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Magnetization curves of nanoparticle with single-ion uniaxial anisotropy
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 310 (2007) 2364–2366 www.elsevier.com/locate/jmmm
Magnetization curves of nanoparticle with single-ion uniaxial anisotropy E.E. Kokorina, M.V. Medvedev Institute of Electrophysics, Ural Division, Russian Academy of Sciences, Amundsena,106, Ekaterinburg 620016, Russia Available online 28 November 2006
Abstract The calculations of the magnetization of superparamagnetic nanoparticles were carried out in a model of isotropic intraparticle exchange interaction as well as in a model with isotropic exchange interaction and single-ion anisotropy. These models take into account thermal excitations of atomic magnetic moments, of which the total magnetic moment of nanoparticle consists. r 2006 Published by Elsevier B.V. PACS: 64.60.Cn; 75.10.Jm; 75.20.g Keywords: Superparamagnetism; Single-ion anisotropy; Nanoparticle
In the Neel theory of superparamagnetism [1,2] the magnetization of a nanoparticle is assumed to undergo a sort of Brownian rotation as a rigid whole. Thereby up to the present [3], no consideration is given to the possibility of thermal deviations of individual atomic magnetic moments from direction of the total magnetic moment of this nanoparticle. In order to take into account these spin deviations we consider the Hamiltonian of a nanoparticle which is composed from N atoms with a nearest-neighbour ferromagnetic exchange coupling J40 and an uniaxial single-ion anisotropy with parameter D6¼0, N X z N N X X 1 X H¼ J S n S nþD D ðSzn Þ2 gmB Sn H a . 2 n¼1 D¼1 n¼1 n¼1
coordinate system for the case D ¼ 0: hX ¼ Im sin y cos j; hY ¼ I m sin y sin j; hZ ¼ I m cos y þ g mB H a .
Here I ¼ Jz (z is the nearest-neighbour number), m is the average value of spin projection onto the total magnetic moment direction, and the direction of the total moments is given by the angle variables y, j. In the molecular field approximation we obtain the partition function Z Z¼ sin y dy dj TrS expðbH IS MF ðy; jÞÞ Z 1 1 1 sh½ðS þ 1=2ÞbQðxÞ N ¼ exp bNIm2 dx , 2 2 1 sh½ð1=2ÞbQðxÞÞ ð3Þ
(1) We suppose that inside the nanoparticle there is a molecular field due to the isotropic exchange interaction which supports the parallelism of spin array in the course of Brownian rotation of the total magnetic moment and has the following projection onto the laboratory Corresponding author. Tel.: +7 343 267 8823; fax: +7 343 267 8794.
E-mail address: [email protected] (E.E. Kokorina). 0304-8853/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.jmmm.2006.11.107
ð2Þ
where N X X 1 2 H IS ha ðy; jÞS a ðnÞ, MF ðy; jÞ ¼ NIm 2 n¼1 a¼X ;Y ;Z
QðxÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I 2 m2 þ 2ImgmB H a x þ ðgmB H a Þ2 .
(4)
(5)
ARTICLE IN PRESS E.E. Kokorina, M.V. Medvedev / Journal of Magnetism and Magnetic Materials 310 (2007) 2364–2366
From minimization of the free energy F ¼ b1 ln Z we find the self-consistent equation for the order parameter m Z 1 1 1 Im þ gmB H a x m ¼ Z 1 exp bNIm2 dx 2 2 1 QðxÞ N sh½ðS þ 1=2ÞbQðxÞ bS ðbQðxÞÞ ð6Þ sh½ð1=2ÞbQðxÞ (bS(x)—the Brillouin modified function). It is seen from Eq. (6) that at Ha ¼ 0 the order parameter m ¼ bS(bIm) disappears at the critical temperature Tc0 ¼ S(S+1)I /3 kB. The magnetization of nanoparticle (per atom) is equal to Z 1 M 1 Imx þ gmB H a 1 2 1 m¼ ¼ Z exp bNIm dx NgmB 2 2 1 QðxÞ N sh½ðS þ 1=2ÞbQðxÞ bS ðbQðxÞÞ . ð7Þ sh½ð1=2ÞbQðxÞ We calculated the magnetization of a nanoparticle from formulas (7)–(8) at T/Tc0 ¼ 0.5 and 0.8 (solid curves in Fig. 1) and compared with Neel’s results for the magnetization m ¼ M/NgmB ¼ L(NgmBHa) (dotted curves in Fig. 1), where L(x) is the Langevin function. At both the temperatures the curves from Neel theory run above our curves and the difference between the corresponding curves becomes more profound with temperature increase up to T/Tc0 ¼ 0.8. It means that the temperature dependence of atomic magnetic moments is of considerable importance as the critical temperature region is approached. For the case of a nonzero anisotropy parameter D6¼0 the magnitude of the average value of atomic spin projection onto the rotating total moment direction depends upon the angle between the anisotropy axis and the direction of the total magnetic moment. Therefore, at D6¼0 we cannot describe the spin ordering along the rotating total moment by the only parameter [4]. Then, we introduce two order parameters mz and m? and generalize the approximation for
the molecular field projections (2) as hX ¼ Im? sin y cos j, hY ¼ Im? sin y sin j; hZ ¼ ImZ cos y.
ð8Þ
A contribution gmB Ha from the magnetic field can be added either to hX or to hZ. It leads to the partition function [4] Z 1 Z ¼ sin y dy dj exp bNIðm2Z cos2 y þ m2? sin2 yÞ Z N 1S , 2 (9) where Z1S ¼ TrS expfbðhX S xn þ hY S yn þ hZ S zn þ DðS z Þ2 Þg
(I) mz6¼0, m? ¼ 0—the superparamagnetic state of Z type; (II) mz ¼ 0, m?6¼0—the superparamagnetic state of XY type; (III) mz6¼0, m?6¼0—the superparamagnetic state of ellipsoidal type. Self-consistent equations for the order parameters mz and m? and magnetization for the parallel and perpendicular directions of the external magnetic field in relation to the anisotropic axis can be obtained from Eq. (9). We calculated the magnetization of nanoparticles with N ¼ 104 magnetic atoms (spin S ¼ 1) in the superparamagnetic ellipsoidal phase (at D/I ¼ 0.0025, T/Tc ¼ 0.8 and Tc were calculated for anisotropic case—Fig. 2) for two directions of the magnetic field and compared with corresponding curves of the Neel theory (with consideration of the anisotropy)
M / NgµB
0.8
0.6 Isotropic magnetization at T/Tc0 = 0.8 Isotropic magnetization at T/Tc0 = 0.5
0.2
In the Neel model of "rigid" superparamagnet Isotropic magnetization at T/Tc0 = 0.8 Isotropic magnetization at T/Tc0 = 0.5
0.0 0.000
0.001
(10)
and to the occurrence of one of the three following types of anisotropic superparamagnetic states (depending on the sigh and magnitude of D/I ratio and temperature)
1.0
0.4
2365
0.002 gµBHa / l
0.003
0.004
Fig. 1. Isotropic magnetization of nanoparticle with N ¼ 104 magnetic atoms (spin S ¼ 1) vs. magnitude of applied field gmBHa/I.
ARTICLE IN PRESS 2366
E.E. Kokorina, M.V. Medvedev / Journal of Magnetism and Magnetic Materials 310 (2007) 2364–2366
1.0 T / Tc = 0.8 D/l = 0.0025
M / NgµB
0.0
0.6
0.4 Parallel and perpendicular magnetization in the ellipsoidal phase Parallel and perpendicular magnetization for a "rigid" superparamagnet Parallel and perpendicular magnetization in a modified Neel theory
0.2
0.0 0.000
0.002
0.004
0.006
gµBHa/l Fig. 2. Anisotropic magnetization of nanoparticle with N ¼ 104 magnetic atoms (spin S ¼ 1) vs. magnitue of applied field gmBHa/I at T/Tc ¼ 0.8.
and those curves of a modified Neel theory (dashed curves in Fig. 2) that are obtained when the temperature dependence of the macromoment of a nanoparticle Mg ¼ NgmBm is roughly approximated with the Brillouin function (here m ¼ 0.680 for S ¼ 1 and T/Tc ¼ 0.8). The calculated curve for the parallel magnetization always runs below the magnetization curve of a ‘‘rigid’’ superparamagnet and coincides practically with the dashed line for m ¼ 0.680. However, the calculated curve for the perpendicular magnetization in the ellipsoidal phase runs initially above the analogues curve of the ‘‘rigid’’ superparamagnet and runs below only at higher magnetic fields. As for the curve with the approximation of the macromoment via the Brillouin function (the lower dashed line in Fig. 2), it always
runs below our calculated curve for the perpendicular magnetization. Summing up we can assert that the consideration of the thermal excitations of individual atomic magnetic moments can essentially modify the results of the Neel theory for superparamagnets in the temperature region above T/Tc0.5. References [1] [2] [3] [4]
L. Neel, Ann. Geophys. 5 (1949) 99. L. Neel, Adv. Phys. 4 (1955) 191. J.L. Dormann, D. Fiorani, E. Tronc, Adv. Chem. Phys. 98 (1997) 283. E.E. Kokorina, M.V. Medvedev, Phys. Met. Metallogr. 99 (2005) 236.
Journal of Magnetism and Magnetic Materials 310 (2007) 2364–2366 www.elsevier.com/locate/jmmm
Magnetization curves of nanoparticle with single-ion uniaxial anisotropy E.E. Kokorina, M.V. Medvedev Institute of Electrophysics, Ural Division, Russian Academy of Sciences, Amundsena,106, Ekaterinburg 620016, Russia Available online 28 November 2006
Abstract The calculations of the magnetization of superparamagnetic nanoparticles were carried out in a model of isotropic intraparticle exchange interaction as well as in a model with isotropic exchange interaction and single-ion anisotropy. These models take into account thermal excitations of atomic magnetic moments, of which the total magnetic moment of nanoparticle consists. r 2006 Published by Elsevier B.V. PACS: 64.60.Cn; 75.10.Jm; 75.20.g Keywords: Superparamagnetism; Single-ion anisotropy; Nanoparticle
In the Neel theory of superparamagnetism [1,2] the magnetization of a nanoparticle is assumed to undergo a sort of Brownian rotation as a rigid whole. Thereby up to the present [3], no consideration is given to the possibility of thermal deviations of individual atomic magnetic moments from direction of the total magnetic moment of this nanoparticle. In order to take into account these spin deviations we consider the Hamiltonian of a nanoparticle which is composed from N atoms with a nearest-neighbour ferromagnetic exchange coupling J40 and an uniaxial single-ion anisotropy with parameter D6¼0, N X z N N X X 1 X H¼ J S n S nþD D ðSzn Þ2 gmB Sn H a . 2 n¼1 D¼1 n¼1 n¼1
coordinate system for the case D ¼ 0: hX ¼ Im sin y cos j; hY ¼ I m sin y sin j; hZ ¼ I m cos y þ g mB H a .
Here I ¼ Jz (z is the nearest-neighbour number), m is the average value of spin projection onto the total magnetic moment direction, and the direction of the total moments is given by the angle variables y, j. In the molecular field approximation we obtain the partition function Z Z¼ sin y dy dj TrS expðbH IS MF ðy; jÞÞ Z 1 1 1 sh½ðS þ 1=2ÞbQðxÞ N ¼ exp bNIm2 dx , 2 2 1 sh½ð1=2ÞbQðxÞÞ ð3Þ
(1) We suppose that inside the nanoparticle there is a molecular field due to the isotropic exchange interaction which supports the parallelism of spin array in the course of Brownian rotation of the total magnetic moment and has the following projection onto the laboratory Corresponding author. Tel.: +7 343 267 8823; fax: +7 343 267 8794.
E-mail address: [email protected] (E.E. Kokorina). 0304-8853/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.jmmm.2006.11.107
ð2Þ
where N X X 1 2 H IS ha ðy; jÞS a ðnÞ, MF ðy; jÞ ¼ NIm 2 n¼1 a¼X ;Y ;Z
QðxÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I 2 m2 þ 2ImgmB H a x þ ðgmB H a Þ2 .
(4)
(5)
ARTICLE IN PRESS E.E. Kokorina, M.V. Medvedev / Journal of Magnetism and Magnetic Materials 310 (2007) 2364–2366
From minimization of the free energy F ¼ b1 ln Z we find the self-consistent equation for the order parameter m Z 1 1 1 Im þ gmB H a x m ¼ Z 1 exp bNIm2 dx 2 2 1 QðxÞ N sh½ðS þ 1=2ÞbQðxÞ bS ðbQðxÞÞ ð6Þ sh½ð1=2ÞbQðxÞ (bS(x)—the Brillouin modified function). It is seen from Eq. (6) that at Ha ¼ 0 the order parameter m ¼ bS(bIm) disappears at the critical temperature Tc0 ¼ S(S+1)I /3 kB. The magnetization of nanoparticle (per atom) is equal to Z 1 M 1 Imx þ gmB H a 1 2 1 m¼ ¼ Z exp bNIm dx NgmB 2 2 1 QðxÞ N sh½ðS þ 1=2ÞbQðxÞ bS ðbQðxÞÞ . ð7Þ sh½ð1=2ÞbQðxÞ We calculated the magnetization of a nanoparticle from formulas (7)–(8) at T/Tc0 ¼ 0.5 and 0.8 (solid curves in Fig. 1) and compared with Neel’s results for the magnetization m ¼ M/NgmB ¼ L(NgmBHa) (dotted curves in Fig. 1), where L(x) is the Langevin function. At both the temperatures the curves from Neel theory run above our curves and the difference between the corresponding curves becomes more profound with temperature increase up to T/Tc0 ¼ 0.8. It means that the temperature dependence of atomic magnetic moments is of considerable importance as the critical temperature region is approached. For the case of a nonzero anisotropy parameter D6¼0 the magnitude of the average value of atomic spin projection onto the rotating total moment direction depends upon the angle between the anisotropy axis and the direction of the total magnetic moment. Therefore, at D6¼0 we cannot describe the spin ordering along the rotating total moment by the only parameter [4]. Then, we introduce two order parameters mz and m? and generalize the approximation for
the molecular field projections (2) as hX ¼ Im? sin y cos j, hY ¼ Im? sin y sin j; hZ ¼ ImZ cos y.
ð8Þ
A contribution gmB Ha from the magnetic field can be added either to hX or to hZ. It leads to the partition function [4] Z 1 Z ¼ sin y dy dj exp bNIðm2Z cos2 y þ m2? sin2 yÞ Z N 1S , 2 (9) where Z1S ¼ TrS expfbðhX S xn þ hY S yn þ hZ S zn þ DðS z Þ2 Þg
(I) mz6¼0, m? ¼ 0—the superparamagnetic state of Z type; (II) mz ¼ 0, m?6¼0—the superparamagnetic state of XY type; (III) mz6¼0, m?6¼0—the superparamagnetic state of ellipsoidal type. Self-consistent equations for the order parameters mz and m? and magnetization for the parallel and perpendicular directions of the external magnetic field in relation to the anisotropic axis can be obtained from Eq. (9). We calculated the magnetization of nanoparticles with N ¼ 104 magnetic atoms (spin S ¼ 1) in the superparamagnetic ellipsoidal phase (at D/I ¼ 0.0025, T/Tc ¼ 0.8 and Tc were calculated for anisotropic case—Fig. 2) for two directions of the magnetic field and compared with corresponding curves of the Neel theory (with consideration of the anisotropy)
M / NgµB
0.8
0.6 Isotropic magnetization at T/Tc0 = 0.8 Isotropic magnetization at T/Tc0 = 0.5
0.2
In the Neel model of "rigid" superparamagnet Isotropic magnetization at T/Tc0 = 0.8 Isotropic magnetization at T/Tc0 = 0.5
0.0 0.000
0.001
(10)
and to the occurrence of one of the three following types of anisotropic superparamagnetic states (depending on the sigh and magnitude of D/I ratio and temperature)
1.0
0.4
2365
0.002 gµBHa / l
0.003
0.004
Fig. 1. Isotropic magnetization of nanoparticle with N ¼ 104 magnetic atoms (spin S ¼ 1) vs. magnitude of applied field gmBHa/I.
ARTICLE IN PRESS 2366
E.E. Kokorina, M.V. Medvedev / Journal of Magnetism and Magnetic Materials 310 (2007) 2364–2366
1.0 T / Tc = 0.8 D/l = 0.0025
M / NgµB
0.0
0.6
0.4 Parallel and perpendicular magnetization in the ellipsoidal phase Parallel and perpendicular magnetization for a "rigid" superparamagnet Parallel and perpendicular magnetization in a modified Neel theory
0.2
0.0 0.000
0.002
0.004
0.006
gµBHa/l Fig. 2. Anisotropic magnetization of nanoparticle with N ¼ 104 magnetic atoms (spin S ¼ 1) vs. magnitue of applied field gmBHa/I at T/Tc ¼ 0.8.
and those curves of a modified Neel theory (dashed curves in Fig. 2) that are obtained when the temperature dependence of the macromoment of a nanoparticle Mg ¼ NgmBm is roughly approximated with the Brillouin function (here m ¼ 0.680 for S ¼ 1 and T/Tc ¼ 0.8). The calculated curve for the parallel magnetization always runs below the magnetization curve of a ‘‘rigid’’ superparamagnet and coincides practically with the dashed line for m ¼ 0.680. However, the calculated curve for the perpendicular magnetization in the ellipsoidal phase runs initially above the analogues curve of the ‘‘rigid’’ superparamagnet and runs below only at higher magnetic fields. As for the curve with the approximation of the macromoment via the Brillouin function (the lower dashed line in Fig. 2), it always
runs below our calculated curve for the perpendicular magnetization. Summing up we can assert that the consideration of the thermal excitations of individual atomic magnetic moments can essentially modify the results of the Neel theory for superparamagnets in the temperature region above T/Tc0.5. References [1] [2] [3] [4]
L. Neel, Ann. Geophys. 5 (1949) 99. L. Neel, Adv. Phys. 4 (1955) 191. J.L. Dormann, D. Fiorani, E. Tronc, Adv. Chem. Phys. 98 (1997) 283. E.E. Kokorina, M.V. Medvedev, Phys. Met. Metallogr. 99 (2005) 236.