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Theoretical Visualization of Atomic-Scale Phenomena in Inhomogeneous Superconductors

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Theoretical Visualization of Atomic-Scale Phenomena in Inhomogeneous Superconductors
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Choubey, Peayush Kumar
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Doctorate ( Ph.D.)
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University of Florida
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Physics
Committee Chair:
HIRSCHFELD,PETER J
Committee Co-Chair:
KUMAR,PRADEEP
Committee Members:
MUTTALIB,KHANDKER A
STEWART,GREGORY R
NINO,JUAN C

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bdg -- cuprate -- stm -- superconductor -- wannier
Physics -- Dissertations, Academic -- UF
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government publication (state, provincial, terriorial, dependent) ( marcgt )
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Physics thesis, Ph.D.

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Abstract:
Scanning tunneling microscopy (STM), with its unique capabilities to image the electronic structure of solids with atomic resolution, has been extensively used to study superconductors. The tunneling conductance measured by STM is proportional to the local density of states (LDOS) at the STM tip position and, hence, provides indispensable information about the superconducting gap in the energy spectrum. Many of the unconventional superconductors are layered materials where, generally, the layer exposed to the STM tip is different from the 'active' layer responsible for superconductivity. The intervening layers between STM tip and the active layer can provide indirect tunneling paths and, hence significantly alter the conclusions drawn from assuming a direct tunneling. To addressed this issue, we have developed a novel 'BdG+W' method which combines the solution of the widely used lattice Bogoliubov- de Gennes (BdG) equations and first principles Wannier functions. Here, the BdG equations are solved on a two-dimensional (2D) lattice in the active layer, and the effect of other degrees of freedom are included through Wannier functions, enabling calculation of LDOS at STM tip position with sub-unit cell resolution. As the first application of the BdG+W method, we study the response of the superconductor FeSe to a single point-like impurity substituting a Fe site. Using the solution of the 10-Fe-orbital BdG equations in conjunction with the first principles Wannier functions, we show that that the atomic-scale dimer-like states observed in STM experiments on FeSe and several other Fe-based superconductors can be understood as a consequence of simple defects located on Fe sites due to hybridization with the Se states. Next, we study the effects of Zn and Ni impurities on the local electronic structure in cuprate superconductor Bi$_2$Sr$_2$CaCuO$_8$ (BSCCO). Modeling Zn as a strong on-site potential scatterer, we obtain LDOS maps at a typical STM tip height using BdG+W method showing excellent agreement with the STM measurements, resolving the long-standing 'Zn-paradox'. Moreover, we show that the LDOS obtained using BdG+W framework treating Ni in a simple model of a magnetic impurity shows excellent agreement with the STM results. Finally, we study the charge ordered states in the extended t-J model using a renormalized mean-field theory. We show that the nodal pair density wave state supported by this model has characteristics very similar to that observed in the STM experiments on the underdoped cuprates. ( en )
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Thesis (Ph.D.)--University of Florida, 2017.
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Adviser: HIRSCHFELD,PETER J.
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Co-adviser: KUMAR,PRADEEP.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2017-11-30
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by Peayush Kumar Choubey.

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THEORETICALVISUALIZATIONOFATOMIC-SCALEPHENOMENAININHOMOGENEOUSSUPERCONDUCTORSByPEAYUSHK.CHOUBEYADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2017

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c2017PeayushK.Choubey

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Tomymother

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ACKNOWLEDGMENTSFirstandforemost,IwouldliketoexpressdeepestgratitudetomyadvisorProf.PeterHirschfeldforhisconstantsupport,understanding,andguidance.Noneofthiswouldhavebeenpossiblewithouthiswonderfulinsights,patience,andencouragement.ManythankstoProf.DmitriiMaslovforhisinsightfulandcrystal-clearlectures.IamespeciallyindebtedtomyfriendandcollaboratorDr.AndreasKreiselforhishelpandsupportatvariousstagesofmydoctoralwork.IwouldliketothankmycollaboratorsProf.BrianAndersen,Prof.Ting-KuoLee,Prof.WeiKu,Dr.TomBerlijn,andWei-LinTu.WithoutyourhelpIcouldnothavepublished.Iamthankfultoformerandpresentgroupcolleagues,Dr.VivekMishra,Dr.YanWang,Dr.WenyaRowe,Dr.AndreasLinschied,Dr.SaurabhMaiti,XiaoChen,ShinibaliBhattacharyya,andJasdipSidhu.Iwouldliketothankthemembersofmysupervisorycommittee,Prof.PradeepKumar,Prof.KhandkarMuttalib,Prof.GregStewart,andProf.JuanNinofortheirtimeandexpertisetoimprovemywork.IamthankfultoDavidHansen,BrentNelson,andClintCollinsfortheirhelpwithvariousaspectsofcomputingandothertechnicalsupport.IthankPhysicsDepartmentstafortheirassistanceoverpastyears,especiallyIexpressmygratitudetoKristinNichola,TessieColson,andPamMarlinfortheirsupport.IwouldalsoliketoacknowledgethehelpandassistanceofallmyfriendsattheUniversityofFlorida,especiallyVarunRishi,AvinashRustagi,TathagataChoudhuri,NaweenAnand,PanchamGupta,AkhilAhir,andAkashKumar.Finallyandmostimportantly,Iwouldliketothankallmyfamilymembersfortheirhelp,understandingandconstantsupport.Ithankmywife,Pallavi,forbeingacontinuoussourceofencouragementandsupporttome. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 11 1.1HistoricalBackground ............................. 11 1.2BCSTheory ................................... 12 1.3OverviewofCuprateSuperconductors .................... 15 1.4OverviewofIron-BasedSuperconductors ................... 18 1.5STMStudiesofSuperconductors ....................... 22 2BOGOLIUBOV-DEGENNES-WANNIERAPPROACH ............. 26 2.1ImpuritiesinSuperconductors ......................... 26 2.2Mean-FieldTheory ............................... 27 2.2.1Hamiltonian ............................... 27 2.2.2BdGEquations ............................. 29 2.3CalculationofSTMObservables ........................ 31 2.3.1LatticeLDOS .............................. 31 2.3.2ContinuumLDOS ............................ 32 2.4NumericalImplementation ........................... 33 3IMPURITYINDUCEDSTATESINIRONBASEDSUPERCONDUCTORS .. 35 3.1Motivation .................................... 35 3.2NormalStatePropertiesofFeSe ........................ 37 3.3HomogeneousSuperconductingState ..................... 40 3.4EectsofanImpurity ............................. 44 3.5Conclusion .................................... 48 4IMPURITYINDUCEDSTATESINCUPRATES ................. 50 4.1Motivation .................................... 50 4.2NormalState .................................. 52 4.3HomogeneousSuperconductingState ..................... 54 4.4EectsofaStrongNon-MagneticImpurity .................. 58 4.5EectsofaMagneticImpurity ......................... 60 4.6QuasiparticleInterference ........................... 65 4.7Conclusion .................................... 68 5

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5CHARGEORDERINCUPRATES ......................... 70 5.1Motivation .................................... 70 5.2Extendedt-JModel ............................. 75 5.2.1RenormalizedMean-FieldTheory ................... 75 5.2.2Quasi1DBdGEquations ........................ 78 5.2.3CalculationofSTMObservables .................... 81 5.3UnidirectionalChargeOrderedStatesinExtendedt-JModel ....... 83 5.3.1Anti-PhaseChargeDensityWaveState ................ 86 5.3.2NodalPairDensityWaveState .................... 90 5.4Conclusion .................................... 98 6SUMMARYANDFINALCONCLUSIONS .................... 100 APPENDIX ADERIVATIONOFBOGOLIUBOV-DEGENNESEQUATIONS ......... 106 BSUPERCELLMETHOD ............................... 112 CANALYTICALPROOFOFU-SHAPEDCONTINNUMLDOSINSUPERCONDUCTINGBSCCO .......................... 115 DMICROSCOPICJUSTIFICATIONOFTHE"FILTER"THEORY ....... 121 REFERENCES ....................................... 124 BIOGRAPHICALSKETCH ................................ 136 6

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LISTOFFIGURES Figure page 1-1Tcversusyearofdiscoveryforvarioussuperconductors. .............. 13 1-2Unitcellofatypicalcuprate. ............................ 15 1-3Schematicphasediagramofhole-dopedcuprates. ................. 16 1-4UnitcellandFermisurfaceofpnictides. ...................... 19 1-5Schematicphasediagramofiron-basedsuperconductors. ............. 20 1-6STMset-upandworking. .............................. 22 1-7Znimpurity-inducedresonancestatesinBi2Sr2CaCu2O8+. ............ 23 3-1STMtopographyofFeSethinlm. ......................... 36 3-2FeSeWannierorbitals. ................................ 37 3-3PropertiesofthenormalstateofFeSe. ....................... 39 3-4PropertiesofthehomogeneoussuperconductingstateofFeSe. .......... 43 3-5LatticeLDOSinthevicinityofanimpurity. .................... 46 3-6ContinuumLDOSmapsatvariousbiasesandheightsfromFe-plane. ...... 47 3-7Topographyofimpuritystates. ........................... 48 4-1Cu-dx2)]TJ /F5 7.9701 Tf 6.587 0 Td[(y2WannierfunctioninBi2Sr2CaCu2O8+. ................. 52 4-2PropertiesofthenormalstateofBi2Sr2CaCu2O8+. ................ 53 4-3PropertiesofthehomogeneoussuperconductingstateofBi2Sr2CaCu2O8+. ... 55 4-4Bi2Sr2CaCu2O8+Wannierfunctionataheightz5AabovetheBiOplane. 57 4-5LDOSspectruminthevicinityofaZnimpurityinBSCCO. ........... 59 4-6LDOSmapsattheZnimpurityinducedresonantenergy. ............. 60 4-7LDOSspectruminthevicinityofalocalmagneticimpurity. ........... 62 4-8LDOSspectruminthevicinityofanextendedmagneticimpurity. ........ 63 4-9LDOSmapsatthemagneticimpurity-inducedresonantenergies. ........ 64 4-10QPIpatternsobtainedfromlatticeandcontinuumLDOS. ............ 66 4-11QPIZ-mapsand-mapsobtainedfromlatticeandcontinuumLDOS. ...... 67 7

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5-1Peirelsdistortionin1Dmetals. ........................... 71 5-2Characteristicsofthechargeorderincuprates. .................. 72 5-3Energypersiteatvariousholedopingsforhomogeneoussuperconductingstate,APCDW,andnPDWstates. ............................. 84 5-4CharacteristicfeaturesoftheAPCDWstate. .................... 86 5-5ContinuumLDOSmapsintheAPDWstate. .................... 88 5-6ContinuumLDOS,formfactors,andspatialphasedierenceintheAPCDWstate. .......................................... 89 5-7CharacteristicfeaturesofthenPDWstate. ..................... 91 5-8ContinuumLDOSmapsinthenPDWstate. .................... 92 5-9ContinuumLDOSspectruminthenPDWstate. .................. 93 5-10Biasdependenceoftheintra-unitcellformfactorsinthenPDWstate. ..... 94 5-11BiasdependenceofaveragespatialphasedierenceinthenPDWstate. ..... 96 5-12ImportanceofthePDWcharacterofthenPDWstate. .............. 97 B-1Supercellset-up .................................... 113 C-1Nodalcoordinates ................................... 118 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTHEORETICALVISUALIZATIONOFATOMIC-SCALEPHENOMENAININHOMOGENEOUSSUPERCONDUCTORSByPeayushK.ChoubeyMay2017Chair:PeterHirschfeldMajor:PhysicsScanningtunnelingmicroscopy(STM),withitsuniquecapabilitiestoimagetheelectronicstructureofsolidswithatomicresolution,hasbeenextensivelyusedtostudysuperconductors.ThetunnelingconductancemeasuredbySTMisproportionaltothelocaldensityofstates(LDOS)attheSTMtippositionand,hence,providesindispensableinformationaboutthesuperconductinggapintheenergyspectrum.Manyoftheunconventionalsuperconductorsarelayeredmaterialswhere,generally,thelayerexposedtotheSTMtipisdierentfromthe'active'layerresponsibleforsuperconductivity.TheinterveninglayersbetweenSTMtipandtheactivelayercanprovideindirecttunnelingpathsandhencesignicantlyaltertheconclusionsdrawnfromassumingadirecttunneling.Toaddressthisissue,wehavedevelopedanovel'BdG+W'methodwhichcombinesthesolutionofthewidelyusedlatticeBogoliubov-deGennes(BdG)equationsandrstprinciplesWannierfunctions.Here,theBdGequationsaresolvedonatwo-dimensional(2D)latticeintheactivelayer,andtheeectofotherdegreesoffreedomareincludedthroughWannierfunctions,enablingcalculationofLDOSatSTMtippositionwithsub-unitcellresolution.AstherstapplicationoftheBdG+Wmethod,westudytheresponseofthesuperconductorFeSetoasinglepoint-likeimpuritysubstitutingonaFesite.Usingthesolutionofthe10-Fe-orbitalBdGequationsinconjunctionwiththerstprinciplesWannierfunctions,weshowthatthattheatomic-scaledimer-likestatesobservedinSTM 9

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experimentsonFeSeandseveralotherFe-basedsuperconductorscanbeunderstoodasaconsequenceofsimpledefectslocatedonFesitesduetohybridizationwiththeSestates.Next,westudytheeectsofZnandNiimpuritiesonthelocalelectronicstructureincupratesuperconductorBi2Sr2CaCuO8(BSCCO).ModelingZnasastrongon-sitepotentialscatterer,weobtainLDOSmapsatatypicalSTMtipheightusingBdG+WmethodshowingexcellentagreementwiththeSTMmeasurements,resolvingthelong-standing'Zn-paradox'.Moreover,weshowthattheLDOSobtainedusingBdG+WframeworktreatingNiinasimplemodelofamagneticimpurityshowsexcellentagreementwiththeSTMresults.Finally,westudythechargeorderedstatesintheextendedt-Jmodelusingarenormalizedmean-eldtheory.WeshowthatthenodalpairdensitywavestatesupportedbythismodelhascharacteristicsverysimilartothatobservedintheSTMexperimentsontheunderdopedcuprates. 10

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CHAPTER1INTRODUCTION 1.1HistoricalBackgroundSuperconductivity-characterizedbyzeroresistivityandperfectdiamagnetismbelowacriticaltemperature(Tc)-representsoneofthemostremarkablemanifestationsofquantummechanicsatthemacroscopicscale.ItwasdiscoveredbyKamerlinghOnnesin1911withtheobservationofasuddendropintheresistivityofmercurywhenitwascooledbelow4.2K[ 1 ].Twenty-twoyearslater,MeissnerandOchsenfeldshowedthatthesuperconductorismorethanjustaperfectconductorbydiscoveringthe"Meissnereect",thecompleteexpulsionofmagneticeldwhenamaterialiscooledbelowTcregardlessofitspasthistory[ 2 ].ItledtheLondonbrotherstoproposetheexistenceofamacroscopicquantum-mechanicalcondensatewhichcarriesthesupercurrent[ 3 ].In1950,GinzburgandLandauintroducedanorderparameterdescriptionofthesuperconductingcondensateandproposedthefamous"Ginzburg-Landau(GL)"equationleadingtoasuccessfulphenomenologicaldescriptionofsuperconductivity[ 4 ]asamacroscopicquantumphenomenon.Inthesameyear,thediscoveryoftheisotopeeect-achangeinTccausedbyisotopicsubstitutionoflatticeions-pavedthewayforthemicroscopictheoryofsuperconductivity[ 5 6 ].Itshowed,forthersttime,thatthelatticedegreesoffreedomplayimportantroleinthesuperconductivityofsimplematerials.Soonafter,BardeenandFrohlichshowedthatthelatticevibrationscanmediateaneectiveattractionbetweenelectrons,evenifCoulombrepulsionistakenintoaccount[ 7 { 9 ].Inanotherimportantdevelopment,CoopershowedthattheFermiseaisunstableagainsttheformationofboundpairsofelectrons,called"Cooperpairs",inpresenceofeectiveelectron-electronattractiveinteraction,nomatterhowweakitis[ 10 ].Withthesetwoimportantingredients,Bardeen,Cooper,andSchrieerproposedthefamousBCStheoryin1957[ 11 ],explainingthemicroscopicoriginofsuperconductivityforwhichtheywereawardedtheNobelprizein1972.BCStheoryexplainedmostoftheexperimental 11

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observationsinelementalsuperconductorsandisregardedasoneofthecornerstonesofthecondensedmatterphysics.InthenexttwodecadesoftheBCStheory,evenafterintenseexperimentaleorts,Tccouldnotberaisedbeyond23K.AmajorbreakthroughwasachievedbyBednorzandMullerin1986[ 12 ].TheydiscoveredsuperconductivityindopedperovskiteLa2)]TJ /F5 7.9701 Tf 6.587 0 Td[(xBaxCuO4withrecord-breakingTcof35K;atrulyremarkableresultgiventhattheundopedcompoundisaninsulator.Withinayear,intenseexperimentaleortsacrossmanylaboratoriesresultedintodiscoveryofsimilarcompoundswithincreasingTcsuchasYBa2Cu3O7)]TJ /F5 7.9701 Tf 6.587 0 Td[((Tc=93K),Ba2Sr2CaCu2O8+(Tc=85K),andTlBa2Ca2CuO8+(Tc=110K).ThesecompoundshaveCuO2layersasacommonstructuralunitandhencetheyarecalledcuprates.Insection1.3,wereviewsomeoftheimportantpropertiesofthesematerials.Thesecondclassofhigh-Tcsuperconductorswasdiscoverediniron-basedmaterialsbyHosonoandco-workerswhoreported26Ksuperconductivityinuorine-dopedLaFeAsOinearly2008[ 13 ].Soonafterthat,highestTcinthisclassofmaterialsreachedto55KbythediscoveryofSmO1)]TJ /F5 7.9701 Tf 6.587 0 Td[(xFxFeAs[ 14 ]and65-70KinmonolayerFeSegrownonSrTiO3substrate[ 15 ].Insection1.3,wereviewsomeoftheimportantpropertiesoftheseFe-basedmaterials.Figure 1-1 summarizesthehistoryofhigh-temperaturesuperconductivity. 1.2BCSTheoryHere,wewillreviewsomemainresultsoftheBCStheory.Detaileddiscussionsandderivationscanbefoundinmanyexcellenttextbooks[ 17 { 19 ].BCSconsideredthefollowingpairingHamiltonianasaminimalmodeltoproduceasuperconductinggroundstate: HSC=Xkkcykck+Xkk0Vkk0cyk0"cy)]TJ /F11 7.9701 Tf 6.587 0 Td[(k0#c)]TJ /F11 7.9701 Tf 6.586 0 Td[(k#ck"(1{1)Here,cykcreatesanelectronwithmomentumkandspin=";#.ThersttermintheaboveHamiltonianrepresentsthekineticenergyofelectrons,withkbeingthesingle-particleenergyrelativetotheFermienergy.Thesecondtermrepresentsan 12

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Figure1-1. Superconductingtransitiontemperature(Tc)versusyearofdiscoveryforvarioussuperconductors.ReprintedbypermissionfromMacmillanPublishersLtd:Nature[ 16 ],copyright2015. attractiveinteractionwhichscattersapairofelectronsinstate(k";)]TJ /F9 11.9552 Tf 9.299 0 Td[(k#)tostate(k0";)]TJ /F9 11.9552 Tf 9.299 0 Td[(k0#)withanamplitudeVkk0<0.ThisHamiltoniancanbestudiedbyavarietyofapproachessuchasvariationalmethod,mean-elddecomposition,andGreensfunctionmethods.Originally,BCSusedthevariationalapproach,andhypothesizedthefollowingtrialwavefunction ji=Ykuk+vkcyk"cy)]TJ /F11 7.9701 Tf 6.586 0 Td[(k#j0i(1{2)Here,j0irepresentsthevacuumstate.ukandvkarethevariationalparameterswithrespecttowhichthegroundstateenergyE=hjHSCjihastobeminimizedundertheconstraintjukj2+jvkj2=1asrequiredforthenormalization.TheBCSwavefunctionrepresentsacoherentsuperpositionofCooperpairswithjvkj2(jukj2)beingtheprobabilityofpairstatejk";)]TJ /F9 11.9552 Tf 9.298 0 Td[(k#itobeoccupied(empty).OneofthekeyquantitiesintheBCStheoryisthesuperconductinggapparameterk,denedas: k=Xk0Vkk0hc)]TJ /F11 7.9701 Tf 6.586 0 Td[(k0#ck0"i=)]TJ /F10 11.9552 Tf 11.291 11.357 Td[(Xk0Vkk0uk0vk0(1{3) 13

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Theexcitationenergyofaquasi-particlewithmomentum~kinthesuperconductingstatedependsonk,andisgivenby Ek=q 2k+jkj2(1{4)ItshowsthatanenergygapisopenedattheFermilevel(k=0)andthattheminimumenergytoexciteaquasi-particlewithwavevectorkisjkj.ThesuperconductinggapparameterdependsonthepairpotentialVkk0andisobtainedbysolvingthesocalled"BCSgapequation" k=)]TJ /F10 11.9552 Tf 11.291 11.358 Td[(Xk0Vkk0k0 2Ek0tanhEk0 2(1{5)where=1=kBT,withkBandTbeingtheBoltzmannconstantandtemperature,respectively.Itisahighlynon-linearintegralequationandhasatrivialsolutionk=0representingthehigh-temperaturenormalstate.Atsucientlylowtemperatures,anon-trivialsolutioncanexist,implyingasuperconductingstate.Equation 1{5 ishardtosolveforageneralVkk0.BCSusedasimpliedmodelofthephonon-mediatedattractiveinteractiontogetaclosed-formsolutionofthegapequation: Vkk0=8>><>>:)]TJ /F4 11.9552 Tf 9.299 0 Td[(V;ifjkj;jk0j~!D0;otherwise(1{6)WhereVisapositiveconstantand!DreferstotheDebyefrequency.UsingthissimpliedmodelinEquation 1{5 ,wendthatthesuperconductinggapisconstantforkwithinanenergy~!DoftheFermienergy,anditszero-temperaturevalue,intheweak-couplinglimit,isgivenby (0)=2~!Dexp)]TJ /F1 11.9552 Tf 9.298 0 Td[(1 NFV(1{7)Here,NFrepresentstheelectronicdensityofstates(DOS)attheFermilevelandNFV1.Takingthelimit(T)!0inEquation 1{5 yieldsTc: kBTc=1:13~!Dexp)]TJ /F1 11.9552 Tf 9.298 0 Td[(1 NFV(1{8) 14

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Figure1-2. (a)CrystalstructureofstoichiometriccuprateLa2CuO4(b)SchematicofCuO2plane.RedarrowsindicateapossiblealignmentofspinsonCuionsintheantiferromagneticgroundstateofLa2CuO4.TheshadedregionindicatesthehybridizationofCu-dx2)]TJ /F5 7.9701 Tf 6.586 .001 Td[(y2,O-pxandO-pyorbitalswhichleadstoaneective1-bandsystemincuprates.Bothguresarefrom[ 22 ].ReprintedwithpermissionfromAAAS. Theuniversalratio(0)=kBTc=1:764predictedbyBCStheory(compareEquations 1{7 and 1{8 )hasbeenfoundtobeingoodagreementwiththeexperiments[ 20 21 ]. 1.3OverviewofCuprateSuperconductorsCupratescontainCuO2layersasacommonstructuralunit,generallyseparatedbyinsulatingspacerlayers.Asanexample,Figure 1-2 (a)showsthecrystalstructureofstoichiometriccuprateLa2CuO4whichisthe"parentcompound"ofthersthigh-TccupratefamilyLa2)]TJ /F5 7.9701 Tf 6.587 0 Td[(xSrxCuO4alsoknownasLSCO.IntheCuO2planeCuformasquarelatticeandhasfour-foldcoordinationwithOatomsasshowninFigure 1-2 (b).ItwasrealizedearlybyAndersonthattheCuO2planeplaysthemostimportantroleinthephysicsofcuprates.SpacerlayersarethoughttoactasthechargereservoirswhichdopetheCuO2plane.Intheparentcompounds,CuhasavalencestateCu2+andelectronicconguration3d9leadingtoanoccupancyofoneelectronperCuO2unitcell.Forsuchahalf-lledsystem,bandtheorypredictsametallicstate.However,cuprateparentcompoundsare"Mott"insulatorswithachargegapoforder2eV.Inthehalf-lledconguration,anyhoppingofanelectronwillproduceadoublyoccupiedsite.DuetostrongCoulomb 15

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Figure1-3. Schematicphasediagramofcupratesasafunctionoftemperatureandholedoping.TN,TSDW,andTCDWrepresentsthetemperaturesatwhichantiferromagnetic,spin-densitywaveandchargedensitywaveordersset-in.T?representsthecrossovertemperaturebetweenstrangemetalandpsuedogapphases.Tc;onset,TSDW;onset,andTSC;onsetindicatesthetemperatureatwhichuctuationsofcharge,spinandsuperconductingorderbecomesapparent.Quantumcriticalpointsforsuperconductingandchargeorderareindicatedbyarrows.ReprintedbypermissionfromMacmillanPublishersLtd:Nature[ 16 ],copyright2015. repulsion,theenergycost(U)ofsuchadoubleoccupancyishugecomparedtothehoppingenergyt,prohibitingthehoppingofelectronsandthusblockingthechargeconduction.AlthoughchargeuctuationsareprohibitedintheMottphase,spinsarefreetohavedynamics.Infact,thevirtualhoppingofelectronsproducesaneectiveantiferromagneticcoupling(J)betweenspins,amechanismknownas"super-exchange"[ 23 ].ThisleadstoasimpleantiferromagneticorderingofspinsasshowninFigure 1-2 (b).Dopingtheparentcompoundwithholes,i.e.loweringthenumberofelectronsperunitcellfrom1to1)]TJ /F4 11.9552 Tf 12.6 0 Td[(p,wherepdenotesthedopinglevel,producesanincrediblyrichphasediagramasshowninFigure 1-3 takenfromRef.[ 16 ].Withincreasingp,theNeeltemperature(TN)decreasessharplyandatacriticaldopingp=pc(0.02inLSCO 16

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[ 22 ])antiferromagneticordervanishes.Almostimmediatelyafterwards,superconductivitysetsinatp=pmin(0.06inLSCO[ 24 ]).Withincreasingdoping,theTclinetracesadome-likeshape.ThemaximumTcoccursatthe"optimal"dopingp=poptdividingthephasediagrambyconventionintotworegionsnamelyunderdoped(ppopt).Ithasbeenestablishedbyphase-sensitiveexperiments[ 25 26 ]thatthesuperconductinggap(k)incuprateshasad-wavesymmetry,meaningthatthegapchangessignunder90orotation.Furthermore,angle-resolvedphotoemissionspectroscopy(ARPES)experiments[ 27 ]revealthatthegaphasdx2)]TJ /F5 7.9701 Tf 6.587 0 Td[(y2structurewhich,initssimplestform,canbeexpressedask(coskx)]TJ /F1 11.9552 Tf 12.589 0 Td[(cosky).ThegapmagnitudevanishesatfourpointsontheFermisurfacenear(=2;=2)callednodes,andachievesamaximumoneightpointsnear(;0)and(0;)calledanti-nodes.Thepresenceoflowenergy"nodal"excitationsleadstoverydierenttemperaturedependenceofthermodynamicandtransportobservables,suchasspecicheat,thermalconductivityandsuperuiddensity,comparedtothefully-gapedconventionalsuperconductors[ 28 ].Ontheoverdopedside,manyofthepropertiesofthesuperconductingstatecanbephenomenologicalyexplainedinasimpleBCStheoryframeworkwithadx2)]TJ /F5 7.9701 Tf 6.586 0 Td[(y2gap.ThisapproachissubstantiatedbytheARPESobservationofwelldenedquasiparticlesandahole-likeFermisurface,similartothebandtheorypredictions,suggestingaFermiliquid-likenormalstateinoverdopedcuprates[ 27 ].WeusethisframeworkinChapter 4 tostudytheeectsofmagneticandnon-magneticimpuritiesinover-to-optimallydopedcuprates.Theunderdopedside,ontheotherhand,isconsideredtobemuchmorestronglycorrelatedduetoitsproximitytotheMottinsulatorphase,andnotsuitableforadescriptionintermsofasimpleBCSframework.Inthiscontext,asimplebutextensivelystudiedmodelwhichcapturestheideaofsuperconductivityoriginatingfrom"dopingaMottinsulator"[ 24 ]isthet)]TJ /F4 11.9552 Tf 12.702 0 Td[(Jmodel[ 29 ].Asthestrongcouplinglimit(U!1)oftheHubbardmodel,itexcludesallcongurationsofstateswithanydoubleoccupancy 17

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ofsites.Althoughanexactsolutionisnotknownyet,variousapproximatetreatmentssuchasvariationalMonte-Carlomethods[ 30 ],Gutzwillermean-eldtheory[ 30 31 ],andslave-bosonapproach[ 24 32 33 ]havefoundasuperconductinggroundstatewithdx2)]TJ /F5 7.9701 Tf 6.587 0 Td[(y2symmetry.InChapter 5 ,weuseanextendedversionofthismodel,includingnext-nearesthoppingt0,tostudythecoexistencephaseofsuperconductivityandchargeorderwhichhasbeenobservedacrossthecupratefamilyintheunderdopedregion(seeFigure 1-3 ).Variousexperimentalprobessuchasnuclearmagneticresonance(NMR)[ 34 ],scanningtunnelingmicroscopy(STM)[ 35 ],andresonantX-rayscattering[ 36 ]revealthatthechargeorderisshort-range,uni-directional,andincommensuratewithad)]TJ /F1 11.9552 Tf 9.298 0 Td[(wavetypeintra-unitcellsymmetry.Moreover,itgenerallydoesnotaccompanyamagneticorder[ 36 ]andisthoughttobedistinctfromthe"stripeorder",whichconsistsofcombinedunidirectionalmodulatingmagneticandchargeorder,observedprimarilyinLa2)]TJ /F5 7.9701 Tf 6.587 0 Td[(xBaxCuO4nearp=1=8doping[ 37 ].BycrossingtheTclineintheunderdopedregimeoneentersintothe"psuedogap"regionofthephasediagram.Thepsuedogapischaracterizedbythesuppressionoflowenergysingle-particlespectralweightandisobservedthroughvariousexperimentalmeansincluding,butnotlimitedto,NMR,ARPESandspecicheatmeasurements(see[ 24 27 38 ]forreview).Inparticular,ARPESndsopeningofanincoherentgapintheanti-nodalregionofBrillouinzoneanddisconnectedarc-likefeatures,called"Fermiarcs",inthenodalregion[ 27 ].ThecrossoverlinedenotedbyT?inFigure 1-3 separatesthepsuedogapregimefromanotherexoticregime,namelythe"strangemetal"whichischaracterizedbylackofthewell-denedquasiparticlesandresisitivityvaryinglinearlywithtemperature.Thesetworegimesconstituteleastunderstoodandmostcontentiouspartofthecupratephasediagram. 1.4OverviewofIron-BasedSuperconductorsIronbasedsuperconductors(FeSCs)aredividedintotwocategories:Fe-pnictides(e.g.LiFeAs)andFe-chalcogenides(e.g.FeSe).Figure 1-4 (a)showscrystalstructureof 18

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Figure1-4. (a)Crystalstructureofrepresentativemembersofthevariousfamiliesofiron-basedsuperconductors.Featomsareshowninredandpnictogen/chalcogenatomsareshowningold.(b)Schematicofiron-pnictogen/iron-chalcogenlayer.Here,Featoms,showninred,formasquarelatticeandpnictogen/chalcogenatoms,showningold,arelocatedaboveandbelowtheFe-planeinanalternatingmanner.(a)and(b)arereprintedbypermissionfromMacmillanPublishersLtd:Nature[ 39 ],copyright2010.(c)AgenericFermisurfaceofiron-basedsuperconductorsinnormalstate.Greendottedlinesdenotetheboundaryofthe1-FeBrillouinzone.)]TJ 0 -14.446 Td[(representstheBrillouinzonecenter.XandYrepresenttheBrillouinzoneedgecenters.Reproducedfrom[ 40 ],withthepermissionoftheAmericanInstituteofPhysics. representativemembersofthefourmoststudiedfamiliesofironbasedsuperconductors(FeSCs)namely11,111,122and1111,reectingtheformulasoftheirstoichiometricparentcompounds.AllthesestructuresarecharacterizedbyacommonFePnorFeChlayers,Pn(Ch)beingthepnictogen(chalcogen)elementssuchasAs(Se),separatedbyspacerlayers.LikeCuO2layersincuprates,FePn/FeChlayersplaythemostimportantroleinthephysicsofFeSCs.Thespacerlayerprimarilyactsasareservoirofchargecarriers,asincuprates,orasourceofchemicalpressure.AsshowninFigure 1-4 (b),FeformsasquarelatticeandPnorChionsarelocatedaboveandbelowthecenterofsquaresinalternatingfashion.Apartfromthesebulkmaterials,monolayersofFeSe, 19

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Figure1-5. Schematicphasediagramofiron-basedsuperconductorsasafunctionoftemperatureanddoping.SDWandSCreferstospindensitywaveandsuperconductingphases,respectively.Reproducedfrom[ 40 ],withthepermissionoftheAmericanInstituteofPhysics. epitaxiallygrownonstrontiumtitanate(STO)substrate,haveattractedenormousattentionrecently[ 41 ]owingtotheirhighTc(110Kfromresistivemeasurements[ 42 ]and70KfromARPESmeasurements[ 43 ])whichisanorderofmagnitudelargerthanthebulkFeSe(Tc8K).Inparentcompounds,FehasavalencestateFe2+andelectronicconguration3d6.Unlikecuprates,whereonlyoned-orbital(dx2)]TJ /F5 7.9701 Tf 6.586 0 Td[(y2)contributestotheFermisurface,rst-principlesbandstructurecalculations[ 44 ]andARPESresults[ 45 ]showthatallved-orbitalscontributetotheFermisurfaceofFeSCparentcompoundswithdxy,dxz,anddyzhavingthelargestweight.Moreover,itisfoundthatthehole-likeandelectron-likebandscrosstheFermienergyneartheBrillouinzonecenter()-326(point)andBrillouinzoneboundary(XandYpoints),leadingtoaFermisurfacewithnearlycircular"holepockets"atthe)-327(pointandelliptical"electronpockets"attheXandYpointsasshowninFigure 1-4 (c).Thismulti-bandnatureplaysacentralroleinthephysicsofFeSCs.AschematicphasediagramofarepresentativeFepnictideasafunctionofelectronandholedopingisshownFigure 1-5 .Thehightemperaturephaseofundopedparentcompoundisaparamagneticmetalwithatetragonallattice.Loweringthetemperature 20

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resultsinanalmostsimultaneousstructuralandmagnetictransitiontoanorthorhombicspindensitywave(SDW)phase[ 46 ].Inthisphase,generally,orderedspinsforma"stripe"pattern,characterizedbyferromagneticorderinonedirectionandantiferromagnetic(Neel-type)orderinother.Indopedcompounds,thestructuraltransitionprecedesthemagnetictransitioni.e.Ts>TN,whereTsandTNrepresentsstructuralandmagnetictransitiontemperatures,respectively.TheblueregionbetweenTsandTNphaselinesasshowninFigure 1-5 iscalledthe"Nematic"phase,borrowingtheterminologyfromliquid-crystalswhereitcorrespondstoaphasewithbrokenrotationalsymmetrybutpreservedtranslationalsymmetry.ThenematicphaseinFe-basedmaterialsisbelievedtohaveanelectronicorigin[ 47 ],asthelargeanisotropyobservedinvariousexperiments,suchasin-planeresistivitymeasurements[ 48 ],STM[ 49 ],ARPES[ 50 ],andtorquemagnetometry[ 51 ],cannotbeexplainedbytheverysmallorthorhombiclatticedistortion.Atlowtemperatures,dopingtheparentcompoundwithelectronsorholestypicallyleadstotheappearanceofthesuperconductingphase.TheTclinetracesanasymmetricdome-likeshapewithmaximumTctakingvaluesaslargeas57Kin1111materials[ 52 ].NMRexperimentsonmanyFeSCs[ 53 ]stronglyindicatethatthesuperconductingstateisspinsinglet,makingacaseforevenparitysuperconductinggapsuchass-waveandd-wave.However,unlikecuprates,thereisnodenitiveexperimentalevidenceofaparticulargapsymmetryinFeSCs.Atmoderatedopinglevels,mostoftheexperimentspointtowardsamulti-bands-wavetypegapwithtwopossiblescenarios,namelys++[ 54 ]ands[ 55 ].Ins++(s)pairingthesuperconductinggaphassame(opposite)signonelectronandholepockets.Although,thereisno"smokinggun"evidencetofavoroneovertheother,siswidelybelievedtobethepairingsymmetryinmostoftheweak-to-moderatelydopedFeSCs.InChapter 3 ,westudytheimpurityboundstatesinthesuperconductingFeSewithspairing. 21

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Figure1-6. (a)Schematicofascanningtunnelingmicroscope.AbiasvoltageVisappliedbetweenasharpmetallictipandthesample,whichresultsinameasurabletunnelingcurrent.(b)Schematicoftip-sampletunneling.Whenapositivebiasisappliedonthesample,itsFermilevellowersbyanamounteVcomparedtothetipFermilevel.Electronstunnelprimarilyfromlledstatesinthetiptoemptystatesinthesample.Reprintedfrom[ 56 ].CopyrightIOPPublishing.Reproducedwithpermission. 1.5STMStudiesofSuperconductorsScanningtunnelingmicroscopy(STM),withitsuniquecapabilitiestoimagetheelectronicstructureofsolidswithatomicresolution,hasbeenextensivelyusedtostudysuperconductors[ 56 { 59 ].TheSTMapparatusconsistsofanatomicallythinmetallictipseparatedfromaconductingsamplesurfacethroughavacuumbarrierasshowninFigure 1-6 (a).Thepositionofthetip(r)canbecontrolledwithasub-Aresolutionusingpiezoelectricactuators.Applyingabiasvoltagebetweenanappropriatelyplacedtipandthesampleresultsinameasurabletunnelingcurrentthroughthevacuumbarrier.Undervariousjustiableassumptionssuchaslowtemperature,energy-independenttunnelingmatrixelement,andfeaturelessdensityofstatesinthetipmaterial,dierentialtunnelingconductanceturnsouttobeproportionaltothelocaldensityofstates(LDOS)ofthesample[ 56 57 ], dI dV/s(r;eV)(1{9) 22

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Figure1-7. (a)ComparisonoftunnelingconductancespectradirectlyaboveZnimpuritysiteandatasitefarawayfromimpurityinsuperconductingBi2Sr2CaCu2O8+dopedwithZn.(b)SpatialconductancemaparoundZnimpurityatresonantenergy()]TJ /F1 11.9552 Tf 9.299 0 Td[(1:2meV)showingintensitymaximumrightabovetheimpuritysite.BothguresarereprintedbypermissionfromMacmillanPublishersLtd:Nature[ 60 ],copyright2000.(c)Localdensityofstatesaroundimpuritysiteattheresonantenergyresultingfromasimpletheoreticalmodelofastrongnon-magneticimpurityinad-wavesuperconductor. Here,Iisthetunnelingcurrent,Vistheappliedbias,sisthesampleLDOSandeistheelectroniccharge.Thus,conductancemeasuredthetippositionrandabiasvoltageVisequivalenttothesample'sLDOSatthespatialpositionrandenergy!=eV.Thesub-AresolutionspectroscopicimagingofLDOSprovidesindispensableinformationaboutthesuperconductinggap.Furtherinformationaboutthenatureofsuperconductivitycanbeobtainedbystudyingtheatomic-scaleelectronicstructuresaroundimpuritiesinsuperconductors.Finally,momentumresolveddetailsoftheFermisurfaceandsuperconductinggapcanbeobtainedusing'quasi-particleinterference(QPI)'techniquewhichessentiallyinvolvestheFouriertransformoftheconductancemaps.InlayeredsuperconductorssuchascupratesandFeSCs,generally,thelayerexposedtotheSTMtipwherematerialcleavesturnsouttobedierentfromthe'active'layer 23

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responsibleforthesuperconductivity(CuO2layerincupratesandFelayerinFeSCs).Inthisscenario,adirectcomparisonofSTMresultswiththeoreticalcalculationswhichonlytaketheactivelayer'sdegreesoffreedomintoaccountcanleadtomisinterpretationofdata.Thisisnicelyillustratedbythefamous"Zn-paradox"inthecupratecompoundBSCCOwhichcleavesattheBiOplane,twolayersabovetheCuO2plane.STMconductancespectratakenoveraZnimpurityinsuperconductingBSCCO[ 60 ]revealthatZninducesanin-gapboundstateasshowninFigure 1-7 (a).Theconductancemaptakenattheboundstateenergy()]TJ /F1 11.9552 Tf 9.299 0 Td[(1:2meV)showsintensitymaximaattheZnsite(Figure 1-7 (b))whereastheoreticalcalculationsofimpurity-inducedstatesinad-wavesuperconductor(basedonasimple,on-sitenonmagneticpotential)predictLDOSminimaattheZnsite[ 61 ].ItwasproposedthatthisdiscrepancycanbereconcilediftheBiOlayerisassumedtoactasa"lter"providinganon-trivialtunnelingpathfromSTMtiptoZnatom[ 62 ].Toaddresstheaforementionedissue,Ihavedevelopedanovel'BdG+W'methodwhichcombinesthesolutionofBogoliubov-deGennes(BdG)equations,widelyusedtostudyinhomogenoussuperconductivity,andrstprinciplesWannierfunctionstoyieldLDOSatSTMtippositionwithsub-Aresolution.Here,theBdGequationsaresolvedself-consistentlyonatwo-dimensionallatticeintheactivelayer,andtheeectofatomsinotherlayersisincludedthroughWannierfunctionsdenedin3Dcontinuumspace.InChapter 2 ,Idescribethemathematicalframeworkbehindthe'BdG+W'method.Astherstapplication,inChapter 3 ,Istudytheeectsofasinglenon-magneticimpurityinthesuperconductingFeSe,andshowthatthegeometricaldimerstatesobservedinSTMexperiments[ 63 ]canbeunderstoodasaconsequenceofsimpledefectslocatedonFesitesduetohybridizationwiththeSestate.InChapter 4 ,IcomputetheimpurityinducedresonantLDOSandQPIpatterns,withintheBdG+Wframework,insuperconductingBSCCOdopedwithZnandNiimpuritiesandshowthattheresultsareinexcellentagreementwiththeSTMexperiments[ 60 64 ].InChapter 5 ,usingrenormalized 24

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mean-eldtheoryandWannierfunctionbasedanalysis,Ishowthattheextendedt)]TJ /F3 11.9552 Tf 11.956 0 Td[(JmodelsupportsincommensuratechargeorderedstateswhichdisplaycharacteristicsverysimilartothechargeorderedstatesobservedthroughSTMexperiments[ 65 66 ]inunderdopedcuprates.Finally,inChapter 6 ,Ipresentthesummaryandconclusionsofmydissertationwork. 25

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CHAPTER2BOGOLIUBOV-DEGENNES-WANNIERAPPROACHInthischapter,IwilldiscussthemathematicalformulationbehindtheBdG+WschemeintroducedinSection 1.5 .Someofthematerialpresentedhereisbasedonapublishedpaper[ 67 ].Allthepublishedcontents(excerptsandgures)arereprintedwithpermissionfrom PeayushChoubey,T.Berlijn,A.Kreisel,C.Cao,andP.J.Hirschfeld,Phys.Rev.B90,134520(2014) ,copyright2014bytheAmericanPhysicalSociety. 2.1ImpuritiesinSuperconductorsStudyingcontrolleddisorderinacorrelatedsystemcanprovideimportantinsightsaboutitsgroundstate[ 68 ].Eectsofdisorderinconventionals-wavesuperconductors,toagreatextent,areencapsulatedin"Anderson'stheorem"[ 69 ]andtheAbrikosov-Gorkov(AG)theory[ 70 ].Theformerstatesthatthenon-magneticimpuritiesdonotaectTcorthesuperconductinggapduetotime-reversalinvariance.Magneticimpurities,ontheotherhand,breaktime-reversalsymmetryandleadtothesuppressionofTc,andtherateofsuppressionisdescribedbytheAGtheoryforweakscatterers.Forstrongerpotentials,impurity-inducedin-gapboundstatesarecreated,oftenreferredastheYu-Shiba-Rusinovstates[ 71 { 73 ].Anderson'stheoremdoesnotapply,however,ifthesuperconductinggaphasastrongmomentumdependence,whichisoftenthecasewiththeunconventionalsuperconductors.Insuchascenario,evennon-magneticimpuritiescansuppressTcandproducein-gapstates.Thus,anexperimentalobservationofTcsuppressionandthepresenceofin-gapstates(oritsindirectmanifestation)canbeanindicationofunconventionalpairing.Earlierstudiesofdisordereectsweremainlyfocusedonthebulkpropertiesofsuperconductors[ 74 ]suchasTc,penetrationdepth,specicheatandplanartunneling.Inrecenttimes,however,thefocushasshiftedtowardslocalstudiesofimpurityeects[ 56 75 ]mostlyowingtorapidtechnologicaladvancementsinSTMtechniques.Therstobservationofthein-gapboundstateslocalizedaroundamagneticimpurityina 26

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conventionalsuperconductorwasreportedbyYazdanietal.in1997[ 76 ].Subsequently,detectionofthevirtualboundstatesinducedbyanon-magneticimpurity(Zn)[ 60 ]andamagneticimpurity(Ni)[ 64 ]inthecupratecompoundBi2Sr2CaCu2O8+,conrmedtheearliertheoreticalpredictions[ 61 ]ofsuchstatesinad-wavesuperconductor.Thein-gapboundstatesobservedinmanybulkFeSCs[ 63 77 { 79 ]haveindicated,althoughnotconclusively,thatasignchangingsisthemorelikelypairingsymmetrycomparedtothenon-signchangings++.Morerecently,theabsence(presence)ofboundstatesinducedbynon-magnetic(magnetic)impuritiesinthemonolayerFeSegrownonSTOsubstratehasbeentakenasevidencethatthepairingstateinthissystemshouldbeisotropics++-wave[ 80 ].Thisisnotnecessarilycorrect,however,ifthepairinginvolvesstatesawayfromtheFermilevel[ 81 ].Here,wewillpresentasimpleschemetostudytheeectsofanimpurityinasuperconductorinthecontextofSTMexperiments.WeassumethataBCS-likemean-elddescription,withpropergapsymmetry,candescribethesuperconductingstate.Althoughsuchadescriptionneglectscorrelationeectsortreatsthemataverycrudelevel(e.g.throughanoverallbandrenormalization),ithasbeenusedsuccessfullytocapturequalitativeaspectsoftheimpurityeectsinsuperconductors[ 75 ].Furthermore,thenormalstateofamaterialunderstudyisassumedtobewelldescribedbyatight-bindingmodelderivedfromthedensityfunctionaltheory(DFT)calculations.ThisisjustiedincaseofoverdopedcupratesandmanyFeSCswhereARPESdeterminedFermisurfacedisplaysstrongsimilaritywiththecorrespondingDFTresults[ 27 45 ].Forstrongly-correlatedmaterialssuchasoptimal-to-underdopedcuprates,itshouldstillbeapplicableforsymmetry-relatedquestions. 2.2Mean-FieldTheory 2.2.1HamiltonianThesimplestHamiltoniandescribingeectsofimpuritiesinasuperconductorconsistsofthreeterms,namelyakineticenergytermH0describingthenon-interactingelectronsin 27

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thenormalstate,aBCStypepairinginteractiontermHBCSdescribingsuperconductivity,andanimpuritytermHimp. H=H0+HBCS+Himp;(2{1)ToexpresstheHamiltonianinatight-bindingmodel,werstconstructasingleparticlebasisofWannierfunctionswiusingtheDFTwavefunctions.Here,irepresentsanunitcellwiththelatticevectorRiandlabelstheorbitaldegreesoffreedomwithintheunitcell.Subsequently,hoppingparametersareobtainedleadingtothetight-bindingmodelforthenormalstate, H0=Xij;;tijcyicj)]TJ /F4 11.9552 Tf 11.955 0 Td[(0Xi;;cyici:(2{2)Here,cyi(ci)creates(destroys)anelectronwithspinintheWannierorbitallocatedintheunitcelli.tijistheamplitudeofhoppingfromtheunitcelliandorbitaltotheunitcelljandorbital.Theaverageelectronllingcanbesetbytuningthechemicalpotential0.OftentheWannierfunctioninformationisdiscardedafterconstructingthetight-bindingmodel,butinourformalismitplaysacrucialroleaswewillshowinSection 2.3.2 .ThesuperconductingstateisaccountedforviaaBCS-likepairingtermas HBCS=)]TJ /F10 11.9552 Tf 11.871 11.357 Td[(Xij;Vijcyi"cyj#cj#ci";(2{3)whereVij>0isthereal-spacepairpotentialbetweenorbitalintheunitcelliandorbitalintheunitcellj.Vijcanbeeitherputin"byhand"toyieldaparticulargapmagnitudeandsymmetry,oritcanbedeterminedfromamicroscopictheoryofpairing.Forexample,inChapter 3 weusepairpotentialsdeterminedfromaspinuctuationtheoryofpairinginFeSCs,andinChapter 4 weputitin"byhand",choosinganon-zerovalueonlyforthenearest-neighborinteraction,toyieldad-wavegapwiththeexperimentallyobservedmagnitude. 28

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Defectsinamaterialcanbeofmanytypessuchastwinboundaries,grainboundaries,interstitialatoms,andsubstitutionalatoms.Here,weconsiderthesimplestkindofdefect:substitutionalimpurityatoms.ImpurityatomsinteractwiththeconductionelectronsthroughthescreenedCoulombpotentialwhich,inthelimitofperfectscreening,canbeaccountedforviathefollowingHamiltonian, Himp=XVimpcyi?ci?;(2{4)where,i?istheimpurityunitcellandVimpistheimpuritypotentialtakentobeproportionaltotheidentitymatrixinorbitalspaceforthesakeofsimplicity.Thismodelcanbeeasilygeneralizedtoincludeanextendedrangepotential,orbitaldependence,aswellasmagneticscattering.WewillreturntothelatterinSection 4.5 2.2.2BdGEquationsTodiagonalizetheHamiltonianH(Equation 2{1 ),weusethemean-eldapproximationanddecomposethequarticoperatorinEquation 2{3 intotwobilinearones,namelythepaircreationoperatorcyi"cyj#andthepairdestructionoperatorcj#ci".Thefollowingistheresultingmean-eldHamiltonian: HMF=Xij;;tijcyicj)]TJ /F4 11.9552 Tf 11.955 0 Td[(0Xi;;cyici)]TJ /F10 11.9552 Tf 12.534 11.358 Td[(Xij;hijcyi"cyj#+H.c.i+X;Vimpcyi?ci?;(2{5)whereH.c.impliesHermitianconjugate.Thesuperconductingorderparameterisgivenby ij=Vijhcj#ci"i:(2{6)ThemeaneldHamiltonianisquadraticinelectronoperatorsandcanbediagonalizedusingthefollowingspin-generalizedBogoliubovtransformation, ci=Xnhunin+vniynicyi=Xnuniyn+vnin;(2{7) 29

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whererepresentstheoppositeofspin,theoperatorsareBogoliubovquasiparticleeldsthatsatisfythefermioncommutationrules,andthecoecientsuandvsatisfyPnjunij2+jvnij2=1.WhenappliedtoHMF(Equation 2{5 )theabovetransformationyieldsaHamiltoniandiagonalintheoperators: HMF=E0+XnEnynn(2{8)Here,E0isthegroundstateenergyandEnisthequasiparticleenergylabeledbyindicesnand.ItshouldbenotedthatthesuminEquation 2{8 runsover(n;)correspondingtopositiveeigenvalues(En>0).Comparingthecommutator[HMF;ci]obtainedfromthetwodenitionsofmean-eldHamiltonian(Equation 2{5 andEquation 2{8 )yieldstwosetsoftheBdGequations,relatedtoeachotherbytheparticle-holesymmetry(seeAppendix A fordetails).Wechoosetoworkwiththefollowingset Xj0B@ij")]TJ /F1 11.9552 Tf 9.298 0 Td[(ij)]TJ /F1 11.9552 Tf 9.299 0 Td[(ji)]TJ /F4 11.9552 Tf 9.299 0 Td[(ij#1CA0B@unj"vnj#1CA=En"0B@uni"vni#1CA:(2{9)Here,ij=tij)]TJ /F4 11.9552 Tf 13.127 0 Td[(0ij)]TJ /F4 11.9552 Tf 13.127 0 Td[(VimpiiijwhererepresentstheKroneckerdeltafunction.Mean-eldssuchastheorbital-resolvedelectrondensityni=hcyiciiandsuperconductinggapijcanbeexpressedas ni"=Xnjuni"j2f(En")ni#=Xnjvni#j2(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(f(En"))ij=VijXnuni"vnj#f(En"):(2{10)Here,frepresentstheFermifunctionandthesumrunsoverallvaluesofn.ThederivationofEquations 2{9 and 2{10 isstraightforwardbutlengthy,andisprovidedinAppendix A .TheentriesoftheBdGmatrixinEquation 2{9 dependimplicitlyupontheeigenvaluesandeigenvectorsthroughthemean-eldsdenedinEquation 2{10 ;thus 30

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theequationmustbesolvedself-consistently.Todoso,werstguesstheinitialvaluesofmean-elds(n";n#;)andchemicalpotential0.ThentheeigenvaluesandeigenvectorsoftheBdGmatrixareobtained,which,inturn,areusedtocomputemean-elds(Equation 2{10 ).Theprocessisiterateduntilthesuperconductinggapandelectrondensityconvergeuptoadesiredaccuracy. 2.3CalculationofSTMObservablesAsexplainedinSection 1.5 ,underawidesetofassumptions,theSTMtunnelingconductancemeasuredattheappliedbiasV=!=eisproportionaltothesampleLDOS(r;!)whereristheSTMtipposition.Thus,tocomparewiththeSTMresultswemustcalculatethelocaldensityofstatesinthe3Dcontinuumspace.Inthefollowing,werstdescribethecalculationoflatticeLDOS,thequantitymostoftenusedinliteraturetocomparewiththeSTMobservations.Then,wederivetheexpressionforthecontinuumLDOSwhichisamoreappropriatequantitytocompare. 2.3.1LatticeLDOSUsingtheself-consistentsolutionoftheBdGequations(Equation 2{9 ),latticeGreen'sfunctionsGij(!),forthepropagationbetweensitesi,jofarealspacelatticecanbeconstructedusingthefollowingformula(seeAppendix A forthederivation) Gij(!)=Xn>0uniunj !)]TJ /F4 11.9552 Tf 11.955 0 Td[(En+i0++vnivnj !+En+i0+;(2{11)where0+isthearticialbroadening,takentobemuchsmallerthansmallestphysicalenergyintheproblem,andn>0indicatesthatthesumistobeperformedovereigenstateswithpositiveeigenvaluesonly.Foreasiernumericalimplementation,theaboveexpressioncanbefurthersimpliedasbelow(seeAppendix A forthedetails) Gij"(!)=Xnuni"unj" !)]TJ /F4 11.9552 Tf 11.956 0 Td[(En"+i0+Gij#(!)=Xnvni#vnj# !+En"+i0+(2{12) 31

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Here,thesumrunsoverallvaluesofn.Theorbitally-resolvedlatticeLDOScanbeobtainedusingtheimaginarypartofthediagonallatticeGreen'sfunctions Ni(!)=)]TJ /F1 11.9552 Tf 11.102 8.088 Td[(1 Im[Gii(!)](2{13)whereImrepresentstheimaginarypart.ThetotallatticeLDOScanbefoundbysummingoverallorbitalsi.e.Ni(!)=PNi(!). 2.3.2ContinuumLDOSThecontinuumLDOSisrelatedtotheretardedcontinuumGreen'sfunctionG(r;r0;!),denedinthe3Dcontinuumspace,as (r;!)=X)]TJ /F1 11.9552 Tf 11.102 8.088 Td[(1 Im[G(r;r;!)](2{14)whererisacontinuumrealspaceposition.ThecontinuumGreen'sfunctionitselfcanbedenedinthetimedomainintheusualway G(rt;r0t0)=)]TJ /F4 11.9552 Tf 9.299 0 Td[(i(t)]TJ /F4 11.9552 Tf 11.955 0 Td[(t0)h[(rt);y(r0t0)]+i;(2{15)whereisthestepfunction,[]+representstheanti-commutator,and(rt)denotestheeldoperatorthatannihilatesanelectronwithspinlocatedatpositionrattimet.Now,totransformthesingle-particlebasisfromthecontinuumspacertothelatticespace(i;),weutilizethefollowingrepresentationoftheeldoperators (rt)=Xi;wi(r)ci(t);(2{16)wherewilabelstheWannierorbitallocatedintheunitcelli.PuttingEquation 2{16 inEquation 2{15 yields G(rt;r0t0)=Xij;Gij(t;t0)wi(r)wj(r0);(2{17)whereGij(t;t0)=)]TJ /F4 11.9552 Tf 9.299 0 Td[(i(t)]TJ /F4 11.9552 Tf 12.711 0 Td[(t0)h[ci(t);cyj(t0)]+iisthelatticeGreen'sfunctioninthetimedomain.TakingFouriertransformwithrespecttotimeonbothsidesleadstothe 32

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frequencydomainresult G(r;r0;!)=Xij;Gij(!)wi(r)wj(r0):(2{18)ThususingEquation 2{18 withr0=rinconjunctionwithEquation 2{14 ,thecontinuumLDOScanbeobtained.NotethatthelatticeLDOS(Equation 2{13 )iscompletelydeterminedbythelocalanddiagonal(inorbitalspace)latticeGreen'sfunction,whereasthecontinuumLDOSincludescontributionsfromthenon-localando-diagonallatticeGreen'sfunctionsG6=i6=j;(!)too.Thenon-localtermscanleadtointerferenceeects,asthesignoftheirimaginarypartisnotxed.SucheectscancauseaqualitativechangeinthespectralfeaturesofthelatticeLDOS.WediscussthisissueindetailinSection 4.3 inthecontextofoverdopedcuprates. 2.4NumericalImplementationTostudytheeectsofanimpurityonthelocalelectronicstructureinasuperconductorwithintheBdG+Wframework,werstneedtheDFTderivedWannierfunctionsandtight-bindingparameterscharacterizingthenormalstateofthematerial.WannierfunctionscanbeconstructedfromtheDFTwavefunctions,spanninganenergywindowofseveraleV,usingaprojectionanalysis[ 82 ]oramaximallocalizationroutinesuchasWannier90[ 83 ].Thensuperconductivityisintroducedusingreal-spacepairpotentials(Vij)thatcanbeeitherobtainedfromamicroscopictheoryorcanbeputin"byhand"toyieldanexperimentallyrelevantgapmagnitudeandsymmetry.Theimpuritypotentialisusuallyintroduced"byhand",butamoreaccurateestimatecanbeobtainedfromtherst-principlescalculations.ThenextstepistosolvetheinhomogeneousBdGequationsinaself-consistentway.WetakeaNNsquarelatticewitheachlatticesiterepresentingaunitcellwithNorbWannierorbitals,andsolveEquation 2{9 inconjunctionwithEquation 2{10 onthislatticeusingthefollowingsteps: 33

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1. Chooseinitialguessesforelectrondensityn,superconductinggap,andchemicalpotential0. 2. ConstructtheBdGmatrixinEquation 2{9 andnditseigenvaluesandeigenvectors. 3. Findmean-eldsusingEquation 2{10 .Letthevaluesthusobtainedbennewandnew. 4. Checkconvergence.Ifjnnew)]TJ /F4 11.9552 Tf 12.352 0 Td[(nj,jnew)]TJ /F1 11.9552 Tf 12.353 0 Td[(j,jn0)]TJ /F4 11.9552 Tf 12.352 0 Td[(navgj<,thenconvergenceisachievedandthereisnoneedforfurtheriterations;otherwisegotothestep5.Here,isthedesiredaccuracy,navgistheaverageelectrondensityateachiteration,andn0isthedesiredelectrondensity. 5. Updatethemean-eldsandchemicalpotential. n!nnew+(1)]TJ /F4 11.9552 Tf 11.955 0 Td[()n!new+(1)]TJ /F4 11.9552 Tf 11.955 0 Td[()0!0+(n0)]TJ /F4 11.9552 Tf 11.955 0 Td[(navg)(2{19)where0<;<1.Here,mean-eldshavebeenmixedwiththeirvaluesinthepreviousiteration.Suchanupdateschemeisveryoftenneededfortheproperconvergenceoftheself-consistentsolution. 6. Gotostep2.Usingtheself-consistentsolutionoftheBdGequations,thelatticeGreen'sfunctionmatrix(Equation 2{12 )isconstructedina"supercell"set-uptoachieveabetterspectralresolution.ThesupercellmethodisdescribedindetailinAppendix B .LocalanddiagonalelementsofthelatticeGreen'sfunctionmatrixyieldthelatticeLDOSasinEquation 2{13 .TondthecontinuumLDOS,valuesoftheWannierorbitalsattheSTMtipposition,usuallyafewAngstromsabovetheexposedlayer,arerstextracted.ThenusingEquations 2{15 and 2{14 thecontinuumLDOSisobtained. 34

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CHAPTER3IMPURITYINDUCEDSTATESINIRONBASEDSUPERCONDUCTORSInthischapter,Iwilldiscusstheeectsofasinglenon-magneticimpurityinthesuperconductingFeSe.Mostofthematerialpresentedhereisbasedonapublishedpaper[ 67 ].Allthepublishedcontents(excerptsandgures)arereprintedwithpermissionfrom PeayushChoubey,T.Berlijn,A.Kreisel,C.Cao,andP.J.Hirschfeld,Phys.Rev.B90,134520(2014) ,copyright2014bytheAmericanPhysicalSociety. 3.1MotivationImpurityinducedstateshavebeenobservedinSTMstudiesonvarietyofFeSCsincludingFeSe[ 63 84 85 ],LiFeAs[ 77 86 ],NaFeAs[ 49 ],LaOFeAs[ 87 ]andNa(Fe0:97)]TJ /F5 7.9701 Tf 6.587 0 Td[(xCo0:03Cux)As[ 79 ].Asanexample,Figure 3-1 (a)showstheSTMtopographyofFeSelmgrownonSiC(001)substrate.FeSecleavesattheSelayerandtheFelayerresidesbelowit1.Defectsappearasatomic-scalebrightdumbbelllikestructures("geometricdimers"),labeledasand,andaccompanyinglonger,unidirectionaldepressionsofLDOS("electronicdimers").Theelectronicdimersareorientedat45withrespecttogeometricdimers,andextendtoalength16aFe-Fe,whereaFe-FedenotestheFe-Febondlength,asshowninFigure 3-1 (c).Similarfeatureshavebeeninterpretedasemergentdefectstatesintheorderedmagneticphase[ 88 ].Azoomed-inviewofgeometricdimers(Figure 3-1 (b))showsthemtobecenteredattheFeatomicsitesinthesubsurfaceFelayerwithbrightlobesextendingtotheneighboringSeatomsinthetop-mostSelayer.DuetomutuallyorthogonalarrangementofSeatomsintoplayeraroundtwoinequivalentFeatomsintheunitcell,geometricdimersappearintwoorthogonalorientationslabeledasandinFigure 3-1 .Itisnotknownconclusivelywhetherthesedefectstructuresarecausedbyadatoms,Fevacancies,orsiteswitching,however,arecentrst-principlesstudyclaimsthattheFevacancyisthemostprobablecandidate[ 89 ]. 1seeFigure 1-4 (a)forthedetailsoftheFeSecrystalstructure 35

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Figure3-1. (a)STMtopographyofFeSethinlmgrownonSiC(001)substrate.Defectsappearasbrightatomic-scaledumbbells("geometricdimers"),surroundedbylarger"electronicdimers"characterizedbyunidirectionaldepressionsinthedensityofstates.Atwinboundaryrunsalongthediagonalofthegureacrosswhichthecrystallineaxes,denotedbyaandb,switchtheirorientation.(b)Zoomed-inviewofdefects.Themutuallyorthogonalgeometricdimers,labeledasand,arecenteredattwoin-equivalentFeatomicsites,inthesubsurfaceFelayer,anditsbrightlobesextendtotheneighboringSeatomsinthetop-mostSelayer.(c)Alarger,zoomed-inviewofdefectsshowingelectronicdimers,markedbyyellowdottedlines,whichextendtoalength16aFe-Fe.SubsurfaceFeatomsaremarkedbybluedotsingures(b)and(c).Reprintedwithpermissionfrom[ 63 ],copyright2014bytheAmericanPhysicalSociety. TheSTMexperimentsclearlysuggestthatthegeometricdimerstatesinvolvecouplingoftheimpuritywithSe(orAsincaseofFe-pnictides[ 49 77 79 ])statesinthetopmostsurfaceexposedtotheSTMtip,andhence,itcannotbecapturedbyaFe-latticeonlytheoreticalcalculation,mostoftenemployedtostudytheimpuritystatesinFeSCs[ 90 { 93 ].TheBdG+WapproachdescribedinChapter 2 ,ontheotherhand,canaccountforthelocalC4symmetrybreakingduetoFe/Seatomsviatherst-principlesWannierfunctions.Here,usinga10-orbitaltightbindingmodelforFeSeandcorrespondingWannierfunctionsderivedfromtheDFTcalculations,weshowthatthegeometricdimerstatescanbeunderstoodastheconsequenceofimpurityinducedin-gapboundstatesinanssuperconductorhybridizingwithSestates.Inthefollowingsections,werstdescribethenormalstateelectronicstructureofFeSewithinaDFTderivedtight-bindingmodel.Next,weusepairpotentialsobtained 36

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Figure3-2. (a)TopandsideviewoftheisosurfaceplotsofFeSedxyWannierorbitalonFe(I)andFe(II)siteobtainedatisovalue0.03bohr)]TJ /F6 7.9701 Tf 6.586 0 Td[(3=2.RedandblueindicatesignoftheWannierorbital.(b)IsosurfaceplotsofremainingFe-dWannierorbitalsononFe(I)siteatthesameisovalue. fromthespin-uctuationtheorytorealizethespairingsymmetryanddescribethepropertiesofthehomogeneoussuperconductingstate.Then,withinBdG+Wframework,westudytheresponseofasinglepoint-likesubstitutionalimpurityandcalculatethecontinuumLDOSatatypicalSTMtippositionaswellasatopographicmapoftheimpuritystateshowingtheemergenceofgeometricdimers. 3.2NormalStatePropertiesofFeSeFeSecrystallizesinthesymmetrygroupP4=nmmwithlatticeconstantsa=b=7:13bohr,c=10:44bohrandfractionalpositionofSeatomsz=0:265c[ 94 ].Startingwiththisexperimentalcrystalstructureinformation,theelectronicstructureofFeSewascalculatedusingtheWIEN2Kpackage[ 95 ].DFTresultsshowthatthebandsclosetotheFermienergyhavedominantcontributionsfromFed-orbitalswithsmallweightsofSep-orbitals.Subsequently,a10-orbitalWannierbasiswasconstructedbyprojectingtheDFTwavefunctions[ 82 ]withintheenergyrange[-2.5eV,3eV]toFed-orbitalscorrespondingtothetwoin-equivalentFeatomsnamelyFe(I)andFe(II).Figure 3-2 (a)showsthetopandsideviewsofdxy-WannierorbitalsonFe(I)andFe(II)sites.The 37

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WannierorbitalisexponentiallylocalizedontheFesiteswithfeaturesthatcanbeassociatedwiththecorrespondingatomicorbital.Moreover,ithassignicantweightsonothernearbyatomssuchasFeandSeatomsinthesameunitcellandFeatomsinthetwoadjacentunitcells.Furthermore,atheightsseveralAngstromsabovetheSeplane,wheretheSTMtipwouldbetypicallyplaced,thedominantcontributiontotheWannierorbitalsisderivedfromtheSe-pstates,asclearlyseeninthesideviewoftheisosurfaceplots(Figure 3-2 (b),andbottomrightplotsinFigure 3-2 (a)).InSection 3.4 ,weshowthatthisparticularfeatureoftheWannierorbitalsgivesrisetothegeometricdimerimpuritystates.TheFe(I)andFe(II)Wannierorbitalsarerelatedtoeachotherbyasymmetrytransformation,namelytranslationbyaFe)]TJ /F5 7.9701 Tf 6.587 0 Td[(FeandsubsequentreectioninFe-plane.ThusinFigure 3-1 (b)weonlyshowtherestoftheWannierorbitalscenteredonFe(I).Figure 3-3 (a)showsthatthebandstructureobtainedfromdownfoldingtothe10-orbitalWannierbasiscomparesverywellwiththeDFTbands.Toavoidthecomputationalcomplexityassociatedwitha3Dcalculation,weignorethehoppingsinthez-directionandworkwitha2Dtight-bindingmodelofthenormalstatedescribedbyEquation 2{2 withthellingsetton=6electronsperFesite.Theorbitally-resolvedDOS,andtheFermisurfacewithcolor-codedorbitalcharacterforthis2DmodelisshowninFigure 3-3 (b)and 3-3 (c),respectively.TheDOSismostlyataroundtheFermilevelandhaslargestcontributionsfromFe-dxy,dxz,anddyzWannierorbitals.TheFermisurfacehasthreeholepocketscenteredaround)-326(point(Brillouinzonecenter)andtwoelectronpocketscenteredattheMpoint(Brillouinzonecorners)inthe2-FeBrillouinzoneasshowninFigure 3-3 (c).Theinnerandouterholepocketshavedyz/dxzcharacterswhereasmiddleholepockethasdxycharacter.Theouterandinnerelectronpocketshavedxyanddyz/dxzcharacters,respectively.Figure 3-3 (d)showsaveryrecentARPESdeterminedkz=(inreciprocallatticeunits)cutofthequasi-2DFermisurfaceofFeSeatT=7K[ 96 ].Notethatsincethemeasurementsweretakeninthenematicphase(T
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Figure3-3. (a)ElectronicstructureofFeSeshowingDFTandWannier-ttedbands.(b)FeSeFermisurfacewithcolor-codedorbitalcharacter.(c)Orbital-resolveddensityofstatesinthenormalstate.(d)TheARPESdeterminedFermisurfaceofFeSeatT=7K.Reprintedwithpermissionfrom[ 96 ],copyright2016bytheAmericanPhysicalSociety. shownherealsocontainsduplicateelectronandholepocketsrotatedby90,arisingfromtwindomains.Clearly,theholepocketsatBrillouinzonecenterandelectronpocketsatthecornersshowsignicantshrinkagecomparedtotheDFT-determinedFermisurfacebyalmostafactorof5[ 97 ].Moreover,ARPESobservesonlyoneholepocketattheBrillouinzonecenterwithdxz=dyzcharacter,andtwoelectronpocketsattheBrillouinzonecornerswithouter(inner)pockethavingdxy(dxz=dyz)character[ 96 ],incontrasttothethreeholepocketsandtwoelectronpocketsintheDFT-derivedFermisurface.Atthetimeourworkwasdone,therewerenoavailableARPESdataduetolackofclean 39

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FeSesamples.Hence,weassumedthatthenormalstateoftheFeSeisreasonablywellrepresentedbytheDFT-derivedbandstructure.Moreover,wedidnottakeintoaccountthesmallorthorhombicdistortionofthelatticeatlowtemperatures(T
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theintra-bandrepulsionthenitturnsoutthatthestrongAFMspin-uctuationsatq=Qleadtoaneectiveelectron-electroninteraction(repulsive)intheCooperchannel(Vkk0)whichpeaksatk)]TJ /F9 11.9552 Tf 12.937 0 Td[(k0=Q[ 99 ].Forsuchinteraction,BCSgapequation(Equation 1{5 )admitsanon-trivialsolutiononlyifthegapchangessignbetweenelectronandholepockets,thusyieldingansstate.However,thissimplisticscenarioexcludestheeectsoforbitalcontentoftheFermisurfacethatplaysacrucialroleindeterminingtheanisotropyofthesuperconductinggapontheFermisurface[ 98 102 103 ].Abetteralternativeistousethemulti-orbitalspinuctuationtheorywithDFT-determinedbandstructure[ 98 102 104 105 ].Infollowing,webrieydescribethecalculationofreal-spacepairpotentials(Vij)fromspin-uctuationtheorywithintherandomphaseapproximation(RPA).ThedetailsofcalculationcanbefoundinRefs.[ 92 98 102 105 ].WestartwiththefollowingHubbard-HundHamiltonianinconjunctionwiththenon-interactingHamiltonianH0(Equation 2{2 ), Hint=UXi;ni"ni#+U00Xi;
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ThedenitionsofmatricesUsandUccanbefoundinRef.[ 102 ].WithinRPA,thechargeandspinsusceptibilitiesaregivenas RPA1(q)1234=n0(q)1)]TJ /F1 11.9552 Tf 13.953 3.022 Td[(Us0(q))]TJ /F6 7.9701 Tf 6.586 0 Td[(1o1234;RPA0(q)1234=n0(q)1+Uc0(q))]TJ /F6 7.9701 Tf 6.587 0 Td[(1o1234;(3{3)where0(q)isthebaresusceptibilitydenedexplicitlyinRef.[ 102 ].Thereal-spacepairpotentialscanbecalculatedbyprojectingthemomentum-spaceverticestothespin-singletchannelfollowedbyaFouriertransformtorealspace. Vij=1 2Xk[)]TJ /F5 7.9701 Tf 10.566 -1.793 Td[((k;)]TJ /F9 11.9552 Tf 9.299 0 Td[(k)+)]TJ /F5 7.9701 Tf 26.285 -1.793 Td[((k;k)]e)]TJ /F5 7.9701 Tf 6.587 0 Td[(ik(Ri)]TJ /F11 7.9701 Tf 6.586 0 Td[(Rj);(3{4)AsshowninFigure 3-4 (a),thepairpotentialsthusobtainedformacheckerboardpatternandshowarapidspatialdecay.Thepatternisdominatedbytheon-siteintra-orbital(Vii)repulsionvaryinginarangeof1.2eVto4eV.Thelargestvaluesoftheattractivepairpotentialoccuratnearestneighbor(NN)andnext-nearestneighbor(NNN)siteswithVNN>VNNN.UsingVijasaninput,wesolve10-orbitalBdGequationsonasquarelatticewith1515unitcellsusingtheself-consistencyschemedescribedinSection 2.4 .Theresultingsuperconductinggapparameterijisrathershort-ranged,andhaslargestvaluesinthesubspacespannedbytheorbitals;=(dxy;dxz;dyz)asshowninFigure 3-4 (b).ItsorigincanbetracedbacktothefactthatthesethreeorbitalshavethelargestDOSattheFermilevel.Moreover,thespatialpatternofijreectsthesymmetriesoftheunderlyingorbitals.Forexample,ijfor=dxydisplaysaC4symmetrywhereasfor=dxz;dyzitshowsaC2symmetry.Furthermore,duetodegeneracyofdxzanddyzDOSattheFermisurface,thecorrespondinggapvaluesarerelatedtoeachotherbya=2rotation.ThemomentumspacestructureofthegapcanbefoundbytheFouriertransformingthereal-spacegapfollowedbyanorbital-to-bandbasistransformationusingthenormalstateeigenvectors.Theresulting(k)isdisplayedinFigure 3-4 (c).Clearly,(k)preserves 42

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Figure3-4. (a)Pairpotentials(Vij)inreal-andorbital-space.EachpixelrepresentsanFesite.DiagonaltermsVii,varyinginarangeof1.2eV-4eV,arenotshownheretoemphasizesmaller,o-diagonalpairpotentials.(b)Superconductinggapijobtainedfromtheself-consistentsolutionofBdGequations.(c)Superconductinggap((k))inmomentum-space.(b)and(c)havebeenplottedonasquare-rootscaletoemphasizesmallervalues.(d)Orbitallyresolveddensityofstatesinthesuperconductingstate. allsymmetriesofasquarelattice,butchangessignbetweenouterholepocketscenteredattheBrillouinzonecenter()-327(point)andelectronpocketscenteredattheBrillouinzonecorner(Mpoint),yieldinganssymmetry.Notethatthegapontheinnerholepocketalsochangessignwithrespecttotheouterholepockets.Theoccurrenceofthelargestgapvaluesonthemiddleholepocketandtheouterelectronpocketcanbetracedbacktotheirdominantdxycharacter.Figure 3-4 (d)showsthelow-energyorbital-resolvedDOSinthesuperconductingstatecomparedwiththenormalstateDOS.ThetotalDOSin 43

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superconductingstateshowstwo-gaplikefeatureswithinnercoherencepeaksat!=5:5meVandoutercoherencepeaksat!=16:5meV.Theinnercoherencepeakshavedominantcontributionfromdegeneratedxzanddyzorbitalswhereastheouterpeaksismainlycontributedfromdxyorbitals.Wedeliberatelychosetheinteractionparameterstoyieldalargergapmagnitude(16:5meV)comparedtothemuchsmallervalue(3meV)observedintheSTMexperiments[ 84 ]inordertoclearlyvisualizetheimpurityboundstatewithinournumericalresolution. 3.4EectsofanImpurityAsdiscussedinSection 2.2.1 ,wecanmodelanon-magneticimpurityinthesimplestpossiblewaybyanon-sitepotentialVimpwhichisdiagonalandidenticalforallorbitalchannels.Thisapproximationissupportedbytherst-principlesstudiesoftheimpuritypotentialsinseveralFe-basedmaterialsndingaverysmallorbitaldependence[ 107 { 109 ].Theexistenceofanon-magneticimpurity-inducedin-gapboundstateinanssuperconductorhasbeenfoundtobeverysensitivetothesignandmagnitudeoftheimpuritypotential,detailsoftheunderlyingbandstructure,andthegapfunction[ 91 92 98 ].WechoosearepulsiveimpuritypotentialVimp=5eVtorealizeanin-gapboundstateat=2meV.Westressthatthenumericalvalueoftheimpuritypotentialusedherehasnospecialsignicancebeyondthefactthatitinducesanin-gapboundstate.Wearearejustinterestedinexaminingthespatialformoftheboundstate;anyvalueofimpuritypotentialwhichinducessuchstateatanyin-gapenergyisanequallygoodchoice,andqualitativeconclusionsdrawninfollowingparagraphswillstaythesame.AsinthecaseofhomogeneoussystemdescribedinSection 3.3 ,weworkwitha1515unit-cellslatticeandreplaceoneFeatominthetheunitcell[8,8]byanimpurity.Aftersolvingthe10-orbitalBdGequationsself-consistently,weconstructthelatticeGreen'sfunctionusing2020supercellswhichyieldaspectralresolutionoforder0:5meV.Figure 3-5 (a)showsthelatticeLDOS(Ni(!))spectrum(calculatedusingEquation 2{13 )farfromtheimpurity,attheimpuritysite,ontheNNsite,andontheNNNsite. 44

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Sincetheimpuritypotentialiscomparabletothebandwidth,itactsasastrongrepulsivepotentialscatterer,heavilysuppressingtheLDOSattheimpuritysite.Theimpurityinducedin-gapboundstatesat=2meVareclearlyobservedintheLDOSspectrumattheNNandNNNsites.Figure 3-5 (b)and(c)showthelatticeLDOSmaparoundtheimpuritysiteattheresonanceenergies.WendthatthespatialpatternaroundtheimpuritypreservestheC4symmetryoftheunderlyingsquarelattice,asitshould,andisverydierentfromtheC2symmetricdimer-likepatternsobservedintheSTMexperiments[ 63 ].ItisnecessarytotakeintoaccountSestatesthatbreakstheC4symmetryinamannerconsistentwiththeorientationoftheexperimentallyobservedgeometricdimers.ThiscanbeachievedbythecalculatingcontinuumLDOSabovetheSesurface,exposedtotheSTMtip,usingWannierorbitalsasdescribedinSection 2.3.2 .Figure 3-6 showsthecontinuumLDOS(r=(x;y;z);!)mapscomputedfromEquations 2{14 and 2{18 atvariousheightszmeasuredfromFe-plane,atvariousbiases.TheresonancepatternisC4symmetricintheFe-plane(z=0),asshowninFigure 3-6 (d).Moreover,itresemblescloselythecorrespondinglatticeLDOSmapplottedinFigure??(c).MovingabovetheFe-planelowersthelocalsymmetrytoC2duetoplacementoftheSeatoms.Consequently,theLDOSmapstakenrightabovetheFe-plane(Figure 3-6 (f)),andabovetheSe-plane(Figure 3-6 (a)-(c))areC2symmetric.Moreimportantly,theLDOSmapsabovetheSe-planeexhibitageometricdimer-likestructurewithbrightlobespositionedatthetwoNN(up)SeatomsasshowninFigure 3-6 (a)-(c).Suchapatternisvisibleatallenergies,however,withavariationintheintensityatNNN(up)Sepositions.Theoriginofthesedimer-likepatternscanbeunderstoodasaconsequenceoftheparticularshapeoftheWannierorbitalsabovetheSe-layer,andthespatialpatternofthelatticeLDOS.UsingEquations 2{13 2{14 ,and 2{18 ,wecanwrite (r;!)=Xi;Ni(!)jwi(r)j2+o-diagonalterms(i6=j;6=)(3{5) 45

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Figure3-5. (a)LatticeLDOSspectruminthesuperconductingstate,farfromtheimpurity,attheimpuritysite,ontheNNsite,andontheNNNsite.Insetshowsdensityofstatesinthehomogeneoussuperconductingandnormalstates.(b)Real-spacepatternoflatticeLDOSattheboundstateenergy=2meVand(c)=)]TJ /F1 11.9552 Tf 9.298 0 Td[(2meV.EachsquarepixelrepresentsanFesite. Now,Figure 3-2 showsthatfortheheightsweareconsidering,theWannierorbitalswi(r)havesignicantweightsonlyaroundtheup-Seatoms.Moreover,thelatticeLDOS,shownasacartooninFigure 3-6 (a)-(b),islargestattheNNandNNNsites.Thus,ignoringo-diagonaltermsinEquation 3{5 ,wendthattheLDOShasthelargestvaluesat(up)NNand(up)NNNSe-atomsleadingtotheobservedpatterns.Theo-diagonaltermsinvolvetheproductofexponentiallylocalizedWannierfunctionsatdierentsitesand/orwithdierentorbitalsymmetries,andtheirsignisnotxed,hencetheircontributionisusuallysmallerthanthediagonalterms.ThisisparticularlytrueinthecaseoftheLDOSmapsshowninFigure 3-6 (a)-(c).However,neglectingthesetermscanleadtowrongresultsasillustratedinFigure 3-6 (e)and(f).Theformershowsthatignoring 46

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Figure3-6. ContinuumLDOSmapsatheightz=0:045c(aboveSe-plane),andenergies(a)!=)]TJ /F1 11.9552 Tf 9.298 0 Td[(2meV,(b)2meV,and(c)30meV,andatheightz=0c(attheFe-plane)(d),z=0:5c(closetotheFe-plane)[(e),(f)]andthesameenergy!=2meV.AllmapsareobtainedfromEquation 2{18 except(e)whichincludesonlylocalanddiagonal(i=j;=)terms.ColorbarvaluesareintheunitsofeV)]TJ /F6 7.9701 Tf 6.586 0 Td[(1bohr)]TJ /F6 7.9701 Tf 6.586 0 Td[(3.Thepositionofz-cutsrelativetotheunit-cellisvisualizedbyagreenlineintheschematicoftheside-viewofFeSeunitcell.RedcirclesandyellowtrianglesrepresentFeandSeatomsrespectively.ThecartoonoflatticeLDOSpatternsat!=2meV(compareFigure 3-5 (b)and(c))isshownin(b)and(c)toexplaintheparticularshapeofcontinuumLDOSpatterns. theo-diagonaltermsleadstoaC4symmetricLDOSmapjustabovetheFe-plane.ThecorrectC2symmetryisrecoveredonlyifthesetermsareincludedasshowninFigure 3-6 (f).Atthetimethisworkwasdone,theexperimentalresultsforthebiasdependenceofthetunnelingconductancemapsaroundtheimpuritywerenotpublished.Instead,Ref.[ 63 ]reportedthetopographyoftheFeSesurfaceasdescribedinSection 3.1 .To 47

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Figure3-7. Theoreticaltopographofimpuritystatesat6meVobtainedfrom(a)BdGcalculations(b)BdG+Wmethod.In(a)eachpixelrepresentsanFesite.(c)Experimentaltopographofimpuritystates,reproducedfrom[ 63 ],rotatedtomatchorientationoflatticevectorsin(a)and(b).Reprintedwithpermissionfrom[ 63 ],copyright2014bytheAmericanPhysicalSociety. comparewiththeexperimentalresult,wecalculatethetopographicmapusingtheenergyintegratedcontinuumLDOS.TheSTMtunnelingcurrentatagivenbiasVcanbeapproximatedas[ 56 ] I(x;y;z;V)=)]TJ /F1 11.9552 Tf 10.494 8.088 Td[(4e ~t(0)jMj2ZeV0(x;y;z;)d;(3{6)Here,isthecontinuumLDOS,(x;y;z)istheSTMtipposition,t(0)isthedensityofstatesofthetipattheFermienergy,andMisthetunnelingmatrixelement.AtopographatbiasVandset-pointcurrentI0isdenedbytheconditionI(x;y;z;V)=I0whichisequivalenttoReV0(x;y;z;)d=constant.Weobtaintheheightz(x;y)satisfyingthiscondition,withenergyintegratedLDOSandbiassetto3:4510)]TJ /F6 7.9701 Tf 6.586 0 Td[(7eV)]TJ /F6 7.9701 Tf 6.586 0 Td[(1bohr)]TJ /F6 7.9701 Tf 6.586 0 Td[(3and6mV,respectively.TheresultisshowninFigure 3-7 .Clearly,aBdG-onlylatticecalculation(Figure 3-7 (a))yieldsaC4symmetrictopographwhichisverydierentfromtheexperimentalresultshowninFigure 3-7 (c).However,thetopographcalculatedusingBdG+Wapproachyieldsresultverysimilartotheexperimentalobservation. 3.5ConclusionAstherstapplicationoftheBdG+WmethodintroducedinChapter 2 ,westudiedtheresponseofthesuperconductorFeSetoasinglepoint-likeimpurityonaFesite.WestartedwithaDFTderived10-orbitaltight-bindingmodeloftheFeSenormalstateandcorrespondingWannierbasis.Thessuperconductingstateisintroducedthrough 48

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thereal-spacepairpotentialsobtainedfromthespin-uctuationtheorycalculationperformedwithinrandomphaseapproximation.Bysolvingthe10-orbitalBdGequationsself-consistently,wefoundthatastrongnon-magneticimpurity,modeledsimplyasanon-siterepulsivepotential,inducesboundstatesinthessuperconductinggap.UsingWannierfunctions,weobtainedhigh-resolutioncontinuumLDOSmapsatvariousbiasesandheightsfromtheFe-plane,anddemonstratedthattheBdG+WanniermethodyieldsaqualitativeimprovementovertheconventionalBdGresultsbycapturingthelocalsymmetryinternaltotheunitcell.Moreover,weshowedthatthegeometricaldimerstatesobservedinSTMexperimentsonFeSeandseveralotherFeSCoriginatefromthehybridizationofimpurities,locatedatFesites,withtheSe(As)states.Finally,weshowedthatthetopographicmapcalculatedusingcontinuumLDOScomparedverywellwiththeSTMtopographobtainedatthesamebias. 49

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CHAPTER4IMPURITYINDUCEDSTATESINCUPRATESInthischapter,Iwilldiscusstheeectsofsinglenon-magneticaswellasmagneticimpurityinacupratesuperconductorBi2Sr2CaCu2O8+(BSCCO).Someofthematerialpresentedhereisbasedonapublishedpaper[ 110 ].Allthepublishedcontents(excerptsandgures)arereprintedwithpermissionfrom A.Kreisel,PeayushChoubey,T.Berlijn,W.Ku,B.M.Andersen,andP.J.Hirschfeld,Phys.Rev.Lett.114,217002(2015) ,copyright2015bytheAmericanPhysicalSociety.IcontributedtoallofthefollowingsectionsexceptSection 4.6 4.1MotivationTheexperimentalobservationofZnimpurity-inducedresonancestatesinthecupratecompoundBSCCO[ 60 111 { 113 ]wasastrongindicatorofunconventionalpairinginthisclassofsuperconductors.Simpletheoreticalcalculationstreatingimpuritiesaspoint-likepotentialscatterersinad-wavesuperconductorhadalreadypredictedtheexistenceofsuchstates[ 61 ].However,thedetailsoftheSTMconductancespectradisplayedtwoimportantdeviationsfromthetheoreticalpredictions[ 60 68 75 ].First,theconductancemaximumwasfoundattheimpuritysitewheretheoryhadpredictedaconductanceminimum.Second,thelong-rangeintensitytailswerefoundtobeorientedalongCu-Obonddirections,45rotatedfromthetheoreticallypredictedCu-Cubonddirectionalongwhichlow-energynodalquasiparticlesexist,andcanleadtosuchlongrangetails[ 75 ].SimilardiscrepanciesalsooccurredincaseofNi(amagneticimpurity)dopedBSCCO[ 64 68 75 ].Severalremedieswereproposedtoreconcilewiththeexperiments,includingextendedimpuritypotentials[ 114 { 116 ],Kondo-likeeects[ 117 ],Andreevscatteringatimpurities[ 118 ],and"lter"eects[ 62 119 ].ThelastpostulatedthattheinterveninglayersbetweenSTMtipandCuOlayerprovideanalternatetunnelingpathmakingthetipsensitivetothenearest-neighborCusitesinsteadoftheimpuritysitebeneathit.First-principlescalculationsoftheimpurityinducedstatesinthenormalstateof 50

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BSCCO[ 120 ],andananalysisbasedonseveralatomic-likeorbitals[ 121 ]supportedthephenomenologicalltertheories,butitwasnotclearhowtoincludesuperconductivityaswellaseectofinterveninglayerstogetherinpresenceofdisorder.Here,weshowthatthecontinuumLDOSobtainedinthevicinityofastrongnon-magneticimpurityandamagneticimpurityusingtheBdG+WframeworkshowsexcellentagreementwiththecorrespondingSTMresultsforZn[ 60 ]andNi[ 64 ]dopedBSCCO.WendthattheSTMconductancepatternscanbeeasilyexplainedbyaccountingfortheCu-dx2)]TJ /F5 7.9701 Tf 6.586 0 Td[(y2Wannierfunctiontailswithsignicantweightcomingfromtheapicaloxygenatomsinthenearestneighborunitcells,puttingthecrudeltertheoriesonarmmicroscopicfooting.Inthefollowingsections,werstdescribethepropertiesofthenormalandsuperconductingstateofBSCCOusingaDFTderivedtight-bindingHamiltonianandnearest-neighborpairpotentialputin"byhand"toyieldad-wavegapwithmagnitudesimilartothatobservedintheSTMexperiments[ 60 ].ThenwestudytheZnimpurityproblembytreatingitasastrongpoint-likepotentialscattererinad-wavesuperconductor,andcomputecontinuumLDOSatatypicalSTMtipheight,demonstratinganexcellentagreementbetweentheoryandtheexperiment[ 60 ].Finally,westudytheNiimpurityproblemusingaverysimplemodelofamagneticimpurity[ 68 75 122 ]andshowthatitinducestwospinpolarizedin-gapresonancestatesaspredicted[ 122 ].Moreover,thecontinuumLDOSpatternsattheresonanceenergiesandatatypicalSTMtipheightagreesverywellwiththeexperiment[ 64 ].However,therelativeweightsofthetworesonancestatesarefoundtobereversedcomparedtotheexperimentalresult.Weshowthatincludingnearestneighborpotentialandmagneticscatteringyieldsthecorrectrelativeweightsoftheresonancepeaks,however,themicroscopicoriginofsuchanimpuritymodelisnotclearyet. 51

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Figure4-1. (a)ElementarycellusedintheDFTcalculationtoobtainelectronicstructureofBSCCO.IsosurfaceplotsofCu-dx2)]TJ /F5 7.9701 Tf 6.586 0 Td[(y2WannierfunctioninBSCCOatisovalue(b)0.05(b)0.005(c)0.0002bohr)]TJ /F6 7.9701 Tf 6.587 0 Td[(3=2.Arrowspointtotheapicaloxygentailsinthenearest-neighborunitcells.RedandbluerepresentthesignoftheWannierfunction. 4.2NormalStateWerstobtainaone-bandtight-bindingmodelandthecorrespondingWannierfunctionsforBSCCOusingrstprinciplescalculations.TheelementarycellusedinbandstructurecalculationsisshowninFigure 4-1 (a).TheBSCCOcrystalusedinexperimentslikeSTM[ 57 ]andARPES[ 27 ]cleavesattheBiOlayer.Hence,itismoreappropriatetoobtaintheBSCCOWannierfunctionsfortheBiOterminatedsurfaceinsteadofthebulk.Accordingly,wetakeabody-centeredtetragonalunit-cellterminatedattheBiOlayerwitha18.5AvacuumslabasshowninFigure 4-1 (a),andperformelectronicstructurecalculationsusingtheWIEN2Kpackage[ 95 ].Thecrystalstructureparameterswere 52

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Figure4-2. (a)Bandstructureand(b)Fermisurfacecorrespondingtotherenormalized1-bandtight-bindingmodelofBSCCO.(c)Densityofstatesinthenormalstate. adoptedfromtheRef.[ 123 ].Subsequently,theWannierfunctionwasconstructed,usingaprojectedWanniermethod[ 82 ],byprojectingCu-dx2)]TJ /F5 7.9701 Tf 6.586 0 Td[(y2orbitalonDFTwavefunctionsinanenergywindowof[-3,3]eV.TheisosurfaceplotsoftheresultingWannierfunctionareshowninFigure 4-1 (b)-(d).ForthelargeisovaluestheWannierfunctionislocalizedinCuOplaneandreectsastronghybridizationbetweenCu-dx2)]TJ /F5 7.9701 Tf 6.587 0 Td[(y2statesandin-planeO-pstatesasshowninFigure 4-1 (b).Thecontributionsofotheratoms,especiallytheapicalOatomsfromtheneighboringunitcells,becomesapparentwithdecreasingisovalues.ThelobesassociatedwithapicalO-pstates(markedwitharrowsinFigure 4-1 (c))arethemainsourceofthepeculiarstructureofthisparticularWannierfunction(Figure 4-1 (d))attheheightsfewAabovetheBiOlayerwhereSTMtipwilltypicallyreside.Attheseheights,WannierfunctionhaszeroweightdirectlyabovethecenterCuandlargestweightabovethenearest-neighbor(NN)Cuatoms.Thetight-bindingparametersobtainedintheWannierbasisdescribedaboveyieldabandstructurewithmuchlargerFermivelocitycomparedtothatobservedintheARPESexperiments[ 27 124 ],aconsequenceoftheinadequatetreatmentofcorrelationsintheDFTcalculations.Onacrudelevel,wecanaccountforthecorrelationeectsbyusinganoverallrenormalizationoftight-bindingparametersbyafactor1=Z.Wedividealltight-bindingparametersbyZ=3toapproximatelymatchtheexperimentally 53

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observedFermivelocity[ 27 ].Also,wextheelectronllingperCusitetoben=0:85correspondingtooptimaldoping.Thebandstructure,Fermisurface,andDOScorrespondingtotherenormalizedtight-bindingparametersandaforementionedllingareshowninFigure 4-2 (a),(b),and(c),respectively.Avan-Hovepeakat!=)]TJ /F1 11.9552 Tf 9.298 0 Td[(63meV,arisingasaconsequencetheatbandneartheXpoint,iseasilyobservedinthespectrum. 4.3HomogeneousSuperconductingStateThesuperconductingstatewithd-wavesymmetryisobtainedbyputtinginattractivepair-potentials(Vij)"byhand"suchthatVij=V0,withifsitesiandjarenearestneighborsandzerootherwise.ChoosingV0=150meVandsolving1-orbitalBdGequationsona3535squarelatticeyieldsad-wavesuperconductinggapthatcanberepresentedinthemomentumspaceas(k)=0 2(coskx)]TJ /F1 11.9552 Tf 11.966 0 Td[(cosky)with033meV.Thevalueofthespectralgapthusobtainedisverysimilartotheexperimentallyobservedvalue[ 57 60 ].ThedensityofstatesinthesuperconductingstateiscomparedtothenormalstateinFigure 4-3 (a).Atlowenergies(!!0),theDOSvarieslinearlywithj!j,resultinginaV-shapespectrum.ThislinearvariationisaresultoftheexcitationsatthefournodalpointsontheFermisurfacealongtheBrillouinzonediagonals,wheresuperconductinggaporderparametervanishes(seeAppendix C forthederivation).InFigure 4-3 (b),weplotthecontinuumLDOSaboveaCusiteataheightz5AabovetheBiOplane.TheprominentdierencefromthelatticeLDOS(Figure 4-3 (a))istheU-shapedspectrumatlowenergies.Thechangeinthelowenergyspectralfeaturescanbeunderstoodfrombothreal-spaceandk-spaceperspectives.InthelatticeLDOS,onlydiagonalGreen'sfunctionsappear,forwhichthesignofimaginarypartisxedbycausalityrequirements.However,incaseofthecontinuumGreensfunction,o-diagonallatticeGreen'sfunctioncontributionstoEquation 2{18 ,forwhichthesignofimaginarypartisnotxed,cangiverisetoasuppressedLDOSatcertainenergies.ThusthespectralpropertiesoflatticeandcontinuumLDOScandier.ThisisexplicitlyshownintheFigure 4-3 (c),whereweplotthelatticeLDOS(sameasinFigure 4-3 (a)),the 54

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Figure4-3. (a)LatticeDOS(redsolidline)inthehomogeneoussuperconductingstatewithd-wavegapcomparedwiththenormalstateDOS(blackdashedline).(b)ContinuumLDOSspectrumaboveaCusiteatheightz5AabovetheBiOplaneinthehomogeneoussuperconductingstate(redsolidline)andnormalstate(blackdashedline).(c)LatticeDOS(reddashedline),continuumLDOSobtainedusingfullGreensfunctionmatrixinEquation 2{18 (bluedashedline),comparedwiththecontinuumLDOScalculatedusingonlydiagonalGreensfunctioninEquation 2{18 (blacksolidline).(d)TheaverageconductancespectrainsixBSCCOsampleswithholedopingrangingfromundertooverdopedregime.ReprintedbypermissionfromMacmillanPublishersLtd:Nature[ 125 ],copyright2008.(e)AveragetunnelingconductancespectrainaBSCCOsampleinregionswithdierentgapsizes(blackcircles)andcorrespondingts(blacksolidlines)toamodelwithd-wavesuperconductinggapandaquasiparticlescatteringratevaryinglinearlywithenergy.ReprintedbypermissionfromMacmillanPublishersLtd:Nature[ 126 ],copyright2008. 55

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continuumLDOS(sameasinFigure 4-3 (b)),andalsothecontinuumLDOSwhereonlythediagonaltermsofthelatticeGreensfunctionhavebeentakenintoaccount.Thelatterisnotaphysicalquantity,butshowsnicelythattheresultingcontinuumLDOShasthesamespectralshapeasthelatticeLDOSbecauseitisjustmultipliedwiththeWannierfunctionswhichdoesnotdependonenergy.Clearly,thecontinuumLDOS,withcontributionsfrombothdiagonalando-diagonalGreensfunctions,showsasuppressionoftheLDOSat!!0whencomparedwiththeLDOSwithcontributionsonlyfromthediagonalGreensfunctions.AmoreconcreteexplanationoftheU-shapeinthecontinuumLDOScanbegivenfromthek-spacecalculations.ThedetailedderivationisprovidedintheAppendix C ;here,wewillsummarizetheargumentinshort.TransformingthebasistomomentumspaceinEquation 2{18 ,itcanbeeasilyshownthatthecontinuumLDOSforahomogeneoussystemwillbegivenby (r;!)=XkA(k;!)jWk(r)j2:(4{1)Here,Wk(r)=Piwi(r)eikRiistheFouriertransformoftheWannierfunction,andA(k;!)isthespectralfunction.IfwesetWk(r)=1,thenEquation 4{1 willyieldthelatticeLDOSN(!)whichcanbeshowntovarylinearlywithj!jas!!0(seeAppendix C foraderivation),resultingintothewell-knownV-shapedspectrum.IncaseofthecontinuumLDOSthenon-trivialmomentumdependenceofWk(r)changesthelowenergyshapeofthespectrum.AtheightsseveralAabovetheBiOlayerwhereSTMtipwouldtypicallyreside,theWannierfunctionaboveaCusiteisshownintheFigure 4-4 .Clearly,thetheWannierfunctionabovethecentralCusite(w0)iszero,asdictatedbythedx2)]TJ /F5 7.9701 Tf 6.586 0 Td[(y2-wavesymmetry,andhaslargestmagnitudew1aboveNNsites.Thus,forthecontinuumpositionabovetheCusite,r0=[0;0;z],atheightszseveralAabovetheBiOplane,k-spaceWannierfunctioncanbeapproximatedasWk(r0)w0+2w1(coskx)]TJ /F1 11.9552 Tf 9.336 0 Td[(cosky).Althoughw0=0,westillkeepittofacilitatediscussionsinthenextparagraphs.Now,for 56

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Figure4-4. Cu-dx2)]TJ /F5 7.9701 Tf 6.586 0 Td[(y2BSCCOWannierfunctionatheight5AabovetheBiOplaneplottedina77CuO2unitcellarea.Blue(red)representspositive(negative)signoftheWannierfunctionvalue. !!0,performingthemomentumspaceintegralresultsinto(seeAppendix C fordetails) (r0;!)=a0j!j+a1j!j3;(4{2)wherea0/w20anda1/w21.Thus,thedx2)]TJ /F5 7.9701 Tf 6.587 0 Td[(y2-wavesymmetryresultsintovanishinglinearin!contribution,andaU-shapedspectrumarisingfrom!3term.Althoughithasbeenneveraddressedexplicitlyasfarasweknow,afterdoinganextensiveliteraturesearch,wefoundthatthespectralshapeofthemeasuredtunnelingcurrentindeedshowssomevariation,frommoreV-shapeintheoptimal-to-underdopedsamplestoamoreU-shapeinoverdopedsamples.Inaddition,U-shapedspectrahavebeenmeasuredwiththesameSTMtiponsamplesthatshowadistributionofgapmagnitudes,attributedtolocaldopingconcentration[ 126 127 ].Similarobservationshavebeenmadein[ 125 128 { 130 ],butsofarnotcompletelyunderstoodtheoretically.Figure 4-3 (d)-(e)showthetransitionfrommoreV-shapedtunnelingconductancetoU-shapedspectraldependencewhenmovingfromoptimallydopedsamples(sampleareas)tooverdopedsampleswhereourtheoryshouldbeapplicable.Figure 4-3 (d),showingoptimaltounderdopedsamples,iscompiledfrom[ 125 ],whileFigure 4-3 (e)istakenfrom[ 126 ],wheretheblackcurverepresentatofthemeasuredspectra(blackcircles)toamodelwithd-wavesuperconductinggapandadditionalquasiparticlescatteringrate,and 57

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thedierentdatasetscorrespondtoregionsinonesamplewithdierentgapsizes.Notethatthesmall-gap("overdoped")regionshavethemoreU-shapedcurves.Thetheoreticalpicturepresentedaboveshouldbevalidonlyfortheweakly-correlatedoverdopedcuprateswherewecansafelyjustifytheuseofWannierfunctionobtainedfromDFTcalculations.Foroptimal-to-underdopedcuprates,theWannierfunctionshapemightchangeresultingintoaV-shapedspectra.Infact,fromEquation 4{2 wecaneasilyseethatiftheWannierfunctionhasnon-vanishingweightabovethecentralCusite,thencontinuumLDOShasalinear-in-!termyieldingaV-shapedspectra.SuchascenariowillarisewhenelectroniccorrelationsresultintoaCu-dz2contributiontotheFermisurface,apossibilitywhichhasbeendiscussedextensivelyintheliterature[ 131 { 135 ],andrecentlyusedtoexplainthevariationofTcamongvariouscupratefamilies[ 136 137 ].However,tomakeamoreconcretestatementaboutthecontinuumLDOSinunderdopedcuprates,wemustsystematicallystudytheevolutionoftheWannierfunctionwithincreasingelectroniccorrelations[ 138 { 140 ]whichweleaveforafutureproject.AnotherpromisingscenariowhichcanexplaintheV-shapedcontinuumLDOSspectrainunderdopedcupratesisthecoexistenceofchargeorder.InSection 5.3.2 ,weshowthataparticularchargeorderedphasecoexistingwithsuperconductivityresultsintoaV-shapedcontinuumLDOSspectra,verysimilartothatobservedintheSTMexperimentsonunderdopedcuprates. 4.4EectsofaStrongNon-MagneticImpurityWemodeltheZnimpurityreplacingtheCuatomasastrongpoint-likepotentialscattererwithVimp=)]TJ /F1 11.9552 Tf 9.298 0 Td[(5eV,verysimilartothevaluefoundinourrst-principlescalculations.BdGequations(Equation 2{9 and 2{10 )weresolvedself-consistentlyona3535squarelatticewithimpurityinthemiddle.Subsequently,thelatticeGreen'sfunctionmatrixwasconstructedusing2020supercells.Figure 4-5 (a)showsthelatticeLDOSspectrumatafarawaysite,attheimpuritysite,andtheNNsite,calculatedusingEquation 2{13 .TheLDOSisheavilysuppressedattheimpuritysiteasexpectedgiventhattheimpuritypotentialismuchlargerthanthebandwidth.Asharpin-gapresonance 58

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Figure4-5. (a)LatticeLDOSspectrumaroundZnimpuritymodeledbyastrongon-sitepotentialVimp=)]TJ /F1 11.9552 Tf 9.298 0 Td[(5eV.Spectrumcalculatedusing2020supercellswitharticialbroadeningof1meV,attheimpuritysite,nearestneighborsite,nextnearestneighborsite,andasitefarawayfromtheimpurity,areshownindashedblack,red,blue,andsolidblack,respectively.(b)ContinuumLDOSspectrumataheightz5AabovetheBiOsurface.ShownarepositionsdirectlyaboveaCuatomfarfromimpurity(black,solidline),attheimpurity(black,dashedline),onthenearestneighborposition(red,solidline),andonthenext-nearestneighborposition(blue,solidline).ResultsaretobecomparedwiththeexperimentalresultofRef.[ 60 ],showninFigure 1-7 (a). stateat=3:6meVisclearlyobservedintheLDOSspectrumattheNNsitewithstronglyelectron-holeasymmetricweights.Thisisincontrastwiththeexperimentalobservation[ 60 ]ofresonancepeakrightabovetheimpuritysite.ThespatialpatternoftheresonancestateismoreclearinthelatticeLDOSmapat=)]TJ /F1 11.9552 Tf 9.299 0 Td[(3:6meVshowninFigure 4-6 (a).ThisdiscrepancybetweentheoryandtheexperimenthasarisenbecausewearecomparinglatticeLDOSevaluatedattheCusiteswiththeSTMconductancemapobtainedfewAabovetheexposedBiOlayer.AsexplainedinSection 2.3 ,therightquantitytocomparewiththeSTMresultisthecontinuumLDOS(r;!)evaluatedattheSTMtipposition.Indeed,wendthatthecontinuumLDOScomputedataheightz5AfromBiOplane,whereSTMtipwouldbetypicallyplaced,displaysexcellentagreementwiththeexperimentasevidentfromcomparingFigure 4-6 (b)and(c).Inadditionto 59

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Figure4-6. (a)LatticeLDOSmapattheresonantenergy=)]TJ /F1 11.9552 Tf 9.298 0 Td[(3:6meVcomputedfromBdGsolutionsusingEquation 2{13 .EachpixelrepresentaCusite.(b)ContinuumLDOSmapatthesameenergy,andataheightz5AaboveBiOplanecomputedusingEquation 2{14 .(c)STMconductancemapattheimpurityresonanceenergyrotatedtomatchtheorientationin(b)and(c),andcroppedto1111unitcells.Impurityislocatedatthecenter.ReprintedbypermissionfromMacmillanPublishersLtd:Nature[ 60 ],copyright2000. theLDOSmaximumabovetheimpuritysiteandsub-maximumabovetheNNNsite(seealsoFigure 4-5 (b)),thecontinuumLDOSmapat=)]TJ /F1 11.9552 Tf 9.299 0 Td[(3:6meValsocapturesthelong-rangeintensitytailsorientedalongCu-Odirectionasseenintheexperiment[ 60 ].Thus,BdG+WapproachcapturesallqualitativefeaturesoftheZn-impurityinducedstatesinBSCCOasseenintheSTMexperiments.InAppendix D ,weshowthatthephenomenological"lter"theory[ 62 ],describedinSection 4.1 ,canbeobtainedasarst-orderapproximationoftheBdG+Wequations. 4.5EectsofaMagneticImpurityAsshownintheFigure 4-7 (a),theSTMexperimentsonNi-dopedBSCCO[ 64 ]foundtwospin-resolvedin-gapvirtualboundstatesatenergies1;2inthevicinityoftheNiatomssubstitutingCuatoms.Impurity-inducedresonancepeakswereobservedtobeparticle-likeattheimpuritysiteandnextnearestneighborsites,andholelikeatthenearestneighborsites.Thespatialpatternsat+1;2resembledcross-shapedandXshapedat)]TJ /F1 11.9552 Tf 9.299 0 Td[(1;2.Observationoftheseresonancestatesisconsistentwiththemodelsofcombinedpotentialandmagneticscatteringofquasiparticlesinad-wavesuperconductor[ 122 ].However,likethecaseofZn,spatialpatternsdeviatefromthepredictions. 60

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Amagneticimpurityactsasthesourceofpotentialaswellasmagneticscattering,andcanbesimplymodeledbyfollowingHamiltonian[ 68 75 ]. Himp=XjUi?jcyi?cj+XjJi?jcyi?cj+H.c.;(4{3)where=1,andH.c.referstotheHermitianconjugate.ThersttermintheaboveHamiltonianaccountsforthepotentialscatteringattheimpuritysitei?andUi?jistheimpuritypotentialwhichcanbewrittenasU0i?j,whererepresentstheKroneckerdeltafunction,foracompletelylocalimpuritymodel.ThesecondtermintheaboveHamiltonianaccountsforthemagneticscatteringduetoexchangeinteractionbetweenimpuritymagneticmoment,approximatedasaclassicalobject,andconductionelectrons.Ji?jrepresentstheextendedexchangecouplingwhichcanbeexpressedasJ0i?jforthespecialcaseofcompletelylocalexchangeinteraction.TheBdGequationsdescribedinSection 2.2.2 canbeeasilygeneralizedtostudythemagneticimpuritymodeledbyEquation 4{3 bymakingfollowingmodicationstotheBdGmatrixelements ij"=tij)]TJ /F4 11.9552 Tf 11.955 0 Td[(0ij)]TJ /F1 11.9552 Tf 11.955 0 Td[((U0+J0)i?jij;ij#=tij)]TJ /F4 11.9552 Tf 11.955 0 Td[(0ij)]TJ /F1 11.9552 Tf 11.955 0 Td[((U0)]TJ /F4 11.9552 Tf 11.955 0 Td[(J0)i?jij(4{4)NiisknowntobeaweakerimpuritythanZnasitdoesnotdisruptsuperconductivityinitsvicinity,andthepuremagneticscatteringisthoughttobesub-dominanttothepurepotentialscattering[ 64 ].Accordingly,inthesimplestscenario,wemodelNiascompletelylocalimpurity,withon-sitepotentialU0=0:6eVandexchangecouplingJ0=0:3U0,andsolvetheBdGequationsona3535lattice.ThevaluesofU0andJ0havebeenchosenjusttoclearlydemonstratethespin-splittingofimpurity-inducedresonancepeaksduetoon-sitemagneticscattering.Wenotethatinthissimplemodel,itisnotpossibletoobtainsharpresonancepeaksclosetothegapedgeasseenintheexperiment(Figure 4-7 (a))[ 75 ].Figure 4-7 (b)showsthelatticeLDOSspectrum,computedwith2020supercells,atdierentsitesaroundsuchimpurity.Resonancepeaksat01=2:6meV 61

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Figure4-7. (a)DierentialtunnelingconductancespectraabovetheNiatomsitesubstitutingforaCusiteinBSCCO(solidcircles),abovetherstnearestneighborCuatomposition(opencircles),abovethesecondnearestneighborCuatom(squares),andatypicalspectrumfarfromtheNiatom(triangles).Impurityinducedresonancepeaksareobservedat1=9meVand1=19meV.ReprintedbypermissionfromMacmillanPublishersLtd:Nature[ 64 ],copyright2001.(b)LatticeLDOSspectrumaroundamagneticimpuritywithon-sitepotentialU0=0:6eVandon-siteexchangecouplingJ0=0:3U0.Spectrumcalculatedusing2020supercellswitharticialbroadeningof1meV,attheimpuritysite,nearestneighborsite,next-nearestneighborsite,andasitefarawayfromtheimpurity,areshowninblue,red,greenandblack,respectively.(c)ContinuumLDOSspectrumataheightz5AabovetheBiOsurface.ShownarepositionsdirectlyaboveaCuatomfarfromimpurity(black),attheimpurity(blue),onthenearestneighborposition(red),andonthenext-nearestneighborposition(green). 62

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Figure4-8. (a)LatticeLDOSspectrumaroundamagneticimpuritywithon-sitepotentialU0=0:6eV,nearest-neighborpotentialUNN=0:3eV,andnearest-neighborexchangecouplingJNN=UNN.Spectrumcalculatedusing2020supercellswitharticialbroadeningof1meV,attheimpuritysite,nearestneighborsite,next-nearestneighborsite,andasitefarawayfromtheimpurity,areshowninblue,red,greenandblack,respectively.(b)ContinuumLDOSspectrumataheightz5AabovetheBiOsurface.ShownarepositionsdirectlyaboveaCuatomfarfromimpurity(black),attheimpurity(blue),onthenearestneighborposition(red),andonthenext-nearestneighborposition(green). and01=7:6meVareeasilyobservedintheLDOSspectrumattheimpuritysiteanditsnearestneighbor(NN);however,spectrumatthenextnearestneighbor(NNN)sitesimplyfollowsthebulkLDOSspectrum.Resonancepeaksat+01and)]TJ /F1 11.9552 Tf 9.298 0 Td[(02havedown-spinpolarizationwhereasthoseat+02and)]TJ /F1 11.9552 Tf 9.298 0 Td[(01haveup-spinpolarization.Inabsenceofmagneticscattering,resonancepeaksarespindegenerate.Asmallon-siteexchangecouplingJ0liftsthisdegeneracyastheeectiveimpuritypotentialexperiencedbyelectronsbecomesspindependent,Veimp=U0+J0,andleadstotwospinpolarizedin-gapresonancestates.Figure 4-7 (c)showsthecontinuumLDOSspectruminaplanelocatedatz5AaboveBiOplane,atpositionswhicharedirectlyabovetheimpurity,NNsite,NNNsite,andadistantsite.SimilartotheSTMresults[ 64 ]continuumLDOSshowsdoublepeakstructureatnegative(positive)energiesatthepositionsdirectlyaboveNNsite(impurityandNNsites).NotethatthistrendiscompletelymissinginthelatticeLDOSspectrum 63

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Figure4-9. (a),(d)LatticeLDOSmaparoundthemagneticimpurityinaregioncomprising1111unitcells,attheresonantenergies01=2:6meV.ColorbarvaluesareintheunitsofeV)]TJ /F6 7.9701 Tf 6.587 0 Td[(1.(b),(e)ContinuumLDOSmapsat01andz5AaboveBiOplanewithsameareaasin(a)and(d).ColorbarvaluesareintheunitsofeV)]TJ /F6 7.9701 Tf 6.587 0 Td[(1bohr)]TJ /F6 7.9701 Tf 6.587 0 Td[(3.(c),(f)STMconductancemapsat1=9meVrotatedtomatchtheorientationin(b)and(c),andcroppedto1111unitcellswiththeimpuritylocatedatcenter.ReprintedbypermissionfromMacmillanPublishersLtd:Nature[ 64 ],copyright2001. (Figure 4-7 (b)).Althoughthesplittingofpeaksduetomagneticscatteringiscapturedinthissimplemodelofapoint-likeimpurity,acarefulcomparisonwiththeexperimentalconductancespectrumshownintheFigure 4-7 (a)showsthattherelativeheightsofthepeaksarereversed.Inthesimpleon-siteimpuritymodel,theheightoftheresonancepeakdecreasesandwidthincreasesasitmovesawayfromthemid-gapduetoincreasinghybridizationbetweenbulkandimpuritystates[ 75 ].Thustheexperimentallyobservedrelativepeakheightscannotbeexplainedbysuchmodel.Tothisend,wendthatanextendedimpuritymodelwithon-sitepotentialU0=0:6eV,NNpotentialUNN=0:3eV,andNNexchangecouplingJNN=UNNcanleadtothedesiredtrend.Thisimpuritymodelyieldssharpresonancepeaksat001=4:2meVand002=18:6meVinthelatticeandcontinuumLDOSspectrumasshowninFigure 4-8 (a)and(b),respectively.Resonancepeaksat+001and)]TJ /F1 11.9552 Tf 9.299 0 Td[(002(+002and)]TJ /F1 11.9552 Tf 9.299 0 Td[(001)arefoundtohaveup(down)spin 64

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polarization.Clearly,resonancepeaksat002arehigherthanthoseat001.Moreover,similartotheexperimentandalsotothecaseofon-siteimpuritymodel,continuumLDOSshowninFigure 4-8 (b)displaysswitchingofresonancepeaksfrompositivetonegativebiasesandthenbacktopositivebiasesasonemovesfromimpuritytoNNandthentoNNNsites.Thus,theextendedimpuritymodelcapturesmostofthefeaturesofSTMresults[ 64 ];however,themicroscopicoriginofsuchamodelstillneedstobeinvestigated.Now,weturntothespatialLDOSmapsattheresonanceenergiesforthepoint-impuritymodel.TheLDOSmapsfortheextendedimpuritymodelisfoundtobequalitativelythesame.Figure 4-9 (a)showsthelatticeLDOSmapat!=+01aroundanimpurityinaregioncomprising1111unitcells.ThevanishingLDOSattheimpuritysiteandmaximumattheNNsiteobservedhereiscompletelyoppositetotheexperimentalconductancemapat!=+1reproducedinFigure 4-9 (c).However,thecontinuumLDOSmap(r;+01),ataheightz5AaboveBiOplane,showninFigure 4-9 (b)comparesverywellwiththeexperiment.Asimilartrendholdsatthenegativeenergyresonancepeak.ThecontinuumLDOSmapat!=)]TJ /F1 11.9552 Tf 9.298 0 Td[(01showninFigure 4-9 (e)showsexcellentagreementwiththeexperimentalconductancemapat!=)]TJ /F1 11.9552 Tf 9.298 0 Td[(1reproducedinFigure 4-9 (f).ThecontinuumLDOSmapsat!=+02()]TJ /F1 11.9552 Tf 9.299 0 Td[(02)isfoundtobeverysimilartothatat!=+01()]TJ /F1 11.9552 Tf 9.299 0 Td[(01). 4.6QuasiparticleInterferenceTheSTMexperimentscanprovidemomentum-spaceinformationabouttheenergydispersioninamaterialviatheFouriertransformspectroscopyorquasiparticleinterference(QPI).ThebasicconceptbehindQPIistheFriedeloscillation:AdefectinotherwisehomogeneousmetalscatterselectronselasticallyfromtheBlochstatek1tok2resultingintooscillationsinthelocaldensitywithwavevectorq=k1)]TJ /F9 11.9552 Tf 13.219 0 Td[(k2;q=2kFwherekFistheFermimomentum.SuchoscillationscanbeobservedintheSTMtunnelingconductanceinanenergy-resolvedway,andFouriertransformationofreal-space 65

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Figure4-10. (a)IsoenergycontoursforquasiparticleenergyEkinthed-wavehomogeneoussuperconductingstateobtainedwithinourmodel.QPIpatternsobtainedbyFouriertransformofthe(b)latticeDOS(BdG)(c)continuumLDOS(BdG+W)obtainedat!=24meVforaweakimpuritymodeledwithon-sitepotentialVimp=0:3eV. conductancemapsyieldthedispersionq(!)fromwhichimportantinformationaboutthehomogeneoussystemcanbeextracted[ 141 142 ].Inthecupratesuperconductingstatewherethequasiparticlebandstructureishighlyanisotropic,QPIpatternshavebeenunderstoodintermsofthesocalled'octetmodel'[ 93 ].Duetoelasticscatteringatdefects,thewavevectorswhichappearinQPIpatternatabiasV=!=emustcorrespondtothevectorsconnectingtwopointsontheisoenergycontourk(!).ForsuperconductingcuprateswithadispersionEk=p 2k+2kwherekisthebandenergyrelativetotheFermienergyandkisthed-wavesuperconductinggaporderparameter,theisonergycontoursform"banana-like"shapesasshowninFigure 4-10 (a).Moreover,theeightpointscorrespondingtothetipsofthebananasatagivenbiashavelargestDOS,hencethescatteringprocessesconnectingthesetipsshouldhavelargestprobabilityresultingintolargeQPIsignalsatthecorrespondingwavevectors.Thesevenwavevectorsq1toq7fromoneoftheeightpointsareshowninFigure 4-10 (a).TheimpuritieswhichcauseQPIinBSCCOarebelievedtobethenativedefectslikeout-of-planeoxygeninterstitials,andBi-Srsite-switching.Wemodelsuchdefectsbyweakpoint-likeimpuritieswithon-sitepotentialVimp=0:3eVandobtainlattice 66

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Figure4-11. QPIZ-mapsobtainedat!=24meVusing(a)latticeDOS(BdG)(b)continuumLDOS(BdG+W)(c)STMexperiment[ 143 ].Energyintegrated-mapsobtainedfrom(d)latticeLDOSusingT-matrixmethod[ 143 ],(e)latticeLDOSusingBdGcalculation,(f)continuumLDOSusingBdG+Wapproach,and(g)STMexperiment.(c),(d),and(g)arereproducedfrom[ 143 ].ReprintedwithpermissionfromAAAS. andcontinuumLDOSfollowingaprocedureverysimilartothatoutlinedinSection 4.4 .Subsequently,weFouriertransformthelatticeandcontinuumLDOSmapstondtheQPIpatterns.Correspondingresultsforbias!=24meVareshowninFigure 4-10 (b)and(c),respectively.SincethelatticeLDOSinformationisavailableonlyforlengthslargerthanthelatticeconstanta,theQPIpatternobtainedfromlatticeLDOSshowswavevectorsonlyintherstBrillouinzone[)]TJ /F4 11.9552 Tf 9.299 0 Td[(=a;=a],thus,whencomparedwiththeoctetmodelpredictions(shownbyredcircles)onlyfewwavectorsarereproduced.However,theexperimentalpatternshavenosuchrestrictions,andQPIsignalsareobservedinhigherBrillouinzonestoo.TheQPIpatternobtainedfromthecontinuumLDOSshowamuchbetteragreementwiththeoctetmodelwhereallqvectorsarecapturedduetoavailabilityoftheintra-unitcellLDOSinformation.ForaclosercomparisonwiththeexperimentalQPIpatternswefollow[ 143 ],andcomputetheFouriertransformofZ-mapsZ(q;!)andtheenergyintegratedmap(q) 67

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denedas Z(q;!)=FT[Z(r;!)]=FT(r;!) (r;)]TJ /F4 11.9552 Tf 9.298 0 Td[(!)(4{5) (q)=Z00d!Z(q;!);(4{6)whereFTrepresentstheFouriertransform.Figure 4-11 (a)and(b)showtheQPIZ-mapsat!=24meVobtainedfromlatticeLDOSandcontinuumLDOS,respectively.Clearly,thelattershowsamuchbetteragreementwiththeexperimentalmapatthesamebiasshowninFigure 4-11 (c).Particularly,thearc-likefeature(reminiscentoftheFermisurface)nearq=[;]andrelativeintensityatdierentqvectorscompareverywellwiththeexperimentalplot.The(q)mapcalculatedfromthelatticeLDOSobtainedusingT-matrixmethod[ 143 ]andBdGmethod,continuumLDOSobtainedfromourBdG+Wmethod,andthecorrespondingexperimentalresult[ 143 ]areshowningures 4-11 (d),(e),(f),and(g),respectively.Whencomparedwiththeexperimentalresult,itcanbeeasilyobservedthattheBdG+WmethoddemonstratesaqualitativeimprovementovertheT-matrixandBdG-onlymethodsusedpreviously[ 144 { 151 ]tostudytheQPIpatterns. 4.7ConclusionInthischapter,weusedtheBdG+WschemetostudytheimpurityinducedstatesinZn-andNi-dopedBSCCO,acupratesuperconductorthathasbeenwidelystudiedinSTMandARPESexperiments.Westartedwitha1-bandtight-bindingmodelandcorrespondingCu-dx2)]TJ /F5 7.9701 Tf 6.586 0 Td[(y2Wannierfunctionderivedfromtherstprinciplescalculations.Subsequently,d-wavesuperconductivitywasintroduced"byhand"viaNNpair-potentialswithmagnitudechosentoyieldthespectralgapobservedinSTMexperiments[ 57 60 ]onoptimally-dopedBSCCO.WeshowedthatthecontinuumLDOSspectrumaboveaCusiteobtainedataheightz5AabovetheBiOplane,whereanSTMtipwouldbetypicallyplaced,showsamoreU-shapelikefeatureatlowenergiesincontrasttothewell-knownV-shapedspectrumofthelatticeDOS.Weshowedthatthischangeinthelowenergyspectralfeaturesisadirectconsequenceofthedx2)]TJ /F5 7.9701 Tf 6.587 0 Td[(y2-symmetryoftheWannier 68

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function.Furthermore,wementionedfewexperimentalevidencesofU-shapedspectraintheoverdopedregionofthecupratephasediagramwhereourtheoryshouldbevalidandpresentedsomescenarioswhichcanleadtothetransitiontoaV-shapedspectragoingfromover-to-underdopedregime.ThenwestudiedtheZnimpurityprobleminBSCCObymodelingZn,basedonabinitiocalculations,asastrong,attractive,point-likepotentialscatterersubstitutingaCuatom.BysolvingtheBdGequationsonalargesquarelattice,wefoundthatitinducessharpin-gapboundstatesasexpectedfromprevioustheoreticalandnumericalworks[ 68 75 ].Moreimportantly,bycalculatingthecontinuumLDOSwithinBdG+Wframework,weresolvedthelong-standingdiscrepancybetweentheoreticalpredictionsoftheimpurityinducedspatialpatterns[ 61 ]oftheLDOSandcorrespondingSTMobservations[ 60 ].WeshowedthatthecontinuumLDOSobtainedatatypicalSTMtipheightabovetheBiOplanedisplaysexcellentagreementwiththeexperiment[ 60 ],andtheintensitymaximumabovetheimpuritysightcanbeeasilyexplainedbyaccountingforthetailsofCudx2)]TJ /F5 7.9701 Tf 6.587 0 Td[(y2Wannierfunctiontailswithsignicantweightcomingfromtheapicaloxygenatomsinthenearestneighborunitcells.Next,westudiedtheNiimpurityprobleminBSCCObyusingasimplemodelofmagneticimpuritywithweakon-sitepotentialandexchangescattering,andshowedthatsuchanimpurityinducedtwospin-polarizedin-gapresonancestatesaspredictedbyearliertheories[ 122 ].Moreover,thecontinuumLDOSmapsattheresonanceenergies,computedattypicalSTMtipheightdisplaysexcellentagreementwiththecorrespondingSTMresults[ 64 ].Furthermore,weproposedanextendedimpuritymodeltoexplaintherelativeresonancepeakheightsasobservedintheexperiment.Finally,weshowedthattheQPIpatternsobtainedusingtheFouriertransformofthecontinuumLDOSmapsdemonstratedaqualitativeimprovementovertheT-matrixorBdGresults,capturingnearlyallwavevectorspredictedbytheoctetmodel,andshowedanexcellentagreementwiththeSTMexperiments[ 143 ]. 69

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CHAPTER5CHARGEORDERINCUPRATESInthischapter,Iwilldiscussthechargeorderedstatesintheextendedt)]TJ /F3 11.9552 Tf 11.955 0 Td[(Jmodelforunderdopedcuprates.Mostofthematerialpresentedhereisbasedonapublishedpaper[ 152 ].Allthepublishedcontents(excerptsandgures)arereprintedwithpermissionfrom PeayushChoubey,Wei-LinTu,T.K.Lee,andP.J.Hirschfeld,NewJ.Phys.90,013028(2017) ,copyright2017bytheInstituteofPhysicsPublishing. 5.1MotivationThechargeorder(CO)orchargedensitywave(CDW)statereferstoatranslationalsymmetrybrokenphasewheretheelectronicchargedensityformsamodulatingstructurewithaperiodicitydierentfromtheunderlyinglattice.FirstpredictedbyRudolfPeierls[ 153 ]asaninstabilityofone-dimensional(1D)metals,COhasbeenobservedinvarietyofmaterialssuchasquasi1D[ 154 155 ]and2D[ 156 ]organiccompounds,transition-metalchalcogenides[ 155 ]andoxides[ 157 { 159 ].TheFermisurfacedrivenmechanismofCDWformationinquasi1DmetalsisexplainedinFigure 5-1 .TheperfectFermisurfacenestingatthewavevectorq=2kFleadstoadivergingchargesusceptibilityatthesamewavevector,whichinturnmakestheelectrongasunstablewithrespecttoformationofamodulatingelectrondensitywithperiodicity0= kF.ThecostofCoulombrepulsioncanbeovercomebytheopeningofagapattheFermilevel.Iftheperiodicityofthechargedensityhappens(not)tobeanintegermultipleofthelatticeconstantthentheCDWstateiscalledcommensurate(incommensurate).Incuprates,therstobservationofchargeorderoccurredintheformof"stripeorder"[ 37 161 { 163 ]throughneutronscatteringexperimentsontheLa-basedcupratesLa2)]TJ /F5 7.9701 Tf 6.587 0 Td[(xBaxCuO4andLa2)]TJ /F5 7.9701 Tf 6.587 0 Td[(x)]TJ /F5 7.9701 Tf 6.586 0 Td[(yNdySrxCuO4[ 164 165 ].Stripeorderreferstoastatecharacterizedbyacombinationofaunidirectional,incommensurateCDWandaspindensitywave(SDW),wherethewavelengthofthelatteristwiceoftheformer,i.e.SDW=2CDW.Themostprominenteectsofthestripeorderareobservedaround 70

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Figure5-1. (a)SchematicoftheFermisurfaceofaquasi1Dmetal,illustratingnestingatthewavevectorq.(b)Thechargesusceptibility(q)ofthefreeelectrongasinone,two,andthreedimensionsasafunctionofq.In1D(q)divergesatq=2kF,wherekFistheFermimomentum,makingthesystemsusceptibletochargeordering.(c)Dispersionof1Delectrongasbefore(after)thePeierlstransitionshowninblue(red).Energygapopensupatq=2kF.(d)Uniform(modulating)electrondensitybefore(after)thePeierlstransition.Reprintedfrom[ 160 ].CopyrightIOPPublishing.Reproducedwithpermission. holedopingofx=1=8,wherestripesshowacommensuratenaturewithCDW=4a0[ 164 ],a0beingthein-planelatticeconstant.TheTcisobservedtobestronglysuppressed,indicatingacompetitionbetweenthestripephaseandd-wavesuperconductivity[ 166 ].Moreover,thestripewavevectorisobservedtoincreasewithincreasingholedoping[ 36 ].ApartfromtheLa-basedfamily,thereislittleevidenceofsimilarstripeorderinothercuprates.Instead,aunidirectionalincommensuratechargeorder,withoutanyaccompanyingmagneticorder,isobservedinvarietyofcupratessuchasYBa2Cu3O6+x(YBCO)[ 34 167 168 ],Bi2Sr2CaCu2O8+x(BSCCO)[ 35 142 ],Bi2Sr2)]TJ /F5 7.9701 Tf 6.587 0 Td[(xLaxCuO6+(LaBSCCO)[ 169 ],HgBa2CuO4+(HBCO)[ 170 171 ],andNaxCa2)]TJ /F5 7.9701 Tf 6.587 0 Td[(xCuO2Cl2(NaCCOC)[ 35 ],usingarangeofexperimentaltechniqueslikenuclearmagneticresonance(NMR)[ 34 172 ],STM[ 35 169 173 { 175 ],X-raydiraction[ 168 176 ],andresonantX-rayscattering[ 36 167 177 178 ].Inmostofthesecompounds,thechargeorderingwavevectorisfoundtodecreasewithincreasingholedoping[ 36 ],atrendoppositetothatinstripeorderedcuprates.InSTMexperiments,chargeorderappearsasdomainsofunidirectionalconductancemodulationswithshortcoherencelengths(30A[ 36 ])asshowninFigure 5-2 (a).ThemodulationsaremostprominentonOsites,asreectedinFigure 5-2 (b).In 71

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Figure5-2. (a)TheSTMR-map(ratiooftunnelingcurrentmapsatbiasV)of12%hole-dopedNaCCOCtakenatV=150mVandT=4:2K.(b)Zoomed-inviewoftheR-mapintheareaboundedbythebluesquarein(a).ThelocationsofCuatomsaremarkedbycross.DensitymodulationsareveryweakonCusitesandstrongonOsites.Bothguresarefrom[ 35 ].ReprintedwithpermissionfromAAAS.(c)ResonantX-rayscatteringmeasurementsinunderdopedNd1:2Ba1:28Cu3O7showingquasi-elasticpeakatq[)]TJ /F1 11.9552 Tf 9.298 0 Td[(0:31;0]inreciprocallatticeunits,asasignatureoftheunidirectionalchargeorderatthesamewavectorq.Reprintedfrom[ 167 ]withpermissionfromAAAS.(d)SchematicrepresentationofchargedensitymodulationsonCuandOsitesinthechargeorderedstateswithpures-,s0-,andd-formfactor,respectively.ReprintedbypermissionfromMacmillanPublishersLtd:Nature[ 179 ],copyright2015.(d)STMdeterminedbiasdependenceofformfactorsin8%hole-dopedBSCCOatT=4:2K.ReprintedbypermissionfromMacmillanPublishersLtd:Nature[ 66 ],copyright2015. resonantX-rayscatteringexperiments,chargeorderisidentiedbytheappearanceofaquasi-elasticpeakinthephotonenergylossspectrumataparticularmomentumqparalleltotheCu-Obonddirection[ 36 ].Forexample,inNd1:2Ba1:28Cu3O7[ 167 ],suchapeakisobservedatq[)]TJ /F1 11.9552 Tf 9.299 0 Td[(0:31;0](inreciprocallatticeunits)asshowninFigure 5-2 (c). 72

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Anotherintriguingpropertyoftheaforementionedchargeorderisitsintra-unitcellsymmetry.Bytakingtheintra-unitcelldegreesoffreedomcorrespondingtotheO-px,O-py,andCu-dx2)]TJ /F5 7.9701 Tf 6.587 0 Td[(y2orbitalsintoaccount,theCDWorderparameterCDW(k;Q)=Dcyk+Q=2ck)]TJ /F11 7.9701 Tf 6.587 0 Td[(Q=2E,withQbeingtheorderingwavevector,canberepresentedasalinearcombinationofthreecomponents[ 36 179 180 ] CDW(k;Q)=s+s0(coskx+cosky)+d(coskx)]TJ /F1 11.9552 Tf 11.955 0 Td[(cosky)(5{1)Thersttermcorrespondstoapures-formfactorCDWwithamplitudeswherechargemodulationoccursonlyattheCu-dx2)]TJ /F5 7.9701 Tf 6.587 0 Td[(y2orbital.Thesecondandthirdtermscorrespondtos0-andd-formfactorCDWstates,respectively,wherechargemodulationoccursonlyattheO-px;pyorbitals.Informer(latter)themodulationsonO-pxandO-pyorbitalsarein-phase(outofphase).TheschematicrepresentationoftheseformfactorsisshownintheFigure 5-2 (d).TheSTMexperimentsonthesuperconductingunderdopedBSCCO[ 35 65 175 ]andNaCCOC[ 35 65 ]revealthatthedominantchargemodulationoccursattheOsiteswithaphasedierencebetweenthetwoinequivalentOatomsresultingintoadominantd-formfactor(seeFigure 5-2 (b)).Moreover,theresonantx-rayscatteringexperiments[ 179 ]onLaBSCCOandYBCOconcludethesame.Takingastepfurther,Hamidianetal[ 66 ]mappedthebiasdependenceofformfactorsusingaphase-resolvedFourieranalysis[ 65 ]ofreal-spaceSTMdatawithintra-unitcellresolution.TheresultisreproducedinFigure 5-2 (e).Atlowenergies,thes0-formfactordominates,whereasathigherenergiesthed-formfactorhasthelargestweightwithpeakaroundthepsuedogapenergyscale.Moreover,intherangewherethed-formfactordominates,theconductancemapsaboveandbelowFermienergydisplayaspatialphasedierenceof.Furthermore,usingJosephsontunnelingspectroscopy[ 181 ],thesamegrouphasreportedtheobservationofpairdensitywaves-astatewithspatiallymodulatingsuperconductinggaporderparameter[ 182 ]. 73

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Theoretically,chargeorderincuprateshasbeenstudiedfromboththeitinerantandlocalizedperspectives.Intheweak-couplingtreatmentoftheitinerantspin-fermionmodel[ 183 ],ad-formfactorchargeorderarisesasconsequenceoftheinstabilityinthechargechannelcausedbystrongantiferromagneticspin-uctuationsatthewavevectorsconnectingthehot-spotsontheFermisurface[ 180 184 { 187 ].Allsuchcalculationsexcept[ 185 ]ndtheorderingwavevectortobeorientedalongtheBrillouinzonediagonal,45awayfromtheexperimentallyobserveddirection[ 36 ].Wangetal.[ 185 ]foundthecorrectorientationofthewavevectorbutastrong-couplingEliashbergliketreatmentoftheirmodelresultedinastrongsuppressionofthechargeorderalongthesewavevectors[ 188 ].Fromthelocalizedperspectiveof"dopingaMottinsulator"[ 24 ],thechargeorderincuprateshasbeenextensivelystudiedusingt)]TJ /F3 11.9552 Tf 11.956 0 Td[(Jtypemodels[ 24 29 30 ]throughavarietyoftechniquessuchasGutzwillermean-eldtheory[ 189 190 ],variationalMonteCarlo[ 191 { 194 ],andtensornetworkmethods[ 195 ].Mostofthesestudies,withanexceptionof[ 189 ]and[ 190 ],werefocusedonthestripeorder,appropriatefortheLa-basedcupratefamily.Theauthorsof[ 190 ]consideredchargeorderwithandwithoutanyaccompanyingmagneticorderandshowedthatthet)]TJ /F3 11.9552 Tf 11.955 0 Td[(Jmodel,withinGutzwillermean-eldapproximation,supportsseveralunidirectionalandbidirectionalchargeorderedstateswithnearlydegenerateenergies.Ofthese,theanti-phasechargedensitywave(APCDW),aunidirectionalcommensurateorderwithperiodicityof4latticeconstants,andnodalpairdensitywave(nPDW),anincommensurateorderwithwavevector[Q;0]or[0;Q]withQ0:3inunitsof2 a0,exhibitadominantd-formfactorandexistinadopingrangeverysimilartothatinwhichchargeorderhasbeenexperimentallyobservedincuprates.Moreover,thenPDWstate,whichintertwinesuniformd-wavesuperconductivity,pair-densitywave(PDW),andCDWorder,showsanodalstructureintheLDOSspectrumatlowenergies,similartotheexperimentalobservations[ 35 66 ].TheSTMexperimentsdeterminetheintra-unitcellformfactorsbydirectlyimagingtheCuandOsites,andtheresultsthusobtainedexhibitasignicantbiasdependence 74

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asshownintheFigure 5-2 (e).However,inmostoftheprevioustheoreticalstudies[ 15 180 186 190 196 ],theformfactorshavebeendeterminedfromtherenormalizationsofthenearest-neighbor(NN)hoppingsbetweentheCulatticesitesalongthex-andy-directions.SuchanapproachdisregardstheOdegreesoffreedomandyieldsbias-independentformfactors.Thesedecienciescanbealleviatedusingthecalculationsbasedonthethreebandmodelofcuprates[ 197 ];however,treatingtheno-doubleoccupancycriteriainsuchmodelsisverydicult[ 198 199 ].Alternatively,wecanusetheWannierfunctionbasedapproachdescribedintheSection 2.3 tocalculatetheSTMobservables,avoidingthecomplicationsofathreebandmodelbutstillincludingtheOdegreesoffreedom.Inthischapter,weaddresstheSTMndingsofHamidianetal.[ 66 ]bycalculatingthebiasdependenceofformfactorsandspatialphasedierenceintheAPCDWandnPDWstatesusingtheWannierfunctionbasedapproach.Werstsolvethet)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0)]TJ /F3 11.9552 Tf 11.955 0 Td[(Jmodelwithintherenormalizedmean-eldtheory(Gutzwillerapproximation)andobtaintheAPCDWandnPDWstates.Exploitingtheunidirectionalnatureofthechargeorderedstates,wehaveemployedapartialFouriertransformbasedapproachtostudylargesystemsinacomputationallyecientway.Next,weobtaincontinuumLDOSaboveCuanOsitesatatypicalSTMtipheightusingtheWannierfunctionbasedapproachdescribedintheSection 2.3 ,andcalculatethebias-dependentformfactorsandthespatialphasedierence.WendthattheresultsforthenPDWstatecompareverywellwiththeexperiment[ 66 ].Finally,weshowthatthePDWcharacterofthechargeorderedstatesplaysthekeyroleinyieldingthebiasdependenceofvariousquantitiessimilartotheexperimentalobservations. 5.2Extendedt-JModel 5.2.1RenormalizedMean-FieldTheoryTheextendedt-J(ort)]TJ /F3 11.9552 Tf 11.956 0 Td[(t0)]TJ /F3 11.9552 Tf 11.955 0 Td[(J)model,includingnext-nearestneighbor(NNN)hoppingt0=)]TJ /F1 11.9552 Tf 9.299 0 Td[(0:3ttoappropriatelymodeltheFermisurfaceofcuprates,isgivenona2D 75

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latticeby HtJ=)]TJ /F10 11.9552 Tf 14.249 11.357 Td[(X(i;j);PGtijcyicj+H.c.PG+JXhi;jiSiSj:(5{2)Here,H.c.referstoHermitianconjugate,cyicreatesanelectronatlatticesiteiwithspin=,andSidenotesthespinoperator.hi;jirepresentNNsites,whereas(i;j)referstobothNNandNNNsites.PG=Qi(1)]TJ /F4 11.9552 Tf 12.979 0 Td[(ni"ni#),withni=cyicithespindependentnumberoperatoratsitei,representstheGutzwillerprojectionoperator[ 200 ]whichensuresthatallstateswithdoubleoccupancyofanysiteareprojectedoutfromtheHilbertspace.Thus,therstterminthet)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0)]TJ /F3 11.9552 Tf 11.956 0 Td[(JHamiltonianrepresentshoppingofelectronsinsuchawaythatnodoublyoccupiedsiteiscreated.ThesecondtermrepresentstheantiferromagneticexchangeinteractionbetweenspinsattheNNsiteswithJ=0:3t.TheprojectionoperatorcanbetreatedexactlyusingthevariationalMonte-Carlotechnique[ 30 ]andapproximatelyusingslave-bosonmethods[ 32 33 ]orrenormalizedmean-eldtheory[ 30 ]withinGutzwillerapproximation[ 201 ],wheretheprojectionoperatorisreplacedbyGutzwillerfactorsdeterminedfromclassicalcountingarguments[ 30 201 { 203 ].ApplicationoftheGutzwillerapproximationtotheHamiltonianinEquation 5{2 leadstothefollowingrenormalizedHamiltonian H=)]TJ /F10 11.9552 Tf 14.249 11.357 Td[(X(i;j);gtijtij(cyicj+H.c.)+Xhi;jiJ"gs;zijSziSzj+gs;xyij S+iS)]TJ /F5 7.9701 Tf -.696 -8.012 Td[(j+S)]TJ /F5 7.9701 Tf -.696 -8.012 Td[(iS+j 2!#:(5{3)Here,Szirepresentsthez-componentofthespinoperator,andS+i(S)]TJ /F5 7.9701 Tf -.696 -8.012 Td[(i)representstheraising(lowering)spinoperator.TheGutzwillerfactorsforhoppingrenormalization(gtij),longitudinal(gs;z),andtransverse(gs;xy)super-exchangerenormalizationdependonthelocalvaluesofholedensityi,magnetizationmi,paireldvij,andbondeldvij 76

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denedby i=1)-222(h0jXnij0imi=h0jSzij0ivij=h0jcicjj0ivij=h0jcyicjj0i(5{4)where,=)]TJ /F4 11.9552 Tf 9.298 0 Td[(andj0idenotestheunprojectedwavefunction.Thesuperscriptvinvijandvijisusedtoemphasizethatthesequantitiesarenotthephysicalorderparameters.TheGutzwillerfactorsaregivenas[ 189 ] gtij=gtigtj;gs;xyij=gs;xyigs;xyjgti=s 2i(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(i) 1)]TJ /F4 11.9552 Tf 11.955 0 Td[(2i+4m2i1+i+2mi 1+i)]TJ /F1 11.9552 Tf 11.955 0 Td[(2migs;xyi=2(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(i) 1)]TJ /F4 11.9552 Tf 11.955 0 Td[(2i+4m2igs;zij=gs;xyij"2(2ij+2ij))]TJ /F1 11.9552 Tf 11.956 0 Td[(4mimjX2ij 2(2ij+2ij))]TJ /F1 11.9552 Tf 11.955 0 Td[(4mimj#Xij=1+12(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(i)(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(j)(2ij+2ij) q (1)]TJ /F4 11.9552 Tf 11.955 0 Td[(2i+4m2i)(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(2j+4m2j);(5{5)where,ij=1 2Pvijandij=1 2Pvij.Sinceweareinterestedinthechargeorderedstate,whichdoesnotaccompanyanymagneticorder,wecansimplifytheGutzwillerfactorsbyputtingmi=0as gti=r 2i 1+igs;xyi=2 1+igs;zij=gs;xyij=gsij:(5{6)Bythemean-elddecompositionofthequarticoperatorsintherenormalizedHamiltonian(Equation 5{3 )wecanobtainaquadraticHamiltonianHH)]TJ /F5 7.9701 Tf 6.586 0 Td[(F.However,duetothedependenceoftherenormalizationfactorsonthemean-eldsthemselves,adirectmean-elddecompositiondoesnotyieldthelowestgroundstateenergy 77

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Eg=h0jHH)]TJ /F5 7.9701 Tf 6.587 0 Td[(Fj0i.Instead,Egshouldbeminimizedwithrespecttoj0iundertheconstraintofxedelectronllingNeandnormalizationofthewavefunction.Equivalently,wecanminimizefollowingfunction W=h0jHj0i)]TJ /F4 11.9552 Tf 19.261 0 Td[((h0j0i)]TJ /F1 11.9552 Tf 19.261 0 Td[(1))]TJ /F4 11.9552 Tf 11.955 0 Td[((Xini)]TJ /F4 11.9552 Tf 11.956 0 Td[(Ne);(5{7)whereandaretheLagarangemultipliers.EnforcingW h0j=0leadstothefollowingmean-eldHamiltonian[ 189 190 204 ] HMF=X(i;j);ijcyicj+h:c:+Xhi;ji;Dijcicj+h:c:)]TJ /F10 11.9552 Tf 11.955 11.358 Td[(Xi;ini;(5{8)where ij=)]TJ /F4 11.9552 Tf 9.299 0 Td[(gtijtij)]TJ /F4 11.9552 Tf 11.955 0 Td[(ij;hiji3 4JgsijvijDij=)]TJ /F1 11.9552 Tf 10.494 8.087 Td[(3 4Jgsijviji=+3J 4Xj0jvij0j2+jvij0j2dgsij dni+tijXj0)]TJ /F4 11.9552 Tf 5.48 -9.684 Td[(vij0+vij0dgtij dnidgsij dni=)]TJ /F4 11.9552 Tf 26.837 9.167 Td[(gtj (1+i)2;dgtij dni=2gtj p 2i(1+i)3=2:(5{9)Here,ij;hiji=1ifiandjareNNsand0otherwise.NotethatEquation 5{9 istrueonlyforthenon-magneticstate(mi=0).Foramagneticstate,theexpressionsfortheparametersinEquation 5{8 aremuchmorecomplicatedandcanbefoundin[ 190 204 ].Thephysicald-wavesuperconductinggaporderparameterandthebondorderalongx(y)directionaregivenby i=1 8X(gti;i+^xvi;i+^x;+gti;i)]TJ /F6 7.9701 Tf 7.088 0 Td[(^x;vi;i)]TJ /F6 7.9701 Tf 7.088 0 Td[(^x;)]TJ /F4 11.9552 Tf 11.955 0 Td[(gti;i+^yvi;i+^y;)]TJ /F4 11.9552 Tf 11.955 0 Td[(gti;i)]TJ /F6 7.9701 Tf 7.175 0 Td[(^yvi;i)]TJ /F6 7.9701 Tf 7.175 0 Td[(^y;)(5{10) i;i+^x(^y)=1 2X)]TJ /F4 11.9552 Tf 5.48 -9.683 Td[(gti;i+^x(^y)vi;i+^x(^y);+H.c.(5{11) 5.2.2Quasi1DBdGEquationsThechargeorderobservedinvariouscupratesisunidirectionalwithmodulationsalongeitheroftheCu-Obonddirections[ 36 ].Accordingly,westudyonlystatesofthis 78

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type,byassumingchargemodulationsalongx-direction,andutilizethetranslationalinvariancealongthey-directiontoconvertthe2DHamiltonianinEquation 5{8 intoaquasi-1DHamiltonian,substantiallyreducingthecomputationalcostofdiagonalization.Wechangethebasis(ix;iy)to(ix;k)usingthefollowingtransformation cyi=1 p NXkcyix(k)e)]TJ /F5 7.9701 Tf 6.586 0 Td[(ikRiy:(5{12)Here,cyix(k)createsanelectronatquasi-1Dsiteixwithtransversemomentumkandspin,Nisthenumberoflatticesitesiny-direction,andRiyisthey-componentofthelatticevectorcorrespondingtositei.Usingthistransformation,themean-eldHamiltonianinEquation 5{8 becomes HMF=Xhix;jxi;k;ixjx(k)cyix(k)cjx(k)+H.c.+Xhix;jxi;k;Dixjx(k)cix(k)cjx()]TJ /F4 11.9552 Tf 9.298 0 Td[(k)+h:c:)]TJ /F10 11.9552 Tf 14.323 11.358 Td[(Xix;k;ixnix(k);(5{13)where ixjx(k)=Xiyixiyjx0e)]TJ /F5 7.9701 Tf 6.587 0 Td[(ikRiy;Dixjx(k)=XiyDixiyjx0e)]TJ /F5 7.9701 Tf 6.586 0 Td[(ikRiy;ix=(ix;0):(5{14)HMFisquadraticinelectronoperatorsandcanbediagonalizedusingthefollowingspin-generalizedBogoliubovtransformation, cix(k)=Xnhunix(k)n(k)+vnix(k)yn(k)icyix()]TJ /F4 11.9552 Tf 9.299 0 Td[(k)=Xnhvnix(k)n(k)+unix(k)yn(k)i(5{15)ProceedinginamannerverysimilartothederivationofBdGequationsinSection 2.2.2 ,weobtainthefollowingBdGequationsforthemean-eldHamiltonianinEquation 5{13 Xjx264ixjx"(k))]TJ /F4 11.9552 Tf 9.299 0 Td[(Dixjx"()]TJ /F4 11.9552 Tf 9.299 0 Td[(k))]TJ /F4 11.9552 Tf 9.299 0 Td[(Djxix"(k))]TJ /F4 11.9552 Tf 9.299 0 Td[(ixjx#()]TJ /F4 11.9552 Tf 9.299 0 Td[(k)375264unjx"(k)vnjx#(k)375=En"(k)264unix"(k)vnix#(k)375:(5{16) 79

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Here,ixjx(k)=ixjx(k))]TJ /F4 11.9552 Tf 12.231 0 Td[(ixixjx.Theelectrondensity,paireld,andbondeldsaregivenbythefollowingequations: nix"=1 NXknix"(k)=1 NXnkjunix"(k)j2f(En"(k))nix#=1 NXknix#(k)=1 NXnkjvnix#(k)j2[1)]TJ /F4 11.9552 Tf 11.955 0 Td[(f(En"()]TJ /F4 11.9552 Tf 9.299 0 Td[(k))]vixjx"(k)=Xnunix"(k)vnjx#(k)[1)]TJ /F4 11.9552 Tf 11.955 0 Td[(f(En"(k))]vixjx"(k)=Xnunix"(k)unjx"(k)f(En"(k))vixjx#(k)=Xnvnix#()]TJ /F4 11.9552 Tf 9.299 0 Td[(k)vnjx#()]TJ /F4 11.9552 Tf 9.299 0 Td[(k)[1)]TJ /F4 11.9552 Tf 11.955 0 Td[(f(En"()]TJ /F4 11.9552 Tf 9.298 0 Td[(k))];(5{17)wherefdenotestheFermifunctionandthesumrunsoverallvaluesofn.Equations 5{16 and 5{17 aresolvedself-consistentlyusingthefollowingprocedure,verysimilartothatoutlinedinSection 2.4 1. Chooseinitialguessesforthechemicalpotential0,holedensityi,paireldvijonNNsites,bondeldijonNNandNNNsites.Wetakepaireldasasinusoidalfunctionwithamplitude0andwavenumberQ0,keepingtheholedensityandbondeldsuniform: i=0vi;i+^x;"(k)=0cos[2Q0(ix+1=2)]vi;i+^y;"(k)=)]TJ /F1 11.9552 Tf 9.298 0 Td[(0cos(2Q0ix)vi;i+^x(^y)"=vi;i+^x(^y)#=NN0vi;i+^x^y"=vi;^x^y#=NNN0(5{18)Themodulatingpaireldinducesmodulationsinchargedensity[ 205 ],astheyarecoupledtoeachotherthroughtheBdGequations(Equations 5{13 and 5{17 ). 2. ComputetheGutzwillerfactors,theirrelevantderivatives,andlocalchemicalpotentialiusingEquations 5{6 and 5{9 3. Foreachk=m N(inunitsof2 a0);m=0;1;2;:::;N)]TJ /F1 11.9552 Tf 12.151 0 Td[(1(a)constructtheBdGmatrixinEquation 5{16 usingthedenitionsofmatrixelementsfromEquation 5{14 (b)FindtheeigenvaluesEn(k)andeigenvectors[unix"(k);vnix#(k)].(c)Findthek-spacemean-eldsnix(k),vixjx"(k),andvixjx(k)usingEquation 5{17 80

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4. Findthelatticemean-eldsusingfollowingequations: ni=1 NXknix(k)vi;i+^x;"(k)=1 NXkvix;ix+1;"(k)vi;i+^y;"(k)=1 NXkvix;ix;"(k)eikvi;i+^x;=1 NXkvix;ix+1;(k)vi;i+^y;=1 NXkvix;ix;(k)eikvi;i+^x+^y;=1 NXkvix;ix+1;(k)eik:(5{19) 5. Checkconvergence.Ifmeaneldsobtainedinstep4donotchangefromtheirvaluesinpreviousiterationsuptoadesiredaccuracythenconvergenceisachievedandthereisnoneedforfurtheriterations;otherwisegotostep6. 6. Updatethemean-eldsandchemicalpotentialbymixingthemeaneldsfromcurrentandpreviousiterations. 7. Gotostep2. 5.2.3CalculationofSTMObservablesInmostprevioustheoreticalwork[ 15 180 186 196 ],theintra-unitcellformfactors(s;s0,andd)ofthechargeorderhavebeendeterminedfromtheFouriertransformofthechargedensityonCulatticesitesandtheNNbondorderparameteri;i+^x(^y)whichcanbethoughtasameasureofchargedensityontheCu-Obonds.Mathematically, D(q)=FT(~i;i+^x)]TJ /F1 11.9552 Tf 13.346 0 Td[(~i;i+^y)=2S0(q)=FT(~i;i+^x+~i;i+^y)=2S(q)=FT(1)]TJ /F1 11.9552 Tf 12.496 3.154 Td[(~i):(5{20)Here,FTrepresentstheFouriertransformand~referstothesubtractionofthespatialaverageofthecorrespondingquantity.Clearly,thereisnobiasdependenceoftheformfactorsdenedinthisway.However,theSTMexperiments[ 66 ]usedphase-resolvedLDOS 81

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[ 65 ]obtainedatCuandOsublatticestodenetheformfactorsandcalculatedtheirbiasdependence.UsingtheWannierfunctionbasedmethoddescribedintheSection 2.3 ,wecanadoptasimilarapproach.TherststepistoconstructthetheGreen'sfunctionmatrixinlatticespace,usingtheself-consistentsolutionoftheBdGequations(Equations 5{13 and 5{17 ),throughfollowingequations. Gij(!)=1 NXkgtijGixjx(k;!)eik(Riy)]TJ /F5 7.9701 Tf 6.586 0 Td[(Rjy)Gixjx(k;!)=Xn>0unix(k)unjx(k) !)]TJ /F4 11.9552 Tf 11.955 0 Td[(En+i0++vnix(k)vnjx(k) !+En(k)+i0+:(5{21)Here,0+isthearticialbroadeningandn>0indicatesthatthesumistobeperformedovereigenstateswithpositiveeigenvaluesonly.WithlatticeGreen'sfunctionsinhand,thecontinuumLDOS(r;!)atacontinuumpositionrandagivenbias!canbeobtainedusingEquations 2{14 and 2{18 .Followingtheexperimentalprocedure[ 66 ],weobtainthecontinuumLDOSmapatatypicalSTMtipheightz5AabovetheBiOplaneatpositiveandnegativebiases,andconstructtheZ-mapsdenedbelow[ 35 ], Z(r;!>0)=P(r;!) P(r;)]TJ /F4 11.9552 Tf 9.298 0 Td[(!):(5{22)Subsequently,wesegregatetheZ-mapsinthreesublattices:CuZ(r;!),OZx(r;!)andOZy(r;!)correspondingtoaCuandtwoinequivalentOatomsalongmutuallyorthogonalCuObonddirectionsxandyintheunitcell.TheformfactorscanbeobtainedbytakingproperlinearcombinationsoftheFouriertransformsofthesublatticeZ-mapsasfollows[ 66 ] DZ(q;!)=(~OZx(q;!))]TJ /F1 11.9552 Tf 14.64 3.022 Td[(~OZy(q;!))=2S0Z(q;!)=(~OZx(q;!)+~OZy(q;!))=2SZ(q;!)=~CuZ(q;!):(5{23)Inadditiontothebiasdependenceoftheintra-unitcellformfactors,Hamidianetal.[ 66 ]alsoobtainedthethebiasdependenceoftheaveragespatialphasedierence 82

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()betweenthed-formfactormodulationsaboveandbelowtheFermienergyandfoundthatitswitchesfrom0toclosetotheenergyscalewherethed-formfactorbecomesdominant.Toaddressthisintriguingobservation,wecomputefollowingtheexperimentalprocedure[ 66 ].Werstobtainthed-formfactorDg(q;!)usingthesublatticecontinuumLDOSmaps(org-maps).Subsequently,thed-formfactormodulationvectorQdislteredoutusingaGaussianlter,followedbytheinverseFouriertransformtoyieldaspatialmapD(r;!).Finally,itsphase(r;!)iscalculatedat!andtheirdierenceisaveragedoverspacetoobtain.Mathematically, Dg(q;!)=(~Ogx(q;!))]TJ /F1 11.9552 Tf 14.64 3.022 Td[(~Ogy(q;!))=2D(r;!)=2 (2)2ZdqeiqrDg(q;!)e)]TJ /F7 5.9776 Tf 7.782 4.623 Td[((q)]TJ /F16 5.9776 Tf 5.756 0 Td[(Qd)2 22(r;!)=arctan(Im[D(r;!)]=Re[D(r;!)])=h(r;!))]TJ /F4 11.9552 Tf 11.955 0 Td[((r;)]TJ /F4 11.9552 Tf 9.298 0 Td[(!)i:(5{24) 5.3UnidirectionalChargeOrderedStatesinExtendedt-JModelAlthoughthet)]TJ /F3 11.9552 Tf 11.955 0 Td[(Jmodelsupportsavarietyofchargeorderedstates[ 189 190 ],herewewillfocusontheAPCDWandnPDWstateswhichareshowntohavedominantd-formfactor[ 190 ].ToobtaintheAPCDWstatewithperiodicityoffourlatticeconstants,weworkwithasystemsizewhichisamultipleof8andinitializetheBdGequations(Equations 5{13 and 5{17 )withasinusoidald-wavepaireldmodulatingwithwavenumberQ0=1=8(inunitsof2 a0).Although,itisimpossibletoobtainatrueincommensuratechargeorderedstateinanitelatticecalculation,wecanstillobtainaquasi-incommensuratestatewithaperiodicityequaltothesystem'slength.SuchastatecanbethoughtasamixtureofseveralcommensuratestateswithdierentlatticeperiodicitiesandcanbeobtainedbyinitializingBdGequationswithamodulatingpair-eldwithQ06=m N,wheremisaninteger.WeemphasizethattheinitializingBdGequationswithapairdensitywavedrivesthechargeordering.ThiscanbeeasilyseenthroughEquation 5{9 whichshowsthatthelocalchemicalpotentialidependsonthe 83

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Figure5-3. Energypersite(E/t)atvariousholedopings(x)forhomogeneoussuperconductingstate,APCDW,andnPDWstates.Inset:Variationofsuperconductingorderparameterinthehomogeneousstateasafunctionofholedoping.VerticallinesmarkthedopingrangeinwhichAPCDWandnPDWstatesarerealized. squareofpaireldsjvijj2.Thus,withamodulatingcomponentQ0intheo-diagonaltermsoftheBdGmatrixinEquation 5{13 inducesamodulatingcomponent2Q0inthediagonalterms,resultingintoa2Q0modulatingcomponentintheelectrondensitywhentheself-consistentsolutionconverges.Similarconclusionsweredrawnin[ 205 ],however,usingtheGreen'sfunctionsmethod.Thedopingdependenceoftheenergypersite(E=t)fortheuniformsuperconductingstate,APCDWstatewithchargemodulationwavevector[Q;0];Q=0:25,andnPDWstatewithQ=0:26and0:3isshowninFigure 5-3 .Theinsetdepictsaplotofd-waveorderparameterintheuniformsuperconductingstateasfunctionofholedoping(x)showingadomelikeshapeinrangex=0:01)]TJ /F1 11.9552 Tf 12.376 0 Td[(0:48.Thechargeorderedstatesoccurinanarrowdopingrangeof0:09)]TJ /F1 11.9552 Tf 12.243 0 Td[(0:17.WendthattheAPCDWandnPDWstates,withdierentwavevectors,havealmostidenticalenergypersiteatalldopings.Atthispoint,itisnotclearwhetherthisneardegeneracyofchargeorderedstatesisanintrinsicfeatureofthet)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0)]TJ /F3 11.9552 Tf 11.956 0 Td[(JmodelorjustanartifactoftheGutzwillerapproximation.Although 84

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wejustshowtwonPDWstateswithdierentwavevectors,wendthattherearemanypossiblenPDWstateswithQintherangeof0.2-0.35.WenotethattheSTMandx-rayscatteringexperimentsndthechargeorderincupratesexistingintheadopingandwavevectorrangeverysimilartoournding[ 36 66 ].However,wecannotaddressthedopingdependenceoftheorderingwavevector[ 36 ]withinourmodel,asthenPDWstateswithdierentQarealmostdegenerate.ItisclearfromFigure 5-3 thattheuniformstatehaslowestenergyatalldopings.Asimilarconclusionwasdrawninthepreviousstudiesofthet)]TJ /F3 11.9552 Tf 11.955 0 Td[(Jlikemodelsusingdierentnumericalschemes[ 189 { 192 195 ].However,theenergydierencebetweenthechargeorderedanduniformstateisverysmall,O(1meV)persite,andcanbeovercomebyothereectssuchaselectron-phononcouplinganddisorder,thus,stabilizingchargeorderedstatesasthegroundstate.TheSTM[ 59 ]andresonantx-rayscattering[ 36 ]observationofshortcoherencelength(30A[ 36 ])indeedpointstowardstheimportantroleofdisorder.Weleavethesystematicstudyofchargeorderinginthedisorderedt)]TJ /F3 11.9552 Tf 11.955 0 Td[(Jmodelforafutureproject.Theapplicationofamagneticeldoforder10T(or1meV)mightbeanotherpossibleroutetotiptheenergybalanceinfavorofchargeorderedstates.Infact,thechargeorderedstateswithlargecoherencelength(>100latticeconstants)havebeenrecentlyobservedinYBCOinthepresenceofamagneticeldofO(30T)[ 206 ].Weplantostudytheeectsofthemagneticeldonthechargeorderedstatesrealizedintheextendedt)]TJ /F3 11.9552 Tf 11.955 0 Td[(Jmodelinafutureproject.Inwhatfollows,wedescribethecharacteristicsoftheAPCDWandnPDWstatesandshowthatthelatterexhibitbiasdependenceoftheLDOS,formfactors,andspatialphasedierence,verysimilartothatobservedintheSTMexperiments[ 35 66 ].AlthoughmanypropertiesoftheAPCDWstatesuchasitscommensuratenature,biasdependenceofthecontinuumLDOSandthespatialphasedierencedonotconformtotheexperimentalobservations[ 36 66 ],westilldiscussitindetailtosetthestageforamorecomplicatednPDWstate. 85

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Figure5-4. CharacteristicfeaturesoftheAPCDWstate.(a)Variationoftheholedensity()andgaporderparameter()withlatticesitesinthecentralregionofa5656system.y-axisinleft(right)correspondsto().(b)Fouriertransformoftheholedensity((q))andgaporderparameter((q)).Theq=0componentoftheholedensitymodulation,notshownintheplot,is0.125.(c)Densityofstatesinthehomogeneoussuperconductingstateand,APCDWstateoveraperiodoflatticesites.(d)Intra-unitcellformfactorsinAPCDWstatecomputedusingEquation 5{20 5.3.1Anti-PhaseChargeDensityWaveStateFigure 5-4 showsthecharacteristicsoftheAPCDWstateobtainedfora5656latticesystematthedopinglevelx=0:125.Theholedensity()andsuperconductinggaporderparameter()modulatealongthex-axiswithaperiodicityof4and8latticeconstants,respectively,asshowninFigure 5-4 (a).changessignafteraperiodofchargemodulation,hencethename'anti-phase'pairdensitywave.Themaximumofholedensityoccursatthedomainwallsiteswherevanishes.TherealspacendingsareechoedintheFourierdomainwherewendthatthedominantchargeandgapmodulationwavevectorsareQ=0:25andQ=0:125,respectively,asshowninFigure 5-4 (b).The 86

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relationQ=2QisalsofoundintheGinzburg-LandautheoriesoftheCDWdrivenbyaPDW[ 163 ].Figure 5-4 (c)showstheLDOSatasetoffourinequivalentlatticesitesintheAPCDWstate.LDOSintheuniformd-wavesuperconductingstateisalsoshownforthecomparison.IntheAPCDWstate,LDOSisnon-vanishingatallsitesandallenergiesandhasaU-shapewithtwosetsofcoherencepeaks,withpeaksatthehigherenergiesalmostcoincidingwiththeuniformd-wavecoherencepeaks.ThecoherencepeaksintheAPCDWstatearisefromtheAndreevboundstates(ABS)[ 189 ].ABSoccuratthethedomainwallsiteswherethesuperconductingchangessignandformone-dimensionalbandduetotranslationalinvarianceiny-direction.Moreover,thehybridizationbetweentheAndreevstatesattheneighboringdomainwallsleadstothebroadeningofABSpeaks,andashiftintheABSenergyawayfromthechemicalpotential[ 189 207 ].Figure 5-4 (d)showsthewavevectordependenceoftheintra-unitcellformfactorsintheAPCDWstate,calculatedfromthebondorderparameterijusingEquation 5{20 .Clearly,allformfactorspeakatq=Qwithd-formfactorhavingthelargestweight.However,asdiscussedinSection 5.2.3 ,tondthebiasdependenceoftheformfactorsinamannersimilartotheSTMexperiments[ 66 ]wemustndthecontinuumLDOSatCuandOsitesandcalculatetheformfactorsusingEquation 5{23 .WetaketheWannierfunctionforBSCCO(discussedindetailinSection 4.2 )asaninputandcalculatethecontinuumLDOSataheightz5AabovethetheBiOplaneusingEquations 5{21 2{14 ,and 2{18 .Veryrecently,wefoundthattheshapeofNaCCOCWannierfunctionattheheightsfewAabovethesurfaceexposedtotheSTMtip,isverysimilartotheBSCCO,andhenceweexpectthatbothshouldhavesimilarrealspacepatternofcontinuumLDOS.Indeed,theSTMexperimentsndaverysimilarladder-likepatterns(Figure 5-2 (a-b))inbothcompounds[ 35 ].Figure 5-5 (a)showsthecontinuumLDOSmapat!=0:25tina2020unitcellarealocatedinthecentralregionof5656system.Clearly,thecontinuumLDOSexhibitsaperiodicmodulationinthex-direction 87

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Figure5-5. ContinuumLDOSmapat!=0:25tandataheightz5AabovetheBiOplaneintheAPCDWstate.(a)LDOSmapat!=0:25tinarangeof2020unitcells.(b)Zoomed-inviewoftheareamarkedbysquarein(a).BlackdotsandopencirclesrepresentpositionofCuandOatoms,respectively,intheCuOplaneunderneath.(c)LDOSmapat!=)]TJ /F1 11.9552 Tf 9.299 0 Td[(0:25tinthesameregionasin(b). withawavelengthoffourlatticeconstants.Azoomed-inviewoftheregionboundedbytheblacksquareisshowninFigure 5-5 (b).BlackdotsandcirclesmarkthelocationsofCuandOsites,respectively,underneath.WendthattheLDOSshowsmodulationonbothCuandOsites,suggestingthatallformfactors(s,s0,andd)havenon-zeroweightatthisparticularbias.Moreover,whentheLDOSonaOxsiteinaunitcellislargethentheLDOSonOysiteinthesamecellissmall,implyingthatthemodulationsonOxandOysitesareoutofphase.Thus,atthisparticularbiasthed-formfactormusthavelargerweightthanthes0-formfactor.Figure 5-5 (c)showsthecontinuumLDOSmapatthenegativebias!=)]TJ /F1 11.9552 Tf 9.299 0 Td[(0:25tinthesameregionasinFigure 5-5 (b).Comparingbothgures,wendthatthesiteswithlargerLDOSatthepositivebiashavesmallerLDOSatthenegativebias,suggestingaspatialphasedierenceofbetweenthetwobiases.Now,wewillexaminethebiasdependenceofthecontinuumLDOS,formfactors,andspatialphasedierencesintheAPCDWstate.Figure 5-6 (a)showsthecontinuumLDOSspectrumattheCuandtwoinequivalentOsites(OxandOy)intheunitcell[27;27]ataheightz5AabovetheBiOplane.TheinsetshowsthepositionoftheCuandOsites,markedbyablackdotandcircles,respectively,withreferencetothecontinuum 88

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Figure5-6. (a)ContinuumLDOSspectrumregisteredaboveCu,OxandOysitesintheunitcell[27,27]attheheightz5AabovetheBiOplane.ThelocationoftheunitcellcanbeinferredfromFigure 5-5 (b).Biasdependenceofthe(b)intra-unitcellformfactorsand(c)averagespatialphasedierenceatdopingx=0:125. LDOSmapshowninFigure 5-5 (b).Thelow-energyU-shapefeature,similartothelatticeLDOSspectrum,canbeeasilyobserved.ThisisverydierentfromtheSTMconductancemeasurements[ 35 66 ]whichndaV-shapednodalstructurearound!=0(seeFigure 5-9 (c)).Figure 5-6 (b)showsthebiasdependenceoftheformfactorsevaluatedatthewavevectorq=[0:25;0].Thes0-formfactorshowsapeakatanenergy!=0:17twhereasthed-formfactorpeaksatahigherenergy!=0:24t.Moreover,thes-formfactorhastheleastweightatallenergiesbelow0:3t.Although,thesefeaturesresembletheexperimentalobservation[ 66 ](seeFigure 5-10 (b)),thebiasdependenceofthespatialphasedierence()shownintheFigure 5-10 (c)exhibitsaqualitativedeparturefromtheexperiment[ 66 ].IntheAPCDWstateisnon-zeroatallenergiesinthewindow[0;0:4t],anddisplaysasteepincreasefromasmallvalueat!=0toclosetoat!=0:17t.TheSTMexperiments,however,ndthatatlowenergiesremainsnearlyzerouptoan!whichfallsbetweenthes0-andd-formfactorpeaks(compareFigures 5-11 (b)and 5-10 (b)).Inthefollowingsection,weshowthatthenPDWstatecorrectlycapturesallqualitativefeaturesofthebiasdependenceoftheformfactorsaswellasthespatialphasedierence.Toobtainanquasi-incommensuratenPDWstate,wesolvetheBdGequations(Equations 5{13 and 5{17 )self-consistentlyona6060lattice.Asaninitialguess, 89

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wetakeapaireldmodulatingwithwavenumberQ0=0:154keepinguniformvaluesfortheholedensityandbondeld.Figure 5-7 showsthecharacteristicsofthenPDWstateatthedopinglevelx=0:125.Thevariationoftheholedensity()andthed-wavesuperconductinggaporderparameterwithlatticesitesisshownintheFigure 5-7 (a).Wendthattheholedensityshowsarootmeansquaredeviationof=0:01holesfromtheaveragevalueof0.125,similartotheNMRresults[ 34 ]showingaholedensityvariationof=0:030:01inthechargeorderedphaseofthe10:8%hole-dopedYBCO.Moreover,theholedensityisobservedtobemaximumatthedomainwallsiteswherechangessign.AsshowninFigure 5-7 (b),theFouriertransformoftheholedensitycontainsmanyFouriercomponents,withlargestcontributionfromQ=0:3,reectingthequasi-incommensuratenature.ThegaporderparametertoohasmanyFouriercomponentswithdominantcontributionfromtheQ=0:15.TherelationsQ=2QholdsasinthecaseofAPCDW.However,incontrasttotheAPCDW,thenPDWstatehasanon-vanishinguniform(q=0)componentofthed-wavegapparameter.Thus,thenPDWstateintertwinesincommensurateCDW,incommensuratePDW,anduniformd-wavesuperconductivity.TheeectofaniteuniformcomponentcanbeseenastheappearanceofaV-shapenodalstructureinthelatticeLDOSatlowenergiesasshownintheFigure 5-7 (c).AsinthecaseofAPCDW,wendthetwosetsofcoherencepeaksattheenergies0:21tand0:37twhichcanbeattributedtotheAndreevboundstatesasexplainedinSection 5.3.1 .InFigure 5-7 (d),weplotthewavevectordependenceoftheformfactorsobtainedfromthebondorderparameterusingEquation 5{20 .Wendthatallformfactorspeakatq=Q=0:3withthed-formfactorhavingthelargestweight. 5.3.2NodalPairDensityWaveStateTocomparewiththeSTMndingsofHamidianet.al.[ 66 ],weobtaincontinuumLDOSinthenPDWstateanduseittondthebiasdependenceoftheformfactorsandspatialphasedierence.Figure 5-8 (a)showsthecontinuumLDOSmapat!=0:25tandaheightz5AabovetheBiOplaneina2020unitcellarealocatedinthe 90

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Figure5-7. CharacteristicfeaturesofnPDWstate.(a)Variationofholedensity()andgaporderparameter()withlatticesitesinthecentralregionof6060system.y-axisonleft(right)correspondsto().(b)Fouriertransformoftheholedensity((q))andgaporderparameter((q)).Theq=0componentofholedensitymodulation,notshownintheplot,is0.125.(c)DensityofstatesinthehomogeneoussuperconductingstateandnPDWstateonfourconsecutivelatticesites.(d)Intra-unitcellformfactorsinnPDWstatecomputedusingEquation 5{20 centralregionofthe6060unitcellsystem.Twokindsofmodulatingstripescanbeseenhere.Azoomed-inviewofoneofthesestripesintheareamarkedbytheblacksquareisshowninFigure 5-8 (b).BlackdotsandopencirclesmarktheCuandOsites,respectively,underneath.LDOSmodulationsareobservedonbothCuandOsites,suggestingnon-zeroweightsforallformfactors.Also,themodulationsattheOxandOysitesareoutofphase,indicatingthatthed-formfactorhasalargercontributionatthisbiascomparedtothes0-formfactor.Moreover,acomparisonwiththeLDOSmapatthenegativebias!=)]TJ /F1 11.9552 Tf 9.299 0 Td[(0:25tplottedintheFigure 5-8 (b)showsaspatialphasedierenceofbetweenthem,similartothecaseofAPCDWstatediscussedinSection 5.3.1 .Inthefollowingparagraphs,wemakethesestatementsmorequantitativebycalculatingthebias 91

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Figure5-8. ContinuumLDOSmapat!=0:25tandattheheightz5AaboveBiOplane.(a)LDOSmapat!=0:25tinarangeof2020unitcells.(b)Zoomed-inviewoftheareamarkedbysquarein(a).BlackdotsandopencirclesrepresentpositionofCuandOatoms,respectively,intheCuOplaneunderneath.(c)LDOSmapat!=)]TJ /F1 11.9552 Tf 9.299 0 Td[(0:25tinthesameregionasin(b). dependenceofformfactorsandspatialphasedierenceusingEquations 5{23 and 5{24 ,respectively.Figure 5-9 (a)showsthecontinuumLDOSspectrumattheCu,Ox,andOysitesintheunitcell[25;25]ataheightz5AabovetheBiOplane.TheinsetshowsthepositionoftheCuandOsites,markedbyablackdotandcircles,respectively,withreferencetothecontinuumLDOSmapshownintheFigure 5-8 (b).Thebreakingoffour-foldrotationalsymmetryisevidentfromthedierentLDOSspectrumatOxandOysites.Moreover,AsmallV-shapestructureispresentnear!=0asaconsequenceoftheuniformcomponentofthed-wavepairingorderparameter.Furthermore,twosetsofcoherencepeaksattheenergies0:21tand0:37tarepresentasinthecaseoflatticeLDOS(Figure 5-7 (c)).Atpresent,thecontinuumLDOSdoesnotbearacloseresemblancetoexperimentaldata(seeFigure 5-9 (c)).However,thehigherenergycoherencepeaksareexpectedtobesmearedbytheinelasticscatteringprocesses[ 126 ]consideredtobeintrinsicallypresentintheoptimal-to-underdopedcuprates[ 208 209 ].Following[ 126 ],wemodelsuchprocessesbyalinear-in-!scatteringrate)-394(=j!j.Theconstantwasdeterminedtobeintherangeof0:25)]TJ /F1 11.9552 Tf 12.358 0 Td[(0:33forthelocalspectralgaplyingintherange80)]TJ /F1 11.9552 Tf 11.962 0 Td[(100meVintheunderdopedBSCCO[ 126 ].Wereplacethearticialbroadeningi0+in 92

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Figure5-9. ContinuumLDOSspectrumregisteredaboveCu,OxandOysitesintheunitcell[25,25]ataheightz5AabovetheBiOplane(a)without,and(b)with)-277(=j!jinelasticscattering;=0:25asextractedin[ 126 ].ThelocationoftheunitcellcanbereferredfromFigure 5-8 (b)asshownintheinset.DotsandopencirclesrepresentCuandOatoms,respectively.(c)AtypicaltunnelingconductancespectrumoveraCusiteinBSCCOsamplewith6%holedoping.ReprintedbypermissionfromMacmillanPublishersLtd:Nature[ 66 ],copyright2015. theGreen'sfunction,denedinEquation 5{21 ,byi0++i)-326(toincorporatetheeectsoftheinelasticscatteringinouranalysis.Figure 5-9 (b)showsthecontinuumLDOSspectrumthusobtainedfor=0:25.Clearly,thehigherenergycoherencepeaksaresignicantlybroadenedandcannotberesolvedanymore.ThespectraareverysimilartotheSTMconductancespectratakenovervariousCuandOsitesin[ 35 ].Foracomparison,weshowthetypicaltunnelingconductancespectratakenoveraCusiteinthechargeorderedstateofthesuperconductingBSCCOat6%hole-dopinginFigure 5-8 (c)[ 66 ].Here,0isthenodalsuperconductinggapbeyondwhichequilibriumBogoliubovquasiparticlesceasetoexist,whereas1isconsideredtobethepsuedogap.Thecorrespondingfeaturesinourmodel,asseeninFigure 5-8 (b),areassociatedwiththeuniformcomponentofthesuperconductinggapandthelowerPDWcoherencepeak.Interestingly,theenergyscalesinthetwoguresarealsosimilar,giventhatthebareNNhoppingtinBSCCOis300)]TJ /F1 11.9552 Tf 11.955 0 Td[(400meV[ 124 ].Thebiasdependenceoftheintra-unitcellformfactorsatwavevectorq=[0:3;0],derivedfromthecontinuumLDOSinthenPDWstateusingEquation 5{23 ,isplottedin 93

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Figure5-10. (a)Biasdependenceoftheformfactorsatx=0:125computedfromEquation 5{23 .(b)BiasdependenceoftheformfactorsasobservedintheSTMexperiment.ReprintedbypermissionfromMacmillanPublishersLtd:Nature[ 66 ],copyright2015.Dopingdependenceof(c)energyatwhichd-formfactorpeaks(d)and(d)correspondingmagnitude(DZmax). Figure 5-10 (a).Thecorrespondingexperimentalresult[ 66 ]isreproducedintheFigure 5-10 (b).Wendthatthes0-andd-formfactorsdisplaypeaksatalower(s0=0:11t)andahigherenergy(d=0:21t),respectively,inagreementwiththeexperiment[ 66 ].ComparingwithlatticeLDOS(Figure 5-7 (c))andcontinuumLDOSspectra(Figure 5-9 (a)),wendthatthed-formfactorpeaksattheenergycorrespondingtothelowercoherencepeakintheLDOSoccurringduetohybridizationofABSasexplainedintheSection 5.3.1 .Thus,wecanconcludethatthed-formfactornatureofthechargeorderisintimatelyrelatedtothePDWcharacter.Amoreconvincinganalysistosupportthisclaimispresentedattheendofthissection.Wenotethattherelativemagnitudeofform 94

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factorscloseto!=0isdierentfromtheexperimentalresults.Also,wendthatthes-formfactoriscomparable,butloweratallbiases,tothes0-andd-formfactors,whereasintheexperimentitismuchmoresuppressed.Thedopingdependenceoftheenergyatwhichthed-formfactorpeaks(d)andthecorrespondingpeakvalue(DZmax)isplottedinFigures 5-10 (b)and(c).Wendthatddecreasesmonotonicallywithincreasinghole-doping(x)whereasDZmaxshowsanon-monotonicbehavior;rst,itincreasesslowlywithx,attainsamaximumatx=0:13,anddropsrapidlywithaslightincreaseindoping.ThedopingdependenceofDZmaxisverysimilartotheexperimentallydeterminedintensityofthedensitymodulationwavevector[ 65 ],whichcanbeconsideredasananalogueofDZmax.Wenotethatnoothertheoryhasbeenabletocalculatebiasdependenceofintra-unitcellobservables.Figures 5-10 (c)and(d)arepredictionsofourmodelwhichcanbeeasilytestedbyrepeatingtheexperimentalworkpresentedin[ 66 ]forseveralholedopings.InFigure 5-11 (a),weplotthecalculatedbiasdependenceoftheaveragespatialphasedierence()betweend-formfactormodulationsatpositiveandnegativebiasesinthenPDWstate,evaluatedusingEquation 5{24 .Thecorrespondingexperimentalresult[ 66 ]isreproducedinFigure 5-11 (b).Inagreementwithexperiment,wendthatatlowenergies!<0:08t,is0,andsharplyincreasesafterwardsachievingthevalueat!0:12t.Thetransitionoccursneartheenergywherethed-formfactorbecomesdominant(comparewithFigure 5-10 (a)).ThedopingdependenceoftheenergyatwhichthespatialphasechangestoisshowninFigure 5-11 (c).Wendthatdecreasesmonotonicallywithhole-doping.Thisresultisinagreementwiththeextractedfromthebias-dependenceofatadditionaldopingsx=0:06;0:17presentedinthesupplementaryinformationof[ 66 ].Hamidianetal.[ 66 ]attributedtheparticularbiasdependenceoftheformfactorsandspatialphasedierencetothed-formfactornatureofchargeorder,whereanti-phasemodulationsoccursatthetwoinequivalentOatomsintheCuO2unitcell.However,in 95

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Figure5-11. (a)Biasdependenceofaveragespatialphasedierence()denedinEquation 5{24 .(b)BiasdependenceofasobservedintheSTMexperiments[ 66 ].ReprintedbypermissionfromMacmillanPublishersLtd:Nature[ 66 ],copyright2015.(c)Dopingdependenceoftheenergyatwhichtheinitialphasejumpintakesplace. ouranalysis,wendthatthechargeorderisdrivenbythePDWorder.Moreover,thecorrespondencebetweentheenergyatwhichthed-formfactorpeaks,andtheparticularcoherencepeaksintheLDOSspectrumarisingduetothePDWorder,stronglysuggestthatthelatterispivotaltogetthebiasdependenceoftheformfactorsandspatialphasedierenceasseenintheexperiment.Wefurthersubstantiatethisclaimbydisentanglinguniformsuperconductivity,CDW,andPDWordersintertwinedinthenPDWstate"byhand".First,wekeeponlyPDWorderandturnotheCDWorder.Thisisdonebytakingthemean-eldsinthenPDWstateandreplacingthespatiallyvaryingholedensity(i)andbondelds(vij)intheBdGequation(Equation 5{16 )bythecorrespondingquantities(0andv0)intheuniformd-wavesuperconductingstate,i.e.i!0andvij!v0.Thepaireldvijiskeptthesame(andthusmodulating)asinthenPDWstate.Thechemicalpotentialisadjustedtoyieldthecorrectaverageelectronlling.WesolveEquation 5{16 withthesechangesandcalculatethelatticeLDOS,formfactors,andthespatialphasedierence;resultsareshowninFigures 5-12 (a),(b),and(c),respectively.ComparingFigures 5-12 (a)and 5-7 (c),weconcludethecoherencepeakscomefromthePDWorderandarticiallysettingthechargemodulationtozerohasasmalleectonit.Moreover,thepeakind-form 96

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Figure5-12. (a)-(c)LatticeLDOS,formfactorsandaveragespatialphasedierence()inthecasewhennPDWchargeandbondmodulationsaresettozero"byhand"keepingonlypaireldmodulations.(d)-(f)LatticeLDOS,formfactorsandaveragespatialphasedierence(),respectively,inthecasewhennPDWpaireldmodulationsaresettozero"byhand"keepingchargeandbondmodulations. factorsurvivesevenintheabsenceoftheCDWorder.Furthermore,thebiasdependenceofisalsoverysimilartothenPDWstate(seeFigure 5-11 (a)).Thus,wendthatthequalitativefeaturesofthebiasdependenceofvariousquantitiesinthenPDWstateremainsintacteveniftheCDWorderisarticiallyturnedo.Finally,wedothesimilarexerciseagain,butnowsettingthePDWordertozero"byhand"andkeepingtheCDWanduniformsuperconductivity.Thisisachievedbytakingthemean-eldsinthenPDWstateandreplacingthemodulatingpaireldvijinEquation 5{16 bythecorrespondingquantity(v0)intheuniformd-wavesuperconductingstatewhilekeepingthemodulatingbondeldandholedensityunchanged.ResultsareshowninFigures 5-12 (d)-(f).Clearly,thelatticeLDOSdoesnotshowmuchdierencefromtheuniformd-wavesuperconducting 97

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state.Moreimportantly,thebiasdependenceoftheformfactorsandspatialphasedierencelookverydierentfromthenPDWstateandtheexperimentalobservation.Thus,wecanconcludethatthePDWorderiscrucialtoyieldthebiasdependenceofLDOS,formfactors,andspatialphasedierenceasobservedintheSTMexperiments. 5.4ConclusionInthischapter,westudiedchargeorderedstatesinthet)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0)]TJ /F3 11.9552 Tf 11.955 0 Td[(Jmodel,whichhasbeenwidelyusedinpasttounderstandthephenomenologyoftheunderdopedcupratesfromastrongcouplingperspective.Usingtherenormalizedmean-eldtheory(Gutzwillerapproximation),alongwithapartialFouriertransformschemesubstantiallyreducingthecomputationalcost,weobtainedthecommensurateandincommensuratechargeorderedstateswhicharenearlydegeneratetotheuniformd-wavesuperconductingstate.TheincommensuratenPDWstateintertwinesCDW,PDW,anduniformsuperconductivityandhaspropertiesverysimilartothechargeorderedstatesobservedincupratesinSTMexperiments[ 35 36 66 ].Particularly,theladder-likereal-spacepatternofthecontinuumLDOSaresimilartowhathasbeenobservedinBSCCOandNaCCOC[ 35 ].Moreover,thebiasdependenceofcontinuumLDOS,intra-unitcellformfactors,andspatialphasedierenceinthenPDWstateareingoodagreementwiththeexperiments[ 66 ].WehaveusedaWannierfunctionbasedmethodtocalculatetheCuandOsublatticeLDOSandobtainedthebias-dependentformfactorsandspatialphasedierenceinamannersimilartotheexperiments[ 66 ].ThisisaqualitativeimprovementoverthetheoreticalschemesusedpreviouslyinliteraturewhichconsideredstaticformfactorsderivedfromtherenormalizationoftheNNhoppings.WefoundthatinthenPDWstatethes0-formfactordominatesatlowerenergies,whereasthed-formfactordominatesathigherenergies,inagreementwiththeSTMresults[ 66 ].Moreover,thed-formfactorpeakamplitudeshowsanon-monotonicbehaviorwithrespecttotheholedoping,whereastheenergycorrespondingtothepeakdecreasesmonotonicallywithincreasingdoping.TheaveragespatialphasedierenceinthenPDWstateshowsajumpfrom0toatanenergy 98

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scalewherethed-formfactorexceedsthes0-formfactorasseenintheSTMexperiments[ 66 ].Furthermore,theenergyatwhichthephaseshiftoccursdecreasesmonotonicallywithincreasingholedoping,similartoexperiment.Finally,weshowedthatthePDWcharacterofthenPDWstateiscrucialtoobtainthebiasdependenceofvariousquantitiesasseenintheexperiments.AlthoughthenPDWstateisnotthegroundstatewithinGutzwillerapproximation,itsenergydierencewiththeuniformd-wavestatewhichisthegroundstateisextremelysmallandcanbeovercomebysmallperturbationslikeelectron-phononcoupling.WediscussedthepossibilitythatthedisorderwhichisintrinsicallypresentincupratescanstabilizethenPDWstatelocally,yieldingashortcoherencelengthasseeninmanySTMexperiments.Moreover,amagneticeldofO(10T)cantilttheenergybalanceinfavorofthenPDWstateandresultintoachargeorderedstatewithalargecoherencelengthasseenintheexperiments.Weleavethesetwoscenariosforthefutureprojects. 99

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CHAPTER6SUMMARYANDFINALCONCLUSIONSInthiswork,wehaveproposedanovel'BdG+W'method,combiningsolutionsofthelatticeBogoliubov-deGennesequationsandrstprinciplesWannierfunctions,totheoreticallycomputethecontinuumLDOSattypicalSTMtippositionswhichcanbedirectlycomparedwiththeexperimentalSTMconductance.Thismethodyieldssub-unitcellspatialresolutionandcapturestheeectsofallatomsintheunitcellthroughWannierfunction,withoutanysubstantialincreaseofcomputationalcomplexitycomparedtotheconventionalBdGequations.AlthoughtheBdG+Wmethodisquitegeneral,wehavemainlyfocusedonitsapplicationtotheproblemofinhomogeneousstatesincupratesandFeSCs.ThesearelayeredmaterialswherethelayerexposedtotheSTMtipisdierentfromthe'active'layer(CuOlayerincupratesandFe-layerinFeSCs)responsibleforsuperconductivity.Theeectofinterveninglayers,whichisoftenignoredbutcanleadtoanon-trivialtunnelingpaths,iseasilyaccountedfortheBdG+WmethodviarstprinciplesWannierfunctions.InChapter 2 ,wederivedanelementaryexpressionforthecontinuumGreen'sfunctionintermsoflatticeGreen'sfunctions,usingabasistransformationfromthelatticeto3DcontinuumspacewithWannierfunctionsasmatrixelements,whichcanbeusedtoobtaincontinuumLDOSattheSTMtipposition.Moreover,wepresentedanecientnumericalschemetosolvethemulti-orbitalBdGequationsonthelatticeandcombinetheirsolutionwithWannierfunctiontoobtainthecontinuumLDOS.AsarstapplicationoftheBdG+Wmethod,westudiedthenon-magneticimpurity-inducedstatesintheFeSesuperconductorinChapter 3 .WestartedwithaDFTderived10-orbitaltight-bindingmodelandacorrespondingWannierbasistodescribethenormalstateofFeSe.Next,tointroducethesuperconductivity,wecalculatedpair-potentialsviaaspin-uctuationtheorycalculationwithintherandomphaseapproximation.Thesuperconductingstatethusobtainedwasfoundtohavesgap 100

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symmetry.Further,bysolvingthe10-orbitalBdGequationsself-consistently,wefoundthatastrongnon-magneticimpurity,modeledsimplyasanon-siterepulsivepotential,inducesboundstatesinthessuperconductinggap.Moreover,thecontinuumLDOSmapandtopographobtainedabovetheSelayerattheboundstateenergyyieldsanatomic-scaledimerlikepatternaroundtheimpuritypositionthatisverysimilartothegeometricdimerstatesobservedinFeSe[ 63 ]andotherFeSCs[ 49 77 79 ].Thisistherstandonlyexplanationofthesestatestoourknowledge.InChapter 4 ,weusedtheBdG+WmethodtostudyZnandNiimpurity-inducedstatesinthesuperconductingBSCCO.Werstobtainedthe1-bandtight-bindingmodelandcorrespondingCu-dx2)]TJ /F5 7.9701 Tf 6.587 0 Td[(y2Wannierfunctionusingrstprinciplescalculations.Weshowedthatthedx2)]TJ /F5 7.9701 Tf 6.586 0 Td[(y2symmetryoftheWannierfunctionleadstoachangeofthelowenergyspectralfeaturesinthehomogeneousd-wavesuperconductingstatefromaV-shapedspectruminthelatticeLDOStoamoreU-shapedspectruminthecontinuumLDOS.Thisindeedcorrespondstoexperimentaldataintheliteratureforoptimal-to-overdopedcuprates,althoughithadneverbeennotedbefore,toourknowledge.Next,modelingZnasastrongattractiveon-sitepotentialscattererinad-wavesuperconductor,wesolvedthelatticeBdGequationsandfound,asothershadbeforeus,thatitinducesasharpin-gapresonancestate.Further,thelatticeLDOSshowedaminimumrightattheZnsiteandmaximaatthenearest-neighborsites,aspredictedinearliertheories,butincontrasttotheSTMexperiments[ 60 ]whichshowanoppositeintensitypattern.WeresolvedthislongstandingparadoxbycalculatingcontinuumLDOSmapafewAaboveBiOplane,wheretheSTMtipwouldbetypicallylocated,whichdisplayedexcellentagreementwiththeSTMresults[ 60 ].WefoundthattheintensitymaximumattheimpuritysiteinBSCCOcanbeattributedtotheelectrontransferfromnearest-neighborunitcellsviaapicaloxygenatoms.Finally,westudiedtheNiimpurityprobleminBSCCObyusingasimplemodelofamagneticimpuritywithweakon-sitepotentialandexchangescattering,andshowedthatsuchanimpurityinduced 101

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twospin-polarizedin-gapresonancestates.Moreover,asinthecaseofZnproblem,thecontinuumLDOSmapsattheresonanceenergiesdisplayedexcellentagreementwiththecorrespondingSTMresults[ 64 ].InChapter 5 ,weaddressedaveryrecentSTMexperiment[ 66 ]onthechargeorderedstateinBSCCOthathascapturedtheattentionofthehigh-Tccommunity.Bysolvingthet)]TJ /F3 11.9552 Tf 11.955 0 Td[(t0)]TJ /F3 11.9552 Tf 11.955 0 Td[(Jmodelwithinrenormalizedmean-eldtheory(Gutzwillerapproximation),weobtainedtwokindsofunidirectionalchargeorderedstates,namelythecommensurateanti-phasechargedensitywave(APCDW),andincommensuratenodalpairdensitywave(nPDW)states,whichhadbeenearliershowntopossesadominantd-formfactor[ 190 ].UsingtheWannierfunctionbasedapproachdevelopedinChapter 2 ,wecalculatedthecontinuumLDOSattypicalSTMtippositions,andsubsequentlyobtainedthebiasdependenceoftheintra-unitcellformfactorsandspatialphasedierenceintheAPCDWandthenPDWstate,closelyfollowingtheexperimentalanalysis[ 65 66 ].WefoundthatthatthenPDWstatewhichintertwineschargedensitywave,pairdensitywave(PDW),anduniformsuperconductivity,hascharacteristicsverysimilartotheSTMobservations[ 66 125 ].ThecontinuumLDOSmapsinthenPDWstateshowsladder-likespatialpatternssimilartotheexperiments[ 125 ].Moreover,thecontinuumLDOSaboveCusiteshasanodalspectrum,arisingfromtheuniformcomponentofd-wavesuperconductingorderparameter,thatresemblesverycloselytheSTMtunnelingconductancespectrum[ 66 125 ]oncetheeectsofinelasticscattering[ 126 ]areproperlytakenintoaccount.Furthermore,inthenPDWstate,thes0-formfactordominatesatlowerenergies,whereasthed-formfactordominatesathigherenergies,inagreementwiththeSTMresults[ 66 ].Finally.theaveragespatialphasedierenceinthenPDWstateshowsajumpfrom0toatanenergyscalewherethed-formfactorexceedsthes0-formfactorasseenintheSTMexperiments[ 66 ].Inthiswork,wenotonlyexplainedthepastSTMobservationsthathaddeedamicroscopicexplanationforalongtime,forexampleZnandNiimpuritypatternsin 102

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BSCCO,andgeometricdimerstatesinFeSCs,butalsomadenewpredictionswhichcanbetestedexperimentallytovalidateourtheory.InSection 4.3 ,weshowedthatthecontinuumLDOSinsuperconductingoverdopedBSCCOatatypicalSTMtipheightandbias!!0variesasj!j3,incontrasttothecommonlyassumedlinear-in-j!jvariation.Withsub-meVresolutionofthemodernSTMapparatus,thisresultcanbedirectlycheckedbycomparinggoodnessoftforthecubicandlinearcurvestothemeasuredlow-biasdierentialtunnelingconductancespectrainoverdopedBSCCO.InSection 5.3.2 ,wepredictthatthebiasatwhichthed-formfactoracquiresthelargestvalue,andthebiasatwhichthespatialphasedierencerstchangesfrom0to,shoulddecreasewithincreasinghole-doping,whereastheamplitudeofthed-formfactorshouldrstincrease,achieveamaximum,anddroprapidlyuntilthechargeordervanisheswithfurtherholedoping.Again,thesepredictionscanbeeasilytestedbyrepeatingtheexperimentalanalysispresentedin[ 66 ]forBSCCOsampleswithdierentholedopings.ThepredictivepoweroftheBdG+WapproachisalsoreectedinarecentcombinedtheoreticalandSTMstudy[ 109 ]ofsuperconductingLiFeAs,awidelystudiedFeSC,wheresomeoftheauthorsrstpredictedtheregistrationofLiorAsstatesintheSTMtopographyasafunctionoftip-sampledistanceandsetpointcurrentusingtheWannierfunctionbasedapproach,whichwaslaterveriedbyotherco-authorsviaactualSTMmeasurements.Finally,inaveryrecentinvestigation,whichisstillaworkinprogress,wendthattheWannierfunctioninanothercupratecompoundNaCCOChasaverysimilarshapetothatinBSCCO,inspiteofaverydierentcrystalstructure.ItstronglysuggeststhatiftheSTMexperimentswithZnandNidopingarerepeatedforNaCCOC,thesamespatialpatternoftheimpuritiesshouldbeobservedasinBSCCO,despitethefactthattheinterveninglayersbetweenthesurfaceandtheCuO2planearequitedierentinthetwomaterials.Thisisanothernon-trivialpredictionwhichcanbeeasilytestedbySTMexperiments.TheBdG+WmethodpresentedinthisthesisrepresentsaqualitativeimprovementovertheconventionalmethodssuchaslatticeBdGequationsandT-matrix,used 103

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extensivelyinliteraturetounderstandtheSTMexperimentsonlayeredsuperconductorslikecupratesandFeSCs.Firstandforemost,itallowsthecalculationofthecontinuumLDOSatatypicalSTMtipposition,whichcanbedirectlycomparedwiththeSTMtunnelingconductanceatlowtemperatures.Incontrasttothepreviouslyproposedmethods[ 62 120 121 ],whicharesimilarinspiritbuttailor-madeforaparticularcompound(BSCCO)anddonottreatinhomogeneitiesandsuperconductivityonthesamefooting,theBdG+Wmethodprovidesasystematicframeworktostudyanysuperconductoratthemean-eldlevelinthecontextofSTMexperiments.Itcapturesthelocalsymmetryinternaltotheunitcellandyieldsasub-unitcellspatialresolution,comparabletotheSTMresults,enablingabettervisualizationoftheatomic-scalephenomenaininhomogeneoussuperconductors.Furthermore,theQPIpatterncalculatedusingBdG+WmethodcaptureswavevectorsinallBrillouinzones,incontrasttotheconventionallatticecalculationswhichyieldwavevectorsonlyintherstBrillouinzone,leadingtoaqualitativeimprovementwhencomparedwiththeexperiment.LookingbackonthehistoryofSTMexperimentsoncupratesandFeSCs,onendsmanyinstanceswherethecrudeleveloftheoryledtomisunderstandingofexperimentaldata.Weanticipatethatwiththeadventofcomputationallyinexpensiveanduser-friendlyabinitioplatforms,experimentalgroupswillincreasinglyadoptourapproachtounderstandtheirresults.Therearesomeimportantdirectionsinwhichtheworkpresentedinthisthesiscanbeexpanded.ThemethoditselfcanbeimprovedtoincludetheeectsofcorrelationsandimpuritiesontheWannierfunctions.Asaninitialapproximation,weassumedthattheWannierfunctionsaroundimpuritysitesareunaectedbytheimpuritysubstitution,andignoredtheeectsofelectroniccorrelationsinthenormalstate.TherstassumptioncanberelaxedbyobtainingWannierfunctionsaswellasimpuritypotentialsfromtherst-principlescalculationforasingleimpurityinthenormalstatewithinthesupercellapproximation[ 210 ].ThecorrelationeectsontheWannierfunctioncanbeaccounted 104

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forviaLDA+UandLDA+DMFTmethods[ 138 211 212 ].Apartfromimprovingthemethod,therearefewopenquestionsfromtheworkpresentedinChapter 4 and 5 thatIwouldliketoaddressinfuture.First,howdoesthecontinuumLDOSchangefromamoreU-shapedtoV-shapedspectraldependencewhenmovingfromoverdopedcupratesamples(sampleareas)tooptimallydopedsamples(seeSection 4.3 fordetails)?Second,candisorderormagneticeldstabilizethechargeorderedstatesinthet)]TJ /F3 11.9552 Tf 11.955 0 Td[(Jmodelasthegroundstate(seeSection 5.3 fordetails)?Ifthelattercanbeshowntobecorrect,wewillhaveestablishedaconsistentpictureofthechargeorderedphasesovertheentirecupratephasediagram,whichwouldconstituteamajorstepforwardinunderstandinghightemperaturesuperconductivity. 105

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APPENDIXADERIVATIONOFBOGOLIUBOV-DEGENNESEQUATIONSFollowingthediscussioninSection 2.2.2 ,wecomputethecommutator[HMF;ci]forthetwodenitionsofmean-eldHamiltonianasinEquations 2{5 and 2{8 .IfthetransformationgiveninEquation 2{7 diagonalizesthemean-eldHamiltonian,thenthetwocommutatorsmustbeequalandbycomparingthecoecientsofoperatorswewillgettheBdGequations. H0MF;ci000="Xijtijcyicj)]TJ /F4 11.9552 Tf 11.955 0 Td[(0Xicyici;ci000#=Xij)]TJ /F4 11.9552 Tf 5.48 -9.684 Td[(tij)]TJ /F4 11.9552 Tf 11.955 0 Td[(0ijhcyicj;ci000i=Xij)]TJ /F4 11.9552 Tf 5.48 -9.684 Td[(tij)]TJ /F4 11.9552 Tf 11.955 0 Td[(0ij()]TJ /F4 11.9552 Tf 9.298 0 Td[(ii000cj)=)]TJ /F10 11.9552 Tf 11.291 11.358 Td[(Xjt0i0j)]TJ /F4 11.9552 Tf 11.955 0 Td[(0i0j0cj0: (A{1) HBCSMF;ci000=")]TJ /F10 11.9552 Tf 11.291 11.357 Td[(Xijijcyi"cyj#+H.c.;ci000#=)]TJ /F10 11.9552 Tf 11.291 11.357 Td[(Xijijhcyi"cyj#;ci000i+?ij[cj#ci";ci000]=)]TJ /F10 11.9552 Tf 11.291 11.358 Td[(Xijijcyi"ncyj#;ci000o)]TJ /F1 11.9552 Tf 11.955 0 Td[(ijncyi";ci000ocyj#=)]TJ /F10 11.9552 Tf 11.291 11.357 Td[(Xijijcyi"ji00#0)]TJ /F1 11.9552 Tf 11.955 -.001 Td[(ijii00"0cyj#:Thus, HBCSMF;ci00"=Xj0i0jcyj#HBCSMF;ci00#=Xj0ji0cyj";(A{2) hHimpMF;ci000i="XVimpcyi?ci?;ci000#=VimpXhcyi?ci?;ci000i 106

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=VimpX()]TJ /F4 11.9552 Tf 9.299 0 Td[(i?i000ci?)=)]TJ /F4 11.9552 Tf 9.298 0 Td[(Vimpi?i0ci000: (A{3) FromEquations A{1 A{2 ,and A{3 [HMF;ci00"]=)]TJ /F10 11.9552 Tf 11.291 11.357 Td[(Xjt0i0jcj"+Xj0i0jcyj#)]TJ /F4 11.9552 Tf 11.956 0 Td[(Vimpi?i0ci00"=)]TJ /F10 11.9552 Tf 11.291 11.358 Td[(Xj0i0jcj")]TJ /F1 11.9552 Tf 11.956 0 Td[(0i0jcyj#; (A{4) where,ij=tij)]TJ /F4 11.9552 Tf 11.955 0 Td[(ij0+i?iijVimp.Similarly, [HMF;ci00#]=)]TJ /F10 11.9552 Tf 11.291 11.358 Td[(Xjt0i0jcj#)]TJ /F10 11.9552 Tf 11.955 11.358 Td[(Xj0ji0cyj")]TJ /F4 11.9552 Tf 11.956 0 Td[(Vimpi?i0ci00#:=)]TJ /F10 11.9552 Tf 11.291 11.357 Td[(Xj0i0jcj#+0ji0cyj": (A{5) Now,wewillexpresscommutatorsintermsofoperatorsusingEquation 2{7 [HMF;ci00"]=)]TJ /F10 11.9552 Tf 11.291 11.357 Td[(Xj"0i0j Xnunj"n"+vnj"yn#!)]TJ /F1 11.9552 Tf 11.955 0 Td[(0i0j Xnunj#yn#+vnj#n"!#=Xnjh)]TJ /F4 11.9552 Tf 9.299 0 Td[(0i0junj"+0i0jvnj#n"+0i0junj#)]TJ /F4 11.9552 Tf 11.956 0 Td[(0i0jvnj"yn#i; (A{6) [HMF;ci00#]=)]TJ /F10 11.9552 Tf 11.291 11.357 Td[(Xj"0i0j Xnunj#n#+vnj#yn"!+0ji0 Xnunj"yn"+vnj"n#!#=Xnjh)]TJ /F4 11.9552 Tf 9.298 0 Td[(0i0junj#)]TJ /F1 11.9552 Tf 11.955 0 Td[(0ji0vnj"n#+)]TJ /F1 11.9552 Tf 9.299 0 Td[(0ji0unj")]TJ /F4 11.9552 Tf 11.955 0 Td[(0i0jvnj#yn"i: (A{7) ComputingcommutatorsusingdiagonalizedHamiltonian(Equation 2{8 ), [HMF;ci000]=Xnn0hEnynn;un0i000n00+vn0i000yn0 0i=Xnn0Enun0i000ynn;n00+Envn0i000hynn;yn00i=Xnn0hEnun0i000()]TJ /F4 11.9552 Tf 9.298 0 Td[(nn00n)+Envn0i000)]TJ /F4 11.9552 Tf 5.48 -9.684 Td[(nn0 0yni=Xnh)]TJ /F2 11.9552 Tf 5.479 -9.684 Td[()]TJ /F4 11.9552 Tf 9.299 0 Td[(En0uni000n0+)]TJ /F4 11.9552 Tf 5.479 -9.684 Td[(En 0vni000yn 0i; 107

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[HMF;ci00"]=Xnh)]TJ /F2 11.9552 Tf 5.48 -9.684 Td[()]TJ /F4 11.9552 Tf 9.298 0 Td[(En"uni00"n"+)]TJ /F4 11.9552 Tf 5.48 -9.684 Td[(En#vni00"yn#i:(A{8) [HMF;ci00#]=Xnh)]TJ /F2 11.9552 Tf 5.48 -9.684 Td[()]TJ /F4 11.9552 Tf 9.298 0 Td[(En#uni00#n#+)]TJ /F4 11.9552 Tf 5.48 -9.684 Td[(En"vni00#yn"i:(A{9)ComparingEquations A{6 and A{8 andchangingindicesfromi00toi, Xj)]TJ /F4 11.9552 Tf 5.48 -9.684 Td[(ijunj")]TJ /F1 11.9552 Tf 11.955 0 Td[(ijvnj#=En"uni";(A{10) Xj)]TJ /F1 11.9552 Tf 5.479 -9.684 Td[(ijunj#)]TJ /F4 11.9552 Tf 11.955 0 Td[(ijvnj"=En#vni":(A{11)ComparingEquations A{7 and A{9 andchangingindicesfromi00toi, Xj)]TJ /F4 11.9552 Tf 5.48 -9.684 Td[(ijunj#+jivnj"=En#uni#;(A{12) Xj)]TJ /F2 11.9552 Tf 5.48 -9.684 Td[()]TJ /F1 11.9552 Tf 9.298 0 Td[(jiunj")]TJ /F4 11.9552 Tf 11.955 0 Td[(ijvnj#=En"vni#:(A{13)Equations A{10 and A{13 canbecombinedintoamatrixequation Xj0B@ij)]TJ /F1 11.9552 Tf 9.298 0 Td[(ij)]TJ /F1 11.9552 Tf 9.299 0 Td[(ji)]TJ /F4 11.9552 Tf 9.299 0 Td[(ij1CA0B@unj"vnj#1CA=En"0B@uni"vni#1CA:(A{14)Similarly,Equations A{12 and A{11 read Xj0B@ijjiij)]TJ /F4 11.9552 Tf 9.299 0 Td[(ij1CA0B@unjvnj1CA=En0B@univni1CA:(A{15)Equations A{14 and A{15 formtwosetsofBdGequations.Ifweapplythefollowingparticle-holetransformationinEquation A{14 thenwegetEquation A{15 0BBBB@unj"vnj#En"1CCCCA!0BBBB@vnj"unj#)]TJ /F4 11.9552 Tf 9.298 0 Td[(En#1CCCCA:(A{16) 108

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Thus,weneedtosolveonlyEquation A{14 andbyusingabovecorrespondence,wecanobtainthesolutionofEquation A{15 .AllsubsequentexpressionswillbeobtainedintermsoftheeigenvaluesandeigenvectorsofEquation A{14 .Now,weturntothederivationforthemean-eldquantities(Equation 2{10 )intermsoftheBdGsolutions.Theelectrondensitycanbeexpressedas ni=hcyicii=Xnn0h)]TJ /F4 11.9552 Tf 5.48 -9.684 Td[(uniyn+vnin un0in0+vn0iyn0 i=Xnn0uniun0ihynn0i+vnivn0ihn yn0 i=Xnn0uniun0if(En)nn0+vnivn0i(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(f(En ))=Xn;En>0junij2f(En)+Xn;En >0jvnij2(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(f(En )): (A{17) UsingEquation A{16 ni"=Xn;En"i0juni"j2f(En")+Xn;En#>0jvni"j2(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(f(En#))=Xn;En">0juni"j2f(En")+Xn;)]TJ /F5 7.9701 Tf 6.587 0 Td[(En">0juni"j2(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(f()]TJ /F4 11.9552 Tf 9.298 0 Td[(En"))=Xn;En">0juni"j2f(En")+Xn;En"<0juni"j2f(En")=Xnjuni"j2f(En"): (A{18) HeresumrunsoveralleigenvaluesofEquation A{14 ,bothpositiveandnegative.Similarly, ni#=Xn;En#>0juni#j2f(En#)+Xn;En">0jvni#j2(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(f(En"))=Xn;)]TJ /F5 7.9701 Tf 6.587 0 Td[(En">0jvni#j2f()]TJ /F4 11.9552 Tf 9.299 0 Td[(En")+Xn;En">0jvni#j2(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(f(En"))=Xn;En"<0jvni#j2(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(f(En"))+Xn;En">0jvni#j2(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(f(En")) 109

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=Xnjvni#j2(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(f(En")): (A{19) Thesuperconductinggaporderparameter,denedinEquation 2{6 ,canbeexpressedas ij=Vijhcj#ci"i=VijXnn0hunj#n#+vnj#yn"un0i"n0"+vn0i"yn0#i=VijXnn0unj#vn0i"hn#yn0#i+un0i"vnj#hyn"n0"i=VijXnn0unj#vn0i"(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(f(En#))nn0+un0i"vnj#f(En")nn0=VijXn;En#>0unj#vni"(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(f(En#))+VijXn;En">0uni"vnj#f(En")=VijXn;)]TJ /F5 7.9701 Tf 6.587 0 Td[(En">0vnj#uni"(1)]TJ /F4 11.9552 Tf 11.955 0 Td[(f()]TJ /F4 11.9552 Tf 9.298 0 Td[(En"))+VijXn;En">0uni"vnj#f(En")=VijXnuni"vnj#f(En"): (A{20) Equations A{14 A{18 A{19 ,and A{20 formBdGequationsthathastobesolvedself-consistentlyusingtheproceduredescribedinSection 2.4 .Once,solutionisknownwecanconstructsingleparticleGreen'sfunctiondenedbelow, Gij(t;t0)=)]TJ /F4 11.9552 Tf 9.299 0 Td[((t)]TJ /F4 11.9552 Tf 11.955 0 Td[(t0)h[ci(t);cyj(t0)]+i: (A{21) Here,isthestepfunction,and[]+representstheanti-commutator.Tondthetimeevolutionofcoperatorswehavetondtimeevolutionoftheoperators.Sincetheoperatorsarethefreequasiparticles(seeEquation 2{8 ),theirtimeevolutionisgivenbyfollowingequation, n(t)=ne)]TJ /F5 7.9701 Tf 6.586 0 Td[(iEnt:(A{22)UsingEquation 2{7 andEquation A{22 ,weget ci(t)=Xnhunie)]TJ /F5 7.9701 Tf 6.587 0 Td[(iEntn+vnieiEn tyn i;cyi(t)=XnunieiEntyn+vnie)]TJ /F5 7.9701 Tf 6.587 0 Td[(iEn tn :(A{23) 110

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UsingEquations A{23 and A{21 ,wend Gij(t;t0)=)]TJ /F4 11.9552 Tf 9.298 0 Td[(i(t)]TJ /F4 11.9552 Tf 11.955 0 Td[(t0)hXnn0uniun0je)]TJ /F5 7.9701 Tf 6.586 0 Td[(iEnteiEn0t0nn0+vnivn0jeiEn te)]TJ /F5 7.9701 Tf 6.587 0 Td[(iEn0 t0nn0i=)]TJ /F4 11.9552 Tf 9.298 0 Td[(i(t)]TJ /F4 11.9552 Tf 11.955 0 Td[(t0)hXnuniun0je)]TJ /F5 7.9701 Tf 6.586 0 Td[(iEnteiEn0t0nn0+vnivn0jeiEn te)]TJ /F5 7.9701 Tf 6.587 0 Td[(iEn0 t0nn0i=)]TJ /F4 11.9552 Tf 9.298 0 Td[(i(t)]TJ /F4 11.9552 Tf 11.955 0 Td[(t0)Xnhuniunje)]TJ /F5 7.9701 Tf 6.587 0 Td[(iEn(t)]TJ /F5 7.9701 Tf 6.586 0 Td[(t0)+vnivnjeiEn (t)]TJ /F5 7.9701 Tf 6.587 0 Td[(t0)i: (A{24) TakingtheFouriertransformofEquation A{24 yieldstheGreen'sfunctioninfrequencyspace, Gij(!)=Z1d(t)]TJ /F4 11.9552 Tf 11.955 0 Td[(t0)Gij(t;t0)ei!(t)]TJ /F5 7.9701 Tf 6.586 0 Td[(t0);=Xn>0uniunj !)]TJ /F4 11.9552 Tf 11.955 0 Td[(En+i0++vnivnj !+En +i0+: (A{25) Here,0+isthearticialbroadening,takentobemuchsmallerthansmallestphysicalenergyintheproblem,andn>0indicatesthatthesumistobeperformedovereigenstateswithpositiveeigenvaluesonly.Now,wewillexpresstheaboveGreensfunctiononlyinthetermsofthesolutionsofEquation A{14 Gij"(!)=Xn;En">0uni"unj" !)]TJ /F4 11.9552 Tf 11.956 0 Td[(En"+i0++Xn;En#>0vni"vnj" !+En#+i0+=Xn;En">0uni"unj" !)]TJ /F4 11.9552 Tf 11.956 0 Td[(En"+i0++Xn;)]TJ /F5 7.9701 Tf 6.587 0 Td[(En">0uni"unj" !)]TJ /F4 11.9552 Tf 11.956 0 Td[(En"+i0+=Xnuni"unj" !)]TJ /F4 11.9552 Tf 11.955 0 Td[(En"+i0+: (A{26) Here,wehaveusedEquation A{16 inthesecondstep.Now,thesumrunsoverallvaluesofn.Similarly, Gij#(!)=Xn;)]TJ /F5 7.9701 Tf 6.586 0 Td[(En">0vni#vnj# !+En"+i0++Xn;En">0vni#vnj# !+En"+i0+=Xnvni#vnj# !+En"+i0+: (A{27) 111

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APPENDIXBSUPERCELLMETHODThesolutionoftheinhomogeneousBdGequations(Equation 2{9 )onaNNlatticewithNoorbitalsperlatticesiteyields2N2Noeigenvalues,whichveryoftenturnsouttobeinsucientforobtainingasmoothLDOSspectrum.Insuchascenario,thesupercellmethodiswidelyusedtoincreasethespectralresolutionwithoutincreasingthenumberoflatticesites.Inthismethod,rst,theBdGequationsonNNlatticeareself-consistentlysolved,thentheNNlatticeispicturedasa'supercell',abigunitcellwithN2Noorbitals,whichisrepeatedMMtimestomakealargerlattice.Thenperiodicboundaryconditionsareimposedonthissystem.Thus,thesystemnowbecomestranslationallyinvariantandcanbedescribedinmomentumspacewiththefollowingsupercellBdGequation(writteninmatrixformtoavoidtheclutteringofsymbols): 0B@H0(K))]TJ /F1 11.9552 Tf 9.298 0 Td[((K))]TJ /F1 11.9552 Tf 9.298 0 Td[(y(K))]TJ /F4 11.9552 Tf 9.298 0 Td[(H0(K)1CA0B@Un"(K)Vn#(K)1CA=En"(K)0B@Un"(K)Vn#(K)1CA;(B{1) H0(K)=XITI;0eiKRI;(B{2) (K)=XII;0eiKRI:(B{3)Here,TI;0andI;0aretheN2NoN2NohoppingandsuperconductinggapmatrixbetweenthesupercellatoriginandsupercellwithlatticevectorRI,KisthesupercellmomentumvectorwithcomponentsKx;Ky=n M;m=0;1;2;:::;M)]TJ /F1 11.9552 Tf 12.253 0 Td[(1(inunitsof2 Na0),Un"(K)andVn#(K)arecolumnvectorswithN2Noelementsuni"andvni#,respectively,wherei=1;2;::;N2and=1;2;:::;No.TI;0andI;0canbefoundfromthehoppingelementsti0(fortheoriginalunitcell),andthechemicalpotentialandthegapeldobtainedfromtheself-consistentsolutionoflatticeBdGequations(Equation 2{9 )asshownschematicallyintheFigure B-1 .Clearly,thereareonlyninesupercellsfromwhichhoppingtothecentralcellispossible.These 112

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FigureB-1. Schematicofthesupercellset-up.ThesupercellisschematicallyrepresentedbybiggersquarescontainingNNunitcells(hereN=7)denotedbysmallersquares.Animpurityatthecenterofsupercellisrepresentedbyaredstar.ThissupercellisrepeatedMMtimesasindicatedbythedottedlines.Afterperiodicboundaryconditionsareimposed,thesystemcanbedescribedbyhoppingandsuperconductinggapeldmatricesbetweenthecentralcelland8neighboringsupercellsasdiscussedinthetext. includetheon-site,fournearestneighbor(NN),andfornextnearestneighbor(NNN)supercells.Theon-sitematricesT0;0and0;0haveelementssameelementsijandijasinEquation 2{9 withonedierencethattheperiodicboundaryconditionsarenotimposedwhilendingtheelements.TheothersupercellmatricesTI;0andI;0canbeobtainedinasimilarwaybyconsideringthehoppingandgapeldbetweenorbitalsinthecentralandneighboring(NNandNNN)supercells.Thenon-vanishingmatrixelementswillonlyoccurfortheorbitalsaroundtheedgesofthecentralsupercellsandtheyareschematicallyindicatedbydouble-headedarrowsintheFigure B-1 .NotethatforK=0,thesupercellBdGequations(Equation B{1 )reducetothelatticeBdGequations(Equation 2{9 ).Withsupercellhoppingandgapeldmatricesinhand,thesupercellBdGequation(Equation B{1 )issolvedforeachK.NotethatforK=0,thesupercellBdGequations 113

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reducetothelatticeBdGequations(Equation 2{9 ).UsingtheeigenvaluesEn(K)andeigenvectorsUn"(K)andVn"(K)thelatticeGreensfunctionmatrixiscalculatedviafollowingequations Gij"(!)=1 M2Xn;Kuni"(K)unj"(K) !)]TJ /F4 11.9552 Tf 11.955 0 Td[(En"(K)+i0+;Gij#(!)=1 M2Xn;Kvni#(K)vnj#(K) !+En"(K)+i0+:(B{4)ComparingwithEquation 2{12 ,wecanseethatwithsupercellmethodM2)]TJ /F1 11.9552 Tf 12.114 0 Td[(1additionaleigenvaluesareobtainedwhichhelpstoincreasethespectralresolutionleadingtoasmoothlatticeandcontinuumLDOSspectrum. 114

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APPENDIXCANALYTICALPROOFOFU-SHAPEDCONTINNUMLDOSINSUPERCONDUCTINGBSCCOInthisappendix,weshowthatthecontinuumLDOSforBSCCOaboveaCusite,atheightseveralAabovetheBiOplanewheretheSTMtipistypicallyplaced,atlowenergiesvariesas(r;!!0)j!j3yieldingaU-shapeinthespectrum.Weworkinunitswhere~=1.Inatranslationallyinvariantsystem,thelatticeGreensfunctionGijcanberepresentedas Gij(!)=XkG(k;!)eik(Ri)]TJ /F11 7.9701 Tf 6.586 0 Td[(Rj):(C{1)Here,wehaveconsidereda1-bandsystem,and,hence,theorbitalindicesaresuppressed.UsingEquations C{1 and 2{18 weobtain G(r;!)=XkG(k;!)jWk(r)j2;(C{2)where Wk(r)=Xiwi(r)eikRi:(C{3)Here,wi(r)istheWannierfunctioncenteredatthelatticevectorRi,andWk(r)canbecalledasthek-spaceWannierfunction.Now,usingEquations 2{14 and C{2 ,wecanwritecontinuumLDOS(r;!)as (r;!)=XkA(k;!)jWk(r)j2;(C{4)wherethek-spacespectralfunctionA(k;!)canbeexpressedas A(k;!)=)]TJ /F1 11.9552 Tf 11.102 8.088 Td[(1 Im[G(k;!)]=)]TJ /F1 11.9552 Tf 11.102 8.088 Td[(1 Imjukj2 !)]TJ /F4 11.9552 Tf 11.955 0 Td[(Ek+i0++jvkj2 !+Ek+i0+=jukj2(!)]TJ /F4 11.9552 Tf 11.955 0 Td[(Ek)+jvkj2(!+Ek): (C{5) 115

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Here,istheDiracdeltafunction.Thecoherencefactorsukandvk,andthequasiparticleenergyinthesuperconductingstateEkaregivenbyfollowingexpressions jukj2=1 21+k Ek=1)-222(jvkj2;(C{6) Ek=q 2k+2k:(C{7)Here,kisthebandenergyrelativetotheFermienergy.Forcuprates,thed-wavesuperconductinggaporderparameterkcanbeexpressedas k=0(coskx)]TJ /F1 11.9552 Tf 11.955 0 Td[(cosky):(C{8)UsingEquations C{4 C{5 ,and C{6 ,wecanwrite (r;!)=1 2Xk1+k EkjWk(r)j2(!)]TJ /F4 11.9552 Tf 11.956 0 Td[(Ek):(C{9)Infollowingparagraphs,wewillderiveexpressionforcontinuumLDOSas!!0onlyforpositiveenergies.Theresultscanbeeasilyextendedtothenegativeenergiesusingparticle-holesymmetry.ForthecontinuumpositionabovetheCusite,r0=[0;0;z],atheightszseveralAabovetheBiOplane,thedominantcontributiontothesumintherighthandsideoftheEquation C{3 comesfromthetheWannierfunctionatthenearestneighborsites;itcanbeeasilyinferredfromtheWannierfunctioncutattheheightz5AshownintheFigure 4-4 .Thus,thek-spaceWannierfunctionatsuchpositionscanbeapproximatedas: Wk(r0)w0+2w1(coskx)]TJ /F1 11.9552 Tf 11.955 0 Td[(cosky)(C{10)where,w0andw1arethevaluesoftheWannierfunctionatr=r0,andr=r0+a0^x,wherea0isthein-planelatticevector.Thedx2)]TJ /F5 7.9701 Tf 6.587 0 Td[(y2-wavesymmetryoftheWannierfunctiondictatesthatw0=0,however,westillkeepittofacilitatediscussionspresentedlatterinthisappendix.Also,thesamesymmetryassuresthattheallhigherorderterms,whichmightariseinEquation C{10 fromthecontributionsoftheotherneighbors,musthave 116

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thedx2)]TJ /F5 7.9701 Tf 6.586 0 Td[(y2-wavesymmetry.Forexample,contributionsfromNNNsiteswouldbezero,andthatfrom3rdnearestneighborswouldbew3(cos2kx)]TJ /F1 11.9552 Tf 12.923 0 Td[(cos2ky).AllsuchhigherordertermswillnotqualitativelychangeournalconclusionsregardingtheshapeofthecontinuumLDOSspectrumatlowenergies.UsingEquations C{9 and C{10 ,wecanexpressthecontinuumLDOSabovetheCusiteas (r0;!)=00+10+1100=1 2w20Xk1+k Ek(!)]TJ /F4 11.9552 Tf 11.955 0 Td[(Ek)10=2w0w1Xk1+k Ek(coskx)]TJ /F1 11.9552 Tf 11.955 0 Td[(cosky)(!)]TJ /F4 11.9552 Tf 11.956 0 Td[(Ek)11=2w21Xk1+k Ek(coskx)]TJ /F1 11.9552 Tf 11.955 0 Td[(cosky)2(!)]TJ /F4 11.9552 Tf 11.955 0 Td[(Ek):(C{11)AtlowenergiesthemomentumspaceintegralswillhavedominantcontributionsfromthenodalregionsoftheBrillouinzone.Accordingly,wecanexpandkandkaroundthenodalpointk0=[k0x;k0y],withk0x=k0y=1 p 2k0,andshiftthek-spaceorigintothesametofacilitatethecomputationofintegral.Thenewcoordinateaxesk1andk2,givenbyfollowingtransformations,areshownintheFigure C-1 k1=1 p 2(ky)]TJ /F4 11.9552 Tf 11.955 0 Td[(kx)k2=1 p 2(ky+kx)+k0:(C{12)Linearizingthespectrumaroundthenodalpointyields k=vFk2+O(k2);(C{13)wherevFistheFermivelocityatthenode.NotethatforaNNtight-bindingmodelvF=)]TJ /F1 11.9552 Tf 9.299 0 Td[(2p 2tsink0x.Wecanalsolinearizethegapfunctionbyexpandingaroundthenodeas k=vk1+O(k2);v=p 20sink0x:(C{14) 117

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FigureC-1. Coordinatesystem(k1;k2)withaxesparallelandperpendiculartoanodaldirectionandoriginlocatedatanodek0denotedbythegreenarrow. FromEquations C{8 C{13 ,and C{14 ,wehave Ek=q v2k21+v2Fk22:(C{15)Now,usingEquations C{11 and C{13 ,andconvertingthesumoverkinformertoanintegral 00=1 2Mw20ZZdk (2)2(1+vFk2 p v2k21+v2Fk22)(!)]TJ /F10 11.9552 Tf 8.634 14.409 Td[(q v2k21+v2Fk22);(C{16)whereisacircularregionofradius)-326(aroundthenodalpointk0,andM=4isthenumberofnodes.Theaboveintegralcanbesimpliedbyscalingthecoordinatesask01=vk1andk02=vFk2,followedbyatransformationtothepolarcoordinateswithk01=k0cos0andk02=k0sin0.Followingistheresultingexpression. 00=1 2Mw201 vFv1 (2)2Z)]TJ -5.313 -23.91 Td[(0Z20k0dk0d0(!)]TJ /F4 11.9552 Tf 11.955 0 Td[(k0)+Z)]TJ -5.313 -23.91 Td[(0Z20k0dk0d0(!)]TJ /F4 11.9552 Tf 11.955 0 Td[(k0)sin0=1 2Mw201 vFv1 (2)2[2!+0]:(C{17)Hence, 00=w20 vFv!:(C{18) 118

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NotethatsettingtheWannierfunctionfactorto1inEquation C{4 yieldsthelatticeLDOSN(!)=PkAk(!),suggestingthatwecansimplyobtaintheexpressionforlatticeLDOSas!!0bysettingw0=1andw1=0.Hence,usingEquation C{18 withw0=1,wegetthelatticeLDOSatlow(positive)energiesas N(!)=1 vFv!:(C{19)Whichshowsthatas!!0latticeLDOSvarieslinearlyyieldingaV-shape.However,incaseofthecontinuumLDOSw0=0,asexplainedearlier,thus,thislineartermdoesnotcontributetothelowenergyDOS.Infollowingweshowthatthe10vanishesand11yieldsa!3variation,thus,yieldingaU-shape.UsingEquations C{11 C{13 C{14 ,and C{15 ,weget 10=2w0w1MZZdk (2)2(1+vFk2 p v2k21+v2Fk22)(2p 2tsink0xk1)(!)]TJ /F10 11.9552 Tf 8.634 14.409 Td[(q v2k21+v2Fk22):(C{20)Rescalingthecoordinates,followedbyatransformationtopolarcoordinatesinaboveintegralyieldsangularintegralsofcos0andcos0sin0over=[0;2].Theseangularintegralsvanishresultinginto 10=0:(C{21)Finally,weevaluate11usingEquations C{11 C{13 C{14 ,and C{15 11=2w21MZZdk (2)2(1+vFk2 p v2k21+v2Fk22)(2p 2tsink0xk1)2(!)]TJ /F10 11.9552 Tf 8.634 14.409 Td[(q v2k21+v2Fk22):(C{22)Rescalingthecoordinates,followedbyatransformationtopolarcoordinatesinaboveintegralyields 11=4Mw21sin2k0x (2)2vFv3Z)]TJ -5.314 -23.911 Td[(0Z20k0dk0d0(!)]TJ /F4 11.9552 Tf 11.955 0 Td[(k0))]TJ /F4 11.9552 Tf 5.48 -9.684 Td[(k02cos20+sin0cos20=4Mw21sin2k0x (2)2vFv3!31 +0:(C{23) 119

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Hence, 11=4w21sin2k0x 3vFv3!3:(C{24)CombiningEquations C{11 C{18 C{21 ,and C{24 ,wegettheexpressionforlowenergycontinuumLDOSas (r0;!)=a0!+a1!3;a0=w20 vFv;a1=4w21sin2k0x 3vFv3:(C{25)Thecoecientofthelinearin!termvanishesasw0=0duetodx2)]TJ /F5 7.9701 Tf 6.587 0 Td[(y2symmetryoftheWannierfunction.However,thecubicin!termisnon-zero,andyieldsaU-shapedspectrum. 120

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APPENDIXDMICROSCOPICJUSTIFICATIONOFTHE"FILTER"THEORYToexplainthespatialpatternoftheZnimpurity-inducedresonancestatesinBSCCO[ 60 ](seeSection 1.5 ),Martinetal.[ 62 ]proposedaphenomenological"lter"wavefunctionresultingfromthenon-trivialtunnelingpathprovidedbytheBiOlayer.TheauthorsarguedthatthedirecttunnelingbetweentheSTMtipandthe3dx2)]TJ /F5 7.9701 Tf 6.586 0 Td[(y2orbitaloftheCuatomunderneathisprohibitedduetoitsvanishingoverlapwithaxialorbitals(Bi-6pzandapicalO-2pz)inBiOplane.Instead,theyproposed,anindirecttunnelingoccursthroughtheCu-4s(orCu-3dz2)orbitalwhichoverlapswiththenearest-neighborCu-3dx2)]TJ /F5 7.9701 Tf 6.586 0 Td[(y2orbital,resultinginthetunnelingintensityonthelatticesitei=(ix;iy),expressedintermsoftheimpuritystatewavefunctionix;iy, ABiOix;iy/jix+1;iy+ix)]TJ /F6 7.9701 Tf 6.586 0 Td[(1;iy)]TJ /F1 11.9552 Tf 11.955 0 Td[(ix;iy+1)]TJ /F1 11.9552 Tf 11.955 0 Td[(ix;iy)]TJ /F6 7.9701 Tf 6.586 0 Td[(1j2;(D{1)whichisverydierentfromthetunnelingintensityACuOix;iy/jix;iyj2obtainedbyassumingadirecttunneling.Here,weshowthattheBdG+Wschemeprovidesamicroscopicjusticationfortheaforementioned"lter"argument.Settingorbitalindices;=1inEquation 2{11 thelatticeGreensfunctionmatrixforthe1-bandBSCCOcanbewrittenas Gij(!)=Xn>0uniunj !)]TJ /F4 11.9552 Tf 11.955 0 Td[(En+i0++vnivnj !+En+i0+:(D{2)ThecontinuumGreensfunction,incaseofoneorbitalpersite,willbegivenbyEquation 2{18 G(r;!)=XijGij(!)wi(r)wj(r)=Xn>024(Piuniwi(r))Pjunjwj(r) !)]TJ /F4 11.9552 Tf 11.955 0 Td[(En+i0++(Pivniwi(r))Pjvnjwj(r) !+En+i0+35=Xn>0jun(r)j2 !)]TJ /F4 11.9552 Tf 11.955 0 Td[(En+i0++jvn(r)j2 !+En+i0+; (D{3) 121

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whereun(r)=Piuniwi(r)andvn(r)=Pivniwi(r).AsevidentfromFigure 4-4 ,atacontinuumpointr=r0directlyabovetheimpuritysitei?locatedataheightafewAabovetheBiOplane,theon-siteWannierfunctioniszero,whereastheNNWannierfunctionshavethelargestmagnitude.Thus,ignoringsmallcontributionsfromtheWannierfunctionslocatedatdistantsites,wecanwrite un(r0)w1uni?+^x;+uni?)]TJ /F6 7.9701 Tf 7.088 0 Td[(^x;)]TJ /F4 11.9552 Tf 11.955 0 Td[(uni?+^y;)]TJ /F4 11.9552 Tf 11.955 0 Td[(uni?)]TJ /F6 7.9701 Tf 7.176 0 Td[(^y;;vn(r0)w1vni?+^x;+vni?)]TJ /F6 7.9701 Tf 7.088 0 Td[(^x;)]TJ /F4 11.9552 Tf 11.956 -.001 Td[(vni?+^y;)]TJ /F4 11.9552 Tf 11.955 -.001 Td[(vni?)]TJ /F6 7.9701 Tf 7.176 0 Td[(^y;;(D{4)wherew1isthemagnitudeoftheWannierfunctionattheNNsite,andwehaveproperlyaccountedforthesignchangeoftheWannierfunctionwith90rotationintheaboveexpression.Equation D{4 isequivalenttothe"lter"wavefunctionproposedin[ 62 ]butitnowhasaprecisedenitionandmicroscopicjustication.Now,usingEquations D{3 and D{4 ,thecontinuumLDOSatr=r0is (r0;!)=)]TJ /F1 11.9552 Tf 11.102 8.088 Td[(1 Im[G(r0;!)]=jw1j2X=^x;^y)]TJ /F1 11.9552 Tf 11.102 8.087 Td[(1 Im"Xn>0juni?+;j2 !)]TJ /F4 11.9552 Tf 11.956 0 Td[(En+i0++jvni?+;j2 !+En+i0+#+interferenceterms=jw1j2X=^x;^yNi?+;(!)+interferenceterms; (D{5) wherewehaveusedthedenitionoflatticeLDOSNi(!)asstatedinEquation 2{13 ,inconjunctionwithEquation D{2 .Equation D{5 showsthatthecontinuumLDOSabovetheimpuritysiteataheightfewAabovetheBiOplaneisequaltothesumofthelatticeLDOSattheNNsitesandinterferenceterms.Since,latticeLDOSabovetheNNsiteshavelargemagnitudeattheresonanceenergy(=)]TJ /F1 11.9552 Tf 9.298 0 Td[(3:6meV)(seeFigure 4-5 (a))andthesignofinterferencetermsisnotxed,Equation D{5 impliesthatthecontinuumLDOSdirectlyabovetheimpuritysitemusthavealargemagnitudeatthesameenergy,whichisindeedthecase(seeFigure 4-5 (b)).However,WenotethattheinterferencetermsareimportanttoaccountforthefullbiasdependenceofthecontinuumLDOS.Forexample,simplyaddingNNlattice 122

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LDOSsuggeststhatthereshouldbeanobservableweightat!=+3:6meV;however,thecontinuumLDOSplottedinFigure 4-5 (b)showsnosuchfeature. 123

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BIOGRAPHICALSKETCHPeayushChoubeywasborninBallia,India.Hecompletedhisschoolinginhishometown,andadmittedtotheIndianSchoolofMines,Dhanbad,India,in2004,wherehegraduatedwithBachelorofTechnologydegreeinElectronicsEngineeringin2008.Subsequently,HeworkedattheCenterforAirborneSystems,Bangalore,India.Infall2010,hewenttotheUniversityofFlorida,Gainesville,USA,andcompletedhisDoctorofPhilosophyinphysicsinspring2017. 136