H.H.Wu·S.G.Cao·J.M.Zhu·T.Y.Zhang
The frequency-dependent behavior of a ferroelectric single crystal with dislocation arrays
Received:9January2015/Revised:13April2015
?Springer-Verlag Wien2015
Abstract Phase?eld simulations are conducted to investigate the frequency-dependent behavior of a ferro-electric single crystal with and without dislocation arrays.For the dislocation-free ferroelectric,both coercive ?eld and remnant polarization increase with the increase in the applied electric?eld frequency.For the fer-roelectric with dislocation arrays,however,the variation of remnant polarization with frequency depends on the amplitude of the applied alternating electric?eld.When the applied electric amplitude ranges from0.3to 0.7,called the low?eld,the remnant polarization increases?rst and then decreases.On the other hand,if the applied?eld amplitude is higher than0.8,the remnant polarization does not change too much at low frequency (3.13×10?4–1.56×10?2),while it decreases sharply at high frequency(1.56×10?2–0.156).For various applied electric amplitudes ranging from0.3to1.5,the overall trend of the coercive?eld is to increase in the low-frequency range,while it varies in the high-frequency range.The frequency-dependent properties are attributed to the generalized pinning and depinning of the dislocation arrays to polarizations,which is endorsed by the corresponding domain structures.
1Introduction
Ferroelectric materials have attracted much attention due to their distinguished ferroelectric and piezoelectric properties,which are closely related to the polarization switching and domain structures in the materials[1,2]. An important characteristic of ferroelectric materials is the hysteresis loop of polarization versus applied electric ?eld(P–E curve),which describes the switching behavior of polarization.The switching dynamics depends on the loading frequency,thereby making the remnant polarization and the coercive?eld frequency-dependent. Experimental investigations show the complex nature of switching dynamics.For example,Bozgeyik studied the frequency(ranging from1to100kHz)dependence of ferroelectric hysteresis loops in BaZrO3(BZ) and Sr0.8Bi2.2Ta2O9(SBT)thin?lms[3]and found that the coercive?eld(2E c)increased and the remnant polarization(2P r)decreased with increasing applied?eld frequency.Similar results have been found by other researchers[4–6].The opposite trend in the frequency-dependent behavior has also been observed experimentally[7–11].For instance,both remnant polarization and coercive?eld of the mixed ferroelectric KTa1?x Nb x O3decreased with increasing frequency in the range of10–1000Hz[11].
H.H.Wu·S.G.Cao·J.M.Zhu·T.Y.Zhang()
Department of Mechanical and Aerospace Engineering,Hong Kong University of Science and Technology,Clear Water Bay, Kowloon,Hong Kong,China
E-mail:zhangty@https://www.doczj.com/doc/234032192.html,;mezhangt@ust.hk
Tel.:+852********
Fax:+8522358-1543
H.H.Wu et al.
Theoretical modeling [12,13]and computer simulations [14–16]have been employed to study the frequency-dependent phenomena.Based on the time-dependent Ginzburg–Landau (TDGL)equation,the effects of frequency on the hysteresis loops of polarization versus electric ?eld were simulated by a phase ?eld model for Bi 4Ti 3O 12(BIT)ferroelectric single crystal with the dimensionless frequency ranging from 0.0063to 0.05[15].The simulations [15]showed that the coercive ?eld increased with increasing frequency,while the remnant polarization was frequency-insensitive.Obviously,the in?uence of dislocations on the switching dynamics has not been studied https://www.doczj.com/doc/234032192.html,ing phase ?eld simulations,we [17–19]have investigated the effects of bias voltage,applied alternating electric amplitude and temperature on the properties of a ferro-electric single crystal with dislocation arrays.Following the previous work,we investigate here the effect of dislocation arrays on the frequency-dependent behavior by phase ?eld simulations.
2Methodology
Two-dimensional (2D)phase ?eld simulations were conducted under the plane strain condition.The detailed information of the phase ?eld model can be found in the previous publications [17–19].Figure 1shows a schematic drawing of the used simulated cells of 64×64discrete grids (a normalized cell size of x ?1= x ?2=0.8)with dislocation arrays,where the x 1and x 2axes are set along the pseudocubic crystallographic [100]and [010]directions,respectively.A dislocation array composed of periodically distributed dislocations with the same Burgers vector of x ?12[100]is allocated at the middle height of the simulated cell.The dislocation spacing and the simulated cell dimension are denoted by D and L ,respectively,giving the dislocation linear density (DLD)of L /D .The periodic boundary condition is adopted in both x 1and x 2directions,implying that the simulated cell is within an in?nitely large 2D single crystal with periodically distributed dislocation arrays.The used normalized formula and the input material constants of PbTiO 3single crystal at room temperature are all listed in Ref.[20].The semi-implicit Fourier-spectral method was employed to solve the partial differential equation [20].A random distribution of initial polarizations with the maximum magnitude less than 0.001was assigned to the simulated system to trig the polarization evolution,which resulted in the initial domain structure after 20,000-step evolution with a dimensionless time step of t ?=0.04.Then,a dimensionless
Fig.1Schematic illustration of a simulated 2D ferroelectric single crystal with dislocation arrays
The frequency-dependent behavior of a ferroelectric single crystal
external electric?eld,E?2=E?0sin(2πf?t?)with frequency f?,was applied along the x2direction.The average polarization along the electric?eld direction was taken as the macroscopic response of the simulated ferroelectric single crystal.
3Results and discussion
3.1Dislocation-free ferroelectric
Figure2shows the frequency-dependent behavior of a dislocation-free(DLD=0)ferroelectric single crystal. At the applied alternating electric amplitude E?0=0.6,no typical hysteresis loops show up when the frequency is higher than f?=0.0125.That is why all simulation results shown in Fig.2are with a maximum frequency at each applied electric?eld amplitude.Figure2a,b presents the P–E curves at the applied electric?eld amplitudes E?0=0.4and E?0=0.6,respectively.As shown in Fig.2a,no typical P–E hysteresis loop forms at a wide range of frequency for the applied electric amplitude E?0=0.4is lower than the coercive?eld.When the applied electric?eld amplitude E?0increases to0.6,as shown in Fig.2b,typical P–E hysteresis loops form during the normalized loading frequencies ranging from f?=0.0003125to f?=0.0125.Figure2c illustrates that both remnant polarization and coercive?eld,determined from the phase?eld simulations, basically increase with the increase in frequency,as plotted,which is consistent with the reported results [9,13,15,16].Correspondingly,the hysteresis loop area essentially increases with the increasing frequency from3.13×10?4to0.156,as shown in Fig.2d.
The domain structures in Fig.3correspond to points of the P–E hysteresis loops in Fig.2b,showing that the typical domain structures throughout the polarization switching process are90?domain wall motions.Overall, two kinds of domain structures appear during the alternating electric?eld loading process of a2D model:One is a domain de?ned as P1=0,while P2=0,and another is c domain as P1=0and P2=0.With the loading
Fig.2The frequency-dependent behavior of a dislocation-free(DLD=0)ferroelectric single crystal.P–E hysteresis loops at the applied electric?eld amplitude of a E?0=0.4and b E?0=0.6,c the remnant polarization and coercive?eld basically increase with increasing frequency,d the hysteresis loop area essentially increases with increasing frequency
H.H.Wu et al.
Fig.3Domain structures corresponding to the points of P–E hysteresis loops in Fig.2b with respect to different frequencies f?=0.0125,0.00125and0.0003125,respectively
frequency f?decreases from0.0125to0.0003125,the fraction of c domains always decrease at the applied electric?eld E?2=0.6(a1–c1),?0.4243(a3–c3)and?0.6(a4–c4),respectively,while the fraction of both a and c domains are all the same at the applied electric?eld E?2=0(a2–c2).Obviously,the change in domain structure agrees well with the phenomena illustrated by Fig.2c that both remnant polarization and coercive ?eld increase with the increase in loading frequency.
3.2Ferroelectrics with dislocation arrays
Figure4a,b shows the hysteresis loops at the alternating applied electric amplitudes E?0=0.5and E?0= 1.0,respectively,indicating that the switching dynamics depends on both applied?eld frequency and?eld amplitude.Figure4c illustrates that under E?0=0.5,the remnant polarization increases and then decreases with the increase in frequency and thus a peak occurs at f?=0.02083,whereas the coercive?eld still increases essentially with the increase in frequency although some?uctuation happens around f?=0.10417.Under E?0=1.0,the increase and decrease rates in remnant polarization are much slower and faster,respectively,than the corresponding ones under E?0=0.5when the frequency increases from0.0003125to0.01563and further to 0.15625.P–E hysteresis loop areas usually re?ect the degree of energy dissipation during polarization switching process.As shown in Fig.4d,with the increase in loading frequency from0.0003125to0.0625,the hysteresis loop area always increases?rst and then decreases.This behavior of hysteresis loop area versus frequency in the dislocated ferroelectrics is in contrast to that shown in Fig.2d of the dislocation-free ferroelectrics.
Figure5gives the domain structures corresponding to the points indicated in Fig.4b of the applied electric ?eld amplitude E?0=1.0and loading frequencies f?=0.0625,0.0125and0.0003125,respectively.The preexisting dislocations play important roles in the ferroelectric switching,acting as nucleation sites of new domains or/and pinning the domain wall motion.The strain mismatch between the vertical and horizontal
The frequency-dependent behavior of a ferroelectric single crystal
Fig.4Simulation results for the model with DLD=21.a Hysteresis loops at the applied electric?eld amplitude E?0=0.5,b hysteresis loops at the applied electric?eld amplitude E?0=1.0,c the remnant polarization and coercive?eld versus the applied ?eld frequency,d the hysteresis loop area shows a non-monotonic behavior with the increasing frequency
domains at the dislocation array is partially or completely released by the dislocations,especially near the dislocations,which makes the underneath domain almost steady to be a single domain with most polariza-tions along the horizontal direction.The domain structure below the dislocation array contributes slightly, through polarization rotation,to the macroscopic responses of the P–E loops.Above the dislocation array, most polarizations are along the vertical direction as multi-domain structures,which is responsible for the diverse hysteresis P–E loops in Fig.4b.At the loading frequency f?=0.0625,shown in Fig.5a1–a4,two domains always exist to be antiparallel to the external electric?eld,indicating that the domain is pinned by the dislocation array.The domain pinning is distinguished from the classical domain wall pinning that dislocations prevent a domain wall to move through.The domain pinning here just means that a domain with polarizations antiparallel to an external applied electric?eld can survive due to the presence of dislocations.Although the applied electric?eld amplitude E?0is larger than the quasistatic critical depinning electric?eld E?0=0.8in the same DLD=21ferroelectric,which was discussed in our previous work[18],the high loading frequency pro-vides much short time for the domain to depin from the dislocation arrays,directly leading to the low averaged (remnant)polarization and large coercive?eld,as shown in Fig.4b.When the loading frequency f?decreases to0.0125,one antiparallel domain above the dislocation array disappears at the maximum/minimum applied electric?eld,as shown in Fig.5b1,b4,subsidizing the increase in averaged(remnant)polarizations.However, the existing of an antiparallel domain by the dislocation pinning effect renders a large coercive?eld.With the loading frequency further decreasing to a quasistatic level f?=0.0003125,as given in Fig.5c1–c4,the driven force provided by the external electric?eld and the response time are large enough to completely align all polarizations above the dislocation array into a single domain at the maximum/minimum applied electric ?eld E?2=1.0/?1.0.In this case,the antiparallel domains showing-up under high frequency f?=0.0625, 0.0125disappear totally under f?=0.0003125of the same applied electric?eld E?0=1.0.The present phase?eld simulations show that the dislocation arrays may pin some antiparallel domains if the loading frequency is high.The domain depinning occurs when the loading frequency is low even though the applied
H.H.Wu et al.
Fig.5Domain structures corresponding to the points of Fig.4b with loading frequencies f?=0.0625,0.0125and0.0003125, respectively
electric?eld amplitude is larger than the corresponding quasistatic critical depinning electric?eld.It is the frequency-dependent domain pining and depinning behavior that causes the great change in the macroscopic frequency-dependent P–E hysteresis loops.
Crystalline defects like oxygen vacancies,impurity atoms and dislocations.are often introduced to ferro-electric materials during the material fabrication processing.The defects play important roles in domain wall motion,polarization switching and dynamic macro-hysteresis loops.The present phase?eld simulations show that the dislocations impose a signi?cant impact on the dynamic hysteresis.The exhibited domain con?gura-tions provide direct evidence for macro-response of remnant polarization E c,coercive?eld P r and hysteresis area.The simulations on the ferroelectrics with dislocations show the similar dynamic behaviors as those observed in experiments[21–25].This might indicate that the experimentally observed dynamic ferroelectric behaviors could be caused by dislocations,which provides a clue to the material characterization.
4Conclusions
In summary,phase?eld simulations were conducted to explore the frequency-dependent behavior of a ferro-electric single crystal with/without dislocation arrays.The simulation results show that without dislocations, the ferroelectric single crystal exhibits the monotonic variations of remnant polarization,coercive?eld and hysteresis loop areas as functions of applied?eld frequency.However,with the dislocation arrays,the ferroelec-tric behavior becomes much more complex,and the corresponding domain structures shed physical light onto the non-monotonic behavior of the remnant polarization and coercive?eld versus the loading?eld frequency, i.e.,dislocation pinning on polarizations and dislocation-assistant new domain nucleation.The current work presents a deeper understanding of the dynamic behavior of ferroelectrics,which is helpful for the application of ferroelectric materials.
The frequency-dependent behavior of a ferroelectric single crystal
Acknowledgments This work was supported by the Hong Kong Research Grants Council under the General Research Fund, 622813.
References
1.Scott,J.F.,Araujo,C.A.P.:Ferroelectric memories.Science246,1400–1405(1989)
2.Garcia,V.,Bibes,M.:Electronics:inside story of ferroelectric memories.Nature483,279–281(2012)
3.Bozgeyik,M.S.:Frequency dependent ferroelectric properties of BaZrO3modi?ed Sr0.8Bi2.2Ta2O9thin?lms.Chin.J.
Phys.51,327–336(2013)
4.Horiuchi,S.,Kagawa,F.,Hatahara,K.,Kobayashi,K.,Kumai,R.,Murakami,Y.,Tokura,Y.:Above-room-temperature
ferroelectricity and anti-ferroelectricity in https://www.doczj.com/doc/234032192.html,mun.3,1308(2012)
5.Yang,S.M.,Jo,J.Y.,Kim,T.H.,Yoon,J-G.,Song,T.K.,Lee,H.N.,Marton,Z.,Park,S.,Jo,Y.,Noh,T.W.:AC dynamics of
ferroelectric domains from an investigation of the frequency dependence of hysteresis loops.Phys.Rev.B82,174125(2010) 6.Hu,W.J.,Juo,D.-M.,You,L.,Wang,J.,Chen,Y.-C.,Chu,Y.-H.,Wu,T.:Universal ferroelectric switching dynamics of
vinylidene?uoride-tri?uoroethylene copolymer?lms.Sci.Rep.4,4772(2014)
7.Chaipanich,A.,Potong,R.,Rianyoi,R.,Jareansuk,L.,Jaitanong,N.,Yimnirun,R.:Dielectric and ferroelectric hystere-
sis properties of1–3lead magnesium niobate–lead titanate ceramic/Portland cement composites.Ceram.Int.38,S255–S258(2012)
8.Tang,M.H.,Dong,G.J.,Sugiyama,Y.,Ishiwara,H.:Frequency-dependent electrical properties in Bi(Zn0.5Ti0.5)O3doped
Pb(Zr0.4Ti0.6)O3thin?lm for ferroelectric memory application.Semicond.Sci.Technol.25,035006(2010)
9.Ortega,N.,Kumar,A.,Scott,J.F.,Chrisey,D.B.,Tomazawa,M.,Kumari,S.,Diestra,D.G.B.,Katiyar,R.S.:Relaxor-
ferroelectric superlattices:high energy density capacitors.J.Phys.Condens.Matter.24,445901(2012)
10.Chand,P.,Gaur,A.,Kumar,A.:Study of optical and ferroelectric behavior of ZnO nanostructures.Adv.Mater.
Lett.4,220(2013)
11.Knauss,L.A.,Pattnaik,R.,Toulouse,J.:Polarization dynamics in the mixed ferroelectric KTa1?x Nb x O3.Phys.Rev.
B55,3472–3479(1997)
12.Picinin,A.,Lente,M.H.,Eiras,J.A.,Rino,J.P.:Theoretical and experimental investigations of polarization switching in
ferroelectric materials.Phys.Rev.B69,064117(2004)
13.Morozovska,A.N.,Eliseev,E.A.,Remiens,D.,Soyer,C.:Theoretical description of ferroelectric and pyroelectric hystereses
in the disordered ferroelectric-semiconductor?lms.J.Appl.Phys.100,014109(2006)
14.Liu,J.M.,Li,Q.C.,Wang,W.M.,Chen,X.Y.,Cao,G.H.,Liu,X.H.,Liu,Z.G.:Scaling of dynamic hysteresis in ferroelectric
spin systems.J.Phys.Condens.Matter.13,L153–L161(2001)
15.Zheng,X.-J.,Lu,J.,Zhou,Y.-C.,Wu,B.,Chen,Y.-Q.:Evolution of domain structure and frequency effect on ferroelectric
properties in BIT ferroelectrics.Trans.Nonferrous Met.Soc.China17,s64–s68(2007)
16.Sivasubramanian,S.,Widom,A.,Srivastava,Y.:Equivalent circuit and simulations for the Landau–Khalatnikov model of
ferroelectric hysteresis.IEEE Trans.Ultrason.Ferroelectr.Freq.Control50,950–957(2003)
17.Wu,H.H.,Wang,J.,Cao,S.G.,Zhang,T.Y.:Effect of dislocation walls on the polarization switching of a ferroelectric single
crystal.Appl.Phys.Lett.102,232904(2013)
18.Wu,H.H.,Wang,J.,Cao,S.G.,Chen,L.Q.,Zhang,T.Y.:Micro-/macro-responses of a ferroelectric single crystal with
domain pinning and depinning by dislocations.J.Appl.Phys.114,164108(2013)
19.Wu,H.H.,Wang,J.,Cao,S.G.,Chen,L.Q.,Zhang,T.Y.:The unusual temperature dependence of the switching behavior in
a ferroelectric single crystal with dislocations.Smart Mater.Struct.23,025004(2014)
20.Wang,J.,Shi,S.Q.,Chen,L.Q.,Li,Y.,Zhang,T.Y.:Phase-?eld simulations of ferroelectric/ferroelastic polarization switch-
ing.Acta Mater.52,749–764(2004)
21.Liu,J.M.,Yu,L.C.,Yuan,G.L.,Yang,Y.,Chan,H.L.W.,Liu,Z.G.:Dynamic hysteresis of ferroelectric Pb(Zr0.52Ti0.48O3
thin?lms.Microelectron.Eng.66,798–805(2003)
22.Sarjala,M.,Sepp?l?,E.T.,Alava,M.J.:Dynamic hysteresis in ferroelectrics with quenched randomness.Phys.B403,418–
421(2008)
23.Liu,J.M.,Xiao,Q.,Liu,Z.G.,Chan,H.L.W.,Ming,N.B.:Dynamic hysteresis stability of ferroelectric Pb(Zr0.52Ti0.48)O3
thin?lms.Mater.Chem.Phys.82,733–741(2003)
24.Yu,G.,Chen,X.,Cao,F.,Wang,G.,Dong,X.:Dynamic ferroelectric hysteresis scaling behavior of40BiScO3–60PbTiO3
bulk ceramics.Solid State Commun.150,1045–1047(2010)
25.Chen,D.P.,Liu,J.M.:Dynamic hysteresis of tetragonal ferroelectrics:the resonance of90?-domain switching.Appl.Phys.
Lett.100,062904(2012)