PDF Frequency dispersion of low-field dynamic hysteresis in model system

Frequency dispersion of low-field dynamic hysteresis in ( η ² ) ³ model system

H.B. Zuo

^a

, H. Yu

^b

, M.F. Zhang

^a

, J.C. Han

^a

, J.-M. Liu

^b,c,∗

aCentre for Composite Materials and Structure, Harbin Institute of Technology, Harbin, China

bNanjing National Laboratory of Microstructures, Nanjing University, Nanjing 210093, China

cInternational Center for Materials Physics, Chinese Academy of Sciences, Shenyang, China Received 14 September 2005; received in revised form 30 November 2005; accepted 6 December 2005

Abstract

A numerical simulation on dynamic Landau–Ginzburg (η²)³ model system is performed to investigate the dynamic hysteresis and external field-induced phase transitions. The dynamic symmetry breaking of the free energy is discussed. Centro-symmetry-broken hysteresis is found when the external field is low, which is very different from the high-field behaviors of the hysteresis dynamics. A resonant peak and a saddle back of the hysteresis dispersion curve are observed for the low-field case, which may correspond to two oscillating modes and two resonant characteristic times.

© 2006 Elsevier B.V. All rights reserved.

PACS: 75.60.Ej; 77.80.Dj; 75.40.Gb

Keywords: Dynamic hysteresis; Resonance; Frequency dispersion; Symmetry breaking

1. Introduction

Hysteresis is the central feature of ferroic materials below the Curie point and the macroscopic hysteresis should be the natural outcome of the microscopic description [1]. When a time-oscillating external magnetic or electric field is applied to a magnetic or ferroelectric system, a time lag of the system response to the external field occurs, resulting in the hystere- sis behavior if one plots the system order parameter against the external field. In fact, such a hysteresis typically represents the first-order switching or reversal transitions of domained order parameter upon the varying external field.

The hysteresis loop contains rich information on the dynamic response of the system order parameter to external field, such as coercive field and remnant magnetization (polarization). The area of a closed loop represents the energy loss or dispersion of the hysteresis in one period. The up to date high-speed memory industry arouses interests to study the principles of dynamic hys- teresis. Besides, the intriguing physics of hysteresis underlying

∗Corresponding author. Tel.: +86 25 83596595; fax: +86 25 83595535.

E-mail address:liujm@nju.edu.cn (J.-M. Liu).

field-induced phase transitions and non-equilibrium dynamics is still challenging.

In the recent ten years, both of the dynamic symmetry break- ing and frequency dispersion of the hysteresis in ferroic systems have been focused on, which was fully reviewed in[2]. Tak- ing magnetic system as example, for the former issue[3–7], a parameter defined asQ

mdt, wheremis the magnetization andtthe time in one period, was introduced to describe the sym- metry of a hysteresis loop. The dynamic phase transition refers to a transition from aQ= 0 phase to aQ=0 phase, which is a manifestation of the coercivity property. For the later topic, the area of a closed loop is a function of temperature and external field. The areaA

mdh, wherehis the external field, namely the energy loss, is sensitive to the frequency of the ac external fieldhof frequencyfand amplitudeH0, when other conditions are fixed. This hysteresis dispersion was firstly systematically studied by Rao et al.[8,9]. A power-law scaling for the disper- sion against bothfandH0in either low-for high-fregime was argued. Dhar and Thomas[10,11]and Sides et al.[12]started from the classic nucleation-and-growth concept and studied this problem in small-sized systems under small amplitudeH0. They predicted that the frequency dispersion over an extremely-low-f range is logarithmic. Besides, the frequency dispersion exhibits

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doi:10.1016/j.mseb.2005.12.017

a resonant behavior. A time parameterτ= 1/fr, wherefris the frequency corresponding to the maximum of the area as a func- tion of field frequency, is related to the characteristic time of domain reversal in such systems[13].

The motivation of this paper is to investigate the hysteresis dynamics in low field amplitude and high frequency. In memory technologies and for the high speed and low-energy-consume demands, the material behaviors in the condition of high fre- quency and low field amplitude are especially important. In the present work, we study hysteresis dispersion behavior using anO(N)-symmetric Landau–Ginzburg (η²)³model. It has been demonstrated that a double-power law in different temperature ranges exists in this kind of system[15]. However, it is found that in low H0 and highf, the pattern of hysteresis loop and dispersion curve is different from that in high H0 case. The centro-symmetry-broken loop appears and the dispersion behav- ior of high frequency does not obey a simple power law. We bring forward a double-resonant-modes mechanism to explain this phenomenon: one is the resonance of domain-reversal oscil- lating, the other is the resonance of domain-canted oscillating. In our earlier work[16], we studied the results of Monte Carlo sim- ulation and mean-field calculation based on an Ising model and found double resonant peaks and asymmetric loop. The present work provides a theoretical support on this phenomenon within the framework of Ginzburg–Landau theory.

2. Model

The O(N→ ∞)-symmetric (η²)² and (η²)³ models were firstly discussed by Mazenko and Zannetti[14]and introduced to study relaxation behaviors associated with phase transitions of magnetic and ferroelectric systems by Rao et al.[8,9]. The order parameter motion equation takes the Langevin form:

∂η_α

∂t = −Γ δF

δηα +ϕ_α (1)

with the Gaussian white noise satisfying

< ϕ_α(x, t)>=0,

< ϕ_α(x, t)ϕ_β(x, t)>=2Γδ_αβ(x−x)δ(t−t) (2) whereα,β= 1, 2,. . .,N, represents the orientation in the spin space,xis the spatial coordinate,Γ the mobility for the spin- lattice relaxation,Frepresents the free energy under an external field, which can be written as:

F =

d³x 1

2J(∇η_α· ∇η_α)+ r

2(η_αη_α)+ u

4N(η_αη_α)² + v

6N²(ηαηα)³−√ NHαηα

(3) whereηis anN-component vector andJthe interaction between two components,u andvare the prefactors of the non-linear terms. The phase-diagram in r–u plane can be found in [9], and hereu=−25.59 andv=105.28 are taken if not noticed elsewhere. Such a choice ofuandvrefers to a consideration that the system exhibits enough large magnetization and static coercive field for calculation reliability. The external magnetic

field is in theα= 1 direction, i.e.H_α=Hδ_α,1. In theN→ ∞limit, this infinite hierarchy of differential equations is truncated. We substitute Eq.(3)into Eq.(1), and obtain the following coupled integro-differential equations:

dM(t) dt = 1

2[A(t)M(t)+H(t)]

A(t)= −(r+uM²+uS+vM⁴+2vM²S+uS²) S(t)= 1

2π²

₁

0 q²C⊥(q, t) dq d

dtC_⊥(q, t)= −

q²−A(t)

C_⊥(q, t)+1 H(t)=H0 sin(2πft)

(4)

wheretis time,fthe frequency,Han external ac magnetic field, AandSare mediate variables, magnetization (order parameter) M is alongα= 1 direction, andC(q,t) is the correlation func- tion which has the transverse componentC_⊥(q,t) (α=1) and longitudinal componentC11(q,t) (α= 1):

M(t)= Φ1(q, t),

C⊥(q, t)= Φ_α(q, t)Φ_α(−q, t), α=1, C11(q, t)= Φ1(q, t)Φ1(−q, t)

(5)

The numerical simulation procedure was given by Rao et al.

[8,9] and is utilized to investigate the hysteresis behaviors in different temperature (r), amplitude (H0) and frequency (f) of ac external field (magnetic field here). We concern more with the behaviors at lowH0and highf. Here, we give a brief descrip- tion of this numerical method. Eqs. (4)are a set of non-linear intergraodifferential equations which cannot be solved analyti- cally. Therefore, we solve these equations numerically. Firstly, the differential variables should be converted to difference form.

For instance, dM∼M, dt∼t, wheremeans a finite small change of these variables. Then, these differential equations can be dealt with through programming compute. The integral Eqs.

(4)were evaluated by using Gauss quadrature routines. The fre- quencyfand time steptdetermine the consumption of CPU time. For low frequency and then a long-time circle, the CPU time required for solving the equations is extremely long. We performed a testing of the time steptso that a further decreas- ing oftwill no longer change the evaluated value ofM, which guarantees the reliability of the numerical simulation. The ini- tial value ofMis 0 andS is 1/(4π²). In computing, after tens of circles of the hysteresis, the system turns to be stable. The effective simulation data should be taken from the stable state.

For more details about this numerical procedure, please refer to [8,9].

3. Results and analysis

We study the simulation results of hysteresis loop and fre- quency dispersion behaviors in various conditions (r,H0, andf).

The behaviors for low- and high-H0cases are distinct from each other in high frequency regime, which may indicate different dynamic response mechanism.

3.1. Shape evolution of hysteresis loop

InFig. 1, the shape evolution of hysteresis loop under small H0and differentfof external ac field is a kind of dynamic phase transition. With increasingf, a pinched loop (Fig. 1(a)) turns to be saturated and squarish, and the coercivity increases, until the loop reaches its maximum area, as shown inFig. 1(b). For the low-floops of type (a) and (b), the variation of coercivity con- tributes to the change of the area. If the system has a positive remnant magnetizationMr, the loop in high-fcase will evolve to an inclined ellipse in the upper plane ofMaxes. It is noticed that the centro-symmetry is broken, as shown inFig. 1(c) and (d). However, given a fixed frequency even if it is high, a sym- metry center for a loop is possible as long as the field amplitude is high enough, though it is also in the upper plane ofMaxes.

Fig. 1(e) is a loop obtained at extremely-highf, where the sys- tem cannot respond to the applied field at all and then rests onMr.

3.2. Symmetry breaking of free energy

The dynamic phase transition basically stems from the broken symmetry of the system free energy. The (η²)³model is essen- tially a free-energy phenomenological theory, and therefore, it can be utilized to analyze the energy evolution of the system.

Fig. 2shows the diagram of free energy as a function ofMat different frequencies for calculating the loops shown inFig. 1.

In low-frequency regime, as shown inFig. 2(a) and (b), the free energy curve is symmetric for the positive and negative field

Fig. 1. Hystresis loops at various frequencies atr= 1,H0= 0.1: (a)f= 0.01; (b) f= 0.05; (c)f= 0.23; (d)f= 0.28; (e)f= 2.1; (f)f= 20.

Fig. 2. System free energy vs.Mat different frequencies atr= 1,H0= 0.1: (a) f= 0.01; (b)f= 0.05; (c)f= 0.2; (d)f= 0.23; (e)f= 0.28; (f)f= 2.1.

and has two minimums corresponding to the double stationary state. With increasingf, the stationary state in the negative-M side turns to be metastable, and the hysteresis loop evolves into the pattern as shown inFig. 1(c). At higherf, the negative-M stationary state disappears, and the curve only has one positive minimum. At ultra-highf, the curve becomes close to a straight line at a fixedMr. The curves change greatly from (c) to (e), which are near the inflection point to be discussed later. It is a dynamic effect that the pattern of free energy evolves from bilateral symmetric to asymmetric. The symmetry transition of free energy results in the shape evolution of hysteresis loop as shown inFig. 1.

3.3. Peak and saddle back of the dispersion curve

In Fig. 3, we present a log–log plot of the hysteresis dis- persions at different H0, namely, the area of the loop versus frequencyf, given the value ofH0. There is one peak of each curve showing the maximum of the energy loss. The peak is a resonant state where the frequency of external periodic field is equal to the eigen-frequency of the domain reversal at a given H0. It is the point that the energy loss of the domain reversal is maximal. The peak has a blue shift whenH0increases. It was demonstrated that, for high-field case, there is double-power law in the low and high frequency limit for (η²)³model[15]. The power laws correspond to the straight lines in log–log plots, as shown inFig. 3, curves e–g. However, for the low-H0case, as shown by curves a–d, the situation is different. An obvious con- trast is that there is a saddle back between two inflection points in high frequency regime. An increasing ofH0gradually elimi-

Fig. 3. Log–log plots of hysteresis area vs. frequencyfat differentH0, form a–g,H0= 0.05, 0.1, 0.2, 0.5, 1, 2, 5.

nates the saddle-back part and then the high-fpart becomes close to a power law. For an understanding of this saddle-back fea- ture, we choose one dispersion curve at lowH0inFig. 4(a) for a more detailed discussion.Fig. 4(b) shows the first-order and second-order derivatives of the dispersion with respect to logf:

the point where the first derivative d log(A)/d log(f) = 0 is the

Fig. 4. (a) A log–log plot ofA–fforr= 1 andH0= 0.1; there is one peak and one saddle-back in the curve. (b) Solid line: the first derivative d log(A)/d log(f);

dot line: the second derivative d²Alog/dflog2. The point that d log(A)/d log(f) = 0 is the peak and the points that d²log(A)/d log(f)²= 0 are the two ends of the saddle-back.

resonant peak withf= 0.05 (Figs. 1(b) and 2(b)), and the points where the second derivative d²log(A)/d log(f)²= 0 are the inflec- tion points withf= 0.28 (Figs. 1(d) and 2(e)) and 2.1 (Figs. 1(e) and 2(f)), between which is a saddle back.

The centro-symmetry-broken hysteresis loop at lowH0con- tributes to the saddle-back feature of the dispersion curve. It was well established that the frequency dispersion of a centro- symmetric loop obeys a simple power law both in low- and high-f limit. This may suggest the mechanism underlying the evolution of this kind of loop is single. But for an irregular loop, because of the two asymmetric ends of the loop, the area evolution becomes complicated and does not obey a power law.

3.4. Mechanism and discussion

We propose that there are two kinds of resonant mechanisms coupled into the phase transitions in the low-H0region: one is the domain-reversal resonance, the other is the domain-canted res- onance. For the latter, the polarized domains do not reverse, but oscillate around the equilibrium position. This kind of resonance has a higher eigen-frequency than the former one.Fig. 5(a) is a sketch map of this mechanism, where (I) is the reversal oscillat- ing and (II) is the canted one. The two resonant peaks of these two oscillating modes are roughly plotted as a hint inFig. 5(b).

Fig. 5(c) shows the combination of them and the second peak is coved up by the first oscillating mode to become a saddle-back part, considering the fact that the first mechanism (domain rever- sal resonance) is dominant. Single mode of domain oscillating will bring a simple power-law dispersion relation. However, if the system is determined by the competition of two oscillating modes, the hysteresis loop will lose its centro-symmetry. The irregular and centro-symmetry-broken loop is the result of the co-existed oscillating modes. In another word, the physics pic- ture is that in low-H0middle-frange some of the domains are reversed but others are not, but keeping oscillation in small angle.

InFig. 3, we have noticed that the first resonant peak has a blue shift with the increasing amplitude of the applied field. In high- field case, the first resonant peak is dominant and fully blankets the second resonant peak. Consequently, the saddle-back part will never appear in the dispersion curve.

In our recent early work, the results of mean-field calculations and Monte-Carlo simulations on Ising model showed the double- peak frequency dispersion and the possibly different dynamic mechanism when the hysteresis is symmetric or asymmetric [16]. The similarity between the results of the present paper and our earlier related work suggests that the double-peak (or quasi double-peak) frequency dispersion and the corresponding dynamic mechanisms may be observable for the low field ampli- tude cases in many systems. The major challenge for experimen- tal identification of this behavior is that the data for the hysteresis in low field amplitude cases may not be reliable to resolve the canted oscillating mechanism. The Ising model is a microscopic and statistical spin model but Landau–Ginzburg equation is a macroscopic phenomenological theory. Both results support the similar phenomenon, which allows us to argue that the physics of low-field hysteresis in magnetic and ferroelectric system is general and model independent.

Fig. 5. Sketch map of two oscillating modes. (a) (I) the domain-reversal oscillating; (II) the domain-canted oscillating. (b) Two resonant peaks corresponding to (I) and (II). (c) The combine of two mechanism forms a typical low-H0dispersion curve.

4. Concluding remarks

In conclusion, the hysteresis frequency dispersion is simulated utilizing O(N)-symmetric (η²)³ model based on Landau–Ginzburg equation. The shape evolution of the hystere- sis loop is studied and centro-symmetry-broken loop is found.

We also present the diagram of free-energy symmetry broken to illustrate the dynamic phase transition. The low-field and high-frequency behaviors are focused and abnormal dispersion curve is discovered. There are one peak and a saddle-back part in a low-field curve. The peak is a domain-reversal res- onant peak and the saddle-back part between two inflection points suggests the exist of another resonance mode. We pro- pose that the second resonance mode belong to domain-canted oscillating.

Acknowledgement

The authors would like to acknowledge the financial sup- port of the National Key Project for Basic Researches of China

(2002CB613303), NSFC through the innovative group project and projects 50332020, 10021001 and 10474039.

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