PHYSICAL REVIEW B VOLUME 59, NUMBER 21 1 JUNE 1999-I Collinear spin-density-wave ordering in Fe/Cr multilayers and wedges R. S. Fishman Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6032 Zhu-Pei Shi Read-Rite Corporation, R & D Division, 345 Los Coches Street, Milpitas, California 95035 Received 6 April 1998; revised manuscript received 5 February 1999 Several recent experiments have detected a spin-density wave SDW within the Cr spacer of Fe/Cr multi- layers and wedges. We use two simple models to predict the behavior of a collinear SDW within an Fe/Cr/Fe trilayer. Both models combine assumed boundary conditions at the Fe-Cr interfaces with the free energy of the Cr spacer. Depending on the temperature and the number N of Cr monolayers, the SDW may be either commensurate C or incommensurate I with the bcc Cr lattice. Model I assumes that the Fe-Cr interface is perfect and that the Fe-Cr interaction is antiferromagnetic. Consequently, the I SDW antinodes lie near the Fe-Cr interfaces. With increasing temperature, the Cr spacer undergoes a series of transitions between I SDW phases with different numbers n of nodes. If the I SDW has n m nodes at T 0, then n increases by one at each phase transition from m to m 1 to m 2 up to the C phase with n 0 above TIC(N). For a fixed temperature, the magnetic coupling across the Cr spacer undergoes a phase slip whenever n changes by one. In the limit N , TIC(N) is independent of the Fe-Cr coupling strength. We find that TIC( ) is always larger than the bulk NeŽel transition temperature and increases with the strain on the Cr spacer. These results explain the very high IC transition temperature of about 600 K extrapolated from measurements on Fe/Cr/Fe wedges. Model II assumes that the I SDW nodes lie precisely at the Fe-Cr interfaces. This condition may be enforced by the interfacial roughness of sputtered Fe/Cr multilayers. As a result, the C phase is never stable and the transition temperature TN(N) takes on a seesaw pattern as n 2 increases with thickness. In agreement with measurements on both sputtered and epitaxially grown multilayers, model II predicts the I phase to be unstable above the bulk NeŽel temperature. Model II also predicts that the I SDW may undergo a single phase transition from n m to m 1 before disappearing above TN(N). This behavior has recently been confirmed by neutron- scattering measurements on CrMn/Cr multilayers. While model I very successfully predicts the behavior of Fe/Cr/Fe wedges, a refined version of model II describes some properties of sputtered Fe/Cr multilayers. S0163-1829 99 03021-0 I. INTRODUCTION models for the SDW within the Cr spacer. Both models com- bine assumed boundary conditions at the Fe-Cr interfaces The original discovery of giant magnetoresistance1 in with the free energy of the Cr spacer. Model I takes the Fe/Cr multilayers inspired an intensive investigation into interfaces to be perfect and the Fe-Cr interactions to be an- their magnetic and electrical properties. Although giant mag- tiferromagnetic. Consequently, only collinear C and I phases netoresistance was soon found in other multilayers with non- are stable and the I SDW antinodes lie close to the interfaces. magnetic spacers, Fe/Cr heterostructures have continued to Model II takes the I SDW nodes to lie precisely at each hold the interest of the physics community. Due to the com- Fe-Cr interface. This condition restricts the SDW wave vec- petition between the spin-density wave SDW ordering in tor to quantized values. As a result, the NeŽel temperature the Cr spacer2 and the Fe-Cr interactions at the interfaces, changes discontinuously whenever the number of I SDW Fe/Cr multilayers and wedges provide new insights into the nodes changes by one. While model I quite successfully de- physics of transition-metal magnets. scribes the properties of Fe/Cr wedges, model II may par- In bulk Cr, commensurate C or incommensurate I tially describe the behavior of sputtered Fe/Cr multilayers. SDW's are stabilized in different ranges of doping. For Because SDW formation in bulk Cr relies on the balance Fe/Cr multilayers, neutron-scattering measurements3,4 reveal between magnetoelastic and Coulomb energies,6 the ampli- that the I phase is stable when the number of monolayers tude and wave vector of the SDW are notoriously sensitive ML's N inside the Cr spacer is greater than 30 or when the to both doping and pressure.2 The d bands of Mn and V temperature is lower than the NeŽel temperature 310 K of contain one extra or one fewer electron than Cr. So Mn or V pure Cr. By contrast, scanning electron microscopy impurities are often used to control the level of the chemical measurements5 on Fe/Cr/Fe wedges indicate that the I phase potential and the wave vector of the SDW. Cooled below its is stable for N 23 ML and up to at least 550 K. As the NeŽel temperature of 310 K, pure Cr enters an I state with a thickness of the wedge increases, the Fe-Fe coupling alter- node-to-node distance of 27 ML. When the Mn concentra- nates between ferromagnetic F and antiferromagnet AF tion exceeds 0.3%, Cr1 xMnx enters a C state with an en- with phase slips every 20 ML at room temperature. To ex- hanced NeŽel temperature. By contrast, doping with V makes plain these measurements, we have introduced two different the SDW more incommensurate and shortens the distance 0163-1829/99/59 21 /13849 12 /$15.00 PRB 59 13 849 ©1999 The American Physical Society 13 850 R. S. FISHMAN AND ZHU-PEI SHI PRB 59 between nodes. With 2% V impurities, the NeŽel temperature tion by placing a node at each interface. Assuming that a is reduced to about half that of pure Cr and the node-to-node node is fixed at each interface, the SDW wave vector is distance at TN is shortened to about 13 ML. obtained by minimizing the free energy with respect to the Remarkably, applying pressure to bulk Cr has the same total number n 2 of SDW nodes. This model is developed effect as doping with V: the NeŽel temperature decreases and in Sec. IV. the SDW becomes more incommensurate. A volume com- Unfortunately, model II only partially describes the prop- pression of 1.5% corresponds to doping with about 1% V erties of sputtered multilayers. Of course, the C phase is impurities.7 Alternatively, a pressure of 2.8 GPa corresponds never stabilized by this model because a C SDW has no roughly to lowering the electron concentration by 1%.8 Since nodes. In agreement with Fullerton et al.,4 model II predicts Fe has a slightly smaller lattice constant than Cr, the lattice that the paramagnetic P to I transition always occurs below strain exerted on the Cr spacer in an Fe/Cr wedge should the bulk NeŽel transition. As very recently observed in lower9 the bulk NeŽel temperature T CrMn/Cr multilayers,15 model II permits phase transitions N and make the SDW more incommensurate. Below T between SDW's with different numbers of nodes. However, N , model I implies that the distance between phase slips approaches the node-to-node this model also predicts that the NeŽel temperature suddenly distance for the bulk SDW as N increases. Hence, the mea- drops when n increases by 1. The resulting seesaw pattern sured distance between phase slips of 20 ML at room tem- for TN(N) is quite unlike the monotonically increasing NeŽel perature confirms that the SDW of the Cr spacer in an Fe/Cr temperature observed in sputtered multilayers.4 Even after wedge is more incommensurate than in pure Cr. the positions of the SDW nodes are allowed to shift a few Strain is far less significant in Fe/Cr multilayers. Low- ML's from each interface, the first large drop in the NeŽel temperature measurements by Fullerton et al.4 reveal that the temperature from n 2 to n 3 survives. SDW period is about 59 Ć , corresponding to a node-to-node This paper is divided into five basic sections. A brief tu- distance of 20 ML. This is quite close to the period of pure torial on the SDW in bulk Cr is presented in Sec. II. Sections Cr Ref. 2 at low temperatures. III and IV develop models I and II, respectively, and present The difference between Fe/Cr multilayers and wedges their results. A discussion and conclusion is given in Sec. V. raises an intriguing puzzle. Although the large strain in Fe/Cr We obtain the large N dependence of the magnetic coupling wedges should reduce the NeŽel temperature of the Cr spacer for model I in Appendix A. Also in the large N limit, the IC far below its value in pure, unstrained Cr, the measured IC transition temperature of model I is derived in Appendix B. transition temperature of the wedge is at least 550 K, far The basic ideas of this work were first presented in three higher than the transition temperature of relatively strain-free short papers: Refs. 16 and 17 were devoted to model I while multilayers. But as shown in Sec. III for large N, the IC model II was first developed in Ref. 18. transition temperature TIC of model I is independent of the size of the Fe-Cr coupling constant. Unlike the NeŽel tem- II. SPIN-DENSITY WAVE AND FREE ENERGY perature TN of the bulk alloy, TIC increases with V doping OF BULK Cr and with strain. Because the distance between interfacial steps is much The SDW instability19 in Cr alloys is produced by the shorter in sputtered multilayers than in epitaxially grown Coulomb attraction U between electrons and holes on nearly multilayers, the behavior of the SDW in epitaxial and sput- perfectly nested20,21 electron a and hole b Fermi surfaces, tered Fe/Cr multilayers is quite different. While both roughly octahedral in shape.22 The electron Fermi sur- experiments10,11 on epitaxially grown trilayers report that the face centered at and the hole Fermi surface centered at the Fe-Cr interaction is antiferromagnetic, measurements by Ful- zone boundary H are separated by wave vector G/2, where G lerton et al.4 on sputtered multilayers suggest that the SDW is a reciprocal lattice vector with magnitude 4 /a. Together nodes lie close to the Fe-Cr interfaces. Consequently, the Cr with the electron pockets around the X points, the a Fermi spacer does not magnetically couple the neighboring Fe lay- surface forms part of the so-called ``electron jack.'' Also ers in a sputtered multilayer. present are hole pockets at the N points. Both the electron The precise nature of the IC transition is also in some and hole pockets play an ancillary role in the formation of doubt. In the sputtered mutlilayers studied by Fullerton et al., the SDW and are often grouped together into an electron a remnant C phase is observed at low temperatures below 30 ``reservoir'' that supplies electrons to the a and b Fermi sur- ML but does not appear at higher thicknesses above TN . But faces once the quasiparticle gap opens below TN . in the epitaxially grown multilayers studied by Schreyer et However, these pockets may directly affect the magnetic al.,3 an I SDW coexists with a noncollinear, helical H SDW coupling across an Fe/Cr/Fe trilayer. As discussed in the next between 200 and 300 K. The reported H SDW is believed to section, the periodicity of that magnetic coupling is deter- couple neighboring Fe moments at 90° angles, with the Fe mined by the extremal points of the Fermi surface.23,24 For moments returning to the same orientation every other this reason, oscillatory coupling has also been observed in layer.12 Recent work13,14 indicates that a H SDW is produced Fe/Cu and Fe/Ag multilayers,23 where the Cu or Ag spacer is by the well-separated interfacial steps of epitaxially grown paramagnetic. Quasiparticle transitions across the necks of multilayers. the electron jack25 or, alternatively, across the hole pockets,23 Many of the unique features of sputtered Fe/Cr multilay- are associated with a long period oscillation of the magnetic ers can be attibuted to either interfacial roughness or the coupling in Fe/Cr multilayers and wedges. intermixing of Fe within the first few ML's of the spacer. Because the electron Fermi surface is slightly smaller than Both roughness and intermixing frustrate the antiferromag- the hole Fermi surface, there are two different nesting wave netic Fe-Cr interaction.13 An I SDW can avoid such frustra- vectors Q that translate four faces of one Fermi surface PRB 59 COLLINEAR SPIN-DENSITY-WAVE ORDERING IN . . . 13 851 onto four faces of the other. Since the average of Q is G/2, the nesting wave vectors may be written as Q (G/2)(1 ), where 0.04 is a measure of the size difference be- tween the electron and hole Fermi surfaces. Unlike the condensate of a superconductor, which con- tains pairs of electrons with zero total momentum, the con- densate of an I SDW contains pairs of electrons and holes with nonzero total momentum. In the I phase of the SDW, the condensate contains two types of electron-hole pairs: one with pair momentum Q (G/2)(1 ) and the other with pair momentum Q (G/2)(1 ). Because 0 , the ordering wave vectors Q of the SDW lie closer26,27 to G/2 than the nesting wave vectors Q . Whereas Q and are fixed by the band-structure topology, Q and are solved by minimizing the nesting free energy F and generally de- pend on temperature. When 0 and Q G/2, the SDW is commensurate. When 0, the I SDW has a period of FIG. 1. I and C SDW's with 0 and 0. For better visu- a/ , corresponding to a node-to-node distance of 1/ alization, the node-to-node distance is substantially shorter than in ML's. For pure Cr just below its NeŽel temperature,2 pure Cr. 0.037 and 1/ 27. The three sets of possible ordering wave vectors Q cor- 1 respond to the three possible orientations of the nesting wave F g, ,T,z 0 ehg2ln T eh g2 vectors along the 100 , 010 , or 001 directions. When T l 1/2 N * l 0 pure Cr is cooled below its NeŽel temperature TN 310 K, six types of domains form.2 In each domain, the spin polar- T dz ln D g, ,i l , 2 ization m lies along one of two possible directions perpen- D 0, ,i l dicular to one of the three sets of wave vectors Q . Replacing the strongly peaked Bloch wave functions by delta functions at every lattice site R, the general form for D g, ,i l i l z i l z0/2 z 2 z0 /2 2 the Cr spin at R can be simply written as g2 2i l z0 2z , 3 S R where l (2l 1) T are the Matsubara frequencies and z m s g T cos Q *R cos Q *R eh is the density of states of the nested portions of the a and 1 2Rz /am b Fermi surfaces. When g 0, F 0 as expected. The vari- sg T cos 2 /a Rz /2 , 1 able of integration in F is z vF(k*n kF), where n is normal to one of the octagonal faces of the a Fermi surface where s and s are constants, g(T) is the temperature- and vF (kF) is the Fermi velocity momentum . dependent order parameter, and . While is Doping affects the bulk free energy through the energy arbitrary in the I state, /2 Ref. 28 in the C state with mismatch z0 4 vF / 3a between the a and b Fermi sur- 0. Hence, the amplitudes of the I and C SDW's are faces. While V impurities increase z0, Mn impurities lower given by sg(T) and sg(T)/ 2, respectively. Across a the mismatch between the Fermi surfaces. When the Mn second-order IC phase transition with the same order param- concentration exceeds about 0.3%, the mismatch is suffi- eter g on both sides, the SDW amplitude drops by a factor of ciently small to stabilize the C SDW phase with 0 at 1/ 2 but the rms magnetic moment is continuous. In the C TN . In units of TN* , the triple point where the P, C, and I phase at low temperatures,2 the Cr moment is approximately phases meet is given by z0 4.29TN* 430 meV. 0.8 B and the C SDW amplitude sg(0)/ 2 is about 0.4. Below TN , the electron and hole energies are hybridized For pure Cr at low temperatures, sg(0) 0.3 corresponding by the Coulomb attraction U. The resulting quasiparticle en- to a magnetic moment of 0.6 B . Both I and C SDW's are ergies (z) are obtained from the condition D(g, , ) 0. sketched in Fig. 1, where the period of the I SDW is some- In the C state with 0, lower and upper bands are sepa- what shorter than in pure Cr. rated by the energy gap 2 2 2g. At low temperatures, The Coulomb interaction U between the electrons and 2 is about 370 meV. The quasiparticle spectrum of the I holes on the a and b Fermi surfaces never explicitly appears phase is somewhat more complicated, with two identical en- in the free energy. It only enters implicitly through the ficti- ergy gaps of roughly 120 meV separated by a third band of tious NeŽel temperature TN* 100 meV of a perfectly nested quasiparticle states.29 alloy with 0. For an alloy with 0, the actual NeŽel The normalization of the free-energy difference F in Eq. temperature TN must be less than TN* . In terms of TN* , the 2 is chosen so that for a perfectly nested C alloy with z0 free-energy difference between the P and SDW phases can 0 at T 0, F(0) eh (0)2/4. Since eh/4 is the den- be written26 sity of states for electrons on the a Fermi surface with a 13 852 R. S. FISHMAN AND ZHU-PEI SHI PRB 59 single spin but both spin states are paired, this result is the spite some residual ordering near the interfaces at z a/2 and analogue of the T 0 BCS free energy30 for Cooper pairing. z Na/2. Below the bulk NeŽel temperature, the SDW ampli- Due to the lattice mismatch between Cr and Fe, the lattice tude will be enhanced near the interfaces. But the pair coher- constant of Cr inside the Fe/Cr/Fe wedge is about 0.6% ence length30 of the I phase 0 vF / g is about 10 Ć, so smaller5 than in bulk. As discussed in the Introduction, lat- the SDW order parameters g and are expected to be modi- tice strain has the same qualitative effect as V doping. There- fied only within 5 or 6 ML from each interface. This has fore, the effects of lattice strain can be modeled by choosing been confirmed by recent first-principles calculations32 and the energy mismatch z0 to yield the observed periodicity of observed by x-ray magnetic circular dichroism.31 Even above the SDW. A node-to-node distance of 20 ML at room tem- the bulk NeŽel temperature of the spacer but below the para- perature is obtained with a mismatch of z0 6.4TN* , which is magnetic transition temperature of the multilayer, the equi- substantially larger than the mismatch z0 5T librium value of the SDW amplitude which scales like 1/N) N * in pure Cr. However, recent work by Marcus et al.6 indicates that lat- should be reached within a coherence length or so from the tice strain does not significantly alter the sizes of the Fermi interfaces. Hence, the IC phase boundary evaluated from this surfaces. Nevertheless, strain does play a crucial role in sta- model should be qualitatively accurate. bilizing the SDW of pure Cr. We use the energy mismatch to The free energy of an Fe/Cr/Fe trilayer may be obtained model the effects of lattice strain simply because changes in in one of two ways. First, the energy E can be evaluated for pressure have qualitatively the same effects as changes in the either ferromagnetically or antiferromagnetically oriented Fe electron concentration. moments. The Fe moments may be fixed in one orientation or the other in an Fe/Cr/Fe trilayer containing permanently III. MODEL I: ANTIFERROMAGNETIC INTERACTIONS magnetized Fe whiskers. Alternatively, we can allow the Fe AT THE Fe-Cr INTERFACES moment on one side of the trilayer to find its lowest-energy configuration. Of course, this is the case for Fe/Cr wedges, Model I assumes that the antiferromagnetic Fe-Cr interac- where the thin Fe overlayer is not permanently magnetized. tions have the form ASI,II Fe *S(z) at interfaces I (z a/2) and II Then the stable magnetic configuration F or AF has the (z Na/2) with coupling constant A 0. Such an antiferro- lower energy. We shall examine the behavior of Fe/Cr/Fe magnetic interaction would be expected for microscopically trilayers from both perspectives in the following discussion. smooth interfaces and is clearly warranted in Fe/Cr wedges. After fixing the magnetic configurations of the Fe layers, Most measurements on epitaxially grown multilayers3,10,11 the SDW order parameters g and , as well as the arbitrary and even some measurements on sputtered multilayers31 ob- phase ,33 are chosen to minimize the energy E in Eq. 4 . tain antiferromagnetic interactions at the interfaces. But The corresponding F and AF energies of the trilayer are other measurements on sputtered multilayers4 suggest that surface roughness interferes with the magnetic coupling be- 1 tween neighboring Fe and Cr layers, at least below T EF 2A sgSFe cos N . An- 2 F g, ,T a3 N 1 , 5 tiferromagnetic interfacial interactions were confirmed in the first-principles calculations of Mirbt et al.32 1 For simplicity, we assume that the Fe moments are either EAF 2A sgSFe sin 2 F g, ,T a3 N 1 , F or AF aligned with SI II I II Fe SFe or SFe SFe , both parallel to 6 the interface. The SDW will then be transversely polarized with respect to the ordering wave vectors along the z axis. where ( /2)(N 1)(1 ). The SDW order parameter While the Fe moments in Fe/Cr wedges satisfy this assump- is restricted to values below the bulk maximum of gmax tion, the Fe moments in Fe/Cr multilayers may not. The mea- 1.246TN* , which is achieved in the C SDW phase of a bulk surements of Schreyer et al.3 on epitaxially grown Fe/Cr Cr alloy at T 0. Note that the number n of SDW nodes multilayers indicate that interfacial steps produce a 90° inside the Cr spacer is approximately given by (N 1) . coupling13,12 between adjacent Fe moments, which are joined For comparison with previous papers, Ref. 16 used the defi- by a helical modulation of the Cr moment. We shall return to nition / . this possibility in the final section. Because the nesting free energy F is proportional to With antiferromagnetic interactions at the interfaces, the ehTN*2, the total free energy E depends only on the dimen- free energy of the multilayer or wedge for an interfacial area sionless constant of a2 and spacer width L (N 1)a/2 may be written as16 A sSFe 1 , 7 E A SI II V/N Fe*S a/2 SFe*S Na/2 ehTN * 2 Fa3 N 1 , 4 which represents the average coupling strength between Fe which assumes that the SDW is rigid with order parameters g and Cr at the interfaces. It can be estimated either from first- and independent of z but see Ref. 33 . Since the interfa- principles calculations or by comparison with the experimen- cial energies always induce some SDW ordering with g 0 tal data. For example, a value of 3-which will be used no matter how high the temperature, the P state is unstable later in this section to model the phase diagram of Fe/Cr within this model. wedges-corresponds to an average Fe-Cr exchange interac- In a more realistic, albeit far more complex, model, the tion of 6.8 meV. In bulk Fe, the Fe-Fe interaction is of order SDW amplitude g(z) would vanish inside the spacer above 100 meV. So if the Fe-Cr exchange energy at a perfect in- the paramagnetic transition temperature of the multilayer de- terface is the same order as the Fe-Fe interaction, then the PRB 59 COLLINEAR SPIN-DENSITY-WAVE ORDERING IN . . . 13 853 Appendix A demonstrates that Jcoup decreases like 1/N2. This behavior was predicted24 when isolated, extremal points of the Fermi surface are nested but is unexpected for our idealized octahedral Fermi surfaces, where finite regions are nested. Indeed, van Schilfgaarde et al.24 predicted a 1/N1.25 dependence in this case. Quasiparticle transitions across the necks of the ``electron jack'' or across the hole pockets are believed to be responsible23 for a magnetic coupling with a 1/N2 falloff and a long period of 10­12 ML.5 A short, 2-ML period coupling with a 1/N2 falloff would also result from Ruderman-Kittel-Kasuya-Yosida RKKY coupling24 across a paramagnetic Cr spacer. So above TN , the RKKY and nesting contributions to the magnetic coupling cannot be dis- tinguished by their dependence on N. We emphasize that the predicted dependence of Jcoup on N only holds asymptotically. Pierce et al.38 have found that the short-period coupling of an Fe/Cr multilayer grown at a substrate temperature of 350 °C but measured at room tem- perature can be fit by a 1/N dependence for N 40. But as clearly seen in Fig. 2 b , the predicted 1/N2 falloff for T TN may only be recovered for values of N above 100 or so, FIG. 2. Model I: Bilinear magnetic coupling in meV as a func- particularly when T is not far above the bulk NeŽel tempera- tion of spacer thickness for z ture. 0 /TN * 5, 3, and a T 0.5TN or b T 1.2TN . For N 28, Fig. 2 a reveals that the magnetic coupling with the lowest free energy is F for odd N and AF for even Fe-Cr interface interaction in Fe/Cr multilayers and wedges N. The stable coupling then alternates between F and AF is about 1/15 that at a perfect interface. This agrees with until N 28, when a phase slip occurs. For both N 27 and recent experiments34 and with model calculations35 that the N 28, the stable coupling is F. Until the next phase slip at measured Fe-Cr coupling is substantially smaller than ex- N 46, the stable coupling is F for even N and AF for odd N. pected for a perfect interface. For example, Venus and This series of phase slips was observed in the NIST Heinrich34 found that the measured coupling is about 1/30 measurements.5 Each time a phase slip occurs, the number of smaller than the coupling given by first-principles nodes within the stable SDW increases by one. So the stable calculations36,24 for a perfect interface. Possible explanations SDW is commensurate prior to the first phase slip, contains for this suppression are surface roughness and intermixing. one node for N between 28 and 45, and two nodes for N Throughout the remainder of this section and into the between 46 and 71. next, we shall take the nesting parameter to be 0.043 Compared to the first-principles predictions of Stoeffler when z0 /TN* 5. At TN , this yields a bulk value for the and Gautier,36 the results of Figs. 2 a and 2 b for J SDW wave vector of 0.037, corresponding to the node- coup are about 50% too small. But even when the energy mismatch is to-node distance of 27 ML observed in pure Cr. For larger enhanced to account for the strain in Fe/Cr multilayers, our values of the energy mismatch in strained Cr, is assumed results are still roughly 30 times larger than the experimen- to increase linearly. So 0.055 when z0 /TN* 6.4, which is tally measured coupling strengths.39,40 This suggests that the used to model Fe/Cr wedges. All energies will be scaled by effects of interdiffusion and atomic steps are too complex to TN* 100 meV. be modeled by one or two fitting parameters. Once EAF and EF are found,37 the magnetic coupling The results of Fig. 2 can be more easily appreciated from Jcoup EAF EF may be evaluated as a function of tempera- the vantage of Fig. 3, which plots the magnetic phase dia- ture T and thickness N. Taking 3, z0 /TN* 5, (V/N) eh gram for unstrained (z0 5TN*) and strained (z0 6.4TN*) 3.7 states/Ry atom,22 and T 0.5TN or 1.2TN , we plot Fe/Cr/Fe trilayers with 3. We also display the number n Jcoup as a function of spacer thickness in Fig. 2. As expected, of SDW nodes for the stable magnetic phase as a function of Jcoup oscillates between F 0 and AF ( 0) values with a thickness and temperature. The thick solid curves denote the short 2-ML period. Below the NeŽel temperature, the mag- IC transition while the thinner curves denote the transitions netic coupling decays slowly with the size of the spacer as between I phases with different n. At a fixed temperature, shown in Fig. 2 a . This behavior is easily understood in phase slips occur whenever a solid curve is crossed. Away terms of the competing energies in Eq. 4 . In a large spacer, from a phase slip, the stable magnetic coupling alternates the wave-vector parameter is more constrained by the between F and AF with increasing thickness N. On either bulk free energy F(g, )a3(N 1)/2. Hence, the SDW side of a phase slip, the stable magnetic coupling AF or F is cannot deform as easily to maximize the antiferromagnetic the same. Fe-Cr coupling at the interfaces. We prove in Appendix A Returning to Figs. 2 a and 2 b , we find that a phase slip that Jcoup falls off like 1/ N below TN . occurs every time a phase boundary in Fig. 3 a is crossed at As shown in Fig. 2 b , the magnetic coupling falls off T 0.5TN or 1.2TN . At the higher temperature, the phase much more rapidly above the NeŽel temperature. For large N, boundaries are shifted to the right and further apart. For N 13 854 R. S. FISHMAN AND ZHU-PEI SHI PRB 59 FIG. 4. Model I: Energy in meV versus thickness for an I SDW with n nodes, with T/TN 0.5 and a z0 5TN* or b z0 6.4TN* . FIG. 3. Model I: Phase diagram of Fe/Cr multilayers and wedges for 3 and a z0 5TN* and TN 0.384TN* or b z0 Fixing T 0.5TN , we plot energy versus N for the differ- 6.4TN* and TN 0.282TN* . The thick solid curve denotes the IC ent SDW solutions in Fig. 4. The region of stability for a transition while the thin solid curves separate different I phases with SDW with n nodes corresponds exactly to the region be- n nodes. tween the solid curves in Figs. 3 a and 3 b . Taking the difference between the lowest-energy solution and the one 45, the I SDW phase with n 2 is stable at very low tem- just above it in Fig. 4 a yields the amplitude of J peratures, but gives way to an I SDW phase with n 1 be- coup plotted in Fig. 2 a . The C SDW solution with n 0, drawn as a tween 0.345TN and 1.505TN , and finally to a C SDW phase solid curve, is much more robust for z with n 0 above 1.505T 0 5TN * than for N . If the phase slips occur between thicknesses N 6.4TN* . i and Ni 1, then the distance between phase slips shall be denoted In Fig. 5, we plot the distance s2 between the second and by s third phase slips as a function of temperature. Below T i Ni 1 Ni . While the SDW is C before the first phase N , s2 slip at N is almost constant and very close to the bulk distance 1/ 1, the I SDW has n i nodes for Ni N Ni 1 1. bulk The distance s between SDW nodes. Above T 1 between the first two phase slips is always N , s2 begins to increase rap- the smallest. For large Cr spacers, the bulk free energy F idly with temperature. As expected, s2 diverges as T ap- dominates the interfacial energies. So below TN , si proaches TIC . This figure bears a striking resemblance to the 1/ measured phase slip distance5 in Fe/Cr wedges. Since the bulk as i . In other words, the distance between phase slips approaches the distance between nodes of the nesting wave vectors do not change with temperature, the bulk SDW. In addition, the distance N1 to the first phase slip is always less than 1/ bulk and only reaches this value as . Above the bulk NeŽel temperature and close to the IC phase boundary, the phase slip distances si become more disparate with the higher si's diverging most rapidly as T TIC . For a larger mismatch z0, the bulk SDW period is smaller so the phase boundaries in Fig. 3 b are closer together than in Fig. 3 a . At low temperatures, the critical thickness sepa- rating the C (n 0) and I (n 1) phases shifts downwards as z0 increases. For both Figs. 3 a and 3 b , the temperature is normalized by the bulk NeŽel temperature for that particular energy mismatch. As the mismatch increases, the bulk NeŽel temperature decreases: TN 0.384TN* when z0 /TN* 5, while T FIG. 5. Model I: Distance s2 between the second and third phase N 0.282TN * when z0 /TN* 6.4. In units of TN* , the large N slips versus temperature for z limits for the IC transition temperatures are 0.651T 0 /TN * 5 and 6.4. Inset is the IC N * and transition temperature T 0.834T IC /TN * thick solid curve versus energy N * for z0 /TN* 5 and 6.4, respectively. So paradoxi- mismatch z0 /T cally, the IC transition temperature is larger for z N * for large N. Also plotted in the inset is the NeŽel 0 6.4TN * temperature TN /TN* thin solid curve of bulk Cr. The triple point is than for z0 5TN* . denoted by a dot. PRB 59 COLLINEAR SPIN-DENSITY-WAVE ORDERING IN . . . 13 855 temperature dependence of s2 is completely due to the tem- g/T perature dependence of the bulk SDW free energy. N * falls off like /N above TN. Nonetheless, the IC phase boundary for N does not depend on .17 The dependence of the SDW amplitude and wave vector Although T on thickness was discussed in Refs. 16 and 17. Each time the IC( ) is independent of , the critical thick- ness N number of nodes increases by one, the SDW amplitude de- 1(T 0) below which the C SDW is stable at T 0 strongly depends on the coupling strength. As shown in Ref. creases discontinuously. With increasing N, both g and 17 for z approach their bulk values and the oscillations about the bulk 0 /TN * 5, N1(0) increases from 16 to 53 as in- creases from 1 to 6. For the value of 3 used in Fig. 3 a , values become narrower. As N increases between phase N slips, the number of SDW nodes remains the same but the 1(0) 28. SDW stretches to maximize the antiferromagnetic coupling IV. MODEL II: SDW NODES AT THE Fe-Cr INTERFACES at the interfaces. When n increases by 1, the SDW period suddenly contracts with the addition of another node. We Neutron-scattering measurements on multilayers have not refer the reader to the references above for a more detailed followed the same pattern as the NIST measurements on discussion of this behavior. wedges. Although Fig. 3 a for z0 5TN* predicts an IC tran- These results indicate that the C phase is stable for small sition temperature of about 1.7TN 530 K, the I phase is N or large temperatures. This may be easily understood in observed to disappear above about 300 K in both epitaxially terms of the competition between the interface coupling, grown3 and sputtered4 Fe/Cr multilayers. As the temperature which maximizes the SDW amplitude at the boundaries, and increases for a fixed N, multilayers do not exhibit the pre- the intrinsic antiferromagnetism of the spacer, which favors dicted series of I-to-I phase transitions with decreasing num- the bulk values of the SDW amplitude and wave vector. bers of nodes from n m to n m 1 on up to n 0. The While the SDW gains energy 2A sSFeg cos F or different behavior of Fe/Cr wedges and multilayers may be 2A sSFeg sin AF due to the interactions at interfaces, it ascribed to the interfacial disorder in multilayers. Noncol- forfeits energy F(g, ) F(gbulk , linear SDW ordering13,14 may be produced by the well- bulk ) a3(N 1)/2 due to the changes in the order parameters of the spacer. separated interfacial steps in epitaxially grown multilayers. When 0, cos 1 and sin 0 for odd N, while Nearby atomic steps in sputtered multilayers may establish cos 0 and sin 1 for even N. Hence, the interactions SDW nodes near the interfaces,4 in which case neighboring at the interfaces with F AF moments prefer a C I SDW in Fe moments are not magnetically coupled. a spacer with odd N and an I C SDW in a spacer with even If the SDW nodes lie precisely at the Fe-Cr interfaces, N. If F 0, the interface coupling always favors a C SDW then is restricted to the values n (n 1)/(N 1), where state with cos or sin equal to one. n 2 is the number of SDW nodes including the two at the So for odd N, the C SDW is stabilized with F coupling at interfaces. We evaluate n by minimizing the nesting free a high enough temperature that the bulk free energy energy F(g, n ) with respect to both g and n. Like model I, F(g, 0) is sufficiently small. For even N, the C SDW this model also assumes that the SDW is rigid. Hence, the is favored with AF coupling at a sufficiently high tempera- SDW amplitude and wave vector do not depend on the loca- ture. The same considerations apply for small N: the C SDW tion z inside the spacer. is favored with F coupling for odd N and AF coupling for Because the C SDW does not contain any nodes, the C even N. phase is never stabilized by model II. In Fig. 6 a , the NeŽel When N is large, the IC transition temperature is remark- temperature TN and phase boundaries are normalized by the ably independent of the Fe-Cr coupling constant . In Ap- bulk NeŽel temperature TN,bulk , which is evaluated by allow- pendix B, we prove that TIC(N ) is implicitly given by ing to be a continuous parameter. As in the previous section, we take z0 5TN* and 0.043. So the bulk value of at T N,bulk is 0.037, corresponding to a node-to-node dis- Re 1 0, 8 tance of 27 ML. These parameters are different than the ones n 0 X3n used in Ref. 18. For T/TN,bulk 0.2, the SDW order param- eter and wave vector are plotted versus N in Fig. 7. where Xn n 1/2 iz0/8 TIC( ). As a consequence, As N decreases below 41 ML, increases and the SDW TIC( ) depends only on the energy mismatch z0 and is in- period decreases as a half wavelength of the SDW tries to dependent of . Both TIC( ) and the bulk NeŽel temperature squeeze into the Cr spacer. When N 27, is larger than its TN are plotted versus the energy mismatch z0 /TN* in the inset bulk value so that the SDW period is smaller than in bulk. to Fig. 5. Precisely at the triple point z0 4.29TN* where the For N 20 ML's, a half wavelength of the SDW cannot CI and paramagnetic phase boundaries of bulk Cr meet, squeeze into the Cr spacer without a prohibitive cost in free TIC( ) TN . With increasing z0 , TIC( ) increases but TN energy and the NeŽel temperature vanishes. As N increases, decreases. So the IC transition temperature of an Fe/Cr the SDW goes through cycles of expansions followed by multilayer always exceeds the NeŽel temperature of bulk Cr. sudden contractions with the addition of another node to the Recall from our previous discussion that lattice strain en- SDW. The SDW amplitude and wave vector plotted in Figs. hances the effective value for the energy mismatch z0. 7 a and 7 b are correlated: decreases as g grows larger. The independence of TIC from in the large N limit In other words, the cyclical expansion and contraction of the raises an intriguing question: How is bulk behavior recov- SDW follow the same pattern as found for model I in Ref. ered as 0? Stabilized by the interfacial coupling energy, 16. Only now these cycles also produce a seesaw pattern in a remnant SDW survives above the bulk ordering tempera- TN . The NeŽel temperature reaches a maximum whenever n ture TN . As N increases, the bulk free energy dominates and passes near its bulk value of 0.037. 13 856 R. S. FISHMAN AND ZHU-PEI SHI PRB 59 6 a only allows a single I-to-I phase transition from n m to n m 1 before the SDW disappears above TN . For narrow ranges of thicknesses, phase transitions are allowed between SDW's with different numbers of nodes as a function of temperature. Such a phase transition occurs for N 122, when the SDW transforms from n 6 to n 5 with increas- ing temperature. As in model I, the SDW amplitude jumps up when n decreases by one.17 Transition between SDW's with n differing by one in CrMn/Cr multilayers may be eas- ily observed because the neutron-scattering profiles of SDW's with odd and even n are quite different.4 Very re- cently, Fullerton and Robertson15 observed a phase transition from n 5 to n 4 with increasing temperature in a CrMn/Cr multilayer with L 200 Ć . It is clear that forcing the SDW nodes to lie at the Fe-Cr interfaces generates a seesaw pattern in TN(N), g, and . The shift in along one of the seesaws with fixed n may be difficult to observe due to limitations in experimental reso- lution roughly 10% and the effects of surface roughness, which averages over several values of N. For example, the predicted change in from N 68 to 93 at T/TN,bulk 0.2 corresponds to a variation in the SDW period from 22 to 31 lattice constants, all with n 4. The average SDW period, FIG. 6. Model II: NeŽel temperature thick solid curve and phase however, is very close to the bulk value of 27 lattice con- boundaries thin solid curve versus N for z0 /TN* 5. The number stants. of SDW nodes is given by n. In a SDW nodes are fixed at each For a bulk SDW with (TN,bulk) 1/27, Fig. 6 a predicts interface; in b the nodes can shift by 3 ML from each interface. that the I phase becomes unstable below 20 ML. But the Notice that T measured critical thickness41,4 of 30 ML is much larger. This N , g, and all approach their bulk values as N . With increasing N, the oscillations about the bulk could be caused by the displacement of the SDW nodes values become narrower and the seesaw patterns become away from the interfaces. Surface roughness may be ex- flatter. For large N, the maxima in T pected to suppress the SDW ordering within a pair coherence N are separated by 1/ length 0 5 ML from the interfaces. If the region within 5 bulk (TN,bulk) 27 ML. Unlike the more complex phase diagrams of Fig. 3, Fig. ML from each interface is paramagnetic, then the observed critical thickness of 30 ML would correspond to a ``true'' critical thickness of 30 10 20 ML, equal to the predicted value. For N 30 or temperatures greater than 300 K, the residual antiferromagnetic coupling at the Fe-Cr interfaces may be sufficient to stabilize a C SDW in some regions of the Cr spacer, as found by Fullerton et al.4 Even if the first 5 ML from the Fe-Cr interface are para- magnetic, however, the NeŽel temperature would still be ex- pected to contain a deep minimum at 39 10 49 ML or 74 Ć. None has been observed. This sudden drop in the NeŽel temperature would be softened if the positions of the SDW nodes vary within a few ML from each Fe-Cr interface. For example, imagine that the SDW nodes can shift by 3 ML from each interface. Then for a given spacer thickness N, the SDW amplitude and wave vector would be chosen among seven possible SDW's with lengths N between N and N 6. The SDW with the smallest free energy F(N 1) would determine the order parameters of the multilayer. So the SDW would pay a price in condensation energy in order to move its nodes away from the interfaces. This program was implemented in Fig. 6 b . The first and last SDW nodes lie a minimum distance of N 6 ML apart and a maximum distance of N ML apart. As shown, this freedom allows the NeŽel temperature to linger close to its FIG. 7. Model II: a SDW order parameter and b wave vector bulk value. Compared to the NeŽel temperature of Fig. 6 a , versus N for T/T the size of the oscillations about T N,bulk 0.2 for the same parameters as in Fig. 6, N,bulk are smaller and the with nodes fixed at each interface. The bulk values are indicated by first dip in the NeŽel temperature is much weaker. The dis- the dashed lines. placement of the SDW nodes from the interfaces is largest PRB 59 COLLINEAR SPIN-DENSITY-WAVE ORDERING IN . . . 13 857 for spacer thicknesses with a depressed NeŽel temperature ment with model II, the observed NeŽel temperature41,4 when N N. In addition, the phase boundaries between TN(N) of sputtered Fe/Cr multilayers shows no sign of the SDW's with neighboring n are more slanted than in Fig. dips and peaks associated with the cyclical expansion and 6 a . contraction of the SDW. The measurements by Fullerton et al.4 on sputtered mul- On the other hand, model II correctly predicts that the tilayers provide some evidence for this behavior. Fits to their SDW may undergo a transition from n m to n m 1 data reveal that the SDW nodes lie very close to the Fe-Cr nodes with increasing temperature before entering the para- interfaces except for N 35, corresponding to a SDW with magnetic state. This behavior, which was recently observed n 2 near the predicted depression in TN when N N. For in CrMn/Cr multilayers,15 is quite different than the series of this SDW, Fullerton et al. find that the antinodes rather than phase transitions from n m to n m 1 to n m 2 and on the nodes lie close to the Fe-Cr interfaces. However, their up to n 0 predicted by model I. A refined version of model data for N 35 can be equally well described by a SDW with II, which no longer ties the SDW nodes to the interfaces, nodes displaced 7 ML from each interface. produces a smoother NeŽel temperature TN(N) and may ex- plain most properties of sputtered multilayers. V. DISCUSSION AND CONCLUSION But even such a refined model cannot stabilize the H SDW observed by Schreyer et al.3 in epitaxially grown mul- This paper has presented two very different models for tilayers. For small thicknesses and low temperatures, a H the formation of a SDW in an Fe/Cr/Fe trilayer. Within SDW is believed to couple adjacent Fe moments at a 90° model I, the Fe-Cr interfacial interactions are assumed to be angle. The Fe moment returns to its original orientation ev- antiferromagnetic. As a result, the SDW antinodes lie near ery other Fe layer.12 Since a H SDW does not occur in bulk the interfaces. Even above the bulk NeŽel temperature, the Cr, it must be stabilized by the interfacial energy. In the interfacial interaction stabilizes a SDW within the Cr spacer. presence of well-separated interfacial steps, a H SDW is Surprisingly, this model predicts that the IC transition tem- found to have a lower free energy above TN than either the I perature is always larger than the bulk NeŽel temperature. By SDW predicted by model I or the P phase predicted by contrast, model II assumes that the SDW nodes lie precisely model II.14 at the Fe-Cr interfaces, although this requirement is some- To summarize, we have evaluated the phase diagram of what relaxed in Fig. 6 b . As a result, both the NeŽel tempera- Fe/Cr trilayers using two different methods. While model I ture and SDW wave vector undergo oscillations with in- assumes that the magnetic interactions at the Fe-Cr interfaces creasing spacer thickness. are antiferromagnetic, model II assumes that the SDW nodes Measurements by Unguris et al.5 on Fe/Cr/Fe wedges lie at the interfaces. The results of model I are in good agree- closely follow the scenario depicted in Fig. 3 b for model I. ment with measurements on Fe/Cr wedges, where interfacial In terms of the NeŽel temperature TN 0.384TN* 310 K of disorder is minimized. Sputtered multilayers may be ad- unstressed Cr, the IC transition of the stressed film is given equately described by a refined version of model II, which by 2.1TN 650 K. Although the measurements of Ref. 5 allows the SDW nodes to shift away from the interfaces with only go up to about 550 K, 650 K is just slightly larger than some cost in condensation energy. However, neither model the IC transition temperature, which can be extrapolated satisfactorily describes the properties of epitaxially-grown from the NIST data. Unguris et al. observed a very uniform Fe/Cr multilayers. pattern of phase slips with the same si depending only on We would like to thank Dr. Eric Fullerton, Dr. Daniel temperature. The values of 3 and z0 6.4TN* used in Fig. Pierce, Dr. Lee Robertson, Professor Andreas Schreyer, and 3 b were chosen to give the smallest possible variation of si Dr. Mark Stiles for helpful discussions. This research was and a bulk value of 1/ 19 at TN 227 K, slightly smaller supported by Oak Ridge National Laboratory managed by than the observed phase slip distance of 20 ML at 300 K. Lockheed Martin Energy Research Corp. for the U.S. De- For T 300 K, the first predicted phase slip at N1 partment of Energy under Contract No. DE-AC05- 13 ML in Fig. 3 b occurs earlier than the first observed5 96OR22464. phase slip at 24 ML in an Fe/Cr/Fe wedge. Accounting for the intermixing of Fe and Cr within the first 5 ML of the wedge,42 an initial phase slip at 5 19 24 ML can be ob- APPENDIX A: BEHAVIOR OF Jcoup tained using a somewhat larger coupling constant of 6. Intermixing within the first few ML's is also necessary to This appendix uses model I to evaluate the behavior of explain the reversal5 of the expected F and AF couplings. Jcoup EAF EF for large N. For notational convenience, we Probably due to the restricted temperature range of their set ehTN* 1 and TN* 1 so that both g and F are dimen- measurements and the small size of their wedge (N sionless. Minimizing the AF and F energies of Eqs. 5 and 80 ML), Unguris et al. did not observe the phase slip pat- 6 with respect to g and , we find tern to become nonuniform at high temperatures. Doping the Cr spacer with a small concentration of Mn impurities (x F g, ,T 0.3%) could lower the IC transition temperature below 2 cos g N 1 0 F , A1 550 K and permit this behavior to be observed. Some evidence suggests that Fe/Cr multilayers cannot be described by either model. The disappearance of the I phase F g, ,T above about 300 K Refs. 3 and 41 rules out model I, which 2 sin g N 1 0 AF , predicts TIC to be substantially larger than TN . In disagree- A2 13 858 R. S. FISHMAN AND ZHU-PEI SHI PRB 59 F g, ,T g sin sgn cos 0 F , cos sgn sin AF Ny 1 2 AF . A14 A3 Then using Eqs. A5 ­ A8 and assuming x 2, we find F g, ,T gAF 1/N, AF 1/N, gF 1/Nx 1, and F 1/Nx 1 g cos sgn sin 0 AF , with y 2. So long as x 2, the leading order term in J coup is given by A4 where (N 1)(1 )/2. 1 As N , both g and approach their bulk values. To Jcoup 2 gF . A15 Nx 1 first order in g g gbulk and bulk , the above relations become It only remains to evaluate the exponent x. Near the envelope maximum, the unstable F energy with 2 cos N 1 F11 g F12 0 F , N ML's and n 1 nodes is nearly equal to the unstable F A5 energy with N 2 ML and n 1 nodes. Expanding Eq. 5 in powers of gF , F , and F /Nx 1, we obtain 2 sin N 1 F11 g F12 0 AF , A6 2 g (N) 2 bulk gF 1 F 2/ 8N2(x 1) Fbulk N 1 g 1 bulksin sgn cos F12 g F22 0 F , (N)2 (N) A7 12 F11 gF F12 gF F (N) 2 F22 F (N)2 gbulkcos sgn sin F12 g F22 0 AF , N 1 A8 2 g (N 2) where F bulk gF 11 2 F/ g2, F12 2 F/ g , and F22 2 F/ 2 are evaluated at g 2 bulk and 1 2/ 8 N 2 2(x 1) F bulk . For large N, a F bulk N 1 very small change in the SDW wave vector is required to optimize the interfacial coupling with cos 1 F and 1 (N 2)2 F (N 2) sin 1 AF . So sin 0 and cos 0 in these two 2 F11 gF 12 gF F (N 2) cases. 1 To obtain the behavior of Jcoup as N , we examine the coupling near a local maximum in J 2 F22 F (N 2)2 N 1 , A16 coup(N), roughly midway between phase slips. We assume that the stable coupling is where F AF with n SDW nodes and that the unstable coupling is F bulk is the bulk free energy evaluated at gbulk and (N) (N)2 with n 1 nodes. For example, the thickness N 85 in Fig. 2 bulk . Since gF F (N) , we conclude that gF F satisfies this condition with n 3. Consequently, the wave bulk /N. So below TN , x 3/2 and Jcoup 1/ N. Above T vectors of the SDW's can be written N , Fbulk 0 and gbulk 0. Then Eq. A16 can be used to show that gF 1/N with x y 2. To order 1/N, the F and AF order parameters are identical with n 1 F F N 1 , A9 Nx 2 F g 22 AF gF N , A17 F 2 11F22 F n 12 AF AF N 1 , A10 Ny 2 F 12 AF F . A18 2 where 1 x y below T N F N . Since the AF coupling is assumed 11F22 F12 to be stable, n is odd when N is odd and even when N is Therefore, J even. So as expected, cos 1 for F coupling and sin coup vanishes to order 1/N and its leading-order behavior is given by J 1 for AF coupling. coup 1/N2. For large N, it is easy to show that APPENDIX B: DERIVATION OF TIC 2 2 At the IC phase boundary of model I, a C SDW with n cos 1 F N2(x 1) 8 F , A11 0 and order parameter g0 has the same free energy as an I SDW with n 1 and order parameters g1 and 1 . As N 2 2 , g0 , g1, and 1 all tend to zero. Therefore, the free sin 1 AF energy F(g, ) may be expanded in powers of g and u N2(y 1) 8 AF , A12 z0 /8 T : 1 sin sgn cos F F g, g2 ln T S g4S Nx 1 2 F , A13 1 u2S3 8 2T2 3 , B1 PRB 59 COLLINEAR SPIN-DENSITY-WAVE ORDERING IN . . . 13 859 where 2sin z0 2 2S 1 sgn cos 1 2g1 1 S3 2u1 5 1 8 T S 1 Re 1 , B2 n 0 Xn n 1/2 1 g3 z0 2 1 8 T 1 S5 . 2T2 S 2m 1 3 Re 1 , B3 B9 n 0 X2m 1 n The last relation implies that 1 z X 0 1 1 n n 2 i 8 T . B4 1 N 1 S3 ***, B10 N 1 3 If N is even, then the stable C phase above TIC is AF while the stable I phase below T which agrees with Eq. A9 when x 3 and the ferromag- IC is F. Their energies are netically coupled SDW has n 1 1 node. Hence, sin 1 2 2 S 2 3 /N2 and cos 1 1 S3/N4. E Using Eqs. B7 and B8 , it is simple to show that a3 AF F g0,0 N 1 g0 , B5 1 2 2 g 2 3 3 1 g0 ln T S1 u1S3g1 S3 g1 g0 . E 2 2T2 a3 F F g1 , 1 N 1 g1 cos 1 , B6 B11 where Consequently, g 1 (N 1)(1 1 )/2. 1 g0 is of order S3 /N3. But to order 1/N4, The minimization conditions for E the equality E F and EAF with respect F EAF requires to g and are given by 2 2 g2 2 2 0 ln T S1 u1S3g1 . 1 N 1 g1 g0 g1 g 3 0 ln T S1 S B12 2 2T2 3g0 N 1 , B7 Substituting Eqs. B7 and B8 , we conclude that g1 g0 is also of order 1/N4. So S 1 3 must be of order 1/N and vanish as g 2 3 N . Therefore, the condition for the IC transition tem- 1 ln T S1 u1S3 S 2 2T2 3g1 N 1 cos 1 , perature in the limit of large N is given by Eq. 8 , indepen- B8 dent of . 1 M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. 12 S. Adenwalla, G. P. Felcher, E. E. Fullerton, and S. D. Bader, Petroff, P. Eitenne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. 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