- Email: [email protected]
Single-domain particle hysteresis for a Random Anisotropy Ising System with exchange and magnetostatic interactions
ARTICLE IN PRESS Journal of Magnetism and Magnetic Materials 322 (2010) 1368–1372
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Single-domain particle hysteresis for a Random Anisotropy Ising System with exchange and magnetostatic interactions Cristian Enachescu , Alexandra Dobrinescu, Alexandru Stancu Department of Physics, Alexandru Ioan Cuza University, Blvd Carol I, nr. 11, 700506 Iasi, Romania
a r t i c l e in f o
a b s t r a c t
Available online 3 August 2009
In this paper we present an analysis on the hysteretic behaviour of magnetic nanoparticles described by a Random Anisotropy Ising Model. We have carried out an extensive study on the size, kinetics and inter-atomic interactions strength influences on the width and shape of the hysteresis loops of spherical particles composed by atoms situated in a cubic crystalline structure and interacting by the way of magnetostatic and exchange interactions. & 2009 Elsevier B.V. All rights reserved.
Keywords: Size effect Ising model Hysteresis Relaxation
1. Introduction
2. The model
The problem of size effects and kinetic behaviour in magnetic media is of increasing importance since the use of nanoparticulate media in recording and data storage [1,2]. Among many other methods used to describe the effect of surface magnetic moments and the rate dependent properties of these media, Ising type models are the most commonly used for studying the hysteresis [3]. They have been used previously mainly to characterize thermal effects [4], but also to investigate disorder and interactions inside the system by the way of FORC distributions of nanoparticulate media [5]. In order to study the effect of the surface anisotropy on the hysteretic properties, a micromagnetic model, based on solutions of the Landau Lifshitz equation derived from the anisotropic Dirac Heisenberg model has been presented in [1,2], but the high computing time needed limits the number of atoms in the particle. In this paper we present an analysis of the size effects influence on the hysteresis properties and relaxation curves of a system described by a Random Anisotropy Ising Model (RAIM), that we propose here as a variant of Random Anisotropy Model from Ref. [6]. While in the RAM, all the atoms have the same anisotropies with different orientations, in the RAIM the anisotropies are parallel but they have different values.
In the Random Anisotropy Ising model every atom is characterized by a spin s ¼ 71 and by an anisotropy field Hk randomly calculated according to a lognormal or Gaussian distribution. The fundamental brick in our Ising type model is a hysteron with a rectangular hysteresis loop and consequently with coercivity, while in the classical Ising model the spin has no anisotropy. The coercive field is given by the intrinsic anisotropy of the atom. The anisotropy field for every atom is oriented along the z axis; in case of a spherically shaped particle, the anisotropy of surface particles was also considered normal to the surface (radial), but the differences between the two situations were minor. Here we mainly refer to spherically shaped particles, with free boundary conditions, cut out of a cube with volume (N a)3, where N is the number of atoms on a single axis and a is the distance between consecutive atoms. The size of atoms is negligible compared to a. A particular atom belongs to the spherical particle with (N1)a diameter if situated inside or exactly on that sphere. The surface of the particle consists of atoms situated precisely at the border of the system (with incomplete set of nearest neighbours). For large enough values of N (Z10), the number n of atoms contained in a (N1)a diameter particle approaches p/6 (N1)3 [1]. In our simulations, we considered spherical particles with diameter varying from 7 particles (total 123 particles, 73.1% on surface) to 31 particles (total 14,127 particles, 15.9% on surface). In Fig. 1 one can see such a particle with surface clusters of atoms of positive and negative magnetizations. According to the classical Stoner–Wohlfarth model [7], if H is the local field and k the anisotropy constant, the energy barriers
Corresponding author. Tel.: +40 232201175; fax: +40 232 201205.
E-mail address: [email protected] (C. Enachescu). 0304-8853/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2009.07.062
ARTICLE IN PRESS C. Enachescu et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 1368–1372
1369
interactions, because we have noticed that both interaction type present similar effect relative to system size and shape. We shall discuss explicitly the role of exchange interactions only when their effect presents noticeable differences compared to magnetostatic interactions. The external field is changed in small steps and the corresponding atom states are calculated according to a standard Monte-Carlo Metropolis procedure [14]. For every external field, we randomly verify the atoms state, then calculate the local field. The individual energy barriers are computed accordingly to Eq. (1). A random number is generated for each chosen atom and compared to the expression exp(Eb/T). A Monte-Carlo step is concluded when a number of atoms equal to the total number of atoms in the system have been checked. In all the simulations ¯ k. presented in this paper we considered T ¼ 0.6 Tc, with Tc ¼ k H As one expects, the hysteresis width diminishes for increasing temperature, but we shall not insist here on this aspect. Fig. 1. A spherical particle during magnetization processes. Clusters of s ¼ +1 atoms (closed circles) and s ¼ 1 atoms (open circles) can be observed.
3. Size effects influence on hysteresis loops 3.1. System shape
H 2 Eb ¼ k Hk 1 for s ¼ 1 and H4 Hk Hk
ð1Þ
and 0 in all others situations. The local field is the sum of the external applied field and the interaction field. The main effort in the most applications of Ising models was concentrated in explaining the transition from paramagnetic to ferromagnetic states taking into account only interactions between the nearest neighbours and therefore neglecting the magnetostatic interactions [8]. However, in some nanoparticle assemblies magnetostatic interactions have been evidenced by long range magnetic correlations [9]. They are non-negligible in various modern materials, such as nanocomposite magnets prepared by the mixing powder technique [10] or hybrid organic–inorganic ferromagnets with large interlayer spacing [11]; their effect has been studied in fine particle systems [12] as well as in magnetic nanoparticles [13]. In the present RAIM model, we took into account both magnetostatic and exchange interactions. As their relative contributions in above cited systems are variable, in the present paper we take into account similar order of magnitude for both magnetostatic and exchange interactions. The magnetostatic interactions are positive and decrease proportionally with the cube of the distance between atoms. The local magnetostatic interaction field Hi for the atom i is taken as the sum of magnetostatic interaction fields between all the atoms: 1 X mj ð2Þ Him ¼ 4p j rij3 where rij is the distance between the atoms i and j and mj ¼ 71 the magnetic moment of the atom j. The exchange interactions are considered only with the nearest neighbours and are proportional with their magnetic moment, giving the interaction field: X mi ð3Þ Hie ¼ J i
For the sake of simplicity, in most simulations presented below, we insist mainly on the effect of the magnetostatic
Before analyzing the various aspects of size effects for spherical shape particles, we have compared the hysteresis loops obtained for several shapes of the particle. In Fig. 2 one can see the hysteresis loops for three particles containing approximately the same number of atoms: a spherical system cut out from a 27 dimension cube (9171 atoms), a 21 21 21 cubic system (9261 atoms) and a wire-like system with surface formed by 25 atoms and with 367 atoms length (9175 atoms). We notice that for cubic and spherical systems, the hysteresis loops are quite similar, with a just visible smaller width for spherical particles, while in case of wire-like system, the hysteresis width is much smaller. As we shall see later, this fact can be correlated to a smaller average interaction inside the wire-like system, which has the highest percentage of edge molecules of the three systems. 3.2. System size The main aim of the present study is to analyze the hysteretic behaviour of the spherical particles in the frame of RAIM model as a function of their size (diameter). In Fig. 3 we present hysteresis loops for various particle diameters, with a 20 Monte-Carlo steps
1.0
0.5
m
have the expressions H 2 Eb ¼ k Hk 1 þ for s ¼ þ1 and HoHk Hk
cubic system wire sphere
0.0
-0.5
-1.0 -4
-2
0
2
4
6
Fig. 2. Hysteresis loop for three 3D systems with approximately the same number of atoms.
ARTICLE IN PRESS 1370
C. Enachescu et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 1368–1372
1.0
1.0
m
m
0.5
bulk 0.0
0.5
surface
-0.5 -1.0 -5.0
-5.0
0.0 0.0
-2.5
2.5
H -2.5
5.0
7.5 31 27 19 13 7
-0.5
0.0
15.9% 18.2% 31.4% 43.1% 73.1%
2.5
5.0
H surface surface surface surface surface
-1.0 Fig. 3. Dependence of hysteresis loops of a RAIM spherical monoparticle on its diameter. Inset: Hysteresis loop for bulk and surface atoms.
1 .0
1 .0
m 0 .5 d= 2 d= 3 d= 25
0 .0
0 .5
-0 .5
m
-1 .0 -5
-5.0
-2.5
0.0
2.5
-0 .5
5.0
0
7.5
5
H
H
20 MC steps 50 MC steps 100 MC steps 500 MC steps 1000 MC steps
-1 .0 Fig. 4. Dependence of hysteresis loops of a RAIM spherical particle on number of Monte Carlo steps. (inset) Dependence of the hysteresis loops with the packing factor for a spherical particle.
waiting time at different external fields. The simulations show that the coercivity diminishes for a decreasing size of the system. A stabilization is observed for a sphere diameter of 31 atoms, which corresponds to a percentage of 15.9% atoms on surface, close to real particles [1]. Not only does the coercivity vary with the diameter, but also the squareness of the hysteresis loop: the larger the diameter of the particle, the steeper the hysteresis branches. This can be correlated with the clusters created during transition since they develop easily in particles with larger volume, while in small particles, this evolution will be limited by the size. In other words, in larger particles, the atoms will have a higher probability to influence one another than in case of small particles, where the edge atoms will act as a barrier. This is clearly illustrated by the inset in Fig. 3: the hysteresis loop for surface atoms has a smaller width and is less steep than that of the bulk atoms.
3.3. System kinetics In several magnetic systems, and especially in those composed from nanoparticles, an important influence of the field rate on measurements can be noticed, i.e. for reptation, magnetic after effect, magnetic viscosity or accommodation effects [15]. Like other Ising models, the RAIM model is strongly influenced by the sweeping rate of the output parameter and the simulated hysteresis is highly rate dependent [16,17]. In Fig. 4, we present hysteresis loops calculated for different field sweeping rates, i.e. different waiting times (number of Monte-Carlo steps) at every field step. It can be noticed that the coercivity diminishes for longer waiting times. A short waiting time will not allow the system to reach equilibrium position; consequently, the calculated magnetic moment of the system will correspond to a point along the relaxation curve, which is
ARTICLE IN PRESS C. Enachescu et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 1368–1372
2
0 1
M
s ep st rl o e ) Ca al te s c on (log
10
100 10
5
15
30
25
20
re Sphe
35
us
Radi
Coercitive fiels
4
gradually approaching the equilibrium position as the number of Monte-Carlo steps per spin increases. The hysteresis loop reaches the equilibrium only for higher Monte-Carlo waiting time, i.e. more than five hundreds Monte-Carlo steps at every temperature. Opposite to the situation in Fig. 3, the hysteresis widths is steeper as its width decreases; this feature is determined by the fact that longer waiting times allow the development of clusters, so more atoms will change their states simultaneously. As the dependencies on size and kinetics are complex, we represent in Fig. 5 the simultaneous dependence of the coercive field on number of Monte-Carlo steps and on the particle diameter for both interaction types. We notice that for every Monte-Carlo waiting time, the dependency of the coercive field in function of the particle diameter is the same, for both dipolar (up) and exchange interactions (down). It shows a fast increase for smaller number of atoms and then a long, almost asymptotic, approach to the value corresponding to the bulk. Actually, while increasing the particle diameter the bulk becomes more important and the contribution of the surface atoms diminishes, as suggested by the inset in Fig. 3. Here we can detect a qualitative difference between the role played by magnetostatic and exchange interactions for the hysteresis widths dependence on the sphere radius. In case of exchange interaction the hysteresis width changes with the sphere radius only for small radius, then it reaches saturation, while for magnetostatic interactions the change is continuous, even for higher values of the sphere radius. 3.4. Packing density dependencies
2
0 1
s ep st rlo e) Ca al te sc on (log
M
10
100
10
5
15
25
20
30
35
dius
e Ra pher
S
Fig. 5. Simultaneous dependence of the coercivity on the sphere diameter and on the number of Monte Carlo steps for magnetostatic (up) and exchange interactions (down).
d=1.5 d=1.75 d=2 d=2.5 d=25
5
correlation factor
Another important dependence is that of the hysteresis width in function of the strength of interactions. The interactions amplitude can be modelled by the packing density (the distance between two consecutive atoms) in the magnetostatic system and by the value of exchange coupling constant in case of exchange interactions. As the magnetostatic interactions are decreasing with the 3rd power of the distance between atoms, their effect diminishes fast for decreasing the packing density. In the inset of Fig. 4 we present hysteresis loops for different packing densities. The hysteresis disappears when the distance between the atoms is large enough that their influence on the neighbours is negligible and becomes wider and steeper when the packing density is higher. The results are similar with those obtained in the frame of a Monte-Carlo model with exchange and dipolar interactions for magnetic nanoparticle systems, including complete thermal fluctuations [18].
4
5 4 J=1 J=0.5 J=0.25 J=0.1 J=0
3 2
3 2 1
1 0 -1.0
correlation factor
Coercitive fiels
4
1371
-0.5
0.0
0.5
magnetic moment
-0.5
0.0
0.5
0 1.0
magnetic moment
Fig. 6. The correlation factor evolution with the average magnetic moment of the particle for different packing densities.
ARTICLE IN PRESS 1372
C. Enachescu et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 1368–1372
Besides the net magnetization m, the macrostates of the systems can be characterized by the nearest neighbour correlation P factor, with the expression s ¼ i,jmimj/n. In this way, a plot in the (m, s) space adequately describes the interaction distribution inside a particulate system. We use the correlation factor as defined above in order to study the domain evolution, i.e. clusters of positive and negative magnetized atoms, their dependence on interaction strength and their impact on the hysteresis loop of the systems. In order to achieve this, we calculate the correlation factor of the particles in different situations. In perfect correlated states (such as negative or positive saturation), all the moments are aligned and the correlation factor is equal to the average number of neighbours for every atom (that is six for a bulk atom, and four or three for an atom situated on the sphere surface). For perfect non-correlated states, such as a non-interacting system at zero external field, the correlation factor is zero. If the correlation factor is not zero, then domains appear, larger for higher correlation factor. In Fig. 6 we present the evolution of correlation factor with respect to magnetization m for different interaction strengths; all the curves have the same departure point (somewhat lower than six for reasons described above), but their minimum values depend strongly on the interactions, higher for higher interactions and descending to zero for non-interacting atoms. The results are similar for both magnetostatic and exchange interactions.
4. Conclusion In conclusion, we notice that the RAIM model proved to be tractable to describe the magnetization processes in nanoparticle systems. The results are compatible with those obtained in the
frame of the micromagnetic model [1,2], but the computation time is considerably reduced. Further research will concentrate on the study of small size particles, interacting by magnetostatic interactions. Acknowledgement Work was supported by the Romanian CNCSIS Complex Ideas program, grant 239/2008. References [1] D. Garanin, H. Kachkachi, Phys. Rev. Lett. 90 (065504) (2003). [2] M. Dimian, H. Kachkachi, J. Appl. Phys. 91 (10) (2002) 7625–7627. [3] R.A. dos Anjos, J.R. Viana, J.R. de Sousa, J.A. Plascak, Phys. Rev. E 76 (2007) 022103. [4] D.P. Landau, Phys. Rev. B 13 (1976) 2997–3011. [5] C. Enachescu, A. Stancu, IEEE Trans. Magn. 42 (2006) 3156. [6] C. Jayaprakash, S. Kirkpatrick, Phys. Rev. B 21 (9) (1980) 4072. [7] E.C. Stoner, E.P. Wohlfarth, Philos. Trans. R. Soc. London, Ser. A 599 (1948). [8] J.P. Sethna, K. Dahmen, S. Kartha, J.A. Krumhansl, B.W. Roberts, J.D. Shore, Phys. Rev. Lett. 70 (1993) 3347–3350. [9] D.F. Farrell, Y. Ijiri, C.V. Kelly, J.A. Borchers, J.J. Rhyne, Y. Ding, S. Majetich J. Magn. Magn. Mater. 303 (2006) 318. [10] R. Sato Turtellia, P. Kerschlb, H. Fukunagac, Y. Kawazoec, A. Shintanid, R. Grossinger, J. Magn. Magn. Mater. 272 (2004) e497. [11] M. Drillon, P. Panissod, J. Magn. Magn. Mater. 188 (1998) 93. [12] R.W. Chantrell, G.N. Coverdale, M. El Hilo, K. O’Grady, J. Magn. Magn. Mater. 157 (1996) 250. [13] M. Blanco-Mantecon, K. O’Grady, J. Magn. Magn. Mater. 296 (2006) 124. [14] K. Binder, Phase transition and critical phenomena. In: C. Domb, M.S. Green (ed.), vol. 5B. Academic Press, London. [15] L. Neel, Ann. Geophys. 5 (1949) 99. [16] G. Bertotti, Academic Press, San Diego, 1998. [17] C. Enachescu, R. Tanasa, A. Stancu, F. Varret, J. Linare s, E. Codjovi, Phys. Rev. B 72 (2005) (054413). [18] H.F. Du, A. Du, Phys. Status Solidi 244 (2007) 1401.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Single-domain particle hysteresis for a Random Anisotropy Ising System with exchange and magnetostatic interactions Cristian Enachescu , Alexandra Dobrinescu, Alexandru Stancu Department of Physics, Alexandru Ioan Cuza University, Blvd Carol I, nr. 11, 700506 Iasi, Romania
a r t i c l e in f o
a b s t r a c t
Available online 3 August 2009
In this paper we present an analysis on the hysteretic behaviour of magnetic nanoparticles described by a Random Anisotropy Ising Model. We have carried out an extensive study on the size, kinetics and inter-atomic interactions strength influences on the width and shape of the hysteresis loops of spherical particles composed by atoms situated in a cubic crystalline structure and interacting by the way of magnetostatic and exchange interactions. & 2009 Elsevier B.V. All rights reserved.
Keywords: Size effect Ising model Hysteresis Relaxation
1. Introduction
2. The model
The problem of size effects and kinetic behaviour in magnetic media is of increasing importance since the use of nanoparticulate media in recording and data storage [1,2]. Among many other methods used to describe the effect of surface magnetic moments and the rate dependent properties of these media, Ising type models are the most commonly used for studying the hysteresis [3]. They have been used previously mainly to characterize thermal effects [4], but also to investigate disorder and interactions inside the system by the way of FORC distributions of nanoparticulate media [5]. In order to study the effect of the surface anisotropy on the hysteretic properties, a micromagnetic model, based on solutions of the Landau Lifshitz equation derived from the anisotropic Dirac Heisenberg model has been presented in [1,2], but the high computing time needed limits the number of atoms in the particle. In this paper we present an analysis of the size effects influence on the hysteresis properties and relaxation curves of a system described by a Random Anisotropy Ising Model (RAIM), that we propose here as a variant of Random Anisotropy Model from Ref. [6]. While in the RAM, all the atoms have the same anisotropies with different orientations, in the RAIM the anisotropies are parallel but they have different values.
In the Random Anisotropy Ising model every atom is characterized by a spin s ¼ 71 and by an anisotropy field Hk randomly calculated according to a lognormal or Gaussian distribution. The fundamental brick in our Ising type model is a hysteron with a rectangular hysteresis loop and consequently with coercivity, while in the classical Ising model the spin has no anisotropy. The coercive field is given by the intrinsic anisotropy of the atom. The anisotropy field for every atom is oriented along the z axis; in case of a spherically shaped particle, the anisotropy of surface particles was also considered normal to the surface (radial), but the differences between the two situations were minor. Here we mainly refer to spherically shaped particles, with free boundary conditions, cut out of a cube with volume (N a)3, where N is the number of atoms on a single axis and a is the distance between consecutive atoms. The size of atoms is negligible compared to a. A particular atom belongs to the spherical particle with (N1)a diameter if situated inside or exactly on that sphere. The surface of the particle consists of atoms situated precisely at the border of the system (with incomplete set of nearest neighbours). For large enough values of N (Z10), the number n of atoms contained in a (N1)a diameter particle approaches p/6 (N1)3 [1]. In our simulations, we considered spherical particles with diameter varying from 7 particles (total 123 particles, 73.1% on surface) to 31 particles (total 14,127 particles, 15.9% on surface). In Fig. 1 one can see such a particle with surface clusters of atoms of positive and negative magnetizations. According to the classical Stoner–Wohlfarth model [7], if H is the local field and k the anisotropy constant, the energy barriers
Corresponding author. Tel.: +40 232201175; fax: +40 232 201205.
E-mail address: [email protected] (C. Enachescu). 0304-8853/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2009.07.062
ARTICLE IN PRESS C. Enachescu et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 1368–1372
1369
interactions, because we have noticed that both interaction type present similar effect relative to system size and shape. We shall discuss explicitly the role of exchange interactions only when their effect presents noticeable differences compared to magnetostatic interactions. The external field is changed in small steps and the corresponding atom states are calculated according to a standard Monte-Carlo Metropolis procedure [14]. For every external field, we randomly verify the atoms state, then calculate the local field. The individual energy barriers are computed accordingly to Eq. (1). A random number is generated for each chosen atom and compared to the expression exp(Eb/T). A Monte-Carlo step is concluded when a number of atoms equal to the total number of atoms in the system have been checked. In all the simulations ¯ k. presented in this paper we considered T ¼ 0.6 Tc, with Tc ¼ k H As one expects, the hysteresis width diminishes for increasing temperature, but we shall not insist here on this aspect. Fig. 1. A spherical particle during magnetization processes. Clusters of s ¼ +1 atoms (closed circles) and s ¼ 1 atoms (open circles) can be observed.
3. Size effects influence on hysteresis loops 3.1. System shape
H 2 Eb ¼ k Hk 1 for s ¼ 1 and H4 Hk Hk
ð1Þ
and 0 in all others situations. The local field is the sum of the external applied field and the interaction field. The main effort in the most applications of Ising models was concentrated in explaining the transition from paramagnetic to ferromagnetic states taking into account only interactions between the nearest neighbours and therefore neglecting the magnetostatic interactions [8]. However, in some nanoparticle assemblies magnetostatic interactions have been evidenced by long range magnetic correlations [9]. They are non-negligible in various modern materials, such as nanocomposite magnets prepared by the mixing powder technique [10] or hybrid organic–inorganic ferromagnets with large interlayer spacing [11]; their effect has been studied in fine particle systems [12] as well as in magnetic nanoparticles [13]. In the present RAIM model, we took into account both magnetostatic and exchange interactions. As their relative contributions in above cited systems are variable, in the present paper we take into account similar order of magnitude for both magnetostatic and exchange interactions. The magnetostatic interactions are positive and decrease proportionally with the cube of the distance between atoms. The local magnetostatic interaction field Hi for the atom i is taken as the sum of magnetostatic interaction fields between all the atoms: 1 X mj ð2Þ Him ¼ 4p j rij3 where rij is the distance between the atoms i and j and mj ¼ 71 the magnetic moment of the atom j. The exchange interactions are considered only with the nearest neighbours and are proportional with their magnetic moment, giving the interaction field: X mi ð3Þ Hie ¼ J i
For the sake of simplicity, in most simulations presented below, we insist mainly on the effect of the magnetostatic
Before analyzing the various aspects of size effects for spherical shape particles, we have compared the hysteresis loops obtained for several shapes of the particle. In Fig. 2 one can see the hysteresis loops for three particles containing approximately the same number of atoms: a spherical system cut out from a 27 dimension cube (9171 atoms), a 21 21 21 cubic system (9261 atoms) and a wire-like system with surface formed by 25 atoms and with 367 atoms length (9175 atoms). We notice that for cubic and spherical systems, the hysteresis loops are quite similar, with a just visible smaller width for spherical particles, while in case of wire-like system, the hysteresis width is much smaller. As we shall see later, this fact can be correlated to a smaller average interaction inside the wire-like system, which has the highest percentage of edge molecules of the three systems. 3.2. System size The main aim of the present study is to analyze the hysteretic behaviour of the spherical particles in the frame of RAIM model as a function of their size (diameter). In Fig. 3 we present hysteresis loops for various particle diameters, with a 20 Monte-Carlo steps
1.0
0.5
m
have the expressions H 2 Eb ¼ k Hk 1 þ for s ¼ þ1 and HoHk Hk
cubic system wire sphere
0.0
-0.5
-1.0 -4
-2
0
2
4
6
Fig. 2. Hysteresis loop for three 3D systems with approximately the same number of atoms.
ARTICLE IN PRESS 1370
C. Enachescu et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 1368–1372
1.0
1.0
m
m
0.5
bulk 0.0
0.5
surface
-0.5 -1.0 -5.0
-5.0
0.0 0.0
-2.5
2.5
H -2.5
5.0
7.5 31 27 19 13 7
-0.5
0.0
15.9% 18.2% 31.4% 43.1% 73.1%
2.5
5.0
H surface surface surface surface surface
-1.0 Fig. 3. Dependence of hysteresis loops of a RAIM spherical monoparticle on its diameter. Inset: Hysteresis loop for bulk and surface atoms.
1 .0
1 .0
m 0 .5 d= 2 d= 3 d= 25
0 .0
0 .5
-0 .5
m
-1 .0 -5
-5.0
-2.5
0.0
2.5
-0 .5
5.0
0
7.5
5
H
H
20 MC steps 50 MC steps 100 MC steps 500 MC steps 1000 MC steps
-1 .0 Fig. 4. Dependence of hysteresis loops of a RAIM spherical particle on number of Monte Carlo steps. (inset) Dependence of the hysteresis loops with the packing factor for a spherical particle.
waiting time at different external fields. The simulations show that the coercivity diminishes for a decreasing size of the system. A stabilization is observed for a sphere diameter of 31 atoms, which corresponds to a percentage of 15.9% atoms on surface, close to real particles [1]. Not only does the coercivity vary with the diameter, but also the squareness of the hysteresis loop: the larger the diameter of the particle, the steeper the hysteresis branches. This can be correlated with the clusters created during transition since they develop easily in particles with larger volume, while in small particles, this evolution will be limited by the size. In other words, in larger particles, the atoms will have a higher probability to influence one another than in case of small particles, where the edge atoms will act as a barrier. This is clearly illustrated by the inset in Fig. 3: the hysteresis loop for surface atoms has a smaller width and is less steep than that of the bulk atoms.
3.3. System kinetics In several magnetic systems, and especially in those composed from nanoparticles, an important influence of the field rate on measurements can be noticed, i.e. for reptation, magnetic after effect, magnetic viscosity or accommodation effects [15]. Like other Ising models, the RAIM model is strongly influenced by the sweeping rate of the output parameter and the simulated hysteresis is highly rate dependent [16,17]. In Fig. 4, we present hysteresis loops calculated for different field sweeping rates, i.e. different waiting times (number of Monte-Carlo steps) at every field step. It can be noticed that the coercivity diminishes for longer waiting times. A short waiting time will not allow the system to reach equilibrium position; consequently, the calculated magnetic moment of the system will correspond to a point along the relaxation curve, which is
ARTICLE IN PRESS C. Enachescu et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 1368–1372
2
0 1
M
s ep st rl o e ) Ca al te s c on (log
10
100 10
5
15
30
25
20
re Sphe
35
us
Radi
Coercitive fiels
4
gradually approaching the equilibrium position as the number of Monte-Carlo steps per spin increases. The hysteresis loop reaches the equilibrium only for higher Monte-Carlo waiting time, i.e. more than five hundreds Monte-Carlo steps at every temperature. Opposite to the situation in Fig. 3, the hysteresis widths is steeper as its width decreases; this feature is determined by the fact that longer waiting times allow the development of clusters, so more atoms will change their states simultaneously. As the dependencies on size and kinetics are complex, we represent in Fig. 5 the simultaneous dependence of the coercive field on number of Monte-Carlo steps and on the particle diameter for both interaction types. We notice that for every Monte-Carlo waiting time, the dependency of the coercive field in function of the particle diameter is the same, for both dipolar (up) and exchange interactions (down). It shows a fast increase for smaller number of atoms and then a long, almost asymptotic, approach to the value corresponding to the bulk. Actually, while increasing the particle diameter the bulk becomes more important and the contribution of the surface atoms diminishes, as suggested by the inset in Fig. 3. Here we can detect a qualitative difference between the role played by magnetostatic and exchange interactions for the hysteresis widths dependence on the sphere radius. In case of exchange interaction the hysteresis width changes with the sphere radius only for small radius, then it reaches saturation, while for magnetostatic interactions the change is continuous, even for higher values of the sphere radius. 3.4. Packing density dependencies
2
0 1
s ep st rlo e) Ca al te sc on (log
M
10
100
10
5
15
25
20
30
35
dius
e Ra pher
S
Fig. 5. Simultaneous dependence of the coercivity on the sphere diameter and on the number of Monte Carlo steps for magnetostatic (up) and exchange interactions (down).
d=1.5 d=1.75 d=2 d=2.5 d=25
5
correlation factor
Another important dependence is that of the hysteresis width in function of the strength of interactions. The interactions amplitude can be modelled by the packing density (the distance between two consecutive atoms) in the magnetostatic system and by the value of exchange coupling constant in case of exchange interactions. As the magnetostatic interactions are decreasing with the 3rd power of the distance between atoms, their effect diminishes fast for decreasing the packing density. In the inset of Fig. 4 we present hysteresis loops for different packing densities. The hysteresis disappears when the distance between the atoms is large enough that their influence on the neighbours is negligible and becomes wider and steeper when the packing density is higher. The results are similar with those obtained in the frame of a Monte-Carlo model with exchange and dipolar interactions for magnetic nanoparticle systems, including complete thermal fluctuations [18].
4
5 4 J=1 J=0.5 J=0.25 J=0.1 J=0
3 2
3 2 1
1 0 -1.0
correlation factor
Coercitive fiels
4
1371
-0.5
0.0
0.5
magnetic moment
-0.5
0.0
0.5
0 1.0
magnetic moment
Fig. 6. The correlation factor evolution with the average magnetic moment of the particle for different packing densities.
ARTICLE IN PRESS 1372
C. Enachescu et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 1368–1372
Besides the net magnetization m, the macrostates of the systems can be characterized by the nearest neighbour correlation P factor, with the expression s ¼ i,jmimj/n. In this way, a plot in the (m, s) space adequately describes the interaction distribution inside a particulate system. We use the correlation factor as defined above in order to study the domain evolution, i.e. clusters of positive and negative magnetized atoms, their dependence on interaction strength and their impact on the hysteresis loop of the systems. In order to achieve this, we calculate the correlation factor of the particles in different situations. In perfect correlated states (such as negative or positive saturation), all the moments are aligned and the correlation factor is equal to the average number of neighbours for every atom (that is six for a bulk atom, and four or three for an atom situated on the sphere surface). For perfect non-correlated states, such as a non-interacting system at zero external field, the correlation factor is zero. If the correlation factor is not zero, then domains appear, larger for higher correlation factor. In Fig. 6 we present the evolution of correlation factor with respect to magnetization m for different interaction strengths; all the curves have the same departure point (somewhat lower than six for reasons described above), but their minimum values depend strongly on the interactions, higher for higher interactions and descending to zero for non-interacting atoms. The results are similar for both magnetostatic and exchange interactions.
4. Conclusion In conclusion, we notice that the RAIM model proved to be tractable to describe the magnetization processes in nanoparticle systems. The results are compatible with those obtained in the
frame of the micromagnetic model [1,2], but the computation time is considerably reduced. Further research will concentrate on the study of small size particles, interacting by magnetostatic interactions. Acknowledgement Work was supported by the Romanian CNCSIS Complex Ideas program, grant 239/2008. References [1] D. Garanin, H. Kachkachi, Phys. Rev. Lett. 90 (065504) (2003). [2] M. Dimian, H. Kachkachi, J. Appl. Phys. 91 (10) (2002) 7625–7627. [3] R.A. dos Anjos, J.R. Viana, J.R. de Sousa, J.A. Plascak, Phys. Rev. E 76 (2007) 022103. [4] D.P. Landau, Phys. Rev. B 13 (1976) 2997–3011. [5] C. Enachescu, A. Stancu, IEEE Trans. Magn. 42 (2006) 3156. [6] C. Jayaprakash, S. Kirkpatrick, Phys. Rev. B 21 (9) (1980) 4072. [7] E.C. Stoner, E.P. Wohlfarth, Philos. Trans. R. Soc. London, Ser. A 599 (1948). [8] J.P. Sethna, K. Dahmen, S. Kartha, J.A. Krumhansl, B.W. Roberts, J.D. Shore, Phys. Rev. Lett. 70 (1993) 3347–3350. [9] D.F. Farrell, Y. Ijiri, C.V. Kelly, J.A. Borchers, J.J. Rhyne, Y. Ding, S. Majetich J. Magn. Magn. Mater. 303 (2006) 318. [10] R. Sato Turtellia, P. Kerschlb, H. Fukunagac, Y. Kawazoec, A. Shintanid, R. Grossinger, J. Magn. Magn. Mater. 272 (2004) e497. [11] M. Drillon, P. Panissod, J. Magn. Magn. Mater. 188 (1998) 93. [12] R.W. Chantrell, G.N. Coverdale, M. El Hilo, K. O’Grady, J. Magn. Magn. Mater. 157 (1996) 250. [13] M. Blanco-Mantecon, K. O’Grady, J. Magn. Magn. Mater. 296 (2006) 124. [14] K. Binder, Phase transition and critical phenomena. In: C. Domb, M.S. Green (ed.), vol. 5B. Academic Press, London. [15] L. Neel, Ann. Geophys. 5 (1949) 99. [16] G. Bertotti, Academic Press, San Diego, 1998. [17] C. Enachescu, R. Tanasa, A. Stancu, F. Varret, J. Linare s, E. Codjovi, Phys. Rev. B 72 (2005) (054413). [18] H.F. Du, A. Du, Phys. Status Solidi 244 (2007) 1401.