Theory of Gap Symmetry and Structure in Fe-Based Superconductors

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Theory of Gap Symmetry and Structure in Fe-Based Superconductors
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english
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Wang, Yan
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University of Florida
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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:
INGERSENT,J KEVIN
Committee Members:
MASLOV,DMITRII
TANNER,DAVID B
STEWART,GREGORY R
PHILLPOT,SIMON R

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Subjects / Keywords:
chalcogenide -- disordered -- gap -- impurity -- pairing -- pnictide -- rpa -- spinfluctuation -- structure -- sts -- superconductivity -- superconductor -- symmetry -- vortex
Physics -- Dissertations, Academic -- UF
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Physics thesis, Ph.D.
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Abstract:
We consider a series of problems related to determining the origin of superconductivity in the recently discovered iron pnictide and chalcogenide materials, where the common layer in the crystal structure with Fe atom at the square lattice site and with pnictogen or chalcogen atoms buckling above or below is believed to be responsible for the high Tc. In many experiments, these Fe-based superconductors also share similar physical properties such as multiple superconducting gaps exhibiting nontrivial structure with anisotropy and nodes that evolve with doping, a generic phase diagram with antiferromagnetic metal phase at zero doping, and the superconducting dome (or domes) with electron or hole doping or with pressure. A sign-changing s+- superconducting state (that is, in an oversimplified term, two order parameters with the opposite sign) is proposed based on the electronic structure, Fermi surface and magnetic properties as measured by experiments and predicted by spin-fluctuation calculations. In the first few chapters, we phenomenologically explain the experimental observations and their interpretations on the symmetry of pairing state and claim a consistency with s+- state after the subtle features of the superconducting state are considered in theory. We first discuss the Volovik effect in a highly anisotropic s+- wave multiband superconductor, specifically the optimally doped BaFe2(As1-xPx)2. The square-root magnetic field dependence for the specific heat coefficient at a low field (so-called Volovik effect) and the linear dependence at high field can be understood from a multiband calculation in the quasiclassical approximation assuming gaps with different momentum dependence on the hole- and electron-like Fermi surface sheets. Next, we examine the quasiparticle vortex bound states in LiFeAs. The "unexpected" (assuming an anisotropic s+- wave pairing) tails of low energy density of states measured by scanning tunneling spectroscopy are reconciled by taking account of anisotropy of the Fermi surface and a cautionary message for the analysis of scanning tunneling spectroscopy data on the vortex state on Fe-based superconductors is sent to the experimentalists. In the next chapter, we have investigated the Tc suppression rate for s+- and s++ gap structure for Fe-based superconductors. The rate of Tc suppression is shown to vary dramatically according to details of the impurity model considered. A two-band model calculation with realistic parameters for BaFe2As2 with nonmagnetic impurities suggests a probable s+- wave state with small inter- to intra-band scattering rate ratio. We thus propose that observation of particular evolution of the penetration depth, nuclear magnetic resonance (NMR) relaxation rate, or thermal conductivity temperature dependence with disorder would suffice to differentiate s+- and s++ gap in experiments. In the Chapter 4, we use the multiband Hubbard-Hund Hamiltonian generalized to microscopically investigate the symmetry of superconductivity in Fe-based superconductors (FeBS) by spin-fluctuation theory. The spin-fluctuation calculation of the superconducting instability is done for LiFeAs with three-dimensional Fermi surface. The pairing instabilities explored with full 10-orbital model suggest important three-dimensional effects that can be verified by experiments, such as angle-resolved photoemission spectroscopy (ARPES) and NMR. In the LiFeAs system, a comparison of density functional theory (DFT) derived model and ARPES derived model shows a strikingly good agreement between calculated results and principal ARPES measured gaps. We explain the only discrepancy of gaps on the small inner hole-like pockets. Finally, we close with brief discussion on recent experiments and theoretical work on AFe2Se2 and monolayer FeSe, two exciting families that challenge standard arguments in favor of the s+- pairing state with their unique electronic structure and magnetic properties.
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by Yan Wang.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
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Adviser: HIRSCHFELD,PETER J.
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Co-adviser: INGERSENT,J KEVIN.

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THEORYOFGAPSYMMETRYANDSTRUCTUREINFE-BASED SUPERCONDUCTORS By YANWANG ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2014

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c r 2014YanWang 2

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Tomyparents 3

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ACKNOWLEDGMENTS IwouldliketoexpressmygreatestthankstomyadviserProf. PeterJ.Hirschfeld forhismentoringandsupportinmydoctoralstudy.Hehasgui dedmecloselyatevery stageofmyresearchthroughcountlessinspiringandpatien tdiscussions,andprovided enormoussuggestionsandencouragementsasamentorandafr iend.Iwouldlike furthertothankDrs.TomBerlijn,SiegfriedGraser,Andrea sKreisel,Chia-HuiLin, ThomasMaier,VivekMishra,DouglasScalapino,GregorySte wart,IlyaVekhterand LiminWangforvaluablediscussionsanddedicatedcollabor ationsinvariousprojects. I'malsogratefultomycommitteemembersDrs.KevinIngerse nt,DmitriiMaslov,David Tanner,SimonPhillpotandGregoryStewartfortheirtimean dexpertisedevotedto scrutinizingmydissertation. IacknowledgethehospitalityofDr.WeiKuatBrookhavenNat ionalLaboratory, Dr.ThomasMaieratOakRidgeNationalLaboratoryandDr.Ros erValentiatGoethe UniversityFrankfurtforinvitingmetotheirlaboratories oruniversity. IacknowledgethepartialnancialsupportfromInstitutef orFundamentalTheoryat UniversityofFlorida. Iamdeeplythankfultomyformerandcurrentgroupcolleague sMaximKorshunov, GregBoyd,LexKemper,PeayushChoubey,andWen-YaRowe,tom yfellowgraduate studentsPanZheng,JueZhang,Iek-HengChu,XiangguoLi,Li liDengandXiaochang Miao,andtoPhysicsDepartmentstafffortheirprofessiona lassistance. FinallyIwouldliketothankmyparentsfortheirunconditio nalsupport,understandingandlovethatcheermeupanytimeandanywhere. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ..................................4 LISTOFTABLES ......................................7 LISTOFFIGURES .....................................8 ABSTRACT .........................................10 CHAPTER 1INTRODUCTION ...................................13 1.1TheChronicleofSuperconductivity:EarlyHistory ..............14 1.2TheChronicleofSuperconductivity:AfterBCS ...............18 1.3TheSymmetryPropertiesofFe-BasedSuperconductors ..........25 1.3.1CrystalStructureandCrystalSymmetry ...............25 1.3.2GapSymmetryandGapStructure ..................26 2INHOMOGENEOUSSUPERCONDUCTIVITY ..................29 2.1QuasiclassicalApproximation .........................30 2.2VolovikEffectinMultibandSuperconductorBaFe 2 (As 0.7 P 0.3 ) 2 .......34 2.2.1Motivation ................................34 2.2.2ExperimentResults ...........................36 2.2.3Two-BandModel ............................38 2.2.4Results .................................43 2.2.5Conclusions ...............................47 2.3QuasiparticleVortexBoundStatesinFeBS:Application toLiFeAs ....49 2.3.1Motivation ................................49 2.3.2Model ..................................54 2.3.3Results .................................56 2.3.4Conclusions ...............................59 3DISORDERINSUPERCONDUCTORS ......................61 3.1Motivation ....................................61 3.2Model ......................................63 3.3 T c Suppression .................................65 3.4ResidualResistivity ..............................66 3.5Results .....................................66 3.5.1 T c SuppressionvsResistivity .....................66 3.5.2DensityofStates ............................70 3.5.3NonmonotonicDependenceofResidualDOSonDisorder .....71 3.5.4RealisticImpurityPotentials ......................73 3.6Conclusions ...................................73 5

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4SPIN-FLUCTUATIONPAIRINGINFE-BASEDSUPERCONDUCTORS ....75 4.1Motivation ....................................75 4.2Ten-OrbitalTight-BindingFitsandFermiSurfaces .............78 4.3FluctuationExchangePairingModel .....................80 4.4ResultsforthePairingState ..........................83 4.4.1ResultsfortheARPES-DerivedFermiSurface ............83 4.4.2AnalysisofGapSizesinTermsofPairingVertex ..........85 4.4.3Discussion:ToyModelforGapSizes .................89 4.5Conclusions ...................................90 5FINALCONCLUSIONS ...............................99 APPENDIX ASPIN-FLUCTUATIONCALCULATIONFORDFT-DERIVEDFERMISURF ACE .102 A.1ElectronicStructureofLiFeAsfromDensityFunctional Theory ......102 A.2PairingStateforDFT-DerivedFermiSurface .................102 BFITTINGPARAMETERSFORTEN-ORBITALTIGHT-BINDINGMODEL H ARPES 0 106 REFERENCES .......................................111 BIOGRAPHICALSKETCH ................................120 6

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LISTOFTABLES Table page 1-1Irreduciblerepresentationsandgapfunctionsfortetr agonalsymmetry .....28 2-1Differentmodelsforthecouplingmatrixandgapanisotr opyonelectronpockets 44 A-1LiFeAsDOSattheFermilevelfromten-orbitalDFTandARP ESbasedmodels 103 7

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LISTOFFIGURES Figure page 1-1Meissnereffectdemonstration ...........................15 1-2CriticaleldandmagnetizationoftypeIandtypeIIsupe rconductors ......16 1-3Schematicphasediagramsandactivelayersofcupratesa ndFeBS ......21 1-4FermisurfaceofFeBS ................................25 1-5CrystalstructuresofFeBS ..............................27 2-1Specicheatcoefcient r vshigheld H onBaFe 2 (As 0.7 P 0.3 ) 2 .........37 2-2Specicheatcoefcient r vsloweld H onBaFe 2 (As 0.7 P 0.3 ) 2 ..........38 2-3SpatiallyaveragedZDOSfornodelessandnodalsupercon ductors .......40 2-4GapelddependenceandspatiallyaveragedZDOS ...............43 2-5Measurednormalizedspecicheatcoefcientandthetwo -bandcalculations .45 2-6Specicheatcoefcientfromgapswithdeepminimaandac cidentalnodes ..48 2-7ConductancemapandZDOSaroundavortexcorefora d x 2 y 2 -wavegap ...51 2-8FermisurfaceofLiFeAsat k z =0 andtheFermivelocities ............52 2-9SketchofARPESmeasuredgapsandSTSconductancemapofL iFeAs ....54 2-10ZDOSfordifferentmodelswithcircularFermisurfaceo rLiFeAs r pocket ...57 2-11LDOS N ( r ) = N 0 vsenergy fordifferentgapmodelsandFermisurfaces ..60 3-1Sketchofthetwo-bandmodelwithconstantimpurityscat tering .........63 3-2 T c = T c 0 vsdisorder-inducedresistivitychange 0 forisotropic s -wavepairing 68 3-3 T c = T c 0 vs 0 forvariousvaluesoftheinter-tointrabandscatteringrat io ..70 3-4Theresistivityathalfsuppression 1 = 2 asafunctionoftheratio = u = v ...71 3-5SchematicevolutionofgapsandDOSwithincreasingdiso rder;DOSvs 0 .72 4-1Comparisonofthetight-bindingbandsandARPESdata .............92 4-2Sketchofthegap j ( ) j asseeninrecentARPESexperiments .........93 4-3FermisurfaceofLiFeAsfrom H ARPES 0 ,gapfunctions g ( k ) andspinsusceptibility 95 4-4BandstructuresforARPES-derivedmodelwithandwithou tspin-orbitcoupling 96 8

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4-5Comparisonbetweengap j ( ) j bycalculationandARPESexperiment ....97 4-6Componentsofpairingvertex ij ( k k 0 ) matrixatlling n =6.00 and n =5.90 .98 A-1FermisurfaceofLiFeAsfrom H DFT 0 ,gapfunctions g ( k ) andspinsusceptibility .105 9

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy THEORYOFGAPSYMMETRYANDSTRUCTUREINFE-BASED SUPERCONDUCTORS By YanWang May2014 Chair:PeterJ.HirschfeldMajor:Physics Weconsideraseriesofproblemsrelatedtodeterminingtheo riginofsuperconductivityintherecentlydiscoveredironpnictideandc halcogenidematerials, wherethecommonlayerinthecrystalstructurewithFeatoma tthesquarelattice siteandwithpnictogenorchalcogenatomsbucklingaboveor belowisbelievedtobe responsibleforthehigh T c .Inmanyexperiments,theseFe-basedsuperconductors alsosharesimilarphysicalpropertiessuchasmultiplesup erconductinggapsexhibiting nontrivialstructurewithanisotropyandnodesthatevolve withdoping,agenericphase diagramwithantiferromagneticmetalphaseatzerodoping, andthesuperconducting dome(ordomes)withelectronorholedopingorwithpressure .Asign-changing“ s ” superconductingstate(thatis,inanoversimpliedterm,t woorderparameterswith theoppositesign)isproposedbasedontheelectronicstruc ture,Fermisurfaceand magneticpropertiesasmeasuredbyexperimentsandpredict edbyspin-uctuation calculations. Intherstfewchapters,wephenomenologicallyexplainthe experimental observationsandtheirinterpretationsonthesymmetryofp airingstateandclaima consistencywith s stateafterthesubtlefeaturesofthesuperconductingstat eare consideredintheory.WerstdiscusstheVolovikeffectina highlyanisotropic s -wave multibandsuperconductor,specicallytheoptimallydope dBaFe 2 (As 1 x P x ) 2 .The square-rootmagneticelddependenceforthespecicheatc oefcientataloweld 10

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(so-calledVolovikeffect)andthelineardependenceathig heldcanbeunderstood fromamultibandcalculationinthequasiclassicalapproxi mationassuminggapswith differentmomentumdependenceonthehole-andelectron-li keFermisurfacesheets. Next,weexaminethequasiparticlevortexboundstatesinLi FeAs.The“unexpected” (assumingananisotropic s -wavepairing)tailsoflowenergydensityofstatesmeasure d byscanningtunnelingspectroscopyarereconciledbytakin gaccountofanisotropyof theFermisurfaceandacautionarymessagefortheanalysiso fscanningtunneling spectroscopydataonthevortexstateonFe-basedsupercond uctorsissenttothe experimentalists.Inthenextchapter,wehaveinvestigate dthe T c suppressionratefor s and s ++ gapstructureforFe-basedsuperconductors.Therateof T c suppression isshowntovarydramaticallyaccordingtodetailsoftheimp uritymodelconsidered.A two-bandmodelcalculationwithrealisticparametersforB aFe 2 As 2 withnonmagnetic impuritiessuggestsaprobable s -wavestatewithsmallinter-tointra-bandscattering rateratio.Wethusproposethatobservationofparticulare volutionofthepenetration depth,nuclearmagneticresonance(NMR)relaxationrate,o rthermalconductivity temperaturedependencewithdisorderwouldsufcetodiffe rentiate s and s ++ gapin experiments. IntheChapter 4 ,weusethemultibandHubbard-HundHamiltoniangeneralize d tomicroscopicallyinvestigatethesymmetryofsupercondu ctivityinFe-basedsuperconductors(FeBS)byspin-uctuationtheory.Thespin-uc tuationcalculationof thesuperconductinginstabilityisdoneforLiFeAswiththr ee-dimensionalFermi surface.Thepairinginstabilitiesexploredwithfull10-o rbitalmodelsuggestimportant three-dimensionaleffectsthatcanbeveriedbyexperimen ts,suchasangle-resolved photoemissionspectroscopy(ARPES)andNMR.IntheLiFeAss ystem,acomparison ofdensityfunctionaltheory(DFT)derivedmodelandARPESd erivedmodelshowsa strikinglygoodagreementbetweencalculatedresultsandp rincipalARPESmeasured gaps.Weexplaintheonlydiscrepancyofgapsonthesmallinn erhole-likepockets. 11

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Finally,weclosewithbriefdiscussiononrecentexperimen tsandtheoretical workonAFe 2 Se 2 andmonolayerFeSe,twoexcitingfamiliesthatchallengest andard argumentsinfavorofthe s pairingstatewiththeiruniqueelectronicstructureand magneticproperties. 12

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CHAPTER1 INTRODUCTION NearlyadecadeagoIheardof Woodstockofphysics inmySolidStatePhysics class.AtthattimeIwasdeeplyimpressedandawedbytheenth usiasmandexcitement, asdescribedbymyprofessor,ofthosephysicistsattending themarathonsessionof theAmericanPhysicsSociety'sMarchmeetingin1987,which featuredpresentations onthen-newlydiscoveredceramicsuperconductors,nowkno wnascuprates,from K.AlexM¨uller,PaulChu,andmanyothers.ThatK.AlexM¨ull erandJ.GeorgBednorz discovered 35K superconductorlanthanumbariumcopperoxide[(La 1 x Ba x ) 2 CuO 4 orBa-dopedLa214]in1986andlaterPaulChuandcolleaguesd iscovered 93K superconductoryttriumbariumcopperoxide(YBa 2 Cu 3 O 7 ,Y123orYBCO)in 1987wassoearthshakingthatabout2000participantspacke dtheaforementioned “Woodstock”sessionbeginningat7:30PMandmanyofthemdid n'tleaveuntil3:00 AMinthenextmorning.Thisdiscoveryofcupratesystemison eofthemostnotable milestonesinthehistoryofthesuperconductivityandprob ablythemostdramaticone amongotherssuchasthediscoveryofsuperconductivityin1 911,theBCStheoryin 1957,andthediscoveryofironpnictideandchalcogenidesu perconductorsin2008. Forphysicists,theexperimentalandtheoreticalchalleng esofsuperconductivitysince itsdiscoveryhavebeenmuchmoreessentialandmotivatingt hantheattentionand popularitygraduallygainedinthegeneralpublicormassme dia.Inthischapter,I willrstbrieylistthecornerstonesofsuperconductivit y,includingtheexperimental discoveriesofvarioussuperconductorclassesandthepres tigiousBCStheory, nextintroducethedevelopmentsofBCStheoryandnewtheori esthataccountfor unconventionalandnontrivialsuperconductivityfoundin newclassesofsuperconductors,andlastfocusonthebasicexperimentalandthe oreticalaspectsofiron pnictideandchalcogenidesuperconductors(Fe-basedsupe rconductorsorFeBS),which 13

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setsthestagefortheremainingchapters.Recenthistorica lreviewscanbefoundin Refs.[ 1 – 3 ];generalreviewsonFeBScanbefound,forexample,inRefs. [ 4 – 10 ]. 1.1TheChronicleofSuperconductivity:EarlyHistory In1911,usingtheheliumrefrigerationtechniqueheinvent ed,Dutchphysicist HeikeKamerlinghOnnesdiscoveredinmercury(Hg, T c =4.2K )superconductivity, aphenomenonwheretheresistivitytotheelectricalcurren tsuddenlyvanishes whenthematerialiscooledbelowacriticaltemperature( T c ).Afterhavingaclose lookatKamerlingh-Onnes'slabnotebooks,vanDelftandKes [ 11 ]piecedtogether thefascinatingtruestoryaboutthediscoveryofsupercond uctivity,includingthe actualexperimentproceduresandtheoverlookedobservati onofsuperuidtransition (Bose-Einsteincondensation)ofhelium( 4 He)at T c =2.2K In1933,asthethirdresearchgroupwhohadmanagedtoliquif yheliuminthat time(J.C.McLennanfromTorontowasthesecondin1923),Wal therMeissnerand RobertOchsenfeldfromBerlinobservedthediamagneticbeh aviorofsuperconductors, i.e.,thecompleteexpulsionofmagneticeldfromthesuper conductor[ 12 ].Perfect diamagnetismandperfectconductivityaretwohallmarksof asuperconductor.However, perfectdiamagnetismisnotequivalenttoperfectconducti vitybecauseifitweremerely amaterialundergoingaperfect-conductivitytransition, themagneticuxoriginally insidethematerialwouldbetrappedinduetoLenz'slawasex pectedbyclassical electrodynamics.Thereversible Meissnereffect manifeststhatthesuperconductivity isaquantumeffectonamacroscopicscale.Iftheexternale ldissmallerthana temperaturedependentcriticalvalue H c ( T ) ,thesystemcanloweritsfreeenergyby goingintothesuperconductingstateatthecostofexpellin gthemagneticeldfrom thebulk.Diamagnetismissuitableformagneticlevitation [ 13 ],andanoften-seen demonstrationofthediamagnetismofasuperconductoristo levitateapermanent magnetbyasuperconductor,asshowninFig. 1-1 .Incontrasttothe“yingfrog”[ 13 ]or thelike,thedemonstrationwithsuperconductoriseasiert oaccomplishbecausetheso 14

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Figure1-1.Apieceofpermanentmagnetlevitatingabovethe superconductorcooledby liquidnitrogen,demonstratingtheMeissnereffect.Photo takenbytheauthor attheM2Sconference,2012. calledtypeIIsuperconductor(usuallywithhigher T c )isused,wherethemagneticux lineshelptostabilizethelevitationandevenmakethesusp ensionofasuperconductor belowapermanentmagnetpossible[ 14 ].IntypeIIsuperconductor,thequantized magneticux,withtheuxoidquantum 0 = hc 2 e ,penetratesthesuperconductorat temperaturebelow T c inamagneticeldrangebetweenthelowercriticaleld H c 1 and uppercriticaleld H c 2 ,beinginthesocalled vortexstate .Below H c 1 ,theMeissnereffect follows.Fig. 1-2 showsthecomparisonbetweentypeIandIIsuperconductor. In1935,thebrothersF.andH.Londonsuggestedatheory[ 15 ]toexplainthe Meissnereffect,aphenomenologicaltheorydescribingthe electrodynamicpropertiesof thesuperconductor.FollowinganunpublishedtheoremofBl och(groundstateshould havezeronetmomentumwithoutexternaleld),theyshowedt hatthecurrentdensity J = n s e h v s i = n s e 2 mc A = ( c = 4 2L ) A ,where L =( mc 2 = 4 e 2 n s ) 1 = 2 isthepenetration depthofthemagneticeldat T =0 and A isthevectorpotential.TheCoulombgauge div A =0 isusuallyimposedastheparticulargaugechoice.LaterPip pardintroduced thecoherencelength andnon-localgeneralizationofLondontheorytoaccountfo rthe actualexperimentallymeasuredpenetrationdepth[ 16 ]. In1950,the isotopeeffect T c / 1 = p M ( M isionmass),wasseeninexperiments bySerin[ 17 ]andMaxwell[ 18 ],indicatingtheessentialroleplayedbylatticevibratio ns 15

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Meissner Normal H H c T T c Meissner Vortex State H H c1 T T c Normal H c2 M H H c H H c1 H c2 M (a) (b) (c) (d) Figure1-2.(a),(c)Criticaleldtemperaturedependencef ortypeI(a)andtypeII(c) superconductor.(b),(d)Magnetizationelddependencefo rtypeI(b)and typeII(d)superconductor. insuperconductivity.Inthesameyear,theGinzburg–Landa utheorybasedonLandau's theoryofsecond-orderphasetransitionsfurtherextended theLondontheory,wherethe orderparameter ( x T ) describingthesuperconductingtransitionwasintroduced [ 19 ]. Thetheoreticalbreakthroughinsuperconductivitycamefr omBardeen,Cooper andSchrieffer(BCStheory)in1957[ 20 ].Thiselegantmicroscopictheorysuccessfully explainsthesuperconductivityintermsof Cooperpairs ,thatis,pairedelectronswith oppositemomentumandspin ( k k # ) ,andtheBCSpairingwave-function j G i thegroundstateformany-bodysysteminthesuperconductin gstate,whichisevidently relatedtotheorderparameterinGinzburg–Landautheory[ 21 ]andisproportionalto theenergygap seenintheexponentialtemperaturedependence e = k B T inphysical quantitiessuchasthespecicheatatlowtemperatures T .ThepivotalobjectinBCS theoryisthe“Cooperpair”ofelectrons,whichsurprisingl yextendsinrealspaceover adistance muchlargerthantheinter-atomdistanceandthereforestro nglyoverlaps 16

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withotherCooperpairs.Coopershowedthattwoelectronsad dedtotheFermisea j F i atzerotemperaturecanformaboundstate,lowingthetotals ystemenergy,nomatter howweaktheeffectiveattractionbetweenthemis.Whilethe effectiveattractioncanbe thoughtofastheneteffectoftwoelectronsinteractingwit hthelatticeatsamelocation butdifferenttimes 1 ,ittookgreatimaginationandintuitionforSchrieffertow ritedown theformforthegroundstatewave-function j G i = Y k ( u k + v k c y k c y k # ) j 0 i (1–1) wherethevariationalparameters u k and v k satisfy j u k j 2 + j v k j 2 =1 and j 0 i isthe vacuumstate(zeroparticle).TheforminEq.( 1–1 )impliesaprobability j v k j 2 forthepair ( k k # ) tobeoccupied.Thenalstepistodeterminetheseprobabili tyamplitudesby minimizingthegroundstateenergy h G jHj G i wherethe pairingHamiltonian or reduced Hamiltonian is H = X k ( k ) c y k c k + X kl V kl c y k c y k # c l # c l (1–2) Herethechemicalpotential isincludedasaLagrangemultipliertoxthemean numberofparticles,and V kl istheeffectiveelectron-electroninteractionduetophon on exchange.Althoughwewon'tshowthederivation,itisworth whiletomentionthatby assuminganattractiveinteraction V kl = V ( V > 0 )foronlystates k l inashellnearthe Fermisurfacewithenergy ( E F ~ c ) ,otherwise 0 ,aboundedgroundstatewithlower energythanthatofthenormalmetalcanbefound[ 22 ].Here c isthecut-offfrequency andinBCStheoryitapproximatelyequalstheDebyefrequenc y.Thesuperconducting gap, V P k h c k # c k i where j ( k k F ) v F j
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beaconstantinmomentumspace,andhencewecallit s -wavepairing.BCSfound the universal ratio = k B T c = e r 1.76 ,whichmatchesexperimentvaluesinthe elements,suchasCd,Al,andSn,remarkablywell. 1.2TheChronicleofSuperconductivity:AfterBCS BCStheoryimmediatelyttheexperimentalmeasurementsof theenergygap,the Meissnereffect,thecriticaleldandmanyothersinsimple elementalsuperconductors foundatthattimeandgainedgreatacceptancethroughthefo rmalimprovementsby Anderson,Bogoliubov,Gor'kov,Abrikosov,NambuandElias hbergusingthequantum eldtheoryapproach[ 23 24 ].However,problemscamewhentheexperimentsshowed absenceofisotopeeffectinsometransitionmetalsupercon ductors,suchasRh[ 25 ], where T c / M 0 0.1 .Thisinfactstimulatedthequantitativeconsiderationof the phononeffectthatwasoversimpliedinBCStheoryandthest rongelectron-phonon couplingtheorywasformulatedbyEliashbergin1960withth eMigdaltreatment ofelectron-phononinteractions[ 26 ].Thestrong-couplingtheoryascribesthenet effectiveattractionbetweenelectronsinaCooperpairtot hesumoftheattractive electron-phononinteraction(characterizedbythedimens ionlessquantity )and therepulsivescreenedCoulombinteraction(characterize dby ).Thistheorystill inheritstheparadigmofthepairingmechanisminBSCtheory :twoelectronsbind intoCooperpairundertheeffectiveattraction 2 throughthe dynamicscreening mechanism[ 27 ],thatis,twoelectronsatthesamesite“attract”eachothe ratdifferent times.AsageneralizedformincludingtheoriginalBCStheo ryastheweak-coupling limit,todatethestrong-couplingtheory(dynamicscreeni ng)is,ascommentedby Anderson[ 27 ],“oneofthebest-attestedtruthsofquantummaterialsthe ory.”Itreveals theessentialphysicsofall conventional superconductors,andsuccessfullyexplains 2 Theinteractionisrepulsiveforthetimelessthantheorder of ~ = E F andgoes attractivefrom ~ = E F to 1 = n ,where n isatypicalphononfrequency.SeeRef.[ 10 ]. 18

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experimentsonthesematerials.Neverthelessthetheoryba sedontheelectron-phonon mechanismgivesadiscouragingpredictionofthehighest T c thatcouldbeachievedin conventionalsuperconductors:around 40K [ 28 – 30 ].MgB 2 discoveredbythegroupof Akimitsu[ 31 32 ]in2001has T c =39K ,whichisclosesttothepredictedlimitamongall knownconventionalsuperconductors. Backtothehistory:thingsstartedtochangein1972whenthe groupofDavid Lee,DouglasOsheroffandRobertRichardsondiscoveredtha ttheliquid 3 Hebecome superuidat 2.5mK [ 33 ],aphenomenonanalogoustosuperconductivity,thatis,th e absenceofviscosityat T < T c 3 Sincesuperuidityof 3 Hecanbeexplainedwitha p -wavepairingstate,whichfrommanyaspectsisdifferentfr omtheBCSformulation, wecallit unconventional pairing.Afterwards,aturningpointforsuperconductivit y researchcamewhentheheavyfermionsuperconductorclassw asdiscoveredin 1979[ 36 ].ThisclassnowincludesCe-basedcompounds(forexample, CeCu 2 Si 2 CeCoIn 5 ,CePt 3 Si),U-basedcompounds(forexample,UBe 13 ,UPt 3 ,URu 2 Si 2 )andother actinide-basedcompounds(PuCoGa 5 ,PuRhGa 5 andNpPd 5 Al 2 )[ 37 ].Mostofthese materialsexhibitelectronicpropertiesconsistentwithe xtremelylargeeffectivemasses, m = m e 100 – 1000 ,asshownbyexperimentssuchasspecicheatatlowtemperat ure andquantumoscillations.Theexperimentsindicatethehea vyfermionclasshas d p or f -wavepairing.Soonthesecondclassofunconventionalsupe rconductors,organic superconductors,wasdiscoveredin1980.However,arealch allengewasposedby 3 Although 3 Heisaneutralparticlewhiletheelectronisachargedparti cle(hencethe phenomenonintheformeriscalledsuperuidityandthelatt ersuperconductivity),both arefermionsandcanpairinmomentumspace.Thestartlingpa rtaboutsuperuidityin 3 Heisthatthepairingwave-functionis p wave[ 34 ],ananisotropicpairingwave-function, makingittherstunconventionalsuperuidityandtheprot otypefor unconventional superconductivity.Infact12yearsbeforethediscoveryan dacoupleyearsafterBCS theory,theorists,includingLevPitaevskii,EmeryandSes sler,AndersonandMorel, Vdovin,andBalianandWerthamer([ 35 ]andreferencestherein),hadpredictedthe anisotropicpairingstateinliquid 3 He,insteadoftheisotropic s -wavestate. 19

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thesensationaldiscoveryofcupratesuperconductorclass in1986,asmentionedat thebeginningofthischapter.Thephasediagramof T c withrespecttoelectronor holedoping,theantiferromagneticMottinsulatorstateat halflling,theextremely high T c (withcurrentrecordsof 134K atambientand 164K underhighpressurefor cuprates)andmanyotherexperimentalfactsalldefythecon ventionalmechanismfor superconductivity.Finally,thefourthunconventionalsu perconductorclass,iron-based superconductors(FeBS),wasannouncedin2008[ 38 39 ].Thehighest T c nowachieved inFeBSisaround 56K .Unliketheheavyfermionclasswith T c lessthanorequalto 2.3K (CeCoIn 5 ),thecupratesandFeBSbothhaveunusuallyhigh T c andhencetheir unconventionalsuperconductivityisveryeasytorecogniz e;thustheyimmediately attractedworldwideattentionfromsuperconductivitycom munity.Thefurtherdiscussions onthepropertyofFeBSwillbedeferredtonextsection.Neve rthelessitisusefulto showbasicsofthecrystalstructureandthephasediagramof FeBSincomparison withcupratesbeforewediscussthepairingmechanismandth eoreticalmodelfor unconventionalsuperconductivity.AsshowninFig. 1-3 ,theactivelayersofcuprates andFeBSarebothtwo-dimensionalsquarelattices,whichar eresponsibleforthe superconductivityasbelievedbymostresearchers.Theirp hasediagramsarealso similar—bothdisplayingantiferromagneticphasesatzero dopingand“superconducting domes”withholeandelectrondoping.Thedifferencesareas follows.(i)Thecuprates haveapureplanaractivelayerwhiletheFeBShavethepnicto genorchalcogenatoms bucklingaboveandbelowtheFelayer.Therefore,forcuprat esaneffectiveone-band ( d x 2 y 2 bandisdominant)2Dmodelcancrudelydescribetheelectron icstructure,while forFeBSallthree t 2 g bands( d xy d xz d yz )contributestatesneartheFermilevelandin somecases, e g bandshavetobeincludedaswelltoaccountfora3DFermisurf ace. (ii)TheundopedcupratecompoundisaMottinsulatorindica tingstrongelectronic correlation,whiletheundopedFeBScompoundisusuallyame tal,whoseaccurate Fermisurface,bandstructure,andtheone-particleHamilt oniancanbereasonably 20

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Cu O FeAs (a) (b) (c) (d) Doping Doping Figure1-3.(a)Schematicphasediagramsofhole-doped(e.g .,La 2 x Sr x CuO 4 )and electron-doped(e.g.,Nd 2 x Ce x CuO 4 )cuprates.(c)Schematicphase diagramsofhole-doped(e.g.,Ba 1 x K x Fe 2 As 2 )andelectron-doped(e.g., Ba(Fe 1 x Co x ) 2 As 2 )Fe-basedsuperconductors.Bothreprintedbypermission fromMacmillanPublishersLtd: Nature [ 40 ],copyright2010.(b),(d)The activelayerofcupratesandFe-basedsuperconductors(rep rintedwith permissionfrom[ 10 ],copyright2012bytheAmericanPhysicalSociety). wellobtainedfromthedensityfunctionaltheory(DFT)calc ulation.(iii)Thefamous pseudogapphaseisseenincuprates,andmayberelatedtocha rge/spinandother competingordersinthenormalstatewhiletheFeBSdonotevi dentlyshowsuch behavior.(iv)TherearemorediversewaystodopetheFeBSth anthecuprates.Inthe latter,thedopantcaneitherreplacethespacerionsorbead dedasextraout-of-plane oxygen,whileintheFeBS,thedopantcangointoboththespac erionplaneandthe activelayer. Nowwediscusstheprobablepairingmechanismandpairingsy mmetryforthe unconventionalsuperconductivity incupratesandFeBS,sincetheirsuperconductivity can'tbeexplainedbytheconventionalBCStheory,wherethe Cooperpairevadesthe 21

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“instantaneous” 4 Coulombrepulsionthroughtheconventionaldynamic-scree ning mechanism,thatis,thepairedelectronsinteractbyexchan gingthequantaofthe latticevibrations(thephonon)toavoideachotherintime. Tofullytakeadvantageof theeffectiveattraction,thepairingstatetakesasymmetr icform—relative s -wavestate. Ontheotherhand,inunconventionalsuperconductivitythe electronsintheCooper paircanavoideachotherinspace,resultinginananisotrop icpairingstate.Physically thismechanismisunconventionalbutsimplerthanthersto neinthesensethatit doesn'tinvolvethephononandthusispossibletoexplainth ehigh T c thatisrestricted bythelatticeinstabilityinconventionalsuperconductiv ity;however,itisalsocomplicated becauseinthispurelyelectronicmechanismtheelectronsb eingpairedandthebosons beingexchangedarethesameparticles,andthereisnoclear separationoftimescales. Inaddition,theinducedeffectiveattractionisitselfafu nctionofthepairingstate,which makesthefullpairingproblembelow T c extremelydifculttosolve.Consequentiallya correctminimalmodelHamiltonianisessentialtosolvesuc hacomplexproblemasin high T c ofcupratesorFeBS.Andersonwastherstonetoproposethat thedeceptively simple2DsinglebandHubbardmodelcouldaccountforthesup erconductivityof cuprates[ 41 ].ThefamoussinglebandHubbardHamiltonian[ 42 ]reads H = X ij t ij c y i c j + U X i n i n i # (1–3) where t ij aretight-bindingone-electronhoppingparametersbetwee nsites i and j ,which areadjustedtotthebandstructure,and U isanon-siteCoulombinteraction.The HubbardHamiltonian,Eq.( 1–3 ),canproduceavarietyofphasesseenincuprates, despitehavingonlyoneparameter t = U totune.Forexample,when U issmall,Eq.( 1–3 ) describesasimplemetal;when U islarge,wehaveantiferromagneticMottinsulator 4 Thisisalegitimateapproximationincondensedmatterphys ics.Physically,Coulomb interactionisalsoretardedinnatureduetothecausality. 22

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athalflling.However,therearenowell-controlledanaly ticaltechniquestosolvethis modelintheentiredopingrangeinhigherdimensions,altho ughmanysemi-analytical methods,suchasrandomphaseapproximations(RPA),renorm alizedmean-eldtheory, conservinguctuationexchange(FLEX),andslave-bosonap proximations,havebeen applied(seeRef.[ 10 ]andreferencestherein).Inthissituation,thenumerical simulation mightbetheonlyapproachfeasibletojustifywhetherthism odelcharacterizes cupratesandgivesrisetotheunconventionalsuperconduct ivityataproperdoping, butunfortunatelyquantumMonteCarloapproachesarelimit edtohighertemperatures duetothefermionsignproblem[ 43 ].Recently,inthe U = t 0 limit,thissimplemodel wasshowntohaveatransitionto d x 2 y 2 -wavesuperconductingphase[ 44 ]inarigorous weak-couplingapproach. In1994,theextrabrokensymmetryof d x 2 y 2 -wavestate(under 90 rotationabout theprincipalaxisofthelattice)incuprateswasdenitive lyobservedbytheingenious tricrystaltunnelingexperiment[ 45 ]andlaterconrmedinawidedopingrangeincluding electrondoping.However,thepairingsymmetryforFeBSiss tillhighlydebatedsince theinescapablemultibandfeatureblendsmoresubtletiesi ntotheproblemanditis ratherchallengingtogiveauniedpictureforthecontinuo uslyexpandingfamiliesof chemicalcompounds.Sincethesuperconductingphaseisoft enincloseproximitytoan antiferromagneticsemimetalphaseatzerodopinginthepha sediagramsofFeBS,itis intuitivetoproposean s -wavestate[ 46 ]basedonantiferromagneticspin-uctuation theory,wherethegapchangessignbetweentheportionofele ctronandholeFermi sheetsconnectedbythewavevector Q (seeFig. 1-4 ).Theessenceoftheargument forthe s -wavestategivenbythespin-uctuationtheoryisphysical lystraightforward:if therepulsivepairinginteraction V kp = V ( k p ) inthespinsingletchannelduetothe spinuctuationshasastrongmomentumdependence,theusua lBCSgapequation k = P p V ( k p ) p = E p requiresthegapfunction k tochangesignaccordingly;in FeBSwithmultipleFermisurfacesheets(pockets),thepeak ofspinsusceptibilityat Q 23

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dictatesasign-changinggap k+Q = k where Q isthenestingvectorbetweenthe holeandelectronpockets,andfurthermorethegaponeachpo cketisisotropic s -wave formoranisotropicextended s -waveform.(Incuprates,duetotherelativelysimple largeFermisurface,thegapfunctionacquiresastronglyan gulardependenceinthe d -waveform[ 47 ].)Thisantiferromagneticspin-uctuationmediated s -wavepairing isconsistentwithanumberofexperiments;nevertheless,w henorbital-uctuations arestrong,anothertheory[ 48 ]suggestsan s ++ -wavegap,wherethegap(orgap averagedovereachFermipocket)hasthesamesigneverywher e.Furthermore,other symmetries,suchas d waveand s + id wave,arealsoproposedforsomecompounds ofFeBS.Ref.[ 8 ]includesacomprehensivereviewongapsymmetriesinFeBS. Nowit isgenerallybelievedthatinbothcupratesandFeBSthepair ingstateisunconventional, andthescreenedCoulombinteractionthatactuallycausest hepairingisattractiveat relativeseparationsoforderalatticespacingormore[ 10 ]. Tosumup,exceptfortheagreementthatanelectronicmechan ismisthemajor actorinthepairing,wearestillfarfromreachingaconsens usontheunconventional pairingmechanismafternearlytwodecades'researchforcu pratesandahalfdecade forFeBS,becauseformulatingacompletepairingmechanism forunconventional superconductivityisaratherdeepandhardproblem.Onlywh enitissolved,willthe theoristsbeabletopredictnewhigh T c superconductors,willtheexperimentalists designhigh T c superconductorsfroma“blueprint”,andwillthescientist srevolutionize thetechnologywiththe“HolyGrail”ofhigh T c .Completelysolvingthisfundamental problemisbeyondthescopeofthisdissertation.Instead,w ewilldiscussthetheory ofgapsymmetryandgapstructure,ahintforunderstandingt hehigh T c inFeBS.In thefollowingchapters,wephenomenologicallyexplainthe experimentalobservations andtheirinterpretationsonthesymmetryofpairingorderp arameter.Inthenal chapter,weusethemultibandHubbard-HundHamiltoniangen eralizedfromEq.( 1–3 )to microscopicallyinvestigatethepairingsymmetryinFeBSb yspin-uctuationtheory. 24

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* M X Q Y Figure1-4.ThegenericFermisurfaceofFeBSinthetwo-FeBr illouinzone(gray square).Theblacksquareisone-FeBrillouinzone.Twohole pocketsat andtwoelectronpocketat X or Y areshown.The s gapisillustrated: ( k )= 0 for k near and ( k + Q )= 0 for k + Q near X or Y (redcolor forpositivegapandbluecolorfornegativegap).Thegapcha ngessign betweenthepocketsconnectedby Q =( ,0) 1.3TheSymmetryPropertiesofFe-BasedSuperconductors 1.3.1CrystalStructureandCrystalSymmetry Weinvestigatethecrystalstructureandcrystalsymmetryo fFeBSsincethepairing state(seenextsection),thesuperconductinggroundstate ofthesystemHamiltonian, mustbealinearcombinationofthebasisfunctionsofanirre duciblerepresentationof thesymmetrygroupofthesystemHamiltonian.Ifwesimplyta kethesymmetrygroup tobethepointgroupofthecrystal,thepairingstatecanthe nbeclassiedbydifferent irreduciblerepresentationsofthepointgroup. 5 TheFeBSincludedifferentfamilies 5 Infactthesymmetrygroup G ofthesystemHamiltonianconsistsofthepermutation group(theexchangesymmetryoffermions),thespacegroup G ofthecrystal,the spin-rotationsymmetrygroup SU (2) ,thetime-reversalsymmetrygroup K ,andthe gaugesymmetrygroup U (1) [ 49 ].Withoutspin-orbitcoupling,theaboveindividual groupscanbetreatedindependentlyandwewillmostlyconsi derthissituation. 25

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whicharenamedaccordingtothechemicalformulaofthepare ntcompound(i.e.zero doping)as1111,122,111,11forthemajorfamilies,asshown inFig. 1-5 .Thespace groupofasingleFe-As/Selayeris P 4 = nmm ,anonsymmorphicgroup.Nonsymmorphic groupsarenonsimplespacegroupscontainingglideplanesa ndscrewaxes.These aresymmetryoperations(groupelements)combiningareec tionorrotationwitha non-integertranslation.FromFig. 1-5 ,onecanseethattherearetwowaystostack theFe-As/Selayer,i.e.,in-phaseasin1111,111and11anda nti-phaseasin122.The spacegroupofthecrystalconstructedintherstwayisstil l P 4 = nmm whilethespace groupof122issymmorphicspacegroup I 4 = mmm .Botharespacegroupsoftetragonal latticeandthecorrespondingpointgroupcanbederivedfro mdihedralgroup D 2 or D 4 asincuprates.1.3.2GapSymmetryandGapStructure Mostlyweconsiderasingletpairingstate,wherethepairin ggap k haseven parity(underthechange k k )sincethetotalwave-functionisantisymmetricfor fermions.Fortheevenparitypart,thepointgroup D 4 h forthetetragonalsystemhas fourone-dimensionalirreduciblerepresentations, A 1 g A 2 g B 1 g ,and B 2 g ,andone two-dimensionalirreduciblerepresentation E 2 g (here g denotesevenparity).One cansimplyreadacharactertablefortheeigenvaluesofirre duciblerepresentations underdifferentsymmetryoperations.Forsimplicityweonl ylistinTable 1-1 the irreduciblerepresentations,somebasisfunctions,andth enomenclatureofthegap Furthermore,thespacegroup G ofthecrystalcanbefactorizedintothetranslation groupandrotationgroup T n R forsymmorphicspacegroupswhilethisisimpossiblefor nonsymmorphicgroupsandinsteadthequotientgroup G = T isused.Therotationgroup R orthequotientgroup G = T isthegroupofwavevector (0,0,0) ;thegroupsofother wavevectors k aresubgroupsof R forsymmorphicspacegroupsbutspecialtreatment isnecessaryfornonsymmorphicgroups[ 50 51 ].Whenclassifyingthesymmetryofthe pairingstate,werefertothegroupofwavevectorsor,loose lyspeaking,thepointgroup ofthecrystal. 26

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FeSe T c = 8 K High Pressure T c = 37 K LiFeAs NaFeAs T c = 18 K BaFe 2 As 2 SrFe 2 As 2 T c = 26 K LaFeAsO CaFeAsF T c = 56 K Figure1-5.CrystalstructuresofFeBSfor11(FeSe),111(Li FeAs),122(BaFe 2 As 2 ), 1111(LaFeAsO)families(reprintedbypermissionfromMacm illan PublishersLtd: Nat.Phys. [ 5 ],copyright2010).Someofthelabeled transitiontemperaturesareachievedbydopingorpressure functionsanalogoustoeigenfunctionsintheisotropiccas e.Thediscussionhere appliestobothcupratesandFeBS.However,inFeBStheFermi surfaceincludes electronandholepockets,asshowninFig. 1-4 .Theexistenceofseparatepockets givesuniquepropertiestoFeBSundertherestrictionofthe symmetry.Thegap k ondifferentpocketscanhavedifferentsignsandvalues,ma kingFeBSmultigap superconductors.ThezerosofthegapfunctionsontheFermi surface,i.e.,the nodes,affecttheexperimentsmeasuringthelowenergyexci tationsandchangethe exponentialtemperaturedependence e = T topowerlawdependence T .From thesymmetryargument,a d -wavegapmusthavenodesontheholepocketsinthe directiondeterminedbythesymmetry;whilean s -wavegapcaninprinciplealsohave “accidental”nodesatsomedirectionsnotnecessarilyrela tedtothesymmetry. s -and s ++ -wavepairingaresuggestedforthesuperconductivityinFe BSbyantiferromagnetic spin-uctuationtheoryandorbitaluctuationtheory,res pectively.Belongingtothe 27

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Table1-1.Irreduciblerepresentationsandgapfunctionsf orsingletpairinginthe tetragonalsymmetry irreduciblerepresentationbasisfunction k A 1 g 1 cos k x +cos k y s wave A 2 g sin k x sin k y (cos k x cos k y ) B 1 g cos k x cos k y d x 2 y 2 wave B 2 g sin k x sin k y d xy wave E 2 g sin k x sin k z sin k y sin k z same A 1 g irreduciblerepresentation,theyarepredictedbydiffere ntmicroscopic mechanisms,anditiscrucialtodistinguishthemforunders tandingthesuperconductivity ofFeBS.Currently,threeexperimentsofferindirectevide ncesupportingthe s pairing: thenearlyubiquitousobservationofneutronspinresonanc efeaturesininelastic neutronspectroscopy(INS)[ 52 – 57 ],aquasiparticleinterferencescanningtunneling spectroscopy(STS)experimentinamagneticeld[ 58 ],andaphase-sensitive experimentonapolycrystallinesamplewhichreliesonsign icantstatisticalanalysis [ 59 ].Here,Iwillbrieydiscusswhyneutronspinresonanceexp erimentssupport s pairing.Neutronscatteringmeasuresthedynamicalspinsu sceptibility s ( q ) which canbecalculatedwithRPAas s ( q )=[1 U s 0 ( q )] 1 0 ( q ) ,where 0 ( q ) isthebareelectron-holebubble.Inthesuperconductingst ate,duetothepresence ofnitegap, Im 0 ( q ) hasasuddenjumpfromzerotoanitevalueproportionalto thecoherencefactor P k h 1 k k+q E k E k+q i atathresholdfrequency n c .Theleadstoa subgappeakcalledthespinresonancepeakintheRPAdynamic alsusceptibilityatthe antiferromagneticwavevector.Itisonlyvisibleintheneu tronscatteringspectrumwhen thecoherencefactorisnonzero,indicating sgn k 6 =sgn k+q ,where q isthescattering wavevectorwhere s ( q ) ispeaked.Clearly,the s statesatisesthisconditionwhile the s ++ doesn't. 28

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CHAPTER2 INHOMOGENEOUSSUPERCONDUCTIVITY Somepartsofthischapterhavebeenpublishedas“Volovikef fectinahighly anisotropicmultibandsuperconductor:experimentandthe ory,”YanWang,J.-S.Kim, G.Stewart,P.J.Hirschfeld,Y.Matsuda,T.Shibauchi,S.Gr aser,andI.Vekhter,Phys. Rev.B 84 ,184524(2011),and“Theoryofquasiparticlevortexbounds tatesinFe-based superconductors:applicationtoLiFeAs,”YanWang,P.J.Hi rschfeld,andI.Vekhter, Phys.Rev.B 85 ,020506(2012). Inhomogeneoussuperconductivityisaquitebroadtopicsin cetheinhomogeneity inthesuperconductingstatearisesfromdifferentaspects andvariousscenarios suchasappliedmagneticeld,lowdimensionalityorsizeef fectandtheexistenceof surfaces,interfacesordefects.Accordingly,thehomogen eoussuperconductorcan undergoatransitiontothemixed(orintermediate)statein TypeIsuperconductors,to thevortexstateinTypeIIsuperconductors,ortotheFuldeFerrell-Larkin-Ovchinnikov (FFLO)statein,forinstance,heavy-fermionandorganicsu perconductors.Theorder parameter ( x ) oftheseinhomogeneouspairingstateshasvariationsinrea lspace x Theconsequencesandbehaviorsofpairbreakingduetothema gneticeldinthese superconductorsaremorecomplicatedthanthatduetodisor der:Theformerresults innewnontrivialpairingstateswithsuppressionofthemag nitudeoforderparameter inthecoherencelengthscale orsignchangingorderparameterinthescaleof 1 = k F whilethelatter,thepairbreakingduetodisorder(thefocu sofnextchapter),suppresses theorderparameterinrealspaceuniformlyinthesenseofdi sorderaverage.(However, withexperimentaltechniquesinatomicorsub-atomicscale suchasscanningtunneling spectroscopy,theinhomogeneoussuperconductivitydueto disorderonshortdistance scalescanbemappedoutandusedasfurtherprobesforthepai ringinteractionand pairingsymmetry[ 60 ]).Indealingwithsuchanextensivetopic,wewillconcentr ateon thesuperconductivityinthevortexstateinthischapter.F irst,wepresentmeasurements 29

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ofthespecicheatcoefcient r C = T inthelowtemperaturelimitasafunction ofanappliedmagneticeldfortheFe-basedsuperconductor BaFe 2 (As 0.7 P 0.3 ) 2 by G.R.Stewart'sgroup,wherebothalinearregimeathighere ldsandalimitingsquare root H behavioratveryloweldsarefound.ThecrossoverfromaVol ovik-like p H toalinearelddependencecanbeunderstoodfromamultiban dcalculationinthe quasiclassicalapproximationassuminggapswithdifferen tmomentumdependenceon thehole-andelectron-likeFermisurfacesheets.Next,wec onsiderthevortexbound statesandshowthatthehighintensitytailsinthescanning tunnelingspectroscopy (STS)ontheselow-energystatesmayindicateeitherthegap anisotropyortheFermi surfaceanisotropyinthemomentumspace.IftheFermisurfa ceanisotropydominates, preventingdirectobservationofsuperconductinggapfeat ures,onemustbecautious toanalyzetheSTSdataonFe-basedsuperconductorsinthevo rtexstate,inparticular LiFeAs,whichwetreatexplicitly. 2.1QuasiclassicalApproximation Thequasiclassical(Eilenberger)approximation[ 61 – 63 ]isapowerfultoolto describetheelectronicpropertiesofthesuperconducting stateonlargescales comparedtothelatticespacing,providedthequasiclassic alcondition k F 1 is satised.Here k F istheFermimomentumand thecoherencelength.Sinceinthislimit wecanthinkofquasiparticlesaspropagatingcoherentlyal ongawell-denedtrajectory inrealspace,thismethodisparticularlywellsuitedtoadd resstheinhomogeneous situations,suchasthevortexstateoftype-IIsuperconduc tors(SCs).Analternativeand frequentlyusedapproachtothevortexstateistotakeintoa ccountthe(classical)shift ofthequasiparticleenergyduetothelocalsupercurrento w.Suchanapproximation, oftenreferredtoastheDoppler-shiftapproach,isvalidfo rnodalSCswithconsiderable weightofextendedquasiparticleexcitationsoutsidethev ortexcores.Usingthis method,Volovikshowedthatforsuperconductorswithlinen odestheseextended quasiparticleexcitationsleadtoanon-linearmagnetice lddependenceofthespatially 30

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averagedresidualdensityofstates N ( =0, H ) / N 0 p H = H c 2 ,aresultknownasthe Volovikeffect [ 64 ].Thisbehaviorwasrstconrmedbymeasurementsofthespe cic heat[ 65 66 ]andbysubsequentcalculationswithinthequasiclassical approximation forbothasinglevortexina d -waveSC[ 67 68 ]andforavortexlattice[ 69 70 ].Both quasiclassicalandDoppler-shiftmethodsfailatthelowes ttemperaturesduetoquantum effects[ 71 ],butinknownsystemswith T c E F theseeffectsarenegligibleinpractice. Bothmethodshavesuccessfullyexplainedatasemiquantita tivelevelthemagnetic elddependenceofthespecicheatandthermalconductivit yinawidevarietyof unconventionalsuperconductors[ 72 ].Itwasalsoshownthattheaccuratelycalculated quasiparticleexcitationspectrumisconsistentwithSTMs tudiesoftheelectronic structurearoundavortexcore[ 69 ]. Manyexperimentaltechniqueswhicharesensitivetothelow -energydensityof states,suchasthermalconductivity,specicheat,andNMR relaxationrate,canbe usedtodrawconclusionsaboutthepossibleexistenceandth emomentumdependence ofquasiparticleexcitationsinthebulkofiron-basedsupe rconductorsandthusabout thestructureofthesuperconductinggapandthedistributi onofgapnodes.Thelow T limitoftheSommerfeldcoefcientinanappliedmagnetice ld, r ( H ) ,isdirectly proportionaltothespatiallyaveragedlocaldensityofsta tes(LDOS)attheFermilevel. TheDoppler-shiftmethodwasusedtocalculatetheLDOSfora two-bandSCwithtwo isotropicgapsofunequalsize S 6 = L andtogiveaninterpretationoftheexperimental dataavailableatthattime[ 73 ].However,theDoppler-shiftapproachcannotaccount properlyforthecontributionsfromthestatesinthevortex corethathaveaverylarge weightinthenetDOSandhencegivesaquantitativelyandsom etimesqualitatively inaccuratedescriptionoftheelectronicstructureofthev ortex.Forexample,inasimple d -wavesuperconductorthespatialtailsofthelow-energyde nsityofstatesaround thevortexarealignedinthewrongdirections[ 74 ].Toobtainaquantitativettothe specicheatdatabyG.R.Stewart'sgroupandtoallowforamo redecisiveconclusion 31

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aboutthegapstructureofBaFe 2 (As 0.7 P 0.3 ) 2 ,wewillthereforeusethequasiclassical approximation,whichwewillbrieyreviewinthefollowing paragraphs. Inthequasiclassicalmethod,theGor'kovGreen'sfunction sareintegratedwith respecttothequasiparticleenergymeasuredfromtheFermi level.Thenormaland anomalouscomponents g ( r , i n ) and f ( r , i n ) oftheresultingpropagator ^ g obeythe coupledEilenbergerequations h 2 i n + e c v F A ( r ) + i ~ v F r i f ( r , i n )=2 ig ( r , i n )( r ), h 2 i n + e c v F A ( r ) i ~ v F r i f ( r , i n )=2 ig ( r , i n ) ( r ), (2–1) thathavetobecomplementedbythenormalizationcondition ^ g 2 = ^1 ,where ^ g 0B@ gf f g 1CA (2–2) Here ( r ) istheorderparameter, A ( r ) thevectorpotential, v F istheFermivelocity atthelocationontheFermisurfacelabeledby ,and n =(2 n +1) k B T arethe fermionicMatsubarafrequencies.Fortwo-dimensionalcyl indricalFermisurfacessuch asconsideredbelow, v F = v F ^k where ^k =(cos ,sin ) and istheanglemeasured fromthe[100]direction.Inthatcaseitisnaturaltowritet hepositionvectorincylindrical coordinates, r =( , z ) ,where isthewindinganglearoundthevortexinrealspace. Makinguseofthesymmetries[ 75 ]ofthequasiclassicalpropagator f ( r k F i n )= f ( r k F i n ), (2–3) f ( r k F i n )= f ( r k F i n ), (2–4) g ( r k F i n )= g ( r k F i n ), (2–5) thediagonalpartofthenormalizationconditioncanbewrit teninamoreexplicitform as [ g ( r , i n )] 2 + f ( r , i n ) f ( r + i n )=1 .Notethatournotationof g f and f differsfromtheoneusedinRef.[ 75 ].Underthetransformation g i g 32

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f f ,and f f thenotationinRef.[ 75 ]passesintoournotation.Instead ofsolvingthecomplicatedcoupledEilenbergerequationse verywhereinspace,we followRefs.[ 68 75 ]andparameterizethequasiclassicalpropagatoralongrea lspace trajectories r ( x )= r 0 + x ^ v F byasetofscalaramplitudes a ( x ) and b ( x ) ^ g ( r ( x ))= 1 1+ a ( x ) b ( x ) 0B@ 1 a ( x ) b ( x )2 a ( x ) 2 b ( x ) 1+ a ( x ) b ( x ) 1CA (2–6) TheseamplitudesobeynumericallystableRiccatiequation swhichfollowfrom substitutionintoEq.( 2–1 ), v F @ x a ( x )+[2~ n + ( x ) a ( x )] a ( x ) ( x )=0, v F @ x b ( x ) [2~ n +( x ) b ( x )] b ( x )+ ( x )=0. (2–7) Forthesinglevortexproblemthespatialdependencevanish esfarawayfromthevortex core,andhencewehavetheinitialconditions a ( 1 )= ( 1 ) n + p 2 n + j ( 1 ) j 2 b (+ 1 )= (+ 1 ) n + p 2 n + j (+ 1 ) j 2 (2–8) Herewehaveset ~ =1 andwehaveintroducedthemodiedMatsubarafrequencies i ~ n ( x )= i n +( e = c ) v F A ( x ) .SincethemodicationoftheMatsubarafrequencies duetotheexternaleldisoftheorderof 1 = 2 where = L = istheratiooftheLondon penetrationdepthandthecoherencelength,thetermpropor tionalto A ( x ) inEq.( 2–7 ) canbeneglectedforstrongtype-IIsuperconductorswithla rge AfterananalyticcontinuationoftheMatsubarafrequencie stotherealaxis, i n ! + i ,thelocaldensityofstatescanbecalculatedastheFermisu rfaceaverageofthe quasiclassicalpropagator N ( r )= N 0 Z 2 0 d 2 Re 1 ab 1+ ab i n ! + i (2–9) 33

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where N 0 isthenormaldensityofstatesattheFermienergy.Toobtain stablenumerical solutionsweuseasmallimaginarypart =0.02 T c intheanalyticalcontinuation,where T c isthecriticaltemperatureofthesuperconductor. 2.2VolovikEffectinMultibandSuperconductorBaFe 2 (As 0.7 P 0.3 ) 2 2.2.1Motivation Thesymmetryanddetailedstructureofthegapfunctioninth erecentlydiscovered ironpnictide[ 39 ]andchalcogenide[ 76 ]hightemperaturesuperconductorsisstill underdiscussion.Acrossanincreasinglynumeroussetofma terialsfamilies,aswell aswithineachfamilywheresuperconductivitycanbetunedb ydopingorpressure, experimentalindicationsarethatthereisnouniversalgap structure[ 6 7 ].Instead, thesuperconductinggapappearstoberemarkablysensitive todetailsofthenormal stateproperties.This“intrinsicsensitivity”[ 77 ]maybeduetotheunusualFermi surfacetopology,consistingofsmallholeandelectronpoc kets,andtotheprobable A 1 g symmetryofthesuperconductinggapwhichallowsacontinuo usdeformationofthe orderparameterstructurefromafullygappedsystemtoonew ithnodes(forareview see,e.g.Ref.[ 8 ]).Itisimportanttokeepinmind,though,thatanotherposs ibilityto accountfortheobservedvariabilityisthatdifferentexpe rimentsonthesamematerial mayprobeselectivelydifferentFermisurfaceregionsandh encedifferentgapswithinthe system. TheBa-122familyofmaterialshasbeenintensivelystudied becauselarge highqualitysinglecrystalsarerelativelyeasytoproduce [ 6 78 ].Withinthisfamily, theisovalentlysubstitutedsystemBaFe 2 (As 1 x P x ) 2 withamaximum T c of 31K is particularlyintriguingbecauseitexhibitsaphasediagra mandtransportproperties remarkablysimilartotheheterovalentlydopedsystemBa(F e 1 x Co x ) 2 As 2 anddisplays manysignaturesofapparentquantumcriticalbehavioratop timaldoping[ 78 – 80 ].In thesuperconductingstate,penetrationdepth[ 81 ],NMRspin-latticerelaxation[ 82 ], thermalconductivitytemperaturedependence[ 81 ],andthermalconductivityangular 34

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eldvariation[ 83 ]showclearindicationsofnodalbehavior.Surprisingly,a lineareld dependenceofthespecicheatSommerfeldcoefcient r wasmeasured[ 84 ]on optimallydopedsamplesfromthesamebatch.Suchabehavior isexpectedforafully gappedsinglebandsuperconductorsincethefermionicexci tationsfromthenormal coresofvorticesprovidetheonlycontributionto r atlow T ,andthenumberofthese vorticesscaleslinearlywiththeeld H .ItwasarguedinRef.[ 84 ]thatthespecicheat measurementmightbeconsistentwiththeotherexperiments suggestingnodesif theheavyholesheetsinthematerialwerefullygapped,whil ethegapsonthelighter electronsheetswerenodal.Insuchacasethe r p H behaviorwouldbedifcultto observeinexperiment. InRef.[ 84 ],theG.R.Stewartgroupreportedexperimentaldataonthem agnetic elddependenceofthespecicheatofoptimallydopedBaFe 2 (As 1 x P x ) 2 samples, upto 15T .Moreprecisemeasurements[ 85 ]atloweldsrevealedthepresenceofa Volovik-like p H termwhichpersistsroughlyoverarangeof 4T ,crossingovertoalinear behaviorabovethisscale. 1 Theobservationofthisterm,consistentwithnodesinthe superconductinggap,thereforesupportedclaimsmadeinea rlierwork[ 81 – 83 ],without theneedtoassumeanextremelylargemassontheholepockets TheoreticalestimatesusingtheDopplershiftmethodforis otropicgapsgiven inRef.[ 73 ]wereoversimplied,butdidshowtheneedforamorethoroug hanalysis ofanisotropicmultibandsystems.Thetheoreticaldifcul tiescanbeseeneasilyby consideringasimpletwo-bandmodelwithtwodistinctgaps 1 and 2 ,wherewe assumeforthemomentthat 2 > 1 .Ifthetwobandsareuncoupled,thetwogaps correspondtotwoindependentcoherencelengths i v F i = ( i ) ,where i =1,2 ,and 1 IncontrasttoBaFe 2 (As 1 x P x ) 2 ,recenthigheldmeasurementsonunderdoped( x = 0.045 )andoverdoped( x =0.103 )Ba(Fe 1 x Co x ) 2 As 2 havefoundthatthespecicheat coefcientvariesapproximatelyas H 0.7 allthewayupto H c 2 (0) [ 86 ]. 35

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twoindependent“uppercriticalelds” H c 2, i .Vortexcorestatesofthelargegap 2 are connedtocoresofradius 2 .Foreldsintherange H c 2,1 H H c 2,2 ,thevortex coresofthesmallgapwilloverlap,whilethelargegapcores willstillbewellseparated. Notethatif 1 isverysmall(theseconsiderationsalsocrudelydescriben odalgaps),this eldrangecanbewideandextendtoquitelowelds.Ontheoth erhand,methodsof studyingquasiparticlepropertiesinsuperconductorsare typicallyadaptedtocalculating near H c 1 or H c 2 ,i.e.inthelimitofwidelyseparatedornearlyoverlapping vortices.The currentproblemapparentlycontainselementsofbothsitua tions.Intheabsenceof interbandcoupling,ofcourse,onecanusedifferentmethod s,correspondingtothe appropriateeldregimes,forthedistinctbands.Forcoupl edFermisurfaces,however, suchanapproachisnotviable.Intheimmediatevicinityoft hetransition,wherethe Ginzburg-Landauexpansionisvalid,thereisasinglelengt hscalecontrollingthevortex structure[ 87 ].Atlowtemperatures,wherethemeasurementsarecarriedo ut,however, thedistinctlengthscaleslikelysurvive,althoughtheyar emodiedbythestrength oftheinterbandcoupling,seebelow.Possibleanisotropyo fthegapononeormore Fermisurfacesheetscomplicatesthepictureevenfurther. WeshowedinRef.[ 85 ]that judicioususeofthequasiclassicalapproximationevenwit hsimplifyingassumptions aboutthevortexstructurecanprovideageneralframeworkf orthedescriptionofthis problem,andasemiquantitativeunderstandingofthenewda taontheBaFe 2 (As 1 x P x ) 2 system. Inthefollowing,werstpresentexperimentalresultsonth eBaFe 2 (As 1 x P x ) 2 systeminSec. 2.2.2 obtainedbytheG.R.Stewartgroup.InSec. 2.2.3 wediscussthe two-bandquasiclassicalmodelweusetostudythesystem,an dinSec. 2.2.4 wegive ourtheoreticresults.FinallyinSec. 2.2.5 wepresentourconclusions. 2.2.2ExperimentResults Preparationofthecrystalsandexperimentalsetupforspec icheatmeasurements wasdescribedinRef.[ 85 ].Thespecicheatcoefcient r C = T ofBaFe 2 (As 0.7 P 0.3 ) 2 36

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0 5 10 15 20 25 30 35 40 45 0 3 6 9 12 15 18 21 black lines: linear fit for 0 T < H < 35 T BaFe2(As0.7P0.3)2 2 K 1.5 K from C/T = + T2 + T4 for 1.5 K < T < 5 K 2 K high fields from C/T = + T2 + T4 for 2 K < T < 7 K, high fields (T = 0 K, 1.5 K, 2 K)H (T) mJ/(molK 2 ) Figure2-1.Theoriginalspecicheatdata[ 84 ]onBaFe 2 (As 0.7 P 0.3 ) 2 asafunctionofeld upto 15T (solidsymbols)withdatabetween 15T and 35T (opensymbols). Notetheagreementbetweenthelinear, C = T / H ,extrapolationofthe 15T (coloredlines[ 84 ])and 35T (blacklines,presentwork)results.Weextract r fromthedatausingtwo(equivalent)methods:(a)bymakinga n extrapolation C = T = r + T 2 + T 4 from 2K andabove,or(b)bytakingthe smoothedvalueof C = T at 1.5 and 2K foundbyttingapproximately10data pointsaroundthesetemperaturestoobtain C = T ( 1.5K )and C = T ( 2K )with decreasedscatter.Thetemperaturerestrictioneliminate sboththeinuence oftheanomalyandtheeld-inducednuclearcontribution,n egligiblefor H 4T above 1K .Theabsoluteaccuracyofthesedatais 5% (errorbars arenotshownatloweldssincetheyareapproximatelythesa mesizeasa datapoint)whiletheprecisionofthedataisapproximately 2% .Inaddition, additionaldatawithnergradationsinthemeasuredeldsu pto 4T were takentoexploretheloweldnon-linearbehavior.Thesedat aareshownon anexpandedscaleinFig. 2-2 for 0 H 35T isshownbytheopentrianglesinFig. 2-1 .Thereisasmalllow temperatureanomalyinthespecicheatdatabelowabout 1.4K (discussedindetail inRef.[ 84 ]).SuchanomalieshavebeenobservedinotherFe/Pnsamples [ 88 ],and insomecases,e.g.,inBa(Fe 1 x Co x ) 2 As 2 ,theyshowaratherstrongmagneticeld dependence[ 88 ].However,asdiscussedinRef.[ 84 ]forthedataupto 15T ,the anomalyinBaFe 2 (As 0.7 P 0.3 ) 2 isapproximatelyeldindependent.Notethatthesmall 37

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0 1 2 3 4 2 4 6 8 C/T = 1.58 + 0.86 H0.53C/T = 4.21 + 0.73 H0.59C/T = 6.07 + 0.71 H0.66BaFe2(As0.7P0.3)2 2 K 1.5 K C/T = + T2 + T4 for 1.5 K < T < 5 K (T = 0 K, 1.5 K, 2 K)H (T) mJ/(molK 2 ) Figure2-2.Loweld r dataupto 4T fromFig. 2-1 onanexpandedscalefor T =2K (blue), 1.75K (red)and 1.5K (blacksymbols).Greensymbolsareasymptotic lim T 0 C = T determinedovertherange 1.5K < T < 5K .Thettingfunctions ofthedataarelabeledbesidethecurves.Bestpowerlawtst oeld dependenceareshownineachcase. anomalyinthespecicheatappearstovanishabove 1.4K ,i.e.,doesnotaffectthe estimatefor r showninFigs. 2-1 and 2-2 usingdatafrom 1.5K andabove. Inordertohaveacloserlookatthelowelddependenceofthe specicheat,these dataareshownonanexpandedscaleinFig. 2-2 .Inouranalysisbelow,wefocuson theasymptotic T 0 behaviorsinceitisdirectlyrelatedtothedensityofstate satthe Fermilevel,whichiseasytocalculatereliably,andsincei tgivesessentiallythesame elddependenceasthenonzero T data. 2.2.3Two-BandModel TheFermisurfaceoftheoptimallydopedBaFe 2 (As 0.7 P 0.3 ) 2 consistsofmultiple Fermisurfacesheets.DFTcalculationsshowedthattherear ethreeconcentrichole cylindersinthecenteroftheBrillouinzone( point)andtwoelectronpocketsatthe zonecorner( X point)[ 89 ].LaserARPESmeasurements[ 90 ]foundasuperconducting orderparameterthatisfullygappedwithcomparablysizedg apsoneachhole pocketoftheorderof h = k B T c 1.7 .Takingintoaccounttheresultsfromthermal 38

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conductivity[ 81 83 ]andNMRmeasurements[ 82 ]aswellasthemeasurementsof thespecicheatcoefcientinloweldspresentedabove,th atallconsistentlyreport evidenceforlow-energyquasiparticles,thisARPESresult isconsistentwithanodalgap ontheelectronpockets. Fornumericalconvenience,weadoptbelowatwo-bandmodel, distinguishingonly betweenelectronandholepockets.InclusionofallFermisu rfacesheetsthenonly entersasaweightingfactorfortheelectronandholepocket contributions,aswediscuss inthefollowingsection.Wetakethegapsontheelectronand holepocketsintheform 1,2 ( )= e h 0 1,2 ( ) ,wheretheangle parameterizestheappropriateFermisurface, assumedtobecylindrical.Weassumeananisotropicgaponth eelectronpocket[ 91 ] 1 ( )=(1+ r cos2 ) = p 1+ r 2 = 2 ,andanisotropicgaparoundtheholeFermisurface, 2 ( )=1 .Iftheanisotropyfactor r > 1 ,thesuperconductinggapintheelectronband, 1 ( ) ,hasaccidentalnodes;if r =0 1 ( ) isisotropiclike 2 ( ) Firstweassume e0 = h0 ,asisoftenfoundbyARPES.Sinceweconsiderwell separatedelectronandholebands,wecansolvetheRiccatie quations,Eqs.( 2–7 ), forthetwopropagatorsseparately,andtheonlycouplingof thepocketsisviathe self-consistencyequationsontheorderparameter(seebel ow).Withthisinmindwe normalizetheenergyandlengthfortheelectronandholeban dsbythegapamplitudes e0 and h0 ,andthecoherencelengths e 0 = v e F = e0 and h 0 = v h F = h0 respectively. Fermivelocitiesthereforeappearasaninput.DFTcalculat ionsforacomparable Ba-122system[ 92 ]give v h F =1.979 10 5 m = s and v e F =3.023 10 5 m = s ,i.e., v h F = v e F = h 0 = e 0 =0.65 .Inouranalysiswekeepthisratiobutreducethevalueofbot h Fermivelocitiesbyafactorof5toapproximatelyaccountfo rthemassrenormalization ofthissystemnearoptimaldoping[ 80 93 ].Thisreductionalsogivesaroughlycorrect valueofthe c -axisuppercriticaleld H c 2 50T .Inthelimitofnegligiblecoupling betweenthebands,theuppercriticaleld H c 2 isdeterminedbytheoverlapofthe 39

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0.0 0.1 0.2 0.3 0.4 0.0 0.2 0.4 0.6 0.8 1.0 space average N ( = 0, H )/ Nno rmal 0H / Hc 2 clea n s -wave clea n d -wave ta nh ( / ) ta nh ( / ) ( ) ta nh ( / ) ta nh ( / ) ( ) ( ) = ( 1+ r c o s 2 )/ ( 1+ r2/2) r = 1.3 Figure2-3.Thespatiallyaveragedzeroenergydensityofst ates N ( =0, H ) normalizedtothenormalstatevalue N ( =0) = N 0 foranodeless(orange) andanodal(blue)single-bandsuperconductor.Thedashedl inesshowthe idealizedlinear H and p H behaviorforaclean s -waveand d -waveSC, respectively.Thesymbolsarenumericalresultsforasingl ebandSCwithan isotropic s -wavegap(circles)andastronglyanisotropicnodalgap (triangles).Additionallywecompareresultswith(solids ymbols)andwithout (opensymbols)takingintoaccountthevortexcorereductio nduetothe Kramer-Pescheffect.Herewehaveignoredtheelddependen ceofthe superconductinggap,i.e., ( H )= 0 vorticeswithsmallestcoresize, R min f e 0 h 0 g = R h 0 = r H c 2 H (2–10) where R isthesinglevortexradiusundermagneticeld H .Belowwesolvethe Eilenbergerequationsanddeterminethedensityofstatesf oranisolatedvortexandfor eachbandseparately.Inatwo-bandsystemthespatialprol eofthequasiparticlestates ontheelectronandholebandsiscontrolledbytherespectiv ecoherencelengths,and thereforespatialaveragingweighsthecontributionsofth ebandsdifferentlycompared totheDOSofasystemwithasingleortwoequalcoherenceleng ths.Thisisthemost signicantdifferencecomparedtoasingle-bandmodel. 40

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Thesuperconductingorderparametersinthetwobandsarere latedbythe interbandcomponentofthepairinginteraction.Weconside rageneralcouplingmatrix inthefactorizedform, ( 0 )= ( ) ( 0 ) ,where =1,2 and V N Here V 11 = V e and V 22 = V h aretheintrabandpairinginteractionsintheelectronand theholeband,respectively,while V 12 = V eh istheinterbandinteraction. N isthenormal densityofstatesattheFermilevel.Thenthegapequationfo raninhomogeneous superconductoris ( r )=2 T X =1,2 c X n > 0 h ( ) f ( r , i n ) i (2–11) Here ( r ) isthemomentumindependentpartofthegapfunction; 1,2 = e h 0 at T =0 and H =0 Inthevortexstatetheself-consistentdeterminationofth espatiallydependentorder parameterisacomplextask.Sinceweareinterestedinrelat ivelylowelds,whenthe vorticesarewellseparated,wesolvetheEilenbergerequat ionsfortheorderparameter thatisassumedtohaveasinglevortexform, e ( ~ H ; )= 1 ( H )tanh 0.1 e 0 1+ r cos2 p 1+ r 2 = 2 h ( ~ H )= 2 ( H )tanh 0.1 h 0 (2–12) Here ~ =( ) isthetwo-dimensionalprojectionoftheradiusvectorincy lindrical coordinates,andafactorof 0.1 isintroducedtoapproximatetheshrinkingofthecore sizeintheself-consistenttreatmentatlowtemperatures( Kramer-Pescheffect[ 94 95 ]).Thissinglevortexansatzprovidesaqualitativelycorr ectdescriptionofthe low-eldstate,closetowhatisfoundbyfullnumericalsolu tion[ 74 ].Toaccountfor thesuppressionofthebulkorderparameterbythemagnetic eld,wedeterminethe coefcients 1,2 ( H ) fromtheBrandt-Pesch-Tewordtapproximation[ 96 97 ],whereinthe presenceofanAbrikosovlatticethediagonalcomponentsof theGreen'sfunctionare 41

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replacedbytheiraveragesoveravortexunitcellofthevort exlattice.Thisapproximation hasbeenproventogivereliableresultsoveraconsiderable rangeofmagneticeldsand isincorporatedintoourapproach. Notethatouransatzfortheorderparameterbecomesquantit ativelyinaccuratefor stronginterbandcouplingintheregimeofapplicabilityof theGinzburg-Landautheory sincethecoresizesofthetwobandsapproacheachother[ 98 ].Weveriedinafully self-consistentcalculationthatintheparameterrangeth atweuse,thecorresponding effectonthespecicheatisoforder 1% orlessandhencecanbeneglected.We thereforeuseEq.( 2–12 )hereafter. Toproceed,wesubstituteEq.( 2–12 )intotheRiccatiEq.( 2–7 ),solvefor a ( x ) and b ( x ) ,anduseEq.( 2–9 )tondthelocaldensityofstates N ( ~ H ) .Toapproximatethe specicheatcoefcient,weevaluatethespatialaverageof thezeroenergylocaldensity ofstates N ( H )= Z 2 0 d Z R 0 d N ( ~ H ) R 2 N 0 (2–13) wheretheintervortexdistance R dependson H asdescribedbyEq.( 2–10 ).Thetotal densityofstatesisthengivenas N ( H ) tot = w e N e ( H )+ w h N h ( H ) w e + w h (2–14) where w e = w h =2 N e 0 = N h 0 =2 ifweconsider,forexample,twoelectronFermisurface sheetsinthefoldedBrillouinzoneanddenote N e 0 = N h 0 = v h F = v e F =0.65= 21 = 12 .The specicheatSommerfeldcoefcient r ( H ) inthesuperconductingstateisnowobtained as r ( H ) r 0 r n r 0 = N ( H ) tot ,where r n and r 0 aretwoconstantsfromtheexperiment.Sincethe vortexdensityiscontrolledbytheexternaleld,integrat ionuptotheintervortexspacing R p 0 = H correctlyaccountsfortheeldeffectwithinthesinglevor texapproximation. Theintegrationthusincludesnotonlythecontributionofe xtendedquasiparticlestatesto thespecicheat,butalsothelocalizedquasiparticlesint hecore. 42

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0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.4 0.8 1.2 1.6 ( 1) h ( 2) h ( 3) h ( 4) h( 1) ( 2) ( 3) ( 4) e( ) =( + r c o s 2 )/ ( 1+ r2/ ) r = 0.9 0.9 1.3 1.3 e ,h / T cH / Hc 2 ( 1) e ( 2) e ( 3) e ( 4) e(a ) 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 spatially averaged Ne( = 0, H )/ Ne0(b ) H / Hc 2 ( 1) e ( 2) e ( 3) e ( 4) e ( 1) ( 2) ( 3) ( 4) e( ) =( + r c o s )/ ( + r2/ ) r = 0.9 0.9 1.3 1.3 0.00 0.05 0.10 0.15 0.20 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 (c ) ( 1) h ( 2) h ( 3) h ( 4) hspatially averaged N h ( = 0, H )/ N h0H / Hc 2 Figure2-4.Resultsofquasiclassicalcalculationsforthe parametersinTable 2-1 .(a) Magneticelddependenceofthegapsinthetwo-bandmodelca lculated withinthePeschapproximation(Refs.[ 96 97 99 ])forCase1–4.We assume e ( H =0)= h ( H =0) here.Thefoursetsofcouplingconstants ij arelistedinTable 2-1 .(b)FielddependenceofthespaceaverageZDOS N e ( H ) ontheelectronpocketforthefourcaseswithanisotropicga pwith angularvariation e ( )=(1+ r cos2 ) = p 1+ r 2 = 2 .(c)Fielddependenceof thespaceaverageZDOS N h ( H ) forthefourcaseswithisotropicgapalong theholepocket. 2.2.4Results ToillustratethatthesalientfeaturesofthevortexstateD OSarecapturedinour approach,inFig. 2-3 weshowtheelddependenceofthespatiallyaveragedzero energylocaldensityofstates(ZDOS)foraone-bandSCwithe itheranisotropic s -wave gaporastronglyanisotropicnodalgap( r =1.3 ).Notethat,whiletheelddependences inboththenodalandfullygappedcasesclearlyttheantici patedpowerlawsatlow elds, p H and H ,respectively,thereisasignicantinuenceonthemagnit udeofthe DOScausedbythesizeofthecore,withthesmallercoresizey ieldingsmallerZDOS. Inparticular,intheabsenceoftheKramer-Pescheffect,fo rthenodalcasetheZDOS wouldexceedthenormalstatevalueateldsfarbelow H c 2 ,whichisunphysical. Belowweconsider r =0.9 and r =1.3 tomimicagapwithdeepminimaand accidentalnodes,respectively.Toshowdifferenttypesof behaviorallowedwithinour microscopicmodelwechosefoursetsofcouplingconstants, twoforeachvalueof r ,as showninTable 2-1 .InCases1and3,theinterbandpairing 12 isstrongandcloseto theintrabandparameter 11 ,whileinCases2and4, 12 11 22 43

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Table2-1.Thedifferentmodelsforthecouplingmatrixandt hegapanisotropyonthe electronpocketsconsideredinthiswork. 11 12 21 22 rT c = K H c 2 = T case10.510.510.330.650.93154case21.000.020.0130.810.93147case30.510.510.340.641.33154case41.000.0230.0150.771.33142 InFig. 2-4 (a)weshowtheself-consistentlydeterminedmagnitudesof thebulkgaps inthevortexstate 1,2 ( H ) asdenedinEq.( 2–11 )and( 2–12 ). H c 2 40 – 50T .Inthe caseswithonlyweakinterbandpairing,Cases2and4,thegap ontheelectronFermi surfacedeviatesconsiderablyfromthephenomenologicalf orm ( H )= 0 p 1 H = H c 2 Figs. 2-4 (b)and(c)showthespatiallyaveragedZDOScorrespondingt oeachband. For N e ( H ) andfor r =1.3 the p H behavioroftheVolovikeffectisclearlyvisibleat lowereldsupto H = H c 2 =0.2 .ComparingFig. 2-4 (b)toFig. 2-3 wendthatwithinthe two-bandmodelthedensityofstatesoftheelectronband N e ( H ) reachesaquasi-linear behavioralreadyatsmallereldsthanthecorrespondingde nsityofstatesforthe one-bandcase.InFig. 2-3 alinearbehaviorisneverobserved,andmightonlybet oversomeintermediateeldrangefor H = H c 2 > 0.2 ,whileinthemultibandcase N e ( H ) displaysaclearlinearbehavioralreadyfor H = H c 2 > 0.1 Itistemptingtointerpretthelow-eldcrossovertoaquasi lineareldvariationas evidenceforasmallenergyscale sm e0 (1 r ) = p 1+ r 2 = 2 ontheelectronband;this, however,seemsunlikely.Provided sm e0 ,thegapstillincreaseslinearlyalongthe Fermisurfaceawayfromthenodalpointsabovethisenergysc ale,simplywithadifferent slope.ThenwithintheusualVolovikargumentationthecont ributionsfromextended statesattheseintermediateenergiesgiverisetoa p H contributionevenif sm E H max ,where E H / p H istheaverageDopplershiftand max e0 (1+ r ) = p 1+ r 2 = 2 is themaximumgap.ThereisthereforenotruelinearH behaviorarisingfromtheelectron bandwithgapnodes.Consequently,weinterpretthiscrosso verastheconsequenceof 44

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0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 clean s -wave clean d -wave ( ( H ) 0 )/ ( n 0 )H / Hc2 Experiment (Q a ) (Q b) ( Q c) Figure2-5.Comparisonoftheexperimentallymeasurednorm alizedspecicheat coefcient(largepinkdots,adaptedfromFig. 2-1 )todifferenttheoretical resultsforthespatiallyaveragedZDOS.Thedottedvioleta ndsolidorange curvesarethepredictionsforthespatiallyaveragedZDOSf oraclean s -waveand d -waveSC.Thebluesquares(CaseQa)andgreendiamonds (CaseQb)arethedifferentlyweightedsumsof N e ( H ) and N h ( H ) evaluated forcase(4)ofFigs. 2-4 (b)and(c).Theblackline(CaseQc)isobtained usingtheformula r tot = a 1 N e ( H )+ a 2 N h ( H ) where a 1 =3.2mJ = (moleK 2 ) a 2 =10.3mJ = (moleK 2 ) aredeterminedwiththeleastsquaretto experimentaldatabelow 30T .Note“ d -wave”and“ s -wave”curvesrepresent simpleextrapolationsofthelow-eld p H and H termsupto H c 2 .Theerror barshowncorrespondstotheabsoluteaccuracyofthedatadi scussed aboveinFig. 2-1 thetwo-bandbehaviorcoupledwithagraduallyincreasingc ontributionofcorestates whichisnearlylinearineld.Fig. 2-4 (c)clearlyshowsthatthedensityofstatesofthe holeband N h ( H ) ,assumedheretobefullygapped,isalwayslinearasafuncti onof eldandtheresultsforthetwodifferentcouplingmatrices consideredherearevery similar.However,asmentionedbefore,theslopeissmaller thantheonepredictedforan idealized s -waveSC. UsingEq.( 2–14 ),thespatiallyaveragedZDOSontheelectronandtheholeba nd areaddedwithdifferentweights.Usingtheresultspresent edinFigs. 2-4 (b)and(c)as Case4,weinvestigateseveralscenarios.Sincetherearetw oelectronpockets,and 45

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assumingthatonlyoneholepocketcontributessignicantl ytothelowenergydensity ofstates(orthatanaiveaverageovertheholepocketsissuf cient),thenetDOSand theelddependenceoftheSommerfeldcoefcientareonlyfu nctionsoftheratioofthe densitiesofstatesoftheelectronandholesheets.Inthefo llowingwewillstudythree scenariosderivedfromCase4,whichwewillabbreviatewith “Q”indicatingtheuseof thequasiclassical,orEilenberger,approach: CaseQa:weassumethatonlyoneholepocketcontributescons iderablytothe lowenergyDOS,andusetheweights w e = w h =2 N e 0 = N h 0 takenfromtheDFT calculation, N e 0 = N h 0 =0.65 ,seeRef.[ 92 ]; CaseQb:Weonceagainx N e 0 = N h 0 =0.65 ,butadoptamodelforwhichthenormal DOSforallthreeholepocketsofBa 2 Fe 2 (As 0.7 P 0.3 ) 2 arethesameandforwhichall threepocketscontributeequallytothelowenergyDOS,henc e w e = w h =2 N e 0 = 3 N h 0 ; CaseQc:Wedonotholdtheratio N e 0 = N h 0 xed,butinsteadcalculatetheweights fortheelectronpockets a 1 andfortheholepockets a 2 byaleastsquarest totheexperimentaldatausingtheformula r tot = a 1 N e ( H )+ a 2 N h ( H ) .Ifwe normalizeittothepresumedcontributionofthesupercondu ctingfraction, r n r 0 14mJ = (moleK 2 ) ,where r 0 istheextraneousterm(seebelow),wend w e = ( w e + w h )= a 1 = ( r n r 0 ) and w h = ( w e + w h )= a 2 = ( r n r 0 ) and a 1 = a 2 = w e = w h InFig. 2-5 wecomparetheresultsforallthreecasestotheexperimenta lly measuredspecicheatcoefcient(pinkdots).Theexperime ntalvaluesareobtained byextrapolatingthemeasuredspecicheatcoefcient r atvarioustemperaturesto T =0 .Theuppercriticaleld H c 2 istakentobe52T,seeRef.[ 83 ].Thenormal state r n =16mJ = (moleK 2 ) canbeobtainedbyextrapolating r to H c 2 .Asubstantial residual[ 84 ] r 0 =1.7mJ = (moleK 2 ) inthesuperconductingstate,presumeddueto disorder,issubtractedintheplotsoftheelddependencef romtheexperimentaldata (pinkdots)tocomparewithourquasiclassicalcalculation inthecleanlimit(bluesquares andgreendiamonds).Notethatsubtractingoftheresidual C = T tendstoenhancethe scatterinthelow-TdataofFig. 2-2 FromFig. 2-5 ,weseethattheresultsderivedformodelQbwiththreeequal mass holepocketsandtwoequalmasselectronpocketsareingooda greementwiththe 46

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experimentaldata:bothexperimentandtheoryshowa“Volov ikeffect”atthelowest eldsandthenacrossovertoalinear H dependenceatintermediateelds.While modelQahasthesamequalitativebehavior,therelativewei ghtsofholeandelectron bandsareapparentlynotconsistentwiththenormalizedexp erimentaldata,andthetis muchpoorer.ComparedtoQb,theleastsquarestQctotheexp erimentaldata(black line)isonlymarginallyimproved,andgivestheratio N e 0 = N h 0 =0.47 withtwoelectron pockets/threeholepocketsor 0.16 withtwoelectronpockets/oneholepocket,thesame orderasobtainedfromDFTcalculation. Asisusuallythecasewiththemeasurementsthatprobetheam plituderatherthan thephaseofthegap,itisdifculttodistinguishthedeepmi nimafromthetruenodes. Inthiscasewendthatwithourcurrentuncertaintyintheba ndparameters,andthe scatterinthedata,itisimpossibletoassertthenodalbeha viorpurelyfromthecurrent data.Fig. 2-6 showsthecomparisonofCases1and4ofTable 2-1 ,correspondingto r =1.3 and 0.9 ,i.e.,withandwithouttruenodes,withtheweightsofCaseQ b.Even thoughthenodaltappearsbetteratthelowestelds,highe r H dataareinbetweenthe twocases.Thereforetheconclusionaboutthetruenodecome sfromthedataonother experiments,suchaspenetrationdepth.2.2.5Conclusions AmongthevariousfamiliesofFe-basedsuperconductors,Ba Fe 2 (As 1 x P x ) 2 may beakeysystemforunderstandingtheoriginsofsuperconduc tivity.Inpartthisis because,aloneamongthematerialsthoughttodisplaynodes inthesuperconducting gap,itpossessesaratherhigh T c of 31K ,andhencetheinterplayofthepairing mechanismandFermisurfaceshapeandparametersindetermi ningthegapanisotropy isunderspecialscrutiny.ThelackofanobservableVolovik effectinearlierspecicheat measurementswasacautionarynoteinanotherwiseconsiste ntarrayofmeasurements insupportofgapnodes.Wehavepresentedexperimentaldata atbothlowerand highereldsthanpreviousmeasurements,andfoundthatthe initiallyreportedlinearH 47

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0 5 10 15 20 25 0 1 2 3 4 5 6 7 8 gap f o rm o f electr on gap ( 1) Qb r = 0.9 ( no no des deep mi n ima ) ( 4) Qb r = 1.3 (accide n tal no des ) Experime n t ( 1) Qb Nt o t = 2 electr on FS + 3 ho le FS ( 4) Qb Nt o t = 2 electr on FS + 3 ho le FS ( H ) 0 (mJ m o le 1 K 2 )H (T )Experime n t Hc 2 52 T ( 1) Qb Hc 2 54 T ( 4) Qb Hc 2 42 T Figure2-6.Experimentallymeasuredspecicheatcoefcie nt(largepinkdots,adapted fromFig. 2-1 )comparedtocalculationswithdeepgapminima(Case1, r =0.9 ,bluetriangles)andaccidentalnodes(Case4, r =1.3 ,green squares).Inbothcasestheweightofelectronandholepocke tcontributions hasbeenchoseninagreementwithCaseQb. behaviorextendsupto 35T ,butthatatlowelds( H 4T )moreprecisemeasurements withsmallergradationsinthechangeofeldbetweendatapo intsarenowclearly consistentwithaVolovik-typeeffect.Theresidual T 0 Sommerfeldcoefcient r ( T 0) isabout 1.7mJ = (moleK 2 ) ,consistentwithpossiblenanoscaledisorderin thesample.Thelow-eldsublineardependenceoftheSommer feldcoefcientisa strongindicationthatnodes(ordeepminima)arepresent,a ndprovidesthesought-after consistencywithotherprobeswithouthavingtomakeextrem eassumptionsabout theratioofmassesonelectronpocketstothoseonholepocke ts,aswasproposedin Ref.[ 84 ]. Itisneverthelessstrikingthatindicationsofnodalbehav ioronthesamesamplesis somuchweakerinthespecicheatmeasurementsascomparedt othermalconductivity andpenetrationdepth.Thisisclearlyindicatingthatthen odesarelocatedonthe pocketswithsmallermassesand/orlongerlifetimes,aswas pointedoutinRef.[ 84 ]. 48

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Wehaveattemptedtoputthisstatementonasemiquantitativ ebasisbypresentinga quasiclassical(Eilenberger)calculationofthedensityo fstatesandspecicheatofa two-bandanisotropic s superconductor.ComparisonwiththeDopplershiftmethod allowedustoarguethatthequasiclassicalcalculationiss uperiorforsemiquantitative purposes.WendthattheunusuallysmallrangeofVolovik-t ypebehavior,followed byalargerangeoflinearH behavior,isduetothesmallgapandweaknodesonthe smallmass(presumablyelectron)sheet[ 83 84 ].Goodtstothedataareobtained foraverageholeandelectronmaximumgapsofapproximately equalmagnitude, intheweakinterbandcouplinglimit.Thesuccessofthists houldnot,however, temptonetodrawdenitiveconclusionsabouttherelativem agnitudesofthepairing interactions.Theproliferationofparametersinthetheor yduetothemultibandnatureof thesystemmakesitdifculttodeterminegapmagnitudes,de nsityofstatesratios,and nodalpropertieswithanyquantitativecertainty.Equally goodtscanbeobtained,for example,withsubstantiallysmallerfullgapsthananisotr opicgaps;thenodescontrol thelow-eldbehavior,andthesmallfullgapgivesrisetoal argelinearterm.Whatis importantisthatwehaveshownthatatcanbeobtained,with reasonablevaluesofthe parameters,thatitcanonlybeobtainedifnodes(ordeepmin ima)existononeofthe Fermisheets,andthatitrequiresgoingbeyondthesimpleDo pplershiftpicture.Itisour hopethattheresultsofthiscalculationandtwilleventua llyleadtoamorequantitative rst-principles-basedcalculation. 2.3QuasiparticleVortexBoundStatesinFeBS:Application toLiFeAs 2.3.1Motivation Bulkexperimentssuchasspecicheatandthermalconductiv ityoscillationsin anexternalmagneticeld[ 100 101 ]canalsoprobetheorientationsofgapnodes iftheyexist.PerformedontheFe(Te,Se)system[ 102 ]andP-doped122family[ 83 ] respectively,theseexperimentsreportedoscillationpat ternsconsistent[ 83 101 103 49

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104 ]withananisotropicgapwithminimaalongthe – X axis(intheunfoldedBrillouin zone),aspredictedbyspinuctuationtheories(see,e.g., Ref.[ 8 ]). Orderparameterstructureisalsoreectedinthelocalprop ertiesofinhomogeneous superconductingstates.Inhomogeneitiesmayariseduetoi mpurities,andtheresulting quasi-boundstatesinnodalsuperconductorshavetailstha t“leakout”inthenodal directions[ 105 ],providingasignatureoftheamplitudemodulationoftheg ap.The interpretationoftheseimpuritystatesiscomplex:disord erpotentialscanbeoftheorder ofelectronvolts,andhencerelativelyhighenergyprocess escontroltheformationof suchstates,aswellastheircontributiontoscanningtunne lingspectroscopy(STS) images[ 106 ]. Underanappliedmagneticeld,inhomogeneoussuperconduc tivityalsoarises duetomodulationoftheorderparameterinavortexlattice, andboundstateslocalized aroundthevortexcoresappear.Inthiscase,relevantenerg yscalesareoftheorderof thegaporlowerandtheboundstatespropertiesaredetermin edbytheshapeofthe gapandthebandfeaturesneartheFermisurface.Thedecayle ngthofthecorestates isoforderof BCS = v F = ,where v F istheFermivelocityand isthegapamplitude. Consequently,variationofthegapwithdirection b k attheFS, ( b k ) 6 = const,directly inuencestheshapeofthecorestatesinrealspace,leading tothe“tails”extending alongnodesorminima.Sincethedecayofthesestatesisexpo nentialindistance from thecenterofthevortex(exceptalongtruenodeswhereitfol lowspowerlawsinwhich casethereexistnotrulylocalizedboundstates[ 107 ]),thesetailsshouldbeseeninlocal measurements,forinstance,theconductancemapbySTS,and canbeusedtoprobe thegapshapesuchasthefourfoldsymmetric d x 2 y 2 -wavegapincuprates.Theoretical calculationsusingEilenbergerformalismclearlyshowedt hesetailsinthelocaldensity ofstatesnearthevortexcoreina d -wavesuperconductor[ 67 68 ]buttheSTSonthe vortexcoreofYBa 2 Cu 3 O 7 [ 108 ]andBi 2 Sr 2 CaCu 2 O 8 [ 109 ]couldonlysuggestthe fourfoldsymmetryofthegap(seeFig. 2-7 ).Althoughdifcultiesofinterpretationexist 50

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(a) X/ [0 Y/ [0 -5 -5 0 0 5 5 1.5 1 0.5 0 (b) Figure2-7.(a)ConductancemapofavortexcoreofBi 2 Sr 2 CaCu 2 O 8 at 6mV revealing asquarepatteraroundthevortexcenter(reprintedwithper missionfrom [ 109 ],copyright2005bytheAmericanPhysicalSociety).(b)Zer oenergy localdensityofstatesshowingtailsalong y = x directionsfora d x 2 y 2 -wavegap(reprintedwithpermissionfrom[ 67 ],copyright1996bythe AmericanPhysicalSociety). incuprates[ 110 ],wherethecoherencelengthisshortandthecoresmaynucle ate competingorder(see,e.g.,Ref.[ 111 ]),inmostFe-basedsuperconductors(FeBS)these complicationsarelesssevereorabsentoverawiderangeofe xperimentallytunable parameters. Ontheotherhand,acomplexaspectofFeBSarisesduetotheir multibandnature. Thedirectionaldependence v F ( b k ) alsoaffectsthedecaylengthofthecorestates, especiallywhencombinedwithdifferentgapamplitudesond ifferentFermisurface sheets.InFeBS,theFermisurfacetypicallyconsistsoftwo orthreeholepocketsand twoelectronpockets,asrepresentedintheBrillouinzonec orrespondingtoone-Fe unitcell(seeFig. 2-8 ).Thesizeandshapeofthesepocketsvariesconsiderablyfr om familytofamily.Anaturalquestioniswhetheritisthenorm alstatebandstructureand theFermisurface,ortheorderparametershapethatdetermi nethesalientfeatures ofthevortexcorestatesasseeninexperiment,andwhethero necandrawreliable conclusionsaboutthedirectionsofthegapnodesorminimab asedontherealspace structureofthesestates.Wewilladdressthisquestionbel owinthischapter. 51

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0 0 k x k y 1 2 1 2 0 0 1 2 1 2 kxky circularFSLiFeAs 0 4 2 3 4 0 4 2 3 4 vF (a)(b) Figure2-8.(a)FermisurfaceofstoichiometricLiFeAsat k z =0 intheunfoldedone-Fe “effective”BrillouinzonefromDFT.TheFermivelocitiesf ordifferentsheets areindicatedbythearrowspointingtothehigher E ( k ) .Welabeltwoinner holepockets 1 2 ,oneouterholepocket r andtwoelectronpockets 1 2 (b)TheFermivelocitydirection v F vsthemomentum k azimuthalangle for theLiFeAs r pocketandthecircularFermisurface(shownasinsets). Thecompetitionbetweenthetwoeffectshasbeenexplorednu mericallyinother contexts.Forexample,thesixfoldpatternobservedin2H-N bSe 2 corestates[ 112 ]can beexplainedeitherassumingaweakgapanisotropyorusingt heangle-dependent densityofstatesaroundtheFermisurface[ 113 ].Inpnictides,itwasarguedboththat thevortexcorestatesarecontrolledbytheorderparameter shape[ 114 ],andthatthe locationofthepeakintheDOSisdeterminedbytheproximity tothebandedgeinthe electronorholebands[ 115 ].Togainqualitativeinsightintothisissue,weconsidera simplemodelwithboththeorderparameterandbandanisotro pycharacteristicofthe Fe-basedsuperconductors,andndthatintheabsenceofstr ongnodestheFermi velocityanisotropycandominatethereal-spaceshapeofth evortexcorestates.These stateswereobservedinearlySTSexperiments[ 116 117 ],albeitwithoutthespatial resolutionnecessarytoanalyzetheorderparameterstruct ure. WefocusontheLiFeAssystem,whichisidealforSTSmeasurem entsdue toitsnonpolarsurfaces.Accordingtocalculations[ 118 ]usingdensityfunctional 52

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theory(DFT),theFermisurfaceofthismaterialhasthreeho lepocketsandtwo electronpockets(seeFig. 2-8 ).Theouterholepocketislargeandquitesquare, accordingtobothDFTresultsandARPES[ 119 ]anddeHaas-vanAlphen(dHvA)[ 120 ] measurements.Both r and 2 holepocketshavesmallFermivelocitiesandtherefore largenormalstateDOS.ARPEShasidentiedsuperconductin gleadingedgegaps oforder 1.5 – 2meV fortheholepockets,and 3meV fortheelectronpockets[ 119 ]. TheLondonpenetrationdepthdata[ 121 ]andspecicheatmeasurements[ 122 ] ruledouttheexistenceofgapnodesandwerettomodelswith twoisotropicgaps with ( 1 2 ) (3meV,1.5meV) and (2meV,0.5meV) ,respectively.Thissuggests moderategapanisotropy,whichisnoteasilydetectedbythe bulkmeasurements,but cansubstantiallyaffecttherealspacestructureofthecor estates.Borisenko etal. [ 123 ] indeedfoundsubstantialgapanisotropyaroundbothelectr onandholeFermisurfaces inangle-resolvedphotoemission(ARPES)experiment.Umez awa etal. reported similarbutnotquantitativelyidenticalresultsforaniso tropicgaps[ 124 ].Spin-uctuation theory[ 125 ]predictssimilargapanisotropyastheseARPESexperiment s.InFig. 2-9 (a) wesketchtheexperimentaldatainRef.[ 123 ].Sincethegapon r Fermipocketwitha relativelylargedensityofstatesisthesmallestandARPES [ 123 124 ]andSTM[ 126 ] suggestthattheminimumofthe r gapisalongFe-Fedirection,onewouldexpectthat thetailsoflowenergyquasiparticlesextendinthisdirect ion(the k x k y directionsin Fig. 2-8 ).Onthecontrary,thesetailsaroundasinglevortexmeasur edinzeroenergy conductancemapwithSTS[ 127 ]areactuallyalongAs-Asdirection(the k x = k y directionsinFig. 2-8 );seeFig. 2-9 (b). ForcircularFermisurfacesthelow-energycoreboundstate sextendfurthestin thedirectionofthesmallestgap,butforrealisticbandsth eFermivelocityanisotropy playsasignicantrole.Sincethecross-sectionsofthe 1 and 2 electronpocketsrotate byafull 180 alongthe k z direction,andsincethesegapsarelarger,itisunlikelyth at thesesheetscontributesubstantiallytothespatialaniso tropy.Wethereforefocusonthe 53

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(b) (a) S S S T 5 nm Figure2-9.(a)SketchofARPESmeasuredgapsofLiFeAsonele ctronandhole pockets[ 123 ].(b)ZeroenergySTSconductancemaparoundasinglevortex ofLiFeAs(reprintedwithpermissionfrom[ 127 ],copyright2012bythe AmericanPhysicalSociety). possibleanisotropyofthegapontheholeFermisurfaces.Th emostlikelycandidate fortheanisotropicgapthatdominatesthelow-energyvorte xboundstatesisthe r pocket.Theorbitalcontentofthispocketisexclusively d xy ,anditcouplesonlyweakly totheprimarily d xz and d yz electronpocketswhichprovidethemainpairingweightin theconventionalspinuctuationapproach[ 8 ].Itisalsonearlysquare,withweakly dispersiveparallelsurfacesorientedalongthe[110]dire ctionintheone-Fezone,and withsignicantvariationsoftheFermivelocitybetween[1 00]and[110]directions. HencewerstneglectotherFermisurfacesheets,andcontra sttheresultsobtainedfor the r sheetalonewiththoseforasinglecircularFermisurface. 2.3.2Model WefollowthethequasiclassicalmethodintroducedinSec. 2.1 .Specically,inthe EilenbergerEqs.( 2–1 )theFermivelocity, v F ( ) ,isalongthe2Dunitvector b k forthe circularFermisurface,andiscomputedforthe r -bandinLiFeAsusingtheQuantumESPRESSO[ 128 ],asinRef.[ 129 ].Intheloweldregime,weconsidertheproblemof anisolatedvortexandassumeaseparablemomentumandcoord inatedependence oftheorderparameter ( b k )= 0 ( )tanh ( = r 0 ) ,where 0 isthebulkgapvalue intheabsenceoftheeldand ( ) describesthegapshapeontheFermisurface, 54

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s =1 d = p 2cos2 ,and s ani =(1 r cos4 ) = p 1+ r 2 = 2 with r =0.3 ,forthe isotropic s -wave,nodal d -wave,andextended s -wavegapsrespectively[ 130 131 ].The coherencelengthis 0 = ~ v F,rms = 0 where v F,rms = q hj v F ( b k ) j 2 i FS ,andthebrackets denotethenormalizedaverageovertheFermisurface, hi FS = 1 N I FS dk k j v F ( b k ) j = Z 2 0 d 2 ~ ( ) (2–15) where N H FS dk k j v F ( b k) j and ~ ( ) istheangle-dependentdensityofstates.Thefactor r accountsfortheshrinkingofcoresizeatlowtemperature(K ramer-Pescheffect[ 94 95 ]),andweset r =0.1 correspondingto T 0.1 T c .Inafullyself-consistent calculation,thegapanisotropyinmomentumspacewillindu ceweakcoreanisotropy inrealspace[ 67 ],whichweignoreheresincetheeffectissmallevenfornoda l systems[ 67 ]. WesolveEq.( 2–1 )usingtheRiccatiparametrization[ 74 ]andintegratingalong classicaltrajectories, r ( x )= r 0 + x b v F toobtainthefunctions g and f atMatsubara frequencies.TheLDOSisfoundafteranalyticcontinuation fromretardedpropagators, N ( r )= N 0 h Re g R ( k F r + i ) i FS .Ateachpoint r =( ) theLDOSisobtainedby summationoverthequasiclassicaltrajectoriespassingth rough r .Eachtrajectoryfollows thedirectionoftheFermivelocityatagivenpointontheFS, b v F ( b k ) ,andsamplesthegap ( r ( x ), b k ) .Trajectoriessamplingregionsofsmallorderparameterco ntributetothelow energyLDOS.Thisoccursifthetrajectoryeitherpassesint hevicinityofthecorewhere theorderparameterissuppressedinrealspace, ( ) 0 (smallimpactparameter, dominantforisotropicgaps),orisalongthedirectionwher ethegaphasanodeora deepminimuminmomentumspace, ( b k ) 0 (dominantfornodalsuperconductivity). TheinuenceoftheFSshapeisthenclear:thenumberoftraje ctorieswithagiven impactparameterdependsonthebandstructure.Denotethea nglebetween b v F and k x axisas v F .ForacircularFS, v F = ,andquasiclassicaltrajectoriesindifferent directions v F areequallyweightedinFSaveraging.Incontrast,foraniso tropiccases, 55

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suchasthesquare r -sheetinLiFeAs,largepartsoftheFShavethe v F alongthe diagonals(seeFig. 2-8 b),andthereforetheaverageoverthetrajectoriesisheavi ly weightedtowardsthatdirectionaswell. Foranisotropicgap ( b k )= const,thelargestcontributiontothelowenergyLDOS at r =( ) comesfromthetrajectoriespassingthroughthecore, v F = or + .For acylindricalFSparameterizedbyangle thiscorrespondstotwopointssince v F = OnananisotropicFS,suchasthe r pocketinLiFeAs,manydifferentmomentumangles correspondto v F 4 ,andquasiparticlesfromalargeportionoftheFStravel alongthesedirections.Forrealspacedirection = 4 ,allthesetrajectoriessamplethe coreregionandcontributetothelowenergyLDOS.For awayfromthesedirections thesetrajectorieshaveanonzeroimpactparameterandther eforesmallweightatlow energies.Fortheextended s -wavegapmodelwith r > 0 intheformfactor s ani ,this impliesthattheregionsoflargegapwillbeemphasizedduet opreferentialdirectionsof v F ,andthereforetheFSeffectscompetewiththegapshapeinde terminingthespatial proleofthevortexcorestates.Simplyassumingthatthedi rectionofthesmallestgap in k spaceyieldstheorientationofthetailsoftheboundstatew avefunctionneednotbe correct,andmaybewrongwithastronglyanisotropicFermis urface. 2 2.3.3Results Fig. 2-10 showsthezeroenergydensityofstates(ZDOS)ofacircularF ermi surface[Figs. 2-10 (a)– 2-10 (c)]andLiFeAs r pocket[Figs. 2-10 (d)– 2-10 (f)].Comparing Figs. 2-10 (a)and 2-10 (d)fortheisotropicgap,weseethattherotationsymmetryo f ZDOSinFig. 2-10 (a)isbrokenduetotheanisotropyof r pocketandFermivelocity;at thesametimetheZDOSstillpreservesthecrystalfour-fold symmetry.Inthe d -wave 2 Fora d -wavegapalongacircularFermisurface,nearthenodaldire ctions 4 theenergyspectrumisnotstronglyrestrictedtozeroimpac tparameter.Aslongas = v F 4 ,theLDOSisenhancedandthereforethiscasehaswidertails alongdirections 4 56

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x/ y / 2 1 0 1 2 2 1 0 1 2 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2log 10 ( N / N 0 ) (a) s -wavegap x/ y / 2 1 0 1 2 2 1 0 1 2 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2log 10 ( N / N 0 ) (d) s -wavegap x/ y / 2 1 0 1 2 2 1 0 1 2 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1log 10 ( N / N 0 ) (b) d -wavegap x/ y / 2 1 0 1 2 2 1 0 1 2 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1log 10 ( N / N 0 ) (e) d -wavegap x/ y / 2 1 0 1 2 2 1 0 1 2 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2log 10 ( N / N 0 ) (c)anisotropic s -wave x/ y / 2 1 0 1 2 2 1 0 1 2 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2log 10 ( N / N 0 ) (f)anisotropic s -wave Figure2-10.NormalizedZDOSina 2.5 0 2.5 0 regionaroundthecenterofthesingle vortexfordifferentgapmodelswithacircularFermisurfac e(a)–(c)and LiFeAs r pocket(d)–(f):(a),(d)anisotropic s -wavegap 0 ;(b),(e)anodal d -wavegap 0 p 2cos2 ;(c),(f)extended s -wavegap 0 (1 r cos4 ) = p 1+ r 2 = 2 r =0.3 .Thegapbulkvalueistakentobe 0 =1.76 T c .Theinsetoneachpanelrepresentsacartoonofthe correspondinggapalongtheFermisurface.Whitecontourli nesshown correspondto 0.025 N 0 57

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case,Fig. 2-10 (b),foracircularFermisurface,werecoverwell-knownres ultsforthe ZDOS,includingthedoubletailsalongthenodaldirections forcedbythevanishing oftheboundstatewavefunctionexactlyalongthe 45 directionsinthequasiclassical theory[ 69 ].Whilethisfeatureremains,itbecomesessentiallyinvis ibleinthecaseof thesquareFermisurfaceshowninFig. 2-10 (e),astheFermisurfaceconcentratesthe quasiparticletrajectoriesevenmoreinthenodaldirectio ns.Ourprimaryresultsare nowcontainedinFig. 2-10 (c)andFig. 2-10 (f).Theextendeds state s ani hasbeen chosendeliberatelytohavegapminimaalongthe 0 directions(alongtheFe-Febond intheFeBScase).Thisisclearlyvisibleinthecaseofaniso tropicpocket,Fig. 2-10 (c), asthetails,whilenotaswell-denedasinthetruenodalcas e,extendclearlyalong thesedirectionsinrealspace.Thesedirectionsrotateby 45 ,however,whenthe samegapexistsonthesquareLiFeAs r pocket,asinFig. 2-10 (f).Infact,theZDOSin Fig. 2-10 (f)stronglyresemblesthestructureobservedbyHanaguri etal. inrecentSTS measurementsonLiFeAs[ 127 ]. TheresultsinFig. 2-10 stronglychallengethecommoninterpretationofSTS imagesofvortices,whichassigngapminimatothedirection softheextendedintensity inrealspace.Thisisprobablyreasonableinthecaseoftrue nodes,asindicatedby the d -waveexamplesshown,butfailsiftheseminimaarenotsufc ientlydeepdue tothecompetitionwiththeFermisurfaceeffects.Nowthatt hebasicstructureofthis competitioninthecaseoftheZDOShasbeenunderstood,itis interestingtoaskwhat mayhappeninthecaseofniteenergies 6 =0 .Fig. 2-11 showsthecalculatedLDOS N ( r ) asafunctionofenergyatthevortexcorecenter[Fig. 2-11 (a)– 2-11 (c)]and onecoherencelengthawayfromthecenterinthe 0 direction[Fig. 2-11 (d)– 2-11 (f)] and 45 direction[Fig. 2-11 (g)– 2-11 (i)].ThespectrumisquiteinsensitivetotheFermi surfaceshapeatthevortexcorecenterwheretheresultsfor thecircularFSandLiFeAs r pocketarealmostthesame.Awayfromthevortexcenter,thed irection-dependent LDOS N ( r =0) reectsthecompetitionbetweengapandFermisurfaceaniso tropy. 58

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Thehigher/lowerLDOSoftheLiFeAs r pocket/circularFSatzeroenergyinFig. 2-11 (i) thanthatinFig. 2-11 (f)isequivalenttoourresultshowninFig. 2-10 .Thequasiclassical theoryincorporatestheFSpropertiessolelyvia v F ,andthusdoesnotaccountfor thepossiblechangesintheshapeoftheconstantenergysurf acesforSTSbiases awayfromzero.Providedthebandshapevariesveryslowlyon thescaleof T c ,this neglectshouldnotsignicantlyaffecttheshapeofvortexb oundstatesatnonzero energy,however.Ontheotherhand,evenwithinthecurrentm odel,amoreimportant effectmaybeincluded.InouranalysisofLiFeAs,wehaveunt ilnowneglectedall Fermisurfacepocketsexcepttheouter( r )holepocket,duetoitssquareshapeand becauseitseemslikelytohavethesmallestgap.Whenthebia sisincreased,higher energyquasiparticlestates,includingthoseassociatedw ithlargergaps,willbeprobed. Withinspinuctuationtheory[ 8 ],boththehighdensityofstates 2 pocket,andthe electronpockets,tendtohavegapminimaalongthe 0 directions.Thusashigher energiesareprobed,itispossiblethat rotations oftheboundstateshapemaytake placeasthebalancebetweengapstructureandFermisurface anisotropyisaltered. Unfortunatelyevenqualitativestatementsdependonthede tailsofthesizesofgaps andgapanisotropiesoneachsheet,aswellasonthevariousF ermivelocitiesfor eachband.TheLiFeAssystemisquiteclean,however,andift hecurrentcontroversy betweenARPES[ 119 ]anddHvA[ 120 ]regardingtheFermisurfacecanberesolved, spectroscopiesofboundstatesonthissystemshouldprovid eenoughinformationto determinefairlydetailedstructureofthegap.2.3.4Conclusions Wehaveusedquasiclassicalmethodstocalculatethevortex boundstateswithin asinglevortexapproximation,andhighlightedthecompeti tionbetweengapandFermi surfaceanisotropyinthedeterminationoftheshapeofSTSi magesofvortexbound statesinLiFeAs.IftheFermisurfaceanisotropyislargeen ough,wehaveshownthat thetailsofvortexboundstatesatlowenergyneednotcorres pondtothesmallestgaps 59

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(a) (b) (c) (d) (e) (f) (g) (h) (i) 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 -3-2-10123 /Tc N (0 ,, )-3-2-10123 /Tc N ( 0 )-3-2-10123 /Tc N ( 45 ) Figure2-11.NormalizedLDOS N ( r ) = N 0 N ( , ) vsenergyfordifferentgap modelswithacircularFermisurface(redsymbols)andLiFeA s r pocket (blueline):(a),(d),(g)isotropic s -wavegap 0 ;(b),(e),(h)nodal d -wave gap 0 p 2cos2 ;(c),(f),(i)extended s -wavegap 0 (1 r cos4 ) = p 1+ r 2 = 2 r =0.3 .Thegapbulkvalueistakentobe 0 =1.76 T c r =( )=(0, ) for(a)–(c); ( ,0 ) for(d)–(f); ( ,45 ) for (g)–(i). isthecoherencelength. inthesystem,ifthosegapsarenottruenodes.TheZDOSshape measuredbySTS inexperimentsontheLiFeAssystemwithverycleansurfaces iswellreproducedby numericalcalculation.Withinourmodel,weattributethet ail-likespectrumtotheeffect ofthenon-uniformlydistributionofFermivelocitydirect ionontheFermisurfaceofthe LiFeAs r holepocket.Furthermeasurementsoftheenergydependence ofboundstate shapemayfurtherhelpidentifythegapanisotropy. 60

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CHAPTER3 DISORDERINSUPERCONDUCTORS Somepartsofthischapterhavebeenpublishedas“Usingcont rolleddisorder todistinguish s and s ++ gapstructureinFe-basedsuperconductors,”YanWang, A.Kreisel,P.J.Hirschfeld,andV.Mishra,Phys.Rev.B 87 ,094504(2013). 3.1Motivation Determiningthesymmetryandstructureofthesuperconduct ingorderparameterin Fe-basedsuperconductors(FeBSs)isoneofthemainchallen gesinthisneweld[ 8 9 ]. Thesign-changing s andsinglesign s ++ gapdescribedintheIntroductionchapter aretwopromisingcandidatesforFeBSwithatypicalnestedF ermisurface(FS) includingtwoorthree [=(0,0)] -centeredholepocketsandtwo [ M =( )] -centered electronpocketsinthetwo-FezonecomposedprimarilyofFe 3 d states.Surprisingly, ithasprovenratherdifculttodenitivelydistinguishth esetypesofgapstructures experimentally,inpartbecausephase-sensitiveexperime ntsarechallengingdueto surfaceproperties;becauseofthemultibandnatureofthee lectronicstructure;and becausethe s and s ++ “states”aresymmetryequivalent,transformingbothaccor ding tothe A 1 g representationofthecrystalpointgroup.Asmentionedint helastpartof Chapter 1 ,onlyahandfulofexperimentsofferindirectevidenceinfa vorofthe s state [ 52 – 59 ]. Ontheotherhand,alternativeexplanationshavebeenoffer edforallthese measurements;inparticular,KontaniandOnarihaveprovid edanalternateexplanation [ 48 ]fortheneutronresonancefeatureswithinan s ++ scenarioviaapostulatedenergy dependenceofthequasiparticlerelaxationtime.Inadditi on,severalreferences [ 132 – 136 ]havecalledattentiontoa“slow”decreaseof T c inchemicalsubstitution experiments[ 135 – 139 ],whichisthenascribedtothenaturalrobustnessagainst nonmagneticdisorderofan s ++ superconductor.Itisthisissuewhichwestudyhere. 61

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Itisimportanttounderstandwhatismeantby“slow”and“fas t” T c suppression inthiscontext.Atoneextremewehavesituationsinwhich T c isnotsuppressed bynonmagneticdisorderatall.AccordingtoAnderson'sthe orem[ 140 ],thecritical temperatureofanisotropicconventional s -wavesuperconductorwithasingleband ofelectronsisunaffectedbynonmagneticscatterers.From thisstatementitfollows immediatelythatthesameoccursfortwobandsinanisotropi c s ++ state(withequal gaps),butalsoinan s statewithnointerbandscattering.Attheotherextreme,we knowthatmagneticscatterersinaconventionalisotropics uperconductorsuppress T c accordingtotheAbrikosov-Gor'kov(AG)law[ 141 ];itiswellknownthat nonmagnetic scattererssuppress T c atthesamefastAGrateinatwo-band s state, provided the twodensitiesofstates N a = N b andtwogaps a = b areequalinmagnitude,and thescatteringispurely interband innature.Anydeviationfromtheseassumptionswill slow the T c suppressionraterelativetotheAGrate.Thereforebetween thesetwo extremesliemanypossibilitiesfor T c suppressionbehaviorwhichdependondetailsof theelectronicstructureandtherelativeamplitudesofint er-andintrabandscattering. Severaltheoreticalcalculationsof T c suppressionhavediscussedthepairbreaking effectsofnonmagneticscatterersonmodelmultibandsuper conductorswithgeneralized s -waveorder[ 91 132 133 142 – 152 ].Infactthesituationisgenerallyevenmore complicatedthandiscussedaboveorintheseworks,sincech emicalimpuritiesmay domorethansimplyprovideascatteringpotential:theymay dopethesystem,oralter thepairinginteractionitselflocally.Wethereforebelie ve(seealsoRef.[ 8 9 ])that measurementsof T c suppressionrelativetotheamountofchemicaldisorderare not particularlyusefultodeterminethegapstructureinmulti bandsystems.Toimprovethe situation,onerstneedstondawaytocreatepointlikepot entialscatteringcenters, soastocreatedisorderedsystemstowhichtheabovetheoret icalworksapply.The closestapproachtothisidealisachievedwithlow-energye lectronirradiation,whichis thoughttocreateinterstitial-vacancypairs.Experiment softhistypearebeingperformed 62

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Figure3-1.Sketchofthetwo-bandmodelwithlinearizedban ddispersionsontheFermi sheets a and b andconstantimpurityscattering v (intraband)and u (interband),togetherwithapossiblenodal s -wavegaponthebandsinthe superconductingstate. currently,anditisoneofthegoalsofthisworktomakepredi ctionstoguidetheanalysis ofsuchdata. Theotherneededimprovementsaretheoretical:rst,thepa irbreakingtheory mustbeextendedtorelate T c onlytodirectlymeasurablequantities,likethechange inresidual( T 0 )resistivitycausedbythedisorder,ratherthantoanytheo retically meaningfulbutempiricallyinaccessiblescatteringratep arameter.Second,sincethe theoryinvolvesmanyparameters,therobustnessofanyclai medtmustbetested bythesimultaneouspredictionofotherquantitieswhichde pendondisorder,suchas thelow-temperaturepenetrationdepth,nuclearmagneticr esonance(NMR)relaxation rate,orthermalconductivity.Finally,itwouldbeusefult ohave abinitio calculationsof vacancyandinterstitialpotentialstoconstraintheimpur ityparametersused.Thishas beenattemptedforchemicalsubstituents[ 153 – 155 ]recently. 3.2Model Weconsiderasystemwithtwobands a and b withlinearizeddispersionclosetothe Fermilevelthatleadtodensitiesofstate N a and N b inthenormalstate;seeFig. 3-1 63

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The t -matrixequationinthetwo-bandmodelhastheform ^ = n imp ^ t (3–1) ^ t = ^u + ^u^g ^ t (3–2) where n imp istheconcentrationofimpurities, ^ t ( n imp )= P 3i =0 t ( i ) n ^ i ^g ( n imp )= g 0 n ^ 0 + g 1 n ^ 1 and n representsaproductofband(bold)andNambu(caret)matric es. g 0 = diag( g 0 a g 0 b ) and g 1 =diag( g 1 a g 1 b ) arelocalGreen'sfunctionsinthe 0 and 1 channels (wehaveassumedparticle-holesymmetryinordertoneglect g 3 ),where ^ i denotePauli matricesinNambuspace.Duetothetranslationalinvarianc eofthedisorder-averaged system, ^g isdiagonalinbandspace.Wenowassumeasimplemodelforimp urity scatteringwherebyelectronsscatterwithineachbandwith amplitude v andbetween bandswithamplitude u ^ u = 0B@ vuuv 1CA n ^ 3 (3–3) The t -matrixcomponentsarefoundfromEq.( 3–2 )tobe t (0) aa = h g 0 b u 2 + g 0 a v 2 g 0 a u 2 v 2 2 g 2 b i D t (1) aa = h g 1 b u 2 + g 1 a v 2 g 1 a u 2 v 2 2 g 2 b i D (3–4) and t ( i ) bb = t ( i ) aa ( a $ b ) ,where D =1 g 2 a + g 2 b v 2 + g 2 a g 2 b u 2 v 2 2 2 u 2 ( g 0 a g 0 b g 1 a g 1 b ) (3–5) withtheabbreviation g 2 = g 2 0 g 2 1 64

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3.3 T c Suppression Thelinearizedmultibandgapequationnear T c is(see,e.g.,Ref.[ 91 ]) ( k )=2 T n = c X k 0 , n > 0 V kk 0 ~ ( k 0 ) ~ 2 + 2 (3–6) where isthelinearizeddispersionofband ,andweintroducedtheshiftedgaps andfrequencies, ~ ( k 0 )= ( k 0 )+ (1) and ~ = n + i (0) .Wewillsimplifythe modelabovefurtherinthatweadoptagapstructuresimilart othatobtainedfromspin uctuationtheories:Thegaponthe(hole)pocket a isisotropic, a ,andthegaponthe (electron)pocket b maybeanisotropic, b = 0b + 1b ( ) ,where isthemomentum anglearoundthe b pocketand R d 1b ( )=0 .Thepairingpotentialisthentakenas V kk 0 = V ( k ) ( k 0 ) ,with =1+ r b cos2 ,and istheanglearoundtheelectron pocket.Theparameter r controlsthedegreeofanisotropy,andcreatesnodesif r > 1 Thisansatzthengivesthreecoupledgapequationsfor ( a 0b 1b ) T .Inthe basiswecanwritethegapequationsinthecompactform =ln 1.13 c T c M L 0 M (3–7) wherethematrix M =(1+ V R 1 X R ) 1 V andtheconstant L 0 =ln 1.13 c T c were introduced.Here V istheinteractionmatrixintheabovebasis. R istheorthogonal matrixwhichdiagonalizesthematrix = n imp D N 266664 N b u 2 N b u 2 0 N a u 2 N a u 2 0 00 N b v 2 + N a u 2 377775 (3–8) where D N =1+2 u 2 2 N a N b +( u 2 v 2 ) 2 4 N 2 a N 2 b + v 2 2 ( N 2 a + N 2 b ) (3–9) 65

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isEq.( 3–5 )evaluatedinthenormalstatewherethelimit 0 hasbeentakeninthe localGreensfunctions. X isamatrixwithonlydiagonalelements, X ii = L 0 1 2 + c 2 T c + i 2 T c 1 2 + i 2 T c (3–10) where isthedigammafunctionand i aretheeigenvaluesofthematrix .The maximumeigenvalue [ max ( T c )] ofthematrix M determines T c via T c =1.13 c e 1 = max 3.4ResidualResistivity Themostdirectobservablemeasureofscatteringin T c suppressionexperiments istheresidualresistivitychange 0 ,i.e.,thechangeintheextrapolated T 0 value oftheresistivitywithdisorder.Wewillassumethatinterf erenceeffectsbetweenelastic andinelasticprocessesarenegligible,i.e.,thattheeffe ctonthe ( T ) curvewhenthe systemisdisorderedisessentiallya T -independentshiftupward.Wethereforecalculate 0 withinthesameframeworkasabove,assumingthatalldefect sarepointlike.In thezerofrequencylimit,therearenointerbandtransition s,andthetotalconductivity inthe x directionisthesumoftheDrudeconductivitiesofthetwoba nds, = a + b with =2 e 2 N h v 2 x i ,where v x isthecomponentoftheFermivelocityinthe x directionand thecorrespondingsingleparticlerelaxationtimeobtaine dfromthe self-energyinthe t -matrixapproximation, 1 = 2Im (0) .Notethat 1 contains contributionsfromboththeintrabandandinterbandimpuri tyscatteringprocesses.The transporttimeandsingle-particlelifetimeareidentical withinthismodelbecauseofour assumptionofpointlike s -wavescatterers,whichimpliesthatcorrectionstothecur rent vertexvanish.Anitespatialrangeofthescatteringpoten tialwilltendtosteepenthe T c vs 0 curve[ 156 157 ]. 3.5Results 3.5.1 T c SuppressionvsResistivity WenowsolveEqs.( 3–7 )for T c andcalculatesimultaneouslythechangein resistivity 0 at T 0 .Unlike T c vs n imp orvariousscatteringrates, T c vs 0 canthen 66

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becompareddirectlytoexperiment.Clearly,theresultswi llbeparameterdependent, however,soweherespecifyourpreciseassumptionsregardi ngtheelectronicstructure. Forconcreteness,wefocusontheBaFe 2 As 2 (Ba122)systemonwhichthelargest numberofmeasurementshavebeenreported.Wechoosevalues fortheFermi velocitiesanddensitiesofstatesattheFermilevelthatar ecompatiblewithbothdensity functionaltheory(DFT)calculations[ 4 ]andangle-resolvedphotoemissionspectroscopy (ARPES)measurements[ 158 159 ].WeassumeadensityofstatesoneachFermi surfacesheetof N a =3.6 and N b =2.7 = V c = eV = spin( V c istheunitcellvolume),for the“effective”holeandelectronpockets,respectively,t hatapproximatelydescribesthe imbalanceinthedensitiesofstatesthatalsohasbeenseenw ithARPES[ 158 – 160 ], andisconsistentwiththedensityofstatesofBa122arising fromFe d -orbitalsaccording toDFTcalculations[ 4 ]withaneffective-massrenormalizationof z =3 .Wetake theroot-mean-squareFermivelocitiesas v F a =2 = 3 10 5 m = s and v F b =10 5 m = s from v F ? inTableIofRef.[ 92 ],andrenormalizethembythesamefactorof z =3 toapproximatelymatchthevelocitiesfoundinARPESexperi ments[ 158 – 160 ].Inthe transportcalculation,thecomponentoftheFermivelociti esinthedirectionofthecurrent istakentobe h v 2 F x i =1 = 2 v 2 F duetothequasi-cylindricalFermisurface.Thepairing potentialschosenforthemaintextare V aa = V bb =0.05 and V ab = V ba = 0.04 Usingtheseparameters,weobtainfortheisotropiccase( r =0 )thezero temperaturegapvaluesof 0a 0 = 1.79 T c 0 and 0b 0 =1.73 T c 0 ,whereasforthe nodalcase( r =1.3 )theseare 0a 0 = 1.22 T c 0 and 0b 0 =1.23 T c 0 withthecritical temperaturechosenas T c 0 =30K .Wehavexedtheintrabandscatteringpotential atanintermediatestrengthvalueof v =0.25 ,butshowresultsforothervaluesinthe Appendix.PotentialsaregivenineVandweset ~ = k B =1 InFig. 3-2 ,wenowexhibit T c suppressionvsthecorrespondingchangeinresidual resistivity 0 asdenedabove,bothforafullyisotropic s gap( r =0 ),andfora gapwhichhasnodesontheelectronpockets( r =1.3 ),forarangeofratios u = v .Itis 67

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0 20 40 60 80 -1 -0.8 -0.6 -0.4 -0.2 0 a =1 a =0.7 a =0.5 a =0.2 a =0 0 500 1000 -1 -0.5 0 0 0 : 2 0 : 4 0 : 6 0 : 8 1 0 0 : 5 1 05001000Tc=Tc 0 0( n cm) 0 20 40 60 80 -1 -0.8 -0.6 -0.4 -0.2 0 0 500 1000 -1 -0.5 0 0 0 : 2 0 : 4 0 : 6 0 : 8 1 020406080 0 0 : 5 1 05001000Tc=Tc 0 0( n cm)Tc=Tc 0Tc=Tc 0 0( n cm) (a) (b) Figure3-2.(a)Normalizedcriticaltemperature T c = T c 0 vsdisorder-inducedresistivity change 0 forisotropic s -wavepairingforvariousvaluesoftheinter-to intrabandscatteringratio u = v .Inset:Samequantityplottedoveralarger 0 scale.(b)As(a)butforananisotropic(nodal)gapwithanis otropy parameter r =1.3 clearthatawidevarietyofinitialslopesandcriticalresi stivities c0 forwhich T c 0 ispossible,dependingonthescatteringcharacteroftheim purity.Thevariabilityof thesuppressionratewiththeratioofinter-tointrabandsc atteringhasbeennoted byvariousauthors[ 91 152 ]beforethis.Infact,Efremov etal. [ 152 ]haveshownthat thevarious T c suppressioncurvesoftheisotropic s gapfallontouniversalcurves dependingonwhethertheaveragepaircouplingconstant h i < ,=, > 0 whenplotted againsttheinterbandscatteringrate(whichisnotdirectl ymeasurable,however). Here h i = 1 N a + N b P 2f a b g N V N .Wehaveusedavalue h i =0.037 0 in ourinvestigations.Wehaveexaminedotherparametersetsw ithnegative h i ,and 68

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foundnoessentialdifferencein T c whenplottedagainsttheresidualresistivity 0 whichofcoursedependsonbothintra-andinterbandscatter ing.Otherworkshave madecomparisonswiththeresistivitychanges(forexample Refs.[ 135 136 ]),buthave typicallypresentedresultsfor s statesonlyforasinglesetofimpurityparameters correspondingtothefastestrateof T c suppression.Suchassumptionsleadalwaysto critical 0 valuescomparabletothesmallestonesseeninFig. 3-2 ,ofordertensof ncm .Hereweseethatmoregeneralvaluesoftheparameterscanea silyleadtomuch slower T c vs 0 suppressionratesbydisorder,withcriticaldisorder c0 valuesoforder mncm .AsdiscussedbyLi etal. [ 135 ],suchvaluesaretypicalofchemicalsubstitutions onvariousdifferentlatticesites;hereweseethatsuchslo w T c suppressiondoesnot ruleoutthe s state,evenwithintheassumptionsofourpotential-scatte ring-onlymodel. Tocheckhowrobustourconclusionsare,wetakedifferentva luesfortheimpurity parametersandpairpotentialparameters:whenweincrease theintrabandscattering potential v to v =1.25eV V c keepingallotherparametersidenticaltothoseofFig. 3-2 the T c suppressionsignicantlyslows,asseeninFig. 3-3 ,withtheexceptionofthe value =1 ,whichplaysaspecialroleinthetheoryoftwo-band s superconductivity, ascanbeeasilycheckedanalytically.WhileinRef.[ 48 ]itwasarguedthattheinterband scatteringpotential u shouldbegenericallylargeforanychemicalsubstituent,t here isnoreasontoexpect =1 toholdexactly,andthereforeweseethatlargecritical resistivities c0 areevenmorelikelytobefoundforstrongerimpurities(the unitaritylimit v !1 withxed ispathologicalinthismodel[ 152 ]andwehavenotconsideredit here).Thespecialroleofthevalue =1 canbeillustratedbyplottingtheresistivity 1 = 2 atwhichthecriticaltemperatureissuppressedbyhalf, T c =0.5 T c 0 ,asshownin Fig. 3-4 ,whichmaybecomparedwithexperiments.Notethat 1 yieldsthefastest T c suppressionindependentoftheimpuritypotentialintheph ysicalregime v & u 69

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0 50 100 150 200 -1 -0.8 -0.6 -0.4 -0.2 0 a =1 a =0.7 a =0.5 a =0.2 a =0 0 5000 10000 -1 -0.5 0 0 0 : 2 0 : 4 0 : 6 0 : 8 1 0 0 : 5 1 0500010000T c =T c 0 0 ( n cm) 0 50 100 150 200 -1 -0.8 -0.6 -0.4 -0.2 0 0 5000 10000 -1 -0.5 0 0 0 : 2 0 : 4 0 : 6 0 : 8 1 050100150200 0 0 : 5 1 0500010000T c =T c 0 0 ( n cm)T c =T c 0 T c =T c 0 0 ( n cm) (a) (b) Figure3-3. T c = T c 0 vs 0 forvariousvaluesoftheinter-tointrabandscatteringrat io u = v with v =1.25eV V c .(a)forisotropic s wavepairingand(b)foran anisotropic(nodal)gapwithanisotropyparameter r =1.3 3.5.2DensityofStates Arealunderstandingoftheeffectsofdisorderinagivensit uationwillprobably dependoncorrelatingtheresultsofseveralexperiments.O therquantitieswhichare quitesensitivetodisorderarethetemperaturedependence ofthelowT London penetrationdepth ( T ) andthenuclearmagneticspin-latticerelaxationtime T 1 1 WithinBCStheory,thesequantitiesarecontrolledbythelo w-energydensityofstates. Inthepuresystem,thenodalstructurethendeterminesthep owerlawoftemperature, andonegenericallyexpects ( T ) T forgaplinenodesexceptinveryspecial situations[ 161 ].Inthepresenceofasmallamountofnonmagneticdisorder, anite densityofstatesiscreated[ 162 163 ]whichleadsautomaticallytoa T 2 terminthe 70

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10 1 10 2 10 3 10 4 D r1/2 ( m W cm) 0 0.5 1 1.5 10 1 10 2 D r1/2 ( m W cm)a =u/v v=0.125 v=0.25 v=0.5 v=1.25 v=2.5 Figure3-4.Theresistivityathalfsuppression 1 = 2 asafunctionoftheratio = u = v for variousintrabandimpuritypotentials v (measuredineV V c );theother parametersaretakenasinthemaintextfortheisotropic s wavepairing (top)andforananisotropic(nodal)gap(bottom). penetrationdepth[ 161 164 ].Ifthestateisof s character,thegapnodesarenot symmetryprotectedandcanbeliftedbyfurtheradditionofd isorder[ 91 165 ]. 3.5.3NonmonotonicDependenceofResidualDOSonDisorder Inthisworkwenoteafurtherpossibilityinthedisorderevo lutionofthelow-energy densityofstates(DOS)ofanodalmultiband s -wavesuperconductor,namely,thata reentrantbehaviorof N (0) canoccurafterliftingofthenodes.Thereasonisthat,in asituationdominatedbyintrabandscatteringbutwithnonz erointerbandscattering, anisotropyofthegapsoneachindividualsheetwillquickly beaveragedonintroduction ofintrabanddisorder.Ifthestateis s ,amidgapimpuritystatecanthenbecreatedby interbandscattering,andgrowuntilitoverlapstheFermil evel,asshownschematically inFig. 3-5 (a).SuchmidgapstatesaretheanalogsoftheYu-Shibabound states 71

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w w D(f f w w D(f)f D(f f w w n w w D(f f 0 0.5 N( w =0)/Nb s 0 10 20 30 40 0 0.5 Dr0 ( mW cm) s++ (b) a =0.5 a =0.6 a =0.7 a =0.8 a =0.9 Figure3-5.(a)Schematicevolutionoftheorderparametera nddensityofstateswith increasingdisorderforasystemwithintra-andinterbands cattering.(b)Top: Fermileveldensityofstates N b (0) (nodalband)asshowninFig. 3-2 (b)vs 0 forvariousvaluesofscatteringratio u = v inananisotropic s state. Bottom:Fermi-leveldensityofstatesforanisotropic s ++ statewith V ab identicalinmagnitudetotheabovepanel,butpositive.Ani sotropyparameter r =1.3 inbothcases. createdbymagneticimpuritiesinconventionalsupercondu ctors,andcanappearfor nonmagneticimpuritiesifthesuperconductinggapchanges sign[ 106 ].Theresidual densityofstates N (0)= Im P k Tr ^ G ( k =0) = (2 ) ( ^ G istheNambuGreen'sfunction) effectivelydeterminesthelow-energythermodynamicbeha vior,sowehaveplotteditfor theanisotropicbandasafunctionofincreasingdisorderin Fig. 3-5 ,forboth s and s ++ states.Intheformercasethereentrantbehaviorisclearly seen. 72

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Thecorrespondingsequencewithincreasingdisorderinthe s penetrationdepth ( T ) wouldbe T T 2 exp( n G = T ) T 2 ,where n G istheminimumgapinthe system.Thissequenceoflow T penetrationdepthshasrecentlybeenobservedbythe Kyotogroup(T.Shibauchi,privatecommunication).Forthe NMRspin-latticerelaxation rate T 1 1 ,theanalogousevolutionshouldbe T 3 T exp( n G = T ) T .The residuallinear T terminthethermalconductivity, ( T 0) = T ,shouldvanishandthen reappearwithincreasingdisorder.Inthe s ++ case,thelaststepineachsequenceis entirelyabsent,sinceinterbandscatteringcannotgiveri setolow-energyboundstate formation.3.5.4RealisticImpurityPotentials Itisclearfromtheaboveanalysisthatwehaveestablishedt hatthereisawide rangeofpossibilitiesforthebehaviorof T c inan s superconductor,aswellasfor low-temperaturepropertieslikethepenetrationdepth,wh endisorderissystematically increased.Tomakemoreprecisestatements,oneneedstohav esomeindependent waytoxthescatteringpotentialofagivenimpurity,andin particulartherelative proportionofinter-tointrabandscattering.Kemper etal. [ 153 ]foundtheratiobetween inter-andintrabandscatteringtobeoforder =0.3 forCoinBa122,whichwould leadaccordingtoFig. 3-2 toacriticalresistivitystrengthofabout 300 ncm ,roughlyin accordwithexperiment[ 135 136 ].OnariandKontani[ 134 ]havemadetheimportant pointthatthe“natural”formulationforamodelimpuritypo tential,i.e.,diagonalinthe basisoftheveFe d orbitals,automaticallyleadstosignicantinterbandsca tteringif onetransformsbacktothebandbasis.However,simpleestim atesshowthatdepending ondetails foron-siteFesubstituentscanvarybetween0.2and1,again leadingas seeninFig. 3-2 toawidevarietyofpossible T c suppressionscenarios. 3.6Conclusions Wehavearguedthat s pairingcannotberuledoutsimplybecausethe T c suppressionisslowaccordingtosomearbitrarycriterion. Thedenitiveexperiments 73

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alongtheselineswillmostprobablyinvolveelectronirrad iation,whereonecanbe reasonablysurethatthedefectscreatedactonlyaspotenti alscatterers.Inthiscase wendcriticalresistivitiesforthedestructionofsuperc onductivitywhichvaryover twoordersofmagnitudeaccordingtotheratioofinterbandt ointrabandscattering. Resultsforthe s statearethennotinconsistentwithexperimentaldata,but proof ofsignchangeoftheorderparameterreliesonknowledgeoft heimpuritypotential, whichrequiresfurther abinitio calculationsforeachdefectineachhost.Asan alternativeapproach,wehaveproposedthatsystematicvar iationofdisordercould giverisetoaclearsignatureof s pairinginthelow-energyFermilevelDOS N (0) Inan s state, N (0) couldincreasewithdisorder,vanishagainduetonodelifti ng, andincreaseagainafterwardduetoimpurityboundstatefor mation.This“reentrant” behavioroftheDOSwillbereectedinthetemperaturedepen denceoflow-temperature quasiparticlepropertieslikethepenetrationdepth,nucl earspinrelaxationtime,or thermalconductivity.Forsomematerials(withgapnodes), thiscouldbea“smoking gun”experimentfor s pairing. 74

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CHAPTER4 SPIN-FLUCTUATIONPAIRINGINFE-BASEDSUPERCONDUCTORS Somepartsofthischapterhavebeenpublishedas“Supercond uctinggapin LiFeAsfromthree-dimensionalspin-uctuationpairingca lculations,”YanWang, A.Kreisel,V.B.Zabolotnyy,S.V.Borisenko,B.B¨uchner,T .A.Maier,P.J.Hirschfeld,and D.J.Scalapino,Phys.Rev.B 88 ,174516(2013). 4.1Motivation ThecompoundLiFeAsisan 18K superconductorthatpresentsseveralnovel featuresrelativetotheotherfamiliesofFepnictides[ 6 ].High-qualitycrystalswith atomicallyatnonpolarsurfacesarenowstraightforwardt oprepare,andthesurface electronicstructurehasbeenshowntobethesameasinthebu lk[ 166 ],suggestingthat thissystemandrelated111materialsareidealonestoapply surfacespectroscopies likeangle-resolvedphotoemission(ARPES)andscanningtu nnelingmicroscopy (STM)[ 127 ].ARPESexperiments[ 119 124 167 ]andelectronicstructurecalculations withindensityfunctionaltheory(DFT)[ 166 – 168 ]reportedearlyonaFermisurfacevery differentfromtheconventionalsetofholeandelectronpoc ketspredictedbyDFTfor theotherFe-basedsuperconductors(Figs. 4-1 and 4-2 ).Inparticular,lessclearnesting ofholeandelectronpocketswasobserved,leadingtothesug gestionthatthiswasthe reasonfortheabsenceofmagnetisminthisparentcompound[ 119 ].Morerecently,de Haas-vanAlphen(dHvA)measurements[ 120 ]showedreasonableagreementwithbulk DFTfororbitsontheelectronpockets. Onecontinuingpuzzlehasbeenthesmalltonegligiblesizeo ftheinner( 1 2 )holepocketsobservedbyARPEScomparedtotherelativelyl argesizesfound inDFT.Recently,local-densityapproximation(LDA)+dyna micalmean-eldtheory (DMFT)calculationshavepresentedapicturewhichsuggest sthatthe111materials areconsiderablymorecorrelatedthan,e.g.,thewell-stud ied122materialsandhave arguedthatstrongerinteractionsleadtoashrinkageofthe innerholepocketsbut 75

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maintenanceoftheelectronpocketsizeandshape[ 169 – 171 ].Thispicturewouldthen accountforbothARPESanddHvAresults,includingveryrece ntdHvAmeasurements whichdetectedverysmallholelikeorbits[ 172 ].However,theextentoftheagreement ofLDA+DMFTtheoryandexperimentfortheholepocketsisobs curedsomewhat bydisagreementsamongthevariouscalculationsastothesi zeoftheinnerpockets, aswellasbythechallengesofresolvingthenear-grazing -centeredholebandsin ARPES. Withinthespin-uctuationmodelforpairingintheFe-base dmaterials,thestructure oftheFermisurfaceiscrucialforsuperconductivityaswel lasmagnetism.Sincethe usualargumentsleadingto s pairing[ 46 ]invokeinterbandpairscatteringbetween electronandholepocketsenhancedbynesting,theabsenceo fnestinginthismaterial wouldseemtoundercutthecaseforan s superconductingstate.Asecondaspect ofthisdiscussionrelatestothespinsymmetryoftheorderp arameter.Whileearly NMRworkreportedastronglytemperature-dependentKnight shiftand 1 = T 1 below T c consistentwith s -wavepairing[ 173 ],Baek etal. [ 174 ]reportedaKnightshiftinsome magneticelddirectionswithno T dependence,suggestiveofequalspin-tripletpairing, whichwouldthenbeconsistentwiththeoreticalanalysispr oposingtripletpairingfor thissystem[ 175 ].Neutronexperimentshavethusfarnotprovidedconclusiv eevidence onewayoranother.Aweakincommensuratespinresonancewas observedininelastic neutronscatteringexperiments[ 176 ]andassociatedwithaprobable s state,butit shouldbenotedthattheexistenceofaspinresonancedoesno tdenitivelyexclude tripletpairing[ 177 ]. Morerecently,someauthors[ 123 ]reporteddetailedARPESmeasurementsof thesuperconductinggapinLiFeAs.Thesemeasurementswere remarkableinthe sensethatwhiletheyshowedthatthesystemhasafullgap,co nsistentwithother low-temperatureprobes[ 119 121 127 178 179 ],theyalsoexhibitedsubstantialgap anisotropyaroundbothelectronandholeFermisurfaces.Si milarbutnotquantitatively 76

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identicalresultsforanisotropicgapswerereportedbyUme zawa etal. [ 124 ]The reportsofstronglyangle-dependentgapsarerelativelyra reamongthemanyARPES measurementsonFe-basedsuperconductors(forexceptions seeRefs.[ 180 ]and[ 181 ]), whereisotropicgapsareoftenreportedevenforthosesyste mswhereitisbelieved fromlow-temperaturetransportmeasurementsthatgapnode sexist(foradiscussion ofthisso-called“ARPESparadox,”seeRef.[ 8 ]).Theexistenceofanisotropyaround someoftheFermi-surfacepocketsofLiFeAswasalsoreporte dbyAllan etal. [ 126 ], whoperformedhigh-resolution,low-temperatureSTMmeasu rementstogetherwith aquasiparticleinterference(QPI)analysiswhichfoundas mallgapnearlyidentical toARPESonthelargeouterhole( r )pocket,withgapminimaalongtheFe-Febond direction(assuggestedinRef.[ 182 ]).Asecond,largergap,alsowithmoderate anisotropy,wasreportedandattributedtoaninner 1,2 holepocket. ToillustratethetypesofgapsfoundbytheARPESexperiment s,wepresentin Fig. 4-2 aschematicrepresentationofthesedatatofamiliarizethe readerwiththe qualitativefeaturesreported.Onecanseethatseveralasp ectsstandout:(a)oscillatory gapsontheouterhole( r )andelectron( )pockets,(b)smallestgaponthe r pocket, and(c)largegapsofroughlyequalaveragesizeoninnerhole ( )andelectron( ) pockets.Therelativephasesofthegaposcillationsonthet wo pocketsarealso striking.Wenoteherethatthemeasurementofthegaponthe pocketisparticularly delicatesincethisbandbarelycrossestheFermilevelnear the Z point,andmaynot crossnear atall. Itisessentialtotheunderstandingofsuperconductivityi nFe-basedsuperconductorstodecidewhetherLiFeAstsintotheusualframe work,withpairing drivenbyspinuctuations,orrepresentsdifferentphysic s.Testingtoseeifthe variousqualitativeandquantitativefeaturesofthegapsr eportedinexperimentcan bereproducedisthereforeanimportantchallengetotheory .Inthischapterwecalculate theeffectivepairingvertexwithintheuctuationexchang eapproximationforthefull 77

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three-dimensional(3D)FermisurfaceofLiFeAsandcompare ourresultsforthe superconductingstateswhichbecomestableatthetransiti ontoprevioustheoryand toexperiment.Tounderstandhowrobusttheseresultsare,w eperformthecalculation forabandstructurettotheARPESresults,whichdifferpri marilyfromDFTdueto themuchlargersizeoftheinnerholepocketinthelatter,as discussedabove,aswell aslargeshiftsintheorbitalcharacteroftheFermisurface s.Inaddition,wecompare ourresultstoaslightlyhole-dopedsystemtosimulatethee ffectofmissingLiatthe LiFeAssurfaceandtocalculationswitha“standard”DFTban dstructure.Wend thatmostaspectsofthesuperconductinggapareremarkably wellreproducedbythe theoryusingtheARPES-derivedelectronicstructuremodel .Ourconclusionisthat thesuperconductivityinLiFeAsisverylikelytobeofthe“c onventional” s type,with signicantanisotropyonbothholeandelectronpockets. 4.2Ten-OrbitalTight-BindingFitsandFermiSurfaces Ourapproachheretothepairingcalculationdifferssomewh atfromthoseperformed formaterialswhereDFTandARPESwereinqualitativelygood agreement.Since thespin-uctuationpairingtheoryinvolvesstatesverycl osetotheFermisurface,the disagreementbetweenDFTandARPESsuggeststhatstrongele ctroniccorrelations mustbeaccountedforatsomelevel.Thesimplestmodicatio noftheusualapproachis toadoptabandstructurewhichtsexperimentwell,aproced urewhichisnotuniquely denedduetothemultibandnatureofthesystem.Wehavechos entobeginwith aten-Feorbitaltight-bindingHamiltonian H ARPES 0 ,ttomeasuredARPESdataona high-qualityLiFeAssample[ 123 ]usingthemethodofRef.[ 183 ],whichwereferto astheARPES-derivedbandstructure.Thehoppingparameter sandthedispersions aregivenintheAppendix B ,andthecomparisonofthetightbindingbandsandFermi surfacecutsareshowninFig. 4-1 .ThefullFermisurfacefromthismodelisthenshown inFigs. 4-3 (a)and 4-3 (e)fortwodifferentdopings, n =6.00 and n =5.90 .Thelatter resultsarepresentedtomimicthepossibleeffectsofLide ciencywhichareknownto 78

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bepresentinthesampleandbecausetheFermi-surfacetopol ogychangesabruptly near n =6.00 .Wendthatthesechangesarepotentiallyquiteimportantf orthe superconductivity,asdiscussedbelow. ItisinterestingtorstcomparetheARPES-derivedFermisu rfaceinFig. 4-3 (a) totheDFTFermisurfacediscussedintheAppendix A.1 sincetheDFTresultsare essentiallythoseusedinearliertwo-dimensional(2D)spi n-uctuationcalculations[ 184 ]. BoththeDFT-andtheARPES-derivedFermisurfacesincludes imilarlargeholepockets ( r )andinnerandouterelectronpockets( in out ).The r pocketsareofcomparablesize andaresimilarinshape.IntheDFT-derivedmodel,theinner andouter pocketscross eachotheralonghighsymmetrydirectionsequivalentto X Y intheone-Fezone.They alsoapproacheachothercloselyatnonzero k z valuesawayfromthehigh-symmetry directionsduetothehybridizationoftheDFTelectronband s,andthisleadstosome k z distortionsandabruptchangesintheirorbitalcharacters with k z .Bycontrast,the k z -dispersionsoftheARPES-derivedelectronpocketsarewea k.Thepocketsonly approacheachotheratthehighsymmetrydirections(wheret heycrossintheabsence ofspin-orbitcoupling),andtheyretaintheirorbitalchar actersalong k z ,asmeasuredby theARPESexperiment. 1 Themaindifferencebeyondtheseshiftsoforbitalcharacte rs andshapeoftheouter Fermisheetsisthemuchsmaller 1,2 holepocketsandthe closingofthe 2 holepocketintheARPES-derivedinner-holeFermisheets.T hedensity ofstates(DOS)attheFermilevelisshowninTable A-1 inAppendix A.1 .Withina scalingfactor r =0.5 ,thetotaldensitiesofstatesandpartialdensityofstates arequite consistentbetweenthesetwomodels. Thecalculatedcarrierconcentrationinthecompensated( n =6.00 )case(where numberofelectrons/Fe=numberofholes/Fe)from H ARPES 0 isroughlyconsistent with 0.18 electrons/Feand 0.2 holes/FefromtheARPESexperimentbyUmezawa 1 Borisenko(unpublished). 79

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etal. [ 124 ].Itisinterestingtonotethatthedifferenceinholeandel ectroncarriersin Ref.[ 124 ]isalreadyahintthatthesurfaceofthesamplemaycontains omeLivacancies andthereforebeslightlyholedoped.Forthe n =5.90 casewehavechosenherefor illustration'ssakecorrespondingelectronandholedensi tiesthatare 0.16 and 0.26 respectively. Ingeneral,theARPES-derivedtight-bindingmodelisaclos ettotheARPES datainRef.[ 123 ]andreproducestheorbitalcharactersonallpockets.Onea pparently minordiscrepancy(whichmayplayamoreimportantrolethan expectedatrstsight; seebelow)isthatduetothecrystalsymmetry,thetwoholeba ndsdispersingnear Z in thetight-bindingHamiltonian H ARPES 0 aredegenerateatthe Z pointandthereforeina nonrelativisticcalculationmustbothcrossorneithercro sstheFermisurface,asshown inFig. 4-4 (toppanel).Apartfromthelarge r pocket,ARPESobservesonlyasingle holelikeband( 2 )crossingtheFermisurfacenear Z ,whileasecondholelikeband( 1 ) ispushedbelow.Thissuggeststhatspin-orbitcoupling,wh ichwillsplitthetwohole bandsasshowninFig. 4-4 (bottompanel),mayberelevanthere.Forthemomentwe neglectthisdistinctionandfocusonthenonrelativisticb andstructure,butwewillreturn toitinthediscussionbelow. 4.3FluctuationExchangePairingModel Withthetight-bindingHamiltonian H 0 intheprevioussection,weincludethelocal interactionviatheten-orbitalHubbard-HundHamiltonian H = H 0 + U X i ` n i ` n i ` # + U 0 X i ` 0 <` n i ` n i ` 0 + J X i ` 0 <` X 0 c y i ` c y i ` 0 0 c i ` 0 c i ` 0 + J 0 X i ` 0 6 = ` c y i ` c y i ` # c i ` 0 # c i ` 0 (4–1) wheretheinteractionparameters U U 0 J J 0 aregiveninthenotationofKuroki et al. [ 185 ].Here ` istheorbitalindexcorrespondingtoFe 3 d -orbitals,and i istheFe atom 80

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site.Thespectralrepresentationoftheone-particleGree n'sfunctionisgivenas G ` 1 ` 2 ( k i n )= X a ` 1 ( k ) a ` 2 ( k ) i n E ( k ) (4–2) wherethematrixelements a ` ( k )= h ` j k i arespectralweightsoftheBlochstate j k i withbandindex andwavevector k intheorbitalbasisand n arethefermionic Matsubarafrequencies.IntermsoftheGreen'sfunction,th eorbitallyresolved noninteractingsusceptibilityis 0` 1 ` 2 ` 3 ` 4 ( q i m )= 1 N X k, i n G ` 4 ` 2 ( k i n ) G ` 1 ` 3 ( k + q i n + i m ), (4–3) where N isthenumberofFeatomsites, =1 = T istheinversetemperatureand m are thebosonicMatsubarafrequencies.Aftersummingthefermi onicMatsubarafrequencies followingtheanalyticcontinuationtotherealaxisofboso nicMatsubarafrequencies,we obtaintheretardedsusceptibility 0` 1 ` 2 ` 3 ` 4 ( q )= 1 N X k, a ` 4 ( k ) a ` 2 ( k ) a ` 1 ( k + q ) a ` 3 ( k + q ) + E ( k ) E ( k + q )+ i 0 + f f [ E ( k )] f [ E ( k + q )] g (4–4) Forthe3D k -sumweusea( 47 47 31 )-point k meshfortheARPES-derivedmodelin theone-FeBrillouinzone(1Fe-BZ);weinterpolatethestat icnoninteractingsusceptibility 0` 1 ` 2 ` 3 ` 4 ( q =0) fromdirectlycalculatedsusceptibilityvaluesona( 20 20 8 )-point q meshinthe1Fe-BZtoperformtheexpensivenumericalcalcul ationwithanepatched Fermisurfaceinsolvingthepairingeigenvalueproblem.Wi thintherandom-phase approximation(RPA)wedenethespin-uctuation( RPA1 )andorbital-uctuation( RPA0 ) partsoftheRPAsusceptibilityas RPA1, ` 1 ` 2 ` 3 ` 4 ( q )= 0 ( q )[1 U s 0 ( q )] 1 ` 1 ` 2 ` 3 ` 4 (4–5) RPA0, ` 1 ` 2 ` 3 ` 4 ( q )= 0 ( q )[1+ U c 0 ( q )] 1 ` 1 ` 2 ` 3 ` 4 (4–6) 81

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suchthattheRPA-enhancedspinsusceptibilityisthengive nbythesum s ( q )= 1 2 X ` 1 ` 2 RPA1, ` 1 ` 1 ` 2 ` 2 ( q ). (4–7) Theinteractionmatrices U s and U c inorbitalspacehavematrixelementsconsistingof linearcombinationsoftheinteractionparameters,andthe irexplicitformsaregiven,e.g., inRef.[ 77 ]. Next,wedenethesingletpairingvertexinbandspace, ij ( k k 0 )=Re X ` 1 ` 2 ` 3 ` 4 a ` 1 i ( k ) a ` 4 i ( k ) [ ` 1 ` 2 ` 3 ` 4 ( k k 0 =0)] a ` 2 j ( k 0 ) a ` 3 j ( k 0 ), (4–8) where k 2 C i and k 0 2 C j arequasiparticlemomentarestrictedtotheelectron orholeFermi-surfacesheets C i and C j and i and j arethebandindicesofthese Fermi-surfacesheets.Thevertexfunctioninorbitalspace ` 1 ` 2 ` 3 ` 4 describestheparticle scatteringofelectronsinorbitals ` 2 ` 3 into ` 1 ` 4 whichisgivenbytheRPAinthe uctuationexchangeformalism[ 186 ]as ` 1 ` 2 ` 3 ` 4 ( k k 0 )= 3 2 U s RPA1 ( k k 0 ) U s + 1 2 U s 1 2 U c RPA0 ( k k 0 ) U c + 1 2 U c ` 1 ` 2 ` 3 ` 4 (4–9) Thesuperconductinggapcanbefactorizedintoanamplitude ( T ) anda normalizedsymmetryfunction g ( k ) .Near T c ,thepairingsymmetryfunction g ( k ) is thestablesolutionmaximizingthedimensionlesspairings trengthfunctional[ 131 ] [ g ( k )] ,whichdetermines T c .Viathevariationalmethod,thisisequivalenttosolvinga n eigenvalueproblemoftheform 1 V G X j I C j dS 0 j v F j ( k 0 ) j ij ( k k 0 ) g ( k 0 )= g ( k ), (4–10) where V G isthevolumeof1Fe-BZ, v F j ( k )= r k E j ( k ) istheFermivelocityonagiven Fermisheetand dS istheareaelementoftheFermisheet.Theeigenfunction g ( k ) correspondstothe theigenvalue andgivesthestructureofthegapatthetransition. 82

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Dening k ? =( k ? ,0) inthecylindricalcoordinates k =( k ? , k z ) andusing dS j v F (k) j = k 2 ? j k ? v F (k) j d dk z isconvenientfordiscretizingtheFermisheetinparameter form k ? = k ? ( k z ) intosmallpatches[ 187 ].Adense( 24 12 )-pointmeshinparameter space f gnf k z g isusedforeachFermipocketinnumericalcalculations,imp lying altogether n k 2500 k pointsdistributedonallFermipockets.Afterchoosingthe lattice constant a asthelengthunit, eV astheenergyunit,and a eV = ~ asthevelocityunit,the eigenvalueproblemEq.( 4–10 )reads 1 16 3 X j I C j ij ( k k 0 ) k 0 2 ? d 0 dk 0 z j k 0 ? v F ( k 0 ) j g ( k 0 )= g ( k ), (4–11) wherethenormalized 2 eigenfunctions g ( k ) aresolvednumericallybytransformingthe integrationkernel(forallFermisheets C i )intoan n k n k matrix.Here,wehaveused thesymmetricpairingvertex ij 1 2 [ ij ( k k 0 )+ ij ( k k 0 )] foraspin-singletpairing statesincewewanttorstexaminewhethertheunconvention alsuperconductingstate oftheLiFeAscompoundandotherFe-basedsuperconductorsi suniversalandcanbe explainedinthesameantiferromagneticspin-uctuationt heorybeforeanyconsideration oftripletpairingorotherapproaches. 4.4ResultsforthePairingState 4.4.1ResultsfortheARPES-DerivedFermiSurface WenowpresentoursolutionstoEq.( 4–11 )fortheleadingpairingeigenvectors (gapfunctions).ForthisworkwexHubbard-Hundparameter s U =0.75eV and J =0.37 U andassumespin-rotationalinvariancetodetermine U 0 and J 0 .These parametersarerelativelystandardintheliteraturemakin guseoftheRPAapproach tothepairingvertex,andwefoundthatchangingthemwithin alimitedrangedoesnot changethequalitativeaspectsofourresultsforthesuperc onductingstate.TheRPA 2 Thenormalizationischosensothat 1 V G P j H C j dS 0 j v F j (k 0 ) j [ g ( k 0 )] 2 =1 83

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susceptibilitythenshowsanenhancedincommensuratepeak around q = (1,0.075, q z ) [Fig. 4-3 (c)]or q = (1,0.175, q z ) [Fig. 4-3 (g)], 3 andthepeakdecreasesweaklyfrom q z =0 to q z = .ThemostimportantresultasshowninFigs. 4-3 (b)and 4-3 (f)and inanotherrepresentationinFigs. 4-3 (d)and 4-3 (h)isthat,usingtheARPES-based bandstructureforbothllingsconsidered,wendan s -wavestatewithanisotropic butfullgapsontheelectron(negativegap)andhole(positi vegap)pockets.Theother leadingeigenvaluecorrespondsinbothcasestoa d xy -wavestatewhichisclosely competing[ 131 ]butisinconsistentwithexperiments,suchthatwedonotin vestigateit furtherhere. Ifwenowconsiderthegapfunctionsfoundonthevariouspock etsindetail,we noticeanumberofstrikingsimilaritiestotheexperimenta lresultssketchedinFig. 4-2 Thefulldetailsofthe s gapfunctionsobtainedareshowninFigs. 4-3 (b)and 4-3 (f) and 4-3 (d)and 4-3 (h),butforthereaderwishingamoreconcisesummary,wehav e showninFig. 4-5 aschematiccomparisonofthecalculatedgapsofthe s states foundat k z = versustheexperimentaldata,usingtheangleconventionde nedin Fig. 4-2 .Takingrstthelarge and r pockets,weseefromFig. 4-5 thattheaverage gapmagnitudeislargeronthe pocketsbyafactorof2orsocomparedto r ,and theaveragegapontheinner pocketisabout20%largerthanthatontheouter pocket,asinexperiment.Thegapson r and pocketsexhibitsignicantanisotropy. Theminimaandmaximaonthe r pocketareatthesameangularpositionsasin experiment,andaresimilartothosefromtheDFT-basedmode ldiscussedinthe Appendix A.1 .Thesegapminimaareparticularlyimportantastheywillde terminethe momentaofthequasiparticleswhichdominatelow-temperat uremeasurements if ,as seeninARPES,thegapon r isthesmallestforthissystem.Theirlocationalongthe 3 InFig. 4-3 (g)forlling n =5.90 ,anotherpeakinthetotalmagneticsusceptibilityis at q = (0.8,0, q z ) 84

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Fe-Febonddirection(ortheequivalentplanein k space)isconsistentwithARPES measurements[ 123 124 ]aswellaswiththequasiparticleinterference[ 126 ]and scanningtunnelingmicroscopyexperiments[ 127 ],accordingtotheinterpretationof thelatterprovidedinRef.[ 182 ].Thegaposcillationsonthe pocketsareingood agreementwithexperimentonthe d xy -orbital-dominatedinnersheetsbutareapparently 180 outofphasewithexperimentalresultsontheouterelectron sheets.Wecomment ontheoriginofthisdiscrepancybelow. Wenowdiscussthegapsonthe holepockets.ARPES[ 123 ]seesonlyone bandcrossingtheFermilevelverycloseto Z ,withalargegapoforder 6meV ,which weassignto 2 .Inourcurrenttight-bindingband,whichobeysthesymmetr iesofthe nonrelativisticDFTapproach,wehavealwaystwo pocketsornone,asmentioned above.Itmaythereforeberoughlyappropriatetospeakofan averagegaponthe 1,2 pocketsintherstanalysis.Withinourcalculationwithth eARPES-derivedband structure,thelargestdiscrepancywithexperimentisseen forour n =6.00 calculation, wherethe gapisfoundtobethesmallestofallthegapsintheproblem.I nthe hole-dopedcase n =5.90 ,thesizeofthegapon 1 increasessignicantly,becoming comparabletoexperiment,butthegapon 2 remainssmall.Itisinterestingtonotethat theDFTcalculation(Appendix A.2 ),whiledisagreeingwithARPESontheexistenceofa -centeredholepocket,producesalargegaponboth 1 and 2 4.4.2AnalysisofGapSizesinTermsofPairingVertex Toanalyzetheoriginoftheremainingdiscrepancieswithex periment,weinvestigate thestructureoftheeffectivepairingvertexbypresenting inFig. 4-6 agraphical representationofthepairingvertex ij ( k k 0 ) matrix.Eachblock ( i j ) intheimage representsamatrix ( k k 0 ) consistingofentrieswhichcorrespondtothevertex ij ( k k 0 ) with k 2 C i and k 0 2 C j .ThemajorityorbitalcharactersalongtheFermisurfaces C i arealsoindicatedintheFig. 4-6 .Inthetablesbelowtheplots,thedensitiesofstates 85

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summedover3Dpocketsandscatteringvertexcomponentsave ragedover ( k k 0 ) onthe k z cutarealsoshown. Forbothdopingsshown,thebrightestsetofblocksisthatre presentingscattering processesamongthethreelargestpockets, r in,out .Onaverage,itisclearvisuallyin Fig. 4-6 andalsofromtheintegratedintensitiesthatthedominants catteringprocesses withinthissetofpocketsoccurfor r in and r r and,toalesserextent, r out Thereareseveralinterestingconclusionstobedrawnfromt hissimpleobservation. First,oneofthecrucialdifferencesbetweenLiFeAsandthe “canonical”1111systems whichwereoriginallyusedtodeducegeneralprinciplesabo utpairingintheFe-based superconductorsistheexistenceofapocket( r )withverylargedensityofstates (Fig. 4-6 )ofdominant d xy character.Thispocketnestsverypoorlywiththe pockets, aspointedoutinRef.[ 119 ],butneverthelessproducestheprimarypairinginteracti on leadingtosuperconductivityinpartduetotheunusuallyla rge d xy contentofthe pocketsintheARPES-derivedbandstructure.Thisisentire lyconsistentwiththe suggestionthatwhilelong-rangemagnetismissuppressedb ythelackofnesting (althoughthiseffectneednotrelyexclusivelyonstatesex actlyattheFermilevel),strong magneticuctuationsremainandareavailableforpairing, whichisof s character becausethe r interactionsarerepulsive.Itisinterestingtonotethatw hilethepair scatteringprocessesconnectingthe r pocketstotherestoftheFermisurfaceare large,thegaponthe r pocketitselfisnot.Thisisaconsequence,withinthetheor yof multibandsuperconductivity,ofthelargedensityofstate sonthe r pocket,asdiscussed inChapter 4.4.3 Second,wenotethattheintrabandscattering r r isalsoquitestrong.These aresmallq processeswhichmayberesponsibleforthetendencytoferro magnetism seeninthesesystems. 4 Althoughwedonotseeenhancementofthe total magnetic 4 B.B¨uchner(unpublished). 86

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susceptibilitynear q =0 (Fig. 4-3 ),thereareevidentlyintraorbitalsusceptibilities including xy xy xy xy whicharelargeatsmall q ,andthepartialdensityofstates N r (0) is thelargestamongallpockets. Finally,wenotethatthestrongangulardependenceoftheve rtexisinducedby thevariationoftheorbitalcontentingeneral,andthe d xz = d yz contentinparticular.As discussedinRefs.[ 188 ]and[ 77 ],thereisastrongtendencyforpairscatteringbetween likeorbitalstobeenhanced,othereffectsbeingequal,acc ountingforthelarge r in scattering.Buteveninthiscasesubdominant xz = yz orbitalsarepresentonthe in sheetswhichleadtotheobservedmodulationviathematrixe lementswhichoccurin Eq.( 4–8 ). Tounderstandtheangularoscillationswithinaphenomenol ogicalpicture,Maiti etal. [ 189 ]ttedthegapsontheelectronpocketsmeasuredbyUmezawa etal. [ 124 ] withtheangledependence inner ( )= 0 (1+ r 2 j cos2 j + r 4 cos4 ), (4–12) outer ( )= 0 (1 r 2 j cos2 j + r 4 cos4 ), (4–13) where isdenedinthecaptionofFig. 4-2 (measuredfromdashed-linedirection),and theyfound(i) r 2 > 0 and(ii) r 4 > 1 4 r 2 .Point(i)isequivalentto inner > outer ,whichis measuredbybothARPESexperiments,andourresultsfrombot htight-bindingmodels alsoagreewithpoint(i).Point(ii)isrelatedtothein-pha sefeatureandtheorientationof gapmaximaonbothpocketsbecause,rst,at =0 d inner = d =0 and d outer = d =0 and,second, d 2 inner = d 2 = 4( r 2 +4 r 4 ) and d 2 outer = d 2 =4( r 2 4 r 4 ) r 4 > 1 4 r 2 meansbothinnerandouterpocketshavemaximaat =0 ,andhencetheyarein phase.Considering r 2 > 0 andthegapontheouterpocketoscillatesstrongerthanthe innerpocket(largercurvatureat =0 )inourresults,areasonablerangefor r 4 atall k z is r 4 1 4 r 2 .Thesignof r 2 isdeterminedbytheangledependenceofthepairing interactionandispositiveinthecasewheretheinteractio nbetweenelectronandhole 87

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pocketsisdominant[ 189 ].Ournumericalresultssuggestthesameconclusionasthe ARPESexperiments.Thediscrepancyinthephaseoftheoscil lationsontheouter pocketisattributabletothe“wrong”signofthemoresensit iveparameter[ 189 ] r 4 obtainedwithinourcalculations. Wenowturntothemoredelicateissueofthepairing-vertexc omponentsconnecting the pocketstotherestoftheFermisurface.Itisclearfromboth theplotsandtable correspondingtoFigs. 4-6 (a)and 4-6 (b)thatthesearenegligibleinthecompensated case n =6.00 ,inFig. 4-6 (a)simplybecause H ARPES 0 containsno pocketsat k z =0 andinFig. 4-6 (b)becausethedensitiesofstatesontheseclosed3Dpocket sare extremelysmall.In2Dmodels,wheredensitiesofstatesten dtobeweaklydependent onpocketsize,theseeffectsaresuppressed.Wediscussthe connectionofthesmall gaponthe pocketstothecorrespondingcomponentsofthevertexbelow .Forthe moment,wenotesimplythattheeffectofholedopingto n =5.90 showninFigs. 4-6 (c) and 4-6 (d)clearlyenhancesthescatteringofpairsonthe pocketstothe pockets, particularlyto in .AsseeninFigs. 4-3 (a)and 4-3 (e),holedopingbyasmallamount (5%Fe)transformsthesmall Z -centered pocketsintotwonarrowconcentrictubes andtherebyenhancestheDOSonthe pockets.Whilethe n =5.90 caseisnominally inconsistentwiththeARPESobservationofno -typeFermisurfacesat k z =0 ,itis clearthatthedeterminationoftheholedispersionnear Z becomesquitechallenging whenthebandsaregrazingtheFermilevel.Itissignicantt hattheresultsfortheDFT analysisgiveninAppendix A.2 alsogivelargegapsonthe pockets,althoughthe Fermisurfaceoftheholepocketsdisagreesqualitativelyw ithARPES.Takentogether, theseresultssuggestthatthe interactionisenhancedandthegaponthe pocket islargeonlyifstatesnear of xz = yz charactercontributestronglytopairing.Thisoccurs whentheFermisurfaceincludesanopen(cylindrical) 1 pocketandalsowhenthe rangeofpairingisexpandedtoincludestatesawayfromtheF ermilevel,asdiscussed below. 88

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4.4.3Discussion:ToyModelforGapSizes TounderstandtherelativesizesofthegapsonthevariousFe rmisurfacesheets, oneneedstocombineknowledgeofthepairingvertexfunctio ndiscussedabovewith thedensitiesofstatesoneachsheet.Here,asimpliedpict urecanhelpusunderstand whycertaingapsarelargeandothersaresmall.Weneglectfo rthisdiscussionthe momentumdependenceofthegapeigenfunctions,densitieso fstates,andverticesover theindividualFermisurfacesheets.Ifweareprimarilyint erestedingap sizes ,agood approximationtothegapequation( 4–11 )isthengivenby g i X j g j N j ij (4–14) where g i nowdenotesthegapand N i thedensityofstatesattheFermilevelinthe i thband.Webeginbydiscussingthequestionofhowthegapont iny Z -centered(or -centered)holepocketscanbecomelarge,asseenbyARPES[ 123 ].Wereintraband scatteringprocessesdominant,thetinyDOSonthe pocketswouldgenericallycreate anextremelysmallgap.Sinceinterbandscatteringismorei mportant,inthesituation wheretheDOSonthe pocketsissmall,thegapon willbedeterminedbyscattering fromtheothermajorbands,inparticular r and in,out asseeninFig. 4-6 Insuchasituation,wehaveapproximately g g r N r r X g N (4–15) where sumsoverbothinnerandouter pockets.Sincethestateisan s state drivenbyrepulsiveinterbandinteractions,therstandse condtermstendtocancel eachother.Largegapscanthenbeobtainedifparametersare chosensuchthatthe contributionfromthe r pocketisminimized.Aswehaveseenabove,however,inthe currentARPES-derivedmodel,whilethescatteringof statestothe pocketsis muchstronger,the r densityofstatesissignicantlylarger,suchthatthetwot ermsin Eq.( 4–15 )arecomparableandthereforemostlycanceleachother.Asc anbeseenby 89

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comparingthehole-dopedcasewiththecompensatedcasewit hthetablesfor in Fig. 4-6 ,themaineffectoftheholedopingonthebalanceofthetwote rmsinEq.( 4–15 ) isduetotheenhancementof in byafactorof2. 4.5Conclusions Wehaveperformed3Dcalculationsofthesuperconductingpa irstateintheLiFeAs compound,oneofthefewmaterialswhereARPESexperimentsi ndicatesignicantgap anisotropy,possiblyduetoreduceddiffusescatteringfro mtheveryclean,nonpolar surface.SincetheinnerholepocketsoftheFermisurfaceof thismaterialarethoughtto bestronglyrenormalizedbyinteractions,weusedastheinp utatight-bindingmodeltto ARPESdatareproducingboththebandstructureandorbitalp olarizationmeasurements attheFermisurface.Ourcalculationsndagapstructurewh ichchangessignbetween theholeandelectronpocketsandreproducesemiquantitati velytherelativegapsizes onthethreelargestpockets,alongwiththeoscillatorybeh aviorseen.Weperformeda carefulanalysisofthestructureofthepair-scatteringve rtextounderstandthestructure ofthesepairstates.ThegapfunctionobservedbyARPESonth emainpocketscan thenbeunderstoodentirelyintermsoftherepulsiveinterb andinteractionswithinthe spin-uctuationapproach.Ontheouterelectronpocket,ad ifferenceinthesignofthe oscillationswithrespecttoexperimentcanbetracedtoate rminthephenomenology ofMaiti etal. [ 189 ]whichdependsverysensitivelyonthebalancebetweenintr a-and interpocketinteractions. Ourresultsdifferfromexperimentinoneimportantrespect ,namelythesmallsize ofthegapontheinnerhole( )pocketswend,incontrasttothelargegapobserved inRef.[ 123 ].Wehavediscussedhere,andinAppendicesAandB,variousm odel Fermisurfaceswhichtendtowardsgivingsignicantlylarg er gapsanddeducedthat inclusionofthe xz = yz statesinthepairingnearthe pointoftheBrillouinzoneappears tobeessential.Whilethesemodelsdonotappeartobefullyc onsistentwiththeFermi surfacefoundbyARPES,theypointthewaytowardsidentifyi ngmissingingredientsin 90

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thetheory.Inparticular,sincethe pocketsinthismaterialaretinyandveryclosetoa Lifschitztransition,itmaybenecessaryinthissystemtoa ccountforstatesslightlyaway fromtheFermilevelinordertoreproducetheoverallgapstr ucture. Duetotheremarkableagreementoftherobustpartofthegaps tructureonthe mainpockets,weconcludethatthepairinginLiFeAshasesse ntiallythesameoriginas inotherFe-basedsuperconductors,despitethefactthatth ereisnonestingapparent attheFermisurface.Wepointoutthatthemaindifferencebe tweenLiFeAsandthe paradigmatic1111systemsisthepredominanceofthescatte ringbetweenthehole r Fermipocketandtheelectron pockets,allofwhichhavesubstantial xy orbital character;pure xz = yz scatteringissubdominant.Astrong d xy intrapocketinteraction mayexplaintheferromagneticcorrelationsobservedinexp eriment,despitethelackofa q =0 peakinthetotalmagneticsusceptibility. 91

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k x (1 / ) Bi nd i ng en er gy (eV ) 0.0 0.2 0.0 2 .0 1 .5 1 .0 0.5 k y (1 / ) 0.0 0.5 1.0 0.5 2.0 1.5 Mo m en t u m k a) b) c) d) e) f) b c d e f Figure4-1.Comparisonofthetight-bindingbandsandARPES databoth(a)atthe Fermisurfaceand(b)–(f)inenergy–momentumcutsfor k z = = 2 .Theblack arrowsin(a)denotethepositionsofseveralrepresentativ eenergy momentumcuts.FordemonstrationpurposesBorisenko etal. [ 125 ]used oneofthehigh-qualityFermi-surfacemapsfromRef.[ 123 ],althoughto recoveradditionalinformationon k z dispersionmoredatawithvarious h wereused.ForfurtherdetailsseeAppendix B 92

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(a) (b) (c) Figure4-2.(a)ThecutoftheFermisurfaceoftheARPES-deri vedtight-bindingmodel (lling n =6.00 )at k z = toshowthedenitionofthevariouspocketsand theangle thatparametrizesthesurfacepoints.Sketchoftheresults ofthe gap j ( ) j asseeninrecentARPESexperimentscompiledfromthets providedin(b)Ref.[ 124 ]and(c)Ref.[ 123 ]. 93

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d xz d yz d xy d yz d xz d yz d xy d xz d xy k x k y S S k x k z k y S S S S k x k z k y S S S S k x k y S S d xz d yz d xy d yz d xz k x k y S S k x k z k y S S S S J J E out E out E in E in D D D D k x k z k y S S S S k x k y S S d yz d xy d xz d xy J J E out E out E in E in D D D D q x q y S S S S F 0 ( q x q y q z ) q x q y S S S S F 0 ( q x q y q z S ) q x q y S S S S F 0 ( q x q y q z ) F RPA ( q x q y q z ) q x q y S S S S F 0 ( q x q y q z S ) q x q y S S S S F RPA ( q x q y q z S ) F RPA ( q x q y q z ) q x q y S S S S F RPA ( q x q y q z S ) q x q y S S S S q x q y S S S S -1.2 0 1.2 a 1 a 2 g b out b in -1.2 0 1.2 -1.2 0 1.2 k z = k z =0 : 5 k z =0 02 02 02 02 02 -1.2 0 1.2 a 1 a 2 g b out b in -1.2 0 1.2 -1.2 0 1.2 k z = k z =0 : 5 k z =0 02 02 02 02 02 k pole z =0 : 52 (a)ARPES-derivedFermisurface, n =6 : 00(e)ARPES-derivedFermisurface, n =5 : 90 (b) n =6 : 00, g ( k ): 1 =1 : 04 ;U =0 : 75eV ;J =0 : 37 U (f) n =5 : 90, g ( k ): 2 =0 : 62 ;U =0 : 75eV ;J =0 : 37 U (c) (g) (d) n =6 : 00, g ( k ): 1 =1 : 04 ;U =0 : 75eV ;J =0 : 37 U (h) n =5 : 90, g ( k ): 2 =0 : 62 ;U =0 : 75eV ;J =0 : 37 U 94

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Figure4-3.FermisurfaceofLiFeAsfrom H ARPES 0 at(a)lling n =6.00 and(e) n =5.90 plottedinthecoordinatesoftheone-FeBrillouinzoneastw osets,outer (left)andinner(right)pockets.Majorityorbitalweights arelabeledbycolors, asshown.Notethesmallinnermost,holepocket 1 withrotationaxis Z ( M A )hasbeenarticiallydisplacedfromitspositionalongthe k x axisfor betterviewingin(a)and(e).(b)and(f)arethegapsymmetry functions g ( k ) correspondingtotheleadingeigenvalues( s wave)andinteraction parametersshowninthegure.(c)and(g)arethecorrespond ing noninteractingspinsusceptibilityandRPAspinsusceptib ilityat q z =0, .In theRPAsusceptibilityplot,athinwhitelineisplottedalo ngthepath ( q y q z =0) or ( q y q z = ) ,itsprojectiononthe q y s planeisplottedas athickorangeline,andtheredtriangleindicatesthepeakp osition.(d)and (h)aretheangledependenceof g ( k ) onthepocketsindicatedat k z =0,0.5 .In(d)thegapvalueon pocketsatthepoleisplottedsince thesepocketsdonotextendto k z =0.5

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-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 M¡ZA E [eV]n =6 : 00 n =5 : 95 n =5 : 90 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 M¡ZA E [eV]n =6 : 00 n =5 : 95 n =5 : 90 dxz dyz dxy dyz dxz Figure4-4.Thebandstructuresalongthe M Z A pathintheone-FeBrillouinzonefor (top)theARPES-derivedmodeland(bottom)thesamemodelwi ththe approximatespin-orbitcouplingterm[ 190 ] 3 d Fe P i L zi S z i ,with 3 d Fe =0.025eV Thecolorencodesthemajororbitalcharacters,asindicate dbythe horizontalcolorbar.Thedashedlinesmarkthecorrespondi ngFermienergy atlling n =6.00 5.95 ,and 5.90 96

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(a) (b) Figure4-5.Comparisonbetweenthegapfunctionpredictedi nthepresentwork(solid lines)at k z = andtheexperimentalndingsof j ( ) j fromRef.[ 123 ] (dashedlines).ResultoftheARPES-derivedmodelat(a)ll ing n =6.00 and (b)atlling n =5.90 97

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J EoutEinDDSSSSS JEoutEinDDJ EoutEinDDSSSSS JEoutEinDD J E out E in D D SSSSS JE out E in D D J E out E in D D SSSSS JE out E in D D SSSSS (a) n =6.00 k z =0 (b) n =6.00 k z = (c) n =5.90 k z =0 (d) n =5.90 k z = ARPES-basedmodel,lling n =6.00 bandDOSgap pairingvertex N (0) g 1 2 r out in 1 0.010.19 0.450.440.320.620.76 2 0.150.16 0.440.460.190.600.38 r 0.700.32 0.320.192.141.813.11 out 0.34 0.46 0.620.601.810.550.77 in 0.15 0.79 0.760.383.110.771.38 ARPES-basedmodel,lling n =5.90 bandDOSgap pairingvertex N (0) g 1 2 r out in 1 0.020.52 0.430.370.490.821.52 2 0.240.20 0.370.410.200.650.30 r 0.610.26 0.490.201.991.292.44 out 0.36 0.37 0.820.651.290.500.67 in 0.14 0.69 1.520.302.440.671.21 Figure4-6.Componentsofthepairingvertex ij ( k k 0 ) matrixresultinginthepairing functionplottedinFig. 4-3 ,fromARPES-basedmodelat(a)and(b)lling n =6.00 and(c)and(d)lling n =5.90 ,wherethevalueisproportionalto thebrightnessofthecolor.Therowsandcolumnsofthetiles of(a)–(d) correspondtoFermipoints k 2 C i and k 0 2 C j where C i j arethe k z cutsof Fermisheets 1,2 r at and out in atthe X point.Here k z = k 0 z =0 for(a) and(c)and k z = k 0 z = for(b)and(d).Theangulardependenceofthe majororbitalcharactersoftheseFermipointsarelabeledb ycoloras d xz (red), d yz (green),and d xy (blue),asshowninthehorizontalandvertical colorbarsattachedtoeachpanel.Thetablesshowthedensit yofstates summedoverthreedimensions(3DDOS),theangularaveraged pairing vertex P k,k 0 ( k k 0 ) = n k = n k 0 at k z = (where n k isthenumberof k pointsinthesum),andtheapproximatedmodelgapsolvedfro mlinearized gapequationsusingthe3DDOSandangularaveragedpairingv ertexat k z = 98

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CHAPTER5 FINALCONCLUSIONS Inthisworkwehaveexploredthegapsymmetryandgapstructu reinFe-based superconductorsbytheoreticalcalculationsthataccount fordifferentexperiments. Wehaveconsideredthelowenergyexcitationsinasupercond uctingstateincluding thepossibilityofgapnodes(Volovikeffectinthemagnetic elddependenceofthe specicheatcoefcientofoptimallydopedBaFe 2 (As 1 x P x ) 2 ),therelationbetween thegapstructureandextendedlowenergy“bound”statesnea rasinglevortexcore (scanningtunnelingspectroscopy(STS)onLiFeAs),thepoi ntlikeimpurityeffects onthesuperconductingtransitiontemperatureandlowener gydensityofstates( T c suppressionrateontheelectronirradiatedBaFe 2 As 2 ),andthesuperconductinggap magnitudesbyangle-resolvedphotoemission(ARPES)(spin -uctuationcalculation ofthegapsymmetryfunctionofLiFeAscomparingwithARPESm easurements). Theseexperimentsandcalculationsconsistentlyassertth atthegapinFe-based superconductorshasnontrivialstructureinmomentumspac e,thengerprintof unconventionalandhightemperaturesuperconductorswher e“aconventionalnotion thatthe s -wavegapisnodeless,the d -wavehasfournodes,etc.,”doesn'tapply[ 9 ] (seethediscussionbelow).Theantiferromagneticspin-u ctuationpairingtheory predictsanisotropic s gap[ 46 ]basedonthenestingbetweentheholelikeand electronlikepocketsconnectingby ( ,0) wave-vectorwithorwithoutnodesdepending onthedopingandtheresultingnestingcondition.Ourspinuctuationcalculationon LiFeAs[ 125 ]furtherafrmsthatthesamepairingmechanismworksdespi tealack ofgoodFermisurfacenesting,givingaleading s -wavepairinginstability.However, thesuperconductivityinthealkalimetalironselenidefam ily(e.g.,K x Fe 2 y Se 2 ;see Ref.[ 191 ]forarecentreview)doesstimulatemoreercedebatesonth ispairing mechanismbasedontheFermisurfacenestingsincethereisn oholepocketatthe Brillouinzonecenter point[ 192 ],althoughaholepocketat Z pointwithanodeless 99

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gapisfoundbyARPES[ 193 ].Spin-uctuationandfunctionalrenormalizationgroup calculationsintwodimensionspredictedanodeless d -wavegap[ 130 194 195 ],but lateritwasshownthatthedifferentcrystalsymmetryofthe 122compoundalsoenforces horizontalorverticalnodallines[ 196 197 ]inthe d -wavestate,andthespin-uctuation calculationinthreedimensions[ 190 ]hasfoundsucha d -wavestateistheleading instabilityandthenodallinesaresonarrowontheFermisur facethatitmaycause difcultytomeasuretheminexperiment(forexample,theaf orementionedARPES[ 193 ] foundanodelessgapontheholepocketsuggestingan s -wavestate).Mazin[ 196 ]also proposedanother s -wavestateforantiferromagneticspin-uctuationpairin gwithonly electronpockets,thebonding-antibonding s -wavestatewithsignchangebetweeninner andouterelectronpocketsinthetwo-Fezone,whichisexplo redinRefs.[ 190 197 ]asa competingstatewith d wavewithrespecttothestrengthofthehybridizationparam eter dependingontheout-planehopping,spin-orbitcoupling,e tc.Finallysinglesign s ++ stateisanotherpossibilitywhentheorbitaluctuationsa reincludedaswell[ 198 ].The complexityofnodalstructureinthesecompoundswithonlye lectronpocketsrequires morecarefulexperimentstotellwhetherthegapis s waveor d waveorwhetherthere existsatransitionfrom s waveto d wave,thatis,atransitionfrom A 1 g to B 1 g symmetry, uponelectrondoping,andtheexperimentresultswillsurel ytestthetheoryofthepairing mechanism. ThesuperconductivityinthemonolayerFeSelmonSrTiO 3 substratedeserves furtherattentionsinceithasthesimpleststructurewhile settingtherecordofthe highest T c ofFe-basedsuperconductors,around 65K observedbySTS[ 199 ]and ARPES[ 200 – 202 ](the exsitu transportmeasurementsfound T c 40K inamonolayer FeSelmcoveredbynon-superconductingFeTelayersforpro tection[ 203 ]).Sucha high T c inthemonolayerFeSe/SrTiO 3 isremarkablesincethebulkFeSehasamuch lower T c 8K atambientpressure[ 76 ].SimilartoK x Fe 2 y Se 2 ,thesuperconducting monolayerFeSelmhasonlyelectronpocketsatthezonecorn erasshownbythe 100

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aforementionedARPESexperiments.The T c denedbyARPESgapclosingcanbe renedupto 75K usingdifferentsubstrates[ 204 205 ].RecentARPESexperiment[ 206 ] identiedinmonolayerFeSe/SrTiO 3 theevolutionofaMottinsulatingparentcompound toasuperconductorwiththeincreasingdopingcausedbydif ferentannealingsequences invacuum,whichestablishedacloserlinktocuprates.This suggestssomecommon physicsforsuperconductivityincupratesandFe-basedsup erconductors. Tosumup,AFe 2 Se 2 andmonolayerFeSeindicatetheexistenceofmore terrae incognitae inFe-basedsuperconductors.BytheprincipleofWilliamof Ockham,it isreasonabletobelievethereisacommon(essentialorqual itative)mechanismfor differentfamiliesofFe-basedsuperconductorswhichalso appliestocuprates.Thegap structureandgapsymmetryisthengerprintforsuchamecha nismwhilepredicting newsuperconductorsistherealtouchstoneforthetheoryof suchamechanism. 101

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APPENDIXA SPIN-FLUCTUATIONCALCULATIONFORDFT-DERIVEDFERMISURFA CE A.1ElectronicStructureofLiFeAsfromDensityFunctional Theory ThebandstructurefromDFTfortheLiFeAsparentcompoundis calculatedusing thequantumESPRESSOpackage.Theexperimentallydeterminedlatticeparameter s usedinthecalculationaretakenfromTableIinRef.[ 207 ],includinglatticeconstants a =3.7914 A c =6.3639 A andtheinternalcoordinatesfortheLiatoms z Li = 0.8459 andtheAsatoms z As =0.2635 .NextweobtaintheDFTderivedten-orbital tight-bindingHamiltonianmodel H DFT 0 byprojectingthebandsneartheFermienergy ontheten 3 d -orbitalsofthetwoFeatomsintheprimitivecelloftheLiFe Ascrystal usingmaximallylocalizedWannierfunctionscomputedusin gtheWANNIER90package. TheFermisurfacefromthismodelisshowninFig. A-1 (a),wherethecolorsencode theorbitalcharacter.TheFermisurfacesheetsoftheten-o rbitalmodelareplotted usinga repeated-zone schemeofthetwo-FeBrillouinzone(2Fe-BZ)inthecoordina tes ( k x k y k z ) ofthe1Fe-BZ,andthecubein k spaceinFig. A-1 (a)enclosesthevolumeof the1Fe-BZ.Thisrepresentationisconvenientforlatercal culationsincethesusceptibility isonlyaperiodicfunctioninthe1Fe-BZ.Fortheconvenienc eoflaterdiscussion,we denotethetwoholepocketsatthe (0,0,0) [or M ( ,0) ]pointas 1 / 2 andtwo electronpocketsatthe X (or Y )pointas out / in .TheDOSattheFermilevelisshownin Table A-1 ,incomparisonwiththatofARPES-derivedmodel. A.2PairingStateforDFT-DerivedFermiSurface AlthoughtheFermisurfacepredictedbyDFTdiffersinsomee ssentialrespects fromthatfoundinARPES,itisneverthelessusefultocalcul atethegapfunctionswhich arisewithinthespin-uctuationtheoryforthiselectroni cstructuretogetasenseof howmuchthegapvariesforsmallchangesintheelectronicst ructureandtocompare withearlier2DtheoreticalcalculationsusingaDFT-deriv edFermisurface[ 184 ].As showninFig. A-1 ,for U =0.88eV and J =0.25 U ,wendan s -wavestate( 1 = 102

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TableA-1.LiFeAsdensityofstates(twoFeatoms,twospins) attheFermilevelfrom(a) theten-orbitalDFT-basedmodel H DFT 0 and(b)theARPES-basedmodel H ARPES 0 alongwith(c)thedensityofstatesin(b)scaledbyafactor r =0.5 1 2 r out in total (a) 0.0400.5540.6600.6100.3772.241 (b) 0.0380.5922.7821.3770.5945.383 (c) 0.0190.2961.3910.6890.2972.692 0.237 )withanisotropicfullgapsontheelectron(negativegap)a ndhole(positivegap) pockets,asshowninFigs. A-1 (b)and A-1 (d).(Thesecondeigenvalue 2 =0.1006 isa d x 2 y 2 -wavestate.)The s -wavestateisdrivenbytheenhanced commensurate peakat q =( ,0, q z ) intheRPAsusceptibility,seeFig. A-1 (c).Thispeakvaluehasamoderate q z dependenceandbecomessmallerat q z = ,whichmeansthegapstructurewillnot changetoomuchalong k z Thegapsontheholepockets 2 and r exhibitmoderate k z dependence.Thegap minimaonthe r pocketareinthe k x k y orFe-Fedirections.Thegapontheclosed 1 pocketisamongthelargestones,althoughtheDOSofthe 1 pocketisthesmallest, andthisgapshowsstrong k z dependencenearthepoleofthepocket.Next,thegapson theinner in andouter out pocketsseemtobeintertwinedandcorrelated:near k z =0 thegapontheinnerpocketissmallerthanthegapontheouter pocket,whilenear k z = theorderisipped[seeFig. A-1 (d)forgapsat k z =0, ],buttheycoincideatthe pointswheretwoFermipocketstoucheachother.Last,while thegapmagnitudesfrom ourfull3Dcalculationaresubstantiallysimilartothoseo btainedusing2Dfunctional renormalizationgroupcalculationsbyPlatt etal. [ 184 ]at k z =0 ,at k z = wend qualitativelydifferentholepocketgaps,indicatingthei mportanceof3Dpair-scattering processes. 103

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d xz d yz d xy d yz d xz d yz d xy d xz d xy k x k y S S k x k z k y S S S k x k z k y S S S S k x k y S S S J J E out E out E in E in D D D D q x q y S S S S F 0 ( q x q y q z ) F RPA ( q x q y q z ) q x q y S S S S F 0 ( q x q y q z S ) q x q y S S S S F RPA ( q x q y q z S ) q x q y S S S S -1.2 0 1.2 a 1 a 2 g b out b in -1.2 0 1.2 -1.2 0 1.2 k z = k z =0 : 5 k z =0 02 02 02 02 02 (a)DFT-derivedFermisurface, n =6 : 00 (b) g ( k ): 1 =0 : 237 ;U =0 : 88eV ;J =0 : 25 U (c) (d) g ( k ): 1 =0 : 237 ;U =0 : 88eV ;J =0 : 25 U 104

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FigureA-1.(a)FermisurfaceofLiFeAsfrom H DFT 0 plottedinthecoordinatesofthe one-FeBrillouinzoneastwosets,outer(left)andinner(ri ght)pockets. Majorityorbitalweightsarelabeledbycolorsasshown.Not ethesmall innermost,holepocket 1 withtherotationaxis Z (or M A )hasbeen articiallydisplacedfromitspositionalongthe k x axisforbetterviewing.(b) Thegapsymmetryfunctions g ( k ) correspondingtotheleadingeigenvalues ( s wave)withinteractionparametersshowninthegure.(c)Th e correspondingnoninteractingspinsusceptibilityandRPA spinsusceptibility at q z =0, .(d)Theangledependenceof g ( k ) onthepocketsindicatedat k z =0,0.5

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APPENDIXB FITTINGPARAMETERSFORTEN-ORBITALTIGHT-BINDINGMODEL H ARPES 0 Inthefollowing,wegivetheHamiltonianmatrixofthetight -bindingmodel H ARPES 0 fromRef.[ 183 ](withcorrectionsandminorchanges)andthehoppingparam eters t rst `` 0 whichareobtainedbyttingtheARPESmeasuredbandstructu reforLiFeAsusingthat tight-bindingmodel.Thehoppingparametersweretunedtoo ptimallyreproduceavast setofexperimentaldatameasuredalonghighsymmetrydirec tionsaswellascomplete Fermisurfacemaps,cuttingthebandstructureatarbitrary anglestocrystallographic axes.OnesuchmapisshowninFig. 4-1 .Topindown k z dispersions,highsymmetry cutsmeasuredwithdifferentexcitationenergieswereused .Here ` ` 0 areorbitalindices with 1= d xy 2= d x 2 y 2 3= id xz 4= id yz 5= d z 2 fortherstFewithintheunitcell and 6= d xy 7= d x 2 y 2 8= id xz 9= id yz 10= d z 2 forthesecondFe. r s t are integersdenotingahoppingdistance r T x + s T y + t R 3 where R 1 R 2 R 3 arelattice basisvectorsand T x T y arebasisvectorsfortheone-Feunitcell.Specically,weh ave T x = 1 2 ( R 1 R 2 ) T y = 1 2 ( R 1 + R 2 ) ,andaccordinglyinthereciprocalspace,wehave k 1 = k x + k y k 2 = k x + k y k 3 = k z ,wherethewave-numbercomponentsarescaledby choosingthelatticeconstant a =1 .Theentirecalculationisdonewith k 1,2,3 (in2Fe-BZ) andthenplottedwith k x y z (in1Fe-BZusingarepeated-zonescheme),suchas,for example,inFig. 4-3 (a). H ARPES 0 isgivenintheblockmatrixformasfollows: H ARPES 0 = 0B@ H ++ H + H + H ++ 1CA (B–1) Hereanasterisk( )meanscomplexconjugate.Eachelementof H ++ H + isgivenin twoparts:the2Dpartandthe3Dpart. Forthe2DpartoftheHamiltonian, H ++ 11 = 1 +2 t 11 11 (cos k 1 +cos k 2 )+2 t 20 11 (cos2 k x +cos2 k y ), H ++ 12 =0, 106

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H ++ 13 =2 it 11 13 (sin k 1 sin k 2 ), H ++ 14 =2 it 11 13 (sin k 1 +sin k 2 ), H ++ 15 =2 t 11 15 (cos k 1 cos k 2 ), H ++ 22 = 2 +2 t 11 22 (cos k 1 +cos k 2 ), H ++ 23 =2 it 11 23 (sin k 1 +sin k 2 ), H ++ 24 =2 it 11 23 ( sin k 1 +sin k 2 ), H ++ 25 =0, H ++ 33 = 3 +2 t 11 33 (cos k 1 +cos k 2 )+2 t 20 33 cos2 k x +2 t 02 33 cos2 k y +4 t 22 33 cos2 k x cos2 k y H ++ 34 =2 t 11 34 (cos k 1 cos k 2 ), H ++ 35 =2 it 11 35 (sin k 1 +sin k 2 ), H ++ 44 = 3 +2 t 11 33 (cos k 1 +cos k 2 )+2 t 02 33 cos2 k x +2 t 20 33 cos2 k y +4 t 22 33 cos2 k x cos2 k y H ++ 45 =2 it 11 35 (sin k 1 sin k 2 ), H ++ 55 = 5 H ++ ji =( H ++ ij ) H + 16 =2 t 10 16 (cos k x +cos k y ) +2 t 21 16 [(cos k 1 +cos k 2 )(cos k x +cos k y ) sin k 1 (sin k x +sin k y )+sin k 2 (sin k x sin k y )], H + 17 =0, H + 18 =2 it 10 18 sin k x H + 19 =2 it 10 18 sin k y H + 1,10 =0, H + 27 =2 t 10 27 (cos k x +cos k y ), 107

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H + 28 = 2 it 10 29 sin k y H + 29 =2 it 10 29 sin k x H + 2,10 =2 t 10 2,10 (cos k x cos k y ), H + 38 =2 t 10 38 cos k x +2 t 10 49 cos k y +2 t 21 38 [(cos k 1 +cos k 2 )cos k x (sin k 1 sin k 2 )sin k x ] +2 t 21 49 [(cos k 1 +cos k 2 )cos k y (sin k 1 +sin k 2 )sin k y ], H + 39 =0, H + 3,10 =2 it 10 4,10 sin k y H + 49 =2 t 10 49 cos k x +2 t 10 38 cos k y +2 t 21 49 [(cos k 1 +cos k 2 )cos k x (sin k 1 sin k 2 )sin k x ] +2 t 21 38 [(cos k 1 +cos k 2 )cos k y (sin k 1 +sin k 2 )sin k y ], H + 4,10 =2 it 10 4,10 sin k x H + 5,10 =0. (B–2) Forthe3DpartoftheHamiltonian, H ++ 11 = H ++ 11 +[2 t 001 11 +4 t 111 11 (cos k 1 +cos k 2 ) +4 t 201 11 (cos2 k x +cos2 k y )]cos k z H ++ 13 = H ++ 13 4 t 201 14 sin2 k y sin k z H ++ 14 = H ++ 14 4 t 201 14 sin2 k x sin k z H ++ 33 = H ++ 33 +[2 t 001 33 +4 t 201 33 cos2 k x +4 t 021 33 cos2 k y ]cos k z H ++ 44 = H ++ 44 +[2 t 001 33 +4 t 021 33 cos2 k x +4 t 201 33 cos2 k y ]cos k z H + 16 = H + 16 +4 t 101 16 (cos k x +cos k y )cos k z +2 t 121 16 f [cos( k 1 + k y )+cos( k 1 + k x )]exp( ik z ) +[cos( k 2 + k y )+cos( k 2 k x )]exp( ik z ) g 108

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H + 18 = H + 18 +4 it 101 18 sin k x cos k z 4 t 101 19 sin k y sin k z +2 it 211 19 [sin( k 1 + k y )exp( ik z ) sin( k 2 + k y )exp( ik z )], H + 19 = H + 19 +4 it 101 18 sin k y cos k z 4 t 101 19 sin k x sin k z +2 it 211 19 [sin( k 1 + k x )exp( ik z )+sin( k 2 k x )exp( ik z )], H + 38 = H + 38 +4( t 101 38 cos k x + t 101 49 cos k y )cos k z +2 t 121 38 [cos( k 1 + k x )exp( ik z )+cos( k 2 k x )exp( ik z )] +2 t 121 49 [cos( k 1 + k y )exp( ik z )+cos( k 2 + k y )exp( ik z )], H + 39 = H + 39 +4 it 101 39 (cos k x +cos k y )sin k z H + 49 = H + 49 +4( t 101 49 cos k x + t 101 38 cos k y )cos k z +2 t 121 49 [cos( k 1 + k x )exp( ik z )+cos( k 2 k x )exp( ik z )] +2 t 121 38 [cos( k 1 + k y )exp( ik z )+cos( k 2 + k y )exp( ik z )]. (B–3) Thenumericalvaluesforhoppingparametersinunitsof eV areasfollows.Forthe 2Dpart, 1 =0.020, 2 = 0.2605, 3 = 0.0075, 5 = 0.3045, t 11 11 =0.030, t 10 16 = 0.0185, t 20 11 = 0.010, t 21 16 =0.0035, t 11 13 = 0.0635 i t 10 18 =0.155 i t 11 15 = 0.090, t 10 27 = 0.2225, t 11 22 =0.070, t 10 29 = 0.1925 i t 11 23 = 0.010 i t 10 2,10 =0.1615, t 11 33 =0.152, t 10 38 =0.050, t 20 33 = 0.004, t 21 38 =0.040, t 02 33 = 0.051, t 10 49 =0.210, t 22 33 = 0.005, t 21 49 = 0.053, t 11 34 =0.090, t 10 4,10 =0.0995 i t 11 35 =0.1005 i 109

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Forthe3Dpart, t 101 16 = 0.004, t 001 11 =0.0105, t 111 11 =0, t 201 11 =0, t 201 14 =0, t 001 33 = 0.003, t 201 33 =0, t 021 33 =0.0105, t 121 16 =0, t 101 18 =0, t 101 19 =0, t 211 19 =0, t 101 38 =0.0115, t 121 38 =0, t 101 39 =0, t 101 49 =0, t 121 49 =0. Somehoppingparameters t rst `` 0 arepurelyimaginarynumbersbecausethe d xz and d yz orbitalsaremultipliedbytheimaginaryunitfactortogett herealHamiltonian matrix.However,ifonewereinterestedinorbitalresolved susceptibilityorpairing vertexfunction,realorbitalsaremoremeaningful[ 77 ],sowecanintroduceagauge transformationtorealorbitalsbythematrix S =diag(1,1, i i ,1,1,1, i i ,1) ,andthe transformedHamiltonianis ~ H ARPES 0 = S 1 H ARPES 0 S 110

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BIOGRAPHICALSKETCH YanWangwasborninJiangsu,Chinain1985.HeobtainedhisBa chelorofScience degreeinphysicsfromFudanUniversityinShanghai,Chinai n2008.Inthesameyear, heenrolledinthegraduateschoolatUniversityofFlorida, Gainesville,UnitedStates. HereceivedhisDoctorofPhilosophydegreeinphysicsinthe springof2014.Hehas experiencedandenjoyedtheculturalclashandharmonywhil epursuingtherigorous knowledgeinscience.Hevaluestheyearsofstudyanddoctor alresearchtrainingmost asaneducationinmethodologyofconductingscienticrese arch. 120