PHYSICAL REVIEW B VOLUME 57, NUMBER 1 1 JANUARY 1998-I Equilibrium configuration of magnetic trilayers T. L. Fonseca and N. S. Almeida Departamento de Fi´sica Teo´rica e Experimental, Universidade Federal do Rio Grande do Norte, 59072-970, Natal, RN, Brazil Received 27 June 1997 We use a simple and realistic theoretical model to investigate the equilibrium configuration of ferromagnetic-nonmagnetic-ferromagnetic trilayer systems. We assume the ferromagnetic films have a crys- talline anisotropy and interact via bilinear and biquadratic coupling. We consider the system is always in the configuration that gives the absolute minimum to the energy to construct field-dependent phase diagrams for the case that the external dc field (H0) is applied parallel to the surface of the films. We show that for a given value of the dc field, the equilibrium configuration has a peculiar dependence on the parameters that describe the anisotropy (Ha), bilinear (Hx), and biquadratic (Hb) coupling. We present general results for different values of Hb /Hx and Ha /Hx and we specialize our numerical calculation to display theoretical results for physical properties of systems that have values of Ha , Hb , and Hx , suitable to fit the experimental parameters of Fe/Cr/Fe magnetic trilayers. We predict interesting behavior of the magnetization with the strength of the dc field when it is not applied parallel to an easy axis. S0163-1829 98 00601-8 Very often in the last decade, the experimental results of Recently Azevedo et al.13 presented a series of experi- physical properties of systems consisting of two magnetic mental results obtained by MOKE, FMR, and BLS for a layers separated by a nonmagnetic spacer suggested physical Fe 40 Å /Cr 15 Å /Fe 40 Å sample, grown by magnetron behavior that challenged our basic knowledge of such simple sputter deposition in a UHV chamber onto MgO 001 . From systems. The oscillatory dependence of the coupling between their results one can appreciate the rich variety of magnetic magnetic films on the thickness of the nonmagnetic spacer,1,2 phases of this system. They use a simple model to interpret the alignment of the magnetic moments at 90° with respect their experimental data and, with the same set of parameters, to each other observed in different metallic trilayers,3,4 they successful fitted the data obtained in all experiments among others, have motivated several authors to look for mentioned above. In their model they assume that, besides explanation for these intriguing physical properties.5­10 On the crystalline anisotropy and interfilm exchange interaction the experimental side, the techniques frequently used for of the Heisenberg form Hx(n1*n2) , the films also interact characterization of these systems are magneto-optical Kerr through a biquadratic coupling Hb(n1*n2)2, Hb 0. The bi- effect MOKE , ferromagnetic resonance FMR , and Bril- quadratic coupling was observed some years ago by Ru¨hring louin light scattering BLS . It should be remarked that, et al.3 and, shortly after, Slonczewski10 proposed intrinsic while MOKE gives information on the equilibrium configu- and extrinsic mechanisms that lead to this kind of coupling. ration of the system, FMR and BLS supplement this infor- When the biquadratic coupling was first observed, it was mation with details of the dynamical behavior of the system. found to be rather weak, compared to the bilinear exchange H In other words, MOKE gives information on the ``magnetic x . However, it has been found that the Fe/Cr 211 struc- tures might have the ratio h phase'' of the system and through the data obtained by FMR b Hb /Hx quite large. This fact suggests that one may synthesize samples for which h and/or BLS the dynamical behavior of the system can be b can be adjusted to have the value appropriate for any particular analyzed. purpose. The first step to understand the dynamical behavior of the We should mention that the system studied here is equiva- system is to learn how the equilibrium configuration is af- lent to an infinite magnetic superlattice with similar param- fected by of the environment; in other words, how the equi- eters. The aim of this work is to present a theoretical study of librium configuration is modified by changes of the tempera- the influence of the combined effects of the biquadratic cou- ture, applied field, etc. It is known that, if the temperature of pling and crystalline anisotropy on the phase diagram of a magnetic multilayer system is modified, the magnetic cou- trilayer system. We will also show that the magnetic phases pling changes and the experimental data obtained by BLS or of these systems can be completely characterized by mea- FMR must give information on the modifications of the in- surement of the magnetization and/or resonances of the sys- trinsic parameters.11,12 It is also well known that, even if the tem. This paper can be seen as an extension of Ref. 14, temperature remains constant, an external dc magnetic field where the influence of the biquadratic exchange on the phase can modify the behavior of the system, and may give it en- diagram of magnetic finite and infinite superlattice was stud- tirely different physical properties. For a convenient choice ied. In that paper the authors considered that the magnetic of the parameters thickness of the films and spacer , a small multilayer systems had an uniaxial anisotropy and they pre- variation of the direction and/or strength of an externally dicted striking field-dependent magnetic phase diagrams. applied field may modify significantly the response of the However, some of these systems have a very weak uniaxial system to an external input. This might be of the interest in anisotropy15 or it does not appear at all.13 On the other hand, the development of devices like magnetic sensors. it has been observed that in Fe/Cr/Fe structures grown onto 0163-1829/98/57 1 /76 4 /$15.00 57 76 © 1998 The American Physical Society 57 BRIEF REPORTS 77 GaAs 100 or onto MgO 100 , a crystalline anisotropy is al- ha ways present. Therefore, to have a better comprehension of sin 1 2 hbsin 2 1 2 4 sin 4 1 these system, the contribution of the crystalline anisotropy for the internal energy must be included in the energy func- h0sin 1 H 0 2a tional of Ref. 14. In order to investigate the effects of this anisotropy on the magnetic phase diagram, we replace the and term of the uniaxial anisotropy in Eq. 2.1 of Ref. 14, by h sin a 1 2 hbsin 2 1 2 4 sin 4 2 Ha x y x z y z h 2 ni ni 2 ni ni 2 ni ni 2 . 0sin 2 H 0. 2b i 1,2 In Eqs. 2 , hb Hb /Hx , ha Ha /Hx , and h0 H0 /Hx . The presence of this anisotropy gives to the system a totally The system of equations is solved by using the same ap- different symmetry and consequently a quite different be- proach used in Ref. 14 and here we will just mention the havior and it is responsible for the main characteristics of main steps. First we introduce the variables cos( 1 2) the physical properties reported in this paper. and ha/2 sin(4 1) h0sin( 1 H) to rewrite Eq. 2a as The system under investigation is constituted by two in- 1 2 1 4h 2 2 2. 3 finitely extended thin ferromagnetic films a few atomic b 4hb monolayers , with static magnetization in-plane, separated For a given 1 one can find values for 2( 1) by solving Eq. by a nonmagnetic spacer. We define n i as a unit vector in the 3 for . Solutions for the problem are found when the val- direction of the magnetization of the ith film (i 1,2), and ues of 1 and 2 , obtained from Eq. 3 , are also the solution then we write the energy functional as for Eq. 2b . The next step is to find out which solutions are the stable minimum. To do that we construct the matrix M E n i j 2E/ i j , (i 1,2) and we search, among the solu- 1 ,n 2 Hxn 1*n 2 Hb n 1*n 2 2 H0n H* n 1 n 2 tions, what are the angles that satisfy Eq. 2 and also give H positive values for both eigenvalues of the matrix M. We a x y x z notice that, for finite values of h 2 ni 2 ni 2 ni 2 ni 2 b , in general, there is more i 1,2 than one pair ( 1 , 2) that satisfy this requirement. To con- ny z struct the field-dependent phase diagram we choose the so- i 2 ni 2 2 ms n 1 2 n 2 2 . 1 lution that gives, for the fixed values of the parameters, the absolute minimum for E(n 1 ,n 2). The borders of the phase In Eq. 1 the energy is given in units of magnetic field and are, in general, obtained numerically. However there are the first term is the regular bilinear exchange which gives to some special cases that analytical expressions can be found. the system a antiferromagnetic ferromagnetic character if For example, if H0 is applied parallel to an easy axis, the H 2 x is positive negative . The second term is the biquadratic antiferromagnetic phase is stable if H0 2Ha(Hx 2Hb) exchange that has been found always positive (H 2 b 0). The Ha, while the ferromagnetic phase is stable for H0 2Hx third term is the Zeeman energy where we are assuming that 4Hb Ha . The boundaries of the region where the spin- the external dc magnetic field is applied parallel to n H . The flop configuration is stable can be obtained from the calcu- fourth term is the crystalline anisotropy that defines the x, y, lation of the maximum and minimum values of H0 , that give and z directions as the easy axis of the system. Finally, the real solutions for in the equation 2(ha 4hb)cos3( ) (ha last term is the surface anisotropy. In this term ms denotes 4hb 2)cos( ) h0 0, and also give 8(ha 4hb)cos4( ) the saturation magnetization of the layer and (m sn i ) is the 4(1 2ha 8hb)cos2( ) h0cos( ) ha 4hb 0. component of the magnetization perpendicular to the surface In Figs. 1 a ­1 d we show, for different values of ha of the layer i. The approach used here can be used for any Ha /Hx , the magnetic phase diagram for the case where the particular values of Hx , Hb , and Ha but in our numerical external dc field is applied parallel to the x axis one of the calculations we will always consider all of them positive. easy axis . It should be said that the characteristic value of We assume the demagnetization field generated by tip- Ha for Fe films is 0.55 kG. The external dc field H0 , as well ping the magnetization out of plane is strong enough to sup- as the biquadratic exchange field Hb , are given in units of press any tendency of the equilibrium magnetization to tilt the bilinear exchange parameter Hx . In these figures is ob- out of plane. Therefore, in the equilibrium, the magnetization served that, for small values of hb Hb /Hx and h0 of both films lie in the plane parallel to the surface and we H0 /Hx , the system has the magnetic moments aligned define the direction perpendicular to the film as the y direc- symmetrically with respect to the easy axis. This configura- tion to write the unit vectors n 1 x cos( 1) z sin( 1), n 2 tion characterizes the spin-flop phase. In this region of the x cos( 2) z sin( 2), n i 0 (i 1,2) , and n H x cos( H) h0-hb diagram, the spin-flop angles are between /4 and z sin( H). Here the x and z directions are parallel to the /2; i.e., the moments align between the hard axis and the surface of the films and the angles are measured from the x easy axis perpendicular to the external field z axis . In the axis. diagrams we name this phase as SF1. For small but finite To obtain the equilibrium configuration, first we require values of hb , when h0 is increased the system goes from the that the energy functional be an extremum. Therefore, we spin-flop configuration to the nonsymmetric phase AS with must find solutions for the nonlinear set of equations one of the magnetic moment aligned between the hard axis 78 BRIEF REPORTS 57 FIG. 2. h0-hb magnetic phase diagram of a trilayer system for ha 4, and the angle between H0 and the x axis equal to a 20°, b 45°. The labels are defined in the text. ration in which both magnetic moments point in the same direction is reached after another first-order phase transition at higher value of h0 . It should be remarked that, in this phase, the angle between the magnetic moments and the ex- ternal field is finite and decreases when the strength of the external field is increased. This angle goes to zero for a very high value of h0 . A singular behavior is observed in the phase diagram when the external field is applied parallel to the hard axis. We depict it in Fig. 2 b . In this arrangement, for small values of h0 and hb , the system is in the SF1 phase FIG. 1. h but this phase is degenerated; the configurations 1 2 0-hb magnetic phase diagram of a trilayer system for H and 0 parallel to the x axis and a ha 1, b ha 2, c ha 3, and d 1 /2 , 2 /2 have the same energy. ha 4. The labels SF1, SF2, FM, and AS are defined in the text. When h0 is increased the system goes, after a first-order phase transition, to a configuration where the angles between and the external field, while the other remains in a direction the external field and the magnetic moments are equal between the hard and z axis. The transition from SF1 to AS (n 1*n H n 2*n H). We name this phase as SF3. The configu- was always found to be of the first order. For h ration that has the moments aligned parallel to the field is a 3 and h reached at high value of h0 and the phase transition is of the b 0.75, when h0 is increased, both moments become aligned parallel to the external field after another first-order second order. phase transition from the nonsymmetric AS to a ferromag- There are several ways to observe the phase diagrams netic phase FM . We emphasize that there are values of h shown in Figs. 1 and 2. Probably the simplest experiments b for which the system goes to another spin-flop phase SF2 , would be the direct measurement of the magnetization after a first-order transition, before reaching the ferromag- through MOKE and the measurement of the resonance fre- netic phase. This spin-flop configuration SF2 is similar to quency FMR . The magnetization of the sample can easily SF1 but in this case both moments point in directions be- be calculated from the equilibrium configuration. The reso- tween the dc field and the hard axis. For H nance frequencies can be obtained from the equations of mo- b smaller than Hx , this SF2 phase should be observed at a relatively low exter- tion: nal dc field in samples for which ha 3. The phase transition from SF2 to FM when it does exist , is always of the second dmi i order. In the figures, dashed lines represent the frontier be- dt mi Heff 4 tween phases that the phase transition is of the second order. It should be noted that, in all diagrams showed, there are with tricritical points at particular values of hb . We notice that these systems have interesting physical behavior around Hi H x ny 2 nz 2 x these points but it will not be discussed in this paper. eff x 2Hb n 1*n 2 n j Ha ni i i The phase diagrams have a quite different form when the ny x z z x y i ni 2 ni 2 y ni ni 2 ni 2 z external dc field is applied in an oblique direction with re- spect to an easy axis. We illustrate this fact by showing in 4 m sn i H0 , 5 Fig. 2 a the phase diagram for the case where the angle between H0 and the x axis is 20°. In Fig. 2 a as well as in where n i is the unit vector parallel to the magnetization of Fig. 2 b we used H a /Hx 4. In this geometry the system is the film i and n i is the unit vector perpendicular to the also in the SF1 phase for small values of h0 and hb . When surface of the layer. A laborious but straightforward calcula- the external field is increased the system goes from SF1 to tion gives the following expression for the resonance fre- the AS phase after a first-order phase transition. The configu- quencies (H0 Hx ): 57 BRIEF REPORTS 79 2 2R 2 A1B1 A2B2 2CD A1B1 A2B2 2CD 2 4 A1A2 C2 B1B2 D2 1/2 , 6 where Ai Hxcos 1 2 2Hbcos2 1 2 H a 2 1 cos2 2 i H0cos i 4 ms , 7a Bi Hxcos 1 2 2Hbcos 2 1 2 Hacos 4 i ] H0cos i , 7b FIG. 3. For the case ha 2, hb 0.55, we show the variation of the difference C H R between the frequencies of the acoustical and x 2Hbcos 1 2 , 7c optical modes with the external field h0 applied parallel to the x and axis full line and y axis and the behavior of the magnetization with h0 under the same conditions. D Hxcos 1 2 2Hbcos 2 1 2 . 7d by considering that the system is always in the configuration We should notice that the frequencies given by Eq. 6 that gives the absolute minimum for the energy. In a labora- correspond to modes where the magnetic moments oscillate tory the values of these fields should be slightly different if in phase acoustical mode and in opposite phase optical the transition is of the first order. mode . We choose h We have shown that the magnetic-nonmagnetic-magnetic a 2, hb 0.55 to show, in Fig. 3, the behavior of the component of the magnetization parallel to system with crystalline anisotropy and biquadratic exchange, H in addition to the bilinear coupling, exhibits a rich variety of 0 dashed line, right-hand y axis and the absolute values of the difference between the frequencies of the acoustical configurations induced by the external field. This kind of and optical modes ( system has been grown, with very high quality, elsewhere R) full line, left-hand y axis with the external field. From this picture one can see that, when and we hope the present calculation stimulate a detailed the strength of the dc field approaches the values that make study of the remarkable capability of this system to display the system change the phase, the magnetization as well as different physical properties. R exhibit singular behavior. In particular, when the sys- tem exhibits a first-order phase transition, a discontinuity is The authors thank Dr. A. S. Carric¸o, Dr. A. Azevedo, and observed in both quantities. Once again we should remark C. Chesman for helpful discussions. This research was par- that the transition fields shown in the pictures were obtained tially supported by the Brazilian Agency CNPq. 1 S. S. Parkin, N. More, and K. P. Roche, Phys. Rev. Lett. 64, 304 Edwards, Phys. Rev. B 51, 12 876 1995 . 1990 . 9 S. Demokritov, E. Tsymbal, P. Gru¨nberg, W. Zinn, and I. K. 2 See articles in, Ultrathin Magnetic Structures I and II, edited by Schuller, Phys. Rev. B 49, 720 1994 . 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