PHYSICAL REVIEW B VOLUME 54, NUMBER 22 1 DECEMBER 1996-II Magnetoresistance and magnetization of Fe/Cr 001... superlattices with noncollinear magnetic ordering V. V. Ustinov, N. G. Bebenin, L. N. Romashev, V. I. Minin, M. A. Milyaev, A. R. Del, and A. V. Semerikov Institute of Metal Physics, Ural Division of the Russian Academy of Sciences, GSP-170, Ekaterinburg, 620219, Russia Received 6 May 1996; revised manuscript received 12 August 1996 We study the magnetization and magnetoresistance of superlattices with biquadratic exchange. The equilib- rium states are analyzed on the base of the free energy expression that includes all terms up to the fourth order in components of magnetizations of magnetic sublattices. The correlation between the magnetization curve and the magnetoresistance for various orientation of magnetic field relative to the film plane is established. The experimental studies are made on the samples of molecular-beam epitaxy grown Fe/Cr superlattices. The positive magnetoresistance was found for the perpendicular-to-plane magnetic field. It is shown that this effect as well as the characteristic features of the magnetization curves are connected with the noncollinear magnetic alignment, which exists in our samples, and the fourth order magnetic anisotropy of unfamiliar type. S0163-1829 96 08745-0 I. INTRODUCTION It is to be noted that FM and AFM alignments do not exhaust all possible magnetic structures in multilayers. The Since the discovery of the giant magnetoresistance effect noncollinear 90° coupled magnetic profile in Fe/Cr trilayer in the Fe/Cr superlattices,1 the artificially made materials was reported in Ref. 7. The coupling angle of 50° between consisting of alternating layers of magnetic and nonmagnetic the magnetizations of neighboring Fe layers have been dis- metals attract a great attention. The stimulus is perspectives covered in Fe/Cr superlattices by spin polarized neutron of applications.2 The main efforts are made to understand reflectometry.8 The noncollinear magnetic ordering in the how magnetoresistance MR depends on the thickness of MBE grown Fe/Cr superlattices has also been found and the magnetic layers as well as the nonmagnetic spacer, tem- studied by magneto-optical methods.9 It has also been dem- perature, and magnetic field strength. The effect of orienta- onstrated in Ref. 9 that the angle 0 between the magnetiza- tion of magnetic field on MR remains less studied although it tions of neighboring Fe layers in zero magnetic field varies has been established already in Ref. 1 that this dependence is with Cr interlayer thickness. essential; the results of the FMR experiments on Co/Ru/Co Phenomenological description of a magnetic alignment trilayers3 also confirm that the measurements at the various deviated from a collinear one is performed usually in terms field orientation can be very instructive. of biquadratic exchange. In contrast to a bulk crystal, in a MR reflects the magnetic state of a multilayer-generally multilayer the usual bilinear exchange can be made very in a very complicated manner. It is believed, however, that weak and hence the biquadratic exchange, reported in Ref. 7 for the trilayer, may play an important role. The microscopic the magnetoresistance r is mainly a function of the angle origin of noncollinear ordering remains unclear, perhaps, it is between the magnetization of neighboring magnetic layers different in different samples. Thus according to only, i.e., of the absolute value of the relative with respect to Slonczewski,10 the biquadratic exchange can result from the the saturation value magnetization m of the superlattice: r interlayer thickness variation in the sample plane that leads (m). As long as it is true, the problem of understanding to the fluctuation of the usual bilinear exchange near zero the orientation dependence of MR is reduced to two sepa- level. As a consequence, the intermediate coupling angle be- rated problems: the determination of (m) and the deter- tween the magnetizations of neighboring magnetic layers re- mination of m in a given magnetic field. sults. There are also other theoretical models which predict The first problem is far from being solved; fortunately, in appearance of biquadratic coupling due to peculiarities of the some cases it is sufficient to know only the basic properties interlayer magnetic11 or electronic12­17 structure; the last of (m), which are more or less simple, rather than its ex- mechanisms are, however, too weak to explain the strong plicit form. As for the second, the magnetic alignment in a biquadratic coupling observed in Fe/Cr superlattices. In the superlattice may be very specific. The point is that the ex- present paper we will not discuss the nature of biquadratic change coupling between magnetic layers is governed by the exchange; a brief review of the theoretical results has been spacer thickness, so that one can create multilayers with vari- made recently by Slonczewski.18 ous magnetic ordering. It was found that this coupling oscil- The existence of relatively strong biquadratic exchange lates with interlayer thickness between ferromagnetic FM indicates that in multilayers the terms of the fourth order in and antiferromagnetic AF . The system Fe/Cr 001 was the free energy expansion in components of the layers mag- found to exhibit AF exchange coupling.4 Oscillation behav- netizations may be essential although in a bulk crystal these ior of exchange interaction was discovered in Fe/Cr and terms are, as a rule, of no interest. Hence one may think that other multilayers.5 The evidence of FM and AF ordering in unfamiliar magnetic interactions as well as magnetic states Fe/Cr was demonstrated in SEMPA studies of trilayers with of unusual type can be found in multilayers. a wedge shaped Cr interlayer.6 The aim of the present article is to show that in our MBE 0163-1829/96/54 22 /15958 9 /$10.00 54 15 958 © 1996 The American Physical Society 54 MAGNETORESISTANCE AND MAGNETIZATION OF . . . 15 959 anisotropy has been reported in Refs. 19 and 20 but in those articles the biquadratic exchange has not been taken into consideration. The expression 1 includes all terms up to the fourth order in components of M1 and M2. The first two terms in 1 are bilinear and biquadratic exchange, respec- tively, the third and the fourth appear because of the anisot- ropy within magnetic layers, the next five are due to the anisotropic interaction between magnetic layers through a spacer, the last two describe the demagnetization and the interaction with external magnetic field. We shall not find out the microscopic mechanism that leads to 1 and shall treat J's, K's, and L's as a phenom- enological quantities that depend on the layers materials, val- FIG. 1. Scheme of magnetic superlattice. ues of thickness, etc. The expression for F can be simplified by introducing the grown Fe/Cr superlattice samples with noncollinear mag- new variables m and l defined by netic ordering, the anisotropic interaction of the fourth order, which has never been observed and analyzed, really exists M1 M2 M1 M2 and that this interaction results in positive MR in magnetic m 2M , l 2M , field directed perpendicular to the layers plane. The article is 0 0 organized as follows. In Sec. II the theoretical model is for- m2 l2 1, ml 0. 2 mulated and the free energy expression is written down in explicit form. In Sec. III the equilibrium states in zero- and It follows from 1 and 2 non-zero-magnetic field are analyzed for the cases of the in-plane and the perpendicular-to-plane magnetization. In A A B B C F 1 2 1 2 2 4 1 2 Sec. IV we describe the main properties of (m) and show 2 m2 4 m4 2 mz 4 mz 2 m2mz how MR can depend on the field strength at the various field orientation. Section V is devoted to some experimental de- C D F F 2 2 2l2 1 2 2 4 hm. 3 tails. The results of measurements and their interpretation are 2 m2lz 2 mz z 2 lz 4 lz given in Sec. VI. Section VII is the Conclusions. Here h 2dmM0H; A's, B's, etc., are linear combinations of J's, K's, and L's, for example, A1 4J1 8J2 and II. FREE ENERGY A2 16J2 . The complete set of the relations between co- Let us consider the superlattice consisting of the magnetic efficients in 1 and those in 3 are given in Appendix A. layers of thickness d We have omitted in 3 the constant J m , separated by the layers of nonmag- 1 J2 which is of no netic metal; see Fig. 1. We assume that our superlattice can interest. be treated as a sum of two magnetic sublattices. The magne- The right-hand side of 3 contains the anisotropy terms tization of a layer belonging to the first sublattice is M1, the C C magnetization of a layer of the second one is M 1 2 2 2 2 , and , 4 M 2 m2mz 2 m2lz 1 M2 M 0 . The free energy per unit area and per one magnetic cell of the superlattice placed in a constant mag- which are rather unfamiliar. As there are both isotropic m2 netic field H directed under the angle with respect to the and anisotropic m 2 or l 2 multipliers, one may call this layers plane can be written as z z anisotropy the fourth-order-exchange-uniaxial anisotropy. Of course, 2 is applicable to any uniaxial antiferromag- M M M2 M2 F J 1M2 1M2 2 1z 2z net. However, in usually investigated antiferromagnetic crys- 1 M2 J2 4 K1 2 0 M0 M0 tals, m and mz are very small. In a superlattice the biqua- dratic exchange interaction between layers can be relatively M4 M4 M M K 1z 2z 1zM 2z 1zM 2z 2 strong, so the effects related to the exchange-uniaxial anisot- 2 M4 L1 2 L2 4 0 M0 M0 ropy may be observed. M M 2 M2 L 1M2 M 1zM 2z 1M2 M 1z 2z III. STABLE STATES 3 M4 L4 4 0 M0 To describe the magnetization process, one has to find M 2 M2 minimum s of the free energy. The general analysis is too L 1zM 2z M 1z 2z 2 2 5 M M4 2 dm M1z 2z involved, and hence it is desirable to simplify the problem. 0 Notice that the demagnetization field acting upon the mag- d netic moments makes them to lie, as a rule, in the film plane mH M1 M2 . 1 if the external field is absent. So we shall restrict ourselves We suppose that the z axis of coordinate system is per- by considering an easy plane sample and set lz 0 every- pendicular to interfaces and that magnetocrystalline anisot- where. The results of experiments described in Sec. VI show ropy in (x,y) plane can be neglected. A study of magnetic that this constraint is valid for our samples. Of course, there behavior of multilayer systems of cubic and uniaxial in-plane may exist the multilayers with biquadratic exchange and 15 960 V. V. USTINOV et al. 54 sufficient for a minimum of to be at one of the points mentioned, are given in Appendix B. One can easily verify that if A2 0, 0, the minimum is unique whether M0 belongs to the triangle or not. If A2 or is negative, the number of minimum is one, two, or three but not more, because if reaches its minimum at a vertex, there is no minimum at an adjacent side of the triangle. B. H 0 For simplicity we shall assume that a magnetic field is applied either along the film plane or perpendicular to the plane, so that hm hm or hm hmz ; then F(m,mz ,h) turns out to be a polynomial. Unfortunately even in these cases a general analysis involves many parameters and is too cum- bersome to be instructive. In this situation we restrict our- FIG. 2. Domain of definition of F(m,mz) and possible points of selves without much originality by considering some par- the free energy minimum in zero magnetic field. ticular cases. with l Let us suppose that a minimum of F m,m2,h 0 can lie z 0. Such a situation has been demonstrated to occur if the uniaxial anisotropy renders the z direction an easy only on OP side of OPQ. This means that both M1 and direction,21 and we are referring a reader to that article to M2 lie in the xy plane. At first we consider the in-plane find the detailed analysis. magnetization. Equating mz to zero, we obtain from 3 In what follows we shall for brevity write F(m,mz ,h) A A instead of the more correct F(m,m 1 2 z ,lz 0,h). F The domain of definition of F(m,m 2 m2 4 m4 hm. 7 z ,h) on (m,mz) plane is OPQ with vertices O O(0,0), P P(0,1), and A simple examination of 7 leads to the following conclu- Q Q(1,1). The free energy reaches a minimum either at an sions. internal point of the triangle or/and at a vertex, or/and at an 1 When A1 0, 2A1 A2 0 or, in other notations, internal point of a side of OPQ; see Fig. 2. For the sake of J1 0, J1 2J2 0 , the equilibrium state at H 0 is antifer- simplicity we assume that all the points mentioned are dif- romagnetic one. Corresponding region is shown in Fig. 3 as ferent. a sum of three regions: AFM1, AFM2, and AFM3. AFM1 region. If A1 0, A1 3A2 0 i.e., J1 2J2 0, A. H 0 J1 10J2 0 , the equilibrium state at H 0 is antiferromag- If h 0, it is convenient to consider the free energy netic one. If h h1 , the relative magnetization m obeys the F(m,m relation z ,h 0) F0(m,mz) as a function of m2 and m 2z rather than m and mz . From 3 it follows h A1m A2m3, 8 A A B B where h F m,m 1 2 1 2 1 A1 A2 ; if h exceeds h1 , then m 1. The mag- z , 2 2 4 2 2 4 netization curve, sketched in Fig. 4, is convex downward, if A2 is positive, and upward if it is negative. C AFM2 region. When A 1 1 3A2 0, A1 A2 0 i.e., 2 . 5 J1 10J2 0, J1 2J2 0 , the local minimum appears at m 1 if h exceeds h The domain of definition of on , plane is O P Q 1 . The state with m 1 remains, how- ever, stable if h h with O O (0,0), P P (0,1), and Q Q (1,1). Our 2 with aim is to determine the location of minimum s of . h 2 23 A1 A1/3 A2 . 9 Let us define the surface S over the , plane by the relation , . It is a second degree surface. If Thus the first order phase transition takes place at the field A 2 2B2 C 1 0, S is an elliptic paraboloid; if 0, S is a lying between h1 and h2, the width h h2 h1 of the hys- hyperbolic paraboloid. Partial derivatives / and / teresis loop being determined by the isotropic interaction. are equal to zero at M0 ( 0 , 0) with coordinates AFM3 region. When inequalities A1 0, A1 A2 0 J1 2J2 0, J1 2J1 0 are satisfied, there are two mini- B A mums in zero magnetic field: one is at m 0 and the second 1C1 A1B2 1C1 A2B2 0 , 0 . 6 is at m 1. If A1 A2 0, 2A1 A2 0, J1 2J2 0, J1 0 , then at h 0 the antiferromagnetic state is realized because it M0 corresponds to the bottom of S if A2 0, 0, the top of has a lower energy than ferromagnetic one. If in this case h S if A2 0, 0, and a saddle point if 0. If M0 corre- is increased from zero to a value which is less than h2 and sponds to the bottom of S and M0 belongs to O P Q , the after that the field is decreased back to zero, the system re- minimum of is at this point: otherwise a minimum lies at turns into the initial antiferromagnetic state. But if h reaches the boundary of the domain, i.e., at an internal point of a side h2, the system jumps into the ferromagnetic state. Once and/or at a vertex of O P Q . The conditions, which are there, the superlattice cannot leave this state even 54 MAGNETORESISTANCE AND MAGNETIZATION OF . . . 15 961 FIG. 4. Typical magnetization curves in the case of the in-plane magnetic field. m A 2 1 A2m2 C1mz 0, 12 B 3 1mz B2mz C1m2mz H 0. 13 It is assumed here that m mz , m 1. Since m 0, it follows FIG. 3. Phase diagram on (J1 ,J2) and (A1 ,A2) planes, a and from 12 b , respectively. C 2 1 2 after the field is switched off because this state is stable at m2 m0 A mz . 14 2 arbitrary field strength. As a consequence after the jump the multilayer looks like an ordinary ferromagnet whatever the Substituting this into 13 , we obtain field strength although the true equilibrium state is an anti- ferromagnetic one. A2B1 A1C1 3 In the event A h mz mz. 15 1 0, 2A1 A2 0. J1 0, J1 2J2 0 the A2 A2 ferromagnetic state always has a lower energy. When inequalities A The linear term in the right-hand side of 15 is positive 1 0, A1 A2 0 are satisfied, i.e., J while the sign of the second term coincides with the sign of 2 J1 /2, the noncollinear canted state with magnetiza- tion . These formulas show that the equilibrium point moves on m (m,m 0 A1 /A2 J1 2J2 /4 J2 10 z) plane as magnetic field is increased. A trajectory consists of two parts; see Fig. 5. The first one begins at M exists at h 0, the angle 1 0 between M1 and M2 being equal on OP and ends at h h to 2a cos m c at the second order phase transition 0 . The magnetization versus magnetic field is point M again given by Eq. 8 until h h c which lies on OQ or PQ; this part of the trajectory 1 . is described by the relations 12 ­ 15 . Farther the point of The magnetic susceptibility for h 0 is given by equilibrium moves along the corresponding side of the tri- m 1 angle towards the vertex Q where the movement terminates. 0 If C h 2 A 1 0, the first part of the trajectory is the vertical h 0 1 . 11 straight line. It means that the magnetization vectors of the sublattices go out of the film plane with increasing magnetic At last, if both A1 and A2 are negative J1 2J2 0, field in such a way that the angle between these vectors J2 0 , the system is in the ferromagnetic state. remains fixed until the plane to which the vectors belong Let us proceed to consider the perpendicular-to-plane becomes perpendicular to the film plane; further decreases magnetization. We shall discuss only one case which is the till zero. When the equilibrium point moves along OQ, the most interesting to us; namely, we shall assume that the non- magnetization m m collinear state is realized at h 0 and that there is no meta- z obeys the equation stable state. It implies that inequalities A1 0, A1 A2 0, h A 3, 16 A 1 B1 mz A2 B2 mz 2B1 C1A1 0 are satisfied see Appendix B . The equilib- rium conditions are given by which is valid until m mz 1. 15 962 V. V. USTINOV et al. 54 R r H R0 R , 20 0 where RH denotes the resistance in the presence of magnetic field, R0 RH 0. Generally, the magnetoresistance depends on the direction of the electric current as well as that of the magnetization of the multilayer. In the most experiments, including those described in the next section, the current flows in the film plane. In this case, r is nearly unaffected by varying the orientation of the current. It is not our aim here to discuss this anisotropy and, as is stated in the Introduction, we ignore it assuming that r is a function of m only: r (m), where m m(H, ). Both theory and experiments say that (m) is a monotone decreasing function. The features of magnetic state of a multilayer are then clearly reflected in H dependence of FIG. 5. The trajectories of the equilibrium point on (m,mz) magnetoresistance. For example, if at H 0 the noncollinear plane in the case of the perpendicular-to-plane magnetic field: 1 C 2); 4 C magnetic ordering exists, in the case 0 the magnetoresis- 1 0; 2 C1 0; 3 C1 0, C1 A2(1 m 0 1 0, C 2). tance is a monotone decreasing function of H until saturation 1 A2(1 m 0 because m monotonically increases. If C On the contrary, in the case of the perpendicular-to-plane 1 0, the trajectory of the point of equilibrium devi- ates from the vertical straight line to the left. The critical magnetization the magnetoresistance may be a monotone de- value m creasing function of H until saturation , or be a non- c m(hc) of the magnetization can be found from 14 . One gets increasing one, or even have a maximum at the second order phase transition point when h hc , as it follows from the m theoretical results described above. Another interesting point m z c is the following. The saturation of the magnetoresistance and . 17 1 C1 /A2 that of the magnetization of a multilayer may take place ei- ther in one and the same magnetic field or the magnetoresis- If h exceeds hc , which is found from 15 by setting tance is saturated in a lower field than magnetization; the last mz mc , the magnetization curve is described by the relation case occurs if C 2 1 0 and C1 A2(1 m0 ). If the measurements of the magnetoresistance are per- h A 3 1 B1 mz A2 B2 2C1 mz . 18 formed at the angles which are not necessarily equal to 0° or Notice that in the case C 90°, the functions r(H, ) for various 's may not be com- 1 0 the angle between the magne- tization vectors first increases and only if h h pletely independent. The question is how to calculate r for c this angle decreases with increasing field. arbitrary if the magnetoresistance for some certain 's is If C known. The answer can be formulated as follows. Let us fix 1 0, the point of equilibrium deviates from the ver- tical straight line to the right and reaches OQ or PQ depend- the value r of magnetoresistance. Then the relation ing on whether C 2 1 is less or more than A2(1 m 0). In the m H, ... r 21 later case vanishes at Mc , so that at h hc the magneti- zation vector of the superlattice turns to the symmetry axis defines the implicit function Hr . It follows that just like in an ordinary ferromagnet. If C 2 1 A2(1 m0 ), the equation 18 is valid; otherwise m Hr m H 0. 22 h B 3 1 C1 mz B2mz 19 Finding m/ and m/ H by making use of the expression until the magnetization is parallel to the z axis. for the free energy, one can obtain a differential equation for The examples considered above show that the magnetiza- Hr and hence find this function. This program has been tion curves of a multilayer with sufficiently large terms of realized in Ref. 22 for the case when the anisotropy terms of the fourth order may be rather unusual. The interesting point the fourth order are absent. The equation for Hr has the is that there can be the metastable states because the exist- form ence of these states can lead to an incorrect conclusion about 1 the equilibrium magnetic ordering. But even if these states 1 2 0. 23 are absent, one can hit upon such unfamiliar fact as the de- sin 2 Hr crease of absolute value of the magnetization vector of the The result of solving 23 can be written as superlattice with increasing magnetic field. sin2 H 1/2 . 24 IV. MAGNETORESISTANCE r cos2 H2 2 r 0° Hr 90° When magnetic field is applied, the resistance of a sample The solution Hr contains two arbitrary constants which changes. The magnetoresistance r is defined by are nothing but the values of this function Hr 0° and 54 MAGNETORESISTANCE AND MAGNETIZATION OF . . . 15 963 FIG. 6. Low-angle x-ray diffraction pattern of the Fe 23 Å /Cr 8 Å 30 sample. FIG. 7. Magnetization curves of the Fe 23 Å /Cr 8 Å 30 sample for the in-plane and the perpendicular-to-plane magnetic Hr 90° at the boundary of the domain of definition. It is to field. be noted that 24 does not contain any parameters that char- acterize the interaction between adjacent layers. displacement is equal to m0 0.39. If H is perpendicular to Obviously, m must be a continuously differentiable func- the plane, i.e., 90°, the magnetization curve tion and m/ H must be nonzero everywhere in order for mH(H, 90°) mz(H) solid squares has a weak pecu- this method to be valid. For example it is inapplicable for liarity near 6 kOe one can easily confirm the existence of the those values of the field strength for which the magnetore- peculiarity by plotting mr/ H vs H curve. No jumps on a sistance has an extremum. magnetization curve were observed. The magnetization curves at 77 and 290 K are practically identical. V. EXPERIMENT Figure 8 shows the magnetic field strength dependence of magnetoresistance for the in-plane r and the The Fe/Cr 30 samples were grown by the molecular-beam perpendicular-to-plane r magnetization. In the first case epitaxy method on MgO 100 substrate. The chromium the resistance does not depend on the angle between H and buffer was about 100 Å. In different samples the Fe layers the current and we restrict ourselves by presenting the data varied from 10 to 30 Å in thickness; the Cr spacer was from referred to the case H j. As magnetic field grows up, the 10 to 30 Å in thickness. Every sample was covered with a longitudinal magnetoresistance r is negative and decreases protective Cr layer. until saturation. The perpendicular magnetoresistance r is The vibrating sample magnetometer VSM was used in positive if H 8 kOe and has a maximum at H 6 kOe. The measuring magnetization. The magnetometer allows us to saturation value of r is equal to that of r . The saturation determine the projection mH of the magnetization m onto fields found from the VSM data and from the MR are equal magnetic field direction. The resistance was measured by the to each other. standard four-probe method. The temperature was 290 and Using the data for r and the VSM data for 77 K. mH(H, 0°) we have determined (m). The result is pre- The low-angle x-ray diffraction pattern of a typical sample presented in Fig. 6 clearly indicates that our multi- layers have a layered structure of satisfactory quality. VI. RESULTS AND INTERPRETATION We present here the results concerning the properties of only one sample Fe 23 Å /Cr 8 Å 30 at room temperature which demonstrates all the features we would like to discuss. The detailed analysis of the dependence of the physical prop- erties on the layers thickness and temperature will be pub- lished elsewhere. Shown in Fig. 7 is the magnetic field dependence of mH. If H lies in the film plane, the magnetization curve mH(H, 0°) m(H) solid circles is smooth and convex upward. It is to be noted that in a weak field, H 150 Oe, the magnetization is determined by the domain walls displace- ment and by the in-plane anisotropy, therefore this region is excluded from our consideration. The value of magnetization FIG. 8. The longitudinal 0° and perpendicular 90° m(H) near H 0 but outside the region of domain walls magnetoresistance. 15 964 V. V. USTINOV et al. 54 FIG. 9. Longitudinal MR vs m2. Dots are extrapolation to small FIG. 10. r H, 90° and r H, 0° at low magnetic field. (m m0) values of m. 3 The resistivity r grows in the range 0 H 6 kOe sented in Fig. 9 by circles. It is evident that (m) deviates despite the fact that r decreases monotonically and substan- from the simplest one (m) m2. The possible model for tially in this range. describing such a deviation has been developed in Refs. 23 4 Magnetization m(H, 90°) decreases with growing and 24. It has been shown there that (m) is proportional to H over the range 0 H 6 kOe. m2 only in the case of antiferromagnetic ordering in zero Comparing these facts with the theoretical results of Sec. field for small values of m while for arbitrary m in the case IV we may conclude that a noncollinear magnetic ordering of noncollinear ordering exists in the sample in zero magnetic field. To confirm this statement, we have calculated the magne- m m tization curve for the case of the in-plane H in accordance r 0 r s 1 , 25 m with 8 . It is convenient to rewrite 8 as 0 where r Hs s is the saturation value of r 2 , H , 27 1 m2 m m2 m0 0 m2 m4 m where H 1 s (A1 A2)/2dmM 0 is the saturation field. The up- 1 m2 . 26 per line in Fig. 7 is the result of calculations in accordance with 27 . The saturation field has been taken to be 11 kOe, The parameters and depend on the conduction electrons and m0 0.39; these values correspond to A1/2dmM0 2 spin scattering inside the layers and at the interfaces. Solid kOe and A2/2dmM0 13 kOe or J1/2dmM0 1.15 kOe and curve in Fig. 9 is the approximation of experimentally ob- J2/2dmM0 0.82 kOe. One can see that the calculated tained (m) by expressions 25 and 26 with 0.41 and curve agrees well with the experimental points. The devia- 0.42. Formally, the function (m) extracted from the tion exists only near Hs . data for r According to the consideration of Sec. IV, the decrease of and the VSM data for m H( H , 0 ° ) is defined only for m m0 . The approximation of (m) by expression m(H, 90°) in the range 0 H 6 kOe implies that C1 is 25 gives us the possibility to calculate magnetoresistance positive and that the critical field Hc is close to 6 kOe. Tak- for m m0. In fact we have used the quadratic extrapolation ing into account 14 and the fact that for H 6 kOe the of (m) to the region m m 2 0 because m0 0.15 is small. magnetization mz is approximately proportional to H, one Knowing (m) over the whole range 0 m 1 we are able to may expect r to be proportional to H2 if H 6 kOe. It is calculate the absolute value of magnetization m(H, 90°) easy to see from Fig. 10 that it is just the case. from the MR data for r presented in Fig. 9. The result of We have calculated also the magnetization curves this procedure is shown in Fig. 7 by triangles. We have ob- mH(H, 90°) and m(H, 90°) for the case of the tained a rather unusual picture: at first m(H, 90°) de- perpendicular-to-plane H in accordance with 14 , 15 , and creases with growing H. Minimum of m(H, 90°) takes 18 with A1 and A2 as in the case of the in-plane H. The best place at H 6 kOe. fitting see Fig. 7 corresponds to Hc 6.6 kOe, So, we have the following experimental facts. B1/2dmM0 21 kOe, B2/2dmM0 14 kOe, and 1 The magnetization mH(H, 0°) is nonzero in the C1/2dmM0 5 kOe. Now we easily find mc 0.33 and obtain nearest vicinity of H 0. the estimation C1/A2 0.38. C1 is small in comparison with 2 The perpendicular resistivity r does not change no- A2 and B1; therefore the corresponding summand ticeably and is positive in a wide region of magnetic field up C 2 1m2m z in the free energy expression 3 is insignificant to 8 kOe. if the equilibrium point is far enough from the critical one. 54 MAGNETORESISTANCE AND MAGNETIZATION OF . . . 15 965 manuscript of papers8 prior to publication. The research de- scribed in this publication was made possible by Grant No. 95-02-04813 from the Russian Foundation of Basic Re- searches and Grant-in-Aid No. 1-053/2 for Research Pro- gram from the Ministry of Science of Russia. APPENDIX A The relations between coefficients that appear in Eq. 1 and in Eq. 3 are given by the following formulas: A1 4J1 8J2 , A1 A2 16J2 , A2 B 2 1 4K1 2L1 2L3 4L4 8 dmM 0, A3 B2 8K2 4L2 8L5 , A4 C FIG. 11. Magnetoresistance of the Fe 23 Å /Cr 8 Å 1 4L3 8L4 , A5 30 sample at various orientation of magnetic field. The solid lines are the result of calculations in accordance with 24 . C2 4L3 8L4 , A6 The term B 4 D 24K 2m z is unessential as compared with the second 2 4L2 , A7 order anisotropy term B 2 1m z at mz 1. Thus one may ex- pect that the relation 24 hold in the range 0 r r 2 s . Figure F1 4K1 2L1 2L3 4L4 8 dmM0, A8 11 shows r versus H measured for some 's; the solid lines are the results of calculations in accordance with 24 . One F2 8K2 4L2 8L5 . A9 can see that the theoretical curves are indeed in agreement with the experiment. APPENDIX B VII. CONCLUSIONS Here we present the sufficient conditions for a minimum of to exist at a point mentioned. The local minimum is at 1 In our MBE grown Fe/Cr superlattice samples the bi- M quadratic exchange interaction between magnetic layers re- 1 ( 1,0) with 0 1 1 when sults in the noncollinear magnetic ordering. A1 0, A2 0, A1 A2 0, A2B1 C1A1 0. 2 New type of magnetic anisotropy in superlattices has B1 been discovered. Taking into account all terms of the fourth order in the free energy expression, we have analyzed pos- M2 (1, 2) with 0 2 0 is the point of minimum in the case sible equilibrium states in zero magnetic field as well as the B features of the magnetization curves for the in-plane and the 2 0, B1 C1 0, B1 B2 C1 0. B2 perpendicular-to-plane magnetization. It has been found reaches the minimum at M3 provided that theoretically and observed in experiment that in the last case the specific second order phase transition takes place in a A1 B1 0, A2 B2 2C1 0, B3 multilayer with the noncollinear ordering. In our samples the fourth order exchange-uniaxial anisotropy results in decreas- A1 B1 A2 B2 2C1 0, C1 A2B1 B1C1 A1B2 . ing the absolute value of magnetization with increasing mag- B4 netic field. The conditions in order for a minimum of to be at a vertex 3 At low magnetic field the in-plane magnetoresistance of the triangle can be formulated as follows. A minimum is of the superlattice with noncollinear structure is proportional at A 0,0 if to the field strength H whereas the perpendicular-to-plane magnetoresistance varies as H2. A 4 The exchange-uniaxial anisotropy in our MBE grown 1 0, A1 B2 0; B5 Fe/Cr gives rise to the positive magnetoresistance in the at B 1,0 when perpendicular-to-plane magnetic field, the maximum of mag- netoresistance taking place at the second order phase transi- A1 A2 0, B1 C1 0; B6 tion point. at last a minimum is at C 1,1 in the case ACKNOWLEDGMENTS B1 B2 C1 0, A1 A2 B1 B2 2C1 0. B7 The authors are grateful to Professor H. 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