Infrared Study of Magnetic and Electric Excitations in Novel Complex Oxides

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Infrared Study of Magnetic and Electric Excitations in Novel Complex Oxides
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Miller, Kevin H
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Physics
Committee Chair:
Tanner, David B
Committee Members:
Hebard, Arthur F
Biswas, Amlan
Stanton, Christopher Jay
Norton, David P

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magnons -- multiferroic
Physics -- Dissertations, Academic -- UF
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Physics thesis, Ph.D.
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Abstract:
The contents of this thesis report the characterization of novel complex-oxide single crystals that are candidates for the discovery of new multiferroic materials and the realization of strong magnetoelectric coupling. The primary method of characterization is infrared spectroscopy in either reflection or transmission geometry, as governed by the material's optical response over a given frequency range. Optical properties are estimated via Kramers-Kronig relations and by fits to a Lorentz oscillator model. In certain materials, infrared results have motivated further characterization techniques: namely, magnetic susceptibility, x-ray diffraction, Raman spectroscopy, and terahertz spectroscopy. The specific materials studied are identified along with a short statement describing the major experimental findings resulting from each study. The Cu$_2$OSeO$_3$ system exhibited anomalous behavior of its infrared active phonons across the ferrimagnetic ordering temperature (T$_c$=60~K), which contributed to an abrupt change in the dielectric constant at the onset of magnetic order. The FeTe$_2$O$_5$Br system displayed a highly anisotropic phonon spectrum, which was corroborated by theoretical lattice dynamical calculations. In the Cu$_3$Bi(SeO$_3$)$_2$O$_2$Cl system, 16 new infrared phonons were observed below 115~K despite the lack of a structural transition, and 2 magnetic excitations were discovered below the long range magnetic ordering temperature (T$_c$=24~K). The Cu$_3$(SeO$_3$)$_2$Cl system, which is still a work in progress, has shown drastic phonon anomalies near both 80~K and 40~K (suspected magnetic ordering temperature) suggesting the existence of a rich phase diagram.
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by Kevin H Miller.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
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Adviser: Tanner, David B.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-11-30

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INFRAREDSTUDYOFMAGNETICANDELECTRICEXCITATIONSINNOVE L COMPLEXOXIDES By KEVINH.MILLER ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2013 1

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c r 2013KevinH.Miller 2

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Tomyfamily 3

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ACKNOWLEDGMENTS Mycompletionoftheworkinthisdissertationwouldnothavebeenpos siblewithout thehelpofnumerouspeople.Iowemydeepestgratitudetomyadvis or,Prof.DavidB. Tanner,whoseknowledgehasneverceasedtoimpressme,andwho sestyleofmentoring promotedtheindependentandproblem-solvingstyleofthinkingnec essarytothrivein thiseldofwork.Dr.Tanner 0 sstyleofmentoringisundeniablycollaborativeatthesame time,withgraduatestudentshelpingeachotherwheneverneeded .Iamalsothankfulfor theguidanceprovidedbytheotherfourmembersofmycommittee: ProfessorsArthur Hebard,AmlanBiswas,ChristopherStanton,andDavidNorton. IwillforeverbeindebttoHelmuthBergeroftheEPFLinLausanne,S witzerland whogrewallthecrystalsdescribedinthisdissertation.Thetimelines sofourcollaboration withHelmuthBergercombinedwiththehigh-qualityofthecrystalspr ovidedallowedme to\hitthegroundrunning"followingmyrsttwosemestersofcour sework. Additionalacknowledgementisdirectedtowardthemanycollaborat orswho haveprovidedmewiththeopportunityofexperimentallearningbey ondtheTanner Lab.AmongsuchindividualsmustbementionedProf.MarkMeiselofU F 0 sPhysics Departmentformagneticsusceptibilitymeasurements,Dr.G.Lar ryCarrandProf. PeterStephensoftheNationalSynchrotronLightSource(Broo khavenNationalLab)for infraredmagneto-opticsandx-raydiraction,Dr.DJArenasoft heUniversityofNorth FloridaforRamanmeasurements,andDr.XiaoshanXuofOakRidgeN ationalLabfor latticedynamicalcalculations.AspecialthanksgoestoProf.Roge rA.LewisandEvan ConstableoftheUniversityofWollongong,Australia.Prof.Lewisgr aciouslyaccepted meintohislaboratoryforthreemonthsasanNSFEAPSIfellowtopur sueanambitious project.DuringmystayinAustralia,Prof.Lewisalsoprovidedmewit hsomemuch needednancialaid,thusgoingaboveandbeyondwhatwasexpect edofhim. Iwouldalsoliketoexplicitlythanktwopeersofminewhohavehelpsculpt myskills asanexperimentalphysicist.FirstisDr.CatalinMartin,apostdoct oralresearcherinthe 4

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TannerLabwhosetechnicalsavvyandproblem-solvingskillswerean essentialingredient aidingmytransitionintoalineofworkinwhichunexpectedproblems,alm ostonadaily basis,arecustomary.Secondismyformerocemate,Dr.Xiaoxian gXi,whowasthe drivingforcebehindtheprogressionofmycomputingskills. IwouldliketopubliclyacknowledgemynumerouspeersintheTannerLa bwith whomIhavebeenbothlearnerandteacher.TheseincludeDr.Dimitr iosKoukis,Zahra Nasrollahi,NaweenAnand,BerikUzakbaiuly,EvanThatcher,Chang Long,andLuyi Yan.AdditionalgratitudeisexpressedtowardsthePhysicsDepar tmentsMachineShop, specically,EdStorchandMarcLink.Finally,IwouldliketothankJohn Mockoofthe PhysicsDepartmentfortheequipmentandadvicehehasprovidedf ormynumerous outreachexcursionstolocalgrammarschools;theseevents,inw hichIsharedwith elementaryschoolchildrenmyfascinationforphysics,werearefr eshingbreakfrommy normallaboratorywork. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 10 ABSTRACT ........................................ 12 CHAPTER 1INTRODUCTION .................................. 14 2MULTIFERROICSANDMAGNETOELECTRICS ................ 17 2.1Fundamentals .................................. 17 2.1.1DenitionsandApplications ...................... 17 2.1.2HistoryandRenaissance ........................ 19 2.1.3ContradictingRequirements ...................... 21 2.2ExperimentalRealization ............................ 24 2.2.1InfraredActivityinMultiferroicsandMagnetoelectrics ....... 24 2.2.2Electromagnons ............................. 25 3EXPERIMENTALMETHODOLOGY ....................... 28 3.1Instrumentation ................................. 28 3.1.1FourierTransformInterferometry ................... 28 3.1.1.1Detectors ........................... 34 3.1.1.2Sources ............................ 35 3.1.2TerahertzTwo-colorPhotomixingSystem ............... 37 3.2Analyticaltechniques .............................. 39 3.2.1LorentzModel .............................. 39 3.2.1.1Fittingprocedures ...................... 41 3.2.2KramersKronig ............................. 42 3.2.3SumRules ................................ 45 4MAGNETODIELECTRICCOUPLINGINCu 2 OSeO 3 .............. 47 4.1Overview .................................... 47 4.2ExperimentalProcedures ............................ 48 4.3ResultsandAnalysis .............................. 49 4.3.1Magnetism ................................ 49 4.3.2RerectanceandTransmittanceSpectra ................ 50 4.3.3Kramers-KronigAnalysisandOpticalProperties ........... 52 4.3.4Oscillator-ModelFits .......................... 54 4.4Discussion .................................... 55 6

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4.4.1MagnetodielectricEect ........................ 55 4.4.2AnomalousPhonons ........................... 56 4.4.3AssignmentofPhononModes ..................... 58 4.5Summary .................................... 60 5OPTICALPROPERTIESOFMULTIFERROICFeTe 2 O 5 Br ........... 64 5.1Motivation .................................... 64 5.2ExperimentalProcedures ............................ 66 5.3ResultsandAnalysis .............................. 68 5.3.1X-Raydiraction ............................ 68 5.3.2RerectanceandTransmittanceSpectrum ............... 68 5.3.3Field-DependentTransmittance .................... 70 5.3.4DeterminationofOpticalProperties .................. 72 5.3.5LorentzOscillatorFits ......................... 73 5.4Discussion .................................... 74 5.4.1GrouptheoryandLatticeDynamics .................. 74 5.4.252 B u Modes ............................... 76 5.4.353 A u Modes ............................... 76 5.4.4AssignmentofModes .......................... 78 5.5Summary .................................... 79 6PHONONANOMALYANDMAGNETICEXCITATIONSINCu 3 Bi(SeO 3 ) 2 O 2 Cl 85 6.1Overview .................................... 85 6.1.1CrystalStructure ............................ 85 6.1.2BeautyofInfraredSpectroscopy .................... 86 6.1.3MajorFindings ............................. 86 6.2ExperimentalProcedures ............................ 87 6.3ResultsandAnalysis .............................. 89 6.3.1ZeroFieldRerectanceandTransmittanceSpectra .......... 89 6.3.2Kramers-KronigandOscillator-modelts ............... 90 6.3.3OpticalProperties ............................ 92 6.3.4PowderX-RayDiraction ....................... 93 6.3.5MagneticField-DependentTransmission ............... 95 6.3.6MagneticProperties ........................... 98 6.3.7MagneticField-DependentCapacitance ................ 100 6.4Discussion .................................... 102 6.4.1GroupTheoryandObservedModes .................. 102 6.4.2PowderX-RayDiraction ....................... 104 6.4.3PhononRepulsion ............................ 105 6.4.4MagneticExcitations .......................... 107 6.4.4.1Natureandisotropy ..................... 108 6.4.4.2 H k ^ c elddependence .................... 109 6.4.4.3 H ? ^ c elddependence .................... 112 6.5Summary .................................... 113 7

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7CONCLUSIONS ................................... 116 APPENDIX APRELIMINARYRESULTSONSINGLECRYSTALCu 3 (SeO 3 ) 2 Cl 2 ....... 119 A.1BackgroundandCrystalStructure ...................... 119 A.2MagneticProperties .............................. 119 A.3Room-temperatureInfraredandRamanResults ............... 119 A.4Temperature-dependentInfraredSpectra ................... 121 BCALCULATINGSINGLE-BOUNCEREFLECTANCE .............. 125 B.1Preface ...................................... 125 B.2Formalism .................................... 125 REFERENCES ....................................... 131 BIOGRAPHICALSKETCH ................................ 136 8

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LISTOFTABLES Table page 2-1Mechanismsofinducingferroelectricity ....................... 23 4-1OscillatorparametersforCu 2 OSeO 3 at20K .................... 57 4-2LatticedynamicalcalculationsofCu 2 OSeO 3 .................... 62 5-1OscillatorparametersforFeTe 2 O 5 Brphononsalong^ e A .............. 75 5-2OscillatorparametersforFeTe 2 O 5 Brphononsalong^ e C .............. 81 5-3OscillatorparametersforFeTe 2 O 5 Brphononsalong^ e B .............. 83 6-1OscillatorparametersofCu 3 Bi(SeO 3 ) 2 O 2 Clat7K ................ 115 A-1Pointgroupsymmetrycharactertableof C 2 h .................... 121 9

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LISTOFFIGURES Figure page 2-1Multiferroiccouplingschematic ........................... 18 2-2Spiralspinorder ................................... 24 2-3Firstreportofelectromagnons ............................ 27 3-1SchematicofaMichelsoninterferometer ...................... 29 3-2ThesamplingofaninterferogramusingaBruker113v .............. 33 3-3HorizontalpolarizationofbeamlineU4IRattheNSLSbelow100cm 1 ..... 38 3-4Two-colorphotomixerattheUniversityofWollongong .............. 39 3-5Kramers-Kronigintegrationprocedure ....................... 43 4-1ThedcmagneticsusceptibilityofCu 2 OSeO 3 .................... 50 4-2Temperature-dependentrerectancespectrumofCu 2 OSeO 3 ............ 51 4-3BroadbandrerectanceandopticalconductivityofCu 2 OSeO 3 ........... 52 4-4Temperature-dependentfar-infraredtransmissionofCu 2 OSeO 3 .......... 53 4-5BroadbandtransmissionofCu 2 OSeO 3 at300K .................. 54 4-6Temperature-dependentfar-infraredopticalconductivity ofCu 2 OSeO 3 ..... 55 4-7LorentzoscillatortofCu 2 OSeO 3 rerectance ................... 56 4-8ThedielectricconstantofCu 2 OSeO 3 asextractedfromtheinfrared ....... 58 4-9AnomalousphononparametersinCu 2 OSeO 3 .................... 59 5-1Single-crystalx-raydiractionpatternofFeTe 2 O 5 Brat300K .......... 69 5-2Temperature-dependentrerectancespectrumofFeTe 2 O 5 Br ............ 70 5-3BroadbandopticalconductivityofFeTe 2 O 5 Br ................... 71 5-4Measuredmid-infraredtransmissionofFeTe 2 O 5 Br ................. 71 5-5Calculatedsingle-bouncererectanceinthemid-infraredofFeT e 2 O 5 Br ...... 72 5-6LorentzoscillatortofFeTe 2 O 5 Brrerectance ................... 74 5-7EvidenceforburiedmodesinFeTe 2 O 5 Br ...................... 77 6-1ThecrystalstructureofCu 3 Bi(SeO 3 ) 2 O 2 Cl ..................... 86 10

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6-2Temperature-dependentrerectancespectraofCu 3 Bi(SeO 3 ) 2 O 2 Cl ........ 91 6-3Mid-infraredtransmissionofCu 3 Bi(SeO 3 ) 2 O 2 Clat300K ............. 91 6-4OscillatortofCu 3 Bi(SeO 3 ) 2 O 2 Clrerectance ................... 93 6-5OpticalconductivityandlossfunctionofCu 3 Bi(SeO 3 ) 2 O 2 Cl ........... 94 6-6TemperaturedependenceofCu 3 Bi(SeO 3 ) 2 O 2 Cllatticeparameters ........ 95 6-7ReitveldrenementofCu 3 Bi(SeO 3 ) 2 O 2 Clat295and85K ............ 96 6-8IsotropicmagneticexcitationinCu 3 Bi(SeO 3 ) 2 O 2 Clat33.1cm 1 ......... 97 6-9MagneticelddependenceofexcitationinCu 3 Bi(SeO 3 ) 2 O 2 Clat33.1cm 1 ... 98 6-10ComplementarytechniquestoprobeamagneticexcitationinCu 3 Bi(SeO 3 ) 2 O 2 Cl 99 6-11IsothermalmagnetizationofCu 3 Bi(SeO 3 ) 2 O 2 Clat5K .............. 100 6-12InverseofthemagneticsusceptibilityforCu 3 Bi(SeO 3 ) 2 O 2 Cl ........... 101 6-13Field-dependentcapacitanceofCu 3 Bi(SeO 3 ) 2 O 2 Cl ................. 101 6-14Normalizedtemperature-dependentcapacitanceofCu 3 Bi(SeO 3 ) 2 O 2 Cl ...... 102 6-15EvidenceforphononrepulsioninCu 3 Bi(SeO 3 ) 2 O 2 Cl ............... 106 6-16IsotropicoscillatorstrengthsofthemagneticmodeinCu 3 Bi(SeO 3 ) 2 O 2 Cl .... 109 A-1InfraredrerectanceofCu 3 (SeO 3 ) 2 Cl 2 300K .................... 120 A-2Raman ab spectraofCu 3 (SeO 3 ) 2 Cl 2 300K ..................... 122 A-3Temperature-dependentinfraredrerectanceofCu 3 (SeO 3 ) 2 Cl 2 .......... 123 A-4NewinfraredmodesinCu 3 (SeO 3 ) 2 Cl 2 ....................... 124 B-1Refractiveindexfromboth R s andKramers-Kronig ................ 129 11

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy INFRAREDSTUDYOFMAGNETICANDELECTRICEXCITATIONSINNOVE L COMPLEXOXIDES By KevinH.Miller May2013 Chair:DavidB.TannerMajor:Physics Thisdissertationdescribesthecharacterizationofnovelcomplex -oxidesinglecrystals thatarecandidatesforthediscoveryofnewmultiferroicmaterials andtherealization ofstrongmagnetoelectriccoupling.Theprimarymethodofcharac terizationisinfrared spectroscopyineitherrerectionortransmissiongeometry,asgo vernedbythematerial's opticalresponseoveragivenfrequencyrange.Opticalpropert iesareestimatedvia Kramers-KronigrelationsandbytstoaLorentzoscillatormodel.I ncertainmaterials, infraredresultshavemotivatedfurthercharacterizationtechn iques:namely,magnetic susceptibility,x-raydiraction,Ramanspectroscopy,andterah ertzspectroscopy.The specicmaterialsstudiedareidentiedalongwithashortstatement describingthe majorexperimentalndingsresultingfromeachstudy.TheCu 2 OSeO 3 systemexhibited anomalousbehaviorofitsinfraredactivephononsacrosstheferr imagneticordering temperature(T c =60K),whichcontributedtoanabruptchangeinthedielectriccons tant attheonsetofmagneticorder.TheFeTe 2 O 5 Brsystemdisplayedahighlyanisotropic phononspectrum,whichwascorroboratedbytheoreticallattice dynamicalcalculations. IntheCu 3 Bi(SeO 3 ) 2 O 2 Clsystem,16newinfraredphononswereobservedbelow115K despitethelackofastructuraltransition,and2magneticexcitat ionswerediscovered belowthelongrangemagneticorderingtemperature(T c =24K).TheCu 3 (SeO 3 ) 2 Cl system,whichisstillaworkinprogress,hasshowndrasticphonona nomaliesnearboth 12

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80Kand40K(suspectedmagneticorderingtemperature)sugges tingtheexistenceofa richphasediagram. 13

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CHAPTER1 INTRODUCTION Thisdissertationreportsthecharacterizationofnovelcomplexo xidesinglecrystals grownbycollaboratorsattheEPFLinLausanne,Switzerland.Them otivationforthis researchfallsintothebroadercategoryofmultiferroicandmagne toelectricmaterials. 1 Multiferroicmaterialsexhibitthecoexistenceofatleasttwolongran geferroicorders (e.g.,ferroelectricityandferromagnetism)inasinglehomogeneous crystal.Theorders areoftencoupled,thusgivingrisetopotentialdeviceapplications. Althoughmultiferroic materialswererststudiedinthe1960sintheformerSovietUnion,[ 1 2 ]experimental andcomputationaltoolsatthetimewerenotadvancedsucientlyt oinvestigatefully thewealthofknowledgeandpotentialfutureapplicationsprovided bythesematerials. However,arenaissanceinmultiferroicresearchthatoccurredne artheturnofthe21st century[ 3 { 5 ]unveiledarenewedappreciationformultiferroicmaterialsaidedbya theoreticalunderstandingofthemicroscopicmechanismsnecess aryformultipleorders tocoexistandcoupleinasinglehomogeneouscrystal.Itwassubseq uentlydiscovered thatcomplexinteractions(e.g.,frustratedlatticegeometries)at timesledto\improper" orderingofelectricdipoles,thussidesteppingtheoftencontradic tingrequirementsforthe simultaneousexistenceofmultipleferroicorders(cf.Chapter 2 ).Therefore,additional inspirationforthisworkislinkedtotheongoingquesttodiscoverand characterizenew multiferroicmaterialsfromtheplethoraofpromisingcandidates,int hiscase,novel complexoxidesinglecrystals. Moderninterestinmultiferroicmaterialsisfueledfromboththebasic scienticand theappliedtechnologicalviewpoints.Onthetechnologicalside,suc hmaterialspresent anopportunitytocombinemultipletasks,formallyperformedbysep aratematerials,by 1 Henceforth,thetermmultiferroicwillincludemagnetoelectricmate rials.Thesubtle dierencebetweenthetwotypesofmaterialswillbeelucidatedinCha pter 2 14

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onematerial.Ifoneconsidersferroelectricferromagneticmultife rroics,forexample,the benetswouldstemfrommakingcertainmagnetictechnologieselect ric-eldcontrollable,a clearadvantageconsideringthatelectriceldsaremucheasiertop roduceandmanipulate. Onthescienticside,interestisdrivenbyanadvancementofknowle dgeaboutthe microscopicinteractionsthatfavortheexistenceofmultiferrocit y,aswellasaboutthe mechanismsthatgiverisetothecouplingoftheferroicorders.Fru itfulnessemergingfrom suchtheoreticalunderstandingswillcontinuetoprovidedirection forthedesign[ 6 ]ofnew materialspossessinghigherT c 0 s(criticaltemperaturesatwhichferroicordersmanifest themselves)andstrongercoupling. Infraredspectroscopy,whichistheprimarymethodofcharacte rizationforthis work,representsauniquewaytoprobethephenomenaassociate dwithmultiferroicand magnetoelectricmaterials.Firstofall,infraredspectroscopyisac ontactlessprobeof materials 0 properties;consequently,interfaceeectsassociatedwithelec tricalcontacts canbeavoided.Second,infraredspectroscopyisextremelysens itivetoanumberof signaturesassociatedwiththeferroicorders,namely,magneticr esonancemodesof orderedmagneticdipoles,softmodebehaviorandstructuraltra nsitionsofferroelectrics, andlatticedeformationofferroelastics.Resultsfromtheinfrare dmeasurementsofthe materialsexaminedinthisdissertationhaveattimessuggestedthen eedforalternative experimentaltechniquesprovidingbothcomplementaryandsupple mentaryinformation. Suchtechniquesincludesingle-crystalandpowderx-raydiractio n,terahertzand infraredmagneto-optics,Ramanspectroscopy,anddcmagnetic susceptibility.The independentvariablecommontoallsuchtechniquesutilizedinthewor kpresented hereisthesampletemperature,anecessaryparameterconsider ingthatonlyoneroom temperaturemultiferroiccurrentlyexists,BiFeO 3 andanaloguesthereof.[ 7 ] Thisdissertationisorganizedasfollows:Chapter 2 presentsacomprehensive overviewofmultiferroicphysicsaswellasareviewofrecentliteratu repertainingto thecharacterizationofmultiferroicsbyinfraredspectroscopy. Chapter 3 coversprinciples 15

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ofsomeexperimentalinstrumentationutilizedandalsoanalyticalte chniquesimplemented tointerpretthedata.Chapter 4 ,Chapter 5 ,andChapter 6 containthestudyofthree novelcomplexoxidesinglecrystals.Thereportofeachcrystalsy stemwillbeginwitha briefreviewofallpertinentinformationpreviouslypublished.Thesp ecicexperimental instrumentationusedineachstudywillbestated.Thecomplexoxide singlecrystals studiedinChapter 4 ,Chapter 5 ,andChapter 6 areCu 2 OSeO 3 ,FeTe 2 O 5 Br,and Cu 3 Bi(SeO 3 ) 2 O 2 Clrespectively.Appendix A containsanuncompletedreportofafourth andquiteinterestingsinglecrystalsystem,Cu 3 (SeO 3 ) 2 Cl,whichwasongoingatthetime thisdissertationwascompiled.Appendix B developsthetheoryofacombinedrerection andtransmissionanalysistechnique,rstreportedintheseminalw orkofZiboldetal.[ 8 ] thatwasimplementedtoscrutinizeinfraredresultsinthreeofthef ourcrystalsystems studied. IninfraredandRamanspectroscopy,itisconventionalwisdomtor eportthe frequencyoflightintheunitsofwavenumbers.Wavenumbersaref oundbytakingthe reciprocalofthewavelength(incentimeters)andthustheyhave dimensionalitycm 1 1cm 1 correspondstoanenergyof0.124meVandafrequencyof30GHz. Themajority oftheinfraredandRamanplotsinthisdissertationemployingwavenu mberswillhave atwinaxisoftheappropriatescaleintheunitsofelectronvolts.Allo thermeasured quantitieswillbereportedintheSIunits. 16

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CHAPTER2 MULTIFERROICSANDMAGNETOELECTRICS 2.1Fundamentals 2.1.1DenitionsandApplications Multiferroicsaredenedasmaterialsthatsimultaneouslypossessm ultipleferroic properties(orderparameters).Theprimaryferroicorderpara metersunderconsideration areferromagnetism,ferroelectricity,andferroelasticity;howe ver,thecontemporary denitionofmultiferroicshascometoalsoincludeantiferromagnetis m,ferrimagnetism, andantiferroelectricityaswell.Basicscienceinterestandpotentia ltechnological applicationsarisebecausethesimultaneouslyexistingordersareof tencoupledto oneanother.Forafullappreciationofmultiferroicphysics,letusc onsiderFigure 2-1 Ferroelectricityisdenedbyspontaneouslyorderedelectricdipole sthatareswitchable byanappliedelectriceld.Ferroelasticmaterialsexhibitaspontaneo usdeformation thatiscontrollablebyanexternallyappliedstrain.Ferromagneticma terialsdisplaya spontaneousmagnetizationthatisswitchablebyanappliedmagnetic eld.(Itisworthy ofnotethattheaformementionedferroicordersonlymanifestth emselvesinaniteregion oftemperaturespaceandwhoseonsetissigniedbyaCurieorNeel temperature.)In termsofcoupling,onecouldimagineapplyinganexternalelectriceld andsubsequently harnessmagneticallyordereddipolesorcontrolaspontaneousde formation.Suchcoupling iswidelyreferredtoasmagnetoelectriccoupling.Themostdirectfo rmofmagnetoelectric couplinginmultiferroicmaterialsoccurswhenthespontaneousonse tofoneferroic orderatacriticaltemperaturepromotesasecondordertocrop upatthatverysame temperature.Acommonexampleofthis,whichwillbediscussedinSec tion 2.1.3 ,is magneticallydrivenferroelectricity. Magnetoelectricmaterials,bydenition,alsoexhibitthemagnetoele ctriccoupling eect;however,theyaredierentfrommultiferroicmaterialsinth efollowingsubtle way:Intheabsenceofexternalperturbations,magnetoelectr icmaterialsonlypossess 17

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Figure2-1.Inmultiferroics,spontaneousorderparameterscan bealteredbyapplying externalperturbations(e.g.,externaleldsofconjugateorde rs).When ferroelectricity,ferromagnetism,andferroelasticitycoexist,a spontaneous electricpolarization, P ,canbetunedbynotonlyanexternalelectriceld E butalsoanexternalmagneticeld H ,oranappliedstrain .Figure reproducedfromRef.[ 9 ]. oneferroicorder.Nevertheless,uponapplyinganexternalpert urbationofthesole ferroicorder(e.g.,electriceldforaferroelectric),onecanbring aboutasecondferroic order(e.g.,ferromagnetism).Magnetoelectricsarethusaclosec ousintomultiferroic materials,andconsequently,thefollowingdiscussionofapplications willberelevantto bothmagnetoelectricsandmultiferroics. Thegreattechnologicalinterestinmultiferroicmaterialsrevolvesa roundferroelectric ferromagnetsandisdrivenbyademandforspintronic(utilizingelect ronsspinandcharge) devices.Beforeapplicationscanbefullyappreciated,itisworthwhile toconsiderthe 18

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implementationofferroelectricsandferromagnetsaloneindevices .Ferroelectricsare ubiquitousintheactuatorandsensingindustries,specicallybeingu sedastunable capacitorsandfornon-volatilerandom-accessmemory(Ferroele ctricRAM).Ferromagnets arealmostexclusivelyutilizedinthedesignoftransformers,aswella sfortherecording andstorageofdata.Thepropertiesofmultiferroicscanpotentia llybeexploitedin manyways.Atthemundanelevel,thehopeistoeventuallymakesome ofourcurrent magnetictechnologieselectric-eldcontrollable,forelectriceldsp roducelessheat dissipationandtheyareeasiertogenerate.Furthermore,oneca nimaginenoveldevices thatwouldensue(e.g.,tunnelingmagnetoresistancesensors,mag neticread-headelectronic write-headmultistatememorydevices,andmagneticeldsensorsw herethemagneticeld isdeterminedbymonitoringamaterial 0 selectronicproperties). Thecurrentpushtowardsminiaturizingtechnologicaldevicescorr elateswellwiththe designofmaterialsthatcansimultaneouslyperformmultipletasks.T heimplementation ofmultiferroicmaterialsinspintronicdeviceswouldcertainlyadvance theprogression predictedbyMoore 0 slaw,forsuchdeviceswouldnotbeassusceptibletoheatingeects whentheoveralldimensionalityisreduced.2.1.2HistoryandRenaissance Inthe19thcentury,physicistPierreCurierstpostulatedacoup lingbetween magneticandelectricordersinasinglehomogeneousmaterial.Despit ethisearly postulation,itwasnotuntil1960thatthemagnetoelectriceectw asfortherst timeexperimentallyobservedinthecompoundCr 2 O 3 .[ 1 10 11 ]Subsequentworkon themagnetoelectriceectinthe1960sand1970swerespearhead edbythegroupsof Smolenskii[ 1 ]andVenevtsev[ 2 ](operatingintheformerSovietUnion).Atthetime, theworkstirredmoderateinterestinthephysicscommunity;howe ver,theoretical understandingofthemagnetoelectriceectwassparseandatte mptstoimprovethe couplingstrengthfordeviceapplicationsprovedbleak. 19

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Arenewalofinterestinmultiferroics 1 andmagnetoelectricsfollowingtherstyears ofthe21stcenturyhasbeendrivenbymultiplefactors.Onesuchf actoristhetheoretical workofN.Hill(nowN.Spaldin)[ 3 ]publishedin2000thatilluminatedtheseemingly contradictingnatureofmultipleferroicorders(c.f.Section 2.1.3 ),andthusexplaining thescarcityofmultiferroicandmagnetoelectricmaterialsexistinga tthetime.Thenewly foundedknowledgeusheredina\materialsbydesign"approachinwh ichtheamountsand natureofcertainmaterialconstituentsweretailoredtomeetthe suspectedrequirements ofmultipleorders.[ 6 ]This\materialsbydesign"approachhadtheserendipitousbenet of promotingsynergybetweentheoreticalandexperimentalgroup s,thusspreadinginterestin theeld. Anotherequallyimportantfactorfuelingtherenewedinterestinmu ltiferroicmaterials stemsfromtheadvancementofinstrumentation.Inthepioneerin gworkofSmolenskii andVenevtsev[ 1 2 ]40yearsprior,thetoolswerenotdevelopedyettocapitalizefully onthewealthofknowledgeanddiscoveriesintrinsictomultiferroicma terials.The biggeststridesintherealmoffabricationtechniqueshaveoccurre dinthegrowthofthin lms.Specically,theintroductionofstrainviathelatticemismatchw ithasubstrate andnewgrowthtechniquesemployingpressurizedenvironmentsha vegreatlybeneted theproductionofnewpristinematerials.Moreover,theadventof newexperimental techniquestoprobethemicroscopicphenomenaassociatedwithfe rroicorders(e.g., mappingdomainboundaries)arenowwidespread.Onthetheoretica lside,faster andmorepowerfulcomputingcapabilitieshavepavedthewayforr st-principleand densityfunctionalcalculationstoshedlightonthemicroscopicinter actionsandcoupling mechanismsactiveinthematerials. Breakthroughsthemselves,suchasnewandunexpectedmechan ismsforferroelectricity (c.f.Section 2.1.3 )inmanganese-oxideperoskitecompounds,havestirredcuriosity and 1 Theterm\multiferroic"wasnotcoineduntil1994byH.Schmid.[ 12 ] 20

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stimulatedevengreatervigorintheeld.[ 4 ]Manyunansweredquestions,suchasthose pertainingtomechanismsofcoupling,stillremainhowever.2.1.3ContradictingRequirements Thesynthesisofnewmultiferroicshasproventobeaformidabletas k,considering thatferroicordersfavormaterialpropertiesthatgenerallycon tradictoneanother.For simplicity,letusconsiderferroelectricferromagneticmultiferroics andexaminethe electricalproperties,crystalsymmetry,andelectroncongur ationsindicativeofthetwo orders.Ferroelectricmaterials,bynature,mustbeelectricallyins ulating;otherwise,when electriceldsareapplied,chargeswouldrowratherthanbecomepo larized.Incontrast, manyferromagnetsaremetallic,althoughmetallicityisnotastrictre quirement. Concerningsymmetryrequirements,ferroelectricsmandateano ncentrosymmetric structurebecausetheelectricdipolesmustreversedirectionupo nreversingtheeld polarity.Incontrast,ferromagnetsnecessitateastructuret hatbreakstimereversal-symmetry. Thereexist31pointgroupsthatsatisfythesymmetryrequiremen tofferroelectricity. Ironically,31pointgroupsalsosatisfythesymmetryrequiremento fferromagnetism.Of theoriginal122Shubnikov-Heeschpointgroups,13overlap,perm ittingsimultaneously ferromagnetismandferroelectricity.[ 12 ] Asnotedabove,loopholesexistforbothelectricalpropertiesand symmetry requirements;however,theprevailingconceptsaboutelectronc ongurationssupporting ferroelectrictyandferromagnetismaretrulyincompatible.Itwas empiricallynotedin the1940sbyMatthaisthatferroelectricityisfavoredbyemptydo rbitals.Uponcloser examinationofourclassicalferroelectrics(e.g.,BaTiO 3 andPbTiO 3 ),itwasdiscovered thattheswitchableo-centeringbetweencationandanion,whichd enesferroelectricity, isaresultofahybridizationbetweenthelled2porbitalsoftheoxyge nsandtheempty 3dorbitalsofthetransitionmetalions(titaniuminthisexample).Ont heotherhand, 21

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ferromagnetismrequiresunpairedelectronsinthe3dshellofatra nsitionmetal. 2 This incompatibilityamountstoacompetitionbetweenHundandPauli.Hund 0 srulespromote unpairedspins,whilethoseofPauliwouldliketohaveallspinspairedint hesameorbital. Whenlocalizedelectronsareinvolved,thiscompetitiontendstoleant owardsmagnetism, andtheferroelectricityisforcedtoarisefrom\improper"means. 3 The\improper"orunusualmechanismsofcreatingferroelectricit ywererstreported inTbMnO 3 ,[ 13 ]CuFeO 2 ,[ 14 ]andNi 2 V 3 O 8 .[ 15 ]A2007articlebyCheongandMostovoy[ 4 ] denedimproperferroelectricityasaspontaneouspolarization,w hichisaby-product ofamorecomplexlatticedistortiontowardsanoncentrosymmetric statedrivenbyan alternateorderingphenomena.Table 2-1 detailsthemechanismsofproperandimproper ferroelectricity.Inwhatfollows,theimpropermechanismofmagne ticorderingwillbe discussed,foritisadirectmanifestationofmagnetoelectriccouplin g.Inaddition,electric dipolesinducedbytheorderingofmagneticdipolesremainhighlysusce ptibletoapplied magneticelds,thuspromptingtremendousinterestinthisclassof materials. Magneticfrustrationleadingtoaspiralspinordercaninduceanelec tricpolarization stemmingfromthefreeenergyterm,| P || M | @ | M |,inthefollowingway.Ina cubiccrystal,themagneticallydrivenelectricpolarizationtakesthe form P =[( M r ) M M ( r M )] : (2{1) Considerachainofspinsinwhichnearestneighborsandnext-neare stneighborsare coupledferromagneticallyandantiferromagneticallyrespectively. Thisfrustratedordering 2 Magnetismarisingfrompandfelectronphysicsisexcludedinthisdiscu ssion. 3 Ferroelectricitynecessitatesadistortionofthecrystallatticea titsonset.Proper ferroelectricorderoccurswhenthedistortionisdrivenbythemec hanismofferroelectricity itself.Improperferroelectricorderoccurswhenthelatticedisto rtionisjustaby-product ofamorecomplexorderingphenomena. 22

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Table2-1.Mechanismsofinducingferroelectricity TypeMechanismofinversionsymmetrybreakingMaterials ProperCovalentbondingbetween3 d 0 transitionmetalBaTiO 3 (Ti)andoxygen Polarizationof6 s 2 lonepairofBiorPbBiMnO 3 ,BiFeO 3 Pb(Fe 2 = 3 W 1 = 3 )O 3 ImproperStructuraltransitionK 2 SeO 4 ,Cs 2 CdI 4 `Geometricferroelectrics'hexagonalRMnO 3 ChargeorderingLuFe 2 O 4 `Electronicferroelectrics' MagneticorderingOrthorhombicRMnO 3 `Magneticferroelectrics'RMn 2 O 3 ,CoCr 2 O 4 TableadoptedfromRef.[ 4 ]. hasbeenshowntostabilizeinaspiralmagneticstate(Figure 2-2 )oftheform: S n = S [ e 1 cos Q x n + e 2 sin Q x n ] : (2{2) Intheaboveequation S describesthespin-density-waveordering, e 1 and e 2 areorthogonal unitvectors,and Q isthewavevectorgivenbycos( Q= 2)= J 0 = (4 J ).Traditional magneticorderingisinvariantunderspatialinversion(x n goesto x n ),however,changing thesignofallcoordinatesintheaforementionedspiralcongurat ioninvertsthedirection oftherotationofspinsinthespiral,thusbreakinginversionsymmet ry.Aspontaneous polarizationarisesintheform: P / r ( e 12 J s ) ; (2{3) where J s isthespincurrentfromtwocouplednoncollinearspins( J s / S 1 S 2 )and e 12 isthevectorconnectingthetwomagneticions.Sergienko etal .[ 16 ]postulatedthatthe inverseDzyaloshinskii-Moriya(DM)interactionwasalikelymechanismf ortheinduced polarization.Putbriery,inspiralmagnetstheDMinteractionactst omovenegative oxygenionsinthedirectionnormaltothespinchainformedbythepo sitivemagnetic ions,thusinducinganelectricpolarizationnormaltothechain.[ 16 ] 23

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Figure2-2.Aspiralspincongurationthathasbeenknowntobrea kinversionsymmetry. Thefrustratedspinchainhasferromagneticnearest-neighbour and antiferromagneticnext-nearest-neighbourinteractionsJandJ '. 2.2ExperimentalRealization 2.2.1InfraredActivityinMultiferroicsandMagnetoelect rics Thissectionisintendedtoprovideabriefreviewoftheinfraredsigna turesusedto probemultiferroicandmagnetoelectricmaterials.Infraredactivit yunambiguouslyrelated totheonsetofferroelectricorder,magneticorder,andmagnet oelectriccouplingaswellas anovelexcitationdenotingmultiferroismarediscussedinthissectio n. Whenferroelectricityarisesinacentrosymmetriccrystal,asymme trylowering transitionoccursinwhichtheinversioncenterisremoved.Suchatr ansitionaltersthe distributionofinfrared-activeBrillouin-zone-centerphononmode s.Thecompounds YMnO 3 [ 17 ]andTbMn 2 O 5 [ 18 ]bothshowtheclearappearanceofnewabsorption peaksuponenteringtheferroelectricstate.Eveninmaterialsalre adypossessing noncentrosymmetricstructures,theonsetofferroelectricity correspondstoaphysical displacementthatoftenmanifestsitselfinaphononmodethatsoft enswithtemperature (cf.EuTiO 3 [ 19 ]andKH 2 PO 4 [ 20 ]). Infraredspectroscopyisalsosensitivetoonsetandnatureofma gneticordering.The spin-wavedispersionbranchesassociatedwithlongrangemagnetic ordercanbedetected neartheBrillouinzonecenterbyacouplingtotheacmagneticeldoft helight.More 24

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specically,magnons(quantizedexcitationsofspinwaves)giverise toabsorptionsinthe infrared,typicallyintherange100 eV-10meV.[ 21 ]Thedynamicsofmagnonabsorptions asafunctionofmagneticeldcanhelptodistinguishbetweenantifer romagneticand ferromagneticorder.Specically,antiferromagneticmagnonsar etypicallydegenerate inzeroappliedeld.Moreover,whenamagneticeldisapplied, 4 antiferromagnetic magnonssplitintotwobranchesofopposingdispersion.[ 23 ]Ferromagneticmagnonsare non-degenerateeveninzeroappliedeldandhaveazeroeldinterc eptatzeroenergy.[ 24 ] Magnetoelectriccouplingintheinfrarediscommonlyobservedviathe magnetodielectric eect(achangeinthedielectricconstantinducedbytheonsetofs pontaneousmagnetization). Thedielectricconstant,whichissensitivetotheelectricnatureofa material,canbe estimatedfromIRmeasurementsthroughcausalityrelationsando scillatormodeling(cf. Section 3.2.1 andSection 3.2.2 ).TbMnO 3 [ 25 ]andCu 2 OSeO 3 [ 26 ]bothexhibitanabrupt changeintheinfrared-extracteddielectricconstantcoincidingwit htheonsetofmagnetic order.Moreover,becauseasuddenchangeintheinfrared-extr acteddielectricconstant ofamaterialusuallystemsfromanumberofstronginfraredphono nanomaliesinthe temperaturerangeofinterest,ourdiscussionisextendedtoinclu dematerialsthatshow suchphenomena.CoO,[ 27 ]MnO,[ 28 ]andMnF 2 [ 29 ]allexhibitarenormalizationoftheir respectivephononparametersbelowamagneticorderingtempera ture. 5 2.2.2Electromagnons Thecouplingbetweenelectricandmagneticordersinmultiferroicand magnetoelectric materialshasrecentlymanifesteditselfinanewfundamentalinfrar edexcitationcoined \electromagnon."In2006,Pimenov etal .[ 30 ]rstobservedelectromagnonexcitations 4 Subtletiesinsplittingbehaviordependsontheorientationoftheext ernaleldwith respecttotheeasyaxisofmagnetization.[ 22 ] 5 Inallthreematerials,thephononrenormalizationissuspectedtoo ccurfrommagnetic orderingalone;however,onlyMnF 2 doesnotexhibitastructuraltransitioninthevicinity ofthemagnetictransitiontemperature. 25

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inthedielectricconstantsofmultiferroicTbMnO 3 andGdMnO 3 atanenergyof 3meV(SeeFigure 2-3 ).Electromagnonshavesubsequentlybeenidentiedinnumerous compounds.[ 31 { 34 ]Electromagnonswereoriginallypostulatedtoarisefromtheintera ction ofaspinwavewiththeacelectriceldofthelight;however,thecurr entviewofan electromagnonismuchmorenuanced.Conversely,alargebodyoflit erature(Raman,[ 35 ] neutron,[ 36 ]andinfrared[ 37 ])indicatesthatelectromagnonsareactuallyahybrid magnon-phononresonancemode.Withthismodieddenitioninmind, andconsidering electromagnonsareobservedinthedielectricconstantofamater ial,theresonancemode mustgainspectralweightfromadipole-activeexcitation,themainc andidatesbeing domainrelaxations,phonons,andelectronictransitions.Inother words,theopticalf-sum rule(oscillatorstrengthsumrule),givenby Z 1 0 1 ( ) d! = ne 2 2 m ; (2{4) statesthattheareaunderneaththeopticalconductivityisacon stantregardlessof temperature.Therefore,whenanelectromagnonappears,the spectralweightofanother existingfeaturemustdecreaseaccordingly.Thismethodofspect ralweighttransferhas becometheprimarymethodfordistinguishingelectromagnonsfrom theirseemingly identicalcounterpart,traditionalmagnons,whichdonotcontrib utetotheoptical conductivity.[ 38 ] Additionalfruitfulnessinthestudyofelectromagnonsrelatestot heeasewithwhich thelowfrequencyexcitationcaninduceconsiderablechangesinama terialsdielectric constantandthereforeindexofrefraction( p =n)fromdcuptoterahertzfrequencies. Theabilitytotuneeasilytherefractiveindexofamaterialallowsfort hedesignofanew generationofopticalswitchesandoptoelectronicdevices. 26

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Figure2-3.SpectraoftheelectromagnonsinTbMnO 3 andGdMnO 3 asmanifestedinthe real( 1 )andimaginary( 2 )partsofthedielectricfunction.Thespectraare obtainedwiththeacelectriceldofthelightorientedparalleltothe a axisof bothmaterials.Theelectromagnonexcitationhascharacteristicf requenciesof 23 3cm 1 and20 3cm 1 inGdMnO 3 andTbMnO 3 ,respectively.Figure reproducedfromRef.[ 30 ]. 27

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CHAPTER3 EXPERIMENTALMETHODOLOGY Thischapterwillcoverspecicspectroscopyinstrumentationand analytical techniquesutilizedtoacquireandprocessdatainthisdissertation. Theprinciplesof aFouriertransforminterferometerwillbediscussedalongwithsou rcesanddetectors. Theintricaciesofatwo-colorphotomixingsystemusedtoprobethe terahertzregion willalsobeilluminated.Finally,twomethodstoestimatecomplexrespon sefunctions (e.g.,complexdielectricfunction)fromthemeasuredrerectanceo rtransmittancewill bediscussed.Appendix B providesdetailsonacombinedrerectionandtransmission approach. 3.1Instrumentation 3.1.1FourierTransformInterferometry TheprincipalelementofaFouriertransforminterferometerisaMic helsoninterferometer. 1 AsshownschematicallyinFigure 3-1 ,aMichelsoninterferometerutilizesabeamsplitter (BS)topartitionequalamountsoftheincomingradiationtowardsm irrorsM1(xed)and M2(movable).Themovablemirrorscansbackandforththroughth ezeropathdierence position(ZPD)andthelightisrecombinedattheBS;aportionofther ecombinedlight travelstowardsthedetector.InFigure 3-1 theZPDpositionoccurswhenbothmirrorsare adistance\d"awayfromtheBS.Infraredradiationistypicallymodu latedbythemovable mirrorinthekHzfrequencyrange. Theprinciplesofaninterferometerarebestdevelopedbyrstcon sideringa monochromaticsourceoffrequency (inwavenumbers)andadisplacementofthemovable mirrordenoted x .(Theopticalpathdierenceisgivenby=2 x becausethelighttravels 1 Foradetailedintroductiontothistechnique,seeforexampleRef.[ 39 ]andRef.[ 40 ] 28

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rn Figure3-1.AschematicofaMichelsoninterferometer,whichconst itutestheprinciple elementofaFouriertransforminterferometer. outandbackthedistance x .)Theintensityatthedetectorisgivenby, I det ( x )= 1 2 I source [1+cos(2 x )] : (3{1) When x =0(theZPDposition),lightfromthetwoarmsoftheinterferomete rinterfere constructivelyandtheintensityatthedetectorismaximum.As x deviatesfromzero,the intensityvariessinusoidally,reachingaminimumwhen x issuchthattheargumentofthe cosinetermisanoddintegermultipleof .Becauseweareconcernedwiththedynamics ofaninterferometer,letusonlyconsiderthesecondterm(oscillat oryterm);namely, I 0 det ( x )= 1 2 I source cos(2 x ) : (3{2) 29

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The 1 2 prefactorstemsfromtherealizationthatanidealbeamsplittertr ansmits(rerects) onehalfoftheincidentlight.Therefore,onlyonehalfofthelightfro mthesourcewill impingeonthedetector,withtheotherhalfdirectedbacktowards thesource. 2 Inreality lossesareincurred(e.g.,unwantedabsorptioninbeamsplittersand lightrerectedfrom mirrorsthatdonotproducephaseshiftsequalto180 ,whichinprinciplearefrequency dependent).Inlieuofthisletusreplacetheprefactor 1 2 I source byatermoflesservalue, denotedK( ).Thesignalatthedetectorbecomes S ( x )= K ( )cos(2 x ) : (3{3) Atthispointwecanextendourformalismtothepracticalcaseofab roadbandsourceby integratinginfrequencyspacefromzerotoinnity, S ( x )= Z 1 0 K ( )cos(2 x ) d: (3{4) Fromthesignalasafunctionofretardation,wecanobtainthesign alasafunctionof frequencybyapplyingaFourierTransform: K ( )= Z 1 1 S ( x )cos(2 x ) dx: (3{5) (Notethatwehaveextendedourfrequencyintervaltobesymme tricin x ,thusfacilitating thecomputation.) Inpracticedisplacingthemirroraninnitedistanceoneithersideoft heZPDis unachievable.Thelimitsofintegrationbecomeplusandminusthemaxim umdistance thatthemirrormoves,denotedas x max .Thecorrespondingopticalpathdierence(OPD) is max =2 x max (lighttravelsoutandback).TheOPDisinverselyproportionaltoth e 2 Inpracticeinterferometersareslightlymisalignedonpurposesoth atthelightleaving theinterferometerthatisnotdirectedatthedetectordoesnot modulatethesource. 30

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spacingbetweenstatisticallysignicantdatapointsinthespectrum (resolution 3 )bythe relation =1/2 x max Forcomputationalhandlingoftheinterferogramonemustconver tthecontinuous intensity,measuredasafunctionofOPD,toadiscretesumofvolta gesatspecicvalues of x viaananaloguetodigitalconverter(ADC)board.Themethodofex tractingdiscrete valuesfromacontinuousresponsefunctionistermedsampling.Sam plingmotivates adiscussionoftheNyquistcriterion.AccordingtoNyquist 0 stheorem,thesampling frequencymustbeequaltoorlargerthantwicethebandpassoft hemeasurement ( f samp 2 f max ).IftheNyquisttheoremisnotsatised,thenaliasingoccurswher ethe outlyingfrequenciesarefoldedbackintotheassumedfrequencyd omainresultingin thecreationofunwantedspectralartifacts.Therefore,critic alsampling(attheNyquist frequency)necessitatestheneedforecientanalogueltersto cutsignalbeyondthe samplingrange. Asecondissue,apropossampling,relatestotheaveragingofmultip leinterferograms inordertoimprovethesignal-to-noiseratioofthemeasurement.T obecollectively averaged,allinterferogramsmustbesampled(ordigitized)onthe samegridofdata points(i.e.,samevaluesofretardation, x ).Thisfeatisaccomplishedbyutilizingasecond referenceinterferometer.Thedetailsofthereferenceinterfe rometervaryfordierent instruments,soatthistimewewillrestrictthediscussiontoaBruke r113vFTIR.Here thereferenceinterferometerisseparatefromthemaininterfer ometerbututilizesthesame movingmirror.AwhitelightsourceandaHeNelasersource( HeNe =15798cm 1 )are passedthroughthereferenceinterferometer,resultinginasec ondinterferogramanda sinewaveattheirrespectivedetectors.Theintensityofthethre esources(IRthroughthe 3 Otherparameterssuchasthesourceapertureandthefocallen gthofmirrorsinthe interferometeralsolimittheresolution,butfurtherdetailsarebey ondthescopeofthis discussion. 31

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maininterferometerandWLandHeNethroughthereferenceinter ferometer)fromthe twointerferometersasafunctionoftime(opticaldistanceisdete rminedbythemirror velocity)aredepictedinFigure 3-2 .TheinterferogrampeakfromtheWLsourcealways precedesthemaininterferogrampeakfromtheinfraredsource. Simplyput,whentheWL interferogrampeakoccurs,athresholdisexceededthatsignalst heADCboardtobeing sampling.Thereafter,everyzerocrossing(ormultipleof)oftheH eNesinewavetriggers thesamplingofthemaininfraredinterferogram.Theunwaveringpo sitionoftheWLpeak withrespecttothemaininterferogrampeakcombinedwithastablela sersourceequates torepeatedinterferogramssampledatthesameretardation. BeforetheFouriertransformcanbecomputed,theinterferogr ammustbemodied toaccountforseveralfactors.First,theinterferogrammust betruncatedbyapiecewise functionthatisrealandpositiveintheintervalx max to x max andzeroeverywhereelse. (Inpracticethetruncationfunctionisnotsymmetricabout x =0becauseonlyhalfofthe interferogramlineshapeiscollected.)Asimpleboxcarfunctionthatt akestheform f ( x )= 8><>: 1 j x j x max 0 j x j >x max (3{6) satisesthiscriterion.However,thisboxcarfunctionprovestob einsucientforanother reason;itsFouriertransformisthesincfunctionwhichhas\sidelob es"or\feet"that dropo22%belowzerooneithersideoftheprincipalmaximum.Torem edy,theboxcar functionisthenreplacedbyanapodizationfunction,whichservest osuppresstheside lobesbuthasthedrawbackofbroadeninglines,thusdecreasingth eresolutionofthe measurement.AcommonlyusedapodizationwasderivedbyNortona ndBeerandconsists ofspecializedsumsofpolynomialfunctions: W ( x )= x max X m =0 m (1 x 2 ) m ; (3{7) 32

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Figure3-2.ThesamplingmethodologyofaBruker113vFTIR.Whent hewhitelight interferogram(goldline)reachesthethresholdV trig samplingcommences.The samplingofthemaininterferogram(blueline)occursateveryzeroc rossing (dottedblackline)oftheHeNelaserinterferogram(redline).Ther elative scalingofintensityinthisplothasnotmeaning. subjecttotheconstraintthat x max X m =0 m =1 : 0 : (3{8) AnotherapodizationfunctionistheHapp-Genzelfunctionwhichta kestheform W ( x )=0 : 54+0 : 46cos x x max : (3{9) TheHapp-GenzelfunctionhasacorrespondingFouriertransfor mgivenby, F.T.[ W ( x )]= sin2 x max 2 1 : 08 + 0 : 46 x max =! 0 : 46 x max = 2+ # : (3{10) 33

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Second,aphasecorrectionmustensuetoaccountfornumerous factorsthatactto distorttheinterferogram:o-axisraysintheinterferometer,e rroneousdetermination oftheZPDposition,andfrequencydependentnoisecomingfromele ctricalprocessing components.Tocompensatefortheerror,wecanintroduceaph asetermonthecosine leadingtotheequation S ( x )= Z 1 0 K ( )cos(2 x O ) d; (3{11) with O = A + B + C 2 + D 3 : (3{12) Whenhigherthanrstordertermsexist,theinterferogramissaid tobe\chirped"and thephasehastobecalculatedandsubsequentlyremovedfromthe spectrum.Thephaseis usuallycalculateddirectlyfromtheinterferogram, O =arctan Im ( K ( )) Re ( K ( )) : (3{13) Twomethodologiesofphasecorrectionaredoublesided,wherethe phaseiscalculatedfor allpointsintheinterferogram,andsinglesided,wherethephaseisc alculatedforhalfthe interferogramandthenmirroredtoformtheotherhalf.3.1.1.1Detectors Infrareddetectorsaregroupedintotwocategories:photonde tectorsandthermal detectors.Photondetectorsrelyonphoto-excitedchargecar rierschangingtheelectrical propertiesofthesensingchip.Thesensingchipsaretypicallysemico nductorswith abandgapinthevicinityofthelowenergysideofthefrequencyband thatisbeing measured.Thefrequencyresponseofaphotondetectorisnotc onstant.Lightincidenton aphotondetectormustbeelectricallyormechanicallychoppedtoallo wforelectronhole recombination.Timeconstantsforrecombinationvarybutaregen erallyontheorderof nanoseconds.Typicalphotondetectorsutilizedintheinfraredinc ludeHgCdTe,InGaAs, InAs,andPbSe. 34

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Ontheotherhand,thermaldetectorssensetheheatoftheimpin ginginfrared radiation.Thetemperaturechangecausedbyheatingaltersthes ensingmaterials properties(e.g.,thermoelectricvoltageorelectricalconductivity ).Thermaldetectors typicallyoperateatlowtemperaturestoavoidspuriousructuation sinthethermal propertiesoftheenvironment.Onesuchthermaldetectorutilize dforthemajorityofthe infraredmeasurementsinthisdissertationisaliquid-helium-cooled(L HC)bolometer. LHCbolometersconsistofasemiconductingsensingchip(typicallyge rmaniumorsilicon dopedwithgalliumorphosphorus)cooledatorbelow4.2K.Thermalde tectorsalsoneed timeforrelaxationofthesensingchip 0 sproperties.Thetimeconstantofabolometer detector,givenbytheratiooftheheatcapacityofthechiptoitst hermalconductance withthebath,isaround1ms.Wethereforechopthelightatafrequ encyequivalenttoa fewmultiplesofthetimeconstant.3.1.1.2Sources Sourcesintheinfraredcanalsobepartitionedintotwocategories: continuouswave (CW)sourcesandintermittentsources.ThemostcommonCWsour cesofinfrared radiationarehigh-temperatureblackbodies,whichcontinuouslyem itphotons.Thework inthisdissertationemployedmultipleblackbodysourcestoachievesp ectralcoverage fromthefarinfraredtotheultraviolet(10{50000cm 1 or2meV{8eV).Ahighpressure mercuryarclampwithaneective 4 operatingtemperatureof2000Kcoveredthe far-infrared(10{400cm 1 or2{50meV).Aglobarlamp,whichconsistsofasiliconcarbide rodheatedto1200K,wasusedtocoverthemid-infraredrange(4 00{5000cm 1 or 50{625meV).Thenearinfraredandvisibleranges(5000{50000cm 1 of0.6{8eV)are spannedbytungstenbasedlampsandvariousarclamps(deuterium andxenon). 4 Radiationfromthemercuryplasmaissupplementedbyradiationfrom theglowinghot quartzcasingofthearclamp. 35

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IntermittentsourcesarelessadvantageousthanCWsourcesfo rbroadbandspectroscopy (e.g.,onehastofactorintherepetitionrateofthesourcewhench oosingachopping frequencyforthedetector);however,thedrawbackscanbec ompensatedforbyhigher intensities,widelytunablephotonenergies,andmicrometersizedfo calspots.One suchintermittentsourceutilizedthroughoutthisdissertationissy nchrotronradiation. Synchrotronradiationisreleasedbyelectronsastheychangethe directionoftheir accelerationvector(viabendingmagnets)whiletravelingaroundac ircularstoragering atrelativisticspeeds.Synchrotronradiationexhibitsanintrinsicfr equencyattheheight ofitsbroadspectraldistribution,whichisbestdescribedbyamodi edBesselfunction. 5 Electronsdonotcontinuouslycirculatethestorageringofasynch rotron,butrathertravel inbunches(likecarriagesonacarouselwheel)thuscreatingtheint ermittentnatureofthe source. Adiscussionofpolarizationeectsarisingfromsynchrotronradiat ionisofinterest sincemostopticalmeasurementsinthisdissertationnecessitated linearlypolarizedlight. Synchrotronstorageringsaretypicallythesizeofamoderndaysp ortsarenaandoriented paralleltotheearth 0 ssurface(hereafterhorizontal).Becauseanacceleratingelect ron emitsradiation,withtheelectriceldoftheradiationinthedirectiono ftheelectron 0 s accelerationvector,theintrinsicpolarizationofthesynchrotron lightishorizontal. However,fromtheperspectiveofanobservereitheraboveorbe lowthestoragering, therearealsocircularlypolarizedcomponentsnoticeablethatwould otherwisecancelout whenobservationpointisinthehorizontalplaneofthering.Whenth eobservationpoint isnotentirelyinthehorizontalplane(thecaseformeasurementsin thisdissertation), thedominantcircularlypolarizedportionoftheradiationleadstoave rticalcomponent 5 Thebroadfrequencydistributionofsynchrotronradiationstems fromtheFourier transformoftheshortpulsesofelectriceldinthetimedomainfrom theelectronbunches circulatingthering. 36

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totheoverallpolarizationofthelight.Moreover,foraxedobser vationpointnotin thehorizontalplane,theratiooftheintensitiesofhorizontaltov erticalcomponentsof thelightisnotconstantbutratherdependsonthewavelengthoft helight.Briery,the wavelengthdependencestemsfromadiractioneectoccurringa ttheverybeginningof thebeamlineinwhichthesolidanglecapturingthesynchrotronradiat ionbecomessmaller aswavelengthsincrease(frequencydecreases).Therefore,t herstpartofthebeam 0 s proletobe\trimmed"owillbethetopandbottom,whichconstitut ethecircularly polarizedlightthatgivesrisetothesmallverticalcomponent.Since verylongwavelengths weremeasuredinthisdissertation,thehorizontalpolarizationoft helightbelow100cm 1 forthespecicsynchrotronbeamlineutilized,U4IR(NSLS,Brookh aven),isshownasa functionofincreasingwavelengthinFigure 3-3 Inlieuofthisunusualpolarizationdependence,awiregridpolarizerw asplaced beforethesampletoforcethepolarizationofthelighttobecomeco mpletelyhorizontal. Tochangethegeometryofthepolarization,onemustrotatethes ample(asopposedtothe commonmethodofrotatingthepolarizerwhenblackbodysourcesa reutilized). 3.1.2TerahertzTwo-colorPhotomixingSystem Aterahertz(THz)two-colorphotomixingsystemattheUniversity ofWollongong, Australia,wasusedtomeasurelowerphotonenergies(lowerfrequ encies)thanthose attainablebythesourcesandtechniquesofconventionalFourier transforminfrared spectroscopy.TheTHztwo-colorphotomixingsystemwasinterfa cedtoasuperconducting magnet,thusrepresentinganovelanduniqueexperimentalsetu p.Itmeritsadetailed explanation. ATHztwo-colorphotomixingsystemconsistsoftwotunablenear-in fraredlaser diodesadjustedtohaveadierencefrequencyontheorderof1T Hz.Thelasersare coupledineitherfreespaceorviaanopticalber,andfocusedont oaphotomixer.The photomixerconsistsofalow-temperaturegrown(LTG)GaAswafe rontowhichapattern ofsixinterdigitatedelectrodengersaredepositedbylithographic techniques.The 37

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n n n n n r Figure3-3.Thehorizontalpolarizationpercentageofradiationbe low120cm 1 at beamlineU4IRoftheNSLS. waferissupportedbyaGaAssubstrate.Animageoftheinterdigita tedpatternand correspondingantennageometryisshowninFigure 3-4 .Electron-holepairsgeneratedin theLTGGaAsbyphotonabsorptionareacceleratedtowardsthee lectrodeswhichhavea voltagebiasacrosstheminthemannerseeninFigure 3-4 .Duetothehighcharge-carrier mobilityandsub-picosecondrecombinationtimeoftheLTGGaAs,con ductionband bendingatthemetal-semiconductorinterfaceisnegligible,meaningt heelectrical propertiesofthephotomixermaymostlyberepresentedbyaphot oconductancethat isafunctionoftheabsorbedopticalpower.[ 41 ]Therefore,thedominantphotocurrents inthephotomixer 0 sngersandhenceantennasaregeneratedatthesumanddiere nce 38

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Figure3-4.Imageofthetwo-colorphotomixerutilizedattheUniver sityofWollongong, Australia.Toppanel(a)depictsthegeometryofthegoldantenna pattern. Bottompanel(b)isanenlargedviewoftheseparationbetweenthe two antennas. frequenciesofthenear-infraredpumplasers.Byuseofanoptica llter,thedierence frequencyisselectedoutandcoupledintofreespace,producingC WTHzradiation. 3.2Analyticaltechniques 3.2.1LorentzModel ThissectionderivestheformalismoftheLorentzoscillatormodel.Th eLorentz oscillatormodelisapplicabletodescribetheopticalbehaviorofinsula tors,anditisthus utilizedmultipletimesinthisdissertation. Themodelisbasedonaclassicalharmonicoscillatorthatissimultaneo uslydamped anddriven.Inthemodel,electronsareviewedasbeingboundtoato mswithtinysprings, 39

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andthedampingforcearisesfromvariousscatteringmechanisms( e.g.,phononsand electron-electroninteractions).Theoscillatorydrivingforceont heelectroncomesfrom theacelectriceldofthelight.Theforceonanelectroniswrittenas m x = m! 2 0 x mr x + F ext : (3{14) Theforcefromthedrivingeldis F ext = e E 0 e i!t e i q r : (3{15) Atthispointtwoassumptionsmustbemade.First,weignorethespa tialdependence oftheeld, e i q r ,byrestrictingourselvestowavelengthsoflightthataremuchgre ater thantheradiusofanatom( a 0 ).Second,themicroscopiceld E loc issetequalto themicroscopiceld E .Ingeneral E loc 6 = E ;however,solvingfortheexactrelationshipis anextremelyformidabletask,andtheaforementionedassumption turnsouttobequite robustinpreservingtheessentialfeaturesnecessarytodescr ibetheopticalpropertiesofa material. With x = x 0 e ( i!t ) ourrelationbecomes m! 2 x 0 i!mr x 0 + m! 2 0 x = e E : (3{16) Thefactorizedexpressionifgivenby x 0 = e=m 2 0 2 i!r E : (3{17) Thedisplacementoftheelectronwithrespecttothedrivingeldcan begroupedinto threecharacteristicregions.As 0,thedisplacementisapproximatelyin-phasewith thedrivingeld.Attheresonancefrequency, = 0 ,thedisplacementis90 out-of-phase withthedrivingeld.Inthelimit !1 thedisplacementis180 out-of-phasewiththe drivingeld. 40

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Assumingthedisplacement x issucientlysmallsothatalinearrelationshipbetween dipolemomentandelectriceldisaccurate,wecanwritethat p = e x 0 p E : (3{18) Theatomicpolarizibilityisthengivenby = e 2 =m 2 0 2 i!r : (3{19) Thecomplexdielectricfunctionforasystemofnelectronstakesth eform =1+4 n p =1+ 4 n e 2 m 2 0 2 i!r : (3{20) Lettheplasmafrequencyofboundelectronsbedenedas p = q 4 ne 2 m andconsiderthat anumber( j )ofLorentzianoscillatorsareneededtomodelaccuratelymostins ulating materials.TheLorentzdielectricfunctionthenbecomes, = 1 + 1 X j =1 2 pj 2 j 2 i!r j : (3{21) Notethatwehavereplaced1withthevalue 1 1 encapsulatesalltheabsorption processesatfrequencieshigherthanthosemeasuredinatypical experiment.Ifallthe absorptionprocessesaremeasuredandmodeledbyLorentzianos cillators(typically needingdatauptox-raywavelengths)then 1 takesthevalueofunity. 3.2.1.1Fittingprocedures Opticalmeasurementsinthisdissertationweremodeledwiththediele ctricfunction giveninEq.( 3{21 ).EachindividualLorentzoscillatorisdenedbyaresonancefrequ ency 0 ,alinewidth r ,andastrength p .Aseriesof j Lorentzoscillatorswereinitiallytto theexperimentalspectrabysight,andthenfurtheroptimizedby aleastsquarestting routinedevelopedbyBevington.[ 42 ]Figure 4-7 ,Figure 5-6 ,andFigure 6-4 showresults ofthettingroutineandoptimization.Withanaccuraterepresent ationofthecomplex dielectricfunctionfromtheLorentzmodel,itisthenonlyamatterof algebratoarrive 41

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atexpressionsforotherfrequency-dependentcomplexrespon sefunctions(e.g.,optical conductivityandrefractiveindex).3.2.2KramersKronig Kramers-Kroniganalysisisasecondmethodtoestimatethecomplex dielectric function(andothercomplexresponsefunctions)fromthemeasu redrerectanceor transmittance.Inshort,Fouriertransforminfraredspectros copy(FTIR)measures powerrerectanceandtransmittance;therefore,noinformatio naboutthephaseofthe lightisobtained.Thephaseistheimaginarypartofeitherresponse, anditisnecessaryto calculatethecomplex(realplusimaginary)dielectricfunction.Kram ers-Kronigrelations, viacausality,provideamethodtoestimatetheimaginarypartofare sponsefunctionwith knowledgeoftherealpartandviceversa. TodeveloptheKramers-Kronigformalismletusbeginbydeningalinea rresponse function.Considertherelation X ( t )= Z 1 1 G ( t t 0 ) f ( t 0 dt 0 ) ; (3{22) where X ( t )representstheresponseofasystem(e.g.,conductivity), f ( t )istheexternal stimulus(e.g.,adrivingelectriceld),and G ( t )istheresponsefunction(e.g.,susceptibility). Itshouldbenotedthatweareneglectingthespatialdependence, r,inthisdiscussionby restrictingourselvestowavelengthsmuchlongerthanthelengths calesofthemicroscopic interactionsgivingrisetosuchresponses.InEq.( 3{22 )weseethattheresponseis eectedbyapreviouslyappliedstimulus,similartothewaytheinstant aneousspeed ofafreefallingobjectdependsonthedurationofthefreefall.Ont heotherhand,the principleofcausalitystatesthattheresponsefunction, G ( t ),cannotdependontwhentis afuturetime: G ( t t 0 ) 0 ;t
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Figure3-5.Asdescribedinthetext,theKramers-Kronigintegral onlynecessitates integrationintheupperhalfofthecomplexplane.Segments1and3a re principalvalueintegralsandsegment2canbeevaluatedanalytically. frequencies = 1 + i! 2 G ( )= Z 1 1 G ( t t 0 ) e i! 1 ( t t 0 ) e 2 ( t t 0 ) dt: (3{24) Theterm e i! 1 ( t t 0 ) isboundedforallfrequenciesinthecomplexplane.Theterm e i! 2 ( t t 0 ) isonlyboundedwhenitsargumentisnegative;therefore,when 2 isnegativetheterm ( t t 0 )mustalsobenegative.Recallingourcausalityrelation,itfollowsthat G =0 when 2 isnegative,thusrestrictingourintegrationofEq.( 3{24 )totheupperhalfofthe complexplane(SeeFigure 3-5 ).NextweuseCauchy 0 sresiduetheoremfortheintegration oftheanalyticfunction G ( )intheupperhalfplane,namely, I G ( ) 0 d! =0 : (3{25) Theintegralaroundtheclosedcontourcanbebrokenintofourse gmentsasshownin Fig. 3-5 .Thecontributionofsegment4isdisregardedbecause G ( ) zeroas 43

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innity.Segments1and3areprincipalvalueintegrals(denotedby P )alongthereal axis,andsegment2canbeevaluatedanalyticallyfortheresults iG ( 0 ).Theresultof Cauchy 0 stheoremthenbecomes, iG ( 0 )= P Z 1 1 G ( ) 0 d!: (3{26) Nowwecansplit G ( )intoitsrealandimaginarycomponentstoobtain, Re [ G ( 0 )]= 1 P Z 1 1 Im [ G ( )] 0 d!: (3{27) Im [ G ( 0 )]= 1 P Z 1 1 Re [ G ( )] 0 d!: (3{28) Wehavenowprovedthattherealandimaginarypartsoftherespo nsefunction G ( )are notindependentbutratherconnectedviaKramers-Kronigrelatio ns. TheKramers-Kronigrelationsofinterestinthisdissertationconne ctthererectance, r ,andthephase, .Thecomplexrerectanceisgivenby r = j r j e i : (3{29) wherethepowerrerectance, j r 2 j ,isthetypicalmeasuredquantity.Takingthenatural logarithmofEq.( 3{29 ),wecanseparatetherealandimaginarypartsofthecomplex rerectance.DerivingtheKramersKronigrelationsforalogarithmic functionisa challengingtaskbecause j ln r ( ) j!1 as j j!1 .Foracompletederivation theinterestedreaderisreferredtoaveryinformativediscussion oftheKramers-Kronig relationswrittenbyFredrickWooten.[ 43 ]TheresultingKramers-Kronigrelationbecomes ln r ( )= 2 Z 1 0 ( ) 0 ( 0 ) 2 2 0 d!: (3{30) ( )= 2 Z 1 0 ln j r ( ) j ln j r ( 0 ) j 2 2 0 d!: (3{31) Inpractice,sinceweonlymeasuredaniteintervaloffrequencies ,extrapolationsmust beemployedtoextendthemeasuredrerectancetowardsthelimits ofintegrationin 44

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Eq.( 3{30 ).Belowourlowestmeasurablefrequency,thecalculatedrerecta ncefrom Lorentzoscillatormodelingofourdataisusedtoextendedtowards zerofrequency.At highfrequencies,twoavenuesopenforuse:eitherpowerlawextr apolationsappropriate forthesuggestedbehaviorofopticalconstantsareused,orxrayrerectancespectraare estimatedfromtheconstituentatomsandbridgedtothemeasure drangewithasingle Lorentzoscillator. 6 3.2.3SumRules SumrulescanbederivedfromKramers-Kronigrelationsandareimpo rtanttools foranalyzingopticaldata.Oneparticularsumrulestatesthatthe spectralweight(area undertherealpartoftheopticalconductivity)isconserved.Th isruleessentiallystates theconservationofchargeandisexpressedas Z 1 0 1 ( ) d! = 2 p 8 = 2 ne 2 m : (3{32) Thissumruleisoftenappliedtounderstandchangesintheopticalco nductivityofa materialwithrespecttotemperature.Awell-knownexampleisthelo ssofspectralweight atlowfrequencywhenasuperconductinggapopens.Themissingsp ectralweightinthe superconductingstateispushedintoadeltafunctionatzerofreq uency,whichrepresents theresponseofthecondensate. Infrared-activephononsoftenexhibitdynamicalbehaviorintheo pticalconductivity withrespecttotemperature.Atlowtemperaturesthephononst ypicallyresemblesharp Lorentzianoscillators.Astemperatureincreases,thelinewidthof atypicalphonon broadens.Onecanoftenextendtheabovesumruletoasinglephon on.Ifthephononis notcoupledtoanotherphonon,transition,orbroadcontinuum,t henitsareashouldbe conservedwithtemperature.Thismethodologyhasbeenrecently extendedtoidentifythe formationofelectromagnons,whichhavetogainspectralweighta ttheexpenseofexisting 6 FordetailsonthelatterapproachseeSection 5.3.4 45

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dipoleactivetransitions.Electromagnonsareusuallyobservedtob orrowspectralweight fromanearbyphonon. 46

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CHAPTER4 MAGNETODIELECTRICCOUPLINGINCu 2 OSeO 3 4.1Overview Themagnetodielectriceectreferstoachangeinthedielectriccon stantinducedby anexternalmagneticeldorbytheonsetofspontaneousmagnet ization. HerewereportourinfraredstudiesonasinglecrystalofCu 2 OSeO 3 ,apiezoelectric withaferrimagnetictransitiontemperatureof T c 60K.[ 44 ]Aperviousstudyon Cu 2 OSeO 3 byBos etal. [ 44 ]reportedamagnetodielectriceect(anomalousjumpin dielectricconstant)attheferrimagnetictransitiontemperature ,asobservedthrough dielectriccapacitancemeasurements.Althoughwefoundnodrast icanomaliesacross T c ,athoroughinspectionofthedatacombinedwithsomemodelingleadu stoa magnetodielectriceectaswell.RecentlyGnezdilov etal. [ 45 ]havepresentedaRaman studyofCu 2 OSeO 3 preparedinthesamewayasourcrystal.Theyobservedtheabrup t appearanceof3newlinesinthespectrauponcoolingbelow T c ,andanadditional2lines thatappearedbelow20K.Gnezdilov etal. alsogaveadetaileddescriptionofthecrystal andmagneticsymmetryofthiscompound. CrystalStructure: EenbergerandPertlik[ 46 ]solvedthecrystalstructureusing single-crystalX-raydiraction.Thecompactcrystalstructur econsistsofthreebasic buildingblocks,squarepyramidalCuO 5 ,trigonalbipyramidalCuO 5 ,andalonepair containingtetrahedralSeO 3 unit.Theoxygenatomsintheunitcellaresharedamongst thethreebuildingblocks.ThesquarepyramidalCuO 5 unitsexistina3-to-1ratioto thetrigonalbipyramidalCuO 5 unitswithintheconventionalunitcell.Thisratiowill beimportantsubsequentlywhenexplainingthemagneticstructure .Allcopperions ReprintedwithpermissionfromK.H.Miller,X.S.Xu,H.Berger,E.S. Knowles,D. J.Arenas,M.W.Meisel,andD.B.Tanner,Phys.Rev.B 82 ,144107(2010). 47

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possessa+2oxidationstate.Moredetaileddescriptionsofthecry stalstructurearefound elsewhere.[ 44 47 ] ThematerialcrystallizesintheP2 1 3cubicspacegroupandhasbeenshownto remainmetricallycubicwithnoabnormalchangeinthelatticeconstan tthroughthe Curietemperatureanddownto10K.[ 44 ]Theonsetofmagneticorderdoeshavethe eectofreducingthecrystalsymmetrytoR3.Fullcubicsymmetr ywouldrequireall copperionstofeelthesameCoulombinteractionfromnearestneig hborcopperspins.The proceedingexplanationisthecaseforferromagnetismandantifer romagnetismbutnot forferrimagnetism,whichiswhyareductionfromcubicsymmetrymu staccompanythis ordering. 4.2ExperimentalProcedures SinglecrystalsofCu 2 OSeO 3 weregrownbyastandardchemicalvaporphase method.MixturesofhighpurityCuO(Alfa-Aesar,99.995%)andSeO 2 (Alfa-Aesar, 99.999%)powderinmolarratio2:1weresealedinquartztubeswithelec tronicgradeHCl asthetransportgasforthecrystalgrowth.Theampouleswere thenplacedhorizontally intoatubulartwo-zonefurnaceandheatedslowlyby50 C/hto600 C.Theoptimum temperaturesatthesourceanddepositionzonesforthegrowth ofsinglecrystalshave beenfoundtobe610 Cand500 C,respectively.Aftersixweeks,manydarkgreen, indeedalmostblack,Cu 2 OSeO 3 crystalswithamaximumsizeof8 6 3mm 3 were obtained.X-raypowderdiraction(XRD)analysiswasconductedo naRigakuX-Ray diractometerwithCuK radiation( =1 : 5418 A).Anelectronmicroprobewasusedfor chemicalanalysisofallsolidsamples. Thetemperaturedependent(5{300K)rerectanceandtransmit tancemeasurements employedaBruker113vFourierTransforminterferometerinconj unctionwithahelium CrystalgrowthwascarriedoutbyHelmuthBergerattheEPFLinLa usanne, Switzerland 48

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cooledsiliconbolometerdetectorinthespectralrange30{700cm 1 andwithanitrogen cooledMCTdetectorfrom700{5,000cm 1 .Roomtemperaturemeasurementsfrom 5,000-40,000cm 1 wereobtainedwithaZeissmicroscopephotometer.Aftermeasurin g thebulkrerectanceovertheentirespectralrange,acrystalw aspolishedtoathickness of194 mfortransmittancemeasurements.Allmeasurementswereperf ormedusing unpolarizedlightatnear-normalincidence,withtheelectriceldoft helightinthe h 111 i crystalplane.Thecubicnatureofthematerialforbidsanisotrop yintheopticalspectra. MagneticmeasurementswereperformedinacommercialSQUIDmag netometer(Quantum DesignMPMS-XL7)onasinglecrystalsamplemountedwiththe[111]a xisparalleltothe appliedeld.Aftercoolingthesampleinzeroeldto50K,magnetizatio nwasmeasured inanappliedeldof10Gwhilewarmingto70K.Thedcsusceptibilitywasca lculated inthelow-eldlimitas ( T )= M ( T ) =H .Inaddition,theisothermalmagnetizationasa functionofappliedeldwasmeasuredatatemperatureof2K,whiles weepingtheeld fromzeroto2kGandbacktozero. 4.3ResultsandAnalysis 4.3.1Magnetism Recentstudieshavemeasuredthemagneticsusceptibilityofpowde redsamples Cu 2 OSeO 3 ,ndingorderingtemperaturesof T inrection c =55K[ 48 ]and T onset c = 60K.[ 44 ]Becauseanomaliesintheinfraredspectrumatthetransitiontemp erature areimportant,anaccuratedeterminationof T c forthesinglecrystalofinterestwas desired.Themeasureddcsusceptibilityasafunctionoftemperatu re, ( T ),isshownin Fig. 4-1 .Takingthetransitiontemperaturetobewherethesusceptibilityt urnsupward, T onset c =60Kisfound,consistentwiththeobservationsofBos etal. [ 44 ]At2K(Fig. 4-1 inset),wellwithintheorderedstate,themagnetizationsaturate sinaeldof800Gat 1.0 N B ,whichishalfoftheexpectedsaturationvaluefora S =1 = 2system,indicating aferrimagneticorderinginathree-upandone-downconguration .Nocoerciveeldwas 49

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n n n c r m #$ %& '( ) Figure4-1.Thedcsusceptibility, ( T ),ofCu 2 OSeO 3 neartheorderingtemperature ( T c =60K)inanappliedeldof10Gasmeasuredwhilewarmingafter zero-eldcoolingto50K.Theinsetshowstheisothermalmagnetiza tion, M(H),asafunctionofappliedeldat2K,wheretheeldwassweptup to 2kGbeforebeingreducedtozero.Inallinstances,thelinesconne ctingthe datapointsareguidesfortheeyes.Theschematicshowstheorien tationofthe singlecrystalwithrespecttotheeldappliedparalleltothe[111]dir ection. measured;however,aninrectionpointwithsomeslighthysteresisw asobservednear 400G,whichisalsoconsistentwiththendingsofBos etal. [ 44 ] 4.3.2RerectanceandTransmittanceSpectra Thetemperature-dependentrerectancespectrumofCu 2 OSeO 3 between30and1,000 cm 1 (4{120meV)isshowninFig. 6-2 .Astrongsharpeningofmanyphononmodes isobservedwithdecreasingtemperature.Itshouldbenotedthat therearenodrastic anomaliesinthefar-infraredspectrum,suchasthepresenceofn ewmodesorthesplitting ofexistingmodes.Becauseinfraredspectroscopyisextremelyse nsitivetochangesin dipolemomentsandforceconstants,theabsenceoftheseanoma liesgivesstrongsupport totheassertionofnolatticedistortionsat T c asinitiallydeterminedbyX-raydiraction measurements.[ 44 ] 50

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n n n r Figure4-2.Temperature-dependentrerectancespectrumofC u 2 OSeO 3 ThetoppanelofFig. 4-3 showsthe300Krerectanceupto40,000cm 1 .The onsetofelectronicabsorptionisindicatedbytheupturnaround26 ,000cm 1 (3.2eV). Thetransmittancespectra,asdepictedinFigure. 5-4 andFigure. 4-5 ,areingood agreementwiththererectancemeasurements.Atfrequenciesb elowthestrongphonon modes( < 80cm 1 ),thecrystaltransmits.Theperiodicoscillationsinthisrangeare theFabry-Perotinterferencefringesduetomultipleinternalrer ections.Transmission gapsopenbetweentheinfraredphononmodes.Theseregionsbec omeincreasinglymore evidentastemperatureisloweredandthemodessharpen.Thelowfrequencytransmission spectrumexhibitsaweakphononwitharesonancefrequencyof68 cm 1 thatrstappears asasmallstructurearound120Kandstrengthenswithdecreasin gtemperature.The modehasasmalloscillatorstrength,thusexplainingwhyitwasnotob servedinrerection. Theweakphononshowsnoresponsetoeldsof10Tappliedperpend iculartothe crystalsurface(insetofFigure. 5-4 ).Incontrast,thehighfrequencyspectra(Figure 4-5 ) possessestwosharpdips,1530and2054cm 1 ,whicharetoohightobesinglephonon peaks.Consequently,wearenotabletomakeanassignmentofthe sefeatures. 51

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r n r n Figure4-3.Theupperpanel(a)showsthe300Kbroadbandrerec tanceoutto40,000 cm 1 .Thelowerpanel(b)depictstheopticalconductivityatroom temperatureoverthesamefrequencyregion. 4.3.3Kramers-KronigAnalysisandOpticalProperties Kramers-Kroniganalysiscanbeusedtoestimatetherealandimagin aryparts ofthedielectricfunctionfromthebulkrerectance R ( ).[ 43 ]Beforecalculatingthe Kramers-Kronigintegral,thelowfrequencydatawereextrapolat edasaconstantfor 0asbetsaninsulator.Athighfrequenciesthererectancewasas sumedtobe constantupto1 10 7 cm 1 ,afterwhich R ( ) 4 wasassumedastheappropriate behaviorforfreecarriers.Theopticalpropertieswerederivedf romthemeasured 52

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n n n r n n n n Figure4-4.Temperaturedependenttransmissionspectrumof19 4 mthickCu 2 OSeO 3 singlecrystalbelow100cm 1 .Thisregionishighlightedbyaweakphonon ( 68cm 1 )thatbeginstoappeararound120K.Theinsetshowsthe magneticelddependenceoftheweaklow-energyabsorptionwitha resolution of0.25cm 1 .Itseemstohavenoresponsetomagneticeld. rerectanceandtheKramers-Kronig-derivedphaseshiftonrere ction;inparticular,we estimatedtherealpartoftheopticalconductivity, 1 ( ). Figure 4-3 bshowstheopticalconductivityatroomtemperatureoutto40,00 0 cm 1 .Thelowabsorptionintheinfraredisconsistentwiththeinsulatingna tureofthe compound.Ourtemperature-dependentmeasurementsceased at5,000cm 1 .Accordingly, thedynamicsofinterbandtransitionswithtemperatureintheregio nofhighphoton energydonotcontributetoouroverallndings.Figure 4-6 depicts 1 ( )at20K(below T c )and70K(above T c )from30to1,000cm 1 .Theprincipaleectisasharpeningof mostmodes,yieldingalargerconductivityattheresonantfrequen cy. ThephononmodesinrerectanceappearasLorentzianlinesintheop ticalconductivity, makingthemintuitiveformodelingasharmonicoscillators.Thevanishin glysmallstatic limitof 1 ( )andthelowbackgroundlevelofconductivitythroughouttheinfr ared regionsisfurtherevidenceoftheinsulatingnatureofthecompoun d. 53

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n r Figure4-5.BroadbandtransmissiononthethinnedpieceofCu 2 OSeO 3 crystalat300K measuredouttothe40,000cm 1 4.3.4Oscillator-ModelFits TheDrude-Lorentzmodelwasusedtotthererectanceandobt ainasecondestimate ofthecomplexdielectricfunctionintheinfraredrange.Themodelc onsistsofadamped oscillatorforeachputativephononinthespectrumplusahighfrequ encypermittivity 1 thatdescribesthecontributionofallelectronicexcitations.Them odelhasthefollowing mathematicalform: = 1 X j =1 S j 2 j j 2 2 i!r j + 1 ; (4{1) where S j j ,and r j representtheoscillatorstrength,centerfrequency,andlinewid thof the j thdampedoscillator.Thecomplexdielectricfunctionprovidedbythe Drude-Lorentz modelisusedtocalculatethererectivity,whichagreeswellwithour originalmeasured quantity.Figure 4-7 comparesthecalculatedrerectivityandthemeasuredrerectivity at 70K.Similarqualitytswereobtainedatallothertemperatures. 54

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n n r r Figure4-6.FarInfraredopticalconductivityCu 2 OSeO 3 at20K(red)70K(black). 4.4Discussion 4.4.1MagnetodielectricEect Equippedwithoscillatorparameterstodescribeeachoftheinfrare dphononsatall measuredtemperatures,oneisnowinpositiontocloselymonitorthe subtledynamics ofthephononstructuresacrossthetransitiontemperature.D espitethelackofdrastic changesinthephononspectrumat T c ,itisworthwhileexaminingwhetheracombination ofmanysmallanomaliesinthephonondynamicsmightsumtogiveanove ralleect.At thispointitislogicaltoexaminethestaticdielectricconstantbecaus eitisthesumof parametersthatdescribethedielectricnatureofthecompound( thesidetowhichthe infraredismostsensitive).TakingthezerofrequencylimitoftheDr ude-Lorentzformula, onearrivesatthefollowingsimpleexpressionforthestaticdielectric constant: o = 1 X j =1 S j + 1 : (4{2) 55

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n n r n Figure4-7.Experimentalrerectance(bluepoints)andcalculated rerectance(redline) fromtheDrude-LorentzmodeloftheCu 2 OSeO 3 70Kdielectricfunction. Thecalculatedstaticdielectricconstantattemperaturesnear T c ,isshowninFig. 4-8 Thereisananomalousjumpat60K.Itshouldbenotedthatthismagn etodielectriceect inCu 2 OSeO 3 waspreviouslyobservedbyBos etal. [ 44 ]throughdielectriccapacitance measurements.Whilethevalueofthedielectricconstantmeasured hereisconsiderably smallerthanthatinthepreviousreport,thedirectionandmagnitud eoftheanomalyat T c areingoodagreement.Itisalsoworthwhiletonotethatthesystem aticchangesin rerectancenear T c overthemidinfraredband(1,000{5,000cm 1 )correspondedtochanges inthemidinfrareddielectricfunctionthatwereoneorderofmagnitu delessthanour magnetodielectriceect,thusidentifyingphononsasthemaincont ributortothiseect. 4.4.2AnomalousPhonons TheobservedphononsinCu 2 OSeO 3 maybedividedintotwocategories:conventional andanomalous.Conventionalphononsshowaslighthardeningofth eirfrequencieswhen cooledtolowtemperatures,accompaniedbyasignicantreduction inlinewidthandat mostmodestchangesinoscillatorstrength.Moreover,thetempe raturevariationissmooth withnosuddenchangesinslopeorvalue.Anomalousphononsviolateo neormoreof 56

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Table4-1.OscillatorparametersforCu 2 OSeO 3 (at20K)andtheircorrespondingassignmentsforafewstrongmo desinthe farinfrared. IndexOscStrengthCenterFreqLinewidthAnomalousAssignment S (cm 1 ) r (cm 1 )TypeA,B,C,orNo 10.00367.71.1No-20.50184.90.6NoVibrationofCuO 5 units 30.33791.20.6No-40.233100.70.5No-50.073126.10.8No-60.134147.51.2TypeA-70.544184.81.5TypeB-80.071204.82.4TypeC-90.521212.02.9NoSeO 3 vibratingagainstCuO 5 100.038273.95.5TypeC-110.123293.27.6TypeC-120.042311.76.2TypeB-130.107335.83.6TypeCInternalin-planevibrationofCuO 5 units 140.188385.52.0NoInternalbendingmodeofSeO 3 units 150.063399.43.6No-160.095437.45.9TypeCInternalout-of-planevibrationofCuO 5 units 170.007454.14.3No-180.445504.57.4No-190.224537.96.5TypeC-200.091551.33.8TypeC-210.094592.53.4TypeC-220.291717.115.8TypeBAntisymmetricstretchofSeO 3 units 230.024753.111.6No-240.054781.713.9TypeAAntisymmetricstretchofSeO 3 units 250.017813.86.6No-260.016831.35.1NoRadialbreathingmodeofSeO 3 units 57

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n nr Figure4-8.ThestaticdielectricconstantascalculatedfromtheDr ude-Lorentzmodelat temperaturesbetween20and100K,including T c theseexpectations.Wehaveidentied13anomalousphononsinour spectra,whichshow 26totalmodes.Anumberingschemehasbeenusedwhichcorrespo ndstothesequence ofappearanceofoscillatorsintheinfraredspectrumstartingwith #1(lowestfrequency mode)andendingwith#26(highestfrequencymode) Figure 4-9 displaysthetemperaturedependenceofthethreeoscillatorpara meters for3ofthe13anomalousphonons(#13,#19,and#20)intheinfrar edspectrum.The oscillatorstrength,representedby S j ,wascalculatedfromthespectralweightandcenter frequencyofeachphononusingtherelation S j =( pj =! oj ) 2 .Thebehaviorof S j directly resultsfromchangesin p .Theoscillatorparametersforoneconventionalphonon(#2) areshownforcomparison.Itwasobservedthatanomalousbehav iorusuallycouldbe foundinallthreeoscillatorparameters.4.4.3AssignmentofPhononModes Becausethemagnetodielectriceectisobservedthroughlatticed ynamics,wemake apartialanalysisofthephononspectrum.Thenumberofphononm odesexpectedin Cu 2 OSeO 3 canbefoundbyspacegroupanalysis.UsingtheSMODESprogram,[ 49 ]we 58

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0 100 200 300 Temperature (K) 84.6 84.7 84.8 Tc #2 334 335 336 #13 536 537 538 #19 547 549 551 #20 0 100 200 300 Center Frequency + (cm,1 )a 0 100 200 300 Temperature (K) 0.80 0.85 0.90 0.95 1.00 1.05Normalized Oscillator Strength, S j ( T ) /S j (20 K ) Tc b #2 #13 #19 #20 0 100 200 300 Temperature (K) 0 5 10 15Line Width (cm,1 ) c Tc #2 #13 #19 #20 Figure4-9.The(a)centerfrequency,(b)normalizedoscillatorst rength,and(c)linewidth offouroscillatorsasafunctionoftemperature.Oscillator#2isaty pical conventionalphonon,whereastheotherthreeoscillatorsshowa nomalous behaviorastemperatureisloweredacross T c arriveatthefollowingdistributionofmodes: optical =14 A ( R ) +14 E ( R ) +41 T ( R ; IR ) ; (4{3) where(R)and(IR)denoteRamanactiveandinfraredactivemodes respectively.We thereforehavethepotentialof41totalinfraredactivemodes, allofwhichpossess threefolddegeneracyasindicatedbytheirirreduciblerepresenta tion.However,only 26modesintheinfraredspectrumaredetected.Theoscillatorpar ameters(at20K)ofthe 26experimentallyobservedmodesarelistedinTable 6-1 .Basedonourlatticedynamical calculationofthepositionofthe41predictedmodesandtheexperim entallinewidthofthe 26observedmodes,wesuspectthatweakerphononsareburiedw ithinstrongerphonons andmergeinthespectrum.TheLorentzanalysisformergedphono nswouldresultinan averageofoscillatorparametersweightedbyoscillatorstrength. 59

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Morespecically,thelatticedynamicalcalculationemployedhereinar ebasedon areal-spacesummationofscreenedcoulombinteractionsinvolvinga sphericalcut-o boundary.[ 50 ]Frequency,modeintensity,aswellasdisplacementpatternwere calculated basedonthestructureandvalenceasreportedbyBos etal. [ 44 ]Thecenterfrequency, modeintensity,andspecicnatureofatomicvibrationofallcalculat edmodesarelisted inTable 4-2 .Wehaveindicatedgroupsofthecalculatedmodeswhereapotentia lmerging mayhaveoccurredintheexperimentalspectra.Wewerealsoablet oassigneightstrong modesintheinfraredspectrumbycomparingthecalculatedandexp erimentalspectra. Table 4-2 detailsthemodeassignmentsmadeusingtheadoptednumbering schemeandcenterfrequencyofeachoscillatorforidentication. Inaddition,wehave distinguishedbetweenoscillatorsexhibitinganomalousbehaviorinone parameter(type A),twoparameters(typeB),andallthreeparameters(typeC) .Itshouldbenotedthat oscillators#13and#16,whichexhibitanomalousbehavioratthetran sitiontemperature inallthreeoscillatorparameters(#13isshowninFig. 4-9 ),areassociatedwithvibrations ofoxygenaroundthecentralcopper,theionresponsibleformag neticordering. 4.5Summary Far-infraredmeasurementsofsinglecrystalrerectancefromC u 2 OSeO 3 revealno drasticanomaliesinthephononspectrumattheferrimagnetictran sitiontemperature. However,acloserinspectionofthedynamicsofthephononspectr um,asmodeledthrough Drude-Lorentztting,uncoverananomalousjumpinthedielectric constantnear T c .Itis observedthat13ofthe28totalfar-infraredphononscontribu tetothismagnetodielectric eect.Afewstrongfar-infraredphononshavebeenassignedto motionoftheCuO 5 and SeO 3 unitsviaalatticedynamicalcalculation.Itisnoteworthythat2ofth e13modes exhibitinganomalousbehavioracross T c havebeenassignedtothemotionsofoxygen aroundthecentralcopper,theionresponsibleformagneticorde ring.Aweakphononthat wasnotresolvedinrerectanceisobservedbelow120Kinthetransm issionspectrum.A magneticoriginforthisstructurehasyettoberuledout. 60

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OurinfraredresultsagreewiththeRamanstudiesofGnezdilov etal. [ 45 ]inthatwe alsoobservedanumberofphononmodesthatexhibitanomaliesinthe irstrength,center frequency,orlinewidth,butwedieronotherissues.Forexample, wedonotobserveany newmodesbelowthemagnetictransitionwhereasGenzdilov etal. [ 45 ]dodetectsome additional(ratherweakandbroad)features.Theyreportthre enewmodesappearingin theRamanspectrabelowthetransitiontemperatureatfrequenc iesof 261,270,and 420cm 1 .Theonlystructureweobserveinthesethreespectralregionsis at270cm 1 where1ofthe13reportedanomalousphononsispresent.Theabs enceofatypical modeat420cm 1 givesfurthersupporttotheclaimbyGenzdilov etal. thatthenew lineappearingintheirspectraatthisfrequency\unambiguouslyhas magneticorigin." Genzdilov etal. [ 45 ]alsoreporttwonewlinesoriginatingbelow20KintheRaman spectraat 86and203cm 1 .Ourinfraredstudiesrevealastrongrathertypicalmode at 86cm 1 andaweakanomalousmodeat203cm 1 .Wehaveassignedthetypical modeat 86cm 1 tothecollectivevibrationoftheedgesharingCuO 5 units.Ifanynew infraredfeaturespossessedthesamerelativeintensitiesasrepo rtedfortheRamanspectra, wewouldhaveobservedthemclearly. 61

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Table4-2.Centerfrequencyandintensityofthe41predictedinfr aredactivemodesinCu 2 OSeO 3 CenterFreqDipoleMomentExperimentallyObservedAtomicMotion! (cm 1 )(Arb.units)Exp.indexDescription 87.1 ? 0.02-GeneralmotionofCuO 5 units 89.9 ? 0.05-VibrationofCuO 5 units 92.50.162VibrationofCuO 5 units 104.50.02-GeneralmotionofCuO 5 units 112.70.08-VibrationofCuO 5 trigonalbipyramidalvsCuO 5 squarepyramidal 114.30.13-VibrationofCuO 5 trigonalbipyramidalvsCuO 5 squarepyramidal 127.9 ? 0.05-GeneralmotionofCuO 5 units 130.8 ? 0.04-GeneralmotionofCuO 5 units 143.90.11-GeneralmotionofCuO 5 units 159.7 ? 0.03-GeneralmotionofCuO 5 units 173.8 ? 0.03-GeneralmotionofCuO 5 units 213.60.159SeO 3 unitsvibratingagainstCuO 5 units 232.20.12-GeneralmotionofSeO 3 andCuO 5 units 277.9 ? 0.09-VibrationofSeO 3 units 289.2 ? 0.01-GeneralmotionofSeO 3 units 305.60.20-GeneralmotionofSeO 3 units 317.80.2313Internalin-planevibrationofCuO 5 units 374.30.57-Internalin-planevibrationofCuO 5 units 386.71.8514InternalbendingmodeofSeO 3 units 401.30.35-GeneralmotionofCuO 5 units 408.90.71-GeneralmotionofCuO 5 units 412.00.80-GeneralmotionofCuO 5 units 423.80.70-Internalout-of-planevibrationofCuO 5 units 435.30.7316Internalout-of-planevibrationofCuO 5 units 444.40.90-GeneralmotionofOxygenatoms450.2 y 0.22-GeneralmotionofOxygenatoms Theterm\generalmotion"referstomodeswheretheatomsinmot ionareknown,butspecicmotionisunclear.The annotation ? indicatesmodesthatarepossiblynotresolvedfrommergingduetow eakdipolemomentsonadjacentphonons. Theannotation y indicatesphononsthatarepossiblynotresolvedduetomerging(br oadexperimentallinewidths).62

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Table 4-2 .Continued. CenterFreqDipoleMomentExperimentallyObservedAtomicMotion! (cm 1 )(Arb.units)Exp.indexDescription 453.3 y 0.63-GeneralmotionofOxygenatoms 469.0 ? 0.07-GeneralmotionofOxygenatoms 485.6 ? 0.06-GeneralmotionofOxygenatoms 492.10.24-GeneralmotionofOxygenatoms498.70.43-GeneralmotionofOxygenatoms512.60.68-InternalvibrationofSeO 3 units 517.70.49-GeneralmotionofOxygenatoms525.50.17-GeneralmotionofOxygenatoms557.40.46-GeneralmotionofOxygenatoms716.21.3022AntisymmetricstretchingofSeO 3 units 750.70.56-GeneralmotionofSeO 3 units 782.4 y 0.8324AntisymmetricstretchingofSeO 3 units 790.0 y 0.57-AntisymmetricstretchingofSeO 3 units 803.30.34-GeneralmotionofSeO 3 units 830.70.3026RadialbreathingmodeofSeO 3 units 63

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CHAPTER5 OPTICALPROPERTIESOFMULTIFERROICFeTe 2 O 5 Br 5.1Motivation Atechnologicalshortcomingofmagneticallydrivenferroelectricsh asbeenthesmall valueofinducedspontaneouselectricpolarizationobserved.Much largerspontaneous polarizationshavebeenpredictedinmaterialswhereferroelectricit yarisesfromcollinear magneticorderingofspins.[ 51 52 ]Incollinearmagneticordering,momentsmayvaryin amplitudebutdonotvaryindirection;therefore,onedoesnotexp ecttheaforementioned spin-orbit-relatedinteractionstobeimportant,[ 53 ]enablingthemuchstrongerexchange strictioninteractionasthemechanismbywhichferroelectricityisind uced. Thischapterrepotsinfraredstudiesonsinglecrystalsofthemultif erroicFeTe 2 O 5 Br compound,amagneticallydrivenferroelectricwithnearly-collinearin commensurate spinorder.ApreviousstudybyPregelj etal. [ 54 ]hasreportedsignicantchangesinthe magneticandelectricorderswhenexternalmagneticeldsareapp liedalongthedierent crystallographicdirections.Strikingly,forB k b theyobservedthateldsgreaterthan4.5T destroyedtheelectricpolarizationcompletely.Inourinvestigation ofmagnetoelectric couplingwemeasuredtransmissioninthefar-infrareddownto15cm 1 (2meV)andin externalmagneticeldsupto10Torientedalongallthreecrystallo graphicaxes.The signicantmagnetoelectriccouplingpreviouslyshownbytheapplicat ionofexternal magneticeldsdidnotsurfaceinthefarinfraredtotheextentwee xaminedit.However, wepresentacomprehensivereportoftheexcitationspectrumins inglecrystalFeTe 2 O 5 Br. Themodelingoftheinfraredactivephononsaswellaslatticedynamic alcalculations haveleadustomakeaninterestingcomparisontoapreviousinfrare dstudyonasimilar ReprintedwithpermissionfromK.H.Miller,X.S.Xu,H.Berger,V.Cr aciun, X.Xi,C.Martin,G.L.Carr,andD.B.Tanner,submittedtoPhys.R ev.B,eprint arXiv:1301.5881(2013). 64

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geometricallyfrustratedspin-clusteroxyhalidecompound,FeTe 2 O 5 Cl,[ 55 ]whereanequal numberofmodeswaspredictedbutasignicantlylowernumberwasr eported. CrystalStructure: Becker etal. [ 56 ]solvedthecrystalstructureusingsingle-crystal x-raydiraction.TheFeTe 2 O 5 Brcompoundcrystallizesinalayeredstructurewherethe individuallayerscomprisethe bc planeinthemonoclinicsystem(P2 1 / c ).Thelayersare weaklyconnectedviavanderWaalsforcesandslipatan18 angleastheystackalongthe normaltothe bc plane,thusdeningthemonoclinc a axis.Asinglelayeriscomposedof [Fe 4 O 16 ] 20 groupsconnedontopandbottomby[Te 4 O 10 Br 2 ] 6 groups.Thethreegroups containcommonoxygenswhichconnectthemandalsoservetocrea techargeneutrality withinthelayers.The[Fe 4 O 16 ] 20 groupsconsistoffouredge-sharing[FeO 6 ]distorted octrahedra.Allironionspossessa+3oxidationstate.Moredetaile ddescriptionsofthe crystalstructurearefoundelsewhere.[ 56 ] TheFeTe 2 O 5 Brcompoundexhibitsnearly-collinear(7 ofcantingbetweentheFe1 andFe2sites)incommensurateantiferromagneticorderingofitsm omentsbelowT N = 10.6K.[ 57 58 ]Theamplitude-modulatedmagneticorderisdescribedwiththewave vector q =(1/2,0.463,0).Simultaneously,aferroelectricpolarizationisinduce dperpendicularto q andtheFe 3+ moments.[ 57 ]Theferroelectricorderisattributedtothehighlypolarizable Te 4+ lonepairelectrons.Single-crystalx-raydiractionmeasurement sintheorderedstate didnotdetectanychangeincrystalsymmetryfromthehightempe raturephase;however, clearlydistinguishablechangesinFe-TeandFe-Ointeratomicdistanc eswereobserved thatsigniedexchangestrictionasthemeansbywhichtheinversion centerisremoved, thusallowingforferroelectricitytoarise.[ 58 ]Exchangestrictionisthemechanismbywhich magneticionsshiftawayfromtheircentrosymmetricpositionstoma ximizetheirexchange interactionenergies.[ 4 59 ] 65

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5.2ExperimentalProcedures SinglecrystalsofFeTe 2 O 5 Brhavebeengrownbystandardchemicalvaporphase method.MixturesofanalyticalgradepurityFe 2 O 3 ,TeO 2 andFeBr 3 powderinmolar ratio1:6:1weresealedinthequartztubeswithelectronicgradeHBra sthetransport gasforthecrystalgrowth.Theampouleswerethenplacedhorizo ntallyintoatubular two-zonefurnaceandheatedveryslowlyby50 C/hto500 C.Theoptimumtemperatures atthesourceanddepositionzonesforthegrowthofsinglecrysta lshavebeen500 Cand 440 C,respectively.Aftersixweeks,manydarkyellow,almostorangeF eTe 2 O 5 Brcrystals withamaximumsizeof8 12 1mm 3 wereobtained.Thepowderx-raydiraction patternobtainedbyusingaRigakux-raydiractometershowsthe monoclinicspacegroup P2 1 / c forallFeTe 2 O 5 Brcrystals. Thezero-eldtemperature-dependent(5{300K)rerectancea ndtransmittance measurementswereperformedona1 : 3 8 6mm 3 singlecrystal(crystal1)using aBruker113vFouriertransforminterferometerinconjunctionw itha4.2Ksilicon bolometerdetectorinthespectralrange25{700cm 1 anda300KDTGSdetector from700{7,000cm 1 .Thecrystalwascooledusingarowcryostat.Roomtemperature measurementsfrom7,000{33,000cm 1 wereobtainedwithaZeissmicroscopephotometer. Appropriatepolarizerswereemployedtospanthedesiredspectra lrange. Field-dependenttransmissionmeasurementsinthefarinfraredat aresolutionof 0.25cm 1 wereperformedona0 : 3 8 5mm 3 singlecrystal(crystal2)atbeam lineU4IRoftheNationalSynchrotronLightSource,BrookhavenN ationalLaboratory. ThemeasurementsemployedaBrukerIFS66-v/Sspectrometerin conjunctionwitha 10TOxfordsuperconductingmagnetanda1.8Ksiliconbolometerdet ector.Afree CrystalgrowthwascarriedoutbyHelmuthBergerattheEPFLinLa usanne, Switzerland 66

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standingwiregridpolarizerwasorientedalongthepreferentialpola rizationdirectionof thesynchrotronbeam. Atthispointitisimportanttoestablishwithcondencethatwecooled oursingle crystalbelow10KinboththeOxfordsuperconductingmagnetand theJanisrow cryostat.Thesuperconductingmagnetisequippedwithavariablet emperatureinsert thatcoolsinliquidHevapors.Thissamesystemhasbeenusedtoaccu ratelymeasure thepropertiesofsuperconductingmaterialsdownto2K.[ 60 ]Theaccuracyofthecooling capacityoftherowcryostatwascorroboratedbycomparingrat iosof T s / T n forthe superconductorNb 0 : 5 Ti 0 : 5 Nwiththosesameratiosmeasuredinthesuperconducting magnet.Withthesuperconductorfastenedtothecoldngerofo urrowcryostatwitha copperclamp,thecoolingcapabilities,below10K,agreedtowithin 1K. Inmaterialswherethecrystalsymmetryislowerthancubic,electr icalexcitations stronglydependontheorientationoftheelectriceld.Fortheselo wsymmetrycrystals, thedielectricconstantisa3 3tensorandwillgenerallyhaveodiagonalcomponents. Foreverycrystalsymmetry,however,thereexistsasetofort hogonalCartesianaxes, calledtheprincipaldielectricaxes,whichdiagonalizethedielectricten sor.[ 61 ]Inan opticsexperiment,theprincipaldielectricaxesarefoundbyrotat ingtheorientation ofthesamplewhichisplacedbetweentwopolarizersthatarecrosse dandremain xed.Throughoutarevolutionof360 ,fourorthogonalorientationswillresultinzero transmittedlightbeyondthesecondpolarizer.Thesefourdarkor ientationscorrespondto theelectriceldoftheincominglightbecomingalignedparalleltoaprinc ipaldielectric axisofthesystem. Foramonoclinicsystem,thecrystallographic b axisisuniquebecauseitisperpendicular toboththe a and c axes.Accordingly,oneprincipaldielectricaxisofthesystemisxed alongthecrystallographic b axis,buttheothertwoaxescanorientthemselvesarbitrarily aslongastheyremainorthogonal.Consequently,weperformedsin glecrystalx-ray diractiononcrystal1withknowledgeofhowtheprincipaldielectric axesfellinrelation 67

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tothecrystalfacetsinordertodeterminetheirorientationwithr especttothe a and c crystallographicdirections. X-raydiractionmeasurementswerecollectedat300Konafour-c irclePanalytical X'PertMRDinaparallelbeamgeometrywithK radiation.A -2 scanwasusedto checkthatthesamplesurfacecorrespondedtothe bc planewhilepoleguremeasurements wereperformedtondthespatialorientationofthe a axiswithrespecttothe bc plane. 5.3ResultsandAnalysis 5.3.1X-Raydiraction Thediractionexperimentshowedthatthecrystallographic c axisalsocoincideswith aprincipaldielectricaxisofthesystem.Furthermore,thethirdpr incipaldielectricaxis makesan18 anglewiththecrystallographic a axis.TheXRDpatternsacquiredfrom thesamplewiththesurfacealignedperpendiculartothediractionp lanearedisplayed inFig. 5-1 .Onlyverynarrow(h00)diractionlineswereobserved,conrming thatthe samplewasahighqualitysinglecrystal,withthesurfacealignedparalle ltothe bc plane. Severalacquiredpolegureshelpedidentifytheorientationofthe a axiswithrespectto the bc plane.TheinsettoFig. 5-1 containstheresultsofourXRDinvestigation,namely, asketchofcrystal1withtheprincipaldielectricaxes(^ e A ,^ e B ,and^ e C )superimposedon themonocliniccrystalaxes.(Notethat^ c ,^ a ,^ e C ,and^ e A alllieinaplane.) 5.3.2RerectanceandTransmittanceSpectrum Thetemperature-dependentrerectancespectrumofFeTe 2 O 5 Brforelectriceld orientedparallel^ e A ,^ e B ,and^ e C inthefrequencyinterval30{1000cm 1 (4{120meV)is showninFig. 6-2 .Astrongsharpeningofmanymodesaccompaniedbyahardeningof resonancefrequenciesisobservedwithdecreasingtemperature .Nodrasticdeviationsfrom theroom-temperaturespectrumwereobservedalonganyofthe threedirectionswhen coolingto5K.Particularattentionwasgiventotheexcitationspect rumslightlyabove andbelowthe10.6Kmultiferroicorderingtemperature.Theabsenc eofanomaliesinthe spectrauponcoolingbelow10.6Kgivesstrongsupporttotheobser vationofnocrystal 68

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n r n Figure5-1.TheXRDpatternforthesamplesurfacealignedperpen diculartothe diractionplane.Theinsetdepictstheorientationoftheprincipald ielectric axesandcrystallographicaxeswithrespecttothefacetsofcrys tal1. symmetrychangeintheorderedstate;however,asubtlecontra dictionexists.[ 57 ]The hightemperatureP2 1 / c spacegroupiscentrosymmetric,butferroelectricity,forwhich noncentrosymmetryisastrictrequirement,isobservedbelowT N .Thisinturnsuggests thatnewinfraredmodesshouldaccompanythetransition.Howeve r,thesmallvalueof polarizationmeasured(8.5 C/m 2 )leadsonetosuspetthatasubtletransitionoccursand additionalinfraredmodesarelikelybelowourexperimentalresolutio n( < 0.5cm 1 ). Figure 5-3 displaystheopticalconductivityalongthethreeprincipaldielectric axesin thefrequencyinterval30{33,000cm 1 .Thestronganisotropyobservedinthefarinfrared persistsathighfrequenciesresultinginthreedistinctabsorptione dges.Theinsetof Fig. 5-3 showsthe300Krerectanceupto33,000cm 1 along^ e A ,^ e B ,and^ e C Crystal1exhibitstransmissionatfrequenciesabovetheinfrared phononmodesand belowtheabsorptionedge.Thedimensionsofthecrystalonlyallowe dfortransmission withlightpolarizedalong^ e B and^ e C .AsshowninFig. 5-4 ,asharpdipat1850cm 1 as wellasabroadabsorptioncenteredat3250cm 1 arepresentalongbothpolarizations. Structuresattheseparticularfrequenciescalltomindtheabsor ptionbandsofH 2 O, whichinturnsuggestwaterofhydrationinthecrystalstructure .Buttheanisotropic strengthsoftheseabsorptionscanbeusedtoruleoutH 2 O.Thereforethestructures suggestabsorptionsthatareintrinsictoFeTe 2 O 5 Br.Thedierencesinenergyofthehigh 69

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r n r n r Figure5-2.Temperature-dependentrerectancespectrumofF eTe 2 O 5 Bralong^ e A ,^ e B ,and ^ e C frequencytransmissionedgealong^ e B and^ e C reinforcetheanisotropyseenintheoptical conductivityathighfrequencies(Fig. 5-3 ). 5.3.3Field-DependentTransmittance Crystal2transmitsatfrequenciesbelowthestronginfraredpho nons,inthegaps in-betweenthephonons,andinthemidinfraredbetweenthehighes tphononandthe onsetofinterbandtransitions.Particularattentionwasgiventot hetransmissionbelow thestronginfraredphonons.Noappreciablechangeinthespectr awereobservedfor transmittanceat5Kmeasuredalong^ e B and^ e C withmagneticeldsof10Tappliedalong allthreecrystallographicdirections.Aclassoflowenergymagnetic excitationscalled antiferromagneticresonancemodes(AFMR)arepredictedtoexis tatnitefrequencies inzeroeldandshiftupontheapplicationofexternalelds.AFMRmo desinthe FeTe 2 O 5 Brhavebeenreportedinarecentelectronspinresonancestudyb elow400GHz 70

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r n Figure5-3.Theopticalconductivityinthefrequencyinterval30{ 33,000cm 1 .Theinset showsthe300Kbroadbandrerectanceoutto33,000cm 1 fromwhichthe opticalconductivitywasextractedviaKramers-Kronigrelations. n n r r r Figure5-4.Transmittanceofcrystal1inthemid-infraredandnea r-infraredregions. ( 13cm 1 ).[ 62 ]However,inanexternaleldof10Tthemodesremainslightlybelowo ur measurablefrequencyrange(15cm 1 ). 71

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n n r n n n Figure5-5.Single-bouncererectance(red)asapproximatedfro mthescaledmeasured rerectance(black)andthescaledmeasuredtransmittance(blue )ofcrystal1. Theelevatedlevelofmeasuredrerectancethroughoutthetran smissioninterval isexpectedduetoadditionallightrerectedfromthebacksurface ofthe crystal. 5.3.4DeterminationofOpticalProperties Wehaveusedthemeasuredrerectanceandtransmittancespect raofcrystal1to extractthecomplexdielectricfunction.BecauseKramers-Kronig analysisrequiresa single-bouncererectancespectrumoverabroadfrequencyreg ion,wemustrstcorrect themeasuredrerectancespectruminregionswherethecrystal transmitsbyusing acombinedrerectionandtransmissionanalysis.Anattempttosolve forthesingle bouncererectancebyinvertingthefullrerectionandtransmissio nequationsleadstoa transcendentalequationwithinnitelymanysolutions.Neverthele ss,usingthemethod employedbyZibold etal. ,[ 8 ]i.e.,assumingthat k issmallintheregionofinterest,leads toanapproximationofrerectionandtransmissionthatcanbesolve dnumericallyfora singlesolution. Themeasuredrerectance(notshown)andtransmittance(Fig. 5-4 )inthefrequency interval1500{10000cm 1 weresignicantlylowerthanexpectedthroughoutthisregion oflowabsorption(Fig. 5-3 ).Thelowmeasuredvalueswereattributedtothescattering 72

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oflightfromfacetsexistingonthebacksurfaceofcrystal1.Toc ompensate,weassumed amaskonthebacksurfaceofcrystal1whichleadtoscalingthemea suredrerectance andtransmittanceupbyafactorof1.4intheinterval1500{10000 cm 1 ,thusexplaining thediscrepancybetweenFig. 5-4 andFig. 5-5 .Thescaledrerectanceandtransmittance yieldedasinglebouncererectancewhichagreedwellwiththemeasur edbulkrerectancein regionswherecrystal1didnottransmit,asshowninFig. 5-5 Kramers-Kroniganalysiscanbeusedtoestimatetherealandimagin aryparts ofthedielectricfunctionfromthebulkrerectance R ( ).[ 43 ]Beforecalculatingthe Kramers-Kronigintegral,thelowfrequencydata(0.1{30cm 1 )wereextrapolatedusing parametersfromthecomplexdielectricfunctionmodeldescribedin Sec.IIIE.Athigh frequencies(8 10 4 {2 : 5 10 8 cm 1 )thererectancewasapproximatedfromknownx-ray scatteringfunctionsoftheconstituentatoms.Thegapbetween ourhighestmeasurable frequencyandthex-rayrerectancedatawasbridgedwithaLore ntzoscillatorat 60,000cm 1 .Subsequently,thephaseshiftonrerectionwasobtainedviaKram ers-Kronig analysis,andtheopticalpropertieswerecalculatedfromtherere ctanceandphase. 5.3.5LorentzOscillatorFits ThemeasuredrerectancewastwithaDrude-Lorentzmodeltoo btainasecond estimateofthecomplexdielectricfunctionintheinfraredrange.Th emodelassignsa lorentzianoscillatortoeachdistinguishablephononinthespectrump lusahighfrequency permittivity, 1 ,toaddressthecontributionofelectronicexcitations.Themodel hasthe followingmathematicalform: = 1 X j =1 S j 2 j j 2 2 i!r j + 1 ; (5{1) where S j j ,and r j signifytheoscillatorstrength,centerfrequency,andthefullwid that halfmax(FWHM)ofthejthlorentzianoscillator.TheDrude-Lorent zcomplexdielectric functionisusedtocalculatethererectivity,whichiscomparedtoth eoriginalmeasured 73

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r n n Figure5-6.Experimentalrerectance(bluepoints)andcalculated rerectance(redline) fromtheDrude-LorentzmodeloftheFeTe 2 O 5 Br100K^ e B dielectricfunction. quantityinFig. 5-6 .Similarqualitiesoftswereobtainedforallothertemperaturesan d polarizations. 5.4Discussion 5.4.1GrouptheoryandLatticeDynamics Duetothelackofanomaliesuponcoolingbelow10.6K,weturnourfocu sto thehighlyanisotropicfar-infraredexcitationspectrum.Thenumb erofphononmodes expectedinFeTe 2 O 5 Brcanbefoundbygroup-theoryanalysis.UsingtheSMODES program,[ 49 ]wearriveatthefollowingdistributionofmodes: optical =54 A ( R ) g +54 B ( R ) g +53 A ( IR ) u +52 B ( IR ) u ; (5{2) where(R)and(IR)denoteRamanactiveandinfraredactivemodes respectively.The 53 A u modesareexpectedtoliealongtheunique b axis(^ e B )ofthemonoclinicsystem. The52 B u modesareexpectedtolieinthe ac plane,andoneshouldbeabletoresolveall 52modesusingthetwoorthogonalpolarizationspectrameasured inthisplane.However, themergingofweakermodesnearstrongermodescanhinderthisc ount.Wetherefore 74

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Table5-1.Oscillatorparametersforexperimentallyobservedphon onsalong^ e A (at20K). Themotionsofatomsassociatedwithgroupsofphononshavebeen assigned whenthepatternwasapparent.Theatomsarelistedindescending orderof theircalculatednetatomicdisplacement. IdxOscStrCtrFreqFWHMDegenAsgn S (cm 1 ) r (cm 1 )idxin^ e C 10.20745.52.01Br21.32079.62.15-30.35688.41.16Br,Te,O,Fe40.04298.11.08Br,Te,O,Fe50.056107.52.19Br,Te,O,Fe60.048137.11.912Br,Te,O,Fe70.015157.61.614Te,Br,O,Fe80.023166.02.215Te,Br,O,Fe90.354184.91.316Te,O,Fe100.519194.32.417Te,O,Fe110.159211.72.0Te,O,Fe120.007236.44.922-130.084247.81.723-140.009256.54.024-150.169291.91.827Fe,O,Te160.030302.83.4Fe,O,Te170.033321.63.628Fe,O,Te180.006328.02.4Fe,O,Te190.003348.52.029Fe,O,Te200.102370.02.5-210.008387.12.6-220.030401.34.1-230.005418.64.0-240.005426.22.630-250.090435.03.931-260.119463.65.432O,Fe,Te270.004481.01.4O,Fe,Te280.029493.24.933O,Fe,Te290.350575.37.535O,Fe300.203666.18.037O,Fe310.257730.513.9O,Fe320.018784.09.242employedlatticedynamicalcalculationstodeterminetherelativeres onancefrequency, modeintensity,aswellasdisplacementpatternforthe105infrare dactivemodes.The calculations,whichutilizedthestructureandvalenceasreportedb yBecker etal. ,[ 56 ]were 75

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basedonareal-spacesummationofscreenedcoulombinteractions involvingaspherical cut-oboundary.[ 63 ]Thefollowingtwosectionsdescribethemethodusedtocompareth e observedandcalculatedspectrumforthe A u and B u modesrespectively. 5.4.252 B u Modes All52 B u modespredictedlieinthe ac planeandshouldpossessaniteintensity alongbothmeasuredpolarizationsinthisplaneunlessthedirectionof theinduceddipole momentforaparticularvibrationisparalleltoeither^ e A or^ e C .(Smallangledeviations fromtheparallelcasemaybehardtodetect.)Forallothermodeso nemustsortout whichexcitationsalong^ e A and^ e C correspondingtothesamephysicalvibrationsothat theyarenotcountedtwice.Tocountthemodes,weusedtheDrud e-Lorentzparameters foreachphononat20Kaswellastheresonantfrequencyandmod eintensityfromour latticedynamicalcalculation.Withknowledgeoftheopticalconduct ivityspectraalong ^ e A and^ e C at20K,modesweregroupedtogetheriftheywereobservedinthe samesmall frequencyinterval(ontheorderofatypicallinewidth)ofoneano therinthetwospectra. Next,wecomparedtheexperimentallinewidthsofthegroupedmod estoensurethat theywouldoverlapifplottedside-by-side.Astwonalcriteria,wec omparedboththe ratioofexperimentaloscillatorstrengthswiththecomputedoscilla torstrengths,andalso comparedthedisplacementpatternsgeneratedbyourcalculation tomakesuretheywere ingoodagreementastowhichexcitationscorrespondedtothesam ephysicalvibrations. All52 B u modespredictedinthe ac planewereobservedintheexperimentalspectrum. Lorentzoscillatorparametersforallmodesaswellasmodesprese ntalongboth^ e A and^ e C polarizationscanbefoundinTable 5-1 andTable 5-2 5.4.353 A u Modes Countingtheexperimentallyobserved A u modesisstraightforward;weobserve 43ofthe53predictedmodesalongthisdirection.Webelievethat10m odeswerenot resolvedasaresultofweakermodesbeingburiedwithinstrongermo desandmerging togetherinthespectrum.Intheopticalconductivity,allsingleph ononexcitations 76

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n n n n r n n r Figure5-7.Theleftpanel(a)showsunaccountedforspectralw eightonthehighfrequency sideoftheasymmetricoscillatorat85cm 1 alongwithattoasymmetric oscillator(79cm 1 ).Wesuspecttheasymmetricshoulderistheresultofa buriedmode.Therightpanel(b)depictstheappearanceofaonce buried mode( 276cm 1 )uponcooling. shouldberepresentedbysymmetriclorentzianoscillators.Figure 5-7 leftpaneldepicts aslightlyasymmetriclorentzianoscillatorwitharesonanceof85cm 1 .Ourattemptto ttheresonanceat85cm 1 withasymmetriclorentzianoscillatoryieldsunaccounted forspectralweightonthehighfrequencysideoftheresonance. Asymmetriclorentzian oscillatorcenteredat79cm 1 andcorrespondingtarealsoincludedasareference.(Note thattheadditionalspectralweightbetweentheresonancesat7 5cm 1 and85cm 1 is attributedtoabsorptionbetweenthebandsandthereforenott obemistakenforalattice vibration).Wesuspectthatthisasymmetricshoulderisduetoawea kburiedmodethatis notfullyresolved.Tofurthersupportourinterpretationofburie dmodes,therightpanel ofFig. 5-7 depictstheappearanceofaonceburiedmode( 276cm 1 )uponcooling.The Lorentzoscillatorparametersofthe52 B u modesat20KarefoundinTable 5-3 77

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5.4.4AssignmentofModes Theassignmentofaparticularphonon,observedintheexperiment alspectrum,to itspreciseatomicdisplacementpatterncannotbemadewithoutsac ricingobjective certaintybecauseofthedensityofthephononspectrum.Assign mentstogroupsof modes,however,canbemadethroughthedisplacementpatterns generatedbyourlattice dynamicalcalculations.Usingasimpliedmodelofaphononasadriven oscillatorwith resonantfrequencyinverselyproportionaltomass,weproceed tolisttheconstituent atomsinthestructurefromheaviesttolightest:Te,Br,Fe,andO .Normally,thelow frequencymodesinvolvetheheaviestatoms,butintheFeTe 2 O 5 BrstructuretheBratoms haveaweakinteractionwiththeotheratomsandthereforearere sponsibleforthelowest frequencyphonons.AtacertainfrequencythemotionofBratom swillcease.Likewise, atahigherfrequencythevibrationofatomsinvolvingTewillcease,le avingthemotion ofFeandOtobeexpectedinthehighestfrequencyphonons.Acco rdingly,thephonon spectrumcanbedividedintothreeclustersthatisrerectedinFig. 6-2 (namely,below 180cm 1 ,between180cm 1 and530cm 1 ,andabove530cm 1 ).Theclusteringisalso observedinthecalculatedspectrumviathedisplacementpatterns produced,intermsof rsttheBrandsubsequentlytheTeceasingtovibrate.Withineach clustertrendsare tobenotedonthefollowingprinciple:atomsarelistedcorresponding totheirnetatomic displacement.Thisapproach,inturn,hasallowedustoidentifyanot hercharacteristicof atomicdisplacementwithinthethreeclusters.Theatomoffocusina cluster(e.g.,Brin therstcluster)hasanetatomicdisplacementeverdiminishing.Ces sationdoesnotarrive abruptly.ThethreeclustersareproledineachofTable 5-3 ,Table 5-1 ,andTable 5-2 AproposthethirdclusterthatfocusesonFeandO,anadditionalr esourceisprovided bythefamiliarorthoferritefamilyofcompoundsbecausetheFeO 6 octrahedralbuilding blockthatispresentinFeTe 2 O 5 Brisitselfadeningunitoftheorthoferritefamily. Giventheabundantliteratureonvibrationalassignmentsmadetot heFeO 6 ochrahedral 78

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unit,[ 64 65 ]weareconsequentlyabletomapthesewellrecognizedassignment sontoour experimentalndings,thuscorroboratingbothperspectives. 5.5Summary Far-infraredmeasurementsonmultiferroicsingle-crystalFeTe 2 O 5 Brrevealnodrastic anomaliesintheexcitationspectrumbelowthemultiferroicorderingt emperature. Nonetheless,athoroughinspectionofthephononspectrahaslea dtotheidenticationof all52predictedmodesinthedegenerate ac planeand43ofthe53predictedmodesalong theunique b axis.Theabsenceof10modesalongthe b axisispresumablyduetothe mergingofweakermodesnearstrongermodesinthedenseexcitat ionspectrum. Themotionsofindividualatomshavebeenassignedtogroupsofpho nonsvia thedisplacementpatternsproducedthroughourlatticedynamica lcalculation.The assignmentsdividetheoverallphononspectrumintothreecluster s,whicharecharacterized byanatomthatiscommontoeveryvibrationinthatcluster.Weobse rvedthatas frequencyincreases,vibrationsinvolvingthemotionofBratomswill progressivelycease. Likewise,athigherfrequenciesthevibrationsinvolvingTeatomswillc easeinthesame manner,leavingthemotionofFeandOtobeexpectedinthehighestf requencyphonons. Moreover,theparticularatomsinvolvedinavibrationarelistedindes cendingorderofnet atomicdisplacement,whichinturnmakesclearatrendwithineachclus ter(namely,the atomoffocusinaparticularclusterhasanetatomicdisplacementev erdiminishing). ItremainstocompareourresultstoarecentinfraredstudybyPf uner etal. [ 55 ]on thesimilarFeTe 2 O 5 Clcompoundwhereanequalnumberofphononswerepredicted. Ouranalysisofthephononspectrumagreeswiththeinfraredstud yofPfuner etal. in thatwealsobelievemodesresonatingatequivalentenergyscalesca noverlap,preventing themfrombeingresolvedseparately,butwedieronotherissues. Pfuner etal. report acombined38modesalongthe^ e B and^ e C directions.Weobservemoremodesalongthe unique b axisalone,aswellasmodesbelow50cm 1 ,whereasthestudybyPfuner etal. doesnotreportmeasurementsbelowthisfrequency. 79

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Wealsoobservedrasticallydierentandmorestronglyanisotropich ighfrequency opticalproperties.Pfuner etal. [ 55 ]reportanabsorptionedgearound4,000cm 1 in FeTe 2 O 5 Cl,whereasweobservelowabsorptionaccompaniedbysubstantial transmission throughtheFeTe 2 O 5 Brcrystalinthisregion.Inspeculatingaboutthedierencesinhigh frequencyopticalpropertiesofthetwosimilarcompounds,wepoin toutthatspurious changesofslopeofthererectancethroughoutthemid-infrared region,suchasthose resultingfromthecontributionofbacksurfacererection(blacklin einFig. 5-5 ),can drasticallychangetheKramers-Kronig-derivedopticalpropertie s. Transmissionmeasurementsaswellasrerectancealong^ e A werenotreportedinthe studybyPfuner etal. .[ 55 ]SimilartotheFeTe 2 O 5 Brsystemreportedhere,theFeTe 2 O 5 Cl systemalsoshowednoresponsetomagneticeldsaboveorbelowth emagneticordering temperature. 80

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Table5-2.Oscillatorparametersforexperimentallyobservedphon onsalong^ e C (at20K). Themotionsofatomsassociatedwithgroupsofphononshavebeen assigned whenthepatternwasapparent.Theatomsarelistedindescending orderof theircalculatednetatomicdisplacement. IdxOscStrCtrFreqFWHMDegenAsgn S (cm 1 ) r (cm 1 )idxin^ e A 14.27546.70.21Br21.33553.90.5Br30.95165.81.0-40.21671.70.8-50.43581.20.92-60.20589.30.93Br,Te,O,Fe70.06197.00.8Br,Te,O,Fe80.13099.01.14Br,Te,O,Fe90.533109.20.85Br,Te,O,Fe100.294112.81.0Br,Te,O,Fe110.369128.50.8Br,Te,O,Fe120.127135.40.96Br,Te,O,Fe130.411140.60.9-140.857157.00.57Te,Br,O,Fe150.107166.81.08Te,Br,O,Fe160.212184.50.69Te,O,Fe170.039194.81.210Te,O,Fe180.066204.01.4-190.014206.31.1-200.085221.71.1-210.101225.40.9-220.065232.61.012-230.107248.01.513-240.291254.20.614-250.239274.41.1-260.195284.91.2-270.633294.81.415Fe,O,Te280.138319.01.917Fe,O,Te290.173349.02.219Fe,O,Te300.514426.72.424-311.562437.02.325-320.088459.33.426O,Fe,Te330.006488.54.328O,Fe,Te340.067509.69.6O,Fe,Te350.260573.55.629O,Fe360.096626.75.2O,Fe370.015662.03.230O,Fe380.061672.05.7O,Fe 81

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Table 5-2 .Continued. IdxOscStrCtrFreqFWHMDegenAsgn S (cm 1 ) r (cm 1 )idxin^ e A 390.182693.58.9O,Fe400.172707.45.2O,Fe410.101749.89.4-420.014780.46.632-430.033814.76.982

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Table5-3.Oscillatorparametersforexperimentallyobservedphon onsalong^ e B (at20K). Themotionsofatomsassociatedwithgroupsofphononshavebeen assigned whenthepatternwasapparent.Theatomsarelistedindescending orderof theircalculatednetatomicdisplacement. IdxOscStrCtrFreqFWHMAsgn S (cm 1 ) r (cm 1 ) 11.38835.70.8Br20.33043.51.4Br30.89953.20.8-40.28960.50.9Br,Te,O,Fe50.11979.11.3Br,Te,O,Fe60.07585.01.2Br,Te,O,Fe70.05291.21.0Br,Te,O,Fe80.13893.31.3-90.25998.00.9-101.359102.01.1-110.073106.90.7-120.170123.70.7Te,Br,O,Fe130.092126.30.9Te,Br,O,Fe140.074131.41.1Te,Br,O,Fe151.066184.00.4Te,O,Fe160.128190.20.7Te,O,Fe170.491198.60.8Te,O,Fe180.440210.00.3-190.023215.31.3-200.053222.01.1-210.058240.11.3-220.028249.00.9-230.274273.90.9Fe,O,Te240.102275.81.4Fe,O,Te250.661295.51.3Fe,O,Te260.366317.61.1Fe,O,Te270.101324.31.5Fe,O,Te280.261341.71.2Fe,O,Te290.808393.72.6Fe,O,Te300.371418.04.1-310.009437.73.5-320.209462.83.2O,Fe,Te330.173478.03.3O,Fe,Te340.012499.04.6-350.009508.31.2-360.047595.36.9O,Fe370.080626.63.6O,Fe380.323652.04.9O,Fe 83

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Table 5-3 .Continued. IdxOscStrCtrFreqFWHMDegenAsgn S (cm 1 ) r (cm 1 )idxin^ e B 390.018682.18.2O,Fe400.217728.17.8O,Fe410.027746.27.4-420.031779.923.5-430.071836.09.684

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CHAPTER6 PHONONANOMALYANDMAGNETICEXCITATIONSINCu 3 Bi(SeO 3 ) 2 O 2 Cl 6.1Overview 6.1.1CrystalStructure Cu 3 Bi(SeO 3 ) 2 O 2 Clisageometrically-frustratedlayeredmaterialpossessing magneticorder.P.Millet etal. [ 66 ]determinedtheroomtemperaturecrystalstructure ofCu 3 Bi(SeO 3 ) 2 O 2 Clusingsinglecrystalx-raydiraction.TheCu 3 Bi(SeO 3 ) 2 O 2 Cl compoundcrystallizesinalayeredstructurewheretheindividuallay ersstackalong the c axisoftheorthorhombiccell( Pmmn ).ThemagneticCu 2+ ionsformahexagonal arrangementwithineachplanethatisreminiscentofaKagomelattice ,andthusimplies magneticfrustration.Withinanyonehexagonthereexisttwouniqu ecoppersites,Cu1 andCu2,whichpossessdierentout-of-planeoxygenbonding.Bo thcoppersitesform distinct[CuO 4 ]unitsthatarelinkedbySe 4+ andBi 3+ ions.TheCl atoms,which sandwichtheplanesformedbythecopperhexagons,positionthem selvesalongthe axisthatdenestheparallelstackingofthecopperhexagons.An imageofthecrystal structureemphasizingthehexagonalarrangementofcopperion sisshowninFigure 6-1 Additionalpicturesandmoredetaileddescriptionsofthecrystals tructurearefound elsewhere.[ 66 67 ] P.Millet etal. [ 66 ]alsoreportedmagneticsusceptibilitymeasurementsonapowder sampleofCu 3 Bi(SeO 3 ) 2 O 2 Cl.Near150Ktheyobservedachangeinslopeof1/ ( T ) thatresultedintwodistinctandpositiveWeisstemperatures.Tore concilethisanomaly in1/ ( T ),theyperformedlinearbirefringenceonasinglecrystalanddedu cedthata second-orderstructuraltransitionwaslikely,buttheywerenot abletodeterminetheexact ReprintedwithpermissionfromK.H.Miller,P.W.Stephens,C.Martin ,E. Constable,R.A.Lewis,H.Berger,G.L.Carr,andD.B.Tanner,P hys.Rev.B 86 174104(2012). 85

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Figure6-1.Oneunitcellisboxedinthelowerleft.Thehexagonalarr angementofcopper ions(bluespheres)isemphasized. natureofthetransition.P.Millet etal. alsoreportedferromagnetic-likebehaviorbelowT c 24K. 6.1.2BeautyofInfraredSpectroscopy Thetechniqueofinfraredspectroscopylendsitselfwelltotheinve stigationof geometricallyfrustratedmagneticmaterials.Forexample,thespin -drivenJahn-Teller eectintheCrspinelcompoundsCdCr 2 O 4 andZnCr 2 O 4 actstoliftthedegenerate groundstatearisingfromcompetingmagneticinteractionsviaacou plingtothe latticedegreesoffreedom;theensuinglatticedistortionisunambig uouslyobservedin theinfraredasasplittingofinfrared-activephononmodes.[ 68 69 ]Furthermore,the a.c.electric(magnetic)eldoftheinfraredlightcaninteractwithor deredmomentsand exciteanelectromagnon(magnon)excitation,whichinturncanpro videinformationabout thesymmetryandnatureofthemagneticorder.[ 30 70 ] 6.1.3MajorFindings HerewepresentourinfraredstudiesonsinglecrystalCu 3 Bi(SeO 3 ) 2 O 2 Cl.Weobserve 16newmodesinthephononspectraoriginatingbelow115K.Strikingly ,oursubsequent 86

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powderx-raydiractionmeasurementsrevealthesame300Kstr uctureexistingat85K. PreliminaryRamanmeasurementssuggestthatalossofinversionsy mmetryislikely.In addition,weobservenewexcitationsintheinfraredarisinginthemag neticallyordered state(below24K)thatshowisotropicinfraredpolarizationdepend encebutanisotropic externalmagneticelddependence.Inlightofthenovelexcitatio nsobservedinthe magneticallyorderedstate,weperformedd.c.susceptibilitymeasu rementstoexamine theanisotropicmagneticpropertiesofCu 3 Bi(SeO 3 ) 2 O 2 Cl.Theresultsofourmagnetic susceptibilitymeasurementsareinagreementwitharecentreport [ 67 ]ofthemagnetic propertiesinthesimilarCu 3 Bi(SeO 3 ) 2 O 2 Brcompound(T c =27.4K).(Thismanuscriptis accompaniedbysupplementarymaterialincludingadditionalgures referredtointhetext andtheresultsoffullRietveldrenements.) 6.2ExperimentalProcedures SinglecrystalsofCu 3 Bi(SeO 3 ) 2 O 2 Clweregrownbystandardchemicalvapor-phase method.MixturesofanalyticalgradepurityCuO,SeO 2 andBiOClpowderinmolar ratio3:2:1weresealedinquartztubeswithelectronicgradeHClasth etransportgasfor thecrystalgrowth.Theampouleswerethenplacedhorizontallyint oatubulartwo-zone furnaceandheatedat50 C/hto450 C.Theoptimumtemperaturesatthesourceand depositionzonesforthegrowthofsinglecrystalswerefoundtobe 480 Cand400 C, respectively.Afterfourweeks,manytabulargreenCu 3 Bi(SeO 3 ) 2 O 2 Clcrystalswitha maximumsizeof15 12 1mm 3 wereobtained,whichwereindentiedassynthetic francisiteonthebasisofx-raypowderdiractiondata. Zero-eldtemperature-dependent(7{300K)rerectanceandt ransmittancemeasurements werecollectedona6 : 7 4 : 8 0 : 7mm 3 singlecrystal(crystal1)usingaBruker113v Fouriertransforminterferometerinconjunctionwitha4.2Ksiliconb olometerdetector CrystalgrowthwascarriedoutbyHelmuthBergerattheEPFLinLa usanne, Switzerland 87

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inthespectralrange25{700cm 1 andanitrogencooledMCTdetectorfrom700{5,000 cm 1 .Roomtemperaturemeasurementsfrom5,000{33,000cm 1 wereobtainedwitha Zeissmicroscopephotometer.Magneticeld-dependenttransmis sionmeasurementsin thespectralrange15{100cm 1 wereperformedona7 : 5 3 : 8 0 : 2mm 3 singlecrystal (crystal2)atbeamlineU4IRoftheNationalSynchrotronLightSo urce,Brookhaven NationalLaboratory,utilizingaBrukerIFS66-v/Sspectrometer .Thecrystalwasplaced ina10TOxfordsuperconductingmagnetandthetransmittedinten sitiesweremeasured usinga1.8Ksiliconbolometerdetector.Additionalmagneticeld-dep endenttransmission measurementswereperformedona5 : 1 3 : 5 0 : 3mm 3 singlecrystal(crystal3)inthe spectralrange10{45cm 1 usingamodiedPolytecFIR25spectrometerinterfacedto a7TOxfordsplit-coilsuperconductingmagnetattheUniversityof Wollongong.The transmittedintensitiesweresubsequentlyconvertedtoabsorpt ioncoecientsusinga modiedBeer-Lambertlawthataccountedforrerectionlossesat thesurfaces.Linearly polarizedlightwasusedinallinfraredmeasurementsandorientedalo ngthethree principaldielectricaxesofthesystem.Foranorthorhombicsyste m,thethreeprincipal dielectricaxescoincidewiththethreecrystallographicaxes.[ 61 ]Inthefollowingreport, thesymbol\ E "alwaysreferstothepolarizationoftheincominglight,while\ H "isthe orientationoftheexternaleld(whenapplicable). Powderx-raydiractionmeasurementswereperformedatbeamlin eX16Cofthe NationalSynchrotronLightSource,BrookhavenNationalLabor atory.Asmallcrystalwas crushedinamortarandpestle,mixedwithasmallamountofSipowde r(NISTStandard ReferenceMaterial640c)androughlythreetimesitsvolumeofgro undcork,andloaded inathin-walledglasscapillaryof1mmnominaldiameter.Thecorkserve dtodilute thesamplesothatitcouldllthecross-sectionofthex-raybeamw ithoutdrastically absorbingit;measuredtransmissionatthecenterofthecapillaryw as12%.X-raysof nominalwavelength0.6057 Awereselectedbyachannel-cutSi(111)monochromator beforethesample;smalldriftsinthex-raywavelengthwerecorre ctedviatheSiinternal 88

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standard.ThediractedbeamwasanalyzedbyaGe(111)crystal anddetectedbya commercialNaI(Tl)detector.Dataweretypicallycollectedinsteps ofdiractionangle2 of0 : 005 .Forlatticeparametermeasurementsasafunctionoftemperatu re,thesample wasinsideaBeheatshieldinaclosedcycleHerefrigerator,whichwasr ockedseveral degreesateach2 step.Dataforcrystallographicrenementswerecollectedwithth e samplecontinuouslyspinningseveralrevolutionsforeachpoint,ina nOxfordCryostream samplecooler.Powderx-raydiractiondatawereanalyzedusingTo pas-Academic software.[ 71 ] Magneticmeasurementswereperformedinacommercialsupercon ductingquantum interferencedevicemagnetometeroncrystal2. Magneticeld-dependentcapacitancemeasurementswereperfo rmedatSCM2at theNationalHighMagneticFieldLaboratory,Tallahassee,Florida.A singlecrystalof dimension4 : 2 4 : 0 0 : 2mm 3 (crystal4)wasutilizedforthismeasurement.Aluminum electrodesweredepositedonboth ab facesofcyrstal4.Theelectrodespossessedan approximateareaof 4.8mm 2 .Thecrystalwasinsertedinan18Tmagnetequipped witha 3 Heinsert.Measurementswererecordedwithacommerciallyavailable AH2700A Ultra-precisionCapacitanceBridge. 6.3ResultsandAnalysis 6.3.1ZeroFieldRerectanceandTransmittanceSpectra Thetemperature-dependentrerectancespectraofCu 3 Bi(SeO 3 ) 2 O 2 Clalongthe a b ,and c axesinthefrequencyinterval30{1,000cm 1 (4{120meV)isshowninFig. 6-2 Remarkably,manynewphononmodesareobservedinthererectan cespectraalongall threecrystaldirectionsuponcoolingfrom120Kto110K.Thearro wsinFig. 6-2 indicate thepositionwherenewmodesoccur.Meticuloustemperatureswee ps(every1K)between 120Kand110Kindicatethatallthenewmodesariseat115K;thistre ndwasveried uponcoolingandwarming. 89

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Inaddition,astrongsharpeningofmanymodesisobservedwithdec reasing temperature.Inregionswherenewmodesappear,thetypicalso fteningofresonance frequencieswithincreasingtemperatureisnotsystematicallyobse rved.Thetypical softeningofresonancefrequencieswithincreasingtemperature ,whichiscorrelatedto theexpansionofthelattice,canbecounteractedbyrepulsiondue tophononmixing.An explanationoftherepulsionofphonons,whicharebosonicexcitatio ns,willbepresented inthediscussionsection.Inadditiontothenewphonons,weobserv ethesoftening(at 5Kto 30%ofthe300Kvalue)ofthelowestfrequencymodealongthe ^ b direction, anoccurrencewhichisnotassociatedwithphononrepulsion.Theso ftmodebehavior exhibitedbythislowfrequencyphononisreminiscentofthatoccurr inginadisplacive ferroelectricmaterial. Cu 3 Bi(SeO 3 ) 2 O 2 Clisopaquebetween 40and800cm 1 duetothestrongoptical phononabsorptionsinthisregion;however,abovetheopticalabs orptionslightis observedtotransmit.Figure 6-3 displaysthezeroeld300Ktransmissionspectra ofCu 3 Bi(SeO 3 ) 2 O 2 Clalongthe a and b axesinthemid-infraredandnear-infrared regions.(Duetothegeometryofthesamplenolightwasobservedt otransmitalong the c axis.)Theanisotropy,whichisprominentintheinfraredrerectance spectra,is alsoobservedintransmissionathighfrequencies.Sharpbutweaka bsorptionfeatures existat1950cm 1 alongthe a axisandat2000and2050cm 1 alongthe b axis.The aforementionedabsorptionsexhibitaminimalstrengtheningwithde creasingtemperature. Thedownturnintransmissionat9,000cm 1 ( 1.1eV)indicatestheonsetofelectronic absorptions.Theenergyoftheobservedgapisconsistentwithth einsulatingnatureofthe material.Transmissionbelowtheopticalabsorptions(below40cm 1 )isthesubjectof sectionIIIE.6.3.2Kramers-KronigandOscillator-modelts Alongthe a and b axeswehaveutilizedacombinedrerectionandtransmission analysistoextractthesinglebouncererectancenecessaryfort heKramers-Kronig 90

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r n r r n r Figure6-2.Thetemperature-dependentrerectancespectrao fCu 3 Bi(SeO 3 ) 2 O 2 Clalongthe a b ,and c axes.Crimsonarrowsindicatedthepositionsofnewphonons arisingat115K. r n Figure6-3.Transmissionalongthe a and b axesofcrystal1inthemid-infraredand near-infraredregions.Theinsetisasketchofcrystal1withthe a b ,and c axesindicated. transformation.Morespecically,inthefrequencyregionwheret hecrystaltransmits alongthe a and b axes,themeasuredrerectancehasanadditionalcontributionfr omthe backsurface.UsingthemethodemployedbyZibold etal. ,[ 8 ]i.e.,assumingthat k issmall intheregionofinterest,weareabletoextractthesinglebouncere rectanceofthesample. Sincethe c axisdoesnotexhibittransmission,themeasuredrerectancewasa ssumedtobe singlebouncererectancealongthisdirection. 91

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Therealandimaginarypartsofthedielectricfunctionwereestimat edfromthe singlebouncererectance, R s ( ),usingtheKramers-Kronigtransformation.[ 43 ]Before calculatingtheKramers-Kronigintegral,thelowfrequency(0.1{30 cm 1 )datawere approximatedusingthedielectricfunctiondeterminedfromthett ingproceduredescribed below.Athighfrequenciesthererectancewasassumedtobecons tantupto1 10 7 cm 1 afterwhich R ( ) 4 wasassumedastheappropriatebehaviorforfreecarriers.The opticalpropertieswereobtainedfromthemeasuredrerectance andtheKramers-Kronig derivedphaseshiftonrerection. ThesinglebouncererectancewastwithaLorentzoscillatormodel toobtaina secondestimateofthecomplexdielectricfunctionintheinfraredra nge.Themodel assignsaLorentzianoscillatortoeachphononmodeinthespectrum plusahighfrequency permittivity, 1 ,toaddressthecontributionofelectronicabsorptions.Themode lhasthe followingmathematicalform: ( )= 1 X j =1 S j 2 j j 2 2 i!r j + 1 ; (6{1) where S j j ,and r j signifytheoscillatorstrength,centerfrequency,andthefullwid th athalfmax(FWHM)ofthej th Lorentzianoscillator.Thererectivityisthencalculated usingtheDrude-Lorentzcomplexdielectricfunction.Acomparison ofthecalculatedand measuredrerectivityisshowninFigure 6-4 6.3.3OpticalProperties TheopticalpropertiesextractedfrombothKramers-Kronigana lysisandourts ofrerectancecanbeusedtoinvestigatefurthertheappearanc eofnewphononmodes intheinfraredspectraat115K.TheupperpanelofFig. 6-5 depictstherealpartof theopticalconductivity, 1 ( ),alongthe b axisinthefrequencyrange250{285cm 1 ThelowerpanelofFig. 6-5 depictsthelossfunction,Im(-1/ ( )),inthesameregion. BothopticalpropertiesshownarefromKramers-Kroniganalysiso fthesinglebounce rerectance.Thepeaksobservedin 1 ( )andIm(-1/ ( ))closelycorrespondrespectively 92

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r n n Figure6-4.ThecalculatedrerectancefromtheDrude-Lorentzm odel(redline) superimposedonthemeasuredrerectance(bluepoints)ofCu 3 Bi(SeO 3 ) 2 O 2 Cl alongthe ^ b directionat100K.Similarqualitiesoftswereobtainedatall othermeasuredpolarizationsandtemperatures.. totheTOandLOphononfrequencies.Theinsetofthelowerpanelo fFig. 6-5 depicts theplasmafrequency(n s = p S j 2 0 )associatedwithbothphononsasafunctionof temperatureasobtainedbyourDrude-Lorentztting.Theequa tiondeningtheplasma frequencyofionicvibrationsissimilarinformtothatwhichdescribest hefreecarrier responseinmetallicsystemsaftertheelectronicmassandchargea rereplacedwiththe reducedmassofthenormalmodeandioniccharge.Theplasmafreq uencyisofparticular interestherebecauseitssquareisproportionaltothespectral weightassociatedwitha particularphonon.Thenewphononmodethatappearsat 276cm 1 below115Kgains spectralweightattheexpenseoftheexistingmodeat 256cm 1 .Thesmoothshiftof spectralweightwithdecreasingtemperaturefromtheexistingmo detothenewmodeis reminiscentofasecondordertransition.6.3.4PowderX-RayDiraction Thechangesintheinfraredspectra,namelytheappearanceofne wvibrationalmodes below115K,suggestareductioninlatticesymmetry.Toinvestigate thelowtemperature 93

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n nn n n r rn "!%& r ##%$' ( # r # n "& n n rn n n "$%" r n n "$%" n r n Figure6-5.Thetemperaturedependentopticalconductivity 1 ( )(upperpanel)andloss functionIm(-1/ ( ))(lowerpanel)alongthe b axisinthefrequencyrange 250{285cm 1 .Theinsetsoftheupperandlowerpanelsdepictrespectively thetemperaturedependenceoftheresonanceandplasmafrequ enciesofthe twomodes. structureofCu 3 Bi(SeO 3 ) 2 O 2 Cl,weperformedpowderx-raydiractionmeasurements between30and300K.Thelatticeparametersasafunctionoftemp eratureextractedfrom RietveldtsofthediractionspectraareshowninFig. 6-6 .Thereexistssomestructure inthecurvesoflatticeparameters vs. temperature,butthereisnothingthatcouldbe regardedasconclusiveevidenceofaphasetransition.Thenegativ ethermalexpansion observedforthe a latticeparameterbelow100Kisnotunusualforanioniccompound oforthorhombicsymmetrywhereitmaybeenergeticallyfavorablef ortheunitcellto contractinonedirectionwhileexpandinginotherdirections.TheRiet veldrenements ofthepowderx-raydiractionpatternat295and85Kareshownin Figure. 6-7 .Both renementswereconsistentwiththepublishedstructureofP.Mille t etal. inspace group Pmmn .Fulldetailsoftherenements,includingbondinggeometry,mayals obe foundinthesupplementaryinformation.Wesawnoevidenceforsplit tingorbroadening ofdiractionlinesnortheappearanceofnewdiractionlinesatlowte mperature.We 94

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nnn r Figure6-6.Latticeparametersandunitcellvolume vs. temperature.Errorbarsrerect thereproducibilityofindependentlymeasureddatasetsatselecte d temperatures.Solidlinesaredrawnasguidestotheeye.Latticepa rametersat 30Kare a =6.3463(2) A, b =9.6277(3) A, c =7.2186(3) A,Volume= 441.10(2) A 3 concludethatthereisnodirectevidenceforaloweringofcrystallog raphicsymmetry near115K.Thisisaseriouspuzzlebecausetheappearanceof16ne winfraredmodes below115Kimpliesthatthesymmetryofthecrystalislowerthan Pmmn belowthat temperature.6.3.5MagneticField-DependentTransmission Far-infraredtransmissionasafunctionoftemperatureandexte rnalmagneticeld wasmeasuredoncrystal2andcrystal3withlightpolarizedalongth e a axis, b axis, andat45 toboththe a and b axes.Uponcoolingto5Kanexcitationwasobserved at33.1cm 1 thatonlyexistedinthemagneticallyorderedstate(below24K).The excitationwasobservedinallfourpolarizations.Figure. 6-8 depictstheexcitation,at5K, ineachofthefourpolarizationsmeasured,aswellastheevolutiono ftheexcitationasthe externaleldisrampedto10T( H k ^ c geometry).Whentheexternalmagneticeldwas appliedparalleltothe c axis( H k ^ c )andtheeldwasrampedto10T,theexcitationat 95

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n r n r Figure6-7.RietveldtsofthepowderdiractionpatternsofCu 3 Bi(SeO 3 ) 2 O 2 Clat295K (upperpanel)and85K(lowerpanel).Reddotsrepresentdata,blu elinethe Rietveldt.Thedierencecurveisnormalizedtothestatisticalunc ertaintyof eachdatapoint.Thetshave 2 valuesof2.21(295K)and1.70(85K). 33.1cm 1 increasedlinearlywithincreasingeld(insetFig. 6-9 a).Nohysteresiseects wereobserveduponrampingthemagneticeld.Toestablishconde ncethattheobserved excitationonlyexistedinthemagneticallyorderedstate,thecryst alwaswarmedupto 40Kandtheexternaleldwasrampedfrom0to10T.Theexcitation wasnotobserved. TheresultsareshowninFig. 6-9 afortheE k ^ b +45 polarization. Inaddition,forelds1Tandgreaterappliedparalleltothe c axis,asecondmagnetic excitationwasobservedwhichalsopossessedisotropicpolarization dependenceinthe ab plane.Theexcitationincreasedlinearlyfrom10.5cm 1 at1Tto19.3cm 1 at 10T.Becauseofthelowsignaltonoiselevelitisnotknownwhethert heexcitation disappearedbelow1Toritwasjustunresolved.Theexcitationispict uredforthe E k ^ a polarizationinFig. 6-10 .ThespectrainFig. 6-10 isaconjunctionofdata 96

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n r n n r n n n r n n n r n Figure6-8.Field-dependentabsorptionobtainedfromthetransm ittedintensitiesoffour polarizationsinthe ab plane.Theexternalmagneticeldwasorientedparallel tothe c axisforallspectrashown. fromtwodierentinterferometersontwoseparatecrystalsofC u 3 Bi(SeO 3 ) 2 O 2 Cl(see ExperimentalProceduressection).Thecomplimentarytechnique sonseparatecrystals establishcondenceintheexistenceoftheexcitation. Whentheexternalmagneticeldwasappliedperpendiculartothe c axis( H ? ^ c ) andrampedto10T,theexcitationat33.1cm 1 decreaseditsresonancefrequency quadraticallywitheld(insetFig. 6-9 b).Theelddependenceoftheexcitationforthe E k ^ a +45 polarizationisdepictedinFig. 6-9 b.Itshouldbenotedthatthedetailsof ourexperimentalsetuprequiredthattheexternalmagneticeld beappliedparalleltothe polarizationoftheincominglightinthe H ? ^ c geometry.Noadditionalmodesinthis orientationofexternaleldweredetected. Similarexperimentsinmagneticeldswerecarriedoutinrerectanceg eometry; however,theexcitationsweremuchtooweaktogiverisetorerect ancebands. 97

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n r n n n r n r n n r Figure6-9.Experimentalsupport(a)forclaimingthatthemagnet icresonanceobserved at33.1cm 1 existsbelowthemagneticorderingtemperature(T c 24K). Externalelds,whichacttosharpentheresonance,werealsoap pliedaboveT c forfurthervericationthattheexcitationwasnotobserved.In theinset,the dashedredlinedenotedH MM indicatestheregionwherethemetamagnetic transitionoccursintheH k ^ c geometry.Theisotropicmagneticexcitation's elddependenceinthe H ? ^ c geometry(b)withlightpolarizedalongthe E k ^ a +45 direction. 6.3.6MagneticProperties Theanisotropicresponsetoexternalmagneticeldsofthemagne ticexcitation observedat33.1cm 1 intransmissionhasinspiredaninvestigationoftheanisotropic magneticpropertiesofCu 3 Bi(SeO 3 ) 2 O 2 Clthroughd.c.susceptibilitymeasurements. Theresultsofourd.c.susceptibilitymeasurementsaredepictedinF ig. 6-11 .Isothermal magnetizationmeasurementstakenat5Kexhibitthestronganisot ropicresponsetothe directionofanexternaleldpreviouslynotedintransmissionmeasu rements.Asshown inFig. 6-11 a,with H ? ^ c ,theresultingmagnetization vs. eldloopresemblesthatofan antiferromagnet.Onthecontrary,with H k ^ c (Fig. 6-11 b),antiferromagneticbehavioris observedfrom0to0.1T,followedbyametamagnetictransitionfrom 0.1to0.8T,and thenferroorferrimagneticbehaviorpersistsfrom0.8to5T.Figur e 6-11 cisanenlarged 98

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n n r n Figure6-10.Thelowerfrequencymagneticresonancemodeobser vedintheH k ^ c geometry foreldsof1Tandgreater.Thetwopanelsresultfromcomplement ary techniquesontwoseparatecrystalsofCu 3 Bi(SeO 3 ) 2 O 2 Cl.Experimental limitationspreventedastudyofthemodebelow1Telds. viewofthemetamagnetictransitioninthe H k ^ c geometry.Thehysteresisobservedupon sweepingtheeldislikelyassociatedwiththemetamagnetictransition .Toverifyfurther theantiferromagnetictoferroorferrimagnetictransitionoccur ringinthe H k ^ c geometry, magnetizationwasmeasuredasafunctionoftemperatureineldso f0.01and1T.The resultsareshowninFig. 6-11 d.Thelowtemperaturecancellationofoppositelyaligned momentsexpectedforanantiferromagnetisobservedat0.01T,w hereasastrongferro orferrimagneticmagnetizationisobservedatlowtemperaturefor the1Tmagnetization data. Ourresultsandinterpretationareinqualitativeagreementwithare centreportby M.Pregelj etal. [ 67 ]onthesimilarCu 3 Bi(SeO 3 ) 2 O 2 Brcompound( T c =27.4K).Small dierencesinthemagnetizationsaturizationvaluewith H k ^ c islikelylinkedtodefects arisingfromdierentgrowthconditions.Wenormalizeoursusceptib ilitycurvestomoles ofCuwhileM.Pregelj etal. normalizetomolesofformulaunit(scalingfactorofthree dierence). Theoriginoftheanomalyin1/ ( T ),rstobservedinpowdersamplesbyP.Millet etal. ,[ 66 ]andsubsequentlyobservedbyusalongmultipleaxesofsinglecryst alsof Cu 3 Bi(SeO 3 ) 2 O 2 Cl(Fig. 6-12 )andCu 3 Bi(SeO 3 ) 2 O 2 Br,[ 67 ]remainsanunresolvedissue. 99

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n n n r n n r n n r n n r Figure6-11.Isothermalmagnetizationmeasurementsat5K,whic hiswellwithinthe magneticallyorderedstate,for H ? ^ c (a)and H k ^ c (b,c).Themagnetization asafunctionoftemperature(d)for H k ^ c ateldsaboveandbelowthe metamagnetictransitionasmeasuredwhilewarmingaftereldcooling to 5K. Below150K,theplotof H=M exhibitsaslightcurvature.Wethereforeconcludethatthe dataisnolongeraccuratelydescribedbytheCurie-Weisslawbelow15 0K(adashedline tbelow150KremainsinFig. 6-12 toexemplifythediscrepancy).Thisbehaviorcouldbe indicativeofmorecomplexspin-correlationfunctionsbetween150a nd24Kduetothelow dimensionalandfrustratednatureofthespinsystem. Weobservedtheanomalyforexternaleldsof1Tand0.01Tinbothz eroeld coolingaswellaseldcoolingmeasurements.6.3.7MagneticField-DependentCapacitance Below24Kandwithanexternalmagneticeldorientedparalleltothe c axis,the capacitanceofCu 3 Bi(SeO 3 ) 2 O 2 Clexhibitsanabrupttransitionateldscoincidingwith thatofthemetamagnetictransitionalongthisorientation(0.1{0.8T ).Uponsweepingthe magneticeld,thetransitiondisplayssignicanthysteresis(SeeFig 6-13 ).Proofthatthe capacitancetransitionisassociatedwiththemetamagnetictransit ionisgivenbybothits 100

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r n r n r r n Figure6-12.Plotsof H=M asafunctionoftemperaturemeasuredwitha1Tmagnetic eldorientedparalleltothe a axis(leftpanel),andalsoparalleltothe c axis (rightpanel). nn n n n n nr Figure6-13.Field-dependentcapacitanceofCu 3 Bi(SeO 3 ) 2 O 2 Clat300mKwith H k ^ c temperaturedependence(Fig. 6-14 )anditsdisappearanceuponrotatingthedirectionof theexternallyappliedmagneticeld(notshown). Theexactnatureofthecapacitancetransitionatthetimeofcomp ilingthis dissertationisunknown.Briery,metamagnetictransitionsareoft enrst-orderphase transformationsresultingindiscontinuitiesofamaterial 0 stheunitcellvolume.[ 72 ].The compoundsMn 1 : 8 Co 0 : 2 Sb,[ 73 ]Ca 1 : 8 Sr 0 : 2 RuO 4 ,[ 74 ]andUNiGa[ 75 ]allshowanabrupt 101

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rrr rrrn rrr rrr rrrr n r Figure6-14.Normalizedtemperature-dependentcapacitanceof Cu 3 Bi(SeO 3 ) 2 O 2 Clwith H k ^ c .Upanddownrefertothedirectiontheeldwasswept. changeinvolumeateldsneartheirrespectivemetamagnetictrans itions.Further analysisandexperimentalinvestigations(e.g.,eld-dependentdilat ometry)areneededto determinewhetherthecapacitancetransitionismagnetostrictive innature,orrathera directsignatureofmagnetoelectriccouplinginthiscompound. 6.4Discussion 6.4.1GroupTheoryandObservedModes ThenumberofopticalmodesintheCu 3 Bi(SeO 3 ) 2 O 2 Clcompoundcanbedetermined bygrouptheoryanalysis.UsingtheSMODES[ 49 ]program,wearriveatthefollowing distributionofmodes: optical =14 B ( IR ) 1 u +14 B ( IR ) 2 u +11 B ( IR ) 3 u +12 A ( R ) g +6 B ( R ) 1 g +9 B ( R ) 2 g +12 B ( R ) 3 g +9 A 1 u where(R)and(IR)denoterespectivelyRamanactiveandinfrared activemodes.TheB 3 u B 2 u ,andB 1 u modesareinfraredactivealongthe a b ,and c crystalaxesrespectively.The 9A 1 u modesaresilent.The12A g modesareRamanactiveradialbreathingmodesand 102

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requireparallelincomingandoutgoingpolarizationsaccordingtothe selectionrulesof the Pmmn spacegroup.TheB 3 g ,B 2 g ,andB 1 g modesarealsoRamanactive,butrequire crossedpolarizationofthelight( bc ac ,and ab respectively). At300Kweobserveall11ofthe11predictedB 3 u modesalongthe a axis,all14of the14predictedB 2 u modesalongthe b axis,and11ofthe14predictedB 1 u modesalong the c axis.Thediscrepancybetweenobservedanddetectedmodesalon gthe c axisislikely anexperimentalshortcomingarisingfromthelowsignaltonoisewhe nmeasuringalong thethindimensionofthecrystal.At115Kwedetect8additionalmod esalongthe a axis, 6additionalmodesalongthe b axis,and2additionalmodesalongthe c axis.Tostudy theobservedmodes,thereaderisreferredtoTable 6-1 wherethemodesat7Kalongall threecrystalaxesareidentiedaswellastheirrespectiveLoren tzoscillatorparameters. AnasteriskaftertheTOfrequencyindicatesanewmodethatarise sat115K. Inmaterialswithaninversioncenter,the k =0opticalmodesasmeasuredthrough Ramanandinfraredrerectancearemutuallyexclusive.Whentheinv ersioncenterina structureisremoved,thelocalcentersofsymmetryaboutwhich alltheRamanmodes havezeronetdipolemomentwillberemoved,andallRamanmodeswillt herefore becomeinfraredactive.[ 76 ]Tofurtherinvestigatethe16newmodesobservedat115K, preliminary300KRamanmeasurementswererecordedinthe ab planeofcrystal1. Ramanmodesobservedat172.9,323.5,and484.4cm 1 closelycorrespondtothreeof thenewinfraredmodesobservedbelow115K.Ifwetakeintoaccou ntthefactthat Ramanmodescanslightlyincreasetheirresonancefrequenciesupo ncoolingtolow temperatures,then300KRamanmodesat538.3and583.1cm 1 drawcloseredolence totwoadditionalnewmodesobservedintheinfraredbelow115K.We thereforeturnto acloserexaminationofapossiblecentrosymmetrictonon-centros ymmetrictransitionin Cu 3 Bi(SeO 3 ) 2 O 2 Clnear115K. 103

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6.4.2PowderX-RayDiraction Anoutstandingproblemintheinterpretationofourresultsisthefa ctthatthe powderx-raydiractionmeasurementsdidnotndanyevidencefo raphasetransition accompanyingthedramaticappearanceof16infrared-activemod esbelow115K.Intheir 2001report,P.Millet etal. [ 66 ]soughttodetectapotentialstructuralphasetransition byopticalbirefringence,andtheyplacedanupperlimitof1 onthepossiblerotation ofopticalaxis,thusindicatingnodistortiontomonoclinicsymmetrya tthatlevel.The presentpowderdiractionmeasurementswouldbesensitivetoamo noclinicdistortionon theorderof0.001 ,andnoneisobserved.Theiranalysisoflinearbirefringencesugges t thatthenatureofthesuspectedtransitionissecond-order,wh ichisconsistentwith whatwehavefoundinouranalysisofshiftsinspectralweightfrome xistingmodesto newmodes(insetlowerpanelFig. 6-5 ).Inwhatfollows,weconsiderlower-symmetry non-centrosymmetricorthorhombicstructuresthatmightbeap otentialhostfor Cu 3 Bi(SeO 3 ) 2 O 2 Cl.Inaddition,wescrutinizethepossibilityofatransitiontoan incommensuratelattice,andwediscusstheimplicationsoftherecen tneutrondiraction reportonCu 3 Bi(SeO 3 ) 2 O 2 Br.[ 67 ] Thecentrosymmetric Pmmn spacegrouphasextinctionclass P -n ,i.e.,itobeys theconditionthat h + k mustbeevenfor( hk 0)rerections.Acontinuoustransitionto anon-centrosymmetricorthorhombicsubgroupofthesamelattic edimensionscouldlead tospacegroups P 2 1 2 1 2, Pmm 2, Pm 2 1 n ,or P 2 1 mn .Thetwoformerchoiceswouldallow additionalpowderx-raydiractionpeaks.Wehavecarefullysearc hedforthemthroughout thetemperaturerangebelow115Kwithoutsuccess.Thetwolatte rspacegroupshave thesameextinctionclassas Pmmn ,andsotheyrepresentamechanismforbreakingof inversionsymmetrywithoutproducingaqualitativechangeinthepow derdiraction pattern.Suchadistortionmightnotbeeasytorecognizefrompow derdiractiondata becausetheatomswouldpresumablymoveasmalldistancefromthe irundistorted locations,andsothe Pmmn modelwouldstillgiveareasonablyaccuratedescriptionof 104

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thedataintheacentricphase.Onepossibilityistolookforunusualb ehaviorinthermal displacementparameters,similartothemethodusedtodecodecom plicateddistortionsin magneticallyfrustratedspinels.[ 77 ]Wehavenotbeenabletodetectsuchaneectfrom thedataathand. NeutronpowderdiractionmeasurementsonCu 3 Bi(SeO 3 ) 2 O 2 Brat60Kwere renedin Pmmn ,[ 66 67 ]butthathasonlylimitedbearingonthepresentissue.First, itisnotknownthatdatafromthebromidematerialsuggeststhesa melossofinversion symmetrythatwehavereportedhereforthechloridematerial.Se cond,asnotedabove, thedierencebetweencentricandacentricstructurescouldbev erydiculttoseein neutronorx-raypowderdiraction.Indeed,itisworthpointingou tthatthemagnetic transitionstudiedinthebromideanalogmaywellhaveoccurredfrom asymmetrylower than Pmmn .[ 66 67 ] AnotherpossibilityisthatanincommensuratetransitionoccursinCu 3 Bi(SeO 3 ) 2 O 2 Cl below115K.Incommensuratephasesacquiresatellitex-raydirac tionlinesaround theBraggpeaksofthesymmetricphase.[ 78 ]Wedidnotobservesatellitepeaks,but itispossibletheyexistandweretooweaktobedetectedintheprese ntpowderx-ray diractionmeasurements.Weplantoperformmoresensitivemeasu rements(e.g.,single crystalorneutronpowderdiraction)toresolvethisissue.6.4.3PhononRepulsion Thetypicaldispersionofphononresonancefrequencies(i.e.,soft eningwithincreasing temperature)asdictatedbytheanharmonicterminthelatticepot entialisnotobserved foranumberofthenewmodesandexistingmodesincloseproximityto thenewmodes appearingbelow115K.AccordingtoourpresumptionthatRamanmo desbecomeinfrared activewithalossofinversionsymmetry,weexaminedthephysicsbeh indphononsof similarstrengthsresonatingatcontiguousfrequencies.Figure 6-15 depictsanewmode appearingaround 328cm 1 alongthe a axisthatisincloseproximitytoanexisting modeat 323cm 1 .AsfurtheremphasizedintheinsetofFig. 6-15 ,thetwomodes 105

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n n r Figure6-15.Therealpartoftheopticalconductivityalongthe a axisinthefrequency range300{340cm 1 .At115Kanewmodeappearsaround 328cm 1 that isincloseproximitytoanexistingmodeat 323cm 1 .Therepulsionofthe twomodesastemperatureisdecreasedisemphasizedintheinsetwh erethe Drude-Lorentzresonancefrequenciesareplottedasafunction oftemperature. stronglyrepeloneanotherastemperaturedecreases,whichse emsuncharacteristicof bosonicexcitations.Therepulsionisduetophononmixingandcanbeu nderstoodbyan analogytotheclassicalsystemoftwocoupledidealharmonicoscillat ors.Theclassical problemamountstosolvingfortheeigenvaluesofamatrixwithodiag onaltermsarising fromthecouplingbetweentheoscillators.Ifbothoscillatorsaregiv enthesameinitial frequency(analogoustobothphononsresonatingatthesameen ergy)andthecoupling isturnedon,thentheoscillatorswillrepeloneanother.Therepuls ionincreaseswith increasingcoupling.Applyingthesameconcepttothecaseofphono nmixing,onecan seethatthecouplingbetweenthephononsisincreasedwithdecrea singtemperature, aneectwhichiswellunderstoodviathethermalcontractionofth elattice.Itisalso worthwhilenotingthatthenewphononmodedepictedinFig. 6-5 at 276cm 1 causes arepulsionoftheresonantfrequencyoftheexistingmodeat 256cm 1 below115K. Thetwomodesappeartobeinitiallynon-degeneratebecauseofthe irseparationinenergy ( 20cm 1 );however,asseenintheinsetofFig. 6-5 lowerpanel,thereiscertainlyan interactionbetweenthetwomodes.Thecouplingisveriedfurther bytheshiftsin oscillatorstrengthsbetweenthetwomodes(lowerpanelFig. 6-5 inset). 106

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6.4.4MagneticExcitations Magneticexcitationsstimulatedbyinfraredlightcanbegroupedinto oneoftwo categories:(a)traditionalmagnonsthatareexcitedbythea.c.m agneticeldofthelight, and(b)novelelectromagnonsthatareexcitedbythea.c.electric eldcomponentofthe light.Single-magnonabsorptions,whichatinfraredfrequenciesar ecommonlyobserved inantiferromagnetsandthuslytabbedantiferromagneticresona ncemodes(AFMR), aremagneticdipoletransitionsthatoccurwhentheoscillatingfrequ encyofthelight correspondstothe k =0frequencyofaspinwave.Electromagnonswererstproposed asstronglyrenormalizedspinwaveswithdipolarmomentum.[ 30 ]Sincetheirdiscovery, electromagnonshavebeenextensivelystudiedinrareearthmanga nitecompounds, namelyinTbMnO 3 ,wherealargebodyofrecentliterature(Raman[ 35 ],neutron[ 36 ], andinfrared[ 37 ])hastiedtheexcitationtothelatticeitself.Theworkhasresultedin ageneralizedhybridmagnon-phononmodepictureofelectromagno ns.Challengesarise indierentiatingthetwoaforementionedexcitationsbecausethey resonateinthesame generalfrequencyintervals.Acommonsolutionistomeasurethed ierentfacesofa crystalwhilerotatingthepolarizationoftheincominglight. ThegeometryoftheCu 3 Bi(SeO 3 ) 2 O 2 Clcrystalonlyallowedfortransmission measurementswiththe k vectoroftheincominglightalignedperpendiculartothe ab plane.However,athoroughpolarizationstudywithinthe ab planewascarriedoutandit hasleadtotheobservationofanisotropicmagneticexcitationat33 .1cm 1 .Theseresults contradictthepreviouslyreportedanisotropicnatureofmagnon sandelectromagnons. Moreover,whenexternalmagneticeldsareappliedinthe H k ^ c and H ? ^ c geometries, the33.1cm 1 resonanceshiftstohigherandtolowerfrequenciesrespectively. Inwhat followswewillexaminethenatureoftheexcitationaswellaspropose areasonforits isotropicbehavior.Externalelddependentspectrawith H k ^ c and H ? ^ c willbe discussedseparately.(Thefollowingsectionwillfocusonthe33.1cm 1 excitationbecause itistheonlymodeobservedinzeroeld.) 107

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6.4.4.1Natureandisotropy ToelucidatethenatureofthemagneticexcitationobservedinCu 3 Bi(SeO 3 ) 2 O 2 Clat 33.1cm 1 ,wecomparethestrengthoftheobservedexcitationtotheexte nsiveliterature onmagnonsandelectromagnonsintheinfrared.Ourmagneticexcit ationcreatesapeak in ( )thatis 30cm 1 abovethebaseline.Takingtheaveraged.c.indexofrefraction tobe3inthe ab plane,wedeterminethatourmagneticexcitationcorrespondstoa n opticalconductivityofabout0.23n 1 cm 1 ( 1 = c 4 n ),whichisroughlythesame strengthasotherreportedsingle-magnonexcitations.[ 79 { 81 ]Electromagnons,whichhave beenextensivelystudiedinanumberofrare-earthmanganites,ar eobservedtopossess opticalconductivitiesofatleastafactorof10larger.[ 30 82 ]Althoughthismethodof comparisondoesnotyieldobjectivecertainty,wecanfurthersup portitsconclusion byemployingopticalsumruleanalysisonourmeasuredrerectanced ata.Sincean electromagnoncontributestothedielectricconstant,itmustgain spectralweightfrom adipoleactiveexcitation,themaincandidatesbeingdomainrelaxation s,phonons,or electronictransitions.Thererectivityspectrameasuredat7Kan d30Kdonotshowany changeassociatedwiththemagneticexcitation.Therefore,them agneticexcitationdoes notgainspectralweightfromthelowfrequencyinfrared-activep hononmodes. Theisotropiccharacterofthemagneticexcitationobservedat33 .1cm 1 is depictedinFig. 6-16 wheretheoscillatorstrengthoftheexcitationisplottedversus thepolarizationangleofthelightinthe ab plane.(Itisworthwhilenotingthatinfrared rerectancespectrameasuredoncrystal2revealedthesamest ronganisotropyasdepicted inFig. 6-2 ,thusexcludingtwinningofthesurface.)ArecentreportonTbMnO 3 by Pimenov etal. [ 83 ]detailstheobservationofamagnonandanelectromagnon,active alongperpendiculardirections,resonatingatthesamefrequency .Thisoccurrence givestheillusionofanisotropicmagneticexcitationandisworthyofco nsideration inCu 3 Bi(SeO 3 ) 2 O 2 Cl.Stronglyopposingthisargumentarethenearlyequivalent oscillatorstrengthsmeasuredinanytwoorthogonalpolarizations ,asseeninFig. 6-16 108

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Figure6-16.Apolarplotoftheoscillatorstrengthsassociatedwith the33.1cm 1 modeat 0Tforallfourpolarizationsmeasuredinthe ab plane. Electromagnonstypicallyhaveoscillatorstrengthsthatareatleas toneorderofmagnitude greaterthantraditionalmagnons,asnotedinthecomparisonofo scillatorstrengthsabove. Asecondmoreplausibleexplanationistheoccurrenceofweakmagno nsattwo orthogonalpolarizations.N.Kida etal. [ 82 ]observedasimilarphenomenainDyMnO 3 namely,weakexcitationsarisingwiththea.c.magneticeldorienteda longboththe a and c crystalaxes.Theysupportedtheirinterpretationoftwoortho gonalmagnonsby inelasticneutronscatteringexperimentsonasimilarrare-earthma ganitewhereitwas reportedthattwomagnondispersionscurvesfromorthogonala xescrossed k =0atthe sameenergy.InCu 3 Bi(SeO 3 ) 2 O 2 Cl,wesuspectthatmagnondispersioncurvesfromthe [100]and[010]directionscross k =0at33.1cm 1 ( 4meV);however,inelasticneutron scatteringmeasurementsareneededtosupportourhypothesis 6.4.4.2 H k ^ c elddependence Toanalyzethe H k ^ c spectra(i.e.,Hparalleltotheeasyaxis),wewillutilize ourd.c.susceptibilitymeasurementsaswellasgeneralizethemagne ticstructure determinedforCu 3 Bi(SeO 3 ) 2 O 2 BrtoCu 3 Bi(SeO 3 ) 2 O 2 Cl.Atzeroeld,sixdistinct magneticsublatticescanbeidentied,whichwouldpresumablyleadto sixdistinct 109

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magnonbranches;however,duetothecantednatureofthecop perionsoccupyingthe 4(c)sites,theexactnumberofbranchescoulddierfromsix.Atz eroeld,weonly resolvedonemagneticexcitation(33.1cm 1 ).Noclearsignatureofthemetamagnetic transitioncanbeidentiedwhentrackingthisexcitationateldsbet ween0and1T.In a1Teld,Cu 3 Bi(SeO 3 ) 2 O 2 Clhasalreadyundergoneametamagnetictransitionwhere magneticmomentsoneverysecondlayerrip,resultinginferromagn eticinterlayerand cantedferrimagneticbehavioroverall.[ 67 ](Itshouldbenotedthatthecaxisremains theeasyaxisafterthetransition).Ironically,althoughthemetam agnetictransition eectivelyreducesthenumberofmagneticsublatticesfromsixtot hree,weobservean additionalmagneticexcitationappearingateldsof1Tandgreater .Atthispoint,two logicalquestionsarise:First,whywouldthe33.1cm 1 excitationpresentintheloweld antiferromagnetically-orderedstatepersistsmoothlythrought hetransitionstateandinto thehigh-eldferroorferrimagneticallyorderedstate?Second,w hyaremoremagnetic excitationspresentinthehigheldphasedespiteareductioninthen umberofmagnetic sublattices? Astotherstquestion,magnonsresultingfromferromagneticre sonancehave beenobservedatinfraredfrequencies;however,theytypicallyr esonateatmuchlower frequenciesbecausetheymustextrapolatelinearlytozerofrequ encyatzeroapplied magneticeld.Itisthereforeveryunlikelytoobserveaferromagn eticresonanceat 33.1cm 1 ina1Texternaleld.Butthereisanexceptionalcasethatwasrs tobserved byJacobsinFeCl 2 aproposmetamagnetictransitions.[ 84 ]Whenthemetamagnetic transitionoccursinFeCl 2 ,whichsignalsatransitionfromatwo-sublatticeantiferromagnet toaferromagnet,theAFMRlinedisappearsinfavorofaferromagn eticresonanceline aroundthesamefrequency.Thehighfrequencyoftheferromag neticresonancelinein FeCl 2 isexplainedbylargeanisotropyeldsinthematerial,andtheoretical calculations supportitsexistence.[ 85 ]WesuspectthatasimilarsituationoccursinCu 3 Bi(SeO 3 ) 2 O 2 Cl thatexplainsthesmoothmovementoftheexcitationat33.1cm 1 frombelowtoabove 110

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themetamagnetictransition.Likewise,wesuspectthatthemagne ticexcitationat10.5 cm 1 (1T)alsooriginatesfromanantiferromagneticresonanceline,whic hwouldpossess azeroeldresonancefrequencyof9.5cm 1 (slightlybelowourmeasurablerange). Aproposthesecondquestion,thatistosay,thepresenceofmor eexcitationsabove themetamagnetictransitionversusbelowitcontradictsthereduc tioninmagnetic sublatticesfromsixtothree.Wesuspectthatmoreexcitationsdo existbeneaththe metamagnetictransitionthatareeitherbelowourmeasurablefreq uencyrangeorthatare tooweakforustoresolve(resolution0.3cm 1 ).Wehavethoroughlyinspectedthelow frequencyspectraaswellasthe33.1cm 1 excitationbetween0and1T,andweobserve noclearsignatureofnewresonancesorthesplittingoftheexisting 33.1cm 1 excitation. Itispossiblethatthe10.5cm 1 excitation(at1T)splitsbeneaththemetamagnetic transition;however,below1Texperimentallimitationspreventedu sfromtrackingthe excitation.Futurestudiesinvolvingahigh-resolution,low-frequen cysourceareneededto investigatefurther. Intheinterval1{10Twecanestimatetheeective g factorfromtheslopeof theobservedresonancelinesat10.5and33.1cm 1 usingtheformula ~ = g B H eff .[ 24 ] Chosinganeectiveeldcorrespondingtoaplanesamplewithextern aleldperpendicular totheplane(i.e., H eff = H 4 M s )wearriveatanequationinwhichthe g factorcan independentlybedeterminedfromboththeslopesandinterceptso fthetworesonance lines;however,becausethematerialisnolongerferromagneticbe low0.8T,weutilized themeasuredslopesbetween1and10T.Theslopeoftheresonanc elineat10.5cm 1 correspondstoa g factorof2.16.Theslopeoftheresonancelineat33.1cm 1 varied slightlywiththepolarizationangleinthe ab plane(seesupplementaryinformation). The g factorsandcorrespondinguncertaintiesfortheresonancelinea t33.1cm 1 as measuredalongthe a axis, b axis, a axis+45 ,and b axis+45 are0.24 0.05,0.44 0.11, 0.35 0.04,and0.42 0.05respectively.Thelow g factorobtainedforthe33.1cm 1 resonancelinecanbereconciledinpartbyassociatingitwiththemagn eticmomentson 111

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the4csites,whicharecanted 50 from c towards b .[ 67 ]Theenergyofadipoleis ~m ~ H ormHcos .Assumingthemagneticmomentsonthe4csitesarethesameastho seonthe 2asites,onecanexpectareductioninenergy(slope)ofthereson ancelinebycos50 =0.64. Theresulting g factorestimateisincreased,butnotbyenoughtoobtaintheexpe cted valueforCu 2+ (i.e.,2.0). 6.4.4.3 H ? ^ c elddependence Again,weuseourd.c.susceptibilitymeasurementsandtheneutron scatteringresults onCu 3 Bi(SeO 3 ) 2 O 2 Brtoconcludethatantiferromagneticinteractionsexistwithinthe ab planeatallmeasurableeldsinthe H ? ^ c geometry.Thetheoryofantiferromagnetic resonancesforuniaxialorcubicantiferromagneticcrystalswasw orkedoutbyKeerand Kittel,[ 23 ]whoshowedthatwhentheexternaleldisorientedperpendicular totheeasy axis,thetwo k =0magnonbranchescorrespondtofrequencies !=r = [2 H E H A + H 2 0 ] 1 = 2 Inthepreviousequation, r = ge= 2 mc where g isthespectroscopicsplittingfactor,and H 0 H A ,and H E representthestatic,anisotropy,andexchangeeldsrespectiv ely.Generalizing thistheory,whichwasworkedoutfortwosublattices,toasituatio nwithmorethantwo sublattices,wecanseethatthedispersionoftheexcitationweobs erve(insetofFig. 6-9 b) isinqualitativeagreement(i.e.,botharenon-linear)withthelowerfre quencybranch predictedbytheAFMRtheory.(Furtherexperimentsareneeded todetermine H A H E and g andsubsequentlyttheformulatothedatatoeithervalidateorun dermineour generalization.)Whenthereisnoexternaleld,thetwoabsorption sarepredictedtobe degenerate.However,inantiferromagnetswithstrongmagnetic anisotropy,whichisthe caseforCu 3 Bi(SeO 3 ) 2 O 2 Cl,thedegeneracyofmagnonbranchesinzeroeldislifted(e.g., MnO,[ 70 ]NiO,[ 70 ]andNiF 2 [ 79 ]).Followingourpreviousreasoningthattheresonance observedinCu 3 Bi(SeO 3 ) 2 O 2 Cl,whichmovestolowerfrequencywithincreasingeld,is thelowerfrequencybranchofthe k =0AFMR,weconcludethatthehigherfrequency branchbecomesmasked,thusunobservable,behindthestrongp hononabsorptionsstarting 40cm 1 .AFMRtheoryalsopredictsthatwhenthelowerfrequencybranch reacheszero 112

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theexternaleldisabletorotatethespins,attheexpenseofthe anisotropyeld,away fromtheeasyaxis( c axis)andintothe ab plane.[ 21 ]Extrapolatingthelowerfrequency branchtowardszero,whiletakingintoaccounttheuncertaintyint hedatapoints,we estimateazerofrequencycrossingsomewherebetween18.5and2 0.1T.Ourpredictedeld rangeatwhichthespinsrotateawayfromtheeasyaxisisslightlyhigh er,butremains inroughagreementwiththevaluespredictedontheCu 3 Bi(SeO 3 ) 2 O 2 Branalogue.[ 67 ] ButunlikeCu 3 Bi(SeO 3 ) 2 O 2 Br,wecannotidentifyintermediateandhardaxesbecause themovementofourobservedmodedoesnotseemtodependonth eorientationofthe externaleldwithinthe ab plane. 6.5Summary Thenovelgeometrically-frustratedlayeredcompoundCu 3 Bi(SeO 3 ) 2 O 2 Clhasbeen characterizedusinginfraredspectroscopy,powderx-raydira ction,andd.c.magnetic susceptibilitymeasurements.Far-infraredrerectancemeasure mentshaverevealed16new infrared-activephononmodesbelow115K.Theplethoraofnewmod esobservedstrongly suggestarearrangementofatomicpositionswithintheunitcell;how ever,oursubsequent powderx-raydiractionmeasurementsarecompletelyconsistent withthesame300K structure( Pmmn )existingat85K.PreliminaryRamanspectratakenat300Koncryst al 1haverevealedvephononmodesatfrequenciesclosetoveofth enewmodesobserved intheinfraredbelow115K.Theresultssuggestalossofinversionsy mmetrybelow115K. Uponfurtherinvestigationwehaveidentiedtwonon-centrosymm etricorthorhombic spacegroups( Pm 2 1 n and P 2 1 mn )thathavethesameallowedBraggrerectionpeaksas the300K Pmmn structure.Thereforewesuspectthatasubtlesecond-ordert ransition from Pmmn toeither Pm 2 1 n or P 2 1 mn occursnear115Kthatisbelowtheresolutionof ourpowderx-raydiractionexperiment.Weplantoperformmores ensitivemeasurements ( e.g., singlecrystalorneutronpowderdiraction)toresolvetheuncer taintyofthisissue. Inaddition,anisotropicmagneticexcitationisobservedat33.1cm 1 at5K.We havetentativelyassignedthemagneticexcitationtoamagnonbase donanalysisof 113

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previouslyreportedoscillatorstrengthsofmagnonsandelectrom agnons.Theisotropic behavioroftheexcitationwithinthe ab planeispotentiallyduetotheBrillouinzone centercrossingoftwomagnondispersioncurvesalongorthogona ldirections,butinelastic neutronscatteringmeasurementsareneededtoinvestigatefur thertheexcitation's seeminglyisotropicexistence. Theresonancefrequencyofthe33.1cm 1 excitationstronglydependsonthe orientationofthestaticmagneticeld.Ananisotropicresponseto theorientationofa staticmagneticeldisalsoseenind.c.susceptibilitymeasurementson Cu 3 Bi(SeO 3 ) 2 O 2 Cl, aswellasneutrondiractionmeasurementsonthesimilarCu 3 Bi(SeO 3 ) 2 O 2 Brcompound. Whentheexternalmagneticeldisappliedparalleltothe c axis( H k ^ c ),theresonant frequencyofthe33.1cm 1 excitationincreaseslinearlywithincreasingeld.Forelds of1Tandgreaterappliedalongthe c axisanadditionallinearly-increasingmagnetic excitationisobserved(10.5cm 1 at1T). Whentheexternalmagneticeldisappliedperpendiculartothe c axis( H ? ^ c ),the resonantfrequencyofthe33.1cm 1 excitationdecreasesquadraticallywithincreasing eld.Theresultsareinagreementwiththebehaviorofanantiferro magneticresonance lineinthepresenceofstrongmagneticanisotropy. 114

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Table6-1.Oscillatorparametersfortheinfraredobservedmodes ofCu 3 Bi(SeO 3 ) 2 O 2 Cl(at7K)alongallthreecrystalaxes. Thenewmodesarisingbelow115Kareindicatedwithanasterisknextt otheircorrespondingTOfrequencies. ^ a ^ b ^ c OscStrTOFreqLOFreqFWHMOscStrTOFreqLOFreqFWHMOscStrT OFreqLOFreqFWHM S (cm 1 ) (cm 1 ) r (cm 1 )S (cm 1 ) (cm 1 ) r (cm 1 )S (cm 1 ) (cm 1 ) r (cm 1 ) 3.94752.857.71.726.22136.355.61.90.03553.253.61.51.29169.9 72.41.92.38968.378.81.32.22899.8112.93.9 4.09689.0101.91.90.03499.8 100.21.00.055115.1 118.42.8 0.126101.1 110.75.70.468115.2126.22.40.059144.8145.83.0 0.029137.6138.21.40.071128.9 130.50.90.023161.5162.01.9 0.433161.9164.82.60.154133.5138.21.30.210204.0208.11.60.223172.3 174.11.90.744185.8196.21.20.035273.8 274.71.5 1.540191.6201.41.00.211256.9261.91.20.204284.4289.82.40.115202.1228.33.10.106276.1 279.22.00.093337.6340.52.0 0.085320.0 323.11.80.062300.3302.31.60.154433.5439.34.7 0.045331.1333.62.10.038313.9315.21.60.586528.4547.44.80.067422.9426.04.70.291456.3466.41.90.145554.4579.412.40.055470.2471.612.20.156484.7 489.93.80.136794.9812.76.3 0.139542.4 546.15.00.336507.0528.03.7 0.293557.3577.54.00.059542.3550.23.10.066587.2 587.43.90.045571.4 575.722.0 0.564688.2669.17.10.122716.1 725.95.3 0.122703.8 705.516.80.062730.0767.610.3 0.005737.0 774.019.90.024811.3815.016.1 0.030825.0834.47.9 115

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CHAPTER7 CONCLUSIONS Thisdissertationhasreportedcomprehensiveexperimentalresu ltsforthreenovel complexoxidesinglecrystals.Theinvestigationsweremotivatedbya searchfornew multiferroicsandlargemagnetoelectriccoupling.Theprimarytoolo fcharacterization wasinfraredspectroscopy;however,infraredresultsmotivate dfurtherinvestigationusing complementaryexperimentaltechniquesforthethreematerialsin vestigated.Inwhat follows,anoverviewofthemajorexperimentalndingsforeachma terialwillbediscussed. Therelationshipofourndingstomultiferroicandmagnetoelectricp henomenaaswellas futureaimsofstudywillbehighlightedforeachmaterial. Chapter4detailstheexperimentalndingsofsingle-crystalCu 2 OSeO 3 .Cu 2 OSeO 3 waspreviouslyreportedtoshowmagnetoelectriccouplingviaanano malousjumpinthe dielectricconstant(obtainedbydielectriccapacitancemeasureme nts)attheferrimagnetic orderingtemperature( T c 60K).InthisdissertationthedielectricconstantofCu 2 OSeO 3 asafunctionoftemperaturewasobtainedbyanalternativemetho d.Briery,bytting theinfraredrerectancespectrawithaDrude-Lorentzmodel,th edielectricconstantwas estimatedbytakingthezerofrequencylimitoftheDrude-Lorentz dielectricfunction. Ananomalyintheinfrared-obtaineddielectricconstantwasobserv ed,anditagreedwell inbothmagnitudeanddirectionwiththedielectricanomalyreportedb ycapacitance measurements.Furthermore,latticedynamicalcalculationswere carriedouttofurther analyzethenatureofanomalousphononsthatcontributedtothe jumpindielectric constant.Itwasdeterminedthat2ofthe13modesexhibitinganom alousbehaviornear theferrimagneticorderingtemperaturewerelinkedtothemotiono fcopperions,theions responsibleformagneticordering. SincethetimeoftheinfraredstudyonCu 2 OSeO 3 reportedinthisdissertation,a numberofintriguingreportsofexoticmagnetoelectricphenomena inthetitlematerial havesurfaced.Forexample,Seki etal. [ 86 ]observedmagneticskyrmionsinCu 2 OSeO 3 116

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andalsodeemedthematerialtobemultiferroic.Asubsequentrepo rtbyWhite etal. [ 87 ] detailedhowtheapplicationofanexternalelectriceldorientedpar alleltothe[111]axis andmagneticeldorientedparalleltothe[1 10]axiscouldbeusedtocontrollablyrotate theskyrmionlatticearoundthemagneticeldaxis.Afutureinfrare dstudyutilizingthe newndingsofanexternallycontrollableskyrmionlatticeinCu 2 OSeO 3 (detailedabove) wouldbeofgraveinterest.Specically,measuringtheinfraredpho nonspectrawhile simultaneouslyapplyingexternaleldstoCu 2 OSeO 3 couldpotentiallyshedfurtherinsight intothemicroscopicatomicmotionsresponsiblefortherotationoft heskyrmionlattice. Chapter5reportsonmultiferroicsinglecrystalFeTe 2 O 5 Br.FeTe 2 O 5 Brwaspreviously showntoexhibitthemostdirectformofmagnetoelectriccoupling,n amely,thecasein whichaferroelectricpolarizationarisesasaby-productofalattice distortiondrivenby theonsetofcomplexmagneticordering(T n =10.6K).Theinfraredresultspresented inthisdissertationdidnotcontainthemagnetoelectriccouplingprev iouslyreportedin FeTe 2 O 5 Brdespitetheapplicationofexternalmagneticelds(at5K)orient edalongthe variouscrystaldirections;however,uponorientingasinglecryst alofFeTe 2 O 5 Brusing x-raydiraction,acomprehensivestudyoftheanisotropicphono nspectraofthetitle materialwascarriedout.Withtheaidoflatticedynamicalcalculation s,all52infrared activemodespredictedinthemonoclinic ac planewereaccountedfor.Alongtheunique b axisofthemonocliniccell,43ofthe53modeswereobserved.Inaddit ion,thelattice dynamicalcalculationsrevealedinterestingtrendsrelatingtothem otionoftheconstituent atomsasthefar-infraredwavelengthswerespanned.Furtherm ore,acombinedrerection andtransmissionanalysiswasutilizedthroughoutthemid-infraredr egiontoaccurately obtainthecomplexdielectricfunctionandothercomplexopticalpro perties.Theresults oftheaforementionedanalysisshedlightonthecontradictingmeas urementsofabsorption edgesinFeTe 2 O 5 Br,andtherelatedanalogue,FeTe 2 O 5 Cl. FuturestudiesonFeTe 2 O 5 Brinvolvehigh-resolutionmagneto-terahertzspectroscopy toexaminetheplethoraoflowenergymagneticexcitationspredicte dforfrustrated 117

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antiferromagneticcompounds.Additionally,high-resolutioninfrar edspectroscopyabove andbelowthemultiferroicorderingtemperature(10.6K)areneede dtoobservethe expectedsplittingofphononmodesaccompanyingthecentrosymm etrictonon-centrosymmetric crystalstructurerearrangementmandatedbytheonsetoffe rroelectricity. Chapter6detailstheveryintriguingandnovelCu 3 Bi(SeO 3 ) 2 O 2 Clsystem.The infraredresultsreportedinthisdissertationgivestrongevidence forasymmetry-lowering structuraltransitionexistingbelow115Kviatheplethoraofnewph ononmodesarising alongallthreecrystalaxesoftheunitcell.Surprisingly,powderxraydiractionresults indicatethatthe300Korthorhombicstructurepersistsdownto3 0K.Room-temperature Ramanspectrarevealphononmodesresonatingatenergiescorr espondingtothe energiesofthenewphononmodesintheinfraredbelow115K,thusg ivingstrong experimentalsupportforthelossofinversionsymmetryat115K. Inaddition,magnetic excitationsareobservedat33.1and10.5cm 1 intransmissiongeometryat5K.The excitationsdisappearabovethemagneticorderingtemperature( T c =24K).Whena magneticeldisappliedparalleltothe c axis,ametamagnetictransitionistriggered intheeldrange0.1{0.8T.Themetamagnetictransitionsigniesacha ngefrom antiferromagneticallyorderedspinstoferromagneticallyordered spins.Veryrecent dielectriccapacitancemeasurementshaverevealedananomalyofe lectricnaturecoinciding withthemetamagnetictransition.Itisstillyet-to-bedeterminedw hetherthetransitionis ferroelectricinnatureorratherduetomagneto-strictivepheno mena. FuturestudiesonCu 3 Bi(SeO 3 ) 2 O 2 Clinvolvehighresolutionneutrondiraction toinvestigatefurtherthesuspectedstructuraltransitionat1 15K.Inaddition,the ferroelectricpropertiesofCu 3 Bi(SeO 3 ) 2 O 2 Clwillbemeasuredtoprovidemoreinsightinto thedielectriccapacitanceanomalyobservedtocoincidewiththemet amagnetictransition. 118

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APPENDIXA PRELIMINARYRESULTSONSINGLECRYSTALCu 3 (SeO 3 ) 2 Cl 2 A.1BackgroundandCrystalStructure WorkonsinglecrystalsofCu 3 (SeO 3 ) 2 Clwasongoingatthetimeofcompiling thisdissertation.Alongwithbeingapotentialcandidateforanewmu ltiferroicand magnetoelectricmaterial,Cu 3 (SeO 3 ) 2 Cl 2 isadditionallyofinterestbecauseitwas previouslyreportedtoexistintwodistinctcrystalstructures(t riclinicandmonoclinic). In2000P.Millet etal. [ 88 ]synthesizedCu 3 (SeO 3 ) 2 Cl 2 bymeansofchemicaltransport reaction.Singlecrystalx-raydiractionwasemployedtodetermin ethatthematerial possessedatriclinicstructurewithspacegroup P 1.In2007HelmuthBerger[ 89 ] synthesizedCu 3 (SeO 3 ) 2 Cl 2 alsobychemicalvaportransportreactionand,surprisingly, thematerialwasreportedtocrystallizeinthemonoclinicspacegrou p C 2/ m .Moreover, the2007reportclaimedthatCu 3 (SeO 3 ) 2 Cl 2 possessedalayeredstructureasopposedto theseeminglythree-dimensionalstructurereportedin2001.Its houldbenotedthatthe crystalsofCu 3 (SeO 3 ) 2 Cl 2 reportedinthisdissertationweregrownbyHelmuthBerger. Ourinitialmotivetodeterminecrystalstructuresolelybyopticalm eansisdiscussedin Section A.3 A.2MagneticProperties OurpreliminarymagneticsusceptibilitymeasurementsonCu 3 (SeO 3 ) 2 Cl 2 indicate theonsetoflongrangeorderbelow40K.ThesusceptibilityofCu 3 (SeO 3 ) 2 Cl 2 (not shown)isinqualitativeagreementwithapreviousreportontheisost ructuralcompound Cu 3 (TeO 3 ) 2 Br,whichexhibitslongrangemagneticorderingatT c =70K. A.3Room-temperatureInfraredandRamanResults Thestudyoftheplate-likegeometryoftheCu 3 (SeO 3 ) 2 Cl 2 crystalsallowedonlyfor in-planeinfraredmeasurements.Wehaveidentiedthetwoorthog onalprincipaldielectric axesin-plane(hereafterdenoted ^1and ^2).The300Kinfraredrerectancealongboth axesisshowninFigure A-1 .Weobserve17phononmodeswiththelightpolarized 119

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n n n r FigureA-1.InfraredrerectanceofCu 3 (SeO 3 ) 2 Cl 2 at300Kwithlightpolarizedalongthe twoin-planeprincipaldielectricaxes. paralleltotheprincipalaxis ^1,and22phononmodesareobservedalong ^2.Group theoreticalcalculationsbasedonthetworeportedcrystalstru cturesresultinthefollowing distributionofmodes: monoclinic =11 A ( R ) g +7 B ( R ) g +7 A ( IR ) u +11 B ( IR ) u (A{1) and triclinic =39 A ( R ) g +36 A ( IR ) u (A{2) where(R)and(IR)denoteRamanactiveandinfraredactivemodes respectively.We observemanymorephononmodesthanpredictedbythemonoclinics pacegroupandthree morethanpredictedbythetriclinicspacegroup.Sincetheplate-like natureofthecrystals resemblesalayeredmaterials(layeredmaterialsaretypicallythinalon gthelayeringaxis), wecannotruleoutthepossibilityofadisorderedmonoclinicstructur e. 120

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TableA-1.Thepartialpointgroupsymmetrycharactertableof C 2 h C 2 h EC 2 i h A g 1111 R z x 2 y 2 z 2 xy B g 1-11-1 R x R y xz yz Ramanspectrosocpywasalsoutilizedtoexaminethephononspectr aoftheexact crystalmeasuredintheinfrared.Focusingonthemonoclinicspace group,wenowdiscuss theRamanselectionrulesofthe11A g and7B g modes.Thecharactertableofthe C 2 h pointgroupisshowninTable A-1 .Thestandardcrystallographicconventionis toassociate x withthe a axis, y withthe b axis,and z withthe c axis.Followingthis logic,onegeometrytoselectthe11A g occurswhenboththeincidentlightandscattered lightarepolarizedalongthe a axis(hereafterweadoptthenotation aa ).Tosummarize thetable,A g modesareselectedwith aa bb cc ,and ab .TheB g modesareselected with ac and bc .The2007reportstatesthatthe c axisisthelayeringaxis;therefore,if crystalisinfactmonoclinic,thenthein-planefaceconstitutesthe ab plane.Sincex-ray analysistoorientthecrystalislacking,wehaveassumedthe ab planecorrespondsto thein-planeorientationforpreliminaryRamanmeasurements.Room temperature ab RamanspectrumisshowninFigure A-2 .Weobserve12Ramanactivephononsinthe ab geometry,whichis1morethanpredictedforthemonoclinicspacegr oup.Althoughmore modesareobservedthanpredictedforthemonoclinicspacegroup (sameconclusionasthe infrared),adisorderedmonoclinicstructurestillcannotberuledo ut.Noclearselection rulesexistforthe39A g Ramanmodesinthetriclinicstructure,andthereforetheywill notbediscussed.Powderx-raymeasurementsareexpectedinth enearfuturetofurther investigatetheroom-temperaturestructureofourexactCu 3 (SeO 3 ) 2 Cl 2 crystal. A.4Temperature-dependentInfraredSpectra Infraredrerectanceandtransmittancehavebeenmeasuredbe tween4and300K. Strikingly,asthecrystalofCu 3 (SeO 3 ) 2 Cl 2 iscooled,manynewphononmodesappearin theinfrared.Withlightpolarizedparalleltothe ^1direction10newphononsareobserved. Paralleltothe ^2direction30newphononsarise.Thetemperature-dependentinf rared 121

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n n r FigureA-2.RamanspectraofCu 3 (SeO 3 ) 2 Cl 2 withincidentlightpolarizedparalleltothe a axisandscatteredlightpolarizedparalleltothe b axis.The ab and ba geometriesareidentical,asisthestandardcase. spectraalongthetwoin-planeprincipaldielectricaxesareshowninF igure A-3 .Crimson arrowsdenotetheenergiesofnewmodesarisingintheinfrared. Itisalsonoteworthythatnewinfraredphononsariseintwodieren ttemperature ranges:around80Kandaround40K.Thenewmodesarisingaround 40Ksuggestthat asymmetry-loweringstructuraltransitionaccompaniestheonse toflongrangemagnetic ordering.Figure A-4 depictsaregionalongthe ^2directionwherenewmodesappearin bothtemperatureranges.Magneto-infraredtransmissionmeas urementsineldsupto 5T,orientedbothparallelandperpendiculartotheratfaceofthe crystal,havealsobeen carriedoutattheUniversityofWollongong,Australia.At5Kthemod esdonotshowan observableelddependence. Temperature-dependentpowderx-raydiractionmeasurement sareanticipatedin thenearfuturetodeterminethelowtemperaturestructureand theexistenceofmultiple transitions. 122

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n n n n n r FigureA-3.Temperature-dependentinfraredrerectanceofCu 3 (SeO 3 ) 2 Cl 2 at300Kwith lightpolarizedalongthetwoin-planeprincipaldielectricaxes.Crimsonarrowsdenotetheenergiesofallnewmodesarisingintheinfrared. 123

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n n n n r n FigureA-4.Thefrequencyinterval235{280cm 1 alongthe ^2directionwherenewinfrared modesariseoneithersideofanexistingmodeat 256cm 1 .Theresonance frequenciesofthenewmodesareestimatedfromrerectancepea ksandplotted intheinset.Blackmarkerscorrespondtotheexistingmode,redma rkersto newmodesaround80K,andbluemarkerstonewmodearound40K. 124

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APPENDIXB CALCULATINGSINGLE-BOUNCEREFLECTANCE B.1Preface Certainionicinsulatorsexhibitlowabsorptioninthefrequencyrange above theinfraredactivephononsandbelowtheonsetofelectronictran sitions(typically mid-infraredfrequencies).Dependingoncrystalgeometryandt hickness,thetransmittance cantakeonnitevaluesinthisregion,andthereforethererectan cecontainsanadditional contributionfromthebacksurfaceofthecrystal(cf.Figure 5-5 ).Ourspectralanalysis, whichinvolvesKramers-Kronigrelationstoestimatethephaseshift uponrerectionand subsequentlyinvertingrerectionandphasetocalculatethecomple xresponsefunctions, assumesthatasinglelayerofmaterialisdoingthererection(singlebouncererectance). Briery,formulti-layerrerectionthephaseestimatedfromKramer s-Kronigisstillcorrect; however,invertingrerectanceandphasearenon-trivialproces ses.Analternativeoptionis toestimatethesingle-bouncererectanceusingthemeasurednon -single-bouncererectance andnon-single-passtransmittance.Themethodutilizedinthisdisse rtationissummarized inajournalarticlebyZibold etal. [ 8 ]andisfullydevelopedinthisappendix.Inwhat follows,wewillrestrictourselvestotheincoherentlimitoftransmiss ion(samplethickness muchgreaterthanthewavelengthofthelight)whichisjustiedbye xaminingthe dimensionsreported. B.2Formalism Thecomplexamplitudeoftransmittancethroughaslabofthickness d isgivenby ~ t = p T ( ) e i ( ) =(1 ~ r 2 )~ a (1+~ a 2 ~ r 2 +~ a 4 ~ r 4 + ) ; (B{1) Ionicinsulatorsalsoexhibittransmissionatfrequenciesbelowtheinf rared-active phononsandatfrequencyintervalsbetweentheinfrared-active phonons. 125

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whichcanbere-writtenas ~ t = ~ a (1 ~ r 2 ) 1 ~ a 2 ~ r 2 (B{2) byassuming|~ a ~ r |<1andusingthegeometricseries.Intheaboveequationsthe amplitudeattenuationcoecientisgivenby ~ a = e i 2 ~ nd (B{3) andthecomplexrerectanceisgivenby ~ r = 1 ~ n 1+~ n = p R s ( ) e i ( ) : (B{4) Wenowcomputethepowertransmittance, T = ~ t ~ t ,forthatisthequantityweactually measureinanexperiment.Todosoitisusefultore-writeEq.( B{2 )as ~ t =(~ r 1 ~ r )[(~ a ~ r ) 1 (~ a ~ r )] : (B{5) Forsimplicitywelet~ a = ae i a and~ r = re i r .Utilizingthedoubleangleformulawearrive atthefollowingexpressionforthepowertransmittance: T = r 2 + r 2 2cos2 a ( ar ) 2 +( ar ) 2 2cos2 r +2cos a : (B{6) Usingtherelationcos2 x =1 2sin 2 x wecanputEq.( B{6 )inthefollowingveryuseful from: T = a 2 (1 r 2 ) 2 +4 a 2 sin 2 a (1 a 2 r 2 ) 2 +4 a 2 r 2 sin 2 a +cos r : (B{7) FollowingthemethodologyofE.E.Bell[ 90 ]weintegrateEq.( B{7 )overacycleof a by thestandardtechnique:thetrigonometricfunctionof a isreplacedbyafunctionofa complexvariableandtheresultingintegrandisintegratedaroundth eoriginonacircleof Eq.( B{2 )isutilizedtocomputeabsorptiondirectlyfromtransmissionwhile accountingforrerectionlossesatthesurfaces. 126

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unitradius.Thevaluesoftheintegralsaredeterminedbytheresidu esofthesingularities withintheunitcircle.Theaveragevalueofpowertransmittancetha tresultsisgivenby: T = a 2 (1 r 4 ) 2 a 2 r 2 cos2 r 1 a 4 r 4 : (B{8) Wenowmakethecriticalassumptionthattheimaginarypartofther efractiveindex, k ,is small.Thisassumptionisjustiedbyexaminingtheargumentoftheex ponentialinthe followingequation: e c kd = e 2 kd : (B{9) Asmentionedabove,weareworkingintheincoherentlimit( d>> )so k mustbesmall tokeeptheargumentoftheexponentialnottoobigsoastoprodu ceanitetransmission. Neglecting k allowsthefollowingapproximationtothephaseofrerectance[cf.Eq .( B{4 )]: ~ r = p R s ( ) e i ; (B{10) andhence ~ r ~ r = r 2 = R s : (B{11) With r equalto radiansanddening A ( )=~ a ~ a = e 4 kd wearriveatthefollowing expressionforthepowertransmittance: T ( )= A ( )[1 R s ( )] 2 1 A 2 ( ) R 2 s ( ) : (B{12) Atthispointarelationisnecessarythatexpressesthemeasuredr erectance, R ( ),asa functionofboththesingle-bouncererectance, R s ( ),andmeasuredpowertransmittance, Eq.( B{12 ).ArelationcanbeobtainedbyfollowingtheworkofTinkham[ 91 ]onthin lms.Todoso,thenormalizedadmittance~ y = Z 0 / ~ Z = y 1 iy 2 ( y 1 and y 2 arereal) isintroduced.Therelation ~ Z =1/~ d relatesthecomplexadmittancetothecomplex opticalconductivity.Themeasuredpowerrerectanceandtrans mittancecanbeexpressed 127

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intermsoftherealandimaginarypartoftheadmittanceby R = y 2 1 + y 2 2 ( y 1 +2) 2 + y 2 2 (B{13) and T ( )= 4 ( y 1 +2) 2 + y 2 2 : (B{14) FollowingthepioneeringworkofTinkham[ 91 ]weinvertEq.( B{13 )andEq.( B{14 )to solveforthe y 1 and y 2 bywayoffactoringintheindexofrefractionofthematerial,which canbewrittenintermsof R s ( )(cf.Eq.( B{18 )).Onearrivesatthefollowingdesired relation: R ( )= R s ( )[ T ( ) A ( )+1] : (B{15) Inpracticeonecannowinvert R ( )and T ( )toobtain R s ( )and A ( ).Specically, wesolvefor A ( )inEq.( B{12 )and R s ( )inEq.( B{15 )toobtain A ( )= (1 R s ( )) 2 + p (1 R s ( )) 4 +4( T ( ) R s ( )) 2 2 T ( ) R s ( ) 2 (B{16) and R s ( )= R ( ) T ( ) A ( )+1 : (B{17) Boththemeasuredrerectanceandtransmittance, R ( )and T ( ),mustbeinterpolated tothesamesetofdatapoints(frequencyvalues)tocarryoutth enumericalcalculationof arraysthatensues.Atwocolumnarraywithrstcolumnequaltot hefrequencyvalues determinedinthepreviousstepandsecondcolumnvaluesallinitialized tounityisthen fabricated[label A ( )].Wethencomputethearray R s ( )(Eq.( B{18 ))andusethisresult tore-assignvaluesto A ( )[Eq.( B{16 )].Aniterativeprocessensuesinwhichtheloopis notbrokenuntilaconvergencelimitisreached.Typicallytheconver genceissigniedby reachingalowerlimitofthechangein A ( )fromiterationtoiteration[ A ( )=1 10 6 hasbeenusedsuccessfullyinthepast]. 128

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r "% !#$ nnnn % FigureB-1.Therefractiveindex, n ,extracteddirectlyfromthecomputationof R s ( )(red dashedline)andfromKramers-Kroniganalysisof R s ( )(bluesolidline). Althoughthecalculatedsingle-bouncererectanceshowninFigure 5-5 seemsto completelyeliminatetheelevatedlevelofrerectancestemmingfrom theadditionalback surfacecontribution,itisalwaysassuringtofurthervalidateassu mptionsmade.Wenow examinetherefractiveindex, n ,ascomputedfrombothKramers-Kroniganalysisofthe single-bouncererectanceandalsoasdirectlyextractedfromthe abovecomputationby invertingtherelation R s ( ) ( n 1) 2 ( n +1) 2 : (B{18) (Torecall,wehaveneglected k .)Thetwocalculationsof n arecomparedinFigure B-1 The n extracteddirectlyfromcomputing R s ( )isonlymeaningfulintheregionwherethe crystaltransmits( 1600{10000cm 1 ). Theproceduredetailedinthisappendixcanbesummarizedasfollows. Themeasured rerectanceandtransmittanceinfrequencyintervalswhereboth quantitiesexhibit nitevaluescanbeusedtosolvefortherealpartoftherefractiv eindex, n ,underthe 129

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assumptionthattheimaginarypartoftherefractiveindex, k ,isnottoobig.Thereal partoftherefractiveindexcanbeusedtocalculatethebulk(single -bounce)rerectance throughoutthefrequencyintervalbyinvertingtheapproximatio ngiveninEq.( B{18 ). Thebulkrerectancecanthenbemergedwiththemeasuredrerect anceinregionswhere thetransmissioniszero.(Inregionswherethetransmissioniszero k isnotnecessarily smallandtypicallythemeasuredrerectanceisequaltothesingle-b ouncererectance.) Kramers-Kronigrelationsarethenusedtoextracttherealandim aginarypartsofthe dielectricfunction(andotheropticalproperties)intheusualman ner. 130

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BIOGRAPHICALSKETCH KevinMillerwasborninWashingtonDCin1985.Themajorityofhisearlyy ears werespentinPhiladelphia.Followingatraditionofthemalesinhisfamily,h eattended LaSalleCollegeHighSchool.Subsequently,hewasawardedascholars hiptoSaint BonaventureUniversityinOlean,NewYorkwherehemajoredinPhys ics.SinceSaint Bonaventuredoesnotconductgraduatestudiesinthenaturals ciences,Kevin 0 sonly experiencewithacademic-styleresearchcameduringaresearche xperienceforundergrads (REU)atNotreDameinthesummerof2007(laboratoryofProf.Ja cekFurdyna). FollowinghisgraduationfromSaintBonaventureinMay2008,Kevinde cidedtopursue graduatestudiesinphysics,andindoingso,movedfarawayfromth ecoldrainandsnow ofwesternNewYork.HematriculatedattheUniversityofFloridaint heFallof2008and promptlyjointhegroupofProf.DavidB.Tanner.Inthelastveye ars,hehasmetmany wonderfulpeopleandestablishedbondingswiththemthatwillstren gthenhisworkin physicsforyearstocome.KevinMillerwillbeawardedaDoctorofPhilo sophyinPhysics inMay2013. 136