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Conventional and Time-Resolved Spectroscopy of Magnetic Properties of Superconducting Thin Films

Permanent Link: http://ufdc.ufl.edu/UFE0043583/00001

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Title: Conventional and Time-Resolved Spectroscopy of Magnetic Properties of Superconducting Thin Films
Physical Description: 1 online resource (186 p.)
Language: english
Creator: Xi, Xiaoxiang
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: electrodynamics -- infrared -- magnetic -- pair-breaking -- properties -- quasiparticle -- recombination -- spectroscopy -- superconducting -- thinfilms -- vortex-state
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The magnetic properties of two thin-film type-II superconductors are investigated by optical spectroscopy. We first performed conventional Fourier-transform infrared spectroscopy on the Nb0.5Ti0.5N and NbN samples in a magnetic field. When the field is parallel to the film surface, it breaks the time-reversal symmetry of the Cooper pairs in the thin sample and causes spatial variation of the order parameter in the thick sample, both resulting in pair-breaking effects. The extracted optical conductivity is consistent with the pair-breaking theory. When the field is perpendicular to the film surface, it creates vortices in both samples and weakens the strength of superconductivity. The optical conductivity data of the two samples are consistent with the Maxwell-Garnett theory. We also demonstrated that the pair-breaking effects should be included for an accurate description of the effective electrodynamic response in the mixed state. After elucidating these magnetic-field-dependent equilibrium state properties, we performed time-resolved infrared spectroscopy to study the charge dynamics in these superconducting thin films after photo-excitation. We found that the quasiparticle recombination process is significantly slowed by a parallel magnetic field. The effective recombination rate scales linearly with the photo-induced excess quasiparticle signal, with the slope showing strong field dependence. Our recombination model explains the field dependence though the field-induced pair breaking.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Xiaoxiang Xi.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Tanner, David B.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043583:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043583/00001

Material Information

Title: Conventional and Time-Resolved Spectroscopy of Magnetic Properties of Superconducting Thin Films
Physical Description: 1 online resource (186 p.)
Language: english
Creator: Xi, Xiaoxiang
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: electrodynamics -- infrared -- magnetic -- pair-breaking -- properties -- quasiparticle -- recombination -- spectroscopy -- superconducting -- thinfilms -- vortex-state
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The magnetic properties of two thin-film type-II superconductors are investigated by optical spectroscopy. We first performed conventional Fourier-transform infrared spectroscopy on the Nb0.5Ti0.5N and NbN samples in a magnetic field. When the field is parallel to the film surface, it breaks the time-reversal symmetry of the Cooper pairs in the thin sample and causes spatial variation of the order parameter in the thick sample, both resulting in pair-breaking effects. The extracted optical conductivity is consistent with the pair-breaking theory. When the field is perpendicular to the film surface, it creates vortices in both samples and weakens the strength of superconductivity. The optical conductivity data of the two samples are consistent with the Maxwell-Garnett theory. We also demonstrated that the pair-breaking effects should be included for an accurate description of the effective electrodynamic response in the mixed state. After elucidating these magnetic-field-dependent equilibrium state properties, we performed time-resolved infrared spectroscopy to study the charge dynamics in these superconducting thin films after photo-excitation. We found that the quasiparticle recombination process is significantly slowed by a parallel magnetic field. The effective recombination rate scales linearly with the photo-induced excess quasiparticle signal, with the slope showing strong field dependence. Our recombination model explains the field dependence though the field-induced pair breaking.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Xiaoxiang Xi.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Tanner, David B.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043583:00001


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CONVENTIONALANDTIME-RESOLVEDSPECTROSCOPYOFMAGNETIC PROPERTIESOFSUPERCONDUCTINGTHINFILMS By XIAOXIANGXI ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011 1

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c r 2011XiaoxiangXi 2

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Tomyparents,HuahongXiandLipingZhu 3

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ACKNOWLEDGMENTS IowemydeepestgratitudetomyadvisorDr.DavidB.Tanner,whop rovidedme thisexcitingresearchopportunityandsupportedmethroughout mygraduatecareer.His knowledge,advice,andencouragementhavealwaysbeeninvaluable tome.Iamdeeply indebtedtoDr.G.LawrenceCarrattheBrookhavenNationalLab oratory.Ithasbeen agreathonortoworkwithhim,andtolearnfromhimmanyexperiment alskillsand intuitiveunderstandingsofphysics.IwouldliketothankDr.Christo pherJ.Stanton, Dr.DavidH.Reitze,Dr.PeterJ.Hirschfeld,Dr.ArthurF.Hebard ,andDr.David P.Nortonforservingonmysupervisorycommitteeandfortheirins ightfulcomments. AcknowledgementsalsogotoDr.MarkW.Meiselforstimulatingdiscu ssions. Itisagreatpleasuretothankthosewhohelpedmemakingtheseexp eriments possible.Inparticular,IamgratefultoDr.JungseekHwangfore nlighteningmeatthe initialstageofthiswork.IwouldliketoexpressmyappreciationtoDr .CatalinMartin forcarryingoutsomeoftheexperimentstogether.Thesamplesw ereprovidedbyP. BoslandandE.JacquesatCEASaclay.Thefour-proberesistivitym easurementson thesesamplesweredonewiththehelpofDr.Ju-HyunPark,Dr.Dav idE.Graf,Mr. TimothyP.Murphy,andDr.StanleyW.TozerattheNationalHighMa gneticField Laboratory.Dr.RicardoLoboprovidedvaluabledataacquisitionpr ogramsforthe time-resolvedexperiment.IwouldalsoliketothankmycolleaguesinTa nnerLabfrom whomIlearnedalot.AmongthemareDanielArenas,RichardOttens ,DimitriosKoukis, ZahraNasrollahi,KevinMiller,andNaweenAnand. Thisprojectcouldnothaveprogressedsosmoothlywithoutthehe lpofmany supportingsta.TheusersupportgroupattheNationalSynchr otronLightSource,Gary NintzelandRandySmithinparticular,providedtechnicalsupportd uringallmyvisits. Iwouldliketoexpressmygratitudetothemachineshopmembersoft heUFPhysics Department,whomadeseveralsampleholderswhichwereessentia linmyexperiments.I 4

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amalsogratefultotheelectronicshoptechniciansfortheirhelp,a ndtothesecretarysta formanagingmytripstoBrookhaven. Finally,Iwouldliketoexpressmysincerestthankstomyfamilyandmyf riends.My parentshavealwaysbeensteadfastbelieversinme.Itwasonlywith theirconstantlove andcontinuoussupportthatIcouldhavecompletedthiswork. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 10 LISTOFSYMBOLS .................................... 13 ABSTRACT ........................................ 14 CHAPTER 1INTRODUCTION .................................. 15 2FUNDAMENTALSOFSUPERCONDUCTIVITY ................ 18 2.1HistoryofSuperconductivity .......................... 18 2.2LondonEquations ................................ 19 2.3BCSTheory ................................... 20 2.3.1MicroscopicOriginofSuperconductivity ............... 20 2.3.2EnergyGap ............................... 21 2.3.3CriticalTemperature .......................... 22 2.3.4ExcitationSpectrum .......................... 22 2.3.5PenetrationDepthandCoherenceLength ............... 23 2.3.6OpticalConductivity .......................... 24 2.4Ginzburg-LandauTheory ............................ 26 2.4.1Ginzburg-LandauEquations ...................... 27 2.4.2EectivePenetrationDepth ...................... 28 2.4.3Ginzburg-LandauCoherenceLength .................. 28 2.4.4TypeIandTypeIISuperconductors ................. 29 2.4.5VortexState ............................... 30 2.4.6Thin-FilmSuperconductors ....................... 32 2.4.6.1Criticalcurrent ........................ 32 2.4.6.2Parallelcriticaleld ..................... 33 2.4.6.3Fielddistributioninparallelmagneticeld ......... 34 3EXPERIMENTALTECHNIQUESANDSAMPLES ............... 35 3.1FourierTransformInfraredSpectroscopy ................... 35 3.1.1Principles ................................ 35 3.1.2SourcesinFT-IR ............................ 39 3.1.3DetectorsinFT-IR ........................... 40 3.2Time-ResolvedInfraredSpectroscopy ..................... 42 3.2.1Principles ................................ 42 3.2.2Laser-PumpSynchrotron-ProbeSpectroscopyatNSLS ....... 42 6

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3.2.2.1Lasersystem ......................... 43 3.2.2.2VUVringatNSLS ...................... 44 3.2.2.3Timingschemeandsynchronization ............. 47 3.2.2.4Dierentialtechnique ..................... 51 3.3SampleParameters ............................... 52 3.3.1CriticalTemperature .......................... 53 3.3.2Normal-StateSheetResistance ..................... 54 3.3.3PenetrationDepthandCoherenceLength ............... 55 3.3.4CriticalFields .............................. 56 4MAGNETO-SPECTROSCOPYOFPAIR-BREAKINGEFFECTS ....... 59 4.1Magnetic-Field-InducedEectsonSuperconductivity ............ 59 4.2Motivation .................................... 60 4.3Experimental .................................. 61 4.3.1ConsiderationofFieldOrientation ................... 61 4.3.2TransmissionandRerectioninParallelField ............. 63 4.4Analysis ..................................... 64 4.4.1ExtractionofOpticalConductivity .................. 64 4.4.2AnalysisoftheZero-FieldData .................... 68 4.4.3AnalysisoftheField-DependentData ................. 71 4.4.4Pair-BreakingParameter ........................ 72 4.4.5Pair-CorrelationGapandEectiveSpectroscopicGap ........ 75 4.4.6Sum-RuleAnalysis ........................... 76 4.4.7ComparisonwithHomes'sLaw ..................... 80 4.4.8DensityofStates ............................ 83 4.4.9ConsistencyCheckoftheFits ..................... 84 4.5Summary .................................... 84 5MAGNETO-SPECTROSCOPYOFVORTEXSTATE .............. 87 5.1VortexStateinSuperconductors ........................ 87 5.2Motivation .................................... 88 5.3Experimental .................................. 88 5.4Analysis ..................................... 91 5.4.1ExtractionofOpticalConductivity .................. 91 5.4.2AnalysisofZero-FieldData ...................... 93 5.4.3TheoriesofEectiveOpticalConductivity .............. 94 5.4.3.1Maxwell-Garnetttheory ................... 95 5.4.3.2Bruggemaneectivemediumapproximation ........ 97 5.4.3.3Coey-Clemmodel ...................... 98 5.4.3.4Comparisonoftheories .................... 100 5.4.4AnalysisoftheField-DependentData ................. 102 5.4.5Pair-BreakingEects .......................... 105 5.4.6ConsistencyCheckoftheFits ..................... 108 5.4.7FittingParameters ........................... 108 7

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5.4.8SuperruidDensity ............................ 110 5.4.9DiscussionsofFlux-PinningandFlux-FlowRegimes ......... 113 5.5Summary .................................... 115 6TIME-RESOLVEDMAGNETO-SPECTROSCOPYOFCHARGEDYNAMICS 117 6.1IntroductiontoNon-EquilibriumSuperconductivity ............. 117 6.2Motivation .................................... 118 6.3Experimental .................................. 119 6.4TemperatureDependence ........................... 120 6.5FieldDependence:FieldParalleltoSampleSurface ............. 122 6.5.1Exponential-DecayFit ......................... 123 6.5.2UniversalScalingBehavior ....................... 129 6.5.3RecombinationModel .......................... 131 6.5.3.1Recombinationratecoecient ................ 134 6.5.3.2Spin-polarizationfactor ................... 138 6.5.3.3Constantofproportionality ................. 140 6.5.3.4Totaleectofmagneticelds ................ 142 6.6FieldDependence:FieldPerpendiculartoSampleSurface ......... 143 6.7Summary .................................... 147 7CONCLUSIONS ................................... 150 APPENDIX AUSEFULPHYSICALCONSTANTSANDUNITCONVERSIONS ....... 153 BINTERACTIONOFLIGHTWITHMATTER .................. 154 CTHINFILMOPTICS ................................ 157 C.1FresnelEquations ................................ 157 C.2FresnelEquationsforThinFilmonThickSubstrate ............. 159 C.3ExternalRerectanceandTransmittanceforThinFilmonThickSu bstrate 162 C.3.1MethodofSummation ......................... 162 C.3.2MatrixMethod ............................. 166 DCORRECTIONOFREFLECTIONDATA ..................... 169 D.1CorrectionforStrayLight ........................... 169 D.2CorrectionforFiniteAngleofIncidence ................... 170 D.3TemperatureDependenceoftheOpticalConductivityforNbTiN ..... 171 EDATANOTDISCUSSED .............................. 174 REFERENCES ....................................... 177 BIOGRAPHICALSKETCH ................................ 186 8

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LISTOFTABLES Table page 3-1Sampleparameters .................................. 53 4-1Pair-breakingparameter,eectivespectroscopicgap,andp air-correlationgap 73 4-2Superruiddensityinparallelelds ......................... 78 5-1Normal-volumefraction f .............................. 110 5-2Superruiddensityinperpendicularelds ...................... 111 6-1Fittingparametersforthetemperature-dependentdecayd ataofNbTiN ..... 121 6-2FittingparametersfortheNbTiNdecaydatainparalleleldsatv ariousruences 129 6-3FittingparametersfortheNbNdecaydatainparalleleldsatva riousruences 130 A-1ConversiontableforCGSandSIunits ....................... 153 B-1Relationsbetweenopticalconstants ......................... 156 9

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LISTOFFIGURES Figure page 2-1BCSgapanddensityofstates ............................ 22 2-2OpticalconductivityfromBCSMattis-Bardeentheory .............. 25 3-1InterferometerusedinFT-IR ............................ 36 3-2Laserfocusingoptics ................................. 44 3-3Experimentalset-upfortime-resolvedpump-probespectro scopyatNSLS .... 47 3-4Timingschemefortime-resolvedpump-probespectroscopyat NSLS ....... 48 3-5Resistancevstemperature .............................. 53 3-6TransmittanceofNbTiNandNbNat20K ..................... 55 3-7Paralleluppercriticalelds H k c 2 ........................... 57 3-8Perpendicularuppercriticalelds H ? c 2 ....................... 58 4-1Sampleholdersformeasurementsinparallelmagneticelds ........... 62 4-2 T s = T n and R s = R n inparallelelds ......................... 65 4-3Comparisonof T s = T n T ext ;s = T ext ;n R s = R n ,and R ext ;s = R ext ;n ........... 67 4-4Zero-eldopticalconductivityobtainedinparallel-eldcongur ation ...... 68 4-5OpticalconductivityofNbTiNinparallelelds .................. 69 4-6OpticalconductivityofNbNinparallelelds ................... 70 4-7 1 = n fromSkalski'stheory ............................. 72 4-8Fielddependenceofthepair-breakingparameter .................. 74 4-9Fielddependenceofthegaps ............................ 76 4-10 2 = n vs1 =! inparallelelds ............................ 79 4-11Missingspectralweightinparallelelds ...................... 80 4-12ComparisonwithHomes'slaw ............................ 81 4-13(), T c (),andn G () ............................... 82 4-14Fielddependenceofthedensityofstates ...................... 83 4-15 T s = T n ofNbTiNinparallelelds .......................... 86 10

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5-1Sampleinaperpendicularmagneticeld ...................... 89 5-2Opticallayoutformeasurementsinaperpendicularmagnetice ld ........ 90 5-3 T s = T n and R s = R n inperpendicularelds ...................... 92 5-4Opticalconductivityinperpendicularelds .................... 93 5-5Zero-eldopticalconductivityobtainedinperpendicular-eldc onguration ... 94 5-6Topologyofatwo-componentcompositematerial ................. 95 5-7Comparisonoftheoriesforeectiveopticalconductivity ............. 101 5-8OpticalconductivityofNbTiNinperpendiculareldsttedusing s (0T) ... 103 5-9OpticalconductivityofNbNinperpendiculareldsttedusing s (0T) ..... 104 5-10OpticalconductivityofNbNinperpendiculareldsttedusingp air-breaking s 107 5-11 T s = T n ofNbNinperpendicularelds ........................ 109 5-12Fielddependenceofthenormal-volumefraction f ................. 110 5-13 2 = n vs1 =! inperpendicularelds ......................... 112 5-14Superruiddensityinperpendicularelds ...................... 113 5-15EectiveopticalconductivitycalculatedfromtheCoey-Clem model ...... 115 6-1Illustrationofthequasiparticlerelaxationprocess ................. 118 6-2Typicaldierentialsignalandintegratedsignal .................. 120 6-3TemperaturedependenceofthedecayforNbTiN ................. 121 6-4FittingparametersofNbTiNtemperaturedependentdecay ............ 122 6-5NbTiNdecaydatainparalleleldsatvariouslaserruences ........... 124 6-6NbTiNdecaydatainparalleleldsatvariouslaserruencesonsem i-logscale .. 125 6-7NbNdecaydatainparalleleldsatvariouslaserruences ............. 126 6-8FittingparametersofNbTiNdecaydatainparalleleldsatvariou slaserruences 127 6-9FittingparametersofNbNdecaydatainparalleleldsatvarious laserruences 128 6-10EectiveinstantaneousrelaxationrateforNbTiNandNbNinpa rallelelds .. 131 6-11Exactsolutionoftherecombinationequation ................... 134 6-12Numericalcalculationoftherecombinationrate R ................. 137 11

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6-13MajorityspinfractionofNbTiNandNbNintheparamagneticlimit ....... 139 6-14ConstantofproportionalityforNbTiNinparallelelds .............. 141 6-15SlopevsspectroscopicgapforNbTiNandNbNinparallelelds ......... 142 6-16NbTiNdecaydatainperpendiculareldsatlowandhighlaserruen ces ..... 144 6-17NbNdecaydatainperpendiculareldsatvariouslaserruences ......... 145 6-18EectiveinstantaneousrelaxationrateforNbTiNinperpendic ularelds .... 146 6-19EectiveinstantaneousrelaxationrateforNbNinperpendicu larelds ...... 149 C-1Incident,rerectedandtransmittedwavesattheinterfaceo ftwomedia ..... 158 C-2Incident,rerectedandtransmittedwavesforthinlmonasub strate ...... 160 C-3Incident,rerectedandtransmittedwavesattheinterfaceo fthreemedia .... 163 C-4 T s = T n athighresolution ............................... 168 D-1Multiplicationfactorforthecorrectionofniteangleofincidenc e ........ 171 D-2Temperaturedependenceof T s = T n and R s = R n forNbTiN ............. 172 D-3TemperaturedependenceoftheopticalconductivityforNbT iN ......... 173 E-1 T s = T n and R s = R n forNbTiNatmoretemperatures ................ 174 E-2Temperaturedependenceof T s = T n and R s = R n forNbN .............. 174 E-3 T s = T n ofNbTiNatdierenteldorientations ................... 175 E-4 T = T ofNbTiNatdierentruences,elds,anddelaytimes .......... 176 12

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LISTOFSYMBOLS Penetrationdepth Coherencelength 0 Single-particlegapatzerotemperatureandzeromagneticeld Opticalconductivity T Transmittance R Rerectance Subscript s Usedtodenotethesuperconductingstate Subscript n Usedtodenotethenormalstate R Sheetresistance 0 Magneticruxquantum(approximately2 : 068 10 15 Wb) Z 0 Impedanceoffreespace(approximately376.730n) Pair-correlationgap(orderparameter) 2n G Eectivespectroscopicgap Pair-breakingparameter n s Superruiddensity f Normalvolumellingfraction N Totalquasiparticledensity N th Thermalquasiparticledensity N ex Excessquasiparticledensity S Photo-inducedtransmissionsignal(integratedsignal) dS=dt Dierentialsignal R Recombinationratecoecient R Intrinsicquasiparticlelifetime B Phononpair-breakinglifetime(lifetimeofphononsgivingrisetothebo ttleneckeect) r Phononescapingtime(lifetimeofphononsescapingthesamplewithou tpairbreaking) F Laserruence 13

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy CONVENTIONALANDTIME-RESOLVEDSPECTROSCOPYOFMAGNETIC PROPERTIESOFSUPERCONDUCTINGTHINFILMS By XiaoxiangXi December2011 Chair:DavidB.TannerMajor:Physics Themagneticpropertiesoftwothin-lmtype-IIsuperconductor sareinvestigated byopticalspectroscopy.WerstperformedconventionalFour ier-transforminfrared spectroscopyontheNb 0 : 5 Ti 0 : 5 NandNbNsamplesinamagneticeld.Whentheeldis paralleltothelmsurface,itbreaksthetime-reversalsymmetry oftheCooperpairsin thethinsampleandcausesspatialvariationoftheorderparamete rinthethicksample, bothresultinginpair-breakingeects.Theextractedopticalcon ductivityisconsistent withthepair-breakingtheory.Whentheeldisperpendiculartothe lmsurface,it createsvorticesinbothsamplesandweakensthestrengthofsup erconductivity.The opticalconductivitydataofthetwosamplesareconsistentwithth eMaxwell-Garnett theory.Wealsodemonstratedthatthepair-breakingeectssho uldbeincludedforan accuratedescriptionoftheeectiveelectrodynamicresponseint hemixedstate.After elucidatingthesemagnetic-eld-dependentequilibriumstateprope rties,weperformed time-resolvedinfraredspectroscopytostudythechargedynam icsinthesesuperconducting thinlmsafterphoto-excitation.Wefoundthatthequasiparticler ecombinationprocess issignicantlyslowedbyaparallelmagneticeld.Theeectiverecomb inationratescales linearlywiththephoto-inducedexcessquasiparticlesignal,withthes lopeshowingstrong elddependence.Ourrecombinationmodelexplainstheelddepend encethoughthe eld-inducedpairbreaking. 14

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CHAPTER1 INTRODUCTION Inaconventionalsuperconductor,superconductivityoriginate sfromthepair correlationofelectronsthroughelectron-phononinteraction,a well-knownmechanism fromtheBardeen-Cooper-Shrier(BCS)theory[ 1 2 ].Eachpairconsistsofelectrons ofoppositemomentumandspin( k ; k # ),thereforehavingtime-reversalsymmetry. Conventionalsuperconductorsarecategorizedintotwotypesa ccordingtotheirresponse toamagneticeld[ 3 ].AtypeIsuperconductorexhibitsperfectdiamagnetismupto acriticalmagneticeld H c ,abovewhichsuperconductivityisdestroyed.AtypeII superconductorshowsperfectdiamagnetismbelowalowercritical eld H c 1 ,andreverts tothenormalstateaboveanuppercriticaleld H c 2 .Intheintermediateregimewhere H c 1
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sensitivelyprobed.Suchexperimentstypicallyyieldtheopticalcons tantsofmaterials throughmodel-independentanalyis,whichcanbeusedasastringen ttesttoexisting modelsandtheoriesforthematerialsunderstudy[ 11 ].Far-infraredspectroscopyis particularlyusefulforstudyingsuperconductivity.Becausethe typicalenergyscaleof superconductingenergygapsfallsinthefar-infraredrange,this techniqueiscapable ofcapturingthissignatureofsuperconductivity[ 11 12 ].Itisalsohighlysensitiveto condensationformationorloss,becausetheexcitationsrelevant totheformationof thesuperconductingstateexistinthefar-infared.Combinedwith variableexperiment conditionssuchastemperatureandmagneticeld,itprovidesvalua bleinsighttothe pairingmechanism.Suchadvantagesareempoweredbyusingsynch rotronradiation asthesourceofspectroscopy.AttheNationalSynchrotronLig htSource,Brookhaven NationalLaboratory,wehaveaccesstoamagneto-opticalsetupthatdeliversinfrared lightorders-of-magnitudebrighterthanconventionalsourcesa ndreachesthelowerend ofthefar-infraredspectralrange.Moreover,thepulsedfeat ureofsynchrotronradiation combinedwithafastinfraredlaserisidealforstudyingnon-equilibriu mprocessesin superconductors. Inthisdissertation,wepresentspectroscopicstudiesofmagnet icpropertiesoftype-II superconductors.ThesamplesareNb 0 : 5 Ti 0 : 5 N(abbreviatedasNbTiNhereafter)andNbN thinlmsondielectricsubstrates,suitablefortransmissionandrer ectionmeasurements. Thin-lmsamplesarealsointerestingbecauseoftheanisotropydue tothesample geometry.Ontheonehand,whenamagneticeldisappliednormalto thelmsurface, vorticesarecreated.Thinlmsamplesinthiscasearethereforees sentiallyequivalent tobulksamples.Theelectrodynamicsresponseofsuperconducto rsinthevortexstateis investigatedtotestvarioustheoriesdescribingthisphaseofmatt er.Ontheotherhand, whenamagneticeldisappliedparalleltothelmsurface,onemayminim izeoreven avoidvortex-inducedeects,sothatanyotherinteractionsbet weenthemagneticeldand thesuperconductingelectronscanbestudied.Tunnelingtechniqu ewasappliedforthis 16

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kindofstudyinthepast[ 13 14 ],butopticalstudiesarerare.Ourgoalistoestablisha reliablemagneto-opticaltechniquethattacklestheseproblemsfr omanewperspective. Thisworkisalsomotivatedbyunderstandingthequasiparticledynam icsina superconductorundertheinruenceofamagneticeld.Theeldwa sthoughttocreate vorticesasextrachannelsforquasiparticlerelaxation,expediting therelaxationprocess. Ourexperimentfoundunexpectedslowingoftherelaxation.Wers tdeterminehow magneticeldsaecttheequilibriumsuperconductingstate.Suchin formationassistsour understandingofthenon-equilibriumstate. Thisdissertationisorganizedinthefollowingfashion.Itbeginswithan introduction tothefundamentalsofconventionalsuperconductivityinChapte r2.Manyconcepts andwell-developedfactsaresummarizedtobeappliedinlaterchapte rs.Chapter3 explainstheexperimentaltechniquesandcharacterizesthesamp lesbydeterminingthe materialparameters.InChapter4andChapter5weemployFourie rtransforminfrared spectroscopytoextracttheopticalconductivityoftwothin-lm samplesinbothparallel andperpendicularmagneticelds.Thedataareinterpretedinterm softhepair-breaking eectsintheformercase,andtheeectiveelectrodynamicrespo nseofthevortexstate inthelatter.InChapter6weprobethequasiparticlerecombination dynamicsusing time-resolvedpump-probespectroscopy.Thenon-equilibriumres ultsareanalyzedbased ontheequilibrium-statepropertiesstudiedinChapter4andChapte r5.Chapter7 summarizesthemainresultsandpointsouttopicsforfuturestudie s. Ininfraredspectroscopyitisconventionaltousewavenumbersa stheunitof frequency.Thewavenumberisdenedasthereciprocalofthewa velengthorthenumber ofwavelengthperunitdistance,withtheunitcm 1 .1cm 1 correspondstoawavelength of1cmandafrequencyof30GHz.Intheeldofsuperconductivity CGSunitsare preferred.Theseconventionsarefollowedthroughoutthisdisse rtation.Conversionsfrom CGSunitstoSIunitsareincludedinAppendix A forreference. 17

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CHAPTER2 FUNDAMENTALSOFSUPERCONDUCTIVITY 2.1HistoryofSuperconductivity Onecenturyago,soonafterheestablishedthetechniquesoflique fyinghelium,H. KamerlinghOnnesdiscoveredsuperconductivityinmercury[ 15 ].Themysteryofthis phenomenonhasattractedgreatattentionofscientistseversin cethen.Twodecadeslater, WaltherMeissnerandRobertOchsenfeldfoundthatsuperconduc torsrepelmagnetic elds[ 16 ];theperfectdiamagnetismisreferredtoastheMeissnereect.S ubsequently, superconductivitywasdiscoveredinavarietyofmetals,alloys,and compounds,some withthecriticaltemperature( T c )above10K,e.g.Ref.[ 17 ].Meanwhiletheoretical understandingofthemechanismofsuperconductivitywasunderp rogress.In1935, F.andH.Londonproposedtwoequationstodescribetheelectrod ynamicproperties ofsuperconductors[ 18 ],whichexplainedtheMeissnereect.In1950Ginzburgand Landauproposedaphenomenologicaltheorythatsuccessfullyde scribedthemacroscopic propertiesofsuperconductingelectrons[ 19 ].Atruebreakthroughwasthetheoryby Bardeen,Cooper,andSchrierin1957[ 1 2 ],whichforthersttimeexplainedthe microscopicoriginofsuperconductivityasthecondensationofelec tronpairsmediated byelectron-phononinteraction.Theeldwasrevitalizedin1986whe nIBMresearchers BednorzandMullersynthesizedaceramiccompoundcomposedofla nthanum,barium, cooper,andoxygen,whichsurprisinglysuperconductsat30K[ 20 ].Oneyearlater whenresearcherssubstitutedyttriumforlanthanum,asuperco nductorwith T c of 92Kwasdiscovered[ 21 ],beingtherstmaterialthatsuperconductsatatemperature greaterthanliquidnitrogen.Thissignicantlyreducesthecostofc ooling,making high-temperaturesuperconductorspopularcandidatesforvar iousapplications[ 22 23 ]. Thetermhigh-temperaturesuperconductorswascommonlyused interchangeablywith cuprates,whichstandsforcooper-oxidesuperconductors,un tilafewyearsagoanew familyofsuperconductorswasfoundinlayeredironarsenidemater ialswith T c ashigh 18

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as56K[ 24 25 ].Thevariousfamiliesofhigh-temperaturesuperconductorsshar esome commonfeaturesbutarealsodramaticallydierentinmanyaspects [ 26 ].Uptonowthere isnouniedtheoryonthemechanismofhigh-temperaturesuperco nductors. Inthisdissertationwewilldealwithconventionalmetallics-wavesup erconductors. Thereforesomeoftheirbasicpropertiesrelatedtoourstudieswill bereviewedinthis chapter.Sincethetheoriesofconventionalsuperconductorsa rerelativelycomplete,the followingreviewistheory-based,citingimportantconclusionsforth euseinlaterchapters. MoredetailsofthissubjectcanbefoundinRefs.[ 3 27 { 31 ]. 2.2LondonEquations TheLondonequationsaretwoequationsrelatingthecurrentande lectromagnetic wavesinasuperconductor.F.andH.Londonadoptedthetworuid modelofC.J.Gorter andH.B.G.Casimir[ 32 ],inwhichitwasproposedthatthetotaldensityofelectrons iscomprisedoftwocomponents,oneofwhichiscondensedinto\sup erruid"thatis responsibleforthesuperconductingproperties,andtheotherf ormaninter-penetrating ruidof\normal"electrons.Thesuperconductingelectronsaretr eatedasanidealelectron gaswithdensity n s ,whichislimitedbythetotaldensityofconductionelectrons n inthe systemandhastheempiricaltemperaturedependence n s ( t )= n 1 T T c 4 # : (2{1) TheLondonbrothersarguedthatanexternalelectriceld E couldfreelyaccelerate thesuperconductingelectrons,makingthemrowwithoutresistan ceandcreatinga supercurrentofcurrentdensity J s .TherstLondonequationdescribesthisphenomenon ofperfectconductivity, @ J s @t = n s e 2 m E : (2{2) ThisequationisreminiscentofOhm'slaw, J = E =( ne 2 =m ) E ,where n isthefree electronnumberdensityand istheelectronicmeanfreetimebetweenioniccollisions. 19

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ItisevidentfromthisequationthattousetheLondonequations,t heresponseofthe superconductingelectronstotheexternaleldsshouldbelocal. ThesecondLondonequation, r J s = n s e 2 mc H ; (2{3) togetherwith r H =(4 =c ) J s fromMaxwell'sequations,yields r 2 H = H 2L : (2{4) H representsthelocalvalueofthemagneticeld,whichvariesovert helengthscale L = s mc 2 4 n s e 2 ; (2{5) calledtheLondonpenetrationdepth,typicallytensofnanometers inmetallicsuperconductors [ 28 ].Eq.( 2{4 )thereforeexplaintheMeissnereect:theexternalmagnetice ldisshielded fromtheinteriorofbulksuperconductors.Since n s n L hastheminimumvalueat 0K, L (0)= p mc 2 = 4 ne 2 ,anddivergesat T c accordingtothetemperaturedependenceof n s givenbyEq.( 2{1 ), L ( T )= L (0) p 1 ( T=T c ) 4 : (2{6) 2.3BCSTheory 2.3.1MicroscopicOriginofSuperconductivity TheBCStheorysuccessfullyexplainsthemicroscopicmechanismofs uperconductivity. Superconductivityarisesbecauseelectronsformpairsandconde nseinthegroundstate. Thetheorydoesnotrelyonthespecicmechanismofpairing,aslong astheattractive interactionexists.Inreality,thepairingbetweenelectronsismedia tedbyphonons: oneelectronpolarizesthesurroundingmediumbyattractingpositiv eionsfromthe lattice,whichinturnattractsasecondelectron.Theeectiveatt ractionbetween thesetwoelectronsoverridestheirCoulombrepulsion,andaCoope rpairisformed. 20

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Superconductivitycanonlyoccuratsucientlylowtemperaturesw henelectronsnearthe FermisurfacebecomeunstableagainsttheformationofCooperp airs.Quantitativeresults oftheBCStheoryareintheweakcouplinglimitwhentheattractiveint eractionbetween pairedelectronsissmall.2.3.2EnergyGap BCStheoryndsanenergygapbetweenthegroundstateandthe quasiparticle excitationstate.Theexistenceofagapinthesuperconductingst atewashintedbythe e a=k B T ( a> 0)temperaturedependenceoftheelectronicspecicheatatlowt emperature, andconrmedbyelectromagneticabsorptionspectrum.Inthegr oundstateat0K,the gap 0 isobtainedfrom 1 N (0) V = Z ~ c 0 d p 2 + 20 =sinh 1 ~ c 0 ; (2{7) where N (0)isthedensityofstatesattheFermienrgyforelectronsofon espinorientation, V istheparingpotential,and ~ c isthecut-oenergycharacterizingthepotential V abovewhichthepairingpotentialiszero. ~ c isoftheorderoftheDebyeenergy ~ D that characterizesthecut-oofthephononspectrum.Inmetalsitist ypicallytensofmeV.In theweakcouplinglimit N (0) V 1, 0 2 ~ c e 1 =N (0) V : (2{8) Thisresultindicatesthat 0 isonlyaverysmallfractionoftheDebyeenergy. Thetemperaturedependenceofthegap( T )isobtainedfromtheintegral 1 N (0) V = Z ~ c 0 tanh p 2 + 2 = 2 k B T p 2 + 2 d: (2{9) ( T )issolvednumericallyandplottedintheleftpanelofFigure 2-1 .Itcanbewell approximatedbytheanalyticalform 0 = vuut cos 2 T T c 2 # : (2{10) 21

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Figure2-1.Left:temperaturedependenceoftheBCSgap.Right :BCSdensityofstates. remainsverycloseto 0 atsucientlylowtemperatureswhen T=T c 1.Itdrops monotonicallyasthetemperatureincreases,anddrasticallyappro acheszeroatthe transitiontemperature T c .Notethatas T T c ,( T ) / p 1 ( T=T c ) 2 according totheapproximationgivenbyEq.( 2{10 ),whiletheexactformfromEq.( 2{9 )yields ( T ) / p 1 T=T c 2.3.3CriticalTemperature Thecritical(ortransition)temperature T c canbedeterminedfromEq.( 2{9 )by settingtozero.Ityields k B T c 1 : 13 ~ c e 1 =N (0) V : (2{11) FromthisandEq.( 2{8 )weseethatintheweakcouplinglimit, 0 k B T c 1 : 76 ; (2{12) ingoodagreementwithexperimentalvalues.2.3.4ExcitationSpectrum Theexistenceofelectronpairsandthegaprequiresthatgenera tionsofexcited quasiparticlesarealsocreatedinpairs,consumingenergyof2(te rmedthespectroscopic gap).Thetermquasiparticleistodistinguishitfromnormalfreeelec trons,becausein 22

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superconductorsthereareinteractionsbetweentheseexcited electronsandtherestofthe system.Thesequasiparticleshaveenergy E k = p 2 k + j k j 2 ,where k isthesingle-particle energyrelativetotheFermienergy,and k playstheroleofanenergygap.Thethermal distributionofquasiparticlesobeystheFermi-Diracfunction f ( E k )= 1 e E k =k B T +1 ; (2{13) with E k isequaltoorgreaterthanthesingle-particlegap. Thesingle-particledensityofstates N s ( E )isgivenas N s ( E ) N (0) = 8><>: E p E 2 2 ( E> ) 0( E< ) (2{14) shownintherightpanelofFigure 2-1 .Thegureshowsthatquasiparticlesonlyexist withenergygreaterthanthesingle-particlegap.Thedensityofs tatesdivergesatthe gap,indicatingthatquasiparticlestendtoaccumulatejustabovet hegap. 2.3.5PenetrationDepthandCoherenceLength ItwasmentionedbeforethattheLondonpenetrationdepth L isonlyvalidinthe locallimit.TheBCStheorygivesasolutiontothemoregeneralpenet rationdepth thatcanbeappliedtoboththelocalandnon-locallimit.Thenon-loca llimitapplies tomaterialswithhighFermivelocity v F ,low T c ,andlongmeanfreepath l ,andwas discussedbyPippard[ 33 ]beforetheBCStheory.BCStheorynds ( 2L 0 ) 1 = 3 inthis limit,where 0 istheBCScoherencelength 0 ~ v F 0 : (2{15) 0 isameasureofthedistancebetweenthetwoelectronsinaCooperp air. Boththepenetrationdepthandthecoherencelengthdependont hepurityofthe superconductingmaterials.Fortype-IIsuperconductors(de nedbelowinSection 2.4.4 )in 23

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thedirtylimit, 1 withelectronmeanfreepath l ,thepenetrationdepthandthecoherence lengthare[ 28 ] L 0 l 1 = 2 ; (2{16) ( 0 l ) 1 = 2 : (2{17) 2.3.6OpticalConductivity MattisandBardeeenderivedtheopticalconductivityofaBCSsupe rconductorin thedirtylimit,wellknownastheMattis-Bardeentheory[ 34 ].Denethecomplexoptical conductivityas = 1 + i 2 .Accordingtothistheory,therealandimaginarypartsofthe opticalconductivityofdirty-limitsuperconductorsare 1 n ( )= 2 ~ Z 1 [ f ( E ) f ( E + ~ )] E p E 2 2 E + ~ p ( E + ~ ) 2 2 1+ 2 E ( E + ~ ) dE + 1 ~ Z ~ [1 2 f ( E + ~ )] j E j p E 2 2 j E + ~ j p ( E + ~ ) 2 2 1+ 2 E ( E + ~ ) dE; (2{18) 2 n ( )= 1 ~ Z ~ !; [1 2 f ( E + ~ )] E p 2 E 2 E + ~ p ( E + ~ ) 2 2 1+ 2 E ( E + ~ ) dE: (2{19) Here n isthenormal-stateopticalconductivity,isthetemperature-de pendent gapgivenbyEq.( 2{9 ),and f ( E )istheFermi-Diracfunction.Thesecondintegralin Eq.( 2{18 )doesnotappearunless ~ !> 2,inwhichcasethelowerlimitintheintegralof Eq.( 2{19 )shouldbe .ThesecondintegralinEq.( 2{18 )isthereforethecontribution fromphoto-excitedquasiparticles,andtherstintegralinEq.( 2{18 )isthecontribution fromthermally-excitedquasiparticles.NumericalresultsofEqs.( 2{18 )and( 2{19 )at T =0KareshowninFigure 2-2 .Therealpartincreaseswithaniteslopejustabove 1 Theimpurityconcentrationinasuperconductorischaracterizedb ytheratio l= 0 .If l= 0 1,thematerialissaidtobeclean,whilefor l= 0 1,thematerialisdirty. 24

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Figure2-2.Left:real(solidline)andimaginary(dashedline)partso ftheoptical conductivityat T =0KfromtheBCSMattis-Bardeentheory.Right:the imaginarypartoftheopticalconductivity(solidline),withonepartr elatedto the functionin 1 (dashedline),andtheotherpartrelatedtothenite frequencypartof 1 (dottedline).Theinsetshowsthesamequantitiesplotted on1 =! scale. theopticalgap2 0 ,eventhoughatthesingle-particlegapthedensityofstatesdiver ges. Thisisbecausethecoherencefactoroftheform1+ 2 =E ( E + ~ )inEq.( 2{18 ) increasesgraduallyjustabovethesingle-particlegap.Suchafact orarisesbecausethe superconductingstateconsistsofphase-coherentsuperposit ionofstates( k ; k # ) occupiedorunoccupiedasaunit.Whencalculatingtheelectromagne ticabsorptionasa scatteringprocess,interferencetermsappearinthetransition probability,whichisabsent inthenormalstate. Theoscillator-strengthsumrule Z 1 0 1 ( ) d! = ne 2 2 m ; (2{20) requiresthattheareaunder 1 ( ),denedasthespectralweight,shouldbeconserved forboththenormalandsuperconductingstates.Here n isthetotalelectrondensity, and e and m arethechargeandmassoftheelectron.Theratio 1 = n inFigure 2-2 is alwayslessthanunity.The\missingarea"(orthemissingspectralw eight) A = n s e 2 = 2 m 25

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condensestoa functionatzerofrequency,whichisameasureofthecondensate density [ 35 ].Therealpartcanthereforebewrittenas 1 ( )= A (0)+ 1 ;> ( )= n s e 2 2 m (0)+ 1 ;> ( ) ; (2{21) where 1 ;> ( )denotesthepartof 1 atnitefrequencyplottedintheleftpanel ofFigure 2-2 .Theimaginarypartisrelatedtotherealpartthroughthefollowing Kramers-Kronigrelation, 2 ( )= 2 P Z 1 0 1 ( 0 ) 0 2 2 d! 0 ; (2{22) where P denotestheprincipalvalueoftheintegral.Accordingtothisequat ion,the imaginarypartoftheopticalconductivityhasthefollowingform, 2 ( )= A + 2 ;> ( )= n s e 2 2 m! + 2 ;> ( ) ; (2{23) where 2 ;> ( )denotesthepartof 2 relatedto 1 ;> ( ).Thecontributionfromthetwo termsin 2 arecomparedintherightpanelofFigure 2-2 .Clearly,thetermcomingfrom the functionin 1 dominatesthebehaviorof 2 belowthegap,andisresponsiblefor thealmostfrequencyindependentpenetrationdepth = c= p 4 2 .Thesepropertiesof 1 and 2 areobservedinmostmetallicsuperconductors(Sn,In,Pb,Hg,et c.),although strong-couplingeectsaresometimesnecessaryforquantitativ eagreement[ 36 { 40 ]. TheopticalconductivityofBCSsuperconductorswitharbitrarye lectronmeanfree pathisdiscussedbyW.Zimmermann etal .inRef.[ 41 ]. 2.4Ginzburg-LandauTheory GinzburgandLandauintroducedacomplexorderparameter ( r )asthepseudo wavefunctionofthesuperconductingelectrons. ( r )isproportionaltothegapparameter ( r ). j ( r ) j 2 representsthelocalsuperruiddensity n s ( r ).IntheBCStheory,thegapis homogeneous.Ginzburg-Landautheorydealswithamoregeneral caseinwhichspatial inhomogeneityexists,e.g.inhomogeneityinducedbyanexternalma gneticeld,which 26

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couldbepresentinourelddependentstudies.Thegapcanvarywit hposition,andis generallycomplex.Gor'kovprovedthattheGinzburg-Landautheo ryisalimitingcaseof theBCStheory[ 42 ]. 2.4.1Ginzburg-LandauEquations Thetheorybeginswiththeconstructionofthefreeenergyofasu perconductor.When issmallandvariesslowlyinspace,thefreeenergydensity f hastheform f = f n 0 + j j 2 + 2 j j 4 + 1 2 m ~ i r e c A 2 + H 2 8 ; (2{24) where f n 0 isthenormal-statefreeenergydensityintheabsenceofmagnetic elds, e =2 e m =2 m A isthevectorpotentialcorrespondingtothemicroscopicmagnetic eld H and H 2 = 8 representsmagneticenergyinvacuum.Thecoecients and arediscussed below,givenbyEqs.( 2{34 )and( 2{35 ).Minimizing f withrespecttoructuationsinthe orderparameter andvectorpotential A yieldstheequation, + j j 2 + 1 2 m ~ i r e c A 2 =0 ; (2{25) whichdeterminestheorderparameter = ( ;; A ( r )).Withthissolution,the supercurrentdensitycanbeevaluated, J s = e ~ i 2 m ( r r ) e 2 m c j j 2 A : (2{26) Eqs.( 2{25 )and( 2{26 )arethefamousGinzburg-Landauequations. Becausebothequationsspecializetoaposition r ,theGinzburg-Landautheory isalocaltheory.NotethatintheLondongaugetheLondonequatio nsEqs.( 2{2 ) and( 2{3 )canbewrittenasasingleequation, J s = n s e 2 A =mc [ 28 ].Comparingthis withEq.( 2{26 ),weseethattheGinzburg-Landautheorypredictsanextrater minthe supercurrentdensitywhichoriginatesfromthespatialvariationo ftheorderparameter. Eq.( 2{26 )thereforereducestotheformgivenbyLondonequationsinthelim itthatthe orderparameterisspatiallyhomogeneous. 27

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2.4.2EectivePenetrationDepth AccordingtoEq.( 2{25 ),intheweakeldlimit,totherstorderin H 2 canbe approximatedasitszero-eldvalue j 1 j 2 = ; (2{27) whichisspaceindependent.TakingthecurlofEq.( 2{26 )andusing r H =(4 =c ) J s yields, r 2 H = 4 e 2 j j 2 m c 2 H : (2{28) Thereforethemagneticeldisscreenedaccordingtoaneectivep enetrationdepth e = s m c 2 4 e 2 j j 2 = r m c 2 4 e 2 : (2{29) Asthetemperatureapproaches T c j j 2 isdrivento0,and e diverges.Notingthat m =2 m e =2 e ,and j j 2 = n s = n s = 2astheCooperpairdensity, e isconsistentwith theLondonpenetrationdepth L givenasEq.( 2{5 ). 2.4.3Ginzburg-LandauCoherenceLength Intheabsenceofeldsorcurrents,andforreal withtheform = 1 y ,Eq.( 2{25 ) becomes 2 d 2 y dx 2 + y y 3 =0 ; (2{30) where istheGinzburg-Landaucoherencelength = ~ p 2 m j j : (2{31) FromEq.( 2{30 )itisclearthat isthecharacteristiclengthforthevariationof y (or equivalently ).Inotherwords,itgivesanapproximatespatialdimensionofCoop erpairs. Thethermodynamiccriticaleld H c isdenedfromthesuperconducting-statefree energydensity f s andthenormal-statefreeenergydensity f n as H 2 c 8 = f s f n = 2 2 ; (2{32) 28

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whichhastheempiricaltemperaturedependence H c ( T ) H c (0) 1 T T c 2 # : (2{33) Eqs.( 2{32 ),( 2{27 ),and( 2{29 )determinethecoecients and intermsofthe measuredquantities e and H c = 2 e 2 mc 2 H 2 c 2e ; (2{34) = 16 e 4 m 2 c 4 H 2 c 4e : (2{35) SubstitutingEq.( 2{34 )intoEq.( 2{31 )yields ( T )= 0 2 p 2 H c ( T ) e ( T ) ; (2{36) inwhich 0 = hc= 2 e =2 : 068 10 15 Wbistheruxoidquantum(ormagneticrux quantum).TheGinzburg-Landaucoherencelength diersfromtheBCScoherence length 0 giveninEq.( 2{15 ),butthetwoarerelated.UsingEq.( 2{15 ), H 2 c (0) = 8 = 1 2 N (0) 20 ,and N (0)=3 n= 4 E F forfreeelectrons,onecanderivethat ( T ) 0 = 2 p 3 H c (0) H c ( T ) L (0) e ( T ) : (2{37) ItturnsoutthatatverylowtemperaturetheGinzburg-Landauc oherencelengthis approximatelythesameastheBCScoherencelength.2.4.4TypeIandTypeIISuperconductors TypeIsuperconductorsaretotallycharacterizedbytheMeissne reect,andhavea criticalelddenedasthethermodynamiccriticaleld H c givenbyEq.( 2{32 ).Stronger perturbationsinducedbytheeldgreaterthan H c destroyssuperconductivity.Simple metallicsuperconductorssuchaspuresamplesoflead,mercury,a ndtinbelongtothis type.Theyusuallyhavesmallpenetrationdepth (forsimplicitythesubscript\e" isomittedhereafter,and meanstheeectivepenetrationdepth e unlessotherwise stated),buthighFermivelocitythuslargecoherencelength,mak ing < 29

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Intheoppositelimitwhen ismuchgreaterthan ,itwouldbeenergetically favorableforhighmagneticeldstopenetratepartiallythesuperc onductorandreduce themagneticenergywithoutreducingthecondensationenergy.A teldsbelowthelower criticaleld H c 1 thesuperconductorstillexhibitstheMeissnereect.Above H c 1 and belowtheuppercriticaleld H c 2 ,eldspartiallypenetratethesuperconductorinthe formofquantizedruxtubes(vortices),eachcarryingaruxof 0 alongthedirection oftheappliedeld.Theoreticallythesetubesformaregulararray, butinpractice symmetriesoftheunderlyingcrystalstructureorthedefectsin thematerialsdetermine thevortexdistribution.Thecoreregionofavortexissimpliedasac ylinderwith aradiusofthecoherencelength ,inwhichthesuperconductingstateisquenched. Thesurroundingregionofradius r< istheelectromagneticregionwhereeldand supercurrentspersist.Thewholematerialisthereforeinamixeds tateinthisintermediate regime H c 1 1 = p 2aretypeIIsuperconductors. 2.4.5VortexState UsingtheGinzburg-Landautheory,Abrikosovpredictedtheexist enceoftype-II superconductorsandthemixedstatewhen > 1 = p 2,wellbeforesuchphenomenawere experimentallyobserved[ 43 ].FortypeIIsuperconductorstheGinzburg-Landautheory 30

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nds H c 1 inthelimit 1tobe H c 1 = ln p 2 H c ; (2{39) and H c 2 inthegeneralcasetobe H c 2 = 0 2 2 = 4 H 2 c 2 0 = p 2 H c : (2{40) IngettingthelasttwoequalitiesinEq.( 2{40 ),Eq.( 2{36 )and( 2{38 )areused.If H c 1 < H
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2.4.6Thin-FilmSuperconductors Thinlmsuperconductorsdierfrombulksuperconductorsintheir magnetic properties.Considerathinlmsuperconductorofthickness d inthelimit d .Assume thespatialvariationoftheamplitudeofthepseudo-wavefunction j j canbeneglected,so that ( r )= j j e i ( r ) .TheGinzburg-Landautheorycanbeusedtocalculatethecritical currentandtheparallelcriticaleldofthethinlm.Theelddistribu tionisalsosolved below.2.4.6.1Criticalcurrent Letasupercurrentdensity J s rowintheplaneofthelm.Wealsoconnethe discussiontothelimit d sothat J s canbeassumedconstant.Thesupercurrent density J s fromEq.( 2{26 )intheselimitsis J s = 2 e m ~ r 2 e c A j j 2 =2 e j j 2 v s (2{41) where v s =( ~ r 2 e A =c ) =m isthevelocityoftheparticledescribedbythe pseudo-wavefunction .Thefreeenergyinthesamelimitsis f = f n 0 + + 2 j j 2 + 1 2 m v 2 s j j 2 + h 2 8 : (2{42) Minimizingthefreeenergywithrespectto j j yields + j j 2 + 1 2 m v 2 s =0 ; (2{43) whichgivesthesolution j j 2 = 1 m v 2 s 2 j j = j 1 j 2 1 m v 2 s 2 j j : (2{44) Thissaysthatthesuperruiddensitydecreasesquadraticallyasth evelocity v s increases. ThesupercurrentdensityEq.( 2{41 )becomes J s =2 e j 1 j 2 1 m v 2 s 2 j j v s : (2{45) 32

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At v s = p 2 j j = 3 m J s hasitsmaximumvalueas J c J c =2 e j 1 j 2 2 3 r 2 j j 3 m = 4 3 p 3 e ~ j 1 j 2 m = cH c 3 p 6 ; (2{46) whereingettingthelasttwoequalitiesEqs.( 2{31 ),( 2{34 ),( 2{35 ),and( 2{36 )areused. Atcurrentdensityhigherthan J c thereisnosolutionof v s toEq.( 2{45 );thethinlm becomesnormal.Hence J c iscalledthecriticalcurrent. 2.4.6.2Parallelcriticaleld Applyanexternalmagneticeld H paralleltothethinlmsurface.Takethe directionnormaltothelmsurfaceas z axis.WechoosetheLondongaugesothatthe vectorpotentialonlyhascomponent A x Hz .Thephase isconstantinthisgauge, making v s = 2 e A =m c .ByminimizingtheGibbsfreeenergyofthelm,itcanbefound that j j 2 = j 1 j 2 1 d 2 H 2 24 2 H 2 c : (2{47) Notingtheproportionalitybetweenthegapand ,theaboveequationmeansthatlow magneticeldssuppressthegapalmostquadratically.Theparallelc riticaleld H c k is denedasthevalueatwhich 0, H c k =2 p 6 H c d : (2{48) ForatypeIIsuperconductor, .Therefore d .Theparallelcriticaleldof athin-lmtypeIIsuperconductorcanbesignicantlyhigherthanit sthermodynamic criticaleld.Eq.( 2{48 )isvalidaslongasthelmthickness d< p 5 .Belowthiscritical thickness,themagnetictransitiontothenormalstateat H c k isofsecondorder;aboveit thetransitionbecomesrstorderasinbulksamples.Thetransition becomesrstorder againforextremelythinsampleswithsmallspin-orbitscattering,inw hichthePauli paramagnetismdominatesoverdiamagnetisminthedeterminationof thecriticaleld, knownastheChandrasekhar-Clogstonlimit[ 44 45 ]. 33

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2.4.6.3Fielddistributioninparallelmagneticeld Theelddistribution h ( z )forathinlminahomogeneousparallelmagneticeld H a canbesolvedanalytically.Here h denotesthemicroscopiceld,whichdistinguishesit fromthespatiallyaveragedmacroscopiceld H .TheGinzburg-Landauequationsdescribe thelocalresponse.Inthelimit j j isconstanttheyreducetotheLondonequations. Eq.( 2{4 )thereforeapplies,withthepenetrationdepthgivenbyEq.( 2{29 ).Assumethat thethinlmoccupiesthespacefrom z = d= 2to z = d= 2.SolvingEq.( 2{4 )usingthe boundaryconditions h ( d= 2)= h ( d= 2)= H a yields h ( z )= H a cosh( z= ) cosh( d= 2 ) : (2{49) Themacroscopiceld H canbefoundbyaveragingthis h overthelmthickness, H = 1 d Z d= 2 d= 2 h ( z ) dz = H a 2 d tanh d 2 : (2{50) ThissolutioncomesdirectlyfromLondonequationsandtheboundar ycondition,thus donotrequire d .Howevertorequiretheeldtobeequaltotheappliedeld H a at bothsurfaces,thesampleshouldbethinenoughsothattheeldisn otdisturbedatthe interfaces.Inthelimit d ,onends H H a 1 d 2 12 2 : (2{51) Themacroscopiceldinsidethethinlmisthereforeveryclosetothe appliedeld. 34

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CHAPTER3 EXPERIMENTALTECHNIQUESANDSAMPLES 3.1FourierTransformInfraredSpectroscopy Toobtainmid-infraredandfar-infraredspectra,Fourier-trans forminfraredspectroscopy (FT-IR)canbeused. 1 Itsimultaneouslydetectsallfrequency,thereforeishighlyecien t comparedtodispersiveinstruments(multiplexadvantage).Ithas muchhigherthroughput thanconventionalspectrometersrequiringaslitaperturetoach ieveresolution(Jacquinot Advantage).Italsoprovidesmeasurmentswithhighresolution.3.1.1Principles ThemostcrucialcomponentofanFT-IRspectrometerisaMichelso ninterferometer, showninFigure 3-1 .Abeamsplitterbisectstheincidentcollimatedbeamfromthe source.Onebranchofthebeamisrerectedtowardsaxedmirror ,andtheotherbranch istransmittedtowardsamovablemirrororientedperpendiculartot hexedone.The twobranchesarererectedbythetwomirrorsandmeetatthebea msplitterwithan opticalpathdierence(orretardation) introducedbythemovablemirror,resultingin interference.Foramonochromaticsourceofwavenumber withintensity I ,atanoptical pathdierence thedetectedsignalis I 0 ( )= 1 2 I [1+cos(2 )] : (3{1) Asthemovablemirrorchanges ,thedetectedsignalchangessinusoidally.Whenthe twobeamspropagatingtothedetectorinterfereconstructively ,thedetectedsignalis I equaltothesourceintensity.Whentheyinterferedestructively ,thedetectedsignalis0; alltheincidentpowerreturnstothesource.Theoscillatoryparto f I 0 ( )isdenedas interferogram, I ( )= 1 2 I cos(2 ) : (3{2) 1 Foradetailedintroductiontothistechnique,seeforexampleRef.[ 46 ] 35

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Infrared WhieLightSourceHeNeLaser FixedMirror MovableMirror GePhotodiode SiPhotodiode InfraredDetedctor SecondaryInterferometer BeamSplitter Source Figure3-1.InterferometerusedinFT-IR.Thesecondaryinterf erogramforsamplingis alsoshown. Inreality,thedetectedsignalisalwaysaectedbybeamsplittere ciency,detector responseandampliercharacteristics.Incorporatingtheseee ctsintoEq.( 3{2 )doesnot changethecosineterm,butmodiesitspre-factor I toanewvaluedenotedas B ( ).The practicallydetectedinterferogramistherefore S ( )= B ( )cos(2 ) : (3{3) Theaboveequationcanbereadilygeneralizedtothecaseofabroad bandsource, S ( )= Z + 1 1 B ( )cos(2 ) d: (3{4) Thelowerintegrationlimitisextendedto 1 foreasiermathematicaltreatment.In reality B ( )isunderstoodtobezerofornegative .TheFourier-transformofEq.( 3{4 )is thesingle-beamspectruminthefrequencydomain, B ( )= Z + 1 1 S ( )cos(2 ) d: (3{5) 36

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Therefore,toobtainasingle-beamspectrumisatwo-stepproces s:collecttheinterferogram andconvertittothefrequencydomain. Inrealityitisimpracticaltoscanthemovablemirroroveraninnitelylo ngdistance. TheupperandlowerlimitsoftheintegralinEq.( 3{5 )aredeterminedbythelongest distancethemovablemirrorcantravel,denotedas0 : 5.Thecorrespondingopticalpath dierence =determinestheniteresolutionofthespectrum, =1 = ,usually callednominalresolution.Anotherpracticalproblemformeasuring theinterferogramis thatitisimpossibletoobtain S ( )atallvaluesbetween and.Theinterferogram canonlybediscretelysampledatniteincrements.Thechallengeisto choosean optimalsamplingmethodbasedonwhichthetheoreticalfunctionca nbeunambiguously reconstructed.AccordingtotheNyquistsamplingtheorem,thes amplingrateshouldbe atleasttwiceofthebandpassofthespectrum.TheNyquistfrequ encyisdenedashalf ofthesamplingrate.Forexample,ifthespectrumisintherangebet ween0and max ,a minimumsamplingrateof2 max isrequired,orequivalentlythesamplingintervalshould notbegreaterthan1 = 2 max .Ifthespectrumliesbetween min and max ,thesampling intervalshouldnotbegreaterthan1 = 2( max max ).Moreover,anysignalatfrequency higherthantheNyquistfrequencywillbefoldedbackintothesamplin grangeifitis detected,causingaliasing.Electronicandopticalltersshouldthe reforebeusedtocuto thesignaloutsideofthesamplingrange. Typically,toobtainonespectrumtheintereferogramissampledrep eatedly, N times,toincreasethesignal-to-noiseratio,whichisproportionalt o p N .Toaddthe interferogramscoherently,correspondingpointsintherepeate dinterferogramsmust besampledatthesameretardation.Themethodforachievingthisd ependsonthe specicFT-IRspectrometer.Thediscussionthatfollowsisspecic totheBruker113. Inthismodel,coherentlyaddinginterferogramsisaccomplishedwith thehelpofa secondaryinterferometer(Figure 3-1 ),whichhasitsownbeamsplitterbutsharesthe samescannerwiththemaininterferometer.Awhitelightgenerates aninterferogram 37

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withanarrowcenterburst.Oncethisisdetected,samplingisinitiate d.Insteadof usingtheNyquistfrequency,inpracticeitismoreconvenientandac curatetousethe interferogramofahelium-neonlaserasthereferenceforsampling .Thisavoidsthe demandingtoleranceonthescannervelocityforaccuratesampling .Thehelium-neon laser,whichismonochromaticatwavenumber HeNe =15798cm 1 ,generatesasinusoidal interferogramdetectedindependentlybyaSiphotodiode.Samplin gistriggeredatevery otherzerocrossingoftheinterferogram.Thisallowsspectrumto bemeasuredupto onehalfofthelaserfrequency, max =7899cm 1 ,sucientforthefar-infraredand mid-infrared.Oversamplingateveryzerocrossingdoublestheupp erlimitto15798cm 1 InBruker66thislimitcanbeextendedtoabout47400cm 1 Oncetheinterferogram S ( )iscollected,itisconvertedtoasingle-beamspectrum throughFouriertransformalgorithmsbyacomputer.Foranaccu rateconversion,two complicationshavetobeaddressed. Firstly,theniteretardationmodiesEq.( 3{5 )byatruncationfunction, B 0 ( )= Z + 1 1 S ( )( )cos(2 ) d = B ( ) f ( ) ; (3{6) wheretheboxcartruncationfunction( )isunityfor and0otherwise; f ( ) isitsFourier-transformpair f ( )=2sinc(2 ) : (3{7) ThelaststepinEq.( 3{6 )istheresultoftheconvolutiontheorem.Thefunction f ( ) haslargesidelobesandcancausespectralleakageatthosefrequ encies.Tosuppressthe magnitudeoftheseoscillatorysidelobes,aremedyistouseapodizat ionfunctionsinstead oftheboxcarfunction( ).AcommonlyusedapodizationfunctionistheNorton-Beer function.Eventhoughtheapodizationfunctionscanbringaninter ferogramsmoothly downtozeroattheedgeofthesampledregion,thetrade-oisthe broadeningofthelines andthereforethedecreasingoftheresolution. 38

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Secondly,aphasecorrectionhastobeapplied.Thisistocompensat eforpossible errorsindeterminingthezeropathdierence,andforfrequency -dependenterrorsinduced byelectronicsandvariousopticalcomponents.Theseerrorsinco mbinationresultina frequency-dependentphaseerror ( )intheinterferogram, S ( )= Z + 1 1 B ( )cos[2 ( )] d: (3{8) Correctionsaretypicallymadebycalculatingthephaseerror fromtheinterferogram, ( )=arctan Im( ~ B 0 ( )) Re( ~ B 0 ( )) # (3{9) whereImandRedenotetheimaginaryandrealpartsof ~ B 0 ( ),thecomplexFourier transformoftheinterferogram.Awidely-usedschemeforphase correctionistheMertz method[ 47 ].Usingthismethod,thephase-correctedsingle-beamspectrumis B 00 ( )=Re( ~ B 0 ( ))cos( ( ))+Im( ~ B 0 ( ))sin( ( )) : (3{10) Thisisthenalsingle-beamspectrumusedindataanalysis.3.1.2SourcesinFT-IR Thecommonlyusedinfraredradiationsourceishigh-temperatureb lackbody,modied bythematerialemissivity.Dependingonthefrequencyrange,die renttypesofsources areused.Typically,high-pressuremercurylampisusedinthefar-in frared(10{400cm 1 ). Globar(aresistivelyheatedsiliconcarbiderod,operatingat 1300K)isusedinthe mid-infrared(400{4000cm 1 ).Inthenear-infrared(4000{12800cm 1 ),commonsources aretungstenlampandquartz-tungsten-halogenlamp[ 46 ]. Thefar-infraredregionisespeciallydiculttocoverbyablackbodys ource,because thespectralenergydensitydecreasesprecipitouslyinthelong-w avelengthlimit,according toPlanck'slaw.Inprinciple,Globarcanalsobeusedinfar-infraredsp ectroscopy. Howeveritsemissivitybecomestoolowbelow100cm 1 ,wherethemercurylampis preferred[ 48 ].Theplasmaofthemercurygasdischargecanreachaneectivete mperature 39

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of 5000Kandhashighemissivity.Theplasmaisenclosedinaquartzenvelo pewhichis opaqueabove 100cm 1 ,makingmercurylampcomparabletoGlobarinthefar-infrared above100cm 1 [ 49 ].Inthefar-infrared,theradianceofsynchrotronradiationcan be three-orders-of-magnitudehigherthanconventionalthermal sources,makingitthebest sourceforfar-infraredspectroscopy[ 50 ]. Synchrotronradiationislightproducedfromrelativisticelectronst raversingthe magneticeldinasynchrotronstoragering.Theelectronscirculat einthering,producing sharpelectriceldpulseswhichgiverisetobroadbandradiation.The radiationis extractedandcarriedtoendstationsaroundthestoragering,c alledbeamlines,where experimentsareperformed.Althoughmostsynchrotronlightsou rcesareoptimizedto produceX-rayandvacuumultravioletradiation,radiationintheinfr aredregionismuch moreintensethanthatfromconventionalthermalsources.Bec ausethelightoriginates fromasmallpacketofelectrons,thesourcecanbetreatedasap ointsource.Thus, infraredlightfromasynchrotroncanbeeasilycollimatedandfocuse dtodiraction limitedspotsizes,allowinghighspatialresolutionforinfraredspectr omicroscopyandhigh spectralresolution.Furthermore,theoperatingprinciplesfort hestorageringrequirethe electronstocirculateinshortbunches,thustheobservedradiat ionatagivenbeamlineis pulsed.Thisuniquefeatureofthesynchrotronallowstimingmeasur ementsoftypically nanosecondtimescale,whichisdiscussedinSection 3.2.2 3.1.3DetectorsinFT-IR Infrareddetectorscanbegenerallycategorizedintotwotypes: thermaldetectors andphotondetectors.Duetofundamentallydierenttypesofno iseoccurringduringthe detectoroperation,thesetwoclassesofdetectorshavediere ntdetectivitycharacteristics. Photondetectorsarefavoredinthenearandmid-infraredspect ralrange,andthermal detectorsarefavoredinthefar-infrared.Goodintroductionst oinfrareddetectoroperating principlesandperformancecharacteristicscanbefoundinRefs.[ 51 { 53 ]. 40

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Thermaldetectorsoperatebysensingthetemperaturechange duetotheheating fromIRradiation.Thistemperaturechangeinduceschangesinelec tricalconductivity, thermoelectricvoltageorpyroelectricvoltage,whichareusedtog eneratesignaloutput. Thistypeofdetectorsarefundamentallylimitedbytemperatureru ctuationnoisearising fromradiantpowerexchangewiththesurroundingbackground.C ommonlyusedthermal detectorsaredopedsiliconbolometers,dopedgermaniumbolomete rs,andpyroelectric detectorssuchasDTGS(deuteratedtriglycinesulfate). Photondetectorssensetheelectromagneticradiationasphoton sinsteadofwaves. Theyaremostlymadeofsemiconductors,andarelimitedbygenerat ion-recombination noisearisingfromphotonexchangewithradiationbackground.Int hemid-infrared, commonlyusedphotondetectorsincludeMCT(mercurycadmiumtellu ride,HgCdTe)and InSb,butthermaldetectorssuchasDTGSarealsouseful.Inthe near-infrared,avariety ofphotondetectorsareavailable,e.g.,InSb,InAs,Ge,InGaAs,I nAs,PbSe,PbSe,andSi. Forthedetectionoffar-infraredradiation,liquid-helium-cooledbo lometersprovide highsensitivitybecausethethermalandbackgroundructuationn oiseissignicantly reducedwhenthebolometerelementiscooledtotheliquidheliumtempe rature(4.2K). Theworkdiscussedinthisdissertationwasmainlydoneinthefar-infr ared,usinga doped-siliconbolometerfromInfraredLaboratories.Theheliumre servoircanbepumped sothattheliquidheliumundergoesaphasetransitiontosuperruidhe liumIIatthe lambdapoint(2.17K).Continuouspumpingreducestheoperatingte mperatureto1.6K, almostdoublingthedetectivitywhencomparedtoabolometerdesign edtooperateat 4.2K[ 48 ].Thespectralresponseislimitedbyafar-infraredcut-ontypelt eratthe entranceoftheconeleadingtothesiliconelement.Thislteriscooled bythehelium reservoirandsignicantlyimprovesthedetectivity. 41

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3.2Time-ResolvedInfraredSpectroscopy 3.2.1Principles Withtheadvanceinultrafastlasertechnology,time-resolvedspec troscopyhas becomeapowerfultoolforthestudyofexcitationsinmaterials.I nthesimplestform thepulsetrainfromalaserisdividedintotwo,onegeneratingexcitat ionsinthesample manifestedaschangesinthesample'sspectralproperties,andth eotherprobingthe inducedchanges.Acontrollabledelayisintroducedbetweenthetwo pulsetrainsso thatthesample'sspectralpropertiesinthevariousphasesofthe relaxationbacktothe equilibriumstatecanbestudied.Thisiscalleddegeneratepump-prob espectroscopy. Thetechniquecanalsobeoperatedinanon-degeneratecongura tion,inwhichthe pumpandprobepulsetrainscomefromdierentsourcesbutarealw ayssynchronized. Thisaddstherexibilityofprobingthesample'sopticalpropertiesinth einterested frequencyrange,whichcouldbedierentfromthelaserfrequenc y.Thetypicallymeasured propertiesarethephoto-inducedchangesintransmission,rerec tion,luminescence,Raman scattering, etc. Thepulsewidthoftheprobebeamisusuallythelimitingfactorofthe timeresolution,whichdeterminesthetimescaleofmeasurabledynam icalprocesses. State-of-the-artfemtosecondlasersmakeitpossibletostudyd ynamicsoffemto-second scale[ 54 ].Thetechniqueprovestobeapowerfultooltounderstanddyna micalprocesses andexcited-statepropertiesofconventionalsuperconductor s[ 55 { 57 ]andtouncoverthe complexelectronicstructureandpairingmechanisminhigh-tempera turesuperconductors [ 58 { 61 ]. 3.2.2Laser-PumpSynchrotron-ProbeSpectroscopyatNSLS Thetimingstructureofsynchrotronradiationallowslaserpulsesof highrepetition ratetobesynchronizedtothesynchrotronpulsesfromastorag eringtoperform time-resolvedpump-probespectroscopy.Wehaveaccesstosuc hapump-probesetup attheNationalSynchrotronLightSource(NSLS),BrookhavenN ationalLaboratory.A near-infraredlaserexcitesthesamples.Thesynchrotronradiat ionisusedastheprobe, 42

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makinguseofthefeaturethatpulsetrainsareradiatedbyelectro nbunchescirculatingin thering.Thoughnotappropriateforthestudyofultrafastphen omenaduetothelimited timeresolution,thislaser-synchrotronsystemhasafewadvanta ges.Thebroadband natureofsynchrotronradiationallowstheprobebeamtospanane xceedinglywide frequencyrange.Inthefar-infarred,synchrotronradiationa lsohasbothbrightness andpoweradvantagesoverconventionalthermalsources,incr easingthesensitivityof themeasurements.Thissectiondescribesthelasersystemandth esynchrotronfacility atNSLS,aswellasthetechniqueofsynchrotron-lasersynchron izationusedinthe time-resolvedpump-probeexperiment.Manydescriptionsinthisse ctionarefromprivate correspondencewithG.LawrenceCarr.AnothersourceisRef.[ 62 ]. 3.2.2.1Lasersystem ATi:sapphirelaserfromCoherentLaserGroupisusedtophoto-exc itesamples. Ithasatunablefrequencyrangeinthenear-infraredfrom700nm to950nm.When mode-locked,itproducespulsesofabout2psindurationand20nJp eakenergyperpulse withapulserepetitionfrequency(PRF)of105.8MHz.Thislaserisopt icallypumpedby acontinuous-waveNd:VO 4 laseroperatingat532nmwithamaximumpowerof10.5W. ThesetwolasersarehousedinsidetheU6Beamlinelaserhutch,witha ninterlocksystem implementedtoensuresafetyoflaseroperations.Thepulsesfrom theTi:sapphirelaser arecoupledintoopticalbercableusingastandardbercoupler,a ndtransportedto BeamlineU4IRwiththeberendingseveralcentimetersawayfromt hewindowonthe OxfordMagnetCryostat.Thelightexitingtheberdivergesatave rtexangleofabout 32 .Atwo-lenssystemisusedtofocusthelaserbeamontothesample( Figure 3-2 ).The rstlens(C)hasa30mmfocallengthandapproximatelycollimatesth elaserbeamfrom theber.Asecondlens(F)with 250mmfocallengththenfocusesthelightatthe samplelocation(G)inthemagnet.Whenperfectlyfocused,thesma llestspotsizeisabout 0.5mm.Thelenstubewiththesecondlensisheldinavacuumcompressio n-typetting. AnARcoatedBK-7glasswindow(E)upstreamofthesecondlenssep aratesairfromthe 43

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A B C F G E D Figure3-2.Schematicforthelaserfocusingoptics(notdrawntos cale).Thelaserbeamis shownastheredrays.Therstlens(C)collimatesthebeamfromth eber. Thesecondlens(F)focusesthebeamatthesamplelocation(G).Af ocusing collar(A)allowstheadjustmentofthedistancefromthebertoLe nsC.The X-Ystage(B)allowssteeringofthebeambytranslatingtheberco upler. Thelargesquarebox(D)isusedtoplacepolarizers.Eisaglasswindow to separateairfromvacuum. roughbeampipevacuum.Afocusingcollar(A)ontheberallowsthed istancefromthe bertotherstlenstobeadjusted,thusvaryingthefocaldista nceinsidethemagnet. Thisenablesustodefocusthelaserspottoapproximatelyllthesa mpleaperture,which istypically 6mmindiameter.Thelaserbeamcanbesteeredontothesampleusing an X-Ystage(B)fortherstlens.Polarizerscanbeplacedinthesqua rebox(D)without aectingthefocusbecausethebeamiscollimatedbetweenthetwole nses.Toensure thatthelaserbeamisdirectedtothesampleandllthesampleapert ure,alow-power reddiodelaser(0.95mW,670nm)isusedforvisuallyaligningandfocusin gthebeam beforetheopticalberisaxedinfrontofthemagnetwindow.The laserpowercan bemeasuredintheU6LaserhutchandatBeamlineU4IRwithapowerm eter.Dueto couplinglosses,thepowerdeliveredtothebeamlineisapproximately7 0%oftheinput value.3.2.2.2VUVringatNSLSOperationmodes TheNationalSynchrotronLightSourcehastwostoragerings, anX-rayringandaVUV(vacuumultraviolet)ring.Experimentspres entedinthis dissertationweremostlyperformedatBeamlineU4IRontheVUVring .Theelectrons intheVUVstorageringareacceleratedbyradio-frequency(RF)c avitiestorestore theirenergylostthroughsynchrotronradiation.Duetotheoscilla tionoftheRFeld, 44

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onlyelectronsarrivingatsomeparticulartimesareaccelerated,re sultinginelectrons movinginbuncheswithbunchlengthsrangingfrom 1nsdowntoafew10sofps.The emittedsynchrotronradiationisalsopulsedasaconsequence.The RFacceleratingsystem operatesat52.9MHz,makingtheminimumspacingbetweenadjacent electronbunches 18.9ns.Theringwasdesignedtohaveacircumferenceof51.0m.With theoperating energyoftheringat808MeV,electronscirculatearoundtheringa lmostatthespeedof light.Theorbitingperiodistherefore170ns.Thisprovides9equallyspacedlocations, eachoneknownasan\RFbucket",whereelectronbunchescanex istandreceivethe properaccelerationtoremainonastableorbit.TheVUVringelectro ninjectionsystem canllspecicRFbuckets,allowingfordierentbunchpatterns.M oreover,thebunch length,whichdeterminesthepulsewidthofsynchrotronradiation, canbecontrolledbya secondhigherfrequencyRFsystem,byadjustingtheringmagnet system,orbyadjusting theelectronbeamenergy.Thellingpatternfornormaloperation isatrainof7bunches ofapproximatelyequal-currentfollowedbytwoemptyRFbuckets. Thismodeallowsthe highestaveragecurrentof1Aimmediatelyafterinjectingelectron sintothering,which isthemoststableandmeetsmostusers'requirements.Thepulsew idthis1.2{2.4nsand thePRFis52.9MHz. 2 Specialmodesofoperationscanbeachieved,withdierentlling patternsandbunchlengths.Forexample,twootheroperationmo deshavingthesame llingpatternandPRFasthenormalmodearethe7-bunchdetuned modewith800mA ofmaximumaveragecurrentand0.6{1.0nsofpulsewidth,andthe7bunchcompressed modewith200mAmaximumaveragecurrentand300{500psofpulsew idth.These specialoperations,thoughhavinglowerintensitythanthenormal mode,providebetter timeresolutioncriticalinatime-resolvedexperiment.Theringcanals obeoperatedat 2 Infacttwobunchesareemptyinthismodeandother7-bunchmode s,sothatthere arenosynchrotronpulsesemittedfromthem.Butthepump-prob eexperimentisnot aected,because2outofthe9laserpulsessychronizedtothesy nchrotronpulsesare simplynotusedintheexperiment. 45

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a3-bunchsymmetricmodewithaPRFof17.6MHzandpulsewidthof0.7{ 1.4ns,and ata1-bunchmodewithasingleRFbucketlled,sothatthePRFis5.9MH z,again withloweraveragecurrent.Theseotherllpatternsareuseful inthestudyofdynamical processesoflongertimescale.BeamlineU4IR OurworkwasdoneatBeamlineU4IRoftheNationalSynchrotron LightSource.Atlongwavelength,theemissionangleofsynchrotro nradiationislarge, makingtheextractionofinfraredbeamdicult.Uniqueextractiono ptics(90mrad horizontal 90mradvertical)atU4IRoershigh-brightnessbeamsinmidand far-infraredregions,almost100{1000timesbrighterthanconve ntionalsources[ 63 ]. Moreover,inthefar-infraredbeyond100cm 1 synchrotronradiationhashigherpower thanthermalsources[ 63 ].Inthefar-infrared,thespotsizeofthebeamisdiraction limited(approximately8timesofthewavelengthatthesamplelocation ,afterdemagnied bytheopticalsystemfocusingtheinfraredbeamintothemagnet) .Above1000cm 1 theVUVsourceisnolongerdiractionlimited,andthephysicalelectr onbeamsize controlsthesourcesize.Thebeamlinehastwoparts:anultra-high vacuum(UHV) sectionthatisdirectlyconnectedtothestorageringincludingthee xtractionoptics,and aroughvacuumsectioncontainingopticsthatdirectslighttothesp ectrometer.The twosectionsareseparatedbyadiamondwindow,whichismostlytran smissivefromthe far-infraredtonearultra-violet.Thebeamlinecoverstheenergy rangefrom2meVto 2.5eV(20{20000cm 1 ).Acombinationofbeamsplittersanddetectorsinprincipleallows FT-IRinthewholeinfraredrange.AnOxfordSpectromag10Tmagn etisavailablefor magneticstudies.TheincidentbeampassesthroughtwoZ-cutcry stalquartzwindows. Therstwindow(6mmthick)separatestheroughbeampipevacuum fromthecryostat highvacuum.Thesecondwindow( 1mmthick)separatesthehighvacuumfromthe VTIsamplespace.Thesequartzwindows[ 64 ]onthemagnetandthecooledlterin thebolometerlimittheusablefar-infraredrangetobeblow 110cm 1 .Thelightis 46

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n r n r n n n n "# $% n & n' n !"#%& "#%&( )n r n n )n +, n n / "#n "# n "n0n n $%) +1, Figure3-3.Experimentalset-upforlaser-pumpsynchrotron-p robespectroscopyatNSLS. Ifthedivide-by-Npulsepickeranditselectronicdriveareused,the bunch timingsignalfrequencyandthePRFoftheoutputlaserpulsesfrom thepulse pickeraredierentfromthoseshowninthegure. predominantlyverticallypolarizedasitentersthemagnet,especially atlowfrequency below50cm 1 3.2.2.3Timingschemeandsynchronization Thelaserandsynchrotronpulsesneedtobesynchronizedandana djustabledelayis introducedbetweenthetwoinalaser-pumpsynchrotron-probee xperiment.Theset-up andthetimingschemeareillustratedinFigure 3-3 andFigure 3-4 ,respectively.Detailsof howthesystemworksareexplainedbelow. Tosynchronizethelaserandsynchrotronpulses,areferencesig nalgeneratedfrom thesynchrotronringissenttothelasercontrolsysteminU6laser hutch.Specically,a 47

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n r n r !"#$%& ( )( % + ( /,% % n n"0 % "! /,% 1 r ) .2 3 # !0 1 n r 4 5 r 65 4 ( ( % )( 1 7 r( r 'r ( 4 r ( r 1 r ) Figure3-4.Timingschemeforlaser-pumpsynchrotron-probespe ctroscopyatNSLS.Theratiometeraccountsforthedriftin thedetectedsignalduetothedecayofthesynchrotronringcur rent,byconstantlymonitoringthebeamcurrent anddividingoutthedrift.ConOptics360-40EOMandConOptics350160EOMarethedivide-by-2and divide-by-Npulsepickers;ConOpticsModel10andConOpticsModel 25Daretheirdriveelectronics.ConOptics Model305CountdownorHP8082AcontrolstheelectronicdriveCon OpticsModel25D.Theoutputfrom Photodiode1isusedintheSynchro-locksystem.Photodiode2isfor monitoringthelaserpulsesbeforetheyare coupledtotheopticalbercable,whichdeliversthemtophoto-exc itesamples.48

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bunchtimingsignalisgeneratedfromastriplineelectrodeinashorts traightsectionof thestoragering.Theelectriceldofeachpassingbunchofelectro nsgeneratesavoltage signalinthestriplinethatissenttoelectronicsforturningeachbun chsignalintoa constantamplitudepulse.Twosetsofpulsesignalsareproduced:o neisapulseforeach electronbunchinthebunchtrainwiththePRFdependingontheoper ationmode,and theotherisasinglepulseforthefullorbitwithaPRFof5.9MHz.The5.9 MHzsingle pulseisderivedfromoneparticularelectronbunchinthering,whiche xistsforallmodes ofoperationsincludingthe1-bunchmode.Thecorrespondingpulse signalistherefore called\1-bunchclocksignal"evenwhenotherRFbucketsarelled.I tisusedinthe synchronizationprocesswhenthesynchrotronringisoperatedin the1-bunchmodeand inall7-bunchmodes.Forthe3-bunchsymmetricmode,theactual bunchsignalwitha PRFof17.6MHzisused.Sincethismodewasnotusedinourexperiment ,thefollowing descriptionsfocusontheschemewiththe1-bunchclocksignalast hereference. The1-bunchclocksignalissenttoanHP81101pulsegenerator,wh ichdrivesthe pulsegenerator'strigger.Anewpulseisgeneratedaccordingtoth etime-delaysettingon thepulsegenerator.TheoutputpulseissenttoU6laserhutchwhe reitisdividedbya powersplitter.Onepartgoestothedivide-by-Npulsepickerelectr onics.Theotherpart goestoa52.9MHznarrowbandpasslterthatseparatesoutthe9 thharmonicofthe5.9 MHzpulsetrain,whichmatchespreciselytheRFfrequency.ARFamp lierbooststhe 52.9MHzsignalsothatitcanbesplitintwo,withonepartgoingtothed ivide-by-2pulse pickerandtheotherservingasthereferencefortheCoherent\ Synchro-lock"systemfor synchrotron-lasersynchronization.TheSynchro-locksystemm ixesthepulsedlaseroutput signaldetectedbyaphotodiode(Photodiode1inFigure 3-4 )withthereferencesignal. AnerrorsignalisproducedtocorrectthecavitylengthoftheTi:sa pphirelaser,which determinesthelaserPRF.Asteppermotoronthecavityendmirror ,aglassprismpairon agalvanometermovement,andapiezoelectrictransducerononeo fthecavityfoldmirrors worktogethertominimizethiserrorsignal.TheSynchro-locksyste misofaphase-locked 49

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looptype,sothatwhentheerrorsignalisreducedtozerothelase rpulsesarenotonly attheexactsamerepetitionfrequency,theyalsohaveaxed(an dcontrollable)time delayrelativetothesynchrotronpulses.Theaccuracyofthesyn chronizationcanbe checkedbyplacingahigh-speedphotodiodeatthesamplelocationat BeamlineU4IR tosimultaneouslydetectthesynchrotronandlaserpulses.Afast oscilloscopeisusedto displaythesignaloutputfordiagnosis. ThePRFoftheTi:sapphirelaseris105.8MHz,twiceoftheringRFsyste m frequency.Laserpulseshavetobeselectedtomatchthebunchp atternsofthesynchrotron radiation.ThisisachievedbyusingpulsepickersfromConOptics,whic hareelectro-optic modulators(EOMs)madeofPockelscells.ThePockelscellisbasicallya voltage-controlled opticalretardercombinedwithalinearpolarizer.Thelaserlightislinea rlypolarizedalong thehorizontaldirectionwhenitentersthecell.Atoneparticularvo ltage,thePockels cellactsasahalf-waveplatethatconvertsthepolarizationtover ticalandthelaser lightisrejectedbythelinearpolarizer;atanothervoltagethecella ctsasafull-wave plateandthepolarizationisunaected,thusthelaserlightcomesth rough.Thepulse pickingprocessiscontrolledbyvaryingthevoltageappliedtothecell betweenthesetwo voltagesettings.Tomatchthe52.9MHzPRFofthe7-bunchmodes, adivide-by-two pulsepicker(ConOptics360-40EOM)isusedtoselecteveryotherla serpulse.The electronicdriveforthispulsepicker,ConOpticsModel10,takesina 52.9MHzreference signal,andthenusesaRFamplierandaDCpowersupplytoalternate theoutput voltagebetweenthetwoswitchingvoltagesatthatfrequency.Ad ivide-by-Npulsepicker (ConOptics350-160EOM)isusedtomatchthebunchpatternswhe ntheringisoperated atothermodes.Forexample,the1-bunchmoderequiresthedivide -by-Npulsepicker tooperateasdivide-by-9.TheelectronicdriveConOpticsModel25 Dworkssimilarly asConOpticsModel10,butitsoperationrequiresaninputfromana dditionalpulse generator,ConOpticsModel305Countdown(onlyworksforsymm etricmodes)orHP 8082A(worksforanymodes),whichsetsthevoltagepulsepatter nusedbythedrive 50

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accordingtothesynchrotronoperationmode.Ineithercase,all rejectedpulsesare recovered,delayed,adjustedtorestoretheoriginalpolarizatio n,andre-introducedintothe pulsetrainfromthelaser. TheHP81101pulsegeneratorsetsanadjustabledelaybetweenth elaserand synchrotronpulses,thereforeservesasaprogrammablevariab ledelay.Whenitshifts thereferencepulsesignalbycertaindelaytime,boththelaserand thedivide-by-2pulse picker(orthedivide-by-Npulsepickerifitisused)shifttogether.F orthelaserpulsesto beoptimallytransmittedormaximallyrejected,theyhavetoarrivea tthepulsepicker atthecorrectmomentwhenthevoltageonthepulsepickerjustch angestooneofthe twooperatingvalues.Forthedivide-by-2pulsepickerthiscanbeac hievedbyadjusting thepulsedelayofthelaserontheSynchro-lockelectronicssoftwa re.Forthedivide-by-N pulsepickeradelaycanbesetonthepulsegenerator(ConOpticsCo untdownModel305 orHP8082A)thatcontrolstheelectronicdrive.Whenalltheseare successfullydone, laserpulseswithcorrecttimestructurearesynchronizedtothes ynchrotronpulses,witha controllabletimedelaybetweenthetwo.3.2.2.4Dierentialtechnique Toimprovesensitivityandsingleoutthesignalduetothetime-depen dentchangeof thesampleproperties,adierentialtechniqueisemployedinthedat aacquisitionprocess. TheSRS830lock-inamplier'sinternaloscillatorintroducesaphasem odulationon theHP81101pulsegenerator'striggersignalthroughabias-tee, 3 seeFigure 3-4 .This sinusoidallydithersthearrivaltime t ofthelaserpulsewithrespecttothesynchrotron pulsewithasmallamount dt ,whichissettobeslightlysmallerthanthesynchrotron pulsewidth.Themodulationsignalalsoservesasthereferencefor thelock-inamplier, whichdetectsthesignalasthedierenceinthesampleresponseat thetime t dt= 2 3 TheDCbiasteepassesthehigh-frequencyRFsignalwithaDCoset suppliedfrom thelock-inamplier,withminimumdisturbingintheRFsignal. 51

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and t + dt= 2.When dt issucientlysmall,thedetectedsignalisaderivativesignalof thesampleresponse,denotedas dS=dt .Integratingitovertime,weobtainthesample response S ( t ). 3.3SampleParameters WestartedwithasetofNbTiNandNbNthinlmsofvaryingthicknesse sand substratematerials,andselectedoneNbTiNlmonquartzsubstra teandoneNbNlm onMgOsubstrateforourmagnetic-eldstudy.Aselectioncriteria wasstrongspin-orbit scattering ~ = so 1[ 65 ],wherethespin-orbitscatteringtime so =3 : 0 10 14 sis takentobethatofNbTi[ 66 ]andisthesingle-particlegap.TheNbTiNlmwasgrown byreactivemagnetronsputteringinargonandnitrogengaswithaN bTicathode,andthe NbNlmbyreactivesputteringofNbinN 2 atmosphere[ 67 ].Table 3-1 liststhesample parameters,includingthelmthickness d ,opticalgap 0 ,criticaltemperature T c ,sheet resistance R at20K,penetrationdepth ,coherencelength ,zero-temperatureparallel uppercriticaleld H k c 2 ; 0 ,zero-temperatureperpendicularuppercriticaleld H ? c 2 ; 0 ,and therefractiveindexofthesubstrate n .Determinationofsomeoftheseparameterswillbe presentedinthefollowingsections.Theopticalgapisdeterminedfr omtheopticaldatato bediscussedinSection 4.4.2 ThequartzandMgOsubstrateshavenegligibleabsorptioninthespe ctralrangeof interest(10{110cm 1 ).Bothmaterialshavetheirrefractiveindicesalmostindependent offrequencyinthisspectralrange.Forquartz n isabout2.123at300Kand2.119at 1.5K[ 68 ],showingveryweaktemperaturedependence.Weuseaconstant valueof2.12 below20K.TherefractiveindexofMgObelow10Kisnotdirectlyavailab leinliterature, butwefounditvariesfrom2.78at20Kto2.72at50Kinonestudy[ 69 ],andfrom 3.09at10Kto3.10at65Kinanother[ 70 ].Bothstudiesindicateweaktemperature dependencebelow65K,thoughtheabsolutevaluesfromthemared ierent.Wechoosean intermediatevalueof2.90astherefractiveindexofMgObetween2K and20K. 52

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Table3-1.Sampleparameters Sample d 0 T c R H k c 2 ; 0 H ? c 2 ; 0 n (nm)(cm 1 )(K)(n)(nm)(nm)(T)(T) NbTiN/quartz1012.810.2117200-4003.8-5.022.0212.502.12 NbN/MgO7017.912.848180-5004.0-7.021.8349.552.90 3.3.1CriticalTemperature Todeterminethecriticaltemperatureofbothsamples,weperfor medfour-probe resistivitymeasurementsatStationSCM2attheNationalHighMagn eticFieldLaboratory. A2mm 2mmpiecewascutfromeachsampleforsuchmeasurementsbecaus eofthe requirementsoftheexperimentalset-up.Electricalcontactsw eremadebygluinggold wirestothethinlmswithsilverpaint.Thetwosamplesweremounteds imultaneously onarotatingprobewhichallowsaccuratealignmentofthesamplesur facewithrespect tothemagneticeldorientation.Resistivitywasmeasuredfrom300 Kdownto5K, showninFigure 3-5 .Ifthecriticaltemperatureisdenedasthetemperatureatwhic h theresistancedropstozero,weestimate T c tobeabout10.2KforNbTiNand12.8Kfor NbN. 10 12 14 16 18 20 Temperature [K] 0 20 40 60 80 100 120Resistance [] NbTiN 10 100 Temperature [K] 0 20 40 60 80 100 120Resistance [] 12 13 14 15 16 17 18 19 20 Temperature [K] 0 20 40 60 80 100 120 140Resistance [] NbN 10 100 Temperature [K] 0 20 40 60 80 100 120 140Resistance [] Figure3-5.ResistancevstemperatureforNbTiNandNbNfromfou r-proberesistivity measurements. T c ,ifdenedasthetemperatureatwhichtheresistancedrops to0,isabout10.2KforNbTiNand12.8KforNbN. 53

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3.3.2Normal-StateSheetResistance Thenormal-statesheetresistance R =1 = n d ,where n isthenormal-stateoptical conductivity,canbedeterminedfromthenormal-statetransmitt ance.InEq.( C{29 )in AppendixC,setting 1 = n and 2 =0,andinvertingtheequation,wehave R = Z 0 p 4 n= T n n 1 ; (3{11) where T n isthenormal-statetransmittanceatnormalincidence,and Z 0 377nisthe vacuumimpedance.IngettingEq.( 3{11 )wemadetheassumptionthatthescattering rate1 = ismuchgreaterthanthefar-infraredfrequency,sothattheDr udeconductivity n = n; 0 1 i! (3{12) canbeapproximatedasitdcvalue n; 0 Todeterminetheirnormal-statesheetresistance,thetwosample sweremounted onsampleholdersandloadedintoanOxfordmagnet,thencooledto2 0Kinhelium vaporatwhichtransmissionwasmeasured.Anemptysampleholdero fthesamesizewas measuredasthereferencetocalculatetransmittance.Thedata areshowninFigure 3-6 Thenormal-statetransmittanceforbothsamplesisratinthefarinfrared,atalevelof about0.211forNbTiNand0.083forNbN.Thisratbehaviorconrmst hat 1isa validassumptionforbothsamplesinthefar-infrared.Usingthevalu eof n inTable 3-1 R iscalculatedtobe117n = forNbTiNand48n = forNbN.Whencomparingthese valueswiththoseobtainedfromtheresistivitydatashowninFigure 3-5 ,wefounda goodagreementforNbTiN,butabigdiscrepancyforNbN.Werepea tedtheoptical measurementstwomoretimesandreproduced R withinadierenceof10%,butdidnot haveachancetorepeatthefour-proberesistivitymeasurement .Thereforeweusethe R valueslistedinTable 3-1 inouranalysis.ThediscrepancyforNbNmightbeduetothe changeofcontactresistanceduringthecoolingprocess. 54

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0 20 40 60 80 100 120 Frequency [cm1 ] 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40Transmittance NbTiN NbN Figure3-6.Normal-statetransmittanceofNbTiNandNbNat20K,m easuredwitha resolutionof4cm 1 .Thesolidlinesindicatetheaverageofthedatapoints, whichis0.211forNbTiNand0.083forNbN. 3.3.3PenetrationDepthandCoherenceLength Thereisagreatamountofliteratureonthepenetrationdepthand coherencelength ofNbNthinlmsgrownbydierentmethodsunderdierentcondition s.Forexample,the approximatevaluesofthepenetrationdepth(mostlyatlowtemper ature)werereportedas 200nmbyShapoval etal. [ 71 ],187nmbyKawakami etal. [ 72 ],tobeconsistentlygreater than300nmat10KbyHu etal. [ 73 ],270nmbyVillegier etal. [ 74 ],400nmbyFeenstr etal. [ 75 ],194nmbyKomiyama etal. [ 76 ],400nmbyPambianchi etal. [ 77 ],370nm byOates etal. [ 78 ],280-380nmbyStern etal. [ 79 ],and300-500nmbyKubo etal. [ 80 ]. Thecoherencelengthwasreportedtobeabout5nmbyChockalinga m etal. [ 81 ],4.3nm byThakur etal. [ 82 ],4.0nmbyBell etal. [ 83 ],6.9nmbyShoji etal. [ 84 ],and4-7nmby Irie etal. [ 85 ].InsummarythepenetrationdepthofNbNisbetween180nmand50 0nm, andthecoherencelengthisbetween4nmand7nm. ForthesimilarcompoundNbTiN,thepenetrationdepthatlowtemper aturewas determinedtobe200-400nmandcoherencelengthtobe4nmbyYu etal. [ 86 ].In anotherstudyYu etal. determinedthecoherencelengthtobe3.8nm[ 87 ].Kau etal. [ 88 ] 55

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foundthecoherencelengthtobe5nm.Thereportedvalueofcohe rencelengthistherefore between3.8nmand5.0nm.3.3.4CriticalFields H c 1 forbothsamplesislow,atleastbelow1T,whichisthelowestappliedeld weuseinourstudy.Mathur etal. [ 89 ]reported H c 1 tobe4mTforNbNthinlmand 9.3mTforabulksampleat4.2K.AccordingtoLamura etal. [ 90 ] H c 1 shouldbeless than200mTattemperatureclosetozero. H k c 2 and H ? c 2 atvarioustemperaturesare determinedfromfour-proberesistivitymeasurements,showninF igure 3-7 andFigure 3-8 respectively.Wellbelowthecriticaltemperature,theresistance wasmeasuredbyscanning themagneticeldbetween0Tand16Tatselectedtemperatures.C losetothecritical temperature,theresistancewasmeasuredbysweepingtempera turebetween5Kand20K atselectedloweldvalues.Sincethetransitionisbroad,forconven ienceweassumethat thetransitionoccurswhentheresistancedropsto1/2ofits20Kv alue. Toestimatethevalueof H k c 2 atlowtemperature,whichisbeyondthemeasurement range,thedatapointsarettedtotheform H k c 2 = H k c 2 ; 0 r 1 t 2 1+ t 2 ; (3{13) where t = T=T c isthereducedtemperature.Theaboveequationisbasedonthefa ct thatforathinlmsuperconductorofthickness d H k c 2 =2 p 6 H c =d accordingto Eq.( 2{48 ),wherethethermodynamiccriticaleld H c / 1 t 2 andthepenetrationdepth / 1 = p 1 t 4 givenbyEqs.( 2{33 )and( 2{6 ).ForNbTiN,thetyields H k c 2 ; 0 22 : 02T and H k c 2 20 : 25Tat3K.ForNbN,thetyields H k c 2 ; 0 21 : 83Tand H k c 2 21 : 30Tat 2K. Thetemperaturedependenceof H ? c 2 hasadierentform.AccordingtoEq.( 2{40 ) H ? c 2 / H 2 c 2 ,andassumingthesametemperaturedependenceof H c and mentionedin 56

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0 2 4 6 8 10 12 14 16 Magnetic Field [T] 0 20 40 60 80 100 120Resistance [] 4.0K 5.0K 6.0K 7.0K 8.0K 8.5K 9.0K 9.5K 10.0K 10.5K 11.0K 11.5K 12.0K 9.0 9.5 10.0 10.5 11.0 11.5 12.0 Temperature [K] 0 20 40 60 80 100 120Resistance [] 0.2T 0.5T 1.0T 1.5T 2.0T 3.0T 4.0T 0 2 4 6 8 10 12 Temperature [K] 0 5 10 15 20 25Hc 2 [T] NbTiN data fit 0 2 4 6 8 10 12 14 16 Magnetic Field [T] 0 20 40 60 80 100 120 140Resistance [] 11.0 11.5 12.0 12.5 13.0 13.5 14.0 Temperature [K] 0 20 40 60 80 100 120 140Resistance [] 0 2 4 6 8 10 12 14 Temperature [K] 0 5 10 15 20 25Hc 2 [T] NbN data fit Figure3-7.Four-proberesistivitydatainparalleleldsforNbTiN( rstrow)andNbN (secondrow).Leftcolumn:resistancevsparalleleldatdierenttemperatures.Middlecolumn:resistancevstemperatureatdier entparallel elds.Rightcolumn:circlesare H k c 2 extractedfromdata;solidlinesarets usingEq.( 3{13 ).Inthersttwocolumnsthedashedlinesindicatethe resistanceat20Kandahalfofthatvalue. thepreviousparagraph,wehave H ? c 2 = H ? c 2 ; 0 1 t 2 1+ t 2 : (3{14) Usingthisequationtotthedata,wefound H ? c 2 ; 0 12 : 50TforNbTiNand H ? c 2 ; 0 49 : 55TforNbN. TheextractedvaluesshowthatforNbTiN H k c 2 >H ? c 2 ,whileforNbN H k c 2
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0 2 4 6 8 10 12 14 16 Magnetic Field [T] 0 20 40 60 80 100 120Resistance [] 2.0K 3.0K 4.0K 5.0K 6.0K 7.0K 8.0K 9.0K 9.5K 10.0K 11.0K 12.0K 7 8 9 10 11 12 Temperature [K] 0 20 40 60 80 100 120Resistance [] 0.2T 0.5T 1.0T 2.0T 0 2 4 6 8 10 Temperature [K] 0 2 4 6 8 10 12 14Hc 2 [T] NbTiN data fit 0 2 4 6 8 10 12 14 16 Magnetic Field [T] 0 20 40 60 80 100 120 140Resistance [] 11.0 11.5 12.0 12.5 13.0 13.5 14.0 Temperature [K] 0 20 40 60 80 100 120 140Resistance [] 0 2 4 6 8 10 12 14 Temperature [K] 0 10 20 30 40 50Hc 2 [T] NbN data fit Figure3-8.Four-proberesistivitydatainperpendiculareldsforN bTiN(rstrow)and NbN(secondrow).Leftcolumn:resistancevsperpendicularelda tdierent temperatures.Middlecolumn:resistancevstemperatureatdier ent perpendicularelds.Rightcolumn:circlesare H ? c 2 extractedfromdata;solid linesaretsusingEq.( 3{14 ).Inthersttwocolumnsthedashedlines indicatetheresistanceat20Kandahalfofthatvalue. measurements.Onepreviousstudyofa200nmNbNthinlmfound H k c 2 ; 0 32Tand H ? c 2 ; 0 44T[ 91 ]. 58

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CHAPTER4 MAGNETO-SPECTROSCOPYOFPAIR-BREAKINGEFFECTS 4.1Magnetic-Field-InducedEectsonSuperconductivity Magneticeldshavedramaticeectsonthesuperconductingstat e;whentheyare strongerthantheuppercriticaleld,superconductivityisdestr oyed.Fieldsbelowthis criticalvalueactonthespinandorbitalmotionofquasiparticlestat es. Anappliedeldliftsthespin-degeneracyofeachelectronicstate,p otentiallycausing aPauliparamagneticshiftofquasiparticledensityofstates[ 14 92 ]whichwouldgive alinearshiftofthespectroscopicgapwitheld[ 93 ].Theeectisnoticeableonlyfor materialswithaverysmallspin-orbitscatteringrate,inwhichthesp inisa\good" quantumnumber.InthepurePaulilimitwheretheeectofthemagn eticeldonthe electronorbitalmotioncanbeignored,Clogston[ 44 ]andChandrasekhar[ 45 ]proposed thatatransitiontothenormalstateshouldoccuratacriticaleld whentheZeeman splittingoftheCooperpairsbecomeslargerthanthesuperconduc tingcondensation energy.Thetransitionisofrstorder,andthecriticaleldisgiven as H p = 0 = p 2 B IncertainmaterialsinwhichthePauliparamagnetismdominates,whe ntheeldisstrong enough,aphasecalledtheFulde-Ferrel-Larkin-Ovchinnikov(FFLO )statecanexist,in whichCooperpairsareformedwithnon-zerototalmomentumandt heorderparameteris spatiallyinhomogeneous[ 94 95 ]. Theeldalsoalterstheorbitalsofsingle-particlestatesfromwhicht heBCSground stateisformed,breakingthetime-reversalsymmetryofthecon densatepairing.Theresult isanitelifetimeforagivenCooperpairandanoverallweakeningofth esuperconducting state.Thisweakeningisdirectlyrevealedbyareductioninthesingleparticlegapand formsthebasisofthepair-breakingtheoryoriginallyproposedbyA brikosovandGor'kov [ 96 97 ]todescribetheeectofmagneticimpuritiesonsuperconductivity .Thedepairing phenomenacanbecharacterizedbyasinglepair-breakingparamet er,,thatdepends onwhetherthetheorydescribesexternalmagneticelds,super currents,spinexchange, 59

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orothereects.Maki[ 98 ]showedthatathinlmsuperconductorinthedirtylimitwill exhibitpairbreaking,equivalenttothatcausedbymagneticimpuritie s,whensubjected toahomogeneousmagneticeld.Healsodemonstratedthatforat hinlmtype-II superconductorinthevortexstate,theeld-inducedperturba tioncausesspatialvariation oftheorderparameter,whichgivesrisetoanotherdistincttypeo fpairbreaking[ 5 ]. Ingeneral,bothparamagnetismandorbitalpairbreakingcanaec tthespectroscopic gap.HoweverthePauliparamagnetismis\smearedout"inmaterials withlargespin-orbit scattering[ 65 ],makingtheorbitaleectdominant.Ifthesampleisoftheformofa thin lm,thelmthicknessalsodeterminesthespinandorbitaldepairingr egimes.Asthelm thicknessisdecreased,theelectronorbitalmotionbecomesmore compressed.Theorbital eectsofthescreeningcurrentgiveswaytothespinPauliparama gnetism[ 93 99 100 ],at athresholdthickness,e.g.,approximately10nmforaluminumlms. 4.2Motivation Opticalspectroscopyprobesdirectlyasuperconductor'sexcita tionspectrum,making itidealforstudyingthegapevolutionunderanappliedmagneticeld. Ordinarymetallic superconductorshaveagapintheiropticalspectrum[ 34 ],requiringaminimumof2of photonenergytobreakCooperpairs.AsdiscussedinSection 2.3.6 ,thegapmakesthe T =0realpartoftheopticalconductivitybezeroforphotonenerg iesbelowthegap. Themissingspectralweightin 1 ( )appearsasadeltafunctionatzerofrequency.By theKramers-Kronigrelations,thedeltafunctiongivesadominant1 =! formto 2 ( ).By determiningboththerealandimaginarypartsoftheeld-depende ntopticalconductivity, onecantesttheoriesforthemagneticeldsuppressionofthegap Wenditsomewhatsurprisingthatmagnetic-eld-inducedpair-bre akingeects havenotbeenconvincinglyveriedbyopticalstudies.Sucheects havebeenobserved intunnelingspectra[ 101 ]andarehintedatbyabsorptiondata[ 93 ].Inaddition,the eectofmagneticimpuritieshavebeenstudiedindetail[ 102 103 ].Inthischapterwe reportfar-infraredtransmissionandrerectionspectraofNbTiN andNbNunderexternal 60

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magneticeld,appliedparalleltothelmsurface.Theextractedop ticalconductivity 1 demonstratesasuppressionofthegapbytheeld,inquantitative agreementwith pair-breakingtheory.Thisisthersttimethatopticalabsorption hasbeenemployedto testquantitativelythetheoryofpairbreakingbyanexternalmag neticeld. 4.3Experimental Wewouldliketodeterminetheopticalgapfromtheopticalconductiv ity,which canbeextractedfromtransmissionandrerection.Infraredtra nsmissionandrerection measurementswereperformedatBeamlineU4IRoftheNationalSy nchrotronLight Source,BrookhavenNationalLaboratory,usinghigh-brightnes sbroadbandsynchrotron radiationasthespectroscopicsource.Thesamplesweremounted ina 4 HeOxfordcryostat equippedwitha10Teslasuperconductingmagnet.Thespectrawer ecollectedusinga BrukerIFS66-v/Sspectrometerandahighsensitivity,largearea compositeSibolometer operatingat1.6K.Cooledltersinthebolometerandthequartzwind owonthemagnet limitedtheupperfrequencyto110cm 1 4.3.1ConsiderationofFieldOrientation Theelddirectionisimportantwhenconsideringthebehaviorofthes etypeII superconductors.Foreldperpendiculartothin-lmsamples,vor ticesappearabove H c 1 andformadenselatticeoflossycorematerialasitapproaches H c 2 .Weintendedtoavoid thisvortexregimebyorientingtheeldparalleltothelmsurface. InthecaseofNbTiN,becausethelmthicknessismuchsmallerthant hepenetration depthandsomewhatclosetothecoherencelength,asignicantde nsityofvorticesis unlikely.Considerthepossibilityofthepresenceofvortices.Assum ingatriangular vortexlattice,thevortexspacing a 4 =1 : 075 p 0 =H is48.9nmat1Tand15.5nmat 10T.Evenatthehighestappliedeldof10Tthevortexspacingislarg erthanthelm thickness.Thereforewedonotexpectvortex-inducedeectst obesignicant.Inaddition, 61

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Figure4-1.Sampleholdersfortransmissionandrerectionmeasure mentsinparallel magneticelds.Thesampleisthethinblackslabnearthemiddle.Thearrowedlinesshowthepathofthebeams.Theleftholderisusedfortransmissionmeasurementatnormalincidence.Themiddleoneisfortransmissionat30 angleofincidence.Therightoneisforrerectionat30 angleofincidence.Thescrewsontopareusedtoattachthesample holdersto aprobethatgoesintothemagnet. accordingtoEq.( 2{51 )theaveragemagneticeldinthelmisapproximately H H a 1 d 2 12 2 0 : 999 H a ; (4{1) where H a istheappliedeld.Hence,theeldisnearlyuniforminthethinlmand almostthesameastheappliedeld. TheNbNlmismuchthickerthanNbTiN.Evenat1Titisunlikelytoavoidth e presenceofvortices.Howevertheaverageeldinsidethethinlmis morethan0.987of theappliedeldaccordingtoEq.( 2{51 ).Theeldisthereforenearlyhomogeneousand equaltotheappliedeld. Wedesignedspecialsampleholderstomeasuretransmissionandrer ectionina paralleleld. 1 SketchesareshowninFigure 4-1 .FortheNbTiNsample,theangleof incidenceinthetransmissionmeasurementwasnearnormal,shownin theleftpanelof 1 Thesampleholdersfortransmissionmeasurementatnormalinciden ceandrerection measurementat30 angleofincidenceweredesignedbyG.LawrenceCarrandJungseek Hwang.Thesampleholderfortransmissionmeasurementat30 angleofincidencewas designedbytheauthor. 62

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Figure 4-1 :apolishedaluminummirrorrerectstheinputbeamverticallyontothe sample, andanothermirrordirectsthetransmittedbeambacktotheorigin aldirectionatan elevatedlevel.Theangleofincidenceinthererectionmeasurementw asabout30 ,shown intherightpanelofFigure 4-1 :apolishedaluminumroof-typemirrorrerectstheinput beamontothesampleata30 angleandthenredirectsthererectedbeambackontothe originalopticalpath.Toanalyzedata,thererectionneedstobec orrectedforthe30 angleofincidence. WebeganstudyingtheNbNsampleaftertheanalysisoftheNbTiNsam plewas completed.Duringtheanalysiswerealizedthatwecanmatchtheang leofincidencein bothtransmissionandrerectionmeasurements.Whenmeasuringt ransmissionforNbN, wemodiedthetransmissionsampleholder,showninthemiddlepanelo fFigure 4-1 ,so thattheangleofincidencewasalso30 4.3.2TransmissionandRerectioninParallelField Ourgoalistoextracttheopticalconductivityofthethinlmsuper conductorsfrom rerectionandtransmissionmeasurements.Beginningwiththepione eringworkofPalmer andTinkham[ 36 ],thisapproachhasbeenusedanumberoftimestostudybothmeta llic andcupratesuperconductors[ 104 105 ].Inaconventionaltransmittanceorrerectance measurement,onemeasuresseparatelythesampleandareferen cehavingknownoptical properties|typicallyanopenaperturewithnosamplefortransmitt anceandaknown metalforrerectance.Sampleexchangecanleadtoerrors,espe ciallyfortheabsolute rerection,wheresampleorientationiscritical.Toavoidsampleexcha ngeerrors,we usedthesampleinthenormalstate,andat H =0T,forourreference.Specically,we measuredthesamplespectrum(transmissionorrerection)atdie renteldsrangingfrom 0to10Tinthesuperconductingstate( T =3KforNbTiNand T =2KforNbN), usingthenormalstate( T =20K),zero-eldspectrumforthereference.Therelative measurementscanbemadeabsolutebymeasuringthenormal-stat etransmittanceand rerectanceorbycalculatingthemfromtheDrudemodelinthelimit 1 = ,avery 63

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goodassumptionforourlmsasthemeasurednormal-statetrans mittanceisalmost constantinthemeasuredfrequencyrange(Figure 3-6 ).Thedirectlyacquireddataare thereforetheratiosoftransmittance T s = T n andrerectance R s = R n ,wherethesubscripts s and n denotesuperconductingstateandnormalstaterespectively.F igure 4-2 shows thedataatvariouselds.TheNbTiNwasmeasuredwitharesolutiono f3.5cm 1 ,which doesnotfullyresolvetheinterferencefringesfrommultiplererect ionsinsidethesubstrate. TheNbNwasmeasuredwitharesolutionof4cm 1 .Thepeakin T s = T n shiftstolower frequencyaseldincreases,suggestingthesuppressionofthee nergygapduetotheeld. ForNbTiN,thererectiondatawerecorrectedforthemeasureds traylightandforthe30 angleofincidence(Appendix D )beforecalculatingtheopticalconductivity. 4.4Analysis 4.4.1ExtractionofOpticalConductivity Theanalysisforthethinlmopticalconductivity = 1 + i 2 ofNbTiNbeginswith theexpressionsforthenormal-incidencetransmissionthrough,a ndrerectionfrom,the frontlmsurfaceofthesample[ 36 ], T s = 4 n ( Z 0 1 d + n +1) 2 +( Z 0 2 d ) 2 ; (4{2) R s = ( Z 0 1 d + n 1) 2 +( Z 0 2 d ) 2 ( Z 0 1 d + n +1) 2 +( Z 0 2 d ) 2 ; (4{3) where Z 0 377nisthevacuumimpedance, d isthelmthickness,and 1 2 aretherealandimaginarypartsoftheopticalconductivityofthe thinlminthe superconductingstate(thinlmopticsisdiscussedindetailinAppen dix C ).The normal-statetransmittanceandrerectancecanbederivedfrom Eqs.( 4{2 )and( 4{3 ) bysetting 1 = n and 2 =0because 1 = inthefar-infrared, T n = 4 n ( Z 0 n d + n +1) 2 ; (4{4) R n = ( Z 0 n d + n 1) 2 ( Z 0 n d + n +1) 2 : (4{5) 64

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n r n n r n n r n n r n Figure4-2.Measuredsuperconducting-statetonormal-stater atiosoftransmission T s = T n andrerection R s = R n atvariousparallelmagneticelds.Theweakoscillatory featuresshownintheNbTiNdataarepartially-resolvedmultipleinter nal rerectionsinthesubstrate. i istheangleofincidence. Here n isrelatedtothenormal-statesheetresistanceofthethinlm R =1 = n d ,which wehavedeterminedfromthenormal-statetransmittance. Inpractice,wemeasurethecombinationoflmandsubstrate,givin gtheexternal transmittance T ext andexternalrerectance R ext .Ifthesubstratesurfacesareparallel onthescaleofthewavelengthandthemeasurementresolutionishig henough,these quantitiestypicallyshowfringesduetopartiallycoherentmultipleinte rnalrerections insidethesubstrate.Atypicalspectrummeasuredwithhighresolu tionisshownin Figure C-4 inAppendixC.Smoothinghighresolutiondataortakingmeasurement swith alowresolutionproducestheincoherentspectrum,whereonemay addintensitiesrather 65

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thanamplitudes.Inthiscase, T ext = T f (1 R u ) e x 1 R u R 0f e 2 x ; (4{6) R ext = R f + T 2 f R u e 2 x 1 R u R 0f e 2 x ; (4{7) where R u ( n 1) 2 = ( n +1) 2 isthererectanceofthequartzsurface, istheabsorption coecientofthequartz, x isthethicknessofthequartzsubstrate,and R 0f isthelm rerectionfrominsidethesubstrate, R 0f = ( Z 0 1 d n +1) 2 +( Z 0 2 d ) 2 ( Z 0 1 d + n +1) 2 +( Z 0 2 d ) 2 : (4{8) Quartzhasnegligibleabsorptionanddispersionoverthespectrala ndtemperaturerange ofinterest[ 68 ].Thuswetake =0and n =2 : 12,yielding R u 0 : 13. Ourmeasurementsgiveustheexternaltransmissionandrerectio nratios, T ext ;s = T ext ;n and R ext ;s = R ext ;n thatincludethesubstrate.Forrangeofconductivityvaluesexpe cted forthelm,wendthat,toaverygoodapproximation, T ext ;s = T ext ;n = T s = T n and R ext ;s = R ext ;n = R s = R n .Figure 4-3 comparesthesuperconducting-statetonormal-state transmittanceandrerectanceratios,calculatedusingEqs.( 4{2 )and( 4{3 ),andEqs.( 4{6 ) and( 4{7 ).TheopticalconductivityaretakentobethatofNbTiNat2K,0T, calculatedfromtheMattis-Bardeentheory,shownintheleftpane lofFigure 4-4 .The approximations T ext ;s = T ext ;n = T s = T n and R ext ;s = R ext ;n = R s = R n seemtoworkwell,even thoughsomesmalldiscrepanciesexistatlowfrequency. Wemeasured T n andusedittocalculate R (Section 3.3.2 ),whichinturndetermines R n .Weuse T n and R n tocalculate T s and R s fromourmeasuredratios.Thenweinvert Eqs.( 4{2 )and( 4{3 )tond 1 n = nR Z 0 1 R s T s T s ; (4{9) 2 n = R Z 0 4 n T s ( Z 0 1 d + n +1) 2 1 = 2 : (4{10) 66

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0 20 40 60 80 100 Frequency [cm1 ] 0.0 0.5 1.0 1.5 2.0 2.5 3.0Ratio of Transmittance T s / T n T ext ,s / T ext ,n 0 20 40 60 80 100 Frequency [cm1 ] 0.0 0.5 1.0 1.5 2.0 2.5 3.0Ratio of Reflectance R s / R n R ext ,s / R ext ,n Figure4-3.Left:comparisonof T s = T n and T ext ;s = T ext ;n .Right:comparisonof R s = R n and R ext ;s = R ext ;n .MaterialparametersarethoseoftheNbTiNsample. InthestudyoftheNbNsample,sincethetransmissionandrerectio nweremeasured at30 angleofincidence,theequationsareslightlydierent.Thesynchro tronradiation ispredominantlyverticallypolarizedasitentersthemagnet.Forthe sampleholdersused tomeasuretransmissionandrerectioninaparalleleld,thepolariza tionendsupparallel tothesamplesurface,i.e.p-polarized.Thesuperconducting-sta tetransmittanceand rerectanceinthispolarizationatanangleofincidence i arederivedinAppendix C ,given byEqs.( C{24 )and( C{25 ), T s = 4 n cos t cos i ( n + cos t cos i + Z 0 1 d cos t ) 2 +( Z 0 2 d cos t ) 2 ; (4{11) R s = ( n cos t cos i + Z 0 1 d cos t ) 2 +( Z 0 2 d cos t ) 2 ( n + cos t cos i + Z 0 1 d cos t ) 2 +( Z 0 2 d cos t ) 2 ; (4{12) where t istheangleofrefraction.Thecorrespondingnormal-statetrans mittanceand rerectanceare T n = 4 n cos t cos i ( n + cos t cos i + Z 0 n d cos t ) 2 ; (4{13) R n = ( n cos t cos i + Z 0 n d cos t ) 2 ( n + cos t cos i + Z 0 n d cos t ) 2 ; (4{14) 67

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n r! #% !&"#"$"% n r! #% !&"#"$"% Figure4-4.TheopticalconductivityofNbTiNandNbNinzeroeldat3 Kand2K, respectively,normalizedtotheirnormal-stateconductivities.The solidlines aretswiththeMattis-Bardeentheory.Thedashedlinesarecalcu lationswith thetheory. whichareusedtocalculate T s (30 )and R s (30 )fromthemeasured T s (30 ) = T n (30 )and R s (30 ) = R n (30 ).TheopticalconductivitycanbeinvertedfromEqs.( 4{11 )and( 4{12 ) directlywithoutcorrectionsfortheangleofincidence, 1 n = nR Z 0 cos t 1 T s R s T s ; (4{15) 2 n = R Z 0 cos t 4 n T s cos t cos i Z 0 R 1 n cos t + n + cos t cos i 2 # 1 = 2 : (4{16) WemadesimilarapproximationthattheeectfromtheMgOsubstrat ecanbeneglected. Theopticalconductivityatvariouseldsnormalizedtothenormal-s tatevalue n areshowninFigure 4-4 ,Figure 4-5 ,andFigure 4-6 2 = n hassomedatapointsmissing becausethetermsunderthesquarerootinEqs.( 4{10 )and( 4{16 )arenotguaranteedto bepositiveforthemeasuredtransmissionandrerectionwhennoise isincluded. 4.4.2AnalysisoftheZero-FieldData Thetemperature-dependentopticalconductivityofdirty-limitty pe-IIsuperconductors arewellknownfromtheMattis-Bardeentheory,givenbyEqs.( 2{18 )and( 2{19 )in Section 2.3.6 .Usingastheonlyttingparameterwettedourzero-eld 1 = n 68

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n n n n n %!r"!#!$ n n n n n n "$ Figure4-5.Thereal(circles)andimaginary(triangles)partsofth e T =3Koptical conductivityofNbTiNatdierentappliedparallelmagneticelds,nor malized tothenormal-stateconductivity.Thesolidlinesaretsto 1 = n usingthe pair-breakingtheory.Thedashedlinesshowthecorresponding 2 = n as determinedbyaKramers-Kronigtransformoftherealpart. 69

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n n n n n %!r"!#!$ n n n n n n "$ Figure4-6.Thereal(circles)andimaginary(triangles)partsofth e T =2Koptical conductivityofNbNatdierentappliedparallelmagneticelds,norm alizedto thenormal-stateconductivity.Thesolidlinesaretsto 1 = n usingthe pair-breakingtheory.Thedashedlinesshowthecorresponding 2 = n as determinedbyaKramers-Kronigtransformoftherealpart. 70

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shownasthesolidlinesinFigure 4-4 .Theimaginarypart 2 = n wascalculatedfromthe Mattis-Bardeentheoryusingthettedvalueof,shownasdashe dlines.Thettedvalue ofis12.8cm 1 forNbTiNand17.9cm 1 forNbN.Sincethetemperatureismuchlower than T c ,weassumethat= 0 ,thezero-eldzero-temperaturegap,whichgivesthe valuesof 0 inTable 3-1 .Thesevalueswillbeusedintheanalysisoftheeld-dependent data.4.4.3AnalysisoftheField-DependentData ThesolidlinesinFigure 4-5 andFigure 4-6 aretstothedatausingthepair-breaking theoryasextendedbySkalski etal. [ 106 ]tocalculate 1 = n at0K: 1 n = 1 Z n G + != 2 n G != 2 dq [ n ( q + != 2) n ( q != 2)+ m ( q + != 2) m ( q != 2)](4{17) for 2n G andzerootherwise,where n ( q )=Re u p u 2 1 ; (4{18) m ( q )=Re 1 p u 2 1 : (4{19) u isthesolutionto u = q + i u p u 2 1 ; (4{20) withthepair-correlationgap.This,inturn,canbedeterminedfro mthepair-breaking parameterandthezero-eldsingle-particlegap 0 using ln 0 = 4 (4{21) for < .n G intheintegrationlimitsistheeectivespectroscopicgap, n G = 1 2 = 3 # 3 = 2 (4{22) for < .Using 0 determinedfromthe H =0Tresults,weproceededtot 1 = n for H> 0Tusingonlyasanadjustableparameter.Theimaginarypartofth econductivity 71

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n n rr n n n rr n Figure4-7.Fitofthereal-partopticalconductivityinparalleleldf orNbTiNandNbN, takenfromFigure 4-5 andFigure 4-6 .Thezero-eldopticalconductivitiesare fromFigure 4-4 (dashedlinesinFigure 4-5 andFigure 4-6 )wascalculatedbyaKramers-Kronigtransform oftherealpartusingEq.( 2{22 ).Thetemperatureislowenoughthatthegaphasreached itszero-temperaturevalue.Thermalcontributionto 1 isexpectedtobeverysmallatlow reducedtemperature.The0KresultEq.( 4{17 )isexpectedtobevalidforourdatataken at T=T c 1. Figure 4-7 showsthetted 1 = n atdierentelds.Clearly,theabsorptionedge movestolowerenergyastheeldincreases.Theeld-inducedchan geismoresignicant inNbNthaninNbTiN.4.4.4Pair-BreakingParameter Thequantitydescribesthestrengthofpairbreakingandisdeter minedbythe mechanismofpairbreaking.Typicalmechanismsareparamagneticim purities,exchange interactions,magneticelds,andelectriccurrents.Inthecaseo fmagnetic-eld-induced pairbreaking,thespecicmechanismofpairbreakingdependsonth esamplegeometry andtheeldorientationwithrespectivetothesample[ 5 ]. Inathinlmsuperconductor,foranyperturbingHamiltonianthatb reakstime-reversal symmetry,isproportionaltothesquareoftheperturbingHamilt onianinthedirtylimit 72

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Table4-1.Pair-breakingparameter,eectivespectroscopicgap andpair-correlationgapof NbTiN(Column2{4)andNbN(Column5{7). H (T)(cm 1 /T 2 )n G (cm 1 )(cm 1 )(cm 1 /T 2 )n G (cm 1 )(cm 1 ) 00.00012.80012.8000.00017.90017.90010.08012.09212.7370.40015.50917.58320.11511.89112.7090.68014.44317.35830.21011.42012.6341.20012.77316.93140.35510.81312.5181.70011.38016.50950.52010.21012.3852.20010.11816.07660.7109.58512.2292.9008.51115.44671.0008.72511.9883.3507.55415.02481.2807.96911.7503.7506.74614.63891.6207.12311.4544.2005.87814.186 102.0306.18211.0854.7004.96213.662 [ 107 ].Inthecasewheretheperturbationiscausedbyaparallelmagnet iceld,andin thelimitthesampleissothinthattheorderparametercanbetreate dasconstant,is quadraticineld[ 3 5 ], k = 1 6 De 2 d 2 ~ c 2 H 2 ; (4{23) where D = 1 3 tr v 2 f (4{24) isthematerialdiusionconstant.Here tr isthetransportcollisiontimeand v f isthe Fermivelocity.Whenthemagneticeldisperpendiculartothethinlm ,insteadof breakingthetime-reversalsymmetry,itcausesspatialvariation oftheorderparamter whichinducesanitepair-breakingparameter v .Thispair-breakingparameterislinear ineld, v = 1 c DeH: (4{25) Thepair-breakingparamtersofNbTiNandNbN,asextractedfrom thedataat dierentelds,arelistedinTable 4-1 andplottedinFigure 4-8 Thepair-breakingparameterofNbTiNisquadraticineld,asexpect edforthe caseofathinlminparalleleld.AquadratictoftheformgivenbyEq .( 4{23 )yields De 2 d 2 = 6 ~ c 2 =0 : 020cm 1 /T 2 .Weestimate tr from n = ne 2 tr =m and R =1 = n d tr = 73

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n rr rr Figure4-8.Fielddependenceofthepair-breakingparameter,de terminedfromthe experimentalopticalconductivity(circlesforNbTiNandsquaresf orNbN). Thesolidlineisaquadratictandthedashedlinealineart. m=R dne 2 1 : 89 10 16 s,where R =117n = d =10nmand n 1 : 61 10 23 cm 3 is theelectrondensityofNbN[ 81 ]similartothatofNbTiN.IfwetaketheFermivelocityto bethatofNbN[ 81 ], v f 1 : 95 10 8 cm/s,then, De 2 d 2 = 6 ~ c 2 =0 : 049cm 1 = T 2 ,consistent withthettedvalueof0.020cm 1 /T 2 withintheuncertaintyofthematerialsparameters. Incontrast,thepair-breakingparameterofNbNislinearineld,co nsistentwith Eq.( 4{25 ).Thissuggeststhatthepairbreakingiscausedbythespatialvar iationof theorderparameter.AlineartoftheformgivenbyEq.( 4{25 )totheeld-dependent pair-breakingparameterofNbNyields De=c =0 : 464cm 1 /T.Using R =48n = d =70nmand n 1 : 61 10 23 cm 3 ,weestimate tr = m=R dne 2 6 : 58 10 17 s. Assumingagain v f 1 : 95 10 8 cm/sforNbN,wehave De=c =0 : 671cm 1 /T,in agreementwiththettedvaluewithintheuncertaintyofthemater ialsparameters.This consistencysupportsthatthepair-breakingmechanismisdieren tintheNbNsample fromthatintheNbTiNsample.EventhoughtheeldisparalleltotheN bNlmsurface, thelmthicknessisnotthinenoughtoavoidtheenteringofvortices ,makingthepair breakingequivalenttothatcausedbyaperpendicularmagneticeld 74

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Atthispointed,itshouldbepointedoutthattheopticalconductivit ygivenby Skalski etal. isvalidforthinlmsuperconductorswiththicknesssosmallthatthe order parametercanbeassumedtobeconstantacrossthelm.Itisthe reforenotdirectly applicabletothecasewherespatialvariationoftheorderparamet erispresent,asinthe caseoftheNbNsamplediscussedabove.However,Makiexplicitlyde rivedthatinthe higheldregion,suchasinthegaplessregime,forthecaseinwhichth eelectriceld oftheelectromagneticwaveisparalleltotheexternalstaticmagn eticeld,theoptical conductivityhasthesameformasthatgivenbySkalski etal. [ 5 108 ].Equivalenceofthe homogeneousandinhomogeneoussituationsisalsopointedoutby,e .g.,Anthore etal. [ 101 ].TheanalysisbasedontheopticalconductivitygivenbySkalski etal. istherefore validonthisground,buttheorderparameterisunderstoodtobea spatially-averaged one,i.e., p hj ( r ) j 2 i av .Thisalsoappliestoboththepair-correlationgapandtheeective spectroscopicgapoftheNbNsample,tobediscussedinthefollowing section. 4.4.5Pair-CorrelationGapandEectiveSpectroscopicGap Theshiftoftheexcitationenergygap2n G duetotheapplicationofmagneticeld hasalreadybeendiscussed,andcanbeseenfromtheabsorptione dgein 1 = n .This gapn G andthepair-correlationgaparecalculatedusingEqs.( 4{21 )and( 4{22 ).The resultsarelistedinTable 4-1 ,andcomparedinFigure 4-9 .Bothn G anddropas eldincreases,butthereductionofn G ismuchgreateratanygiveneld.Bothsamples atthehighestattainableeldof10Tarestillfarawayfromthegaple ssregionwhere n G vanishes.Theexperimentalandtheoreticalvaluesofn G andthepair-correlation gapareinexcellentagreement,asshowninFigure 4-9 .Theelddependenceofthe spectroscopicgapdataofNbTiNisalmostquadratic,whilethatofNb Nisalmostlinear. Alinearshiftcanbepossiblyduetoparamagneticeect,havingthef ormn G = 0 B H [ 93 ].ButthiscannotbethecasefortheNbNsample,becauseiftheeld -dependentn G ofNbNisttedwithastraightline,theslopeisapproximately 4 : 12cm 1 /T,havinga muchgreatermagnitudethan B = 0 : 4659cm 1 /T. 75

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0 2 4 6 8 10 Magnetic Field [T] 0.0 0.2 0.4 0.6 0.8 1.0Gap/0 NbTiN /0 nG /0 A ( H ) / A (0T) 0 2 4 6 8 10 Magnetic Field [T] 0.0 0.2 0.4 0.6 0.8 1.0Gap/0 NbN /0 nG /0 A ( H ) / A (0T) Figure4-9.Thepaircorrelationgap(circles)andtheeectivespe ctroscopicgapn G (squares)atdierentmagneticelds,normalizedto 0 .Thesolidlinesare theoreticalpredictionsusingthepair-breakingtheory.Thetrian glesarethe squarerootofthemissingspectralweight A (proportionaltotheorder parameter)calculatedusingEq.( 4{27 ),normalizedtoitszero-eldvalue. Asmentionedintheprevioussection,thepair-correlationgapandt heeective spectroscopicgapofNbNarespatiallyaveragedvalues h ( r ) i av and h n G ( r ) i av .The dimensionoftheeld-inducedspatialvariationisontheorderofthe coherencelength, whichismuchsmallerthanthewavelengthoffar-infraredradiation. Thesampletherefore behavesasacontinuum,andthegapsareprobedasspatiallyavera gedones. 4.4.6Sum-RuleAnalysis Theoscillator-strengthsumrule Z 1 0 1 ( ) d! = ne 2 2 m ; (4{26) where n istheelectrondensityand e and m arethechargeandmassoftheelectron, requirestheareaunder 1 ( )tobethesamefornormalandsuperconductingstates.The ratio 1 = n inFigure 4-7 isalwayslessthanunity;themissingspectralweightcondenses toa functionatzerofrequency,whichisameasureofpaircondensate densityandis directlyrelatedtothepair-correlationgap.Figure 4-7 thereforeshowsaweakeningof superconductivityastheeldincreases.Thereisalimitinwhichtheab sorptionedge 76

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approaches0,whilethemissingspectralweightremainsnite.Thes uperconductorenters a\gapless"regionbutstillmaintainssuperconductingproperties. Specically,themissingspectralweight A canbecalculatedusingthesumrule, A = Z 1 0 n; 1 d! Z 1 0 1 d! = Z 1 0 n; 1 (1 1 n; 1 ) d!; (4{27) where n; 1 = n; 0 = (1+ 2 2 )istherealpartoftheDrudeconductivityinthenormal state.Here n; 0 isthedcconductivityinthenormalstate,and istakentobethe previously-mentionedtransportcollisiontime, tr 1 : 89 10 16 sforNbTiNand tr 6 : 58 10 17 sforNbN.Notethatwhen approaches100n G 1 isalready approximatelyequalto n; 1 .Eq.( 4{27 )iscalculatedbychangingtheupperlimitof theintegralto100n G ,andusingEq.( 4{17 )for 1 = n; 1 .Wecalculatedthemissing spectralweight A atvariouseldsforbothsamples.Theresultsarecomparedwithth e pair-correlationgapandeectivespectroscopicgapn G inFigure 4-9 .Weplotthe squarerootofthemissingspectralweight,whichisproportionalt otheorderparameter (recallthat A / n s and n s /j ( r ) j 2 ).Forbothsamples,theelddependenceof p A isverysimilartothatofthepair-correlationgapratherthantoth espectroscopicgap n G .Thisisconsistentwiththefactthatitisthepair-correlationgapra therthanthe spectroscopicgapthatcharacterizesthestrengthofsuperco nductivity. The functionin 1 givesrisetoafunctionoftheform A=! thatdominatesthe behaviorof 2 .Themissingspectralweight A canthereforebeestimatedfrom 2 .In Section 2.3.6 wediscussedthetwotermsin 2 ,expressedinEq.( 2{23 ).Noticethatinthe insetintherightpanelofFigure 2-2 2 canbeapproximatedas A=! belowtheoptical gap(2 0 = ~ !> 1),eventhoughsuchanapproximationcouldslightlyoverestimates the valueof A becauseofthesteeperslope.Abovetheopticalgap, A=! dropsrapidlyand becomescomparabletotheterm 2 ;> .Foramoreaccurateanalysis,onecancalculate 77

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Table4-2.Superruiddensity(in10 19 cm 3 )forNbTiN(Column2{3)andNbN(Column 4{5).Column2andColumn4arecalculatedusingsumrule.Column3andColumn5arecalculatedfrom 2 .TheaverageerrorofColumn2is0.85,and theaverageerrorofColumn5is0.48. H (T) n SR,NbTiN n SI,NbTiN n SR,NbN n SI,NbTiN 03.696.061.821.7713.665.771.741.7323.655.621.701.6533.635.671.641.6643.575.571.571.5253.515.461.511.5263.445.351.421.3773.335.261.351.4183.255.171.301.2693.144.951.241.29 102.954.831.161.21 2 ;> ( )from 1 ;> ( ),andsubtractitfrom 2 ( )toget A=! .UsingEq.( 2{22 ), 2 ;> ( )= 2 P Z 1 0 1 ;> ( 0 ) 0 2 2 d! 0 : (4{28) Theintegralcanbeevaluatedbysplittingitintotwoparts.Therst partisintherange between0and100n G ,inwhichweusethepair-breakingopticalconductivityfor 1 ;> ( ), shownassolidlinesinFigures 4-5 and 4-6 .Thesecondpartisintherangeabove100n G inwhich 1 ;> ( )canbewellapproximatedas n; 1 = n; 0 = (1+ 2 2 ).Thetotalintegral issubtractedfromthe 2 data,andcomparedwiththeoriginal 2 datainFigure 4-10 plottedvs1 =! .Thisdierencebetween 2 and 2 ;> isthenttedtotheform A=! to extractthevalueof A ,shownassolidlinesinFigure 4-10 .ItworkswellforNbTiN,but forNbNasmally-interceptisneededtogetagoodlineart.InFigur e 4-11 ,thetted valueof A arecomparedwiththosecalculatedusingthesumrule.Thetwometh odsseem togiveconsistentelddependence,thoughtheabsolutevalueof A diers,especiallyfor NbTiN. Togetasenseoftheorderofmagnitudeofthesuperruiddensity n s ,letususe thevalueofthemissingspectralweight A foranestimation.Thelineartof 1 = n for 78

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n r n n n n n n r n n n n n Figure4-10.Theimaginarypartoftheopticalconductivityinparalle leldsplottedvs 1 =! forNbTiN(rstcolumn)andforNbN(secondcolumn).Thetriangles arethe 2 = n data.Thecirclesarethedierencebetween 2 = n and 2 ;> = n Thesolidlinesarelineartstothecircles. 79

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r nn r nn Figure4-11.Fielddependenceofthemissingspectralweight,norm alizedtoitszero-eld value.Thetrianglesareresultscalculatedfromthesumrule,andth ecircles areestimatedfromtheimaginarypartoftheopticalconductivity 2 .The errorbarsarefromthelineartsinFigure 4-10 NbTiNat3K,0Tyieldsthevalueoftheslope A= n = n s e 2 = 2 m n =33 : 02cm 1 Converttotheunitofs 1 n s e 2 = 2 m n =9 : 91 10 11 s 1 .Thenormal-stateoptical conductivitycanbecalculatedfromthesheetresistance R =117nandthelm thickness d =10nm, n =1 =R d =8 : 55 10 3 n 1 cm 1 ,inpracticalunits.Converting toCGSunits, n =8 : 55 10 3 9 10 11 s 1 =7 : 70 10 15 s 1 .Thisdetermines n s e 2 =m =1 : 53 10 28 s 2 .ConvertingtoSIunits,wehave n s e 2 = 4 0 m =1 : 53 10 28 s 2 Therefore n s =6 : 06 10 19 cm 3 .Similarly,forNbNat2K,0T,thelineartin Figure 4-10 yields n s e 2 = 2 m n =27 : 86cm 1 .Using n =2 : 98 10 3 n 1 cm 1 ,we estimated n s =1 : 77 10 19 cm 3 .Thevaluesofsuperruiddensity n SR calculatedfrom thesumruleandthevaluesofthesuperruiddensity n SI calculatedfrom 2 arelistedin Table 4-2 4.4.7ComparisonwithHomes'sLaw Withtheestimatedvalueofthesuperruiddensity n s ,itisinterestingtocomparethe datawithHomes'slaw.Homes etal. [ 109 ]discoveredasimplescalingrelationbetween thesuperruiddensityandtheproductof T c andthed.c.conductivityjustabove T c .If thesuperruiddensityisconvertedtotheassociatedspectralwe ight A ,therelationcanbe 80

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n r Figure4-12.ComparisonwithHomes'slaw.ThecirclesareNbTiNdata. Thesquaresare NbNdata.ThesolidlineisHomes'slawfordirty-limitweak-couplingBCSsuperconductors.ThedashedlineisHomes'slawforcuprates. writtenas A n; 0 T c ; (4{29) where A isincm 2 n; 0 and T c areincm 1 (1cm 1 = = 15n 1 cm 1 =1 : 44K).Within errors,theconstant isapproximately4.4forcupratehigh-temperaturesuperconduct ors, regardlessofthedoping,crystalstructureanddisordertype. Therelationholdsbothin thea-bplainandalongthecaxisofcuprates.ForBCSweak-coupling superconductorsin thedirtylimit, isabout8.1[ 110 ].Thesetwoscalingrelationsareshownasdashedline andstraightlineinFigure 4-12 Foroursamples,thespectralweightassociatedwiththesuperru idatdierent eldsisobtainedintheprevioussection.Weusethevaluescalculated fromthesum ruleintheanalysishere.Thed.c.conductivity n; 0 at20Karealsoestimatedinthe previoussection,anditisindependentofthemagneticeld.Thecrit icaltemperature T c atdierentparalleleldsisdeterminedfromtheresistivitydatasho wninFigure 3-7 Herewedene T c asthetemperatureatthemiddleofthetransitionintheresistance vs 81

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nr Figure4-13.Left:thepair-correlationgap,thecriticaltemper ature T c ,andthehalf eectivespectroscopicgapn G asafunctionofthepair-breakingparameter. Thedashedlineis 2 .Allquantitiesarenormalizedtotheirzero-eldvalues. Right: 2 vs T c ,comparedtoastraightline. temperatureplot.Thespectralweightdata A areplottedvs n; 0 T c inFigure 4-12 ,showing consistencywiththescalingrelationfordirty-limitweak-couplingBCS superconductors. Thelinearrelationbetweenthespectralweight A andthecriticaltemperature T c canbeverieddirectlyfromthepair-breakingtheory.Notingthec orrelationbetween thesquarerootofthemissingspectralweight p A andthepair-correlationgapshown inFigure 4-9 ,weassumethat p A / ,orequivalently A / 2 .Thepair-correlation gap()issolvedfromEq.( 4{21 ).Thecriticaltemperatureasafunctionofthe pair-breakingparameterissolvedfromthefollowingequation[ 106 ], ln T P c T c = + 1 2 1 2 ; (4{30) inwhich = = 2 k B T c T P c isthecriticaltemperatureat=0,and ( x )isthedigamma function, ( x )= d dx ln( x ) ; (4{31) where( x )isthegammafunction.Thecriticaltemperature T c asafunctionofthe pair-breakingparameteriscomparedwiththepair-correlationgap andtheeective 82

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n Figure4-14.FielddependenceofthedensityofstatesforNbTiN,c alculatedfrom Eq.( 4{32 )usingthepair-breakingparametersinFigure 4-8 spectroscopicgapn G intheleftpanelofFigure 4-13 .Alsoshownis 2 ,whichfollows closelywith T c .Therelationbetween 2 and T c isplottedintherightpanelofFigure 4-13 Thequasi-linearrelationbetweenthetwoqualitativelyagreeswithHo mes'slaw. 4.4.8DensityofStates Itisinstructivetocomparethemagnetic-eld-modiedquasipartic ledensityofstates totheBCSdensityofstatesgivenbyEq.( 2{14 ).Thesingle-particledensityofstates givenbythepair-breakingtheoryis N ( ) N (0) =Re u p u 2 1 = n ( q ) ; (4{32) where N (0)isthesingle-particledensityofstatesattheFermilevel,and u issolvedfrom Eq.( 4{20 ).UsingthettedvalueofofNbTiNatdierentelds,wecalculated its densityofstates,showninFigure 4-14 .Foreachcurve,thereisanenergythresholdabove whichquasiparticlescanbecreated.Thisisexactlythespectrosco picgap,whichshifts witheld.Theeld-inducedpairbreakingalsosmearsoutthegap-ed gesingularityinthe quasiparticledensityofstates[ 106 ],sothattheinitialriseof 1 becomeslessabruptfor increasingelds.Thisslowerincreaseisevidentwhencomparingther esultsinFigure 4-7 83

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4.4.9ConsistencyCheckoftheFits Tochecktheconsistencyofthetstotheopticalconductivity,w ecanusethe tted 1 = n and 2 = n tocalculatethetransmissionorrerection,andthencompare thatwithourrawdata.Thebestquantitytocompareisthetransm issionofNbTiN, becauseitistheonlyonemeasuredatnormalincidence.Inthiscase themeasurement isthemostreliable,andthedatadonotinvolvecomplicationsduetoth eangleof incidence. T s = T n iscalculatedfromthettedopticalconductivityshowninFigure 4-5 usingEqs.( 4{2 ),( 4{3 ),( 4{4 ),and( 4{5 ).Theresult,showninFigure 4-15 ,isingood agreementwiththerawdata,conrmingthevalidityoftheanalysis. 4.5Summary Inconclusion,wemeasuredfar-infraredtransmissionandrerect ionoftwothin-lm superconductorswithdierentthicknessinamagneticeldparallel tothelmsurface. Therealandimaginarypartsoftheopticalconductivityarederive dfromthesedata, theformershowingtheabsorptionedgesuppressedduetotheap pliedeld.Theoptical conductivitiescanbettedwellbythepair-breakingtheory,andt hedegreeofgap suppresionisingoodagreementwiththetheory.Thecoecientsint heeld-dependent pair-breakingparameterarecomparedwiththosecalculatedfro mmaterialparameters. Theopticalconductivitiesfromthetareusedtocalculatetransm ission,whichisthen comparedwiththerawtransmissiondata.Bothareconsistent,de monstratingthevalidity oftheanalysis. Theelddependenceofgapsuppressionisdierentinthetwosample s.The pair-breakingparameter,whichdescribesthedegreeofpairbre akingandgapreduction, isalmostquadraticineldforthethinnerNbTiNsample,andislinearine ldfor thethickerNbNsample.Thissuggestsdierentmechanismsofpairb reakinginthe twosamples.TheNbTiNlmisthinenoughtoavoidthevortexregimeev enatthe highestattainableeldof10T.Whenaparallelmagneticeldisapplied, itbreaks thetime-reversalsymmetryofthecondensateparingandinduce spairbreakingwith 84

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quadarticineld.TheNbNlmismuchthickerthantheNbTiNsothatev enwhenthe appliedeldisparalleltothelmsurface,atthelowesteldof1Tvort icescanexist inthesample.Thiscausesspatialvariationoftheorderparameter ,whichinducespair breakingwithlinearineld. 85

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0 20 40 60 80 100 0 1 2 3 H = 1 T 0 20 40 60 80 100 0 1 2 3 H = 2 T 0 20 40 60 80 100 0 1 2 3 H = 3 T 0 20 40 60 80 100 0 1 2 3 H = 4 T 0 20 40 60 80 100 0 1 2 3T s / T n H = 5 T 0 20 40 60 80 100 0 1 2 3 H = 6 T 0 20 40 60 80 100 0 1 2 3 H = 7 T 0 20 40 60 80 100 0 1 2 3 H = 8 T 0 20 40 60 80 100 0 1 2 3 H = 9 T 0 20 40 60 80 100 0 1 2 3 H = 10 T Frequency [cm1 ] Figure4-15.Superconducting-statetonormal-statetransmiss ionratioofNbTiNinparallel eldsat T =3K.Thecirclesaredata.Thesolidlinesarecalculationsusing thetted s = n fromthepair-breakingtheory. 86

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CHAPTER5 MAGNETO-SPECTROSCOPYOFVORTEXSTATE 5.1VortexStateinSuperconductors Type-IIsuperconductorsinmagneticeldsabovethelowercritica leld H c 1 enterthe vortexstate.Inthepreviouschapter,wehaveintentionallychos entheeldorientation tobeparalleltothethinlmsuperconductors,sothatsuchstate mightbeavoidedor atleastvortex-inducedeectscanbeminimized.Inthischapterwe applythemagnetic eldperpendiculartothelmsurfacetoinvestigatethevortex-st ateelectrodynamicsof superconductors. Inthevortexstate,magneticeldspartiallypenetratesupercon ductorsintheform ofquantizedruxlines.Aroundthevortexcores,supercurrents circulateandscreen themagneticruxfromthecoreinterior;superconductivitypersis tsinbetweenthe vortexcores.Thepropertiesofvorticesandtheirinteractionsw iththesuperconducting condensatehavebeenstudiedsincethediscoveryofconventiona lsuperconductors.An earlymodelbyBardeenandStephen[ 111 ]discussedthecurrent-drivenfreevortex motionanditsinduceddissipation.Suchmotionisrestrainedbypinning forcesdueto spatialinhomogneitiesinthesuperconductors.Thecrossoverfr omthevortex-pinning regimetotherux-rowregimewasrstobservedbyGittlemanandRo senblum[ 112 113 ]inthemicrowaverangefortype-IIsuperconductors.Interes tinthiseldwas revivedbythediscoveryofhigh-temperaturesuperconductors [ 114 ].Flux-pinning andrux-creepcontributionstothevortexdynamicswereaddres sed,forexample,by CoeyandClem[ 115 ],Brandt[ 116 ],andDulcicandPozek[ 117 118 ].Ingeneral,the appliedeldcouldmodifythebehaviorsofboththecondensateandt hequasiparticles. Whilethemodicationofthequasiparticlestatesduetothevortices wasobservedin high-temperaturesuperconductors[ 119 ],itwaspointedoutthatvortexdynamicsmainly aectthecondensateintype-IIsuperconductors[ 120 ]. 87

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5.2Motivation Theunderstandingofthevortexstateintype-IIsuperconduct orsismeaningful inthesensethatthemechanismofsuperconductivityiswellknownf romBCStheory, whichmaysimplifytheproblem.Thepropertiesofsuchamixtureofsu perconducting materialandvorticescanbeprobedbyitselectrodynamicrespons e.Withtheestablished descriptionoftheMiessner-stateelectrodynamicsofconvention alsuperconductorsbyBCS Mattis-Bardeentheory[ 34 ],thevortex-inducedchangetotheresponsefunctionsshould manifestitselfinvariousobservablequantities.Microwavetechniqu eshavebeenthe prevailingtoolsintestingthetheoreticalmodelsofvortexdynamics [ 121 { 126 ].Duetothe muchlowerenergyscaleofmicrowavescomparedtothesupercond uctingenergygaps,such techniquesprovedtobeexcellentforthedetectionoftheconden sateresponse.Infrared experiments,sensitivetotheresponseofboththecondensatea ndthequasiparticles,and capableofcapturingtheshiftsofthespectralweight,arehowev erscarceinliteraturefor thestudyofvortexdynamics.Time-domainterahertzspectrosc opy[ 127 ]andfar-infrared transmissionatasinglefrequency[ 128 ]wererecentlyemployedtostudyNbNthinlmsin thevortexstate.Herewereportfrequency-dependentfar-in fraredtransmissionofNbTiN andNbNthinlmspenetratedbyastaticperpendicularmagneticeld .Thedataare comparedtotheMaxwell-Garnetttheory,theBruggemaneectiv emediumapproximation, andtheCoey-Clemmodel.Wedemonstratethatforthinlmsitisnec essarytoconsider themagnetic-eld-inducedpairbreakingonthesuperconductinge lectronsoutsideofthe vortexcores. 5.3Experimental ThesamplesarethesameNbTiNandNbNthin-lmsuperconductorss tudiedinthe previouschapters.TheirparametersarelistedinTable 3-1 .Far-infraredtransmission andrerectionmeasurementsonthesesampleswereperformedat BeamlineU4IRofthe NationalSynchrotronLightSource,BrookhavenNationalLabor atory.Thebeamlineis equippedwithanIFS66-v/SvacuumFT-IRspectrometerfromBru kerOptics,modied 88

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H EMwaves Figure5-1.Thinlmsampleinaperpendicularmagneticeld.Incidente lectromagnetic wavesarealsonormaltothelmsurface.Thealignmentisachievedb y rotatingthesampleholderalongtheaxisshownasthedashedline. tousethesynchrotronradiationasanexternallightsource,and toincludea 4 Hecryostat alongwithasuperconductingmagnetfromOxfordInstrumentfor low-temperature magneto-spectroscopy.AcompositesiliconbolometerfromInfra redLaboratories, operatingat1.6Kbypumpingtheheliumcoolant,detectsthefar-inf raredradiation withhighsensitivity.Thesystemallowsmeasurementstobeperform edinthespectral rangeof10{110cm 1 ,thetemperaturerangeof1.8{300K,andthemagneticeldrange of0{10T.Bothsamplesweremeasuredintheirsuperconductingst ateat2K,withthe magneticeldappliedperpendiculartothelmsurfacesandincrease dfrom0to10T. Normal-statetransmissioninzeroeldwasmeasuredat20Kasaref erence. ThesampleswereloadedatthebottomofaprobeintheFaradaycon guration showninFigure 5-1 (externalmagneticeldsperpendiculartolmsurface).Westart ed withtransmissionmeasurements.Intheexperimentlayoutshownin Figure 5-2 ,theplane mirror(A)forcollectingthererectedlightwastakenoutfromthef our-waycross-shaped range,sothattheincidentbeam(showninred)cancompletelypass through.The transmittedsignal(showningreen)isdetectedbythebolometerp lacedatPositionP1,as showninFigure 5-2 .Toapplytheeldperpendiculartothelm,werstroughlyloaded thesamplesintheapproximateanglerange,thencarefullyrotated theprobetomaximize thetransmissionsignal,asillustratedinFigure 5-1 .Thepositionwascarefullymarkedfor laterreference.Theerrorofthealignmentisexpectedtobewithin 5 89

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IncidentBeam fromSpectrometer Magnet Bolometer Bolometer A B C D E F G H A:PlaneMirrorB,C,G,H:ParaboloidD,F:QuartzWindowE:Sample atPositionP1 atPositionP2 Figure5-2.Opticallayoutfortransmissionandrerectionmeasure mentsinaperpendicularmagneticeld(thisisasimplied versionofthedesignmadebyG.LawrenceCarr).Theredraysare theincidentbeam,thegreenraysthe transmittedbeam,andtheblueraysthererectedbeam.Aplanemir rorisinstalledinthefour-waycross-shaped rangetodirectbeamsinthererectionmeasurement,andistakeno uttoallowthefullbeamtopassinthe transmissionmeasurement.ThebolometerisplacedatPositionP1fo rtransmissionandmovedtoPositionP2for rerectionmeasurement.90

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Thererectionmeasurementwasdoneimmediatelyafterthetransm issionmeasurement wasnished.Thesamplewaskeptatthesameposition,withtheorien tationofthe sampleprobeunchangedsothatthemagneticeldwasperpendicula rtothelmsurface. Theplanemirrorwasinstalledtodirectthererectedsignalfromthe sample(shownin blueinFigure 5-2 )tothebolometer,whichwasmovedfromPositionP1toPosition P2downstreamoftheplanemirror.Thererectionmeasurementisc omplicatedbythe rerectionfromthequartzwindowonthemagnet,whichhasarerec tanceofabout23% (thisisthecalculatedaveragererectanceforaquartzwindowofr efractiveindex n =2 : 12, includingtheeectsfrombothsurfaces),andthereforecauses asignicantamountof straylight.Thisstraylightwasmeasuredbyrotatingthesamplepro be45 fromthe markedpositionmentionedinthepreviousparagraph,andsubtrac tedfromallthesingle beamrerectancespectraofthesamples.Tominimizethestraylight ,theplanemirror(A) andtheparaboloid(B)weremanipulatedtominimizetheratioofthest raysignalandthe signalfromthesampleatthemarkedposition(truesignalplusstra ysignal). Bothtransmissionandrerectionweremeasuredataresolutionof4 cm 1 toavoid interferencefringesduetothemultipleinternalrerectionsofthe incidentwavesinthe substrate.Thedataatselectedeldsforbothsamplesareshown inFigure 5-3 .For bothsamples,whenthemagneticeldincreases,thesuperconduc ting-statetransmission approachesthenormal-statetransmission,suggestingtheweak eningofsuperconductivity. Thepeakposition,asanindicationoftheopticalgapfrequency,do esnotshowsignicant shiftwhentheeldincreases.Thetwosamplesshowessentiallythes amebehaviorinthe T s = T n and R s = R n ratios.NbTiNhasamuchloweruppercriticaleldthanNbNsothat at10Titisalreadyclosetothenormalstate. 5.4Analysis 5.4.1ExtractionofOpticalConductivity Fromthemeasuredtransmissionandrerection,wecanextractth eopticalconductivity usingthesamemethodemployedinthepreviouschapter.Thenorma l-statetransmittance 91

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0 20 40 60 80 100 Frequency [cmr1 ] 0.0 0.5 1.0 1.5 2.0 2.5 3.0T s / T n NbTiN, 2 K, H 0 T 2 T 4 T 6 T 8 T 10 T 0 20 40 60 80 100 Frequency [cmr1 ] 0.5 1.0 1.5 2.0 2.5R s / R n NbTiN, 2 K, H 0 T 2 T 4 T 6 T 8 T 10 T 0 20 40 60 80 100 Frequency [cmr1 ] 0.0 0.5 1.0 1.5 2.0 2.5 3.0T s / T n NbN, 2 K, H 0 T 2 T 4 T 6 T 8 T 10 T 0 20 40 60 80 100 Frequency [cmr1 ] 0.5 1.0 1.5 2.0 2.5R s / R n NbN, 2 K, H 0 T 2 T 4 T 6 T 8 T 10 T Figure5-3.Measuredsuperconducting-statetonormal-stater atiosoftransmission T s = T n andrerection R s = R n atvariousperpendicularmagneticelds.Straylighthas beensubtractedfromthererectionsinglebeamspectrabeforec alculatingthe ratios. ismeasuredandthethinlmnormal-statesheetresistance R iscalculatedfrom Eq.( 3{11 ).Thevalueofthesubstraterefractiveindex n islistedinTable 3-1 .The normal-statererectanceisthencalculatedfromEq.( 4{5 ). T n and R n aremultipliedto themeasured T s = T n and R s = R n toobtainthesuperconducting-statetransmittance T s and rerectance R s .Theopticalconductivity isthenextractedfromthemusingEqs.( 4{9 ) and( 4{10 ).Theprocedureinthecongurationusedheredoesnotinvolveco mplications duetoniteangleofincidence,becausebothtransmissionandrere ctionweremeasuredat nearnormalincidence.Theeld-dependentopticalconductivities areshowninFigure 5-4 92

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0 20 40 60 80 100 Frequency [cm1 ] 0.0 0.5 1.0 1.5 2.0Normalized Optical Conductivity NbTiN 0 T 2 T 4 T 6 T 8 T 10 T 0 20 40 60 80 100 Frequency [cm1 ] 0.0 0.5 1.0 1.5 2.0Normalized Optical Conductivity NbN 0 T 2 T 4 T 6 T 8 T 10 T Figure5-4.TheopticalconductivityofNbTiNandNbNextractedfr omtransmissionand rerectionmeasuredinperpendicularmagneticelds,normalizedtot heir normal-stateconductivities.Thesolidsymbolsare 1 = n ,andtheopen symbolsare 2 = n Theextractedopticalconductivityisaquantitycharacterizingth eeectiveelectrodynamic responseofthevortexstate.Intheremainingpartofthischapt erwewillusethis quantitytotesttheoriesofvortex-stateelectrodynamicsfort ype-IIsuperconductors. 5.4.2AnalysisofZero-FieldData Webeginouranalyiswiththezero-eldopticalconductivity.Forour dirty-limit type-IIsuperconductingsamples,theMattis-Bardeentheoryp rovidesexplicitexpressions fortherealandimaginarypartsoftheopticalconductivity,given byEqs.( 2{18 ) and( 2{19 ).AssumingBCStemperaturedependenceofthegapandusing 0 asthe onlyttingparameter,wettedtherealpartoftheopticalcond uctivity.Theimaginary partwascalculatedfromtheMattis-Bardeentheoryusingthett edvalueof 0 .Thet andcalculation,showninFigure 5-5 ,areingoodagreementwiththedata.Thevalue ofthettingparameter 0 agreeswiththosefoundinSection 4.4.2 .Theconsistency ofthezero-eldopticalconductivityforbothsamplesobtainedus ingthetwodierent measurementcongurations(asshowninthepreviouschapteran dinthischapter) supportsthereliabilityofbothexperimentalmethods. 93

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n r! #% !&"#"$"% n r! #% !&"#"$"% Figure5-5.TheopticalconductivityofNbTiNandNbNinzeroeldat2 K,normalized totheirnormal-stateconductivities.ThesolidlinesaretswiththeMattis-Bardeentheory;thedashedlinesarecalculationswiththet heory. 5.4.3TheoriesofEectiveOpticalConductivity Toexplainthemagnetic-elddependenceshowninFigure 5-4 ,weapplyafew theoriestostudytheeectiveopticalresponseofthemixtureof vorticesandasuperconducting material.Thetheoriescomparedherearetwoeectivemediumtheo riesandthe Coey-Clemmodel. Abasicassumptionoftheeectivemediumtheoriesisthatthegrains aresmall comparedtothewavelengthofradiation,sothattheinhomogeneo usmediumappears tobeuniformtotheexternalelectromagneticwaves.Becauseth evortices,treatedas grainsembeddedinasuperconductingmedium,aregenerallyregard edascylindricaltubes withradiusofthecoherencelength(oftheorderofnanometers) ,andbecausethelength ofthesecylindricaltubesarelimitedbythethinlmthickness(lessth an100nm),this assumptioncanbesatisedinthefar-infrared. Thetwomostwidely-usedapproachesofthetheoriesaretheMaxw ell-Garnetttheory (MGT)andBruggemaneectivemediumapproximation(EMA)[ 129 130 ].TheMGT approachtreatsthegrainstobeembeddedinthesurroundingmed ium.Itisbestsuitable forthedescriptionofcompositeswiththeso-called\cermettopolo gy"proposedbyLamb 94

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Figure5-6.Twotypesoftopologiesofatwo-componentcomposite material.Left:cermet topology,withonecomponentdispersedintheother.Right:aggre gate topology.FigureadaptedfromRef.[ 131 ]. etal. (leftpanelinFigure 5-6 ),inwhichinclusionsarewellseparated[ 131 ].Indilute mixturessuchatopologycanusuallybeachieved.Theeectiveresp onsefunctionsvary smoothlywiththevolumefractionofthegrains.Quitedierently,th eEMAapproach treatsallconstituentsinanequivalentway.Thisisvalidinsystemswit hthe\aggregate topology"(rightpanelinFigure 5-6 ),inwhichallconstituentsareconnected,makingit ambiguoustodenethehostandtheinclusion.Assumingthesurrou ndingmediumto becharacterizedbytheeectivepropertiesoftheinhomogeneou ssystem,itiscapableof describingmixtureswithpercolationoccurringatcertainmixturefr actions.Inourcase, weexpectthesampletobestillpartiallysuperconductingevenathig hvolumefractionof vortices,sothatpercolationdoesnotoccur.Moreover,thevor ticescorrelatetostayapart becauseoftherepulsiveforcebetweenthem.Hence,everyvort exissurroundedbythe superconductingfraction,whichseemstobemoreconsistentwith thepictureofcermet topology.WethereforeexpectMGTtobeabetterdescriptionofo urcase.Boththeories aredetailedbelow.5.4.3.1Maxwell-Garnetttheory Consideraninhomogeneousmediummadeoftwocomponents,grain a withvolume fraction f embeddedinthesurrounding b withvolumefraction1 f .Theelectromagnetic wavestraversingthemediumcanbetreatedasaspatially-average deldwithitselectric 95

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componentexpressedas h E i = f E a +(1 f ) E b : (5{1) Theresponsefunction D issimilarlyspatiallyaveraged,expressedas h D i = f a E a +(1 f ) b E b : (5{2) Theeectivedielectricfunctionisdenedas e = h D i h E i : (5{3) TheMaxwell-Garnettheoryconsiderstheelectriceld E a tobethelocaleldacting onthegrain a .Itconsistsoftheexternaleldandtheeldcausedbythepolariz ed chargesonthesurfaceofanarticialcavityinwhichthegrainresid es.Inconsideringthis localeld,thetheoryassumesthattheseparationbetweengrain sarelargeenoughsothat individualgrainsscatterlightindependently.Furthermore,thee ldinthesurrounding medium b isassumedtobeunaectedbythepresenceofthegrains[ 132 133 ].Basedon theseassumptions,theMaxwell-Garnetttheorygivestheeectiv edielectricfunctionfor orientedellipsoidgrainsas MGT = b + b f ( a b ) g (1 f )( a b )+ b ; (5{4) where g isthedepolarizationfactordeterminedbytheshapeoftheellipsoid. Usingthe relation =1+4 i=! (Appendix B ),andwriting a = n and b = s ,wehave MGT n = s n + 4 i n + s n f (1 s n ) g (1 f )(1 s n )+ 4 i n + s n : (5{5) Forourthinlmsamples,theterm != 4 n ismuchlessthan 2 = n inthefrequencyrange [10 ; 100]cm 1 .Forexample,theNbTiNsamplehas R =117n = andlmthickness d =10nm.Thereforeinpracticalunits prac n =1 =R d =8 : 55 10 3 n 1 cm 1 .Converting tocgsunits, esu n =7 : 69 10 14 s 1 .Thefrequencyrange[10 ; 100]cm 1 correspondsto 96

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! 2 [3 10 11 ; 3 10 12 ]s 1 ,sothat != 4 esu n 2 [3 : 11 10 5 ; 3 : 11 10 4 ].Thereforewecan simplifytheaboveequationto MGT n = s n + s n f (1 s = n ) g (1 f )(1 s = n )+ s = n : (5{6) Thisisequivalenttoreplacingall 'swith 'sinEq.( 5{4 ).Similarsimplicationscanbe madefortheNbNsample. Vorticesaregenerallyregardedascylindricaltubeswithanormalc oreofradiusofthe coherencelength ,eachcarryingaquantumofmagneticrux 0 [ 134 ].Outsideofthecore regions,thesuperconductormaintainssuperconductingproper ties.Setting g =1 = 2inthe aboveequationforcylindricalgrains,theeectiveopticalconduc tivityis MGT n = s n (1+ f )(1 s = n )+2 s = n (1 f )(1 s = n )+2 s = n ; (5{7) where f isthevolumefractionofthevorticescharacterizedby n 5.4.3.2Bruggemaneectivemediumapproximation Bruggemanproposedamethodtoaddresstheissuethatthegrain sandhostmaterial inMaxwell-Garnetttheoryaretreatedasymmetrically.Becauseof thepresenceofgrains withdierentpropertiesfromthesurroundingmedium,theelectric eldintheregion aroundthegrainsaremodied,andtheelectricruxdeviatesfromt hatwhensuchgrains areabsent.Bruggemanarguedthattheaverageruxdeviationfo rthewholemedium shouldvanish[ 132 ].Hethensuggestedthatanadequatechoiceofaself-consistent localeldcansatisfythiscondition.Thisleadstotheconsiderationo faneective mediuminwhichallinclusionsaretreatedonanequalbasis,andtheav eragerux deviationiszero[ 133 ].Equivalently,theeectivedielectricfunctioncanbecalculated usingEqs.( 5{1 ),( 5{2 ),and( 5{3 ).ThedierencefromtheMaxwell-Garnetttheoryisthat thereisnohostmediuminBruggemaneectivemediumapproximation. Both E a and E b arecalculatedassumingtheconstituents a and b aregrainsimmersedinaneective mediumcharacterizedbyaneectivedielectricfunction. 97

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Foraninhomogeneousmediummadeoftwocomponents,grain a ofvolumefraction f embeddedinthehostmaterial b ofvolumefraction1 f ,Bruggemaneectivemedium approximationgivestheeectivedielectricfunctionfororientedellip soidgrainsasthe solutionofthefollowingequation, f a EMA g a +(1 g ) EMA +(1 f ) b EMA g b +(1 g ) EMA =0 : (5{8) EMA isoneofthetwosolutionsofthisquadraticequation,withtheimagina rypart greaterthanorequaltozero.Sincewetreatthevortexascylind ricalparticles,weset g =1 = 2.Thesolutionis EMA = 1 2 h (2 f 1)( a b )+ p (2 f 1) 2 ( a b ) 2 +4 a b i : (5{9) Onlythisoneoutofthetwosolutionsofthequadraticequation( 5{8 )ischosen,because itreducestothecorrectvaluesinthelimit f 0and f 1.Usingtherelation =1+4 i=! ,andwriting a = n and b = s ,wehavetheeectiveoptical conductivity EMA n + i 4 n = 1 2 (2 f 1)(1 s n )+ r (2 f 1) 2 (1 s n ) 2 +4( s n + i 4 n )(1+ i 4 n ) : (5{10) Since != 4 n 2 [3 : 11 10 5 ; 3 : 11 10 4 ]forNbTiNandsimilarlynegligibleforNbN,the aboveequationcanbesimpliedas EMA n = 1 2 (2 f 1)(1 s n )+ r (2 f 1) 2 (1 s n ) 2 +4 s n : (5{11) 5.4.3.3Coey-Clemmodel Theeectivemediumtheoriesdiscussedabovetreatvorticiesasst aticgrains embeddedinasuperconductingsurrounding.CoeyandClem[ 115 135 ]proposeda theorytocalculatetherf(radio-frequency)surfaceimpedance oftype-IIsuperconductors undertheinruenceofvortexdynamics.Thetheorygeneralizedth etwo-ruidmodel 98

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toself-consistentlycouplethesuppercurrentdensitywiththevo rtexdisplacements.It considerstheLorentzforceexertedonthevorticesduetothet wo-ruidcurrentdensity J = J n + J s ,where J n an J s arethenormal-ruidcurrentdensityandsupercurrentdensity, respectively.Thisforceredistributesthevortices,inducesaccu rrentsduetothechange ofthevortex-inducedelds,andresultsinthechangeofthetota lcurrentdensity J .The centralresultisthefrequency,eld,andtemperaturedepende ntcomplexpenetration depth ~ ( !;H;T ).Forthecaseofathinlmonasubstrateinaperpendiculareld H ,and fornormally-incidentrfwaveswithfrequency [ 135 ], ~ ( !;H;T )= 2 ( H;T )+( i= 2) ~ 2 vc ( !;H;T ) 1 2 i 2 ( H;T ) = 2 nf ( !;H;T ) # 1 = 2 ; (5{12) where ( H;T )isthepenetrationdepthassociatedwiththesupercurrent, ~ vc isthe complexeectiveskindepthduetovortexmotionandruxcreep,an d nf isthenormal-ruid skindepth. ( H;T )isrelatedtothezero-eldpenetrationdepth (0 ;T )through ( H;T )= (0 ;T ) = (1 b ) 1 = 2 ,where b = H=H ? c 2 isthereducedeldwithrespectto theperpendicularuppercriticaleld H ? c 2 ,whichwetakeasthevortexvolumefraction f ~ vc =(2~ v = 0 ) 1 = 2 where~ v isthecomplexeectiveresistivityassociatedwiththelocal vortex-motion-inducedelectriceld, ~ v ( ) f = +( ) 2 + i (1 ) 1+( ) 2 ; (5{13) inwhich f = n f = f= n istheruxrowresistivity, istherux-creepfactor,and isthecharacteristicrelaxationtimeofthevortexmotion.Theelda ndtemperature dependenceofthenormal-ruidskindepthtakestheform 2 nf = 2 n = [1 (1 t 4 )(1 b )], where n =(2 n = 0 ) 1 = 2 isthenormal-stateskindepth,and t = T=T c isthereduced temperature.Notetherelationbetweenthecomplexpenetration depthandthecomplex opticalconductivity CC ~ = i= 0 ~ 2 ,andassumeasimilarrelationbetweenthe zero-eldpenetrationdepth (0 ;T )andthesuperconductingopticalconductivity s s = i= 0 2 (0 ;T ),weobtainthecomplexopticalconductivityinthelowtemperature 99

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limit( t 1), CC n (1 f ) s = n + f f (1 f ) s = n +1 : (5{14) isdenedas ~ v f 1 1 i! 0 =! ; (5{15) where 0 =1 = isthecharacteristicfrequencythatdistinguishestherux-pinning and rux-rowregimes,calleddepinningfrequency.Eq.( 5{14 )reducestocorrectvaluesinthe limits f 0and f 1.InEq.( 5{15 )theeectofthermalcreephasbeenneglectedin thelowtemperaturelimitbysetting 0. 5.4.3.4Comparisonoftheories ThetheoriesdiscussedabovearecomparedinFigure 5-7 .Theeectiveoptical conductivitiesarecalculatedusingEqs.( 5{7 ),( 5{11 ),and( 5{14 )withvariousvalues ofthenormal-volumefraction f s = n istakentobethatofNbTiNat0T,shownin Figure 5-5 .InEq.( 5{14 )oftheCoey-Clemmodelthedepinningfrequency 0 issetto 300cm 1 ,assumingtherux-pinningregime. Theopticalconductivitiescalculatedfromthethreetheoriesshow verysimilar behaviorabovethegap,butaredistinctivebelowthegap.Themost signicantdierence isintherealpartof e = n .Re( e = n )calculatedfromboththeMaxwell-Garnetttheory andtheCoey-Clemmodelapproachesaconstantlevelasthefre quencyisloweredto zero.Incontrast,Re( e = n )calculatedfromBruggemaneectivemediumapproximation tendstoincreasewithdecreasingfrequencybelowthegap.Theas ymptoticbehavior ofRe( e = n )asthefrequencyapproacheszerodistinguishestheBruggeman eective mediumapproximationfromtheothertwotheories. TheMaxwell-GarnetttheorycanbedistinguishedfromtheCoey-C lemmodelby noticingtwofeatures.Asthefrequencyisloweredfromabovetob elowthegap,thereis aminimumintheRe( e = n )calculatedfromtheMaxwell-Garnetttheory,buttheone calculatedfromtheCoey-Clemmodelsimplybecomesaconstant.F urthermore,asthe normal-volumefraction f increases,thenear-zero-frequencyconstantlevelinRe( e = n ) 100

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n rr n rr n rr n n rr Figure5-7.TheeectiveopticalconductivitycalculatedfromtheM axwell-Garnetttheory (MGT),theBruggemaneectivemediumapproximation(EMA),andt he Coey-Clemmodel. f isthenormal-volumefraction. approachesonemuchfasterinMaxwell-Garnetttheorythaninthe Coey-Clemmodel. Forexample,inthelastsub-plotinFigure 5-7 ,theconstantlevelisalmostatonefor theMaxwell-Garnetttheory,butitisataround0.8fortheCoey-C lemmodel.In fact,onecanderivefromEqs.( 5{7 )and( 5{14 )thatasfrequencyapproacheszero, Re( MGT = n ) 4 f= (1+ f ) 2 intheMaxwell-Garnetttheory,andRe( CC = n ) f inthe Coey-Clemmodel.Thisisagainundertheassumptionoftherux-pinn ingregime,where thedepinningfrequency 0 ismuchgreaterthantheupperlimitofourspectrum.Note 101

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thatthelimitingconstantlevelobtainedfromtheCoey-Clemmodel isindependentof thedepinningfrequency 0 intherux-pinningregime. Whenwecomparethesefeaturesinthetheoriestothoseinourdat ashownin Figure 5-4 ,weconcludethattheMaxwell-Garnetttheoryistheclosestinterp retation ofourdataamongthethree.Thereasonsarethefollowing.Firstly ,theRe( e = n ) dataneverexceedoneforbothsamples.ThisrulesouttheBrugge maneectivemedium approximation.Secondly,asthefrequencyisloweredfromabovet obelowthegap, aminimumfeatureisapparentintheRe( e = n )datainbothsamples.Thisisnot accountedforbytheCoey-Clemmodel,butisconsistentwiththeM axwell-Garnett theory. Intheremainingpartofthischapter,wewillfocusontheMaxwell-Ga rnetttheory anduseittoanalyzetheeld-dependentopticalconductivityshow ninFigure 5-4 5.4.4AnalysisoftheField-DependentData WeusedtheeectiveopticalconductivitygivenbyEq.( 5{7 )totthedatashownin Figure 5-4 .Theeectiveopticalconductivity,expressedas MGT n = s n (1+ f )(1 s = n )+2 s = n (1 f )(1 s = n )+2 s = n ; (5-4) hastwovariables. s = n ,theopticalconductivityofthesuperconductingfraction,is assumedtobethesameasthezero-eldopticalconductivity[ 3 ]showninFigure 5-5 .For convenienceweusedthettedresultsinFigure 5-5 intheanalysispresentedhere.The normal-volumefraction f istheonlyttingparameterthatisvariedtottherealpart oftheopticalconductivityatdierentelds.Alltswereeye-balle d.Onceanoptimal valueof f wasfoundatagiveneld,itwasusedtocalculatetheimaginaryparto fthe opticalconductivityusingEq.( 5{7 ).TheresultsareshowninFigure 5-8 andFigure 5-9 forNbTiNandNbN,respectively. Forbothsamples,thetsarequantitativelyconsistentwiththeda ta.Thedipfeature locatedaroundthegapfrequencyintherealpartoftheopticalc onductivityiswell 102

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n n n n n %!r"!#!$ n n n n n n "$ Figure5-8.Thereal(circles)andimaginary(triangles)partsofth e T =2Koptical conductivityofNbTiNatdierentappliedperpendicularmagneticeld s, normalizedtothenormal-stateconductivity.Thesolidlinesaretst o 1 = n usingtheMaxwell-Garnetttheory.Thedashedlinesshowthecorre sponding 2 = n ascalculatedfromthesametheory. 103

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n n n n n %!r"!#!$ n n n n n n "$ Figure5-9.Thereal(circles)andimaginary(triangles)partsofth e T =2Koptical conductivityofNbNatdierentappliedperpendicularmagneticelds normalizedtothenormal-stateconductivity.Thesolidlinesaretst o 1 = n usingtheMaxwell-Garnetttheory.Thedashedlinesshowthecorre sponding 2 = n ascalculatedfromthesametheory. 104

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capturedbythets.However,forbothsamples,thedipappears atalowerfrequencyin thedatathaninthets,especiallyforNbN.Moreover,forNbNthe dipseemsshallower inthedatathaninthets.Accordingtotheoscillator-strengthsu mruleexpressedby Eq.( 2{20 ),thesetwofactspointtotheevidencethattheextractedreal partoftheoptical conductivityhaslessmissingareacomparedtothatinthettedopt icalconductivity, indicatingweakersuperconductivity.Anotherevidencetosuppor tthispointofviewis fromthebelow-gapbehavioroftheimaginarypartoftheopticalco nductivity,whichtakes theform A=! withtheconstant A proportionaltothecondensatedensity.Theimaginary partoftheopticalconductivitydata,especiallythatofNbN,devia tefromthetasthe eldincreases,indicatingmoreweakeningofsuperconductivitytha ninthets.Therefore, onemightarguethatusingthezero-eld s = n isnotadequate,andadditionalweakening ofsuperconductivityispresent.Thiswillbediscussedinthefollowing section. 5.4.5Pair-BreakingEects Intheanalysiswehavepresentedsofar,wemadeawidely-usedass umptionthatthe superconductingstateoutsideofthevortexcoresisunaected bytheexternalmagnetic eld.Whilethismaybevalidforbulksamples,itcouldbeinadequatefort hinlms throughwhichmagneticeldscaneasilypenetrate.Magneticeldsh avebeenproven tobreakthetime-reversalsymmetryoftheCooperpairsinbotht ype-Iandtype-IIthin superconductingsamples,inducingpair-breakingeects[ 136 137 ].Sucheectssmearout thesingularityinthequasiparticledensityofstatesandmodiesthe opticalconductivity s .Onemanifestationofsucheld-inducedmodicationisthesuppres sionoftheoptical gaprevealedintherealpartof s ,whichhasbeendemonstratedinourstudyofthe NbTiNandNbNthinlmsplacedinparallelmagneticelds,discussedinCh apter4.This motivatesustore-evaluateouranalysisbasedontheMaxwell-Garn etttheory,usingthe pair-breakingopticalconductivityfor s inEq.( 5{7 )insteadoftheunperturbedoneat zeroeld. 105

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TheNbNsampleisabetterexampletotesttheinruenceofthepair-b reakingeects thantheNbTiNsamplefortworeasons.Firstly,theweakeningofsu perconductivity duetovorticesneedstobesmallenoughsothatthepair-breaking eectscanplaysome noticeablerole.WehaveestimatedinSection 3.3.4 thatat2K, H ? c 2 oftheNbNsample isabout49.55T,and H ? c 2 oftheNbTiNsampleisabout12.50T.Intheeldrange of0{10T,theamountofvorticescanbesmallintheNbNsample,but issignicant intheNbTiN.Thisisevidentintheeld-dependenttransmissionandre rectiondata showninFigure 5-3 : T s = T n and R s = R n forNbTiNquicklyapproachesoneastheeld isincreasedto10T,buttheystillshowrobustsuperconductingfe aturesat10Tfor NbN.Secondly,toincludetheeectsofpairbreakingintheanalysis, itisimportant toidentifythemechanismofpairbreakingandtocorrectlycalculate thepair-braking opticalconductivity.Becausethetransmissionandrerectionmea surementspresentedin thischapterweredoneinperpendicularelds,weexpectpairbreak inginducedbythe spatialvariationoftheorderparameter.Inthepreviouschapte r,wehavefoundsuch eectsintheNbNsample.Wemayusethoseresultstoanalyzetheda tainthischapter. FortheNbTiNsample,weobservedadierenttypeofpair-breaking induceddirectly bytime-reversalsymmetrybreaking.Thecorrespondingoptical conductivitycannotbe appliedtotheanalysishere. Usingthenormalizedpair-breakingopticalconductivityshowninFigu re 4-6 as the s = n inEq.( 5{7 ),andusingthevalueof f yieldingthetsoftheNbNdatain Figure 5-9 ,wecalculatedtheeectiveopticalconductivityfromEq.( 5{7 ).Theresultsare showninFigure 5-10 assolidlines.Thedashedlinesaretheeectiveopticalconductivity calculatedusingthezero-eldopticalconductivity,takendirectly fromFigure 5-9 for comparison. Theinclusionofthepair-breakingeectssignicantlyimprovestheq ualityofthets, especiallyathigheldswherethepair-breakingeectsarethemost signicant.Thedip inthecalculatedopticalconductivitymatcheswellwiththatintheda ta.Moreover,the 106

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n n n n n %!r"!#!$ n n n n n n "$ Figure5-10.Thereal(circles)andimaginary(triangles)partsoft he T =2Koptical conductivityofNbNatdierentappliedperpendicularmagneticelds normalizedtothenormal-stateconductivity.Thesolidlinesaretsu singthe Maxwell-Garnetttheorywiththeinclusionofpair-breakingeects. The dashedlinesaretswithoutthepair-breakingeects,takendirec tlyfrom Figure 5-9 107

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below-gappartofthecalculatedIm( MGT = n )describesthebehaviorofthedatavery well.Theimprovementofthetssupportstheargumentthatitisne cessarytoconsider thepair-breakingeectsonthesuperconductingfractionofthe vortexstatewhenthe eectiveelectrodynamicresponseofthethinlmsuperconductor iscalculated. 5.4.6ConsistencyCheckoftheFits Aswhatwedidinthepreviouschaptertocheckthequalityofthets totheoptical conductivitydata,wecansimilarlyusethettedopticalconductivit yweobtainedabove tocalculatethetransmissionratio,thencomparewiththerawdata .Transmissionis pickedbecauseitiseasiertomeasurethanrerection,thusmorere liable.Weusethe ttedeectiveopticalconductivityshowninFigure 5-9 andFigure 5-10 tocalculate thetransmissionratioaccordingtoEqs.( 4{2 )and( 4{4 ).Theresultsareshownin Figure 5-11 .Inclusionofthepair-breakingeectsclearlymakesthetsconsis tentwiththe data,conrmingtheconclusionreachedattheendofSection 5.4.5 5.4.7FittingParameters Theanalysispresentedaboveyieldsthevaluesofthettingparame ter,the normal-volumefraction f .Itsvalueatdierenteldsforbothsamplesarelistedin Table 5-1 andshowninFigure 5-12 .Itistypicallyassumedthat f = H=H ? c 2 .Alinear tofthe f vs H datatotheform f = aH yieldsanestimateof H ? c 2 .Suchtstothe f obtainedfrombothsamplesareshownasstraightlinesinFigure 5-12 .Thetted valuefortheslopeis0.092T 1 forNbTiNand0.039T 1 forNbN.Thesecorrespond toanuppercriticaleldof10.87TforNbTiNand25.64TforNbN.ForN bTiN,the estimatedvalueisconsistentwiththevaluelistedinTable 3-1 .ForNbN,however,the H ? c 2 extrapolatedto2Kfromthefour-proberesistivitymeasurement maynotbeaccurate. Thereforeitisdiculttomakeareliablecomparison.Theelddepende nceofthe normal-volumefraction f ofNbNcanbettedbetterwithasquare-rootform f = a p H with a =0 : 11T 1 = 2 ,thoughsuchaformisunexpected. 108

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0 20 40 60 80 100 0 1 2 3 H = 1 T 0 20 40 60 80 100 0 1 2 3 H = 2 T 0 20 40 60 80 100 0 1 2 3 H = 3 T 0 20 40 60 80 100 0 1 2 3 H = 4 T 0 20 40 60 80 100 0 1 2 3T s / T n H = 5 T 0 20 40 60 80 100 0 1 2 3 H = 6 T 0 20 40 60 80 100 0 1 2 3 H = 7 T 0 20 40 60 80 100 0 1 2 3 H = 8 T 0 20 40 60 80 100 0 1 2 3 H = 9 T 0 20 40 60 80 100 0 1 2 3 H = 10 T Frequency [cm1 ] Figure5-11.Superconducting-statetonormal-statetransmiss ionratiodataofNbNin perpendicularelds(circles).Thesolidlinesarecalculationstakingint o accountthepair-breakingeects,whilethedashedlinesdonotinclu desuch eects. 109

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Table5-1.Normal-volumefraction f forNbTiNandNbN. H (T) f ofNbTiN f ofNbN 00.0000.00010.1400.09020.2400.15030.3300.18040.4000.22050.4800.24060.5600.26070.6400.28580.7100.30090.8000.315 100.8900.320 r n nn nn Figure5-12.Fielddependenceofthenormal-volumefractionforNb TiNandNbN.The solidanddashedlinesaretstotheform f = aH ,yielding a =0 : 092T 1 for NbTiNand a =0 : 039T 1 forNbN.Thedash-dottedlineisttothe f of NbNtotheform f = a p H ,yielding a =0 : 11T 1 = 2 5.4.8SuperruidDensity Tolearnhowmuchthemagneticeldisweakeningthesuperconductin gstate,wecan alsoestimatethesuperruiddensityfromtheimaginarypartoftheo pticalconductivity. ThemethodofestimationisthesameasthatusedinSection 4.4.6 110

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Table5-2.Superruiddensity(in10 19 cm 3 )forNbTiN(Column2)andNbN(Column3), calculatedfrom 2 .TheaverageerrorofColumn2is0.24,andtheaverage errorofColumn3is0.01. H (T) n s; NbTiN n s; NbN 06.291.7914.971.2923.991.1233.360.9842.670.8652.380.8362.040.8171.910.7681.550.7091.010.69 100.200.65 Firstly,wecalculate 2 ;> ( )from 1 ;> ( ), 2 ;> ( )= 2 P Z 1 0 1 ;> ( 0 ) 0 2 2 d! 0 : (5{16) Theintegralcanbeevaluatedbysplittingitintotwoparts.Therst partisintherange between0and100n G ,inwhichweusetheeectiveopticalconductivityfor 1 ;> ( ),shown assolidlinesinFigures 5-8 and 5-10 .Thesecondpartisintherangeabove100n G ,in which 1 ;> ( )canbewellapproximatedas n; 1 = n; 0 = (1+ 2 2 ). Secondly,theintegralEq.( 5{16 )issubtractedfromthe 2 data,andcomparedwith theoriginal 2 datainFigure 5-13 ,plottedvs1 =! .Thisdierencebetween 2 and 2 ;> is thenttedtotheform A=! toextractthevalueof A .Thetsareshownassolidlines inFigure 5-13 .ItworkswellforNbTiN,butforNbNasmally-interceptisneededt oget agoodlineart.Thevalueof A isthenconvertedtothesuperruiddensity n s usingthe relation A = n s e 2 = 2 m .TheresultsarelistedinTable 5-2 ,andplottedinFigure 5-14 .As theeldincreases,thesuperruiddensityinNbTiNdropsmoredrast icallythaninNbN becausetheuppercriticaleldismuchlowerintheformer.At10T,N bTiNalmosthas nosuperruidleft,whileNbNstillhasalmost40%ofsuperruidascompa redto0T. 111

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n r n n n n n n r n n n n n Figure5-13.Theimaginarypartoftheopticalconductivityinperpe ndiculareldsplotted vs1 =! forNbTiN(rstcolumn)andforNbN(secondcolumn).Thetriangles arethe 2 = n data.Thecirclesarethedierencebetween 2 = n and 2 ;> = n Thesolidlinesarelineartstothecircles. 112

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r nn r nn Figure5-14.Fielddependenceofthesuperruiddensity n s normalizedtoitszero-eld value,estimatedfromtheimaginarypartoftheopticalconductivit y 2 .The errorbarsarefromthelineartsinFigure 5-13 5.4.9DiscussionsofFlux-PinningandFlux-FlowRegimes WefocusedontheMaxwell-Garnetttheoryandusedittotouropt icalconductivity data.Indoingsowehaveimplicitlyassumedthatthevorticesaresta tionary,because theMaxwell-Garnetttheorydoesnottakeintoaccountpossibledy namicsofthegrains. Itisthereforeimportanttoknowifoursamplesareintherux-pinnin gregimeunderthe experimentconditions. Asimilarinfraredspectroscopicstudyofa80-nm-thickNbNlm[ 128 ]assumedtheir sampletobeintherux-pinningregime.Butthedepinningfrequencyr eportedonsome otherconventionalsuperconductorsbymicrowavemeasuremen tsaresmallerthanthe lowerlimitofourspectrum.Forexample,Sarti etal. [ 123 ]determinedtheupperlimit of 0 fora100-nm-thickMgB 2 lmtobe0.33cm 1 .Janjusevic etal. [ 124 ]foundthat decreasingthethicknessoftheirNbthinlmsfrom160nmto10nminc reases 0 from 0.03cm 1 to0.66cm 1 .Theysuggestedthatforthinlmsplacedinaperpendicular magneticeld,vortexpinningcouldbedominatedbysurfacepinningc enters,whichmay causethepinningofvortices.Thismayhappeninoursamples. 113

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Weestimatethedepinningfrequency 0 forourthinlmsamples,usinganexpression givenbyGittlemanandRosenblum[ 113 ], 0 = 2 cJ c n H c 2 r H 0 ; (5{17) where J c isthecriticalcurrent, 0 isthemagneticruxquantum,and H c 2 istheupper criticaleld.Therelationbetweenthethermodynamiccriticaleld H c andtheparallel uppercriticaleld H k c 2 forasuperconductingthinlmisgivenbyEq.( 2{48 )as H k c 2 = 2 p 6 H c =d ,where isthepenetrationdepthand d isthelmthickness.Four-probe resistivitymeasurementsoftheNbTiNthinlminparalleleldsupto16 Testimated H k c 2 at2Ktobe22.0T.NbTiNtypicallyhasapenetrationdepth 300nm(Table 3-1 ). Since d =10nm,wehave H c 0 : 15T.This,accordingtoformulaofthecriticalcurrent ofathinlmsuperconductorbyEq.( 2{46 ) J c = cH c = 3 p 6 ,gives J c 21 : 62MA/cm 2 Notingthat n =1 =R d =8 : 55 10 3 n 1 cm 1 ,and H c 2 11 : 2Tinperpendicular elds,fromEq.( 5{17 )wehave 0 rangingfrom16.5cm 1 to52.2cm 1 between1Tand 10T.UsingthematerialparametersoftheNbNsample,thecalculat ed 0 rangesfrom 109.3cm 1 to345.6cm 1 between1Tand10T.ItisthereforepossiblethattheNbTiN sampleisintherux-rowregime,andtheNbNinrux-pinningregime. ToseeiftakingintoaccountruxrowcanimprovetheCoey-Clemthe oryasa modeltoexplainourdata,wecalculatedtheeectiveopticalcondu ctivity CC = n from Eq.( 5{14 )usingdierentvaluesofthedepinningfrequency 0 .Theresultsareshown inFigure 5-15 .Asthedepinningfrequencyisdecreasedbelowtheupperlimitofour spectrum,theopticalconductivitybehavesverydierently.The above-gappartofthe Re( CC = n )isreducedbyagreatamount,makingitsmallerthanobservedinthe data showninFigure 5-4 .Farintotherux-rowregime,e.g.,thecase 0 =3cm 1 ,crossing ofRe( CC = n )occursbelowthegapasthenormal-volumefractionincreases.In the intermediateregimewhereruxrowandruxpinningarediculttodistin guish,e.g., thecase 0 =30cm 1 ,wedonotseeacleardipfeatureinRe( CC = n )aroundthegap 114

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r n n n n r n n n n r n n n n r n n n n Figure5-15.EectiveopticalconductivitycalculatedfromtheCo ey-Clemmodel.The rstrowshowstherealandimaginarypartsforthreevaluesofde pinning frequency 0 ,calculatedatthenormal-volumefraction f =0 : 4.Thesecond rowshowsresultsforthesamequantitiesat f =0 : 8. frequency.ThesefactsruleouttheCoey-Clemtheoryintherux -rowregimeasagood explanationofouropticalconductivitydata. 5.5Summary Inconclusion,wemeasuredfar-infraredtransmissionandrerect ionofNbTiNand NbNthinlmsinthevortexstatewhentheeldwasperpendiculartot hesamples. Signicantweakeningofsuperconductivitywasobservedinbothsa mples.Therealand imaginarypartsoftheopticalconductivityhavebeenextracted. Theopticalconductivity datacanbedescribedreasonablywellbyMaxwell-Garnetttheory, showingfeatures 115

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whichcannotbeaccountedforbyBruggemaneectivemediumappr oximationandthe Coey-Clemmodel. WhenMaxwell-Garnetttheoryisusedtoanalyzetheeld-dependen toptical conductivity,ifthesuperconductingstateoutsideofthevortex coresisassumedtobe unperturbedbytheexternaleld,thetsyieldlessweakeningofs uperconductivitythan showninthedata.Inclusionofthepair-breakingeectsimprovest hequalityofthets. Thisleadstotheconclusionthattheappliedmagneticeldnotonlycre atesnormalcores inthevortices,butalsoinducespair-breakingeectsonthesuper conductingfraction outsideofthevortices. 116

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CHAPTER6 TIME-RESOLVEDMAGNETO-SPECTROSCOPYOFCHARGEDYNAMICS 6.1IntroductiontoNon-EquilibriumSuperconductivity Whensuperconductorsaresubjecttoexternalexcitationssuc hasphotons,phonons, andinjectedquasiparticlecurrents,thepairedelectronsforming thecondensatearebroken intohigh-energyquasiparticles.Thesequasiparticlesfaraboveth eFermilevelequilibrate amongthemselveswithinfemtoseconds,andthenrelaxtothesupe rconductinggapedge throughelectron-electron,electron-phononscatteringwithinp icoseconds.Thegap-edge quasiparticlesnallyrecombinetoformpairsandemitenergyasphon ons[ 138 { 140 ]. Theserecombination-generatedphononswerecentraltothest udybecausetheycanbreak additionalCooperpairsbeforetheyescapefromthesupercondu ctortothesurroundings, dominatingtheeectivequasiparticlelifetime[ 141 { 144 ].Figure 6-1 illustratesthewhole process.TherelaxationprocesshasbeenwidelystudiedinBCSsupe rconductorsto testnon-equilibriumtheoriesofmany-bodysystems[ 55 { 57 ],andinhigh-temperature superconductorstogainnewperspectiveoftheirpairingmechanis m[ 58 { 61 ]. Experimentally,Testardishowedthatstronglaserexcitationsco ulddrivesuperconducting thinlmsnormal,andthatthiseectwasnotduetolatticeheating[ 145 ].Toexplainthis phenomenon,OwenandScalapino[ 146 ]proposedaneectivechemicalpotentialmodel todescribethenon-equilibriumstate.Theyarguedthattheenerg ydistributionofthe non-equilibriumquasiparticlesischaracterizedbythelatticetemper ature T andan additionalchemicalpotential .Thismodelpredictsarst-orderphasetransitiontothe normalstateatasucientlylargedensityofexcessquasiparticles .Debatingthatthis transitionwasnotobservedinexperiments,Parker[ 147 ]proposedamodiedheating theory,inwhichthenon-equilibriumstateischaracterizedbyamodi edtemperature T determinedbythephononswithenergygreaterthantheopticalg ap.Thetwomodelsare indistinguishableinthelowperturbationlimit. 117

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excess quasiparticles excess2 phonons highenergy( E E F )electronicexcitations(quasiparticles) lowenergyquasiparticlesand E Debye phonons pairedelectronsandthermalquasiparticles femtoseconds picoseconds phononescape (atrate 1 r ) bottleneck nanoseconds h 2 1 R 1 B Figure6-1.Illustrationofthequasiparticlerelaxationprocess. Therecombinationprocesswastheoreticallystudiedfurtherbyco nsideringthe distributionsofqusiparticlesandphonons.Kaplan etal .[ 148 ]calculatedthequasiparticle scattering,recombination,andbranch-mixinglifetimesandphonon pair-breakingand scatteringlifetimesinthenear-equilibriumcondition,undertheassu mptionthat quasiparticlesandphononsobeythesamedistributionsasintherma lequilibrium. Beginningwiththecoupledquasiparticleandphononkineticequations andtheBCSgap equation,ChangandScalapino[ 149 ]obtainedthemodiedquasiparticleandphonon distributionsandthechangeinthegapparameter.Experimentallyt herecombination processwasinvestigatedthroughthedependenceofthequasipa rticlelifetimeson temperature,thinlmthicknessandexcitationstrength[ 150 151 ]. 6.2Motivation Anopenquestionishowanexternalmagneticeldcouldchangether ecombination process.InChapter 4 andChapter 5 weobservedtheeectsoftheparallelandperpendicular magneticeldsontheopticalpropertiesofthinlmsuperconducto rs.Inthischapterwe willinvestigatehowtheseeectscouldaectthequasiparticlereco mbinationdynamics inthesesamples.Wereportatime-resolvedspectroscopicexperim entthatusesexternal 118

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magneticeldstomanipulatethequasiparticlerecombination.Wend thatthemagnetic eldallowsustotunetherecombinationrate.Suchphenomenumisex plainedthrough eld-inducedpairbreaking. 6.3Experimental ThequasiparticlerecombinationdynamicsintheNbTiNandNbNthin-lm superconductorsisstudiedbytime-resolvedpump-probespectr oscopy.Specicsof thistechniquearediscussedinSection 3.2 .Apumpbeamwithphotonenergygreater thanthesuperconductinggapexcitesthesampleandgeneratesq uasiparticles;aprobe beammeasuresthesample'smodiedopticalpropertiesduetothee xcitedquasiparticles atdierentdelaytimerelativetothearrivalofthepumpbeam.Oure xperimentswere performedusinganovelpump-probeset-upillustratedinFigure 3-3 .Thesampleswere mountedina 4 HeOxfordcryostatequippedwitha10Tsuperconductingmagnet. They arephoto-excitedbynear-infraredlaserpulsesfromaTi:sapphire laser,andprobedby far-infraredradiationproducedatBeamlineU4IRoftheNationalS ynchrotronLight Source,BrookhavenNationalLaboratory.Thefrequencyrang eoftheprobebeam (10{100cm 1 )isoptimalforstudyingtype-IIsuperconductorsbecauseofth eirgap energyscales.Utilizingthepulsedfeatureofthesynchrotronrad iation,wegenerated laserpulsesthatmatchestherepetitionfrequencyofthesynchr otronpulsestoexcite thesamples,andthenmeasuredthelaser-inducedchangeinthetr ansmissionofthe synchrotronprobebeamthroughthesamples.Toimprovesensitiv ity,weemployeda dierentialtechniquewherethepumppulsewasphase-modulateds othatitsarrivaltime wassinusoidallyditheredasmallamplitude dt withrespecttotheprobebeam.The sampleresponsewasthenthedierentialchangeofthetransmiss ionsignal dS=dt .This signalwasdetectedusingahighsensitivityB-dopedSibolometerop eratingat1.6Kby pumpingthe 4 Hecoolantfordetectingthefar-infraredradiation.Integrating itovertime yieldsthephoto-inducedtransmission S ( t ),whichcloselyrelatestothephoto-induced 119

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r nr Figure6-2.Typicaldierentialsignal(squares)andintegrateds ignal(circles),measured onNbTiNat1.9K,0T,atlowlaserruence. quasiparticledensity[ 152 ].Typicaldierentialsignalandintegratedsignalareshownin Figure 6-2 6.4TemperatureDependence Beforediscussingtheeld-dependentphenomena,itisinstructive tolookatthe temperaturedependentrecombinationprocessbecauseitisbett erunderstood.Figure 6-3 showsthetemperature-dependentphoto-inducedtransmission ofNbTiNinzeromagnetic eldatapumplaserruenceof0.37nJ/cm 2 .Inthislow-ruenceregimethesample responsecanbewelldescribedassingleexponential, r ( t )= Ae t= (6{1) for t> 0andzerootherwise,where A istheamplitudeoftheexponentialdecayand is thecharacteristiclifetime.Thedetectedsignal S ( t )istheconvolutionoftheexponential decayandthesynchrotronprobepulsewithaGaussianproleanda pulsewidth p ( t )= 1 p 2 e t 2 = 2 2 (6{2) Thedetectedsignalis 120

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r n Figure6-3.TemperaturedependenceofthedecayforNbTiNmeas uredat0Tandlow laserruence.ThesolidlinesaretsusingEq.( 6{3 ). S ( t )=( r p )( t )= Z 1 1 r ( t 0 ) p ( t t 0 ) dt 0 = Z 1 0 Ae t 0 = 1 p 2 e ( t t 0 ) 2 = 2 2 dt 0 = A 2 exp 2 2 2 t erfc p 2 t p 2 ; (6{3) inwhicherfc( x )=1 erf( x )=2 = p R 1 x e t 2 dt isthecomplementaryerrorfunction. FitsusingEq.( 6{3 )areshowninFigure 6-3 assolidlines,with A and asthetting parameters.Table 6-1 liststhevaluesofthesettingparameters. Fromthettedamplitude A wecanestimatethethermalquasiparticledensityvia N th / Q 1 1where Q = A=A T 0K [ 153 ].Weextrapolatethevalueof A to0Kanduse Table6-1.Fittingparametersforthetemperature-dependentd ecaydataofNbTiN T (K) A (arb.unit) (ns) (ns) 1.94.133.580.284.23.772.750.295.63.651.350.297.02.600.600.17 10.11.650.550.28 121

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nr r nr r nr r Figure6-4.Left:temperaturedependenceoftheexponentiald ecayamplitude A ;thesolid lineisaquadratict,extrapolatingto T =0Kwithavalueof4.36.Center: Theestimatedthermalquasiparticledensity N th ;thesolidlineisatusing Eq.( 6{4 ).Right:temperaturedependenceofthelifetime thatas A T 0 .Theestimated N th isttedto[ 153 ] N th = N (0) p 2 k B Te =k B T ; (6{4) with N (0),theelectronicdensityofstatesperunitcellattheFermileve l,asthetting parameter.TheresultisshowninthecenterpanelofFigure 6-4 .WeusedtheBCS temperaturedependenceforthegap.Theagreementoftheda taandthetindicates thatthesampleisinthestrongphonon-bottleneckregime.Thesimila rcompoundNbN isexpectedtobeinthesameregime.FitsinFigure 6-3 alsoyieldtheFWHM(fullwidth athalfmaximum)ofthesynchrotronpulses(approximately2.35 ),whichdeterminesour besttimeresolutioninthetime-resolvedexperimenttobe 300ps.Thedataat7.0K wastakenwiththesynchrotroninthe7-bunchcompressedmodew ithanarrowerpulse width,whileotherdataweretakeninthe7-bunchdetunedmode. 6.5FieldDependence:FieldParalleltoSampleSurface Tostudytheeectofmagneticeldsontherecombinationdynamics ,wemeasured theelddependentphoto-inducedtransmissionforNbTiNandNbNa tvariouslaser ruences,withthesamplessubmergedundersuperruid 4 Heat T 2Ktoavoidlaser heating.Theeldwasappliedparalleltothethinlmsurfacestominimiz ethecomplexity 122

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ofvortexdynamics(discussedinSection 4.3.1 ).TheresultsareshowninFigures 6-5 6-6 and 6-7 6.5.1Exponential-DecayFit Thesemi-logplotsinFigures 6-6 and 6-7 showthatatlowruencethedecayissimple exponential,similartothatshowninFigure 6-3 .Athighruencethedecayisinitiallyfast, andthenslowsdownastheexcessquasiparticlesrecombine.Inbot hregimes,theslopes ofthesemi-logplotssuggestthatthelifetimeincreaseswithincreas ingeld.Earlystudies ofnon-equilibriumphysicsinsuperconductingSnthinlmsinparallelma gneticelds nearthetransitiontemperaturefoundshortenedquasiparticlelif etimewithappliedeld [ 154 155 ]. WefoundthatforNbTiN,thedatacannotbettedwithasingleexpo nentialdecay. Weusethefollowingmodelwithtwoexponentialdecaystotthedat a, S ( t )= A 1 2 exp 2 2 2 1 t 1 erfc p 2 1 t p 2 + A 2 2 exp 2 2 2 2 t 2 erfc p 2 2 t p 2 ; (6{5) whichisaconvolutionoftheprobepulse p ( t )givenbyEq.( 6{2 )andasampleresponseof theform r ( t )= A 1 e t= 1 + A 2 e t= 2 (6{6) for t> 0andzerootherwise,where A 1 and A 2 areamplitudes,and 1 and 2 are characteristiclifetimes.Fitoftheeld-dependentdataforNbTiNu singthismodel areshownassolidlinesinFigure 6-5 ,withttingparameterslistedinTable 6-2 ForNbN,thedatacanbedecentlydescribedbythesingleexponent ialdecaymodel Eq.( 6{3 ).ThetsareshowninFigure 6-7 assolidlines,withttingparameterslistedin Table 6-3 .Thetwo-exponential-decaymodelEq.( 6{5 )doesnotsignicantlyimprovethe t. TheanalysisofNbTiNandNbNdatashowssomecommonfeatures,as shownin Figure 6-8 andFigure 6-9 .Theeectivelifetimeincreaseswithincreasingeld,whichis 123

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! n r n ! n ! n Figure6-5.NbTiNdecaydatainparalleleldsatvariouslaserruence sand T 2K.The solidlinesaretsusingEq.( 6{5 ). 124

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! n r n n n Figure6-6.NbTiNdecaydatainparalleleldsatvariouslaserruence sand T 2Kon semi-logscale. 125

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! n n n n n n n n n n n n r n Figure6-7.NbNdecaydatainparalleleldsatvariouslaserruences and T 2K.The rightcolumnarethesamedatashownintherstcolumnplottedonasemi-logscale.ThesolidlinesaretsusingEq.( 6{3 ). 126

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n n n n n n n n n n n n n n n n r n r Figure6-8.FittingparametersofNbTiNdecaydatainparalleleldsa tvariouslaser ruences.Fromtherstrowtothelastrow,theruencesare:0.37 ,0.80,1.60, 2.67,5.35,and10.70nJ/cm 2 127

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n n n n r n r Figure6-9.FittingparametersofNbNdecaydatainparalleleldsat variouslaser ruences.Fromtherstrowtothelastrow,theruencesare:2.35 ,4.56,9.05, and18.10nJ/cm 2 128

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Table6-2.FittingparametersfortheNbTiNdecaydatainparallele ldsatvarious ruences Fluence(nJ/cm 2 )Field(T) A 1 (arb.unit) 1 (ns) A 2 (arb.unit) 2 (ns) (ns) 0.370.01.7221.5162.6275.2810.3250.372.51.3151.1863.2044.6370.3160.375.01.7171.2613.2855.4090.3650.377.54.4135.2800.1435.2810.3210.3710.00.0001.4605.6756.7140.3030.800.03.4220.7625.4993.9960.3370.802.53.9180.9924.8884.4670.3240.805.04.1621.5584.1355.4270.3560.807.55.9903.1502.18410.0170.3160.8010.09.0164.9470.89710.0460.2961.600.07.6250.6748.6933.3850.3381.602.57.5350.8347.3284.0190.3181.605.09.3641.2146.3255.5410.3491.607.57.0041.03210.1374.9230.3311.6010.012.5353.3723.81010.1500.2932.670.014.4880.51012.6603.0000.3382.672.521.4610.35511.9293.0830.4292.675.014.6901.0249.2904.3810.3462.677.513.4750.97012.7724.7480.3252.6710.019.9872.7015.19510.0180.2895.350.031.2620.46719.5482.3280.3335.352.538.7150.38118.4902.5800.4145.355.029.6050.87813.5233.7520.3445.357.523.3430.85718.2353.6100.3215.3510.033.0922.1337.16910.0360.289 10.700.054.7180.35436.1851.4390.33910.702.569.7060.38427.4042.1770.40210.705.064.1690.43829.7462.2990.37110.707.540.6230.63029.7412.5740.33210.7010.041.9950.86130.8223.7060.312 alreadyevidentinthesemi-logplotoftherawdata.ForNbTiN,botht hefastandthe slowcomponentsdecayslowerwhentheeldisincreased.Thereisno cleartrendinthe elddependenceofthedecayamplitudes.Itmightbedecreasingas theeldincreases, especiallyforNbNathighruence. 129

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Table6-3.FittingparametersfortheNbNdecaydatainparalleleld satvariousruences Fluence(nJ/cm 2 )Field(T) A (arb.unit) (ns) (ns) 2.350.06.2221.9890.3032.352.05.9582.1130.3052.354.06.2773.0360.3182.356.05.9463.1200.3134.560.011.2431.6400.3084.562.010.5062.2640.2864.564.010.9972.4820.3134.566.011.2092.4580.3489.050.021.3241.3980.3049.052.020.9341.4730.3429.054.019.5261.9880.3089.056.019.2442.0160.333 18.100.035.1451.2650.30818.102.036.5341.2880.32618.104.032.3781.7340.28018.106.031.9631.7640.28318.108.030.8391.9230.29218.1010.029.3432.1870.289 6.5.2UniversalScalingBehavior Theelddependenceofthequasiparticlerecombinationcanbedemo nstratedmore clearlybydeninganeectiveinstantaneousrelaxationratefromt hedierentialsignal dS=dt andtheintegratedsignal S atthesamedelaytime t 1 e 1 S dS dt : (6{7) Foreachsetofdataatagivencondition(sameeldandsamelaserru ence),weskipthe rstfewpointswherethebehaviorisdominatedbytheGaussianpro bebeamprole;the restisusedtodenetheeectiverate 1 e .Wefoundthatateacheld 1 e scalesalmost linearlywith S .Moreover,ateacheld,whenplotting 1 e vs S ,dataatdierentruence overlapverywell,showingauniversalscalingbehavior.Theresultsf orbothsamples areshowninFigure 6-10 .The 1 e dataareslightlysmoothedwiththemovingaverage methodsothatthelinearbehaviorbecomesclear.However,curva turesexistintheplots showninFigure 6-10 ,especiallywhen S islarge,whichcorrespondstohighruenceorthe 130

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n rr n rr n Figure6-10.EectiveinstantaneousrelaxationrateforNbTiNand NbNinparallelelds. ForNbTiN,dataateacheldincluderuencerangingfrom0.37to10.70nJ/cm 2 .ForNbN,dataat8Tand10Tareonlyattheruenceof 18.10nJ/cm 2 ,butrangefrom2.35to18.10nJ/cm 2 atotherelds.The straightlinesarelineartstothedata. initialstageofquasiparticlerecombination.NotethatinFigure 6-10 ,foradecaytraceat agiveneldandruence, 1 e appearsathigher S attheinitialstageoftherecombination. Asrecombinationcontinues, S decreases,and 1 e movestowardssmaller S onthestraight line.Foragiveneld, 1 e atdierentlaserruencestartsatdierentlevelonthestraight line.Inthefollowing,wewilldiscusstheelddependenceshowninFigur e 6-10 6.5.3RecombinationModel Inthissectionweproposearecombinationmodeltoexplaintheeldd ependence showninFigure 6-10 Thequasiparticlerecombinationinasuperconductorincludingphono nbottleneck eectwasrstdiscussedinaphenomenologicaltheorybyRothwar fandTaylor[ 141 ]using twocoupledrateequations, dN dt = I 0 N 2 + 1 B n; (6{8) dn dt = J 0 1 2 1 B n + 1 2 N 2 1 r ( n n th ) : (6{9) 131

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Here N and n arethedensitiesofquasiparticlesandhigh-energyphonons, I 0 and J 0 are theinjectionratesofquasiparticlesandhigh-energyphononsinto thesystem, isthe intrinsicquasiparticlerecombinationratecoecient,and 1 B istherateofpairbreaking bythehigh-energyphonons.ThelastterminEq.( 6{9 )representsthephononescaping process,inwhich n th isthehigh-energyphonondensityatthermalequilibrium,and 1 r is thephononescapingrate.Thequadratictermsinthesetwoequat ionsisbasedonthefact thattwoquasiparticlesneedtobepresentforarecombinationtoh appen.Thecoecients 1/2inEq.( 6{9 )areincludedforthereasonthatwhentwoquasiparticlesrecombin e,only onehigh-energyphononiscreated. Inprinciple,thesetwocouplednon-linearequationsarediculttoso lve.However wearemoreinterestedinthequasiparticles,becausetheirrecomb inationgivesrisetothe changeofopticalpropertiesobservedinourexperiments.Wepro poseasimplermodel thatcancapturethenatureofbimolecularrecombinationshownby Eq.( 6{8 ),meanwhile takingintoaccountthephononbottleneckeect.Inasupercond uctor,whenquasiparticles recombine,aspin-upquasiparticleandaspin-downquasiparticlenee dtobeavailable, justlikethecaseofelectron-holerecombinationinasemiconductor .Motivatedbythis analogy,weuseanequationsimilartotheband-to-bandrecombinat ionequationina semiconductortodescribethequasiparticlerecombination, dN dt = dN dt + dN # dt = 2 R N N # N th N # th : (6{10) Here N isthetotalquasiparticlenumberdensityasinEqs.( 6{8 )and( 6{9 ),whichconsists ofspin-upandspin-downpopulations, N = N + N # .Similarly N th = N th + N # th forthe thermaldensity N th .Thephononbottleneckeectisintroducedintothemodelthroug h therecombinationratecoecient R .Theaboveequationisbasedontheobservationthat becausetherecombinationrequiresthepresenceoftwoquasipar ticlesofoppositespin,the recombinationrateisexpectedtobeproportionaltotheproduct of N and N # .Noting 132

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thatatthermalequilibriumthereshouldbenonetchangeofthequa siparticledensity,a thermalterm N th N # th shouldbesubtractedfrom N N # Eq.( 6{10 )canberewrittenbyusingthefollowingrelations.Firstly, N consistsofthe thermaldensity N th andtheexcessdensity N ex N = N th + N ex .Secondly,bydeningthe spin-upandspin-downquasiparticlefractions P and P # ,thecorrespondingdensitycanbe expressedas N = P N and N # = P # N .Ataconstanttemperaturewherethethermal termdoesnotvarywithtime,Eq.( 6{10 )canberewrittenas 1 N ex dN ex dt =2 R ( P P # )( N ex +2 N th ) : (6{11) Thisordinarydierentialequationcanbesolvedexactly,withthefo llowingsolution, N ex ( t )= N ex ; 0 2 e t= 1 2+ N ex ; 0 N th (1 e t= 1 ) ; (6{12) where N ex ; 0 = N ex ( t =0),and 1 =1 = 4 RN th ( P P # ).Atlowruence, N ex ; 0 =N th 1. Thesolutionreducestoasingleexponential,withalifetimegivenby 1 .Athighruence, Eq.( 6{12 )predictsafastdecayfollowbyaslowdecay.Numericalresultsare compared inFigure 6-11 .Thesefeaturesareconsistentwithourdata,showninFigure 6-6 and Figure 6-7 .Eq.( 6{12 )canthereforebeusedtotthesedatausingtheconvolutionmod el. Butthetrequiressimultaneouslyadjustingtwoparameters,the ratio N ex ; 0 =N th and thelifetime 1 ,whichmakesitdiculttosettlethevalueforeitherofthem.Instea dof performingsuchat,wedecidedtofocusonthescalingbehaviorsh owninFigure 6-10 TolinkourmeasuredquantitytotherecombinationmodelEq.( 6{11 ),weassume thephoto-inducedtransmission S ( t )measuredateachdelaytimeisproportionalto thephoto-inducedexcessquasiparticledensity, S ( t )= CN ex ,where C isaconstantof proportionality.Amoreprecisedenitionof C is r ( t )= CN ex where r ( t )isthesample responsefunction.HoweverwhenmakingFigure 6-10 wehaveskippedtheinitialpart ofthetracecharacterizingtheGaussianprobebeam. r ( t ) S ( t )inthislimit.The 133

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 t/1 0.0 0.2 0.4 0.6 0.8 1.0N ex ( t ) /N ex 0 N ex /N th 0 01 0 1 1 10 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 t/1 10 -3 10 -2 10 -1 10 0 N ex ( t ) /N ex 0 N ex /N th 0 01 0 1 1 10 100 Figure6-11.ExactsolutionEq.( 6{12 )oftherecombinationequationatvariousruences, plottedonlinearscale(left)andsemi-logscale(right).Athighruenc e,this solutionpredictsafastdecayfollowedbyaslowdecay. recombinationequationcanbefurtherwrittenas 1 S dS dt =2 R C ( P P # )( S +2 S th ) ; (6{13) where S th = CN th .ThequantityontheleftofEq.( 6{13 ), ( dS=dt ) =S ,istheeective instantaneousrelaxationrate 1 e wedenedintheprevioussection.Eq.( 6{13 )is consistentwithFigure 6-10 ,iftheelddependenceinthegurecanbeaccountedforby thecoecient2( R=C )( P P # ). NotethatthedatainFigure 6-10 deviatefromthestraightlinesinthehigh-ruence limitwhen S islarge,possiblybecausethehigh-ruencelaserpulsesbreakasigni cant amountofCooperpairs,whichinducesfurtherweakeningofthesu perconductivity [ 145 146 ].Wewouldliketointerprettheeld-dependentlinearbehavior,ther efore focusonthelowruencedatainthefollowingdiscussion.Intheslope2 R ( P P # ) =C in Eq.( 6{13 ),theconstant C isessentiallyindependentofeld.Wewilldiscusstheeld dependenceof P P # R ,and C separatelyinthefollowingthreesections. 134

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6.5.3.1Recombinationratecoecient Therecombinationratecoecient R describestherateofrecombination,withthe phononbottleneckeectsincluded.Thissectiondiscussesitseldd ependenceinthe pair-breakinglimit. Thequasiparticles,interactingwithphononsinthesystem,fullydec aywithan eectiverate 1 .Inthelow-ruencelimit,GraylinearizedtheRothwarf-Taylorequat ions toderiveaneectiverate[ 156 157 ], 1 = 1 2 p q p 2 8 1 R 1 r ; (6{14) where p =2 1 R + 1 B = 2+ 1 r .Here 1 R istheintrinsicrecombinationrate, 1 B isthe phononpair-breakingrate,and 1 r isthephononescapingrate.Thephononescapingrate isdeterminedbythethinlmthicknessandtheacousticmatchbetwe enthelmandthe environment[ 158 ],andisthereforeeldindependent.Foralmofafewnanometerso na dielectricsubstrate,itistypicallyontheorderofhundredsofpicos econd[ 159 ].Weassume foroursamplesitisabout300ps.Theintrinsicrecombinationrate 1 R andphonon pair-breakingrate 1 B arediscussedbyKaplan etal .[ 148 ]undernear-thermal-equilibrium condition, 1 R ( )= 1 0 ( kT c ) 3 1 1 f ( ) Z 1 + d nn 2 n p (n ) 2 2 1+ 2 (n ) [ n (n)+1] f (n ) ; (6{15) 1 B (n)= 1 0 ; ph 0 Z n d! p 2 2 n p (n ) 2 2 1+ 2 (n ) [1 f ( ) f (n )] ; (6{16) where f and n aretheFermi-DiracandBose-Einsteindistributionfunctions,resp ectively. 0 and 0 ; ph arethecharacteristiclifetimesofthequasiparticlesandphonons, 0 = ~ Z 1 (0) 2 b ( k B T c ) 3 ; (6{17) 0 ; ph = ~ N 4 2 N (0) h 2 i av 0 : (6{18) 135

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Z 1 (0)isthequasiparticlerenormalizationfactor,whichisabout2.5. N isthedensity ofions.UsingthemassdensityofNbN(8.38g/cm 3 fromRef.[ 160 ])anditsmolecular mass(64.95fromRef.[ 161 ]),weestimate N 7 : 78 10 22 cm 3 N (0)isthenormal-state quasiparticledensityofstatesatthefermilevel,whichisapproxima tely2 : 00 10 28 states/m 3 eV forNbN[ 81 ]. b and h 2 i av arefromtheelectron-phononspectralfunction 2 (n) F (n), where 2 (n)isthesquareofthematrixelementoftheelectron-phononinte raction,and F (n)isthephonondensityofstates. h 2 i av isdenedbyKaplan etal .as h 2 i av 1 3 Z 1 0 2 (n) F (n) d n : (6{19) Usingthetunneling 2 (n) F (n)dataforNbN[ 162 ],wend h 2 i av 5 : 11meV.The factor b isintroducedbyassumingthattheelectron-phononspectralfun ctionhastheform 2 (n) F (n)= b n 2 .AccordingtoEq.( 6{19 ), h 2 i av = 1 3 Z 1 0 2 (n) F (n) d n 1 3 Z k B D 0 b n 2 d n= 1 9 b ( k B D ) 3 ; (6{20) where D istheDebyetemperature,whichisapproximately331KforNbN[ 163 ].From theaboveequation,weestimate b 1 : 98 10 3 meV 2 .FromTable 3-1 wehave T c =12 : 8Kand 0 =17 : 9cm 1 forNbN.Withtheseparameters,weestimatefrom Eqs.( 6{17 )and( 6{18 )that 0 9 : 84 10 11 sand 0 ; ph 7 : 99 10 12 s.NbTiN shouldhavesimilarmaterialparameters.Usingallthesameparamet ersexceptfor T c and 0 ,whichwetakefromTable 3-1 ,weestimateforNbTiN 0 1 : 94 10 10 sand 0 ; ph 1 : 12 10 11 s. NotethatEqs.( 6{15 )and( 6{16 )involvethegap,thequasiparticledensity ofstates,andthecoherencefactor.Whenamagneticeldisapplie dparalleltothe thinlms,thepair-breakingeectscouldaectallofthem.Werepla cetheinthe integrationlimitsbythehalfeectivespectroscopicgapn G .Themodiedquasiparticle densityofstatesisgivenbyEq.( 4{32 )asafunctionofthepair-breakingparameter.We didnotndexplicitdiscussionsofthecoherencefactorinthepair-b reakinglimit.Asa 136

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rstorderapproximation,weassumethisfactorstilltakesthesa meform,butwith replacedbyn G .Weconsiderthequasiparticlesatthegapedgewithenergy =n G andthephononsjustenergeticenoughtobreakmorepairs(take n=2 : 001n G forthe convenienceofcalculation).Thecalculatedlifetimesarecomparedin Figure 6-12 .At zeroeld, B iscomparableto r ,bothbeingsmallerthan R .Astheeldincreases,the inducedpair-breakingeectsdrasticallyincrease B ,makingitmuchgreaterthan R and r .Thephononescapingrateismuchfasterthanthephononpair-br eakingrate,sothat thephononbottleneckbecomesunimportant.Theeectivelifetime isdominatedbythe intrinsicrecombinationtime R Inthelowruencelimit,assumingnospinpolarization,wehavefromEq. ( 6{11 )that 1 e RN th : (6{21) ThisequationtogetherwithEq.( 6{14 )yields R = 1 N th ; (6{22) where N th =2 Z 1 0 N ( E ) f ( E ) ; (6{23) withthesingle-spinquasiparticledensityofstates N ( E )givenbyEq.( 4{32 ).Then G dependenceof N th aswellas R areplottedinFigure 6-12 inthesecondcolumn,withthe lattershowinganalmostquadraticrelation.6.5.3.2Spin-polarizationfactor Thespin-polarizationfactor P P # willstronglydependonthemagneticeldifthe eldmainlyactsontheelectronspin,requiringthespin-orbitscatte ringtobenegligible. ThiscorrespondstothePauliparamagneticlimit.AccordingtotheB CStheory,electrons formspin-singletpairscondensedinthegroundstate;thespinsus ceptibilityvanishes asthetemperatureapproaches0.Agreatamountofworkonsup erconductorspin susceptibilitywasdoneonthinlmssampleswiththicknesssosmalltha ttheeect 137

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r r r r n Figure6-12.Firstcolumn:thecalculatedintrinsicrecombinationtime R ,phonon pair-breakingtime B ,andtheeectivelifetime asafunctionofn G .The phononescapingtime r istakentobe0.3ns.Secondcolumn:thermal quasiparticledensity N th andtherecombinationratecoecient R asa functionofn G ofamagneticeldontheelectronorbitscanbeneglected.Paramag neticsplittingof thequasiparticledensityofstateswasobservedin5nmaluminumlms inaparallel magneticeld[ 14 ].vanBentumandWyder[ 93 ]studiedfar-infraredabsorptionofthin superconductingaluminumlmswithdierentthicknessesinparallelm agneticeld.They foundthat10nmthinlmsareinthepair-breakinglimit,whileultrathin lmslessthan 5nmareintheparamagneticlimit. 138

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n r! n r !" Figure6-13.MajorityspinfractionofNbTiNandNbNintheparamagn eticlimit.The circlesandsquaresareextractedfromtheslopesinFigure 6-10 ,allscaledso thatthemajorityspinfractionis1/2at0T.Thesolidanddashedlines are theoreticalcalculationsusingEqs.( 6{24 ),( 6{25 ),and( 6{26 ). Inthelow-ruencelimit,theexcessquasiparticledensityissmallcomp aredto thethermalpopulation.Themajorityspinfraction P # canbecalculatedfromthe paramagneticmodel,usingFermi-Diracdistributionofquasiparticles f ( E )andthe quasiparticledensityofstatesDOS( E )fromtheBCStheory, N # =2 Z 1 0 f ( E )DOS( E + B H ) dE; (6{24) N =2 Z 1 0 f ( E )DOS( E B H ) dE: (6{25) Fromthese, P # canbecalculatedas P # = N # N + N # : (6{26) Ontheotherhand,ifweassumethattheelddependenceweseeinF igure 6-10 is onlythroughtheproduct P P # ,wecanextractthemajorityspinfractionatdierent elds.TheresultsarecomparedwiththecalculationusingEq.( 6{26 )inFigure 6-13 139

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Apparently,Pauliparamagnetismgivesmorepolarizationthanwhat thedata show.Thiscouldbeduetothestrongspin-orbitscatteringinNbTiNa ndNbN.Tedrow andMeserveyobservedthespin-statemixinginthinaluminumlmsdue tospin-orbit scattering[ 164 ].Theydenedaspin-orbitscatteringparameter b ~ = 3 so todescribe thestrengthofspin-orbitscattering.Theycalculatedthatas b isincreasedto0.5,the spin-upandspin-downquasiparticledensityofstatesmixupcomplet ely,leavingnoclear signatureofthetwo-peakfeatureinthedensityofstatesdueto Zeemansplitting.We foundforasimilarcompoundNbTi, so =3 : 0 10 14 s[ 66 ].Usingthe 0 ofNbTiNand NbN,weestimatedthat b =4 : 2and3.3forNbTiNandNbN,respectively.Thisindicates strongspin-statemixingintheNbTiNandNbNsamples,iftheirspin-or bitscattering time so isclosetothatofNbTi.Basedonthesearguments,weexpectthat thespin polarizationfactor P P # isweaklydependentontheeld. 6.5.3.3Constantofproportionality The\constant"ofproportionality C introducedthrough S = CN ex cannotbe perfectlyeldindependentunderallconditions.Thephoto-induce dtransmission S ( t ) measuredatdierenteldsarerelatedtotheeld-dependenttra nsmissionshownin Figure 4-2 .Theexcessquasiparticledensity N ex dependsontheeld-dependentenergy gapn G .The\constant" C ,denedastheratio S=N ex ,ispossiblyalsoelddependent.In thissectionwewillshowthat C isalmosteld-independentinthelowruenceregime. Assumethatallthelaserenergyiseventuallydistributedtothegap -edgequasiparticles afterrelaxation.Foralaserpulsewithruence F illuminatedonathinlmsamplearea a withlmthickness d ,themaximumexcessquasiparticledensityis N ex,max = F a = n G a d = F n G d : (6{27) Here isthepercentageoflaserpowerthateventuallygoesintothephot o-excitation process.Itconsistsofthreefactors,allevaluatedatthelaser frequency(near-infrared). Therstfactoristhetransmittanceofquartzwindowonthemagn et,whichisabout 140

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90%[ 165 ].Thesecondfactoristhererectanceofthealuminummirrorthatd irectslaser beamontothesample(leftpanelofFigure 4-1 ),whichisabout88%[ 166 ].Thethird factoristheabsorptionofthethinlmsuperconductor,whichisab out30%,calculatedas 1 T s R s .Therefore =24%.Asanestimate,at10.7nJ/cm 2 ,forNbTiNat2Kand 0T,n G =14 : 3cm 1 and d =10nm,wecalculated N ex,max =9 : 03 10 18 cm 3 .Comparing tothesuperruiddensityweestimatedearlierinSection 4.4.6 n s =6 : 06 10 19 cm 3 ,this isabout14.9%ofthetotalsuperruiddensity n s .Atthelowestruenceof0.37nJ/cm 2 ,the correspondingpercentageisabout0.5%. Weintroducedaconstantofproportionality C tolinkthemeasuredphoto-induced transmissionsignaltotheexcessquasiparticledensity, S = CN ex .Thereforewehave S max = CN ex,max .UsingEq.( 6{27 ),thisbecomes S max = C F n G d : (6{28) Thereforeaplotof S max vs F = n G d (orequivalently S max vs N ex ; max )atdierentelds shoulddemonstratetheelddependenceof C .SuchaplotforNbTiNisshownin Figure 6-14 .Inthecalculation,thevaluesof S max arereaddirectlyfromdata,and thevaluesofelddependentn G arefromTable 4-1 .TheleftpanelofFigure 6-14 showsatrendofsaturationathighruence.Overall,thereisanalmo stsquareroot relationbetween S max and N ex ; max .Thetracesatvariouseldsdiersignicantlyas S max increases.Thelowruencepartoftheleftpanelisshownintheright panel.The almostlinearrelationindicatesthat C isapproximatelyconstant.Tracesatdierentelds overlapreasonablywell,suggestingthat C isweaklyelddependent.Thisjustiesour approximationthatinthelowruenceregime,wheretheexcessquas iparticledensityis muchlessthanthetotalsuperruiddensity, S = CN ex and C isaconstant. 6.5.3.4Totaleectofmagneticelds Nowwehavedeterminedthat C and P P # arealmostindependentofeld.Theeld dependenceoftheslope2( R=C )( P P # )isdominatedbytheeectiverate R .Thispredicts 141

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r n n n n n n r n n n n n n Figure6-14. S max vs N ex ; max forNbTiNinparallelelds.Theratiobetweenthetwois proportionalto C .Therightpanelshowsthelowruencepartoftheleft panel.Theupperscaleofthex-axisshowstheratioof N ex ; max andthe superruiddensityat0T,2K. analmostquadraticrelationbetweentheslopeandthegap.Thecalc ulated R vsn G for bothNbTiNandNbNarecomparedtotheslopevsgapdatainFigure 6-15 ,suggesting qualitativeagreement.IntheleftpanelofFigure 6-10 onlythedatawith S smallerthe magnitudeof20(i.e.,inthelow-ruencelimit)areusedfortheextract ionoftheslope.n G isthecalculatedvaluetakenfromthesolidlineintheleftpanelofFigur e 4-9 ,ignoringthe dierencebetween2Kand3Kbecausethegapisfullydeveloped.Int herightpanelof Figure 6-10 onlythedatawith S smallerthemagnitudeof10areusedfortheextraction oftheslope.Theerrorbarsinbothplotsarethestandarddeviatio nsoftheslopeasthe ttingparameterofthelineartinFigure 6-10 Someissuesremaininourinterpretationsoftheelddependence.A lthoughthe predicted R vsn G isconsistentwiththeslopevsn G data,thepredictedeectivelifetime vsn G fromtheKaplan-typecalculationpredictsadierenttrendfromth atshownbythe decaydata.Thisinconsistencyneedstobeaddressed.Thecoher encefactorinamagnetic eldisexpectedtobemodiedfromtheBCSform.Thereforeitshou ldbederivedin thepair-breakingregime.Itwouldalsobeusefulifhighermagnetic eldscanbeused tosuppressthegapevenfurther,sothatonecanverifywhatre allyhappenswhenn G 142

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n r nr n Figure6-15.Circles:slopevsn G forNbTiNandNbNinparallelelds.Theerrorbarsare calculateddeviationsoftheslopefromthelineartsinFigure 6-10 .Dashed lines:thecalculatedrelationbetween R andn G ,takenfromFigure 6-12 approacheszero.Iftheeldisincreasedmore,eventuallyonecan achievethegapless regime,whenn G isdriventozero.HowdoesalaserpulsebreaktheCooperpairsinthis regime?Howdothequasiparticlesrecombine?Thesecouldbeinteres tingproblemstolook at.However,theyareexperimentallychallenging,becausethelowe rsideofthefar-infrared regionisdiculttoaccessduetovariouslimits,e.g.,inthesource,det ector,andthebeam splitter. 6.6FieldDependence:FieldPerpendiculartoSampleSurfac e Intheprevioussection,wehaveseenthatthemaineectofapara llelmagneticeld istosuppressthespectroscopicgap,whichresultsintheslowingof thequasiparticle recombination.Whathappensiftheexternalmagneticeldisapplied perpendicularto thesamplesurfacesothatvorticesarecreated?Suchexperimen tswereperformedon theNbTiNandNbNsamplesusingthesamepump-probetechnique.Th ephoto-induced transmissionsignalatvariousmagneticeldsandlaserruencesare plottedinFigure 6-16 forNbTiNandinFigure 6-17 forNbN.Thesamplesweresubmergedundersuperruid helium( T 2K)toreducelaserheating. 143

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$ n r# "! n $ n r# n n "! n n $ n r# "! $ n r# "! n Figure6-16.NbTiNdecaydatainperpendiculareldsatlowandhighlas erruencesand T 2K.Thesecondrowshowsthesamedataintherstrowplottedona semi-logscale. 144

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$ n $ n n "! $ n $ n n $ n $ n r# Figure6-17.NbNdecaydatainperpendiculareldsatvariouslaserr uencesand T 2K. Thesecondcolumnshowsthesamedataintherstcolumnplottedon a semi-logscale. 145

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ForNbTiN,thedataatlowandhighruenceshowsimilarfeatures.InF igure 6-16 atlowelds,thedecayismoreorlesssingleexponential.Astheeldisin creasedabove 8T,thedecaycurveclearlychangesitsshape,indicatingthatther ecombinationspeeds upasquasiparticlesrecombine,whichisnotobservedintheparallel eldconguration.In boththelowandhighruenceregimes,thedecayseemstoslowdowna stheeldincreases, whichagreesqualitativelywiththeparalleleldconguration. ForNbN,theelddependenceatlowandhighruencearecompletelyd ierent. Thelowruencedata(intherstandsecondrowofFigure 6-17 )areverysimilarto thoseofNbNintheparalleleldconguration.Thedecaysaresingle -exponentiallike. Therecombinationisslowedastheeldisincreased.Athighruence(t hethirdrowin Figure 6-17 ),thedecayspeedsupastheeldincrease.Thisisclearlydierentf romthe paralleleldconguration,anddierentfromtheNbTiNdataathigh ruenceshownin Figure 6-16 Fromthemeasuredphoto-inducedtransmission S ( t )andthedierentialsignal dS=dt wecalculatedtheeectiveinstantaneousrecombinationrate 1 e ,denedbyEq.( 6{7 ). Plottingitvsthephoto-inducedtransmissionatvariousruencesan delds,weobtained Figure 6-18 forNbTiNandFigure 6-19 forNbN. FortheNbTiNdatashowninFigure 6-18 ,wedonotobservetheclearlinearrelation between 1 e and S asintheparalleleldconguration,especiallyatlowruenceandlow elds.Notethatthereisacurvatureinthedataat1.34nJ/cm 2 and0T.Intheparallel eldconguration,however,wedidnotseethisfeatureat0Tandv ariousruences.The curvaturemightbeduetotheinaccuracyofthedataatlowvalueso f S .Figure 6-18 isdierentfromtheparallelcaseshowintheleftpanelofFigure 6-10 intwoaspects. Firstly,thedataatlowandhighruencesinthetwopanelsofFigure 6-18 donotoverlap. Secondly,athighelds(9Tand10T),itisclearthattherecombinatio nbecomesfaster astheprocesscontinues(notethatasthequsiparticlesrecombin e,thephoto-induced signal S decreases).Howeverthereisonesimilarityinthe 1 e and S plotbetweenthe 146

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!$ # n r r r !$ # r rn r Figure6-18.EectiveinstantaneousrelaxationrateforNbTiNinpe rpendiculareldsat lowruence(left)andhighruence(right). parallelandperpendiculareldcongurations:atagivenlevelof S ,astheeldincrease, theeectiveratedecreases,i.e.,therecombinationslowsdownath igherelds. FortheNbNdatashowninFigure 6-19 ,therelationbetween 1 e and S isalmost linear,similartotheNbNdataintheparalleleldcongurationshownin therightpanel ofFigure 6-10 .Atlowruence(0.40nJ/cm 2 and1.34nJ/cm 2 ),theelddependenceis qualitativelythesameasintheparalleleldconguration:atagivenle velof S ,the eectiverecombinationrateissmallerathigherelds.Athighruence (10.70nJ/cm 2 ),at thesame S ,theeectiverecombinationrateishigherathigherelds.Thisiscom pletely dierentfromthecasesshowninFigure 6-10 andFigure 6-18 .Thedataatdierent ruencesandthesameeldinthethreepanelsofFigure 6-19 donotoverlap. Thesenewfeaturesintheelddependencesuggestdierentphys icsinvolvedin therecombinationprocesswhenvorticesarepresent.Theycann otbeexplainedbythe recombinationmodeldescribedinSection 6.5.3 ,thereforerequirenewtypesofanalysis. ThefasterrecombinationathighereldsinNbNinthehighruencereg imesuggests vorticesmightprovideextrachannelsfortheexcessphononstor elax. 147

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6.7Summary Inconclusionweusedtime-resolvedlaser-pumpsynchrotron-pro bespectroscopyto studythequasiparticlerecombinationdynamicsinNbTiNandNbNthin lms.Parallel andperpendicularmagneticeldswereobservedtochangetherec ombinationprocess dierently. Whenaparallelmagneticeldwasapplied,signicantslowingoftherec ombination wasobservedinbothsamples.Thiscanbedirectlyseenfromthedec aytraces,orcan beexplicitlyshownbyttingthedatawithasingle-exponentialortwo -exponential decaymodel.Weinterpretedthedecaydatabydeninganeective recombination rate,andfoundalinearrelationbetweenthisrateandthephoto-in ducedtransmission signal.Theslopeofthestraightlinesshowsdramaticelddependenc e.Weproposed arecombinationmodeltoexplaintheelddependence.Theanalysiss uggeststhatthe magnetic-eld-inducedpair-breakingeectsmodifythequasipart icledensityofstates andsuppresstheenergygap.Theconsequenceisadramaticslowin gofthequasiparticle recombination.Wedidnotneedtoincludetheeectofthespin-polar izationfactoronthe recombination,probablyduetothestrongspin-orbitscatteringin thesematerials. Whenaperpendicularmagneticeldwasapplied,thetwosamplesshow dierent behaviors.ForNbTiN,whenthemagneticeldisincreasedabove8T, inboththe lowandhighruenceregimesthequasiparticlerecombinationrateincr easesasthe recombinationprocesscontinues.ForNbN,athighruence,there combinationisfasterat highelds.Thesefeaturesarenotexplainedbytherecombinationm odelweproposed. 148

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n r n n n r n n n n r n Figure6-19.EectiveinstantaneousrelaxationrateforNbNinper pendiculareldsatlow ruence(rstandsecondrow)andhighruence(lastrow). 149

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CHAPTER7 CONCLUSIONS Inthisdissertation,conventionalandtime-resolvedinfraredspe ctroscopywere employedtostudythemagneticpropertiesoftype-IIsupercond uctingthinlmsofNbTiN andNbN. Therstpart(Chapter4andChapter5)dealtwithequilibrium-stat epropertiesin magneticelds.Transmissionandrerectionmeasurementswereta kenatlowreduced temperatureinbothparallelandperpendicularmagneticelds,usin ghigh-brightness synchrotronradiationasthesource.Opticalconductivitieswere extracteddirectlyfrom thesedata. Whentheeldwasappliedparalleltothesamplesurface,wefoundth eoptical conductivitysignicantlymodied.Theenergygap,shownastheab sorptionedgein therealpartoftheopticalconductivity,isclearlysuppresseddu etothemagneticeld. Boththeopticalconductivityandthedegreeofgapsuppressiona reingoodagreement withAbrikosovandGor'kov'spair-breakingtheory.Thedata,when comparedtothe theory,suggesttwodistinctmechanismsofpairbreakinginthetwo samples.Forthe 10nmNbTiNthinlm,theorderparameterisalmostconstantthroug houtthesample. Theeld,whenactingontheelectronorbitalmotion,mainlybreakst hetime-reversal symmetryoftheparing.Weobservedthatthestrengthofpairbr eakingisalmost quadraticineld.Forthe70nmNbNlm,aparallelmagneticeldinevita blycreates vorticesinthesample,causingspatialvariationoftheorderparam eter.Suchaneect inducespairbreakingwithitsstrengthalmostlinearineld,asconrm edbyouranalysis. Thisisthersttimethatopticalabsorptionhasbeenemployedtote stquantitativelythe theoryofpairbreakingbyanexternalmagneticeld. Whentheeldwasappliedperpendiculartothesamplesurface,weob servedmore dramaticweakeningofsuperconductivitycomparedtotheparalleleldcase.Three models,theMaxwell-Garnetttheory,theBruggemaneectivemed iumapproximation, 150

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andtheCoey-Clemmodel,wereusedtocalculatetheeectiveoptic alconductivity ofatype-IIsuperconductorinthemixedstate.Ouropticalcond uctivitydataclearly ruleoutthelattertwoandsupporttheMaxwell-Garnetttheoryas adecentdescription ofthemixedstate.Furthermore,thedataofNbNsuggestadditio nalweakeningof superconductivityotherthanthatcausedbyvortices.Inclusion ofthepair-breaking eectsfoundinChapter4intotheMaxwell-Garnetttheorymakest hemodelmore consistentwiththedata.Thesendingsleadustotheconclusionth atthemixed-state electrodynamicresponsecanbewelldescribedbytheMaxwell-Garn etttheory.Contrary tothegeneralassumptionthatthesuperconductingfractionint hevortexstateis unperturbedbytheexternalmagneticeld,ourdatashoweviden ceofpairbreaking onthesesuperconductingelectrons. Thesecondpart(Chapter6)discussedthenon-equilibriumquasipa rticlerecombination dynamicsafterphoto-excitation,undertheinruenceofmagnetic elds.Thetwosamples studiedinChapter4andChapter5wereexcitedbyinfraredlaserpu lses,andprobed withsynchronizedsynchrotronradiation.Thephoto-excitationinducedtransmission changerevealsthequasiparticlerecombinationdynamicsinthesesa mples.Byapplying aparallelmagneticeld,wefoundthattherecombinationprocessis signicantlyslowed. Thereisalinearrelationbetweentheeectiverecombinationratean dthephoto-induced transmissionsignal,withtheslopeshowingstrongelddependence. Ourrecombination modelisconsistentwiththeinterpretationthattheslowingisdueto theeld-induced pair-breakingeects,whichwehaveestablishedinChapter4.Them odelalsosuggests anotherregimewheretheelddependenceisdominatedbyPaulispin paramagnetism.We didnotneedtoincludesucheectstoexplainourdata,probablybec auseofthestrong spin-orbitscatteringinoursamples. ThestudiesinChapter4onthepair-breakingeectsmaybefurthe rextended. Ourstudyfocusedontwodierenttypesofmaterials.Clearly,the lmthicknessisan importantparameterthatdistinguishesdierentregimesofpairbr eaking.Itwouldbe 151

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interestingtosystematicallyidentifydierentregimesusingthesam emagneto-optical techniquepresentinChapter4,byvaryingthethicknessofthesa metypeofmaterial.In general,thesameexperimentaltechniqueanddataanalysisproce durescanbeappliedto studyanykindofsuperconductorintheformofathinlmonasubst rate. ThestudiesinChapter6canalsoberenedandextended.Inthepa ralleleld conguration,thecorrectformofthecoherencefactorinthep air-breakinglimitwillbe useful.Itwouldalsobeinterestingtosuppressthegapfurther,e venintothegapless regime.Thecorrespondingeectiverecombinationratecanbeuse fultoverifythevalidity ofourrecombinationmodel.Thedierentfeaturesobservedinthe perpendiculareld congurationrequiresnewmodelstounderthequasiparticlerecom binationinthepresence ofvortices. 152

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APPENDIXA USEFULPHYSICALCONSTANTSANDUNITCONVERSIONS Wavenumber =1 = ispredominantlyusedtodenotefrequencythroughout thisdissertation.Itisthereforehelpfultoknowthecommonly-us edunitsandphysical constantsexpressedintermofwavenumbers. 1cm 1 =3 10 10 Hz=1 : 9878 10 23 J=1 : 2424 10 4 eV : (A{1) Fromthis,theBoltzmannconstant k B =8 : 6173 10 5 eV/K=0 : 6936cm 1 /K,andthe Bohrmagneton B =5 : 7884 10 5 ev/T=0 : 4659cm 1 /T. EquationsthroughoutthisdissertationuseGaussianCGSunits.In manyoccasions, quantitiesarereportedinpracticalunits.Aconversiontabletake nfromRef.[ 3 ]is includedhereforconvenience.Aparticularlyusefulconversionus edinthisdissertationis fortheopticalconductivity.Numerically, CGS =9 10 11 prac ,where CGS isins 1 and prac isinpracticalunitsn 1 cm 1 TableA-1.Conversiontableforelectromagneticquantitiesbetwee nCGSandSIunits. QuantityCGSSI Velocityoflight c 1 = p 0 0 Magneticinduction B p 4 = 0 B Magneticeld H p 4 0 H Magnetization M p 0 = 4 M Chargedensity(orcharge,current, (or Q;I; J ; P ) = p 4 0 (or Q;I; J ; P ) currentdensity,polarization)Electriceld(orpotential,voltage) E (or ;V ) p 4 0 E (or ;V ) Displacement D p 4 = 0 D Conductivity = 4 0 Resistance(orimpedance,inductance) R (or Z )4 0 R (or Z ) Inductance L 4 0 L Capacitance CC= 4 0 Permeability = 0 Dielectricconstant = 0 153

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APPENDIXB INTERACTIONOFLIGHTWITHMATTER Discussionsofinteractionbetweenelectromagneticwavesandmat tercanbefoundin varioustextbooks,e.g.[ 167 168 ].Hereashortsummaryisincludedforquickreference. Considerahomogeneousisotropicmediumofdielectricconstant 1 ,permeability andconductivity 1 .Thesubscript\1"istodenotetherealpartofthequantity;sub script \2"denotestheimaginarypart.Thematerialequationsrelatethe electriccurrentdensity J totheelectriceld E ,theelectricdisplacement D to E ,andthemagneticinduction B tothemagneticvector H J = 1 E ; (B{1) D = 1 E ; (B{2) B = H : (B{3) Intheabsenceofexternalfreecharges,theMaxwell'sequations read r D =0 ; (B{4) r B =0 ; (B{5) r E = 1 c B ; (B{6) r H = 4 c J + 1 c D : (B{7) NotingtherelationsEqs.( B{1 ){( B{3 )andeliminating H fromEq.( B{6 )usingEq.( B{7 ) yieldsthewaveequation r 2 E = 1 c 2 E + 4 1 c 2 E : (B{8) Ifwenowspecializetothecaseofmonochromaticplanewavesofthe form E = E 0 e i ( k r !t ) ,Eq.( B{8 )becomes r 2 E + 2 c 2 1 + 4 i 1 E =0 : (B{9) 154

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DenethequantityintheparenthesisofEq.( B{9 )asthecomplexdielectricconstant = 1 + 4 i 1 : (B{10) FromthewaveequationEq.( B{9 )thecomplexwavevectorhasitsmagnitudedenedby k 2 = 2 c 2 : (B{11) Thecorrespondingcomplexrefractiveindexisdenedas N = n + i; (B{12) where n istherefractiveindexand istheextinctioncoecient.Since N = ck=! N 2 = c 2 k 2 = = 1 + 4 i 1 : (B{13) Comparingwith N 2 =( n + i ) 2 wehave n 2 2 = 1 ; (B{14) 2 n = 4 1 2 : (B{15) Therefore 1 =( n 2 2 ) =; (B{16) 2 =2 n= =4 1 =!; (B{17) and n = 1 2 q 21 + 22 + 1 1 = 2 ; (B{18) = 1 2 q 21 + 22 1 1 = 2 : (B{19) Theimaginarypartoftheconductivityisdenedthroughtherelatio n =1+ 4 i (B{20) 155

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sothat 1 =1 4 2 : (B{21) Nowconsiderthepropagationofthewave, E = E 0 e i ( k r !t ) = E 0 e c ^ k r e i ( !n c ^ k r !t ) ; (B{22) where ^ k istheunitvectoralongthewavepropagationdirection.Therstex ponential termintheaboveequationdescribestheattenuationoftheelectr omagneticwavesinside themedium.Theabsorptioncoecient isdenedas =2 c (B{23) sothattheenergyofthewavefallsto1 =e afterittravelsadistance1 = Thefollowingtablesummarizestherelationsbetweentheseopticalc onstants. TableB-1.Relationsbetweenopticalconstants = 1 + i 2 = 1 + i 2 N = n + i = 1 + i 2 { 1 =1 4 2 1 =( n 2 2 ) = { 2 = 4 1 2 =2 n= = 1 + i 2 1 = 4 2 { 1 = 4 2 ( n; ) 2 = 4 (1 1 ) { 2 = 4 [1 1 ( n; )] N = n + i n =[ 1 2 ( p 21 + 22 + 1 )] 1 = 2 n = n ( 1 ( 2 ) ; 2 ( 1 )) { =[ 1 2 ( p 21 + 22 1 )] 1 = 2 = ( 1 ( 2 ) ; 2 ( 1 )) { 156

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APPENDIXC THINFILMOPTICS ThinlmopticsisessentialintheanalysisperformedinChapter 4 andChapter 5 Detailedderivationsofthemainequationsusedinthosetwochapter swillbepresentedin thisappendix. C.1FresnelEquations Beforediscussingthinlmoptics,wewillreviewtheclassicalproblemo fndingthe transmissionandrerectioncoecientsattheboundaryoftwomed ia.Thesearesolved fromtheFresnelequations.Thesamemethodwillthenbeusedfor ourproblemofthin lmonathicksubstrate. TheFresnelequationsdescribehowlighttransmitsandrerectsat theinterfaceof twolinear,isotropic,andhomogeneousmediaofrefractiveindices n i and n t ,wherethe subscripts i and t denoteincidenceandtransmission,respectively.Theequationsca nbe foundinmoststandardtextbooksonoptics,e.g.,[ 169 ],fors-polarizedandp-polarized light(s-polarizedreferstopolarizationperpendiculartotheplaneo fincidence;p-polarized referstopolarizationparalleltotheplaneofincidence).Thecombin ationofthesetwo explainsanyothercasesofpolarization.Thetwoboundaryconditio nsusedinthe derivationoftheFresnelequations,i.e.,thetangentcomponents oftheelectriceld E andmagneticeld H withrespecttotheinterfacearecontinuous,arefromtwoofthe Maxwell'sequationsEqs.( B{6 )and( B{7 ), r E = 1 c B ; (C{1) r H = 4 c J f + 1 c D ; (C{2) byintegratingthemalonganinnitesimalStokesianloopacrossthein terface[ 170 ].Here J f isthefreechargecurrentdensity. Assumetheincident,rerectedandtransmittedwavesareallplane waves, E i = E 0 i e i ( k i r i t ) E r = E 0 r e i ( k r r r t + r ) E t = E 0 t e i ( k t r t t + t ) ,where r and t denotephase 157

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As-polarized Bp-polarized FigureC-1.Illustrationsoftheincident,rerectedandtransmitte dwavesattheinterface oftwomedia. shifts.The B eldsarerelatedto E by n ^ k E = B .Tosatisfytheboundaryconditions, i = r and n i sin i = n r sin r .ThelatterisSnell'slaw. Inthecaseofs-polarizedlightdepictedbyFigure C-1 (A),thetwoboundary conditionsgive E 0 i + E 0 r = E 0 t ; (C{3) n i i ( E 0 i E 0 r )cos i = n t t E 0 t cos t : (C{4) Theamplitudererectioncoecientandamplitudetransmissioncoec ientaresolvedas r ? = E 0 r E 0 i = n i i cos i n t t cos t n i i cos i + n t t cos t ; (C{5) t ? = E 0 t E 0 i = 2 n i i cos i n i i cos i + n t t cos t : (C{6) Similarly,inthecaseofp-polarizedlightdepictedbyFigure. C-1 (B), ( E 0 i E 0 r )cos i = E 0 t cos t ; (C{7) n i i ( E 0 i + E 0 r )= n t t E 0 t ; (C{8) 158

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whichgivetheamplitudecoecients r k = n t t cos i n i i cos t n t t cos i + n i i cos t ; (C{9) t k = 2 n i i cos i n t t cos i + n i i cos t : (C{10) Byconsideringtheproductoftheintensityandthecross-section alareaofthebeams, thererectanceandtransmittancearerespectively R = I r A cos r I i A cos i = j r j 2 ; (C{11) T = I t A cos t I i A cos i = n t cos t n i cos i j t j 2 ; (C{12) validforbothpolarizations.Theintensityisthetime-averagedmagn itudeofthePoynting vector I = h S i = nc 0 E 2 = 2. A istheareaofthebeamontheinterface,whichcancelsout inthecalculations. Whendealingwithnon-magneticmaterials,wecantake i = t =1.Wewillusethis assumptioninthenextsection.Underthiscondition,thererectan ceandtransmittance forthecaseofnormalincidencecanbederivedforeitherthes-po larizedorthep-polarized incidentlight, R = n i n t n i + n t 2 ; (C{13) T = 4 n i n t ( n i + n t ) 2 : (C{14) C.2FresnelEquationsforThinFilmonThickSubstrate Nowweconsiderabeamcomesinamediumwithrefractiveindex n i ,andincidents onathinlmdepositedonasubstratewithrefractiveindex n t .Thelmthicknessis d andischaracterizedbyitsopticalconductivity .Wewanttocalculatethererectance fromthethinlmandthetransmittancethroughthethinlmintothe substrate. 159

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As-polarized Bp-polarized FigureC-2.Illustrationsoftheincident,rerectedandtransmitte dwavesattheinterface ofamedium n i andathinlmonasubstrate n t Theboundaryconditionsneedtobere-evaluated.Sincefreechar gescanbepresentat theinterfaceduetothethinlm,Eq.( C{2 )isnowdierent, r H = 4 c J f = 4 c E t : (C{15) Wehaveassumedthattheelectriceldisthesameinthethinlmandju stafteritenters thesubstrate,sincethethinlmisextremelythincomparedtothew avelength. Fors-polarizedlight,thetwoboundaryconditionsyield E 0 i + E 0 r = E 0 t ; (C{16) H 0 i cos i H 0 r cos r H 0 t cos t = 4 c dE 0 t : (C{17) ThesecondequationisobtainedbyintegratingEq.( C{15 ).Using H = nE ,thesolutions are r ? = n t cos t n i cos i + 4 c d n t cos t + n i cos i + 4 c d ; (C{18) t ? = 2 n i cos i n t cos t + n i cos i + 4 c d : (C{19) 160

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Forthecaseofp-polarization,wehave ( E 0 i E 0 r )cos i = E 0 t cos t ; (C{20) n i ( E 0 i + E 0 r ) n t E 0 t = 4 c dE 0 t cos t : (C{21) Thesolutionsare r k = n t cos i n i cos t + 4 c d cos i cos t n t cos i + n i cos t + 4 c d cos i cos t ; (C{22) t k = 2 n i cos i n t cos i + n i cos t + 4 c d cos i cos t : (C{23) Intheaboveequations,wecanwrite4( =c ) d as Z 0 =R ,where Z 0 =4 =c 377n = is thevacuumimpedanceand R =1 =d isthethinlmimpedance. Consideraspeciccaseinwhichlightisincidentfromvacuumonathinlm with = 1 + i 2 andthickness d .Thesubstratehasrefractiveindex n .Forp-polarized incidentwaves,forinstance,thererectanceandtransmittance are R k = ( n cos t cos i + Z 0 1 d cos t ) 2 +( Z 0 2 d cos t ) 2 ( n + cos t cos i + Z 0 1 d cos t ) 2 +( Z 0 2 d cos t ) 2 ; (C{24) T k = 4 n cos t cos i ( n + cos t cos i + Z 0 1 d cos t ) 2 +( Z 0 2 d cos t ) 2 : (C{25) Forthecaseofnormalincidence, i = t =0.Supposelightisincidentfromvacuumtoa thinlmonasubstratewithrefractiveindex n .Theseequationsreducetothewell-known equationsderivedbyTinkhamandPalmer[ 38 ], r f = n 1+ Z 0 d n +1+ Z 0 d ; (C{26) t f = 2 n +1+ Z 0 d ; (C{27) R f = ( n 1+ Z 0 1 d ) 2 +( Z 0 2 d ) 2 ( n +1+ Z 0 1 d ) 2 +( Z 0 2 d ) 2 ; (C{28) T f = 4 n ( n +1+ Z 0 1 d ) 2 +( Z 0 2 d ) 2 : (C{29) 161

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Wehavedroppedthepolarizationsymbolbecausesandppolarizatio nsareindistinguishable atnormalincidence.Insteadweputan f asthesubscript,meaningtransmittanceacross andrerectancefromthethinlm .Theobservedrerectanceandtransmittanceareusually dierentfrom R f and T f becausesubstratesareinvolved. C.3ExternalRerectanceandTransmittanceforThinFilmonT hick Substrate InSection C.2 wehavederivedthererectancefromthethinlmandalsothe transmittanceintothesubstrate,withoutconsiderationofthein terferenceeectsinthe substrate.Practicallyithastobetakenintoaccount,becauseth equantitiesmeasured aretheexternalrerectionandtransmissiondeterminedbybotht hethinlmandthe substrate.C.3.1MethodofSummation ConsiderthecongurationshowninFigure C-3 .Itissimilartotheproblemofnding thererectanceandtransmittanceofaFabryPerotinterferom eter[ 171 ],thedierence beingthepresenceofthethinlminbetweenMedium1andMedium2.At hinlmof thickness d ,depositedonasubstrateofthickness d s andrefractiveindex n 2 ,issandwiched betweentwomediaofrefractiveindices n 1 and n 3 .Inourexperimenttheyarevacuum, sowetake n 1 = n 3 =1.Wewrite n 2 = n forsimplicity.Weconsiderthegeneralcase whenabsorptionispresentinthesubstrate,describedbytheext inctioncoecient orthe absorptioncoecient =2 !=c .TheamplitudererectioncoecientbetweenMedium 1andMedium2is r f andtheamplitudetransmissioncoecientis t f .Dependingonthe polarizationtheyaregiveneitherbyEqs.( C{18 )and( C{19 )orEqs.( C{22 )and( C{23 ). Theamplitudererectionandtransmissioncoecientsforlightpropa gatingfromMedium 2toMedium1aredenotedas r 0 f and t 0f ,respectively.Theycanbeevaluatedbysimply swappingtherefractiveindices n i and n t inEqs.( C{18 ),( C{19 ),( C{22 ),and( C{23 ). TheamplitudererectioncoecientbetweenMedium2andMedium3isde notedas r 23 162

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r f t f t f r 23 t f t 23 t f r 23 r 0 f t f r 23 2 r 0 f t f r 23 2 r 0 f 2 t f r 23 t 0f t f r 23 2 r 0 f t 0f t f r 23 3 r 0 f 2 t f r 23 3 r 0 f 2 t 0f t f r 23 r 0 f t 23 t f r 23 2 r 0 f 2 t 23 n 1 =1 n 2 = n n 3 =1 d d s 1 2 3 FigureC-3.Illustrationoftheincident,rerectedandtransmitted wavesattheinterfaceof threemedia.Thecenteroneisathinlmonathicksubstrate.Multipleinternalrerectionsinthesubstratearetakenintoaccount. and t 23 fortransmission.Bytracingthebeam,wecandeterminetheamplitu deofeach wave.TherstfewarelabeledinFigure C-3 Theresultantamplitudetransmissionandrerectioncoecientsare thesummation ofallthetransmissionandrerectionamplitudes.Howeverallbeams shouldbemeasured alongalineperpendiculartothedirectionofnalpropagation.Take thersttworerected beamsforexample.Thesecondrerectedbeamtravels r 2 =2 d s = cos 2 inMedium2,but isthenretardedfromtherstrerectedbeambyadistance r 1 =2 d tan 2 sin 1 inMedium 1.Assumeallwavestreatedhereareplaneswavesoftheform E = E 0 e i ( kr !t ) .Thephase factorofthesecondrerectedbeamdiersfromthatoftherst rerectedbeambyan amount e k 2 r 2 e k 1 r 1 = e i c ( n 2 + i 2 )2 d s = cos 2 e i c n 1 2 d s tan 2 sin 1 = e 2 c 2 d s = cos 2 e i c (2 n 2 d s = cos 2 2 n 1 d s tan 2 sin 1 ) = e d s = cos 2 e i c 2 n 2 d s cos 2 ; (C{30) where =2 2 =c =2 !=c istheabsorptioncoecientofthesubstrate,i.e.,Medium 2.IngettingthesecondexponentialtermweusedSnell'slaw, n 1 sin 1 = n 2 sin 2 .To 163

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simplifythenotation,wedenote d s = cos 2 as x and( !=c )2 n 2 d s cos 2 as = c 2 n 2 d s cos 2 =(4 nd s cos 2 ) : (C{31) Theaboveexponentialfactorbecomes e x e i Uptoaconstantphasefactor,thetotalamplitudererectioncoe cientis r = r f + t f r 23 t 0f e x e i + t f r 23 2 r 0 f t 0f e 2 x e i 2 + t f r 23 3 r 0 f 2 t 0f e 4 x e i 4 + ::: = r f + r 23 t f t 0f e x e i [1+ r 23 r 0 f e x e i + r 23 2 r 0 f 2 e 2 x e i 2 + ::: ] = r f + r 23 t f t 0f e x e i 1 r 23 r 0 f e x e i ; (C{32) andthetotalamplitudetransmissioncoecientis t = t f t 23 e x= 2 + t f r 23 r 0 f t 23 e 3 x= 2 e i + t f r 23 2 r 0 f 2 t 23 e 5 x= 2 e i 2 + ::: = t f t 23 e x= 2 [1+ r 23 r 0 f e x e i + r 23 2 r 0 f 2 e 2 x e i 2 + ::: ] = t f t 23 e x= 2 1 r 23 r 0 f e x e i : (C{33) Thererectanceandtransmittanceare R = j r j 2 = r f 2 + r 23 2 t f 2 t 0f 2 e 2 x 1+ r 23 2 r 0 f 2 e 2 x 2 r 23 r 0 f e x cos +2 r f r 23 t f t 0f e x cos r 23 r 0 f e x 1+ r 23 2 r 0 f 2 e 2 x 2 r 23 r 0 f e x cos ; (C{34) T = n 3 cos 3 n 1 cos 1 j t j 2 = j t j 2 = t f 2 t 23 2 e x 1+ r 23 2 r 0 f 2 e 2 x 2 r 23 r 0 f e x cos : (C{35) Intheseequations, r f isgivenbyEq.( C{18 )orEq.( C{22 ), t f isgivenbyEq.( C{19 ) orEq.( C{23 ), r 23 isgivenbyEq.( C{5 )orEq.( C{9 ), t 23 isgivenbyEq.( C{6 )or Eq.( C{10 ), r 0 f and t 0f areobtainedfromEq.( C{18 )andEq.( C{19 )orEq.( C{22 )and Eq.( C{23 )byswappingtherefractiveindexofthesubstrateandvacuum. Thecosinetermdescribestheoscillatoryfringesinthespectrumdu etothecoherent multipleinternalrerectionsinthesubstrate.Ifthemeasurement resolutionissolowthat 164

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thefringesarenotresolved,incoherentspectraareobtained.T hesecanbecalculatedfrom Eqs.( C{34 )and( C{35 )byaveragingthespectraoverthecyclesofthefringes,notingt hat theFresnelcoecientsareslowlyvaryinginfrequencyandcanbet reatedasconstantin thecyclesusedintheaveraging.Averagingoverahalfofthefringe period T = 2 4 nd s cos 2 = 1 2 nd s cos 2 (C{36) issucient.Makinguseofthefollowingintegrationresult, Z dx a + b cos( mx ) = 1 m 1 p a 2 b 2 arcsin a cos( mx )+ b b cos( mx )+ a ( a 2 >b 2 ) ; (C{37) onends 1 T= 2 Z T= 2 0 d 0 1+ r 23 2 r 0 f 2 e 2 x 2 r 23 r 0 f e x cos(2 0 =T ) = 1 1 r 23 2 r 0 f 2 e 2 x : (C{38) Theincoherentrerectanceandtransmittanceare R incoherent = 1 T= 2 Z T= 2 0 R [cos(2 0 =T )] d 0 = r f 2 + r 23 2 t f 2 t 0f 2 e 2 x 1 r 23 2 r 0 f 2 e 2 x ; (C{39) T incoherent = 1 T= 2 Z T= 2 0 T [cos(2 0 =T )] d 0 = t f 2 t 23 2 e x 1 r 23 2 r 0 f 2 e 2 x : (C{40) Fornormalincidence, R f = r f 2 givenbyEq.( C{28 ), R 23 = r 23 2 =( n 1) 2 = ( n +1) 2 T 23 =( n 3 cos 3 =n 2 cos 2 ) t 23 2 = t 23 2 =n =1 R 23 T f =( n 2 cos 2 =n 1 cos 1 ) t f 2 = nt f 2 t 0f = nt f ,and R 0f = r 0 f 2 = (1 n + Z 0 1 d ) 2 +( Z 0 2 d ) 2 (1+ n + Z 0 1 d ) 2 +( Z 0 2 d ) 2 : (C{41) Theincoherentrerectanceandtransmittanceare R incoherent,normal = R f + R 23 T f 2 e 2 x 1 R 23 R 0f e 2 x ; (C{42) T incoherent,normal = T f (1 R 23 ) e x 1 R 23 R 0f e 2 x : (C{43) 165

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C.3.2MatrixMethod Anotherapproachofsolvingthisproblemisthematrixmethod[ 167 ].Itprovidesa generalprocedurefortreatingstratiedmediumwhosepropert iesareconstantthroughout eachplaneperpendiculartoaxeddirection.Multilayersystemsbelo ngstothiscategory. Consideramultilayersystemwith m layerssandwichedbetweentherstlayer denotedas f andthelastlayerdenotedas l .Eachlayerishomogeneousandischaracterized byitcomplexrefractiveindex N j andthickness h j .LightpropagatesfromLayer f to Layer l .TheangleofincidencefromLayer j 1toLayer j is j .Wewouldliketo calculatethetransmittancetoLayer l andrerectancetoLayer f Thecharacteristicmatrixthatrelatestheelectromagneticwaves enteringandexiting the j thlayeris M j = 264 cos(2 N j h j cos j ) i p j sin(2 N j h j cos j ) ip j sin(2 N j h j cos j )cos(2 N j h j cos j ) 375 ; (C{44) where p j = N j cos j fors-polarizedlightand p j =cos j =N j forp-polarizedlight.The amplitudererectionandtransmissioncoecientsare r = ( m 11 + m 12 p l ) p f ( m 21 + m 22 p l ) ( m 11 + m 12 p l ) p f +( m 21 + m 22 p l ) ; (C{45) t = 2 p f ( m 11 + m 12 p l ) p f +( m 21 + m 22 p l ) ; (C{46) where m 11 m 12 m 21 ,and m 22 arethecomponentsofthefollowingmatrix, M = M 1 M 2 M m = 264 m 11 m 12 m 21 m 22 375 : (C{47) p f and p l aredeterminedbytherstandthelastlayer.Thererectanceand transmittance are R = j r j 2 ; (C{48) T = p l p f j t j 2 : (C{49) 166

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Applythismethodtoourtwo-layeredsystem( m total =2).Thelayer m =1is thethinlmwithopticalconductivity andthickness d .Thecorrespondingcomplex refractiveindexcanbecalculatedfrom usingTable B-1 .Thelayer m =2isthe substratewithrefractiveindex n 2 andthickness d s .Sinceweareconsideringthe transmissionandrerectionintovaccumonthetwosidesofthetwolayeredsystem, therstandlastlayersarevacuum, n f = n l =1inthecalculationof p f and p l .With theseconditionsthetransmittanceandrerectancecanbereadily calculated. Figure C-4 showsthesuperconducting-statetransmittanceofNbTiNonaqu artz substrateat2K,0T,normalizedtoitsnormal-statevalue.Theres olutionofthe measurementis1cm 1 ,whichenablesustoresolvethemultipleinternalrerectionsin thequartzsubstrate.Thesmoothcurveistheincoherentexter naltransmittanceratio calculatedfromEq.( C{43 ),with 1 and 2 formtheMattis-Bardeentheory(leftpanel ofFigure 4-4 ).Thecurvewithoscillatoryfeaturesiscalculatedusingthematrixm ethod, withthesame 1 2 ,andtherefractiveindexofquartz.Inexperimentsdiscussedin Chapter4andChapter5,wechosealowerresolution,typically4cm 1 ,sothatthe oscillatoryfeaturesdonotappearinthespectra.Thisavoidsthec omplicationsduetothe substrateandsimpliesthedataanalysis. 167

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n rr r FigureC-4. T s = T n data(circles)ofNbTiNat2Kmeasuredwitharesolutionof1cm 1 ThesmoothcurveisacalculationusingEq.( C{43 ),with 1 and 2 formthe Mattis-Bardeentheory.Thecurvewithoscillatoryfeaturesiscalc ulatedusing thematrixmethodwiththesame 1 2 ,andtherefractiveindexofquartz. 168

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APPENDIXD CORRECTIONOFREFLECTIONDATA Beforebeingusedforanalysis,thererectanceofNbTiNmeasured usingthesample holdershownintherightofFigure 4-1 mustberstlycorrectedforthestraylightand thenfortheniteangleofincidence.Themethodsforthesetwoco rrectionsareinthis appendix. D.1CorrectionforStrayLight Thestraylightconsistsoftwoparts,onefromthererectionatth ebrassconeholding thesample(themiddlepieceshownintherightpanelofFigure 4-1 )andtheotherfrom thererectionatthebrasssurfacejustbehindthesample.Thelat tercontributionis importantbecausethesampleispartiallytransmissiveinourfrequen cyrangeofinterest. Tomakethecorrection,werstmeasuredtheemptysampleholder includingthebrass piecebehindthesample.Thespectrumisdenotedas R e .Wethenmountedandmeasured thespectrumofapieceofblacksandpaperwitharoughsurface.T hemeasuredspectrum is R k .Thethirdspectrum,denotedas R 0n ,wasmeasuredwiththesamplemounted. Allspectraweretakenatroomtemperature. R 0n isthereforethemeasured normal-state rerection.Theprimesymbolistodistinguishitasadirectlymeasured quantityrather thantheonewewouldmeasureintheabsenceofthestraylight(den otedas R n ). Thererectanceofbrassisalmosttemperatureindependentinthe far-infrared. R k shouldthenbeaconstantinallmeasurements.Thererectionfrom thetopbrasspiece (denotedas R b )intheabsenceofsampleis R b = R e R k ,whichisalsoindependentof temperature.Consideringallcontributionstothemeasuredquan tity, R 0n = R n + R k + R b T 2 n ; (D{1) where T n isthenormal-statetransmittanceofthesample,calculatedfromE q.( 4{4 ). Similarlythemeasured superconducting-statererectionis R 0s = R s + R k + R b T 2 s ; (D{2) 169

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inwhich T s isthesuperconducting-statetransmittancederivedfromtheme asuredratio T s = T n andthecalculated T n .Fromthetwoequationsabovewecanrelatetheratioof actual rerectance R s = R n totheratioofmeasuredrerectance R 0s = R 0n R s R n = R 0s R 0n R k R 0n R e R 0n R k R 0n T 2 s 1 R k R 0n R e R 0n R k R 0n T 2 n : (D{3) Werealizedlaterthattheabovecorrectioncanbesimpliedif R 0n and R 0s were measuredwiththesandpaperleftinbetweenthesampleandthebra sssurface.The contributionsfrom R b canbeneglectedinEqs.( D{1 )and( D{2 ).Asaresult, R s R n = R 0s R 0n R k R 0n 1 R k R 0n : (D{4) Thisisusedforsomeofthedatameasuredwiththesandpaperbehin dthesampleto eliminatethestraylight R b D.2CorrectionforFiniteAngleofIncidence ThecorrectionforangleofincidencecanbecalculatedfromtheFre snelequation derivedintheAppendix C .Thesynchrotronradiationispredominantlyparallel-polarized withrespecttotheplaneofincidence.Forangleofincidence i andthecorresponding angleofrefraction t determinedbySnell'slaw,thethinlmsuperconducting-state rerectanceisgivenbyEq.( C{24 ), R f;s = n cos t cos i + Z 0 cos t R 1 n 2 + Z 0 cos t R 2 n 2 n + cos t cos i + Z 0 cos t R 1 n 2 + Z 0 cos t R 2 n 2 : (D{5) Thenormal-statevalue R f;n canbeobtainedbysetting 1 = n and 1 =0.Usingthe approximationdiscussedinSection 4.4.1 ,theratiooftheexternalrerectionsare R s ( i ) R n ( i ) R f;s ( i ) R f;n ( i ) = n + cos t cos i + Z 0 cos t R 2 n cos t cos i + Z 0 cos t R 2 n cos t cos i + Z 0 cos t R 1 n 2 + Z 0 cos t R 2 n 2 n + cos t cos i + Z 0 cos t R 1 n 2 + Z 0 cos t R 2 n 2 : (D{6) 170

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n r r r n r n r n r FigureD-1.(a) atdierenttemperaturescalculatedforNbTiN. 1 and 2 atdierent temperaturesarecalculatedfromtheMattis-Bardeentheory.( b)Relative changeof atdierenttemperaturescomparedtothatat2K,calculatedas 1 ( T ) = (2K). Thererectionmeasurementswereperformedwith i =30 .Toconverttonormal-incidence rerection,thererectionratiodatashouldbemultipliedbyafactor = R s (0 ) = R n (0 ) R s (30 ) = R n (30 ) (D{7) calculatedfromEq.( D{6 ).Howeverthecalculationrequirespriorknowledgeofthe superconducting-stateopticalconductivity,whichisthequantit yofourobjective.We maketheassumptionthattotherst-orderapproximationwecan usethelow-temperature (2K)zero-eldopticalconductivitycalculatedfromtheMattis-Ba rdeentheory.Athigher temperature,thedeviationofthismultiplicationfactor fromits2-Kvalueissmallinthe frequencyrangeofinterest,showninFigure D-1 .Themagnetic-eld-induceddeviationis expectedtobeevensmaller,ascanbeseenbycomparingtheeld-d ependenttransmission andrerectiontothetemperature-dependenttransmissionandr erection. D.3TemperatureDependenceoftheOpticalConductivityfor NbTiN Toverifythemethodforcorrectingthererectiondatadescribed above,weapplyit tostudythetemperaturedependenceoftheopticalconductivit yforNbTiN.Theresult shouldbeconsistentwiththeMattis-Bardeentheory. 171

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rr rr n n n n n FigureD-2.Superconducting-statetonormal-statetransmissio nandrerectionratioof NbTiNatdierenttemperaturesandzeroeld. Thesuperconducting-statetonormal-statetransmissionratioa ndrerectionratioare showninFigure D-2 fordierenttemperatures,allmeasuredusingthesameexperime ntal proceduresfortheNbTiNsampledescribedinSection 4.3.2 .Thererectiondataare therawexperimentdatawithoutanycorrection.Boththetransm issionratioandthe rerectionratioindicatethegapreductionastemperatureincreas es,manifestedasthepeak shiftin T s = T n andthedipshiftin R s = R n UsingthemethoddescribedaboveinSection D.1 andSection D.2 ,wecorrectedfor thestraylightandthe30 angleofincidenceforthererectiondata.Wethenextracted 1 and 2 fromthe T s = T n and R s = R n datafromEqs.( 4{9 )and( 4{10 ),usingthemethod describedinSection 4.4.1 .TheresultsareshowninFigure D-3 .Thesolidlinesinthe leftpanelaretstotheMattis-Bardeentheory,andthoseinthe rightpanelarethe corresponding 2 = n calculatedfromthetheory.Werstttedthe2Kdatawiththe zero-temperatureopticalgap2 0 astheonlyttingparameter.Thenwekeptthisvalue xedandvariedtemperaturetotthedataathighertemperatur es.Themethodworks wellfor 1 upto8K,butisnotreliableforextracting 2 .Oneofthedicultiesliesin thefactthatthetermunderthesquarerootinEq.( 4{10 )cannotbeguaranteedtobe positiveduetomeasurementerrorsinthedata.Thisexplainsthemis singdatapoints 172

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rrn rrn FigureD-3.Realpart(left)andimaginarypart(right)oftheoptic alconductivityfor NbTiNatdierenttemperatures,normalizedtothenormal-statev alue n ThesolidlinesintheleftpanelaretsusingtheMattis-Bardeentheo ry.The solidlinesintherightpanelarethecorrespondingcalculatedimaginar ypart. for 2 atfrequenciesabove40cm 1 intherightpanelofFigure D-3 .Forthecalculated 2 atalltemperatures,astrongcouplingcorrectionisincluded.At10 Ktheextracted opticalconductivityhaspoorquality,possiblybecausetheerrorin theangle-of-incidence correctionistoosignicanttobeignored,especiallyatlowfrequenc ynearthegap. Wealsofoundthatthettedtemperaturesforthedataabove2K areconsistently 0.1K-0.6Khigherthanthereadingsrecordedfromthetemperatur esensor.Thismay bebecausethetemperaturesensorwasnotdirectlyattachedto thesample,causinga temperaturegradientbetweenthesensorandthesample. Overall,thecorrectionmethodisreliableforthetemperature-dep endencestudyat temperaturesnottoocloseto T c .Weexpectitshouldworkwellfortheeld-dependence studyatlowtemperature. 173

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APPENDIXE DATANOTDISCUSSED Thisappendixincludesomedatanotdiscussed.Forcomparison,som edatadiscussed inthemainpartofthedissertationarealsoincluded. 0 20 40 60 80 100 Frequency [cm1 ] 0.0 0.5 1.0 1.5 2.0 2.5T s / T n (20K) NbTiN 0 20 40 60 80 100 Frequency [cm1 ] 0.0 0.5 1.0 1.5 2.0 2.5R s / R n (20K) 2K 3K 4K 5K 6K 7K 8K 9K 10K 11K 12K FigureE-1.Temperaturedependenceof T s = T n and R s = R n forNbTiN.Thenormalstateis at20K.Themeasurementresolutionis4cm 1 .Thedataat2K,4K,6K, 8K,and10KarediscussedinAppendix D.3 0 20 40 60 80 100 Frequency [cm1 ] 0.0 0.5 1.0 1.5 2.0 2.5 3.0T s / T n (20K) NbN 0 20 40 60 80 100 Frequency [cm1 ] 0.0 0.5 1.0 1.5 2.0 2.5 3.0R s / R n (20K) 2K 3K 4K 5K 6K 7K 8K 9K 10K 11K 12K 13K 14K FigureE-2.Temperaturedependenceof T s = T n and R s = R n forNbN.Thenormalstateis at20K.Themeasurementresolutionis4cm 1 174

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0 20 40 60 80 100 Frequency [cm1 ] 0.0 0.5 1.0 1.5 2.0 2.5T s / T n H film surface 3 K 0 T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 10 T 0 20 40 60 80 100 Frequency [cm1 ] 0.0 0.5 1.0 1.5 2.0 2.5T s / T n H 5 from film surface 2 K 0 20 40 60 80 100 Frequency [cm1 ] 0.0 0.5 1.0 1.5 2.0 2.5T s / T n 2 K H film surface FigureE-3. T s = T n ofNbTiNwiththemagneticeldparallel,atasmallangle( 5 ),and perpendiculartothelmsurface.ThedatainparalleleldsandperpendiculareldsarediscussedinChapter 4 andChapter 5 ,respectively. 175

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r n r n r n r n FigureE-4. T = T ofNbTiNatdierentruences,elds,anddelaytimes.Theeldis paralleltothelm. T = T 0 T ,where T 0 isthetransmissionatthe specieddelaytimeafterlaserexcitation,and T isthetransmissionwell beforethearrivalofthelaserpulsessothatnophoto-excitation sarepresent. 176

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BIOGRAPHICALSKETCH XiaoxiangXiwasborninJiangsuProvince,Chinain1984.Hegrewupwit ha greatcuriosityinscience.HeattendedNanjingHighSchoolinthesa metownhewas raised.In2003heattendedNanjingUniversity,majoringinastron omy.In2005he visitedtheDepartmentofPhysicsatHongKongUniversityofScienc eandTechnology asanexchangestudent,whereherstlearnedaboutcondensed matterphysics.After graduatingfromNanjingUniversityin2007withaBachelorofScience degree,hebegan hisgraduatestudyattheDepartmentofPhysics,UniversityofFlo rida.Heobtained aMasterofSciencedegreein2009.HejoinedDr.DavidTanner'sgro upin2008,and workedinclosecollaborationwithDr.G.LawrenceCarrattheNation alSynchrotron LightSource,BrookhavenNationalLaboratoryonresearchinth eeldofconventionaland time-resolvedspectroscopyofsuperconductorsandsemicondu ctingnano-structures.He wasawardedaDoctorofPhilosophydegreeinDecember2011. 186