Ferroelectric Properties of Polycrystalline Ceramics with Dipolar Defect Simulated from the

CHIN.PHYS.LETT.Vol.25,No.4(2008)1442 Ferroelectric Properties of Polycrystalline Ceramics with Dipolar DefectSimulated from the Potts–Ising Model∗ZHANG Yan-Fei(张艳飞)∗∗,WANG Chun-Lei(王春雷),ZHAO Ming-Lei(赵明磊),LI Ji-Chao(李吉超), ZHANG Rui-Zhi(张睿智),LIU Jian(刘剑),MEI Liang-Mo(梅良模)School of Physics and Microelectronics,State Key Laboratory of Crystal Materials,Shandong University,Jinan250100(Received10November2007)Physical properties of polycrystalline ferroelectrics including the contributions of thefixed dipolar defects and the average grain size in the Potts–Ising model are simulated by using the Monte Carlo method.Domain pattern,hysteresis loop and switching current of the polarization reversal process are obtained.Two processes are considered in our simulation.In thefirst one,the grain texture of ferroelectric ceramics are produced from the Potts model,and then the Ising model is implemented in the obtained polycrystalline texture to produce the domain pattern,hysteresis loop and switching current.It is concluded that the defect has the ability to decrease the remnant polarization P r as well as the coercivefield E c.The back switching is obviously observed after the electricfield is off,and it shows some variation after introducing thefixed dipolar defect.Meanwhile,the spike of the switching current is found to lower with the increasing defect concentration and the decreasing average grain size.PACS:77.80.−e,61.72.Bb,05.50.+qFerroelectric(FE)materials have been extensively studied in recent years because of their promising properties for nonvolatile random access memories as well as other practical applications.The defect ef-fect has been one of the fundamental physical is-sues in ferroelectrics.[1−7]The defect is known to influence the remnant polarization,motion of do-main wall,polarization reversal and leakage current. Therefore,the defect in ferroelectrics has been sub-ject of many studies in experimental and theoretical aspects.[1−11]On the other hand,the conventional theory-experiment[12−21]basis for scientific research has been expanded to include computer simulation, which is an increasingly important component of many research lines.Thus interest in computer simulation of physical properties in ferroelectrics has grown recently due to the application of the switching in ferroelec-tric ceramics for nonvolatile date storage.[8−11,22−27] Among them,the Monte Carlo(MC)simulation has been proven to be an effective method used to study the physical properties of ferroelectrics.Wu et al.[26] have studied the influence of defect-induced internal field on the dielectric response of relaxors.Based on these results,the ferroelectric system with two-type defects has been further investigated.[10]It was found that thefirst-type off-centre dipolar defect will cause two90◦successive rotations of the direction of the dipoles during the switching process,while the switching behaviour of the ferroelectric system with the second-typefixed-dipolar defect shows almost a single180◦flipping.In this Letter,the dipolar defect influence on the properties of polycrystalline is studied using the MC method on the Potts–Ising model.Wefirst simulate the polycrystalline texture from the Potts model using the Monte Carlo method,which is based on the energy minimization of an ensemble of electric dipoles on the lattice configuration.Then the Ising model is imple-mented to investigate the effects of the proposed defect on the ferroelectric properties.To our best knowledge, this is thefirst attempt to study the effect of dipolar defect on the hysteresis loop and switching behaviour in ferroelectric ceramics from the Potts–Ising model.A two-dimensional square lattice of size200×200 under periodic boundary conditions is employed in our simulations.The periodic boundary conditions imply that the simulated domain structure reflects a domain structure within an infinitely large polycrystalline ce-ramic,in which there are no free edges and thus the edge effect is not considered.Monte Carlo step(MCS) is used as time scale and one MCS represents200×200flip events.The Potts model isfirst proposed as a general-ization of the Ising model for simulation the critical transitions in magnetic materials with more than two degenerate states.[28]Hamiltonian of a q-state Potts model isH=−ijδσiσj,(1)whereσi=0,1···q−1stands for the different grain orientation;δis the Kronecker operator,when σi=σj,δ=1,otherwiseδ=0.The Potts model has been widely used to simulate the grain growth in metal materials.[17,18]It has also been applied to de-scribe ferroelectric behaviour,particularly in the con-text of grain boundaries,anti-phase boundaries in in-commensurate ferroelectrics and considering the grain size,surface nucleation,and related effects in general-∗Supported by the National Natural Science Foundation of China under Grant No10474057.∗∗Email:zhangyf@c 2008Chinese Physical Society and IOP Publishing LtdNo.4ZHANG Yan-Fei et al.1443izing the switching theories.[19−21]The simulation procedure involves the flipping of the orientation of a randomly chosen site to any of the other neighbouring orientations.This procedure is found to be very inefficient since the probability of the acceptance of a spin is very small if q value is very large.However,for low q values,large occur-rences of grain coalescing are observed.Since such coalesce events are strictly forbidden in grain growth,an appropriate large value of q is required to mini-mize the chances of two grains of the same crystal orientation meeting and coalescing.[29]Therefore,the suitable number of states in the Potts model is q =68from our trial simulating processes.This value of q represents the maximum value of grain orientation.The orientation of each individual grain is assigned by a q value,and it does not change with time in the following domain switching simulations.After we obtain the polycrystalline texture with desired average grain size from the Potts model,the spin-1/2Ising model is implemented in the simulation.The two spin states represent the two dipole orienta-tion states in ferroelectrics,and the dipole could be reversed by the external electric field.The Hamilto-nian of the Ising model has the following form:H =− ijJ ij S z i S z j −2ηE (t )iS zi cos θi ,(2)where S z i =±1stands two opposite directions of the dipoles,J ij measures the interaction of pair interac-tions,here only the nearest-neighbouring couplings are taken into account in the simulation.The ex-change strength J ij =J cos(θij ),where θij is the an-gle between orientation of σi and σj .When the two dipoles are within the same grain,we have σi =σj .Otherwise,σi =σj ,that is,the two dipoles lies in the grain boundary.The angle between the two grains is defined as θij =2π(σi −σj )/q .Here ηis the dipole moment,E (t )is the applied external field.An electric field in sinusoidal form is applied to reverse polariza-tion,E (t )=E 0sin(ωt ),(3)where E 0is the amplitude of the applied field,whichis set to be larger than the coercive field in our sim-ulation,E 0=6;ωis the frequency of the applied field,ω=0.001MCS −1is used in our simulation.The global polarization is taken the average of the dipole in the whole lattice,i.e.,P = i S zicos(θi )N ×N.(4)Monte Carlo simulation with Metropolis sampling is employed for the Ising system to obtain the global polarization and hysteresis loops.Temperature T is set to 2.0J in the simulation.The Curie tempera-ture of the two-dimensional infinite Ising-1/2lattice is T c =2.2692J ,so our simulation is at temperatureT =0.88T c .The fixed dipolar defect is presumed that the di-rections of these selected dipoles can not be reoriented by the thermal distribution and the applied electric field.In order to make the simulation more realistic,both the position and the direction of these fixed dipo-lar defects are all assigned randomly.Therefore,do-main patterns,hysteresis loops as well as the switching current are obtained for polycrystalline texture with different defect concentrations and different average grain sizes.Figure 1shows the grain textures simulated from the Potts model and the corresponding domain pat-terns from the Ising model with different defect con-centrations.Figures 1(a),1(b)and 1(c)represent the case of no defect,1%defect concentration and 3%de-fect concentration,respectively.The grain size is con-trolled by the MCS.The average grain area is defined as the total lattice site 200×200divided by the total number of grains.Thus the unit of grain area is the number of lattice site.Then the average grain size is the square root of the average grain area.From the obtained polycrystalline texture pattern,the average grain size in Fig.1is of 22lattice sites.The defect concentration is defined as the ratio of the number of defect sites to the total lattice site number 200×200.The defects are marked as dark dots in the diagrams.To visualize the simulation results,the domain orien-tation is presented by different grey scales.The dark colour stands for one direction of polarization,and the white colour represents the opposite direction.The domain patterns are obtained under an electric field at the saturated polarization.We can find that all grains are in single domain.The different grey scale of the domains,i.e.,the different orientation of the domain is due to the different grain orientations.The number of the domain orientations is more than that simulated in single crystal ferroelectrics.[9,10,22]Figure 1(d)shows the domain nucleation under the external electric field for systems without defect (left panel)and with defect concentration of 3%(right panel).It can be easily seen that the nucleation occurred mainly at the grain boundary without defect as the electric field applied to the system.However,for the system with defects,the domain nucleation occurred mainly around the dipolar defect sites in the grains.The hysteresis loops for different defect concentra-tions are shown in Fig.2(a)and for different average grain sizes in Fig.2(b),respectively.It can be seen from Fig.2that the effect of defect and the average grain size on the coercive field and remnant polariza-tion in the hysteresis loops.Both the coercive field and the remnant polarization decrease with the increase of the defect concentration as shown in Fig.2(a).The increasing average grain size enhances the remnant polarization,see Fig.2(b),but slightly increases the coercive field.1444ZHANG Yan-Fei et al.Vol.25 Fig.1.Polycrystalline textures and corresponding domain patterns for different defect concentrations:(a)no defect,(b) 1%defect concentration,(c)3%defect concentration.The average grain size is about R=22.(d)The domain nucleation under the external electricfield for the system without and with defects.Fig.2.(a)Hysteresis loops for different defect concentra-tion of average grain size R=20.(b)Hysteresis loops of different average grain sizes without defects.The remnant polarization P r and the coercivefield E c,which are taken from the hysteresis loops as such in Fig.2,are plotted in Figs.3(a)and3(b)against av-erage grain size R for different defect concentrations. The remnant polarization P r increases with the grain size monotonically for all the three defect concentra-tions,and its value is much depressed by the defect effect.This is consistent with the previous theoretical calculations that the increase of defect concentration leads to a decline of polarization hysteresis loops and brings about a decrease in P r.[30]The coercivefield E c slightly increases with the average grain size for sam-ple without defects introduction,see the curve shown in Fig.3(b).It is interesting that E c becomes less dependent on the average grain size when the dipo-lar defects are introduced,see curves of1%and3% defect concentration.Also the coercivefield E c is re-duced by the effect of the dipolar defects.Especially, with the high defect concentration to3%,the coercive field E c is much depressed by the defect effect.This is consistent with thesimulation results in single crys-tal ferroelectrics[10]and the experimental results.[31] In Ref.[10],it was found that dipolar defects decrease the coercivefieldand saturate polarization compared with the defect-free case.Thus we can conclude that the dipolar defects have similar effect on the hysteresis loop in both the polycrystalline and the single crystal ferroelectrics.Fig.3.(a)The remnant polarization Pr and(b)the co-ercivefield Ec as a function of average grain size R for different defect concentrations.Fig.4.Pulsed electricfield applied to the ferroelectric polycrystalline texture.The switching current is then generated from the global average polarization P as i(N)=P(N)−P(N−1),when the pulsed electricfield applied,here N is the time scale represented by the MCS.The applied pulse electricfield as is shown in Fig.4.The strength of the electricfield is set as E0=5,which is larger than theNo.4ZHANG Yan-Fei et al.1445 coercivefield strength.The response of the polariza-tion and the switching current is shown in Figs.5and6respectively for different defect concentration anddifferent average grain size.In Fig.5(a),the arrows mark the back switching after the electricfield is off, i.e.,after the removal of the electricfield,a gradual decrease of the polarization is observed,however,hav-ing different ratios depending on defect concentration. It is noticed that lower defect concentration results in a smaller and faster polarization decay.An opposite behaviour is noticed for higher defect concentration. We can also see that thefirst pulse switches down the polarization,and the secondpulse reverses the polar-ization upwards.The corresponding switching current is shown in Fig.5(b).From the curve without de-fect,we can see that after the removal of the applied negative electricfield,a spontaneous deploring pro-cess occurs,indicated by positive current response as shown in the inset of Fig.5(b).The switching current exhibits a maximum when the electricfield immedi-ately resumes to its peak value from zero,i.e.,the cur-rent shows spike.The current maximum corresponds to the reorientation contribution of the ferroelectric dipoles.The value of maximum current that corre-sponds to the second negative appliedfield is lower than for thefirst one.It means that the electrical response in the ferroelectrics is partly determined by the signal that has been previously applied to the sam-ple.The simulation result agrees well with the exper-imental result obtained in MAPBB crystals.[32]The spikes of the switching currents lower in the next pulse with increased defect concentration from0%to3%as shown in Fig.5(b).In contrast,the peak values of the back switching current increase as seen in the inset of Fig.5(b).Under these conditions,a single cycle takes 120MCS.Fig.5.Response of polarization(a)and switching cur-rent(b)for polycrystalline texture with average grain size R=25for different defect concentrations.In addition,Fig.6shows the switching behaviour obtained for different average grain size with no de-fect concentration.These data reveal almost similar changes in comparison to those obtained considering different values for defect concentration as presented in Fig.5.From Fig.6(a),it is obvious that both the polarization and the back switching increase with the increasing average grain size.One can clearly see in Fig.6(b)that the spike of the switching current in-creases as the average grain size increases,whereas the back switching current slightly changes.Fig.6.Responses of polarization(a)and switching cur-rent(b)of different average grain sizes for the polycrys-talline texture without defects.In conclusion,we have performed the Monte Carlo simulation on the domain,hysteresis loop and the switching behaviours of ferroelectric ceramics with randomly distributedfixed dipolar defects using the Potts–Ising model.Our results reveal that introduc-ing defects to ferroelectric ceramics leads to a decrease of both the remnant polarization P r and the coercive field E c.Wefind the back switching after the pulsed electricfield is offand the dependence of the defect and the average grain size on the polarization. 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