Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 E!ect of roughness, frustration, and antiferromagnetic order on magnetic coupling of Fe/Cr multilayers D.T. Pierce*, J. Unguris, R.J. Celotta, M.D. Stiles Electron Physics Group, National Institute of Standards and Technology, Gaithersburg, MD 20899-8412, USA Received 22 February 1999 Abstract The interplay between interfacial disorder and the antiferromagnetic order in Cr leads to complex behavior in Fe/Cr multilayers. Measurements of interlayer coupling are discussed for samples with di!erent amounts of disorder ranging from optimally fabricated trilayers of Fe/Cr/Fe on Fe(0 0 1) whiskers, to trilayers with increasing degrees of interfacial roughness, and "nally to superlattices of Fe/Cr. The coupling of ferromagnets through noble-metal spacer layers can be described by a model that consists of bilinear coupling averaged over thickness #uctuations and extrinsic biquadratic coupling induced by the thickness #uctuations. This, the conventional model, also describes much of the behavior observed for Fe/Cr multilayers. However, in this case, the antiferromagnetism in Cr leads to results not explained by the conventional model. For nearly ideal interfaces, the Fe}Cr coupling can induce order in Cr, modifying the temperature dependence of the interlayer coupling. In addition, interfacial disorder can frustrate the antiferromagnetic order in the Cr, leading to a variety of ordered states which have been observed by neutron scattering. Each of these ordered states, in turn modi"es the interlayer coupling in unexpected ways. The di!erent ways in which the systems minimize the frustration can explain the experimental results. 1999 Elsevier Science B.V. All rights reserved. Keywords: Magnetic multilayers; Interlayer exchange coupling; Interfacial disorder; Spin density wave 1. Introduction the "rst systems to show oscillations in the coup- ling between the layers as the Cr thickness was Fe layers separated by Cr spacer layers have varied [4]. The existence of short-period as well as been at the center of many important discoveries long-period oscillations was "rst seen in Fe/Cr/Fe related to magnetic coupling and transport proper- trilayers [5,6]. In spite of the intense study of the ties. Fe/Cr/Fe trilayers exhibited the "rst evidence Fe/Cr system as indicated by these many discove- of antiferromagnetic coupling of two ferromagnetic ries, there are still many unanswered questions and layers through a transition-metal spacer layer [1]. apparent discrepancies between experiments. Giant magnetoresistance was discovered in Fe/Cr For noble-metal spacer layers, the interlayer ex- multilayers [2,3]. Fe/Cr superlattices were among change coupling is well described by quantum well models where the coupling properties are deter- mined by the Fermi surface of the spacer-layer * Corresponding author. Fax: #1-301-926-3746. material and the re#ection amplitudes for electrons E-mail address: daniel.pierce@nist.gov (D.T. Pierce) scattering at the interfaces between the spacer layer 0304-8853/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 3 1 9 - 4 D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 291 and the ferromagnetic layers [7}10]. Good agree- techniques like scanning tunneling microscopy ment has been obtained between quantum well (STM), re#ection high-energy electron di!raction model calculations and measured periods and (RHEED) and X-ray di!raction (XRD). We discuss strength of the oscillatory exchange coupling the magnetic coupling and Cr magnetic order in the [11,12]. In contrast, measurements of Fe/Cr/Fe samples measured with a variety of techniques: trilayers and Fe/Cr superlattices yield contradic- scanning electron microscopy with polarization tory results on the following topics: observation of analysis (SEMPA), magneto-optic Kerr e!ect short-period versus long-period oscillatory coup- (MOKE), Brillouin light scattering (BLS), neutron ling, coupled layers versus uncoupled Fe layers, scattering, and perturbed angular correlation spec- collinear versus non-collinear coupling, commen- troscopy (PACS). surate versus incommensurate antiferromagnetic The organization of the paper is as follows. The order in the Cr, etc. The disparate results are con- next three sections give background information on nected with the unique magnetic nature of Cr and the special characteristics of Cr as a spacer layer the sensitivity of the Cr magnetic order and the (Section 2), the important length scales and origin interlayer exchange coupling to a variety of struc- of spin frustration (Section 3), and the conventional tural details. model for describing interlayer exchange coupling The purpose of this paper is to review selected (Section 4). The next six sections come in pairs Fe/Cr multilayer coupling measurements, which presenting "rst the experimental results and then may appear at "rst glance to be contradictory, and the interpretation for each of the three classes interpret them in a consistent framework. We at- of samples: optimal trilayers (Sections 5 and 6), tempt to take into account di!erences in sample trilayers with varying interfacial roughness (Sec- structure and to synthesize various explanations tions 7 and 8), and superlattices (Sections 9 and 10). into a coherent picture of the physics that shows Conclusions are presented in Section 11. We how the disparate experimental results can be list some symbols used frequently in this paper in understood. Rather than attempting an encyclo- Table 1. pedic review, we select experimental results to illus- trate main points and point out areas of agreement and disagreement in light of current models. For Table 1 this paper, we concentrate on the coupling of the Fe Commonly used symbols. List of symbols that are commonly layers through Cr up to a thickness of about 10 nm, used in the paper. Unaveraged coupling strengths are de"ned for discrete thicknesses nd, and averaged coupling strengths are which is roughly the maximum distance over which de"ned for continuous average thickness t exchange coupling is still observed. Because interface structure is believed to be very Symbol De"nition important in determining the magnetic coupling, we will focus on three general classes of samples: (1) d Cr layer spacing n Number of layers &optimal trilayers', i.e., trilayer samples fabricated rms roughness on Fe whiskers under conditions for optimum $ Roughness at lower trilayer interface growth to approach, as closely as possible, ideal ! Roughness at upper trilayer interface interfaces, (2) &rougher trilayers', i.e., trilayers fab-  Standard deviation of Cr thickness distribution ricated on Ag-bu!ered GaAs substrates and tri- R Mean island spacing ¸ Terrace length layers fabricated on Fe whiskers at lower l Lateral response length of Fe temperatures, both of which have rougher Fe/Cr J(n) Unaveraged bilinear coupling interfaces, and (3) &superlattices', i.e., Fe/Cr superla- JM(t), JM Averaged bilinear coupling ttices with still rougher interfaces. After presenting JM(t), JM Averaged biquadratic coupling the experimental facts for each of these classes, we J1(n), J1 Unaveraged short-period coupling JM discuss the theories and plausible explanations of 1 Averaged short-period coupling J the di!erent behavior. Knowledge of the sample 1 Di!erence in J1(n) for thicknesses di!ering by one layer interfaces comes from sample characterization 292 D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 2. Special characteristics of Cr as a spacer layer In its paramagnetic state, Cr has considerable portions of its Fermi surface nearly parallel, or nested, as seen in Fig. 1 and labeled with the wave vector Q. The unenhanced susceptibility (q) is therefore peaked at q"Q. When electron}electron interactions are included we get the enhanced sus- ceptibility (q)" (q)/[1!I (q)], (1) where I, the enhancement factor, accounts for ex- change and correlation [13]. The large susceptibil- ity at Q leads to a transition from paramagnetic Cr to antiferromagnetic order as the temperature de- creases below the NeHel temperature ¹,. The nest- Fig. 1. A slice through the paramagnetic Cr Fermi surface for an ing wave vector Q is slightly incommensurate with interface in the (0 0 1) direction. The wave vector Q connects the lattice wave vector /d, i.e., Q" (1! )/d parallel &nested' regions of the Fermi surface. where is the incommensurability parameter [13]. The layer spacing is d which equals a/2"0.144 nm, where a is the lattice constant. When Cr orders antiferromagnetically, a small gap opens at the Fermi level and that part of the nested Fermi sur- face connected by Q in Fig. 1 disappears. This behavior is associated with resistivity anomalies in bulk Cr [13,14]. The antiferromagnetic order of Cr has complex variations. Commensurate antiferromagnetic order is the simplest, as illustrated in Fig. 2a. The mag- netic moments of the Cr atoms are all of the same magnitude and alternate in direction with each Cr layer in a [0 0 1] direction. This type of AF order is seen in Cr alloyed with, for example, small amounts of Mn [13,14]. In pure bulk Cr, the antiferromag- netic order leads to an incommensurate spin den- sity wave (ISDW) with a periodic modulation of the Cr magnetic moment, Fig. 2. (a) Commensurate antiferromagnetic order is " shown where the solid arrows and dashed arrows represent the  cos(Qz)" (!1)L cos(n  ), (2) Cr moments on corner atom and body-center atom sites, where respectively. (b) One period of an incommensurate spin density "0.62 for bulk Cr at zero temperature, and the SDW ordering wave vector is Q" wave (ISDW) is illustrated showing the variation of the Cr (1! )/d. The SDW ordering wave vectors are moments. always closer to commensuration than the nesting wave vectors Q, i.e., 0( ( [15]. The distance between nodes is d/ . The Cr moments can be observed in bulk Cr at temperatures below 123 K. perpendicular to Q to form a transverse SDW as Commensurate, transverse ISDW and longitudinal illustrated in Fig. 2b. The moments can also be ISDW are also referred to as AF0, AF1, and AF2 parallel to Q forming a longitudinal SDW that is order, respectively. D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 293 3. Length scales and spin frustration standard deviation of the thickness #uctuations can be greater or lesser than the rms roughness of the We "rst consider how to describe the roughness upper interface. at an interface. A measure of the lateral distribution The spin con"guration in the Fe and Cr layers is of the roughness can be obtained by calculating the a!ected by interface roughness. In a local moment height}height correlation function, for example model, the Fe}Fe interactions favor ferromagnetic from STM images. The height}height correlation is alignment of spins, while the Cr}Cr and Cr}Fe de"ned as G(r)"1 h(r#r) h(r)2 where h(r)" interactions favor antiferromagnetic alignment. h(r)!1h(r)2 is the deviation of the local height, For perfect interfaces, there are spin con"gurations, measured here in monolayers (ML), from the aver- as in Fig. 3a, in which all pairs of spins have their age height, and 1- - -2 denotes the spatial average preferred alignment. If there is roughness at the over all points r within the region of interest. The interface as shown in Fig. 3b}Fig. 3d, it is not rms roughness at the interface is "(G(0)). The possible to obtain the preferred alignment for all "rst peak in G(r) gives the mean separation between pairs of spins. Some pairs will necessarily not be in typical features, R, which for simplicity we will refer their minimum energy con"guration, that is, the to as the mean island separation. An important coupling will be &frustrated'. For the same structure length for the discussion of the coupling is the there can be many plausible spin con"gurations average terrace length ¸. For a given R, the average that are local minima of the energy [16}18]. In Fig. terrace length ¸, in a simple model, decreases with 3b, the Fe}Fe and Fe}Cr interactions are satis"ed, increasing , that is ¸JR/ . but the Cr}Cr interactions are frustrated through We distinguish between interface roughness and the Cr "lm at the position of the steps in the thickness #uctuations. If there is a step at the bot- interface. The frustration of the Fe}Cr interaction tom interface of the Cr spacer layer that is rep- at the interface is shown in Fig. 3c and in Fig. 3d the licated at the upper interface, the roughness due to frustration is taken up in the Fe layer. this step is fully correlated and there is no thickness The energy minimization that determines where #uctuation. If the lower interface is #at over the the frustration occurs will depend on the relative region of interest, as may be the case with an Fe sizes of several length scales such as the thickness of whisker, the thickness #uctuations are completely the Fe and Cr layers and the vertical and lateral speci"ed by the of the upper interface. For arbit- extent of the interfacial defects. It will also depend rary interfaces, the standard deviation of the thick- on the strength of the interactions and on the ness distribution, , depends on the roughness of temperature since the interactions are temperature both interfaces which can be correlated to varying degrees. For simplicity, consider a trilayer with roughness at the lower interface, $ , and at the upper interface, !. Then, in general, " $ # !!21 h$ h!2, where we have suppressed the spatial arguments that are averaged over. We con- sider two limiting cases: (1) the roughness at the upper interface and the roughness at the lower inter- face are not correlated, 1 h$ h!2"0, so that " $ # !, or (2) the Cr thickness #uctuations and the roughness at the lower interface are not correlated, 1 h$ t2"0, so that 1 h$ h!2" 1 h$ ( h$ # t)2" $ and " !! $ . Thus, for these two simple cases, the standard deviation of Fig. 3. Relieving spin frustration at an Fe/Cr interface. (a) Per- the thickness #uctuations is either "( !# $ ) fect interface, no frustration. (b) Frustration caused by a step is or relieved by a wall in the Cr. (c) Frustration relieved at the "( !! $ ). Depending on the growth- induced correlation between the interfaces, the interface. (d) Frustration relieved by walls in the Fe. 294 D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 dependent. It is expected that the Cr}Cr interaction constant, A will be more temperature sensitive since the bulk Cr "2.1;10\ J/m, and the cubic crys- tal anisotropy constant, K"4.7;10 J/m. The ex- NeHel temperature, ¹,"311 K, is much smaller change energy tends to increase the wall width to than the Fe Curie temperature, ¹!"1043 K. Pre- achieve a slow spatial variation of spin direction. cise values are not available for the strengths of the The anisotropy, on the other hand, tends to de- Fe}Fe coupling, the Fe}Cr coupling, and the crease the wall width to reduce the number of spins Cr}Cr coupling. Typical estimates of the relative pointing in hard directions. In the Fe/Cr multi- strengths of the Fe}Fe, Fe}Cr, and Cr}Cr coupling layer, the interlayer coupling behaves like an an- are 1: !0.3: !0.18 [15] and 1: !0.55: !0.3 [19]. isotropy that favors the Fe magnetization in a given Roughly speaking, these estimates have the Fe}Fe direction. When the interlayer coupling dominates coupling about two to three times the antifer- other anisotropies, one can write [20] l" romagnetic Fe}Cr coupling which in turn is about  (A t$ / J two times the Cr}Cr coupling. This suggests that it (n)) where t$ is the thickness of the Fe layer and J costs less energy for the interface frustration region (n) is the interlayer coupling strength at the discrete Cr layer thickness nd [20]. For a thin Fe of Fig. 3c to be in the Cr rather than right at the "lm in the Fe/Cr/Fe trilayer, the magnetization will interface [16]. The lines drawn in Fig. 3 to repres- turn in a plane parallel to the interface as a NeHel ent the frustration are only schematic. Generally, wall. In a more detailed calculation of the wall not much is known about these regions; for width, the anisotropy, the interlayer coupling, and example, we do not know whether the change is any other contributions to the energy must be in- fairly abrupt or spread out over many lattice con- cluded. RuKhrig and Hubert calculated an Fe wall stants. width of 150 nm for 30 nm thick Fe layers separ- Local moments coupled to each other, as shown ated by 13 ML of Cr; this value agreed well with schematically in Fig. 3, can be a good description of their measurements [21]. Cr with commensurate antiferromagnetic order. Less is known about magnetic transition lengths However, it should be remembered, that Cr is an in Cr. Some calculations "nd that the frustration itinerant antiferromagnet. This distinction is parti- can reduce the moments, reducing in turn the cularly important when Cr is in an incommensur- length scale required to relieve frustration ate order state. The intineracy may lead to more [16,17,22]. In other cases frustration may lead to variation in the moments than is expected from rotation of the moments [23}25]. Determinations local moment models. This variability of the size of of the minimum energy magnetic order in the pres- the moments complicates even further the deter- ence of frustration will depend strongly on the mination of the minimum energy state for a frus- model used to describe the Cr. While the model trated system. calculations of defect structures done to date give There are also magnetic length scales that must an indication of what might happen, they are not be considered. Of particular importance is the de"nitive because they do not include all the impor- length over which the magnetization in Fe can tant physics. In particular, these model calculations reverse its direction, which we call the lateral re- do not describe antiferromagnetism in Cr su$- sponse length l. This length can be estimated, in the ciently accurately to produce incommensurate or- simplest approximation, as the domain wall width der in bulk Cr. in bulk Fe. The Bloch wall width in bulk Fe is given by l" (A /K)+66 nm, where the exchange 4. The conventional model  The Cr}Cr estimate is the mean "eld result for a Heisenberg We present brie#y the model most commonly antiferromagnet. On the other hand, an estimate of the Cr}Cr used in describing the exchange coupling of mag- coupling from the spin wave velocity is eight times larger. These values emphasize the di$culty of estimating the Cr}Cr coupling netic layers in order to provide a structure for our in an itinerant spin wave system. (R.S. Fishman, private com- later discussion of numerous experimental results. munication.) In the conventional model the coupling is described D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 295 by a bilinear coupling term and a biquadratic term. assuming two contributions to the oscillatory The model assumes a paramagnetic spacer layer in coupling, as observed, J an itinerant electron picture. Thickness #uctuations  can be written of the spacer layer average the coupling, so short- J(n)"J1(n)#J*(n) period couplings are not observed if the thickness "(1/nd)A #uctuations are too large. The thickness #uctu- 1 sin(2 nd/ 1# 1) ations also lead to the biquadratic coupling. Begin- #(1/nd)A* sin(2 nd/ *# *), (4) ning in Section 6, we will discuss departures from where this conventional model owing to the special nature 1 and * are the short and long periods, respectively, of Cr. 1 and * are the phases, and the amplitudes A The total coupling energy per unit area, E 1 and A* include all the Fermi surface , is geometry and interface re#ection probabilities. The described in a phenomenological model that was 1/nd thickness dependence for the short-period os- proposed to explain certain experimental observa- cillation is unique to Cr because there is full planar tions [26,27]. nesting. Not shown in Eq. (4) are additional factors E arising from e!ects of temperature and disorder "!JMm( ) m(!JM(m( ) m() that further decrease the coupling with increasing "!JM cos( )!JM cos( ). (3) spacer layer thickness [30]. This model only ap- plies to paramagnetic Cr. When the Cr is antifer- The "rst term in this equation is the Heisenberg- romagnetic, a gap opens at the Fermi level [13] and like exchange term. The bar is used to emphasize quantum well models can no longer be used to that the measured quantities are averaged values. describe the two-layer short-period coupling. In Depending on the sign of JM, the magnetization this case, the antiferromagnetic order determines directions of the two Fe layers, given by unit vec- the short-period coupling. tors m( and m(, will be parallel or antiparallel. The The short- and the long-period parts of the coup- coupling depends on m()m(, i.e., it is bilinear in the ling, J magnetization directions. The second term, called 1(n) and J*(n), are de"ned only at each dis- crete thickness nd. In realistic spacer layers, there the biquadratic coupling term since it is biquad- are thickness #uctuations which act, within a re- ratic in m( and m(, leads to canted or non-collinear gion de"ned by the lateral response length l of the coupling, that is di!erent from 03 or 1803, when magnetic layer, to average the coupling contribu- JM(0 and "JM"(!2JM. Minimizing E with re- tions from regions of di!erent thickness. Thus, one spect to gives the angle of the canted coupling as measures an average coupling, cos "!JM/2JM. To "nd the minimum energy state of a multilayer, in the general case, it is neces- sary to include not just the terms in Eq. (3) but also JM(t)"JM1#JM*" P(t, n) J1(n)# P(t, n) J*(n), other terms such as the anisotropy energies of the L L (5) magnetic "lms [26,28,29]. In the conventional model, the materials are where P(t, n) is the fraction of the interlayer area treated in an intinerant electron picture. For that is n layers thick when the average thickness is t. a paramagnetic spacer layer, the interlayer coup- Thus, short-period oscillatory coupling will be ling is determined by the spin-dependent re#ections more rapidly averaged out by the thickness #uctu- at the interfaces between the spacer layer and the ations than the long-period coupling. ferromagnet layers [7}10]. The periods of the oscil- When the average bilinear coupling JM latory coupling are determined from the critical  becomes small enough, as a result of spacer-layer thickness spanning vectors of the spacer-layer Fermi surface. #uctuations, the multilayer "nds its minimum The strength of the coupling depends on the Fermi energy state when the magnetic moments of the surface geometry and the re#ection amplitudes of Fe layers turn into a direction perpendicular to electrons at the interfaces between the spacer layer each other. This is the basis of the model proposed and the ferromagnetic layers. For paramagnetic Cr, by Slonczewski [31] that takes into account the 296 D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 #uctuations J(n) in the bilinear coupling as the there are thickness #uctuations so that both odd coupling J(n) changes sign from one discrete layer and even Cr thicknesses are present, the energy is thickness to the next. In the case of Cr, the minimized by the Cr moments in the region with an unaveraged J1(n) is larger than unaveraged J*(n) odd number of layers winding like a torsion spring and dominates the contribution to the biquadratic with one sense, and regions with an even number of coupling so J(n) J1(n). When the overlayer layers with the opposite sense, to reach the same thickness t$ is small compared to the characteristic average direction of the top Fe layer [23]. The length scale ¸ of the terraces producing the thick- coupling per unit area is given by ness #uctuations, the leading contribution to the biquadratic term in this model is E "J #J  (" "! ), (7) where ! ( ( and J JM  and J   are the coup- J!( J)¸/A t$ "!( J1(n))¸/A t$ , ling functions associated with areas where the num- (6) ber n of Cr layers is odd and even, respectively [23}25]. For n odd, Eq. (7) is minimized for "0 where A  is the intralayer exchange coupling and the Fe layers are ferromagnetically coupled. which hinders magnetization reversals, as would be For equal regions of odd and even layers, " /2. dictated by #uctuations in the bilinear coupling, The general case of Cr thickness #uctuations in this over the lateral response length l. The model is model leads to non-collinear coupling. Another invalid when ¸'l. consequence of this model is that for any small Other models have been proposed to explain thickness #uctuations there are both odd and even biquadratic coupling. Intrinsic theories that con- thicknesses present. In this case, the magnetization sider ideal systems with perfect interfaces predict remains at a "nite (not 0 or ) angle for all applied biquadratic contributions that are much smaller "elds, giving hysteresis curves with a gradual ap- than what is observed [32]. The dipole "elds result- proach to saturation. ing from rough interfaces in the layered system We have just described two models for the inter- provide another extrinsic mechanism that is always layer coupling, the conventional model and the present to some degree. Biquadratic coupling from torsion model, based on the simplest possible ap- this mechanism is independent of the material proximations for the variation of the energy as parameters for non-magnetic spacer layers. An esti- a function of the relative orientation of the mo- mate of its size for 0.5}1 nm roughness with a ments of the two magnetic layers, given by the characteristic length scale of 20}50 nm gives angle . In the conventional model we assume that JM"0.01 mJ/m. This contribution decreases for for ideal interfaces, the energy as a function of this smaller ¸ and larger spacer-layer thickness [32,33]. angle of the moments varies as !J While it must always be considered, it is likely  cos( ), with the sign of J smaller than other contributions in the samples  depending on whether parallel or antiparallel alignment is preferred. In the torsion considered here. model, the coupling energy is assumed to vary as As an alternative to the conventional model, Eqs. J( ) or J(" "! ) depending on whether parallel (3)}(6), we mention the torsion or proximity model or antiparallel alignment is favored. The correct that depends on the intrinsic antiferromagnetic model will have to describe both the itineracy of the sti!ness of a spacer layer like Cr or Mn and is electrons as represented by the conventional model sometimes invoked to explain the resulting special and the atomic-like correlations as represented by behavior [23]. A strong Fe}Cr interaction is as- the local moments in the torsion model. Calcu- sumed such that the proximity of the Fe leads to lations of the variation of the energy as a function a commensurate antiferromagnetic structure in the of relative magnetization angle using tight-binding Cr that persists even above its bulk ¹,. For an odd [24,25] for six layers of Cr gave a result consistent or even number of Cr layers, the minimum energy with the form assumed in the torsion model. state has the Fe layers coupled with magnetization For less than six Cr layers, these calculations directions parallel or antiparallel, respectively. If gave signi"cant deviations from that form. While D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 297 calculations have not been carried out for thicker "lms, the functional form is also likely to be more complicated than either of the simple limits de- scribed above, particularly when the incommensur- ate state of Cr becomes energetically competitive with the commensurate state. Since there are no calculations for thick Cr spacer layers that allow twisting of moments and the formation of the in- commensurate state, the correct angular variation of the energy is not known. In this paper, we will analyze the results using the form !J cos( ). We will show that there are very few results that are inconsistent with this form, which only shows that it can be di$cult to di!erentiate between these forms. Whatever the form of the angular variation, it is necessary to consider the di!erent possible ways to relieve the frustration that will be present with disorder, as discussed in Section 3. Fig. 4. An STM image of 3.7 ML of Cr evaporated on an Fe whisker at 573 K. The large single atom high islands show 5. Optimal trilayer structures: measurements layer-by-layer growth [37]. 5.1. Sample preparation determined from a height}height correlation analy- The starting point for nearly ideal Fe/Cr/Fe tri- sis to be 85$10 nm. layer structures is an Fe single-crystal whisker. Fe However, at the optimum temperature for layer- whiskers are typically a few tenths of a mm wide by-layer growth, there is some interchange of the and 10}20 mm long with 11 0 02 faces. The deposited Cr atoms and the Fe substrate atoms at whiskers are cleaned by ion bombardment and the interface leading to an interfacial alloy. This can annealing [34]. Whisker surfaces can have approx- be seen in the STM image of 0.4 ML Cr deposited imately 1 m terraces between single atom steps on the Fe shown in Fig. 5a [38,39]. A single atom [35]. This corresponds to a misalignment from high island is evident, as are many little bumps both a perfect (0 0 1) surface of less than 0.013. There is on the substrate and the island. Unique surface some variation in step density between whiskers states on Fe(0 0 1) and Cr(0 0 1) allow positive iden- and over a given whisker; terraces a few hundred nm ti"cation by scanning tunneling spectroscopy of the wide have also been observed. For growth of the Cr bumps as spectroscopic features derived from Cr "lm nearest to the layer-by-layer ideal, RHEED stud- atoms [40]. Thus, the islands contain Fe as well as ies of intensity oscillations and di!raction spot width Cr and there are Cr atoms in the Fe substrate. have shown that the optimum temperature range of Angle-resolved Auger studies of 0.5 ML Cr depos- the Fe whisker substrate is 550(¹ (590 K ited at 570 K on Fe(0 0 1) found that about half of [36]. Fig. 4 shows an STM image of a 3.7 ML thick the Cr deposited goes into the "rst two layers of the Cr "lm grown at ¹"573$20 K [37]. The substrate and about half remains in the "rst adlayer layer-by-layer character of the Cr growth is clearly as shown in Fig. 5b [36,41,42]. The alloying can be evident. The mean island separation, R, in Fig. 4, is reduced by growing the "rst layer or "rst few layers of Cr at a reduced temperature followed by increas- ing the temperature to ¹  [36]. Proton-induced  Uncertainties reported in this paper represent one standard Auger spectroscopy has also shown evidence of deviation and include both statistical and systematic errors. alloying at the interface [43]. Scanning tunneling 298 D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 Fig. 6. A schematic exploded view of the wedge trilayer sample structure showing the Fe(0 0 1) single-crystal whisker substrate, the evaporated Cr wedge, and the Fe overlayer. The arrows in the Fe show the direction of magnetization in each domain. The vertical scale is expanded many times. alloying with the Cr spacer layer [36]. In any case, the growth of an Fe overlayer at room temperature is adequate for completing a good trilayer. Results from two types of optimal trilayers are presented. The MOKE and BLS measurements [36] discussed in this section were made on Fe/Cr/Fe(0 0 1) trilayers with Cr layers of uniform thickness using RHEED to monitor the completion of each full layer. The SEMPA measurements, on the other hand, were carried out on a trilayer struc- ture where the average Cr spacer thickness in- creases linearly over a distance of approximately Fig. 5. (a) A rendered perspective STM image of 0.4 ML depos- 1 mm as shown in Fig. 6 [5]. This wedge-shaped Cr ited on Fe at 563$10 K. The small bumps on the Fe whisker spacer provides a sample that contains a linearly substrate and on the one-atom high islands (much expanded varying range of thicknesses, all prepared under the vertical scale) are Cr atoms which have interchanged with Fe same growth conditions. The slope of the Cr wedge atoms to create an interfacial alloy [38]. (b) The results of angle-resolved Auger measurements of the substrate temper- is typically such that the Cr thickness increases ature dependence of the fraction of deposited Cr atoms in the 1 ML over 10 m. No changes in the magnetic adlayer ( ), in the "rst (surface) Fe layer of the whisker (z) and in properties were observed for wedges twice as steep the second (subsurface) Fe layer ( ) [36]. or ten times less steep [44]. spectroscopy measurements indicate that the sec- 5.2. SEMPA observations of short-period oscillations ond layer deposited is predominantly Cr [38,39]. in the magnetic coupling When Heinrich et al. intentionally deposited a mixed layer of Fe and Cr at the interface, they Short-period oscillations in the magnetic coup- found that it behaved as if the Fe}Cr alloy was part ling are strikingly displayed by the SEMPA image of the Fe "lm for Fe concentrations '15% [36]. of the magnetization, along the length of the whi- The consequences of interfacial alloying for the sker, shown in Fig. 7a [45]. The SEMPA image is coupling are discussed below. It has been suggested formed by measuring the spin polarization of the that the Fe overlayer is not as susceptible to secondary electrons as the SEM beam is rastered D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 299 Cr layer up to 24 layers. The 24th and 25th layers are both coupled ferromagnetically and only at the 26th layer does the coupling switch to antifer- romagnetic. At room temperature, where this was measured, this phase slip in the coupling is repeated each subsequent 20 layers as noted by the arrows in Fig. 7. Before the Fe overlayer was deposited for the measurements displayed in Fig. 7a and Fig. 7b, the thickness and magnetization of the bare Cr were measured. It is possible to scan the SEM beam along the wedge at grazing incidence and observe RHEED intensity oscillations as it moves from a thickness where the top Cr layer is partially "lled, to a position where it is "lled, and so on [5]. The RHEED intensity oscillations measured in this way are shown in Fig. 7c. The decrease in intensity with increasing thickness correlates with the expected increase in roughness. The RHEED oscillations not Fig. 7. (a) SEMPA image of the component of magnetization, only help characterize the quality of "lm growth, MV, in the Fe overlayer along the Fe whisker. The arrows mark but act as a very accurate ruler to give the thickness the Cr spacer-layer thicknesses where phase slips in the short- at each position of the wedge to $0.1 layer. This, period oscillations of the magnetization occur. (b) A line scan along with the observation of the short-period os- through (a) showing the measured spin-polarization pro"le of the overlayer. (c) Spatial RHEED intensity oscillations along cillations in the coupling over many periods, al- the Cr wedge before depositing the Fe overlayer give an accurate lowed us to accurately determine the short period, determination of Cr thickness. (d) The spin polarization of the Cr layer P(Cr), before depositing the Fe overlayer, after subtract- 1"2.105$0.005d [48]. The SEMPA measurement of the bare Cr polar- ing the background from the whisker. ization P(Cr) is shown in Fig. 7d after subtracting an exponential to reduce the background from the across the sample surface. SEMPA is a surface- Fe whisker that is signi"cant for about the "rst sensitive technique that gives a polarization image 10 ML of Cr. Note that the magnitude of P(Cr) is proportional to the magnetization in the top few much smaller than the polarization measured for layers of the specimen [46,47]. The opposite con- the Fe overlayer P(Fe). The 1/e sampling depth in trast in the top and bottom half of the magneti- Cr for the electrons measured by SEMPA is zation image of the Fe overlayer in Fig. 7a results 3.8$0.3d [45] so even though the top Cr layer from the coupling through the Cr spacer layer to dominates P(Cr), subsurface moments with alter- the Fe whisker which has two domains with mag- nating directions reduce the measured value. Com- netization in opposite directions as illustrated in paring P(Cr) and P(Fe) in Fig. 7b and Fig. 7d, it can Fig. 6. The Fe overlayer is seen to be coupled be seen that the polarization of the Fe overlayer is ferromagnetically to the Fe whisker substrate for opposite to that of the Cr at Cr thicknesses of 5 ML the "rst four layers, and then the coupling begins to and above. This is consistent with antiparallel oscillate, changing from ferromagnetic to antifer- coupling at the top Fe}Cr interface assuming that romagnetic (overlayer magnetization antiparallel the Cr polarization direction does not change on to the whisker magnetization) and back as the Cr the addition of an Fe overlayer. Antiparallel Fe}Cr increases by two additional layers. This can be seen coupling was also found in spin-polarized photo- clearly in Fig. 7b, which shows the pro"le of the emission measurements [49,50]. polarization from Fig. 7a. This change in the direc- Like P(Fe), we see that P(Cr) changes sign with tion of the coupling continues with each additional each single layer increase in Cr thickness except for 300 D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 the phase slips at 24}25, 44}45, and 64}65 layers. Up to the "rst phase slip, for antiferromagnetic Fe}Cr coupling at both interfaces and antifer- romagnetic stacking of the Cr, we expect the Fe layers of the trilayer to be coupled ferromagneti- cally for an odd number of Cr layers and antifer- romagnetically for an even number of Cr layers. Just the opposite is observed in Fig. 7b. This one layer o!set has been attributed to the alloyed re- gion approximately one layer wide at the interface observed for this high-temperature growth of Cr on Fe [36,38,39,41,42,51]. 5.3. Temperature dependence of the phase slips The SEMPA measurements described thus far were made at room temperature, that is, in the neighborhood of the Cr bulk NeHel temperature, ¹,"311 K. It was found that as the sample tem- perature during the SEMPA measurement of a bare Cr wedge on the Fe(0 0 1) whisker was varied between room temperature and 1.8¹,, the Cr thickness at which phase slips occurred varied reversibly [45]. The amplitude of the oscillations of P(Cr) changed less than 20% on heating to 1.8¹,. The displacement by 14 layers of the position of the "rst phase slip from its position at 24}25 at 310 K to 38}39 layers at 550 K is displayed in Fig. 8a. The phase of the oscillations below 24 layers was not observed to change in this temperature range. Also Fig. 8. (a) The temperature dependence of the number of layers shown in Fig. 8a is the distance between phase slips between phase slips for bulk Cr [52] and the change in position in bulk Cr measured by neutron scattering [52]. of the phase slip in Cr/Fe(0 0 1) with temperature. The NeHel The change in the thickness where the "rst phase temperature for bulk Cr is marked by the arrow. (b) Temper- slip occurs can also be seen in the SEMPA ature dependence of SEMPA images indicating the bilinear coupling in Fe/Cr/Fe(0 0 1). The phase slips measured on the measurements of an Fe/Cr/Fe trilayer at a series of bare Cr are shown by the solid gray line; the dashed line is measurement temperatures shown in Fig. 8b [53]. the estimated position of the next phase slip. Note that, where It is somewhat more di$cult to locate the phase visible, the short-period oscillations have opposite direction slips as a function of temperature in the trilayer above and below these lines. data compared to bare Cr on Fe data because the short-period coupling strength drops o! more rap- idly with temperature than the long period. Only tization direction is reversed. The dashed line is the short-period oscillations are seen in the polariza- same curve displaced 20 layers. There is some evid- tion P(Cr) of the bare Cr. The heavy line marking ence of the magnetization reversal at 46}48 layers, the change in the position of the phase slip in the e.g., compare the room-temperature data with that trilayer with temperature is taken from the P(Cr) above the dashed line at approximately 425 K. All data of Fig. 8a. Where short-period oscillations can of these measurements are completely reversible be seen above and below the phase slip line, for and are not due to an irreversible roughening of the example at a Cr thickness of 30 layers, the magne- trilayer structure. D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 301 5.4. Observation of non-collinear coupling of the Fe explain the hysteresis loops observed in Co/Cu layers structures [27]. The SEMPA image of Fig. 7a shows the Fe 5.5. Strength of the interlayer exchange coupling overlayer magnetization component along the whisker parallel to the magnetization of the The strength of the interlayer exchange coupling, whisker substrate, de"ned here as MV. A similar i.e. the coupling energy per unit area E image from a di!erent trilayer wedge sample is in Eq. (3), can be determined by varying the magnetic "eld shown in Fig. 9a [5]. The varying width of the applied to the trilayer structure and measuring black and white contrast in Fig. 9a is evidence of the BLS spectra or MOKE magnetization curves the long (12 ML) period coupling superimposed on [54]. These optical techniques have increased the short-period coupling. Simultaneously mea- the sensitivity to the overlayer as compared sured with such an image are the intensity image, to conventional magnetometry where the signal which gives topography information, and the image from the much larger Fe whisker would overwhelm of the orthogonal in-plane component of the mag- that from the thin Fe overlayer. By assuming netization, MW. The magnetization M lies in the that the form of Eq. (3) holds, the BLS measure- plane of the Fe "lm and has constant magnitude, ments allow the separation of JM "M""(M  and JM when V#MW). The direction of M is given by the coupling is antiferromagnetic [54]. The bilinear the angle, "tan\(MW/MV). Fig. 9b and Fig. 9c and biquadratic coupling strengths, JM show enlarged angle maps of the magnetization  and JM, of an optimally grown trilayer are shown in Fig. 10 direction from the Cr thickness regions outlined in taken from Heinrich et al. [36]. The separation Fig. 9a. In the thinner part of the wedge, Fig. 9b at 10 and 12 layers in Fig. 10, where the coupling shows that the Fe overlayer does not alternate is ferromagnetic, was made assuming that between parallel and antiparallel, but instead be- JM tween canted and roughly antiparallel. This canted  was the same as for 9, 11, and 13 layers. The coupling strength was reported to be very coupling observed in the thin part of the Cr wedges sensitive to slight di!erences in sample fabrication varies from sample to sample due to slight di!er- conditions [36,54]. ences in preparation. A variation in the coupling With increasing Cr thickness, JM angle across the whisker is visible in Fig. 9b and  changes from ferromagnetic to antiferromagnetic at four layers highlighted by the line scans of Fig. 9d. In the and oscillates around an antiferromagnetic o!set thicker part of the wedge, Fig. 9c shows that where until the short-period coupling increases and there MV becomes small as it goes through zero and is a crossover to ferromagnetic coupling at the 10th reverses direction, there are regions of orthogonal layer, with oscillating sign of the coupling after magnetization, MW shown in red and blue, i.e., there that. Even though the polarization pro"le P(Fe) of is 903 coupling. Of interest for later discussion, is Fig. 7b looks tantalizingly similar to the strength the fact that the 903 coupling regions become nar- measurements of Fig. 10, P(Fe) is proportional not rower at a Cr thickness near 24 ML where the to the bilinear coupling strength but to M phase slip occurs. V of the top Fe layer. The fact that P(Fe) does not saturate The "rst observation [26] of such regions of 903 suggests a biquadratic component nearly equal to coupling was in Kerr microscopy studies of Fe/ the bilinear coupling. On the other hand, BLS Cr /Fe trilayers grown on Ag-bu!ered measurements "nd "JM GaAs substrates. These structures exhibited long- "'2"JM", except near zero crossings of JM period coupling. In the transition region between , and the coupling is not canted but collinear, either ferromagnetic or antiferromagnetic the ferromagnetically and antiferromagnetically [36]. We attribute this discrepancy between these coupled regions, the coupling of the two Fe layers BLS measurements and the SEMPA measurements was at 903. These experiments led to the addi- of Fig. 7 to slightly rougher Cr in the Fe/Cr tion of the biquadratic coupling term to obtain  /Fe trilayer as indicated by the less than optimal Eq. (3) [26]. A similar equation was proposed to RHEED oscillations in Fig. 7c. 302 D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 Fig. 9. (a) SEMPA image showing the oscillatory magnetic coupling in an Fe/Cr /Fe(0 0 1) trilayer. (b) and (c) Enlarged angle maps from the regions outlined in (a). The colors give the direction of the magnetization. Canted non-collinear coupling is evident in (b). 903 biquadratic coupling regions, shown as red and blue, of varying width are seen in (c). (d) There is some variation of the canted coupling as seen in the line scans at two positions across the whisker. D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 303 Fig. 11. MOKE signal proportional to magnetization versus Fig. 10. BLS measurements of bilinear JM magnetic "eld for two Fe(20 ML)/Cr(11 ML)/Fe(0 0 1) whisker  and the biquadratic !JM trilayers. The increase in H  coupling in optimally fabricated Fe/Cr/Fe(0 0 1) trilayers  and H indicate a larger bilinear as a function of Cr thickness [36]. coupling strength JM in the sample shown on the right for which the "rst ML of Cr was deposited at a substrate temperature of 453 K and subsequent layers at ¹  compared to the other trilayer shown on the left for which the "rst 7 ML of Cr were The coupling strength shown in Fig. 10 measured deposited at 519 K and the remaining layers at ¹  [42]. from optimum samples is still over an order of magnitude smaller than predicted theoretically [55,56]. The sensitivity of the coupling strength to the substrate temperature during the deposition of pected for perfect antiferromagnetic stacking. Hein- the "rst few Cr layers suggests that it is a!ected by rich et al. also induced variations in the coupling interfacial alloying. An indication that interface strength by depositing 1}3 ML of Cu, Ag, or Mn at alloying a!ects the coupling strength is seen from one of the Fe}Cr interfaces [36]. The results were the very di!erent H and H in Fig. 11 for samples compared to recent calculations [57], but further grown at di!erent temperatures [42]. For applied discussion here is beyond the scope of this paper. "elds slightly above H, the magnetic moments in Atomic-scale defects, for example due to inter- the Fe whisker and the Fe overlayer are parallel. facial alloying or steps, cause di!use scattering of For "elds slightly below H the overlayer and electron states and cause frustration. These e!ects whisker moments are antiparallel [36]. A decrease tend to reduce the coupling for each discrete thick- in JM by a factor of three from !1.23 to ness, J !0.41 mJ/m was calculated from the hysteresis (n). This discrete thickness coupling strength is used to determine the biquadratic coup- curves when the interface was formed at substrate ling strength, Eq. (6), and the lateral response temperatures of 453 and 519 K, respectively, and length l. While the steps that cause the thickness the rest of each Cr "lm was deposited at ¹  [42]. #uctuations also lead to di!use scattering and frus- There are limits on how much the substrate tem- tration, in general, they have a more important perature during interface formation can be reduced e!ect. The thickness #uctuations average the coup- to decrease the e!ect of alloying on the coupling ling at discrete thicknesses over the growth front to strength. A substrate temperature of at least 370 K give a reduced average coupling strength JM was found to be necessary to obtain reasonable (t), Eq. (5). This averaging explains the reduced coup- growth [36]. Even with this care taken to minimize ling strength in systems like Fe/Au where no interfacial alloying, these samples showed the re- alloying is believed to occur [12]. The thickness versed phase of the oscillations in the thinner Cr #uctuations also cause the biquadratic coupling in regions. As seen in the SEMPA measurements, the conventional model. In contrast, the interfacial antiferromagnetic coupling is present for an odd alloying occurs on a lateral length scale of atomic number of Cr layers instead of the ferromagnetic dimensions (see Fig. 5a) and only a!ects JM coupling for an odd number of Cr layers as ex-  indirect- ly through the e!ect of di!use scattering on J1(n). 304 D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 The critical "elds in Fig. 11 are determined from "tting the hysteresis curves in the conventional model taking into account the micromagnetic re- sponse of the Fe whisker as well as the Fe overlayer [28]. The antiferromagnetic alignment of the Fe "lms is expected to show a jump at H and a well- de"ned kink at H. The remanence observed in Fig. 11 is that of the Fe whisker. Heinrich et al. [36] argue that a variation in exchange coupling strengths JM and JM by $10% over length scales larger than the lateral response length is su$cient to explain the rounding of the hysteresis curves as well as the di!erences between H and H deter- mined by BLS and MOKE on the same samples. The hysteresis curves can be explained in the con- ventional model. The variation of the coupling strength out to thicker Cr layers is shown in Fig. 12a for an Au(10 ML)/Fe(15 ML)/Cr /Fe whisker sample. The "rst two layers of the wedge were grown at 403$10 K and the rest at 623$20 K. This "gure is a series of MOKE images, each acquired at a di!erent applied "eld [53]. The dark vertical bands are the antiferromagnetically coupled re- gions that switch at di!erent applied "elds. For this 15 layer Fe "lm, switching the magnetization from antiferromagnetic to ferromagnetic in an applied "eld of 100 kA/m corresponds to a coupling strength JM of 0.47 mJ/m. At a Cr thickness of 11 Fig. 12. (a) A series of MOKE images from an layers and below, the "eld available in these Au(10 ML)/Fe(15 ML)/Cr experiments was insu$cient to switch the Fe. The  /Fe(0 0 1) whisker taken at vari- ous applied magnetic "elds showing the "eld dependence and Cr interpretation of the fading contrast of the antifer- thickness dependence of the reversal of the antiferromagnetic romagnetic peaks at Cr thicknesses of 13 and regions (dark bands). Below 11 ML the "eld is insu$cient to 15 ML would require a measurement of the hyster- switch the antiferromagnetic regions. The exchange coupling strength, proportional to the switching "eld, reaches a minimum esis loops at these points. Possible explanations at the thickness of the phase slip, 24}25 ML. An SEMPA image include either a distribution of coupling strengths of the same trilayer at zero applied "eld is shown at the bottom or the slow approach to saturation expected from for reference. (b) The transition width, i.e., the range of Cr the torsion model. This series of images graphically thickness where biquadratic coupling is observed when the shows how the exchange coupling depends on Cr bilinear coupling goes through zero, is seen to be a minimum at the phase slip where the short-period bilinear coupling is min- thickness with a clear minimum at the phase slip for imum. Data are presented for two Fe/Cr a Cr thickness of 24}25 layers.  /Fe trilayers: (1) one with the Cr grown at 620 K where the short-period It is also possible from SEMPA data to get some coupling dominates, and (2) one with the Cr grown at RT where idea of the variation of the strength of the biquad- the long-period coupling dominates. ratic coupling JM from the width of the transition region between thicknesses of opposite bilinear transition regions where the averaged bilinear coupling as was seen in the MV image in Fig. 9c. In coupling, JM samples, like wedges, where the thickness of the (t), changes from ferromagnetic to antifer- romagnetic going through zero. In these transition spacer layer varies continuously, there are regions, the biquadratic coupling, JM, becomes D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 305 larger than JM(t)/2, and the minimum energy con"g- thermodynamic phase transition [58]. The coup- uration for the Fe magnetizations is non-collinear. ling can be understood as the response of the Fe The width of this transition region is the distance atoms at one interface to the electrons in the between the points where JM(t)"2JM, and JM(t)" Cr spacer that are polarized by the magnetized !2JM. In Fe/Cr multilayers, the unaveraged short- states in the other Fe layer. This description also period coupling is larger than the unaveraged long- applies for paramagnetic spacer layers. There, the period coupling, so the strength of the biquadratic Fe induces a (weak) spin density that mediates the coupling is proportional to the square of the coupling. change in the short-period coupling strength from A paramagnetic description of Cr was the basis one layer to the next, see Eq. (6), JMJ( J1). This of the RKKY-like calculation used by Wang et al. dependence is independent of whether thickness [59,60] to treat the magnetic coupling in Fe/Cr/Fe. #uctuations obscure the short-period coupling The striking feature of the calculation was the from JM(t) or not. The constant of proportionality short-period oscillatory coupling with a period will vary with the interface quality, represented in given by the Fermi surface nesting vector Q of the Eq. (6) by the terrace length ¸. Near the thickness paramagnetic Fermi surface shown in Fig. 1. At the t where the coupling changes sign, the averaged time of the calculation, only the long-period coup- bilinear coupling varies as JM(t)JJ(t!t). Thus ling oscillations had been observed [4] and Wang we expect the transition width to vary like et al. [59,60] showed how interface roughness wJ( J1)/J. Near a phase slip in the short-period could average out the short-period oscillations and coupling, the envelope of the unaveraged short- bring their results into better agreement with that period coupling goes through zero linearly, experiment. J1J(t!t ). As it does, the biquadratic coup- The conventional model, which includes such an ling goes through zero. Based on the simple argu- RKKY calculation, treats Cr in an itinerant elec- ments presented here, we would expect to see the tron or band picture and ignores the electron} width of the transition region, w, go to zero near the electron interactions in the Cr that stabilize the phase slip either linearly if the short-period coup- antiferromagnetic state. At the other extreme is ling survives the thickness #uctuations or quadrati- the localized moment picture in which atomic Cr cally if it does not. The results in Fig. 12b are moments are antiferromagnetically coupled by consistent with these expectations, but insu$cient a Heisenberg exchange. The proximity model [23] for a quantitative comparison to the model. of Eq. (7) is of this type. These models do not describe the Cr incommensurate spin density wave. Between these extremes are calculations based on 6. Optimal trilayer structures: interpretation di!erent models for treating the electron}electron interactions. Examples include calculations such as In principle, a perfect multilayer structure com- those of Mirbt et al. [55] and of van Schilfgaarde posed of thin Fe and Cr "lms results in a coherent and Herman [56] based on the local spin density structure with one thermodynamic phase transition approximation, calculations of the Strasbourg for the whole structure at a temperature between group [61] based on the tight binding approxima- the Curie temperature of bulk Fe and the NeHel tion and on-site Coulomb interactions, and the temperature of bulk Cr [58]. In the multilayer, we calculations of Shi and Fishman [62] based on expect the Fe to induce antiferromagnetic order in models for the free energy of bulk Cr in di!erent the Cr up to the transition temperature of the ordered states. In most calculations, the interlayer multilayer. The degree of order in the Cr depends exchange coupling is taken as the di!erence be- on the temperature, the Cr thickness, and the dis- tween the calculated energy of the structure for the tance from the interface. When the Cr becomes ferromagnetic layers aligned parallel and the en- thick enough, we expect it to behave as if it had ergy for antiparallel alignment. It is possible a transition at the bulk NeHel temperature even in local spin density approximation calculations to though the transition is broadened and not a true modify the treatment of the electron}electron 306 D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 interactions so as to suppress the formation of antiferromagnetism in the Cr. This allows an ap- proximate comparison of the coupling in both paramagnetic and antiferromagnetic SDW Cr. Short-period oscillatory coupling is found in all cases, but the strength varies. At a Cr thickness of 11 ML, corresponding to the maximum coupling measured by Heinrich et al. [36], the strength of the calculated coupling through SDW Cr ranges from approximately 60 to 80 mJ/m and is roughly three times stronger than that calculated for paramag- netic Cr [55,56]. The experimental coupling strength, of order 1 mJ/m, is reduced by the inter- facial alloying and also, we believe, by Cr thickness #uctuations. These are di$cult to avoid even in Fig. 13. Calculated [63] curve showing the transition (heavy carefully optimized growth. line) from commensurate order (n"0) to incommensurate or- In contrast to local magnetic moments of a rare- der, ISDW with one node (n"1), as a function of temperature earth metal or of a conventional antiferromagnet, and Cr thickness. The lighter curves show the transitions be- Cr is an itinerant antiferromagnet with a spin den- tween ISDWs with di!erent number of nodes. sity wave such that ordered magnetic moments at each lattice site can vary in magnitude. Shi and Fishman have presented a model of Fe/Cr/Fe that smaller node-to-node distance of 20 ML at 300 K treats the competition between the SDW antifer- for the Fe/Cr romagnetism of the Cr spacer layer and the antifer-  /Fe compared to the 27 ML for bulk Cr as shown in Fig. 8a, is attributed to the romagnetic coupling of the Cr and Fe at the 0.6% smaller lattice constant for Cr on Fe than for interfaces [62]. For ideal interfaces, the interface Cr in the bulk. Using this lattice constant, the coupling tends to increase the SDW amplitude at model gets the node-to-node distance correct, but the interfaces whereas the intrinsic antiferromag- the phase slips occur at smaller Cr thicknesses than netism of the Cr favors the temperature-dependent in experiment. bulk SDW values for the amplitude and wave The SEMPA measurements are not directly sensi- vector. When the bulk contribution from the Cr tive to the presence of antiferromagnetic order in the spacer is su$ciently small compared to the inter- Cr. However, the qualitative agreement between the face energies, as is the case for a thin Cr spacer or at model calculation and the measured temperature higher temperature, Shi and Fishman show that the dependence of the phase slips is strong evidence that commensurate SDW (CSDW) is favored over the the Cr layer in these experiments is in an antifer- incommensurate SDW (ISDW) [62,63]. The thick- romagnetic state. Within the conventional model, ness and temperature dependence of the transition the change in the phase slip would have to come from the commensurate phase with n"0 nodes to from the temperature dependence of the Fermi sur- the "rst incommensurate phase with n"1 node in face. However, the temperature dependence of the the SDW is shown by the heavy line in Fig. 13 [63]. Cr Fermi surface is too weak to give the observed The variation of the commensurate-to-incommen- variation in the incommensurability. The agree- surate transition with temperature shown by the ment also implies that the antiferromagnetic order heavy line in Fig. 13 is consistent with the measured that exists well above the bulk NeHel temperature of change in the position of the phase slip depicted by Cr is due to the proximity of the Fe. the solid line in Fig. 8b. The n"1 to 2 transition in Also in accord with the model of Shi and Fish- Fig. 13 corresponds to the second phase slip shown man [62] is the fact that the coupling strength in by the dashed curve in Fig. 8b. The increased in- Fig. 12a decreases with Cr thickness up to the commensurability of Cr on Fe, indicated by the phase slip at the node in the SDW at a Cr thickness D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 307 ation on the Fe whisker surface is much larger than the Cr thickness of less than 10 nm. The frustration caused by these Fe steps for a number of complete Cr layers is therefore expected to be taken up in the Cr "lm as shown in Fig. 3b, rather than as a very long Fe}Cr interface wall as would be required in Fig. 3c. Even for perfect growth on a very #at Fe whisker, at other than a perfectly completed layer, there will be thickness #uctuations in the Cr spacer layer on a length scale ¸, which is at least an order of magnitude less than the step spacing on the whisker at the growth temperatures used. Over the lateral response length l(l'¸) that it takes the Fig. 14. The energy of a spin density wave in Cr with 0, 1, 2, 3, or Fe magnetization in the overlayer to change direc- 4 nodes as a function of Cr thickness calculated for a temper- ature of 100 K [64]. The boundaries of the shaded regions show tion, the Fe overlayer responds to the average of the variation of the coupling strength. the coupling strengths for each Cr thickness as in Eq. (5). The magnetization of the Fe overlayer is constant. This causes the frustration at the Cr}Fe of 24}25 layers and then increases again. The family overlayer interface that is likely relieved by an of curves in Fig. 14 gives the energy of a spin interface wall of the type shown in Fig. 3c. density wave in Cr for n"0, 1, 2, or 3 nodes, One possible consequence of such frustration is calculated with the same model parameters as canted coupling as is often observed for the thin Fig. 13 [64]. The ground state of a trilayer is part of the Cr wedge, as seen in Fig. 9b for example. commensurate with alternating ferromagnetic and A plausible explanation of the results can be given antiferromagnetic alignment of the Fe "lms up to in terms of Slonczewski's #uctuation model [31]. a thickness where the n"0 and 1 lines cross. After The angle of the coupling comes from the competi- this point the ground state has incommensurate tion between JM SDW order. The phase slip occurs at the crossover  and JM, see Eq. (3), which is highly sensitive to sample properties. The decrease of the where the two types of SDW have the same energy. biquadratic coupling relative to the short-period In the region of the commensurate SDW, reversing bilinear coupling after a dozen or so layers can be the orientation of the Fe layers from the low-energy understood from Eq. (6). The biquadratic coupling con"guration at a given thickness, introduces decreases as the square of the bilinear coupling, so a node in the SDW, raising the energy to that given it decreases faster than the bilinear coupling as the by the n"1 curve. The coupling energy of the latter decreases. In the thicker parts of the Cr system, E$!E$, is just the di!erence between the wedge, the thickness #uctuation model of biquad- two curves that form the boundary of the shaded ratic coupling explains the 903 coupling in the area. This gives the envelope of the coupling transition regions where the averaged bilinear strength in this model. Comparing Fig. 12a and coupling, JM Fig. 14 shows that this result is in qualitative agree- (t), changes from ferromagnetic to anti- ferromagnetic going through zero. ment with what is measured. In summary, the magnetic coupling in optimized With this overall picture of the short-period Fe whisker trilayers can be described by Eq. (3) coupling in place, we will now discuss the inevitable where the bilinear coupling consists of a short- magnetic frustration at imperfect interfaces. Steps period oscillatory coupling that dominates the in the Fe whisker substrate and roughness at the long-period coupling. To correctly explain the upper Fe}Cr interface due to the Cr growth lead to short-period coupling, it is necessary to go beyond frustration that is likely relieved in two di!erent the conventional model and include a treatment of ways. We "rst consider frustration due to steps in the electron}electron interactions of the type the whisker substrate. The typical 1 m step separ- that stabilize the incommensurate order in Cr. The 308 D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 observed non-collinear coupling can be explained to both the interface structure and the magnetic satisfactorily by the Slonczewski thickness #uctu- coupling. Results will "rst be presented for Fe ation model [31]. Agreement with the trends of whisker trilayers with varying degrees of interfacial that model does not imply that other models may roughness caused by di!erent Cr growth temp- not be developed that give better descriptions of the eratures. We then discuss the results from the non-collinear coupling; in fact, the best description GaAs-based trilayers. STM measurements of may change with the Cr thickness. the roughness and mean island separation are dis- cussed for both types of samples and summarized in Table 2. 7. Trilayers with varying interfacial roughness: Trilayers grown on Fe whiskers are unique be- measurement cause the interface at the whisker is exceptionally smooth. Because the Fe(0 0 1) whisker surface is The degree of interface roughness in Fe/Cr/Fe very #at, the roughness of the Cr layer determines trilayers has a very strong in#uence on the mag- the thickness #uctuations, i.e., netic coupling. The roughness of a Cr spacer layer  is assumed equal to grown on an Fe whisker substrate is very depen- !. STM images of approximately 5 ML Cr "lms grown on Fe whisker substrates at 323$20 dent on the temperature of the substrate during and 488$20 K are shown respectively at the top evaporation of the Cr [5,36,48]. A GaAs(0 0 1) sub- left and right in Fig. 15a and Fig. 15d [37,48]. The strate with an Ag bu!er layer has also been used for height}height correlation function was computed the epitaxial growth of Fe/Cr/Fe trilayers which for each image to obtain the mean island separ- exhibit long- and short-period oscillations in the ation R and the rms roughness shown at the magnetic coupling [65}69]. bottom of Table 2. There is a strong correlation We discuss the similarities and di!erences in the between the growth temperature of the Cr spacer Fe whisker and GaAs-based trilayers with regard layers and the observed oscillatory coupling in the Table 2 Interface parameters of Fe/Cr/Fe trilayers. The mean island separation, R, the rms roughness, , and the standard deviation of the thickness distribution, , are given for di!erent "lms grown at temperatures, ¹1, on GaAs and Fe whisker substrates. Note that the roughness measurements of 17.4 ML thick Cr "lms on GaAs substrates and approximately 5 ML thick Cr "lms on an Fe whisker cannot be compared directly because roughness increases with thickness. For example, using the power law dependence of ! from a previous analysis [48], one estimates !"0.86 ML for a 17.4 ML Cr "lm grown at 488 K on an Fe whisker Notation Layer ¹1 (K) Interfaces Spacer layer  (ML) R (nm) (ML) GaAs substrate, t!"17.4 ML RT Top Fe 300 } } } Cr 300 6.8 1.25 1.94 Bottom Fe 300 6.1 1.46 } MT Top Fe 520 } } } Cr 520 22.4 1.32 1.88 Bottom Fe 100/570 19.7 1.32 } MT Top Fe 520 } } } Cr 520 15.4 1.11 1.46 Bottom Fe 100/520 10.1 0.90 } Fe whisker, t!+5 ML RT Cr 323 10$0.5 0.86 0.86 Intermediate ¹ Cr 488 31$1 0.47 0.47 High ¹ Cr 573 85$10 &0 &0 D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 309 Fig. 15. STM images [48] of approximately 5 ML of Cr grown on an Fe(0 0 1) whisker at temperatures of 323 and 488 K shown in (a) and (d), respectively. Note change in lateral scale. SEMPA images of Fe/Cr /Fe(0 0 1) trilayers where the Cr was grown at 303 and 473 K are shown in (b) and (e), respectively. The relative magnetization from SEMPA images of (b) and (e) is shown in (c) and (f), respectively, normalized to the saturation value MH in a range of Cr thickness from 20 to 30 ML. rougher trilayers grown on Fe whiskers. The The starting point for trilayer growth on GaAs is SEMPA image of an Fe/Cr wedge/Fe(0 0 1) trilayer a substrate quite di!erent from the Fe whisker. The structure grown at 303$10 K and a magnetization GaAs-based samples are typically prepared by "rst pro"le from this image are shown in Fig. 15b and depositing a 1 nm Fe seed layer on the GaAs(0 0 1) Fig. 15c respectively. The corresponding "gures for surface followed by a 150 nm Ag bu!er layer, all at the Fe/Cr /Fe(0 0 1) grown at 473$10 K are 373 K, followed by a 1 h anneal at 573 K. Fe/Cr/Fe shown in Fig. 15e and Fig. 15f [48]. For the lower trilayers subsequently deposited at 293 K showed temperature growth, the magnetic coupling shows long-period oscillations quite similar to those ob- primarily long-period coupling with some "ne struc- served for the trilayer grown at RT on the Fe ture at 6 layers. Although the trilayer with the Cr whisker [67]. When the trilayer was deposited at wedge grown at 473 K exhibits primarily a long- 523 K, except for the "rst few Fe layers which were period coupling, short-period oscillations can be deposited at room temperature to minimize di!u- observed out to a thickness of 18 Cr layers. As in sion of the Ag into the Fe, short-period oscillations Fig. 7b and Fig. 9b, the fact that the magnetization in the coupling strength were observed, but no does not correspond to complete ferromagnetic or phase slips [67,68]. The coupling for growth at antiferromagnetic alignment, means that there is an 523 K exhibited a strong biquadratic component in MWcomponent and thatthe magnetization is canted. addition to the bilinear component. At this growth 310 D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 temperature, a negative (antiferromagnetic) bias of the magnetic coupling strength. The sample prep- the coupling up to 25 ML Cr was observed, which aration di!ered slightly from those studied by the decreased with increasing measurement temper- GruKnberg group [67], but the magnetic coupling ature [67]. results are similar. The combination of structural as Recently, Schmidt et al. [69] combined a careful well as magnetic measurements on these samples STM characterization of trilayers fabricated on makes them of particular interest for closer examina- Ag-bu!ered GaAs substrates with measurements of tion. In samples denoted as room-temperature (RT) Fig. 16. Structural characterization and magnetic coupling measurements of Fe/Cr /Fe(0 0 1) trilayers grown on Ag-bu!ered GaAs(0 0 1) [69]. (a) STM overview on left and detail image on right of "rst Fe layer grown at RT. (b) Similar images of the Cr layer grown at RT at a thickness of 17.4 ML. (c) MOKE measurement of the magnetic coupling showing the long-period oscillatory behavior. (d}f) Similar "gures for the MT trilayer. Short-period oscillations of the coupling are now evident. D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 311 samples, the entire trilayer was fabricated at room temperature. A second designation, mixed temper- ature (MT2), corresponds to samples where the "rst 2 nm of the "rst Fe layer were deposited at 100 K to inhibit segregation of Ag to the surface, and the remaining 3 nm of the 5 nm bottom Fe layer were deposited at temperature ¹ which was 520 or 570 K. The Cr layer and the 5 nm Fe over- layer were both deposited at 520 K. STM images were acquired after deposition of the bottom Fe layer and then again after deposition of the Cr layer. From the STM images the rms roughness, , and mean island separation, R, were calculated. The results are summarized at the top of Table 2. From Fig. 17. MOKE hysteresis loop from an MT trilayer at a Cr spacer-layer thickness of 17.4 ML shows that the magnetic the inequivalence of R for Fe and Cr and from coupling is predominantly biquadratic. modeling, Schmidt et al. [69] concluded that the roughness of the Fe bottom layer was not corre- lated with the roughness at the top of the Cr layer. On the basis of this argument, they calculated the bilinear antiferromagnetic coupling is apparent in rms thickness #uctuation of the Cr spacer layer, the hysteresis curves [70]. In contrast to the MT  R"( $ # !) also shown in Table 2. samples, the MT Interesting correlations are found between the  coupling data showed only very weak short-period oscillations. magnetic coupling and the growth morphology as characterized by the STM. The STM images and the magnetic coupling results for RT and MT 8. Trilayers with varying interfacial roughness: structures are compared in Fig. 16. Fig. 16a}Fig. 16c interpretation show respectively an STM image of the bottom Fe layer, an STM image of the Cr layer at a thickness The long-period oscillatory coupling is clearly of 17.4 ML (2.5 nm), and the magnetic coupling seen in trilayers on Fe whiskers with the Cr grown curve, determined from MOKE hysteresis loops, at room temperature, Fig. 15b and Fig. 15c, and for the RT Fe/Cr /Fe(0 0 1) trilayer. The STM in trilayers on Ag-bu!ered GaAs(0 0 1), Fig. 16c. images include both low- and high-resolution im- Unlike the short-period coupling that is believed ages. Fig. 16d}Fig. 16f contain the corresponding to have its origins in the antiferromagnetism of Cr, images for the MT trilayer. The magnetic coup- we believe the long-period coupling can be de- ling from the RT trilayer exhibits long-period oscil- scribed using a quantum well model in the same latory coupling with a hint of structure at a Cr way as is used to describe the coupling through thickness of six layers. The MT sample shows noble metals. However, even accepting this model, a long-period oscillation modulated by short-peri- the important part of the Fermi surface is still od oscillations. A MOKE hysteresis curve for the controversial, as discussed elsewhere in this volume MT sample at a Cr thickness of 17.4 ML is [11]. One proposal is that the appropriate spann- shown in Fig. 17. The plateaus in the magneti- ing vectors for the long-period oscillatory coupling zation, nearly equal to half the saturation magnet- would be located at the N-centered ellipses in ization for zero applied "eld, are indicative of 903 Fig. 1 [71}73]. The critical spanning vectors of the coupling for this multilayer with Fe layers of equal N-centered ellipses are similar for the (0 0 1), (1 1 0) thickness. In fact, for both mixed temperature (MT) and (2 1 1) interfaces. Since this part of the Fermi trilayers, biquadratic coupling dominated over surface is not believed to be strongly dependent on most of the thickness range. For 7 ML and below, the presence or absence of antiferromagnetic order, 312 D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 the paramagnetic Fermi surface should be appro- For interpreting the magnetic coupling results priate. In this case, the long-period coupling would from trilayers with interfacial roughness, it is useful be expected to be relatively insensitive to temper- to keep in mind how di!erent quantities vary with ature and disorder. the standard deviation of the thickness distribution, When there are sizable thickness #uctuations, as seen in the STM images of Fig. 15a and Fig. 16b, , and the average terrace length, ¸. Over the ranges of the long-period coupling is observed, because the  and ¸ encountered in the trilayers considered in this section, that is short-period coupling JM  less than 2 ML 1 is reduced by thickness and ¸ less than 80 nm, we can say the following: (1) #uctuations as described in Eq. (5). An example of The average long-period coupling strength JM the e!ect of thickness #uctuations on the short- * re- mains nearly constant, (2) the average short-period period coupling is shown in Fig. 18a. This plot of coupling JM the averaged coupling, JM 1 decreases dramatically with increasing 1, was generated by assum- ing a coupling J  while J1 remains constant, and (3) there is a wide 1 oscillating in sign with each one variation in the biquadratic coupling. In the thick- layer change in Cr thickness, and adding together ness #uctuation model of biquadratic coupling, Eq. the coupling contributions from all the layers in (6), JM the growth front. Distributions of the discrete  varies quadratically with the terrace width ¸ as shown in Fig. 18b. thickness #uctuations were generated with varying We call attention to an additional experimental standard deviation, , by sampling Gaussian distri- result from RT trilayers on Fe whiskers that places butions at integer layer thicknesses and appro- constraints on models of biquadratic coupling. In priately normalizing the discrete distributions. these samples, ¸ is small giving a relatively small Similar distributions were found to be a good ap- JM proximation for rough growth of thin layers of Cr , but biquadratic coupling is still observed when the long-period coupling goes through zero. This on Fe whiskers [48]. The plot is normalized to 1 for can be understood in terms of the thickness #uctu- ideal interfaces. For a thickness distribution with ation model of biquadratic coupling; even though a standard deviation "1 ML the value of the average short-period coupling JM normalized JM 1 is small for low- 1 has decreased to 0.014. As the temperature growth, it is the unaveraged J strength of the averaged short-period coupling be- 1, which is not small, that contributes to the biquadratic comes weaker than competing energies such as coupling. This idea is supported by the fact that the those of the biquadratic coupling, the anisotropy, widths of the biquadratic coupling regions from an or the long-period coupling, it will become more SEMPA image, see Fig. 12, are smallest where J di$cult to observe. 1 is at a minimum in the vicinity of the phase slips in the short-period coupling. A signi"cant feature of the magnetic coupling in the Fe whisker trilayer which we want to be able to explain is the canted biquadratic coupling ob- served up to a Cr thickness of 18 ML in Fig. 15f. Unlike optimal trilayers on Fe whiskers where the short-period bilinear oscillations dominate for Cr thicker than about 10 ML, for Cr growth on whiskers at 488 K the short-period coupling is much reduced. The canted coupling can be inter- Fig. 18. Model dependence of coupling strengths on structured preted as a competition between biquadratic and parameters. (a) The thickness #uctuations (or equivalently the bilinear coupling. The variation in the canted Cr roughness if there is one #at interface at the whisker) reduce coupling with Cr thickness can be understood in the short-period coupling J1, which alternates in sign with each the thickness #uctuation model of biquadratic Cr layer, to JM1, normalized to one for ideal interfaces, "0. (b) The biquadratic coupling JM coupling. The canted coupling dies out above  varies quadratically with the terrace length ¸ in Slonczewski's thickness #uctuation model, about 18 ML Cr thickness because, as inter- Eq. (6) [31]. face roughness and thickness #uctuations increase D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 313 with Cr thickness, ¸ decreases, and thus the bi- for the MT quadratic coupling JM  sample and the MT sample. The  decreases. MT Turning to the trilayers on GaAs substrates [69],  exhibited short-period oscillations in the coupling that were four times larger than those of Table 2 shows that the roughness of the Cr grown the MT at room temperature and at 520 K in the MT  sample. This correlates with the higher  proportion of pillars with a large cross section and sample is about the same. However, the mean is- with the larger R for the MT land spacing R is much larger for high-temperature  sample. In summary, the magnetic coupling appears growth, as also can be seen by comparing the STM quite similar for the rougher RT Cr growth on images in Fig. 16b and Fig. 16e. The magnetic either the Fe whiskers or the Ag-bu!ered GaAs. coupling is clearly very di!erent. The striking STM There is long-period oscillatory bilinear coupling in images of the MT Fe and Cr surfaces show both cases. For higher temperature Cr growth, mesa-like structures with deep canyons between. the observed magnetic coupling is di!erent for tri- This morphology is thought to result from relieving layers on Fe whiskers and on Ag-bu!ered GaAs. the strain in the Fe "lm grown on Ag with a 0.8% Both types of samples probably have contributions lattice mismatch. Similar structure is seen in the Cr from both long- and short-period bilinear coupling growth, but the mean island spacing R increases over as well as biquadratic coupling. For Cr growth at that of the Fe "lm by 52 and 14% for the MT and 488 K on the Fe whisker, the long-period bilinear the MT samples, respectively. It is di$cult to coupling dominates with short-period, non-col- analyze these results in terms of R, and  that we linear oscillations, up to a Cr thickness of 18 ML, have used to analyze other results because these due to the short-period coupling and the biquad- three parameters do not describe all of the impor- ratic coupling. For Cr growth at 520 K on the tant properties of this morphology. The canyons GaAs, above a Cr thickness of 7 ML, the biquad- between the mesas dominate the measured rough- ratic coupling dominates [70]. The origin of this ness. The height distribution of the mesas alone has di!erence lies in the structural di!erences at the a much smaller than the surface as a whole. Also, interfaces as observed by STM. the average terrace length ¸ on the mesas is much larger than the average ¸ derived from R/ . Schmidt et al. [69] recognized that it is not 9. Fe/Cr superlattices: measurements possible to understand the coupling strengths of the Fe/Cr/Fe trilayers on GaAs only in terms of the The magnetic ordering within the Cr layers and measured  of the thickness distribution as was the interlayer coupling in Fe/Cr superlattices is possible for the SEMPA data from trilayers on complex and sensitive to interfacial roughness. whiskers [48]. The largest areas over which the top Neutron scattering has been applied to the study of and bottom interfaces of the Cr "lm are #at, i.e., Fe/Cr multilayers in part because it is directly sen- regions of constant Cr thickness, called pillars in sitive to the magnetic order. Neutron scattering at the model of Schmidt et al., are found in the mesa small angles, referred to as neutron re#ectivity, is regions. Regions of constant thickness with a large sensitive to magnetic structure on length scales of ¸ contribute most strongly to the biquadratic a few nm, such as a superlattice period. For coupling. These regions may also dominate the example, neutron re#ectivity distinguishes between short-period coupling, if there is an imbalance of ferromagnetic and antiferromagnetic ordering of the area associated with pillars of an odd rather the Fe layers in Fe/Cr superlattices. Neutron scat- than even number of Cr layers. Such an imbalance tering at high angles, referred to as neutron di!rac- would be possible if the thickness of these regions is tion, is sensitive to magnetic structure on an atomic distributed with a  that is much smaller than the length scale such as the antiferromagnetic ordering  of the whole spacer layer. Schmidt et al. empha- of the Cr moments. Additionally, using incident sized the importance of the pillar size [69]. From polarized neutrons and polarization analysis, it is their analysis, they concluded that even for the possible to determine the orientation of the mo- mesas, the thickness #uctuations were very similar ments in the plane of the layer. However, neutron 314 D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 scattering requires large samples, in particular, superlattices. Since these structures are grown thicker and with more interfaces than the trilayers discussed in the preceeding sections, they tend to have more disorder. Two groups have carried out most of the neutron scattering investigations of Fe/Cr superlattices. We "rst review magnetic coupling, transport, and neu- tron scattering measurements of Fullerton and coworkers as a function of temperature on a series of superlattices di!ering in the thickness of the Cr layers. The superlattices were epitaxially deposited at tem- peratures from 350 to 450 K on a 10 nm Cr bu!er on Fig. 19. A phase diagram summarizing the magnetic structure MgO(0 0 1) by DC magnetron sputtering [74,75]. of the Cr spacer layers as a function of temperature and Cr Long-period oscillatory coupling was observed up thickness for the measurements on superlattices discussed in to a Cr layer thickness of 45 ML in saturation the text. The boundary ¹' is from transport measurements of Fullerton et al. [76]. Diamonds denote CSDW, squares denote magnetization measurements at room temperature ISDW, and circles denote paramagnetic regions. The measure- and magnetoresistance measurements at room tem- ments are from Fullerton et al. [78] (solid symbols), Schreyer perature and 4.2 K. For larger Cr thicknesses, et al. [81,83,84] (open symbols), and Meersschaut [88] biquadratic coupling was observed. The biquad- (cross-hatch symbol). The shaded region is the transition region ratic coupling in an [Fe(10 ML)Cr(51 ML)] found by Schreyer et al. [83]. The rise of the ¹  , ! boundary at smaller Cr thickness (dashed line) was found in recent superlattice was characterized in detail by polari- measurements [86]. zed neutron re#ectivity (PNR), magnetization and magnetotransport measurements [75,76]. Below the transition temperature for this thickness, ¹,"187 K, the biquadratic coupling was not anomalies like those measured for the superlattice found and the Fe layers became decoupled. We are seen at the NeHel temperature in bulk Cr. In note in passing that in a set of magnetization and some dilute alloys of Cr, there are similar resistance magnetoresistance experiments with a Cr(2 1 1) anomalies associated with transitions from incom- spacer carried out in parallel to those measure- mensurate to commensurate states. This type of ments with the Cr(0 0 1) spacer, the same phase, transition was observed by subsequent neutron period and strength was found for the long-period scattering measurements as discussed below. We oscillatory coupling thereby adding an important label the transition ¹ constraint on theoretical explanations of the long- ' to indicate loss of incommen- surate order. period coupling [74,77]. The neutron di!raction measurements for Cr Anomalies in the resistivity and in magnetic thicknesses greater than about 30 ML and for tem- properties were used to determine a transition tem- peratures below ¹ perature ¹ ', as determined from the resistiv- ' as a function of the thickness of the Cr ity anomaly, con"rmed that the Cr has an spacer with a series of Fe(10 ML)/Cr(t!) superlatti- incommensurate transverse SDW with Q perpen- ces [76]. The transition temperature is plotted as dicular to the interfaces [78]. The ISDW period the solid line in Fig. 19, which shows Cr order as was independent of Cr thickness and near that of a function of Cr thickness and temperature. Anti- the bulk Cr ISDW. The best "t to the data was ferromagnetic order was not observed for t!( obtained assuming that the nodes in the ISDW 29 ML. Above this thickness, ¹' rises rapidly and were near the superlattice interfaces. The scattered then asymptotically approaches the value for thick neutron intensity for a superlattice with Cr layer "lms. The temperature ¹' was originally attributed thickness of 21 ML, although weak, could be to a transition from an ordered incommensurate quantitatively "t assuming commensurate antifer- state to a paramagnetic state [76]. Resistivity romagnetic order [78]. The results for superlattices D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 315 on MgO are summarized on the phase diagram of RT growth) found for superlattice growth at elev- Fig. 19. Fullerton et al. [76,78,79] extensively ated temperatures [80,81,85]. discussed how the magnetic frustration at rough Two transition temperatures were identi"ed interfaces could cause the observed behavior as we when PNR studies were carried out over an extended examine further below. temperature range for two superlattices, [Fe(14 ML)/ Schreyer and coworkers investigated the mag- Cr(56 ML)] netic state of Cr and the magnetic coupling in  and [Fe(13 ML)/Cr(29 ML)], grown on Al superlattices grown on two di!erent types of sub- O substrates [83]. For a Cr thick- ness of 56 ML, ISDW order was found for low strates: (1) Fe/Cr superlattices were grown by mo- temperatures. From 175 to 310 K a gradual lecular beam epitaxy (MBE) on Ag-bu!ered GaAs transition was observed that was characterized by at room temperature and at 523 K in the same way a superposition of a double-peak ISDW spectrum as for the trilayers [67,80,81] and (2) Fe/Cr superla- and a single-peak CSDW spectrum of changing ttices were grown by MBE at 570 K on relative weight [86]. Finally, at still higher temper- Cr(0 0 1)/Nb(0 0 1)/ Al0(1 1 0 2) as described else- atures, a transition to paramagnetic Cr is observed where [82,83]. at ¹ The superlattices grown on Ag-bu!ered GaAs(0 0 1) , !. These two transitions are distinguished in Fig. 19, the transition from paramagnetic to CSDW had relatively few bilayer repeats. The superlatti- Cr that takes place at ¹ ces, [Fe(36 ML)/Cr(6 ML)] , !, and the transition from  and [Fe(36 ML)/ CSDW to ISDW order that takes place in the Cr(12 ML)], were grown at room temperature and shaded region around ¹ 523 K, respectively. X-ray di!raction showed '. For the superlattice with 29 ML Cr, small-angle similar correlated roughness for both superlattice PNR and magnetization measurements found non- types, presumably due to the starting Ag bu!er collinear ordering of the Fe layers below ¹ layer surface. At a given growth temperature, , !. Additionally, using high-angle neutron di!raction, uncorrelated roughness, which a!ects the coupl- a modulation of the commensurate antiferromag- ing, was found to increase with an increasing netic structure of Cr was observed with a period number of layers in the superlattice. The estimated twice the superlattice period caused by a spiral-type terrace lengths were much smaller for the growth modulation of the Cr layers [83]. The non-collinear of Cr at RT than at 523 K. For this reason, the coupling of the Fe layers was thus associated with bulk of the neutron scattering measurements were the spiral antiferromagnetic order of the Cr layers. done on the superlattices grown at the higher Above ¹ temperature. , ! where the long-range Cr order vanish- es, MOKE hysteresis loops showed that the Fe From polarized neutron re#ectivity measure- layers were no longer coupled [83]. For thin Cr ments, Schreyer et al. [81] found collinear fer- layers, ¹ romagnetic coupling for the RT superlattice , ! increases due to the larger in#uence of the proximity of the Fe on the Cr layer. consistent with the RT trilayer results of GruKnberg Perturbed angular correlation spectroscopy et al. [66]. However, for the 523 K superlattice, (PACS) measurements have been used to determine PNR measurements at 297, 200 and 42 K showed the direction of the Cr moments and indirectly the that the coupling of the Fe layers was non-collinear direction of Q. In early measurements of superlatti- and at an angle of 50$43 near remanence ces grown by MBE on MgO at 420 K (in contrast [80,81,84]. The data points representing this non- to the sputtered superlattices of Fullerton et al. collinear coupling, which was associated with com- [74]), the Cr was found to be non-magnetic for mensurate antiferromagnetic order in Cr, are thicknesses below 42 ML [87]. When the superla- included in Fig. 19. For this superlattice with Fe ttice Cr layers were thicker than 42 ML, Meer- layers of equal thickness, the observed non-zero sschaut et al. [87] found a longitudinal ISDW with remanence and high saturation "eld indicate non- the Cr moments out of the "lm plane, that is the collinear coupling [81]. The occurrence of the Cr moments were found to be perpendicular to non-collinear coupling is correlated with long ter- the Fe moments. Recently, the experiments have race lengths (as opposed to short terrace lengths for been repeated on an [Fe(12 ML)/Cr(58 ML)] 316 D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 superlattice grown at 580 K [88]. The behavior of 10. Fe/Cr superlattices: interpretation the resistivity anomaly and hysteresis curves for this superlattice were measured and found to be The coupling in superlattices is strongly in- similar to the work of Fullerton et al. [76]. Above #uenced by the presence of a high degree of dis- the transition temperature ¹'"200$10 K, iden- order and the resulting spin frustration. There are ti"ed by the resistivity anomaly, the hysteresis many possible states for Cr in these disordered curve gave evidence of biquadratic coupling. At superlattices. Fullerton et al. [76,78,79] interpret lower temperatures the Fe layers were uncoupled. their results in terms of a transition between Also in these samples, in contrast to the samples a high-temperature paramagnetic state in which grown at 420 K, PACS measurements at 77 K [88] there is no antiferromagnetic order and a low- showed in-plane Cr moments corresponding to temperature ISDW state with the nodes near the a transverse ISDW "tting onto the phase diagram, interfaces. In the ISDW state, the frustration is Fig. 19, of the neutron measurements. This is believed to be taken up by interface domain walls a striking demonstration that the growth condi- discussed in connection with Fig. 3c. These walls tions are decisive in determining the magnetic or- connect the Fe steps as illustrated schematically by dering of the Cr in Fe/Cr superlattices. It would be the thick lines in Fig. 20. While nearly perfect very interesting to know what structural changes in interfaces appear to favor antinodes at the interfa- the superlattice were caused by the 160 K increase ces, it is plausible that disordered interfaces favor in Fe/Cr growth temperature. nodes close to the interface because then the mo- There are a few remaining discrepancies in the ments are reduced where there is frustration in the neutron and PACS measurements of the magnetic coupling with the Fe. The domain walls parallel to order of the Cr layers in Fe/Cr superlattices grown the interface in the Cr essentially decouple Cr anti- at elevated temperatures as summarized in Fig. 19. ferromagnetic order from the Fe ferromagnetic or- PACS measurements have yet to report other than der, as illustrated by the dotted line in Fig. 20. The paramagnetic Cr below a Cr thickness of 42 ML resulting coupling between Fe layers is small, pre- [87,88]. In this thickness region, neutron measure- sumably because none of the Fe layers are coupled ments of superlattices on AlO show that the Cr is to the Cr. in a commensurate spiral state that leads to non- Bulk Cr makes a transition into the paramag- collinear coupling of the Fe layers [83]. Other netic state as the temperature is increased above the neutron measurements are not inconsistent with NeHel temperature. The resistance anomaly found this. The superlattice on Ag-bu!ered GaAs did not have su$cient layers to produce a large enough signal to determine the existence of the spiral struc- ture. The unpolarized neutron measurements of superlattices on MgO were not able to determine if the commensurate structure they observed was from non-collinear Cr order. At thicknesses above 45 ML where long-period coupling is no longer observed, and at low temperatures where Cr is in an ISDW state, the Fe "lms are not magnetically coupled. In this thickness region, Schreyer et al. [83] found a transition to commensurate order with increasing temperature while Fullerton et al. Fig. 20. A representation of the possible relief of spin frustration [76,78] found a transition to paramagnetic order. in antiferromagnetic Cr spacers in superlattices leading to the This is the major remaining discrepancy in the ISDW state. The heavy lines schematically indicate domain results; it can be attributed to di!erences in the walls terminated at interfacial steps. The spin frustration is interface structure of the sputtered and MBE- relieved by these walls near the interface leaving a region, shown grown samples. by the dashed line, of ISDW ordered Cr [79]. D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 317 by Fullerton et al. [76] is consistent with this type of transition. As the thickness decreases, the NeHel temperature also decreases, either due to the de- coupled antiferromagnetic state behaving like a thin "lm, or due to the increasing spin frustration due to the closer interfaces. Below a certain thick- ness, it is no longer favorable to "t in a half period of the ISDW, and the Cr goes into a di!erent state [78]. There are indications of weak commensurate antiferromagnetic order for these thin Cr layers. The Cr may be paramagnetic in parts of the sample and commensurate in others, or may be in a strong- ly disordered commensurate state. Fishman [89] has derived a phase diagram consistent with this by considering a model in which the Cr moment is constrained to be zero at both interfaces. With this rigid constraint, he predicts an oscillatory compon- ent to the transition temperature. These oscillations have not been seen. Relaxing the strict constraint of the moments being zero exactly at the interface may weaken these oscillations [89]. Above the transition temperature, Fullerton et al. [74] ob- serve a combination of long-period coupling and biquadratic coupling which can be interpreted in terms of a thickness #uctuation mechanism [31]. As an alternate to taking up the frustration in domain walls parallel to the interface, the frustra- tion could be taken up in domain walls perpendicu- lar to the interface, allowing the Cr moments to twist, as shown in Fig. 21. The resulting helical state Fig. 21. Schematic illustration of non-collinearly coupled Fe is yet another possible state for the Cr that is found layers showing how the energy is minimized in the presence of theoretically to be favorable in some situations a Cr thickness #uctuation by a spiral rotation of the Cr mo- [24,25,90]. In this case, regions of thickness that ments. The empty and "lled small arrows indicate an opposing di!er by one layer of Cr favor coupling in opposite sense of rotation of the Cr moments [83]. directions. In these regions, the twist has di!erent senses of rotation leading to a non-collinear coup- ling of subsequent Fe layers as described by Slon- state in a temperature range between the incom- czewski's torsion or proximity magnetism model mensurate state and the paramagnetic state. In [23]. Whether domain walls parallel or perpen- a certain range, they "nd a coexistence of incom- dicular to the interfaces are favored depends on mensurate order and commensurate order, presum- many di!erent properties of the samples including ably in di!erent parts of the sample. It may be that the temperature and the average step spacing. the terrace lengths in these samples of Schreyer et Schreyer et al. [83] interpret their results in terms al. [83] are larger than those in the samples of of the incommensurate state described above at Fullerton et al. [76,78], allowing the commensur- large thicknesses and low temperature and the ate, helical state to develop for certain samples and paramagnetic state at high temperatures, both con- temperatures. In the samples of Fullerton et al., sistent with the results of Fullerton et al. [76,78]. smaller terrace lengths could cause frustration at However, they observe a commensurate, helical Cr the interface su$cient to keep the Fe from inducing 318 D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 antiferromagnetic order in the Cr. Alternatively, preted in terms of models that describe di!erent there could be CSDW order in domains, but if the ways of minimizing the frustration. structure causes the domains to be su$ciently The magnetic coupling in Fe/Cr/Fe(0 0 1) tri- small, they are averaged over by the neutrons, layers grown on Fe whiskers, where the growth is and the CSDW order would not be observed [91]. optimized to approach ideal layer-by-layer growth, In either case a resistivity anomaly is expected is best understood. In this case, the interface mor- [13,14,92]. phology is relatively simple and for the most part Schreyer et al. [83] support the description of can be taken into account. There is strong evidence their results in terms of the torsion model by noting in the temperature dependence of the phase slip that in magnetic hysteresis measurements by that the short-period bilinear coupling is tied to MOKE they observe the gradual approach to satu- the SDW state of the Cr spacer. The experimental ration predicted by Slonczewski's torsion model results are consistent with CSDW order up to a Cr [23]. However, the gradual approach to saturation thickness where the "rst phase slip occurs and is only observed on the superlattices, not on ISDW order beyond that. The short-period oscilla- trilayers prepared the same way on either GaAs or tory coupling dominates the long-period coupling. AlO substrates [86]. Two explanations for The biquadratic coupling observed when the aver- this di!erence between the superlattices and the aged short-period coupling goes to zero can trilayers are the following. If there were variations be described by the Slonczewski thickness #uctu- in JM or JM due to variations in thickness of one ation model, which ignores possible antiferromag- spacer layer to the next, or if there were lateral netic order in the Cr [31]. However, there is variations in JM or JM within the MOKE laser spot no direct measurement of the Cr moments in these size, both of which might not be unexpected for trilayers. Since the Cr is likely in an ordered a superlattice, such variations would have the e!ect state, its moments are likely to have a more com- of rounding the hysteresis curves as previously de- plicated behavior than is implied by the conven- scribed [36]. Alternatively, the interface structure tional bilinear/biquadratic model of the coupling. of the superlattices may be su$ciently di!erent Particularly when the Fe moments are not col- from the trilayers requiring a di!erent description linear, the Cr moments are also likely to be non- of the coupling that includes aspects of the torsion collinear. Thus, a complete description of the model. behavior of these samples will require treatment of non-collinear Cr moments [23}25,90]. On the other hand, the simple torsion model used 11. Conclusions to explain other measurements does not describe incommensurate order and hence cannot explain Interlayer exchange coupling through a Cr the existence of phase slips in the coupling or spacer layer is special because the Cr can be in the behavior of the biquadratic coupling in these various states of magnetic order itself. The coupling samples. depends intimately on the ordering in the Cr, When the roughness at the interfaces increases whether it has ISDW order, CSDW order, or is due to di!erent growth conditions or substrate paramagnetic. The roughness distribution at the conditions, the coupling changes dramatically. For interface, both vertically and laterally, also strongly room temperature growth, which produces rough- a!ects the magnetic coupling. In particular, the ness with a short terrace lengths, long-period oscil- interface roughness frustrates the preferred align- latory bilinear coupling dominates. The long- ments of the magnetic moments in the multilayer. period coupling is consistent with the conventional Even when the system has found its lowest energy model for coupling and is associated with a part of con"guration, some pairs of moments are frus- the Cr Fermi surface that is largely insensitive to trated. Using what is known about the multilayer, the presence or absence of antiferromagnetic order particularly about the interface structure, measure- in the Cr. Over the lateral response length l of the ments of the magnetic con"guration can be inter- Fe layer, thickness #uctuations greatly reduce the D.T. Pierce et al. / Journal of Magnetism and Magnetic Materials 200 (1999) 290}321 319 contribution of the short-period oscillatory coup- Acknowledgements ling. The biquadratic coupling, which can be observed for Fe whisker samples near zeros in the We wish to acknowledge very helpful commun- long-period coupling, varies with the strength of ications with D.E. BuKrgler, A. 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