VOLUME 81, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 30 NOVEMBER 1998 Helical and Incommensurate Spin-Density Waves in Fe Cr Multilayers with Interfacial Steps R. S. Fishman Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6032 (Received 28 May 1998) Although absent in bulk transition metals, a noncollinear, helical (H) spin-density wave (SDW) is stabilized by steps at the interfaces in Fe Cr multilayers. Using the random-phase approximation, we evaluate the phase boundary between the H SDW and the collinear, incommensurate (I) SDW found in bulk Cr. In agreement with neutron-scattering results, the I-to-H transition temperature TIH is always lower than the bulk Néel temperature TN and the nodes of the I SDW lie near the Fe-Cr interfaces. While a H SDW with a single 6p 2 twist has lower free energy than a I SDW above TN, H SDW's with larger twists are stable between TIH and TN. [S0031-9007(98)07739-4] PACS numbers: 75.30.Fv, 75.30.Ds, 75.70.Cn Despite its absence in bulk transition metals, recent Remarkably, a H SDW was the first SDW predicted neutron-scattering measurements [1,2] on Fe Cr multilay- by Overhauser [13] in 1960. Soon afterwards, however, ers suggest that a helical (H) spin-density wave (SDW) polarized neutron-scattering measurements [14] revealed appears inside the Cr spacer at high temperatures or small that the SDW in pure Cr was collinear. Cr thicknesses N. By contrast, extensive measurements All three SDW states are produced by the nearly perfect on bulk Cr alloys [3] only reveal collinear, incommen- nesting [15,16] of electron and hole Fermi surfaces which surate (I), or commensurate (C) SDW's. Although pre- are roughly octahedral in shape. Because the hole Fermi dicted [4] to be stabilized by the steps at Fe-Cr interfaces, surface is slightly larger than the electron Fermi surface, the precise conditions required to support a H SDW have the nesting wave vectors Q6 G 2 1 6 d differ been unknown. This Letter uses a simple model to com- from G 2 2p a. To maximize the nesting on both pare the free energies of the C, H, and I SDW's in an sides of the Fermi surfaces [17], the ordering wave vectors Fe Cr trilayer with interfacial steps. of the SDW Q06 G 2 1 6 d0 are slightly closer to Early measurements [5] on Fe Cr multilayers indicated G 2 than the nesting wave vectors with 0 # d0 , d. that the magnetic coupling between adjacent Fe layers survives up to about 500 K, far above the bulk Cr Néel temperature of TN 310 K. Since then, the role of the SDW in Fe Cr multilayers has been intensely debated [6]. Only recently have neutron-scattering measurements [1,7] confirmed the presence of a SDW in Fe Cr multilayers. Measurements by Schreyer et al. [1] strongly suggest that a noncollinear, H SDW produces the observed [8] 90± or biquadratic coupling [9] between adjacent Fe moments for thicknesses below 30 monolayers (ML's) or temperatures above TN. The biquadratic coupling gradually disappears for larger thicknesses or smaller temperatures, as the H SDW is replaced by an I SDW with nodes close to the Fe- Cr interfaces [7]. Schreyer et al. [1] find that the H and I SDW phases coexist in a region of thicknesses above 30 ML's and for temperatures between 200 and 300 K. Based on a tight-binding approximation, Stoeffler and Gautier [10] first argued that a H SDW would be stable in a perfect Fe Cr trilayer when the orientation of the Fe moments frustrates C ordering. In the presence of steps at the Fe-Cr interface, a single C SDW domain would be totally decoupled from the neighboring Fe moments. Yet as shown in Fig. 1, two sets of H SDW domains with opposite helicity [4] can maintain their antiferro- magnetic coupling with Fe moments that are oriented 90± apart [11]. FIG. 1. A sketch of two H SDW's, one right handed (m 1) A H SDW is unstable in bulk Cr [12] because its free and the other left handed (m 21), coupling Fe moments 90± energy is always higher than that of a C or I SDW. apart due to a step at the interface. 0031-9007 98 81(22) 4979(4)$15.00 © 1998 The American Physical Society 4979 VOLUME 81, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 30 NOVEMBER 1998 Since the Bloch wave functions are sharply peaked at an I SDW, the distance between nodes is 1 d0 ML's. the lattice sites, they may be replaced by delta functions For a H SDW, this is the distance for a p twist. Just in the spin density. With Q06 along the z axis, the local below the Néel temperature of pure Cr, the SDW nodes spins in the I and H SDW phases can be written µ are separated by 1 d0 27 ML's [3]. 2p The mismatch between the electron and hole Fermi S p I z masg 21 2z a cos d0z 2 u , (1) a surfaces is measured by the energy z0 4pdyF 3a, µ 2p where yF is the Fermi velocity. As z0 and d are decreased SH z asg 21 2z a x cos d0z 2 u by doping with Mn or Fe, d0 is also diminished. At aµ æ some critical value of z0, d0 drops to zero and the 2p 1 y sin d0z 2 u , (2) SDW becomes commensurate with Q06 G 2. In the a limit d0 ! 0, SI z and SH z reduce to C SDW's with where as is a constant, m is the polarization of the I SDW, amplitude asg. u is an arbitrary phase, and g T is the order parameter. Within the random-phase approximation, the change in At low temperatures in bulk Cr [3], asg 0.6mB. For the bulk SDW free energy below TN is given by [17,18]: µ T X DFI g, d0, T, z0 rehg2 ln 1 r T eh N n 0 Z 1 Ç Çæ 3 g2 1 2 T d´ ln 1 2 g2 2ivn 2 z0 1 2´ , (3) n 1 1 2 2 ivn 2 ´ ivn 2 z0 2 1 ´ 2 2 z0d0 2d 2 µ T X DFH g, d0, T, z0 rehg2 ln 1 r T eh N n 0 Ç µ 1 Z 1 2g2 3 g2 1 2 T d´ ln 1 2 n 1 1 2 2 2 ivn 2 ´ ivn 2 z0 2 1 ´ 2 z0d0 2d µ Çæ 2g2 3 1 2 , (4) ivn 2 ´ ivn 2 z0 2 1 ´ 1 z0d0 2d where vn 2n 1 1 pT are the Matsubara frequencies, out each domain. Indeed, the inherent "softness" of bulk reh is the density of states of the nested portions of Cr-as evidenced by its rotational and translational Gold- the Fermi surfaces, and T N 100 meV is the Néel stone modes [20]-is broken by its interactions with the temperature of a perfectly nested alloy with d 0 and two interfaces. But as mentioned latter, some softness z0 0. We shall use a value for the mismatch energy of may be retained by an I SDW in a large spacer. In the ab- z0 5T N, which is appropriate for pure, unstressed Cr. sence of interface steps, Shi and Fishman [21] employed In the limit d0 ! 0, DFI and DFH reduce to the same C this model to evaluate the magnetic phase diagram of an free energy. The bulk values of the SDW order parameter Fe Cr wedge [22] with nearly perfect interfaces. and wave vector are obtained by minimizing these free To calculate the coupling energy, we assume that the energies with respect to g and d0. Both H and I SDW states regions of the spacer with thicknesses N and N 1 1 (or have the same Néel temperature and the same period 1 d0 N 2 1) are the same. For the H phase, this implies that at TN [19]. Below TN, however, a H SDW has a shorter adjacent Fe moments lie 90± apart [4]. For the collinear period than an I SDW. For any fixed z0 and T , TN, the I phase, adjacent Fe moments are either parallel (F) or minimum value of DFH exceeds the minimum value of antiparallel (AF). Minimizing Ecoup with respect to the DFI so that a H SDW (with d0 . 0) always has a higher arbitrary phase u of the I SDW for F or AF Fe moments, free energy than an I SDW. Nonetheless, the stability we find of the H phase in the presence of interfaces is possible E F only because the H SDW state already provides a local coup 2AasgSFej cos f 2 cos pd0 2 f j , (5) minimum of the bulk Cr free energy. E AF coup 2AasgSFej sin f 1 sin pd0 2 f j , (6) The total energy E of the multilayer is modeled by sim- where f pL a 1 1 d0 . As expected, both cou- ply adding the free energy DFa2L of the spacer (with pling energies vanish in the C phase with d0 0. In the area a2) and the interfacial coupling energy Ecoup I phase, the coupling energies are minimized when the A SIFe ? S 0 1 SII Fe ? S L , where L N 2 1 a 2 is SDW nodes lie precisely at the Fe-Cr interfaces. How- the width of the Cr spacer. In accord with the observa- ever, the actual spin configuration is obtained by mini- tions of Fullerton et al. [7], we assume that the SDW is mizing the total energy EF,AF E F,AF coup 1 DFIa2L with rigid with the same amplitude and wave vector through- respect to g and d0. 4980 VOLUME 81, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 30 NOVEMBER 1998 Despite the rotation of the SDW in the H phase, vector in the plane of the multilayer. This figure clearly neighboring Cr and Fe moments are still assumed to be indicates that the I-to-H phase transition occurs below antiparallel [23] at the interfaces. Hence, the H coupling the bulk Néel temperature of the Cr spacer. For perfect energy in each domain is given by Ecoup 22AasgSFe. interfaces, on the other hand, the I-to-C phase transition The boundary conditions imposed on the H SDW in a predicted by Ref. [21] happens above the bulk TN. domain with thickness N restrict the wave vector to values With increasing thickness, the IH transition temperature d0 m, N m 2 N 2 1 , where m 2n 1 1 is an odd passes through consecutive valleys. The maxima in TIH integer and the helix rotates through the angle mp 2 occur as jmj changes by 2. The minima in TIH occur from z 0 to z L. For m . 0 and d0 . 0, Eq. (2) when the H SDW wave vector passes closest to its bulk indicates that the helix is right handed; for m , 0, the value. Similar oscillations in the transition temperature helix is left handed. Because d0 m, N d0 m, N 6 1 are produced by a model [24] which forces the I SDW for large N, the bulk free energies DFHa2L of H SDW's nodes to lie at the Fe-Cr interfaces. For larger values of in the two sets of domains with thicknesses N or N 6 1 g, the lower critical thickness at T 0 increases and TIH are taken to be the same. decreases as the larger coupling energy of the H SDW Since the bulk free energies are proportional to rehT 2 N , gains it more of an advantage over the I SDW. the total energy E depends on the single dimensionless pa- Even with steps at the interface, the I SDW continues rameter g AasSFe V N rehT N. For a perfect Fe-Cr to magnetically couple adjacent Fe layers below TIH. interface with AS2Fe 100 meV [21], g 12. However, However, Jcoup EAF 2 EF is roughly an order of interfacial roughness and interdispersion suppress g by an magnitude smaller than in the absence of interfacial steps unknown amount. The phase diagram of an Fe Cr wedge [21]. The magnetic coupling and order parameters of is fit rather well with g 3 [21]. the I SDW at T TN 0.05 are plotted in Fig. 3. With Interfacial steps have two important effects within this increasing thickness, the coupling alternates between F model. First, steps frustrate C ordering so that a C and AF except on either side of the nearly vertical lines SDW does not gain any coupling energy at the interfaces. in Fig. 2, when the magnetic coupling repeats. As in Second, steps reduce the coupling between an I SDW and Fe Cr wedges, these phase slips are roughly separated the neighboring Fe moments to the point that a H SDW by the bulk value of 1 d0 26.4 ML's. Between phase has the lower total free energy for high temperatures or slips, the SDW stretches to keep its nodes near the small thicknesses. interfaces. Across a phase slip, both the SDW amplitude In Fig. 2, the IH phase boundary for g 2 is plotted and period change discontinuously as the SDW suddenly in the solid curve. Above TIH, a H SDW with twist contracts. A similar series of oscillations about the bulk parameter jmj has the lowest free energy between the order parameters, except with much larger magnitudes, thin dashed lines. So a helix with a single 6p 2 twist was predicted for Fe Cr wedges [21]. is stable for thicknesses below 25 ML's. Higher-order Unlike a H SDW, an I SDW in a large spacer can adjust helixes with jmj . 1 become stable as N increases. to the presence of lattice defects by shifting the position Different jmj states may be distinguished by polarized of its nodes with very little cost in free energy. Higher- neutron-scattering measurements with the scattering wave order helixes with jmj . 1 may be especially frustrated by the presence of defects in the Cr spacer. Assuming that only jmj 1 helixes are stable, the IH phase boundary Tjmj 1 IH is plotted in the thick dashed curve of Fig. 2. As expected, Tjmj 1 IH is always larger than TIH but still lies below TN. For realistic Fe Cr multilayers, we expect that the I SDW and higher-order H SDW's coexist in the region between TIH and Tjmj 1 IH . With increasing g, this coexistence region becomes even larger as TIH decreases, so this calculation may explain the IH transition region observed by Schreyer et al. [1] between 200 and 300 K in epitaxially grown multilayers. By contrast, higher-order helixes may be unable to form in rougher, sputtered multilayers. Then the thick dashed FIG. 2. Phase diagram of an Fe Cr multilayer with g 2 curve would correspond to the monotonically increasing and z transition temperature measured by Fullerton et al. [25]. 0 T N 5. The thick solid curve gives the IH transition temperature between an I SDW and a H SDW with an mp 2 Fitting this curve to the expression 1 2 Tjmj 1 twist. The thick dashed curve gives the phase boundary IH N TN b N 2 N between an I SDW and a H SDW with a single 6p 2 0 2l0 yields the exponent l0 0.86, consistent twist. Below T with the result l0 0.8 6 0.1 of the Argonne group [25]. IH, the magnetic coupling between adjacent Fe moments experiences a phase slip across the thin, nearly Because the coupling energy of the I SDW is so small vertical lines. in the presence of a step, almost exactly the same IH phase 4981 VOLUME 81, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 30 NOVEMBER 1998 for helpful discussions. This research was supported by Oak Ridge National Laboratory managed by Lockheed Martin Energy Research Corp. for the U.S. Department of Energy under Contract No. DE-AC05-96OR22464. [1] A. Schreyer et al., Europhys. Lett. 32, 595 (1995); Phys. Rev. Lett. 79, 4914 (1997). [2] P. Bödeker et al., Phys. Rev. Lett. 81, 914 (1998). [3] E. Fawcett, Rev. Mod. Phys. 60, 209 (1988); E. Fawcett et al., Rev. Mod. Phys. 66, 25 (1994). [4] J. C. Slonczewski, J. Magn. Magn. Mater. 150, 13 (1995). [5] S. S. P. Parkin, N. More, and K. P. Roche, Phys. Rev. Lett. 64, 2304 (1990). [6] M. 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(1997); R. S. Fishman and Z. P. Shi, J. Phys. C 10, L277 A series of H SDW's are stabilized by the interfacial (1998). coupling at small thicknesses or high temperatures. In [22] J. Unguris, R. J. Celotta, and D. T. Pierce, Phys. Rev. Lett. an intermediate range of temperatures between T 67, 140 (1991); 69, 1125 (1992). IH and Tjmj 1 [23] Relaxing this approximation and allowing the angle IH , the I SDW and H SDW may coexist due to the between neighboring Cr and Fe moments to deviate from frustration experienced by higher-order H SDW's in the 180± lowers the IH transition temperature slightly but does presence of lattice defects. not affect any of our conclusions. The author would like to thank Dr. Sam Liu, Dr. Lee [24] R. S. Fishman, Phys. Rev. B 57, 10 284 (1998). Robertson, Dr. Andreas Schreyer, and Dr. Zhupei Shi [25] E. E. Fullerton et al., Phys. Rev. Lett. 75, 330 (1995). 4982