PHYSICAL REVIEW B VOLUME 58, NUMBER 1 1 JULY 1998-I Magnetic trilayers with bilinear and biquadratic exchange couplings: Criteria for the measurement of J1 and J2 C. Chesman,* M. A. Lucena, M. C. de Moura, A. Azevedo, F. M. de Aguiar, and S. M. Rezende Departamento de Fi´sica, Universidade Federal de Pernambuco, 50670-901 Recife, Brazil S. S. P. Parkin IBM Research Division, Almaden Research Center, 650 Harry Road, San Jose, California 95120-6099 Received 9 February 1998 We have analytically calculated phase boundaries of magnetic trilayers with bilinear and biquadradic ex- change couplings in order to investigate possible phase transitions in these systems as the external magnetic field, applied either along an easy or a hard magnetization axis, is varied. A simple scheme is obtained for both antiferromagnetic and ferromagnetic couplings that is shown to be consistent with magnetization curves pre- viously measured in different systems. In addition, experimental data regarding static and dynamic responses in sputtered 100 Fe 40 Å /Cr(tCr)/Fe 40 Å are reported for the Cr thickness in the range 15 Å tCr 35 Å. As a result, our model calculations indicate that the bilinear and biquadratic exchange coupling constants, J1 and J2, cannot be accurately determined from a fit to the experimental data when the ratio J2 / J1 1, and if only the main magnetization axes are considered. S0163-1829 98 02725-8 Layered films of ferromagnetic metals exchange coupled action leads to first-order phase transitions in the magnetiza- through a nonferromagnetic spacer layer are of considerable tion's configuration, and numerical approaches are usually fundamental and technological interest. Remarkable findings required to circumvent this problem. Here we show that, in in these systems include interlayer antiferromagnetic AF simple yet usual situations, one can obtain analytical expres- coupling,1 the concurrent giant magnetoresistance GMR ,2 sions for the boundaries between phases, which are quite oscillations in the interlayer exchange coupling and GMR as useful in interpreting the experimental data. Since the details a function of the spacer layer thickness,3 and biquadratic of our analysis may be found elsewhere,9­12 we present here exchange coupling.4,5 On the other hand, magnetoelectric de- only the main assumptions aiming at the steps for obtaining vices based on the GMR have been widely considered for the expressions for the critical fields. applications in information storage technology.6­8 The rela- We consider a very thin trilayer, so that the dipolar inter- tionship between the interlayer exchange coupling and the action can be neglected, and could be described in terms of a GMR thus makes the measurement of the former of primary free energy per unit area E EZ EA EE, where the three importance. The cheapest and most widely used technique in terms on the right-hand side are, respectively, the Zeeman, this regard is the magneto-optical Kerr effect MOKE , the fourfold magneto-crystalline anisotropy, and the ex- which is useful for extracting the bilinear exchange constant change bilinear and biquadratic energies. We assume that only if the coupling is AF. Other techniques, such as ferro- the ferromagnetic layers have the same thickness d, and that magnetic resonance FMR and Brillouin light scattering the magnetizations are uniform in both layers, with the same BLS , are needed in the case of ferromagnetic coupling. saturation value MS . For the external magnetic field H0 ap- We have recently presented phenomenological model cal- plied in the film plane, this energy can be written as9 culations where the coupling between the magnetic films is 2 fully taken into account through bilinear and biquadratic ex- E H change and magnetic dipolar interactions, together with sur- 0cos i H 18 HAsin2 2 i i 1 face, in-plane uniaxial and cubic anisotropies.9 The calcula- tions were previously shown to provide good quantitative Hblcos 1 2 Hbqcos2 1 2 , 1 agreement with MOKE, magnetoresistance, FMR, and BLS where the effective fields are given by HA 2K1 /MS (K1 experiments in several trilayer systems,10­12 with the advan- effective anisotropy constant , Hbl J1 /dMS , and Hbq tage of treating both static and dynamic responses on an J2 /dMS . The variables H , 1 , and 2 are, respectively, equal footing. In this report we present further investigations the angles of the applied field and of the equilibrium magne- of phase transitions in sputtered 100 Fe/Cr/Fe trilayers ex- tizations with respect to an easy axis. Consider a system hibiting bilinear and biquadratic exchange couplings, making grown in the 100 plane, with a spacer layer thickness such use of phase diagrams to help in understanding the behavior that the bilinear and biquadratic interactions are of the same of the magnetizations as the external magnetic field is varied. order of magnitude. Furthermore, assume that the external Previous studies on phase diagrams either did not take into field is applied along the 001 easy magnetization axis. The account the biquadratic interaction,13 nor looked carefully at equilibrium configuration can be determined by equating to the so-called 90° phase,14 where the magnetizations in the zero the derivatives of the free energy with respect to the two magnetic films are nearly perpendicular to each other. angles 1 and 2. In the case of AF coupling, there are three As has been demonstrated previously,10 the biquadratic inter- possible phases, namely, AF, 90°, and saturated.10 The ex- 0163-1829/98/58 1 /101 4 /$15.00 PRB 58 101 © 1998 The American Physical Society 102 BRIEF REPORTS PRB 58 FIG. 1. Calculated phase diagram Hbq / Hbl vs H0 / Hbl for FIG. 2. a Calculated magnetization curve for a trilayer with antiferromagnetic a and ferromagnetic b coupling for a symmet- Hbl 40 Oe and Hbq 40 Oe solid line and Hbl 27 Oe ric thin-film trilayer. The arrows indicate the relative positions of and Hbq 54 Oe dashed line . b Same as a , with Hbl 40 the magnetizations in the two magnetic films in each phase. The Oe and Hbq 40 Oe solid line and Hbl 40 Oe and Hbq three horizontal lines in the inset correspond, from top to bottom, to 120 Oe dashed line . the sweep of the external magnetic field in the MOKE experiments described in Refs. 10, 16, and 17, respectively. phase diagram of Fig. 1 a . One can see that if the ratio H pression for the critical field H bq / Hbl J2 / J1 is smaller than one, the values of the ef- C1 that separates the AF phase fective exchange fields can be determined from the measured from the 90° phase can be determined from the boundary values of H condition C1 and HC2 through Eqs. 2 and 3 , namely, Hbl (HC1 HC2)/2, and Hbq (HC2 HC1)/2. To illus- trate the usefulness of this diagram, we show by the horizon- E 1 90°, 2 90°, H 0 tal lines in the inset of Fig. 1 a , our classification of three E magnetic-field sweeps in MOKE measurements in 1 0, 2 90°, H 0 , Fe/Cr/Fe. The uppermost line corresponds to our own result that gives described in Ref. 10, while the mid and lower lines fit the data in Refs. 16 and 17, respectively. We show in Fig. 1 b HC1 Hbl Hbq . 2 the phase diagram for the ferromagnetic coupling, in the Actually, same range. In this case, there is only one transition at a 1 and 2 have a weak dependence on the mag- netic field within each region,9,10 but, as far as the determi- critical field given by Eq. 3 , if 1. That happens when nation of the critical field is concerned, the values taken at the coupling is AF and 1, as well. Therefore, only the the frontier are excellent approximations to the real ones. difference Hbq Hbl would be available from a measurement Similarly, for the boundary H of HC2 in this case 1 . There are important consequences C2 between the 90° and the saturated phases, we assume that following this restriction, to which little attention seems to 1 0 and 2 90° in the 90° phase, while we take have been paid. First, we point out that the authors in Refs. 1 2 0 in the saturated one. By applying the boundary condition, we obtain the critical field 18 and 19 were most probably working in this regime, and that might explain why they were unable to provide a precise H measurement of both bilinear and biquadratic exchange pa- C2 Hbl Hbq . 3 rameters from their experiments. To better illustrate this Notice that in this case the effective anisotropy field has no point, we show in Fig. 2 plots of MZ /MS (cos 1 influence on HC1 and HC2. However, it can be shown15 that cos 2)/2, a quantity that is proportional to the MOKE the situation described here is possible only if HA 2 Hbl . signal. The solid line in Fig. 2 a was obtained with Thus, the transitions predicted by Eqs. 2 and 3 may not Hbl 40 Oe and Hbq 40 Oe 1 , while the dotted occur if the magnetic films are too thin. The boundaries line corresponds to Hbl 27 Oe and Hbq 54 Oe given by Eqs. 2 and 3 are shown by the solid lines in the 2 . Thus, quite different values of Hbl , Hbq , and PRB 58 BRIEF REPORTS 103 FIG. 4. Same as Fig. 3, for tCr 29 Å a , 33 Å b , and 35 Å c . FIG. 3. MOKE data in 100 Fe 40 Å /Cr tCr)/Fe 40 Å for tCr rately, as previously explained. Surprisingly, the same be- 15 Å a and tCr 25 Å b , as described in the text. havior were observed for all samples around the second AF peak. In Fig. 4 we show the magnetization curves observed 1 , can result in almost identical magnetization curves. in the samples with t Another ambiguous issue might happen if 1: as shown in Cr 29 Å Fig. 4 a , 33 Å Fig. 4 b , and 35 Å Fig. 3 c , with shapes qualitatively similar to that Fig. 2 b , where the solid dotted line corresponds to Hbl in Fig. 3 b , and thus, with the same restriction regarding the 40 Oe 40 Oe and Hbq 40 Oe 120 Oe , reason- coupling fields. We have also used the FMR and BLS tech- able fits to a hypothetically measured magnetization curve niques in some of these samples, with equally limited results. could be obtained for both AF solid line and ferromagnetic For instance, the symbols in Fig. 5 are the acoustic and optic dotted line coupling. Which coupling would be the right modes in the sample with t one? Finally, if 1, one might attempt to measure H Cr 25 Å, as measured by BLS, bl and with the same configuration described in Ref. 10. The solid Hbq by applying the magnetic field along the hard axis, as and dashed lines are numerical fits with parameters 4 M well. In this case, there is a second-order transition between S 20.5 kG, H a spin-flop and the saturated phase,11 at the critical field A 0.55 kOe, as for the sample with tCr 15 Å, and 1.0 and 1.2, respectively. The fits are almost identi- HC3 2Hbl 4Hbq HA . Thus, a measurement of HC2 cal, in spite of the different ratios used. and HC3 would allow one to determine Hbl and Hbq, pro- vided HA is known. However, the fitting parameter HA is usually much larger than Hbq , and it is difficult to measure HC3 precisely, given that this transition is of second-order nature. In the remainder of the paper, we apply the results above to the Fe 40 Å /Cr(tCr)/Fe 40 Å system. The samples were grown onto 100 MgO substrates by sputter deposition in a UHV chamber, and belong to the same batch as the sample with tCr 15 Å discussed in Ref. 10. Figure 3 shows room- temperature MOKE measurements in two representative samples located close to the first and second AF peaks,10 namely, tCr 15 Å Fig. 3 a and tCr 25 Å Fig. 3 b , re- spectively. Figure 3 a is clearly a situation in which 1, and the measured values of HC1 and HC2 yield Hbl 150 Oe and Hbq 50 Oe, i.e., 0.33, consistent with the phase FIG. 5. Magnon frequencies for q 1.22 105 cm 1 vs external diagram in Fig. 1 a . The reduced value of Hbl is apparent field H0, applied along an easy magnetization axis in 100 Fe 40 for the sample with tCr 25 Å, which exhibits only the tran- Å /Cr 25 Å /Fe 40 Å . Symbols are BLS data: circles for the optic sition between the 90° and the saturated phases. In this case mode and triangles for the acoustic mode. The lines are theoretical 1, and we cannot determine the two coupling fields accu- fits Ref. 9 with Hbq / Hbl 1.0 solid line and 1.2 dashed line . 104 BRIEF REPORTS PRB 58 We thank N. S. Almeida for valuable discussions on Federal Agencies CNPq, CAPES, PADCT, and FINEP, and phase diagrams and for sending us a preprint of Ref. 15 prior by the Pernambuco State Agency FACEPE. The work at to publication, and K. P. Roche for technical assistance. IBM was partially supported by the Office of Naval Re- The work at UFPE has been supported by the Brazilian search. *Present address: Departamento de Fi´sica Teo´rica e Experimental, Aguiar, and S. S. P. Parkin, J. Appl. Phys. to be published . Universidade Federal do Rio Grande do Norte, Caixa Postal 1641, 10 A. Azevedo, C. Chesman, S. M. Rezende, F. M. de Aguiar, X. 59072-970 Natal, Brazil. Bian, and S. S. P. Parkin, Phys. Rev. Lett. 76, 4837 1996 . 1 P. Gru¨nberg, R. Schreiber, Y. Pang, M. O. Brodsky, and H. Sow- 11 S. M. Rezende, M. A. Lucena, F. M. de Aguiar, A. Azevedo, C. ers, Phys. Rev. Lett. 57, 2442 1986 . Chesman, P. Kabos, and C. E. Patton, Phys. Rev. B 55, 8071 2 M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. 1997 . Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, 12 M. A. Lucena, F. M. de Aguiar, S. M. Rezende, A. Azevedo, C. Phys. Rev. Lett. 61, 2472 1988 . Chesman, and S. S. P. Parkin, J. Appl. Phys. 81, 4770 1997 . 3 S. S. P. Parkin, N. More, and K. P. Roche, Phys. Rev. Lett. 64, 13 B. Dieny and J. P. Gavigan, J. Phys.: Condens. Matter 2, 187 2304 1990 . 1990 ; W. Folkerts, J. Magn. Magn. Mater. 94, 302 1991 . 4 M. Ruhrig, R. Shafer, A. Hubert, R. Mosler, J. A. Wolf, S. 14 M. Maccio and M. G. Pini, Phys. Rev. B 49, 3283 1994 ; N. S. Demokritov, and P. Gru¨nberg, Phys. Status Solidi A 125, 625 Almeida and D. L. Mills, ibid. 52, 13 504 1995 ; H. J. Elmers, 1991 . G. Liu, H. Fritzche, and U. Gradmann, ibid. 52, R696 1995 . 5 U. Ko¨bler, K. Wagner, R. Wiechers, A. Fuss, and W. Zinn, J. 15 T. L. Fonseca and N. S. Almeida, Phys. Rev. B 57, 76 1998 . Magn. Magn. Mater. 103, 236 1992 . 16 M. Schafer, S. Demokritov, S. Mu¨ller-Pfeiffer, R. Schafer, M. 6 W. J. Gallagher, S. S. P. Parkin, Yu Lu, X. P. Bian, A. Marley, K. Schneider, P. Gru¨nberg, and W. Zinn, J. Appl. Phys. 77, 6432 P. Roche, R. A. Altman, S. A. Rishton, C. Jahnes, T. M. Shaw, 1995 . and Gang Xiao, J. Appl. Phys. 81, 3741 1997 . 17 P. Gru¨nberg, S. Demokritov, A. Fuss, M. Vohl, and J. A. Wolf, J. 7 J. M. Daughton, J. Appl. Phys. 81, 3758 1997 . Appl. Phys. 69, 4789 1991 . 8 B. A. Gurney, V. S. Speriosu, D. R. Wilhoit, H. Lefakis, R. E. 18 A. Fuss, S. Demokritov, P. Gru¨nberg, and W. Zinn, J. Magn. Fontana, Jr., D. E. Heim, and M. Dovek, J. Appl. Phys. 81, 3998 Magn. Mater. 103, L211 1992 . 1997 . 19 R. J. Hicken, C. Daboo, M. Gester, A. J. R. Ives, S. J. Gray, and 9 S. M. Rezende, C. Chesman, M. A. Lucena, A. Azevedo, F. M. de J. A. C. Bland, J. Appl. Phys. 78, 6670 1995 .