Dipole alignment and dielectric susceptibility of defective ferroelectric: Monte-Carlo simulation

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Dipole alignment and dielectric susceptibility of defective ferroelectric: Monte-Carlo simulation

J.-M. Liu

^a,b,c,*

, K.F. Wang

^a

, S.T. Lau

^b

, H.L.W. Chan

^a,b

, C.L. Choy

^a,b

aLaboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China

bDepartment of Applied Physics, Hong Kong Polytechnic University, Kowloon, Hong Kong, China

cInternational Center for Materials Physics, Chinese Academy of Sciences, Shenyang, China

Abstract

We present a Monte-Carlo simulation on the dipole alignment conﬁguration and dielectric susceptibility of a defective ferroelectric lattice with randomly distributed lattice defects in the framework of the Ginzburg–Landau theory.

These defects are assumed to suppress locally the electric dipoles. It is found that with increasing defect concentration the dipole lattice conﬁguration evolves from a normal ferroelectric state to a two-phase coexisted state, where highly polarized regions are embedded in a paraelectric matrix. The dielectric susceptibility in lattices with various defect dis- tributions is studied. The simulated results are used to explain the measured dielectric response in ferroelectric copolymers containing defects induced by proton irradiation.

PACS:77.80.Bh; 77.22.Gm; 77.80.Dj

Keywords:Dielectric susceptibility; Random defect model; Ferroelectric relaxor

1. Introduction

The role of defects in ferroelectrics (FEs) has been one of the fundamental issues in the physics of ferroelectrics. A comprehensive knowledge of these defects and their impact on the materials

property would be important for materials pro- cessing and property optimization [1]. Here, we focus on a special type of crystal defects which are believed to be responsible for the relaxor-like behaviors in some doped ferroelectrics[2–8]. These defects can be either impurity atoms distributed randomly in the lattice or off-center dopant ions which generate the so-called internal random fields or random bonds [7]. Although it remains chal- lenging to identify directly the core configuration and physical behaviors of these defects, extensive

doi:10.1016/j.commatsci.2004.12.034

*Corresponding author. Address: Department of Applied Physics, Hong Kong Polytechnic University, Kowloon, Hong Kong, China. Tel.: +86 25 359 6595; fax: +86 25 359 5535.

E-mail address:[email protected](J.-M. Liu).

www.elsevier.com/locate/commatsci

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studies on relaxor ferroelectrics (RFEs) have pro- vided a sound basis for establishing the roles of these defects[7–11].

The role of defects mentioned above is consid- ered in the compositional inhomogeneity model for RFEs in which the type of defects can be impurity atom or dopant ion [9]. For the former, the impurity atoms in the lattice are viewed as disor- dered (random) static defects coupled locally with the stable dipole. The defects may change the mag- nitude of local dipoles from site to site, by sup- pressing the dipole moment. For the latter, if the lattice is doped by off-center dopants, a local dipole will appear and the local dopant may occupy one of the crystallographically equivalent off-center sites around the unit cell center, and the resul- tant dipole moment may align along one of the equivalent vectors [7]. We focus on the first class of defects with the defect model by Semenovskaya et al. [12] and the Ginzburg–Landau thermody- namic description of the ferroelectric phase transition[13].

In this paper, we summarize our Monte-Carlo (MC) simulation on the lattice configuration of electric dipoles and the dielectric property in a FE lattice with local lattice defects[14]. We shall study in detail the effects of induced defects on the equilibrium lattice configuration and the dielectric susceptibility. We also compare our simulations with the experimental data on proton-irradiated poly(vinylidene fluoride-trifluoroethylene) 70/30 mol% copolymer (P(VDF-TrFE)) [15].

2. Model and procedure of simulation

The MC simulation is performed on a two- dimensional (2D) L·L lattice with the periodic boundary conditions applied, where the PE phase takes the square configuration and FE phase the rectangular one [14]. We once employed a 3D 16·16·16 cubic lattice for a pre-simulation and did not find significant difference of the simulated results (e.g. dielectric susceptibility to be defined below) from those we obtained for a 2D lattice of L64. Therefore, all of our simulations re- ported below will be performed on a 2D lattice.

For each lattice site, a dipole vectorP= (Px(r), Py(r)) where Px and Py are the two components along x-axis and y-axis, respectively, is imposed with its moment and orientation deﬁned by the energy minimization. We consider the contributions from the Landau double-well potential, the long- range dipole–dipole interaction and gradient energy associated with domain walls. The long-range elastic energy may not be important for RFEs. The Landau double-well potential fLcan be written as [13]:

f_LðPiÞ ¼A₁ðP²_xþP²_yÞ þA₁₁ðP⁴_xþP⁴_yÞ

þA12P²_xP²_yþA111ðP⁶_xþP⁶_yÞ ð1Þ where subscriptirefers to lattice sitei,A1,A11,A12

and A111 are the energy coeﬃcients, respectively.

In the present model, the dipole vector is assumed to take one of the four orientations: [±1, 0] and [0, ±1].

If there exists a spatial distribution of the dipoles (refer to moment or orientation), a so-called gradients of the polarization ﬁeld is introduced [12,13]:

f_GðPi;jÞ ¼¹₂G₁₁ðP²_x;xþP²_y;yÞ þG₁₂P_x;xP_y;y

þ¹₂G44ðPx;yþPy;xÞ²þ¹₂G⁰₄₄ðPx;yPy;xÞ² ð2Þ wherePi,j=oPi/oxj. Since parametersG11,G12,G44

and G⁰₄₄are all positive, in most cases this energy term is positive, favoring a ferroelectric ordering.

The dipole–dipole interaction is long-ranged. In the SI unit, the energy for site ican be written as [13]:

fdipðPiÞ ¼ 1 8pe0v

X

hji

PðriÞ PðrjÞ jr_ir_jj³

"

3½PðriÞ ðrir_jÞ½PðrjÞ ðrir_jÞ jrirjj⁵

# ð3Þ

where hji represents a summation over all sites within a separation radius R centered at site i, parametersri,rj,P(ri) andP(rj) here should be vectors, riand rjare the coordinates of sites iand j, respectively. An eﬀective cut-oﬀ of R at R= 8 is taken in our simulation (the as-induced error is less than 2%). It is clearly seen that a minimizing

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offdipfavors the alignment of dipoles in the head- to-tail form. Finally, the electrostatic energy induced by an external electric ﬁeld is:

f_EðPiÞ ¼ PiE ð4Þ whereEis external electric ﬁeld, andPiandEare vectors. In our simulation, vectorEtakes the [1, 0]

direction. The Hamiltonian for the system is H ¼P

hiiðfLþfGþfdipþfEÞ.

As for the eﬀect of defects, besides the Landau energy fL, the other three terms fG, fdip and fE

may be affected too. Introduction of randomly distributed defects imposes a spatial distribution for the coefficients A1, A11, A12 and A111in the Lan- dau energy Eq. (1). Following the argument of Semenovskaya et al., we also assume that only A1 is affected by the defects and the other three coefficients remain unchanged. That is[12]:

A₁ðriÞ ¼A₁₀þb_mc

A₁₀¼aðT T⁰Þ; a>0 ð5Þ where a> 0 is a materials constant,Tis the temperature,A₁₀is the coefficientA₁in Eq.(1),T⁰is the critical temperature for a normal FE crystal with first-order phase transition features, c takes 0 or 1 to represent a perfect site or a defective site, bm is the coefficient characterizing the influence of defects on T⁰ and can be written as bm= adT⁰(C0)/dC0, where C0 is the average concentration of defects. Here, a positivebmis taken because we focus on the defect-induced suppression of polarization in the lattice.

With a set of given system parameters, the MC simulation is performed via the following procedure. A defect is attached to a site and the proba- bility is determined byC0. A random number R1

between [0, 1] is generated and randomly a defect is chosen to attach to this site if R₁<C₀, and is not attached otherwise. The simulation begins at an extremely high temperature T= 8.0 at which no freezing eﬀect is retained within the period of simulation (we choose T⁰= 3.0). For a site i, a new dipole with its moment and orientation chosen randomly is assigned to replace the original dipole at this site, and the total energy diﬀerence is assumed to beDH. A probabilitypfor the new dipole choice is calculated by the Metropolis algorithm. Then one cycle of simulation is completed.

Consequently the simulation is performed until a huge number of Monte-Carlo steps (mcs, one mcs represents L·L cycles) is reached in order to approach the equilibrium dipole conﬁguration.

The data presented below represent an averaging over four runs with diﬀerent seeds for random number generator of the initial lattice and defect distribution.

Under an external ac-electric ﬁeld E¼E₀þ E_msinð2pxtÞ, where E0is the dc-bias ﬁeld, Em

is the ac-signal amplitude, is the frequency, t is time, applied to the lattice, the dielectric suscepti- bilityvcan be written as[6]:

v⁰¼ C NT

X^N

i

1 1þ ðxs=x0Þ²

* +

v⁰⁰¼ C NT

X^N

i

xs=x0

1þ ðxs=x0Þ²

* + ð6Þ

wherehirepresents the conﬁguration averaging,v⁰ and v⁰⁰ are the real and imaginary parts of v (we focus solely on the real part), x0is the polariton frequency which is a material constant, s is the averaged time for dipole switching from one state to another which is scaled by x0, N=L² and C is a temperature-dependent constant. In our simulation here,x0= 1 is assumed for simpliﬁcation.

What should be pointed out here is that a kinetic MC algorithm may be employed in order to correlate linearly the time of simulation (unit:

mcs) with the real time. With this algorithm, the dipole-exchange between the nearest-neighbor site-pair is performed, in which the order parameter (dipole) over the whole lattice should be con- served. However, driven under an ac-ﬁeld, the dipole switching at a site from one state to another results in breaking of the conservation law. The time of simulation in the kinetic MC algorithm may not represent the real time. Therefore, a simulation on the frequency-dependence of the dielectric susceptibility may not possible. In the present simulation, we do not simulate this frequency effect. Instead, only the dielectric susceptibility as a function of temperature and ﬁeld-amplitude E0is simulated. The other lattice parameters used in the simulation are chosen and the dimensionless normalization of them is done following the works

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by Hu et al. on the dynamics of domain switching in BaTiO3system[14]. These parameters are given inTable 1.

3. Result of simulation

We first look at the equilibrium dipole alignment in the lattice as shown inFig. 1for different C₀ at b_m= 10 and T/T⁰= 0.5 in the left column andT/T⁰= 0.2 in the right column, where the value ofT⁰remains fixed. For C0= 0, a normal FE configuration with a multi-domained structure at both values of T is observed. The parallel dipole alignment within each domain and the head-to-tail pattern at the domain boundaries (90-domains dominate here) can be clearly identified. For a non-zeroC0, the lattice inhomogeneity is remark- able and one sees the moment shrinking (depolarization effect) for the dipoles along the domain boundaries although the defects are randomly distributed in the lattice. These regions can be viewed as the PE phase. At C0= 0.2 and T/T⁰= 0.5 (as shown inFig. 1), the PE phase is predominant in the lattice and tends to isolate the FE regions.

The lattice consists of local FE regions embedded in the matrix of PE phase, a typical picture for RFEs. With increasing C0, the FE regions con- tinue to shrink both in size and in volume fraction.

It is seen that only very small dipole clusters exist atC0= 0.6 (T/T⁰= 0.5).

At a lowerT(T/T⁰= 0.2), similar effects of the defects on the configuration are observed. Com- paring the two columns one can see the influence of thermal fluctuation on the lattice inhomogeneity. At C0= 0.2, the FE phase still dominates at T/T⁰= 0.2, but it becomes nearly unidentifiable at T/T⁰= 0.5. The depolarization effect along the domain boundaries seems much weaker at T/T⁰= 0.2 than at T/T⁰= 0.5. Some well-defined

FE regions, in spite of their small size, can be observed in the lattice even at C0= 0.6.

v as a function of T for diﬀerent values of C0

under a given E is evaluated and the results are plotted inFig. 2(a). It is seen that the introduction of defects indeed affects the dielectric property of the system. AtC0= 0 (normal FE), a Curie–Weiss type single-peaked v–T relation is generated with Tc2.0. As C0 increases, the single-peaked v–T curve shifts toward the low-Tside and also down- wards slightly. In addition, the normal FE shows a sharp decrease invat a temperature just belowTc, but this feature is weakened for the lattice with defects, i.e. the FE transition becomes diffused with increasingC0. The effect ofbmonvis quite similar to the effect of C0on v, and details of the results are not shown here.

To understand these effects of defects on the dielectric susceptibility, one refers to the definition of v, Eq. (6), where v is roughly proportional to T¹ands²(assumingxs/x01 is always satis- fied). However, the relaxation time s can be roughly viewed as a constant at TT⁰, while at TT⁰it depends exponentially onT because of a significant slowing-effect of the dipole relaxation time in the FE state. In a very rough sense, one has:

v/ 1=T; atT T⁰

1

T expða=TÞ; a>0; at T T⁰ (

ð7Þ

whereais a positive constant. Introduction of defects into the lattice shifts the FE transition to a lower T. Since thev–Tpeak position corresponds roughly toT⁰, the entirev–Tcurve must shift leftward asC0orbmincreases. As for the downward shifting of thev–Tcurve, it is noted thatsis longer at a lower T. For the present model the leftward and downward shifting of thev–Tcurve to a lower Tand a smallervare correlated with each other, as C0orbmincreases.

It is interesting to investigate the effect of external electrical field onv. This effect was investigated experimentally for relaxor-like systems [16,17].

However, only the eﬀect of E0on v is simulated.

Fig. 2(b) shows thev–Tcurves for diﬀerent values of E0 at x= 0.1 for a system with C0= 0.4 and bm= 10. The eﬀect of the dc-bias E0 is opposite

Table 1

System parameters used in the simulation

Parameter Value Parameter Value Parameter Value

T⁰ 3.0 a 1.0 A11 0.5

A12 9.0 A111 0.8 G11 1.0

G14 0.2 G44 1.0 L 64

bm 012 C0 01.0 R 8

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Fig. 1. Snapshoted dipole conﬁguration of mode ferroelectric lattice with diﬀerent defects concentration C0.bm= 10.0, E0= 0.0, Em= 0.2 andx= 0.1. Left column:T/T⁰= 0.2, right column:T/T⁰= 0.5. The areas covered by the cycles are the FE regions.

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to that ofEm(not shown here). It is found that a higher Em results in a leftward and upward shift of the v–T curve, while the susceptibility at T>T⁰ is little aﬀected. The physics underlying it is quite simple. A dc-bias favors the FE ordering due to the reduced electrostatic energy, and consequently makes the relaxation time much longer.

4. Experimental relevance and remarks

A comparison of these simulated behaviors with experimental data would help providing a justiﬁca- tion for the defect model. Nevertheless, it is not easy to search for an experimental system where the defect concentration can be controlled in a de-

fined manner in order to understand the role of defects. Recently, it has been repeatedly verified that tremendous variations in microstructure and physical property of a variety of ferroelectric polymers may be generated by irradiation with high-energy electrons or protons [18,19]. The most interesting features associated with such an irradiation is the significant improvement of the electrostrictive properties of the polymers. Several experiments revealed that the high-energy particles injected into the copolymers convert the single FE phase into a two-phase coexisted microstructure with a FE phase and a PE phase. In fact, one may argue that the high-energy particle irradiation introduces randomly distributed point-like defects into the sample and disrupts the stability of the FE phase.

The microstructural details are determined by the energy level and the dose of the particles. Given an energy level, the defect concentration C0is directly related to the dose.

The above argument provides justification for the application of the present model to explain the results of the irradiation experiments. While several careful experiments on the effect of irradiation were performed, in this section, we compare our model simulation with our earlier experiments on the dielectric susceptibility of proton-irradiated poly(vinylidene fluoride-trifluoroethylene) 70/30 mol% copolymer (P(VDF-TrFE)). For details of the experiments, refer to our earlier report [15].

Fig. 3shows the measured dielectric constant at 1 MHz as a function ofTin a cooling run for several samples with the same initial state but irradiated at diﬀerent dose levels. Our simulation is performed for the cooling run too. While the non-irradiated sample shows the typical ﬁrst-order FE phase transitions at 70C, the irradiated samples exhibit a broader transition peak and the peak height becomes smaller and the peak position shifts to a lower T, as the irradiation dose increases. These features are well reproduced in our simulations, as shown inFig. 2(a). In fact, hys- teresis measurements indicated that a relaxor-like two-phase microstructure is formed in irradiated ferroelectric copolymers. What should be mentioned here is that the peak-shift of the peak position becomes quite small when the dose is higher than 150 Mrad and this is not consistent with

Fig. 2. Simulated dielectric susceptibility v as a function of temperature kT for (a) diﬀerent defect concentration C0

(bm= 10.0, E0= 0.0, Em= 0.2 and x= 0.1), (b) diﬀerent dc- electric biasE0(Em= 0.2,C0= 0.4,bm= 10.0 andx= 0.1).

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our simulations where a signiﬁcant shift continues at a very high defect concentration (C₀> 0.6).

However, it remains a big challenge to obtain quantitative consistency between the model predic- tion and observed properties of real materials. The present model employs a square lattice and the energy formulation which are not strictly applicable to the copolymers. Moreover, it was found experimentally that the copolymer exhibits a structural change upon irradiation which is argued to be responsible for the giant electrostrictive eﬀect.

The as-induced elastic energy (may not be long- ranged) is not included in the present model, which remains a problem to be focused on. As for the simulation method, a kinetic MC algorithm applicable to the present model may need a further clariﬁcation.

5. Conclusion

In conclusion, we have presented a Monte-Car- lo simulation on the dielectric and ferroelectric properties of a ferroelectric lattice of randomly distributed point-like defects and compared the simulated results with the properties of proton- irradiated poly(vinylidene fluoride-trifluoroethylene) 70/30 mol% copolymer (P(VDF-TrFE)). The simulation has revealed that the introduction of defects results in an evolution of the dipole config-

uration from a normal multi-domain ferroelectric lattice to a relaxor-like two-phase coexisted microstructure consisting of ferroelectric regions embedded in the matrix of a paraelectric phase. Our simulations show qualitatively good consistency with the measured dielectric behaviors of irradiated P(VDF-TrFE) by using the dose level as a measure of defect concentration.

Acknowledgements

This work is supported by the Centre for Smart Materials of the Hong Kong Polytechnic Univer- sity and the Hong Kong Research Grants Council (PolyU 5147/02E). The authors acknowledge the ﬁnancial support of the National Key Project for Basic Researches of China (2002CB613303), NSFC through the innovative group project and projects 50332020, 10021001 and 50172020, as well as the K.C. Wong Education Foundation.

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