Journal of Magnetism and Magnetic Materials 198}199 (1999) 391}395 Invited paper The role of Cr antiferromagnetism on interlayer magnetic coupling in Fe/Cr multilayered systems Zhu-Pei Shi *, R.S. Fishman R & D Division, Read-Rite Corporation, 44100 Osgood Road, Fremont, CA 94539, USA Solid State Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6032, USA Abstract Many experiments have veri"ed the presence of a spin-density wave (SDW) within the Cr spacer of Fe/Cr multilayers and wedges. We review the recently proposed interlayer magnetic coupling mediated by a SDW. Unlike previously proposed mechanisms, this magnetic coupling is strongly temperature-dependent. Depending on the temperature and the number N of Cr monolayers (ML), the SDW may be either commensurate (C) or incommensurate (I) with the BCC Cr lattice. 1999 Elsevier Science B.V. All rights reserved. Keywords: Multilayers; RKKY coupling mechanism; SDW order parameter 1. Introduction polarization of the intervening conduction electrons and the associated magnetic scattering of conduction elec- There is a great deal of current interest in the proper- trons with their moments in the magnetic layer. But even ties of magnetic multilayered structures consisting of after intensive studies of interlayer magnetic coupling alternating thin ferromagnetic and non-ferromagnetic [5], Fe/Cr heterostructures have continued to surprise metallic layers. The antiferromagnetic coupling "rst ob- scientists with their unique properties. served between the Fe layers separated by a thin Cr Due to the competition between the spin-density wave spacer [1] and the giant magnetoresistance (GMR) found (SDW) ordering in the Cr spacer [6,7] and the Fe}Cr in this system [2] have led to intense theoretical and interactions at the interfaces, Fe/Cr multilayers and experimental studies of an extraordinary range of struc- wedges have provided new insight into the physics of tures over the past few years. In metallic magnetic layers, transition-metal magnets. Depending on temperature the magnitude of the GMR e!ect oscillates as the thick- and spacer thickness, the SDW may be either commen- ness of the non-ferromagnetic spacer layers [3]. This surate or incommensurate with the BCC Cr lattice. For oscillatory behavior can be explained by applying the Fe/Cr multilayers [8], the C SDW phase is stabilized general exchange theory of Ruderman}Kittel}Kasuya} when the number of monolayers N inside the Cr spacer is Yosida (RKKY) to the problem of the interlayer coupling less than 30 or when the temperature exceeds the Neel [4]. This theory provides a physically transparent ex- temperature 310 K of pure Cr. By contrast, SEMPA planation of the measured oscillatory periods in terms of measurements [9,10] on Fe/Cr/Fe wedges indicate that the topological properties of the spacer Fermi surface. It the I SDW phase is stable for N'23 ML and up to at is now well accepted that interlayer coupling between least 550 K. As the spacer thickness increases, the Fe}Fe separated magnetic layers occurs because of the spin coupling alternates between ferromagnetic (F) and anti- ferromagnet (AF) with phase slips every 20 ML at room * Corresponding author. Tel.: #1-510-6837246; fax: #1-510- temperature. Here we review a new type of exchange 6836075. mechanism in which the coupling between ferromagnetic E-mail address: zhupei.shi@readrite.com (Z.P. Shi) layers is mediated by the SDW of the Cr spacer [11,12]. 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 8 ) 0 1 1 4 0 - 8 392 Z.-P. Shi, R.S. Fishman / Journal of Magnetism and Magnetic Materials 198}199 (1999) 391}395 The organization of the paper is as follows. Section 2 For simplicity, we assume that the Fe moments are provides the general formalism of interlayer magnetic either F or AF aligned with S'$ "S''$ or S'$ "!S''$ , coupling mediated by a SDW. As shown in Section 3, this both parallel to the interface. The SDW will then be coupling is strongly temperature-dependent. Section 4 transversely polarized with respect to the ordering wave describes the I-to-C SDW transition in the Cr spacers. vectors along the z-axis. With antiferromagnetic interac- Section 5 illustrates the stretching and relaxing cycle of tions at the interfaces [16], the total energy of the multi- the SDW as the spacer thickness is increased. Possible layer or wedge for an interfacial area of a and spacer e!ect of interface roughness on the SDW ordering and width (N!1)a/2 may be written as [11,12] Neel temperature are addressed in Section 6. Section 7 contains some "nal remarks. E"AS'$ )S(1)#AS''$ )S(N)# Fa(N!1)/2, (2) which assumes that the SDW is rigid with order para- 2. Interlayer coupling mediated by spin-density waves meters g and  independent of z. Here F is the free energy of the SDW phases in Cr spacer. It is a function of Interlayer magnetic coupling across noble metal space- g, Y, ¹, and energy mismatch  between the electron and rs can be well understood in terms of the RKKY coupling hole Fermi surfaces [11,12]. mechanism and the oscillation periods can be accurately After "xing the magnetic con"gurations of the Fe determined by ab initio calculations [5]. The interlayer layers, the SDW order parameters g and Y as well as the coupling in Fe/Cr/Fe multilayers, however, show some arbitrary phase are chosen to minimize the energy E in unusual features such as an antiferromagnetic bias at Eq. (2). The corresponding F and AF energies of the ultra thin Cr spacer and strong temperature dependence trilayer are [13]. Shi et al. [14,15] proposed a coupling mechanism E based on the s}d mixing interactions at Fe/Cr interfaces. $"!2A 1gS$ "cos "# F(g, , ¹, )a(N!1)/2, The RKKY oscillatory terms are superimposed upon an (3) antiferromagnetic background in order to interpret the E interlayer magnetic coupling in Fe/Cr multilayers. How- $"!2A 1gS$ "sin "# F(g, , ¹, )a(N!1)/2, ever, the SDW ordering observed in Fe/Cr multilayers (4) [8}10] and the strongly temperature-dependent inter- layer coupling challenge all RKKY-like coupling mecha- where "( /2)(N!1)(1# ). The SDW order para- nisms and zero-temperature ab initio calculations. The meter is restricted to values below the bulk maximum of SDW-mediated coupling [11,12] reviewed here provides g "1.246 ¹H,, which is achieved in the C SDW phase a new theoretical framework to address the unusual of a bulk Cr alloy at ¹"0 (¹H,&80 meV). magnetic properties observed in Fe/Cr systems. Because the nesting free energy F is proportional to The SDW instability in Cr alloys is produced by the ¹H , , the total free energy E depends only on the Coulomb attraction between electrons and holes on dimensionless constant nearly perfectly nested electron and hole Fermi surfaces, A both roughly octahedral in shape [4]. Because the elec- " 1S$ , (5) tron Fermi surface is slightly smaller than the hole Fermi (<"N) ¹H, surface, there are two di!erent nesting wave vectors which represents the average coupling strength between Q! which translate four faces of one Fermi surface onto Fe and Cr at the interfaces. four faces of the other. The nesting wavevectors may be Once E written as Q $ and E$ are found, the magnetic coupling !"2 /a(1$ ), where &0.05 is a measure J of the size di!erence between the electron and hole Fermi "E$!E$ may be evaluated as a function of tem- perature ¹ and spacer thickness N. surfaces, and a is the BCC lattice constant. In the I phase of the SDW, the condensate contains two types of elec- tron-hole pairs with momenta Q!"2 /a(1$ ). When 3. Strongly temperature-dependent magnetic coupling "0, the SDW is commensurate with underlying lat- tice. If the SDW wave vector lies along the z-direction Taking "1, normal to the multilayer interface, the spin at each /¹H,"5, and ¹"0.5¹, or 1.2¹,, we plot J atomic layer can be written  as a function of spacer thickness in Fig. 1. As expected, J  oscillates between F('0) and AF ((0) values with a short 2 ML period. As shown in Fig. 1a, S(z)"m( 1g(!1)X?cos 2 z! a , (1) above the Neel temperature, the magnetic coupling falls o! rapidly with the size of the spacer. For large N, we where 1 is a constant, is an arbitrary phase, g is an show in Ref. [17] that J  decreases like 1/N. So above order parameter, and 1g"0.6 for bulk Cr at zero ¹,, the RKKY and nesting contributions to the magnetic temperature. coupling cannot be distinguished by their dependence on Z.-P. Shi, R.S. Fishman / Journal of Magnetism and Magnetic Materials 198}199 (1999) 391}395 393 Fig. 2. Magnetic energies as functions of temperature for "1 and N"25 ML. The solid dot denotes the I-to-C transition for a SDW with F coupling. Fig. 1. Bilinear magnetic coupling as a function of spacer thick- ness for "1 and (a) ¹"0.5¹, or (b) ¹"1.2¹,. N. Below the Neel temperature, the magnetic coupling decays slowly with the size of the spacer as shown in Fig. 3. SDW wave vector parameter  as a function of spacer Fig. 1b. J thickness for ¹"0.5¹  falls o! like 1/N  below ¹, [17]. This , and "2.5. The dotted line is for decay is slower than indicated by either a Kohn-anomaly F arranged magnetic layers and the solid line is for the stable analysis (J magnetic con"guration (F or AF) with the lowest energy. &1/N) [18] or density-functional total energy calculations (J &1/N  at ¹"0 [19]). Fixing the thickness of the spacer, we evaluate the magnetic coupling as a function of temperature. For Cr spacers. This prediction [11,12] has been con"rmed N"25 and "1 and ¹"0.5¹,, J  is plotted as the by recent neutron scattering measurements of Fe/Cr solid curve in Fig. 2. Notice that antiferromagnetic coup- multilayers [22]. ling at low temperature decreases by a factor of 2 as the The proximity Fe layers will modify the SDW order temperature increases to ¹,, and becomes weakly parameters of the Cr spacer. In the absence of interface ferromagnetic coupling above 1.41¹,. Because the coupling ( "0), the bulk values of the SDW amplitude temperature is much less than the Fermi energy, the g and wave vector  are evaluated by minimizing F conventional RKKY coupling mediated by spin-polariz- [11,12]. If ¹"0.5¹,, then g  "0.647¹H, and ed electronic states is only weakly temperature-depen-   "0.0378. When '0, the SDW order parameter dent [20]. At least qualitatively, our model explains the g always exceeds its bulk value. After "xing the magnetic rapid decrease in the coupling strength of Fe/Cr multi- con"gurations of the proximity magnetic layers, we layers above ¹, [13] and the disappearance of the AF found that both order parameters oscillate as a function coupling above 320 K in Co/Mn multilayers [21]. of the spacer thickness with a 2 ML period and approach their bulk values as N goes to in"nity. For F arranged Fe layers with ¹"0.5¹, and "2.5,  is given by the 4. Incommensurate to commensurate SDW transitions dotted line in Fig. 3. For AF arranged magnetic layers, in Cr spacers the oscillation patterns of  plotted in Fig. 3 are shifted to the right by 1 ML (with the same shift for the SDW For N"25, "1, and ¹"0.5¹,, the energies of order parameter g). The order parameter  correspond- F and AF magnetic con"gurations are individually plot- ing to the lowest energy magnetic con"gurations (F or ted as dashed and dotted curves in Fig. 2. The solid dot AF) of the proximity Fe layers is also plotted in Fig. 3, on the F curve indicates an I-to-C SDW transition in the a thick solid curve. So for N between 24 and 39, the 394 Z.-P. Shi, R.S. Fishman / Journal of Magnetism and Magnetic Materials 198}199 (1999) 391}395 F (AF) con"gurations are stable for even (odd) N while 6. Oscillatory Neel temperature in Fe/Cr systems for N between 40 and 61, F (AF) con"gurations are stable for odd (even) N. The order parameter g has the same Sputtered Fe/Cr multilayers may have thickness #uc- behavior as that of . The steps on the stable line of tuations or atomic steps at the interfaces. Such #uctu- the order parameters at the spacer thickness of 23, 39, ations may establish the SDW nodes near the interfaces and 61 ML correspond to the nodes of J  [11,12]. The [8], in which case the Fe moments within the multilayers "rst step on the &stable line' in Fig. 3 agrees with recent are not magnetically coupled. Assuming that the SDW neutron scattering measurements [8], where a C-to-I nodes lie precisely at the Fe}Cr interfaces,  is restricted SDW transition between 21 and 35 ML is observed. The to the values L"(n!1)/(N!1), where n*2 is the other steps on the &stable line' stand for I-to-I SDW phase number of SDW nodes including the two at the interfaces transitions which are clearly indicated by the NIST [23]. We evaluate n by minimizing the nesting free energy measurements [9,10]. F(g, L) with respect to both g and n. Because the C SDW does not contain any nodes, the C SDW phase is never stabilized in this case. In Fig. 5, 5. Stretching and relaxing cycle of SDW in Cr spacers the Neel temperature ¹, and phase boundaries are normalized by the bulk Neel temperature ¹,  , which While the SDW order parameters g and  jump be- is evaluated by allowing  to be a continuous para- tween lower and higher values with a period of 2 ML, meter. Here we take "5¹H, and "0.043. So the their oscillation patterns shift at spacer thicknesses of 34, bulk value of  at ¹,  is 0.037, corresponding to 51, and 74 ML, as shown in Fig. 3. This striking behavior a node-to-node distance of 27 ML's. For ¹/¹,  "0.2, can be explained by the competition between the inter- the SDW order parameter  are plotted versus N in face coupling, which maximizes the SDW amplitude at Fig. 6. As N decreases below 41 ML,  increases and the boundaries, and the intrinsic antiferromagnetism of the SDW period decreases as a half-wavelength of the Cr spacer, which favors the bulk values of the SDW the SDW tries to squeeze into the Cr spacer. When amplitude and wave vector. As the Cr spacer thickness N(27,  is larger than its bulk value so that the increases for odd or even N in Fig. 3, the SDW "rst SDW period is smaller than in bulk. For N(20 ML, stretches to optimize the interface coupling and then a half-wavelength of the SDW cannot squeeze into suddenly relaxes to lower the bulk free energy. For the Cr spacer without a prohibitive cost in free energy example, the SDW with N"34 ML drawn as the solid and the Neel temperature drops to zero. As N increases, curve in Fig. 4 contains a single node. As even N in- the SDW goes through cycles of expansions followed creases, the SDW stretches until it attains the pro"le of by sudden contractions with the addition of another the dotted curve for N"50 ML. With the addition of node to the SDW. As  plotted in Fig. 6 decreases, two more ML's, two new nodes appear in the SDW the amplitude g grows. In other words, the cyclical (dashed curve) and the SDW amplitude drops towards its expansion and contraction of the SDW follows the bulk value. As N increases further, the cycle of stretching same pattern as depicted by Fig. 3. Only now these cycles and relaxing repeats with a period close to the also produce an oscillatory pattern in ¹,. Whenever wavelength &40 ML of the bulk SDW. For odd N, the  passes near the bulk value of 0.037, ¹, reaches a same cycle is o!set by about 20 ML. So the jumps in maximum. the SDW order parameters at 34, 51, and 74 ML are also separated by about 20 ML. Fig. 4. SDW pro"les in the spacer for N"34 (solid), 50 (dotted), Fig. 5. Neel temperature versus N. The number of SDW nodes and 52 (dashed) for the same parameters as in Fig. 3. is given by n. Z.-P. Shi, R.S. Fishman / Journal of Magnetism and Magnetic Materials 198}199 (1999) 391}395 395 a magnetic phase transition from an I to a C SDW below a critical spacer thickness or at high temperatures. These theoretical results agree with recent polarized neutron scattering measurements. References [1] P. Grunberg et al., Phys. Rev. Lett. 57 (1986) 2442. [2] M.N. Baibich et al., Phys. Rev. Lett. 61 (1988) 2472. [3] S.S.P. Parkin, N. More, K.P. Roche, Phys. Rev. Lett. 67 (1991) 3598. Fig. 6. SDW wave vector versus N for ¹/¹,  "0.2. The bulk [4] P. Bruno, C. Chappert, Phys. Rev. Lett. 67 (1991) 1602. values are indicated by the dashed lines. [5] For a review, see for an example, B. Heinrich, J.F. Coch- ran, Adv. Phys. 42 (1993) 523. [6] E. Fawcett, Rev. Mod. Phys. 60 (1998) 209. The measurements by Fullerton et al. [8] provide [7] E. Fawcett et al., Rev. Mod. Phys. 66 (1994) 25. some evidence for this behavior. Fits to their data reveal [8] E.E. Fullerton, S.D. Bader, J.L. Robertson, Phys. Rev. that the SDW nodes lie very close to the Fe}Cr interfaces Lett. 77 (1996) 1382. except for N"35, corresponding to a SDW with n"2 [9] J. Unguris, R.J. Celotta, D.T. Pierce, Phys. Rev. Lett. 67 near the predicted depression in ¹ (1991) 140. ,. For this SDW, Fullerton et al. "nd that the antinodes rather than the [10] J. Unguris, R.J. Celotta, D.T. Pierce, Phys. Rev. Lett. 69 nodes lie close to the Fe}Cr interfaces. However, their (1992) 1125. [11] Z.P. Shi, R.S. Fishman, Phys. Rev. Lett. 78 (1997) 1351. data for N"35 can be equally well described by a SDW [12] R.S. Fishman, Z.P. Shi, J. Phys.: Condens Matter 10 (1998) with nodes displaced 7 ML from each interface. L277. [13] E.E. Fullerton et al., Scr. Metall. Mater. 33 (1995) 1637. [14] Z.P. Shi, P.M. Levy, J.L. Fry, Phys. Rev. Lett. 69 (1992) 7. Conclusions 3678. [15] Z.P. Shi, P.M. Levy, J.L. Fry, Europhys. Lett. 26 (1994) Based on a simple model with antiferromagnetic inter- 473. actions at the Cr}Fe interfaces, the role of Cr antifer- [16] T.G. Walker et al., Phys. Rev. Lett. 69 (1992) 1121. romagnetism on the exchange coupling and magnetic [17] R.S. Fishman, Zhu-Pei Shi, Phys. Rev. B, in press. phase diagram of Fe/Cr multilayers has been reviewed. [18] D.D. Koelling, Phys. Rev. B 50 (1994) 273. [19] M. van Schilfgaarde et al., Phys. Rev. Lett. 74 (1995) 4063. A SDW mediates a new type of interlayer magnetic [20] Z. Zhang et al., Phys. Rev. Lett 73 (1994) 336. coupling which is strongly temperature-dependent. The [21] Y. Henry, K. Ounadjela, Phys. Rev. Lett. 76 (1996) 1944. properties of this coupling are quite di!erent from the [22] A. Schreyer et al., Phys. Rev. Lett. 79 (1997) 4914. well-known RKKY coupling. The Cr spacer undergoes [23] R.S. Fishman, Phys. Rev. B 57 (1998) 10 284.