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Magnetostatic interactions in planar ring-like nanoparticle structures
Thin Solid Films 515 (2006) 731 – 734 www.elsevier.com/locate/tsf
Magnetostatic interactions in planar ring-like nanoparticle structures Yu. A. Koksharov a,*, G.B. Khomutov a, E.S. Soldatov a, D. Suyatin b, I. Maximov b, L. Montelius b, P. Carlberg b b
a M. V. Lomonosov Moscow State University, 119992, Moscow, Russia Lund University, Solid State Physics/Nanometer Consortium, Box 118, SE-221 00, Lund, Sweden
Available online 20 January 2006
Abstract Numerical calculations of equilibrium state energies and local magnetic fields in planar ring-like nanoparticle structures were performed. The dipole – dipole, Zeeman and magnetic anisotropy interactions were included into the model. The result of their competition depends on the value of the external magnetic field, magnetic parameters of an individual nanoparticle, size and shape of the structures. Flux-closed vortexes, single domain, two-domain ‘‘onion’’-like, ‘‘hedgehog’’-like and more complex spin structures can be realized. The critical field, providing a sharp transition from the flux-closed vortex to the ‘‘onion’’-like state, can be regulated by a variation of the particle magnetization and anisotropy constant, their easy directions, and particle space arrangement. D 2005 Elsevier B.V. All rights reserved. PACS: 78.75.+a; 81.07. b Keywords: Magnetism; Nanoparticles; Two-dimensional; Ring-like structures
1. Introduction Nanoscale magnets attract much attention due to their promising technological applications in magnetic recording [1], spintronics [2], medicine [3], etc. For example, arrays of lithographically defined nano-sized magnetic elements can be used in non-volatile magnetic random access memory (MRAM) [2,4]. Recently, ring-shaped magnetic nano-structures have been proposed as basic elements of MRAM [5,6]. Existence of several different stable magnetic states is one of the advantages of the ring shaped magnets. These states could allow for more than one bit storage in each element [4,7,8]. The ring-like nanostructures can be successfully fabricated by numerous lithography methods [4,9] and self-assembled processes [10]. In ferrofluids magnetic particles form various structures, such as separate and branched chains, circles and other ring-like structures, clusters and fractal-like complexes [11 – 13]. It was shown [14] that in magnetically dilute ferrofluids dipole – dipole interactions between magnetic parti* Corresponding author. M. V. Lomonosov Moscow State University, Faculty of Physics, General Physics Department, 119992 Moscow, Russia. Tel.: +7 95 939 29 73; fax: +7 95 939 14 89. E-mail address: [email protected] (Yu.A. Koksharov). 0040-6090/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2005.12.177
cles promote a formation of flexible ring-like nanostructures. In weak magnetic fields, these ring-like structures are stable and orient perpendicular to the field. In strong magnetic fields the rings are destroyed, and the particles form linear chains [14]. The use of biological molecules (like DNA) and their molecular-recognition properties to guide the assembly of nanoparticle structures on a solid surface looks attractive [15,16]. In this case particles should form essentially rigid structure, so that the distances between them are invariant. As it is shown in our work, magnetic properties of such rigid ringlike structures depend strongly on the dipole – dipole and Zeeman magnetostatic interactions, and can be sufficiently modified by the single-particle magnetic anisotropy. The magnetic properties of ring-like planar nanoparticles were studied theoretically and experimentally in a number of works [1,2,4 – 10,20]. In nano-sized magnetic elements (continuous circular or elliptical metallic dots) magnetic vortex structures appear from a competition between exchange and magnetostatic energies, while anisotropy energy is usually considered to be negligible. In the present work different ringlike nanoparticle structures are studied. Dipole –dipole (magnetostatic), Zeeman, and anisotropy interactions are the most important in these structures, whereas the exchange energy does not play any significant role. Similar to ring-like
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continuous nanostructures, nanoparticle structures, studied in our work, show various spin configurations — flux-closed vortexes, ‘‘onion’’-like and more complex ones. 2. Model We consider a system of N spherical equivalent magnetic particles (with magnetic moments m i and a diameter d) located in vertexes of a regular polygon with a distance R between the polygon center and each particle (Fig. 1). The total energy W of classical interactions is the sum of Zeeman W Z, dipole –dipole W DD, and single-particle magnetic anisotropy Wanis terms (see, for example, [21], Eqs. B2, E3, E4, E9). The system energy can be regarded as a function of N parameters, e.g. angles a i between the vectors m i and the Xaxis. Using a so called ‘‘dynamical’’ version [17] of a conjugate graduate method [18] we performed the total energy minimization procedures. In order to escape the local minima we varied initial spin configurations (the examples are mV1 mW1, mV2 mW2, mV3 mW3 in Fig. 1). Besides the ideal circle structures we investigated also oblate and oblong ellipses, and circle-like structures without some particles. In that way a semi-circle and arcs of different length were considered with the same R and d values. Using a classical formula ([21], Eq. E4) we computed also dipole– dipole local magnetic fields acting on the nanoparticles and the local field in the system center (the point O in Fig. 1). In the calculations we used magnetic parameters for magnetite (Fe3O4): saturation magnetization M s = 480 emu/ cm3, and the volume anisotropy constant |K anis| å 105 erg/cm3 [19]. Magnetite is often used in biomedical nanoparticle
Fig. 1. Schematic illustration of arrangement of magnetic nanoparticles in vertexes of a regular planar polygon, where R is the distance between the polygon center (O) and its vertex, d is the particle diameter. Easy directions (solid line segments A1V – AW1, AV2 – AW2 as an example) are determined by the angle h, dotted lines denote the tangents to the circumcircle in the vertexes. Thick vertical arrow shows the direction of the external magnetic field H ext. Pairs of arrows (mV1; mW1), (m 2V ; mW2), (mV3; mW3) show directions of neighboring magnetic moments in basic configurations, which were used in the energy minimization procedures.
applications due to its robustness, low toxicity, and relatively high magnetization. Other parameters for the model were chosen as follows: d = 2 nm, R = 10 nm. In this case the maximum number of particles should be equal to ¨2pR / d å 30. We did not take into account any magnetic exchange interactions between neighboring particles since for real nanoparticles, which are in direct contact, the interparticle short-range forces are very complicated and can not be reduced to a simple isotropic bi-spin exchange. 3. Results of calculations and discussion Spin configurations in circular nanoparticle structures can be rather different. Some of the configurations are shown in Fig. 2. Let us consider the case K anis = 0. If H ext = 0, the fluxclosed vortex structure has a minimal energy (Fig. 2a). In this case we get W Z å 0 and W å W DD. With H ext increased the spins smoothly turn to the external field direction, but while H ext < H cr they remain approximately directed along tangents to the circumcircle and the flux-closed structure is not disturbed significantly. At H ext = H cr the vortex structure drastically transforms to ‘‘onion’’-like structure, which is characterized by an existence of two ‘‘domains’’ separated by a ‘‘domain’’ wall (Fig. 2b). The type of a preferable spin structure is determined by the energy gain, which the system obtains after a transition from one spin configuration to another. If H ext increases more, the ‘‘onion’’-like structure gradually transforms into a single-domain structure, in which all spins are parallel to the external field. An intermediate situation between the uniform single-domain state and the two-domain ‘‘onion’’-like state is presented in Fig. 2c. More complex spin behavior takes place in case of non-zero magnetic anisotropy. If the value of the angle h (Fig. 1) is close to zero, the anisotropy energy stabilizes the flux-closed vortex state (Fig. 3d). However, with the angle h increased, the critical field H cr changes significantly (Fig. 3d). A qualitative transformation of the spin structure can be observed for h å p / 2. Firstly, the ‘‘wall’’ between ‘‘onion’’ domains can become more extended (Fig. 2d,e). Secondly, if N is relatively small and the maximum value of Wanis,max (which is realized for h å p / 2) is comparable with the maximum value of W DD,max (which is realized for tangent to the circumcircle spins), the ground state structure is not flux-closed vortex, but ‘‘hedgehog’’-like (Fig. 2f) even in case of zero external magnetic field. With W DD,max / Wanis,max ratio increase the zero-field ground state structure will become a flux-closed vortex, irrespective of h value. This happens, e.g. for N = 20 and N = 30, H ext = 0 and K anis 105 erg/cm3. The transition from the flux-closed vortex to the ‘‘onion’’like state is accompanied by drastic changes in various parameters. The Zeeman and dipole– dipole energies undergo sharp transformations at H cr, while the total energy has a kink at this point. Average magnetic field, acting on a nanoparticle due to magnetic dipole moments of other particles and its dispersion are changed abruptly at H cr. The same behavior is typical for the local dipole –dipole magnetic field in the circle center. It is important that the local field dispersion is very
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Fig. 2. Examples of different spin configurations in the circle nanoparticle structure: (a) flux-closed vortex state; (b) two-domain ‘‘onion’’-like state; (c) intermediate state between homogeneous ‘‘single-domain’’ state and ‘‘onion’’-like state; (d) and (e) ‘‘onion’’-like states with extensive ‘‘domain boundary’’; (f) ‘‘hedgehog’’-like state. Model parameters N, H ext, K anis and h are shown in the figure.
Fig. 3. Variation of the critical magnetic field H cr depending on the interparticle distances (a) – (c), and the easy direction (d). In cases (a) – (c) H cr is plotted as a function of the average value of 1 / r ij3, where r ij is the interparticle distance. The value of r ij was changed: (a) by the variation of the particle number (N = 10, 20, 30) in the ideal circle structure; (b) by making a symmetrical ‘‘gap’’ in the circle structure, removing some particles, but keeping the distance between neighbor particles constant, the figures in (b) show the number of residuary particles; (c) by distorting of the structure shape from circular to elliptical one, the corresponding ratios of horizontal and vertical semi-axes lengths are indicated in (c), the minimal axis length is equal to R. In the case (d) ideal circle structures with N = 10, 20 and 30 are considered, K anis = 105 erg/cm3, the meaning of h is clear from Fig. 1.
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small in the flux-closed vortex state, that is, all particles are in nearly the same conditions. This is the feature of the fluxclosed state. The critical field H cr depends on the averaged interparticle distance. In the case of ideal circle structures, the value of H cr increases linearly with averaged 1 / r ij3 (Fig. 3a). However, for more complex structures, e.g. the circles with a ‘‘gate’’ (Fig. 3b), the critical field changes non-monotonously with 1 / r ij3. The ‘‘gap’’ between spins breaks the dipole– dipole interaction chain, typical of the flux-closed vortex state, and makes this state less stable in the external magnetic field (left part of the curve in Fig. 3b). But, H cr grows with < 1 / r ij3 > increase again when the ‘‘gate’’ in the circle becomes sufficiently large (Fig. 3b). It should be noted that the spin configurations of the ‘‘defective’’ circles (Fig. 3b) can be in some ways different from those presented in Fig. 2. However, the critical field can be easily determined by observing the sharp changes in local magnetic fields. The transformation of the ideal circle into ellipses results in a significant reduction of the critical field (Fig. 3c). For the ellipse stretched along the external magnetic field axes, this reduction is clearly less pronounced than for the ellipse elongated perpendicular to H ext. The reduction of H cr can be explained by a significant increase in the average interparticle distance (compare X-axis scales in Fig. 3a and c). Furthermore, the flux-closed vortex configuration should be more stable for the oblong ellipse than for the oblate one, since in the former case the particles with parallel magnetic moments are situated closer and therefore effectively magnetize each other. It is interesting to compare the dipole – dipole energy with a thermal energy W T ; nBT å 5 I 10 14 erg (T = 300 K). In case of d = 2 nm, and R = 10 nm the magnetic moment m of the magnetite nanoparticle is equal to ¨200 l B and a maximal value of W DD is estimated as ¨N I m 2 / d 3 å 5 I 10 5 erg (N = 30). Since m ¨ d 6, then we get W DD ¨ d 3, assuming N to be a constant. In case of nanoparticles with d = 4 nm, the dipole – dipole and thermal energies will be nearly equal. Therefore magnetic vortex-like structures, considered in this work, will be stable at room temperature in spite of superparamagnetic fluctuations if d > 5 –6 nm. 4. Conclusions We have performed numerical calculations of magnetostatic energy and local magnetic fields in various planar ring-like nanoparticle structures. Without an external magnetic field, when only the dipole – dipole and magnetic anisotropy energies compete, the flux-closed vortex and ‘‘hedgehog’’-like configurations appear. The former is stable if the anisotropy energy is small or the particles magnetization easy directions are tangent to the circumcircle. The ‘‘hedgehog’’-like spin configuration is stable if the magnetic anisotropy is sufficiently large and the particles magnetization easy directions are perpendicular to the circumcircle tangents. Without single-particle anisotropy we found three basic spin configurations. These depend on external magnetic field
strength and intensity of dipole– dipole interactions. If the external field is high enough, a ‘‘single-domain’’ uniform structure is realized. In this case, magnetic moments are parallel to external magnetic field. With external magnetic field decrease a two-domain ‘‘onion’’-like structure gradually appears without sharp changes of magnetic properties. There is a critical value for the magnetic field (H cr) at which the sharp transition from ‘‘onion’’-like to vortex configuration occurs. Our study demonstrated that ring-like nanoparticle structures have rather intricate magnetic properties. The results obtained in the present work can be useful for prediction of different properties of planar magnetic nano-systems which will be developed in future for various magneto-electronic applications. Acknowledgements The work was partially supported from the Royal Swedish Academy of Sciences (KVA) within the project ‘‘Fabrication and Properties of Regular Arrays of Magnetic Nanoparticles’’. Yu. A. K. appreciates also the partial support from Russian Foundation for Basic Researches (Grants 05-03-32756; 05-0332083, 05-03-08215, 04-03-32090, 06-04-620). References [1] R. Skomski, J. Phys., Condens. Matter 15 (2003) R841. [2] S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daughton, S. von Molna´r, M.L. Roukes, A.Y. Chtchelkanova, D.M. Treger, Science 294 (2001) 1488. [3] P. Tartaj, M. del Puerto Morales, S. Veintemillas-Verdaguer, T. Gonza´lezCarren˜o, C.J. Serna, J. Phys., D Appl. Phys. 36 (2003) R182. [4] R.P. Cowburn, J. Phys., D Appl. Phys. 33 (2000) R1. [5] J.G. Zhu, Y.F. Zheng, G.A. Prinz, J. Appl. Phys. 87 (2000) 6668. [6] W. Jung, F.J. Castan˜o, C.A. Ross, R. Menon, A. Patel, E.E. Moon, H.I. Smith, J. Vac. Sci. Technol., B 22 (2004) 3335. [7] R.P. Cowburn, D.K. Koltsov, A.O. Adeyeye, M.E. Welland, D.M. Tricker, Phys. Rev. Lett. 83 (1999) 1042. [8] R.P. Boardman, H. Fangohr, S.J. Cox, A.V. Goncharov, A.A. Zhukov, P.A.J. de Groot, J. Appl. Phys. 95 (2004) 7037. [9] J. Aizpurua, P. Hanarp, D.S. Sutherland, M. Ka¨ll, G.W. Bryant, F.J. Garcı´a de Abajo, Phys. Rev. Lett. 90 (2003) 057401. [10] S.L. Tripp, S.V. Pusztay, A.E. Ribble, A. Wei, J. Am. Chem. Soc. 124 (2002) 7914. [11] P. Jund, S.G. Kim, D. Toma´nek, J. Hetherington, Phys. Rev. Lett. 74 (1995) 3049. [12] W. Wen, F. Kun, K.F. Pa´l, D.W. Zheng, K.N. Tu, Phys. Rev., E 59 (1999) R4758. [13] A. Ghazali, J.-C. Le´vy, Phys. Rev., B 67 (2003) 064409. [14] J.Y. Cheng, W. Jung, C.A. Ross, Phys. Rev., B 70 (2004) 064417. [15] L.M. Demers, S.J. Park, T.A. Taton, Z. Li, C.A. Mirkin, Angew. Chem., Int. Ed. Engl. 40 (2001) 3071. [16] D. Gerion, W.J. Parak, S.C. Williams, D. Zanchet, C.M. Micheel, A.P. Alivisatos, J. Am. Chem. Soc. 124 (2002) 7070. [17] V.A. Il’ina, P.K. Silaev, ‘‘Numerical Calculations for Theorist Physicists’’, Moscow-Izhevsk, ‘‘Institute of Computer Research’’, 2003 (in Russian). [18] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, Cambridge University Press, 1995. [19] C.M. Sorensen, in: K.J. Klabunde (Ed.), ‘‘Magnetism’’, in Nanoscale Materials in Chemistry, John Willey and Sons, Inc., 2003, p. 169. [20] T. Okuno, K. Mibu, T. Shinjo, J. Appl. Phys. 95 (2004) 3612. [21] J.L. Dormann, D. Fiorani, E. Tronc, Adv. Chem. Phys. 98 (1997) 283.
Magnetostatic interactions in planar ring-like nanoparticle structures Yu. A. Koksharov a,*, G.B. Khomutov a, E.S. Soldatov a, D. Suyatin b, I. Maximov b, L. Montelius b, P. Carlberg b b
a M. V. Lomonosov Moscow State University, 119992, Moscow, Russia Lund University, Solid State Physics/Nanometer Consortium, Box 118, SE-221 00, Lund, Sweden
Available online 20 January 2006
Abstract Numerical calculations of equilibrium state energies and local magnetic fields in planar ring-like nanoparticle structures were performed. The dipole – dipole, Zeeman and magnetic anisotropy interactions were included into the model. The result of their competition depends on the value of the external magnetic field, magnetic parameters of an individual nanoparticle, size and shape of the structures. Flux-closed vortexes, single domain, two-domain ‘‘onion’’-like, ‘‘hedgehog’’-like and more complex spin structures can be realized. The critical field, providing a sharp transition from the flux-closed vortex to the ‘‘onion’’-like state, can be regulated by a variation of the particle magnetization and anisotropy constant, their easy directions, and particle space arrangement. D 2005 Elsevier B.V. All rights reserved. PACS: 78.75.+a; 81.07. b Keywords: Magnetism; Nanoparticles; Two-dimensional; Ring-like structures
1. Introduction Nanoscale magnets attract much attention due to their promising technological applications in magnetic recording [1], spintronics [2], medicine [3], etc. For example, arrays of lithographically defined nano-sized magnetic elements can be used in non-volatile magnetic random access memory (MRAM) [2,4]. Recently, ring-shaped magnetic nano-structures have been proposed as basic elements of MRAM [5,6]. Existence of several different stable magnetic states is one of the advantages of the ring shaped magnets. These states could allow for more than one bit storage in each element [4,7,8]. The ring-like nanostructures can be successfully fabricated by numerous lithography methods [4,9] and self-assembled processes [10]. In ferrofluids magnetic particles form various structures, such as separate and branched chains, circles and other ring-like structures, clusters and fractal-like complexes [11 – 13]. It was shown [14] that in magnetically dilute ferrofluids dipole – dipole interactions between magnetic parti* Corresponding author. M. V. Lomonosov Moscow State University, Faculty of Physics, General Physics Department, 119992 Moscow, Russia. Tel.: +7 95 939 29 73; fax: +7 95 939 14 89. E-mail address: [email protected] (Yu.A. Koksharov). 0040-6090/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2005.12.177
cles promote a formation of flexible ring-like nanostructures. In weak magnetic fields, these ring-like structures are stable and orient perpendicular to the field. In strong magnetic fields the rings are destroyed, and the particles form linear chains [14]. The use of biological molecules (like DNA) and their molecular-recognition properties to guide the assembly of nanoparticle structures on a solid surface looks attractive [15,16]. In this case particles should form essentially rigid structure, so that the distances between them are invariant. As it is shown in our work, magnetic properties of such rigid ringlike structures depend strongly on the dipole – dipole and Zeeman magnetostatic interactions, and can be sufficiently modified by the single-particle magnetic anisotropy. The magnetic properties of ring-like planar nanoparticles were studied theoretically and experimentally in a number of works [1,2,4 – 10,20]. In nano-sized magnetic elements (continuous circular or elliptical metallic dots) magnetic vortex structures appear from a competition between exchange and magnetostatic energies, while anisotropy energy is usually considered to be negligible. In the present work different ringlike nanoparticle structures are studied. Dipole –dipole (magnetostatic), Zeeman, and anisotropy interactions are the most important in these structures, whereas the exchange energy does not play any significant role. Similar to ring-like
732
Yu.A. Koksharov et al. / Thin Solid Films 515 (2006) 731 – 734
continuous nanostructures, nanoparticle structures, studied in our work, show various spin configurations — flux-closed vortexes, ‘‘onion’’-like and more complex ones. 2. Model We consider a system of N spherical equivalent magnetic particles (with magnetic moments m i and a diameter d) located in vertexes of a regular polygon with a distance R between the polygon center and each particle (Fig. 1). The total energy W of classical interactions is the sum of Zeeman W Z, dipole –dipole W DD, and single-particle magnetic anisotropy Wanis terms (see, for example, [21], Eqs. B2, E3, E4, E9). The system energy can be regarded as a function of N parameters, e.g. angles a i between the vectors m i and the Xaxis. Using a so called ‘‘dynamical’’ version [17] of a conjugate graduate method [18] we performed the total energy minimization procedures. In order to escape the local minima we varied initial spin configurations (the examples are mV1 mW1, mV2 mW2, mV3 mW3 in Fig. 1). Besides the ideal circle structures we investigated also oblate and oblong ellipses, and circle-like structures without some particles. In that way a semi-circle and arcs of different length were considered with the same R and d values. Using a classical formula ([21], Eq. E4) we computed also dipole– dipole local magnetic fields acting on the nanoparticles and the local field in the system center (the point O in Fig. 1). In the calculations we used magnetic parameters for magnetite (Fe3O4): saturation magnetization M s = 480 emu/ cm3, and the volume anisotropy constant |K anis| å 105 erg/cm3 [19]. Magnetite is often used in biomedical nanoparticle
Fig. 1. Schematic illustration of arrangement of magnetic nanoparticles in vertexes of a regular planar polygon, where R is the distance between the polygon center (O) and its vertex, d is the particle diameter. Easy directions (solid line segments A1V – AW1, AV2 – AW2 as an example) are determined by the angle h, dotted lines denote the tangents to the circumcircle in the vertexes. Thick vertical arrow shows the direction of the external magnetic field H ext. Pairs of arrows (mV1; mW1), (m 2V ; mW2), (mV3; mW3) show directions of neighboring magnetic moments in basic configurations, which were used in the energy minimization procedures.
applications due to its robustness, low toxicity, and relatively high magnetization. Other parameters for the model were chosen as follows: d = 2 nm, R = 10 nm. In this case the maximum number of particles should be equal to ¨2pR / d å 30. We did not take into account any magnetic exchange interactions between neighboring particles since for real nanoparticles, which are in direct contact, the interparticle short-range forces are very complicated and can not be reduced to a simple isotropic bi-spin exchange. 3. Results of calculations and discussion Spin configurations in circular nanoparticle structures can be rather different. Some of the configurations are shown in Fig. 2. Let us consider the case K anis = 0. If H ext = 0, the fluxclosed vortex structure has a minimal energy (Fig. 2a). In this case we get W Z å 0 and W å W DD. With H ext increased the spins smoothly turn to the external field direction, but while H ext < H cr they remain approximately directed along tangents to the circumcircle and the flux-closed structure is not disturbed significantly. At H ext = H cr the vortex structure drastically transforms to ‘‘onion’’-like structure, which is characterized by an existence of two ‘‘domains’’ separated by a ‘‘domain’’ wall (Fig. 2b). The type of a preferable spin structure is determined by the energy gain, which the system obtains after a transition from one spin configuration to another. If H ext increases more, the ‘‘onion’’-like structure gradually transforms into a single-domain structure, in which all spins are parallel to the external field. An intermediate situation between the uniform single-domain state and the two-domain ‘‘onion’’-like state is presented in Fig. 2c. More complex spin behavior takes place in case of non-zero magnetic anisotropy. If the value of the angle h (Fig. 1) is close to zero, the anisotropy energy stabilizes the flux-closed vortex state (Fig. 3d). However, with the angle h increased, the critical field H cr changes significantly (Fig. 3d). A qualitative transformation of the spin structure can be observed for h å p / 2. Firstly, the ‘‘wall’’ between ‘‘onion’’ domains can become more extended (Fig. 2d,e). Secondly, if N is relatively small and the maximum value of Wanis,max (which is realized for h å p / 2) is comparable with the maximum value of W DD,max (which is realized for tangent to the circumcircle spins), the ground state structure is not flux-closed vortex, but ‘‘hedgehog’’-like (Fig. 2f) even in case of zero external magnetic field. With W DD,max / Wanis,max ratio increase the zero-field ground state structure will become a flux-closed vortex, irrespective of h value. This happens, e.g. for N = 20 and N = 30, H ext = 0 and K anis 105 erg/cm3. The transition from the flux-closed vortex to the ‘‘onion’’like state is accompanied by drastic changes in various parameters. The Zeeman and dipole– dipole energies undergo sharp transformations at H cr, while the total energy has a kink at this point. Average magnetic field, acting on a nanoparticle due to magnetic dipole moments of other particles and its dispersion are changed abruptly at H cr. The same behavior is typical for the local dipole –dipole magnetic field in the circle center. It is important that the local field dispersion is very
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733
Fig. 2. Examples of different spin configurations in the circle nanoparticle structure: (a) flux-closed vortex state; (b) two-domain ‘‘onion’’-like state; (c) intermediate state between homogeneous ‘‘single-domain’’ state and ‘‘onion’’-like state; (d) and (e) ‘‘onion’’-like states with extensive ‘‘domain boundary’’; (f) ‘‘hedgehog’’-like state. Model parameters N, H ext, K anis and h are shown in the figure.
Fig. 3. Variation of the critical magnetic field H cr depending on the interparticle distances (a) – (c), and the easy direction (d). In cases (a) – (c) H cr is plotted as a function of the average value of 1 / r ij3, where r ij is the interparticle distance. The value of r ij was changed: (a) by the variation of the particle number (N = 10, 20, 30) in the ideal circle structure; (b) by making a symmetrical ‘‘gap’’ in the circle structure, removing some particles, but keeping the distance between neighbor particles constant, the figures in (b) show the number of residuary particles; (c) by distorting of the structure shape from circular to elliptical one, the corresponding ratios of horizontal and vertical semi-axes lengths are indicated in (c), the minimal axis length is equal to R. In the case (d) ideal circle structures with N = 10, 20 and 30 are considered, K anis = 105 erg/cm3, the meaning of h is clear from Fig. 1.
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small in the flux-closed vortex state, that is, all particles are in nearly the same conditions. This is the feature of the fluxclosed state. The critical field H cr depends on the averaged interparticle distance. In the case of ideal circle structures, the value of H cr increases linearly with averaged 1 / r ij3 (Fig. 3a). However, for more complex structures, e.g. the circles with a ‘‘gate’’ (Fig. 3b), the critical field changes non-monotonously with 1 / r ij3. The ‘‘gap’’ between spins breaks the dipole– dipole interaction chain, typical of the flux-closed vortex state, and makes this state less stable in the external magnetic field (left part of the curve in Fig. 3b). But, H cr grows with < 1 / r ij3 > increase again when the ‘‘gate’’ in the circle becomes sufficiently large (Fig. 3b). It should be noted that the spin configurations of the ‘‘defective’’ circles (Fig. 3b) can be in some ways different from those presented in Fig. 2. However, the critical field can be easily determined by observing the sharp changes in local magnetic fields. The transformation of the ideal circle into ellipses results in a significant reduction of the critical field (Fig. 3c). For the ellipse stretched along the external magnetic field axes, this reduction is clearly less pronounced than for the ellipse elongated perpendicular to H ext. The reduction of H cr can be explained by a significant increase in the average interparticle distance (compare X-axis scales in Fig. 3a and c). Furthermore, the flux-closed vortex configuration should be more stable for the oblong ellipse than for the oblate one, since in the former case the particles with parallel magnetic moments are situated closer and therefore effectively magnetize each other. It is interesting to compare the dipole – dipole energy with a thermal energy W T ; nBT å 5 I 10 14 erg (T = 300 K). In case of d = 2 nm, and R = 10 nm the magnetic moment m of the magnetite nanoparticle is equal to ¨200 l B and a maximal value of W DD is estimated as ¨N I m 2 / d 3 å 5 I 10 5 erg (N = 30). Since m ¨ d 6, then we get W DD ¨ d 3, assuming N to be a constant. In case of nanoparticles with d = 4 nm, the dipole – dipole and thermal energies will be nearly equal. Therefore magnetic vortex-like structures, considered in this work, will be stable at room temperature in spite of superparamagnetic fluctuations if d > 5 –6 nm. 4. Conclusions We have performed numerical calculations of magnetostatic energy and local magnetic fields in various planar ring-like nanoparticle structures. Without an external magnetic field, when only the dipole – dipole and magnetic anisotropy energies compete, the flux-closed vortex and ‘‘hedgehog’’-like configurations appear. The former is stable if the anisotropy energy is small or the particles magnetization easy directions are tangent to the circumcircle. The ‘‘hedgehog’’-like spin configuration is stable if the magnetic anisotropy is sufficiently large and the particles magnetization easy directions are perpendicular to the circumcircle tangents. Without single-particle anisotropy we found three basic spin configurations. These depend on external magnetic field
strength and intensity of dipole– dipole interactions. If the external field is high enough, a ‘‘single-domain’’ uniform structure is realized. In this case, magnetic moments are parallel to external magnetic field. With external magnetic field decrease a two-domain ‘‘onion’’-like structure gradually appears without sharp changes of magnetic properties. There is a critical value for the magnetic field (H cr) at which the sharp transition from ‘‘onion’’-like to vortex configuration occurs. Our study demonstrated that ring-like nanoparticle structures have rather intricate magnetic properties. The results obtained in the present work can be useful for prediction of different properties of planar magnetic nano-systems which will be developed in future for various magneto-electronic applications. Acknowledgements The work was partially supported from the Royal Swedish Academy of Sciences (KVA) within the project ‘‘Fabrication and Properties of Regular Arrays of Magnetic Nanoparticles’’. Yu. A. K. appreciates also the partial support from Russian Foundation for Basic Researches (Grants 05-03-32756; 05-0332083, 05-03-08215, 04-03-32090, 06-04-620). References [1] R. Skomski, J. Phys., Condens. Matter 15 (2003) R841. [2] S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daughton, S. von Molna´r, M.L. Roukes, A.Y. Chtchelkanova, D.M. Treger, Science 294 (2001) 1488. [3] P. Tartaj, M. del Puerto Morales, S. Veintemillas-Verdaguer, T. Gonza´lezCarren˜o, C.J. Serna, J. Phys., D Appl. Phys. 36 (2003) R182. [4] R.P. Cowburn, J. Phys., D Appl. Phys. 33 (2000) R1. [5] J.G. Zhu, Y.F. Zheng, G.A. Prinz, J. Appl. Phys. 87 (2000) 6668. [6] W. Jung, F.J. Castan˜o, C.A. Ross, R. Menon, A. Patel, E.E. Moon, H.I. Smith, J. Vac. Sci. Technol., B 22 (2004) 3335. [7] R.P. Cowburn, D.K. Koltsov, A.O. Adeyeye, M.E. Welland, D.M. Tricker, Phys. Rev. Lett. 83 (1999) 1042. [8] R.P. Boardman, H. Fangohr, S.J. Cox, A.V. Goncharov, A.A. Zhukov, P.A.J. de Groot, J. Appl. Phys. 95 (2004) 7037. [9] J. Aizpurua, P. Hanarp, D.S. Sutherland, M. Ka¨ll, G.W. Bryant, F.J. Garcı´a de Abajo, Phys. Rev. Lett. 90 (2003) 057401. [10] S.L. Tripp, S.V. Pusztay, A.E. Ribble, A. Wei, J. Am. Chem. Soc. 124 (2002) 7914. [11] P. Jund, S.G. Kim, D. Toma´nek, J. Hetherington, Phys. Rev. Lett. 74 (1995) 3049. [12] W. Wen, F. Kun, K.F. Pa´l, D.W. Zheng, K.N. Tu, Phys. Rev., E 59 (1999) R4758. [13] A. Ghazali, J.-C. Le´vy, Phys. Rev., B 67 (2003) 064409. [14] J.Y. Cheng, W. Jung, C.A. Ross, Phys. Rev., B 70 (2004) 064417. [15] L.M. Demers, S.J. Park, T.A. Taton, Z. Li, C.A. Mirkin, Angew. Chem., Int. Ed. Engl. 40 (2001) 3071. [16] D. Gerion, W.J. Parak, S.C. Williams, D. Zanchet, C.M. Micheel, A.P. Alivisatos, J. Am. Chem. Soc. 124 (2002) 7070. [17] V.A. Il’ina, P.K. Silaev, ‘‘Numerical Calculations for Theorist Physicists’’, Moscow-Izhevsk, ‘‘Institute of Computer Research’’, 2003 (in Russian). [18] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, Cambridge University Press, 1995. [19] C.M. Sorensen, in: K.J. Klabunde (Ed.), ‘‘Magnetism’’, in Nanoscale Materials in Chemistry, John Willey and Sons, Inc., 2003, p. 169. [20] T. Okuno, K. Mibu, T. Shinjo, J. Appl. Phys. 95 (2004) 3612. [21] J.L. Dormann, D. Fiorani, E. Tronc, Adv. Chem. Phys. 98 (1997) 283.