Scaling of hysteresis dispersion in a model spin system

Scaling of hysteresis dispersion in a model spin system

J.-M. Liu*

Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China;

Department of Applied Physics, Hong Kong Polytechnic University, Kowloon, Hong Kong;

and Laboratory of Laser Technologies, Huazhong University of Science and Technology, Wuhan 430074, China H. L. W. Chan and C. L. Choy

Department of Applied Physics, Hong Kong Polytechnic University, Kowloon, Hong Kong C. K. Ong

Department of Physics, National University of Singapore, Singapore 119260 共Received 22 March 2001; published 3 December 2001兲

We present a calculation of the magnetic hysteresis and its area for a model continuum spin system based on three-dimensional (⌽²)²model with O(N) symmetry in the limit N→⬁, under a time-varying magnetic field.

The frequency dependence of the hysteresis area A( f ), namely, hysteresis dispersion, is investigated in detail, predicting a single-peak profile which grows upwards and shifts rightwards gradually with increasing field amplitude H0. We demonstrate that the hysteresis dispersion A( f ) over a wide range of H0can be scaled by scaling function W(␩⁾⬀␶1A( f ,H₀), where␩⫽log₁₀(f␶1) and␶1is the unique characteristic time for the spin reverse, as long as H₀is not very small. The inverse characteristic time␶1⫺1

shows a linear dependence on amplitude H₀, supported by the well-established empirical relations for ferromagnetic ferrites and ferroelectric solids. This scaling behavior suggests that the hysteresis dispersion can be uniquely described by the charac- teristic time for the spin reversal once the scaling function is available.

DOI: 10.1103/PhysRevB.65.014416 PACS number共s兲: 75.60.Ej, 75.40.Gb, 75.10.Hk

I. INTRODUCTION

When a spin system below its Curie point T_cis submitted to a periodic time-varying magnetic field H, say, a sinusoid field H(t)⫽H₀sin(2␲ft), where t is time, H₀ is the ampli- tude, and f is the frequency, a looplike magnetic hysteresis is observable as plotting the system average ordering parameter 共magnetization兲M against field H.^1,2It has been well estab- lished that the hysteresis is dynamic in origin,³ i.e., the shape, symmetry, and area of the hysteresis are all f and H₀ dependent. The problem of dynamic hysteresis was not em- phasized until the recent ten years. For a comprehensive re- view of this subject, one may refer to the article of Chakra- barti and Acharyya and references therein.³

In the framework of first-order phase transitions, the dy- namic hysteresis is generated because of the spin-ordered domain reversal through irreversible domain wall migration 共irreversible nucleation and growth兲, assisted by the field- induced static magnetic energy.² The hysteresis area A thus represents the energy dissipation共loss兲in one cycle of such reversal. From a more general point of view, the hysteresis is formed due to the relaxational delay of the system respond- ing to the external field.³It has been assumed that either the nucleation-and-growth mode or the relaxational delay mechanism can be described by a characteristic time that is mainly H₀ dependent. As the system responds to the time- varying external field whose characteristic time is the inverse frequency, the dynamic hysteresis is essentially determined by the two competing time scales. An understanding of the dynamic hysteresis for either real magnetic materials or model spin system is thus of interest from the point of view of basic research. On the other hand, for recording or

memory applications of magnetic materials, knowledge of dynamic hysteresis enables us to understand the kinetics of domain reversal.² The pattern of the hysteresis and the pa- rameters such as remanence and coercivity are essential for evaluating the materials performance. In particular, knowl- edge of high-frequency hysteresis is useful because high- speed spin electronics has attracted special interest nowadays.⁴

Extensive studies of the dynamic hysteresis in the past ten years have focused on two problems: the dynamic transi- tions and hysteresis dispersion. For the former, an increasing frequency f will break the symmetry of the hysteresis loop observed at low frequency for a given H₀, producing an asymmetric loop around the origin. The dynamic order pa- rameter Q⫽f养M (t)dt, where M (t) is the system average magnetization and t is time, becomes nonzero with increas- ing f, indicating interesting dynamic transitions in such non- equilibrium driven systems. This problem has been exten- sively investigated^5–9and comprehensively reviewed.³Since it is irrelevant to the present work, no details will be pre- sented here. For the latter problem, the dependence of hys- teresis area A as a function of f and H₀, A( f ,H₀), has been studied for various magnetic systems. We present a brief re- view of the works along this line. The earliest work can be referred back to the well-known empirical Steinmetz law for ferrites.¹⁰Subsequently, the work of Rao et al. represents the first systematic study of the hysteresis dispersion.^11,12 They studied O(N)-symmetric (⌽²)² and (⌽²)³ theories at N

→⬁ and provided a detailed analysis of the dispersion over extremely-low- and extremely-high-f ranges, respectively. It was predicted that A( f ) over the low- and high-f ranges exhibits the following power-law behaviors, respectively:

0163-1829/2001/65共1兲/014416共9兲/$20.00 65 014416-1 ©2001 The American Physical Society

A共f ,H₀兲⬀H₀^2/3f^1/3 as f⇒0, 共1a兲 A共f ,H₀兲⬀H₀²/ f as f⇒⬁. 共1b兲 Either following or in parallel to the work of Rao et al., intensive studies on the hysteresis dispersion relationships or A( f ,H₀) for different systems were carried out. These in- clude the mean-field approaches and extensive Monte Carlo simulations based on Ising-like Hamiltonians as well as ex- perimental checking of the predicted A( f ,H₀) behaviors.³ For example, Dhar and his co-workers,^13,14Sides et al.,¹⁵and Rikvold et al.¹⁶ started from the classic nucleation-and- growth concept and studied this problem in small-sized sys- tems under small amplitude H₀. They predicted that the dis- persion over an extremely-low-f range is logarithmic.

However, the dispersion for the relatively high-frequency range can be better fitted with a power law, particularly when temperature T is close to the Curie point. Either when the system size is large or when H₀ is higher, the power-law behavior is followed by dispersion in a more reasonable manner. This prediction was confirmed by Monte Carlo simulations.^8,17 On the other hand, several theoretical approaches^{18 –21} to the dispersion overall frequency range were developed too, most of which started from solving the mean-field equation of motion for the average magnetization, predicting

A⫽A₀⫹H₀^af^bg

冉

^Hf0c

冊

^{, 共2兲}

where A₀ is the area in the f⇒0 limit counting the effect from nondynamic origins, a, b, and c are the scaling expo- nents which take different values as reported from different sources, and g is a nonmonotonic function which meets g(x)⇒0 as x⇒0 or⬁. When taking the thermal fluctuations into account, Acharyya and Chakrabarti^3,8 obtained the fol- lowing dispersion for T⬎T_c:

A共f ,H₀,T兲⬀H₀^aT^⫺mg

冉

^H0cfTn

冊

^,

g

冉

^f˜⫽H0cfTn

冊

^{⬀f˜bexp共⫺f˜2/␴兲, 共3兲}

where m and n are scaling exponents. For f⇒0, Eq. 共3兲 reduces to a power law. The exponents depend on the system dimensionality and differ from those given in Eq.共1兲. Simi- lar behavior was predicted for systems under linearly varying fields.²²

In the meantime, several experiments on thin-film magnets,^23–26 including Co films on Cu substrates and Fe films on W共110兲 and Au共001兲 surfaces, were performed re- cently in order to investigate the dynamic hysteresis. Indeed, a strong dynamic contribution to the hysteresis dispersion in these systems has been demonstrated. The evaluated data on A( f ,H₀) can be reasonably fitted by Eq.共2兲, but the evalu- ated values for exponents a, b, and c are different from one system to another. The scattering of these data may be attrib- uted to the difference in coercivity for these thin films, which

is obviously not included in Eq.共1兲. A quantitative compari- son of the experimentally evaluated data with the simulated exponents seems not sufficient.

Besides the works on ferromagnetic and Ising systems reported above, the problem of dynamic hysteresis in ferro- electrics is also of interest because of the high similarity between ferroelectrics and ferromagnetics in the phenomeno- logical sense.²⁷A similar mean-field approach was developed by Acharyya and Chakrabarti.²⁸ In addition, a phenomeno- logical theory of the hysteresis dispersion in typical ferro- electrics, based on the nucleation-and-growth model, was proposed by Orihara and co-workers^29,30 and a power-law behavior over the low-f range was predicted. Nevertheless, for advanced ferroelectric applications, attention should be paid to the dispersion over the extremely-high-f range, which remains challenging to us.

From all of the above description, we understand that the physical mechanism underlying the dynamic hysteresis is the competition of the two time scales. Although it is well ac- cepted that the dynamic response at any fixed H₀ exhibits some characteristic time scale, the uniqueness of this time scale remains to be identified. From the general point of view, the evolution of some physical quantity associated with the system order parameter, no matter whether it is con- served or not, may be scaled by a generalized scaling function.^2,31Such a scaling behavior predicts the existence of a unique characteristic parameter to describe the evolution.

In this paper, we study the scalability of the hysteresis dis- persion. Let us discuss the magnetic hysteresis under a time- varying magnetic field from the point of view of the nucleation-and-growth concept. It is believed that spin rever- sal contributes dominantly to hysteresis generation, unless f is extremely high 共typically 10⁷ Hz for ferrites兲. A direct argument is that the dispersion A( f ) under different H₀ should be scalable if a unique characteristic time␶1 for the spin reversal is available and no other mechanism besides the spin reversal contributes to the hysteresis. This picture is physically quite similar to the dynamic scaling in diffusion- limited precipitation in supersaturated systems in which the correlation length of the second phase is a unique character- istic quantity.³¹Therefore, if there exists a one-variable scal- ing applicable to the hysteresis dispersion, the characteristic time scale for the system response should be unique.

The present paper focuses on the scaling behavior in spin systems. We calculate the dispersion relation for the model continuum spin system based on the three-dimensional (⌽²)² model with O(N) symmetry. Our results demonstrate the scalability of the hysteresis dispersion in this system. The remaining part of this paper is organized as follows: in Sec.

II we introduce the model continuum spin system and the numerical algorithm. The calculated dispersion and proposed scaling analysis will be presented in Sec. III, together with a discussion of the experimental relevance. A brief conclusion is given in Sec. IV.

II. MODEL AND NUMERICAL CALCULATION We start from the N-component (⌽²)² model with O(N) symmetry in three dimensions as responding to field H

⫽H₀sin(2␲^{ft). Although this model was introduced} previously,¹¹a brief description is presented here for clarifi- cation. Because the magnetization is not conserved, its relax- ation in due course is described by the nonconserved order parameter dynamics. This model is exact in the limit N

→⬁. The evolution of the system order parameter set ⌽ obeys the Langevin equation

⳵⌽_␣

⳵^{t ⫽⫺⌫}

␦^F

␦⌽_␣⫹␩␣, 共4兲 with the Gaussian white noise␩␣:

具␩␣共x,t兲典⫽0,

具␩␣共x,t兲␩␤共x

⬘

^,t

⬘

兲典⫽2⌫␦␣␤␦共x⫺x

⬘

兲␦共t⫺t

⬘

兲, 共5兲 where␣^,␤⫽1,2, . . . ,N, represent the orientation in the spin space, respectively; x is the spatial coordinate,⌫ is the mo- bility for the spin-lattice relaxation 共⬃10⁷Hz for ferrites兲, and F is the free-energy function关(⌽²)² type兴,

F⫽

冕

^d3x

冋

^{12J具ⵜ⌽␣兲共ⵜ⌽␣兲}

⫹r

2共⌽␣⌽_␣兲⫹ u

4N共⌽␣⌽_␣兲²⫺

冑

^NH␣⌽_␣

册

^{, 共6兲}

where ⌽is an N-component vector and J is the interaction between two components; r⫽T⫺T_c^TFwhere T_c^TFis the mean field T_c with T_c⬍T_c^TFin the general case; u is the prefactor and counts the contribution of the second-order nonlinear interaction, and u⫽⫺2␲^2(Tc⫺T_c^TF). Since⌽_␣⌽_␣ scales as N, each term in the bracket scales as N, and therefore so does the free energy. We assume the external field H_␣⫽H␦␣,1, pointing to axis ␣⫽1. Equation共4兲is equivalent to an infi- nite hierarchy of differential equations for the cumulants of

⌽_␣. In the N→⬁ limit, this infinite hierarchy of differential equations is truncated and the following coupled integrodif- ferential equations are obtained:^11,32

d M共t兲 dt ⫽1

2关M共t兲K共t兲⫹H₀sin共2␲^{f t}兲兴, K共t兲⫽⫺关r⫹u M²共t兲⫹uS共t兲兴,

S共t兲⫽ 1

2␲²

冕

^{01q2CT共q,t兲dq,}

dC_T共q,t兲

dt ⫽⫺关q²⫺K共t兲兴C_T共q,t兲⫹1, 共7兲 where M (t) is the component of the order parameter M along spin direction␣⫽1, i.e., magnetization, and C(q,t) is the correlation function which has the transverse component C_T(q,t) (␣⫽1) and longitudinal component C_L(q,t) (␣

⫽1):

M共t兲⫽具^⌽1共q,t兲典^,

C_T共q,t兲⫽具⌽_␣共q,t兲⌽_␣共⫺q,t兲典^, ␣⫽1,

C_L共q,t兲⫽具⌽1共q,t兲⌽1共⫺q,t兲典^. 共8兲 The numerical procedure for the hysteresis given by Rao et al.¹¹is utilized in our calculation in which various values for r and u are taken. The time step⌬t as a replacement of dt is 10^⫺7with a unit of (2⌫)^⫺1at low f (⬃10^⫺5) and reduced with increasing f, until a further reduction of ⌬t does not produce any variation of the output data within our numeri- cal uncertainty.

III. RESULTS AND SCALING ANALYSIS A. Shape evolution of hysteresis

The hysteresis loops as r and u take different values are evaluated. Figures 1共a兲–1共c兲present the calculated hysteresis at different H₀, respectively, as r⫽⫺1.0 and u⫽1.0. For each given H₀ six loops obtained at different frequencies FIG. 1. Hysteresis loops as calculated at different frequencies and amplitudes, r⫽⫺0.1 and u⫽1.0. a: f⫽1.5⫻10^⫺5. b: f

⫽1.0⫻10^⫺4. c: f⫽6.0⫻10^⫺4. d: f⫽6.0⫻10^⫺3. e: f⫽0.06.

g: f⫽0.6.

covering 10^⫺5– 10⁰ are plotted. A considerable dependence on f of the hysteresis in shape and area is clearly revealed.

Take Fig. 1共a兲where H₀⫽1.0 as an example. The loops are well saturated and show thin squarish shape as f is low. With increasing f, the loop expands along the H₀ axis, producing increasing coercivity. The high-field magnetization remains saturated. The loops show fat squarish or rhombic pattern.

With further increase in f, the loop has no longer saturated M at maximum field and a corner-rounded elliptical pattern is observed. At this stage the loop still remains symmetric around the origin. At an even higher f, the loop becomes asymmetric around the origin and a positive bias appears. At an extremely high f, the calculation produces no more loop- like hysteresis, but only a slightly tilted line.

Such a pattern evolution of the hysteresis is repeated as H₀ takes higher values关H₀⫽4.0 in Fig. 1共b兲 and H₀⫽10.0 in Fig. 1共c兲兴, while the transition from one shape to another appears at a higher f. For instance, for the same value of f 共loop e兲, the hysteresis in Fig. 1共a兲is a seriously asymmetric one, but it becomes symmetric in Fig. 1共b兲 until a well- defined and saturated one in Fig. 1共c兲. Furthermore, the evo- lution sequence remains quite similar as the temperature pa- rameter r is different. In fact, at a lower temperature 共more negative r兲one just sees higher coercivity and remanence as well as a more squarish shape. A detailed description of this evolution and related stability diagram has been given previously.¹¹

Such evolution of hysteresis with increasing frequency can be qualitatively explained in terms of spin-reversal ki- netics. Keeping in mind the simple assumption that the ki- netics of spin reversal can be characterized by a characteris- tic time, say, ␶, one understands that the shape and area of the hysteresis are fully determined by the relative dominance between␶^{and f⫺1}. Surely, the results shown in Fig. 1 tell us that this characteristic time depends on H₀. If␶Ⰶf^⫺1, the spin reversal in the system can be sufficient, resulting in near-equilibrium共quasistatic兲hysteresis. As␶Ⰷf^⫺1, the spin reversal cannot catch up in kinetics with the field oscillation such that an unsaturated elliptical loop or even geometrically nonconverged loop is generated.

B. Hysteresis dispersion

If the above argument on the characteristic time is true, the hysteresis dispersion must exhibit a single-peaked pat- tern. The calculated hysteresis dispersion A( f ) at various H₀ is presented in Fig. 2 for r⫽⫺1.0 and u⫽1.0 and in Fig. 3 for r⫽⫺3.0 and u⫽1.0, respectively, in which the f axis is in a logarithmic scale. It is clearly indicated that A( f ) indeed exhibits the single-peak pattern which is slightly tilted to- wards the high-frequency side. What should be mentioned here is that a preevaluation by taking much more data dots does not show any tail of the second peak if any. As H₀ increases, the peak position shifts gradually rightwards the high-f side and the peak value increases too. Furthermore, all curves remain similar in shape from one to another, thus predicting the possibility of one-parameter scaling.

Comparing the calculated data at different temperatures, r⫽⫺1.0 and⫺3.0, allows us to conclude that the dispersion

behavior remains the same and no qualitatively identifiable difference between them can be found. A careful comparison advises us that given a value of H₀ the dispersion curve shows a lower peak value but a higher-f position as the tem- perature is higher 共r is bigger兲. This is understandable be- cause a shorter characteristic time ␶and a lower coercivity are expected at a higher temperature, while the magnetiza- tion is lower too.

In addition, the field-dependence analysis allows us to conclude that the hysteresis dispersion indeed shows power- law behaviors over the low- and high-f ranges, well consis- tent with Eq.共1兲. While the results remain the same as those reported previously,¹¹ no more detailed description on the power-law behaviors will be given here. We shall come back to this point in Sec. III E.

C. Scaling analysis

In order to check the existence of a characteristic time ␶ applicable to spin reversal, we perform the one-variable scal- ing analysis.³³ To evaluate an arbitrary nth scaling momen- tum of the dispersion, i.e., S_n⫽兰0⬁fⁿA( f )d f , the high- FIG. 2. Hysteresis dispersion A( f ) under various H₀as labeled at r⫽⫺1.0 and u⫽1.0.

frequency decaying of A( f ) must be faster than the term f^(n⫹1). Referring to Eq.共1b兲, we understand that the zeroth momentum S₀is already diverse. A modified definition of the scaling parameter such as S_n is thus required. In fact, it is more reliable to replace variable f with log₁₀(f ). One may define several scaling parameters:

␥⫽log₁₀共f␶0兲

S_n共H₀兲⫽

冕

⫺⬁

⬁ ␥^nA共␥^,H0兲d␥^{, n}⫽0,1,2, . . . ,

␥n共H₀兲⫽S_n共H₀兲/S₀共H₀兲,

n₂共H₀兲⫽S₂共H₀兲/S₁²共H₀兲, 共9兲 where ␶0 is a time constant chosen arbitrarily 共10^⫺7 used here兲,␥is the modified frequency, S_n is the nth momentum as defined above,␥nis the nth characteristic frequency, and n₂ is the scaling factor. Note here that for the high-f range, one has A( f )⬀f^⫺1⬀10^⫺␥ so that the integral兰0⬁␥^nA(␥^)d␥ is always converged as long as n is finite. For the low-f range, one has, from Eq. 共1a兲, A( f )⬀f^1/3⬀10^␥/3. The inte- gral 兰_⫺⬁⁰ ␥ⁿ⫻10^␥/3d␥ also converges to a finite value, no matter how big the integer n is. Therefore, the scaling param- eters as given in Eq.共9兲are mathematically definable.

When our data over f⫽10^⫺6– 10² in place of 0⬍f⬍⬁

are used for evaluating the above parameters, the as- produced uncertainties are less than 0.01. These parameters as a function of H₀ each are plotted in Fig. 4 for r⫽⫺1.0 and u⫽1.0. Apart from the cases where H₀ is very small (H₀⬍1.0), a perfectly linear S_n(H₀) is obtained. The param- eter ␥1 shows a gradual growth with increasing H₀, but the scaling factor n₂ remains unchanged within the calculation

uncertainty. For other temperatures, the same conclusion is obtained. The independence of n₂ on H₀ over a wide range of H₀indicates that the dispersion curves at different H₀can be scaled using a one-parameter scaling function.

To construct such a scaling function, one assumes that a unique characteristic time for spin reversal exists, which scales the kinetics of spin reversal at a given H₀. If the scaling behavior is approved, this assumption becomes true.

Because the time scale is definable only in one-dimensional space, i.e., the possible exponent for time is 1, the scaling function can be constructed by multiplying the characteristic time by the hysteresis dispersion. The scaling function may take the following form:

W共␩兲⫽␶1/␶0A共␥^,H0兲, 共10兲 with

␩⫽log₁₀共f⫻␶1兲, log₁₀共␶0/␶1兲⫽␥1,

␶1⫽␶0⫻10^⫺␥1, 共11兲 being the scaling variables 共i.e., scaled frequency兲 and the effective characteristic time for the spin reversal. Corre- spondingly, we can define the effective characteristic fre- quency f₁⫽␶0/␶1, so that Eq.共10兲can be rewritten as

W共␩兲⫽f₁^⫺1A共␥^,H0兲. 共12兲 FIG. 3. Hysteresis dispersion A( f ) under various H0as labeled

at r⫽⫺3.0 and u⫽1.0.

FIG. 4. Scaling variables Sn, ␥1, and n2as a function of am- plitude H₀at r⫽⫺1.0 and u⫽1.0.

Plotting all calculated dispersion curves A( f ) after trans- forming them according to Eqs.共10兲and共11兲produces Figs.

5 and 6 for r⫽⫺1.0 and⫺3.0 and u⫽1.0, respectively. It is clearly shown that apart from the cases where H₀ is very small 共typically H₀⬍1.0兲, all dispersion curves A( f ) fall onto the same curve within the numerical uncertainties, dem- onstrating the scaling property of the hysteresis dispersion.

This indicates that for spin reversal in the model continuum spin system, there indeed exists a unique characteristic time which is either␶1or a time proportional to␶1, by which the hysteresis dispersion effect can be uniquely characterized.

It is interesting to compare this scaling behavior for the hysteresis dispersion with the scaling for the diffusion- limited precipitation 共DLP兲.^2,31 For the latter, one under- stands that the structure function S(q,t), where q is the spa- tial wave vector for the system, also shows the single-peaked pattern and is proportional to the spatial correlation between the compositional variable. S(q,t) at different t can be scaled using the scaling transform

W共q/q₁兲⫽q₁³S共q,t兲, 共13兲

where q₁ is the characteristic wave vector to uniquely scale the time evolution of the structure function. Here the expo- nent for wave vector q₁ is 3 because q₁ is defined in three- dimensional space. A surprising similarity between Eqs.共12兲 and共13兲 is shown. At the same time, the DLP problem can be described by a Langevin equation similar to Eq. 共4兲with similar order parameter of nonconservation.²

Our calculation confirms too that scaling function, Eq.

共10兲, applies over a wide range of temperature r. The only difference lies in the magnitude of function W(␩^).

D. Field dependence of time␶1

Let us look at the characteristic time ␶1 as a function of H₀, as presented in Fig. 7 in a double-logarithmic scale. The solid line represents an inversely linear relationship between

␶1 and H₀: i.e., the exponent for f₁ is 1:

␶1⬀H₀^⫺1,

f₁⬀H₀. 共14兲

FIG. 5. Scaling function W(␩) as evaluated by scaling trans- form, Eq.共8兲, applied to all hysteresis dispersion curves A( f ). Here r⫽⫺1.0 and u⫽1.0.

FIG. 6. Scaling function W(␩) as evaluated by scaling trans- form, Eq.共8兲, applied to all hysteresis dispersion curves A( f ). Here r⫽⫺3.0 and u⫽1.0.

Our data at two temperatures reveal clearly that the char- acteristic time␶1is linearly dependent of H₀as long as H₀is not very small. In the other words, the relationship between

␶1 and H₀ becomes linear once the dispersion reaches the scaling state or vice verse. When H₀⭐1.0, a superficial de- viation of the data from the linear relation is observed, an explanation of which will be given below. Equation共14兲pre- dicts that␶1is shorter and f₁is higher if H₀is higher. As for the temperature dependence, a shorter ␶1 for a higher tem- perature is indicated in Fig. 7, a well-accepted conclusion.

The unity exponent as defined in Eq. 共14兲 is general for scaling phenomena for the first-order phase transitions. For the DLP phenomena mentioned above, the same exponent applies if correlating q₁and time for the evolution of S(q,t):

q₁³⬀t^⫺1, 共15兲 which has been well evidenced, as long as t is not very small.³⁴

E. Power-law behaviors of the scaling function It is of interest to check the frequency dependence of the scaling function over the low- and high-f ranges, respec- tively. In fact, we see clearly that the power-law behaviors for the hysteresis dispersions, as predicted in Eq.共1兲, remain unaffected by the scaling transform, Eq.共10兲. We rewrite Eq.

共1兲as

W共f⫻␶1兲⬀共f⫻␶1兲^1/3 as f⇒0, 共16a兲 W共f⫻␶1兲⬀共f⫻␶1兲^⫺1 as f⇒⬁. 共16b兲 As an example, we present in Fig. 8 all rescaled disper- sions W( f⫻␶1) at different H₀ as a function of f⫻␶1 and a linear behavior over the low-f range is shown for each case.

The power law, Eq. 共16a兲, is confirmed. The same is appli- cable to Eq.共16b兲.

What should be mentioned here is that Eqs. 共16兲 are ac- tually another form of Eqs. 共1兲. Substituting Eqs. 共12兲 and 共14兲into Eqs. 共16兲, we obtain Eqs.共1兲once more.

F. Experimental relevance

Up to date there have been no sufficient data for ferro- magnetic solids to check the scalability of the hysteresis dis-

persions. We consider the linear relationship between␶1 and H₀, Eq.共14兲, as derived from the scaling analysis. Equation 共14兲 is not a new theoretical prediction. For ferrite solids, typical ferromagnetics, it was experimentally reported 30 years ago that the following empirical relationship holds if spin reversal takes place predominantly as a result of irre- versible domain wall migration:¹⁰

共H₀⫺H_f兲␶

⬘

⫽const, 共17兲 where H_f is a constant slightly smaller than the stationary coercivity and ␶

⬘

is the time defined as that for a half- reversal of the magnetization, i.e., some characteristic time for spin reversal. Since H_f is quite small compared to H₀, unless the latter is so low that no regular hysteresis is ob- tained 共no irreversible spin reversal occurs兲, Eq. 共17兲 is equivalent to Eq. 共12兲. Also, as H₀ is very close to H_f, ␶

⬘

has to be bigger than the prediction from the linear relation- ship H₀␶

⬘

^⫽const. Therefore, this empirical relation explains the superficial deviation of the data from the straight lines, as shown in Fig. 7.

Nevertheless, it should be pointed out that as H₀⬍H_f, the fast and reversible domain rotation rather than irreversible FIG. 7. Characteristic time␶1for the spin reversal as a function

of amplitude H₀. Here u⫽1.0.

FIG. 8. Power-law dependence of scaling function W on fre- quency f⫻␶1over the low-f range, with an exponent of 1/3. Here r⫽⫺3.0 and u⫽1.0.

domain wall motion is responsible for the hysteresis genera- tion. Such a fast domain rotation is highly related to thermal fluctuation-activated spin switching, which is reversible and thus very rapid. Consequently, a negative deviation of the characteristic time from the inversely linear relationship, Eq.

共14兲, is possible. Unfortunately, reversible spin switching at H₀⬍H_f seems not reachable by the present model.

On the other hand, we may consider similar empirical relations established for typical ferroelectric oxides such as BaTiO₃ 共BTO兲 and KNbO₃ 共KNO兲, although the present continuum model as applied to ferroelectric polar systems does not predict any ferroelectric transition.³⁵ However, the domain reversal through irreversible domain boundary mo- tion in ferroelectric solids remains similar to that in ferro- magnetic ones. We consider the case in which both multi- nucleation and domain boundary motion occur concurrently.³⁶Once the applied electric field is not very low, the new domain nucleation rate in BTO can be expressed as p(1/ms)⬀E₀^2/3, where E₀ is the field magnitude, so that a characteristic time ␶n⬀E₀^⫺2/3 can be obtained. Furthermore, the domain boundary motion velocity as a function of E₀ takes the form ␯⬀E₀^4/3, from which a second characteristic time␶␯⬀E₀^⫺4/3is predicted. The domain reversal can thus be characterized by an effective time

␶

⬘

⫽

冑

␶n␶_␯⬀E₀^⫺1, 共18兲 which as a function of E₀ takes the same form as Eq. 共12兲. Note that there were no high-f data available to confirm this relation. For KNO, the measured domain switching time as a function of E₀ was reported by Scott.³⁷ The fitted results over a wide range of E₀ confirmed the linear relationship too.

The scaling behavior as revealed presents us with a clear and simple physical picture with which the empirical rela- tions, Eqs. 共17兲 and 共18兲, work indeed, at least for ferrite- based ferromagnetic solids and ferroelectric BTO and KNO.

Although the experiments on various systems and by differ- ent researchers may show variation from one to another, the as-derived relationships are not very different from Eq.共14兲.

G. Remarks

The scaling behavior as revealed in the present model spin system relies on the assumption that the hysteresis is completely attributed to the spin-reversal mechanism, with- out contribution from any others. This assumption is ques- tionable as f is extremely high where internal induction be- comes serious with significant loss. Also, the dielectric effect should be taken into account too for realistic systems, espe- cially for insulating magnetic solids. As for ferroelectric sol- ids, the contribution over the extremely high frequency may be mainly from the electron or ion polarization, which is not considered here at all.

Although we demonstrate the scaling behavior for the present model system, no sufficient experimental evidence is available up to date. Also, a mathematical form of the scaling function W(␩) and its analytical dependence on temperature r and nonlinear correlation u have not yet been derived out.

These issues seem not easy, considering the fact that the Langevin-type equation共4兲has no analytical solution.

IV. CONCLUSION

In summary, we have presented a systematic calculation of the hysteresis dispersion in the model continuum spin sys- tems based on the three-dimensional (⌽²)² model with O(N) symmetry in the limit N→⬁. The scaling behavior for the single-peak dispersion relation has been demonstrated for this model spin system once the amplitude of the external field is not very small. This scaling effect allows us to predict the existence of a characteristic time for the irreversible spin reversal that is responsible for the hysteresis generation, by which the hysteresis dispersion is uniquely predictable. The characteristic time shows an inversely linear dependence on the field amplitude, well consistent with the well-evidenced empirical relation for ferrites and ferroelectric solids.

ACKNOWLEDGMENTS

The authors acknowledge support from the Materials Re- search Centre of the Hong Kong Polytechnic University, the NSFC, and the National Key Projects for Basic Research of China as well as LSSMS of Nanjing University.

*Electronic address: [email protected]

1G. Bertotti, Hysteresis in Magnetism 共Academic, New York, 1998兲.

2J. D. Gunton, M. San Miguel, and P. S. Sahni, in Phase Transi- tions and Critical Phenomena, edited by C. Domb and J. L.

Lebowitz共Academic, London, 1983兲, Vol. 8, p. 1.

3B. K. Chakrabarti and M. Acharyya, Rev. Mod. Phys. 71, 847 共1999兲.

4G. Bate, J. Magn. Magn. Mater. 100, 413共1991兲.

5T. Tome and M. J. de Oliveira, Phys. Rev. A 41, 4251共1990兲.

6W. S. Lo and R. A. Pelcovits, Phys. Rev. A 42, 7471共1990兲.

7M. Acharyya, Phys. Rev. E 59, 218共1999兲.

8M. Acharyya and B. K. Chakrabarti, Phys. Rev. B 52, 6550 共1995兲.

9M. Acharyya, J. K. Bhattacharjee, and B. K. Chakrabarti, Phys.

Rev. E 55, 2392共1997兲.

10J. Smit and H. P. J. Wijn, Ferrites共Wiley, New York, 1959兲.

11M. Rao, H. R. Krishnamurthy, and R. Pandit, Phys. Rev. B 42, 856共1990兲.

12M. Rao and R. Pandit, Phys. Rev. B 43, 3373共1991兲.

13D. Dhar and P. B. Thomas, J. Phys. A 25, 4967共1992兲.

14P. B. Thomas and D. Dhar, J. Phys. A 26, 3973共1993兲.

15S. W. Sides, P. A. Rikvold, and M. A. Novotny, Phys. Rev. E 57, 6512共1998兲.

16P. A. Rikvold and B. M. Gorman, in Annual Reviews of Compu- tational Physics, edited by D. Stauffer 共World Scientific, Sin- gapore, 1994兲, Vol. I, p. 149.

17M. Acharyya and D. Stauffer, Eur. Phys. J. B 5, 571共1998兲.

18P. Jung, G. Gray, R. Roy, and P. Mandel, Phys. Rev. Lett. 65, 1873共1990兲.

19C. N. Luse and A. Zangwill, Phys. Rev. E 50, 224共1994兲.

20A. Hohl, H. J. C. Linden, R. Roy, G. Goldsztein, F. Broner, and S.

H. Strogatz, Phys. Rev. Lett. 74, 2220共1995兲.

21G. H. Goldsztein, F. A. Broner, and S. H. Strogatz, J. Appl. Math.

Mech. 57, 1163共1997兲.

22G. P. Zheng and J. X. Zhang, J. Phys.: Condens. Matter 10, 275 共1998兲.

23P. Bruno, G. Bayreuther, P. Beauvillain, C. Chappert, G. Luget, D.

Renard, J. P. Renard, and J. Seiden, J. Appl. Phys. 68, 5759 共1990兲.

24Q. Jiang, H. N. Yang, and G. C. Wang, Phys. Rev. B 52, 14 911 共1995兲.

25Q. Jiang, H. N. Yang, and G. C. Wang, J. Appl. Phys. 79, 5122 共1996兲.

26J. S. Suen and J. L. Erskine, Phys. Rev. Lett. 78, 3567共1997兲.

27J.-M. Liu, H. P. Li, C. K. Ong, and L. C. Lim, J. Appl. Phys. 86, 5198共1999兲.

28M. Acharyya and B. K. Chakrabarti, in Annual Reviews of Com- putational Physics, edited by D. Stauffer共World Scientific Pub-

lishers, Singapore, 1994兲, Vol. I, p. 107.

29H. Orihara, S. Hashimoto, and Y. Ishibashi, J. Phys. Soc. Jpn 63, 1031共1994兲.

30S. Hashimoto, H. Orihara, and Y. Ishibashi, J. Phys. Soc. Jpn 63, 1610共1994兲.

31H. Furukawa, Adv. Phys. 34, 703共1985兲.

32G. Mazenko and M. Zannetti, Phys. Rev. B 32, 4565共1985兲.

33A. Craevich and J. M. Sanchez, Phys. Rev. Lett. 18, 1308共1981兲.

34T. M. Rogers, K. R. Elder, and R. C. Desai, Phys. Rev. B 37, 9638 共1988兲.

35T. Koma and H. Tasaky, Phys. Rev. Lett. 74, 3916共1995兲.

36H. L. Stadler and P. L. Zachmanids, J. Appl. Phys. 34, 3255 共1963兲; 35, 2895共1964兲.

37J. F. Scott, Ferroelectr. Rev. 1, 1共1998兲.