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Phenomenological theory of surface-induced interactions in magnetic nanostructures
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) 705–707
Phenomenological theory of surface-induced interactions in magnetic nanostructures . U.K. RoXler*, A.N. Bogdanov, K.-H. Muller . Leibniz-Institut fur und Werkstoffforschung Dresden, Postfach 270116, Dresden D-01171, Germany . Festkorper.
Abstract Within discretized and continuous phenomenological models the equilibrium distribution of the induced uniaxial anisotropy in a ferromagnetic nanolayer can be calculated as function of the layer thickness. The theory explains experimentally observed nonlinear and nonmonotonic thickness dependencies of the total magnetic anisotropy in thin nanoscale ferromagnetic layers with out-of-plane magnetization. r 2003 Elsevier B.V. All rights reserved. PACS: 75. 70.-I; 75.70.Cn; 75.75.+a Keywords: Thin film; Anisotropy; Surface effects; Micromagnetism
Modern depth-resolving experimental techniques show that in magnetic nanostructures the surface/ interface-induced interactions (different forms of exchange coupling, Dzyaloshinsky–Moriya interactions, magnetic anisotropy) considerably spread into the depth of the magnetic constituents [1]. These effects may stabilize specific magnetic phases and are responsible for a number of remarkable phenomena observed in nanostructures [2,3]. In particular, the magnetization processes and spin reorientation in epitaxial systems as Ni/Cu(0 0 1) [4] and Co/Au(1 1 1) [5] are due to the spatially distributed character of induced uniaxial anisotropy within the nanostructure [3]. Recently, we have formulated a phenomenological theory which describes ferromagnetic thin layer systems with surface-induced couplings by using phenomenological continuum theory of ordered media [2,3]. While this approach is suitable for micromagnetic modelling, discretized formulations in a more atomistic spirit are desirable, in particular to make contact with calculations of magnetic properties for magnetic nanoscale layers from first-principles electron-theory [6].
*Corresponding author. Tel.: +49-351-4659-542; fax: +49351-4659-537. . E-mail address: [email protected] (U.K. RoXler).
Here, we present a discretized model for surfaceinduced perpendicular anisotropy in thin films. We consider a ferromagnetic layer consisting of N atomic planes with induced uniaxial anisotropy in an applied magnetic field h. The energy of such systems in the simplest discretized approximation for a micromagnetic energy functional is N 1 N X X 1 Wm ¼ Ji mi miþ1 þ Ki m2iz h mi hd mi ; ð1Þ 2 i¼1 i¼1 where mi is the normalized magnetization of the ith plane, hd is the demagnetizing field, Ji are exchange constants, and Ki are coefficients of the induced uniaxial anisotropy (the axis z is along the normal of the layer). The finite extension lengths of the surface-induced interactions into the depth of the nanolayer can be described by a corresponding phenomenological interaction functional [2,3]. In the discretized version, this can be expressed by a cluster-expansion for a functional describing the phenomenological coefficients in the micromagnetic energies of type (1). Again using the simplest form allowed by symmetry, this functional for Ki can be written as WA ¼ A
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.569
N X n¼1
Kn2 B
N1 X n¼1
Kn Knþ1 ;
ð2Þ
ARTICLE IN PRESS U.K. Ro. X ler et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 705–707
706
where A is a stiffness constant and B describes a ‘‘coupling’’ between the planes. In the continuum limit energy (2) is reduced to a Ginzburg–Landay-type functional [3] # Z d=2 " 2 dK 2 2 * WA ¼ þl0 K ðzÞ dz; ð3Þ dz d=2 where d ¼ NDffi is the layer thickness. l0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D 0:5B=ðA BÞ is a characteristic length. For symmetric boundary conditions K1 ¼ KN ¼ K0 ; typical profiles Ki are shown in Fig. 1. Correspondingly, minimization of Eq. (3) yields (Fig. 2) KðzÞ ¼
K0 coshðz=l0 Þ : cosh½d=ð2l0 Þ
N X ð2p þ Ki Þ
¼ 2pN þ
2z/d
- 2π Fig. 2. Continuous distribution of local anisotropy calculated from Eq. (3): curves from top to bottom for l0 in the range from 0:05d (‘‘thick’’ films) to 2:5d (‘‘thin’’ films).
φN q>1
4
8
12
16
20 N
q<1
The function FN describes the thickness dependence of the uniaxial anisotropy. They are qualitatively different if parameter q ¼ K0 =ð2pÞ is larger or smaller than unity (Fig. 3). For qo1 (weak induced anisotropy) the demagnetization fields (‘‘shape anisotropy’’) play the dominant role forcing the magnetization into the layer plane. In systems with strong induced anisotropy (q > 1)
N=3 N=5 N=7 N=12 N=25
Ki / K0
-d/2
+1
ð5Þ
Ki :
i¼1
1
-1
0
i¼1 N X
K0 - 2π
ð4Þ
Depending on the materials parameters in Eqs. (2) and (3), the distributions of induced anisotropy in the layer may be strongly confined to the surface or may assume smooth profiles with nearly constant value throughout the film (Figs. 1 and 2). For laterally homogeneous distributions of the magnetization, the stray field energy can be expressed rigorously by an anisotropy-like term wd ¼ hd m=2 ¼ 2pm2iz [7]. The experimentally observable effective anisotropy includes this ‘‘shape anisotropy’’ wd and may be written as FN ¼
K
0
+d/2
Fig. 1. Example for localized uniaxial anisotropy calculated from discretized model Eq. (2) and various values of N:
Fig. 3. Example of effective total uniaxial anisotropy calculated for discretized models from Eq. (4) for various values of N and different strength of surface-induced contribution q:
the function FN is markedly nonlinear and nonmonotonic and becomes positive below a certain critical thickness (for small N). Both types of thickness dependencies have been observed experimentally [4,5]. Finally, the anisotropy distributions calculated from Eq. (2) in the discretized version quantitatively agree with results from the continuum version Eq. (3) of the energy (1) [3] even for few magnetic layers. In conclusion, the phenomenological analysis based on model (1) shows that reducing the thickness of a magnetic film to the atomic limit yields (i) a substantial redistribution of the induced anisotropy from the surface-like contribution for ‘‘thick’’ films (limit of N!eel’s theory for surface anisotropy [8]) to smooth distributions for ‘‘thin’’ films (Figs. 1 and 2), and (ii) an increasing tendency to states with out-of-plane magnetization (Fig. 3).
ARTICLE IN PRESS U.K. Ro. X ler et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 705–707
References [1] B. Stahl, et al., Phys. Rev. Lett. 84 (2000) 5632; S. Kim, J.B. Kortright, Phys. Rev. Lett. 86 (2001) 1347. . [2] A.N. Bogdanov, U.K. RoXler, Phys. Rev. Lett. 86 (2001) 037203; . A.N. Bogdanov, U.K. RoXler, K.-H. Muller, . J. Magn. Magn. Mater. 242 (2002) 594. . [3] A.N. Bogdanov, U.K. RoXler, K.-H. Muller, . J. Magn. Magn. Mater. 238 (2002) 155.
707
[4] G. Bochi, et al., Phys. Rev. B 53 (1996) 1729; M. Ciria, et al., Phys. Rev. B 67 (2003) 024429. [5] F.J.A. den Broeder, et al., J. Magn. Magn. Mater. 93 (1991) 562; M. Dreyer, et al., Phys. Rev. B 59 (1999) 4273. [6] M. Alouani, H. Dreyss!e, Curr. Opin. Solid State Mater. Sci. 4 (1999) 499. [7] A. Hubert, R. Sch.afer, Magnetic Domains, Springer, Berlin, 1998. [8] L. N!eel, J. Phys. Radium 15 (1954) 225.
Journal of Magnetism and Magnetic Materials 272–276 (2004) 705–707
Phenomenological theory of surface-induced interactions in magnetic nanostructures . U.K. RoXler*, A.N. Bogdanov, K.-H. Muller . Leibniz-Institut fur und Werkstoffforschung Dresden, Postfach 270116, Dresden D-01171, Germany . Festkorper.
Abstract Within discretized and continuous phenomenological models the equilibrium distribution of the induced uniaxial anisotropy in a ferromagnetic nanolayer can be calculated as function of the layer thickness. The theory explains experimentally observed nonlinear and nonmonotonic thickness dependencies of the total magnetic anisotropy in thin nanoscale ferromagnetic layers with out-of-plane magnetization. r 2003 Elsevier B.V. All rights reserved. PACS: 75. 70.-I; 75.70.Cn; 75.75.+a Keywords: Thin film; Anisotropy; Surface effects; Micromagnetism
Modern depth-resolving experimental techniques show that in magnetic nanostructures the surface/ interface-induced interactions (different forms of exchange coupling, Dzyaloshinsky–Moriya interactions, magnetic anisotropy) considerably spread into the depth of the magnetic constituents [1]. These effects may stabilize specific magnetic phases and are responsible for a number of remarkable phenomena observed in nanostructures [2,3]. In particular, the magnetization processes and spin reorientation in epitaxial systems as Ni/Cu(0 0 1) [4] and Co/Au(1 1 1) [5] are due to the spatially distributed character of induced uniaxial anisotropy within the nanostructure [3]. Recently, we have formulated a phenomenological theory which describes ferromagnetic thin layer systems with surface-induced couplings by using phenomenological continuum theory of ordered media [2,3]. While this approach is suitable for micromagnetic modelling, discretized formulations in a more atomistic spirit are desirable, in particular to make contact with calculations of magnetic properties for magnetic nanoscale layers from first-principles electron-theory [6].
*Corresponding author. Tel.: +49-351-4659-542; fax: +49351-4659-537. . E-mail address: [email protected] (U.K. RoXler).
Here, we present a discretized model for surfaceinduced perpendicular anisotropy in thin films. We consider a ferromagnetic layer consisting of N atomic planes with induced uniaxial anisotropy in an applied magnetic field h. The energy of such systems in the simplest discretized approximation for a micromagnetic energy functional is N 1 N X X 1 Wm ¼ Ji mi miþ1 þ Ki m2iz h mi hd mi ; ð1Þ 2 i¼1 i¼1 where mi is the normalized magnetization of the ith plane, hd is the demagnetizing field, Ji are exchange constants, and Ki are coefficients of the induced uniaxial anisotropy (the axis z is along the normal of the layer). The finite extension lengths of the surface-induced interactions into the depth of the nanolayer can be described by a corresponding phenomenological interaction functional [2,3]. In the discretized version, this can be expressed by a cluster-expansion for a functional describing the phenomenological coefficients in the micromagnetic energies of type (1). Again using the simplest form allowed by symmetry, this functional for Ki can be written as WA ¼ A
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.569
N X n¼1
Kn2 B
N1 X n¼1
Kn Knþ1 ;
ð2Þ
ARTICLE IN PRESS U.K. Ro. X ler et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 705–707
706
where A is a stiffness constant and B describes a ‘‘coupling’’ between the planes. In the continuum limit energy (2) is reduced to a Ginzburg–Landay-type functional [3] # Z d=2 " 2 dK 2 2 * WA ¼ þl0 K ðzÞ dz; ð3Þ dz d=2 where d ¼ NDffi is the layer thickness. l0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D 0:5B=ðA BÞ is a characteristic length. For symmetric boundary conditions K1 ¼ KN ¼ K0 ; typical profiles Ki are shown in Fig. 1. Correspondingly, minimization of Eq. (3) yields (Fig. 2) KðzÞ ¼
K0 coshðz=l0 Þ : cosh½d=ð2l0 Þ
N X ð2p þ Ki Þ
¼ 2pN þ
2z/d
- 2π Fig. 2. Continuous distribution of local anisotropy calculated from Eq. (3): curves from top to bottom for l0 in the range from 0:05d (‘‘thick’’ films) to 2:5d (‘‘thin’’ films).
φN q>1
4
8
12
16
20 N
q<1
The function FN describes the thickness dependence of the uniaxial anisotropy. They are qualitatively different if parameter q ¼ K0 =ð2pÞ is larger or smaller than unity (Fig. 3). For qo1 (weak induced anisotropy) the demagnetization fields (‘‘shape anisotropy’’) play the dominant role forcing the magnetization into the layer plane. In systems with strong induced anisotropy (q > 1)
N=3 N=5 N=7 N=12 N=25
Ki / K0
-d/2
+1
ð5Þ
Ki :
i¼1
1
-1
0
i¼1 N X
K0 - 2π
ð4Þ
Depending on the materials parameters in Eqs. (2) and (3), the distributions of induced anisotropy in the layer may be strongly confined to the surface or may assume smooth profiles with nearly constant value throughout the film (Figs. 1 and 2). For laterally homogeneous distributions of the magnetization, the stray field energy can be expressed rigorously by an anisotropy-like term wd ¼ hd m=2 ¼ 2pm2iz [7]. The experimentally observable effective anisotropy includes this ‘‘shape anisotropy’’ wd and may be written as FN ¼
K
0
+d/2
Fig. 1. Example for localized uniaxial anisotropy calculated from discretized model Eq. (2) and various values of N:
Fig. 3. Example of effective total uniaxial anisotropy calculated for discretized models from Eq. (4) for various values of N and different strength of surface-induced contribution q:
the function FN is markedly nonlinear and nonmonotonic and becomes positive below a certain critical thickness (for small N). Both types of thickness dependencies have been observed experimentally [4,5]. Finally, the anisotropy distributions calculated from Eq. (2) in the discretized version quantitatively agree with results from the continuum version Eq. (3) of the energy (1) [3] even for few magnetic layers. In conclusion, the phenomenological analysis based on model (1) shows that reducing the thickness of a magnetic film to the atomic limit yields (i) a substantial redistribution of the induced anisotropy from the surface-like contribution for ‘‘thick’’ films (limit of N!eel’s theory for surface anisotropy [8]) to smooth distributions for ‘‘thin’’ films (Figs. 1 and 2), and (ii) an increasing tendency to states with out-of-plane magnetization (Fig. 3).
ARTICLE IN PRESS U.K. Ro. X ler et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 705–707
References [1] B. Stahl, et al., Phys. Rev. Lett. 84 (2000) 5632; S. Kim, J.B. Kortright, Phys. Rev. Lett. 86 (2001) 1347. . [2] A.N. Bogdanov, U.K. RoXler, Phys. Rev. Lett. 86 (2001) 037203; . A.N. Bogdanov, U.K. RoXler, K.-H. Muller, . J. Magn. Magn. Mater. 242 (2002) 594. . [3] A.N. Bogdanov, U.K. RoXler, K.-H. Muller, . J. Magn. Magn. Mater. 238 (2002) 155.
707
[4] G. Bochi, et al., Phys. Rev. B 53 (1996) 1729; M. Ciria, et al., Phys. Rev. B 67 (2003) 024429. [5] F.J.A. den Broeder, et al., J. Magn. Magn. Mater. 93 (1991) 562; M. Dreyer, et al., Phys. Rev. B 59 (1999) 4273. [6] M. Alouani, H. Dreyss!e, Curr. Opin. Solid State Mater. Sci. 4 (1999) 499. [7] A. Hubert, R. Sch.afer, Magnetic Domains, Springer, Berlin, 1998. [8] L. N!eel, J. Phys. Radium 15 (1954) 225.