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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
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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
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position by the value [2,3] dependence on the roughness rate for the frustrated
system: ferromagnetic filmantiferromagnetic substrate.
47 75
šA
B af Kaf Ž1=2S2af When finding the distributions of the order para-
EB ; š1Ž
Ma meters in the film and in the substrate we suppose that
49 77
where B is
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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 yiši ¼ f; afŽ
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
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ARTICLE IN PRESS
2 A.I. Morosov, A.S. Sigov / Journal of Magnetism and Magnetic Materials ] (]]]]) ]]]]]]
1 Dy 57
i ¼ 0 š2Ž
3 with the following boundary conditions:(a) 59
5 * qy 61
Dy f
f ¼ 0; š3Ž
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 Siž1 distances are given in lattice constants. The ratio between 69
i ¼ 7 sinšy
qn J i yiž1Ž; š4Ž
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 ž Jf;af Sf Bb: š7Ž 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],
išx; zŽ 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 šp cŽ2 c2
37 domain state. w ¼ ž 2Z cosšc jŽ ; š8Ž
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 š9Ž 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 : š10Ž
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; š5Ž equilibrium c value and the quantities M8 ¼ M cosšc
47 103
where jŽ and M> ¼ M sinšc jŽ:
If the external magnetic field is parallel to the
49 J 105
g ¼ fS2f b1; š6Ž spontaneous magnetization of the film šj ¼ p=2Ž; 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 ½6šjZj 1Ž 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
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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: š11Ž
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