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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
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