Dynamical Systems Analysis for Polarization in 2006

Dynamical systems analysis for polarization in ferroelectricsA. K. Bandyopadhyaya�

Government College of Engineering and Ceramic Technology, West Bengal University of Technology, 73,A. C. Banerjee Lane, Calcutta-700010, India

P. C. Rayb�

Department of Mathematics, Government College of Engineering and Leather Technology, LB Block,Sector III, Salt Lake, Calcutta-700098, India

Venkatraman Gopalanc�

Materials Research Institute, Pennsylvania State University, University Park, Pennsylvania 16802and Department of Materials Science and Engineering, Pennsylvania State University, University Park,Pennsylvania 16802

�Received 6 January 2006; accepted 15 September 2006; published online 6 December 2006�

The nonlinear hysteresis behavior in ferroelectric materials, such as lithium tantalate and lithiumniobate, may be explained by dynamical systems analysis. In a previous work, the polarization“domain wall width” was studied in terms of only spatial variation and eventually critical values ofpolarization were determined to derive the stability zone in the context of Landau-Ginzburg freeenergy functional. In the present work, the temporal dynamics of the domains themselves areconsidered by taking the time variation through Euler-Lagrange dynamical equation of motion,which gives rise to a Duffing oscillator differential equation as a governing equation. From thisnonlinear Duffing oscillator equation, three cases are studied theoretically: First, with no electricfield with and without any damping; secondly, taking the external field as static with damping; andfinally, taking an oscillatory electric field with damping. After giving perturbation at the coercivefield, the eigenvalues deduced through a Jacobian transformation of the perturbed matrix showinteresting cases of stability and instability of polarization for different values of electric field. Thepossibility of chaos at high oscillatory electric field is also briefly explored merely as a limiting casein terms of the Lyapunov exponents spectrum in our particular ferroelectric system. © 2006American Institute of Physics. �DOI: 10.1063/1.2388124�

I. INTRODUCTION

At Curie point, there is a spontaneous polarization in adipolar system, which is known as ferroelectricity. Ferroelec-tric materials have a spontaneous polarization, which has avariety of applications such as in nonvolatile memory1 andwhich remains finite even when the field is withdrawn.2 Thismakes the nonlinear behavior of polarization �P� versus theexternal electric field �E� quite interesting. In a previousstudy, we used the Landau-Ginzburg polynomial expansionfor free energy functional to explain this interesting behaviorby giving perturbations on P, E, and Landau free energy �G�by taking the spatial variation of these parameters in terms ofordinary differential equations.3 By treating it as an eigen-value problem, the critical values of polarization �Pc� wereestimated within the zone of stability through a linear Jaco-bian transformation, which showed the possibility of a largememory. The corresponding limits of domain wall widthwere also found in the case of lithium tantalate and lithiumniobate ferroelectric crystals, whose switching and hysteresisbehaviors have been studied in detail.

In Ref. 3, only the spatial variation of the domain wallthickness was considered through a governing equation, as

described by Lines and Glass4 and as also used in Ref. 2. Fora better understanding of the memory function and conse-quent switching phenomenon, it is also important to considerthe polarization domains themselves and study them as afunction of time,5 which leads to a dynamical situation. Thisapproach is quite different from our previous work,3 sincetemporal behavior of polarization is studied near domainwalls. Then the concept of dynamical equation in terms ofEuler-Lagrange equation of motion can be used in formulat-ing a Duffing oscillator equation in order to get a better pic-ture of the nonlinear behavior of polarization.

In ferroelectrics literature, the width of the domain wallshas been very controversial, with many authors claiming thatthe walls can be very wide.2,6 It is pertinent to mention thatdomain walls are highly unusual in lithium niobate because itis, unlike most ferroelectrics, not ferroelastic. Hence, thereare only 180° domain walls, which in ideal static conditionsare predicted to be a few lattice units wide. The work ofZhirnov is worth mentioning, where the domain wall widthof an important ferroelectric such as barium titanate was cal-culated analytically in terms of elastic constants throughLandau-type continuum theories.6 For a 180° domain wall,which is the commonly allowed wall in lithium niobate typeferroelectrics, the wall width was found to be 0.5–2.0 nm.6

However, an extensive first principles calculation by Vander-bilt and co-workers7,8 �see the references therein� showedthat this apparent width results instead from time-averaging

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data, which are based on rather wide spatial excursions takenby such domain walls. Recently, Floquet and Velot9 did somex-ray diffraction �XRD� work on barium titanate and relatedthe presence of twin walls to different concentrations ofpoint defects.

In this connection, it should be mentioned that there isan extensive work on phase transitions in ferroelectric mate-rials by Girshberg and Yacoby in terms of temperature de-pendence of the wall widths10 �see the references therein�. Aphenomenological model of ferroelectricity was developedthat quantitatively accounts for both displacive type andorder-disorder type properties of perovskites.10 Here, an im-portant work that also should be mentioned is that by Royt-burd, who originally formulated the more general concept ofelastic domains, indicating that the domains do form in fer-roelestic materials by minimizing the internal elasticenergies.11

We have already done a preliminary study of such do-main walls in lithium niobate3 based on a static soliton solu-tion of Lines and Glass.4 Here, the lower limit of wall widthcorresponds well with the result of Padilla et al.7 and thehigher limit agrees well with that found by Kim et al.2 Themotivation of the present work emerged from the desire tostudy the “polarization” in a dynamic situation, wherein theDuffing oscillator equation was considered to be ideal, andwhere some damping can also be applied.12 As we are doingthis investigation around the coercive field �Ec�, it indicates astudy of temporal behavior of polarization at or near thedomain walls.

It is worthwhile to note that the main purpose of theDuffing oscillator equation has traditionally been to simulatechaos. There are other unrelated equations that producechaos as well, such as the Lotka-Volterra equations13 �see thereferences therein�, Van der Pol equation,14 Lorenz equationfor biological systems,14 etc. Actually, in a hysteresis study, itis interesting to see what happens to the stability of polariza-tion at different points of interest in terms of the applied field�E�. It is of fundamental importance to study the stability orinstability in the ferroelectric dynamic system near domainwalls. It is also important to study a situation when the ex-ternal field �E� is increased beyond the saturation value �i.e.,beyond Ec�, which could lead to failure of the ferroelectricmaterials. In this way, the stability picture in a dynamicalsystem can be mapped and some ideas on the route to chaoscan be obtained. Here, first of all, the field is considered to bezero with and without damping, and then it is taken as astatic field obviously with a damping term. Finally, an oscil-latory electric field is also considered with a damping termnaturally included. In all these cases, the stability pictureneeds to be explored.

The equations are taken in the differential form. Aftergiving perturbation at a given electric field, the eigenvaluesderived through linearization by a Jacobian transformationshould show interesting features on stability. It is interestingto give perturbation, since it reveals significant informationon the stability aspects of the ferroelectric materials. Basedon a detailed analysis of the eigenvalues, the realm of stabil-

ity of polarization at different values of the external electricfield is revealed and the stability diagrams show interestingbehavior.

It may be pertinent to mention the atomistic modelingscheme for ferroelectrics, where there is a description of areasonable mechanical model for a crystal structure and theuse of classical mechanics to solve its dynamics, though itsapplication to polarization dynamics has been littlestudied.15,16 Here, we present a detailed theoretical analysisin order to elucidate different “stability pictures” and alsopresent some ideas on the onset of chaos with the help of theDuffing oscillator equation.

II. THEORETICAL DEVELOPMENT

The relation of the free energy �G� with the order param-eter �P� is described by the Landau-Ginzburg functional, ne-glecting the higher order terms, as

G = �−�1

2P2 +

�2

4P4� − EP. �1�

Here, EP is the energy due to the applied electric field �E�,and �1 and �2 are the expansion coefficients, which are bothgreater than zero and which are also important parametersfor ferroelectric materials. It should be clearly pointed outthat it has been a common practice to take only the “first”two terms, as in Eq. �1� above. Two terms are also found tobe sufficient to describe the lithium niobate and lithium tan-talate ferroelectrics, which are the ferroelectric materials infocus here. The addition of the higher order terms does notappear to make a significant difference to the basic physicsof the problem presented here, since the higher order termsare positive and the shape of the potential wells remainssimilar except for being sharper or deeper.

A. Hamiltonian formulation

In a Hamiltonian system such as the dipolar system,since we are interested in the time evolution of the polariza-tion �P�, we are inclined to take a series of N number ofrectangular boxes with uniform polarization as an “array ofdomains” in the direction of spatial x coordinate, and thenstudy the temporal evolution of polarization of the ith do-main �i.e., Pi�. These domains are stacked sideways. Now, letus write the Hamiltonian of such a system of domains interms of the momentum �pi� in the kinetic part and considerEq. �1� for free energy functional in the potential formulationas

H = �i=1

N � 1

2md�pi

2 + �i=1

N ��−�1

2Pi

2 +�2

4Pi

4� − EPi . �2�

The momentum can be defined in terms of the order param-eter �Pi� as

pi =�

�t�md

QdPi� = �md

Qd�Pi

•

. �3�

Here, we define md=mass of dipoles per unit volume �massdensity� and Qd=charge of dipoles per unit volume �chargedensity�. The concept of mass and charge in a dipolar system

114106-2 Bandyopadhyay, Ray, and Gopalan J. Appl. Phys. 100, 114106 �2006�


is lucidly given by Todorov.17 Equation �3� can be used towrite the Hamiltonian in terms of the order parameter �Pi� as

H = �i=1

Nmd

2Qd2 �Pi�2 + �

i=1

N ��−�1

2Pi

2 +�2

4Pi

4� − EPi . �4�

In terms of the order parameter �Pi�, the Euler-Lagrangeequation of motion can be easily formulated by taking theLagrangian density �L� as

L = �i=1

Nmd

2Qd2 �Pi�2 − �

i=1

N ��−�1

2Pi

2 +�2

4Pi

4� − EPi , �5�

d

dt� �L

�Pi� −

�L

�Pi= 0, �6�

�md

Qd2�Pi = �1Pi − �2Pi

3 + E . �7�

Equation �7� appears to be in a suitable form of a non-linear Duffing oscillator without any damping term by con-sidering the terms on the right hand side as the potentialformulation. Now, let us introduce a damping coefficient ��,which should exhibit the natural forces of resistance to os-cillations within the crystal system of ferroelectric materials.

B. Damping term

With the damping term, let us write Eq. �7� as

�md

Qd2�Pi + �Pi − �iPi + �2Pi

3 − E = 0. �8�

For the convenience of analysis, different participatingvariables are taken in the nondimensional form as

P� =P

Ps, �9a�

E� =E

Ec, �9b�

t� =t

tc, �9c�

where Ps=saturation polarization, Ec=coercive field in theusual hysteresis curve, and tc=critical time scale. By takingthe external electric field as static and �2�1 / Ps

2 from therelation �1�, Eq. �8� can now be written as

� mdPs

Qd2Ectc

2�d2Pi�

dt�2 + ��Ps

Ectc�dPi�

dt�− ��1Ps

Ec�Pi� + ��1Ps

Ec�Pi�

3

− E� = 0. �10�

Let us drop the prime notation for the sake of simplicity andwrite the final form of the nonlinear Duffing oscillator dif-ferential equation with a damping term with all the variablesin the nondimensional form henceforth as

d2Pi

dt2 + �dPi

dt− �1Pi + �2Pi

3 − E = 0. �11�

Here, we define tc= �1/Qd��mdPs /Ec s and �=�Ps / tcEc, �1

= �2=�1Ps /Ec are the nondimensional parameters. Differentimportant parameters including the damping coefficient arethus defined, which will be useful later in the stability analy-sis for polarization. The terms �1 and �2 for lithium niobateand lithium tantalate ferroelectric crystals will be estimatedlater to show the physical meaning for polarization values,which are needed in the construction of the stability dia-grams for dynamic analysis in our ferroelectric system. Now,let us use different cases with and without an electric field tosee the stability situation.

C. Various types of electric field

1. Without electric field and with damping

Here, let us deal with a situation in which there is noelectric field, but there is damping. Now, let us use Eq. �11�by assuming

dPi

dt= Qi, �12�

dQi

dt= − �Qi + �i�Pi − Pi

3� . �13�

In this situation, the stationary points are Qi=0 and Pi

=0, ±1. Now, if the electric field �E� is brought to zero froma very high value, then a small perturbation in E �i.e., �E�will cause a perturbation or fluctuation in P �i.e., �P�, exceptthat these perturbations are given with respect to time as avariable on an array of domains. These perturbation termscan be written as

d

dt��Pi� = �Qi, �14�

d

dt��Qi� = − ��Qi + �1�1 − 3Pi

2��Pi. �15�

At the stationary or equilibrium point at Qi=0 and Pi= Pic,these perturbation terms can be put in a matrix form in aperturbed state from the above system of equations and alinear Jacobian transformation is made as follows:

�d

dt��Pi�

d

dt��Qi� = � 0 1

�1�1 − 3Pic2 � − �

��Pi

�Qi . �16�

From the above perturbed Jacobian matrix, the characteristicdeterminant must vanish at the stationary points, i.e., at Qi

=0 and Pic=0, ±1, as

� − � 1

�1�1 − 3Pic2 � − � − �

� = 0. �17�

The eigenvalues can be deduced as

�2 + �� − �1�1 − 3Pic2 � = 0, �18a�



� =− � ± ��2 + 4�1�1 − 3Pic

2 �2

. �18b�

Two distinct cases can be described as follows:Case C1-I. When Pic=0, then �= �−�±��2+4�1� /2 at

the stationary point �Pi=0, Qi=0�; the polarization is un-stable under this situation.

Case C1-II. When Pic= ±1, then �= �−�±��2−8�1� /2at the stationary points �Pi= ±1, Qi=0�. In this particularcase, there are three different possibilities.

Case C1-II(a). When �2=8�1, the polarization is stable,since both eigenvalues are negative.

Case C1-II(b). When �2�8�1, the system is stable,since both eigevalues are negative.

Case C1-II(c). When �2�8�1, both eigenvalues arecomplex with the real part as negative, and hence, the polar-ization is stable though slowly.

The stability diagrams corresponding to these cases willbe shown in Sec. III.

2. Without electric field and without damping

Here, first of all, we deal with a situation where there isstill no electric field and we assume the damping term to bezero. Since the analysis in terms of perturbed matrix andcharacteristic determinant is similar to the above case, let usstraightway write the eigenvalue equation as

�2 − �1�1 − 3Pic2 � = 0. �19�

Two distinct cases can be described as follows:Case C2-I. When Pic=0, then �= ±��1 at the stationary

point �Pic=0, Qi=0�; the polarization is unstable.Case C2-II. When Pic= ±1, then �= ± i�2�1 at the sta-

tionary points �Pic= ±1, Qi=0�.Thus, from the eigenvalues, we get a complex oscillatory

solution, which indicates that the fluctuation in polarizationis oscillatory with respect to the variation of time in thissituation without any electric field and without any dampingterm, but this fluctuation is not high.

3. Perturbation in a static field

Here, the electric field �E� is assumed to be static. Now,from the nonlinear Duffing oscillator equation �11�, we get

dPi

dt= Qi, �20�

dQi

dt= − �Qi + �1Pi − �1Pi

3 + E . �21�

From the equilibrium point, i.e., at Qi=0, we get

�1Pi − �1Pi3 + E = 0. �22�

Here, we get three �approximate� solutions for critical valuesof polarization as Pic= +1.001 41, −0.998 58, and −0.002 83,as estimated by a computer, with �1=3.534102 and E�=1�i.e., E=Ec� for lithium niobate. This value has been calcu-lated by taking �=1.8849�109 V m/C from Kim et al.2 andPs=0.75 C/m2 and Ec=40 kV/cm also from Ref. 2. Forlithium tantalate,2 the value of �1=4.2058�102, which is

quite close to that of lithium niobate. Hence, these values ofcritical polarization �Pic� only for lithium niobate will beused later in our stability analysis.

Now, let us give perturbation at E�=1 �i.e., E=Ec�, andwe get the same set of perturbed equations such as �14� and�15�. As before, we can make a linear Jacobian transforma-tion on these equations, and the perturbed matrix will looklike Eq. �16�, and hence, these equations are not shownagain.

From this perturbed Jacobian matrix, the characteristicdeterminant must vanish at the stationary points, i.e., at Qi

=0 and Pic=−0.002 83, +1.001 41, −0.998 58, and we get theeigenvalues as

�2 + �� − �1�1 − 3Pic2 � = 0, �23a�

� =− � ± ��2 + 4�1�1 − 3Pic

2 �2

. �23b�

Two distinct cases can be described as follows:Case C3-I. When Pic=−0.002 83, then �= �−�± �2

+3.99�1� /2 at the stationary point �Pic=−0.002 83, Qi=0�;the polarization will be unstable under this situation.

Case C3-II. When Pic= +1.001 41, −0.998 58, then �

= �−�±��2−8.0339�1� /2 and �= �−�±��2−7.9659�1� /2 at the stationary points �Pic= +1.00141, Qi=0� and �Pic

=−0.998 58, Qi=0�, respectively.In this particular case, there are three different possibili-

ties.Case C3-II(a). When �2�8.0339�1, the polarization is

stable, since both eigenvalues are negative.Case C3-II(b). When �2=8.0339�1, the system is stable,

since both eigenvalues are negative and equal.Case C3-II(c). When �2�8.0339�1, both eigenvalues

are complex with the real part as negative. Although this caseleads to an oscillatory situation, it indicates the stability atthe equilibrium point, where the perturbation was applied.All these values clearly demonstrate the extent of stability orinstability in a ferroelectric system, which is revealed underperturbation. These eigenvalues will be used to derive thestability diagrams in Sec. III for various cases.

Similar analysis can be made when �= �−�

±��2−7.9659�1� /2.

D. With oscillatory electric field

Here, the perturbation aspect is dealt with by taking theexternally applied electric field �E� as oscillatory, and obvi-ously, the damping term is also included. Here, we aremainly interested in the effect of this oscillatory field onpolarization, after giving perturbation, relevant for switchingbehavior of ferroelectric materials. Now, from the abovenonlinear Duffing oscillator equation �11�, we get

dPi

dt= Qi, �24�

dQi

dt= − �Qi + �1Pi − �2Pi

3 + E0 cos��t� . �25�



Since Eq. �25� is not autonomous, we define a new vari-able R=�t. Then, Eq. �25� becomes

dQi

dt= − �Qi + �1Pi − �2Pi

3 + E0 cos�R� , �26�

where

dR

dt= � . �27�

It is not possible to find the “stationary points” under theabove situation, and hence the eigenvalues cannot be de-duced. In our system of ferroelectrics such as lithium nio-bate, the Lyapunov exponents are determined by time seriesanalysis through the Gram-Schmidt reorthonormalizationmethod, as described by Wolf et al.18 We have numericallycomputed the spectrum of all the three Lyapunov exponents,as will be shown later in Fig. 9.

A stability analysis should be carried out in terms of aresponse function at different values of the electric field,which has been done by Machado et al.19 for a very similarsituation with an oscillatory electric field. Since our systemis quite similar to that of Machado et al.,19 we are not in-clined to give a repetitive description. In this reference, ofcourse, there was chaos with the increase of the drivingforce, i.e., electric field. In our case, one of the Lyapunovexponents is gradually tending towards positive values as theamplitude of the electric field �Eo� increases, indicating theonset of chaos in our system of ferroelectrics �Fig. 9�. How-ever, we are merely pointing out the stability aspect of sucha system within the proper hysteresis zone in this paper,since here we are mainly concerned with the development ofa “governing equation” for polarization without any interac-tion term between the domains.

III. RESULTS AND DISCUSSION

It is necessary to visualize the stability diagrams at dif-ferent values of critical polarization �Pic� for an “array” ofparallel domains with uniform polarization in a ferroelectricmaterial, as assumed in the present work. Such diagramshave been constructed with different values of relevant andconcerned parameters. Different cases are discussed as fol-lows:

First of all, it should be mentioned that all the results andthe consequent discussion are presented in different partscorresponding to various cases involved, from item numberA1 to A12. The summary of all the results are tabulated inTable I to bring clarity. The case number is referred to thecases C1, C2, C3, and D of Sec. II, and some high fieldcases, as described in the following text itself. The equationnumbers are mostly the eigenvalue equations, except for Eqs.�1�, �20�–�22�, �24�, and �25�, as described in Sec. II, and thestability behavior is given briefly in this table.

�A1� A potential diagram is drawn, as shown in Fig. 1,wherein two symmetric potential wells are clearly shown forthe Landau-Ginzburg type of potential, with E=1 and twowells at ±1, which has been used in the Hamiltonian formu-lation and consequent Euler-Lagrange equation of motion inSec. II A.

Here, our entire stability analysis will be presented onlyfor lithium niobate, where the value of �1=3.534�102 istaken for all the stability diagrams shown below. It can beassumed that this stability analysis will also be valid for arange of different ferroelectric materials. As already men-tioned in Sec. II C 3, the physical significance of this expan-sion coefficient is to be able to calculate the polarizationvalues for the stability diagrams of actual ferroelectrics in thedynamical analysis.

�A2� In case C1-I with Pic=0 for no electric field but

TABLE I. Stability behavior for different cases.

Item No. Case No. Figure No. Equation No. Stability behavior

A1 Sec. II A 1 �1� Potential diagram at E=0, 1 and 100A2 C1-I 2 �18b� Spiral formA3 C1-II�a� and II�b� Not shown �18b� Stable nodesA4 C1-II�c� Not shown �18b� Stable focus �as inside Fig. 3�A5 C2-I 3 �19� Saddle point �darker�A6 C2-II 3 �19� Stable circle insideA7 C3-I Not shown �23b� Saddle point �as in Fig. 3�

4 �23b� Oscillations of P vs t5 �23b� Damping of oscillationsNot shown �23b� Stable orbit with damping

A8 C3-II�a� and II�b� Not shown �23b� Stable nodesC3-II�c� Not shown �23b� Stable focus

A9 Sec. II C 3 6 �22� Equilibrium hysteresis curve of P vs EA10 Sec. II C 3 7 �22� Stability at high fieldA11 Sec. II C 3 Not shown �20� and �21� Center shifted to Pi=1.1192

Not shown Saddle point as in Fig. 3Not shown Center shifted to Pi=−0.8054

A12 Sec. II D 8 and 9 �24� and �25� Chaotic instability and Lyapunov exponents



with a finite value of damping, i.e., for Pi=0.01, Qi=0,E=0, and �=0.5, the phase plane diagram shown in Fig. 2seems to indicate a stable behavior.

�A3� In cases C1-II�a� and C1-II�b� with Pic= ±1 for noelectric field but with a finite value of damping, i.e., for Pi

=0.9991, Qi=0,E=0, and �=0.5, the phase plane diagramshows “stable nodes.”

�A4� In case C1-II�c� with Pic= ±1 for no electric fieldbut with a finite value of damping, i.e., for Pi=0.9991, Qi

=0,E=0, and �=0.5, the phase plane diagram is very similarto that shown in the middle of Fig. 3, which indicates thestable focus.

�A5� In case C2-I with Pic=0 for no electric field and nodamping, i.e., for Pi=0.01, Qi=0,E=0, and �=0, the Qi ver-sus polarization �Pi� diagram is shown in Fig. 3. It is seenthat the stationary point is a “saddle point” in the phase plane�Pi ,Qi� indicating instability. It should be also mentionedthat this “stationary point” is situated in terms of the poten-tial well diagram at the centre, as shown in Fig. 1, with apotential maximum that indicates instability.

�A6� In case C2-II with Pic= ±1 for no electric field andno damping, i.e., for Pi=0.9991, Qi=0,E=0, and �=0, thephase plane diagram shown in Fig. 3 in the middle portionindicates that the stationary points behave like “centres,” i.e.,periodic solution around these two stationary points at ±1,when the situation is stable.

�A7� In case C3-I with Pic=0 for a static electric field,the situation is represented first without any damping forclarity, i.e., for Pi=0.01, Qi=0,E=1, and �=0, and the phaseplane diagram shows a saddle point which is very similar tothat shown in Fig. 3 �darker portions�.

The corresponding plot of Pi versus time �t� for such asituation with �=0 is shown in Fig. 4. It is clearly seen thatthe amplitude of fluctuation of polarization is increasing withtime. It is quite remarkable to note that this fluctuation goesbeyond the limits of ±1 at t=0.

This amplitude needs to be damped, which is shown inFig. 5 in the plot of Pi versus time �t� with �=0.5, where itis seen that the fluctuation quickly dies down, say, after anondimensional value of time �t� of about 20 and the situa-tion becomes stable without any oscillation thereafter.

The corresponding phase plane diagram with damping�i.e., for Pi=0.01, Qi=0, E=1, and �=0.5� is not shownhere, where the situation gives rise to a stable periodic orbit,which is naturally expected in case of damping.

�A8� In cases C3-II�a�–C3-II�c� with Pic=0, ±1 and Qi

=0, with a static electric field �E=1� and a finite value ofdamping ��=0.5�, the phase plane diagram shows a similarbehavior as shown above in cases C1-II for �a� stable nodes,�b� stable nodes, and �c� stable focus.

�A9� Now, it is interesting to see the hysteresis behaviorof a ferroelectric systems, which throws more light on the

FIG. 1. �Color online� Landau-Ginzburg potential for E=0 and E=1 �over-lapping� and for E=100 for the value of �1=353.43 for lithium niobateferroelectric crystal.

FIG. 2. �Color online� Phase plane diagram at E=0 and �=0.5 showing twocircling orbits with a stable focus with centers at ±1.

FIG. 3. �Color online� Phase plane diagram at E=0 and �=0 showingsaddle point �darker regions� and two circling orbits with centers at ±1.

FIG. 4. �Color online� The polarization �Pi� vs time �t� plot showing oscil-lations for E=1 and �=0.



realm of stability in the context of dynamical systems analy-sis that is important for switching behavior. It is shown inFig. 6 in a P vs E diagram considering Landau-Ginzburg�LG� potential of Fig. 1. Basically, it is described as an equi-librium diagram �XYZOABC curve� in the literature.20 Thisequilibrium curve makes it possible for us to observe thestate of the ferroelectric system as the independent variable Echanges. In this case, the equilibrium curve is given by E= �1�−P+ P3�, which is obtained from the criteria of station-ary points of Duffing oscillator Equation.

It is interesting to note that on either side of the curve, atthe stable points at Ec= ±129.07, the polarization jumps to astable value if the electric field is further increased, i.e., thepoints Z and A separate the stable intervals from the unstablezone. This is what is called the “points of bifurcation,” whichdoes not have to lead to chaos.20 This jump is clearly markedby two longer arrows in Fig. 6 to indicate how polarizationbypasses the unstable zone and goes over to a stable zone ina hysteresis curve, which is very relevant for switching be-havior of ferroelectric materials. On the top of the right handside of the curve and on the bottom of the left hand side ofthis curve, the stability part is shown by two convergingsmall arrows, whereas the unstable portions are shown bytwo diverging arrows in the middle of the diagram for clarity.If the shape of this hysteresis curve needs to be changed,then the Landau-Ginzburg potential needs to be suitablymodified.

�A10� For a high static electric field, where the symme-try of the potential wells break �Fig. 1� and the right handwell is deeper than that on the left hand side, Fig. 7 showsthe phase plane diagram of the above system only at E=−129.07 with no damping. This point is just before theonset of a breakup of the double well potential to a singlewell. It is clearly seen that although it starts at the right handside of the curve, it ultimately converges to a stable point onthe left hand side at the center of a circling orbit. The casewith the damping is not shown here, but it should obviouslygive rise to more stability.

�A11� Various cases of stability of polarization at E=0and E=Ec=1 with and without any damping term have beenshown in the above diagrams. All these cases of stabilityhave not possibly been observed in ferroelectric materials. Atleast, the theoretical foundation has not been created so far,except in an important work by Wagner and co-worker onpiezoceramics involving strains and electric enthalpy densityfunction in a completely different context at very weakfields.21,22 In order to undertake a detailed work towards un-derstanding the switching behavior of ferroelectric materialsfor nonvolatile memory applications, the notion of stabilityor instability of polarization in the context of fluctuation isimportant, which has been done in the above analysis tosome extent. However, in order to get a complete picture, weneed to study a situation where the electric field is furtherincreased, or rather to study what happens to a ferroelectricmaterial at very high electric field.

Here, the potential well diagram, as shown in Fig. 1 forE=100 �nondimensional value�, indicates that the left handside of the potential well at −0.8054 is quite shallow with apeak near the center at −0.3138 denoting instability, but theright hand side of the potential well at +1.1192 is quitedeeper indicating a higher degree of stability, and therebyshowing asymmetry. The corresponding diagram for E=1shows a highly symmetric two-well �i.e., potential minima�picture. It is pertinent to mention that there could be asymmetry-breaking mechanism in the ferroelectric material,as we go from E=1 to E=100, which could be slow butcontinuous. The relation between this sort of behavior withthat of stability is beyond the scope of this paper, and thisdefinitely merits further attention. Therefore, the stability

FIG. 5. �Color online� The polarization �Pi� vs time �t� plot showing theeffect of damping on oscillations for E=1 and �=0.5 at about t20.

FIG. 6. �Color online� The equilibrium hysteresis curve of Pi vs E showingthe stability points and bifurcation points at Z and A.

FIG. 7. �Color online� Phase plane diagram at E=129.07 and �=0 showingstability on the left hand side with a center near −1.



analysis at E=100 assumes significance. For this value of E,there are three values of critical polarization at +1.1192,−0.3138, and −0.8054, respectively.

For the phase plane diagram �not shown here� for a situ-ation for Pic= +0.9991, Qi=0, E=100, and �=0, the circularorbit is created with the center shifted to +1.1192. The sameplot with the same values of the parameters as above butwith Pic=−0.3138 is not shown here, which simply shows asaddle point behavior �like Fig. 3� with two orbiting circles,except that the orbit on the left hand side is much smallerthan that on the right hand side. This is obviously due to theshallow potential well near −0.8054. The figure with thesame values of the parameters as above but with Pic

=−0.8050 �also not shown here� simply shifts the center ofthe orbit at −0.8054.

When the damping is applied to such a situation, i.e., forPic=−0.8050, Qi=0, E=100, and �=0.5, it has been ob-served that the stability diagram forms “spiral-like” circularorbits, which end at the center at the critical value of Pic

=−0.8054.�A12� The above analysis clearly shows the stability and

instability of polarization in the realm of a dynamical sys-tem. Here, for an oscillatory electric field, instability maylead to a chaotic situation, particularly at E0=130, �=1, and�=0.5. It is shown in Fig. 8, which clearly indicates a cha-otic behavior. Although the onset of chaos is not shown here,it can be safely concluded that with increasing electric field,the chaos ultimately occurs in such systems, which should beclear from the Lyapunov exponents spectrum. For oscillatoryelectric field, Fig. 9 shows the Lyapunov exponents spectrumfor a ferroelectric such as lithium niobate. As described inSec. II D, one of the exponents is gradually tending towardspositive values after a value of Eo of about 136 �nondimen-sional value� indicating the onset of chaos. From the natureof the curve in Fig. 9, it is also seen that the same Lyapunovexponent remains in the positive domain at or near Eo=130,where chaos has already been observed as shown above.

Finally, it is pertinent to mention that this paper is nei-ther about the theory of chaos, which is available in manyimportant textbooks on chaos,14,20 nor about a chaotic situa-tion in real ferroelectric materials, wherein no referencesseem to exist. Since we are doing dynamical analysis in fer-

roelectrics at different values of electric field, it was defi-nitely of scientific interest to see what happens when theelectric field is further increased, which shows in our case achaotic instability. In ferroelectrics, within the concernedhysteresis parameters, the stability and instability need to bestudied seriously, which has precisely been done in thiswork. To that end, the governing equation needs to be devel-oped, which has also been done in the present work. Futurework will focus on extracting an expression for the spa-tiotemporal polarization wall width in ferroelectrics.

IV. CONCLUSION

The present study on the evolution of polarization withtime gives rise to a nonlinear Duffing oscillator equation asthe governing equation, taking the ferroelectric material as aHamiltonian system and using the equation of motion ofEuler-Lagrange. For the Duffing oscillator, different situa-tions are tackled: with zero electric field with and withoutany damping, with a static field obviously with a dampingterm, and also with an oscillatory field with damping. Withthe objective of studying the dynamic behavior of such asystem, the stability analysis is carried out in terms of differ-ent eigenvalues, which are obtained by giving perturbationthrough linearization by Jacobian transformation of the per-turbed matrix. These eigenvalues show very interesting casesof stability and instability for various cases of critical valuesof polarization at different points of interest in the usual P vsE hysteresis curve. Such information for any ferroelectricmaterial will be useful in understanding their switching be-havior.

ACKNOWLEDGMENTS

The authors would like to thank Professor E. Klotins,Institute of Solid State Physics �Latvia�, for an interestingdiscussion on Duffing oscillator equation. One of the authors�V.G.� would like to acknowledge support from the National

FIG. 8. �Color online� Phase plane diagram showing chaos in an oscillatoryfield with Eo=130, �=0.5, and �=1.

FIG. 9. �Color online� Lyapunov exponents spectrum showing chaos at highoscillatory field, at the amplitude of the applied electric field, once at 130and again after about 136.



Science Foundation �Grant Nos. DMR-0122638, DMR-0507146, DMR-0512165, DMR-0349632, and DMR-0103354�, and NSF-MRSEC center at Penn State �DMR-0213623�.

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Dynamical Systems Analysis for Polarization in 2006