PHYSICAL REVIEW B VOLUME 57, NUMBER 10 1 MARCH 1998-II Field-induced spin-reorientation transitions in magnetic multilayers with cubic anisotropy and biquadratic exchange V. V. Kostyuchenko Institute of Microelectronics, Russian Academy of Science, Universitetskaya 21, Yaroslavl, 150007, Russia A. K. Zvezdin Institute of General Physics, Russian Academy of Science, Vavilov 38, Moscow, 117942, Russia Received 7 May 1997 A theoretical investigation of field-induced spin-reorientation phase transitions in magnetic multilayer struc- tures is presented. We suppose that biquadratic and Heisenberg exchange energies between adjacent layers are of the same order of magnitude. The anisotropy energy is cubic. Our consideration is analogous to the Fe/Cr 100 superlattice system. We show that two phases with a noncollinear orientation of the magnetization in adjacent layers can represent the ground state. In our case an external magnetic field aligned perpendicular to the layers' direction induces spin-reorientation transitions in the plane of layers. A full list of such transitions is given. Other types of anisotropy are briefly discussed. S0163-1829 98 03409-2 I. INTRODUCTION change tends to align the magnetization of adjacent layers in perpendicular directions and Heisenberg exchange in collin- At the present time the exchange interaction of magnetic ear directions. The joint action of the Heisenberg and biqua- thin layers via a nonmagnetic spacer is the subject in a con- dratic exchange brings into existence phases with a noncol- siderable amount of literature. It is well known that the main linear orientation of the magnetization in neighboring layers. term of this interaction has a Heisenberg form J1nini 1, In the next section we examine the phase diagram for the where ni is a unit vector in the direction of magnetization in spin-valve structure at zero field. Only four phases can sat- the ith magnetic layer. The value of J1 is strongly dependent isfy the conditions of energy minimization and two of them on the magnetic and nonmagnetic layers thicknesses and are noncollinear. changes sign as the spacer thickness increases.1­3 A Heisen- As shown in Sec. III the action of an external magnetic berg interaction tends to align the magnetization in adjacent field in the direction perpendicular to the layers gives rise to layers parallel for J1 0) or antiparallel for J1 0). But it the renormalization of the exchange and anisotropy constants has been experimentally established that a noncollinear ori- which determine the orientation of magnetization. Thus per- entation of magnetization in adjacent layers is also pendicular to the plane the magnetic field induces changes of possible.4­7 Therefore some other mechanism must be in- the magnetization orientation in the plane of layers. volved to account for the experimental data. For this reason a large number of different models for the non-Heisenberg in- teraction between adjacent layers is proposed. However, II. ENERGY FUNCTIONAL AND PHASE DIAGRAM none of the theoretical models can quantitatively account for AT ZERO FIELD the experimental data.14,15 In most cases the non-Heisenberg Let us consider N magnetic layers mediated by N 1 exchange interaction is usually reported in terms of a biqua- spacer layers with an antiferromagnetic interaction between dratic exchange form J2(nini 1)2 see Refs. 4 and 8­10 adjacent layers. In the case of an infinite layer number N although other expressions are used also see, e.g., Ref. 11 . 1 or a spin-valve structure N 2 the energy functional has In the case being considered in the zero anisotropy ap- the form of a two-sublattice magnet.16 It differs from that of proximation phase transitions in magnetic multilayers are a classical antiferromagnet by the biquadratic exchange term. studied in Ref. 12. Nevertheless, the results obtained in Ref. For definiteness sake we take the z axis perpendicular to the 12 have only a qualitative meaning. In real multilayer mag- layer plane and the x axis along one of two easy axes in the netic materials the anisotropy energy has at least the same layer plane. order of magnitude as the exchange energy. A uniaxial an- The energy functional of that system can be written as isotropy in the plane of magnetic layers is taken into account in Ref. 13. But this paper discusses only phase transitions under the action of an external field parallel to the magnetic 2 m layers. We examine the case of a cubic anisotropy and bi- F k nx y z y x z z z i ni 2 ni ni 2 ni ni 2 2 h ni quadratic exchange. The case under consideration is analo- i 1 2 ni gous to the Fe/Cr 100 superlattice structure. In our paper we 1 1 present some theoretical consequences of this model which can be verified rather easily, namely, the field-induced phase 2 J1 nini 1 2 J2 nini 1 2. 1 transition in magnetic multilayers. According to the theoretical treatment and experimental Here J1 is the Heisenberg exchange energy, J2 the biqua- observations J2 is always positive. Thus biquadratic ex- dratic exchange energy, k the energy of the cubic anisotropy, 0163-1829/98/57 10 /5951 4 /$15.00 57 5951 © 1998 The American Physical Society 5952 V. V. KOSTYUCHENKO AND A. K. ZVEZDIN 57 TABLE I. Properties of phases which minimize the energy functional 2 . Phase Stability condition Energy I 1 2 0 2k J1 2J2 (J1 J2)/2 II 1 0, 2 2k J1 2J2 (J2 J1)/2 J 2 1 J 4kJ III 1 2 4k2 1 2 12 arccos2(k J 2 2) 2 J1 J2 k 8(k J2) J IV 1 2 1 2 2 12 arcsin2(k J 2(k J2) J1 J1/8(k J2) 2) h the Zeeman energy in an external magnetic field, and m the analytical solution of this problem in the general case. In demagnetizing energy. All energies are measured in units of actual conditions an exchange field and anisotropy field are the magnetic field. far less than a demagnetization field. For the superlattice It is convenient to change the variables from rectangular Fe/Cr, as an example, anisotropy and exchange fields are less ni to polar coordinates i and i , where i measures the than or equal to 1 kOe and the demagnetization field is on angle between the z axis and vector ni and the azimuthal the order of 10­20 kOe. Then we can find from the equa- angle i measures the angle between the ni projection on the tions F/ 1 0 and F/ 2 0 to a good approximation x-y plane and x axis. Thus the energy functional depends on four variables 1, 1, 2, and 2. 1 2 arccos h/m . 3 At zero external field a strong demagnetization field pre- vents any deviation of magnetization from the plane. Thus Substitution of these expressions into Eq. 1 gives an energy substituting in Eq. 1 functional in the form of Eq. 2 , where J1, J2, and k are 1 2 /2 we take replaced correspondingly by renormalized values 1 1 1 2 F h2 h2 2 J1cos 1 2 2 J2cos2 1 2 4 k sin22 i . i 1 J1 h 1 J1 2J2 , 2 m2 m2 It is easy to verify that only four phases can satisfy the h2 2 h2 2 conditions for the global minimum of the functional 2 . Two J2(h) 1 , k h k 1 . 4 m2 m2 of them are collinear ferromagnetic phase I and antiferro- magnetic phase II and two others are noncollinear with n1 By this means the equilibrium values of 1(h) and 2(h) and n2 symmetrical with respect to the easy axes phase III can be defined by using the expressions in Table I, provided and hard axes phase IV . The energy functional 2 has its that instead of J1, J2, and k expressions 4 are taken. As minimum when a set of 1, 2 is equal to one of four sets h m, k(h)/2J1(h) and J2(h)/2J1(h) tend to zero. Then presented in the Table I. Minimum conditions and energy values for each set are displayed in Table I also. Figure 1 shows the phase diagram in the variables J2 /k and J1 /k. This phase diagram may be interpreted as following. Due to the cubic anisotropy, phases I and II with 1 2 0, and phase IV with 1 2 /2 have nearly the same anisotropy energy. Hence it follows that the ground state phase is determined by the J1 and J2 ratio. If it is granted that J1 J2, the collinear phase II or I is less energetic than the canted phase IV; otherwise, the phase IV is energetically preferable. It is interesting that for small an- isotropy energy the transition from a collinear phase to a canted phase IV occurs via an intermediate phase, namely, the canted phase III with a nearly collinear orientation of the magnetization in adjacent layers. III. FIELD-INDUCED SPIN-REORIENTATION TRANSITIONS There is evidence that magnetization departs from the layer plane under the action of an external field directed in FIG. 1. Phase diagram in variables J2 /k and J1 /k. Roman num- the perpendicular direction. As this takes place the problem bering of phases is the same as in Table I. The solid line corre- of the energy functional 1 minimization becomes consider- sponds to the first-order phase transition and the dashed line corre- ably more complex. It has not proved feasible to obtain an sponds to the second-order phase transitions. 57 FIELD-INDUCED SPIN-REORIENTATION . . . 5953 according to the data in Table I, 1(h) 2(h) 0, as h APPENDIX: FULL LIST OF FIELD-INDUCED PHASE m. If 1(m) and 2(m) differ from 1(0) and 2(0), TRANSITIONS FIELD PERPENDICULAR TO then the applied external magnetic field causes a spin- THE PLANE OF MAGNETIC LAYERS... reorientation transition as h increased from 0 to m. Investigation of field-induced phase transitions becomes There are no field-induced phase transitions if J1, J2, and easy to grasp through the use of the phase diagram in Fig. 1. k satisfy the following conditions: The state of the multilayer system is completely determined by the two variables J 1 J 1 J1(h)/k(h) and J 2 J2(h)/k(h). 1 0, J2 3k, 2J2 J1 k k k 2J1 . Due to the condition J2(h)/k(h) const see Eq. 4 the trajectory J 2 J1 0, J2 3k, 2J2 2k J1 . 2( J 1) has the form of a straight line parallel to the J2 axis. As h is increased from 0 to m, J 1 changes from J 3 J1 0, J1 2J2 0. 1 /k to for J1 2J2 0 and to for J1 2J2 0. The intersection of this straight line and the lines of phase transitions in Fig. 1 shows possible field-induced phase tran- Otherwise the following phase transitions occur as h sitions. changes from 0 to m: A full list of possible field-induced spin-reorientation phase transitions is given in the Appendix. It is interesting to 1 J2 3k, 2J2 J1 k k k 2 J1 . note that the total number of field-induced transitions in some instances can be even 4 as h changes from 0 to m. First order at h h1. The most interesting case is J1 0 and J2 J1 /2. Under 2 2 these conditions J 2 J2 3k, 2J2 2k2 J1 . 1(h) 0 at h h* m J1 /2J2. Thus the effective Heisenberg exchange ``disappears'' at h h* and the exchange field in the plane of layers becomes purely a First order at h h2. biquadratic. It is also vital to note that J b Second order at h h 1(h) changes sign at 3. h h*. 2 2 The main conclusion 3 J 2k2 J . 1(h) 2(h) as h re- 1 0, J2 3k, 2J2 2k J1 , 2J2 1 mains unchanged for other types of anisotropy for J1 2J Second order at h h 2 0). This fact can be explained as follows. If the 3. angle between ni and ni 1 exceeds /2, then Heisenberg 2 2 exchange acts as a repulsive force and biquadratic exchange 4 J1 0, J2 3k, 2J2 2k2 J1 , J1 2J2 k. as an attractive force. Otherwise both of these forces act as a repulsive force. Thus, if deviation of the magnetization from a First order at h h4. the basal plane exceeds /4, then exchange forces tend to b First order at h h2. align the magnetization of adjacent layers so that the angle c Second order at h h3. between the projection of magnetization in adjacent layers equals . Our calculations for uniaxial anisotropy give sup- 5 J1 0, J2 3k, 0 J1 2J2 k. port for this view. a Second order at h h5. IV. CONCLUSIONS b First order at h h4. In this paper we have presented a theoretical investigation c First order at h h2. of the field-induced phase transitions in magnetic multilayers d Second order at h h3. with cubic anisotropy and biquadratic exchange. The case of field perpendicular to the layer plane has been examined. 6 J2 3k, 2J2 J1 k k k 2J1 , The orientation of magnetization in the plane of layers is determined by the effective constants of anisotropy and ex- 2J2 J1 0, J1 0. change. The values of these constants are dependent on the value of the magnetic field. Thus a magnetic field perpen- a First order at h h6. dicular to the layers changes the orientation of the magneti- b First order at h h1. zation in the plane of layers. Here the values hi have the following meaning: In the particular case of ferromagnetic Heisenberg ex- change between layers the value of the magnetic field can be m chosen so that effective exchange interaction between mag- h1 2J2 2J2 J1 netic layers becomes purely biquadratic. To the authors' 2J2 4J2 3k minds experimental examination of these phase transitions can help to reveal the nature of the non-Heisenberg exchange k 4J2 J1 J1 2J2 k k J2 1/2, A1 interaction between layers. m 2 ACKNOWLEDGMENT h2 2 k2 J J 2 k2 J2 2 1J2 2 The work was partially supported by the Russian Founda- tion for Fundamental Research Grant 96-02-16250 . J 2 1 2J2 2 J2 k2 1/2, A2 5954 V. V. KOSTYUCHENKO AND A. K. 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