Journal of Magnetism and Magnetic Materials 200 (1999) 322}337 Interlayer exchange coupling M.D. Stiles* Electron Physics Group, National Institute of Standards and Technology, Bldg 220 Rm B206 Gaithersburg, MD 20899, USA Received 20 January 1999; received in revised form 2 March 1999 Abstract The extensive research done on interlayer exchange coupling in transition metal multilayers has resulted in a deep understanding of this coupling and a remarkable agreement between theoretical results and measurements. The coupling between two magnetic layers separated by a non-magnetic spacer layer is mediated by the electrons of the spacer layer. The coupling, which oscillates in sign as a function of the thickness of the spacer layer, is closely related to the well-known RKKY interaction between magnetic impurities. Due to the existence of many high-quality measurements, it has been much more fully developed theoretically than the interaction between impurities. Theory predicts that the periods of the oscillatory coupling should depend on critical spanning vectors of the Fermi surface belonging to the spacer-layer material. There is remarkable agreement for the measured periods and those predicted from the Fermi surfaces. There is also substantial agreement between theory and experiment on the strength of the coupling. This review presents the comparison between theory and experiment in some detail. 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Interlayer exchange coupling; Spacer-layer thickness; Fermi surfaces; RKKY interaction 1. Introduction transition metal multilayers [4] and in rare-earth multilayers [5,6]. Shortly thereafter, a large change When magnetic "lms are separated by a non- in resistance with changing alignment of the layer magnetic spacer layer, the magnetizations of the magnetizations, known as giant magnetoresistance, layers are coupled to each other by an exchange was discovered in Fe/Cr multilayers [7,8]. This interaction through the electrons of the spacer discovery led to a great deal of interest because of layer. As the thickness of the spacer layer is varied, commercial applications. In 1990, systematic stud- the coupling can oscillate in sign; as many as 60 ies of the giant magnetoresistance in several sign changes have been observed as a function of transition metal multilayers found that the coup- thickness. This coupling is closely related to the ling oscillates as a function of spacer-layer thick- oscillatory coupling, known as the RKKY interac- ness [9]. This review article will describe our tion [1}3], between magnetic impurities in a non- current understanding of the interlayer exchange magnetic host. It was "rst observed in 1986, both in coupling in transition metal multilayers, based on the extensive research done since 1990. Much has been written about interlayer ex- * Corresponding author. Tel.: #1-301-975-3745; fax: #1- change coupling, including several reviews [10}13]. 301-926-2746. This review emphasizes the current status of the E-mail address: mark.stiles@nist.gov (M.D. Stiles) comparison between theory and experiment. I will 0304-8853/99/$ - see front matter 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 3 3 4 - 0 M.D. Stiles / Journal of Magnetism and Magnetic Materials 200 (1999) 322}337 323 not attempt to describe the related coupling found is essentially a single-particle e!ect. All electrons in rare-earth multilayers, which have been the sub- scatter from the interface or impurity. The interfer- ject of several reviews [14,15]. Similarly, I will not ence between the incoming and scattered waves discuss quantum-well states in magnetic multi- gives rise to oscillatory probability densities for layers, even though their observation has played an each electron. Since the interfaces and impurities important role in con"rming our understanding of are magnetic and scatter spin-up and spin-down interlayer exchange coupling. These results have electrons di!erently, "lling all states below the been recently reviewed in Refs. [16,17]. Finally, Fermi energy gives an oscillatory spin density. I will only consider the bilinear coupling in these Since di!erent waves are characterized by vectors systems. I will not describe the biquadratic coup- for all the states, each of their states contributes to ling which is almost always present in addition to oscillations with di!erent periods. However, all of the bilinear coupling. The biquadratic coupling is the oscillations cancel out, except for those at the believed to arise from extrinsic e!ects [18], like Fermi energy, where there is a sharp cut-o! from interface roughness, and the comparison between completely "lled states to completely un"lled theory and experiment is much more ambiguous. states, leaving an oscillation characterized by the A recent review [19] describes what is known. Fermi surface. The second interface or impurity Particular forms of biquadratic coupling are asso- couples to the spin density set up by the "rst. Since ciated with antiferromagnetism in the spacer layer, the spin density oscillates as a function of the Cr and Mn being two examples. These e!ects are spacer-layer thickness or impurity separation, the described elsewhere in this volume [20] and will coupling oscillates as well. not be considered here. Several early models [21}25] of the interlayer In the next section, I will describe the physical exchange coupling were based explicitly on using origin of interlayer exchange coupling. This under- the RKKY interaction for impurities. These models standing is embodied in several di!erent models for consider the interaction between two two-di- the coupling. I will describe these models, empha- mensional sheets of impurities embedded in a sizing their common features. Some experimental non-magnetic host. This model is based on a local- predictions, like the periods of the coupling, are moment approximation to describe the magnetic common to all of the models. Others, like the coup- material. While such an approximation may ling strength, vary from model to model. In Section [26,27] or may not [28] be valid for magnetism in 3, I will describe some of the di$culties in compar- rare-earth multilayers, it is not valid for magnetism ing theory to experiment. Finally in Section 4, I will in transition metals where the bandwidths of the compare experimental results with theoretical re- d-electrons, which make up the magnetic moments, sults on a system by system basis. are on the order of 10 eV wide. While RKKY-based models may not use an ap- propriate description of the magnetism in 2. Bilinear coupling models transition metals, they point out some important general features of models for the interlayer ex- Interlayer exchange coupling between magnetic change coupling. They point out that the periods of layers mediated by a non-magnetic spacer has es- the oscillatory coupling are determined by critical sentially the same physical origin as the RKKY spanning vectors of the Fermi surface of the mater- coupling between magnetic impurities in a non- ial that makes up the spacer layer [22,29]. Critical magnetic host. In both cases, localized and spin- spanning vectors, see Fig. 1, are vectors in the polarized disturbances, interfaces in one case and direction of the interface normal, that connect two impurities in the other, are coupled to each other by sheets of the Fermi surface that are parallel to each their in#uence on the electrons in the spacer or other at the endpoints of the vector. Critical spann- host, respectively. First consider a single interface ing vectors determine the coupling periods in all or impurity. It sets up an oscillatory polarization in models of interlayer exchange coupling. Compar- the non-magnetic spacer or host. This polarization ing measured periods with those predicted by 324 M.D. Stiles / Journal of Magnetism and Magnetic Materials 200 (1999) 322}337 various models is the most robust aspect of com- One feature of free-electron models is that the paring theory and experiment in these systems. strength of the interlayer coupling depends on the In these models, the critical spanning vectors are spin di!erence of the re#ection amplitudes for elec- properties of the Fermi surface of the bulk material trons in the spacer layer re#ecting from the interfa- that makes up the spacer, rather than the Fermi ces with the magnetic material. This dependence surface of the spacer layer itself. The fact that the transfers to models with general band structures spacer layer is part of a multilayer, and cannot have [37}39]. Spin-dependent re#ection from the inter- an independent Fermi surface is beside the point. faces gives quantum con"nement in the spacer The (possibly "ctitious) bulk material that makes layer [29], setting up spin-dependent quantum-well up the spacer does have a well-de"ned Fermi sur- states, both true bound states and resonances, due face that is useful for understanding the interlayer to the interference resulting from multiple re#ection exchange coupling. The electron states in the multi- from the interfaces. As mentioned above, these layer can be described in terms of linear combina- quantum-well states have been seen experimentally tions of bulk states of the material that makes up in photoemission and inverse photoemission each layer. These states are matched together at the [40}43]. The "lled quantum-well states give rise to interfaces to construct the scattering states of the the oscillatory polarization described earlier. As the full system. Even though the bulk materials may be thickness of the spacer layer is changed, the quan- arti"cial constructs, they still provide a useful basis tum-well states move up or down in energy depend- for a description of the states in the system. ing on the details of the spacer-layer band Soon after the discovery of the interlayer ex- structure. The oscillatory interlayer exchange coup- change coupling, it was thought, based on free- ling is determined by the energy changes associated electron models, that oscillation periods would be with "lling and emptying these states as they cross much shorter than those that were observed experi- the Fermi energy when the thickness of the spacer mentally. It was quickly pointed out by several layer is varied. The stronger the spin-dependent groups [30}32] that it is important to account for re#ection, the stronger the con"nement and the the lattice of the spacer layer. With a lattice, it is stronger the oscillatory coupling. impossible to talk of oscillations that are faster than the lattice spacing. Such oscillations get &aliased' &&&&&&&&&&&&&&&&&&&&&&&&&&& with the lattice spacing to produce oscillations that Fig. 1. Critical spanning vectors and interface re#ectivities. For are slower than the lattice spacing. The period of an a series of spacer layers, magnetic materials, and interface ori- oscillation on a lattice with layer spacing d is deter- entations, organized in rows, the middle panels show slices mined from a critical spanning vector q though the Fermi surface of each spacer layer material for , by "nding the shortest possible equivalent spanning vector, kV"0. The interface normal is the z direction in all cases. Superimposed in red on the Fermi surfaces are some of the "q,!2 n/d", where n is some integer. critical spanning vectors. Each critical spanning vector is labeled While models based on the RKKY interaction by its associated coupling period in monolayers as determined capture much of the essential physics and correctly from the experimental Fermi surfaces [22,114]. In the left and predict possible coupling periods, they do not cor- right panels, the Fermi surface is projected onto the kX"0 rectly predict the strength of the coupling because plane. It is color-coded based on the probability for an electron incident from the spacer layer material to re#ect from the inter- they do not adequately describe magnetism in face with the magnetic material. Probabilities for electrons with transition metals. An alternate approach, which is spins parallel to the majority and minority spin directions are also not quantitative but captures much of the shown in the left and right panels, respectively. The locations of essential physics, is to make a free-electron approx- the critical spanning vectors are labeled by red circles centered imation in each layer with exchange split bands in at the critical point. The Cu Fermi surface projected into a (1 1 0) interface and the Cr Fermi surface projected into the ferromagnetic material [33}36]. The two simple a (0 0 1) interface have multiple sheets. To present these overlap- models, RKKY and spin-split free-electron bands, ping sheets, each is only shown in a faction of the interface represent two extremes in the description of mag- Brillouin zone. The full Fermi surface can be reconstructed by netism, local-moment models and itinerant models rotating the various partial sheets into to the other symmetric respectively. parts of the zone. M.D. Stiles / Journal of Magnetism and Magnetic Materials 200 (1999) 322}337 325 326 M.D. Stiles / Journal of Magnetism and Magnetic Materials 200 (1999) 322}337 All these models, RKKY [22], quantum-con"ne- critical point (maximum, minimum, saddle point). ment [29], free-electron [33}36], and interface-re- In this approach, the critical spanning vectors are #ection [37}39] predict that for large spacer-layer identi"ed, and the spin-dependent re#ection ampli- thicknesses, D, the coupling should be given by tudes for the states at those points on the Fermi a sum of terms of the form surface are computed (see Fig. 1). These calcu- lations are much less demanding computationally J? J(D)" sin(q? and give insight into what aspects of the band ,D# ?). (1) ? D structures, Fermi surfaces, and electronic wave There is a contribution from each critical point, functions are important. The asymptotic form re- labeled by , with critical spanning vector q? sults from an approximation that ignores the en- ,, coup- ling strength, J? and phase ?. For large thick- ergy and parallel wave-vector dependence of the nesses, this form is independent of the model used to re#ection amplitudes and assumes that the Fermi describe the interlayer coupling. For small thick- surface is strictly quadratic near the critical points nesses, other terms, called pre-asymptotic correc- of the Fermi surface. These approximations may tions, become important. In all models, the periods not be appropriate for particular systems [44}46], are determined by the critical spanning vectors of as will be discussed below. For small thicknesses, the spacer-layer Fermi surface, ¸?"2 /q? the pre-asymptotic corrections both can change the ,. Thus, the best way to compare measured coupling peri- decay of the oscillations and can modify their e!ec- ods with theory is to use the critical spanning tive period. However, the simple asymptotic form vectors found from analyzing experimental measure- provides a useful context in which to understand ments of the Fermi surfaces [22]. the further details. All pre-asymptotic corrections Model-independent comparisons are not pos- are automatically included in total-energy calcu- sible for comparing coupling strengths. Two main lations that are adequately converged. approaches have been used for computing coupling A "nal theoretical issue is related to self-consist- strengths. The "rst approach is to calculate the ency in the electronic structure calculations. All of total energy of the multilayer by computing and the asymptotic calculations and many of the total- "lling all the electron states below the Fermi en- energy calculations ignore the e!ect of the elec- ergy. These calculations can be self-consistent, tron}electron interaction in the spacer layer on the allowing the potential to vary in response to vari- spin-density wave that gets set up there. Ignoring ations in the densities, or not. Even if not done the electron}electron interaction is analogous to self-consistently, these calculations are computa- using the bare susceptibility, , as opposed to the tionally very demanding. They have been imple- Stoner enhanced susceptibility, " /(1!J ). mented using di!erent approximations for the band For noble metal spacer layers, this approximation structure. The two most common approximations appears to be good. For transition metal spacer are tight-binding (TB) approximations or the lo- layers, the situation is less clear. In Fe/Cr multi- cal-density approximation (LDA). layers, the coupling is found to be a superposition The second approach is to calculate the coupling of a short period and a long period. As discussed strength, J?, in the asymptotic expansion for each elsewhere in this issue [20], ignoring the elec- contribution to the sum, Eq. (1) from tron}electron interaction is not a good approxi- mation for the short-period component of the J? v? coupling. On the other hand, it may be a good sin(q? , ? Im[ r? D ,D# ?)"4 D  r? e O?,"e Q?], (2) approximation for the long-period contribution, as discussed below. Unfortunately, it is di$cult to where v?, is the component of the e!ective group determine the importance of the electron}electron velocity in the interface direction, ? is the radius of interactions in other systems. No other transition curvature of the Fermi surface, r? is the spin metals are well lattice-matched to transition-metal di!erence in the re#ection amplitude for the left ferromagnets, a condition that is necessary for (right) interface, and ? is a phase from the type of meaningful determinations. Several calculations M.D. Stiles / Journal of Magnetism and Magnetic Materials 200 (1999) 322}337 327 based on the local-density approximation have The coherent-potential approximation can be tried various approximations for turning o! the used to treat more localized and uniformly distrib- electron}electron interaction [47,48] and com- uted defects, like bulk defects, including alloying pared the results with those from treating the elec- [50,51], or interdi!usion at interfaces [52,53]. tron}electron interactions self-consistently within Physically, scattering from defects reduces the co- the local-density approximation. herent scattering fraction, this scattering reduces the amplitude of the quantum-well states, reducing in turn the size of the oscillatory coupling. Only in 3. Di7culties in comparing theory and experiment the case of intentional alloying is the concentration of defects known well enough that quantitative Several issues complicate the comparison be- comparisons between theory and experiment can tween theory and experiment for interlayer ex- be made. These will be discussed below. change coupling. Many of these are related either Finite temperature also reduces the amplitude of to growing an experimental structure that is close the interlayer coupling. Most models predict a spe- enough to ideal that the system can be treated ci"c form of the temperature dependence [22,29], theoretically or to doing a calculation that is su$- an additional factor of the form ciently complicated that it provides an adequate description of the actual experimental system. The (2 k ąD/ v?,) most di$cult issue to address theoretically is a lack (3) sinh(2 k of periodicity in the interface plane. Without this ąD/ v?,) periodicity, the asymptotic approximations are not associated with each critical spanning vector. How- valid and the total-energy calculations are intrac- ever, none of these models include an accurate table. Thus, good comparisons between theory and description of the temperature dependence of the experiment are only possible for systems that are ferromagnetic magnetization. Since the primary close enough to lattice matched that is possible to temperature dependence in ferromagnets at low grow coherent structures, structures in which all temperature comes from the thermal excitation of layers have the same in-plane lattice net. Unfortu- spin waves, the temperature dependence of the ex- nately, this consideration greatly restricts the num- change coupling will depend on the behavior of ber of systems for which meaningful comparisons correlated spin waves in both materials. The spin are possible. waves in both materials become correlated by the Other possible types of disorder can at least be exchange coupling across the interface. I am un- treated theoretically at some level of approxima- aware of any quantitative treatment of this temper- tion. However, to do so it is necessary to know ature dependence for interlayer exchange coupling what the disorder is, and how much of it there is. (see Ref. [54] for a treatment of tunneling mag- For all but a few measurements, this information is netoresistance). There have been some studies in not available. The simplest type of disorder to ac- which the form of the measured temperature count for is the presence of #uctuations in the dependence is consistent with that predicted by thickness of the spacer layer. While there are addi- various models. In many systems the temperature tional e!ects due to di!use scattering at the steps, dependence is weak, so the comparisons discussed the gross e!ect of thickness #uctuations can be below will ignore the temperature for the most part. accounted for by averaging the coupling over the A theoretical di$culty is that the local-density distribution of thicknesses due to the growth front approximation only reproduces the band structure [21]. It is possible to vary the roughness of the to within some unknown accuracy. Where the interface by varying the growth conditions, to Fermi surface is known experimentally, the accu- measure the growth front with scanning tunneling racy can be checked. However, this inaccuracy is microscopy, and to compare the resulting coupling impossible to correct for within total-energy calcu- with averages over the growth front [49]. In this lations. Thus the periods of the oscillations in these case, the comparison appears to be quite good. calculations will have some inherent inaccuracy, 328 M.D. Stiles / Journal of Magnetism and Magnetic Materials 200 (1999) 322}337 and because the Fermi surfaces of the constituent would aid in determining the parameters. However, materials are also slightly inaccurate, there is an in most measurements these sharp features are unknown uncertainty in the resulting coupling broadened out, presumably by disorder. The deter- strengths that cannot be eliminated. Most tight- mination of the model parameters is sensitive to binding band structures are constructed by "tting how the disorder is treated [57,58]. If the measure- to band structures computed within the local-den- ment involves reversal of the magnetization, it is sity approximation [55]. It is possible to adjust the also necessary to have a model for the reversal tight-binding parameters to bring the Fermi surface process. Most often, the models for reversal are into better agreement with experiment [56], but solved in one of two limits, either the global min- this is an uncontrolled approximation and may imum energy limit, which gives no hysteresis, or the make other aspects of the tight-binding description Stoner}Wohlfarth limit, which gives maximal worse. hysteresis. Reality is usually in between. Accurate Asymptotic calculations give J?, total-energy cal- determinations of the coupling require accurate culations give J(D), and experiments give J(D) determinations of the magnetic moments of each of modi"ed by whatever disorder is present. Even if the layers. Often, the magnetic moment is estimated there were no disorder in the experiment, the coup- from the thickness of the ferromagnetic "lms and ling at large thicknesses cannot be directly com- the bulk magnetization. Then, the accuracy de- pared with that from a total-energy calculation pends on how accurately the thickness is known, because the periods in the calculation will be di!er- and how close the thin-"lm magnetization is to that ent than those in the physical system. Ideally, both of the bulk. total-energy calculations and experiment are "t to parameterized forms, like the asymptotic form, Eq. (1), and the parameters compared. In this case, 4. Comparison between theory and experiment the parameters are also easily compared with the results of asymptotic calculations. Unfortunately, it In this section, I describe the comparison be- has been shown that the asymptotic form is not tween theory and experiment for speci"c systems. always a good description of at least the theoretical Overall, the agreement between theory and experi- results (see the discussion below for Co/Cu(0 0 1)). ment is quite substantial, much more so than for If the results are not "t, experimental periods are similar comparisons for the bulk RKKY interac- usually determined from the separation of several tion between impurities. One of the outstanding peaks in the coupling. Without "ts, coupling points of agreement comes from the periods pre- strengths are compared through the strengths of dicted from the experimental Fermi surfaces [22] a peak at small thickness. These coupling strengths and their precise experimental determination for are di!erent dimensionally from the parameters the Ag/Fe(0 0 1) [58] and the Au/Fe(0 0 1) [59] found in asymptotic calculations, J?. To make systems. The coupling periods for other systems meaningful comparisons it is useful to report the have been found less precisely, but all are in sub- results of asymptotic calculations in the form of stantial agreement with critical spanning vectors of J?/(1 nm), as will be done below. the experimental Fermi surfaces. The coupling A "nal experimental di$culty is determining the strengths for Cu/Co(0 0 1) which have been mea- coupling strengths from whatever properties are sured [60}64] and calculated [39,44}46,65}71] by measured. The coupling strength is determined by several groups are in qualitative agreement when computing the properties from a model and vary- some of the di$culties in the comparison are taken ing the parameters of the model, including the into account. Comparisons for the coupling coupling strength, to "t the measured properties. strength of Cu/Co(1 1 1) [56,71}76] and Au/ This procedure is strongly dependent on having the Fe(0 0 1) [71,77] are also in agreement, although correct model for all of the important energies in studied less extensively. Several alloy studies the system. In many model calculations, there are [78,82] show agreement between periods predicted sharp features in the calculated properties that from approximate calculations of the alloy Fermi M.D. Stiles / Journal of Magnetism and Magnetic Materials 200 (1999) 322}337 329 surfaces and the measured periods as the alloy totic result [46]. In fact, the apparent period cha- concentration is varied. Finally, the models de- nges as well. In addition, the re#ection amplitude scribed above predicted that the strength of the for "nite Co thickness is very sensitive to the Co interlayer coupling should oscillate as a function of thickness [83]. the thickness of every layer in the multilayer, e.g. For the short period, there is a gap for the Co not only the spacer-layer thickness, but also the minority electrons with the same symmetry as the thicknesses of the magnetic layers, and any capping Fermi surface electrons in the Cu at the critical layers. All these variations have been observed. point. However, the gap is narrow in energy. Since the phase of the re#ection amplitude changes by 4.1. Lattice-matched noble-metal spacers across the gap, the re#ection amplitude is strong- ly energy dependent, even though its modulus is 4.1.1. Co/Cu(0 0 1) constant [45,46,90]. The energy dependence of the The most extensively studied system, both theor- re#ection amplitude in this case gives a large reduc- etically [39,44}46,56,65}71,83] and experimentally tion of the coupling for thin spacer layers compared [60}64,84], has been FCC Co grown on Cu(0 0 1). to the value expected from the asymptotic result. These two materials are quite well lattice-matched These deviations from the asymptotic behavior for and high-quality multilayers can be grown. How- both critical points are born out in total-energy ever, there are substantial complications in the calculations [45,66}68]. comparison between theory and experiment. Per- The periods measured experimentally are in haps these complications are apparent simply be- good agreement with those expected from the criti- cause this system has been so extensively studied. cal spanning vectors of the experimental Fermi The coupling is expected to involve a combination surface [22]. Johnson et al. [62] measured the of two periodicities [22]. One period is long and is coupling as a function of thickness and "t the associated with the belly of the free-electron-like Cu results to a combination of two periods with Fermi surface. The other is short and is associated the asymptotic form, Eq. (1). Since the theoretical with the necks of the Fermi surface. Experi- models indicate that the asymptotic behavior is not mentally, the ratio of the two coupling strengths expected to hold for the thicknesses measured in depends quite strongly on the growth [64,84]. The- the experiment, these "ts need to be treated with oretically, it has been found [44}46] that the caution. The resulting "ts give periods of asymptotic approximation does not hold in the 2.60$0.05 ML and 8.0$0.5 ML, while the peri- region of experimental interest. Finally, in both ods extracted from the experimental Fermi surfaces calculations and experiment [85,86] the coupling are 2.56 and 5.88 ML. The short periods are in strength is found to depend on the thickness of the good agreement, and the increase of the long period Co layers and even Cu capping layers [87,88]. in the pre-asymptotic region is seen in total-energy In Co/Cu(0 0 1), theoretical di$culties arise for calculations [45,66}69], and expected based on several reasons [46]. For the long period, which is analysis of the pre-asymptotic corrections [46]. associated with a critical point at the zone center, M, Weber et al. [64] measure the sign of the coupling the re#ection amplitude is small for both minority and "t the zero crossing to the sum of two asymp- and majority electrons. So the resulting asymptotic totic contributions. With the same caution about coupling strength is small for in"nitely thick Co. the applicability of such "ts they "nd short periods However, for the minority electrons, the re#ection in the range of 2.58}2.77 ML and long periods in amplitude increases rapidly with parallel mo- the range 6.00}6.17 ML over the four samples they mentum around the critical point up to nearly studied. In all four samples, the zero crossings were complete re#ection at points still close to the zone very di!erent, showing the sensitivity of the results center [71,89] (see Fig. 1). Thus, for thinner spacer to the quality of the growth. layers, in which the coupling is sensitive to larger Given the large pre-asymptotic behavior in areas of the Fermi surface, the coupling is much this system, there is substantial agreement between stronger than would be expected from the asymp- di!erent calculations of the coupling strength. 330 M.D. Stiles / Journal of Magnetism and Magnetic Materials 200 (1999) 322}337 Table 1 Coupling energies in Co/Cu(0 0 1) multilayers. All coupling energies are in mJ/m. The asymptotic values are given at a spacer thickness of 1 nm. Entries with a * have been estimated by the author from "gures in the given references. The total-energy calculations are compared with each other and experiment by determining the coupling energy at a peak close to 1 nm spacer thickness. The position of the peak, D is given in ML Reference Method J*/(1 nm) J1/(1 nm) D J(D ) [71] Asymptotic (LDA) 0.12 11 [56] Asymptotic (TB) 0.021 6.7 [89] Asymptotic (TB) 0.14 0.57 [44] Total-energy (TB) 0.01H 12H 6 1.2H [69,70] Total-energy (LDA) 13H 9 4.6H [66}68] Total-energy (LDA) 7 1.42 [63] Experiment 5.2 0.39 [85] Experiment 5.2 0.24 [61] Experiment 6 0.16 Representative values of coupling energies are com- because the energy di!erence during growth be- pared in Table 1. It is possible from some of the tween FCC-like and HCP-like structures is very total-energy calculations to calculate or estimate small. It is easy to nucleate islands of both struc- the asymptotic behavior, which is found to agree tures during growth. These di!erent growth struc- with strictly asymptotic calculations. The total-en- tures lead to highly defective multilayers. ergy calculations can be compared with each other Theoretically, this system is interesting because in terms of the strength of the coupling at a peak there is not a free-electron-like critical spanning close to 1 nm spacer thickness. The calculated vector for this orientation of the Cu Fermi surface, coupling strengths are at least a factor of three only one spanning vector that bridges the neck at larger than the measured values. This di!erence is an oblique angle [22] (see Fig. 1). There is only likely due to the (unmeasured) thickness #uctu- weak evidence of an oscillatory coupling. In most ations in the samples. Lang et al. [66}68] demon- experiments, one strong antiferromagnetic coup- strate that they can average the calculated coupling ling peak is seen and evidence for another peak at over a reasonable growth front and get values very some greater thickness is seen. Asymptotic calcu- close to what is measured. lations of the coupling strength give There have been several studies of the phase of J/(1 nm)"0.59 [56] and 0.67 mJ/m [71]. These the interlayer coupling as a function of the alloying values compare well with the coupling strengths in the magnetic layers [91,92]. However, these measured at thicknesses of about 1 nm of 0.54 [74], studies only observed the long-period contribution 1.1 [72], and 0.4 mJ/m [76]. to the coupling. Since the short-period coupling may contribute for thin "lms, and the amount it 4.1.3. Co/Cu(1 1 0) contributes may vary as a function of the alloy For the (1 1 0) orientation of FCC Co on Cu, the composition of the magnetic layers, any compari- coupling is expected to be a superposition of four son with theory is di$cult. periods [22]. Three of these are short periods, one of which is expected to be stronger than those from 4.1.2. Co/Cu(1 1 1) all other critical spanning vectors for all orienta- The growth of FCC Co on Cu(1 1 1) has also tions of Co/Cu [56,71]. However, only the long been studied extensively [72}76,93}95], in part to period has been observed [72,97]. The coupling understand the di!erences in growth by sputtering strength at about 1 nm is measured to be 0.7 mJ/m and molecular beam epitaxy. For a review of [72], which is close to the asymptotic values of growth studies see Ref. [96]. The problem arises 1.0 [56] and 1.3 mJ/m [71] for the long-period M.D. Stiles / Journal of Magnetism and Magnetic Materials 200 (1999) 322}337 331 coupling strength, but much smaller than the coup- thickness of the wedge as a function of position, ling strength calculated for the short-period, 38 and Unguris et al. [59] were able to observe 60 changes 27 mJ/m, respectively, in the two calculations. in sign of the coupling as the Au thickness varied While the coupling strengths calculated for the over 80 monolayers. This allowed them to deter- (1 1 1) and the (1 1 0) orientations of Co/Cu multi- mine the periods of the two oscillatory contribu- layers appear to be in good agreement with mea- tions quite precisely. The measured periods of sured coupling strengths, these orientations have 2.48$0.05 and 8.6$0.3 ML are in remarkable not been studied as extensively as has the (0 0 1) agreement with those predicted for the system orientation. The growth of these multilayers is not based on the critical spanning vectors of the experi- as well controlled as it is for the (0 0 1) orientation. mental Fermi surface by Bruno and Chappert [22] Theoretically, only asymptotic calculations have 2.51 and 8.60 ML. been done. Either total-energy calculations need to In a subsequent experiment, Unguris et al. [77] be done, or at least an analysis of the pre-asymp- measured both the coupling strength as a function totic corrections. of the spacer-layer thickness and, through the RHEED intensity oscillations, the width of 4.1.4. Fe/Cu(0 0 1) the growth front. This allowed them to correct the If Cu is grown on Fe, it can be forced to grow in measured coupling strength for the averaging of the a BCC structure up to some thickness. There have thickness #uctuations. They "t the resulting coup- been several studies of this system which is of inter- ling strength to a combination of two oscillations est because the BCC Cu Fermi surface, while still assuming the asymptotic form, Eq. (1), holds for free-electron-like, has very di!erent critical spann- all thicknesses. The resulting coupling strengths, ing vectors than the FCC Cu Fermi surface. Calcu- J1/(1 nm)"1.29$0.16 and J*/(1 nm)"0.18$ lations [62,98,99] show that the coupling is a 0.02 mJ/m, are reasonably close to calculated combination of two short-period components and asymptotic coupling strengths [71], 2.0 and a long period, with one of the short periods domi- 1.1 mJ/m. While this agreement is comparable to nant. The experimental results are at odds with that for Co/Cu(0 0 1), the measured results for this each other. Johnson et al. [62] "nd a short-period system have been corrected for thickness #uctu- oscillation for spacer-layer thicknesses between 11 ations. Thus the agreement is not completely satis- and 18 ML. Celinski et al. [100] "nd much stronger factory. The strength of the short-period oscillation coupling but with at least a strong contribution is in good agreement, but that of the long period is from a component with a much longer period. o! by a signi"cant amount. At the critical point associated with the short-period oscillation, the re- 4.1.5. Fe/Au(0 0 1) #ection amplitudes for both minority and majority While BCC Fe and FCC Au have very di!erent electrons are signi"cant, and varying with parallel lattice constants, when the (0 0 1) orientation of Au wave vector [71] (see Fig. 1). The errors in the is rotated by 453 with respect to the (0 0 1) orienta- alignment of the Fermi surface due to the local- tion of Fe, the two layers have an in-plane lattice density approximation are likely to lead to signi"- match that is better than 1% [101,102]. This close cant uncertainty in the value of this contribution to lattice match allows for very good growth of multi- the coupling. On the other hand, at the critical layers of these materials, particularly on Fe point associated with the long-period oscillation, whiskers [59,77]. While the in-plane lattice is well only the re#ection probability for the minority elec- matched, out-of-plane there is a huge mismatch. trons is signi"cant. However, the re#ection prob- Thus, any steps on the substrate lead to growth ability does vary with parallel wave vector, and defects that propagate through the layer. The very since it is in a symmetry gap, its phase does change low step densities of iron whisker surfaces make with energy. It remains to be seen whether pre- them quite useful substrates for these systems. asymptotic corrections [46] of the type important By growing a wedge of Au on an iron whisker for Co/Cu(0 0 1), will also be important in this case. and using RHEED oscillations to calibrate the Additionally, there have also been indications 332 M.D. Stiles / Journal of Magnetism and Magnetic Materials 200 (1999) 322}337 [103] that the coupling strength for this system can tions consistent with those expected from the criti- depend on Fe thickness. cal spanning vector of the Fermi surface [22]. There have been several total-energy calcu- lations for the Fe/Au(0 0 1) system [104,105]. Both 4.2. Other lattice-matched spacer layers "nd coupling energies that are the same order as expected from the asymptotic calculation [71]. 4.2.1. Fe/Cr(0 0 1) Costa et al. [104] "nd a strong short-period oscilla- Fe/Cr multilayers, particularly in the (0 0 1) ori- tion and a weaker long-period oscillation. While it entation, have been quite widely studied. They were is di$cult to make quantitative comparisons based the "rst transition-metal multilayers to show on the published results, this trend is closer to what antiferromagnetic coupling [4], giant magneto- is seen experimentally than the asymptotic results. resistance [7,8], oscillatory coupling [9], and short- The authors compare their results with those of an period oscillatory coupling [108}110]. Cr is the earlier experiment for Au/Fe multilayers grown on only transition metal that is well lattice matched to GaAs [107]. While the overall scale factor dis- either Fe or Co, so Fe/Cr multilayers are parti- agrees by about an order of magnitude, the calcu- cularly interesting to test whether the understand- lations give excellent agreement of peak heights ing of interlayer exchange coupling extends beyond and positions. Unfortunately, neither the measured noble metal spacer layers. Unfortunately, high- peak positions, nor their strength agree with the quality Fe/Cr multilayers are dominated by the later measurements on Fe whiskers. This disagree- antiferromagnetic order in the Cr, which masks the ment highlights the di$culty in comparing total- coupling of the type that applies to noble metals. energy calculations directly with experiment. Since The short-period coupling in these systems and the the local-density approximation makes some error antiferromagnetism in the Cr are discussed else- in the Fermi surface, these calculations necessarily where in this issue [20] and will not be discussed get the oscillatory periods wrong, and hence the further here. peaks in the wrong places. The long-period coupling in Fe/Cr(0 0 1) is still controversial. Experimentally, it is well established 4.1.6. Fe/Ag(0 0 1) [49,97,108,110,111], that in samples where the Since silver has a lattice constant very close to thickness #uctuations are large enough that the that of gold, the same considerations hold for the short-period coupling is averaged out, a long-peri- growth of Ag on Fe(0 0 1) as for Au. There is the od component with a period of about 12 ML is similar agreement between the oscillation periods observed. Theoretically, the origin of this long- measured on Fe whiskers [58], 2.37$0.07 and period oscillation is controversial. Even among 5.73$0.05 ML, and those predicted based on the authors who believe that the coupling comes from experimental Fermi surface [22], 2.38 and 5.58 ML. essentially the same mechanism as the coupling in Measurements [106] of the coupling strength have noble metal spacer layers, there is disagreement not been analyzed in terms of two asymptotic con- over what part of the Fermi surface is responsible. tributions, but both sets of measurements [58,106] Several di!erent parts of the Fermi surface have show that the long-period oscillation is relatively been suggested. Mirbt et al. [48] suggest that the stronger for Ag spacers than it is for Au spacers. long-period contribution comes from a short- This same trend is seen in both asymptotic calcu- period contribution at the center of the interface lations [71] and total-energy calculations [104]. Brillouin zone. This short-period oscillation gets While direct comparison is di$cult, the measured aliased to a long-period oscillation by beating coupling strengths [106] are consistent with the against the doubled unit cell due to antiferromag- magnitudes of the coupling found in both calcu- netic order. Based on supercell calculations with Fe lations. layers two atomic layers thick, van Schilfgaarde There has been one study of the hexagonal (1 1 1) et al. [79,112] suggest that the long period comes face of Ag grown on the pseudo-hexagonal face of from an aliasing of the second harmonic of the Fe (1 1 0) [107]. That measurement found oscilla- short-period coupling. From analyses of the critical M.D. Stiles / Journal of Magnetism and Magnetic Materials 200 (1999) 322}337 333 spanning vectors of the paramagnetic Cr Fermi long-period contribution to the interlayer coupling surface, Koelling [113] has suggested that a spann- comes from the N-centered ellipsoids and that the ing vector of the lens part of the Fermi surface has periods, when extracted from the experimental the correct size. Several calculations [114}116] of Fermi surfaces are very close to what is measured in the asymptotic contributions to the coupling have all three cases. found that the strongest coupling comes from spanning vectors of the ellipsoids at the N-points of 4.2.3. Fe/V(0 0 1) and Fe/Al(0 0 1) the Brillouin zone (see Fig. 1). These various Fe and V are close enough to being lattice- mechanisms have been discussed in detail in Refs. matched that there is some hope of growing co- [114,117]. herent multilayers that could be compared with There are no de"nitive experiments that distin- theory. Oscillatory coupling has been seen in guish between the di!erent suggestions. Perhaps V/Fe(0 0 1) multilayers grown on Fe whiskers the best evidence comes from alloy studies [82] in [119], but the growth is signi"cantly worse than for which the Cr is doped with V. The period of the Cr, Au, or Ag, and the oscillatory coupling is only coupling is found to change with alloying. The observed over a much smaller range of spacer di!erent spanning vectors that have been suggested thicknesses. The coupling that is observed in this all change di!erently with doping. The changes in system bears little resemblance to that calculated the spanning vectors of the N-centered ellipsoids [47] for it. It may be that the strain in the V is great are the most consistent with the experiment. This enough that the structure considered in the calcu- evidence is indirect. Ideally, this issue could be lation di!ers too signi"cantly from that present in settled by photoemission experiments like those of the measurement. It is also possible to observe Li et al. [118]. Unfortunately, establishing the im- antiferromagnetic coupling in multilayers grown portant part of the Fermi surface by photoemission with thin layers of Fe, so that the V is less strained is quite di$cult. To establish a part of the Fermi [120]. However, since only one antiferromagnetic surface as the origin of the coupling requires dem- coupling peak is observed, it is impossible to dis- onstrating that there are quantum-well states there, cuss oscillatory coupling. Multilayers of this type that they are spin-polarized, that they are at a criti- can be reversibly loaded with atomic hydrogen cal point, and most di$cult, that there are no [121]. This changes the lattice somewhat, but also quantum-well states on some other part of the signi"cantly changes the electronic structure. At the Fermi surface that "t these criteria. No photoemis- same time the coupling in these multilayers changes sion studies of interlayer exchange coupling on any quite signi"cantly [122]. system have systematically studied the whole Fermi While Al is also close to lattice matched with Fe, surface. it does not grow well [107,119]. There are indica- tions of oscillatory coupling in some measurements 4.2.2. Fe/Cr(2 1 1) [107] and biquadratic coupling in others [123] but Several studies of Fe/Cr(2 1 1) multilayers nothing to compare with theoretical calculations. [97,111] have found remarkable similarities be- tween the long-period coupling for these systems 4.3. Alloy studies and the Fe/Cr(0 0 1) multilayers. The period found in both systems is very close to that found in sput- Additional evidence that the coupling is deter- tered multilayers that are believed to be predomi- mined by the spacer-layer Fermi surface comes nantly (1 1 0) textured [9]. This orientation from comparing measurements [78,79,82,102,124] independence suggests that there might be a di!er- and calculations [79}82] of the oscillatory periods ent mechanism for the coupling than the quantum- as a function of alloy concentration in the spacer well mechanism described above. However, layer. Alloying in the spacer layer changes the oscil- calculations of the asymptotic contributions to the latory coupling in two ways. First, it changes the coupling show that for both the (2 1 1) [114] and band structure, the Fermi surface, and hence the (1 1 0) [114}116] interface directions, the dominant critical spanning vectors. These changes can be 334 M.D. Stiles / Journal of Magnetism and Magnetic Materials 200 (1999) 322}337 captured in models based on the virtual-crystal to the systematic study by Parkin of most of approximation. In these models, the material is the transition metals sandwiched between Co imagined to have no disorder, but a "ctitious, frac- and grown by sputtering [135]. There have been tional, nuclear charge on each site. Second, the fewer studies of multilayers with Fe as the magnetic alloy disorder leads to di!use scattering, which layer, Fe/Pd [136,137], Fe/Nb [138], and Fe/Mo reduces the amplitude of the coupling. These cha- [139]. nges can be captured in models based on the coher- There are two trends that come out of these ent-potential approximation. In these models, the studies that demand explanation. First, with the electronic states gain a width due to the di!use exception of Os (1.5 nm) and Cr (1.8 nm), all of the scattering. observed periods are in the range 0.9}1.2 nm. Sec- For small concentrations of Ni in Cu, the Ni ond, there is a very strong trend of increased coup- remains non-magnetic, and the Fermi surface con- ling strength as the spacer material moves to the tracts due to the lower electron density in Ni right in the periodic table. For any interface ori- compared to Cu. For (1 1 0) and (1 1 1) oriented entation, the Fermi surfaces of transition metals multilayers, Ni doping gives an increase in the have many critical spanning vectors. Assuming that oscillation period of the long period, while for the sputtered multilayers have the low-index ori- (0 0 1) multilayers, it gives a decrease. An increase, entation, there are critical spanning vectors that consistent with the Fermi surface properties has match the observed periods for all of the measured been observed for (1 1 0) multilayers [124] and for systems [38]. The issue then becomes why are other sputtered multilayers with a (1 1 1) texture [78]. periods not observed. A possible explanation is that Alloying Cu and Au only weakly changes the a common period is observed because the experi- oscillation periods because both are noble metals. mental sensitivity function is peaked for periods However, there is still alloy scattering due to the with the observed value. Shorter-period oscillations disorder. Annealing an alloy with 55% Au [102] are averaged out by thickness #uctuations as has gives a large increase in the coupling energy. Pre- been observed for lattice matched systems. Longer- sumably this is due to ordering in the alloy which period oscillations are obscured both by the small reduces the di!use scattering. On the other hand, range of thicknesses measured and by measurement the increase in the unit cell size in the interlayer of properties, like magnetoresistance, that do not may give rise to additional critical spanning vectors change sign. While this explanation may be correct, that can change the coupling in less systematic a more physical explanation would provide more ways. insight. Alloying the Cr in Fe/Cr(1 1 0) multilayers with Mathon et al. [140] explain the trend in the V [79] gives an increase in the period of the oscilla- coupling strengths as coming from trends in the tory coupling that matches an increase found in band structure as a function of position in total-energy calculations of supercells of this ori- the periodic table. They consider a simple cubic entation. There is good agreement over the whole d-band tight-binding model and consider trends as alloy range between the results of the calculation a function of band "lling. For band "llings close to and the measured periods. The alloying of V in to the "lling of the ferromagnet they "nd strong re#ec- Cr in Fe/Cr(1 0 0) multilayers [82] has been dis- tion for one spin and weaker re#ection for the spin cussed above. for which the band structure is aligned with that of the spacer material. The resulting large spin asym- 4.4. Lattice mismatched systems metry gives rise to strong coupling. Then, moving left in the periodic table, as the band "lling becomes There have been many studies of multilayers that very di!erent from that of the ferromagnet, both are not as well lattice matched as those described spins have strong re#ection, and the coupling is above. Multilayers studied include Co/Au weaker. An alternative explanation may be related [125,126], Co/Ru [127}129], Co/Rh [130], Co/Ir to trends in structural quality as a function of [131], Co/Re [132,133], Co/Os [134], in addition position in the periodic table. M.D. Stiles / Journal of Magnetism and Magnetic Materials 200 (1999) 322}337 335 5. Summary [3] K. 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