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Fluctuation, Disorder and Inhomogeneity in Unconventional Superconductors

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Title:
Fluctuation, Disorder and Inhomogeneity in Unconventional Superconductors
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MISHRA,VIVEK
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[Gainesville, Fla.]
Florida
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University of Florida
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english
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
Hirschfeld, Peter J
Committee Members:
Muttalib, Khandker A
Dorsey, Alan T
Stewart, Gregory R
Phillpot, Simon R
Graduation Date:
4/30/2011

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Subjects / Keywords:
Critical temperature ( jstor )
Doping ( jstor )
Electrons ( jstor )
Fermi surfaces ( jstor )
Impurities ( jstor )
Inhomogeneity ( jstor )
Low temperature ( jstor )
Magnetic fields ( jstor )
Superconductors ( jstor )
Thermal conductivity ( jstor )
DISORDERED -- FERMI -- IRON -- MODELS -- NODE -- SUPERCONDUCTING -- SUPERFLUID -- THERMAL -- UNCONVENTIONAL -- UNDERDOPED
Physics -- Dissertations, Academic -- UF
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Physics thesis, Ph.D.

Notes

Abstract:
In this dissertation, I present the results of theoretical investigations of the effect of fluctuations, disorder and inhomogeneity in unconventional superconductors. Copper oxide based and iron pnictide based high temperature superconductors are the systems of primary interest. Both the materials have been subject of massive experimental and theoretical investigations since their discovery. The mechanism of superconductivity is one of the most challenging problems of contemporary physics, and it is hoped that by comparing the two classes of materials one can gain insight into this question. Both materials have many similarities, yet they are different. Both the materials are layered compounds and have intimate relation with antiferromagnetism, but the electronic structure is fundamentally different. Copper oxide based superconductors have only one band near the Fermi surface, while iron pnictides are multiband systems. Most of these materials are intrinsically dirty, since they become superconductors when doped with either electrons or holes. There is substantial evidence from experiments that these systems have extremely anisotropic order parameters in contrast with conventional superconductors like Sn or Hg. Such an anisotropy in the order parameter makes disorder a very important parameter in the problem. Additional complexity comes in iron pnictides due to the multiple bands and orbitals involved. In this dissertation, I discuss the results of our work on disorder in multiband systems in the context of iron pnictides. We calculate the low temperature transport properties and spectral properties in the presence of disorder for models appropriate to various iron pnictides, and compare our results with experiments. Another aspect of these materials is inhomogeneity. Especially in copper oxide based superconductors, scanning tunneling spectroscopy measurements have revealed nanoscale inhomogeneity in the order parameter. Some researchers have argued that these inhomogeneities may play a crucial role in mechanism of pairing. We study a simple toy model to gain some understanding about the correlation between inhomogeneity and superconductivity. The normal states of these materials represent an equally interesting and challenging problem as the superconducting state. The fluctuations of the order parameter above the transition temperature can significantly affect the normal state properties. In copper oxide based superconductors, disconnected Fermi surfaces have been observed in experiments. We study the effect of d-wave order parameter fluctuations above the critical temperature for model systems relevant to copper oxide materials, and try to relate them to the observed normal state electronic structure. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Hirschfeld, Peter J.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-04-30
Statement of Responsibility:
by VIVEK MISHRA.

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Copyright MISHRA,VIVEK. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
4/30/2013
Resource Identifier:
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FLUCTUATION,DISORDERANDINHOMOGENEITYINUNCONVENTIONALSUPERCONDUCTORSByVIVEKMISHRAADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011VivekMishra 2

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Tomyparents 3

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ACKNOWLEDGMENTS IowemydeepestgratitudetoProfessorPeterHirschfeldforhissupportandguidanceduringmygraduatestudyatUniversityofFlorida.Hehasbeenawonderfulpersontoworkwith,manythanksforthenumerousdiscussions,advice,encouragementandpatience,withoutwhichIwouldnothaveachievedmygoals.Ithankmembersofmysupervisorycommittee,Prof.GregStewart,Prof.K.Muttalib,Prof.AlanDorseyandProf.SimonPhillpot,fortheirtimeandexpertisetoimprovemywork.IamespeciallygratefultomycollaboratorsYu.S.Barash,SiegfriedGraser,IlyaVektherandAntonVorontsovfortheirvaluablediscussionsandassistanceduringthework.Ithankmyteachers,friends,andcolleagueswhoassisted,advisedmeovertheyears,especiallyMengxingCheng,ChungweiWang,TomoyukiNakayama,RajivMisraandRiteshDasandmanyotherfriendsattheDepartmentofPhysics.Imustalsoacknowledgeformerandpresentgroupcolleagues,WeiChen,GregBoyd,LexKemper,MaksimKorshunov,WenyaWang,YanWangandLiliDeng.IalsothankDavidHansenandBrentNelsonfortheirtechnicalsupportandthestaffoftheUniversityofFloridaHighPerformanceComputingCenter,wheremostofthecomputationalworkincludedinthisdissertationwasperformed.IwouldalsoliketothankRobertDeSerioandCharlesParkfortheirconsiderationandassistanceduringmyteaching.IthankPhysicsDepartmentstafffortheirassistanceoverpastfewyears,especiallyIneedtoexpressdeepappreciationtoKristinNicholaforherhospitalityandsupport.IacknowledgenancialsupportfromDepartmentofEnergy,NationalScienceFoundationandRussellFoundation.Finally,Iwouldliketothankmyparentsandsistersfortheirsupportandloveinallaspectsofmylifeandalwaysencouragingmetofollowmytemptationforscience. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 1.1HistoricalBackground ............................. 13 1.2Cuprates:Overview .............................. 14 1.3Pnictides:Overview .............................. 17 2INHOMOGENEOUSPAIRING ........................... 21 2.1Motivation .................................... 21 2.2Model ...................................... 23 2.3CriticalTemperature .............................. 26 2.4QuasiparticleSpectrumandPhaseDiagram ................ 26 2.5SuperuidDensity ............................... 28 2.6OptimalInhomogeneity ............................ 33 2.7Conclusion ................................... 35 3DISORDERINMULTIBANDSYSTEMS ...................... 37 3.1Introduction ................................... 37 3.2Formalism .................................... 38 3.3NodeLiftingPhenomenon ........................... 41 3.4EffectOnCriticalTemperature ........................ 48 3.5Conclusion ................................... 52 4TRANSPORTI-THERMALCONDUCTIVITY ................... 55 4.1Introduction ................................... 55 4.2BasicFormalism ................................ 56 4.2.1ThermalConductivity .......................... 56 4.2.2T-MatrixAndImpurityBoundStates .................. 58 4.3Modelfor1111Systems ............................ 62 4.3.1ThesWaveState ........................... 62 4.3.2StatewithDeepMinima ........................ 65 4.3.3NodalState ............................... 69 4.4ModelFor122Systems ............................ 73 4.4.1ZeroTemperatureResidualTerm ................... 78 5

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4.4.2MagneticFieldDependence ...................... 82 4.5ThreeDimensionalFieldRotation ....................... 90 4.6Conclusion ................................... 93 5TRANSPORTII-PENETRATIONDEPTH ..................... 94 5.1Introduction ................................... 94 5.2Modelfor1111Systems ............................ 96 5.3Modelfor122Systems ............................ 99 5.4Conclusion ................................... 105 6SUPERCONDUCTINGFLUCTUATIONSANDFERMISURFACE ....... 106 6.1IntroductionandReview ............................ 106 6.2Formalism .................................... 107 6.3SpectralFunction ................................ 111 6.4FluctuationinDirtyLimit ............................ 114 6.5EffectofMagneticFluctuations ........................ 117 6.6Conclusion ................................... 118 APPENDIX AT-MATRIXFORTWOBANDSYSTEM ....................... 120 A.1BasicFormalism ................................ 120 A.2SpecialCases ................................. 123 A.2.1OneBand ................................ 123 A.2.2BornLimitofTwoBandCase ..................... 123 A.2.3UnitaryLimit ............................... 123 BGENERALEXPRESSIONFORPENETRATIONDEPTH ............. 125 B.1Introduction ................................... 125 B.2NormalStateCancellation ........................... 130 B.3TotalCurrentResponse ............................ 132 B.4SpectralRepresentationofTotalCurrentResponse ............. 134 REFERENCES ....................................... 135 BIOGRAPHICALSKETCH ................................ 146 6

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LISTOFTABLES Table page 1-1PhaseorderingtemperatureTandpenetrationdepthcindifferentsuperconductors. ................................... 17 4-1Basicparametersusedincalculationforcases1to5. .............. 78 4-2ResiduallinearterminlowTthermalconductivityinW=K2cmforvariouspnictidecompounds. ................................. 79 5-1Zerotemperaturepenetrationdepth0forallcases. ............... 105 6-1Valueofindifferentsuperconductors. ...................... 110 7

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LISTOFFIGURES Figure page 1-1Schematicphasediagramofcuprates.XaxisisdopingofholesperCuO2planeandYaxisistemperatureinKelvin. ..................... 15 1-2Phasediagramforironpnictides. .......................... 18 1-3FermisurfaceofBaFe2As2. ............................. 19 2-1Unitcellofsquarebipartitelattice .......................... 23 2-2Criticaltemperatureasafunctionofinhomogeneity. ............... 25 2-3Thequasiparticleenergyinthemomentumspace ................ 27 2-4Thelocaldensityofstatesoneachsublattice. ................... 29 2-5ZerotemperaturemeaneldphasediagraminthespaceofthetwocouplingconstantsgAandgB. ................................. 30 2-6Superuiddensityns=mvs.inhomogeneitywithxedaveragecouplingg=tatdifferenttemperatures,T=0(solid),0.1t(dashed),and0.18t(dottedline). 32 2-7Superuiddensityandthecurrentcurrentcorrelationfunctionasafunctionoftemperature. ..................................... 33 2-8Meaneldcriticaltemperatureandthephase-orderingtemperatureasafunctionofinhomogeneity. .............................. 34 3-1DiagramsincludedintheT-Matrixapproximation. ................. 40 3-2Schematicpictureoforderparameter. ....................... 42 3-3Spectralgapontheholeandtheelectronpockets. ................ 45 3-4Thedensityofstatesfordifferentanisotropies. .................. 46 3-5Spectralgapontheholeandtheelectronpocketsinthepresenceofmagneticimpurities. ....................................... 47 3-6Thedensityofstatesinthepresenceofinterbandscattering. .......... 49 3-7Variationofthecriticaltemperaturewithdisorder. ................. 52 3-8SpectralgapasafunctionofTcreduction. .................... 53 3-9SpectralgapformicroscopicorderparameterasafunctionofTcreduction. .. 53 4-1Theeffectofinterbandscatteringonimpurityresonanceenergy. ........ 60 8

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4-2DOSandthermalconductivityforanisotropicsstateforvariousvaluesofinterbandscatterings. ................................ 63 4-3DOSandthermalconductivityforanisotropicsstatefordifferentimpurityconcentrationinisotropicdisorderlimit. ...................... 64 4-4DOSandthermalconductivityforstatewithdeepminimaontheelectronsheetinthepresenceofpureintrabandimpurityscattering. ............... 65 4-5DOSandthermalconductivityforstatewithdeepminimaontheelectronsheetinisotropicdisorderlimit. .............................. 67 4-6Zerotemperaturelimitoftheresiduallineartermforthestatewithdeepminimaontheelectronsheet. ................................ 68 4-7Theeffectofdifferentkindsofdisorderonthetransitiontemperature. ...... 69 4-8DOSforstatewithaccidentalnodesontheelectronsheetforvariousvaluesofpureintrabanddisorder. .............................. 70 4-9Thermalconductivityforthenodalstatefordifferentvaluesofdisorder. ..... 71 4-10Zerotemperaturelimitlinearterminthethermalconductivityforthenodalstate. 72 4-11ZerotemperaturelimitofnormalizedresiduallineartermasafunctionofCodopingforBa(CoxFe1)]TJ /F7 7.97 Tf 8.47 0 Td[(x)2As2inH=0andinH=Hc2/4. .............. 74 4-12Fermivelocitiesforcases1to5. .......................... 76 4-13Fermisurfacesforcase1to5,withrespectivenodalstructures. ......... 77 4-14ResiduallinearterminT!0limitasafunctionofdisorderforcases1and2. 79 4-15ResiduallinearterminT!0limitasafunctionofdisorderforcases3and4. 80 4-16ResiduallinearterminT!0limitasafunctionofdisorderforcase5. ...... 82 4-17Themalconductivityanisotropyforallcasesasafunctionofnormalizeddisorder. ........................................ 83 4-18ResiduallinearterminT!0limitasafunctionofmagneticeldforcases1and2. ......................................... 85 4-19ResiduallinearterminT!0limitasafunctionofmagneticeldforcases3and4. ......................................... 86 4-20ResiduallinearterminT!0limitasafunctionofmagneticeldforcase5. .. 87 4-21VariousFermisurfacesobtainedbyEq. 4 ................... 88 9

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4-22NormalizedlineartermasafunctionofaringparametervmaxFc=vaveFc,insetshowsthesamequantitiesinthepresenceof10Tmagneticeld. ............ 89 4-23Normalizedlineartermasafunctionofmagneticeld'spolarangleZandazimuthalangle. .................................. 91 4-24Normalizedspecicheatcoefcientasafunctionofmagneticeld'spolarangleZandazimuthalangle. ............................. 92 5-1Penetrationdepthasafunctionoftemperaturefornodal1111systems. .... 97 5-2Penetrationdepthasafunctionoftemperatureforanisotropics1111systems. 98 5-3ResidualDOSforcases1to5. ........................... 100 5-4Penetrationdepthasafunctionoftemperatureforcases1and2. ........ 101 5-5Penetrationdepthasafunctionoftemperatureforcases3and4. ........ 102 5-6Penetrationdepthasafunctionoftemperatureforcase5. ............ 103 5-7Penetrationdepthanisotropyforallcasesincleananddirtylimit. ...... 104 6-1Fluctuationpropagatordiagrams. .......................... 108 6-2Fermiarcs ...................................... 112 6-3Arclengthdependenceontemperature ...................... 113 6-4Dopingdependence ................................. 114 6-5Arclengthdependenceontemperature,forgapfunctionwithhigherharmonicterm. .......................................... 115 6-6DisordereffectonFermiarcs ............................ 116 6-7Spectralfunctioninthepresenceofmagneticuctuations. ............ 119 A-1DiagramsfortheselfenergywithintheT-Matrixapproximation. ......... 121 A-2Diagramsinvolvingscatteringwithinthesecondband. .............. 122 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyFLUCTUATION,DISORDERANDINHOMOGENEITYINUNCONVENTIONALSUPERCONDUCTORSByVivekMishraMay2011Chair:PeterJ.HirschfeldMajor:PhysicsInthisdissertation,Ipresenttheresultsoftheoreticalinvestigationsoftheeffectofuctuations,disorderandinhomogeneityinunconventionalsuperconductors.Copperoxidebasedandironpnictidebasedhightemperaturesuperconductorsarethesystemsofprimaryinterest.Boththematerialshavebeensubjectofmassiveexperimentalandtheoreticalinvestigationssincetheirdiscovery.Themechanismofsuperconductivityisoneofthemostchallengingproblemsofcontemporaryphysics,anditishopedthatbycomparingthetwoclassesofmaterialsonecangaininsightintothisquestion.Bothmaterialshavemanysimilarities,yettheyaredifferent.Boththematerialsarelayeredcompoundsandhaveintimaterelationwithantiferromagnetism,buttheelectronicstructureisfundamentallydifferent.CopperoxidebasedsuperconductorshaveonlyonebandneartheFermisurface,whileironpnictidesaremultibandsystems.Mostofthesematerialsareintrinsicallydirty,sincetheybecomesuperconductorswhendopedwitheitherelectronsorholes.ThereissubstantialevidencefromexperimentsthatthesesystemshaveextremelyanisotropicorderparametersincontrastwithconventionalsuperconductorslikeSnorHg.Suchananisotropyintheorderparametermakesdisorderaveryimportantparameterintheproblem.Additionalcomplexitycomesinironpnictidesduetothemultiplebandsandorbitalsinvolved.Inthisdissertation,Idiscusstheresultsofourworkondisorderinmultibandsystemsinthecontextofironpnictides.Wecalculatethelowtemperaturetransportpropertiesand 11

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spectralpropertiesinthepresenceofdisorderformodelsappropriatetovariousironpnictides,andcompareourresultswithexperiments.Anotheraspectofthesematerialsisinhomogeneity.Especiallyincopperoxidebasedsuperconductors,scanningtunnelingspectroscopymeasurementshaverevealednanoscaleinhomogeneityintheorderparameter.Someresearchershavearguedthattheseinhomogeneitiesmayplayacrucialroleinmechanismofpairing.Westudyasimpletoymodeltogainsomeunderstandingaboutthecorrelationbetweeninhomogeneityandsuperconductivity.Thenormalstatesofthesematerialsrepresentanequallyinterestingandchallengingproblemasthesuperconductingstate.Theuctuationsoftheorderparameterabovethetransitiontemperaturecansignicantlyaffectthenormalstateproperties.Incopperoxidebasedsuperconductors,disconnectedFermisurfaceshavebeenobservedinexperiments.Westudytheeffectofd-waveorderparameteructuationsabovethecriticaltemperatureformodelsystemsrelevanttocopperoxidematerials,andtrytorelatethemtotheobservednormalstateelectronicstructure. 12

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CHAPTER1INTRODUCTION 1.1HistoricalBackgroundOnApril08,1911,KamerlinghOnnesdiscoveredthatbelow4.2Ktheelectricalresistivityofpuremercuryabruptlybecomesunmeasurable.Thiseffect,nowknownassuperconductivity,hasattractedtheattentionofmanyphysicistsforroughlyacentury[ 1 3 ].Nearlyfourdecadesafterthediscovery,GinzburgandLandauproposedaphenomenologicaltheorybasedonLandau'sgeneraltheoryofsecondorderphasetransitions.Theyintroducedthedensityofsuperconductingelectronsasanorderparameter.Nearly50yearsafterOnnes'sdiscovery,in1957,Bardeen,CooperandSchriefferproposedthefamousmicroscopicpairingtheoryofsuperconductivity(BCStheory)[ 4 ].Theyshowedthatevenasmallattractionbetweenelectronsmediatedbyphononsmakesthenormalstateunstableandthisleadstoformationofpairsofelectronswithoppositespinandmomentum(Cooperpairs).Thekeypredictionofthismodelwastheformationofanenergygapinthequasi-particleexcitationspectrum.Thistheoryalsopredictedthattheratioofthisgaptothetransitiontemperature(Tc)is1.76andtheratioofspecicheatjumpatthetransitiontothenormalstatevalueis1.43,inagreementwithexperiments.TheBCSmodelprovidedabasicframeworktocalculatedifferentphysicalquantities.Thequestforthesuperconductorswithhighertransitiontemperaturecontinuedformanyyears,andamajorbreakthroughcamein1986,whenBednorzandMullerdiscoveredsuperconductivityinLa2-xBaxCuO4(LBCO)at35K[ 5 ].TheparentcompoundLa2CuO4isaninsulatorandbecomessuperconductingwhendopedeitherwithholesorwithelectrons.Veryquickly,othergroupsfoundsuperconductivityinYBa2Cu3O7-(YBCO),Ba2Sr2CaCu2O8+(BSCCO)andTl2Sr2CaCu2O8+(TBCCO)andTccrossedthevalueof150K,muchhigherthanthetemperatureofliquidnitrogen.ThecommoningredientinallthesesuperconductorsistheCuO2planeanditiswidelyacceptedthattheseplanesarecrucialforsuperconductivity. 13

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DuetothepresenceoftheseCuO2planes,thesematerialsarecollectivelyknownascuprates.In2008,anewclassofironbasedsuperconductors(pnictides)wasdiscovered,withageneralformulaROFFePn(1111family).HereRisarareearthelementandPniselementfrompnictogengroup[ 6 ].TheSmbasedsuperconductorshowsaonsetTc=55K,whichishighestamongnon-cupratesuperconductors.ThesesystemsalsohavelayersofFewithapnictogenontopandbottomofitalternatively.Soonafterthediscoveryoftheof1111family,superconductivitywasobservedinaBaFe2As2basedcompound(122family)[ 7 8 ],inFeSe[ 9 10 ]andLiFeAs[ 11 ]. 1.2Cuprates:OverviewTheparentcompoundsofallcupratesareantiferromagneticinsulatorsandbecomesuperconductorsuponholeandelectrondoping.FirstTcincreaseswithdoping,attainsamaximumvalueandthenstartstodecreaseandeventuallydisappears.Figure 1-1 showstheschematicphasediagramforcuprates[ 12 14 ].Withdopingofholes,thelongrangeantiferromagneticphasedisappearsandsuperconductivityemerges.Althoughthereisnoconsensusregardingthepairingmechanism,itiswellestablishedthatitisnotphononmediated.Cupratesareverydifferentfromconventionalsuperconductors.Thekeydifferenceisamomentumdependentsuperconductingorderparameter.Itiswellestablishednowthattheorderparameter,orCooperpairwavefunction,hasd-wavesymmetry[ 12 ],whichmeanstheorderparameterbecomeszeroatsomepointsink-space.Thesepointsarecallednodes,andthepointswithmaximumvalueoforderparametersareknownasanti-nodes.Sothepresenceofthesenodalpointsmakestheminimumenergytoexciteaquasiparticlezero,hencethesuperconductorsaregapless.Thisleadstocompletelydifferentbehavioroflowtemperaturethermodynamicandtransportproperties.Apartfromanisotropicorderparameter,anotherdifferenceisalargeenergygapcomparedtotheconventionalsuperconductors.Incuprates,=TcismuchlargerthantheweakcouplingBCSvalueof2.15forad-waveorderparameter.ThemaximumTcoccursatdopingofroughly0.16holesperCuO2plane. 14

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Thisoptimalvalueofdopingseparatestheunderdopedandoverdopedregimesinthephasediagram.Thenormalstateofthecupratesuperconductorsisalsovery Figure1-1. Schematicphasediagramofcuprates.XaxisisdopingofholesperCuO2planeandYaxisistemperatureinKelvin[ 15 ]. interesting.Theunderdopedcupratesshowapseudo-gapphaseinthenormalstate,wheretheFermisurfaceispartiallygappedeveninthenormalstate.Angleresolvedphotoemissionspectroscopy(ARPES)experimentshavefounddisconnectedFermisurfaces,whicharenowknownasFermi-arcs[ 16 18 ].ThepresenceofanenergygapontheFermisurfaceleadstopoormetallicbehaviorintransportmeasurements[ 19 20 ].Indifferentexperimentalprobesthisenergygapstartstoappearataslightlydifferenttemperatures.ThistemperatureisusuallydenotedasT,anditisstillnotclearwhetheritisassociatedwithasecondorderphasetransitionornot[ 20 21 ].Asystematicdoping 15

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dependentstudyoftheNernsteffecthasrevealedthatuctuationsinsuperconductingchannelarenotrelatedwiththispseudo-gaptemperatureT[ 22 ].Recentpolarizedneutronscatteringmeasurements[ 23 24 ]andpolarKerreffectexperiments[ 25 ]havefoundevidenceforanovelmagneticphaseandbrokentimereversalsymmetrybelowatemperaturescalesimilartoT.Therearemanymodelsproposedforthisphase.Oneschoolofthoughtbelievesinthepresenceofcompetingorders[ 26 27 ],whilesomepeoplethinkofthisphaseasareectionofpaiructuations.TheideaofpaiructuationsisbasedonpreformedpairsaboveTc,whichleadstoreductionofthesingleparticledensityofstates.EmeryandKivelsonhavearguedthatinthebadmetals,uctuationspreventlongrangeorder[ 28 ].Theyproposedthatthepseudogapisaphasewithpreformedpairs,butwithoutphasecoherence.TheirtheorycorrectlygivesthelinearrelationbetweenthesuperuiddensityandTc,whichisobservedinmanyexperimentsonunderdopedcuprates[ 29 ].Forthephasecoherence,theydeneaphaseorderingtemperature, T=A~2ns` 4m,(1)wherensisthesuperuiddensityatzerotemperature,mistheeffectivemass,`isalengthscaleandAisamodeldependentconstantoforder1.IfT'Tcthenphaseuctuationsbecomeveryimportant.Inunderdopedcuprates,thesuperuiddensityisverysmall.Table 1-1 showstheratioofT=Tcforsomesystems.Fortheunconventionalsuperconductorsitiscloseto1,whileintheconventionalsuperconductorsitisverylarge.InChapter6,Idiscusstheroleofuctuationsindetail,anduseTc=Tasaperturbationparameter.Nexttothispseudo-gapphaseisthestrangemetalphase,whereresistivityislinearintemperaturefromtemperaturesoffewKelvintoathousandKelvin[ 20 ]andontheoverdopedsidenormalstateisapparentlyaconventionalFermiliquid.ThecrossoverbetweenthestrangemetalphaseandFermiliquidphaseinilliustratedbyadottedlinewithaquestionmarkinFig. 1-1 .Thequestionmarkonthedottedlineis 16

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Table1-1. PhaseorderingtemperatureTandpenetrationdepthcindifferentsuperconductors. MaterialcinnmTcinKT=Tc Pb3972105Nb3Sn64182103Tl2Ba2CuO6+200802Bi2Sr2CaCu2O8185841.5La2)]TJ /F7 7.97 Tf 6.59 0 Td[(xSrxCaCuO4+370281YBa2Cu2O8260800.7Ba1)]TJ /F7 7.97 Tf 6.59 0 Td[(xKxFe2As21350380.5 toindicatecurrentexperimentalambiguityaboutthecrossover.Thesuperconductingstateontheoverdopedsideisqualitativelysimilartoameaneldd-wave,butthesamephysicsintheunderdopedregimedeviatessignicantlyfromameaneldsuperconductor.Theenergyscaleassociatedwiththesuperconductivityisnothomogeneousoverananometerlengthscaleatleastinsomesystems.Tunnelingexperimentshaverevealednanoscaleinhomogeneityinthesematerialsandobservedstripeandcheckerboardlikepatternsintunnelingconductance[ 30 ];theoriginoftheseinhomogeneitiesarenotunderstood.Someresearchersbelievethattheseinhomogeneitiesarenotintrinsicandcausedbydopantatoms[ 31 ],butmanyhavearguedthatthismightplayaimportantroleinpairingandhightransitiontemperature.Wehaveconsideredasimpletoymodeltostudytheeffectofinhomogeneity,whichisdiscussedinchapter 2 1.3Pnictides:OverviewDespitemanysimilaritieswithcuprates,Fepnictidesarefundamentallydifferent.Thephasediagramsofcupratesarequalitativelysimilarforallfamilies,butforpnictidesdifferentfamiliesshowverydifferentphasediagrams.Someparentcompoundsareantiferromagneticmetalswith(,0)ordering,ondopingthetransitiontemperaturetotheantiferromagneticphase(TSDW)decreases.Somesystemsgothroughastructural 17

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Figure1-2. Phasediagramforironpnictides.Filled()indicatesstructuralphasetransition,lled(4)showsSDWtransitiontemperatureand()showssuperconductingtransitiontemperature.(a)isforLaFeAsO1)]TJ /F7 7.97 Tf 6.59 0 Td[(xFx[ 32 ],(b)isforCeFeAsO1)]TJ /F7 7.97 Tf 6.59 0 Td[(xFx[ 33 ],(c)isforSmFeAsO1)]TJ /F7 7.97 Tf 6.59 0 Td[(xFx[ 34 ]and(d)isforBaFe2)]TJ /F7 7.97 Tf 6.59 0 Td[(xCoxAs2F2[ 35 ]. transitionbeforethe(,0)spindensitywave(SDW)transition.Insomesystems,theSDWphasedisappearsbeforeemergenceofthesuperconductivity[ 32 33 ]andinsomesystemsSDWandsuperconductivityco-existuptosomedoping[ 34 35 ].ThereareafewstoichiometricsuperconductorslikeLaFePO[ 6 ]andLiFeAs[ 11 ]withsimilarelectronicstructuretoLaFeAsO.Unlikeotherpnictides,thenormalstateofthesecompoundsisnonmagnetic.Figure 1-2 showsthephasediagramsforvariousmaterials. 18

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IncaseofLaFeAsO,magneticorderabruptlydisappearsasthesuperconductingphaseonsets.InCeFeAsO,themagneticphasedisappearssmoothlybeforetheemergenceofthesuperconductivity.ForSmFeAsOandCodopedBaFe2As2,SDWandsuperconductivitycoexistinasmallregionofphasediagram.Anotherkeydifference Figure1-3. FermisurfaceofBaFe2As2. betweenFe-basedsuperconductorsandcupratesistheelectronicstructure.ThereismorethanonebandneartheFermisurfacewithbothelectronandholelikedispersions.TherearetwoholelikeFermisheetscenteredatthe)]TJ /F1 11.955 Tf 10.09 0 Td[(pointandelectronlikeFermisheetsarelocatedattheMpoint.TheelectronicstructureisverysensitivetotheheightofpnictogenatomaboveFeplaneandplaysaveryimportantroleinthetransitiontemperature[ 36 37 ].Figure 1-3 illustratestheFermisurfaceofBaFe2As2.Theouterholesheetcenteredatthe)]TJ /F1 11.955 Tf 10.1 0 Td[(pointshowssignicantdispersionalongthe`c'axis.Thisthreedimensionalnatureisuniquetothe122family.The1111familyismorelikethecupratesintermsofthedimensionality,butthe122systemsaremorethreedimensional[ 38 40 ].Thisdifferenceinthedimensionalityisreectedinthetransportpropertiesoftwofamiliesinthenormalandthesuperconductingstate[ 41 44 ].Theoreticalinvestigationofthesuperconductingstateforthe1111familypredictasignchangings-wavestatebetweenelectronandholepocket[ 45 47 ].Somegroupshavefoundveryanisotropicgapstructuresontheelectronsheets,potentiallyleadingtogap 19

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nodes.Thesenodesarereferredasaccidentalbecausetheyareduetothedetailsofthepairinginteractionratherthanbeingrequiredbysymmetry.Inaddition,athreedimensionalnodalgapstructurehasbeensuggested[ 40 48 ].Inchapter 4 and 5 ,Idiscussthephenomenologicalmodelsofthegapstructuretoexplainsomeoftheexperimentaldata.Thesymmetryoftheorderparameterisstillanunsettledissue.Someexperimentssuggestafullygappedsystemandsomendsignaturesoflowlyingexcitations.Thepresenceofmultiplebandsmakesdataanalysismorecomplicated,becauseinmostoftheexperimentsmeasuredquantitiesrepresentsumsoverallbands.Manyofthematerialsareintrinsicallydirtyandsamplequalityhasimprovedslowly.Inmultibandsystems,disorderplaysaverycrucialrole.Disordercanscatterbetweenandwithinthebands,suchthattherelativestrengthofinterbandandintrabanddisorderisaveryimportantparameterinsuchsystems.AccordingtotheAndersontheorem,inanisotropics-wavesuperconductor,nonmagneticimpurityscatteringisnotpairbreaking[ 49 ],butinasignchangings-wavesystemthepresenceofniteinterbandnonmagneticdisordercanproducepairbreaking[ 50 52 ].Allenhasshowedthatananisotropicsuperconductorismathematicallyequivalenttoamultibandsuperconductor,henceitisnotviolationoftheAnderson'stheorem[ 53 54 ].TheeffectofimpurityscatteringhasbeenincludedwithintheextendedframeworkofAbrikosov-Gorkovtheory[ 55 56 ]inthesameformalismusedbyAllen,wherethemomentumsummationisperformedovereachindividualband[ 54 ].Inchapter 3 ,Idiscusstheformalismofthemultibandproblemindetail.Inpnictides,theroleofdisordercannotbeignoredtounderstandtheexperimentaldata.Inchapter 3 ,Ifocusontheeffectofdisorderonthecriticaltemperatureandspectralfunction,inchapter 4 ,Idiscusstheeffectofdisorderonthelowtemperaturethermalconductivityforthemodelsrelevanttothe1111andthe122familiesofpnictides.Inchapter 5 ,Ipresentourstudyoflowtemperaturepenetrationdepthforthemodelsintroducedinchapter 4 .Finally,inchapter 6 ,Idiscusstheeffectofdwavesuperconductinguctuationabovethetransition. 20

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CHAPTER2INHOMOGENEOUSPAIRINGMostofthischapterhasbeenpublishedasSublatticemodelofatomicscalepairinginhomogeneityinasuperconductor,VivekMishra,P.J.HirschfeldandYu.S.Barash,Phys.Rev.B78,134525(2008). 2.1MotivationRecently,scanningtunnelingspectroscopy(STS)experimentshaveobservedinhomogeneityinthelocaldensityofstates(LDOS)ofcuprates,whichhasbeenrelatedtothepresenceofaninhomogeneousenergyscale[ 30 ].Inelasticneutronscatteringmeasurementshavealsofoundsignaturesofspinandchargedensitywaves[ 57 ].Theoriginoftheseinhomogeneitiesisnotyetclear.Someresearchersbelievethattheseinhomogeneitiescomefromtheprocessofselforganizationduetocompetingorders[ 58 ],whilesomeothersattributethistodisorderinducedbyrandomdistributionofdopantatoms[ 31 59 60 ].Manytheoreticalinvestigationshavearguedthattheseinhomogeneitiescanplayacrucialroleinthemechanismofpairing,whichleadstosuchhighcriticaltemperature[ 61 63 ].Martinetal.studiedaHubbardmodelwithaneffectiveinhomogeneouspairingpotentialintheweakcouplingBCSframework.Aonedimensionalperiodicmodulationinpairingpotentialwasincludedwithmodulationlengthscale`&a,whereaisthelatticespacing.TheyfoundenhancementofthecriticaltemperatureandthemaximumincreaseinTcwasobtainedfor`'0,where0isthecoherencelengthforthehomogeneoussystem.InasimilarworkbyLohetal.[ 64 ],atwodimensionalXYmodelwasstudied,withnearestneighborcouplingandincludinginhomogeneityintheexchangetermbyaddingaperiodicmodulation.TheyfoundamonotonicenhancementoftheBerezinskii-Kosterlitz-Thoulesstransitiontemperature(TBKT)[ 65 66 ]withincreasingmodulationlengthscale,butatthecostofreductioninsuperuidstiffness.TBKTistheorderingtemperatureforpuretwodimensionalsystems,wherelongrangeorderinthethermodynamiclimitisforbidden[ 67 ],butordercanoccur 21

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atlengthscalesmuchlargerthanthesamplelengthscales.Aryanpouretal.[ 68 69 ]studiedanattactiveHubbardmodelonsquarelatticewithcheckerboard,stripeandrandompatternsofpairingpotentialforanorderparameterwithswavesymmetry,withinameaneldapproximation,andfoundsuperconductingandchargedensitywavegroundstates.TheyalsofoundenhancementofTcforcertainvalueofdopingforinhomogeneousditributionofthepairingpotentialincomparisionwithhogogeneoussystems.InconnectionwiththeSTMexperiments,Nunneretal.[ 31 ]consideredamodelfordwavesuperconductor,withvariationofthepairingpotentialinducedbydopantatomsoveraunitcelllengthscaleandexplainedthecorrelationbetweenmeasuredlocalgapandmanyobservablesinBa2Sr2CaCu2O8+.Maskaetal.[ 70 ]haveshownthatadisorderinducedshiftofatomiclevelsalsoenhancesthepairingpotentialforasinglebandHubbardmodel.Recently,Foyevtsovaetal.havefoundsimilarresultsforathree-bandHubbardmodel,whichisappropriateforcuprates[ 71 ].IncomplexmaterialssuchaslayeredHTSCorpnictideswithlargeunitcells,thepairinginteractioncanvarywithinaunitcell.Henceunderstandingthecorrelationbetweeninhomogeneityandsuperconductivityisveryimportant.Realsystemsareverydifculttohandleevennumerically,sothestudyofsimplemodelHamiltonianisveryusefultogainunderstandingofinhomogeneouspairing.Earlierstudies[ 62 64 68 69 ]weredoneusingpurelynumericalmethods,butinsuchmethodsitissometimesdifculttodrawsimplequalitativeconclusions.HereweconsiderasimpletoymodelonabipartitelatticeintwodimensionswithtwodifferentvaluesofeffectivecouplingconstantgontwointerpenetratingsublatticesasshowninFigure 2-1 .MontorsiandCampbell[ 72 ]havefoundasuperconductinggroundstateonabipartitelatticeinarbitrarydimensionwithanattractivehomogeneouspairinginteraction.Inourstudy,wecalculatethegroundstatepropertiesofthissystemforanyvalues(gA,gB),whereAandBrepresentthetwosublattices.Weconsidertwopossiblecases,rstwhenboththesublatticeshave 22

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Figure2-1. Unitcellofsquarelattice,wheresquaresandcirclesdenotesitesoftypeAandB,andthelatticespacingisdenotedby`a',whichwehavesetto1.ThecoordinatesystemusedintheFouriertransformationisshownwithtwodashedlines. attractiveinteraction(gA,gB>0)andasecondcasewithamixtureofattactiveandrepulsiveinteractions(gA,)]TJ /F4 11.955 Tf 9.3 0 Td[(gB>0).Thisisreminiscentofcalculationsinthecontextofimpuritiesindwavesuperconductor,wheremodelswithabruptsignchangeinthepairinginteractionhavebeenstudied[ 73 74 ].Innextsection,Idiscussthedetailsofthevariousmodelsandapproaches. 2.2ModelTheHamiltonianonthebipartitesquarelatticeisgivenbyEq.( 2 ),whereci,cyiaretheelectronannihilationandcreationoperatorsonsitei.Werestrictourselvestonearestneighborhoppingbetweenthesiteswithhoppingenergytandwithonlyonsitepairingi,whichisthemeaneldorderparameteratithsite.ThesumintheEq. 2 isoverallsitesandspinsdenotedby.Weconsiderisotropicswavepairingandthecaseofhalf-lling,forwhichthechemicalpotentialissettozero. 23

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H=Xi,,)]TJ /F4 11.955 Tf 9.3 0 Td[(tcyi,ci+,+icyi,cyi,)]TJ /F5 7.97 Tf 6.59 0 Td[(+h.c..(2)TheHamiltoniancanbeexpressedasamatrixinFourierspacespannedbythestaggeredNambubasis~ck=(cA)]TJ /F8 7.97 Tf 6.59 0 Td[(k,cB)]TJ /F8 7.97 Tf 6.59 0 Td[(k,cAyk,cByk), H=Xk~cykM~ck,(2)with M=2666666640kA0k00BA00)]TJ /F6 11.955 Tf 9.3 0 Td[(k0B)]TJ /F6 11.955 Tf 9.3 0 Td[(k0377777775,(2)wherethedispersionrelationisgivenby, k=)]TJ /F10 11.955 Tf 9.3 0 Td[(4tcos(kx p 2)cos(ky p 2)(2)inthereducedBrillionzone.AfterdiagonalizingtheHamiltonian,wendthequasiparticleenergiesE1,2, E1,2=AB+q (A+B)2+42k 2,(2)whichreducetotheusualEk=p 2k+2for!.Self-consistentgapequationsforthegapsoneachsublatticeare A=gAXk)]TJ /F4 11.955 Tf 9.3 0 Td[(x21 x21+1tanhE1 2+x22 x22+1tanhE2 2B=gBXk1 x21+1tanhE1 2+)]TJ /F10 11.955 Tf 9.29 0 Td[(1 x22+1tanhE2 2, (2) wherex1=2aretheparameters,denedas x1,2=)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(A+B+q (A+B)2+42k 2k.(2) 24

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Figure2-2. CriticaltemperatureTc=tplottedvs.differenceofsublatticecouplingconstants(gA)]TJ /F4 11.955 Tf 11.95 0 Td[(gB).Redcurve:windowdensityofstates.Theaveragevalueovertheunitcellgist.Bluecurve:exact. Inourconvention,apositivegcorrespondstoanattractiveon-siteinteraction.`'is(kBT))]TJ /F8 7.97 Tf 6.59 0 Td[(1,whereBoltzmann'sconstant`kB'issetto1.Forobtainingtheanalyticalresults,weestimatetheintegralsinvolvedusingtheapproximatewindowdensityofstates(wDOS). (!)=1=8t)]TJ /F10 11.955 Tf 11.95 0 Td[(4t!4t0elsewhere,(2)whichisagoodapproximationforthetightbindingmodelforqualitativepurposes[ 75 76 ].Thisproblemisalsoequivalenttoatwobandsuperconductorwithanelectronandaholeband.UsingthewDOS,wecalculatemanypropertiesanalyticallyandcomparewithfullnumericalcalculation.Thedifferenceinthepairingpotentials(gA)]TJ /F4 11.955 Tf 12.9 0 Td[(gB)oftwosublatticesistheparameterthatcontrolstheinhomogeneity.Forahomogeneoussystem,iszero.Theaveragevalueofpairingpotential(g)iskeptconstantinordertoenableustoseparatetheeffectoftheinhomogeneityfromthetrivialeffectsarisingfromchangingtheoverallpairingstrength. 25

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2.3CriticalTemperatureTocalculateTc,welinearizethegapequationsinandsolvethesystemoftwoequationsforthemaximumeigenvalue.Withinthewindowapproximation, kBTc'8te exp")]TJ /F10 11.955 Tf 13.18 8.85 Td[(8t)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(gA+gB)]TJ /F10 11.955 Tf 11.96 -.17 Td[((8t)2 4gAgB)]TJ /F10 11.955 Tf 11.95 0 Td[(8t(gA+gB)# (2) =8te exp)]TJ /F10 11.955 Tf 20.6 8.08 Td[(16tg)]TJ /F10 11.955 Tf 11.95 0 Td[(64t2 4g2)]TJ /F6 11.955 Tf 11.96 0 Td[(2)]TJ /F10 11.955 Tf 11.96 0 Td[(16tg, (2) where(=0.577)isEuler'sconstant.Fig. 2-2 showsthecriticaltemperatureasafunctionofinhomogeneityparameter,allenergyscalesarenormalizedtothehoppingenergyt.ThequalitativebehaviorofthefullnumericalsolutionandthewDOSapproximationissimilar,butthereisabigquantitativedifference.ThisdifferenceisduetothevanHovesingularityattheFermienergyforasimple2Dtighbindingdispersion,buttheenhancementofthecriticaltemperaturewithincreasinginhomogeneityisinagreementwithpreviousstudies.Forsmallinhomogeneity,weget Tc Thomoc=1+ 2g2ln8te kBThomoc1 4t=g)]TJ /F10 11.955 Tf 11.95 0 Td[(1.(2)HereThomocisthetransitiontemperatureforthehomogeneouscaseandinthephysicalrangethatweconsiderheretheaveragevalueofinteractiongisalwayssmallerthanthebandwith4t,whichisthelargestenergyscaleintheproblem. 2.4QuasiparticleSpectrumandPhaseDiagramThequasiparticledispersioninsuperconductingstatewithnitevalueoftheorderparameterisgivenbyEq.( 2 ),whichisafunctionofkxandky.Forvariouscases,weplotthequasiparticleenergiesasafunctionofkxforaxedvalueofky=0inFig. 2-3 .Panel(a)showsthequasiparticleenergyforthehomogeneouscase,whenboththesublaticehaveequalvalueoftheorderparameterwithnormalstatedispersionasdashedline.Ontheotherhand,intheinhomogeneouscasethebandssplitsandthisleadstofourdistinctbandswithgapontheFermilevelasshowninpanel(b).Whenoneofthesublatticeshaszerocouplingconstant,thentwoenergylevels 26

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Figure2-3. Thequasiparticleenergy(inunitsof4t)inthemomentumspacealongkx(onx-axis),withky=0.(a)spectrumwhenboththegapsarezero(b)gapshavesamesignandequalmagnitude(c)gapswithsamesignsbutdifferentmagnitudes(d)gapswithoppositesignsandunequalmagnitudes. remaingapless,asshowninpanel(c).Panel(d)showsthedispersionforthecase,whenoneofthepairinginteractionsisrepulsive.Inthiscase,alevelcrossingoccursinsidetheBrillouinzone.Thisleadstoacontourofzeroenergyexcitations,whichisequivalenttoasuperconductorwithalinenode.InFig. 2-3 (d),E2isalwayspositive,butE1ispositiveexceptinanarrowrange)]TJ /F15 11.955 Tf 9.3 10.23 Td[(p AjBjp AjBj.ThegaplessphaseinisotropicswavesuperconductorsinthepresenceofmagneticimpuritieswasrstfoundbyAbrikosovandGorkov[ 55 ].Aboundstateisformednearthemagneticimpurityandoverlapofboundstatesfromeachrandomlydistributedimpurityleadstoformationofanimpurityband[ 77 78 ].TheresidualDOSontheFermisurfaceis 27

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proportionaltothebandwidthofimpurityband.SimilargaplessphasescanappearduetoinhomogeneoussignchangingorderparameterduetolowenergyAndreevboundstatesnearoffdiagonalimpurities[ 73 74 ].Gaplessphasescanalsoappearinamultibandsuperconductorswithsign-changingorderparametersduetointerbandnonmagneticscattering[ 50 79 ].Inmultibandsystemswithsignchangingisotropics-wavestates,pureintrabandmagneticandoffdiagonalorphaseimpuritiescanalsoleadtoimpurityresonances.Forthepresentcase,onecanimaginethatsuchoffdiagonalphaseimpuritiesaredenselydistributedoverthesystem,andinterferenceofboundstatesduetothesephaseimpuritiesleadstothegaplesssuperconductivity.Toobservethespectralfeaturesforeachsublattice,wecalculatethelocaldensityofstatesonAandBsites, A(!)=Xkx21 1+x21[(!)]TJ /F4 11.955 Tf 11.96 0 Td[(E1)+(!+E1)]+x22 1+x22[(!)]TJ /F4 11.955 Tf 11.95 0 Td[(E2)+(!+E2)] (2) B(!)=Xk1 1+x21[(!)]TJ /F4 11.955 Tf 11.96 0 Td[(E1)+(!+E1)]+1 1+x22[(!)]TJ /F4 11.955 Tf 11.95 0 Td[(E2)+(!+E2)]. (2) Fig. 2-4 exhibitstheLDOSonthetwosublattices.Inpanels(a)and(b),whenboththecouplingsarepositive,weclearlyseeafullspectralgap.Butwhenoneofthecouplingsbecomeszero,asin(c),asharppeakdevelopsattheFermilevelontheassociatedsublattice.Thesystemremainsgaplessasthecouplingconstantonthissublatticeismadenegative,asin(d).Withinthistwoparameter(gA,gB)model,weconstructthephasediagramofthegroundstateofthemodelHamiltonian,showninFig. 2-5 2.5SuperuidDensityWhenoneofthecouplingconstantsiszeroornegative,thecorrespondingphaseisgaplessandhaslowenergyexcitations.Theselowenergyexcitationsleadtonovel 28

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Figure2-4. Thelocaldensityofstates(LDOS)atsitesA(blue)andB(red).(a)LDOSforhomogeneouscase,=0,(b)=0.4t,(c)=2t,(d)=2.4t.gistforallcases.Thedashedlinerepresentsthenormalstate2Dtightbindingband. behaviorinthetransportandthethermodynamicpropertiesofthesystem.Theorderparameterisconstant,butduetoaphasedifferenceofbetweenthetwosublattices,onegetsagaplessphase.AsAryanpouretal.haveemphasized[ 69 ],modelsofinhomogeneouspairingcanleadtoavarietyofgroundstates,includinginsulatingones.Toshowthatthethestatesthatweconsiderhereareindeedsuperconducting,weusethecriteriadevelopedbyScalapinoet.al.[ 80 ]forthelatticesystems.Wecalculatethesuperuiddensityns,whichisthesumofthediamagneticresponseofthesystem(kineticenergydensity)andparamagneticresponse(current-currentcorrelationfunction) ns m=h)]TJ /F4 11.955 Tf 13.95 0 Td[(kxi)]TJ /F10 11.955 Tf 19.26 0 Td[((!=0,q!0).(2) 29

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Figure2-5. ZerotemperaturemeaneldphasediagraminthespaceofthetwocouplingconstantsgAandgB.Thesolidlinesrepresenttransitionsbetweenthenormalmetaldenedbyzeroorderparameter,gappedandgaplesssuperconductingphases.Thedashed-dottedlinerepresentsthelineofconstantaveragepairingg=t.ThedashedlinerepresentsthehomogeneousBCScase.Finally,thedottedlinerepresentsthenormal-gaplesssuperconductingtransitionlineascalculatedanalyticallyusingthewindowdensityofstates. 30

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Thecurrent-currentcorrelationfunctionisdenedas, xx(q,i!m)=Z0dei!mjPx(q,)jPx()]TJ /F16 11.955 Tf 9.3 0 Td[(q,0),(2)wherethecurrentattheithsiteisgivenby, jPx(i)=itXcyi+x,ci,)]TJ /F4 11.955 Tf 11.95 0 Td[(cyi,ci+x,.(2)Afterevaluatingtheexpectationvalue,wendthefollowingrelativelysimpleexpressionfortheanalyticcontinuationofthestatichomogeneousresponse: xx(!=0,q!0)=2Xk4tsin(kx p 2)cos(ky p 2)2f(E1))]TJ /F4 11.955 Tf 11.95 0 Td[(f(E2) E1)]TJ /F4 11.955 Tf 11.95 0 Td[(E2.(2)IftheEi(k)donotchangesignovertheBrillouinzone,asisthecaseforthegappedphasegA,gB>0,itisclearthatthisexpressionvanishesasT!0asinthecleanBCScase.ThisisnolongerthecaseinthegaplessregimegB0,whereanitevalueof,correspondingtoaresidualdensityofquasiparticles,isfoundatzerotemperature.Theothertermweneedtoevaluateistheexpectationvalueofthelatticekineticenergydensityoperator,whichinthehomogeneousBCScaseisdirectlyproportionaltothesuperuidweightns=m=h)]TJ /F4 11.955 Tf 13.95 0 Td[(kxiatT=0.Thekineticenergyoperatorkxisdenedas, kx(i)=)]TJ /F4 11.955 Tf 9.3 0 Td[(tXcyi+x,ci,+cyi,ci+x,.(2)Theexpectationvalueofthex-kineticenergydensityisgivenas, hkx(i)i=2Xkkx1 1+x21tanhE1 2+x2 1+x22tanhE2 2.(2)ForthecasewhengA,gB>0atT=0,wecansimplifythisexpressionas, h)]TJ /F4 11.955 Tf 13.94 0 Td[(kx(i)i=2Xk2k q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(A0+B02+42k,(2)whichisalwayspositive;hence,thesystemisasuperconductoranddisplaysaconventionalMeissnereffectatT=0.Theexpressionalsoshowsexplicitlythat 31

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Figure2-6. Superuiddensityns=mvs.inhomogeneitywithxedaveragecouplingg=tatdifferenttemperatures,T=0(solid),0.1t(dashed),and0.18t(dottedline). thesuperuiddensityoneachsitecorrespondstothatoftheaveragesuperconductingorderparameteroverthesystem. Fig. 2-6 showsthevariationinthesuperuidweightasafunctionofinhomogeneity.AtT=0,nsforthismodelisaconstantandequaltothevalueforthehomogeneoussystem,wheneverthesystemisfullygapped,sincetheaveragegapremainsthesame.Thesuperuiddensityisindependentofinhomogeneityinthegappedphase,butthetransitiontemperatureincreasesmonotonicallywithinhomogeneity(seeFig. 2-2 ).Asoneincreasestheinhomogeneityfurtherwithxedaveragecoupling,thereisacriticalvalue1forwhichthesystementersthegaplessphase,atwhichthesuperuiddensitydropsabruptly.Thetemperaturedependenceofsuchacaseisshown,alongwithothercases,inFig. 2-7 ,andweseethatthisdiscontinuitycorrespondstothecreationofa 32

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Figure2-7. Theuppergraphshowsthesuperuiddensitynsasafunctionoftemperatureforvariousinhomogeneityparameters=t=0(black),1.2(red),2(green)andg=t(magenta).Thelowerpanelshowsthecurrentcurrentcorrelationxx(T). niteresidualDOSattheFermisurface.Thereisthenasecondcriticalvalue2,beyondwhichthesuperuiddensityvanishes,identicaltothevaluebeyondwhichnosolutionwithA,B6=0isfound.TheseconsiderationsdeterminethelabelingofthephasediagramshowninFig. 2-5 2.6OptimalInhomogeneity Wheninhomogeneityisincreasedfurtherinthegaplessphasewhilekeepingtheaveragepairingpotentialgxed,thetransitiontemperaturemonotonicallyincreases,butthezerotemperaturesuperuiddensitycontinuouslygoesdownwithlargerinhomogeneity.Insuchsituations,largephaseuctuationsduetolowsuperuiddensity 33

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Figure2-8. MeaneldcriticaltemperatureTc=t(dashed-dottedline)andthephase-orderingtemperatureT=t(dashedline)vsinhomogeneity=tforaxedaverageinteractiong=t.Thesolidlineistheminimumofthetwocurvesatany=t. prohibitthelongrangeorder[ 28 ].Toestimatethetemperaturewhenlongrangeorderisdestroyedbyphaseuctuations,weusethecriterionproposedbyEmeryandKivelson[ 28 ]anddeneacharacteristicphaseorderingtemperature, T=A~2ns` 4m(2)whichdecreasesinthegaplessphasefollowingthereductioninzerotemperaturesuperuiddensity.Forquasitwo-dimensionalsuperconductors,thelengthscale`isthelargerofthetwolengths,theaveragespacingbetweenlayers`d'andc,wherecisthesuperconductingcoherencelengthperpendiculartothelayers.Anexactdeterminationof`isbeyondoursimplemeaneldmodel,butitisqualitativelyirrelevantinthepresenceoftheabruptdropofthesuperuiddensity.Takingsimply` 34

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a,whichiswithinafactorof2)]TJ /F1 11.955 Tf 9.3 0 Td[(3ofbothdandcinthecuprates,wemaynowobtainacrudemeasureoftheoptimalinhomogeneityneededtomaximizethetrueorderingtemperatureinthismodel.Wehavesetthevalueof`A'tounity.InFig. 2-8 ,wenowplotboththemeaneldTcandthephase-orderingtemperatureTasfunctionsofinhomogeneity=t.Weseethatthetwocurvescrossveryclosetothephaseboundarycat=2tinthegureduetothesteepdropinsuperuiddensitythere.Itthereforeappearsthattheoptimalinhomogeneitywithinthethismodeloccursforacheckerboardlikepairinginteractionwithattractiveinteractionononesublatticeandzerointeractionontheother. 2.7ConclusionWehaveconsideredasimplemeaneldmodelonasquarebipartitelatticewithinhomogeneouspairinginteraction.Westudiedthephysicalpropertiesofthissystembykeepingtheaveragepairinginteractionxedoveraunitcellandchangingthelocalpairinginteractiononeachsublattice.Wendtwodistinctkindsofsuperconductingphasesfullygappedforattractiveinteractionsonbothsublattices,andgaplessphaseinothercases.Thecriticaltemperaturemonotonicallyincreaseswithinhomogeneity,whichisconsistentwithearliernumericalworkbyseveralgroups.Wendthatasthesuperconductingstatebecomesgapless,thezerotemperaturesuperuiddensitystartstodecreasewithinhomogeneityandphaseuctuationsstarttodeterminethetruetransitiontemperature.ThisgivesanoptimalvalueofinhomogeneityformaximumTc.WendthatthisoptimalvalueofTcforthemodelconsideredherecorrespondstoacheckerboardlikepattern,whenoneofthesublatticehaszeropairinginteractionandthecorrespondingsuperconductingstateisgapless.Whileourresultshavebeenobtainedwithinthemeaneldapproximation,wedonotexpectuctuationstochangethemsignicantlybecausethelengthscaleassociatedwithinhomogeneityissmallandthesystemishomogeneousoverunitcelllengthscales.Ourresultssuggest 35

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thatmodulatedpairinginteractionattheatomicscalemayprovidearoutetohightemperaturesuperconductivity. 36

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CHAPTER3DISORDERINMULTIBANDSYSTEMSMostofthischapterhasbeenpublishedasLiftingofnodesbydisorderinextended-sstatesuperconductors:applicationtoferropnictides,V.Mishra,G.Boyd,S.Graser,T.Maier,P.J.HirschfeldandD.J.Scalapino,Phys.Rev.B79,094512,(2009). 3.1IntroductionRealmaterialsalwayscarryrandomlylocateddefects,whichgivesrisetoaneffectivedisorderpotential.Electronsandholesareinelasticallyscatteredfromthesedefects,hencedisorderplaysaveryimportantroleinthephysicalpropertiesofmaterialsatlowertemperatures.Evenintheorderedphasesofmatterlikesuperconductorsormagnets,disordercannotbeignored.Thesimplestmodelofdisordertreatsimpuritiesaspointlikescatteringcentersrandomlydistributedoverthesample.Thiscanbedonebytreatingthemasdeltafunctionpotentialswhichcouldbemagneticand/ornonmagnetic.Physicalpropertiescanbecalculatedbydoinganensembleaveragingoverallcongurationsofimpurities.TheeffectofdisorderisstudiedinsuperconductorsintheframeworkofAbrikosov-Gorkovtheory[ 55 ],whichcanbegeneralizedtoamultibandsystemslikepnictidesorMgB2.Insuchsystems,impuritiescanscattertheelectrons/holeswithinasamebandorinbetweentwodifferentbands.AsimplemodelHamiltonianfordisorderpotentialis, Hdisorder=XnZdrXijUij(r)]TJ /F16 11.955 Tf 11.96 0 Td[(Rn)yi(r)j(r).(3)Herei,jdenotebandsand~Rnistheimpurityposition.IntheFourierspaceforadeltafunctionkindofimpuritypotential, Hdisorder=Xij,k,k0Uijyi,kj,k0. (3) 37

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Therelativestrengthoftheinterandintrabanddisorderpotentialsisaveryimportantparameter,whichistakenasphenomenologicalinputintherestofthediscussion.Bothpureintrabandscatteringandcompletelyisotropicscatteringarediscussed.Inthenextsection,Idiscussthebasicformalismofthedisorderprobleminatwobandsuperconductor. 3.2FormalismWeusedthefollowingmodelHamiltonianrstproposedbySuhletal.[ 81 ], H=Xk,,ii,kcyi,k,ci,k,+Xk,k0,,i,jVijkk0cyi,k,cyi,)]TJ /F8 7.97 Tf 6.59 0 Td[(k,)]TJ /F5 7.97 Tf 6.59 0 Td[(cj,)]TJ /F8 7.97 Tf 6.58 0 Td[(k0,)]TJ /F5 7.97 Tf 6.58 0 Td[(cj,)]TJ /F8 7.97 Tf 6.59 0 Td[(k0,(3)Herecy,carethecreationandtheannihilationoperators.i,jarethebandindex,kisthemomentumandisthespin.k,,iisbanddispersionforithbandandVijkk0isthepairingpotential.Sinceonlytheformoftheorderparameterisimportantforunderstandingthepropertiesofsuperconductors,weuseasimpleseparableformofthepairingpotentialforgeneratingthedesiredmomentumdependenceoftheorderparameter,whichisgivenas, Vkk0=V0Y(k)Y(k0)(3)whereV0isstrengthofpairingandY(k)isthefunctionwhichdenesthemomentumdependenceoftheorderparameter.Withinthemeaneldapproximation,theMatsubaraGreen'sfunctionisdiagonalinthebandbasisandwrittenas, G =264G100G2375=264)]TJ /F5 7.97 Tf 10.49 5.26 Td[(!n^1+1^1++1^3 !2n+21+2100)]TJ /F5 7.97 Tf 10.5 5.26 Td[(!n^1+2^1++2^3 !2n+22+22375(3)Here!n[(2n+1)T]isthefermionicMatsubarafrequency,1=2ismeaneldorderparameterand^1is22identitymatrixintheNambubasisofeachindividualband.Theself-consistentequationfortheorderparameteris, i(k)=2T!n=!cXk0,j,!n=0Vijkk0j(k0) !2n+2j+2j(3) 38

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Here!cisthecut-offenergyscale.Theeffectofdisorderisincludedbycalculatingthedisorderselfenergy.Forlowimpurityconcentrations,onecanignoretheprocesseswhichinvolvescatteringfrommultipleimpuritysites[ 82 ].Withinthissinglesiteapproximation,wesumallpossiblescatteringeventstocalculatetheT-matrix,whichisrelatedtotheselfenergyas =nimpT ,(3)withnimptheimpurityconcentration.Foratwo-bandsuperconductor,anotherchannelofscatteringcomesthroughscatteringbetweenthebandsbytheimpurities.Fig. 3-1 showstheimpurityaverageddiagramsuptothirdorder.Anyprocesswhichinvolvesanoddnumberofinterbandscatteringsdoesnotcontributetotheselfenergy,becauseattheendthenalstatebelongstotheotherband.Inthecontextofpnictides,wefocusonnonmagneticimpuritypotentials.Amicroscopiccalculationoftheimpuritypotentialinducedbyadopantatomsuggestsstrongintrabandscatteringpotential,whileinterbandscatteringandmagneticscatteringisrelativelyweak[ 83 ].Dopingisonesourceofdisorder,butthedopingconcentrationmaynotreallybeproportionaltoimpurityconcentration,sincedopingcanalsomodifythebandstructureandmodifythepairinginteraction.Thereforeweconsidertwolimitingcases,thestrongscatteringunitarylimitwhenUijNj1andtheweakscatteringBornlimitwithUijNj1.NjisthedensityofstatesattheFermilevelfortheithband.Inthischapter,IfocusprimarilyontheBornlimittounderstandtheeffectofdisorderonthespectralfunctionandcriticaltemperature.Somedetailsoftheproblemforarbitaryimpuritypotentialstrengtharegivenintheappendix A .IntheBornlimit,onlythelowestorderdiagramscontributeto 39

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Figure3-1. Thesearetheimpurityaverageddiagrams,whichcontributetotheselfenergyoftherstbandGreen'sfunction.Heretheinterbandcontributioncomesthroughprocesses,whichinvolveanevennumberofinterbandscatterings.Thediagramsalsotakesintoaccounttheorderofinterandintrabandscatterings.Uijistheimpuritypotentialstrength.i=jdenotesintrabandandi6=jdenotesinterbandscattering. theselfenergyandallkindsofscatteringareimportant.Theselfenergyiswrittenas, 1,0=nimph(U211+~U211)g1,0+(U212+~U212)g2,0i, (3) 1,1=)]TJ /F4 11.955 Tf 9.3 0 Td[(nimph(U211)]TJ /F10 11.955 Tf 13.42 2.65 Td[(~U211)g1,1+(U212)]TJ /F10 11.955 Tf 13.42 2.65 Td[(~U212)g2,2i, (3) 2,0=nimph(U222+~U222)g2,0+(U221+~U221)g1,0i, (3) 2,1=)]TJ /F4 11.955 Tf 9.3 0 Td[(nimph(U222)]TJ /F10 11.955 Tf 13.42 2.66 Td[(~U222)g2,2+(U221)]TJ /F10 11.955 Tf 13.42 2.66 Td[(~U221)g1,2i. (3) Hereinquantitiesi,andgi,,therstsubscriptiisforbandsandthesecondsubscriptdenotestheNambucomponentoftheGreen'sfunction.gi,istheenergyintegrated 40

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Green'sfunction,denedas gi,0=)]TJ /F15 11.955 Tf 11.29 16.27 Td[(Zdii!n !2n+2i+2i, (3) gi,1=)]TJ /F15 11.955 Tf 11.29 16.27 Td[(Zdii !2n+2i+2i. (3) Inaparticleholesymmetricsystemgi,3iszero.ThefullGreen'sfunctionisevaluatedselfconsistentlywithdisorderselfenergies hG(i~!n,~n)i)]TJ /F8 7.97 Tf 6.58 0 Td[(1=G0(i!n,))]TJ /F8 7.97 Tf 6.59 0 Td[(1)]TJ /F10 11.955 Tf 11.96 0 Td[((i~!n,~n).(3)HereG0isthebareGreen'sfunctionandGisthefullGreen'sfunction,denedintermsofrenormalizedenergy~!nandrenormalizedorderparameter~nas, G=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(i~!n^1+~^1++^3 ~!2n+2+~2(3)Theselfconsistencyconditioncanberewrittenas, ~!i(!n)=!n)]TJ /F10 11.955 Tf 11.95 0 Td[(i,0(!n),(3) ~i(!n)=i+i,1(!n),(3) i=2T!n=!cXk0,j,!n=0Vijkk0~j(k0) ~!2j+2j+~2j.(3)Herethesubscriptsi,jdenotethebands.OncethefullGreen'sfunctionisknown,anyoneparticlephysicalquantitycanbecalculatedeasily.Innextsection,Idiscusstheeffectofdisorderonthespectralfunctionfororderparametersrelevantfortheironpnictides. 3.3NodeLiftingPhenomenonKnowledgeofthesymmetryoftheorderparameterofasuperconductoriscrucialtounderstandthemechanismofsuperconductivityandtodesigndevicesusingthatparticularmaterial.ThesymmetryofthenewlydiscoveredFepnictdesisstillacontroversialissue,andtherearecontradictoryresultsfromexperiments.Someof 41

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Figure3-2. Schematicpictureoforderparameter.RedandBluedenotedifferentsignsoftheorderparameteroverFe-pnictideFermisurface.Thereisaphasedifferenceofbetweentheorderparametersoftwoelectronpockets()locatedat(,0)and(0,).ThefullsystemhasA1gsymmetry.Theorderparameteronholepockets()at(0,0)isveryisotropic[ 45 ]. thetransportmeasurementssuggestthepresenceoflowenergyexcitations[ 84 94 ].Theoreticalinvestigationshavefoundthatsignchangingswavestateappearstobemoststableinthesematerials[ 45 47 95 98 ].SomeresearchershavefoundnearlyisotropicorderparameterontheFermisurface,whilesomeothershavefoundstronganisotropyonelectronlikeFermisurfacesheets.Forsomesetofparameters,orderparameterswithaccidentalnodesontheelectronsheetshavebeenreported.Someoftheexperimentalresultscanbethenexplainedwithisotropicswavestatesbyincludingstronginterbandscattering[ 50 51 99 100 ],whichcancreatealowenergy 42

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impuritybandinthegap.Fromtheearlierworkontheeffectofdisorderontwobandsuperconductors,itisknownthatnonmagneticimpurityscatteringbetweenthebandsispairbreakingforsuperconductorswithsignchangingorderparameter[ 50 51 ].Suchscatteringleadstoformationofmid-gapimpuritybands,whichaffectsthelowtemperatureproperties.Thisexplainssomeexperiments,buttherearesomenuclearmagneticrelaxation(NMR)andspecicheatdatawhichcannotbeexplained.Kobayashietal.havemeasuredNMRrelaxationrate1=T1forLaFeAsO1-xFxforvaiousvaluesofxandtheyfoundTnbehaviorwithn3forsomedopings[ 101 ].T3behaviorofNMRrelaxationrateisexpectedforasuperconductorwithlinenodeandlinear,ifthesystemisdirty.Muetal.measuredmagneticelddependentspecicheatcoefcientforLaFeAsO1-xFx,andtheyreportedp Hbehavior,whichisanindicationofnodalgap[ 102 ].TheseNMRandelddependentspecicheatdataforLaFeAsO1-xFxfavorsthepresenceofnodalgapandcannotbeexplainedbyisotropicsorderparameterswithdisorder.OneofthestoichiometriccompoundsLaFePOclearevidenceofshowsalinearterminthepenetrationdepth,whichcanonlybeexplainedbynodesintheorderparameters[ 103 104 ].Angleresolvedphotoemissionspectroscopy(ARPES)experimentsdon'tndorderparameternodesontheFermisurface[ 105 113 ],butthismaybeasurfaceeffect.Weconsideradifferentmodelhere,wherewestartwithasystemwithaccidentalnodesandbyaddingdisordertoit,weshowthatthethenodesdisappear.Thisphysicsissimpletounderstand:intrabandscatteringwillaveragethenodalgapovertheFermisurfacewithniteaveragehki,leadingtoanisotropicgapatsufcientlylargedisorder.Theorderparametersforoursimplemodelare, hole=1, (3) electron=2(1rcos2). (3) 43

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Here`'istheangleoneachFermisurfacesheet,measuredfromthecenterofthatsheetand`r'istheparameter,whichcontrolsanisotropy.Avalueoflargerthanunityyieldsanorderparameterwithnodes.Forgainingqualitativeunderstanding,weconsideronlyoneholesheetandoneelectronsheet.Figure 3-2 showsaschematicpictureoforderparameterfoundbyGraseretal.[ 45 ]for1111family.Theeffectofdisorderonthetransportpropertiesisdiscussedindetailinnexttwochapters.Herewelookatthespectralfunction,whichismeasuredinARPESmeasurements.ThespectralfunctionisdirectlyrelatedtotheretardedGreen'sfunctionas, A(k,!)=)]TJ /F10 11.955 Tf 10.89 8.08 Td[(1 ImG(!n!!+i0+,k).(3)Here!isarealfrequency.Welookatthespectralgap,whichisdenedtobetheenergybetweenthepeakandtheFermilevelwhereA(k,!)fallstohalfitspeakvalue.Thespectralgapcontainsfullinformationaboutthemomentumdependenceandforanorderparameterk,itisproportionaltotheabsolutevaluejkj.Tomodeltheorderparameterof1111familypnictides,weusefollowingpairingpotential, V11(k,k0)=)]TJ /F4 11.955 Tf 9.3 0 Td[(V1, (3) V12(k,k0)=V0, (3) V22(k,k0)=)]TJ /F4 11.955 Tf 9.3 0 Td[(V2(1+rcos2)(1+rcos20). (3) Usingthissetofpairingpotentials,wecalculatethefullGreen'sfunctionwiththeselfconsistencyconditions 3 3 and 3 .Pairingpotentialswithnegativesignsareattractive,andpositiveinterbandpairingpotentialsarerepulsive.Arepulsiveinterbandpotentialisnecessarytogetasetoforderparameterswithoppositesigns.Werstlookatthespectralfunctioninthepresenceofonlyintrabandscattering.Forisotropicswavesuperconductors,pureintrabandscatteringisnotpairbreaking.SuchasystemisequivalenttoaswavesuperconductorwithnonmagneticimpuritiesandobeysAnderson'stheorem[ 49 ].Butinthepresenceofanisotropy(nonzeror),impurity 44

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Figure3-3. SpectralgapasfunctionofanglefortheholepocketG()andfortheelectronpocketG().Theparameter)]TJ /F1 11.955 Tf 10.1 0 Td[(isthenormalstateaverageimpurityscatteringratemeasuredinunitsofthecleanlimitcriticaltemperatureTc0.Theanisotropyparameter`r'is1.3forthisgureandtheratioofdensityofstates(N/N)forthetwobandsis1.25. scatteringcausessuppressionoftheanisotropiccomponent,withinthistypeoftheory,whichneglectsthelocalizationeffects[ 52 ].Forlargevaluesofdisorder,theanisotropiccomponentoforderparameterbecomesverysmall,butthereisnocriticalvalueofdisorderwhichcankillthesuperconductivity,withinthistheorywherelocalizationeffectsareneglected.Figure 3-3 showsthespectralgap(=G)forthehole()andtheelectron()sheets.Wecanseethatthechangeintheorderparameteroftheholepocketisveryweak.Thesmallsuppressionshowningure 3-3 isduetotheinterbandpairingpotential,whichcouplesthetwobands.Anotherveryimportantquantitywhich 45

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Figure3-4. Densityofstatesisshownfortwodifferentanisotropies.TheupperpanelshowsDOSforr=1.3andlowerpanelshowDOSforr=1.1.ForthesameamountofdisorderthegapinDOSislargerforthelessanisotropic(r=1.1)system. 46

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Figure3-5. SpectralgapasfunctionofanglefortheholepocketG()andfortheelectronpocketG()inpresenceofmagneticintrabandimpurities.Theanisotropyparameter`r'is1.3forthisgureandtheratioofdensityofstates(N/N)forthetwobandsis1.25. clearlyshowsthenodeliftingisthedensityofstates(DOS),whichis N(!)=ZdkA(k,!). (3) Fig. 3-4 showstheDOSfortwodifferentanisotropiesasafunctionofenergyforvariousdisorder.AnitegapappearsintheDOSassoonasthenodesdisappear.Thecriticalvalueofdisorderthatliftsthenodesdependsontheanisotropyoftheorderparameter.Forlargeranisotropy(r1)nodesvanishmoreslowly,whileaccidentalnodes(r1)vanishforverysmallconcentrationsofimpurities.Thepresenceofweakmagneticimpuritiescannotberuledoutinthesesystems.Wehavealsolookedattheeffectofmagneticimpurities.Inanordinaryswavesuperconductorsmagneticimpuritiescancreateagaplessphasewherethespectral 47

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gapiszero[ 55 ],theyarepairbreakersinbothisotropicandanisotropicsuperconductors.Asweexpect,forthetwobandsuperconductorunderconsiderationhere,bothisotropicandanisotropiccomponentsgetsuppressed.Asshowningure 3-5 ,magneticimpuritiessuppresstheorderparameterontheholepocketandontheelectronpocketsignicantly.Sincetheisotropicandanisotropiccomponentsaresuppressedequally,thenodesdon'tdisappear.Forverylargevaluesofdisorder,superconductivityeventuallydisappears.Anotherimportantquestionis,whatistheeffectofgeneralnonmagneticimpurities?Thesizeoftheinterbandimpuritypotentialisstillanunansweredproblem.IfwethinkofanimpurityasascreenedCoulombpotential,wewouldexpecttheinterbandscatteringtobesmallerduetolargemomentumtransferintheprocess.Orbitalphysicsmayhoweverplayavitalroleinproducinglargeinterbandscattering[ 114 ],soweconsidertheeffectqualitativelyhere.Figure 3-6 showstheDOSinthepresenceofinterbandscattering.Inthepresenceofweakinterbandscattering,nodesgetliftedmoreslowlycomparetothepureintrabandcase.Forlargerinterbandscattering,nodesdonotvanishatall,andtheisotropiclimit(interband=intraband)isstronglypairbreaking.Weakinterbandscatteringleadstoformationofmidgapimpuritystates,whichforstronginterbandscatteringmovetowardstheFermilevel.ForisotropicscatteringtheimpuritybandisformedontheFermilevel.Idiscusstheformationofimpuritybandsinthenextchapter 4 ,whereIusefullTmatrixwhichcapturesthephysicsofboundstatesaccurately.Inthenextsection,Idiscusstheeffectofdisorderonthecriticaltemperature. 3.4EffectOnCriticalTemperatureTheeffectofthedisorderoncriticaltemperaturestronglydependsonthesymmetryoftheorderparameter.Inisotropicswavesuperconductors,Tcdoesnotchangewithnonmagneticimpurities[ 49 ],butTcgoestozeroforacriticalvalueofdisorderfornonmagneticimpurities[ 55 ].Inanisotropicsuperconductors,Tcdecreasesevenwithnonmagneticimpurities.Whenaisotropiccomponentisalsopresentinorderparameter,thenTcgoesdownwithdisordertilltheanisotropiccomponentgoestozero 48

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Figure3-6. DOSinthepresenceoftheinterbandscattering.TheblackcurveshowstheDOSforanodalstatewithr=1.3andpureintrabandscatteringrate)]TJ /F1 11.955 Tf 6.78 0 Td[(=2.5Tc0.Thenodeshavedisapperedduetopureintrabandscatteringandinterbandscatteringisaddedtothestatecorrespondingtotheblackcurve.Legendsshowthestrengthoftheinterbandscattering,whichchangesfrom0)]TJ /F1 11.955 Tf 9.3 0 Td[(100%ofintrabandscattering. andafterthatitsaturatestoitsvalueatthiscriticalscatteringratebecausethesystemagainobeystheAnderson'stheorem[ 52 ].Herewelookattheeffectofanisotropyandadditionalbands.TocalculateTc,welinearizetheselfconsistentequations 3 3 and 3 innearTc.NearTc, gj,0=Zdk)]TJ /F4 11.955 Tf 23.97 8.09 Td[(i~!n ~!2n+2j, (3) gj,1=Zdk)]TJ /F10 11.955 Tf 27.25 10.74 Td[(~j ~!2n+2j. (3) Sincethedisorderselfenergydependsonlyonenergy,notonmomentum,theanisotropiccomponentoftheorderparameterdoesnotdependonenergy.Thechange 49

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intheanisotropiccomponentcomesthroughthegapequations,whichnearTcreadas i=2T!n=!cXk0,j,!n=0Vijkk0~j(k0) ~!2j+2j.(3)Inthebasis^(,iso,ani),wecanwritethegapequationsinthecompactform ^=ln1.13!c Tc(^1+^V^R)]TJ /F8 7.97 Tf 6.59 0 Td[(1^X^R))]TJ /F8 7.97 Tf 6.59 0 Td[(1^V^=ln1.13!c TcM^.(3)Here^Vistheinteractionmatrixin^basis,whichcanbeobtainedbywritingthegapequationsin^and^1is33identitymatrix.^Ristheorthogonalmatrix,whichdiagonalizesthematrix^, ^=nimp266664NU2NU20NU2NU2000)]TJ /F4 11.955 Tf 9.3 0 Td[(NU2377775.(3)Theconcentrationofimpuritiesisdenotedbynimp,and^Xisadiagonalmatrixwhoseelementsare ^Xij=ij(1 2+!c 2Tc))]TJ /F10 11.955 Tf 11.95 0 Td[((1 2+!c 2Tc+i 2Tc),(3)whereisthedigammafunctionandijistheKroneckerdeltafunction.Thei'saretheeigenvaluesofthematrix^.Themaximumeigenvalue(max(Tc))ofthematrixMisrelatedtoTc, Tc=1.13!cexp()]TJ /F10 11.955 Tf 30.63 8.09 Td[(1 max(Tc)).(3)ThesolutionofEq. 3 givesthevalueofTc.Welookattheeffectofdisorderanditscorrelationwiththeanisotropy,whichcanprovidesomeunderstandingabouttheorderparameter.Systematicdisorderinasystemcanbeintroducedbyirradiationorbydoping.SuppressionofTcintheLaFeAsO0.9F0.1irradiatedbyneutronsshowssuppressionofTc,whichisweakerthand-wavesuperconductorYBC0[ 115 ].SimilarresultshavebeenfoundforNdFeAsO0.7F0.3,whichsuggestaTcsuppressionweakerthantheoreticalmodelofswavewithinterbanddisorder[ 116 ].Anotherexperimenton 50

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LaFe1-yZnyAsO1-xFxwithdifferentdopings(x),whereZndoping(y)isusedtointroducedisorder,providesadopingdependentinformationabouttheorderparameters.Underdopedandoptimallydopedsamplesshowverylittlevariation;ontheotherhandoverdopedsamplesshowveryrapidsuppressionofTc[ 117 ].Thisindicatesthattheorderparameterisnotuniversallikethecupratesbutitchangesacrossthephasediagram.SimilarsubstitutionexperimentsonoptimallydopedKdoped122systems,ndverycontrastingresults,ZndoesnotchangesTcwhileMndopingsuppressesTcrapidly[ 118 ].Thereissomeevidencethatimpuritiesdonotfullysubstituteinthemelt,soTcshouldbecarefullycorrelatedwithresidualresistivity.Takenatfacevalue,however,theseexperimentssuggestthattheanisotropyisaminimumatoptimaldoping.Dopingdependentirradiationexperimentsonelectrondoped122familyalsoshowsveryweakTcsuppressionforoptimaldopingandstrongersuppressionforoverdopedsamples.WerstlookattheeffectofanisotropyonTcsuppressioninpresenceofnonmagneticimpurities.Figure 3-7 showstheeffectofdisorderonTcfordifferentanisotropy.Forisotropicorderparameters(r=0),pureintrabandscatteringdoesn'tmakeanychange.Oncetheinterbandchannelofscatteringopensup,Tcgoesdownandeventuallybecomeszero.Ontheotherhand,foranisotropicsystemspureintrabandscatteringleadstoTcsuppressionandtherateofsuppressionincreaseswithlargeranisotropy.Next,welookatTcsuppressioninthecontextofnodelifting.Figure 3-8 showstheminimumspectralgapasafunctionofTcsuppression.Thenodesforasystemwithr=1.3disappearforroughly10%Tcreduction.Microscopiccalculationsoftenndr1andforsmallerrnodesgoawaymuchfaster.Thepresenceofadditionalbandsincreasetherate,becauseTcdecreasesslowlywithmorebandsinpresenceofintrabanddisorder.TheDOSofeachbandalsoplaysarole,whichisillustrateding. 3-8 .ThiseffectonmicroscopicallyobtainedorderparameterbyGraseretal.[ 45 ]withallbandsisshowningure 3-9 ,wherenodesareliftedforapproximately5%suppression 51

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Figure3-7. Tcsuppressionasafunctionoftotalnormalstatescatteringrate)]TJ /F1 11.955 Tf 6.77 0 Td[(.Tc0iscleanlimitcriticaltemperature,whichisusedasanenergyscale.Fourdifferentpanelsshowtheeffectofinterbandscattering,wherepanel(a)pureintraband,(b)25%,(c)50%,(d)100%oftheintrabandscattering. ofTc.Oncethenodesarelifted,thesystemshowsfullygappedbehaviorbelowtheminimumspectralgap. 3.5ConclusionWeconsideredanalternatescenariotoexplaintheexperimentalresultsonpnictides,wherewestartedwithacleannodalsystem.Insuchsystems,intrabandimpuritynonmagneticscatteringcanliftthenodesandcreateagapinDOS.Thepresenceofinterbandscatteringactsagainstthisprocess,henceamorerealisticmodelofimpuritiesshouldprovidemoreunderstanding.Lowtemperaturethermalconductivityputsfurtherconstraintsonthenatureofdisorderinthesesystems,whichIdiscussindetailinthenextchapter.Systematicintroductionofdisorderbyirradiationorbydopingalsosuggeststhattheorderparameterhasanisotropy,andstructureoftheorder 52

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Figure3-8. SpectralgapasafunctionofTcreduction. Figure3-9. SpectralgapformicroscopicorderparameterasafunctionofTcreduction. 53

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parameterisnotuniversalintheFepnictides.Kurokietal.[ 36 ]havepointedouttheimportanceofstructuralparametersinthesematerials.Inamorerealistictreatmentofdisorder,onemusttakecareofanystructuralchangecausedbyirradiationordoping,anditseffectonthepairinteraction. 54

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CHAPTER4TRANSPORTI-THERMALCONDUCTIVITY 4.1IntroductionThermalconductivityisaveryusefulbulkprobewhichhasplayedanimportantroleinidentifyingthesymmetryoftheorderparameterinunconventionalsuperconductors[ 119 120 ].Lowtemperaturethermalconductivity(T<5K)providesvitalinformationaboutthelowenergyelectronicexcitationsinthesystem;athighertemperatures,phononsalsomakeasignicantcontribution.Thermalconductivityisverysensitivetonodesandshowsastrongresponsetothemagneticeld.Thermalconductivitymeasurementsinthepresenceofanin-planemagneticeldhavealsobeenusedtoidentifythelocationofnodesinunconventionalsuperconductors[ 119 120 ].Innodalsuperconductors,thermalconductivityislinearatlowtemperaturesandisassociatedwithresidualquasiparticlesinducedbydisorderorthemagneticeld.Recently,manylowTthermalconductivitymeasurementshavebeenperformedontheBaFe2As2based122familyandonthestoichiometric1111compoundLaFePO.InLaFePO,ahugelinearterm(3000Wcm)]TJ /F8 7.97 Tf 6.58 0 Td[(1K)]TJ /F8 7.97 Tf 6.59 0 Td[(2[ 121 ])hasbeendetected,whichismuchlargerthancuprates(e.g.inYBCO190Wcm)]TJ /F8 7.97 Tf 6.59 0 Td[(1K)]TJ /F8 7.97 Tf 6.59 0 Td[(2[ 122 ]).InBaFe2(As1)]TJ /F7 7.97 Tf 6.58 0 Td[(xPx)2thelineartermis250Wcm)]TJ /F8 7.97 Tf 6.58 0 Td[(1K)]TJ /F8 7.97 Tf 6.59 0 Td[(2[ 123 ],whichclearlysuggeststhepresenceofnodesinthesesystems.InholedopedBa1)]TJ /F7 7.97 Tf 6.59 0 Td[(xKxFe2As2,averyweaklineartermhasbeendetected,whileelddependenceofthelineartermsuggestsstronganisotropyintheorderparameters[ 124 ].InNidoped122,nolineartermhasbeendetectedandtheelddependenceisalsoveryweak,whichindicatesthepresenceofanenergygap[ 125 ].Ontheotherhand,inanotherelectrondopedmaterialBa(Fe1)]TJ /F7 7.97 Tf 6.58 0 Td[(xCox)2As2,alineartermorderof1Wcm)]TJ /F8 7.97 Tf 6.59 0 Td[(1K)]TJ /F8 7.97 Tf 6.59 0 Td[(2hasbeenfoundinabplanemeasurementsandsimilarsizetermshavebeendetectedforthethermalconductivityalongthecaxis[ 42 126 ].Anothergroupclaimsalineartermoforder100Wcm)]TJ /F8 7.97 Tf 6.59 0 Td[(1K)]TJ /F8 7.97 Tf 6.59 0 Td[(2forthesamematerial[ 127 ].Themagneticelddependenceissimilartonodalsuperconductorsinthese 55

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samples.Innextsection,Ibrieyreviewthetheoryofthermalconductivityandthendiscusstheresultsfor122and1111familiesinsubsequentsections. 4.2BasicFormalism 4.2.1ThermalConductivityThethermalconductivityiscalculatedusingKubo'sformulafortheheatcurrentasoriginallyproposedbyAmbegaokarandGrifninthecontextofswavesuperconductors[ 128 ].TheheatcurrentoperatorforananisotropicsuperconductorderivedbyDurstandLee[ 129 ]is, jh(q,)=Xk,!!+ 2vFcyk^3ck+q)]TJ /F4 11.955 Tf 13.15 8.09 Td[(d dkcyk^1ck+q.(4)ThisisgeneralizationoftheheatcurrentobtainedbyAmbegaokarandGrifn[ 128 ].Thethermalconductivityisgivenas, T=)]TJ /F10 11.955 Tf 13.43 0 Td[(lim!0ImP(q,) .(4)HereisthethermalconductivityandP(q,)isheatcurrent-currentcorrelationfunction,whichisabubblediagramwithheatcurrentvertex.Afterastraightforwardcalculation,wecanwrite, T=nk2B 16~dXiZ10d!sech2! 2T (4) *mi1 Re[q ~2i)]TJ /F10 11.955 Tf 12.68 0 Td[(~!2i] v2i)]TJ /F10 11.955 Tf 9.74 2.96 Td[(+j~!j2iv2i+)-221(j~j2iv2i)]TJ ET q .478 w 263.33 -453.64 m 347.65 -453.64 l S Q BT /F2 11.955 Tf 281.89 -466.27 Td[(j~!2i)]TJ /F10 11.955 Tf 13 2.66 Td[(~2ij!+,kz.HerewehaveaccountedforthelayeredtwodimensionalstructureofthesystemforwhichDOSfortheithbandismi=2~2d,where`mi'isthebandmassoftheithbandand`d'isthecaxisunitcelllength.InEq. 4 ,hf(kz,)ikz,denotestheaverageovertheFermisurface,`n'isnumberoflayers,=a,cisthedirectionofthethermalconductivity,vFiistheFermivelocityontheithband,andv2iv2Fi(@k,i)2.~!and~aretheenergyandtheorderparameterrespectivelyrenormalizedbyimpurityscattering.Inthelimit 56

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EF,wecanignorethegapvelocityterm;theexpressionthensimpliesto, T'nk2B 16~dXiZ10d! T!2 T2sech2! 2T*mi(vFi^)21 Req ~2i)]TJ /F10 11.955 Tf 12.67 0 Td[(~!2i"1+j~!ij2)-222(j~ij2 j~!2i)]TJ /F10 11.955 Tf 13 2.66 Td[(~2ij#+kz,. (4) Inthezerotemperaturelimit,theintegrandinEq. 4 issharplypeakedat!=0duetothesech2term.Sowecanapproximatetheintegralbysubstitutingtherestoftheintegrandbyits!=0valueandintegratetheresttoget limT!0 T'nk2B 16~d42 3Xi*mi(vFi^)2)]TJ /F8 7.97 Tf 6.77 4.33 Td[(20i ~2i+)]TJ /F8 7.97 Tf 18.73 4.12 Td[(20i3=2+kz,. (4) Hereweused~!(!=0)=i)]TJ /F8 7.97 Tf 6.78 -1.8 Td[(0,where)]TJ /F8 7.97 Tf 6.78 -1.8 Td[(0istheresidualscatteringrateduetoimpuritiesas!!0.Inthezerotemperaturelimit,themajorcontributiontotheintegrandinEq. 4 comesfromtheregionsclosetothenodes.Welinearizetheorderparameternearthenodesas (k)=d dkk=km(k)]TJ /F11 11.955 Tf 11.95 0 Td[(km)v(k)]TJ /F11 11.955 Tf 11.95 0 Td[(km),(4)wherekmisthepositionofthenodeandvisthenodalgapvelocity.Usingthisnodalapproximation,wecanperformtheFermisurfaceaveragingoftheintegrandinEq. 4 andget, T'nk2B 16~d42 3Xi,kmmi(vFi^)2nodal v.(4)ThisisaverygoodapproximationaslongastheFermivelocityisnotchangingrapidlynearthenodes.Onethingtonotehereistheabsenceofsingleparticlescatteringrate)]TJ /F8 7.97 Tf 6.78 -1.8 Td[(0inthiszerotemperaturelimit.Theeffectofdisorderentersonlythroughrenormalizationofthegapvelocity.Inthecaseofdwavesuperconductors,thegapvelocitydoesn'tgetrenormalizedbydisorder,hencethisresidualthermalconductivityisindependentofdisorder(universal),whichisconrmedbyexperiments[ 121 ].Inthepreviouschapter 3 ,IdiscussedtheeffectofdisorderwithintheBornapproximationin 57

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thecontextofthespectralfunction,butfromthepointofviewoftransportproperties,boundstatesformedbyimpuritiesplayaveryimportantroleaswell.Inthenextsection 4.2.2 ,IreviewtheT-Matrixapproximation,whichcapturesthephysicsofimpurityboundstatesexactlyinthelimitofweakimpurityconcentrationin3D. 4.2.2T-MatrixAndImpurityBoundStatesTheT-Matrixapproximationinvolvesallthediagramswhichcontributestoscatteringfromasingleimpurity(seeFig. 3-1 ).Forlowimpurityconcentrations(nimp1),thescatteringprocesseswhichinvolvemorethanoneimpurityareverysmallin3D.ThegeneralexpressionfortheT-Matrixselfenergiesinpresenceofbothintrabandandinterbandscatteringare, 1,0=nimpU211g1,0+U212g2,0)]TJ /F4 11.955 Tf 11.95 0 Td[(g1,0(U212)]TJ /F4 11.955 Tf 11.95 0 Td[(U11U22)2(g22,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g22,1) D, (4) 1,1=)]TJ /F4 11.955 Tf 9.3 0 Td[(nimpU211g1,1+U212g2,1)]TJ /F4 11.955 Tf 11.96 0 Td[(g1,1(U212)]TJ /F4 11.955 Tf 11.95 0 Td[(U11U22)2(g22,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g22,1) D, (4) 2,0=1,0(1,2!2,1), (4) 2,1=1,1(1,2!2,1). (4) HereU12istheinterbandscatteringandU11,U22aretheintrabandscatteringpotentialsforband1and2respectively,theg'sareenergyintegratedGreen'sfunctiondenedbyEq. 3 and 3 andthedenominatoroftheT-MatrixDis, D=1)]TJ /F10 11.955 Tf 11.95 0 Td[(2U212(g1,0g2,0)]TJ /F4 11.955 Tf 11.95 0 Td[(g1,1g2,1))]TJ /F4 11.955 Tf 11.96 0 Td[(U222(g22,0)]TJ /F4 11.955 Tf 11.95 0 Td[(g22,1))]TJ /F4 11.955 Tf 11.95 0 Td[(U211(g21,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g21,1) (4) +(U212)]TJ /F4 11.955 Tf 11.95 0 Td[(U11U22)2(g22,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g22,1)(g21,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g21,1).Wefocusonthestrongscatteringunitarylimit,wheretheimpuritypotentialismuchlargerthantheFermienergy.Therearetwointerestinglimitingcasesfortheunitarylimit.Therstcaseiswhentheinterbandscatteringissmallerthantheintrabandscatteringandthesecondcaseistheisotropiccasewhenthesetwokindsofscatteringshaveequalmagnitude.Intherstcase,itturnsoutthatintheunitarylimittheintraband 58

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scatteringcompletelydominatesandnoscatteringoccursbetweenthebands.Fornonmagneticimpuritiestheselfenergyisgivenas, 1,0=nimp)]TJ /F4 11.955 Tf 9.29 0 Td[(g1,0 (g21,0)]TJ /F4 11.955 Tf 11.95 0 Td[(g21,1),(4) 1,1=nimpg1,1 (g21,0)]TJ /F4 11.955 Tf 11.95 0 Td[(g21,1),(4) 2,0=nimp)]TJ /F4 11.955 Tf 9.29 0 Td[(g2,0 (g22,0)]TJ /F4 11.955 Tf 11.95 0 Td[(g22,1),(4) 2,1=nimpg2,1 (g22,0)]TJ /F4 11.955 Tf 11.95 0 Td[(g22,1).(4)TheseselfenergiesarethesameasoriginallyobtainedbyHirschfeldetal.forasingleband[ 82 ].TheenergyoftheboundstateformedbyimpurityisgivenbythepoleoftheT-Martixforasingleimpurity.Superpositionofboundstatesfrommanyisolatedimpuritiesleadstoformationofanimpurityband,whosebandwidthisgivenbytheimaginarypartoftheimpurityaveragedT-matrix.SuchimpuritybandscanbeseenclearlyintheDOS.Forarealisticimpurity,interbandandintrabandscatteringareunlikelytobeequalandthereforewerestrictourdiscussiontononmagneticimpurities.Formagneticimpurities,theexactnatureofthemagneticimpuritypotentialbecomesimportant.Thequantummechanicalnatureofimpurityspinandkondoscreeningcannotbeignored[ 130 ].Ontheotherhand,inthelimitofisotropicnonmagneticscatteringtheselfenergiesbecomethesameforallbandsandarewrittenas, 1,0=)]TJ /F4 11.955 Tf 9.3 0 Td[(nimpg1,0+g2,0 (g1,0+g2,0)2+(g1,1)]TJ /F4 11.955 Tf 11.95 0 Td[(g2,1)2, (4) 1,1=nimpg1,1+g2,1 (g1,0+g2,0)2)]TJ /F10 11.955 Tf 11.96 0 Td[((g1,1+g2,1)2, (4) 2,0=1,0, (4) 2,1=1,1. (4) 59

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Figure4-1. Impurityresonancepositionasafunctionofscatteringphaseshiftc(=1=NU)fordifferentratiosofinterandintrabandscatterings.Blackcurvesrepresentsymmetricswavestate,with1=)]TJ /F10 11.955 Tf 9.3 0 Td[(2=0andredcurvespresentsasymmetricswavestatewith1=0,2=)]TJ /F10 11.955 Tf 9.3 0 Td[(1.40.Thesolidlinerepresentsisotropicscatteringlimit. Togainsomeunderstandingintotheimpurityboundstates,Istartwithequationforthepole(D=0)forthesingleimpurityT-Matrix, 0=1)]TJ /F10 11.955 Tf 11.96 0 Td[(2U212(g1,0g2,0)]TJ /F4 11.955 Tf 11.95 0 Td[(g1,1g2,1))]TJ /F4 11.955 Tf 11.96 0 Td[(U222(g22,0)]TJ /F4 11.955 Tf 11.95 0 Td[(g22,1))]TJ /F4 11.955 Tf 11.95 0 Td[(U211(g21,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g21,1) (4) +(U212)]TJ /F4 11.955 Tf 11.95 0 Td[(U11U22)2(g22,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g22,1)(g21,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g21,1).Inoursimplemodel,wetakeU11N1=U22N2=UandU12N1=2=V.Firstweconsiderthepopularmodelfor1111family,theswavestate,wherethetwobandshaveisotropicbutoppositesigngaps(1=0,2=)]TJ /F4 11.955 Tf 9.3 0 Td[(y0)andwetakeequalDOS(N1=N2)forboththebands,whichdoesn'taffecttheresultsqualitatively.The 60

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parameteryisadimensionlessconstant,whichcontrolstherelativemagnitudesoftheorderparametersinthetwobands.Forthiscase,wecansolveEq. 4 togettheresonanceenergy, !0 0=vuut [(y+1)2+(y)]TJ /F10 11.955 Tf 11.95 0 Td[(1)2])]TJ /F15 11.955 Tf 11.95 14.92 Td[(q [(y+1)2+(y)]TJ /F10 11.955 Tf 11.96 0 Td[(1)2]2)]TJ /F10 11.955 Tf 11.96 0 Td[(16y2 4,(4)whereis, =(1+2U2+(U2)]TJ /F4 11.955 Tf 11.95 0 Td[(V2)2)2+4V2 (1+2U2+(U2)]TJ /F4 11.955 Tf 11.95 0 Td[(V2)2)2)]TJ /F10 11.955 Tf 11.96 0 Td[(4V2.(4)Forthespecialcaseofequalmagnitudeandoppositesigngaps,wecansimplifytheresonanceenergy!0as, !0 =s 1+2(U2)]TJ /F4 11.955 Tf 11.96 0 Td[(V2)+(U2)]TJ /F4 11.955 Tf 11.95 0 Td[(V2)2 1+2(U2+V2)+(U2)]TJ /F4 11.955 Tf 11.95 0 Td[(V2)2.(4)Herewecanclearlyseethatwithintrabandscattering(V=0),theresonanceenergyisequaltothegapenergyandonlyintheisotropicscatteringunitarylimit(V!1withU=V),theresonanceoccursattheFermilevel.Figure 4-1 ,showstheresonanceenergyasafunctionofdisorderforvariousinterbandandintrabandscatteringsforsymmetric(y=1)andasymmetric(y=1.4)swavestates.OnlyintheisotropiclimittheimpurityresonancestateformedattheFermilevel,andincaseofintermediateinterbandscatteringtheboundstateenergyisclosetothegapedgeandabovetheFermilevel;infactitisfoundtomovetothegapedgeintheunitarylimit.Thisbehaviordoesnotdependontherelativemagnitudesofthetwogaps.Inthecaseofananisotropicstatewithdeepminimainoneofthebands,theseresultsarequalitativelyvalid,becauseatlowtemperaturesonlythedeepminimumenergyscalemattersandtheproblemisequivalenttoastatewithdeepminimaontheanisotropicband.ThestatesattheFermilevelareimportantinthecontextoftheresiduallinearterminthermalconductivity.Inthenextsection,Idiscusstheresultsonin-planethermal 61

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conductivityforthe1111family.Thestrengthofdisorderismeasuredwithaveragenormalstatescatteringrate)]TJ /F1 11.955 Tf 6.77 0 Td[(,whichisdenedas )-277(=nnimp Nave,(4)whereNaveistheaverageDOSandthestrengthofdisorderpotentialismeasuredwithscatteringphaseshiftci=1=NiUii,wherethesubscriptdenotesthebands.Intheunitarylimit,theparameterciiszero,andforthefollowingdiscussionfor1111systems,wetakeci=0.07. 4.3Modelfor1111SystemsFor1111systems,weconsiderthreedifferentsetoforderparametersandattempttoputaconstraintonthepossibleorderparameterforthesesystemsbycomparingthemtoexperiments.Therstcaseweconsideristhepopularsymmetricswavestate[ 46 ],whichIdiscussrst. 4.3.1ThesWaveStateTheorderparameterforswavestateis, 1=)]TJ /F10 11.955 Tf 9.3 0 Td[(2=0. (4) WersttakeequalDOSatFermienergyandequalFermivelocityfortwobands.Wetakeequalintrabandscattering`Ud'forboththebandsandlookattheeffectofdifferentinterbandscattering`U12'.FromthenumericalsolutionoftheBCSgapequations,thevalueof0is1.76Tc0inthecleanlimit,whereTc0isthecriticaltemperatureforthecleansystem.Withswavesuperconductorwithonlynonmagneticintrabanddisorder,thethermalconductivityisexponentiallysmall(exp()]TJ /F10 11.955 Tf 9.3 0 Td[(0=T))inthelowtemperaturerange,thesameasaconventionalfullygappedswavesuperconductor.ThelowerpanelofFig. 4-2 ,shows=Tasafunctionoftemperature.Amoreinterestingsituationariseswiththepresenceofinterbandscattering.Withniteinterbandscattering,theresiduallineartermremainszerointhezerotemperaturelimit,becausetheimpuritybandis 62

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Figure4-2. TheDOS(top)andthermalconductivity(bottom)asafunctionoftemperatureforanisotropicsstatewith1=)]TJ /F10 11.955 Tf 9.3 0 Td[(2=1.7Tc0,shownforUd=U11=U22(intrabandscattering)andscatteringrateparameter)]TJ /F1 11.955 Tf 6.78 0 Td[(=0.3Tc0incases(i)weakintrabandscatteringonly,Ud=U12=0(solidline);(ii)pairbreakingscatteringwiththemidgapimpurityband,U12=Ud=0.98(dashedline);(iii)pairbreakingscatteringwithimpuritybandoverlappingtheFermilevel,U12=Ud=1.0(dottedline).ThetemperatureismeasuredinunitsofthecriticaltemperatureTc. 63

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Figure4-3. Densityofstates(insert)andthermalconductivityforanisotropicsstatewithisotropicscatteringandsameimpuritypotentialasFig. 4-2 butwithdifferentimpurityconcentrationwhichgives)-277(=0.2,0.25,and0.3Tc0. formedawayfromtheFermilevel.OnlythelimitofisotropicimpurityscatteringleadstotheimpuritybandattheFermilevel,asshowninthetoppanelFig. 4-2 .TheniteDOSattheFermilevelgivesrisetoanitelinearterminthezerotemperaturelimit.Wenextlookatthelineartermincaseofisotropicdisorder.Hereweincreasetheimpurityconcentration(nimp),whichincreasesthequasiparticleDOSattheFermilevel.ThisincreaseintheFermilevelDOSisdirectlyreectedinthezeroTlimitoftheresiduallineartermshowninFig. 4-3 .Thereisanitelineartermincaseofisotropicimpurityscattering,butitisverysmallcomparedtothenormalstatelinearterm. 64

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Figure4-4. (Top)Normalizedthermalconductivity((T)=T)=(n=Tc)vs.T=Tcforthedeepminimastateonthesecondband(Eq.( 4 )).Resultsareshownforvariousvaluesofintrabandscatteringrate)]TJ /F6 11.955 Tf 6.78 0 Td[(=Tc0.(Bottom)Thedensityofstatesforthecorrespondingcases.Theorderparametersare1=-1.1Tc0and2=1.3Tc0(1+0.9cos2). 4.3.2StatewithDeepMinimaInthissection,Ilookastatewithdeepminimaononeofthebands.Spinuctuationcalculationshavefoundstatewithdeepminimaorwithaccidentalnodesontheelectronsheet.Theorderparameterforoneofthebandsistakentobeisotropicandfortheotherbandis, 2=(iso+anicos2), (4) 65

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whereaniiso.Wetake1=)]TJ /F10 11.955 Tf 9.3 0 Td[(1.1Tc0,iso=1.3Tc0andani=iso=0.9inthecleanlimit,whereTc0isthecleanlimittransitiontemperature.Insuchsystems,thedeepgapminimaenergyscale(min(iso)]TJ /F10 11.955 Tf 12.41 0 Td[(ani))playsaveryimportantroleinlowtemperaturetransport.ForthetemperaturesT
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Figure4-5. DensityofstatesN(w)=N0(top)andnormalizedthermalconductivity(T)Tc=(nT)vs.T=Tcfortwobandanisotropicmodelwithdeepgapminimaononeband.Theorderparametersare1=-1.1Tc0and2=1.3Tc0(1+0.9cos2). 67

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Figure4-6. =TinT!0limt,forthestatewithdeepgapminima.00isthermalconductivityforpured-wavestate.Thisgureshowsthetotalthermalconductivityforthetwobands. disorder.Thesizeofthelineartermismeasuredin00=T,whichisthelineartermforadwavesuperconductorwithorderparameteranicos2givenas, 00 TNv2F2 31 ani, (4) Obtaininganitelineartermrequireslargeinterbandscatteringfortheisotropicscase.SuchstronginterbandscatteringcausesahugesuppressionofTc,whichisshowninFig. 4-7 .Thisdoesn'tseemtobeconsistentwithresultsonthematerialLaFePO,whichisstoichiometricandthereforeveryclean[ 104 ].ThismaterialalsoexhibitsalinearTpenetrationdepth,whichcannotbeproducedbydisorder[ 103 ].So 68

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Figure4-7. Effectofintrabandandisotropic(equalstrengthinterandintrabandimpurityscattering)onTcintheunitarylimit,forthecasewithdeepminimaononeofthebands. welooktoanotherpicture,whereonebandhasaccidentalnodes,whichIdiscussinthenextsection. 4.3.3NodalStateAnA1gstatewithaccidentalnodessimilartoFig. 3-2 isachievedbythesameorderparameterdescribedbyEq. 4 ,butwiththeconditionani>iso.Thisgivesasetoftwinnodesaround==2,3=2.Forthefollowingdiscussion,wesetani=iso=1.3andinthecleanlimit1=)]TJ /F10 11.955 Tf 9.29 0 Td[(1.1Tc0,iso=1.2Tc0,whereTc0isthecriticaltemperatureforthecleansystem.TheeffectofdisorderontheorderparametersisincludedbysolvingthegapequationsselfconsistentlywithrenormalizedGreen'sfunction.We 69

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Figure4-8. DensityofstatesnormalizedtototalnormalstateDOS(N(!)/N0)forthenodalcase.Resultsareshownforvariousvaluesofintrabandscatteringrate)]TJ /F6 11.955 Tf 6.78 0 Td[(=Tc0.)]TJ /F1 11.955 Tf 6.77 0 Td[(=0.4Tc0correspondstonodeliftedstateandshowsagapof0.03Tc0inDOS.Theorderparametersare1=-1.1Tc0and2=1.2Tc0(1+1.3cos2). onlyconsiderpureintrabandscatteringforthenodalstate,keepinginmindLaFePO,whichisastoichiometriccompoundandestimatedmeanfreepathforthesamplesare94nm[ 131 ].Inthecleanlimit,thereisnitelinearterm,whichincreaseswithdisorderandthenabruptlygoesawayassoonasthenodesareliftedbydisorder.Fig. 4-8 showstheDOSforvariousvaluesofdisorder,andFig. 4-9 showstheresiduallinearthermalconductivityasafunctionoftemperatureforvariousvaluesofdisorder.Theresiduallineartermdependsondisorderforsystemswithaccidentalnodesand 70

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Figure4-9. Forthenodalstatenormalizedthermalconductivity((T)=T)=(n=Tc)vs.T=Tcforvariousvaluesofintrabandscatteringrate)]TJ /F6 11.955 Tf 6.78 0 Td[(=Tc0.)]TJ /F1 11.955 Tf 6.77 0 Td[(=0.4Tc0representsfullygappedsystem,withnolinearterminthezerotemperaturelimit.Theorderparametersare1=-1.1Tc0and2=1.2Tc0(1+1.3cos2). isnotuniversallikedwavesuperconductors.Thebreakdownofuniversalityisduetotherenormalizationofthegapvelocityv.Theformofthelineartermisotherwisestillindependentofsingleparticlescatteringrate)]TJ /F1 11.955 Tf 6.77 0 Td[(,asinthedwavecase.Fig. 4-10 ,showsthelineartermasafunctionofdisorder.WecanestimatethelineartermusingEq. 4 .Inthiscase,mostofthecontributiontothelineartermcomesfromnodes,hencewecanlinearizetheorderparameternearthenodesas 2=)]TJ /F10 11.955 Tf 9.29 0 Td[((iso+anicos2), (4) =)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(v()]TJ /F6 11.955 Tf 11.96 0 Td[(0)2, (4) 71

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Figure4-10. Magnitudeoflinear-TterminthethermalconductivityinlimitT!0forcasewithnodesr=1.3.Solidline:exactnumericalresult.Dashedline:analyticalestimatefromtext,Eq.( 4 ).Residuallineartermismeasuredinunitsof00,whichisthermalconductivityforpured-wavestate(Eq. 4 ). wherevis v=q 2ani)]TJ /F10 11.955 Tf 13 2.66 Td[(~2iso, (4) andwiththisnodalapproximationwecangetanestimateforT=0limitof=T, Tk2B~22 3N0v2F q 2ani)]TJ /F10 11.955 Tf 13 2.65 Td[(~2iso=k2B ~d 3mev2F q 2ani)]TJ /F10 11.955 Tf 13 2.65 Td[(~2iso. (4) 72

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HerewehavetakenintoaccountthelayeredstructureofLaFePObytakingDOSN0=m=2~2d,where`d=0.85nm'iscaxisunitcelllengthand`me'istheelectronmass,.ToestimatetheFermivelocity,weusethecoherencelength(0~vF=2Tc),whichis18nmestimatedfromtheuppercriticaleld[ 104 ].ThisgivesvF=105m=sandfortheorderparameterwetakeiso=Tc=7Kandani=1.3iso.Thesenumbersgive=T=0.25W=K2m,whichisveryclosetotheexperimentallymeasuredvalueof=T=0.3W=K2m.Inthenextsection,Iwilldiscussthemodelusedforstudyingthe122systems. 4.4ModelFor122SystemsThe122familybasedontheparentcompoundBaFe2As2hasattractedmanyresearchersbecausegoodsinglecrystalswithreasonablygoodsurfacescanbegrowneasily.ManygroupshavemeasuredlowTtransportandthermodynamicpropertiesforboththeholedopedBa1-xKxFe2As2systemsandelectrondopedBa(CoxFe1-x)2As2,Ba(NixFe1-x)2As2.Themajordifferencebetweenthese122compoundsand1111compoundsisdimensionality.The122familyseemtobemuchmorethreedimensional,whileontheotherhandthe1111isverytwodimensionallikecuprates[ 42 44 ].Bandstructurecalculationshavefoundsignicantcaxisdispersionforboththeholedopedandtheelectrondopedsystems[ 132 135 ].Thereisasignicantquantitativediscrepancybetweendifferentbandstructuregroups,butthequalitativenatureof3Dcharacterforholepocketshavebeenfoundbyall.ARPESmeasurementsonBa(CoxFe1-x)2As2havealsoreportedstrong3Dcharacterofholepockets[ 136 139 ],andboththeARPESmeasurementsandbandstructurecalculationsuggeststhat3Dcharacterofholesheetsincreaseswithelectrondoping.Microscopicspinuctuationcalculationsofgapsymmetryhavefound3Dnodalstatesonholepockets[ 40 48 ]forthe122family.InelasticneutronscatteringexperimentsonelectrondopedBa(NixFe1-x)2As2alsosuggestthreedimensionalnodalstructure[ 140 ].Furtherevidenceof3Dnodalstatescomefrompenetrationdepthandthermalconductivity 73

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Figure4-11. ZerotemperaturelimitofnormalizedresiduallineartermasafunctionofCodopingforBa(CoxFe1)]TJ /F7 7.97 Tf 8.47 0 Td[(x)2As2inH=0andinH=Hc2/4.Hc2istheuppercriticaleldforrespectivedopings[ 42 ]. 74

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measurements[ 42 44 ].Reidetal.haveperformedasystematicdopingdependentmeasurementofthezerotemperaturelimitresiduallineartermonBa(CoxFe1)]TJ /F7 7.97 Tf 8.47 0 Td[(x)2As2[ 42 ].Intheabsenceofamagneticeld,theyreportedzeroresiduallineartermforin-planeheatcurrentwithinexperimentalresolution,butalongthecaxisforsomedopingstheyfoundaresiduallineartermoforder1W=K2cm.Inthepresenceofmagneticeldbothdirectionsshowsimilarvalueswhennormalizedtotheirrespectivenormalstatevalues.Fig. 4-11 illustratesthenormalizedlineartermasafunctionofdopingwithandwithoutthepresenceofthemagneticeld.Tomodelthesesystems,weconsiderthreedifferentpossiblekindsofFermisurfaces,keepinginmindthequantitativedisagreementbetweendifferentgroupsregardingtheamountofcaxisdispersion.WeneedtheFermivelocitiesandFermimomentumforcalculatingthetransportpropertiesandweusedtheFermisurfacescalculatedusingdensityfunctionaltheory(DFT)showninFig. 4-12 .Fortheelectronsheet(S2inFig. 4-13 ),weconsiderananisotropicstatewithdeepminima, S2=0(1+rcos2).(4)Thegapvaluesaretakenasphenomenologicalinputheretomatchourresultswithexperiments.Weset2=1.5meVandr=0.9forallthecases.Fortheholesheets,weconsidertwodifferentkindsoforderparameters, 1S1=0[1+rcos4(1)]TJ /F10 11.955 Tf 11.96 0 Td[(coskz)], (4) whichisqualitativelysimilartotheorderparameterobtainedbyGraseretal.[ 40 ],intheirspinuctuationcalculations.Theotherorderparameterconsideredherefortheholesheetis, 2S1=0[1+rcoskz]. (4) 75

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Figure4-12. TherootmeansquareinplaneandoutofplaneFermivelocitiesvF,xyandvF,zrespectivelyareplottedvs.ckz=2forthevedifferentcases,weconsider.Fermivelocitiesusedincalculationsarerenormalizedbyafactorof4totakeintoaccounttheeffectivemass. 76

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Figure4-13. FermisurfacesS1fortheholepocketforcases1-5withtheirrespectivenodalstructuresindicatedbydarkbluelineTheelectronsheetS2consideredforallcases,withlocationofgapminima(darkbluedashedlines)shownbydashedblueline. Fig. 4-13 showsdifferentFermisurfacesandthenodalstructureofthetheoreticallyproposedorderparameters.Weconsidervecases:ThersttwocasesarewithFermisurfacewithweakkzdispersionandorderparameterfortheholesheetwithVshapednodes1S1and2S1respectively.ThenexttwocasesarecombinationsofmorearedFermisurfacewith1S1and2S1respectively.Thelastcaseiswiththeclosedholepocketandhorizontallinenode2S1.ExperimentsonCodopedBaFe2As2havefoundanitelineartermalongcaxis,butnolineartermforthein-planedirection.Keepingthisinmind,wechoosethelocationsofhorizontallinenodestogetmaximumcaxistransport.ThisispossibleifthenodesarelocatedinaplacewherethecaxisFermivelocityislarge(SeeTable 4-2 ).Forallthecases,weusesameelectronpocketS2,withtheorderparametergivenbyEq. 4 .Table 4-1 summarizesthebasicparametersusedfor 77

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Table4-1. Basicparametersusedfordifferentmodelsincalculationsforcase1to5.0isthemagnitudeoforderparameterinmeV,ristheanisotropyparameter,vave=nodalF,xy=zaretheaverageandnodalunrenormalizedFermivelocitiesforintheplane(vF,xy)andoutoftheplane(vF,z)directionsintheunitof104m/s. S1(Case1)S1(Case2)S1(Case3)S1(Case4)S1(Case5)S2(Forall) 0(meV)-8.69.1-8.6-9.4-8.41.5r0.9-1.40.91.21.10.9vaveF,xy3.153.152.902.902.774.81vaveF,z0.470.472.032.031.090.90vnodalF,xy3.353.233.252.461.7-vnodalF,z0.160.690.234.131.26calculationofthelinearthermalconductivity.Wefocusonthezerotemperaturelimitoftheresiduallinearterm=Twithonlyintrabandimpurityscatteringintheunitarylimit. 4.4.1ZeroTemperatureResidualTermTheT=0limitofthelinearTthermalconductivitytermisevaluatedusing T'nk2B 16~d42 3Xi*kFi,? vFi,?(vFi^)2)]TJ /F8 7.97 Tf 6.78 4.34 Td[(20i ~2i+)]TJ /F8 7.97 Tf 18.73 4.12 Td[(20i3=2+kz,. (4) HerekFi,?andvFi,?aretheinplanecomponentsoftheFermimomentumandtheFermivelocityrespectivelyfortheithband,kFi,?=vFi,?istheeffectivemasstermforananisotropiclayeredsystem.Therenormalizedorderparametersandthescatteringratesare ~i=i(k)+i,1(!=0), (4) )]TJ /F7 7.97 Tf 6.77 -1.8 Td[(i=)]TJ /F10 11.955 Tf 9.3 0 Td[(Imi,0(!=0), (4) whereselfenergiesarecalculatedusingtheequations 4 4 4 and 4 .Werstcalculatethelinearterminabsenceofanexternalmagneticeldandinthenextsection 4.4.2 ,Idiscusstheeffectofthemagneticeldindetail.Werstlookatcases1and2,forwhichtheFermisurfacehasweakdispersionalongthecaxis.Fig. 4-14 showstheresidualthermalconductivitybothinabsoluteunitsandthenormalizedvalues.Thelatterareobtainedbytakingtheratioofabsolutevalueswithrespective 78

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Table4-2. ResiduallinearterminlowTthermalconductivityinW=K2cmforvariouspnictidecompounds. MaterialInplanecaxisReferences LaFePO1900[ 104 ]BaFe2(As1)]TJ /F7 7.97 Tf 6.59 0 Td[(xPx)2250[ 123 ]FeSe0.8216[ 141 ]BaFe1.916Co0.084As202.3BaFe1.904Co0.096As210.20.9BaFe1.852Co0.148As210.9[ 42 ]BaFe1.772Co0.228As203.8BaFe1.746Co0.254As2175.6 Figure4-14. ResiduallinearterminT!0limitasafunctionoftotalnormalstatescatteringrate)]TJ /F1 11.955 Tf 10.1 0 Td[(inmeV,forcase1(a,b)andcase2(c,d).Thesolidlineshowstheinplanedirectionanddashedlineshowscaxis.PanelsaandcshowabsolutevalueofthelinearterminWK)]TJ /F8 7.97 Tf 6.58 0 Td[(2cm)]TJ /F8 7.97 Tf 6.59 0 Td[(1forcase1and2respectivelyandpanelsbanddshowlineartermnormalizedtonormalstatethermalconductivityN=Tcforcase1and2. 79

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Figure4-15. ResiduallinearterminT!0limitasafunctionoftotalnormalstatescatteringrate)]TJ /F1 11.955 Tf 10.1 0 Td[(inmeV,forcase3(a,b)andcase4(c,d).Thesolidlineshowstheinplanedirectionanddashedlineshowscaxis.PanelsaandcshowabsolutevalueofthelinearterminWK)]TJ /F8 7.97 Tf 6.58 0 Td[(2cm)]TJ /F8 7.97 Tf 6.59 0 Td[(1forcase3and4respectivelyandpanelsbanddshowlineartermnormalizedtonormalstatethermalconductivityN=Tcforcase3and4. normalstatevalues(N)atTcgivenas N= 12nk2BTc ~dXikFi,? vFi,?(viF.^a)2=)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(i,kz, (4) wherePiissumoverallthebands.Weseequalitativelydifferentbehaviorofthein-planelineartermforVshapedandhorizontallinenodes.ForVshapedlinenodes,thelinearterm=Tisroughlyconstantuptoacriticalvalueofdisorder()]TJ /F7 7.97 Tf 6.78 -1.79 Td[(NL)andthenrapidlygoestozeroasshowninpanel(a)ofFig. 4-14 .Thiscriticalvalueofdisorderisthenodeliftingpoint.Inpresenceofpureintrabandscattering,theanisotropic 80

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componentoftheorderparametergoesdownandeventuallythesystembecomesfullygapped,asdiscussedinthelastchapterindetail.Forhorizontallinenodes,=Tincreaseswithdisorderandthengoestozeroat)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(NL.Ontheotherhand,caxisvaluesmonotonicallydecreasewithdisorder.Thecaxisvaluesareextremelysmallforthesecasescomparedtoin-planevalues,andnormalizedvalueforbothdirectionsareverysmall.Inthiscase,smallmeansbothsmallcomparedtothevalueexpectedfortheuniversalresultwithfullydevelopedlinenodes,andcomparableorsmallerthanexperimentalresolution.Wenextconsidercase3and4,ofFig. 4-13 whichhavemuchstrongerdispersionalongcaxiscomparetocases1and2.Thequalitativebehaviorforin-plane=Tissameastheearliertwocases,butweseeahugedifferenceinthemagnitudesofabsoluteandnormalizedvalues.Inthesecases,thecaxistransportiscomparablewithin-planetransportandnormalizedvaluesforcaxisarelargerthanin-planenormalizedvalues,asseenintheexperimentsbyReidetal.Thecaxis=Tincreaseswithdisorder,beforegoingtozeroat)]TJ /F7 7.97 Tf 6.77 -1.79 Td[(NL.ThehorizontallinenodesproducesmorequasiparticlesthantheVshapednodes,buttheygoawayatafasterratethanVshapednodeinthepresenceofintrabandscattering.Incase5,whereweconsiderhorizontallinenodesonaclosedFermisurface,thein-planelineartermdecreaseswithdisorderbecausethenodemovetowardsthezonefaceandthein-planeFermivelocitynearthenodesgoesdown.Inthiscase,thelinearterminbothdirectionsiscomparablebutthenormalizedvaluesforcaxisareverylargecomparetoinplanenormalizedvalues.Themainproblemwithcase5isthatthe)]TJ /F1 11.955 Tf 10.1 0 Td[(estimatedforexperimentalsamplescaneasilyliftthenodesandcreateafullygappedsystem.Thepresenceofniteinterbandscatteringintheunitarylimitdoesnotmakeanydifference,butforisotropicscatteringwhenbothintrabandandinterbandareequallystrong,theimpurityboundstateswillbeontheFermilevel.Insuchsituation,theresiduallineartermwillbelargeandanisotropybetweenthein-planeandthecaxisdirectionwillbesameasinthenormalstate,butthispictureisagainsttheexperimentalresults[ 42 ].Anotherquantity 81

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Figure4-16. ResiduallinearterminT!0limitasafunctionoftotalnormalstatescatteringrate)]TJ /F1 11.955 Tf 10.1 0 Td[(inmeV,forcase5.Thesolidlineshowstheinplanedirectionanddashedlineshowscaxis.PanelaandbshowsabsolutevalueofthelinearterminWK)]TJ /F8 7.97 Tf 6.58 0 Td[(2cm)]TJ /F8 7.97 Tf 6.59 0 Td[(1andlineartermnormalizedtonormalstatethermalconductivityN=Tcforcase5respectively. whichtheexperimentalistshavestressedisthenormalizedanisotropyofthermalconductivity, =c=cN ab=abN.(4)Forcaxiscurrents,thenormalstatethermalconductivityissmallcomparedtotheab-plane,sothenormalizedconductivity=Nislargeforthecaxisevenwhentheabsolutevaluesforthecaxisaresmall.Sothevalueofiscloseto10for122systems.Fig. 4-17 showsthermalconductivityanisotropyasafunctionofdisorderforallcasesinconsideration.Nextweconsidertheeffectofexternalmagneticeld. 4.4.2MagneticFieldDependenceTheeffectofmagneticeldisstudiedwithintheDopplershiftapproximation,whichisqualitativelyagoodapproximationatlowtemperatureandlowelds[ 142 144 ].Thisisasemiclassicalapproach,wheretheenergyofthequasiparticlemovinginthesuperuidvelocityeldvs(r)ofthevortexlatticeexperiencesashiftofenergy!givenby, !=mvFvs,(4) 82

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Figure4-17. Themalconductivityanisotropyforallcasesasafunctionofnormalizeddisorder.)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(NLiscriticalvalueofdisorder,whichliftsthenodes. wherevFistheFermivelocityandvsisthesupercurrentvelocityaroundthevortexcore,whichisgivenas, vs=~ 2mr^.(4)Hereristhedistancefromthevortexcoreand^isthecirculardirectionperpendiculartothedirectionofappliedeldHandtangentialtothecircleofradiusrfromthevortexcore.Foramagneticeldalongthecaxis,theDopplershiftenergybecomes, !=~vF 2rsin()]TJ /F6 11.955 Tf 11.96 0 Td[()=EH sin()]TJ /F6 11.955 Tf 11.95 0 Td[(),(4) 83

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whereisthevortexwindingangle,istheangleofvFinmomentumspaceandtheenergyscaleEHis~vF=2RH.RHisthemagneticscaledenedas RH=s 0 H,(4)where0isthemagneticuxquantumandHistheexternaleld.isaconstantoforderunityanddependsonstructureofthevortexlattice.Wehavetakenatriangularvortexlatticeforwhichis0.433.ThisDopplershiftapproximationisvalidaslongastheappliedeldismuchsmallerthantheuppercriticaleld(Hc2)(see[ 145 ]foracomparativestudyofvarioussemiclassicalmethods).TheGreen'sfunctioninthepresenceofthemagneticeldbecomes G=(~!)]TJ /F6 11.955 Tf 11.95 0 Td[(!)^1+~k^1+k^3 (~!)]TJ /F6 11.955 Tf 11.96 0 Td[(!)2)]TJ /F10 11.955 Tf 13 2.66 Td[(~2k+2k.(4)AlllocalphysicalquantitiesarenowcalculatedbyusingtheDopplershiftedGreen'sfunctionandthenlocalquantitiesareaveragedoverthevortexlattice.Theexactwayofaveragingdependsonthenatureofthephysicalquantity[ 143 ].Forspecicheatordensityofstatestheaveragingisdoneinfollowingway, hF(H,T)i=1 R2HZRH0drrZ20dF(r,;H,T).(4)Thisparticularwayofaveragingisappropriateforthecaxisthermalconductivity,whentheeldisalongthecaxis(heatcurrentktotheeld).Forthermalcurrentsin-planewitheldalongthecaxis,differentregionsofvortexdonotcontributeequally,buttheymaybeconsideredtoformaneffectiveseriescombination.Forsuchsituationsthefollowingaverageismoreappropriate, F)]TJ /F8 7.97 Tf 6.59 0 Td[(1(H,T)=1 R2HZRH0drrZ20d1 F(r,;H,T).(4)Forsystemswiththreedimensionalnodesandstrong3dimensionalFermisurface,theactualsituationismorecomplicatedandiscombinationofbothseriesandparallel 84

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Figure4-18. ResiduallinearterminT!0limitasafunctionofappliedmagneticeldHinTesla,forcase1(a,b)andcase2(c,d).Thesolidblacklineshowsseriesaveraginganddashedblacklinesshowsparallelaveragingfortheinplanedirectionandsolidredlineshows=Tcaxis.PanelsaandcshowtheabsolutevalueofthelinearterminWK)]TJ /F8 7.97 Tf 6.59 0 Td[(2cm)]TJ /F8 7.97 Tf 6.59 0 Td[(1forcase1and2respectivelyandpanelsbanddshowthelineartermnormalizedtonormalstatethermalconductivityN=Tcforcase1and2. combinations.Hencehereweconsiderboththeseriesh)]TJ /F8 7.97 Tf 6.59 0 Td[(1iandtheparallelaveraginghiaslimitstoamoresophisticatedaveragingprocedure.WenotethattheparallelaverageappearstobeclosertotheresultsoftheBrandt,PeschandTewordt(BPT)approach,validathigherelds[ 79 146 149 ].Wefocusonthequalitativeaspectsoftheelddependenceinlowmagneticeldinthezerotemperaturelimit,henceusetheDopplershiftmethod,withthefurtherassumptionthattheelddoesnotchangetheorderparametermuch,andkeeptheuppercriticaleldHc2'40T.Thevalueof 85

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Figure4-19. ResiduallinearterminT!0limitasafunctionofappliedmagneticeldHinTesla,forcase3(a,b)andcase4(c,d).Thesolidblacklineshowsseriesaveraginganddashedblacklinesshowsparallelaveragingfortheinplanedirectionandsolidredlineshows=Tcaxis.PanelsaandcshowtheabsolutevalueofthelinearterminWK)]TJ /F8 7.97 Tf 6.59 0 Td[(2cm)]TJ /F8 7.97 Tf 6.59 0 Td[(1forcase3and4respectivelyandpanelsbanddshowthelineartermnormalizedtonormalstatethermalconductivityN=Tcforcase3and4. thedisorderscatteringrateischosentoxthenormalstatevalueofN=Tcclosetoexperimentalvalues.Forthecases1and2)]TJ /F1 11.955 Tf 10.09 0 Td[(is0.1meV,forcase3and4)]TJ /F1 11.955 Tf 10.09 0 Td[(is0.3meVandforcase5)]TJ /F1 11.955 Tf 10.1 0 Td[(is0.01meV.Forcase5,suchasmallvalueofdisorderischosentokeepthenodes.Anyexperimentallyrelevantvalueissufcienttoremovethenodes.Forthersttwocases,thein-planedirectionhasastrongerresponsetothemagneticeldthantheoutofplanedirection.Ontheotherhand,thecaxisshowsastrongresponseinnormalizedplotsasshowninFig. 4-18 ;thisisduetothesmallernormalstatevalue 86

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Figure4-20. ResiduallinearterminT!0limitasafunctionofappliedmagneticeldHinTeslaforcase5.Panel(a)andpanel(b)showtheabsolutevalueofthelinearterminWK)]TJ /F8 7.97 Tf 6.59 0 Td[(2cm)]TJ /F8 7.97 Tf 6.59 0 Td[(1andthelineartermnormalizedtonormalstatethermalconductivityN=Tcrespectively.Thesolidblacklineshowsseriesaveraginganddashedblacklinesshowsparallelaveragingfortheinplanedirectionandsolidredlineshows=Tcaxis. ofN=Tc.ThebigdifferencebetweentheseriesandtheparallelaveragingisalsoevidentfromFig. 4-18 .TheseriesaveraginghasacurvatureatlowH,buttheparallelaveraginggivesverylinearbehavior.Nextwelookatcases3and4,whichhavemorearedFermisurfaces;theresultsareshowninFig. 4-19 .Inzeroeld,thelineartermsarecomparableforbothdirections,butinthepresenceofamagneticeld,theabplanedirectionhasastrongeldresponse.Thisisduetoaorderparameterwithdeepminimaontheelectronsheet.AssoonastheDopplershiftenergybecomescomparabletothedeepminimaenergyscale,electronsheetsmakesignicantcontributionduetolargeFermivelocities.Forthesecases,bothkindsofaveragingshowcurvatureatloweldwhichisobservedinexperiments.Forthesecases,normalizedlineartermsarecomparable,whichisalsoconsistentwiththeexperiments.Forcase5,asshowninFig. 4-20 ,themagneticeldresponseoftheabplaneisweakerthanthecaxis,becausethenodesmovetoveryclosetothetopoftheFermisurface,wheretheabplaneFermivelocityisverysmallbutthecaxiscomponentoftheFermivelocityislarge.Again 87

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Figure4-21. VariousFermisurfacesobtainedbyEq. 4 inthiscase,thereissignicantdifferencebetweentheseriesandparallelaveraging.Thesetwoaveragingproceduresgivetwolimitsoftheactualsituation.ItappearsthatexperimentalresultsfavoraaredFermisurfaceandthearingisincreasingtowardshigherdoping.Tomodelthis,weassumethattheFermisurfaceischanginglinearlyandtheFermisurfacesusedincases1and2andcase3and4aretwoendpointstothisline.WecalculateotherFermisurfacesbyusingthisequationfortheFermimomentumandtheFermivelocities.Theequationforthislineis, Xn(k)=X1(k)+t 0.15(X2(k))-222(X1(k)].(4) 88

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Figure4-22. NormalizedlineartermasafunctionofaringparametervmaxFc=vaveFc,insetshowsthesamequantitiesinthepresenceof10Tmagneticeld. HereXdenotesthevariousFermimomentaorvelocities,andtistheparametertogeneratetheevolution.Thisisaverycrudeapproach,butwecangainsomeunderstandingaboutthedopingdependencethisway.Fig. 4-21 ,showssomeoftheFermisurfacesalongthisline.Wecalculatetheresiduallinearterminzeroeldandin10Teslaeld,showninFig. 4-22 .WedenevmaxFc=vaveFcasaparameterofaring.Thisisnotamonotonicfunctionofparametert.Thearingparameteris1.49(t=0)fortheFermisurfaceusedincase1and2.44fortheFermisurfaceusedincase3.WeusethesameVshapedorderparameterusedincase1and3.Disorderischosentokeeptheinplanenormalstatelinearterm240WK)]TJ /F8 7.97 Tf 6.59 0 Td[(2cm)]TJ /F8 7.97 Tf 6.58 0 Td[(1.Fig. 4-22 shows 89

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thenormalizedlineartermasfunctionofaring.ThecomparisontoFig. 4-11 isquitecompelling,andsuggeststhatdopingincreasesthearing.ItshowsveryclearlythataaredFermisurfaceisconsistentwithexperimentalresults.AtH=10T,foraredsurfacenormalizedlineartermsarecomparableintwodirections.ThemagneticeldcanbeusedtondthelocationsofnodesontheFermisurface[ 119 120 ].Inthenextsection,weproposeamethodtondthenodalstructureofinsuchthreedimensionalsystems. 4.5ThreeDimensionalFieldRotationThemagneticeldangledependencemeasurementofthethermalconductivityorspecicheatisaverypowerfulmethodoflocatingthenodesinunconventionalsuperconductorsandhasbeenusedextensivelybymanyresearchgroups[ 144 150 ].Recently,angledependentspecicheatmeasurementshavebeenmadebyZengetal.[ 151 ]onFeSe0.45Te0.55,whichhasminimainspecicheatalongthe)]TJ /F4 11.955 Tf 6.77 0 Td[(Mdirectionatlowtemperatures.Theseminimagetinvertedathighertemperatures[ 152 153 ].Hereweproposeageneralizationofthismethodtondthestructureoftheorderparameterbyrotatingthemagneticeldwithrespecttothecrystalaxes.Weconsiderthecaxisthermalconductivityandcalculateitasafunctionofelddirection.Measurementofcaxisthermalconductivityispracticallyverydifcultforageneraldirectionofmagneticeld,butmeasurementofthespecicheatcoefcientatlowtemperatureisarelativelyeasiertask.ForamagneticeldwhichmakesanangleZwiththecaxisandfromaaxis,theDopplerenergyisgivenby, EH(kz,)=1 2RHvabF(kz)[)]TJ /F10 11.955 Tf 11.29 0 Td[(sincosZcos()]TJ /F10 11.955 Tf 11.95 0 Td[()+cossin()]TJ /F10 11.955 Tf 11.96 0 Td[()] (4) +vcF(kz)sinsinZ].Hereisthevortexwindingangleandistheangleofthequasiparticlemomentumfromkxaxis.Forthissection,wetake0,hole=0.3meVand0,electron=1.3meV,whichisappropriateforhighlyoverdopedsystemsatH=10T.Wetaketwoelectronpockets 90

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Figure4-23. Normalizedlineartermasafunctionofmagneticeld'spolarangleZandazimuthalangle. torestorethefourfoldsymmetrywithr=0.9.Theminimumoftheorderparameterislocatedalongthe(0,0)to(,0)and(0,0)to(0,)lines.TheholepocketVshapednodesarerotatedby=4comparetoourpreviouscalculationstobringthenodalVshapesalongtheFe-Febonddirectionintherealspace[ 40 ].Thisrotationdoesnotaffectanyoftheearlierresults.Fig. 4-23 showsthecaxisresiduallineartermasafunctionofpolar(Z)andazimuthalangle()fromthecrystalaxes.Weperformthiscalculationinthecleanlimit,sincedisordertendstosuppresstheanisotropy.=450islocationoftheholesheetnodesintherealspacewithrespecttocrystalaxisa.Thethermalconductivityalongthecaxisdoesnotshowmuchvariationasafunction 91

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Figure4-24. Normalizedspecicheatcoefcientasafunctionofmagneticeld'spolarangleZandazimuthalangle. azimuthalanglebutalongpolaranglethereisaminimumatZ=0o.Wehaveusedincase1.ForacylindricalFermisurface,thedirectionofthenodesclearlyshowupinthemagneticeldrotation;butforthreedimensionalaredFermisurface,themomentumdependenceofthecaxisFermivelocityandthebandmassstronglyaffectsthelinearterm.ThishappensbecausethethermalconductivityisproportionaltothesquareoftheFermivelocity.Wenextconsiderthespecicheatatlowtemperature,whichiscalculatedbyevaluatingthefollowingintegral, C(H,T)=)]TJ /F10 11.955 Tf 10.5 8.09 Td[(1 2Z1d!!2 T2sech2! 2TZdk1 vortex.(4) 92

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Here<>vortexistheaverageoverthevortexunitcell.Fig. 4-24 showsthespecicheatcoefcientC=Tinthezerotemperaturelimitnormalizedtoitsnormalstatevalue.ThisgureshowsaweakereffectoftheanisotropicFermisurfaceandclearlyshowsminimaat=45onearZ=90o.Athighelds,thecontributionsfromtwoelectronpockets,dominatethespecicheat.Onlywithveryhighexperimentalresolutioncanthesesmallfeaturesinthespecicheatbeobserved. 4.6ConclusionInthischapter,weconsideredmodelsforexplainingthethermalconductivitymeasurementsonthe1111and122familiesofpnictides.Forthe1111familycompoundLafePO,wendthatanorderparameterwithaccidentalnodesonelectronpocketscanbeusedtoexplaintheexperimentaldata.Ontheotherhand,weconsiderednodeswiththreedimensionalstructureontheholesheetandfoundthattheVshapednodessuggestedbymicroscopictheoryareconsistentwiththeexperiments,butthestructureoftheFermisurfaceisequallyimportant.Itappearsthattheholepocketsarethreedimensionalandtheychangewithdoping.Wealsoproposeamethodtodetectthethreedimensionalnodalstructurebyusinganangledependentspecicheatandthermalconductivitymeasurement.Althoughthesemethodsarequitedifcultexperimentally,withrecentadvancesinexperimentaltechniquestheymaybepossible.Furtherinvestigationisrequiredtounderstandandusethis3Drotationmethods.Inthenextchapter,Idiscussthepenetrationdepthforthemodelsconsideredinthischapterandcomparethemwithexperimentalmeasurements. 93

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CHAPTER5TRANSPORTII-PENETRATIONDEPTH 5.1IntroductionLowtemperaturepenetrationdepthisanotherbulkprobelikethermalconductivitywhichprovidesinformationaboutlowenergyquasiparticles.Typicallytheabsolutevalueofpenetrationdepth()isnotmeasured,butthechangeinitsvaluefromthezerotemperaturevalue0ismeasuredasafunctionoftemperature[ 154 ].Theoretically,theabsolutevalueofpenetrationdepthdependsonthedetailsofbandstructureforalloccupiedstates,buttherelativechangewithtemperaturedependsonlyonthelowenergyquasiparticlesandisproportionaltoquasiparticledensityofstatesatlowtemperatures(TTc).Forconventionalfullygappedswavesuperconductors,isexponentiallysmallinthelowtemperatureregime.OntheotherhandfordwavesuperconductorsitislinearintemperatureinthecleanlimitandshowsaT2behaviorindirtylimit[ 155 ].Formultibandsystems,likepnictidesorMgB2,experimentsmeasuretheoverallcontributionofallFermisurfacesheets,soanalysisofsuchsystemsismorecomplicated.Inthesesystems,multiplebandscanleadtosomekindofeffectivepowerlawinaspecictemperaturerange.Bandswithlargeisotropicgapdonotaffectthelowtemperature.Thepenetrationdepthisrelatedtothecurrentresponseofthesystemtoanexternaleldas, 1 2=0K(q=0,!=0).(5)Here0isthepermeabilityoffreespaceandKistotalelectromagneticresponseofthesystem,denedas, j(q,!)=K(q,!)A(q,!),(5)whereAistheexternalvectorpotentialandjisthecurrent.Thecurrentoperatorisgivenas, j(q)=eXk,vcyk-q,ck,.(5) 94

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Here y, arethecreationandannihilationoperators,respectively,andvisthequasiparticlevelocity.AgeneralderivationforanysystemisgiveninAppendix B .Forthemodels,weareinterestedinonecanreplacevbytheFermivelocity,sinceonlythelowenergyquasiparticleclosetotheFermisurfacecontributeto.Theelectromagneticresponseiswrittenas, K(q,i)=)]TJ /F4 11.955 Tf 9.3 0 Td[(TX!n,kd2k dk2Tr[(^0+^3)G(k,i!m)]+v2Tr[G(k+q,i!m)G(k,i!m)]TJ /F4 11.955 Tf 11.95 0 Td[(i)].(5)Here!nandaretheMatsubarafrequencies,^0,^aretheidentityandthePaulimatrixrespectively,anddenotesthedirectionofthecurrentassumingthattheresponsetensorisdiagonal.GinEq. 5 isimpuritydressedGreen'sfunctionwithintheT-Matrixapproximation.Formultibandsystems,thetotalpenetrationdepthisissumofcontributionsfromeachband, 1 2=Xi1 2i.(5)Foralayeredsystem,afterevaluatingtheexpressiongivenbyEq. 5 andtakingtheanalyticcontinuationtorealfrequencies,wendforthepenetrationdepth, 1 2=n0e2 d~2XiZ10d! *tanh! 2Tmi"(viF.^)2Im ~2i (~2i)]TJ /F10 11.955 Tf 12.67 0 Td[(~!2i)3 2! (5) )]TJ /F10 11.955 Tf 21.65 0 Td[(2(viF.^vi.^)Im ~i~!i (~2i)]TJ /F10 11.955 Tf 12.67 0 Td[(~!2i)3=2!#+,kz.Thisexpressionincludesselfenergycorrectionsduetodisorderbutnotthevertexcorrections.Thesehavebeenshowntovanishforisotropicscatterersandaspinsingletorderparameter[ 82 ],andarethereforenotexpectedtobeimportanthere.InEq. 5 ,isthedirectionofthecurrent,iisthebandindex,miisthebandmass,nisnumberoflayers,disthecaxisunitcelllengthandvisthegapslopedk=dk.InthelimitTcEF,thecontributionofthesecondterminEq. 5 canbeignored.Weareinterestedintherelativechangeofthepenetrationdepthandnotinthezero 95

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temperaturevalueof0,whichrequiresanintegrationovertheentireBrillouinzone,whichisnumericallyveryexpensivetask.Inthenextsection,Iconsideramodelfor1111systems. 5.2Modelfor1111SystemsThelowtemperaturepenetrationdepthresultsarenotuniversalfortheRFeAsO1-xFx1111family.ForR=Sm,Prbasedmaterialsanexponentialbehaviorhasbeenreported,whichisanindicationoffullygappedsystem[ 156 157 ].Ontheotherhand,ForR=La,Nd,annonexponentialpowerlawTnhasbeenfound,withn2[ 43 ].InanothermaterialLaFePO,alinearpenetrationdepthismeasured[ 88 103 ],whichisaclearevidenceoflinenodes.Fromthetheoreticalpointofview,theisotropicswavemodelwithstronginterbanddisorderhasbeenproposedbyVorontsovetal.[ 158 ].ThismodelcangivepowerlawsTnwithn>1,butcannotexplainthelinearbehaviorobservedinLaFePO.Itisalsoinconsistentwiththecaxistransportobservedforthe122family.Anorderparameterwithastatewithdeepminimaisqualitativelysimilartothesmodel,becauseitalsohasagap.Hereweconsiderastatewithaccidentalnodesontheelectronpocketandanisotropicgapontheholepocket,asdiscussedinSec. 4.3 .Fortheorderparameter,wetakehole=)]TJ /F10 11.955 Tf 9.3 0 Td[(1.1Tc0fortheholesheetandelectron=1.5Tc0(1+1.3cos2)fortheelectronsheet.Firstwelookattheeffectofpureintrabandscattering,showninFig. 5-1 .Forsimplicity,wetakeequalDOSandFermivelocityforboththebands.Inthecleanlimit,isnotlinearbuthasaslightcurvature.ApowerlawtgivesT1.19,andthishappensduetoachangeofslopecausedbythelowenergyscaleani)]TJ /F10 11.955 Tf 12.46 0 Td[(iso.TheadditionofalittledisordermakesthecurvemorelinearandthepowerlawbecomesT1.04;asthedisorderincreasesnodesdisappearandweseeanexponentialbehavioratverylowtemperatureasanindicationoffullgap,withanaeffectivepowerlawofT1.77.Wegiveexponentsherewhichtoveranitelowtemperaturerange(0)]TJ /F10 11.955 Tf 12.86 0 Td[(0.2Tc),notbecausetheyareinanysenseuniversal,buttocomparewiththeexponentswhichhavebeenreportedbyexperiments 96

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Figure5-1. Normalizedchangeinpenetrationdepthasafunctionofnormalizedtemperature.0iszerotemperaturepenetrationdepthandTcisthecriticaltemperature.)]TJ /F1 11.955 Tf 10.1 0 Td[(isimpuritylifetime,measuredincleanlimitcriticaltemperature.Theorderparametersare1=-1.1Tc0and2=1.5Tc0(1+1.3cos2). overthisrange.Onfurtherincreasingdisorder,theeffectivepowerincreasesandfor)-382(=1.5Tc0,thepowerbecomes2.6.Nextweconsiderthestatewithdeepminimaontheelectronpocket.Forthiscaseweusethesameorderparameterfortheholebandasinnodalcase,butforelectronpocketwetakeelectron=1.5Tc0(1+0.8cos2).Fig. 5-2 showsresultsforthisanisotropicsstatewithdeepminima.Forthiscase,inthecleanlimitwendaneffectivepowerlawT2.59inthelowtemperaturerange.Theadditionofpureintrabanddisorderincreasesthispowerto3.34,becausepureintrabandscatteringlowerstheanisotropiccomponentoforderparameterandincreasesthe 97

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Figure5-2. Normalizedchangeinpenetrationdepthasafunctionofnormalizedtemperaturefortheanisotropicswavestatewithdeepminima.0iszerotemperaturepenetrationdepthandTcisthecriticaltemperature.)]TJ /F1 11.955 Tf 6.77 0 Td[(=0.1Tc0forallcasesandlegendsshowthestrengthofinterbandscatteringwithpowerlawsforeachcurve.Theorderparametersare1=-1.1Tc0and2=1.3Tc0(1+0.8cos2). deepminimumenergyscale.Moreinterestingfeaturescomewithinterbandimpurityscattering.Withaninterbandscatteringwhichis90%oftheintrabandscattering,weseeareductionintheexponent.Thetemperatureexponentgoesto2.14from2.59ofthepureintrabandscatteringlimit.ThishappensduetotheformationofimpuritymidgapstateawayfromtheFermilevelandthismakesthegapinDOSverysmall.Intheisotropicscatteringlimit,whenboththeinterbandandintrabandscatteringsareequal,thenthispoweris2.46,whichissmallerthanthecleancaseandslightlylargerthanthe 98

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weakinterbandscatteringcase.TheproblemwithinterbandscatteringisthatitcausesastrongsuppressionofcriticaltemperatureasshowninFig. 4-7 ,andincontradictiontoexperiments.Inthisisotropicscatteringlimit,theimpuritybandisformedontheFermilevelwhichcausesasmallerpowerthanthecleancase.Nextweconsiderthe122systems. 5.3Modelfor122SystemsPnictidesbasedonBaFe2As2havebeenstudiedextensivelybymanygroups.Boththeelectronandholedopedsystemsshowpowerlawsforinplanepenetrationdepthinthelowtemperatureregimewithpowern'2[ 41 44 86 87 159 ].AnotherexperimentbyKimetal.[ 160 ]onelectrondopedsystemswithirradiationwithheavyionsshowspowerlawsintherange2.2to2.8forthein-planepenetrationdepth,withadecreaseinexponentwithincreasingTcsuppression[ 160 ].TheKimetal.[ 160 ]experimentwasexplainedwithanisotropicswavemodelassuminginterbandscattering.Ontheotherhand,recenttransportmeasurementalongthecaxishavecontradictedthisisotropicswavemodel,whichwillgivesamenormalizedbehaviorinbothaandcdirectionsincontradictiontoexperiments.PenetrationdepthmeasurementsalongthecaxisforNidopedsystemsfoundlinearbehaviorandquadraticbehavioralongtheplanes[ 44 ].ThiswouldbeimpossibleifthesepowerlawsextendedtoarbitrarilylowT,andthereforesuggestsamorecomplicatedinterplayofgapstructureandmultibandphysics.ArecentstudyofthermalconductivityforCodopedsystemsbyReidetal.hasalsoprovidedstrongevidenceforthreedimensionalnodalstructure[ 42 ].Motivatedbytheseexperiments,weconsiderathreedimensionalmodelfor122systems,whichwasintroducedinSec. 4.4 .WeconsiderallvecasesconsideredinSec. 4.4 andlookatbothinplaneandcaxispenetrationdepths.TheparametersofthemodelsarelistedinTable 4-1 .Firstweconsidercases1and2,whichhaveFermisurfaceswithweakcaxisdispersion.Case1hasVshapednodes,whilecase2hashorizontallinenodes.We 99

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Figure5-3. ResidualdensityofstatesN(!=0)normalizedtothetotalnormalstatedensityofstateattheFermienergyN0vs.totalunitaryintrabandscatteringrateparameter)]TJ /F1 11.955 Tf 10.1 0 Td[(inmeVforcases1to5,respectively. 100

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Figure5-4. RelativechangeinpenetrationdepthinnmasafunctionoftemperatureTinKelvinforcase1panelsa,bandforcase2panelsc,d.Solidlinesshowsforinplanedirectionanddashedlineshowthecaxis.Bluelinesdenotesthecleansystemandredlinesshowsdirtysystemwith)]TJ /F1 11.955 Tf 6.77 0 Td[(=0.9)]TJ /F7 7.97 Tf 22.82 -1.79 Td[(NL,where)]TJ /F7 7.97 Tf 6.78 -1.8 Td[(NListhecriticalvalueofdisordertoliftthenodes.Thepowerlawsareindicatedforinplanedirection. considertwolimitingcases,thecleanlimitandthedirtylimit.Inthdirtylimitthetotalnormalstatescatteringrateis0.9)]TJ /F7 7.97 Tf 22.82 -1.8 Td[(NL,where)]TJ /F7 7.97 Tf 6.77 -1.8 Td[(NListhecriticaldisorderwhichremovesthenodes(SeeFig. 5-3 ).Fig. 5-4 showsthepenetrationdepthchangeforcases1and2.Inthecleanlimit,boththecasesgivepowerlawvariationwithpower2.08forthein-planepenetrationdepth,butthecaxisisverylinear.Withtheadditionofpureintrabandimpurityscattering,theexponentincreasesfortheab-planeandthecaxisalsodeviatesfromlinearbehaviorton>1powerlawforVshapednodes.Forthe 101

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Figure5-5. RelativechangeinpenetrationdepthinnmasafunctionoftemperatureTinKelvinforcase3panelsa,bandforcase4panelsc,d.Solidlinesshowsforinplanedirectionanddashedlineshowthecaxis.Bluelinesdenotesthecleansystemandredlinesshowsdirtysystemwith)]TJ /F1 11.955 Tf 6.77 0 Td[(=0.9)]TJ /F7 7.97 Tf 22.82 -1.8 Td[(NL,where)]TJ /F7 7.97 Tf 6.78 -1.79 Td[(NListhecriticalvalueofdisordertoliftthenodes.Thepowerlawsareindicatedforinplanedirection. horizontalnodes,thecaxisremainslinearexceptatverylowtemperatures.Cases3and4areshowninFig. 5-5 andcase5isshowninFig. 5-6 .Othercasesalsoshowqualitativelysimilarbehaviortocases1and2.Forallcases,theelectronpocketshavesameorderparameterandtwoelectronpocketsstartmakingsignicantcontributionatverylowtemperatures.Allcasesgivequalitativelydifferentbehaviorforinplaneandcaxispenetrationdepths,similartotheexperimentaldataonNidopedBaFe2As2.Suchbehaviorcannotbeobtainedforanisotropicswavestate.Atlowtemperaturesthe 102

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Figure5-6. RelativechangeinpenetrationdepthinnmasafunctionoftemperatureTinKelvinforcase5panelaandb.Solidlinesshowsforinplanedirectionanddashedlineshowthecaxis.Bluelinesdenotesthecleansystemandredlinesshowsdirtysystemwith)]TJ /F1 11.955 Tf 6.77 0 Td[(=0.9)]TJ /F7 7.97 Tf 22.81 -1.79 Td[(NL,where)]TJ /F7 7.97 Tf 6.77 -1.79 Td[(NListhecriticalvalueofdisordertoliftthenodes.Thepowerlawsareindicatedforinplanedirection. leadingordertemperaturedependenceisgivenby, /Z10d!sech2! 2T*(vF^)2Im"~2 (~!2)]TJ /F10 11.955 Tf 13 2.65 Td[(~2)3=2#+FS.(5)Here<>FSistheaverageovertheFermisurface.~and~!arethegapandenergyrenormalizedbythedisorderrespectively.Therenormalizationduetothedisorderdoesnotdependonthemomentum,hencewecanrewriteEq. 5 /F(T)(vF^)2FS, (5) F(T)=Z10d!sech2! 2TIm"~2 (~!2)]TJ /F10 11.955 Tf 13 2.65 Td[(~2)3=2#. (5) Soforisotropicsorderparameters,thetemperaturedependenceisthesameinalldirections.Foranothercompoundof122family,PdopedBaFe2As2,linearTbehaviorhasbeenreportedintheplane[ 123 ].MicroscopiccalculationshavefoundanextremelyisotropicstateinthismaterialandVshapednodesontheholesheet[ 48 ],similartothenodalstructurefoundbyGraseretal.forholedopedBaFe2As2[ 40 ].Insuchasystem,electronpocketwillnotaffectthelowTpenetrationdepthanditwillbepurely 103

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Figure5-7. PenetrationdepthanisotropyforallcasesincleananddirtylimitasafunctionoftemperatureinKelvin. determinedbynodalholesheetwithlinearbehavior.QualitativelythissystemwillbeequivalenttonodalmodelconsideredinSec. 5.2 for1111family.Wenowinvestigateanotherimportantquantity,whichispenetrationdepthanisotropydenedas, =c ab.(5)Hereinvolvesknowledgeofabsolutevaluesofpenetrationdepth,whichrequirefullinformationofthebandstructure.Fig. 5-7 showsthepenetrationdepthanisotropyforallcasesincleananddirtylimit.TheT=0valuesofpenetrationdepthcalculatedforourmodelarelistedinTable 5-1 .Thepenetrationdepthanisotropyisroughlyconstant 104

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Table5-1. Zerotemperaturepenetrationdepth0forallcases. Caseab(nm)c(nm) 1,22188363,42187265218825 overtheentirelowtemperaturerangeiscloseto4.Itisroughlyclosetotherootmeansquareratioofthein-planeandthecaxisFermivelocities. 5.4ConclusionWeconsideredmodelsintroducedinchapter 4 forthe1111andthe122familiesforcalculatingthelowtemperaturepenetrationdepth.WefoundthattheorderparameterwithaccidentalnodescangivelinearpenetrationdepthasobservedinLaFePOandBaFe2(As1-xPx)2.Disorderinthissystem,togatherwiththemixtureofdifferentdensitiesofquasiparticlesondifferentbands,canleadtoeffectivepowerlawsobservedfordifferentmaterialsinthe1111family.Wealsoinvestigatedamodelofananisotropicsstate,whichcangivepowerlawclosetoquadratic.Additionofpureintrabanddisorderincreasestheexponent,whileadditionofniteinterbanddisorderreducestheexponent.SimilarresultswereobtainedbyKimetal.usingisotropicswavestateinthecontextofelectrondopedBaFe2As2[ 160 ],butthisexplainationappearstoberuledoutbythecaxispenetrationdepth.For122family,westudygapswiththreedimensionalstructureandndqualitativelydifferentbehaviorforinplaneandoutofplanepenetrationdepth. 105

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CHAPTER6SUPERCONDUCTINGFLUCTUATIONSANDFERMISURFACE 6.1IntroductionandReviewARPESmeasurementsonunderdopedBi2212cuprateshaverevealeddisconnectedarclikeFermisurfacescenteredaroundthenodaldirectionsinsteadofclosedFermisurfacesheets,whichareexpectedfornormalmetals[ 16 18 ].SimilarfeatureswerealsofoundinYBCOsamplesbyin-situARPESmeasurements[ 161 ].Ontheotherhand,quantumoscillationexperimentsonYBCOhavesuggestedsmallpocket-likeFermisurfacesinhighelds[ 162 166 ].Recently,Menget.al.havereportedtheexistenceofFermisurfacepocketsinBi2201,butwhetheritsoriginisthestructuralsupermodulationornot,isaissueofdebate[ 167 169 ].Amongmanyothertheoreticalproposals,Pereg-Barneaet.al.haveshownthatthequantumoscillationexperimentscanbeexplainedwithaphenomenologicalmodelofFermiarcswithinasemiclassicaltreatmentofmagneticeld[ 170 ].TheformationofFermiarcisassociatedwithalinearintemperaturequasiparticlelifetimeabovethetruephasecoherencetemperature[ 171 ].Withinthismodel,thenormalstateisassumedtobeameanelddwavesuperconductor,butinelasticscatteringduetolackofphasecoherenceleadstoformationofarcs.Inthischapter,wetreatthisproblemfromadifferentpointofview.Westartwithanormalstatewithoutanysuperconductingorderparameterandlookathowtheorderparameteructuationaffectstheelectronicstructure.Weconsideranphenomenologicalmodeltogetadwavesuperconductinggroundstateandstudytheeffectofuctuationabovemeaneldtransitiontemperatureinthelongwavelengthlimit.FormationofCooperpairsaboveTcleadstoareductionofthenormalstateDOS.ThisresultwasrstobtainedbyAbrahamset.al.[ 172 ]inthecontextoftunnelingexperimentsongranularAl,whereDOSsimilartoagaplesssuperconductorisobservedmuchabove 106

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thecriticaltemperature[ 173 ].HerewefocusontheuctuationeffectontheFermisurfaceinasimilarframework,butforthed-wavecase. 6.2FormalismAsimplemodelHamiltoniantogetd-wavesuperconductivityis, H=Xk,kcykck)]TJ /F4 11.955 Tf 11.95 0 Td[(gXk,k0,q(k)(k0)cyk+q"cy)]TJ /F21 7.97 Tf 6.58 0 Td[(k#c)]TJ /F21 7.97 Tf 6.58 0 Td[(k0#ck0+q".(6)Heregisaeffectivecouplingconstantand(k)isafactortogetthed-wavesymmetry,whichinagasmodeliscos(2)andonalatticeis(cos(kx))]TJ /F10 11.955 Tf 13.19 0 Td[(cos(ky))=2.SuchaHamiltoniancanbederivedfromthemeaneldtheoryofthet)]TJ /F4 11.955 Tf 12.82 0 Td[(Jmodel.kisthefermionenergywithrespecttotheFermienergy.Inthemeaneldtheory,onerestrictsoneselftotheq!0limit,i.e.pairsonlytime-reversedstates,andgetsthefollowingresultforthesuperconductingtransitiontemperatureforad-wavesuperconductor, kBTc=1.13!ce)]TJ /F19 5.978 Tf 13.73 3.26 Td[(2 gN0,(6)whereN0isthedensityofstates(DOS)attheFermisurfaceand!cisthecutoffenergy,whichdependsonthemicroscopicdetailsofthepairingmechanism.MostofthephysicalquantitiesmeasuredinexperimentsaremeasuredinunitsofTc,henceknowledgeof!cisnotnecessary.Theorderparameterinthesuperconductingstateisdenedas, (k)=)]TJ /F4 11.955 Tf 9.3 0 Td[(g(k)Xk0(k0)hcyk"cy)]TJ /F21 7.97 Tf 6.59 0 Td[(k0#i=0(k).(6)Inthenormalstate,thisanomalousexpectationvalueiszero,whichafterthesymmetrybreakingbelowTcbecomesnite.Butinthenormalstateuctuationsdoexist.Togetsomequalitativeunderstanding,wedothemeanelddecompositionofinteractiontermintheHamiltonian 6 ,whichgives, Hint=Xk,q(k)cyk+q"cy)]TJ /F21 7.97 Tf 6.59 0 Td[(k#0(q!0)+h.c. (6) 107

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Figure6-1. Left:Lowestorderselfenergydiagramduetosuperconductinguctuations.Right:Diagramscontributingtothevertexfunction Nowtostudytheuctuations,wesubstitute0(q)!MF0+bqandusethefactthatinthenormalstateMF0=0,wegetaneffectiveinteractionfortheelectronsandthebosonicsuperconductinguctuationeldbq. He=Xk,q(k+q)(k)cyk+q"cy)]TJ /F21 7.97 Tf 6.59 0 Td[(k#bq+h.c.(6) He=Xk,q(k)cyk+q"cy)]TJ /F21 7.97 Tf 6.58 0 Td[(k#bq+h.c.(6)Theuctuationeldisdenedas, bq=Xk)]TJ /F4 11.955 Tf 9.3 0 Td[(g(k)c)]TJ /F21 7.97 Tf 6.58 0 Td[(k#ck+q".(6)FromtheeffectiveuctuationfermionHamiltonianwecanseetwothings.First,theuctuationfermionvertexismomentumdependentandcomeswithafactorof(k),wherekisthefermionmomentum.Thistellsus,thattheselfenergyisgoingtohaveamomentumdependencecorrespondingtothesquareoftheorderparametersymmetry.ThisisillustratedinFigure 6-1 .Thisisconsistentwiththeangledependenceofscatteringrateangularmagnetoresistanceoscillation(AMRO)experimentsdoneonthecuprates[ 174 175 ],wherethescatteringrateisaminimumalongthenodaldirectionandmaximumalongtheantinodaldirectionsandthisanisotropicscatteringratehasastrongcorrelationwithTc,whichsuggeststhatitisrelatedtosuperconductivity[ 176 ].ThesecondthingthatwegetfromEq. 6 isthattheuctuationpropagatorisrelated 108

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tothekdependentpartofthefullvertexfunction.ThisisalsoconsistentwiththeearliertheorybyLarkin[ 177 ]andNarozhny[ 178 ].Toevaluatetheuctuationpropagator,ifweincludeonlytheladderdiagramsintheBethe-Salpeterequationforthevertexfunction,itcanbewrittenasaseriesofbubblesasshowningure 6-1 .Thesumofthebubblescanbewrittenas, \(k,k0,q)=(k)(k0) )]TJ /F4 11.955 Tf 9.3 0 Td[(g)]TJ /F8 7.97 Tf 6.59 0 Td[(1+P(q,i)=(k)(k0)L(q,i).(6)Thisequation 6 issimilartothevertexfunctionobtainedbyNarozhny[ 178 ].Thed-waveuctuationpropagatorisLandPis P(q,i)=TXk,!n(k)(k)G(k+q,i!n+i)G()]TJ /F11 11.955 Tf 9.3 0 Td[(k,)]TJ /F4 11.955 Tf 9.3 0 Td[(i!n),(6)where!nisthefermionicMatsubarafrequencyandisthebosonicMatsubarafrequency.GisthenormalstateGreen'sfunctiondenedas G(k,i!n)=1 i!n)]TJ /F6 11.955 Tf 11.95 0 Td[(k.(6)Theuctuationpropagatorcanbewrittenas L)]TJ /F8 7.97 Tf 6.59 0 Td[(1(q,)=)]TJ /F4 11.955 Tf 9.3 0 Td[(g)]TJ /F8 7.97 Tf 6.59 0 Td[(1+N0 21 2+!c 2T+ 4T (6) )]TJ /F10 11.955 Tf 20.45 0 Td[(1 2+ 4T+(vFq)2 2D(4T)2001 2+ 4T,whereDisthedimension,vFistheFermivelocity,istheDigammafunctionand00isthesecondderivativeoftheDigammafunction.Inderivingthisexpression,weassumedthatqvF<<2T.Thepoleoftheuctuationpropagatorinthelimitq!0givesthemeaneldtransitiontemperatureTc.Theretardedpropagatorforsmallcanbeobtainedbydoingananalyticcontinuation(i!+i0+)andaTaylor'sseriesexpansionoftheDigammafunctionsandusingEq. 6 toeliminateg.Weobtain L(q,)=2=N0 i 8T)]TJ /F10 11.955 Tf 11.96 0 Td[(log(T=Tc))]TJ /F10 11.955 Tf 11.96 0 Td[(2q2v2F T2.(6) 109

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Table6-1. Valueofindifferentsuperconductors. Material Pb10)]TJ /F8 7.97 Tf 6.59 0 Td[(6Nb3Sn10)]TJ /F8 7.97 Tf 6.59 0 Td[(4Tl2Ba2CuO6+x0.3)]TJ /F10 11.955 Tf 11.96 0 Td[(0.5Bi22120.7)]TJ /F10 11.955 Tf 11.96 0 Td[(1.0YBCO0.7)]TJ /F10 11.955 Tf 11.96 0 Td[(1.4 Here2is7(3) 162D,where(3)isthezetafunction.WeareinterestedintheselfenergyneartheFermisurface; (kF,!,T)=2(^kF)(!,T) (6) (!,T)=)]TJ /F4 11.955 Tf 9.3 0 Td[(TXZdq (2)DL(q,i)G()]TJ /F16 11.955 Tf 9.29 0 Td[(k+q,)]TJ /F4 11.955 Tf 9.29 0 Td[(i!+i). (6) ThiscanberewrittenafterperformingtheMatsubarasummationas (!,T)=)]TJ /F15 11.955 Tf 11.3 16.27 Td[(Z10dx Zdq (2)DL00(q,x))]TJ /F4 11.955 Tf 9.3 0 Td[(xtanh()]TJ /F8 7.97 Tf 6.59 0 Td[(k+q=2T)+(!+)]TJ /F8 7.97 Tf 6.59 0 Td[(k+q)coth(x=2T) x2)]TJ /F10 11.955 Tf 11.96 0 Td[((!+)]TJ /F8 7.97 Tf 6.59 0 Td[(k+q)2.(6)TounderstandtheeffectoflongwavelengthuctuationsontheFermisurface,whicharemostimportant,wecandroptheqdependenceoftheelectronGreen'sfunctionandselfenergyreducesto (!,T)=)]TJ /F15 11.955 Tf 11.29 16.28 Td[(Z10dx Zdq (2)DL00(q,x)(!)coth(x=2T) x2)]TJ /F10 11.955 Tf 11.96 0 Td[((!)2,(6)wheretheimaginarypartoftheuctuationpropagatorL00 L00(q,x)=)]TJ /F10 11.955 Tf 9.3 0 Td[(2N)]TJ /F8 7.97 Tf 6.59 0 Td[(10x=8T (=8T)2x2+("+20v2F=T2q2)2,(6)where"islog(T=Tc).WecanrewriteEq. 6 afterintegratingoutthemomentumas, T,T=T2 22N0v2F2Z10dy 2)]TJ /F10 11.955 Tf 11.96 0 Td[(tan)]TJ /F8 7.97 Tf 6.59 0 Td[(1"8T !(!=T)coth(y=2) y2)]TJ /F10 11.955 Tf 11.96 0 Td[((!=T)2.(6)ThedenominatorinEq. 6 hasdimensionsoftemperature,henceweintroducea 110

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temperaturescaleT, T=22N0v2F=7(3) 16DN0v2F.(6)ForalayeredtwodimensionalsystemN0v2Fissd=m,wheredistheinterlayerdistance.UsingthiswerewriteTas, T=7(3) 16Dsd m=Asd 4m,(6)whereAis1.1058.Asimilaranalysisforthreedimensionsgives, T=As` 4m,(6)withA=1.6391and`=p vF=0.EmeryandKivelsonproposedasimilartemperaturescaleknownasthephaseorderingtemperaturewithA=0.9for2DandA=2.2foranisotropic3Dsystem.Thiswecanrewritetheselfenergyas =Tc=T2 2T2cZ10dy 2)]TJ /F10 11.955 Tf 11.96 .01 Td[(tan)]TJ /F8 7.97 Tf 6.58 0 Td[(1"8 y(!=T)coth(y=2) y2)]TJ /F10 11.955 Tf 11.96 0 Td[((!=T)2,(6)whereisTc=T.ThedimensionlessparameterforsomesuperconductorsislistedinTable 6-1 .Theimaginarypartoftheselfenergycanbecalculatedanalyticallyandisgivenas, 00(!=T,T) Tc=)]TJ /F6 11.955 Tf 9.3 0 Td[(T2 T2c 2)]TJ /F10 11.955 Tf 11.96 0 Td[(tan)]TJ /F8 7.97 Tf 6.59 0 Td[(1"8T !coth(!=2T) 4.(6)TherealpartcomesfromtheprincipalvalueoftheintegralinEq. 6 .Innextsection,Idiscusstheeffectofuctuationsontheone-electronspectralfunction. 6.3SpectralFunctionThespectralfunctionisrelatedtotheimaginarypartoftheretardedGreen'sfunctionandreadsas, A(k,!)=)]TJ /F10 11.955 Tf 10.89 8.09 Td[(1 ImG(k,!).(6)Innormalmetals,thespectralfunctionispeakedattheFermienergyandinthesuperconductingstatethepeakshiftstothegapenergyscale.TotracktheFermiarc, 111

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Figure6-2. FermiarcsaboveTcfor=1.5forvarioustemperatures. welookatthepeakpositionofspectralfunctionusingtheARPEScriterion,wherehalfofpeakvalueismeasuredfromareferenceFermilevel.Fig. 6-2 shows,thespectralgapdeterminedusingtheARPEScriterionfordifferenttemperaturesaboveTc.Thevalueofistakentobe1.5,whichiscomparableforslightlyunderdopedBi2212samples.TheFermiarcistheangularregionaroundthenodewherethespectralgapiszero.Nextwelookatthetemperaturedependenceofthearclength,whichisshowninFig. 6-3 ,withexperimentaldatafromKanigelet.al.[ 17 ].TheexperimentsclaimauniversaltemperaturedependencewithpointlikeFermisurfaceforsampleswithTcofzeroKelvin,andthearcclosingtemperatureismuchhigherthanmostofthepseudogaptemperaturescalescomingfromotherscales.Pured-waveuctuationsseemtogiveaqualitativelygoodtfortheweaklyunderdopedcase,butcannotaccountforthebehaviorofstronglyunderdopedsamples.WecalculatethevalueoffordifferentdopingsusingthedatafromStoreyet.al.[ 179 ],andconstructaphasediagramforthe 112

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Figure6-3. ArclengthasafunctionoftemperaturenormalizedtoarcclosingtemperatureTF.TheexperimentaldataistakenfromKangielet.al.[ 17 ].Theendpointofthegreencurveis1.05Tc,thearclengthgoestozeroatTcandwithinourmodel,wedonotconsiderthesuperconductingstate.Thecurveisfor=1.5. arcclosingtemperature(TF)frompured-waveuctuations.Fig. 6-4 showsthedopingdependenceofarclengthclosingtemperaturewithinthemodelconsideredhere.Thereisalsosomeevidencefortheexistenceofhigherharmonictermsinthegapfunction,whichcouldaffectthetemperaturedependenceofarc,becausetheuctuationselfenergycarriesthefullinformationofthegapstructure.Tomodelthis,weconsiderasimplemodelgapfunctionwithd-wavesymmetry,writtenas, ()=0(cos2+B6cos6),(6)hereB6isaparametertocontrolthestrengthofthehigherharmonicterm.InFig. 6-5 ,weplottheFermiarclengthasafunctionoftemperaturenormalizedtothearcclosing 113

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Figure6-4. DopingdependenceofarclengthclosingtemperatureTFwithtemperaturescaleTc(from[ 179 ]),pseudogaptemperatureT(from[ 180 ])andNernsteffect'sonsettemperatureT(from[ 22 ]). temperature.Thisclearlyshowsthatthearclengthclosingstronglydependsonthegapfunctionsymmetry.Inthenextsection,Idiscusstheeffectofmagneticuctuations. 6.4FluctuationinDirtyLimitInthepresenceofimpurities,theimpuritylifetimeentersthesingleparticleGreen'sfunctionandfermionuctuationvertexgetrenormalized.ThenormalstateelectronGreen'sfunctionbecomes, G(k,i!)=1 i!+i1 2sgn!)]TJ /F6 11.955 Tf 11.95 0 Td[(k,(6)whereisthesingleparticlelifetime.Therenormalizedvertexwithintheladderapproximationiswrittenas, (q,~!1,~!2)=j~!1)]TJ /F10 11.955 Tf 12.67 0 Td[(~!2j j!1)]TJ /F6 11.955 Tf 11.95 0 Td[(!2j+()]TJ /F10 11.955 Tf 10.02 0 Td[(~!1~!2)D=j~!1)]TJ /F10 11.955 Tf 12.68 0 Td[(~!2j3.(6) 114

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Figure6-5. Arclengthdependenceontemperature,forgapfunctionwithhigherharmonicterm. HereDisthediffusionconstant(v2F=2)and(x)istheHeavysidethetafunctionwithintheapproximationj(~!1)]TJ /F10 11.955 Tf 13.21 0 Td[(~!2)jmax(T,)]TJ /F8 7.97 Tf 6.59 0 Td[(1)and~!is!+1=2sgn!.Theuctuationpropagatortakestheform, L(q,)=2=N0 i 8T)]TJ /F10 11.955 Tf 11.95 0 Td[(log(T=Tc))]TJ /F10 11.955 Tf 11.96 0 Td[(2q2v2F T,(6)whereTcisthecriticaltemperatureforthedirtysystem.Theselfenergyiscalculatedtolowestorderincludingthevertexcorrection,andforthedirtylimitthiscanbecalculatedinclosedformbytakingonlytheleadingcontributionsintheMatsubarasum.ItcomesouttobeexactlythesameasoriginallycalculatedbyAbrahamet.al.[ 172 ].Theonlydifferenceduetod-wavesymmetrycomesinthemomentumdependenceofselfenergy.Inthecleanlimitquantumuctuationsdominate,butinthedirtycaseclassical 115

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Figure6-6. Fermiarclengthasafunctionofdisorder,)]TJ /F8 7.97 Tf 6.59 0 Td[(1isthesingleparticlescatteringtimeandTcisthecriticaltemperatureforthecorrespondingsystem. uctuationsplaydominantrole.Theuctuationselfenergycanbewrittenas, (!)=~!2 i~!+kZ10qdq 21 "+2q2v2F T1 h8T" +28q2v2F +2!+q2v2F 2i2.(6)Here!isthefermionicMatsubarafrequency.Wecanintegrateoutq,andontheFermisurfacetheselfenergybecomes, (!)=)]TJ /F4 11.955 Tf 9.3 0 Td[(i~!2 TN0v2F24log2(16"T+4!) e"((16T2+T) (42!)]TJ /F6 11.955 Tf 11.96 0 Td[("T)2+162"T+"T (42!)]TJ /F6 11.955 Tf 11.96 0 Td[("T)2(16"T+4!)35.(6)Tocomparewithcleanlimitbehavior,IreplaceNv2FinEq. 6 with4T=1.1.Fig. 6-6 showstheeffectofdisorderontheFermiarcsfor=1.5.WecanclearlyseethatthearcclosingtemperatureTFstillcanbemuchhigherthanTcfordirtiersystems,althoughthetemperaturedependenceisqualitativelydifferentfromtheexperimental 116

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observations.AnotherpointtonotehereisTcgoingtozerolimitofthecurve,whichappearstobenite.Inthecontextofcuprates,disorderdoesn'tseemtobeveryimportantbecausethecoherencelength0ismuchshorterthanmeanfreepath`,whichisequivalenttothecondition)]TJ /F8 7.97 Tf 6.59 0 Td[(1Tc. 6.5EffectofMagneticFluctuationsTheproximitytoantiferromagnetismmakestheroleofantiferromagneticuctuationsimportantinnormalandsuperconductingstatesofcuprates,althoughthereisnoclearevidenceofcoexistingmagneticandsuperconductinggroundstatesinholedopedcuprates.Magneticuctuationswillscatterfermionsandcannotbeignored.Wetreatthemagneticuctuationsinasimilarwayasthesuperconductinguctuations.TheSDWuctuationpropagatorortheeffectiveantiferromagneticspinuctuationpotentialiswrittenas[ 181 182 ], LSDW(q,)=2=N0 i 8T)]TJ /F10 11.955 Tf 11.96 0 Td[(log(T=TSDW))]TJ /F10 11.955 Tf 11.95 0 Td[(2q2~v2F T2,(6)whereqisneartheordervectorQ,~visthevelocityatk=kF+QandthemeanSDWtransitiontemperatureTSDWisgivenas, TSDW=1.13EFe)]TJ /F8 7.97 Tf 6.59 0 Td[(1=N0U.(6)HereUistheeffectiveelectronelectroninteraction.Thelowestorderselfenergyiscalculatedinasimilarwayasinlastsectionbutinthissectionweconsideratightbindingdispersion, k=)]TJ /F10 11.955 Tf 9.3 0 Td[(2t(coskx+cosky)+1.6tcoskxcosky+1.1t.(6)Wefocusthequalitativeaspectofco-existingmagneticuctuation,henceusethesamepropagatorasintheelectrongasmodel.Theorderingvectorthatweconsideris(,).Atightbindingdispersionwillgiveslightlydifferentprefactorforqintheuctuationpropagator,butthequalitativeformwillremainsameforsmallquctuations.Intheself 117

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energycalculation,weassumethatqkF+Qandsett=1.ThesuperconductingtransitiontemperatureTcistakenas0.1tandSDWtransitiontemperatureistakentobesmallerthanTc(TSDW=0.9TC).Itisassumedthatsuperconductivityistheleadinginstabilityandsystemrstgoesthroughasuperconductingtransition.Wedonotaddressthequestionofwhethertherewillbeaco-existingphaseofmagneticandsuperconductingorderbelowTc,becauseweareprimarilyinterestedinthetemperaturerangeaboveTc.Theselfenergyiswrittenas, (k,!,T)SDW=)]TJ /F15 11.955 Tf 11.29 16.28 Td[(Z10dx Zdq (2)DL00SDW(q,x) (6) )]TJ /F4 11.955 Tf 9.3 0 Td[(xtanh(k+Q+q=2T)+(!+k+Q+q)coth(x=2T) x2)]TJ /F10 11.955 Tf 11.96 0 Td[((!+k+Q+q)2.Wealsocalculatetheselfenergyforthesuperconductingstatewithtight-bindingdispersionandthefullGreen'sfunctionis, G(k,!)=[!)]TJ /F6 11.955 Tf 11.95 0 Td[(k)]TJ /F10 11.955 Tf 11.96 0 Td[(SC)]TJ /F10 11.955 Tf 11.95 0 Td[(SDW])]TJ /F8 7.97 Tf 6.58 0 Td[(1.(6)Fig. 6-7 showsthespectralfunctiononkspacefor!=0,withthenormalstateFermisurface.Panel(a)showstheeffectofpureSDWuctuations,wecanclearlyseethattheFermiSurfacemovesawayfromitsoriginalposition,andatransitiontoSDWstateleadstoareconstructionoftheFermisurfacewithsmallpockets.IncaseofSCuctuations,wehaveanarclikeFermisurfacealongtheoriginalFermisurfaceasshowninthepanel(b).Inpresenceofbothuctuations,wegetagainarclikeFermisurface,whichisshiftedfromtheoriginalFermisurfaceanditslargerthanthearcfrompureSCuctuations. 6.6ConclusionWeconsideredtheeffectofsuperconductingorderparameteructuationsonspectralfunctionabovetransition,andwefoundthatthed-wavesuperconductinguctuationsinthenormalstatecanleadtoarc-likeFermisurfacesegmentsasobservedinBi2212cuprates.Ourstudiesshowthatcleanlimitresultsareinbetterqualitativeagreementwithexperiments.Wealsoconsideredeffectofmagneticuctuationsalong 118

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Figure6-7. SpectralfunctionA(k,!=0)for(a)SDWuctuations(b)dwaveSCuctuationand(c)inpresenceofbothkindsofuctuations.Thetemperatureforallthreeplotsis1.1TcandTSDWis0.9Tc.ThereddashedlineshowstheFermisurfacewithoutanyuctuationeffect. withsuperconductinguctuationsgiveverysmallqualitativedifferencebetweenthetwocases.OurworkisingoodagreementwithweaklyunderdopedBi2212systems,butnotforextremelyunderdopedregimes. 119

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APPENDIXAT-MATRIXFORTWOBANDSYSTEM A.1BasicFormalismForlowimpurityconcentrations,onecanignoretheprocesseswhichinvolvescatteringfrommultipleimpuritysites.Withinthissinglesiteapproximation,wesumallpossiblescatteringeventsfromasinglesitetocalculatetheT-matrix,whichisrelatedtotheselfenergyas =nimpT ,(A)wherenimpistheimpurityconcentration.A denotesaquantityinthetwobandNambubasis~ck=(c)]TJ /F8 7.97 Tf 6.59 0 Td[(k,cyk,c)]TJ /F8 7.97 Tf 6.59 0 Td[(k,cyk).Here,denotethebands,candcyareannihilationandcreationoperatorsrespectively.Inamultibandsuperconductor,animpuritycanscatterwithinthesamebandorinbetweentwodifferentbands.Fig. A-1 showstheimpurityaverageddiagramsuptothirdorder.Anyprocesswhichinvolvesoddnumberofinterbandscatteringsdoesnotcontributetotheselfenergy,becauseattheendthenalstatebelongtotheotherband.Thesumofallthediagramsfromasingleimpuritysitecanbeexpressedas T1=1 1)]TJ /F4 11.955 Tf 11.95 0 Td[(Ue11G1(Ue11).(A)IntheNambubasis, Uij=Uij^3, (A) hGiik=gi,0^0+gi,1^1 (A) Theeffectiveimpurityscatteringpotentialisgivenas, Ue11=U11+U12G21 ^1)]TJ /F4 11.955 Tf 11.95 0 Td[(U22G2U12(A)ThesecondtermintheEq. A isduetoscatteringinthesecondbandbeforecomingbacktotheinitialband,andcontainstermslikeU12G2U22G2U12.Thisisgoingtomodify 120

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FigureA-1. Thesearetheimpurityaverageddiagrams,whichcontributetotheselfenergyoftherstbandGreen'sfunction.Heretheinterbandcontributioncomesthroughprocesses,whichinvolveevennumberofinterbandscatterings.Thediagramsalsotakesintoaccounttheorderofinterandintrabandscatterings.Uijistheimpuritypotentialstrength,wherei,jarethebandindexes.i=jdenotestheintrabandandi6=jdenotestheinterbandpotentialstrength.GiisthebareGreen'sfunctionandTiisT-matrixforithband. theeffectiveinterbandvertex,whichisillustrateding. A-2 .InEq. A ,U11,U22aretheIntra-bandimpuritypotentialstrengthsinband1and2respectively.U12istheInter-bandimpuritypotentialstrength,whichscattersbetweenthebands.AfterdoingthePaulimatrices()algebra,wecanwritetheexpressionfortheselfenergyintermsofgi,,theGreen'sfunction,wherethemomentumhasbeenintegratedout.Selfenergies 121

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FigureA-2. DiagramscontributingtosecondtermofEq. A ,whichcanberegardedasamodiedeffectivevertex)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(1fortheinterbandscattering,whichtakesintoaccountfortheintrabandscatteringprocessinthesecondbandaftergettingscatteredbytherstbandandbeforecomingbacktotherstbandagain. 'scanbewrittenas, D=1)]TJ /F10 11.955 Tf 11.95 0 Td[(2U212(g1,0g2,0)]TJ /F4 11.955 Tf 11.95 0 Td[(g1,1g2,1) (A) )]TJ /F4 11.955 Tf 19.27 0 Td[(U222(g22,0)]TJ /F4 11.955 Tf 11.95 0 Td[(g22,1))]TJ /F4 11.955 Tf 11.96 0 Td[(U211(g21,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g21,1)+(U212)]TJ /F4 11.955 Tf 11.95 0 Td[(U11U22)2(g22,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g22,1)(g21,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g21,1),1,0=nimpU211g1,0+U212g2,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g1,0(U212)]TJ /F4 11.955 Tf 11.96 0 Td[(U11U22)2(g22,0)]TJ /F4 11.955 Tf 11.95 0 Td[(g22,1) D, (A) 1,1=)]TJ /F4 11.955 Tf 9.29 0 Td[(nimpU211g1,1+U212g2,1)]TJ /F4 11.955 Tf 11.96 0 Td[(g1,1(U212)]TJ /F4 11.955 Tf 11.96 0 Td[(U11U22)2(g22,0)]TJ /F4 11.955 Tf 11.95 0 Td[(g22,1) D. (A) Therstsubscriptini,representsthebandindexandthesecondsubscriptdenotestheNambucomponents. 122

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A.2SpecialCases A.2.1OneBandForaonebandsuperconductor,thereisnointerbandscattering,soU12=0andwecansimplifytheexpressionsfortheself-energiesas, 1,0=nimpU2g1,0 1)]TJ /F4 11.955 Tf 11.95 0 Td[(U2(g21,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g21,1), (A) 1,1=)]TJ /F4 11.955 Tf 9.3 0 Td[(nimpU2g1,1 1)]TJ /F4 11.955 Tf 11.95 0 Td[(U2(g21,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g21,1). (A) HereintrabandscatteringstrengthU11hasbeenreplacedby\U"[ 82 ]. A.2.2BornLimitofTwoBandCaseIntheBornlimit,wewillkeepthetermsuptosecondorderin\Uij",sothedenominatorbecomes1andweget, 1,0=nimp(U211g1,0+U212g2,0), (A) 1,1=)]TJ /F4 11.955 Tf 9.3 0 Td[(nimp(U211g1,1+U212g2,2), (A) 2,0=nimp(U222g2,0+U212g1,0), (A) 2,1=)]TJ /F4 11.955 Tf 9.3 0 Td[(nimp(U222g2,2+U212g1,2). (A) A.2.3UnitaryLimitDuetothepresenceofanadditionalband,anewparametercomesintoplayintheunitarylimit.Thisparameteristheratiooftheinterbandscatteringtotheintrabandscattering.Ifoneassumesthattheinterbandscatteringisweakerthanintrabandscattering,thenintheunitarylimit,thereisnorolefortheinterbandscattering,butifinterbandscatteringisnotsmallandgrowswiththeintrabandscattering,thentheirrelativestrengthcouldbeimportant.Inthisrstscenario,wedenetheunitarylimitas,U11N11,U22N21,(U212N1N2)U11N1and(U212N1N2)U22N2.N1isthedensityofstates(DOS)attheFermisurfaceoftherstbandandN2istheDOSattheFermi 123

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surfaceforthesecondband.Thismeansintheunitarylimit,1=U211!0,1=U222!0andU212=U211!0.Sotheselfenergiesbecome 1,0=)]TJ /F4 11.955 Tf 9.3 0 Td[(nimpg1,0 (g21,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g21,1), (A) 1,1=nimpg1,1 (g21,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g21,1), (A) 2,0=)]TJ /F4 11.955 Tf 9.3 0 Td[(nimpg2,0 (g22,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g22,1), (A) 2,1=nimpg2,1 (g22,0)]TJ /F4 11.955 Tf 11.96 0 Td[(g22,1). (A) Weobtainexactlysameresultsasaboveforthecasewheninterbandscatteringisstrongerthantheintrabandscattering.Thishappensduethefactthatonlyanevennumberofinterbandscatteringprocessescontributetotheselfenergyandintheunitarylimittwosuccessiveinterbandscatteringeventsbetweentwobandsareequivalenttooneintrabandscatteringevent.Nowweconsiderthespeciallimitingcaseofisotropicunitaryscattering,wheninterbandandintrabandscatteringshaveequalpotentialstrength.ForthisU11=U22=U12limitweget, 1,0=)]TJ /F4 11.955 Tf 9.3 0 Td[(nimpg1,0+g2,0 (g1,0+g2,0)2)]TJ /F10 11.955 Tf 11.96 0 Td[((g1,1+g2,1)2, (A) 1,1=nimpg1,1+g2,1 (g1,0+g2,0)2)]TJ /F10 11.955 Tf 11.96 0 Td[((g1,1+g2,1)2, (A) 2,0=1,0, (A) 2,1=1,1. (A) Inthislimit,theselfenergiesarethesameforallthebands,becausetheisotropicscatteringmixesallmomentumstates,notjustthestateswithinthesameband. 124

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APPENDIXBGENERALEXPRESSIONFORPENETRATIONDEPTH B.1IntroductionInthisAppendix,IdiscusssomedetailsinvolvedinthederivationofthecurrentresponseforananisotropicsingletsuperconductorinthecaseofalatticeHamiltonian,inawaygeneralenoughtoincludeallkindsofscatteringprocessesexcludingthevertexcorrections.ThekineticenergyterminatightbindingHamiltonianisgivenas, H0=Xi,jtijcyicj+h.c.,(B)wherei,jdenotesthesitesandtijisthehopingmatrixelementbetweentheithandjthsites.cyi,ciarethecreationandannihilationoperatorsrespectivelyontheithsite.Inthepresenceofavectorpotentialtij!tijexp(ieArij).where~=c=1andrij=rj)]TJ /F11 11.955 Tf 12.38 0 Td[(ri[ 80 183 ].WemakeaTaylorseriesexpansionforthephasefactorduetovectorpotentialandonlykeepthetermwhichhasA^x. H(A^x)=H0+Xi,jtijcyicj)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(ieA^xrij^x)]TJ /F10 11.955 Tf 11.96 0 Td[((eA^xij^x)22)+h.c.,(B)whereH0isthepartoftheHamiltonianwhichisindependentofA^x.Notewehavechosenthe^xdirectionarbitrarilyasbeingthedirectionofthecurrent,thustheonlytermsintheHamiltonianwhichwillcontributearethosewhichgiveprojectionsontothex-direction.Toobtainthetotalcurrent,weoperatewith)]TJ /F5 7.97 Tf 16.87 4.71 Td[( A^x.Thisgives jxtotal=Xi,j2^xtijcyicj)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F4 11.955 Tf 9.3 0 Td[(ierij^x+(erij^x)2A^x+h.c.,(B)wherewedene jxdia=Xi,j2^x(rij^x)2tijcyicj+h.c.,(B)and jp=Xi,j2^xirij^xtijcyicj)]TJ /F4 11.955 Tf 11.95 0 Td[(h.c..(B) 125

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Nowwehavetocalculatetheexpectationvalueofthetotalcurrent.ThesecondtermistheEq.( B )isalreadylinearinexternaleld.Sowecandirectlycalculatetheexpectationvaluewiththeunperturbedgroundstate,becauseanycorrectiontothegroundstatewavefunctionduetotheexternalpotentialwillnotcontributeinrstorder.FortheparamagneticcurrentwhichisindependentofA^x,weneedtoconsidertheeffectofanexternaleldonwavefunctionuptorstorderintheeld.SointheinteractionrepresentationthewavefunctionacquiresafactorTexph)]TJ /F4 11.955 Tf 9.3 0 Td[(iRtH0(t0)dt0ij0i(seeanymany-bodytext,e.g.[ 75 ]).Inthiscase,thezerothordercontributioniszero,becausethereisnocurrentintheabsenceofaeld,andwekeeptermswhicharerstorderinA^x.WewritedowntheresponsefunctionKij,whichisdenedas: hjtotali=Ji(q,!)=)]TJ /F4 11.955 Tf 9.3 0 Td[(e2Kij(q,!)Aj(q,!),(B)whereKijis, Kij(q,!)=h)]TJ /F4 11.955 Tf 13.95 0 Td[(jdiaiij)]TJ /F15 11.955 Tf 11.96 16.27 Td[(Z0dhTjpi(q,)jpj()]TJ /F4 11.955 Tf 9.3 0 Td[(q,0)i=Kdiaij(q,!)+Kparaij(q,!). (B) Eq. B hastwoterms,thediamagneticterm Kdiaij(q,!)=h)]TJ /F4 11.955 Tf 13.95 0 Td[(jdiaiij,(B)andtheparamagneticterm, Kparaij(q,!)=)]TJ /F15 11.955 Tf 11.29 16.27 Td[(Z0dhTjpi(q,),jpj()]TJ /F4 11.955 Tf 9.3 0 Td[(q,0)i,(B)withthisdenitionofresponsefunctionsuperuidweightDs=e2==ns=misgivenby[ 2 80 184 ], Ds e2=ns m=Kij(q!0,!=0).(B) 126

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NowthetermsofrstorderinexternaleldinEq.( B )willgivethediamagneticresponse( B ).Forthediamagnetictermwecaneasilytaketheexpectationvalue, jxdia=Xi,jtijcyicj+h.c.)]TJ /F2 11.955 Tf 5.48 -9.69 Td[()]TJ /F10 11.955 Tf 9.3 0 Td[((rxij)2.(B)Takingtij=)]TJ /F4 11.955 Tf 9.3 0 Td[(t,rxi,j=afornearestneighborandtij=)]TJ /F4 11.955 Tf 9.3 0 Td[(t0,rxi,j=aandwewillsumonlyalongthedirectionwhererxi,j6=0.SotheexpressioninFourierspacebecomes, jxdia=Xkcykck)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F10 11.955 Tf 9.3 0 Td[((a)2()]TJ /F10 11.955 Tf 9.3 0 Td[(2tcos(kx))]TJ /F10 11.955 Tf 11.95 0 Td[(4t0cos(kx)cos(ky))(B)Herewenoticethat, ()]TJ /F10 11.955 Tf 9.3 0 Td[(2tcos(kx))]TJ /F10 11.955 Tf 11.95 0 Td[(4t0cos(kx)cos(ky))=)]TJ /F4 11.955 Tf 9.3 0 Td[(a2d2k dk2x,(B)Thistermhasdimensionsofinversemass,andthisisingeneraltrueforanytightbindingHamiltonian.Forageneralhopingmatrixelementtij,wecanwrite tij(=i+a)=tx(x+a)+ty(yb)+t0(x+a)(yb)+...(B)Onlytermswhichhaveprojectionalong^xwillcontributetotheresponse.Inthepresenceofanexternaleld,tij!tijexp(iAxrij).Soforthetermlinearinexternaleld,along+^xitbecomes Xktijcykck0exp(ikri+ik0rj)=Xkcykck"Xnantnxexp(ikxan) (B) +Xn,n0an0tn0xyexp(ikxan0)(2cos(kyn0b))#.Herethesuperscriptndenotestheneighbordistancealongtheaandbaxisandn0alongthediagonalsinthelattice,e.g.n=1isnearestneighbor,n0=1isnextnearestneighborandsoon.Forthediamagneticcurrentterm,weneedtij+tij,whichfollows 127

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fromEq.( B ), Xktijcykck0exp(ikri+ik0rj)+h.c.=Xk,n,n0cykck(na)2tnx(2cos(kxan)) (B) +(n0a)2tn0xy(2cos(kxan0))(2cos(kyn0b))i,andforthecorrespondingtightbindingdispersion, k=tx(2cos(kxa)+ty(2cos(kya)+t0(2cos(kxa)(2cos(kyb)) (B) +Xntnx(2cos(kxan))+Xntny(2cos(kxbn))+Xntnxy(2cos(kxan))(2cos(kynb)),UsingEq.( B ),wecanwritethediamagnetictermas Xktijcykck0exp(ikri+ik0rj)=Xkcykck)]TJ /F4 11.955 Tf 10.49 8.09 Td[(d2k dk2x(B)Thisexpressionistrueingeneralforanylatticedispersion.Nowforthediamagneticterm,theexpectationvalueof=<^nk>isequalto1=Pi!mG(i!,k)exp(i!m0+).Sothediamagneticcurrentoperatorbecomes jxdia=)]TJ /F15 11.955 Tf 11.29 11.36 Td[(Xkcykck)]TJ /F4 11.955 Tf 10.49 8.09 Td[(d2k dk2x,(B)andthenalexpressionforthediamagneticresponseis =2TXi!mXkd2k dk2xG(i!m,k),(B)whereGisthediagonalMatsubaraGreen'sfunctioninthesuperconductingstate, G(i!n,k)=i!n+k (i!n)2)]TJ /F6 11.955 Tf 11.96 0 Td[(2k)]TJ /F10 11.955 Tf 11.96 0 Td[(2k.(B)Nextwecalculatetheparamagneticresponse.TheparamagneticcurrentcomesfromtherstterminEq.( B ).ThistermisindependentofAx.SowewillusetheKuboformula,whichrequiresthecalculationofthecurrent-currentcorrelationfunction.In 128

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general,theparamagneticcurrentonalatticeis jp=Xi,jtijcyicj)]TJ /F4 11.955 Tf 11.95 0 Td[(h.c.)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(irxij,(B)whichfornearestneighborhopingisgivenas, jpnn=Xk)]TJ /F10 11.955 Tf 9.3 0 Td[(2tsin(kx)cykck()]TJ /F4 11.955 Tf 9.3 0 Td[(a),(B)andfornextnearestneighborhopingis, jpnnn=Xk)]TJ /F10 11.955 Tf 9.3 0 Td[(4t0sin(kx)cos(ky)cykck()]TJ /F4 11.955 Tf 9.3 0 Td[(a).(B)Soingeneralitisthevelocityandisexpressedas, jp=Xk,dk dkxcykck,(B)IntheNambubasis,wedenecyk=fcyk,"c)]TJ /F21 7.97 Tf 6.59 0 Td[(k,#gandtheparamagneticcurrentoperatormaybewrittenas, jp=Xk,dk dkxcy0c(B)Nowweneedtocalculatethecurrentcurrentcorrelationfunctiondenedas (q,i)=Z0exp(i)(B)soweneed.WithintheHartree-Fockapproximation,wekeeponlytermscorrespondingtoBCSpairing,suchthattheproductmaybewrittenTr[G (k+q,)G (k,)]TJ /F6 11.955 Tf 9.3 0 Td[()].InFourierspacethecurrentcurrentcorrelationfunctionbecomes, (q,i)=Z0dexpiT2X!m,!nexp)]TJ /F7 7.97 Tf 6.59 0 Td[(i!mexpi!nTr[G (k+q,!m)G (k,!n)],(B) 129

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andafterintegratingout,weget, (q,i)=T2X!m,!n(+!n=!m)Tr[G (k+q,!m)G (k,!n)],(B)whichcanbesimpliedas (q,i)=TX!mTr[G (k+q,!m)G (k,!m)]TJ /F6 11.955 Tf 11.96 0 Td[()].(B)WherewehaveintroducedtheNambu(matrix)Green'sfunction G =0B@G(k,i!n)F(k,i!n)Fy(k,i!n)G()]TJ /F4 11.955 Tf 9.29 0 Td[(k,i!n)1CA,(B)andGandFaredenedas G(k,i!n)=i!n+k (i!n)2)]TJ /F6 11.955 Tf 11.95 0 Td[(2k)]TJ /F10 11.955 Tf 11.95 0 Td[(2k, (B) F(k,i!n)=k (i!n)2)]TJ /F6 11.955 Tf 11.95 0 Td[(2k)]TJ /F10 11.955 Tf 11.95 0 Td[(2k. (B) Sonallytheexpressionforthecurrentcurrentcorrelationfunctionbecomes, (q,i)=TX!m,kTr[G (k+q,!m)G (k,!m)]TJ /F6 11.955 Tf 11.95 0 Td[()]dk dkx2.(B)Inthelastexpressionthenite'q'termsinvelocityhavebeenignoredbecausethemaincontributiontothisintegralcomesfromtheregionneartheFermisurface,andinthesmallqlimitk+q'k.Inthenextsection,Ishowthatthesuperuiddensityiszerointhenormalstate. B.2NormalStateCancellationInthenormalstatetheGreen'sfunctionreducestoasimplenormalstateGreen'sfunction.Soforthediamagneticcomponentwecanwrite, =2TXi!mXkd2k dk2x1 i!)]TJ /F6 11.955 Tf 11.95 0 Td[(k.(B) 130

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SummationoverMatsubarafrequenciesgives, =2Xkd2k dk2xf(k),(B)andnowweperformintegrationbypartsonthediamagneticterm, =21 42Zkydk dkxf(k))]TJ /F5 7.97 Tf 6.59 0 Td[( (B) )]TJ /F10 11.955 Tf 19.27 0 Td[(21 42ZkyZkxdk dkxdf(k) dkx.ThersttermiszeroinEq.( B )andinthesecondtermdf(k) dkx!df(k) dkdk dkx.Sonallywehave, =)]TJ /F10 11.955 Tf 9.29 0 Td[(21 42ZkyZkxdk dkx2df(k) dk. (B) Nowintheparamagneticterm,it'sveryimportanttorstseti=0andthentakingthelimitq!0,forthecancellationinthenormalstate[ 80 ].Thisgives, (q!0,i=0)=21 42ZkTX!m1 i!m)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q1 i!m)]TJ /F6 11.955 Tf 11.95 0 Td[(kdk dkx2,(B)whichcanbewrittenafterdoingsummationovertheMatsubarafrequencies (q!0,i=0)=21 42Zkf(k+q))]TJ /F4 11.955 Tf 11.96 0 Td[(f(k) k+q)]TJ /F6 11.955 Tf 11.96 0 Td[(kdk dkx2,(B)whichinlimitq!0becomes, (q!0,i=0)=21 42Zkdf(k) dkdk dkx2,(B)whichexactlycancelsthediamagneticresponsegivenbyEq. B .Inthenextsection,Iexpressthesuperuiddensityintermsofspectralfunctions,whichiseasiertohandleinthepresenceofcomplicatedscatteringprocesses. 131

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B.3TotalCurrentResponseThediamagnetictermcanbewrittenas, =)]TJ /F10 11.955 Tf 9.3 0 Td[(2TXi!mZXkd2k dk2xA(k,) i!m)]TJ /F6 11.955 Tf 11.96 0 Td[(=)]TJ /F10 11.955 Tf 9.3 0 Td[(2ZXkd2k dk2xA(k,)f(),(B)wheref()istheFermifunctionandA(k,)isthespectralfunctiondenedas, A(k,!)=)]TJ /F10 11.955 Tf 10.89 8.09 Td[(1 Im!++k !2+)]TJ /F4 11.955 Tf 11.95 0 Td[(E2k.(B)WhereEk=p 2k+2k,!+is!+i)]TJ /F1 11.955 Tf 10.1 0 Td[(and)]TJ /F1 11.955 Tf 10.1 0 Td[(isthesingleparticlescatteringrate,assumedheretomodifyonlythe!termintheGreen'sfunction.Theparamagnetictermcanbeexpressedusing (q,i)=TX!mTr[G (k+q,!m)G (k,!m)]TJ /F6 11.955 Tf 11.95 0 Td[()]dk dkx2,=TX!mZ1,2Tr[Im(G (k+q,1))Im(G (k,2))] (i!m)]TJ /F6 11.955 Tf 11.95 0 Td[(1)(i!m)]TJ /F4 11.955 Tf 11.95 0 Td[(i)]TJ /F6 11.955 Tf 11.95 0 Td[(2)dk dkx2,=Z1,2,kTr[Im(G (k+q,1))Im(G (k,2))] (1)]TJ /F6 11.955 Tf 11.96 0 Td[(2)]TJ /F4 11.955 Tf 11.96 0 Td[(i)[f(1))]TJ /F4 11.955 Tf 11.96 0 Td[(f(2)]dk dkx2, (B) andspecicallyforthelimiti!0,wehave (q,0)=Zd1 2d2 2Tr[Im(G (k+q,1))Im(G (k,2))]f(1))]TJ /F4 11.955 Tf 11.95 0 Td[(f(2) (1)]TJ /F6 11.955 Tf 11.95 0 Td[(2)dk dkx2, (B) =2Zd1 2d2 2[A(k+q,1)A(k,2) (B) +B(k+q,1)B(k,2)]f(1))]TJ /F4 11.955 Tf 11.96 0 Td[(f(2) (1)]TJ /F6 11.955 Tf 11.96 0 Td[(2)dk dkx2,whereB(k,!)is, B(k,!)=)]TJ /F10 11.955 Tf 10.89 8.09 Td[(1 Imk !2+)]TJ /F4 11.955 Tf 11.96 0 Td[(E2k.(B)Thisisusuallysufcientinad-wavesuperconductor,butnotcompletelygeneral.Forsuperconductorsotherthandwave,isreplacedbyarenormalizedandparticle-holeasymmetricsystemsthesingleparticleenergykmustberenormalizedaswell.Wecan 132

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furthersimplifythediamagneticterm, Kdia=2TX!Xkd2k dk2xi!+k (i!)2)]TJ /F6 11.955 Tf 11.95 0 Td[(2k)]TJ /F10 11.955 Tf 11.95 0 Td[(2k.(B)Nowweperformanintegrationbypartsoverthevariablekx,Thediamagnetictermis Kdia=2TX!"Zkydk dkxi!+k (i!)2)]TJ /F6 11.955 Tf 11.96 0 Td[(2k)]TJ /F10 11.955 Tf 11.95 0 Td[(2k)]TJ /F5 7.97 Tf 6.59 0 Td[( (B) )]TJ /F15 11.955 Tf 20.46 11.35 Td[(Xkdk dkx1 (i!)2)]TJ /F6 11.955 Tf 11.95 0 Td[(2k)]TJ /F10 11.955 Tf 11.96 0 Td[(2kdk dkx+i!+k [(i!)2)]TJ /F6 11.955 Tf 11.95 0 Td[(2k)]TJ /F10 11.955 Tf 11.95 0 Td[(2k]2dE2k dkx#.Thersttermintheaboveexpressioniszero,andthesecondtermcanbewrittenas: Kdia=2TX!Xk)]TJ /F4 11.955 Tf 9.3 0 Td[(V2x1 (i!)2)]TJ /F6 11.955 Tf 11.96 0 Td[(2k)]TJ /F10 11.955 Tf 11.95 0 Td[(2k)]TJ /F10 11.955 Tf 11.95 0 Td[(2i!+k ((i!)2)]TJ /F6 11.955 Tf 11.95 0 Td[(2k)]TJ /F10 11.955 Tf 11.95 0 Td[(2k)2)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(kVx+kVx.(B)HereVx=dk dkxandVx=dk dkx.Nowtheparamagnetictermintotalcurrentresponseis Kpara=2TX!XkV2x(i!)(i!0)+2k+2k+k(i!+i!0) ((i!)2)]TJ /F4 11.955 Tf 11.96 0 Td[(E2k)((i!0)2)]TJ /F4 11.955 Tf 11.96 0 Td[(E2k), (B) =2TX!XkV2x1 (i!)2)]TJ /F6 11.955 Tf 11.95 0 Td[(2k)]TJ /F10 11.955 Tf 11.96 0 Td[(2k (B) +(i!)(i!0))]TJ /F10 11.955 Tf 11.96 0 Td[((i!0)2+2E2k+k(i!+i!0) ((i!)2)]TJ /F4 11.955 Tf 11.96 0 Td[(E2k)((i!0)2)]TJ /F4 11.955 Tf 11.96 0 Td[(E2k).Nowwecancombinebothterms,notingthatthedivergentcontributioncomeswithoppositesignsinboththeequation,hencecanceled.Nowwecanalsoset!0;!=!0.Theremainingtermscanbewrittenas, K=2TX!XkV2x22k ((i!)2)]TJ /F4 11.955 Tf 11.96 0 Td[(E2k)2)]TJ /F4 11.955 Tf 11.96 0 Td[(VxVx2k(i!+k) ((i!)2)]TJ /F4 11.955 Tf 11.96 0 Td[(E2k)2(B)Inpresenceofselfenergycorrections,!!!)]TJ /F10 11.955 Tf 12.2 0 Td[(0and!+1.Eq. B isvalidaslongastheselfenergiesarenotmomentumdependent. 133

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B.4SpectralRepresentationofTotalCurrentResponseThetotalcurrentresponsecanbeexpressedintermsofspectralfunctionsas K=4XkZ1dxZ1dyV2xB(k,x)B(k,y) (B) )]TJ /F4 11.955 Tf 20.46 0 Td[(VxVxB(k,x)A(k,y)f(x))]TJ /F4 11.955 Tf 11.96 0 Td[(f(y) x)]TJ /F4 11.955 Tf 11.95 0 Td[(y+i,B(k,x)=)]TJ /F10 11.955 Tf 10.89 8.09 Td[(1 Imk x2)]TJ /F4 11.955 Tf 11.95 0 Td[(E2k, (B) A(k,x)=)]TJ /F10 11.955 Tf 10.89 8.09 Td[(1 Imx+k x2)]TJ /F4 11.955 Tf 11.95 0 Td[(E2k, (B) Vx=dk dkx, (B) Vx=dk dkx. (B) 134

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BIOGRAPHICALSKETCH VivekMishrawasborninFebruary1983,inJagdalpur,India.HeobtainedhisHigherSecondarySchoolCerticatewithmathematics,physicsandchemistryinhishometown,andadmittedtoIndianInstituteofTechnology,Kanpur,Indiain2001,wherehegraduatedwithMasterofScience(Integrated)inphysicsin2006.In2006,hewenttoUniversityofFlorida,Gainesville,UnitedStatesAmericaandcompletedhisDoctorofPhilosophyinphysicsin2011. 146