Effects of space charge distribution on ferroelectric hysteresis loops

Effects of space charge distribution on ferroelectric hysteresis loops considering the inhomogeneous built-in electric field: A phase field simulationX. L. Wang, B. Li, X. L. Zhong, Y. Zhang, J. B. Wang et al.Citation: J. Appl. Phys. 112, 114103 (2012); doi: 10.1063/1.4767702View online: /10.1063/1.4767702View Table of Contents: /resource/1/JAPIAU/v112/i11Published by the American Institute of Physics.Additional information on J. Appl. Phys.Journal Homepage: /Journal Information: /about/about_the_journalTop downloads: /features/most_downloadedInformation for Authors: /authorsDownloaded 16 Jun 2013 to 221.212.116.50. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: /about/rights_and_permissionsEffects of space charge distribution on ferroelectric hysteresis loops considering the inhomogeneous built-in electric field:A phase field simulationX.L.Wang,1,2B.Li,2X.L.Zhong,1,a)Y .Zhang,1J.B.Wang,2,a)and Y .C.Zhou 11Key Laboratory of Low Dimensional Materials and Application Technology of Ministry of Education,Xiangtan University,Xiangtan 411105,Hunan,China 2Faculty of Materials,Optoelectronics and Physics,Xiangtan University,Xiangtan 411105,China(Received 17February 2012;accepted 31October 2012;published online 3December 2012)A phase field model based on the time-dependent Ginsburg-Landau equation is developed to investigate the space charge distribution effects on the ferroelectric hysteresis characteristics of the ferroelectric thin film,in which the inhomogeneity of the built-in electric field (E i b )distribution along the vertical direction of ferroelectric thin film and the corresponding effect on the domain kinetics are taken into account.The hysteresis loops of the ferroelectric thin films with different space charge distributions are calculated by using this model.The results indicate that symmetrically (including homogeneously)distributed space charges lead to the hysteresis loop constriction because the E i b weakens the total electric field to make the ferroelectric film form a stable monodomain structure or the 180 strip domain.However,asymmetrically distributed space charges cause not only the shift but also the distortion of the hysteresis loop which is pinched in one head and loses its centrosymmetry.This is attributed to that the negative E i b changes the totalelectric field and activates domain nucleation at the interface.VC 2012American Institute of Physics .[/10.1063/1.4767702]I.INTRODUCTIONFerroelectric thin films have been extensively studied for their enormous potential in nonvolatile memory devices application.1–3It is reported that large amounts of charged defects such as ion impurities,oxygen vacancies,and dislo-cations may be produced during the manufacturing pro-cess.4–6As a result,a built-in electric field (E i b )along the film thickness is formed due to space charge distribution in ferroelectric thin films.The E i b in some regions of the film,especially at the interfaces,can be so large that it is even comparable to the external applied read/write voltage of the memory cells.This will lead to the enhancement of the power consumption and the write/read error rate for ferro-electric memory device.Several models have been developed to study the effects of space charges on hysteresis loops in ferroelectric film and superlattice.7–12In most of the above-mentioned models,however,researchers only consider the homogeneous distri-bution of space charge,and the E i b induced by space charges is artificially taken to be constant along the film thickness.7–11In fact,the distribution of space charges is a function of the position of ferroelectric thin film,which will lead to an inho-mogeneous E i b .Generally,the E i b at the interfaces of the ferroelectric thin film is much larger than that in the center of the ferroelectric film.13Recently,Misirlioglu et al.investigate the effect of inhomogeneous E i b due to asymmetric space charges on hysteresis loop of ferroelectric thin film by a non-linear thermodynamic method and reveal the hysteresischaracteristics can be altered dramatically under asymmetric space charges.12As we know,the prepared ferroelectric thin film is multi-domain structure usually and its domain kinetics should be focused.12However,the effect of E i b on domain kinetics is not considered for the limit of method in previous works.7–12Moreover,experiment results show that the space charge distribution is time-dependent after the application of an electric field.14Therefore,a kinetics model is desired to describe the effect of space charge on the evolution of ferro-electric domain and related properties.The phase field method is one of the useful tools to study the domain kinetics of ferro-electric.15,16Artemev et al.apply the phase field method to investigate the domain structure evolution and hysteresis loop in ferroelectric films with deadlayers,17which provides some guidance for our work from the view of the domain kinetics.In order to study the effect of space charges on the domain kinetics and the ferroelectric characteristics of ferroelectric film and its corresponding mechanism,in this work,a phase field model incorporating the inhomogeneous E i b due to the inhomogeneous space charges is developed.It is expected that this model will be useful to study the mechanism of the space charge effects and improve the electrical properties of ferroelectric films.II.THEORY AND METHODOLOGYThe phase field method is applied to describe the polar-ization evolution in the ferroelectric thin film under an exter-nal electric field and the E i b induced by space charges.The spontaneous polarization vector,P ¼(P 1,P 2,P 3),is taken as the order parameter.The time-dependent Ginsburg-Landau (TDGL )equationa)Authors to whom correspondence should be addressed.Electronic addresses:xlzhong@ and jbwang@.0021-8979/2012/112(11)/114103/6/$30.00VC 2012American Institute of Physics 112,114103-1JOURNAL OF APPLIED PHYSICS 112,114103(2012)@P iðr;tÞ@t ¼ÀLd Fd P iðr;tÞði¼1;2;3Þ(1)is employed to describe the temporal evolution of the sponta-neous polarizationfield,where L is the kinetic coefficient,F is the total free energy,t is the time,r¼(x,y,z)is the spatial vector,and d F/d P i(r,t)represents the thermodynamic driv-ing force.We consider the total free energy of thefilm asF¼ðV½f LðP iÞþf GðP i;jÞþf EðP i;e ijÞþf eleðP i;E iÞ dV;(2)where V,f L,f G,f E,and f ele are thefilm volume,the bulk free energy density,the gradient energy density,the elastic energy density,and the electrostatic energy density,ing the Landau phase transformation theory,the bulk free energy density in a given grain is expanded into terms of polarization components as follows:F LðP iÞ¼a1ðP21þP22þP23Þþa11ðP41þP42þP43Þþa12ðP21P22þP22P23þP23P21Þþa111ðP61þP62þP63Þþa112½P21ðP42þP43ÞþP22ðP43þP41ÞþP23ðP41þP42Þþa123P21P22P23;(3)where a1,a11,a12,a111,a112,and a123are the phenomenolog-ical Landau expansion coefficients.In the Ginzburg-Landau theory,the total free energy also depends on the gradient of the order parameter.For simplicity,we use the lowest order of the gradient energy,i.e.,F GðP i;jÞ¼12G11ðP21;xþP22;yþP23;zÞþG12ðP1;x P2;yþP2;y P3;zþP3;z P1;xÞþ12G44½ðP1;yþP2;xÞ2þðP2;zþP3;yÞ2þðP3;xþP1;zÞ2 þ12G044½ðP1;yÀP2;xÞ2þðP2;zÀP3;yÞ2þðP3;xÀP1;zÞ2 ;(4)where G ij is the gradient energy coefficients and P i,j denotes the spatial differentiation of the polarization components. The third term on the right hand of Eq.(1)is the elastic energy density,which is determined to beF elasðe ijÞ¼12C11ðe211þe222þe233ÞþC12ðe11e22þe22e33þe33e11Þþ2C44ðe212þe223þe213Þ;(5)where C ij is the second order elastic stiffness tensor,and e ij is the elastic strain,which is calculated bye ij¼e t ijÀe0ij;(6) where e t ij represents the total strain.The spontaneous straine0 ij ,which is also called eigenstrains,is associated with thespontaneous polarizations through the electrostrictive coeffi-cient Q ijkl,e0ij¼Q ijkl P k P l;(7)where the repeated subscripts imply the Einstein summa-tion.18The total strain is composed of the inhomogeneousand homogeneous strains.In ferroelectric materials,the in-homogeneous strain arises from the macroscopic deforma-tions in bulk material.For the case of ferroelectric thinfilm,the total strain is attributed mainly to the homogeneous straindue to the thickness of the ferroelectric thinfilm,which isseveral grades smaller than that of the substrate.19The valid-ity of neglecting the inhomogeneous strains in ferroelectricthinfilm has been demonstrated.19,20Thus,we can neglectthe inhomogeneous part of the total strain,and the total straine tijis considered equal to the homogeneous strain e ij,which isset to bee11¼ e22¼l m; e12¼ e13¼ e23¼0; e33¼À21Àl m;(8)where and l m are the Poisson ratio and the in-plane misfitstrain caused by the lattice mismatch,respectively.The elec-trostatic energy density is strongly depended on the spacecharge distribution.It can be expressed asF ele¼ÀE a iþ12E d iþE b iP i;(9)where E i a is the applied electricfield,E i d is the depolariza-tionfield,and E i b is the built-in electricfield.Then,we cancalculate the depolarizationfield asE d i¼ÀP iÀ P ie;(10)where e is the dielectric constant of the ferroelectric thinfilm.In Baudry’s model,the electrostatic interactions areestablished through the built-in polarization due to the spacecharges and the spontaneous polarization.10This modelassumes the ferroelectric thinfilm to be composed of“sections”or“layers”with different polarization valueswhen a constant space charge is discretely inserted at a givenposition.In this article,the space charges are position-dependent and continuously distributed along the thicknessof thefilm,and hence the space charge density in the ferro-electric thinfilm is solely the function of the position.Fora better understanding of the space charge effects in ferro-electric thinfilms,we consider four kinds of space chargedistributions,i.e.,homogeneous distribution,asymmetric dis-tribution with the space charge density exponentially decay-ing to zero from the top interface,asymmetric distributionwith the space charge density exponentially decaying to zerofrom the bottom interface,and symmetric distribution withthe space charge density exponentially decaying to zero atthe center of thefilm from the both interfaces.For the caseof homogeneous space charge distribution,the space chargedensity in the wholefilm is q sc(z)¼C,which means that thespace charge density is a constant value in the monolithic ferroelectricfilm.However,for the case of asymmetric space charge distribution,the space charge density in the ferroelec-tricfilm is taken to be q sc(z)¼q maxÁj z or q sc(z)¼q maxÁj d-z, where j is the base of the exponential function and d is the thickness of the ferroelectric thinfilm.For the case of sym-metric space charge distribution,q sc(z)¼q maxÁj z(0<z<d/2) and q sc(z)¼q maxÁj d-z(d/2<z<d).The E i b is taken to be a two-dimensional(2D)distribution instead of an average value.The corresponding electric potential is calculated by solving Poisson’s equation@2uðzÞ@z ¼Àq scðzÞe;(11)where q sc(z)and u(z)are the space charge density and corre-sponding electric potential at coordinate point z,respec-tively.Then the E i b at coordinate point z is obtained byE scðzÞ¼Àr/ðzÞ;(12) where r denotes the gradient operator.The lead titanate(PbTiO3)ferroelectric thinfilm is cho-sen as an example in the present simulations.The magnitude of the spontaneous polarization at room temperature P0is set to be0.757C/m2,the reference value of the gradient energy coefficients G110¼1.73Â10À10m4N/C2,and the value of a0¼a1,25 C¼(25–479)Â3.8Â105m2N/C2.21The sponta-neous polarization component P2and the electricfield com-ponent E2are treated to be zero.We employ64Â16discrete grid points to represent an infinitely large value of the width-to-thickness ratio of the ferroelectric region.The grid spac-ing is chosen to be D x*¼D z*¼1nm,and the time step is set as D t*¼0.02.The normalized external electricfield E*is first decreased from3.1to0,then continually brought down toÀ3.1,andfinally increased to3.1again.The whole pro-cess consists of126steps,with the change in the electric field linearly and homogenously.To ensure a stable electric domain state,a relaxation of20D t*is adopted at each step of different external electric level.All the electricfield compo-nents along the x direction are neglected,and the averaged polarization along the z direction is recorded as the total rem-nant polarization.For convenience,the set of dimensionless variables described in Ref.22are employed here to solve Eq.(1)PüP=P0;rüffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij a0j=G110pr;tüj a0j Lt;aÃ1¼a1=j a0j;aÃ11¼a11P20=j a0j;aÃ12¼a12P20=j a0j;aÃ111¼a111P40=j a0j;aÃ112¼a112P40=j a0j;aÃ123¼a123P40=j a0j; QÃ11¼Q11P20;QÃ12¼Q12P20;QÃ44¼Q44P20;GÃ11¼G11=G110;GÃ12¼G12=G110;GÃ44¼G44=G110; G0Ã44¼G044=G110;EüE=ðj a0j P0Þ;eÃ0¼e0j a0j(13) where the superscript asterisk*denotes the normalized variables.The values of this dimensionless parameters used in the simulation are listed in Table I.Periodic boundary condition for the polarization is applied along the x axis.Extrapolative polarization boundary condition along the z axis is given by dP/dn¼ÀP/d,19where n refers to a unit length in the outward normal direction of the surface,and d is the extrapolation length and is taken to be8nm in this sim-ulation.23For convenience,the space charge density varia-tion along the x axis is neglected and the Diriclet electric potential boundary conditions along the z axis are adopted in order to introduce the effect of the external bias u(0)¼V ext, u(z)¼0.To ensure c-domain with either positive or negative polarization along the out-of-plane axis in the whole simula-tion region,the misfit strain l m is taken asÀ0.012.24III.RESULTS AND DISCUSSIONFigs.1(a)–4(a)illustrate the schematic diagrams of the ferroelectricfilm sandwich structure with various space charge distributions,i.e.,homogeneous distribution,symmet-ric distribution with the space charge density exponentially decaying to zero at the center of thefilm from the both inter-faces,asymmetric distribution with the space charge density exponentially decaying to zero from the top interface,and asymmetric distribution with the space charge density expo-nentially decaying to zero from the bottom interface.The color change of ferroelectric layer in the schematic diagrams represents the variation of space charge density,and deeper color means higher space charge concentration.For the case of homogeneous space charge distribution,we consider four different space charge density q¼0.01,0.03,0.05,and 0.075C/m2as shown in Fig.1(b).For the case of inhomoge-neous space charge distribution,three different maximum space charge densities(q max),i.e.,0.05,0.075,and0.1C/m2 are adopted as shown in Figs.2(b),3(b),and4(b).The corre-sponding E i b distributions and ferroelectric hysteresis loops under four above-mentioned cases are shown in Figs.1(c)–Fig.4(c)and Figs.1(d)–4(d),respectively.From Fig.1(d),it can be seen clearly that the hysteresis loops show restriction effects when space charges are present in thefilms,and the restriction extent remarkably increases with the increment of space charge density from0.01to 0.075C/m2.Similar results have been reported by other researchers’works.9,10When the space charge density reaches0.075C/m2,polarization changes almost linearly with the applied electricfield.To investigate the origins of this strong constriction effect on the hysteresis loop,we try to analyze the domain evolution process within an entire period of the hysteresis loop.And180 strip domain(see Fig.5(a))is observed when the space charge density is 0.075C/m2,which can be explained as follows.The E i b pro-duced by the homogeneous space charges is linearly distrib-uted as illustrated in Fig.1(c),so it has considerable magnitudes even in the body of the ferroelectricfilm.In TABLE I.Dimensionless parameters used in the simulation.aÃ1aÃ11aÃ12aÃ111aÃ112GÃ11GÃ12À1À0.24 2.50.49 1.220 GÃ44G0Ã44QÃ11QÃ12QÃ44l m 110.05À0.0152À0.0120.3consequence,the total electric field in the ferroelectric film is not sufficient to saturate the polarization and form a stable monodomain structure,thus forming the 180 strip domain.19As a result,the total remnant polarization is dramatically decreased and the hysteresis loop is greatly constricted.For the symmetric but inhomogeneous distribution of space charge,the hysteresis loops are slightly constricted as shown in Fig.2(d).The restriction extent increases with the increase of q max from 0.05to 0.1C/m 2,but it is much less severe than that in the case of homogeneous space charge distribution shown in Fig.1(d).This is because that significant internal electric field is produced only at the inter-faces while the E i b in the film decreases rapidly.The total electric field is so large that it is sufficient to form stable monodomain as shown Fig.5(b).Thus only slight constric-tions happen in the hysteresis loops.For the case of asymmetric space charge distribution illustrated in Figs.3and 4,strong E i b are generated at the top and bottom interfaces,respectively.As shown in Figs.3(d)and 4(d),the hysteresis loops exhibit not only a horizontal shift but also a severe distortion,which are pinched in one head and lose their centrosymmetry.The restriction extent increases with the increase of q max from 0.05to 0.1C/m 2.This distorted phenomenon is also observed in thepreviousFIG.1.(a)Schematic diagram,(b)space charge distributions,(c)built-in electric field distributions,and (d)hysteresis loops of the ferroelectric film with homo-geneous space chargedistribution.FIG.2.(a)Schematic diagram,(b)space charge distributions,(c)built-in electric field distributions,and (d)hysteresis loops of the ferroelectric film with the space charge density exponentially decaying to zero at the center of the film from the both interfaces.experiments.25,26In order to explore the distortion mechanism of the hysteresis loops due to the inhomogeneous distribution of space charge,the time-dependent domain structures under different densities of inhomogeneous space charge are simu-lated and analyzed.The hysteresis loop and domain structures in ferroelectric thin film with the asymmetrically space charge density q max ¼0.1C/m 2are taken as an example,as shown in Fig.6.From Fig.3(c),it can be seen that a large negative E i b is induced by the asymmetrically distribution space charges at the top interface area of the film,while the E i b at the bottom interface closes to zero.When a positive electrical field is applied to the film,the negative E i b decreases the magnitudeof the external electric field,which leads to left shift of the loops in the ferroelectric film and the polarization becomes unsaturated,as shown in the right part of Fig.6.This is because the negative E i b activates the opposite domain nucle-ation at the top interface,as shown in the insets (a)and (b)of Fig.6.On the other hand,when a negative electrical field is applied to the film,the negative E i b increases the magnitude of the external electrical field,which leads to the left shift of loops and the increase of polarization for complete domain switching (the insets (c)and (d)of Fig.6).The above results reveal that the E i b due to the inhomo-geneous space charge can strongly affect the domainkinetics,FIG.3.(a)Schematic diagram,(b)space charge distributions,(c)built-in electric field distributions,and (d)hysteresis loops of the ferroelectric film with the space charge density exponentially decaying to zero from the topinterface.FIG.4.(a)Schematic diagram,(b)space charge distributions,(c)built-in electric field distributions,and (d)hysteresis loops of the ferroelectric film with the space charge density exponentially decaying to zero from the top interface.thus altering the hysteresis loop of ferroelectric film.This indicates that it is meaningful to take the inhomogeneity of the E i b into consideration in our model.In addition,this model can also study the promoting and hindering effect of the ferroelectric domain nucleation and domain wall motion caused by the space charges accumulated at the ferroelectric interface,which can provide us a theoretical reference on the analysis of ferroelectric memory failure.IV.CONCLUSIONA phase field model considering in the inhomogeneity of the built-in electric field is developed to investigate the effects of different space charge distributions along the thin film thickness on the ferroelectric hysteresis loop.The resultsindicate that symmetrically (including homogeneously)dis-tributed space charges lead to hysteresis loop constriction,while asymmetrically distributed space charges cause not only shift but also distort of the hysteresis loop.These ferro-electric hysteresis loop anomalies are attributed to the effect of built-in electric field due to space charges on domain kinetics in ferroelectric thin film.This model is useful to study the mechanism of the space charge effects and improve the electrical properties of ferroelectric films.ACKNOWLEDGMENTSThis work was supported by the National Natural Science Foundation of China (Nos.11072208and 11032010)and the Hunan Provincial Natural Science Foundation of China (No.12JJ1007).1J.F.Scott and C.A.Paz de Araujo,Science 246,1400(1989).2L.Liao,H.J.Fan,B.Yan,Z.Zhang,L.L.Chen,B.S.Li,G.Z.Xing,Z.X.Shen,T.Wu,X.W.Sun,J.Wang,and T.Yu,ACS Nano 3,700(2009).3E.B.Song,B.Lian,S.M.Kim,S.Lee,T.K.Chung,M.Wang,C.Zeng,G.Xu,K.Wong,Y.Zhou,H.I.Rasool,D.H.Seo,H.J.Chung,J.Heo,S.Seo,and K.L.Wang,Appl.Phys.Lett.99,042109(2011).4K.T.Lia and V.C.Lo,J.Appl.Phys.97,034107(2005).5Y.T.Pu,J.L.Zhu,X.H.Zhu,Y.S.Luo,M.S.Wang,X.H.Li,J.Liu,J.G.Zhu,and D.Q.Xiao,J.Appl.Phys.109,044102(2011).6V.Koval,M.J.Reece,and A.J.Bushby,J.Appl.Phys.101,024113(2007).7C.K.Wong and F.G.Shin,J.Appl.Phys.96,6648(2004).8Y.Zhou,H.K.Chan,m,and F.G.Shin,J.Appl.Phys.98,024111(2005).9P.Zubko,D.J.Jung,and J.F.Scott,J.Appl.Phys.100,114112(2006).10L.Baudry,J.Appl.Phys.86,1096(1999).11C.J.Brennan,Mater.Res.Soc.Symp.Proc.243,141(1992).12I.B.Misirlioglu,M.B.Okatan,and S.P.Alpay,J.Appl.Phys.108,034105(2010).13A.K.Tagantsev and G.Gerra,J.Appl.Phys.100,051607(2006).14A.N.Morozovska and E.A.Eliseev,J.Phys.:Condens.Matter.16,8937(2004).15J.Slutsker,A.Artemev,and A.Royturd,Phys.Rev.Lett.100,087602(2008).16B.Li,J.B.Wang,X.L.Zhong,F.Wang,and Y.C.Zhou,J.Appl.Phys.107,014109(2010).17A.Artemev,Philos.Mag.90,7(2010).18L.Q.Chen,J.Am.Ceram.Soc.91,1835(2008).19L.Hong,A.K.Soh,Y.C.Song,and L.C.Lim,Acta Mater.56,2966(2008).20G.B.Stephenson and K.R.Elder.J.Appl.Phys.100,051601(2006).21J.Wang and T.Y.Zhang,Phys.Rev.B 73,144107(2006).22J.Wang,Y.L.Li,L.Q.Chen,and T.Y.Zhang,Acta Mater.53,2495(2005).23K.Ishikawa and T.Uemori,Phys.Rev.B 60,11841(1999).24Y.L.Li,S.Y.Hu,Z.K.Liu,and L.Q.Chen,Acta Mater.50,395(2002).25F.Chen,X.L.Tan,Z.Huang,X.F.Xuan,and W.B.Wu,Appl.Phys.Lett.96,262902(2010).26Y.Lin,B.R.Zhao,H.B.Peng,Z.Hao,B.Xu et al .,J.Appl.Phys.86,4467(1999).FIG.5.The domain structures of ferroelectric thin film with (a)homogene-ous space charge distribution and (b)symmetric but inhomogeneous distri-bution of space charge when the space charge density is 0.075C/m 2.FIG.6.The hysteresis loop of ferroelectric thin film with the asymmetrically space charge density q max ¼0.1C/m 2.The insets are the corresponding do-main structures.。