Journal of Magnetism and Magnetic Materials 195 (1999) 514}519 Spin con"gurations in the absence of an external magnetic "eld in a magnetic bilayer with in-plane cubic or uniaxial anisotropies Ming-hui Yu *, Zhi-dong Zhang Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110015, People's Republic of China International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110015, People's Republic of China Received 23 October 1998 Abstract The spin con"gurations in the absence of an external magnetic "eld have been systematically investigated for a magnetic bilayer system consisting of two ferromagnetic layers separated by a non-magnetic layer with interlayer exchange coupling. Based on a phenomenological model, the conditions for the existence of collinear and non-collinear spin structures were derived for three kinds of magnetic bilayers with di!erent combinations of in-plane cubic and uniaxial anisotropies for the two ferromagnetic layers. The phase diagrams of the spin con"gurations at zero "eld were drawn, taking into account the lowest-order anisotropy parameters of both the ferromagnetic layers. The values of the canting angle have been derived analytically and then numerically plotted. 1999 Elsevier Science B.V. All rights reserved. PACS: 75.60.!d; 75.70.!i; 75.25.#z Keywords: Spin con"gurations; Ferromagnetic layers; Uniaxial anisotropies; Coupling energy 1. Introduction obtained with nearly perfect crystals and layered structures. Film thickness may reach a few or even The ultrathin "lms, multilayers and superlattices one atomic layer. It is natural, that such a high have attracted extensive attention over the last dec- quality of the new magnetic systems results in the ade on both experimental and theoretical aspects discovery of many static magnetic properties which [1}5]. By growth techniques such as molecular contain valuable information about the intrinsic beam epitaxy (MBE), an extraordinarily high qual- magnetism of layered structures. One feature of ity of thin magnetic "lms and layered system can be these layer structures is the well-known oscillatory interlayer exchange coupling between the adjacent * Corresponding author. Tel.: #86-24-2384353155857; fax: ferromagnetic layers through the non-magnetic #86-24-23891320. layer [6,7]. Several mechanisms have been pro- E-mail address: dygeng@imr.ac.cn (M.-h. Yu) posed for this oscillation, including the dipolar and 0304-8853/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 1 0 1 - 8 M.-h. Yu, Z.-d. Zhang / Journal of Magnetism and Magnetic Materials 195 (1999) 514}519 515 Ruderman}Kittel}Kasuya}Yosida (RKKY) inter- single crystal and that the easy axes of magneti- actions as well as the quantum well [8}10]. The zation of both layers are parallel to each other. strength of the interlayer coupling is quite weak as Moreover, it is assumed that strong shape and/or compared with that of the intersublattice exchange negative perpendicular anisotropy con"nes the in bulk materials. But the ratio of interlayer ex- magnetic moments of both layers to the "lm planes. change energy to anisotropy energy lies in the range Fig. 1 schematically represents the con"guration of 10\}10. Therefore, the competition between the this magnetic bilayer model. The assumptions cor- exchange energy and the anisotropy energy also respond to the situation in magnetic bilayer with exists in the layer structures, which plays an impor- (0 0 1)-type growth. is the angle between the mo- tant role in determining the spin arrangements. ments of the two layers. E Dieny et al. [11,12] have phenomenologically  and E are the anisot- ropy energies of the layers A and B, respectively. studied the magnetic behaviour of magnetic bilayer The anisotropy energy is set by K systems of cubic and uniaxial in-plane anisotropies  cos sin for the cubic case and K with assumed antiferromagnetic coupling between S sin for the uniaxial case with the angle of the magnetization of each layer adjacent identical ferromagnetic layers. What with respect to the easy direction. If we use a con- would happen if the magnetic bilayer consists of vention allowing for negative values of two distinct ferromagnetic layers, which possess  and , then di!erent magnetizations and anisotropies? This can be substituted by # , so the model can be studied in terms of only two independent work extends the magnetic bilayer system to a gen- variables, eral case, and systematically investigates the spin  and . The equilibrium state is found by minimizing con"gurations in the absence of an external mag- Eq. (1) with respect to the angles netic "eld for this bilayer system, which is depen-  and . Detect- ing the minima of Eq. (1) involves the "rst and dent on the competition between the strengths of second partial derivatives of the free energy with the exchange coupling and the anisotropies. More- respect to the angles over, the phase diagrams for di!erent spin con"g-  and : urations have been drawn out for three kinds of *E "0, (2) magnetic bilayers with di!erent combinations of *  in-plane cubic and uniaxial anisotropies. This *E method has even been successfully used in our "0, (3) previous work on the two-sublattice system * [13,14]. *E *E " ! *  *E  *  * * '0, (4) 2. Model *E'0. (5) *  The magnetic bilayer system to be studied in  what follows can be de"ned by the free-energy expression E"JMM cos #E( )#E ( ), (1) where J represents the interlayer exchange coup- ling between two ferromagnetic layers. A positive value of J favours antiparallel alignment (ferrimag- netic coupling), whereas a negative J favours a parallel one (ferromagnetic coupling). M and Fig. 1. Model of a magnetic bilayer system consisting of two M denote the magnitude of the magnetization of ferromagnetic layers A and B, separated by an interlayer. On the layers A and B, respectively. It is postulated that right side is the vertical view, and the dashed lines are the easy each ferromagnetic layer is single-domain and axes of magnetization. 516 M.-h. Yu, Z.-d. Zhang / Journal of Magnetism and Magnetic Materials 195 (1999) 514}519 The solutions of Eqs. (2) and (3) satisfying the A collinear spin con"guration will always satisfy inequalities (4) and (5) correspond to the local Eqs. (7) and (8), because for "0, and minima of the free energy. We shall choose the one corresponding to the lowest minima of the "0, /4, /2, 3 /4, every term in both equations vanishes. To investigate whether such solutions in- free energy to determine the resulting moment deed correspond to energy minima one has to turn orientations. to the criterion '0 and *E/* '0, which en- sures the existence of a minimum. If under certain conditions, multiple solutions satisfy the criterion, 3. Spin con5gurations the one corresponding to the lowest energy min- imum will be e!ective. Therefore, provided In this section, we investigate the possible stable K'0, combining inequalities (10) and (11) for spin con"gurations of the magnetic bilayer model "0, and "0, /4, /2, 3 /4 gives collinear in three cases which correspond to the possible con"gurations in four regions. combinations of the cubic and uniaxial anisot- ropies for each layer. In each case, only the lowest- I. For "0, "0, /2: order term of the anisotropy energies is taken into y'!1 and x(0 and x(2y/(1#y); account. II. For "0, " /4, 3 /4: y(!1 and x(!2y/(1#y); 3.1. Two cubic in-plane anisotropies III. For " , "0, /2: y'!1 and x'0 and x'!2y/(1#y); First, we consider the case where both the layers IV. For " , " /4, 3 /4: have cubic in-plane anisotropy. Eq. (1) can be y(!1 and x'2y/(1#y). rewritten as follows: In the case of K E"JM (0, we can also observe these M cos #K cos  sin  four regions, but the collinear con"gurations in #K them are interchanged symmetrically about the ori-  cos( # ) sin( # ). (6) gin. In Fig. 2 these results are summarized, four For simplicity, the relative magnitude of the inter- curves and the positive y-axis separate the four layer exchange and the anisotropy energies and the regions. In each case, there are always two di!erent ratio of the two anisotropy energies are de"ned as spin con"gurations as the consequence of the bi- x and y, respectively: axial symmetry of the cubic anisotropy. The central part of the phase diagram delineated by the border- JM K x" M , y"  . (7) lines of the collinear regions undoubtedly repre- K K sents non-collinear spin arrangements where one shall be only interested in the value of the canting Making a convenience of this notation, Eqs. (2) and angle . In order to reveal the dependence of the (3) and inequalities (4) and (5) can be expanded and canting angle on x and y, one has to solve Eqs. (8) reduced to and (9). To get the solutions of Eqs. (8) and (9), we sin 4 "rst subtract Eq. (9) from Eq. (8) to obtain #y sin 4( # )"0, (8) !2x sin #y sin 4( # )"0, (9) sin 4 #2x sin "0 (12) "2K[!x cos cos 4  and use this result to eliminate  from Eq. (9), and #y(2 cos 4 !x cos )cos 4( # )]'0, remove the common term sin (sin O0) to obtain (10) x[1#y(1!2 sin2 )] *E"2K *  [cos 4 #y cos 4( # )]'0. (11)  "2y cos ((1!4x sin )(1!sin2 ). (13) M.-h. Yu, Z.-d. Zhang / Journal of Magnetism and Magnetic Materials 195 (1999) 514}519 517 Fig. 3. Dependence of the canting angle on the values of x and Fig. 2. Phase diagrams of the di!erent spin con"gurations of y, which are de"ned in Eq. (7). Both the layers have the cubic a magnetic bilayer in the absence of an external "eld (K in-plane anisotropy (K '0). '0). Both the layers take the cubic in-plane anisotropy. Only the lowest-order anisotropy constants of the two layers are taken into account. The de"nition of the axes x and y is in Eq. (7). one, which results in the following free energy ex- pression: After performing further manipulations, one can E"JMM cos #K sin  get a cubic equation in terms of cos 2 : #K cos( # )sin( # ). (15) !2y cos2 #2y(2x!y)cos2 Accordingly, the de"nitions of x and y are changed #x(1!y)"0. (14) into This cubic equation determines the value of the JM K canting angle in the region for non-collinear spin x" M , y"  . (16) K K con"gurations. The required root must be a real,   not complex, so it is very di$cult to "nd the true Eqs. (2) and (3) and inequalities (4) and (5) should be root from the three roots of this cubic equation expanded and reduced to [15]. But we can numerically determine the value of the canting angle from Eq. (14). The dependence of 2 sin 2 #y sin 4( # )"0, (17) the canting angle on x and y is represented in Fig. 3. It is worth noting that there is a gap of ( /4, !2x sin #y sin 4( # )"0, (18) 3 /4) (for x"0) for the canting angle. The reason "2K shall also be attributed to the biaxial symmetry [!x cos cos 2 #y(2 cos 2  of the cubic anisotropy, which can force the canting !x cos )cos 4( angle to lie in the range of 0! /4 (or 3 /4! ) in # )]'0, (19) order to diminish the interlayer exchange coupling *E energy. "2K *  [cos 2 #y cos 4( # )]'0.  (20) 3.2. One uniaxial, the other cubic For "0, and "0, /2, the solutions of In this subsection, we let one layer take the Eqs. (17) and (18), combining inequalities (19) uniaxial in-plane anisotropy, the other the cubic and (20) would lead to four possible collinear 518 M.-h. Yu, Z.-d. Zhang / Journal of Magnetism and Magnetic Materials 195 (1999) 514}519 con"gurations. #4xy(1!2x)cos #x(xy!y#1)"0. I. For "0, (21) "0: K'0 and y'!1 and x(0 and x(2y/ (1#y); Fig. 6 shows the dependence of the canting angle II. For "0, on x and y in the case of K " /2: '0, which is K numerically derived from Eq. (21). The gap of ( /4, (0 and y(1 and x'0 and x'2y/ (1!y); 3 /4) for the canting angle results from the cubic III. For " , "0: K'0 and y'!1 and x'0 and x'!2y/ (1#y); IV. For " , " /2: K(0 and y(1 and x(0 and x(2y/ (y!1). These results are displayed in Figs. 4 and 5 for K'0 and K(0, respectively. The existence of the collinear structures are very sensitive to the sign of the anisotropy coe$cient of layer A. Moreover, the boundaries between collinear and non-collinear con"gurations are di!erent in Figs. 4 and 5. Ac- cording to the procedure described in the above subsection, we can obtain a six-order equation in terms of the cosine of the canting angle: 16y cos !16xy cos !16y cos Fig. 5. Phase diagrams of the di!erent spin con"gurations of a magnetic bilayer in the absence of an external "eld (K #8xy(3x!1)cos #y(4!x)cos (0). One layer takes the uniaxial in-plane anisotropy, the other cubic one. Only the lowest-order anisotropy constants of the two layers are taken into account. The de"nition of the axes x and y is in Eq. (16). Fig. 4. Phase diagrams of the di!erent spin con"gurations of a magnetic bilayer in the absence of an external "eld (K'0). One layer takes the uniaxial in-plane anisotropy, the other cubic one. Only the lowest-order anisotropy constants of the two Fig. 6. Dependence of the canting angle on the values of x and layers are taken into account. The de"nition of the axes x and y, which are de"ned in Eq. (16). One layer takes the uniaxial y is in Eq. (16). in-plane anisotropy (K'0), the other cubic one. M.-h. Yu, Z.-d. Zhang / Journal of Magnetism and Magnetic Materials 195 (1999) 514}519 519 anisotropy of layer B. For K(0, the dependence coupling energy and the two anisotropy energies, of the canting angle on x and y has a feature result in collinear or non-collinear con"gurations. similar to that for K'0, however, correspond- The symmetry of the anisotropy determines the char- ing to Fig. 5. It is a central symmetry of Fig. 6 with acter of the spin arrangement in the layer structures. respect to the origin. 3.3. Two uniaxial in-plane anisotropies Acknowledgements This case in a magnetic bilayer system is very This work has been supported by the project No. similar to that in a two-sublattice system, when 59725103 of the National Sciences Foundation of only the second-order anisotropy constants of the China and by the Science and Technology Com- two sublattices are taken into account. Substituting mission of Shenyang and Liaoning. the c-axis and the intersublattice molecular-"eld coe$cient n in Ref. [13] with the a-axis and the interlayer exchange coupling J will enable the re- References sult obtained in Ref. [13] to be applicable in this subsection. Figs. 1 and 2 in Ref. [13] showed the [1] M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petro!, P. Eitenne, G. Creuzet, A. Friederich, phase diagram of the di!erent spin con"gurations J. Chazelas, Phys. Rev. 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Lett. 71 (1993) 3870. We have systematically studied the spin con"g- [11] B. Dieny, J.P. Gavigan, J.P. Rebouillat, J. Phys.: Condens. urations in the absence of an external magnetic Matter 2 (1990) 159. [12] B. Dieny, J.P. Gavigan, J. Phys.: Condens. Matter 2 (1990) "eld in a highly idealized bilayered model consist- 187. ing of two ferromagnetic layers of uniaxial or cubic [13] Z.-d. Zhang, T. Zhao, P.F. de Cha(tel, F.R. de Boer, anisotropies intervening by a non-magnetic layer. J. Magn. Magn. Mater. 147 (1995) 74. The results obtained indicate that the model can [14] Z.-d. Zhang, T. Zhao, P.F. de Cha(tel, F.R. de Boer, describe a wealth of possible relative spin arrange- J. Magn. Magn. Mater. 174 (1997) 269. [15] R.S. Burington, Handbook of Mathematical Tables and ments between the two ferromagnetic layers. The Formulas, "fth ed., McGraw-Hill, New York, 1972, competitions between three energies, the exchange pp. 12}14.