JournaloftheKoreanPhysicalSociety,Vol. 46,No. 1,January2005,pp. 59Landau-KhalatnikovSimulationsforFerroelectricSwitchinginFerroelectricRandomAccessMemoryApplicationTae Kwon SongDepartment of Ceramic Science and Engineering,ChangwonNational University, Changwon, Kyungnam641-773(Received5August2004)Landau-Khalatnikovsimulation,adynamicalversionofLandau-Devonshiretheory,forferroelec-tricswitchingpropertiesisshowntobeveryusefulinunderstandingtheoperationofferroelectricrandomaccess memorydevices. The simulatedferroelectric hysteresis loops andpulse-currentresponses agreedwell withexperimental results. Partial switchingresponses andexternal stresseectsarestudied. Therelationbetweentheshapeof hysteresisloopsandswitching-currentre-sponsesisalsostudied. Thesimulatedswitching-currentresponsesagreedwell withconventionalKolmogorov-Avrami-Ishibashiswitchingtheoryindependentofhysteresis-loopshape.PACSnumbers: 77.80.Fm,78.20.Bh,77.80.Dj,77.22.Ej,77.65.-jKeywords: Ferroelectricswitching,Landau-Khalatnikovsimulation,Hysteresisloop,Polarization,StressI.INTRODUCTIONFerroelectric switching properties have received muchattention since ferroelectric materials were discovered. Aferroelectric material has spontaneous polarization with-outexternal electriceld, andthepolarizationshouldbeswitchedinreversewithanelectriceld. Domaindynamics havebeenstudiedexperimentallyandtheo-retically to understand ferroelectric switching properties[14].In the last decade, switching properties of ferroelectricthin lm capacitors have been studied intensively for fer-roelectric random access memory (FeRAM) applications,sinceultra-fastwritingandreadingoperationisoneofthemeritsof FeRAMdevices[5, 6]. PhenomenologicalLandau-Devonshire(LD)theoryhasbeenstudiedwell,to understand the ferroelectric phase transition and sizeeectsof ferroelectrics[7, 8]. TheLandau-Khalatnikov(LK)equationisadynamicalversionofLDtheory[911].Inthis paper, theapplicabilityof theLKequationfor the ferroelectric-switching hysteresis loop and pulse-switching current response is demonstrated. Partialswitchingandexternal stresseectsarestudied. Theswitching dynamics of dierently shaped hysteresis loopsare compared.II.SIMULATIONE-mail: tksong@changwon.ac.kr;Fax: +82-55-262-6486The phenomenological investigations of dynamicalproperties of ferroelectric materials are studied with theaid of the LK Equation: dPdt= d GdP , (1)whereis a kinetic coecient. This equation expressesthe fact that the regression dP/dt of a given polarizationuctuationtowardsequilibriumisthefaster,thelargerthe thermodynamic force, G(P, E)/P, that is ap-plied [9].The LD free energy density functional ( G) is expressedas expansions of the order parameter of the ferroelectricphase, polarization (P) [4]:Fig. 1. Simulatedhysteresis loopfor BaTiO3at 20CfromLKequation.-5--6- JournaloftheKoreanPhysicalSociety,Vol. 46,No. 1,January2005G(P, E) = g0 + 12g2P2+ 14g4P4+ 16g6P6 12EP.(2)Theparameters g0, g2, g4, andg6inEq. (2)havebeendecidedfromtheexperimental datasuchastem-peraturedependent dielectricconstants andhysteresisloop shapes,i.e.PsatandPr. For the well-known ferro-electric single crystals such as BaTiO3 and PbTiO3, theparameters have been reported [2,3].Equation (1) can be rewritten with Eq. (2) asdPdt

=12E g2P g4P3g6P5, (3)where dimensionless time t

=t/ is used. Eq. (3)issolvednumericallybyusingaRunge-Kuttamethod.With a given electric-eld value of E(t

) and a given po-larizationvalueof P(t

), thechangeofpolarizationdPiscalculatedfromEq. (3)afteraveryshorttimein-terval dt

. Thepolarizationvalueafter dt

isgivenasP +dP= P(t

+dt

). As the electric eld changes sinu-soidally, the ferroelectric hysteresis loop is obtained fromthe polarization as a function of electric eld.Thehysteresis-loopobtainedagreedwellwithexper-imentalresults, exceptfortheelectriceld. Thesimu-lated coercive eld is much smaller than the experimen-tal result, sothat theelectriceldis re-scaledinthesimulatedresults [12]. Thediscrepancybetweensim-ulatedandexperimentalcoerciveeldsinLDtheoryiswell known and attributed to the ferroelectric domain dy-namics [13]. In LD theory, all the ferroelectric dipoles areassumed to be switched simultaneously together. In realexperiments, however, the ferroelectric materials consistof domainswhichareswitchedeasilyinamuchlowerelectriceld, sothattheexperimental coerciveeldismuchsmallerthanthetheoretical one. Inthesimula-tion, the last term of Eq. (2), 12EP, is multiplied bythe electric-eld parameter, , in view of this issue, sothat Eq. (2) is rewritten asG(P, E) = g0 + 12g2P2+ 14g4P4+ 16g6P6 12EP,(4)and Eq. (3) asdPdt

=12E g2P g4P3g6P5. (5)III.RESULTS1. Applicationof LKequationinferroelectricswitchingFigure 1 shows a simulated hysteresis loop calculatedfromtheparameters of BaTiO3. Parameters usedinthissimulationareg2= 5.94 107Jm/C2, g4= 2.23109Jm5/C4, andg6 = 3.961010Jm9/C6[2].The polarization-reversal switching responses withpulse electric eld, which are crucial properties forFeRAM application, are also simulated as shown in Fig-ure 2. The applied pulse trains are shown in the inset ofFigure2. Fortheswitchedresponse, thewritepulseisappliednegativelyandthereadpulseispositive. Posi-tive write pulse was used for the non-switched response.Pulse-current responses were obtained at a point denotedA.Thesimulatedresultsagreedwellwithexperimen-tal results [1]. From the dierence between the switchedandnon-switchedresponses, theferroelectricswitchingresponsesareobtained. Thesimulatedhysteresisloopsand switching-current responses agree well with experi-mental results [7,12].2. Partial-switchingresponseOn decreasing the operating voltage of an FeRAM de-vice, full switching is hardly obtainable. For full switch-ing, the pulse height should be about 2 or 3 times largerthanthecoerciveeld. However, evenwithasmallerelectric eld, a part of the polarization is switched. Theswitching responses withpulses of whichthe electriceldsarearoundthecoerciveeldcouldbesimulatedwith the LK equation as shown in Figure 3 as an exam-ple. The parameters used in this example areg2 = 1,g4 = 1, andg6 = 1 [14].3. External stresseectsFerroelectric thinlmcapacitors are inevitablyex-posedtoavarietyofstressesinfabricationanddevicestructure, so that the simulation of external stress eectsinFeRAMdevicesisessential tooptimizetheFeRAMdevice design,and electromechanical properties of PZTFig. 2. Simulated pulse-switching current response forBaTiO3at20 C.Insetshowsapplied-electric-eldproles.Landau-KhalatnikovSimulationStudy TaeKwonSong -7-Fig. 3. Simulatedhysteresis loops byLKequationwithvariouselectricelds.Fig. 4. (a) Ferroelectric hysteresis loops of aPb(Zr0.52Ti0.48)O3thin-lmcapacitorwithandwithoutex-ternal stresses. (b) Simulated hysteresis loops with LK equa-tionwithexternalstresses.lmsareimportantformicroelectromechanical-systemsapplication [15]. The eects of external stresses onthe ferroelectric properties of Pb(Zr0.52Ti0.48)O3(PZT)thin-lm capacitors have been reported [16]. Under com-pressive stress the polarization and coercive eld are in-creased, buttheyaredecreasedundertensilestressasshown in Figure 4 (a).Thesimulatedhysteresisloopswithandwithoutex-Fig. 5. Ferroelectric hysteresis loops of various shapes,simulatedwithparametersshowninTable1.ternal stresses of 240 MPa agreed well with experimentalresults, as shown in Figure 4 (b) [17]. In this simulation,the coupling between external stress and polarization isconsidered as a termQ

12(21 + 22)P2in the free energydensity functionalG(E, P, ).G(E, P, ) = g0 + 12g2P2+ 14g4P4+ 16g6P6 12EP Q

12(21 + 22)P2, (6)whereg2= 4.48 108Jm/C2, g4=1.3 1010Jm5/C4, andg6=0Jm9/C6. Theelectrostrictivepa-rameterQ

12is9.23 1010m7/JC2. Nonlinearcou-plingof 2P2betweenstress andpolarizationP isimposed, instead of linear coupling of P2[2]. The non-linear electrostrictive parameter obtained from the per-ovskite BaTiO3 single-crystal result well explains the re-sults for PZT thin lm [18].In experimental results, 6 % change of polarizationunder tensile and + 7 % under compressive stresses arereported, and9%and+8%changesaresimulatedwith240MPa. However, thecoercive-eldresultsarefar dierent: less than 8 % changes are observed in ex-periment, butmorethan25%changesaresimulated.Thisdiscrepancymaycomefromthemeasurementsofhysteresis loops. The measured hysteresis loops at higheldaredierentfromsimulatedonesbecauseofleak-age currents as shown in Figure 4. The measured coer-civeeldwithleakagecurrentisusuallyoverestimatedin comparison with the real coercive eld.4. Therelationbetweenhysteresis loopshapeandpulse-currentresponseAnalternativepolarizationreversal switchingmodelof nucleation-limitedswitching (NLS) was introduced-8- JournaloftheKoreanPhysicalSociety,Vol. 46,No. 1,January2005Table1. SimulationparametersforLKequationforthehysteresisloopsinFig. 5.g2(Jm/C2) g4(Jm5/C4) g6(Jm9/C6) A;Square 2.01092.41094.210102000B 2.01032.41094.210102000C;S-shape2.01032.41094.21010500Fig. 6. Switching-currentresponsesof dierently-shapedhysteresisloops.recently to explaina strong qualitative disagreementof experimental results with the predictions of theKolmogorov-Avrami-Ishibashi (KAI)approach[19, 20].In the KAI model, the switching kinetics is basically gov-ernedbythedynamicsof domaincoalescenceandtheswitching is performed in a 1-decade interval of time. Inthe experimental results,however,the switching is per-formed in an 8-decade interval of time. The NLS modelassumesthattheferroelectriclmconsistsof indepen-dently switching regions and polarization reversal in eachregion is limited by domain nucleation.Therelationbetweentheferroelectric-hysteresis-loopshape and pulse-switching response is studied in connec-tionwiththeNLSmodel. Asquare-shapedhysteresisloopwouldhavealmostthesamecoerciveeldineachregion,but an S-shaped hysteresis loop for ferroelectricmaterials consists of regions where the coercive elds arefardierentfromeachother. Eachregionisswitchedatdierentcoerciveelds, anddierentcoerciveeldsresultintheS-shapedhysteresisloop. Underagivenelectriceld, theregionsof lowercoerciveeldswouldbeswitchedrapidly, buttheregionsof highercoerciveelds would be switched slowly, so that all the ferroelec-tric lm would be switched in the several-decades intervalof time.Varioushysteresisloopsof dierentshapeswereob-tainedwithparametersshowninTableI. Thesquare-shapedhysteresisloopA isobtainedwithalmostthesame parameters of BaTiO3. The hysteresis-loop shapestrongly depends on the parameterg2. With a dierentFig. 7. Normalized switching polarizations calculated fromtheswitching-currentresponsesofFig. 6.g2 value, the S-shaped hysteresis loop B is obtained, butthe coercive eld is far dierent, so that the electric-eldparameter is also changed and the hysteresis loop Cis obtained.Figure 6 shows pulse-switching responses for the vari-ous hysteresis loops simulated with the same method asinFigure2. Asexpected, theS-shapedhysteresisloopswitchedmuchmoreslowlythanthesquarehysteresisloop, eventhoughtheapparentcoerciveeldsarethesame. Switching dynamics, i.e.the time evolution of po-larizations, look like similar results, as shown in Figure 7(a). In the log-scaled plot of Figure 7 (b), however, eachresponsehasadierentswitchingtimebutisswitchedin a 1-decade interval of time.From LK simulation results, it is decided that the KAImodel explains the switching dynamics well independentof hysteresis-loopshape. Anothermechanism, suchaspoling, would be a reason for the long switching intervalin the NLS model.IV.DISCUSSIONLK simulation is an easy and useful method to under-standtheswitchingpropertiesforFeRAMapplication.Theswitchingphenomenainvariouselectriceldsareeasily obtained and external stress eects are well stud-ied. Since LK simulation is based on a phenomenologicalLandau-KhalatnikovSimulationStudy TaeKwonSong -9-theory, it has some limits. This phenomenological simu-lation does not say anything about the microscopic mech-anisms of various properties of ferroelectric thin lms.For the long-term reliability issues such as fatigue, re-tention,andimprint,newcouplingtermsshouldbein-troduced. Pulse-width-dependent ferroelectricproper-ties such as the poling process are hard to treat in thissimulation. Toinvestigatethesereliabilityissues, newterms such as odd-power polarization P3were introduced[7,10].V.SUMMARYFerroelectric switchingproperties suchas hysteresisloops and pulse current responses are simulated with theaid of the LK equation. The LK simulation is proved tobe an easy and useful method to study switching prop-ertiesofferroelectricsandtounderstandFeRAMoper-ationundervariousconditions. Partial switchingandexternal stress eects are studied. The relation betweenthe hysteresis-loop shape and pulse-switching responsesis investigated. The switchingresponses are well ex-pressedwiththeconventional KAImodel independentof hysteresis-loop shape.ACKNOWLEDGMENTSThis work was supported by Grant No. R01-2000-000-00029-0 from the Basic Research Program of the KoreaScience and Engineering Foundation.REFERENCES[1] T. K. Song, J. Ahn, B. Yang, S. Aggarwal and R.Ramesh,J.KoreanPhys.Soc.32,S1721(1998).[2] N.A.Pertsev,A.G.ZembilgotovandA.K.Tagantsev,Phys.Rev.Lett.80,1988(1998).[3] N.A.PertsevandV.G.Koukhar,Phys.Rev.Lett.84,3722(2000).[4] E. Fatuzzo and W. J. Merz, Ferroelectricity (North-Holland,Amsterdam,1967),Chap.3.[5] S.E.Moon,E.-K.Kim,S.H.Lee,J.-K.Lee,H.J.Joo,M. S. Jang, B. S. Kang, J.-G. Yoon, T. W. Noh, S.-I.KwunandT. K. Song, J. KoreanPhys. Soc. 42, S1117(2003).[6] YoungHyunLee,Jung-KunLeeandKugSunHong,J.KoreanPhys.Soc.42,S1395(2003).[7] C.L.WangandS.R.P.Smith,SolidStateComm.99,559(1996).[8] D. Ricinschi, Y. Ishibashi, M. Iwata, L. Mitoseriu, M.Noda and M. Okuyama,J. Korean Phys. Soc. 42,S1232(2003).[9] R. BlincandB. Zeks, Soft ModesinFerroelectricsandAntiferroelectrics(North-Holland, Amsterdam, 1974), p.21.[10] C. L. Wang, L. Zhang, W. L. ZhongandP. L. Zhang,PhysicsLettersA254,297(1999).[11] VengCheongLo,J.Appl.Phys.94,3353(2003).[12] TaeKwonSong,Ferroelectrics259,157(2001).[13] S.Ducharme,V.M.Ridkin,A.V.Bune,S.P.Palto,L.M. Blinov, N. N. Petukhova and S. G. Yudin, Phys. Rev.Lett.84,175(2000).[14] TaeKwonSong,Ferroelectrics259,163(2001).[15] Seung-Hyun Kim, Jeong-Suong Yang, Chang Young KooandJung-HoonYeom, J. KoreanPhys. Soc. 42, S1101(2003).[16] W.Lim,J.Ahn,Y.S.Kim,J.Lee,S.O.ParkandS.I.Lee,Ferroelectrics259,251(2001).[17] T. K. Song, J. S. Kim, M. H. Kim, W. Lim, Y. S. KimandJ.Lee,ThinSolidFilms424,84(2003).[18] P.W.ForsberghJr.,Phys.Rev.93,686(1954).[19] A. K. Tagantsev, I. Stolichnov, N. Setter, J. S. Cross andM.Tsukada,Phys.Rev.B66,214109(2002).[20] I. Stolichnov, A. Tagantsev, N. Setter, J. S. CrossandM.Tsukada,Appl.Phys.Lett.83,3362(2003).