- Email: [email protected]
Domain wall propagation in adiabatic regime
ARTICLE IN PRESS
Physica B 403 (2008) 386–389 www.elsevier.com/locate/physb
Domain wall propagation in adiabatic regime Y. Kostyka, R. Vargaa,, M. Vazquezb, P. Vojtanika a
Institute of Physics, Faculty of Sciences, Park Angelinum 9, 04154 Kosice, Slovakia b Instituto de Ciencia de Materiales, CSIC, 28049 Madrid, Spain
Abstract The domain wall dynamics in the adiabatic regime has been studied. It is shown that the domain wall velocity in the low-field range (when the domain wall interacts with the distributed defects) satisfy the power law: v ¼ S0 (HH0)b, where H0 is the critical field. The temperature dependence of the power exponent b is treated in terms of the change of the domain wall shape from rigid to flexible one. In addition, the mobility exponent S0 is shown to be field independent and is proportional to the domain wall mobility S in the viscous regime. r 2007 Elsevier B.V. All rights reserved. Keywords: Barkhausen effect; Domain wall propagation; Magnetization process
1. Introduction Domain wall propagation is used in a number of new devices such as MRAM, spintronics, and sensor devices [1–3]. The domain wall motion is driven both electronically and magnetically. However, the speed of such devices depends on the velocity of the magnetic domain wall. As a result of the stresses introduced during their fabrication, amorphous microwires with positive magnetostriction exhibit a unique domain structure with a single domain in the center and small closure domains at the ends of the wire [4,5]. Such domain structure results in a magnetization process determined by the depinning and subsequent propagation of a single domain wall from a closure structure that enables one the study of the single domain wall propagation in a constant applied magnetic field [6,7]. Very interesting results have been previously reported which somehow do not fit in the actually accepted theory [7]. Although the velocity of the domain wall has been shown to be proportional to the applied magnetic field (according to a theory), the negative critical propagation field was found during the propagation of the domain wall.
In the present contribution, we report a detailed study on the single domain wall propagation at low field, when the domain wall propagates at low velocity interacting with the local defects distributed along the sample. 2. Experimental We have studied the domain wall dynamics by the classical Sixtus–Tonks method [8] in the amorphous glass-coated microwire with nominal composition Co68Mn7Si15B10. The diameter of the metallic nucleus is 8 mm and the total diameter is 20 mm. The length of the sample taken for measurements has been 105 mm. The sample holder consists of four coaxially placed coils. The primary coil (100 mm long) is fed by a 33 Hz square-shaped signal resulting in a constant and homogeneous magnetic field during the domain wall propagation. The nucleation coil (12 mm) is placed at the very end of the primary coil in order to nucleate a reverse domain, and finally, two pickup coils (3 mm long and 60 mm in between) are placed symmetrically so that two maxima appear when the domain wall passes through them. 3. Results and discussion
Corresponding author. Tel.: +421 55 62 211 28; fax: +421 55 62 221 24. E-mail address: [email protected] (R. Varga).
0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.08.056
The dependence of the domain wall velocity, v, as a function of the amplitude of the applied field, H, is shown
ARTICLE IN PRESS Y. Kostyk et al. / Physica B 403 (2008) 386–389
in Fig. 1. At high fields, the velocity is linearly proportional to the applied field H according to the theory of the domain wall motion in the viscous regime [9]: v ¼ SðH H 0 Þ,
(1)
where S is the domain wall mobility and H0 is the critical propagation field. However, the extrapolation into the lowfield regime leads to a negative critical propagation field. This has been also reported for microwires [7,10]. At low fields, the domain wall interaction with the local defects distributed along the sample should be taken into account. Generally, the critical field H0 consists of two contributions: long-range contribution Hl arising from the magnetostatic and magnetoelastic interaction of the domain wall with the stresses. On the other hand, the domain wall interacts during its propagation with the local defects giving rise to the short-range contribution to the critical field Hp, which corresponds to the pinning field of the domain wall on the defects distribution. Finally, the domain wall dynamics is described by Yang and Erskine [11] v ¼ S½H ðH 1 þ H P Þ.
(2)
As a result, the domain wall moves in small intermittent jumps at a velocity determined by the local distribution of the defects according to Eq. (2). Such a regime of the domain wall motion is called adiabatic. The domain wall moves slowly, having time to interact with the defects. The distribution of the pinning field Hp can be approximated by a gaussian function with width R [12]. According to the Random Field Ising model, the magnetization change DM during the small domain jump is given by the power law [12]: R Rc b DM , (3) Rc
where Rc is the critical distribution width, below which the small intermittent domain wall jumps do not appear, and b is the critical exponent. The average velocity of the domain wall finally scales as [13] v ¼ S 0 ðH H 00 Þb ,
where S is the so-called domain wall mobility parameter and H 00 is the critical field, below which the domain wall motion cannot be observed. Fitting the velocity dependence on the magnetic field given in Fig. 1 gives results in b ¼ 0.53, in reasonable agreement with theory expectation of 12 [12,13]. Measurements of the domain wall dynamics in the wide temperature range give us additional information about the process. Fig. 2 shows the evolution of the domain wall velocity v with the applied field H in the temperature range from 173 to 373 K. To check the validity of the power law in a wide temperature range, the linear dependence of log(v) as a function of the logðH H 00 Þ is shown. The power exponent b changes with temperature as shown in Fig. 3. Starting from the value of 0.25 at 77 K, it increases with temperature until it saturates at 200 K at the value 0.55. Such behavior can be explained in terms of the domain wall shape. It comes out from the Random Field Ising model [14], that for a planar wall, the power exponent b ¼ 1/2, which is reasonably close to our values obtained at high temperature. However, the power exponent b was calculated to be lower than 12 for the flexible domain wall [12,13]. The domain wall energy is proportional to the domain wall area. On the other hand, the interaction energy of the domain wall with the defects should be taken into account when the defects are present in the sample. The amorphous glass-coated microwires are metastable in nature because of their amorphous character, containing a lot of the defects in the form of the local fluctuation of
173 K 223 K 273 K 323 K 373 K
3.0
800
log (v)
v (m/s)
2.5 600
(4)
0
1000
v = S (H-H0)
387
2.0
400 v = S'(H-H'0)q
200
1.5
0 0
50
100 H (A/m)
150
Fig. 1. Domain wall velocity v as a function of a constant applied field H, measured at the temperature T ¼ 273 K. Full-lines corresponds to the fit according to Eqs. (1) and (4).
-1.0
-0.5
0.0
1.0 0.5 log (H-H'0)
1.5
2.0
2.5
Fig. 2. The linear dependence of the log(v) on the log(HH0) approves the validity of the power law given by Eq. (4) in the wide temperature range.
ARTICLE IN PRESS Y. Kostyk et al. / Physica B 403 (2008) 386–389
388
350 0.6
10
S* 300
S
S* (m/s)
250
0.4
6
200
S (m2/As)
8
0.5
4
150 0.3
2
100 0.2
200
100 100
200
300
400
300
400
T (K)
T (K) Fig. 3. The temperature dependence of power exponent b.
density (interstitials, vacancies). The interaction energy of the domain wall with the defects is strongly temperature dependent [15,16]. At high temperature, the domain wall interacts with the defects weakly. So, the domain wall prefers to take planar configuration in order to decrease its energy. However, at low temperature, the mobile defects loose their mobility, stabilizing the local anisotropy direction, which finally leads to the increase of the domain wall interaction energy with the defects. The presence of strong interacting pinning centers inside the wall favors a flexible wall to reduce its energy. As a consequence, b takes reduced value down to 0.25 that reasonably agrees with that expected from the Random Field Ising Model [12]. Such behavior has been already confirmed in glasscoated microwires by the measurements of the temperature dependence of the switching field [16] and domain wall damping [7,17]. Finally, the domain wall mobility parameter S0 , given in Eq. (4), has been considered. From one point of view, the dependence of the domain wall velocity on the applied magnetic field given in Fig. 1 can be treated in terms of the field dependence of the domain wall mobility S given in Eq. (1), keeping the power exponent equal to 1. On the other hand, considering Eq. (3) and the fact that the domain wall velocity v is given by the magnetization change DM with time t (v ¼ DM/t) [12,13], the mobility parameter S0 can be expressed as S0 ¼ Sn =H 0 b0 . The constant S and the domain wall mobility S exhibit the same temperature dependence (see Fig. 4). They differ just by a constant, from the dimension of the field (SHmaxS). The value of this field (given in Table 1 as Hmax) is of the order of the transient field at which the domain wall dynamics changes from adiabatic to viscous regime. Moreover, its temperature dependence corresponds to that of the switching field, already measured in CoMnSiB microwires [16], which points to the fact that both fields are governed by the same mechanism.
Fig. 4. Temperature dependencies of the domain wall mobility S and the constant S.
Table 1 Parameters of the fitting according to Eqs. (1) and (4) T (K)
S0
H0 (A/m)
q
S (m2/A s)
Hmax (A/m)
77 98 123 148 173 198 223 248 273 298 323 348 373
87.7 56.4 53.0 43.2 50.9 47.5 44.7 52.8 68.1 118.0 89.5 80.3 109.3
10.1 4.6 6.8 7.3 7.4 7.2 4.6 5.3 4.4 7.1 7.2 7.5 7.3
0.25 0.38 0.44 0.53 0.51 0.56 0.61 0.58 0.53 0.41 0.51 0.58 0.54
2.07 1.72 2.89 2.47 2.68 3.91 3.26 3.79 4.56 5.73 4.92 6.15 9.81
75.6 58.7 42.6 50.3 53.2 36.6 34.8 36.7 32.6 46.1 50.1 41.5 32.5
Finally, the mobility parameter S0 can be expressed as S0 ¼
SH max H 0 b0
.
(5)
However, all parameters given in Eq. (5) are field independent. So, the mobility parameter S0 should be field independent, too. 4. Conclusion The domain wall dynamics in the adiabatic regime has been measured. It is shown that the domain wall velocity v in the low-field regime can be described as vðHÞ ¼ ðSH max =H 0 b0 ÞðH H 00 Þb . The temperature dependence of the power exponent b is treated in terms of the domain wall shape from the planar one at high temperatures to the flexible one at low temperatures as a result of the strong domain wall interaction with the defects distributed along the sample. The domain wall mobility parameter is shown to be field independent.
ARTICLE IN PRESS Y. Kostyk et al. / Physica B 403 (2008) 386–389
Acknowledgments Y. Kostyk acknowledges the support from The Orange Konto. This work was supported by scientific Grants VEGA 1/3035/06 and APVT-20-007804.
[9] [10] [11] [12]
[13]
References [1] [2] [3] [4] [5] [6] [7]
D. Atkinson, et al., Nat. Mater. 2 (2003) 85. G.S.D. Beach, et al., Nat. Mater. 4 (2005) 741. D.A. Allwood, et al., Science 309 (2005) 1688. M. Vazquez, Physica B 299 (2001) 302. H. Chiriac, T.A. Ovari, G. Pop, Phys. Rev. B 52 (1995) 10104. R. Varga, et al., Appl. Phys. Lett. 83 (2003) 2620. R. Varga, K.L. Garcı´ a, M. Va´zquez, P. Vojtanik, Phys. Rev. Lett. 94 (2005) 017201. [8] K.J. Sixtus, L. Tonks, Phys. Rev. 42 (1932) 419.
[14] [15]
[16] [17]
389
H.J. Williams, W. Shockley, C. Kittel, Phys. Rev. 80 (1950) 1090. R. Varga, et al., Phys. Rev. B 74 (2006) 212405. S. Yang, J.L. Erskine, Phys. Rev. B 72 (2005) 064433. J.P. Sethna, K.A. Dahmen, O. Perkovic, Random field ising models of hysteresis, in: G. Bertotti, I. Mayergoyz (Eds.), The Science of Hysteresis, Academic Press, New York, 2006, p. 107. G. Durin, S. Zapperi, Barkhausen effect, in: G. Bertotti, I. Mayergoyz (Eds.), The Science of Hysteresis, Academic Press, New York, 2006, p. 181. S. Zapperi, G. Durin, Comput. Mater. Sci. 20 (2001) 436. H. Kronmuller, M. Fanhle, in: Micromagnetism and the Microstructure of Ferromagnetic Solids, Cambridge University Press, Cambridge, 2003, p. 274. R. Varga, K.L. Garcı´ a, M. Va´zquez, A. Zhukov, P. Vojtanik, Phys. Rev. B 70 (2004) 024402. R. Varga, A. Zhukov, N. Usov, J.M. Blanco, J. Gonzalez, V. Zhukova, P. Vojtanik, J. Magn. Magn. Mater. 316 (2007) 337.
Physica B 403 (2008) 386–389 www.elsevier.com/locate/physb
Domain wall propagation in adiabatic regime Y. Kostyka, R. Vargaa,, M. Vazquezb, P. Vojtanika a
Institute of Physics, Faculty of Sciences, Park Angelinum 9, 04154 Kosice, Slovakia b Instituto de Ciencia de Materiales, CSIC, 28049 Madrid, Spain
Abstract The domain wall dynamics in the adiabatic regime has been studied. It is shown that the domain wall velocity in the low-field range (when the domain wall interacts with the distributed defects) satisfy the power law: v ¼ S0 (HH0)b, where H0 is the critical field. The temperature dependence of the power exponent b is treated in terms of the change of the domain wall shape from rigid to flexible one. In addition, the mobility exponent S0 is shown to be field independent and is proportional to the domain wall mobility S in the viscous regime. r 2007 Elsevier B.V. All rights reserved. Keywords: Barkhausen effect; Domain wall propagation; Magnetization process
1. Introduction Domain wall propagation is used in a number of new devices such as MRAM, spintronics, and sensor devices [1–3]. The domain wall motion is driven both electronically and magnetically. However, the speed of such devices depends on the velocity of the magnetic domain wall. As a result of the stresses introduced during their fabrication, amorphous microwires with positive magnetostriction exhibit a unique domain structure with a single domain in the center and small closure domains at the ends of the wire [4,5]. Such domain structure results in a magnetization process determined by the depinning and subsequent propagation of a single domain wall from a closure structure that enables one the study of the single domain wall propagation in a constant applied magnetic field [6,7]. Very interesting results have been previously reported which somehow do not fit in the actually accepted theory [7]. Although the velocity of the domain wall has been shown to be proportional to the applied magnetic field (according to a theory), the negative critical propagation field was found during the propagation of the domain wall.
In the present contribution, we report a detailed study on the single domain wall propagation at low field, when the domain wall propagates at low velocity interacting with the local defects distributed along the sample. 2. Experimental We have studied the domain wall dynamics by the classical Sixtus–Tonks method [8] in the amorphous glass-coated microwire with nominal composition Co68Mn7Si15B10. The diameter of the metallic nucleus is 8 mm and the total diameter is 20 mm. The length of the sample taken for measurements has been 105 mm. The sample holder consists of four coaxially placed coils. The primary coil (100 mm long) is fed by a 33 Hz square-shaped signal resulting in a constant and homogeneous magnetic field during the domain wall propagation. The nucleation coil (12 mm) is placed at the very end of the primary coil in order to nucleate a reverse domain, and finally, two pickup coils (3 mm long and 60 mm in between) are placed symmetrically so that two maxima appear when the domain wall passes through them. 3. Results and discussion
Corresponding author. Tel.: +421 55 62 211 28; fax: +421 55 62 221 24. E-mail address: [email protected] (R. Varga).
0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.08.056
The dependence of the domain wall velocity, v, as a function of the amplitude of the applied field, H, is shown
ARTICLE IN PRESS Y. Kostyk et al. / Physica B 403 (2008) 386–389
in Fig. 1. At high fields, the velocity is linearly proportional to the applied field H according to the theory of the domain wall motion in the viscous regime [9]: v ¼ SðH H 0 Þ,
(1)
where S is the domain wall mobility and H0 is the critical propagation field. However, the extrapolation into the lowfield regime leads to a negative critical propagation field. This has been also reported for microwires [7,10]. At low fields, the domain wall interaction with the local defects distributed along the sample should be taken into account. Generally, the critical field H0 consists of two contributions: long-range contribution Hl arising from the magnetostatic and magnetoelastic interaction of the domain wall with the stresses. On the other hand, the domain wall interacts during its propagation with the local defects giving rise to the short-range contribution to the critical field Hp, which corresponds to the pinning field of the domain wall on the defects distribution. Finally, the domain wall dynamics is described by Yang and Erskine [11] v ¼ S½H ðH 1 þ H P Þ.
(2)
As a result, the domain wall moves in small intermittent jumps at a velocity determined by the local distribution of the defects according to Eq. (2). Such a regime of the domain wall motion is called adiabatic. The domain wall moves slowly, having time to interact with the defects. The distribution of the pinning field Hp can be approximated by a gaussian function with width R [12]. According to the Random Field Ising model, the magnetization change DM during the small domain jump is given by the power law [12]: R Rc b DM , (3) Rc
where Rc is the critical distribution width, below which the small intermittent domain wall jumps do not appear, and b is the critical exponent. The average velocity of the domain wall finally scales as [13] v ¼ S 0 ðH H 00 Þb ,
where S is the so-called domain wall mobility parameter and H 00 is the critical field, below which the domain wall motion cannot be observed. Fitting the velocity dependence on the magnetic field given in Fig. 1 gives results in b ¼ 0.53, in reasonable agreement with theory expectation of 12 [12,13]. Measurements of the domain wall dynamics in the wide temperature range give us additional information about the process. Fig. 2 shows the evolution of the domain wall velocity v with the applied field H in the temperature range from 173 to 373 K. To check the validity of the power law in a wide temperature range, the linear dependence of log(v) as a function of the logðH H 00 Þ is shown. The power exponent b changes with temperature as shown in Fig. 3. Starting from the value of 0.25 at 77 K, it increases with temperature until it saturates at 200 K at the value 0.55. Such behavior can be explained in terms of the domain wall shape. It comes out from the Random Field Ising model [14], that for a planar wall, the power exponent b ¼ 1/2, which is reasonably close to our values obtained at high temperature. However, the power exponent b was calculated to be lower than 12 for the flexible domain wall [12,13]. The domain wall energy is proportional to the domain wall area. On the other hand, the interaction energy of the domain wall with the defects should be taken into account when the defects are present in the sample. The amorphous glass-coated microwires are metastable in nature because of their amorphous character, containing a lot of the defects in the form of the local fluctuation of
173 K 223 K 273 K 323 K 373 K
3.0
800
log (v)
v (m/s)
2.5 600
(4)
0
1000
v = S (H-H0)
387
2.0
400 v = S'(H-H'0)q
200
1.5
0 0
50
100 H (A/m)
150
Fig. 1. Domain wall velocity v as a function of a constant applied field H, measured at the temperature T ¼ 273 K. Full-lines corresponds to the fit according to Eqs. (1) and (4).
-1.0
-0.5
0.0
1.0 0.5 log (H-H'0)
1.5
2.0
2.5
Fig. 2. The linear dependence of the log(v) on the log(HH0) approves the validity of the power law given by Eq. (4) in the wide temperature range.
ARTICLE IN PRESS Y. Kostyk et al. / Physica B 403 (2008) 386–389
388
350 0.6
10
S* 300
S
S* (m/s)
250
0.4
6
200
S (m2/As)
8
0.5
4
150 0.3
2
100 0.2
200
100 100
200
300
400
300
400
T (K)
T (K) Fig. 3. The temperature dependence of power exponent b.
density (interstitials, vacancies). The interaction energy of the domain wall with the defects is strongly temperature dependent [15,16]. At high temperature, the domain wall interacts with the defects weakly. So, the domain wall prefers to take planar configuration in order to decrease its energy. However, at low temperature, the mobile defects loose their mobility, stabilizing the local anisotropy direction, which finally leads to the increase of the domain wall interaction energy with the defects. The presence of strong interacting pinning centers inside the wall favors a flexible wall to reduce its energy. As a consequence, b takes reduced value down to 0.25 that reasonably agrees with that expected from the Random Field Ising Model [12]. Such behavior has been already confirmed in glasscoated microwires by the measurements of the temperature dependence of the switching field [16] and domain wall damping [7,17]. Finally, the domain wall mobility parameter S0 , given in Eq. (4), has been considered. From one point of view, the dependence of the domain wall velocity on the applied magnetic field given in Fig. 1 can be treated in terms of the field dependence of the domain wall mobility S given in Eq. (1), keeping the power exponent equal to 1. On the other hand, considering Eq. (3) and the fact that the domain wall velocity v is given by the magnetization change DM with time t (v ¼ DM/t) [12,13], the mobility parameter S0 can be expressed as S0 ¼ Sn =H 0 b0 . The constant S and the domain wall mobility S exhibit the same temperature dependence (see Fig. 4). They differ just by a constant, from the dimension of the field (SHmaxS). The value of this field (given in Table 1 as Hmax) is of the order of the transient field at which the domain wall dynamics changes from adiabatic to viscous regime. Moreover, its temperature dependence corresponds to that of the switching field, already measured in CoMnSiB microwires [16], which points to the fact that both fields are governed by the same mechanism.
Fig. 4. Temperature dependencies of the domain wall mobility S and the constant S.
Table 1 Parameters of the fitting according to Eqs. (1) and (4) T (K)
S0
H0 (A/m)
q
S (m2/A s)
Hmax (A/m)
77 98 123 148 173 198 223 248 273 298 323 348 373
87.7 56.4 53.0 43.2 50.9 47.5 44.7 52.8 68.1 118.0 89.5 80.3 109.3
10.1 4.6 6.8 7.3 7.4 7.2 4.6 5.3 4.4 7.1 7.2 7.5 7.3
0.25 0.38 0.44 0.53 0.51 0.56 0.61 0.58 0.53 0.41 0.51 0.58 0.54
2.07 1.72 2.89 2.47 2.68 3.91 3.26 3.79 4.56 5.73 4.92 6.15 9.81
75.6 58.7 42.6 50.3 53.2 36.6 34.8 36.7 32.6 46.1 50.1 41.5 32.5
Finally, the mobility parameter S0 can be expressed as S0 ¼
SH max H 0 b0
.
(5)
However, all parameters given in Eq. (5) are field independent. So, the mobility parameter S0 should be field independent, too. 4. Conclusion The domain wall dynamics in the adiabatic regime has been measured. It is shown that the domain wall velocity v in the low-field regime can be described as vðHÞ ¼ ðSH max =H 0 b0 ÞðH H 00 Þb . The temperature dependence of the power exponent b is treated in terms of the domain wall shape from the planar one at high temperatures to the flexible one at low temperatures as a result of the strong domain wall interaction with the defects distributed along the sample. The domain wall mobility parameter is shown to be field independent.
ARTICLE IN PRESS Y. Kostyk et al. / Physica B 403 (2008) 386–389
Acknowledgments Y. Kostyk acknowledges the support from The Orange Konto. This work was supported by scientific Grants VEGA 1/3035/06 and APVT-20-007804.
[9] [10] [11] [12]
[13]
References [1] [2] [3] [4] [5] [6] [7]
D. Atkinson, et al., Nat. Mater. 2 (2003) 85. G.S.D. Beach, et al., Nat. Mater. 4 (2005) 741. D.A. Allwood, et al., Science 309 (2005) 1688. M. Vazquez, Physica B 299 (2001) 302. H. Chiriac, T.A. Ovari, G. Pop, Phys. Rev. B 52 (1995) 10104. R. Varga, et al., Appl. Phys. Lett. 83 (2003) 2620. R. Varga, K.L. Garcı´ a, M. Va´zquez, P. Vojtanik, Phys. Rev. Lett. 94 (2005) 017201. [8] K.J. Sixtus, L. Tonks, Phys. Rev. 42 (1932) 419.
[14] [15]
[16] [17]
389
H.J. Williams, W. Shockley, C. Kittel, Phys. Rev. 80 (1950) 1090. R. Varga, et al., Phys. Rev. B 74 (2006) 212405. S. Yang, J.L. Erskine, Phys. Rev. B 72 (2005) 064433. J.P. Sethna, K.A. Dahmen, O. Perkovic, Random field ising models of hysteresis, in: G. Bertotti, I. Mayergoyz (Eds.), The Science of Hysteresis, Academic Press, New York, 2006, p. 107. G. Durin, S. Zapperi, Barkhausen effect, in: G. Bertotti, I. Mayergoyz (Eds.), The Science of Hysteresis, Academic Press, New York, 2006, p. 181. S. Zapperi, G. Durin, Comput. Mater. Sci. 20 (2001) 436. H. Kronmuller, M. Fanhle, in: Micromagnetism and the Microstructure of Ferromagnetic Solids, Cambridge University Press, Cambridge, 2003, p. 274. R. Varga, K.L. Garcı´ a, M. Va´zquez, A. Zhukov, P. Vojtanik, Phys. Rev. B 70 (2004) 024402. R. Varga, A. Zhukov, N. Usov, J.M. Blanco, J. Gonzalez, V. Zhukova, P. Vojtanik, J. Magn. Magn. Mater. 316 (2007) 337.