JOURNAL OF APPLIED PHYSICS VOLUME 89, NUMBER 4 15 FEBRUARY 2001 Magnetization in quasiperiodic magnetic multilayers with biquadratic exchange coupling C. G. Bezerra Departamento de Fi´sica, Universidade Federal do Rio Grande do Norte, 59072-970, Natal-RN, Brazil J. M. de Arau´jo Departamento de Cie ncias Naturais, Universidade Estadual do Rio Grande do Norte, 59610-210, Mossoro´-RN, Brazil C. Chesman and E. L. Albuquerquea) Departamento de Fi´sica, Universidade Federal do Rio Grande do Norte, 59072-970, Natal-RN, Brazil Received 17 July 2000; accepted for publication 20 November 2000 A theoretical study of the magnetization curves of quasiperiodic magnetic multilayers is presented. We consider structures composed by ferromagnetic films Fe with interfilm exchange coupling provided by intervening nonferromagnetic layers Cr . The theory is based on a realistic phenomenological model, which includes the following contributions to the free magnetic energy: Zeeman, cubic anisotropy, bilinear, and biquadratic exchange energies. The experimental parameters used here are based on experimental data recently reported, which contain sufficiently strong biquadratic exchange coupling. © 2001 American Institute of Physics. DOI: 10.1063/1.1340600 I. INTRODUCTION to the bilinear exchange coupling. Therefore, the biquadratic The study of the properties of magnetic multilayers has coupling can play a remarkable role in the properties of mag- been one of the most investigated fields in the last decade. netic multilayers. The understanding of a number of intriguing results became On the other hand, from an experimental point of view, an exciting challenge from both a theoretical and experimen- due to the rapid development of the crystal growth tech- tal point of view. In their pioneer work, Gru¨nberg and niques, it is now possible to tailor a wide class of magnetic collaborators1 reported evidence of an antiferromagnetic bi- multilayers, whose film thickness is extremely well con- linear exchange coupling in Fe/Cr/Fe structures. After that, trolled. As a consequence, there are magnetic phases and Baibich et al.2 noticed a sudden fall in the electrical resis- properties which are not shared by the constituent materials. tance of Fe/Cr magnetic multilayers when an external mag- It is known that magnetic properties can depend strongly netic field was applied. The effect was so striking that it was on the stacking pattern of the layers. Under this aspect, the called giant magnetoresistance, and recently it has been physical properties of a class of artificial material, the so- widely considered for applications in information storage called quasiperiodic structures recently became an attractive technology.3 Through magnetoresistance measurements, Par- field of research. Quasiperiodic structures, which can be ide- kin et al.4 observed an oscillatory behavior of the exchange alized as the experimental realization of a one-dimensional coupling in magnetic metallic multilayers as a function of the quasicrystal, are composed by the superposition of two or nonmagnetic spacer thickness. This work was seminal to a more building blocks that are arranged in a desired manner. number of experimental studies on Fe/Cr/Fe structures with They can be defined as an intermediate state between an different nonmagnetic spacer thickness. Later on, in 1991, ordered system a periodic crystal and a disordered one an Ru¨hrig et al.5 showed evidence of a noncolinear alignment amorphous solid .9,10 One of the most interesting features of (90°) between ferromagnetic layers in Fe/Cr magnetic mul- these systems is that the long range correlations, induced by tilayers, for nonmagnetic spacer thickness, where the bilinear the construction of the systems, are reflected in their various exchange coupling was small. This behavior could not be spectra. In fact, many physical properties of quasiperiodic explained considering only the usual bilinear exchange cou- systems have been studied such as light propagation,11 pling in the free magnetic energy. In fact, the inclusion of a phonons,12 electronic transmission,13 polaritons,14 and biquadratic exchange term in the free magnetic energy of the magnons.15 In all of these situations, despite the diversity of system allows the stabilization of noncolinear alignments. the systems a common feature is present, namely, a fractal Until recently, it was found that the biquadratic exchange spectra of energy, which can be considered as their basic coupling was too small when compared to the bilinear ex- signature.16,17 However, only very recently were efforts change coupling. However, Azevedo and co-workers6­8 pre- taken towards the understanding of the properties of quasip- sented a number of experimental results in Fe/Cr/Fe samples eriodic magnetic multilayers.15,18 which show the biquadratic exchange coupling comparable The main aim of this article is a contribution to the un- derstanding of the effects of the quasiperiodic arrangement a Author to whom correspondence should be addressed; electronic mail: on the magnetization curves in magnetic multilayers. We are ela@dfte.ufrn.br interested in magnetic phases and alignments that are only 0021-8979/2001/89(4)/2286/7/$18.00 2286 © 2001 American Institute of Physics Downloaded 28 Feb 2001 to 148.6.169.65. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html J. Appl. Phys., Vol. 89, No. 4, 15 February 2001 Bezerra et al. 2287 In a given generation SN , the total number of letters is given by the Fibonacci number FN , which is obtained by the rela- tion FN FN 1 FN 2 , with F0 F1 1. Also, FN 1 and FN 2 are the number of letters A and B, respectively. As the generation order increases (N 1), the ratio FN /FN 1 ap- proach to (1 5)/2, an irrational number which is known as the golden mean. It is also possible to obtain the number of letters A and B for a given generation by the substitution matrix of the Fibonacci sequence MF from,20 nN 1 N A nA M . 1 nN 1 F N B nB Here (nN 1 N 1 A ,nB ) are the number of letters A and B in the (N 1)th generation, and (nN N A ,nB ) are the number of letters A and B in the Nth generation. The explicit form of the substitution matrix for the Fibonacci sequence is, FIG. 1. The third and fifth Fibonacci generations and their magnetic coun- terpart. M F 1 1 , 2 1 0 whose first eigenvalue is the golden mean . due to the quasiperiodicity of the system. We have studied In Fig. 1, we show the third and fifth Fibonacci genera- Fe/Cr 100 structures which follow a Fibonacci and a double tions and their magnetic counterparts. Note that the third period or generalized Fibonacci quasiperiodic sequences. Fibonacci generation corresponds to a trilayer Fe/Cr/Fe, and The layout of the article is as follows: In Sec. II we in the fifth Fibonacci generation there is a double Fe layer. It discuss the physical model used here, with emphasis in the is easy to show that the Fibonacci magnetic multilayers, for description of the quasiperiodic sequences. In Sec. III we any generation, are composed by single Cr layers, single Fe define the contributions to the magnetic energy. The numeri- layers, and double Fe layers. The number of Fe single layers cal methods, used to obtain the equilibrium configuration, is 1 FN 2 , the number of Fe double layers is 1 FN 1 are described in Sec. IV. In Sec. V, the results are presented FN 2 and the number of Cr layers is FN 2 . It should be and discussed. Finally, we draw the conclusions in Sec. VI. observed that only odd Fibonacci generations have a mag- netic counterpart they start and finish with an Fe building II. PHYSICAL MODEL layer . A quasiperiodic structure can be experimentally con- B. The double period magnetic multilayers structed juxtaposing two building blocks or, as considered here, building layers following a given quasiperiodic se- The Nth generation of the double period sequence can be quence. We choose Fe as the building layer associated with obtained from the relations, the letter A, and Cr as the building layer associated with the S , 3 letter B see Fig. 1 . Therefore, we only take into account the N SN 1SN 1 generation of sequences that start and finish with an Fe with building layer, which means an even number of Fe layers, to S S guarantee a real magnetic counterpart. In this way we also N N 1SN 1 N 2 , 4 avoid the intriguing behavior found when even and odd num- The initial conditions are S0 Ae S1 AB. We can, alterna- bers of Fe layers are considered.19 In this article we have tively, use the substitution rules A AB, B AA. The considered two quasiperiodic sequences, namely, the Fi- double period generations are bonacci and the double period sequences. S0 A , S1 AB , S2 ABAA , etc. A. The Fibonacci magnetic multilayers In a given generation SN , the total number of letters is 2N, The Nth generation of the Fibonacci sequence can be and the number of letters A and B for consecutive genera- determined appending the N 2 generation to the N 1 one, tions can be related by the substitution matrix of the double i.e., S period sequence M N SN 1SN 2 (N 2). This algorithm construction re- dp , i.e.,20 quires initial conditions which are chosen to be S0 B and N 1 N S1 A. The Fibonacci generations can also be alternatively nA nA obtained by an iterative process from the substitution rules M . 5 nN 1 dp nN or inflation rules , A AB, B A. The Fibonacci genera- B B tions are Here (nN 1 N 1 A ,nB ) are the number of letters A and B in the S N N 0 B , S1 A , S2 AB , S3 ABA , etc. (N 1)th generation, and (nA ,nB) are the number of letters Downloaded 28 Feb 2001 to 148.6.169.65. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html 2288 J. Appl. Phys., Vol. 89, No. 4, 15 February 2001 Bezerra et al. within a given ferromagnetic film. Therefore, we can repre- sent the ferromagnetic films as classical magnetizations M , composed by the real spins within the films, which are strongly coupled by the intrafilm exchange coupling. These classical magnetizations interact through the interfilm ex- change coupling and they can present some anisotropy de- pending on the structure studied. It should be noted that this system is isomorphous to a one-dimensional chain of classi- cal spins. The global behavior of this system is well described by a realistic phenomenological theory in terms of the free mag- netic energy,7 i.e., ET Ez Eca Ebl Ebq . 7 Here Ez is the Zeeman energy between the ferromagnetic films and the external applied magnetic field , Eca is the cu- bic crystalline anisotropy energy which we consider present in the ferromagnetic films and Ebl and Ebq are the bilinear and the biquadratic exchange coupling energies between the ferromagnetic films , respectively. The explicit form of the free magnetic energy can be written as n n t E iKca T tiM i*H i 1 i 1 M i 4 M2 2 2 2 2 2 ixM iy M ixM iz M iyM iz FIG. 2. Same as Fig. 1 for the second and fourth double period generations. n 1 M n 1 M J i* M i 1 i* M i 1 2 bl Jbq . 8 i 1 M i M i 1 i 1 M i 2 M i 1 2 A and B in the Nth generation. As the generation number increases (N 1), the ratio between the number of letters A Here, H is the external magnetic field which is applied in the and B tends to 2. The explicit form of the substitution matrix film plane, ti is the thickness of the ith Fe layer, M i is the for the double period sequence is classical magnetization of the ith Fe layer, and Kca is the cubic anisotropy constant. Also, Jbl and Jbq are the bilinear M and the biquadratic exchange couplings, respectively. This F 1 1 . 6 2 1 expression, after a tedious but straight calculation, takes the In Fig. 2 we show the second and fourth double period form, generations and their magnetic counterparts. The double pe- n 1 riod magnetic multilayers are composed by single Fe layers, E T tiMiH cos i H 4 tiKca sin2 2 i double Fe layers, triple Fe layers, and single Cr layers. It i 1 should be observed that, contrary to the Fibonacci case, only n 1 even double period generations have a magnetic counterpart. Jbl cos i i 1 Jbq cos2 i i 1 . 9 i 1 III. MAGNETIC ENERGY Here i is the angular orientation of the magnetization of the ith Fe layer and We consider magnetic multilayers whose constituents H is the angular orientation of the magnetic field. From this point we consider are Fe ferromagnetic films, separated by Cr nonmagnetic H 0, which means that the magnetic field is applied along the easy axis. It is usual to films. We take the xy plane as the film plane and the z axis as write the total free magnetic energy in terms of experimental the growth direction. We consider that the magnetic films are parameters, like uniformly magnetized and that they behave as mon- odomains. We also consider that they do not present dynami- 2Kca cal excitations and that the very strong demagnetization field, Hca M , 10 S generates by tipping the magnetization out of the plane, will suppress any tendency for the magnetization to tilt out of J H bl , 11 plane. Therefore, the degrees of freedom of the magnetiza- bl tMS tions are restricted to the xy plane. The interfilm exchange couplings between the ferromagnetic films are weak when J H bq . 12 compared to the strong exchange couplings between spins bq tMS Downloaded 28 Feb 2001 to 148.6.169.65. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html J. Appl. Phys., Vol. 89, No. 4, 15 February 2001 Bezerra et al. 2289 In this way we obtain a final expression for the free magnetic i Choose an initial point in parameter space, correspond- energy per unit area ing to an initial configuration j , and calculate the associ- ated energy E E n j . T 1 ii Choose a second point in parameter space, correspond- tM ti /t H0 cos i S i 1 8 Hca sin2 2 i ing to a second configuration j 1 , and calculate the as- n 1 sociated energy Ej 1 . H iii If E Ej 1 Ej 0, j 1 is the new configuration bl cos i i 1 i 1 of the system. iv If E 0, we define the probability p exp Hbq cos2 i i 1 . 13 ( E/kBT) and choose a random number 0 x 1. If x Here t is the thickness of a single Fe layer which is consid- p, j 1 is the new configuration of the system. Other- ered to be the basic tile, M wise, i is assumed to be equal to M S j is maintained as the configuration of the system. the saturation magnetization , and H v This procedure is executed again and again until the ca is the cubic anisot- ropy field which turns the 100 direction an easy direction. equilibrium is reached. Hbl is the bilinear exchange coupling field which favors an- tiferromagnetic alignment when negative, and ferromagnetic alignment when positive. Hbq is the biquadratic exchange B. The gradient method coupling field which is experimentally found to be positive and favors a noncolinear alignment 90° between two adja- The second method that we have used was the so-called cent magnetizations. gradient method.23 This method is based on the directional Once the free magnetic energy is determined, we can derivative of the cost function the magnetic energy in the calculate the equilibrium configuration for specific values of search of its global minimum.23 In this way, we need to the experimental parameters as a function of the external calculate the gradient of ET with relation to the set applied field. In simple situations, the equilibrium configura- n tion can be analytically obtained by equating to zero the E E T derivatives of the magnetic energy with respect to the angle T i . 15 i 1 i . However, in most cases this leads to transcendental equa- tions which can not be analytically solved. From a numerical From this relation we execute the following algorithm to find point of view, many methods have been proposed to calcu- the equilibrium configuration, late the equilibrium positions of the magnetizations. In the i We generate a configuration in the parameter space j next section we describe the methods used in this article. from which we calculate the associated energy Ej and the gradient of the cost function. ii A second point in the parameter space is generated by IV. NUMERICAL METHODS j 1 j ET . Here controls the size of the dis- placement in the direction E In this section we want to find the global minimum of T . iii The energy of the second point is calculated and if the cost function Ej 1 Ej , the parameter the size of the displacement is E divided by two and we go back to ii . Otherwise, we instead T ET 1 , 2 , . . . , n , 14 generate a new configuration from where j 1 . n can assume values in the range 0,2 and it de- fines a n-dimensional space. When the dimension of this In the last step the reduction of is limited by the pre- space is high, the cost function has a rough surface, i.e., there cision value required for . This limit is reached when are many local minima which make this difficult to find the ET . global minimum. There are many numerical methods to We have used the two methods discussed above to ob- solve this problem.21 In our specific case two methods were tain the equilibrium positions of the magnetizations. Each successfully used, namely, simulated annealing and the so- method was applied for each value of the applied magnetic called gradient method. field and for each set of experimental parameters. We choose the configuration with the lowest energy furnished by both A. Simulated annealing method methods as the equilibrium configuration. Introduced by Kirkpatrick et al.22 simulated annealing comes from the fact that the heating annealing and slowly cooling a metal, brings it into a more uniformed crystalline V. NUMERICAL RESULTS state, which is believed to be the state where the free energy of bulk matter takes its global minimum. The role played by In this section we present the numerical results obtained the temperature is to allow the configurations to reach higher for the magnetization curves of quasiperiodic magnetic mul- energy states with probability given by Boltzmann's expo- tilayers. In all situations we have considered the cubic an- nential law. Then they can overcome energy barriers that isotropy effective field Hca 0.5 kOe which corresponds to would otherwise force them into local minima. In general, a Fe 100 with t 30 Å. In our calculations we have used two simulated annealing technique can be written as follows: sets of experimental values for the bilinear and biquadratic Downloaded 28 Feb 2001 to 148.6.169.65. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html 2290 J. Appl. Phys., Vol. 89, No. 4, 15 February 2001 Bezerra et al. FIG. 3. Magnetization vs applied field for the third a , fifth b , and seventh FIG. 4. Same as Fig. 3, but for Hbq / Hbl 1.0, corresponding to a realistic c Fibonacci generations with H sample whose Cr thickness is about 25 Å. bq / Hbl 0.10, corresponding to a realis- tic sample whose Cr thickness is about 10 Å. We have considered the cubic anisotropy effective field Hca 0.5 kOe, which corresponds to Fe 100 with t 30 Å. Fe layer thickness, for the fifth and seventh generations, the magnetization is not zero even for zero magnetic field. In Fig. 4 we show the results for the second set of parameters. exchange coupling: i the first one with H For the third generation, due to the strong biquadratic field, bl 1.0 kOe and H there is no antiparallel phase in the low field region. Two bq 0.1 kOe. It lies in the region of the first antiferromag- netic peak of the bilinear exchange coupling, corresponding magnetic phases are present: 90° (H 72 Oe and saturated to a realistic sample whose Cr thickness is about 10 Å; ii (H 72 Oe . The fifth generation presents three magnetic the second set with H phases: i 90° (H 72 Oe ; ii almost saturated (72 Oe bl 0.035 kOe and Hbq 0.035 kOe. It is in the region of the second antiferromagnetic peak of the H 0.14 kOe ; and iii saturated (H 0.14 kOe . The sev- bilinear exchange coupling, corresponding to a realistic enth generation presents four magnetic phases from 90° (H sample whose Cr thickness is about 25 Å. 36 Oe to the saturated regime (H 0.14 kOe . All transi- tions are of first order. Note the striking self-similar pattern A. Fibonacci magnetic multilayers shown by the magnetization profile in this figure see the The magnetization curves for the first set of parameters windows . of the Fibonacci magnetic multilayers are shown in Fig. 3. For the third generation which corresponds to the well B. Double period magnetic multilayers known Fe/Cr/Fe trilayer , in the low field region, the magne- Figures 5 a and 5 b show our results for the double tizations are antiparallel. As the field increases, they continu- period magnetic multilayers, considering the first set of pa- ously rotate toward the field direction second order phase rameters. For the second generation, due to the double Fe transition and the saturation is reached when the external layer, the magnetization has about 1/3 of its saturation value magnetic field H 1.91 kOe. For the fifth generation there for zero magnetic field. There is a first order phase transition are two first order phase transitions at H 0.71 kOe and H from antiparallel to an asymmetric phase at H 0.69 kOe. In 0.87 kOe, respectively. The saturation is reached at H this phase, the magnetizations are asymmetrically oriented 2.93 kOe. For the seventh generation, there are three first along the magnetic field. When H 1.34 kOe the saturated order phase transitions at H 0.28 kOe, H 0.96 kOe and phase emerges. For the fourth generation, for zero magnetic H 1.06 kOe, respectively. The saturation is reached at H field, the magnetization has about 10% of its saturation value 3.03 kOe. For this set of parameters the majority of the due to the different thickness of the Fe layers. There is a first transitions are of second order. Note that due to the different order phase transition at H 0.29 kOe and the saturation is Downloaded 28 Feb 2001 to 148.6.169.65. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html J. Appl. Phys., Vol. 89, No. 4, 15 February 2001 Bezerra et al. 2291 region, due to the strong biquadratic field. For the second generation, the magnetization is about 2/3 of its saturation value when H 0. There are two magnetic phases: i 90° (0 H 72 Oe and ii saturated (H 72 Oe . For the fourth generation, the magnetization for zero magnetic field is about 1/2 of its saturation value. Four magnetic phases are present, from the 90° (0 H 38 Oe to the saturated phase (H 0.14 kOe . As in the Fibonacci case, a self-similar pattern is also present in the magnetization curves see the window . VI. CONCLUSIONS We have studied quasiperiodic magnetic multilayers, composed by ferromagnetic Fe layers separated by nonmag- netic Cr layers, arranged according to the Fibonacci and double period quasiperiodic sequences. We consider that the Fe layers are linked by bilinear and biquadratic exchange couplings through Cr layers and present cubic anisotropy. The external magnetic field is applied in the plane of the layers and along an easy axis. We have used two numerical methods to determine the equilibrium configurations of the layers's magnetizations. The magnetization curves of these artificial structures were calculated considering two sets of experimental parameters recently reported.6­8 Our results FIG. 5. Magnetization vs applied field for the second a and fourth b show that quasiperiodic magnetic multilayers exhibit a rich double period generations with Hbq / Hbl 0.10. The cubic anisotropy ef- variety of configurations induced by the external magnetic fective field is again Hca 0.5 kOe. The Cr thickness is 10 Å. field. In particular two points may be emphasized: i the effect of different thickness of Fe layers and ii the effect of reached at H 3.27 kOe. All other phase transitions are of the biquadratic exchange coupling. second order. For the second set of parameters see Fig. 6 , The effect of different thickness of Fe layers is evident on the contrary, all transitions are of first order. For this set in the low field region. In that region, due to these differ- of parameters, there is no antiparallel phase in the low field ences, there is a net magnetization even if the alignment is antiparallel and the external magnetic field is zero. Besides, the nature of the phase transitions are changed by the differ- ent thickness Fig. 3 a shows only second order phase tran- sitions, while Fig. 5 a presents an additional first order phase transition . These results suggest that, varying the thickness of Fe layers, it is possible to tailor magnetic mul- tilayers to present desired specific phase transitions and criti- cal fields. However, as the thickness of Fe layers increases, the crystalline anisotropy of Fe 100 films on Cr 100 also increases. Fortunately, as a characteristic of the quasiperiodic multilayers arrangements considered here, the maximum number of joint Fe layers is two for the Fibonacci case and three for the double period case , no matter the value of their generation numbers. Besides, from a thickness greater than 40 Å, the crystalline anisotropy reaches saturation.7 On the other hand, the biquadratic exchange coupling plays a remarkable role in the features of the magnetization curves. For example, when the bilinear exchange coupling prevails, the majority of the transitions are of second order character see Figs. 3 and 5 . However, when the biquadratic exchange is compared to the bilinear one, in the presence of a stronger crystalline anisotropy,8 the transitions are charac- terized by discontinuous jumps in the magnetization that in- dicate first order phase transitions. This can be considered as the basic signature of the biquadratic exchange coupling see FIG. 6. Same as Fig. 5, but for H Figs. 4 and 6 , although for the case where there is no biqua- bq / Hbl 1.0, corresponding to a realistic sample whose Cr thickness is about 25 Å. dratic term, a first order phase transition appears due to the Downloaded 28 Feb 2001 to 148.6.169.65. Redistribution subject to AIP copyright, see http://ojps.aip.org/japo/japcpyrts.html 2292 J. Appl. Phys., Vol. 89, No. 4, 15 February 2001 Bezerra et al. anisotropy.24 Furthermore, as shown by the windows in these 3 W. J. Gallagher et al., J. Appl. Phys. 81, 3741 1997 . figures, the magnetization curves of higher generations re- 4 S. S. P. Parkin, N. More, and K. P. Roche, Phys. Rev. Lett. 64, 2304 produce the magnetization curves of lower generations. This 1990 . 5 self-similar behavior is a general characteristic of quasiperi- M. Ruhrig, R. Schafer, A. Hubert, R. Mosler, J. A. Wolf, S. Demokritov, and P. Gru¨nberg, Phys. Status Solidi A 125, 635 1991 . odic systems, although it is not present when the bilinear 6 A. Azevedo, C. Chesman, S. M. Rezende, F. M. de Aguiar, X. Bian, and exchange prevails Figs. 3 and 5 . A possible explanation for S. S. P. Parkin, Phys. Rev. Lett. 76, 4837 1996 . these different behaviors is because the biquadratic exchange 7 S. M. Rezende, C. Chesman, M. A. Lucena, A. Azevedo, F. M. de Aguiar, coupling induces long range correlations that emphasize the and S. S. P. Parkin, J. Appl. Phys. 84, 958 1998 . quasiperiodicity of the system. These long range correlations 8 C. Chesman, M. A. Lucena, M. C. de Moura, A. Azevedo, F. M. de make the whole structure seeing its quasiperiodicity, which Aguiar, and S. M. Rezende, Phys. Rev. B 58, 101 1998 . 9 D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, is reflected in the magnetization curves. This argument is 1951 1984 . reinforced by previous works on the correlation lengths of 10 P. J. Steinhardt and S. Ostlund, The Physics of Quasicrystals World Sci- magnetic systems presenting biquadratic exchange coupling entific, Singapore, 1987 . see, for example, Sørensen and Young25 . 11 M. S. Vasconcelos, E. L. Albuquerque, and A. M. Mariz, J. Phys.: Con- The most appropriate experimental technique for study- dens. Matter 10, 5839 1998 . 12 M. Quilichini and T. Janssen, Rev. Mod. Phys. 69, 277 1997 . ing the magnetization curves of magnetic films is the 13 P. M. C. de Oliveira, E. L. Albuquerque, and A. M. Mariz, Physica A 227, magneto-optical Kerr effect MOKE .8 However, because 206 1996 . the MOKE measurements provide surface sensitivity on the 14 M. S. Vasconcelos and E. L. Albuquerque, Phys. Rev. B 57, 2826 1998 . scale of the optical penetration depth ( 10 Å , it is neces- 15 C. G. Bezerra and E. L. Albuquerque, Physica A 245, 379 1997 ; 255, sary to use also a superconductor quantum interface device 285 1998 . 16 magnetometry.19 The two techniques prove complementary M. Kohmoto, B. Sutherland, and K. Iguchi, Phys. Rev. Lett. 58, 2436 1987 ; M. Kohmoto, B. Sutherland, and C. Tang, Phys. Rev. B 35, 1020 in understanding the switching behavior of the multilayer 1987 . films, as far as the magnetization curves are concerned. We 17 C. G. Bezerra, E. L. Albuquerque, and E. Nogueira, Jr., Physica A 267, hope that the present results can stimulate experimental stud- 124 1999 ; M. S. Vasconcelos, E. L. Albuquerque and E. Nogueira, Jr. , ies of these structures. ibid. 268, 165 1999 . 18 C. G. Bezerra, J. M. de Arau´jo, C. Chesman, and E. L. Albuquerque, Phys. Rev. B 60, 9264 1999 . ACKNOWLEDGMENTS 19 R. W. Wang, D. L. Mills, E. E. Fullerton, J. E. Matson, and S. D. Bader, The authors would like to thank the Brazilian Research Phys. Rev. Lett. 72, 920 1994 . 20 Council CNPq for financial support and CESUP-RS where L. Turban, P. E. Berche, and A. B. Berche, J. Phys. A 27, 6349 1994 . 21 Handbook of Global Optimization, edited by R. Horst and P. M. 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