Dynamic response of the dielectric and electro-opt

arXiv:cond-mat/020500730Apr2002Dynamicresponseofthedielectricandelectro-opticpropertiesofepitaxialferroelectricthinfilmsB.H.Hoerman,B.M.Nichols,andB.W.WesselsDepartmentofMaterialsScienceandEngineeringandMaterialsResearchCenter,NorthwesternUniversity,Evanston,IL60208April24,2002AbstractAnanalysisofthedynamicdielectricandelectro-opticrelaxationresponseofthinfilmferroelectricsispresented.Theanalysisisbasedupontherelaxationofferroelectricdomainswithacontinuousdistributionofsizesgivenbypercolationtheory.TheresultingtemporalresponseisdescribedbytheexpressionΦ(t)∝t-mexp(-(t/τ)β).TheanalysiswasappliedtoKNbO3thinfilms.Measurementsofthepolarization,birefringenceanddielectrictransientsshowqualitativeagreementwiththemodelover11ordersofmagnitudeintime.I.INTRODUCTIONThedynamicresponseofthedielectricandnon-linearopticalpropertiesofferroelectricsisoffundamentalinterestforelucidatingthemicroscopicmechanismsoftheferroelectric(FE)state.Bothofthesepropertiesareinfluencedbythedomainstructure,whichisinturndependentonthemicrostructure.InthinfilmFEsthesecondharmonicgeneration(SHG)1andelectro-optic(EO)2coefficientsaretypicallyanorderofmagnitudelowerthaninbulkmaterials.Thereductionisattributedtothepresenceofacomplexdomainstructure.1,2Uponapplicationofanelectricfieldthesedomainsalign,leadingtoanincreaseinthenon-linearopticalsusceptibilities.1-4Thispolingprocesshasbeenobservedtooccurontimescalesrangingfrom1to1000sforSHG,1spontaneousbirefringence,5andtheEOeffect.2Moreover,thealignmentprocessinthinfilms,incontrasttosinglecrystalFEs,hasbeenobservedtobecompletelyreversible1uponremovalofthepolingfield.2-4Thecompetitionofelectrostaticandelasticinteractionsbetweendomainsisbelievedtobethesourceofthereversibilityofthepolingprocess.1,4,6Recentlywehaveexaminedtheinfluenceofdomaindynamicsonthetransientresponseofthenon-linearopticalpropertiesofFEthinfilmsontimescalesfrom6nsto100ms.3,7ThetransientresponseoftheEOeffectfollowsapowerlaw,~t-m.Thisdependencewastentativelyattributedtotherelaxationofferroelectricdomains.AsimilarpowerlawtimeresponsehasbeenobservedforthedielectricpropertiesofthinfilmFEs.8,9Thetimeresponseoftherelaxationcurrentresponsibleforthedielectriclosshasbeenobservedtofollowapowerlawtimeresponse,t-m,orfrequencyresponse,ωm-1,with0m1.Thepowerlawdependenceoftherelaxationcurrenthasbeenobservedtoextendover12ordersofmagnitudeinfrequency.9ThephysicaloriginofthepowerlawrelaxationinthinfilmFEsisnotunderstood,althoughprocessesinvolvingFEdomainsarebelievedtoplayanimportantrole.10Toexplainthedielectricrelaxationorswitchingcurrentofferroelectrics,amodelbasedonthenucleationandgrowthofFEdomainsunderanappliedelectricfieldhasbeenpreviouslydeveloped.11Themodelassumes:(1)arandomdistributionofpre-existing,aligneddomainswithauniformradius,RC;(2)aconstantnucleationrateofaligneddomainswitharadiusRC;(3)thegrowthofthedomainboundaryproceedsataconstantvelocityandcanbecategorizedas1-,2-,or3-dimensional;and(4)nocooperativeinteractionbetweendomains.Themodelpredictsastretchedexponentialgrowthofthevolumefractionofthealigneddomains,1-exp(-(t/τ)β),wheretisthetime,andβandτareconstants.Thevalueofβdependsonthedimensionalityofthedomaingrowthandcantakeonintegervaluesof1,2,or3.Theexpressiondescribesexperimentalswitchingcurrentdatainmanycases,althoughnon-integervaluesofβareoftenrequired.11,12Fortτ,thestretchedexponentialreducestoapowerlaw,(t/τ)β.However,thetheorycannotaccountforthefractionalvaluesofβtypicallyobserved.11,12Thustodescribesuchcasesadifferentexplanationofthepowerlawresponsemustbefound.Whilethepreviousmodelassumespre-existingdomainsofuniformradius,thedomainstructureinthinfilmsisverydifferent.IncontrasttothelargesizesandregularshapesofbulkFEdomains,irregularlyshaped,nano-scaledomainswithadistributionofsizeshavebeenobservedinthinfilms.1,10,13,14TheEOresponseofthesedomainstomicrowaves(2–4GHz)hasbeenstudied.10,14Measurementsindicatetheswitchingprocessofthenanodomainstakeplaceoververyshorttimescales.12TofillinthegapbetweenexperimentandexistingtheorywehavedevelopedamodelforthedynamicdielectricandEOresponseinthinfilmFEsthattakesintoaccountthedistributionofdomainsizesobservedinthinfilmsystems.ThemodelpredictsatemporalresponsethatisdescribedbytheexpressionΦ(t)∝t-mexp(-(t/τ)β).ThemodeliscomparedtotransientmeasurementsoftheEOcoefficient,thepolarization,andthedielectricconstantofepitaxialKNbO3thinfilms.Thesemeasurementssupporttheproposedmodelinthatthetemporalresponseofallthreepropertiesagreeswiththepredictedresponseovertimescalesrangingfrom≤6nsto~103s.II.THEORETICALANALYSISTheproposedmodeldescribesthedynamicresponseintermsofadistributionofrelaxationtimes,15incontrasttoa“normal”FEsystemwheretheresponseisdescribedbyasinglerelaxationtime.Sincetherelaxationratedependsondomainsize,thebroadenedresponseoccursasaresultofthedistributionofdomainsizesobservedinthethinfilms.16Therelaxationfunction,Φ(t),forthesystemconsistingofΩdynamicallycorrelateddomainsisassumedtobeoftheform:15∑Ω=⎟⎟⎠⎞⎜⎜⎝⎛-=Φ0exp)()(iiitVtτϕ(1)wheretisthetime,andϕ(Vi)isthemagnitudeoftheresponseforadomainwithvolumeVi