3B2v7:51c ED:Chanakshi GML4:3:1 MAGMA : 8771 Prod:Type:COM pp:123ðcol:fig::NILÞ PAGN: bvr SCAN: BINDU ARTICLE IN PRESS 1 3 Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]­]]] 5 7 Interface roughness and unidirectional anisotropy of thin 9 ferromagnetic film on uncompensated surface of 11 antiferromagnet 13 A.I. Morosov*, A.S. Sigov 15 Moscow State Institute of Radioengineering, Electronics and Automation, Technical University, 78, Vernadski Avenue, 177454 Moscow, Russia 17 19 Abstract 21 Magnetization curves of a ferromagnetic film on antiferromagnetic substrate were investigated with account of the 23 frustrations produced by interface roughness. The conditions of appearance of unidirectional anisotropy were obtained and its dependence upon roughness of film­substrate interface was found. r 2002 Published by Elsevier Science B.V. 25 Keywords: Anisotropy, unidirectional; Thin films, bilayer; Interface structure; Magnetization, curves 27 The unidirectional anisotropy reveals itself in the shift magnetization of ferromagnetic film, and a is the film 29 57 of magnetization curve of ferromagnetic film deposited thickness. on the surface of antiferromagnet. The phenomenon of A real interface is never ideally flat, but contains 31 59 unidirectional anisotropy has been considered in a atomic steps changing the substrate local thickness by number of papers (see e.g., the review [1]). one atomic layer. On different side of the step the 33 61 Let us consider the case when the spins of the orientation of spins in upper atomic layer of antiferro- antiferromagnet atomic plane parallel to the interface magnet is opposite, therefore the steps on the interface 35 63 between ferromagnet and antiferromagnet are not give rise to spin frustrations regardless of the sign of compensated. In the framework of the simple model exchange integral J 65 f;af between the spins of the film and 37 with ideally flat interface, the exchange interaction the substrate. The phase diagram of such a frustrated between spins of the film and spins of the substrate system has been investigated in the framework of the 39 67 causes the film magnetization direction exceptioning. continuum model [4]. Remagnetization of the film in an external magnetic field The origins of the unidirectional anisotropy for the 41 69 gives rise to formation of domain wall in the anti- case of compensated spins of antiferromagnet atomic ferromagnetic substrate [2,3]. plane parallel to the interface are discussed in Refs. [2,5]. 43 71 Energy consumption in wall generating leads to the The goal of the present paper is to study magnetiza- shift of magnetization curve from the field-symmetrical tion processes and to find the unidirectional anisotropy 45 73 position by the value [2,3] dependence on the roughness rate for the frustrated system: ferromagnetic film­antiferromagnetic substrate. 47 75 ðA B af Kaf Þ1=2S2af When finding the distributions of the order para- EB ; ð1Þ Ma meters in the film and in the substrate we suppose that 49 77 where B is 51 79 53 *Correspo UNCORRECTED PROOF magnetic induction, A both the magnetization vector, and the vector of af is exchange stiffness of antiferromagnet, K antiferromagnetism lies in the plane parallel to the af is its anisotropy constant in the plane parallel to the film­substrate interface, M is interface and are characterized by the angle yiði ¼ f; afÞ between the order parameter vector and a certain given 81 axis in the plane. Minimization of the exchange energy nding author. Tel.: +7-095-433-0311; fax: +7- in the system film­substrate leads to the equations: 55 83 095-434-8665. E-mail address: morosov@eot.mirea.ru (A.I. Morosov). 0304-8853/01/$ - see front matter r 2002 Published by Elsevier Science B.V. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 1 3 6 1 - 0 MAGMA : 8771 ARTICLE IN PRESS 2 A.I. Morosov, A.S. Sigov / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]­]]] 1 Dy 57 i ¼ 0 ð2Þ 3 with the following boundary conditions:(a) 59 5 * qy 61 Dy f f ¼ 0; ð3Þ qn 7 at the free film surface, where *D is two-dimensional 63 Laplacian in the film plane, and q=qn denotes the 9 derivative in the direction of the outer normal to the 65 interface plane;(b) 11 Fig. 1. Order parameter distributions in the vortex phase. The 67 zero ordinate corresponds to the film­substrate interface. All 13 * qy J Dy i f;af Siþ1 distances are given in lattice constants. The ratio between 69 i ¼ 7 sinðy qn J i yiþ1Þ; ð4Þ iSi hatching and the yi value in radians is shown in the inset. 15 at the film­substrate interface, where J 71 i is the exchange integral, and S optimum (0 for one side of the step and p for the other) i is the mean spin of the atom within the 17 ith layer;(c) minimizing the surface energy equals 73 J b 19 daf af Saf þ Jf;af Sf Bb: ð7Þ 75 y 0 E af ¼ 0 Jf;afSf 21 in the substrate volume far from the interface. All At all other part of the interface with the width Rbdaf0; 77 distances are normalized to the lattice constant b; which the angle yf yaf equals to its optimum value. If the yaf 23 is assumed to be the same in all layers. value at the interface is nonzero, then it changes 79 The solution of the system of differential Eq. (2) with continuously from this value to the zero one over the 25 boundary conditions (3) and (4) giving the order length R into the substrate volume. 81 parameter distributions in the structure considered has If the mean film magnetization vector makes an angle 27 been found by numerical method for the case of periodic c with the antiferromagnetic order parameter in the 83 set of rectilinear atomic step edges parallel to each other. substrate volume, then the value of yaf changes from c 29 The x-axis of the coordinate system lies in the layer to zero in the vortex occupying the first type region, 85 plane and is perpendicular to the step edges, and the z- whereas in the vortices occupying the second type 31 axis is perpendicular to the layer plane (two-dimensional regions the value of yaf changes from c p to zero. 87 case). The functions y By analogy with the ``magnetic proximity'' model [6], iðx; zÞ have been obtained by 33 Fourier expansion with respect to x variable in the the surface energy density of the film­substrate system in 89 region jxjoL with periodic boundary conditions. an external magnetic field B0 directed parallel to the film 35 Let us consider the remagnetization process in the and at a j angle to the axis can be written as 91 range of parameters where the film retains its single- C ðp cÞ2 c2 37 domain state. w ¼ þ 2Z cosðc jÞ ; ð8Þ 2 2 2 93 The atomic steps divide the whole interface into 39 regions of two types: in the regions of the first type the where 95 surface energy takes its minimum for parallel orientation CEJafS2 41 of ferromagnetic and antiferromagnetic order para- af =Rb ð9Þ 97 meters and in the regions of the second type the energy and dimensionless parameter Z equals 43 takes its minimum for antiparallel orientation. B0Ma B0MaRb 99 If the characteristic distance R between atomic steps Z ¼ E : ð10Þ C JafS2af 45 at the interface is less than some critical value 101 Minimizing the energy (8), it is easy to find the Rc ¼ dfEga; ð5Þ equilibrium c value and the quantities M8 ¼ M cosðc 47 103 where jÞ and M> ¼ M sinðc jÞ: If the external magnetic field is parallel to the 49 J 105 g ¼ fS2f b1; ð6Þ spontaneous magnetization of the film ðj ¼ p=2Þ; then JafS2af for ZX 1 the angle c ¼ p=2 and M8 ¼ M: For Zo 51 107 the film 53 UNCORRECTED PROOF remains in a single domain state and static spin 1; jZj 151; the square root anomaly takes place: vortices arise near the interface in the substrate (Fig. 1). 109 Two lengths characterize the vortex. The width daf p 0 of c ¼ ½6ðjZj 1Þ 1=2: 55 the region around the step edge at the film­substrate 2 111 interface where the value of yf yaf differs from its The quantity M8 has zero value at Z ¼ p=2; so the MAGMA : 8771 ARTICLE IN PRESS A.I. Morosov, A.S. Sigov / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]­]]] 3 1 metastable. The difference between the energies of the 29 metastable and the stable states at saturation exceeds the 3 domain wall energy. So the domain wall in the 31 antiferromagnet can arise, that results in the antiferro- 5 magnetism vector rotation near the interface by p and in 33 diminishing the vortex energy. 7 Thus, one can deduce that the unidirectional aniso- 35 tropy of single domain ferromagnetic film on uncom- 9 pensated surface of antiferromagnet is caused by 37 appearance of static spin vortices at a rough film­ 11 substrate interface, its value being inversely proportional 39 to the mean distance between the atomic steps at the 13 interface. 41 Fig. 2. Magnetization curve of the single domain film. Mag- This work is partly supported by Russian Foundation 15 netic field is directed parallel to the spontaneous magnetization for Basic Research, grant 00-02-17162. 43 of the film. 17 45 magnetization curve is shifted to negative field region References 19 (Fig. 2). The unidirectional anisotropy field equals the 47 value [1] J. Nogues, I.K. Schuller, J. Magn. Magn. Mater. 192 (1999) 21 pC 203. 49 BE ¼ pR 1: ð11Þ 2Ma [2] A.P. Malozemoff, Phys. Rev. B 35 (1987) 3679. 23 For jZjb1; Zo0; the M [3] D. Mauri, H.C. Siegmann, P.S. Bagus, E. Kay, J. Appl. 51 jj magnitude approaches its Phys. 62 (1987) 3047. saturation valueFM according to the jB0j 1 law. At the [4] V.D. Levchenko, A.I. Morosov, A.S. Sigov, JETP Lett. 71 25 saturation, in the regions of the first and the second 53 (2000) 373. types the antiferromagnetic order parameter rotates by [5] N.C. Koon, Phys. Rev. Lett. 78 (1997) 4865. 27 the angle 3p=2 and p=2; respectively. Such a state is [6] J.C. Slonczewski, J. Magn. Magn. Mater. 150 (1995) 13. 55 UNCORRECTED PROOF