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Explicit solutions to phenomenological models of magnetization reversal of thin ferromagnetic films in the presence of a sawtooth magnetic field
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Journal of Magnetism and Magnetic Materials 303 (2006) 49–53 www.elsevier.com/locate/jmmm
Explicit solutions to phenomenological models of magnetization reversal of thin ferromagnetic films in the presence of a sawtooth magnetic field Eshel Faraggi,1 Physics Department, Ilya Prigogine Center for Studies in Statistical Mechanics and Complex Systems, The University of Texas at Austin, Austin, Texas 78712, USA Received 2 March 2005; received in revised form 9 October 2005 Available online 21 November 2005
Abstract Explicit solutions are derived for several phenomenological models of magnetization reversal in thin ferromagnetic films driven by a sawtooth magnetic field. For a domain wall velocity that is linear in the magnetic field, it is found that the dynamic coercive field follows a square-root power-law in the slope of the magnetic field, shifted by the depinning field. For a more general domain wall velocity different power-law exponents are found, yet the overall form for the scaling of the area of the hysteresis loop remains a power-law shifted by the depinning field. This shifted power-law could be interpreted to be a crossover between adiabatic and dynamic regimes. r 2005 Elsevier B.V. All rights reserved. PACS: 75.40.Gb; 75.60.d; 75.70.Ak Keywords: 2D ferromagnetism; Coercive field; Hysteresis; Dynamic scaling; Dissipation
1. Introduction The hysteresis curve for thin ferromagnetic films has attracted some attention in the literature recently, with particular attention paid to the dependence of the area of the hysteresis loop on the driving field [1–29]. In what follows an explicit solution to a class of phenomenological models of magnetization reversal will be presented. The first two phenomenological models studied here were analyzed numerically with a sinusoidal magnetic field by Ruiz-Feal et al. [23]. As explained in an earlier paper [29], for a sawtooth driving magnetic field the mature [30], statistically averaged, hysteresis loop depends only on a single driving parameter, H 0 O, where H 0 is the amplitude of the field and O is its frequency. The reason for this dependence on a Tel.:+1 3053486661; fax:+1 3053486700.
E-mail address: [email protected]. Present address: Physics Department, Florida International University, Miami, Florida 33199, USA. 1
0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.10.228
single parameter is that after maturation the magnetization curve is a single valued function of the magnetic field, and for the conditions outlined above two sawtooth fields with equal slopes will have identical profiles between the maturation points. Henceforth, a sawtooth magnetic field will be assumed. This study concerns the dissipation in the hysteresis cycle, and its dependence on system parameters, most specifically the temporal derivative or slope of the driving field. As was reported in earlier studies this dissipation was either modeled as a power-law behavior over a limited range [1–16,19,24–27], or as ½lnðH 0 OÞ1 by Sides et al. [17,18]. The main controversy regarding the power-law behavior is summarized by Zhong et al. [15] and Erskine et al. [16]. This study and a related one [28] show that this controversy can be resolved by considering the region in which the scaling of hysteresis is calculated, each region with its own apparent scaling exponent. In the case of slowly varying driving fields the adiabatic limit, accounted for by the depinning field, would produce an apparent scaling exponent approaching zero in the
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E. Faraggi / Journal of Magnetism and Magnetic Materials 303 (2006) 49–53
limit of small slopes. However, for larger slopes of the driving field dissipation due to dynamic lagging of the domain walls becomes dominant over adiabatic losses and a power-law behavior with a finite scaling exponent would appear. Hence, this shifted power-law could be interpreted to be a crossover between adiabatic and dynamic regimes. 2. Models For a statistically averaged hysteresis curve one branch of the magnetization can be obtained from the other by using the reversal symmetries h+ h and m+ m, with h and m the magnetic field and magnetization, respectively. Hence, only the magnetization curve for the transition from the negative state to the positive state will be considered. The down branch of magnetization, i.e., for the transition from the positive state to the negative state can be expressed exactly as md ðhÞ ¼ mu ðhÞ; with md the down branch of magnetization, and mu the up branch of magnetization. Henceforth the symbol m will be used for the magnetization normalized by the saturation magnetization. The driving magnetic field for the up transition can be expressed as hðtÞ ¼ H 0 þ h0 t.
(1)
The model assumes that the ferromagnetic sample is a two-dimensional collection of domains of two orientations, with a zero-width domain wall separating the two states. The perpendicular velocity of the domain wall is designated by v, and is assumed to have the form ( 0; hohdp ; vðhÞ ¼ (2) mðh hdp Þ; hXhdp ; where m is the domain wall mobility, and hdp is the depinning field. Initially, the sample is considered to be in the negative state, with a density r of reversed domains.
found that 8 0; > > > < 2mpffiffiffi r ðh hdp Þ dh; dm ¼ > h0 > > : 0;
hohdp ; hdp phphs ;
(4)
h4hs :
One can obtain hs as the field value where saturation is obtained. Considering that here ms ¼ 1, and Z hs pffiffiffi Z ms pffiffiffi 2m r m r dm ¼ ðh hdp Þ dh ¼ ðhs hdp Þ2 , 2¼ h0 h0 ms hdp (5) it is found that sffiffiffiffiffiffiffiffiffiffi 2h0 hs ¼ hdp þ pffiffiffi. m r The magnetization curve for negative saturated state to the obtained from Eq. (4) is 8 ms ; > > > pffiffiffi < m r ðh hdp Þ2 ; mðhÞ ¼ ms þ > h0 > > :m ; s
(6) the transition from the positive saturated state hohdp ; hdp phphs ;
(7)
h4hs :
The magnetization curve for the reverse transition can be obtained by using the reversal symmetry. The dynamic coercive field is obtained from the condition mðhc Þ ¼ 0, sffiffiffiffiffiffiffiffiffiffi h0 (8) hc ¼ hdp þ pffiffiffi. m r
2.2. Two-dimensional model 2.1. One-dimensional model In this case the domain wall is assumed to be a line sweeping through the film, hence this case can be reduced to a one-dimensional problem. The average area that a domain wall would have to cover to reverse the sample is 1=r, hence for a square domain the domain wall would pffiffiffi have to cover an average distance of rs ¼ 1= r. If we designate by dr ¼ v dt a differential displacement perpendicular to the domain wall, the change in the normalized magnetization can be expressed as pffiffiffi dm ¼ 2 r dr. (3) In the integration of Eq. (3) one should consider that once the magnetization saturates, no further change is possible for the magnetization as the field increases. Hence, there is a field value hs such that for h4hs , dm ¼ 0. Since dh=dt ¼ h0 , with h0 the constant slope of the driving magnetic field, dr ¼ ðv=h0 Þ dh. Thus, for this model it is
In this case the domain walls are assumed to be circles. For a domain wall of radius r the differential area element covered by a wall displacement dr is 2pr dr. Hence, the change in the normalized magnetization dm ¼ 4prr dr.
(9)
Writing as before dr ¼ v dt ¼ ðv=h0 Þ dh, it is found for this model 8 0; hohdp ; > > > < 2prm2 ðh hdp Þ3 dh; hdp phphs ; (10) dm ¼ 2 > h > > 0 : 0; h4hs ; where mðhs Þ ¼ ms as before, i.e., sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffi 2h0 ms . hs ¼ hdp þ m pr
(11)
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Integrating Eq. (10) the normalized magnetization is found to be 8 ms ; hohdp ; > > > < 2 prm ðh hdp Þ4 ; hdp phphs ; (12) mðhÞ ¼ ms þ 2 > 2h > 0 > : ms ; h4hs :
With hs obtained from the condition rs ¼ rðhs Þ, sffiffiffiffiffiffiffiffiffiffi 2h0 hs ¼ hdp þ pffiffiffi. m r
The magnetization curve for the reverse transition can be obtained from the symmetry conditions. As before the dynamic coercively field is found from the condition of zero magnetization and is given by ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u sffiffiffiffiffiffi uh 2 0 hc ¼ hdp þ t . (13) m pr
Eq. (19) has a single solution for rc 2 ½0; rs , rs rc ¼ . (20) 2 Hence, the condition for the dynamic coercive field hc is found to be Z hc h0 r s ¼ vðhÞ dh. (21) 2 H 0
2.3. Saturation model
2.4. Velocity dependence
The previous models consider the average area covered by the domains during magnetization reversal and assume an artificial cutoff in the change in magnetization once reversal is complete. One can relax this artificial constraint by considering that the final part of magnetization reversal is obtained by coalescing domains. This type of behavior is different than the initial growth stage, where reversed domains expand in regions of antialligned spins. As a first approximation, these two types of evolution for the initial and final stages can be modeled by replacing pffiffiffi 2pr dr in Eq. (9) with krðrs rÞ dr, where rs ¼ 1= r is the distance between nucleating domains and k is a proportionality constant,
The dependence of the domain wall velocity on the magnetic field has been studied by several authors [20–22]. In this subsection a general power-law is assumed for the domain wall velocity,
dm ¼ 2krrðrs rÞ dr.
(14)
The constant k can be obtained by considering the saturation condition. The maximum allowed change in magnetization, Dm ¼ 2 is obtained integrating over r from 0 to rs , Z rs Z 1 r3 (15) dm ¼ 2krrðrs rÞ dr ¼ 2kr s . 2¼ 6 0 1 Using the fact that r ¼ 1=r2s the result is k ¼ 6=rs , i.e., kr ¼ 6r3=2 . The magnetization is obtained integrating Eq. (14), mðrÞ ¼ 1 þ 2r
3=2 2
r ð3rs 2rÞ,
(16)
where rðhÞ can be obtained from the velocity of the domain wall, Eq. (2), using the general condition for a sawtooth driving magnetic field, dh ¼ h0 dt, i.e., dr ¼ v dt ¼ vðhÞ dh=h0 . Integrating this last expression we obtain 8 0; hohdp ; > > < m 2 ðh hdp Þ ; hdp phphs ; (17) rðhÞ ¼ 2h0 > > : h4hs : rs ;
(18)
The dynamic coercive radius rc is defined by the condition mðrc Þ ¼ 0 obtaining 0 ¼ 1 þ 2r3=2 r2c ð3rs 2rc Þ.
vðhÞ ¼ mq ðh hdp Þq ,
(19)
(22)
with hdp phphs . For a given qX0 the resulting equation for the dynamic coercive field obtained from Eq. (21) is mq h0 rs ðhc hdp Þqþ1 ¼ . qþ1 2 Hence, the dynamic coercive field is found to be sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðq þ 1Þrs h0 qþ1 hc ¼ hdp þ . 2mq
(23)
(24)
This last expression shows the dependence between the domain wall velocity and the resulting functional form of the dynamic coercive field. Since the area of the hysteresis loop can be approximated as A 4ms hc , it can be expected that Aðh0 Þ has a similar functional form to the one of hc ðh0 Þ. 3. Analysis Representative hysteresis plots for the models described earlier are given in Fig. 1. In all hysteresis loops the transition from the positive saturation state to the negative saturation state was obtained from the symmetry of the system with respect to reversal of the magnetization and the magnetic field. Fig. 1(a) is for lower r and m than Fig. 1(b), resulting in a higher coercive field for the models. Typically it is found that the saturation model with linear domain wall velocity (model (3)) behaves in a most similar way to the one-dimensional model (model (1)), while the two-dimensional model (model (2)) undergoes a faster transition. Also, the saturation model with quadratic
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1
(3) (4)
(2)
1000 (1) (1),(3) hc* [Oe]
m
0.5
0
(2) 100 (4)
-0.5
-1 -150
-100
-50
(a)
0 h [Oe]
50
100
10 10
100
1000 h0 [Oe/sec]
(a)
10000
100000
100
1
(3)
(1)
(2) (4)
hc* [Oe]
0.5
m
150
0
(1),(3) (2) (4)
-0.5
-1 -30 (b)
10 10 -20
-10
0 h [Oe]
10
20
30
Fig. 1. Hysteresis loops obtained from the models described in the text: (1) one-dimensional model, (2) two-dimensional model, (3) saturation model with linear domain wall velocity, and (4) saturation model with quadratic domain wall velocity. For both plots h0 ¼ 6283 Oe/s, and hdp ¼ 14 Oe. Part (a) was obtained using H 0 ¼ 150 Oe, r ¼ 104 m2 , and m ¼ 0:01 (m/s)/Oe. Part (b) was obtained using H 0 ¼ 25 Oe, r ¼ 105 m2 , and m ¼ 0:9 (m/s)/Oe.
domain wall velocity (model (4)) is studied. It is found that this model undergoes the most rapid transition. In the models presented it is found that the assumption for the dependence of the velocity of the domain walls on the driving field will determine the dependence of the dynamic coercive field on the driving parameter h0 . The general form for hc in these phenomenological models is a power-law in h0 shifted by the depinning field. Log–log plots of the dynamic coercive field are given in Fig. 2 for the two cases presented in Fig. 1. As is evident from these plots, the power-law behavior of hc obtained exactly, appears skewed due to the presence of hdp . Note that the one-dimensional model and the saturation model with a linear velocity produce the same expression for hc . Finally, it should be noted that for the H models presented the area of the hysteresis loop, A ¼ h dm can also be
100
1000 h0 [Oe/sec]
(b)
10000
100000
Fig. 2. Dynamic coercive field obtained from the models described in the text, same labels as in Fig. 1. For both plots hdp ¼ 14 Oe. Part (a) was obtained using r ¼ 104 m2 , and m ¼ 0:01 (m/s)/Oe. Part (b) was obtained using r ¼ 105 m2 , and m ¼ 0:9 (m/s)/Oe.
exactly calculated. As an example the calculation is performed for the two-dimensional model. From the mentioned symmetry of the two branches R H of the hysteresis loop, A can also be written as A ¼ 2 H0 0 mðhÞ dh. Using Eq. (12) this can be expressed as A ¼ 2
Z
Z
hdp
H0
mðhÞ dh þ
ms dh þ H 0
Z
hs hdp
ms dh.
(25)
hs
Integrating Eq. (25) leads to
A prm2 ¼ 2ms hs þ ðhs hdp Þ5 . 2 10h20
Using Eq. (11) in Eq. (26) it is found that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffi! 4ms 2h0 ms 3hdp þ 4 A¼ . 5 m pr
(26)
(27)
As expected, the area of the hysteresis loop has the same power-law dependence on h0 as the dynamic coercive field.
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4. Conclusions
References
Explicit solutions have been derived for several phenomenological models of magnetization reversal in thin ferromagnetic films driven by a sawtooth magnetic field. As mentioned, for the cases presented here, the hysteresis loop depends only on the slope of the driving magnetic field. For a linear domain wall velocity it was found that the dynamic coercive field, defined for the hysteresis phenomenon, follows a square-root power-law in the slope of the sawtooth magnetic field shifted by the depinning field. For a more general assumption for the domain wall velocity different power-law exponents are found. This study concerns the dissipation in the hysteresis cycle, and its dependence on system parameters. It was shown here and in an accompanying experimental study [28] that different rates of change of the magnetic field would produce different apparent scaling exponents due to the shifted power-law form of Eq. (24). For slowly varying driving fields the adiabatic limit, accounted for by the depinning field, would produce an apparent scaling exponent approaching zero in the limit of small slopes. For larger slopes of the driving field dissipation due to dynamic lagging of the domain walls becomes dominant over adiabatic losses and a power-law behavior with a scaling exponent equal to 1=1 þ q would appear. Hence, this shifted power-law could be interpreted to be a crossover between adiabatic and dynamic regimes.
[1] M. Rao, H.R. Krishnamurthy, R. Pandit, Phys. Rev. B 42 (1) (1990) 856. [2] M. Rao, R. Pandit, Phys. Rev. B 43 (4) (1991) 3373. [3] W.S. Lo, R.A. Pelcovits, Phys. Rev. A 42 (12) (1990) 7471. [4] S. Sengupta, Y. Marathe, S. Puri, Phys. Rev. B 45 (14) (1992) 7828. [5] F. Zhong, J.X. Zhang, G.G. Siu, J. Phys.: Condens. Matter 6 (38) (1994) 7785. [6] F. Zhong, J.X. Zhang, Phys. Rev. Lett. 75 (10) (1995) 2027. [7] D. Dhar, P.B. Thomas, J. Phys. A 25 (19) (1992) 4967. [8] M. Acharyya, B.K. Chakrabarti, Phys. Rev. B 52 (9) (1995) 6550. [9] C.N. Luse, A. Zangwill, Phys. Rev. E 50 (1) (1994) 224. [10] A.M. Somoza, R.C. Desai, Phys. Rev. Lett. 70 (21) (1993) 3279. [11] Q. Jiang, H.-N. Yang, G.-C. Wang, Phys. Rev. B 52 (20) (1995) 14911. [12] Y.-L. He, G.-C. Wang, Phys. Rev. Lett. 70 (15) (1993) 2336. [13] J.-S. Suen, J.L. Erskine, Phys. Rev. Lett. 78 (18) (1997) 3567. [14] J.-S. Suen, M.H. Lee, G. Teeter, J.L. Erskine, Phys. Rev. B 59 (6) (1999) 4249. [15] F. Zhong, J. Don, D.Y. Xing, Phys. Rev. Lett. 80 (5) (1998) 1118. [16] J.L. Erskine, J.-S. Suen, Phys. Rev. Lett. 80 (5) (1998) 1119. [17] S.W. Sides, P.A. Rikvold, M.A. Novotny, J. Appl. Phys. 83 (11) (1998) 6494. [18] S.W. Sides, P.A. Rikvold, M.A. Novotny, Phys. Rev. E 59 (3) (1999) 2710. [19] G.P. Zheng, J.X. Zhang, Physica A 264 (4) (1999) 515. [20] R.P. Cowburn, J. Ferre, S.J. Gray, J.A.C. Bland, Appl. Phys. Lett. 74 (7) (1999) 1018. [21] I.F. Lyuksyutov, T. Nattermann, V. Pokrovsky, Phys. Rev. B 59 (6) (1999) 4260. [22] T. Nattermann, V. Pokrovsky, V.M. Vinokur, Phys. Rev. Lett. 87 (19) (2001) 197005. [23] I. Ruiz-Feal, T.A. Moore, L. Lopez-Diaz, J.A.C. Bland, Phys. Rev. B 65 (5) (2002) 054409. [24] B. Raquet, R. Mamy, J.C. Ousset, Phys. Rev. B 54 (6) (1996) 4128. [25] W.Y. Lee, B.-Ch. Choi, Y.B. Xu, J.A.C. Bland, Phys. Rev. B 60 (14) (1999) 10216. [26] T.A. Moore, J. Rothman, Y.B. Xu, J.A.C. Bland, J. Appl. Phys. 89 (11) (2001) 7018. [27] M.P. Sharrock, J.T. McKinney, IEEE Trans. Magn. MAG-17 (6) (1981) 3020. [28] C. Nistor, E. Faraggi, J.L. Erskine, Phys. Rev. B 72 (1) (2005) 014404. [29] E. Faraggi, L.E. Reichl, D.T. Robb, Magnetic behavior of partially filled finite Ising surfaces, Phys. Rev. B (2002) submitted for publication. [30] A mature hysteresis loop will have its two branches of magnetization coincide before the maximum (or minimum) driving magnetic field value.
Acknowledgements The author would like to thank James Erskine for pointing out the original reference to the Ruiz-Feal et al. paper, James Erskine and Corneliu Nistor for comparative experimental work, and Linda Reichl, James Erskine, Dan Robb, and Corneliu Nistor for helpful discussions. The author would also like to thank Linda Reichl for partial support through the Welch Foundation Grant no. F-1051 and the Engineering Research Program of the Office of Basic Energy Sciences at the US Department of Energy, Grant no. DE-FG03-94ER14465. Finally the author would like to thank Natali Teszler for general support.
Journal of Magnetism and Magnetic Materials 303 (2006) 49–53 www.elsevier.com/locate/jmmm
Explicit solutions to phenomenological models of magnetization reversal of thin ferromagnetic films in the presence of a sawtooth magnetic field Eshel Faraggi,1 Physics Department, Ilya Prigogine Center for Studies in Statistical Mechanics and Complex Systems, The University of Texas at Austin, Austin, Texas 78712, USA Received 2 March 2005; received in revised form 9 October 2005 Available online 21 November 2005
Abstract Explicit solutions are derived for several phenomenological models of magnetization reversal in thin ferromagnetic films driven by a sawtooth magnetic field. For a domain wall velocity that is linear in the magnetic field, it is found that the dynamic coercive field follows a square-root power-law in the slope of the magnetic field, shifted by the depinning field. For a more general domain wall velocity different power-law exponents are found, yet the overall form for the scaling of the area of the hysteresis loop remains a power-law shifted by the depinning field. This shifted power-law could be interpreted to be a crossover between adiabatic and dynamic regimes. r 2005 Elsevier B.V. All rights reserved. PACS: 75.40.Gb; 75.60.d; 75.70.Ak Keywords: 2D ferromagnetism; Coercive field; Hysteresis; Dynamic scaling; Dissipation
1. Introduction The hysteresis curve for thin ferromagnetic films has attracted some attention in the literature recently, with particular attention paid to the dependence of the area of the hysteresis loop on the driving field [1–29]. In what follows an explicit solution to a class of phenomenological models of magnetization reversal will be presented. The first two phenomenological models studied here were analyzed numerically with a sinusoidal magnetic field by Ruiz-Feal et al. [23]. As explained in an earlier paper [29], for a sawtooth driving magnetic field the mature [30], statistically averaged, hysteresis loop depends only on a single driving parameter, H 0 O, where H 0 is the amplitude of the field and O is its frequency. The reason for this dependence on a Tel.:+1 3053486661; fax:+1 3053486700.
E-mail address: [email protected]. Present address: Physics Department, Florida International University, Miami, Florida 33199, USA. 1
0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.10.228
single parameter is that after maturation the magnetization curve is a single valued function of the magnetic field, and for the conditions outlined above two sawtooth fields with equal slopes will have identical profiles between the maturation points. Henceforth, a sawtooth magnetic field will be assumed. This study concerns the dissipation in the hysteresis cycle, and its dependence on system parameters, most specifically the temporal derivative or slope of the driving field. As was reported in earlier studies this dissipation was either modeled as a power-law behavior over a limited range [1–16,19,24–27], or as ½lnðH 0 OÞ1 by Sides et al. [17,18]. The main controversy regarding the power-law behavior is summarized by Zhong et al. [15] and Erskine et al. [16]. This study and a related one [28] show that this controversy can be resolved by considering the region in which the scaling of hysteresis is calculated, each region with its own apparent scaling exponent. In the case of slowly varying driving fields the adiabatic limit, accounted for by the depinning field, would produce an apparent scaling exponent approaching zero in the
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E. Faraggi / Journal of Magnetism and Magnetic Materials 303 (2006) 49–53
limit of small slopes. However, for larger slopes of the driving field dissipation due to dynamic lagging of the domain walls becomes dominant over adiabatic losses and a power-law behavior with a finite scaling exponent would appear. Hence, this shifted power-law could be interpreted to be a crossover between adiabatic and dynamic regimes. 2. Models For a statistically averaged hysteresis curve one branch of the magnetization can be obtained from the other by using the reversal symmetries h+ h and m+ m, with h and m the magnetic field and magnetization, respectively. Hence, only the magnetization curve for the transition from the negative state to the positive state will be considered. The down branch of magnetization, i.e., for the transition from the positive state to the negative state can be expressed exactly as md ðhÞ ¼ mu ðhÞ; with md the down branch of magnetization, and mu the up branch of magnetization. Henceforth the symbol m will be used for the magnetization normalized by the saturation magnetization. The driving magnetic field for the up transition can be expressed as hðtÞ ¼ H 0 þ h0 t.
(1)
The model assumes that the ferromagnetic sample is a two-dimensional collection of domains of two orientations, with a zero-width domain wall separating the two states. The perpendicular velocity of the domain wall is designated by v, and is assumed to have the form ( 0; hohdp ; vðhÞ ¼ (2) mðh hdp Þ; hXhdp ; where m is the domain wall mobility, and hdp is the depinning field. Initially, the sample is considered to be in the negative state, with a density r of reversed domains.
found that 8 0; > > > < 2mpffiffiffi r ðh hdp Þ dh; dm ¼ > h0 > > : 0;
hohdp ; hdp phphs ;
(4)
h4hs :
One can obtain hs as the field value where saturation is obtained. Considering that here ms ¼ 1, and Z hs pffiffiffi Z ms pffiffiffi 2m r m r dm ¼ ðh hdp Þ dh ¼ ðhs hdp Þ2 , 2¼ h0 h0 ms hdp (5) it is found that sffiffiffiffiffiffiffiffiffiffi 2h0 hs ¼ hdp þ pffiffiffi. m r The magnetization curve for negative saturated state to the obtained from Eq. (4) is 8 ms ; > > > pffiffiffi < m r ðh hdp Þ2 ; mðhÞ ¼ ms þ > h0 > > :m ; s
(6) the transition from the positive saturated state hohdp ; hdp phphs ;
(7)
h4hs :
The magnetization curve for the reverse transition can be obtained by using the reversal symmetry. The dynamic coercive field is obtained from the condition mðhc Þ ¼ 0, sffiffiffiffiffiffiffiffiffiffi h0 (8) hc ¼ hdp þ pffiffiffi. m r
2.2. Two-dimensional model 2.1. One-dimensional model In this case the domain wall is assumed to be a line sweeping through the film, hence this case can be reduced to a one-dimensional problem. The average area that a domain wall would have to cover to reverse the sample is 1=r, hence for a square domain the domain wall would pffiffiffi have to cover an average distance of rs ¼ 1= r. If we designate by dr ¼ v dt a differential displacement perpendicular to the domain wall, the change in the normalized magnetization can be expressed as pffiffiffi dm ¼ 2 r dr. (3) In the integration of Eq. (3) one should consider that once the magnetization saturates, no further change is possible for the magnetization as the field increases. Hence, there is a field value hs such that for h4hs , dm ¼ 0. Since dh=dt ¼ h0 , with h0 the constant slope of the driving magnetic field, dr ¼ ðv=h0 Þ dh. Thus, for this model it is
In this case the domain walls are assumed to be circles. For a domain wall of radius r the differential area element covered by a wall displacement dr is 2pr dr. Hence, the change in the normalized magnetization dm ¼ 4prr dr.
(9)
Writing as before dr ¼ v dt ¼ ðv=h0 Þ dh, it is found for this model 8 0; hohdp ; > > > < 2prm2 ðh hdp Þ3 dh; hdp phphs ; (10) dm ¼ 2 > h > > 0 : 0; h4hs ; where mðhs Þ ¼ ms as before, i.e., sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffi 2h0 ms . hs ¼ hdp þ m pr
(11)
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Integrating Eq. (10) the normalized magnetization is found to be 8 ms ; hohdp ; > > > < 2 prm ðh hdp Þ4 ; hdp phphs ; (12) mðhÞ ¼ ms þ 2 > 2h > 0 > : ms ; h4hs :
With hs obtained from the condition rs ¼ rðhs Þ, sffiffiffiffiffiffiffiffiffiffi 2h0 hs ¼ hdp þ pffiffiffi. m r
The magnetization curve for the reverse transition can be obtained from the symmetry conditions. As before the dynamic coercively field is found from the condition of zero magnetization and is given by ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u sffiffiffiffiffiffi uh 2 0 hc ¼ hdp þ t . (13) m pr
Eq. (19) has a single solution for rc 2 ½0; rs , rs rc ¼ . (20) 2 Hence, the condition for the dynamic coercive field hc is found to be Z hc h0 r s ¼ vðhÞ dh. (21) 2 H 0
2.3. Saturation model
2.4. Velocity dependence
The previous models consider the average area covered by the domains during magnetization reversal and assume an artificial cutoff in the change in magnetization once reversal is complete. One can relax this artificial constraint by considering that the final part of magnetization reversal is obtained by coalescing domains. This type of behavior is different than the initial growth stage, where reversed domains expand in regions of antialligned spins. As a first approximation, these two types of evolution for the initial and final stages can be modeled by replacing pffiffiffi 2pr dr in Eq. (9) with krðrs rÞ dr, where rs ¼ 1= r is the distance between nucleating domains and k is a proportionality constant,
The dependence of the domain wall velocity on the magnetic field has been studied by several authors [20–22]. In this subsection a general power-law is assumed for the domain wall velocity,
dm ¼ 2krrðrs rÞ dr.
(14)
The constant k can be obtained by considering the saturation condition. The maximum allowed change in magnetization, Dm ¼ 2 is obtained integrating over r from 0 to rs , Z rs Z 1 r3 (15) dm ¼ 2krrðrs rÞ dr ¼ 2kr s . 2¼ 6 0 1 Using the fact that r ¼ 1=r2s the result is k ¼ 6=rs , i.e., kr ¼ 6r3=2 . The magnetization is obtained integrating Eq. (14), mðrÞ ¼ 1 þ 2r
3=2 2
r ð3rs 2rÞ,
(16)
where rðhÞ can be obtained from the velocity of the domain wall, Eq. (2), using the general condition for a sawtooth driving magnetic field, dh ¼ h0 dt, i.e., dr ¼ v dt ¼ vðhÞ dh=h0 . Integrating this last expression we obtain 8 0; hohdp ; > > < m 2 ðh hdp Þ ; hdp phphs ; (17) rðhÞ ¼ 2h0 > > : h4hs : rs ;
(18)
The dynamic coercive radius rc is defined by the condition mðrc Þ ¼ 0 obtaining 0 ¼ 1 þ 2r3=2 r2c ð3rs 2rc Þ.
vðhÞ ¼ mq ðh hdp Þq ,
(19)
(22)
with hdp phphs . For a given qX0 the resulting equation for the dynamic coercive field obtained from Eq. (21) is mq h0 rs ðhc hdp Þqþ1 ¼ . qþ1 2 Hence, the dynamic coercive field is found to be sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðq þ 1Þrs h0 qþ1 hc ¼ hdp þ . 2mq
(23)
(24)
This last expression shows the dependence between the domain wall velocity and the resulting functional form of the dynamic coercive field. Since the area of the hysteresis loop can be approximated as A 4ms hc , it can be expected that Aðh0 Þ has a similar functional form to the one of hc ðh0 Þ. 3. Analysis Representative hysteresis plots for the models described earlier are given in Fig. 1. In all hysteresis loops the transition from the positive saturation state to the negative saturation state was obtained from the symmetry of the system with respect to reversal of the magnetization and the magnetic field. Fig. 1(a) is for lower r and m than Fig. 1(b), resulting in a higher coercive field for the models. Typically it is found that the saturation model with linear domain wall velocity (model (3)) behaves in a most similar way to the one-dimensional model (model (1)), while the two-dimensional model (model (2)) undergoes a faster transition. Also, the saturation model with quadratic
ARTICLE IN PRESS E. Faraggi / Journal of Magnetism and Magnetic Materials 303 (2006) 49–53
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1
(3) (4)
(2)
1000 (1) (1),(3) hc* [Oe]
m
0.5
0
(2) 100 (4)
-0.5
-1 -150
-100
-50
(a)
0 h [Oe]
50
100
10 10
100
1000 h0 [Oe/sec]
(a)
10000
100000
100
1
(3)
(1)
(2) (4)
hc* [Oe]
0.5
m
150
0
(1),(3) (2) (4)
-0.5
-1 -30 (b)
10 10 -20
-10
0 h [Oe]
10
20
30
Fig. 1. Hysteresis loops obtained from the models described in the text: (1) one-dimensional model, (2) two-dimensional model, (3) saturation model with linear domain wall velocity, and (4) saturation model with quadratic domain wall velocity. For both plots h0 ¼ 6283 Oe/s, and hdp ¼ 14 Oe. Part (a) was obtained using H 0 ¼ 150 Oe, r ¼ 104 m2 , and m ¼ 0:01 (m/s)/Oe. Part (b) was obtained using H 0 ¼ 25 Oe, r ¼ 105 m2 , and m ¼ 0:9 (m/s)/Oe.
domain wall velocity (model (4)) is studied. It is found that this model undergoes the most rapid transition. In the models presented it is found that the assumption for the dependence of the velocity of the domain walls on the driving field will determine the dependence of the dynamic coercive field on the driving parameter h0 . The general form for hc in these phenomenological models is a power-law in h0 shifted by the depinning field. Log–log plots of the dynamic coercive field are given in Fig. 2 for the two cases presented in Fig. 1. As is evident from these plots, the power-law behavior of hc obtained exactly, appears skewed due to the presence of hdp . Note that the one-dimensional model and the saturation model with a linear velocity produce the same expression for hc . Finally, it should be noted that for the H models presented the area of the hysteresis loop, A ¼ h dm can also be
100
1000 h0 [Oe/sec]
(b)
10000
100000
Fig. 2. Dynamic coercive field obtained from the models described in the text, same labels as in Fig. 1. For both plots hdp ¼ 14 Oe. Part (a) was obtained using r ¼ 104 m2 , and m ¼ 0:01 (m/s)/Oe. Part (b) was obtained using r ¼ 105 m2 , and m ¼ 0:9 (m/s)/Oe.
exactly calculated. As an example the calculation is performed for the two-dimensional model. From the mentioned symmetry of the two branches R H of the hysteresis loop, A can also be written as A ¼ 2 H0 0 mðhÞ dh. Using Eq. (12) this can be expressed as A ¼ 2
Z
Z
hdp
H0
mðhÞ dh þ
ms dh þ H 0
Z
hs hdp
ms dh.
(25)
hs
Integrating Eq. (25) leads to
A prm2 ¼ 2ms hs þ ðhs hdp Þ5 . 2 10h20
Using Eq. (11) in Eq. (26) it is found that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffi! 4ms 2h0 ms 3hdp þ 4 A¼ . 5 m pr
(26)
(27)
As expected, the area of the hysteresis loop has the same power-law dependence on h0 as the dynamic coercive field.
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4. Conclusions
References
Explicit solutions have been derived for several phenomenological models of magnetization reversal in thin ferromagnetic films driven by a sawtooth magnetic field. As mentioned, for the cases presented here, the hysteresis loop depends only on the slope of the driving magnetic field. For a linear domain wall velocity it was found that the dynamic coercive field, defined for the hysteresis phenomenon, follows a square-root power-law in the slope of the sawtooth magnetic field shifted by the depinning field. For a more general assumption for the domain wall velocity different power-law exponents are found. This study concerns the dissipation in the hysteresis cycle, and its dependence on system parameters. It was shown here and in an accompanying experimental study [28] that different rates of change of the magnetic field would produce different apparent scaling exponents due to the shifted power-law form of Eq. (24). For slowly varying driving fields the adiabatic limit, accounted for by the depinning field, would produce an apparent scaling exponent approaching zero in the limit of small slopes. For larger slopes of the driving field dissipation due to dynamic lagging of the domain walls becomes dominant over adiabatic losses and a power-law behavior with a scaling exponent equal to 1=1 þ q would appear. Hence, this shifted power-law could be interpreted to be a crossover between adiabatic and dynamic regimes.
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Acknowledgements The author would like to thank James Erskine for pointing out the original reference to the Ruiz-Feal et al. paper, James Erskine and Corneliu Nistor for comparative experimental work, and Linda Reichl, James Erskine, Dan Robb, and Corneliu Nistor for helpful discussions. The author would also like to thank Linda Reichl for partial support through the Welch Foundation Grant no. F-1051 and the Engineering Research Program of the Office of Basic Energy Sciences at the US Department of Energy, Grant no. DE-FG03-94ER14465. Finally the author would like to thank Natali Teszler for general support.