VOLUME 84, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 21 FEBRUARY 2000 Origin of Biquadratic Exchange in Fe Si Fe G. J. Strijkers,* J. T. Kohlhepp, H. J. M. Swagten, and W. J. M. de Jonge Department of Applied Physics and COBRA, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands (Received 13 September 1999) The thickness and temperature dependences of the interlayer exchange coupling in well-defined mo- lecular beam epitaxy-grown Fe Si Fe sandwich structures have been studied. The biquadratic coupling shows a strong temperature dependence in contrast to the bilinear coupling. Both depend exponentially on thickness. These observations can be well understood in the framework of Slonczewski's loose spins model [J. Appl. Phys. 73, 5957 (1993)]. No bilinear contribution of the loose spins to the coupling was observed. PACS numbers: 75.70.­i, 75.50.Bb, 75.60.­d Recently, an exceptional type of coupling was found in stabilized Knudsen cell. The thicknesses were controlled Fe Si Fe, strongly antiferromagnetic (AF) and varying ex- by calibrated quartz crystal monitors. The layers were ponentially with the spacer layer thickness [1]. This rather grown at room temperature on Ge(001) substrates, which unique behavior of the coupling is mediated by c-Fe12xSi were cleaned by several Ar1 sputter (700 ±C) and anneal formed by Fe diffusion into the Si spacer layer [2] and (780 ±C) treatments until a sharp Ge 001 - 2 3 1 LEED can be qualitatively understood in terms of the Bruno elec- pattern was observed. tronoptics model [3] with imaginary extremal Fermi vec- Previous studies have shown that in these Fe Si Fe tri- tors or by the Anderson sd-mixing model [4]. layers the Si spacer transforms to metallic c-Fe12xSi by Fe Apart from bilinear exchange coupling also a bi- and Si interdiffusion, and the whole stack grows epitaxially quadratic contribution to the coupling was observed in bcc(001)-like on the Ge(001) substrate [1,2]. However, sputtered Fe Si multilayers [5­10]. However, a quali- this recrystallization only slightly alters the effective tative and quantitative interpretation of the biquadratic thickness of the spacer layer, and therefore we will refer to coupling in these multilayers is impossible due to lateral the nominal Si spacer layer thickness for the remainder of and vertical variations of the coupling properties [8,10]. this Letter. The temperature dependence of the coupling Nevertheless, the Fe Si system is unique and particularly was studied in six samples of the following composition: attractive for studies on biquadratic coupling. Unlike Ge 001 1 60 Å Fe 1 tSi Si 1 45 Å Fe 1 30 Å Si, with the oscillatory RKKY interactions, the monotonically tSi 14, 14.5, 15, 15.25, 16, and 16.25 Å. The spacer (exponentially) decaying intrinsic AF exchange coupling layer thickness dependence was measured in a wedge- limits the possible mechanisms for biquadratic exchange shaped sample composed of Ge 001 1 60 Å Fe 1 and allows a flexible and large range of spacer thicknesses 7 17 Å Si wedge 1 45 Å Fe 1 30 Å Si. to be studied. Figure 1 shows two normalized magneto-optical Kerr In this Letter we present a study of biquadratic ex- effect (MOKE) hysteresis curves with the field applied change coupling in well-defined molecular beam epitaxy along the 100 easy axis, at 300 and 100 K for a nominal (MBE)-grown Fe Si Fe trilayers which do not suffer from Si thickness tSi 15.25 Å. These magnetization curves the complications reported for multilayers. Based on com- are representative for all of the other loops. At 300 K the bined detailed measurements of the temperature and thick- magnetization loop can be characterized by two switching ness dependence of the bilinear as well as the biquadratic fields, indicated by H1 and H2 in the figure (defined in the coupling constants, it is shown that biquadratic exchange middle of the hysteresis). Going from high to low fields, in Fe Si Fe is due to a small concentration of (super)para- H1 corresponds to a reorientation of the magnetic moments magnetic Fe clusters in the spacer layer, which act as of the two Fe layers from a parallel to a perpendicular so-called "loose spins." By exploiting the unique exponen- alignment, and at H2 the alignment of the magnetizations tial character of the coupling, we are able to demonstrate changes from perpendicular to antiparallel. The difference that J1 and J2 are caused by the same interaction poten- between H1 and H2 is a measure of the biquadratic cou- tial, which is a basic ingredient for modeling the loose pling strength, J2, and the sum is characteristic for the spins coupling. Surprisingly, however, a contribution of bilinear coupling strength, J1. When the biquadratic cou- the loose spins to the bilinear coupling seems to be absent. pling strength is larger than the bilinear coupling strength, The Fe Si Fe layers were grown at room temperature as is the case in Fig. 1(b) for 100 K, only one step is ob- in a molecular beam epitaxy system (VG-Semicon V80M) served in the magnetization loop. The perpendicular align- with a base pressure of 2 3 10211 mbar. An e-gun source ment of the moments of the Fe layers is then maintained with feedback control of the flux was used for the deposi- down to zero field. In the latter case J1 and J2 can be evalu- tion of Fe, whereas Si was evaporated from a temperature ated only from a fit of the experimental curves. We note 1812 0031-9007 00 84(8) 1812(4)$15.00 © 2000 The American Physical Society VOLUME 84, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 21 FEBRUARY 2000 proximately 1.8 3 104 J m3 at 300 K. The temperature dependences of J1 and J2 resulting from the fits are plot- ted in Figs. 2(a) and 2(b), respectively, for the six different nominal Si thicknesses. J1 is antiferromagnetic and decreases slowly with in- creasing temperature for all Si thicknesses. In a previous paper by de Vries et al. [1] it was shown that the origin of the exponential thickness dependence of J1 can be un- derstood within the Anderson sd-mixing model applied to Fe Si by Shi et al. [12] or within the framework of Bruno's theory introducing the concept of an imaginary critical Fermi wave vector [3]. Both models predict a different temperature dependence. However, the tempera- ture dependence of J1 is rather small, and we cannot rule out completely that the small temperature dependence ob- served is a result of a decrease of the (surface) magne- FIG. 1. MOKE hysteresis curves for t tization of the Fe layers. Nevertheless, at this point the Si 15.25 Å with the field along the 100 easy axis at (a) 300 K and (b) 100 K, and temperature dependence of J1 seems to be in favor of the corresponding fits (right hand side) of the experimental curves sd-mixing model by Shi et al., which predicts a decreasing with Eq. (1). The insets show the relative orientations of the coupling strength with increasing temperature, in contrast magnetic moments for different fields (dashed lines are the easy to Bruno's model. axes and solid lines are the hard axes). The biquadratic coupling J2, shown in Fig. 2(b), has a that sometimes small steps in the magnetization loops are much stronger temperature dependence than J1. There are observed at low fields, as also can be seen in Fig. 1(b), a number of possible mechanisms for this biquadratic cou- which are magneto-optical artifacts caused by a small mis- pling. First of all, J2 may be of intrinsic origin as was alignment of the easy axis with respect to the field. claimed recently [9]. We reject this possibility because the The magnetization hysteresis curves can be fitted by considering the phenomenological expression for the areal energy density of the two magnetic layers 0.20 (a) E 2m0MsH t1 cos f1 2 fH 1 t2 cos f2 2 fH )2 0.15 1 Kt1 cos2 f1 sin2 f1 1 Kt2 cos2 f2 sin2 f2 J/m 2 J1 cos f1 2 f2 2 J2 cos2 f1 2 f2 , (1) (m 0.10 with M J 1 s the saturation moment of layers 1 and 2 with - thicknesses t1 and t2. Here f1 and f2 are the angles be- 0.05 tween the magnetization of layers 1 and 2 and the 100 easy axis, respectively, while fH is the angle between the field H and the 100 axis. The cubic anisotropy con- 0.00 stant K was assumed equal for layers 1 and 2. An extra 0.20 (b) t (Å) Si uniaxial anisotropy as is sometimes observed for epitax- 14.0 ial ultrathin Fe(001) films on semiconductors like Ge and )2 0.15 14.5 GaAs (see, e.g., [11]), was not observed in the relatively 15.0 thick Fe films [.30 ML (monolayer)] used in our experi- J/m 15.25 ments and thus neglected. J1 is the bilinear coupling 0.10 (m 16.0 constant (,0 for AF coupling) and J2 is the biquadratic J 2 16.25 - coupling constant (,0 for 90± coupling). By a minimal- 0.05 ization of Eq. (1) as function of the applied field H, the magnetization loops can be reproduced in a satisfactory way by choosing the correct combination J 0.00 1 and J2 as is 0 50 100 150 200 250 300 demonstrated on the right hand side of Fig. 1. Uncertain- ties in the determination of the coupling constants when T (K) 2J2 . 2J1 are overcome by combining easy and hard axes loops. The anisotropy K, evaluated from the shape FIG. 2. Temperature dependence of (a) J1 and (b) J2 for six Si thicknesses as indicated in the figure. The solid lines in (b) of easy and hard axes loops, decreases with increasing are fits to the experimental data with Eq. (5), as explained in temperature from about 3.5 3 104 J m3 at 10 K to ap- the text. 1813 VOLUME 84, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 21 FEBRUARY 2000 magnitude of an intrinsic second order term J2 is generally applied to explain the biquadratic coupling in the RKKY orders of magnitude smaller than J1, much smaller than systems Fe Al Fe, Fe Au Fe, and Fe Ag Fe [16­19]. observed experimentally [3]. Furthermore, the tempera- Especially in the Fe Ag Fe system the loose spins model ture dependence should be less dramatic. Two other ex- has been proven to explain the biquadratic coupling di- trinsic mechanisms for biquadratic coupling are proposed rectly by depositing an ultrathin Fe layer in the center of by Slonczewski. the Ag spacer. Without the ultrathin Fe layer this system The fluctuation mechanism for biquadratic exchange shows no loose spins behavior. [13] predicts a J2 when spatial fluctuations of the Unlike the other mechanisms for biquadratic coupling, interlayer thickness cause a competition between ferro- the loose spins model predicts a strong temperature magnetic and antiferromagnetic coupling for neighboring dependence of J2. Indeed, the huge increase of J2 shown regions. The resulting frustration can lead to a perpendicu- in Fig. 2(b) with decreasing temperature is a very strong lar alignment of the magnetic moments, when the size of indication for a loose spins origin. Figure 2(b) is sup- the thickness fluctuations is below the size of the domain plemented with fits of J2 T with the loose spins model walls. For Fe Si, however, the bilinear coupling J1 always (solid lines), assuming S 1 and U U1 U2, which favors an antiparallel alignment of the magnetic layers means that the loose spins are atoms near the midplane of and therefore lateral thickness variations do not lead to the spacer or randomly distributed. The areal loose spins a frustration of the coupling here [14]. Furthermore, the density c and the interaction potential U were adjusted for fluctuation mechanism predicts a temperature dependence the fit. As can be seen in the figure, the loose spins model J2 T ~ J1 T 2. As shown in Fig. 2(a), J1 is decreased is in very good agreement with the experimental data. The only by 20% to 30% at 300 K compared to 10 K, while density of loose spins following from the fits converged J2 decreases by a factor of 19, ruling out an interpretation consistently to approximately 1% for all thicknesses, and in terms of the fluctuation model. We also note that U kB 343, 334, 292, 266, 222, and 199 K for tSi magnetic dipole fields created by roughness can result 14, 14.5, 15, 15.25, 16, and 16.25 Å, respectively. In in a biquadratic alignment of the moments [15]. The another approach one can assume c 2 and allow for magnitude of this contribution is however small. U1 fi U2, which describes the case of two loose spins The second mechanism proposed by Slonczewski is bi- layers near the edge of the ferromagnetic interfaces [16]. quadratic coupling mediated by paramagnetic atoms in the However, with these assumptions our data cannot be spacer layer [16]. These so-called "loose spins" can couple described, because this leads to a plateau in J2 for lower to both ferromagnetic layers via an indirect exchange, temperatures, not observed experimentally. which also is responsible for J1 (we will come back to In the loose spins model the interaction potential U this point later). The total interaction potential U between is driven by the intrinsic bilinear coupling J1 between loose spins and ferromagnetic layers is the vector sum of the ferromagnetic layer and the loose spins in the spacer. the interaction U1 and U2 of the loose spins with ferromag- Qualitatively this can be proven by measuring the thickness netic layers 1 and 2, respectively, and can be expressed as dependences of both the bilinear and the biquadratic cou- U u U2 pling. Figure 3 shows J1 and J2 at room temperature as a 1 1 U22 1 2U1U2 cos u 1 2, (2) function of the nominal Si spacer layer thickness measured where u is the angle between the two moments. The free on the wedge-shaped sample. We have plotted the coupling energy per loose spin is parameters only for t ! Si between 12.8 and 16.25 Å, when sinh 1 1 2S 21 U u k f T, u 2k BT BT ln , sinh U u 2SkBT 1 (3) ) -J1 2 -J2 with S the atomic spin and T the temperature. The macro- scopic free energy per unit spacer area F ca22f, with c the areal density of loose spins and a the nearest neighbor 0.1 distance between atoms, can be expanded in F u J0 2 J1 cos u 2 J2 cos2 u 1 . . . , (4) 0.01 where Coupling Strength (mJ/m 1 J2 T 2 ca22 f T, 0 1 f T, p 2 2f T, p 2 12 13 14 15 16 17 2 (5) Nominal Si Thickness (Å) FIG. 3. Thickness dependence of J is the loose spins contribution to the biquadratic coupling. 1 (open circles) and J2 (solid squares) at room temperature. The solid lines illustrate Previously, this loose spins model has been successfully the exponential decay with the Si thickness. 1814 VOLUME 84, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 21 FEBRUARY 2000 J1 and J2 could be separated unambiguously. Indeed, both authors acknowledge valuable discussions with Dr. J. C. J1 and J2 decrease exponentially with the Si spacer layer Slonczewski. thickness with approximately the same decay length. This shows that indeed both J1 and J2 are intimately related and find their origin in the same mechanism. Furthermore, the unique exponential thickness depen- *To whom correspondence should be addressed. dence of the intrinsic exchange interaction enables us to Present address: Department of Physics and Astronomy, perform a more quantitative analysis of J1. From the fits The Johns Hopkins University, Baltimore, MD 21218. for J2 tSi we can obtain the strength of the driving intrin- Electronic address: gustav@pha.jhu.edu sic bilinear interaction J1 under the assumption that the [1] J. J. de Vries, J. Kohlhepp, F. J. A. den Broeder, R. Coe- loose spins are located at the midplane. Given the expo- hoorn, R. Jungblut, A. Reinders, and W. J. M. de Jonge, nential character of J Phys. Rev. Lett. 78, 3023 (1997). 1 as shown in Fig. 3 and characterized by l, we can calculate the intrinsic J [2] G. J. Strijkers, J. T. Kohlhepp, H. J. M. Swagten, and 1 tSi at 0 K between the magnetic electrodes: W. J. M. de Jonge, Phys. Rev. B 60, 9583 (1999). [3] P. Bruno, Phys. Rev. B 52, 411 (1995). J1 tSi, 0 K a22e2tSi 2lU . (6) [4] Z.-P. Shi, P. M. Levy, and J. L. Fry, Europhys. Lett. 26, 473 This results in J (1994). 1 0.13, 0.10, 0.072, 0.058, 0.035, and 0.028 mJ m2 for t [5] Y. Saito, K. Inomata, and K. Yusu, Jpn. J. Appl. Phys. 35, Si 14, 14.5, 15, 15.25, 16, and 16.25 Å, respectively, in good agreement with the L100 (1996). [6] E. E. Fullerton and S. D. Bader, Phys. Rev. B 53, 5112 actually measured low temperature bilinear coupling [see (1996). Fig. 2(a)]. This surprising result, given the simplicity of [7] J. Kohlhepp and F. J. A. den Broeder, J. Magn. Magn. the model, leads to two important conclusions: First of Mater. 156, 261 (1996). all, the intrinsic J1 is apparently the only contribution [8] J. Kohlhepp, F. J. A. den Broeder, M. Valkier, and A. van to the overall bilinear coupling and virtually no bilinear der Graaf, J. Magn. Magn. Mater. 165, 431 (1997). loose spins contribution is observed. Qualitatively this [9] Y. Endo, O. Kitakami, and Y. Shimada, Phys. Rev. B 59, conclusion is also reflected in the relatively small tem- 4279 (1999). perature dependence of J [10] J. Kohlhepp, M. Valkier, A. van der Graaf, and F. J. A. 1 as already discussed earlier. Second, this is the first direct proof of one of the basic den Broeder, Phys. Rev. B 55, R696 (1997). assumptions of Slonczewski's loose spins model that the [11] G. W. Anderson, P. Ma, and P. R. Norton, J. Appl. Phys. interaction potential driving the coupling via loose spins 79, 5641 (1996); P. Ma and P. R. Norton, Phys. Rev. B 56, 9881 (1997). is the same as the intrinsic interaction mechanism. [12] Z.-P. Shi (unpublished). The absence of a bilinear loose spins contribution in [13] J. C. Slonczewski, Phys. Rev. Lett. 67, 3172 (1991). systems where J2 originates from loose spins is well docu- [14] P. A. A. van der Heijden, C. H. W. Swüste, W. J. M. mented and is not restricted to our rather unique Fe Si sys- de Jonge, J. M. Gaines, J. T. W. M. van Eemeren, and tem. Although the available data are scarce in literature, K. M. Schep, Phys. Rev. Lett. 82, 1020 (1999). also in the RKKY-driven Fe Ag Fe system the bilinear [15] S. Demokritov, E. Tsymbal, P. Grünberg, W. Zinn, and I. K. loose spins contribution is only 20% of the value expected Schuller, Phys. Rev. B 49, 720 (1994). from the Slonczewski model [19]. Apparently, further re- [16] J. C. Slonczewski, J. Appl. Phys. 73, 5957 (1993). finements of the theory seem to be necessary. The prospect [17] C. J. Gutierrez, J. J. Krebs, M. E. Filipkowski, and G. A. of predicting the bilinear loose spins contribution has been Prinz, J. Magn. Magn. Mater. 116, L305 (1992). questioned already by Slonczewski because of scattering [18] A. Fuss, J. A. Wolf, and P. A. Grünberg, Phys. Scr. T45, 95 (1992). effects of the loose spins atoms on the electron waves ne- [19] M. Schäfer, S. Demokritov, S. Müller-Pfeiffer, R. Schäfer, glected in the original model [20]. M. Schneider, P. Grünberg, and W. Zinn, J. Appl. Phys. 77, The work of G. J. S. was supported by the Founda- 6432 (1995). tion for Fundamental Research on Matter (FOM). The [20] J. C. Slonczewski, J. Magn. Magn. Mater. 150, 13 (1995). 1815