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Magnetic characteristics of dipole clusters
ARTICLE IN PRESS
Physica B 372 (2006) 239–242 www.elsevier.com/locate/physb
Magnetic characteristics of dipole clusters Ja´nos Fu¨zia,b, a
Neutron Spectroscopy Department, Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary b Electrical Engineering Department, Transilvania University, Brasov, Romania
Abstract The dipole–dipole interaction between magnetic moments of monodomain particles is simulated for the zero-temperature case. Random and ordered clusters are investigated. Magnetization characteristics are plotted for spherical, toroidal and ellipsoidal-shaped sets of such particles in alternating, respectively rotational external field. Rate dependence is introduced in order to account for dynamic effects. r 2005 Elsevier B.V. All rights reserved. PACS: 75.10.Hk; 02.50.Ng; 05.50.+q Keywords: Micromagnetics; Dipolar interactions; Rate dependence
1. Introduction Zero temperature Monte-Carlo simulation is performed to investigate the behavior of monodomain magnetic particle assemblies [1,2] connected through the dipole–dipole interaction. Such situation occurs when ferromagnetic nanocrystals are embedded in a non-magnetic matrix [3]. First the magnetization characteristics are determined for spherical clusters placed in a uniform external field. In this case the demagnetizing effect is quite important. Two ways are explored to reduce this effect without having to consider an exceedingly large number of particles: hypothetical toroidal clusters are considered, magnetized in azimuthal direction by a current passing through the loop (appropriate for study of scalar hysteresis), respectively a first neighbor exchange-like interaction is added in case of spherical or ellipsoidal clusters to compensate the demagnetizing field. Clusters with magnetic moments placed in the vertices of simple cubic (SC), body-centered (BCC), face-centered cubic lattice (FCC), respectively are compared to ‘‘amorphous’’ structures obtained by ranCorresponding author. Neutron Spectroscopy Department, Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary. Tel.: +36 1 392 2222 1738; fax: +36 1 392 2501. E-mail address: [email protected].
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.10.057
domly leaving out a number of particles (leading also to various filling factors). 2. Computation method The magnetic field sensed by a particle of the cluster is N 1 X mk ðmk rkl Þrkl H l ¼ H ext 3 ; 4p k¼1 r3kl r5kl kal
l ¼ 1; N,
ð1Þ
where H l is the field strength at the location of particle l, consisting of H ext —the externally applied field and the field created by the other members of the cluster, where mk is the magnetic dipole moment of particle k and rkl the relative position vector of particles k and l. The condition ml H l ! max;
l ¼ 1; N
(2)
is used to determine the orientation of each particle. To enhance convergence, the orientations of the magnetic moments are confined to a number of 92 equally distributed fixed directions. The magnetic moments of randomly selected particles are oriented parallel to the allowed direction closest to that of the field created by all the other particles, added to a given external field. This process is repeated until no change is encountered (all
ARTICLE IN PRESS J. Fu¨zi / Physica B 372 (2006) 239–242
240
moments find their equilibrium orientation). The average magnetization is the vector sum of individual moments divided by the volume of the cluster.
Dynamic effects are modeled by a supplementary variable (H m ) as input instead of the actual applied field strength H, delayed with respect to the latter,
Table 1 Spherical cluster dimensions (nm) Structure
SC
BCC
FCC
Lattice constant Cluster radius Particle radius Number of particles Filling factor
3.5 21 1.75 739 0.428
4.4 21 1.9 701 0.519
5.6 21 2 675 0.583
Structure
SC
BCC
FCC
Lattice constant Mean radius Tube radius Particle radius Number of particles Filling factor
3.5 30 10 1.75 924 0.350
4.4 30 10 1.9 900 0.437
5.6 30 10 2 820 0.464
Table 2 Toroidal cluster dimensions (nm)
1.2
1.2 70%
-1.2
-1
0 0 H [T]
1
70%
100% 70%
40%
40% 0 M [T]
0 M [T]
0 M [T]
40% 0
1.2
100%
100%
0
-1.2
-1
0 0 H [T]
1
0
-1.2 -1
0 0 H [T]
Fig. 1. Hysteresis loops of spherical cluster in SC, BCC and FCC configuration, respectively.
Fig. 2. Hysteresis loops of toroidal cluster in SC, BCC and FCC configuration, respectively.
1
ARTICLE IN PRESS J. Fu¨zi / Physica B 372 (2006) 239–242
according to:
241
the resulting magnetization when the dipole orientations are computed with H m as input. First neighbor exchange interactions are included using
dH m dB ; ¼ aðH H m Þ b dt dt B ¼ m0 ðH þ MðH m ÞÞ,
ð3Þ
where a and b are model parameters (a is the inverse of the delay time constant, b is responsible for increased lag in case of large magnetization variation rate) and MðH m Þ is
H l ¼ H ext þ
N X
J lk mk
k¼1 kal
N 1 X mk ðmk rkl Þrkl 3 ; 4p k¼1 r3kl r5kl kal
l ¼ 1; N,
0.3
ð4Þ
where J lk is the exchange coefficient, different from zero only if dipoles l and k are in first neighbor locations. 3. Results
0 M [T]
50kHz
0
5Hz 50Hz
5kHz 500Hz
-0.3 -1
0 0 H [T]
Fig. 3. Dynamic magnetizing loops of toroidal cluster.
1
The investigated cluster configurations are given in Tables 1 and 2 (dimensions in nm). The different lattice constant values have been chosen to ensure roughly equal filling factors for sake of comparison. Fig. 1 shows the hysteresis loops obtained for spherical clusters with SC, BCC and FCC structure and ‘‘amorphous’’ configurations obtained by leaving out dipoles from randomly selected positions of the respective lattices. The dominant effect of the demagnetizing field is observable. Hysteresis loops obtained for toroidal clusters with similar structures are plotted in Fig. 2. More rectangular loops are obtained since the demagnetizing effect does not appear in azimuthal direction (the structure is infinite). Throughout the
Fig. 4. Rotational magnetization of ellipsoidal cluster, cubic structure, 70% dipoles present.
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J. Fu¨zi / Physica B 372 (2006) 239–242
Fig. 5. Rotational magnetization of ellipsoidal cluster, cubic structure, all dipoles present.
simulations the magnetic polarization of the particles has been considered to be 2 T. Dynamic magnetizing curves obtained with parameter values a ¼ 10; 000 s1 , b ¼ 10; 000 m=H in Eq. (3) are plotted in Fig. 3, for a toroidal cluster in SC structure, 40% of dipoles present in randomly chosen locations. Figs. 4 and 5 show the results of rotational magnetizing process simulations of an ellipsoidal cluster, (semiaxis lengths 12.5, 20, 28.5 nm) of 1.75 nm diameter monodomain nanocrystals (m0MS ¼ 2 T), first neighbor exchange coefficient between crystals 0.004 nm3, in a cubic array, 70% of the crystals present in randomly chosen sites (366 dipoles remaining—Fig. 4), respectively all crystals present (Fig. 5).
clusters placed in external magnetic field. While the magnetization characteristics of spherical clusters are determined mainly by the demagnetizing effect, toroidal clusters magnetized in azimuthal direction exhibit hysteresis. The shape of the hysteresis loop is square for ordered structures and more rounded for random distribution of particles in the cluster. Dynamic effects can be simulated by an input delaying technique. Shape anisotropy can be investigated by computations carried out for ellipsoidal clusters.
References 4. Conclusion Zero temperature Monte-Carlo simulation is an appropriate tool for qualitative study of the behavior of dipole
[1] Gy. Szabo´, Gy. Ka´da´r, Phys. Rev. B 58 (9) (1998) 5584. [2] J. Fu¨zi, Gy. Ka´da´r, Physica B 343 (2004) 293. [3] J. Fu¨zi, L.K. Varga, Physica B 343 (2004) 159.
Physica B 372 (2006) 239–242 www.elsevier.com/locate/physb
Magnetic characteristics of dipole clusters Ja´nos Fu¨zia,b, a
Neutron Spectroscopy Department, Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary b Electrical Engineering Department, Transilvania University, Brasov, Romania
Abstract The dipole–dipole interaction between magnetic moments of monodomain particles is simulated for the zero-temperature case. Random and ordered clusters are investigated. Magnetization characteristics are plotted for spherical, toroidal and ellipsoidal-shaped sets of such particles in alternating, respectively rotational external field. Rate dependence is introduced in order to account for dynamic effects. r 2005 Elsevier B.V. All rights reserved. PACS: 75.10.Hk; 02.50.Ng; 05.50.+q Keywords: Micromagnetics; Dipolar interactions; Rate dependence
1. Introduction Zero temperature Monte-Carlo simulation is performed to investigate the behavior of monodomain magnetic particle assemblies [1,2] connected through the dipole–dipole interaction. Such situation occurs when ferromagnetic nanocrystals are embedded in a non-magnetic matrix [3]. First the magnetization characteristics are determined for spherical clusters placed in a uniform external field. In this case the demagnetizing effect is quite important. Two ways are explored to reduce this effect without having to consider an exceedingly large number of particles: hypothetical toroidal clusters are considered, magnetized in azimuthal direction by a current passing through the loop (appropriate for study of scalar hysteresis), respectively a first neighbor exchange-like interaction is added in case of spherical or ellipsoidal clusters to compensate the demagnetizing field. Clusters with magnetic moments placed in the vertices of simple cubic (SC), body-centered (BCC), face-centered cubic lattice (FCC), respectively are compared to ‘‘amorphous’’ structures obtained by ranCorresponding author. Neutron Spectroscopy Department, Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary. Tel.: +36 1 392 2222 1738; fax: +36 1 392 2501. E-mail address: [email protected].
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.10.057
domly leaving out a number of particles (leading also to various filling factors). 2. Computation method The magnetic field sensed by a particle of the cluster is N 1 X mk ðmk rkl Þrkl H l ¼ H ext 3 ; 4p k¼1 r3kl r5kl kal
l ¼ 1; N,
ð1Þ
where H l is the field strength at the location of particle l, consisting of H ext —the externally applied field and the field created by the other members of the cluster, where mk is the magnetic dipole moment of particle k and rkl the relative position vector of particles k and l. The condition ml H l ! max;
l ¼ 1; N
(2)
is used to determine the orientation of each particle. To enhance convergence, the orientations of the magnetic moments are confined to a number of 92 equally distributed fixed directions. The magnetic moments of randomly selected particles are oriented parallel to the allowed direction closest to that of the field created by all the other particles, added to a given external field. This process is repeated until no change is encountered (all
ARTICLE IN PRESS J. Fu¨zi / Physica B 372 (2006) 239–242
240
moments find their equilibrium orientation). The average magnetization is the vector sum of individual moments divided by the volume of the cluster.
Dynamic effects are modeled by a supplementary variable (H m ) as input instead of the actual applied field strength H, delayed with respect to the latter,
Table 1 Spherical cluster dimensions (nm) Structure
SC
BCC
FCC
Lattice constant Cluster radius Particle radius Number of particles Filling factor
3.5 21 1.75 739 0.428
4.4 21 1.9 701 0.519
5.6 21 2 675 0.583
Structure
SC
BCC
FCC
Lattice constant Mean radius Tube radius Particle radius Number of particles Filling factor
3.5 30 10 1.75 924 0.350
4.4 30 10 1.9 900 0.437
5.6 30 10 2 820 0.464
Table 2 Toroidal cluster dimensions (nm)
1.2
1.2 70%
-1.2
-1
0 0 H [T]
1
70%
100% 70%
40%
40% 0 M [T]
0 M [T]
0 M [T]
40% 0
1.2
100%
100%
0
-1.2
-1
0 0 H [T]
1
0
-1.2 -1
0 0 H [T]
Fig. 1. Hysteresis loops of spherical cluster in SC, BCC and FCC configuration, respectively.
Fig. 2. Hysteresis loops of toroidal cluster in SC, BCC and FCC configuration, respectively.
1
ARTICLE IN PRESS J. Fu¨zi / Physica B 372 (2006) 239–242
according to:
241
the resulting magnetization when the dipole orientations are computed with H m as input. First neighbor exchange interactions are included using
dH m dB ; ¼ aðH H m Þ b dt dt B ¼ m0 ðH þ MðH m ÞÞ,
ð3Þ
where a and b are model parameters (a is the inverse of the delay time constant, b is responsible for increased lag in case of large magnetization variation rate) and MðH m Þ is
H l ¼ H ext þ
N X
J lk mk
k¼1 kal
N 1 X mk ðmk rkl Þrkl 3 ; 4p k¼1 r3kl r5kl kal
l ¼ 1; N,
0.3
ð4Þ
where J lk is the exchange coefficient, different from zero only if dipoles l and k are in first neighbor locations. 3. Results
0 M [T]
50kHz
0
5Hz 50Hz
5kHz 500Hz
-0.3 -1
0 0 H [T]
Fig. 3. Dynamic magnetizing loops of toroidal cluster.
1
The investigated cluster configurations are given in Tables 1 and 2 (dimensions in nm). The different lattice constant values have been chosen to ensure roughly equal filling factors for sake of comparison. Fig. 1 shows the hysteresis loops obtained for spherical clusters with SC, BCC and FCC structure and ‘‘amorphous’’ configurations obtained by leaving out dipoles from randomly selected positions of the respective lattices. The dominant effect of the demagnetizing field is observable. Hysteresis loops obtained for toroidal clusters with similar structures are plotted in Fig. 2. More rectangular loops are obtained since the demagnetizing effect does not appear in azimuthal direction (the structure is infinite). Throughout the
Fig. 4. Rotational magnetization of ellipsoidal cluster, cubic structure, 70% dipoles present.
ARTICLE IN PRESS 242
J. Fu¨zi / Physica B 372 (2006) 239–242
Fig. 5. Rotational magnetization of ellipsoidal cluster, cubic structure, all dipoles present.
simulations the magnetic polarization of the particles has been considered to be 2 T. Dynamic magnetizing curves obtained with parameter values a ¼ 10; 000 s1 , b ¼ 10; 000 m=H in Eq. (3) are plotted in Fig. 3, for a toroidal cluster in SC structure, 40% of dipoles present in randomly chosen locations. Figs. 4 and 5 show the results of rotational magnetizing process simulations of an ellipsoidal cluster, (semiaxis lengths 12.5, 20, 28.5 nm) of 1.75 nm diameter monodomain nanocrystals (m0MS ¼ 2 T), first neighbor exchange coefficient between crystals 0.004 nm3, in a cubic array, 70% of the crystals present in randomly chosen sites (366 dipoles remaining—Fig. 4), respectively all crystals present (Fig. 5).
clusters placed in external magnetic field. While the magnetization characteristics of spherical clusters are determined mainly by the demagnetizing effect, toroidal clusters magnetized in azimuthal direction exhibit hysteresis. The shape of the hysteresis loop is square for ordered structures and more rounded for random distribution of particles in the cluster. Dynamic effects can be simulated by an input delaying technique. Shape anisotropy can be investigated by computations carried out for ellipsoidal clusters.
References 4. Conclusion Zero temperature Monte-Carlo simulation is an appropriate tool for qualitative study of the behavior of dipole
[1] Gy. Szabo´, Gy. Ka´da´r, Phys. Rev. B 58 (9) (1998) 5584. [2] J. Fu¨zi, Gy. Ka´da´r, Physica B 343 (2004) 293. [3] J. Fu¨zi, L.K. Varga, Physica B 343 (2004) 159.