFM ordering of the Cr. Recent found to be much lower than that of the bilinear coupling. A studies of the Fe-Cr interface using scanning tunneling problem with some of the intrinsic theories has been that the microscopy9 STM and angle-resolved Auger size of the biquadratic coupling predicted is too small to spectroscopy10 show that strong exchange interdiffusion of account for the experimentally observed values,17,18 and in Fe and Cr occurs. This process may be responsible for the some cases it is too small by orders of magnitude.19­21 A phase slip observed at low Cr thicknesses, a view which is consequence of the different phase and period of the biqua- supported by calculations of exchange coupling in inter- dratic coupling is that at certain interlayer thicknesses, where mixed Fe/Cr interfaces using a tight-binding scheme.11 the bilinear coupling passes through zero, the biquadratic 0163-1829/97/55 18 /12428 11 /$10.00 55 12 428 © 1997 The American Physical Society 55 OSCILLATORY BIQUADRATIC COUPLING IN Fe/Cr/Fe 001 12 429 coupling may be larger than the bilinear coupling, even when the amplitude of the oscillations in the biquadratic coupling is many times smaller than it is for the bilinear coupling. One extrinsic theory due to Demokritov et al.,22 in which biqua- dratic coupling arises as a result of the magnetic-dipole field created by magnetic layers with roughness, predicts an expo- nential falloff of the biquadratic coupling with no oscillation as a function of interlayer thickness. Two well-known theories of biquadratic coupling are those of Slonczewski. The first of these, called the fluctua- tion mechanism, is an extrinsic mechanism associated with spatial fluctuations of bilinear coupling due to terraced varia- tions of spacer thickness in nonideal specimens with roughness.23 The second is an intrinsic mechanism, called the loose-spins model, which postulates that the biquadratic exchange coupling is mediated by localized atomic-electron states at the interfaces of the spacer layer.24 Both these theo- ries were shown to predict magnitudes of biquadratic cou- pling strengths that were in good agreement with experimen- tally observed values. Very strong near-90° coupling has recently been reported in CoFe/Mn/CoFe sandwich struc- tures, without any evidence for bilinear coupling.25 In this case, the field dependence of the magnetization was found to be well fitted by an extrinsic model assuming a type of cou- pling energy of the form E C ( 1 2)2 C ( 1 2 )2. So far, while many theories have been proposed to ac- count for the biquadratic coupling, very little experimental data has been published to show how the biquadratic cou- pling varies in strength with nonmagnetic interlayer thick- ness in real magnetic/nonmagnetic multilayer systems. In the present paper, we seek to redress this imbalance by describ- ing how polar Kerr measurements have been used to inves- tigate the dependence of the biquadratic coupling on Cr in- terlayer thickness in a Fe/Cr/Fe trilayer grown on Ag/ GaAs 001 . The Cr thickness dependence of the biquadratic coupling obtained using polar Kerr measurements is com- pared first with previously published estimates of the biqua- FIG. 1. a The in-plane easy axis saturation field as determined dratic coupling strength obtained from in-plane Kerr and from in-plane Kerr magnetometry is plotted as a function of Cr thickness for the trilayer. b The perpendicular saturation fields Brillouin light-scattering BLS measurements on the same H closed circles and H open circles are plotted versus Cr sample;26 second, with the published experimental results of s s interlayer thickness for the trilayer. c The normalized perpendicu- Ko¨bler et al.;16 and finally with the results of a theoretical lar saturation fields H closed circles and H open circles analysis by Edwards, Ward, and Mathon17 arising from an s s are plotted versus Cr interlayer thickness for the trilayer. intrinsic biquadratic coupling mechanism. The Fe/Cr/Fe trilayer studied here was grown with struc- ture Cr 20 Å /Fe 20 Å /Cr 0­40 Å /Fe 20 Å /Ag 600 Å / polar Kerr data. Two regions of AFM coupling exist: the first Fe 15 Å /GaAs 001 . The Cr spacer layer was grown with extends from a Cr thickness of 4 Å 2.8 ML to 15 Å 10.4 the substrate held at room temperature. Although this is ML , while the second begins at a Cr thickness of 20 Å 13.9 known to result in less well-defined interfaces and in the ML and continues to the end of the wedge where the Cr suppression of the short-wavelength oscillations in A layer is 40 Å 27.8 ML thick. The form of the coupling is 12 , such structures can offer valuable insights into the origin of the predominantly due to the long-period coupling oscillations, coupling behavior. In particular, the comparison with the be- but vestiges of the short period coupling may be seen as havior obtained for structures prepared at elevated tempera- shoulders on both sides of the main AFM coupling peak at 7 ture is important. The procedures used for the ultrahigh Å 4.9 ML . From the separation of the maxima of the first vacuum growth of the trilayer, together with easy-axis in- and second bilinear coupling peaks in Fig. 1 a , a value of plane Kerr magnetization curves and BLS measurements ob- about 18 Å 12.5 ML was obtained for the long period of tained as a function of Cr thickness, have been described oscillation in the coupling, which agreed well with the value previously in Ref. 26. The variation of the in-plane easy-axis of 12 1 ML (17.3 1.4 Å) deduced by Pierce et al.27 for saturation field with Cr thickness, showing the oscillatory samples grown on Fe whisker substrates. For Cr thicknesses nature of the coupling, has however been reproduced again less than 4 Å 2.8 ML and between 16 and 20 Å 11.1 and in Fig. 1 a , since it will be needed for comparison with the 13.9 ML , the easy-axis loops are square, indicating that the 12 430 A. J. R. IVES, J. A. C. BLAND, R. J. HICKEN, AND C. DABOO 55 Fe layers are either ferromagnetically FM coupled or un- from the polar Kerr magnetization curves are plotted in Fig. coupled. 1 b . For the first three data points on the left-hand side of the plots of H s and Hs the Cr interlayer has zero thickness II. POLAR KERR MAGNETOMETRY so the effective Fe layer thickness is 40 Å. The Cr wedge begins after the third data point, and between the third and Polar Kerr magnetization curves were obtained as a func- tion of Cr thickness from the Fe/Cr/Fe trilayer using an ex- fifth points Hs and Hs both fall abruptly by about 1.6 kOe. perimental arrangement shown in Ref. 28. For the 20-Å Cr As a partial layer of Cr is introduced into the middle of the layers in the sample studied here, the normal to the film 40 Å Fe layer, the saturation field decreases due to the in- surface is a hard direction of magnetization so that large creased interface anisotropy fields associated with the extra applied fields are necessary to saturate the Fe layers in the interfaces with the Cr layer. The saturation field continues to perpendicular direction. The analysis of the polar Kerr mag- decrease with Cr spacer thickness until antiferromagnetic netization curves is thus simplified by the fact that magneti- coupling begins to be established. zation can nearly always be assumed to proceed by coherent It is interesting to note that a minimum in the saturation rotation, and by the fact that the remanence is always zero or field occurs at approximately 1 ML 1.44 Å Cr thickness. almost zero. Davies et al.9 deduced that FM coupling persists up to The 12 mm long sample was placed in air and at room 3 ML 4.3 Å Cr thickness but were unable to probe the temperature at the end of an insert tube, close to the center of variation in coupling strength since no fields could be ap- a 7 T superconducting magnet, with the magnetic field di- plied in the scanning electron microscopy with polarization rected perpendicular to the sample surface. An intensity sta- analysis experiment that they performed. For a sharp inter- bilized He-Ne laser beam was focused on the sample down face, ferromagnetic coupling of the Fe layers would be ex- to a spot size of 0.2 mm, at near normal incidence, and pected to occur at a Cr thickness of 1 ML since a monolayer moved across it using a plane and a concave mirror both of Cr is thought to order antiferromagnetically with a neigh- mounted on a micrometer stage. This arrangement was de- boring Fe layer.30 However, the STM and Auger spectros- signed so that the laser light did not have to pass through any copy studies in Refs. 9 and 10, respectively, indicate that the windows or lenses in the vicinity of the field, thus eliminat- spacer layer will correspond to a mixture of Cr and Fe at ing problems due to Faraday rotation or birefringence. Polar 1 ML thickness. Stoeffler and Gautier11 have shown that Kerr measurements were performed as a function of position along the wedge in order to sample the magnetization curves for a 2 ML 2.88 Å period ordered alloy, consisting of al- at the different Cr interlayer thicknesses. Nearly all the per- ternating layers of Fe0.75Cr0.25 and Fe0.25Cr0.75, the layers are pendicular magnetization curves obtained using the polar ferromagnetically aligned. Thus the minimum we observe in Kerr effect have a background contribution which varies lin- the polar saturation field at 1 ML is likely to be due to early with field, and which is always subtracted off before ferromagnetic coupling between the Fe layers mediated by a the curves are normalized to the saturation value of the mag- ferromagnetically aligned FeCr spacer layer of monolayer netization. In addition, a very slight distortion of the magne- thickness. For growth close to room temperature, only the tization curves can occur because the relationship between first Cr layer is significantly intermixed according to the Au- the intensity recorded after the analyzing polarizer and the ger spectroscopy studies of Heinrich et al.10 Our results magnetization, is not perfectly linear. This distortion is as- show that as the spacer thickness is further increased beyond sumed to have a negligible affect on the magnetization 1 ML, the ferromagnetic coupling strength is rapidly re- curves for the purposes of the analysis carried out here. duced, first becoming AF at about 2.8 ML 4 Å . This is consistent with a corresponding rapid increase in the Cr con- The conventional perpendicular saturation field H s has centration of the spacer layer, as expected from the intermix- been estimated for the wedged trilayer by taking the inter- ing studies for Cr growth close to room temperature. The section of the perpendicular magnetization curve with second and successive Cr layers within the spacer should be M/Ms , where is chosen to be the highest value of almost 100% Cr and therefore the antiferromagnetic ordering M/M s at which the values of Hs obtained for all thicknesses will become established, while the moment of the first Cr of the wedged layer are not seriously affected by noise in the layer will remain parallel to the bottom Fe layer magnetiza- magnetization curves see Refs. 28, 29 , and for the Fe/Cr/Fe tion. This leads to short-period oscillations in the coupling trilayer has been chosen to have the value 0.96. This pro- strength which are only weakly seen in our sample due to the cedure was necessary because of the asymptotic approach of rougher interfaces in comparison with those of structures the magnetization to saturation and because of noise fluctua- grown at elevated temperatures. The existence of such an tions in the data, although it does lead to H s being an un- intermixed first layer is consistent with the phase slip which derestimate of the true saturation field. As in Refs. 28 and leads to the peak in the bilinear coupling occurring at 5 ML 29, a second saturation field H s 1/ 0 has also been used ( 7 Å), seen more clearly in Fe-whisker samples with near- to analyze the magnetization curves, where 0 is the initial perfect interfaces.6,8 magnetization gradient of the perpendicular magnetization It should be noted that the behavior of the perpendicular curve calculated using reduced units of M/Ms . Evaluating saturation fields we observe is strikingly different from that this saturation field should, in principle, allow further infor- of the easy-axis in-plane saturation field in Fig. 1 a , for mation to be extracted from the perpendicular magnetization which only a small change in the coercive field was observed curve, as its dependence on the quantities which vary with at the start of the Cr wedge. The dramatic fall in saturation the wedge thickness is different from that of H s . field observed using polar Kerr magnetometry shows that The perpendicular saturation fields H s and Hs obtained this technique is far more sensitive to the effect of a 55 OSCILLATORY BIQUADRATIC COUPLING IN Fe/Cr/Fe 001 12 431 magnetic/nonmagnetic interface than the more conventional in-plane Kerr magnetometry. After the fifth data point, both saturation fields H s and H s increase as the bilinear coupling changes from being ferromagnetic to antiferromagnetic. It is observed that after the fifth data point, H s starts to increase dramatically a few data points before H s starts to increase. This is significant, and it will be seen later on that the resulting large difference in the values of H s and Hs at dCr 5 Å 3.5 ML is due to a peak in the biquadratic coupling at this Cr thickness. The first AFM coupling region is clearly shown by a large peak centered on about dCr 7 Å 4.9 ML for the plots of both H s and Hs , in approximate agreement with the in-plane data in Fig. 1 a . The second FM coupling region can also be FIG. 2. A schematic diagram showing the magnetizations of the identified by the minimum in the plots of H s and Hs be- two magnetic layers and the angles they make with the field, H, tween about 15 and 20 Å 10.4 and 13.9 ML , again in ap- applied along the film normal. The dashed lines indicate the plane proximate agreement with the in-plane data in Fig. 1 a . The which includes the surface normal and a fixed direction in the film second AFM coupling region is not well defined, however, in plane, to which the magnetizations are assumed to be confined for the plots of H the purposes of calculation. s and Hs , with the values of Hs and Hs both increasing unexpectedly towards the end of the wedge where the Cr thickness is largest. Indeed the values of H system is shown in Fig. 2. The zero of s 1 and 2 is assumed and H to be the surface normal, with both angles being allowed to s are generally much larger than expected for the Cr thicknesses after the first AFM coupling peak. It will be seen vary from to . in the following section that, in addition to being sensitive to The existence of any in-plane uniaxial anisotropy is ig- the coupling, H nored, but the presence of a first-order cubic anisotropy is s and Hs are also sensitive to any changes that might occur in the magnetocrystalline anisotropies or in assumed, denoted by K1,j for layer j 1,2. One of the three the larger in this case interface anisotropies as the inter- cubic easy axes is parallel to the film normal, while the other layer thickness varies. This is different to the situation for two are in the plane of the film. It will be assumed that for in-plane Kerr measurements in which the saturation field de- each magnetic layer, one of the in-plane cubic easy axes is pends only on the coupling and the relatively small magne- aligned with the plane through the surface normal to which tocrystalline anisotropies. Since the coupling is weak at large the magnetizations are taken to be confined. Thus the cubic Cr thickness, the unexpected increases in H anisotropy energy per unit area, for magnetization perpen- s and Hs are dicular to the plane, for a 001 surface for layer j 1,2, is probably associated with variations in interface anisotropy with Cr thickness that are associated with the details of the K structure of the Fe/Cr interface between the spacer and the E 1,jd cubic,001, j upper Fe layer. 4 sin22 j . 1 A demagnetizing energy per unit area for layer j, and an III. RELATING POLAR SATURATION FIELDS interface anisotropy energy per unit area are assumed, having TO COUPLING CONSTANTS the forms It is possible to find approximate relationships between 1 the exchange coupling strengths A 2 12 and B12 and the satura- Edemag,j 2 0Mjdjcos2 j tion fields H s and Hs by assuming a coherent rotation model for magnetization reversal, as we shall now describe. and The calculations presented here are reproduced from Ives' Ph.D. thesis Ref. 31 . Einterface,j 2Ki,jcos2 j , 2 where K A. H i, j is the interface anisotropy constant per interface s and coupling strengths for layer j. These two terms together are equivalent to a In order to derive the relation between the conventional uniaxial anisotropy with the hard-axis direction along the perpendicular saturation field, H film normal. s , and the coupling strength for the 001 film plane, a trilayer film is considered, consist- The presence of both bilinear and biquadratic exchange ing of magnetic layers with thicknesses d coupling across the nonmagnetic interlayer, is assumed. For 1 and d2 separated by a nonmagnetic interlayer. The applied magnetic field, ferromagnetic layers j and j 1 separated by a nonmagnetic H, is assumed to be perpendicular to the film surface. The interlayer of thickness tj , the contribution of the coupling angles between the surface normal and the directions of the between layers j and j 1 to the energy of the system per magnetizations M1 and M2 are denoted by 1 and 2 , where unit area is for ease of calculation the magnetizations are assumed to be confined to a plane which includes the surface normal and a E ,j 1 2A12 tj M j*M j 1 2B12 tj M j*M j 1 2, fixed direction in the film plane. A schematic diagram of the 3 12 432 A. J. R. IVES, J. A. C. BLAND, R. J. HICKEN, AND C. DABOO 55 where M j and M j 1 are the unit vectors of the magnetiza- tions of layers j and j 1, respectively. For the moment, the E ajcos j bjcos2 j cjsin22 j j 1,2 coupling constants A12 and B12 will be allowed to take on any real values both positive and negative. 2A12cos 1 2 2B12cos2 1 2 . 6 The general equation for the energy per unit area for a This equation may be differentiated with respect to trilayer in a 001 plane for perpendicular magnetization is 1 and then 2 , giving E a1sin 1 b1sin2 1 2c1sin4 1 E 2 1 0M jd jH cos j 1 dj 2Ki,j cos2 j j 1,2 2 0M j 2A12sin 1 2 2B12sin2 1 2 , 7 K 1,jdj E 4 sin22 j 2A12cos 1 2 a2sin 2 b2sin2 2 2c2sin4 2 2 2B12cos2 1 2 . 4 2A12sin 1 2 2B12sin2 1 2 . 8 To simplify the expressions which follow, the following sub- Stationary points exist for E/ 1 E/ 2 0, and there- stitutions will be made fore for 1 2 0. In order to determine the perpendicular saturation field, the condition which defines the field at which the stationary points at 1 2 0 become unstable is a 2 required. This is j 0M jd H, bj 12 0Mjdj 2Ki,j , 2E 2 2E and 2E2 2 0, 1 2 1 2 K evaluated for 1 2 0. 9 c 1,jd j j 4 . 5 Substitution of the second derivatives of Eq. 6 into Eq. 9 , making use of Eqs. 5 , and replacing H by H s , leads di- The energy per unit area then becomes rectly to the relation 1 A 1 12 2B12 2 0M 1d1 Hs M 1 4Ki,1 / 0M 1d1 2K1,1 / 0M 1 1 1 . 10 0M 2d2 Hs M 2 4Ki,2 / 0M 2d2 2K1,2 / 0M 2 The perpendicular saturation fields of layers 1 and 2 in the 001 , H H , all M absence of coupling would be s1 s2 1 , all M 2 , 4K 2K all d1 , all d2 , all A12 , all B12 . 12 H i,1 1,1 s1 M 1 0M 1d1 0M1 If the two magnetic layers have different saturation fields and such that Hs1 Hs2 , but equal thicknesses and magnetiza- tions defined by d1 d2 d and M1 M2 M, this reduces 4K 2K to H i,2 1,2 s2 M 2 , 001 only. 11 0M 2d2 0M2 1 1 1 Thus, provided the two magnetic layers have different satu- A12 2B12 2 0Md 1 , 001 , H s Hs1 Hs Hs2 ration fields such that H s1 Hs2 , it is found from Eq. 10 that the following relation applies for both positive and nega- H H , M tive values of A s1 s2 1 M 2 , d1 d2 , all A12 , all B12 . 12 and B12 13 1 A 1 12 2B12 2 B. H and coupling strengths 0M 1d1 Hs Hs1 s In order to determine the initial gradient perpendicular 1 1 saturation field, H 1/ , s 0 , the quantity 0 must be evalu- 0M 2d2 Hs Hs2 ated according to the definition 55 OSCILLATORY BIQUADRATIC COUPLING IN Fe/Cr/Fe 001 12 433 B12 as well as on the field and the anisotropies. In this case, substitutions which would apply for all magnitudes of A12 and B12 cannot be made, and the relation between A12 and B 12 and Hs cannot then be determined by this method. If A12 0 and B12 0, the assumption that the magnetiza- tions move within a plane which passes through the surface normal is likely to be valid in most cases and for all values of A12 and B12 . If A12 0 and B12 0, however, it can be shown, by differentiating the bilinear and biquadratic energy terms with respect to an in-plane angle between the magne- tizations for all values of the out-of-plane angle, that this FIG. 3. Diagram showing the relationship between the large assumption is valid for A12 2B12 0 but not for values of angles 1 and 2 and the small angles 1 and 2 , for small mag- B12 of greater magnitude. This restriction on the values of netic fields applied perpendicular to the film plane. The film plane is B12 allowed with this model has been verified to be correct shown by the horizontal dotted line, and the magnetizations are using numerical simulations of the magnetization curves. assumed to be confined to the plane of the page. The sizes of the As illustrated in Fig. 3, the following substitutions are angles 1 and 2 have been exaggerated in the diagram. The solid therefore made: arrows labeled M1 and M2 show the likely arrangement of the magnetizations if both A12 and B12 are negative. The dashed arrow shows the alternative likely position of magnetization M1 if A12 and 1 2 1, 2 2 2, 15 B12 are instead positive. where the upper signs in the second equation refer to the M cases where i A 1d1cos 1 M 2d2cos 2 12 2B12 0 or ii A12 0 and B12 0; the 0 H . 14 M1d1 M2d2 lower signs refer to the case A H 0 12 0 and B12 0; and the case where A12 0 and B12 0 is not catered for here. With The same energy equation as in Eq. 4 will be used here, these substitutions one obtains together with the substitutions made in Eqs. 5 . The first derivatives of Eq. 6 , given in Eqs. 7 and 8 , together with the conditions E/ cos 1 cos sin 1 1 , 1 E/ 2 0, give the values of 1 2 1 and 2 corresponding to stationary points for all values of H, including those points near H 0 that are required for evaluating cos sin 0 . It is not possible, however, to directly extract 2 cos 2 2 2 2 , the exact dependences of cos 1 and cos 2 on H from these equations. It is therefore necessary to calculate approximate sin2 values of cos 1 sin 2 1 sin2 1 2 1 , 1 and cos 2 , using the fact that the magneti- 16 zations in the two magnetic layers make very small angles with the film plane close to H 0. Substitutions for sin2 2 sin 2 2 sin2 2 2 2 , 1 and 2 need to be made such that the substituted angles will have small values near H 0. cos As above, it will be assumed that M 1 2 cos 2 1 2 2 1 and M2 are confined to a plane which includes the in-plane easy axes of the two 1 1 layers and the surface normal. For positive values of H, as- 2 1 2 2, suming no hysteresis, M1 and M2 may therefore occupy one cos2 of two quadrants in this plane, and, according to the defini- 1 2 1 12 1 2 2 2 1 1 2 2. tions of 1 and 2 , these angles may have positive values in Using these relations in Eq. 6 leads to one quadrant and negative values in the other. In order to make appropriate substitutions for 2 2 2 2 1 and 2 , it is therefore E a1 1 b1 1 4c1 1 a2 2 b2 2 4c2 2 2A12 necessary to know whether M1 and M2 are in the same quad- rant or in adjacent quadrants near H 0. If both A A 12 and 12 1 2 2 2B12 2B12 1 2 2. 17 B12 have negative values, then M1 and M2 will be in adja- Differentiation of this expression with respect to cent quadrants for small positive values of H, as shown in 1 and 2 gives Fig. 3, and appropriate substitutions can be made for 1 and 2 . This is also the case if A12 is negative and B12 is posi- E tive, since a positive B a 12 favors parallel and antiparallel 1 2b1 1 8c1 1 2A12 1 2 alignment of the magnetizations equally, whereas a negative 1 A12 favors antiparallel alignment. If both A12 and B12 have 4B12 1 2 , 18 positive values, then M1 and M2 will be in the same quad- rant see Fig. 3 and substitutions can again be made. If E A a 12 is positive and B12 is negative however, then whether 2 2b2 2 8c2 2 2A12 1 2 M 2 1 and M2 will be in the same quadrant or in adjacent quadrants depends on the relative magnitudes of A12 and 4B12 1 2 . 19 12 434 A. J. R. IVES, J. A. C. BLAND, R. J. HICKEN, AND C. DABOO 55 Using the fact that stationary points are given by E/ 1 It is now possible to write E/ 2 0, results in the following set of simultaneous equations in 2 1 and 2 : 2b1 8c1 0M1d1 4Ki,1 2K1,1 a 4K 2K 1 2b1 8c1 2A12 4B12 1 2A12 4B12 2 , M i,1 1,1 20 0M 1d1 1 0M1d1 0M1 a 2 2b2 8c2 2A12 4B12 2 2A12 4B12 1 . 0M1d1Hs1 , 23 These equations are solved for 1 and 2 , and setting 2 2A 2b d 12 4B12 D12 , the following expressions are obtained: 2 8c2 0M 2 2 4Ki,2 2K1,2 a 4Ki,2 2K1,2 1 2b2 8c2 D12 a2D12 M 1 0M 2d2 2 2b 2 , 0M2d2 0M2 1 8c1 D12 2b2 8c2 D12 D12 21 . a 0M 2d2Hs2 2 2b1 8c1 D12 a1D12 2 2b 2 . Using the fact that a 1 8c1 D12 2b2 8c2 D12 D12 1 0M 1d1H and a2 0M2d2H, and writing 1 M1d1 and 2 M2d2 , Eqs. It is now necessary to introduce the quantities H s1 and 21 can be written as H s2 , which are equal to the saturation fields derived from the initial magnetization gradient of layers 1 and 2, respec- 2 0 1 2Hs2 D12 0 1 D12 0 2 tively, in the absence of coupling. These fields are similar to 1 H D D 2 , the conventional saturation fields of layers 1 and 2 in the 0 1Hs1 12 0 2Hs2 12 D12 24 absence of coupling, H s1 and Hs2 , defined above, but differ 2 in the sign of the term from the cubic anisotropy. They are D 0 1 2Hs1 12 0 2 D12 0 1 given by 2 H 2 . 0 1Hs1 D12 0 2Hs2 D12 D12 4K 2K The initial gradient saturation field H s can now be evalu- H i,1 1,1 s1 M 1 ated, which, from the definition in Eq. 14 , is given by 0M 1d1 0M1 and 1 H H 1 2 s , 25 4K 2K 0 1 1 2 2 H 0 H i,2 1,2 s2 M 2 , 001 only. 22 0M 2d2 0M2 and becomes H 2A H 1 2 0 1 2Hs1 s2 12 4B12 1Hs1 2Hs2 s 0 1 2 2Hs1 1Hs2 2A12 4B12 1 2 2 . 26 Rearranging this equation gives 1 H H A 2 0 1 2 Hs 2Hs1 1Hs2 s1 s2 1 2 12 2B12 , 001 ; H H ; all M H s1 s2 1 , M 2 ; all d1 , d2 ; s 1 2 2 1 2 1Hs1 2Hs2 upper sign i A12 2B12 0, ii A12 0 and B12 0; lower sign A12 0 and B12 0. 27 If the two magnetic layers have equal thicknesses and mag- extremely small exchange coupling between the Fe layers, netizations so that 1 2 , but different values of Ki,j or when taken together suggested that the thicknesses and mag- K1,j , then for the case A12 0, Eq. 27 reduces to netizations of the two Fe layers were identical, as expected, but that the two Fe layers possessed different interface 2H H anisotropies. This meant that the conventional perpendicular A s1 s2 12 2B12 14 0Md Hs , 001 , H saturation fields, H and H , of the individual Fe layers in s1 Hs2 s1 s2 the absence of coupling, were not quite the same, and that H the initial gradient saturation field, H and H , of the s1 Hs2 , M 1 M 2 , d1 d2 , s1 s2 individual layers in the absence of coupling, were also not i A identical. Thus in this case the appropriate relation between 12 2B12 0, ii A12 0 and B12 0. 28 the conventional perpendicular saturation field H s and the BLS measurements and polar Kerr magnetization curves, coupling strengths is given by Eq. 13 , while the appropriate performed on the trilayer at a Cr thickness corresponding to relation between the initial gradient perpendicular saturation 55 OSCILLATORY BIQUADRATIC COUPLING IN Fe/Cr/Fe 001 12 435 field H s and the coupling strengths, for either A12 2B12 est observed value of Hs from the plots of both Hs and 0 or A 12 0 and B12 0, is given by Eq. 28 . From those Hs . The plots of the normalized saturation fields Hs and equations it can be seen that the saturation fields H s and Hs are shown in Fig. 1 c . H s will in general both have a nonlinear dependence on the Values of A12 may then be estimated by adding the plots coupling strengths, and since the values of H s1 , Hs2 , of Hs and Hs , while values of B12 may be estimated, H s1 , and Hs2 are not known, evaluation of A12 and B12 in principle, by subtracting the plot of Hs from that of from Eqs. 13 and 28 is virtually impossible. H s according to A reasonable estimate of A12 and B12 can however be made, by making the approximation that the two Fe layers 0Md are identical. With this assumption H A12 Hs , 34 s1 Hs2 , Hs1 Hs2 , 8 Hs and so 0Md 4 B 12 Hs . 35 H 16 Hs s Hs1 0Md A12 2B12 , 29 These equations are only valid for A12 2B12 0 or A12 0 4 and B12 0, as explained above. The region we are most H interested in is that of the first AFM coupling peak however, s Hs1 0Md A12 2B12 . 30 in which A12 is negative and the condition A12 2B12 is The expressions for H well satisfied in our data. This means that we can only s1 and Hs1 , which were given in Eqs. 11 and 22 , depend on the magnetocrystalline and inter- strictly rely on the values of A12 and B12 obtained using Eqs. face anisotropies of the magnetic layers, and on demagnetiz- 34 and 35 for Cr thicknesses in the range 3 to 15 Å, ing effects. In theory these quantities would be expected to which is the range of the first AFM coupling region shown be independent of any changes in nonmagnetic interlayer by the polar Kerr measurements in Fig. 1 b . thickness and could be calculated for a particular sample. For a sample such as the Fe/Cr/Fe trilayer studied here, however, IV. COUPLING BEHAVIOR DEDUCED we have found that H FROM POLAR KERR MAGNETOMETRY s1 and Hs1 are not entirely independent of the Cr interlayer thickness, which we believe is due to a In Fig. 4 a , the values of A variation of the interface anisotropy with Cr thickness, and 12 estimated from polar Kerr measurements using Eq. 34 are plotted versus Cr thickness. as a consequence H s1 and Hs1 cannot be calculated with The first bilinear coupling peak at about dCr 7 Å 4.9 ML sufficient accuracy to allow the coupling to be deduced. can be clearly seen in Fig. 4 a . The second bilinear peak, In order to estimate values for the coupling strengths, val- although out of range of the validity of our theoretical inter- ues of H s1 and Hs1 can be chosen from the plots of Hs and pretation using Eq. 34 which applies for 3 dCr 15 Å, is H s versus Cr thickness, and this has been done in two anyway obscured by the increase in the saturation fields oc- steps. The first step is to consider that if the cubic anisotropy, curring towards larger Cr thicknesses believed to be caused which causes a small curvature in the magnetization curves, by variations in interface anisotropies with Cr thickness. is ignored, the values of H s and Hs should be equal at the The bilinear and biquadratic coupling strengths estimated point just before the Cr wedge begins since no interlayer from in-plane Kerr and BLS measurements were described in coupling can exist at this point. The plots of H detail in Ref. 26, but are reproduced again in Figs. 4 b and s and Hs have thus been shifted in field relative to each other until 5 b for comparison with the polar Kerr data. From Fig. 4 b , they coincide at d the maximum bilinear coupling strength appears to occur at Cr 0. The second step is to produce a plot of normalized satu- about dCr 7 Å, having the value 0.15 mJ m 2 at this ration fields so that for d point. In comparison, a somewhat larger maximum value of Cr 0 the values of the normalized fields are in theory due to coupling only. If the normalized A12 0.21 mJ m 2, estimated from the peak at 7 Å in Fig. perpendicular saturation fields are defined by H 4 a , is obtained using polar Kerr magnetometry, and the s and H difference would appear to result from the underlying in- s , then crease in the polar Kerr plot of A12 with increasing Cr thick- 4 ness. These values for the bilinear coupling strength are H s Hs Hs1 much smaller than those obtained by some other 0Md A12 2B12 , 31 researchers,8,16,32,33 however, and it is believed that increased roughness at the Fe-Cr interfaces may be responsible for at- 4 H tenuating the total bilinear plus biquadratic coupling s Hs Hs1 0Md A12 2B12 . 32 strength in our sample. Heinrich et al.10 have shown that the overall coupling strength is strongly affected by the growth Since the result of step 1 is that the smallest value of H s is temperature, being greater for higher growth temperatures. below that of H s , the second step is achieved by putting This indicates that the reduced coupling strength in our samples is due to the growth at room temperature of the H s1 Hs1 smallest observed value of Hs . 33 Fe/Cr/Fe structure, as opposed to the higher growth tempera- tures used by other research groups. The plots of the normalized saturation fields H s and From Fig. 5 b it is seen that the biquadratic coupling H s are thus obtained in this case by subtracting the small- strength B12 estimated from in-plane Kerr magnetometry and 12 436 A. J. R. IVES, J. A. C. BLAND, R. J. HICKEN, AND C. DABOO 55 FIG. 4. a Shown plotted as a function of Cr thickness are the FIG. 5. a Shown plotted as a function of Cr thickness are the values of A12 as deduced from the normalized perpendicular satu- values of B ration field plots in Fig. 1 c . b The values of A 12 as deduced from the normalized perpendicular satu- 12 as deduced from ration field plots in Fig. 1 c . b The values of B in-plane Kerr magnetometry circles , from BLS squares , and 12 as deduced from in-plane Kerr magnetometry circles , from BLS squares , and from BLS on a second trilayer with 0 dCr 20 Å triangles are from BLS on a second trilayer with 0 d plotted versus Cr thickness for comparison. The dashed curve is a Cr 20 Å triangles are plotted versus Cr thickness for comparison. The dashed curve is a scaled version of the curve in Fig. 1 a and serves only to guide the d 1.4 fit to the data. eye. Cr with increasing Cr thickness. For Cr thicknesses outside the BLS is largest at a Cr thickness of about 5 Å 3.5 ML and range 3 d decreases for larger Cr thicknesses, approximately following Cr 15 Å the plot is strictly invalid due to the limitations put on A a d 1.4 12 in the theory. Also, the values of B12 Cr dependence for Cr thicknesses between 5 and 30 Å shown for the larger Cr thicknesses are not reliable, unlike 3.5 and 20.8 ML . The fluctuation of some data points, how- those shown for lower thicknesses, because of the increases ever, in particular around dCr 10 Å 6.9 ML , means that that were observed in the saturation fields at larger Cr thick- the existence of oscillations in B12 are not ruled out by the nesses. Above 15 Å however, the plot of B12 estimated from in-plane Kerr and BLS measurements. In Fig. 5 a , the val- polar Kerr measurements does suggest a much more gradual ues of B12 estimated from polar Kerr measurements using falloff than the approximate d 1.4 dependence of B Eq. 35 are plotted versus Cr thickness. It was observed Cr 12 in Fig. 5 b . above when the polar Kerr saturation fields H s and Hs It should be noted that the absolute values of A12 and were plotted in Fig. 1 b that after the fifth data point H s B12 presented in Figs. 4 a and 5 a are subject to some ex- started to increase dramatically a few data points before tent to the method chosen for normalizing the saturation field H s started to increase. This resulted in a large difference in plots of Hs and Hs in Fig. 1 b . A different method of the values of H s and Hs at dCr 5 Å. Thus when Hs is normalization would result in all the values of A12 and all the subtracted from H s in Eq. 35 to obtain an estimate of values of B12 having a constant value added or subtracted to B12 , a peak is observed in the plot of B12 versus Cr thickness them, although the shape of the plots of A12 and B12 would at dCr 5 Å, which can be clearly seen in Fig. 5 a . This remain unchanged. peak occurs in almost exactly the same place as the maxi- mum value of B12 observed using in-plane Kerr magnetom- V. CONCLUSION etry and BLS in Fig. 5 b , and also its magnitude is almost exactly the same as that of the peak in B12 in Fig. 5 b . To summarize, the polar Kerr measurements described The plot of B12 estimated from polar Kerr measurements here strongly suggested an oscillatory behavior for the biqua- suggests that before the peak at 5 Å 3.5 ML , B12 increases dratic coupling as a function of Cr thickness in the range from zero near zero Cr thickness. It also suggests that after dCr 3 to 15 Å. The in-plane Kerr and BLS measurements, the peak at 5 Å there is a trough at d 1.4 Cr 8 Å 5.6 ML described in Ref. 26, suggested a dCr thickness dependence followed by a second peak in B12 at dCr 12 Å 8.3 ML , for the biquadratic coupling for Cr thicknesses greater than which is evidence that B12 falls off in an oscillatory fashion the maximum value at 5 Å 3.5 ML , but the fluctuation of 55 OSCILLATORY BIQUADRATIC COUPLING IN Fe/Cr/Fe 001 12 437 some data points did not rule out the presence of oscillations. dCr 12 Å 8.3 ML . Apart from a scaling factor of approxi- The polar Kerr measurements also clearly indicated a peak in mately 3.4, probably caused by the greater amount of inter- the biquadratic coupling at dCr 5 Å, having the same peak face roughness in our sample, the behavior of the coupling height as that observed using in-plane Kerr magnetometry strengths that is reported by Ko¨bler et al. is thus very similar and BLS. The polar Kerr measurements suggested addition- to some of the polar Kerr data reported here, and the ratio of ally that B12 increases monotonically from zero near zero Cr the maximum value of B12 to the maximum value of A12 has thickness to the maximum at 5 Å. They also suggested that the same value here as in Ref. 16, about 0.41. after the peak at 5 Å there is a trough at dCr 8 Å 5.6 ML Finally, we compare the coupling behavior observed here followed by a second peak in B12 at dCr 12 Å 8.3 ML , with that predicted theoretically by Edwards, Ward, and which is evidence that B12 falls off in an oscillatory fashion. Mathon17 using an intrinsic model to obtain the biquadratic The polar Kerr data at higher Cr thicknesses was not thought coupling. In Ref. 17, a plot showing how A12 and B12 are to be so reliable because of the increases in the saturation both expected to oscillate as a function of spacer thickness field data at these thicknesses which were thought to be due gives the height of the first maximum in B12 to be 0.56 of the to a variation of interface anisotropy with Cr thickness, and height of the first maximum in A12 , which compares quite this data was also outside the range of 3 dCr 15 Å for favorably with the ratio of 0.41 found in this work. The which our theoretical interpretation was valid. position of the first maximum in B12 is shown in the theoret- We now compare the coupling behavior observed for our ical plot in Ref. 17 to occur at a spacer thickness equal to Fe/Cr/Fe trilayer with that reported by Ko¨bler et al.16 The half the spacer thickness at which the first maximum occurs maximum value of A12 reported by Ko¨bler et al. was in the plot of A12 . This is quite similar to the ratio of 0.71 0.50 mJ m 2, which occurred at dCr 7 Å 4.9 ML . This arising from the positions of dCr 5 Å 3.5 ML and dCr value of A12 is 3.3 times greater in magnitude than the maxi- 7 Å 4.9 ML observed here for the first biquadratic and mum value of A12 0.15 mJ m 2 obtained from Fig. 4 b bilinear coupling peaks, respectively. from in-plane Kerr magnetometry and BLS more reliable The second maximum of B12 in the theoretical plot of than the maximum value from Fig. 4 a , but it is interesting Ref. 17 occurs just before the bilinear coupling crosses over that the position of the maximum of A12 reported by Ko¨bler from antiferromagnetic to ferromagnetic, as the Cr thickness et al. is the same as the position of the maximum in Figs. increases beyond the first maximum in A12 , which is also the 4 a and 4 b which is dCr 7 Å. The maximum value of case for our plot of B12 from the polar Kerr measurements in B12 reported by Ko¨bler et al. was 0.21 mJ m 2, and this Fig. 5 a . The height of the second maximum in B12 in Ref. occurred at a Cr thickness of 6 Å 4.2 ML . This value of 17 is, however, a significantly smaller fraction of the height B12 is 3.5 times greater in magnitude than the maximum of the first maximum in B12 in Ref. 17 than is our second value of B12 0.06 mJ m 2 obtained for our Fe/Cr/Fe maximum in B12 in Fig. 5 a as a fraction of the correspond- trilayer from both the polar Kerr measurements in Fig. 5 a ing first maximum in B12 . The very rapid decay of the bi- and the in-plane Kerr and BLS measurements in Fig. 5 b . quadratic coupling strength with spacer thickness in the Ed- The position of the maximum in B12 reported by Ko¨bler wards model, contrasting with the less rapid decay observed et al. is however very similar to the position of the maximum here and by Ko¨bler et al.,16 may result from the fact that the in Figs. 5 a and 5 b which is dCr 5 Å 3.5 ML . There are model used in Ref. 17 was quite a simple one, and more no values of A12 or B12 given for Cr thicknesses less than 6 advanced theories of the biquadratic coupling may resolve Å in the paper by Ko¨bler et al., so we cannot compare the such discrepancies. Nevertheless, the fact that the Edwards behavior of B12 in their sample with the behavior of B12 theory, like many other theories of the biquadratic exchange suggested by the polar Kerr measurements in this low Cr coupling,15,16 predicts that B12 should decay with interlayer thickness range. For Cr thicknesses between 6 and 16 Å 4.2 thickness in an oscillatory fashion, tends to support the re- and 11.1 ML , however, there is some evidence of a gradu- sults from the polar Kerr measurements in Fig. 5 a . ally decaying oscillatory behavior in their values of B12 with increasing Cr thickness, which is essentially the behavior ACKNOWLEDGMENTS observed here using polar Kerr magnetometry in this thick- ness region. A second maximum in the value of B12 is seen We would like to thank S. J. Gray and M. Gester for by them at about dCr 13 Å 9.0 ML , whereas a second growing the Fe/Cr/Fe trilayer and to acknowledge financial maximum is observed in Fig. 5 a at a similar thickness of support from the EPSRC and the Toshiba Corporation. 1 P. Gru¨nberg, R. Schreiber, Y. Pang, M. Brodsky, and H. Sowers, Jager, J. van de Stegge, W. B. Zeper, and W. Hoving, Phys. Rev. Phys. Rev. Lett. 57, 2442 1986 . Lett. 67, 903 1991 . 2 Y. Wang, P. M. Levy, and J. L. Fry, Phys. Rev. Lett. 65, 2732 6 J. Unguris, R. J. Celotta, and D. T. Pierce, Phys. Rev. Lett. 69, 1990 . 1125 1992 . 3 E. Fawcett, Rev. Mod. Phys. 60, 209 1988 . 7 E. E. Fullerton, M. J. Conover, J. E. Mattson, C. H. Sowers, and 4 J. Unguris, R. J. Celotta, and D. T. Pierce, Phys. Rev. Lett. 67, S. D. Bader, Phys. Rev. B 48, 15 755 1993 . 140 1991 . 8 B. Heinrich, M. From, J. F. Cochran, L. X. Liao, Z. Celinski, C. 5 S. T. Purcell, W. Folkerts, M. T. Johnson, N. W. E. McGee, K. M. Schneider, and K. Myrtle, in Magnetic Ultrathin Films, Mul- 12 438 A. J. R. IVES, J. A. C. BLAND, R. J. HICKEN, AND C. DABOO 55 tilayers and Surfaces/Interfaces, and Characterization, edited by 21 J. C. Slonczewski, J. Magn. Magn. Mater. 126, 374 1993 . B. T. Jonker et al., MRS Symposia Proceedings No. 313 Ma- 22 S. Demokritov, E. Tsymbal, P. Gru¨nberg, W. Zinn, and I. K. terials Research Society, Pittsburgh, 1993 , p. 119. Schuller, Phys. Rev. B 49, 720 1994 . 9 A. Davies, J. A. Stroscio, D. T. Pierce, and R. J. Celotta, Phys. 23 J. C. Slonczewski, Phys. Rev. Lett. 67, 3172 1991 . Rev. Lett. 76, 4175 1996 . 24 J. C. Slonczewski, J. Appl. Phys. 73, 5957 1993 . 10 B. Heinrich, J. F. Cochran, D. Venus, K. Totland, C. Schneider, 25 M. E. Filipkowski, J. J. Krebs, G. A. Prinz, and C. J. Gutierrez, and K. Myrtle, J. Magn. Magn. Mater. 156, 215 1996 . Phys. Rev. Lett. 75, 1847 1995 . 11 D. Stoeffler and F. Gautier, Phys. Rev. B 44, 10 389 1991 . 26 R. J. Hicken, C. Daboo, M. Gester, A. J. R. Ives, S. J. Gray, and 12 M. Ru¨hrig, R. Scha¨fer, A. Hubert, R. Mosler, J. A. Wolf, S. J. A. C. Bland, J. Appl. Phys. 78, 6670 1995 . Demokritov, and P. Gru¨nberg, Phys. Status Solidi A 125, 635 27 D. T. Pierce, J. A. Stroscio, J. Unguris, and R. J. Celotta, Phys. 1991 . Rev. B 49, 14 564 1994 . 13 A. Fuss, S. Demokritov, P. Gru¨nberg, and W. Zinn, J. Magn. 28 A. J. R. Ives, J. A. C. Bland, T. Thomson, P. C. Riedi, M. J. Magn. Mater. 103, L221 1992 . Walker, J. Xu, and D. Greig, J. Magn. Magn. Mater. 154, 301 14 Z. Celinski, B. Heinrich, and J. F. Cochran, J. Magn. Magn. 1996 . Mater. 145, L1 1995 . 29 A. J. R. Ives, R. J. Hicken, J. A. C. Bland, C. Daboo, M. Gester, 15 B. Rodmacq, K. Dumesnil, P. Mangin, and M. Hennion, Phys. and S. J. Gray, J. Appl. Phys. 75, 6458 1994 . Rev. B 48, 3556 1993 . 30 C. Turtur, G. Bayreuther, Phys. Rev. Lett. 72, 1557 1994 . 16 U. Ko¨bler, K. Wagner, R. Wiechers, A. Fuss, and W. Zinn, J. 31 A. J. R. Ives, Ph. D. thesis, University of Cambridge, United Magn. Magn. Mater. 103, 236 1992 . Kingdom, 1995. 17 D. M. Edwards, J. M. Ward, and J. Mathon, J. Magn. Magn. 32 P. Gru¨nberg, A. Fuss, Q. Leng, R. Schreiber, and J. A. Wolf, Mater. 126, 380 1993 . Magnetism and Structure in Systems of Reduced Dimension, 18 J. Barnas´, J. Magn. Magn. Mater. 123, L21 1993 . NATO Advanced Study Institute, Series B Physics Plenum, 19 R. P. Erickson, K. B. Hathaway, and J. R. Cullen, Phys. Rev. B New York, 1992 . 47, 2626 1993 . 33 S. Demokritov, J. A. Wolf, and P. Gru¨nberg, Europhys. Lett. 15, 20 P. Bruno, J. Magn. Magn. Mater. 121, 248 1993 . 881 1991 .