PHYSICAL REVIEW B VOLUME 60, NUMBER 13 1 OCTOBER 1999-I Self-similar magnetoresistance of Fibonacci ultrathin magnetic films C. G. Bezerra Departamento de Fi´sica Teo´rica e Experimental, Universidade Federal do Rio Grande do Norte, 59072-970, Natal-RN, Brazil J. M. de Arau´jo Departamento de Fi´sica Teo´rica e Experimental, Universidade Federal do Rio Grande do Norte, 59072-970, Natal-RN, Brazil and Departamento de Cie ncias Naturais, Universidade Estadual do Rio Grande do Norte, 59610-210, Mossoro´-RN, Brazil C. Chesman and E. L. Albuquerque Departamento de Fi´sica Teo´rica e Experimental, Universidade Federal do Rio Grande do Norte, 59072-970, Natal-RN, Brazil Received 28 June 1999 We study numerically the magnetic properties magnetization and magnetoresistance of ultrathin magnetic films Fe/Cr grown following the Fibonacci sequence. We use a phenomenological model which includes Zeeman, cubic anisotropy, bilinear, and biquadratic exchange energies. Our physical parameters are based on experimental data recently reported, which contain biquadratic exchange coupling with magnitude comparable to the bilinear exchange coupling. When biquadratic exchange coupling is sufficiently large a striking self- similar pattern emerges. S0163-1829 99 07437-8 The discovery of quasicrystals in 1984 Ref. 1 aroused a sistance in granular systems. On the other hand, from a tech- great interest, both theoretically and experimentally, in qua- nological point of view as we will show later in this letter siperiodic systems. One of the most important reasons for the BEC associated with quasiperiodicity permits us to con- that is because they can be defined as an intermediate state trol magnetic field regions, where magnetoresistance remains between an ordered crystal their definition and construction almost constant before saturation. follow purely deterministic rules and a disordered solid The aim of this work is to investigate the influence of many of their physical properties exhibit an erraticlike quasiperiodicity on the magnetic properties of ultrathin mag- appearance .2 On the theoretical side, a wide variety of par- netic films. In particular, we are interested in Fe/Cr 100 ticles, namely, electrons,3 phonons,4 plasmon-polaritons,5 structures, which follow a Fibonacci sequence, whose ex- spin waves,6 etc., have been studied. A quite complex fractal perimental magnetic parameters were recently reported by energy spectrum, which can be considered as their basic sig- Rezende et al.16 nature, is a common feature of these systems. On the experi- A Fibonacci structure can be grown experimentally by mental side, the procedure to grow quasiperiodic superlat- juxtaposing two building blocks A and B following a Fi- tices became standard after Merlin et al.,7 who reported the bonacci sequence. In our specific case we choose Fe as the realization of the first quasiperiodic superlattice following building block A and Cr as the building block B. A Fibonacci the Fibonacci sequence by means of molecular beam epitaxy sequence SN is generated by appending the N 2 sequence MBE . to the N 1 one, i.e., SN SN 1SN 2 (N 2). This construc- Parallel to these developments in the field of quasicrys- tion algorithm requires initial conditions which are chosen to tals, the properties of magnetic exchange interactions be- be S0 B and S1 A. The Fibonacci generations are S0 tween ferromagnetic films separated by nonmagnetic spacers B , S1 A , S2 AB , S3 ABA , etc. Therefore, the have been also widely investigated.8 The discovery of physi- well known trilayer Fe/Cr/Fe is the magnetic counterpart of cal properties such as antiferromagnetic exchange coupling,9 the third Fibonacci generation (A/B/A). We remark that giant magnetoresistance GMR ,10 oscillatory behavior of the only odd Fibonacci generations have a magnetic counterpart, exchange coupling,11 and biquadratic exchange coupling because they start and finish with an A Fe building block. BEC ,12 made these films excellent options for technologi- Figure 1 shows schematically the third and fifth Fibonacci cal applications and attractive objects of research. For ex- generations and their magnetic counterpart, where t d is the ample, GMR in magnetic films has been widely considered thickness of a single Fe layer single Cr layer . It is impor- for applications in information storage technology.13 tant to note a double Fe layer whose thickness is 2t in the It is known that GMR also occurs in nonperiodic granular fifth generation corresponding to a double letter A. It is easy systems, such as Cu-Co alloys, consisting of ultrafine Co- to show that the quasiperiodic magnetic films, for any Fi- rich precipitate particles in Cu-rich matrix.14 Due to the fact bonacci generation, will be composed by single Cr layers, that precipitate particles of these heterogeneous alloys have single Fe layers and double Fe layers. an average diameter and an average spacing similar to mag- We consider the ferromagnetic films with magnetization netic films, the origin of GMR in granular systems is also in the plane xy and take the z axis as the growth direction similar to the one found in magnetic films.15 Therefore, qua- see Fig. 1 . The very strong demagnetization field, generates siperiodic systems which present magnetoresistive properties by tipping the magnetization out of plane, will suppress any can be a first step for a better understanding of magnetore- tendency for the magnetization to tilt out of plane. The glo- 0163-1829/99/60 13 /9264 4 /$15.00 PRB 60 9264 ©1999 The American Physical Society PRB 60 BRIEF REPORTS 9265 FIG. 1. The third and fifth Fibonacci generations and their mag- FIG. 2. Magnetization a and magnetoresistance b versus netic counterpart. magnetic field for the third Fibonacci generation with Hbl 150 Oe and Hbq 50 Oe. In a the arrows indicate the relative positions bal behavior of the system is well described by a simple of the magnetizations in each phase. theory in terms of the magnetic energy per unit area,16 i.e., ever, we got results in sufficiently large generations to infer E important informations about the effect of the quasiperiodic- T EZ Ebl Ebq Ea , 1 ity. where EZ is the Zeeman energy, Ebl is the bilinear energy, Theoretically, the spin-dependent scattering is accepted as Ebq is the biquadratic energy and Ea is the cubic anisotropy responsible for the GMR effect.15 It has been shown that energy. It is usual to write the total magnetic energy in terms GMR varies linearly with cos( ) when electrons form a of experimental parameters or effective fields of each inter- free-electron gas, i.e., there are no barriers between adjacent action, films.18 Here, is the angular difference between adjacent magnetizations. In metallic systems such as Fe/Cr this angu- E n T 1 lar dependence is valid and once the set i is found, we tM ti /t H0 cos i H obtain normalized values for magnetoresistance, i.e., S i 1 8 Ha sin2 2 i n 1 n 1 H R H 1 cos bl cos i i 1 Hbq cos2 i i 1 , 0 /R 0 i i 1 /2 n 1 , 3 i 1 i 1 2 where R(0) is the resistance at zero field. Now we present numerical calculations for the magneti- where t is the thickness of a single Fe layer and we assume zation and the magnetoresistance curves for Fibonacci ultra- Mi MS . Hbl is the conventional bilinear exchange coupling thin magnetic films. The physical motivation for that is be- field which favors antiferromagnetic alignment ferromag- cause the Fibonacci quasiperiodic structure can exhibit netic alignment if negative positive . We are concerned magnetic properties not found in the periodic case,6 and the here with the case Hbl 0 because magnetoresistive effects long range correlations induced by the construction of this occur only for this case. Hbq is the BEC field, which is re- system are expected to be reflected someway in the magne- sponsible for a 90° alignment between two adjacent magne- toresistance curves. We have considered physical parameters tizations and is experimentally found to be positive.12 Ha is based on realistic values of the magnetic coupling fields, the cubic anisotropy field which renders the (100) direction whose experimental data were recently reported.16 We as- an easy direction. H0 is the external in-plane magnetic field sume the cubic anisotropy field Ha 0.5 kOe, corresponding and H is its angular orientation. From now on we consider to Fe 100 with t 30 Å growth by sputtering.8 We choose H 0, which means that the magnetic field is applied along the bilinear and the biquadratic fields Hbl and Hbq , such that the easy axis. The thickness and the angular orientation of their values lie in three regions of interest: i close to the the ith Fe layer are given by ti and i , respectively. region of first antiferromagnetic-ferromagnetic transition The equilibrium positions of the magnetizations i are where Hbl is moderate;12 ii near to the maximum of first numerically calculated by minimizing the magnetic energy antiferromagnetic peak, where Hbl reaches its maximum given by Eq. 2 . It should be remarked that it has proved value;10 and iii in the second antiferromagnetic peak, where difficult for us to generate accurately configurations for Hbl is small and equal to Hbq .19 larger structures, mainly when the BEC is strong.17 How- In Fig. 2 we show the curves of the normalized magneti- 9266 BRIEF REPORTS PRB 60 FIG. 3. Magnetoresistance for the fifth a and seventh b Fi- FIG. 4. Magnetoresistance for the third a , fifth b , and seventh bonacci generations with the same parameters of Fig. 2. In a the c Fibonacci generations with Hbl 1.0 kOe and Hbq 0.1 kOe. relative positions of magnetizations are indicated by the arrows, and the Fe double layer is indicated by the bigger arrow. 2( Hbl 2Hbq). However, when the ratio between Hbq and zation and magnetoresistance versus the magnetic field, for Hbl is increased ( Hbl Hbq 35 Oe , we observe again a the third Fibonacci generation corresponding to the striking self-similar pattern see Fig. 5 , where each new Fe/Cr/Fe trilayer . We assumed Hbl 0.15 kOe and Hbq transition occurs for a value of magnetic field which is about 0.05 kOe ( Hbl Hbq). These parameters correspond to a a half of the previous one. For this set of parameters the realistic sample with Cr thickness equal to 15 Å . From there magnetoresistance is approximately 1/2 its value at zero one can identify two first order phase transitions at H magnetic field, because the magnetizations of the adjacent Fe 1 100 Oe and H films are nearly perpendicular to each other due to the strong 2 220 Oe. Also there are three magnetic phases presented: i an antiferromagnetic phase (H biquadratic field. For the third generation, Fig. 5 a , there is 0 100 Oe ; ii a 90° phase 100 Oe H only one transition at H 0 220 Oe ; and iii a 1 70 Oe and two magnetic phases: a saturated phase (H0 220 Oe . We remark that our numeri- cal calculations indicate that a first order phase transition occurs when Ha 2( Hbl 2Hbq). Since the transition mag- netic fields are the same for both the magnetization and the magnetoresistance, from now on we concentrate our discus- sion on the magnetoresistance curves, because it is easier to investigate their self-similar pattern. Figure 3 shows the normalized magnetoresistance curves for the fifth a and seventh b Fibonacci generations with the same experimental parameters considered in Fig. 2. In Fig. 3 a we can identify four first order phase transitions, where each one is due to a 90° jump of magnetization. This behavior is always displayed when the BEC is present in the magnetic energy. Previous works on phase diagrams have looked carefully at the origin and features of the so-called 90° phase.17,19 For the seventh generation there are eight first order phase transitions and nine magnetic phases are present from the antiferromagnetic phase (H0 38 Oe to the satu- rated one (H0 440 Oe . Note a clear self-similar pattern of magnetoresistance curves by comparing Figs. 2 and 3, i.e., the pattern of the trilayer Fe/Cr/Fe is always present in the next generations. On the contrary, when Hbl 1.0 kOe and Hbq 0.1 kOe ( Hbl Hbq), which correspond to a sample FIG. 5. Magnetoresistance for the third a , fifth b , and seventh with Cr thickness equal to 10 Å, the self-similarity is not c Fibonacci generations with Hbl Hbq 35 Oe, which corre- observed, as it is shown in Fig. 4. For this set of parameters, spond to a sample with Cr thickness equal to 25 Å . Note a striking the majority of phase transitions are of second order and we self-similar pattern. In a and b the arrows indicate the relative have found numerically that this occurs when Ha positions of the magnetizations in each phase. PRB 60 BRIEF REPORTS 9267 90° phase at H0 70 Oe and a saturated phase at H0 70 first system which presents magnetoresistance with self- Oe.19 In the fifth generation, Fig. 5 b , there are two transi- similar properties. In addition, from a technological point of tions at H1 70 Oe and H2 140 Oe, respectively. For the view, magnetoresistance with almost constant regions Figs. seventh generation, as one can see from Fig. 5 c , there are 3 and 5 opens new perspectives in information storage tech- three transitions at H1 35 Oe, H2 70 Oe, and H3 140 nology by the possibility of a recording system with more Oe. than two states. Certainly Fibonacci ultrathin magnetic films From the numerical results discussed above, we can infer can be realized experimentally following the procedures of that the magnetoresistance exhibits a self-similar behavior Refs. 12 or 20 to grow the samples. when a Hbq is comparable to Hbl and b there is a first order phase transition see Figs. 3 and 5 . A possible expla- We would like to thank A. M. Mariz, N. S. Almeida, and nation for that is because the BEC reinforces the quasiperi- G. M. Viswanathan for fruitful discussions, and the Brazilian odic order, which is responsible by the self-similarity in qua- Research Council CNPq for partial financial support. We are siperiodic systems. This is an unexpected effect of this also grateful to A. Albino, Jr. for Fig. 1 and CESUP-RS unusual exchange coupling and, as far as we know, this is the where part of the numerical calculations was done. 1 D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. 11 S. S. P. Parkin, N. 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