PHYSICAL REVIEW B VOLUME 59, NUMBER 17 1 MAY 1999-I Oscillatory exchange coupling between iron layers separated by chromium A. T. Costa, Jr.,* J. d'Albuquerque e Castro, and R. B. Muniz Instituto de FiŽsica, Universidade Federal Fluminense, NiteroŽi 24210-340, Brazil Received 30 July 1998; revised manuscript received 8 October 1998 The exchange coupling J between Fe layers separated by nonmagnetic Cr is calculated for Fe/Cr/Fe 001 trilayer structures as a function of the spacer thickness N for several temperatures T. It is shown that for perfectly sharp interfaces J(N,T) is entirely dominated by short period oscillations for 0 K T 500 K and N varying from 5 to 50 atomic planes. At zero temperature the amplitude of J decays as N 3/2 for large values of N. This behavior is caused by the particular type of singularity in the nesting of the Cr Fermi which is responsible for one of the dominant short-period oscillations of J(N). A strong temperature dependence of the coupling strength is obtained for some values of N, in excellent agreement with experiments. The effect of interface mixing on J(N) reduces the overall coupling strength, as well as the relative importance of the short period oscillatory components, and causes a phase shift in the oscillations of J(N). S0163-1829 99 03317-2 I. INTRODUCTION standing of the interlayer coupling in Fe/Cr systems, there are still several aspects which deserve clarification. In many Fe/Cr systems have been at the forefront of research in Fe/Cr specimens, it has been observed that J(N) oscillates magnetic multilayers in recent years. About a decade ago it basically with a long period, having maximum antiferromag- was observed that the exchange coupling J between Fe layers netic AF amplitude 1 mJ/m2. Short-period oscillations separated by Cr changes sign for different Cr thicknesses.1 of 2 monolayers are seen only in carefully grown Later, it was found that J generally oscillates as a function of samples.9,24­26 On the other hand, all existing calculations the spacer thickness N in metallic multilayers.2 The rather assuming perfect interfaces find that J(N) in Fe/Cr/Fe 001 large period of 18 Ć observed in early measurements of trilayers is dominated by the short-period oscillations. Addi- J(N) in Fe/Cr caused surprise and greatly stimulated the tionally, the coupling amplitude calculated across AF Cr in- development of theories and experiments.3­7 In addition, the variably comes out much larger than that mediated by non- striking ``giant'' magnetoresistance effect was discovered in magnetic Cr, and the latter is, in turn, much larger than those Fe/Cr multilayers.8 The importance of interface quality and measured so far.19,27,28 These discrepancies are rightfully at- crystal ordering of the intervening layer in highlighting dif- tributed to interface inhomogeneities and lack of crystal or- ferent periods of J(N) was experimentally evidenced in Fe/ dering in the spacer. Experimentally it has been recognized Cr/Fe wedge structures.9­11 Some of the earliest observations that interface mixing always occurs even in the best-grown of noncollinear spin structures corresponding to 90° cou- Fe/Cr samples. Its amount and extent depend on the substrate pling of the magnetic layers were made on Fe/Cr/Fe temperature during growth, and are difficult to be accurately sandwiches,12­14 and the search for the physical origin and determined.21,29 relative importance of this type of coupling in different sys- There are general rules which provide a systematic way of tems has further stimulated theories and experiments.15,16 determining the oscillation periods of J(N) for sufficiently It is well known that the magnetic ground state of Cr is large spacer thicknesses. According to Ruderman-Kittel- rather delicate. Bulk Cr exhibits a spin-density-wave SDW Kasuya-Yosida RKKY -like theories they are given by the antiferromagnetism, which is not commensurate with its lat- extremal spanning vectors of the spacer Fermi surface tice. The SDW in Cr originates from Fermi-surface nesting, FS .30,31 Quantum-well theory, however, predicts additional which causes a maximum in the noninteracting static spin possibilities, especially when the spacer FS has more than susceptibility at a wave vector close to 2 /a, where a is the one sheet,32 as in the case of Cr. Because of the complexity Cr lattice constant. Such nesting can be modified by alloying of its FS, a complete analysis of all possible individual com- Cr with different materials, in some cases drastically affect- ponents of the coupling mediated by Cr is rather involved. ing its NeŽel temperature TN 311 K.17 The presence of the First, because of the large number of Cr FS extrema. Second, Fe layers in Fe/Cr/Fe 001 trilayers poses magnetic bound- one would need investigate the possibility of occurrence of ary conditions which affect both the magnetic state and TN of non-RKKY contributions, coming from critical points asso- the Cr layer. Interfacial inhomogeneities can also largely af- ciated with integer linear combinations of the various Cr FS fect J and the magnetic properties of the Cr spacer layer, sheets. Besides, for those extrema associated with nesting especially for small Cr thicknesses.15,18­21 For sufficiently between different bands, the stationary phase approximation, thick Cr layers the SDW is likely to settle. However, long- usually employed in this type of analysis, is somewhat range magnetic order may be suppressed in relatively thin Cr subtler, and in some cases not applicable.32 layers, due to frustrations generated by the presence of Contributions coming from several extremal points of the roughness, interdiffusion, vacancies, and steps at the Fe/Cr Cr FS have been calculated.33,34 Nevertheless, the determina- interface.19,20,22,23 tion of the most important periodic components of J(N) in In spite of the remarkable progress made in our under- Fe/Cr/Fe trilayers has not been fully settled. The origin of the 0163-1829/99/59 17 /11424 8 /$15.00 PRB 59 11 424 ©1999 The American Physical Society PRB 59 OSCILLATORY EXCHANGE COUPLING BETWEEN IRON . . . 11 425 18 Ć period has been the subject of recent theoretical de- bate. There are strong arguments indicating that it comes from extremal dimensions associated with the N-centered el- lipsoids of the Cr FS.33,34 However, since its amplitude is so much smaller than those of the short-period contributions, one is led to ask: why does it experimentally dominate the coupling in so many different samples? What kind of inho- mogeneity and how much of it would be required to suppress the short-period oscillations so that only a long period is observed? Another conjecture is related to the fact that, in principle, each oscillatory component of J behaves differ- ently with temperature.35,36 Would it be possible that in the case of Fe/Cr temperature effects could suppress the short- period components, revealing others with smaller ampli- tudes? To help answering such questions it is necessary to calculate J(N,T) in these systems. For spacers with relatively simple Fermi surfaces, the quantum-well theory predicts that the contributions to J(N,T) coming from single point singularities s of the FIG. 1. Calculated exchange coupling at T 0 K for a spacer FS are asymptotically i.e., for N 1) given by35 Fe/Cr/Fe 001 trilayer as a function of the number of Cr atomic planes. A J N,T skBT sin sN s . 1 when a line of coincidence occurs. Therefore, it is necessary s N sinh kBT BsN Cs to know what the real asymptotic behavior of J(N,T) is in At T 0 it reduces to Fe/Cr, and beyond what value of N it actually sets in, before one can use equations such as Eqs. 1 or 2 to analyze the data. J N As sin sN s / BsN2 CsN , 2 In this paper we show that for perfectly sharp interfaces, s and at T 0 K, the amplitude of the coupling between Fe where N is the spacer thickness measured in number of layers across noninteracting Cr, in Fe/Cr/Fe 001 trilayers, atomic planes. B decays as N 3/2 for large values of N. The asymptotic be- s and s depend only on geometrical aspects of the spacer FS around those singularities; i.e., on the FS havior of J(N) sets in only for rather large Cr layer thick- velocities and extremal radii along the direction perpendicu- nesses: N 30 atomic planes. Therefore, the use of Eqs. 1 lar to the layers, respectively. On the other hand, As , s , or 2 to analyze numerical or experimental values of the and C coupling for smaller Cr thicknesses is, strictly speaking, in- s depend also on the degree of confinement of the carriers in the spacer caused by the magnetic layers, hence, correct. We have also calculated J(N,T) for several tempera- on the matching of the electronic states across the interfaces. tures, and have found that for perfect interfaces it oscillates Asymptotic expressions similar to Eqs. 1 and 2 have been as a function of N with short-period oscillations for all tem- extensively used to analyze both theoretical and experimen- peratures investigated. For some values of N a strong tem- tal results for J(N,T). However, it is noteworthy that ap- perature dependence of the coupling strength is obtained in proaches which rely on fits to results of numerical calcula- excellent agreement with experiments. The effect of Fe/Cr tions or to experimental values of the coupling to obtain interfacial mixing has also been investigated, and we show periods, amplitudes, phases, and decay rates of J(N,T) must that for sufficiently large mixing the short-period oscillations be viewed with caution for the following reasons: the ex- tend to be suppressed and the long period begins to show up pected asymptotic regime 1/N2, obtained from isolated FS as expected. singularities, applies solely to zero temperature, ordered spacers, and depending on the values of Cs , it sets in only II. RESULTS FOR PERFECT INTERFACES for relatively large values of N (N 20­30 atomic planes at least . At finite T, the envelope functions of the oscillatory We have used the formalism developed in Refs. 7 and 35 components vary exponentially with N. For Co/Cu/Co 001 to calculate J(N,T), defined as the total-energy difference trilayers at room temperature, for instance, there is no range per surface atom between the antiferromagnetic and ferro- of N in which the coupling amplitude can be correctly de- magnetic configurations of the trilayer. We show results for scribed by a dependence 1/N2.35­37 Furthermore, these fit- the bilinear exchange coupling term J1(N,T) which, for per- tings usually involve several parameters and, in some cases, fectly smooth Fe/Cr 001 interfaces, is virtually equal to they are not unique. J/2.7 To calculate J we have used a tight-binding model with For spacers with more complicated Fermi surfaces, such s,p,d orbitals, and hopping up to second-nearest neighbors. as Cr, where nesting between sheets occurs, the asymptotic The tight-binding parameters for noninteracting Cr were decay of J(N) may be different from the usual 1/N2 behav- taken from Ref. 38, and those for ferromagnetic Fe were ior, depending on the nature of the dominant singularity.31 obtained as in Ref. 39. As discussed in Ref. 31, the expected rate of decay is 1/N Results for J1(N,T 0) in Fe/Cr/Fe 001 trilayers are for the extreme case of perfect planar nesting, and 1/N3/2 shown in Fig. 1. It is evident that short-period oscillations 11 426 COSTA, d'ALBUQUERQUE e CASTRO, AND MUNIZ PRB 59 FIG. 3. Discrete Fourier transform of N3/2J1(N) for 50 N 100, showing the periods of the most important oscillatory com- ponents of J1. large values of N. The result is shown in Fig. 3, and it clearly demonstrates that, asymptotically, J1 is dominated mainly by three periods which are identified by the major peaks in the figure. Short-period oscillations are the most important ones. FIG. 2. Dependences of J1(N) N a , J1(N) N2 b , and They come from the same FS nestings that cause the SDW in J1(N) N3/2 c , on the number N of Cr atomic planes. J1 has been calculated at zero temperature, and is measured in mRy/surface Cr, involving the electron and hole octahedralike FS pieces atom. centered around the and H points of the bcc Brillouin zone, respectively. Normally, such nesting is described as dominate the calculated coupling at zero temperature for all perfectly planar, but this is clearly an approximation. In fact, values of N. Our values agree very well with those obtained by observing it along the 001 direction, one realizes that it in Ref. 28 for N 20 atomic planes, but they are 20­60 times has critical points in the lines kx ky , ky 0, and kx 0, larger than the experimentally observed first AF peaks. It is where x, y, and z are the usual cubic directions. The critical noteworthy that the coupling calculated across interacting Cr points located in the lines kx ky are maxima, and those is even larger.28,19 On the other hand, curiously, the maxi- located in ky 0, and in kx 0 are minima. These stationary mum coupling amplitude measured in good-quality samples points are responsible for the short-period oscillations in showing short-period oscillations is 3 times smaller than J(N,T); the maxima and the minima lead to oscillatory com- that observed in specimens where only a long period is ponents of J(N) having periods of 2.05, and 2.01 atomic apparent.24 For thin Cr spacers the observed short oscilla- planes, respectively. tions show up superimposed to the long-period oscillation, A revealing result is shown in Fig. 4 where the calculated suggesting that some degree of interfacial disorder is still nesting surface is depicted around its critical points in order present in those samples. to determine the character of each singularity. The maximum Before discussing the possible reasons for these disagree- leads to a contribution which, at T 0 K, has the usual N 2 ments between theory and experiments, we first address the asymptotic behavior. On the other hand, it is clear from Fig. question of how the coupling across noninteracting Cr really 4 b , that the minimum associated with the nesting singular- behaves asymptotically. To determine the correct asymptotic ity at ky 0 is extremely shallow along one of the principal behavior of J1(N,0) we have calculated the coupling for axis directions. As a result, the corresponding effective mass large values of N, and plotted J1(N) N, J1(N) N2, and is nearly infinite, thereby justifying the N 3/2 asymptotic de- J1(N) N3/2 as functions of N. It is clear from Fig. 2 that, at cay rate found in Fig. 2. We notice in Fig. 2 c , however, that T 0 K, the oscillatory coupling across noninteracting Cr in the envelope bound of the first group of oscillations is clearly Fe/Cr/Fe 001 trilayers decays asymptotically as N 3/2, smaller than the asymptotic limiting value. This means that rather than N 2 or N 1 as assumed earlier.28,27,40 The pos- the asymptotic dependence 1/N3/2 is reached for Cr thick- sibility of finding a N 3/2 decaying rate has been envisaged nesses 30 atomic planes. Consequently, in practice, this by Koelling, provided the dominant contribution to J comes dependence may not be observed, because the SDW antifer- from a particular type of nesting where a line of coincidence romagnetism is expected to settle for smaller Cr occurs.31 Below, we shall show that this is precisely what thicknesses,23,20 presumably affecting the N 3/2 behavior. happens in noninteracting Cr. Before doing so, since we Therefore, assuming perfect interfaces, it is not correct, or at know the asymptotic behavior of J1 we can estimate the least not rigorous, to use asymptotic expressions derived for relative amplitude of its most important oscillatory compo- noninteracting Cr to analyze numerical or experimental val- nents, by taking the discrete Fourier transform41 of N3/2J1 for ues of J(N) in Fe/Cr/Fe 001 trilayers for Cr layer thick- PRB 59 OSCILLATORY EXCHANGE COUPLING BETWEEN IRON . . . 11 427 FIG. 5. Temperature dependence of J1(N) for selected Cr thick- nesses: N 5 squares , N 10 full circles , and N 23 triangles . effects in both calculations come solely from the temperature dependence of the Fermi function. No spin fluctuations in the ferromagnetic layers are taken into account, which is a rea- sonable approximation for the range of temperatures consid- ered, namely T much smaller than the Fe Curie temperature. To verify our numerical approach we have thoroughly tested both the energy and Brillouin-zone numerical integra- FIG. 4. Nesting of the electron and hole octahedralike Cr FS tions involved in our calculations of J(N,T). We have per- sheets along the 001 direction see text , calculated around its formed the energy integration in the complex plane, using extrema, located in the lines kx ky a and ky 0 b , respectively. Matsubara frequencies at finite temperatures and Gauss- Legendre quadrature at T 0 K. For the two-dimensional nesses 30 atomic planes. It is also unwise to use them to Brillouin-zone integration we have used the number of spe- analyze experimental results for J(N), showing short period cial k points43 necessary to ensure a relative precision of oscillations, for N 30 atomic planes because for such thick- 10 2 in our final results. The short-period contributions are nesses the Cr spacer layer is probably magnetic. The fact that not suppressed in our calculations for T 300 K, and this is it is possible to find good fits of calculated or experimental supported by the results of Ref. 28, which compare ex- results using Eqs. 1 and 2 does not mean that the param- tremely well with ours, particularly for N ranging between eters obtained are necessarily trustworthy. However, in those 10 and 20 atomic planes. The results of Ref. 28 were ob- samples where the short period contributions are suppressed tained for a temperature comparable to 235 K using spin- and the long period is prominent, J(N) is no longer expected density-functional theory to calculate the change in total en- to decay asymptotically as N 3/2 as we have found at T ergy between the ferromagnetic and antiferromagnetic 0 K. In this case, it is possible that J(N,T) follows Eq. configurations of the trilayer. 1 for sufficiently large values of N because, as mentioned There is consensus about the fact that, at T 0 K, the earlier, there are strong indications that the long-period con- short-period oscillations dominate the coupling in Fe/Cr/Fe tribution comes from isolated FS singularities. Nevertheless, 001 trilayers with perfect interfaces, for all values of before using Eq. 1 to analyze the coupling, one must be N.19,27,34,44 Estimated ratios between the short- and long- certain that, in practice, the asymptotic regime has been period amplitudes vary from 5 to 10. In fact, from the reached and that a SDW state is not present in the Cr spacer heights of the peaks in Fig. 1 b we obtain a ratio of 7, for layer for such thicknesses. large values of N. Therefore, in order to become impercep- We now turn to the temperature dependence of the cou- tible at T 300 K for N 7, as found in Ref. 42, the short- pling. We have calculated J1 for several temperatures and Cr period contribution would need to decay much faster than the spacer thicknesses. For temperatures between 0 and 500 K, long period one. The temperature dependence of J is gov- and Cr layer thicknesses varying from 5 to 50 atomic planes, erned by geometrical aspects of the spacer Fermi surface and we have found, assuming perfect interfaces, that the coupling by the confining strength of the magnetic layers. These quan- is entirely dominated by short-period oscillations. Our results tities are determined by the electronic structure of the contrast with those of Ref. 42, where it was found that for trilayer, which seems to be reasonably well described in all temperatures T 300 K the long period oscillation prevails these calculations. It is unlikely that small differences be- for Cr thicknesses 7 atomic planes. This is very surprising, tween the band structures could lead to such a radical change since their theoretical framework is the same as ours, and the of behavior. Therefore, the reason why our results and those band structures used are only slightly different. Temperature of Ref. 28 differ from Ref. 42 remains a mystery. 11 428 COSTA, d'ALBUQUERQUE e CASTRO, AND MUNIZ PRB 59 FIG. 6. Calculated exchange coupling considering different interface mixings. Results are obtained at T 300 K as a function of the number N of Cr atomic planes. a Interface mixing restricted to two atomic planes, comparison between different concentrations: Fe/Cr1 pFep /Fe1 pCrp /Cr/Fe 001 trilayers with p 1 triangles , p 0.9 squares , and p 0.75 filled circles ; b Interface mixing with different spatial extents: mixing confined to two atomic planes with p 0.75 filled circles , and four atomic planes, i.e., Fe/Cr1 pFep /Cr1 qFeq /Fe1 qCrq /Fe1 pCrp /Cr/Fe 001 trilayers, with p 0.80 and q 0.75 open circles . c Comparison with perfect interfaces corresponding to p q 1 triangles and p 0.80, q 0.75 open circles . As shown in Fig. 5, we have obtained a very strong re- theories,3,6 except for the C term, which has been incorpo- duction of the coupling as a function of temperature for some rated later and shown to be very important in Co/Cu values of N. Our results agree very well with experimental systems.36 In fact, the experimental data obtained in Ref. 46 observations.45,46 It should be pointed out, however, that the were very well fitted by this expression even though it was latter were made on samples showing basically a long period unnecessarily assumed that C 0. oscillatory J(N). In this case one may argue that it would be reasonable to analyze the experimental data with Eq. 1 , since the conditions for using it seem to be at least partially III. EFFECT OF INTERFACIAL MIXING satisfied. It is clear from Eqs. 1 and 2 that when the cou- The quality of interfaces in multilayer systems depends on pling shows only one periodic component, coming from an the substrate temperature during layer deposition. Experi- isolated singularity of the spacer FS, its temperature depen- mentally, it is very difficult to avoid the occurrence of inter- dence can be written, for sufficiently large values of N, as face inhomogeneities in these systems. As far as the oscilla- J(N,T)/J(N,0) (T/T 1 0)/sinh(T/T0), where T0 kB(BN tory coupling between magnetic layers is concerned, we C). The same temperature dependence is obtained by know that variation of interfacial quality alters the degree of Ruderman-Kittel-Kasuya-Yosida and earlier quantum-well carrier confinement in the spacer, causing a phase change PRB 59 OSCILLATORY EXCHANGE COUPLING BETWEEN IRON . . . 11 429 and an overall reduction of the coupling strength. It may also modify the relative contributions of the different periodic components of J(N). In fact, it has been explicitly shown that the amplitude of the short-period oscillations of J(N) in Fe/Cr can be strongly attenuated by interfacial roughness.18 The temperature of optimal layer-by-layer growth varies from system to system, and for Fe/Cr 001 multilayers it is T 300 °C. Nevertheless, even under optimal conditions, the Cr growth on Fe 001 leads to formation of a Cr-Fe alloy in approximately three interfacial atomic planes.21 The alloying is an asymmetric effect which happens only when Cr is de- posited on Fe but not vice versa. According to scanning tun- neling studies, for the 300 °C growth condition, the Cr con- centration in Fe seems not to exceed 10%. However, Auger spectroscopy estimates that it can be as much as 40%.29 It is reasonable to attribute the large difference between calculated and measured values of the coupling in Fe/Cr sys- tems to the probable existence of inhomogeneities in the samples. However, one must quantitatively verify their effect in the calculated results. With this purpose, we have investi- FIG. 7. Comparison between the values of J1(N), calculated at gated the effect of mixing at the Fe/Cr interface by treating T 300 K for Fe/Cr1 pFep /Cr1 qFeq /Fe1 qCrq /Fe1 pCrp /Cr/ the interfacial Fe/Cr atomic planes as disordered alloys com- Fe 001 trilayers, with p 0.80 and q 0.75 open circles , as a patible with a given concentration profile. We restrict our- function of N, and the corresponding average values obtained by selves to moderate interfacial admixtures, and assume that it J1(N) J1(N 1) 2J1(N) J1(N 1) /4 filled diamonds . takes place at either two or four atomic planes of the Fe/Cr These results give an idea of the combined effect of interface mix- interface. The effect of disorder is treated within the average ing and the presence of steps at the Fe/Cr interface. t-matrix approximation which, in the dilute limit, is equiva- lent to the coherent-potential approximation used in Refs. 47 tablish reasonable agreement with experimental data. One and 48. Our results are presented in Fig. 6. First, we calculate possibility to close such a gap would be to assume larger the coupling as a function of the Cr spacer thickness assum- amounts of interface mixing. However, one should note that ing that the interface mixing is confined to two atomic the presence of steps, which has been ignored in our calcu- planes, considering Fe/Cr alloys with different concentra- lations, but are probably present in real samples, also con- tions at the interface. More specifically, in Fig. 6 a we ex- tributes to diminish the interlayer coupling amplitude, and its amine Fe/Cr1 pFep /Fe1 pCrp /Cr/Fe(001) trilayers, with p short-period oscillatory components.48,49 If the steps are suf- 1.0, 0.9, and 0.75. It is evident that the overall amplitude ficiently large it is possible to assess their effect by taking of the coupling decreases, and the short-period oscillations averages of the coupling calculated for different spacer thick- tend to be progressively washed out with increasing interfa- nesses. A rough estimate, assuming thickness fluctuations of cial mixing. However, the reduction in the coupling ampli- 1 atomic plane, can be obtained by taking a simple aver- tude in Fe/Cr is not so dramatic as found in Co/Cu age J(N) J(N 1) 2J(N) J(N 1) /4, as depicted multilayers.48 In Fig. 6 b we investigate the effect of broad- in Fig. 7. We notice that the presence of steps at the Fe/Cr ening the region in which the interface mixing occurs. We interface, combined with interface mixing, practically re- compare results of the previous calculation corresponding to moves the residual short-period oscillations, reducing the p 0.75 with the case in which the mixing is confined to four coupling amplitude even further, in agreement with what has atomic planes, more precisely with Fe/Cr been found in Ref. 18. Nevertheless, the calculated value at 1 pFep / Cr the first AF peak remains a factor of five larger than those 1 qFeq /Fe1 qCrq /Fe1 pCrp /Cr/Fe 001 trilayers, with p 0.80 and q 0.75. We note that upon increase of the measured. spatial extent of the interface mixing, the overall amplitude of the coupling decreases further, and a phase shift in the IV. CONCLUSIONS oscillations becomes apparent. This is expected because, as mentioned earlier, both the amplitudes and phases of the os- We have calculated the coupling between Fe layers sepa- cillatory components of the interlayer coupling depend on rated by noninteracting Cr in Fe/Cr/Fe 001 trilayers as a the degree of carrier confinement in the spacer caused by the function of the Cr layer thickness for several temperatures. magnetic layers, which is obviously affected by interface We have found that for perfectly sharp interfaces the short- mixing. Finally, to assess the overall effect of interface mix- period oscillations are the dominant contributions to J(N,T) ing we compare in Fig. 6 c the latter results corresponding for 0 T 500 K and 1 N 30 atomic planes. We have to p 0.80 and q 0.75 with those for perfect interfaces. It is also shown that at zero temperature the amplitude of J(N) clear that the short-period oscillations are substantially sup- decays as N 3/2 for large values of N, rather than as N 2 or pressed by interface mixing. However, the calculated cou- N 1 as usually assumed. This behavior is caused by the par- pling amplitude is effectively reduced only by approximately ticular type of singularity in the nesting of the Cr FS along a factor of 3 for the maximum amount of interface mixing the lines ky 0 and kx 0) which is responsible for one of which we have considered. This is still not sufficient to es- the dominant short-period oscillatory contributions to J(N). 11 430 COSTA, d'ALBUQUERQUE e CASTRO, AND MUNIZ PRB 59 It must be pointed out, however, that the N 3/2 asymptotic interface mixing reduces the overall coupling strength, as decay, predicted for noninteracting Cr, happens for such well as the relative importance of its short-period oscillatory large values of N that, in practice, the Cr layer would de- components, and also causes a phase shift in the oscillations velop a SDW antiferromagnetism which presumably affects of the interlayer coupling. However, our results indicate that this type of behavior. Our results obtained at finite tempera- moderate interface mixing alone does not seem to be suffi- tures show a strong temperature dependence of the coupling cient to bring theory in accord with experiment. The com- strength for some values of the Cr spacer thickness, in very bined effect of the presence of steps and moderate mixings at good agreement with experiments. the Fe/Cr interface leads to better, but still far from perfect All existing calculations, including ours, for Fe/Cr/ agreement. Therefore, either there is something important Fe 001 trilayers with perfect interfaces have found inter- missing in all existing calculations or currently the samples layer coupling strengths which are at least an order of mag- have far more inhomogeneities than they seem to. We stress nitude larger than those experimentally observed at the first that our results are for trilayers and, therefore, apply to mul- AF peaks. On the other hand, it is currently very difficult, if tilayers with sufficiently thick Fe layers only. not impossible, to avoid the appearance of inhomogeneities in Fe/Cr interfaces. The absence of strong short-period oscil- ACKNOWLEDGMENTS latory components, and the existence of a relatively large biquadratic contribution in the measured coupling are strong We have benefitted from very helpful discussions with J. evidence of the presence of inhomogeneities in the sample. Mathon, D. M. Edwards, M. S. Ferreira, and A. Umerski. We have assessed the reduction of the coupling amplitude This work has been financially supported by the CNPq and due to their presence at the interfaces. We have found that FINEP of Brazil. *Permanent address: Departamento de Cie ncias Exatas, Univer- Yakhmi, Rev. Mod. Phys. 66, 25 1994 . sidade Federal de Lavras, 37200-000, Lavras, MG, Brazil. 18 Y. Wang, P. M. Levy, and J. L. Fry, Phys. Rev. Lett. 65, 2732 1 P. Grušnberg, R. Schreiber, Y. Pang, M. B. Brodsky, and H. Sow- 1990 . ers, Phys. Rev. Lett. 57, 2442 1986 . 19 D. Stoeffler and F. Gautier, Phys. Rev. B 44, 10 389 1991 ; J. 2 S. S. P. Parkin, Phys. Rev. Lett. 67, 3598 1991 . Magn. Magn. Mater. 104-107, 1819 1992 ; A. Vega, D. Stoef- 3 D. M. Edwards, J. Mathon, R. B. Muniz, and M. S. Phan, Phys. fler, H. DreysseŽ, and C. Demangeat, Europhys. Lett. 31, 561 Rev. Lett. 67, 493 1991 ; 67, 1476 E 1991 ; J. Phys.: Con- 1995 ; M. Freyss, D. Stoeffler, and H. DreysseŽ, Phys. Rev. B dens. Matter 3, 3941 1991 . 54, 12 677 1996 . 4 C. Chappert and J. Renard, Europhys. Lett. 15, 553 1991 . 20 E. E. Fullerton, S. D. Bader, and J. L. Robertson, Phys. Rev. Lett. 5 R. Coehoorn, Phys. Rev. B 44, 9331 1991 . 77, 1382 1996 , and references therein. 6 P. Bruno and C. Chappert, Phys. Rev. Lett. 67, 1602 1991 ; 67, 21 A. Davies, J. A. Stroscio, D. T. Pierce, and R. J. Celotta, Phys. 2592 E 1991 . Rev. Lett. 76, 4175 1996 . 7 J. d'Albuquerque e Castro, M. S. Ferreira, and R. B. Muniz, Phys. 22 E. E. Fullerton, K. T. Riggs, C. H. Sowers, S. D. Bader, and A. Rev. B 49, 16 062 1994 . Berger, Phys. Rev. Lett. 75, 330 1995 . 8 M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen van Dau, F. 23 J. Meersschaut, J. Dekoster, R. Schad, P. Belien, and M. Rots, Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 75, 1638 1995 . Phys. Rev. Lett. 61, 2472 1988 . 24 9 S. Demokritov, J. A. Wolf, P. Grušnberg, and W. Zinn, in Mag- J. Unguris, R. J. Celotta, and D. T. Pierce, Phys. Rev. Lett. 67, netic Surfaces, Thin Films, and Multilayers, edited by S. S. P. 140 1991 . 10 S. T. Purcell, W. Folkerts, M. T. Johnson, N. W. E. McGee, K. Parkin et al., MRS Symposia Proceedings No. 231 Materials Jager, J. aan de Stegge, W. B. Zeper, W. Hoving, and P. Grušn- Research Society, Pittsburgh, 1992 , p. 133; S. T. Purcell, W. berg, Phys. Rev. Lett. 67, 903 1991 . Folkerts, M. T. Johnson, N. W. E. McGee, K. Jager, J. aan de 11 S. Demokritov, J. A. Wolf, and P. Grušnberg, Europhys. Lett. 15, Stegge, W. B. Zeper, W. Hoving, and P. Grušnberg, Phys. Rev. 881 1991 . Lett. 67, 903 1991 . 12 25 M. Rušhrig, R. Schašfer, A. Hubert, R. Mosler, J. A. Wolf, S. A. Azevedo, C. Chesman, S. M. Rezende, F. M. de Aguiar, X. Demokritov, and P. Grušnberg, Phys. Status Solidi A 125, 635 Bian, and S. S. P. Parkin, Phys. Rev. Lett. 76, 4837 1996 . 1991 . 26 E. E. Fullerton, M. J. Conover, J. E. Mattson, C. H. Sowers, and 13 B. Heinrich, J. F. Cochran, M. Kowalewski, J. Kirschner, Z. Ce- S. D. Bader, Phys. Rev. B 48, 15 755 1993 . linski, A. S. Arrott, and K. Myrtle, Phys. Rev. B 44, 9348 27 M. van Schilfgaarde and F. Herman, Phys. Rev. Lett. 71, 1923 1991 . 1993 . 14 A. Schreyer, J. F. Ankner, Th. Zeidler, H. Zabel, C. F. Majkrzak, 28 S. Mirbt, A. M. N. Niklasson, B. Johansson, and H. L. Skriver, M. Schašfer, and P. Grušnberg, Europhys. Lett. 32, 595 1995 ; Phys. Rev. B 54, 6382 1996 . Phys. Rev. B 52, 16 066 1995 , and references therein. 29 B. Heinrich, J. F. Cochran, T. Monchesky, and K. Myrtle, J. 15 J. C. Slonczewski, Phys. Rev. Lett. 67, 3172 1991 ; J. Appl. Appl. Phys. 81, 4350 1997 . Phys. 73, 5957 1993 ; J. Magn. Magn. Mater. 150, 13 1995 , 30 M. D. Stiles, Phys. Rev. B 48, 7238 1993 . and references therein. 31 D. D. Koelling, Phys. Rev. B 50, 273 1994 . 16 D. M. Edwards, J. M. Ward, and J. Mathon, J. Magn. Magn. 32 M. S. Ferreira, J. d'Albuquerque e Castro, D. M. Edwards, and J. Mater. 126, 380 1993 . Mathon, J. Magn. Magn. Mater. 154, L1 1996 ; J. Phys.: Con- 17 E. Fawcet, H. L. Alberts, V. Yu. Galkin, D. R. Noakes, and J. V. dens. Matter 8, 11 259 1996 . PRB 59 OSCILLATORY EXCHANGE COUPLING BETWEEN IRON . . . 11 431 33 M. D. Stiles, Phys. Rev. B 54, 14 679 1996 , and references 41 J. Kudrnovsky, V. Drchal, I. Turek, and P. Weinberger, Phys. therein. Rev. B 50, 16 105 1994 ; V. Drchal, J. Kudrnovsky, I. Turek, 34 L. Tsetseris, B. Lee, and Y.-C. Chang, Phys. Rev. B 55, 11 586 and P. Weinberger, ibid. 53, 15 036 1996 . 1997 . 42 L. Tsetseris, B. Lee, and Y.-C. Chang, Phys. Rev. B 56, R11 392 35 J. Mathon, M. Villeret, A. Umerski, R. B. Muniz, J. 1997 . d'Albuquerque e Castro, and D. M. Edwards, Phys. Rev. B 56, 43 S. L. Cunningham, Phys. Rev. B 10, 4988 1974 . 11 797 1997 . 44 M. van Schilfgaarde, F. Herman, S. S. P. Parkin, and J. 36 J. d'Albuquerque e Castro, J. Mathon, M. Villeret, and A. Umer- Kudrnovsky, Phys. Rev. Lett. 74, 4063 1995 . ski, Phys. Rev. B 53, R13 306 1996 . 45 E. E. Fullerton, J. E. Mattson, C. H. Sowers, and S. D. Bader, Scr. 37 J. Mathon, M. Villeret, R. B. Muniz, J. d'Albuquerque e Castro, Metall. Mater. 33, 1637 1995 . and D. M. Edwards, Phys. Rev. Lett. 74, 3696 1995 . 46 38 B. G. Almeida, V. S. Amaral, J. B. Sousa, J. Colino, and I. K. R. H. Victora, in Magnetic and Electronic Properties of Low- Shuller, J. Magn. Magn. Mater. 177-181, 1170 1998 . Dimensional Systems, edited by L. M. Falicov and J. L. Moran- 47 Lopez Springer, Berlin, 1986 , p. 25. P. Bruno, J. Kudrnovsky, V. Drchal, and I. Turek, Phys. Rev. 39 A. T. Costa, Jr., J. d'Albuquerque e Castro, R. B. Muniz, M. S. Lett. 76, 4254 1996 . 48 Ferreira, and J. Mathon, Phys. Rev. B 55, 3724 1997 . J. Kudrnovsky, V. Drchal, I. Turek, M. Sob, and P. Weinberger, 40 M. van Schilfgaarde and W. A. Harrison, Phys. Rev. Lett. 71, Phys. Rev. B 53, 5125 1996 . 49 3870 1993 ; M. van Schilfgaarde, F. Herman, S. S. P. Parkin, P. Lang, L. Nordstrom, K. Wildberger, R. Zeller, P. H. Dederichs, and J. Kudrnovsky, ibid. 74, 4063 1995 . and T. Hoshino, Phys. Rev. B 53, 9092 1996 .