PHYSICAL REVIEW B VOLUME 57, NUMBER 17 1 MAY 1998-I Magnetic phase diagram of interfacially rough Fe/Cr multilayers R. S. Fishman Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6032 Received 3 September 1997 Neutron-scattering measurements have revealed that the spin-density wave SDW inside the Cr spacer of Fe/Cr magnetic multilayers may be either commensurate (C) or incommensurate (I) depending on the tem- perature and Cr thickness N. We theoretically evaluate the magnetic phase diagram of Fe/Cr multilayers under the assumption that the nodes of the SDW lie at the Cr-Fe interface, as found by recent neutron-scattering measurements. While the C phase is never stable under this assumption, the paramagnetic to I transition temperature TN(N) takes on a seesaw pattern as the SDW wave vector switches between different allowed values. S0163-1829 98 07717-0 Fe/Cr multilayers have been the object of intense scrutiny F(g, ) evaluated8 from a three-band model for the qua- since the discovery of giant magnetoresistance in 1988.1 siparticle energies. However, neutron-scattering techniques have only recently2,3 The material parameters of any Cr alloy enter the model been used to investigate their magnetic-phase diagram. free energy F through the energy mismatch z0 These studies reveal that the spin-density wave4 SDW 4 vF / 3a (vF is the Fermi velocity between the elec- within the Cr spacer may be either commensurate (C) or tron and hole Fermi surfaces. Unlike the wave-vector param- incommensurate (I) with the bcc lattice. The C phase is eter , which depends on temperature, both the nesting mis- stabilized when the number of monolayers ML N inside the match and the energy mismatch z0 are constants. If TN* is Cr spacer is less than 30 or when the temperature exceeds the the NeŽel temperature of a perfectly nested alloy with 0 NeŽel temperature 310 K of pure Cr. For N 30 and T and z0 0, then the bulk free energy F depends only on the 310 K, the I phase is stable. Although other measurements ratio z0 /TN* . We use the value z0 /TN* 5, which is appropri- on multilayers5 and wedges6 suggest that the SDW is anti- ate for slightly strained Cr. At the bulk NeŽel temperature ferromagnetically coupled with the neighboring Fe layers, TN,bulk , this value for the energy mismatch corresponds to a the neutron-scattering measurements of Fullerton et al.3 in- SDW with the wave-vector parameter 0.043 or with a dicate that the nodes of the I SDW lie just inside the Cr-Fe period of 1/ 23 lattice constants. interfaces. We find that a simple model based on the assump- For a microscopically smooth interface, the Cr-Fe inter- tion that the SDW nodes lie at the Fe/Cr interfaces has some action is expected to be antiferromagnetic with the form dramatic consequences. While the C phase is never stable, ASFe*S(z) at interfaces I (z 1) and II (z N). Indeed, such the paramagnetic (P) to I transition temperature T an antiferromagnetic interaction was observed in some N(N) as- sumes a seesaw pattern as the SDW wave vector undergoes multilayers5 and was obtained from first-principles transitions between different allowed values. calculations.9 However, the recent neutron-scattering data by Since the magnetic form factor of Cr is strongly peaked at Fullerton, Bader, and Robertson3 suggests that the SDW the atomic sites,4 the SDW within the Cr spacer may be nodes actually lie 4­5 ML inside the Cr-Fe interfaces. This approximated by the form may be caused by surface roughness, which frustrates the antiferromagnetic interaction due to steps and interdiffusion at the interfaces. Because the SDW nodes lie near the inter- S z m sg 1 2z/acos 2 a z , 1 faces, there is no long-range magnetic ordering of the Fe moments within the multilayer. Assuming that the SDW nodes lie precisely at the inter- where a is the bcc lattice constant, m is the polarization faces, the wave-vector parameter is restricted to the val- direction, s is a constant, is an arbitrary phase, g(T) is a ues n n/(N 1), where n 1 0 is the number of nodes temperature-dependent order parameter, and sg(0) inside the spacer. We evaluate n by minimizing the nesting 0.6 B for bulk Cr at zero temperature. Adjacent ML of Cr free energy F(g, n ) with respect to both g and n. As con- are separated by a distance of a/2. Because the hole Fermi firmed experimentally by Fullerton, Bader, and Robertson,3 surface of dilutely doped Cr alloys is slightly larger than the this model assumes that the SDW is rigid so that the SDW electron Fermi surface, the nesting wave vectors Q amplitude and wave vector do not depend on the location z (2 /a)(1 ) differ from 2 /a. For pure Cr, the nesting inside the spacer. Since the pair coherence length10 of the I mismatch is approximately4 0.05. The SDW ordering wave phase 0 vF / g is about 10 Ć, the SDW order param- vectors Q (2 /a)(1 ) are obtained by minimizing eters g and are expected to be modified only within 5 or the nesting free energy7,8 F(g, ) with respect to g and . 6 ML from each interface. In terms of , the period of the SDW is given by a/ . The results of this calculation are provided in Figs. 1 and When 0, this period diverges and the SDW is C. For 2. Because the C SDW does not contain any nodes, the C bulk Cr alloys, 0 so that the SDW ordering wave phase is never stable. So the phase boundary in Fig. 1 sepa- vectors are always closer to commensuration than the nesting rates the I and P phases. The NeŽel temperature TN is nor- wave vectors. This paper employs the nesting free energy malized by the bulk NeŽel temperature TN,bulk , which is 0163-1829/98/57 17 /10284 3 /$15.00 57 10 284 © 1998 The American Physical Society 57 BRIEF REPORTS 10 285 transitions do not occur for N smaller than 104 ML. They may be observed either directly through the change in the SDW wave vector or indirectly through the change in neutron-scattering intensity, which is proportional to the square of the SDW amplitude g. So far, no such transition has been observed in Fe/Cr multilayers. Although the seesaw pattern of TN(N) is an inevitable consequence of forcing the SDW nodes to lie at the inter- faces, the NeŽel temperature measured by Fullerton and co-workers11,3 seems to be a smoothly increasing function of spacer thickness N. While their experimental resolution is probably inadequate to discern the predicted oscillations of the wave vector, the oscillation pattern of TN(N) should be readily observable. Another discrepancy is that the measured critical thickness of 30 ML below which the I phase be- FIG. 1. The NeŽel temperature vs N with the number of SDW comes unstable is much larger than the predicted critical nodes given by n 1. The SDW wave-vector parameter is plot- thickness of 17 ML separating the P and I phases. ted vs N in the inset with specific thicknesses denoted. One possible explanation for the latter discrepancy is that the multilayer interfaces are sufficiently rough to place the evaluated by allowing to be a continuous parameter. For SDW nodes inside the multilayer spacer. Surface roughness N 17 ML, a half-wavelength of the SDW cannot be may be expected to suppress the SDW ordering within a pair squeezed into the Cr spacer so the NeŽel temperature drops to coherence length zero. As N increases, the SDW goes through cycles of ex- 0 6 ML from the interfaces. As men- tioned earlier, Fullerton, Bader, and Robertson3 find that the pansions followed by sudden contractions with the addition SDW nodes lie to the inside of the Cr-Fe interfaces by 4­5 of another node to the SDW. This appears as the seesaw ML, which is suggestively close to the value of pattern in both the NeŽel temperature and the wave-vector 0. If the region within parameter . In the plot of T 0 6 ML from the interface is paramagnetic, N /TN,bulk , we indicate the then the observed critical thickness of 30 ML would corre- number n 1 of SDW nodes for each portion of the graph. spond to a ``true'' critical thickness of 30 12 18 ML, The NeŽel temperature goes through maxima when passes very close to the predicted value of 17 ML. For thicknesses through its bulk value of 0.043. Notice that both TN and less than 30 ML or temperatures greater than 300 K, the approach their bulk values as N . residual antiferromagnetic coupling at the Cr-Fe interfaces Near the discontinuous changes in the SDW wave vector may be sufficient to stabilize a C SDW in some regions of with increasing N, two I SDW's with neighboring values of the Cr spacer, as found by Fullerton, Bader, and Robertson.3 n have free energies that are almost equal. Just prior to the Even if the first 6 ML from the Cr-Fe interface are para- jump in n, the SDW may transform between different I magnetic, however, the NeŽel temperature would still be ex- states with increasing temperature. This behavior is shown in pected to show a deep minimum at 34 12 46 ML or 66 Fig. 2 for N 106 at the top of a seesaw with n 5 at TN) Ć. But our calculation of T and N 105 and 104 at the bottom of a seesaw with n 4 at N has assumed that the first and last SDW nodes lie precisely N ML apart. If the SDW nodes TN). The SDW envelopes for n 4 and 5 are sketched in the can adjust their positions within inset to this figure. If N 105 or 104, the SDW wave vector 0 from each Cr-Fe inter- face, then the sudden drop in the NeŽel temperature might be shifts from n 5 to n 4 with increasing temperature. Such somewhat weakened. Clearly, surface roughness plays a crucial role in con- straining the SDW ordering within Fe/Cr multilayers. By contrast, the smoother interfaces and stronger antiferromag- netic interfacial coupling in Fe/Cr/Fe wedges allow the direct application of a model12 with antiferromagnetic coupling at the Cr-Fe interfaces. In agreement with this model, the I-C transition temperature is elevated far above the bulk NeŽel temperature of pure Cr. Other consequences of this model are discussed elsewhere.12 To summarize, we have developed a model with the SDW nodes at the Cr-Fe interfaces. Whereas this model explains some features of Fe/Cr multilayers, a second model with SDW antinodes at the Cr-Fe interfaces may be more appro- priate for Fe/Cr/Fe wedges. We would like to thank Dr. Robert Celotta, Dr. Sam Liu, FIG. 2. The SDW order parameter g versus normalized tem- and Dr. Lee Robertson for helpful discussions. This research perature T/TN for N 106 solid , 105 long dash , and 104 short was supported by Oak Ridge National Laboratory managed dash . The lower portions of the curves correspond to a wave vector by Lockheed Martin Energy Research Corp. for the U.S. with n 5 while the higher portions correspond to n 4. The cor- Department of Energy under Contract No. DE-AC05- responding envelopes of the SDW are sketched in the inset. 96OR22464. 10 286 BRIEF REPORTS 57 1 M. N. Baibich et al., Phys. Rev. Lett. 61, 2472 1988 . 7 P. A. Fedders and P. C. Martin, Phys. Rev. 143, 245 1966 . 2 A. Schreyer et al., Europhys. Lett. 32, 595 1995 ; A. Schreyer 8 R. S. Fishman and S. H. Liu, Phys. Rev. B 48, 3820 1993 . et al. unpublished . 9 S. Mirbt, A. M. N. Niklasson, B. Johansson, and H. L. Skriver, 3 E. E. Fullerton, S. D. Bader, and J. L. Robertson, Phys. Rev. Lett. Phys. Rev. B 54, 6382 1996 . 77, 1382 1996 . 10 See, for example, A. Fetter and J. Walecka, Quantum Theory of 4 E. Fawcett, Rev. Mod. Phys. 60, 209 1988 ; E. Fawcett et al., Many-Particle Systems McGraw Hill, New York, 1971 , ibid. 66, 25 1994 . p. 426. 5 T. G. Walker, A. W. 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