- Email: [email protected]
Field- and anisotropy-induced evolution of magnetization distributions in ultrathin cobalt films
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 300 (2006) e301–e304 www.elsevier.com/locate/jmmm
Field- and anisotropy-induced evolution of magnetization distributions in ultrathin cobalt films V. Zablotskiia,b,, M. Kisielewskia, A. Maziewskia, T. Polyakovac a
Institute of Experimental Physics, University of Bialystok, Lipowa 41, 15-424 Bialystok, Poland b Institute of Physics ASCR, Na Slovance 2, 18221 Prague 8, Czech Republic c Donetsk National University, St. Universitetskaya 24, Donetsk, 83055 Ukraine Available online 16 November 2005
Abstract The thickness- and field-driven evolutions of magnetization distributions in ultrathin cobalt films are studied by micromagnetic simulations. For laterally infinite ultrathin films we suggest simple formulas allowing calculations of the domain periods in wide scales of the thicknesses and in-plane applied fields. For laterally limited samples the edge effect was shown to extend the existence of out-of-plane magnetizations states behind the reorientation phase transition thickness (or field). New types of magnetization distributions such as the edge domains structure and out-of-plane domains patterned vortex and leaf were found in semi-infinite films and nanodisks, respectively. r 2005 Elsevier B.V. All rights reserved. PACS: 75.70Kw; 75.30Gw Keywords: Magnetic domains; Ultrathin films; Magnetic anisotropy
Domain patterns of ultrathin films of thickness less than the exchange length have many distinctive features: unusually sharp thickness dependence of domain sizes [1–3], strong reconstruction of magnetization distributions in domains related to anisotropy changes [4], the dendritic growth [5], etc. which are of great interest and essential for an understanding of magnetism on the nanoscale length. Generally, one can expect ultrathin film in a practically monodomain state—large magnetic domains with geometry determined mainly by coercivity rather than magnetostatic forces. However, domain size drastically decreases down to a sub-micrometer scale while approaching the field or thickness-induced reorientation phase transition (RPT). As the film thickness or in-plane magnetic field increases the transition from the perpendicular magnetization state into the in-plane state takes place [1,2]. In laterally infinite films with uniaxial anisotropy, under the assumption of homogenous magnetization distributions, the RPT takes place at the thickness defined by the condition: Corresponding author. Tel.: +48 85 745 72 28; fax: +48 85 745 72 23.
E-mail address: [email protected] (V. Zablotskii). 0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.10.106
K eff ðd ¼ d 1 Þ ¼ 0, where K eff ¼ K 1 2pM 2S , K1 is the first uniaxial anisotropy constant and MS is the saturation magnetization. An open problem here is the influence of the domains and the sample edge on the critical RPT thickness. It was recently shown that the existence of sinusoidal domains shifts the critical thickness d1 to a larger value, d , e.g. for gold enveloped cobalt films d 1 ¼ 1:79 nm (in the terms of the quality factor, Q ¼ K 1 =2pM 2S the RPT condition is Qðd 1 Þ ¼ K 1 ðd 1 Þ=2pM 2S ¼ 1) and d ¼ 1:86 nmðQðd Þo1) [4]. Squeezing the sample lateral size one can expect a larger shift of the RPT thickness because of the edge influence: cutting-off the demagnetizing field [6]. For nanoplatelets with Q ¼ 0:9 such a phenomenon was recently demonstrated by Monte-Carlo simulations in [7]. In the present work by micromagnetic simulations we demonstrate that the edge effect manifests itself in the appearance of new types of out-of-plane magnetization distributions at the edge of cobalt films and in nanosized disks of thickness dXd . Utilizing the OOMMF software [8] we simulate magnetization distributions in gold enveloped cobalt samples with the thickness dependent uniaxial anisotropy described by
ARTICLE IN PRESS V. Zablotskii et al. / Journal of Magnetism and Magnetic Materials 300 (2006) e301–e304
It follows from Eq. (1) that huge changes of the domain period take place as the film thickness changes. These changes are caused by the both thickness dependencies of magnetostatic and anisotropy contributions. Increasing the thickness from d ¼ 1 nm the domain period drops in several orders of magnitude and reaches its minimal value: p =l ex ¼ 8pl ex =d þ 2pd=l ex at the RPT [4]. Near the RPT, in the thickness range d=d 1 ¼ 121:04, the DS periods obtained by simulations coincide with periods given by the last formula. Under influence of in-plane applied magnetic field magnetization distributions and domain size evolve in a similar way. Here the same sequence of magnetization distributions was found by simulations [4]. On the basis results obtained by simulations we deduce the field dependence of the DS period as pðHÞ H ln þ 6:16, (2) ¼ 4:16 l ex H Aeff where H Aeff ¼ 2K 1 =M S 4pM S ¼ 6:66 kOe is the effective anisotropy field for ultrathin cobalt film with d ¼ 1 nm. Approaching the field-induced RPT (at H ¼ H Aeff ) the sinusoidal domains appear with period p ¼ p . Similarity and difference in the field and thickness driven evolutions of the magnetization distributions are seen in Fig. 1, where we show the normalized normal component of the film magnetization as a function of the normalized in-plane filed and thickness. The solid line represents the coherent rotation of the magnetization with the field described as mz ¼ ð1 ðH=H Aeff Þ2 Þ1=2 [10]. The last formula shows the RPT takes place at H ¼ H Aeff . The simulated dependencies mz(H/HAeff) and mz(d/d1) are shown by the dashed lines with diamond and circles symbols, respectively. Comparing these curves and keeping in the mind Eqs. (1) and (2) one can conclude: (i) the
1
mz
QðdÞ ¼ ðK 1v þ 2K 1S =dÞ=2pM 2S , with the bulk and surface anisotropy constants: K 1v ¼ 1:9 MJ=m3 and K 1S ¼ 0:57 mJ=m2 [9]. Let us consider the thickness driven evolution of magnetization distributions in the laterally infinite film at zero-applied field. To find the magnetization distributions we performed 1D-simulations dividing samples into cells with lateral sizes less than the cobalt exchange length (l ex ¼ ðA=ð2pM 2S ÞÞ1=2 3 nm (for details see Ref. [4]), where A is the exchange constant. It was found that the increase of the cobalt thickness of from 1 nm up to 1.86 nm leads to the following sequence of magnetization distribution: (i) perpendicularly magnetized stripe domains with negligible narrow domain walls; (ii) structure of out-ofplane stripe domains with walls width comparable with the domain size; (iii) sinusoidal domain structure with small normal component of magnetization vector for dEd1, and (iv) the in-plane state for dXd . The obtained thickness dependence of the domain period could be fitted by pðdÞ d ln ¼ 27:9 exp 2:675 . (1) l ex l ex
0.5
0 0
0.2
0.4
0.6 d/d1, H/HAeff
1
0.8
1.2
Fig. 1. Normalized normal component of the film magnetization as a function of the normalized in-plane field and thickness. The solid line describes the coherent rotation of the magnetization to the field direction. The simulated dependences mz(H/HAeff) and mz(d/d1) are shown by the dashed lines with diamonds and circles, respectively.
1 d=1.9 nm d=2.0 nm d=2.1 nm d=2.2 nm
0.5 mz
e302
0 0
100
200 2800
2900
3000
x, nm Fig. 2. Magnetization distributions in the edge areas of 3000 nm wide samples of different thickness, d4d .
domain like magnetization distributions exist in wide ranges of the field and thickness below d1; (ii) the domains magnetostatic contribution shifts the both thicknessinduced and field-induced RPTs to d 4d1 and H 4HAeff, accordingly; (iii) the thickness-induced RPT is more abrupt than that induced by the in-plane magnetic field. Thus, in laterally infinite ultrathin films the domains magnetostatic contribution shifts the RPT to a higher value of the applied in-plane field (H 4HAeff) and/or lower quality factor (Qo1). Let us consider the edge influence on the magnetization distribution in semi-infinite films of d4d . Note, an infinite film has no domains for d4d . By simulations we demonstrate that domains with mz a0 exist in semi-infinite samples of d4d . In Fig. 2 we show magnetization distributions in the edge area of the semi-infinite films of different thickness. The RPT occurs by means of a damped sinusoidal-like domain structure in which the magnetization profile—mz(x)-function-decays with the x-coordinate (the point x ¼ 0 is placed at the film edge). These
ARTICLE IN PRESS V. Zablotskii et al. / Journal of Magnetism and Magnetic Materials 300 (2006) e301–e304
magnetization profiles could be fitted by the function: 2px x=Ld cos mz ðxÞ ¼ mz0 e , (3) p where Ld and p are the decay length and preriod of the edge domain structure, and mz0 is the edge amplitude. Thus, the edge domain structure (EDS) is characterized by these three parameters. The shown in Fig. 2 magnetization distributions demonstrate the edge effect: mz-amplitude at the edge is much higher than that in the interior. At the edges the magnetization distribution is distorted by the demagnetizing field cutting-off while in the sample interior the magnetization is in-plane. The internal in-plane magnetization state has almost zero energy and therefore could be easily ‘‘squeezed’’ by decreasing the sample lateral size. For the sample with d ¼ 1:9 nm (which is slightly above d ) the magnetization oscillates with the period p ¼ p [4]. Hovewer, the period decreases as d increases. In Fig. 3, we plotted the decay length and period and as functions of the sample thickness. Both Ld and p decreases with the sample thickness. The shown dependences could be fitted by c l ex d , (4) d d where c is a fitting parameter (cE1). It is a rather complicated problem to calculate the three equilibrium EDS parameters from an approach based on the total energy functional minimization. Here, on the basis of the simulated magnetization distributions, we suggest the fitting function Eq. (4) which satisfies the limiting case: Ld-N when d-d . Thus, the edge effect manifests itself in the appearance of the EDS which decays with in the sample interior on the length scale approximately determined by Eq. (4). Following is the qualitative explanation of the edge effect. At the film edge, the demagnetize field is half of its value in the center of the film. This implies that at the film edge area the effective anisotropy constant is larger than that in the film interior. The edge-enhanced effective anisotropy favors an out-of-plane magnetization resulting Ld ðdÞ ¼
in the existence of the perpendicular magnetization component in the edge areas even for d4d . We simulate magnetization distributions in disks of the thickness d ¼ 1:86 nm which is just below d . For this thickness perpendicular anisotropy is rather weak (Qo1), and therefore, in infinite films j mz ðxÞ j o0:38 (see Fig. 1). However, in such disks, due to the edge effect, the magnetization amplitude mz increases. To demonstrate the edge effect performance we choose cobalt disks of d ¼ d ¼ 1:86 nm and radius R ¼ 238 nm. The obtained magnetization distributions are shown in Figs. 4(a–c). The shown distributions were obtained starting with the different initial distributions: (a) domain-like distribution, (b) pure in-plane vortex, (c) homogeneous perpendicular distribution. In the all distributions a non-zero mzcomponent exists at the sample edges, additionally to the in-plane components. The energies of these structures are: E a ¼1 240 584 J/m3, E b ¼1 237 105 J=m3 , and E c ¼ 1 237 009 J=m3 (the ground state). The edge effect is clearly visible for each of the samples (Fig. 4). In the patterned vortex and leaf (see Figs. 4b,c) the spatial period of the quasi-domains is pE120 nm which is close to the period of the sinusoidal DS in an infinite film (p ¼ 150 nm for d ¼ d ¼ 1:86). Thus, the laterally limited samples may keep a few out-of-plane domains with the period slightly below the DS period inherent infinite films of the same thickness. In conclusion, in laterally infinite ultrathin films the domains contribution increases the both applied in-plane field (H4HAeff) and anisotropy (Qo1) ranges of the
150
30
10
p, nm
200
50
Ld, nm
e303
100 1.8
2
2.2 d, nm
Fig. 3. Decay length and EDS period as a function of the film thickness.
Fig. 4. Simulated magnetization distributions in disks of d ¼ d ¼ 1:86 and 2R ¼ 476 nm: (a) multivortex state; (b) domain patterned vortex, and (c) the ground state-domain patterned leaf. The simulations were performed on the mesh consisting of cubic-shaped cells with size 1.86 nm.
ARTICLE IN PRESS e304
V. Zablotskii et al. / Journal of Magnetism and Magnetic Materials 300 (2006) e301–e304
multidomain state existence. In low-anisotropy laterally limited samples of the magnetization distributions are strongly influenced by the edge cutting-off of the demagnetizing field. Acknowledgements The work was supported by Marie Curie Fellowships for ‘‘Transfer of Knowledge’’ (‘‘NANOMAG-LAB’’, N 2004003177) and the Polish State Committee for Scientific Research (Grant No.: 4 T11B 006 24). References [1] M. Speckmann, H.P. Oepen, H. Ibach, Phys. Rev. Lett. 75 (1995) 2035.
[2] H.P. Oepen, M. Speckmann, Y. Millev, J. Kirschner, Phys. Rev. B 55 (1997) 2752. [3] M. Kisielewski, A. Maziewski, V. Zablotskii, T. Polyakova, J.M. Garcia, A. Wawro, L.T. Baczewski, J. Appl. Phys. 93 (2003) 6966. [4] M. Kisielewski, A. Maziewski, T. Polyakova, V. Zablotskii, Phys. Rev. B. 69 (2004) 184419. [5] W. Stefanowicz, M. Tekielak, V. Bucha, A. Maziewski, V. Zablotskii, L.T. Baczewski, A. Wawro, Mater. Sci., to appear. [6] M. Kisielewski, A. Maziewski, V. Zablotskii, J. Magn. Magn. Mater. 290–291 (2005) 776. [7] E.Y. Vedmedenko, H.P. Oepen, J. Kirschner, Phys. Rev. B 67 (2003) 012409. [8] M. Donahue, D. Porter, Object oriented micromagnetic framework, http://math.nist.gov/oommf, free software for micromagnetic simulations. [9] M. Kisielewski, A. Maziewski, M. Tekielak, A. Wawro, L.T. Baczewski, Phys. Rev. Lett. 89 (2002) 087203. [10] A. Hubert, R. Scha¨fer, Magnetic Domains, Springer, Berlin, 1998.
Journal of Magnetism and Magnetic Materials 300 (2006) e301–e304 www.elsevier.com/locate/jmmm
Field- and anisotropy-induced evolution of magnetization distributions in ultrathin cobalt films V. Zablotskiia,b,, M. Kisielewskia, A. Maziewskia, T. Polyakovac a
Institute of Experimental Physics, University of Bialystok, Lipowa 41, 15-424 Bialystok, Poland b Institute of Physics ASCR, Na Slovance 2, 18221 Prague 8, Czech Republic c Donetsk National University, St. Universitetskaya 24, Donetsk, 83055 Ukraine Available online 16 November 2005
Abstract The thickness- and field-driven evolutions of magnetization distributions in ultrathin cobalt films are studied by micromagnetic simulations. For laterally infinite ultrathin films we suggest simple formulas allowing calculations of the domain periods in wide scales of the thicknesses and in-plane applied fields. For laterally limited samples the edge effect was shown to extend the existence of out-of-plane magnetizations states behind the reorientation phase transition thickness (or field). New types of magnetization distributions such as the edge domains structure and out-of-plane domains patterned vortex and leaf were found in semi-infinite films and nanodisks, respectively. r 2005 Elsevier B.V. All rights reserved. PACS: 75.70Kw; 75.30Gw Keywords: Magnetic domains; Ultrathin films; Magnetic anisotropy
Domain patterns of ultrathin films of thickness less than the exchange length have many distinctive features: unusually sharp thickness dependence of domain sizes [1–3], strong reconstruction of magnetization distributions in domains related to anisotropy changes [4], the dendritic growth [5], etc. which are of great interest and essential for an understanding of magnetism on the nanoscale length. Generally, one can expect ultrathin film in a practically monodomain state—large magnetic domains with geometry determined mainly by coercivity rather than magnetostatic forces. However, domain size drastically decreases down to a sub-micrometer scale while approaching the field or thickness-induced reorientation phase transition (RPT). As the film thickness or in-plane magnetic field increases the transition from the perpendicular magnetization state into the in-plane state takes place [1,2]. In laterally infinite films with uniaxial anisotropy, under the assumption of homogenous magnetization distributions, the RPT takes place at the thickness defined by the condition: Corresponding author. Tel.: +48 85 745 72 28; fax: +48 85 745 72 23.
E-mail address: [email protected] (V. Zablotskii). 0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.10.106
K eff ðd ¼ d 1 Þ ¼ 0, where K eff ¼ K 1 2pM 2S , K1 is the first uniaxial anisotropy constant and MS is the saturation magnetization. An open problem here is the influence of the domains and the sample edge on the critical RPT thickness. It was recently shown that the existence of sinusoidal domains shifts the critical thickness d1 to a larger value, d , e.g. for gold enveloped cobalt films d 1 ¼ 1:79 nm (in the terms of the quality factor, Q ¼ K 1 =2pM 2S the RPT condition is Qðd 1 Þ ¼ K 1 ðd 1 Þ=2pM 2S ¼ 1) and d ¼ 1:86 nmðQðd Þo1) [4]. Squeezing the sample lateral size one can expect a larger shift of the RPT thickness because of the edge influence: cutting-off the demagnetizing field [6]. For nanoplatelets with Q ¼ 0:9 such a phenomenon was recently demonstrated by Monte-Carlo simulations in [7]. In the present work by micromagnetic simulations we demonstrate that the edge effect manifests itself in the appearance of new types of out-of-plane magnetization distributions at the edge of cobalt films and in nanosized disks of thickness dXd . Utilizing the OOMMF software [8] we simulate magnetization distributions in gold enveloped cobalt samples with the thickness dependent uniaxial anisotropy described by
ARTICLE IN PRESS V. Zablotskii et al. / Journal of Magnetism and Magnetic Materials 300 (2006) e301–e304
It follows from Eq. (1) that huge changes of the domain period take place as the film thickness changes. These changes are caused by the both thickness dependencies of magnetostatic and anisotropy contributions. Increasing the thickness from d ¼ 1 nm the domain period drops in several orders of magnitude and reaches its minimal value: p =l ex ¼ 8pl ex =d þ 2pd=l ex at the RPT [4]. Near the RPT, in the thickness range d=d 1 ¼ 121:04, the DS periods obtained by simulations coincide with periods given by the last formula. Under influence of in-plane applied magnetic field magnetization distributions and domain size evolve in a similar way. Here the same sequence of magnetization distributions was found by simulations [4]. On the basis results obtained by simulations we deduce the field dependence of the DS period as pðHÞ H ln þ 6:16, (2) ¼ 4:16 l ex H Aeff where H Aeff ¼ 2K 1 =M S 4pM S ¼ 6:66 kOe is the effective anisotropy field for ultrathin cobalt film with d ¼ 1 nm. Approaching the field-induced RPT (at H ¼ H Aeff ) the sinusoidal domains appear with period p ¼ p . Similarity and difference in the field and thickness driven evolutions of the magnetization distributions are seen in Fig. 1, where we show the normalized normal component of the film magnetization as a function of the normalized in-plane filed and thickness. The solid line represents the coherent rotation of the magnetization with the field described as mz ¼ ð1 ðH=H Aeff Þ2 Þ1=2 [10]. The last formula shows the RPT takes place at H ¼ H Aeff . The simulated dependencies mz(H/HAeff) and mz(d/d1) are shown by the dashed lines with diamond and circles symbols, respectively. Comparing these curves and keeping in the mind Eqs. (1) and (2) one can conclude: (i) the
1
mz
QðdÞ ¼ ðK 1v þ 2K 1S =dÞ=2pM 2S , with the bulk and surface anisotropy constants: K 1v ¼ 1:9 MJ=m3 and K 1S ¼ 0:57 mJ=m2 [9]. Let us consider the thickness driven evolution of magnetization distributions in the laterally infinite film at zero-applied field. To find the magnetization distributions we performed 1D-simulations dividing samples into cells with lateral sizes less than the cobalt exchange length (l ex ¼ ðA=ð2pM 2S ÞÞ1=2 3 nm (for details see Ref. [4]), where A is the exchange constant. It was found that the increase of the cobalt thickness of from 1 nm up to 1.86 nm leads to the following sequence of magnetization distribution: (i) perpendicularly magnetized stripe domains with negligible narrow domain walls; (ii) structure of out-ofplane stripe domains with walls width comparable with the domain size; (iii) sinusoidal domain structure with small normal component of magnetization vector for dEd1, and (iv) the in-plane state for dXd . The obtained thickness dependence of the domain period could be fitted by pðdÞ d ln ¼ 27:9 exp 2:675 . (1) l ex l ex
0.5
0 0
0.2
0.4
0.6 d/d1, H/HAeff
1
0.8
1.2
Fig. 1. Normalized normal component of the film magnetization as a function of the normalized in-plane field and thickness. The solid line describes the coherent rotation of the magnetization to the field direction. The simulated dependences mz(H/HAeff) and mz(d/d1) are shown by the dashed lines with diamonds and circles, respectively.
1 d=1.9 nm d=2.0 nm d=2.1 nm d=2.2 nm
0.5 mz
e302
0 0
100
200 2800
2900
3000
x, nm Fig. 2. Magnetization distributions in the edge areas of 3000 nm wide samples of different thickness, d4d .
domain like magnetization distributions exist in wide ranges of the field and thickness below d1; (ii) the domains magnetostatic contribution shifts the both thicknessinduced and field-induced RPTs to d 4d1 and H 4HAeff, accordingly; (iii) the thickness-induced RPT is more abrupt than that induced by the in-plane magnetic field. Thus, in laterally infinite ultrathin films the domains magnetostatic contribution shifts the RPT to a higher value of the applied in-plane field (H 4HAeff) and/or lower quality factor (Qo1). Let us consider the edge influence on the magnetization distribution in semi-infinite films of d4d . Note, an infinite film has no domains for d4d . By simulations we demonstrate that domains with mz a0 exist in semi-infinite samples of d4d . In Fig. 2 we show magnetization distributions in the edge area of the semi-infinite films of different thickness. The RPT occurs by means of a damped sinusoidal-like domain structure in which the magnetization profile—mz(x)-function-decays with the x-coordinate (the point x ¼ 0 is placed at the film edge). These
ARTICLE IN PRESS V. Zablotskii et al. / Journal of Magnetism and Magnetic Materials 300 (2006) e301–e304
magnetization profiles could be fitted by the function: 2px x=Ld cos mz ðxÞ ¼ mz0 e , (3) p where Ld and p are the decay length and preriod of the edge domain structure, and mz0 is the edge amplitude. Thus, the edge domain structure (EDS) is characterized by these three parameters. The shown in Fig. 2 magnetization distributions demonstrate the edge effect: mz-amplitude at the edge is much higher than that in the interior. At the edges the magnetization distribution is distorted by the demagnetizing field cutting-off while in the sample interior the magnetization is in-plane. The internal in-plane magnetization state has almost zero energy and therefore could be easily ‘‘squeezed’’ by decreasing the sample lateral size. For the sample with d ¼ 1:9 nm (which is slightly above d ) the magnetization oscillates with the period p ¼ p [4]. Hovewer, the period decreases as d increases. In Fig. 3, we plotted the decay length and period and as functions of the sample thickness. Both Ld and p decreases with the sample thickness. The shown dependences could be fitted by c l ex d , (4) d d where c is a fitting parameter (cE1). It is a rather complicated problem to calculate the three equilibrium EDS parameters from an approach based on the total energy functional minimization. Here, on the basis of the simulated magnetization distributions, we suggest the fitting function Eq. (4) which satisfies the limiting case: Ld-N when d-d . Thus, the edge effect manifests itself in the appearance of the EDS which decays with in the sample interior on the length scale approximately determined by Eq. (4). Following is the qualitative explanation of the edge effect. At the film edge, the demagnetize field is half of its value in the center of the film. This implies that at the film edge area the effective anisotropy constant is larger than that in the film interior. The edge-enhanced effective anisotropy favors an out-of-plane magnetization resulting Ld ðdÞ ¼
in the existence of the perpendicular magnetization component in the edge areas even for d4d . We simulate magnetization distributions in disks of the thickness d ¼ 1:86 nm which is just below d . For this thickness perpendicular anisotropy is rather weak (Qo1), and therefore, in infinite films j mz ðxÞ j o0:38 (see Fig. 1). However, in such disks, due to the edge effect, the magnetization amplitude mz increases. To demonstrate the edge effect performance we choose cobalt disks of d ¼ d ¼ 1:86 nm and radius R ¼ 238 nm. The obtained magnetization distributions are shown in Figs. 4(a–c). The shown distributions were obtained starting with the different initial distributions: (a) domain-like distribution, (b) pure in-plane vortex, (c) homogeneous perpendicular distribution. In the all distributions a non-zero mzcomponent exists at the sample edges, additionally to the in-plane components. The energies of these structures are: E a ¼1 240 584 J/m3, E b ¼1 237 105 J=m3 , and E c ¼ 1 237 009 J=m3 (the ground state). The edge effect is clearly visible for each of the samples (Fig. 4). In the patterned vortex and leaf (see Figs. 4b,c) the spatial period of the quasi-domains is pE120 nm which is close to the period of the sinusoidal DS in an infinite film (p ¼ 150 nm for d ¼ d ¼ 1:86). Thus, the laterally limited samples may keep a few out-of-plane domains with the period slightly below the DS period inherent infinite films of the same thickness. In conclusion, in laterally infinite ultrathin films the domains contribution increases the both applied in-plane field (H4HAeff) and anisotropy (Qo1) ranges of the
150
30
10
p, nm
200
50
Ld, nm
e303
100 1.8
2
2.2 d, nm
Fig. 3. Decay length and EDS period as a function of the film thickness.
Fig. 4. Simulated magnetization distributions in disks of d ¼ d ¼ 1:86 and 2R ¼ 476 nm: (a) multivortex state; (b) domain patterned vortex, and (c) the ground state-domain patterned leaf. The simulations were performed on the mesh consisting of cubic-shaped cells with size 1.86 nm.
ARTICLE IN PRESS e304
V. Zablotskii et al. / Journal of Magnetism and Magnetic Materials 300 (2006) e301–e304
multidomain state existence. In low-anisotropy laterally limited samples of the magnetization distributions are strongly influenced by the edge cutting-off of the demagnetizing field. Acknowledgements The work was supported by Marie Curie Fellowships for ‘‘Transfer of Knowledge’’ (‘‘NANOMAG-LAB’’, N 2004003177) and the Polish State Committee for Scientific Research (Grant No.: 4 T11B 006 24). References [1] M. Speckmann, H.P. Oepen, H. Ibach, Phys. Rev. Lett. 75 (1995) 2035.
[2] H.P. Oepen, M. Speckmann, Y. Millev, J. Kirschner, Phys. Rev. B 55 (1997) 2752. [3] M. Kisielewski, A. Maziewski, V. Zablotskii, T. Polyakova, J.M. Garcia, A. Wawro, L.T. Baczewski, J. Appl. Phys. 93 (2003) 6966. [4] M. Kisielewski, A. Maziewski, T. Polyakova, V. Zablotskii, Phys. Rev. B. 69 (2004) 184419. [5] W. Stefanowicz, M. Tekielak, V. Bucha, A. Maziewski, V. Zablotskii, L.T. Baczewski, A. Wawro, Mater. Sci., to appear. [6] M. Kisielewski, A. Maziewski, V. Zablotskii, J. Magn. Magn. Mater. 290–291 (2005) 776. [7] E.Y. Vedmedenko, H.P. Oepen, J. Kirschner, Phys. Rev. B 67 (2003) 012409. [8] M. Donahue, D. Porter, Object oriented micromagnetic framework, http://math.nist.gov/oommf, free software for micromagnetic simulations. [9] M. Kisielewski, A. Maziewski, M. Tekielak, A. Wawro, L.T. Baczewski, Phys. Rev. Lett. 89 (2002) 087203. [10] A. Hubert, R. Scha¨fer, Magnetic Domains, Springer, Berlin, 1998.