PHYSICAL REVIEW B VOLUME 57, NUMBER 13 1 APRIL 1998-I Quantum approach for magnetic multilayers at finite temperatures Lei Zhou Institute for Materials Research, Tohoku University, Sendai 980-77, Japan and Department of Physics, Fudan University, Shanghai 200433, People's Republic of China Liangbin Hu and Zhifang Lin Department of Physics, Fudan University, Shanghai 200433, People's Republic of China Yoshiyuki Kawazoe Institute for Materials Research, Tohoku University, Sendai 980-77, Japan Ruibao Tao Center for Theoretical Physics, Chinese Center of Advanced Science and Technology (World Laboratory), P.O. Box 8730, Beijing 100080, People's Republic of China and Department of Physics, Fudan University, Shanghai 200433, People's Republic of China Received 5 September 1997; revised manuscript received 3 November 1997 The Green's function technique has been employed to examine the finite-temperature properties of a mag- netic multilayer both in aligned and in canted spin configurations. A local coordinates system is introduced to describe complicated spin configurations which are determined by minimizing the free energy of the magnetic multilayer. Thermal averages of Szm m in each layer are selected to be independent variables, and the necessarily self-consistent equations are obtained successfully based on the random phase approximation. The temperature dependences of the coercivity, the hysteresis, and the magnetic multivalued recording have been discussed, and an application of the present approach to the canted spin state is given in a double-film system. S0163-1829 98 02413-8 I. INTRODUCTION to discuss a magnetic multilayer within the whole tempera- Magnetic multilayers received more and more attention in ture region. The GF method was introduced into magnetic recent years because they showed high application potentials systems in 1959 by the pioneer work of Bogolyubov and in magnetic devices and strong theoretical interests.1 Many Tyablibov who studied the thermodynamic properties of the exciting phenomena, such as the giant magnetoresistance spin-12 ferromagnetic systems.18 In 1962, Tahir-Kheli and ter Haar extended that technique to arbitrary spin cases effect,2 the giant magneto-impedance effect,3 the magneto- successfully.19 Since then, many authors have used this ap- optical recording,4 the spin reorientation phase transition,5 proach to discuss various kinds of magnetic systems.20,21 The etc., were discovered based on the magnetic orders in quasi- most remarkable merit of the GF method is its approximate two-dimensional systems at finite temperatures even near validity within the entire temperature region, which the other the Curie temperature . Recently, many efforts were devoted approaches such as spin-wave theory, molecular-field theory, to complicated coupled magnetic multilayers to improve the and high-temperature expansion theory22 did not possess. field sensitivity of the reading device,6 to enhance the In this paper, we would like to discuss the temperature memory densities of the recording media,7 and so on. Theo- retically, a basic model which considers the exchange en- dependences of several interesting properties in magnetic ergy, the uniaxial anisotropy, and the Zeeman energy has multilayers with the help of the GF technique based on the been popularly adopted to discuss various properties of the simple model.8­14 Since the magnetic easy axes may be dif- magnetic multilayers.8­16 Recently, a quantum theory based ferent from layer to layer in an arbitrarily layered system, a on this model was established to calculate the hysteresis loop local coordinate system will be introduced into the system to and the coercivity of a magnetic multilayer,14 and was ap- optimize the spontaneous magnetized directions of each plied to give a basic consideration of the magnetic multival- layer which are determined by minimizing the free energy. ued MMV recording in double-film structures15 and in The temperature dependences of the coercivity and different magnetic granular film.16 Some authors have tried to extend thermal effects to the magnetic multivalued recording are their methods to finite temperature, for example, using the discussed. Finally, an application to the canted spin configu- molecular-field approximation which is good for the high- ration is presented. temperature case,9 naively applying the Bose-Einstein statis- tics which is better for the very-low-temperature case11 or some other assumptions.12 On the other hand, some authors II. FORMULAS OF THE GREEN'S FUNCTION were devoted to making the model more realistic by includ- APPROACH ing the long-ranged dipolar interactions in their Hamiltonian.17 For a general L-layer magnetic structure, a simple model The Green's function GF technique seems to be helpful Hamiltonian can be given as 0163-1829/98/57 13 /7863 7 /$15.00 57 7863 © 1998 The American Physical Society 7864 LEI ZHOU et al. 57 1 For the on-site interactions to which the above ordinary H 2 Im,m r,r Sm r *Sm r h* Sm r RPA cannot be applied, we adopt the decoupling scheme of m,m r,r m,r Lines20 as follows 0 D zm m S r 2, 1 S r,t Szm r,t Szm r,t S r,t zm S r,t , m r m m m m m m Sm m 7 where Im,m (r,r ) is the exchange interaction between the spins. z0 where m are the magnetic easy axes for each layer and they need not be the same in general. For simplicity, let us assume that z0 3 Szm 2 S m axes in each layer are in the x-z plane, and use m m m Sm 1 to denote the angle between the z0 0 m z . 8 m and z axes: z m*z S m 2 cos m m . h is the applied field. In this paper, we only study the case that the external field is applied perpendicularly. It is Abbreviations are used as follows: helpful to introduce the local coordinate LC transformation13 Sz zm xm x z z m(r) cos mS 2 m (r) sin mSm (r), Sm(r) S¯ m m m Sm r , S¯m Sm r ...2 . 9 cos xm zm mSm (r) sin mSm (r), where m denote the sponta- neous magnetized direction which are different from layer to Variational parameters m should be determined by layer and should be determined later by minimizing the free minimizing the free energy, energy. After the LC transformation, the Hamiltonian can be divided into the following three parts: f KT*ln Zp H H m m 1 H2 H3 , 2 where 1 H Z n e H n p n m 1 H1 2 Im,m r,r Sm r *Sm r , 3 m r,r H 0, 10 m 1 H zm zm which, based on the RPA, takes the following form: 2 2 Im,m r,r cos m m Sm r S r m m,m r,r I Sxm xm ym ym zm m,m sin m m S¯mS¯m hsin mS¯m m r S r S r S r h cos r m m m mS m m,r m 1 2 Sx 2 0. 11 D zm m m cos2 m m S r 2 2 Dmsin 2 m m S¯m m,r m It is interesting to note that the above condition is the same sin2 xm m m Sm r 2 , 4 as S m ,H3 0 on the basis of the RPA, which implies that the excitations are stable. It should be noted that Eq. 10 is 1 H x correct only if H/ m zm m commute with H. We show in the 3 2 Im,m r,r sin m m Sm r S r m Appendix that this is true based on the RPA. m,m r,r With the help of the commutators and a usual Fourier 1 transformation, we arrive at Szm xm m r S r D m 2 msin 2 m m m,r S k ,H F n,m m ,k S k Gn,m m ,k Sxm zm zm xm xm n m m m m r Sm r Sm r Sm r h sin mS r . m,r m 5 S m k , 12 The equations of motion can be obtained with the help of where Fn,m and Gn,m are some functions of m ,k and can be the spin operator commutations. After that, the following calculated straightforwardly, and S m is defined by S m random phase approximation18,19 RPA can be applied: S m/ 2S¯m. Introducing a Bogolyubov type (U ,V ) transformation to S zm zm m r,t S r ,t S r,t S , m m or r r , m m m Eq. 12 , it can be proved that if U , iV T are selected as 6 the eigenvectors of the matrix where the notation *** means thermal averages and is de- fined by (1/Zp) n n e H*** n in which Zp H k F k iG k , 13 n n e H n is the partition function. iG k F k 57 QUANTUM APPROACH FOR MAGNETIC MULTILAYERS . . . 7865 where F(k) and G(k) in the matrix H(k) are two L L matrices whose elements are defined by Fm,m (k, ) and Gm,m (k, ), we have A m k ,H Em k Am k , A m k ,H Em k Am k , 14 where Em(k), Em(k) are corresponding eigenvalues of the matrix H(k) see Ref. 14 for a comparison . According to Refs. 18 and 19, after a careful calculation, one can get similar self-consistent equations as the bulk spin system, S FIG. 1. Hysteresis loops for model 1 at low temperature dotted S¯ m m 1 m 2Sm 1 Sm 1 m m 2Sm 1 line , at moderate temperature dashed line , and near the Curie m , 1 m 2Sm 1 m 2Sm 1 temperature solid line . 15 result, one can calculate the magnetic properties tracing the S¯2m Sm Sm 1 1 2 m S¯m , 16 present spin configuration m 0 until the external field except for different forms of comes to a field when the excitation gap approaches zero. m : This field is just the coercive field. Thus, the total hysteresis 1 U loops at finite temperatures can be determined. m,l 2 Vm,l 2e El k For the model m N . 17 l,k e El k 1 Therefore, we have 3L parameters 2 m ,S ¯m ,S¯m; m model 1: L 4, Im,m r,r J, D/J 1.0, 1,2, . . . ,L , and 3L self-consistent equations, Eqs. 11 , 15 and 16 , so that we can uniquely determine a state by the hysteresis loops of a magnetic thin film at different tem- solving these equations. When the temperature approaches peratures are shown in Fig. 1 for comparison. When the tem- zero, one may find that the present approach will automati- perature increases, the coercivity of the system is weakened, clly recover spin wave theory which is believed to be the the hysteresis loops are smoothened, and the induced mag- best for the zero-temperature case.8,14 netization is more sensitive to the temperature. Especially in The poles of the Green's function are the elementary ex- the case that the temperature approaches the Curie tempera- citations Em(k), while the gap is defined as the minimum ture, the present ferromagnetic system is very similar to a value of the excitations: (h) min Em(k) . According to paramagnetic system-the coercivity is nearly zero and the Ref. 14, a spin configuration determined by Eq. 11 may be magnetization is very sensitive to the external field. A very a stable or metastable state if there is a positive gap (h) small negative field can turn the spins over Fig. 1 . The 0... in elementary excitations to suppress the strong thermal temperature dependence of the coercivity is shown in Fig. 2. fluctuations. The negative field hcoer at which the gap comes This phenomenon has been well known and been applied in to zero is understood as the coercive force, (hcoer) 0, technological magnetism successfully. It usually costs a very since the state cannot be metastable any longer at this field strong magnetic force to write a message onto a domain due to the strong fluctuations. Thus, all the physically inter- which already has one at room temperature. To facilitate the esting properties, such as the spin configurations, the surface process, the engineer heats the magnetic material so as to magnon modes, the temperature dependence of the coerciv- lower the coercivity, and then a small negative magnetism is ity, the Curie temperature of the layered structures, etc., can be obtained by some simple numerical calculations. III. TEMPERATURE DEPENDENCES OF THE COERCIVITY AND THE HYSTERESIS First, we consider the coercivity and the hysteresis loop, which can be obtained analogous to the zero-temperature case.14 When an external magnetic field is decreasing from a positive saturation value, the spin configuration must be m 0 , and the induced magnetization of the system can be calculated by M(h) (1/L) mS¯m with the help of Eq. 15 . As a applied field decreases across zero, the spins do not turn over at once although the configuration m should actually be the ground state with a lower energy. In fact, in the state m 0 , the magnon excitation still pos- FIG. 2. The temperature dependences of the coercivity for sesses a positive gap so that it is a metastable state. As a model 1. 7866 LEI ZHOU et al. 57 enough to turn the spins over. After that, the engineer cools the system to room temperature and such a message can be fixed successfully. By the way, in this paper, we have chosen the uniaxil anisotropy to be much larger than those of the real material since we only want to show the qualitative picture of the results. Actually, we have made the calculations for the smaller anistropy case and find that the physical pictures are completely the same. IV. MAGNETIC MULTIVALUED RECORDING In this section, we will discuss how the thermal fluctua- tions influence the magnetic multivalued recording.7,15 The basic idea of the MMV recording is to find such materials whose hysteresis loops are multistep shaped. A quantum model for the MMV recording has been proposed in Ref. 15 where the zero-temperature case is studied. It is a double- film structure where the main parameters are Im,m r,r Jm , Im,m r,r I, Dm 2Sm 1 D m , m 0. It is argued in Ref. 15 that the MMV recording can be achieved when the interlayer interaction I is not very strong compared to the anisotropy Dm and the coercivities of the two magnetic thin films must not be very close to each other. FIG. 3. a The critical values h 1 2 A , hB , and hB as a function of Without losing generality, it is assumed that D the temperature for model 2. b Comparison of the magnetic mul- 1 D 2. According to Ref. 15, the possible metastable states are tivalued recording for model 2 at low dashed line and high solid the following four ones: A( line temperatures. 1 2 0), B( 1 , 2 0), C( 1 0, 2 ), and D( 1 2 ). The metastable re- gions of the four states can be found following the steps that the difference of the two coercivities are enlarged. To be described in the last section. One may find that the meta- specific, the following model parameters are chosen: stable region of spin state A is h hA , the metastable region for spin state B is h1 2 Model 2: S1 1, S2 2, D 1 0.2, D 2 0.3, B ,hB , and straightforwardly, the metastable regions for spin states C and D are h2 1 B ,hB J and ( , h 1S1 3, J2S2 4, I 0.01. A , respectively.15 According to Ref. 15, the condition for realizing the MMV recording is that the exist- The critical values h 1 2 A , hB , and hB are shown in Fig. 3 a as ing regions of the metastable states should overlap with each a function of the temperature, and the hysteresis loops of the other. systems at low and high temperatures are compared in Fig. In the zero-temperature case, it is found that the intralayer 3 b . It can be clearly seen that the thermal fluctuations can- interaction Jm has nothing to do with these critical fields, and not change the main feature of the MMV recording so that the condition for realizing the MMV recording is fully deter- such materials are good candidates for MMV recording. mined by the interlayer interaction I as well as the anisotropy D m .15 However, in the finite-temperature case, the thermally B. Abnormal case excited magnons depend strongly on the intralayer interac- On the other hand, if J tion. As a result, if the Curie temperatures T 1S1 J2S2, thermal fluctuations are C of the two nontrivial. Although the coercivities of the two films will materials depending on JmSm) have a large difference, the both decrease as the temperature increases, the velocities of thermal fluctuations must affect the two films in distinct the decrement are, however, different see Fig. 2 for refer- ways, and there may be some new and interesting results. ence . Sooner or later, the coercivity of the first layer will Subsequently, we show two different cases. become larger than the second one. At that time, different interesting phases are possible to appear. The following A. Normal case model is studied as an illustration: If the magnetic material which has a small coercivity de- Model 3: S termined by D 1 1, S2 2, D 1 0.2, D 2 0.3, m Dm(2Sm 1) also possesses a low Curie temperature, that is, J1S1 J2S2, the thermal fluctuations do J not change the basic picture of the MMV recording. In this 1S1 5, J2S2 2, I 0.01. case, the coercive force of the first film decreases more h 1 2 A , hB , and hB are shown together in Fig. 4 a . Three tem- quickly than the second one when temperature increases, so perature regions are found for the MMV recording, and the 57 QUANTUM APPROACH FOR MAGNETIC MULTILAYERS . . . 7867 FIG. 5. The spin configuration 1 up triangles and 2 circles as a function of the temperature for model 4. axes to another one, the total double-film structure may dis- play many new and interesting effects.4,6 We would like to discuss the following double-layer system as an illustration. Suppose the first film has an in-plane magnetic easy axis 1 and the second film a perpendicular magnetic easy axis 2 0. In this case, unlike the last two sections, the nonlinear equations 11 are never trivial. The following model is studied: model 4: D1 0.2, D2 0.2, J1 1, J2 2, I 0.1, FIG. 4. a The critical values h 1 2 A , hB , and hB as a function of the temperature for model 3. b Comparison of the magnetic mul- S1 S2 1. tivalued recording for model 3 at low dashed line , moderate dot- ted , and high solid line temperatures. The spin configuration m is shown in Fig. 5 as a func- tion of temperature. It can be clearly seen that there is a spin hysteresis loops in different cases are shown in Fig. 4 b for reorientation phase transition in a critical temperature T1c . comparison. When the temperature increases from zero, the The physics can be understood as follows: The spin configu- multistep shape is obscured. At some temperatures, the two rations m are determined by the competition of the effec- coercivities are so close to each other that the multistep tive anisotropy and the effective interlayer exchange interac- shape disappears in the hysteresis loop. As the temperature is tions Eq. 11 . However, the temperature dependences of high enough, caused by different sensitivities to thermal fluc- those terms are determined by the Curie temperature TC , tuations, it is possible to show another kind of multistep- and different materials will have distinct temperature depen- shaped hysteresis loop in which the second film turns over dences see Fig. 2 . For the present system, the capping film earlier than the first one. In a technological process, these has a low TC and the recording one has a high TC . When the kinds of materials must be avoid being used as the recording temperature increases, the capping film becomes softer and media for MMV recording. softer the coercivity and effective anisotropy decreases and decreases . As a result, at a critical temperature when the V. APPLICATION TO THE CANTED SPIN effectively in-plane anisotropy in capping film cannot com- CONFIGURATION pete with the effectively perpendicular anisotropy in record- ing film as it does in the zero-temperature case, the spins are In this section, we would like to give an application of the all aligned in the z direction so that the spin reorientation present method to the canted spin configuration case by dis- transition occurs. One can easily understand another type of cussing the temperature sensitivities of the spin configuration spin reorientation transition: If the capping film has a higher of a double-film structure. Curie temperature, the spins in the other film will be even- Experimentally, a spin reorientation transition in ultrathin tually aligned along the magnetic easy axis of the capping film was observed as the temperasture increases,5 which is film. believed to be the result of a competition between the per- pendicular anisotropy in the surface and the dipolar interac- VI. CONCLUSION tions which favor the spins to lie in the plane. This phenom- enon has been appropriately explained by Ref. 17 through To summarize, in this paper, we have combined the introducing the dipolar interaction term in their Hamiltonian Green's function technique with the previous local coordi- and using a molecular-field approximation. We will not dis- nates transformation to discuss some finite-temperature prop- cuss this effect in this paper. On the other hand, the capping erties of magnetic multilyer systems. The nonlinear equa- technique has been popularly used in experimental magne- tions for determining the spontaneously magnetized tism. By capping a magnetic thin film with different easy directions in each layer have been derived by the minimiza- 7868 LEI ZHOU et al. 57 tion of the free energy, and the necessarily self-consistent Szm ,H equations have been obtained successfully following the m 1 0, A2 standard Green's function technique. Some applications of RPA the method have been presented. The temperature depen- Szm ,H 0, A3 dences of the coercivity and the hysteresis loop for layered m 2 structures have been shown, and the thermal fluctuations to the magnetic multivalued recording have been discussed, in Szm zm ym ym m ,H3 Im,m sin m m Sm Sm hsin mSm which two cases are found to display different sensitivities to m the thermal fluctuations. Finally, we study a more compli- 1 cated model: the double-film structure composed of two ymSzm zm ym m Sm Sm magnetic films with different magnetic easy axes. The spin 2 Dmsin 2 m m Sm configurations are shown to have interesting sensitivities to RPA the temperature. Im,m sin m m S¯m m ACKNOWLEDGMENTS 1 y This research was partly supported by the National Natu- hsin m m 2 Dmsin 2 m m S¯m m Sm . ral Science Foundation of China, Project No. 19774020. This research was also partly supported by the National Education A4 of China and Hitachi Maxell Ltd. Dr. N. Ohta and Professor zm A. Itoh are thanked for helpful discussions. One of us L.Z. According to Eq. 8 , we find that the condition Sm ,H 0 would like to acknowledge partial support from the Japan coincides with Eq. 11 . In other words, only when one has Science and Technology Corporation. chosen a spin configuration which satisfies Eq. 11 can the condition Szm m ,H 0 be automatically ensured under the APPENDIX RPA. If that is the case, it is not difficult to show that In this appendix, we will show that Eq. 10 is correct H2 / m ,H 0 A5 under the random phase approximation, Eqs. 6 ­ 9 . under the RPA. Equation 10 should be correct if H/ m commute with By the way, we would like to point out that the contribu- H. Since the other terms have no contributions to Eq. 10 , tions from the terms (S )2 (S )2 have been neglected as we need only consider H m m 2 / m which is an approximation which has also been adopted by other H authors.8­15 Actually, those terms have a spin-state mixing 2 zm zm zm effect which is very important for ``easy-plane'' single-ion Im,m sin m m Sm S hsin mSm m m m anistropy case.23 For the present easy-axis case, the effect is 1 significant only in very few cases.23 To incorporate this ef- zm x 2 Dmsin 2 m m Sm 2 Sm 2 . A1 fect, a characterestic angle CA method has been established for spin-1 and spin-3 Noting that under the same RPA as Eqs. 6 ­ 9 , the com- 2 cases.23 Since the method has the limit of no university, we did not apply the CA method in the mutations of Szm m with Hamiltonian are present paper. 1 For a review work in this area, see R. E. Camley and R. L. J. Magn. Soc. Jpn. 19, Suppl. S1, 429 1995 . Stamps, J. Phys. Condens. Matter 5, 3727 1993 . 8 L. L. 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