'Soft Electronic Matter', Magnetoelectric Coupling, and Multiferroism in Complex Oxides

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Title:
'Soft Electronic Matter', Magnetoelectric Coupling, and Multiferroism in Complex Oxides
Physical Description:
1 online resource (137 p.)
Language:
english
Creator:
Mickel,Patrick R
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
Hebard, Arthur F
Committee Members:
Rinzler, Andrew G
Biswas, Amlan
Hershfield, Selman P
Dempere, Luisa A

Subjects

Subjects / Keywords:
dielectric -- film -- magnetodielectric -- magnetoelectric -- manganite -- multiferroic -- phase -- soft -- strain -- thermodynamics -- thin
Physics -- Dissertations, Academic -- UF
Genre:
Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
This thesis has focused on the electronic and magnetic properties of thin-film oxide crystals. Oxides are home to some of the richest physics in condensed matter, producing complex features in response to changes in temperature, electric/magnetic fields, and strain. Three of these features have gained particular prominence, and are among the most active research topics today: phase separation, magnetoelectric coupling, and multiferroism. ``Phase separation" describes the state of materials containing neighboring regions with distinct electronic and magnetic properties - an important phenomenon associated with some of the most exotic material properties known: colossal magnetoresistance, multiferriosm, and high-temperature superconductivity. Phase separation is commonly explained by disorder and strain, resulting in static and stationary phases. However, It is shown here that competing dielectric phases dynamically transform into one another over macroscopic lengths and long time periods ($10^{-3}-10^{-5} $ seconds, and 1 $\mu$m$^{2}$), indicating the phases are far from static. These results argue for a fundamental reinterpretation of the physics of phase separation from localized rigid structures to wave-like thermodynamic entities. Magnetoelectric coupling describes the induction of electric (magnetic) polarization via magnetic (electric) fields, and has myriad applications from sensors to data storage. Lattice strain is commonly proposed as a mediating mechanism, but these conjectures have remained primarily phenomenological. However, this thesis introduces a first principles strain-based microscopic model that describes the measured magnetoelectric coupling of competing dielectric phases. The results of this model accurately reproduce the effects of magnetic fields on the capacitive properties of both dielectric phases, as well as predict additional results seen in the literature. These results provides a direct experimental check of strain's role in magnetoelectric coupling of single phases, and marks an important step forward in understanding the mechanisms producing magnetoelectric coupling. Multiferroics are materials that display two separate orderings, typically spontaneous magnetization (ferromagnetism) and electric polarization (ferroelectricity). These materials promise technological revolutions and are arguably the most intensely researched subject in materials science today. This thesis describes the single-phase multiferroic, BiMnO$_{3}$, providing important evidence for the growing debate concerning its ferroelectric nature. It is shown that BiMnO$_{3}$ displays relaxor ferroelectricity, and that its remanent polarization is highly tunable, decreasing by as much as 10\% in 7 T magnetic fields, and increasing by almost 50\% under small externally induced strains.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Patrick R Mickel.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Hebard, Arthur F.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-08-31

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APh.D.isalongroadthatrequirescommittedsupportfromnumerouspeople,bothprofessionallyandpersonally.Asthesayinggoes,ittakesavillagetoraiseagraduatestudent-andmycaseisnoexception.Itisimpossibletothankeveryonewhohashepledmealongtheway,butbelowaremyattempts.Firstandforemost,IamforevergratefulfortheopportunitiesthatArthasprovidedme.Icametohimlostintheworldofbio-physics,andhegavemetheguidanceandsupportIneeded,helpingmeturnmygraduatecareer180around.Thestimulatingprojects,andpositiveenvironmentthatsurroundhimhavetrulychangedmylife.ArtprovidedinvaluablefeedbackandinsightintoeveryproblemIpresentedhim,andIwillforeveraspiretounderstandphysicsassimplyanddeeplyashedoes.Mycommitteemembershaveallhelpedmereachthispointsuccessfully.Amlan,myunofcialsecondadvisor,hasprovidedmeindispensablesupport.Ourdailyconversations,whichoftenendinlaughter,areacornerstoneinmygraduateeducation.Withouthim,noneofthisworkwouldbepossible.Dr.Rinzlerhasalsohelpedshapemygraduatecareer,andIamverygratefulforhissupportduringmylabtransition.Asmyteacher,Dr.Hersheldhasgreatlyimprovedmyunderstandingofquantummechanics.Hisdoorhasalwaysbeenopenforquestionsconcerningbothclassandresearch.IamalsoverygratefulforallthewaysIhavelearnedwithDr.Dempere.ThroughherclassonSEM,andtheopportunitiessheprovidedmeworkingatMAIC,Ilearnedvaluablecharacterizationtechniquesandconceptsthatstillhelpmetoday.IhavealsobenetedgreatlyfrommanydiscussionsandrelationshipswithothermembersoftheUniversityofFlordiaPhysicsDepartment.First,mycollaborationwithDr.PradeepKumarwasquitefruitful,ashetaughtmeagreatdealaboutmagnetoelectriccoupling.Thetheoreticalunderstandingofmagnetoelectriccouplinginthisthesiswouldnothavebeenpossiblewithouthim.ThetimeIspentinDr.TomMareci'slabwasalsovaluable,asIlearnedtheinnerworkingsofDiffusionTensor 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 12 CHAPTER 1BasicPhysicsReview ................................ 14 1.1MixedValenceManganites .......................... 14 1.1.1Introduction ............................... 14 1.1.2StructureandEnergyDiagrams .................... 15 1.1.3EffectsofDopingandCationSubstitution .............. 18 1.1.4La1xCaxMnO3,Pr1xCaxMnO3,and(La1yPry)1xCaxMnO3 21 1.2Multiferroics ................................... 28 1.2.1Introduction ............................... 28 1.2.2Ferromagnetism ............................ 29 1.2.3Ferroelectricity ............................. 34 1.2.4MagnetoelectricMultiferroics ..................... 43 1.3MagnetoelectricCoupling ........................... 48 1.3.1Introduction ............................... 48 1.3.2MaxwellEquationsvs.MagnetoelectricCoupling .......... 50 1.3.3FreeEnergy ............................... 52 2ExperimentalTechniques .............................. 54 2.1SampleFabrication ............................... 54 2.1.1GrowthMethods ............................ 54 2.1.2StructuralandCompositionalCharacterizations ........... 55 2.2TemperatureandMagneticFieldControl ................... 56 2.3Resistance ................................... 57 2.4CapacitanceMeasurements .......................... 58 2.4.1CapacitanceBridgeandStick ..................... 58 2.4.2DielectricElectrodes .......................... 60 2.4.3InterdigitalCapacitance ........................ 63 2.4.4BandwidthTemperatureSweeps ................... 65 2.5FerroelectricMeasurements .......................... 67 2.5.1Sawyer-TowerCircuit .......................... 67 2.5.2PrecisionLC:FerroelectricTester ................... 69 2.5.3RemanentPolarization ......................... 70 6

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......................... 73 3.1Introduction ................................... 73 3.2TransportProperties .............................. 74 3.2.1Resistance ............................... 74 3.2.2ComplexCapacitance ......................... 74 3.3CompetingDielectricPhases ......................... 79 3.3.1Modeling ................................. 80 3.3.2TemperatureDependenceofModelParameters ........... 81 3.4`SoftElectronicMatter' ............................. 85 3.4.1PolaronsandDetailedBalance .................... 85 3.4.2TestingDetailedBalanceConstraints ................. 87 3.4.3LatticeRelaxationRates ........................ 90 3.4.4ChargeDensityWaves ......................... 91 3.5Summary .................................... 93 4StrainMediatedMagnetoelectricCouplingin(La1yPry)1xCaxMnO3 95 4.1Introduction ................................... 95 4.2DielectricConstantTuning ........................... 96 4.2.1ExperimentalResults .......................... 96 4.2.2Modeling ................................. 97 4.3ActivationEnergyTuning ........................... 99 4.3.1ExperimentalResults .......................... 99 4.3.2ComparisonofMagnetoelectricCouplings .............. 101 4.4FilmThicknessStudy ............................. 101 4.5Summary .................................... 103 5MultiferroisminBiMnO3 105 5.1Introduction ................................... 105 5.2Characterizations ................................ 105 5.2.1StructuralCharacterization ....................... 106 5.2.2MagneticCharacterization ....................... 107 5.2.3ResistiveCharacterization ....................... 109 5.2.4FerroelectricCharacterization ..................... 109 5.2.5DielectricCharacterization ....................... 112 5.3NatureofFerroelectricity ............................ 115 5.3.1RelaxorReview ............................. 115 5.3.2Comparison ............................... 116 5.3.3PulseSequencing ........................... 118 5.3.4IslandGrowth .............................. 119 5.4Summary .................................... 121 7

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122 6.1Introduction ................................... 122 6.2Strain:ExternalandIslandEdges ...................... 123 6.2.1ExternalStrain ............................. 123 6.2.2ElectrodeLensing ........................... 125 6.2.3IslandEdgeStrainGradients ..................... 125 6.3MagnetoelectricCouplinginBiMnO3 126 6.3.1RemanentPolarizationTuning ..................... 126 6.3.2ReorientationTime-Scales ....................... 127 6.3.3ConnectiontoLatticeTransition .................... 128 6.4Summary .................................... 130 REFERENCES ....................................... 131 BIOGRAPHICALSKETCH ................................ 137 8

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Figure page 1-1CubicABO3PerovskiteManganiteStructure ................... 16 1-2OrbitalEnergyLevels:CrystalFieldSplittingandJahn-TellerDistortions .... 16 1-3CubicandJahn-TellerDistrotionofMnO6Octahedra ............... 18 1-43dOrbitals:egandt2g 19 1-5PolaronDepiction .................................. 19 1-6La1xCaxMnO3PhaseDiagram ........................... 22 1-7Pr1xCaxMnO3PhaseDiagram ........................... 23 1-8(La1yPry)1xCaxMnO3PhaseDiagram ...................... 25 1-9Dark-FieldElectronDiffractionImageofPhaseSeparation ........... 26 1-10MagneticForceMicroscopy ............................. 27 1-11MultiferroicCouplingSchematic ........................... 30 1-12TypesofMagneticOrdering ............................. 31 1-13M-HLoopsforDifferentMagneticOrderings .................... 32 1-14StonerBandTheoryofFerromagnetism ...................... 35 1-15Polarizationvs.ElectricFieldLoopsforDifferentElectricOrderings ....... 36 1-16FerroelectricBananas ................................ 37 1-17FerroelectricUnitCell ................................ 39 1-18FerroelectricEnergyDiagrams ........................... 40 1-19AntisymmetricDzyaloshinskii-MoriyaInteraction ................. 42 1-20CompositeMultiferroicGeometries ......................... 45 1-21Bi6sLonePair .................................... 47 1-22MagnetoelectricRevival ............................... 49 1-23MagnetoelectricMultiferroicVennDiagram .................... 50 2-1PLDSchematicandImage ............................. 55 2-2PPMSSampleChamberSchematic ........................ 57 9

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........................ 58 2-4HP4284Circuitry ................................... 59 2-5DielectricElectrodeCircuits ............................. 61 2-6Maxwell-WagnerCircuitSimulation ......................... 63 2-7InterdigitalCapacitanceFundamentals ....................... 64 2-8InterdigitalCapacitanceFabrication ........................ 65 2-9Multi-FrequencyConsistencyCheck ........................ 67 2-10Sawyer-TowerCircuit ................................ 68 2-11PrecisionLCCircuitry ................................ 70 2-12RemanentPolarizationPulseSequence ...................... 72 3-1DCResistancevs.Temperature .......................... 75 3-2ComplexCapacitancevs.Frequency ........................ 76 3-3Cole-ColePlot .................................... 77 3-4LogarithmicParametricSlope ............................ 78 3-5CompetingDielectricPhaseAnsatz ........................ 79 3-6CircuitModel ..................................... 81 3-7ModelResults .................................... 82 3-8vs.Temperature .................................. 83 3-9rampvs.Temperature ................................. 84 3-10ArrheniuisPlotofRelaxationTime-Scales ..................... 84 3-11PolaronDepiction .................................. 85 3-12DetailedBalance3StateModel ........................... 86 3-13rampvs.Temperature ................................. 88 3-14Energy/PopulationSchematic ............................ 89 3-15ThicknessDependenceofDielectricConstatns .................. 91 3-16Energy/PopulationSchematic ............................ 92 3-17ChargeDensityWaveSchematic .......................... 93 10

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...................... 97 4-2MagneticTuningofActivationEnergies ...................... 100 4-3ModelingResultsforMultipleThicknessFilms ................... 102 4-4StrainDependenceofMagnetoelectricCoupling ................. 103 5-1MonoclinicUnitCellofBiMnO3 106 5-2BiMnO32Scans ............................... 108 5-3MagneticCharacterization .............................. 110 5-4RemanentHysteresisLoops ............................ 111 5-5FrequencyDependenceofImaginaryCapacitance ................ 113 5-6ArrheniusPlotofRelaxtionTime-Scales ...................... 114 5-7TemperatureDependenceofRealCapacitance .................. 115 5-8DielectricPredictionofFerroelectricTC 118 5-9DualPulseSequenceResults ............................ 120 6-1ExternalStrainGeometry .............................. 124 6-2StrainTuningofFerroelectricity ........................... 124 6-3ElectrodeLensing ................................. 126 6-4IslandStrain ..................................... 127 6-5MagnetoelectricCouplinginBiMnO3 128 6-6PulseSequenceinMagneticFields ........................ 129 6-7CorrelationofCouplings ............................... 129 11

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1.1.1IntroductionThebasicelectricandthermalpropertiesofmanganiteswererstinvestigatedasearlyasthe1950's[ 1 ],wherealargemagnetoresistancewasdiscoveredneartheferromagneticCurietemperature,TC,producingmoderateinterestinthephysicscommunity.In1994,however,researchinmanganitessurgedinresponsetoanew-foundphenomena:colossalmagnetoresistance,wheremagneticeldswerefoundtoinduceadecreaseinresistancebymorethanafactorof103[ 2 ].Theinitialdreamwastosomedayreplacetheubiquitousgiant-magnetoresistance(GMR)effect,whichhadquicklybecomethestandardinthemagneticinformationstorageindustryforread-heads.GMRisbasedonspin-valvemechanismsandresultsinafewtensofpercentchangeinresistance,socolossalmagnetoresistanceprovidedahugepotentialforimproveddeviceperformance.Asresearchprogressed,however,thevisionofmanganiteschangedradicallyfromwhatseemedastraightforwardapplicationinthemagnetic-information-storageindustry,toacolossalchallengetocondensed-matterphysics.Soonafterthereemergenceofmanganites,itwasrealizedthattheirinitialtheoreticaldescription(i.e.doubleexchange,seesection 1.1.3 )wasincapableofquantitativelyreproducingthenew-foundcolossalproperties,andthatthesesystemsweremuchmorecomplexthanoriginallythought.Manganitesarenowthoughtofasaquintessentialcomplex-oxidesystemwherethesimultaneousinterplayofspin,charge,orbital,andlatticedegreesoffreedomspawnsomeofthemostcomplicatedandexoticmaterialpropertiesandphysicsincondensed-mattertoday.Thesepropertiesinclude:insulator-to-metaltransitions,chargeandorbitalordering,ferroandanti-ferromagneticordering,chargedensitywaves,andmultiferroism.Additionally,manganitesareextrememlysensitivetoexternal 14

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1-1 showsthebasiccubicstructureofthemanganiteunitcell.TheAsiteatomsoccupythecornersoftheunitcellandactprimarilyaschargereservoirsfordopingandspacellersforstructuralintegrity.TheMnatomsoccupythecenterofMnO6oxygenoctahedra,thenetworkofwhichistheprimarymediatorofelectricalconductivityandmagneticstructurewithinthecrystal.TheAsiteatomsindirectlycontroltheelectricandmagneticpropertiesbyinuencingthevalenceandbondanglesoftheMnatoms(whichnecessarilyaffectsitsmagneticmoment).Specically,theelectronicandmagneticpropertiesofthecrystalarecontrolledbythe3delectronsoftheMnatom.Therefore,itisusefultoconsidertheenergiesoftheseorbitals.Inisolation,theMn3delectronsshareavefolddegeneracy:thedxy,dyz,dxz,dz2,anddx2y2orbitals.However,whentheMnatomsarebroughtnearneighboringions,theambientelectriceldscanaltertheenergylevelsoftheelectronicorbitals.Thiseffectiscalledcrystaleldsplitting,[ 3 ]andincubicperovskitesitresultsinsplittingthe5degenerate3dorbitalsintotwogroupsofenergylevels:3low-energyt2gorbitalsand2high-energyegorbitals.Figure 1-2 illustratesthisdegeneracysplitting.TheenergylevelsplittingcanbeunderstoodintermsofCoulombinteractionsbetweentheO2pelectronswhichliealongthex,y,andzaxisoftheMnO6octahedra.TheCoulombpotentialislargestfororbitalsalongtheseaxes,raisingtheenergyofthedz2,anddx2y2orbitals(eg),andloweringtheenergyofthedxy,dyz,anddxzoffaxisorbitals(t2g). 15

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Inthecubicperovskitemanganitestructure,theBsiteMnatom(red)isencagedinanOoctahetron(black).Asiteatoms(blue)constitutethecubicshellthatsupportstheoctahedronstructurally.Alternatively,thestructuremaybeviewedintermsofMnO2planesthatareseparatedbyAOplanes,wheretheAsiteatomslieinthesameplaneastheapicalOatomsofeachoctahedron. The3dorbitalenergylevelsdisplaytomodications:crystaleldsplittingandJahn-Tellerdistortions.Ontheleftarethe5degenerage3delectronorbitalsforanatominisolation.Whentheatomisplacedinthepresenceofambientelectriceldsfromionswithinacrystalstructurethedegeneracyisbroken(middlesection).Andwhenoneegorbitalisoccupiedbyasingleelectron,aspontaneousJahn-TellerdeformationlowerstheenergyofthesytembyEJTbyelongatingthez-axisoftheunitcellandloweringtheCoulombinteractionofthedz2orbitalwhichisalignedwiththeneighboringO2porbital.SeeFigs. 1-4 and 1-3 16

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1-2 and 1-3 ).BecausetheoverallColoumbicpotentialdoesnotincrease,theoverallvolumeoftheunitcellremainsconstant,resultinginacontractioninthex-yplanewhichincreasestheCoulombinteractionofthedx2y2andO2porbital,raisingitspotentialenergy.Thet2gorbitalsarealsoeffected,withtheyzandxzlevelsnowlowerthanthexyorbitals,againbecauseoftheproximityoftheO2pelectrons.ThisspontaneousenergyloweringdistortioniscalledaJahn-Tellerdistortion[ 4 ],andisdepictedinFigs. 1-2 and 1-3 .WenoteherethatthisenergygainthroughspontaneousdistortionisnotavailableinMn4+systemsbecauseallegorbitalsareemptyandsotherearenooccupiedelectronstatesthatdecreaseinenergy,sincethetotalenergyofthet2glevelsisconstant.Thistypeofdistortion,thatisdependentonthepresenceofanelectroniscalledapolaron.APolaronisaquasi-particlethatencompassesanelectronandtheinducedlatticedistortionsurroundingit,seeFig. 1-5 .ThecrystaleldandJahn-Tellerenergeticadjustmentsareimportantbecausetheyhaveadirecteffectonthemagneticandelectrictransportpropertiesofmanganites.Theloweringofthet2genergylevelsresultsinalocalized3/2spinwhichcanbetreatedasaclassicalcorespinthatistiedtothelattice,providingabasisforallthemagneticorderingspresent.TheformationoftheJahn-Tellerpolaronandloweringoftheegenergylevelmodiestheelectronicconductionbylocalizingtheegelectronsinaselfpotentialwell,resultinginaMottlikeinsulatingstate-sincetheconventionalbandpicturedictatesthatLaMnO3(theprototypicalMn3+manganite)withit'ssinglyoccupiedegstateshouldbeconducting. 17

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A)Theoxygenoctahedraofthecubicperovskitestructureisshown.B)TheJahn-Tellerdistorsionisshown.TheegorbitalisoccupiedbyasingleelectronallowingittobecomeenergeticallyfavorablefortheunitcelltodistortfromacubicMnO6octahedratoanoctahedrathatiselongatedalongthez-axis(Jahn-Tellerdistorted).ThisreducestheorbitaloverlapandresultingCoulombpotentialenergybetweentheMndz2andO2porbitals.SeeFig. 1-4 for3dorbitalorientations,andFig. 1-2 fortheorbitalenergylevelmodications. 18

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The53dorbitalsaresplitintotwogroups:eg(2)andt2g(3).The2egorbitalsareorientedtowardtheOatomsonthex,y,andzaxes,whilethe3t2gorbitalsarenot.TheJahn-Tellerdistortion(seeFigs. 1-2 and 1-3 )causesanelongationoftheunitcellalongthez-axisandacontractioninthex-yplanemovingO2porbitalsfurtherawayandcloser,respectively.Thedecreasedoverlapalongthez-axislowerstheenergyofthedz2,dxz,anddyzorbitals,andraisestheenergyofthedx2y2anddxyorbitals. Apolaronisaquasi-particlethatisdenedbyanelectronandthecloudofdistortionsitinducesinthesurroundinglatticesites. 19

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5 ]andwaslaterreformulatedbyAndersonandHasegawain1955[ 6 ],andconsistsofthesimultaneoustransferofanelectronfromaMn3+sitetoanO2siteandthetransferofanelectronfromanO2sitetoaMn4+site.Inthechargetransferprocess,thehoppingelectrons/polaronsarecoupledmagneticallytothe3/2corespinofthet2gorbitalsthroughalargeHundcoupling(>1eV)thatenergeticallyrequiresthattheegelectronshavethesamespinorientationasthecorespinatthenewlocation.ThisresultsinaneffectivehoppingmatrixbetweenthetwoMnatomsofthe(simplied)form[ 6 ]: 7 ].AnadditionalconsequenceofthelargeHundcouplingisthatatlowtemperatureshalf-dopedmanganitesarehalfmetalswithnear100%spinpolarizationoftheconductionelectrons,makingthemprimecandidatesforspintronicapplications[ 8 ].Doubleexchangeprovidesabasicdescriptionofthemagnetoresistanceobservedinmanganites,nearTCtheinducedalignmentofcorespinsbyexternaleldsfacilitateshoppingtherebyincreasingtheconductivityresultingintheobservednegativemagnetoresistance.Thisdescriptionisonlyqualitative,however,failingtoreproduce 20

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9 11 ].InadditiontothevalenceoftheAsiteatoms,theirsizealsohasastrongeffectontheelectricandmagneticpropertiesofthecrystal.VaryingtheionicradiioftheAsiteatomscaninduceadditionaldistortionsoftheMnO6octahedrabyallowingtheO-Mn-Obondstobuckleawayfrom180o.Thiseffectiscommonlyquantiedusingthetolerancefactor: 11 ]).Additionally,bucklingtheO-Mn-Obondangleencumbersdoubleexchangebyloweringthehoppingintegral,preventingthealignmentofcorespinsanddelayingtheferromagneticandinsulator-to-metaltransitionstolowertemperaturesoreliminatingthemaltogether. 1.1.3 ).LPCMOiscomposedofanincommensurate(inhomogeneous)mixtureofLCMOandPCMO,thereforetounderstanditsproperties,itisnecessarytorevieweachparentcompound.ThetwolimitingcompoundsinLCMO'sphasediagram(seeFig. 1-6 )aretheMn3+LaMnO3andtheMn4+CaMnO3,however,thepropertiesofLCMOareconsiderably 21

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TheLa1xCaxMnO3phasediagramshowstheeffectofCadoping,whichintroducesMn4+intothecompletelyMn3+systemLaMnO3.Intermediatedopingsproducephysicsnotseenineitherparentcompound(LaMnO3andCaMnO3):.Atintermediatedopingsbelow50%aferromagneticmetallicphaseisformedatlowtemperatures,andintermediatedopingsabove50%acharge-orderedinsulatorwithaspatialmodulationofthechargedistributiononMnsitesisformed(Mn4+Mn3+Mn4+Mn3+...).Here,CAFstandsforcantedantiferromagnet,FIforferromagneticinsulator,andCOforchargeordered.TheCAFandFIcouldhavespatialinhomogeneitywithbothferro-andantiferro-magneticstatespresent.FigureadaptedfromRef.[ 12 13 ]. differentthanasimpleinterpolationbetweenthepropertiesofthelimitingcompounds.LaMnO3andCaMnO3arebothparamagneticinsulatorsathigh-temperatureandtransitiontocanted-antiferromagneticinsulatorsatlowtemperature.Thehigh-temperatureparamagneticinsulatingstateremainsatallcompositions,however,atintermediatemixingsLCMOdevelopsentirelynewlow-temperatureelectromagneticphases.AsCadopingincreasestobetween5%and20%thelower-temperaturephasetransitionstoaferromagneticinsulator.ThenasCaincreasesfurthertobetween20%and50%thelow-temperaturephasebecomesaferromagneticmetal,withthedevelopment 22

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ThePr1xCaxMnO3phasediagramshowstheeffectsofCadoping.CadopingintroducesMn4+intothecompletelyMn3+systemPrMnO3,andalsointroducesstructuraldistortionsbecausetheionicradiiofCaissignicantlysmallerthanPr.LikeLa1xCaxMnO3,Pr1xCaxMnO3developschargeorderingatintermediatedopings,butunlikeLa1xCaxMnO3,remainsinsulatingforalldopings.ThePI,PM,andCIdenotetheparamagneticinsulating,paramagneticmetallic,andcantedinsulatingstates,respectively.TheFIandFMdenotetheferromagneticinsulatingandferromagneticmetallicstates,respectively.TCandTNdenotetheferromagneticCurieandantiferromagneticNeeltemperatures,respectively.FigurereproducedfromRef.[ 14 ] ofaninsulator-to-metaltransitionmediatedbydoubleexchange.Above50%Cadopingresultsinalow-temperatureantiferromagneticcharge-orderedinsulatingphasewherethereisaspatialmodulationoftheMnchargedistribution(Mn4+Mn3+Mn4+Mn3+...).Above7/8thsCadopingthesystemtransitionsbacktotheoriginalcantedantiferromagneticphase.ItshouldbenotedthatCaandLahavecomparableionicradii,sotherichphysicsembodiedinLCMO'sphasediagramareprimarilytheresultofthebalanceofthemixedvalences,Mn3+andMn4+. 23

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1.1.3 ).ThesedistortionscausealargeralternatingtiltingoftheMnO6octahedrawhichreducestheone-electronbandwidththerebyhinderingdoubleexchange[ 14 ].Asaresult,PCMO'sconductionisinsulatingoveritsentirephasediagram(seeFig. 1-7 ).However,therearestillcomplexlow-temperaturephasesthatarisewithincreasingCadoping.At15%Cadopingalowtemperatureferromagneticinsulatingphasedevelops,andabove30%Cadopingtherearecharge-orderedantiferromagneticandcanted-antiferromagneticphases.WhilePCMOisnaturallyinsulatingoveritsphasediagram,itisimportanttonotethattheapplicationofmagneticeldsissabletomeltthecharge-orderedinsulatingphasesandinducealow-temperatureinsulator-to-metaltransition[ 15 ].Naturally,thephasediagramofLPCMOisevenmorecomplexthanthephasediagramsofitstwoparentcompounds.Inthisthesis,wefocusonthestoichiometrywithanequalmixtureofLCMOandPCMO:(La1yPry)1xCaxMnO3,withy=0.5andx=0.33.AsimpliedphasediagramisshowninFig. 1-8 .Atlowtemperaturesandforthecompositionx=0.33,LCMOisaferromagneticmetalandPCMOisacharge-orderedinsulator.Combiningthesecompoundsatequalratios(y=0.5)resultsinacoexistenceandcompetitionbetweenthesetwodissimilarphases.Thiscoexistencehasbeentermedphase-separationandhasbeenshowntooccuroverquasi-macroscopiclengthscalesapproaching1m.Figures 1-9 and 1-10 showthemostconvincingevidenceofphaseseparationinmanganites.Figure 1-9 isadark-eldelectron-diffractionimagetakenatasecond-orderBraggreectionpeak.ThebrightspotsaretheresultoftheconstructiveinterferenceofthespatialmodulationofthechargeorderingofMn3+andMn4+,whereasthedarkregionsarecharge-disorderedregionswhicharebelievedtobetheferromagneticmetallicphase(regardlessofwhatphasetheyrepresent,theimage 24

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The(La,Pr,Ca)MnO3phasediagramshowsacombinationofthephasediagramsoftheparentcompounds(La,Ca)MnO3and(Pr,Ca)MnO3.Theredlinedenotesthex=0.33Cadopingconcentration,andthegreyboxdenotesarangeofcompositionswhichexhibitphase-separation,however,thisworkfocusesexclusivelyonthecenterofthisregionaty=0.5.Inthephaseseparatedregion,thecharge-orderedinsulatingphaseofPCMOcompeteswiththeparamagnetic-insulatingphaseofLCMOatintermediatetemperatures.AtlowtemperaturestheferromagneticmetallicphaseofLCMOcompeteswiththecharge-orderedphaseofPCMO.IllustrationprovidedbyDr.AmlanBiswas. stilldemonstratesphaseseparationbetweenchargeorderedandchargedisorderedphasesonmlengthscales).Figure 1-10 isamagnetic-force-microscopy(MFM)imagewhichprovidesdirectevidenceofthepercolationandtheevolutionofferromagneticmetallicphaseastemperatureissweptthroughtheinsulator-to-metaltransition.ForthecompositionofLPCMOwithy=0.5andx=0.33,athightemperaturestheentirecrystalisintheparamagneticinsulating(PMI)phase.Thenatintermediatetemperaturesphaseseparationoccursasaportionofthesamplebecomescharge-orderedinsulating(COI)near240K.Finallyatlowtemperatures(below115Kfor30nmlms)theferromagneticmetallic(FMM)phaseisformed,whichsupplantsthePMIphaseandcompeteswiththeCOIphase.Thecompetitionbetweenthesethreephases(PMI,COI, 25

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Dark-eldimagesforLa5=8yPryCa3=8MnO3areobtainedbyusingasuper-latticepeakcausedbychargeorder(CO)Panelashowsthecoexistenceofcharge-ordered(insulating)andcharge-disordered(FMmetallic)domainsat20Kfory=0.375.Thecharge-disordereddomain(darkarea)ishighlightedwithdottedlinesforclarity.ThecurveddarklinespresentinCOregionsareantiphaseboundaries,frequentlyobservedindark-eldimagesforthecommensurateCOstatesofLa0.5Ca0.5MnO3.Panelsbandc,obtainedfromthesameareafory=0.4at17Kand120K,respectively,showthedevelopmentofnanoscalecharge-disordereddomainsatTTC.Thecurvedlinesina,bandcsignifythepresenceofanti-phaseboundariesoftheCOdomains.FigureandcaptionreproducedfromRef.[ 16 ] andFMM)hasbeenthesubjectofintenseexperimentalandtheoreticalinvestigationsincetheirdiscovery.ThepercolativeonsetoftheFMMphasehasreceivedparticularattention,producingcolossalchangesinresistance[ 2 ]andcapacitance[ 18 ]byinducinganearlyinsulator-to-metaltransitionthroughitsmagneticelddependentstabilizationathigherandhighertemperatures.Inthisthesis,however,wewillshowthatthecompetitionbetweenthePMIandCOIphasesisalsooffundamentalinterest,asitprovidesauniqueperspectiveintothebasicnatureofphaseseparationitselfincomplexoxides.Bymeasuringthefrequencydependenceofthecomplexcapacitanceofthin 26

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Amagneticforcemicroscopy(MFM)imageofphaseseparationisshown.Thetemperature-dependentMFMimagesequence(A)forcoolingand(C)forwarming,andtheresistivity(B)oftheLa0.33Pr0.34Ca0.33MnO3thinlmoverathermalcycle.Theblueseriescorrespondstocooling,theredseriestowarming.Thecenterofeachimageisalignedhorizontallywiththetemperaturescalefrom(B)atthetimeofimagecapture.Scannedareasare6mby6mforallcoolingimagesand7.5mby7.5mforallwarmingimages.Allcoolingimagesweretakenatoneareaofthesample,andallwarmingimagesweretakenatanotherarea.FigureandcaptionreproducedfromRef.[ 17 ] LPCMOlms,weshowthatthecompetitionbetweenthesedielectricphases(PMI,COI)providestherstevidenceforelectronicallysoftphases. 27

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1.2.1IntroductionMultiferroics-denedasmaterialspossesingatleasttwoferroicorderings[ 19 ]-havequicklybecomeoneofthemostwidelyresearchedtopicsincondensedmatterphysicstoday,bothfortheirpotentialapplicationsandfortheircomplexphysicalorigins.Ferromagnetism,ferroelectricity,andferroelasticityaretheclassicferroicorders,however,contemporaryfocushasplacedlittleemphasisonferroelasticproperties,andthemagneticferroicrequirementshavebeenbroadenedtoincludeantiferromagnetismandferrotoroidicorderings.Therstattemptstocombinemultipleferroicpropertiesintoonematerialstartedinthe1960'sbySmolenskiiandVenevtsev[ 20 21 ].Initially,theseresultsinspiredmoderateinterestinthephysicscommunity,butmultiferroicshaverecentlyundergoneanintenserenaissance[ 22 26 ].Themultiferroicrenaissancehasbeenfueledbymultiplefactors.Firstin2000,aseminalpaperhighlightedthecurious(andinconvenient)lackofoverlapbetweenferromagneticandferroelectricmaterials,resultinginadearthofsingle-phasemultiferroics[ 27 ].Thispaperineffectissuedagrandchallengetomaterialsdevelopmentwhich-thankstorecentadvancementsinboththeoreticalandexperimentaltools-hasbeenaggressively(andsucessfully)pursued.Experimentally,thinlmcrystalgrowthhasprogressedsignicantlysincetheinitialmultiferroicinterestofthe1960's,withtheadventofstrainengineeringthroughepitaxiallatticemismatchandnewhigh-pressuregrowthtechniques[ 28 29 ].Additionally,newexperimentaltechniquesforobservingelectricandmagneticdomainshavedeveloped[ 30 ].Theoretically,improvementsinrst-principlesanddensityfunctionaltheory(DFT)computationaltechniqueshaveprovidedinsightintorelevantmicroscopicmechanismspromotingferromagnetism,ferroelectricity,andtheircouplings.Mostimportant,however,isthegrowingintersectionofexperimentandtheoryinmultiferroics-wherethenewfoundattainabilityofhighqualitysamplescreatessynergythroughthedirectfeedbackbetweenbothdisciplines. 28

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1-11 ).Couplingbetweenelasticandferroicpropertiesiswidelyobservedintheformofpiezoelectricityandpiezomagnetism,wherestraincaninduceelectricormagneticpolarization(andviceversa).However,themostinterestingcouplingisbetweentheelectricandmagneticorderingsthemselves:magnetoelectriccoupling-foracompletediscussionofmagnetoelectriccouplingseeSec. 1.3 Thissectionwillcoverthebasicphysicsofmultiferroics,beginningwiththetwomostpopularferroicorderings:ferromagnetismandferroelectricity.Thisleadstoadiscussionofd0-ness,andtheirseeminglyincompatiblemechanisms.Finallywediscussnovelapproachestocombineferromagnetismandferroelectricityinasinglematerial. 1-12 ).Diamagneticsystemsexhibitmagnetizations(coherentorientationsofinternalmagneticmoments)whichcanbeinducedunderexternalelds,withthemagnetizationlinearlyproportionaltothemagnitudeofappliedeldandalignedanti-paralleltotheeld.Paramagneticsystemsexhibitinducedmagnetizationsunderexternalelds,withthemagnetizationlinearlyproportionaltothemagnitudeofapplied 29

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Theferroicordersofmultiferoicscanbecontrolledbyexternalperturbations.TheelectriceldE,magneticeldH,andstresscontroltheelectricpolarizationP,magnetizationM,andstrain,respectively.InaferroicmaterialP,M,andarespontaneouslyformedtoproduceferromagnetism,ferroelectricity,orferroelasticity,respectively.Inamultiferroic,thecoexistenceofatleasttwoferroicformsoforderingleadstoadditionalinteractions.Inamagetoelectricmultiferroic,amagneticeldmaycontrolPoranelectriceldmaycontrolM(greenarrows).FigurereproducedfromRef.[ 24 ] eldandalignedparalleltotheeld.Forthesesystems,oncetheexternaleldisremovedthemagneticmomentsre-randomizecancelingthemacroscopicmagnetization(seeFig. 1-12 a).Inferroicmagneticsystems,however,thereisaninherentcouplingbetweenspinsthatpromotesacoherentalignmentofthemagneticmomentsevenintheabsenceofanexternalmagneticeld,oftenresultinginaspontaneousmagnetization(seeFig. 1-12 b).Inantiferromagneticsystems,however,thiscouplingpromotesananti-parallelalignmentofmagneticmomentsresultinginzeromagnetization(seeFig. 1-12 c).Ferrimagnetismisacombinationofferromagnetismandantiferromagnetism,wheresublatticesofspinsexhibitferromagneticcouplinginternallybutantiferromagneticcouplingtoneighboringsublattices(seeFig. 1-12 e).Bydenitionthesesublatticeshaveunequalmagnetizations(otherwisethesystemwouldbeantiferromagnetic),resultinginanoverallweakferromagneticbehavior.Cantedantiferromagneticsarefrustratedantiferromagnetswhereitisenergiticallyfavorableforthealignmentofthesublatticesto 30

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Asamplingofmagneticorderisshown.A)Paramagnetism:Magneticmomentsarerandomizedfornonetmagnetizationinzeroexternaleld.B)Antiferromagnetism:Magneticmomentsareorderedinsublatticeswhichareanti-alignedwitheachotherresultinginnonetmagnetization.C)Ferromagnetism:Magneticmomentsarealignedparallelproducingalargenetmagnetizationinzeroexternaleld.D)Canted-antiferromagnetism:Magneticmomentsareorderedinsublatticeswhichareonlypartiallyanti-aligned,producinganetmagnetization(totherighthere).E)Ferrimagnetism:Magneticmomentsareorderedinsublatticeswhichareanti-aligned,butunequal,resultinginanetmagnetization. skewfromthe180oanti-parallelalignment,resultinginasmallnetmagnetization(seeFig. 1-12 d).Athightemperatures,ferroicmagneticsystemsaretypicallyparamagneticbeforeundergoingatime-reversalinvariancebreakingtransitionatlowertemperatures(theCurietemperature,TC,forferromagnetics;theNeeltemperature,TN,forantiferromagnetics)withtheonsetofmagneticcouplingandthemanifestationofspontaneousordering.Experimentally,inferromagnetsthistransitionisobservedbytheopeningofmagnetizationvs.magneticeld(M-H)hysteresisloops,seeFig. 1-13 .Initiallythesystemordersintolocaldomainswhichcancelgloballytozeromacroscopicmagnetization.Whenaeldofsufcientstrengthisapplied,thedomainsalignandremainalignedaftertheremovalof 31

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M-Hloopsareshownfordifferentmagneticorderings.A)Diamagnetism:M-Hloopisclosed,withalinearlyinducedmagnetizationthatopposestheappliedeld.B)Paramagnetism:M-Hloopisclosed,withalinearlyinducedmagnetizationalignedwiththeappliedeld.C)PM-FMtransition:NearTCM-Hloopsbegintoopenwiththeonsetofspontaneousmagnetization.D)Ferromagnetism:M-Hloopsareopen,asthereisspontaneousmagnetizationatzeroexternaleld(MS).MScanbereorientatedunderacoerciveeld,HC.Ferrimagnetsandcanted-antiferromagnetshaveM-Hloopssimilartobutsmallerthanferromagnets. theexternaleldresultinginalargeremanentspontaneousmagnetization,MS(seeFig. 1-13 ).Ferrimagnetsandcantedantiferromagnetsdisplayreducedmagnetichysteresisloops,however,pureantiferromagnets(withzerospontaneousmagnetization)havenomagnetichysteresis.Thus,ferromagnetsarethemosttechnologicallyrelevantmagneticmaterials,andaccordinglytheyhavereceivedthemostattentioninresearch.Twophenomenologicaltheorieshavesuccessfullyreproducedmanyofthepropertiesofferromagnetism:theCurie-Weisslocal-momenttheory,andtheStonerbandtheoryofferromagnetism. 32

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33

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1-14 .Themagnitudeoftheshiftisuniformforallwavevectors,resultinginarigiddisplacementbetweenminorityandmajorityspincarriers.WhentheFermilevellieswithinthe3dbandthisresultsinanincreasedpopulationofmajorityspincarriers,andaspontaneousmagnetization:M=B(n"-n#),wheren"andn#arethemajorityandminoritypopulations,respectively(seeredandblueareasinFig. 1-14 ),andBistheBohrmagneton.ByincorporatingFermistatistics,thismodelsuccinctlyresolvestheobservationthatmagneticmomentsdonotcorrespondtointegernumbersofelectrons,aswellasthepotentialchangeofmagneticmomentsasenergieschangeduringphasetransitions.Themodelalsoexplainsthetrendofferromagnetismintransitionmetals:inlatertransitionmetalstheFermilevelrisesabovethe3dbandscausingbothspinbandstobeoccupiedequally,cancelingthenetmagnetization.Hence,fortransitionmetalions,ferromagnetismrequiresapartiallyoccupied3dband. 31 ],andsincethennumeroussimilaritiestoferromagnetismhavebeendocumented.Analogoustotheenergygainfromexchangecouplinginferromagnetism,ferroelectricityiscommonlydrivenbyanenergygainassociatedwiththehybridizationofionicorbitals.Likeinmagnetism,therearemultipleelectronic 34

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TheStonerBandTheoryofFerromagnetismisdepicted.Exchangeinteractionsraisetheenergyofanti-alignedspins,shiftingthebandofminorityspin-downcarriers(blue).Thisresultsinanincreasedpopulationofmajorityspin-upcarriers(red),andanetmagnetization.3dbandsproducelargemagnetizationsduetotheirlargedensityofstates,D(E),whichproducealargepopulationdifferencebetweenspinsforthesmallshiftinducedbyexchangeinteractions.The4sband(green)hasalowdensityofstatesanddoesnotcontributetothemagnetization. orderings:dielectric,paraelectric,ferroelectric,ferrielectric,antiferroelectric,andcantedantiferroelectric.Thediscussionoftheseorderingsisalmostidenticaltotheirmagneticcounterparts,withthesimplereplacementofmagneticdipolemomentswithelectricdipoles.Dielectricandparaelectricsystemsdisplayinducedpolarizationsunderexternalelds,withthedipolesre-randomizingoncetheeldisremoved.Onesmalldifferenceisthatthedistinctionbetweendielectricandparaelectricisalinearlyandnon-linearlyinducedpolarizationinexternaleld,respectively-asopposedtolinearlyinducedwithanti-parallelandparallelalignmentfordiamagneticandparamagnetic,respectively.Inferroicelectricsystems-justasinferroicmagneticsystems-thereisaninherentcouplingthatpromotesacoherentalignmentofthedipolemomentsevenintheabsence 35

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Polarizationvs.electriceldloopsareshownfordifferentelectricorderings.Dielectric:P-Eloopisclosed,withalinearlyinducedpolarizationthatalignedwiththeappliedeld.Paraelectric:P-Eloopisclosed,withanon-linearlyinducedpolarizationalignedwiththeappliedeld.PE-FEtransition:NearTCP-Eloopsbegintoopenwiththeonsetofspontaneouspolarization.Ferroelectric:P-Eloopsareopen,asthereisspontaneouspolarizationatzeroexternaleld(PS).PScanbereorientatedunderacoerciveeld,EC.Ferrielectricsandcanted-antiferroelectricshaveP-Eloopssimilartobutsmallerthanferroelectrics. ofanexternaleld,oftenresultinginspontaneouspolarization.Thedenitionsforferroelectric,ferrielectric,antiferroelectric,andcantedantiferroelectricaredirectlyanalogoustomagneticferroicsystems,seesection 1.2.2 andFig. 1-12 .Thethermodynamicpropertiesofferroelecctricsarealsoanalogoustomagneticsystems.Ferroelectricsareparaelectricathightemperaturesbeforeundergoingatransitionatlowertemperatures(alsocalledtheCurietemperature,TC)withthebreakingofspatialinversionsymmetryandtheonsetoflongrangeorderamongelectricdipoles.Thetransitioncanbebothrstandsecondorder,asbothdisplaciveandorder/disordertransitionshavebeenobserved(discussedbelow).Asinmagnetics,the 36

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FerroelectricBananas.A)Chargeversusvoltagelooptypicalforalossydielectric,inthiscasetheskinofabananaB)electrodedusingsilverpaste.ThehysteresisloopforatrulyferroelectricmaterialsuchasBa2NaNb5O15C)isshowninD)ferroelectrichysteresiscurveforceramicbariumsodiumniobate.FigureandcaptionreproducedfromRef.[ 32 ]. mostcommontechniqueforobservingferroelectrictransitionsishysteresisloops,herepolarizationvs.electriceld(P-Eloops),whichopennearTC.Ferroelectricdipolesalsoinitiallyorderintolocaldomains,whichcanbealignedunderastrongelectriceldresultinginaremanentspontaneouspolarization,PS,whentheeldisremoved,seeFig. 1-15 d.Thedipolemomentsinantiferroelectricsareanti-alignedresultinginzeroremanentpolarization,whileferrielectricsandcantedantiferroelectricsdisplayweakbutopenP-Eloops.Unlikeferromagnetism,however,ferroelectrichysteresisloopsareconstructedfromtransportmeasurements,makingthemsusceptibletomultiplepotentialartifactssuchasleakageanddielectricloss.Todrivehomethispoint,recentlyoneresearcherhumorouslydemonstratedthattransportmeasurementsonabananacouldproduceopenP-Ehysteresisloopssimilartothosereportedintheliterature,despitetheobviousabsenceofinherentferroelectricty[ 32 ]. 37

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1-17 ).Ferroelectricityistheresultofadelicatebalancebetweenshort-rangerepulsionswhichfavornon-polarcubicstates,andlong-rangeColoumbforceswhichstabilizeferroelectricdistortions.Densityfunctionaltheoryhasprovidedasignicantcontributiontotheunderstandingofthisbalance,asithasbeenclearlydemonstratedthattheoff-centershiftsaretheresultofthehybridizationofBsite3dorbitalswithO2porbitals,whichisessentialtoweakenshort-rangerepulsionsandlowerstheenergyofthedistortedferroelectricstate(seeFigs. 1-18 and 1-17 ).Theenergygainassociatedwiththishybridizationcanalsobedescribedanalytically[ 26 33 ].Upondistortion,thehybridizationmatrixelementtpdmodiestotpd(1+gu)whereuisthedistortionandgisthecouplingconstant.Therstordertermsinthehybridizationenergycancel,withthesecondorderapproximationproducinganenergygain: 1-18 a,wherethetwoO2pelectronsoccupyalowerenergyhybridizedbondingstate.However,theenergygainassociatedwiththishybridizationisdependentonthevalanceofthe3dorbital.Ifthe3dorbitalcontainsanelectron,thenoneelectronisforcedtooccupythehigherenergy,anti-bondinghybridizedstate-loweringtheenergygainand 38

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Aferroelectricunitcell,anditsdistortions,areshown.Theoff-centershiftoftheBsitecation(hashedtosolidredsphere)isaferroelectricdistortionwhichinducesadipolemoment(solidblackarrow).ThehybridizationoftheO2porbitalswiththe3dorbitalsoftheBsitecation,denotedherebygreyOatomsandconnections,providesanenergygainwhichstabilizestheferroelectricdistortion.Inperovskitesthedistortioncommonlytakesplacealongthe<111>bodydiagonals,shownherebythegreenarrows. destabilizingtheferroelectricdistortion.Accordingly,thisdistortioniscalledthesecondorderJahn-Tellereffect:secondorderbecausethelineartermscancel,andJahn-TellerbecausetheenergygainfromthedistortionisdependentonthevalenceoftheBsiteatom(seesection 1.1.2 ).Thedistortion'sstabilityisdependentontheoverallenergygainfromhybridization,however,thiscanbenegatedbyanelasticenergycost.ThispointmakesthechoiceoftheAsiteatomparticularlyimportant,asitsownsizeandbondingwithOatoms(eithercovalentorionic)cantunetheelasticpropertiesofthelatticeandthereforetheferroelectricityaswell.AsshowninFig. 1-17 ,theBsiteatomcanhybridizewiththreeoxygenatomsatonce,withitsdisplacementorientedalongthe<111>bodydiagonals.Distortingineitherdirectionalongthe<111>axisandhybridizingwitheithersetofOatoms 39

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FerroelectricEnergyDiagrams.A)ThishybridizationenergydiagramshowstheenergygainfromhybridizingO2pandBsite3dorbitalswhenthe3dorbitalisempty.Whenthe3dorbitalisoccupied(shownherewithdashedarrows),itselectron(s)mustoccupytheanti-bondinghybridizationstate,loweringtheenergygain.B)Thispotentialenergydiagramshowsthedoublewellassociatedwithhybridizingwithbothsetsoxygenalongthe<111>bodydiagonal. resultsinanidenticalenergygain,leadingtoadoublewellpotential(seeFig. 1-18 b).Thisdoublewellresultsinthecharacteristicswitchableferroelectricstates,asoppositedistortionsinverttheinduceddipolemoment(andthereforethebulkpolarizationvector).Intheorder/disorderinterpretation,theBsitecationsaredisplacedalongthebodydiagonals,producing(microscopic)spontaneousdipolemomentsateverytemperature.Athightemperatures,alleightdipoleorientationsarestable-whichwhenaveragedacrossthesampleresultsinzeronetpolarization.ThennearTCthedipolesadopteitherthesameorientation(rhombohedralsymmetry)ortwoorthreepreferredorientations(tetragoalororthorhombicsymmetry).Theorder/disordermodelthereforepredictsasecondordertransition.Alternatively,thesoft-modemodelpredictsarstordertransition.Inthesoft-modeinterpretation,BsitedisplacementsareonlystablebelowTC.Athighertemperatures,phononmodesprovidearestoringforcethateliminatesthe 40

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34 ].Charge-frustrationbasedferroelectricityhasbeendemonstratedincharge-orderedsystemswithtriangularlatticeswhichintroducegeometricfrustration,herethelatticeiscentrosymmetric,anditistheasymmetricchargedistributionthatbreaksinversionsymmetry[ 29 ].Itwasalsoshownthatthecoexistenceofbond-centeredandsite-centeredchargeordersinhalf-dopedPr1xCaxMnO3leadstoanon-centrosymmetricchargedistributionandanetelectricpolarization[ 35 ].Additionally,Asitecationswithlone-pairshavebeenshowntoinduceferroelectricityevenwhentheircorrespondingBsitecationshavepartiallylled3dorbitals(seeSection 1.2.2 ).However,despitethesepromisingavenues,magneticferroelectricmechanismshavereceivedthemostattention.Magneticmechanismsbeganreceivingattentionafterthepopularreportofaspin-optransitioninTbMnO3whereamagneticeldof5Twasshowntorotatetheferroelectricpolarization90ofromthea-axistothec-axis,aswellasincreasethedielectricconstantasmuchas500%[ 28 ].Interestingly,thesephenomenawerelinkedtothemagneticfrustration.Ithasbeenshownthatinhomogeneousmagneticorderallowsforthird-orderfree-energytermsoftheformPM@M.Incubiccrystalsthisresults 41

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ThisgureshowstheeffectsoftheantisymmetricDzyaloshinskiiMoriyainteraction.TheinteractionHDM=D12[S1S2].TheDzyaloshinskiivectorD12isproportionaltospin-orbitcouplingconstant,anddependsonthepositionoftheoxygenion(opencircle)betweentwomagnetictransitionmetalions(lledcircles),D12/x^r12.Weakferromagnetisminantiferromagnets(forexample,LaCu2O4layers)resultsfromthealternatingDzyaloshinskiivector,whereas(weak)ferroelectricitycanbeinducedbytheexchangestrictioninamagneticspiralstate,whichpushesnegativeoxygenionsinonedirectiontransversetothespinchainformedbypositivetransitionmetalions.FigureandcaptionreproducedfromRef.[ 22 ] inpolarizationoftheform[ 36 ]: 42

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1 ,thisproducesapolarizationorthogonaltobothqand^z:Pjj^zq.Alikelymicroscopicmechanismforthisferroelectricpolarizationistheanti-symmetricDzyaloshinskii-Moriya(DM)interaction.TheDMinteractionisarelativisticcorrectiontosuperexchange,andcanbewritten:Dn,n+1SnSn+1,whereDn,n+1istheDzyaloshinskiivector-whichisproportionaltoxrn,n+1,wherern,n+1isaunitvectoralongthelineconnectingthemagneticions,andxisthedisplacementoftheoxygenionfromthisline.Inspiralmagnets,theproductSnSn+1intheDMinteractionhasthesamesignforallpairsanduniformlypushesnegativeoxygenionsinonedirectionperpendiculartothespinchaincomposedofpositivemagneticions,thuscreatingpolarizationperpendiculartothechain. 1.3 ).Accordingly,foramaterialtodisplaybothorderingsitissubjecttothephysical,structural,andelectricalconstraintsrequiredforbothferroicproperties.Theseconstraintsinclude:symmetry,electricalconduction,andorbitalchemistry.Ferroelectricpolarizationrequiresalowsymmetrystructurewhichbreaksinversionsymmetry,andferromagneticpolarizationrequiresalowsymmetrystructurewhichbreakstime-reversalsymmetry.Thereare31pointgroupswhichallowspontaneousferroelectricpolarization,andtherearealso31pointgroupswhichallowspontaneousferromagneticpolarization.Ofthesetwosetsofpointgroups,13overlap-allowingbothelectricandmagneticspontaneouspolarizations[ 19 ].WiththeinitialnumberofShubnikovpointgroupsat122,reducingto13pointgroupsappearstobeastronglimitation.However, 43

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1.2.2 and 1.2.3 orbitalchemistryplaysavitalroleintheprimarymechanismsforbothferromagnetismandferroelectricity.IntheStonerbandtheoryofferromagnetism,apartiallyfull3dorbitalwithexchangeinteractionsbetweenspinsshiftsthepopulationbalanceofspin-upandspin-downelectronsresultinginanetmagnetization.Ferroelectricity,however,typicallyreliesonthehybridizationofempty3dorbitalswithO2porbitalsproducinganoff-centershiftoftheBsitecationwhichbreaksinversionsymmetryandinducesanetspontaneouselectricdipolemoment.Therefore,thefundamentalmechanismsforthetwoferroicorderingsseemtobemutuallyexclusive,requiringbothempty3dorbitalsforferroelectricity(termedd0-ness)andparitallyfull3dorbitalsforferromagnetism.Whileithasbeendemonstratedthatferroelectricitycanalsobeestablishedvia`improper'mechanisms(seeSection 1.2.3 ),itiscertainlytruethattheseconictingprocessesstronglylimitthecoexistenceofferromagnetismandferroelectricitymakingsingle-phasemagnetoelectricmultiferroicsexceptionallyrare.Therstattemptstobypassthismutualexclusioninvolvedconstructingelaboratematerialswhichincludedseparatestructuralunitstoproducetheindividualferroicproperties.TheseattemptscenteredaroundmaterialswithBO3groups,suchasGdFe3(BO3)4andNi3B7O13I,andweresuccessfulinproducingmultiferroicproperties[ 37 ].However,duetotheisolationoftheferromagneticandferroelectriccomponents,couplingbetweentheferroicorderswasextremelylimited.Withtherecentimprovement 44

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CompositeMultiferroicGeometries.A)Compositemultiferroiccanbegrowninahorizontallaminatedstructureofepitaxiallayersgrownsuccessively.b)Compositemultiferroiccanbealsogrowninaverticalcolumnarstructurewithselforganizingcolumnsofoneferroicmaterialinsideaparentmatrixofanother. inthinlmgrowthtechniques,moderneffortshavefocusedonnanoscaleheterostructures.Bothhorizontalheterostructurescomposedofalternatinglayersofferromagneticandferroelectriccompounds,andverticalheterostructurescomposedofself-assemblednano-pillarsinsideparentmatriceshavebeeninvestigated[ 38 39 ].Whentheferromagneticandferroelectriccompoundsarealsopiezoelectricandpiezomagnetic(orelectrostrictiveandmagnetostrictive)thisgeometryproducesefcientmagnetoelectriccouplingmediatedbystrain.Here,appliedelectric(magnetic)eldsinduceamechanicalstraininthepiezoelectric(piezomagnetic)layerswhich,throughtheepitaxialgrowthconditions,additionallystrainsthepiezomagnetic(piezoelectric)layersproducingamagnetic(electric)polarization.Thecompositeapproachhasprovensuccessfulforseveralnichetechnologicalapplicationssuchasmicrowaveapplicationsandhigh-resolutionmagneticeldsensors,however,itfailstoencompassthefullvisionofmagnetoelectricmultiferroics. 45

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1.2.3 ).Frustratedmagnetswherethespatialvariationinmagnetizationbreaksinversionsymmetryhaveemergedasaferroichotbed.Here,becausetheferroelectricityiscausedbythemagneticordering,itisparticularlysensitivetomagneticeldsleadingthesesystemstohavedisplayedthelargestmagnetoelectriccouplingobservedinsinglephasestodate.Unfortunately,thecouplingisunidirectionalaselectriceldshavenoeffectonmagneticproperties,limitingtheirpotentialapplication.However,anotherapproachhasalsoprovedpromising:Bibasedmultiferroics.InBibasedmultiferroics,theBiioncontains 46

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Theisosurface(atavalueof0.75)ofthevalenceELFofmonoclinicBiMnO3projectedwithinaunitcellisshown.Bluecorrespondstoalmostnoelectronlocalization,andwhitecorrespondstocompletelocalization.TheprojectionononeofthecellfacesisofthevalenceELF,colorcodedasinthebarbythesideofthegure.Theviewofthecrystalisnearlydownthebaxis.FigureandcaptionreproducedfromRef.[ 40 ]. 6selectronswhichdonotbondandinsteadformastereochemicallyactivelone-pair.Thelonepairisextremelypolarizable,andhelpsinduceferroelectricdistortionsintheunitcell.BiMnO3andBiFeO3aretwosuccessfulexamplesoftheBilone-pairapproach,asbothhavedisplayedferromagneticandferroelectricproperties,aswellasmagnetoelectriccoupling.BiFeO3ispopularbecauseitspropertiesexistatroomtemperature,however,itslimitationisthatitsmagnetizationistheresultofacanted-antiferromagneticstateandisquiteweak.Ontheotherhand,BiMnO3displaystrueferromagnetismandferroelectricityatlowertemperatures.Accordingly,BiMnO3isamodelsystemtostudyinordertounderstandthecoexistenceofferroicproperties,andithasbeendeemedthe`hydrogenatom'ofmultiferroics.ThemultiferroicportionofthisthesisfocusesonthepropertiesofBiMnO3,seeChapters 5 and 6 47

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1.3.1IntroductionMagnetoelectriccouplinghasalsoundergoneanintenserevival[ 41 ],asshowninFig. 1-22 ,wherethenumberofpublicationsciting`magnetoelectric'asakeywordisshowntohaveincreasedexponentiallyoverthepast20years.Initially,interestinmagnetoelectriccouplingstartedmodestly,withtherstexperimentalobservationsandtheoreticalpredictionsdatingasfarbackasthe1800's.In1888,Rontgenfoundthatwhenadielectricwasplacedinmotioninanexternalelectricelditbecamemagnetized-soonfollowedbytheobservationofthereverseeffect(polarizationofamovingdielectricinamagneticeld)[ 42 43 ].Thenin1894Curieprovidedthersttheoreticaldescriptionofthepotentialforstaticmagnetoelectriccouplingonthebasisofsymmetry.Muchlater,itwasrealizedthatmagnetoelectriccouplingwasonlypossibleinmaterialswhichbreaktime-reversalsymmetry,suchas:materialsinmotion,materialsinthepresenceofmagneticelds,ormaterialswithintrinsicmagneticordering.Finally,inthelate1950'salinearmagnetoelectriceffectbasedontheviolationoftime-reversalsymmetrywaspredictedbyDzyaloshinskiiinaspecicstaticmaterial,Cr2O3,whichwasfollowedshortybytheexperimentalobservationofitselectriceldinducedmagnetization.Thesendingsgalvanizedthephysicscommunitybriey,however,ageneralweaknessofthecoupling,adearthofsystemsdisplayingit,andalimitedunderstandingofthemicroscopicmechanismsledtoadecreasedinterestinmagnetoelectricphenomena.Aftera20yearlull,theeldhasbeenreignitedinresponsetherecentdevelopmentsinmultiferroicresearch.Magnetoelectriccouplingofasmanyas5ordersofmagnitudelargerthanthatobservedinCr2O3hasbeenachievedincompositesofpiezoelectricandpiezomagneticmaterials,withstraininducingpolarizationsineachcomponent.Single-phasemultiferroicshavealsoshowngreatpotential,wherefundamentallimitationsonthemagnitudeofmagnetoelectriccouplinghavebeenshowntobe 48

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Thisgureshowsthepublicationsperyearwith`magetoelectric'asakeywordaccordingtotheWebofScience.FigureandcaptionreproducedfromRef.[ 41 ] loosenedinthesesystems(seeSection 1.3.3 ).However,asshownbythevenndiagramofFig. 1-23 ,magnetoelectriccouplingisnotlimitedtoferroicmaterials,andthismagnetoelectric`momentum'hasalsospilledoverintoadditionalmaterialsresearchareas.Inparticular,theeffectofmagneticeldsontheelectricpropertiesofcorrelatedelectronsystemshasbecomeanextremelyactiveareaofresearch.Spinoffsofmagnetoelectriccouplingstermed`magnetocapacitance'and`magnetodielectric'havebecomeubiquitous,exempliedbyreportofcolossalmagnetoresistanceandmagnetocapacitanceinmixedvalencemanganites[ 18 ].Magnetocapacitancewasevenshowntobepossiblenon-magneticmediaaslongasitisinhomogeneous[ 44 ].Thus,thesearchformagneticeldinducedelectricpolarizationhasbeenreinvigorated,andhasexpandedtomultiplenewlandscapes.Finally,itshouldbenotedthatthecurrentrevivalisalsoduetoalonglistofpotentialtechnologicalapplicaitonsformagnetoelectriccoupling.Althoughmanyoftheseconceptswereconceivedfollowingtheinitialresearchsurge,therecentprogresshasmadetherealizationoftheirpotentialtantalizinglyclose.Inparticular,magnetoelectriccouplingcouldonedayleadtothewritingandreadingofmagneticdatawithelectricelds,acapabilitythatwoulddecreasethepowerusageandincreasethe 49

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Thisgureshowstherelationshipbetweenmultiferroicandmagnetoelectricmaterials.Ferromagnets(ferroelectrics)formasubsetofmagnetically(electrically)polarizablematerialssuchasparamagnetsandantiferromagnets(paraelectricsandantiferroelectrics).Theintersection(redhatching)representsmaterialsthataremultiferroic.Magnetoelectriccoupling(bluehatching)isanindependentphenomenonthatcan,butneednot,ariseinandofthematerialsthatarebothmagneticallyandelectricallypolarizable.Inpractice,itislikelytoariseinallsuchmaterials,eitherdirectlyorindirectlyviastrain.FigureandcaptionreproducedfromRef.[ 25 ] speedofnearlyeverydeviceinvolvingmemory.Additionaldevicesproposedincludehigh-resolutionmagneticeldsensors,electricallytunablemicrowaveapplicationssuchaslters,oscillatorsandphase-shifters,andspintronicapplicationssuchasspin-wavegeneration,amplication,andfrequencyconversion. 50

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51

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20i,jEiEj+1 20i,jHiHj+i,jEiHj+i,j,k 1.3.2 ).Thesemechanismscanthenbemappedontothephenomenologicalexpansionforsystematiccomparisons.Analysisofthefreeenergyhasalsoprovidedimportantinsightandguidelinestoresearchers.Inparticular,itwasshownthatenforcingastabilityconditiononi,jandi,jbyrequiringthesumoftherstthreetermsinEq. 1 tobegreaterthanzero(ignoringhigherordercoupling),thelinearmagnetoelectriccouplingconstant,i,j,isboundedby 52

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53

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2.1.1GrowthMethodsThisthesisincludesdatafromtwotypesofthinlmcomplexoxides:(La1yPry)1xCaxMnO3(LPCMO)andBiMnO3(BMO).Bothlmsweregrownviapulsedlaserdeposition(PLD)inDr.Biswas'lab.Thegrowthisepitaxial(latticematched),wherethestructureofathin(0.5mm)singlecrystalsubstrateactsasatemplateforthecrystalstructureofthelm.ThesubstratesusedinthisthesiswereNdGaO3andSrTiO3forLPCMOandBMOrespectively.ThePLDsystemiscomposedofaKrFexcimerlaser(248nm),avacuumchamber,atargetmaterial,andasubstrateheater.Thelaserispulsedonandoff,ablatingthetargetandcreatingastoichiometricplumeofthematerial'selementsthatextendstojustabovethesubstrate,resultinginthedepositionoflessthanonemonolayerperpulse.Theheatersuppliesthethermalenergynecessarytoallowtheelementstoshiftintothemostenergeticallyfavorableconguration,resultinginthecrystallinestructure.Carefuloptimizationofmultiplegrowthparameterswasrequiredtoobtainhigh-quality,stoichiometric,epitaxial,andcrystallinesamples.ThisoptimizationwasprimarilyperformedbyDr.TaraDhakalandHyoungjeenJeenofDr.Biswas'researchgroupforLPCMOandBMOrespectively.FortheLPCMOlmsthesubstratewasheatedto820oCinavacuumof106Torr,thenapartialpressureofoxygenof450mTorrwasappliedduringlmgrowthwithalaserpulsefrequencyof5Hzandlaserenergyof480mJ,withapre-ablationperiodprimingthetargetbeforeashutterguardingthesampleisremoved.FortheBMOlmsthesubstratewasheatedto632oC(alowertemperaturethanforLPCMOwasnecessarytoavoidBievaporation)underavacuumof106,thenapartialpressureofoxygenof37mTorrwasappliedduringlmgrowthwithlaserpulsefrequencyof5Hzandlaserenergyof480mJ.Additionally,theBMOlmsrequireda 54

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A)AschematicofPLDchamberisshownB)Animageillustratingtheplumecreatedbythelaserablatingthetargetduringlmgrowth(white)isshown.Thetipoftheouteredgeoftheplumeverynearlycoincideswiththeheater(red/orange,820C)wherethesampleismountedonthesubstrate,resultinginslowcontrolledgrowth.TheiImageswereprovidedbyDr.Biswas. non-stoichiometrictargetofBi2.4MnO3aswellasanquenchingoxygenpressureof680Torrduringcooling.Filmthicknessesranged30-150nmforLPCMOand30-60nmforBMO.FortheLPCMOsamples,a10-15nmcappingdielectriclayerofAlOxwasgrownusingrfmagnetronsputteringofanalumninatargetinanultra-highvacuumchamberwithabasepressureof109Torr.BothLPCMOandBMOlmsalsohadtopelectrodesdepositedbythermalevaporationinavacuumof106ofhigh-qualitymetalsheldintungstenboats.TheLPCMOelectrodeswereAl,depositedthroughashadow-masktoattaindesignedsizesandshapes.TheBMOelectrodes(Cr/Au)wereaninterdigitalarrayrequiringmultipleprocessingstepswhicharedescribedinsection 2.4.3 45 ].ForBMO,2X-raydiffractionandAugerelectronspectroscopy 55

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46 ]. 2-2 .ThePPMSprovidestemperaturecontrolusingtwoheaters,threethermometers,andaowcontrolvalveseparatingtheliquidHeandamechanicalpump.ThemechanicalpumppullscoldHegasoveracoolingannulusthatsurroundsthesampleatsetrates,providingatemperaturerangeof1.7K
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A)AschematicofthePPMSisshown,withitsinsulationlayers-N2,vacuum,andHe.B)Thisschematicshowsthetemperatureandmagneticeldcontroldesign.FiguresarereproducedfromthePPMSmanualandaQuantum-Designbrochure. capacitancestick(discussedbelow)isused,theelectricalconnectionsaremadedirectlytothestick. 57

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Thefour-pointmeasurementofresistanceismadebetweenvoltagesenseconnections2and3.Thecurrentissuppliedviasourceconnections1and4.Thegurewasprovidedbywikipedia. themeasuredvoltagedropbetweenthesensingleadsisduepurelytothecurrentthroughthesample.Thisgeometryisillustratedin 2-3 .Theresistancemeasurementsdiscussedinthisthesisweremadeusingthefour-terminalgeometryandaKeithley220CurrentSourceandKeithley2182Nanovoltmeter.Contactsweremadeusinggoldwireswitheithersilverpaint,carbonpaint,orpressedindium. 2.4.1CapacitanceBridgeandStickThecapacitancemeasurementsinthisthesisweremadeusingaHewlettPackardLCRmeter,theHP4284.TheHP4284hasaninternalvoltageoscillatorthatisusedtoexcitethedeviceundertest(DUT),andhasawiderangeofoperationalfrequencies,f:8610frequenciesoverthebandwidth20Hz
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TheschematicshowsthecircuitdesignoftheHP4284capacitancebridge,includingthe4terminalgeometry,virtualground,andauto-balancing-bridge.ThegurewasreproducedfromtheHP4284manual. resistor)ismostappropriate,althoughanychosenoutputcaneasilybetransformedintoanyotherformat(CS-RS,Z-,etc).Finally,allofourcapacitancemeasurementshavebeenmadeinsideacustombuiltcapacitanceprobe,whichisdesignedtobeelectricallyisolatedfromitssurroundings,withacoppercanactingasaFaradaycagetocancelambientelectriceldsthatcanbesourcesofnoise.ThesimplestsamplegeometryforcapacitancemeasurementsistheparallelplatecapacitorwheretwoelectrodesofareaAencloseaninsulatingmediumwithdielectricconstant,andthicknessd,whichstoresinducedpolarization(or,equivalentlycharge)accordingto d(2)whereCisthecapacitanceand0isthepermittivityoffreespace.Inourstudies,however,thisgeometryisnotpossiblebecauseourthinlmsaregrownoninsulatingsubstrates,prohibitingtheplacementofanelectrodebelowourlms.Thefollowingtwosectionsdescribetechniqueswhichovercomethislimitation,oneinwhichthedielectricisusedasthebottomelectrodeandanadditionaldielectriclayerdecloaksits 59

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18 ].Analyzingthecompleximpedanceofappropriatecircuitmodels,itwasshownthatifcertainexperimentalconstraintsaremet,theequipotentialplanesintheelectrodewillbeparalleltotheelectrode-dielectriclayerinterface,resultinginsensitivitytothec-axiscapacitanceofthebottomelectrode.Theybeganbymodelingtheentirestructureasaresistanceinseriestoalossyandleakycapacitor,wheretheseriesresistanceisthea-bplaneresistanceoftheLPCMOlm(seeFig 2-5 a)).Thetermlossyherereferstoacomplexcapacitor(C=C1iC2)thatdissipatesenergyfromdipolereorientations,butdoesnotpassdccurrent.Thus,theadditionofashuntingresistanceinparallelresultsinalossy,leakycapacitorthatdoespassdc.InourstructurestheshuntingresistancewasfoundtobeextremelylargeR0>1010,whereasatmaximumRS107,meaningthat99.9%ofadcvoltageisappliedacrossCandthatRScanbeignored.However,whenthevoltageisactherearemultiplecurrentpathsavailable(R0andR2=1=!C2,seeFig. 2-5 b)).Sinceweareinterestedinthedielectricproperties,wechooseourfrequencyrangesothatR2<
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A)Thecircuitequivalentofthetwo-terminalmeasurementcongurationwhereRSistheseriesresistanceoftheLPCMOsampleandtheparallelcombinationofacomplex(lossy)capacitorC(!)witharesistorR0representstheimpedanceoftheLPCMOinserieswiththealuminumoxidecapacitorisshown.Inthetwo-terminalconguration,thelongitudinalvoltagedropacrossRscannotbedistinguishedfromtheperpendicularvoltagedropacrosstheparallelcombinationofC(!)andR0.B)ThedecompositionofC(!)=C1(!)iC2(!)intoaparallelcombinationofC1(!)andR2(!)=1/!C2(!)isshown.C),ThecircuitequivalentforthecapacitanceCP(!)andconductance1/RP(!)reportedbythecapacitancebridgeisshown.D)TheMaxwell-WagnercircuitequivalentfortheLPCMOimpedanceinserieswiththeAl/AlOxcapacitorisshown.TheLPCMOmanganitelmimpedanceisrepresentedasalossycapacitorCM(!)shuntedbyaresistorRM.ThereisnoshuntingresistoracrossCAlOxbecausethemeasuredlowerboundonR0is1010,wellabovethehighestimpedanceoftheothercircuitelements.ThegureisreproducedfromRef.[ 18 ] 61

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2-5 c)).UsingsimplecircuitanalysisitcanbeshownthattheCPandRPofthemodelcircuitcanbewrittenintermsofthemodelcircuitcomponentsas, (R2(!)+RS)2+(!RSR2(!)C1(!))2)(2) 2-5 d).TheMaxwell-Wagnercircuitiscommonlyusedtoaccountfortheeffectsofcontactsindielectricmeasurementsandiscomposedoftwoleakycapacitorsinseries-herethemanganitecapacitanceandtheAlOxcapacitance.Fig. 2-6 b)showsaparametricplotofthecompleximpedanceofamodelMaxwell-Wagnercircuitwithfrequencytheimplicitvariable.IfCM<
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AMaxwell-Wagnercircuitsimulationisshown.Thisisaparametricplotofthecompleximepedance(withfrequencytheimplicitvariable)ofasimulatedMaxwell-Wagnercircuit.Notethethreetime-scales,onecharacteristicforeachseriescomponent,andonemarkingthecross-overinthedominantresponse. dielectricresponseoftheelectrode.Becausethetransversevoltagedropisknowntobenegligible,thismeansthemeasurementissensitivetothec-axiscapacitance. 2-7 ).WeusethistechniquetostudythinlmBMO,whichisgrownoninsulatingsubstratesmakingtheplacementofabottomelectrodechallenging, 63

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A)Az-axisviewofaninterdigitalelectrodearrayisshownisthespacingbetweenelectrodengersofthesamevoltage.B)Anx-axisviewofaninterdigicalarrayisshown.Thealternatingvoltageofthengersresultsinequipotentialplanesseparatingthem.Figurereproducedfrom[ 47 ] andthetransverseresistanceofwhichistoolargetoutilizedielectric-electrodeanalysisdescribedinsection 2.4.2 .Ourinterdigitalelectrodeswerefabricatedusingphotolithographyandthermalevaporation.A3:1mixtureofShipley1813andthinnerwasspincoatedat4000rpmfor45seconds,depositinga1.5mphotoresistpolymerlayeronourBMOsamples.Thesampleswerethenpre-bakedat115Cfor45seconds.Next,thesamplewaspositionedunderneathaphotolithographymaskwithapatternsimilartothatshowninFig. 2-7 usingaKarlSussMA-6ContactMaskAligner.OncealignedtheportionsofthesamplevisiblethroughmaskwereexposedtoUVlightfor14.5seconds,andsubsequentlydissolvedinAZ300MIFdeveloperandrinsedindeionizedwater.Finallythesampleswerepost-bakedat115Cfor1.5minutestohardentheminpreparationforfurtherprocessing.Atthispointintheprocessthesamplesnowhaveopenareasintheshapeoftheelectrodes,whichareseparatedbyasnakeofphotoresistthatweavesin-betweentheelectrodengers. 64

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A)Anopticalimageofgoldelectrodesisshown.B)Aschematicshowingrelevantlength-scalesisshown. Next,theelectrodesweredepositedusingthermalevaporation.Thechamberispumpeddowntoabasepressureof106Torr,andthenalargecurrent(100Amp)ispassedthroughatungstenboatwhichholdsasmallamountofhighpuritymetaluntilthemetalissohotthatitevaporates.TherateofdepositionismonitoredwithaInniconQuartzcrystalmonitor,andisstabilizednear5A/s.Theelectrodeshavetwometallayerswhicharegrownsequentiallywithoutbreakingvacuum,anultra-thinlayerofChromium(2nm)followedbythinlayerofgold(50nm).Theelectrodesaredepositedwithoutshadow-masks,allowingthemetaltodepositovertheentiresample.However,thephotoresistissufcientlythick(1.5m)thatthemetaldepositedonitdoesnotconnecttothemetaldepositedondevelopedportions.Thisallowsthemetaldepositedonthephotoresisttoberemovedsmoothlybysonicatingthesampleinacetoneanddissolvingthepolymer.Fig. 2-8 showsthenalresultofourinterdigitalelectrodefabricationprocess. 65

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2-9 showsthecomparisonofmultiple/single-frequencytemperaturessweepsforfrequenciesof500Hzand20kHz,wherethetworesultsareidenticalexceptforaslighttemperatureshiftduetodifferentthermaldriftsfordifferenttemperaturesweeprates.Forreasonabletemperaturesweepratesof0.5K/minitispossibletomeasureupto200frequenciesoverthebandwidth20Hz
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Thecapacitanceandlossdataformulti-frequency(black)andsingle-frequency(red)temperaturesweepsat500Hzand20kHzarecompared.A)Thecapacitanceismeasuredat500Hz.B)Thecapacitanceismeasuredat20kHz.C)Thelossismeasuredat500Hz.D)Thelossismeasuredat20kHz. 2.5.1Sawyer-TowerCircuitTheSawyer-Towercircuitisarguablythemostwidelyacceptedferroelectriccharacterizationtechnique,andremainsthestandardtowhichallothermeasurementmethodsarecompared.TheSawyer-TowercircuitisquitesimpleandconsistsofaDeviceUnderTest(DUT)placedinseriestoahigh-precisionsensecapacitor(seeFig. 2-10 ).Thedesignreliesonthefactthatcapacitorsinseriesmaintainequalcharges,sothatwhateverpolarizationisinducedonthesurfaceoftheDUTisalso 67

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TheSawyer-Towercircuitisthestandardferroelectricmeasurementtechnique.Itincludesanacvoltagesourceandasensecapacitorinseriestoadeviceundertest(DUT).Avoltagemeasurementonthesensecapacitor,thecapacitanceofwhichiswelldened,determinesthetransferedchargefromtheDUT.BackVoltageisdiscussedinthetext. inducedonthesensecapacitor.Thetotaltransferredchargecantheneasilybedeterminedbyasimplemeasurementofthevoltageacrossthesensecapacitor.TheSawyer-Towercircuitistypicallyusedwithacvoltagessuppliedbyfunctiongenerators,withtheoutputreadbyoscilloscopes.Despiteitssimplicity,theSawyer-Towercircuithasmultiplecaveatswhich,ifignored,canresultinsystematicartifacts.Therstconcernisthevoltageonthesensecapacitor.Capacitorsinseriesactasvoltagedividers,soifthesensecapacitorisnotsufcientlylargethiscandecreasethemagnitudeoftheelectriceldacrosstheDUT.Also,backvoltageisaconcern.Backvoltageoccurswhentheappliedvoltagereturnstozero,andthechargecollectedbythesensecapacitorinducesavoltageontheDUTwhichisoppositetothemaximumappliedvoltage(seeFig. 2-10 ).Thedurationofbackvoltageiseeting,however,itsmagnitudecanbesevereenoughtoreprogramthedomainstateofthesample. 68

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2-11 ).Avirtualgroundisanegativefeedbacksystemthatguaranteesthataspecicpointinacircuitmaintainsadesiredpotential,inthiscaseV=0.Thefeedbackisgeneratedbyconnectingthespeciedlocationtotheinvertinginputandagroundtothenon-invertinginputofanop-amp.Ifthespeciedlocationdevelopsapositive(negative)potential,thentheresultoftheinvertinginputistocreateanegative(positive)voltageattheoutput.Becausethisoutputisconnectedtothespeciedlocationthisinversionprovidesfeedbackthatshiftsthepotentialuntilthetwoinputsareequal,andinthiscasegrounded.TheadditionofavirtualgroundtothecircuitdesignelegantlysolvestwoofthecaveatsdescribedintheSawyer-TowerCircuitsectionabove.ByguaranteeingthatthepotentialjustbelowtheDUTismaintainedatground,theissueofbackvoltageis 69

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PrecisionLCcircuitryisreproducedfromtheRadiantmanual. completelycircumvented.Similarly,parasiticcapacitancesarealsogreatlyreducedbecausethereturnpathisalwaysgrounded,thereforebothsidesoftheparasiticcapacitancesareatgroundandnochargecanaccumulate.Chargeonlyaccumulatesontheintegrator,whichisontheothersideofthevirtualground. 70

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2 )mayberewrittenintermsofonlytwocomponents:remanentpolarization,andnon-remanentpolarization, 2-12 showstwohysteresiswaveformswhichutilizethisprinciple.Presettingpulsesaligntheferroelectricdomainsalongthedirectionofthesubsequentelectriceld(forboththepositiveandnegativeportionsofthehysteresisvoltagewaveform),guaranteeingthatanychargetransferredduringthehysteresismeasurementcannotbefromferroelectricdomainreversals-thedomainsarealreadyorientedalongtheeld.Figure 2-12 billustratestheoppositescenario,inthiswaveformpresettingpulsesalignthedomainsanti-paralleltothefollowingelectricelds, 71

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Intheremanentpolarizationpulsesequence,presettingpulses(blue)precedehysteresis-loopvoltagesections(red),guaranteeingeithernoneA)orallB)oftheferroelectricdipolesareavailabletoipandcontributetothemeasuredtransfercharge. guaranteeingthatallofthedomainsareavailabletoipduringthehysteresisvoltagesweeps.Thenon-remanentpolarizationisthesameinbothhysteresismeasurements,however,onlyb)hascontributionsfromferroelectricdomainreversals.Therefore,subtractingthehysteresisloopofwaveforma)fromthehysteresisloopofwaveformb)resultsinapurelyremanenthysteresisloop.Thismeasurementprocedureisusedextensivelyinthediscussionofmultiferroicsinthisthesis. 72

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48 49 ],spinels[ 50 ],multiferroics[ 51 52 ],andmixed-valencemanganites[ 16 17 ].Accordingly,understandingthefundamentalmechanismsofphaseseparation/competitionisastrongpriorityforphysicists,andisnecessaryforthetechnologicalimplementationofthesenextgenerationmaterials.Importantly,thendingsdescribedinthischaptersupportarecenttheorythatpredicted`electronicallysoft'phasescomposedof`charge-density-waves',challengingthestaticdisorderandstrainbasedexplanationsofphaseseparation.Thistheoryhasbroadimplicationsforcomplexoxideswithcoexistingandcompetingphases[ 53 55 ],however,evidencefor`electronicallysoft'phaseshasyettobeprovided. 73

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3.2.1ResistanceAsdiscussedinSec. 1.1.4 ,athightemperaturesLPCMOiscomposedoftwodielectricphases:theparamagneticinsulatingphases(PMI),andthecharge-orderedinsulating(COI)phases.Accordingly,theresistanceisexpectedtohaveinsulatingtemperaturedependence.Figure 3-1 displaysthetemperaturedependenceofthea-bplaneresistanceR(T)ofa30nm-thickpulsedlaserdepositedLPCMOlm(seeSec 2.1.1 ).Withdecreasingtemperature,T,theresistanceincreasessmoothlyuntilaninsulator-to-metaltransitionatT=115K,andthendecreasesasanexpandingferromagneticmetallic(FMM)phaseformsapercolatingconductingnetworkattheexpenseoftheinsulatingdielectricphases[ 16 17 ].InbulkLPCMOsamplesthereisalsoasignaturekinkinR(T)intherange200-220K,whichsignalsthetemperaturewheretheCOIphasebecomeswellestablished[ 56 57 ].TheCOIresistancekinkissystematicallyabsentinthinlms,andthishasbeeninterpretedasthesuppressionoftheCOIphase.However,evidencefordelocalizedcharge-density-waves(CDWs)associatedwiththeCOIphaseofmanganiteshasbeenveriedbymultipleexperimentaltechniques[ 55 58 59 ].Theabsenceofacharge-orderingfeatureintheresistanceofLPCMOlms(seeFig. 3-1 )doesnotmeanthattheCOIphaseisnotpresentinthinlms;rathertheCOItransitionissmearedbytheinherentdisorderandstrainofthinlms[ 58 ],analogoustosimilarbehaviorinCDWsystemsdopedwithlargeimpuritydensities[ 60 61 ].Inthefollowingsections,furtherevidencefortheCDWnatureoftheCOIphasewillbepresented. 2.4.2 inwhichthemanganitelmservesasthebaseelectrode.Iftheseriesresistanceofthe 74

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Thea-bplaneresistance(measureduponcooling)showsapronouncedpeakattheinsulator-to-metaltransition,TIM115K,butlacksaCOIassociatedanomalyseeninbulkmanganitesinthetemperaturerange200-250K[ 56 57 62 ]. electrodeisnottoolargeandiftheleakageacrosstheAlOxdielectricisnegligible,theequipotentialplaneswillbeparalleltothelmsurface,therebyplacingthec-axiscapacitanceoftheelectrodeinserieswiththeinsulatingdielectriclayer,effectivelydecloakingthesmallerelectrodecapacitance[ 18 ].Usingthistechnique,wemeasurethecomplexcapacitanceof30nmthick(La1yPry)0.67Ca0.33MnO3lms(cappedbyAlOxdielectriclayers)overthebandwidth20Hzto200kHz,andthetemperaturerange100K
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Thefrequencydependenceofthecomplexcapacitanceisshowntoqualitativelymatchtypicaldielectricrelaxations.A)Therealcapacitanceshowsalowfrequencyplateau,anddecreaseslogarithmicallyathighfrequency.B)Theimaginarycapacitanceshowsalosspeakwithlogarithmicdecreaseonbothsides. temperaturegridwithstepsof1Kforeachfrequency,allowingeachdielectricspectrumtobeanalyzedatconstanttemperature.Asacheck,theinterpolatedcapacitancevaluesfromthemultiple-frequencytemperaturesweepwerecomparedtosingle-frequencytemperaturesweepsatseveralrepresentativefrequenciesacrossthebandwidth,andwerefoundtobeidentical.Warmingrunswerealsoperformedwithnoqualitativechangeinmodelparametersotherthanahystereticshiftintemperature.SeeSec. 2.4.4 fordetails. 76

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IntheCole-Colerepresentationofthecomplexcapacitance,theimaginarycapacitanceisplottedparametricallyasafunctionoftherealcapacitancewiththemeasuringfrequency,!,variedastheimplicitparameter. Atrstglanceourcomplexcapacitancedata(where~C=C0iC00)displaysallofthequalitativefeaturesofadielectricrelaxation(see 3-2 ):intherealcomponentthereisalowfrequencyplateauwithminimaldispersion,followedbyalogarithmicdecreaseathighfrequencies,whiletheimaginarycomponentdisplaysaclearlosspeakwithlogarithmicdecreasesoneitherside.However,whenthedielectricdataisanalyzedquantitatively,itisnotconsistentwithdielectrictheoriesdescribingsinglephases.AcommonconventionforanalyzingcomplexcapacitancedatautilizestheCole-Colerepresentation.InCole-Coleplots,theimaginarycapacitanceisplottedparametricallyasafunctionoftherealcapacitance,withthemeasurementfrequencyvariedastheimplicitparameter,allowingforasimultaneousanalysisofbothcomponents.Figure 3-3 showsaCole-Coleplotofourcomplexcapacitancedataat200K.Tocompareourdatatostandarddielectrictheories,wefurthercalculatethelogarithmicparametricslope,(@(lnC00)=@!)=(@(lnC0)=@!),ateachpointofFig. 3-3 .Figure 3-4 showsthecalculatedlogarithimcparametricslopeasafunctionof 77

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Thelogarithmicparametric-slope,(C0=C00)(@C00=@C0)(green),isshowntodifferfromtheCole-Cole[ 64 ]dielectricresponseand`Universal'dielectricresponse[ 63 ]. frequency,comparedtoaCole-Coledielectricresponsefunction,and`Universal'dielectricresponse(UDR).Cole-Coledielectricfunctionshavetheform: 63 ].UDRresultsinalogarithmicparametricslopeof1,asbothcomponentsarechangingwithfrequencyatthesamerate.AsseeninFig. 3-4 ,ourdataareinconsistentwithbothdielectrictheories(onlyslopesgreaterthanzeroareshownforclarity).TheCole-ColeresponseincreasesmonotonicallytowardtheUDRslopeof1,whileourdatadisplaysahigh-frequencynon-monotonicanamoly,andthenappearstosaturateforanextendedbandwidth. 78

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Thelogarithmicparametric-slope,(C0=C00)(@C00=@C0)(green),isshowntodifferfromtheCole-Cole[ 64 ]dielectricresponse,providingasignatureofmultiplephases. ThefollowingsectionswilldemonstratethatbothofthesefeaturesaretheresultoftherstordercompetitionbetweenthePMIandCOIdielectricphases[ 65 ].Furthermore,characterizingthetemperaturedependenceoftheircompetitionrevealsahighlycorrelatedcollectivetransportmodeoftheCOIphasedomains,similartothe`coherentcreep'preceding`sliding'inCDWsystems[ 66 ]. 3-5 79

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3-6 displaysacircuitmodelthatwasdevelopedwhichtakesthishypothesisintoaccount.ThecircuitiscomposedofthreeparallelcomponentsallinseriestofractionalareasoftheAlOxdielectriclayer:aCOI,aPMI,andaR(thefractionalareaoftheFMMphase,whichactsasaresistiveshortatlowtemperatures),withtheconstraintPai=1.TheseriesresistanceisanartifactofmeasuringthedielectricresponseofanelectrodeinaMIMstructure,andisdiscussedindetailinSec. 2.4.2 .Placingthecapacitancesofeachphaseinparallelrequiresthatthedomainsofeachphasespanthelmthickness,thusobviatingaseriesconguration.Ourlmthicknessof30nm,however,satisesthisrequirementforthephasedomainsofmanganites,thelength-scalesofwhichareknowntobeontheorderofmicrons[ 17 ].AboveTIM,aR0,andthecircuitisdominatedbyCPMIandCCOIinourfrequencyrange,sothatthetotalcomplexdielectricresponsemaybeapproximatedbythesuperimpositionoftwoCole-Coledielectricfunctions, 3 atxedtemperaturesin1Kstepsbetween100Kand350Kbyvarying!=2fover185frequencies.ThetsareproducedbysimultaneouslyminimizingthedifferencebetweenthemeasuredcomplexcapacitanceandboththerealandimaginarypartsofEq.(2).Inthelow-frequencylimit,~(0)APMI+ACOI,allowingattingvariabletobeeliminatedbyreparameterizingthedielectricamplitudesintermsoftheirratio,ramp=ACOI=APMI,andthemeasured~(0),i.e.,APMI=~(0)=(1+ramp),andACOI=ramp(0)=(1+ramp).AsPMIisdeterminedfrom 80

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Inourcircuitmodel,threeparallelcomponents,CPMI,CCOI,andRmet,areinseriestofractionalareas(aPMI,aCOI,andaRrespectively)oftheAlOxlayer.Abovetheinsulatortometaltransitiontemperature,TIM,aR0. thelosspeakfrequency(seeFig. 3-7 ),vefreevariablesaredeterminedbythets:1,ramp,,,andCOI.Figure 3-7 showsatypicalttoEq. 3 (greencurve)wheretheaveragerelativeerrorislessthan103foreachtemperature.Alsoshownaretheindividualrelaxationsofeachphase,thelowfrequencyPMIphase(red)andthehighfrequencyCOIphase(blue)(theiridenticationsarediscussedbelow).Importantly,Fig. 3-7 explainsthehigh-frequencynon-monotonicanomalyinthelogarithmicparametricslope.Atlowfrequencies,thePMIphasedominatesbothchannels,andathighfrequenciestheCOIdominatesintherealchannel,butnottheimaginarychannel.Thismixingofrelaxationcomponentsathighfrequenciesresultsinthelogarithmicparametric-slopebehaviourseeninFigs. 3-4 and 3-5 ,andthusareasignatureofphaseseparationbetweentwoseparateandreadilyidentiabledielectrics. 81

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A)Themeasuredrealcapacitance(black)iscomparedwithtstoEq. 3 (green)atT=200K.B)theimaginarycapacitanceiscompared.TheindividualcapacitancesofthePMI(reddash)andCOI(bluedot)phasesarealsoshownforbothcomponents. itisknownthattheCOIphaseisahighlycorrelatedandorderedphase.Thus,itisexpectedthatthebroadeningoftheCOIphaseshouldincreaseasthephaseforms.Thetemperaturedependenceofconrmsthisexpectation(seeFig. 3-8 ),matchingtheknowntemperaturedependenceoftheCOIphase-whereCOInanoclustersarereportedinarelatedmaterialtoappearnear280Kwiththephasefullydevelopedbelow240K[ 62 ]-whileremainsrelativelyconstant,therebyidentifyingthehigh-frequencyresponseastheCOIdielectricphase.Theratioofdielectricamplitudes,ramp=aCOICOI=aPMIPMI,corroboratesthisidentication.SinceitisknownfromtheresistancemeasurementsthattheCOIphase 82

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Thetemperaturedependenceof(blue)isshowntomatchthetemperaturedependenceofthecorrelationsofCOIphase,whereCOInanoclustersarereportedinarelatedmaterialtoappearnear280Kwiththephasefullydevelopedbelow240K[ 62 ],identifyingthehigh-frequencyrelaxationastheCOIphase.(red)isshowntodisplaylimitedtemperaturedependenceinthisrange. isnotthedominantphase(absenceofCOIkink),wewouldexpectthattheareaoftheCOIphasewouldbeasmallminority.AsshowninFig. 3-9 ,ramp,isontheorderofafewpercent.Asacaveat,thisargumentrequiresthatthedielectricconstantsofeachphasearecomparable,butthisisprovedtobeaccurateinthefollowingsections.Furthermore,theratiodemonstratesthattheCOIareaisincreasinginthetemperaturerangewherethedielectricbroadeningtellsustheCOIphaseisforming.Thus,themodelprovidesaconsistentidenticationofthePMIandCOIphases.Figure 3-10 showsArrheniusplotsofPMIandCOIoverthetemperaturerange100K
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ThetemperaturedependenceoframpisshowntomatchthetemperaturedependenceoftheCOIphase,whereCOInanoclustersarereportedinarelatedmaterialtoappearnear280Kwiththephasefullydevelopedbelow240K[ 62 ],andidentifyingthehigh-frequencyrelaxationastheCOIphase-consistentwiththedielectricbroadeningdata(). TheArrheniusplotsareshownforCOIandPMI,withEA118meVinthelinearregionforbothdielectricphases,whichidentiesthepolarizationmechanismassmallpolarons.ThenearlyidenticalEA'ssuggestthephasesshareasingleenergybarrier. hoptonewspatiallocations,wherethelocallatticesitesubsequentlyrelaxestothe(lowerenergy)distortedstate.Inmanganites,thisprocessisadiabaticwhereelectronshopquicklyandthelatticerelaxesslowlyaroundtherelocatedelectronwithatime-scaleatleastaslongastheelectronhoppingtime-scale[ 67 ]. 84

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Apolaronisaquasi-particlethatincludesanelectronandthesurroundingcloudoflatticedistortionsinducedbyitspresence. 3.4.1PolaronsandDetailedBalanceThestrikinglysimilaractivationenergiesofthetwophasessuggeststherelaxationsarecoupled,possiblysharingacommonenergybarrier.Crossingthisenergybarrierwouldresultinthetwophasesconvertingintoeachother.Inoursystem,however,eachdielectricalsopolarizesindependentlywithoutconvertingintotheotherphase.Therefore,thephaseswouldhavetobeconnectedthroughacommonexcitedstatefromwhichrelaxationscanoccurtoeitherphase.Thiscommonstateisconsistentwithadiabaticpolaronhoppinginmanganites[ 67 ],wherethelatticerelaxesslowlyinresponsetofastelectronichopping.WemodelthisprocessinoursamplesbythethreestatesystemshowninFig. 3-12 .Theelectronsofthepolaronsofbothdielectricphasesabsorbthermaluctuationsthatactivatethemovertheirhoppingbarrierstoanequivalentexcitedstate:arelocatedelectronsurroundedbyalatticesitethathasyettorelax.Thenewlatticesitehassomeinitialdistortion(eitherPMIorCOI),butasitaccommodatesthenewelectronitcantransform/relaxintodistortionsthatcorrespondtoeitherdielectricphase.Theelectronichoppinghappensatcharacteristicrateswhichwemeasuredirectlyfromlosspeakpositionsinthecomplexcapacitance(PMI=1=PMI,andCOI=1=COI).Thelatticesiterelaxation,however,occursatunknownrates,Eand0E,forthePMIandCOIphases 85

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Anenergylevelschematicofathreestatesystemdescribingthepolaronhoppingprocessanddetailedbalanceispresented.Polaronsofbothdielectricphasesabsorbthermaluctuationsandhoptoanexcitedstateattheircharacteristicrates,PMIandCOI.Theexcitedstateisequivalentforbothphases,correspondingtoanelectronsurroundedbyanundistortedlattice.Thelatticecanthenrelaxintodistortionstatesthatcorrespondtopolaronsofeitherdielectricphase. respectively.Thisprocesseffectivelyresultsintwochannels,oneinwhichpolarizationismanifestedindependentlyineachphasebypolaronsrelocatingwithoutalteringtheirdistortions,andoneinwhichpolaronsrelocateaswellastransformtheirdistortionstate.Sincetheequilibriumpopulationsofeachphaseareconstantintime,therateequationsforthethreestatemodelinFig. 3-12 aregivenby, @t0BBBB@nPMInEnCOI1CCCCA=0BBBB@PE0P(E+0E)C00EC1CCCCA0BBBB@nPMInEnCOI1CCCCA=0BBBB@0001CCCCA,(3)wherenPMI,nCOI,andnEarethepopulationsofeachstate.Solvingthissystemofequationsatequilibrium(therightmostequality)resultsinadetailedbalanceequationoftheform, 86

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3 .ThisAnsatz(Eq. 3 )combinedwiththedetailedbalanceresult(Eq. 3 )allowsustoeliminatetheexponentialfactorandobtaintheresultthattheequilibriumratiooffractionalareascanbewrittenintermsoftheproductoftworatiosoftransitionrates,i.e., 87

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Thetemperaturedependenceoframpisshowntomirrorthatofrintemperature(withaconstantoffsetfactor),conrmingouruseofthedetailedbalanceequation. Weareabletoverifythisequationconstrainsoursystembecauseourmodeldeterminesthevariablesoneithersidetowithinaconstantfactor:r=COI=PMIandramp=(aCOI=aPMI)(COI=PMI).Sincedetailedbalancedoesinfactconstrainoursystem,thentheindependentlydeterminedratiosshouldhaveaconstantoffsetfactoroverthemeasuredtemperaturerangeof,r=ramp=(PMI=COI)(assumingthat0E=E1,whichisdiscussedbelow).Figure 3-13 showsthetemperaturedependenceoftheindependentlydeterminedratios,randramp.Thetworatiosfollowasimilartrendwitharatioofratios, 3 andEq. 3 togetherwiththeresultthat0E=E1leadstotherelationaCOICOIaPMIPMIwhichwiththenormalization,aCOI+aPMI=1, 88

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AschematicdepictionofthetemperatureevolutionofdetailedbalanceandtheenergybarriersseparatingthePMI(red)andCOI(blue)dielectricphasesisshown.AthightemperaturesECOIdecreasingwhileEPMIisconstant,causingitspopulationtoincrease.Atintermediatetemperatures,theenergiesofbothphasesareequalleadingtobalancedpopulations.Thenatlowtemperatures,EPMIincreaseswhileECOIisconstant,whichincreasesthepopulationoftheCOIphase. givestheparticularlysimplerelations, 3-13 and 3-14 presentaphysicallyintuitivepictureoftheevolutionofcompetingphaseswithdecreasingtemperature.InthehigherTregion(T>235K)whereECOIisatitsmaximumvalue,thePMIphaseistrappedinadeeperwell(E=ECOIEPMI>0),implyingthatPMI<
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3 showsaCOIPMI=COIandaPMI1PMI=COI,conrmingthatthefractionalareaoccupiedbytheCOIphaseissubstantiallysmallerthanthatofthePMIphase.Thenatintermediatetemperatures,thetwophasesareatequalenergies(E=0)overthesurprisinglylargetemperaturerangeof100K,makingtheirpopulationstemperatureindependent.Withinthisregiontheenergybarriertoescapetotheexcitedstateisthesameforeachphase(118meV)(Fig. 3-10 .Finally,atlowtemperaturesthePMIphasedestabilizesandEbecomesnegativeasEPMIincreasesrelativetoECOI.AsaresulttheCOI'spopulationincreasesproportionaltoeE=kTasseeninFig. 3-13 3 intoEq. 3 toobtaintheresult, 3 ),weassumedthedielectricconstantsioftheCOIandPMIphasestobetemperatureindependentwithallofthetemperaturedependenceinthenumeratorsoftherespectivedielectricfunctionssubsumedintothefractionalareasai=Ai=i.Accordingly,Eq. 3 thentellsusthattheratiooflatticerelaxationrates0E=Eisatemperature-independentconstantwhichforsimplicityweassumeisunity(discussedfurtherbelow).Withthischoiceequations 3 and 3 canbecombinedtogive 90

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Thedielectricconstants,calculatedfromthedielectricamplitudesandareasofeachphase,i=Ai=ai,areshowntosaturateneartheirbulkvaluesoncesubstratestrainisrelaxed.Theagreementwithbulkdatavalidatestheareascalculatedassumingequallatticerelaxationrates,E0E. henceconrmingthatr=COI=PMIdoes,infact,representtheratioofareas.KnowledgeoftheratiooffractionalareasaCOI=aPMIandtheratioofdielectricamplitudesACOI=APMItogetherwiththerespectiveconstrainingnormalizations,aCOI+aPMI=1andACOI+APMI=(0),allowanexperimentaldeterminationoftherespectivedielectricconstantsCOI=ACOI=aCOIandPMI=APMI=aPMI.Figure 3-15 showsthedielectricconstantsdeterminedinthismannerforfourlmswiththicknessrangingfrom30nmto150nm.Thedielectricconstantsofeachphaseincreaseandsaturateneartheirknownbulkvalues[ 68 70 ]asthesubstratestrainrelaxes.InmanganitesgrownonNGO,thelmisrelaxedatathicknessofd100nm[ 15 18 ]inagreementwiththesaturationofthedatainthegure.Theagreementofthedatawiththeseexpectationstendstovalidateourassumptionthat0E=E1. 91

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Thenormalizeddistributionsofhopping-ratetime-scalesareshowntobenarrowforbothphases,suggestingtemporallycoherenthopping(PMIdottedred,andCOIsolidblue). propagationofthecharge-densitydistributionoftheCOIphase.Theconstraintsoftheparallelmodelrequirethatthehoppingmechanismiscorrelatedoversufcientlylonglengthscalesthatregionsequaltoatleastthelmthicknesshoptogethercollectively,sothatasthephasesconverttheentirephaseboundariesprogresssimultaneouslyina`creep'likemanner.`Creep'istypicallyarandomphenomenon,however,transformingourdielectricbroadeningtoadistributionoftime-scales[ 71 ](shownintheinsetofFig.4)wendanarrowdistributionofhoppingratessuggestinganorderedprocesssimilartothe`temporallycoherentcreep'foundintheCDWsystemNbSe3[ 66 ].Theexactnatureoftheorderisambiguous,withtwolikelyscenarios.Therstpossibilityisthecoherentpropagationofphasedomains,whereasthephaseboundary`creeps'forwardtheregionsbehindsynchronouslyhop,guaranteeingthecontinuityofthephase.ThisscenarioisdepictedintheschematicofFig. 3-17 ,wheretheboundariesoftheMn3+/Mn4+chargeorderedphase(blue)coherentlypropagatewithsuccessivephaseboundaryhops/creeping,supplantingandconvertingintothecharge-disorderedparamagneticinsulatingphase(red).Thesecondscenarioisa`breathing'modeinwhichtheareaofdifferentphasedomainscooperativelyincreaseanddecreaseatacharacteristicfrequency(withtotalareaconserved).Bothscenariosdemonstratethe 92

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PanelsA),B),andC)representsuccessivesnapsotsintimeofthephasecompetitionbetweentheCOI(blue)andPMI(red)phases.TheboundariesoftheMn3+/Mn4+chargeorderedphase(COI,blue)propagatewithcoherentphaseboundaryhops,supplantingandconvertingintothecharge-disorderedparamagneticinsulatingphase(PMI,red)asafunctionofspaceandtime.Thesolidblacklinerepresentstheoscillationofchargedensity. collectiveanddelocalizednatureoftheCOIphaseinwhichitsentirechargedistributionmovescollectivelyandcoherentlyindynamiccompetitionwiththePMIdielectricphase. 93

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72 ])overthedisorderandstrainbasedexplanationswhichresultinstaticphases. 94

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22 33 42 43 ].Theoretically,MEcouplingistypicallydescribedusingGinzburg-Landautheory,whereafreeenergyanalysisprovidesauniedtemperatureandelddependenceofthethermodynamicsandstabilityoftheconstituentelectronicphases.Thefreeenergydescriptionalsoprovidesaninfrastructurefortestingspecicmicroscopiccouplingmechanisms,andmultiplemechanismshavebeenshowntoproduceMEcoupling:singleionanisotropy,symmetricandantisymmetricsuperexchange,dipolareffects,Zeemanenergy,andetc[ 73 ].However,thesecouplingsarefoundtobesystematicallysmall[ 41 ].Experimentally,largemagnetoelectriccouplinghasbeenachievedincompositesbycombiningpiezoelectricandmagnetostrictivematerialswhichcouplethroughstrain.Thecouplingincompositescanbemultipleordersofmagnitudelargerthaninsinglephasematerials,establishingstraincouplingasthemostpowerfulknownmagnetoelectricmechanismandsparkingarevivalinmagnetoelectricresearch[ 41 74 ].However,thetheoreticaldescriptionofthelargecompositecouplinghasprovidedlittleinsightintothethermodynamicsormicroscopicmagnetoelectricmechanisms,asitisestimatedbythesimplephenomenologicalmultiplicationofbulkpropertiesandislimitedstrictlytotheinterfaces[ 41 75 ].Outsideofcomposites,onlyindirectevidenceofstrain-mediatedmagnetoelectriccouplinghasbeendemonstrated.Forexample,anomaliesinthedielectricconstanthavebeenshowntoaccompanystructuraltransitionsresultingfromantiferrodistortionatmagneticorderingtemperatures[ 76 ]. 95

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4.2.1ExperimentalResultsThersttypeofmagnetoelectriccouplingpresentinoursamplesisthedirecttuningofdielectricconstantswithmagneticeld.Figure 4-1 showsthedielectricconstantsforbothphasesasafunctionofmagneticeldataxedtemperatureT=200K,withPMIdecreasingquadratically,andCOIincreasingquadraticallywithincreasingeldinboththe30nmand150nmlms.ThedielectricconstantsarecalculatedbydividingthedielectricamplitudesreturnedfromthemodelintroducedinChap. 3 ,APMI=aPMIPMIandACOI=aCOICOI,bytherespectivefractionalareas,aPMIandaCOI,whicharedeterminedusingadetailedbalanceequationthatwasfoundtoconstrainoursystem,aPMI(1=PMI)=aCOI(1=COI),andtheknownconstraintaPMI+aPMI=1(seeSec. 3.4.2 ).Belowwedescribeanearestneighbormean-eldmodelthatreproducesallofthequalitativefeaturesofthemagnetictuningofthedielectricconstants(quadraticelddependence,andsignofthechangeforeachphase)usingstrainasthecoupling 96

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A)Themagneticelddependentdielectricconstantsforthe30nmareshown.B)Themagneticelddependentdielectricconstantsforthe150nmareshown.Thedielectricconstantsofbothphasesareshowntotunequadraticallywithmagneticeld.Thesequalitativefeaturesarereproducedbyanearest-neighbormeaneldmodeldiscussedinthetext.Thedottedlinesaretsto,=0H2(Eq. 4 ). mechanism.Althoughtherearetwodielectricphasespresent,itisimportanttonotethattheircoexistenceiscoincidentalandthatourmodelassumesnointeractionbetweenthem:thestrainmediationoccurswithintheindividualphases. 97

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2k2.(4)Theenergyofthesystemisthenminimizedbymodifyingthelatticespacingbyanamount, 4-1 showsthetstoEq. 4 forbothphases(solidlines).Themagneticeldrangeswerechosentoguaranteeeachdielectricphaseiswellestablished,i.e.theArrheniusplotislinearoverasufcientlylargeenoughrangetoprovideawelldenedEA.With2,2,andkallpositive,thesignofisdeterminedsolelybyrandrJ.Thefunctionalformofisknownfromtheclassicalresultfortheeldofanelectricdipole, 98

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77 ].Thechangeinxisnegativeincompressiveexperiments(x<0),meaningthatrJPMITPMI=x<0,andrJCOITCOI=x>0,whichcombinedwithr<0resultsinPMI>0andCOI<0,therebypredictingtheexperimentallyobservedsignofthemagnetoelectriccouplingforbothphases. 4.3.1ExperimentalResultsFigure 4-2 displaysthesecondtypeofmagnetoelectriccouplingpresentinoursamples.TheArrheniusplotsofthetime-scalesfortherelaxationofbothdielectricphasesareshownforeld-cooledtemperaturesweepsineldsof0,2,4,and6T.Thelargelinearregionsdemonstratetheprocessisactivated,withactivationenergiesdeterminedfromtsto,=0eEa=kBT(solidblacklines).Thelowtemperaturedeviationsfromlinearityareduetotheonsetofthewellknownlow-temperatureinsulator-to-metaltransitionineachphase.Atzeroeld,EA118meV,consistentwithsmallpolarons,theknownpolarizationmechanisminmanganites[ 67 69 ].Asshown,theEA'sofbothdielectricphasesaresensitivetomagneticelds.TheinsetofFig.3showsthattheEA'sarefoundtodecreasewithmagneticeldaccordingtotheequation, 99

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ThemagneticelddependentEA'sareshownforthe30nmlm.Thelargelinearregionsoftherelaxationtime-scalesdemonstratethehoppingisactivated,solidblacklinesaretstotheArrheniusequation.DeviationsfromlinearityatlowtemperaturesindicatetheonsetoftheFMMphase.Inset:EAisshowntodependlinearlyonHforbothphases.Thesolidlinesaretsto,EA(H)=E0H. whereisdenedasaphenomenologicalmagnetoelectriccouplingcoefcient.Whenpolaronshoptheycarryadistortionofthelatticewiththem,soifthebarriertopolaronichopping(EA)ismodied,itislikelythatthelatticestrainhasbeenmodiedbythemagneticeldsothatdistortionsareeasier/hardertoaccommodate.Insupportofthisinterpretation,wenotethatpolaronicbarriersintheCOIare50%moresensitivetoelds(COI=PMI1.5),whichisconsistentbecauseitslargerinherentdistortionsuggestsitslatticecouplingisstrongerthanthePMIphase.ThelinearorderofthemagnetoelectriccouplinginEAisalsoworthnoting.Anon-zerospontaneousmagnetizationresultsinlinearelddependenceinenergy,similartotherstterminEq. 4 .Thissuggeststhepresenceofspontaneousmagnetizationinthepolaronsofeachphase,consistentwithpreviousreportsofmagneticpolaronswellaboveTCinmagnetoresistiveperovskites[ 78 ].Therelativemagnitudeofthecouplingofeachphaseisalsoconsistentwiththisexplanation,asthemagneticallyorderedCOI 100

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79 ].Asthelmthickness(d)increases,however,thestrain 101

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A)Themodelingresultsareshownforthe30nmlm.B)Themodelingresultsareshownforthe150nmlm.Themeasuredrealcapacitance(black)iscomparedtotheparallelmodelresults(green).TheindividualcapacitancesofthePMI(reddash)andCOI(bluedot)phasesarealsoshown.Insets:Imaginarycapacitance,wherelosspeakpositionsshowthecharacteristicrelaxationtime-scales. relaxesasthelatticeconstantsofthelmapproachtheirbulkvalues,andonlythelayersneartheinterfaceremainstrained.Inmanganites,thickness-dependentstudieshaveshownthatford>125nmthelatticeconstantshavealmostcompletelyrelaxedtotheirbulkvalues[ 15 ].Therefore,varyinglmthicknessinthisrangeprovidesadirectprobeofstraininthinlms.Toinvestigatetheroleofstraininthemagnetoelectriccouplingofoursamples,wehavemeasuredLPCMOlmsoffourthicknesses:30,60,90,and150nm.Figure 4-3 showsthemodelingresultsforthethinnestandthickestlmsinthisrange. 102

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A)Theactivationenergycouplingconstant,,isshowntoincreaseandsaturatewithlmthickness.B)Thedielectricconstantcouplingconstant,,isalsoshowntoincreaseandsaturatewithlmthickness.Bothmagnetoelectriccouplingscorrelatewiththerelaxationofsubstrateinducedmismatchstrain Bothmagnetoelectriccouplingcoefcientsarefoundtodependstronglyonthestrainstateofthelattice.Fig. 4-4 showsthatthemagnetoelectriccouplingcoefcientscorrelatewithstrainrelaxation,increasingandsaturatingneard125nm.Thisprovidesfurtherevidencethatbothmagnetoelectriccouplingsaremediatedthroughstrain.Accordingtoourmodel,themagnetoelectriccouplingresultsfromthemodicationoflatticeconstants.Therefore,asthemismatchstrainrelaxes,theabilityoftheelectricandmagneticpropertiestocouplewouldbeampliedasthelatticesoftenedandwasfreetorespondelasticallyandmodifyitsspacings,whichisconsistentwithourobservations. 103

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104

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80 ].InBiMnO3,asinBiFeO3,the6s2lonepairontheBiionleadstoitsdisplacementfromthecentrosymmetricpositionattheA-siteofaperovskiteunitcell.However,inBiMnO3,theresultantdistortionmodiesthesuperexchangeintegralsandleadstoaferromagneticinteractionbetweentheMnionsattheB-siteinBiMnO3[ 81 82 ].Inbulkform,BiMnO3hasbeenobservedtobebothferromagneticandferroelectric[ 83 ],whileinthinlmfewreportshavedemonstratedsimilarmagneticpropertiesorlowenoughleakagestoallowaclearferroelectricmeasurement[ 84 105

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ThemonoclinicunitcellofBiMnO3isshown.ThegureisreproducedfromRef.[ 46 ] and(2)themetastabilityoftheBiMnO3structureitself.ThestoichiometricunitcellofBiMnO3isthecubicperovskite(seeFig. 1-1 ),however,itishighlydistortedleadingtoalargermonoclinicunitcell(seeFig. 5-1 ).Thelmswerecharacterizedstructurallyusing2scansandAtomicForceMicroscopy,stoichiometricallywithAugerElectronMicroscopy,andmagneticallywithaSuperconducting-Quantum-Interference-Device(SQUID). 5-2 showsthe2x-raydiffractiondatafora60-nm-thickBiMnO3thinlm(sampletype1).TheinsetshowsthattheBiMnO3growswitha111orientationasexpectedfromthestructureofBiMnO3.AsmallpeakcorrespondingtoMn2O3impuritiesisalsovisibleinthesemi-logplot[Fig. 5-2 (b)](theintegratedintensityratiooftheMn2O3peaktotheBiMnO3111peakis0.025).Toremovetheseimpurities,thepost-depositioncoolingrateofthesubstratewasincreasedtoabout40C/mininanO2atmosphereof680Torr.Figure 5-2 balsoshowsthex-raydatafora60nmlmgrownusingthenewcoolingrate(sampletype2),whichconrmsthattheimpuritypeakshave 106

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85 ].InBiMnO3,thedistortioncausedbytheBiionleadstoanferromagneticinteractionbetweenthelayers[ 81 82 ].Hence,BiMnO3hasanoverallmagneticmomentthathasbeenmeasuredtobeashighas3.6B/Mninpolycrystallinesamples(whereBistheBohrmagneton);thisisclosetomaximumpossiblemagnetizationof4B/Mn[ 86 ].Inthinlms,themagneticmomentisreducedquitelikelyduetothesubstrateinducedstrain.TheTCinthinlmsisalsolower 107

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A)The2x-raydiffractionpatternofa60-nm-thickBiMnO3thinlmisshown.TheinsetshowstheBiMnO3(111)peakindetail.B)AsemilogplotshowingasmallamountofMn2O3impurityphaseandasmallBiMnO3(20-3)peak(higherintensityline)andtheimpurityfreelmgrownusingtherapidquenchingtechnique(lowerintensityline)areshown.ThisFigureandcaptionisreproducedfromreference[ 46 ]. thanthevalueofabout105Kobtainedinpolycrystallinesamples[ 83 86 ].Figure 5-3 showstheM-TandM-Hcurvesof60nm-thickBiMnO3thinlms(sampletypes1and2).Themagneticeldwasappliedintheplaneofthelmforthemagneticmeasurements.TheM-TplotrevealsaTCofabout85-65Kforbothsampletypes.Saturationmagnetizationsofabout1.0and1.1B/Mnwereobtainedat10Kinaeldof50kOeforsampletypes1and2,respectively.TheinsetofFig. 5-3 (b)showsthehysteresisintheM-Hplotatdifferenttemperaturesforsampletype2.Acoerciveeldof 108

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83 ].ThereducedmagneticmomentofourthinlmscomparedtobulkBiMnO3isnotduetothepresenceofthenonmagneticimpuritiesbecauseboththesampletypes1and2havesimilarsaturationmagnetizationsandcoerciveelds.Finally,theinsetofFig. 5-3 showsthesurfacemorphologyofsampletype2.Bothtype1and2thinlmsshowthree-dimensional(3-D)islandgrowthmodewithanr.m.sroughnessofabout10nm.Ithasbeenshownthat3-Dislandgrowthleadstononuniformstrainresultinginhighvaluesofstrainsattheislandedges[ 87 88 ].BecausethecrystalstructureofBiMnO3iscloselyrelatedtothatofantiferromagneticLaMnO3,thenonuniformstraindistributioncouldberesponsibleforboththereducedvaluesofTCandthesaturationmagnetization. 25 89 ]However,by140K(below140Ktheresistanceistoohightomeasurewithourinstrumentation),theresistivityincreasestoabout1M-cm,anditwaspossibletomakedirectpolarizationversuselectriceld(P-E)measurementsattemperaturesbelow100K. 109

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A)Themagnetizationvstemperature(M-T)plotfortwo60-nm-thickBiMnO3thinlmsinanin-planeeldof500Oeareshown:sample1(circles)andsample2(triangles).Thefullsymbolsandopensymbolsarethezeroeldcooledandeldcooleddatarespectively.Theinsetshowsa5x5matomicforcemicroscopeimageofthesurfaceofsample2.B)Themagnetizationvsmagneticeld(M-H)plotforsample1(circle)andsample2(triangle)at10Kareshown.TheinsetshowsthereductioninthehysteresisoftheM-Hdataforsample2asthetemperatureisincreased.ThisFigureandcaptionisreproducedfromreference[ 46 ]. 110

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Remanentpolarizationhysteresisloopsareshown.Theferroelectrictransitionisshowntospan100K,openingnear100Kwithamaximumremanentpolarizationof23C/cm2at5K. substrate(STO),wehaveimplementedaninterdigitalcapacitancegeometry.ThecapacitoriscomposedofalternatingV+/Velectrodesuniformlyspacedonthelmsurface,whichleadstoequipotentialplanesintersectingthelmbetweeneachpairofelectrodes,resultinginacapacitancebetweentheprojectedareasofeachelectrodewithinthelm(seeSec. 2.4.3 ).Theprojectedareaswerecalculatedanalyticallyusingconformalmappingandequatingthecapacitorthicknesstohalftheelectrodespatialwavelength[ 47 ].Thepolarizationisthencalculatedbydividingthetransferredchargebythisprojectedarea.Thepolarizationinatypicalhysteresisloopiscalculatedbyintegratingthetotaltransferredchargeduringapplicationofabipolartriangularvoltagewaveform,withcontributionsfromleakagecurrent,capacitance,andferroelectricdomainswitching.However,asdescribedindetailinSec. 2.5.2 ,wemeasureremanenthysteresisloops,wherethecontributionsfromleakageandcapacitancehavebeeneliminatedthroughspecicpulsingsequences. 111

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2.5.3 ),thisbroadtransitionwouldresultintheslowsquare-to-slim-looptransitionintemperaturethatistypicalinrelaxorferroelectrics[ 90 ].Interestingly,theferroelectrictransitionisfoundtocoincidewiththeferromagnetictransition.TheinsetofFig. 5-4 ashowsthatboththeferroelectricpolarizationandmagnetizationincreasefromzerosimultaneously,signalingtheirrespectivetransitionsandsuggestingastrongcouplingbetweentheorderings(seeChap. 6 foracompletediscussion).Theferroelectricpolarizationisalsofoundtobehighlytunable,andismodulatedbybothmagneticelds(magnetoelectriccoupling)andexternalstrain(seesections 6.2 and 6.3 ,respectively).ThestraincouplingismuchstrongerthantheMEcoupling,loweringthecoerciveeldandincreasingtheRPbyasmuchas50%atmodeststrainsoflessthan102%.Thestraincouplingisstrongestatlowtemperatures,anddisappearsaboveT50K.Thestraincouplingisalsofoundtobeextremelysensitivetotheorientationofthestrain(discussedinChap. 5-5 ).The 112

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Thefrequencydependenceoftheimaginarycapacitance,C00,displaystwolosspeaks,shownhereat135Kover200frequenciesbetween20Hzand1MHz.ThesolidanddashedlinesaretstocomplexCole-Coledielectricfunctions. relaxationsaremodeledbyCole-Coledielectricfunctionsoftheform, 5-5 .Plottingthetemperaturedependenceoftherelaxationtime-scalesprovidedbytheCole-ColetsinArrheniusformatrevealsthatthepolarizationmechanismisactivated(seeFig. 5-6 ).Fittingthetime-scalestotheArrheniusequation,i=0,ieEA=kBT,overthelargelinearregionsprovidesawelldenedactivationenergy,EA,andpre-exponential 113

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Arrheniusplotsof1and1areshown.Bothrelaxationtime-scalesarethermallyactivated,withactivationenergiesof205meVand189meV,respectively. factorforeachlosspeak.Theactivationenergiesare2055meVand1894meV,withpre-factorsof2.910110.11011sand3.610130.11013sfor1and2,respectively.Wenotethatthepre-factorsarequitesmallforadielectricrelaxation,andareintherangeofphononfrequencies.Figure 5-7 showsthetemperaturedependenceoftherealcomponentofthecomplexcapacitanceforaselectionoffrequencies:20Hz,200Hz,and2kHz.Athightemperatures,thereisevidenceofabroadmaximum,however,becauseoftheincreasedleakagethetemperaturerangeislimitedandthepeakisnotdirectlyobserved.Themaximumisfrequencydependent,however,occurringathighertemperaturesforhigherfrequencies.Atlowtemperature,thedispersiondisappearsandtherealcapacitancediverges(forreasonswhicharediscussedbelow).Thelackofdispersionsigniesthatthecapacitanceisrepresentativeofthestaticdielectricconstantatourmeasurementfrequencies. 114

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Thetemperaturedependenceoftherealcapacitanceisshownforselectfrequencies.Thehightemperaturerangeislimitedduetoincreasedleakagethere. 5.3.1RelaxorReviewRelaxorferroelectricitycanbeunderstoodbyconsideringarepresentativemodelcalledsuperparaelectricity[ 90 ],whichisdirectlyanalogoustosuperparamagnetism,exceptthatinsuperparaelectricitythemagneticdomainsarereplacedbyferroelectricdomains.Superparamagnetsaretypicallycomposedofnano-sizedmagneticparticleseachofwhichactastheirownferromagneticdomain.Inanappliedeldthemagneticmomentsalignresultinginamagnetizationwhichismuchlargerthantypicalparamagnets.However,oncetheeldisremovedtheindividualdomainsrandomizecancelingthemacroscopicmagnetizationoveranactivatedtime-scaleknownastheNeelrelaxationtime.Belowacertaintemperature,theNeelrelaxationtimeexceedsthemeasurementtime-scaleandthesystemissaidtobeblocked,andbehavessimilartoaregularferromagnet.Relaxorferroelectricsbehavealmostidentically,withdipolesoriginatingfrompolar-nano-regionsorPNRs.PNRsexhibitlargeinducedpolarizations 115

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91 ],andgrowinsizeslowlywithdecreasingtemperatureuntilthefreezingtemperature.Then,atTF,thePNRsgrowrapidlycausingtheuctuationtime-scalestodivergeasaresultofthecoalescingandcouplingofneighboringPNRs[ 92 93 ].Besidestheirspontaneousferroelectricdipolemoments,theseregionshavelargerintrinsicdielectricconstantsaswell,leadingtoanincreaseintheoveralldielectricconstantastheir%areaincreaseatTF.Innormalferroelectrics,theferroelectrictransitionistypicallycausedbyasoftlong-wavelengthphonon-modethatdecreasestozerofrequencyatTC,resultinginastaticlatticedeformationthatextendsthroughoutthecrystal[ 94 ].However,relaxorferroelectricsareinherentlydisorderedbythePNRswhichcriticallydampthezone-centerphononmodesandprohibittheirpropagation[ 95 96 ].Asaresultthetransitionisverybroad(diffuse),andthecorrespondingslowslimtosquarelooptransitioninthehysteresisasthetemperatureisloweredisasignatureofrelaxorbehavior.ThePNRsactaslocalfrozenphononmodesthatvibrateoutofphase.ThesevibrationsarethermallyactivatedandcorrespondtotheippingofPNRdipolemoments.PNRscontributetothedielectricresponseviatwodistinctmechanisms:thethermallyactivatedreorientationoftheirdipolemoments,andthedisplacementoftheirboundaries[ 97 ].ThePNRsarealsobelievedtocausethebroadpeakin0asafunctionoftemperature,whichisaclassicsignatureofrelaxorferroelectricity[ 93 ]. 5-4 showsthattheferroelectrictransitionspansmorethan100K.Iftheseremanentpolarizationloopswereincorporatedintototalpolarizationhysteresisloops(wherechargecontributionsfromleakageandcapacitancewereincluded),itwouldresultina 116

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98 ].Phononmodesthatarecriticallydamped,however,areinsteadrepresentedbyCole-Coledielectricfunctionsinstead(seeEq. 5 )[ 64 ].AsshowninFig. 5-5 ,therelaxationsarebothsuccessfullymodeledbyCole-Colefunctions.Thetemperaturedependenceofthedielectricrelaxationsisalsoinagreementwithrelaxorferroelectricity.Therelaxationtime-scalesareactivatedasexpectedforPNRreorientationtime-scalesinthesuperparaelectricmodel,andttingthetemperaturedependenceoftherelaxationtime-scales(providedbytheCole-Colets)totheArrheniusequation,i=0,ieEA=kBT,resultsinquitesmallpre-exponentialfactorsof0,131011and0,23.51013,clearlyestablishingtheirphononoriginofthePNRs(seeFig. 5-8 .Therealcapacitance,showninFig. 5-7 ,isalsoconsistentwithPNRsandrelaxorferroelectricity.Athightemperaturesweseeevidenceofthetailendofthecharacteristicfrequencydependentmaximum(wearelimitedtolowertemperaturesduetoleakage)-ahallmarkofrelaxorferroelectricity-andatlowtemperaturesweseeafrequencyindependentdivergenceofthedielectricconstantasisexpectedwiththerapidgrowthofthePNRsatthefreezingtemperature. 117

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Therelaxationtime-scalesareextrapolatedusingtheArrheniusequation.Atlowtemperaturesthetime-scaleofthePNRdipolereorientationsisshowntointersectwiththeRPlooptime-scalealmostexactlyatTC=100K.Athightemperaturesthepre-exponentialfactorsareshowntobeintherangeofphononfrequencies. Themostconvincingevidenceforrelaxorferroelectricity,however,istheaccuratepredictionoftheferroelectricTCfromthedielectricdata.AthightemperaturesthePNRsrandomizequicklysothatoncetheferroelectricmeasurementsarenishedtheremanentpolarizationhascompletelyrandomized.Astemperatureislowered,however,therandomizationislessandlesscomplete.Therefore,onewouldexpectthatjustastheferroelectrichysteresisloopsareopeningthatthemeasurementtime-scaleisjustequaltothePNRreorientationtime-scale.AsshowninFig. 5-8 ,extrapolatingtheactivatedtime-scalesviatheArrheniusequation,weseethatthetemperatureatwhichthePNRreorientationtime-scale,1,andtheRPloopmeasurementtime-scaleareequalcorrespondsexactlytothetemperaturewheretheRPloopsbegintoopen(TC),conrmingthePNRsasthesourceofferroelectricity. 5-9 .Therstvoltagepulsepolesthedomains,andthesecondvoltageisappliedafteravariabledelaytime.Afterthesecondpulseis 118

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5-9 )todetermineifanyferroelectricdomainshavereorientedduringthedelaytime.Thechargetransferduetoresistiveleakageandcapacitivechargingshouldbethesameforeachpulseindependentofdelaytime,thereforeanychangeinchargetransferwithdelaytimeisattributedtoreorientedferroelectricdomains.ItisworthnotingthatthisprocedureisdistinctfromtheremanentpolarizationmeasurementprocedurediscussedinSec. 2.5.3 .Intheremanentpolarizationmeasurement,therearepresettingpulseswhichpolethedomains,however,thechargetransferisonlymeasuredduringthefollowingtriangularwaveform.Herethechargetransferismeasuredonlyasaresultofthepulseitself.Inanormalferroelectric,thesecondpulseofourtwopulsewaveformshouldhavenoeffectasthereshouldbeminimaldomainreorientations.Inarelaxorferroelectric,however,thechargetransferedduringthesecondpulseshouldincreaseasthedelaytimebetweenpulsesincreasesallowingtimeformorereorientations.TheinsetofFig. 5-9 showsthatthechargetransferedbythesecondpulseincreaseslogarithmicallywithdelaytime,onceagainconrmingthepresenceofrapidlydisorderingferroelectricdomains,i.e.relaxorferroelectricity. 99 ],andcentrosymmetricstructureshavealsobeenpredictedusingdensityfunctionaltheorycalculations[ 100 ].Thispointisoffundamentalimportantancebecauseanon-centrosymmetriccrystalstructureisessentialforferroelectricity.Therefore,althoughBiMnO3wasrstthoughtaprimeexampleofmultiferroicity,therehasbeenagrowingdebateconcerningwhetheritisanintrinsicferroelectric.IfthecrystalstructureofBiMnO3isindeedcentrosymmetric,thenthepossiblereasonsfortheferroelectricbehaviorofBiMnO3thinlmsare:1structuraldistortionsduetooxygenvacancies[ 99 101 ],acentrosymmetric 119

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Theswitchedpolarizationisshowntoincreasewithdelaytimebetweenpulses,conrmingthequickreorientationofPNRs.Inset:Schematicofthesimplepulsesequence. tononcentrosymmetrictransitionbelowTCthatis,below100K[ 24 99 ],andsubstrateinducedstrain[ 102 ].Although,theAugerelectronspectroscopymeasurementsonourthinlmsrevealanoxygendeciencythatcouldleadtotheferroelectricbehavior,wecannotruleouttheroleofsubstrateinducedstrain.Ifthelmisuniformlystrained,thelatticemismatch,whichis-0.77%(compressive),isnotenoughtobreakthecentrosymmetryasshownbyHattetal[ 102 ].However,ithasbeenshownthatcompressivelatticemismatchstraincouldleadtoanonunifromstraindistributioninthethinlmduetoislandformationandthestrainattheislandedgescouldfarexceedtheaveragelatticemismatchstrain[ 87 88 ].Thegrowthmorphologyofourthinlms(seetheinsetofFig. 5-3 )suggeststhatsuchnonuniformstraindistributionisalsoapossiblemechanismfortheappearanceofferroelectricity.Itisalsoworthnotingthatthisinterpretationisconsistentwithrelaxorferroelectricity.TheindividualislandedgeswouldserveasPNRs,andtheinherent 120

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6.2.1 )andLSATandNGOsubstratesinducemuchlargercompressivestrainsthanSTO.ThereforethethinlmBiMnO3researchedinthisthesisisbelievedtodisplayferroelectricpolarization(potentiallylimitedtoislandedgesonly). 121

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122

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6.2.1ExternalStrainStressisappliedtothethinlmsdirectlyusinganexternalthree-pointbeambendingtechnique.Inthistechniquethelmissupportedonoppositeedgeswhileanexternalforceisappliedinthecenterofthelm.Theresultisabendingofthelmsimilartoaclassicalbeam,wherethestrainisquantiedas, 6-1 ,andisnominallyuniaxial.Thestressisappliedbyturninga60turns/inchscrewincontactwiththesamplesurfacewiththeaidofawormgearprovidingspatialresolutionof1mdisplacements.Theremanentpolarizationwasfoundtodependstronglyonexternallyappliedstress,andincreasesofalmost50%wereobservedatverymodeststrainsoflessthan102%changeinlatticeconstants(seeFig. 6-2 ).Theeffectwasfoundtobeoddinstrain,withthesignofthechangeinpolarizationdependingonwhethercompressiveortensilestrainwasapplied.ThetopinsetofFig. 6-2 showsthatthecouplingislinearinstrainatlowstrains,withthecouplingdiminishingathighstrains.ThebottominsetofFig. 6-2 showthatthestraincouplingdisplaysstrongtemperaturedependence,appearingbelowapproximately50Kanddivergingatlowtemperautres. 123

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Thethree-pointbeambendingtechniqueforapplyingcompressiveandtensilestraintoathinlmisshown.Whenthelmisontheoppositesideoftheappliedstress(orange)thestrainistensile.Whenthelmisonthesamesideastheappliedstress(purple),thestrainiscompressive.Theareabetweentheorangeandpurplelinesrepresentsthesubstrate.Thedashedboxistheundistortedbeam/substrate.Figurereproducedfrom[ 45 ]. Anexternalstrainoflessthan102isshowntoincreasetheFEpolarizationby50%(bluecurve).Insets:(upper)Thestraincouplingisshowntoincreaseasafunctionoftensilestrain.(lower)Thestraincouplingisshowntodecreaseasafunctionoftemperature,disappearingnear50K. 124

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6-3 ).Applyingstrainalongthe(100)and(010)axesproducednochangeinremanentpolarizationregardlessofelectriceldorientation.Strainingalong(110)and(1-10),however,producesthegiantchangesshowninFig. 6-2 .Theincreaseisonlyforelectriceldsparalleltothestrain,withperpendiculareldsinducingnochangeinremanentpolarization(seeFig. 6-3 ).Theinsensitivityoftheperpendiculareldsrulesoutarotationofthepolarization,suggestingthatthestraininducesferroelectricpolarizationbystabilizingthemonoclinicferroelectricdistortionofBiMnO3(seeFig. 6-3 ). 102 ].Furthermore,theexternallyappliedstrainisnotsufcientlylargeeither.Thereforeanotherfactorthatissensitivetostrainmustbepresent:islandedgestraingradients.Compressivelatticemismatchstraincanleadtoanon-unifromstraindistributioninthinlmduetoislandformation,andthestrainattheislandedgescanfarexceedtheaveragelatticemismatchstrain(seeFig. 6-4 )[ 87 88 ].Thegrowthmorphologyofourthinlms(seeFig. 5-3 )suggeststhatsuchnon-uniformstraindistributionisadenitepossiblemechanismfortheappearanceofferroelectricity.Additionally,thismechanismexplainsthehighsensitivitytoexternalstrain.AsshownintheinsetofFig. 5-3 ,thestraingradientsdivergeneartheislandedge.Therefore,itislikelythatourexternallyappliedstrainconvertstheregionsneartheislandedgesjustshortofthecriticalstrainnecessarytoproduceferroelectricpolarization. 125

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A)Theschematicshowstheorientationsoftheelectriceldsandinducedstrains.Theremanentpolarizationisinsensitiveto<100>and<010>strains,however,<110>and<110>strainsinduce/stabilizethemonoclinicferroelectricdistortionofBiMnO3.Onlyelectrodeswithelectriceldsparalleltotheinducedtensilestrainsdetectchanges(increases),meaningtheRPdoesnotrotate,butthatnewferroelectricdipolesarecreatedasthePNRsgrow.B)Arrowsindicatethemonoclinicdistortionsoftheunitcellinducedbystrainingalong<110>and<110>.Biisblue,Oisred,andMnisyellow. 6-5 ,wheretheremanentpolarizationisshowntodecreasebyapproximately10%inaeldof7T.Thecoerciveeldisunchanged,withonlythemagnitudeofthepolarizationeffected,andthecouplingisisotropic-equivalentforallanglesofappliedelds.Themagnetoelectriccouplingisshowntobelinearineld(upperinsetofFig. 6-5 ),withamaximumlinearmagnetoelectriccouplingcoefcientof,=0.1C/cm2Tnear 126

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A)AnAFMimageshowsGeislandsgrownonaSisubstrate.B)Across-sectionTEMimageshowsaGeislandona6monolayerthickGewettinglayeronaSisubstrate.C)TheboundaryofLingsmoundandacontourdiagramshowsthecalculatedstrainxxinthesystem.D)Thevariationofsurfacestrainsisshownalongthesystemsurface.Thegureisreproducedfromreference[ 88 ]. 65K(seeSec. 1.3.3 .Themagnetoelectriccouplingisalsofoundtodecreasewithtemperature,disappearingcompletelybelowT50K(seelowerinsetofFig. 6-5 ). 5.3.3 (whereanintitialpulsepolestheferroelectricdomains,andafteravariabledelaytimeasecondpulsechecksfordomainswhichhavereoriented).Figure 6-6 showstheresultofvaryingthedelaytimebetweenpulsesforboth0Tand7Telds.Asseen,foralldelaytimes,themagneticelddatahavelargermeasuredswitchingpolarizations.Therefore,weconcludethatthemagneticelddecreasesthereorientationtime-scalesothatmorePNRsareavailabletoipduringthesecondpulse.Thedielectricdataalsosupportthis 127

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Amagneticeldof7Tisshowntodecreasetheferroelectricpolarizationbyapproximately10%.Inset:Themagnetoelectriccouplingisshowntobelinearineld.ThedecreaseinremanentpolarizationisbelievedtobecausedbyanincreasedreorientationrateofthePNRs(seeSec. 6.3.2 ). interpretationasadecreaseinEAofafewpercentisseenat7Tinbothrelaxations(notshown).Thismechanismexplainsthenegativesignofthemagnetoelectriccoupling,however,thetemperaturedependenceofthecouplingandhowmagneticeldsincreasethereorientationratearestillnotunderstood.Apossibleexplanationforthisisthattheferroelectricityislinkedtoorpartiallycausedbythespatialvariationofthemagnetization(seeSec. 1.2.3 ).Inthisscenariomagneticeldswouldforcemagnetizationvectorstoalignandwouldeliminateaportionoftheferroelectricregions.Thereductioninsizeoftheferroelectricdomainswouldthenmakethemmoresusceptibletothermaluctuations,therebyreducingthereorientationtime-scalesofthePNRs. 128

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Thedualpulsesequenceshowsthatforeverydelaytimethemagneticelddatahavelargerswitchingpolarization,indicatingthatthereorientationtime-scalesaredecreasedbythe7Teld. Themagnetic,electric,andlatticepropertiesareallshowntocorrelate.Near50K,vesimultaneousphenomenaoccurasthetemperatureislowered.A)Theremanentpolarization(lledcircles)andmagnetization(opencircles)increasesrapidly.B)Thestraincoupling(blue)appearsandthemagnetoelectriccoupling(red)disappears.C)Thestaticdielectricconstant,0,diverges.Thisprovidesevidencethatthemagnetoelectriccouplingislinkedtothestrainstateofthelattice(seetext). themagnetizationincreasesrapidly,thestraincouplingappears,themagnetoelectriccouplingdisappears,and0diverges.Figure 6-7 displaysallofthesesimultaneousphenomena. 129

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6-7 (a)).Thisisalsosuggestedbytheappearanceofthestraincoupling(seeFig. 6-7 (b)),indicatingthatthelatticeisnowrigidenoughtoholdtheinduceddistortions.Thecauseofthechangeinthelatticeisnotclear,however,thecoincidenceoftheincreaseinmagnetizationsuggestsferrodistortionislikely(seeFig 6-7 (a)).TheincreaseindielectricconstantindicatesPNRsarelikelyincreasinginsize(seeFig. 6-7 (c)).Thesephenomenasuggestthat50Kisthefreezingtemperatureknowninrelaxorferroelectrics(whereuctuationtime-scalesdiverge,andthePNRscoalesceandcouple).Thesimultaneousdisappearanceofmagnetoelectriccouplingsuggestsitsmechanismistiedtotheexibilityofthelattice(seeFig. 6-7 (b)),consistentwitharecenttheoreticalpredictionofgiantmagnetoelectriccouplinginducedby'structuralsoftness'[ 103 ]. 130

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PatrickR.Mickelwasbornin1982totwolovingparents,StanandKarenMickel.Hewasbornintoafamilywithtwoolderbrothers,AndyandJeremy,wherehisparentsfosteredalloftheirtalentswithoutpushinginpresetdirections(evidencedbytheirdiversechoices:art,writing,andscience).Asachild,hisafnityfortoolsshinedthrough,ashewasconstantlybuildingtoys.Heenjoyedsomesuccess-afull-scalehalf-pipeforroller-bladingandskateboarding,andamake-shiftBBgun-aswellassomefailures,suchasapressurizedsquirt-gunandbatterypoweredmoped(muchtohisteasingfriends'delight).Althoughheexcelledinmathgrowingup,itwasnotuntillateintohighschoolthathediscoveredhisloveforphysics.Duringthesecondsemesterofhissenioryear,heenrolledinanastronomycourseatWittenbergUniversity,taughtbythephysicsprofessorDr.DanielFleisch-whowasfamousoncampusforhistalentandcharisma.Patrickwouldstayafterclass,sometimesforhours,talkingwithDr.Fleischandlearningaboutalldifferentareasofscience.Hewashooked.HestartedcollegeattheUniversityofNotreDamethefollowingfalllistedasatentativetheology/philosophymajor,butquicklychangedcourseandmajoredinphysics.Duringhisundergraduateyears,hesoughtadiverseexperienceandvolunteeredforresearchinmanydifferentareas:highenergyphysics,optics,statisticalphysics,andcondensedmatter.HissenioryearhedecidedtoapplyforgraduateschoolandpursueaPh.D.inphysics(notsurprisingforthesonoftwoacademics).Then,undertheguidanceofafewtrustedphysicsprofessors,heacceptedafellowshipfromtheUniversityofFlorida.Whileinitiallyfocusedonbiophysics(NuclearMagneticResonancediffusiontensorimgaing),duringhissecondyear,hetookacoursewithDr.ArthurHebardandquicklyrealizedhowgreatanopportunityitwouldbetoworkwithhim.ThenextsemesterhejoinedArt'slab,andbegantheresearchthathasculminatedinthisdissertation.Finally,hegraduatewithhisPh.D.fromtheUniversityofFloridainAugustof2011. 137