Journal of Magnetism and Magnetic Materials 223 (2001) 284}292 Biquadratic exchange coupling in an unequal Fe/Cr/Fe(1 0 0) trilayer P. Vavassori , M. Grimsditch *, Eric E. Fullerton Materials Science Division 223, Argonne National Laboratory, 9700 S. Cass Ave., Argonne, IL 60439, USA IBM, Almaden Research Center, San Jose, CA 95120-6099, USA Received 14 July 2000; received in revised form 1 November 2000 Abstract We have investigated the magnetic properties of a (1 0 0)-oriented unequal trilayer, Fe(45 As)/Cr(30 As)/Fe(15 As), by means of Brillouin light scattering and magnetization measurements. The experimental results show that this sample highlights the e!ect of biquadratic coupling which aligns the magnetization of the Fe layers at 903 to each other. We extracted the bilinear and biquadratic coupling strengths by "tting the experimental results with a theory that treats the static and dynamic responses on an equal footing. Our results con"rm that the model describes both the static and dynamic properties even when the magnetization of the layers is aligned at 903. The coupling strengths, and their temperature dependence, are discussed and compared with other results reported in the literature. 2001 Elsevier Science B.V. All rights reserved. Keywords: Fe/Cr superlattices; Magnetic; Biquadratic; Bilinear; Brillouin scattering; And MOKE 1. Introduction Fe/Cr/Fe(1 0 0) trilayer structures were the "rst systems in which indirect exchange coupling across a nonmagnetic transition-metal spacer layer was observed [1] along with its oscillatory dependence on interlayer thickness [2]. The leading term of the coupling is a Heisenberg-like bilinear term that results in parallel or antiparallel alignment of adjacent Fe layers. The presence of an additional biquadratic coupling term (s )s) is now also well established; this coupling favors 903 alignment of adjacent layers and has been discussed in a number of recent reviews [3}5]. The oscillatory bilinear coupling has two periods for Cr spacer layers; a long (18 As) period and, for atomically smooth surfaces, a two-monolayer period that can be directly related to the antiferromagnetic properties of Cr. Both periods can be related to the Cr Fermi surface, the short period results from the nested Fermi surface and the long period from the N-centered ellipse [6,7]. * Corresponding author. Tel.: #1-630-252-5544; fax: #1-630-252-7777. E-mail address: grimsditch@anl.gov (M. Grimsditch).  Present address: INFM-Dipartamento di Fisica, UniversitaH de Ferrara, via Paradiso 12, I-44100 Ferrara. 0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 1 3 3 4 - 2 P. Vavassori et al. / Journal of Magnetism and Magnetic Materials 223 (2001) 284}292 285 Although the period of the coupling is quantitatively understood, the strength and phase of the oscillatory coupling is less well understood and is found to depend on both the roughness and interdi!usion at the Fe}Cr interfaces. For a recent review, see Pierce et al. [7]. Biquadratic coupling often exhibits a stronger temperature dependence than the bilinear coupling. Although biquadratic coupling can, in principle, be an intrinsic property of indirect exchange coupling [8}11] it is now thought to mostly arise from a variety of extrinsic mechanisms. Several mechanisms have been suggested to explain this behavior as arising from structural imperfections that average over varying bilinear contributions and result in a biquadratic term. As summarized in Ref. [4] they are the #uctuation mechanism [12], loose spin model [13], and the magnetic}dipole model [14]. For Cr interlayers, its intrinsic antiferromagnetic order is also thought to contribute to the biquadratic coupling [3}7,15}19]. This is supported for thicker Cr layers (t!'42As) by the experimental results which show that for Fe/Cr samples, a strong correlation exists between magnetic phase transitions in the antiferromagnetic Cr layers and the biquadratic coupling [17,18]. For thinner Cr layers, the relation between the antiferromagnetic order within the Cr interlayer and biquadratic coupling is less clear but correlations between non-collinear alignment of the Fe layers and Cr antiferromagnetism have been observed by Schreyer et al. [19]. This behavior is probably related to Slonczewski's proximity magnetism model [3]. Clearly, a deeper understanding of biquadratic coupling in this and related systems requires careful measurements of the interlayer exchange terms. In this paper we use an experimental approach for determining these coe$cients based on the "tting of BLS and magnetization measurements. This approach, successfully used in recent investigations [5,15,20}23], is here applied to an Fe(45 As)/Cr(30 As)/Fe(15 As) trilayer in which the two magnetic layers have di!erent thicknesses. The thickness of the Cr interlayer was chosen in order to highlight the e!ects of biquadratic coupling. As will be shown, at certain "elds the two magnetic layers are aligned at 903 and provide an opportunity to investigate the magnetic excitations in such a con"guration and to test the models which are used to interpret the BLS spectra [23]. The values obtained for the coupling strengths and their temperature dependence are discussed and compared with other results reported in the literature. 2. Experimental details The sample was epitaxially grown by DC magnetron sputtering on a polished single-crystal MgO(1 0 0) substrate using the same procedure outlined for superlattices [24]. A 100 As Cr(1 0 0) layer was grown at 6003C. The substrate was then cooled to +753C prior to the growth of the Fe(45 As)/Cr(30 As)/Fe(15 As) trilayer. The complete structure was then capped with a 30 As Cr layer. A calibrated quartz crystal oscillator monitored the thickness of the various layers. Under these conditions, the layers grow along a [1 0 0] direction and exhibit a four-fold in-plane anisotropy with the easy axes along the remaining 11 0 02 directions. The magnetization hysteresis loops were measured by SQUID magnetometry. The spin-wave excitations were measured by BLS using 250 mW of 5145 As radiation from an Ar> laser. The scattered radiation was analyzed with a tandem Fabry}Perot interferometer [25] in 3#2 pass operation. The sample was mounted with its normal along the collection axis and the laser beam was incident at an angle of 503 to the normal. This geometry "xes the magnitude of the wave vector parallel to the surface q, at 0.93;10cm\. The magnetic "eld was applied in the plane of the sample and perpendicular to the scattering plane, i.e. perpendicular to the wave vector of the magnon. The sample could be rotated about the normal, thereby allowing the magnetic "eld to be applied along di!erent in-plane directions. The polarization of the scattered light was analyzed at 903 to the incident polarization in order to minimize the intense signal of the unshifted laser radiation. All BLS measurements were made at room temperature. 286 P. Vavassori et al. / Journal of Magnetism and Magnetic Materials 223 (2001) 284}292 3. Results and discussion Fig. 1 shows the room temperature magnetization results of our sample when H is applied along the in-plane easy-axis of the sample. The magnetization curve in the low-"eld region (0}100 Oe) is characterized by a square hysteresis loop centered at H"0 and with a partial saturation value for M, that is a half of the saturation value M at higher "elds. The qualitative interpretation of this portion of the loop shows that the AF coupling is dominant and the magnetization of the two Fe layers are antiparallel, lie in the "lm plane, and are along the easy axis. The value of M"0.5 M is consistent with the di!erent thicknesses of the two "lms, i.e. M/M"(d!d)/(d#d)"0.5 for our sample with d"15As and d"45As. This assumes the same value for the saturation magnetization in the two "lms. As the "eld increases two "rst-order phase transitions are observed. The "rst occurring at H+100 Oe is consistent with the spins switching from being antiparallel to a situation in which M for the thinner layer is oriented along the other easy axis and is almost at 903 to the "eld and to the other (thick) layer. In this region M/M"d/(d#d)" as expected. The second transition takes place at H+260 Oe and corresponds to parallel alignment of the spins along the "eld direction. The above qualitative behavior indicates that the spins are at 903 to each other over the range 100}260 Oe. Our purpose is to determine the numerical strengths of the anisotropies and biaxial and bilinear coupling which stabilize this con"guration. Our aim is also to ensure that the model that accounts for the magnetization, is also capable of describing the BLS frequencies. As expected for two magnetic layers of these thicknesses, the BLS spectra show two modes which can be viewed as the in-phase (symmetric mode) and out-phase (antisymmetric mode) oscillations of the two Fe layers. The frequencies of the symmetric and antisymmetric modes as a function of the external "eld H are shown in Fig. 2, for H applied along the easy axis and in Fig. 3 for H along the hard axis. Blow-ups of the frequencies in the 0}1 kOe range are shown in Fig. 4 for both "eld orientations. The numerical values of the magnetic parameters have been obtained by "tting the "eld dependence of the magnetization and BLS results. The basics of the model are described in Ref. [23]; a slightly modi"ed version of that approach was used in Ref. [20]. Our approach here follows that in Ref. [20] but includes the obvious changes necessary to account for the two layers having di!erent thicknesses. At any given "eld the minimum of the total energy with respect to the orientation of the magnetization yields the equilibrium magnetic con"guration from which the magnetization can be calculated and compared with the SQUID results. The Fig. 1. SQUID loop measured with the "eld applied along the easy axis. The full line is a guide to the eye. P. Vavassori et al. / Journal of Magnetism and Magnetic Materials 223 (2001) 284}292 287 Fig. 2. Brillouin frequencies as a function of the external "eld applied along the easy axis: experiment (dots) and "t (line) described in the text. Fig. 3. Brillouin frequencies as a function of the external "eld applied along the hard axis: experiment (dots) and "t (line) described in the text. magnon frequencies, obtained as perturbations of the layers from their equilibrium state, are then compared with the BLS results. The total energy per unit area is given by the sum of anisotropy, Zeeman and coupling terms E"E #E#E  "Kd sin(2 )#Kd sin(2 )!MdH cos( ! &)!MdH cos( ! &) #J cos( ! )#J cos( ! ), (1) 288 P. Vavassori et al. / Journal of Magnetism and Magnetic Materials 223 (2001) 284}292 Fig. 4. Blow-up of Figs. 2 and 3 in the region 0}1 kOe. For clarity the hard-axis results have been plotted along negative "eld axis. Fig. 5. Fit (full line) to magnetization data in Fig. 1. where K and K are the cubic anisotropy constants of the two magnetic layers, M and M their magnetization, H is the applied "eld, J and J are the bilinear and biquadratic coupling constants and ,  and & the angles that the magnetization and applied "eld subtend with the easy axis. H along the easy and hard axes corresponds to &"0 and /4, respectively. In order to extract the material parameters we minimize the energy expression for given (numerical) values of K, K, M, M, J, J and H to determine  and . The easy-axis magnetization is then given by (Md cos #M d cos )/(Md#Md). Fig. 5 shows our best "t of the magnetization loop. The best "t parameters are listed in Table 1, where an asterisk indicates that the "t was insensitive to that particular parameter. Therefore, during the "tting of the SQUID results we have "xed the values of the cubic anisotropy constants (K and K) and of the saturation magnetization in the two "lms (M and M) to the values extracted from the "ts to the BLS data. P. Vavassori et al. / Journal of Magnetism and Magnetic Materials 223 (2001) 284}292 289 Table 1 Parameters extracted from the best least-squares "tting of the data shown in Figs. 1}4 K (;10ergscm\) K (;10ergscm\) 4 M (kG) 4 M (kG) J(;10\ergscm\) J(;10\ergscm\) BLS 1.9$0.6 4.5$0.8 20.2$1.3 20.7$1.2 3.85$1.35 0.75$0.55 SQUID 1.9* 4.5* 20.2* 20.7* 4.70$0.20 1.35$0.25 K are cubic anisotropies, 4 M are e!ective magnetizations (which may include contributions from the perpendicular anisotropy as described in the text), and J and J are the inter-layer bilinear and biquadratic coupling strengths, respectively. Subscripts a and b indicate properties of the 15 and 45 As Fe layers, respectively. Also shown are con"dence levels obtained as described in the text. The formalism we use to derive the magnon frequencies is conceptually the same as that used in Ref. [20]. We stress that, since in this formalism the same energy expression yields both the magnetization and the magnon frequencies, it guarantees that any discrepancies between BLS and magnetization results cannot be attributed to inconsistent forms of the energy expression. In the present case, di!erences produced by considering layers with di!erent thicknesses and magnetizations require changes to matrices A5 and A6 of Ref. [20]. (Note that in Eq. (1) we have omitted the out of plane dependence for simplicity, the generalizations to include it are straightforward following [20].) With the notation that E?@ are the second derivatives of the energy with respect to the variables and , these matrices are now EF /d /d /d /d F  EF( !i M EFF  EF(  E( /d /d /d /d  F #i M E((  E(F  E((  . (2) EF /d /d /d /d F  EF(  EFF  EF( !i M E( /d /d /d /d  F  E((  E(F #i M E((  and cos( ! F)dM 0 cos( ! F)cos( ! F)dMM i cos( ! F)dMM 0 !d 2 q M i cos( ! F)dMM dMM , . (3) cos( ! F)cos( ! F)dMM !icos( ! F)dMM cos( ! F)dM 0 !i cos( ! F)dMM !dMM 0 !dM The magnon frequencies are obtained by solving the secular equation of the sum of these two matrices. In analyzing the BLS data we found that simultaneously "tting the easy- and hard-axis data considerably reduced the uncertainties. The results of the "tting are shown by the full lines in Figs. 2 and 3 over the whole "eld range while Fig. 4 shows them in more detail in the range 0}1 kOe in which the spin}#op "rst-order phase transitions take place. The agreement between the experimental data and the "ts is excellent since the modeling accounts for even small details at low "elds. The splitting, observed in the calculations at low "elds, occurs in the regions where the Stokes and the anti-Stokes portions of the spectra are not time reversal invariant of each other. For larger "elds, where the two layers are aligned with the "eld, no such di!erences exist. All the curves presented in Figs. 2}4 correspond to least-squares "ts to the data and the best "t values for the material parameters are listed in Table 1. 290 P. Vavassori et al. / Journal of Magnetism and Magnetic Materials 223 (2001) 284}292 3.1. Determination of errors In our least-squares "tting , the sum of the square of the di!erences between calculation and experiment, is evaluated and the parameters varied to "nd its minimum value. The error estimate for each parameter, which is described in detail in Ref. [20], involves "nding the change such that the  after adjusting all other parameters increases by 50%. Crucial in the comparison of the results extracted from magnetization and BLS summarized in Table 1 are the estimated uncertainties. Because the values extracted for J and J are within the estimated errors, we can claim that the energy given by Eq. (1) provides a self-consistent description of the experimental results. Although this may seem somewhat trivial, we note that since J is introduced into the theory on purely phenomenological grounds, it is not a priori obvious that it will describe all physical properties correctly. In this context it validates the postulate that there is an interaction between magnetic layers, which favors 903 alignment. The approach of independently "tting the results of several experimental measurements has the additional advantage of yielding more reliable estimates of the coupling parameters and their con"dence levels. The value of 4.5;10 ergs cm\ found for K is close to the bulk value and in good agreement with the reported values for 40 As Fe layer [22]. The value of K is smaller than K, as expected for a thinner layer and it is close to that reported in Ref. [20] for similar Fe thicknesses. The magnetization values of both "lms are slightly smaller than for bulk Fe and are consistent with previous determinations [20,22,26}29]. Part of the reduction may be due to the fact that BLS is sensitive to 4 M "4 M!2K /M, where the perpendicu- lar anisotropy is typically induced by the surface. Returning to the values for J and J we observe that the values determined from BLS are smaller than those obtained from SQUID measurements but that they have larger error bars. The larger error bars in the BLS results are related to the larger number of "tting parameters, which allow for a greater degree of interdependence. On the other hand, the SQUID measurements, being sensitive only to the interlayer coupling parameters, produce noticeably smaller error bars. Particularly relevant is the relatively small uncertainty in the value of the biquadratic exchange parameter J that is often determined with large uncertainties since it is di$cult to separate from the usually large J contribution. The accuracy of the present determination is due to two reasons: (i) the reduction of the bilinear coupling interaction through the appropriate choice of the Cr interlayer thickness; (ii) the inequality of the two Fe layer thicknesses. Both these conditions contribute to make the magnetization jumps more evident in the "rst-order spin #op transitions. There is only one other determination of J and J for the (100) orientation and the same Cr thickness [21]. In units of 10\ ergs cm\ our J"4.7$0.2 and J"1.35$0.25 do not agree well with 0.4 and 0.22 from Ref. [21]. The agreement is somewhat better when compared with measurements on an Fe/Cr/Fe(1 0 0) trilayer with a 25 As Cr layer that gave 2.4;10\ ergs cm\ for both J and J [22]. Discrepancies in measured values of the coupling strengths are common and are typically ascribed to the microscopic di!erences in the sample characteristics. Since these e!ects have been extensively discussed [7,15], we will not pursue this issue further. Given the di$culty in interpreting absolute values of the coupling strengths, it is interesting to consider their temperature dependence to probe the origin of the coupling. In recent studies of FeSi/Fe superlattices [30,31] it was suggested (based on the authors' interpretation of the quantum interference model of Bruno [32]) that the observed temperature dependence of J and J could be explained by J"J(¹/¹)/sinh(¹/¹), J"J(2¹/¹)/sinh(2¹/¹), (4) where the superscript 0 indicates the value at 0 K and ¹ is a function of the spacer layer properties and is expected to scale as the inverse of the layer thickness. In Fig. 6 we have plotted the coupling strengths, P. Vavassori et al. / Journal of Magnetism and Magnetic Materials 223 (2001) 284}292 291 Fig. 6. Temperature dependence of J and J obtained from magnetization. The full lines are the "t according to Eqs. (4). obtained from magnetization loops, as a function of temperature. The full lines are "ts to Eq. (4) with, J"8.4;10\ergscm\, J"7.9 10\ergscm\ and ¹"158K. A similar "t to the temperature dependence of J was reported in Ref. [33]; for 11, 13 and 15As Cr thicknesses they obtained ¹"390, 214 and 122 K, respectively. We have also "tted the temperature dependence of the results presented in Ref. [34] for a 16 As Cr layer and obtained ¹"144K. Hence, even though the functional form given by Eq. (4) seems to describe the measured temperature dependence of J and J, the values obtained for ¹ do not scale as the inverse of thickness as predicted by the equations given in Ref. [30]. 4. Conclusions In this study we determined the coupling coe$cients in a (1 0 0)-oriented unequal trilayer Fe(45 As)/Cr(30 As)/Fe(15 As) in which the thickness of the Cr interlayer was chosen in order to highlight the e!ects of biquadratic coupling. To evaluate the coe$cients, we use an experimental approach based on the "tting of the BLS and magnetization measurements. We obtained consistent values of the magnetic parameters by "tting the experimental results with a theory treating the static and dynamic responses on an equal footing. Our results con"rm the theoretical model used in interpreting both the static and dynamic properties even in systems in which the magnetization of the layers is aligned at 903. As found in previous investigations, the values obtained for the bilinear and biquadratic coupling strengths are not in perfect agreement with previous results reported in the literature for the same Cr spacer thickness. This has been interpreted as being due to the details of the interfaces at the atomic level. We also "nd that the temperature dependence of the coupling strengths, although they can be "tted by the equations proposed by Endo et al. [30], produce "tting parameters that do not scale according to the predictions of that theory. Acknowledgements Work at Argonne National Laboratory was supported by the US Department of Energy, Division of Material Sciences, O$ce of Basic Energy Sciences, under contract W-31-109-ENG-38. We thank 292 P. Vavassori et al. / Journal of Magnetism and Magnetic Materials 223 (2001) 284}292 C.H. Sowers for the Fe "lm deposition. P.V. acknowledges support by a research grant from INFM-Istituto Nazionale per la Fisica della Materia. References [1] P. GruKnberg, R. Schreiber, Y. Pang, M.D. Brodsky, H. Sowers, Phys. Rev. Lett. 57 (1986) 2442. [2] S.S. Parkin, N. More, K.P. Roche, Phys. Rev. Lett. 64 (1990) 2304. [3] J.C. Slonczewski, J. Magn. Magn. Mater. 150 (1995) 13. [4] S.O. Demokritov, J. Phys. D: Appl. Phys. 31 (1998) 925. [5] B. Heinrich, J.F. Cochran, Adv. Phys. 42 (1993) 523. [6] M.D. Stiles, J. Magn. Magn. Mater. 200 (1999) 322. [7] D.T. Pierce, J. Unguris, R.J. Celotta, M.D. Stiles, J. Magn. Magn. Mater. 200 (1999) 290. [8] R.P. Erickson, K.B. Hathaway, J.R. Cullen, Phys. Rev. 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