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Hysteresis mediated by a domain wall motion
Physica A 340 (2004) 625 – 635
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Hysteresis mediated by a domain wall motion Thomas Nattermanna , Valery Pokrovskyb; c;∗ a Institute
for Theoretical Physics, University of Cologne, Zulpicher Strasse 77, D-5000 Cologne 41, Germany b Department of Physics, Texas A& M University, College Station, TX 77843-4242, USA c Landau Institute for Theoretical Physics, Chernogolovka, Moscow District 142432, Russia Received 17 February 2004; received in revised form 4 March 2004
Abstract The position of an interface (domain wall) in a medium with random pinning defects is not determined unambiguously by the instantaneous value of the driving force, even on average. Employing the general theory of the interface motion in a random medium, we study this hysteresis, di1erent possible shapes of the hysteresis loop, and the dynamical phase transitions between them. Several principal characteristics of the hysteresis, including the coercive force and the curves of dynamical phase transitions obey scaling laws and display a critical behavior in the vicinity of the mobility threshold. At 3nite temperature the threshold is smeared and a new range of thermally activated hysteresis appears. At a 3nite frequency of the driving force there exists a range of the non-adiabatic regime in which not only the position, but also the average velocity of the domain wall, displays hysteresis. c 2004 Elsevier B.V. All rights reserved. PACS: 46.65.+g; 75:60: − d; 74.50.Qt Keywords: Random phenomena and media; Domain e1ects; Magnetization curves and hysteresis; Vortex lattice; Flux pinning; Flux creep
1. Introduction This is our tribute to the memory of Per Bak, great scientist and an exceptional personality. His main contribution to science is the discovery of a new, very broad class of nonlinear deterministic systems displaying stable chaotic critical behavior, which he called self-organized criticality (SOC). It is di>cult to overestimate the signi3cance ∗
Corresponding author. Department of Physics, Texas A& M University, College Station, TX 77843-4242, USA. E-mail address: [email protected] (V. Pokrovsky). c 2004 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2004.05.014
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of this discovery, which has opened new explanation and uni3ed description of very remote phenomena, such as earthquakes, plastic deformations, hysteresis in magnets, dynamics of the biosphere and 3nancial catastrophes. The notion of avalanches, introduced by Per, is now central to the dynamics of such systems and is most probably a primary source of 1=f noise. Theory of these systems has deserved a wide popularity and Per is one of the most cited physicists. However, his moral inGuence in the scienti3c community was no less important than his scienti3c contribution. He always was surrounded by people seeking new approaches in di1erent 3elds: geologists, biologists, physicians, economists, historians and, certainly physicists. Everyone in this community felt free to criticize everything and to propose new ideas, but Per’s authority and respect for him was extremely high. It was based on his uncompromising pursuit of truth and hate of any falsehood in science as well as in social and personal life. Once he was seen on a major TV station at the JFK airport expressing his indignation at the violation of passengers rights. He was a brave and noble man. He could be very harsh in discussions, a feature that not everybody could tolerate. However, Per was a reliable, friendly, and responsive person ready to help if there was a need for his help, and for this purpose he could spend his time, money and inGuence. We feel that his death has left a vacancy in our community, one that will not be 3lled. In this review we consider a simple, but rather wide-spread phenomenon: the hysteresis mediated by a motion of an interface or domain wall (DW) driven by an external alternating force in a medium with random pinning centers. The interplay between the driving force and elastic and pinning forces develops in time. If the driving force is constant, it can establish an average velocity of the DW after a su>ciently long time interval. For an alternating force this time may be longer or shorter than the period of oscillations, resulting in di1erent regimes. The inGuence of a heat reservoir gives an additional dimension to this phenomenon. The interface is a typical system displaying the SOC [1]: in the absence of an external force it exhibits a fractal structure. A small force results in avalanches on spatial and temporal scales that depend on the force. The stronger the force and the higher the frequency, the weaker is the inGuence of the avalanches, but they are crucial for the range of small frequencies and amplitudes important in experiment. This problem is a part of a very old hysteresis problem [2] occurring in many different dynamical processes: chemical reactions, magnetization reversal, crystal growth, absorption and desorption etc., in which many DW are involved. In this case the interaction between walls must be taken in account. This is beyond the scope of our consideration, but some rough estimates can be made on the basis of one-DW theory. On the other hand, recent experimental development with its tendency to the sub-micron spatial scale has generated many small systems in which the motion of one or a few DWs is absolutely realistic. Moreover, the motion of an individual DW has been experimentally observed and studied by several groups of experimenters [3]. Therefore, a theory of such motion is necessary to understand the observed phenomena and to predict new ones. Although the observation of hysteresis in ferromagnets has a history of more than 100 years, the phenomenon could not be satisfactory explained until the end of 20th
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century, when a theory of the DW and its motion in a random medium was developed. Therefore, we start our review with an introductory section brieGy describing this theory. In the next section we consider a DW moving adiabatically at zero temperature. Adiabatic motion means that the frequency of the driving force is small enough that the instantaneous velocity equals to its stationary value at a constant force equal to the instantaneous value of the AC force. The third section is dedicated to the inGuence of 3nite temperature, but still in adiabatic regime. In the fourth section we consider the non-adiabatic regime and predict a new phenomenon: the hysteresis of velocity (magnetization rate in the case of a ferromagnet). In the last section we review relevant experiments. 2. Domain wall in a random medium 2.1. Zero temperature For de3niteness we consider a DW in an impure ferromagnet with uniaxial anisotropy at zero temperature. As shown in Refs. [4–7], the equation of motion for a DW without overhangs can be written as 1 9Z (1) = ∇2 Z + h + (x; Z) ; 9t where Z(x; t) denotes the interface position; and are the DW mobility and sti1ness, respectively; h is the external driving force. For a ferromagnet h = B HM , where H is the external magnetic 3eld and M is the magnetization. The random forces (x; Z) generated by pinning centers obey Gaussian statistics and have short range correlations:
(x; Z) (x ; Z ) = 2 lD+1 l (x − x )P(Z − Z ) :
(2)
Here l (x) denotes a delta-function smeared out over a distance l, a basic microscopic size (lattice constant) in the direction perpendicular to interface. The initial correlator P0 (Z) is an even analytical function of Z, which decays to zero over a 3nite distance l and has a maximum at z = 0. In the following we assume that the disorder is weak, i.e., that l. Under this assumption the interface is essentially Gat on length scales L Lp (see Ref. [6]), where Lp ≈ l(= l)2=(4−D) l is the so-called Larkin length [8–10]. On larger scales the wall adapts to the disorder and gets pinned for driving 3elds h . hp with hp ≈ lL−2 c =
( l=)D=(4−D) for the pinning threshold. The transverse displacement w(L) of the DW at the scale L is determined by the roughness exponent : w(L) ˙ L ( 6 1). The characteristic time of an avalanche t(L) at the scale L is determined by the dynamic exponent z: t(L) ˙ Lz . If h exceeds hp , the wall starts to move. Close to the depinning transition the velocity, which can be considered to be an order parameter, vanishes according to a power law h − hp v ≈ vp ; h ¿ hp : (3) hc
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Here we introduced the characteristic velocity scale vp = hp . The moving DW conserves the rough (fractal) structure at length scales between Lp and a new scale v = Lp (vp =v)1=(z−) established by small velocity, where the pinning force dominates. At larger scales the bumps on the DW are healed and it becomes smooth. Functional renormalization group calculations [6,7,11] show that 1 ¡ z ¡ 2, i.e., the dynamics close to the depinning transition is super-di
(5)
In the absence of an external driving force the typical free energy Guctuations on the length scale L are of the order F(L) = Tp (L=Lp )" where " = D − 2 + 2. The energy barriers between di1erent metastable states of such a scale have the same order of magnitude. When the driving force h switches on, it changes barriers between neighboring metastable states at the scale L by the value −hw(L) = −hLp (L=Lp ) . Thus,
T. Nattermann, V. Pokrovsky / Physica A 340 (2004) 625 – 635 T
EB(L,h) ~ EB(h)
629
hω hT h(t) (2)
Eω
Tp
creep
v=0
Tp [1-(h/hp)]
linear regime critical regime h(t) (1)
L Lp
~L
ω,
~ Lh
,
h
Lh
hp
Fig. 1. Left: Energy barrier as a function of the length scale l for a given driving force density h. Right: The lines h! and hT as explained in the text.
the total energy barrier is L" EB (L; h) ≈ F(L) − hL wR (L) = Tp " Lp D
1−
L Lh
2−˜ ;
(6)
where we introduced the force length scale Lh = Lp (hp =h)1=(2−) . A schematic graph of EB (L; h) vs. h is shown in Fig. 1. It has a maximum at L = L˜h ∼Lh and vanishes at L = Lh . The maximum height of the barrier is hp "˜ : (7) ; = E B (h) ≡ EB (Lh ; h) ≈ Tp h 2 − ˜ It must be overcome by thermally activated hopping to initiate motion. The creep velocity of the DW follows from vcreep ≈ w(L˜h )='(L˜h ). According to the Arrhenius law, ˜ the hopping time is ' ∼!p−1 eE B (h)=T . Thus, we obtain vcreep (h) ≈ w(L˜h )='(L˜h ) ∼exp[ − Tp =T (hp =h) ]. We have omitted a prefactor, which is beyond our accuracy. This formula is valid for T E˜ B (h) and was found 3rst by Io1e and Vinokur [13] (see also Ref. [14,15]). In the opposite case T E˜ B (h) we expect a linear relation between the driving force and the velocity: v h. The border line between the two cases i.e., the inGection point of the curve v(h), T ≈ E˜ B (h), de3nes a temperature dependent force hT (compare Fig. 1.) 1= Tp : (8) hT = hp T Note that the creep formula is valid only for h hT . Let us now consider the inGuence of thermal Guctuations on the depinning transition. At h 6 hp and T = 0 the velocity is zero, but one has to expect that as soon as thermal Guctuations are switched on, the velocity will become 3nite. Scaling theory predicts in this case an Ansatz [16,17] (generalizing (3)) h − hp =' (9) v(h; T ) ∼ T ( T 1=' with ((x) → const: for x → 0 and ((x)∼x=' for x 1, such that v(hp ; T ) ∼T =' . ' ¿ 0 is a new exponent which still has to be determined.
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It is worthwhile to note that the relevant thermal Guctuations, which unpin the DW, act on scales of the order of Lp , as was 3rst indicated by Middleton [17]. At the critical point h = hp essentially only barriers on the scale L ≈ Lp are left as we saw earlier. It is therefore su>cient to consider only this length scale. A detailed analysis of the form of the e1ective potential on this scale gives ' = 32 . 3. Adiabatic motion of a single domain wall driven by AC force 3.1. Dynamics of a rectilinear domain wall at zero temperature In this section we describe the motion of a DW, rectilinear at large scale [18]. The reason for the motion is the oscillatory driving force, which we assume to have a simple harmonic shape: h(t) = h0 sin !t. In the previous section we demonstrated that the DW roughness can be ignored on a time scale t ¿ tv and the length scale L ¿ Lv . Thus, the bending of the DW can be neglected if !tv 1. Simultaneously this condition means that the process is adiabatic, i.e., that the velocity at any moment of time with high precision is equal to its stationary value corresponding to the value of the driving force h(t) at the same moment of time. The rectilinear (or plane) DW can be characterized by one coordinate only. We denote it as Z(t). In the adiabatic approximation it satis3es an obvious equation: h(t) − hp dZ = hp f ; (10) hp dt where the scaling function f(x) behaves asymptotically as x at small x and as x at large x. Instead of integrating it over time, we integrate it over 3eld by the following change of coordinate: dt = dh=! h20 − h2 Thus, we 3nd the following expression for Z as function of the 3eld h: hp h h − hp dh f : (11) Z(h; h0 ; Z0 ) = Z0 + ! hp hp h2 − h 2 0
This equation is correct for h ¿ hp ; at smaller positive values of h the DW does not move and Z = Z0 = const. Therefore, the magnetization remains constant until the amplitude of the driving 3eld reaches the value hp . This value marks the 3rst dynamical phase transition: the appearance of the hysteresis loop. Now starting from Z0 = −L at h = 0, where L is the half of size of a rectangular sample, and applying positive magnetic 3eld, one can increase Z until it takes its maximum value 2hp h0 h − hp dh f : (12) Zmax = −L + ! hp hp h20 − h 2 This equation is valid as long as the right-hand side of Eq. (12) is less than L. When it becomes larger, the maximum value of Z obviously remains equal to L. The value ht2 of the amplitude h0 , at which Zmax equals L determines the second dynamical phase transition: the complete reversal of magnetization. The hysteresis loop becomes
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M
M
. . h
. .
h
ω h0
(a)
hc =h0 h
hω=hr
(b)
M
M
. hr...
hω (c)
631
hc
h0
. . ..
h
hω (d)
hc hr h0
h
Fig. 2. (a–d) Shapes of the hysteresis loops and dynamical phase transitions.
symmetric with respect to inversion: h → −h; M → −M (Z → −Z) (see Fig. 2b). At smaller values of h0 between hp and ht1 the magnetization reversal is incomplete; the hysteresis loop is symmetric with respect to reGection h → −h, but not to inversion of magnetization (see Fig. 2a). Finally, when Z reaches the value L for a quarter of period or less, i.e., at h 6 h0 , in a range of 3elds hr ¡ h ¡ h0 the magnetization reaches saturation (in a somewhat more realistic model it becomes a single-valued function of the 3eld). We call these parts of the magnetization graph whiskers (see Fig. 2d). The value ht3 of h0 , at which the whiskers 3rst appear determines the third dynamical transition. It can be found from the obvious equation: Z(ht3 ; ht3 ; −L) = L, where we have used the notation of Eq. (11). Note that a similar equation is valid for ht2 : Z(ht2 ; ht2 ; −L) = L=2. From these equation we 3nd scaling relation for the phase transitions 3elds: ht2 ht3 !L !L ; (13) ; =F =F 2hp hp hp hp y where F(x) is a function inverse to G(y) = 1 f(y − 1)dy = y2 − y 2 . Normally the hysteresis loop is characterized by the coercive force hc , i.e., by a value of the driving force at which the magnetization vanishes (Z = 0): Z(hc ; h0 ; −L) = 0. Earlier we have introduced the saturation 3eld hr , whose equation is: Z(hr ; h0 ; −L) = L. Eq. (11) implies that these two 3elds divided by hp obey scaling equations depending on dimensionless arguments, u = !L=hp and v = h0 =hp . The reader can 3nd details in the original article [18]. We should mention the asymptotic behavior at very large u or v: ht2 ≈ !L=2; ht3 ≈ 2ht2 ; hc ≈ ht2 (2h0 − ht2 ); hr ≈ ht3 (2h0 − ht3 ). The critical asymptotics near the mobility threshold (v − 1 1) is more complicated. The details can be found in the same article [18]. Numerical MC simulation of the 2-dimensional Ising model with random bonds and Glauber dynamics [18] has demonstrated that the DWs, initially numerous, quickly merge to a few with linear size of the same order as the size of the sample. This stage of the magnetization reversal is the longest and determines the overall dynamics. Thus, our results are qualitatively correct for larger systems at the last and longest stage of the hysteresis process. The same numerical simulation has reproduced all types of the hysteresis loop predicted by theory of the single DW hysteresis. However, the
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quantitative details may be di1erent. Moreover, since theory of multidomain hysteresis does not yet exist, there is no reliable estimate of the time at which only few domains remain. 3.2. Motion of a domain wall at ?nite temperature As was shown in Section 2.2, the principal di1erence between the dynamics at zero temperature and at 3nite temperature is that, at 3nite T the DW can start to move at an arbitrarily weak driving force, due to thermal activation processes. Thus, strictly speaking, there is no longer a mobility threshold for a constant driving force. At low temperature the threshold will be smeared. In addition the 3nite temperature establishes new scales of length LT , time tT ˙ LzT , force hT and activation energy EB (T; h). In the adiabatic regime it leads to the appearance of an e1ective temperature-dependent threshold for the AC driving force. In this section we follow the work [19]. If the driving force oscillates in time with frequency ! and amplitude h0 , then barriers on the scale L for which !'(L; h) ¿ 1, where '(L; h) = '0 expEB (L; h)=T is the relaxation time for such a Guctuation, cannot be overcome during one cycle of oscillation. Thus, the global motion of the DW may be initiated only when the maximum value !'(L; h) over L becomes of the order of unity. From the condition !'max (h) = 1 we 3nd a new, frequency and temperature dependent driving 3eld h! , which plays the role of a dynamic threshold. It obeys the following equation: 1=
Tp h! h! 1− ; (14) = T0 hp hp where 0 = −ln(!'0 ) (reminder: '0 = !0−1 is a microscopic hopping time: we assume !'0 1). At h ¡ h! , there is no macroscopic motion of the wall, though its segments still transfer between di1erent metastable states with avalanches whose development time is less than 21=! and the length scale is less than L! = Lp (T=Tp ln !p =!)1=" . This process gives rise to dissipation. Drift of the wall as a whole starts at h0 ¿ h! . For !'0 1, h! ¡ hT . Note that Eq. (14) determines h! as a monotonically decreasing function of temperature equal to hp at T = 0. At a 3xed temperature h! is a monotonically increasing function of frequency. Thus, a DW subject to the AC driving force either remains at rest for h0 ¡ h! , or it moves due to thermal activation (creep regime) for h! ¡ h0 ¡ hT , or it moves in the sliding regime for h ¿ hT . At low temperatures these three regimes overlap with the critical behavior near h = hp . A schematic phase diagram of the DW motion in variables T − h is shown on the right of Fig. 1. Having derived the velocity of DW motion as a function of the driving 3eld at 3xed temperature and frequency, we can follow the hysteresis curve using the adiabatic equation of motion similar to that at zero temperature (10), but containing new parameters T and h! : dZ h T h ; (15) f ; ≡ v(h) = dh h! Tp ! h20 − h2
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where f(x; y) is a dimensionless function of dimensionless arguments which is equal zero at x ¡ 0 and 1 at x or y in3nite. The shapes of hysteresis loops at 3nite temperature are rather similar to their shapes at zero temperature. In particular, all three dynamic phase transitions described in Section 3.1 proceed at 3nite temperature as well, but the role of the threshold driving force is played by the 3eld h! . At the 3rst transition the amplitude h0 coincides with h! . At the second and the third transition amplitudes the same values are determined by the equations htn v(h)dh (n − 1)!L = ; n = 2; 3 : (16) 2 2 htn − h2 h! Eqs. (15) and (16) display scaling similar to that at zero temperature: the dimensionless ratios htn =h! are de3ned by the same function of the dimensionless variable (n−1) !L=(2h! ). 4. Non-adiabatic motion of a wall driven by an AC eld 4.1. Zero temperature We again consider the motion caused by an AC driving force h(t) = h0 sin(!t) in Eq. (1) with a frequency ! !p = hp =l, but we will not assume the adiabatic condition. We start at zero temperature. The 3nite frequency ! of the driving force acts as an infrared cuto1 for the propagation of perturbations generated by pinning centers. As follows from (1) these perturbations can propagate during one cycle of the external force up to the (renormalized) di1usion length L! = Lp (=!Lp2 )1=z ≡ Lp (!p =!)1=z . If L! ¡ Lp , (i.e., ! ¿ !p ) there is no renormalization and z has to be replaced by 2. During one cycle of the AC drive, perturbations generated by local pinning centers a1ect the con3guration of the DW only up to a scale L! . If this scale is less than Lp , the resulting curvature force lL−2 ! is always larger than the pinning force and there is no longer any pinning. In the opposite case L! ¿ Lp (i.e., ! ¡ !p ), the pinning forces compensate the curvature forces on length scales larger than Lp . As a result of the adaptation of the DW to the disorder, the pinning forces are renormalized. This renormalization is truncated at L! . Contrary to the adiabatic limit ! → 0, there is no sharp depinning transition at ! ¿ 0. Indeed, a necessary condition for the existence of a sharp transition in the adiabatic case was the requirement that the Guctuations of the depinning threshold in a correlated volume of linear size , hp ≈ hp (Lp =v )(D+)=2 , be smaller than (h − hp ), i.e., (D + ) ¿ 2 [20]. For ! ¿ 0 the correlated volume has a maximum size L! and hence the Guctuations hp are given by hp ≈ hp (Lp =L! )(D+)=2 = hp (!=!p )(D+)=(2z) :
(17)
Thus, di1erent parts of the DW have di1erent depinning thresholds—the depinning transition is smeared over the interval hp . For a better understanding of the velocity hysteresis we consider the coupling between di1erent segments of a DW with average
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ω
0.4
ωp
h0=0.8, ω=0.3
transition smeared
) ω T(h hc
v(t)
δ hc
ω=0.0 -hc
0
hco
-0.2
adiabatic region
-0.4
h
vc
h0=hc , ω=0.3
0.2
-0.6
~ hc
-0.4
-0.2
0
0.2
hc
0.4
0.6
h(t)
Fig. 3. Left: Schematic frequency-3eld diagram for the depinning in an AC external 3eld (with h0 ¿ hp ): For 0 ¡ ! !p the depinning transition is smeared but traces of the ! = 0 transition are seen in the frequency dependency of the velocity at h = hp . This feature disappears for ! !p . Right: Velocity hysteresis of a D = 1 dimensional interface in a random environment.
lateral size L! . Approaching the depinning transition from su>ciently large 3elds, h0 hp (and ! !p ), one observes the critical behavior of the adiabatic case as long as v L! . The equality v ≈ L! de3nes a 3eld hco signaling a crossover to an inner critical region where singularities are truncated by L! . Note that hc0 − hp = hp (!=!p )1=( z) ¿ hp (cf. Fig. 3). A new physical phenomenon occurring in the non-adiabatic regime is the hysteresis of velocity. It is shown on the right of Fig. 3, which was obtained by numerical simulation of a one-dimensional version of Eq. (1). The physical reason for this e1ect is the hampering of the DW by the pinning centers. It stops the average motion when the force becomes small, but non-zero. Then the elastic forces tend to adjust overly stretched intervals of the DW and submit the DW an average velocity opposite to the applied force, as is seen on the same 3gure. This consideration explains the clockwise circulation on the central hysteresis loop and the appearance of secondary loops. The reader is referred for details to Ref. [21]. 5. Experiments In this section we review several experimental works relevant to our topic. We start with the work by Budde et al. [22]. They studied the 3rst order phase transition between a two-dimensional gas and a two-dimensional solid in the 3rst adsorbed monolayer of a noble gas (Ar, Kr, Xe) on the face (1 0 0) of NaCl. The adsorbed layer was in equilibrium with a 3d gas of the same atoms. Varying temperature back and forth linearly with time, they observed a hysteresis of the adsorbate density. The deviation of the temperature from the transition point plays the role of the driving force. They have found that the width of the hysteresis loop, which corresponds to the coercive force hc in magnetic system, scales as (PT )1=2 , where PT is the amplitude of the temperature oscillations. This result agrees with theoretical predictions made in Ref. [18] (see the end of Section 3.1). This result suggests that the experimental parameters corresponded to a regime of the force much larger than the threshold value, so that the pinning force was negligible.
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In the works by Kleemann and coworkers [23,24] the authors studied the dielectric in the vicinity of its spectra in ferroelectric single crystals Sr 0:61−x Ba0:39 Nb2 O6 Ce3+ x transition temperature. They have found di1erent behaviors in the electric susceptibility "˜ vs. frequency. At low frequencies they observed a power-like behavior " ˙ ! with between 0.2 and 0.7. The authors ascribe such behavior to creep motion with a broad distribution of activation energies. At higher frequencies they observed a logarithmic behavior of ", ˜ which they treat as reversible relaxation of the DW segments. The crossover between these two regimes they attribute to the dynamical phase transition at h = h! as predicted in Ref. [19] (see Section 3.2). Acknowledgements Our thanks are due to Dr. W. Saslow for attentive reading of manuscript and useful remarks. This work was supported by NSF under the grant DMR-0321572 and by DOE under the grant DE-FG03-96ER45598. References [1] M. Pazcuski, S. Maslov, P. Bak, Phys. Rev. E 53 (1996) 414–443. [2] C.P. Steinmetz, Trans. Am. Inst. Electr. Eng. 9 (1892) 3. [3] A. Imre, et al., Physica E 19 (2003) 240; M. Klaui, et al., Appl. Phys. Lett. 83 (2003) 105; J. Crollier, et al., Appl. Phys. Lett. 83 (2003) 509; M. Tsoi, R.E. Fontana, S.S.P. Parkin, Appl. Phys. Lett. 83 (2003) 2617. [4] M.V. Feigel’man, Sov. Phys. JETP 58 (1983) 1076. [5] J. Koplik, H. Levine, Phys. Rev. B 32 (1985) 280. [6] T. Nattermann, S. Stepanow, L.-H. Tang, H. Leschhorn, J. Phys. II France 2 (1992) 1483. [7] O. Narayan, D.S. Fisher, Phys. Rev. B 48 (1993) 7030. [8] A.I. Larkin, Sov. Phys. JETP 31 (1970) 784. [9] Y. Imry, S.K. Ma, Phys. Rev. Lett. 35 (1975) 1399. [10] T. Halpin-Healy, Y.C. Zhang, Phys. Rep. 254 (1995) 215. [11] P. Chauve, P. Le Doussal, J.-K. Wiese, Phys. Rev. Lett. 86 (1785) 2001. [12] P. Chauve, T. Giamarchi, P. Le Doussal, Europhys. Lett. 44 (1998) 110. [13] L.B. Io1e, V.M. Vinokur, J. Phys. C 20 (1987) 6149. [14] T. Nattermann, Europhys. Lett. 4 (1987) 1241; T. Nattermann, Rev. Lett. 68 (1992) 3615; T. Nattermann, Phys. Rev. B 46 (1992) 11520. [15] L.V. Mikheev, B. Drossel, M. Kardar, Phys. Rev. Lett. 75 (1995) 1170. [16] D.S. Fisher, Phys. Rev. Lett. 50 (1983) 1486. [17] A. Middleton, Phys. Rev. Lett. 68 (1992) 670. [18] I. Lyuksyutov, T. Nattermann, V. Pokrovsky, Phys. Rev. B 58 (1999) 4260. [19] T. Nattermann, V. Pokrovsky, V. Vinokur, Phys. Rev. Lett. 87 (2001) 197005. [20] T. Nattermann, S. Stepanow, L.-H. Tang, H. Leschhorn, J. Phys. II France 2 (1992) 565 (1483); T. Nattermann, S. Stepanow, L.-H. Tang, H. Leschhorn, Ann. Phys. (Leipzig) 6 (1997) 1. [21] A. Glatz, T. Nattermann, V. Pokrovsky, Phys. Rev. Lett. 90 (2003) 47201. [22] K. Budde, I. Lyuksyutov, H. Pfnur, H. Godzik, H.U. Everts, Europhys. Lett. 47 (1999) 575. [23] W. Kleemann, J. Dec, S. Miga, T. Woike, R. Pankrath, Phys. Rev. B 65 (2002) 220101. [24] W. Kleemann, J. Dec, R. Pankrath, Ferroelectrics 286 (2003) 743.
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Hysteresis mediated by a domain wall motion Thomas Nattermanna , Valery Pokrovskyb; c;∗ a Institute
for Theoretical Physics, University of Cologne, Zulpicher Strasse 77, D-5000 Cologne 41, Germany b Department of Physics, Texas A& M University, College Station, TX 77843-4242, USA c Landau Institute for Theoretical Physics, Chernogolovka, Moscow District 142432, Russia Received 17 February 2004; received in revised form 4 March 2004
Abstract The position of an interface (domain wall) in a medium with random pinning defects is not determined unambiguously by the instantaneous value of the driving force, even on average. Employing the general theory of the interface motion in a random medium, we study this hysteresis, di1erent possible shapes of the hysteresis loop, and the dynamical phase transitions between them. Several principal characteristics of the hysteresis, including the coercive force and the curves of dynamical phase transitions obey scaling laws and display a critical behavior in the vicinity of the mobility threshold. At 3nite temperature the threshold is smeared and a new range of thermally activated hysteresis appears. At a 3nite frequency of the driving force there exists a range of the non-adiabatic regime in which not only the position, but also the average velocity of the domain wall, displays hysteresis. c 2004 Elsevier B.V. All rights reserved. PACS: 46.65.+g; 75:60: − d; 74.50.Qt Keywords: Random phenomena and media; Domain e1ects; Magnetization curves and hysteresis; Vortex lattice; Flux pinning; Flux creep
1. Introduction This is our tribute to the memory of Per Bak, great scientist and an exceptional personality. His main contribution to science is the discovery of a new, very broad class of nonlinear deterministic systems displaying stable chaotic critical behavior, which he called self-organized criticality (SOC). It is di>cult to overestimate the signi3cance ∗
Corresponding author. Department of Physics, Texas A& M University, College Station, TX 77843-4242, USA. E-mail address: [email protected] (V. Pokrovsky). c 2004 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2004.05.014
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of this discovery, which has opened new explanation and uni3ed description of very remote phenomena, such as earthquakes, plastic deformations, hysteresis in magnets, dynamics of the biosphere and 3nancial catastrophes. The notion of avalanches, introduced by Per, is now central to the dynamics of such systems and is most probably a primary source of 1=f noise. Theory of these systems has deserved a wide popularity and Per is one of the most cited physicists. However, his moral inGuence in the scienti3c community was no less important than his scienti3c contribution. He always was surrounded by people seeking new approaches in di1erent 3elds: geologists, biologists, physicians, economists, historians and, certainly physicists. Everyone in this community felt free to criticize everything and to propose new ideas, but Per’s authority and respect for him was extremely high. It was based on his uncompromising pursuit of truth and hate of any falsehood in science as well as in social and personal life. Once he was seen on a major TV station at the JFK airport expressing his indignation at the violation of passengers rights. He was a brave and noble man. He could be very harsh in discussions, a feature that not everybody could tolerate. However, Per was a reliable, friendly, and responsive person ready to help if there was a need for his help, and for this purpose he could spend his time, money and inGuence. We feel that his death has left a vacancy in our community, one that will not be 3lled. In this review we consider a simple, but rather wide-spread phenomenon: the hysteresis mediated by a motion of an interface or domain wall (DW) driven by an external alternating force in a medium with random pinning centers. The interplay between the driving force and elastic and pinning forces develops in time. If the driving force is constant, it can establish an average velocity of the DW after a su>ciently long time interval. For an alternating force this time may be longer or shorter than the period of oscillations, resulting in di1erent regimes. The inGuence of a heat reservoir gives an additional dimension to this phenomenon. The interface is a typical system displaying the SOC [1]: in the absence of an external force it exhibits a fractal structure. A small force results in avalanches on spatial and temporal scales that depend on the force. The stronger the force and the higher the frequency, the weaker is the inGuence of the avalanches, but they are crucial for the range of small frequencies and amplitudes important in experiment. This problem is a part of a very old hysteresis problem [2] occurring in many different dynamical processes: chemical reactions, magnetization reversal, crystal growth, absorption and desorption etc., in which many DW are involved. In this case the interaction between walls must be taken in account. This is beyond the scope of our consideration, but some rough estimates can be made on the basis of one-DW theory. On the other hand, recent experimental development with its tendency to the sub-micron spatial scale has generated many small systems in which the motion of one or a few DWs is absolutely realistic. Moreover, the motion of an individual DW has been experimentally observed and studied by several groups of experimenters [3]. Therefore, a theory of such motion is necessary to understand the observed phenomena and to predict new ones. Although the observation of hysteresis in ferromagnets has a history of more than 100 years, the phenomenon could not be satisfactory explained until the end of 20th
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century, when a theory of the DW and its motion in a random medium was developed. Therefore, we start our review with an introductory section brieGy describing this theory. In the next section we consider a DW moving adiabatically at zero temperature. Adiabatic motion means that the frequency of the driving force is small enough that the instantaneous velocity equals to its stationary value at a constant force equal to the instantaneous value of the AC force. The third section is dedicated to the inGuence of 3nite temperature, but still in adiabatic regime. In the fourth section we consider the non-adiabatic regime and predict a new phenomenon: the hysteresis of velocity (magnetization rate in the case of a ferromagnet). In the last section we review relevant experiments. 2. Domain wall in a random medium 2.1. Zero temperature For de3niteness we consider a DW in an impure ferromagnet with uniaxial anisotropy at zero temperature. As shown in Refs. [4–7], the equation of motion for a DW without overhangs can be written as 1 9Z (1) = ∇2 Z + h + (x; Z) ; 9t where Z(x; t) denotes the interface position; and are the DW mobility and sti1ness, respectively; h is the external driving force. For a ferromagnet h = B HM , where H is the external magnetic 3eld and M is the magnetization. The random forces (x; Z) generated by pinning centers obey Gaussian statistics and have short range correlations:
(x; Z) (x ; Z ) = 2 lD+1 l (x − x )P(Z − Z ) :
(2)
Here l (x) denotes a delta-function smeared out over a distance l, a basic microscopic size (lattice constant) in the direction perpendicular to interface. The initial correlator P0 (Z) is an even analytical function of Z, which decays to zero over a 3nite distance l and has a maximum at z = 0. In the following we assume that the disorder is weak, i.e., that l. Under this assumption the interface is essentially Gat on length scales L Lp (see Ref. [6]), where Lp ≈ l(= l)2=(4−D) l is the so-called Larkin length [8–10]. On larger scales the wall adapts to the disorder and gets pinned for driving 3elds h . hp with hp ≈ lL−2 c =
( l=)D=(4−D) for the pinning threshold. The transverse displacement w(L) of the DW at the scale L is determined by the roughness exponent : w(L) ˙ L ( 6 1). The characteristic time of an avalanche t(L) at the scale L is determined by the dynamic exponent z: t(L) ˙ Lz . If h exceeds hp , the wall starts to move. Close to the depinning transition the velocity, which can be considered to be an order parameter, vanishes according to a power law h − hp v ≈ vp ; h ¿ hp : (3) hc
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Here we introduced the characteristic velocity scale vp = hp . The moving DW conserves the rough (fractal) structure at length scales between Lp and a new scale v = Lp (vp =v)1=(z−) established by small velocity, where the pinning force dominates. At larger scales the bumps on the DW are healed and it becomes smooth. Functional renormalization group calculations [6,7,11] show that 1 ¡ z ¡ 2, i.e., the dynamics close to the depinning transition is super-di
(5)
In the absence of an external driving force the typical free energy Guctuations on the length scale L are of the order F(L) = Tp (L=Lp )" where " = D − 2 + 2. The energy barriers between di1erent metastable states of such a scale have the same order of magnitude. When the driving force h switches on, it changes barriers between neighboring metastable states at the scale L by the value −hw(L) = −hLp (L=Lp ) . Thus,
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EB(L,h) ~ EB(h)
629
hω hT h(t) (2)
Eω
Tp
creep
v=0
Tp [1-(h/hp)]
linear regime critical regime h(t) (1)
L Lp
~L
ω,
~ Lh
,
h
Lh
hp
Fig. 1. Left: Energy barrier as a function of the length scale l for a given driving force density h. Right: The lines h! and hT as explained in the text.
the total energy barrier is L" EB (L; h) ≈ F(L) − hL wR (L) = Tp " Lp D
1−
L Lh
2−˜ ;
(6)
where we introduced the force length scale Lh = Lp (hp =h)1=(2−) . A schematic graph of EB (L; h) vs. h is shown in Fig. 1. It has a maximum at L = L˜h ∼Lh and vanishes at L = Lh . The maximum height of the barrier is hp "˜ : (7) ; = E B (h) ≡ EB (Lh ; h) ≈ Tp h 2 − ˜ It must be overcome by thermally activated hopping to initiate motion. The creep velocity of the DW follows from vcreep ≈ w(L˜h )='(L˜h ). According to the Arrhenius law, ˜ the hopping time is ' ∼!p−1 eE B (h)=T . Thus, we obtain vcreep (h) ≈ w(L˜h )='(L˜h ) ∼exp[ − Tp =T (hp =h) ]. We have omitted a prefactor, which is beyond our accuracy. This formula is valid for T E˜ B (h) and was found 3rst by Io1e and Vinokur [13] (see also Ref. [14,15]). In the opposite case T E˜ B (h) we expect a linear relation between the driving force and the velocity: v h. The border line between the two cases i.e., the inGection point of the curve v(h), T ≈ E˜ B (h), de3nes a temperature dependent force hT (compare Fig. 1.) 1= Tp : (8) hT = hp T Note that the creep formula is valid only for h hT . Let us now consider the inGuence of thermal Guctuations on the depinning transition. At h 6 hp and T = 0 the velocity is zero, but one has to expect that as soon as thermal Guctuations are switched on, the velocity will become 3nite. Scaling theory predicts in this case an Ansatz [16,17] (generalizing (3)) h − hp =' (9) v(h; T ) ∼ T ( T 1=' with ((x) → const: for x → 0 and ((x)∼x=' for x 1, such that v(hp ; T ) ∼T =' . ' ¿ 0 is a new exponent which still has to be determined.
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It is worthwhile to note that the relevant thermal Guctuations, which unpin the DW, act on scales of the order of Lp , as was 3rst indicated by Middleton [17]. At the critical point h = hp essentially only barriers on the scale L ≈ Lp are left as we saw earlier. It is therefore su>cient to consider only this length scale. A detailed analysis of the form of the e1ective potential on this scale gives ' = 32 . 3. Adiabatic motion of a single domain wall driven by AC force 3.1. Dynamics of a rectilinear domain wall at zero temperature In this section we describe the motion of a DW, rectilinear at large scale [18]. The reason for the motion is the oscillatory driving force, which we assume to have a simple harmonic shape: h(t) = h0 sin !t. In the previous section we demonstrated that the DW roughness can be ignored on a time scale t ¿ tv and the length scale L ¿ Lv . Thus, the bending of the DW can be neglected if !tv 1. Simultaneously this condition means that the process is adiabatic, i.e., that the velocity at any moment of time with high precision is equal to its stationary value corresponding to the value of the driving force h(t) at the same moment of time. The rectilinear (or plane) DW can be characterized by one coordinate only. We denote it as Z(t). In the adiabatic approximation it satis3es an obvious equation: h(t) − hp dZ = hp f ; (10) hp dt where the scaling function f(x) behaves asymptotically as x at small x and as x at large x. Instead of integrating it over time, we integrate it over 3eld by the following change of coordinate: dt = dh=! h20 − h2 Thus, we 3nd the following expression for Z as function of the 3eld h: hp h h − hp dh f : (11) Z(h; h0 ; Z0 ) = Z0 + ! hp hp h2 − h 2 0
This equation is correct for h ¿ hp ; at smaller positive values of h the DW does not move and Z = Z0 = const. Therefore, the magnetization remains constant until the amplitude of the driving 3eld reaches the value hp . This value marks the 3rst dynamical phase transition: the appearance of the hysteresis loop. Now starting from Z0 = −L at h = 0, where L is the half of size of a rectangular sample, and applying positive magnetic 3eld, one can increase Z until it takes its maximum value 2hp h0 h − hp dh f : (12) Zmax = −L + ! hp hp h20 − h 2 This equation is valid as long as the right-hand side of Eq. (12) is less than L. When it becomes larger, the maximum value of Z obviously remains equal to L. The value ht2 of the amplitude h0 , at which Zmax equals L determines the second dynamical phase transition: the complete reversal of magnetization. The hysteresis loop becomes
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M
M
. . h
. .
h
ω h0
(a)
hc =h0 h
hω=hr
(b)
M
M
. hr...
hω (c)
631
hc
h0
. . ..
h
hω (d)
hc hr h0
h
Fig. 2. (a–d) Shapes of the hysteresis loops and dynamical phase transitions.
symmetric with respect to inversion: h → −h; M → −M (Z → −Z) (see Fig. 2b). At smaller values of h0 between hp and ht1 the magnetization reversal is incomplete; the hysteresis loop is symmetric with respect to reGection h → −h, but not to inversion of magnetization (see Fig. 2a). Finally, when Z reaches the value L for a quarter of period or less, i.e., at h 6 h0 , in a range of 3elds hr ¡ h ¡ h0 the magnetization reaches saturation (in a somewhat more realistic model it becomes a single-valued function of the 3eld). We call these parts of the magnetization graph whiskers (see Fig. 2d). The value ht3 of h0 , at which the whiskers 3rst appear determines the third dynamical transition. It can be found from the obvious equation: Z(ht3 ; ht3 ; −L) = L, where we have used the notation of Eq. (11). Note that a similar equation is valid for ht2 : Z(ht2 ; ht2 ; −L) = L=2. From these equation we 3nd scaling relation for the phase transitions 3elds: ht2 ht3 !L !L ; (13) ; =F =F 2hp hp hp hp y where F(x) is a function inverse to G(y) = 1 f(y − 1)dy = y2 − y 2 . Normally the hysteresis loop is characterized by the coercive force hc , i.e., by a value of the driving force at which the magnetization vanishes (Z = 0): Z(hc ; h0 ; −L) = 0. Earlier we have introduced the saturation 3eld hr , whose equation is: Z(hr ; h0 ; −L) = L. Eq. (11) implies that these two 3elds divided by hp obey scaling equations depending on dimensionless arguments, u = !L=hp and v = h0 =hp . The reader can 3nd details in the original article [18]. We should mention the asymptotic behavior at very large u or v: ht2 ≈ !L=2; ht3 ≈ 2ht2 ; hc ≈ ht2 (2h0 − ht2 ); hr ≈ ht3 (2h0 − ht3 ). The critical asymptotics near the mobility threshold (v − 1 1) is more complicated. The details can be found in the same article [18]. Numerical MC simulation of the 2-dimensional Ising model with random bonds and Glauber dynamics [18] has demonstrated that the DWs, initially numerous, quickly merge to a few with linear size of the same order as the size of the sample. This stage of the magnetization reversal is the longest and determines the overall dynamics. Thus, our results are qualitatively correct for larger systems at the last and longest stage of the hysteresis process. The same numerical simulation has reproduced all types of the hysteresis loop predicted by theory of the single DW hysteresis. However, the
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quantitative details may be di1erent. Moreover, since theory of multidomain hysteresis does not yet exist, there is no reliable estimate of the time at which only few domains remain. 3.2. Motion of a domain wall at ?nite temperature As was shown in Section 2.2, the principal di1erence between the dynamics at zero temperature and at 3nite temperature is that, at 3nite T the DW can start to move at an arbitrarily weak driving force, due to thermal activation processes. Thus, strictly speaking, there is no longer a mobility threshold for a constant driving force. At low temperature the threshold will be smeared. In addition the 3nite temperature establishes new scales of length LT , time tT ˙ LzT , force hT and activation energy EB (T; h). In the adiabatic regime it leads to the appearance of an e1ective temperature-dependent threshold for the AC driving force. In this section we follow the work [19]. If the driving force oscillates in time with frequency ! and amplitude h0 , then barriers on the scale L for which !'(L; h) ¿ 1, where '(L; h) = '0 expEB (L; h)=T is the relaxation time for such a Guctuation, cannot be overcome during one cycle of oscillation. Thus, the global motion of the DW may be initiated only when the maximum value !'(L; h) over L becomes of the order of unity. From the condition !'max (h) = 1 we 3nd a new, frequency and temperature dependent driving 3eld h! , which plays the role of a dynamic threshold. It obeys the following equation: 1=
Tp h! h! 1− ; (14) = T0 hp hp where 0 = −ln(!'0 ) (reminder: '0 = !0−1 is a microscopic hopping time: we assume !'0 1). At h ¡ h! , there is no macroscopic motion of the wall, though its segments still transfer between di1erent metastable states with avalanches whose development time is less than 21=! and the length scale is less than L! = Lp (T=Tp ln !p =!)1=" . This process gives rise to dissipation. Drift of the wall as a whole starts at h0 ¿ h! . For !'0 1, h! ¡ hT . Note that Eq. (14) determines h! as a monotonically decreasing function of temperature equal to hp at T = 0. At a 3xed temperature h! is a monotonically increasing function of frequency. Thus, a DW subject to the AC driving force either remains at rest for h0 ¡ h! , or it moves due to thermal activation (creep regime) for h! ¡ h0 ¡ hT , or it moves in the sliding regime for h ¿ hT . At low temperatures these three regimes overlap with the critical behavior near h = hp . A schematic phase diagram of the DW motion in variables T − h is shown on the right of Fig. 1. Having derived the velocity of DW motion as a function of the driving 3eld at 3xed temperature and frequency, we can follow the hysteresis curve using the adiabatic equation of motion similar to that at zero temperature (10), but containing new parameters T and h! : dZ h T h ; (15) f ; ≡ v(h) = dh h! Tp ! h20 − h2
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where f(x; y) is a dimensionless function of dimensionless arguments which is equal zero at x ¡ 0 and 1 at x or y in3nite. The shapes of hysteresis loops at 3nite temperature are rather similar to their shapes at zero temperature. In particular, all three dynamic phase transitions described in Section 3.1 proceed at 3nite temperature as well, but the role of the threshold driving force is played by the 3eld h! . At the 3rst transition the amplitude h0 coincides with h! . At the second and the third transition amplitudes the same values are determined by the equations htn v(h)dh (n − 1)!L = ; n = 2; 3 : (16) 2 2 htn − h2 h! Eqs. (15) and (16) display scaling similar to that at zero temperature: the dimensionless ratios htn =h! are de3ned by the same function of the dimensionless variable (n−1) !L=(2h! ). 4. Non-adiabatic motion of a wall driven by an AC eld 4.1. Zero temperature We again consider the motion caused by an AC driving force h(t) = h0 sin(!t) in Eq. (1) with a frequency ! !p = hp =l, but we will not assume the adiabatic condition. We start at zero temperature. The 3nite frequency ! of the driving force acts as an infrared cuto1 for the propagation of perturbations generated by pinning centers. As follows from (1) these perturbations can propagate during one cycle of the external force up to the (renormalized) di1usion length L! = Lp (=!Lp2 )1=z ≡ Lp (!p =!)1=z . If L! ¡ Lp , (i.e., ! ¿ !p ) there is no renormalization and z has to be replaced by 2. During one cycle of the AC drive, perturbations generated by local pinning centers a1ect the con3guration of the DW only up to a scale L! . If this scale is less than Lp , the resulting curvature force lL−2 ! is always larger than the pinning force and there is no longer any pinning. In the opposite case L! ¿ Lp (i.e., ! ¡ !p ), the pinning forces compensate the curvature forces on length scales larger than Lp . As a result of the adaptation of the DW to the disorder, the pinning forces are renormalized. This renormalization is truncated at L! . Contrary to the adiabatic limit ! → 0, there is no sharp depinning transition at ! ¿ 0. Indeed, a necessary condition for the existence of a sharp transition in the adiabatic case was the requirement that the Guctuations of the depinning threshold in a correlated volume of linear size , hp ≈ hp (Lp =v )(D+)=2 , be smaller than (h − hp ), i.e., (D + ) ¿ 2 [20]. For ! ¿ 0 the correlated volume has a maximum size L! and hence the Guctuations hp are given by hp ≈ hp (Lp =L! )(D+)=2 = hp (!=!p )(D+)=(2z) :
(17)
Thus, di1erent parts of the DW have di1erent depinning thresholds—the depinning transition is smeared over the interval hp . For a better understanding of the velocity hysteresis we consider the coupling between di1erent segments of a DW with average
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ω
0.4
ωp
h0=0.8, ω=0.3
transition smeared
) ω T(h hc
v(t)
δ hc
ω=0.0 -hc
0
hco
-0.2
adiabatic region
-0.4
h
vc
h0=hc , ω=0.3
0.2
-0.6
~ hc
-0.4
-0.2
0
0.2
hc
0.4
0.6
h(t)
Fig. 3. Left: Schematic frequency-3eld diagram for the depinning in an AC external 3eld (with h0 ¿ hp ): For 0 ¡ ! !p the depinning transition is smeared but traces of the ! = 0 transition are seen in the frequency dependency of the velocity at h = hp . This feature disappears for ! !p . Right: Velocity hysteresis of a D = 1 dimensional interface in a random environment.
lateral size L! . Approaching the depinning transition from su>ciently large 3elds, h0 hp (and ! !p ), one observes the critical behavior of the adiabatic case as long as v L! . The equality v ≈ L! de3nes a 3eld hco signaling a crossover to an inner critical region where singularities are truncated by L! . Note that hc0 − hp = hp (!=!p )1=( z) ¿ hp (cf. Fig. 3). A new physical phenomenon occurring in the non-adiabatic regime is the hysteresis of velocity. It is shown on the right of Fig. 3, which was obtained by numerical simulation of a one-dimensional version of Eq. (1). The physical reason for this e1ect is the hampering of the DW by the pinning centers. It stops the average motion when the force becomes small, but non-zero. Then the elastic forces tend to adjust overly stretched intervals of the DW and submit the DW an average velocity opposite to the applied force, as is seen on the same 3gure. This consideration explains the clockwise circulation on the central hysteresis loop and the appearance of secondary loops. The reader is referred for details to Ref. [21]. 5. Experiments In this section we review several experimental works relevant to our topic. We start with the work by Budde et al. [22]. They studied the 3rst order phase transition between a two-dimensional gas and a two-dimensional solid in the 3rst adsorbed monolayer of a noble gas (Ar, Kr, Xe) on the face (1 0 0) of NaCl. The adsorbed layer was in equilibrium with a 3d gas of the same atoms. Varying temperature back and forth linearly with time, they observed a hysteresis of the adsorbate density. The deviation of the temperature from the transition point plays the role of the driving force. They have found that the width of the hysteresis loop, which corresponds to the coercive force hc in magnetic system, scales as (PT )1=2 , where PT is the amplitude of the temperature oscillations. This result agrees with theoretical predictions made in Ref. [18] (see the end of Section 3.1). This result suggests that the experimental parameters corresponded to a regime of the force much larger than the threshold value, so that the pinning force was negligible.
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In the works by Kleemann and coworkers [23,24] the authors studied the dielectric in the vicinity of its spectra in ferroelectric single crystals Sr 0:61−x Ba0:39 Nb2 O6 Ce3+ x transition temperature. They have found di1erent behaviors in the electric susceptibility "˜ vs. frequency. At low frequencies they observed a power-like behavior " ˙ ! with between 0.2 and 0.7. The authors ascribe such behavior to creep motion with a broad distribution of activation energies. At higher frequencies they observed a logarithmic behavior of ", ˜ which they treat as reversible relaxation of the DW segments. The crossover between these two regimes they attribute to the dynamical phase transition at h = h! as predicted in Ref. [19] (see Section 3.2). Acknowledgements Our thanks are due to Dr. W. Saslow for attentive reading of manuscript and useful remarks. This work was supported by NSF under the grant DMR-0321572 and by DOE under the grant DE-FG03-96ER45598. References [1] M. Pazcuski, S. Maslov, P. Bak, Phys. Rev. E 53 (1996) 414–443. [2] C.P. Steinmetz, Trans. Am. Inst. Electr. Eng. 9 (1892) 3. [3] A. Imre, et al., Physica E 19 (2003) 240; M. Klaui, et al., Appl. Phys. Lett. 83 (2003) 105; J. Crollier, et al., Appl. Phys. Lett. 83 (2003) 509; M. Tsoi, R.E. Fontana, S.S.P. Parkin, Appl. Phys. Lett. 83 (2003) 2617. [4] M.V. Feigel’man, Sov. Phys. JETP 58 (1983) 1076. [5] J. Koplik, H. Levine, Phys. Rev. B 32 (1985) 280. [6] T. Nattermann, S. Stepanow, L.-H. Tang, H. Leschhorn, J. Phys. II France 2 (1992) 1483. [7] O. Narayan, D.S. Fisher, Phys. Rev. B 48 (1993) 7030. [8] A.I. Larkin, Sov. Phys. JETP 31 (1970) 784. [9] Y. Imry, S.K. Ma, Phys. Rev. Lett. 35 (1975) 1399. [10] T. Halpin-Healy, Y.C. Zhang, Phys. Rep. 254 (1995) 215. [11] P. Chauve, P. Le Doussal, J.-K. Wiese, Phys. Rev. Lett. 86 (1785) 2001. [12] P. Chauve, T. Giamarchi, P. Le Doussal, Europhys. Lett. 44 (1998) 110. [13] L.B. Io1e, V.M. Vinokur, J. Phys. C 20 (1987) 6149. [14] T. Nattermann, Europhys. Lett. 4 (1987) 1241; T. Nattermann, Rev. Lett. 68 (1992) 3615; T. Nattermann, Phys. Rev. B 46 (1992) 11520. [15] L.V. Mikheev, B. Drossel, M. Kardar, Phys. Rev. Lett. 75 (1995) 1170. [16] D.S. Fisher, Phys. Rev. Lett. 50 (1983) 1486. [17] A. Middleton, Phys. Rev. Lett. 68 (1992) 670. [18] I. Lyuksyutov, T. Nattermann, V. Pokrovsky, Phys. Rev. B 58 (1999) 4260. [19] T. Nattermann, V. Pokrovsky, V. Vinokur, Phys. Rev. Lett. 87 (2001) 197005. [20] T. Nattermann, S. Stepanow, L.-H. Tang, H. Leschhorn, J. Phys. II France 2 (1992) 565 (1483); T. Nattermann, S. Stepanow, L.-H. Tang, H. Leschhorn, Ann. Phys. (Leipzig) 6 (1997) 1. [21] A. Glatz, T. Nattermann, V. Pokrovsky, Phys. Rev. Lett. 90 (2003) 47201. [22] K. Budde, I. Lyuksyutov, H. Pfnur, H. Godzik, H.U. Everts, Europhys. Lett. 47 (1999) 575. [23] W. Kleemann, J. Dec, S. Miga, T. Woike, R. Pankrath, Phys. Rev. B 65 (2002) 220101. [24] W. Kleemann, J. Dec, R. Pankrath, Ferroelectrics 286 (2003) 743.