- Email: [email protected]
Transverse susceptibility for single-domain particle with cubic anisotropy
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 266 (2003) 200–206
Transverse susceptibility for single-domain particle with cubic anisotropy Al. Stancua,*, L. Spinub a
Department of Electricity and Physical, Faculty of Physics, ‘‘Alexandru Ioan Cuza’’ University, BD. Copou 11, Iasi 6600, Romania b AMRI & Physics Department, University of New Orleans, LA 70148, USA Received 30 September 2002; received in revised form 7 February 2003
Abstract For the uniaxial single-domain particles a 2D critical approach can be used for the calculation of the transverse susceptibility (TS) signal. The TS for cubic anisotropy is essentially a 3D problem that was solved with a micromagnetic Landau–Lifshitz–Gilbert algorithm. Since the TS experimental method is extensively used in the laboratories for the anisotropy evaluation, the study value consists of the fact that it offers a tool for understanding the problems and errors in the correctinterpretation of the TS curves when particles with cubic anisotropies are involved. r 2003 Elsevier B.V. All rights reserved. PACS: 75.40.Mg; 75.60.Ej; 75.50.Tt Keywords: Transverse susceptibility; Cubic anisotropy; Micromagnetics
1. Introduction The well-known experimental method of transverse susceptibility (TS) is a method for direct determination of the magnetic anisotropy in particulate magnetic systems due to the fact that, as predicted by Stoner–Wohlfarth model, for noninteracting uniaxial single-domain fine particle systems, the field dependence of TS presents characteristic peaks, located at the anisotropy and switching fields [1]. The conditions, as uniaxial anisotropy, single-domain non-interacting *Corresponding author. Tel.: +40-32-20-11-75; fax: +40-3220-12-05. E-mail address: [email protected] (Al. Stancu).
particles, in which the anisotropy can be determined accurately using this method might appear to be very restrictive. In fact, the same conditions that apply to other popular methods for determining the anisotropy in the case of particulate systems, as single point detection techniques or rotation hysteresis methods, are satisfied in many practical cases. For TS experiments an important step in this direction was made by advancing the classical model for TS due to Aharoni [1], by taking into account the influence of the higher order terms of the uniaxial anisotropy [2–4], a fact that is important for many usual magnetic uniaxial materials such as cobalt. In this paper, we present the methods used in the TS evaluation for cubic single-domain particles.
0304-8853/03/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0304-8853(03)00475-X
ARTICLE IN PRESS Al. Stancu, L. Spinu / Journal of Magnetism and Magnetic Materials 266 (2003) 200–206
2. TS for single-domain particles In the TS experiment one applies to the sample a system of two magnetic fields (see Fig. 1), one DC field that is considered on the Oz direction, and a small amplitude AC field, on the Ox direction in Fig. 1. The material easy axis (EA) orientation is given by the spherical angles ðya ; ja Þ and, if one can define a single orientation of the sample magnetization, as in the coherent rotation, model, the orientation of the magnetic moment is given by the radial unit vector in the spherical coordinates system ð~ uR; ~ uy; ~ u j Þ: The angle between the total applied field and the Oz-axis at a certain moment is yh : In the TS experiment, one measures the limit: dMX wt ¼ lim ð1Þ yh -0 dHX that will be referred to as the TS in the experimental setup described in Fig. 1. A systematic analysis has shown [4] that the critical curve approach can be applied for the 2D case quite efficiently. We have compared the results obtained with this method with the micromagnetic method, that is presented below, and a good agreement was found in each case. However, even in the case of uniaxial anisotropy there are
EA
z
M H DC H
uR θa
θh
θ y
O H AC
ϕa
ϕ
x Fig. 1. TS experiment for an uniaxial single-domain ferromagnetic particle.
201
cases in which the bi-dimensional image is not sufficient. For example, uniaxial systems with the first anisotropy constant negative, i.e. K1 o0; for different values of the second anisotropy constant K2 correspond to systems with the easy axes forming a cone (an easy cone). In these cases one observes in certain conditions jumps from one cusp point to another on the critical curve. This is due to the fact that the magnetic moment can move freely around the easy cone surface. Especially in these cases, the critical curve approach has a limited value in the calculation of the TS curves. For cubic anisotropy, the 2D critical curve cannot be used anymore because the applied field direction, the EA direction and the magnetization direction are not in the same plane. The cubic anisotropy case is essentially a 3D problem that needs a full 3D approach. The complexity of the critical approach presented in Refs. [5–7] is an argument in favor of the micromagnetic method that is presented in the next section. 2.1. Micromagnetic algorithm The micromagnetic model we have used is based on the Landau–Lifshitz–Gilbert (LLG) equation ~ [8]. The dynamics of the magnetization vector M ~ of each particle in the applied field H is described by the LLG equation [8]: " !# ~ ~ dM a ~ dM ~ ~ ¼ jgjðM H Þ þ M ; ð2Þ ~j dt dt jM ~ is assumed to be invariable, a is the where M ¼ M phenomenological damping constant assumed to be positive and g is the gyromagnetic factor. With the following notations: ~ H tgM h~ ¼ ; t ¼ ð3Þ M 1 þ a2 and using spherical co-ordinates, Eq. (2) can be written as dy ¼ hj þ ahy ; dt ð4Þ dj hy þ ahj ¼ ; dt sin y where hy and hj are the azimuthal and polar components, respectively, of the effective field
ARTICLE IN PRESS Al. Stancu, L. Spinu / Journal of Magnetism and Magnetic Materials 266 (2003) 200–206
202
h ¼ H=M: One considers a system of monodomain magnetic fine particles for which the contributions of conduction and displacement currents and magnetostriction to the effective field can be neglected. In this case the effective field h can be written as the sum of the effective anisotropy field, ha ; the external applied field, happl and the dipolar interaction (magnetostatic) field, hdip : In the present study one considers only a non-interacting magnetic fine particle system for which hdip ¼ 0: In these conditions the azimuthal and polar components of the effective field can be written as hy ¼ ha;y þ happl;y ; hj ¼ ha;j þ happl;j ;
ð5Þ
where 1 2jK1 j 1 qwa ha;y ¼ ; M m0 M 2 qy 1 2jK1 j 1 qwa ; ha;j ¼ M m0 M 2 sin y qj
and the equivalent anisotropy fields are given by 1 2jK1 j u R Þð~ ua ~ uyÞ ð~ ua ~ ha;y ¼ M m0 M
ha;j
fsgnðK1 Þ þ 2k2 ½1 ð~ ua ~ u R Þ2 g; 1 2jK1 j ¼ u R Þð~ ua ~ ujÞ ð~ ua ~ M m0 M fsgnðK1 Þ þ 2k2 ½1 ð~ ua ~ u R Þ2 g:
2.3. Cubic anisotropy For cubic anisotropy, the anisotropy free energy density can be expresses as a function of the relative orientation of the [1 0 0], [0 1 0] and [0 0 1] axes with respect to the magnetization vector. If one uses the Euler angles to define the orientation of these axes ðja ; ya ; ca Þ; one obtain ~ u ½1 0 0 ¼
ð6Þ
are equivalent anisotropy fields, with wa the anisotropy free energy density. The presented approach of studying the TS using the dynamic LLG equation makes possible to take into account the interparticle interactions in a natural way, through hdip ; as the sum of the dipolar fields created by neighbor particles. A systematic study of the influence of the interparticle interaction on the field dependence of the TS curves will be the object of another paper.
ð8Þ
þcos Ca ð~ u 1 cos ja þ ~ u 2 sin ja Þ þsin Ca ð~ u 1 sin ja cos ya Þ þ~ u 2 cos ja cos ya þ ~ u 3 sin ya ;
~ u ½0 1 0 ¼
sin Ca ð~ u 1 cos ja þ ~ u 2 sin ja Þ þcos Ca ð~ u 1 sin ja cos ya þ~ u 2 cos ja cos ya þ ~ u 3 sin ya Þ;
ð9Þ
~ u 1 sin ja sin ya ~ u 2 cos ja sin ya u ½0 0 1 ¼ ~ þ~ u 3 cos ya ; where ð~ u1; ~ u2; ~ u 3 Þ are the unit vectors of the Cartesian coordinates of the laboratory system. The free energy density in these condition id given by wa ¼ sgnðK1 Þ½ð~ u ½1 0 0 ~ u R Þ2 ð~ u ½0 1 0 ~ u R Þ2 þ ð~ u ½0 1 0 ~ u R Þ2 ð~ u ½0 0 1 ~ u R Þ2 þ ð~ u ½0 0 1 ~ u R Þ2 ð~ u ½1 0 0 ~ u R Þ2
2.2. Uniaxial anisotropy For uniaxial anisotropy, if the EA orientation is given by the unit vector ~ u a ; and the orientation of the magnetization vector is on the direction of the ~ u R unit vector (in spherical coordination), the anisotropy free energy can be expressed as wa ¼ sgnðK1 Þ ½1 ð~ ua ~ u R Þ2
þ k2 ½1 ð~ ua ~ u R Þ2 2
ð7Þ
þ k2 ð~ u ½1 0 0 ~ u R Þ2 ð~ u ½0 1 0 ~ u R Þ2 ð~ u ½0 0 1 ~ u R Þ2 : ð10Þ Using Eq. (10) in Eq. (6) one obtains the equivalent anisotropy fields. The TS process was simulated using a sequence of applied field happl ; identical to those applied in the experiment. At each step, the LLG equation is integrated until the procession motion of the magnetic moments around the stable equilibrium position can be neglected.
ARTICLE IN PRESS Al. Stancu, L. Spinu / Journal of Magnetism and Magnetic Materials 266 (2003) 200–206
3. Simulated TS curves: discussion We have systematically simulated the TS for a cubic single-particle with K1 ¼ 5:103 J/m3, k2 ¼ 0:5 (values characteristic for Ni). The first case analyzed was when the EA [1 1 1] is oriented on the
203
pffiffiffi Oz direction, that is, ya ¼ arctan ð 2Þ; ja ¼ ðp=2Þ; ca ¼ ðp=4Þ: The DC field is applied in the yOz plane, making the angle yf with the axis [1 1 1] (see Fig. 2). The results are compared with the TS curve for an uniaxial with k2 ¼ K2 =K1 ¼ 0 The field is normalized to the anisotropy field, Hk ¼ 2jK1 j=ðm0 MÞ for the uniaxial case. A supplementary normalization factor of (7/9) is used for the single-domain with cubic anisotropy. This is motivated as follows: the free energy density expression when the orientation of the [1 1 1] EA is on the Oz direction (and the stable equilibrium orientation of the magnetization is on the j ¼ 0 semi-plane) is given by " # pffiffiffi 1 1 2 4 3 4 wa ¼ sgnðK1 Þ sin y sin y cos y þ cos y 4 3 3 "
pffiffiffi 1 1 2 5 sin y cos y þ sin6 y sin4 y cos2 y 16 54 12 # pffiffiffi 3 3 1 1 2 3 4 6 cos y ; sin y cos y sin y cos y þ þ 9 27 81 þk2
Fig. 2. Cubic (a) and uniaxial (b) single-domain particles. The surfaces are the representation of the free magneto-crystalline energy density for the two cases.
with sgnðK1 Þ ¼ 1 and k2 ¼ 0:5: The coefficient of the sin2 y is 2ð2=3ÞsgnðK1 Þ ð2k2 Þ=9 ¼ ð7=9Þ ¼ 0:777: With the simplified expression of the free energy one can use the critical curve approach, as described for the uniaxial case. The critical curve is presented in the Fig. 3. The detail presented in the right figure show that the critical curve touches the axis in 0.777. We should mention that the configuration presented by the critical curve, especially for very small angles with the EA [1 1 1], explains the problems for any micromagnetic calculus to determine precisely the critical field in this case. Another observation that can be made is that the minimum of the free energy in the EA direction has a different symmetry in the cubic case as in the uniaxial one as it can be observed in Fig. 4 where it is presented a magnified image of the minimum in the EA direction for the cubic and uniaxial crystals. In Figs. 5 and 6 one presents the TS curves for the cubic and uniaxial cases. One observes that the differences in this experimental configuration are quite small when the DC field is oriented along the EA. When the angle increases the differences
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Al. Stancu, L. Spinu / Journal of Magnetism and Magnetic Materials 266 (2003) 200–206
Fig. 3. (a) The 2D critical curve for the cubic crystal when the EA [1 1 1] is on the Oz direction and the magnetization and the applied field are both in the j ¼ 0 semi-plane. (b) Detail of the critical curve.
are becoming very significant. For angles near 90 , the TS signal for cubic anisotropy is similar to the one at 0 since the system has another free energy minimum and not a maximum as in the uniaxial case. The result obtained for the uniaxial particle is the one known for j ¼ 90 in Ref. [1], as expected. Other simulations were made for an experimental configuration in which the DC field is applied on the Oz direction, the AC field on the Ox direction and the EA is in the xOz plane. One also observes as the angle between the EA and the DC applied field increases the shape of the TS signal is increasingly different (Fig. 7). This is quite
Fig. 4. The magnified image of the free energy minimum for the cubic (a) and uniaxial (b) crystals.
understandable if one takes into account the differences of the free energy surface in the two cases. One can mention that in order to systematically study the TS of a single-domain ferromagnetic particle with cubic anisotropy, the number of distinct situations is much higher that in the uniaxial case. Due to the possibility to use the critical curve approach, the uniaxial case is a very
ARTICLE IN PRESS Al. Stancu, L. Spinu / Journal of Magnetism and Magnetic Materials 266 (2003) 200–206
205
Fig. 5. Comparison between TS signal for cubic and uniaxial single-domain particles: y½1 1 1 ¼ 10 ; ja ¼ 90 (a) and y½1 1 1 ¼ 45 ; ja ¼ 90 (b).
Fig. 6. Comparison between TS signal for cubic and uniaxial single-domain particles: y½1 1 1 ¼ 89 ; ja ¼ 90 (a) and y½1 1 1 ¼ 90 ; ja ¼ 90 (b).
important tool for testing the micromagnetic model. However, our analyses have shown that in certain cases the sensitivity of the micromagnetic algorithm to factors like the AC field amplitude, is much higher that usual. The discussion of these cases is not among the objectives of this paper and will be presented in a further paper on this subject. What can be also observed is that the shape of the TS curves is however similar with the curves calculated for uniaxial case but with different orientation. Seems to be a good idea to find the equivalence between the relative orientation of the AC/DC fields and the [1 0 0], [0 1 0] and [0 0 1] axes for the cubic crystal and the orientation of the same AC/DC fields and the EA of the uniaxial particle. If this will be found possible, it will open the possibility to use the TS analytical expression
for the cubic case as well. That will also allow the analytical calculation of the TS signal for systems of cubic particles.
4. Conclusion In this paper, we have presented a systematic calculation of the TS signal for a single-domain particle with cubic anisotropy and have compared this signal with the similar one calculated for the uniaxial single-domain particle. The micromagnetic algorithm was tested comparing the results with those obtained using the critical curve approach for the uniaxial case. In the cubic case we have found situations in which the critical curve approach can still be used and supplementary test problems can be formulated. In this way
ARTICLE IN PRESS 206
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Fig. 7. Comparison between TS signal for cubic and uniaxial single-domain particles: y½1 1 1 ¼ 10 ; ja ¼ 90 (a) y½1 1 1 ¼ 45 ; ja ¼ 0 (b), and y½1 1 1 ¼ 89 ; ja ¼ 0 (c).
one can assume that one can calculate sufficiently accurate the TS for single-domains with any anisotropy.
References [1] A. Aharoni, E.M. Frei, S. Shtrikman, D. Treves, Bull. Res. Counc. Isr. 6A (1957) 215.
[2] L. Spinu, Al. Stancu, H. Srikanth, C.J. O’Connor, Appl. Phys. Lett. 80 (2002) 276. [3] Al. Stancu, L. Spinu, C.J. O’Connor, J. Magn. Magn. Mater. 242–245 (2) (2002) 1026. [4] L. Spinu, Al. Stancu, H. Srikanth, C.J. O’Connor, Physica B 306 (1–4) (2001) 221. [5] A. Thiaville, J. Magn. Magn. Mater. 182 (1998) 5. [6] A. Thiaville, Phys. Rev. B 61 (18) (2000) 12221. [7] C.-R. Chang, J. Appl. Phys. 69 (4) (1991) 2431. [8] P.R. Gillete, K. Oshima, J. Appl. Phys. 29 (1958) 529.
Journal of Magnetism and Magnetic Materials 266 (2003) 200–206
Transverse susceptibility for single-domain particle with cubic anisotropy Al. Stancua,*, L. Spinub a
Department of Electricity and Physical, Faculty of Physics, ‘‘Alexandru Ioan Cuza’’ University, BD. Copou 11, Iasi 6600, Romania b AMRI & Physics Department, University of New Orleans, LA 70148, USA Received 30 September 2002; received in revised form 7 February 2003
Abstract For the uniaxial single-domain particles a 2D critical approach can be used for the calculation of the transverse susceptibility (TS) signal. The TS for cubic anisotropy is essentially a 3D problem that was solved with a micromagnetic Landau–Lifshitz–Gilbert algorithm. Since the TS experimental method is extensively used in the laboratories for the anisotropy evaluation, the study value consists of the fact that it offers a tool for understanding the problems and errors in the correctinterpretation of the TS curves when particles with cubic anisotropies are involved. r 2003 Elsevier B.V. All rights reserved. PACS: 75.40.Mg; 75.60.Ej; 75.50.Tt Keywords: Transverse susceptibility; Cubic anisotropy; Micromagnetics
1. Introduction The well-known experimental method of transverse susceptibility (TS) is a method for direct determination of the magnetic anisotropy in particulate magnetic systems due to the fact that, as predicted by Stoner–Wohlfarth model, for noninteracting uniaxial single-domain fine particle systems, the field dependence of TS presents characteristic peaks, located at the anisotropy and switching fields [1]. The conditions, as uniaxial anisotropy, single-domain non-interacting *Corresponding author. Tel.: +40-32-20-11-75; fax: +40-3220-12-05. E-mail address: [email protected] (Al. Stancu).
particles, in which the anisotropy can be determined accurately using this method might appear to be very restrictive. In fact, the same conditions that apply to other popular methods for determining the anisotropy in the case of particulate systems, as single point detection techniques or rotation hysteresis methods, are satisfied in many practical cases. For TS experiments an important step in this direction was made by advancing the classical model for TS due to Aharoni [1], by taking into account the influence of the higher order terms of the uniaxial anisotropy [2–4], a fact that is important for many usual magnetic uniaxial materials such as cobalt. In this paper, we present the methods used in the TS evaluation for cubic single-domain particles.
0304-8853/03/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0304-8853(03)00475-X
ARTICLE IN PRESS Al. Stancu, L. Spinu / Journal of Magnetism and Magnetic Materials 266 (2003) 200–206
2. TS for single-domain particles In the TS experiment one applies to the sample a system of two magnetic fields (see Fig. 1), one DC field that is considered on the Oz direction, and a small amplitude AC field, on the Ox direction in Fig. 1. The material easy axis (EA) orientation is given by the spherical angles ðya ; ja Þ and, if one can define a single orientation of the sample magnetization, as in the coherent rotation, model, the orientation of the magnetic moment is given by the radial unit vector in the spherical coordinates system ð~ uR; ~ uy; ~ u j Þ: The angle between the total applied field and the Oz-axis at a certain moment is yh : In the TS experiment, one measures the limit: dMX wt ¼ lim ð1Þ yh -0 dHX that will be referred to as the TS in the experimental setup described in Fig. 1. A systematic analysis has shown [4] that the critical curve approach can be applied for the 2D case quite efficiently. We have compared the results obtained with this method with the micromagnetic method, that is presented below, and a good agreement was found in each case. However, even in the case of uniaxial anisotropy there are
EA
z
M H DC H
uR θa
θh
θ y
O H AC
ϕa
ϕ
x Fig. 1. TS experiment for an uniaxial single-domain ferromagnetic particle.
201
cases in which the bi-dimensional image is not sufficient. For example, uniaxial systems with the first anisotropy constant negative, i.e. K1 o0; for different values of the second anisotropy constant K2 correspond to systems with the easy axes forming a cone (an easy cone). In these cases one observes in certain conditions jumps from one cusp point to another on the critical curve. This is due to the fact that the magnetic moment can move freely around the easy cone surface. Especially in these cases, the critical curve approach has a limited value in the calculation of the TS curves. For cubic anisotropy, the 2D critical curve cannot be used anymore because the applied field direction, the EA direction and the magnetization direction are not in the same plane. The cubic anisotropy case is essentially a 3D problem that needs a full 3D approach. The complexity of the critical approach presented in Refs. [5–7] is an argument in favor of the micromagnetic method that is presented in the next section. 2.1. Micromagnetic algorithm The micromagnetic model we have used is based on the Landau–Lifshitz–Gilbert (LLG) equation ~ [8]. The dynamics of the magnetization vector M ~ of each particle in the applied field H is described by the LLG equation [8]: " !# ~ ~ dM a ~ dM ~ ~ ¼ jgjðM H Þ þ M ; ð2Þ ~j dt dt jM ~ is assumed to be invariable, a is the where M ¼ M phenomenological damping constant assumed to be positive and g is the gyromagnetic factor. With the following notations: ~ H tgM h~ ¼ ; t ¼ ð3Þ M 1 þ a2 and using spherical co-ordinates, Eq. (2) can be written as dy ¼ hj þ ahy ; dt ð4Þ dj hy þ ahj ¼ ; dt sin y where hy and hj are the azimuthal and polar components, respectively, of the effective field
ARTICLE IN PRESS Al. Stancu, L. Spinu / Journal of Magnetism and Magnetic Materials 266 (2003) 200–206
202
h ¼ H=M: One considers a system of monodomain magnetic fine particles for which the contributions of conduction and displacement currents and magnetostriction to the effective field can be neglected. In this case the effective field h can be written as the sum of the effective anisotropy field, ha ; the external applied field, happl and the dipolar interaction (magnetostatic) field, hdip : In the present study one considers only a non-interacting magnetic fine particle system for which hdip ¼ 0: In these conditions the azimuthal and polar components of the effective field can be written as hy ¼ ha;y þ happl;y ; hj ¼ ha;j þ happl;j ;
ð5Þ
where 1 2jK1 j 1 qwa ha;y ¼ ; M m0 M 2 qy 1 2jK1 j 1 qwa ; ha;j ¼ M m0 M 2 sin y qj
and the equivalent anisotropy fields are given by 1 2jK1 j u R Þð~ ua ~ uyÞ ð~ ua ~ ha;y ¼ M m0 M
ha;j
fsgnðK1 Þ þ 2k2 ½1 ð~ ua ~ u R Þ2 g; 1 2jK1 j ¼ u R Þð~ ua ~ ujÞ ð~ ua ~ M m0 M fsgnðK1 Þ þ 2k2 ½1 ð~ ua ~ u R Þ2 g:
2.3. Cubic anisotropy For cubic anisotropy, the anisotropy free energy density can be expresses as a function of the relative orientation of the [1 0 0], [0 1 0] and [0 0 1] axes with respect to the magnetization vector. If one uses the Euler angles to define the orientation of these axes ðja ; ya ; ca Þ; one obtain ~ u ½1 0 0 ¼
ð6Þ
are equivalent anisotropy fields, with wa the anisotropy free energy density. The presented approach of studying the TS using the dynamic LLG equation makes possible to take into account the interparticle interactions in a natural way, through hdip ; as the sum of the dipolar fields created by neighbor particles. A systematic study of the influence of the interparticle interaction on the field dependence of the TS curves will be the object of another paper.
ð8Þ
þcos Ca ð~ u 1 cos ja þ ~ u 2 sin ja Þ þsin Ca ð~ u 1 sin ja cos ya Þ þ~ u 2 cos ja cos ya þ ~ u 3 sin ya ;
~ u ½0 1 0 ¼
sin Ca ð~ u 1 cos ja þ ~ u 2 sin ja Þ þcos Ca ð~ u 1 sin ja cos ya þ~ u 2 cos ja cos ya þ ~ u 3 sin ya Þ;
ð9Þ
~ u 1 sin ja sin ya ~ u 2 cos ja sin ya u ½0 0 1 ¼ ~ þ~ u 3 cos ya ; where ð~ u1; ~ u2; ~ u 3 Þ are the unit vectors of the Cartesian coordinates of the laboratory system. The free energy density in these condition id given by wa ¼ sgnðK1 Þ½ð~ u ½1 0 0 ~ u R Þ2 ð~ u ½0 1 0 ~ u R Þ2 þ ð~ u ½0 1 0 ~ u R Þ2 ð~ u ½0 0 1 ~ u R Þ2 þ ð~ u ½0 0 1 ~ u R Þ2 ð~ u ½1 0 0 ~ u R Þ2
2.2. Uniaxial anisotropy For uniaxial anisotropy, if the EA orientation is given by the unit vector ~ u a ; and the orientation of the magnetization vector is on the direction of the ~ u R unit vector (in spherical coordination), the anisotropy free energy can be expressed as wa ¼ sgnðK1 Þ ½1 ð~ ua ~ u R Þ2
þ k2 ½1 ð~ ua ~ u R Þ2 2
ð7Þ
þ k2 ð~ u ½1 0 0 ~ u R Þ2 ð~ u ½0 1 0 ~ u R Þ2 ð~ u ½0 0 1 ~ u R Þ2 : ð10Þ Using Eq. (10) in Eq. (6) one obtains the equivalent anisotropy fields. The TS process was simulated using a sequence of applied field happl ; identical to those applied in the experiment. At each step, the LLG equation is integrated until the procession motion of the magnetic moments around the stable equilibrium position can be neglected.
ARTICLE IN PRESS Al. Stancu, L. Spinu / Journal of Magnetism and Magnetic Materials 266 (2003) 200–206
3. Simulated TS curves: discussion We have systematically simulated the TS for a cubic single-particle with K1 ¼ 5:103 J/m3, k2 ¼ 0:5 (values characteristic for Ni). The first case analyzed was when the EA [1 1 1] is oriented on the
203
pffiffiffi Oz direction, that is, ya ¼ arctan ð 2Þ; ja ¼ ðp=2Þ; ca ¼ ðp=4Þ: The DC field is applied in the yOz plane, making the angle yf with the axis [1 1 1] (see Fig. 2). The results are compared with the TS curve for an uniaxial with k2 ¼ K2 =K1 ¼ 0 The field is normalized to the anisotropy field, Hk ¼ 2jK1 j=ðm0 MÞ for the uniaxial case. A supplementary normalization factor of (7/9) is used for the single-domain with cubic anisotropy. This is motivated as follows: the free energy density expression when the orientation of the [1 1 1] EA is on the Oz direction (and the stable equilibrium orientation of the magnetization is on the j ¼ 0 semi-plane) is given by " # pffiffiffi 1 1 2 4 3 4 wa ¼ sgnðK1 Þ sin y sin y cos y þ cos y 4 3 3 "
pffiffiffi 1 1 2 5 sin y cos y þ sin6 y sin4 y cos2 y 16 54 12 # pffiffiffi 3 3 1 1 2 3 4 6 cos y ; sin y cos y sin y cos y þ þ 9 27 81 þk2
Fig. 2. Cubic (a) and uniaxial (b) single-domain particles. The surfaces are the representation of the free magneto-crystalline energy density for the two cases.
with sgnðK1 Þ ¼ 1 and k2 ¼ 0:5: The coefficient of the sin2 y is 2ð2=3ÞsgnðK1 Þ ð2k2 Þ=9 ¼ ð7=9Þ ¼ 0:777: With the simplified expression of the free energy one can use the critical curve approach, as described for the uniaxial case. The critical curve is presented in the Fig. 3. The detail presented in the right figure show that the critical curve touches the axis in 0.777. We should mention that the configuration presented by the critical curve, especially for very small angles with the EA [1 1 1], explains the problems for any micromagnetic calculus to determine precisely the critical field in this case. Another observation that can be made is that the minimum of the free energy in the EA direction has a different symmetry in the cubic case as in the uniaxial one as it can be observed in Fig. 4 where it is presented a magnified image of the minimum in the EA direction for the cubic and uniaxial crystals. In Figs. 5 and 6 one presents the TS curves for the cubic and uniaxial cases. One observes that the differences in this experimental configuration are quite small when the DC field is oriented along the EA. When the angle increases the differences
ARTICLE IN PRESS 204
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Fig. 3. (a) The 2D critical curve for the cubic crystal when the EA [1 1 1] is on the Oz direction and the magnetization and the applied field are both in the j ¼ 0 semi-plane. (b) Detail of the critical curve.
are becoming very significant. For angles near 90 , the TS signal for cubic anisotropy is similar to the one at 0 since the system has another free energy minimum and not a maximum as in the uniaxial case. The result obtained for the uniaxial particle is the one known for j ¼ 90 in Ref. [1], as expected. Other simulations were made for an experimental configuration in which the DC field is applied on the Oz direction, the AC field on the Ox direction and the EA is in the xOz plane. One also observes as the angle between the EA and the DC applied field increases the shape of the TS signal is increasingly different (Fig. 7). This is quite
Fig. 4. The magnified image of the free energy minimum for the cubic (a) and uniaxial (b) crystals.
understandable if one takes into account the differences of the free energy surface in the two cases. One can mention that in order to systematically study the TS of a single-domain ferromagnetic particle with cubic anisotropy, the number of distinct situations is much higher that in the uniaxial case. Due to the possibility to use the critical curve approach, the uniaxial case is a very
ARTICLE IN PRESS Al. Stancu, L. Spinu / Journal of Magnetism and Magnetic Materials 266 (2003) 200–206
205
Fig. 5. Comparison between TS signal for cubic and uniaxial single-domain particles: y½1 1 1 ¼ 10 ; ja ¼ 90 (a) and y½1 1 1 ¼ 45 ; ja ¼ 90 (b).
Fig. 6. Comparison between TS signal for cubic and uniaxial single-domain particles: y½1 1 1 ¼ 89 ; ja ¼ 90 (a) and y½1 1 1 ¼ 90 ; ja ¼ 90 (b).
important tool for testing the micromagnetic model. However, our analyses have shown that in certain cases the sensitivity of the micromagnetic algorithm to factors like the AC field amplitude, is much higher that usual. The discussion of these cases is not among the objectives of this paper and will be presented in a further paper on this subject. What can be also observed is that the shape of the TS curves is however similar with the curves calculated for uniaxial case but with different orientation. Seems to be a good idea to find the equivalence between the relative orientation of the AC/DC fields and the [1 0 0], [0 1 0] and [0 0 1] axes for the cubic crystal and the orientation of the same AC/DC fields and the EA of the uniaxial particle. If this will be found possible, it will open the possibility to use the TS analytical expression
for the cubic case as well. That will also allow the analytical calculation of the TS signal for systems of cubic particles.
4. Conclusion In this paper, we have presented a systematic calculation of the TS signal for a single-domain particle with cubic anisotropy and have compared this signal with the similar one calculated for the uniaxial single-domain particle. The micromagnetic algorithm was tested comparing the results with those obtained using the critical curve approach for the uniaxial case. In the cubic case we have found situations in which the critical curve approach can still be used and supplementary test problems can be formulated. In this way
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Fig. 7. Comparison between TS signal for cubic and uniaxial single-domain particles: y½1 1 1 ¼ 10 ; ja ¼ 90 (a) y½1 1 1 ¼ 45 ; ja ¼ 0 (b), and y½1 1 1 ¼ 89 ; ja ¼ 0 (c).
one can assume that one can calculate sufficiently accurate the TS for single-domains with any anisotropy.
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